shared earth modeling

Shared Earth Modeling

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Shared Earth Modeling

John R. Fanchi

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Fanchi, John R. Shared earth modeling / John R. Fanchi

p. cm. Includes bibliographical references and index. ISBN 0-7506-7522-5 1. Geology--Mathematical models. 2. Geology--Computer simulation. I. Title

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Table of Contents

Preface xi

Chapter 1 Introduction to Shared Earth Modeling 1

1.1 Porous Media 1

1.2 Reservoir Heterogeneity and Reservoir Scales 4

1.3 Shared Earth Modeling Defined 7

1.4 The Shared Earth Modeling Process 11

1.5 The Value of Shared Earth Modeling 14

CS-1 Valley Fill Case Study: Introduction 16

Exercises 16

Chapter 2 Geology 18

2.1 Geologic History of the Earth 18

2.2 Rock Formation 22

2.3 Formations and Facies 25

2.4 Structures and Traps 26

2.5 Petroleum Occurrence 27

2.6 Stratigraphy 28

CS-2 Valley Fill Case Study: Geologic Model 31

Exercises 32

Chapter 3 Petrophysics 33

3.1 Elastic Constants 33

3.2 Elasticity Theory 38

3.3 Acoustic Velocities 42

3.4 Gassmann's Equation 44

3.5 Moduli from Acoustic Velocities 45

CS-3 Valley Fill Case Study: Bulk Moduli 50

Exercises 51

Chapter 4 Well Logging 52

4.1 Principles of Logging 52

4.2 Direct Measurement Logs 54

4.3

4.4

4.5

4.6

4.7

CS-4

Lithology Logs

Porosity Logs

Resistivity Logs

Other Types of Logs

Reservoir Characterization Issues

Valley Fill Case Study: Well Logs

Exercises

Chapter 5 Geophysics

5.1 Physics of Waves

5.2 Propagation of Seismic Waves

5.3 Acoustic Impedance and Reflection Coefficients

5.4 Seismic Data Acquisition, Processing and Interpretation

5.5 Seismic Resolution

CS-5 Valley Fill Case Study: Vp/V s Model

Exercises

Chapter 6 Fluid Properties

6.1 Description of Fluid Properties

6.2 Classification of Petroleum Fluids

6.3 Sources of Fluid Data

6.4 Representation of Fluid Properties

CS-6 Valley Fill Case Study: Fluid Properties

Exercises

Chapter 7 Measures of Rock-Fluid Interactions

7.1 Darcy's Law

7.2 Permeability

7.3 Directional Dependence of Permeability

7.4 Capillary Pressure

7.5 Relative Permeability

CS-7 Valley Fill Case Study: Permeability

Exercises

Chapter 8 Applications of Rock-Fluid Interactions

8.1 Frontal Advance Theory

56

57

60

63

64

66

67

69

69

71

75

78

81

84

86

87

87

93

97

101

105

107

108

108

115

117

122

127

132

132

133

133

8.2 Welge's Method

8.3 Transition Zones

8.4 Transition Zone Volumetrics and Numerical Simulation

CS-8 Valley Fill Case Study" Rock-Fluid Interaction Data

Exercises

Chapter 9 Fluid Flow Equations

9.1 Material Balance

9.2 Continuity Equation

9.3 Convection-Dispersion Equation

9.4 Navier-Stokes Equation

9.5 Integrated Flow Model Equations

CS-9 Valley Fill Case Study: Conceptual Areal Model

Exercises

Chapter 10 Fundamentals of Reservoir Characterization

10.1 Flow Units

10.2 Traditional Mapping

10.3 Computer Generated Maps

10.4 Visualization Technology

CS-10 Valley Fill Case Study: Reservoir Structure

Exercises

Chapter 11 Modern Reservoir Characterization Techniques

11.1 Geostatistics

11.2 Geostatistical Modeling

11.3 Time-Lapse (4-D) Seismology

11.4 Case Study: N.E. Nash Unit, Oklahoma

CS-11 Valley Fill Case Study: Time-Lapse Response

Exercises

Chapter 12 Well Testing

12.1 Pressure Transient Testing

12.2 Oil Well Pressure Transient Testing

12.3 Gas Well Pressure Transient Testing

12.4 Well Test Capabilities

138

140

146

147

149

150

150

152

156

161

162

168

168

170

170

174

176

178

179

180

182

182

184

190

196

197

198

199

199

201

215

221

CS-12 Valley Fill Case Study: Well Pressures

Exercises

Chapter 13 Production Analysis

13.1 Decline Curve Analysis

13.2 Produced Fluid Ratios

13.3 Tracer Tests

13.4 Tracer Test Design

CS-13 Valley Fill Case Study: Production

Exercises

Chapter 14 Reservoir Flow Simulation

14.1 Reservoir Flow Modeling

14.2 Data Acquisition and Evaluation

14.3 Gridding and Upscaling

14.4 Flow Model Calibration- History Matching

14.5 Predictions

CS-14 Valley Fill Case Study: History Match

Exercises

Chapter 15 Reservoir Management

15.1 Reservoir Management Process

15.2 Multidisciplinary Integration

15.3 Economics

15.4 Environmental Issues

CS-15 Valley Fill Case Study: Base Case Prediction

Exercises

Chapter 16 Improved Recovery

16.1 Recovery Efficiency

16.2 Production Stages

16.3 Drilling Technology

223

225

227

228

229

232

237

242

244

245

245

247

249

251

254

256

258

259

259

264

266

268

269

271

272

272

275

278

CS-16 Valley Fill Case Study: Waterflood Prediction 280

Exercises 280

References

Index

283

299

PREFACE

The primary goal of Shared Earth Modeling is to introduce the reader to the processes and concepts needed to develop shared earth models. A Shared Earth Model (SEM) is a multidisciplinary representation of a

subsurface resource. Shared Earth Modeling describes the process of integrating static and dynamic data from two or more disciplines to construct a model of a subsurface resource. Reservoir structure and fluid production are examples of static and dynamic data, respectively. The SEM

evolves as static and dynamic data are acquired and integrated into the

model. This book is suitable for geoscientists and engineers with an interest in

shared earth modeling, and professionals working in multidisciplinary asset management teams. It can be used in multidisciplinary upper division undergraduate and graduate classes, continuing education classes, distance learning (remote or Intemet-based) courses, short courses, or self-study. The reader is exposed to topics from several disciplines that have a significant impact on the development of a SEM.

My colleagues in industry and academia, and students in multidisciplinary classes at the Colorado School of Mines, helped me

identify important and relevant topics that cross disciplinary lines. I am, of course, responsible for the final selection of topics. I would like to thank

Drs. Matt Cole, Craig Van Kirk and Mark Miller for their support and for Mark's contributions to Chapter 2. I would especially like to thank Kathy Fanchi for her efforts in the preparation of this manuscript.

John R. Fanchi, Ph.D.

Golden, CO

June 2002

xi

Chapter 1 Introduction to Shared Earth Modeling

Shared earth modeling is an essential aspect of the reservoir management

process. Properly constructed shared earth models integrate all available information to yield the most accurate representation of a reservoir possible. The shared earth model is a quantitative representation of a subsurface and

can be used to determine the size of the resource and to predict fluid flow performance. Our goal is to introduce the process and concepts needed to develop shared earth models using language that is understandable from a multidisciplinary perspective. This chapter introduces the notions of reservoir scales and reservoir heterogeneity, and provides a working definition of shared earth modeling and reservoir characterization. The value of shared earth modeling is then discussed.

1.1 Porous Media ]

Reservoirs are examples of porous media. They consist of volumes of rock grains and pore space. Figure 1-1 illustrates a cube of rock with grains of sand filling the cube. Bulk volume is the volume of the cube and includes both grain volume and the volume of space, or pore volume. Bulk volume VB of the porous medium is the product of area A times gross thickness H:

2 Shared Ear th M o d e l i n g

V B = A H (1.1.1)

The volume that is not occupied by

grains of sand is the pore space avail-

able for occupation by fluids such as oil, gas, and water.

Porosity is defined as the ratio of

pore volume to bulk volume. Pore volume Vp is the volume remaining when the volume of grains is sub- tracted from the bulk volume, thus

F i g u r e 1-1. P o r e s a n d P o r e S p a c e

~) - - (1 .1 .2)

where ~ is total porosity, V s is bulk volume and V c is the volume of the grains. While there are several kinds of porosity, porosity can simply be thought of as the void space in a rock. Upon rearrangement, we see from

Equation (1.1.2) that pore volume Vp is the product of bulk volume and porosity ~:

V e = d O V s (1.1.3)

When sedimentary rocks are being deposited, the pore space is filled with water. The pores of a rock can be filled at a later time with commercially important fluids ranging from potable water to oil and gas. As a rule, we are

interested in void spaces that are connected with other void spaces. Con- nected pore spaces form a conduit for fluid flow. Permeability is a measure of the connectivity of pore spaces.

The above definitions of bulk volume and pore volume are measures of the gross volume in a system. To determine the volume of the system that

is commercially significant, the gross volume must be adjusted by introducing the concept of net thickness.

Net thickness h is the thickness of the commercially significant

formation. For example, if the gross thickness includes 5 ft of impermeable

John R. F a n c h i 3

shale and 15 ft of permeable sandstone, the gross thickness is 20 ft and the

net thickness is 15 ft. If all of the permeable sandstone is not in communica-

tion with a production well, then the net thickness can be reduced further.

It is often necessary to justify the well log interval that is identified as net

thickness.

The net to gross ratio r I is the ratio of net thickness h to gross thickness H:

r I = h / H , 0_< r 1 _< 1 (1.1.4)

The inequality highlights the fact that net thickness is always less than or

equal to gross thickness. Net and gross thicknesses are ordinarily determined

using well logs. The volume of net pay is the product of pore volume and net to gross ratio:

V.a -- n n - (1.1.5)

The saturation S~ of phase Q is the fraction of the volume of pore space

occupied by phase ~. The volume V~ of phase Q in the pay interval is the product of net pay volume and phase saturation:

Ve - SeVp~y - S e h A f (1.1.6)

The sum of the saturations in the pay interval must equal one. If the system

has NQ phases, the saturation constraint is

N~

1- ~ S e (1.1.7) e= 1

For an oil-water-gas system, the saturation constraint is So + S,, + Sg = 1

where the subscripts {o, w, g} refer to oil, water and gas respectively.

When fluids are produced from a reservoir, they travel through the reservoir rock from a place of higher energy to a place of lower energy. The

we|lbore of a producing well is a point of.lower energy. The route that the

fluid takes can be a straight path, or very circuitous. The property of the rock

4 Shared Earth Modeling

that measures the length of the path

from one point A to another point B relative to a straight line is called

tortuosity. If the path is a straight line,

which can occur in a fracture, the

tortuosity is 1. In most cases, the flow path between points A and B will be

longer than a straight line so that Figure 1-2. Tortuosity tortuosity is greater than 1. Figure 1-2

illustrates the concept of tortuosity.

[ 1.2 Reservoir Heterogeneity and Reservoir Scales ]

The description of reservoir heterogeneity depends on how the values of the

property of interest vary spatially in the reservoir, and the size of samples

used to measure the rock property. Every measurement of any rock property

is made on a sample of some specific size. For example, porosity may be

measured using core plugs of various sizes. The value of the rock property

depends on the number of measurements made, which depends on the num-

ber of samples. Thus the

accuracy of a characterization of the reservoir

depends on the number

of samples. Given these

concepts, we can think of a reservoir as a col-

lection of elements.

Each element has the

volume of one of our Figure 1-3. Ensemble of Volume Elements samples (Figure 1-3).

Rock property values have meaning only for samples of rock of some

macroscopic size which contain large numbers of pores and grains. These

values will not be unique because of random events in the depositional

process and subsequent diagenetic history. We can expect to find a statistical distribution of values associated with measurements of any rock property

on many neighboring samples within a given depositional structure. A

John R. Fanchi 5 ,,,

depositional structure can be defined as homogeneous in a statistical sense if the same frequency distribution of a rock property is found throughout the structure. Given this definition of homogeneity, we define heterogeneity

as the absence of homogeneity. The determination of the heterogeneous quality of a depositional

structure requires the comparison of frequency distributions of any rock property in spatially distinct regions of the depositional unit. Collins [ 1961 ] showed that every measured value of the rock property should be on a sample of the same size because the form of the frequency distribution is determined in large part by the sample size for each measurement. We can illustrate this concept by a special and relatively simple case. Suppose we measure porosity

on samples as small as a sand grain. The porosity measurements will exhibit one of two values, zero (all grain) or one (all pore space). The observed frequency distribution is therefore bi-modal with peaks at ~ = 0 and ~ = 1. For larger samples, such as core plugs, the distribution changes from a discrete bi-modal distribution to a continuous Gaussian distribution. If we wish to be consistent when we compare frequency distributions, we

should use measurements from samples of comparable size. The depositional environment determines the statistical heterogeneity

of rock when viewed on any size scale. A depositional unit might be

statistically heterogeneous on the scale of core plugs but be statistically homogeneous on a larger scale. In this case, rock properties within the depositional unit could be treated on the larger scale as deterministic functions of spatial position {x, y, z}. As deterministic functions, the rock

properties would be adequately represented by smooth functions and it would be unnecessary to introduce statistical fluctuations. A critical sample volume should exist at the larger scale that would be large enough to mask statistical fluctuations in values of the rock property among adjacent samples. We may still see regional trends, or deterministic variations in rock property values

as a function of spatial position, but the observed values of the rock property could be contoured with reasonable accuracy at the larger scale. The resulting

contour map would provide a deterministic representation of the rock property as a function of spatial position in the depositional unit.

The scale associated with data measurements depends on the type of measurement, the sampling technique, and size of the sample. For example, core and well log information sample a very small part of the reservoir. A /

~eismic section expands the volume of the subsurface environment that is


sampled, but the interpretation of seismic data is less precise. Surface seismic data can cover the entire region of interest, but surface seismic data are often viewed as "soft data" because surface seismic measurements are remote, indirect measurements. The reliability of surface seismic data can be improved when correlated with "hard data" such as core and well log measurements.

The importance of sample size in characterizing rock properties has been stressed by many authors. As early as 1961, Collins [ 1961 ] showed that variance in porosity distribution depended on sample size. Bear [ 1972] stressed the need to identify an appropriate averaging volume for assigning a macroscopic property to a porous medium. Haldorsen and Lake [1989]

defined fourconceptualscales Figure 1-4. Reservoir Scales (after to account for variations in Haldorsen and Lake, 1989; reprinted by therange ofdataapplicability, permission of the Society of Petroleum Their scheme is schematically Engineers)

illustrated in Figure 1-4. The giga scale includes information associated with geophysical

measurement techniques. An example of a giga scale measurement is the imaging of reservoir architecture. Theories of regional characterization, such as plate tectonics, provide an intellectual framework within which giga scale measurement techniques, like seismic and satellite data, can be interpreted. The mega scale is comparable in size to the reservoir being characterized. Mega scale measurement techniques include well logging, well testing and 3-D seismic analysis. Data sampling at the level of core analysis and fluid property analysis provides macro scale information. Thin section analysis and measurements of grain size distribution provide pore size information

John R. Fanchi 7

atthe micro scale. Fayers and Hewett [ 1992] have noted that scale definitions

are not universally accepted, but can be used to illustrate the relative scale

associated with reservoir property measurements. The shared earth model

will integrate information from all scales.

[ 1.3 Shared Earth Modeling Defined ]

A shared earth model is a multidisciplinary representation of the subsurface

resource. The shared earth model is developed from static geological

information such as reservoir structure, and dynamic information such as

fluid production performance. A shared earth model may change as static

and dynamic data are acquired and integrated into the model. The correct relationship between data measured at two different scales

may be difficult to determine. For example, permeability is often obtained

from both pressure transient testing and routine core analysis. The respective

permeabilities, however, may appear to be uncorrelated because they

represent two different measurement scales. The integration of data obtained

at different scales is often referred to as the "scale-up" problem [for example,

see Oreskes, et al., 1994].

An important task of the scale-up problem is to develop a detailed

understanding of how measured parameters vary with scale. The focus on

detail in one or more aspects of the reservoir modeling process can obscure

the fundamental character of the reservoir in a flow model study. One way

to integrate available data within the context of a "big picture" is to apply

the concept of flow unit. Ebanks defined the flow unit as "a volume of rock

subdivided according to geological and petrophysical properties that

influence the flow of fluids through it" [Ebanks, 1987].

All rock properties measured for use in reservoir characterization are

measured on samples of a specific size and scale. The measured values are usually averages over smaller elements of rock making up the sampled

region. For example, core plug measurements are macro scale averages over

micro scale elements, while pressure transient testing provides a mega scale

average over both macro scale elements and, on a finer scale, micro scale elements. The four scales defined above are related to averaging volume in Table 1-1.


Table 1-1 Reservoir Characterization Scales

Micro Scale

Macro Scale

Mega Scale

Giga Scale

the size of a few pores only

the size of conventional core plugs

the size of grid blocks in full field flow models

the total formation or regional scale

We can see how scale influences the analysis of a porous medium by

considering a statistical description of porosity. Let us begin with a porous

medium, such as a core plug, that has the bulk volume V. We subdivide the

volume V into a number n of elements each with volume e so that

V = ne (1.3.1)

The volume and associated elements are shown in Figure 1-5. Each element

is sampled to determine if it is either grain or pore space. This means the

elements are small enough to be considered micro scale samples.

Figure 1-5. Elements of Volume

Letx be the number of elements that are pore space. Porosity is expressed

in terms o fx as

d~- x/n (1.3.2)

and pore volume is

John R. Fanchi 9

x Vp - - - ne - xe (1.3.3)

n

The distribution of samples in this formulation is described by the binomial

distribution since the sampling of each element yields one of two possible

outcomes: either grain or pore space.

The probability that in n independent trials an element of pore space is

encountered x times is given by the binomial distribution. Let p denote the

probability that pore space is encountered in a single trial. Then the

probability that grain is encountered is q = 1 -p . The resulting binomial

distribution is

fb (X)=(; lpXq "-x (1.3.4)

with the binomial coefficient

n _ n!

x!(n - x) ! (1.3.5)

The DeMoivre-Laplace Central Limit theorem says that the binomial

distribution becomes the normal distribution

1 [ l(x fN(X)= " ~ 0 exp - -~ o (1.3.6)

when the number of elements n is large, which is reasonable, even for a core

plug. The distribution in Equation (1.3.6) has mean la = np and standard

deviation o = ~/npq, which gives the average porosity 9 - p and o

becomes

10 S h a r e d E a r t h M o d e l i n g

a - p q = Tq~( 1_ ~) (1.3.7)

The effect of sample size on the distribution can be estimated using Equation (1.3.7) and the following procedure.

Suppose we compare two micro scale distributions with sample sizes 8 a , 8 b . A relationship between the number of samples in each volume and sample size is obtained by recognizing that bulk volume is constant, thus

V = n a b a = rib8 b (1.3.8)

A standard deviation for each distribution can be obtained from Equation (1.3.7). The ratio of standard deviations is

Oao I:ba '39'

The distribution with larger samples ea > eb has a smaller standard deviation Oa < Ob, as shown in Figure 1-6. The distribution with larger samples is more sharply peaked about the average porosity than the distribution with smaller samples. Figure 1-6 is based on a micro scale analysis. The samples are assumed to be small enough at the micro scale that each sample is either a pore or a grain. The analysis would have to be

,,j \ _ , o ~ % o _

Porosity

Large

Element

. . . . Small

Element

Figure 1-6. Effect of Sample Size on Porosity Distribution at the Micro Scale [Fanchi, 2001a]

John R. Fanchi 11

modified if it was conducted on a different scale [for example, see Collins,

1961, Chapter 1 ].

The above discussion of porosity illustrates a general observation: the

value of a rock property is a

function of the averaging vol-

ume. This observation is illus-

trated in Figure 1-7, which

shows that the value of a rock

property can vary within a scale

and from one scale to another.

An important task of reservoir

characterization and associated

reservoir flow modeling is to

find a suitable scale for develop-

M i c r o - I - - M a c r o ---t-- M e g a

gS~i l l / / ' S-- . . . . I " �9 - - " . . . . I ' G i g a

~s

A v e r a g i n g V o l u m e

ing a flow model that can be Figure 1-7. Rock Property as used to prepare reliable perfor- Function of Averaging Volume

mance predictions.

] 1~4 The Shared Earth Modeling Process , ]

Shared earth modeling is the process of integrating static and dynamic data

from two or more disciplines into a model of a subsurface resource. Static

data are data acquired at a single point in time, such as reservoir structure

from a seismic survey or a well log. Dynamic data are data that change with

time, such as reservoir pressure and production performance. An essential

element of the shared earth modeling process is reservoir characterization.

Reservoir characterization is the process of preparing a quantitative

representation of a reservoir using data from a variety of sources and

disciplines. All of the information collected at various scales in the reservoir

characterization process must be integrated into a single, comprehensive, and

consistent representation of the reservoir. The reservoir characterization

process should include the acquisition and analysis of static and dynamic

data. The model that results from the reservoir characterization process is

a shared earth model.

Shared earth modeling may be viewed as a reservoir characterization

process that integrates both static and dynamic data. Shared earth models

are refined as additional data are acquired and incorporated into the reservoir


description. In addition to representing our best understanding of the subsurface resource, shared earth models are used in reservoir flow simulators to predict production performance.

A depiction of the general features of the reservoir, even as a simple sketch, is a valuable resource for validating the ideas being quantified in a reservoir model. Richardson, et al. [ 1987a, b] sketched several common types of reservoir models: a deep-water fan, a sand-rich delta, a deltaic channel contrasted with a deltaic bar, etc. A sketch can be used to confirm that people from different disciplines share the same concept of a reservoir before

extensive resources are used to prepare a detailed model. Sketches are simple visual aids that enhance communication.

Reservoir flow simulators combine the reservoir model with fluid flow equations. The fluid flow equations are derived from classical mass, energy and momentum conservation laws and empirical relationships. The equations are coupled systems of nonlinear, time-dependent partial differential equations in three dimensions. The reservoir is subdivided into cells and flow is calculated between cells. Input data sets for reservoir flow simulators are referred to in this book as reservoir flow models to avoid confusion with the term "reservoir model." The term "reservoir model" refers to the static representation of the reservoir and is incapable of modeling fluid flow. The

results of reservoir flow modeling can be combined with economic parameters to provide an economic model of a reservoir management scenario.

Flow modelers cannot effectively use all of the information being provided by computer-based geologic models containing millions of cells. Geoscientists are combining detailed lateral information from 3-D surface

seismic with detailed vertical resolution from well logs to create models of the subsurface that contain millions of cells of data. One way to manage this

huge database is to apply the concept of flow unit. Ebanks defined the flow unit as "a volume of rock subdivided according to geological and petrophysical properties that influence the flow of fluids through it" [Ebanks, 1987]. It is still necessary to coarsen detailed geologic models into representative flow units for use in flow simulators.

The petroleum industry is beginning to rely on sophisticated 3-D and

4-D visualization technology [Tippee, 1998] to display the large volumes of data being generated by computer-based geologic modeling packages. Sophisticated visualization technology that can create a virtual reality representation of the subsurface can be a practical tool for developing shared

John R. Fanchi 13

earth models based on data from different disciplines. One objective of a shared earth model is to help organize the work of multidisciplinary asset management teams. A shared earth model evolves as data is acquired and integrated into the model. A new generation of software is being developed to optimize the seamless integration of data from multiple disciplines into a shared earth model.

Tobias [ 1998] has pointed out that the development and implementation of 3-D, model-centric methods based upon computer-generated 3-D representations of the earth are changing the way the industry characterizes reservoirs. Tobias described a reservoir characterization process called the decouple-recouple work flow. The work flow begins with data acquisition, and proceeds to a decoupling step in which the acquired data is analyzed in four tasks:

A. seismic velocity modeling; B. structural correlation and modeling; C. stratigraphic correlation and modeling; and D. petrophysical modeling.

Several sets of rules are developed from the products of these tasks. Tasks A, B and C create time-depth conversion rules. The time it takes a seismic disturbance to propagate from a source and back to a signal recorder is called two-way travel time. This time is converted to a depth using a seismic velocity model. The seismic velocity model depends on the properties associated with the stratigraphic column. Tasks B and C create framework modeling rules. Well log-seismic property mapping rules are created from Tasks C and D. Each of the interpretations developed in the process should be consistent with all available data.

Once the rules have been created, the recoupling step begins. The time- depth conversion rules and the framework modeling rules are combined to characterize the reservoir framework. The framework is then combined with the well log-seismic property mapping rules to complete characterization of the reservoir.

The decouple-recouple work flow described by Tobias illustrates a modem reservoir characterization process which is focused on a static description of the reservoir. An integrated reservoir characterization process


must account for dynamic data as well. Dynamic data is included when a

flow model is developed, as outlined in the next section.

[ 1.5 The Value of Shared Earth Modeling ]

Reservoir management is often defined as the allocation of resources to

optimize hydrocarbon recovery from a reservoir while minimizing capital investments and operating expenses [Wiggins and Startzman, 1990; Satter,

et al., 1994; A1-Hussainy and Humphreys, 1996; Thakur, 1996]. A good

reservoir description is needed for optimum reservoir management. The value

of shared earth modeling within the context of reservoir management is

outlined here. We begin by briefly reviewing the stages in the life of a reservoir.

A reservoir lifetime can be discussed in terms of stages by relating the stage to a particular function (see Table 1-2). One of the objectives of each of the exploration, discovery and delineation stages is to obtain information

that can be used to prepare a reservoir model. The reservoir model plays a central role in the preparation of a development plan and the subsequent production of the resource. Even in abandonment, it is necessary to under-

stand the reservoir so that all wells can be plugged and abandoned in an

environmentally acceptable manner.

Table 1-2

Stages in the Life of a Reservoir

Stage

Exploration

Discovery

Delineation

Development

Production

Abandonment

Function

Identify resource prospects

Find resource

Determine size of resource

Prepare strategy for extracting resource

Produce resource

Leave resource location

A fundamental task of the reservoir characterization process is determining the spatial distribution of reservoir parameters in a format that is suitable for use in a reservoir flow simulator. The reservoir characterization process

John R. Fanchi 15

provides a representation of a reservoir that begins with the acquisition of seismic data and data at control points such as wells. The acquired data must be processed, interpreted, contoured, and digitized. The resulting set of digitized maps can be used as part of the input data set for a reservoir flow model.

A detailed description of the geological area of interest is usually required for a reservoir flow model study. If a flow model is being used to evaluate flow concepts, the geologic model can be less detailed than a flow model designed to identify bypassed fluids and plan drilling and workover schedules. In addition to improved knowledge and understanding, the most significant commercial product of the shared earth modeling process is a model o f the reservoir that can be used in reservoir flow simulators to perform reservoir management studies.

A simulation study generally consists of two tasks: matching field history and making predictions. The flow model begins with an initial reservoir

description and is then used to match reservoir performance. The reservoir description may have to be modified within reasonable limits to achieve an acceptable match of reservoir history. The history matching phase of the study is an iterative process that makes it possible to integrate reservoir geoscience and engineering data. Once a model has been history matched, it can be used to forecast the behavior of the reservoir under a variety of reservoir management strategies. The steps of a flow model study are discussed in more detail in later chapters.

The integration of data from different disciplines is being enhanced by the development of flow models that include petrophysical calculations [Fanchi, 2000a]. An integrated flow model combines a traditional flow model with a petrophysical model. The petrophysical model can estimate geomechanical properties. Integrated flow models simplify the data transfer process between disciplines, enhance consensus building, and provide performance predictions in a format that is familiar to reservoir managers. They are a natural extension of the model-centric methods that are being used to optimize subsurface resource management decisions.

The shared earth modeling procedure presented in the following chapters is designed to complement a reservoir management study using an integrated flow model. A case study, the Valley Fill case study, is used to illustrate the shared earth modeling process. A case study section and many of the exercises at the end of each chapter apply the principles presented in that


chapter to a synthetic but realistic reservoir study. The Valley Fill case study illustrates the reservoir management process from geologic conceptualization to performance predictions.

[ CS-I. Valley Fill Case Study: Introduction , , [

The primary purpose of the Valley Fill case study from a pedagogical perspective is to show how to integrate reservoir characterization concepts using a valley fill example. The incised valley model is useful for describing reservoirs in both mature and frontier basins around the world [e.g. Bowen, et al., 1993; Peijs-van Hilten, et al., 1998]. The reservoir of interest is an undersaturated oil reservoir that has been producing for a year. Wells in the field are shown in Figure CS-1A.

A reservoir characterization study is needed to prepare a reservoir model that can be used in a flow simulator to make performance predictions. The predictions will be used to guide the preparation of an optimized reservoir management plan. The flow simulator will be used to quantitatively verify our understanding of the reservoir by matching historical performance, and then evaluate various reservoir operating strategies.

[ Exercises ]

1-1.

1-2.

1-3.

Make a directory on your computer called SEM\VALLEY.

Go to the website http://www.bh.com/companions/0750675225 and copy the zip file. Extract all files to SEM\VALLEY.

List several questions you would want to have answered if you were trying to decide how to manage the Valley Fill reservoir.

John R. Fanchi 17

�9 8 n

2 1

4 1 6 7 �9 1

3 ~ �9

9 5

Figure CS-1A. Well locations in area that is 6000 ft long by 3000 ft wide. Productive Well o; Dry Hole

Chapter 2 Geology

Geological principles can be used to prepare a description of the reservoir

structure and the regional context. Familiarity with the theory of the earth's

formation and maturation is an important prerequisite to understanding the

state of hydrocarbon-bearing reservoirs and the regional geology within

which they are embedded. Theories of the earth's formation and plate

tectonics are outlined, and concepts from development geology are briefly

reviewed. For more details, see such references as Press and Siever [ 1982;

2001 ], Montgomery [ 1990], Levin [ 1991 ] and Selley [ 1998].

I 2.1 Geologic History of the Earth ]

Many scientists believe that the earth first began to form about 4 to 5 billion

years ago. It is believed that the earth was a large body of hot, gaseous matter that whirled through space for several hundred million years. This body of matter slowly condensed and cooled. A solid crust gradually formed around

the molten interior. The cross-section of the earth consists of an iron core wrapped inside

a mantle of rock with a thin crust at the surface. The density of rock in the

mantle is higher than in the crust because of an abundance of iron and

18

John R. Fanchi 19

magnesium. Rock density in the mantle increases with depth as you move from the shallower upper mantle to the deeper lower mantle.

The thickness of the crust is small compared to the diameter of the earth.

The crust consists of oceanic crust and continental crust. The mobile part

of the upper mantle and crust is called the lithosphere. Lithospheric plates drift on a denser, partially molten material called the asthenosphere.

As the earth cooled from its hot, gaseous state, the surface was subjected

to forces that caused great changes in its topography, including the formation of continents and the uplift of mountain ranges. Pressure from the earth's

interior could crack the sea floor and allow less dense molten material to flow onto the sea floor. Such cracks in the earth's crust are called subsea ridges.

Studies of material extruded from a subsea ridge show that the material

spread laterally on each side of the ridge. The symmetry of the material

spread on each side of the ridge supports the contention that the material was

in a molten state as it gradually moved outward from the ridge. As the material cooled, magnetic constituents within the molten material aligned

themselves in conjunction with the polarity of the earth's magnetic field at

the time that the material solidified. Several periods of polarity have been identified and dated.

Satellite measurements of the earth's gravitational field have identified

boundaries between continents. The shapes of the boundaries are indicative

of vast plates. These plates are referred to as tectonic plates, and their behavior is the subject of plate tectonic theory.

In the theory of plate tectonics, the entire crust of the earth is seen as a

giant, ever-shifting jigsaw puzzle. Tectonic plates move in relation to one another at the rate of up to 4 inches per year. Tectonic plates are often associated with continental land masses. Many of the plate boundaries can be directly observed by gravimetric surveys of the earth's surface from satellites.

The movement of tectonic plates is responsible for much of the geologic

heterogeneity that can be found in hydrocarbon bearing reservoirs. Figure

2-1 shows the hypothesized movement of tectonic plates during the past 225

million years. It begins at the time that all surface land masses were thought

to be coalesced in a single land mass referred to as Pangaea. Pangaea was not the initial state of surface land masses. Geoscientists believe that Pangaea

was formed by the movement of tectonic plates, and the continued movement

20 Shared Earth Modelin~

Figure 2-1. Tectonic Plate Movement [USGS website, 2001]

John R. Fanchi 21

of plates led to the break up of the single land mass into the surface features we see today. Tectonic plates are driven by forces which originate in the earth's interior. As the plates pull apart or collide, they can cause such geologic activities as volcanic eruptions, earthquakes and mountain range formation.

Along with the occasional impact of a meteor or asteroid, the movement and position of tectonic plates has been theorized to cause extensive environmental changes. These environmental changes are not just the local volcano creating an island where none was before, but include global sea level and atmospheric changes. Plate tectonics can lower the sea level, creating a period of vast erosion and deposition. The biosphere is also affected by these changes. Plants and animals may thrive under one set of conditions, and readily become extinct when the conditions change. Plate movement also provides a mechanism to help explain the geographic distribution of organisms around the world. Based on these changes,

geologists have found that the geologic history of the earth can be broken

into convenient periods. The most encompassing segment of time is the eon. Eons are subdivided

into eras, which are further subdivided into periods. On a finer scale, periods are separated into epochs. Table 2-1 shows an abridged version of the geologic time scale since the earth's formation. The acronym MYBP stands

for millions of years before the present. The starting time of selected intervals is reported from two references [Levin, 1991; Ridley, 1996]. A comparison of the geologic time scales reported in the literature indicates that there is still uncertainty in the actual chronology of the earth.

The oldest terrestrial rocks formed during the Precambrian Eon. Microscopic organisms are believed to have originated during this eon. Organisms with cellular nuclei (eukaryotes) appeared during the Proterozoic Era of the Precambrian. The Paleozoic is the era when life began to blossom. The demise of trilobites during the Permo-Triassic extinction marked the end of the Permian Period during the Paleozoic Era. The Mesozoic Era includes the age of the dinosaurs, and its end was marked by the Cretaceous-Tertiary (K-T) extinction. Cenozoic rocks are relatively new, being less than 70

million years old. Mammals began to flourish during the Cenozoic. The variations in life forms give rise to variations in the fossil record that can help

geoscientists characterize the stratigraphic column.


Table 2-1 Geologic Time Scale

Eon Era

Phanerozoic

Precam-

brian

Cenozoic

Mesozoic

Paleozoic

Proterozoic

Archean

Hadean

Period

Quatemary

Tertiary

Cretaceous

Jurassic

Triassic

Permian

Carboniferous

Devonian

Silurian

Ordovician

Cambrian

Epoch

Recent

(Holocene)

Pleistocene

Pliocene

Miocene

Oligocene

Eocene

Paleocene

Pennsylvanian

Mississippian

Approximate Start

of Interval MVBP)

0.01

2

5

24

37

54 - 58

65 - 66

144

208 - 213

245 - 248

286

320

360

408

438

505

570- 590

2500

3800

[ 2.2 Rock Formation I

The movement of tectonic plates across the surface of the globe generated

forces that can cause rocks to form. We can think of the process of rock

creation as a cycle. The beginning of the cycle occurs with the cooling of

molten magma and subsequent hardening into rock. Typically, the formation

of new rock occurs at plate boundaries, but it can also occur over"hot spots"

John R. Fanchi 23

within the earth' s mantle. As plates collide, pressure and heat may cause part of the plate to melt, and result in molten rock being thrust to the surface. After cooling, surface rock is subjected to atmospheric phenomena.

Chemical and physical processes cause exposed rock to break into smaller and smaller particles. Wind and water transport these particles from their source location in a process called erosion. The particles continually become finer and finer as they collide with other objects during the transport process. When the energy of the wind or water dissipates to the point where there is not enough energy to transport the particle, the particle will be deposited along with other particles. The accumulation ofparticles becomes

thicker and thicker. Slowly, over millions of years, tectonic plates move up and down relative

to sea level, alternately causing erosion and deposition. Deposition can range from thousands of feet of sediment in an area to none at all. Erosion can carve canyons, level once jagged mountains, or remove all traces of a formation that was once hundreds of feet thick. High pressure and tempera-

ture can cause rocks to change character in a process called metamorphism. Particles may become fused together to form considerably larger objects. Given enough time, pressure and heat, rocks will melt and start the cycle

again. Based on this rock cycle, geologists recognize three primary types of

rocks: igneous, sedimentary, and metamorphic. Igneous rocks are formed by the cooling of material that has been molten. Sedimentary rocks form when materials at the surface of the earth are weathered, transported, deposited, and cemented together. If rocks are subjected to heat and pressure, metamorphic rocks may form. Rocks are primarily classified by the sizes

and types of minerals present [Travis, 1955]. Sedimentary rocks are usually the most interesting to professionals

working to characterize commercially important reservoirs such as petroleum

reservoirs. Geologists have determined several key attributes that make up the classification of a sedimentary rock. These attributes are the mineral

composition, grain size, color, and structure. When a rock is described, it is desirable to convey exactly what the rock

looks like. A well-sorted, well-rounded, coarse-grained, quartz sandstone would probably make a better reservoir than a poorly-sorted, angular, fine- grained arkose. Arkose is a sandstone with both quartz and feldspar. Each of these descriptive terms tells us something about the rock.


Sorting refers to the uniformity of grain size. Fluids will typically flow better through a well-sorted rock than a poorly-sorted rock. The ability to flow is related to a rock property known as permeability, which is discussed later. If the grains have sharp edges, the grains making up the rock probably didn't get transported very far. Rounded grains indicate a longer period of transport. Rocks made up of rounded grains may have better permeability than rocks composed of grains that are fiat or have sharp edges. A coarse- grained sandstone is made up of particles that are approximately 0.5 to 1.0 mm in diameter, while fine-grained sandstone particles are between 0.125

and 0.25 mm. As a rule, larger particles allow easier passage of fluids through interstitial pore spaces than smaller particles.

The mineralogy of a rock is the collection of minerals within the rock. A mineral is a naturally occurring, inorganic solid with a specific chemical and crystalline structure. Mineral content is another very important character- istic of a rock. For example, quartz is much less reactive than the feldspars of an arkose, which is why quartz withstands weathering so well. A quartz grain may be able to withstand multiple cycles of erosion and deposition. As another example, the presence of clays in the pore space can cause reduced productivity if a change in the salinity of the formation causes clay swelling. Drilling mud and other fluids introduced into the reservoir through

the wellbore can react with clays to swell and plug clay-bearing formations.

The grains which form sedimentary rocks are created by weathering processes at the surface of the earth. Weathering creates particles that can be practically any size, shape, or composition. A glacier may create and transport a particle the size of a house, while a desert wind might create a uniform bed of very fine sand. The particles, also known as sediments, are transported to the site of deposition, usually by aqueous processes. Some- times the particles are transported very far. In these cases, only the most

durable particles survive the transport. The grains of sand roll and bump

along the transport pathway. Grains which started out as angular chunks of rock slowly become smaller and more rounded. A grain of quartz, being

fairly hard, may even be able to withstand multiple cycles of deposition and erosion. This leaves a grain that is very rounded. The minerals that make up

a sedimentary rock will depend on many factors. The source of the minerals, the rate of mineral breakdown, and the environment of deposition are important factors to consider in characterizing the geologic environment.

John R. Fanchi 25

I 2.3 Formations and Facies I

The environment under which a rock forms is called the environment of

deposition. As the environment, such as a shoreline, moves from one location

to another, it leaves a laterally continuous progression of rock which is

distinctive in character, for example a quartz sandstone. These progressions

of rocks can extend for hundreds of miles. If the progression is large enough

to be mapped, it can be called a formation.

Formations are the basic descriptive unit for a sequence of sediments.

The formation represents a recognizable, mappable rock unit that was

deposited under a uniform set of conditions at one time. Formations should

represent a dominant set ofdepositional conditions even though a formation

may consist of more than one rock type. If the different rock types within

the formation are mappable, they are referred to as members.

Formations can be a few feet thick, or can be hundreds of feet thick. The

thickness is related to the length of time an environment was in a particular

location, and how much subsidence was occurring during that period.

Each environment of deposition causes a different type of rock sequence

to be deposited. A fluvial (river) environment of deposition may cause a

sandstone to be deposited. A deltaic environment will also cause a sandstone

to be deposited. However, the sandstone that each environment produces has

a very different character. For example, a fluvial system can produce rocks

that meander along, while a delta tends to stay in one place and deposit lots

of sediment. In a fluvial system, the sandstones have certain characteristics.

The grain size of sand deposited by a fluvial system becomes finer at

shallower depths in a process called fining upwards. In addition, there is a

coarser deposit at the base of the formation that distinguishes the fluvial

system from a delta deposit. The surrounding rock types for a particular

environment are also predictable. Lying above the sandstones of a meander-

ing river are typically mudstones that may have dessication cracks and traces

of roots. The sandstones are interpreted to be formed in the river channel,

while the mudstones are thought to be floodplain deposits.

Geologists often talk about facies. The term facies refers to those

characteristics of rocks that can be used to identify their depositional environ-

ment. In the fluvial environment of deposition example, the sandstone would

be considered one facies, while the mudstone would be another. The


characteristics of a sandstone that indicate its environment of deposition also define it as a facies. A facies is distinguished by characteristics that can be observed in the field [Walker, 1992]. Ifa pebbly sandstone is seen independ- ently of surrounding rocks, it could easily be classified into a fluvial environment. Adding information that it is surrounded by hemipelagic (deep sea) mud, a geologist may revise the interpretation to be a turbidite depositional environment in which sediments are suspended in a fast-flowing current. The turbidite depositional environment is associated with high- energy turbidity currents. Integration of all of the available information will

lead to better characterization of the rock.

[ 2.4 Structures and Traps I

After hydrocarbons are generated, they migrate along pathways. Typically hydrocarbons are lighter than water and will tend to migrate to the surface. Petroleum will accumulate in traps. Traps are locations where oil and gas can no longer migrate to the surface. There are two primary types of traps: structural and stratigraphic. Structural traps occur where the reservoir beds

are folded and perhaps faulted into shapes which can contain commercially valuable fluids like oil and gas. Anticlines are a common type of structural trap.

Stratigraphic traps are the other principal kind of trap. Stratigraphic traps occur where the fluid flow path is blocked by changes in the character of the formation. The formation changes in such a manner that hydrocarbons can no longer move upwards. Types ofstratigraphic traps include a sand thinning out, or porosity reduction because ofdiagenetic changes. Diagenesis refers to processes in which the lithology of a formation is altered at relatively low pressures and temperatures when compared with the metamorphic formation

of rock. Diagenesis includes processes such as compaction, cementation and dolomitization. Dolomitization is the process of replacing a calcium atom in calcite (calcium carbonate) with a magnesium atom to form dolomite. The resulting dolomite occupies less volume than the original calcite and results in the formation of additional porosity in carbonate reservoirs. This porosity is called secondary porosity.

In addition to structural and stratigraphic traps, there are many examples of traps formed by a combination of structural and stratigraphic features.

John R. Fanchi 27

These traps are called combination traps. An example of a combination trap

is the Prudhoe Bay Field on the North Slope of Alaska [Selley, 1998]. It is an anticlinal trap that has been truncated and bounded by an impermeable

shale.

[ 2.5 Petroleum Occurrence ]

Petroleum reservoirs are normally found in sedimentary rocks. Rarely,

petroleum is found in fractured igneous or metamorphic rocks. Igneous and

metamorphic rocks originate in high pressure and temperature conditions

that do not favor the formation of petroleum reservoirs. They also do not

ordinarily have the interconnected pore space needed to form a conduit for

petroleum to flow to a wellbore. While a metamorphic rock may have originated as a piece of sandstone, it has been subjected to heat and pressure.

Any petroleum fluid that might once have occupied the pores is cooked

away. Sedimentary rocks are not the only element needed for a productive

petroleum reservoir.

Several key ingredients must be present for a hydrocarbon reservoir to

develop. First, a source for the hydrocarbon must be present. It is most

commonly thought that oil forms from the remains of single celled aquatic

life. When the remains are baked as they become buried, oil and gas are

formed. As with baking a cake, the mixture can be underdone or overcooked.

In either case, the formation of oil will not be optimal. Second, there needs

to be a conduit from the source rock to a reservoir rock. Typically, hydrocar-

bons are formed in rocks which are not very amenable to modern production

techniques. For hydrocarbons to be producible, they must be able to flow

into wells. The flow rate must be large enough to make the wells economi-

cally viable. Two important characteristics control the economical viability of the reservoir: porosity and permeability.

Porosity is the ratio of the volume of void space to the total volume.

Porosity is a factor that defines the capacity of the reservoir to store fluid.

Permeability is a measure of the ability of fluids to flow. Both porosity and permeability are discussed in greater detail later.

Once hydrocarbon fluid is in a suitable reservoir rock, a trapping

mechanism becomes important. If the hydrocarbon fluid is not stopped from

migrating, buoyancy and other forces will cause it to move toward the


surface. Overriding all of these factors is timing. Without a correct timing sequence, nothing can happen. A source rock can provide billions of barrels of oil to a reservoir, but if the trap does not form until a million years after the oil has passed through the reservoir, not much will be found except perhaps an oil-stained rock.

[ 2.6 Stratigraphy ]

Finding the most likely locations for new fields is typically the domain of geologists and geophysicists. While some early fields were found by luck, collecting seismic data increased the probability of success. As mile after mile of seismic data were collected, a need arose for a model which allowed geoscientists to interpret some of the large scale sedimentological features that were seen in the data. Two widely used interpretation techniques are discussed here.

Seismic Stratigraphy Structural features, like salt domes and anticlines, have been recognized

from the beginning of seismic data acquisition. In the 1970's, geoscientists devised a way to interpret large packages of sediment being deposited in relation to the sea level [Vail, et al., 1977]. The method, called seismic stratigraphy, allows a global framework of deposition to be developed by interpreting seismic lines that show how sediments were deposited in a given region. Seismic measurements are discussed in more detail in Chapter 5. By

interpreting the conditions under which a sedimentary rock formation was deposited, a geoscientist can infer the surrounding rock types and obtain a better understanding of the reservoir using seismic stratigraphy.

As an example, suppose a shoreline moves inland during periods of high sea levels, while during low sea levels the shoreline moves seaward. The inland movement of a shoreline is referred to as transgression, while the seaward movement of a shoreline is called regression. During a period of transgression, the land mass may be flooded by an encroaching body of water, such as a sea or ocean. The resulting deposition of sediments is a transgressive sequence that often consists of sandstone on the bottom grading upward into mud and then limestone [Montgomery, 1990, Chapter 6]. By contrast, in a period of regression, the land mass rises relative to sea level

John R. Fanchi 29

and the result is a receding shoreline with subsequent exposure of additional land mass to weathering. When the sea is inland, called a high stand, sediments are more likely to be deposited. A low stand yields conditions where erosion is taking place. The Case Study in Section CS-2 is an application of this depositional concept.

Sequence Stratigraphy A representation of the physical structure of the reservoir, the reservoir

architecture, requires the specification of areal boundaries, elevations or structure tops, gross thickness, net to gross thickness and, where appropriate, descriptions of faults and fractures. One of the most useful techniques for developing a model of reservoir architecture is sequence stratigraphy.

Sequence stratigraphy began in the 1960's and 1970's with the acquisition of high-quality offshore seismic data [Selley, 1998; Mulholland, 1998]. Van Wagoner, et al. [ 1988] formalized many ofthe definitions and concepts used in sequence stratigraphy. A sequence is a genetically related succession of strata that is bounded by unconformities or their correlative conformities. Following Hubbard, et al. [1985], Selley [1998] presented the seismic sequence stratigraphic analysis in four steps:

1. Identify sequence boundaries. 2. Define sequence geometry and interpret the depositional environ-

ment.

3. Identify seismic reflector continuity and shape. Define the internal structure of the sequence.

4. Identify seismic reflector shapes and amplitude. Further refine the internal structure of the sequence.

Sequences are controlled by changes in relative sea level. Sequence boundaries are created by changes in eustatic sea level. Eustatic sea level is the global sea level relative to the center of the earth. The eustatic sea level changes when the volume of water in the oceans changes. The volume of oceanic water depends on factors such as glaciation, ocean temperature, and groundwater volume. The accumulation of sediment within a sequence depends on changes in eustatic sea level, the rate of subsidence of the continental lithosphere relative to the asthenosphere due to tectonic forces (tectonic subsidence), and climatic effects.


Sequence stratigraphy is the study of rock relationships within a chronostratigraphic framework. A chronostratigraphic framework is defined by chronostratigraphic surfaces, where a chronostratigraphic surface is a surface which has younger rocks above and older rocks below the surface. Chrono- stratigraphic surfaces are useful time markers because they are correlatable over large lateral distances. Sequence boundaries are potentially correlatable chronostratigraphic surfaces.

Sequence stratigraphy may be applied at scales that are related to global changes of sea level [Selley, 1998; Mulholland, 1998]. First-order eustatic cycles span at least 50 million years and apply to time-scales on the order of the fossil-bearing Phanerozoic age (Table 2-1). Second-order eustatic cycles span from 3 to 50 million years and apply to time-scales on the order of geologic eras. Third-order eustatic cycles span from 300,000 to 3 million years and apply to time-scales commonly associated with seismic stratigraphy.

The application of sequence stratigraphy to reservoir characterization has many advantages. Mulholland [ 1998] observed that chronostratigraphic interpretation of seismic lines makes it possible for geophysicists to estimate the geologic age of seismically observed strata to at least the level of a geologic period; improve the accuracy of facies identification; identify probable source rock intervals and the location of probable reservoir facies; and develop both tectonic and sedimentation histories of new or poorly understood basins. Geologists can apply sequence stratigraphy to both clastic and carbonate systems. Sequence stratigraphy can be used to develop more accurate surfaces for mapping and correlating facies; predict reservoir, source and sealing facies; identify stratigraphic traps; and project reservoir trends into areas with limited data.

Despite its many advantages, sequence stratigraphy is based on some fundamental assumptions that are subject to dispute [Selley, 1998]. For example, one assumption of sequence stratigraphy is that changes in sea level are global and can be correlated. Another assumption is that seismic reflectors are time horizons, even though seismic reflections occur at the interface between two different lithologies, rather than changes in time. If the assumptions are not satisfied, the application of sequence stratigraphy can yield inaccurate results.

John R. Fanchi 31

[ CS-2. Valley Fill Case Study: Geologic Model ]

The valley fill model presented here illustrates the application of an integrated flow model to a valley fill system. An incised valley is formed

by the incision and fluvial erosion of an existing facies [Peijs-van Hilten, et al., 1998; Weimer and Sonnenberg, 1989]. The incised valley forms during

a fall in relative sea level. The receding sea level exposes older deposits to incisement by drainages. The base of the incised valley is a sequence boundary that is referred to as the LSE, or Lowstand Surface of Erosion.

If the sea level starts to rise again, the initial deposition into the incised valley is typical of flooded systems. During this period of transgression, the incised valley is filled by a variety of fluvial, estuarine and marine environments. When the period of transgression ends, the surface of the filled valley is covered by a new depositional layer associated with flooding. The top of the valley fill is a second sequence boundary that is referred to as the TSE, or Transgressive Surface of Erosion. A typical incised valley is characterized by a set of fluvial system tracts bounded below by an LSE and above by a TSE. The LSE and TSE are key surfaces in the description of the geologic system.

The Valley Fill reservoir is a water-wet sandstone. It is interpreted as a meandering channel through an incised valley that has a regional dip. Six

wells are productive in the channel. Six other wells and seismic data help delineate the channel boundaries. The location of wells in the area are shown

in Figure CS-1A in Chapter 1. Wells 1 through 6 are productive. The top and base of the reservoir in each productive well are listed in Table CS-2A.

Table CS-2A. Top and Base of the Valley Fill Reservoir

Well Depth to Top of Reservoir (feet)

8435

8430

8450

8440

8455

8440

Depth to Base of Reservoir (feet)

8555

8550

8570

8560

8575

8560


Details supporting the geologic interpretation are presented in subsequent

sections.

I Exercises I

2-1.

2-2.

2-3.

2-4.

Tectonic plates move relative to one another at the rate of up to 4

inches per year. How far apart would two plates move in 135 million

years if moving at the maximum rate? Express your answer in miles.

The temperature of the earth's crust increases by about 1 ~ for every

100 feet of depth. Estimate the temperature of the earth at a depth of

two miles. Assume the temperature at the earth's surface is 60~

Based on the data in Table CS-2A, estimate the gross thickness of the

Valley Fill reservoir.

Prepare a structure contour map of the Valley Fill reservoir using

Figure CS-1A and data in Table CS-2A.

Chapter 3 Petrophysics

The study of the mechanical and acoustical properties of reservoir rocks and fluids is the focus ofpetrophysics. Petrophysical information is valuable in well logging and of growing importance in time-lapse seismology and reservoir modeling. This chapter defines petrophysical parameters, introduces elasticity theory, identifies sources of petrophysical information, and describes a petrophysical model.

I 3.1 Elastic Constants ]

The behavior of an object when subjected to deforming forces is described by the theory of elasticity. Elasticity is the property of the object which causes it to resist deformation. The deforming force applied to one or more surfaces of the object is called "stress." Stress has the unit of

Yield Point

/ ~ Breaking Point

--Elastic Behavior

Strain

pressure, or force per unit area, Figure 3-1. Stress-Strain Curve for an Elastic Solid

33


and is proportional to the force causing the deformation. The deformation

of the object in response to the stress is called"strain." Strain is a dimension-

less quantity that reflects the relative change of the shape of the object as a

result of the applied stress. Figure 3-1 illustrates the relationship between

stress and strain.

If the stress is not too great, the object can return to its original shape

when the stress is removed. In this case, Hooke's law states that stress is

proportional to strain. The proportionality constant is called the elastic

modulus. Elastic modulus is the ratio of stress to strain:

Elastic modulus = stress

strain (3.1.1)

Dimensional analysis shows that elastic modulus has the unit of pressure.

The elasticity of a substance determines how effective the object is in

regaining its original form.

Stresses are either one or a combination of three basic stresses:

compressional stress, tensile stress, and shear stress. The corresponding

strains are compressional strain, tensile strain, and shear strain respectively.

The corresponding elastic moduli are bulk modulus, Young's modulus, and

shear modulus. Bulk modulus is a measure of the resistance of an object to

a change in its volume. Young's modulus is a measure of the resistance of

an object to a change in its length. Shear modulus is a measure of the

resistance of an object to the movement of the plane of contact between two

contiguous parts of the object in a direction parallel to the plane of contact.

A compressional or

volume stress is a deform-

ing force applied to the

entire surface of an object

(Figure 3-2). A negative

compressional stress is

sometimes referred to as an

" e x p a n s i o n a l " stress.

Compressional stresses can

change the volume and

shape of the object.

c==:>

Compressing Force

Figure 3-2. Compressional Stress

John R. Fanch i 35

Compressional stress is the ratio of the magnitude F of the force applied perpendicularly to the surface of the object divided by the surface area A of the object. The corresponding compressional strain is the ratio of the change in bulk volume A VB of the object divided by the original bulk volume VB of

the object. The ratio of compressional stress to compressional strain is the bulk

modulus K of the object, thus

F / A - A P A P

X = - A V B / V B = A V B / V ~ = - V s A V s (3.1.2)~

where-Ap is the pressure applied to the surface of the object. The negative sign is inserted to assure that K is always a positive number.

Bulk modulus measures the opposition of the object to compressional stresses. The smaller the change in volume A VB for an applied stress Ap, the larger the bulk modulus. The reciprocal of bulk modulus is bulk compressibility cB, thus

I A V B 1 c B = - v 8 AP - K (3.1.3)

Compressibility can be used to estimate subsidence of a layered reservoir. Suppose the reservoir is divided into a sequence of permeable and impermeable layers. Assuming the area of each layer does not change, the bulk compressibility becomes CB =- A h / h A P where h is thickness and Ah is the change in thickness h. The sum of the changes in thickness of each layer subject to pressure depletion is the subsidence of a formation. The layers of interest are those layers that are both permeable and in communication with the source of pressure depletion.

Tensile stress is defined as the magnitude F of the force applied along an elastic rod divided by the cross-sectional area A of the rod in a direction that is perpendicular to the applied force. The corresponding tensile strain is the ratio of the change in length AL of the rod to the original length L. Young's modulus E is defined by the ratio


tensile stress F/A E = tensile strain AL/L (3.1.4)

The smaller the deformation for a given tensile stress, the larger the value of Young's modulus.

/ Aw

AL b--q

m

I

I i=z~ I

I

I

Extensional Force

Figure 3-3. Tensile Stress

Suppose the rod is a bar. Tensile stresses can change the length of the

bar by either stretching or compressing the bar (Figure 3-3). When the length

L of the bar changes, the width w also changes. Poisson's ratio o is a measure

of the relative magnitude of these changes. It is the ratio of the fractional change in width Aw/w to the fractional change in length AL/L, thus

Aw/w cy - AL/L (3.1.5)

If the rod is a cylinder instead of a bar, the width of the bar is replaced by the diameter of the cylinder in Equation (3.1.5).

Another important measure of solid deformation is shear modulus. Shear

modulus is also called rigidity modulus. As with the other moduli, it is the

ratio of a stress to a strain. Shear stress is the ratio of the magnitude of the

tangential force F to the areaA of the face of the object being sheared. Shear strain is the ratio of the horizontal distance Ax that the sheared face moves

to the height h of the object. These terms are illustrated in Figure 3-4 for a

John R. Fanchi 37

Force Ax

eight h I

I Fixed Face with Area A

Figure 3-4. Shear Stress

simple geometric object. Notice that one face of the object is fixed, so that the applied tangential force acts on the unrestrained surface of the object. In terms of these definitions, shear modulus I~ is given by

shear stress F/A shear strain Ax/h (3.1.6)

Bulk modulus K and shear modulus ~ are related to Young's modulus E and Poisson's ratio o by the expressions

E K - 3(1 - 2o) (3.1.7)

and

E - (3.1.8)

~t 2 + 2 o

Values of Poisson's ratio o range from 0.05 for very hard, rigid rocks to about 0.45 for soft, poorly consolidated materials [Telford, et al., 1990, Chapter 4]. The values of E, K and ~ range from 1.4 • 106 to 28.9 • 106 psia (10 to 200 Gpa). Telford, et al. noted that Young's modulus E usually had the largest value while shear modulus Ix had the smallest value of the three


quantities E, K and I ~. Young's modulus and Poisson's ratio are usually

derived from well log or seismic survey measurements of seismic velocities.

The calculation of moduli from seismic velocities is discussed below.

[ 3.2 Elasticity Theory I

An elastic body is a solid body that can be deformed by the application of

an external force. The deformation of an elastic body may be described by

analyzing the displacement of a particle in a general anisotropic medium

from its initial position. Our discussion of elasticity will refer to mathemati-

cal objects known as tensors.

A tensor in an n-dimensional space is a function whose value at a given

point in space does not depend on the coordinate system used to cover the

space. Examples of tensors include temperature, porosity, velocity and

permeability. Temperature and porosity are scalar tensors, or tensors of rank

zero. Velocity is a tensor of rank one, and permeability is a tensor of rank

two. A tensor of rank one in 3-D can be written as a 3 x 1 matrix or column

vector with three rows and one column. A tensor of rank two in 3-D can be

written as a 3 x 3 square matrix, that is, a matrix with three rows and three

columns. The permeability tensor and the stress tensor may be written as 3

x 3 matrices.

In Figure 3-5, we let

denote the initial distance to

a particle at point P0 from

the origin of the coordinate

system and ~ denote the dis-

placement of the particle to

point P from P0. Application

of external forces changes

the position of the particle

from P0 to Pl. The change of position

P~ relative to P0 is expressed

by the displacement

Pl

zjr

Figure 3-5. Displacement from Po to P~

John R. Fanchi 39

3

-S = 2 e j s j j=l

(3.2.1)

where { ~j } are unit vectors (f, .],/~ ) and {sj} are the components of the displacement vector. The components {sj} are functions of space and time. If we write the spatial coordinates {x, y, z} as {ut, u 2, u3}, the change in the displacement vector ~ with respect tothe spatial coordinates is

3 3 OSj 3 OSj

k=l j= 1 OUk "= k=l OUk - ( a a .

(3.2.2)

The derivative {OS/OUk} are the components of a tensor of rank two. We can rewrite c3~ in terms of symmetric and antisymmetric terms by writing the ith component as

l ( O s k 1 OS~ OSklSU k - Os~ ) 5 u k c3 u k

(3.2.3)

o r

5si = 11 ;kS u k - { ,.kS uk (3.2.4)

The antisymmetric terms {~ik} represent a rotation, while the symmetric terms {rlik} represent a distortion or strain.

Element ik of the strain tensor is

1 ( Os~ Osk ] (3.2.5)

The strain tensor is symmetric because rli k = ~ k i . The diagonal elements I =k


of {]]ik} represent distortions, while the off-diagonal elements I ~ k of {]]ik} represent shear strains.

Hooke's Law

The strain tensor may be written as the matrix

~ 1 1 T] 12 'q 13

'[] -- ]'! 21 'q 22 ~ 23 (3.2.6)

]] 31 l] 32 'q 33

Similarly, the stress tensor may be expressed as the matrix

a 11 ~ 12 (Y 13

CY - ~ 21 ~ 22 CY 23 (3.2.7)

31 (Y 32 (Y 33

The elements of the stress tensor have units of pressure, namely force per

unit area. Normal stresses are given by the diagonal elements {oi, for I = k},

and tensional stresses are given by off-diagonal elements {Oik for I e k}. Like

the strain tensor ~, the stress tensor o is symmetric, i.e., oi, = oki.

If the deformation is small, Hooke's Law states that strain 1"1 is propor-

tional to stress o. Hooke's Law lets us express stress in terms of strain as

3 3

o~ - Z 2 courl u (3.2.8) k=l l=1

where the 81 elements of the stress-tensor {co.u} are called elastic constants.

Alternatively, we can write strain in terms of stress by using the expression

3 3

1] u - Z Z ~/ Ou~ O " (3.2.9) i=1 j = l

when the 81 elements of {Y0,~} are called elastic compliances.

J o h n R. F a n c h i 41

Symmetries may be used to reduce the number of nonzero elastic constants. If the elastic body is isotropic, the elastic constants have only two independent components ~, and It. The components ~, and ].t are called Lam6

constants. The stress-strain tensor for an isotropic, elastic body may be written in terms of the Lam6 constants as

Co.kl - ~ 5 0.5 kl Jr ~[ (S ikS jl + 5 il5 j k ) (3.2.10)

where 8 ~ is the Kronecker delta function. The Kronecker delta function 8 ~

= 1 if a = [3, and 6~1~ = 0 if a ~ ~. The Lam6 constant [.t is called the shear modulus or rigidity and relates

the element of the stress tensor to the strain tensor by

~0 - ~trl0. f o r / ~ j (3.2.11)

In addition to the above relationship, the Lam6 constants ~, and [.t are related to Young's modulus and Poisson's ratio.

Effective Vertical Stress

We illustrate stress tensor concepts by calculating effective vertical stress

[Badri, et al., 2000]. The effective stress tensor o 0 for a porous medium is

the difference between the total stress tensor S o. and the pore pressure P, thus

(Y o - So - P 6 o (3.2.12)

The vertical component of { (3" o. } is

(Y 33 = S33 - P (3.2.1 3)

The pore pressure P is internal pressure, and the confining stress S33 is the external pressure. In the case of universal compaction in the vertical

direction, the vertical component of total stress $33 at depth h relative to a

reference depth, or datum, href represents the weight of the fluids and the


formation in the interval href <- z <_ h. If the density p of the fluid-filled medium is known as a function of depth for this interval, then S33 is given by the integral

S33- g p(z )dz ,-ef

(3.2.14)

and g is the acceleration of gravity [Badri, et al., 2000; Economides, et al., 1994]. Formation density P may be obtained from a density log. If density is approximately constant (p(z) - Pc) in the interval hre f < z < h, then

$33-Ocg(h-h ee) (3.2.15)

Pore pressure may be estimated using pressure transient analysis or seismic velocities.

I 3.3 Acoustic Velocities I

Seismic compressional velocity (lip) and shear velocity (Vs) for an isotropic, homogeneous medium are calculated using the equations [Mavko, et al., 1998; Sch6n, 1996; McQuillin, et al., 1984]:

I 4 S S - g s a t + - 3 ~ Vp- OB (3.3.1)

and

P~ (3.3.2)

where

John R. Fanchi 43

S stiffness

gsa t bulk modulus of rock with pore fluid shear modulus

PB bulk density

Stiffness S in Equation (3.3.1) is a measure of the rock frame stiffness and pore fluid stiffness. Inspection of Equation (3.3.1) shows that compressional velocity increases as stiffness increases. Fluid movement in a reservoir can change the bulk modulus in that region. A change in bulk modulus changes both stiffness and compressional velocity.

Bulk density for a porous rock is given by 98 = (1 - ~))Pm + ~) Of where

P m density of rock matrix grains

Py fluid density = Po So + pw Sw + Pg Sg porosity

Bulk and fluid densities and porosities are usually derived from a combination of the neutron porosity and density logs. Oil, water and gas densities

(Po, Pw, Pg) and saturations (So, Sw, Sg) can be obtained from a variety of sources, including laboratory measurements on fluid samples and well logs. Porosity depends on fluid pressure P and the porosity compressibility q = (1/d~)(8~/O P ) r at constant temperature T. The relationship be-

tween porosity, fluid pressure, and porosity compressibility is - ' l *( ..)]'.Thesubscript"init" re~rerstoinitialvaluesof ~ -- ~tntt l + c P - eiimt

the associated variables. A decrease in fluid press re relative to its initial value results in a decrease in porosity.

Measurements ofcompressional and shear velocities are often reported using the ratio Ve/Vs. Examples are presented by Mavko, et al. [ 1998]. The advantage of working with the ratio is the elimination of bulk density, which may be a poorly known quantity. This can be seen by taking the ratio of Equations (3.3.1) and (3.3.2) to find

I 4, VP_ Ksat+ 3 = + Ksat Vs ~t ~t

(3.3.3)


Noting that the moduli Ksa, and I.t are greater than zero, we see from Equation (3.3.3) that the ratio Vp/Vs >-- x/4/3.

I 3.4 Gassmann's Equation ]

A widely used expression for bulk modulus K was derived by Gassmann [ 1951 ] from the theory of elasticity of porous media [also see Mavko, et al., 1998; Sch6n, 1996; McQuillin, et al., 1984]. Gassmann's expression for the saturated bulk modular K~,,t is

K~,~t - Kdo , + Km (3.4.1) * i 1 - * Ka,-y

KU K m K 2

where

Kd~y bulk modulus of porous matrix ("dry flame") K m bulk modulus of rock matrix grains Kf bulk modulus of fluid

The bulk modulus of the fluid in Equation (3.4.1) is the inverse of fluid compressibility: K I - 1/c I. For a mixture, fluid compressibility can be estimated from a volume average of the phase compressibilities (Co, Cw, Cg), thus c I - Co So + cwSw + cg Sg for an immiscible oil, water and gas mixture.

Once K~a, is known, it is possible to estimate compressional and shear velocities for an isotropic, homogeneous medium using Equations (3.3.1) and (3.3.2). Fluid movement can change the bulk modulus in a region of the reservoir according to Equation (3.4.1). The change in Ksa, leads to a change in stiffness and compressional velocity, as illustrated in Equation (3.3.1). By contrast, Equation (3.3.2) shows that shear velocity does not depend on bulk modulus. Therefore, shear velocity is relatively insensitive to pore fluid content. Measurements ofcompressional velocity and shear velocity can be used to distinguish structure from fluid content. The technology for

John R. Fanchi 45

monitoring both compressional and shear velocities is referred to as full vector wavefield, or multicomponent, imaging.

Gassmann' s equation represents perfect coupling between the pore fluid and the solid skeleton of the porous medium [Sch6n, 1996]. This relation is strictly valid only for isotropic, homogeneous, monomineralic media. Other petrophysical models can be used in integrated flow models, such as a frequency dependent theory presented by Biot [ 1956], and Geertsma and Smit [1961]. Gassmann's equation is the zero frequency limit of more general equations presented by Biot and Geertsma. Consequently, Equation (3.4.1) is sometimes called the BGG equation or Biot-Geertsma-Gassmann equation. Gassmann's equation is widely used in rock physics because of its relative simplicity and limited data requirement.

At least four petrophysical parameters must be included in an integrated flow model (IFM) to allow the use of Gassmann's equation: the dry frame bulk modulus of the porous matrix Kdry, the bulk modulus of the rock matrix grains Km, the effective shear modulus ~, and the matrix density 9m" Each of the petrophysical parameters {Kd, y, Kin, kt, 9m} can vary throughout the reservoir, so the IFM should be able to include these parameters as spatially dependent variables. For example, the bulk modulus of the dry frame porous matrix Kd,-y and the shear modulus kt depend on lithology and spatially dependent porosity. The bulk modulus of the rock matrix grains Km depends on lithology, and should equal the bulk modulus of the porous matrix Kd,.y when porosity equals zero. Variations of the petrophysical model can require additional input data, such as effective stress and clay content.

In addition to rock properties such as porosity and moduli, Gassmann's model depends on phase properties and fluid saturations. All of the required input can either be directly derived or at least estimated from well logs. A more detailed discussion of data sources follows.

I 3.5 Moduli from Acoustic Velocities [

Measurements of shear and compressional velocities provide a direct means of determining bulk moduli. Equations (3.3.1) and (3.3.2) can be rearranged to express moduli in terms of measured acoustic velocities, thus the bulk modulus of rock with pore fluid is


E 41 gsat =pB m2p -5V2 (3.5.1)

Similarly, measurements of shear velocity can be combined with measure-

ments of bulk density to yield the effective shear modulus.

g = 98V 2 (3.5.2)

Recall that bulk density requires both matrix density and fluid density. When velocities, particularly shear velocities, are not available, estimates

can be made from general correlations. An example of a correlation between

acoustic velocities in sandstone, porosity, and clay content C is given by

Castagna, et al. [1985]:

V e = 5 .81- 9.42 ~ - 2.21 C (3.5.3)

and

V s = 3.89 - 7.07 ~ - 2.04 C (3.5.4)

The units are km/s for acoustic velocities, while clay volume and porosity are in fractional content. Notice the linear dependence of acoustic velocity on porosity. Similar correlations are available from such sources as Han, et al. [ 1986], Vemik [ 1998] and Mavko, etal. [ 1998]. Nolen-Hoeksema [2000] discuss predictions of low-frequency elastic wave response to fluids based on Gassmann's equation and a range of modulus-porosity relations.

The grain modulus Km is often estimated from lithology. For example, Km for quartz is 38 GPa (5.5• 106 psia). Alternatively, measurements of shear velocity and compressional velocity at zero porosity yield an effective bulk

modulus that equals grain modulus, thus

K m = K a r y - K s a t at ~ - 0 (3.5.5)

This result can be derived using Gassmann's equation and recognizing that bulk modulus equals grain modulus as porosity goes to zero. Bulk modulus

John R. Fanchi 47

can be calculated from measured acoustic velocities, fluid properties, matrix density and grain modulus using the equation

1-~) / gsa t (~ -1

Kf K m (3.5.6) l-d? 2 gsa t

Kf Km Km K2m

This equation was derived by solving Gassmann's equation for dry flame bulk modulus, and calculating saturated bulk modulus using Equation (3.5.1).

Correlations ofmoduli with parameters such as porosity and clay content have been published in the literature. For example, Castagna, et al. [ 1985] present a correlation for the moduli ofclastic silicate rocks. Figure 3-6 shows moduli for quartz sandstone data obtained by Murphy, et al. [ 1993]. The linear correlation in Figure 3-6 is approximately correct for porosity values less than 35%. The grain modulus Km equals the dry frame bulk modulus Kay when porosity equals zero.

6.00E+06 . , - ~

r ~

r ~

~ 4.00E+06

o

2.00E+06

0.00E+00 I I I I I I I I 0.00 0.05 0.10 0.15 0,20 0.25 0.30 0.35

Porosity

--o- Bulk (psia) ~ Shear (psia)

Bulk and Shear Moduli for Quartz Sand Correlation Valid for Porosity < 35%

8.00E+06

Figure 3-6. Illustration of the Relationship between Bulk and Shear Moduli


A variation of the Han-Eberhart-Phillips model [see Section 7.5 of Mavko, et al., 1998] is worth considering in more detail because it expresses moduli as functions of porosity (D, effective pressure Pe, and clay content volume fraction C. Effective pressure is the difference between confining (overburden) pressure and pore pressure P

Peff = P~o, - a P (3.5.7)

with a correction factor

~ = 1 - ( 1 - * ) \K= (3.5.8)

that depends on porosity, saturated bulk modulus K,~, and grain modulus K~. Confining pressure P~o, may be estimated as P~o, = YoB z where z is depth and Y os is average overburden pressure gradient. The dry frame bulk modulus in the HEP model has the functional dependence

Ka~y = a o + a I Pe e' + a2d? + a3r 2 + a4r ee e2 + a 5 (3.5.9)

with regression coefficients {a0, at, a2, a3, a4, as e~, e2}. Grain modulus K~ is calculated from dry frame bulk modulus Kary when porosity equals zero, thus

K m = a o + a I P : ' + a s , f - C (3.5.10)

The functional dependence of shear modulus is

~ = ( X 0 +13f, l e : 1 +C/ ,2 r162 2 +13f ,4r 2 + 0~5%/"C (3.5.11)

with regression coefficients {0~ o, 0~1, 0~2, 0~3, 0~4, 1~5, el. 1~2 }. Grain density p ~ may be approximated as a function of clay content, thus

John R. Fanchi 49

9m = bo + b lC + b 2C 2 (3.5.12)

where {b0, bl, b2} are regression coefficients. Petrophysical modeling requires data that is not usually included in a

traditional reservoir simulation study. In particular, the new data are Kay, Kin, ~t, and 9m" The references give values that may be used if the data are not available from well logs such as shear wave logging tools, or laboratory measurements of parameters such as acoustic velocities or the dry frame Poisson's ratio.

Before leaving this discussion, it is worth noting that several reservoir geomechanical attributes can be estimated once compressional and shear velocities are known [Tiab and Donaldson, 1996; Mavko, et al., 1998]. Poisson's ratio v is expressed in terms of seismic velocities using Equations (3.1.7), (3.1.8), (3.5.1) and (3.5.2). Combining these equations and simplifying gives

o.svy - vy v - V2_ Vs 2 (3.5.13)

The value of v for fluids is 0.5 because fluids have no resistance to shear, therefore the shear modulus for fluids is 0 and the corresponding shear velocity is 0 [Telford, et al., 1990, Chapter 4]. The dynamic Young's modulus E is calculated from dynamic Poisson's ratio as

E =2(1+ v)~t (3.5.14)

where I ~ is effective shear modulus. More accurate calculations of geomechanical properties are obtained from static measurements of Young's modulus and Poisson's ratio [Tiab and Donaldson, 1996].

Uniaxial compaction Ah may be estimated using

1/1+ v) A h - -~ 1- v ~ qhnetAP (3.5.15)

50 Shared Earth Model ing

where hne t is net thickness, ~ is porosity, and c4, is porosity compressibility. The change in pore pressure is A p = p - Pinit where P is current pore pressure and Pinit is initial pore pressure. Pore pressure is approximately equal to the fluid pressure of the dominant phase in the pore space. Porosity compressibil-

ity is defined in terms of fluid pressure P and porosity ~ as

l t g ~ c , - ~ a P (3.5.16)

Confining pressure Pco, may be estimated from depth z and an average

overburden pressure gradient yoB such that ,~ = YoB z. Fracture gradient

is estimated as

V Pcon V ~ ' r - 1 - v z ~ 1 - v ~'~ (3.5.17)

I CS-3. Valley Fill Case Study" Bulk Moduli I

The area of interest in the Valley Fill case study is the area containing productive wells shown in Figure CS-1A. This area has an average effective porosity of 22% and an average permeability of 150 md. Core from the

productive interval was water-wet sandstone. No measurements of moduli were available, therefore correlations

published by Murphy, et al. [ 1993] and shown in Figure 3-5 were used to determine grain, bulk and shear moduli for sandstone with a porosity of 22%.

These moduli were then used to compute shear and compressional velocities. The bulk frame modulus is 16.2 GPa (2.35 x 106 psia), the grain modulus is 36.0 GPa (5.22 x 106 psia), the shear modulus is 18.5 GPa (2.68 • 106 psia), and the grain density is 2.67 g/cc (167 Ibf/cu ft). The unit lbf refers to pounds of force. These petrophysical properties were applied throughout

the area of interest. Initial water saturation in cores from the productive channel ranged from

30% to 100%. Cores from wells outside of the productive channel were water

saturated.

3-1.

3-2.

3-3.

3-4.

3-5.

3-6.

3-7.

John R. Fanchi 51

Exercises ]

Use Figure 3-6 to estimate the bulk and shear modulus of sand with

22% porosity. What is the grain modulus?

Calculate Young's modulus and Poisson's ratio using the bulk and shear moduli in Exercise 3-1 and Equations (3.1.13) and (3.1.14).

Suppose p(z) = a + bz" where a, b, n are constants. Calculate ff 33

using Equation (3.2.14).

Derive Equation (3.5.6) from Gassmann's equation for bulk modulus

and Equation (3.5.1).

Derive Equation (3.5.13) from Equations (3.3.1) and (3.3.2).

Assume the Valley Fill channel has an oil saturation of 70%. Use the

area in Figure CS-1A and the gross thickness from Exercise 2-3 to estimate the bulk volume of the channel.

Use the bulk volume from Exercise 3-6, a porosity of 22% and a net

to gross ratio of 100% to estimate the amount of oil that is in the Valley Fill channel. Express the volume of oil in barrels and cubic

meters.

Chapter 4 Well Logging

Well logs provide valuable information about the formation within a few feet of the wellbore. The properties of the formation are inferred from the well log response. This chapter describes the different types of logs that are available, explains why combinations of logs are used, and discusses techniques and limitations of well log interpretation. The technology of well logging is continuing to evolve, therefore this discussion focuses on the more common logs. A thorough review of well logs is beyond the scope of this book, and the interested reader should consult the references, e.g., Asquith and Gibson [1982], Bassiouni [1994], Felder [1994] and Selley [1998], or contact appropriate service companies for details of specific logs.

[ 4.1 Principles of Logging I

The giga scale is the scale that is comparable in size to the reservoir, but the giga scale is too coarse to provide the detail needed to design a reservoir development plan. The mega scale is the scale at which we begin to integrate well log and well test data into a working model of the reservoir. Several types of logs can contribute to the mega scale level of reservoir characterization. Many of these logs are discussed below in the context of formation evaluation.

52

John R. Fanchi 53

Formation evaluation is the acquisition of data and quantification of parameters needed for drilling, production, reservoir characterization and reservoir engineering. The data may be determined by direct measurement or by remote sensing. Data from direct measurements include drill cuttings, core samples, fluid properties and production testing. Data obtained by remote sensing ranges from electromagnetic and sonic wave signals, to the detection of elementary particles and the monitoring of pressure changes in the reservoir.

Well logs are obtained by running a tool into the wellbore. The tool can detect physical properties such as temperature, electrical current, radioactivity, or sonic reflections. Logging tools are designed to function best in certain types of environments. The environment depends on a variety of factors, including temperature, lithology, and fluid content. The theoretical analysis of log signals is usually based on the assumption that the formation is infinite in extent with homogeneous and isotropic properties. Tool performance will not be optimal in other environments.

Figure 4-1. Schematic of Invasion Zones

The measurement environment is usually represented by an idealized representation that includes several zones. Figure 4-1 illustrates the distribution of drilling fluids in the vicinity of a newly drilled wellbore. Four


zones can be identified near the wellbore: the mud cake, the flushed zone,

the zone of transition, and the uninvaded zone. The pressure difference

between the borehole and the formation can force drilling mud into the

permeable formation. Large particles in the drilling mud will be filtered out

at the sandface and create a mud cake. The remaining filtered liquid is called

the mud filtrate. The pressure differential forces the mud filtrate into the

formation. The mud cake can reduce the productivity or inj ectivity of a well.

A measure of the degree ofwellbore damage caused by the mud cake is the "skin" of the well.

Pore space in the flushed zone has been completely swept by mud filtrate

from the drilling operation. Beyond the flushed zone is the zone of transition

between the flushed zone and the uninvaded zone. The volume of pore space

that is swept by the mud filtrate ranges from completely swept at the interface

between the flushed zone and the zone of transition to completely unswept

at the interface between the zone of transition and the uninvaded zone. The

flushed zone and the zone of transition are considered invaded zones because

original in situ fluids have been displaced by fluids from the borehole. The

uninvaded zone is that part of the formation which has not been altered from

its original state by the drilling operation. Reservoir rock and fluid properties

in the uninvaded zone are needed for accurate reservoir characterization and

are the target of near wellbore remote sensing tools.

Depth of Investigation The depth of investigation is a measure of the volume of the formation

that is primarily responsible for the well log signal. If we assume the

formation has a uniform cylindrical shape for a formation with thickness h,

then the volume investigated is 7zr2h where the radius r is the depth of

investigation into the formation. The depth of investigation of a well log is

usually characterized as shallow, medium, or deep. Depth of investigation

can range from a few inches to several feet.

[ 4.2 Direct Measurement Logs I

Several logs provide direct information about the borehole environment.

They are described below.

John R. Fanchi 55

Driller's Logs The driller's log is a record of observations made by the driller during

the course of drilling the borehole. The driller's log includes such information as rate of penetration of the drill bit, rock and fluid types encountered as a function ofborehole depth, and any other information that is considered worth recording by the driller. The rate of penetration can be used to distinguish between hard and soft formations. In addition to the driller's log, mud logs are recorded while the well is being drilled. They provide measurements of parameters such as the rate of penetration, the detection of natural gas, and the types of cuttings that are encountered at various depths along the wellbore. Combining this information with rock type observations and correlations with other logs provides information that can be used to identify formations.

Caliper Logs Ideally, the borehole shape will be a uniform cylinder with the diameter

of the drill bit used to drill the borehole. In practice, the diameter and expected cylindrical shape of the borehole may differ substantially from a

uniform cylinder with the diameter of the drill bit. The actual borehole shape depends on the type of formation, and it can be measured with a caliper log.

Sample Logs Sample logs are collections of physical samples of cores and cuttings

that are extracted from the borehole. Cuttings are bits of rock that have been produced by the action of the drill bit during the drilling operation. Cuttings are returned to the surface by the circulation of drilling mud. Once at the surface, they can be analyzed to indicate the lithology of the drill bit environment. Cuttings are not always representative of the current position

of the drill bit because bits of rock from shallower formations may be broken off by the circulating mud and brought to the surface with the cuttings. Nevertheless, cuttings are the first solid evidence of lithology obtained at the well site.

Core samples are obtained by a coring procedure in which the drill bit is replaced by a coring bit. Since coring requires a separate bit, the amount of core depends on the cost and relative benefit of acquiring core from a particular borehole. Not all wells will be cored. Those wells that are cored


are usually cored over a limited depth interval, such as the depth interval associated with the productive formation. Cores can be used for a variety of reservoir characterization purposes because they provide a small sample of the reservoir. Analyses of core samples should recognize that some characteristics of the core may change during the extraction process because of changes in temperature, pressure and borehole fluid invasion. Native state coring may be used to obtain core samples at conditions that are close to original in situ conditions, but the added expense limits the applicability of native state coring to special situations.

[ 4.3 Lithology Logs I

We consider two types of logs that may be considered lithology logs: the SP log and the gamma ray log. They are considered lithology logs because they can be used to determine rock type.

Spontaneous Potential Logs The spontaneous potential (SP) log is probably the most common well

log. The SP log is an electric log that provides information about permeable beds. It records the direct current (DC) voltage difference, or electrical potential, between two electrodes as a function of depth. One of the electrodes has a fixed position, and the other electrode is allowed to move. The spontaneous potential Esp can be estimated from the empirical relationship [Selley, 1998, pg. 56]

Rmf Esp - K: log Rmf - ( a + a 2 T) log (4.3 1)

Rw l Rw �9

where K is a linear function of temperature Twith parameters a~ and a2, Rmf is the resistivity of the mud filtrate and Rw is the resistivity of formation water. The temperature dependence of resistivity must be taken into consideration when using the above equation for Esp.

The electrical potential is associated with the presence of ions in the vicinity ofthe well. The number of ions in in situ fluids depends on the type

John R. Fanchi 57

of fluid occupying the pore space. In situ brine usually contains several inorganic salts that ionize in water. The potential difference is a function of the difference in salinity between borehole fluid and formation water. If the formation water salinity is greater than the mud filtrate salinity, the SP will deflect in a negative direction. A small SP log deflection implies the formation does not contain high salinity fluids, such as impermeable shales. A large SP log deflection implies the formation does contain high salinity fluids, such as permeable beds. If the SP log encounters a shale, the small SP deflection associated with the shale beds is used to establish a "shale baseline." An increase in the SP deflection relative to the shale baseline is often used as an indicator of more permeable beds.

SP 10gs can be used for a variety ofpurposes. They can detect permeable beds and indicate their thickness. An examination of the SP log yields information about both net and gross thickness. The shape of the SP log can yield information about the depositional environment. Quantitative estimates of formation water resistivity and formation shaliness can be determined from the SP log.

SP logs have limitations. If the resistivity of formation water equals the resistivity of the mud filtrate, there is no SP. If the borehole fluid is oil, gas

or air, the SP does not exist. Corrections must be applied to the SP log to account for thin beds and large borehole diameters.

Gamma Ray Logs Gamma rays are photons (particles of light) with energies ranging from

104 ev (electron volts) to 107 ev. Gamma ray logs are used to detect in situ

radioactivity from naturally occurring radioactive materials such as potassium, thorium and uranium. In general, shale contains more radioactive materials than other rock types. Consequently, the production of gamma rays by radioactive decay is greater in the presence of shale. A high gamma ray response implies the presence of shales, while a low gamma ray response implies the presence of clean sands or carbonates.

I 4.4 Porosity Logs I

Well logs that can be used to obtain porosity include density logs, acoustic logs, and neutron logs. Each of these logs is described below.


Density Logs The formation density log is useful for determining the porosity of a

logged interval. The porosity is given by

~) _ Pm~-- Pb (4.4.1)

Pma -- Pf

where ~) is porosity, and p~, Pb and Pl are rock matrix, bulk and fluid densities.

Density logs rely on Compton scattering to determine bulk density. In Compton scattering, a gamma ray photon transfers some of its energy to an electron in the inner orbital of an atom. The loss of energy of the photon changes the frequency of the gamma ray photon and can cause the ejection of the electron from the atom. The density log measures electron density by detecting gamma rays that undergo Compton scattering. The intensity of scattered gamma rays is proportional to electron density. Electron density is the number of electrons in a volume of the formation. Electron density is proportional to bulk density. If the measurement of gamma rays is large, it implies a large electron density and correspondingly large bulk density.

Density logs are useful for determining hydrocarbon density and for detecting hydrocarbon gas with low density compared to rock matrix or liquid densities. A small density implies high hydrocarbon gas content, while a high density suggests a low hydrocarbon gas content.

Acoustic Logs Acoustic or sonic logs provide another technique for measuring porosity.

The acoustic log emits a sound wave and measures the transit time associated with the propagation of the sound wave through the medium to a recorder.

The speed of sound depends on the density of the medium, and the density of the medium depends on the relative volumes of rock and pore space. In particular, the bulk density PB of the medium is

(4.4.2)

where ~) is porosity, and p ~ and Pl are rock matrix and fluid densities,

John R. Fanchi 59

respectively. The bulk volume of a formation with large pore space contains more fluid in the bulk volume than a formation with small pore space. Since fluid density is typically several times smaller than matrix density, the bulk density of a system with relatively large pore space is less than the bulk density of a system with relatively small pore space.

The speed of sound v through the formation is approximately given by Wyllie's time average equation

1 ~ ( l- if) - +

V Vf Vma (4.4.3)

where v I is the speed of sound in interstitial fluids, and Vma is the speed of sound in the matrix. Wyllie' s equation can be written in terms of the interval transit time At as

A t - ~ mtf + (1- ~))Atma (4.4.4)

since the speed of sound is inversely proportional to transit time and the subscriptsfand ma again denote fluid and matrix. A long transit time implies a slow speed of sound propagation. The presence of hydrocarbons increases the interval transit time. Solving the transit time form ofWyllie 's equation for porosity gives

A t - Atma ~) = (4.4.5)

A t y - Atma

Corrections for compaction can be made to Wyllie's equation for unconsolidated sands. The compaction factor is an empirical factor that can be estimated from the interval transit time for an adjacent shale.

If we note that sound waves propagate faster in rock matrix than in fluid, we can make some general observations about acoustic log response. Long travel times imply slow speed of sound propagation and large pore space. Conversely, short transit time implies a high speed of sound propagation and small pore space.


Acoustic logs are used for a variety of purposes in addition to porosity

determination. For example, an acoustic log can be compared directly to seismic measurements. The acoustic log therefore serves to constrain the

velocity model used to convert seismic travel times to depths.

Neutron Logs The neutron log can be used to determine the hydrogen content of the

logged interval by counting captured gamma rays or neutrons counted at a

detector. Fast neutrons emitted by a radioactive source in the neutron log are

slowed by collisions to thermal energies. The ensuing thermal neutrons are

captured by nuclei, which then emit detectable gamma rays. Hydrogen nuclei

are especially effective because hydrogen has a relatively large capture cross-

section for thermal neutrons. The neutron log response indicates the

concentration of hydrogen in the fluid filled pore space. Oil and water have about the same concentration of hydrogen, while gas

has a relatively low hydrogen concentration. The presence of a significant amount of hydrogen will appear as a large gamma ray response. A small

response suggests a low hydrogen concentration. If we note that natural gas

such as methane (CH4) has a small hydrogen concentration relative to other

molecules, such as water (H20), a small gamma ray response can be

interpreted as indicating the presence of gas.

I 4.5 Resistivity Logs [

Resistivity logs are the oldest type of well log. They were first applied to formation evaluation in 1927 by Conrad and Marcel Schlumberger and Henri

Doll. We consider some electrical properties of rocks before discussing

specific resistivity logs.

Resistivity is proportional to electrical resistance Re. If we consider a conductor of length L and cross-sectional areaA, resistivity of the conductor

R~ is given by

A g c - R e y (4.5.1)

John R. Fanch i 61

The resistivity R0 of a porous material saturated with an ionic solution is equal to the resistivity Rw of the ionic solution times the formation resistivity factor F of the porous material [Collins, 1961 ], thus

R o = F R w (4.5.2)

Formation resistivity factor F, which is sometimes referred to as the formation factor, can be estimated from an empirical relationship between formation resistivity factor F and porosity ~. The empirical relationship is

F = a~ -m (4.5.3)

where the cementation exponent m varies from 1.14 to 2.52 and the coefficient a varies from 0.35 to 4.78 [Bassiouni, 1994] for sandstones. Both parameters a and m depend on pore geometry: a depends on tortuosity and m depends on the degree of consolidation of the rock.

If a porous medium is partially saturated by an electrically conducting, wetting phase with saturation Sw, the formation resistivity R t of the partially saturated medium is given by Archie's empirical relationship

R t = R o g e r n (4.5.4)

where n is called the saturation exponent. Note that R t = R o when the porous medium is completely saturated by the wetting phase, that is, at Sw = 1.

Combining the above relationships and solving for wetting phase saturation in terms of resistivities and formation resistivity factor gives

wl Rw) n R,

(4.5.5)

The above equation is Archie' s equation for wetting phase saturation, which is usually water saturation.

Resistivity logs measure electrical resistivity in the borehole. The resistivity of formation fluid depends on the concentration of ionized


particles in the fluid. Formation water is a brine with a substantial concentration of ionizing molecules, such as sodium chloride and potassium chloride. The ions in solution decrease the resistivity of the fluids in the formation. Hydrocarbons, on the other hand, generally do not contain comparable levels of ions, and therefore have a larger electrical resistivity. Resistivity differ-

ences between brine and hydrocarbons make it possible to use resistivity logs to distinguish between brine and hydrocarbon fluids. The ability of a rock to support an electric current depends primarily on fluid content in pore space because rock grains are usually nonconductive. Thus, high resistivity

suggests the presence of hydrocarbons, while low resistivity implies the presence of brine. We consider two types of resistivity logs: electrode and induction logs.

Electrode Logs Electrodes in the electrode tool are connected to a generator. Electrical

current passes from the electrodes, through the borehole fluid, into the formation and to a remote reference electrode. Depth of investigation is controlled by the spacing of electrodes and, more recently, by focusing electrical current using logs such as the spherically focused log. The observed current gives information about the resistivity of the formation. Electrode logs must be used with conductive ("salty") mud to allow current to pass through the borehole fluid.

Induction Logs Induction logs measure formation conductivity, which is the inverse of

resistivity, induced by a focused magnetic field. Transmitting coils in the tool emit a high frequency alternating current (AC) signal that creates an alternating magnetic field in the formation. Induced secondary currents are created in the formation by the alternating magnetic field. The secondary currents create new magnetic fields which are recorded by receiver coils. The transmitting and receiving coils are on opposite ends of the tool.

The induction log provides information that is proportional to conductiv-

ity. Induction logs are most accurate when used with nonconductive or low

conductivity mud. They can therefore be used with "fresh" mud, or boreholes filled with air or oil. Induction logs can be used for fluid type evaluation and to identify coals or other nonconducting materials.

John R. Fanchi 63

[ 4.6 Other Types of Logs ]

New logs are being developed as technology evolves and the needs of industry change. Some of these logs do not fit into the major categories presented in previous sections. A few of the logs are described here.

Photoelectric Logs In the photoelectric effect, a low energy gamma photon that collides with

an atom can transfer all of its energy to an inner orbital electron and cause the ejection of the electron from the atom. The amount of energy required to eject the electron depends on the atomic number. Some of the gamma photon energy will be used to eject the electron, and the rest is transferred as kinetic energy. Photoelectric logs measure the absorption of low energy gamma photons by atoms in the formation. They provide information about the atomic number of atoms in the formation that can be used to infer the composition of a formation.

Dipmeter Logs and Borehole Imaging The dipmeter log is a tool for measuring the direction of dip associated

with beds that are adjacent to the borehole. Three or four microresistivity measurements are recorded simultaneously. The measurements are made on different sides of the borehole and then recombined by computer to infer bedding dip. Similar information can be obtained using acoustic borehole imaging tools. Borehole imaging measurements can be used for thin-bed evaluation, fracture identification and analysis, structural and stratigraphic dip analysis, sedimentary facies analysis and textural analysis [Felder, 1994].

MWD and LWD Most logs are run in open holes or cased holes. Some logs may be run

while the well is still being drilled and drill pipe is present in the borehole. These logs are referred to as measurement while drilling (MWD) or logging while drilling (LWD) logs. Logs that can be run during drilling include resistivity logs, gamma ray logs, density logs, neutron logs and electrode logs. These logs are especially useful in long, directionally drilled wells for providing real-time information on borehole environment during the drilling operation.


I 4.7 Reservoir Characterization Issues I

Several issues arise in the application of well logs to reservoir characteriza-

tion. Table 4-1 summarizes the major applications of the most common log

types. Some of the more common issues are discussed in this section.

Table 4-1 Major Applications of Common Log Types

(after Selley [1998, pg. 801)

Log Type

ELECTRIC

SP

Resistivity

RADIOACTIVE

Gamma Ray

Neutron

SONIC

Density

DIPMETER

Lithology

X

X

X

X

Hydrocarbons

X

X

X

X

Porosity Dip

X

X

X

X

Well Log Legacy Well logging technology has developed over many decades. The

improvement in technology from one generation of well logs to another has

been remarkable. This is both an asset and a liability. It is an asset for new

fields where the most modem technology can be applied. It is a liability for

old fields where old logs must be combined with new logs in the analysis

of a field. Care must be taken when working with logs from different generations of technology to be sure that analytical techniques are appropriate for each well log.

Cutoffs The volume of reserves VR is the product of pore volume, hydrocarbon

saturation Sh, and recovery factor RF. Writing pore volume as the product of porosity (D, area A and net thickness h, VR is given by


V n - ~ A h S h R F (4.7.1)

Porosity and saturation are usually obtained from well logs. The estimate of these parameters is often accompanied by the specification of porosity and saturation cutoffs. A cutoff specifies the minimum value of the parameter that is considered a part of the productive formation. Cutoffs may be used for permeability in addition to porosity and saturation.

Cross-plots Cross-plots of well log data may be used to determine such factors as

porosity, lithology and gas saturation. A cross-plot is a plot of one well log parameter against another. For example, the Pickett cross-plot is a cross-plot of porosity versus resistivity. It is motivated by Archie' s equation for wetting phase saturation expressed in the form

R t = ~-mRwSwn (4.7.2)

where the coefficient a = 1 in the empirical relationship for formation resistivity factor F. Taking the logarithm of both sides of the above equation

gives

log R t = - m log ~ + log R w - n log Sw (4.7.3)

It can be seen from this equation that a log-log plot ofporosity and resistivity should give a straight line with slope -m and an intercept that depends on

Rw and Sw.

Another example of a useful cross-plot is the plot of neutron log porosity versus density log porosity. The resulting cross-plot can be used to provide an estimate of the shaliness (shale content) of the formation. The porosity log estimate of shale content can then be used to validate the usually more reliable estimate of shale content obtained from gamma ray logs [Bassiouni, 1994, pg. 318].

Correlations Between Wells Two of the most important uses of well logs is the determination of

formation extent and continuity. Correlations between wells are used to


define formations and productive intervals. An example of a correlation technique is the fence diagram.

A fence diagram is prepared by aligning well 10gs in their proper spatial position and then drawing lines between well logs that show the stratigraphic correlation. Fence diagrams illustrate correlations between wells and can show formation pinchouts, unconformities, and other geologic discontinuities.

Log Suites Modem logging techniques combine logging tools to obtain a more

reliable representation of formation properties. A combination of well logging tools is usually needed to minimize ambiguity in log interpretation, as discussed by Brock [ 1986]. For example, the combination neutron-density log is a combination log that consists of both neutron log and density log measurements. Possible gas producing zones can be identified by the log traces of the combination neutron-density log. The presence of gas increases the density log porosity and decreases the neutron log porosity [Bassiouni, 1994, pg. 329]. Ifa sonic log is added to the log suite, quantitative information about lithology can be estimated using cross-plots, and the log suite can be used to calibrate seismic data.

Sonic log interpretation depends on lithology. In particular, the interval transit time in carbonates depends on the relative amount of primary and secondary porosity. Primary porosity is associated with the matrix, and secondary porosity is associated with features such as fractures and vugs. Subtracting sonic porosity from total porosity recorded using neutron or density logs gives an estimate of secondary porosity.

One more log, the gamma ray log, is usually added to the suite of logs used to evaluate gas bearing formations. The gamma ray log measures natural radioactivity in a formation. It provides a measurement of shale content, and can be used for identifying lithologies, correlating zones and correcting porosity log results in formations containing shale.

! CS-4. Valley Fill Case Study: Well Logs ]

Synthetic well logs are presented in Figure CS-4A for wells 7, 3, and 9 as the figure is viewed from left to right. Again moving from left to right for

John R. Fanchi 67

each well, Figure CS-4A shows the spontaneous potential (SP) log, seismic reflection coefficients (RC), and the resistivity (Res.) log. The SP log and seismic reflection coefficients can be used to estimate regional dip by estimating the dip of the productive interval. The resistivity log shows that the upper part of the productive interval in Well 3 has a higher resistivity than the other wells. This information is consistent with the observation that Well 3 is an oil producer while Wells 7 and 9 are not. Furthermore, the depth where there is a change in resistivity in Well 3 from a higher to a lower value within the productive interval denoted by the SP log is an indication of the depth of the oil-water contact in the productive interval. The combination of well logs supports the valley fill interpretation.

I Exercises I

4-1. Estimate the regional dip using the well logs in Figure CS-4A. Hint: determine the dip angle using the RC trace of all three wells to define the top of the formation.

L r - . . ~

-~ W e n 7 - S P

. . . . . . . W e l l 7 - R C

-- W e l l 7 - R e s .

A W e l l 3 - S P

. . . . . . . W e l l 3 - R C

x W e n 3 - R e s .

W e l l 9 - S P

. . . . . . . W e l l 9 - R C

-- W e l l 9 - R e s .

Figure CS-4A. Well Logs for Valley Fill Reservoir


4-2. Estimate the oil-water contact using the resistivity log for Well 3 shown in Figure CS-4A.

Chapter 5 Geophysics

Developing a picture of the large scale structure of the reservoir is the

primary objective of geophysics within the context of reservoir management.

An image of the reservoir structure is obtained by initiating a disturbance

that propagates through the earth's crust and is reflected at subsurface

boundary interfaces. The reflected signal is acquired, processed and

interpreted. This chapter discusses the physics of waves, describes the

propagation of a seismic wave through the subsurface medium, and discusses

the analysis of seismic data.

I 5.1 Physics of Waves ]

Seismic waves are vibrations, or oscillating displacements from an undis-

turbed position, that propagate from a source, such as an explosion or

mechanical vibrator, through the earth. The energy released by the distur-

bance propagates away from the source of the disturbance as seismic waves.

The seismic waves that propagate through the earth are called body waves

and are either P waves or S waves. P waves are longitudinal waves while

S waves are transverse waves. Longitudinal waves are a class of waves in

which the particles of the disturbed medium are displaced in a direction that

69


is parallel to the direction of propagation of the wave. Transverse waves are

a class of waves in which the particles of the disturbed medium are displaced

in a direction that is perpendicular to the direction of propagation of the

wave. The velocity of P waves depends on the elastic properties and density

of the medium. The velocity of S waves depends on the shear modulus and

density of the medium. S waves do not travel as fast as P waves and Swaves

do not propagate through fluids. Other names exist for P waves and S waves.

P waves are sometimes called compressional, primary or pressure waves,

while S waves are sometimes called shear or secondary waves.

Seismic waves can be detected by a seismometer, which is often referred

to as a geophone on land or a hydrophone in a marine environment. The

output from the seismometer is transmitted to a recording station where the

signals are recorded by a seismograph. The graph representing the motion

of a single seismometer, such as a geophone or hydrophone, is called the

trace of the seismometer. The display of the output is called the seismogram

or seismic section.

A prerequisite to understanding seismic waves is a familiarity with the

physics of waves. Figure 5-1 shows a single wave. The length of the wave

from one point on the wave to an equivalent point is called the wavelength.

For example, the distance from point A to point C is one wavelength. The

number of waves passing a particular point, say point B in Figure 5-1, in a

specified time interval is the frequency of the wave.

. 4 One Wavelength I ~ I

I I

',

Direction of ~ave Motion " ~ -

Figure 5-1. W a v e M o t i o n

The concept of a wave is easily illustrated. Suppose we tie a rope to the

doorknob of a closed door. Hold the rope tight and move it up and then back

John R. Fanchi 71

to its original position. A pulse like that shown in Figure 5-2a should proceed

toward the door. When the pulse strikes the door it will be reflected back

towards us as shown in Figure 5-2b. This pulse is one half of a wave and it

has one half a wavelength. To make a whole wave, move the rope up, back

to its original position, down, and then up to its original position. All of these

motions should be made smoothly and continuously. The resulting pulse should look something like Figure 5-2c. It is a complete wave. We can make

many of these waves by moving the rope up and down rhythmically. The

ensuing series of waves is called a wavetrain. The mathematical equation

for describing the motion of the rope has the same form as the equation for describing the motion of a vibration propagating through the earth. The wave

equation for seismic vibrations is discussed in the next section.

f

a. A Pulse I I

b. Reflection I ~ ~'1 of the Pulse ~ /

J

c. A Wave

One Wave leng th ~ "

Figure 5-2. Creating a Wave

I 5.2 Propagation of Seismic Waves I

Seismic wave propagation is an example of a displacement propagating through an elastic medium. The equation for a wave propagating through an elastic, homogeneous, isotropic medium is

a'u + 2 )v(v. u)- (v• u) (5.2 19 Ot 2 -

where p is the mass density of the medium, ~, and ~ are properties of the


elastic medium called Lam4's constants, and u measures the displacement of the medium from its undisturbed state [Tatham and McCormack, 1991 ]. The following derivation shows how a displacement of an isotropic medium propagates through the medium. The velocities for two modes of propagation

are derived, and the elastic properties X and i ~ are related to elastic properties

discussed in Chapter 3. If the displacement u~ is irrotational, then ui satisfies the constraint

V x u I - 0 (5.2.2)

and Equation (5.2.1) becomes

02UI (;~ + 2g) Ol2 - p V ( V . Ul) (5.2.3)

The vector identity

v (v . , , ) - v • (v • #)+ v '# (5.2.4)

for an irrotational vector is

V(V" " , ) - - V2U/ (5.2.5)

so that Equation (5.2.1) becomes the wave equation

~2 Ul C3t 2

2 v 2 (5.2.6) - - V I U I

The velocity of wave propagation vz for an irrotational displacement uz is

v,- I( +p (5.2.7)

John R. Fanch i 73

A solution of Equation (5.2.6) that is irrotational is the solution for a

longitudinal wave propagating in the z-direction with amplitude u0, frequency ~ and wave number k:

U I -- uoe i ( k z -~ (5.2.8)

If the displacement Us is solenoidal, then the displacement Us satisfies the constraint

V .u~ = 0 (5.2.9)

and Equation (5.2.1) becomes

~2 -s -"-v x(Vxus)

C3 t 2 - 9 (5.2.10)

The vector identity in Equation (5.2.4) reduces to

v x(vx ,,s)-

for a solenoidal vector. Substituting Equation (5.2.11) into Equation (5.2.10) yields the wave equation

0 2 Us

Ot 2 - v~V 2u~ (5.2.12)

The velocity of wave propagation v s for a sinusoidal displacement u s is

Vs -- ~ < 121 (5.2.13)

A solution of Equation (5.2.12) that is solenoidal is the solution for a transverse wave propagating in the z-direction:

74 S h a r e d Ear th M o d e l i n g

uoei( " u s - ~ - o t ) k (5.2.14)

The irrotational displacement/gl represents a longitudinal P (primary) wave, while the solenoidal displacement u~ represents a slower transverse S (secondary) wave. The S-wave corresponds to the shear wave with velocity given by Equation (3.3.2), and the P-wave corresponds to the compressional wave with velocity given by Equation (3.3.1).

Comparing the S- and P-wave velocities in Equations (5.2.7) and (5.2.13) with Equations (3.3.1) and (3.3.2) shows that Lam6's constants are related to bulk and shear moduli by the relations

4 ~, + 2~t = K ~ , + -7 ~t (5.2.15)

.5

and

( L a m e ' ) = ~ ( S h e a r ) (5.2.16)

Using Equation (5.2.16) in (5.2.15) and solving for Lam6's constant ~, gives

2 ~ = Ksa t - -~ ~ (5.2.17)

The saturated bulk modulus gsa t and shear modulus ~ can be obtained using a variety of techniques that were discussed in Chapter 3.

It was noted at the beginning of the above derivation that Equation (5.2.1) was based on the assumptions that the medium was elastic, homogeneous, and isotropic. In practice, these assumptions often yield useful results. Relaxing the assumptions by, for example, attempting to model an anisotropic system requires more complex analysis. The greater complexity is found in nature, however, in systems such as heterogeneous and fractured reservoirs. Naturally fractured reservoirs can give rise to both a fast shear wave and a slow shear wave. The additional data associated with anisotropic shear waves provides information about fracture orientation that is useful in reservoir characterization and can be used to provide more accurate images of the subsurface.

John R. Fanchi 75

I 5.3 Acoustic Impedance and Reflection Coefficients I

Seismic waves are displacements that propagate through the earth. The

displacement can be caused by a mechanical source. These waves have many attributes, such as frequency content and transit time, but our focus here is on reflection amplitude. When the seismic wave encounters a reflecting surface, it is partially transmitted and partially reflected. A seismic reflection occurs at the interface between two regions with different acoustic impedances. Acoustic impedance Z is defined as the product of bulk density and compressional velocity, thus

z - O.Vl (5.3.1)

The acoustic impedance Z is also called compressional impedance because of its dependence on compressional velocity. A similar definition can be made using shear velocity to define shear impedance as the product of bulk density and shear velocity, thus

Z s - PsVs (5.3.2)

The reflection coefficient RC at the interface between two layers with acoustic impedances Z~ and Z 2 is given by

Z 2 - Z 1 R C = (5.3.3)

Z2 + Z,

Equation (5.3.3) assumes zero-offset, that is, the incident plane wave is propagating in a direction that is normal, or perpendicular, to a horizontal reflecting interface [Gadallah, 1994]. Seismic images are indicators of acoustic contrasts because the reflection coefficient depends on the difference between acoustic impedances in two adjacent layers. The transmission coefficient TC is

2Z1 T C = (5.3.4)

Z 2 + Z 1


Notice that the sum of reflection and transmission coefficients is identically one, thus R C + TC = 1.

Nonzero reflection coefficients occur when a wave encounters a difference between acoustic impedances. The acoustic impedance changes if there is a change in either bulk density or compressional velocity as the wave propagates from one medium to another. If the change in acoustic impedance is large enough, the reflected wave can be detected at the surface and the seismic method is said to have measured the contrasts in acoustic impedance. Similar remarks apply to the reflection of shear waves and the analysis of shear impedance.

Gas reservoirs are especially good reflectors because of the large density difference between an overlying water-bearing seal and the underlying gas-

bearing reservoir. The resulting impedance contrast can make it difficult to image the structure of the gas-bearing reservoir because changes in impedance due to lithology changes are relatively small compared to the change in impedance due to fluid phase differences. One way to improve the image is to work with shear waves.

An analysis of shear waves can yield information about reservoir structure for reservoirs containing gas because fluids have no resistance to

shear and the shear modulus for a fluid is zero. The reflected shear waves are therefore imaging only the structure. In some physical systems, such as naturally fractured reservoirs, the incident compressional wave splits into two reflected shear waves with different propagation velocities. An analysis of slow and fast reflected shear waves can provide information about the

system that would not be available by analyzing only compressional wave information. Multicomponent seismology is an emerging technology that is designed to extract information from shear wave behavior.

Seismic Inversion

Seismic inversion is the process of transforming seismic measurements into quantitative estimates of rock properties that are needed for reservoir engineering calculations. Acoustic impedance is especially useful for performing well tie analysis and rock property analysis because acoustic

impedance is often directly related to reservoir porosity. The dependence of acoustic impedance on porosity can be shown by combining Equations (3.3.1), (3.4.1) and (5.3.1) to find

John R. Fanchi 77

9B

(1 -- ~)~ m "+" (l)Pf ]

[]2 1- KB

Ka 4g Kd~y+ +

(5.3.5)

A similar, but simpler, expression can be obtained for shear impedance:

/ Zs - p~Vs - P~.,l t.t

v P~

-- ~/[(1- (l))Dm "+- (DPf ]~

(5.3.6)

We saw in Chapter 3 that bulk and shear moduli may be correlated to porosity. The result is an explicit theoretical dependence of impedance on porosity. In addition to a dependence on porosity, moduli depend on other factors as well, such as effective stress and clay content.

Seismic inversion attempts to determine rock properties by correlating seismic attributes to rock properties. For example, Equation (5.3.6) shows a relationship between porosity and acoustic impedance. It is important to use physically meaningful correlations to guide the distribution of rock properties between wells. At the same time, seismic inversion should preserve rock properties at wells. Correlations are obtained by cross-plotting seismic attributes with groups of rock properties. Examples of cross-plots for a formation with permeability K and net thickness h,e, include acoustic impedance versus porosity ~, seismic amplitude versus flow capacity (Khnet)

or rock quality(~Kh,et), and seismic amplitude versus oil productive capacity

(So(~Khnet). One of the earliest examples of seismic inversion that included a field

test of the technique was published by De Buyl, et al. [ 1988]. They presented an estimate of rock properties using seismic lithologic parameter estimation.


They forecast rock properties and then compared the forecasts with measure-

ments obtained directly at wells. A similar comparison was made between

measurements at wells and forecasts prepared using only well logs. The

forecasts made with seismic inversion were at least as accurate as forecasts

made with well log data only, and were more accurate in some cases.

5.4 Seismic Data Acquisition, Processing and Interpretation I

The primary role of geophysics in the oil and gas industry has historically

been mapping structure. Structure typically refers to the controlling folds,

faults and dips of subsurface formations. It may also include stratigraphic

features such as unconformities and pinchouts. Three steps are required to

prepare a structural model using seismic measurements: data acquisition, data

processing, and interpretation.

In reflection seismology, subsurface features are mapped by measuring

the time it takes an acoustic signal to propagate from an energy source to a

reflector and back to a receiver. Acoustic energy appears in the form of

waves or wave trains. Associated with each wave is a frequency. Observable

seismic frequencies are typically in the range of 10 to 60 Hz.

Energy sources such as dynamite or weight-dropping equipment are used

to initiate vibrations in the earth. The two most common types of vibrations

are compressional (P-) waves and shear (S-) waves. P-waves are the waves

that have been measured by traditional seismic methods. S-waves are

insensitive to fluid properties and can be used to image structures that may

be difficult or impossible to image using P-waves. For example, gas

chimneys and gas caps disrupt P-wave continuity and make it difficult to

observe structural features in the associated reservoir. By contrast, S-waves

are virtually unaffected by fluid properties and may yield a structural image

through the gas. The dependence of compressional and shear velocities on

rock and fluid properties is discussed in more detail in Chapter 3.

Receivers at the surface or in wellbores are used to detect the vibrations

generated by controlled sources. The travel time from the source to the

receiver is the primary information recorded by the receivers during the data

acquisition phase of a seismic survey. Other information such as amplitude

and signal attenuation can also be acquired and used in the next step: data

processing.

John R. Fanchi 79

Data processing is used to transform travel time images to depth images. This requires the conversion of time measurements to depths, which in tum depend on the velocity of propagation of the acoustic signal through the earth. Acoustic velocity contrast at the interface between two subsurface zones is indicated by the difference in acoustic impedance described above. The contrast between acoustic impedances associated with two adjacent layers provides the best means of obtaining an acoustic image of the subsur-

face. One of the most effective methods of improving the signal-to-noise ratio of the measured travel times associated with reflections from subsurface layer interfaces is to use common depth point (CDP) data stacking.

CDP data stacking requires the superpositioning, or summing, of overlapping reflections from a number of source points at each subsurface point. The number of source points that generate overlapping signals is referred

to as "fold." Thus, a 30-fold data gather uses data from 30 sources at each subsurface point. The superpositioning of the vibrations from multiple sources results in constructive interference to enhance the signal and destructive interference to minimize the random noise. The result is a significant enhancement of the signal-to-noise ratio.

Once a satisfactory measurement of travel time is obtained, the travel time must be converted to depths using the process of depth migration. The relationship between travel time and depth requires the preparation of a velocity model. The velocity model provides a distribution of velocity as a function of depth. A model of the acoustic velocity in each layer from the source to the reflector must be defined. The velocity distribution depends on the stratigraphic description of the subsurface geology.

The quality of velocity models can be improved by using vertical seismic profiles (VSP) or checkshots in well bores. The checkshot survey is obtained

by initiating a vibration at the surface and recording the seismic response in a borehole sonde. If the vibration source is vertically above the seismic receiver, the checkshot is a VSP with zero offset. If the source is not vertically above the receiver, the checkshot is a VSP with offset. A reverse

VSP is obtained by placing the source in the wellbore and the receiver on

the surface. The VSP and checkshot information can be compared with well

logs and core samples to validate the velocity model used for the conversion of travel times to depths. Once the seismic image has been transformed from the time to a depth representation, it is ready for the third step in the seismic survey process: interpretation.


The subsurface structure obtained from seismic measurements is part

of the observational information that is used to develop a geologic model.

Dorn [1998] estimated the volume of data V to be interpreted from a 3-D

seismic survey using the formula V = L • C • S • D where L is the number

of lines, C is the number of cross-lines, S is the number of time samples, and

D is the number of data types. The number of lines and cross-lines can range

from 2000 to 3000 lines each in a marine survey. The geologic model is an

interpretation of the seismic data volume or seismic image as a function of

depth. Optimum reservoir characterization is achieved by integrating seismic

information with all other available information, such as regional geology

and well information. Well information can include well logs, cores, and

production or injection information.

Geologic models based on 3-D seismic surveys consist of millions of

cells of information. The number of cells is related to the bin size, which

depends on the number of shot points and receivers. The bin size of a 3-D

seismic survey is the product of one half the interval between receivers AR

and the interval between shot points AS, thus bin size = (AR/2) x AS. Each

cell is a bin and contains all available information, such as acoustic imped-

ance and velocities, lithology, moduli, porosity, permeability and net

thickness. The number of cells in the geologic model depends on how many

bins are needed to cover the volume of interest. Ideally, the set of cells should

cover the entire subsurface volume of interest.

Computers are needed to manipulate the large volume of information

in a 3-D seismic survey. Computer visualization is the most effective way

to view the large volume of data. Several applications of 3-D seismic surveys

to exploration and production cases have been presented in Weimer and

Davis [ 1996].

Most modem flow models are unable to handle the large data volume

used to characterize the reservoir. Flow simulators are usually able to solve

large grid systems, but the time it takes to complete the calculations using

available computers is often considered prohibitive. Consequently, it is

necessary to average the information in several cells to prepare a smaller set

of data with larger cells. Cell size in a flow model is usually much larger than

seismic bin size. The need to transform a large data set for a relatively fine

grid to a smaller data set for a relatively coarse grid is referred to as the

upscaling problem in reservoir flow modeling. The upscaling problem can

be avoided if sufficient computer resources are available, but most practical

John R. Fanch i 81

problems require a coarsening of the geophysical data volume for use in reservoir flow models.

[ 5.5 Seismic Resolution ]

The ability to distinguish two features that are very close together is the essence of resolution. This becomes important in seismic resolution when

there is an interest in imaging two reflecting surfaces in either the vertical or horizontal directions. These reflecting surfaces can represent facies changes, fluid contacts, or any change in acoustic impedance that is relevant to reservoir characterization. The magnitude of seismic resolution will determine the usefulness of seismic surveys.

Vertical Resolution The ability to distinguish two features in the vertical direction depends

on interference effects and the wavelength associated with the seismic event.

The dominant seismic wavelength at a reflector ~'d is the ratio of the interval velocity vi to the dominant frequencyfd:

d -- V i / f d (5.5.1)

The dominant frequency is one over the dominant period of the seismic event. Thus, if the dominant period of the seismic event at a reflector is 50 milliseconds, or 0.050 seconds, the dominant frequencyfd is 20 Hz. If the local interval velocity is 14,000 ft/s, the corresponding dominant wavelength is 700 ft. An analysis of the interference of waves shows that the maximum vertical resolution 8Zv is one fourth of the dominant wavelength ~,d:

d Vi 5Zv - 4 - 4 f a (5.5.2)

An increase in the dominant frequency will cause a decrease in the dominant wavelength and a decrease in the maximum vertical resolution of the seismic measurement. In our example, this corresponds to a resolution of approximately 175 ft.


The relatively coarse vertical resolution available using surface seismic imaging has discouraged the routine application of reservoir geophysics in the reservoir management process. It is often necessary in reservoir management to identify thin features that cannot be identified using surface seismic methods. Examples of hard to image thin features include high permeability intervals and fractures. Crosswell seismic imaging and vertical seismic profiles (VSP) can be used to improve vertical resolution. A crosswell seismic image can be obtained by placing a vibration source in one well and a receiver in another well. Although crosswell seismic imaging is usually performed between vertical wells, advances in technology have made it possible to conduct crosswell seismic imaging in wells that may be deviated or horizontal. The resulting resolution for crosswell imaging is on the order of feet, rather than tens of feet [Williams, et al., 1997].

Lateral Resolution

The ability to distinguish two features in the lateral, or horizontal, direction depends on the reflection of a spherical wave from a fiat surface. The maximum horizontal resolution 8z/~ is approximately equal to the radius rF of the first-order Fresnel zone

Vi I At 5ZH- rF - -2 .)Ca (5.5.3)

where At is the two-way travel time. It is instructive to show how to derive Equation (5.5.3) as an illustration of how seismic resolution is determined.

The radius r of the first Fresnel zone is found from the Pythagorean relation

/ 2 F2+Z 2- Z+ (5.5.4)

where z is the depth from the recording device to the reflecting interface and ~.d is the dominant frequency introduced earlier. Expanding the left hand side of Equation (5.5.4) and simplifying gives

John R. Fanchi 83

2 r - +

2 16 (5.5.5)

Now we use z = viAt/2 and ~'d = 12i/fd in Equation (5.5.5) to find

v/i t 11 r - 2 ~-a + 4 f 2

1/2

(5.5.6)

Keeping only first order terms in the dominant frequency gives Equation (5.5.6).

For our vertical resolution example, if At is 0.6 seconds, the Fresnel radius rF is about 1200 ft. Thus, horizontal features such as pinch outs could only be resolved to approximately 1200 ft. Improvements in horizontal resolution, corresponding to reductions in 8z/~, can be made by increasing the dominant frequency.

Exploration versus Development Geophysics The resolution associated with development geophysics tends to be more

quantitative than the resolution associated with exploration geophysics. Wittick [2000] has pointed out that the difference in resolution is due to the role of calibration. Pennington [2001 ] drew a similar distinction between exploration geophysics and reservoir geophysics. Reservoir geophysics and development geophysics may be viewed as equivalent in the context of reservoir management. Seismic surveys in exploration geophysics acquire seismic data in areas where no wells have penetrated the target horizon. By contrast, well information should be available for calibrating seismic surveys in development geophysics. The lack of well-seismic ties in exploration geophysics results in seismic data that is not calibrated.

By contrast, development geophysical surveys are conducted in fields where wells have penetrated the target horizon. The amount of seismic data is as great or greater than exploration geophysics, and it is possible to tie seismic lines to well information, such as well logs. The result is a set of seismic data which has been calibrated to "hard data" from the target horizons. The availability of well control data makes it possible to extract more detailed information from seismic data.


The calibration of seismic data can be used to improve the apparent resolution of the data and prepare a tool for estimating reservoir properties at inter-well locations. Wittick [2000] identified two steps in the calibration process:

A. Correlate well logs, cores, and other well data with seismic attributes to determine how changes in a reservoir property alter the acoustic properties of the reservoir.

B. Use synthetic seismograms and seismic models to determine how

changes in the acoustic properties of the reservoir affect seismic attributes.

When seismic data has been calibrated to measured reservoir properties, it is possible to use the seismic data to make reasonably accurate quantitative estimates of reservoir properties. De Buyl, et al. [ 1988] published one of the earliest examples of the seismic calibration process and provided a prediction of reservoir properties at two wells. Discussions of some of the first

applications of reservoir geophysics are collected in Sheriff [ 1992].

I CS-5. Valley Fill Case Study" Ve/V ~ Model I

The distribution of the ratio Vp/V s of compressional to shear velocities is shown in Figure CS-5A for the top of the Valley Fill reservoir. The lines of the grid in Figure CS-5A intersect the wells and are 200 feet apart.

Traditionally, seismic data has provided information about the structure of a reservoir. This is true in the Valley Fill example where the Vp/Vs ratio presented in Figure CS-5A shows which parts of the area of interest are in the reservoir, and which are not. The Valley Fill model in Figure CS-5A is

interpreted as a meandering channel in an incised valley with regional dip. Region 1 is the region contained in the meandering channel, and Region 2 is the region outside of the meandering channel.

Figure CS-5A presents the Vp/Vs ratio for an idealized realization of a valley fill reservoir. Figure CS-5B shows a realization of the Vp/V s ratio that includes random heterogeneity or data uncertainty. The heterogeneity or data

uncertainty generates a random variability in the Vp/Vs distribution. Values of Vp/Vs in Figure CS-5B vary from a constant average value in the valley

John R. Fanchi 85

by • 2%. A similar variability applies in the bounding lithology adjacent to the valley. In this case, it is much more difficult to identify the meandering channel using only seismic attributes.

Figure CS-5A. Vp/Vs Distribution

Figure CS-5B. Vp/Vs Distribution with • Variability


[ Exercises [

5-1.

5-2.

5-3.

5-4.

5-5.

Verify that Equations (5.2.8) and (5.2.14) are solutions of Equations (5.2.6) and (5.2.12) respectively.

Use Equations (5.2.16) and (5.2.17) to show that g s a t - - (3 ~ + 21.t)/3.

Use Figure CS-5A to revise your structure map of the Valley Fill reservoir.

Use the new structure map prepared in Exercise 5-3 to determine a new volumetric estimate of oil in place in the channel.

Compare your estimate ofoil in place in Exercise 5-4 with the estimate you obtained in Exercise 3-7. Explain any differences.

Chapter 6 Fluid Properties

The previous chapters have established a representation of the static structure of the reservoir. Reservoir pore spaces are occupied by fluids. The fluids range from in situ water to hydrocarbon liquid and gas. Some of the fluids may be the target of commercial activity, and they are the fluids of interest here. The properties of fluids are needed to determine the amount of fluid present in the reservoir, and to determine their flow characteristics. This chapter defines the concepts needed to understand and classify fluid properties, shows how to identify sources of fluid data, and explains how to prepare a quantitative representation of fluid properties.

[ 6.1 Description of Fluid Properties !

Several concepts are needed to provide a description of fluid properties that can be used for geoscience and engineering calculations. Some of the most fundamental concepts of fluid properties are defined below for ease of reference. For more details, see the extensive literature on fluid properties represented by such references as Amyx, et al. [1960], Pederson, et al. [ 1989], McCain [ 1990, 1991 ], Dake [ 1994], Ahmed [2000], and Whitson and Bml6 [2000].

87

88 S h a r e d Ear th M o d e l i n g

Temperature Temperature is a measure of the average kinetic energy of a system. The

most commonly used temperature scales are the Fahrenheit and Celsius scales. The relationship between these scales is

5 32) (6.1.1) - -

where Tc and TF are temperatures in degrees Celsius and degrees Fahrenheit respectively.

Applications of equations of state described below require the use of absolute temperature scales. Absolute temperature may be expressed in terms of degrees Kelvin or degrees Rankine. The absolute temperature scale in degrees Kelvin is related to the Celsius scale by

T K - T c + 273 (6.1.2)

where TK is temperature in degrees Kelvin. The absolute temperature scale in degrees Rankine is related to the Fahrenheit scale by

T R = T F + 460 (6.1.3)

where TR is temperature in degrees Rankine.

Density Density p is the mass of a substance M divided by the volume V it

occupies: p = M/V. Fluid density depends on pressure, temperature and composition. Density, temperature and pressure are examples of intensive

properties. An intensive property is a fluid property which is independent of the amount of material. For example, suppose we subdivide a cell of gas

with volume Vinto two halves by a vertical partition. If the gas was initially in an equilibrium state, the gas in each half of the cell after inserting the partition should have the same pressure and temperature as it did before the partition was inserted. The mass and volume in each half of the cell will be one half of the original mass and volume, but their ratio, the density, will

John R. Fanchi 89

remain unchanged. Mass and volume are examples of extensive properties. An extensive property is a property that depends on the amount of material.

Composition The composition of a fluid is determined by the types of molecules that

comprise the fluid. A pure fluid consists of a single type of molecule, such as water or methane. A fluid mixture contains several types of molecules. Petroleum is a mixture of hydrocarbon compounds, and in situ water usually contains dissolved solids and may contain dissolved gases. The composition of a fluid is specified by listing the molecular components contained in the fluid and their relative amounts.

The relative amount of each component in a mixture is defined as the concentration of the component. Concentration may be expressed in such units as volume fraction, weight fraction, or molar fraction. The unit of concentration should be clearly expressed to avoid errors. Concentration is often expressed as mole fraction, but it is wise to confirm the unit if a unit is not explicitly specified. The symbols {Xg, Yi, Zi} a r e often used to denote the mole fraction of component i in the liquid phase, gas phase, and wellstream respectively.

The amount of component i in the gas phase relative to the oil phase is expressed as the equilibrium K value, which is the ratio

g i - Yi / x i (6.1.4)

The possible range of the equilibrium K value may be determined by considering two special cases. If component i exists entirely in the oil phase, then the gas phase mole fractiony~ is 0 and K/is 0. Conversely, if component i exists entirely in the gas phase, then the liquid phase mole fraction xi is 0 and K~ approaches infinity. Thus, the equilibrium K value for component i may range from 0 to infinity. An equilibrium K value can be calculated for each distinct molecular species in a fluid, including both inorganic and organic molecules.

Pressure Suppose an external force is applied to a surface. The component of the

force that is acting perpendicularly to the surface is the normal force. The


total normal force applied to the surface divided by the area of the surface

is the average pressure on the surface.

Pascal's law says that pressure applied to an enclosed fluid will be

transmitted without a change in magnitude to every point of the fluid and

to the walls of the container. The pressure at any point in the fluid is equal

in all directions. If the fluid is at rest in the pore space of a rock, the pressure

is equal at all points in the fluid at the same depth. The pressure in the pore

space is often referred to as the pore pressure. If fluid is injected or with-

drawn from the pore space, the rate of transmission of the change in pore

pressure throughout the enclosed system can be used to obtain information

about the system.

Compressibility Compressibility is a measure of the change in volume resulting from the

external pressure applied to the surface of an object. The fractional volume

change of an object is the ratio of the change in volume A V to the initial

volume Y. The ratio A V/V may be estimated from

AV - c A P (6.1.5)

V

where c is the compressibility of the object, and AP is the applied external

pressure. The minus sign is needed to assure that an increase in the applied

external pressure will result in a decrease in the volume of the object.

Similarly, a decrease in the applied external pressure will result in an increase

in the volume of the object.

Formation Volume Factor Formation volume factor is a measure of the ratio of the volume occupied

by a fluid phase at reservoir conditions divided by the volume occupied by

the fluid phase at surface conditions. Surface conditions are typically stock

tank or standard conditions.

The volume of a fluid phase can have a sensitive dependence on changes

in pressure and temperature. For example, gas formation volume factor is

often determined with reasonable accuracy using the real gas equation of

state P V - ZnRT where n is the number of moles of gas in volume V at

John R. Fanchi 91

pressure P and temperature T. The gas is an ideal gas if gas compressibility

factor Z = 1. If Z ~ 1, then the gas is a real gas. The volume of a petroleum mixture depends on changes in composition

as well as changes in temperature and pressure. For example, a barrel of oil

at reservoir conditions (relatively high pressure and temperature) will shrink as the barrel is brought to the surface (relatively low pressure and temperature). The shrinkage, or reduction in volume of the barrel of oil as it moves from reservoir to surface conditions, is due to the release of solution gas as the pressure and temperature of the oil decline. Shrinkage and expansion of fluids can take place within the reservoir as a result of changes in reservoir pressure, temperature, or composition during the life of the reservoir.

Specific Gravity Specific gravity is the ratio of the density of a fluid divided by a

reference density. Gas specific gravity is calculated at standard conditions using air density as the reference density. The specific gravity of gas is

Ma(gas) Ma(gas)

~[g = Ma(air ) ~ 29 (6.1.6)

where Ma is apparent molecular weight. Apparent molecular weight is

calculated as

Ma - ~ YiMi (6.1.7) i=1

where Nc is the number of components, y~ is the mole fraction of component

i, and Mi is the molecular weight of component i. Oil specific gravity is calculated at standard conditions using the density

of fresh water as the reference density. The American Petroleum Institute characterizes oil in terms of API gravity. API gravity is calculated from oil specify gravity Yo at standard temperature and pressure by the equation

141.5 API - - 131.5 (6.1.8)

~O

92 Shared Earth Modeling , ,

Heavy oils do not contain much gas in solution and have a relatively large

molecular weight and specific gravity Yo. By contrast, light oils typically contain a large amount of gas in solution and have a relatively small

molecular weight and specific gravity Yo. The equation for API gravity shows

that a heavy oil has a relatively low API gravity because it has a large Yo, while light oils have a relatively high API gravity.

Heating Value The heating value of a gas can be estimated from the composition of the

gas and heating values associated with each component of the gas. The

heating value of the mixture Hm is defined as

nm- E Yi Hi (6.1.9) i=1

where Arc is the number of components, Yi is the mole fraction of component

i, and H i is the heating value of component i. The heating value of a typical natural gas is often between 1000 BTU/SCF to 1200 BTU/SCF. Heating

values of molecular components in a mixture are tabulated in reference

handbooks.

Gas-Liquid Ratio The gas-liquid ratio is the ratio of a volume of gas divided by a volume

of liquid at the same temperature and pressure. The gas-liquid ratio, or GLR,

is useful for characterizing the behavior of a reservoir. The choice of GLR depends on the fluids in the reservoir. The two most commonly used gas-

liquid ratios are gas-oil ratio (GOR) and gas-water ratio (GWR).

The ratio of gas volume to water volume at the same temperature and

pressure, or gas-water ratio, is a sensitive indicator of the behavior of a gas

reservoir connected to a water source, such as an aquifer or injected water.

Gas-oil ratio, orthe ratio of gas volume to oil volume at the same temperature

and pressure, provides information about the behavior of an oil reservoir.

Viscosity The coefficient of viscosity is a measure of resistance to flow ofthe fluid.

In general, gas viscosity is less than liquid viscosity. The inverse of viscosity

John R. Fanchi 93

is called fluidity [McCain, 1990]. Thus, a fluid with a large viscosity has a low fluidity.

Two types of viscosity are commonly used: dynamic viscosity I.t and kinematic viscosity v. Dynamic viscosity is related to kinematic viscosity by the equation ~ = pv where P is the density of the fluid. The unit of dynamic viscosity ].t is centipoise. If fluid density P has the unit ofg/cc, then kinematic viscosity v has the unit of centistoke. Thus, 1 centistoke equals 1 centipoise divided by 1 g/cc.

Dynamic viscosity I.t is used in Darcy's law to calculate the rate of fluid flow in porous media. The relationship between viscosity and flow rate defines the rheology of the fluid. A fluid is considered a non-Newtonian fluid i f the viscosity of the fluid depends on flow rate. If the viscosity does not depend on flow rate, the fluid is called a Newtonian fluid.

I 6.2 Classification of Petroleum Fluids I

Petroleum fluids are typically mixtures of organic, and often inorganic, molecules. The elemental composition of petroleum is primarily carbon (84 - 87% by mass) and hydrogen (11 - 14% by mass). Petroleum contains other elements as well, including sulphur, nitrogen, oxygen and various metals. A petroleum fluid may be called "sweet" if it contains only negligible amounts of sulphur compounds such as hydrogen sulfide (H2S) or mercap- tans. If the petroleum fluid contains a sulphur compound such as HaS or a mercaptan, it is called "sour"

Petroleum fluids are predominantly hydrocarbons. The most common hydrocarbon molecules are paraffins, napthenes, and aromatics. These molecules are relatively stable at pressures and temperatures commonly found in reservoirs. A paraffin molecule such as methane and ethane has a single bond between carbon atoms and is considered a saturated hydrocarbon. Paraffins have the general chemical formula Cff-/2, + 2" Napthenes have the general chemical formula C, H2n and are saturated hydrocarbons with a tinged structure, as in cyclopentane. Aromatics are unsaturated hydrocarbons with a ringed structure. Benzene is the best known example of an aromatic. Aromatics have multiple bonds between the carbon atoms, and their unique ring structure makes aromatics relatively stable and unreactive.


The phase behavior of a fluid is generally presented as a function of the three variables: pressure, volume and temperature. The resulting PVT diagram is often simplified for petroleum fluids by preparing a pressure-temperature (P- T) projection. Figure 6- I is an example of a pressure- temperature (P-T) diagram.

The P-T diagram in

Single-Phase Region I

oA [ oB I Critical Point

Cricondentherm

Two-Phase Region /

Figure 6-1. P-T Diagram: (A) Black Oil and Figure 6-1 displays both (B) Condensate single-phase and two- phase regions. The curve separating the single-phase region from the two- phase region is called the phase envelope. The pressures associated with the phase envelope are called saturation pressures. A petroleum fluid at a temperature below the critical point temperature T~ and at pressures above the saturation pressure exists as a single phase liquid. Saturation pressures at temperatures below T~ are called bubble point pressures. If the pressure drops below the bubble point pressure, the single-phase liquid will make a transition to two phases" gas and liquid. At temperatures below Tc and pressures above the bubble point pressure, the single-phase liquid is referred to as a black oil (point A in Figure 6-1). If we consider pressures in the single-phase region and move to the right of the diagram by letting temperature increase toward the critical point, we encounter volatile oils.

The behavior of the petroleum fluid at temperatures above the critical point depends on the location of the cricondentherm. The cricondentherm is the maximum temperature at which a fluid can exist in both the gas and liquid phases. If the temperature is less than the cricondentherm but greater than T~, reservoir fluids are condensates (point B in Figure 6-1). When temperature exceeds the cricondentherm, we encounter gas reservoirs, A summary of these fluid types is given in Table 6-1 [Fanchi, 2001 a]. Separator gas-oil ratio (GOR) is a useful indicator of fluid type. The unit MSCF/STB equals thousand standard cubic feet of gas per stock tank barrel of oil.

John R. Fanchi 95

Fluid Type

Dry gas

Wet gas

Condensate

Volatile oil

Black oil

Heavy oil

Table 6-1 Rules of Thumb for Classifying Fluid Types

Separator GOR (MSCF/STB)

No surface liquids

> 100

3- 100

1.5 - 3

0.1 - 1.5

NO

Pressure Depletion Behavior in Reservoir

Remains gas

Remains gas

Gas with liquid drop out

Liquid with significant gas

Liquid with some gas

Negligible gas formation

Changes in phase behavior as a result of changes in pressure can be anticipated using the P-T diagram. Suppose a reservoir contains hydrocarbons at a pressure and temperature corresponding to the single-phase black oil region. If reservoir pressure declines at constant temperature, the reservoir pressure will eventually cross the bubble point pressure curve and enter the two-phase gas-oil region. A free gas phase will form in the two-phase region.

Similarly, if we start with a single-phase condensate and allow reservoir

pressure to decline at constant temperature, the reservoir pressure will eventually cross the dew point pressure curve to enter the two-phase region. In this case, a free phase liquid drops out of the condensate gas. Once liquid drops out, it is very difficult to recover. If the pressure declines further, some hydrocarbon compositions will undergo retrograde condensation.

The P-T diagram may also be applied to temperature and pressure

changes in a wellbore. Reservoir fluid moves from relatively high temperature and pressure at reservoir conditions to relatively low temperature and

pressure at surface conditions. As a result, it is common to see single-phase reservoir fluids become two-phase fluids by the time they reach the surface.

If the change from single-phase to two-phase occurs quickly in the wellbore, which is common, then the fluid is said to have undergone a flash from one

to two phases. The P-T diagram in Figure 6-2 shows two-phase envelopes for four types

of fluids [Pederson, et al., 1989]. Reservoir fluids can change from one fluid type to another during the life of a reservoir. As an example, suppose we


inject dry gas into a black oil reservoir. Dry gas injection increases the

relative amount of low molecular weight component s in the black oil and causes the two-phase envelope to rotate in a counterclockwise direction in

the P-T diagram as the relative amount of lower molecular weight compo-

nents increases. Reservoir management, or the way the reservoir is operated,

has a significant impact on fluid behavior.

C ~ 1 7 6 1 7 6 1 7 6 1 7 6

o ~ �9 . ~

�9 �9

. " ~ 1 7 6

. . . ' " ' G a s ...'" C . ~ ~ ~ ~ , " 4 ~ ~

~ ~ ...r.. , - ~ ~ ~ J "" Gas Condensate~ . . "

~

- - _C

C

Temperature ~-

Figure 6-2. Typical two-phase P-T envelopes for different fluid types

Different compositions for typical fluid types are shown in Table 6-2. The notation CN is used to indicate that there are N carbon atoms in the

molecule. For example, methane C~ has one carbon atom, ethane C2 has two

carbon atoms, and so on. The notation C6§ refers to the set of molecules with six or more carbon atoms. The molar composition in each column should

add up to 100%.

Methane content (C~) is a good indicator of fluid type. It tends to

decrease as fluids change from dry gas to black oil. As we move from left

to right in the table, we also see an increase in higher molecular weight

components. Dry gas usually contains lower molecular weight components,

especially methane, while the addition of higher molecular weight compo-

nents such as organic molecules with 3 or more carbon atoms will eventually

yield black oil. Volatile oil and black oil have significant amounts of

intermediate and high molecular weight components.

John R. Fanchi 97

Table 6-2 Typical Molar Compositions of Petroleum Fluid Types

[after Pedersen, et al., 1989]

Component Gas Gas Condensate Volatile Oil Black Oil

N 2 CO 2 C1 C 2 C3 iC4+nC 4 iCs+nC5 iC6+nC 6

C7 C8 C9 C~o Cll C12

C13 C14 C15 C16 Cl7

C18 Cl9 C2o

0.3 1.1

90.0 4.9 1.9 1.1 0.4

C6+: 0.3

0.71 8.65

70.86 8.53 4.95 2.00 0.81 0.46

0.61 0.71 0.39 0.28 0.20 0.15

0.11 0.10 0.07 0.05

C~7+: 0.37

1.67 2.18

60.51 7.52 4.74 4.12 2.97 1.99

2.45 2.41 1.69 1.42 1.02

C12 +" 5.31

0.67 2.11

34.93 7.00 7.82 5.48 3.80 3.04

4.39 4.71 3.21 1.79 1.72 1.74

1.74 1.35 1.34 1.06 1.02

1.00 0.90

C20+: 9.18

I 6.3 Sources of Fluid Data ]

The goal of fluid sampling is to obtain a sample that is representative of the original in situ fluid. A well should be conditioned before the sample is taken. A well is conditioned by removing all nonrepresentative fluid from within and around the wellbore. Nonrepresentative fluid such as drilling mud is removed by producing the well until the nonrepresentative fluid is replaced by original reservoir fluid flowing into the wellbore.

Surface sampling is easier and less expensive than subsurface sampling. Surface sampling is often accomplished by displacing one fluid by another in a sampling cylinder. Ifa surface sample is taken, the original in situ fluid


must be reconstituted by combining separator gas and separator oil samples. The recombination step assumes accurate measurements of flow data at the surface, especially gas-oil ratio. Subsurface sampling from a properly conditioned well avoids the recombination step, but is more difficult and costly than surface sampling, and usually provides a smaller volume of sample fluid.

Subsurface sampling requires lowering a pressurized container to the production interval and subsequently trapping a fluid sample. Subsurface samples are often obtained using drill stem testing, especially when access to surface facilities is limited. Surface sampling allow the capture of a larger volume of fluid than subsurface sampling, but surface samples are usually not as representative of in situ fluids.

The validity of fluid property data depends on the quality of the fluid sampling procedure. Several problems with sampling are possible: the displacement may be incomplete; mixing of in situ fluids with cylinder fluids may change the apparent composition of the in situ fluid; and corrosive gases in a gas sample may be absorbed by water or the cylinder walls. In addition, sample quality may be degraded during transport of the cylinder from the field to the laboratory, especially if the cylinder is leaking.

Gas reservoir fluid samples require the removal of liquid condensed in the tubing between the bottom of the well and the surface. The removal of condensed liquid in a wellbore can be achieved by producing the gas well at a sufficiently high gas rate. A nomograph published by Turner, Hubbard and Dukler [ 1969] can be used to calculate the minimum flow rate needed for continuous removal of liquid from gas wells.

Once a sample has been acquired, it is necessary to verify the quality of the sample. This can be done by performing a compositional analysis and measuring such physical properties as density, molecular weight, viscosity and interfacial tension. Compositional analysis of a fraction of the sample can include such tests as gas chromatography or low and high temperature distillation. The presence of cylinder leaks can be detected at the laboratory by measuring the cylinder pressure and verifying that it has not changed during transport.

After sample integrity is verified, several experiments may be performed to measure fluid properties that are suitable for reservoir engineering studies. The most common experiments include a combination of one or more of the

John R. Fanchi 99

following expansion tests: constant composition expansion (CCE); differential liberation (DL); or constant volume depletion (CVD). Other experimental

tests include separator tests; swelling tests; and multiple contact tests. These

tests are discussed in more detail by Pederson, et al. [ 1989], and Whitson

and Brul6 [2000].

Constant Composition Expansion A constant composition expansion (CCE) test provides information

about pressure-volume behavior of a fluid without changes in fluid composition. The CCE test begins with a sample of reservoir fluid in a high-

pressure cell at reservoir temperature and at a pressure in excess of the

reservoir pressure. The cell pressure is lowered in small increments and the

change in volume at each pressure is recorded. The procedure is repeated

until the cell pressure is reduced to a pressure that is considerably lower than the saturation pressure. The original composition of the fluid in the cell does

not change at any time during the test because no material is removed from

the cell. The fluid may be either an oil or a gas with condensate. If the fluid

is an oil, the saturation pressure is the bubble point pressure. If the fluid is

a gas with condensate, the saturation pressure is the dew point pressure.

Constant Volume Depletion The constant volume depletion (CVD) test provides information about

a fluid system that is subjected to changes in pressure, composition and phase

volume. It is used to study the formation of a second phase, such as the

evolution of gas from an oil sample or the formation of a condensate from

a wet gas sample. The test cell is charged with a reservoir fluid sample at reservoir

temperature and the saturation pressure of the sample. If the fluid is a gas

or gas condensate, the saturation pressure is the dew point pressure. If the

fluid is an oil, the saturation pressure is the bubble point pressure. The pressure is then lowered to a predetermined value by expanding the volume

of the cell. The cell pressure is kept constant while the resulting liquid and

gas phases equilibrate. Gas is withdrawn until the cell volume is restored to

its initial volume at the new pressure. The composition of the produced vapor is determined, and the liquid volume is reported as a fraction of the initial

cell volume.

100 SharedEarthModeling

Differential Liberation The differential liberation (DL) test is used to determine the liberation

of gas from a live oil, that is, an oil containing dissolved gas. A live oil

sample is placed in a PVT cell at reservoir temperature and at a pressure

above the bubble point pressure. The pressure is lowered in small increments

and the evolved gas is removed at each stage. The volume of the evolved

gas and the oil remaining in the cell are recorded.

Separator Test The separator test is used to study the behavior of a fluid as it flashes

from reservoir to surface conditions. A flash is the one step change from a

relatively high pressure and temperature environment to a relatively low

pressure and temperature environment. The primary difference between a

flash and a differential process is the magnitude of the pressure differential

between stages. The pressure differential is generally much smaller in the

differential process than in the flash process.

The PVT cell in a separator test is charged with a carefully measured

volume of reservoir fluid at reservoir temperature and saturation pressure.

The cell pressure and temperature are then changed. Each change in pressure

and temperature corresponds to a separator stage, and one or more stages may

be used in the test. The volume of gas from each separation stage and the

volume of the liquid remaining in the last stage are measured.

Swelling Test The swelling test simulates the behavior of reservoir oil during gas or

solvent injection. Gas of known composition is added to a sample of

reservoir oil in a series of steps. Gas addition begins at the bubble point

pressure of the reservoir oil. After each gas addition step, the cell pressure

is increased until the gas phase disappears. The pressure in the cell corre-

sponds to the new saturation pressure of the mixture.

Multiple Contact Miscibility (MCM) Test The MCM test simulates the continuous multiple contact process

associated with gas injection into oil. The MCM test measures vapor and

liquid phase compositions after oil and gas are mixed in a PVT cell and

allowed to reach equilibrium. Ifa condensation process is being studied, the

John R. Fanchi 101

vapor phase is purged and the equilibrium liquid is again mixed with a flesh batch of gas. In the case of a vaporization process, the liquid phase is purged and the equilibrium gas is mixed with a fresh batch ofoil. The condensation process is analogous to phase behavior near the gas injection well. The vaporization process is analogous to phase changes occurring in the vicinity of the gas-oil interface. The procedure described above is repeated at several pressures and reservoir temperature until only one phase remains in the cell after equilibration. This is the point when multiple contact miscibility is achieved and the corresponding pressure is considered the minimum miscibility pressure.

I 6.4 Representation of Fluid Properties ]

Black oil models and compositional models are the two most common types of reservoir fluid models. The two different types of fluid models are described here.

Black Oil Model Typical gas and oil properties for a standard black oil model are sketched

in Figures 6-3 and 6-4. Gas and oil properties in a black oil model depend on pressure only. Gas phase properties are gas formation volume factor (Bg), gas viscosity (~g), and liquid yield (rs). Oil phase properties are oil

r~ ~'-- Bg ~ ~ g

Pressure ~ Pressure

Figure 6-3. Gas Phase Properties

Pressure - - -~

formation volume factor (Bo), oil viscosity (I.to), and solution GOR (R~o). These terms were described previously. Phase changes occur at the saturation


pressures. Single-phase oil be-

comes two-phase gas-oil when pressure drops below the bubble

point pressure (Pb), and single- phase gas becomes two-phase

gas condensate when pressure drops below the dew point pres-

sure (Pal)" Both saturated and undersaturated curves are func-

tions of pressure only. Flow simulators are most

efficient when fluid property

data are smooth curves. Numer-

ical difficulties can arise if a discontinuity is present in a

Saturated - - - Undersatura ted

Pb Pressure

Pb Pressure

Pb Pressure

fluid property curve. Realistic Figure 6-4. Oil Phase Properties fluid properties are ordinarily

smooth functions of pressure except at points where phase transitions occur.

Slope discontinuities between saturated and undersaturated conditions are

shown at the bubble point pressure Pb in Figure 6-4. As a practical matter,

it is usually wise to plot input PVT data to verify the smoothness of the data. Most simulators reduce the nonlinearity of the gas formation volume factor

Bg by using the inverse bg = 1/Bg to interpolate gas properties.

Oil properties from a laboratory experiment must usually be corrected

for use in a black oil simulator [Moses, 1986]. The justification for the

correction is based on the argument that flow in the reservoir is a relatively

slow, or differential, process. In a differential process, pressure changes in

small increments over a period of time. Reservoir fluid flow corresponds to

a differential process in the laboratory. When oil is produced through the

wellbore, however, it is subjected to a rapid change in pressure. Flow up a

wellbore is considered a flash process. Corrections to fluid properties used

in the model are designed to more adequately represent fluids as they flow

differentially in the reservoir prior to being flashed to surface conditions as

the fluid moves up the wellbore. The corrections alter solution gas-oil ratio and oil formation volume factor. Given separator oil formation volume factor

Bo and solution gas-oil ratio R~o, the corrections are:

John R. Fanchi 103

Bo(P ) - Boa(P ) B~176 Bodbp (6.4.1)

and

R~o ( P ) - R~o~p - (Rsodb p - R~o d ( P ) Bofbp

Bodbp (6.4.2)

where P is pressure. Subscript d refers to differential liberation data;f refers to flash data; and bp refers to the bubble point.

Water properties are almost always needed in a flow simulator. Ideally water properties will be obtained from laboratory analyses ofproduced water samples. Correlations are often sufficiently accurate for describing the behavior of water if water samples are not readily available.

If reliable fluid data is missing for one or more of the reservoir fluids, fluid properties from analogous fields or from correlations can be used. For a review of the use of correlations to describe fluid properties, see McCain [1991].

Compositional Model Compositional models use equations of state and represent hydrocarbon

phases in terms of components or pseudocomponents. Fluid composition is

usually expressed in terms ofwellstream composition zi of component i, or in terms of the mole fractions of component i in the liquid (xi) and gas (yi) phases. The ratio yi/xi is the equilibrium K value (Ki) for component i, and is a measure of how component i is distributed between gas and liquid phases at equilibrium conditions.

Fluids are usually described in one of two ways: by a multicomponent equation of state, or as a pseudocomponent system [Pedersen, et al., 1989; Whitson and Brul6, 2000]. A pseudocomponent is a mixture of pure

components that is treated as a single component to expedite computing. In general, fluid behavior is most often represented using a multicomponent equation of state with one or more pseudocomponents. Table 6-3 shows some cubic equations of state (EoS) used in commercial compositional simulators.


Equations of state used to quantitatively represent fluid properties are

based on the thermodynamic postulate that all macroscopic properties of a fluid system can be expressed in terms of pressure (P), temperature (T), and composition only. The EoS shown in Table 6-3 are examples of EoS that can be used to model the behavior of oil and gas. They relate pressure (P), volume (V), the gas constant R, temperature (T), and a set of adjustable parameters {a, b} which may be functions of temperature and composition. The EoS in Table 6-3 are called "cubic" because they yield a cubic equation for the compressibility factor Z = PV/RT. In the case of an ideal gas, Z= 1. Following van der Waals, the parameter b adjusts for the finite size of an atom or molecule in the first term on the right hand side of each EoS. The

second term accounts for interactions.

Table 6-3 Examples of Cubic Equations of State

Redlich-Kwong

Soave-Redlich-Kwong

p

p

e m

e

Peng-Robinson

Zudkevitch-Joffe

RT a / T 1/2 m

V-b v(v+b)

RT a(T)

V-b V(V+b)

RT a(T) m

v - b v ( v + b) + b (V - b)

RT a( T) / T '/2 m

v - b(r) v [ v + b(v/]

Equations of state are used to calculate equilibrium relations in a compositional model. This entails tuning parameters such as EoS parameters {a, b} in Table 6-3. For a mixture with Arc components, the parameters {a,

b} have the form

NcN~ ( ) a - Y~ Y~ aiajxix j 1- ~0 (6.4.3)

i=1 j=l

and

John R. Fanchi 105

Nc b - Z bix i (6.4.4)

i=1

where {ai, bi} refer to equation of state values for the ith component with mole

fraction xi, and 60. is a symmetric array of numbers called the binary interaction parameters. The binary interaction parameter 60. represents the interaction between component i and component j. In principle, binary interaction parameters are determined by fitting EoS parameters to fluid

property measurements on a mixture of two components. In practice, they are determined by a regression technique.

Several regression techniques exist for tuning an EoS. They usually differ in the choice of EoS parameters that are to be varied in an attempt to match lab data with the EoS. The modification of EoS parameters is called tuning the EoS. The justification for tuning an EoS is that the EoS parameters

are determined for systems with one or two components only, while petroleum is generally a mixture with many components. The EoS parameter

adjustments attempt to match the multicomponent behavior of the fluid system.

A black oil model can be thought of as a compositional model with two pseudocomponents. The number of pseudocomponents in a compositional

model typically ranges from six to ten. For comparison, EoS models in process engineering require on the order of 20 components or more to model fluid behavior in surface facilities. Compositional model costs increase

dramatically with increases in the number of specified components, but the

additional components make it possible to calculate more accurate fluid properties.

I CS-6. Valley Fill Case Study: Fluid Properties [

The Valley Fill reservoir is initially an undersaturated oil reservoir. Oil, water and gas fluid property date are shown in Table CS-6A. The oil contains 480

SCF/STB dissolved gas at the bubble point pressure. Saturated oil property data in Table CS-6A have been corrected from differential to flash form using the separator test information provided in Table CS-6B [Moses, 1986]. Undersaturated oil properties are shown in Table CS-6C.


Table CS-6A

Valley Fill Fluid Properties

Oil* Gas Water

Pressure Vis FVF Rso , Vis FVF Vis FVF

RB/ SCF/ RCF/ RB/

psia cp STB STB cp SCF cp STB �9 �9 �9 �9 i m un

14.7 1.04 1.06 1 0 0 .9358 0.5 1.019

514.7 0.910 1.111

1014.7 0.830 1.192

89 0.0112 0.0352 0.5005

208 0.0140 0.0180 0.5010

1.0175

1.0160

1514.7 0 . 7 6 5 i l . 2 5 6 309 0.0165 0.0120 0.5015 1.0145

2014.7 0.695 1.320 392 0.0189 0.0091 0.5020 1.0130

2514.7 0.641 1.380 457 0.0208 0.0074 0.5025 1.0115

3014.7 0.594 1.426 521 0.0228 0.0063 0.5030 1.0100

4014.7 0.510 1.472 586 0.0260 0.0049 0.5040 1.0070

5014.7 0.450 1.490 622 0.0285 0.0040 0.5050 1.0040

6014.7 0.410 1.500 650 0.0300 0.0034 0.5060 1.0010

* Corrected from differential to flash form.

Selected Units: RB = Reservoir Barrels; STB = Stock Tank Barrel;

SCF = Standard Cubic Feet; RCF = Reservoir Cubic Feet

Table CS-6B

Valley Fill Separator Test

(Flash from a saturation pressure of 6000 psig to 0 psig)

Variable

B ofbp

Rsofbp Bodbp Rsodbp

Value

1.5 RB/STB

650 SCF/STB

1.63 RB/STB

760 SCF/STB

John R. Fanchi 107

Table CS-6C Valley Fill Undersaturated Oil Properties

Pressure (psia)

2015

4015

Corrected Boo~, (RB/STB)

1.3200

1.2740

~0 (cp)

0.695

0.787

Remarks

Bubble Point

Undersaturated

Values

6-1.

6-2.

6-3.

6-4.

Exercises [

Calculate the specific gravity of a gas with the following composition.

Component

Methane (C1)

Ethane (C2)

Propane (C3)

n-Butane (n - C4)

Total

Mole Fraction

0.85

0.09

0.04

0.02

1.00

Molecular Weight

16

30

44

58

Typical reservoir values for formation, oil, water and gas compress-

ibilities are ci= 3 • 10 .6/psia, Co = 10 • 10 .6/psia, Cw = 3 • 10 .6/psia,

and cu = 500 • 10 .6/psia. Estimate the fractional volume change of

each fluid for a pressure difference Ap = ef inal " Pinitial " - - - 100 psia. The minus sign indicates a reduction in pressure.

Calculate the laboratory differential oil formation volume factor Boa(P ) as a function of pressure using data in Tables CS-6A and CS-6B. The

correction is discussed in Section 6.4.

Plot the PVT data in Table CS-6A as a function of pressure.

Chapter 7 Measures of Rock-Fluid Interactions

The distribution and flow of fluids in the reservoir depends on the interaction between the fluids and the rock structure. Small scale laboratory measurements of fluid flow in porous media show that fluid behavior depends on the properties of the solid material. The interaction between rock and fluid is modeled using a variety of physical parameters that include Darcy's law, permeability, relative permeability and capillary pressure [Collins, 1961; Dake, 1978; Koederitz, et al., 1989; Craft, et al., 1991; Ahmed, 2000]. This chapter discusses measures of rock-fluid interactions. Applications are presented in the next chapter.

I 7.1 Darcy's Law I

The basic equation describing fluid flow in porous media is called Darcy's Law. Darcy's equation for linear, horizontal, single phase flow is

K A A P Q= -0.001127 ~ (7.1.1)

Ax

The physical variables are defined in oilfield units as

108

John R. Fanchi 109

Q flow rate [bbl/day] K permeability [md] A cross-sectional area [ft 2] P pressure [psi] 1~ fluid viscosity [cp] Ax length [ft]

Equation (7.1.1) shows that the movement of a single phase fluid through a porous medium depends on cross-sectional area, pressure difference Ap, length Ax of the flow path, and viscosity of the flowing fluid. The minus sign indicates that the direction of fluid flow is opposite to the direction of increasing pressure: the fluid flows from high pressure to low pressure in a horizontal (gravity-free) system. The proportionality constant in Equation (7.1.1) is called permeability.

If we rearrange Equation (7.1.1) and perform a dimensional analysis, we see that permeability has dimensions of L 2 (area) where L is a unit of length:

K rate x viscosity x length

area x pressure

(tLm3e) ( force x time) L2 L

L2( f~ L 2 J

__ L 2

(7.1.2)

The areal unit (L 2) is physically related to the cross-sectional area of pore throats in rock. The size of a pore throat depends on grain size and distribution. For a given grain distribution, the cross-sectional area of a pore throat will increase as grain size increases. Relatively large pore throats imply relatively large values of L 2 and correspond to relatively large values of permeability.

Darcy's law shows that flow rate and pressure difference are linearly related. The pressure gradient from the point of fluid injection to the point of fluid withdrawal is found by rearranging Equation (7.1.1):


AP ( q . )g (713) A x - - 0.001127A K ""

Superficial Velocity and Interstitial Velocity Superficial velocity is the volumetric flow rate q in Darcy' s law divided

by the cross-sectional area A normal to flow [Bear, 1972; Lake, 1989], thus u = q/A in appropriate units. Superficial flow rate is not the same as the rate of flow through the interstices in the porous medium. The interstitial, or "front," velocity v of the fluid through the porous rock is the actual velocity of a fluid element as the fluid moves through the tortuous pore space. Interstitial velocity v is the superficial velocity u divided by porosity ~), or v = u/~p = q/~pA. Since porosity is a fraction between 0 and 1, interstitial velocity is usually several times larger than superficial velocity.

The Validity of Darcy's Law Darcy's law is valid when fluid flow is laminar. Laminar fluid flow

represents one type of flow regime. Three types of flow regimes may be defined" laminar flow regime with low flow rate; inertial flow regime with moderate rate; and turbulent flow regime with high flow rate. Flow regimes are classified in terms of the dimensionless Reynolds number [Fancher and Lewis, 1933]. Reynolds number is the ratio of inertial (fluid momentum) forces to viscous forces. It has the form

P VDdg NRe = 1 4 8 8 ~ (7.1.4)

where

p fluid density [lbm/ft 3] Vo superficial velocity [ft/sec] dg average grain diameter [ft]

absolute viscosity [cp]

The flow regime is determined by calculating Reynolds number. A low Reynolds number corresponds to laminar flow, and a high Reynolds number

John R. Fanchi 111

corresponds to turbulent flow. The classification of flow regime in terms of Reynolds number is discussed by Gorier [1978, pg. 2-10] and is presented in the following table.

Table 7-1 Classification of Flow Regimes

F L O W REGIME

Laminar

Inertial

Turbulent

DESCRIPTION

Low flow rates (NRe < 1)

Moderate flow rates (1 < NR, < 600)

High flow rates (NR~ > 600)

The linear relationship between pressure gradient and rate in Darcy's law is valid for many flow systems, but not all. Forcheimer observed that turbulent flow in high flow rate gas wells had the quadratic dependence

AP _ _ q g q (7 1.5) Ax 0.001127A K + [39

for fluid with density p and turbulence factor [3. A minus sign and conversion unit is inserted in the first order rate term on the right hand side of Equation (7.1.5) to be consistent with the rate convention used in Equation (7.1.1). Equation (7.1.5) is called the Forcheimer equation.

Virtually all commercial flow simulators are formulated with the assumption that Darcy's law is applicable. The Forcheimer equation is a more accurate relationship between pressure and turbulent flow rate than the linear relationship expressed by Darcy's law. The quadratic dependence on rate in the Forcheimer equation introduces a complexity that is expensive to include in flow simulators for most routine problems.

Darcy's Law correctly describes laminar flow, and may be used as an approximation of turbulent flow. Permeability calculated from Darcy's law is less than true rock permeability at turbulent flow rates.

Radial Flow of Liquids Darcy' s law for steady-state, radial, horizontal, single-phase liquid flow

in a porous medium is


O o 0.00708Kh( Pw - Pe)

~ B In(re~ r~ ) (7.1.6)

where

Q liquid flow rate [STB/D]

r~ wellbore or inner radius [ft]

re outer radius [ft] K permeability [md]

h formation thickness [ft]

Pw pressure at inner radius [psi]

P~ pressure at outer radius [psi]

I.t viscosity [cp]

B formation volume factor [RB/STB] '

The formation volume factor in the denominator on the right hand side of Equation (7.1.6) converts volumetric flow rate from reservoir to surface

conditions. The rate Q is positive for a production well {P~ < Pc} and

negative for an injection well {P~ > Pc}" Different procedures may be used to estimate the outer radius re. The

outer radius r e is equated to the drainage radius of the well when analyzing the pressure at a well. Alternatively, the value ofr e in a reservoir flow model

depends on the size of the grid block containing the flow rate term [Peace- man, 1978; Fanchi, 2001 a]. The flow rate is less sensitive to an error in the estimate of re than a similar error in a parameter like permeability because

the radial flow calculation depends on the logarithm of re. It is therefore possible to tolerate larger errors in re than other flow parameters and still

obtain a reasonable value for radial flow rate.

Radial Flow of Gases

Consider Darcy's Law in radial coordinates for a single phase

2nrhK dP r q~ - - 0 . 0 0 6 3 2 8 ~ (7.1.7)

dr

John R. Fanchi 113

where the radial distance r increases as we move away from the well, and

qr gas rate [rcf/d] r radial distance [ft]

h zone thickness [ft]

tx gas viscosity [cp] K permeability [md]

Pr reservoir pressure [psia]

The cross-sectional area in the numerator of Equation (7.1.7) is the cross- sectional area 2xrh of a cylinder enclosing the wellbore. Subscripts r and

s denote reservoir and surface conditions respectively. To convert from

reservoir to surface conditions, we divide gas rate at reservoir conditions by

the gas formation volume factor, thus

qr q~- Bg (7.1.8)

where

q, gas rate [scf/d]

Be, gas formation volume factor [rcf/scf]

Gas formation volume factor Bg is a function of pressure P, temperature T

and gas compressibility factor Z from the real gas equation of state:

grrZr Bb- Pr Ts Zs (7.1.9)

Substituting Equations (7.1.7) and (7.1.9) into (7.1.8) gives the rate at surface

conditions in the form

rhK P~ T~ Z~ dP~ q s - - 0 . 0 3 9 7 6 ~ - - (7.1 10)

~t Ps Tr Zr dr


and qs has units of scf/d.

If we assume a constant rate, we can rearrange Equation (7.1.10) and integrate from the inner radius to the outer radius to get

ridr re i Pr q * - - - q , g n rw - -0.03976 Ps Tr pw ].t Z r dP r (7.1.11)

Subscripts w and e denote values at the wellbore radius and external radius

respectively. Equation (7.1.11) can be written in a simpler form by introducing the real gas pseudopressure m(P):

P

m ( p ) _ 2 I P ' gZ dP'

P~el

(7.1.12)

where P~eyiS a reference pressure and P ' is a dummy variable of integration.

The integrand of Equation (7.1.12) has a nonlinear dependence on pressure.

It is often necessary to solve the integral numerically because of the nonlinear

dependence of gas viscosity and gas compressibility on pressure. Given m(P),

the radial form of Darcy's law becomes

qs = -0.01988 Xh ___Zs m( wt]

(7.1.13)

Specifying the standard conditions

Z~ =1

Ps = 14.7 psia

Ts = 60~ = 520~

in Equation (7.1.13) gives Darcy's law for the radial flow of gas:

q~ : -0.703

John R. Fanchi 115

re) rw

(7.1.14)

Solving Equation (7.1.14) for real gas pseudopressure at the external radius

gives

1.422 T~ (gn-~-w) (7.1.15)

Equation (7.1.15) shows that m(Pe) is proportional to q~ and inversely

proportional to permeability.

[ 7.2 Permeabi l i ty , !

Permeability has meaning as a statistical representation of a large number of pores. It may be viewed as a mathematical convenience for describing the statistical behavior of fluid flow in a given flow experiment. Well tests that match the pressure response at a well due to changes in flow rate give the best measure of permeability over a large volume. An analysis of the statistical view of permeability was performed by Collins [ 1985] and updated by Fanchi [2000a]. Despite the importance of permeability to the calculation of flow, permeability and its distribution are seldom well known. Seismic data can help determine the distribution of permeability between wells if a good correlation can be found between a seismic attribute and a rock quality measurement that includes permeability.

Permeability depends on rock type. The two most common reservoir rock types are elastic reservoirs and carbonate reservoirs. The permeability in a elastic reservoir depends on pore size and is seldom controlled by secondary solution vugs. Clean, unconsolidated sands may have permeabilities as high as 5 to 10 darcies. Compacted and cemented sandstone rocks tend to have lower permeabilities. Productive sandstone reservoirs usually have


permeabilities in the range of 10 to 1000 md. The permeability in tight gas and coalbed methane reservoirs is less than 1 md.

The presence of clay can adversely affect permeability. Clay material may swell on contact with fresh water, and the resulting swelling can reduce a rock's permeability by several orders of magnitude. The effect of clay swelling needs to be considered whenever water is injected into a clay- bearing formation. Water compatibility tests should be designed to determine the interaction between injected water and the formation.

Carbonate reservoirs are generally less homogeneous than clastic reservoirs and have a wider range of grain size distributions. The typical matrix permeability in a carbonate reservoir tends to be relatively low, and may be as low as 0.1 to 1.0 md. Significant permeability in a carbonate may be associated with secondary porosity features such as vugs and oolites.

Natural or man-made fractures can contribute significant flow capacity in both carbonate and clastic reservoirs. An extensive natural fracture system can provide high flow capacity conduits for channeling flow from the reservoir matrix to a wellbore. Naturally fractured reservoirs are usually characterized by relatively high permeability, low porosity fractures and relatively low permeability, high porosity matrix. Most of the fluid is stored in the matrix, while flow from the reservoir to the wellbore is controlled by permeability in the fracture system.

Porosity-Permeability Correlations Measurements of porosity and permeability distributions in fields around

the world have shown that the statistical distribution of porosity is often the normal (or Gaussian) distribution, and the statistical distribution of permeability is often log normal. These observations suggest that porosity and permeability are correlated. Several correlation schemes between porosity and permeability have been presented in the literature, but the most common correlation is a plot of core porosity versus the log of permeability. Straight line segments on a semi-logarithmic plot ofporosity (Cartesian scale) versus permeability (logarithmic scale) can be used to identify rock types and quantify a correlation between porosity and permeability. The semi- logarithmic plot of porosity versus permeability is often referred to as a phi-k cross-plot. It is often necessary to use linear regression to quantify the straight linesegments of a phi-k cross-plot because there is a considerable amount of scatter in data plotted from real fields.

John R. Fanchi 117

Kiinkenberg's Effect Klinkenberg found that the permeability for gas flow in a porous medium

depends on pressure according to the relationship

kg - kab s 1 + (7.2.1)

where

kg apparent permeability calculated from gas flow tests kab s true absolute permeability of rock P mean flowing pressure of gas in the flow system b Klinkenberg's factor

The factor b is a positive constant for gas in a specific porous medium. Equation (7.2.1)shows that when the factor (1 + b / P ) >__ 1, we have the inequality kg >__ kab~. As pressure increases, the factor (1 + b / - f i ) approaches 1 and the apparent permeability to gas kg approaches the true absolute permeability of the rock kabs.

The dependence of kg on pressure is called Klinkenberg's effect and is attributed to the "slippage" of gas molecules along pore walls. The interaction between gas molecules and pore walls is greater at low pressures than at high pressures. Conversely, slippage along pore walls is greater at high pressures than low pressures. At low pressures, the calculated permeability for gas flow kg may be greater than true rock permeability. Measurements of kg are often conducted with air and are not corrected for Klinkenberg's effect. This should be considered when comparing kg with permeability estimates from other sources such as well tests.

I 7.3 Directional Dependence of Permeability I

The discussion thus far has assumed that flow is occurring in a horizontal plane. We can include the effect of gravity by introducing the concept of phase potential. Phase potential is designed to equilibrate fluid flow relative to a gravitational field. The potential of phase i is


(I) i -- P i - ~ i ( n z ) (7.3.1)

where Az is depth from a datum, Pi is the pressure of phase i, and 'Yi is the

pressure gradient. Darcy's law for single phase flow in the presence of a

constant gravitational field can be expressed as

0.001127KA d~ q = - (7.3.2)

dz

The sign of Az depends on the orientation of the coordinate system. If depth z+Az is deeper than z when Az > 0, the phase pressure at depth z+Az

will be greater than the phase pressure at datum z, but the potential at datum

z should equal the potential at z+Az if the system is in gravitational equilibrium. No vertical fluid movement will occur when the fluid is in

gravitational equilibrium. The condition for gravitational equilibrium is

d~/dz = O.

Permeability can be a complex function of spatial location and orienta-

tion. Spatial and directional variations of a function are described in terms

of homogeneity, heterogeneity, isotropy, and anisotropy. If the value of a

function does not depend on spatial location, it is called homogeneous. The

function is heterogeneous if its value changes from one spatial location to

another. If the value of a function depends on directional orientation, i.e.,

the value is larger in one direction than another, then the function is aniso-

tropic. The function is isotropic if its value does not depend on directional

orientation. Permeability is a function that can be both heterogeneous and

anisotropic. To account for heterogeneity and isotropy, the simple one- dimensional form of Darcy's law must be generalized.

Darcy originally observed that rate is proportional to pressure gradient.

This can be expressed in vector notation for single phase flow as

A c~ - -0 .001127 K - - V ~ (7.3.3)

~t

where the effect of gravity is included in the potential. Equation (7.3.3) is shorthand for the following three-dimensional set of equations:

John R. Fanchi 119

qx = - 0 . 0 0 1 1 2 7 K - - ~

qy - - 0 . 0 0 1 1 2 7 K - - ~

qz = - 0 . 0 0 1 1 2 7 K - - ~

A O~b ~t Ox

A O~b Oy

AO~ ~t Oz

(7.3.4)

We are still treating permeability as homogeneous and isotropic when we

write permeability as a constant in Equation (7.3.4). Equation (7.3.4) is

written in matrix notation as

qx A O~b/Ox] q y - - 0 . 0 0 1 1 2 7 K - - O*/Oy[

o| qz

(7.3.5)

where the terms in square brackets [...] are column vectors.

A more general extension of Equation (7.3.5) that allows permeability

to be anisotropic is to write permeability as a matrix of terms that can depend

on spatial location. The extended form of Darcy 's law becomes

qx A IKx~ Kxy Kx~ a~/ax 1 qy =-O.O01127--Ky x Kyy Ky z ~

qz rtLKzx K~ Kzz O*/Oz]

(7.3.6)

Permeability can be considered either a 3• matrix with 9 elements or a

tensor of rank two. The concept of tensor is related to the properties of a

function when it is transformed from one coordinate system to another. A

vector such as velocity is a tensor of rank one and a scalar like pressure is

a tensor of rank zero. Each element of the permeability tensor is a function

of position in the subsurface. Equation (7.3.6) corresponds to the following

set of equations:


q x - -0.001127--~t Kx~ --~x + Kxy --~y + Kxz -~z

q y - -0.001127--~ K yx --~x + K yy -~y + K y~ -~z

qz - -0.001127--p Kzx --~-x + K~ -~-y+ K= -~z

(7.3.7)

The diagonal permeability elements {Kxx, Kyy, Kzz} represent the usual dependence of rate in one direction on the pressure gradient in the same direction. The off-diagonal permeability elements {Kxy, Kxz, Kyx, Kyz, K,x, Kzy} account for the dependence of rate in one direction on pressure gradients in orthogonal directions.

It is mathematically possible to find a coordinate system {x', y', z'} in which the permeability tensor has the diagonal form [Fanchi, 2000b]

Kx, x, 0 0

0 Ky,y, 0

o o Kz,z,

The coordinate axes {x', y', z'} are called the principal axes of the tensor, and the flow equations reduce to the simpler form

q x , - - O . O 0 1 1 2 7 - - Kx, x, ~t Ox'

A I oy0* 1 qy, - - 0 .001127- - Ky.y. 7_.; ~t

AE q z . - - 0 . 0 0 1 1 2 7 - - Kz. z,

(7.3.8)

John R. Fanchi 121

The form of the permeability tensor depends on the properties of the porous medium. The medium is said to be anisotropic if two or more elements of the diagonalized permeability tensor are different. The permeability of the medium is isotropic if the elements of the diagonalized permeability tensor are equal, that is

I ':y, ,- I':z,z,- X

If the medium is isotropic, permeability does not depend on direction. If the isotropic permeability does not change from one position in the medium to another, the medium is said to be homogeneous in permeability. On the other hand, if the values of the elements of the permeability tensor vary from one point in the medium to another, both the permeability tensor and the medium are considered heterogeneous. Virtually all reservoirs exhibit some degree of anisotropy and heterogeneity, but the flow behavior in many reservoirs can be treated as homogeneous and isotropic.

Flow simulators can account for directional dependence using Equation (7.3.8). In practice, however, the tensor permeability discussed in the literature by, for example, Bear [1972], Lake [1988], and Edwards [ 1995] is seldom incorporated in a flow simulator. Most flow simulators use flow equations that assume the reservoir model is aligned with a coordinate system that corresponds to the diagonalized tensor shown in Equation (7.3.8). This is usually not the case, and can lead to numerical orientation errors [Fanchi, 1983]. Modelers are beginning to realize that the full permeability tensor is needed to adequately represent fluid flow in flow models that employ a coarser geologic representation of the reservoir than reservoir models prepared by geoscientists using well log and seismic data [Fanchi, 2000a].

Vertical Permeability Permeability for flow in a direction that is parallel to the plane of

geologic deposition is usually called horizontal permeability. By contrast, vertical permeability is the permeability for flow in the direction transverse to the plane of deposition and aligned with the direction of the gravitational

field. Vertical permeability can be measured in the laboratory or in pressure

transient tests conducted in the field. In many cases vertical permeability is


not measured and must be assumed. A rule of thumb is to assume vertical

permeability is approximately one tenth of horizontal permeability. These

are reasonable assumptions when there is no data to the contrary. It is

preferable from a technical point of view to make direct measurements of

all relevant reservoir data. Sometimes it is difficult to justify the cost or

logistics of obtaining direct measurements. If it is necessary to use a rule of

thumb or data from an analogous formation to estimate a particular variable,

the sensitivity of the shared earth model to changes in the estimated variable

should be considered. This can be done by preparing alternative realizations

of the shared earth model using values of the estimated variable that are

reasonable but differ from the original estimate. In the case of vertical

permeability, a value of one tenth horizontal permeability is a reasonable

initial estimate, and multipliers of zero (no vertical communication) and one

(vertical permeability = horizontal permeability) are alternative values that

encompass the range of possible values.

The concepts of horizontal and vertical permeability can be significantly

affected if a formation has a steep dip angle. For example, a reservoir with

a 90 degree dip angle is aligned in a direction that is perpendicular to its

original plane of deposition. Steeply dipping reservoirs can occur in areas

where deposition is followed by a period of geologic uplift, such as the

growth of a salt dome in the Gulf of Mexico after a reservoir has been deposited.

I 7.4 Capillary Pressure ]

The concept of capillary pressure is needed to understand the distribution

and displacement of fluid in the reservoir. Before discussing capillary

pressure, it is instructive to first define two additional concepts" interfacial tension and wettability.

Interfacial Tension A surface free energy resulting from electrical forces is present on all

interfaces between solids and fluids, and between immiscible fluids. The

electrical forces cause the surface of a liquid to occupy the smallest possible

area and act like a membrane. The magnitude of the tension between two

fluids is given by the interfacial tension. Interfacial tension (IFT) refers to

John R. Fanchi 123

the tension at a liquid/liquid interface. Surface tension refers to the tension at a gas/liquid interface, but is often used interchangeably with IFT.

Interfacial tension (IFT) is the energy per unit of surface area, or force per unit length. IFT is usually expressed in milli-Newtons/meter or the equivalent dynes/cm. IFT depends on the two fluids at the interface:

Fluid Pair

Air-Brine

Oil-Brine

Gas-Oil

IFT Range (mN/m or dyne/cm)

72-100

15-40

35-65

IFT can be estimated using the Weinaug-Katz variation of the Macleod- Sugden correlation and Fanchi' s procedure for estimating parachors [Fanchi, 1990].

The Weinaug-Katz variation of the Macleod-Sugden correlation is

Nc ( 9L O 1/4 /~1"= echi Xi ML

pv) -Yi Mv (7.4.1)

where

0 interfacial tension [dyne/cm] P~h parachor [(dynes/cm)l/4/(g/cm3)]

ML molecular weight of liquid phase My molecular weight of vapor phase PL liquid phase density [g/cm 3] P v vapor phase density [g/cm 3] xi mole fraction of component i in liquid phase Yi mole fraction of component i in vapor phase

Parachors are empirical parameters. The parachor of component i can be estimated using the molecular weight M; of component i and the empirical regression equation


e c h i - 10.0+ 2.92 M i (7.4.2)

This procedure works reasonably well for molecular weights ranging from 100 to 500. If you need a more accurate procedure, see Fanchi [ 1990].

Wettability Wettability is the ability of a fluid phase to preferentially wet a solid

surface in the presence of a second immiscible phase. Based on laboratory tests, most known reservoirs have intermediate wettability, typically being preferentially water wet. Wettability is quantified using a quantity called the contact angle. Contact angle is always measured through the more dense phase.

Table 7-2 Wettability and Contact Angle

Wetting Condition

Strongly Water-wet

Moderately Water-wet

Neutrally Wet

Moderately Oil-wet

Strongly Oil-wet

Contact Angle, degrees

0-30

30-75

75-105

105-150

150-180

The wetting, or wettability, condition in a rock/fluid system depends on interfacial tension. Contact angle is related to interfacial energies by

a o~ - a ws - ~ ow cos 0 (7.4.3)

where

Oos interfacial energy between oil and solid [dyne/cm] Ow~ interfacial energy between water and solid [dyne/cm] O ow interfacial energy, or IFT, between oil and water [dyne/cm] 0 contact angle at oil-water-solid interface measured through the

water phase [degrees]

John R. Fanchi 125

Changing the type of rock or fluid can change IFT and, hence, the wettability of the system. Adding a chemical such as surfactant, polymer, corrosion inhibitor, or scale inhibitor can alter wettability. This is the basis for a number of enhanced recovery projects.

Several factors can affect laboratory measurements ofwettability using core samples. Wettability can be changed by contact of the core during coting with drilling fluids or fluids on the rig floor; contact of the core during core handling with oxygen and/or water from the atmosphere; or by use of laboratory fluids that are not at reservoir conditions. If reservoir wettability is crucial to a project, it should be determined using native state cores.

Capillary Pressure Capillary pressure is the pressure difference across the curved interface

between two immiscible fluids in contact in a small capillary tube. The pressure difference is expressed in terms of wetting and nonwetting phase pressures, thus:

Pc = P,,w- Pw (7.4.4)

where

Pc capillary pressure [psi] Pnw pressure in nonwetting phase [psi] Pw pressure in wetting phase [psi]

Capillary pressure is related to IFT, contact angle, and pore radius by:

PC "-" 2G cosO

(7.4.5)

where

Pc capillary pressure [dynes/cm 2] r pore radius [cm] o interfacial (or surface) tension [mN/m or dynes/cm]

0 contact angle [degrees]


The above expression shows that capillary pressure in reservoirs depends

on o, the IFT between two immiscible fluids; 0, the contact angle between rock and fluid, which is a function of wettability; and r, pore radius, a

microscopic rock property. An increase in pore radius leads to a decrease in Pc. Thus, high permeability rocks with relatively large pore radii have lower Pc than lower permeability rocks containing the same fluids.

Capillary forces explain why water is retained in oil and gas zones. In water-wet reservoirs, water coats rock surfaces and is preferentially held in

smaller pores. Nonwetting hydrocarbon phases occupy the central space of larger pores.

Capillary pressure cannot be measured directly in a reservoir. Capillary pressure is usually determined by centrifuge experiments that provide a relationship between Pc and water saturation Sw. It is also possible to infer the relationship between capillary pressure and water saturation by matching the water saturation profile obtained from a well log.

Capillary pressure data are used in reservoir engineering primarily for determining initial fluid contacts and transition zones. Capillary pressure is also important in describing fluid flow from fractured reservoirs because

capillary pressure controls the flow of fluids between the fracture and the rock matrix. Equivalent height and transition zones are discussed in more detail in the next chapter.

Capillary Pressure Hysteresis Capillary pressure depends on the history of the rock-fluid system.

Capillary pressure is a function of saturation that depends on the direction of saturation change. The direction of saturation change depends on the historical events that led to the present distribution of fluids in the formation. There are two cases to consider:

�9 Drainage curve: wetting phase saturation decreases as the nonwetting phase saturation increases.

�9 Imbibition curve: wetting phase saturation increases as the nonwetting phase saturation decreases.

The difference in paths between the drainage and imbibition curves is called

hysteresis. Oil trapped in an immobile state at the end of imbibition is residual oil saturation, Sor.

John R. Fanchi 127

Converting Capillary Pressure Measurements Laboratory measurements of capillary pressure Pc should be conducted

with reservoir fluids at reservoir temperature. For example, the capillary pressure for an oil-water system should be determined using reservoir oil and reservoir brine at reservoir temperature. However, Pc measurements are sometimes made using more convenient laboratory fluids, and are almost always made at laboratory conditions. The following relationship can be used to convert laboratory Pc to reservoir Pc:

(~ cosOI)res Pc(res) "- Pc(lab)lqc~176 =- ( [ l)

(3" COS0 lab (7.4.6)

where o is interfacial tension (IFT) and 0 is contact angle [Amyx, et al., 1960]. The problem with the capillary pressure correction shown in Equation (7.4.6) is that it requires data that are often poorly known, namely interfacial tension and wettability contact angle at reservoir conditions.

Interfacial tension can be measured at reservoir conditions in the laboratory or estimated from correlations. The Macleod-Sugden correlation applies to pure compounds and the Weinaug-Katz correlation applies to mixtures [Fanchi, 1990].

Rao and Girard [1997] have described a laboratory technique for measuring wettability using live fluids at reservoir temperature and pressure. Alternative approaches include adjusting capillary pressure curves to be consistent with well log estimates of transition zone thickness, or assuming the contact angle factors out.

I 7.5 Relative Permeability I

Permeability can be thought of as a measure of the cross-sectional area normal to the direction of fluid flow. Absolute permeability is a measure of the capacity of a rock to transmit a single phase fluid. The value of permeability depends on the phase of the fluid flowing through the rock. Effective permeability is the permeability of a particular phase in the interconnected pore space of the rock. In the case of an oil reservoir, effective permeabilities can be defined for oil, water and gas phases, thus


ko effective permeability to oil

effective permeability to water

effective permeability to gas

Absolute permeability is often approximated as the effective permeability to air.

The effective permeability ofmultiphase flow is represented by defining relative permeability in terms of effective permeability and a reference permeability. Relative permeability is defined as the ratio of effective permeability to a reference permeability"

key k r -

kref (7.5.1)

where

keff effective permeability of fluid [md]

kref reference permeability [md]

If the reference permeability is the effective permeability of air, then relative permeability at 100% air saturation should equal one.

If we designate the reference or base permeability as k, we can express the relative permeabilities for oil, water and gas as the ratios

kro k ~ k w kg (7.5.2) - k ' k ~ w - k ' k r g - k

Relative permeability varies between 0 and 1 because keff <- k. The sum of the relative permeabilities over all phases N at the same time and place in the porous medium satisfies the inequality

N

Z kre -< 1 (7.5.3) e - I

John R. Fanchi 129

Multiphase flow is quantitatively described by modifying Darcy's law to include relative permeability. Darcy's law for laminar flow of phase Q becomes

... kre ~e - -0.001127 K A. V �9 e (7.5.4)

~e

where absolute permeability is represented in Equation (7.5.4) as a dyadic to allow for permeability anisotropy. Figure 7-1 presents a typical set of relative permeability curves. The shape of the curves determines the relative flow rates of two or more phases s imulta- neously moving through the same pore space. Initial

fluids-in-place are Figure 7-1. Typical Water-Oil Relative often dependent on Permeability Curves the saturation end points for the relative permeability curves.

Most multiphase flow phenomena involve the movement of two phases through the same interconnected pore space at the same time. The most frequently occurring two-phase relative permeability systems are oil-water relative permeability, gas-oil relative permeability, and gas-water relative permeability.

Three-phase relative permeabilities can be measured in the laboratory, but laboratory measurements can be costly, time-consuming, and inaccurate. Consequently, three-phase k~o are usually estimated from models of three- phase behavior using two-phase kr curves. Stone [ 1973] has provided two widely used three-phase relative permeability methods. Three-phase correlation methods usually estimate oil relative permeability from two sets


of two-phase relative permeability data: water displacing oil; and gas displacing oil at irreducible water saturation Swc.

The shape of a relative permeability curve depends on its history. Relative permeability curves usually increase monotonically as their associated fluid phase saturations increase. As an illustration, the relative permeability of water will increase as water saturation increases. Drainage, imbibition, and wettability can often be surmised from the shape of relative permeability curves.

Relative permeability data should be obtained using experiments that best model the type of displacement that is thought to dominate reservoir flow performance. For example, water-oil imbibition curves are representative ofwaterflooding, while water-oil drainage curves describe the movement of oil into a water zone.

Several procedures exist for averaging relative permeability data [for example, Schneider, 1987; Mattax and Dalton, 1990]. In practice, relative permeability is one of the most useful physical quantities available for performing a history match. As a consequence, the curves that are initially entered into a reservoir model are often modified during the history match. In the absence of measured data, correlations such as Honarpour, et al. [ 1982] give a reasonable starting point for estimating relative permeability.

Permeability and relative permeability describe flow of a particular fluid in a particular rock type. If the fluid system changes or the rock type changes, the appropriate values of permeability and relative permeability must be measured. For example, if a waterflood is planned for an oil reservoir that is being depleted, laboratory measured permeabilities need to represent the injection of water into a core with reservoir oil and connate water. Permeabil- ity measurements for a gas flood would not be consistent with the waterflood system. The permeability distribution and relative permeability curves used in reservoir engineering calculations need to reflect the type ofprocesses that are expected to occur in the reservoir.

Mobility and Mobility Ratio Two frequently encountered concepts in multiphase flow are mobility

and mobility ratio. Mobility is defined as the ratio of effective phase permeability to phase viscosity, or ~ = kQ / I~ where Q denotes the phase. Fluids with large mobilities will move at higher velocities through a porous medium than fluids with small mobilities. Gas, for example, has a small

John R. Fanchi 131

viscosity compared to liquids and will generally have a larger mobility than liquids. Oil and water often have comparable mobilities because they have comparable viscosities.

Mobility ratio can be calculated once mobilities are known. Mobility ratio M is defined as the mobility of the displacing fluid ~,D divided by the mobility ofthe displaced fluid ~'d, orM = ~,z~/~d-For example, the mobility ratio for a waterflood is Xw/)~o where ~,w and ~,o are water and oil mobilities. In a waterflood, water is the displacing fluid and oil is the displaced fluid.

The relative mobility ~'r~ of phase ~ is the relative permeability of the phase divided by the phase viscosity, ~,~ = kr~/~t~. Mobility is equal to the relative mobility times the reference permeability. Notice that mobility ratio can be calculated using relative mobilities because the reference permeability for calculating mobilities is the same permeability for both displacing and displaced fluids.

Fractional Flow

Mobilities may be used to estimate the fractional flow of one phase when two phases are flowing. Suppose we have two phases A and B flowing through a porous medium. The fractional flowfA of phase A is the flow rate qA ofphase A divided by the total flow rate qrofthe two phases A and B, thus

qA qA fA - - (7.5.5)

qr qA + q8

where qB is the flow rate of phase B. Fractional flow may also be expressed in terms ofmobilities. If we assume Darcy's law applies and neglect gravity and capillary pressure, then the fractional flow of phase A is

LA 1

fA - )~A +)~ 8 1 + ) ~ (7.5.6)

where XA and ~.8 are the mobilities of phases A and B respectively. If only one phase (qA) is flowing, then q8 = 0 and fA = 1 as expected. A similar definition of fractional flow applies for phase B.


I CS-7. Valley Fill Case Study: Permeability [

The permeability distribution in the Valley Fill reservoir is assumed to be

isotropic and homogeneous because the spatial dependence of permeability

is not known. Horizontal permeability is 150 md based on a pressure

transient test. Vertical permeability was not measured and is therefore assumed to be one tenth of horizontal permeability.

[ Exercises [

7-1. Consider a linear flow system with area = 25 ft 2. End point A is 5 ft

higher than end point B, and the distance between end points is 15 ft.

Suppose the system contains oil with viscosity = 0.8 cp, gravity = 35 ~ API (yo=0.85), and FVF = 1.0 RB/STB. If the end point pressures are PA = PB = 20 psia, is there flow? If so, how much and in what direction? Use Darcy's law with the gravity term and dip angle 0~:

I J q = -0.001127 kA PA - Pa + pgsinet ~t13 L

7-2.

7-3.

7-4.

Using the data in Exercise 7-1, calculate the pressure PB that would prevent fluid flow.

Suppose the pressure P~ of a water-bearing formation at depth z 1 " - -

10,000 ft is 4000 psia. If the pressure gradient for water is 0.433

psia/ft, calculate the pressure P2 at depth Zz = 11,000 ft. Calculate the

phase potentials ~ and ~2 at depths Zl and z2 respectively.

Use the pressures and potentials in Exercise 7-3 to estimate the derivatives dP/dz and ddP/dz. Will there be vertical flow?

Chapter 8 Applications of Rock-Fluid

Interactions

Laboratory measurements of rock-fluid interaction parameters provide information at the core scale (macro scale) and, in some cases, at the microscopic scale (micro scale). This chapter shows how measures of rock- fluid interactions are applied to the quantitative description of frontal advance and transition zones.

I 8.1 Frontal Advance Theory ]

Injection processes require contact between the injected fluid and the displaced fluid. The interface between injected and displaced fluids is called the flood front. Sweep efficiency is a measure of the fraction of reservoir fluid displaced by the injected fluid. When the injected fluid arrives at the producer, its appearance in the production stream is called breakthrough. Breakthrough times associated with the arrival of injected fluids at producing wells can be used as indicators of sweep efficiency. One of the simplest and most widely used methods for estimating the advance of the front in an immiscible displacement process is the Buckley-Leverett method [1942].

133

134 Shared Earth Model ing

Buckley and Leverett developed a theory that estimates the rate at which

the front of an injected water bank moves through a porous medium. The

approach uses the concept of fractional flow and makes the following

assumptions: flow is linear and horizontal; water is injected into an oil

reservoir; oil and water are both incompressible; the oil and water phases are immiscible; and gravity and capillary pressure effects are negligible. The

following analysis can be found in a variety of sources, such as Collins [ 1961 ] and Dake [ 1978, 1994].

A

[ A x �9

Figure 8-1. Flow Geometry

Frontal advance theory begins with the conservation of mass. Mass

conservation is an accounting requirement that assures that the amount of

mass entering the block minus the amount of mass leaving the block equals

the accumulation of mass in the block. The total flow rate through the volume element depicted in Figure 8-1 is the sum of the flow rates of the injected and displaced phases, thus

q, = qo + qw = total f low rate (8.1.1)

where q is rate and subscripts {o, w, t} refer to oil, water and total

respectively. The rate of water entering the element on the left hand side (LHS) is

q~ f w - entering L H S (8.1.2)

John R. Fanchi 135

with water fractional flowf~. The rate of water leaving the element on the right hand side (RHS) is

qt (fw + Afw)= leaving RHS (8.1.3)

The rate of change of water flowing through the volume element is

rate change - water entering - water leaving

= q , f w - q,(fw + Afw)= -q, Afw (8.1.4)

The rate of change of water saturation in the volume element is the rate

change in Equation (8.1.4) divided by the pore volume of the element, thus

ASw _ -qt Afw - ( 8 . 1 . 5 )

At 6x

where Ax is the length of the volume element with porosity ~ and cross- sectional area A. The distance x is measured from the point of injection at

the left hand face of the volume element. In the limit as At -~ 0 and Ax -~ 0,

we obtain the differential form of Equation (8.1.5):

OSw -qt afw Ot A~ Ox

(8.1.6)

A similar equation can be derived for the oil phase"

aSo -q t Ofo c3---~ = A~ Ox (8.1.7)

Equation (8.1.6) can be written in a form that is amenable to the method of characteristics, a front tracking methodology, by recognizing that the

fractional flow of waterfw depends only on Sw. The partial derivative Offw with respect to the distance x from the injection point can be written as


O~w df w OSw - ( 8 . 1 . 8 )

ax dSw ax

Substituting Ofw/OX into OSw/OX in Equation (8.1.6) yields

OSw - q t dfw OSw = (8.1.9)

cot A~ dS w Ox

It is not possible to solve for the general distribution of water saturation Sw (x, t) in most realistic cases because of the non-linearity of the problem. Water fractional flow is usually a non-linear function of water saturation. This means that Equation (8.1.9) is in general a non-linear partial differential equation. It is therefore necessary to consider a simplified approach to solving Equation (8.1.9).

We begin by expanding the total differential of Sw(x, t) and then dividing by the differential dt, thus

dS w OS w dx OS w ~ = ~ - - + (8.1.10)

dt Ox dt Ot

Equation (8.1.10) is simplified by solving the problem for a surface with constant Sw. In this case, the derivative dSJdt equals 0 and Equation (8.1.10) can be rearranged to give the rate of change of the location of the surface with constant Sw. The result is

_ _ 0______~t (8.1.11)

Ox

We replace the partial derivatives on the right hand side of Equation (8.1.11) by substituting Equations (8.1.8) and (8.1.9) into Equation (8.1.11). The resulting equation is the Buckley-Leverett frontal advance equation and has the form:

John R. Fanchi 137

qt( w) (8 s~ Add dS w Sw

The term (dx/dt)sw is the velocity of the moving plane with constant water saturation Sw, and the derivative (dfJdSw)sw is the slope of the fractional flow curve evaluated at Sw. The integral of Equation (8.1.12) is

i dx d t _ _ qt dfw o Sw o dSw SW

dt (8.1.13)

The solution of Equation (8.1.13) may be written as

( dfw ) (8 1 14) X sw = ~ dS w Sw " "

where

XSw

(df w / dSw) Sw

distance traveled by front with constant Sw [ft] cumulative water injected [cu ft]

slope of fractional flow curve

Equation (8.1.14 ) is a front tracking equation because it follows a front of constant water saturation Sw during the waterflood.

Water Saturation Profile

A plot of water saturation Sw versus distance x using Equation (8.1.14) and typical fractional flow curves leads to the physically impossible situation of multiple values of Sw at a given location. The analysis presented above has introduced an extraneous solution that needs to be removed. A discontinuity in Sw at a cutofflocation Xc will yield a single value of water saturation and provide a volumetric balance for wetting fluids. In 1952, Welge


published a procedure for determining the cutofflocation Xc using material

balance. Welge's approach is presented in the next section.

I 8.2 Welge's Method I

The first step in Welge's method for solving the Buckley-Leverett frontal ad-

vance equation is to plot water

fractional flOWfw versus water saturation Sw. An example plot is shown in Figure 8-2. It is important to let the water saturation and water fractional flow scales range from 0 to 1.

According to Welge's method, a line should be drawn that is tangent to the

water fractional flow curve and intersects the x-axis at the Figure 8-2. Welge's Method original Sw. The tangent line is called the breakthrough tangent, or slope. Water saturation at the flood front

Swl is the point of tangency on thefw curve, and the fractional flow of water at the flood front with water saturation Swl iSfwl. The average water saturation behind the flood front Swb~ is the intercept of the main tangent line with the

upper boundary of the plot drawn atfw = 1.0. In Figure 8-2, Swi is 65%,fwl

is 95%, and average Swbt is 67%. Welge' s approach can be used to calculate more useful information about

the waterflood. The time to water breakthrough at the producer is measured from the beginning of water injection until the water front with water saturation Swl arrives at the producer. The water breakthrough time is

tbt = (8.2 1) qi(df~/dSw)sw '

John R. Fanchi 139

where

qi

Swf

injection rate

slope of main tangent line

linear distance from injection well to production well

Cumulative water injected is given by

1

Oi (dfw / dSw)sw e (8.2.2)

where Qi is the cumulative pore volume of injected water at breakthrough. The actual volume of injected water is determined by multiplying Qi by the pore volume OpLA of the system.

Effects of Capillary Pressure and Gravity Buckley-Leverett displacement is often called piston-like displacement

because the flood front calculated using Buckley-Leverett theory will propagate as a"sharp" step function in the absence of capillary pressure and gravity effects. The presence of capillary pressure or gravity can lead to smearing, or dispersion, of the flood front. The dispersed flood front causes a change in the behavior of produced fluid ratios. Rather than an abrupt increase in water-oil ratio (WOR) associated with piston-like displacement, the WOR will increase gradually as the leading edge of the mobile water reaches the well and is produced.

Applicability to Reservoir Characterization The Buckley-Leverett procedure described above shows how important

the water fractional flow curve is to the calculation of waterflood performance under the assumptions associated with the Buckley-Leverett calculation, such as incompressible linear flow in a homogeneous, isotropic porous medium. If the waterflood does not perform as predicted, the waterflood


performance implies that one or more of the assumptions is incorrect.

Reservoir heterogeneity and anisotropy are two possibilities that would need

to be considered in attempts to understand waterflood performance.

The Buckley-Leverett procedure is an example of a relatively straightfor-

ward analytical technique for modeling reservoir performance. If an

analytical technique does not match actual flow performance, this provides information about the validity of the assumptions that were made in

developing the analytical technique. More sophisticated flow modeling tools

such as reservoir simulators described in Chapter 12 are often needed to

properly model complex flow systems.

[ 8.3 Transition Zones [

Capillary pressure in a reservoir is responsible for the formation of a

transition zone, or a zone in which multiphase flow will occur. Transition

zones may vary in thickness from a few feet to a few thousand feet [Bradley,

1992, Table 27.9]. The size of the transition zone affects estimates of original

hydrocarbons in place and the distribution of recoverable reserves. It is

important to accurately characterize transition zones because of their

potentially large impact on reservoir economics. In one example from the

literature [Heymans, 1997], a 90 foot thick transition zone, in an edge-water drive reservoir with nearly a 1000 foot oil column, contributed more than

30% of the estimated original oil in place. Rigorous volumetrics and material

balance calculations were within 2% agreement. The transition zone was less than 10% of the total thickness of the reservoir, yet the volume of hydrocarbons in a relatively thin transition zone that formed a ring around the

example reservoir had a significant impact on original oil in place estimates.

An oil-water transition zone is generally described from the perspective

of oil recovery as a zone in which both oil and water are produced. The top

of a transition zone in a reservoir is the elevation at which water-free oil can

be produced. It corresponds to the depth at which mobile water saturation

first appears. The bottom of an oil-water transition zone is the shallowest

depth at which oil-free water is produced. Oil saturation at this depth is equal to residual oil saturation. Additional oil is not recoverable from waterflood-

ing once the residual oil saturation to waterflooding is attained.

John R. Fanchi 141

The height of a transition zone depends on capillary pressure and the difference in densities between the wetting and nonwetting phases, thus

h-(Ow-Pnw) (8.3.1)

where

h height of capillary rise [ft]

Pc capillary pressure [psi]

Pw wetting phase density gradient [psi/ft]

Pnw nonwetting phase density gradient [psi/ft]

The wetting phase is usually water for a water-oil transition zone and a water-gas transition zone. For a more detailed discussion of transition zones, see Christiansen, et al. [1999].

The relationship between capillary pressure and elevation is used to establish the initial transition zone in the reservoir. The oil-water transition

zone, for example, is the zone between water-only flow and oil-only flow. It represents that part of the reservoir where 100% water saturation grades into oil saturation with irreducible water saturation. Similar transition zones may exist at the interface between any pair of immiscible phases.

Oil Reserves in an Oi l -Water Transit ion Zone

Data in the literature show that residual oil saturation in the transition

zone (Sor,~) is a function of initial oil saturation in the transition zone (Soi,~). The relationship between ~ortz and Soit2 is referred to as a trapped oil relationship and is illustrated in Figure 8-3 [Pickell, et al., 1966]. Trapped oil relationships can be measured in experiments with partial saturation of a sample with oil. However, such measurements of trapped oil saturations and the associated oil-water relative permeabilities are rarely reported in the literature.

Published results of trapped hydrocarbon measurements, especially for trapped gas saturation, show that hydrocarbons are recoverable from initial hydrocarbon saturations as low as 10% pore volume. These core test measurement results indicate that hydrocarbons are recoverable from a


greater portion of a transition zone than that included in the conventional

method of analysis.

Figure 8-3. Sample Trapped Oil Relationship [Pickell, et al., 1966]

Trapped oil core test measurements show that an empirical relationship

exists between an initial oil saturation (So~) and the associated residual oil

saturation (Sor). Recoverable oil saturation is the difference between So~ and

the associated So~. When trapped oil measurements are related to an oil

saturation profile as a function of height, the relationship shows that Sor decreases with depth within a transition zone. The depth variation of So~ and

Sor results in hydrocarbon recovery being greater at the top of a transition

zone and decreasing with depth and So~. The conventional method for estimating oil reserves in a transition zone

is based on the assumption that residual oil saturation is constant throughout

the transition zone. Including the trapped oil relationship in calculations of

oil reserves is a new method of analysis. The new method provides more

accurate reserve estimates than procedures which neglect the trapped oil

relationship. The effect of the new method is illustrated by using a specific

example.

Figure 8-4 shows two synthetic water saturation profiles (called

"hyperbolic" and "quadratic") as a function of depth together with a constant

John R. Fanch i 143

So~ saturation profile. The depth in Figure 8-4 is measured relative to the top of the transition zone. The hyperbolic curve represents a relatively abrupt

transition zone, and the quadratic curve represents a relatively gradual

transition zone. Most transition zones will be bounded by the hyperbolic and quadratic curves.

Figure 8-4. Synthetic Saturation Profiles

Figures 8-3 and 8-4 contain enough information to estimate the effect

of the trapped oil relationship on the estimate ofoil reserves in the transition

zone. The curves in Figure 8-5 are obtained by integrating the information

in Figures 8-3 and 8-4. The hyperbolic curve in Figure 8-4 is shown in Figure

8-5 together with the saturation profile that accounts for a variable Sor. The

mobile oil saturation is the difference in saturation between the two curves at a common depth.

The fractional increase in estimated oil reserves that is obtained using

the trapped oil relationship rather than the conventional constant Sor assumption is

5 N N . - N c

Nc - N c (8.3.2)


where Nc is mobile oil calculated using the conventional asSumption that Sor is constant in the transition zone, and Nv is mobile oil calculated using the

assumption that So~ varies with depth in the transition zone. Our task is to

calculate 6N/Nc by first calculating Nc and Nv.

-20

-40 N ~ . -60 a

-80

100

120

i ;

I i

i

I

& I . 1 i ' I i , !

~

~

' i . i

~

. I i i

P "&o i

0 0.2 0.4 0.6 0.8 1

S w = 1 - S o

Hyperbo l i c - - . - V a r i a b l e S o r

Figure 8-5. Hyperbolic Saturation Profile and a Variable Sor Profile

The average mobile oil saturation (Som)c in the case with constant Sor is

(Som)c -(Soi)c -Sor (8.3.3)

where (Soi)c is the average initial oil saturation

Z

(Soi)c = ISoidz dz (8.3.4) o

Transition zone thickness z c is measured from the top of the transition zone

to the point of intersection of the initial oil saturation curve (Soi = 1 - Swg) and

Sor. The mobile oil in the transition zone assuming constant Sor is

John R. Fanchi 145

N c - r c [ ( S oi ) c - S or ] (8.3.5)

where the transition zone pore volume is the product of porosity (~, cross- sectional area A and thickness Zc.

The average mobile oil saturation (Som)v in the case with variable So~ is

(Som)v --(Soi)v --(Sor)v (8.3.6)

where (Soi)v is the average initial oil saturation

z z/Z! (Soi)v- ISoi dz (8.3.7)

0

The transition zone thickness Zv is measured from the top of the transition

zone to the point of intersection of the initial oil saturation curve and the

variable Sor c u r v e . In this case, the intersection is at Sor "- 0 for the trapped

oil relationship given in Figure 8-3. Assuming constant Sor, the mobile oil

in the transition zone is the product of the transition zone pore volume and the mobile oil saturation, thus

Nc =(~Azc[(Soi)c - Sot] (8.3.8)

The transition zone pore volume is the product of porosity ~, cross-sectional area A in the horizontal plane, and transition zone thickness Zc.

The fractional increase in reserves for each saturation profile is

~ N [(Soi)v -(Sor)v]Zv [(Soilc - or]Zc - 1 (8.3.9)

Results for the two saturation profiles are presented in Table 8-1 in Section 8.4. The "hyperbolic" saturation profile increased by approximately 11%,

while the "quadratic" saturation profile had almost a 37% increase.


8.4 Transition Zone Volumetrics and Numerical Simulation I

Numerical simulators are powerful tools for enhancing reservoir management

decisions. A numerical simulator of fluid flow in porous media solves the

nonlinear, partial differential equations that describes the physics of fluid

flow. Numerical flow simulators are discussed in more detail in a later chapter. A numerical simulator is used here to validate the analysis of

incremental reserves presented in Section 8.3.

Typically, capillary pressure and relative permeability core test

measurements with a constant residual oil saturation are used in reservoir

model studies. However, a limited set of data in the literature suggests that proper modeling of the trapped oil relationship requires that the flow

simulator account for a depth dependent residual oil saturation. This effect

can be achieved in existing simulators, but the process is arduous. In particular, a layer cake grid consisting of planes of grid blocks is used to

represent the reservoir. Vertical variations in reservoir properties such as

porosity and permeability can still be entered. Each layer is treated as a Rock

Region, that is, each layer is assigned its own set of relative permeability and capillary pressure curves. The flow model can then be initialized.

The analysis presented in Section 8.3 shows that relative permeability curves should account for the variation in residual oil saturation. If not, the

amount of recoverable oil (or gas) in a transition zone will be underestimated.

The variation of residual oil saturation as a function of depth can be achieved

in existing flow simulators by normalizing the measured curves and then

denormalizing the curves for the saturation end points assigned to each Rock Region.

To verify this procedure, a vertical column model of the transition zone

at the base of an undersaturated oil reservoir was run using an extended black

oil simulator. A production well was placed at the top of the column and a

water injection well was assigned to the lowermost grid block. Initial

saturations were based on the saturation profiles presented in Figure 8-4. For

example, the original oil in place in the model with the quadratic saturation

profile was 3.01 MMSTB. The model with a constant residual oil saturation

So~ produced 0.90 MMSTB oil before a limiting water-oil ratio of 5 STB

water/STB oil was reached. The corresponding model with a variable Sor produced 1.24 MMSTB oil before reaching the WOR limit. The fractional

John R. Fanchi 147

increase in oil recovery was approximately 38%, which is in good agreement

with the value predicted in the previous section.

Table 8-1 Fractional Increase in Estimated Reserves for

Two Saturation Profiles

Variable

Zc

Zv

(Soi)c Sot.

(Soi)v (Sor)v 5N/Nc

Hyperbolic Case

113.6 ft

120 ft

0.695

0.340

0.697

0.326

10.7%

Quadratic Case

8Oft

120 ft

0.526

0.340

0.395

0.226

36.8%

I .... CS-8. Valley Fill Case Study: Rock-Fluid Interaction Data [

Two-phase relative permeability curves are shown in Tables CS-8A and CS-

8B for an oil-water system and a gas-oil system respectively. These curves

were calculated using the Honarpour, et al. [ 1982] correlations for a water

wet sandstone with the following end point saturations: initial oil saturation

is 70%, residual oil saturation is 25%, irreducible water saturation is 30%,

and critical gas saturation is 3%. The analytical correlations for relative

permeability curves was used because there are no explicit measurements

of irreducible water saturation, critical gas saturation, relative permeability

curves, or capillary pressure curves. These measurements could be made on

core from the productive interval if core is available. A request to perform

these measurements is reasonable, but the data will not be available in the

time frame of the study. Values of irreducible water saturation and critical

gas saturation are therefore estimated based on analogous fields. Capillary

pressure effects are neglected for now. In a complete study, the sensitivity

of the shared earth model and its flow performance to the rock-fluid

interaction data should be estimated.


Table CS-8A. Oil-Water Relative Permeability

Water Saturation

0.00

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

0.75

0.80

1.00

p o w

1.0

1.0

0.59

0.32

0.18

0.08

0.03

0.01

0.001

0.0001 i

0.0

0.0

0.0

0.0

0.0

0.01

0.02

0.034

0.046

0.068

0.09

0.128

0.166

0.2

0.24

0.24

Table CS-8B. Gas-Oil Relative Permeability

Gas Saturation

0.00

0.03

0.05

0.10

0.15

0.20

0.25

0.30

0.35

Krog 1.0

0.75

0.59

0.32

0.18

0.08

0.03

0.01

0.001

0.0

0.0

0.02

0.09

0.16

0.24

0.33

0.43

0.55

John R. Fanchi 149

Table CS-8B. Gas-Oil Relative Permeability

Gas Saturation

0.40

0.45

0.50

1.00

(cont.)

gro 0.0

0.0

0.0

0.67

0.81

1.0

1.0

Exercises ]

8-1.

8-2.

8-3.

Plot the oil-water relative permeability curves in Table CS-8A as a

function of water saturation. Use fluid property data at 4014.7 psia in

Table CS-6A and relative permeability in Table CS-8A to prepare

plots of oil and water relative mobilities as functions of water satura-

tion.

Calculate and plot the fractional flow of water using the data in

Exercise 8-1.

Plot the gas-oil relative permeability curves in Table CS-8B as a

function of gas saturation. Use fluid property data in Table CS-6A and

relative permeability in Table CS-8B to determine oil and gas relative

mobilities.

Chapter 9 Fluid Flow Equations

Fluid flow equations are the mathematical representation of the physics that is incorporated in computer simulators of fluid flow. The objective of this chapter is to improve our understanding of the assumptions and limitations of fluid flow models by reviewing the assumptions and limitations associated with several widely used fluid flow equations.

[ 9.1 Material Balance ]

Volumetrics provides an estimate of the volume of fluid in a reservoir. The volume Vo of phase Q at surface conditions is estimated by dividing the product of bulk reservoir volume VB, average porosity ~, and average saturation SQ of phase Q, by B~, the formation volume factor of phase t~, thus

v se V e - (9.1.1)

Be

Another way to estimate volumes of fluid in place is to use the law of conservation of mass.

150

John R. Fanchi 151

Mass conservation is the basis of fluid flow equations. If we consider the reservoir as a tank, or 0-D problem, then the conservation of mass provides an accounting of material entering or leaving the tank. An analysis of reservoir pressure changes due to production, injection or influx of fluids into or out of a tank model is referred to as a material balance analysis. The analysis treats the reservoir like a large tank and uses measurable quantities such as cumulative fluid production volumes for oil, water and gas phases; reservoir pressure measurements; and fluid property data obtained from

laboratory analysis of produced fluid samples. The form of the material balance equation depends on the type of fluid

being produced from the reservoir. For example, the volume of original oil in place N for an undersaturated oil reservoir undergoing depletion is

NpBo N = (9.1.2)

B 0 - Boi

where Np is the cumulative volume of oil produced at pressure P and time t, Bo is oil formation volume factor at P and t, and Boi is oil formation volume factor at the initial pressure. Notice that N has a sensitive dependence on oil formation volume factor. For comparison, the material balance equation

for the volume of original gas in place G is

apc(P/Z)i G - (e/z) i - (e /z) t

(9.1.3)

where Gpc is the cumulative volume of gas produced, P is reservoir pressure, Z is real gas compressibility factor. The subscript t indicates that the ratio P/Z is evaluated at the time t that corresponds to Gpc, and subscript Iindicates that P/Z should be evaluated at the initial reservoir pressure.

Material balance calculations provide an independent estimate of original

volumes in place. They also may be used to estimate the relative contribution

of different drive mechanisms, predict future production performance, and estimate cumulative recovery efficiency. Material balance calculations do not provide information about spatially dependent reservoir behavior because


the reservoir is viewed as a tank. To obtain more detailed spatial information,

it is necessary to apply the conservation of mass concept to fluid flow from

one region to another in the reservoir. The remainder of this chapter discusses

some of the ideas associated with reservoir flow modeling. For more

information on material balance, see sources such as Dake [ 1978], Craft, et

al. [ 1991 ], and Ahmed [2000].

! 9.2 Continuity Equation I

Hydrocarbon mixtures contain molecular components with a range of

molecular weights. The following analysis applies to flow of a fluid that may

contain a single component or a mixture of chemical components. It closely

follows the presenta-

tion originally pub-

lished by Fanchi, et al.

[1982]. The flow of

fluid into and out of a

single reservoir block

is illustrated in Figure

9-1 for flow in the x-

direction. The symbol Figure 9-1. Coordinate Convention J denotes fluid flux.

Flux is defined as the rate of flow of mass per unit cross-sectional area

normal to the direction of flow. The term "mass" refers to the mass of a

component in the fluid. In our case, we assume fluid flows into the block at

x with fluid flux Jx and out of the block at x + A x with fluid flux Jx + ax.

Applying the principle of conservation of mass to the system depicted in

Figure 9-l, we have the equality:

mass entering the block - mass leaving the block

- accumulation of mass in the block.

Consider a block with length Ax, width Ay, and depth Az. The bulk volume

of the block is Ax Ay Az. We can write an expression for the mass entering

the block in a time interval At as

John R. Fanchi 153

AxAz + (Jz)z kxAyJAt - M a s s in (9.2.1a)

where we have generalized the expression to allow flux in the y and z

directions as well. In the first term on the left hand side, the notation (J~)x

denotes the x direction flux at locationx and AyAz is the cross-sectional area

that is perpendicular to the direction of fluid flux. Analogous meanings apply

to the remaining terms on the left hand side of Equation (9.2.1 a).

Corresponding to mass entering the block is a term for mass exiting the

block. It has the form

y+ y X Z+( ztz+ z X yl t + qAxAyAzAt = M a s s out

(9.2.1b)

We have added a source/sink term q which represents mass flow into (source)

or out of (sink)the block. The source/sink term represents fluid entering

or leaving the block through an object such as a well or the rock matrix of

a naturally fractured reservoir. Production is represented by q > 0, and

injection by q < 0. Accumulation of mass in the block is the change in concentration of the

mass of a component (C) in the block over the time interval At. If the

concentration C is defined as the total mass of a component in the reservoir

block divided by the block volume, then the accumulation term becomes

[ ( C ) t + A t - (C)t ]AxA yAz - M a s s a c c u m u l a t i o n (9.2.1c)

Using the above equations in the mass conservation equality where (C)t

denotes the concentration of a component at time t, and (C),+at is the

concentration at time t + At.

Mass in- Mass o u t - Mass accumulation

gives

154 SharedEarthModelin~

AxAz + (Jz)z AxAyJAt (Jx)x y z+ (J,) y

- qAxAyAzA, - [(c),+ A, -(c), ]AxAyAz

Dividing Equation (9.2.2) by AxAyAzAt and rearranging gives

(Jx)x+Ax - (Jx)x Ax Ay

(Jz)z+Az-(Jz)z (c)t+ At-Cclt Az - q - At

(9.2.3)

In the limit as Ax, Ay, Az, and At go to zero, Equation (9.2.3) becomes

the continuity equation for a component in the fluid:

OJ OJ OJ OC x Y z q= Ox Oy Oz Ot

(9.2.4)

Equation (9.2.4) can be written in vector notation as

OC - V. J - q - 0 t (9.2.5)

A continuity equation for mass is needed for each component in the block.

Convection-Diffusion Equation Suppose a solute with concentration C is mixing with a solvent. Then

the flux of the solute has the form

John R. Fanchi 155

) - C ~ - D V C (9.2.6)

where D is a scalar diffusion term. In general, the diffusion term can be a

tensor (or dyadic). The form of Equation (9.2.6) uses a diffusion term that

assumes diffusion is homogeneous and isotropic. If we assume there are no

sources or sinks, then q is zero and the continuity equation, Equation (9.2.5),

becomes

OC V . C~ - V . DV C - (9.2.7)

Ot

If v andD are constant, we obtain the Convection-Diffusion (C-D) equation

for the concentration of a solute mixing with a solvent:

~. V C - D V 2 C - 0 C (9.2.8) Ot

Incompressible Flow The continuity equation for the flow of a fluid with density 9, velocity

and no source or sink terms may be written as

Op Ot

- - + V ' ( p V ) - 0 (9.2.9)

If we introduce the differential operator

D 0 = ~ + v . V (9.2.10)

Dt Ot

into Equation (9.2.9), the continuity equation has the form

D9

D t ~ + pV . ~ - 0 (9.2.11)


In the case of an incompressible fluid, density is constant andthe continuity equation reduces to the following condition for incompressible fluid flow:

V. F - 0 (9.2.12)

I 9.3 Convection-Dispersion Equation ]

The one-dimensional Convection-Dispersion (C-D) equation has the form

02C OC D - v ~

Ox 2 cox OC Ot

(9.3.1)

where D is dispersion, v is velocity, and C is concentration. The C-D equation in Equation (9.3.1) is referred to as the Convection-Diffusion equation when D is diffusion. The concentration C(x, t) is a function of space

and time. In this case, we assume that D and v are real, scalar constants. The dispersion term is D'O2C/Ox 2 and the convection term is v'OC/Ox. When the

dispersion term is much larger than the convection term, the C-D equation

behaves like the heat conduction equation, which is a parabolic partial differential equation (PDE). If the dispersion term is much smaller than the convection term, the C-D equation behaves like a first-order hyperbolic PDE.

The C-D equation is especially valuable for modeling fluid flow because the C-D equation can be solved analytically and the C-D equation may be

used to examine two important classes of PDEs (parabolic and hyperbolic). The analytic solution of the C-D equation establishes a standard for determining the validity of a numerical method.

We must specify two boundary conditions and an initial condition for the concentration C(x, t) to solve the C-D equation. We impose the boundary conditions C(0, t) = 1, C(oo, t) = 0 for all times t greater than 0, and the initial condition C(x, 0) = 0 for all values ofx greater than 0. The corresponding solution is [Peaceman, 1977]

1{ C(x,t)- -~ erfc x-vt] el V /DI

2~bTJ + erfc x+vt } 24-

(9.3.2)

John R. Fanchi 157

where the complementary error function erfc (y) is

2 i e_Z 2 e r f c ( y ) - I ] ~ dz (9.3.3)

Abramowitz and Stegun [ 1972] present an accurate numerical algorithm for calculating erfc (y).

We can compare the analytic solution of the C-D equation with a finite difference representation of the C-D equation. The partial derivatives are replaced with finite differences, which are in turn derived from Taylor's series. Indices/, j, k are often used to label grid locations along the x, y, z coordinate axes, respectively. Index n labels the present time level, so that n + 1 represents a future time level. If the finite difference representations of the partial derivatives are substituted into the original flow equations, the result is a set of equations that can be algebraically rearranged to form a set of equations that can be solved numerically. If we replace the time and space derivatives with forward and centered differences, respectively, we obtain the following discretized form of the one dimensional C-D equation:

D At ~ n + l _ 2 c n + l cn-~l _ 2 C n + C n _ l 1 2hx 2 "-7+ 1 + " + Cn- 1

_ A t _ n+l (9.3.4)

where {/, n } denote the discrete levels in the space and time domains, and {Ax, A t} are step sizes in the respective space and time domains. The finite difference representation of the C-D equation leads to a system of linear equations of the form

lcn 1 ( 2kx + D2Ax2 + I + D A x 2 C n +1

v h t _ D At )cn++m 1 C n + D At (C n 2Ax 2hx 2 = 2hx 2 t -1

- 2 c n +cn 1) l +

(9.3.5)


The finite difference equations may be written in matrix form as A u = B where C is the column vector of unknown values {C/"+l}, A is a square

matrix of coefficients from the left-hand side of the equation, and B is a

column vector of terms containing known values { Ci"} from the right-hand

side of the equation. Given both an analytic and a numerical solution of the C-D equation,

we can make a direct comparison of the methods to demonstrate the validity

of the numerical methods. The hyperbolic PDE comparison is shown in Figure 9-2, and the parabolic PDE comparison is shown in Figure 9-3. These

figures show that the finite difference technique does a reasonably good job of reproducing the actual analytical solution. In this case, the parabolic solution is more accurately matched using finite difference methods than is

the hyperbolic solution.

Figure 9-2. Comparison of Hyperbolic PDE Solutions

The most common numerical formulations of fluid flow equations are

Implicit Pressure-Explicit Saturation (IMPES) and Newton-Raphson. The

terms in the finite difference form of the flow equations are expanded in the

Newton-Raphson procedure as the sum of each term at the current iteration

level, plus a contribution due to a change of each term with respect to the

primary unknown variables over the iteration. The three primary unknown

John R. Fanchi 159

Figure 9-3. Comparison of Parabolic PDE Solutions

variables for an oil-water-gas system are oil phase pressure, water saturation,

and either gas saturation or solution GOR. The choice of the third variable

depends on whether the block contains free gas, which depends, in tum, on

whether the block pressure is above or below bubble point pressure. If

reservoir pressure is greater than bubble point pressure, the grid block is in

an undersaturated state. If reservoir pressure is less than bubble point

pressure, the grid block is in a saturated state. Naturally, the choice of

unknowns is different for a gas-water system or a water-only system. The

discussion presented here applies to the most general three-phase case.

To calculate the changes in primary variables, it is necessary to calculate

derivatives, either numerically or analytically, of the flow equation terms.

The derivatives are stored in a matrix called the acceleration matrix or the

Jacobian. The Newton-Raphson technique leads to a matrix equation J " 8X = R that equates the product of the Jacobian J and a column vector 8X of

changes to the primary unknown variables to the column vector of residuals

R. The matrix equation is solved by matrix algebra to yield the column vector

of changes to the unknown primary variables 8X. These changes are added

to the value of the primary variables at the beginning of the iteration. If the changes are less than a specified tolerance, the iterative Newton-Raphson

technique is considered complete and the simulator proceeds to the next


timestep. The Newton-Raphson technique is considered a fully implicit technique because all of the primary variables are calculated at the new time level.

The IMPES procedure is much like the Newton-Raphson technique except that flow coefficients are not updated in an iterative process. The IMPES procedure solves for pressure at the new time level using saturations at the old time level, and then uses the pressures at the new time level to explicitly calculate saturations at the new time level.

Timestep size in a fully implicit model can be much larger than in an IMPES model. Arbitrarily large timestep sizes can lead to smeared spatial gradients of saturation or concentration [Lantz, 1971 ] and grid orientation effects [Fanchi, 1983]. The smearing and grid orientation effects are referred to as numerical dispersion effects, and are due to the discretization of derivatives in time and space. The front from a numerical calculation does not exhibit the same piston-like displacement that is shown by the analytical Buckley-Leverett calculation. The Buckley-Leverett calculation is discussed in several sources, such as Collins, 1961; Wilhite, 1986; Craft, et al., 1991.

Numerical dispersion D ""~ in one spatial dimension may be written in the form

Dnum= V/Ax+ vAt I (9.3.6) 2 ,

with gridblock size Ax, timestep size At, frontal advance velocity v, and porosity ~. The "+" sign in front of the vAt/dp term applies to the fully implicit formulation, and the " - " sign in front of the vAt/d~ term applies to IMPES. An increase in A t in the fully implicit formulation increases D n"m, while the same increase in At decreases D ""m in an IMPES model. Although it appears that a judicious choice of Ax and At could eliminate D "um entirely in an IMPES model, the combination of Ax and At that yields D ""m = 0 violates a numerical stability criterion. Numerical dispersion in an IMPES model is not as large as that associated with fully implicit techniques.

The fully implicit technique is the most advanced flow simulation technology in use today, yet IMPES retains vitality as a relatively inexpensive means of modeling some problems. The IMPES timestep limitation is less of a problem than it might otherwise seem, because simulator timestep

John R. Fanchi 161

sizes are dictated by the need to match historical data. Large timestep sizes in a fully implicit model reduce the ability of the model to track historical variations of rate with time because historical data must be averaged over a longer period of time. The reporting period for well data often controls the frequency with which well data is read during a model run. As a result, the modeler often has to constrain the fully implicit simulator to run at less than optimum numerical efficiency because of the need to more accurately represent the real behavior of the physical system.

[ 9.4 Navier-Stokes Equation I

Suppose a fluid with constant density p and constant viscosity r I is flowing

with velocity ~ in the presence of a pressure distribution P. The equation

of motion for the fluid is derived by applying the principle of conservation of momentum. The resulting flow equation is the Navier-Stokes equation [Sahimi, 1995]

o-b- ; - o -g+ ~.V~) - rlV 2~_ VP+ p~ (9.4.1)

The vector ~ is the acceleration of gravity and the differential operator D/Dt

defined in Section 9.1 is made explicit. The Laplacian operator in Cartesian coordinates has the form

0 2 0 2 0 2 V 2 (9.4.2) -- O X 2 Jr c3 y 2 + O z 2

If viscous effects are negligible the term involving viscosity on the right hand side of Equation (9.4.1) is set to zero and the Navier-Stokes equation becomes Euler's equation

D~ P - - V P + P g (9.4.3)

D t


If inertial effects are negligible, the term pD ~ /Dt is set to zero and the Navier-Stokes equation becomes Stoke's equation

r I V 2g _ V P + p~ - 0 (9.4.4)

I 9.5 Integrated Flow Model Equations I

The fluid flow simulator that was used to prepare the files accompanying this text is an integrated flow model called IFLO [Fanchi, 2000a]. IFLO is a multiphase, multidimensional, pseudocomponent simulator. It is a computer program that is written in FORTRAN 90/95. Most flow simulators are written in a version of FORTRAN because of its ability to solve numerical problems. Other programming languages such as C++ are being considered as the computer language of choice for future flow simulators.

IFLO can be used to solve fluid flow problems in up to three phases. The

flow equations for the pseudocomponents in IFLO are presented below:

Stock Tank Oil

'eo oqo eBo o Posc - Ot ~ (9.5.1/

Water plus Surfactant

Kk rw v ~

~t wBw m

w

qw a

Pw~c Ot (9.5.2)

Surfactant

V " X s

K k r w v ~

~t wBw W X s

qw a

P w~c ~ t (9.5.3)


Soluble Species

�9 IVi

e

Kkrg V d~ i + vi Rio Kkr e V (~) e e

~t i Bi ~t o Bo o + v i R i w ~

qi Pisc -- O t ~vi -~i + Re~ No + Riw

r w v o ~t w B w w

(9.5.4)

for I = {g, 1, ..., N~}. Table 9-1 presents the Nomenclature for the symbols in Equations (9.5.1) through (9.5.4). The superscript e indicates that an effective fluid property is being calculated, and the subscript sc refers to standard conditions.

Table 9-1 Nomenclature for Flow Equations

Symbol Meaning

B~ Formation volume factor of phase Q

K

Ns q

Ri~ s~

vi x~

t-ti

P

Absolute permeability

Relative permeability of phase Q

Number of soluble species

Source/sink flow rate

Solubility of soluble component I in phase Q

Saturation of phase Q i

Volume fraction of soluble component i i

Surfactant volume fraction i

Viscosity of phase Q i

Gas phase viscosity including effects of soluble component i i

Density i

Potential of phase Q

iPorosity

The solution procedure for the flow equations is outlined below. The multidimensional flow model may be run as a material balance program by


setting transmissibility to zero and running the model as a grid with a single grid block. The formulation of fluid flow equations is presented in more detail by Ammer, et al. [1991] and Fanchi [2000a].

Volume Integration and Discretization The fluid flow equations presented above are discretized using volume

integration and finite difference techniques. The volume integration procedure is illustrated by integrating the oil flow equation over a block m with volume Vm, thus

I[ gkr; qo] S~t(S_~o) V. V �9 - dV - dO dV Vm geBo o 9o,~ Vm

(9.5.5)

The divergence theorem is used to replace the volume integral over the convection term on the left hand side of Equation (9.5.5) with a surface integral. Applying the divergence theorem gives

I Kke~ V~ .PtdS-I[q~os~]dV- 0 ! ( S-~o I Sere g ; B o o v, 0 t ~ dV (9.5.6)

where block volume Vm corresponds to the volume V, and the surface S is the external surface Sere of the grid block m. The surface integral represents fluid flow across the block boundaries.

The spatially discretized material balance equation for oil is

dMo - ~ + Qo - AAoA*o (9.5.7)

where the volume integral over rate is

Qo_ i q o d V : qo Vm P o,~ 9 osc

Zm (9.5.8)

John R. Fanchi 165

The volume integral over the accumulation term is

MO ~ So

Vm

(9.5.9)

and the surface integral is

I K k e ~ A A o A ~ o - V dP . ~ d S (9.510)

Sem~.[;B 0 o ~

The term Ao represents oil phase transmissibility and oil phase potential is

A~ "+~ - AP '~+~ - A~," D (9.5.11) o o

The variable P is oil phase pressure, D is depth to the center of the grid block, and ~t o is the specific weight (Pog) of the oil phase. The time derivative in Equation (9.5.7) is replaced with a forward finite difference to obtain

1 [ . + l n] on+l n + l n + l

A t M ~ - M ~ + ~ ~ - A A ~ A ~ o (9.5.12)

The superscript n denotes the present time level t n and the superscript n + 1 denotes the future time level t "+1. Time step size At equals t n+l- t n.

The above formulation is a fully implicit formulation because all variables are assessed at the future time level in Equation (9.5.12). IMPES is invoked by approximating transmissibilities, capillary pressures and densities at time level n + 1 with their values at time level n. The resulting flow equation is

1 At [ Mn+'- Mn]+ ~ 1 7 6 - A A n A * 71 (9.5.13)

Similar equations apply to the other flow equations.


Multi-Variable Newton-Raphson IMPES Procedure The IMPES equations developed above are solved using an iterative

technique that is illustrated by continuing our analysis of the oil flow

equation. The residual form of Equation (9.5.13) is

] --~ M e - M n + Qe o - AA"oA, e 0

(9.5.14)

where the superscript Q denotes the iteration level for the variables that are desired at time level n + 1. The primary variables for a saturated block are

8P, 6Sw, 6Sg and {?3vi: I = 1, ..., N~}. Gas saturation is replaced by bubble point pressure Pb in the set of primary variables for a saturated block. The variable switching logic used to treat blocks undergoing phase transitions is described in Ammer, et al. [ 1991 ]. The solution process is designed to find the values of the primary variables which drive the residuals to zero in all grid blocks for all components. Ammer, et al. [1991] refer to the solution procedure as the multi-variable Newton-Raphson IMPES method.

Well Models

There are many ways to represent wells in flow models [Ertekin, et al., 2001 ]. The simplest well model is a variation of Darcy's law. Darcy's law shows that well flow rate is proportional to pressure change. The relationship between flow rate Q~ of phase Q and pressure change Ap may be written as

Oe - P I A P (9.5.15)

where the proportionality constant is called the productivity index (PI).

Rearranging and using Darcy's law for radial flow into a vertical wellbore, PI can be calculated as

p[ _ Qe _ O.O0708 gehnet A P ~teBe[gn(re/rw)+ S] (9.5.16)

The meaning and appropriate units of each variable in Equation (9.5.16) follow:


t-h = viscosity of phase Q (cp)

B~ = formation volume factor of phase l~ (RB/STB)

re = drainage radius (ft)

rw = wellbore radius (ft)

S = skin

K~ = effective permeability (md) = k~ K,b,

kn = relative permeability of phase Q

K,b, = absolute permeability (md)

hne t = net thickness (ft)

Q~ = rate of phase Q (STB/D)

Relative permeability, viscosity and formation volume factor depend on time- varying pressure and saturation. The remaining variables on the right hand

side of Equation (9.5.16) change relatively slowly or are constant with

respect to time. A value of the effective drainage radius for a vertical well in the center

of a rectangular grid block with cross-sectional area AxAy can be estimated

from Peaceman's formula [ 1978]

r e .~ r ~ = 0.14(Ax 2 + Ay2) ~ (9.5.17)

Equation (9.5.17) applies to an isotropic system, that is, a system in which

lateral permeability does not depend on direction. For a well in a square grid block and an isotropic system, Ax - Ay and r o = 0.2Ax. For a well in

a rectangular grid block and an anisotropic system, the effective permeability can be estimated as

K = ~ / K x K y (9.5.18)

In this case, lateral permeability depends on direction and the directional components of permeability are not equal, thus K~ #Ky. The equivalent well block radius for an anisotropic system must account for the dependence of permeability on direction. The effective drainage radius becomes

168 S h a r e d E a r t h M o d e ! i n g

[(Ky / K~) I/2 Ax 2 +(K~ / Ky)~Ay2] ~ (9.5.19) r e ~ r o = 0.28 /4

(K,/Kx) '/4 +(Kx/K,)

Most flow simulators calculate PI and pressure change, then flow rate. If the magnitude of the flow rate calculated from the PI and pressure change is greater than the magnitude of the pressure change input by the user, the flow rate will usually be set at the user specified flow rate. If the magnitudei of the flow rate calculated from the PI and pressure change is less than the magnitude of the pressure change input by the user, the flow rate will be the simulator calculated value. The reader should consult the technical documentation of a flow simulator to see the details of well model calculations.

[ CS-9. Valley Fill Case Study: Conceptual Areal Model ]

It is often worthwhile to study simplified models of the system of interest to learn more about how displacement mechanisms may behave in the more complex full field model. For example, a conceptual, one layer model of the Valley Fill reservoir is a conceptual areal model. It can provide insight into the volumetrics of the reservoir and the pressure behavior of the field. The conceptual model is considered in more detail in the exercises.

[ Exercises , [

9-1. Find file VFILL_HM.ROF and open it using a text editor. Search the

file for INITIAL FLUID VOLUMES. The file should have been downloaded from website h t t p : / / w w w . b h . c o m / c o m p a n i o n s / O 7 5 0 6 7 5 2 2 5 a s

stated in Chapter 1. A. How much oil is initially in place? B. How much water is initially in place? C. How much gas is initially in place? D. How much of the gas exists in a free gas phase?

9-2.

9-3.

John R. Fanchi 169

Run the visualization program 3DVIEW and load the file VFILL I_HM.ARR. To load the file after 3DVIEW is open, click on the "File" button and select "Open Array File." Select the file called "VFILL 1 HM.ARR" and then click on the "OK" button. Select the oil saturation attribute at the beginning of the run. To select this attribute, click on the "Model" button and select "Select Active Attribute." From the list of options select "SO" for oil saturation. You are looking at the side of the reservoir. To see the top of the reservoir, place the cursor in the black field near the reservoir display, hold the left mouse button down and pull the mouse toward you. You should see the reservoir image rotate. Continue rotating until you see the top of the reservoir. Sketch the image and indicate which part of the image represents the reservoir.

Use the expression for numerical dispersion in one spatial dimension presented in Section 9.2 to fill in the following table. We are assuming a displacement with a frontal advance of 0.5 ft/day, a grid block size of 100 ft, and a porosity of 20%. The results should be expressed for both IMPES and fully implicit formulations.

Timestep size At [days]

1

5

10

30

45

IMPES D hum

[ftZ/day]

Fully Implicit D hum

[ft2/day]

9-4. Derive Equation (9.3.6) from Equation (9.3.1) for the fully implicit procedure.

Chapter 10 Fundamentals of

Reservoir Characterization

Reservoir characterization is the process of preparing a quantitative representation of a reservoir using data from a variety of sources and disciplines. Kelkar [2000, pg. 25] has defined reservoir characterization as the "process of integrating various qualities and quantities of data in a consistent manner to describe reservoir properties of interest at inter-well locations." All of the information collected at various scales in the reservoir characterization process must be integrated into a single, comprehensive, and consistent representation of the reservoir. This chapter describes two fundamental topics in reservoir characterization: flow unit characterization and reservoir mapping.

[ 10.1 Flow Units [

Integrated reservoir characterization requires the acquisition and analysis of static data such as reservoir structure and dynamic data such as production performance. The flow unit concept is an effective means of managing the growing base of data being provided by geoscientists. A flow unit is defined

170

John R. Fanchi 171

as "a volume of rock subdivided according to geological and petrophysical properties that influence the flow of fluids through it" [Ebanks, 1987]. This definition was later modified to state that a flow unit is "a mappable portion of the total reservoir within which geological and petrophysical properties that affect the flow of fluids are consistent and predictably different from the properties of other reservoir rock volumes" [Ebanks, et al., 1993, pg. 282], Typical geologic and petrophysical properties are shown in Table 10-1.

Table 10-1 Properties Typically Needed to Define a Flow Unit

Geologic

Texture Mineralogy

Sedimentary Structure Bedding Contacts

Permeability Barriers

Petrophysical

Porosity Permeability

Compressibility Fluid Saturations

Classic applications of the flow unit concept have been published by Hearn, et al. [ 1985] and Slatt and Hopkins [ 1990]. Hearn, et al. pointed out that conventional geologic maps and cross-sections help develop an understanding of bedding characteristics and depositional environment, but flow model studies require a reservoir zonation that is more closely related to properties that influence fluid flow. From this perspective, a flow unit is a zone or layer with the following characteristics: the unit is continuous over a significant portion of the reservoir; each element of the unit has similar flow properties; and each element of the unit has similar bedding characteristics.

A reservoir is modeled by subdividing its volume into an array of representative elementary volumes (REV). The REV concept is not the same as the flow unit concept. A flow unit is a contiguous part of the reservoir that has similar flow properties as characterized by geological andpetrophysical data. Flow units usually contain one or more REVs. By contrast, the REV is the volume element that is large enough to provide statistically significant average values of parameters describing flow in the contained volume, but small enough to provide a meaningful numerical approximation of the fundamental flow equations [for example, see Bear, 1972]. A reservoir may be visualized as an ensemble, or collection, of representative elementary


volumes. Each REV contains a set of rock properties, such as porosity, permeability, and bulk modulus.

A flow unit is a contiguous part of the reservoir that has similar flow

properties as characterized by geological and petrophysical data. Ebanks,

et al. [ 1993] have identified the following characteristics of a flow unit:

1. A flow unit is a specific volume of reservoir composed of one or

more reservoir quality lithologies; adjacent nonreservoir quality rock types; and associated fluids.

2. A flow unit is correlative and mappable at the interwell scale. 3. Flow unit zonation can be recognized, e.g., on well logs.

4. A flow unit may be in communication with other flow units.

Several flow unit identification techniques are proposed in the literature, such as the modified Lorenz plot used by Gunter, et al. [ 1997]. A simplified

variation of the modified Lorenz plot technique is to identify a flow unit by

plotting cumulative flow capacity as a function of depth. Cumulative flow capacity Fm is calculated as

m

Z kihi i=1 F m = ~ ; m - 1 , . . . ,n ( 1 0 . 1 . 1 )

~ kihi i=1

where n is the total number of reservoir layers. The layers are numbered in order from the shallowest layer at index i = 1 to the deepest layer at index

i = m. The cumulative flow capacity Fm is the value of Equation (10.1.1) at depth

Z m - Z 0 + ~ ~ h i (10.1.2) i=1

where Z0 is the depth to the top of layer 1 from a specified datum. A flow

unit will appear on the plot as a line with constant slope. A change in slope

John R. Fanchi 173

is interpreted as a change from one flow unit to another, as illustrated in Figure 10-1. Slope changes in Figure 10-1 occur at depths of 36 feet, 76 feet, 92 feet, 108 feet, 116 feet, 124 feet, 140 feet, 152 feet, and 172 feet. The largest slope is between 108 feet and 116 feet, and corresponds to a high permeability zone. It is followed immediately by a low permeability zone at a depth of approximately 120 feet.

Figure 10-1. Identifying Flow Units

Another plot that can be used to identify flow units is a plot of cumulative flow capacity F m in Equation (10.1.1) versus a cumulative storage capacity ( I ) m defined by

in

~ihi i=l

(I) m - n , m - 1 , . . . ,n (10.1.3)

~ (~ih/ i=l

Again, n is the total number of reservoir layers and the layers are numbered in order from the shallowest layer i = 1 to the deepest layer i = m. The analyst again looks for changes in slope in the plot of Fm versus ~m"

Flow units usually contain one or more representative elementary volumes (REV). By contrast, the REV is the volume element that is large enough to provide statistically significant average values of parameters describing flow in the contained volume, but small enough to provide a


meaningful numerical approximation of the fundamental flow equations [for example, see Bear, 1972]. Fayers and Hewett [ 1992] have observed that it is "somewhat an act of faith that reservoirs can be described by relatively few REV types at each scale with stationary average properties."

[ 10.2 Traditional Mapping [

The mapping/contouring process is the point where geological and geophysical interpretations have their greatest impact on the representation of the: reservoir in flow models. Mapping and visualization procedures have been discussed by several authors, including Harpole [1985], Tearpock and Bischke [ 1991 ], Valusek [ 1995], and Tippee [ 1998].

The different parameters that must be digitized for use in a flow model grid include elevations or structure tops, permeability in three orthogonal directions, porosity, gross thickness, net to gross thickness and, where appropriate, descriptions of faults, fractures and aquifers. Additional maps are needed for flow models which integrate petrophysics and traditional flow models. In particular, distributions of moduli and bulk density provide important parameters that can be used to calculate seismic velocities, as described in Chapter 3. The resulting maps are digitized by overlaying a grid on the maps and reading a value for each grid block.

The data input into the flow simulator must be consistent. If a project is very large, it is likely that several geologists, geophysicists and engineers have worked to complete the reservoir model. Different geoscientists or engineers may have a different bias or perspective when analyzing data. For example, two or more petrophysicists may differ on defining well log cutoffs, or geologists may disagree on picking the depth of formation tops. These types of inconsistencies introduce variations in the model that may prove significant.

Another type of problem that may appear are errors that arise when incorrect data are entered in a computer model. Errors may range from transposed digits to unit conversion errors. Some of these errors can be spotted by contouring the data. Gross errors will be evident.

The contouring step is another point where geological interpretation is included in the flow model. The following contouring guidelines are worth noting from a technical, rather than scientific, perspective.

John R. Fanchi 175

A. Contour lines do not branch.

B. Contour lines do not cross. C. Contour lines either close or run off the map.

D. Steep slopes have close contour lines. E. Gentle or fiat slopes have contour lines that are far apart.

The first three of these guidelines are illustrated in

Figure 10-2. All of the

guidelines apply to any type of contour map. Discontinu-

ities in contour lines are

possible, but need to be justified by the first inferred existence of geologic dis-

continuities such as faults and unconformities. While

tedious, traditional hand contouring can let a geolo-

gist imprint a vision on the data that many computer Figure 10-2. Examples of Contour Tips

algorithms will miss.

Grid selection is based on need. Flow simulation grids are chosen to

allow adequate representation of the reservoir recovery mechanisms. This

can range from a single grid block for a material balance calculation to a

three-dimensional grid with hundreds of thousands of grid blocks. By comparison, geological models may have several million grid blocks

representing the system of interest. Geological models of reservoir rock may

involve several orders of magnitude more cells than the flow simulation. The

procedure for accurately transferring data between geologic models and flow

models with different grid sizes is an active area of research.

Digitization of the mapped property can be as simple as volumetrically

averaging the four comer points of the grid. Once generated, digitized data

should be checked for consistency. Thicknesses of formations deserve special consideration. Faults and unconformities can cause layers to appear and disappear. Unless care is taken, these geologic features can cause discrepancies in the model.


I 10.3 Computer Generated Maps

An important function of geologic maps is to present values of a spatially distributed property at any point on a surface or within a layer. Examples of spatially distributed properties are structure top, net thickness and porosity, and permeability in three dimensions. Maps of spatially distributed properties can be generated by computer using a variety of techniques.

Each data set has characteristics that cause it to be amenable to one contouring technique over another. If the amount of data is sparse, a technique called kriging may produce an acceptable map. However, with large data sets, kriging may take a lot of computational time and simpler techniques may provide an acceptable representation of the reservoir with less effort. Some techniques, such as the inverse distance method discussed below, will produce many local highs and lows. It is recommended that mappers consider a variety of contouring techniques. Depending on the geological setting, one technique will give a more realistic reservoir representation than another.

One of the simplest algorithms that can be coded in a computer program to generate a map is to distribute property values over a surface or within a layer by using inverse distance weighting of all applicable control point values. The control point values correspond to property values measured at wells or determined by seismic methods that apply to the surface or layer of interest. The formula for inverse distance weighting is

N

~ (Vi/di) i=1 Vx- 2 (1 /d i ) i=1

(10.3.1)

where Vx is the value of the property at x calculated from N known values { V,.} of the property at distances {di} from x. Inverse distance weighting assigns more weight to control points close to location x and less weight to control points further away. The weighting factor is the inverse of control point distance from x. For example, the value at a point x that is at the distances {dA, ds} from two known values { VA, Vs} is


v, +

dA d8 V~= 1 1 (10.3.2)

~ + d A d ,

Figure 10-3 illustrates the inverse distance weighting example in Equation (10.3.2) with two control points. If only one value V c is known (N= 1 ), then

V~ = Vc for all values of x.

~ X A ~

~ B

Figure 10-3. Inverse Distance Weighting with Two Control Points {A, B}

Inverse distance weighting is an example of a technique that uses control points in the neighborhood of an unknown point to estimate the property

value at the point. A more general expression for distributing an attribute using a weighted average is

N

2; A, Aavg _ i= 1

N

i=1

(10.3.3)

where

A avg

A/

W

= weighted average value of attribute

= value of attribute at control point i = weighting function

= distance from the interpolated point to control point i

178 SharedEarthModeling~

R = user specified search radius

N = number of control points

A control point is any spatial location with a known value. It could be a well

location, or a value that is imposed by the mapper using soft data such as

seismic indications of structure boundaries. The search radius R constrains the number of control points N that are used to determine the weighted

average value of the attribute. An example of a weighting function is the

weighting function in Equation (10.3.1), namely 1/di. Another example of

a weighting function with a search radius is

W ( r , R ) - 1 - r �9 (10.3.4)

where the value of the exponent x is entered by the user. After an algorithm has computed a surface, geologists may want to edit

the surface. An easy method is to add data points to force a contour to move to a certain location. More complex computer programs allow trends to be

imposed on the data. The character of the reservoir conceptualized by the

mapper should be adequately represented in the final computer-generated

map. The techniques described above are relatively simple examples of

technology that can be used to generate geologic maps using computer

programs. More sophisticated computer mapping packages exist and are used

to prepare 2-D, 3-D and 4-D maps of spatially distributed parameters.

Geostatistics is an example of a more sophisticated mapping technology that

is based on the spatial distribution of statistically correlated properties.

Visualization technology and geostatistics are introduced in the following

sections.

[ 10.4 Visualization Technology ]

Mapping technology has evolved rapidly in the past decade because of the acceptance of advances in computer mapping and visualization technology.

The last ten years has seen a conversion from 2-D mapping to 3-D and 4-D

John R. Fanchi 179

mapping. In 2-D mapping, planar maps were prepared and then stacked to

form a composite 3-D map. It was very common to find unphysical overlaps

between surfaces because each 2-D map was prepared separately. For

example, a gross thickness map might exceed the gross thickness defined

by the top and base maps of a formation. The excessive thickness would

incorrectly extend into contiguous formations. One of the advantages of 3-D

mapping is that the mapping takes place in three dimensions, so overlaps are

avoided and a more realistic representation of the reservoir can be con-

structed using data displayed in three dimensions.

The use of 3-D mapping became feasible with the advent of computer

systems that could manipulate three dimensional data sets and project a 3-D

image on a two dimensional surface. Improved computer processing speed

and algorithms allow rapid calculation of 3-D views. A viewer can rotate the

reservoir to any desired angle. Slices of the reservoir can be taken to see

features that are influencing fluid flow. Reservoir compartments are easily seen.

Reservoir views can be projected in a manner that lets an audience see

a 3-D image. One 3-D imaging technology requires the audience to wear

colored glasses. Each eye receives a slightly different view, which makes

the image appear to have depth as well as width and breadth. If a sequence

of 3-D images is set in motion, it gives the effect of animation, or 4-D mapping.

Animation can be used to track a variety of physical processes, such as

fluid flow or geologic deposition. Seismic data, for example, can be viewed

as an animated sequence of time slices to watch the formation of geologic

features such as fluvial systems. Arrays of fluid saturations from fluid flow

simulators can be placed into an animated sequence to show the movement

of fluid fronts in a reservoir. These images can be compared with time-lapse

seismic data to evaluate the quality of the flow model.

[ CS-IO. Valley Fill Case Study: Reservoir Structure I

The Valley Fill reservoir is a meandering structure with a regional dip from

north to south. The shape of the meandering structure is indicated by the

compressional to shear velocity ratio distribution shown in Figure CS-5A

in Chapter 5. Figure CS- 10A shows a plan view of the wells in the reservoir.


A Cartesian grid has been overlain on the area of interest. The grid has

30 columns and 15 rows. A comparison of Figures CS-5A and CS-10A shows that the productive wells are in the channel and the dry holes are outside of the channel. The depths to the top of the Valley Fill formation for the six producing wells are given in Table CS-10A.

Table CS-10A. Depths to Top of Structure

Well Depth Column Row (feet)

8435

2 8430

8450

8440

8455

8440

21

24

27

11

12

I Exercises ]

10-1. Sketch a contour map of the top of structure using the data shown in Table CS-10A, the well locations shown in Figure CS-10A, and the seismic data shown in Figure CS-5A. The contour map should cover

the entire area shown in Figure CS-10A. Remember that the Valley Fill reservoir has a regional dip.

10-2. Digitize the contour map prepared in Exercise 10-1. Present the values in array form using the array shown in Figure CS-10A.

10-3. Use the data in Table CS-10A, the distance scale in Figure CS-10A, and inverse distance weighting to estimate the formation top at the

midpoint of the following grid cell locations: A. column 6, row 9; and B. column 15, row 11.

10-4. Compare the values obtained in Exercises 10-2 and 10-3. Explain the differences.

John R. Fanchi 181

o 8i 2 I 1

I / / l l l / l l / / / / / / l / ~ i / / l l l / ~ l l I / l / / / / l / / / l / / / l / n / l i l l / n l | i l l i l l / l l / l l l / / / l / / / l l l / / l i I l l / 0 / / | / / / l / / / l / / / l l l l / / l l mmimimmRimmmmiiiiiimnimmmm

Figure CS-10A. Well locations in 30 x 15 grid. The sides of each square

grid block are 200 feet long. Productive Well 0; Dry Hole ~.

Chapter 11 Modern Reservoir Characterization

Techniques

Shared earth models are quantified and prepared for use in flow simulators by preparing detailed maps of variables that define reservoir structure and fluid distribution in three-dimensional space. This chapter discusses two modern reservoir characterization techniques that contribute to the validity of shared earth models: geostatistics and time-lapse seismology.

[ 11.1 Geostatistics I

Geostatistics is a branch of"applied statistics" that attempts to describe the distribution of a property in space. Geostatistics is also known as spatial statistics. It assumes that a spatially distributed property exhibits some degree of continuity. Porosity and permeability are examples of spatially dependent properties which are suitable for geostatistical description. Much of our discussion of geostatistics is based on publications by Chambers, et al. [2000], Clark and Harper [2002], Kelkar [2000], Deutsch and Journel [ 1998], and Hirsche, et al. [ 1997].

182

John R. Fanchi 183

Geostatistics consists of a set of mathematical tools which employ the assumption that properties are correlated in space and are not randomly distributed. The geological context of the data must be considered in addition to spatial relationships between data. Geostatistical algorithms provide formalized methods for integrating data of diverse type, quality and quantity.

A geostatistical analysis has several goals, including:

A. Acquire an understanding of the spatial relationships and correlations between reservoir properties;

B. Model those relationships with mathematical expressions; C. Develop an understanding of the uncertainty associated with the

reservoir properties and the conceptual geologic model; and D. Determine if a deterministic or stochastic approach is appropriate

for the creation of a reservoir model.

A deterministic model is a single realization, or representation, of reservoir geology. The uncertainty associated with a deterministic model can be estimated by estimating the sensitivity of the model to uncertainties in available data.

A stochastic model is a set of realizations obtained from the probability distributions developed during the geostatistical analysis of data. The shape of a probability distribution is defined by the proximity and quality of local data within the context of a spatial correlation model. By its nature, stochastic modeling propagates the uncertainty of the input parameters.

Stochastic modeling has several goals. The first goal is to preserve the heterogeneity inherent in a geological system as a means of creating more realistic and useful simulation models. The second goal is to quantify the uncertainty in the geologic model by generating many possible realizations. The stochastic model should incorporate multiple data types with varying degrees of quality and quantity, and have been measured at different measurement scales.

The modeling process proceeds in several steps. First, the structural and stratigraphic framework is developed by analyzing available seismic and well data. Once a framework exists, a lithofacies model and petrophysical properties are needed for use in a flow simulator. Multiple realizations may be generated and used to quantify uncertainty in the geologic model. The


process of translating point observations to a conceptual geologic model is a sequential process. It is also an iterative process if a match of time- dependent (dynamic) data is included in the preparation of the final reservoir

model.

I 11.2 Geostatistical Modeling I

Geostatistical modeling refers to the procedure for determining a set of reservoir realizations. The realizations depend on both the spatial relationships between data points and their statistical correlation as a function of separation in space. Chambers, et al. [2000] have identified seven steps in a geostatistical study. The steps are listed in Table 11-1, and are discussed in more detail below.

Step

1

Table 11-1 Steps in a Geostatistical Study

Task

Data mining

2 Spatial continuity analysis and modeling

3 Search ellipse design

4 Model cross-validation

5 Kriging

6 Conditional simulation

7 Uncertainty assessment

Step 1: Data Mining Data mining is the study of available data. Data analysis helps the

modeler locate anomalies and errors in the data and gain familiarity with the data. Familiarity with the data set helps the modeler optimize the usefulness of the data because geostatistics is subject to interpretation and relies on experience. The modeler can use a variety of techniques, many graphical, to analyze the data. The analysis is designed to detect and model patterns of spatial variability and interdependence.

John R. Fanchi 185

Step 2: Spatial Continuity Analysis and Modeling The spatial relationship(s) associated with data are computed and then

modeled. This process is analogous to (1) plotting data on a cross-plot (computing) and then (2) fitting a line to the data with linear regression (modeling). The plotted points make up the experimental variogram, and the line that is fit to the data points is called the model variogram. An example variogram is shown in Figure 11-1.

>

- / ~ * / ' ~ * Experimental Variogram

Variogram Model

Separation distance (lag) F

Figure 11-1. Variogram

A variogram is a plot of variance versus range [Dietrich and Rester, 2000]. Semi-variance is a measure of the degree of dissimilarity between the values of a parameter Z at two different locations, or points in space. The semi-variance y(h) is a function of lag h, or distance of separation, between two observations Z(x) and Z(x + h) of the parameter Z, thus

1 2

' [ ( h ) - 2 N ( h ) [z(xi)- z(xi ~ h)] i=1

(11.2.1)

where N(h) is the number of data pairs that are approximately separated by the lag h.

Figure 11-2 illustrates three important features of the variogram. The sill is the maximum value of the semi-variance for the parameter Z. The sill is also the variance 02 of the measured data, where 0 is the standard deviation.


g(h)/

sill < - . . . .LT.~ -- ~ ,r I I I I I I I I I I I I

$ , !

nugget !

iii

range

Figure 11-2. Characterizing a Variogram

F

h

Another measure of spatial dependence of an attribute called the covariance can be derived from the sill and the variance, in particular, the

covariance C(h) - C(0) - y(h), where C(0) is the sill. Given this definition, an increase in the variance corresponds to a decrease in the covariance. Thus, the covariance is a measure of the spatial correlation between measurements of an attribute.

The range in Figure 11-2 is the minimum separation distance h at which

the spatial correlation ceases to apply. A spatial correlation between values of parameter Z exists at values of the lag less than the range. Parameter values separated by a distance greater than the range are not correlated.

The nugget in Figure 11-2 is the value of the variance at zero lag. A nonzero value of the nugget is due to factors such as sampling error and short-range variability of the parameter.

Several types of variogram models exist. For example, the exponential model has the form

y (h) = 1 - exp a (11.2.2)

where a is a curve-fit parameter. Variogram modeling is achieved by calculating the variance of data and then fitting a variogram model to the plot

of variance versus lag. The resulting variogram is a measure of the spatial dependence of reservoir attributes such as porosity, permeability and net

John R. Fanchi 187

thickness. The variogram model is used to predict the value of the modeled attribute at locations away from control points.

Step 3: Search Ellipse Design The separation between data points is specified by the lag distance.

Variogram calculations can compare data points in all directions or in specified directions. It is possible to study the anisotropy of a reservoir attribute by investigating the correlation of data points in a specified direction. In this case, the lag is a vector with both magnitude and orientation. The search for the number N(h) of data points at a fixed lag h is done by specifying a neighborhood around h. The " ' - . neighborhood is shown in "" s

Figure 11-3, and the orienta- ~ ~ "'" tion of the lag vector with . . magnitude h is specified by "" the angle O. Orientation of lag may be included in the analysis by dividing the areal distribution of properties into a finite number of sectors, such as four quadrants or eight oc- Figure 11-3. Lag Neighborhood tants.

The neighborhood of the lag vector is sometimes referred to as a search bin and the size of the neighborhood is the bin size. Data points within the bin are included in the calculation of the semi-variance of a point at the origin with a point in the bin at lag h. If the search depends on the magnitude of h, but not the orientation, the search is circular with a radius h and the spatial distribution of the parameter will be isotropic. If the search is allowed to depend on the magnitude and orientation of h, the search is elliptical and the spatial distribution of the parameter can be anisotropic.

Step 4: Model Cross-Validation The variograms developed previously are used to estimate the value of

a parameter at an unsampled location. One widely-used estimation technique


is called kriging. Kriging is named after the South African mining engineer D.G. Krige who helped pioneer the development of geostatistical methods in the 1950's. Kriging is a linear weighted-average method similar to inverse distance weighting. The weights used in kriging are based on the variogram model of spatial correlation. The result of applying an estimation technique such as kriging is the determination of a spatial distribution of a parameter at all points of interest. The question then arises: how accurate are the estimated values?

One method of determining the accuracy of the values obtained by an estimation technique is to treat a sampled (known) data point as an unknown point at the test location. The estimation technique is used to calculate the parameter at the test location and the resulting value is compared with the known data point. The accuracy of the estimation process can be quantified by calculating the variance of actual values relative to the estimated values. The resulting variance provides a cross-validation of the original variogram model and provides information about the quality of the estimation technique. Notice that this model cross-validation procedure could be applied to any computer-based estimation technique, including techniques like inverse-distance weighting that do not require geostatistics.

Step 5: Kriging The kriging technique was introduced in Step 4. In this step, kriging or

some other geostatistical mapping technique is used to complete the computer mapping process. Mapping by hand or by computer requires some type of estimation procedure that can interpolate values between control points and extrapolate values in the region beyond control points to provide estimates of reservoir parameters that are needed in the entire region of interest. The kriging method can be computationally intensive when used with large data sets, but can yield maps that exhibit the isotropy associated with the original data set. Kriging is a contouring method which recognizes that data which are close together are more likely to be similar than data that are farther apart.

Step 6: Conditional Simulation The maps that are created in geostatistics are geologic realizations that

match the available data and associated spatial correlations. Each realization

John R. Fanchi 189

is also called a stochastic image. The generation of stochastic images is called stochastic modeling.

There are widely used approaches in stochastic modeling: the pixel-based approach and the object-based approach. The pixel-based approach uses variograms and estimation procedures such as kriging to generate a realization of a spatially distributed variable such as permeability. The object-based or Boolean approach populates 3-D space with distinct geologic bodies, such as sand lenses. In practice, both approaches may be combined.

A stochastic model is considered conditional when it honors both measured data and associated spatial correlations. The spatial correlations are represented mathematically by variograms. If a stochastic model is conditional, we can say that the model is constrained by the measured data or spatial models. Examples of conditioning or constraining data include seismic data, well log data, facies distributions, dimensions of geologic bodies, well test data, and production history. A properly conditioned stochastic model must honor all available constraints. The generation of constrained stochastic images is the goal of conditional simulation.

There are several types of stochastic modeling algorithms. They include simulated annealing, sequential simulation, Boolean or object-based modeling, and so forth. For further discussion of these algorithms, consult references such as Deutsch and Joumel [ 1998] or Clark and Harper [2000].

Step 7: Uncertainty Assessment One of the advantages ofgeostatistical modeling relative to deterministic

modeling is the estimation of uncertainty during the stochastic modeling process. An assessment ofuncertainty can be provided as part of a deterministic study by analyzing the sensitivity of the deterministic model to changes in reservoir parameters within physically meaningful ranges. Many of the techniques involved in stochastic modeling include a measure of uncertainty as a by-product of the calculation procedure.

A geostatistical study is expected to generate many equally probable, or equiprobable, realizations of a reservoir. Each realization should honor all available data. The suite of realizations can be used to provide probabilis- tic estimates of important reservoir management information such as original fluids in place and performance forecasts. The assessment of uncertainty should aid the decision-making process.


The Use and Abuse of Geostatistics Hirsche, et al. [1997; pg. 259] have pointed out that "geostatistical

reservoir characterization should not be done in isolation." Geostatistics is like other reservoir characterization techniques" the technique is most successful when all available data is incorporated into the reservoir characterization process.

The violation of basic geostatistical assumptions can lead to the creation of an inaccurate reservoir model. Inaccuracies in the model appear as errors in associated maps. Limited well control and biased sampling of well information are examples of real world constraints that can violate the underlying assumptions ofgeostatistics. Abrupt changes in reservoir features, such as faults and high permeability channels, are difficult to identify using geostatistics [Fanchi, et al., 1996].

Geostatistics and stochastic modeling can be used to integrate data, provide a realistic representation of reservoir heterogeneity, and quantify uncertainty. On the other hand, the existence of multiple realizations can be confusing and more expensive than the construction of a single deterministic representation of the reservoir. In addition, the stochastic images may look realistic but actually do a poor job of representing flow in the actual reservoir. The process of validating the reservoir model is made more complicated by the existence of multiple realizations. Researchers are presently trying to simplify the stochastic modeling process and improve the reliability of forecasts made using stochastic models [Kelkar, 2000].

I 11.3 Time-Lapse (4-D) Seismology [

Time-lapse seismology is the comparison of 3-D seismic surveys at two or more points in time. Commonly known as 4-D seismic, time-lapse seismic reservoir monitoring has great potential for increasing our ability to image fluid movement between wells. The oil and gas industry has recognized for some time that 4-D seismic can improve the quality of reservoir characterization, identify movements of fluid interfaces, and help operators locate bypassed reserves. The benefits have a price: Anderson, et al. [ 1998] have estimated that 4-D seismic monitoring can add $1/bbl or more to the cost of producing oil from a field.

John R. Fanchi 191

The primary purpose of this section is to introduce the concept of 4-D seismic monitoring. Several case studies have demonstrated the value of 4-D seismic monitoring in the reservoir management process [e.g. Lumley, 2001 a, b; Fanchi, et al. 1999; Jack, 1998].

What Is 4-D Seismic Monitoring? Time-lapse seismology compares one 3-D seismic survey with one or

more repeat 3-D seismic surveys taken in the same geographic location at different times. 4-D seismic monitoring is the comparison of changes in 3-D seismic surveys as a function of the fourth dimension, time. By comparing the differences in measurements of properties such as travel times, reflection amplitudes, and seismic velocities, changes in the elasticity of the subsurface

can be monitored over time. There are two principal elastic parameters that affect seismic waves: the

bulk modulus and the shear (or rigidity) modulus. The bulk modulus is related to rock and fluid compressibility. Shear (S) waves are affected by bulk density and shear modulus, while compressional (P) waves are affected by bulk density and both bulk and shear moduli. The combination of bulk and shear moduli used in the calculation of P-wave velocities is called stiffness, and is a measure of the rock frame stiffness and pore fluid stiffness. Stiffness is quantitatively defined in Equation (3.3.1).

Reservoir elasticity is affected by lithology, fluid content and variations in pore pressure. Seismic velocity (V), attenuation (Q) and reflectivity measurements contain information on the fluid distribution in the reservoir. For example, the ratio of compressional velocity to shear velocity (Vp/Vs) is dependent on bulk modulus, shear modulus and density of the rock. Bulk modulus, and density are functions of porosity and fluid content in the pore space. Seismic monitoring of changes in reservoir elasticity can be linked to properties associated with the movement of fluids in the reservoir. This link yields information that can be used to improve the validity of fluid flow models and the reservoir management decisions that rely on flow model

forecasts. Technological innovations have made it possible to probe the elastic

properties of a reservoir by recording P- and S-waves as they pass through the reservoir. The ability to monitor both P- and S-waves is referred to as full-vector wavefield, or multicomponent, imaging [JPT Staff, 1999]. Time-

192 Shared Earth Modelin 8

lapse, multicomponent seismology is a tool for volume resolution-the ability to sense changes in bulk rock properties and fluid properties of the reservoir as a function of time.

An example of a time-lapse, multicomponent survey is the 4-D, 3C seismic survey. This survey records one vertical and two horizontal velocity components (3C). The recording procedure is similar to that of earthquake seismologists and facilitates the combined recording of P- and S-waves. Comparisons of travel time or seismic velocity measurements, amplitudes and frequencies of P- and S-waves enable the discrimination of rock and fluid properties and their changes over time.

Seismic anisotropy is a measure of the fine scale structure in the reservoir. An anisotropic reservoir exhibits differences in properties as a function of spatial orientation. Horizontal permeability, for example, will be greater in one direction than another. In an anisotropic medium, a shear wave splits into two orthogonally polarized components (S1 and $2). The S 1 wave is faster, and its velocity and attenuation are affected by lithology, porosity and pore saturants. By contrast, the $2 wave is slower, and its velocity and attenuation are affected by reservoir features such as fractures. The different dependencies of the S 1 and $2 waves provide information for determining dynamic reservoir properties such as permeability, porosity and fluid saturations. Multicomponent seismic studies have been especially useful in the characterization of reservoir rock properties, including lithology, porosity and fractures.

Forward Modeling Forward modeling is the calculation of seismic attributes from fundamen-

tal reservoir properties. These attributes are needed to understand changes in seismic variables as a function of time, which is the basis for time-lapse (4-D) seismic analysis. Rock properties include rock frame elastic properties such as bulk modulus and shear modulus. The moduli are used to calculate elastic stiffness. Other rock properties include porosity, shale volume, and grain density. Rock properties can be obtained from well logs, laboratory measurements of core properties, and correlations [e.g., Fanchi and Batzle, 2000; Jack, 1998].

Distributions of pressure (P), temperature (7), fluid composition (Z) and saturation (S) can be obtained from flow simulators. The P-T-Z-S distribu-

John R. Fanchi 193

tions are related to pore fluid properties such as phase densities and compressibilities. In an isothermal flow simulator, the temperature distribution is constant. Temperature is taken into account in the calculation of fluid properties and rock properties. Pore fluid properties may be obtained from the fluid properties model in either the flow simulator or the forward modeling program. The flow simulator uses pore fluid properties to calculate P-T-Z-S distributions in an iterative process. The result is a set of pore fluid properties and P-T-Z-S distributions that are mutually consistent within a user-specified tolerance. By comparison, it is possible to calculate pore fluid properties using P-T-Z-S distributions and fluid property correlations or equations-of-state in the forward modeling program. In this case, the possibility of error arises if the forward modeling pore fluid properties are

not the same as the flow simulator pore fluid properties that were used to determine the P-T-Z-S distributions. This error is avoided in the integrated flow model because all pore fluid properties and P-T-Z-S distributions are generated in the same program.

Seismic attributes are calculated from rock properties, pore fluid

properties, and P-T-Z-S distributions in the rock model. The rock model is an algorithm for calculating seismic attributes such as compressional velocity, shear velocity, acoustic impedance, and the reflection coefficient.

The algorithm discussed in the next section is an example of a rock model. One procedure in use today is a sequential procedure. The sequential

procedure retains the separation between flow model results obtained from a reservoir simulator and forward modeling results. The results of the flow

model are exported to a file that can be imported by the forward modeling program. A recent illustration of the sequential workflow process is presented by Khan, et al. [2000]. The workflow steps can be performed as a sequence of independent workflow actions using separate software packages, or they can be performed in a software package that is designed to simplify the data transfer process. An example of the latter process is to use a software package to control the execution of the flow simulator and the forward

modeling program. The sequential workflow process can be considered integrated only if all properties used in the flow model are also used in the

forward modeling program. The separation of flow and petrophysical calculations in a sequential process is relatively inefficient and special care must be exercised to avoid the use of inconsistentdata sets. The integrated


flow model is a more efficient tool for integrating flow and petrophysical calculations because it makes the forward modeling step a subroutine of the flow simulator, and data consistency is guaranteed.

An alternative to the forward modeling process described above is the integrated flow model. An integrated flow model is a flow simulator that includes a petrophysical algorithm. A prototype integrated flow model (IFLO) is used as the simulator in this book. The flow equations for the simulator are summarized in Chapter 9, and the petrophysical algorithm is based on the Gassmann equation. The simulator IFLO is discussed in greater detail in Fanchi [2000a]. The integrated flow model does not add much overhead to the flow simulation because the properties needed by the petrophysical algorithm are available at the completion of each time step. Thus, the petrophysical algorithm can be accessed as an output option.

Guidelines for Applying 4-D Seismic Monitoring Wang [ 1997] has identified several criteria that can be used to identify

good candidate reservoirs for time-lapse seismic reservoir monitoring. One criterion is the magnitude of bulk and shear moduli. The moduli characterize the elastic frame of a reservoir. Reservoirs with relatively small moduli have weak elastic frames and are good candidates for time-lapse seismic monitoring. Examples include reservoirs with unconsolidated sands or fractured reservoirs. Wang also observed that reservoirs experiencing large compressibility changes in either the rock or pore fluids can exhibit a significant seismic response over time. Some examples include reservoirs in which the gas phase is either appearing or disappearing.

Integrated flow model studies [Fanchi, 1999; 2000a] have found that 4-D seismic should be most effective in regions where the gas phase is appearing or disappearing. This observation is in agreement with Wang's criteria and has been substantiated in the field. For example, Kelamis, et al. [ 1997] noted the importance of similar gas saturation behavior in their study of a elastic oil reservoir with a large gas cap in the Arabian Gulf. The contrast between fluid compressibilities is greatest when one survey images only liquid, while a subsequent survey images liquid and gas.

Very small amounts of gas-as low as one saturation percent-are sufficient to change the total compressibility of a reservoir system by an order of magnitude or more. The change in fluid compressibility with time

John R. Fanchi 195

can generate observable differences in seismic response when comparing one survey with another. The differences in seismic response appear in measurements of attributes like compressional and shear velocities. The ratio ofcompressional to shear velocities is an important parameter for providing information that can be used to improve reservoir characterization and fluid

flow models. The necessity to compare two surveys that have been conducted at

different points in time over the same region leads to the issue of survey repeatability. The signal in a 4-D seismic survey is the magnitude of the change in acoustic response of the reservoir between two surveys taken over the same region at different times. Detection of the signal requires that the differences between time-lapse surveys should be due to actual reservoir changes, and not to differences in data acquisition, processing or interpretation.

Fluid phase behavior, such as the appearance or disappearance of a free gas saturation during the life of the field, can be used to schedule sequential 3-D seismic surveys [Fanchi, 1997]. From a reservoir management perspective, the scheduling of time-lapse seismic surveys should optimize the acquisition of reservoir engineering information.

Anderson, et al. [ 1998] have observed that 4-D seismic is most effective in offshore fields where high quality 3-D seismic surveys exist. They also noted that 4-D seismic works better in soft, unconsolidated sands rather than in hard, carbonate reservoirs. The Vacuum Field case study [Talley, et al., 1998] shows that 4-D multicomponent seismology can be applied effectively in carbonates.

A common factor in most screening criteria is the identification of significant changes to properties that directly influence seismic response. For example, if a reservoir is subjected to large pressure or temperature changes, the resulting change in petrophysical properties can lead to an observable change in seismic response. Case studies [Fanchi, et al., 1999] substantiate the screening criteria outlined here. They demonstrate that 4-D seismic can be an effective tool for reservoir management. It is important to keep in mind, however, the caution pointed out by Lumley and Behrens [ 1997]: "this new technology is not a panacea but rather an exciting emerging technology that requires very careful analysis to be useful." As time-lapse seismology matures, it is continuing to be accepted as an effective reservoir


management tool [Lumley, 2001 a, b; Caldwell, 2000; Fanchi and Penning- ton, 2001 ].

I 11.4 Case Study: N.E. Nash Unit, Oklahoma [

Although considerable work has been done to date, it is not always clear whether the geostatistical approach will generate a more accurate representation of the reservoir than would detailed analyses by experienced geologists and reservoir engineers. The N.E. Nash Unit in Oklahoma was one of the first applications of geostatistical modeling techniques within the context of a full field flow model study [Fanchi, et al., 1996]. The study illustrates some of the advantages of geostatistics as well as its limitations.

Oil production from the Northeast Nash Misener sand began in 1985 and a waterflood was started at the beginning of 1992. The field was initially an undersaturated black oil reservoir that was depleted below the bubble point and then subjected to a waterflood. Although water breakthrough has occurred at several of the production wells, significant parts of the field appeared to be unswept. A flow model study was undertaken to acquire a better understanding of reservoir performance.

Two different mapping algorithms (deterministic versus stochastic) were used to generate a suite of geologic maps for the Nash reservoir. In the deterministic approach, hand-drawn maps were prepared. In the stochastic approach, the structural tops, gross thickness, net/gross ratio, and porosity values at each of 42 well control points were analyzed to determine their spatial continuity. Directional semi-variograms were used to describe spatial continuity changes as a function of distance and direction for each reservoir property. Reservoir flow models were initialized using both the deterministic and stochastic geologic models.

The deterministic model exhibited more homogeneity than the stochastic model. This was an expected result since stochastic models are renowned for their inclusion of random short scale variability associated with large scale trends. A more important distinction between the deterministic and stochastic models was the width of the field. Both models had comparable volumetrics, but the stochastic model had a narrower channel. Adoption of the stochastic model reduced the number of potential flank well drilling

John R. Fanchi 197

locations relative to the deterministic model, thereby reducing the risk of drilling a dry hole. Consequently, the stochastic model was chosen for use in the full field simulation study.

Modifications to geologic maps that were generated using geostatistical methods were needed during the history matching process. The history match process showed a need to incorporate discontinuities such as compartments and high permeability channels. These discontinuities were needed to match tracer data and breakthrough times. The most important modifications are

listed in Table 11-2. Tracer information was used to identify probable sources of produced injection water. The tracers in this study were changes in salinity and total dissolved solids in produced water. Attempts to match breakthrough times required introducing transmissibility reductions that appeared to

channel fluids from an injector to specific producers. This was especially true in the western half of the field. Transmissibility increases, aquifer location and influx rates were more instrumental in matching performance

in the eastern half of the field.

Table 11-2. History Match Modifications for the N.E. Nash Unit Waterflood Study

�9 adjust aquifer locations and influx rates �9 modify relative permeability curves �9 compartmentalize the main channel through the reservoir center �9 increase transmissibilities in select regions �9 add hydrocarbon pore volumes in select regions �9 modify several well productivity indices

CS-11. Valley Fill Case Study: Time-Lapse Response ]

The Valley Fill reservoir is a meandering channel with a regional dip from north to south. We used seismic data in Section CS-5 to view the channel

(Figure CS-5A). It is possible to obtain more information about the reservoir by conducting a second seismic survey later in the life of the field. The feasibility of observing detectable changes in seismic response and the scheduling of the second survey to optimize data acquisition may be studied using the integrated flow model IFLO. The exercises illustrate the process.


Exercises I

11-1. Files VFILL3 WF.* contain the results of an IFLO model of the Valley Fill reservoir. The IFLO model matches historical production

and makes a prediction of field performance when subjected to a waterflood. The file should have been downloaded from website http://www.bh.com/companions/0750675225 as stated in Chapter 1. A. How many days does the run last? B. What is the maximum water

injection rate for the field? Hint: see the time step summary file VFILL3 WF.TSS.

11-2. Run the visualization program 3DVIEW and load the file

VFILL3 WF.ARR. Select the water saturation attribute at the end of the run. Slice the model to obtain a better view of the interior of the model and describe what you see.

11-3. Using the Combine option in 3DVIEW, subtract the initial model (at 0 days) from the final model (at 1460 days). Select a reservoir geophysical attribute such as compressional acoustic impedance (ALP)

or compressional velocity (VP) and examine the change in seismic response. Describe what you see. Does it appear that a reservoir geophysical attribute is detectable at the end of the run? The magnitude of the change can be estimated by selecting the color option in

3DVIEW and noting the values of the attribute that are used to set the color scale.

11-4. Repeat Exercise 11-3 at 730 days and 1095 days. Which of the following times would yield the most information about the reservoir from time-lapse seismology: 730 days, 1095 days, or 1460 days? Justify your answer.

Chapter 12 Well Testing

Well tests can provide information about reservoir structure and the expected flow performance of the reservoir. It is usually necessary to run a variety of well tests as the project matures. Well tests help refine the operator's understanding of the field and often motivate changes in the way the well or the field is managed. Additional information about well testing can be found in literature sources such as Matthews and Russell [1967], Earlougher [ 1977], Sabet [ 1991 ], Economides, et al. [ 1994], Dake [ 1994], and Home.[ 1995]. This chapter discusses a variety of well tests that can be performed on production and injection wells, vertical and deviated/horizontal wells, and combinations of wells. It also describes the information that can be obtained from the well tests and shows how to use well test information in reservoir characterization.

[ 12.1 Pressure Transient Testing ]

Pressure transient testing uses changes in measurable pressure performance to infer reservoir parameters such as flow capacity, average reservoir pressure in the drainage area, reservoir size, boundary and fault locations, wellbore damage and stimulation, and well deliverability. The information from well

199


tests can be combined with data from other sources to obtain additional

reservoir parameters. For example, the estimate of flow capacity from a

pressure transient test can be combined with a well log estimate of net pay

to determine effective permeability in the volume of the reservoir investi-

gated by the well test. A summary of parameters that can be determined by

well tests is given in Section 12.4.

Pressure changes in a well test are induced in a system by changing the

flow rate of one or more wells, and recording the variation in pressure as a

function of time using pressure gauges. Analysis of the pressure response

provides information about reservoir flow capacity in the radius of investiga-

tion of the well. The pressure response is analyzed by plotting pressure and

its derivative as a function of time. Analytical techniques are discussed in

more detail in later sections of this chapter. The techniques depend on the

type of test run, the reservoir geology, and the flowing phase. A few of the

most common tests are discussed in this chapter to illustrate the procedures

and type of information that can be obtained from pressure transient testing

of wells.

A pressure response can be elicited from a well by changing the flow

rate of the well. The pressure response at the well passes through three

stages: early-time wellbore-dominated response; late-time, boundary-

dominated response; and intermediate-time infinite acting response during

a transitional stage between the early- and late-time responses. The three

stages are sketched in Figure 12-1 for a well with initial pressure Pi" The

infinite acting response often behaves as radial flow.

The flow rate of a well can be changed by either increasing the rate or

decreasing the rate. If we recognize that wells are either production or

injection wells, then we see that there are only four basic types of transient

tests: flow rate increases or decreases in a production well, and flow rate

increases or decreases in an injection well. Pressure buildup and drawdown

tests are run on production wells. The buildup test measures pressure

increases after a flowing well has been shut-in. By contrast, a drawdown test

measures pressure decline as the production rate of a well is increased. A

similar set of tests can be run on injection wells. The falloff test measures

pressure decline after an injection well is shut-in, while injectivity tests

measure pressure increase as injection rate is increased. These tests can be

run on both gas and oil wells.

John R. Fanchi 201

Pi i I

I

I

I

r ~

~ , . . . i '

I I I I I I

I

I !

O

O

r ~

Time

Figure 12-1. Schematic of Pressure Decline at Well

Many well tests are based on a discontinuous change in flow rate. For example, in a pressure buildup test, the test is conducted by shutting-in a producing well. Incremental changes from one rate to another, rather than abrupt changes between flow and no-flow conditions, can be used to provide additional information. Deliverability tests in gas wells and pressure pulse

testing are examples of tests that use incremental rate changes to generate observable pressure variations. Gas well deliverability tests tell us about the

flow capacity of gas wells. Pressure pulse tests provide evidence ofpressure communication between wells.

I 12.2 Oil Well Pressure Transient Testing I

The diffusivity equation is the starting point for understanding the behavior of pressure transient tests. The diffusivity equation for a single phase liquid with fluid pressure P is

OZP 1 OP OP +

Or~ r D Or D Ot o (12.2.1)

where dimensionless radius r D and dimensionless time tD are defined as


r D =_ rw ~ ; t D =- 0.000264

k t

(~(~CT)il'2 (12.2.2)

in terms of the variables

t time [hr] r radial distance from well [ft]

k permeability [md] porosity [fraction]

I 3. viscosity of liquid [cp] cr total compressibility [psia -~]

rw wellbore radius [ft]

The subscript i indicates that viscosity I~ and total compressibility cr are evaluated at initial pressure.

The diffusivity equation was derived for a set of assumptions that need to be noted to avoid indiscriminate, and inappropriate, applications of its solutions. The diffusivity equation is based on the assumption that the reservoir is homogeneous and isotropic in porosity, permeability and thickness. The production well is assumed to be completed through the entire thickness of formation to ensure that radial flow is occurring in the formation. The fluid flowing into the well is assumed to be a single phase with constant fluid viscosity. The fluid is considered slightly compressible with a constant compressibility. The total system compressibility is the sum of rock compressibility and the fluid phase compressibilities times phase saturations, thus cr = c~+ coSo + cwSw + cgSg for a system with oil, water and gas phases. The diffusivity equation does not include gravity forces, which corresponds to the assumption that gravity effects are negligible. These assumptions lead to Equation (12.2.1). Although the following discussion is presented in terms of oil wells, it is applicable to any system that satisfies the above assumptions, such as a water production well.

Solutions of the diffusivity equation assume the well is a line source with a constant flow rate at the well. The solutions depend on the imposed boundary conditions. For a reservoir that acts like it does not have an outer boundary, we have the solution for an infinite acting reservoir [Home, 1995]

John R. Fanchi 203

PD(rD tD) -- -- -~ E i ' 4tD

(12.2.3)

where the term Ei(...) is the exponential integral

�9 O0 e _ u

- E i ( - x ) - I u X

m d u (12.2.4)

Dimensionless pressure PD in oilfield units is

Po ~ 141.2QB~t

(12.2.5)

where

K permeability [md] H thickness [ft] P; initial reservoir pressure [psia] Pwy well flowing pressure [psia] Q flow rate [STB/d] B formation volume factor [RB/STB]

viscosity [cp]

The group of parameters K/OP~CT with porosity (~ and total compressibility Cr occurs frequently in the context of the diffusivity equation, and is called the diffusivity coefficient.

Equation (12.2.3) is valid throughout the reservoir, including at the wellbore where the dimensionless radius rD = 1. In many cases, the inequality t D / r 2 > 10 is valid at rD = 1 SO that the exponential integral solution can be approximated by

1 0.80907) + - -i(e to + s (12.2.6)


where S is called the skin and is a measure of well damage or stimulation. A positive value of skin S represents wellbore damage, while a negative value

of S represents a stimulated well. In oilfield units and transforming the logarithm to base 10, Equation (12.2.6) becomes

= - + 0.8686 S - 3.227 (12.2.7) Pwi P/ 162.6 KH logt+log~tcrr2w

Flow regimes are associated with different boundary conditions. Three

flow regimes are usually identified: steady state, pseudosteady state, and

transient state. The flow regime depends on the boundary condition and can

be identified by the rate of change of pressure with time. The steady state flow regime corresponds to a system in which the mass

flow rate is constant everywhere, and pressure is not changing with time

(dP/dt - 0). An example of a system that exhibits steady state flow is a

reservoir connected to an infinite acting aquifer. The corresponding boundary condition is referred to as the constant pressure boundary.

The pseudosteady state regime applies to a system in which both the

wellbore and average reservoir pressure change with time. Pressure changes

at the same, constant rate (dP/dt = constant) in the pseudosteady state regime.

The system behaves like a closed system and there is no fluid movement

across boundaries. An example of this type of reservoir is a reservoir with

closed boundaries and no fluid encroachment from sources such as aquifers or leaking faults.

The final regime is the transient state. The transient state is the flow

regime in which pressure changes as a function of time (dP/dt=f(t)). In this

case, there are no restrictions on fluid movement. Identifying the proper flow

regime is important because the appropriate analysis of a pressure transient test depends on correctly identifying the flow regime.

Diagnostic Analysis of Pressure Transient Tests The pressure transient test analysis begins with a diagnostic analysis.

The first step in the diagnostic analysis is to plot the logarithm of pressure

versus the logarithm of time. The behavior ofpressure on this plot determines the flow regime and the corresponding analytical techniques.

John R. Fanchi 205

The steady state flow regime is characterized by pressures that do not

change with time ( d P / d t - 0). In this case, pressure P equals a constant 0~ such that P - 0~ and 0~ is independent of time. The resulting time derivative

satisfies the time derivative criterion d P / d t - O. The steady state flow regime

appears as a horizontal line on a diagnostic plot of log P versus log t. An

example of the steady state flow regime is the late time pressure response for infinite-acting radial flow in which dimensionless pressure has the form

PD = ln(re/rw) where re is the drainage radius of the well and r w is the wellbore radius.

The pseudosteady state flow regime is characterized by pressures that

change at a constant rate (dP/dt - constant). In this case, pressure P is

proportional to time such that P - [3t where [3 is independent of time. The

resulting time derivative satisfies the time derivative criterion d P / d t - [3. The

pseudosteady state flow regime appears as a straight line with unit slope on

a diagnostic plot of log P versus log t. This can be seen by noting that the

logarithm of P - [~t is log P - log [3 + log t. The geometric characteristics

of the pseudosteady state flow regime on a diagnostic plot are determined by comparing log P = log [3 + log t to the equation for a straight line y = mx

+ b with x the independent variable, y the dependent variable, m the slope

and b they-axis intercept. The relationships between the pseudosteady state

flow regime case and the straight line are x - log t, y - log P, m - 1, and b

- log [~. An example of the pseudosteady state flow regime is the pressure

response for a closed system with area A in which dimensionless pressure

is related to dimensionless time by P9 = 27r, t9 (rwE/A) + "'" with the logarithm

given by log P9 = log t9 + ' .

The transient state flow regime is characterized by pressures that change

as a function of time. Supposef(t) is a function of time such that pressure

P - f ( t ) . The time derivative in this case has the general form dP/dt = df(t)/dt.

The time derivative in the transient state flow regime must satisfy the

inequality dP/dt ~ constant. If dP/dt is a nonzero constant, then we are in

the pseudosteady state flow regime. If d P / d t - 0, then we are in the steady

state flow regime. A diagnostic plot of log P versus log t should exhibit a

nonlinear relationship in the transient state flow regime. The transient state

flow regime can be seen in the early time pressure response for infinite-acting

radial flow in which dimensionless pressure PD is proportional to �89 ln(tD). A semi-logarithmic plot Of PD versus ln(tD) yields a line with slope one hale


Superposition Principle Pressure transient test analyses rest upon an important assumption called

the superposition principle. The superposition principle asserts that the total pressure change at a point in the reservoir is a linear sum of the changes in pressure due to each well in the reservoir. The assumption of linear superposition is a theoretical consequence of the linear diffusivity equation that is substantiated by field performance. The superposition principle implies that a pressure disturbance will propagate through the reservoir even if the source of the disturbance changes or disappears. A quantitative expression of the superposition principle is

A = A ewe,,. + A ewe,,. + A ewe,, (12.2.8)

where the pressure change at a well is given by

Aewell- Pwell- Pinitial (12.2.9)

and Pinitial is the initial reservoir pressure.

Pressure Buildup Test The concepts of pressure transient testing can be clarified by considering

a particular application: the pressure buildup (PBU) test. The PBU test is performed by first flowing the well at a stabilized rate Q for a stabilized flow period tF. In practice, the stabilized rate Q is the last rate prior to shutting in the well, and the flow time tF is given by

cumulative production [STB]] tF=

Q [STB/D] (12.2.10)

The well is shut-in for a duration At after the stabilized flow period while reservoir pressure is recorded as a function of time. The results of the PBU test are then analyzed using the superposition principle.

The superposition principle is applied to the PBU test by recognizing that the shut-in condition is equivalent to adding the pressure response for two wells producing at different rates and at different times from the same

John R. Fanchi 207

location. From this perspective, the original well flows at rate Q for the entire period of the test. The shut-in condition is represented by introducing an image well. An image well is an imaginary well that is introduced for mathematical purposes. The addition of the pressure response due to an image well at the same location as the actual well but producing at rate -Q beginning at time tF gives the shut-in condition. The rates for the PBU test are shown in Figure 12-2, and the PBU analytical procedure is outlined below.

Q �9

0

-e

tF

r Actual Rate

At

Time

Figure 12-2. Rates for the PBU Test

It is useful to recall that dimensionless time to is given by

0.000264Kt t D = r (12.2.11)

Dimensionless pressure PD is a function of dimensionless time. The solution of the diffusivity equation in terms of dimensionless pressure and dimensionless time may be written in the form

Pw~ = P i - 141.2 ~p . ; QB~t [Po(to)+ constant] K H

(12.2.12)


where

Pw~ shut-in pressure [psia] P~ initial pressure [psia] Q stabilized flow rate [STB/D]

Dimensionless pressure changes linearly with the logarithm of time during the infinite-acting time period, thus

l [en(to ) + constant] (12.2.13)

Combining Equations (12.2.12) and (12.2.13) for the PBU case with the superposition principle eliminates the unspecified constant in these equations and gives

Pw~ = P~ -141.2 (12.2.14)

The dimensionless pressures in Equation (12.2.14) can be replaced by Equation (12.2.13) to yield the simplified form

laws- e - mlog(tH) (12.2.15)

where t , is the Homer time

t F + At tH = At (12.2.16)

and the variable m is given by

m --- 1 6 2 . 6 ~ KH

(12.2.17)

John R. Fanchi 209

It can be seen from Equation (12.2.15) that a semi-logarithmic plot of Pw~ versus the logarithm of Homer time will give a straight line with slope -m. This plot is called the Homer plot and the analysis is called the Homer analysis. Figure 12-3 illustrates the main features of the Homer plot.

4000.00

3999.00

3998.00

3997.00

~ 3996.00

~ 3995.00

" 3994.00

r~ 3993.00

3992.00

3991.00

3990.00

Horner Plot

11'o O t

1 10 I00 1000

Horner time

. Pws]

Figure 12-3. Horner Plot

The early time behavior of Figure 12-3 is at the right hand side of the figure, and proceeds to later times on the left hand side of the figure. The rapid buildup of pressure at early times corresponding to Homer time = 25 or greater is the wellbore storage effect. Wellbore storage is discussed in more detail below. The data for the infinite acting period comprises the straight line in the Horner time range from =25 to =4. For Horner times less than 4, the slope of the infinite acting line changes because the reservoir boundary is beginning to influence shut-in pressure response.

A straight line drawn through the infinite acting period of the Homer plot, such as the one in Figure 12-3, has a slope -m where m has units of psia/log cycle. The value of m is approximately 0.8 psia/log cycle for the PBU data shown in Figure 12-3.


Once the value of m is known, it is possible to rearrange Equation (12.2.17) to obtain an estimate of flow capacity in the region of investigation of the well test, thus

QB~t K H = 1 6 2 . 6 ~ (12.2.18)

m

An estimate of net formation thickness H can then be used with Equation (12.2.18) to estimate an effective permeability. Net formation thickness is usually obtained from well log analysis. Notice that permeability in this case represents a larger scale than laboratory measurements of permeability.

Dimensionless skin factor S can be estimated from the Homer plot. The procedure is to extrapolate the straight line drawn through the infinite acting period to the Homer time at a shut-in time At = 1 hour. The corresponding skin factor is

S = 1.151 m

Ktr + 3.22741 (12.2.19) - log (iF+ 1)* la crr2w

where the well flowing pressure P~j(tr) at the end of the stabilized flow period tF is required for the calculation.

Skin factor S accounts for the actual flowing well pressure, which are usually lower than those predicted using the radial flow diffusivity equation. The additional pressure drop is proportional to rate and can be viewed as a zone of reduced permeability, or "skin," in the vicinity of the well. The pressure drop associated with skin is

141.2 QB~t t (12.2.20) (APlskin - S K H

The actual pressure drop is the sum of line source pressure drop and skin

pressure drop: (me)ac tua t = (mP)linesourc e + ( m P ) s k i n. Skin usually ranges from -5 < S < 50. A positive skin corresponds to damage, or reduction in effective permeability near the well, while a negative skin represents stimulation.

John R. Fanchi 211

Wellbore Storage The pressure response to a change in flow rate is made more complicated

by wellbore storage. The effect of the finite volume of the wellbore on pressure response is called the wellbore storage effect. The wellbore pressure drops when the well is first open to flow, as shown in Figure 12-4. Initial fluid production includes expansion of fluid in the wellbore as a result of pressure decline. Wellbore storage is the effect of the finite wellbore volume on well flow response when the well flow rate changes. Wellbore storage prevents the flow rate at the sandface from instantaneously responding to a change in flow rate at the surface.

r

Q ,

0 q

t Surface Rate

y Bottomhole Rate

" ~ Test Begins

Time

Figure 12-4. Effect of Wellbore Storage on Flow Rate

Early-time pressure data is dominated by wellbore storage and is related to the flow test duration t by P = (K/C)t where the pi'oportionality constant K and wellbore storage coefficient C are independent of flow time t. The corresponding time derivative is dP/dt = WC. A diagnostic plot of log P versus log t gives a straight line with unit slope, thus P = (K/C)t implies log P = log t + log (K/C). The straight line with unit slope was also encountered in the pseudosteady state flow regime, but the wellbore storage straight line appears in the period of time immediately following the rate change.

Wellbore storage is known by many names, including afterflow, afterproduction, afterinjection, and wellbore unloading or loading. Two of


the most common types of wellbore storage are compressive storage and changing liquid level.

Compressive storage is due to the compression or expansion of fluid in the wellbore when the wellbore is completely full of a single-phase fluid. The wellbore storage coefficient C is estimated from the compressibility relation

C - VwC (12.2.21)

where

C wellbore storage coefficient [bbl/psia] Vw total wellbore volume [bbl] c average compressibility of wellbore fluid [psia ~]

By contrast, changing liquid level occurs in production wells on pump or gas-lift, and in injection wells taking fluid on vacuum. The wellbore storage coefficient C for changing liquid level is estimated by

C . -

v.

144 gc

(12.2.22)

where

C wellbore storage coefficient [bbl/psia] V, wellbore volume per unit length [bbl/ft] 9 fluid density [lb/ft 3] 9/144 hydrostatic gradient of fluid [psia/ft] g acceleration of gravity [ft/sec 2] gc unit conversion factor [32.17 lbmft/(lbfsec2)]

Wellbore storage from the changing liquid level is usually much larger than compressive storage.

One procedure for determining wellbore storage is to prepare the usual log-log diagnostic plot of pressure versus time. Early time data on the

John R. Fanchi 213

diagnostic plot is dominated by wellbore storage, and should exhibit a unit slope straight line. The analytical procedure begins with a plot of pressure change Ap versus shut-in time At on a log-log graph. Draw a unit slope through the early time data and calculate

- (12.2.23) 24 on the unit slope

where

C wellbore storage coefficient [bbl/psia]

Q flow rate [STB/day] B formation volume factor [RB/STB] At time [hr]

Ap pressure [psia]

Notice that the wellbore storage coefficient is expressed in terms of reservoir barrels of volume per psia of pressure change. The value of the wellbore storage coefficient for the data shown in Figure 12-3 is approximately 0.6 bbl/psia.

Interpreting Pressure Transient Tests There are several "rules of thumb" that can be used to aid in the

interpretation of pressure transient tests. We begin with the preparation of a diagnostic plot that includes both pressure versus time and the derivative

of pressure with respect to time. The "pressure derivative" referred to in the literature is actually the calculation of the product of shut-in time At and the

derivative of pressure with respect to shut-in time, thus At dP/d(At) or dP/d ln(At). This quantity, which we refer to here as the pressure derivative to be consistent with the literature, is a more sensitive indicator of reservoir

characteristics than the pressure response. A procedure for numerically calculating the pressure derivative is discussed by Home [1995]. The following interpretations in Table 12-1 refer to the slope of the pressure derivative curve on a log-log diagnostic plot.


Table 12-1 "Rules of Thumb" for Interpreting

Pressure Transient Tests

Effect

Wellbore Storage

Spherical Flow

Radial Flow

Linear Flow

Bilinear Flow

Slope on Log-Log Plot

Unit slope (rise 1~run 1) Negative half slope (drop 1/run 2)

Zero slope (horizontal)

Positive half slope (rise 1/run 2)

Positive quarter slope (rise 1/run 4)

Pressure transient test interpretation is aided by knowledge from other disciplines, such as the current understanding of reservoir structure and a knowledge of fluid phase behavior. As with other reservoir characterization techniques, it is important to seek a concept of the reservoir that is consistent with data from all sources. Ehlig-Economides and Spivey [2000] pointed out that the worst errors in well test analysis are caused by the application of incorrect geologic models. Spivey, et al. [ 1997] identified two aspects of model selection"

A. Identify the correct reservoir geometry and boundary conditions. B. Properly correlate the features of the well test model pressure

response with the field pressure response.

The selected model should be consistent with all available geoscience and engineering data.

Radius of Investigation The radius of investigation is the distance the pressure transient moves

away from the wellbore in the time interval following the change in flow rate. It may be estimated from a Homer analysis for a PBU test using the equation

~ KAt r i - 0.029 ~ t c r (12.2.24)

John R. Fanchi 215

where

rg radius of investigation [ft] At shut-in time [hr] K permeability [md] ~) porosity [fraction]

viscosity [cp] cr total compressibility [1/psia]

The above equation assumes radial flow, steady-state conditions, an infinite acting reservoir, and single-phase flow.

Although approximate, the radius of investigation can indicate the distance to reservoir features that cause the slope of the pressure transient response to change. For example, the change in slope at late times in Figure 10-2 indicates that a no-flow boundary has been reached. By substituting the time when the infinite acting period changes to a late-time boundary effect, we can estimate the distance from the tested well to the boundary to be about 400 ft. The radius of investigation can also provide information about no-flow barriers such as sealing faults or permeability pinch outs. This type of information should be compared with the geological concept of the reservoir and geophysical indications of structural discontinuities. The most accurate characterization of the reservoir is usually the one that provides a realization of the reservoir that is consistent with all available data from every discipline.

[ 12.3 Gas Well Pressure Transient Testing ]

The analysis of gas well testing is based on many of the same principles as those for oil well testing. The procedure is to change the flow of a well and record the pressure variation with time. The primary purposes of gas well testing are to determine reservoir characteristics using pressure transient testing, and to forecast gas well deliverability using gas deliverability tests. Our focus here is on pressure transient testing. The following discussion uses conversion factors assuming standard temperature and pressure are 60~ = 520~ and 14.7 psia.


Diffusivity Equation A diffusivity equation can be derived for gas wells just as it can for oil

wells. The diffusivity equation for gas wells differs, however, in the treatment of nonlinear fluid properties. In particular, the diffusivity equation for single phase gas flow is expressed in terms of the real gas pseudopressure re(P) [psiaE/cp], thus

02re(P) l Om(P) Ore(P) + =

O rzD r D Or D Ot D (12.3.1)

where dimensionless radius and dimensionless time are defined as

r Kt rD = G ;/o = 0"000264 ~(~c r) 2r

i w

(12.3.2)

and

t time [hr] r radial distance from well [ft] K permeability [md] ~) porosity [fraction]

gas viscosity [cp] cr total compressibility [psia -~] rw wellbore radius [ft]

Real gas pseudopressure m(P) is defined in Chapter 7, Equation (7.1.12). The subscript i in Equation (12.3.2) indicates that ~ and Cr are evaluated at initial pressure.

The general solution of the diffusivity equation for gas wells is

q T K H [mD (t D

where

John R. Fanchi 217

mo dimensionless real gas pseudopressure S skin [dimensionless] D non-Darcy flow coefficient [(Mscf/D) ~] q surface flow rate of gas [Mscf/D] T reservoir temperature [~ K permeability [md] H formation thickness [ft]

Analysis of gas well pressure transient testing is directly analogous to analysis of oil well pressure transient testing with pressure replaced by real gas pseudopressure.

We illustrate gas well test analysis by considering a pressure buildup test. The test begins with the flow of gas from the well at a stabilized rate for a duration tF. Pressure is then recorded as a function of shut-in time At when the well is shut-in after the stabilized flow period.

The superposition principle is applied in the usual way to find dimensionless pseudopressure md as a function of dimensionless time. The solution of the real gas diffusivity equation is written in the form

m(Pw~ ) - m(Pii)+ 1422 vr_r qT constant] (12.3.4) K H [mD (t D ) +

where Pws is well shut-in pressure. Dimensionless pseudopressure increases linearly with the logarithm of

time during the infinite acting period. Using this observation in Equation (12.3.4) gives

m(Pws) - m(Pi)+

s' - s + Dlq [

1637qT

KH I / t 1 log r 2 - 3 . 2 3 + 0 . 8 6 9 S ' (12.3.5)

The skin factor S' for a gas well includes the usual skin factor S plus a factor that is proportional to gas rate q with proportionality constant D. The rate


dependent part of S' accounts for turbulent gas flow. Combining these equations for the pressure buildup case gives

[/1 m(Pw~ ) - m(Pi)+ 1637 r,-r_r qT (AtD)] (12.3.6) + mo

o r

qT (tF+At t m(Pw~)= m(Pi)+1637 KHl~ At (12.3.7)

Horner Analysis Equation (12.3.7) is the gas analog of the Homer equation for a single

phase oil well. It is based on the usual assumptions: radial flow, steady-state conditions, infinite acting reservoir and single-phase flow. The procedure for analyzing gas well tests proceeds as in the liquid case:

1. Plot m(Pw~) versus Homer time (tF+ At)/At. 2. Estimate flow capacity from slope m"

K H - 1637 qT (12.3.8) m

3. Estimate skin using extrapolated shut-in pressure at 1 hour, thus

S ' - 1.151 - log z + 3.23 m d?~crr w

(12.3.9)

Pressure Drawdown Analysis The above procedure applies to pressure buildup tests in gas wells. An

analogous procedure can be used to analyze pressure drawdown tests in gas wells. The change in real gas pseudopressure is

John R. Fanchi 219

A m ( P ) - m ( P / ) - m(Pwf )

[I ] qT log 2 - - 3.23+ 0.869S' (12.3.10) = 1637 K H d?lacrr w

s , - s+ olql

where t is the time the well is flowing at rate q. The analytical procedure for

obtaining meaningful results consists of the following steps:

1. Plot Am(P) versus log t 2. Estimate flow capacity from slope m:

KH = 1637 qT (12.3.11)

3. Estimate skin from

m

S ' - 1 . 5 1 5 m(Plhr)- m

- l o g [ ~ + 3.23 , C rw

(12.3.12)

Two Rate Test If gas flow rate is large, it may be necessary to perform the two rate test.

In this test, two single rate tests at two different flow rates {ql, q2} are conducted. Results from the single rate tests provide enough information to determine two values of skin {S'~, S'2}. The resulting two equations for skin may be solved for the two unknowns S and D in the expression S '= S +

O l q l .

Reservoir Limits Test The drawdown test provides information about the limits of a reservoir.

The reservoir limits test requires that pressure drawdown be continued until pseudo-steady state (PSS) flow is achieved. The beginning of PSS flow is

given by the stabilization time t~:


d~t cr A t~ = 380 K (12.3.13)

where A is drainage area. Drainage area depends on drainage radius, which is uncertain. The drainage area for a radial system may be approximated as A = Xre 2 with re the drainage radius. The uncertainty in drainage radius will introduce uncertainty in the estimate of stabilization time. Consequently, the

onset of PSS flow is only an approximation. The pseudopressure equation for PSS flow has the straight line form

- m ' , + (12.3.14)

when m(Pwf) is plotted against t. The quantity m' is the slope of the line. It

has the form

qT m' = - 2.356 O?l~-c~t c r )~HA (12.3.15)

for a reservoir with thickness H and temperature T. The constant m(eintercept) is the time independent intercept of the infinite-acting straight line. Given

the slope m', we can estimate the drainage volume Vd [CU ft] as

V d = ddHA = - 2 . 3 6 0 qT

m'(~tCr) (12.3.16)

Radius of Investigation The radius of investigation for a pressure transient test in a gas well is

an estimate of the distance the pressure transient moves away from the wellbore in a specified time. It may be estimated for a gas well using

~ Kt r,. - 0.0325 ~ t c r (12.3.17)

John R. Fanch i 221

for r~ <_ re, where

re drainage radius [fl] r~ radius of investigation [ft]

t shut-in time [hr]

K permeability [md]

~) porosity [fraction]

I~ viscosity [cp]

cr total compressibility [ 1/psia]

A comparison of Equation (12.3.17) with Equation (12.2.24) shows that the radius of investigation for gas wells has the same functional dependence as the radius of investigation for oil wells, but the coefficient is larger for gas than for oil.

Before leaving this section, it is worth noting the dependence of stabilization time and radius of investigation on permeability. Tight gas

reservoirs have very low permeabilities (less than 1 md). Consequently, the stabilization time for a tight gas reservoir is very long when compared to the stabilization time of conventional reservoirs with permeabilities greater than 1 md. Similarly, the radius of investigation of a tight gas reservoir is

relatively small when compared to the radius investigation of a conventional reservoir. It is often too expensive to conduct a pressure transient test in a

tight gas well until the reservoir boundary is reached because the stabilization time is too long.

I 12.4 Well Test Capabilities I

There are many types of well tests that can provide information about properties of interest to reservoir characterization. Several are listed in Table 12-2 [Kamal, et al., 1995]. The table illustrates the type of information that can be obtained at the mega scale level from well test data. Many of these tests are performed on a single well, while others require changing rates or monitoring pressures in two or more wells. The wells in many tests can be either vertical or horizontal. The table identifies the properties that can be determined by each test, and notes the time in the life of the project when the test is most likely to be run. It is usually necessary to run a variety of well


tests as the project matures. These tests help refine the operator' s understanding of the field and often motivate changes in the way the well or the field is operated.

Table 12-2 Reservoir Properties Obtainable from Transient Tests

Type of Test

Drill stem tests

Repeat-formation tests / Multiple formation tests

Drawdown tests

Buildup tests

Properties

Reservoir behavior; Permeability; Skin;

Fracture length; Reservoir pressure;

Reservoir limit; Boundaries

Development Stage

Exploration and appraisal wells

Pressure profile


Fracture length; Reservoir limit;

Boundaries


Fracture length; Reservoir pressure;

Exploration and appraisal wells

Primary, secondary and enhanced recovery

Primary, secondary and enhanced

recovery

Reservoir limit; Boundaries

Step-rate tests

Falloff tests

Interference and pulse tests

Formation parting pressure; Permeability; Skin

Mobility in various banks; Skin; Reservoir pressure;

Fracture length; Location of front;

Boundaries

Well communication; Porosity;

Interwell permeability; Vertical permeability

Secondary and enhanced recovery

Secondary and enhanced recovery

Primary, secondary and

enhanced recovery

John R. Fanchi 223

Table 12-2 Reservoir Properties Obtainable from Transient Tests (cont.)

Type of Test

Layered reservoir tests

Properties

Layer properties; Horizontal permeability;

Vertical permeability; Skin; Average layer pressure;

Outer boundaries

Development Stage

Throughout reservoir life

It is beyond the scope of this text to discuss all of these tests in detail. We have sought instead to convey a sense of how well testing can contribute to the reservoir characterization process. Pressure transient testing provides information about individual well performance, wellbore damage, reservoir pressure, and reservoir fluid flow capacity. Perhaps more importantly from

a reservoir characterization perspective, pressure transient testing can be used to estimate the distance to reservoir boundaries, structural discontinuities,

and communication between wells. Kamal, et al. [1995] and Chu [2000] present several examples that show how well tests can be used in reservoir management. The reader should consult the literature for more information

about specific tests.

I CS-12. Valley Fill Case Study: Well Pressures I

Well tests provide information about reservoir continuity, flow capacity, and pressure distribution. Figure 12-3 is a typical Homer plot for wells in the Valley Fill study. Table CS-12A is a tabulation of shut-in pressure versus

time data for the pressure buildup test. The PBU test used a production period of 24 hrs at a stabilized flow rate of 100 STB/D in a well with inner radius

of 0.25 ft. The slope of the infinite-acting period corresponds to a permeability of

approximately 150 md for a formation thickness of 120 ft. For lack of better

data, the permeability distribution is assumed to be isotropic with horizontal permeability equal to 150 md. Vertical permeability is assumed to be one tenth horizontal permeability. No direct measurements of permeability

anisotropy or vertical permeability were made.


Table CS-12A

Pressure Buildup Test Data

Shut-in Time (hrs)

0.2 0.4 0.6 0.8 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 6.0 7.0 8.0 9.0

10.0

Shut-in Pressure

(psia)

3991.05 3994.63 3996.42 3998.21 3998.84 3998.98 3999.08 3999.15 3999.21 3999.26 3999.30 3999.34 3999.37 3999.42 3999.46 3999.47 3999.48 3999.49

Shut-in Time (hrs)

12.0 14.0 16.0 18.0 20.0 25.0 30.0

Shut-in Pressure

(psia)

3999.49 3999.50 3999.50 3999.51 3999.51 3999.51 3999.52

Table CS-12B shows the initial pressures and associated datum depths

at well locations prior to production. The pressures were obtained from a

pressure distribution determined using a reservoir flow model that was

matched to well performance. They could have been measured in the field

using a drill stem test (DST) or a repeat formation test (RFT).

Table CS-12B

Initial Pressures in the Valley Fill Reservoir

Well Pressure (psia)

1 4006

2 4003

Datum Depth (feet)

8543

8538

Table CS-12B Initial Pressures in the

Valley Fill Reservoir (cont.)

Well Pressure Datum Depth (psia) (feet)

3 4012

4 4008

5 4014

6 4008

8558

8548

8563

8548

John R. Fanchi 225

] Exercises I

12-1. Use Equation (12.2.2) to calculate the dimensionless time for a well

that has produced 10,000 STB/D of dry oil for 15 days. Other data are

listed below:

K = 90 md

~ : 0 . 1 7

tx = 13.2 cp

cr = 20.0 x 10 .6 psi -~

rw = 0.5 ft

12-2. Suppose the following physical properties apply to a drill stem test

(DST) in an oil well:

K = permeability = 150 md

= porosity = 0.20

= viscosity = 1.0 cp

cr = total compressibility = 10 x 10 .6 psia -~

Calculate the radius of investigation at shut-in times of 0.5 day, 1 day,

and 2 days using Equation (12.2.24).


12-3. Suppose the following physical properties apply to a pressure transient

test on a gas well:

K = permeability = 1.07 md

= porosity = 0.14

[.t = viscosity = 0.0159 cp

cr = total compressibility = 5.42 x 10 .4 psia -~

Calculate the radius of investigation at shut-in times of 0.5 day, 1 day,

and 2 days using Equation (12.3.17).

12-4. Prepare a Horner plot ofthe PBU data in Table CS- 12A. Find the slope

of the infinite-acting straight line and use it to estimate the permeabil-

ity of the producing formation.

Chapter 13 Production Analysis

Field production data can be used as an independent means of verifying the reservoir description and volumes in place that were obtained using static data. Static reservoir descriptions provide an initial representation of the reservoir that must be evaluated using dynamic flow information. Decline curves, produced fluid ratios and tracer production are examples of dynamic data. A study that purports to be an integrated study must incorporate all available static and dynamic data.

Production data analysis methods include empirical and analytical techniques. Empirical techniques fit a curve to production data, while analytical techniques match a physical model to production data. An empirical technique is decline curve analysis, and analytical techniques include analytical aquifer models, pressure transient test models, and material balance. Semi-analytic techniques combine empirical relationships and physical models. Empirical techniques are widely used to prepare reserves estimates and yield information that can help differentiate between reservoir realizations. Analytic and semi-analytic methods can be used to identify flow regimes and characterize reservoirs.

Dynamic field production data provides information about reservoir heterogeneities like channeling and compartmentalization. It is useful for preparing a flow model, and building confidence in the resulting reservoir

227


representation. This chapter describes a variety of production engineering techniques that have not yet been discussed, including decline curve analysis, the evaluation of produced fluid ratios, and tracer testing in the context of reservoir characterization.

I 13.1 Decline Curve Analysis I

One of the first production analysis methods was decline curve analysis, or the study of the relationship between flow rate q and time t for producing wells [Lewis and Beal, 1918; Arps, 1945]. In the early 1900's, production analysts observed that future production could be predicted by fitting an exponential equation to historical decline rates. The exponential equation worked well for many reservoirs in production at the time, but did not adequately represent the behavior of some producing wells in depletion drive reservoirs. A better fit was obtained using a hyperbolic decline equation for these wells. Assuming constant flowing pressure, a general equation for the empirical exponential and hyperbolic relationships used in decline curve analysis is

dq n+l

dt - - a q (13.1.1)

where a and n are empirically determined constants. The empirical constant n ranges from 0 to 1.

Solutions to Equation (13.1.1) show the expected decline in flow rate q as the production time t increases. Three decline curves have been identified based on the value of n [Economides, et al., 1994].

Exponential Decline (n = 0):

q - qi e-at (13.1.2)

where qi is initial rate and a is a factor that is determined by fitting Equation (13.1.2) to well or field data.

John R. Fanchi 229

Hyperbolic Decline (0 < n < 1):

q-" : na t + q i" (13.1.3)

where qi is initial rate and the factors a and n are determined by fitting Equation (13.1.3) to well or field data.

Harmonic Decline (n = 1):

q-1 _ at + q i 1 (13.1.4)

where qi is initial rate and the factor a is determined by fitting Equation (13.1.4) to well or field data.

Decline curves are fit to actual production data by plotting the logarithm of q versus time t. This typical decline curve plot yields a straight line for exponential decline:

lnq = l n q i - at (13.1.5)

Equation (13.1.5) has the form y = mx + b for a straight line with slope m and intercept b. The factor-a in Equation (13.1.5) is the slope m of the straight line obtained by plotting the logarithm of q versus time t.

13.2 Produced Fluid Ratios

The ratio of one fluid phase to another provides important information for understanding the dynamic behavior of a reservoir. Produced fluid ratios include gas-oil ratio (GOR), gas-water ratio (GWR), and water-oil ratio (WOR) or water cut. Water cut is water production rate divided by the sum ofoil and water production rates. The dominant produced fluid ratio depends on the mechanisms that are active in the reservoir.

Petroleum is initially produced by natural forces called drive mechanisms. Drive mechanisms are discussed in a several reservoir engineering texts, including Craft, et al. [ 1991 ], Dake [ 1994], and Ahmed [2000]. The


most common drive mechanisms are water drive, solution or dissolved gas drive, and gas cap drive. The most efficient drive mechanism for producing oil is water drive. Water from a subsurface source displaces the oil during

the production process. The useful life of natural aquifer energy can be prolonged by balancing oil withdrawal with the rate of water influx. Water drive recovery ranges from 35% to 75% of" original oil in place [Ahmed,

2000]. In a solution gas drive process, gas dissolved in oil at reservoir pressure

and temperature is liberated as reservoir pressure declines during primary production. In addition to oil flow from high pressure regions to lower pressure in the wellbore, gas expansion and movement carries some oil to the production wells. Solution gas drive recovery ranges from 5% to 30%

of original oil in place [Ahmed, 2000]. If an oil reservoir is in communication with free gas, the free gas is a gas

cap that can provide energy to the production process. The resulting drive is referred to as gas cap drive. Oil production through wells completed in the oil zone causes a decline in reservoir pressure that allows gas to expand and displace oil to well completions. When the gas-oil contact reaches the uppermost perforations, large volumes of gas will be produced with the oil. Gas cap drive recovery ranges from 20% to 40% of original oil in place

[Ahmed, 2000]. Recoveries as high as 60% are possible in steeply dipping reservoirs. The flow capacity in these reservoirs must be sufficient to allow good oil drainage to downstructure production wells.

The recovery factor for gas reservoirs is usually much greater than that for oil reservoirs. Recovery factors as high as 80% to 90% of original gas

in place are economically possible for volumetric gas reservoirs undergoing depletion without water-drive [Ahmed, 2000]. The low density and high mobility of gas relative to oil are primarily responsible for the relatively large gas reservoir recovery factors. Recovery from water-drive gas reservoirs tends to be lower, however, because gas is trapped as water encroaches into the gas zone. Typical recoveries range from 50% to 70% of original gas in place [Ahmed, 2000].

Primary production from many reservoirs occurs under conditions when two or more of these drive mechanisms are active simultaneously. This type of drive is referred to as a combination drive. Field production performance depends on which mechanism is dominant at different points during the

John R. Fanchi 231

primary production period. As time evolves, the dominant mechanism may change. Detailed studies using reservoir simulators are often the only cost- effective means of determining the dominant mechanism and predicting the

behavior of combination drive reservoirs. The production performance of certain properties, such as reservoir

pressure and GOR, can be used to identify the drive mechanism if sufficient reservoir production history exists. For example, solution gas drive reservoirs show a significant increase in GOR followed by a decline in GOR as available gas is produced. By contrast, reservoir pressure decline is shallower in water drive reservoirs than in other reservoirs. Water drive is usually the most effective oil recovery mechanism if enough water is available to balance hydrocarbon withdrawal.

A number of injection processes have been developed to supplement natural reservoir energy and significantly increase oil and gas recovery. We have already seen how effective water drive can be. An extension of this observation is the realization that water drive can be an effective means of displacing oil to production wells regardless of the source of water. Thus, if a reservoir does not have significant aquifer support, injection wells can be used to supplement existing natural resources.

Besides water, gas can serve as an injection fluid. The composition of

injected gas can range from re-injection of produced gas or enriched gas, to injection of inorganic gases such as carbon dioxide, nitrogen or even air. The displacement efficiency of each gas depends on its interaction with reservoir

rock and fluids. Industry experience has shown that hydrocarbon recovery is often

optimized when natural energy is supplemented. Alternative operating strategies should be considered early in the life of a field as an important aspect of reservoir management. One of the most useful tools for studying

alternative operating strategies is the reservoir simulator. These topics are discussed in more detail in later chapters.

Production Performance Monitoring Cramer, et al. [2000] observed that in a traditional well test, the operator

tests a well by separating oil, water and gas phases in a test separator and then measuring the volume of each phase separately. The traditional well test requires personnel to visit the site, connect the test separator, and tests


are ordinarily conducted at discrete intervals, such as once a month. Modem well surveillance techniques are providing continuous measurements.

Well production can be monitored using real-time sensors downhole or at the wellhead. A production performance monitoring system should include procedures for well surveillance, quality control of gathered data, data storage, and periodic review and analysis of well performance data. Production monitoring systems should provide data continuously without requiring expensive special tests or surveys.

Production records can be combined with reservoir, petrophysical and well completion data to create a well production model. A properly maintained production model should correctly represent well characteristics, identify wells with diminished deliverability, and generate reasonable production forecasts.

One example of a production monitoring system is Shell's Fieldware Flow Monitor [Cramer, et al. 2000]. It consists of low cost pressure trans- ducers at the wellhead or manifold that transmit signals to a supervisory control and data acquisition (SCADA) system or a distributed control system (DCS). SCADA or DCS software convert incoming pressure transducer signals into estimates of oil, water, and gas flow rates based on the results of previous well test measurements of GOR and BS&W (Basic Sediments and Water).

Effective surveillance provides information about when a well is not producing, producing at abnormally high or low rates, or exhibiting unstable behavior. Surveillance can inform the operator about how the well responds to changes in operating parameters such as gas-lift injection rate, pumping speed or choke setting [Economides, et al. 1994]. In an instrumented oil field [Tura and Cambois, 2001; Lumley, 2001 b], permanently deployed instru- mentation is used to monitor reservoir performance and modify production as needed.

I 13.3 Tracer Tests ]

Tracers are chemicals that are injected into one part of the reservoir and monitored in other parts of the reservoir. They provide information that can be used in a variety of ways. This section discusses the kind of information

John R. Fanchi 233

that can be obtained from tracer testing, identifies several types of tracers,

and outlines tracer test procedures. The following section presents simple

yet practical tracer calculations.

Tracer Test Information

Tracers can provide a wealth of information about reservoir flow paths. A reservoir flow path is the path that fluids follow in moving from one

location to another. A flow path can include uniform movement through a

permeable formation, or channeling through a fracture or high permeability

zone. Tracers can provide a direct connection between one point in the

reservoir (a specific injection well) to another point in the reservoir (a

specific production well). Not only can the path be defined, but also the time

it takes the tracer to traverse the path.

Directional flow trends can be identified by observing differences in

tracer arrival times at production wells. As indicated in previous chapters,

directional flow trends indicate the presence of flow anisotropy that needs

to be identified in reservoir characterization. Different arrival times suggest

preferred flow paths between tracer injection wells and producing wells. Changes in rates can alter areal sweep and influence arrival times.

Rapid interwell communication can be identified if tracer breakthrough

time is short. This may imply channeling or a high permeability streak. If

an injection well does intersect channels or high permeability streaks, vertical conformance treatments may be used to alter permeability.

Tracers may be used to estimate the fraction of the region of interest that has been flooded, or swept, by injected fluids. Sweep efficiency is discussed

in more detail in Chapter 16. Estimates of sweep efficiency require knowledge of injection and production rates and well patterns. This knowledge can

lead to improved estimates of reservoir pore volume and flow paths,

If tracer breakthrough occurs after injecting a small volume, it is possible to infer the existence of a channel, fracture or high permeability streak. By

contrast, if tracer breakthrough does not occur until a large volume of tracer

has been injected, the contacted reservoir is exhibiting a thief zone or more

uniform sweep. Thief zones can be natural or manmade. If thief zones are

discounted, sweep can be estimated from injected volume. A measurement

of tracer production volume can help distinguish a thief zone from uniform

sweep because a thief zone tends to remove tracer from the produced fluids.


Flow barriers may be delineated by delayed tracer response. Although

delayed tracer response could imply the existence of a thief zone, the pres-

ence of flow barriers such as faults or shale breaks could also delay the

appearance of tracer at a producer. Again, the amount of tracer produced

could be used to distinguish between the possibilities of flow barrier or thief

zone. Alternatively, formation evaluation using well logs and geologic

correlations would provide an independent interpretation that could help

evaluate the meaning of delayed tracer response. For example, a fence

diagram based on well logs may show that the thief zone interpretation is

unlikely. In this case, delayed tracer response would be an indication of a

flow barrier. It is possible that the flow barrier could be detected using

seismic surveys.

The sequential application of tracer surveys can be used to evaluate

sweep improvement processes. If a tracer survey is conducted before and

after applying a sweep improvement process, the difference in surveys can

be used to infer the success of the process. Examples of sweep improvement

processes include water flooding, gas flooding, and steam flooding. If the

survey after the process is the same as the survey before the process, it is

likely that the process was ineffective. On the other hand, substantial changes

in the tracer survey performance would suggest that the sweep improvement

process had an effect, although not necessarily positive. The determination

of incremental recovery is one measure that can be used in conjunction with

sequential tracer surveys to assess the merit of the sweep improvement

process.

Tracers may also be used to evaluate the flow of two fluids with different

mobilities, such as water and polymer. One tracer design procedure is to

inject a different tracer with each fluid and then compare breakthrough times

for each tracer. The tracers should have similar characteristics for the

comparison to be meaningful. For example, the breakthrough time should

not be effected by the loss of one tracer due to biodegradation or radioactive

decay while the other tracer is neither biodegradable nor radioactive. If the

breakthrough times of the two tracers are not the same, it can be inferred that

the tracers followed different flow paths through the reservoir. This implies

that the different displacement fluids exhibited different flow characteristics

and contacted (or swept) different parts of the reservoir.

John R. Fanchi 235

Ideal Tracer Features One way to assess the quality of a real tracer is to compare it to an ideal

tracer. According to Greenkorn [ 1962], the ideal tracer would "follow the fluid of interest exactly, traveling at the same velocity as the fluid front." In addition, the ideal tracer would consist of easily detectable material, it would not interact with rock or oil, it would be cheap, and it would be free

of environmental hazards.

Realistic Tracers for Aqueous Systems The appropriate tracer to use in a system depends on the fluid phases in

the system. In most field applications, tracers are expected to move through the aqueous phase. We focus here on a variety of realistic tracers for aqueous

systems. Although none of the traces exhibit ideal tracer behavior, several realistic tracers provide valuable information when properly used.

Radioisotopes such as tritium (an isotope of hydrogen with two neutrons and one proton in the nucleus) are often used as tracers. Radioisotopes are

easily detectable in small concentrations. They have insignificant absorption losses on rock, but they must be handled by licensed personnel and are

closely regulated. Flourescent dyes are also easily detectable in small concentrations. They

should be used only when rapid communication is expected, for example, when breakthrough time is expected within a few days. Flourescent dyes can be applied to the identification of flow paths through thief zones or natural

fractures. Several water soluble salts can be used as tracers. Examples of water

soluble salts include ammonium thiocyanate (NH4SCN), ammonium nitrate (NH4NO3), sodium or potassium Bromide (NaBr, KBr), sodium or potassium Iodide (NaI, KI), and sodium chloride (NaC1). These salts are detectable by noting significant changes in the ionic content ofproduced water. Since many water soluble salts exist in situ, especially sodium chloride, it is important to know the background levels of the salt. Significant deviations from the

background level can then be interpreted as the arrival of an injected salt

tracer. Water soluble alcohols can be inexpensive tracers. Examples of water

soluble alcohols that are appropriate for application as tracers are 2-propanol

(IPA-isopropyl alcohol), methanol (MeOH), and ethanol (EtOH).


Perfluorocarbon Tracers (PFT) One of the more useful modem tracers is perfluorocarbon. Perfluoro-

carbon tracers, or PFT, are fully fluorinated alkyl substituted cycloalkanes.

Their advantages are similar to radioactive tracers [Senum and Fajer, 1992].

PFT have negligible background in atmospheric and subsurface environments. This means that small quantities of PFT may be used and readily detected. A desirable consequence of their chemical composition is the observation that chemically different PFT may be simultaneously deployed,

sampled and analyzed. PFT are environmentally safe because they are non- toxic, non-reactive and non-flammable.

PFT do have some disadvantages. Their flow rate relative to the bulk flow rate of the fluid may be retarded by interphase mass transfer between

the aqueous and hydrocarbon phases. The interphase mass transfer compli-

cates the analysis of PFT travel time when compared with the analysis of the travel time associated with tracers that move at the bulk flow rate of the aqueous phase.

Practical Concerns There are several practical concems that must be considered when using

tracers. These concerns include governmental and environmental issues in

addition to technical issues. Some tracers have significant environmental

issues in addition to technical issues. For example, environmentally safe methods for disposing of radioisotopes must be part of any prudent reservoir

management plan. Similarly, care must be taken to prevent leaks of

potentially toxic tracers such as bromides, iodides and alcohols into potable

aquifers. A tracer design should include a check of local government regulations.

Different issues arise from a technical perspective. A tracer survey

requires the determination of background levels of the tracer in the region

of interest. The claim that a tracer is being produced requires a demonstration that the produced tracer is significantly greater than background levels of

tracer. This is especially applicable to radioisotopes and water soluble salts.

If the tracer is a water soluble alcohol, the tracer survey must recognize

that the water soluble alcohols are susceptible to biodegradation. One way

to combat the loss of water soluble alcohols is to inject bactericides with the

alcohols. An alcohol like 2-propanol is soluble in some oils. This can lead


to the additional retention of 2-propanol and larger than expected losses or delays in appearance of the tracer if solubility is not included in the tracer design.

High background levels of a tracer can mask the detection of the tracer, especially if the deviation of the background level from an average background level is large. This is especially significant with naturally occurring inorganic salts or naturally occurring radioactive materials (NORM). Dietz [1987] and Loder [2000] provide additional information about tracer technology.

I 13.4 Tracer Test Design I

Tracer test design begins with the identification of the pilot or pattern area [Terry, et al., 1981 ]. Determining the volume of the tracer that must be used depends on the volume of the system through which the tracer must pass. An estimate of the volume in the tracer test region can be obtained from volumetrics, material balance, or reservoir model studies. The volume of the region of interest is needed to estimate tracer breakthrough times and the amount of tracer needed for the study, as illustrated below.

A sample estimate of the mass of tracer m r needed for injection is obtained by calculating

m r - (water volume) (density) (concentration)

=

(13.4.1)

where

m r mass of tracer needed for injection [Ibm] r distance between injector and producer [ft] h formation thickness [ft] t~ porosity [frac] Sw water saturation [frac] 9r density of tracer solution [lbm/ft 3] Cr desired tracer concentration at producer [frac]


A typical safety factor ranges from two to five times the estimated mass m r to account for uncertainties in reservoir flow parameters.

One of the most important parameters in Equation (13.4.1) is the desired

tracer concentration. This concentration can be estimated by viewing the motion of a tracer molecule in the porous medium as a random walk process analogous to Brownian motion. According to this view [Scheidegger, 1954; Collins, 1961; and Greenkorn, 1962], the tracer molecule obeys the convection-dispersion (C-D) equation for homogeneous reservoirs in Nd

spatial dimensions, namely

oC

"~i l Di O X i2 -- V i - ~ O t (13.4.2)

where {Xi, Vi, Di} a r e the components of position, Darcy velocity and dispersion along the i th axis. Darcy velocity and dispersion are assumed to be

constants in this analysis. Equation (13.4.2) has the solution

[ 2] exp - x i - t Na 2 D~t --(

Coil i - 1 427tDit

(13.4.3)

where the constant Co is used to normalize the concentration. If length is in feet and time in days, the velocity components are in feet/day and the dispersion components are in feet2/day. Although we refer to D i a s dispersion,

it can also be used to approximate diffusion. Equation (13.4.3) is a more general solution than we need for our tracer

design estimate. It is sufficient for our purposes to consider the C-D equation

in a horizontal plane. A horizontal plane has two space dimensions so Nd = 2. The resulting C-D equation is obtained from Equation (13.4.2) and has

the form

0 2 C 02C OC O f OC - - - - T - v2 -

D1 ax 2 + D2 ax 2 ax I ax 2 at (13.4.4)

John R. Fanchi 239

where the subscripts denote orthogonal Cartesian coordinates in the horizontal plane. The solution of Equation (13.4.4) is

c(x,,x~,,) Co )21 1 ]21 1 v z -- X 1 -- t - X 2 -- t

exp 2Dlt ~ 2D2t -~- (13.4.5)

4 ( 2 ~ ) 2 D 1 D 2 t2

If fluid flow is in the x 1 direction, then D~ is the longitudinal dispersion D t

and D E is the transverse dispersion D r . As an illustration, assume the reservoir

is isotropic so that D t = Dr = D. Equation (13.4.5) becomes

Co [ Vr)21 C ( r , t ) - 2;~9t exp - 2 D t r - T t (13.4.6)

where

r radial distance from injector [ft]

t time after injection [days] D areal dispersion [ftZ/day] Vr Darcy velocity of frontal advance in radial direction [ft/day]

If the tracer is radioactive, Equation (13.4.6) can be modified to account for the radioactive half-life of the tracer by adding an exponential decay term:

El( Co exp _ 2 D t C ( r , t ) - 2 n D t

)2 j v~ 0.693t

r - --~--t - tM

(13.4.7)

where

t~/ tracer half-life [days]


Equations (13.4.6) and (13.4.7) express tracer concentration as a function

of radial position and time. Flow path information within the region of interest can be used to predict

tracer breakthrough locations. Actual tracer performance provides information about the quality of the reservoir characterization that was used to make the pre-test estimates. The following discussion outlines the procedure for

performing a tracer test. The reservoir should undergo waterflooding long enough to fill any void

space in the volume of the reservoir that is going to be swept by the tracer prior to the tracer test. This minimizes the potential loss of tracer in the reservoir. Once the reservoir has been "pressured up," tracers should be

injected as rapidly as possible. Rapid tracer injection forms a slug of tracer that can propagate through

the reservoir. However, the injection rate should be designed so that the solubility of tracer in in situ water is not exceeded. If the solubility of tracer is exceeded, tracer can be lost by oversaturating the system. Tracer solubility can be determined by testing the solubility of the tracer in samples of in situ water. Table 13-1 presents the solubilities for some selected tracers [Terry,

et al., 1981].

Table 13-1

Solubilities of Selected Tracers

Tracer

Ammonium Nitrate

Ammonium Thiocyanate

Potassium Bromide

Sodium Bromide

Potassium Iodide

Sodium Iodide

Sodium Chloride

Solubility in Distilled Water

(lb/bbl)

1280

420

187

278

446

556

125

Recommended Injection

Concentration (lb/bbl)

200

200

100

100

200

100

50

John R. Fanchi 241

It is necessary to implement a sampling program for detecting tracer at producing wells in the region of interest. The sampling frequency should balance two factors: sample often enough to observe breakthrough, and do

not sample so often that costs are excessive. Table 13-2 contains some suggested sampling intervals. If tracer breakthrough is expected to occur shortly after tracer injection begins, a short sampling interval should be implemented. As expected tracer breakthrough time increases, the sampling interval can increase. The sampling interval depends on expected tracer breakthrough times, which can be estimated as follows.

An estimate of frontal advance rate Vr can be obtained by assuming uniform radial flow:

qw v~ = 5 . 6 1 4 6 ~ (13.4.8)

2nrh~

where

v~ radial frontal advance rate [ft/day]

qw injection rate of tracer injector [bbl/day]

h net pay thickness [ft]

r radial distance from injector porosity

The term 2rcrh in the denominator is the cross-sectional area that is

perpendicular to the direction of fluid flow. If we assume uniform, radial flow, the time tb for the tracer to reach the producer is estimated as the ratio of radial distance divided by radial frontal advance rate:

t b - (13.4.9) Vr

where tb is tracer breakthrough time [days]. The effect of sweep efficiency can be included in the estimates of radial frontal advance rate and tracer breakthrough times by modifying the cross-sectional area transverse to fluid flow in the denominator of Equation (13.4.8).


Table 13-2 Suggested Sampling Intervals for Detecting Tracer

Breakthrough

Expected Tracer Breakthrough

I day

2 days

3 days

4 - 7 days

1 - 2 weeks

2 - 4 weeks

1 or more months

Sampling Interval

1 - 2 hours

2 - 3 hours

4 - 8 hours

8 - 16 hours

once a day

every other day

once a week

Data obtained from the sampling program should be promptly evaluated

to determine when enough tracer appears at a well to be significant. Observed

tracer production should exceed background levels of tracer. For each

production well, plot either tracer concentration versus time or tracer

concentration versus pore volume of produced water. Tracer breakthrough

will appear as a significant increase in tracer production followed by a

decline to background levels of tracer as the tracer slug moves through the

reservoir. An estimate of the total amount of tracer reaching each well can

be obtained from a variety of sources, including analytical calculations and

computer flow models.

I CS-13. Valley Fill Case Study: Production [

Six vertical wells are productive in the channel. Their locations are shown

in Figure CS- 1A, and production data for a year are shown in Table CS- 13A.

The wells maintained their initial rates through the first year of history. The

wells are perforated in the upper 72 feet of reservoir. Each well was brought

on-line at an interval of 45 days. Inner radius of tubing is 0.25 feet, and there

is no known skin. The bottomhole flowing pressure Pwl is estimated for an

oil gradient of 0.32 psia per foot and a depth of 8400 feet.

John R. Fanchi 243

Well Well On-Line

days

0

2 46

3 91

4 136

Table CS-13A

Production Well Information

5 181

6 226

Wellbore Skin Pwf Radius

0.25

Initial

Oil Rate

psia STB/D

0 2700 100

0.25 0 2700 100

0.25 0 2700 100

fto

0.25 0 2700 100

0 2700 100

0 2700 100

0.25

0.25

GOR

SCF/STB

392

392

392

392

392

392

Production data for the Valley Fill reservoir is presented in Table CS-

13B. Historical production data covers 360 days. Note that GOR is constant

and water production is very small. Oil rate in Table CS-13B shows the oil

rate at the time reported. For example, oil rate in the field is 200 STB/D at

90 days, and increases to 300 STB/D when Well #3 begins to produce on the 91 st day.

Fieldwide average pressure (Pavg) in Table CS-13B ranges from an

initial pressure of approximately 4000 psia to 2880 psia at 360 days. Since

the bubble point pressure is approximately 2015 psia, the field is continuing

to produce as an undersaturated oil reservoir.

Time

days

5

45

90

135

Pavg

psia

3986

3953

3886

3786

Table CS-13B Field Production Data

Oil Rate Water WOR

Rate

STB/D STB/D STB/STB

100.0 0.0 0.0

100.0 0.0 0.0

200.0 0.0 0.0

300.0 0.1 0.0

Gas

Rate

MCF/D

39.0

39.2

78.4

117.6

GOR

SCF/STB

392.0

392.0

392.0

392.0


Time Pavg

days psia

180 3652

225 3485

270 3284

315 3083

360 2880

Table CS-13B Field Production Data (cont.)

Oil Water WOR Rate Rate

STB/D STB/D STB/STB

400.0 0.2 0.0

500.0 0.5 0.0

600.0 0.8 0.0

600.0 1.1 0.0

600.0 1.4 0.0

Gas

Rate

MCF/D

156.8

196.0

235.2

235.2

235.0

GOR

SCF/STB

392.0

392.0

392.0

392.0

392.0

[ Exercises I

13-1. Tracer Test Design Problem: Suppose we are given the following data:

r = 200 ft, h = 15 ft, ~) = 0.20, Sw = 0.55, pr = 62.4 lbm/ft 3,

cr = 10 ppm, and qw = 500 bbl/day.

Calculate the amount of tracer that is needed for the test and estimate

when tracer breakthrough occurs.

13-2. Assuming that the initial well rates in Table CS-13A are maintained

for 365 days, estimate the cumulative oil recovery for the field after

365 days of production.

13-3. Use the volumetric estimate of oil in place from Exercise 5-3 and the

cumulative oil recovery calculated in Exercise 13-2 to determine the

recovery factor for the Valley Fill reservoir after 365 days.

Chapter 14 Reservoir Flow Simulation

The most sophisticated technology available today for making reservoir performance predictions is reservoir simulation. The process of applying a reservoir flow simulator to the study of a commercial reservoir development project requires a detailed reservoir description. Indeed, the reservoir description developed during the reservoir characterization process is the starting point for reservoir flow modeling. This chapter outlines the reservoir flow modeling process, describes how to acquire and evaluate data, calibrate the flow model (history matching) and make predictions.

[ 14.1 Reservoir Flow Modeling I

A modem, comprehensive reservoir management study is usually quantified using a computer program or set of computer programs that integrate relevant and available data. The reservoir characterization process should provide the reservoir model that is included in the model of fluid flow. Ifa geostatistical study is being used, the reservoir characterization process should provide a set of realizations that can be used in a flow model. The fluid flow model can be thought of as four interacting models: the reservoir model, the well model, the wellbore model, and the surface model.

245


The reservoir model represents fluid flow within the reservoir. The

reservoir is modeled by subdividing the reservoir volume into an array, or

grid, of smaller volume elements (Figure 14-1). Many names are used to

denote the individual volume elements, for example, grid block, cell or node.

The set of all volume elements is known by such names as grid, mesh, or lattice.

Unconformity

Volume Element

Figure 14-1. Subdivide Reservoir

Every practical reservoir simulator includes both a reservoir model and

a well model. The well model is a term in the fluid flow equations that

represents the extraction of fluids from the reservoir or the withdrawal of fluids into the reservoir. The reservoir model and well model can be used

to determine inflow performance relationships that establish the relationship

between flow rate and wellbore flowing pressure. Simpler analytical inflow

performance relationships such as Vogel's method [Economides, et al., 1994;

Ahmed, 2000] are often used instead of flow simulators for production

engineering work, but do not account for all of the effects that are included

in a flow simulator. Full featured commercial simulators also include a

wellbore model and a surface facility model. The wellbore model represents flow from the sandface to the surface. The surface model represents constraints associated with surface facilities, such as platform and separator limitations.

The mathematical algorithms associated with each model depend on physical conservation laws and empirical relationships. Computer simulators

are based on conservation of mass, momentum, and energy. The most widely

John R. Fanchi 247

used simulators assume the reservoir is isothermal, that is, constant temperature. If we are modeling a reservoir where thermal effects are important, such as a recovery process where heat is injected into the reservoir in some form,

then we need to use a simulator that accounts for temperature variation and associated thermodynamic effects. The set of algorithms used to model fluid flow in porous media is sufficiently complex that high speed computers are the only practical means of solving the mathematics associated with a

reservoir simulation study. Coats, one of the pioneers of reservoir simulation, identified a set of

prerequisites that should be satisfied before a model study is undertaken [Coats, 1969]. Reservoir simulation within the context of reservoir manage-

ment is a commercial enterprise. It is therefore important, from a business perspective, to require the existence of a problem of economic importance

before beginning a reservoir simulation study. One of the objectives of a model study usually includes finding a solution to an economic problem.

Once the objectives are defined, the modeler should decide if the

objectives of the study can be achieved using simpler techniques. If less expensive techniques, such as decline curve analysis or the Buckley-Leverett waterflood displacement algorithm can provide adequate results, then more

sophisticated and costly methods are not justified.

[ 14.2 Data Acquisition and Evaluation ]

The modeler should gather all available data and reports relating to the field.

The term "modeler" is used here to refer to either a single individual or an asset management team. The modeler should review the data to see if enough data is available to achieve the objectives of the study. If data is missing, the modeler should determine if missing data can be obtained by a more thorough search of the existing database, by using data from analogous reservoirs, or by using correlations to generate missing data. A complete set

of data must be provided to run the simulator. It is prudent to select data

values that can be justified. Missing data represent gaps in an operator's database, and give the

modeler information about the quality of the results that will be obtained from the model study. It may be difficult to justify the expense of acquiring


more data or delaying the study while additional data is obtained. The

modeler should avoid underestimating the amount of work that may be

needed to prepare an input data set. It can take as long to collect and prepare

the data as it does to conduct the study. If important data is missing, then

model forecasts should be discounted accordingly. In some cases, assumptions about the reservoir may have to be simplified because there is insuffi-

cient data to model the reservoir in greater detail.

After the data has been acquired, the modeler should evaluate the data

with a focus on its quality and suitability for achieving the objectives of the

study. Data should be reviewed to see if it is possible to identify which drive

mechanisms should be included in the model [for example, see Crichlow,

1977; Saleri, et al., 1992]. A material balance study is particularly useful in

this regard. These steps make it easier to select the optimum model for fulfilling study objectives. Three types of models should be considered:

conceptual, window area, or full field model. In many cases all three of these models may need to be used, as illustrated in Fanchi, et al. [ 1996].

A review of geophysical, geological, petrophysical and engineering reports provides a background on the history of the study area and should

provide information about previous and existing interpretations. The new

study may show that a new view of the reservoir is needed based on the best

available information. The modeler will need to explain why a new view is

better than a previously approved interpretation.

Data acquisition is an essential part of model initialization. Model

initialization is the stage when the data is prepared in a form that can be used

by the simulator. The model is considered initialized when it has all the data

it needs to calculate original fluids in place. The reservoir must be character-

ized in a format that can be put in a simulator and that is acceptable to the

commissioners of the study. Reservoir characterization includes the selection

of a grid and associated data for use in the model. It may also require the

study of multiple reservoir realizations in the case of a geostatistical model

study, as discussed in Chapter 11. All fluid data corrections, such as flash

corrections applied to differential PVT data in a black oil simulation, must

be completed during the model initialization process. Acquiring as complete a data set as possible is a necessary part of a

reservoir study. The data set only has value, though, if it is used. The data

must be made available to everyone involved in the study, and its signifi-

John R. Fanchi 249

cance must be understood in the context of the study. New data or changes in the quality of existing data should be incorporated into the data set as they become available. The sharing of data between disciplines and within an asset management team is a fundamental tenet of shared earth modeling.

I 14.3 Gridding and Upscaling I

Reservoir gridding is required for finite difference representations of reservoirs. The need to discretize a physically continuous region into a set of grid blocks is a consequence of the computational procedure used to solve the flow equations for the physical system [Fanchi, 1997]. The equations are coupled systems of nonlinear, time-dependent partial differential equations in three dimensions. Most simulators in practice employ a finite difference solution technique in which derivatives are replaced by finite differences. This approach requires that the region of interest be represented by a discretized region in space called a grid or lattice. The reservoir is a manifold, and the grid corresponds to the coordinate system covering the manifold. Discretization of the time domain is referred to as time stepping. Much of the work in the simulator is the numerical solution of the re- suiting set of algebraic equations.

Several types of reservoir data need to be discretized, as indicated in Chapter 11. Ex- amples include depths, thicknesses, porosity, permeability, and bulk moduli. A digital Figure 14-2. Cartesian Grid

value must be assigned to each block in the grid for each property needed by the flow calculation.


There are many different types of grids [Aziz, 1993; Verma and Aziz, 1997]. The traditional grid is a Cartesian grid with parallelograms, as illustrated in Figure 14-2. More modem grids include flexible grids, local grid refinement, and control volume finite element grids. Flexible, or unstructured, grids are made up of polygons in 2-D, or polyhedra in 3-D, whose shape and size vary from one subregion to another in the modeled region. A locally refined grid is a fine grid defined inside a coarser base grid. The base grid is often a traditional grid such as a Cartesian grid or a grid defined in cylindrical coordinates. Control volume finite element grids are triangular meshes in 2-D and tetrahedral meshes in 3-D. They are used with finite element solutions of flow equations.

An important property of grids is grid orthogonality, which can be either global or local. Global orthogonality is satisfied when grid axes are aligned along orthogonal coordinate directions. Examples include the Cartesianx-y-z grid and the cylindrical r-O-z grid. The grid may be distorted to fit local irregularities such as faults and wells. Local orthogonality is achieved when block boundaries are normal to lines joining the nodes on two sides of each boundary.

Locally orthogonal grids are examples of a more general type of grid called the Voronoi grid. A Voronoi grid block is the region of space that is closer to its grid point than to any other grid point. A Voronoi grid is a PEBI grid. PEBI grids are based on the method of perpendicular bisectors (PEBI) introduced by Heinrich in 1987 and pioneered by Heinemann, et al. in the petroleum industry in the late 1980's.

Gridding should obey several guidelines. Grid block size should change gradually across the grid. Grid block size should be most refined in regions of the reservoir where steep saturation and pressure gradients are expected to occur. The grid should represent structural variations in geology and, whenever possible, the grid should be aligned along stream lines, or expected dominant flow paths. This last criterion is another way of saying that the grid should be aligned along the principal axes associated with permeability anisotropy.

Most modem simulators are based on fluid flow equations that have been defined in terms of a coordinate system that is aligned with the principal axes of the permeability tensor. It has been known for some time that model predictions are affected by the assumption that the permeability tensor is

John R. Fanchi 251

diagonalized [e.g., Fanchi, 1983; Young, 1984]. This approximation is recognized as having an adverse impact on the state-of-the-art of reservoir performance forecasting [Edwards, 1995; Lee, et al., 1997; Young, 1999].

Gridding, or the discretization of model study areas for use in numerical simulation, is an active area of research for a variety of topics that are beyond the scope of this text. We have already noted that grid blocks are usually larger in flow models than in computer generated geologic models. This leads to the problem of how to transform from the finer scale geologic model to the coarser scale flow model, the so-called upscaling problem [Christie, 1996; Stern and Dawson, 1999]. Irregularities in grid shape also affect the calculation of frontal advance and need to be recognized for proper interpretation of flow model results. For a discussion of these and similar topics, the reader should consult the literature.

I 14.4 Flow Model Calibration-History Matching I

A reservoir simulation study consists of two essential elements" matching historical performance to calibrate the flow model; and making predictions [Mattax and Dalton, 1990; Thomas, 1982]. The modeler calibrates the flow model by verifying and refining the reservoir description during the history match. If necessary, the modeler will modify the initial reservoir description by making reasonable changes in input data until an acceptable match is obtained. The history matching phase of the study should be an iterative process that facilitates the integration of reservoir geoscience and engineering data. Predictions generally include preparing a forecast based on the existing operating strategy, and evaluating alternative operating scenarios.

The history matching process begins with clearly defined objectives and a database for the study area. An important task of the data acquisition process is to determine which data should be matched during the history matching process. For example, if a gas reservoir with water influx is being modeled, gas rate is usually specified and water production is matched. By contrast, if an oil reservoir is being modeled, oil rate is specified and water and gas production are matched. The accuracy with which data are matched provides a measure of the quality of the reservoir flow model. A poor match of historical data indicates that the model is not adequately representing an


important mechanism. A good match of historical data means the model provides a sufficient but non-unique match of available historical data,

Although a widely accepted strategy for conducting a history match does not exist, there are some general guidelines that can facilitate the completion of a history match. One set of guidelines with four steps is presented in Table 14-1 [Fanchi, 2001 a]. Steps I and II are considered more important than Steps III and IV. The first two steps provide a global concept of the reservoir and are analogous to a sophisticated material balance study. If Steps I and II cannot be achieved, the model should be considered inadequate and significant revisions may be necessary. There are several reasons why a model may be inadequate. Some of the most frequent problems include poor reservoir characterization and inaccurate or incomplete field data.

Table 14-1 Suggested History Matching Procedure

Step

I

II

III

IV

Remarks

Match volumetrics with material balance and identify aquifer support.

Match reservoir pressure. Pressure may be matched both globally and locally. The match of average field pressure establishes the global quality of the model as an overall material balance. The pressure distribution obtained by plotting well test results at given points in time shows the spatial variation associated with local variability of field performance.

Match saturation dependent variables. These variables include water-oil ratio (WOR) and gas-oil ratio (GOR). WOR and GOR are often the most sensitive production variables, both in terms of breakthrough time and the shape of the WOR or GOR curve.

Match well flowing pressures.

Production rates and cumulative volumes, production ratios and pressure measurements are the data variables most commonly matched in a typical study. Other data, such as injection data, tracer data, well stream composition, or the temperature of produced fluids can be matched if they are available and the appropriate flow simulator has been selected.

John R. Fanchi 253

Pressure is usually the first variable to be matched in the history

matching process. As a first approximation, uncorrected historical pressures can be compared directly with model pressures. Corrections to model

calculated pressures should be applied when fine tuning the history match. Pressure corrections are discussed in the more modem flow modeling texts, such as Mattax and Dalton [ 1990] and Fanchi [2001 a]. The corrections are generally based on Peaceman's pioneering efforts to correctly relate model

pressures and field pressures [Peaceman, 1978]. Production rates are usually obtained from monthly production records.

The modeler specifies one rate or well pressure in the flow model input data set, and then calculates the rates of all other phases. The historical performance of the specified rate is verified by comparing observed cumulative production of the phase or phases that correspond to the specified rate. For example, if we specify oil rate, we adjust model input data to match water and gas rates. To verify that the specified oil rates are correct, we compare model calculated cumulative oil recovery with historical cumulative oil recovery. If the history of reservoir performance is extensive, it is usually

advisable to rely more on the validity of the most recent field data when performing a history match.

The ratios of produced fluid rates, such as GOR and WOR, are sensitive

indicators of model performance. The model should match the following

factors: the time when the ratios begin to increase or decrease; the magnitude of the difference between observed and calculated values; and slopes of

production ratios on plots of production ratio versus time. A match of production ratios provides information about pressure depletion, phase behavior, and front movements. Tracers and time-lapse seismic surveys also provide useful information about front movements, and should be matched if they are available.

Saleri [ 1993] observed that the modeler is more likely to match field performance than individual well performance. The best way to decide if a match is satisfactory is to have a clear understanding of the study objectives. The quality of the match between observed and calculated parameters does not need to be as accurate in a quick and inexact study than it would be in a more detailed study.

History matching generates a non-unique solution since there is usually more than one way to match available data. The solution may be non-unique


for a variety of reasons, such as inaccurate or limited field data, and numerical effects. Data limitations are more difficult to resolve than numerical effects because the system being modeled is inherently under-

determined: we do not have enough information to be sure that the final solution is unique. Geostatistical techniques attempt to quantify the non- uniqueness of the history matching process by generating a set of equally

probable realizations and estimating the statistical uncertainty of the results.

I 14.5 Predictions ]

Reservoir characterization and the history match process are intended to

provide a working model of the reservoir and establish a level of confidence in the validity of a flow model. The purpose of the model study was not to

just build a model, however, but to prepare a tool that would help people make decisions about the prudent management of a subsurface resource. The primary reservoir management objective is to determine the optimum operating conditions needed to maximize the economic recovery of hydrocarbons. This is accomplished by marshaling accessible resources to optimize recovery from a reservoir, and minimize capital investments and operating expenses. The simulation study should produce a forecast of fluid production that can be combined with forecasts of the value of the fluids so

that a cash flow prediction can be prepared. Thus, the reservoir simulation study is usually completed by making field performance predictions.

The prediction process begins with model calibration. If the history match is conducted by specifying a production rate, then the modeler should ensure continuity in well rate when the model is switched from rate control

during the history match to pressure control during the prediction stage of a study. A clear discontinuity in well rate is often observed between the rate at the end of history and the rate at the beginning of prediction. The rate

difference usually arises because the actual well productivity index (PI) is not accurately modeled by the model PI that was used during history

matching. The final historical well rate is usually matched to initial predicted well rate by adjusting well PI in the model at the beginning of the forecast.

The first step in the prediction process after calibration has been

completed is to prepare a base case prediction. The base case prediction is

John R. Fanchi 255

a forecast that will be compared to most of the other forecasts. The base case is usually a continuation of existing operating conditions, but it can be any reservoir management scenario that has been mutually agreed upon by the commissioners of the study. For example, the base case for a newly developed field that is undergoing primary depletion could be a primary depletion case to a user-specified economic limit, or it could be a waterflood because waterflooding has been the optimum reservoir management scenario for analogous fields.

The base case prediction establishes a basis from which to compare alternative operating strategies. The preparation and comparison of predictions is not only important to the reservoir management aspects of the study, but can also be used as a sensitivity analysis to provide insight into the uncertainty associated with model predictions.

Sensitivity analyses are often needed in both the history matching and prediction stages [for example, see Crichlow, 1977; Mattax and Dalton, 1990; Saleri, 1993; and Fanchi, et al., 1996]. Any method that quantifies the uncertainty or risk associated with selecting a particular prediction case may be viewed as a sensitivity analysis [Murtha, 1997]. Risk analysis generates probabilities associated with changes of model input parameters. Parameter changes should be constrained to ranges that are consistent with available data, information from analogous fields, and the experience of the modeler. Each flow model run that uses a complete set of input parameters is a trial. Probability distributions can be generated by conducting a large number of trials.

Modelers should be aware of the validity of model predictions. Although history matching is non-unique, the history match process does help validate the reservoir model. Methods such as decline curve analysis and material balance analysis can generate performance forecasts, but they do not provide the degree of detail that can be obtained from a reservoir flow model.

Performance predictions are valuable for a variety of purposes. Predic- tions can be used to better interpret and understand reservoir behavior and they provide a means of determining model sensitivity to changes in input data. This sensitivity analysis can guide the acquisition of additional data for improving reservoir management.

Predictions enable people to estimate project life by predicting recovery versus time. Project life depends not only on the flow behavior of the


reservoir, but also on commercial issues. Models let the user impose a variety of economic constraints on future reservoir performance during the process of estimating project life. These constraints reflect a range of economic

criteria that will interest management, shareholders, and prospective investors.

Commercial interests are clearly important to the future of a project, and so are technical issues. It is often necessary to compare different recovery processes as part of a study. Since there is only one field, it is unrealistic to believe that many different recovery processes can be evaluated in the field, even as small-scale pilot projects. Pilot projects tend to be substantially more expensive to mn than simulation studies. In some cases, however, it might be worthwhile to confirm a simulation study with a pilot project. This is

especially tree with expensive processes such as chemical and thermal flooding.

In summary, reservoir flow models can corroborate or refute hypotheses about physical systems; identify discrepancies in static models; and perform sensitivity analyses. The preparation of a reservoir management plan is often the single most important motivation for performing a reservoir flow model study. Reservoir management is discussed in more detail in the next chapter.

[ CS-14. Valley Fill Case Study: History Match ]

A 30x15x3 grid was chosen to model the Valley Fill reservoir. It is consistent with the distribution ofcompressional to shear velocity ratio VJVs displayed in previous figures. A plan view of the reservoir grid with well locations is shown in Figure CS-11A.

The homogeneous productive interval was subdivided into 3 layers to allow gravity segregation of fluids. Although gravity segregation is not significant in the case of single phase flow, which is the historical situation in this case, it will become significant in forecasts of reservoir performance.

For example, reservoir pressure in a base case pressure depletion case will drop below the bubble point pressure and result in the formation of a gas cap. The gas will segregate upstructure. This is best represented in a multilayer model. Similarly, waterflooding can result in downstructure water movement, which again is best represented in a multilayer model. The definition of a

John R. Fanchi 257

multilayer model is being done in anticipation of production performance forecasts, and illustrates the need for planning in the performance of a reservoir management study.

Seismic data have traditionally provided information about the structure of a reservoir. Based on the Vp/Vs ratio presented previously, the area of interest in the Valley Fill model has been divided into two regions. Region 1 is the meandering channel, and Region 2 is the region outside of the meandering channel. It is possible that the flow boundary does not coincide with the boundary separating Region 1 from Region 2. To test this possibility, a flow model run was made with the entire volume included in the reservoir description. Volumes in place with and without Region 2 are shown in Table CS-14A. Model I includes both regions, while Model II includes only Region 1. In addition, the net to gross ratio in Model II has been adjusted to obtain a match of reservoir pressure decline. The pore volume in Region 2 of Model II has been set to zero, and some adjustments to pore xkolume along the flanks of the channel were made to match historical pressure performance.

Table CS-14A. Effect of NTG Ratio on Original Fluids in Place

Model Regions

I 1 and2

II 1 only

NTG

Variable

Oil (Million

STB)

36.0

5.4

Water Gas (Million (Billion

STB) SCF)

38.4 14.1

6.5 2.1

Model I has considerably more fluid in place than Model II. Pressure decline in Model I occurs at a much slower rate than Model II.

A comparison of the pressure performance of Model II, the history matched model, with historical performance provides a test of the validity of the model. Similar results from the history matched model should show good agreement between historical and model oil and water production rates, and historical and model gas-oil ratios. The match of GOR is relatively simple because reservoir pressure is still above bubble point pressure at the end of history. Consequently, GOR is determined by a good representation of solution gas versus pressure for a single phase liquid system. The match

258. Shared Earth Modelin 8

of pressure and produced fluids indicates that the channel observed using Ve/V s information has a significant impact on reservoir performance.

Aquifer support was not needed to match either pressure decline or the relatively insignificant water production shown in Table CS-12B.

The difference between the Ve/Vs ratio at 365 days (end of history) and at 0 days in the uppermost layer of the model is negligible. Although a difference does appear in the channel, it is a fraction of a percent. The change in seismic response is a result of historical pressure depletion above the bubble point pressure of the field. The seismic response can change significantly in the future, however, depending on the reservoir management strategy selected for the field.

[ Exercises ]

14-1. History match results are presented in file VFILL3_HM.TSS. Find model calculated oil and water production rates in file VFILL3_HM.TSS and plot them against historical data shown in Table CS-13B. The file should have been downloaded from website http://www.bh.com/companions/0750675225 as stated in Chapter 1.

14-2. Find model calculated GOR in VFILL3_HM.TSS and plot it against historical data shown in Table CS-13B.

14-3. The pressure reported in VFILL3_HM.TSS is the pore volume weighted average reservoir pressure. Plot it against the historical data shown in Table CS- 13B.

14-4. Use the 3-D visualizer 3DView to look at file VFILL3 HM.ARR. Look at oil saturation (attribute SO) and describe what you see.

Chapter 15 Reservoir Management

Reservoir management may be defined as the allocation of resources to optimize hydrocarbon recovery from a reservoir while minimizing capital investments and operating expenses. Reservoir characterization and the shared earth model are essential elements of the reservoir management function. The shared earth model provides information on resource size and complexity. This chapter describes the reservoir management process, discusses how reservoir characterization contributes to the attainment of reservoir management objectives, and describes factors that influence a reservoir management study.

[ 15.1 Reservoir Management Process !

The reservoir management process begins with the determination of strategy and associated objectives. The two outcomes of reservoir management -optimizing recovery and minimizing cost-often conflict with each other.

Determining the relative importance of the conflicting outcomes is an

important task of decision makers charged with managing a reservoir.

Hydrocarbon recovery could be maximized if cost was not an issue, while costs could be minimized if the field operator had no interest in or obligation

259


to prudently manage a finite resource. The primary objec-

tive in a reservoir management study is to determine the optimum conditions needed to maximize the economic recov-

ery o f hydrocarbons from a prudently operated field. The existence of optimum conditions is illustrated in Figure 15-

1, which sketches the dependence of profits on capital investments. A capital invest-

Maximum Profit

Capital Investment

Figure 15-1. Economic Objective

ment which maximizes profit can be found as one possible management criterion. Others are discussed in more detail in Section 15.4.

Reservoir management studies are important when significant choices

must be made. The choices can range from "business as usual" to major changes in investment strategy. For example, decision makers may have to

choose between investing in a new project or investing in an existing project

that requires changes in operations to maximize return on investment. By studying a range of scenarios, decision makers will have information that can help them decide how to commit limited resources to activities that can achieve management objectives.

Reservoir flow modeling is the most sophisticated methodology available for generating production profiles. A production profile presents the volumes of fluid produced as a function of time. Fluid production can be expressed

as flow rates or cumulative production. By combining production profiles

with hydrocarbon price forecasts, it is possible to create cash flow pro- jections. As we saw in Chapter 14, the shared earth model is an essential element of reservoir flow modeling. The combination of production profile from flow modeling and price forecast from economic modeling yields economic forecasts that can be used to compare the economic value of competing reservoir management concepts. This is essential information for the management of a reservoir, and can be used to determine reservoir reserves. The definition of reserves is summarized in Table 15-1 [Staff-JPT, 1997; Fanchi, 2001 a].

John R. Fanchi 261

Table 15-1

SPE/WPC Reserves Definitions

Proved reserves

Unproved reserves

Probable reserves

Possible reserves

0 Those quantities ofpetroleum which, by analysis of geological and engineering data, can be estimated with reasonable certainty to be commercially recoverable, from a given date forward, from known reservoirs and under current economic conditions, operating methods, and govemment regulation.

In general, reserves are considered proved if the commercial producibility of the reservoir is supported by actual production or formation tests. 0 There should be at least a 90% probability (P90) that the quantities actually recovered will equal or exceed the estimate.

Those quantities of petroleum which are based on geologic and/or engineering data similar to that used in estimates of proved reserves; but technical, contractual, economic, or regulatory uncertainties preclude such reserves being classified as proved.

0 Those unproved reserves which analysis of geological and engineering data suggests are more likely than not to be recoverable. 0 There should be at least a 50% probability (Ps0) that the quantities actually recovered will equal or exceed the estimate.

0 Those unproved reserves which analysis of geological and engineering data suggests are less likely to be recoverable than probable reserves. 0 There should be at least a 10% probability (P10) that the quantities actually recovered will equal or exceed the estimate.

Estimates of reserves based on probability distributions depend on the probability distribution function that applies to the region of interest. In the absence of adequate information, we can use the normal distribution as a first order approximation of the probability distribution function for the statistical distribution of reserves. This assumption is based on the observation that normal distributions occur widely in nature. The corresponding probability distribution function (pdf) for a normal distribution of a variable X is


1 i pdf- f ( X ) - exp - 2 (15.1.1) o ~ 2o

where tx and o are the mean and standard distribution of the pdf The pdf obeys the normalization condition

i f ( X ) d X - 1 - 0 0

(15.1.2)

If the average and standard distribution can be obtained from independent

estimates of reserves, the values of proved (P90), probable (Ps0), and possible (P10) reserves can be calculated from the formulas

Pg0 = la - 1 .28o

Ps0 =

P~0 = la + 1.28o

(15.1.3)

As an illustration of this process, suppose studies of a particular field

by five independent groups yielded five models of a particular reservoir. The

Original Oil in Place (OOIP), recovery factor and reserves for each model

are shown in the following table:

Model

OOIP

(MMSTB)

Rec. Fac.

Reserves

(MMSTB)

1 2 3 4 5

700 650 900 450 725

0.42 0.39 0.45 0.25 0.43

294 254 405 113 312

The average value of reserves is 275 MMSTB and the standard deviation is 107 MMSTB. The corresponding proved, probable and possible values

John R. Fanchi 263

of reserves is calculated using the average and standard deviation to find

proved r e s e r v e s (P90) = 139 MMSTB, probable reserves (Ps0) = 275 MMSTB, and possible reserves (P~0) = 412 MMSTB. Figure 15-2 shows the corresponding probability distribution for the OOIP.

0

0.00 200.00 400.00 600.00 800.00 1000.00 1200.00

OOIP

I Probability

Figure 15-2. Normal Distribution of OOIP

The horizontal axis in Figure 15-2 is the probability that the variable of interest will be observed. In this case, the variable is OOIP. To illustrate its

meaning, we observe that the probability that OOIP is one billion barrels or greater is less than 10%. A similar figure can be prepared for reserves. All

of these calculations depend on the quality of the reservoir model that was used in the flow model. Reservoir characterization is fundamental to the development of a reservoir model and contributes directly to the reservoir

management process.

Lessons Learned Kelly [2000] has pointed out that the lessons leamed from a project may

help the organization conduct projects more efficiently in the future if the lessons learned are documented. The documentation of lessons learned can be achieved by assigning a team to gather information about the project, including the process of managing the project. The team must gather, categorize, interpret, prioritize, and summarize information about the project


before communicating the results of the "lessons learned" study to the rest of the organization. Information about the project is acquired by interviewing individuals who were involved in the project. Team members should be trained in interview techniques to optimize the information gathering stage of the "lessons learned" study.

I 15.2 Multidisciplinary Integration I

Many different disciplines contribute to the preparation of an accurate reservoir model. The information is integrated during the reservoir modeling process, and the concept of the reservoir is quantified in the reservoir flow simulator. Figure 15-3 illustrates the contributions different disciplines make to reservoir modeling.

Seismic Petrophysics I Fluid Interpretation Properties

/

Numerical Geological ~ Simulation " Wells

Model Model

Iii li[ Facilities Tubing Model GRID Curves Effects

I I

2 . . . . . . . . . . J . . . . . . . . . I . . . . . . . . . I

' Calibration of Observations & ' I I , Production Data Interpretation , l i m ~ m m m = m ~ m ~ m m m m ~ m m m m m ~ m m ~ m m m m ~ m 1 m 8

Figure 15-3. Disciplinary Contributions to Reservoir Modeling (after H.H. Haldorsen and E. Damsleth, �9 reprinted by permission of the American Association of Petroleum Geologists)

The integration of data from different disciplines is enhanced by flow models that include petrophysical calculations [Fanchi, 1999]. Integrated

John R. Fanchi 265

flow models integrate more reservoir geophysical and petrophysical information than has been required in more traditional flow models. Several examples are discussed in the next section and in the review by Fanchi, et al. [ 1999].

The flow simulator can serve as the point of contact between disciplines. The history match process is a filter that can eliminate many possible

descriptions of the reservoir and begin to focus attention on the most likely representations. The simulator provides an objective appraisal of each reservoir hypothesis, and constrains the power of personal influence described by Millheim [1997].

The flow modeler is often the first to know when a proposed hypothesis about the reservoir is deemed inadequate based on flow model results. One of the modeler's most important tasks is to sort through the proposed reservoir representations and seek consensus among all stakeholders in support of a reservoir representation. Multiple reservoir realizations are

possible because the available data do not uniquely constrain the possible

realizations to a single realization. The dual criteria of reasonableness and Ockham's Razor [Fanchi, 2001a; Jefferys and Berger, 1992] are essential to the history matching process.

Modem simulation studies of major fields are performed by teams that function as project teams or asset management teams in a matrix management organization. Matrix management is synonymous here with Project Manage- ment and has two distinct characteristics [Staff-JPT, 1997]: team members from different disciplines are assigned to a project, and a team member is often accountable to two or more supervisors.

Asset management teams are not a panacea. For example, multiple accountability can lead to difficulties for both team members and their supervisors. In describing changes in the work flow of exploration and

development studies, Tobias [ 1998, pg. 38] observed that "asset teams have their drawbacks. The enhanced teamwork achieved through a team approach often comes at the expense of individual creativity, as group dynamics can and often does inhibit individual initiative [Kanter, 1988]." Tobias

recommended that organizations allow "the coexistence of both asset teams and individual work environments." His solution is the decouple-recouple work flow described in Chapter 1. It allows the "simultaneous coexistence of decoupled individual efforts and recoupled asset team coordination."


Potential problems with teams can be alleviated by noting Tobias' advice and properly constituting the asset management team. Asset management teams should be constituted such that each member of the team is assigned a different task, and all members work toward the same goal. Team members should have unique roles to minimize overlap in responsibility and to

maximize individual accountability.

Visualization Technology The ability ofmultidisciplinary teams to conduct their work and achieve

their objectives depends to a large extent on the ability of team members to effectively communicate with one another. Communication between people with significantly different backgrounds is improved by visualization of a shared model of the system. Visualization technology is being improved by development of computer systems that can project subsurface images into 3-D space. Some systems are designed to allow a person to literally walk through a 3-D shared earth model. Animation software allows the 3-D images

to change as a function of time as well as include a visual depiction of wellbore trajectories. The result is a projection system that can be used to

visualize both 3-D and 4-D systems.

[ 15.3 Economics I

Flow model predictions are frequently combined with price forecasts to estimate how much revenue will be generated by a proposed reservoir management plan. Other methods of predicting reservoir performance could be used, such as decline curve analysis, but flow models based on detailed reservoir characterization provide the most sophisticated technology for

preparing economic forecasts. One crucial economic measure used in reservoir management is net

present value (NPV). Net present value takes into account the time value of money, and may be calculated from the expression

N N P V - ~'~ Po. Qon + Pgn Qg. - CA P E X . - O P E X . - TAX.

. . , (1 + r)" (15.3.1)

John R. Fanchi 267

where

N

Oo~

Ogn

Number of years Oil price during year n Oil production during year n Gas price during year n Gas production during year n

CAPEX,, Capital expenses during year n O P E X n Operating expenses during year n TAX. Taxes during year n r Discount rate

In many cases, resource managers have little influence on taxes and prices. On the other hand, most resource managers can exert considerable influence on production performance and expenses, Several strategies may be used to affect NPV. Some strategies include accelerating production, increasing recovery, and lowering operating costs. The reservoir management challenge is to optimize economic measures like NPV.

Revenue stream forecasts are used to prepared both short and long term budgets. They provide the production volumes needed in the NPV calculation. For this reason, the asset management team may be expected to generate flow predictions using a combination of reservoir parameters that yield a range of recoveries. The sensitivity analysis is a useful process for determining the likelihood that any one set of parameters will be realized. Sensitivity analyses form the basis for estimating the probability distribution of reserves.

The shared earth model determines how much money will be available to pay for wells, compressors, pipelines, platforms, processing facilities, and any other items that are needed to implement the plan represented by the model. The revenue stream is used to pay taxes, capital expenses, and operating expenses. The economic performance of the project depends on the relationship between revenue and expenses [see, for example, Thompson and Wright 1985; Mian, 1992; Bradley and Wood, 1994; Newendorp and Schuyler, 2000]. Several economic criteria may be considered in the evaluation of a project, such as NPV, internal rate of return and profit-to- investment ratio. The selection of economic criteria is typically a management function. Once criteria are defined, they can be applied to a range of


possible operating strategies. The strategies should include assessment of both tangible and intangible factors. A comparative analysis of different operating strategies gives decision making bodies valuable information for making informed decisions.

[ 15.4 Environmental Issues ]

The impact of a project on the environment must always be considered when developing a reservoir management strategy. Environmental studies should consider such topics as pollution evaluation and prevention, and habitat preservation in both onshore and offshore environments [Wilson and Frederick, 1999]. An environmental impact analysis provides a baseline on existing environmental conditions, and provides an estimate of the impact of future operations on the environment. Forecasts of environmental impact often require risk assessment, with the goal of identifying an acceptable risk for implementing a project. Shared earth models can contribute to the preparation of forecasts as well as guiding remedial work to reclaim the environment. Sequestration of carbon dioxide in the subsurface illustrates the value of a shared earth model in environmental planning [Fanchi, 2001 b].

An environmental concern facing society today is global climate change. Many scientists attribute global climate change to the greenhouse effect. The climatic greenhouse effect occurs when greenhouse gases such as methane and carbon dioxide in the atmosphere absorb infrared radiation rather than letting it escape into space. The resulting atmospheric heating is attributed to increasing levels of carbon dioxide in the atmosphere.

The global wanning issue has caused a change in the definition of pollution. For example, it used to be an acceptable practice to release natural gas into the atmosphere by flaring the gas. This practice is now prohibited in some parts of the world and is considered an undesirable practice because natural gas is a greenhouse gas. One proposed method for reducing the climatic greenhouse effect is to collect and store carbon dioxide in geologic formations as part of a process known as COz sequestration. The sequestration o f CO 2 in subsurface formations is a gas storage process that must satisfy the three primary objectives in designing and operating natural gas storage reservoirs. Those objectives are verification of injected gas inventory,

John R. Fanchi 269

monitoring of injected gas migration, and determination of gas injectivity. Fanchi [2001 b] discusses the feasibility of monitoring CO2 sequestration in a mature oil field using time-lapse seismic analysis. The oil field is the East Vacuum Unit in the Vacuum Field, New Mexico. Integrated flow modeling demonstrated that it is feasible to use time-lapse (4-D) seismic technology to monitor the subsurface storage of CO2.

A well-managed field should be compatible with both the surface and subsurface environment. Failure to adequately consider environmental issues can lead to tangible and intangible losses. Tangible losses have more readily. quantifiable economic consequences. For example, near- and long-term economic liabilities associated with potable water contamination can adversely affect project economics. Intangible losses are more difficult to quantify, but can include loss of public support for an economically attractive project. For example, a poor public image in the United States has contributed to political opposition to oil industry development of land regulated by the federal govemment. In some cases, the intangible loss can take the form of active opposition to an otherwise economically viable project. In many parts of the world, it is necessary to provide an environmental impact statement as part of the reservoir management plan.

I CS-15. Valley Fill Cas e Study: Base Case Prediction ]

Two production forecasts are considered" a continuation of the primary depletion scenario and a pressure maintenance program based on waterflooding. The primary depletion scenario is the base case prediction, and waterflooding is an improved recovery process discussed in the next chapter.

The primary depletion forecast continues the current operating strategy using six vertical wells to produce the Valley Fill reservoir. As pressure declines, reservoir fluid makes a transition from single phase oil to two-phase gas and oil mid-way through the second year of depletion. Pressure and GOR performance are shown in Figure CS-15A. Notice that GOR declines when the field pressure first drops below bubble point pressure. Gas comes out of solution when reservoir pressure drops below bubble point pressure. The free gas is not mobile until it forms the critical gas saturation. Free gas begins to flow only when the critical gas saturation has been reached. The presence


of free gas changes the total compressibility of the system and causes the change in slope of the pressure versus time curve in Figure CS-15A.

Figure CS-15A. Pressure and GOR Performance of Pressure Depletion Base Case

Seismic response changes considerably when a free gas phase forms. Figure CS- 15B shows the absolute value of the difference between the Ve/Vs ratio at 730 days and at 0 days for the uppermost layer of the model. Gas has migrated throughout the channel and reflects the reservoir continuity in the

Figure CS-15B. Magnitude of Seismic Velocity Ratio Difference from 0 Days to 730 Days in Pressure Depletion Case

John R. Fanchi 271

reservoir model. A 3-D seismic survey at the end of the second year would confirm or deny this continuity, but the pressure depletion case will not be

the actual strategy adopted for this field. A reservoir management strategy that yields more oil recovery is discussed in Chapter 16.

] Exerc i se s . . . . ]

15-1.

15-2.

15-3.

Prepare a plot ofprobability versus reserves using the data in Section 15.1. Assume the reserves are normally distributed. The resulting plot should be analogous to Figure 15-2.

The flow model results for case study depletion are presented in files with prefix VFILL3_DEPL. A. Plot flow model calculated pressure

and GOR results and compare with historical data. B. When does

free gas saturation begin to form in the reservoir (express in days since beginning of the run)? C. Why does production GOR decline before it begins to increase?

A reservoir has a formation compressibility of6x 10 -6 psia l . A. Use

the definition of compressibility to estimate the subsidence for a reservoir that is 100 feet thick at pressure depletions of 100 psia and 1000 psia. Assume the area of the reservoir does not change. B.

Repeat the exercise for a reservoir that is 500 feet thick.

Chapter 16 Improved Recovery

The principles discussed in previous chapters are fundamental to preparing an optimum reservoir management plan using the best available representation of the reservoir. It was pointed out in Chapter 15 that optimizing recovery is an objective of reservoir management. Recovery optimization requires an understanding of recovery factors, production stages, and reservoir management options. They are the focus of this chapter.

[ 16.1 Recovery Efficiency ]

Recovery efficiency is a measure of the amount of resource recovered relative to the amount of resource originally in place. It is defined by comparing initial and final in situ fluid volumes. An estimate of expected recovery efficiency can be obtained by considering the different factors that contributeto the recovery of a subsurface fluid. Two factors are especially useful: displacement efficiency and volumetric sweep efficiency. Displace- ment efficiency expresses the efficiency of recovering mobile hydrocarbon. Volumetric sweep efficiency expresses the efficiency of fluid recovery in terms of areal sweep efficiency and vertical sweep efficiency. Recovery efficiency is the product of displacement efficiency and areal and vertical

272

John R. Fanchi 273

sweep efficiencies. Each of these terms is discussed below for oil. Similar

definitions can be provided for gas. Displacement efficiency for oil is the ratio of mobile oil to original oil

in place at reservoir conditions. We account for swelling as the fluids flash

to surface conditions by including the formation volume factor. The definition of displacement efficiency including the effects of swelling is

VpSoi VpSor Soi Sor Bog eoi Boo

- V Soi = Soi

Bog Bog

where

Vp initial pore volume

So~ initial oil saturation

Sor residual oil saturation

Boi oil FVF at the beginning of waterflood

Boa oil FVF at the waterflood pressure

A couple of observations are worth making here. We first observe that

displacement efficiency approaches 100% as residual oil saturation is reduced to zero. As a rule, changes to residual oil saturation do not occur during depletion or immiscible displacement, but can occur if the reservoir is subjected to an enhanced oil recovery process such as miscible flooding or

micellar-polymer flooding. Our second observation recognizes a property ofoil formation volume factor: it is a maximum at the bubble point pressure of the oil. Thus, if a waterflood is conducted just above bubble point pressure, the value of Boa approaches a maximum value and the term with residual oil saturation approaches a minimum. The resulting displacement efficiency approaches a maximum for the waterflood. A similar argument could be made for pressure just below the bubble point pressure, but in that case gas comes out of solution and the displacement efficiency is adversely affected by relative permeability effects associated with multiphase flow in a gas-oil-water system.


Displacement efficiency is a measure of the amount of mobile fluid in

the system. In addition to displacement efficiency, recovery efficiency for

oil depends on the amount of oil contacted by injected fluids. Areal and

vertical sweep efficiencies measure the degree of contact between in s i tu and

injected fluids. Sweep efficiencies are defined by

swept area E A = (16.1.2)

total area

and

E v = swept net thickness total net thickness (16.1.3)

where

EA areal sweep efficiency

E v vertical sweep efficiency

Shared earth models and reservoir flow models are useful tools for quantify-

ing both swept area and swept thickness. For example, a cross-section model

can be used to estimate vertical sweep efficiency. Sometimes vertical sweep

efficiency is referred to as vertical conformance.

Volumetric sweep efficiency Evo~ is a measure of the amount of oil contacted by injected fluids. It is the product of areal sweep efficiency and

vertical sweep efficiency:

Evo t = E A x E v (16.1.4)

The product of displacement efficiency and volumetric sweep efficiency

gives recovery efficiency

R E = E D x E vo I = E D x E A x E v (16.1.5)

where R E is recovery efficiency. All efficiencies are fractions between zero and one.

John R. Fanchi 275

Reservoir characterization provides a model of the reservoir that can be used to determine recovery factors and identify sweep and displacement problems. All of the factors in Equation (16.1.5) are fractions that vary from 0 to 1. If one or more of the factors is small, recovery efficiency will be small. On the other hand, each of the factors in Equation (16.1.5) can be relatively large, and the recovery efficiency will still be small because it is a product of factors that are less than one. One of the objectives of reservoir management is to develop a plan for maximizing recovery efficiency by finding strategies to increase one or more of the efficiencies in Equation (16.1.5).

I 16.2 Production Stages I

The stages in the life of a reservoir begin when the first discovery well is drilled. Prior to the discovery well, the reservoir is an exploration target. After the discovery well, the reservoir is a resource that may or may not be economic. Withdrawal of fluid from the reservoir begins the production life of the reservoir.

Production may begin immediately after the discovery well is drilled, or years later after several delineation wells have been drilled. The boundaries of the reservoir are established by seismic surveys and delineation wells. A delineation well may also function as a development well. Develop- ment wells can be either production or injection wells. They are used to optimize resource recovery. Optimization criteria should either be provided by management or take into account management priorities. In addition, the optimization criteria should satisfy relevant governmental regulations. The reservoir management plan should be flexible enough to adapt to changes in the optimization criteria during the life of the reservoir. The reservoir management plan may have to be changed to accommodate technological advances, changes in economic factors, and new information obtained during the life of the reservoir.

The life of the reservoir has traditionally been subdivided into a series of production stages that represent distinct periods [e.g., Dake, 1978; Craft, et al., 1991; Dake, 1994; and Ahmed, 2000]. The stages of reservoir production are described below.


Primary Production Primary production refers to production from the reservoir using only

natural energy sources. It is the first stage of production and is considered

the depletion stage because reservoir pressure can be expected to decline as

fluid is withdrawn without replacement. The natural forces involved in the

displacement of oil during primary production are called reservoir drives.

The most common drives for oil reservoirs are water drive, solution or

dissolved gas drive, and gas cap drive. We discuss oil reservoirs first, and

then comment on gas reservoir recovery. Water drive is the most efficient drive mechanism for an oil reservoir

because the displaced fluid (oil) is being replaced by a displacing fluid (water) with comparable density and viscosity. Recovery from a water drive

reservoir typically ranges from 35% to 75% of original oil in place (OOIP). The solution gas drive relies on the liberation of gas dissolved in the oil

phase as pressure declines. The gas will expand and move to lower pressure zones in the reservoir more quickly than the lower mobility oil. Production

wells need to be properly placed to capture the oil as the fluids redistribute

themselves. Recovery by solution gas drive ranges from 5% to 30% OOIP.

The gas cap drive also relies on gas to displace oil. A gas cap is a volume

of free gas located above a zone ofoil. The point of contact between the gas

cap and the oil zone is the gas-oil contact. If production wells are completed

in the oil zone, production will cause a decline in pressure in the oil zone.

The higher pressure gas cap will move toward the producing wells and drive

oil in front of it. Eventually large volumes of gas will be produced with the

oil. Recovery by gas cap drive ranges from 20% to 40% OOIP. Recoveries

as high as 60% are possible in steeply dipping reservoirs with enough permeability to allow oil to drain to downstructure production wells.

Another primary production mechanism is gravity drainage. Gravity

drainage will occur in highly permeable, steeply dipping reservoirs when

a pressure gradient exists between the top and bottom of the reservoir. If the pressure gradient is large enough, upstructure oil will flow to downstructure production wells.

Two or more drive mechanisms are often active in a reservoir. The

behavior of fluid flow in the reservoir depends on which mechanism is most

important at various times during the life of the field. The behavior of such fields can be predicted using reservoir flow models.

John R. Fanchi 277 ,

Primary production from gas reservoirs depends on water drive and gas

expansion with reservoir pressure depletion. Water drive by aquifer influx

can have an adverse impact on gas recovery before water can trap a

substantial volume of gas. Trapped gas saturation can be on the order of 25%

to 30%. Gas reservoir recovery tends to be much higher than oil recovery

because of the relatively high mobility of gas. Gas recovery can range from

70% to 90% of original gas in place (OGIP).

Secondary Production The term secondary production refers to a period of immiscible fluid

injection following a period of primary pressure depletion. Oil recovery can

be increased substantially by supplementing natural reservoir energy. An

external energy source, such as water or gas injection, provides the supple-

mental energy. The injection of a fluid into the reservoir at immiscible

conditions is sometimes referred to as pressure maintenance because one of

the goals of the injection program is to keep pressure from declining. Pressure maintenance is now implemented at the beginning of production

in some modem reservoirs. In this case pressure maintenance is the first, or "primary production" stage.

Alternative Classifications Primary and secondary recovery processes rely on immiscible displace-

ment to produce oil. Tertiary production processes were designed to improve

displacement efficiency by injecting miscible fluids or heat. These methods

have been referred to in the literature as Enhanced Oil Recovery (EOR)

processes. The term tertiary production was originally used to denote the

third stage ofproduction when an EOR process was implemented following

primary and secondary production. We saw in the case of pressure mainte-

nance that terminology based on the sequence of implementation of a process

is no longer a reliable terminology. Oil recovery from some fields can be

optimized if the enhanced recovery process is implemented before the third

stage in the life of the field. Enhanced recovery processes were often more

expensive than drilling wells in a denser pattern. The term "infill drilling"

was born to denote the drilling of additional wells within an existing well

pattern to reduce well spacing and increase well density. Another term,

improved recovery, is now used to denote EOR and infill drilling.


Improved recovery technology includes traditional secondary recovery processes such as waterflooding and immiscible gas injection, enhanced oil recovery (EOR) processes, and infill drilling [Dyke, 1997]. EOR processes are usually classified as one of the following processes: chemical, miscible, thermal, and microbial. The literature on EOR processes is extensive and should be consulted for more detailed discussions, for example, see Lake [1989] and Green and Willhite [1998].

Accurate shared earth models are essential for optimizing the perform-

ance of EOR processes. In addition, shared earth models are needed for

understanding the behavior of conventional and nonconventional sources of natural gas. Two nonconventional sources of natural gas are coalbed methane [Selley 1998] and gas hydrates [Sloan, 2000].

I 16.3 Drilling Technology I

Advances in drilling technology are having a dramatic impact on reservoir management. Reservoir characterization is both a contributor to and

beneficiary of improved drilling technology. Longer wellbores that follow subsurface formations are providing access to more parts of the reservoir. The additional information is being integrated into reservoir characterization at the same time that more detailed reservoir models are helping guide the longer wellbore trajectories. Three areas of drilling technology are briefly discussed here: infill drilling, multilateral wells, and geosteering.

Infill Drilling Infill drilling is a means of improving sweep efficiency by increasing

the number of wells in an area. Well spacing is reduced to provide access

to unswept parts of a field. Modifications to well patterns and the increase in well density can change sweep patterns and increase sweep efficiency, particularly in heterogeneous reservoirs.

Multilateral Wells and Extended Reach Drilling Multilateral well technology is revolutionizing extraction technology

and reservoir management. Although the use of multilateral wells is considered a relatively recent development, Golan [2000] reported that a type

John R. Fanchi 279

of multilateral well was drilled by the Russians in the Bashkiria Field as long ago as 1955. It had 10 branches and its well schematic was published in the Russian literature [Drilling Journal, December 1955, pg. 87].

The first multilateral wells in the West were wells that were side-tracked to bypass casing problems. Today, multilateral wells make it possible to connect multiple well paths to a common wellbore. Figure 16-2 shows several examples of modem multilateral well trajectories. Multilateral wells have many applications. For example, they are useful in offshore environments where the number of well slots is substantially limited by the amount of space available on a platform. They can also be used to produce highly compart- mentalized reservoirs, and reservoirs with low permeability. Extended reach horizontal or multilateral wells are useful in environmentally or commercially sensitive areas where placing a drilling rig is undesirable or prohibited.

Vertical Plane

r

Horizontal Plane

..... J

Figure 16-2. Multilateral Wells

Geosteering Geosteering is a particularly promising technology for reaching drilling

target locations and is a prerequisite for successful extended reach drilling. Extended reach drilling provides a means of reaching commercial subsurface deposits at great distances from a fixed drilling rig location. Three of the longest applications of extended reach drilling are at the Wytch Farm Oilfield offshore Britain, the Xijiang Field in the South China Sea, and the Ara Field offshore Tierra del Fuego, Argentina [Vighetto, et al., 1999]. These projects have drilled extended reach wells of approximately 8 km of horizontal displacement from the drilling rig. Geosteering and extended reach drilling have many benefits, including reducing costs associated with the construction of new platforms.


CS-16. Valley Fill Case Study: Waterflood Prediction I

An alternative to the pressure depletion base case presented in Chapter 15

is waterflooding. The waterflood is most effective if it begins when reservoir

pressure is just above bubble point pressure of 2015 psia. The pressure at or just above bubble point pressure corresponds to the pressure when the oil

formation volume factor is at or near its maximum value, and oil phase swelling has the most favorable impact on oil recovery.

Water-oil ratio (WOR) increases when water breakthrough occurs. Producing wells were constrained to shut-in when the minimum oil rate

reached 10 STB/D or WOR exceeded 80%. A WOR limit is required to

account for water separation and handling capacities of surface facilities. The

presence of additional water in the system did not have a significant effect on seismic response.

A comparison of cumulative oil recovered by the pressure depletion base

case and the waterflood case shows that the waterflood case is able to produce more oil for a longer period of time than the depletion case because reservoir pressure is maintained above the bubble point pressure of the

reservoir oil. In the waterflood case, GOR does not change because a free

gas phase does not form at pressures above the bubble point pressure.

[ Exercises [

16-1. A. If the initial oil saturation of an oil reservoir is So~ = 0.70 and the residual oil saturation from waterflooding a core sample in the

laboratory is Sor = 0.30, calculate the displacement efficiency E9 assuming the change in formation volume factor is negligible.

B. In actual floods, the residual oil saturation measured in the labora-

tory is seldom achieved. Suppose Sor = 0.35 in the field, and recalculate displacement efficiency. Compare displacement efficiencies.

16-2. A. Calculate volumetric sweep efficiency E~o t and recovery efficiency RE from the following data:

o l

Sor Area Swept Total Area

Thickness Swept Total Thickness

0.70 0.25

700 acres 1000 acres

, , ,

30 feet 50 feet

John R. Fanchi 281

B. Discuss how recovery efficiency could be improved.

16-3. What is the recovery factor at the end of history for the Valley Fill Case Study? Hint: See time step summary file VFILL3_HM.TSS. The file should have been downloaded from website http://www.bh.com/companions/0750675225 as stated in Chapter 1.

16-4. Plot cumulative oil recovery versus time for both the Valley Fill case study depletion prediction and waterflood prediction. The data is available in time step summary files VFILL3_DEPL.TSS and VFILL3_WF.TSS. The cumulative oil recovery scale should range from 0 to 900 MSTB; and the time scale should range from 0 to 1600 days.

16-5. What are the recovery factors at the end of the following prediction runs: pressure depletion and waterflooding? Hint: see time step summary files VFILL3_DEPL.TSS and VFILL3_WF.TSS.

16-6. Plot pressure and WOR versus time for the Valley Fill case study waterflood prediction using data in file VFILL3_WF.TSS. The pressure scale should range from 0 to 4500 psia; the WOR scale should range from 0 to 0.25 STBW/STBO; and the time scale should range from 0 to 1600 days. The WOR spikes at later times in the plot of WOR versus time are showing production wells being shut-in according to pre-defined workover criteria.

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INDEX

absolute permeability 117, 127- 129, 163, 167

accumulation 23, 29, 152, 153, 165 acoustic

impedance 75-77, 79-81,193, 198

log 58 algorithm 157, 178, 193, 194, 247 allocation 14 API 91 aquifer 92, 197, 204, 227, 230,

231,252, 258, 277 Archie 61 areal model 168

base case 280 bin size 80 black oil 94-97, 101, 102, 105,

147, 196, 250 model 101,105 simulator 102, 147

block pressure 159 size 169, 250

borehole imaging 63 boundaries 19, 22, 29-31, 164,

178, 204, 222, 223,250, 275 boundary conditions 156, 202,204,

214 bubble point 94, 95, 99, 100, 102,

103, 105, 107, 159, 166, 196, 243,256-258,269, 273,280

Buckley-Leverett 133, 136, 139, 140, 160, 247

buildup 200, 206, 209, 217, 218, 222-224

bulk

density 43, 46, 58, 59, 75, 76, 174, 191

modulus 34, 35, 37, 43-48, 51, 172, 191,192

volume 1, 2, 8, 10, 35, 51, 59, 152

C-D equation 156-158, 238 calibration 83, 84, 254 caliper log 55 capillary pressure 108, 122, 125-

127, 131,134, 139-141,146, 148 Cartesian 116, 161, 180, 239, 249,

250 cash flow 260 cementation exponent 61 checkshot 79 chemical 23, 93, 125, 152, 236,

256 completion 194, 232, 252 compliances (elastic) 40-41 compositional model 103-105 compressibility

bulk, defined 35 fluid, defined 90

compressional velocity 42-44, 46, 75, 76, 193, 198

computer mapping 178, 188 program 162, 176, 245

concentration 60-62, 89, 153-156, 160, 238, 240, 242

condensate 94, 95, 97, 99, 102 conditional simulation 189 conservation

laws 12, 245 of mass 134, 151,152, 246

299


constant composition expansion 99 volume depletion 99

constraint 3, 72, 73 contact angle 124-127 continuity equation 152, 154-156 contour 5, 32, 175, 178, 180 convection-diffusion 154-156, 164 convection-dispersion 156, 238 core 4-9, 18, 50, 53, 55, 56, 79,

133, 141,142, 146, 148, 192 core analysis 6, 7 cricondentherm 94 critical point 94 cross-section 18, 60, 274 cubic 51,103, 104 cylindrical 54, 55,250

Darcy 93, 108-111, 114, 118, 119, 129, 131,166

data acquisition 13, 28, 78, 195, 197,

232, 247, 248, 251 interpretation (seismic) 78 processing (seismic) 78

datum 41, 118, 172, 224, 225 decline curve 227-229, 247, 255,

266 density

defined 88 gradient 141 log 58

depth migration 79 deterministic 5, 183, 189, 190,

196, 197 development geophysics 83 dew point 95, 99, 102 differential

equations 12, 146, 249

i

liberation 99, 100, 103 diffusivity equation 201, 216 digitize 180 dipmeter log 63 dipping 122, 230 disciplines 11-13, 15, 170, 214,

249 discount rate 267 displacement efficiency 273 drawdown 200, 218, 219, 222 drill stem test 224, 225 driller's log 55 drive 140, 151, 166, 228-231,248,

276-277 dry gas 95, 96 dyadic 129

economics 140, 171,266, 269 effective

permeability 128 pressure 48

elasticity theory 38 electrode log 62 elevation 140, 141 enhanced oil recovery 273,277-

278 environment 5, 24-26, 29, 53-55,

57, 63, 70, 100, 268-269 equation of state 90, 103, 105, 113 equilibration 101, 118 equilibrium 88, 89, 100, 101,103,

104 expenses 14, 254, 259, 267 exploration geophysics 83 extended reach drilling 278

facies 25 facilities 98, 105,246, 267, 280 falloff 200, 222

John R. Fanchi 301

fine grid 80, 250 finite difference 157, 158, 164,

165,249 flash 95, 100, 102, 103, 105, 106,

248,273 flow

capacity 172 equations 12, 120, 121,150, 151,

157, 158, 162-164, 166, 171, 174, 194, 246, 249, 250

regime 111,205 unit 7, 12, 170-173

fluid classification 95 contacts 81, 126 movement 43, 44, 118, 190, 204 properties 47, 53, 54, 78, 87, 98,

101-106, 192, 193,216 sampling 97, 98 type 62, 94-96

fluidity 93 flux 152, 153, 155 Forcheimer 111 formation volume factor 90, 101,

102, 107, 112, 113, 150, 151, 163, 167, 203, 213,273,280

formations 25 fractional flow 131,134-139, 149 fracture 4, 50, 63, 116, 126, 222,

233 frequency, dominant 81 Fresnel

radius 83 zone 82

frontal advance 133, 134, 136, 138, 160, 169, 239, 241,251

full field model 168, 248 fully implicit 160, 161, 165, 169

gamma ray 56, 57, 60, 63-66

gas cap 194, 230, 256, 279 gas-oil 95, 101,123, 129, 147-149,

230,273,276 gas-oil ratio 92, 94, 98, 102, 229,

252 gas-water 92, 129, 159, 229 Gassmann 44 geologic model 15, 31, 80, 183,

184, 251 geology 18, 79,80, 183,200, 250 geometry 29, 61, 134, 214 geophysics 69, 78, 82-84 geostatistics 178, 182-184, 188,

190, 196 giga scale 6, 8, 52 gradient 48, 50, 109, 111, 118, 120,

132, 141,212, 243,276 grain

density 48, 50, 192 modulus 46-48

gravity drainage 276 grid 8, 80, 84, 112, 146, 147, 157,

159, 160, 164-167, 169, 174, 175, 180, 181,246 orientation 160 orthogonality 250 PEBI 250 upscaling 251 Voronoi 250

gridblock 160 gross thickness 1-3, 29, 32, 51, 57,

174, 179, 196

hand-drawn map 196 heating value 92 heavy oil 92, 95 heterogeneity 1, 4, 5, 19, 84, 118,

121,140, 183, 190 historical data 161, 251,252, 258,

271


history matching 15, 197, 245, 251-255,265

Honarpour 130, 147 Hooke's law 40 horizontal permeability 121, 122,

132, 192, 223 Homer

analysis 209, 218 plot 209

hysteresis 126

IFLO 162 igneous 23 immiscible 44, 122, 124-126, 133,

134, 141,273,277-278 IMPES 158, 160, 165, 166, 169 implicit 160, 161, 165, 169 incompressible 134, 139, 155, 156 induction log 62 infill 277-278 influx 151, 197, 230, 251,277 integration 7, 13, 15, 26, 114, 164,

251 interfacial tension 98, 122-124,

127 interstitial velocity 110 irreducible 130, 141, 147, 148 irrotational 72 isothermal 193,247

Jacobian 159

K value 89, 103 Klinkenberg 117 kriging 184, 188

laboratory measurements 43, 49, 108, 125, 127, 129, 133, 192, 210

lag 185

Lam6 41, 74 layer cake 146 lithology log 56 log

correlation 65 crossplot 65 cutoff 64 legacy 64 suite 66

logging 6, 33, 49, 52, 53, 63, 64, 66

LSE 31 LWD 63

Macleod-Sugden 123, 127 macro scale 6-8, 133 mapping 13, 30, 78, 170, 174, 178,

179, 188, 196 mass conservation 151, 153, 154 material

balance 138, 140, 150-152, 163, 164, 175,227, 237, 248, 252, 255

balance equation 151, 164 matrix 38, 40, 43-47, 58, 59, 66,

116, 119, 126, 153, 158, 159, 265

mega scale 6-8, 52, 221 metamorphic 23 micro scale 7, 8, 10, 133 microbial 278 miscibility 100, 101 miscible 273,277-278 mobility 130, 131,222, 230, 276-

277 mobility ratio 130, 131 mole fraction 89, 91, 92, 105, 123 molecular weight 91, 92, 96, 98,

123 momentum 12, 110, 161,246

John R. Fanchi 303

mud cake 53 multidisciplinary 1, 7, 13,264, 266 multilateral wells 278 MWD 63

naturally fractured 74, 76, 116, 153 Navier-Stokes equation 161 net present value 266 neutron 43, 58, 60, 63-66 neutron log 60 Newton-Raphson 158-160, 166 nonlinear 12, 114, 146, 205, 216,

249 normal distribution 9, 261,263 nugget 186 numerical

dispersion 160, 169 simulation 146, 251

objectives 14, 247, 248, 251,253, 259-260, 266, 268, 275

Ockham's Razor 265 offset (seismic) 79 oil productive capacity 77 oil-water 3, 67, 68, 124, 129, 140,

141,147-149, 159, 273

P-T diagram 95-96 parachor 123 partial differential equations 12,

146, 249 Pascal's law 90 performance predictions 11, 15,

16, 245,254, 255 permeability

anisotropic 121 defined 109 homogeneous 121 inhomogeneous 121 isotropic 121

tensor 119 vertical 121

petrophysics 33, 78, 174 phase

behavior 94, 95, 101, 129, 195, 214, 253

envelope 94, 96 potential 118

photoelectric log 63 Pickett crossplot 65 plate tectonics 19 Poisson's ratio 36-38, 41, 49, 51 pore

pressure 48 radius 125, 126 volume 1-3, 9, 64, 135, 139,

141,145,233,242, 257, 258, 273

porosity log 57 prediction 84, 198, 245,254, 255,

269, 280-281 pressure

buildup test 206 derivative 213 drawdown 218 maintenance 269, 277 transient testing 199

primary production 276 productivity index 166 pseudocomponent 103 pulse 71, 201,222 PVT 94, 100, 102, 107, 248

radial coordinates 112 flow 111, 112, 114, 166, 200,

202, 205, 210, 214, 215, 218, 241

radioactive tracer 235,239 radius of investigation 214, 220


real gas pseudopressure 216 realizations 122, 183, 184, 188-

190, 227, 245,248, 254, 265 recovery efficiency 151, 193,272,

274-275,280-281 reflection coefficient 75 regression 28, 48, 49, 105, 116,

123, 185 relative

mobility 131 permeability 108, 127-131, 146-

149, 163, 167, 197, 273 reliability 6, 190 repeat formation test 224 reserves 64, 140-147, 190, 227 reservoir

architecture 6, 29 characterization defined 170 description 12, 14, 15,227, 245,

251,257 engineering 53, 98, 126, 130,

195,229 geophysics 82-84 limits test 219 management defined 259 scale 8 simulation 49, 247, 251,254 structure 7, 11, 18, 69, 76, 170,

179, 182, 199, 214 resistivity 56, 57, 60-65, 67, 68 resistivity factor 61 resistivity log 60 resolution

lateral 82 vertical 81

Reynolds number 110, 111 risk 197, 255,268 rock

quality 77, 115

region 146 rock-fluid interaction 133, 147, 148

saturation constraint 3 defined 3 pressure 94

search ellipse 187 secondary recovery 277-278 sedimentary 23 seismic

inversion 76 methods 78, 82, 176 velocity 13, 191, 192 waves 69-71, 75, 191

semi-variance 185, 187 sensitivity 122, 148, 183, 189, 255,

256 separator test 100 shared earth model defined 7 shear

modulus 34, 36, 37, 41, 43, 45, 46, 48-51, 70, 76, 191, 192

velocity 42, 44, 46, 49, 75, 179, 191, 193,256

sill 185-186 sinusoidal 73 skin 210 slippage 117 solenoidal 73 sonic 53, 58, 64, 66 source/sink 153, 163 SP log 56 spacing 62, 277-278 SPE/WPC Reserves 261 specific gravity 91, 92, 107 spontaneous potential 56, 67 stability 160 stabilization time 219-220

John R. Fanchi 305

stabilized rate 206 standard deviation 9, 10, 185,262-

263 stiffness 43 stochastic 189 Stoke's equation 162 storage capacity 173 strain defined 34 stratigraphy 28 stress

defined 33 effective vertical 41

structures 26 subsidence 25, 29, 30, 35, 271 superficial velocity 110 superposition principle 206 surface model 246 sweep efficiency 133, 233, 241,

272-274, 278, 280 swelling test 100 symmetry 19, 152, 157, 159, 161,

167, 168

tank model 151 team 263-267 temperature scales 88 tensor 38-41, 119-121,250 tertiary production 277 thermal 60, 247, 256, 278 thickness, net and gross 3 three-phase relative permeability

129 time-lapse 190 timestep 160, 161, 169 tracer 197, 227, 228, 232-242, 244,

252 transgression 31 transient tests 122, 200, 201,204,

213, 214, 222, 223

transition zone 127, 133, 140-146 transmissibility 164, 165, 197 transmission coefficient 75 Traps 26 TSE 31

uncertainty 21, 84, 183, 184, 189, 190, 220, 254, 255

undersaturated 16, 102, 105, 107, 147, 151, 159, 196, 243

uniaxial compaction 49

validity 98, 110, 140, 156, 158, 182, 191,253-255, 257

valley fill geology 31 variance 185 variogram 185, 187 vertical conformance 274 viscosity

defined 92 dynamic 93 kinematic 93

visualization technology 178 volatile oil 95-97 volume element 134, 135, 171, 173 volumetric 86, 110, 112, 137, 230,

244, 272, 274, 280 VSP 79

water drive 140, 230, 231,276-277 waterflood 130, 131,137-140, 196-

198, 247, 255,273,280-281 water-oil ratio 147, 229, 252, 280 wave equation 72-73 wave propagation 71 wavelength

defined 70 dominant 81

Weinaug-Katz 123, 127


Welge 137 well

density 277-278 log defined 52 model 166 pattern 277 productivity 197, 254 spacing 277-278

test 6 wellbore storage 209, 211-214 wet gas 95, 99 wettability 122, 124-127, 130 workover 15, 281 Wyllie 59

Young's modulus 35-36, 49

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