a probabilistic approach to jointly integrate 3d/4d seismic

A PROBABILISTIC APPROACH

TO JOINTLY INTEGRATE 3D/4D SEISMIC,

PRODUCTION DATA AND GEOLOGICAL INFORMATION

FOR BUILDING RESERVOIR MODELS

A DISSERTATION

SUBMITTED TO THE DEPARTMENT OF ENERGY

RESOURCES ENGINEERING

AND THE COMMITTEE ON GRADUATE STUDIES

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

Scarlet A. Castro

June 2007

c© Copyright by Scarlet A. Castro 2007

All Rights Reserved

ii

I certify that I have read this dissertation and that, in my opinion, it

is fully adequate in scope and quality as a dissertation for the degree

of Doctor of Philosophy.

Dr. Jef Caers Principal Adviser




Dr. Gerald Mavko




Dr. Louis Durlofsky

Approved for the University Committee on Graduate Studies.

iii

Abstract

Reservoir modeling aims at understanding important static and dynamic components

of the reservoir in order to make decisions about the future of the surface operations.

The practice of reservoir modeling calls for the integration of expertise from differ-

ent disciplines, as well as the integration of a wide variety of data: geological data,

(core data, well-logs, interpretations, etc.), production data (fluid rates or volumes,

pressure data, etc.), and geophysical data (3D seismic data). Although a single 3D

seismic survey is the most common geophysical data available for most reservoirs, a

suite of several 3D seismic surveys (4D seismic data) acquired for production moni-

toring purposes can be available for mature reservoirs. The main contribution of this

dissertation is to incorporate 4D seismic data within the reservoir modeling workflow

while honoring all other available data.

This dissertation proposes two general approaches to include 4D seismic data

into the reservoir modeling workflow. The Probabilistic Data Integration approach

(PDI), which consists of modeling the information content of 4D seismic through a

spatial probability of facies occurrence; and the Forward Modeling (FM) approach,

which consists of matching 4D seismic along with production data. The reservoir

modeling workflow used in this dissertation, follows the “Parallel Modeling Approach”

of perturbing the high-resolution model directly, and also integrates geological and

production data using a probabilistic data integration approach.

The FM approach requires forward modeling the 4D seismic response, which re-

quires to downscale the flow simulation response. This dissertation introduces a novel

dynamic downscaling method that takes into account both static information (high-

resolution permeability field) and dynamic information in the form of coarsened fluxes

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and saturations (solution of the global flow simulation).

The two proposed approaches (PDI and FM approaches) are applied to a promi-

nent field in the North Sea, to model the channel facies of a fluvial reservoir. The PDI

approach constrained the reservoir model to the spatial probability of facies occur-

rence (obtained from a calibration between well-log and 4D seismic data) as well as

other static data while satisfactorily history matching only production data; however,

a probabilistic type of match is achieved rather than an quantitative match of the

4D seismic data. The FM approach achieved a partially good quantitative match on

both production and 4D seismic data; however, the history matching of two data of

very different support was considerably more challenging than the PDI approach.

When high quality 4D seismic data are not available or a complicated 4D seismic

forward modeling may not be carried out due to the lack of rock physics data, the

PDI approach may represent a more robust and less difficult to achieve alternative

to include 4D seismic information into the reservoir modeling workflow. However, if

high quality 4D seismic data are available, the FM approach (although practically

more challenging to apply) could help to understand better the dynamic behavior of

the reservoir as well as to identify valuable static information that may need to be

incorporated into the reservoir model.

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Acknowledgments

I would like to express my gratitude to the Stanford Center for Reservoir Forecasting

(SCRF) research consortium for providing the financial support during my PhD work

at Stanford.

I feel very lucky to have had Prof. Jef Caers as my adviser during all these

years. He is an outstanding professor and he represented an excellent guide to me as

I transitioned from Geophysics to Petroleum Engineering. I am truly grateful for his

support during these years.

I would like to thank Prof. Andre Journel for all his feedback during my presen-

tations at the SCRF seminars. It has been an honor and a privilege to have learned

Geostats from him.

I would like to thank Prof. Lou Durlofsky for carefully reading my thesis and also

for his guidance during my work on the flow-based downscaling procedure.

Also I would like to thank Prof. Tapan Mukerji for being a member of my com-

mittee, and for his guidance on the creation of the Stanford VI reservoir. He was

always willing to discuss and give his feedback on many geophysical questions I had

during my PhD work; I thank him also for that.

I would like to thank Prof. Gary Mavko for being in my reading committee and

Prof. Jerry Harris for chairing my committee.

Creating the Stanford VI reservoir would have not been possible without the useful

feedback I obtained from Prof. Stephan Graham and Dr. Darryl Fenwick. I thank

them for their time to discuss important geological and flow simulation aspects of

that work.

I am also grateful to Dr. Yuguang Chen who provided the 2D two-phase flow

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simulator used by the flow-based downscaling procedure, and also for very useful

discussions.

I appreciate Norsk Hydro and the Oseberg license partners (Statoil, Petoro, Cono-

coPhilips, ExxonMobil and Total) for providing the data I used on testing the two

proposed approaches for incorporating 4D seismic data into reservoir models. Having

the opportunity to work on this data set was invaluable for me; it was a very chal-

lenging project that taught me many things. I would like to thank Eli Zachariassen,

Cecilie Otterlei, Hilde Meisingset, Trond Høye, and Trond Andersen from the Norsk

Hydro Bergen’s Research Center who were always available and willing to help me

with data problems and also with their knowledge about the field. I thank the Norsk

Hydro Bergen’s Research Center for financially supporting part of my trip to Norway

for data collection.

I would also like to thank the ERE staff, especially Ginni Savalli who always help

me through Stanford bureaucratic paper work.

To all SCRF students I am also grateful. During my time at SCRF I always

received help from any SCRF student at any time. In particular I would like to thank

Todd Hoffman and Inanc Tureyen who helped me during my early years at SCRF,

Alex Boucher and Amisha Maharaja who shared with me many difficult times during

the academic life of the PhD.

To my dear officemates and above all extraordinary friends Jenya Polyakova, Lisa

Stright and Whitney Trainor I am deeply grateful. Jenya shared with me the chal-

lenging experience of the first year in the PhD program, I thank her for always being

there to listen. I shared multiple technical discussions about my research with Lisa,

she really knows the details of my work; I thank her for the good feedback and quality

time during those discussions. Also I would like to thank her and Whitney for their

help and sincere support during difficult times of my personal life.

I would like to thank Eddy Romero for convincing me to pursue my PhD in

Petroleum Engineering.

Last but not least I would like to thank my family in Venezuela, especially my

cousin Orglays for her support during tough times, and my parents for have given me

an excellent education and solid moral principles.

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Contents

Abstract v

Acknowledgments vii

1 Introduction 1

1.1 Reservoir Modeling and Data Integration . . . . . . . . . . . . . . . . 2

1.2 Incorporating 4D Seismic Data . . . . . . . . . . . . . . . . . . . . . 7

1.2.1 Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.2.2 Current Approaches . . . . . . . . . . . . . . . . . . . . . . . 14

1.2.3 Proposed Approach . . . . . . . . . . . . . . . . . . . . . . . . 18

1.3 Dissertation Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2 Two Workflows for Integrating 4D Seismic Data 21

2.1 Probabilistic Data Integration Approach . . . . . . . . . . . . . . . . 22

2.1.1 The Tau Representation . . . . . . . . . . . . . . . . . . . . . 24

2.1.2 The Probability Perturbation Method . . . . . . . . . . . . . . 28

2.2 Integrating 4D Seismic Data: PDI Approach . . . . . . . . . . . . . . 31

2.2.1 Modeling the Information Content of 4D Seismic Data . . . . 32

2.3 Integrating 4D Seismic Data: FM Approach . . . . . . . . . . . . . . 35

2.3.1 Modeling the 4D Seismic Response . . . . . . . . . . . . . . . 37

2.3.2 Applying the FM Approach to the Stanford V reservoir . . . . 43

3 Downscaling Saturation to Model 4D Seismic Response 52

3.1 State-of-the-Art Downscaling Methods . . . . . . . . . . . . . . . . . 54

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3.2 Flow-based Downscaling . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.2.1 Governing equations . . . . . . . . . . . . . . . . . . . . . . . 60

3.2.2 Flow on the high-resolution and coarsened grids . . . . . . . . 61

3.3 2D Synthetic Example . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4 Case Study: The Oseberg Field 68

4.1 Alpha North Segment - Upper Ness Formation . . . . . . . . . . . . . 71

4.2 The Reservoir Modeling Workflow . . . . . . . . . . . . . . . . . . . . 74

4.2.1 The High-resolution 3D Geocellular Model . . . . . . . . . . . 74

4.2.2 The 3D Coarsened Model . . . . . . . . . . . . . . . . . . . . 76

4.2.3 The Flow-Simulation Model . . . . . . . . . . . . . . . . . . . 81

4.2.4 Establishing the History Matching Procedure . . . . . . . . . 83

4.3 Integrating 4D Seismic Data: PDI Approach . . . . . . . . . . . . . . 97

4.4 Integrating 4D Seismic Data: FM Approach . . . . . . . . . . . . . . 106

4.4.1 Forward Modeling the 4D Seismic Response . . . . . . . . . . 107

4.4.2 History Matching Results . . . . . . . . . . . . . . . . . . . . 116

5 Conclusions and Future Research 142

A The Stanford VI Reservoir 151

A.1 Structure and Stratigraphy . . . . . . . . . . . . . . . . . . . . . . . . 155

A.2 Petrophysical Properties . . . . . . . . . . . . . . . . . . . . . . . . . 162

A.2.1 Simulation of Porosity . . . . . . . . . . . . . . . . . . . . . . 162

A.2.2 Simulation of Permeability . . . . . . . . . . . . . . . . . . . . 163

A.2.3 Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

A.2.4 P-wave and S-wave Velocities . . . . . . . . . . . . . . . . . . 167

A.2.5 Fluid Substitution . . . . . . . . . . . . . . . . . . . . . . . . 171

A.3 Seismic Attributes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

A.3.1 Mathematical Expressions . . . . . . . . . . . . . . . . . . . . 178

A.3.2 Computation of Seismic Attributes . . . . . . . . . . . . . . . 182

A.4 Reservoir Flow Simulation . . . . . . . . . . . . . . . . . . . . . . . . 189

A.4.1 Upscaling of the Reservoir Model . . . . . . . . . . . . . . . . 190

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A.4.2 Flow Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 191

A.5 4D Seismic Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

B Snesim Parameter File 215

C Results of the PDI Approach 218

Bibliography 237

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List of Tables

4.1 Variograms used for simulating porosity and permeability for each fa-

cies; ranges are shown in meters. . . . . . . . . . . . . . . . . . . . . . 76

4.2 Density and bulk modulus of the fluids in the Upper Ness reservoir, as

a function of pore pressure (from Norsk Hydro). . . . . . . . . . . . . 108

4.3 Reservoir conditions and other gas, water and oil properties (from

Norsk Hydro). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

4.4 P and S-wave velocities as a function of porosity, for a series of effective

pressures (from Norsk Hydro). . . . . . . . . . . . . . . . . . . . . . . 109

4.5 Summary of the behavior of the Objective Function using the FM

approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

4.6 Summary of the matching of production data using the FM approach. 133

4.7 Summary of the matching of 4D seismic data using the FM approach. 133

A.1 Stratigraphic parameters used for the simulation of the facies model

for layer 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157


for layer 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158


for layer 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

A.4 Categories for affinity and angle rotation used for simulating the facies

model for layer 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

A.5 Variogram used for simulating porosity for each facies. . . . . . . . . 164

A.6 κ-variogram used for simulating permeability for each facies. . . . . . 166

A.7 Rock mineralogy for each facies. . . . . . . . . . . . . . . . . . . . . . 167

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A.8 Constant cement model input parameters used in the Stanford VI reser-

voir. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

A.9 Properties of water and oil obtained using Batzle & Wang [8] relations. 174

A.10 Percentage of random noise added to each property, per facies. . . . . 174

A.11 Parameters used to define the “surface seismic” filter. . . . . . . . . . 183

A.12 Oil and Water PVT properties. . . . . . . . . . . . . . . . . . . . . . 192

A.13 Summary of the production schedule. . . . . . . . . . . . . . . . . . . 196

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List of Figures

1.1 Step-by-step workflow for building a high-resolution geo-cellular model

(from Caers [16]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 High resolution model (left) which only matches static data; its flow

response does not match historical water cut at the producer wells. . 4

1.3 Coarsened model (from Figure 1.2) being manually perturbed until

historical water cut at the producer wells is matched. . . . . . . . . . 5

1.4 A facies realization being drawn from a joint conditional probability

distribution which gathers information from all data sources (well-log,

geological information and seismic data) about the unknown sand facies

A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.5 The “Parallel Modeling Approach” applied to the initial facies model

(shown in Figure 1.2); where the high-resolution model is perturbed

in a geologically consistent fashion (using PPM) until historical water

cut at the producer wells is matched. . . . . . . . . . . . . . . . . . . 9

1.6 Geologically consistent perturbation of facies using PPM while honor-

ing all other available data. Perturbations are done iteratively on the

conditional probability from which a model is drawn, rather then on

the model itself. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

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1.7 Change in Vp/Vs versus change in acoustic impedance for each of the

two time-lapse responses: the 1992-1999 time-lapse response (shown in

green) and the 1999-2004 response (shown in red). A change greater

than one means an increase in the attribute over time and a change

smaller that one means a decrease in the attribute over time. The

center of the figure means no change on any of the two attributes. . . 13

2.1 Computation of P (A | D1) for a single grid-block in sequential sim-

ulation with a training image model. The neighboring data (termed

“data event”) near a randomly visited grid-block is extracted. Repli-

cates of this data event are searched for in the training image. The

probability of the central grid-block to be in channel facies (sand) can

be calculated from the set of replicates (modified from Caers [16]). . . 27

2.2 Training image (left), conditional probability P (A | D2) from seismic

data (middle), and an initial realization i0(u) (right). . . . . . . . . . 30

2.3 Realizations simulated using the joint conditional probability distrib-

ution P (A | D1, D2, D3) for rD values of 0.1, 0.3, 0.5, 0.7 and 1. . . . 30

2.4 Reservoir modeling workflow that incorporates the 4D seismic data

through a spatial probability distribution. . . . . . . . . . . . . . . . 32

2.5 Crossplot of Vp/Vs ratio versus acoustic impedance for a typical well

in the field (left). Several logs for this well are shown on the right:

acoustic impedance, gamma-ray, and Vp/Vs. The data points are col-

ored by depth; however, several other color-filled polygons are superim-

posed which correspond to classes defined on the gamma-ray log. The

sands in reservoir are shown inside the red polygon, with low Vp/Vs

and low acoustic impedance (from Andersen et al. [3]). . . . . . . . . 33

2.6 Conceptual sketch of the Base 3D Seismic classification. The green

data points have been obtained from the 3D seismic data, while the

polygons outline the classification of sands into three classes: 5, 10,

and 15 (modified from Andersen et al. [3]). . . . . . . . . . . . . . . . 34

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2.7 Crossplot of the change (time-lapse/base) in Vp/Vs versus the change in

acoustic impedance obtained from the 4D response (left). The center

of this crossplot corresponds to “no change”; however a change to

each quadrant is classified as a “physical phenomenon” occurring over

time (shown on the corners). The same crossplot (shown on the right)

shows the polygons outlining the classification of sand (modified from

Andersen et al. [3]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.8 Table showing the recording of the classes in the combined 3D and 4D

volumes. The figure to the right shows a smoothed sand probability for

the combined volumes. The pink curve shows probabilities for sand and

the red curve shows probabilities for no-sand lithologies(from Andersen

et al. [3]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.9 Reservoir modeling workflow that incorporates 4D seismic data by his-

tory matching it along with production data. . . . . . . . . . . . . . . 37

2.10 General procedure and input data needed for creating the Time-lapse

acoustic impedance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.11 Petro-elastic model to create Vp, Vs and ρ, taking into account pore

pressure and fluid saturation effects. . . . . . . . . . . . . . . . . . . . 41

2.12 3D reference reservoir facies model, shown in depth slices from top (top

left) to the bottom of the reservoir (bottom right). Floodplain facies

in blue, channel facies in green, crevasse facies in red. . . . . . . . . . 44

2.13 Spatial distribution of water saturation in the reservoir after 6 months

of production, as obtained from flow simulation using the reference

model. Shown in depth slices from top (top left) to the bottom of the

reservoir (bottom right). . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.14 4D seismic response (difference average instantaneous amplitude map)

from reference model. . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.15 A slice of the 3D training image used for the facies modeling. Flood-

plain facies in blue, channel facies in green, crevasse facies in red. . . 47

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2.16 The spatial probability distribution of each facies (form left to right:

crevasse, floodplain, channel) obtained from the calibration between

well data and the reference 3D seismic data. . . . . . . . . . . . . . . 47

2.17 3D initial guess model, shown in depth slices from top (top left) to

the bottom of the reservoir (bottom right). Floodplain facies in blue,

channel facies in green, crevasse facies in red. . . . . . . . . . . . . . . 48


of production, as obtained from flow simulation using the initial guess

model. Shown in depth slices from top (top left) to the bottom of the

reservoir (bottom right). . . . . . . . . . . . . . . . . . . . . . . . . . 48


from initial guess reference model. . . . . . . . . . . . . . . . . . . . . 49

2.20 3D best reservoir model found by the history matching algorithm,

shown in depth slices from top (top left) to the bottom of the reser-

voir (bottom right). Floodplain facies in blue, channel facies in green,

crevasse facies in red. . . . . . . . . . . . . . . . . . . . . . . . . . . . 50


of production, as obtained from flow simulation using the best model

found by the history matching algorithm. Shown in depth slices from

top (top left) to the bottom of the reservoir (bottom right). . . . . . . 50


from best model found by the history matching algorithm . . . . . . . 51

2.23 A slice of the water saturation cube after 6 months of production: from

reference model (left), from initial guess model (middle), from history

matched model (right). . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.1 Traditional approach of sub-sampling the single coarse grid block sat-

uration S(v) to all high-resolution grid cells u. . . . . . . . . . . . . . 54

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3.2 Static downscaling of coarse grid block saturation S(v) through a re-

scaling of sub-grid porosity values Φ(u) to saturation values S(u), while

imposing the average of all sub-grid saturation values within the coarse

block to be equal to the coarse grid block saturation. . . . . . . . . . 55

3.3 Comparing well-log and simulator properties: the red curves corre-

spond to the flow simulator, while the blue curves correspond to the

well logs. Left to right: φ =porosity, k =permeability, Sw =water satu-

ration, and Sg/o =simulator gas saturation and well-log oil saturation.

After Sengupta [75]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.4 Downscaling saturations from the flow simulator: (a): Sg taken from

the simulator, (b), (c), (d), (e), (f): Estimations of downscaled Sg.

After Sengupta [75]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.5 Cross plot of time-lapse differential AVO attributes from real data

around the well, and from synthetics corresponding to smooth and

downscaled saturation profiles. The error bars represent the uncer-

tainty in synthetic seismic attributes due to the lack of information

about spatial distribution and total amount of gas. Modified from

Sengupta [75]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.6 Domains for flow on the coarsened and local high-resolution grids.

Lighter lines represent the high-resolution grid and heavier lines the

coarse grid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.7 High-resolution permeability model (isotropic) for a layered reservoir.

The axe correspond to the grid block number. . . . . . . . . . . . . . 64

3.8 Coarsened effective permeability model (anisotropic) for the layered

reservoir: kx (left) and kz (right). The axe correspond to the grid

block number. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.9 Saturation profiles (top row) and corresponding forward-modeled 4D

seismic responses (bottom row): coarse scale (left), flow-based down-

scaled (middle), reference high-resolution (right). . . . . . . . . . . . 66

xviii

3.10 Two 4D seismic traces extracted from the modeled 4D seismic re-

sponses obtained using: the coarse scale saturation map (top left),

the downscaled saturation map (top middle), and the high-resolution

saturation map (top right). The seismic trace on the bottom left cor-

responds to a distance of 1000 meters, and the seismic trace on the

bottom right corresponds to a distance of 3500 meters. . . . . . . . . 67

4.1 Structural overview of the northern North Sea and its oil fields. Detail

of the location of the Oseberg field (modified from Smethurst [77]). . 69

4.2 Outline of the Oseberg Field and its major fault blocks: Alpha, Alpha

North, and Gamma (modified from Johnstad et al. [47]). . . . . . . . 72

4.3 West-East Seismic cross-section from the Alpha North segment, show-

ing the top and base of the Brent Group and the Base Cretaceous

erosion (modified from Rutledal et al. [71]. . . . . . . . . . . . . . . . 73

4.4 Conceptual overview of the four-partite compartmentalization (from

top to the bottom of the reservoir) of the channel development of the

UN on Alpha North (modified from Liestøl et al. [55]). . . . . . . . . 73

4.5 Well-log data available for the UN on Alpha North: binary facies clas-

sification (top left), where zero represents floodplain (red) and one

represents channel facies (blue); porosity (top right) and the base 10

logarithm of permeability (bottom). . . . . . . . . . . . . . . . . . . . 77

4.6 Binary training image showing channels facies in blue and floodplain

facies in gray (provided by Norsk Hydro). . . . . . . . . . . . . . . . . 77

4.7 Channel facies probability cubes provided by Norsk Hydro. The prob-

ability cube shown on the left has been obtained from a calibration

between well-log data and the elastic inversion of the Base 3D seismic

survey (acquired in 1992). The probability cube shown on the right

has been obtained from a calibration between well-log data and the

elastic inversion of the 4D seismic data (surveys acquired in 1992, 1999

and 2004) using the procedure presented by Andersen et al. [3]. . . . 78

4.8 Vertical sand proportion curve (modified from Andersen et al. [3]). . 78

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4.9 Regions of the reservoir where the training image is rotated for simu-

lation purposes; the rotation angle is measured clockwise with respect

to the North. In the top region, shown in blue, the angle of rotation

corresponds to 0◦. In the bottom region, shown in red, the angle of

rotation corresponds to 72◦. . . . . . . . . . . . . . . . . . . . . . . . 79

4.10 Three conditional facies realizations obtained with the “single normal

equation simulation” (snesim) algorithm. Channels facies is depicted

in blue and floodplain facies in gray. . . . . . . . . . . . . . . . . . . . 79

4.11 Porosity realization, simulated first using “sequential Gaussian simu-

lation” (sgsim), for the floodplain facies (left) and the channel facies

(middle). Using a “cookie-cut” approach, the final porosity realization

is obtained (right). The facies realization used for “cookie-cutting” is

the first one (left) shown in Figure 4.10. . . . . . . . . . . . . . . . . 79

4.12 Permeability realization, co-simulated using “sequential Gaussian co-

simulation” (sgcosim), for the floodplain facies (top left) and the chan-

nel facies (middle). Using a “cookie-cut” approach, the final perme-

ability realization is obtained (right); permeability values are shown

as log10(perm). The facies realization used for “cookie-cutting” is the

first one (left) shown in Figure 4.10. . . . . . . . . . . . . . . . . . . . 80

4.13 View of the flow simulation grid (top), colored by oil saturation, and

the active wells in the flow simulation (bottom). Note the two dedi-

cated, long horizontal UN wells. . . . . . . . . . . . . . . . . . . . . . 81

4.14 Oil and water relative permeability curves for regions where only oil

and water are present (left). Oil and gas relative permeability curves

for regions where only oil, gas and connate water are present (right). . 82

4.15 Drainage water-oil capillary pressure. . . . . . . . . . . . . . . . . . . 82

4.16 Historical field production data from the Upper Ness formation in Al-

pha North. Field pressure shown in black, total field oil production

shown in green and total field water production shown in blue. . . . . 85

xx

4.17 Historical field injection data from the Upper Ness formation in Alpha

North. Total field gas injection shown in red and total field water

injection shown in blue. . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.18 Historical production data from well C-19. Total oil production shown

in green and total water production shown in blue. . . . . . . . . . . 86

4.19 Historical production data from well C-17D. Total oil production shown

in green and total water production shown in blue. . . . . . . . . . . 86

4.20 Results of the elastic inversion of the three seismic surveys of the Alpha

North segment. The top row shows Acoustic Impedance at 1992, 1999

and 2004 respectively. The bottom row shows the Vp/Vs ratio at 1992,

1999 and 2004 respectively. . . . . . . . . . . . . . . . . . . . . . . . . 87

4.21 4D seismic data shown as the ratio between the seismic attribute ob-

tained from the new survey and the same attribute obtained from the

old survey. The top row shows the AI ratios [AI]1999/[AI]1992 and

[AI]2004/[AI]1999 respectively. The bottom row shows the Vp/Vs ratios

[Vp/Vs]1999/[Vp/Vs]1992 and [Vp/Vs]2004/[Vp/Vs]1999 respectively. . . . . . 88

4.22 Summary and classification of the 4D field response (modified from

Andersen et al. [3]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.23 Slices of the classified field 4D seismic response from the Alpha North

segment (Upper Ness Formation), from the top to the bottom of the

reservoir (slices shown from left to right). The first row of slices rep-

resents the 4D response between the years 1992 and 1999 (top); the

second row of slices represents the 4D response between the years 1999

and 2004. The colors represent the classes interpreted in Figure 4.22. 93

4.24 Summary and classification of the 4D field response. The circles rep-

resent the percentage of noise (undistinguishable change) used in eval-

uating the reliability of the seismic response in certain areas of the

reservoir (modified from Andersen et al. [3]). . . . . . . . . . . . . . . 94

xxi

4.25 Slices of the classified field 4D seismic response between 1992 and 1999,

from the top to the bottom of the reservoir (slices shown from left to

right). The white areas represent the data points inside the noise

circle and the blue areas represent the data points outside the noise

circle shown in Figure 4.24. Hence, white represents areas that may

not allow discriminating noise from a physical response. Each row of

slices represents the classification after using 2%, 4% and 6% of noise,

respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.26 Slices of the classified field 4D seismic response between 1999 and 2004,

from the top to the bottom of the reservoir. The white areas represent

the data points inside the noise circle and the blue areas represent

the data points outside the noise circle shown in Figure 4.24. Hence,

white represents areas that may not allow discriminating noise from a

physical response. Each row of slices represents the classification after

using 2%, 4% and 6% of noise, respectively. . . . . . . . . . . . . . . 96

4.27 Vertical proportion maps of (from left to right) scenario 1: increase in

pore pressure, scenario 2: waterflooding, scenario 3: decrease in pore

pressure, and scenario 4: gasflooding. The data shown corresponds

only to the areas where the 4D seismic signal is classified as reliable

(using 4% of noise). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

4.28 Initial guess of the high-resolution facies model (left) used as the start-

ing point for the probability perturbation method. High-resolution

facies model (right) obtained after history matching production data

(total oil and water production) from wells C-19 and C-17D. The chan-

nel facies is shown in blue, and the floodplain facies is shown in gray. 100

4.29 Objective function vs. number of flow simulations. The blue curve

shows the value of the objective function for each inner iteration; the

red curve shows the value of the objective function for each outer iter-

ation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

xxii

4.30 Total oil (top) and water production (bottom) from well C-19. His-

torical data is shown in black, the simulated total oil production from

the initial guess model is shown in magenta, and the best match ob-

tained after several flow simulations is shown in green (on the total oil

production plot), and blue (on the total water production plot). . . . 101

4.31 Total oil (top) and water production (bottom) from well C-17D. His-





4.32 Total oil (top) and water production (bottom) from well C-19. Histor-

ical data is shown in black, the other colors represent the best match

obtained for six reservoir models. . . . . . . . . . . . . . . . . . . . . 103


torical data is shown in black, the other colors represent the best match

obtained for six reservoir models. . . . . . . . . . . . . . . . . . . . . 104

4.34 E-type (ensemble average) generated from the six history-matched

reservoir models, using the PDI Approach. . . . . . . . . . . . . . . . 105

4.35 Properties of the fluids in the Upper Ness reservoir as a function of

pore pressure (from Norsk Hydro). . . . . . . . . . . . . . . . . . . . 108

4.36 Seismic velocities of the dry rock as a function of porosity and effective

pressure. Relationships obtained in the lab from measurements made

on cores from the Oseberg Øst field (from Norsk Hydro). . . . . . . . 109

4.37 Seismic velocities of the dry rock as a function of porosity and effective

pressure obtained for Oseberg Øst (black lines) on top of dry rock ve-

locities obtained from Ness well-logs. The color code represents depth

in meters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

4.38 Calibrated seismic velocities of the dry rock as a function of porosity

and effective pressure for the Alpha North Ness formation (black lines)

on top of dry rock velocities obtained from Ness well-logs. The color

code represents depth in meters. . . . . . . . . . . . . . . . . . . . . . 112

xxiii

4.39 Slices of the classified modeled 4D seismic response from the Alpha

North segment (Upper Ness Formation), from the top to the bottom

of the reservoir (slices shown from left to right). The colors represent

the classes interpreted in Figure 4.22. . . . . . . . . . . . . . . . . . . 114

4.40 Vertical proportion maps of scenario 3 (decrease in pore pressure) and

scenario 4 (gasflooding), obtained from the classified volume of the

modeled 4D response between 1992-1999 (shown in Figure4.39). . . . 114

4.41 Slices of the flow simulation results used for modeling the 4D response

shown in Figure 4.39 (top); map of the location of injector and pro-

ducer wells (bottom). The results are shown as differences between the

simulation result at year 1999 and the simulation result at year 1992. 115



facies model (right) obtained after history matching both production

data (cumulative oil and water production from wells C-19 and C-

17D) and 4D seismic data (proportion maps of scenarios 3 and 4). The

channel facies is shown in blue, and the floodplain facies is shown in

gray. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119




ation; the black curve shows the production mismatch for each outer

iteration; the green curve shows the 4D seismic mismatch for each outer

iteration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119






xxiv






4.46 Vertical proportion maps of scenario 3 (decrease in pore pressure)

shown on the top row and scenario 4 (gasflooding) shown on the bot-

tom row. From left to right: the map obtained from the initial guess

reservoir model, the observed map (field data) and the map obtained

from the reservoir model that best matched both production and 4D

seismic data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122







gray. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125





iteration; the green curve shows the value 4D seismic mismatch for

each outer iteration. . . . . . . . . . . . . . . . . . . . . . . . . . . . 125






xxv











seismic data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

4.52 Vertical proportion maps of scenario 4 (gasflooding); From left to right:

the map obtained from the reservoir model that best matches both

production and 4D seismic data using P (A | D3) only in PPM, the

observed map (field data) and the map obtained from the reservoir

model that best matches both production and 4D seismic data using

P (A | D3) and P (A | D4) in PPM. The red circle indicates an area of

considerable mismatch. . . . . . . . . . . . . . . . . . . . . . . . . . . 129

4.53 Vertical average of gas saturation Sg from the flow simulation per-

formed on the best matched model shown in Figure 4.47. From left to

right, snapshots for the years 1992, 1993, 1995, 1997, 1998 and 1999. . 129

4.54 Vertical average of ∆Sg from the flow simulation performed on the

best matched model shown in Figure 4.47. From left to right, ∆Sg is

obtained from subtracting the gas saturation at years 1993, 1995, 1997,

1998 and 1999 from the gas saturation at year 1992. . . . . . . . . . . 130

4.55 Vertical average of ∆Sg from the flow simulation performed on the best

matched model shown in Figure 4.47. From left to right, ∆Sg is the

incremental difference over the years 1993, 1995, 1997, 1998 and 1999. 130

xxvi







gray. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134






















seismic data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

xxvii
















seismic data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140







gray. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141







A.1 Workflow followed to create the Stanford VI data set. . . . . . . . . . 154

xxviii

A.2 Perspective view of the Stanford VI top structure: view from SW (left),

view from SE (right). The color indicates the depth to the top. . . . . 155

A.3 Perspective view of the Stanford VI top and bottom of each of its

layers. The color indicates the depth to the top. . . . . . . . . . . . . 156

A.4 Facies model of Layer 1, which corresponds to sinuous channels: flood-

plain (navy blue), point bar (light blue), channel (yellow), and bound-

ary (red). Stratigraphic grid (left), and cartesian box (right). . . . . . 157

A.5 Facies model of Layer 2, which corresponds to meandering channels:

floodplain (navy blue), point bar (light blue), channel (yellow), and

boundary (red). Stratigraphic grid (left), and Cartesian box (right). . 158

A.6 Facies model of Layer 3 (top), which corresponds to deltaic deposits:

floodplain (navy blue), and channel (yellow). Stratigraphic grid (left),

Cartesian box (right), angle cube (middle), and affinity cube (bottom). 160

A.7 Training Image used for modeling Layer 3. The size of the training

image is 200 × 200 × 5, each slice in the z − direction is shown here

from top to bottom. . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

A.8 Distribution of porosity for each facies in the reservoir. . . . . . . . . 162

A.9 Histogram of porosity for each facies in the reservoir. . . . . . . . . . 163

A.10 Resulting Porosity cube after cookie-cut porosity from each facies’

porosity realization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

A.11 Histogram of the logarithm of permeability, per facies in the reservoir. 165

A.12 Resulting Permeability cube after cookie-cutting permeability from

each facies’ permeability realization. . . . . . . . . . . . . . . . . . . . 166

A.13 Cementing versus sorting trends. . . . . . . . . . . . . . . . . . . . . 168

A.14 P-wave velocity vs. porosity for shales and brine-saturated sandstones. 172

A.15 Resulting density (top), Vp (middle) and Vs (bottom) cubes for the

oil-saturated reservoir. . . . . . . . . . . . . . . . . . . . . . . . . . . 175

A.16 Petrophysical properties crossplots. From left to right: P-wave velocity

vs. porosity, P-wave velocity vs. density, S-wave velocity vs. P-wave

velocity, and porosity vs. density. . . . . . . . . . . . . . . . . . . . . 176

xxix

A.17 P-wave hitting a reflector. The physical properties are different on

either side of the reflector. . . . . . . . . . . . . . . . . . . . . . . . . 179

A.18 Seismic wavefront hitting a reflector. The physical properties are dif-

ferent on either side of the reflector. The part of the P wave striking

at a particular angle-of-incidence (represented by a ray) will have its

energy divided into reflected and transmitted P and S waves. . . . . . 180

A.19 Seismic attributes at the Geostatistical Scale: Acoustic Impedance,

Elastic Impedance, S-wave Impedance, Poisson’s Ratio, Lame coeffi-

cients λ µ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

A.20 Seismic attributes crossplots. From left to right: Acoustic impedance

vs. porosity, Elastic impedance vs. porosity, and Acoustic impedance

vs. Elastic impedance. . . . . . . . . . . . . . . . . . . . . . . . . . . 185

A.21 Seismic attributes crossplots. From left to right: Poisson’s Ratio vs.

Acoustic impedance, Poisson’s Ratio vs. Elastic impedance, and Pois-

son’s Ratio vs. porosity. . . . . . . . . . . . . . . . . . . . . . . . . . 185

A.22 Seismic attributes crossplots. From left to right: S-wave impedance

vs. porosity, S-wave impedance vs. Elastic impedance, and Poisson’s

Ratio vs. S-wave impedance. . . . . . . . . . . . . . . . . . . . . . . . 186

A.23 Lame coefficients λ vs. µ. . . . . . . . . . . . . . . . . . . . . . . . . 186

A.24 AVO Intercept vs. Gradient for oil and brine-saturated sandstones. . 187

A.25 Seismic attributes at the Seismic Scale: Acoustic Impedance, Elastic

Impedance, S-wave Impedance, Poisson’s Ratio, Lame coefficients λ µ,

AVO attributes Intercept and Gradient. . . . . . . . . . . . . . . . . . 188

A.26 Porosity at the high-resolution (left), linearly averaged porosity after

upscaling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

A.27 Effective Permeability after upscaling: kx (top left), ky (top right), kz

(bottom). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

A.28 Oil and Water Relative Permeability curves. . . . . . . . . . . . . . . 192

A.29 Location maps of producer wells (left), and injector wells (right). The

color represents horizon top depth (ft). . . . . . . . . . . . . . . . . . 193

xxx

A.30 Field rates history: Aquifer water influx rate (red line), Oil production

rate (green line), Water injection rate (blue line), Water production

rate (cyan line) and Reservoir pressure (black dotted line). . . . . . . 194

A.31 Field cumulative history: Cumulative aquifer water influx (red line),

cumulative oil production (green line), cumulative water injection (blue

line) and cumulative water production (cyan line). . . . . . . . . . . . 195

A.32 3D view of the reservoir before oil production starts (top), and 30

years after production started (bottom). The color bar represents oil

saturation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

A.33 Constant X = 6151 ft North-South slice of the reservoir before oil

production starts, 10, 20 and 30 years after oil production started. . . 197

A.34 Constant Y = 410ft East-West slice of the reservoir before oil produc-

tion starts, 10, 20 and 30 years after oil production started. . . . . . . 198

A.35 Horizon slice at 100 meters below the top of the reservoir before oil


A.36 Porosity of the “pseudo” high-resolution reservoir model. . . . . . . . 200

A.37 Effective Permeability of the “pseudo” high-resolution reservoir model:

kx (top left), ky (top right), kz (bottom). . . . . . . . . . . . . . . . . 201

A.38 Field rates history: Aquifer water influx rate (red line), Oil production

rate (green line), Water injection rate (blue line), Water production

rate (cyan line) and Reservoir pressure (black dotted line). . . . . . . 202

A.39 Field cumulative history: Cumulative aquifer water influx (red line),

cumulative oil production (green line), cumulative water injection (blue

line) and cumulative water production (cyan line). . . . . . . . . . . . 202

A.40 3D view of the reservoir before oil production starts (top), and 30

years after production started (bottom). The color bar represents oil

saturation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

A.41 Constant X = 6151 ft North-South slice of the reservoir before oil


A.42 Constant Y = 410ft East-West slice of the reservoir before oil produc-

tion starts, 10, 20 and 30 years after oil production started. . . . . . . 205

xxxi

A.43 Horizon slice at 100 meters below the top of the reservoir before oil


A.44 Water cut versus time for well P21: solution from “pseudo” high-

resolution model (red), and solution from upscaled model (blue). Wa-

ter saturation 24 years after oil production started: solution from

“pseudo” high-resolution model (middle), and solution from the up-

scaled model (bottom). . . . . . . . . . . . . . . . . . . . . . . . . . . 207

A.45 Base seismic data set acquired prior to oil production (top left), seismic

data sets acquired after 10 years of oil production (top right), after 25

years of oil production (bottom left), after 30 years of oil production

(bottom right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

A.46 Workflow used to create the 4D seismic response at different times

during oil production. . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

A.47 Water saturation from upscaled model after 10 (top left), 25 (top mid-

dle) and 30 (top right) years of oil production. Seismic amplitude

difference from upscaled model for 10 (middle left), 25 (middle mid-

dle) and 30 (middle right) years of oil production. Seismic amplitude

incremental difference from upscaled model for 10 (bottom left), 25

(bottom middle) and 30 (bottom right) years of oil production. . . . . 213

A.48 Water saturation from “pseudo” high-resolution model after 10 (top

left), 25 (top middle) and 30 (top right) years of oil production. Seis-

mic amplitude difference from “pseudo” fine scale model for 10 (middle

left), 25 (middle middle) and 30 (middle right) years of oil produc-

tion. Seismic amplitude incremental difference from “pseudo” high-

resolution model for 10 (bottom left), 25 (bottom middle) and 30 (bot-

tom right) years of oil production. . . . . . . . . . . . . . . . . . . . . 214

B.1 Main window of the program S-GeMS showing the grids and proper-

ties used in the snesim parameter file, the algorithm panel where the

parameters can be manually input, and the visualization panel with a

simulated facies realization. . . . . . . . . . . . . . . . . . . . . . . . 217

xxxii

C.1 History Match # 1. Total oil (top) and water production (bottom)

from well C-19. Historical data is shown in black, the simulated total oil

production from the initial guess model is shown in magenta, and the

best match obtained after several flow simulations is shown in green (on

the total oil production plot), and blue (on the total water production

plot). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

C.2 Total oil (top) and water production (bottom) from well C-17D. His-





C.3 History Match # 1. Initial guess of the high resolution facies model

(left) used as the starting point for the probability perturbation method.

High resolution facies model (right) obtained after history matching

production data (cumulative oil and water production) from wells C-

19 and C-17D. The channel facies is shown in blue, and the floodplain

facies is shown in gray. . . . . . . . . . . . . . . . . . . . . . . . . . . 221

C.4 History Match # 1. Objective function vs. number of flow simulations.

The blue curve shows the value of the objective function for each inner

iteration; the red curve shows the value of the objective function for







plot). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

xxxiii


from well C-17D. Historical data is shown in black, the simulated to-

tal oil production from the initial guess model is shown in magenta,

and the best match obtained after several flow simulations is shown in

green (on the total oil production plot), and blue (on the total water

production plot). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
















plot). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225






production plot). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

xxxiv
















plot). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228






production plot). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229







xxxv










plot). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231






production plot). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232











xxxvi






plot). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234






production plot). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235



High resolution facies model obtained after history matching produc-

tion data (cumulative oil and water production) from wells C-19 and

C-17D. The channel facies is shown in blue, and the floodplain facies

is shown in gray. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236





xxxvii

xxxviii

Chapter 1

Introduction

Creating a reservoir model is becoming common practice during several stages of

a reservoir’s life. From exploration to field abandonment, reservoir modeling aims

at understanding and predicting important geological, geophysical and engineering

components of the reservoir.

Our knowledge about the properties of the reservoir changes from one stage to the

other as more data becomes available. During the early exploration stages we may

want to estimate the OOIP in the reservoir; however, during later production stages

we may want to forecast production for the next few years, or plan new wells and/or

surface facilities.

Reservoir modeling calls for the integration of expertise from different disciplines,

as well as the integration of data from various sources. Each type of data provides

information about the reservoir heterogeneity on a different scale; therefore, they have

different degrees of accuracy and may be redundant with each other to certain degrees.

The reservoir model needs to simultaneously (not hierarchically) honor all available

data, both static and dynamic, in order to preserve its predictive capabilities.

Static data includes all data that have been measured or interpreted once in time;

such as:

• Core data: porosity, permeability, relative permeability, wave velocities, etc.

• Well-log data: any suite of logs that indicate lithology and fluid types near the

well-bore.

1

2 CHAPTER 1. INTRODUCTION

• Outcrop analog data.

• Sedimentological and stratigraphic interpretation.

• Stratigraphic horizons and faults interpreted from 3D seismic data.

• Seismic attributes.

• Rock physics data.

• PVT data.

On the contrary, dynamic data includes all data that have been measured or inter-

preted over time; such as:

• Production data: Fluid rates or volumes, pressure data.

• 4D seismic data: any suite of 3D seismic attributes computed from each seismic

survey.

1.1 Reservoir Modeling and Data Integration

The state-of-the-art practice of reservoir modeling starts by creating a high resolution

3D geo-cellular model using static data. A hierarchical approach to build the 3D geo-

cellular model is presented by Caers [16] and shown in Figure 1.1. The steps are as

follows:

1. Establish the architecture of the reservoir in terms of horizons and faults that

are determined from 3D seismic data and well-markers.

2. Build a 3D stratigraphic grid from the structural framework.

3. Build a Cartesian grid from the stratigraphic grid. This grid ideally represents

the coordinate system for the original depositional environment. A one-to-one

relationship is established between each grid-cell in the Cartesian grid and in

the stratigraphic grid. All data, well-paths, well-logs and 3D seismic data is

imported in that Cartesian grid.

4. Populate the Cartesian grid with facies rock types. Outcrop data and sedimen-

tological models provide information on the style of facies architecture; well-log,

1.1. RESERVOIR MODELING AND DATA INTEGRATION 3

core and seismic data provide local constraints on the spatial distribution of

these facies types.

5. Populate each facies type with porosity and permeability. Porosity is assigned to

each grid cell of the Cartesian grid based on well-log and core data; permeability

is derived from the porosity model. Porosity is usually determined first since

the data on porosity is more reliable and abundant than permeability data.

6. Map back the petrophysical properties into the stratigraphic grid to provide a

high-resolution 3D geo-cellular model.

Figure 1.1: Step-by-step workflow for building a high-resolution geo-cellular model(from Caers [16]).


The high-resolution 3D geo-cellular model obtained after following these steps

honors static data but does not match the existing production data (Figure 1.2);

therefore, a history matching procedure is applied to perturb the reservoir model

until the flow response of the model matches the field production data.

Figure 1.2: High resolution model (left) which only matches static data; its flowresponse does not match historical water cut at the producer wells.

The geo-cellular model often consists of millions of grid cells, which precludes the

practice of flow simulation and history matching. To make flow simulation feasible,

the number of grid cells needs to be reduced, hence the model is upscaled to man-

ageable grid dimensions. The upscaled model is consequently perturbed (often done

manually) until history match is achieved (Figure 1.3). This commonly used approach

for history matching involves the perturbation of upscaled petrophysical properties

and usually a reasonable match can be achieved (at least at the field scale). However,

perturbing such upscaled properties is done without regard of the high-resolution data

1.1. RESERVOIR MODELING AND DATA INTEGRATION 5

and model which honors well-log, geological information and 3D seismic data. The

final result of this approach is a model that only matches production history, but it is

no longer consistent with any data integrated prior to matching; this often precludes

the prediction of future production since the model has lost most if not all geological

realism.

Figure 1.3: Coarsened model (from Figure 1.2) being manually perturbed until his-torical water cut at the producer wells is matched.

As remarked by Tureyen [86], the main problem with the current practice is that

the high-resolution and coarsened models are treated independently; the coarsened

model is perturbed using only production data, and the high-resolution model is

generated taking into account only static data. Tureyen proposed a “Parallel Mod-

eling Approach” which builds both high-resolution and coarsened models jointly, in


parallel. Additionally, he proposed to perform all model perturbations on the high-

resolution geo-cellular model, which would be immediately reflected on the coarsened

model through a subsequent upscaling. Flow simulation is performed on the coarsened

model, yet these results are used to perturb the high-resolution model.

The advantage of the “Parallel Modeling Approach” is that the coarsened model

is perturbed only as a consequence of the perturbation on the high-resolution model;

hence any perturbation on the high-resolution model must be consistent with static

data. The disadvantage of this approach is its dependency on the particular upscaling

method, since perturbations are done based on the results of the coarse scale model

flow response. In other words, the upscaling method should be able to create a

coarsened model that reproduces the flow response of the high-resolution model fairly

well. When a complex upscaling technique is used to achieve this goal, the main

disadvantage is on the added CPU cost, since upscaling is performed after every

perturbation.

The ultimate goal of reservoir modeling is to build and perturb a high-resolution

geo-cellular model, such that it is constrained to all types of information. The reser-

voir model should match historical production data as well as honor seismic data,

well-log and geological information. Hoffman [44] proposed to integrate all data from

various sources at the same time using a probabilistic approach to address the pos-

sible inconsistency and/or redundancy between data. A detailed explanation of this

approach is presented in Chapter 2 (section 2.1). The high-resolution model used in

Figure 1.2 was built using the probabilistic approach proposed by Hoffman, where

sand facies has been modeled using well-log data, geological information about the

distribution of the sand facies (training image) and seismic data (sand facies proba-

bility obtained from calibration of seismic data with well-logs). Figure 1.4 shows how

a facies realization is drawn from a joint conditional probability distribution which

gathers information from all data sources about the unknown sand facies, as well as

possible inconsistency and/or redundancy between data.

The “Parallel Modeling Approach” can be applied to the initial facies model

(shown in Figure 1.2) which only honors static data, as shown in Figure 1.5. As

mentioned before, the perturbations are done on the high-resolution model and must

1.2. INCORPORATING 4D SEISMIC DATA 7

be geologically consistent with static data. Using a technique called “Probability

Perturbation Method” (PPM), presented by Caers [15], Hoffman proposed to perturb

large scale parameters such as facies distributions while honoring all other available

data. The fundamental principle behind PPM is that perturbations are done on the

conditional probability from which a model is drawn, rather then on the model it-

self (Figure 1.6). The perturbations are done iteratively such that the model’s flow

response is closer to matching the production data.

The reservoir modeling workflow proposed in this dissertation, follows the “Parallel

Modeling Approach” of perturbing the high-resolution model directly, and also uses

the probabilistic data integration approach presented by Hoffman. However, the main

contribution of the workflow proposed here is the inclusion of 4D seismic data, which

had previously not been accounted for.

1.2 Incorporating 4D Seismic Data

Termed “four-dimensional seismology” by Nur [64], 4D seismic data comprise the set

of 3D seismic data acquired at different times over the same area, with the objective

of monitoring changes occurring in a producing hydrocarbon reservoir over time.

4D seismic data record two types of changes: changes in reservoir properties due

to production, and changes in external variables such as ambient noise, recording

equipment, etc. The second type of changes, termed “undesirables” by Jack [45],

represent a research area of its own within the geophysical community; however, it

is out of the scope of this dissertation. The methodology proposed here ignores the

existence of 3D seismic repeatability issues.

Changes in reservoir properties due to production are recorded by 4D seismic data

since seismic velocities and impedance depend on changes in pore fluids, pressure and

temperature [63] [10] [90]. The overall elastic moduli of a rock change with the type

of fluid in the pores, the effective pressure acting on the rock, and the temperature

the rock is subjected to. Due to the change in elastic moduli, the rock becomes more

or less resistant to wave-induced deformations; therefore, seismic velocities experience

an increase or decrease in magnitude. These observations are the basis for using 4D


Figure 1.4: A facies realization being drawn from a joint conditional probabilitydistribution which gathers information from all data sources (well-log, geological in-formation and seismic data) about the unknown sand facies A.


Figure 1.5: The “Parallel Modeling Approach” applied to the initial facies model(shown in Figure 1.2); where the high-resolution model is perturbed in a geologicallyconsistent fashion (using PPM) until historical water cut at the producer wells ismatched.


Figure 1.6: Geologically consistent perturbation of facies using PPM while honor-ing all other available data. Perturbations are done iteratively on the conditionalprobability from which a model is drawn, rather then on the model itself.


seismic data in predicting fluid saturation and pressure changes in the reservoir.

Typically, the difference between two 3D seismic data sets recorded at different

times allows mapping those areas in the reservoir where pressure and/or the distribu-

tion of fluids have changed. Therefore, 4D seismic data not only provides information

about the dynamic process occurring in the reservoir while production takes place; but

also provides information about the spatial lithological heterogeneity where dynamic

changes occur.

1.2.1 Challenges

Providing a spatial distribution of dynamic changes in a producing reservoir, 4D

seismic can be treated as dynamic data, hence can be history matched along with

production data within the reservoir modeling workflow. This quantitative approach

to incorporate 4D seismic data calls for forward modeling the 4D seismic response,

and comparing it with the observed field response.

As production takes place in a reservoir, the spatial distribution of pressure and

saturation in the reservoir changes over time; therefore, the overall elastic moduli of

the reservoir rock changes and 4D seismic is able to record those changes through

“seismic observables”, often called seismic attributes. “Seismic attributes are all the

information obtained from seismic data, either by direct measurements or by logical

or experience based reasoning” [82]. Generally speaking, most seismic attributes are

functions of density, and the velocities of P and S waves as they travel through the

rock; for example, acoustic impedance (a seismic attribute) is the product of density

and P-wave velocity.

Forward modeling the 4D seismic response involves the computation of the seismic

attribute(s) as the spatial distribution of pressure and saturation in the reservoir

changes over time. For the example of acoustic impedance (AI), the change over

time of this seismic attribute can be computed as the set:

[AI]t1 , [AI]t2 , ... [AI]tn

where [AI]ti = [ρVp]ti is the acoustic impedance at time ti, obtained by computing


density (ρ) and P-wave velocity (Vp) of the rock subject to the conditions at time

ti, pressure and saturation at that time. The effect of pressure and saturation on

the density of a rock, as well as their effect on the velocity of P and S waves as

they travel through the rock is computed using rock physics models. A detailed

explanation of this procedure is given in Chapter 2 (section 2.3.1); however, for now

it is important to mention that density, P and S-wave velocities for each grid block of

the high-resolution reservoir model are computed first and the conditions - pressure

and saturation - at time ti are imposed later. The spatial distribution of pressure and

saturation is obtained after flow simulation, which is performed on a coarsened grid.

The 4D seismic response is forward modeled using the high-resolution reservoir

properties since seismic waves are often affected by fine scale details below seismic

resolution [58] [74] [75]; therefore, the first challenge encountered is how to port the

coarsened pressure and saturation into the high-resolution grid.

The 4D seismic response is usually due to a combination of both pressure and

saturation effects. A change in pressure often causes a change in saturation in the

reservoir, especially during early stages of production (primary depletion). Sometimes

one effect may dominate the other; therefore, the second challenge encountered is how

to correctly model those effects using rock physics models and lab measurements of

the dependency of P and S wave velocities on pressure.

Comparing the modeled and the field 4D seismic response is an important step

when history matching both production and 4D seismic data. A measure of similarity

or discrepancy must be established, such as a point-to-point correlation coefficient or

an absolute difference between the forward modeled and the observed 4D responses;

however, regardless of the comparison method used, 4D seismic data must be exam-

ined before hand in order to find the “reservoir signal” to be matched.

Chapter 4 of this dissertation shows in detail a case study from a reservoir in the

North Sea where three 3D seismic surveys have been acquired, respectively in the

years of 1992, 1999 and 2004. From each 3D seismic survey, AI and Vp/Vs ratio are

the seismic attributes studied; the change on both attributes over time is summarized

by Figure 1.7. This figure clearly shows that some areas in the reservoir have changed

more significantly than others; in particular, data points located close to the center of


the crossplot exhibit a very small change that could be undistinguishable from noise

in the data. Consequently, the modeled and the field 4D response should be compared

in areas where the field 4D response is strong. The third challenge encountered is

how significant the change in a seismic attribute over time should be to be treated as

“reservoir signal” rather than noise.

Figure 1.7: Change in Vp/Vs versus change in acoustic impedance for each of the twotime-lapse responses: the 1992-1999 time-lapse response (shown in green) and the1999-2004 response (shown in red). A change greater than one means an increasein the attribute over time and a change smaller that one means a decrease in theattribute over time. The center of the figure means no change on any of the twoattributes.


1.2.2 Current Approaches

A few papers are referenced in the literature for quantitatively matching 4D seismic

along with production data. Basically, two approaches are distinguished:

• The first approach strives to match the change of an interpreted physical prop-

erty from 4D seismic data. Some authors look for matching the change in

saturation and/or pressure [53] [87] [56], while others look for matching the

presence/absence of gas [52]. This approach does not require the forward mod-

eling of the 4D seismic response.

• The second approach pursues the match of the change of a physical seismic

attribute from 4D seismic data [38] [4] [89] [36] [60]; commonly, the change in

acoustic impedance. This approach requires the forward modeling of the 4D

seismic response.

The main difference between the two approaches is the forward modeling of the 4D

seismic response, which is only required by the second approach. Forward modeling

of the 4D seismic response involves the modeling of several 3D seismic data sets, each

one corresponding to the time when the real 3D seismic survey was acquired over the

study area. A full forward modeling of each 3D seismic response is referred in the

Geophysical community to the modeling of the wave propagation that creates seismic

amplitudes as a function of recording time. As explained in more detail in Chapter 2

(section 2.3.1), the simplest process to achieve this goal is termed “the convolutional

model”; however, more rigorous methods, hence computationally expensive, can be

followed such as “3D full wave propagation” which models the entire propagation

and scattering of P and S waves waves as they travel in a 3D earth model. Seismic

amplitudes, however, are functions of several physical attributes such as acoustic

impedance; a procedure called “seismic inversion” is often applied to 3D seismic

amplitude data in order to obtain those physical attributes.

This dissertation considers forward modeling of the 3D seismic response to the

computation of the above mentioned physical attributes. Therefore, when following

the second approach to match the change of a physical seismic attribute from 4D


seismic data, the set of observed 3D seismic surveys must be “inverted” to obtain the

physical attribute(s) of interest to be matched.

Some authors that adopt the first approach basically interpret the 4D seismic data

in terms of changes in saturation (and/or pressure), obtaining a map of ∆S. This map

is then compared with the one obtained from the flow simulation in some sense (e.g.

least squares); the reservoir model is perturbed until a satisfactory match is achieved.

This methodology does not require one to forward model the 4D seismic response since

it directly compares saturations (and/or pressure). There are three main drawbacks

with this approach: 1) it is not clear how the coarse-grid saturations from the flow

simulation are compared to the saturations interpreted from the 4D seismic data; 2)

the changes on physical properties interpreted from the 4D seismic data are assumed

“exact”; 3) the influence of other physical properties, such as pressure or the presence

of multiple fluid phases, is ignored.

The changes in saturation (and/or pressure) interpreted from 4D seismic data

are often obtained after some “inversion” process, which is rarely explained by the

reviewed literature. Furthermore, the resolution of the grid on which the physical

properties are obtained is rarely specified; we could assume that it is the grid on

which the seismic inversion is performed, which is not necessarily the high-resolution

geo-cellular grid on which geostatistical modeling takes place. Any of the two grids

mentioned has a higher resolution than the coarsened grid where flow simulation is

performed; therefore, the question is how a flow simulated property (on a coarse grid)

is compared with the “observed” property which is on a higher resolution grid.

The interpretation of changes in saturation and/or pressure from 4D seismic data,

is not an easy task. The observed 4D seismic response is often due to a combination

of changes in pressure and saturation of multiple fluid phases; while in some specific

cases is due changes in temperature (thermal recovery processes), and compaction

(reduction of the pore space as subsidence takes place when fluids are produced)

among others. Ignoring the special cases of temperature and compaction, changes in

pressure and saturation of multiple fluid phases occur at the same time and decoupling

them is a challenging problem to solve. Moreover, assuming an “exact” change on

any of them (interpreted from 4D seismic data) can be considered as an audacious


statement.

The second approach pursues the match of the change of a physical seismic at-

tribute from 4D seismic data. As mentioned earlier, physical seismic attributes are

obtained after “seismic inversion”; however, this inversion is simpler and perhaps

more reliable than the one used to obtain pressure and saturation from seismic data.

The second approach requires to forward model the 4D response of the physical seis-

mic attribute(s) considered, such as the change in acoustic impedance. The forward

modeled response is compared with the observed change in acoustic impedance ob-

tained after inversion of the 4D seismic data. The main advantage of this approach

is that no complex/unreliable inversion procedure is applied to 4D seismic data to

obtain pressure/and or saturations changes; therefore, the second two drawbacks of

the first approach do not apply to the second one.

Recalling the discussion in section 1.2.1, forward modeling the 4D seismic response

involves the computation of the seismic attribute(s) as the spatial distribution of

pressure and saturation in the reservoir changes over time. The seismic attribute(s)

is computed first for each grid block of the high-resolution reservoir model and the

conditions - pressure and saturation - are imposed later. The spatial distribution of

pressure and saturation is obtained on a coarsened grid after flow simulation, while

forward modeling the 4D seismic response should be performed on the high-resolution

grid since seismic data are often affected by fine scale details below seismic resolution.

The term “grid resolution” is referred to as the physical dimensions of the grid blocks;

however, seismic resolution is referred to as the minimum size of a geological feature

whose boundaries can be identified by the seismic waves. Some of the reviewed

literature that follows the second approach does not mention clearly on which grid

the forward modeling of the physical seismic attribute(s) is performed; however, some

authors “downscale” the coarse-grid saturation and pressure to the high-resolution

grid where the forward modeling is performed. The commonly used downscaling

procedure is “static” and it does not take the physics of fluid flow into account. In

particular, Mezghani [60] uses a 3D interpolation as downscaling procedure.

In order to compare the 4D seismic attribute computed using rock physics mod-

eling with the observed one, the former is filtered to make it comparable with the


observed one. In Mezghani [60], the modeled 4D seismic attribute is computed for

the high-resolution grid, subsequently filtered in the band-width of seismic data and

finally upscaled to the seismic data scale (the upscaling procedure is not described).

The upscaling of 4D seismic attributes implies smoothing of the acoustic contrasts.

Additionally, an upscaled 4D seismic attribute is not comparable with the observed

4D seismic attribute; the observed 4D seismic attribute is affected by fine scale details

below resolution that are not captured by the upscaled 4D seismic attribute.

The reviewed literature that follows the two discussed approaches focus on history

matching dynamic data (4D seismic and production data), and the details of the

optimization algorithm used. The ultimate goal is not only to match dynamic data

but also to honor static data (well-log, geological and geophysical data), since reservoir

models that honor all available data will have a better prediction capabilities of

reservoir behavior than models that only honor some of the available information.

A detailed presentation of the available history matching methods is presented by

Hoffman [44]. He also summarizes their main limitations as follows:

• History matching is often performed without any regard to geologic data. This

limits the predictive capabilities of the “matched” models.

• It is difficult to incorporate important large scale parameters such as NTG, and

reservoir structure in an efficient manner. However, the large scale parameters

often have the most significant impact on production data.

• Most current automatic history matching methods are not practical for real

field applications. They require too many flow simulations or can not handle

the geologic complexity found in most reservoir models.

The alternative method proposed by Hoffman consists of generating reservoir mod-

els constrained to all types of information. The reservoir models will match the his-

torical production data as well as honor 3D/4D seismic data, well log and core data,

and especially large scale geologic information. To achieve this goal reservoir models

are created and perturbed using a data integration approach that takes into account

the possible inconsistency and/or redundancy between data.


1.2.3 Proposed Approach

Two general approaches are proposed to include 4D seismic data into the reservoir

modeling workflow. The first approach is the Probabilistic Data Integration approach

(PDI), which consists of modeling the information content of 4D seismic through a

spatial probability of facies occurrence. It has now become a common practice in

geostatistical modeling of facies distribution to constrain such facies modeling to 3D

seismic by means of a facies probability cube [16]. From 3D seismic data, the seismic

observations made along well-paths that contain logs or have been cored are retained.

Using this information, the probabilistic relationship between the facies present and

seismic data is obtained as follows:

Prob(facies k occurs|local seismic data) = φk(local seismic data)

where∑K

k=1 φk = 1 and 0 ≤ φk ≥ 1, ∀ k

Several methods can be used to estimate the functions φk, such as co-located

seismic-to-well calibration. By applying the functions φk to the entire 3D seismic

data set, a 3D facies probability can be obtained for each facies at each grid block

location. More advanced methods for obtaining a facies probability cube look also

into using 4D seismic data by taking advantage of preferential flow through highly

permeable facies. The works of Wu et al. [91] and Andersen et al. [3] are pioneering

on this subject; in particular, the work of Andersen et al. will be explained in Chapter

2 (section 2.2.1).

The second approach, termed Forward Modeling (FM) approach, presented by

Mezghani et al. for petrophysical models using gradual deformation, and extended

to facies models in Castro and Caers [14], consists of matching 4D seismic along with

production data. This approach requires forward modeling the 4D seismic response,

hence requires reasonable quality seismic data and enough rock physics data to make

this modeling feasible. The forward modeling of the 4D seismic data can be very chal-

lenging, especially when the 4D response is due to a combination of both pressure and

saturation effects occurring in the reservoir. Substantial rock physics knowledge from

1.3. DISSERTATION OUTLINE 19

lab-data on the impact of pressure and saturation changes on petrophysical proper-

ties is required and may not always be easily obtainable, or subject to uncertainty.

The quality of the 4D seismic response is also an important factor, since matching a

noisy or ambiguous 4D response may cause inconsistency with other data. The FM

approach is discussed in detail in Chapter 2 (section 2.3).

The PDI approach constrains the reservoir model to the spatial probability of

facies occurrence as well as other static data while history matching only production

data. This approach is explained in detail in Chapter 2 (section 2.2). When high

quality 4D seismic data is not available or a complicated 4D seismic forward modeling

may not be carried out due to the lack of rock physics data, this approach may

represent a more robust and less difficult to achieve alternative to include 4D seismic

information into the reservoir modeling workflow.

1.3 Dissertation Outline

This dissertation is comprised of five chapters. The reservoir modeling workflow

that uses a probabilistic data integration approach and the probability perturbation

method for history matching is presented in Chapter 2. This chapter also introduces

the two proposed approaches for incorporating 4D seismic data into the workflow as

well as synthetic examples illustrating each one of them.

Chapter 3 introduces a new generalized procedure for downscaling coarse-scale

saturations accounting for both static information and the particular flow problem.

The proposed method is illustrated on a 2D synthetic example.

Chapter 4 presents a case study on a North Sea reservoir where the reservoir

modeling workflow has been applied with the purpose of determining the location of

channel sand facies in the reservoir. The two proposed approaches for including 4D

seismic data are evaluated on this case study.

Chapter 5 discusses the major findings of this dissertation, as well as suggestions

for future work.

Among the appendices, a new synthetic reference data set is presented in Appendix

A: The Stanford VI reservoir is introduced with the purpose of extensively testing new


algorithms for reservoir modeling, reservoir characterization, and forward 4D seismic

modeling.

Chapter 2

Two Workflows for Integrating 4D

Seismic Data

The set of 3D seismic data acquired at different times over the same area, termed 4D

seismic data, is used for monitoring changes occurring in a hydrocarbon reservoir due

to production. Changes in the type of fluid in the pores of a rock, the effective pressure

acting on a rock, and the temperature a rock is subjected to, can cause changes in the

density of a rock as well as changes in the velocities of seismic waves as they travel

through a rock. 4D seismic data is able to detect those changes in density and seismic

velocities, which are used for predicting fluid saturation and pressure changes in the

reservoir.

The interpretation of the changes observed between the 3D seismic data sets

recorded at different times allows mapping those areas in the reservoir where pres-

sure and/or the distribution of fluids have changed. Therefore, 4D seismic data not

only provides information about the dynamic process occurring in the reservoir while

production takes place; but also provides information about the spatial lithological

heterogeneity where dynamic changes occur.

Providing valuable static and dynamic information, 4D seismic data (when avail-

able) can be incorporated into the reservoir modeling process, whose goal is to build

a model of the reservoir while honoring all types of information. Different sources of

information are used for building reservoir models. Well logs and core data provide

21

22CHAPTER 2. TWO WORKFLOWS FOR INTEGRATING 4D SEISMIC DATA

high resolution information about properties near the well bore, seismic data provides

indirect information about the entire field, but the resolution is much poorer than

well data. Another source of information is the geological setting of the reservoir,

i.e., prior knowledge about the depositional system that created the reservoir rock.

This information, termed “static” data, is used by geostatisticians to create a high

resolution geo-cellular model (Figure 1.1). Although this model honors static data, it

may not be able to reproduce the dynamic response of the reservoir. In other words,

it is not able to reproduce the fluid rates and pressure data observed in producing

wells.

Reservoir engineers follow an iterative procedure, termed “history matching”, that

perturbs the reservoir model until it reproduces the observed production data. In

many practical applications of this procedure, the reservoir model is perturbed with-

out keeping consistency with static data. As remarked by Hoffman [44], the ultimate

goal of reservoir modeling is to create a model of the reservoir that honors simultane-

ously all available information (static and dynamic), since those models would have a

better chance to make realistic predictions about the future production performance

of the reservoir. Hoffman proposed an approach for building and perturbing reservoir

models while simultaneously honoring all available data. He proposed a probabilistic

approach to integrate all data from various sources at the same time, while addressing

the possible inconsistency and/or redundancy between data.

This chapter commences by explaining in detail the probabilistic approach intro-

duced by Hoffman, as it is adopted in this dissertation for building and perturbing

reservoir models. Next, two approaches for incorporating 4D seismic data into the

probabilistic approach will be introduced and discussed. This chapter limits itself to

describing the methodology; a full real field case example is given in Chapter 4.

2.1 Probabilistic Data Integration Approach

Data integration for reservoir modeling has been traditionally applied in a two-step

fashion. First, geostatistics is used to build a model by combining static data; second,

a history matching process is carried out in order to incorporate dynamic data. The

2.1. PROBABILISTIC DATA INTEGRATION APPROACH 23

goal of the Probabilistic Data Integration Approach is to combine both static and

dynamic data in a seamless fashion.

The Probabilistic Data Integration Approach follows the general idea of many

geostatistical algorithms which is to populate a grid with property values using a

set of known data. The property or properties being estimated could be porosity,

permeability, facies indicators or any other geological reservoir property. The property

is estimated using the paradigm of sequential simulation [48]. Each uninformed grid

block is visited randomly, and a conditional probability, P (A | D1), is estimated for

that grid block. P (A | D1) is the probability of the unknown property, A, occurring

given some other information D1. For example, A could stand for “channel facies

occurs” or “porosity is less than 15%” and D1 could be well-log data and/or geologic

knowledge of the reservoir of the same type. Subsequently, the property value for that

grid block is randomly drawn from that probability distribution. Once a grid block

has been simulated, its value is used with the known data to create the conditional

probability for the next grid block. This process is repeated until all grid blocks of

the entire grid are simulated. Three properties of sequential simulation are:

1. the desired statistics are reproduced,

2. multiple realizations can be created by changing the random path in which grid

blocks are visited, and

3. integration of different types of data is possible.

The conditional probability distribution P (A | D1), estimated for the particu-

lar grid block being visited by the algorithm, can be determined in different ways.

The sequential Gaussian simulation algorithm [35], defines P (A | D1) as a Gaussian

distribution with mean equals to the kriging mean and variance equals to the krig-

ing variance, after transforming the property being simulated to the Gaussian space.

The single normal equation simulation [81], however, creates P (A | D1) by scanning a

training image for equivalent multiple-point data events. One of the main differences

between these two algorithms is the way the degree of correlation in the model is

defined; sequential Gaussian simulation relies on the variogram (a measure of corre-

lation between two points in the space, as their separation distance increases) while


the single normal equation simulation relies on the training image (an unconditional

and conceptual reservoir analog with the believed geological heterogeneity). Since

the variogram describes the level of correlation between two locations only, it is not

able to model continuous and sinuous patterns such as channels or fractures. For

modeling such geological features a multiple-point approach should be used, where

spatial patterns are inferred from a training image using many spatial locations.

One of the properties of sequential simulation is that it allows for the integration

of different types of data. In other words, the conditional probability distribution

from which the property is drawn can be obtained using other additional types of

data D2, ..., Dn; therefore, instead of drawing from P (A | D1) the algorithm would

draw from the joint conditional P (A | D1, ...Dn).

A conditional probability P (A | Di) can be interpreted as a measure of how

informative the datum Di is about an event A occurring. When dealing with data

from different sources D1, ..., Dn, however, all related to a common event A, the

possible redundancy between data Di must be taken into account. A method to

obtain P (A | D1, ...Dn) from the individual conditional probabilities P (A | Di) while

accounting for redundancy between data is proposed by Journel [49] and is explained

in the following section.

2.1.1 The Tau Representation

The general goal is to model an unknown A (facies or petrophysical property) using

data from different sources D1, ..., Dn. The distance of each datum Di to the unknown

A is defined by Journel [49] as a logistic-type ratio:

xi =1− P (A | Di)

P (A | Di)(2.1)

the joint distance to the unknown is:

x =1− P (A | D1, ..., Dn)

P (A | D1, ..., Dn)(2.2)

Using the paradigm of permanence of ratios, Journel expressed the joint distance x


as:

x

x0

=n∏

i=1

(xi

x0

)τi

(2.3)

where x0 is the “prior” distance to the unknown, given by:

x0 =1− P (A)

P (A)(2.4)

Journel indicates the weights τi in Eqn. 2.3 account for the redundancy of the

n data events Di with regard to modeling event A; however, an expression for these

weights is not provided. Krishnan [51] recognized the weights τi as the most impor-

tant factor in expression 2.3; he reinterpreted the tau representation as a log-linear

estimator of the conditional probability, which allowed him to estimate of the τi

weights:

τi(x1, ..., xi, x0) =ln(

P (Di|A,D1,...,Di−1)P (|A,D1,...Di−1)

)ln(

P (Di|A)P (Di|A)

) ∈ [−∞, +∞] (2.5)

these weights, as remarked by Krishnan, are dependent on the specific values of

A, D1, ..., Dn and are also dependent on the specific data sequence. The tau weight

at position i is impacted only by the data which arrive before it, i.e., from 1 through

i− 1.

In practical situations, however, the τi weights may be quite challenging to com-

pute. Therefore, choosing τi = 1 is the commonly used approach, as it has shown to

provide robust results in many practical situations, particularly in sequential simula-

tion [17]. With τi = 1, the conditional probability P (A | D1, ..., Dn) is calculated as

follows:

P (A | D1, ..., Dn) =1

1 + x=

xn−10

xn−10 +

∏ni=1xi

(2.6)

From Eqn. 2.6 it is observed that the joint conditional probability P (A | D1, ...Dn)


can be computed only from the individual conditional probabilities P (A | Di). These

probabilities reveal how informative the datum Di is about the unknown A; therefore,

the procedure followed to extract the information content of each data source depends

on the specific data source itself. For example, suppose the unknown A is channel

facies present or absent at a certain location u = (x, y, z). D1 is geologic data (well-

log and/or core data, and training image), D2 is geophysical data (3D seismic), and

D3 is historical production data.

In this example, the unknown A is an indicator variable I(u) that describes an

event occurring:

I(u) =

1 if channel facies occurs at u

0 else

Following the sequential simulation approach explained earlier, at each uninformed

grid block (visited randomly), P (A | D1) is estimated by the algorithm itself and com-

bined with the other conditional probabilities P (A | D2) and P (A | D3) using Eqn. 2.6

from which a value is drawn for the grid-block. In particular, since simulating channels

requires a multiple-point algorithm, the single normal equation simulation algorithm

computes P (A | D1) after scanning a training image for equivalent multiple-point

data events. Figure 2.1 shows how this process is performed. In the neighborhood

of an uninformed grid-block, the probability of this grid-block being channel facies

given its specific set of neighboring sand and no-sand data values (D1) is calculated

by scanning the training image (Figure 2.1) for “replicates” of this data event: three

such events are found of which one yields a central sand value, hence, the probability

of having sand is 1/3. By random drawing, a facies category is assigned. The set

of neighboring sand and no-sand data values is represented by previously simulated

values and well-log and/or core data, which are assigned (frozen) facies category val-

ues to those grid-blocks that are intersected by wells. The grid-blocks containing well

constraints are never visited and their facies values never re-considered.

Figure 2.1 shows how the single normal equation simulation algorithm computes

P (A | D1); following is a review of how the conditional probabilities P (A | D2) and

P (A | D3) can be computed.


Figure 2.1: Computation of P (A | D1) for a single grid-block in sequential simulationwith a training image model. The neighboring data (termed “data event”) near arandomly visited grid-block is extracted. Replicates of this data event are searchedfor in the training image. The probability of the central grid-block to be in channelfacies (sand) can be calculated from the set of replicates (modified from Caers [16]).

In this example, the datum D2 is geophysical data: 3D seismic. It is known that

3D seismic data is not able to measure directly rock properties such as porosity, or

rock type; however, 3D seismic data can provide information about these properties

after a calibration with well data. From 3D seismic data, the seismic observations

made along well-paths that contain logs or have been cored are retained. Using this

information, the probabilistic relationship between the facies presence and seismic

data is obtained as follows:

Prob(facies k occurs|local seismic data) = φk(local seismic data)

where∑K

k=1 φk = 1 and 0 ≤ φk ≥ 1, ∀ k

Several methods, either through simple statistics or through rock physics [5], can

be used to estimate the functions φk, such as co-located seismic-to-well calibration

[16]. By applying the functions φk to the entire 3D seismic data set, a 3D facies

probability can be obtained for each facies at each grid-block location.


The datum D3 is historical production data (dynamic data): fluid rates and pres-

sure data observed in producing wells. In order to constrain the reservoir model to

this type of data, an iterative procedure must be followed. Fluid flow in the reservoir

depends on how high and low permeability bodies are spatially distributed in the

reservoir; therefore, an iterative procedure that perturbs the location of these high

and low permeability bodies is a way to achieve the goal. The Probabilistic Data

Integration Approach, rather than directly perturbing an initial realization, uses a

method called “Probability Perturbation Method” [15] which proposes to perturb

the conditional probability P (A | D1, D2) from which the initial realization has been

drawn. This is done by introducing another probability model, P (A | D3), that

depends on dynamic data, D3. The “perturbed” realization is drawn from the joint

conditional probability P (A | D1, D2, D3) obtained using Eqn. 2.3. The following sec-

tion is devoted to explaining the “Probability Perturbation Method” in more detail,

and how P (A | D3) is computed.

2.1.2 The Probability Perturbation Method

Recall the example discussed earlier where the unknown A is channel facies present

or absent at a certain location u = (x, y, z), described by the indicator variable I(u):

I(u) =

1 if channel facies occurs at u

0 else

When a realization i(0)(u) is drawn from P (A | D1, D2) using the single normal

equation simulation algorithm, two cases can be considered while matching dynamic

data D3: the case where a match is achieved, hence the specific realization i(0)(u) is

kept, or, the case where the realization i(0)(u) is far from matching the dynamic data

and should be perturbed considerably.

The goal of the Probability Perturbation Method is to iteratively create the con-

ditional probability P (A | D3) depending on the mismatch between the simulated

and field dynamic data. The probability P (A | D3) also acts as a perturbation of


P (A | D1, D2), and it is defined by Caers [15] as:

P (A | D3) = (1− rD)i(0)(u) + rDP (A) (2.7)

where rD is a parameter that can be chosen between [0, 1], and P (A) is the overall

proportion of the event A occurring independent of location, hence is the marginal

distribution.

The parameter rD determines the magnitude of the perturbation:

• if rD = 0, then P (A | D3) = i(0)(u) and the information D3 is determined to

be fully informative of event A. The flow simulation response of the realization

i(0)(u) matches the production data D3, therefore no perturbation is needed

(rD = 0).

• if rD = 1, then P (A | D3) = P (A), meaning that A is not yet representative of

data D3. The flow simulation response of the realization i(0)(u) does not match

the production data D3, and one retains a different equiprobable realization

i(1)(u).

In other words, the parameter rD defines a perturbation of the initial realization

i(0)(u) towards another independent realization i(1)(u). Each value of rD fully deter-

mines the probability P (A | D3) at every location u. During sequential simulation

P (A | D3) is combined with P (A | D1, D2) to form P (A | D1, D2, D3), from which

simulated values are drawn. The resulting realization simulated in this fashion is

denoted as i(1)rD

(u).

Figure 2.2 illustrates the example being discussed, where an initial realization i0(u)

is drawn from P (A | D1, D2) using the single normal equation simulation algorithm.

Perturbations of the initial realization i(0)(u) towards another independent realization

i(1)(u) are shown in Figure 2.3, achieved by perturbing P (A | D1, D2) with P (A | D3).

Notice the small variations between i(0)(u) (Figure 2.2) and i(1)rD=0.1(u) (Figure 2.3) as

expected for small values of rD.


Figure 2.2: Training image (left), conditional probability P (A | D2) from seismic data(middle), and an initial realization i0(u) (right).

Figure 2.3: Realizations simulated using the joint conditional probability distributionP (A | D1, D2, D3) for rD values of 0.1, 0.3, 0.5, 0.7 and 1.

2.2. INTEGRATING 4D SEISMIC DATA: PDI APPROACH 31

2.2 Integrating 4D Seismic Data: PDI Approach

This chapter started with a discussion about 4D seismic data and why it should be

incorporated into the reservoir modeling process. 4D seismic data comprise a set of

3D seismic data acquired at different times over the same area. In many cases only

one 3D seismic data set is available: the survey acquired during exploration or early

appraisal of the field, often called “Base 3D seismic data”. Subsequent 3D seismic

surveys, acquired later in time, are called in this dissertation “Time-lapse 3D seismic

data”. The change between surveys, expressed as a ratio, difference or any other

measure of change, is denoted in this dissertation as “4D seismic response”.

4D seismic data provides valuable static and dynamic information about the reser-

voir, since some of the dynamic changes (changes in fluid saturation) occur only in a

certain type of rock with high porosity and permeability. Therefore, 4D seismic data

can be used as a constraint for the facies modeling by means of a facies probability

cube.

Following the Probabilistic Data Integration Approach, the integration of 4D seis-

mic data can now be further explained. Recalling the example discussed in the previ-

ous section, where the unknown A is channel facies presence or absence at a certain

location u = (x, y, z), the new set of data after including 4D seismic data would be:

D1 is geologic data (well-log and/or core data, and training image), D2 is geophysi-

cal data (Base 3D seismic), D3 is 4D seismic data (the set of “Time-lapse 3D seismic

data”), and D4 historical production data. Reservoir models are drawn by the sequen-

tial simulation algorithm from the joint conditional probability P (A | D1, D2, D3, D4),

which is obtained from the individual probabilities P (A | D1), P (A | D2), P (A | D3),

and P (A | D4) using Eqn. 2.6.

Figure 2.4 shows a schematic reservoir modeling workflow where 4D seismic data

is proposed to be included as a spatial probability distribution; the workflow uses

the Probabilistic Data Integration Approach proposed by Hoffman [44] as well as

the Parallel Modeling Approach proposed by Tureyen [86]. The workflow consists of

generating a high resolution 3D geo-cellular model, which is then upscaled and flow

simulation is run to check the match of historical production data. The high resolution


model is subsequently perturbed in a fashion that will improve the match and the

entire procedure is repeated until production data are matched. The Probabilistic

Data Integration Approach is used to simulate and perturb the high resolution model,

while simultaneously honoring all static data.

Figure 2.4: Reservoir modeling workflow that incorporates the 4D seismic datathrough a spatial probability distribution.

The information content of 4D seismic data needs to be modeled through a spatial

probability distribution. Although research on this subject is out of the scope of this

dissertation, the works of Wu et al. [91] and Andersen et al. [3] are pioneering on

this matter. In particular, the following section is devoted to explaining the work

of Andersen et al. [3], a particular example of estimating P (A | D2, D3) from rock

physics information.

2.2.1 Modeling the Information Content of 4D Seismic Data

Following the example being discussed in this chapter, when including 4D seismic data

into the Probabilistic Data Integration Approach an additional conditional probability

needs to be computed. The probability P (A | D2), obtained after a calibration of

the Base 3D seismic survey with well-logs [16], summarizes the information that the


Base 3D seismic survey can provide about the unknown A, presence of channel facies

in the example. The new datum D3, the set of Time-lapse 3D seismic surveys, can

provide additional information about the unknown that needs to be summarized by

P (A | D3). The methodology proposed by Andersen et al. [3], however, computes the

joint P (A | D2, D3) directly by merging the independently classified Base 3D seismic

data and the 4D response.

Andersen et al. introduce their methodology on a field case in the North Sea, the

same field where the case study presented in Chapter 4 takes place. The reservoir

under study is a fluvial system, where sands exhibit low Vp/Vs ratio and low AI

(acoustic impedance) according to well-log data (see Figure 2.5).

Figure 2.5: Crossplot of Vp/Vs ratio versus acoustic impedance for a typical well inthe field (left). Several logs for this well are shown on the right: acoustic impedance,gamma-ray, and Vp/Vs. The data points are colored by depth; however, several othercolor-filled polygons are superimposed which correspond to classes defined on thegamma-ray log. The sands in reservoir are shown inside the red polygon, with lowVp/Vs and low acoustic impedance (from Andersen et al. [3]).

An elastic inversion has been applied on both the Base 3D seismic survey and the

Time-lapse 3D seismic survey, in order to obtain Vp/Vs and AI for each of them. The

procedure followed to create a sand facies probability cube from the Base 3D seismic


data and the 4D response is summarized here in three steps:

1. Classify the Base 3D seismic data. From well-log data, the sands in the

reservoir have been identified with low Vp/Vs and low AI; therefore, the same

attributes (Vp/Vs, and AI) from the Base 3D seismic data are classified into

three classes (classes 5, 10 and 15) with increasing confidence for sand as AI

and Vp/Vs get smaller (see Figure 2.6).

Figure 2.6: Conceptual sketch of the Base 3D Seismic classification. The green datapoints have been obtained from the 3D seismic data, while the polygons outline theclassification of sands into three classes: 5, 10, and 15 (modified from Andersen et al.[3]).

2. Classify the 4D response. When crossplotting the change in Vp/Vs versus

the change in AI (change = time-lapse / base), each quadrant in the crossplot

is associated with effects from production (see Figure 2.7). If both AI and

Vp/Vs are reduced over time, then the associated effect may represent a gas

flooding. An increase in AI and a decrease in Vp/Vs may correspond to effects

from pressure decrease (depletion), while a decrease in AI and an increase in

Vp/Vs may correspond to pressure increase. An increase in both AI ratio and

Vp/Vs ratio may correspond to effects from water flooding. Andersen et al.

identified gas flooding as the process mostly related to the presence of sands in

2.3. INTEGRATING 4D SEISMIC DATA: FM APPROACH 35

this reservoir, hence that quadrant (lower left) is classified into four classes (see

Figure 2.7) with increasing confidence for sand as both AI and Vp/Vs are reduced

more over time (as the point moves away from the center in that quadrant).

Figure 2.7: Crossplot of the change (time-lapse/base) in Vp/Vs versus the change inacoustic impedance obtained from the 4D response (left). The center of this cross-plot corresponds to “no change”; however a change to each quadrant is classified asa “physical phenomenon” occurring over time (shown on the corners). The samecrossplot (shown on the right) shows the polygons outlining the classification of sand(modified from Andersen et al. [3]).

3. Merge classes and assign probabilities according to well-log data. In

order to reduce the number of classes and hence increase the number of obser-

vations in each class, the classes from the Base 3D seismic data and the classes

from the 4D response are merged into seven classes (see Figure 2.8). The prob-

ability of observing sand facies for each defined class is computed by comparing

the classification with observations of facies in wells.

2.3 Integrating 4D Seismic Data: FM Approach

The PDI approach, presented in the section 2.2, incorporates the information content

of the 4D response in a probabilistic fashion. An exact match of the 4D response


Figure 2.8: Table showing the recording of the classes in the combined 3D and 4Dvolumes. The figure to the right shows a smoothed sand probability for the com-bined volumes. The pink curve shows probabilities for sand and the red curve showsprobabilities for no-sand lithologies(from Andersen et al. [3]).

is not aimed for, rather a probabilistic type match is achieved. Another approach

to incorporate 4D seismic data is by treating it as dynamic data; in other words,

matching it along with production data. When history matching production data

only, the production response of the reservoir model is computed by performing flow

simulation. Therefore, when including the matching of 4D seismic data, the 4D

response of the reservoir model needs to be computed also by performing a “Forward

Modeling” (FM).

When high quality 4D seismic data is not available, or a complicated 4D seismic

forward modeling may not be carried out due to the lack of rock physics data, the

PDI approach may represent a more robust and less difficult to achieve alternative to

include 4D seismic information into the reservoir modeling workflow.

This approach, like the PDI approach, uses the Probabilistic Data Integration

Approach as well as the Parallel Modeling Approach. Recalling the example discussed

through this chapter, where the unknown A is channel facies presence or absence at

a certain location u = (x, y, z), the new set of data after including 4D seismic data

using the FM approach would be: D1 is geologic data (well-log and/or core data,

and training image), D2 is geophysical data (Base 3D seismic), D3 is dynamic data

(historical production data and the 4D response).


Figure 2.9 shows a schematic reservoir modeling workflow where the 4D seismic

response is proposed to be history matched along with production data. The reservoir

modeling workflow consists of generating a high-resolution 3D geo-cellular model,

then upscale it and run flow simulation to check the historical match to dynamic

data. Within this workflow dynamic data includes both production and 4D seismic

response. The production response of a reservoir model is obtained directly after flow

simulation. However, the 4D seismic response is modeled using the petrophysical

properties of the high resolution 3D geo-cellular model as well as the result of the flow

simulation: the spatial distribution of pressure and saturation. The high-resolution

model is subsequently perturbed in a “probabilistic” fashion, as explained in section

2.1.2.

Figure 2.9: Reservoir modeling workflow that incorporates 4D seismic data by historymatching it along with production data.

2.3.1 Modeling the 4D Seismic Response

In order to forward model 4D seismic data we need to simulate several 3D seismic

data sets, each one corresponding to the time when the real 3D seismic survey was

acquired over the study area. The number of 3D seismic surveys acquired on a study


area varies, usually two surveys are available; however, more than two 3D seismic

surveys could be acquired over the history of a reservoir.

Seismic data is usually forward modeled using a simple convolutional model. The

convolutional model is the most basic definition of a seismic trace:

s(u, t) = r(u, t) ∗ w(t) (2.8)

where s(u, t) is the seismic trace at location u = (x, y), r(u, t) is the reflectivity

series at location u and w(t) is the source wavelet. The convolutional model states

that the seismic trace is the convolution between the reflectivity series of the earth

and the source wavelet. The source wavelet w(t) is the impulse introduced in the

subsurface, hence the subsurface response is s(u, t). The reflectivity series contains

the reflection coefficients or acoustic impedance contrasts from each layer interface

in the subsurface. The contrast in acoustic impedance between two layers - layer 1

immediately above layer 2 - in the subsurface is defined as:

ρ2Vp2 − ρ1Vp1

ρ2Vp2 + ρ1Vp1

(2.9)

where ρ1 and Vp1 are the density and compressional velocity of layer 1 respectively,

while ρ2 and Vp2 are the density and compressional velocity of layer 2 respectively.

The acoustic impedance of a layer is defined as ρVp.

Using the convolutional model we can obtain seismic amplitudes as a function of

time. Therefore, when evaluating the mismatch between the field and the forward

modeled seismic we would be comparing amplitudes. The main advantage of this

approach is that it does not require the seismic inversion of the field data; however,

we would compare amplitudes which measure contrasts of acoustic impedance (zero

offset data).

Another approach would be to forward model and compare acoustic impedance

directly, which does not require us to perform a convolution or to compute the re-

flectivity series in time. Acoustic impedance is computed for the high resolution 3D

geo-cellular model using rock physics models; subsequently, the acoustic impedance


is filtered in order to mimic acoustic impedance data from an actual inversion field

data. The main advantage of this approach is that we would be comparing acoustic

impedance of each layer directly; however, the field seismic data needs to be inverted

to obtain acoustic impedance from the amplitude data. When comparing AI we rely

on the result of a seismic inversion which may not be unique.

Using any of the two approaches explained before to forward model seismic data,

we need to compute acoustic impedance at each of the times that real seismic data

is acquired over the study area.

In order to compute the time lapse acoustic impedance we need the petrophysical

properties modeled on the high resolution 3D geo-cellular model as well as the output

from the flow simulation (see Figure 2.10). Using a petro-elastic model (see Figure

2.11) the acoustic impedance at each time is computed on the high resolution grid;

hence, the output from the flow simulation needs to be downscaled. An approach for

downscaling the output from the flow simulation is presented in Chapter 3.

Upscale

Flow Simulation

Downscale[Pp, So, Sw, Sg] @ t1



Downscale[Pp, So, Sw, Sg] @ tn

realization (φ, k)

Petro-ElasticModeling

Acoustic Impedance @ t1






Acoustic Impedance @ tn

…

Figure 2.10: General procedure and input data needed for creating the Time-lapseacoustic impedance.


The petrophysical properties simulated on the high resolution model are typically

porosity and permeability (φ and k); from porosity we are able to compute ρ, Vp and

Vs for the dry rock (no fluids in the pore space) under certain effective pressure Peff

using rock physics relations - Vp(φ, Peff ) and Vs(φ, Peff ) - obtained in the laboratory

for the specific type of rocks in the study area. The effective pressure Peff is defined

as:

Peff = Poverburden − Ppore (2.10)

where Poverburden is the overburden pressure and Ppore is the pore pressure. The pore

pressure is obtained directly from the flow simulator, while the overburden pressure

could be assumed constant for the entire reservoir:

Poverburden = ρgz (2.11)

where ρ is the average density of the overburden, g is the gravitational constant, and

z is the average depth of the reservoir.

Having computed ρ, Vp and Vs for the dry rock under certain effective pressure

Peff , the same properties are updated according to the fluid present in the pore space

using Gassmann’s transformation [32].

Gassmann introduced a mathematical transformation that allows the calculation

of the elastic moduli of the fully-saturated rock from those of the dry rock:

Ksat = KminφKdry − (1 + φ)KfKdry/Kmin + Kf

(1− φ)Kf + φKmin −KfKdry/Kmin

(2.12)

where Ksat is the fully-saturated rock’s bulk modulus, Kf is the bulk modulus of

the fluid in the pore space, φ is the rock’s porosity, Kmin is the bulk modulus of the

mineral, and µ is the rock’s shear modulus.

Using the compressional Vp and shear Vs wave velocities obtained from the lab-

oratory rock physics relations - Vp(φ, Peff ) and Vs(φ, Peff ) - the dry bulk and shear


Peff = Poverburden - Ppore

Compute Vp and Vs (dry rock)

using Vp(φ,Peff) Vs(φ,Peff)

Compute ρ and k (oil, water and gas)

using [ρ(Ppore), k(Ppore)]o,w,g

Get k and µ for dry rock

Compute k and ρ for the effective fluid

Use Gassmann’s relationto obtain ksat, µ and ρsat for the

saturated sands

Compute Vp and Vs for saturated rockVp

2= (ksat +4µ/3)/ ρsat

Vs2=µ/ ρsat

So, Sw, Sgflow simulation

Pporeflow simulation

Figure 2.11: Petro-elastic model to create Vp, Vs and ρ, taking into account porepressure and fluid saturation effects.


modulus are computed using :

Kdry = ρdry(V2p −

4

3V 2

s ) (2.13)

µdry = ρdryV2s (2.14)

The bulk modulus of the fully-saturated rock Ksat is computed using Gassmann’s

relations (Eq. 2.12); however, the shear modulus µsat remains unchanged µsat =

µdry ≡ µ since shear stress cannot be applied to fluids. The density of the rock is also

transformed and the density of the fully-saturated rock is computed as:

ρsat = ρdry + φρf (2.15)

Having transformed the elastic moduli and the density, the compressional and

shear wave velocities of the rock with the second fluid are computed as:

Vp =

√√√√Ksat + 43µ

ρsat

(2.16)

Vs =

√µ

ρsat

(2.17)

The fluid contained in the rock could be a single fluid like brine or oil, however it

can also be a mixture of fluids. The bulk modulus of the fluid is needed in order to

perform Gassmann’s transformation; when dealing with single fluids, elastic moduli

can be gathered from tables [59], however this is not the case for mixtures and partial

saturations need to be considered.

The most common approach to modeling partial saturation (gas/water or oil/water)

or mixed fluid saturations (gas/water/oil) is to replace the set of phases with a single

“effective fluid”. One approach to obtain the bulk modulus of this “effective fluid”

Kf is using a weighted harmonic average, termed Reuss average in the rock physics

literature:

1

Kf

=Soil

Koil

+Swater

Kwater

+Sgas

Kgas

(2.18)


This model assumes that the fluid phases are mixed at the finest scale; however,

when patches of saturation exist, Brie’s approach [11] could be adopted to obtain the

bulk modulus of the “effective fluid”:

Soil + Swater

Kfluid

=Swater

Kwater

+Soil

Koil

Kf = (Kfluid −Kgas)(Soil + Swater)e + Kgas (2.19)

where e is a calibration parameter. By adjusting e, Brie et al. matched the in-situ

data. The density of the “effective fluid” ρf is computed using:

ρf = Soilρoil + Swaterρwater + Sgasρgas (2.20)

The bulk modulus of each phase (Koil, Kwater, Kgas) and the density of each phase

(ρoil, ρwater, ρgas) are a function of pore pressure; therefore they need to be computed

prior to the computation of the “effective fluid” properties according to the pore

pressure in each grid block of the model.

2.3.2 Applying the FM Approach to the Stanford V reservoir

A 3D reference model (Figure 2.12) is presented on which our approach will be tested.

The geological model corresponds to a fluvial depositional system with three facies:

channel facies (10000 mD), crevasse facies (1000 mD) and mud facies (100 mD). We

assume that permeability values are known, while the location of the high permeability

channels is unknown.

A vertical water injector is located at the lower left corner of the reservoir, injecting

1500 STB/day to an initially oil saturated reservoir; while a vertical producer well

is located at the upper right corner, producing 2000 STB/day. Using a simple black

oil model commercial simulator (ECLIPSE), 6 months of production were simulated

to obtain the water saturation in the reservoir at the end of the simulation time (see

Figure 2.13). Comparing figures 2.12 and 2.13, it is clear how the location of the high

permeability channels in the reservoir affects the sweep efficiency.


Figure 2.12: 3D reference reservoir facies model, shown in depth slices from top (topleft) to the bottom of the reservoir (bottom right). Floodplain facies in blue, channelfacies in green, crevasse facies in red.

Figure 2.13: Spatial distribution of water saturation in the reservoir after 6 monthsof production, as obtained from flow simulation using the reference model. Shown indepth slices from top (top left) to the bottom of the reservoir (bottom right).


Having the reference reservoir model and the water saturation in the reservoir

after 6 months of production, the reference 4D seismic response was created. Using

the facies realization in Figure 2.12, the reference reservoir model was populated

with porosity, compressional velocity and density by using a sequential simulation

algorithm:

• porosity was simulated using a target histogram and variogram.

• compressional velocity was computed using Han’s relations [40] for sandstones

and shaly sandstones, and Gardner’s relation for mudstones [31].

• density was computed from porosity as:

ρ = φρfluid + (1− φ)∑

i

ρmifi

where ρfluid is the density of oil, since the reservoir is initially fully saturated with

oil, ρmiis the density of the mineral mi and fi is the proportion of the mineral mi in

the rock.

Having the reservoir populated with the petrophysical properties as described be-

fore, the 4D seismic response of the reference model was created by forward modeling

two 3D seismic data sets:

• the first 3D seismic data set is forward modeled using the petrophysical prop-

erties of the reservoir fully saturated with oil.

• the second 3D seismic data set is forward modeled using the petrophysical

properties of the reservoir after 6 months of production.

If we were to take the difference of both data sets we would obtain a volume of

the 4D seismic response; however, it has been chosen to take the difference between

a 2D seismic attribute map obtained from each 3D data set. For each seismic trace

the instantaneous amplitude was calculated and the average value was taken. This

process was applied to all traces in each seismic data set (before and after production),

resulting in a map of average instantaneous amplitude for each of them. The difference


between the two maps is referred to from now on in this example as the 4D seismic

response. Therefore, the reference 4D seismic response is depicted in Figure 2.14;

the goal is to find a reservoir model realization which 4D seismic response that best

matches the reference in Figure 2.14.

Figure 2.14: 4D seismic response (difference average instantaneous amplitude map)from reference model.

The history matching procedure started with the initial guess reservoir model

in Figure 2.17, whose corresponding water saturation after production is shown in

Figure 2.18 and 4D seismic response is shown in Figure 2.19. The reservoir modeling

is carried out using the probabilistic approach described earlier in this chapter; the

training image used in this example is shown in Figure 2.15. The spatial probability

distribution of each facies from 3D seismic data was computed from a calibration

between well data and the reference 3D seismic (see Figure 2.16).

The correlation coefficient between the 4D seismic response of the initial guess and

the reference was computed as ρ = 0.18, which means that the value of the objective

function was f(ρ) = 1 − ρ = 0.82. The goal is therefore to decrease the value of

the objective function to its minimum, hence increasing the correlation coefficient

between the reference and the actual 4D seismic response.

The history matching procedure was performed using a maximum of four outer

iterations and 6 inner iterations. The best reservoir model found by this procedure


Figure 2.15: A slice of the 3D training image used for the facies modeling. Floodplainfacies in blue, channel facies in green, crevasse facies in red.

Figure 2.16: The spatial probability distribution of each facies (form left to right:crevasse, floodplain, channel) obtained from the calibration between well data andthe reference 3D seismic data.


Figure 2.17: 3D initial guess model, shown in depth slices from top (top left) to thebottom of the reservoir (bottom right). Floodplain facies in blue, channel facies ingreen, crevasse facies in red.

Figure 2.18: Spatial distribution of water saturation in the reservoir after 6 monthsof production, as obtained from flow simulation using the initial guess model. Shownin depth slices from top (top left) to the bottom of the reservoir (bottom right).


Figure 2.19: 4D seismic response (difference average instantaneous amplitude map)from initial guess reference model.

is shown in figures 2.20, 2.21 and 2.22. The correlation coefficient between the 4D

seismic response of the best model and the reference was computed as ρ = 0.79,

which means that the value of the objective function was f(ρ) = 0.21. This is a

considerable improvement in terms of the objective function that is reflected by the

similarity between the corresponding 4D seismic responses.

Comparing the water saturation predicted by the best model in Figure 2.21 with

the reference water saturation in Figure 2.13, an adequate agreement is achieved.

Using this resulting model, the distribution of fluids in the reservoir can be pre-

dicted, an important decision variable in future well planning. A good estimate of

the water saturation in the reservoir can help us to identify areas where oil has been

bypassed by the injected water, hence reducing the risk involved in placing future

wells to produce oil from such areas (see Figure 2.23).


Figure 2.20: 3D best reservoir model found by the history matching algorithm, shownin depth slices from top (top left) to the bottom of the reservoir (bottom right).Floodplain facies in blue, channel facies in green, crevasse facies in red.

Figure 2.21: Spatial distribution of water saturation in the reservoir after 6 monthsof production, as obtained from flow simulation using the best model found by thehistory matching algorithm. Shown in depth slices from top (top left) to the bottomof the reservoir (bottom right).


Figure 2.22: 4D seismic response (difference average instantaneous amplitude map)from best model found by the history matching algorithm

Figure 2.23: A slice of the water saturation cube after 6 months of production: fromreference model (left), from initial guess model (middle), from history matched model(right).

Chapter 3

Downscaling Saturation to Model

4D Seismic Response

4D seismic data can provide a spatial distribution of dynamic changes occurring in a

reservoir due to production. Dynamic changes (changes in pressure and/or saturation)

are inferred from 4D seismic data through changes in reservoir properties since seismic

velocities and impedance depend on changes in pore fluids, pressure and temperature

[63] [10] [90]. Additionally, 4D seismic data can provide static information about the

reservoir, since some of the dynamic changes (changes in fluid saturation) occur only

in a certain type of rock with high porosity and permeability.

Due to the valuable static and dynamic information it carries, 4D seismic data

should be incorporated into the reservoir modeling workflow. Chapter 2 discussed in

detail two approaches to achieve this goal: the PDI approach and the FM approach.

The PDI approach, presented in section 2.2, incorporates the information content

of 4D seismic data in a probabilistic fashion; as a constraint for the modeling of

a static property of the reservoir such as facies. The FM approach, presented in

section 2.3, incorporates 4D seismic data by treating it as dynamic data; in other

words, matching it along with production data. When history matching production

data only, the production response of the reservoir model is computed by performing

flow simulation. Therefore, when including the matching of 4D seismic data, the 4D

response of the reservoir model needs to be computed also by performing a “Forward

52

53

Modeling” (FM).

Forward modeling the 4D seismic response involves the computation of the seismic

attribute(s) as the spatial distribution of pressure and saturation in the reservoir

changes over time. For example, the change over time of acoustic impedance (AI)

can be computed as the set:

[AI]t1 , [AI]t2 , ... [AI]tn

where [AI]ti = [ρVp]ti is the acoustic impedance at time ti, obtained by computing

density (ρ) and P-wave velocity (Vp) of the rock subject to the conditions at time ti,

pressure and saturation at that time. A detailed explanation on the procedure used

to compute the effect of pressure and saturation on the density of a rock, as well as

their effect on the velocity of P and S waves as they travel through the rock was given

in Chapter 2 (section 2.3.1). Density, P and S-wave velocities for each grid block of

the high-resolution reservoir model are computed first and the conditions - pressure

and saturation - at time ti are imposed later. The spatial distribution of pressure and

saturation is obtained after flow simulation, which is performed on a coarsened grid.

The 4D seismic response is forward modeled using the high-resolution reservoir

properties since seismic waves are often affected by fine details below seismic resolu-

tion [58] [74] [75]. Important details such as “patches” of bypassed oil can affect the

seismic signature. Although sometimes seismic waves are not able to “resolve” those

features (i.e., identify their boundaries) they can “detect” them, since they influence

velocity and impedance. Coarse scale saturations from the flow simulator often show

a very smooth version of those “patches” or simply do not show them at all due to

upscaling. Therefore, coarsened saturations need to be ported into the high-resolution

grid through a downscaling procedure.

This chapter is dedicated to investigate the adequacy of state-of-the-art downscal-

ing methods. Additionally, this chapter presents a new dynamic downscaling method,

based on the work of Gautier et al. [33] and Chen et al. [24], that takes into account

both static information (high-resolution permeability field) and dynamic information

in the form of coarsened fluxes and saturations (solution of the global flow simulation).

54CHAPTER 3. DOWNSCALING SATURATION TO MODEL 4D SEISMIC RESPONSE

3.1 State-of-the-Art Downscaling Methods

Two main approaches have been documented in the literature to downscale coarsened

saturation from the flow simulation to the high-resolution model. The most common

approach consists simply of sub-sampling the single coarse grid block saturation S(v)

to all high-resolution grid cells u (see Figure 3.1).

Figure 3.1: Traditional approach of sub-sampling the single coarse grid block satura-tion S(v) to all high-resolution grid cells u.

The problem with this approach arises when fine details in saturation that are

not captured by the coarsened flow simulation have a strong impact on the seismic

response. The forward modeled “Time Lapse seismic data” is not able to accurately

predict the distribution of fluids since it has been modeled using a smooth or upscaled

version of saturation. Hence, there is a risk that the field 4D seismic data cannot

be accurately matched with this simple model. The solution to this problem cannot

rely on running the flow simulator on the high-resolution model. It is unfeasible, in

practice, to run a flow simulator using a reservoir model that often consists of millions

of grid cells. Moreover, incorporating the modeling of 4D seismic into the process of

history matching involves running tens or hundreds of flow simulations; therefore,

upscaling the high-resolution model cannot be avoided.

The second approach consists of downscaling the coarsened saturation from the

flow simulator guided by high-resolution static information. For example, for a given

coarsened grid block, the sub-grid (high-resolution) porosity and permeability is

known since it has been modeled using geostatistical methods prior to upscaling;

expecting high saturations to occur preferentially in high porosity and permeability

3.1. STATE-OF-THE-ART DOWNSCALING METHODS 55

areas, those static properties could guide a redistribution of the coarsened saturation

value through all the high-resolution grid cells within a coarse grid block. Figure 3.2

shows a schematic example of this static downscaling procedure, where the coarse grid

block saturation S(v) is downscaled by re-scaling of sub-grid porosity values Φ(u) to

saturation values S(u), while imposing the average of all sub-grid saturation values

within the coarse block to be equal to the coarse grid block saturation.

Figure 3.2: Static downscaling of coarse grid block saturation S(v) through a re-scaling of sub-grid porosity values Φ(u) to saturation values S(u), while imposingthe average of all sub-grid saturation values within the coarse block to be equal tothe coarse grid block saturation.

In this regard, previous work has been done by Sengupta [75] in downscaling

saturations from the reservoir simulator by incorporating high spatial frequencies from

well-log data (porosity, permeability, volume of shale). Sengupta (2000), presented a

reservoir monitoring case study of a reservoir that produces oil under water and gas

injection. The downscaling is performed at the well location (Figures 3.3 and 3.4) by

scaling and thresholding the flow simulator saturation profile according to the effective

porosity, high resolution permeability and volume of shale (Vshale) . Figure 3.3 shows

that the simulator model of permeability, porosity, and initial fluid saturations are

smoother than the same properties computed from well logs.


Figure 3.3: Comparing well-log and simulator properties: the red curves correspondto the flow simulator, while the blue curves correspond to the well logs. Left toright: φ =porosity, k =permeability, Sw =water saturation, and Sg/o =simulator gassaturation and well-log oil saturation. After Sengupta [75].


In order to downscale the saturation from the simulator, Sengupta [75] used the

well-log saturations. Figure 3.4 shows the original profile of the gas saturation (Sg)

taken from the flow simulator, and five estimates of the downscaled Sg, each repre-

senting a variation of the same downscaling approach. These estimates are computed

using the smooth Sg profile from the simulator and the initial oil saturation (1− Sw)

at the well location. Assuming gas is most likely to replace oil in the high porosity,

high permeability sands, the five estimates of Sg presented in Figure 3.4 were created

by Sengupta [75] as follows:

• profile (b) is created by setting to zero the smooth flow simulator profile Sg in

the zones of zero effective porosity,

• profile (c) is obtained by scaling profile (b) according to the high-resolution So

profile at the well, based on the assumption that gas is most likely to replace

oil,

• profiles (d) to (f) are obtained by successive hierarchical thresholding of profile

(c). Although it is not explicitly described, Sengupta mentions the thresholding

can be based on various rock physics parameters such as porosity, Vshale or high-

resolution permeability. As seen in Figure 3.4 an increase in vertical resolution

and Sg is seen from (d) to (f).

Using each of the downscaled 1D saturation profiles, the synthetic time-lapse

differential AVO attributes are computed for each profile (Figure 3.5), from the near

offset (5◦−15◦) and the mid offset (15◦−25◦). The time-lapse differential corresponds

to the percent change in root mean square (RMS) amplitude between the two seismic

surveys. The forward modeling of the 4D seismic response is performed at the well

location, no spatial reservoir modeling using geostatistical methods is done in this

example presented by Sengupta.

Figure 3.5 shows a cross plot between the time-lapse differential for near offsets

and mid offsets, where the color filled dots are values computed from the real seismic

data and the empty circles correspond to the synthetic values obtained from the

downscaled Sg profiles shown in Figure 3.4. From this figure, Sengupta concluded


Figure 3.4: Downscaling saturations from the flow simulator: (a): Sg taken from thesimulator, (b), (c), (d), (e), (f): Estimations of downscaled Sg. After Sengupta [75].


that honoring the vertical heterogeneity observed at the well decreases the mismatch

between the real and synthetic seismic.

Figure 3.5: Cross plot of time-lapse differential AVO attributes from real data aroundthe well, and from synthetics corresponding to smooth and downscaled saturationprofiles. The error bars represent the uncertainty in synthetic seismic attributes dueto the lack of information about spatial distribution and total amount of gas. Modifiedfrom Sengupta [75].

The results presented by Sengupta (2000) are very encouraging, however the down-

scaling method used is purely static, no dynamic considerations are taken into ac-

count. The distribution of saturations in the reservoir depends on the particular flow

problem: initial condition of the reservoir and flow boundary conditions. A system-

atic and generalized process is needed for downscaling the coarse scale volume of

saturations from the flow simulator that accounts for both static information and the

particular flow problem.


3.2 Flow-based Downscaling

The method proposed in this section for downscaling the coarsened saturations ac-

counts for both static information and the particular flow problem (initial and bound-

ary conditions). This procedure reconstructs the high-resolution saturation map by

simulating the local flow at every coarse grid block using the sub-grid (high-resolution)

permeability field and approximate local boundary conditions that are consistent with

the results of the flow simulation performed on the coarsened grid.

3.2.1 Governing equations

Consider Darcy’s law for two-phase oil-water flow:

uw =kkrw

µw

∇pw (3.1)

uo =kkro

µo

∇po (3.2)

where p is pressure, u is the Darcy velocity, k is permeability, µ is viscosity, and the

subscripts w and o refer to water and oil. The relative permeabilities of water and

oil are designated krw and kro.

Using these equations along with a statement of mass conservation on a control

volume (or grid block) for each phase, i.e., rate of change of mass in the volume equals

the net influx of mass, the following equations describing oil-water flow are obtained:

φ∂Sw

∂t+∇ · (uw) = 0 (3.3)

∇ · (λTk∇p) = 0 (3.4)

where t is time, λT is the total mobility (λT = krw/µw + kro/µo), and all other

variables are as defined previously. Note that here capillary forces and compressibility

effects are neglected. These two equations, referred to as the saturation and pressure

equations, are solved using an IMPES finite difference procedure. Specifically, the

pressure equation is solved first (implicitly) and then the saturation equation is solved

3.2. FLOW-BASED DOWNSCALING 61

explicitly.

3.2.2 Flow on the high-resolution and coarsened grids

Flow on the upscaled/coarsened grid, is performed using a commercial flow simulator.

The following variables are obtained for each coarse grid block v: water saturation

Scw(v), flux in the x direction qc

x(v) at x oriented (normal in the x direction) block

interfaces and flux in the z direction qcz(v) at z oriented block interfaces. Note that

the subscript c denotes a coarse grid quantity. Each coarse block v corresponds to

a region of the underlying high-resolution scale (geological) grid. We designate as

u the nx × ny high-resolution grid blocks that correspond to a particular v. Using

the information from flow on the coarsened grid, for each coarse block v, a flow

problem (Eqns. 3.3 and 3.4) is solved locally over the high-resolution grid blocks

u that correspond to v (Figure 3.6). For this simulation, the local high-resolution

permeability is used. Boundary conditions for the pressure equation are prescribed in

terms of the fluxes at each of the faces of the boundary blocks u, while the saturation

boundary conditions correspond to the saturation value at the inlet faces. The initial

condition is Sfw(u) = 0 for every high-resolution (superscript f) block.

For a particular coarse block, flux boundary conditions are computed for all high-

resolution cells comprising the boundaries of the coarse block. Following the work

of Gautier et al. [33] and Chen et al. [24], these high-resolution fluxes are assigned

in proportion to the high-resolution grid interblock transmissibility. Specifically, the

local high-resolution flux boundary conditions are computed along x = 0 and z = 0

(note that (x, z) = (0, 0) is the origin of the local grid system) as follows:

(qx)0,j =(Tx)1/2,j∑nzj=1(Tx)1/2,j

qcx j = 1, ..., nz (3.5)

(qz)i,0 =(Tz)i,1/2∑nxi=1(Tz)i,1/2

qcz i = 1, ..., nx (3.6)

where Tx and Tz are the high-resolution interblock transmissibilities in the x and z

directions. This reconstruction guarantees flux continuity across high-resolution cells

in neighboring coarse blocks and accounts for sub-grid heterogeneity, in addition to


Figure 3.6: Domains for flow on the coarsened and local high-resolution grids. Lighterlines represent the high-resolution grid and heavier lines the coarse grid.

forcing the sum of the high-resolution grid fluxes to be equal to the corresponding

coarse grid flux.

Saturation boundary conditions are established for the high-resolution grid inlet

boundaries of the coarse block by using the high-resolution saturation of the adja-

cent grid block (which was previously reconstructed). This ensures continuity at the

boundaries of the coarse grid blocks. The reconstruction starts at well blocks and fol-

lows the direction of flow. Local high-resolution boundary conditions are computed

for the target coarse grid block and local flow is performed until the average of the

reconstructed local high-resolution saturations matches the coarse saturation Scw(v).

The process is repeated for each coarse grid block until the high-resolution saturation

field is reconstructed.

This approach for reconstructing the high-resolution saturation field can be ap-

plied in an adaptive fashion, where local high-resolution flow is simulated only in the

areas of the reservoir with high-resolution saturation effects which impact the seismic

signal (e.g., an advancing water front in a region with a high degree of sub-grid per-

meability variability). With this procedure, areas of the reservoir already swept by

3.3. 2D SYNTHETIC EXAMPLE 63

the injected water would not be reconstructed because they exhibit a high constant

saturation value. This approach could be much more efficient computationally than

reconstructing the high-resolution saturation everywhere, particularly in cases where

the important saturation changes occur over relatively small portions of the reservoir.

This technique will however require some modification of the downscaling procedure

described above, for example in the specification of the inlet saturation boundary

conditions. The success of the adaptive flow-based downscaling method proposed

here depends directly on the method used for upscaling as well as the upscaling ratio

applied to the static data prior to the flow simulation. The upscaling method, and

the chosen upscaling ratio, should be able to keep the overall character and posi-

tion of the saturation front, since the adaptive flow-based downscaling method would

identify the front and work on its refinement.

3.3 2D Synthetic Example

To illustrate the overall methodology, a simple example that effectively demonstrates

the importance of downscaling saturation when modeling 4D seismic data is presented.

The 2D (layered) geological model used for the example is shown in Figure 3.7.

Acoustic properties such as impedance (Z = ρVp) are assumed constant within the

layer interval (ρ = 2.3g/cc, and Vp = 2300m/sec). Using a flow-based upscaling

procedure [27], the high-resolution permeability model is upscaled to obtain coarse

scale permeabilities (kx and ky) for each coarse grid block (Figure 3.8).

Using these upscaled permeabilities, we simulate flow driven by a water injector

at the left boundary of the reservoir and an oil producer at the right boundary. The

injection rate is 400STB/day, the production rate is 415STB/day, and 200 days of oil

production are simulated. The resulting water saturation in the reservoir at the end of

the simulation is downscaled to the high-resolution grid (see Figure 3.9). This figure

shows that the fine details of the saturation front are to some extent restored and

the smooth character has been removed. We reiterate that this is achieved without

running a global high-resolution flow simulation.


Figure 3.7: High-resolution permeability model (isotropic) for a layered reservoir.The axe correspond to the grid block number.

Figure 3.8: Coarsened effective permeability model (anisotropic) for the layered reser-voir: kx (left) and kz (right). The axe correspond to the grid block number.


Using the three saturation maps (coarse, flow-based downscaled, reference high-

resolution), the 4D seismic response is computed using the fluid substitution approach

explained earlier and a traditional zero-offset convolutional model as the seismic for-

ward model (see Figure 3.9). This result shows that using coarsened saturations

directly from the flow simulator to perform fluid substitution on the high-resolution

grid results in a smooth seismic image that does not exhibit the key features apparent

in the reference 4D seismic seismic.

On the other hand, using the flow-based downscaled saturations to perform the

fluid substitution on the high-resolution grid results in a seismic image that exhibits

some of the key features evident in the reference 4D seismic. However, the strong re-

flection amplitudes observed at the left of the reference section are not strong enough

in the modeled 4D seismic response obtained from the flow-based downscaled satu-

rations. This is due to errors in saturation at the left of the downscaled saturation

map, which result in less acoustic contrast.

Perhaps the most important features of the reference 4D seismic response are

the two middle positive reflections (in blue), which are coinciding at a distance of

2500 meters. This feature corresponds to the seismic response of fine details (below

resolution) in the waterfront, and it is completely reproduced by the modeled 4D

seismic response obtained using the flow-based downscaled saturation map.

Figure 3.10 shows two seismic traces extracted from the modeled 4D seismic re-

sponses obtained using the saturation maps shown in Figure 3.9. The seismic traces

are extracted at distances of 1000 meters and 3500 meters. These seismic traces

clearly show how the modeled 4D seismic response obtained using the flow-based

downscaled saturation map (red curve) reproduces the important features observed

in the reference 4D seismic response (blue curve), while the modeled 4D seismic re-

sponse obtained using the coarse scale saturation map (green curve) is inaccurate in

the middle section of the reservoir. The seismic trace extracted at 1000 meters shows

how the modeled 4D seismic response obtained using the flow-based downscaled satu-

ration map reproduces the main seismic reflections both in amplitude and time, while

the modeled 4D seismic response obtained using the coarse scale saturation does not

reproduce the amplitude of the reflections (it only reproduces the reflection from the


top of the reservoir) and misplaces the reflection time of the bottom of the reser-

voir. The seismic trace extracted at 3500 meters shows how the modeled 4D seismic

response obtained using the flow-based downscaled saturation reproduces the ampli-

tude “pinch-out” observed in the reference 4D seismic response, while the modeled

4D seismic response obtained using the coarse scale saturation is unable to identify

the pinch-out and instead introduces two reflections.

Figure 3.9: Saturation profiles (top row) and corresponding forward-modeled 4Dseismic responses (bottom row): coarse scale (left), flow-based downscaled (middle),reference high-resolution (right).

The results of this 2D synthetic example show the impact of fine details of the

saturation front on the 4D seismic response, hence the importance of reconstructing

such fine details from the coarsened saturation through a downscaling procedure. It

has also been shown how the novel flow-based downscaling procedure presented in

this chapter reproduced sufficient details of the saturation field, which enabled the

modeled 4D seismic response to better match the “reference” 4D seismic.


Figure 3.10: Two 4D seismic traces extracted from the modeled 4D seismic responsesobtained using: the coarse scale saturation map (top left), the downscaled saturationmap (top middle), and the high-resolution saturation map (top right). The seismictrace on the bottom left corresponds to a distance of 1000 meters, and the seismictrace on the bottom right corresponds to a distance of 3500 meters.

Chapter 4

Case Study: The Oseberg Field

The Oseberg Field is a large oil and gas accumulation located in the Norwegian sector

of the North Sea, in blocks 30/6 and 30/9, 140 Km west of Bergen (Figure 4.1).

Geologically, the field is located in the transition between the active Viking Graben

to the west and the stable Horda platform to the east. The field was discovered in

1979 and declared commercial in 1983. Hydrocarbons are trapped in three major

eastward rotated fault blocks: Gamma, Alpha, and Alpha North (Figure 4.2). All

three structures exhibit a gas cap.

The hydrocarbons are found trapped in the Middle Jurassic deltaic deposits of the

Brent Group: Oseberg, Rannoch, Etive, Ness and Tarbert formations [79]. These de-

posits represent a regressive/transgressive, large scale, cycle of sedimentation, where

the Ness formation constitutes the upper delta plain depositional unit [30] [37] [41] [46]

[72]. The Ness Formation is one of the main reservoirs of the Alpha North structural

segment of the Oseberg field, and represents the focus of this case study.

Until 1995 the Ness production consisted of oil from channel sands randomly

penetrated by vertical wells dedicated to reservoirs located at the deeper stratigraphic

levels. Since 1995, 14 dedicated Ness wells have been drilled horizontally 3-5 meters

above the oil-water contact (OWC) with pressure support sustained by up-dip gas

injection [55] [1]. Seismic inversion and advanced geosteering tools have led to an

increased accuracy in the capability to penetrate a high percentage of Ness sand

bodies. Additionally, increased experience in drilling into the Ness Formation has led

68

69

Figure 4.1: Structural overview of the northern North Sea and its oil fields. Detail ofthe location of the Oseberg field (modified from Smethurst [77]).

70 CHAPTER 4. CASE STUDY: THE OSEBERG FIELD

to a significant decrease in formation associated drilling problems.

According to Liestøl et al. [55], it is very difficult to predict the reserves associ-

ated with a Ness producer since individual production profiles from Ness wells display

a great variety in production, and low production has been experienced from wells

penetrating up to 70 % channel sands. The best Ness producers have without excep-

tion had some degree of gas-breakthrough, which ensures both pressure support and

efficient sweep of up flank oil. The poorer Ness producers have experienced no gas

or water breakthrough, or only water breakthrough. In addition, some wells struggle

with rapid depletion, reflecting poorly connected volumes.

The main reasons for the production variability and low recovery (26%) experi-

enced from the Ness formation compared to the other shallow-marine reservoirs in the

Oseberg Field are associated with the complexity and heterogeneities of the fluvial

Ness reservoirs [55]. Some of the general observations of Liestøl et al. about the lack

of efficient pressure support from the gas cap in the wells are:

• Large vertical and horizontal heterogeneities in the Ness Formation could cause

nonexistent or poor communication pathways between the channel sands pene-

trated by the well and channel sands in the gas cap.

• The communicating volume around a Ness producer may be so isolated and

limited that the pressure depletion quickly becomes too large to maintain pro-

duction.

• A small part of the sands in the well receives all the pressure support. Pro-

duction from these sands will then dominate and gas flood the well preventing

efficient sweep of sands with poorer pressure support.

As established by Liestøl et al., the distribution of the channel sands in the Ness

reservoir is the parameter that controls production performance in the field. The

distribution of channel sands involves the orientation of the channels, the local pro-

portion of sand and more importantly in this case, the degree of communication

between channel sand bodies. Therefore, the goal of the case study presented in this

chapter is to build several models of the distribution of the channel sands in the Upper

4.1. ALPHA NORTH SEGMENT - UPPER NESS FORMATION 71

Ness Formation of the Alpha North structural segment of the Oseberg field. Each of

these models would be created using the two reservoir modeling workflows, presented

in Chapter 2, to incorporate 4D seismic data: the PDI and FM approaches.

This chapter first introduces the details of the reservoir under study, in the Alpha

North structural segment of the Oseberg field. Subsequently, the available data and

how they are incorporated into the reservoir modeling workflow are presented in

section 4.2. The last two sections of the chapter are devoted to present the results of

the application of the two workflows (PDI and FM approaches) to build a reservoir

model for the Upper Ness in Alpha North; a detailed comparison of the results of the

two approaches is also provided.

4.1 Alpha North Segment - Upper Ness Formation

The Alpha North structural segment, located on the northernmost part of the Oseberg

Field (Figure 4.2), is bounded by normal faults towards the Alpha Main structure in

the South and West and the Theta structure towards north. Cretaceous erosion delin-

eates the western extent of the Brent group reservoirs, while the cap-rock constitutes

late Jurassic and Cretaceous mudstones (Figure 4.3).

One of the main reservoir units of the Alpha North segment is represented by the

Upper Ness Formation (UN). The thickness distribution of the UN on Alpha North

is between 25 and 40 meters TVT (true vertical depth), increasing gently towards

N-NNE, with an average thickness of 33 meters TVT.

The UN corresponds to the upper delta plain depositional unit of the Brent group.

According to Liestøl et al., four sedimentary facies can be identified in the UN: chan-

nel, crevasse, coal and floodplain. The channel facies represent the reservoir unit

in the Ness Formation and is characterized by sharp-based, thick (2.5 - 27 meters)

fining-upwards sandstone units. The properties of the channel sandstones of the UN

on Alpha North gives an average porosity of 24% and and average horizontal perme-

ability of 2600 mD.

Stratigraphically, the distribution of channel-sands forms a four-partition pattern

with two zones of high content of sand alternating with two zones of low sand content


shown in Figure 4.4. According to Liestøl et al., paleocurrent measurements show a

main northwards transport direction, with a deviation span ranging from NNW to

NE, throughout the Ness Formation in the Oseberg area; and this is also supported

by seismic data.

The average individual sandstone body thickness is between 4 and 6 meters; how-

ever, amalgamated multistory sandstone bodies average 27 meters thick in central

parts of the Alpha North segment. An average channel width is in the order of 100

to 125 meters, and an average channel belt width is between 250 and 350 meters [55].

Figure 4.2: Outline of the Oseberg Field and its major fault blocks: Alpha, AlphaNorth, and Gamma (modified from Johnstad et al. [47]).

4.1. ALPHA NORTH SEGMENT - UPPER NESS FORMATION 73

Figure 4.3: West-East Seismic cross-section from the Alpha North segment, showingthe top and base of the Brent Group and the Base Cretaceous erosion (modified fromRutledal et al. [71].

Figure 4.4: Conceptual overview of the four-partite compartmentalization (from topto the bottom of the reservoir) of the channel development of the UN on Alpha North(modified from Liestøl et al. [55]).


4.2 The Reservoir Modeling Workflow

The Alpha North segment of the Oseberg Field exhibits an excellent variety of data,

which has been provided by the team of geologists, geophysicists, geostatisticians

and reservoir engineers of the company that operates the field (Norsk Hydro). The

available data consists of: well-logs for approximately 15 wells; a training image of

the facies distribution in the reservoir; two facies probability cubes, one created from

the Base 3D seismic survey and another one created from the 4D seismic data; the

properties of the fluids in the reservoir; the relationships (obtained in the lab from core

measurements) between seismic velocities, porosity and effective pressure; 4D seismic

data consisting of the results of an elastic inversion applied to each of the three seismic

surveys acquired in 1992, 1999 and 2004 respectively; historical production data and

a flow simulation model built in ECLIPSE 1.

Using the workflow and methodology presented in Chapter 2, data and procedures

from the Alpha North were provided to each of the main modules of the workflow.

The main modules of the workflow are the following:

• Building a high-resolution 3D geocellular model

• Building a 3D coarsened model

• Building a flow-simulation model

• Establishing the history matching procedure

4.2.1 The High-resolution 3D Geocellular Model

The architecture of the reservoir (structural framework) in terms of horizons and

faults, typically determined from 3D seismic data and well-markers, was established

by the team at Norsk Hydro who constructed the 3D stratigraphic grid used in this

case study. The work presented in this section corresponds solely to the population

of the grid with facies and petrophysical properties.

1 c© Copyright 2005 Schlumberger. All rights reserved.

4.2. THE RESERVOIR MODELING WORKFLOW 75

The high-resolution 3D geocellular model is comprised of 96×128×70 grid blocks;

each grid block is 25 meters wide (x and y direction), and approximately 0.8 meters

thick (z direction). The grid is populated with facies and petrophysical properties

simulated using well-log data (see Figure 4.5), the training image of the distribution

of facies in the reservoir (see Figure 4.6), and the facies probability cube created from

3D and/or 4D seismic information (see Figure 4.7). A two-step approach is followed

to populate the high-resolution 3D geocellular model. The first step is to simulate a

facies model, where each facies is populated with porosity and permeability values in

a secondary step.

According to Liestøl et al. [55], four sedimentary facies can be identified in the

Upper Ness: channel, crevasse, coal and floodplain. The channel facies represents the

reservoir unit in the Ness Formation. Due to the strong impact the distribution of

channel facies has on the production performance of the reservoir, the facies to be

modeled have been reduced to two: channel and floodplain.

The facies distribution is simulated using the multiple-point “single normal equa-

tion simulation” (snesim) algorithm. Any simulated facies realization is conditioned

to the facies hard data (see Figure 4.5), follows the geologic concept depicted by the

training image (see Figure 4.6), is conditioned to the facies probability cube being

used, and is also conditioned to the vertical sand proportion curve (see Figure 4.9).

A conceptual overview of the orientation of the channel facies is presented by

Liestøl et al. and shown in Figure 4.4; in order to “mimic” this concept, the simulation

algorithm (snesim) is given certain rotation angles which would be used to rotate

the training image and obtain simulated channels in the orientation expected by the

geologists (conceptualized in Figure 4.4). The reservoir shows channels approximately

oriented E-W towards the top, and channels approximately oriented N-S towards the

bottom. The detailed parameter file used by the multiple-point algorithm snesim

is shown in Appendix B. Figure 4.10 shows three equiprobable realizations obtained

with this algorithm; all conditioned to the same data.

Porosity and permeability are simulated for each facies independently, conditioned

to the hard porosity and permeability data (see Figure 4.5). Porosity is simulated first


using “sequential Gaussian simulation” (sgsim), and permeability is subsequently co-

simulated using “sequential Gaussian co-simulation” (sgcosim) with porosity as sec-

ondary information. The histograms of both porosity and permeability are obtained

from the well-log data and the variograms used for sequential simulation (shown in

Table 4.1) are provided by Norsk Hydro. The final porosity and permeability models

are obtained after “cookie-cutting” the porosity and permeability realizations using

the facies realization (see Figures 4.11 and 4.12).

floodplain channeltype Exponential Exponential

nugget 0.0 0.0ranges 1000/1000/10 1000/250/12.5angles 0/0/0 0/90/0

Table 4.1: Variograms used for simulating porosity and permeability for each facies;ranges are shown in meters.

4.2.2 The 3D Coarsened Model

Although the high-resolution 3D geocellular model only consists of 96×128×70 grid

blocks, less than a million cells; the complexity of the flow simulation model (see

section 4.2.3) precluded performing a single flow simulation in a reasonable compu-

tational time for history matching purposes. The reservoir model is to be iteratively

perturbed in a history matching loop until a satisfactory match of dynamic data is

achieved; usually, multiple (often hundreds) flow simulations are required in order to

achieve such a satisfactory match. Hence, it is imperative to reduce the computational

time to obtain useful results for making decisions about the field.

The high-resolution 3D geocellular model was coarsened from 96× 128× 70 grid

blocks to 48 × 64 × 14 grid blocks, using an upscaling ratio of 2 : 2 : 5 and a single-

phase flow-based upscaling procedure developed by Deutsch in 1985 [27]. The single-

phase flow-based upscaling procedure consists of finding the effective permeability of

a coarse grid block comprising fine scale heterogeneities that would give the same flow

through a homogeneous coarse grid block of the same size.


Figure 4.5: Well-log data available for the UN on Alpha North: binary facies classi-fication (top left), where zero represents floodplain (red) and one represents channelfacies (blue); porosity (top right) and the base 10 logarithm of permeability (bottom).

Figure 4.6: Binary training image showing channels facies in blue and floodplainfacies in gray (provided by Norsk Hydro).


Figure 4.7: Channel facies probability cubes provided by Norsk Hydro. The proba-bility cube shown on the left has been obtained from a calibration between well-logdata and the elastic inversion of the Base 3D seismic survey (acquired in 1992). Theprobability cube shown on the right has been obtained from a calibration betweenwell-log data and the elastic inversion of the 4D seismic data (surveys acquired in1992, 1999 and 2004) using the procedure presented by Andersen et al. [3].

Figure 4.8: Vertical sand proportion curve (modified from Andersen et al. [3]).


Figure 4.9: Regions of the reservoir where the training image is rotated for simulationpurposes; the rotation angle is measured clockwise with respect to the North. In thetop region, shown in blue, the angle of rotation corresponds to 0◦. In the bottomregion, shown in red, the angle of rotation corresponds to 72◦.

Figure 4.10: Three conditional facies realizations obtained with the “single normalequation simulation” (snesim) algorithm. Channels facies is depicted in blue andfloodplain facies in gray.

Figure 4.11: Porosity realization, simulated first using “sequential Gaussian simula-tion” (sgsim), for the floodplain facies (left) and the channel facies (middle). Usinga “cookie-cut” approach, the final porosity realization is obtained (right). The faciesrealization used for “cookie-cutting” is the first one (left) shown in Figure 4.10.


Figure 4.12: Permeability realization, co-simulated using “sequential Gaussian co-simulation” (sgcosim), for the floodplain facies (top left) and the channel facies (mid-dle). Using a “cookie-cut” approach, the final permeability realization is obtained(right); permeability values are shown as log10(perm). The facies realization used for“cookie-cutting” is the first one (left) shown in Figure 4.10.

The single-phase steady-state incompressible flow equation:

∇ · (k∇p) = 0 (4.1)

is solved for each direction parallel to the coarse block coordinate axis, with boundary

conditions:

p = pin (at the inflow boundary)

p = pout (at the outflow boundary)

∂p

∂n= 0 (at the boundaries parallel to flow)

Having the steady-state pressure distribution, the effective permeability is ob-

tained by equating the flux through the coarse heterogeneous grid block with the flux

through the equivalent coarse homogeneous grid block of the same size and under the

same boundary conditions.

The validity of this upscaling approach for this case study has not been checked.

However, more complex upscaling/upgridding techniques could be used as they are

more adequate for channel systems (see Tureyen, 2005 [86]; Chen et al., 2003 [24]).


4.2.3 The Flow-Simulation Model

The flow-simulation model was provided by Norsk Hydro. Eight wells are active

during the flow simulation: three producers (one vertical producer and two dedicated,

long horizontal UN producers close to the OWC), and five injectors (see Figure 4.13).

The producers are controlled by reservoir fluid volume rate (ECLIPSE well control

mode “RESV” under “WCONHIST” keyword) corresponding to the observed phase

flow rates (historical oil, water and gas production), computed using the average

hydrocarbon pressure of the field. The injectors are controlled by surface flow rate of

the fluid being injected (gas or water).

Figure 4.13: View of the flow simulation grid (top), colored by oil saturation, and theactive wells in the flow simulation (bottom). Note the two dedicated, long horizontalUN wells.

Flow simulation starts during the year of 1992 and ends in 2005; during this period

of time oil production takes place in wells C-21, C-19, C-18AT2 and C-17D, while

gas injection takes place in wells C-22, C-13AT2, C-8T2K and C-5. There is a gas

cap with an original GOC located 2497 meters deep and a WOC located 2718 meters


deep.

Flow simulation is performed using a black oil model; three phases are present in

the reservoir with their relative permeability curves shown in Figure 4.14. A drainage

oil-water capillary pressure function is defined and used only to equilibrate the model

(see Figure 4.15); capillary pressure is zero during flow simulation.

Figure 4.14: Oil and water relative permeability curves for regions where only oil andwater are present (left). Oil and gas relative permeability curves for regions whereonly oil, gas and connate water are present (right).

Figure 4.15: Drainage water-oil capillary pressure.

There is a gas cap with an original GOC located 2497 meters deep and a WOC

located 2718 meters deep. The irreducible water saturation in the reservoir Swi is


computed as a function of permeability:

Swi = −0.20068 log(k) + 0.02014 log(k)2 + 0.53173

The flow simulation grid consisted originally of the 3D coarsened model with

two additional layers representing the Tarbert reservoir at the top and the ORELN

(Oseberg - Rannoch - Etive - Lower Ness) at the bottom. In order to improve the

flow simulation time, the two additional layers were inactivated since there was no

communication with UN.

4.2.4 Establishing the History Matching Procedure

Establishing the history matching procedure involves defining: the parameter(s) of

the model to be perturbed, the perturbation method to be used, the dynamic data

to be matched, and the objective function to be minimized.

As observed by Liestøl et al. [55], the distribution of the channel sands in the

Ness reservoir is the parameter that controls production performance in the field.

Therefore, the goal of the history matching procedure is no other than to perturb

the distribution of the channel sands in the reservoir model. As thoroughly discussed

in Chapters 1 and 2, our proposed reservoir modeling workflow follows the “Parallel

Modeling Approach” proposed by Tureyen in 2005 [86], and the “Probabilistic Data

Integration Approach” introduced by Hoffman in 2005 [44]. Therefore, the perturba-

tion is performed on the high resolution 3D geocellular model using a method called

“Probability Perturbation Method” [15] which perturbs the conditional probability

from which the initial facies realization has been drawn; the perturbation is done by

introducing another probability model that depends on dynamic data, as explained

in section 2.1.2.

Dynamic data includes all data that have been measured or interpreted over time;

such as: production data (fluid rates or volumes, pressure data), and 4D seismic data

(any suite of 3D seismic attributes computed from each seismic survey). Both produc-

tion and 4D seismic data are available for the reservoir under study in this chapter;

therefore, the application of the two workflows proposed in Chapter 2, pursues either


matching production data only (PDI approach) or matching both production and 4D

seismic data (FM approach).

Production and 4D seismic data

The production from Alpha North started in 1991 with a two-front production

drive system; gas injection in the initial gas cap and water injection in the water leg

(see Figure 4.17). Until 1995 the Ness production consisted of oil from channel sands

randomly penetrated by vertical wells dedicated to reservoirs located at the deeper

stratigraphic levels. Since 1995, 14 dedicated Ness wells have been drilled horizontally

3-5 meters above the oil-water contact (OWC) with pressure support sustained by

up-dip gas injection [55] [1]. Two of those wells (C-19 and C-17D) are dedicated

Ness producers in Alpha North (see Figures 4.18 and 4.19). The goal of the history

matching procedure would be matching the total oil and water production for these

two dedicated long horizontal wells.

Production started to decline in 1996 (see Figure 4.16) and as IOR initiative, a

time-lapse seismic survey was acquired in 1999 to be compared with the base survey

from 1992. An elastic inversion procedure was applied by Norsk Hydro to both seismic

data sets, providing acoustic impedance (AI) and Vp/Vs ratio for each seismic survey;

the results were extensively used at that time in defining the location of in-fill wells.

In 2004, a new seismic data set was acquired and Norsk Hydro performed a new

elastic inversion (see Figure 4.20). The results of the elastic inversion on the three

seismic surveys were supplied by Norsk Hydro, sampled in the same grid used for the

high-resolution 3D geocellular model.

The 4D seismic data is shown in Figure 4.21 as ratios between the new survey and

the old survey:

[AI]1999

[AI]1992

[Vp/Vs]1999

[Vp/Vs]1992

[AI]2004

[AI]1999

[Vp/Vs]2004

[Vp/Vs]1999


Figure 4.16: Historical field production data from the Upper Ness formation in AlphaNorth. Field pressure shown in black, total field oil production shown in green andtotal field water production shown in blue.

Figure 4.17: Historical field injection data from the Upper Ness formation in AlphaNorth. Total field gas injection shown in red and total field water injection shown inblue.


Figure 4.18: Historical production data from well C-19. Total oil production shownin green and total water production shown in blue.

Figure 4.19: Historical production data from well C-17D. Total oil production shownin green and total water production shown in blue.


Figure 4.20: Results of the elastic inversion of the three seismic surveys of the Al-pha North segment. The top row shows Acoustic Impedance at 1992, 1999 and 2004respectively. The bottom row shows the Vp/Vs ratio at 1992, 1999 and 2004 respec-tively.


Figure 4.21: 4D seismic data shown as the ratio between the seismic attribute obtainedfrom the new survey and the same attribute obtained from the old survey. The toprow shows the AI ratios [AI]1999/[AI]1992 and [AI]2004/[AI]1999 respectively. Thebottom row shows the Vp/Vs ratios [Vp/Vs]1999/[Vp/Vs]1992 and [Vp/Vs]2004/[Vp/Vs]1999

respectively.


The 4D response obtained from the time-lapse datasets is summarized by An-

dersen et al. [3] in a crossplot of the Vp/Vs ratio against AI ratio. This graphical

representation enabled Andersen et al. to classify and interpret the 4D response ac-

cording to the quadrant where data fall into by using fundamental rock physics rules

(see Figure 4.22). The center of the crossplot is referred to as “no change” in both

Vp/Vs and AI since the ratios are equal to one. A data point located away from the

center of the crossplot means that either one or both seismic attributes have changed

during the time span being considered; Vp/Vs and AI could have increased or de-

creased due to changes in pressure and/or saturation. Four possible scenarios are

identified when the two variables (Vp/Vs and AI) increase and/or decrease, they are

as follows:

• Scenario 1 (Vp/Vs ↑ and AI ↓): is interpreted by Andersen et al. as an increase

in pore pressure. When pore pressure increases, the effective pressure decreases

(Eqn. 2.10) causing an increase in Vp/Vs and a decrease of AI. Scenario 1 is

shown in light blue on Figure 4.22.

• Scenario 2 (Vp/Vs ↑ and AI ↑): interpreted by Andersen et al. as a waterflooding

process (water displacing oil), and generalized in this dissertation as a high bulk

modulus fluid (K2) displacing a low bulk modulus fluid (K1) in the pore space

(K2 > K1). Since Vp is more sensitive to fluids than Vs, both Vp/Vs and AI will

increase as the bulk modulus of the fluid in the pore space increases. Scenario

2 is shown in yellow on Figure 4.22.

• Scenario 3 (Vp/Vs ↓ and AI ↑): interpreted by Andersen et al. as a decrease in

pore pressure, reasoned in a way as in the first scenario. Scenario 3 is shown in

orange on Figure 4.22.

• Scenario 4 (Vp/Vs ↓ and AI ↓): interpreted by Andersen et al. as a gasflooding

process (gas displacing oil), and generalized in this dissertation as a low bulk

modulus fluid (K2) displacing a high bulk modulus fluid (K1) in the pore space

(K2 < K1). In such a case, both Vp/Vs and AI will decrease as the bulk modulus


of the fluid in the pore space decreases. Scenario 4 is shown in brown on Figure

4.22.

0.8 0.85 0.9 0.95 1.0 1.05 1.1 1.15 1.2

0.9

0.95

0.85

0.8

1.05

1.1

1.15

1.21999/19922004/1999

Pp ↑

Pp ↓Kf2 < Kf1

Kf2 > Kf1

Figure 4.22: Summary and classification of the 4D field response (modified fromAndersen et al. [3]).

The rock physics based classification and interpretation procedure allows us to

map areas in the reservoir affected by each of the processes mentioned before (see

Figure 4.23). Pressure and saturation changes are not independent events since typ-

ically changes in pressure will cause changes in the distribution of fluids. However,

one effect may dominate over the other depending on the strength of each and also

on the type of lithology, since shales typically do not show saturation effects due to

their low permeability.

The Alpha North segment counts with three seismic surveys which allows analyz-

ing two 4D responses using the rock physics based classification and interpretation

procedure. Data belonging to quadrants NW, NE, SE, and SW (seen in Figure 4.22)

have been classified as scenarios 1 through 4 respectively; the results are shown in

Figure 4.23. The 4D response 1999/1992 shows stronger pressure depletion effects

than fluid effects, while the 4D response 2004/1999 shows all effects with the same


strength. This observation shows clearly the reaction of the reservoir to production;

the first time period is associated with primary depletion of the reservoir (pressure

decrease and consequently gas coming out of solution); while the second time period

is associated with the injection of gas and water.

The classification shown in Figure 4.23 has been obtained after applying the cri-

teria shown in Figure 4.22, which is obtained after crossploting the ratios:

[AI]1999

[AI]1992

[Vp/Vs]1999

[Vp/Vs]1992

[AI]2004

[AI]1999

[Vp/Vs]2004

[Vp/Vs]1999

Some data points exhibit AI and Vp/Vs ratios very close to the unit value (“no

change”). However, these data points, located close to the center of the crossplot,

cannot be distinguished from noise related to the seismic surveys, causing non-physical

responses in the amplitudes. Any possible real subtle change that may appear in the

vicinity of the center of the crossplot is very likely to be misclassified due to the noise;

therefore, it is not certain that points within a circular area of the center are actual

physical responses.

Figure 4.24 shows three different assumptions on the magnitude of the noise in

the data (low, medium, high) that could be used to map the areas with reliable field

4D seismic responses (see Figures 4.25 and 4.26). From these figures the following

observations are relevant:

• The first 4D seismic response (1992-1999) has more reliable information than

the second 4D seismic response (1999-2004). Generally, as the level of noise

increases more data points are classified as noise; however, for a constant level of

noise more data points are classified as noise in the second 4D seismic response.

• The data points classified as noise are generally located near the boundaries of

the reservoir.

• A level of noise of 4% is a reasonable value to use, hence only the informative

data from each 4D seismic response is used.


The goal of the history matching procedure will be to globally match the overall

proportions of each scenario occurring in the field 4D responses observed in Figure

4.22. If such global match can be obtained then, the reservoir static and flow model

adequately match the overall pressure and saturation changes. Matching the overall

global proportions of each scenario on the two 4D responses implies that the model

should reproduce the contribution of each physical process occurring in the reservoir

during the observed time period. For example, during the period 1992-1999 it is

observed more pressure decline rather than pressure increase; hence, it is expected to

obtain a larger proportion of scenario 3 and a rather low proportion of scenario 1.

Locally matching the occurrence of each physical process is also important since it

may provide us with two very important pieces of information about the reservoir: (1)

local information of the type of facies present and, (2) the connectivity of sand bodies

(flow). Pressure will affect both sand channel and mud floodplain; however, fluid flow

takes place only in sand channels. When pressure decreases as oil is produced, gas

may come out of solution in the sand channels and try to migrate up the gas cap as

other fluids such as oil and water remobilize in the reservoir; however, such sand body

may or may not be connected to provide a flow path for gas to migrate to the gas

cap. These effects have been observed in the modeling of the 4D seismic response (see

section 4.4.1); therefore, locally identifying these interpreted physical processes could

bring important insights to the presence of sand bodies as well as their connectivity.

The resolution of the 4D seismic data is about 25 meters, the thickness distribu-

tion of the reservoir is between 25 and 40 meters TVT with an average thickness of

33 meters, and the average individual sandstone body thickness is between 4 and 6

meters with amalgamated multistory sandstone bodies averaging 27 meters thick in

central parts of Alpha North. Clearly, the resolution of the 4D seismic data precludes

the identification or delineation of individual sandstone bodies in the reservoir (par-

ticularly in the vertical direction); therefore, it is interpreted that only an average

vertical response of the reservoir is what is observed in the 4D data. Rather than

matching the classified 3D volumes shown in Figure 4.23, the goal would be to match

2D proportion maps of each scenario.

The proportion map of each scenario occurring in the field 4D responses should be


matched in the areas of the reservoir where the field 4D response is classified as reliable

(blue areas in Figures 4.25 and 4.26). Moreover, since the first 4D seismic response

(1992-1999) has more reliable information than the second 4D seismic response (1999-

2004), as observed in Figures 4.25 and 4.26; only the vertical proportion maps of

each scenario occurring in the first field 4D seismic response (1992-1999) have been

chosen as the 4D seismic data to be matched when applying the FM approach. The

proportion maps of each scenario, obtained from the classified volume of the field 4D

response between 1992-1999 (shown in Figure 4.23) are shown in Figure 4.27. Note

that the data shown corresponds only to the areas where the 4D seismic signal is

classified as reliable.

Figure 4.23: Slices of the classified field 4D seismic response from the Alpha Northsegment (Upper Ness Formation), from the top to the bottom of the reservoir (slicesshown from left to right). The first row of slices represents the 4D response betweenthe years 1992 and 1999 (top); the second row of slices represents the 4D responsebetween the years 1999 and 2004. The colors represent the classes interpreted inFigure 4.22.


0.8 0.85 0.9 0.95 1.0 1.05 1.1 1.15 1.2

0.9

0.95

0.85

0.8

1.05

1.1

1.15

1.21999/19922004/1999

Pp ↑

Pp ↓Kf2 < Kf1

Kf2 > Kf1

2%4%6%

Figure 4.24: Summary and classification of the 4D field response. The circles representthe percentage of noise (undistinguishable change) used in evaluating the reliabilityof the seismic response in certain areas of the reservoir (modified from Andersen etal. [3]).


Figure 4.25: Slices of the classified field 4D seismic response between 1992 and 1999,from the top to the bottom of the reservoir (slices shown from left to right). The whiteareas represent the data points inside the noise circle and the blue areas representthe data points outside the noise circle shown in Figure 4.24. Hence, white representsareas that may not allow discriminating noise from a physical response. Each row ofslices represents the classification after using 2%, 4% and 6% of noise, respectively.


Figure 4.26: Slices of the classified field 4D seismic response between 1999 and 2004,from the top to the bottom of the reservoir. The white areas represent the datapoints inside the noise circle and the blue areas represent the data points outsidethe noise circle shown in Figure 4.24. Hence, white represents areas that may notallow discriminating noise from a physical response. Each row of slices represents theclassification after using 2%, 4% and 6% of noise, respectively.


Figure 4.27: Vertical proportion maps of (from left to right) scenario 1: increase inpore pressure, scenario 2: waterflooding, scenario 3: decrease in pore pressure, andscenario 4: gasflooding. The data shown corresponds only to the areas where the 4Dseismic signal is classified as reliable (using 4% of noise).

4.3 Integrating 4D Seismic Data: PDI Approach

As thoroughly explained in section 2.2, the PDI Approach proposes to integrate 4D

seismic data into the reservoir modeling workflow as a spatial probability distribution.

Figure 2.4 shows the schematic reservoir modeling workflow using the PDI Approach.

The main modules of the workflow are explained in section 4.2 of this chapter.

In order to simulate the high-resolution 3D geo-cellular model, the PDI approach

uses the channel facies probability cube (see Figure 4.7) obtained by Norsk Hydro

from a calibration between well-log data and the results of an elastic inversion on

the 4D seismic data (surveys acquired in 1992, 1999 and 2004) using the procedure

presented by Andersen et al. [3] (reviewed in section 2.2.1).

The goal of the history matching procedure would be matching the total oil and

water production of the two dedicated long horizontal Ness producer wells C-19 and

C-17D (see Figures 4.18 and 4.19). The objective function O(irD(u)) used by the

history matching procedure is the following:

O(irD(u)) =

Nw∑k=1

(∣∣∣∣∣Poil(irD(u))− P obs

oil

max(P obsoil )

∣∣∣∣∣+∣∣∣∣∣Pwater(irD

(u))− P obswater

max(P obswater)

∣∣∣∣∣)

k


where P obsoil and P obs

water are the historical total oil and water production; Poil(irD(u))

and Pwater(irD(u)) are the simulated total oil and water production for realization

irD(u); Nw is the number of wells to be matched (2 wells in this case: C-19 and

C-17D).

An initial guess i(0)(u) of the high-resolution facies model is simulated and shown

in Figure 4.28; using the probability perturbation method, the joint probability dis-

tribution used to simulate the high-resolution facies model is perturbed iteratively.

This perturbation yields a change in the location of the channels; the magnitude of

the perturbation is determined by the parameter rD, described in section 2.1.2. A

one-dimensional optimization is carried out to obtain the rD value that moves the

realization closer to matching production data; the final realization obtained with

the best rD value is denoted as i(1)rD

(u). Using this realization as the initial one, the

process is repeated to obtain i(2)rD

(u) after changing the random seed used for gener-

ating the reservoir model; the process stops when a history match is achieved, hence

the objective function reaches a value below a certain tolerance. The final high-

resolution model obtained with this procedure is shown in Figure 4.28; this model

not only matches production history, but also follows the same geological continuity

model and honors the same well log and 3D/4D seismic probability cube as the initial

model.

The results obtained from the history matching procedure are shown in Figures

4.30 and 4.31; these figures show the actual historical data, the simulated production

from the initial guess model, and the best match obtained after several flow simu-

lations. The decrease of the objective function with the number of flow simulations

(or inner iterations) is shown in Figure 4.29. A maximum of five inner iterations is

performed for each outer iteration; an inner iteration represents an optimization of

the rD value by means of a one-dimensional optimization, while an outer iteration

represents a change in the random seed to generate the high-resolution model.

A satisfactory history match of the total oil and water production is achieved for

well C-19, showing an impressive improvement compared to the simulated production

response of the initial guess model. On the contrary, well C-17D shows a satisfactory

history match on the total oil production only; the factors affecting the mismatch


on the total water production for this well should be further investigated. However,

ignoring completion problems, the proximity of well C-17D to the oil-water contact

(see Figure 4.13) and the observed impact of channel compartmentalization on the

production response of the wells in this field could explain a possible water coning

through a drilled channel sand body that continues through the oil-water contact.

The results of the history matching demonstrate that by changing the distribution

of channels, while remaining consistent with other (static) data, a satisfactory match

of production data was achieved. In other words, the location and connectivity of the

channel facies in this reservoir play an important role in the production response of

the wells, as observed by Liestøl et al. from the field production performance.

Due to the stochastic nature of the snesim algorithm, several reservoir models that

match production data while honoring all other available data, can be generated using

the methodology and workflow presented in this dissertation. Figures 4.32 and 4.33

show the results obtained from six history-matched reservoir models. The individual

results from each history-match are presented in appendix C. Figure 4.34 shows the

E-type obtained from the six history-matched reservoir models, which indicates the

probability of each location in the reservoir being channel facies.


Figure 4.28: Initial guess of the high-resolution facies model (left) used as the startingpoint for the probability perturbation method. High-resolution facies model (right)obtained after history matching production data (total oil and water production)from wells C-19 and C-17D. The channel facies is shown in blue, and the floodplainfacies is shown in gray.

Figure 4.29: Objective function vs. number of flow simulations. The blue curve showsthe value of the objective function for each inner iteration; the red curve shows thevalue of the objective function for each outer iteration.


Figure 4.30: Total oil (top) and water production (bottom) from well C-19. Historicaldata is shown in black, the simulated total oil production from the initial guess modelis shown in magenta, and the best match obtained after several flow simulationsis shown in green (on the total oil production plot), and blue (on the total waterproduction plot).


Figure 4.31: Total oil (top) and water production (bottom) from well C-17D. Histor-ical data is shown in black, the simulated total oil production from the initial guessmodel is shown in magenta, and the best match obtained after several flow simula-tions is shown in green (on the total oil production plot), and blue (on the total waterproduction plot).


Figure 4.32: Total oil (top) and water production (bottom) from well C-19. Historicaldata is shown in black, the other colors represent the best match obtained for sixreservoir models.


Figure 4.33: Total oil (top) and water production (bottom) from well C-17D. Histor-ical data is shown in black, the other colors represent the best match obtained for sixreservoir models.


Figure 4.34: E-type (ensemble average) generated from the six history-matched reser-voir models, using the PDI Approach.


4.4 Integrating 4D Seismic Data: FM Approach

As throughly explained in section 2.3, the FM Approach proposes to integrate 4D

seismic data into the reservoir modeling workflow as dynamic data; in other words,

matching it along with production data. Figure 2.9 shows the schematic reservoir

modeling workflow using the FM Approach. The main modules of the workflow are

explained in section 4.2 of this chapter.

In order to simulate the high-resolution 3D geo-cellular model, the FM approach

uses the channel facies probability cube (see Figure 4.7) obtained by Norsk Hydro

from a calibration between well-log data and the results of an elastic inversion on the

Base 3D seismic data (survey acquired in 1992).

The goal of the history matching procedure would be matching the total oil and

water production of the two dedicated long horizontal Ness producer wells C-19 and

C-17D (see Figures 4.18 and 4.19), as well as 4D seismic data: the proportion maps

of each scenario occurring in the first field 4D seismic response (1992-1999), where

the 4D seismic signal is classified as reliable (shown in Figure 4.27).

The objective function O(irD(u)) used by the history matching procedure is the

following:

O(irD(u)) =

Nw∑j=1

(∣∣∣∣∣Poil(irD(u))− P obs

oil

max(P obsoil )

∣∣∣∣∣+∣∣∣∣∣Pwater(irD

(u))− P obswater

max(P obswater)

∣∣∣∣∣)

j

+ ωNm∑k=1

(1− ρk)

where P obsoil and P obs

water are the historical total oil and water production; Poil(irD(u))

and Pwater(irD(u)) are the simulated total oil and water production for realization

irD(u); Nw is the number of wells to be matched (2 wells in this case: C-19 and C-

17D); ρk is the correlation coefficient between the kth observed and forward modeled

proportion map; ω is a scaling factor that increases the magnitude of the 4D seismic

mismatch to an average order of magnitude of the production mismatch (order of a

thousand); Nm is the number of proportion maps to be matched (2 maps in this case:

maps of scenarios 3 and 4 for period 1992-1999).

When history matching production data only, the production response of the

reservoir model is computed by performing flow simulation. Therefore, when including


the matching of 4D seismic data, the 4D response of the reservoir model needs to be

computed also by performing a “Forward Modeling” (FM).

4.4.1 Forward Modeling the 4D Seismic Response

In order to forward model the 4D seismic response for the Upper Ness Formation the

petro-elastic model presented in section 2.3.1 was followed. The ratio between the

P-wave and S-wave velocities Vp/Vs as well as acoustic impedance AI were computed

using that procedure. The petro-elastic model uses the pore pressure and the oil,

water and gas saturations obtained from the flow simulation to compute Vp, Vs and

ρ of the rock under certain effective pressure Peff , with a specific oil, water and gas

saturation in the pore space. Due to the complexity of the flow problem in this case

(3D three-phase flow), a simple sub-sampling of coarse grid block pore pressure and

saturation to all high-resolution grid cells was used; however, the potential limitations

of this procedure are very well documented in Chapter 3.

Two important pieces of information need to be supplied to the petro-elastic

model: seismic velocities of the dry rock as a function of porosity and effective pressure

(Vp(φ, Peff ) and Vs(φ, Peff )) and properties of the fluids in the reservoir as a function

of pore pressure (density ρ(Ppore), and bulk modulus K(Ppore)).

The properties of the three fluids in the reservoir (oil, water and gas) as a function

of pore pressure were supplied by Norsk Hydro (see Table 4.2). For each grid block of

the reservoir model the properties of each fluid are computed according to the pore

pressure in that grid block. Using these properties, the “effective fluid” properties are

calculated using the saturations of each fluid and the relationships shown in section

2.3.1 (Eqn. 2.19, and Eqn. 2.20). The equilibration parameter e in Eqn. 2.19 was

supplied by Norsk Hydro e = 3.

Vp(φ, Peff ) and Vs(φ, Peff ) relations were supplied by Norsk Hydro; however, they

corresponded to lab measurements made on cores from the Oseberg Øst field located

further east of the Alpha North segment of the Oseberg Field (see Figure 4.1). The

overburden pressure in this field is about 63.7 MPa, while the computed overburden

pressure for the Alpha North Ness Formation corresponds to 54 MPa; this observation


Ppore ρoil ρwat ρgas Koil Kwat Kgas

(MPa) (Kg/m3) (Kg/m3) (Kg/m3) (MPa) (MPa) (MPa)10 649 984 66 397 2484 12.320 658 988 136 500 2562 32.930 666 993 193 606 2639 64.540 673 997 235 711 2715 104.750 680 1001 264 816 2790 150.660 686 1005 287 919 2864 201

Table 4.2: Density and bulk modulus of the fluids in the Upper Ness reservoir, as afunction of pore pressure (from Norsk Hydro).

Temperature 103.9 ◦CInitial Pressure 28 MPaOverburden Pressure 54 MPaSalinity 35000 mg/lGas gravity 0.669Dead Oil density 856.928 Kg/m3

GOR 170 Sm3/Sm3

Table 4.3: Reservoir conditions and other gas, water and oil properties (from NorskHydro).

Figure 4.35: Properties of the fluids in the Upper Ness reservoir as a function of porepressure (from Norsk Hydro).


Peff VP,dry VS,dry

10 −5321φ + 4037 −1542φ + 228020 −4131φ + 4037 −832φ + 228030 −3448φ + 4037 −450φ + 228040 −3057φ + 4037 −245φ + 228050 −2832φ + 4037 −135φ + 228060 −2703φ + 4037 −76φ + 2280

Table 4.4: P and S-wave velocities as a function of porosity, for a series of effectivepressures (from Norsk Hydro).

indicates the data is from a deeper reservoir.

The supplied P and S velocities of the dry rock have been obtained as porosity

dependent relations for a variety of effective pressures (see Table 4.4 and Figure 4.36);

these relations are not specified for a particular lithology. When modeling the 4D

response, the appropriate P and S velocity relation is chosen according to the change

in effective pressure given by the flow simulator over time; Vp and Vs are computed

using the modeled porosity.

Figure 4.36: Seismic velocities of the dry rock as a function of porosity and effectivepressure. Relationships obtained in the lab from measurements made on cores fromthe Oseberg Øst field (from Norsk Hydro).

The relations for P and S velocities were obtained from a different reservoir rock

which may have different sorting and cement. In order to measure the applicability

of those relations to the Alpha North Ness reservoir, they were plotted on top of the


available well data. The supplied relations for Vp and Vs are obtained from dry rock

measurements. To make a comparison possible, the well-log data was transformed to

remove the fluid effect; the transformation was applied to the sand facies only, using

Gassmann’s equation (Eqn. 2.12) to obtain the dry rock velocities.

Figure 4.37 shows the Vp and Vs relations on top of the P and S velocities for the

dry sand facies and shale facies from the well-log data. The figure shows that the

scatter of the well-log data is higher for shales than for sands. Shales do not exhibit

a clear linear relationship between velocity and porosity as sands do; they have more

spread in velocity for their low porosity range. Conversely, sands have lower spread in

velocity for high porosity values; their spread is larger due to the depth changes (see

transition from cold colors to hot colors as velocity increases for a constant porosity

value). This may be an indication of effective pressure increasing with depth altering

velocity.

Figure 4.37 shows that the Vp and Vs relations are not applicable to the Alpha

North Ness reservoir. For the sand facies, the relations are remarkably off with respect

to the Vs well-log data; additionally it is observed that Vp and Vs relations are too high

for the shale facies. The rock physics relationships will always be clean since they are

obtained by fitting a mathematical function on the laboratory rock measurements;

however, they should follow the general trend observed in the data.

Since no core measurements were available for the Alpha North Ness reservoir,

a visual calibration of the Vp(φ, Peff ) and Vs(φ, Peff ) relations using well data was

performed. The results show a better agreement with the Ness reservoir’s well data

(see Figure 4.38). However, there is a large uncertainty on these linear relationships;

besides the wide scatter of the data, well data has been acquired at one point in time

and may not show the entire pressure range the reservoir has experienced through

time.

From a simulated facies model, the 4D seismic response between 1992 and 1999

is modeled using the petro-elastic model presented in section 2.3.1, as well as the

seismic velocities of the dry rock as a function of porosity and effective pressure and

the properties of the reservoir fluids as a function of pore pressure. The obtained AI

and Vp/Vs have been used to classify the modeled 4D seismic responses (see Figure


Figure 4.37: Seismic velocities of the dry rock as a function of porosity and effectivepressure obtained for Oseberg Øst (black lines) on top of dry rock velocities obtainedfrom Ness well-logs. The color code represents depth in meters.


Figure 4.38: Calibrated seismic velocities of the dry rock as a function of porosity andeffective pressure for the Alpha North Ness formation (black lines) on top of dry rockvelocities obtained from Ness well-logs. The color code represents depth in meters.


4.39), using the criteria shown in Figure 4.22.

The classification criterion, based on fundamental rock physics rules, shows that

pressure will affect both sand channel and mud floodplain; however, fluid flow takes

place only in sand channels. Therefore, from Figure 4.39 it is observed that: (1) the

two colors associated with fluid flow (brown and yellow) occur only in sand channel

facies, and (2) the two colors associated with an increase or a decrease in pore pres-

sure (light blue and orange) occur in both sand channel and mud floodplain. The

occurrence of each classification scenario can be explained from the flow simulation

results, shown in Figure 4.41.

Areas of the reservoir shown in brown on Figure 4.39 correspond to areas within

channel facies where ∆So < 0 and ∆Sg > 0 are observed in Figure 4.41; this means

that gas comes out of solution due to the decrease in pore pressure. Areas of the

reservoir shown in yellow on Figure 4.39 correspond to areas within channel facies

where ∆So < 0, ∆Sg ≈ 0, and ∆Sw > 0 are observed in Figure 4.41; this means

that a high bulk modulus fluid displacing a low bulk modulus fluid, in this case water

displacing oil. Figure 4.41 shows that ∆Pp < 0 in the entire reservoir; however,

only some areas in the reservoir are shown in orange on Figure 4.39. Pore pressure

is decreasing in the entire reservoir causing fluid flow; although both pressure and

saturation changes have an effect over the 4D seismic signal, one effect may dominate

the other depending on how strong the dependency of P and S wave velocities on

pressure is modeled.

From the classified modeled 4D seismic response (shown in Figure 4.39), the pro-

portion maps of scenarios 3 and 4 (shown in Figure 4.40) are obtained; the goal of the

history matching procedure is to perturb the model of the reservoir until its modeled

4D seismic response (shown in Figure 4.40) matches the field 4D seismic response

(shown in Figure 4.27).


Figure 4.39: Slices of the classified modeled 4D seismic response from the Alpha Northsegment (Upper Ness Formation), from the top to the bottom of the reservoir (slicesshown from left to right). The colors represent the classes interpreted in Figure 4.22.

Figure 4.40: Vertical proportion maps of scenario 3 (decrease in pore pressure) andscenario 4 (gasflooding), obtained from the classified volume of the modeled 4D re-sponse between 1992-1999 (shown in Figure4.39).


Figure 4.41: Slices of the flow simulation results used for modeling the 4D responseshown in Figure 4.39 (top); map of the location of injector and producer wells (bot-tom). The results are shown as differences between the simulation result at year 1999and the simulation result at year 1992.


4.4.2 History Matching Results

Following the proposed workflow (shown in Figure 2.9) for the FM approach; an

initial guess of the high-resolution facies model is simulated and shown in Figure 4.42;

using the probability perturbation method, the joint probability distribution used to

simulate the high-resolution facies model is perturbed iteratively using P (A | D3),

defined by Eqn. 2.7 in section 2.1.2. Therefore, perturbed facies models are drawn

from P (A | D1, D2, D3), where D1 is geological data (training image and well-log

data), D2 is geophysical data (Base 3D seismic survey), and D3 is dynamic data

(production and 4D seismic data: proportion maps ). The final high-resolution model

obtained with the probability perturbation method (PPM) is shown in Figure 4.42;

this model follows the same geological continuity model (training image), honors the

same well-log data and 3D seismic probability cube as the initial model. Additionally,

this model produced the best match of both historical production data and 4D seismic

data after approximately ninety flow simulations, as seen in Figure 4.43.

The decrease of the total mismatch (production and 4D seismic) is shown by the

red curve in Figure 4.43, while the decrease of the production mismatch and the 4D

seismic mismatch are shown in black and green respectively. These curves show that

the decrease of the total mismatch is mostly driven by the decrease in the production

mismatch, since the decrease in the 4D seismic mismatch is only 13% over the 88 flow

simulations, as opposed to the 90% decrease in production mismatch.


4.44, 4.45 and 4.46; the first two figures show the best match in production data

(wells C-19 and C-17D), while the last figure shows the best match in 4D seismic

data. A satisfactory history match of the total oil and water production is achieved

for well C-19, showing an impressive improvement compared to the simulated pro-

duction response of the initial guess model. On the contrary, well C-17D shows an

unsatisfactory match on the total oil and water production. Figure 4.46 shows an

improvement on the location and magnitude of the main central feature (as indicated

by the ellipse) observed in both maps (high vertical proportion of the “gasflooding”

scenario, low vertical proportion of the “Pore pressure decrease” scenario). The cor-

relation coefficient between the observed and the modeled proportion map of scenario


3 increased from 0.2 (initial guess) to 0.32 (best match); similarly, the correlation

coefficient between the observed and the modeled vertical proportion map of scenario

4 increased from 0.23 (initial guess) to 0.33 (best match).

These results demonstrate that by changing the distribution of channels, while

remaining consistent with other static data, a satisfactory match on the production

data of well C-19 is achieved. As observed from the results of the PDI approach, the

location and connectivity of the channel facies in this reservoir play an important

role in the production response of the wells; however, perturbing the distribution

of channels only is not enough to obtain a considerable decrease in the 4D seismic

mismatch.

The main central feature observed in the field map of scenario 4 (shown in Figure

4.46, bottom center), corresponds to a high vertical proportion of the “gasflooding”

scenario (low vertical proportion of the “Pore pressure decrease” scenario). This

anomaly is directly related to the local net-to-gross (NTG) since fluid flow occurs

preferentially in sand facies (high permeability); in other words, the high vertical

proportion observed in the field map of scenario 4 is an indicator of high NTG.

Therefore, in addition to the perturbation of the distribution of channels obtained

by using P (A | D3), another conditional probability P (A | D4) can be introduced

to either increase or decrease the local NTG as a function of the local 4D seismic

mismatch.

The local 4D seismic mismatch is translated into an indicator variable I4D(u) that

highlights the areas in the reservoir where the local NTG needs to be either increased

or decreased; the indicator variable is defined as:

I4D(u) =

+1 if τ < Sobs

4 (u) and Sobs4 (u) > S4(u)

0 if Sobs4 (u) = S4(u)

−1 if τ > Sobs4 (u) and Sobs

4 (u) < S4(u)

(4.2)

where Sobs4 (u) is the field proportion map of scenario 4, S4(u) is the forward modeled

proportion map of scenario 4, and τ ∈ [0, 1] is the minimum value of Sobs4 (u) above


which the NTG should be increased if S4(u) < Sobs4 (u). The value of τ in our case

is chosen as τ = 0.3, since the high vertical proportion anomaly observed in the field

map of scenario 4 is above that value. The expression of the conditional probability

P (A | D4) is:

P (A | D4) =

F+(u) if I4D(u) = +1

P (A) if I4D(u) = 0

F−(u) if I4D(u) = −1

(4.3)

where P (A) is the overall proportion of the event A occurring independent of location

(the marginal distribution), I4D(u) is the indicator variable defined previously (Eqn.

4.2), F+(u) and F−(u) are two functions that define the local increase or decrease

from P (A).

F+(u) =

1 if i(0)(u) = 1

P (A) + rD |P (A)max − P (A)| if i(0)(u) = 0(4.4)

F−(u) =

0 if i(0)(u) = 0

P (A)− rD |P (A)min − P (A)| if i(0)(u) = 1(4.5)

where rD ∈ [0, 1] is the parameter defined by PPM (see section 2.1.2), P (A)min

and P (A)max define the range within which P (A) can be increased or decreased

(P (A)min < P (A) < P (A)max), and i(0)(u) is the initial facies realization. The

parameter rD determines the magnitude of the perturbation, the same parameter

that defines P (A | D3) (see section 2.1.2); hence, rD = 0 implies no perturbation

while rD = 1 implies maximum perturbation. Within an outer iteration of PPM

both P (A | D3) and P (A | D4) are computed to perturb the facies realization i(0)(u)

towards another realization i(1)(u).


Figure 4.42: Initial guess of the high-resolution facies model (left) used as the startingpoint for the probability perturbation method. High-resolution facies model (right)obtained after history matching both production data (cumulative oil and water pro-duction from wells C-19 and C-17D) and 4D seismic data (proportion maps of scenar-ios 3 and 4). The channel facies is shown in blue, and the floodplain facies is shownin gray.

Figure 4.43: Objective function vs. number of flow simulations. The blue curve showsthe value of the objective function for each inner iteration; the red curve shows thevalue of the objective function for each outer iteration; the black curve shows theproduction mismatch for each outer iteration; the green curve shows the 4D seismicmismatch for each outer iteration.


Figure 4.46: Vertical proportion maps of scenario 3 (decrease in pore pressure) shownon the top row and scenario 4 (gasflooding) shown on the bottom row. From leftto right: the map obtained from the initial guess reservoir model, the observed map(field data) and the map obtained from the reservoir model that best matched bothproduction and 4D seismic data.


Following the workflow for the FM approach (shown in Figure 2.9) and starting

from the same initial guess of the high-resolution facies shown in Figure 4.42, the

joint probability distribution used to simulate the high-resolution facies model is

perturbed iteratively using both P (A | D3) (Eqn. 2.7) and P (A | D4) (Eqn. 4.3) After

several flow simulations, the final high-resolution model obtained with the probability

perturbation method (PPM) is shown in Figure 4.47. This model produced the best

match of both historical production data and 4D seismic data after approximately

250 flow simulations, as seen in Figure 4.48. The decrease of the total mismatch

(production and 4D seismic) is shown by the red curve in Figure 4.48, while the

decrease of the production mismatch and the 4D seismic mismatch are shown in

black and green respectively. The decrease in the production mismatch is 86% while

the decrease in the 4D seismic mismatch is 39% over the 250 flow simulations.


4.49, 4.50 and 4.51; the first two figures show the best match in production data

(wells C-19 and C-17D), while the last figure shows the best match in 4D seismic

data. A satisfactory history match of the total oil and water production is achieved

for well C-19, showing an impressive improvement compared to the simulated pro-

duction response of the initial guess model. On the contrary, well C-17D shows an

unsatisfactory match on the total oil and water production. Figure 4.51 shows an

improvement on the location and magnitude of the main central feature observed

in both maps (high vertical proportion of the “gasflooding” scenario. The correla-

tion coefficient between the observed and the modeled proportion map of scenario

3 increased from 0.2 (initial guess) to 0.53 (best match); similarly, the correlation

coefficient between the observed and the modeled vertical proportion map of scenario

4 increased from 0.23 (initial guess) to 0.53 (best match).

Using the additional local perturbation of NTG through P (A | D4), the 4D seismic

mismatch experienced a considerable decrease during the first five outer iterations of

PPM. The final proportion maps of scenarios 3 and 4 from the best matched high-

resolution facies model (Figure 4.51, top and bottom right) show a better agreement

with the observed maps (field data) than the maps obtained without using P (A | D4)

(shown in Figure 4.46). Figure 4.52 shows a comparison between the field data and the


the maps obtained from the best matched models (with and without using P (A | D4))

for scenario 4; this figure shows how the map obtained from the best matched model

using P (A | D4) shows a better match with the field data outside the red circled

area. The area inside the red circle shows the greatest mismatch with field data; this

area shows a high vertical proportion of the “gasflooding” scenario which has been

related to a high NTG. However, increasing NTG through P (A | D4) was not enough

to obtain gas coming out of solution in this area. The field data shows that more

than 30% of the thickness of the reservoir in this area is classified as “gasflooding”

scenario; therefore, two phenomena could possibly occur in the model: either no gas

is coming out of solution because it is an isolated area (no pressure depletion), or

gas is coming out of solution but migrating up-dip before the second seismic monitor

survey is forward modeled (before the year 1999).

From the flow simulation performed on the best matched reservoir model shown

in Figure 4.47, it was observed that gas came out of solution in this area by the year

1998 but migrated to other areas in the reservoir before 1999. Figure 4.53 shows

several snapshots in time of the vertical average of Sg where a white circle on the

maps corresponding to years 1998 and 1999 indicates the area of greatest mismatch

between the field and the modeled 4D seismic, the same area where gas came out of

solution and migrated before 1999. Similarly, Figures 4.54 and 4.55 show ∆Sg maps

where the gas migration is more clearly seen, in particular note the negative area

circled on the ∆Sg map between the years of 1998 and 1999 in Figure 4.55.

If the signal from the 4D seismic data is given a high level of certainty, further work

should follow this investigation on the geological feasibility of the presence of a fault

(either not modeled or below the seismic resolution) or any other geological barrier

that prevents the gas from migrating. The area of greatest mismatch between the

field and the modeled 4D seismic (see circled area in Figure 4.52) shows a strong field

4D seismic signal interpreted as “gasflooding”; therefore, it is experiencing pressure

depletion as a consequence of oil production down-dip by wells C-19 and C-17D and

it is not isolated from the rest of the reservoir.



Figure 4.48: Objective function vs. number of flow simulations. The blue curve showsthe value of the objective function for each inner iteration; the red curve shows thevalue of the objective function for each outer iteration; the black curve shows theproduction mismatch for each outer iteration; the green curve shows the value 4Dseismic mismatch for each outer iteration.


Figure 4.52: Vertical proportion maps of scenario 4 (gasflooding); From left to right:the map obtained from the reservoir model that best matches both production and4D seismic data using P (A | D3) only in PPM, the observed map (field data) andthe map obtained from the reservoir model that best matches both production and4D seismic data using P (A | D3) and P (A | D4) in PPM. The red circle indicates anarea of considerable mismatch.

Figure 4.53: Vertical average of gas saturation Sg from the flow simulation performedon the best matched model shown in Figure 4.47. From left to right, snapshots forthe years 1992, 1993, 1995, 1997, 1998 and 1999.


Figure 4.54: Vertical average of ∆Sg from the flow simulation performed on thebest matched model shown in Figure 4.47. From left to right, ∆Sg is obtained fromsubtracting the gas saturation at years 1993, 1995, 1997, 1998 and 1999 from the gassaturation at year 1992.

Figure 4.55: Vertical average of ∆Sg from the flow simulation performed on the bestmatched model shown in Figure 4.47. From left to right, ∆Sg is the incrementaldifference over the years 1993, 1995, 1997, 1998 and 1999.


In addition to the results presented earlier in this section, the same workflow for

the FM approach (shown in Figure 2.9) was applied this time using the channel facies

probability cube (see Figure 4.7) obtained by Norsk Hydro from a calibration between

well-log data and the results of an elastic inversion on the 4D seismic data (surveys

acquired in 1992, 1999 and 2004) using the procedure presented by Andersen et al.

[3] (reviewed in section 2.2.1). The results presented earlier followed the workflow for

the FM approach using the channel facies probability cube (see Figure 4.7) obtained

by Norsk Hydro from a calibration between well-log data and the results of an elastic

inversion on the Base 3D seismic data (survey acquired in 1992). A summary of all

these results is shown in Tables 4.5, 4.6, and 4.7.

The results obtained from the history matching procedure using P (A | D3) only,

are shown in Figures 4.58, 4.59 and 4.60; the first two figures show the best match

in production data (wells C-19 and C-17D), while the last figure shows the best

match in 4D seismic data. A satisfactory history match of the total oil and water

production is achieved for well C-19; however, the well C-17D shows a satisfactory

match on the total oil production only. Figure 4.46 shows an slight improvement on

the location and magnitude of the central high vertical proportion of the “gasflooding”

scenario. The correlation coefficient between the observed and the modeled proportion

map of scenario 3 increased from 0.24 (initial guess) to 0.30 (best match); similarly,

the correlation coefficient between the observed and the modeled proportion map of

scenario 4 increased from 0.26 (initial guess) to 0.32 (best match).

The results obtained from the history matching procedure using both P (A | D3)

and P (A | D4), are shown in Figures 4.61, 4.62 and 4.63; the first two figures show the

best match in production data (wells C-19 and C-17D), while the last figure shows

the best match in 4D seismic data. A satisfactory history match of the total oil and

water production is achieved for well C-19. On the contrary, well C-17D shows an

unsatisfactory match on both total oil and water production. Figure 4.63 shows a

considerable improvement on the location and magnitude of the central high vertical

proportion of the “gasflooding” scenario. The correlation coefficient between the

observed and the modeled proportion map of scenario 3 increased from 0.24 (initial

guess) to 0.55 (best match); similarly, the correlation coefficient between the observed


and the modeled vertical proportion map of scenario 4 increased from 0.26 (initial

guess) to 0.55 (best match).

These results were obtained starting from the same random seed as the results

presented earlier (when using the channel facies probability cube obtained from a

calibration between well-log data and the Base 3D seismic data). Similar results are

obtained for the match of production and 4D seismic data; however, figures 4.57 and

4.65 show that a history match is achieved in a shorter number of flow simulations.

When using P (A | D3) only, a history match is obtained in 32 flow simulations,

roughly a third of the number of flow simulation used in the results presented earlier

(88 flow simulations). Similarly, when using P (A | D3) and P (A | D4), a history

match is obtained in 63 flow simulations, roughly a fourth of the number of flow

simulation used in the results presented earlier (250 flow simulations).


Used Used Decreased Decreased Number ofFacies prob. P (A | D4) Production 4D seismic Flow

cube mismatch mismatch simulations1 3D seismic No 90% 13% 882 3D seismic Yes 86% 39% 2503 4D seismic No 92% 10% 324 4D seismic Yes 84% 40% 63

Table 4.5: Summary of the behavior of the Objective Function using the FM approach.

Total Oil Prod. Total Water Prod. Total Oil Prod. Total Water Prod.C-19 C-19 C-17D C-17D

matched matched matched matched1 Yes Yes Yes No2 Yes Yes No No3 Yes Yes Yes No4 Yes Yes No No

Table 4.6: Summary of the matching of production data using the FM approach.

Corr. Coef Corr. Coef Corr. Coef Corr. Coef“Pp decrease” “Pp decrease” “gasflooding” “gasflooding”Initial Guess Best Match Initial Guess Best Match

1 0.20 0.32 0.23 0.332 0.20 0.53 0.23 0.533 0.24 0.30 0.26 0.324 0.24 0.55 0.26 0.55

Table 4.7: Summary of the matching of 4D seismic data using the FM approach.

Chapter 5

Conclusions and Future Research

Important information about static and dynamic components of a reservoir can be

retrieved from the process of reservoir modeling. Estimates of the OOIP, produc-

tion forecasts, and optimized well trajectories are just a few examples of the type of

information a reservoir model can provide. Therefore, creating a reservoir model(s)

calls for the integration of expertise from different disciplines, as well as the simul-

taneous integration of a wide variety of data: geological data (core data, well-logs,

interpretations, etc.), production data (fluid rates or volumes, pressure data, etc.),

and geophysical data (3D seismic data). Some reservoirs exhibit a suite of several 3D

seismic surveys (4D seismic data) acquired for production monitoring purposes. 4D

seismic data not only provides information about the dynamic process occurring in

the reservoir while production takes place; but also provides static information about

the spatial lithological heterogeneity where dynamic changes occur.

The traditional practice of reservoir modeling follows a two-step approach where

static data is used by geoscientists to create a high-resolution geo-cellular model,

which is passed to reservoir engineers who first upscale the geo-cellular model to a

coarsened grid (where flow simulation is computationally feasible) and adjust the

upscaled model to match dynamic data (usually production data). Adjusting such

upscaled properties is often done without regard of the high-resolution data and

model which honors static data. The final result of this approach is a model that

only matches production history, but may no longer be consistent with any data

142

143

integrated prior to matching. This dissertation, however, uses a reservoir modeling

workflow that integrates all data from various sources (both static and dynamic)

at the same time using a probabilistic approach. Additionally, it builds both high-

resolution and coarsened models jointly while perturbing the high-resolution model

in a geologically consistent fashion. The main contribution of this dissertation is the

inclusion of 4D seismic data within this reservoir modeling workflow while attempting

to honor all other available data.

Two general approaches are proposed to include 4D seismic data into the reservoir

modeling workflow: the PDI and FM approaches. The Probabilistic Data Integration

approach (PDI), consists of modeling the information content of 4D seismic through

a spatial probability of facies occurrence that is used as a constraint (along with

conditional probabilities from other data) to create a high-resolution facies model

of the reservoir; production data is the only dynamic data included in the iterative

history matching process. The Forward Modeling (FM) approach uses the facies

information content (through a spatial probability) from the base 3D seismic survey

to create a high-resolution facies model of the reservoir; both production and 4D

seismic data are included in the history matching process.

History matching production data only involves modeling the production response

of the reservoir model by performing flow simulation. Therefore, when including the

matching of 4D seismic data, the 4D response of the reservoir model needs to be com-

puted also by performing a “Forward Modeling” (FM). The 4D seismic response of the

reservoir model is forward modeled using the high-resolution reservoir properties and

the downscaled spatial distribution of fluid saturation obtained from flow simulation.

A novel flow-based downscaling method is proposed in this dissertation, the procedure

takes into account both static information (high-resolution permeability field) and dy-

namic information in the form of coarsened fluxes and saturations (solution of the

flow simulation on the coarsened grid). On a synthetic layered example, it was found

that modeling 4D seismic data using flow-based downscaled saturations resulted in a

better match with the “reference” 4D seismic (flow simulation on the high-resolution

grid was possible for this simplified case), because the flow-based downscaled satu-

rations reproduced the sharpness and key details of the saturation field. Modeling

144 CHAPTER 5. CONCLUSIONS AND FUTURE RESEARCH

4D seismic data using the traditional approach of simply projecting the coarse scale

saturation from the flow simulator onto the high resolution grid was shown to result

in a time-lapse response that did not accurately predict the distribution of fluids in

the reservoir.

The two proposed approaches (PDI and FM approaches) to include 4D seismic

data into the reservoir modeling workflow, were successfully applied on the Oseberg

field in the North Sea, to model channel facies of a fluvial reservoir.

The PDI approach constrained the reservoir model to the spatial probability of

facies occurrence (obtained from a calibration between well-log and 4D seismic data)

as well as other static data while history matching production data. A satisfactory

history match of the total oil and water production is achieved for well C-19, showing

an impressive improvement compared to the simulated production response of the

initial guess model. On the contrary, well C-17D shows a satisfactory history match

on the total oil production only; the large mismatch obtained for the total water

production in this well needs to be further investigated through the flow simulation

model. These results demonstrate that by changing the distribution of channels, while

remaining consistent with other (static) data, a satisfactory match of production data

was achieved. However; only a probabilistic matching of 4D seismic data is achieved,

rather than a deterministic physics-based matching of that data.

The FM approach achieved a satisfactory history match of the total oil and water

production for well C-19; on the contrary, well C-17D shows an unsatisfactory match

on the total oil and water production. Changing the distribution of channels, while

remaining consistent with other static data, a satisfactory match on the production

data of well C-19 was achieved; however, perturbing the distribution of channels only

was not enough to obtain a considerable decrease in the 4D seismic mismatch.

The main central feature observed on the field 4D seismic data, related to a high

vertical proportion of the “gasflooding” scenario (low vertical proportion of the “Pore

pressure decrease” scenario), was interpreted to be directly related to the local net-to-

gross (NTG) since fluid flow occurs preferentially in sand facies (high permeability).

Therefore, in addition to the perturbation of the distribution of channels, perturbation

of the local NTG was obtained through the introduction of a second conditional

145

probability in PPM. This results in either an increase or decrease of the local NTG

as a function of the local 4D seismic mismatch.

Using the local perturbation of NTG through a new conditional probability, the

4D seismic mismatch decreased considerably during the first few outer iterations of

PPM. The 4D seismic mismatch decreased three times more than when the local per-

turbation of NTG is not used; however; the production mismatch decreased roughly

the same amount but over a larger number of flow simulations. The final forward

modeled 4D seismic response showed a better agreement with the observed 4D seis-

mic data than the forward modeled 4D seismic response obtained without using the

local perturbation of NTG. Notably was one area in the reservoir, exhibiting a high

vertical proportion of the “gasflooding” scenario (an indicator of high NTG). How-

ever, the model was not able to fully predict this phenomenon. From flow simulation,

it was observed that gas came out of solution in this area by the year 1998 but mi-

grated to other areas in the reservoir before the year 1999, when the second seismic

monitor survey is forward modeled. Two explanations could possibly apply to this

situation: either the 4D seismic data is indicative of a process (not modeled) other

than “gasflooding”, or some geological information is missing in the model such as

the presence of a fault (either not modeled or below the seismic resolution) or any

other geological barrier that prevents the gas from migrating. The feasibility of each

proposed explanation should be further investigated.

The proposed FM approach uses the channel facies probability cube obtained by

Norsk Hydro from a calibration between well-log data and the results of an elastic

inversion on the Base 3D seismic data (survey acquired in 1992). A proposed hybrid

approach would be to use the channel facies probability cube obtained by Norsk

Hydro from a calibration between well-log data and the results of an elastic inversion

on 4D seismic data (surveys acquired in 1992, 1999 and 2004) while at the same time

matching both production and 4D seismic data iteratively. The results obtained from

this hybrid approach showed a similar quality on the match of production and 4D

seismic data; however, it has been noticed that a history match is achieved in a shorter

number of flow simulations. In particular, when using the local perturbation of NTG

the number of flow simulations required to achieve a history match was reduced by a


factor of four.

When a complicated 4D seismic forward modeling cannot be carried out due to

a lack of rock physics data, the PDI approach may represent a more robust and

less difficult to achieve alternative to include 4D seismic information into the reser-

voir modeling workflow. However, in that case only a probabilistic matching of 4D

seismic data is achieved, rather than a deterministic physics-based matching of that

data. The quality of the 4D seismic data (in terms of signal/noise) is more critical

for the FM approach than for the PDI approach, since a deterministic physics-based

matching of 4D seismic data is pursued. Each approach uses the information content

of 4D seismic data in a different fashion; the PDI approach makes use of the static

information contained in the 4D seismic data, while the FM approach seeks to match

the dynamic information in the 4D seismic data. If high quality 4D seismic data is

available, the FM approach (although practically more challenging to apply) could

help to understand better the dynamic behavior of the reservoir as well as to identify

valuable static information that may need to be incorporated into the reservoir model.

Limitations

Regarding the new flow-based downscaling procedure, it is observed that its suc-

cess depends directly on the method used for upscaling as well as the upscaling ratio

applied to the static data prior to flow simulation. Hence, the upscaling method, and

the chosen upscaling ratio, should maintain the overall character and position of the

saturation front.

This dissertation uses the “Parallel Modeling” approach, introduced by Tureyen

in his PhD dissertation [86], to build both high-resolution and coarsened models

jointly while perturbing the high-resolution model. The limitations of this approach,

as pointed out by Tureyen, also apply to the workflow used here. Perhaps the most

important limitation is the method used for upscaling as well as the upscaling ratio.

If the upscaling procedure is not robust enough to maintain important flow perfor-

mance features from the high-resolution model, the forecast capabilities of the pair

147

high-resolution and coarsened models may be weak. On the other hand, if a sophis-

ticated upscaling procedure is chosen, additional computational cost and complexity

of successive upscaling becomes also a handicap of the approach.

Regarding the two proposed approaches (PDI and FM) to include 4D seismic data

into the reservoir modeling workflow, the following limitations are noticed:

• the PDI approach requires 4D seismic data to be informative about the static

unknown being modeled; facies, for example. This approach uses 4D seismic

data in a “static” fashion where only a probabilistic matching of 4D seismic data

is achieved. The application of this approach to the Alpha North segment of

the Oseberg field was possible because the main dynamic response (gasflooding)

occurs in one particular facies of the binary (channel/no channel) facies model;

hence, the dynamic response was an indicator of the presence of the channel

facies (a static property of the model). In other field applications, it may not

necessarily be possible to translate dynamic changes into information about

static properties.

• the PDI approach only pursues a probabilistic matching of the “static” infor-

mation possibly contained in the 4D seismic data. The PDI approach may not

necessarily achieve a deterministic physics-based matching of the 4D seismic

data since it is not explicitly pursuing that matching like the FM approach

does. This may be of particular relevance in the case where a significant dy-

namic response is observed in the 4D seismic data, and we want the reservoir

model to explicitly reproduce it.

• the FM approach requires forward modeling the 4D seismic response of the

reservoir model. Forward modeling represents a challenge due to several fac-

tors: the joint modeling of pressure and saturation effects while maintaining

the observed dominance of one or the other, the possible lack of rock physics

data (lab measurements on the pressure dependency of the seismic velocities,

for example), the downscaling of the flow response under a complicated flow

scenario (three-phase flow and multiple wells), the forward modeling of the seis-

mic signal (if amplitudes are the seismic attribute being modeled) may imply a


computationally-costly method being iteratively used.

• the FM approach aims at matching simultaneously both production and 4D

seismic data. Several parameters of the high-resolution and/or flow model may

have a significant effect on the matching of either production or 4D seismic

data. For example, in the case study presented in this dissertation, clearly the

matching of production data was influenced by the distribution of the channel

facies in the reservoir; however, the matching of 4D seismic was influenced by

the local NTG in the reservoir and possibly missing structural complexity in

the flow model. The approach used to perturb these parameters should be such

that improving the matching of one datum does not deteriorate the matching of

the other datum. In addition, a thorough analysis for possible inconsistencies

between the available data should be performed prior to history matching.

• the success of such a complete reservoir modeling workflow involves a multidis-

ciplinary knowledge of all data and processes involved. The modeling of the

dynamic (production and 4D seismic) response of the reservoir depends on the

static description of the reservoir, the modeling of flow boundary conditions

(wells, aquifers), the description of fluids in the reservoir, the modeling of the

seismic response, etc.

Future Research

Regarding the new flow-based downscaling procedure presented in this disserta-

tion, the following areas of future research are identified:

• the application of the procedure in an adaptive fashion, where local high-

resolution flow is simulated only in the areas of the reservoir with high-resolution

saturation effects which impact the seismic signal (e.g., an advancing water front

in a region with a high degree of subgrid permeability variability). With this

procedure, areas of the reservoir already swept (or not yet swept) by the injected

water would not be reconstructed because they exhibit a constant saturation

149

value. This approach could be much more efficient computationally than recon-

structing the high-resolution saturation everywhere, particularly in cases where

the important saturation changes occur over relatively small portions of the

reservoir. This technique will however require some modification of the down-

scaling procedure described in this dissertation, for example in the specification

of the inlet saturation boundary conditions. The success of the proposed adap-

tive flow-based downscaling method depends directly on the method used for

upscaling as well as the upscaling ratio applied to the static data prior to flow

simulation. The upscaling method, and the chosen upscaling ratio, should main-

tain the overall character and position of the saturation front, since the adaptive

flow-based downscaling method would identify the front and work on its refine-

ment. An alternative approach (where downscaling may not be needed) is to

use an upscaling adaptive gridding approach where no upscaling is performed

in the areas of the reservoir with high-resolution saturation effects, such as an

advancing water front in a region with a high degree of subgrid permeability

variability.

• the extension of the method to a 3D three-phase flow-based downscaling pro-

cedure. However, only an approximate method may be sufficient; for example,

in a three-phase situation perhaps the most important feature to reconstruct

would be the presence or absence of gas in a grid block (when modeling seismic

attributes that are not sensitive to the particular value of gas saturation).

Some other areas of future research are the following:

• the inclusion of conditioning to a map of local NTG in multiple-point sequential

simulation algorithms. From the results of the case study presented in this

dissertation, it was observed that the local perturbation of NTG through a new

conditional probability caused partial damage of the geological continuity of the

channel facies.

• related to the case study, the factors affecting the observed mismatch on the

water production of well C-17D should be further investigated. In addition, the


geological feasibility of the presence of a fault (either not modeled or below the

seismic resolution) or any other geological barrier that prevents the gas from

migrating in the area of largest mismatch with the 4D seismic should also be

investigated.

• the rock physics modeling used in this dissertation assumes no fluid flow oc-

curring in shales (also inactive during flow simulation), but being affected by

changes in pressure. This may not necessarily be an accurate assumption. On

one hand, Gassmann’s fluid substitution is only applicable to sands; on the

other hand, particularly for the case study presented in this dissertation, the

field 4D seismic data (between 1992 and 1999) shows a strong pressure depletion

response that has been interpreted to be experienced by both sand and shales

due to its vast spatial extension. More rock physics laboratory studies on shales

should be carried out in order to understand better their dynamic behavior.

• another area of fascinating research is to model 4D seismic responses due to

compaction and fault(s) reactivation, occurring separately or simultaneously;

in a hydrocarbon producing reservoir. This area may require intensive research

on dynamic re-gridding of the reservoir model as well as the inclusion of geo-

mechanics.

• the application of the PDI and FM approaches to other case studies should

result in setting a criteria for choosing a probabilistic versus a deterministic but

more physics-based use of 4D seismic data.

Appendix A

The Stanford VI Reservoir

With the purpose of extensively testing algorithms for reservoir modeling or reservoir

characterization, Mao and Journel [57] created an exhaustive 3D reference data set

called Stanford V. Although it has been widely used, this data set is too small to

represent current-day reservoir modeling exercises. Several extensions are proposed

here, while incorporated into a new reference reservoir model, termed Stanford VI.

The proposed reference data set (Stanford VI) exhibits a smooth top and bottom

surface representing a trap in the form of an anticline. It provides an exhaustive

sampling of petrophysical properties. The new reservoir model is represented in a 3D

regular grid of 6 million cells (150 × 200 × 200), with more realistic dimensions for

current-day models (25 m in the x and y directions and 1 m in the z direction).

Following Stanford V, the stratigraphic model corresponds to a fluvial channel

system, and the petrophysical properties computed for this reference reservoir corre-

spond to the classical porosity, density, permeability and seismic P-wave and S-wave

velocities. Although most of these properties are calculated following the standard

procedure and algorithms presented by Mao and Journel, a more appropriate Rock

Physics model is used in the new reference reservoir model to compute P-wave velocity

for sandstones.

Traditionally, P-wave velocity is calculated from empirical expressions obtained

from laboratory data as a function of porosity [92] [67] [40] [84]. The former Stanford

151

152 APPENDIX A. THE STANFORD VI RESERVOIR

V reservoir uses one of these expressions, presented by Han [40], to obtain P-wave ve-

locities from porosity. Strictly speaking, Han’s relations are obtained from sandstone

samples collected from different depths, where porosity is controlled by diagenesis and

cementing. In our case porosity is controlled by sorting and clay content, henceforth,

a more appropriate rock physics model is used, namely, the constant cement model

described by Dvorkin and Nur [29].

Besides petrophysical properties, the new reference reservoir model exhibits a

complete set of physical seismic attributes, which are computed from well-known

mathematical expressions and subsequently filtered and smoothed to obtain realistic

looking data as would have been obtained from actual seismic acquisition and mod-

eling. These realistic seismic attributes provide a filtered view of the true spatial

variation of petrophysical properties.

From this new reference reservoir model 4D seismic data is generated. 4D seismic

data is nothing more than three-dimensional (3D) seismic data acquired at different

times over the same area. 4D seismic is used to assess changes in a producing hy-

drocarbon reservoir with time; changes may be observed in fluid location, saturation,

pressure, and temperature. Consequently, one of the main applications of 4D seismic

data is to monitor fluid flow in the reservoir.

In order to create a 4D seismic response, several 3D seismic data sets are forward

modeled using a simple convolutional model. The first seismic data set is created

using the acoustic impedance of the reservoir prior to production, while three more

seismic data sets are created using the acoustic impedance of the reservoir at different

times during oil production.

The acoustic impedance of the reservoir after a certain time t can be computed

by using a fluid substitution approach. Prior to production the reservoir is filled with

oil, while some years after production starts the reservoir contains a mixture of fluids,

typically water oil and gas. In order to use Gassmann’s equations [32] correctly the

saturations of each fluid at every point in space are needed, therefore a flow simulator

is used to obtain them at any point in time during production.

To create the 3D seismic data sets at different times during oil production, a two-

phase flow simulation is performed. Thirty years of oil production are simulated with

153

an active aquifer below the reservoir and water injector wells that become active after

the aquifer water influx fails to maintain the pressure.

The workflow used to create the Stanford VI reference data set is shown in Figure

A.1. The first step in the generation of Stanford VI property model corresponds to

the modeling of facies. The Stanford VI facies model corresponds to a prograding

fluvial channel system and is modeled using the commercial software SBED 1 and the

multiple-point simulation algorithm snesim.

Subsequently, the facies model is populated with five petrophysical properties:

porosity, density, P-wave velocity, S-wave velocity and permeability. Basically, poros-

ity is simulated first using the sequential simulation algorithm sgsim; density, P-wave

and S-wave velocities are obtained from porosity using well known Rock Physics mod-

els; and permeability is co-simulated conditional to the co-located previously simu-

lated porosity using the algorithm cosgsim.

Having the petrophysical properties, three basic steps are followed: flow simula-

tion, forward model of 4D seismic data and generation of seismic attributes. The

following sections of this appendix explain the details of each step depicted by the

workflow presented in Figure A.1.

1 c© Copyright 2007 Geomodeling Technology Corp. All rights reserved.


Figure A.1: Workflow followed to create the Stanford VI data set.

A.1. STRUCTURE AND STRATIGRAPHY 155

A.1 Structure and Stratigraphy

The structure of the Stanford VI reservoir corresponds to a classical structural oil

trap, an anticline. Specifically, it is an asymmetric anticline with axis N15◦E. As

Figure A.2 shows, the anticline has a different dip on each flank and generally the

dip decreases slowly towards the northern part of the structure. The maximum dip

of the structure is 8◦.

Figure A.2: Perspective view of the Stanford VI top structure: view from SW (left),view from SE (right). The color indicates the depth to the top.

The reservoir is 3.75 Km wide (East-West) and 5.0 Km long (North-South), with

a shallowest top depth of 2.5 Km and deepest top depth of 2.7 Km. The reservoir

is 200 m thick and consists of three layers with thicknesses of 80 m, 40 m and 80 m

respectively (see Figure A.3).

In terms of grid, the Stanford VI reservoir is represented in a 3D regular strati-

graphic grid of 150 × 200 × 200 cells and the dimensions of the grid correspond to

25 m in the x and y directions and 1 m in the z direction. The coordinate system

used corresponds to the GSLIB [28] standard of the stratigraphic coordinate system,

where the z coordinate is measured relative to the top of the reservoir. Due to the

simple structure and stratigraphic grid, an accompanying Cartesian box in which all

of the geostatistical modeling takes place, can easily be constructed.

The stratigraphy of the Stanford VI reservoir corresponds to a prograding fluvial

channel system, where deltaic deposits represented in layer 3 were deposited first and


Figure A.3: Perspective view of the Stanford VI top and bottom of each of its layers.The color indicates the depth to the top.

followed by meandering channels in layer 2 and sinuous channels in layer 1. This

sequence of clastic deposits represents a progradation of a fluvial system into the

basin located in this case toward the north of the reservoir.

In order to model the stratigraphy of Stanford VI, the commercial software SBED

is used to model layer 1 and layer 2, while layer 3 is modeled using the multiple-point

simulation algorithm snesim with local rotation and affinity variation to model the

channel meanders.

The first layer of Stanford VI consists of a system of sinuous channels represented

by four facies: the floodplain (shale deposits), the point bar (sand deposits that

occur along the convex inner edges of the meanders of channels), the channel (sand

deposits), and the boundary (shale deposits). The stratigraphic characteristics of

layer 1 are detailed in Table A.1 and Figure A.4 shows the resulting stratigraphic

model for this layer.

The second layer consists of meandering channels also represented by four facies:

the floodplain (shale deposits), the point bar (sand deposits that occur along the

convex inner edges of the meanders of channels), the channel (sand deposits), and

the boundary (shale deposits). The stratigraphic characteristics of layer 2 are detailed

in Table A.2 and Figure A.5 shows the resulting stratigraphic model for this layer.

The last and third layer of the reservoir consist of deltaic deposits and are repre-

sented by two facies: the floodplain (shale deposits), and the channel (sand deposits).


Facies Proportion Number Avg. thickness (meters) Avg. width (meters)

floodplain 0.68 – – –point bar 0.11 – – 300channel 0.165 8 20 600boundary 0.045 – 1.5 –

Table A.1: Stratigraphic parameters used for the simulation of the facies model forlayer 1.

Figure A.4: Facies model of Layer 1, which corresponds to sinuous channels: flood-plain (navy blue), point bar (light blue), channel (yellow), and boundary (red). Strati-graphic grid (left), and cartesian box (right).

The stratigraphic characteristics of layer 3 are detailed in Table A.3.

The third layer of Stanford VI is modeled using the multiple-point simulation

algorithm snesim with local rotation and affinity variation of the channel meanders.

Traditionally, geostatistical techniques capture geological continuity through a

variogram. A variogram is a two-point statistical function that describes the level

of correlation, or continuity, between any two sample values as separation between

them increases. Since the variogram describes the level of correlation between two

locations only, it is not able to model continuous and sinuous patterns such as channels

or fractures. For modeling such geological features a multiple-point approach should

be used, where spatial patterns are inferred using many spatial locations [81].

In multiple-point geostatistics, the spatial patterns are inferred from a training



floodplain 0.68 – – –point bar 0.14 – – –channel 0.11 4 16 300boundary 0.07 – 1.5 –


Figure A.5: Facies model of Layer 2, which corresponds to meandering channels:floodplain (navy blue), point bar (light blue), channel (yellow), and boundary (red).Stratigraphic grid (left), and Cartesian box (right).

image which represents a conceptual reservoir analog with the expected geological

heterogeneity. Since it is a conceptual model, the training image is not constrained

to any data.

The geostatistical technique that uses a training image to create realizations con-

strained to reservoir data is proposed by Strebelle [81]. The “single normal equation

simulation” (snesim) algorithm is a conditional sequential simulation where the prob-

ability distribution is retrieved from the training image and made conditional to a

multiple-point data event.

The snesim algorithm allows for local rotation and affinity (aspect ratio) variation

of the data event, which enables to create non-stationary realizations from a stationary

training image. Figure A.6 shows the resulting stratigraphic model for layer 3 as well



floodplain 0.56 – – –channel 0.44 – 7-40 70-400


as the rotation, and affinity cubes used, the training image is shown in Figure A.7.

The rotation and affinity cubes are categorical variables and the values assigned to

these categories are shown in Table A.4.

Angle category angle (degree)0, 1, 2, 3, 4, 5, 6, 7, 8, 9 -63, -49, -35, -21, -7, 7, 21, 35, 49, 63

Affinity category affinity [x,y,z]0, 1, 2 [0.5, 0.5, 0.5], [1, 1, 1], [2, 2, 2]

Table A.4: Categories for affinity and angle rotation used for simulating the faciesmodel for layer 3.


Figure A.6: Facies model of Layer 3 (top), which corresponds to deltaic deposits:floodplain (navy blue), and channel (yellow). Stratigraphic grid (left), Cartesian box(right), angle cube (middle), and affinity cube (bottom).


Figure A.7: Training Image used for modeling Layer 3. The size of the training imageis 200× 200× 5, each slice in the z − direction is shown here from top to bottom.


A.2 Petrophysical Properties

Having created the facies model for the three layers of the Stanford VI reservoir, it

is populated with the following petrophysical properties:

• Porosity.

• Permeability.

• Density.

• P-wave velocity.

• S-wave velocity.

A.2.1 Simulation of Porosity

Porosity is simulated first using the sequential Gaussian simulation algorithm sgsim

from GSLIB, conditioned to a reference target distribution and variogram, and inde-

pendently for each facies in the reservoir.

The reference target distribution of porosity in each facies is shown in Figures

A.8 and A.9. Shale deposits in floodplain and boundary facies have distinctively low

porosity values while sand deposits in channel and point bar facies have high porosity

values as expected. The variance for point bar facies is smaller than for channel facies

and their mean is higher since they are typically well sorted sand deposits. Similarly,

boundary facies exhibits very low mean and variance which is translated later into a

flow barrier for fluids.

Figure A.8: Distribution of porosity for each facies in the reservoir.

A.2. PETROPHYSICAL PROPERTIES 163

Figure A.9: Histogram of porosity for each facies in the reservoir.

The reference variogram consisting of a single structure for each facies are specified

in Table A.5, where ranges for x, y and z direction are given in meters.

Having sequentially simulated porosity independently for each facies, a cookie-

cutter approach is used to create the resulting porosity model for the Stanford VI

reservoir (see Figure A.10).

A.2.2 Simulation of Permeability

Typically, the logarithm of permeability is approximately linearly correlated with

porosity, therefore the logarithm of permeability is simulated using a linear correlation

coefficient of 0.7 between both variables.

Permeability for each facies is co-simulated conditional to the simulated poros-

ity, using a Markov1-type model instead of a full model of coregionalization. The


floodplain point bar channel boundary

type Spherical Spherical Spherical Sphericalnugget 0.1 0.1 0.1 0.1ranges 1750/1750/70 5000/2500/10 3000/1750/10 500/500/20angles 0/0/0 90/0/0 90/0/0 0/0/0

Table A.5: Variogram used for simulating porosity for each facies.

Figure A.10: Resulting Porosity cube after cookie-cut porosity from each facies’ poros-ity realization.

algorithm used is cosgsim implemented in the software S-GeMS [68] 2.

Permeability is also simulated independently within each facies. The cookie-cutter

approach is used to merge the permeability simulated for each facies into a single

permeability model.

The implicit assumption of the algorithm used is that both primary and sec-

ondary variables are normally distributed, and the bivariate relationship follows a

(bi)Gaussian distribution. Both variables are transformed to the normal space and

the bivariate relationship is assumed to follow a (bi)Gaussian distribution.

In order to transform the original variables into Gaussian variables, the original

2 c© Copyright 2002-2006 Board of Trustees of the Leland Stanford Junior University. All rightsreserved.


distributions of both porosity and logarithm of permeability are provided to the al-

gorithm. The distribution of porosity was shown in the last section. The distribution

of the logarithm of permeability is obtained (see Figure A.11) by transforming the

distribution of porosity using the well known Kozeny-Carman’s relation [18].

κ =1

72τ

φ3

(1− φ)2d2 (A.1)

where: φ is porosity (fraction), τ is tortuosity (assumed as τ = 2.5), and d is the pore

diameter (in micrometers).

Figure A.11: Histogram of the logarithm of permeability, per facies in the reservoir.

The κ-variogram used for each facies is shown in Table A.6, where ranges for

x, y and z direction are given in meters. The resulting permeability model for the

Stanford VI reservoir is shown in Figure A.12.


floodplain point bar channel boundary

type Spherical Spherical Spherical Sphericalnugget 0.1 0.1 0.1 0.1ranges 1750/1750/70 5000/2500/10 3000/1750/10 500/500/20angles 0/0/0 90/0/0 90/0/0 0/0/0

Table A.6: κ-variogram used for simulating permeability for each facies.

Figure A.12: Resulting Permeability cube after cookie-cutting permeability from eachfacies’ permeability realization.

A.2.3 Density

The rock density is calculated using porosity and the mixing formula:

ρ = φρfluid + (1− φ)ρmatrix (A.2)

where ρfluid is the density of the fluid that fills in the pore space, and ρmatrix is the

density of the rock matrix. Therefore the rock density is calculated as:

ρ = φρfluid + (1− φ)N∑

i=1

fiρmi(A.3)

where fi is the fraction of the mineral mi with density ρmiwhich constitutes part of

the rock matrix mineralogy. The rock mineralogy for each facies is shown in Table


A.7.

mineralmineral density floodplain point bar channel boundary

(g/cc)clay 2.4 0.7 0.0 0.0 0.8

quartz 2.65 0.2 0.70 0.65 0.15feldspar 2.63 0.1 0.2 0.2 0.05

rock fragments 2.7 0.0 0.1 0.15 0.0

Table A.7: Rock mineralogy for each facies.

Using the mineralogy and the mineral densities shown in Table A.7 the rock matrix

density ρmatrix is computed. The rock bulk density ρ is obtained from Eqn. A.3, using

the simulated porosity and water as the saturating fluid.

Typically, density, P-wave and S-wave velocities are calculated first for water

saturated rocks since the rock physics models used for computing velocities have

been obtained in the lab from water-saturated rocks. Therefore, in order to use these

models correctly, the saturating fluid must be water.

For the generation of this data set density, P-wave and S-wave velocities are com-

puted first for water saturated rocks and using a mathematical transformation termed

fluid substitution these properties are obtained for a rock saturated with oil. This

procedure is explained in section A.2.5 of this appendix.

A.2.4 P-wave and S-wave Velocities

The relationship between P-wave velocity (Vp) and porosity is very well known in

Rock Physics. The higher the porosity of the rock the softer the rock is, and the

smaller the P-wave velocity. In other words, when porosity is high, the rock is more

compressible and less resistant to wave-induced deformations, therefore Vp is small.

Similarly, when porosity is low the rock is less compressible and more resistant to

wave-induced deformations, therefore Vp is high.

Many empirical expressions of Vp as a function of porosity have been obtained from

laboratory data [92] [67] [40] [84], and all of them show the inverse proportionality


between these two variables.

The Stanford V reservoir [57] uses Han’s Vp − φ relation to obtain P-wave veloc-

ities from the previously simulated porosity. Strictly speaking, Han’s relations are

obtained from sandstone samples collected from different depths (different levels of

compaction), and they show a steep cementing trend (see Figure A.13) which indi-

cates that porosity is controlled by diagenesis and cementing [6] [5]. The reservoir

model created here is not exhibiting a wide range of depths, and porosity is controlled

more by sorting and clay content (depositional) which means that the cementing trend

should not be steep (see Figure A.13).

Figure A.13: Cementing versus sorting trends.

A more appropriate rock physics model for obtaining Vp from porosity for sand-

stones corresponds to the constant cement model described by Dvorkin and Nur [29].

The theoretical constant cement model predicts the bulk modulus K and shear

modulus G of dry sand with constant amount of cement deposited at grain surface.

The bulk and shear moduli are two elastic moduli that define the properties of a rock

that undergoes stress, deforms, and then recovers and returns to its original shape

after the stress ceases. P-wave velocity is a function of density and these two elastic

moduli:


V 2p =

K + 43G

ρ(A.4)

The equations for the Dvorkin’s constant cement model are as follows:

Kdry =

(φ/φb

Kb + 4Gb/3+

1− φ/φb

Kmin + 4Gb/3

)−1

− 4Gb/3 (A.5)

Gdry =

(φ/φb

Gb + z+

1− φ/φb

Gmin + z

)−1

− z (A.6)

z =Gb

6

9Kb + 8Gb

Kb + 2Gb

(A.7)

where φb is porosity (smaller than φc, the initial depositional porosity, sometimes

referred to as critical porosity) at which contact cement trend turns into constant

cement trend (see Figure A.13). Elastic moduli with subscript “min” are the moduli

of the rock mineral and elastic moduli with subscript “b” are the moduli at porosity

φb. These moduli are calculated from the contact cement theory with φ = φb. The

Dvorkin’s contact cement theory is as follows:

Kdry =n(1− φc)McSn

6(A.8)

Gdry =3Kdry

5+

3n(1− φc)GcSτ

20(A.9)

where n is the coordination number, φc is the critical porosity, Mc is the cement’s

P-wave modulus (M = ρV 2p ), and Gc is the cement’s shear modulus. The constants

Sn and Sτ are computed with the following equations:

Sn = Anα2 + Bnα + Cn

An = −0.024153Λ−1.3646n

Bn = 0.20405Λ−0.89008n


Cn = 0.00024649Λ−1.9864n

Sτ = Aτα2 + Bτα + Cτ

Aτ = −10−2(2.26ν2 + 2.07ν + 2.3)Λ0.079ν2+0.175ν−1.342τ

Bτ = (0.0573ν2 + 0.0937ν + 0.202)Λ0.0274ν2+0.0529ν−0.8765τ

Cτ = −10−4(9.654ν2 + 4.945ν + 3.1)Λ0.01867ν2+0.4011ν−1.8186τ

Λn =2Gc

πG

(1− ν)(1− νc)

(1− 2νc)

Λτ =Gc

πG

α = [(2/3)(φc − φ)/(1− φc)]1/2

where G and ν are the shear modulus and the Poisson’s ratio of the grains (matrix),

respectively; Gc and νc are the shear modulus and the Poisson’s ratio of the cement.

The constant cement model input parameters used in this reservoir are summarized

in Table A.8. A 1% calcite cement is added to the sandstone facies (channel and

point bar).

Having computed Kdry and Gdry for dry sandstones using Dvorkin’s constant ce-

ment model, the following equations are used to obtain Ksat and Gsat for water sat-

urated sandstones. These equations correspond to one form of the Gassmann’s fluid

substitution which is explained in more detail in the next section.

Ksat = Kmin

[φKdry − (1 + φ)KwaterKdry/Kmin + Kwater

(1− φ)Kwater + φKmin −KwaterKdry/Kmin

](A.10)

Gsat = Gdry (A.11)

The P-wave velocity Vp for the sandstones is obtained using Eqn. A.4 with Ksat

and Gsat . The rock physics model used for obtaining Vp for shales corresponds to

the empirical Vp − ρ Gardner’s power law [31]:


Parameter Value

Critical porosity φc 0.38Coordination number n 9Cement’s shear modulus Gc 32 GPaCement’s Poisson’s ratio νc 0.32Cement’s density ρc 2.71 g/ccφb 0.37Effective pressure Peff 0.1 MPa

Table A.8: Constant cement model input parameters used in the Stanford VI reser-voir.

ρ = d V fp (A.12)

where d = 1.75 and f = 0.265 are typical values for shales.

Figure A.14 shows the resulting Vp values as a function of porosity for shales (gray

dots) and brine-saturated sandstones (blue dots); additionally, this figure shows the

previously discussed Dvorkin’s constant cement model for 1% cement (red line), the

Dvorkin’s contact cement model (black line), and two typical Rock Physics bounds

(Hashin-Shtrikman lower and upper bounds) which are shown to demonstrate that

the results are within realistic limits.

Regarding the calculation of S-wave velocities (Vs), we use Vp − Vs relations for

water-saturated sandstones and shales from Castagna et al. [19]. They are as follows:

Vs = 0.862 Vp − 1.172 for shales (A.13)

Vs = 0.804 Vp − 0.856 for sandstones (A.14)

A.2.5 Fluid Substitution

In order to obtain density, P-wave and S-wave velocities for the reservoir saturated

with oil, a mathematical transformation termed fluid substitution introduced by

Gassmann [32] is used to calculate the elastic moduli of the rock as one fluid dis-

places another in the pore space.


Figure A.14: P-wave velocity vs. porosity for shales and brine-saturated sandstones.

The elastic moduli define the properties of a rock that undergoes stress, deforms,

and then recovers and returns to its original shape after the stress ceases. When

the fluid contained in the rock changes the overall elastic moduli of the rock also

changes and the seismic velocities are affected. Intuitively, the less compressible the

fluid in the pore space the more resistant to wave-induced deformations the rock is.

A rock with a less compressible fluid (such as brine) is stiffer than a rock with a more

compressible fluid (such as gas).

Seismic P-wave and S-wave velocities are functions of density and two elastic

moduli, the bulk modulus K and the shear modulus G:

V 2p =

K + 43G

ρ(A.15)

V 2s =

G

ρ(A.16)

Gassmann’s equation shown below is used to obtain the bulk modulus K2 of the

rock saturated with fluid 2, which is oil in this case.


K2

Kmin −K2

− Kfl2

φ(Kmin −Kfl2)=

K1

Kmin −K1

− Kfl1

φ(Kmin−Kfl1)

(A.17)

K1 and K2 are the rock’s bulk moduli with fluids 1 and 2 respectively, Kfl1 and Kfl2

are the bulk moduli of fluids 1 and 2, φ is the rock’s porosity, and Kmin is the bulk

modulus of the mineral.

The shear modulus G2 remains unchanged G2 = G1 only at low frequencies (ap-

propriate for surface seismic), since shear stress cannot be applied to fluids. The

density of the rock is also transformed and the density of the rock with the second

fluid is computed as:

ρ2 = ρ1 + φ(ρfl2 − ρfl1) (A.18)

Having transformed the elastic moduli and the density, the compressional and

shear wave velocities of the rock with the second fluid are computed as:

Vp =

√√√√K2 + 43G2

ρ2

(A.19)

Vs =

√G2

ρ2

(A.20)

For the generation of this data set the properties of water and oil were obtained

using Batzle and Wang relations [8] for pore pressure of 20 MPa and temperature of

85◦C with the result summarized in Table A.9. Batzle and Wang relations summarize

some important properties of reservoir fluids (brine, oil, gas and live oil), as functions

of pressure and temperature among other variables. These relations are mostly based

on empirical measurements and are more appropriate for wave propagation than PVT

data.

One of the fluid properties obtained using Batzle and Wang relations is the adia-

batic bulk modulus, which is believed appropriate for wave propagation. In contrast,

standard PVT data are isothermal and isothermal bulk modulus can be 20% too low


for oil, a factor of 2 too low for gas, while approximately similar for brine [59].

water oildensity (g/cc) 0.99 0.7bulk modulus (GPa) 2.57 0.5Salinity (NaCl ppm) 20,000 —Gravity (API) — 25Gas Oil Ratio (L/L) — 200

Table A.9: Properties of water and oil obtained using Batzle & Wang [8] relations.

The final density, P-wave and S-wave velocities of the reservoir saturated with oil

obtained after performing fluid substitution are shown in Figure A.15.

Figure A.16 shows several crossplots among the computed petrophysical proper-

ties. From this figure a distinction between oil-saturated and both brine-saturated

sandstones and shales is clearly observed.

The scatter of points observed in Figure A.16 is created after the petrophysical

properties are computed by adding a small amount of random noise.

Since density, P-wave and S-wave velocities are computed from mathematical

expressions involving porosity, any crossplot of these properties will reflect their con-

tinuous behavior as it has been computed. However, real data does not show this

behavior and has some scatter. The synthetic data is made more realistic by adding

some random noise that creates the scatter seen in crossplots. The amount of noise

added to each property is not the same and also varies for each facies, Table A.10

summarizes the percentage of random noise used.

Property floodplain pointbar channel boundary

Density 0.5% 0.5% 0.5% 0.5%P-wave velocity 5.0% 2.0% 2.0% 5.0%S-wave velocity 2.0% 2.0% 2.0% 2.0%

Table A.10: Percentage of random noise added to each property, per facies.


Figure A.15: Resulting density (top), Vp (middle) and Vs (bottom) cubes for theoil-saturated reservoir.


Figure A.16: Petrophysical properties crossplots. From left to right: P-wave velocityvs. porosity, P-wave velocity vs. density, S-wave velocity vs. P-wave velocity, andporosity vs. density.

A.3. SEISMIC ATTRIBUTES 177

A.3 Seismic Attributes

“Seismic attributes are all the information obtained from seismic data, either by

direct measurements or by logical or experience based reasoning” [82]. The principal

objectives of the seismic attributes are to provide accurate and detailed information to

the interpreter on structural, stratigraphic and lithological parameters of the seismic

prospect.

Many attributes can be computed from seismic data, however, only those at-

tributes computed from seismic reflection amplitude carry information about elastic

contrast in the subsurface. Seismic inversion attempts to translate this information

into elastic properties, which are functions of density, P-wave and S-wave velocities.

In a real reservoir characterization situation, seismic inversion is performed on

the seismic reflection amplitudes to obtain the elastic properties, also called physical

attributes.

As mentioned before, from elasticity theory we know that these elastic properties

are functions of density, P-wave and S-wave velocities. Since Stanford VI is a synthetic

data set, the petrophysical properties created before are used to compute a typical

set of physical seismic attributes that could be obtained from seismic inversion in a

real situation, although no explicit inversion is performed.

The following list corresponds to the physical seismic attributes computed:

• Acoustic Impedance.

• S-wave Impedance.

• Elastic Impedance.

• Lame coefficients λ and µ.

• Poisson’s Ratio.

• AVO Intercept and Gradient.

These attributes are computed at the point support using several mathematical

expressions. Subsequently, a surface seismic filtering and smoothing is performed to

obtain the same attributes at the seismic scale. In doing so, more realistic seismic

attributes are created that provide a filtered view of the true spatial variation of


petrophysical properties.

A.3.1 Mathematical Expressions

The mathematical expressions for the seismic attributes computed are functions of

density, P-wave and S-wave velocities. Acoustic impedance and S-wave impedance are

the result of the product between density and P-wave or S-wave velocity respectively

(Eqns. A.21 and A.22).

AI = ρ Vp (A.21)

SI = ρ Vs (A.22)

Elastic impedance or pseudo-impedance is a generalization of acoustic impedance

for variable incidence angle θ (Eqn. A.23). The elastic impedance is not an intrinsic

rock property as the acoustic impedance, since it depends on the incidence angle and

is derived from approximations.

When compressional seismic waves (P waves) hit a boundary between two media

of different elastic properties, part of the energy is reflected while part is transmitted.

If the P wave hits the boundary at a zero incidence angle (normal incidence), the

amplitude of the reflected wave is proportional to the contrast in acoustic impedance

between the two media, basically the amplitude depends only on P-wave velocity and

density. However, if the P wave hits the boundary at an angle different from zero,

the amplitude of the reflected wave depends on P-wave velocity, S-wave velocity and

density (see Figure A.17).

How amplitudes change with the angle of incidence for elastic materials are de-

scribed by the “Zoeppritz equations” [94]. Since complicated, various authors have

presented approximations to these equations (e.g., Bortfeld [9]), and elastic impedance

is obtained from one of these approximations of the Zoeppritz equations [25]. Strictly

speaking, elastic impedance is derived from a linearization of the Zoeppritz equations

for P-wave reflectivity [69] that is accurate for small changes of elastic parameters

(Vp, Vs and ρ) and small angles of incidence. The derivation of the equation for the

elastic impedance also assumes that the ratio V 2s /V 2

p is constant.


Figure A.17: P-wave hitting a reflector. The physical properties are different on eitherside of the reflector.

As expected, elastic impedance is a function of P-wave velocity, S-wave velocity,

density and incidence angle. This attribute is typically obtained by inversion of angle

stacks. For this reservoir EI is computed for θ = 30◦ since far offsets (corresponding

to larger incidence angles, θ) are more sensitive to changing saturation than near

ones.

EI(θ) = V 1+tan2θp V −8(Vs/Vp)2sin2θ

s ρ1−4(Vs/Vp)2sin2θ (A.23)

Lame’s coefficients λ and µ (Eqns. A.24 and A.25) have been used as reservoir

indicators. Stewart [80] advised that λ/µ might be less sensitive to lithology and

highlight pore-fill changes; Goodway et al. [34] observed the conversion from velocity

measurements to Lame’s coefficients λ and µ improves identification of reservoir zones,

and Xu and Bancroft [93] shown the moduli ratio of λ/µ is a sensitive hydrocarbon

indicator.

λ = ρV 2p − 2µ (A.24)

µ = ρV 2s (A.25)

Poisson’s Ratio (Eqn. A.26) involves only P and S-wave velocities, it is a very

good indicator of fluid type and can be obtained from AVO Inversion [66].


ν =V 2

p − 2V 2s

2(V 2p − V 2

s )(A.26)

Amplitude variation with offset (AVO) comes from a process called “energy par-

titioning”. When compressional seismic waves (P waves) hit a boundary between

two media of different elastic properties, part of the energy is reflected while part is

transmitted. If the wave hits the boundary at an angle different from zero (incidence

angle), P wave energy is partitioned further into reflected and transmitted P and S

(shear waves) components (see Figure A.18). The amplitudes of the reflected and

transmitted energy depend on the contrast in elastic properties across the boundary,

specifically on P-wave velocity, S-wave velocity and density. But, more importantly

reflection amplitudes also depend on the angle of incidence of compressional seismic

waves.

Figure A.18: Seismic wavefront hitting a reflector. The physical properties are dif-ferent on either side of the reflector. The part of the P wave striking at a particularangle-of-incidence (represented by a ray) will have its energy divided into reflectedand transmitted P and S waves.

How amplitudes change with the angle of incidence for elastic materials are de-

scribed by the “Zoeppritz equations” [94]. One of the most widely used approxima-

tions to the “Zoeppritz equations” is from Shuey [76]:


R(θ) ≈ R0 + Gsin2θ + F (tan2θ − sin2θ) (A.27)

where

R0 =1

2

[∆Vp

Vp

+∆ρ

ρ

](A.28)

G =1

2

∆Vp

Vp

− 2(Vs/Vp)2

[∆ρ

ρ+ 2

∆Vs

Vs

](A.29)

F =1

2

∆Vp

Vp

(A.30)

The expression for the reflection coefficient given in Eqn. A.27 can be interpreted

in terms of different angular ranges [21]. R0 is the normal incidence reflection coeffi-

cient often referred to as the AVO intercept, G describes the variation at intermediate

offsets and is often referred to as the AVO gradient, whereas F dominates at far offsets

near the critical angle (angle at which all the P-wave incident energy is transmitted).

AVO intercept and gradient have been widely used for hydrocarbon detection,

specially gas, and they are obtained by analyzing the amplitudes of pre-stack seismic

data [65] [23] [70] [78] [2] [20] [73] [54].

Note that AVO intercept and gradient are obtained from an approximation to

the exact P-wave reflection coefficient, that is accurate for small changes of elastic

parameters (Vp, Vs and ρ) and small angles of incidence. Additionally, the mathemat-

ical expression for the P-wave reflection coefficient is obtained originally for a single

interface between two semi-infinite layers; in real cases wave propagation occurs in

more complex multilayered media.

For the Stanford VI reservoir, AVO intercept and gradient are computed using the

above equations and the filtered and smoothed density, P-wave and S-wave velocities,

since the compressional wave reflection coefficient (Eqn. A.27) is obtained for a semi-

infinite two layer media.


A.3.2 Computation of Seismic Attributes

Using the mathematical expressions described above, we compute the point-support

seismic attributes (see Figure A.19). Some crossplots show how they discriminate

fluids and lithology.

The relationship between Acoustic Impedance, Elastic Impedance and porosity is

shown in Figure A.20, where it can be seen how Elastic Impedance is an excellent

indicator of the presence of hydrocarbon. Similarly, Figure A.21 also shows how

Poisson’s Ratio discriminates between oil and brine-saturated sandstones. On the

contrary, S-wave impedance by itself is not a good discriminator of either lithology

or fluid (Figure A.22). Figure A.23 shows a crossplot of Lame’s coefficients λ and µ,

where a clear discrimination of both lithology and fluid type is observed.

Finally a typical AVO intercept versus gradient crossplot is created (see Figure

A.24), where oil-saturated sandstones deviate from the background trend followed by

brine-saturated sandstones and shales. According to Castagna’s sand classification

[22] in terms of their AVO response, the Stanford VI sandstones are identified as

“Class III” sands: lower impedance than the overlying shales (classical bright spots),

and increasing reflection magnitude with offset.

Having computed the seismic attributes at the point-support scale, they are fil-

tered and smoothed in order to create seismic attributes at the seismic scale. Note

that this is a simple but robust and economical way for computing the seismic at-

tributes at the seismic scale.

The Born approximation [88] [62] is used to compute the filter using the charac-

teristic transfer function for the surface seismic measurement geometry and assuming

continuous lines of sources and receivers. The parameters used to define such filter

are summarized in Table A.11.

Additionally, the filtered attributes are smoothed using a 3D window averaging

in order to create more lateral smoothing typical of seismic data. The window has

a vertical size of λ/4 and a horizontal size of (Zλ)1/2, which corresponds to the size

of the Fresnel zone. The resulting seismic attributes at the seismic-support scale are

shown in Figure A.25.


minimum signal frequency 10 Hz

maximum signal frequency 40 Hz

source spread -1875 m to 1875 m

receiver spread -1875 m to 1875 m

Table A.11: Parameters used to define the “surface seismic” filter.


Figure A.19: Seismic attributes at the Geostatistical Scale: Acoustic Impedance,Elastic Impedance, S-wave Impedance, Poisson’s Ratio, Lame coefficients λ µ.


Figure A.20: Seismic attributes crossplots. From left to right: Acoustic impedancevs. porosity, Elastic impedance vs. porosity, and Acoustic impedance vs. Elasticimpedance.

Figure A.21: Seismic attributes crossplots. From left to right: Poisson’s Ratio vs.Acoustic impedance, Poisson’s Ratio vs. Elastic impedance, and Poisson’s Ratio vs.porosity.


Figure A.22: Seismic attributes crossplots. From left to right: S-wave impedance vs.porosity, S-wave impedance vs. Elastic impedance, and Poisson’s Ratio vs. S-waveimpedance.

Figure A.23: Lame coefficients λ vs. µ.


Figure A.24: AVO Intercept vs. Gradient for oil and brine-saturated sandstones.


Figure A.25: Seismic attributes at the Seismic Scale: Acoustic Impedance, Elastic Im-pedance, S-wave Impedance, Poisson’s Ratio, Lame coefficients λ µ, AVO attributesIntercept and Gradient.

A.4. RESERVOIR FLOW SIMULATION 189

A.4 Reservoir Flow Simulation

Using the simulated permeability cube for the Stanford VI reservoir model obtained in

section A.2, the next step is to perform a flow simulation. The results from this process

can be potentially used for further research in history matching of both production

and 4D seismic data.

It is well known that reservoir flow simulation provides the means to develop

reservoir management plans to achieve optimal recovery under certain economic con-

straints, since flow simulation allows to predict recovery before production. In order

to do so, flow simulation programs solve mathematical equations that describe the

flow of fluids through a numerical model of the reservoir.

The reservoir model used for solving the flow equations comprises two basic petro-

physical properties: porosity and permeability. Using a discrete 3D reservoir model

with each grid block considered homogeneous and represented by a value of porosity

and permeability, the flow equations often expressed as mass balances are solved for

each grid block under certain boundary conditions.

The number of equations to be solved per block depends on the complexity of the

in situ and injected fluids. Typically, this number varies from 3 (black-oil simulators)

to 15 (compositional simulators). An isothermal black-oil model is used since there

are only two phases in the reservoir (oil and water) and only water is injected at a

certain time during the flow simulation.

Considering two-phase flow (water and oil phases), only 2 equations are to be

solved per grid block; however, the computer work increases rapidly with the number

of blocks in the reservoir model. As mentioned in section A.1 of this appendix, the

size of the high-resolution 3D geo-cellular reservoir model is of 150 × 200 × 200 =

6, 000, 000 grid blocks, which exceeds the capabilities of conventional reservoir simu-

lators.

In order to reduce the size of the simulation model, hence the computational

running time of the flow simulation, upscaling of the reservoir properties is performed

to construct a coarsened reservoir model.


A.4.1 Upscaling of the Reservoir Model

The goal of any upscaling technique is to coarsen geological models to manageable

levels for flow simulation. These coarsened flow models should replicate the high-

resolution behavior in overall flow rate. Usually these techniques are referred to as

flow-based upscaling techniques.

The two reservoir properties that are input to the flow simulation correspond to

porosity and permeability. These two properties are upscaled from a high-resolution

of 150 × 200 × 200 = 6, 000, 000 grid blocks to a coarse scale of 30 × 40 × 40 =

48, 000 grid blocks. Porosity is upscaled using a linear block average, Figure A.26

shows the high-resolution porosity and the resulting coarse scale porosity. Since it

has such a strong impact on flow [26], permeability is upscaled using a flow-based

technique.

Figure A.26: Porosity at the high-resolution (left), linearly averaged porosity afterupscaling.

When upscaling homogeneous and isotropic permeability, the resulting coarse per-

meability or effective permeability becomes anisotropic. In three dimensions and since

the simulation grid follows the reservoir layering (“stratigraphic” grid), three effec-

tive permeabilities are obtained per each coarse grid block: kx, ky and kz. Figure

A.27 shows the resulting effective permeabilities in each direction x, y and z after

upscaling.

The upscaling technique used here produces effective permeabilities kx, ky and kz


by using a single-phase pressure solver [27]. This method corresponds to the GSLIB

program flowsim.

Figure A.27: Effective Permeability after upscaling: kx (top left), ky (top right), kz

(bottom).

A.4.2 Flow Simulation

The flow simulation is performed using the commercial software ECLIPSE 3. A fully-

implicit, three phase, three dimensional, black-oil simulator is used; however, only

two phases (water and oil) are present in the reservoir.

The oil and water PVT properties used for the flow simulation are summarized in

Table A.12. The relative permeability curves shown in Figure A.28 are kept constant

3 c© Copyright 2005 Schlumberger. All rights reserved.


for the entire reservoir, and no capillary pressure is considered in the flow simulation

(Pc = 0).

An active constant flux aquifer is below the reservoir and the water-oil contact

is at 9, 840 ft depth. The constant water inflow rate is 31, 000 STB/day. The flow

simulation starts in January of 1975 with six wells in production (primary production);

a summary of the production schedule is given in Table A.13, and the map location

of the injector and producer wells is shown in Figure A.29.

Property Oil WaterDensity (lb/ft3) 45.09 61.80Viscosity (cp) 1.18 0.325

Formation Volume Factor 0.98 1.0

Table A.12: Oil and Water PVT properties.

Figure A.28: Oil and Water Relative Permeability curves.

The reservoir has 30 years of active production with 31 producers and 15 water

injectors. As indicated in the production schedule table, not all wells start producing

oil or injecting water at the same time, as is typical of an actual reservoir development

where new wells are constantly added. Producer wells are controlled by constant liquid

rate production with a BHP constraint of 2700psia, while injector wells are controlled

by constant water injection rate.


Figure A.29: Location maps of producer wells (left), and injector wells (right). Thecolor represents horizon top depth (ft).

While oil production takes place, the water-oil contact starts to rise and the pro-

ducer wells located far away from the structure axis (see Figure A.29) start producing

both oil and water. For those producer wells, P21 through P28, an economic limit is

set such that they are converted to water injectors after they reach a water cut higher

than 0.5.

Figure A.30 shows a summary of the reservoir flow simulation result in terms of

rates while Figure A.31 shows the simulation history in terms of cumulative quantities.

Figure A.30 also shows reservoir pressure as a function of time where it is observed

how the aquifer constant water influx fails to keep the reservoir pressure constant after

4 years of oil production, and how pressure decreases slowly after water injection starts

(11 years after oil production started).

In general it is observed how the oil production increases with time due to the


Figure A.30: Field rates history: Aquifer water influx rate (red line), Oil productionrate (green line), Water injection rate (blue line), Water production rate (cyan line)and Reservoir pressure (black dotted line).

activation of multiple production wells, keeps constant for 8 years and starts to de-

crease due to the increase in water production. Water is injected in the reservoir

to maintain the pressure, as a consequence the WOC rises reaching producing wells.

The water injection process starts 11 years after oil production started while water

production starts 14 years after oil production started.

Figure A.32 shows a 3D view of the reservoir before and after 30 years of oil

production; it is observed how the WOC has changed due to oil production and

water injection. Another view of the change in the reservoir oil saturation with

time is shown in Figures A.33, A.34 and A.35, where a constant X North-South slice

(Figure A.33), a constant Y East-West slice (Figure A.34), and a horizon slice (Figure

A.35) are shown before production, 10, 20 and 30 years after oil production started.


Figure A.31: Field cumulative history: Cumulative aquifer water influx (red line),cumulative oil production (green line), cumulative water injection (blue line) andcumulative water production (cyan line).

Figure A.32: 3D view of the reservoir before oil production starts (top), and 30 yearsafter production started (bottom). The color bar represents oil saturation.


Date Operation

January 1975 Start primary oil production with wells P1 to P6.

January 1979 Wells P22 and P24 are open to production.

January 1981 Wells P26, P28 and P30 are open to production.

January 1983 Wells P21, P23, P25, P27, P29 and P31 are open toproduction.

January 1986 Wells P7, P9, P11, P13, P15, P17 and P19 are open toproduction.Start water injection in wells I32, I33, I34, I36, I37, I38,I41, I43 and I45.

January 1989 Wells P8, P10, P12, P14, P16, P18 and P20 are opento production.Start water injection in wells I35, I39, I42, I40.

October 1989 Start water injection in wells I44, I46.

January 1995 Increasing production rate of wells P1 to P6.Increasing water injection rate of wells I36, I42, I43, I44,I45 and I46.

January 1998 Increasing production rate of wells P7 to P20.

January 2001 Increasing production rate of wells P1 to P6.

March 2003 Increasing production rate of wells P8, P10, P12, P14,P16, P18 and P20.

March 2005 End of the flow simulation.

Table A.13: Summary of the production schedule.


Figure A.33: Constant X = 6151 ft North-South slice of the reservoir before oilproduction starts, 10, 20 and 30 years after oil production started.


Figure A.34: Constant Y = 410 ft East-West slice of the reservoir before oil produc-tion starts, 10, 20 and 30 years after oil production started.


Figure A.35: Horizon slice at 100 meters below the top of the reservoir before oilproduction starts, 10, 20 and 30 years after oil production started.


Since running the reservoir flow simulator on the fine scale model is not feasible

due to the extremely large size of the high-resolution model (150 × 200 × 200 =

6, 000, 000 grid blocks), a new upscaled version of the reservoir model with 75×100×100 = 750, 000 grid blocks is created to obtain a flow response closer to the real one

and without paying a high computational cost. This “pseudo” high-resolution flow

response is then used as the reference.

Porosity is upscaled using a linear block average (see Figure A.36) and effective

permeabilities kx, ky and kz (see Figure A.37) are obtained using the single-phase

flow-based upscaling technique flowsim.

Figure A.36: Porosity of the “pseudo” high-resolution reservoir model.

Using exactly the same production schedule shown for the upscaled (30×40×40)

reservoir model, the flow simulation is performed and the results are summarized in

Figures A.38, A.39, A.40, A.41, A.42, and A.43.

The result of the flow simulation on the “pseudo” high-resolution reservoir model

is similar to the one obtained from the upscaled model. Comparing the production

rate histories only small changes are noticed, and in general the changes occur during

the last 10 years of production, where the time at which some wells switch from

production to injection differs between the two models. This observation allows us

to conclude that the water front is different for both models, and this is clear to see

when comparing Figures A.32 and A.40, A.33 and A.41, A.34 and A.42, A.35 and

A.43.


To illustrate this important remark the water cut history of well P21 is compared

in both simulations as well as the water front at the well location for the earliest of

the two times (≈ 24 years after production started). The result is shown in Figure

A.44 and it is observed that the water cut is higher in the “pseudo” high-resolution

model and well P21 switches to injection earlier.

This result shows the importance of reservoir heterogeneity in flow, while an early

and high water cut is observed in the field the upscaled reservoir model is unable to

reflect it.

Figure A.37: Effective Permeability of the “pseudo” high-resolution reservoir model:kx (top left), ky (top right), kz (bottom).


Figure A.38: Field rates history: Aquifer water influx rate (red line), Oil productionrate (green line), Water injection rate (blue line), Water production rate (cyan line)and Reservoir pressure (black dotted line).

Figure A.39: Field cumulative history: Cumulative aquifer water influx (red line),cumulative oil production (green line), cumulative water injection (blue line) andcumulative water production (cyan line).


Figure A.40: 3D view of the reservoir before oil production starts (top), and 30 yearsafter production started (bottom). The color bar represents oil saturation.


Figure A.41: Constant X = 6151 ft North-South slice of the reservoir before oilproduction starts, 10, 20 and 30 years after oil production started.


Figure A.42: Constant Y = 410 ft East-West slice of the reservoir before oil produc-tion starts, 10, 20 and 30 years after oil production started.


Figure A.43: Horizon slice at 100 meters below the top of the reservoir before oilproduction starts, 10, 20 and 30 years after oil production started.


Figure A.44: Water cut versus time for well P21: solution from “pseudo” high-resolution model (red), and solution from upscaled model (blue). Water saturation24 years after oil production started: solution from “pseudo” high-resolution model(middle), and solution from the upscaled model (bottom).


A.5 4D Seismic Data

What is referred to as 4D seismic data is nothing more than three-dimensional (3D)

seismic data acquired at different times over the same area to assess changes in a

producing hydrocarbon reservoir with time; changes may be observed in fluid location,

saturation, pressure, and temperature. Consequently, one of the main applications of

4D seismic data is to monitor fluid flow in the reservoir.

In order to create the 4D seismic response, several 3D seismic data sets s(u, tn)

are forward modeled using the simple convolutional model. It is clear the the first

seismic data set s(u, t0) will be created using the acoustic impedance of the reservoir

prior to production, while the following seismic data set s(u, tn) with n > 0 will be

created using the acoustic impedance of the reservoir as it has changed due to the

movement of fluids in the reservoir.

The acoustic impedance of the reservoir at time tn is obtained using the fluid

substitution procedure explained in section A.2.5. However, this procedure requires

one to know the properties (density and bulk modulus) of the fluid in the rock at time

tn and we know from the flow simulation that two fluids are present with different

partial saturations.

The most common approach to modeling partial saturation (gas/water or oil/water)

or mixed fluid saturations (gas/water/oil) is to replace the set of phases with a sin-

gle “effective fluid”. The bulk modulus of this “effective fluid” is computed with a

weighted harmonic average, termed Reuss average in the rock physics literature:

1

Kfl

=∑

i

Si

Ki

(A.31)

where Kfl is the effective bulk modulus of the fluid mixture, Ki denotes the bulk

moduli of the individual fluid phases, and Si represents their saturations. This model

assumes that the fluid phases are mixed at the finest scale.

The density of the “effective fluid” is computed with the mixing formula:

ρfl =∑

i

Siρi (A.32)

A.5. 4D SEISMIC DATA 209

where ρfl is the effective density of the fluid mixture, ρi denotes the density of the

individual fluid phases, and Si represents their saturations.

Using the results from the reservoir flow simulation, three seismic data sets are

computed at different times during the oil production history (Figure A.45). The first

seismic data set s(u, t1) is computed after t1 = 10 years of oil production; this time

corresponds to the end of primary production and the start of waterflooding. The

second seismic data set s(u, t2) is computed after t2 = 25years of oil production; this

time corresponds to 15 years of waterflooding. The last and third seismic data set

s(u, t3) is computed after t3 = 30 years of oil production; this time corresponds to

the end of the reservoir flow simulation.

Figure A.45: Base seismic data set acquired prior to oil production (top left), seismicdata sets acquired after 10 years of oil production (top right), after 25 years of oilproduction (bottom left), after 30 years of oil production (bottom right).

In order to obtain the 4D seismic response due to the rising water front the

difference between the base seismic data set s(u, t0) (computed at time t0 = 0 years,

before oil production starts) and each of the three seismic data sets is computed at


times tn > 0 (n=1,2,3):

∆sn(u, ∆tn) = s(u, tn)− s(u, t0) n = 1, 2, 3 (A.33)

Additionally, the incremental 4D seismic response is computed to observe the

changes between two consecutive seismic surveys:

[∆sn(u, ∆tn)]∆n = s(u, tn)− s(u, tn−1) n = 1, 2, 3 (A.34)

These differences can be directly obtained by subtracting the originally recorded

amplitudes or any seismic attribute such as acoustic impedance. Generally speaking,

s can be considered as any attribute obtained from the seismic data. The difference

between originally recorded amplitudes are computed assuming small changes in ve-

locity due to the movement of fluids in the reservoir. Subtracting amplitudes can

be a wrong approach when large changes in velocity occur due to the stretching or

shrinking of the time axis.

The workflow used to create each of the 4D seismic responses, at times t1 =

10 years, t2 = 25 years and t3 = 30 years, is summarized in Figure A.46. To obtain

the seismic impedance at time tn fluid substitution on the high-resolution model is

performed using sub-sampling of the coarsened saturations to all high-resolution grid

cells (see section 3.1).

Figure A.47 shows the distribution of fluids in the reservoir after t1 = 10, t2 = 25

and t3 = 30 years of oil production, as well as the 4D seismic response ∆s1(u, ∆t1),

∆s2(u, ∆t2), and ∆s3(u, ∆t3), and the incremental 4D response [∆s1(u, ∆t1)]∆n,

[∆s2(u, ∆t2)]∆n, and [∆s3(u, ∆t3)]∆n. This figure shows how the seismic response

changes due to the rising of the water front. In the areas where oil is still in place,

the seismic data shows no difference. In the areas where water is present, the mag-

nitude of the difference increases with time due to an increase in water saturation.

The incremental difference between two consecutive seismic surveys shows the areas

where the distribution of fluids has changed during that time lapse.

The result obtained in Figure A.47 corresponds to the upscaled reservoir model.

The same procedure is followed for the “pseudo” high-resolution reservoir model, and


the results are shown in Figure A.48.

Comparing Figures A.47 and A.48 it has been observed that the 4D seismic re-

sponse at late times (25 and 30 years after oil production started) is different from

each model. The 4D seismic response from the upscaled model exhibits stronger dif-

ferences than the 4D seismic response from the “pseudo” high-resolution model, due

to the differences between the coarse and high-resolution water saturation.


Figure A.46: Workflow used to create the 4D seismic response at different timesduring oil production.


Figure A.47: Water saturation from upscaled model after 10 (top left), 25 (top middle)and 30 (top right) years of oil production. Seismic amplitude difference from upscaledmodel for 10 (middle left), 25 (middle middle) and 30 (middle right) years of oilproduction. Seismic amplitude incremental difference from upscaled model for 10(bottom left), 25 (bottom middle) and 30 (bottom right) years of oil production.


Figure A.48: Water saturation from “pseudo” high-resolution model after 10 (topleft), 25 (top middle) and 30 (top right) years of oil production. Seismic amplitudedifference from “pseudo” fine scale model for 10 (middle left), 25 (middle middle) and30 (middle right) years of oil production. Seismic amplitude incremental differencefrom “pseudo” high-resolution model for 10 (bottom left), 25 (bottom middle) and30 (bottom right) years of oil production.

Appendix B

Snesim Parameter File

The parameter file used in the case study presented in Chapter 4 for the multiple-

point “single normal equation simulation” (snesim) algorithm is shown below. The

main window of the program S-GeMS 1 with the grid and properties defined on the

parameter file is shown in Figure B.1.

<GridSelector_Sim value="Simulation_Grid" />

<Property_Name_Sim value="Facies" />

<Nb_Realizations value="1" />

<Seed value="461004" />

<PropertySelector_Training grid="TI_Grid" property="Training_Image" />

<Nb_Facies value="2" />

<Marginal_Cdf value="0.5 0.5" />

<Max_Cond value="40" />

<Search_Ellipsoid value="50 50 20 0 0 0" />

<Hard_Data grid="WellData_Grid" property="Hard_facies" />

<Use_ProbField value="1" />

<ProbField_properties count="2" value="P_mud_seismic;P_sand_seismic" />

<TauModelObject value="1 1" />

<VerticalPropObject value="Vertical_Proportion_Grid" />

<VerticalProperties count="2" value="VP_mud;VP_sand" />

1 c© Copyright 2002-2006 Board of Trustees of the Leland Stanford Junior University. All rightsreserved.

215

216 APPENDIX B. SNESIM PARAMETER FILE

<Use_Rotation value="1" />

<Use_Global_Rotation value="0" />

<Use_Local_Rotation value="1" />

<Global_Angle value="0" />

<Rotation_property value="Regions_Rotation" />

<Rotation_categories value="72 0" />

<Use_Affinity value="0" />

<Cmin value="1" />

<Nb_Multigrids_ADVANCED value="3" />

<Constraint_Marginal_ADVANCED value="0.5" />

<revisit_nodes_prop value="15" />

<Debug_Level value="0" />

<Subgrid_choice value="1" />

<Previously_simulated value="4" />

<Use_Region value="1" />

<Region_Indicator_Prop value="Reservoir_Region" />

<Active_Region_Code value="1" />

<Use_Previous_Simulation value="0" />

<expand_isotropic value="1" />

<expand_anisotropic value="0" />

<aniso_factor value=" " />

217

Figure B.1: Main window of the program S-GeMS showing the grids and propertiesused in the snesim parameter file, the algorithm panel where the parameters can bemanually input, and the visualization panel with a simulated facies realization.

Appendix C

Results of the PDI Approach

Due to the stochastic nature of the snesim algorithm, several reservoir models that match

production data while honoring all other available data, can be generated using the method-

ology and workflow presented in this dissertation. Section 4.3 has presented the results from

one history-matched reservoir model; however, six history-matched reservoir models have

been obtained using the PDI Approach and the individual results from each history-match

are presented in this appendix.

• The results from the first history-matched reservoir model are shown in Figures C.1

through C.4.

• The results from the second history-matched reservoir model are shown in Figures

C.5 through C.8.

• The results from the third history-matched reservoir model are shown in Figures C.9

through C.12.

• The results from the fourth history-matched reservoir model are shown in Figures

C.13 through C.16.

• The results from the fifth history-matched reservoir model are shown in Figures C.17

through C.20.

• The results from the sixth history-matched reservoir model are shown in Figures C.21

through C.24.

218

219

Figure C.1: History Match # 1. Total oil (top) and water production (bottom) fromwell C-19. Historical data is shown in black, the simulated total oil production fromthe initial guess model is shown in magenta, and the best match obtained after severalflow simulations is shown in green (on the total oil production plot), and blue (on thetotal water production plot).

220 APPENDIX C. RESULTS OF THE PDI APPROACH

Figure C.2: Total oil (top) and water production (bottom) from well C-17D. Historicaldata is shown in black, the simulated total oil production from the initial guess modelis shown in magenta, and the best match obtained after several flow simulationsis shown in green (on the total oil production plot), and blue (on the total waterproduction plot).

221

Figure C.3: History Match # 1. Initial guess of the high resolution facies model (left)used as the starting point for the probability perturbation method. High resolutionfacies model (right) obtained after history matching production data (cumulative oiland water production) from wells C-19 and C-17D. The channel facies is shown inblue, and the floodplain facies is shown in gray.

Figure C.4: History Match # 1. Objective function vs. number of flow simulations.The blue curve shows the value of the objective function for each inner iteration; thered curve shows the value of the objective function for each outer iteration.

223

Figure C.6: History Match # 2. Total oil (top) and water production (bottom) fromwell C-17D. Historical data is shown in black, the simulated total oil production fromthe initial guess model is shown in magenta, and the best match obtained after severalflow simulations is shown in green (on the total oil production plot), and blue (on thetotal water production plot).

225



Figure C.10: History Match # 3. Total oil (top) and water production (bottom)from well C-17D. Historical data is shown in black, the simulated total oil productionfrom the initial guess model is shown in magenta, and the best match obtained afterseveral flow simulations is shown in green (on the total oil production plot), and blue(on the total water production plot).

227




Figure C.13: History Match # 4. Total oil (top) and water production (bottom)from well C-19. Historical data is shown in black, the simulated total oil productionfrom the initial guess model is shown in magenta, and the best match obtained afterseveral flow simulations is shown in green (on the total oil production plot), and blue(on the total water production plot).

229


231


233



235



Figure C.23: History Match # 6. Initial guess of the high resolution facies model (left)used as the starting point for the probability perturbation method. High resolutionfacies model obtained after history matching production data (cumulative oil andwater production) from wells C-19 and C-17D. The channel facies is shown in blue,and the floodplain facies is shown in gray.


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