Apostolos Kantzas, PhD P. Eng.

Jonathan Bryan, PhD, P. Eng.

Saeed Taheri, PhD

[Type the document

subtitle]

1

1. CHAPTER 1........................................................................................................................................... 11

INTRODUCTION ....................................................................................................................................... 11

2. CHAPTER 2........................................................................................................................................... 15

THE POROUS MEDIUM ............................................................................................................................ 15

HOMOGENEITY ................................................................................................................................... 16

ANISOTROPY ....................................................................................................................................... 18

POROSITY ............................................................................................................................................ 18

PORE SIZE DISTRIBUTION .................................................................................................................... 28

SPECIFIC SURFACE AREA ..................................................................................................................... 32

COMPRESSIBILITY OF POROUS ROCKS ................................................................................................ 33

PERMEABILITY ..................................................................................................................................... 37

SATURATION ....................................................................................................................................... 50

FORMATION RESISTIVITY FACTOR ...................................................................................................... 53

MULTI-PHASE SATURATED ROCK PROPERTIES ................................................................................... 60

RELATIVE PERMEABILITY ................................................................................................................... 100

3. CHAPTER 3......................................................................................................................................... 171

MOLECULAR DIFFUSION ....................................................................................................................... 171

Introduction ...................................................................................................................................... 171

........................................................................................................... 171

Diffusion Coefficient ......................................................................................................................... 174

4. CHAPTER 4......................................................................................................................................... 204

Immiscible Displacement ...................................................................................................................... 204

Introduction ...................................................................................................................................... 204

Buckley-Leverett Theory, .................................................................................................................. 204

Water Injection Oil Recovery Calculations ........................................................................................ 215

Vertical and Volumetric Sweep Efficiencies ...................................................................................... 230

5. CHAPTER 5......................................................................................................................................... 242

Miscible Displacement .......................................................................................................................... 242

Introduction ...................................................................................................................................... 242

FLUID PHASE BEHAVIOR .................................................................................................................... 245

First Contact Miscibility Process ....................................................................................................... 255

2

Multiple Contact Miscibility Processes ............................................................................................. 257

Determination of Miscibility Condition............................................................................................. 267

Fluid properties in miscible displacement ........................................................................................ 277

The Equation of Continuity ............................................................................................................... 305

The Equation of Continuity in Porous Media .................................................................................... 306

3

Figure 1-1: World supply of primary energy by fuel type. .......................................................................... 11

Figure 1-2: Simplified Illustrations of Vertical and Horizontal Wells .......................................................... 12

Figure 1-3: Elementary Trap in Sectional View ........................................................................................... 14

Figure 2-1: Dependence of Permeability on Sample Volume ..................................................................... 16

Figure 2-2 A Probability Density Function can be used to find the homogeneity or heterogeneity type of

a porous medium ........................................................................................................................................ 17

Figure 2-3 Microscopic Cross Section Image of a Porous Medium .......................................................... 18

Figure 2-4 Dead-end pore ........................................................................................................................ 20

Figure 2-5 - Storage and connecting pore model for shale or any other type of rock with interconnected

pore systems ............................................................................................................................................... 20

Figure 2-6 - The Various Cross-Sections of Connecting Pores .................................................................... 21

Figure 2-7 a) Three-dimension distribution of connecting; b) Complete storage-connecting pore

system. ........................................................................................................................................................ 21

Figure 2-8 Typical ordered porous medium structures ........................................................................... 23

Figure 2-9 Effect of sorting and grain size distribution on porosity ......................................................... 24

Figure 2-10 Porosimeter Based on Boyle - ..................................................................... 27

Figure 2-11 Colored Thin Section Microscopic Image .............................................................................. 27

Figure 2-12 - Schematic Shape of pore and Pore Throat ............................................................................ 29

Figure 2-13 Sieve Analysis Tools .............................................................................................................. 29

Figure 2-14 - Schematic representation of pores. ...................................................................................... 31

Figure 2-15 Tomographic image. ............................................................................................................. 32

Figure 2-16 Specific Surface Area............................................................................................................. 32

Figure 2-17 Porosity Reduction as an Effect of Compaction Increment by Depth .................................. 34

Figure 2-18 a) Experimental Equipment for Measuring Pore Volume Compaction and Compressibility 36

Figure 2-19 a) Formation Compaction Component of Total Rock Compressibility ................................. 36

Figure 2-20 Flow through a Pipe .............................................................................................................. 38

Figure 2-21 Metallic cast of pore space in a consolidated sand .............................................................. 39

Figure 2-22 Schematic Drawing of Darcy Experiment of Flow of Water through Sand ........................... 39

Figure 2-23 Linear Flow through Layered Bed ......................................................................................... 42

Figure 2-24 Linear Flow through Series Beds ........................................................................................... 44

Figure 2-25 Plot of Experimental Results for Calculation of Permeability ............................................... 46

Figure 2-26 Permeability of Core Sample to Three Different Gases and Different Mean Pressure ........ 48

Figure 2-27 - Effect of Permeability on the Magnitude of the Klinkenberg Effect ..................................... 48

Figure 2-28 ASTM Extraction Apparatus .................................................................................................. 53

Figure 2-29 - The Influence of Pore Structure on the Electrical Conductivity ............................................ 55

Figure 2-30 Formation Resistivity Factor vs. Porosity .............................................................................. 56

Figure 2-31 - Apparent Formation Factor vs. Water Resistivity for Clayey and Clean Sands ..................... 57

Figure 2-32 - Water-Saturated Rock Conductivity as a Function of Water Conductivity ........................... 58

Figure 2-33- Illustration of surface tension (Surface molecules pulled toward liquid causes tension in

surface). ...................................................................................................................................................... 60

Figure 2-34- Simplified Models for Interfacial Tension Determination. ..................................................... 62

Figure 2-35- Pressure relations in capillary tubes ....................................................................................... 62

4

Figure 2-36- Illustration of Wettability ....................................................................................................... 63

Figure 2-37- Equilibrium of Forces at a Liquid-Gas-Solid Interface. ........................................................... 64

Figure 2-38- Rock-Fluid-Fluid Interactions Effect on the Contact Angle ..................................................... 66

Figure 2-39- liquid drop spreading on a solid surface ................................................................................ 67

Figure 2-40- Amott Wettability Test ........................................................................................................... 68

Figure 2-41- Amott Index Ternary Diagram ................................................................................................ 69

Figure 2-41- Amott Index Calculation ......................................................................................................... 69

Figure 2-42- USBM Index Calculation ......................................................................................................... 70

Figure 2-43- Pressure Relation in capillary Tube ........................................................................................ 72

Figure 2-44- Dependency of Water Column to (a). Capillary Radius, (b). Wettability. ............................... 73

Figure 2-45- Principle radii for wetting fluid and spherical grain ............................................................... 74

Figure 2-46- Wetting and non-Wetting fluid distribution about inter grain contact of sphere. ................ 75

Figure 2-47- Non-Wetting fluid entering the capillary tube. ...................................................................... 76

Figure 2-48- Non-Wetting fluid entering the non-uniform capillary tube. ................................................. 76

Figure 2-49- Non-Wetting Fluid enter to a bubble and exit it. ................................................................... 77

Figure 2-50- Capillary Pressure versus wetting phase saturation .............................................................. 78

Figure 2-51- Variation of Capillary Pressure with Permeability .................................................................. 79

Figure 2-52- Flow into a Constriction (cone). ............................................................................................. 79

Figure 2-53- Flow out of a Constriction ...................................................................................................... 80

Figure 2-54- Flow in a Capillary Tube. ......................................................................................................... 81

Figure 2-55- porous diaphragm capillary pressure device. ......................................................................... 83

Figure 2-56- Centrifugal apparatus ............................................................................................................. 84

Figure 2-57- Mercury Injection Method ..................................................................................................... 85

Figure 2-58- Pore size distribution from mercury injection test................................................................. 86

Figure 2-59- Dynamic Measurement of Capillary pressure. ....................................................................... 87

Figure 2-60- Capillary Pressure Curve. ........................................................................................................ 88

Figure 2-61- Contact Angle Hysteresis during the Displacement ............................................................... 89

Figure 2-62- Dynamic Contact Angle Behavior. .......................................................................................... 89

Figure 2.63: Static values of advancing and receding contact angles at rough surfaces versus values at

smooth surfaces (where E refers to smooth surface measurements ........................................................ 90

Figure 2-64- Non-Wetting fluid Enter to a capillary tube with square cross section. ................................ 91

Figure 2-65- Side view after snap-off .......................................................................................................... 91

Figure 2-66- Trapping in a porous media. ................................................................................................... 92

Figure 2-67- Typical non-wetting phase trapping characteristics of some reservoir rocks. ....................... 92

Figure 2-68- Pore Doublet Model. .............................................................................................................. 93

Figure 2-69- Imbibition and Drainage mechanisms in a pore doublet model ............................................ 94

Figure 2-70- Pore doublet model for illustration for displacement and trapping of oil. ............................ 95

Figure 2-71- Trapping of a droplet in a capillary tube. ............................................................................... 97

Figure 2-72- J-function correlation of capillary pressure data in Edwards Jourdanton field...................... 99

Figure 2-73- Typical relative permeability curve. ..................................................................................... 103

Figure 2-74- Typical Gas-Oil Relative Permeability Curve ......................................................................... 105

Figure 2-75- Relative Permeability Curve, (a) Drainage, (b) Imbibition .................................................... 105

5

Figure 2-76- Hafford Relative Permeability Apparatus ............................................................................. 107

Figure 2-77- Fluid Saturation during Steady-State Test ............................................................................ 108

Figure 2-78- Unsteady state apparatus. ................................................................................................... 109

Figure 2-79- (a) Unsteady State Water Flood Procedure, (b) Typical Relative Permeability Curve ......... 110

Figure 2-80- (a) Average Water saturation vs. Water Injection, (b) Injectivity Ratio ............................... 111

Figure 2-81: Amott Ternary Wettability Diagram ..................................................................................... 119

Figure 2-82: Comparison between capillary pressure and relative permeability curves ......................... 122

Figure 2-83 Relative permeability curves of Berea Sandstone. Strongly Water-wet conditions (Sankar

1979) ......................................................................................................................................................... 125

Figure 2-84 Relative permeability of water wet and oil wet systems. ..................................................... 134

Figure 2-85 Comparison between experimental and predicted values (after 3). .................................... 140

Figure 2-86 Compare with results of Corey et al. (after 11) ..................................................................... 145

Figure 2-87 Compare with results of Dalton et al. (after 11) .................................................................... 145

Figure 2- ................... 146

Figure 2- - ........................................ 146

Figure 2-90 Effect of capillary number on relative permeability (after 9) ................................................ 153

Figure 2-91 Low IFT systems (after 21) ..................................................................................................... 155

Figure 2-92 High IFT systems (after 21) .................................................................................................... 155

Figure 2-93 Effects of viscous force on Swir (after 22). .............................................................................. 156

Figure 2-94 Effects of flow rate on relative permeability (after 22) ......................................................... 158

Figure 2-95 Effect of viscosity on relative permeability (after 26) ........................................................... 160

Figure 3-1 - Simple diffusion experiment.................................................................................................. 172

Figure 3-2 Diffusion across a thin film ................................................................................................... 173

Figure 3-3 Prediction overall diffusion from intrinsic diffusion ............................................................. 177

Figure 3-4 Diffusion process in a control volume with a concentration dependent diffusion coefficient

.................................................................................................................................................................. 178

Figure 3-5 In a porous medium fluid generally flowing at about 45o with respect to average direction

of flow ....................................................................................................................................................... 180

Figure 3-6 Pressure decay test cell. ....................................................................................................... 182

Figure 3-7 - Refraction of light at the interface between two media. ...................................................... 183

Figure 3-8 - Sample of light refraction results a) initial time b) after diffusion occurred ......................... 184

Figure 3-9 - (a). Hydrogen nuclei behave as a tiny bar magnets aligned with the spin axes of the nuclei.

(b). Spinning protons with random nuclear magnetic axes in the absence of an external magnetic field.

.................................................................................................................................................................. 185

Figure 3-10 Line up nuclear spins in an external magnetic field. ........................................................... 186

Figure 3-11 Polarization/Relaxation curve. ............................................................................................ 187

Figure 3-12 the Tipping process. ............................................................................................................ 187

Figure 3-13 Net magnetization return to equilibrium by turning off the B1, (the arrow represent net

magnetization) .......................................................................................................................................... 188

Figure 3-14 de-phasing (loss of phase coherence) during T2. ............................................................... 189

Figure 3-15 Spin-echo sequence. ........................................................................................................... 190

Figure 3-16 CPMG pulse sequence. ....................................................................................................... 191

6

Figure 3-17 - The amplitudes of the decaying spin echoes yield an exponentially decaying curve with

time constant T2. ....................................................................................................................................... 191

Figure 3-18 - The echo train (echo amplitude as a function of time) is mapped to a T2 distribution

(porosity as a function of T2). .................................................................................................................... 192

Figure 3-19 Typical NMR spectrum for pure bitumen, pure solvent, and a mixture of them. .............. 194

Figure 3-20 two samples of NMR calibration for bitumen-solvent mixture,. ........................................ 195

Figure 3-21 Diffusion coefficient as a function time, NMR experiment result. ..................................... 197

Figure 3-22 - Schematic view of CAT scanning using x-ray. ...................................................................... 199

Figure 3-23 Calibration curves for the CAT scanner, (a) Liquid calibration curve, (b)Liquid-solid

calibration curve. ...................................................................................................................................... 199

Figure 3-24 Image sample of diffusion process ..................................................................................... 200

Figure 3-25 Medium Domain ................................................................................................................. 201

Figure 3-26 - Sample of diffusion in sand saturated with oil. ................................................................... 202

Figure 3-27 - Average diffusion coefficients for pentane, hexane and octane in heavy oil. .................... 202

Figure 3-28 - Comparison of the diffusion coefficients of pentane in heavy oil in absence/presence sand.

.................................................................................................................................................................. 203

Figure 4-1 Semilog plot of relative permeability ratio versus saturation .............................................. 205

Figure 4-2 Fractional flow curve ............................................................................................................ 206

Figure 4-3 Horizontal bed containing oil and water. ............................................................................. 207

Figure 4-4 Cubic reservoir under active water drive .............................................................................. 208

Figure 4-5 Water fractional flow ant its derivative ................................................................................ 209

Figure 4-6 Fluid Distribution at 60, 120, 240 days ................................................................................. 210

Figure 4-7 - Water saturation distribution as a function of distance, prior to breakthrough .................. 211

Figure 4-8 - Tangent to the fractional flow curve from Sw = Swc ............................................................... 212

Figure 4-9 xD vs. tD for a linear waterflooding. ....................................................................................... 213

Figure 4-10 Saturation Profile at tD = 0.28 ............................................................................................. 214

Figure 4-11 Saturation History at xD = 1, producing face of the medium .............................................. 214

Figure 4-12 - Saturation distribution after 240 days................................................................................. 215

Figure 4-13 - Application of the Welge graphical technique to determine: (a) The front saturation, (b) Oil

recovery after breakthrough .................................................................................................................... 219

Figure 4-14 - Water saturation distribution as a function of distance between injection and production

wells for (a) ideal or piston-like displacement and (b) non-ideal displacement ...................................... 219

Figure 4-15 - (a) Microscopic displacement (b) Residual oil remaining after a water flood .................... 221

Figure 4-16 (a) Relative Permeability Curves, (b) Fractional Flow Curve ............................................... 223

Figure 4-17 Graphical determination of front saturation and water fractional flow. ........................... 223

Figure 4-18 ............................................................................................................................................. 224

Figure 4-19 dimensionless pore volume oil recovery vs. dimensionless pore volume water injection 225

Figure 4-20 - Fractional flow plots for different oil-water viscosity ratios ............................................... 227

Figure 4-21 - Water Saturation Distributions in Systems for Different Oil/Water Viscosity Ratios ......... 228

Figure 4-22 - Typical injection/production well configurations and associated flooding patterns .......... 229

Figure 4-23 - Schematic representation of the two components of the volumetric sweep: (a) areal

sweep; (b) vertical sweep in stratified formation. .................................................................................... 231

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Figure 4-24 a) Bottom coning at oil-water or gas-oil contact, b) Edge coning at oil-water or gas-oil

contact ...................................................................................................................................................... 237

Figure 4-25 - Stable Cone. ......................................................................................................................... 237

Figure 4-26 Flow rate versus time .......................................................................................................... 239

Figure 5-1 Miscible Displacement, a) secondary recovery, b)Tertiary recovery. .................................. 243

Figure 5-2 - A typical phase diagram for a pure component. ................................................................... 245

Figure 5-3 Typical P-T diagram for a multicomponent system. ............................................................. 246

Figure 5-4 (a). Phase diagram of ethane-normal heptane, (b) Critical loci for binary mixtures. ........... 246

Figure 5-5 Typical P-X diagram for the Methane-normal Butane system. ............................................ 248

Figure 5-6 Pressure-composition diagram for mixture of C1 with a liquid mixture of C1-nC4-C10. ..... 249

Figure 5-7 ternary phase diagram for a system consisting of components A, B, and C which are miscible

in all proportions. ...................................................................................................................................... 251

Figure 5-8 All mixture of M1 and M2 would be along line . ...................................................... 252

Figure 5-9 Example 5-2 a) Ternary diagram, b) Chemicals to be mixed. ............................................... 252

Figure 5-10 - Error! Reference source not found.. .................................................................................... 253

Figure 5-11 Ternary Phase Diagram. ...................................................................................................... 254

Figure 5-12 Pressure effect on the miscibility, P1

8

Figure 5-35 Estimated breakthrough recovery as a function of viscosity ratio [32]. ............................. 290

Figure 5-36 Impact of viscous instability on secondary CO2 flood oil recovery efficiency .................... 291

Figure 5-37 Gravity segregation in displacement processes. A) Gr

....................................................................................................................................... 292

Figure 5-38 Flow regimes for miscible displacement in a vertical cross section. .................................. 292

Figure 5-39 Flow regimes in a two-dimensional, uniform linear system (Schematic) [29]. .................. 293

Figure 5-40 volumetric sweep efficiency at breakthrough as a function of the viscous/gravity force

ratio. .......................................................................................................................................................... 294

Figure 5-41 - Model for determining stability criterion in a dipping reservoir ......................................... 295

Figure 5-42 - Schematic of the experimental apparatus. ......................................................................... 298

Figure 5-43 - Plot of mole fraction of solvent in the effluent as a function of the pore volumes of solvent

injected. .................................................................................................................................................... 299

Figure 5-44 - Cumulative recovery for rate reduction and pressure pulsing as a function of pore volumes

of solvent injected .................................................................................................................................... 300

Figure 5-45 - Mole fraction of solvent in the effluent as a function of the pore volumes of solvent

injected. Rate reduction and pressure pulsing case. ................................................................................ 301

Figure 5-46 profile of the solvent effluent concentration produced from a capillary tube in an equal

viscosity and density. ................................................................................................................................ 301

Figure 5-47 Concentration profile for injection of a slug of solvent to displace oil. .............................. 302

Figure 5-48 Mixing of solvent and oil by longitudinal and transverse dispersion. ................................ 303

Figure 5-49 dispersion porous medium being viewed as a series of mixing tank. ................................ 304

Figure 5-50 - Stagnant volume models ..................................................................................................... 304

Figure 5-51 - The effects of capacitance. .................................................................................................. 305

Figure 5-52 Dispersion caused by variation of flow paths in a porous medium. ................................... 305

Figure 5-53 Development of the mixing zone as a function of time during laminar flow in a capillary

(after Nunge and Gill, 1970)...................................................................................................................... 309

Figure 5-54 Summary of the regions of applicability of various analytical solutions for dispersion in

capillary tubes with step change in inlet concentration as a function of dimensionless time and Peclet

number (from Dullien, 1992). ................................................................................................................... 311

Figure 5-55 Dependence of dispersion coefficient on Peclet number in different flow regimes. The scales

on the axes depend on porous medium and other factors. The curve shown approximates longitudinal

dispersion in unconsolidated random packs (after Perkins and Johnston, 1963). ................................... 313

Figure 5-56 Dependence of Peclet number on Reynolds number for an aqueous system (from Perkins

and Johnston, 1963). ................................................................................................................................. 315

Figure 5-57 Range of dispersion coefficients for various sandpacks. The lower curves are for coarse sand

packed to a porosity of about 34%. The upper curves are for finer sand or looser packings (porosity >

34%). Fine sand 200-270 mesh, medium sand 40-200 mesh, coarse sand 20-30 mesh (after Blackwell,

1962). ........................................................................................................................................................ 315

Figure 5-58 Concentration as a function of transformed distance for different values of dispersion

coefficient or time, calculated from eq. (5-72), infinite system. .............................................................. 319

Figure 5-59 Typical probability plot for determination of longitudinal dispersion coefficient. ............... 320

9

Figure 5-60 Effluent concentration profiles for a range of Peclet numbers, calculated from eq. (5-78),

infinite system. .......................................................................................................................................... 321

Figure 5-

boundary conditions (Brenner, 1962). ...................................................................................................... 323

Figure 5-62 Effect of boundary conditions on solutions to the convection-dispersion equation at

different Peclet numbers. ......................................................................................................................... 324

Figure 5-63 The effects of capacitance. ................................................................................................... 326

Figure 5-64 Typical shape of a Langmuir adsorption isotherm. ............................................................... 330

Figure 5-65 Effect of adsorption model parameters on adsorbate effluent concentrations. .................. 332

10

P R E F A C E

The purpose of this manuscript is to provide the reader with the basic principles of flow in

porous media and their association to hydrocarbon production from underground formations.

The intended audience is undergraduate and graduate students in petroleum engineering and

associated disciplines, as well as practicing engineers and geoscientists in the oil and gas

industry.

The material draws from the experiences of the authors in the Western Canadian Sedimentary

Basin and, wherever possible, draws from unconventional resources.

11

1 . C H A P T E R 1

INTRODUCTION

Beginning with the industrial revolution of the early nineteenth century, man has turned more and more to

the use of mineral fuels to supply the energy to operate his machines. The first commercial well drilled

solely for oil was completed in the United States in 1859. Following the success of this well petroleum

production and processing rapidly grew into a major industry in United States. Today, in satisfying the

. Al though their share in the energy mix is

expected to fall, it remains over 80% throughout the period to 2030. The leading role in the energy mix

will continue to be played by oil, with its share remaining above 30%, albeit falling over time (Figure

1-1).

Figure 1-1: World supply of primary energy by fuel type1.

Today petroleum is used not only as a fuel but as a raw material for many industrial materials such as

paint, plastic, rubber, lubricants, and so forth.

What is Petroleum? Petroleum is a mixture of naturally occurring hydrocarbons which may exist in

the solid, liquid, or gaseous state, depending upon the pressure and temperature to which it is

subjected. Virtually all petroleum is produced from the earth in either liquid or gaseous form, and

commonly these materials are referred to as either crude oil or natural gas, depend upon the state of

hydrocarbon mixture.

1 http://www.opec.org/opec_web/en/

http://www.opec.org/opec_web/en/

12

Since the vast majority of oil and gas bearing formations are several hundred meters to several

from the surface. The well-bore region and the collected core (if any) offer a snapshot of the reservoir

properties, in a fashion similar to a line drawn in a three dimensional volume. In other words, results

from core tests do not describe the reservoir accurately but they do help describe the physics.

Up until the late 1980s there was practically no variability in the manner in which a well was drilled.

However, horizontal drilling (a Soviet invention of the 1920s) revolutionized the drilling industry as it

allowed for one well to access a formation at several horizontal locations (Figure 1-2).

Figure 1-2: Simplified Illustrations of Vertical and Horizontal Wells

Origin of Oil

Many theories of the origin of petroleum have been advanced. The theories of the origin of petroleum

can be classified as either organic or as inorganic. The inorganic theory attempts to explain the

formation of petroleum by assuming chemical reaction between water, carbon dioxide and various

inorganic substances such as carbonates, in the earth. The organic theories assume that petroleum

evolved from the decomposition of vegetable and animal organisms that lived during previous

geological ages. Organic theories are commonly acceptable.

Source beds as organic rich formations are the necessity of petroleum generation. Petroleum migration

occurs after formation from source beds toward the reservoir or storage beds. Reservoir rocks have void

spaces and are permeable to fluids, in other words they have interconnected void spaces.

Lithology

Reservoir rocks are categorized as either sandstone or carbonate. Sandstones are formed from grains

that have undergone sedimentation, compaction and cementation. The major characteristics of

sandstone reservoirs are as follows:

Composed of silica grains (mainly quartz and some feldspar),

Consolidated or unconsolidated formations,

May contain shale,

13

May contain minerals (such as iron oxide and iron sulfides),

May include clays (Note: Clays have a negative effect on the reservoir quality).

Carbonates are formed from the remnants of hard-shelled organisms that existed in coral reef

environments. The major characteristics of carbonates are:

Limestone (CaCO3) and/or dolomite (CaMg(CO3)2),

May contain shale,

Minerals, such as pyro-bitumen and anhydrite

Pore space is comprised of areas of dissolution (vugs), fractures and inter-crystalline spaces.

About 60% of the conventional oil reservoir rocks are sandstones and about 39% of them are

carbonates.

Trap

External forces such as buoyancy which force the petroleum to migrate from source rock to reservoir

rock could push oil to reach the surface. So presence of a barrier over the reservoir formation is vital in

accu t

Traps associated with oil fields are complex. Different reservoir according to type of their trap can be

classified as follows (Figure 1-3):

Convex Trap reservoirs which are surrounded by edge water and the trap is due to convexity alone,

Permeability trap reservoir that the barrier is due to the loss of permeability in reservoir rock,

Pinch out trap reservoir, which the periphery partly defined by edge water and partly by the margin

due to the pinch out of reservoir bed,

Fault trap reservoir that has a fault boundary.

14

Figure 1-3: Elementary Trap in Sectional View

Convex Trap Pinchout Trap

Permeability Trap Fault Trap

15

2 . C H A P T E R 2

THE POROUS MEDIUM

Porous materials are encountered literally everywhere in everyday life, in technology and in nature.

With the exception of metals, some dense rocks, and some plastics, virtually all solid and semi-solid

order to qualify as a porous medium:

1. It must contain spaces, so-called voids or pores, free of solids, imbedded in the solid or semi-

solid matrix. The pores usually contain some fluid, such as air, water, oil or a mixture of different

fluids.

2. It must be permeable to a variety of fluids, i.e., fluids should be able to penetrate through one

face of a sample of material and emerge on the other side.

There are many examples where porous media play important roles in technology and, conversely,

many different technologies that depend on porous media. Among the most important technologies

that depend on the properties of porous media are:

1. Hydrology, which relates to water movement in earth and sand structures, such as water flow to

wells from water-bearing formations.

2. Petroleum engineering which is mainly concerned with petroleum and natural gas exploration

and production.

The petroleum engineer is concerned with the quantities of fluid content within the rocks,

transmissibility of fluids through the rocks, and other related properties. These properties depend on

the rock and frequently upon the distribution of character of the fluid occurring within the rock.

Knowledge of the physical properties of the rock and the existing interaction between the hydrocarbon

system (gas, oil and water) and the formation is essential in understanding and evaluating the

performance of a given reservoir.

Rock properties are determined by performing laboratory analyses on cores from the reservoir to be

evaluated. The cores are removed from the reservoir environment through the well during the drilling

operations. There are primarily two main categories of core analysis tests that are performed on core

samples regarding physical properties of reservoir rocks. These are:

Routine core analysis tests

Porosity

Permeability

Saturation

Special core analysis tests

Capillary pressure

Relative permeability

16

Wettability

Surface and Interfacial Tension

Electrical Conductivity

Pore size Distribution

These properties constitute a set of fundamental parameters by which the rock can be quantitatively

described. They are essential for reservoir engineering calculations as they directly affect both the

quantity and the distribution of hydrocarbons and, when combined with fluid properties, control the

flow of the existing phases (i.e., gas, oil, and water) within the reservoir.

HOMOGENEITY

Homogeneous usually means describing a material or system that has the same properties at every

point in space; in other words, uniform without irregularities. It also describes a substance or an object

whose properties do not vary with position. To apply this term on a porous medium we should define a

macroscopic system. A few sand grains cemented together constitute a microscopic rather than

macroscopic porous medium. The properties of a microscopic sample are not expected to be

representative of the macroscopic porous medium from which it was removed.

Let us suppose that a macroscopic pore structure parameter such as porosity or permeability is

determined in a series of samples of increasing size, taking from a large porous sample, and the result is

plotted against the sample size. It could be seen that initially the calculated properties change with

sample size and there are some fluctuations in the results with increasing sample size. With increasing

sample size the amplitude of these fluctuations decreases and gradually diminishes until finally a

smooth line is obtained after a certain sample size (Figure 2-1).

macroscopically representative whenever the macroscopic measured property (such as porosity and

permeability) is not fluctuating any more when including more material around the initial sampling

point, but its variation can 2. When the properties of a porous medium

do not change with changes in the macroscopic representative sample the medium is said to be

macroscopically homogeneous.

Figure 2-1: Dependence of Permeability on Sample Volume

2 F.A.L. Dullien (1979)

17

P

(As a special macroscopic property)

F

(P)

Assume we run a test to determine the homogeneity of the porous medium. In this test along an

arbitrary direction we choose different macroscopic samples and we measure their macroscopic

properties. By increasing the number of samples we can plot a probability density function (PDF) for the

measured properties. The PDF plot could have different shapes according to the homogeneity or

heterogeneity of the medium. The simplest case is when the PDF is a delta function (vertical line) that

shows the property is constant regardless to the sampling position (Figure 2-2.a). Thus we can confirm

the homogeneity of the medium. While proceeding in an arbitrary direction in the medium, we may find

that the measured macroscopic property first remains constant but it suddenly changes to a different

the

discontinuous s (Figure 2-2.b). Processing in any chosen direction in the medium, we may find that

the property continuously changes according to the position of the samples. In this case the medium is

(Figure 2-2.c). Finally it may happen

that the function that representing the variation of parameter with position is piecewise continuous. In

(Figure 2-2.d).

(a)

(c)

(b)

(d)

Figure 2-2 A Probability Density Function can be used to find the homogeneity or heterogeneity type of a porous medium3

a) Macroscopically Homogeneous System; b) Macroscopically Heterogeneous in Discontinuous Sense

c) Macroscopically Heterogeneous in Continuous Sense; d) Macroscopically Heterogeneous In both Continuous and Discontinuous Sense

3 Greenkorn and Kessler (1970)

P

(As a special macroscopic property)

F

(P)

P

(As a special macroscopic property)

F

(P)

P

(As a special macroscopic property)

F

(P)

18

ANISOTROPY

Anisotropy means that some properties of the porous medium do not have the same value in different

directions. In an anisotropic porous medium, the permeability, formation resistivity factor, and

breakthrough capillary pressure depend on the direction. In the most general case these properties are

function of both location in the medium and orientation. Thus the probability density function for each

property can be described with five independent variables (x, y, z) for location and ( , ) for

orientation. If the probability density distribution is independent of the angular coordinates, the

medium is isotropic, otherwise it is anisotropic. In the reservoir formation generally anisotropy can be

caused by periodic layering. So generally we have different properties such as permeability at z direction

compare to the x and y direction.

POROSITY

The rock texture consists of mineral grains of various shapes and sizes and its pore structure is extremely

complex. The most important factors of the pore structure are how much space there is between these

grains and what their shapes are. That is because the spaces between these grains serve to either mainly

transport fluids forming connecting pores, or to store the fluids forming storage pores.

From the reservoir engineering standpoint, porosity is one of the most important rock properties, a

measure of space available for storage of hydrocarbons. Quantitatively, porosity is the ratio of the pore

volume to the total volume (bulk volume). This important rock property is determined mathematically

by the following generalized relationship (Figure 2-3):

(2-1)

Where = porosity

Grain

Pore

Figure 2-3 Microscopic Cross Section Image of a Porous Medium

As the sediments were deposited and the rocks were being formed during geological times, some void

spaces that developed became isolated from the other void spaces by excessive cementation. Thus,

19

many of the void spaces are interconnected while some of the pore spaces are completely isolated. This

leads to two distinct types of porosity, namely:

Absolute porosity

Effective porosity

Absolute porosity

The absolute porosity is defined as the ratio of the total pore space in the rock to that of the bulk

volume. A rock may have considerable absolute porosity and yet have no conductivity to fluid for lack of

pore interconnection. The absolute porosity is generally expressed mathematically by the following

relationship:

(2-2)

Where a = absolute porosity.

Effective porosity

From the standpoint of flow through a porous medium only interconnected pores are of interest, hence

the concept of effective porosity defined as the percentage of interconnected pore space with respect to

the bulk volume, or

(2-3)

Where e = effective porosity.

The effective porosity is used in all reservoir engineering calculations because it represents the

interconnected pore space that contains the recoverable hydrocarbon fluids. Transportation of fluids is

controlled mainly by connected pores. For intergranular materials, poorly to moderately well cemented,

the total porosity is approximately equal to effective porosity. For more cemented materials and some

carbonates, significant difference in total porosity and effective porosity values may occur. Another type

of pores that seem to belong to the class of interconnected pores but contribute very little to the flow,

are dead-end pores or stagnant pockets (Figure 2-4). These pores have just a constricted opening to the

flow path so the fluid in them is practically stagnant. In certain mechanisms of flow such as diffusion and

dispersion it is important to pay attention to the effects of dead-end pores.

20

Figure 2-4 Dead-end pore

A pore-structure concept for rocks (Figure 2-5) was presented (Katsube and Collett, 1973) in the 1970s,

which consisted of total connecting porosity (C) for connecting pores that contribute mainly to fluid

migration and of storage porosity (S) for storage pores that contribute mainly to fluid storage, where

their sum is effective porosity (e):

S + C = e (2-4)

The storage pore shapes can be characterized by vugular or intergranular, as shown in Figure 2-5.

Figure 2-5 - Storage and connecting pore model for shale or any other type of rock with interconnected pore systems4

There are two major types of connecting pores: sheet-like, circular and/or tubular pores. Many more

types of connecting pores can be considered, but these are most representative for describing the

extreme differences between their types. For example, the cross-section of connecting pores can have

many shapes (Figure 2-6). A representative cross-section of a sheet-like pore is shown in Figure 2-6.a.

Cross-sections of circular and/or tubular pores can take many shapes (Figure 2-6.b). The total connecting

porosity ( C) value of a rock includes connecting pores in all three directions (Figure 2-7.a). However,

when considering fluid or electrical current flow through the rock, only pores in two directions are

4 T.J. Katsube (2010)

Dead-end pore

Main flow channel

21

considered for sheet-like pores (Figure 2-7.a), and only in one direction for circular and/or tubular pores

(Figure 2-7.b).

Figure 2-6 - The Various Cross-Sections of Connecting Pores

a) Tortuous Sheet-Like Pore; b) Various Shapes of Tubular Pores

The fluid flow in a rock is controlled mainly by connecting pores. onnecting pores implies all

pores except for isolated pores. However, some of these connecting- or storage- pore systems can be

dead-end. The connecting pores that contribute to electrical current- or fluid flow through the rock have

to be interconnected from one end of the rock to the other, and not include dead-end pores. These are

distinguished as end-to-end connecting pores. The actual porosity of the end-to-end connecting pores

should be smaller than the value of C, since it does not include the porosity of the dead-end pores. The

storage-connecting pore system and its porosity descriptions are shown in Figure 2-7.b for a rock section

that includes vugular storage and isolated pores. The end-to-end connecting porosity of the pore system

in Figure 5 is represented by CF.

Figure 2-7 a) Three-dimension distribution of connecting; b) Complete storage-connecting pore system.5

Where:

5 Bowers and Katsube (2002)

22

Total Porosity , Storage flow porosity

Effective Porosity , Storage blind (Dead-end) porosity

Storage porosity , Connecting flow porosity

Blind porosity , Connecting storage porosity

Connecting porosity , Connecting blind porosity

Porosity may be classified according to the mode of origin as original induced . The original

porosity is that developed in the process of deposition that forms the rock, while induced or secondary

porosity added at a later stage by some geologic and chemical process. The inter-granular porosity of

sandstones and the inter-crystalline and oolitic porosity of some limestones typify original porosity.

Induced porosity is typified by fracture development as found in shales and limestones and by the vugs

or solution cavities commonly found in limestones. Rocks having original porosity are more uniform in

their characteristics than those rocks in which a large part of the porosity is included. Materials having

induced porosity such as carbonate rocks have complex pore configuration. In fact two or more systems

of pore openings may occur in such rocks. The basic rock material is usually finely crystalline and is

referred to as the matrix. The matrix contains uniformly small pore openings which comprise one system

of pores. One or more systems of larger openings usually occur in carbonate rocks as a result of leaching

or fracturing of the primary rock material. Fractures and vugs are highly variable in size and distribution.

Therefore even more than for intergranular materials, laboratory measurements are required for

quantitative evaluation of porosity.

For direct quantitative measurement of porosity, reliance must be placed on formation samples

obtained by coring. Many porous media are made of discrete large and small grains or particles that are

loose (unconsolidated porous media). Consolidated sedimentary rocks are derived from initially

unconsolidated grains that have gone significant cementation at areas of grain contact. Early

investigations of the porosity were conducted to a large extend by investigation in the fields of ground

water geology, chemical engineering, and ceramics. Therefore much interest was centered on the

investigation of the porosity of unconsolidated materials.

The porosity of unconsolidated materials depends on:

Grain shape

Grain packing

Grain sorting

Grain size distribution

Compaction

The porosity of consolidated materials depends mainly on the degree of cementation and consolidation

but also on the above mentioned parameters.

23

Grain shape and packing

Consider simple models, such as a regular packing of uniform sphere or rods. Graton and Fraser (1935)

analyzed the porosity of variable packing arrangements of uniform spheres. The least compact

arrangement of uniform spheres is that of cubical packing with a porosity of 47.6%. The most compact

packing of uniform spheres is the rhombohedra or close-packed, where the porosity is 26.0%. In these

and other cases of sphere of equal size, the porosity is independent of the radius of spheres. Cross view

of the unit cell of two of the mentioned packing are shown in Figure 2-8.

Cubic packing of uniform spheres Rhombohedra or close-packed of uniform spheres

Figure 2-8 Typical ordered porous medium structures

Often porous materials with spherical grains have lower porosity than materials composed of non-

spherical grain.

Example 2-1

Calculate the cubic packing of uniform spheres porosity (Figure 2-8).

Solution

The unit cell is a cube with sides equal to 2r where r is the radius of sphere. Therefore

Since there are 8 (1/8) spheres in the unit cell

The porosity is therefore is

The interesting point is that the radii cancel in the formula and the porosity of packing uniform spheres

is a function of packing only.

Grain size distribution and grain sorting

Naturally occurring materials are composed of a variety of particle sizes. The particle size distribution

may appreciably affect the resulting porosity, as small particles may occupy pores formed between large

particles, thus reducing the porosity (Figure 2-9-a). On the other hand sometimes porosity increases

during a phenomenon called bridging (Figure 2-9-b).

24

a) Porosity reduction (well sorted) b) Porosity increase (bridging)

Figure 2-9 Effect of sorting and grain size distribution on porosity

In naturally occurring materials porosity increases by decreasing the grain sizes. An increase in range of

particle size tends to decrease porosity.

Cementation and compaction

During the cementation process in consolidated rocks as the pore space is filled with cementing

material, significant reduction in porosity may take place.

Because compaction forces vary with depth, porosity will also vary with depth especially in clays and

shales. Krumbein and Sloss (1951) indicate a reduction in sandstone porosity from 52 to 41% and in

shale from 60 to 6% as depth increases from 0 to 2000m. Most of the pore reduction is due to the

inelastic, hence irreversible, effects of intergranular movement. Reservoir rocks may generally show

large variations in porosity vertically but do not show great variations in porosity parallel to the bedding

planes.

LABORATORY POROSITY MEASUREMENT

A great many methods have been developed for determining porosity, mainly of consolidated rocks

having intergranular porosity (encountered in oil reservoir). Most of the methods developed have been

designed for small samples. From the definition of porosity it is obvious that common to all methods is

the need to determine two of three volumes: total or bulk volume of the sample, its pore volume,

and/or the volume of its solid matrix. The various methods based on such volume determination, called

rmined. Other methods are

Examples of such properties are the electrical conductivity of electrically conducting fluid filling the void

space of the sample, or the absorption of radioactive particles by a fluid filling the void space of the

sample. The porosity of the larger portion of rock is determined statistically from the results obtained on

numerous small samples.

Bulk volume

The simplest direct method for determining bulk volume of a consolidated sample with a well design

geometric shape is to measure its dimensions. The method is applicable to cylindrical core with

smoothed flat surfaces. The usual procedure is to determine the volume of fluid displaced by the

sample. This method is particularly desirable for irregular shaped samples. The fluid volume that the

25

sample displaces can be determined volumetrically or gravimetrically. In both methods the displaced

fluid should be prevented from penetrating the pore space of the sample. There are 3 strategies to do

that:

Coating the rock sample with paraffin

By saturating the rock with the fluid into which it is to be immersed

By using mercury, which according to its surface tension and wettability characteristics does not

tend to enter to small pores of most intergranular samples?

Gravimetric determination of the bulk volume can be accomplished by measuring the loss in the weight

of sample when it is immersed in the fluid or observing the change in the weight of pycnometer when is

filled with mercury and when is filled with mercury and core sample.

Example 2-2

Bulk Volume Calculation of coated sample immersed in water.

A = weight of dry sample in air = 30.0 g

B = weight of the sample after coating with paraffin = 31.8 g; Paraffin density = 0.9 g/cm3

C = weight of paraffin = B A = 1.8 g

D = volume of paraffin = 1.8 / 0.9 = 2cm3

E = weight after immersing of the coated sample in the water = 20 g, water density = 1 g/cm3

Volume of water displaced = (B E) / water density = 11.8 cm3

Bulk volume of rock = volume of water displaced volume of paraffin = 9.8 cm3

Example 2-3

Calculate the volume of a dry sample immersed in mercury pycnometer.

A = weight of dry sample in air = 30 g

B = weight of pycnometer filled with mercury at 20oC = 360 g, mercury density = 13.546 g/cm3

C = weight of pycnometer filled with mercury and sample at 20oC = 245.9 g

D = weight of sample + weight of mercury filled pycnometer = A + B = 390 g

E = Weight of mercury displaced = D C = 144.1 g

Bulk volume of rock = E/mercury density = 144.1 / 13.546 = 10.6 cm3.

Grain volume

The grain volume can be determined from the dry sample weight and the grain density. For many

purposes, result with sufficient accuracy can be obtained by using the density of quartz (2.65 g/cm3) as

the grain density.

26

A method of determining the gain volume is crushing the sample after determining the bulk volume,

thus removing all pores including the non-interconnecting ones. The volume of solids is then determined

by fluid displacement in a pycnometer.

Pore volume

There are methods to measure the pore volume of the rock sample directly with no need to determine

the grain volume. Actually, all these methods measure effective porosity. The methods are based on the

extraction of a fluid from the sample or intrusion of a fluid into the pore space of the rock sample.

Mercury injection method:

Both the bulk and pore volume are determined in this method. The tested sample is placed in a chamber

filled to a certain level with mercury, with a known volume of air at a known pressure (e.g. atmospheric

pressure) above it. The volume of mercury displaced by the sample gives the bulk volume. When the

pressure of mercury is increased by a volumetric pump, the mercury penetrates the pore space of the

sample. Total effective pore volume could be determined by gradually increasing the pressure. In

general the method is not suitable for low permeability samples as very high pressure are required.

Gas expansion method:

This method is based on the Boyle-Marriote Gas Law. It may be the most widely used method for

determine porosity. The test usually carried out at the constant temperature. Basically two chambers

with known volumes are connected by a valve (Figure 2-10). The tested sample is placed in the chamber

of volume V1. The pressure in this chamber is P1. The second chamber (volume V2), initially at pressure

P0, is connected to the first one by opening the valve between them, thus permitting the gas to expand

isothermally. If the final pressure is P2, from Boyle-Marriote law we have:

,

(2-5)

Where Vs = grain volume.

Usually, for simplification, helium as an approximately ideal gas, at low pressure, is used and according

the ideal gas assumption Z=1;

WE NEED TO ADD REFERENCES IN EACH SECTION OR AT THE END

27

Figure 2-10 Porosimeter Based on Boyle -

Imbibition method:

Reservoir rocks have the ability of imbibe water spontaneously. This property is used to determine

effective porosity of the rock. In this method the weight of a dry sample is measured and then the

sample is immersed under vacuum in water or any other fluid that rock has the tendency to imbibe.

After enough time, up to several days, the saturated sample is weighted. Utilizing the density of the

liquid we can find the imbibed fluid volume and subsequently the effective porosity of the sample.

Optical methods:

areal porosity is determined on polished thin sections of a sample. It is often necessary to impregnate

the pore with some material such as wax, plastic

visible and/or distinguishing interconnected pore from the isolated pores (Figure 1-1). When

impregnating the sample with a resin only the interconnected pores will be invaded. Whenever there

are very small pores present along with large ones, it is very difficult to make sure that all the small

pores have been accounted for by the measurement. This is one of the reasons why the porosity

measurements by the optical methods may differ significantly from the results obtained by other

methods.

Figure 2-11 Colored Thin Section Microscopic Image

P2 P1

V2 V1

Test Sample Valve

Gas inlet

28

Statistical methods:

A pin is dropped many times in a random manner on an enlarged photomicrograph of a section of

consolidated porous material, the porosity of which is to be determined. It can be shown that the

probability of a random point falling within the pore space of this section is equal to the porosity.

Therefore, as the number of tosses increases, the number s point falls in the pore space

to the total number of tosses approaches the value of the porosity.

X-ray tomography methods:

ADD TEXT

Magnetic resonance methods:

ADD TEXT

WELL LOGGING

Well logging, also known as borehole logging is the practice of making a detailed record (a well log) of

the geologic formations penetrated by a borehole. The log may be based either on visual inspection of

samples brought to the surface (geological logs) or on physical measurements made by instruments

lowered into the wellbore (geophysical logs). Well logging is done during all phases of a well's

development; drilling, completing, producing and abandoning. Logging measurements are quite

sophisticated. The prime target is the measurement of various geophysical properties of the subsurface

rock formations. Of particular interest is porosity. Logging tools provide measurements that allow for

the mathematical interpretation of porosity. There are different types of well logging that used to

estimate the porosity of the formation around the well, such as:

CNL (compensated neutron) logs: also called neutron logs, determine porosity by assuming that the

reservoir pore spaces are filled with either water or oil and then measuring the amount of hydrogen

atoms (neutrons) in the pores. These logs underestimate the porosity of rocks that contain gas.

FDC (formation density compensated) logs: also called density logs, is a porosity log that measures

electron density of a formation and determine porosity by evaluating the density of the rocks. Because

these logs overestimate the porosity of rocks that contain gas they result in "crossover" of the log curves

when paired with Neutron logs.

NMR (nuclear magnetic resonance) logs: may be the well logs of the future. These logs measure the

magnetic response of fluids present in the pore spaces of the reservoir rocks. In so doing, these logs

measure porosity and permeability, as well as the types of fluids present in the pore spaces.

PORE SIZE DISTRIBUTION

determination defines a pore size in terms of a pore model which is best suited to the quantity

measured in the particular experiment. There is the same situation for the definition of void space. For

http://en.wikipedia.org/wiki/Geologichttp://en.wikipedia.org/wiki/Geophysicalhttp://en.wikipedia.org/wiki/Porosity

29

pore space consists of an irregular network of pores, there are terms to distinguish between pore spaces

that are relatively narrow and the interconnected relatively larger spaces. The narrow constrictions that

interconnect relatively larger spaces are called pore throats or pore necks, with pore throats being the

more common term. While the relatively larger pore spaces are called pore bodies, node pores or bulge

pores, with pore bodies being the most commonly used term (Figure 2-12).

Figure 2-12 - Schematic Shape of pore and Pore Throat

In the vast majority of porous media, the pore sizes are distributed over a wide spectrum of values,

giving the distribution of pore volume by a characteristic pore size. If the pores were separated objects

then each pore could be assigned a size according to some consistent definition, and the pore size

distribution would become analogous to the particle size distribution obtained, for example, by sieve

analysis. A sieve analysis (or gradation test) is a practice or procedure used to assess the particle size

distribution (also called gradation) of a granular material. A sieve analysis is performed on a sample of

aggregate in a laboratory. A typical sieve analysis involves a nested column of sieves with wire mesh

cloth (screen) (Figure 2-13). A representative weighed sample is poured into the top sieve which has the

largest screen openings. Each lower sieve in the column has smaller openings than the one above. At the

base is a round pan, called the receiver. The column is typically placed in a mechanical shaker. The

shaker shakes the column, usually for some fixed amount of time. After the shaking is complete the

material on each sieve is weighed. The weight of the sample of each sieve is then divided by the total

weight to give a percentage retained on each sieve. The size of the average particles on each sieve then

being analysis to get the cut-point or specific size range captured on screen.

A mechanical shaker used for sieve analysis Sieves used for sieve test

Figure 2-13 Sieve Analysis Tools

Pore body

Pore Throat

http://en.wikipedia.org/wiki/Particle_size_distributionhttp://en.wikipedia.org/wiki/Particle_size_distributionhttp://en.wikipedia.org/wiki/Sieve

30

The pore in the interconnected pore space, however are not separated objects, and the volume

assigned to a particular pore size depends on both the experimental method and the pore structure

model used.

Methods of measurement

The most popular methods of determining pore size distribution are mercury intrusion porosimetery

sorption isotherm and image analysis. The first of these is used mostly but not exclusively to determine

the size of relatively larger pores whereas sorption isotherms are best suited in the case of smaller

pores. The use of imaging to analyze section of a sample has some advantages over the other two

methods, but the most complete information on pore size distribution may be obtained if all three

methods are used jointly.

Mercury porosimetry:

sample immersed in mercury. The pressure required to int

inversely proportional to the size of the pores, so at the same times it finds pore size distribution.

Theory and key assumption: a key assumption in mercury porosimetry is the pore shape. Essentially all

instruments assume a cylindrical pore geometry using a modified Young-Laplace equation:

(2-6)

It relate the pressure difference across the curved mercury interface (r1 and r2 describe the curvature of

that interface) to the corresponding pore size using the surface tension of mercury () and the contact

angle between solid and mercury. The real pore shape is however quite different and cylinder pore

shape assumption may lead to major differences between reality and analysis. As indicated in equation

(2-5), we need to know surface tension and contact angle for the given sample and then measure the

pressure and the intruded volume in order to obtain the pore volume-pore size relation. Mercury is

completely non wetting phase for each sample and in general, the surface tension of mercury is not of

any great concern with respect to errors in the determination of pore size distribution. A value 0.485

N/m at 25oC is commonly accepted by most researchers. The contact angle is a parameter which clearly

affects the analysis results and numerous paper have demonstrated the wide range of contact angle

between mercury and various different or even very similar solid surface. However in most practical

situations and out of convenience users often apply a fixed value irrespective of the specific sample

material, e.g. 130o or 140o.

Mercury porosimetry also has limitations. One of the most important limitations is the fact that mercury

porosimetry does not actually measure the internal pore size, but it rather determines the largest

connection (throat or pore channel) from the sample surface towards that pore. Thus, mercury

porosimetry results will always show smaller pore sizes compare to the image analysis method results.

For obvious reasons it can also not be used to analyze closed pores, since the mercury has no way of

entering that pore. The smallest pore size, which can be filled with mercury, is limited by the maximum

31

pressure, which can be achieved by the instrument, e.g. 3.5 nm diameter at 400 MPa assuming a contact

angle of 140.

Figure 2-14 - Schematic representation of pores.

Sorption Method:

Sorption (Adsorption Desorption) measurements involve the measurement of the surface area and the

small size pores of a given medium. This method is based on the commencement of condensation with

increasing capillary pressure (Note: condensation starts in the smallest pores and proceeds to

increasingly larger pores). There are many models available for the adsorption of gases onto solids

where the volume adsorbed is a function of pressure with constant temperature.

Basic equation and method: the fundamental equation to find the pore size distribution from capillary

condensation isotherms is follow:

(2-7)

Where VS is the volume of adsorbate at saturation vapor pressure (equal to the total pore volume), V

the volume of adsorbate at intermediate vapor pressure , L(R)dR the total length of pores whose total

length fall between R and R + dR, R pore radius, and (t) the multilayer thickness that is built up at

pressure . This equation states the fact that the volume of adsorbed at pressure is equal to the

volume of pores that has not yet been filled.

Micro-tomography:

A modern tool for the determination of pore properties is x-ray micro-tomography. With this technique

as small sample (less than 1cm in diameter) is exposed to x-rays and a three dimensional reconstruction

of the sample is created typically at a resolution of ~1 micron (Figure 2-15). From the three dimensional

object information about the pore space, connectivity, total porosity and mineral content can be

obtained. The limitation of the technique is related to the resolution of the created images. There are

very powerful micro-tomography systems associated with synchrotron facilities and bench top devices.

Other image analysis methods

Isolated Pore Dead end pores Cross linked pores Throat pores

32

Figure 2-15 Tomographic image.

SPECIFIC SURFACE AREA

The specific surface of a porous material is defined as the interstitial surface area of the voids and pores

either per unit mass (S) or per unit bulk volume (SV) of the porous material (Figure 2-16). The specific

surface based on the solid volume is denoted by SO.

Figure 2-16 Specific Surface Area

For example the specific surface of a porous material made of identical spheres of radius R in a cubic

packing is:

It thus becomes obvious that the fine materials will exhibit much greater specific surface than will

coarse materials. Some fine porous materials contain an enormous specific area. For example the

specific area of sandstone may be in the order of . Carman (1938) gives the range of

for specific surface of the sand.

The specific area of a porous material is affected by porosity, by mode of packing, by the grain size and

by the shape of the grains. For example disc shaped particles will exhibit a much larger specific area than

will spherical ones.

Specific surface plays an important role in a variety of different application of porous media. It is a

measure of the adsorption capacity of various industrial adsorbent; it plays an important role in

determining the effectiveness of catalysts and filters. In petroleum and rheology study it is related to the

fluid conductivity or permeability of porous media.

33

Obviously the specific surface of natural porous media can be determined only by indirect or statistical

methods such as:

Statistical method:

enlarged photomicrograph of a

section of porous material. A count is kept the number of times (

void space and the number of times () the pin intersects the perimeter of pores. The specific surface is

then found from:

(2-8)

This method is considered one of the best methods. Many other matrix properties can be derived with

it.

Adsorption Method:

These are based on the adsorption of a gas or a va

determined from the quantity of gas adsorbed on it, assuming the gas covers the entire surface of the

solid with a uniform monomolecular film.

Fluid Flow:

This method suggests a relation between permeability of a medium and its specific area. Using this

relationship one can obtain the specific area by conducting the experiments leading to the

determination of the permeability of the medium.

COMPRESSIBILITY OF POROUS ROCKS

Under natural conditions, a porous medium volume at some depth in a ground water aquifer or in an oil

reservoir is subjected to an internal stress or hydrostatic pressure of the fluid saturation the medium,

which is a hydrostatic pressure that has the same values at different direction, and to an external stress

exerted by the formation in which the particular volume is surrounded and may have different value at

different directions. The external stress of the formation can lead to the compaction of the porous

medium that is a function of the formation depth. Krumbein and Sluss (1951) showed that porosity of

the sedimentary rocks is a function of the degree of compaction of the rock (Figure 2-17).

34

Figure 2-17 Porosity Reduction as an Effect of Compaction Increment by Depth

Compaction effect on the porosity that leads to porosity reduction is principally due to the packing

rearrangement after compaction. The porosity of shales is greatly reduced by compaction largely

arrangement, rocks also are compressible. Three kinds of compressibility must be distinguished in rocks:

Rock matrix compressibility

Rock bulk compressibility

Pore compressibility

Rock matrix compressibility is the fractional change in volume of the solid rock materials (grains) with a

unit change in pressure. Rock bulk compressibility is the change in volume of the bulk volume of the rock

with a unit change in pressure. Pore compressibility is the fractional change in the pore volume of the

rock with a unit change in pressure.

The depletion of fluids from the pore space of a reservoir rock results in a change in the internal

pressure in the rock while the external pressure in constant, thus results a change in the net pressure.

This change in the net stress could leads to a change in grain, pore and bulk volume of the rock. Pore

volume change is an interesting subject to the reservoir engineer. Bulk volume change is an important

subject in the areas that surface subsidence could cause appreciable property damage. Volume change

under the pressure effect can be expressed as compressibility coefficient. The coefficient of solid matrix

compressibility, pore compressibility and bulk compressibility are defined for of a saturated porous

medium as the fractional change in the volume with a unit change in the pressure:

(2-9)

(2-10)

(2-11)

0

10

20

30

40

50

60

0 1000 2000 3000 4000 5000 6000

Po

rosi

ty (

%)

Depth of burial, ft

Shales

Sandstones

35

The value of (in some literature mentioned as as rock compressibility) can be determined by

saturating the rock with a fluid, immersing the rock in a pressure vessel containing the saturating fluid,

then imposing a hydrostatic pressure on the fluid and observing the change in the volume (or ) of

the rock sample. The compressibility of solid matrix (or ) is considered for most rock to be

independent of the imposed pressure.

But reservoir rocks are under other conditioning of loading than this experiment. A rock buried at depth

is subjected to an overburden load due to the overlying sediments which is in general greater than the

internal hydrostatic pressure of the formation fluids. (Figure 2-18.a) shows an experimental apparatus

that simulate this condition for a sample rock. A core sample is enclosed in a copper jacket which is then

immersed in a pressure vessel and connected to a Jurguson sight glass gauge. The hydraulic pressure

system is arranged so that a saturated core can be subjected to variable internal (or pore) pressure and

external (or overburden) pressure. The resulting internal volume change is indicated by the position of

the mercury slug level in the sight glass. Typical curve are obtained shown if (Figure 2-18.b). The

ordinate is the reduction in pore space resulting from a change in overburden pressure. The slop of the

curve is the compressibility of the form

(2-12)

It may be noted that the slop of the curves can be considered constant over most of the pressure range

above 1000 psi. Hall (1953) ran some similar tests. He designate the compressibility term (2-12) as

formation compaction component as total rock compressibility and develop a correlation of this

function with porosity (Figure 2-19.a). Also he investigated at constant overburden

pressure. This he designated as effective rock compressibility and correlated with porosity (Figure

2-19.b). In Figure 2-19.a and b, it may be noted that compressibility decreases as the porosity increases.

The value of can be determined by measuring the change in the bulk volume of a jacketed sample by

varying the external hydrostatic pressure while maintaining a constant internal pressure. For sandstones

and shale it can be shown that:

(2-13)

(2-14)

36

(a) (b)

Figure 2-18 a) Experimental Equipment for Measuring Pore Volume Compaction and Compressibility

b) Rock compressibility test result6

(a) (b)

Figure 2-19 a) Formation Compaction Component of Total Rock Compressibility

b) Effective reservoir rock compressibility7

6 Carpenter and Spencer (1940)

7 Hall (1953)

37

eq. (2-14) provided that Cr is much less than CB. Therefore

(2-15)

This equation states that total change in volume is equal to the change in the pore volume.

Geertsma (1957) stated that in reservoir only the vertical component of overburden pressure is constant

and the stress components in ho