INTRODUCTION
Petrophysics is the study of rock properties and their interactions
with fluids (gases, liquid hydrocarbons and aqueous solutions). Because
petroleum reservoir rocks must have porosity and permeability, we are
most interested in the properties of porous and permeable rocks. The
purpose of this text is to provide a basic understanding of the physical
properties of permeable geologic rocks and the interactions of the various
fluids with their interstitial surfaces. Particular emphasis is placed on
the transport properties of the rocks for single phase and multiphase
flow.
The petrophysical properties that are discussed in this text
include:
• Porosity
• Absolute permeability
• Effective and relative permeabilities
• Water saturation
• Irreducible water saturation
• Hydrocarbon saturation
• Residual oil saturation
• Capillary pressure
• Wettability
• Pore size
• Pore size distribution
• Pore structure
• Net pay thickness
• Isothermal coefficient of compressibility
• Mineralogy
• Specific pore surface area
• Dispersivity
TABLE OF CONTENTS
Page
1 PETROLEUM RESERVOIR ROCKS ......................................... 1-1
1.1 PETROPHYSICS ............................................................................... 1-1
1.2 PETROLEUM RESERVOIR ROCKS ................................................1-2
1.3 MINERAL CONSTITUENTS OF ROCKS—A REVIEW ...................1-4
1.4 ROCKS............................................................................................... 1-5
1.4.1 Igneous Rocks.................................................................... 1-5
1.4.2 Metamorphic Rocks...........................................................1-6
1.4.3 Sedimentary Rocks ............................................................1-6
1.5 CLASSIFICATION OF SEDIMENTARY ROCKS ............................. 1-7
1.5.1 Clastic Sedimentary Rocks ................................................ 1-7
1.5.2 Chemical Sedimentary Rocks............................................ 1-7
1.5.3 Organic Sedimentary Rocks ..............................................1-8
1.6 DISTRIBUTION OF SEDIMENTARY ROCK TYPES .................... 1-10
1.7 SANDSTONE RESERVOIRS (CLASTIC SEDIMENTARY
ROCK) ............................................................................................. 1-10
1.7.1 Pore Space ....................................................................... 1-12
1.7.2 Compaction and Cementation......................................... 1-15
1.7.3 Classification ................................................................... 1-17
1.8 CARBONATE RESERVOIRS (LIMESTONES AND
DOLOMITES) .................................................................................1-20
1.8.1 Classification ................................................................... 1-21
1.8.2 Pore Space .......................................................................1-22
1.9 FRACTURED RESERVOIRS..........................................................1-28
1.10 RESEVOIR COLUMN.....................................................................1-29
REFRENCES...................................................................................1-32
2 POROSITY AND FLUID SATURATIONS..................................2-1
2.1 DEFINITION OF POROSITY ...........................................................2-1
2.2 FACTORS AFFECTING SANDSTONE POROSITY ........................ 2-2
2.3 FACTORS AFFECTING CARBONATE POROSITY ........................ 2-4
2.4 TYPICAL RESERVOIR POROSITY VALUES.................................. 2-5
2.5 LABORATORY MEASUREMENT OF POROSITY.......................... 2-6
2.5.1 Direct Porosity Measurement by Routine
Core Analysis .................................................................... 2-6
2.5.2 Indirect Porosity Measurement by CT Imaging.............. 2-11
2.6 FLUID SATURATIONS ..................................................................2-16
2.7 INDIRECT POROSITY MEASUREMENTS FROM
WELL LOGS................................................................................... 2-24
2.7.1 Introduction to Well Logging......................................... 2-24
2.7.2 Mud filtrate Invasion...................................................... 2-25
2.7.3 Porosity Logs .................................................................. 2-32
Density Log .................................................................. 2-32
Sonic Log (Acoustic Log) ............................................. 2-36
Neutron Log .................................................................2-41
Combination Porosity Logs ......................................... 2-45
2.7.4 Resistivity Log ................................................................ 2-46
Electric Log ................................................................. 2-54
Induction-Electric Log................................................ 2-56
Dual Induction Laterolog ........................................... 2-58
Focused Electric Log (Guard and Laterolog) ............. 2-62
Microresistivity Logs................................................... 2-65
2.7.5 Lithology Logs ................................................................ 2-68
Spontaneous Potential Log (SP) ................................. 2-68
The Gamma Ray Log (GR)...........................................2-73
2.7.6 Nuclear Magnetic Resonance (NMR) Logs.................... 2-76
Nuclear Spins in a Magnetic Field.............................. 2-76
The Effect of Radiofrequency Pulses -
Resonance Absorption ............................................... 2-79
Relaxation Processes...................................................2-80
Molecular Diffusion Effect.......................................... 2-84
NMR Signal and Corresponding T2 Spectrum ........... 2-84
Pore Size Distribution................................................. 2-89
Estimation of Permeability from
NMR Relaxation Times............................................... 2-95
2.7.7 NMR Imaging of Laboratory Cores................................ 2-97
The Effect of Magnetic Field Gradients...................... 2-98
Slice-Selective Excitation............................................ 2-99
Frequency Encoding ................................................. 2-100
Phase Encoding..........................................................2-101
Image Reconstruction............................................... 2-102
Three-Dimensional NMR Imaging............................2-103
Signal-to-Noise Ratio and Image Contrast............... 2-104
Example NMR Images of Laboratory Cores..............2-105
2.7.8 A Comparison of Various Porosity
Measurements for Shaly Sand....................................... 2-112
2.8 RESERVE ESTIMATION PROJECT ........................................... 2-113
2.8.1 Reserve Estimation........................................................ 2-114
2.8.2 Economic Evaluation..................................................... 2-115
2.8.3 Simulation Procedure.................................................... 2-116
2.8.4 Sampling Procedure ...................................................... 2-116
2.8.5 Simulation Output.........................................................2-124
2.9 PORE VOLUME COMPRESIBILITY............................................2-126
NOMENCLATURE .......................................................................2-135
REFRENCES AND SUGGESTED READINGS.............................2-138
3 PERMEABILITY .....................................................................3-1
3.1 DEFINITION ....................................................................................3-1
3.2 DIMENSIONS AND UNIT OF PERMEABILITY............................ 3-6
3.3 LABORATORY DETERMINATION OF PERMEABILITY...............3-7
3.4 FIELD DETERMINATION OF PERMEABILITY ..........................3-14
3.4.1 Diffusivity Equation for Slightly
Compressible Liquid........................................................3-15
3.4.2 Pressure Drawdown Equation ........................................3-19
3.4.3 Pressure Buildup Equation ............................................ 3-22
3.4.4 Diagnostic Plots.............................................................. 3-24
3.4.5 Skin Factor...................................................................... 3-30
3.4.6 Homogenous Reservoir Model with
Wellbore Storage and Skin............................................. 3-33
3.4.7 Type Curve Matching ......................................................3-37
3.4.8 Radius of Investigation of a Well Test ........................... 3-40
3.4.9 Field Example of Well Test Analysis .............................. 3-40
3.4.10 Welltest Model for Dry Gas Reservoir .......................... 3-52
3.5 FACTORS AFFECTING PERMEABILITY..................................... 3-56
3.5.1 Compaction..................................................................... 3-56
3.5.2 Pore Size (Grain Size) ..................................................... 3-56
3.5.3 Sorting ............................................................................ 3-60
3.5.4 Cementation ................................................................... 3-60
3.5.5 Layering .......................................................................... 3-60
3.5.6 Clay Swelling....................................................................3-61
3.6 TYPICAL RESERVOIR PERMEABILITY VALUES .......................3-61
3.7 PERMEABILITY-POROSITY CORRELATIONS............................3-61
3.8 CAPILLARY TUBE MODELS OF POROUS MEDIA..................... 3-69
3.8.1 Carman-Kozeny Equation .............................................. 3-69
3.8.2 Tortuosity ........................................................................3-75
3.8.3 Calculation of Permeability from Pore
Size Distribution............................................................. 3-79
3.9 STEADY STATE FLOW THROUGH FRACTURES....................... 3-84
3.10 AVERAGING PERMEABILITY DATA .......................................... 3-85
3.11 DARCY’S LAW FOR INCLINED FLOW........................................3-88
3.12 VALIDITY OF DARCY’S LAW....................................................... 3-99
3.13 NON-DARCY FLOW.....................................................................3-101
3.14 DARCY’S LAW FOR ANISOTROPIC POROUS MEDIA............. 3-106
3.14.1 Definition of Homogeneity and Anisotropy ................ 3-106
3.14.2 Darcy’s Law for Homogeneous
and Anisotropic Medium.............................................3-107
3.14.3 Transformation of Permeability Tensor
from One Coordinate system to Another .................... 3-114
3.14.4 Alternative Calculation of the Principal
Values and the Principal Axes of the
Permeability Anisotropy..............................................3-122
3.14.5 Directional Permeability...............................................3-124
3.14.6 Measurement of Transverse Permeability
of a Cylindrical Core ....................................................3-137
3.15 EXAMPLE APPLICATIONS OF PERMEABILITY.......................3-140
3.15.1 Productivity of Horizontal Well ....................................3-140
Introduction............................................................... 3-141
Homogeneous and Isotropic Reservoirs ................... 3-141
Homogeneous and Anisotropic Reservoirs ...............3-145
3.15.2 Productivity of a Vertically Fractured Well ..................3-152
NOMENCLATURE .......................................................................3-155
REFRENCES AND SUGGESTED READINGS.............................3-159
4 HETEROGENEITY ................................................................ 4-1
4.1 INTRODUCTION..............................................................................4-1
4.2 MEASURES OF CENTRAL TENDENCY AND
VARIABILITY (HETEROGENEITY)............................................... 4-3
4.2.1 Measures of Central Tendency......................................... 4-3
Mean.............................................................................. 4-3
Geometric Mean............................................................ 4-3
Median .......................................................................... 4-3
Mode.............................................................................. 4-4
4.2.1 Measures of Variability
(Heterogeneity or Spread) ............................................... 4-4
Variance ........................................................................ 4-4
Dykstra-Parsons Coefficient of Variation..................... 4-5
Lorenz Coefficient ......................................................... 4-8
4.3 MEASURES OF SPATIAL CONTINUITY ...................................... 4-11
4.3.1 Variogram........................................................................4-13
Definition .....................................................................4-13
How to Calculate the Variogram .................................4-16
Physical Meaning of the Variogram............................ 4-27
Variogram Models....................................................... 4-28
Fitting a Theoretical Variogram Model to
an Experimental Variogram ....................................... 4-35
Variogram Anisotropy .................................................4-41
Example Experimental Variograms............................ 4-44
4.3.2 Covariance (Autocovariance) Function...........................4-51
Definition .....................................................................4-51
Physical Meaning of Covariance Function ................. 4-54
4.3.3 Correlation Coefficient Function
(Autocorrelation Function) .............................................4-57
4.4 PROBABILITY DISTRIBUTIONS ................................................. 4-59
4.4.1 Normal (Gaussian) Distribution .................................... 4-60
4.4.2 Log Normal Distribution................................................ 4-72
4.5 ESTIMATION .................................................................................4-75
4.5.1 Introduction ....................................................................4-75
4.5.2 Ordinary Kriging Equations ........................................... 4-86
Derivation in Terms of the
Covariance Function................................................... 4-89
Derivation in Terms of the Variogram ....................... 4-94
Solution of the Kriging Equation in terms
of the Covariance Function......................................... 4-98
Solution of the Kriging Equation in terms
of Variogram..............................................................4-103
4.6 CONDITIONAL SIMULATION....................................................4-132
4.6.1 Introduction ..................................................................4-132
4.6.2 Sequential Gaussian Simulation ...................................4-132
4.6.3 A Practical Application of Sequential
Gaussian Simulation .....................................................4-136
NOMENCLATURE .......................................................................4-148
REFRENCES AND SUGGESTED READINGS.............................4-149
5 DISPERSION IN POROUS MEDIA ..........................................5-1
5.1 INTRODUCTION..............................................................................5-1
5.2 LABORATORY FIRST-CONTACT MISCIBLE
DISPLACEMENTS........................................................................... 5-3
5.3 ORIGIN OF DISPERSION IN POROUS MEDIA.......................... 5-20
5.3.1 Molecular Diffusion.........................................................5-21
5.3.2 Mechanical Dispersion ....................................................5-21
5.4 CONVECTION-DISPERSION EQUATION................................... 5-23
5.4.1 Generalized Equation in Vector Notation...................... 5-23
5.4.2 One Dimensional Convection-Dispersion
Equation ........................................................................ 5-25
5.4.2 Solution of the One-Dimensional
Convection-Dispersion Equation................................... 5-26
5.5 DISPERSION COEFFICENT AND DISPERSIVITY ..................... 5-42
5.6 MEASURMENT OF DISPERSION COEFFICENT
AND DISPERSIVITY ......................................................................5-53
5.6.1 Traditional Laboratory Method with
Breakthrough Curve .......................................................5-53
5.6.2 Laboratory Method of Peters et al. (1996) ..................... 5-56
5.6.3 Field Measurement of Dispersion
Coefficient and Dispersivity ............................................ 5-71
5.7 FACTORS THAT COULD AFFECT DISPERSION
COEFFICENT AND DISPERSIVITY ..............................................5-75
5.8 NUMERICAL MODELING OF FIRST-CONTACT
MISCIBLE DISPLACEMENT .........................................................5-79
5.8.1 Introduction ....................................................................5-79
5.8.2 Mathematical Model of First-Contact
Miscible Displacement ....................................................5-79
5.8.3 Numerical Modeling of Laboratory Experiments.......... 5-82
Experiment 1 ................................................................. 5-84
Experiment 2..................................................................5-91
Experiment 3................................................................. 5-99
Experiment 4................................................................5-106
Experiment 5................................................................ 5-116
Experiment 6................................................................ 5-121
NOMENCLATURE .......................................................................5-126
REFRENCES AND SUGGESTED READINGS.............................5-128
6 INTERFACIAL PHENOMENA AND WETTABILITY................ 6-1
6.1 INTRODUCTION..............................................................................6-1
6.2 SURFACE AND INTERFACIAL TENSIONS................................... 6-2
6.2.1 Surface Tension ................................................................ 6-2
6.2.2 Interfacial Tension .......................................................... 6-11
6.2.3 Measurement of Surface and
Interfacial Tension.........................................................6-20
Capillary Rise Experiment ............................................6-20
Sessile Drop Method..................................................... 6-24
Pendant Drop Method .................................................. 6-26
Ring Method.................................................................. 6-27
Spinning Drop Method ................................................. 6-30
6.3 WETTABILITY................................................................................6-31
6.3.1 Definition.........................................................................6-31
6.3.2 Determination of Wettability ......................................... 6-36
Contact Angle Method ................................................. 6-37
Amott Wettability Test..................................................6-40
United State Bureau of Mines (USBM)
Wettability Test ............................................................. 6-42
6.3.3 Wettability of Petroleum Reservoirs.............................. 6-45
6.3.4 Effect of Wettability on Rock-Fluid Interactions........... 6-46
Microscopic Fluid Distribution at the
Pore Scale ..................................................................... 6-47
Effect of Wettability on Irreducible
Water Saturation .......................................................... 6-47
Effect of Wettability on Electrical
Properties of Rocks ...................................................... 6-48
Effect of Wettability on the Efficiency of an
Immiscible Displacement .............................................6-51
6.3 THERMODYMAMICS OF INTERFACES ..................................... 6-64
6.4.1 Characterization of Interfacial Tension
as Specific Surface Energy.............................................. 6-64
6.4.2 Characterization of Microscopic Pore Level
Fluid Displacements....................................................... 6-66
Case 1. Displacement of a Nonwetting
Phase by a Wetting Phase ............................................ 6-67
Case 2. Displacement of a Wetting Phase
by a Nonwetting Phase ................................................. 6-69
NOMENCLATURE .........................................................................6-71
REFRENCES AND SUGGESTED READINGS .............................. 6-73
7 CAPILLARY PRESSURE .........................................................7-1
7.1 DEFINITION OF CAPILLARY PRESSURE ..................................... 7-1
7.2 CAPILLARY PRESSURE-SATURATION RELATIONSHIP
FOR A POROUS MEDIUM.............................................................. 7-8
7.3 DRAINAGE CAPILLARY PRESSURE CURVE .............................. 7-17
7.4 CONVERSION OF LABORATORY CAPILLARY PRESSURE
DATA TO RESERVOIR CONDITIONS ..........................................7-21
7.5 AVERAGING CAPILLARY PRESSURE DATA ..............................7-21
7.6 DETERMINATION OF INITIAL STATIC RESERVOIR
FLUID SATURATIONS BY USE OF DRAINAGE
CAPILLARY PRESSURE CURVE.................................................. 7-28
7.7 CAPILLARY PRESSURE HYSTERESIS.........................................7-45
7.8 CAPILLARY IMBIBITION..............................................................7-54
7.9 CAPILLARY END EFFECT IN A LABORATORY CORE ...............7-57
7.9.1 Capillary End Effect.........................................................7-57
7.9.2 Mathematical Analysis of Capillary End Effect ..............7-59
7.9.3 Mathematical Model of Capillary End Effect
During Steady State Relative Permeability
Measurement.................................................................. 7-68
7.9.4 Experimental Evidence of Capillary End Effect...............7-70
7.10 CAPILLARY PRESSURE MEASUREMENTS ................................7-76
7.10.1 Restored State Method (Porous Plate Method)..............7-76
7.10.2 Mercury Injection Method .............................................7-77
7.10.3 Centrifuge Method..........................................................7-81
7.11 PORE SIZE DISTRIBUTION......................................................... 7-96
7.11.1 Introduction.................................................................... 7-96
7.11.2 Pore Volume Distribution .............................................. 7-98
7.11.3 Pore Size Distribution Based on Bundle
of Capillary Tubes Model ............................................7-103
7.11.4 Mercury Injection Porosimeter..................................... 7-115
7.12 CALCULATION OF PERMEABILITY FROM DRAINAGE
CAPILLARY PRESSURE CURVE................................................. 7-118
7.12.1 Calculation of Absolute Permeability from
Drainage Capillary Pressure Curve............................. 7-118
7.12.2 Calculation of Relative Permeabilities from
Drainage Capillary Pressure Curve.............................7-132
7.13 EMPIRICAL CAPILLARY PRESSURE MODELS ........................7-133
7.13.1 Brooks-Corey Capillary Pressure Models .....................7-133
7.13.2 van Genuchten Capillary Pressure Model ....................7-143
7.14 CAPILLARY TRAPPING IN POROUS MEDIA........................... 7- 145
7.14.1 Pore Doublet Model of Capillary Trapping................... 7-145
7.14.2 Snap-Off Model of Capillary Trapping ......................... 7-152
7.14.3 Mobilization of Residual Non-Wetting Phase.............. 7-155
7.14.4 Oil Migration.................................................................7-159
7.15 EFFECTS OF WETTABILITY AND INTERFACIAL
TENSION ON CAPILLARY PRESSURE CURVES.......................7-162
NOMENCLATURE .......................................................................7-164
REFRENCES AND SUGGESTED READINGS ............................ 7-168
8 RELATIVE PERMEABILITY................................................... 8-1
8.1 DEFINITION OF RELATIVE PERMEABILITY...............................8-1
8.2 LABORATORY MEASUREMENT OF TWO-PHASE
RELATIVE PERMEABILITY BY THE STEADY STATE
METHOD......................................................................................... 8-6
8.3 THEORY OF ONE DIMENSIONAL IMMISCIBLE
DISPLACEMENT IN A POROUS MEDIUM..................................8-15
8.3.1 Mathematical Model of Two-Phase
Immiscible Displacement................................................8-15
8.3.2 Buckley-Leverett Approximate Solution of the
Immiscible Displacement Equation................................8-21
8.3.3 Waterflood Performance Calculations from
Buckley-Leverett Theory .................................................8-31
Oil Recovery at any Time ...............................................8-31
Oil Recovery Before Water Breakthrough.....................8-31
Oil Recovery at Water Breakthrough............................ 8-32
Oil Recovery After Water Breakthrough ...................... 8-36
Water Production...........................................................8-41
8.4 LABORATORY MEASUREMENT OF TWO-PHASE
RELATIVE PERMEABILITY BY THE UNSTEADY STATE
METHOD........................................................................................8-51
8.5 FACTORS AFFECTING RELATIVE PERMEABILITIES.............. 8-65
8.5.1 Fluid Saturation.............................................................. 8-65
8.5.2 Saturation History..........................................................8-66
8.5.3 Wettability ...................................................................... 8-67
8.5.4 Injection Rate ................................................................. 8-70
8.5.5 Viscosity Ratio ................................................................ 8-73
8.5.6 Interfacial Tension ......................................................... 8-74
8.5.7Pore Structure .................................................................. 8-75
8.5.8 Temperature ................................................................... 8-76
8.5.9 Heterogeneity ................................................................. 8-78
8.6 THREE-PHASE RELATIVE PERMEABILITIES .......................... 8-79
8.4 CALCULATION OF RELATIVE PERMEABILITIES FROM
DRAINAGE CAPILLARY PRESSURE CURVE .............................8-82
NOMENCLATURE .........................................................................8-91
REFERENCES AND SUGGESTED READINGS............................ 8-94
APPENDIX A: A Systematic Approach To Dimensional Analysis .. A-1
Summary.......................................................................................... A-1
Introduction..................................................................................... A-1
Algebraic Theory of Dimensional Analysis......................................A-2
Transformation of the Dimensionless p Groups .............................A-9
Example Problem.............................................................................A-9
Procedure ....................................................................................... A-10
Transformation of the Dimensionless p Groups for
Example Problem........................................................................... A-21
Some Practical Considerations ......................................................A-28
Concluding Remarks...................................................................... A-31
Nomenclature ................................................................................ A-31
References ......................................................................................A-32