Table Of ContentElements of Rock Physics
and Their Application to Inversion
and AVO Studies
Elements of Rock Physics and
Their Application to Inversion
and AVO Studies
Robert S. Gullco and
Malcolm Anderson
First published 2022
by CRC Press/Balkema
Schipholweg 107C, 2316 XC Leiden, The Netherlands
e-mail: [email protected]
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© 2022 Robert S. Gullco and Malcolm Anderson
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Library of Congress Cataloging-in-Publication Data
Names: Gullco, Robert S., author. | Anderson, Malcolm (Mathematician), author.
Title: Elements of rock physics and their application to inversion and AVO studies /
Robert S. Gullco and Malcolm Anderson.
Description: Leiden, The Netherlands ; Boca Raton :
CRC Press/Balkema, 2022. | Includes bibliographical references.
Subjects: LCSH: Amplitude variation with offset analysis. | Rock deformation–Mathematical
models. | Seismology–Mathematics. | Seismic reflection method. | Seismic prospecting. |
Geology, Structural.
Classification: LCC QE539.2.S43 G85 2022 (print) | LCC QE539.2.S43 (ebook) |
DDC 620.1/125–dc23/eng/20211029
LC record available at https://lccn.loc.gov/2021041377
LC ebook record available at https://lccn.loc.gov/2021041378
ISBN: 978-1-032-19993-1 (hbk)
ISBN: 978-1-032-13495-6 (pbk)
ISBN: 978-1-003-26177-3 (ebk)
DOI: 10.1201/9781003261773
Typeset in Times New Roman
by codeMantra
Contents
About the authors ix
Introduction 1
1 Petrophysics review 3
Definition of effective and total porosity, clay and shale 3
The effective porosity model 4
The total porosity model 5
Estimation of the shale point in a Density/Neutron crossplot 6
Calculation of the effective porosity and the shale volume fraction (V )
sh
from the Neutron and Density logs, in oil- or water-bearing sands 8
Using the Gamma Ray log to calculate shale volume fraction:
Comparison with the Neutron/Density approach 8
Evaluating gas-bearing sands using the Neutron, Density and Gamma Ray logs 10
Reference 13
Appendix 1.1: Gas density and hydrogen index at reservoir conditions 13
Hydrogen index of a gas 13
Gas density at reservoir conditions 14
2 Elements of elasticity theory 17
Definition of stress, strain, elasticity and elastic moduli 17
The concept of normal and shear stresses 21
The shear modulus 22
Relationship between seismic velocities and elastic moduli 22
References 23
3 Pore pressure review 25
Introduction 25
Normal and abnormal pressures: Most common causes of abnormal pressure 26
Overburden pressure and Net Overburden Pressure 29
The Gluyas-Cade correlation of porosity vs. depth for clean, uncemented sands 35
Calculation of the pore pressure: The Eaton and Bowers equations 38
The Eaton equation reads 38
vi Contents
The Bowers formula 42
Lithological problems 43
Pore pressure calculations in limestones, and the difficulty of doing this
with velocity data alone 44
Calculation of the fracture pressure 46
The Eaton formula (1969, 1997) for calculating the pore pressure 47
An oil exploration application of pore pressure 49
Sealing and non-sealing faults 49
References 50
4 Incompressibility of rocks and the Gassmann equation 51
Incompressibility moduli and the relationships between them 51
The Gassmann equation 54
The relationship between the porosity and the net overburden pressure 56
Summary 58
Reference 58
5 Fluid substitution 59
The fluid substitution problem 59
Physical properties of fluids 59
A simple fluid substitution exercise 61
1. Calculate the bulk modulus (K ) and the shear modulus (µ) of
b
the wet rock 63
2. Calculate the dry modulus (K ) 63
dry
3. Calculate the effective fluid compressibility and the density
under the new fluid saturation conditions 64
4. Given the new value of K, calculate the new value of the bulk
f
incompressibility K of the rock, assuming 70% gas in the pore space 64
b
5. Calculate the new seismic velocities and the new bulk density 64
Reuss lower bound and Voigt upper bound and Hashin-Shtrikman
upper and lower bounds 65
Marion’s hypothesis 68
Reference 72
6 Forward modelling and empirical equations 73
Forward modelling and empirical equations 73
The Wyllie equation 73
Estimation of the shear velocity from the compressional velocity 77
Input data for the Monte Carlo simulation 80
Estimation of the elastic parameters of the ideal rock using the
Hashin-Shtrikman bounds 81
Correlations used to estimate the bulk and shear moduli 85
The Murphy et al. (1993) correlation 85
The critical porosity hypothesis (Nur, 1992) 86
The Krief et al. (1990) correlation 86
Contents vii
Comparison with real data 87
Summary 90
References 91
Appendix 6.1: Estimation of the incompressibility of the solid part of
the rock in shaley sands 91
7 Applications of rock physics to AVO analyses 95
Reflection coefficients 95
Simulation of the AVO responses 99
Some of the problems in using amplitudes as surrogates for reflection
coefficients 105
Scaling of the amplitudes 107
The possibility of estimating the proportions of lithological types in a
relatively small volume 110
Probability that a point in a gradient-intercept diagram belongs to an
interface of interest 113
General comments and summary 116
References 117
8 Applications of rock physics to inversion studies 119
Introduction 119
Standard outputs of an inversion (apart from the density) 120
Making use of well data to identify facies and assessing the feasibility
of an inversion study 120
Using the properties of the normal distribution 121
Using cluster analysis to identify facies 125
Scaling the well data to make it compatible with the seismic data 130
A quick recapitulation of the steps involved in analysing well data 132
Populating the seismic cube with facies 133
Estimation of the effective porosity of a “sand” facies 138
A theoretical example involving inversion in carbonates 139
References 144
Appendix 8.1: Mixtures of normal distributions 144
Appendix 8.2: Some comments on the use of Bayes’ theorem 146
9 Modelling carbonates using Differential Effective
Medium theory 149
Introduction 149
Preliminary remarks 149
The case of spherical pores 151
The case of penny cracks (representing fractures) 159
Discussion 165
Final remarks 168
References 170
viii Contents
Appendix 9.1: Exact solution of the DEM differential equation in the
case of spherical inclusions filled with fluid 170
The impossibility of simulating a granular medium using
“pure” DEM theory 170
Appendix 9.2: Integration of the DEM equations in the case of penny
cracks, when the pore space is empty (i.e. dry) 174
Appendix 9.3: The probability that penny cracks will be interconnected 177
About the authors
Robert S. Gullco received his Master’s degree in Geological Sciences from the
University of Buenos Aires (Argentina). He has worked as a geologist and petro-
physicist in YPF (then Argentina state oil company), Wapet (now Chevron) in West-
ern Australia, Paradigm Geophysical (in both Australia and Mexico), and CGG
and Citla Energy in Mexico. He has taken courses on geomathematics at Stanford
University and a course on applied mathematics at Curtin University.
Malcolm Anderson took his Master’s degree in Applied Mathematics and Theoreti-
cal Physics at the Australian National University in Canberra before completing a
PhD in Theoretical Astrophysics at the Institute of Astronomy in Cambridge. He
has taught applied mathematics and mathematical physics at the Australian Na-
tional University, the University of New South Wales and Edith Cowan University.
Since 2000, he has been a member of the Mathematics Group at Universiti Brunei
Darussalam.
