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Elements 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] www.routledge.com - www.taylorandfrancis.com CRC Press/Balkema is an imprint of the Taylor & Francis Group, an informa business © 2022 Robert S. Gullco and Malcolm Anderson The right of Robert S. Gullco and Malcolm Anderson to be identified as authors of this work has been asserted in accordance with sections 77 and 78 of the Copyright, Designs and Patents Act 1988. All rights reserved. No part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. Although all care is taken to ensure integrity and the quality of this publication and the information herein, no responsibility is assumed by the publishers nor the author for any damage to the property or persons as a result of operation or use of this publication and/ or the information contained herein. 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.