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New insights into pore structure characterization and permeability measurement of fine-grained sedimentary… Letham, Eric 2018

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NEW INSIGHTS INTO PORE STRUCTURE CHARACTERIZATION AND PERMEABILITY MEASUREMENT OF FINE-GRAINED SEDIMENTARY RESERVOIR ROCKS IN THE LABORATORY AT RESERVOIR STRESS STATES by  Eric Letham  B.Sc. (Honours), Geological Science, The University of British Columbia, 2011  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES (Geological Sciences)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver)   January 2018  © Eric Letham, 2018 ii  Abstract  Fine-grained reservoir rocks are economically important because of the vast quantities of hydrocarbons they contain, but remain poorly understood due to how difficult it is to analyze their highly stress-sensitive, anisotropic, nanometer-scale pore systems in the laboratory. In this thesis, techniques for pore structure characterization and permeability measurement of fine-grained reservoir rocks at reservoir stress states are developed and tested.  Klinkenberg gas slippage measurements provide accurate measurements of pore size. The gas slippage technique has many valuable characteristics, such as the ability to measure pore size at reservoir stress and to quantify anisotropy of pore geometry. In addition to pore size, matrix permeability is quantified when measuring gas slippage. The gas slippage technique developed in this thesis provides important petrophysical information about fine-grained reservoir rocks that cannot be acquired using other commonly applied pore structure characterization techniques.  A technique to semi-quantitatively determine permeability-effective stress law coefficients is developed by analyzing Klinkenberg plots of permeability measurements made at a wide range of confining pressure and pore pressure. This semi-quantitative technique is important because other fully quantitative techniques that are typically applied to coarse-grained reservoir rocks result in erroneous effective stress laws when applied to fine-grained rocks; in the nanometer-scale pores of fine-grained rocks, gas slippage results in significant permeability variation with pore pressure that is independent of changes to pore geometry, and therefore results in erroneous permeability-effective stress laws. iii  A technique for determining the effective permeability of fine-grained reservoir rocks at different fluid saturations is developed by measuring ethane gas permeability at a range of pore pressures up to the saturated vapour pressure of ethane at laboratory temperature. Liquid/semi-liquid ethane saturation increases with increasing pore pressure due to adsorption and capillary condensation, resulting in restricted fluid flow pathways and hence decreased effective permeability to ethane gas. Ethane gas permeability measurements can be made at different stress states to investigate the sensitivity of effective permeability to stress at the range of stress states experienced during production from a fine-grained reservoir.    iv  Lay Summary  Fine-grained sedimentary rocks, such as shale, are economically important because they can contain vast quantities of hydrocarbons. In order to efficiently recover hydrocarbons from shale reservoirs, we need to be able to measure the properties of the rocks that control how much hydrocarbon they store and how easily the hydrocarbons can be produced. However, these rocks are very difficult to analyze because the pores where the hydrocarbons reside are extremely small (~1000 times smaller than the thickness of a human hair), and are very sensitive to stress. In this thesis, I develop new techniques that make it easier to analyze fine-grained reservoir rocks, and therefore to efficiently exploit the resources they contain.  v  Preface  The research program that resulted in this thesis was identified and designed by me. Although the main ideas behind the research program are my own, they would not have come about without the continuous informal dialogue that took place between myself and my supervisor, RM Bustin.  This thesis is divided into six chapters. Contributions that led to each chapter are detailed below, along with publications arising from the work.  Chapter 1 is an introduction written by me with editorial work by RM Bustin.  Chapter 2 is adapted from a paper published in 2015 in Geofluids titled “Klinkenberg gas slippage measurements as a means for shale pore structure characterization” by myself and RM Bustin. The content was modified slightly in order to correct errors identified when performing the research that led to Chapter 4. The corrections have negligible impact on the main conclusions of the published paper, but would have caused confusion if presented in the same document as Chapter 4. I collected all the laboratory data for Chapter 2 except for the source rock analysis, which was performed by Trican Geological Solutions in Calgary, Alberta. Data analysis was performed by me with help from RM Bustin. Mineral phase and abundance from X-ray diffraction was determined by RM Bustin. Pore size distributions from low pressure nitrogen gas adsorption data were determined with major assistance from EO Munson. Both the published paper and Chapter 2 were written by me with major editorial work performed by RM Bustin. vi  Chapter 3 is adapted from a manuscript submitted to Marine and Petroleum Geology titled “Quantitative validation of pore structure characterization using gas slippage measurements by comparison with predictions from bundle of capillaries models” by myself and RM Bustin. The manuscript is identical to the thesis chapter except for formatting edits in order to be consistent with other chapters in this thesis. I collected all the laboratory data for Chapter 3 except for the FESEM images that were collected by Pablo Lacerda Silva with assistance from the staff at the UBC Centre for High-Throughput Phenogenomics. Data analysis was performed by me with help from RM Bustin. Both the manuscript and Chapter 3 were written by me with minor editorial work performed by RM Bustin.  Chapter 4 is adapted from a paper published in 2016 in International Journal of Coal Geology titled “The impacts of gas slippage on permeability effective stress laws: implications for predicting permeability of fine-grained lithologies” by myself and RM Bustin. The published paper is identical to the thesis chapter except for formatting edits in order to be consistent with other chapters in this thesis. I collected all the laboratory data for Chapter 4. Data analysis was performed by me with help from RM Bustin. Both the submitted paper and Chapter 4 were written by me with moderate editorial work performed by RM Bustin.  Chapter 5 is adapted from a manuscript submitted to Geofluids titled “Investigating multi-phase flow phenomena in fine-grained reservoir rocks: insights from using ethane permeability measurements over a range of pore pressures” by myself and RM Bustin. The submitted manuscript is identical to the thesis chapter except for formatting edits in order to be consistent with other chapters in this thesis. I collected all the laboratory data for Chapter 5. Data analysis vii  was performed by me with help from RM Bustin. Both the manuscript and Chapter 5 were written by me with minor editorial work performed by RM Bustin.  Chapter 6 is a conclusion to the thesis written by me with editorial work by RM Bustin. viii  Table of Contents  Abstract .......................................................................................................................................... ii Lay Summary ............................................................................................................................... iv Preface .............................................................................................................................................v Table of Contents ....................................................................................................................... viii List of Tables .............................................................................................................................. xiii List of Figures ............................................................................................................................. xiv List of Symbols ........................................................................................................................... xix List of Abbreviations ................................................................................................................. xxi Acknowledgements ................................................................................................................... xxii Chapter 1: Introduction ................................................................................................................1 1.1 Overview ......................................................................................................................... 1 1.2 Thesis Structure .............................................................................................................. 3 1.2.1 Klinkenberg Gas Slippage Measurements as a Means for Shale Pore Structure Characterization (Chapter 2) ................................................................................................... 3 1.2.2 Quantitative Validation of Pore Structure Geometry Characterization Using Gas Slippage Measurements by Comparison with Bundle of Capillaries Models (Chapter 3) ..... 4 1.2.3 The Impact of Gas Slippage on Permeability Effective Stress Laws: Implications for Predicting Permeability of Fine-Grained Lithologies (Chapter 4) ......................................... 5 1.2.4 Investigating Multiphase Flow Phenomena in Fine-Grained Reservoir Rocks: Insights from Using Ethane Permeability Measurements Over a Range of Pore Pressures (Chapter 5) .............................................................................................................................. 6 ix  Chapter 2: Klinkenberg Gas Slippage Measurements as a Means for Shale Pore Structure Characterization ............................................................................................................................8 2.1 Introduction ..................................................................................................................... 8 2.2 Samples ......................................................................................................................... 13 2.3 Methods/Experiments ................................................................................................... 14 2.3.1 Slippage Measurements ............................................................................................ 14 2.3.2 Ancillary Data ........................................................................................................... 17 2.4 Results ........................................................................................................................... 18 2.4.1 Sample Descriptions ................................................................................................. 18 2.4.2 Slippage Measurements ............................................................................................ 18 2.4.3 MICP and Low Pressure Gas Adsorption Data ........................................................ 26 2.5 Discussion ..................................................................................................................... 28 2.5.1 Klinkenberg Plot Linearity and Effective Stress ...................................................... 28 2.5.2 Slippage Measurement Precision .............................................................................. 31 2.5.3 Slippage Measurements ............................................................................................ 31 2.5.4 Comparison to Pore System Characterization using SEM ....................................... 36 2.5.5 Comparison to Pore System Characterization using MICP ...................................... 38 2.5.6 Comparison to Pore System Characterization Using Low Pressure Gas Adsorption39 2.5.7 Importance of Gas Slippage on Production .............................................................. 40 2.6 Conclusion .................................................................................................................... 40 Chapter 3: Quantitative Validation of Pore Structure Characterization Using Gas Slippage Measurements by Comparison with Predictions from Bundle of Capillaries Models ..........42 3.1 Introduction ................................................................................................................... 42 x  3.2 Background ................................................................................................................... 43 3.2.1 Gas Slippage ............................................................................................................. 43 3.2.2 Gas Slippage Measurements as a Pore Structure Characterization Technique ......... 45 3.2.3 Bundle of Capillaries Models ................................................................................... 47 3.3 Methods and Materials .................................................................................................. 48 3.3.1 Gas Slippage Measurements ..................................................................................... 49 3.3.2 Scanning Electron Microscopy ................................................................................. 52 3.4 Results and Discussion ................................................................................................. 52 3.4.1 Compiled Gas Slippage Data Set .............................................................................. 53 3.4.2 Pore Size Estimates from Gas Slippage Data ........................................................... 56 3.4.3 Comparison with Pore Size Estimates from Bundle of Capillaries Models ............. 58 3.4.3.1 Porosity-Permeability Function ........................................................................ 63 3.4.3.2 Tortuosity-Permeability Function ..................................................................... 64 3.4.3.3 Bundle of Capillaries Models Using Porosity and Tortuosity Functions ......... 67 3.4.4 Significance of the Pore Size-Permeability Trends .................................................. 68 3.5 Conclusion .................................................................................................................... 76 Chapter 4: The Impact of Gas Slippage on Permeability Effective Stress Laws: Implications for Predicting Permeability of Fine-Grained Lithologies ..................................78 4.1 Introduction ................................................................................................................... 78 4.2 Background ................................................................................................................... 79 4.3 Materials and Methods .................................................................................................. 83 4.3.1 Sample Descriptions ................................................................................................. 83 4.3.2 Sample Seasoning ..................................................................................................... 84 xi  4.3.3 Permeability Measurements ...................................................................................... 85 4.4 Results and Discussion ................................................................................................. 87 4.4.1 Impact of Pore Structure Changes on Klinkenberg Plots ......................................... 88 4.4.2 Permeability Effective Stress Law for the Montney Sample .................................... 90 4.4.3 Impact of Ignoring Gas Slippage on Permeability Effective Stress Laws ................ 92 4.4.4 Reanalysis of Data from Heller et al. (2014) ............................................................ 97 4.4.5 Implications for Previous Studies ........................................................................... 103 4.5 Conclusion .................................................................................................................. 106 Chapter 5: Investigating Multi-Phase Flow Phenomena in Fine-Grained Reservoir Rocks: Insights from Using Ethane Permeability Measurements Over a Range of Pore Pressures .....................................................................................................................................108 5.1 Introduction ................................................................................................................. 108 5.2 Methods....................................................................................................................... 111 5.2.1 Sample Suite ........................................................................................................... 111 5.2.2 Permeability Measurements .................................................................................... 112 5.2.3 Permeability Calculation ......................................................................................... 114 5.2.4 Ethane Desorption Rate Calculations ..................................................................... 115 5.2.5 Ethane Gas Slippage Estimates ............................................................................... 116 5.3 Results ......................................................................................................................... 117 5.3.1 Helium Klinkenberg Plots ....................................................................................... 117 5.3.2 Ethane Permeability ................................................................................................ 120 5.3.3 Ethane Desorption Rates ......................................................................................... 122 5.4 Discussion ................................................................................................................... 126 xii  5.4.1 Desorption Rate Data Quality ................................................................................. 126 5.4.2 Ethane Permeability at Low Pore Pressures ........................................................... 128 5.4.3 Ethane Permeability at High Pore Pressures ........................................................... 128 5.4.4 Stress Sensitivity of Ethane Permeability ............................................................... 133 5.4.5 Anisotropy of Ethane Permeability ......................................................................... 133 5.4.6 Permeability Hysteresis .......................................................................................... 134 5.5 Conclusion .................................................................................................................. 135 Chapter 6: Conclusion ...............................................................................................................137 6.1 Overview ..................................................................................................................... 137 6.2 Technique Developments ............................................................................................ 138 6.3 Insights Gained About Fine-Grained Reservoir Rocks .............................................. 143 6.4 Future Direction .......................................................................................................... 144 References ...................................................................................................................................147 Appendix .....................................................................................................................................155 Appendix A Compiled Gas Slippage Data Set ....................................................................... 155  xiii  List of Tables  Table 2.1 Pore pressure (Pp) and confining pressure (Pc) schedule for gas slippage measurements on each subsample. ............................................................................................... 15 Table 2.2 Porosity, XRD, and SRA data for the three samples analyzed in Chapter 2. ............... 18 Table 2.3 Slippage parameter, effective slit width calculated using Eq. 2.2, and effective pore diameter calculated using Eq. 2.3 at all stress states ..................................................................... 25 Table 2.4 Stress conditions for a slippage measurement at 20.68 MPa (3000 psi) simple effective stress .............................................................................................................................................. 29 Table 3.1 Important details of all studies used to generate the compiled data set of gas slippage measurements used in the current study, as well as b conversion factors used to standardize gas slippage measurements to helium at 18 degrees Celsius. ............................................................. 50 Table 4.1 Mineralogical composition of the Montney Formation sample from X-ray diffraction........................................................................................................................................................ 83 Table 4.2 Hypothetical pressure data used for the conceptual analysis presented in Figure 4.4. . 89 Table 4.3 Klinkenberg’s slippage parameter (b, calculated using Equation 6) at each simple effective stress for the Montney sample. ...................................................................................... 97 Table 5.1 Experimental conditions and tabulated data for each sample analyzed in Chapter 5 . 122  xiv  List of Figures  Figure 2.1 Data for Klinkenberg’s sample L for flow of nitrogen and sample A for flow of air  (Klinkenberg, 1941). ..................................................................................................................... 10 Figure 2.2 Velocity profiles for capillaries of different geometries and flowing fluid properties. (Modified from Bravo, 2007). ...................................................................................................... 11 Figure 2.3 Photographs of the three samples analyzed in Chapter 2 ............................................ 13 Figure 2.4 Klinkenberg plots for all samples at all simple effective stress states. ....................... 20 Figure 2.5 Slippage and permeability at different stress states for each subsample ..................... 22 Figure 2.6 Slippage and permeability for samples TEFB7, TEF21, and TEF17 at all stress states....................................................................................................................................................... 24 Figure 2.7 Precision and scalability analysis for sample TEF23Pll. ............................................. 25 Figure 2.8 MICP and low-pressure nitrogen adsorption (LPA in legends) data for all three samples in Chapter 2 ..................................................................................................................... 27 Figure 2.9 Expected result of α values not equal to 1 on the shape of Klinkenberg plots ............ 30 Figure 2.10 Permeability and slippage anisotropy comparison of TEFB7 and TEF17 ................ 34 Figure 2.11 Comparison of slippage measurements from this study to those from previous studies, most of which were for rocks with higher permeability. ................................................. 36 Figure 3.1 Compiled gas slippage data set in b - K∞ space coloured by which study the data are from ............................................................................................................................................... 54 Figure 3.2 Power law fits to the individual data sets comprising the compiled data set. ............. 55 xv  Figure 3.3 Dominant pore sizes calculated from gas slippage measurements in the compiled data set assuming a tubular geometry (black circles in A) and a slot shaped geometry (black circles in B)................................................................................................................................................... 57 Figure 3.4 Relationships between K∞ and tube diameter (calculated using Equation 3.16) for geologically reasonable ranges of Φ and τ ................................................................................... 60 Figure 3.5 Relationships between K∞ and slot-shaped capillary width (calculated using Equation 3.17) for geologically reasonable ranges of Φ and τ .................................................................... 61 Figure 3.6 Portion of the gas slippage dataset for which calculated pore sizes can be captured by at least one of the simple bundle of capillaries models with geologically reasonable inputs of porosity and tortuosity. ................................................................................................................. 62 Figure 3.7 Porosity data for the studies in the compiled data set that included such data ............ 64 Figure 3.8 Upper (red) and lower (green) bounds of tortuosity as a function of porosity calculated using the theoretical predictions of Guo (2015). .......................................................................... 65 Figure 3.9 Tortuosity-permeability relationship developed by combining the porosity-permeability function developed using porosity measurements made on samples from the compiled gas slippage data set (Figure 3.7) with Guo’s (2015) theoretical tortuosity-porosity relationship (Figure 3.8) ................................................................................................................ 67 Figure 3.10 Pore sizes calculated from gas slippage measurements for flow perpendicular to bedding at different simple effective stress states (SES in legends) for the same Montney Formation sample imaged in Figure 3.11 ..................................................................................... 71 Figure 3.11 Backscatter electron images of a Montney Formation sample at ambient stress, oriented such that the imaged surfaces are along bedding planes (gas flowing through the imaged pores would be travelling perpendicular to bedding). .................................................................. 72 xvi  Figure 4.1 Data for Klinkenberg’s sample L for flow of nitrogen (Klinkenberg, 1941) .............. 82 Figure 4.2 Mercury intrusion porosimetry pore size distribution of the Montney sample analyzed in this study. .................................................................................................................................. 84 Figure 4.3 Confining pressure and pore pressure schedule for permeability measurements of the Montney sample ............................................................................................................................ 87 Figure 4.4 Synthetic permeability data calculated for the hypothetical pressure conditions listed in Table 4.2 ................................................................................................................................... 89 Figure 4.5 (A) Klinkenberg plots for the Montney sample at each simple effective stress state. (B) Permeability variation due to gas slippage at each simple effective stress state .................... 91 Figure 4.6 Permeability effective stress law for the Montney sample analyzed in this study ...... 92 Figure 4.7  (A) Permeability variation due to gas slippage at individual simple effective stress states for permeability measurements of the Montney sample made at and above 7 MPa pore pressure. (B) Apparent permeability effective stress law for data fit to a power function ........... 93 Figure 4.8 Permeability of the Montney sample against effective stress for α = 0.64, the α value determined by curve fitting to high pore pressure (≥7 MPa) permeability data ........................... 94 Figure 4.9 Schematic representations of linear (A, B and C) and nonlinear (D) permeability effective stress laws using permeability contours in confining pressure-pore pressure space ..... 95 Figure 4.10 Permeability contour plot in confining pressure-pore pressure space for the Montney sample analyzed in this study ....................................................................................................... 96 Figure 4.11 Klinkenberg plots for the reanalyzed data from Heller et al. (2014) ......................... 99 Figure 4.12 High pore pressure (≥ 7 MPa) permeability data in confining pressure-pore pressure space for the three samples from Heller et al. (2014) ................................................................. 102 xvii  Figure 4.13 Both high and low pore pressure permeability data for the three samples from Heller et al. (2014) plotted in confining pressure-pore pressure space .................................................. 103 Figure 5.1 Isothermal density-pressure relationships for methane (A) and ethane (B) .............. 110 Figure 5.2 Schematic cross sections of pore throats showing adsorbed liquid/semiliquid ethane restricting flow paths for ethane gas at high gas pressure .......................................................... 111 Figure 5.3 Pore pressure schedule for the suite of helium (blue) and ethane (red) permeability measurements made on each sample .......................................................................................... 113 Figure 5.4 Comparison of helium (A) and ethane (B) pulse decay data for sample B5FD2A2 to illustrate how ethane desorption rates were calculated ............................................................... 116 Figure 5.5 Klinkenberg plot for each sample generated using the helium permeability data .... 118 Figure 5.6 Dominant pore size calculated using Equation 5.2 and the helium Klinkenberg plot data .............................................................................................................................................. 119 Figure 5.7 Ethane permeability measurements (coloured markers) and predicted permeability variation due to gas slippage (black markers) calculated using the ethane slippage factor, which was derived from the helium permeability data .......................................................................... 121 Figure 5.8 Ethane desorption rates calculated from the pulse decay pressure data for the five highest permeability samples ...................................................................................................... 124 Figure 5.9 Examples of good quality ethane pulse decay pressure data for a high permeability sample from which accurate desorption rates could be calculated (A and C) and poor quality data for a low permeability sample from which accurate desorption rates could not be calculated (B and D).......................................................................................................................................... 125 Figure 5.10 Liquid/semi-liquid ethane saturation curve for sample TEFB9Pll at 7 MPa simple effective stress ............................................................................................................................. 127 xviii  Figure 5.11 Correlation of difference between predicted ethane gas permeability and measured permeability at 3.45 MPa pore pressure (y axes) and K∞ (A) and d (B) ................................... 130 Figure 5.12 Acyclic pore network model for a pore structure where the pores most responsible for limiting fluid flow are in the macropore range (A and B) and mesopore range (C and D) .. 132 Figure 6.1 Permeability-stress relationships for a single Montney Formation sample ............... 138 Figure 6.2 Dominant pore size calculated from gas slippage measurements at different stress states and flow orientations for the same Montney Formation sample as in Figure 6.1 ............. 140  xix  List of Symbols  𝑏  Klinkenberg’s slip parameter 𝐶  Adzumi’s constant 𝑑  pore diameter 𝑑𝑘𝑖𝑛  kinetic diameter 𝐾  permeability 𝐾𝑎   apparent permeability 𝐾𝑛  Knudsen number 𝐾∞  permeability intercept of Klinkenberg plot 𝐿𝑐  characteristic length 𝑀  gas molecular weight 𝑁𝐴   Avagadro’s constant 𝑃  pressure 𝑃𝑐   confining pressure 𝑃𝑝  pore pressure 𝑄  volume gas flow rate 𝑅  gas constant 𝑟   pore radius 𝑇  temperature 𝑤  slit width α  permeability effective stress law coefficient xx  αT  permeability effective stress law coefficient in the differential form 𝜆  mean free path 𝜇  viscosity 𝜎𝑒𝑓𝑓  effective stress τ  tortuosity 𝛷  porosity xxi  List of Abbreviations BJH  Barrett-Joyner-Halenda FESEM field emission scanning electron microscopy LPA  low pressure adsorption MICP  mercury intrusion capillary pressure SEM  scanning electron microscopy TOC  total organic carbon XRD  X-ray diffraction  xxii  Acknowledgements I would not have embarked on the path to the completion of this thesis without the inspiration and support provided by my supervisor, Marc Bustin. You inspired me as an undergraduate student in your fossil fuels course, where you showed me how fascinating petroleum geology is and how science is applied to solve real-world problems in the field. You opened the door for me to research when you took me on as an undergraduate thesis student, and eventually a graduate student. I appreciate the resources you provided me to pursue my interests, including your guidance and willingness to discuss ideas. I also appreciate the efforts you made to expose me to industry, and your support of me presenting at numerous conferences throughout my degree.  Thank you to the rest of my supervisory committee, Dr. Erik Eberhardt and Dr. Roger Beckie, for the guidance you have provided over the last five years.  The EOAS technical staff provided a great deal of assistance in the laboratory during the completion of my thesis. Special thanks to Jörn Unger for your precision work, out-of-the-box solutions to the many technical problems I encountered, and willingness to teach me new skills, and to David Jones for (partially) demystifying electronics and coding for me. The skills and knowledge I developed thanks to your guys’ encouragement and assistance will serve me for the rest of my life, and might just get me out of a tight spot in the middle of the Pacific Ocean.  Thank you to past and present members of the “Bustin lab” for creating such a great environment to work in. Kristal Li did an excellent job of keeping the lab functioning smoothly, which was no easy task given the multiple users and constant turnover of graduate students and undergraduate xxiii  lab assistants. I am appreciative of the friendship forged with Erik Munson during the early stages of our graduate degrees, and the assistance he has provided me over the years. Your presence has been missed since you graduated, especially behind the blast shield in the lab, and on the Darby’s patio.  My parents, Chris and Tina Letham, and my brother, Bryn Letham, were tremendously supportive during my graduate studies, for which I am grateful.  This thesis would not have been completed without the encouragement and support of my friends, both those who have been reliable constants since prior to me starting graduate school, and those great friends I met during my time at EOAS.  Financial support that made this work possible was provided by the Natural Sciences and Engineering Research Council of Canada, Geoscience BC, and our industry partners. Thank you for your investment. 1  Chapter 1: Introduction  1.1 Overview Fine-grained sedimentary reservoir rocks are economically important primarily due to the vast quantities of hydrocarbons they contain (U.S. Energy Information Administration, 2013). Production of hydrocarbons from fine-grained reservoirs, such as shale oil and shale gas reservoirs, has impacted the global energy market (Maugeri, 2014; Kilian, 2016). Yet, due to challenges associated with analyzing rocks with highly stress-sensitive, anisotropic, nanometer-scale pore structures, these reservoirs are poorly understood relative to their importance. The focus of this thesis is on pore structure characterization and permeability measurement of fine-grained reservoir rocks in the laboratory at reservoir stress states, and quantifying anisotropy of these petrophysical parameters. Being able to quantify these petrophysical parameters at stress is important because finer-grained reservoir rocks are generally more stress sensitive than coarser grained rocks (McLatchie et al., 1958; Vairogs et al., 1971), and measurements at reservoir stress are required for accurate reservoir simulations. Furthermore, fine-grained sedimentary reservoir rocks have markedly heterogeneous petrophysical properties as a result of millimeter-scale laminae that vary in texture and composition, and therefore pore structure (Chalmers et al., 2012a).  Although much petrophysical data can be acquired from the subsurface using wireline logging, many petrophysical parameters cannot be determined from wireline logs alone (e.g. full pore structure characterization) or are substantially less robust and detailed than their laboratory-determined equivalents. Once determined accurately in the laboratory, the petrophysical 2  parameters can be used as inputs for reservoir models in order to predict the behaviour of the reservoirs and thus their economic utility. Laboratory petrophysical measurements can also be used as a calibration tool for the wireline logs (e.g. Chapter 6 in Munson, 2015).   This thesis considers effective stress laws (Robin, 1973) for permeability, characterizing pore structure geometry, and tools for exploring multiphase flow phenomena in fine-grained sedimentary rocks. This thesis addresses the following questions in the context of studying fine-grained reservoir rocks: (1) what are the strengths and limitations of using Klinkenberg gas slippage measurements (Klinkenberg, 1941) as a pore structure geometry characterization tool? (2) do Klinkenberg gas slippage measurements yield quantitatively accurate characterizations of pore structure geometry? (3) how can in situ matrix permeability be predicted from laboratory measurements made at lower than in situ pore pressure? (4) are published predictions of in situ matrix permeability accurate, or are they in error due to gas slippage? (5) can ethane permeability measurements made over a range of pore pressures up to the saturated vapour pressure be used as a tool to investigate multiphase flow characteristics?  This thesis is largely techniques-based; the applicability of selected established techniques for analyzing coarser grained reservoir rocks is investigated for fine-grained rocks, and new techniques are developed to address deficiencies. Ancillary to this thesis however is an increased understanding of the general characteristics of fine-grained sedimentary rocks, derived from the data collected when investigating the different techniques.  3  1.2 Thesis Structure Chapters two through five of this thesis are written as standalone manuscripts that are either published or submitted for publication. Although these four chapters are standalone manuscripts, commonalities link all four of the chapters. A central theme of this thesis is an investigation of gas slippage in fine-grained sedimentary reservoir rocks, both as a tool with unique characteristics that make it well-suited for analyzing these rocks, but also as a source of error. The chapters build on each other, but because of the overlap of gas slippage as a central theme, some background, equations, and ideas are repeated in the standalone chapters. The motivation, scope and content of each chapter is outlined below.  1.2.1 Klinkenberg Gas Slippage Measurements as a Means for Shale Pore Structure Characterization (Chapter 2) Chapter 2 investigates the utility of Klinkenberg gas slippage measurements for pore structure characterization of fine-grained sedimentary rocks. The investigation is warranted because established techniques that have been successfully used to characterize pore systems in coarse-grained reservoir rocks lack the resolution and scalability required to adequately characterize the nano- to micrometer scale pore systems found in fine-grained rocks, and cannot be applied on stressed samples. Different techniques therefore need to be developed and employed.  That gas slippage measurements could be used for pore structure characterization was realized more than half a century ago (Klinkenberg, 1941). However, the technique was initially not widely applied because rocks being exploited for hydrocarbon production were dominantly coarse-grained, conventional reservoir rocks whose pore structures could easily be characterized 4  using optical techniques. For those coarse-grained reservoir rocks, the importance of measuring gas slippage was in correcting laboratory measurements of permeability (Klinkenberg, 1941). Gas slippage measurements were later explored as a tool for investigating the pore structures of tight sands, and the technique was modified so that pore structure characterizations could be made assuming slot-shaped pores instead of circular pores (Randolph, 1984). The technique was applied early on to gas shales (Soeder et al., 1988), but has seen limited application to fine-grained reservoir rocks since (Aljamaan et al. 2013; Cui et al. 2013; Sinha et al. 2013; Heller et al. 2014), and no studies provide a targeted investigation and discussion of the utility of slippage measurements as a tool for pore structure characterization of fine-grained rocks. Chapter 2 addresses this deficiency by presenting and analyzing gas slippage measurements made at different flow orientations and stress states for a suite of samples from a shale gas reservoir. Pore structure geometry is also measured using other techniques for comparison. The analysis reveals many unique, desirable attributes of the gas slippage technique, such as the ability to make measurements on stressed samples and to quantify anisotropy of pore structure geometry.  1.2.2 Quantitative Validation of Pore Structure Characterization Using Gas Slippage Measurements by Comparison with Bundle of Capillaries Models (Chapter 3) The focus of Chapter 3 is investigating the quantitative validity of pore structure characterizations made using gas slippage measurements. To investigate the validity, pore size estimates made using the gas slippage technique are compared to pore size estimates made using bundle of capillaries models (Carman, 1937). The analysis reveals a good match between the two techniques, therefore indicating that the gas slippage technique provides quantitatively valid characterizations of reservoir rock pore structures. This chapter builds on the study of gas 5  slippage measurements as a pore structure characterization tool in Chapter 2 by investigating the quantitative accuracy of the technique. If not quantitatively accurate, the gas slippage technique could still be useful for investigating relative differences between samples. Robust quantitative validation of the technique however provides confidence that gas slippage measurements can be used to predict reservoir phenomena.   1.2.3 The Impact of Gas Slippage on Permeability Effective Stress Laws: Implications for Predicting Permeability of Fine-Grained Lithologies (Chapter 4) The impact of gas slippage on predicting permeability of fine-grained reservoir rocks is investigated in Chapter 4. Chapters 2 and 3 show that gas slippage is advantageous because it can be measured to obtain quantitatively accurate pore size characterizations at reservoir stress states. However, gas slippage is a disadvantageous complication when determining permeability using a gaseous probing fluid (Klinkenberg, 1941). Gas slippage becomes an especially significant complication when measuring matrix permeability of fine-grained reservoir rocks because gas slippage has an increasing impact on flow rate as pore size decreases (Klinkenberg, 1941). The extent of this complication however has not yet been fully realized in the literature.   When analyzing sedimentary reservoir rocks in order to determine their economic utility, of interest is permeability of the rocks at in situ reservoir confining and pore fluid pressures and at the range of pressures experienced during production from the reservoirs. However, replicating these pressures for analysis in the laboratory is more expensive, challenging, and dangerous than making measurements at lower confining and pore fluid pressures and extrapolating to higher pressures. Extrapolation is possible if a permeability effective stress law can be obtained (Robin, 6  1973). Techniques have been developed and implemented to determine such effective stress laws (Zoback and Byerlee, 1975; Morrow et al., 1986; Bernabe, 1987; Bernabe, 1988; Warpinski and Teufel, 1992; Kwon et al., 2001; Al-Wardy and Zimmerman, 2004; Ghabezloo et al., 2009; Li et al., 2009; Heller et al., 2014; Li et al., 2014), but have been implemented without due consideration for errors resulting from gas slippage. How gas slippage impacts permeability effective stress laws is determined in Chapter 4 by measuring permeability of a gas shale to helium at a wide range of confining and pore fluid pressures. The recognizable impacts of gas slippage on permeability effective stress laws found in this investigation are then compared to published permeability effective stress laws for fine-grained rocks in order to assess their validity.  1.2.4 Investigating Multiphase Flow Phenomena in Fine-Grained Reservoir Rocks: Insights from Using Ethane Permeability Measurements Over a Range of Pore Pressures (Chapter 5) Permeability depends not only on stress state but also on the saturation of different immiscible fluids in a reservoir, giving rise to the concepts of effective and relative permeability. Initial fluid saturations are spatially and temporally variable during production due to the change of state variables (e.g. pressure and temperature). Measuring effective permeability at the range of saturations experienced during production is especially important for fine-grained reservoirs due to the nanometer-scale pore structures characteristic of these rocks. The small pores result in very high capillary pressures and sensitivity of flow rate to constrictions imposed by fluid adsorbed to pore walls (Aljamaan et al., 2016). Measuring effective permeability of fine-grained reservoir rocks is challenging due to difficulty in controlling and monitoring fluid saturations and 7  distributions in intact samples, and, because of their inherently low permeability, difficulty in quantifying their characteristic, very low flow rates. New techniques are therefore required to investigate multiphase flow phenomena in fine-grained reservoir rocks. This deficiency is addressed in Chapter 5 by developing a new technique where matrix permeability to ethane is measured over a range of pore pressures up to the saturated vapour pressure of ethane at laboratory temperature. Ethane is a sorptive gas. Liquid/semi-liquid ethane saturation therefore increases with increasing pore pressure due to adsorption and capillary condensation. The liquid/semi-liquid ethane restricts flow paths resulting in decreased effective permeability to ethane gas. Ethane gas permeability is shown to be especially sensitive to pore pressure in samples where gas slippage measurements reveal that the pores most responsible for flow are in the mesopore size range. Capillary condensation causes complete blockage of these pores at pore pressures near to the saturated vapour pressure of ethane (Gregg, 1982), which explains the large effective permeability reductions. That the multiphase flow phenomena experimentally observed in Chapter 5 can be explained by quantitative pore structure characterizations made using gas slippage measurements supports the conclusions of Chapters 2 and 3.  8  Chapter 2: Klinkenberg Gas Slippage Measurements as a Means for Shale Pore Structure Characterization  2.1  Introduction Shale in comparison to conventional petroleum reservoirs is unique in its fine-grained nature and resulting nano- to micrometer scale pore systems. Characterizing pore systems is important as they are the sites where hydrocarbons are stored and transported. The geometry of the pore systems and composition of the pore walls (which controls wettability) influence hydrocarbon production through capillarity and permeability, and by factors recently recognized that include pore size-dependent phase envelopes in nanometer scale pores (Akkutlu and Didar, 2013). Techniques that have been successfully applied to characterize pore systems in conventional reservoir rocks, such as mercury intrusion capillary pressure (MICP), low pressure gas adsorption, and scanning electron microscopy (SEM), lack the resolution and scalability required to adequately characterize pore systems found in shale (Clarkson et al., 2013). New or refined techniques are required to address this issue, which is the motivation for the current study. This study investigates the utility of Klinkenberg gas slippage measurements for analyzing the pore structure of shale.  In 1941, Klinkenberg noted a relationship between pore pressure-dependent gas permeability (higher permeability at lower pore pressures) and mean free path (average distance a gas molecule travels between two successive collisions with other gas molecules) of the flowing gas molecules (Figure 2.1). Klinkenberg attributed this to a process now called gas slippage, 9  whereby the non-zero average velocity in the direction of flow of gas molecules in the immediate vicinity of pore walls contributes considerably to the quantity of gas flowing through the pores (Figure 2.2). This contrasts with liquid flow or gas flow at very high pressure where molecules are in direct contact with one another, resulting in zero velocity at the pore walls and a Poiseuille-type velocity profile (Figure 2.2A). For a given rock, the mean free path of a gas controls how much the non-zero average velocity at the pore walls contributes to gas flow (more contribution with larger mean free path), and therefore controls slippage. More slippage takes place when a gas with a smaller molecular size and/or at lower pressure (and therefore larger mean free path) is flowing than with a gas with a larger molecular size and/or higher pressure (and therefore smaller mean free path). Klinkenberg also noted that permeability has stronger dependence on pore pressure in lower permeability rocks that presumably have smaller pores (Klinkenberg, 1941). For example, two rocks with identical absolute permeability, the first having fewer but bigger capillaries (pores), and the second having more but smaller capillaries (pores), the probability of a gas molecule colliding with a pore wall instead of another gas molecule is lower in the first rock, and hence it experiences less gas slippage than the second (compare Figure 2.2B and 2.2D). This fundamental relationship is what allows use of gas slippage measurements to analyze effective pore size. The utility of the relationship is explored in this chapter. 10   Figure 2.1 Data for Klinkenberg’s sample L for flow of nitrogen (A and B) and sample A for flow of air (C and D) (Klinkenberg, 1941). (A and C) Variation of permeability with pore pressure. (B and D) Linear relationship of inverse pore pressure and permeability. The mean free path of a gas also varies linearly with the inverse of pressure, which means permeability varies linearly with mean free path.  11   Figure 2.2 Velocity profiles for capillaries of different geometries and flowing fluid properties. Vz is velocity in the longitudinal axis direction and r is distance along the radial axis of the capillary. Grey shaded regions represent areas where gas slippage is taking place. (A) Poisseuille type velocity profile for liquid flow or flow of a gas at very high pressure. Note zero velocity at capillary wall. (B) Velocity profile for flow in a capillary with the same geometry as in (A), but for flow of a gas at low pressure and therefore large mean free path and associated gas slippage. Note non-zero velocity at the capillary wall and that the area left of the velocity profile, which after normalizing for compressibility and converting to three dimensional geometry represents volume flow rate, is larger than in (A). (C) Velocity profile for a capillary with the same geometry as in (A) and (B), but for a gas with a larger mean free path than in (B) and therefore more gas slippage. (D) Velocity profile for flow of a gas with the same mean free path as in (B) but in a smaller capillary. Note that the portion of the area left of the velocity profile representing slippage (the area shaded grey) constitutes a larger fraction of the total area left of the velocity profile than in (B) (Modified from Bravo, 2007).  A desirable attribute of slippage measurements for analyzing shale pore structures is the ability to make measurements on stressed samples. It is well known that shale permeability varies markedly with stress as a result of changes to pore structure with compression (Kwon et al., 2001; Pathi, 2009; Kang et al., 2011; Heller et al., 2014). Measuring permeability on stressed samples is common practice. Measuring slippage on such stressed samples provides a further opportunity to characterize pore structure under stress. In contrast, during pore structure characterization using SEM and low pressure gas adsorption, the sample is unstressed. Similarly, characterization of pore structure using MICP approximates hydrostatic stress. During 12  production, pore pressure declines and effective stress increases and thus, to model production it is important to be able to characterize pore structure at different stress states.  Since gas slippage measurements are derived from permeability measurements, they characterize the portion of the pore system responsible for fluid flow; isolated pores do not contribute to the flow of fluids through a rock, and therefore do not contribute to the measured gas slippage. Characterizing only the effective porosity is desirable, as it is the effective porosity that determines hydrocarbon flow and storage properties. Within the effective porosity, the larger and best connected pores will account for most fluid flow and will be more heavily weighted in slippage measurements. For example, in a core sample with a through-going fracture that can transmit fluid much more efficiently than the surrounding intact matrix, fluid flow through the core will dominantly take place in the fracture, and the void spaces between asperities on the fracture planes are what will be characterized by slippage measurements. This also applies to heterogeneous core samples with layers that have drastically different permeability; the high permeability layers are what slippage measurements would characterize.  Very few studies are published that include slippage measurements on stressed shale samples (Soeder, 1988; Sinha et al., 2013; Cui et al., 2013; Aljamaan et al., 2013; Heller et al., 2014), and no studies have provided a targeted investigation and discussion of the utility of slippage measurements as a means of shale pore structure analysis. The current study provides a dataset of slippage measurements on Eagle Ford Shale plugs oriented both perpendicular and parallel to bedding at a variety of stress states. In this study the unique characteristics of slippage measurements in comparison to other pore structure characterization techniques is explored, and 13  how these unique characteristics can contribute to understanding shale flow and storage properties is discussed.  2.2 Samples Three samples (TEFB7, TEF21, and TEF17) from the Upper Cretaceous Eagle Ford Formation, Texas, USA, with different fabric (determined visually (Figure 2.3)) and composition (determined using X-ray diffraction (XRD)) were selected and subsampled for two 3 cm diameter by 3 cm length plugs, one oriented for gas flow parallel to bedding and one perpendicular to bedding. Samples were sealed in plastic upon retrieval and not cleaned or dried once received in the laboratory in order to preserve them at near in situ fluid saturations for future comparative studies. Names for subsamples parallel and perpendicular to bedding are suffixed with Pll and Pd respectively. A fourth sample (TEF23) was subsampled for two plugs oriented for flow parallel to bedding (TEF23Pll-a and TEF23Pll-b) and were used to investigate slippage measurement precision and scalability.   Figure 2.3 Photographs of the three samples analyzed in this study. All are cross sections through bedding revealing fine laminae and millimeter scale heterogeneity.  14  2.3 Methods/Experiments 2.3.1 Slippage Measurements Pressure pulse decay measurements were made using helium as a probing gas and permeability calculated using theory initially developed by Brace et al. (1968) and later modified by Cui et al. (2009). Three calibrated permeameters were used in this study. Permeability measurements were made on two standards (one synthetic metal, one natural) with each permeameter prior to all other measurements to ensure comparability of the results (maximum difference was 2.3% for the natural standard). Plugs from samples TEFB7, TEF21, and TEF17 were run at a range of simple effective stresses (defined herein as the difference between confining pressure and pore pressure) as per the schedule outlined in Table 2.1. Samples were left for three days at each new simple effective stress prior to making any permeability measurements to allow samples to equilibrate at their new stress state. Ideally, prior to making slippage measurements, permeability would have been continuously monitored until permeability creep ceased, but this was prohibited by time constraints. Each subsample took between 1 to 2 months to run, dependent on their permeability. At each simple effective stress, permeability was measured at pore pressures of 1.72, 2.30, 3.45, and 6.89 MPa (250, 333, 500, and 1000 psi) in order to vary mean free path of the helium; permeability measurements for at least two different mean free paths, one of which must be large enough to be in the slip flow regime, are required to quantify slippage. This particular range of pore pressures was chosen after experimentation with a larger range (0.34 to 13.79 MPa), and represents what the authors consider to be an optimization of the time required to make a slippage measurement and the quality of the measurement. Sampling a larger range of pore pressures lessens the significance of non-systematic errors associated with each individual permeability measurement on the determined slippage value. However, low pressure gas 15  permeability measurements take much longer than higher pressure measurements, and measurements at high pressures require large changes in pore pressure (and therefore confining pressure) to achieve a significant change in mean free path of the flowing gas molecules. Large stress changes are undesirable as they could induce poroelastic changes, as discussed in detail later. Table 2.1 Pore pressure (𝑷𝒑) and confining pressure (𝑷𝒄) schedule for gas slippage measurements on each subsample.  Run # Pore Pressure (MPa) Confining Pressure (MPa) Simple Effective Stress (Pc – Pp) (MPa) 1 1.72 8.62 6.89 2 2.3 9.19 6.89 3 3.45 10.34 6.89 4 6.89 13.79 6.89 Stress three days at Pp = 1.72 and  Pc = 43.09 MPa* 5 1.72 8.62 6.89 6 2.3 9.19 6.89 7 3.45 10.34 6.89 8 6.89 13.79 6.89 9 1.72 22.41 20.68 10 2.3 22.98 20.68 11 3.45 24.13 20.68 12 6.89 27.58 20.68 13 1.72 36.2 34.47 14 2.3 36.77 34.47 15 3.45 37.92 34.47 16 6.89 41.37 34.47         *This simple effective stress was chosen to induce inelastic deformation          that would take place during subsequent slippage measurements.  Klinkenberg plots were generated by plotting permeability against the inverse of mean pore pressure for each pore pressure step and applying a linear fit to the data. Slope and intercept of the fit were used to calculate Klinkenberg’s slippage parameter, 𝑏: 16  𝑏 =𝐾𝑎∗𝑃𝑝𝐾∞− 𝑃𝑝          (Eq. 2.1) where 𝑃𝑝 is mean pore pressure and 𝐾𝑎 is apparent permeability for any point along the linear fit and 𝐾∞ is the intercept of the linear fit with the permeability axis. The intercept represents extrapolation to infinite mean pressure where no slippage would take place. Permeability at infinite pressure is therefore taken to be the effective permeability to helium at whatever saturation of water and oil exists in the rock. Although the intercept represents effective permeability, for simplicity it will be referred to in this paper as permeability.  Klinkenberg’s parameter 𝑏 quantifies how much apparent permeability will vary due to slippage at low pressures for a certain gas, which is related to the size of the pores in which the gas flows (Klinkenberg, 1941). From a slippage measurement it is possible to calculate an average effective pore size assuming slot shaped pores using theory from Randolph et al. (1984): 𝑤 =16𝐶𝜇𝑏(2𝑅𝑇𝜋𝑀)12          (Eq. 2.2) where 𝑤 is slit width, 𝐶 is the Adzumi constant (assumed to be 0.9), 𝜇 is gas viscosity, 𝑅 is the gas constant, 𝑇 is temperature, and 𝑀 is molecular weight of the gas.  An alternative is assuming tube shaped pores using the theory from Klinkenberg (1941): 𝑑 =8𝑃𝑝𝐶𝜆𝑏          (Eq. 2.3) where 𝑑 is pore diameter and 𝜆 is mean free path. Effective apertures were calculated in this study using both methods. Some studies have suggested based on microscopic imaging that pores in tight sands and shale are more slit like than tubular (Randolph et al., 1984; Soeder, 1988; Heller et al., 2014). However, both slit shaped and more circular pores can be found in a single 17  shale sample, often with the more circular pores occurring in the organic matter and slit shaped pores associated with the mineral matrix (Jiao et al., 2014). Yet the presence of organic matter does not guarantee organic porosity, even at high thermal maturity (Curtis et al., 2012a). Not having imaged pores in the samples analyzed in this study, it is not known which pore type dominates. Hence average effective pore size was calculated using both methods.  In order to investigate the precision of slippage measurements, slippage was measured multiple times on each of the two subsamples from TEF23, both oriented for flow parallel to bedding.  2.3.2 Ancillary Data For comparison with slippage measurements, pore structure was also analyzed using MICP with a Micromeritics AutoPore IV®. Approximately ten grams of sample was used, and care was taken to sample from rock as close to and along the same bedding planes where plugs were drilled from. Also for comparison with slippage measurements, low-pressure nitrogen gas adsorption was measured at 77 °K using a Quantachrome Autosorb-1®. Pore size distributions were calculated from the adsorption data using the Barrett, Joyner, Halenda (BJH) method (Barrett et al., 1951). Approximately 0.8 g of crushed sample from end trims of the plugs oriented for flow parallel to bedding was used for the nitrogen gas adsorption measurements. Samples were degassed at 383 °K for 24 hours prior to measurement.  Unconfined porosity was determined using bulk density measurements from mercury immersion combined with helium pycnometry on samples crushed to a particle size range of 0.841 to 0.595 mm. Porosity was measured on samples “as received” and after drying in an oven at 383 °K until 18  mass stabilized. Mineral phases in each sample were identified by X-ray diffraction (XRD) using normal-focus CoKα radiation on a Bruker D8 Focus at 35 kV and 40 mA and quantified using the Rietveld method of full-pattern fitting (Rietveld, 1967) using Bruker AXS Topas V3.0 software. Total organic carbon (TOC), retained hydrocarbons (S1), and thermal maturity (Tmax) were determined using a SRAnalyzer – TPH/TOC®.  2.4 Results 2.4.1 Sample Descriptions The samples analyzed in this study are thinly laminated, fossiliferous, carbonate rich shale (Figure 2.3). TEFB7 has the lowest TOC content (1.5%) and highest carbonate content (92%), whereas TEF17 has the highest TOC content (5.9%) and lowest carbonate content (55%). TEF21 is intermediate both in terms of TOC and carbonate content (see Table 2.2). Table 2.2 Porosity, XRD, and SRA data for the three samples analyzed in this study. Sample As Received Helium Porosity (%) Dried Helium Porosity (%) MICP Porosity (%) TOC (%) Tmax (Degrees Celsius) S1 (mg/g) Quartz (wt. %) Total Carbonate (wt. %) Total Clay (wt. %) TEFB7 4.6 5.4 2.9 1.5 478 2.16 4.8 92.5 1 TEF21 4.7 8.3 3.9 3.1 483 3.23 16.7 65.7 14.1 TEF17 5.3 7.4 4.2 5.9 474 5.49 18.7 55.3 19.4 *Samples were dried at 383 °K until mass stabilized  2.4.2 Slippage Measurements Klinkenberg plots for samples TEFB7, TEF21, and TEF17 at all stress states measured in this study are presented in Figure 2.4. All plots follow a near-linear trend with varying amounts of scatter. Subtle concave up or concave down trends are observed in some plots (e.g. TEF21Pll at 19  6.89 MPa simple effective stress, concave-up (Figure 2.4)). No subsamples display a consistent trend of either concave-up or concave-down plots for all measured stress states. The permeameter used to measure slippage on subsample TEFB7Pd developed a leak when making measurements at 34.47 MPa (5000 psi) simple effective stress, which required removal of the sample to fix. The sample broke during removal and therefore measurement at this stress state could not be performed. The effective permeability range for data collection in this study is approximately 10-6 to 10-1 millidarcies (mD; equivalent to 10-21 to 10-16 m2). For higher permeability, pressure decay proceeds too rapidly to acquire adequate data, and for lower permeability, too few data points are acquired in a reasonable time frame, and quality of the collected data can easily be degraded. Samples at stress states where permeability was outside the effective range are noted in Figure 2.4. 20   Figure 2.4 Klinkenberg plots for all samples at all simple effective stress states. Slope and intercept from each plot was used to calculate Klinkenberg’s slippage parameter, 𝒃. The star (*) in the heading for the first column indicates measurements made prior to stressing for 3 days at 41.37 MPa (6000 psi) simple effective stress (see stress schedule in Table 2.1). 𝑷𝒑 is pore pressure (MPa) and 𝑷𝒄 is confining pressure (MPa).  21  For each subsample, Klinkenberg’s slippage parameter, 𝑏 (calculated using Equation 2.1), is plotted against permeability for each stress state in the subplots of Figure 2.5. For all subsamples, permeability decreased with each incremental stress increase. Permeability decreases associated with individual stress steps were between 30 and 90 %. Incremental stress increases caused slippage to increase in some subsamples and decrease in others. Slippage changes associated with individual stress steps were between -30 and 200 %. 22   Figure 2.5 Slippage and permeability at different stress states for each subsample. Number following subsample name in the legend corresponds to the simple effective stress in MPa for that particular slippage measurement. Star (*) indicates measurements made prior to stressing for 3 days at 41.37 MPa (6000 psi) simple effective stress. 𝑷𝒑 is pore pressure and 𝑷𝒄 is confining pressure. Note that permeability axes span three orders of magnitude and slippage axes one order of magnitude in all subplots, allowing comparison of sensitivity of slippage and permeability to stress between the subsamples. A, C, and E are for flow parallel to bedding. B, D, and F are for flow perpendicular to bedding.  23  A compilation of all slippage measurements from TEFB7, TEF21, and TEF17, which represent a limited suite of Eagle Ford Shale samples at a variety of stress states, is presented in Figure 2.6. Measured permeabilities span five orders of magnitude and are associated with two orders of magnitude in slippage variation. Slippage is negatively correlated with permeability. The equation for a least squares best fit line to the data with an R2 value of 0.84 is 𝑏 = 0.026𝐾∞−0.43          (Eq. 2.4) where 𝑏 is in MPa and 𝐾∞ in millidarcys (10-15 m2). 24   Figure 2.6 Slippage and permeability for samples TEFB7, TEF21, and TEF17 at all stress states. Star (*) indicates measurements made prior to stressing for 3 days at 41.37 MPa (6000 psi) simple effective stress. Number following subsample name in the legend corresponds to the simple effective stress in MPa for that particular slippage measurement. Solid line and associated equation is a least squares best fit to all slippage measurements. Dashed lines are parallel to the best fit line and encapsulate all data points for illustrative purposes (see Discussion).  25  Effective slot widths calculated using Equation 2.2 and effective pore diameters calculated using Equation 2.3 are reported in Table 2.3. Slot width ranged from 10 to 2128 nm with a median of 99 nm. Pore diameter ranged from 6 to 1212 nm with a median of 56 nm. Table 2.3 Slippage parameter, effective slot width calculated using Eq. 2.2, and effective pore diameter calculated using Eq. 2.3 at all stress states. Subheadings in second row are simple effective stress in MPa.  Sample b (MPa) Slot Width (nm) Diameter (nm)            Stress 6.89* 6.89 20.68 34.47 6.89* 6.89 20.68 34.47 6.89* 6.89 20.68 34.47 TEFB7Pll 0.44 0.58 1.72 1.31 387 293 99 130 220 167 56 74 TEFB7Pd 1.83 1.48 2.00 ------ 93 115 85 ------ 52 66 48 ------ TEF21Pll ------ 0.11 0.08 0.09 ------ 1548 2128 1892 ------ 882 1212 1078 TEF21Pd 1.73 1.11 1.79 ------ 98 153 95 ------ 56 87 54 ------ TEF17Pll 3.29 5.59 16.4 ------ 52 30 10 ------ 29 17 6 ------ TEF17Pd 3.54 5.84 ------ ------ 48 29 ------ ------ 27 17 ------ ------ * measurements made prior to stressing for 3 days at 41.37 MPa (6000 psi) simple effective stress  Results from multiple slippage measurements on each of the two subsamples of TEF23, both oriented for flow parallel to bedding, are presented in Figure 2.7. For a single subsample, maximum difference in permeability was 13% and maximum difference in slippage was 9%.  Figure 2.7 Precision and scalability analysis for sample TEF23Pll. Note the permeability axis spans three orders of magnitude and the slippage axis one order of magnitude, allowing comparison with subplots of Fig. 2.5.  26  2.4.3 MICP and Low Pressure Gas Adsorption Data Pore size distributions for each sample determined using MICP and low-pressure nitrogen gas adsorption are presented in Figure 2.8. For both analyses, TEF21 and TEF17 have similar pore size distributions whereas the distribution for TEFB7 is shifted towards larger pores. Pore size distributions from low-pressure nitrogen gas adsorption are shifted towards larger pores in comparison to distributions from MICP. Superimposed on the same figure are the effective pore sizes calculated using slippage data (Table 2.3).  27   Figure 2.8 MICP and low-pressure nitrogen adsorption (LPA in legends) data for all three samples in this study. Grey arrows show the range of average effective pore sizes at different stress states calculated using Eq. 2.3 for subsamples oriented for flow perpendicular to bedding and gold arrows for samples oriented for flow parallel to bedding (Y axis has no meaning for these arrows). The range of stress states for which slippage measurements were successfully made is not the same for all subsamples (Fig. 2.4).  28  2.5 Discussion 2.5.1 Klinkenberg Plot Linearity and Effective Stress It was initially assumed in this study that pore structure was not significantly altered by varying the confining pressure by the same amount as the pore pressure in each slippage measurement (maintaining a constant simple effective stress). Some reservoir rocks have permeability effective stress law coefficients (Robin, 1973), α, that are not equal to 1 (e.g. Zoback and Byerlee, 1975; Al-Wardy and Zimmerman, 2004). Permeability is inherently linked to pore structure. It can be concluded from permeability effective stress law coefficients not equal to 1 that pore structures are different at different combinations of pore and confining pressure that equate to the same simple effective stress. In this study it was assumed that the narrow range of pore pressures over which slippage measurements were made (1.72 to 6.89 MPa) was not enough to significantly alter the pore structure.  If the assumption that during slippage measurements pore structure did not change significantly due to poroelastic effects was invalid, then deviations from Klinkenberg plot linearity would be expected. For α values less than one, effective stress would be higher at higher combinations of pore and confining pressure that equate to the same simple effective stress (see Table 2.4). Conversely, for α values greater than one, effective stress would be lower at higher pore and confining pressure combinations. In either case, the outcome is a nonlinear decrease (if α > 1) or increase (if α < 1) in effective stress (column 5 in Table 2.4) with change in inverse pore pressure (column 3 in Table 2.4), resulting in distortion of Klinkenberg plots resulting from two conceptually distinguishable effects (Figure 2.9). One effect is a shift in permeability independent of mean free path of the flowing gas due to changes in pore structure when moving 29  from one effective stress to the next (Figure 2.9A). This will be referred to herein as “stress effects”, following terminology of Heller et al. (2014). The second is a shift in permeability due to changes in slippage as a result of the changes to pore structure (Figure 2.9B). This will be referred to herein as “flow regime effects”, again following terminology of Heller et al. (2014). The combined result of these two effects is nonlinearity of the Klinkenberg plot, except for the seemingly rare case where the two effects would completely counteract one another. A single subsample with a given permeability effective stress law coefficient would display a consistent trend of either concave-up or concave-down Klinkenberg plots if stress effects were sufficiently large. From the lack of consistent concave-up or concave-down Klinkenberg plots for individual subsamples at all stress states in the present study (Figure 2.4), it is concluded here that stress effects did not cause significant pore structure changes during slippage measurements. Table 2.4 Stress conditions for a slippage measurement at 20.68 MPa (3000 psi) simple effective stress. Effective stress is calculated using a hypothetical permeability effective stress law coefficient of α = 0.8. Note nonlinear increase in effective stress with decreasing inverse pore pressure. 𝑷𝒑 is pore pressure and 𝑷𝒄 is confining pressure.  Confining Pressure, Pc (MPa) Pore Pressure, Pp (MPa) Inverse Pore Pressure (MPa-1) Simple Effective Stress (Pc – Pp) (MPa) Effective Stress  (Pc – 0.8Pp) (MPa) 22.41 1.72 0.58 20.68 21.03 22.98 2.3 0.43 20.68 21.14 24.13 3.45 0.29 20.68 21.37 27.58 6.89 0.15 20.68 22.06   30   Figure 2.9 Expected result of α values not equal to 1 on the shape of Klinkenberg plots. This figure is a schematic with calculated synthetic permeability data and the stress conditions in Table 2.4. (A) Deviation from linearity resulting from a nonlinear increase in effective stress with decreasing inverse pore pressure (stress effects). (B) Pore structure is different at each different effective stress. If the effective pore size decreases, more gas slippage take place (crosses). If effective pore size increases, less gas slippage takes place (triangles). These are flow regime effects. 𝝈𝒆𝒇𝒇 is effective stress.  Theoretical studies have been published predicting deviation from linearity unrelated to the changes in effective stress discussed above. Fathi et al. (2012) predicted concave-up Klinkenberg plots, whereas Moghadam and Chalaturnyk (2014) predicted concave-down plots. Neither of these predictions is conclusively supported by the data collected in this study, as both subtly concave-up and subtly concave-down Klinkenberg plots have been observed (Figure 2.4). It is possible that scatter in the present data is obscuring these trends, as is likely also true for the stress effects. Nevertheless, if these effects exist, we conclude their magnitude is small for the samples and experimental conditions of this study and the effects do not detract from utilizing gas slippage measurements to characterize shale pore structure.  31  2.5.2 Slippage Measurement Precision Repeated slippage measurements on single subsamples cluster tightly together (Figure 2.7) relative to the variation in slippage between subsamples (Figure 2.6) and for single subsamples at different stress states (Figure 2.5). The fact that two different 3 cm diameter by 3 cm length plugs from a single sample plot in a relatively tight cluster (Figure 2.7) indicates that, for this particular sample, one subsample is adequate to characterize the sample’s pore structure for flow parallel to bedding using slippage measurements. These results provide confidence in the gas slippage measurements and therefore the information about pore structure deduced from slippage data.  2.5.3 Slippage Measurements The most striking observation from the slippage measurements (Figure 2.5 and Figure 2.6) is the high degree of variability each individual subsample has from one stress state to the next (single subplots of Figure 2.5) while still maintaining the overall trend seen when all slippage measurements are placed on a single log-log plot (Figure 2.6). With increases in applied stress, the intuitive expectation would be all pores decreasing in size, and therefore slippage increasing at each step. However, of the twelve stress steps that the data represent, four resulted in decreases in slippage and therefore increases in effective pore size. This counterintuitive result is attributed to smaller pores most responsible for slippage being cut off from the effective flow paths at high stresses rather than pores increasing in size. For the cases where slippage decreased after pre-stressing (TEFB7Pd, TEF21Pll, and TEF21Pd), it is unclear if inelastic deformation is concentrated in smaller pores, or if pre-stressing opened up previously closed fractures. Decrease in slippage with increase in stress was also reported by Soeder (1988) for a Marcellus Shale 32  sample. The Marcellus sample had near zero liquid saturation (the sample was dried to remove pore water and tested to confirm no liquid hydrocarbon saturation) and produced linear Klinkenberg plots. Soeder also measured slippage on Huron Shale samples with very high liquid hydrocarbon saturations and low effective porosity to gas (<0.18% confined porosity to nitrogen). The result was Klinkenberg plots with very poor fits and dependence of permeability on differential pressure for the Huron Shale. Soeder attributed these results to capillary blockage resulting from the high liquid saturations, and concluded a mobile liquid phase existed during gas permeability measurements. Liquid hydrocarbons were observed on the downstream end of permeameter end caps after gas permeability measurements, supporting his conclusion of a mobile liquid phase.  Samples in the present study were not dried and have approximate liquid saturations determined from the difference between dried and as received helium pycnometry of 15 to 47 percent and effective gas porosity of 4.6 to 5.3% (table 2.2). Due to pore compressibility, liquid saturation should be higher and effective gas porosity lower when samples are under confining stress. Confined porosity to helium measured on TEFB7Pll at 20.68MPa (3000psi) simple effective stress was 3.6% (compared to 4.6% for the crushed sample at ambient stress). Even though the samples in this study have significant liquid saturation, effective gas porosities are much higher than Soeder’s Huron Shale samples. Samples did not change weight when measured before and after being in the permeameter, and no liquid was observed on the downstream permeameter end caps when samples were removed. Linear Klinkenberg plot fits for the samples in the present study are similar in quality to those for Soeder’s Marcellus Shale sample, and no dependence of permeability on differential pressure was observed for the Eagle Ford samples. These lines of evidence suggest a mobile liquid phase 33  did not exist in our experiments, and that our data should be comparable to Soeder’s Marcellus Shale sample.  Significant differences are also found between subsamples of single samples. For example, TEF21 shows a high degree of permeability and slippage anisotropy, and is far less sensitive to stress parallel to bedding than perpendicular to bedding (Figure 2.6). When removed from the permeameter, subsample TEF21Pll was still coherent, but a single fracture oriented parallel to gas flow was observed (this became obvious when the face of the sample was dampened with water). The slippage data suggest that this fracture has an effective pore size of ~2000 nm as calculated using the slot shaped pore model (Equation 2.2). The intact matrix, which sample TEF21Pd is taken to more closely represent, has an effective pore size of ~100 nm, as calculated using the same formula. This indicates that not just tortuosity, but additionally or possibly more so, pore size controls permeability anisotropy between the fracture and the matrix. The lower degree of stress sensitivity for the fracture suggests the asperities on the fracture planes are better able to resist permeability degradation as a result of stress increases than the intact matrix.  Subsamples of the other two samples analyzed in this study show far less permeability anisotropy, and the anisotropy is approximately of the same magnitude for both samples (Figure 2.10). However, sample TEFB7 displays a much higher degree of slippage anisotropy than does sample TEF17. This suggests that permeability anisotropy is largely the result of tortuosity effects for sample TEF17, whereas the larger effective pore size for flow parallel to bedding in comparison to perpendicular to bedding for sample TEFB7 is responsible for the permeability anisotropy observed in that sample. 34   Figure 2.10 Permeability and slippage anisotropy comparison of TEFB7 and TEF17. Both samples show similar magnitude of permeability anisotropy, but TEFB7 shows much less slippage anisotropy than TEF17. All measurements in this plot are at 6.89 MPa (1000 psi) simple effective stress after pre-stressing.  When a single sample is subsampled, it is assumed that the properties of both samples, including pore structure, are identical. For this reason, different slippage values for two subsamples at the same stress but with different flow orientations might seem problematic. However, gas molecules traveling perpendicular to bedding are forced to go through each of the millimeter scale laminae (Figure 2.3), whereas molecules traveling parallel to bedding can remain within one of those layers that are more continuous laterally than vertically for the entire length of the core. Slippage measurements are weighted to the portion of the effective porosity most responsible for flow. Hence, flow parallel to bedding will be more characteristic of higher permeability laminae whereas flow perpendicular to bedding will be more characteristic of lower 35  permeability laminae. Consideration of the scale of heterogeneity is therefore important when determining what portion of the sample is being characterized by permeability and slippage measurements.  The slippage measurements in this study display a convincing trend in permeability-slippage space closely matching those found in other studies (Figure 2.11). However, even for just the limited sample suite in this study, there is much scatter about the trend; permeability can vary more than two orders of magnitude at a given slippage value, and slippage approximately one order of magnitude at a given permeability, as is illustrated by the dashed lines in Figure 2.6. Thus effective pore size as calculated using either Equation 2.2 or Equation 2.3 can vary by a factor of ten for two rocks with the same permeability. The implications of this, in terms of transport properties and therefore production from shale, need to be explored further. The possibility that rocks with similar fabric, composition, and structure might cluster together in permeability-slippage space could explain lithological controls on hydrocarbon production. 36   Figure 2.11 Comparison of slippage measurements from this study to those from previous studies, most of which were for rocks with higher permeability. While Aljamaan et al. (2013) and Heller et al. (2014) used helium as a probing gas, Klinkenberg (1941) and Heid et al. (1950) used air and Sampath & Keighin (1982) used nitrogen. Slippage measurements for air and nitrogen were therefore scaled to the mean free path of helium in order to compare the results.  2.5.4 Comparison to Pore System Characterization using SEM Field emission (FE) SEM allows imaging of pores as small as ~5 nanometers (Chalmers et al., 2012a), and has been applied in attempts to characterize pore structures in many active and prospective shale plays (e.g. Slatt and O’Brien, 2011; Curtis et al., 2012a; Curtis et al., 2012b; Jiao et al., 2014). This technique suffers from the inability to image samples under stress and its resolution limit. The large changes in effective pore size with stress determined in this study 37  using slippage measurements (for instance a five-fold decrease between TEF17Pll at 6.89 and 20.68 MPa simple effective stress (Table 2.3)) indicates that unstressed pores imaged using SEM are not geometrically representative of what exists in the subsurface.   Effective pore sizes calculated in this study (Table 2.3) are above or at the resolution of FESEM (~5 nm). Given that the calculated effective pore size represents some sort of weighted average of all the effective porosity, for samples with effective pore sizes calculated from slippage measurements close to 5 nm, it is possible that a significant portion of the effective porosity is below FESEM resolution. Additionally, 3D imagery using FESEM (e.g. Curtis et al., 2012b) is required to determine what portion of the porosity being imaged is effective porosity. Due to time and cost constraints, typical 3D FESEM studies are restricted to volumes measured in the tens of μm3.  Another limitation of SEM is therefore its lack of scalability (Chalmers at al., 2012a), as it does not capture the heterogeneity inherent to shale. Hundreds of thousands of FESEM 3D volumes would fit into one of the plugs analyzed in this study.  SEM provides some information that slippage analysis cannot. While slippage measurements are characteristic of the effective porosity most responsible for fluid flow, SEM provides a better appreciation of the storage space available to hydrocarbons, albeit incomplete due to the lower limit of FESEM resolution (~5 nm). For instance, from SEM it is possible to distinguish different types of porosity (e.g. intergranular, intragranular, organic), as well as the distribution of different pore sizes (e.g. Jiao et al., 2014). This point is best highlighted by subsample TEF21Pll in this study which had a fracture oriented parallel to flow. Slippage measurements, which in this 38  case characterized the void spaces in the fracture, do not provide insight into the matrix storage properties of that sample, whereas SEM would.  2.5.5 Comparison to Pore System Characterization using MICP MICP is routinely used to estimate the distribution of pore-throat sizes in a sample. However, because commercial Hg porosimeters, such as the one used in this study, are limited to a maximum pressure of ~414 MPa (60 000 psi), only pores with pore throats larger than 3 nm are intruded. Any porosity behind pore throats with diameters less than 3 nm, which can represent a significant portion of shale porosity, is not characterized (Clarkson et al., 2013).   Due to how small the pores are in shale, very high pressures are required before any mercury enters a sample. The stress state a sample is at when intrusion occurs will be dependent on the size of the largest pores. For instance, a sample in which the largest pores are 10 nm will be at a simple effective stress of around 68.95 MPa (10 000 psi) when intrusion first takes place, and will dynamically change throughout the later pressure steps as portions of the porosity are filled with high pressure Hg while others remain void. Void pores will be compressed by the very high capillary entry pressures associated with the smaller pore throat sizes. This could explain why average effective pore sizes calculated using Equation 2.3 at stress states ranging from 6.89 to 20.68 MPa (1000 to 5000 psi) simple effective stress were always greater than the average pore-throat diameter determined using MICP (Figure 2.8). Another explanation could be, due to the weighting of slippage measurements to the pores most responsible for fluid flow, which would likely be the larger pores, slippage analysis would yield higher average effective pore size measurements than MICP.  Comparison of pore size estimates from slippage analysis to pore 39  size estimates from MICP relies on the validity of the models used to convert gas slippage measurements to an equivalent pore size (Equation 2.2 and Equation 2.3). Experimental verification of these models for nanoporous materials would make the comparisons more meaningful.  The amount of sample used for MICP in this study is ~20% the volume of the plugs analyzed in this study. Even though care was taken to get a sample proximal to and therefore representative of the plugs, heterogeneity of the samples coupled with the smaller sample size could cause a discrepancy between the samples being analyzed in each test.  In contrast to MICP, slippage analysis can be used to characterize anisotropy of pore structures and can be applied at the stress states expected during production, both of which are important for evaluating flow properties of a reservoir. However, as with SEM, MICP is better suited for evaluating storage properties.  2.5.6 Comparison to Pore System Characterization Using Low Pressure Gas Adsorption In contrast to SEM and MICP, low pressure gas adsorption can characterize sub-nanometer sized pores (Clarkson et al., 2013). However, as with SEM and MICP, low pressure gas adsorption cannot be applied to stressed samples, it provides no insight into anisotropy, it lacks scalability, and it is not weighted to the pores responsible for flow. One or a combination of these factors likely explains why average effective pore sizes calculated from gas slippage measurements did not always coincide with the dominant pore size in the pore size distributions determined using low-pressure nitrogen adsorption (Figure 2.8). An alternate explanation could be that the small 40  sample size used for low-pressure nitrogen adsorption analyses in comparison to gas slippage measurements, combined with the heterogeneity inherent to these rocks, caused a discrepancy between the samples being analyzed in each test. As with SEM and MICP, low-pressure gas adsorption is better suited for analyzing storage properties than transport properties.  2.5.7 Importance of Gas Slippage on Production Slippage measurements provide valuable information about the efficiency with which gases can be transported in shale at low pore pressures. In the very near wellbore region (skin), a region that is critical to production, reservoir pressure is low enough for the impacts of gas slippage to cause significant increases in apparent permeability. At what pressure gas slippage will become important is dependent on gas composition; dry gas will have a larger mean free path than a mixture of methane and larger gaseous hydrocarbon molecules. Measurements made with helium (such as those in this study) need to be scaled to the mean free path of the gas being produced in order to include the effects of gas slippage in production models.  2.6 Conclusion Klinkenberg gas slippage measurements are a useful tool for analyzing pore structure in shale. They provide insight into the geometry of pores in stressed samples and how pore structures will change with stress during production. Slippage measurements only characterize the effective porosity, which is desirable for evaluating flow and storage properties. Because slippage measurements are derived from permeability measurements, they are weighted to the portion of the effective porosity responsible for fluid flow. This characteristic allows comparison of the geometry of pores and pore throats responsible for flow parallel to bedding to those responsible 41  for flow perpendicular to bedding, but limits the utility of slippage measurements for analyzing storage properties. Slippage measurements in conjunction with SEM, MICP, and low pressure adsorption will collectively contribute to a more complete characterization of pore structures in shale.  The heterogeneity and diversity inherent to shale lends itself to variable responses to stress, likely dependent on myriad of factors including fabric, composition, structure, and flow orientation. In this study, slippage analysis yielded insight into the complex responses of these rocks to stress that other methods of pore structure analysis currently being employed to characterize shale pore systems are blind to. Slippage measurements on three Eagle Ford Shale samples at a range of stress states revealed two orders of magnitude in slippage variation over a five order of magnitude permeability range. Slippage measurements are negatively correlated with permeability and follow similar trends to those found in other studies on higher permeability rocks. The samples had varying degrees of slippage anisotropy, which allowed interpretation of the relative contribution of tortuosity and pore size to permeability anisotropy. Slippage and therefore average effective pore size was found to vary up to one order of magnitude at a given permeability, warranting investigation of the significance this might have on flow properties and ultimately hydrocarbon production from shale. 42  Chapter 3: Quantitative Validation of Pore Structure Characterization Using Gas Slippage Measurements by Comparison with Predictions from Bundle of Capillaries Models  3.1 Introduction Characterizing pore structure geometry is fundamental to understanding and predicting many aspects of porous media behavior. Fully characterizing reservoir rock pore structure is expensive and technically challenging due to the geometrical complexity and small length scale of these media, and is impossible for low permeability reservoir rocks with pore structure length scales below the resolution limits of even the most advanced imaging techniques. Techniques based on simplified models are therefore required for practical applications. Here we investigate the quantitative validity of characterizing pore structure geometry using gas slippage measurements. Compared to other pore structure characterization techniques, such as mercury intrusion porosimetry and CO2 and N2 gas adsorption, the gas slippage technique has the advantageous attribute of being applicable to samples confined at in situ reservoir stress conditions (Letham and Bustin, 2016). This attribute is important because reservoir rock pore structures show varying degrees of stress sensitivity (McLatchie et al., 1958; Vairogs et al., 1971), and pores and pore throats can be significantly smaller at reservoir stress conditions than at ambient surface conditions. Using pore structure characterizations determined at ambient surface stress conditions as inputs for subsurface reservoir models can hence lead to inaccurate predictions of reservoir behavior. It is therefore desirable to use pore structure characterizations determined at subsurface reservoir stress conditions as model inputs.  43  In this study we quantitatively validate the gas slippage technique, which is capable of pore structure characterizations at in situ stress, by comparing pore sizes estimated from gas slippage measurements with pore sizes independently estimated using bundle of capillaries models. The good match found between the techniques provides confidence that pore structure characterizations made using gas slippage measurements can be used as model inputs to understand and predict reservoir behavior. The findings of this study are particularly significant for the successful exploitation of shale gas and shale oil reservoirs that are typically highly stress sensitive and, by definition, have small-scale pore structures that are difficult to characterize with techniques other than gas slippage measurements.  3.2 Background 3.2.1 Gas Slippage In 1941, Klinkenberg published a seminal paper that showed significant pore pressure and gas species dependent permeability variation in conventional reservoir rocks (Klinkenberg, 1941). By varying pore pressure to systematically change mean free path (the average distance a molecule of the probing gas used for permeability measurements travels between two collisions with other gas molecules), he showed that the observed permeability variation was the result of gas slippage; linear relationships between permeability and inverse pore pressure were observed, which were predicted by slip theory because mean free path of a gas is proportional to the inverse of gas pressure. Permeability intercepts of permeability-inverse pore pressure plots, representative of infinite pore pressure and therefore zero mean free path and no gas slippage, were in agreement with liquid permeability measurements (Klinkenberg, 1941). These permeability intercepts were referred to as true permeability in Klinkenberg’s paper, and have 44  since been termed “slip-corrected” or “Klinkenberg-corrected” permeability. 𝐾∞ is used herein to designate permeability from extrapolation to infinite pore pressure.  Gas slippage is a significant control on flow rate when flow takes place in the slip flow regime. The boundaries of the slip flow regime can be defined using the Knudsen number. The Knudsen number is the ratio of mean free path of a gas to the characteristic length scale of the pore structure through which it is flowing (Zhang et al., 2012) 𝐾𝑛 =𝜆𝐿𝑐          (Eq. 3.1) where λ is mean free path and 𝐿𝑐 is characteristic length. Characteristic length is loosely defined in the literature because of the many different possible metrics for pore size (e.g. pore diameter, pore throat diameter). For calculating 𝐾𝑛, the important dimension of the pore structure is the length scale of the restrictions most responsible for limiting fluid flow. Characteristic length is therefore defined as such in this work.   The slip flow regime is defined as 0.001 <  𝐾𝑛 < 0.1 (Zhang et al., 2012). When flow takes place in the slip flow regime, gas molecule-gas molecule collisions are far more frequent than gas molecule-pore wall collisions. However, gas molecule-pore wall collisions are frequent enough that the non-zero flow velocity at the pore walls is significant in comparison to the mean flow velocity (Landry et al., 2016). Hence gas slippage results in significantly λ-dependent permeability in the slip flow regime. In contrast to the Darcy flow regime (𝐾𝑛 < 0.001), permeability is not a constant property of a given pore structure in the slip flow regime, and is therefore referred to as apparent permeability, 𝐾𝑎. 𝐾𝑎 of a given pore structure is controlled by λ, 45  which is dependent on temperature, pressure, and the kinetic diameter of the gas molecules (Loeb, 2004) 𝜆 =  𝑅𝑇√2𝜋𝑑𝑘𝑖𝑛2𝑁𝐴𝑃          (Eq. 3.2) where 𝑅 is the gas constant, 𝑇 is temperature, 𝑑𝑘𝑖𝑛 is the kinetic diameter of a molecule of the gas being considered, 𝑁𝐴 is Avogadro’s constant, and  𝑃 is pressure.  3.2.2 Gas Slippage Measurements as a Pore Structure Characterization Technique Klinkenberg developed an equation to predict 𝐾𝑎 as a function of pore pressure 𝐾𝑎 = 𝐾∞(1 +𝑏𝑃𝑝)          (Eq. 3.3) where 𝑃𝑝 is average pore pressure and 𝑏 is a constant usually referred to as Klinkenberg’s slippage factor. Calculating 𝑏 is a way of quantifying gas slippage, as it is a measure of variation of 𝐾𝑎 with respect to 𝐾∞. 𝑏 is dependent on temperature and gas species (both of which determine λ), as well as the size of the pores through which the gas is flowing. In Klinkenberg’s theoretical derivation of Equation 3.3, he developed the following relationship for 𝑏 𝑏 =4𝑐𝜆𝑃𝑝𝑟          (Eq. 3.4) where 𝑐 is Adzumi’s constant (0.9; Adzumi, 1937) and 𝑟 is capillary radius under the assumption that flow paths can be represented as tubular shaped capillaries.  From Equation 3.4, by calculating 𝑏 using experimental data (Equation 3.3) and knowing the variables determining λ (Equation 3.2), it is possible to calculate 𝑟. Hence gas slippage measurements can be used as a technique for characterizing pore structures. A single 𝑟 value calculated from slippage measurements clearly does not completely characterize the pore 46  structure geometry of a reservoir rock; pore structures are composites of a distribution of different sized pores and pore throats. The calculated 𝑟 represents an average of the smallest pore throats along those flow paths responsible for the bulk of the fluid flux through the porous medium. These pore throats limit the fluid flux. Change in 𝐾𝑎 relative to 𝐾∞ due to gas slippage when mean free path is varied, which is quantified by measuring 𝑏, is therefore controlled by gas slippage at the walls of these smallest pore throats. Hence, 𝑟, which is calculated from 𝑏, is a measure of the size of these smallest pore throats, based on a simplified model that represents pores as cylindrical capillary tubes. Pore (throat) sizes calculated from gas slippage measurements will be referred to as “dominant” pore sizes herein.   That gas slippage measurements only result in the calculation of a dominant pore size is a limitation of the gas slippage technique as compared to techniques capable of measuring a distribution of different pore sizes (e.g. mercury intrusion porosimetry, CO2 and N2 adsorption, and imaging with scanning electron microscopy). However, a key attribute of gas slippage measurements as a pore structure characterization technique is that, because the dominant pore sizes are derived from permeability measurements, the technique allows direct characterization of the pore sizes relevant for fluid flow. Another advantage of gas slippage measurements as a pore structure characterization technique is the ability to easily make measurements at reservoir stress conditions. This is especially important for lower permeability, fine-grained reservoir rocks, which typically have more stress sensitive permeability than higher permeability, coarse-grained rocks (McLatchie et al., 1958; Vairogs et al., 1971). Additionally, the stressed permeability measurements made when quantifying gas slippage are valuable petrophysical data 47  on their own. Letham and Bustin (2016) have provided in depth discussion of the strengths and weaknesses of gas slippage measurements as a pore structure characterization technique.  Assuming a tubular geometry for flow paths, as is the case in Equation 3.4, is unrealistic for reservoir rocks; flow paths transect irregularly shaped pores and pore throats that vary in size. The complex geometry not captured by this simplified geometrical assumption could have an impact on 𝑏 due to capillary end effects not accounted for in Equation 3.4 (Klinkenberg, 1941). The significance of the simplified tubular geometry assumption for pore structure characterization using gas slippage measurements in our study is discussed later (subsection 3.4.4).  Based on images of pores, it has been suggested that for some reservoir rocks slot-shaped capillaries would be a better approximation of pore geometry (e.g. Devonian shales in Soeder, 1988). An equation was developed by Randolph et al. (1984) for calculating pore size from gas slippage measurements assuming slot-shaped capillaries 𝑤 =16∗𝑐∗𝜇𝑏∗ √2𝑅𝑇𝜋𝑀          (Eq. 3.5) where 𝑀 is the molecular weight of the gas and 𝑤 is dominant slot width. As with Equation 3.4 which assumes a tubular geometry, Equation 3.5 does not account for capillary end effects.  3.2.3 Bundle of Capillaries Models For more than a century, researchers have been developing and refining models that represent pore structures as bundles of capillaries (Carman, 1937). The advantage of bundle of capillaries 48  models is that the simplified geometry of the models facilitates applying Poiseuille’s equation to calculate volume flow rate through each individual capillary. Volumetric flow rate through capillaries of different shapes (e.g. slot-shaped or triangular capillaries) can be calculated using simple variations of Poiseuille’s equation. By analogy with Darcy’s law, permeability of a bundle of capillaries can be calculated by summation of volumetric flow rate through all capillaries in a unit volume. For tubular capillaries, permeability is (Al Ismail and Zoback, 2016) 𝐾∞ =𝛷𝑟28𝜏2          (Eq. 3.6) where 𝛷 is porosity. For slot shaped capillaries, permeability is (Randolph et al., 1984) 𝐾∞ =𝛷𝑤212𝜏2          (Eq. 3.7). Equation 3.6 and Equation 3.7 can be used to estimate pore size (𝑟 in Equation 3.6 and 𝑤 in Equation 3.7) from measurements of 𝐾∞, 𝛷 and τ.  3.3 Methods and Materials Equations 3.4 and 3.6 or equations 3.5 and 3.7 provide separate means for estimating pore size. Bundle of capillaries models (the basis of Equation 3.6 and Equation 3.7) have been critically analyzed and refined over the last century to produce the sophisticated Kozeny-Carman type models currently in use (e.g. Civan, 2011). Agreement between pore size characterizations made using gas slippage measurements (Equation 3.4 or Equation 3.5) and bundle of capillaries models would be an indication that gas slippage measurements provide quantitatively accurate pore structure characterizations. To test for this agreement, we here compile a large data set of gas slippage measurements and compare dominant pore sizes calculated from the compiled data set to the pore size-𝐾∞ trends predicted by Equation 3.6 and Equation 3.7. 49   3.3.1 Gas Slippage Measurements The compiled data set of gas slippage measurements used in this study includes measurements made in our laboratory and data from seven previously published studies. The gas slippage measurements made in our laboratory are for a suite of fine grained hydrocarbon reservoir rock samples from the Montney, Doig, Duvernay, and Muskwa formations of the Western Canada Sedimentary Basin, and the Eagle Ford Shale from Texas, USA. Measurements were made on 18 plugs at stress states as low as 6.9 MPa and as high as 48.3 MPa simple effective stress (𝑃𝑐 − 𝑃𝑝, where 𝑃𝑐 is confining pressure). Some plugs were oriented for gas flow perpendicular to bedding and some for flow parallel to bedding. For each slippage measurement, 𝐾𝑎 to helium at 18 °C was measured at 1.72, 2.30, 3.45, and 6.89 MPa 𝑃𝑝 using the pulse decay technique (Brace et al., 1968; Cui et al., 2009). 𝐾𝑎 was then plotted against inverse 𝑃𝑝 (Klinkenberg plot), and the slope and 𝐾𝑎 intercept of the plot were used to calculate 𝑏 using Equation 3.3.  Listed in Table 3.1 are the seven previously published studies from which data were taken to assemble the compiled gas slippage data set. Important details of each study are summarized in the table. Because not all studies employed the same probing gas at the same temperature, 𝑏 conversion factors (last column of Table 3.1) were calculated to convert 𝑏 measurements from the previous studies to what theoretically would have been measured with helium at 18 °C (the measurement conditions of the data collected in our laboratory). The calculated 𝑏 conversion factors are the ratio of λ of helium at 18 °C to the λ of the probing gas used in each previously published study at the temperature of the permeability measurements, both calculated for the 50  same gas pressure. λ for nitrogen, argon, and helium were calculated using Equation 3.2 and kinetic diameters of 0.364 nm, 0.358 nm, and 0.260 nm respectively. Air is a mixture of different gas molecules with different kinetic diameters, so λ cannot simply be calculated using Equation 3.2. Values for λ of air were therefore sourced from the literature (Jennings, 1988). Table 3.1 Important details of all studies used to generate the compiled data set of gas slippage measurements used in the current study, as well as 𝒃 conversion factors used to standardize gas slippage measurements to helium at 18 °C. Study Year Probing gas Temperature (°C) Data points Porosity data 𝒃 conversion factor** Klinkenberg 1941 air 21* 8 no 1.97 Heid et al. 1950 air 26 162 yes 1.94 Jones 1972 nitrogen 21* 99 no 1.94 Sampath and Keighin 1982 nitrogen 21* 19 yes 1.94 Randolph et al. 1984 nitrogen 82 27 yes 1.61 Tanikawa and Shimamoto 2009 nitrogen 21* 330 yes 1.94 Al Ismail and Zoback 2016 argon 38 26 no 2.00 this study ----- helium 18 48 yes 1.00 * room temperature assumed to be 21 °C ** for conversion to Klinkenberg’s slippage factor (𝑏) of helium at 18°C  The previously published studies were performed on a wide range of different reservoir rocks. These rock types are briefly summarized below, as well as other important information about the studies that is not found in Table 3.1. The full dataset is reported in Appendix A.  Klinkenberg (1941) used core samples from what was described as a large variety of fields; exactly which fields were not stated in the paper. Data for their core samples A through H were used in our study (their Table 8).  51  Heid et al. (1950) made measurements on what they called representative sands from different oil-producing areas of the USA. 𝐾∞, 𝑏, and 𝛷 data were taken from their Table 2. Geographical sample locations are listed in their same table.  Jones (1972) made measurements on core plugs ranging in permeability from about 0.01 to 1000 mD. The source of the samples was not stated. 𝐾∞ and 𝑏 values were not presented in a table in their study, so we digitized the data presented in their Figure 9.   Sampath and Keighen (1982) made measurements on Late Cretaceous, low-permeability, fine to very-fine grained sandstones from Uinta County, Utah. Their samples were collected from a restricted depth interval (2512.6 to 2549.9 m) of a single well. Measurements were made at low and high stress states. 𝐾∞, 𝑏, and 𝛷 data were taken from their Table 2.  Randolph et al. (1984) made measurements on western tight sands. All samples were from depths known or expected to be gas producers. 𝐾∞, 𝑏, and 𝛷 data for Mesa Verde samples from four different depositional environments are presented in their Table 3. We used their Mesa Verde data in our data set.  Tanikawa and Shimamoto (2009) made measurements on a suite of outcropping Pleistocene to Miocene sedimentary rocks from the western foothills of Taiwan. 𝐾∞ and 𝑏 values were not presented in a table in their study, so we digitized the data presented in their Figure 7.  52  Al Ismail and Zoback (2016) presented 𝐾∞ and 𝑏 data for a small suite of Utica Shale samples (4 samples) and Permian Shale samples (2 samples). 𝐾∞ and 𝑏 measurements for each sample were made at multiple stress states, and some plugs were oriented for gas flow perpendicular to bedding. Klinkenberg plot permeability intercept (𝐾∞) and 𝑏 data are tabulated in their Table 3.  3.3.2 Scanning Electron Microscopy Field emission scanning electron microscopy (FESEM) was used to investigate the pore morphology of one sample from our gas slippage data set. The pore morphology observed in the FESEM images was compared to the pore morphology predicted from the analysis of gas slippage measurements. Backscattered electron images were collected using a FEI Helios NanoLab 650™ dual beam system. Ultra-polished surfaces were prepared for imaging using focused ion beam milling with gallium ions.  3.4 Results and Discussion In this section we first present the compiled gas slippage data set and discuss trends of the individual data sets in 𝑏 - 𝐾∞ space, as well as the trend of the compiled data set as a whole (Subsection 3.4.1). In Subsection 3.4.2 we convert the gas slippage measurements to dominant pore size estimates using the different geometrical models (Equation 3.4 for cylindrical pores and Equation 3.5 for slot shaped pores). Then we estimate pore size using bundle of capillaries models (Equation 3.6 and Equation 3.7) and compare these pore sizes to the pore sizes calculated from the gas slippage data in Subsection 3.4.3. Implications of the results of the analysis are discussed in Subsection 3.4.4.  53  3.4.1 Compiled Gas Slippage Data Set The compiled data set spans nine orders of magnitude of permeability and five orders of magnitude of 𝑏 (Figure 3.1). The data show a trend of increasing 𝑏 with decreasing 𝐾∞. This trend is expected because finer-grained, lower-permeability rocks typically have smaller pores than coarser-grained, higher-permeability rocks, and gas slippage causes more λ-dependent permeability in rocks with smaller pores (and hence higher 𝑏 values). 54   Figure 3.1 Compiled gas slippage data set in 𝒃 - 𝑲∞ space coloured by which study the data are from. 𝒃 measurements from all datasets were standardized to the measurement conditions in our laboratory (helium at 18 °C).  Six of the eight individual data sets that comprise the compiled data set have power law fits with consistent slopes in log(𝑏) – log(𝐾∞) space (Figure 3.2). The exceptions are the Sampath and Keighin data set with a steeper slope and the Al Ismail and Zoback data set with a shallower slope. These exceptions are for two of the data sets with the fewest number of samples; the 55  Sampath and Keighin data set is for 19 measurements on ten samples from a ~ 37 m depth range in a single well, and the Al Ismail and Zoback data set is for 26 measurements on six samples (see Table 3.1 for comparison with the other data sets). The limited sample suites could be why these two exceptions have different slopes than the consistent slope of the other data sets.   Figure 3.2 Power law fits to the individual data sets comprising the compiled data set. Circular markers truncating trend lines designate the permeability span of the individual data sets and are coloured by study. 𝒃 measurements from all datasets were standardized to the measurement conditions in our laboratory (helium at 18 °C).  56  Of the six data sets with consistent slopes, four fall on a nearly identical trend. The other two are offset from this trend, with the Heid et al. data set showing slightly higher 𝑏 values, and the data set collected in our laboratory showing slightly lower 𝑏 values. These offsets could be an artifact of converting 𝑏 values from other studies to the permeability measurement conditions in our laboratory (helium at 18 °C). These conversions are dependent on recalculations of λ assuming the gas can be modelled as an ideal gas. Alternately, these offsets could reflect systematic differences of the sample suites (e.g. lithological differences). Overall, each data set shows significant overlap with other data sets (Figure 3.1) and the compiled data set forms a well-defined, consistent trend in log(𝑏) – log(𝐾∞) space.  3.4.2 Pore Size Estimates from Gas Slippage Data Dominant pore sizes calculated from gas slippage measurements assuming a tube shaped geometry (Equation 3.4) range from 3 nm at ~10-6 mD 𝐾∞ to 70 μm at ~103 mD 𝐾∞ (Figure 3.3A). Dominant pore sizes calculated from gas slippage measurements assuming a slot shaped geometry (Equation 3.5) range from 5 nm at ~10-6 mD 𝐾∞ to 130 μm at ~103 mD 𝐾∞ (Figure 3.3B). Power law fits to each pore size - 𝐾∞ trend (red lines in Figure 3.3) have the same slope.   57   Figure 3.3 Dominant pore sizes calculated from gas slippage measurements in the compiled data set assuming a tubular geometry (black circles in A) and a slot shaped geometry (black circles in B). Red lines are power law fits to the data sets. Green lines (discussed later in the text) are trends of pore size calculated using a bundle of tube shaped capillaries model (A) and a bundle of slot shaped capillaries model (B), both coupled with porosity and tortuosity functions. Most of the rocks in the compiled data set with the smallest calculated dominant pore sizes are mudrocks that were analyzed in our laboratory, with calculated pore sizes in the nanometers to tens of nanometers length scale. Mudrocks are a very broad, diverse rock type, but pore structure characterization studies using field emission scanning electron microscopy have revealed pore size distributions of the same length scale (e.g. Curtis et al., 2012). Consistency between such studies and dominant pore sizes calculated from gas slippage measurements is an indication that gas slippage analysis yields plausible pore size results.   The rocks in the compiled data set with the largest calculated dominant pore sizes (tens of micrometres) are sandstones. Optical techniques have shown that pores in high-permeability sandstones (e.g. Berea sandstone with permeabilities typically in the range of 102 to 103 mD) are 58  of the same length scale (e.g. Øren and Bakke, 2003), again indicating that gas slippage analysis yields plausible pore size results.  The above comparison of pore size estimates from gas slippage measurements to estimates based on imaging techniques only provides very weak validation of the accuracy of the gas slippage technique. In the following subsection, we seek to establish a more robust quantitative validation of gas slippage measurements as a pore structure characterization technique by comparing the trends of the compiled gas slippage data set with trends predicted by bundle of capillaries models.  3.4.3 Comparison with Pore Size Estimates from Bundle of Capillaries Models Pore size - 𝐾∞ trends calculated using the bundle of tubes model (Equation 3.6) with geologically reasonable input values of porosity and tortuosity for reservoir rocks are presented in Figure 3.4. Geologically reasonable porosity was determined to be greater than 0.5% and less than 40%, based on reported values for reservoir rocks found in a literature survey (e.g. Ehrenberg and Nadeau, 2005). Geologically reasonable tortuosity was determined to be greater than 1.41 and less than 10, based on a modelling study (Guo, 2015). The modelling study was used instead of laboratory data because direct measurements of tortuosity, especially for low permeability rocks, are sparse in the literature. This tortuosity model is discussed in more detail in Subsection 3.4.3.2.  Superimposed on the pore size - 𝐾∞ trends are dominant pore sizes calculated from the compiled gas slippage data set assuming cylindrical-shaped pores (black circles in Figure 3.4). Most data fall within the space defined by the curves for geologically reasonable porosities and tortuosity. At high permeability, some pore sizes calculated from gas 59  slippage measurements fall in space that would require geologically unrealistic tortuosity (less than 1.41) and porosity (greater than 40%) to be accommodated by the bundle of tubes model. However, all but 11 of these high permeability data points fall within the space defined by the curves for geologically reasonable porosity and tortuosity when capillaries are modelled as slot-shaped in cross section and dominant pore sizes from gas slippage measurements are calculated using Equation 3.5 for slot-shaped pores (Figure 3.5). Overall, 99% of the dominant pore sizes calculated from the 722 gas slippage measurements assuming either slot shaped or tubular cross sectional geometry can be captured by at least one of the simple bundle of capillaries models (Equation 3.6 or Equation 3.7) with geologically reasonable inputs of porosity and tortuosity (Figure 3.6). This is quantitative validation of the gas slippage pore structure characterization technique. 60   Figure 3.4 Relationships between 𝑲∞ and tube diameter (calculated using Equation 3.6) for geologically reasonable ranges of 𝜱 and τ. Superimposed are dominant pore sizes of the compiled gas slippage data set calculated assuming a cylindrical shaped geometry (black dots). The thicker, solid red line is a power law fit to these data. 𝑲∞ - tube diameter relationships for tortuosities between 1.41 and 10 and porosities between 0.5% and 40% would fall within the space bounded by the upper, brown, dashed line (𝜱 = 𝟎. 𝟓% and 𝝉 =𝟏𝟎) and the lower, pink, solid line (𝜱 = 𝟒𝟎% and 𝝉 = 𝟏. 𝟒𝟏). 61   Figure 3.5 Relationships between 𝑲∞ and slot-shaped capillary width (calculated using Equation 3.7) for geologically reasonable ranges of 𝜱 and τ. Superimposed are dominant pore sizes of the compiled gas slippage data set calculated assuming a slot-shaped geometry (black dots). The thicker, solid red line is a power law fit to these data. 𝑲∞ - slot width relationships for tortuosities between 1.41 and 10 and porosities between 0.5% and 40% would fall within the space bounded by the upper, brown, dashed line (𝜱 = 𝟎. 𝟓% and 𝝉 = 𝟏𝟎) and the lower, pink, solid line (𝜱 = 𝟒𝟎% and 𝝉 = 𝟏. 𝟒𝟏).  62   Figure 3.6 Portion of the gas slippage dataset for which calculated pore sizes can be captured by at least one of the simple bundle of capillaries models with geologically reasonable inputs of porosity and tortuosity. 99% of the data can be captured by at least one of the models.  No single combination of porosity and tortuosity results in a trend that matches the trend of the gas slippage data set (thick red line in Figures 3.4 and 3.5). Such results are anticipated since porosity and tortuosity systematically vary with permeability (as shown below). Next we therefore generate porosity-permeability and tortuosity-permeability functions to use as inputs for the bundle of capillaries models to see if agreement exists between those models and gas slippage measurements.  63  3.4.3.1 Porosity-Permeability Function Five of the eight data sets used to generate the gas slippage data set include porosity measurements for some or all samples (Table 3.1). The porosity data suggest a trend of decreasing porosity with decreasing slip-corrected permeability (Figure 3.7). A power law function fit to the data provides a tool to make realistic porosity predictions over the wide permeability range. The scatter about the fit and differing trends for individual data sets of measurements on restricted sample suites reflect lithologic variation (i.e. two different rocks with the same porosity might have different pore size and therefore different permeability). The power law fit is the porosity function used later (subsection 3.4.3.3) to determine inputs for the bundle of capillaries models. The porosity function, determined based on a limited data set collected using various techniques, is not a robust function, but establishes a general trend of porosity over the measured range of permeability.  64   Figure 3.7 Porosity data from the studies in the compiled data set that included porosity measurements. The black curve is a power law fit to the data.  3.4.3.2 Tortuosity-Permeability Function No tortuosity data were collected in any of the studies that make up the compiled gas slippage data set. Measuring tortuosity in the laboratory is difficult (Laudone et al., 2015). We therefore used a theoretical study (Guo, 2015) supported by experimental data (Winsauer et al., 1952) to derive the tortuosity-permeability function used to calculate inputs for the bundle of capillaries models.  Guo (2015) presented theoretical, porosity-dependent upper and lower bounds for the tortuosity of isotropic porous materials with random distributions of solid particles and voids (Figure 3.8). 65  Experimental tortuosity measurements for the reservoir rocks studied by Winsauer et al. (1952) plot within or very near the theoretical upper and lower tortuosity bounds and therefore support the theoretical model. Reservoir rocks are generally anisotropic. The rocks in the compiled data set of gas slippage measurements likely have varying degrees of tortuosity anisotropy, and some measurements were made for gas flow parallel to bedding and some measurements for gas flow perpendicular to bedding. In the absence of any measurements of tortuosity or tortuosity anisotropy for the reservoir rocks in the compiled data set, we adopt the isotropic tortuosity model for our analysis while acknowledging that this is a simplification.    Figure 3.8 Upper (red) and lower (green) bounds of tortuosity as a function of porosity calculated using the theoretical predictions of Guo (2015). The dashed blue curve is a trend between the two bounds and is used to calculate inputs for the bundle of capillaries models considered later. Black circles are experimental measurements from Winsaur et al. (1952). 66  The dashed blue curve in Figure 3.8 is the tortuosity-porosity function we use for this study. To determine inputs for the bundle of capillaries models to calculate pore size as a function of permeability, we require a tortuosity-permeability function. The function for the tortuosity-porosity curve is here combined with the porosity-permeability function developed previously (Figure 3.7) to generate the tortuosity-permeability relationship used to calculate inputs for the bundle of capillaries models (dashed blue curve in Figure 3.9). The tortuosity-permeability laboratory data of Winsauer et al. (1952) fit within or near the theoretical upper and lower tortuosity bounds, indicating that the theoretical model combined with our porosity-permeability function yields reasonable values. We are unaware of any reliable tortuosity measurements for low permeability reservoir rocks that could be used to validate the low permeability end of this curve.   67   Figure 3.9 Tortuosity-permeability relationship developed by combining the porosity-permeability function developed using porosity measurements made on samples from the compiled gas slippage data set (Figure 3.7) with Guo’s (2015) theoretical tortuosity-porosity relationship (Figure 3.8). Black circles are experimental measurements from Winsauer et al. (1952).  3.4.3.3 Bundle of Capillaries Models Using Porosity and Tortuosity Functions The bundle of cylindrical capillary tubes model (Equation 3.6) using porosity and tortuosity functions yields a pore size-permeability trend (green line in Figure 3.3A) similar to the trend for dominant pore sizes calculated from gas slippage measurements assuming a cylindrical pore shape (red line in Figure 3.3A). The trend falls within the scatter of the experimental gas slippage data (black circles). The best agreement between the trends exists at the lowest permeability, where pore size predicted by the bundle of tubes model is 1.1 times higher than pore size from gas slippage measurements. The worst agreement between the trends exists at the highest 68  permeability, where pore size predicted by the bundle of tubes model is 2.6 times higher than pore size from gas slippage measurements.   The bundle of slot shaped capillaries model (Equation 3.7) using porosity and tortuosity functions also yields a pore size-permeability trend (green line in Figure 3.3B) similar to the trend for dominant pore sizes calculated from gas slippage measurements, this time assuming slot shaped pores (red line in Figure 3.3B). The trend falls within the scatter of the experimental gas slippage data (black circles). The best agreement between the trends exists at the highest permeability, where pore size predicted by the bundle of slot-shaped capillaries model is 0.9 times the pore size from gas slippage measurements. The worst agreement between the trends exists at the lowest permeability, where pore size predicted by the bundle of slot-shaped capillaries model is 0.4 times the pore size from gas slippage measurements.  3.4.4 Significance of the Pore Size-Permeability Trends Neither the cylindrical nor slot shaped models result in agreement over the entire permeability range between the trends of experimental gas slippage data and bundle of capillaries models (red and green lines in Figure 3.3). The discrepancies in these results, however, are tolerable for many practical applications, especially for fine-grained reservoir rocks where no viable alternative exists for more accurate measurements of pore size at reservoir stress. Possible explanations for the discrepancies include: (1) error in the porosity function (Figure 3.7), which was generated from a data set collected using various techniques (some at ambient stress, some at reservoir stress) and only includes measurements for some of the samples in the compiled data set; (2) error in the tortuosity function (Figure 3.8), which is based on a simple model for isotropic media 69  and includes compound error from combining the porosity-permeability function with the tortuosity-porosity function to obtain the tortuosity-permeability function (Figure 3.9); and (3) only considering two very simple geometries (cylindrical tubes and slot shaped capillaries). Considering these sources of error, lack of statistical agreement over the entire permeability range between either model and the gas slippage data is expected. In addition to the fact that 99% of the gas slippage data result in pore size estimates that are consistent with bundle of capillaries models with geologically plausible inputs of porosity and tortuosity (Figure 3.6), the broadly similar trends of the gas slippage data and the bundle of capillaries models using porosity and tortuosity functions (Figure 3.3) are viewed as further quantitative validation of the gas slippage technique.  At high permeability, better agreement occurs when pores are modelled as slot-shaped in cross sectional geometry than when modelled as circular in cross sectional geometry, which, by extension, suggests high permeability natural reservoir rock pores are often more accurately modelled as slot-shaped. The opposite is true at low permeability, where better agreement is found when pores are modelled as circular in cross section. This could reflect the increased quantity and contribution to flow capacity of organic matter porosity in low permeability rocks; imaging studies have suggested that organic matter porosity is more circular in cross section than other forms of porosity (Loucks et al., 2009; Jiao et al., 2014). Conclusive statements about pore morphology, however, cannot be drawn based on this analysis, which only considers two different geometries.  70  FESEM was employed to assess the plausibility of the hypothesis that increased quantity and contribution to flow capacity of circular organic matter porosity is responsible for better agreement between gas slippage measurements and the bundle of capillaries model when pores in low permeability rocks are modelled as cylindrical. We imaged porosity in a Montney Formation sample for which pore sizes calculated from gas slippage measurements fall directly on the trend predicted by the bundle of tubes model when pores are modelled as circular in cross section (Figure 3.10). Most of the porosity occurs as intraparticle porosity in organic matter, and the morphology of the organic matter porosity is better described as circular in cross section than slot shaped (Figure 3.11), indicating the hypothesis is plausible. However, no conclusive statements about the plausibility of the hypothesis can be drawn from the FESEM images because: (1) the analysis was limited to an extremely small portion of a single sample; (2) the analysis is in two dimensions and therefore says nothing of the connectivity of the porosity; (3) it is not possible to tell which are the smallest pore throats along those flow paths responsible for the bulk of the fluid flux, and therefore are the pores that would be characterized by gas slippage measurements; and (4) the FESEM measurements made at ambient stress are being compared to the gas slippage data collected on stressed samples that therefore have different pore structure.  71   Figure 3.10 Pore sizes calculated from gas slippage measurements for flow perpendicular to bedding at different simple effective stress states (SES in legends) for the same Montney Formation sample imaged in Figure 3.11. (A) Pore sizes calculated assuming a tubular geometry fall on the trend predicted by the bundle of tube shaped capillaries model using porosity and tortuosity functions. (B) Pore sizes calculated assuming a slot shaped geometry do not fall on the trend predicted by the bundle of slot shaped capillaries model using porosity and tortuosity functions. 72   Figure 3.11 Backscatter electron images of a Montney Formation sample at ambient stress, oriented such that the imaged surfaces are along bedding planes (gas flowing through the imaged pores would be travelling perpendicular to bedding). For the section imaged (Panel A), most porosity occurs as intraparticle porosity within organic matter (dark grey in images). Less porosity occurs as intraparticle porosity within mineral grains, or as inter-granular porosity, than as intra-granular organic matter porosity. Panels B and C show higher magnification images of the organic matter pores, which are more circular in cross section than slot shaped. Most organic matter pores are tens of nanometers in diameter. The biggest organic matter pore is ~150 nm in diameter (center of Panel C).  Inaccurate pore size estimates from gas slippage measurements would occur if capillary end effects caused significant deviation of apparent permeability from that predicted for a tube of 73  infinite length (Klinkenberg, 1941; Landry et al., 2016). The capillary end effects would result in disagreement between pore sizes predicted by bundle of tubes models and dominant pore sizes calculated from gas slippage measurements. The analysis presented in Landry et al. (2016) quantifies the effect of finite length capillaries on apparent permeability. Their analysis predicts significant finite length effects at high Knudsen number (early transition flow regime) and low pore length to pore radius ratios (ratios less than 2). Significant end effects would result in variation of 𝑏 with pore pressure, and therefore nonlinear Klinkenberg plots (Klinkenberg, 1941). No deviations from linearity that could not be explained by random error in individual permeability measurements were observed in Klinkenberg plots for the samples analyzed in our laboratory (e.g. Figure 2.4 in Chapter 2). Lack of deviation from linearity and the observed agreement between pore sizes predicted by bundle of tubes models and calculated from gas slippage measurements suggests capillary end effects did not significantly impact pore size calculations. Thus, the measurements were made in the slip flow regime and not the early transition flow regime, and/or the reservoir rocks had large pore length to pore radius ratios.  Although no deviations from linearity were observed in Klinkenberg plots for the samples analyzed in our laboratory, for our lowest permeability samples, calculated pore sizes are small enough that flow could be in the transition flow regime when permeability measurements are made at low pore pressures. If flow is in the transition regime, permeability would vary nonlinearly with inverse pore pressure (Moghadam and Chalaturnyk, 2013: Figure 2; Moghaddam and Jaminolahmady, 2017: Figure 18), resulting in inaccurate calculations of 𝑏 and therefore error in the pore sizes calculated from gas slippage measurements using Equations 3.4 and 3.5. These errors would impact the trend of the dominant pore sizes calculated from gas 74  slippage measurements (red lines in Figure 3.3). Only a small portion of the compiled dataset is potentially impacted by these errors, and the lack of nonlinearity observed for our measurements indicates these errors had a negligible impact on our conclusions. Future studies should seek to determine the magnitude of these errors in reservoir rocks with very small pores, and to determine the measurement conditions (pore pressure, temperature, gas species) required to obtain accurate pore size estimates.  Landry et al. (2016), in their study of reconstructed porous media representing organic matter porosity of mudrocks, concluded that bundle of capillary tubes models yield inaccurate permeability predictions. The inaccuracy was attributed to the inability of the model to capture the complex geometry and pore connectivity of natural porous media. The two reconstructed porous media considered in their analysis have porosity of 58% and 20%, far higher than any of the mudrocks in our compiled data set for which porosity was reported. Additionally, there is discrepancy between the equation used by Landry et al. (2016) to calculate 𝐾∞ using a bundle of capillary tubes model (their Equation 11) and others found in the literature (e.g. Equation 5 in Randolph et al., 1984 and Equation 6.1 in Al Ismail and Zoback, 2016); Landry et al. have tortuosity to the first power in their equation (Equation 11 in their paper), whereas elsewhere in the literature tortuosity is squared, as in our analysis (Equation 3.6 and Equation 3.7). Tortuosity should be squared because pressure gradient and quantity of tubes in the model both scale inversely to tortuosity. It is unclear if this discrepancy is a typographical error in Landry et al. (2016), or if the wrong equation was used in their analysis. Because of this discrepancy and the fact that the highly porous reconstructed porous media analyzed in Landry et al. (2016) is only representative of the organic matter component of mudrocks, not mudrock pore structures as a 75  whole, we discount the concern raised in Landry et al. (2016) about the ability of bundle of capillary tubes models to predict permeability.  Al Ismail and Zoback (2016) analyzed their gas slippage data similarly to our analysis of the compiled gas slippage data set reported here; in pore size - 𝐾∞ space, they compared the trends of their pore size estimates calculated from gas slippage measurements to the slope predicted by the bundle of cylindrical capillary tubes model (Equation 3.6). However, they did not use tortuosity and porosity functions (i.e. they assumed porosity and tortuosity are constant). They state that the slope is nearly parallel to the trends of their measurements made on individual samples at multiple stress states (their Figure 14). They suggest this observation is evidence that flow paths resemble circular, tortuous capillary tubes. However, the slope considered by Al Ismail and Zoback is not expected to match the trend of pore size - 𝐾∞ data, as tortuosity and porosity are not constant properties; even for a single sample at multiple stress states, the pore compression responsible for narrowing of pores and pore throats, and therefore decreased permeability at higher stress states, also results in decreasing porosity, as well as possibly changing tortuosity. When viewed as a component of the larger gas slippage data set considered in this study (Figure 3.1), the data of Al Ismail and Zoback fit within (not as outliers) the gas slippage data set. The trend of dominant pore sizes calculated from the compiled gas slippage data set in pore size – 𝐾∞ space is better matched by the trend of pore sizes calculated using bundle of capillaries models with porosity and tortuosity functions (Figure 3.3) than by any single combination of porosity and tortuosity (Figures 3.4 and 3.5).  76  Al Ismail and Zoback (2016) also concluded that, because the slope in pore size - 𝐾∞ space predicted by the bundle of cylindrical capillary tubes model is nearly parallel to the trends of their measurements made on individual samples at multiple stress states, the flow paths are circular in cross section. However, our analysis shows the slope predicted using the bundle of slot shaped capillaries model is the same as the slope predicted by the bundle of cylindrical capillaries model (Figure 3.3). Our conclusion is better supported; pores that control flow through higher permeability reservoir rocks are more often slot-shaped, and pores in lower permeability rocks tend to be circular in cross section, based on comparison between slot-shaped and circular cross sectional geometry models and the trend of the experimental gas slippage data (Figure 3.3). However, as stated above, conclusive statements about pore morphology cannot be drawn based on either of these analyses that only consider two different pore geometries.  3.5 Conclusion Gas slippage measurements can be used to ascertain quantitatively valid characterizations of reservoir rock pore structures. This is evidenced by (1) the fact that 99% of the gas slippage data result in pore size estimates that are consistent with bundle of capillaries models with geologically plausible inputs of porosity and tortuosity (Figures 3.4 and 3.5); and (2) broadly similar (though not identical) trends of pore sizes calculated from gas slippage measurements in the data set compiled for this paper with the trends of pore sizes estimated using bundle of capillaries models using tortuosity and porosity functions (Figure 3.3). Better agreement is found when flow paths are modeled as slot-shaped in high permeability rocks and tubular in low permeability rocks, which suggests a systematic shift in the morphology of the pores most responsible for limiting fluid flow with changing permeability. 77  Conventional pore geometry characterization techniques, such as mercury intrusion porosimetry, CO2 and N2 adsorption, and scanning electron microscopy, cannot be used in a cost-effective way to characterise pore structure at in situ reservoir stress states, whereas gas slippage measurements can. However, gas slippage measurements only yield an estimate of dominant pore size of the portion of the pore structure most responsible for limiting fluid flow. The conventional techniques that can provide a distribution of the different pore sizes in a reservoir rock are therefore still of value, and in conjunction with gas slippage measurements provide more complete characterization of a sample. 78  Chapter 4: The Impact of Gas Slippage on Permeability Effective Stress Laws: Implications for Predicting Permeability of Fine-Grained Lithologies  4.1 Introduction Quantifying permeability at reservoir stress and at the range of stress states that will be experienced during production from a reservoir is necessary for modelling and ultimately predicting fluid production. Accurate permeability prediction of fine-grained lithologies, such as shale oil and shale gas reservoirs, is especially important because permeability is more sensitive to stress in fine-grained lithologies than in the coarser-grained lithologies of conventional hydrocarbon reservoir rocks (McLatchie et al., 1958; Vairogs et al., 1971). Predicting permeability of fine-grained lithologies in the laboratory is complicated by their nano- to micrometre length-scale pore systems, which can result in significant departure from the Darcy flow regime due to gas slippage (Klinkenberg, 1941) when a gaseous fluid flows through the rock. Laboratory measurements of permeability are commonly made at lower-than-in situ reservoir pore pressures, where gas slippage is more significant.   This paper presents a theoretical discussion supported by experimental data of the potential for errors introduced by not considering gas slippage in permeability predictions derived from laboratory measurements. Specifically, we investigate the impact of gas slippage on experimentally-derived permeability effective stress laws. The theoretical discussion is backed by gas permeability measurements of a Montney Formation shale over a range of confining pressure and pore pressure combinations that include pore pressures low enough to easily 79  recognize gas slippage. Additionally, previously published data for two Eagle Ford Formation shale samples and one Marcellus Formation shale sample are reanalyzed. This paper shows the need to account for gas slippage in fine-grained reservoir rocks, even at high pore pressures where gas slippage is often assumed to be negligible. The results show that in order to determine permeability effective stress laws that can be used to accurately predict permeability at in situ reservoir pressure conditions, corrections for gas slippage must be made when using a gaseous probing fluid.  4.2 Background Permeability of a reservoir rock depends in part on its stress state. Stress state is defined by the confining pressure acting on the rock, which compresses the pore structure, and pore fluid pressure (pore pressure herein), which opposes confining pressure and resists compression of the pore structure. Pore pressure decreases when fluids are produced from a reservoir. Decreasing pore pressure results in a transfer of stress from what were once high-pressure pore fluids, to the matrix framework that hosts the porosity. The result is deformation of the framework and partial collapse of the porosity. As the pore structure collapses, permeability decreases.  Predicting reservoir rock properties that vary as a function of confining pressure and pore pressure (e.g. permeability) can be greatly simplified if an effective stress law (Robin, 1973) can be determined. An effective stress law relates a given rock property, for example permeability (𝐾), to effective stress (𝜎𝑒𝑓𝑓): 𝐾 = 𝑓(𝜎𝑒𝑓𝑓)          (Eq. 4.1) where effective stress is a function of confining pressure (𝑃𝑐) and pore pressure (𝑃𝑝): 80  𝜎𝑒𝑓𝑓 = 𝑓(𝑃𝑐, 𝑃𝑝)          (Eq. 4.2) If such a relationship can be defined for permeability, it is then possible to predict permeability at high confining pressure and pore pressure combinations (representative of in situ reservoir stress) using laboratory measurements made at lower pressures. This is advantageous, because measuring permeability at higher confining pressure and pore pressure combinations is more technically challenging, requires more expensive laboratory equipment, and is more dangerous if a highly compressible gaseous probing fluid is used.  Laboratory methods and data analysis procedures have been developed and applied to determine permeability effective stress laws for a variety of rocks. In some rocks, effective stress was shown to be simply the difference between confining pressure and pore pressure (e.g. Morrow et al., 1986; Kwon et al., 2001): 𝜎𝑒𝑓𝑓 =  𝑃𝑐 − 𝑃𝑝         (Eq. 4.3) In other rocks, effective stress was shown to be a function of confining pressure, pore pressure, and a constant permeability effective stress law coefficient, α (e.g. Zoback and Byerlee, 1975; Bernabe, 1988; Al-Wardy and Zimmerman, 2004): 𝜎𝑒𝑓𝑓 =  𝑃𝑐 − 𝛼𝑃𝑝          (Eq. 4.4).  Many studies have reported nonlinear permeability effective stress laws, for which α is not constant and varies as a function of confining pressure and pore pressure (e.g. Bernabe, 1987; Warpinski and Teufel, 1992; Ghabezloo et al., 2009;  Li et al., 2009; Li et al., 2014). The practical utility of nonlinear permeability effective stress laws is limited, as permeability needs to 81  be measured over the particular pressure range of interest in order to predict permeability within that range (Robin, 1973).  Permeability-stress relationships have been determined for many fine-grained rocks by assuming an effective stress law coefficient of α = 1. (e.g., Kwon et al., 2004; Yang and Aplin, 2007; Dong et al., 2010; Armitage et al., 2011; Metwally and Sondergeld., 2011; Chalmers et al., 2012b). These are not permeability-effective stress laws. In no case has it been experimentally proven that α is in fact equal to 1. Not knowing α means it is uncertain whether or not these permeability-stress relationships could be used to predict permeability at pore pressure and confining pressure combinations other than those measured in the laboratory when developing the relationships. This paper specifically addresses permeability effective stress laws, not permeability-stress relationships.  Incorrect permeability effective stress laws could be determined if a gaseous probing fluid is used for permeability measurements at pore pressures low enough for gas slippage to cause significant flow-rate enhancement relative to that predicted by Darcy’s Law. Gas slippage causes permeability to vary with pore pressure even if the pore structure is not altered by changing stress. Gas slippage enhances apparent permeability when the mean free path of a gas (average distance a gas molecule travels between two successive collisions with other gas molecules) approaches the size of the pores through which it is flowing. Gas slippage causes increased apparent permeability at lower pore pressures (Figure 4.1) because mean free path is inversely proportional to gas pressure (Klinkenberg, 1941).  82   Figure 4.1 Data for Klinkenberg’s sample L for flow of nitrogen (Klinkenberg, 1941). (A) Variation of permeability with pore pressure. (B) Klinkenberg plot showing linear relationship of inverse pore pressure and permeability. The mean free path of a gas varies linearly with the inverse of pressure, which is why permeability varies linearly with inverse pore pressure.  The potentially misleading impacts of gas slippage on experimentally derived permeability-effective stress laws have been recognized in previous studies (Warpinski and Teufel, 1992; Li et al., 2009), and steps have been taken to avoid the impacts. These steps include measuring permeability at high pore pressures, where the impacts of gas slippage on permeability-effective stress laws become negligible. However, the significance of gas slippage is pore structure-dependent; rocks with smaller pores experience more gas slippage than rocks with larger pores. The pore pressure at which the impact of gas slippage on permeability effective-stress laws becomes negligible therefore depends on the sizes of the pores in a rock. A seemingly arbitrary assumption that gas slippage becomes negligible above ~7 MPa pore pressure has been perpetuated in the literature, but the boundary at which this assumption breaks down has not been investigated.  83  4.3 Materials and Methods 4.3.1 Sample Descriptions The sample analyzed in this study is from the Lower Triassic Montney Formation in the Western Canada Sedimentary Basin. The Montney Formation is currently one of the most productive unconventional hydrocarbon reservoirs in Canada (Rivard et al., 2014). The studied sample is a dolomitic siltstone containing detrital, silt sized dolomite grains. One cylindrical plug three centimeters in diameter by three centimeters in length was cut and used for permeability measurements. Mineral phases were identified by X-ray diffraction using normal-focus CoKα radiation on a Bruker® D8 Focus at 35 kV and 40 mA and quantified using the Rietveld method of full-pattern fitting (Rietveld, 1967) using Bruker® AXS Topas V3.0 software. Dominant mineral phases are quartz and dolomite (Table 4.1). Pore size distribution was determined by mercury intrusion porosimetry using a Micromeritics AutoPore IV® (Figure 4.2), and porosity determined from mercury intrusion porosimetry is 4.7 %. Sample material used for both X-ray diffraction and mercury intrusion porosimetry was taken from end trims of the plug used for permeability analysis. Table 4.1 Mineralogical composition of the Montney Formation sample from X-ray diffraction.   84   Figure 4.2 Mercury intrusion porosimetry pore size distribution of the Montney sample analyzed in this study.  Previously published data for the three samples reanalyzed in this study are from Heller et al. (2014) (their samples Eagle Ford 127, Eagle Ford 174, and Marcellus). Permeability data were digitized from their figures because raw permeability data were not presented in their paper. Detailed descriptions of their samples can be found in Heller et al. (2014). Important details of their experimental procedure are reported below where appropriate.  4.3.2 Sample Seasoning To reduce hysteresis effects associated with loading history, the sample analyzed in this study was “seasoned” prior to permeability measurements. Bernabe (1987) showed permeability and α hysteresis associated with loading history in crystalline rocks, and that hysteresis was diminished with repeated loading cycles. Although no published studies have investigated α hysteresis in 85  fine-grained sedimentary rocks, permeability hysteresis associated with loading cycles has been documented (e.g. Kwon et al., 2004). After loading the sample analyzed in this study into a confining cell, balanced confining pressure was applied and cycled two times to 55 MPa confining pressure with pore pressure held constant at 1.7 MPa. The two seasoning cycles were applied over a 7 day period. The maximum difference between confining pressure and pore pressure (𝑃𝑐 − 𝑃𝑝, referred to herein as simple effective stress) during seasoning of 53.3 MPa is greater than the maximum simple effectives stress during permeability measurements (48.3 MPa). Ideally, permeability would have been continuously monitored and seasoning cycles repeated until hysteresis disappeared or stabilized, but this was not practical due to the length of time the experiment would have taken.  The samples in Heller et al. (2014) were also seasoned prior to permeability measurements; for 48-72 hours, they subjected their samples to a simple effective stress 25% greater than what would be experienced in subsequent permeability measurements.  4.3.3 Permeability Measurements Permeability was measured using the pressure pulse decay technique (Brace et al., 1968). The pulse decay technique was chosen over flow through experiments because pressure is easier to monitor than the very small flow rates associated with fine-grained reservoir rocks. Helium was chosen as a probing gas to accentuate gas slippage; at a given pressure, smaller gas molecules have a larger mean free path than larger gas molecules, and will therefore experience more gas slippage (Klinkenberg, 1941). Additionally, helium is commonly used as a probing gas when measuring the permeability of fine-grained, organic rich rocks, such as shale oil and shale gas 86  reservoir rocks, to minimize complications associated with adsorption (Cui et al., 2009). The reanalyzed data from Heller et al. (2014) was also collected using helium as a probing gas.  Permeability was measured over a confining pressure range of 8.6 to 65.5 MPa and a pore pressure range of 1.7 to 17.2 MPa using the pressure schedule presented in Figure 4.3. Permeability measurements at higher confining pressures and pore pressures were not possible due to limitations of the confining cell and permeameter. The pressure schedule was designed so that multiple permeability measurements were made at constant simple effective stresses. Permeability variation due to gas slippage could therefore be analyzed under the assumption that pore structure remained unchanged at a given simple effective stress. The same analysis was possible for the reanalyzed data because Heller et al. (2014) also made multiple permeability measurements at constant simple effective stresses. Their permeability measurements were made over a smaller confining pressure range (8.6 to 55.2 MPa) than in this study, but a larger pore pressure range (1.7 to 27.6 MPa). 87   Figure 4.3 Confining pressure and pore pressure schedule for permeability measurements of the Montney sample.  4.4 Results and Discussion This section first discusses the theoretically expected impact of pore structure change on Klinkenberg plots to show how Klinkenberg plots can be used to assess the value of α (subsection 4.4.1). The Klinkenberg plots for the Montney sample analyzed in this study are then used to determine α, and the sample’s effective stress law corrected for gas slippage is presented in subsection 4.4.2. In subsection 4.4.3, raw permeability data not corrected for gas slippage is used to determine an apparent permeability effective stress law for the Montney sample, which permits a more general discussion on the expected impact of gas slippage on experimentally-derived permeability effective stress laws. Data from Heller et al. (2014) is then reanalyzed in 88  subsection 4.4.4, where it is shown that their effective stress laws are apparent permeability effective stress laws significantly influenced by gas slippage. Last is a review of other published permeability effective stress laws that may have been impacted by gas slippage, and hence are of questionable validity (subsection 4.4.5).  4.4.1 Impact of Pore Structure Changes on Klinkenberg Plots If the pore structure of a given rock is fixed and permeability to gas is measured at a range of pore pressures, apparent permeability variation due to gas slippage will display a linear relationship with inverse pore pressure (Klinkenberg plot, Figure 4.1B). The linear relationship exists because the mean free path of a gas is proportional to the inverse of gas pressure. If pore structure is not fixed but rather varies depending on pressure conditions, nonlinear Klinkenberg plots occur because permeability variation is the result of both pore structure changes and changing mean free path of the gas. This concept is depicted in Figure 4.4, where synthetic permeability data are calculated for the hypothetical pressure conditions listed in Table 4.2. If a rock has an α value less than one and permeability is measured over a range of confining pressure and pore pressure combinations equating to the same simple effective stress, effective stress increases with increasing pore pressure, and concave-down Klinkenberg plots occur (circles in Figure 4.4); when α is greater than one, effective stress decreases with increasing pore pressure, and concave-up Klinkenberg plots occur. When α is equal to one, pore structure remains constant at confining pressure and pore pressure combinations that equate to the same simple effective stress, and linear Klinkenberg plots occur (squares in Figure 4.4).  89   Figure 4.4 Synthetic permeability data calculated for the hypothetical pressure conditions listed in Table 4.2. If α = 1, pore structure is identical at any pressure conditions that equate to the same simple effective stress, and permeability variation due to gas slippage produces a linear Klinkenberg plot (squares). If α ≠ 1, effective stress and therefore pore structure varies at different pressure conditions that equate to the same simple effective stress, and a nonlinear Klinkenberg plot is expected (circles).  Table 4.2 Hypothetical pressure data used for the conceptual analysis presented in Figure 4.4. The fourth column assumes α = 0.8.   Theoretical studies have been published that predict nonlinear Klinkenberg plots resulting from processes other than effective stress and pore structure change. Fathi et al. (2012) predicted 90  concave-up Klinkenberg plots. Their prediction is backed by a limited experimental data set. Moghadam and Chalaturnyk (2014) predicted concave-down Klinkenberg plots. Their prediction is backed by a more robust experimental data set. However, the nonlinearity they predict is most significant at very low pore pressures. Their model predicts very little nonlinearity over the pore pressure range permeability measurements were made at in the present study and in Heller et al. (2014) (pore pressures of 1.72 MPa and higher). In contrast, the stress effects discussed above are expected to result in Klinkenberg plots that are most nonlinear at higher pore pressures (circles in Figure 4.4). Because of this contrast, nonlinearity due to the theoretical predictions of Moghadam and Chalaturnyk (2014), if present, should be distinguishable from nonlinearity due to stress effects.  4.4.2 Permeability Effective Stress Law for the Montney Sample Permeability measurements collected for the Montney sample at each single simple effective stress (𝑃𝑐 − 𝑃𝑝) resulted in linear Klinkenberg plots (Figure 4.5A). The observed permeability variation at a single simple effective stress is hence apparent permeability variation resulting from gas slippage, not effective stress changes. At a given pore pressure, permeability is lower at higher simple effective stress (Figure 4.5B), which is attributed to compression of the pore structure with increasing stress.  91   Figure 4.5 (A) Klinkenberg plots for the Montney sample at each simple effective stress state. (B) Permeability variation due to gas slippage at each simple effective stress state.  The linear Klinkenberg plots presented in Figure 4.5A indicate that α = 1 for the Montney sample. Klinkenberg plots can therefore be extrapolated to infinite pore pressure to determine the slip-free permeability at each simple effective stress. In this sample, effective stress is the same as simple effective stress (α = 1). A plot of slip free permeability against effective stress yields a good power law fit (Figure 4.6). The permeability-effective stress law for the Montney sample is 𝐾 = 0.0031(𝑃𝑐 − 𝑃𝑝)−0.68 (K in mD, 𝑃𝑐 and 𝑃𝑝 in MPa).  92   Figure 4.6 Permeability effective stress law for the Montney sample analyzed in this study.  4.4.3 Impact of Ignoring Gas Slippage on Permeability Effective Stress Laws In subsection 4.2, it was shown that α = 1 for the Montney sample analyzed in this study, and a permeability effective stress law corrected for gas slippage was presented. In this subsection, data is analyzed without correcting for gas slippage to illustrate the impact of ignoring gas slippage on experimentally-derived permeability effective stress laws.  Significant apparent permeability variation takes place due to gas slippage, even at pore pressures equal to or greater than 7 MPa (Figure 4.7A). If not recognized as gas slippage, this variation would be interpreted as α not being equal to one, and therefore changing effective stress at constant simple effective stress. Curve fitting yields an apparent α value of 0.64 if a power law is fit to the data (Figure 4.7B). Gas slippage causes increased apparent permeability at lower pore pressures, and hence, if α = 1, will always result in apparent α values less than one. 93  The apparent permeability effective stress law determined for the Montney sample using measurements made at pore pressures greater than or equal to 7 MPa poorly fits the lower (< 7 MPa) pore pressure permeability data (Figure 4.8). Power law fits to individual high pore pressure (≥ 7 MPa) data clusters indicate systematic variation from the fit, not random scatter around the fit (Figure 4.8). The systematic variation and poor fit to low pore pressure data is an indication that the apparent permeability-effective stress law is nonlinear. It is expected that gas slippage will result in nonlinear apparent permeability-effective stress laws because mean free path, and therefore apparent permeability variation due to gas slippage, varies nonlinearly with pore pressure (mean free path varies linearly with inverse pore pressure).  Figure 4.7 (A) Permeability variation due to gas slippage at individual simple effective stress states for permeability measurements of the Montney sample made at and above 7 MPa pore pressure. (B) Apparent permeability effective stress law for data fit to a power function.  94   Figure 4.8 Permeability of the Montney sample against effective stress for α = 0.64, the α value determined by curve fitting to high pore pressure (≥7 MPa) permeability data.  Permeability effective stress laws can be visualized and quantified using contour plots of permeability in confining pressure-pore pressure space (Figure 4.9), which are useful for visualizing nonlinear permeability effective stress laws. In confining pressure-pore pressure space, permeability contours will be linear and parallel to one another if α is constant (Figure 4.9A, B and C). The slope of the contours is the value of α. Nonlinear permeability effective stress laws will result in nonlinear permeability contours (Figure 4.9D). At a given confining pressure and pore pressure, the slope of a tangent line to the permeability contour is the permeability effective stress law coefficient in the differential form of the permeability effective stress law, αT (Li et al., 2014, their Figure 1): 𝛿𝜎𝑒𝑓𝑓 = 𝛿𝑃𝑐 − 𝛼𝑇(𝑃𝑐, 𝑃𝑝) ∗ 𝛿𝑃𝑝          (Eq. 4.5). 95    Figure 4.9 Schematic representations of linear (A, B and C) and nonlinear (D) permeability effective stress laws using permeability contours in confining pressure-pore pressure space.  A contour plot of all permeability data in confining pressure-pore pressure space shows that the apparent permeability effective stress law for the Montney sample is nonlinear when slippage is ignored (Figure 4.10). Permeability contours become increasingly nonlinear at low pore pressures because mean free path, and therefore gas slippage, increases nonlinearly with decreasing pore pressure. Slope of the permeability contours, and therefore αT, decreases with decreasing pore pressure, becoming values less than zero between 6 and 9 MPa. Negative αT values make no sense in a permeability-effective stress law, as they imply that an incremental increase in pore pressure, which acts to support the pore structure, causes an incremental decrease in permeability. Negative αT values are however explicable in the case of apparent 96  permeability-effective stress laws significantly influenced by gas slippage, as gas slippage results in increasing apparent permeability with decreasing pore pressure. At a given pore pressure, αT decreases with increasing confining pressure, which is attributed to compression of the pore structure with increasing simple effective stress; compression of the pore structure narrows the pores in the rock, resulting in more gas slippage.   Figure 4.10 Permeability contour plot in confining pressure-pore pressure space for the Montney sample analyzed in this study.  For a given gas at a given temperature, how much apparent permeability will vary due to gas slippage for a particular rock at a given stress state can be quantified using Klinkenberg’s (1941) slippage parameter, 𝑏 97  𝑏 =𝐾𝑎∗𝑃𝑝𝐾∞− 𝑃𝑝          (Eq. 4.6) where 𝐾𝑎 is apparent permeability and 𝐾∞ is permeability from extrapolation of Klinkenberg plots to infinite pore pressure (slip-free permeability). The greater the value of 𝑏, the more 𝐾𝑎 will vary with pore pressure relative to 𝐾∞. The slippage parameter for the sample analyzed in this study increases with increasing simple effective stress (Table 4.3), supporting the interpretation that compression of the pore structure with increasing simple effective stress drove αT to lower values at higher confining pressures (Figure 4.10). Table 4.3 Klinkenberg’s slippage parameter (𝒃, calculated using Equation 4.6) at each simple effective stress for the Montney sample.    4.4.4 Reanalysis of Data from Heller et al. (2014) Heller et al. (2014) collected both high pore pressure (≥ 7 MPa) and low pore pressure (< 7 MPa) permeability data for samples Eagle Ford 127, Eagle Ford 174, and Marcellus. They first determined permeability-effective stress laws using only the high pore pressure data under the assumption that gas slippage was negligible (their Figure 8). The effective stress laws were then used in conjunction with the low pore pressure data to quantify gas slippage, with the goal of separating gas slippage from stress effects (their Figure 9). By reanalyzing the data, we show here that their initial assumption of gas slippage being negligible at high pore pressures is 98  invalid, which resulted in invalid permeability-effective stress laws (apparent permeability effective stress laws influenced by gas slippage) and hence inaccurate gas slippage measurements.  The data were first reanalyzed by generating Klinkenberg plots using both the high pore pressure and low pore pressure data (Figure 4.11). Thick, dashed, black lines separate high pore pressure data (to the left) from low pore pressure data (to the right). Klinkenberg plots for Eagle Ford 127 (Figure 4.11C) are linear with very little scatter, similar to Klinkenberg plots for the Montney sample analyzed above (Figure 4.5A). Klinkenberg plots for Marcellus (Figure 4.11A) are linear, but with more scatter than Eagle Ford 127. Klinkenberg plots for Eagle Ford 174 (Figure 4.11B) show some departure from linearity, with concave-down Klinkenberg plots at simple effective stresses of 6.9 and 13.8 MPa. At simple effective stresses of 20.7 and 27.6 MPa, scatter in the data makes it difficult to assess the linearity of the Klinkenberg plots. 99   Figure 4.11 Klinkenberg plots for the reanalyzed data from Heller et al. (2014). Thick, dashed, black lines separate high pore pressure data (≥ 7 MPa, left of the lines) from low pore pressure data (< 7 MPa, right of the lines).  For samples Eagle Ford 127 and Marcellus, which produced linear Klinkenberg plots at each measured simple effective stress, the simplest interpretation is that α = 1 and permeability 100  variation along the Klinkenberg plots is due to gas slippage (same interpretation as for the Montney sample discussed in subsection 4.4.2). In contrast, the interpretation of Heller et al. (2014) implies that the permeability variation along the Klinkenberg plots at low pore pressure is due to gas slippage, but at high pore pressure the mechanism for permeability variation changes to stress effects, even though the data fall on the trend predicted by extrapolating low pore pressure permeability variation due to gas slippage to high pore pressures.  Concave-down Klinkenberg plots for Eagle Ford 174 at 6.9 and 13.8 MPa simple effective stress (Figure 4.11B, blue triangles and red circles) suggest a permeability-effective stress law coefficient less than one, and therefore increasing effective stress with increasing pore pressure (simple effective stress constant). Concavity is greater at high pore pressures than at low pressures, and extrapolation of the data would yield negative permeability intercepts. Neither of these characteristics is predicted by the theoretical model of Moghadam and Chalaturnyk (2014) for Klinkenberg plot nonlinearity unrelated to stress effects. Heller et al. (2014) determined a permeability-effective stress law coefficient less than one for Eagle Ford 174, which is consistent with our interpretation based on Klinkenberg plot linearity. However, their calculated coefficient (α = 0.40) is too low, as the permeability variation their effective stress law describes is due to both gas slippage and stress effects, and gas slippage drives permeability-effective stress law coefficients to lower values. The exact value of the coefficient cannot be determined because permeability variation at a given simple effective stress is the combined result of both pore structure change and gas slippage, and the individual contributions of each mechanism are inseparable in this data set.  101  Gas slippage is ignored and permeability data are reanalyzed to confirm the impact of gas slippage on experimentally derived permeability effective stress laws (investigated in subsection 4.4.3). Only high pore pressure data for each of the reanalyzed samples are plotted in Figure 4.12, along with the slope of linear permeability contours for the effective stress law coefficients determined by Heller et al. (2014). Because these data sets have limited data point density, and the random error associated with each data point is significant relative to the total permeability variation within each data set, contouring is difficult; many different solutions could be determined for the same data set, and therefore many different permeability effective stress laws. To overcome this challenge, previous researchers (Warpinski and Teufel, 1992; Li et al., 2009; Li et al., 2014) employed the response surface method (Box and Draper, 1987) when contouring permeability-effective stress law data. The response surface method can be used to fit a smooth surface to permeability data that has significant random error, without assuming any prior knowledge of how the rock should respond to changes of pore pressure and confining pressure. By assuming gas slippage will take place when using a gaseous probing fluid, an assumption strongly supported by the Klinkenberg plots presented in this paper (Figures 4.5A and 4.11) as well as in other studies (eg. Letham and Bustin, 2016; their Figure 4), low pore pressure data can be used to predict and incorporate permeability variation due to gas slippage into the contour model. This assumption was used to contour all permeability data for each of the reanalyzed samples in confining pressure-pore pressure space (Figure 4.13). The resulting contour plots provide a good fit to the raw experimental data. The contour plots have nonlinear permeability contours with negative slopes at low pore pressures, gradually transitioning to positive slopes at ~7-15 MPa pore pressure; these are the same characteristics of the apparent permeability effective stress law for the Montney sample analyzed above (Figure 4.10). 102   Figure 4.12 High pore pressure (≥ 7 MPa) permeability data in confining pressure-pore pressure space for the three samples from Heller et al. (2014). Slope of permeability contours for α = 1 and for the experimentally derived permeability effective stress law coefficients determined by Heller et al. are plotted with dashed lines.  103   Figure 4.13 Both high and low pore pressure permeability data for the three samples from Heller et al. (2014) plotted in confining pressure-pore pressure space. Solid black lines are permeability contours. Contours in all plots are nonlinear with negative slopes at low pore pressures, transitioning to positive slopes at pore pressures of ~7 to 15 MPa.  4.4.5 Implications for Previous Studies Many permeability-effective stress laws in the literature were determined using liquid probing fluids, and therefore would not be impacted by gas slippage. However, studies that used gaseous 104  probing fluids to measure permeability-effective stress laws for fine grained rocks need to be revisited to confirm their validity. Some such studies are reviewed below.  Warpinski and Teufel (1992) used nitrogen as a probing gas to determine effective stress laws for a suite of tight sandstone and chalk samples. Nitrogen molecules, being larger than helium molecules, have a shorter mean free path at any given temperature and gas pressure. Based on calculations of mean free path for an ideal gas, approximately half as much gas slippage should take place when using nitrogen instead of helium as a probing gas. However, their data analysis resulted in a trend of αT decreasing with decreasing pore pressure and increasing confining pressure (their Figure 11), which is attributed to gas slippage in the present study. Warpinski and Teufel justified ignoring gas slippage at pore pressures at and above 7 MPa because, based on their tests, error from ignoring Klinkenberg corrections at 7 MPa was about 8%, which was within the standard error of their measurements. In their paper, Warpinski and Teufel stated that they used the response surface method because the errors in their permeability measurements were often on the same order as the difference in permeability measured at two different conditions. Significant impacts of gas slippage could easily be obscured by the experimental error of their data. The results and conclusions presented in Warpinski and Teufel (1992) may require revisiting.  Li et al. (2014) also used nitrogen as a probing gas. They used the relationship between 𝑏 and 𝐾∞ determined in Li et al. (2009) to estimate the “relative Klinkenberg correction” for their samples, and found the maximum to be 1.5% for their lowest permeability sample. Their samples have measured permeabilities as low as 0.002 mD (e.g. sample D24-7, their Figure 2B). Assuming 105  𝐾∞=0.002 mD, 𝐾𝑎 at 5 MPa of 0.0022 mD (approximately 10% higher than 𝐾∞) is calculated using the relationship between 𝑏 and 𝐾∞ from Li et al. (2009) and Equation 4.6 rearranged to solve for 𝐾𝑎. The permeability contour plot presented for sample D24-7 (their Figure 2B) is indicative of αT values less than one and decreasing αT with decreasing pore pressure and increasing confining pressure, similar to the contour plot for apparent permeability variation due to gas slippage in the present study (Figure 4.10). Li et al. (2014) also reported permeability effective stress law coefficients less than zero, which they said were likely “unphysical artifacts”. Gas slippage could potentially provide a physical explanation for their negative coefficients.  Letham and Bustin (2016) used helium as a probing gas to determine the permeability-effective stress law of an Eagle Ford Formation sample. Measured slippage parameters at a range of simple effective stress states ranged from 0.44 to 1.72 MPa, and the α value determined was less than one (α = 0.8). Reanalysis of the data indicates the permeability effective stress law was significantly impacted by gas slippage and is not valid. The pressure schedule used for permeability measurements in their study was such that the data could not be reanalyzed to evaluate α based on Klinkenberg plot linearity.  In subsection 4.4.4, it was shown that the permeability effective stress laws presented in Heller et al. (2014) are invalid; they are apparent permeability effective stress laws significantly influenced by gas slippage. Since publication, other authors (Gensterblum et al., 2015; Shi and Durucan, 2016) have used the invalid effective stress laws to validate theoretical considerations. These studies, accordingly, need to be revisited.  106  4.5 Conclusion Gas slippage naturally drives apparent permeability-effective stress law coefficients towards values less than one at lower pore pressures and at higher confining pressures. By how much gas slippage contributes to flow-rate and therefore increases apparent permeability depends on the probing fluid used for permeability measurements, the pore pressure range permeability was measured over, and the pore structure of the rock. The magnitude of permeability variation due to gas slippage in comparison to the stress sensitivity of permeability determines how significant the impact of gas slippage will be on experimentally-derived permeability-effective stress laws.  Fine-grained reservoir rocks, such as shale oil and shale gas reservoirs, inherently have much smaller pores and pore throats than coarse-grained rocks. Smaller pores result in more gas slippage. Experimentally-derived permeability effective stress laws for fine-grained lithologies are therefore more susceptible to the impacts of gas slippage than laws derived for coarse-grained lithologies, and permeability-effective stress laws significantly impacted by gas slippage have been reported in the literature for fine-grained lithologies.   For fine-grained lithologies, the common assumption that gas slippage can be neglected at pore pressures higher than 7 MPa when determining a permeability effective stress law using a gaseous probing fluid needs to be abandoned, and previous studies utilizing this assumption need to be thoroughly re-examined.  Very few robust permeability effective stress laws for fine grained sedimentary rocks have been published. To date, four permeability-effective stress laws with coefficients of α = 1 have been 107  reported (one Wilcox Formation sample (Kwon et al., 2001); one Montney Formation sample (this study); one Eagle Ford Formation sample and one Marcellus Formation sample (data from Heller et al. (2014), reanalyzed in this study)). Analysis of one Eagle Ford Formation sample indicates an effective stress law coefficient less than one (data from Heller et al. (2014), reanalyzed in this study). Determining robust effective stress laws for more fine-grained samples will allow assessment of how likely it is that permeability-stress relationships determined assuming α = 1 can be extrapolated to in situ reservoir pressure conditions from laboratory measurements made at lower-than-in situ pressure. The resulting data set would also be useful for exploring theoretical considerations that could provide more general conclusions about the dynamic behavior of fine-grained sedimentary rocks.  108  Chapter 5: Investigating Multi-Phase Flow Phenomena in Fine-Grained Reservoir Rocks: Insights from Using Ethane Permeability Measurements Over a Range of Pore Pressures  5.1 Introduction Effective matrix permeability is one control on the deliverability of hydrocarbons, and therefore the economics of shale oil and shale gas wells (Bustin and Bustin, 2012). Being able to quantify effective permeability at initial reservoir saturation and the range of saturations experienced during production from these reservoirs is thus paramount to efficient exploitation of the resource. Hydrocarbons in shale oil and shale gas reservoirs travel in the presence of multiple immiscible fluids through small pores in the fine grained matrix prior to reaching a fracture network leading to the wellbore. Immiscible phases include one or both of liquid and gaseous hydrocarbons as well as connate water and imbibed hydraulic fracturing fluid. Capillary pressures across the interfaces of these immiscible phases can be enormous due to how small the pores are in shales (nanometre to micrometre length scale (e.g. Clarkson et al., 2013)). The small pore length scale also means that geometrical constrictions imposed by fluid adsorbed to pore walls can significantly inhibit flow (Aljamaan et al., 2016). Effective permeability of hydrocarbon phases will therefore be very sensitive to the presence and saturation of the different fluid phases (Bennion and Bachu, 2008; Aljamaan et al., 2016).  To date, researchers have had minimal success investigating multiphase flow characteristics of shale due to challenges associated with characterizing reservoir rocks with small length scale 109  pore systems. These challenges include the difficulty in controlling and monitoring fluid saturations and distributions in intact samples, and, because of their inherently low permeability, difficulty in quantifying the very low flow rates characteristic of shales. Combined, these challenges have prevented measurement of effective permeability over a range of fluid saturations and thus impaired our understanding of shale multiphase flow phenomena.  In this study we investigate multiphase flow phenomena in fine grained rocks by measuring effective permeability of a suite of shale samples to gaseous ethane over a range of pore pressures up to the saturated vapour pressure (3.59 MPa at 18 °C). At laboratory temperature, pure ethane can exist as a liquid or a gas, depending on pressure (Figure 5.1B). In contrast, methane, the dominant component of natural gas, is either a gas or a supercritical fluid (Figure 5.1A). At the range of pressures found in natural gas reservoirs, methane typically adsorbs to the pore walls of organic rich rocks to form a monolayer, resulting in Type I (Langmuir) isotherms (Ross and Bustin, 2007). In contrast, multilayer adsorption is expected for ethane at pore pressures approaching the saturated vapour pressure, which is why we chose ethane as a probing gas for this study; a wider range of adsorbed liquid/semi-liquid fluid saturations can be achieved using ethane than if methane is used as a probing gas.  110   Figure 5.1 Isothermal density-pressure relationships for methane (A) and ethane (B) at laboratory temperature (18 °C). SVP is saturated vapour pressure. Data from Lemmon et al., 2017.  Saturation of liquid/semi-liquid ethane increases with increasing pore pressure due to adsorption and capillary condensation, resulting in decreased permeability to ethane gas (Figure 5.2). These permeability decreases vary across a suite of samples with varying pore size. By measuring pore size of the sample suite using gas slippage measurements, we show that the largest drops in ethane relative permeability take place in rocks with the smallest pores. We show that gas slippage measurements can be used to predict relative permeability and therefore aid in predicting the deliverability of hydrocarbons from a shale oil or shale gas reservoir. 111   Figure 5.2 Schematic cross sections of pore throats showing adsorbed liquid/semiliquid ethane (red and grey molecules) restricting flow paths for ethane gas at high gas pressure, resulting in decreased effective permeability to ethane gas.  5.2 Methods 5.2.1 Sample Suite The sample suite includes two gas shales from the Eagle Ford Formation in Texas, USA (TEFB9 and TEF21), and three gas shales from the Montney Formation in British Columbia, Canada (MONT, B5FD1B6, and B5FD2A2). The Eagle Ford and Montney formations are two very important hydrocarbon producers (Rivard et al., 2014; Tunstall, 2014). The sample suite was chosen so that a wide range of matrix permeabilities and pore sizes were investigated. Sample TEFB9 was subsampled for two 3 cm diameter by ~3 cm length plugs, one oriented for fluid flow parallel to bedding (TEFB9Pll) and one for fluid flow perpendicular to bedding (TEFB9Pd). Subsample TEFB9Pll was evaluated at multiple stress states to understand the impact of the shale’s stress sensitive pore structure on its multiphase flow characteristics. For the rest of the samples, a single plug oriented for fluid flow parallel to bedding was evaluated at one stress state to build up a data set large enough that the multiphase flow characteristics of different shales could be compared. 112  5.2.2 Permeability Measurements Permeability measurements to helium and ethane were made on each sample using the pulse decay technique (Brace et al., 1968) at laboratory temperature (18 °C). Helium, an inert gas that does not adsorb to pore walls, was used for petrophysical characterization of pore structures in the absence of adsorbed fluid. Measurements were made at a range of different pore pressures (Figure 5.3) to vary mean free path of the gas such that Klinkenberg plots could be generated (apparent permeability, 𝐾𝑎, against inverse pore pressure). The difference between pore pressure (𝑃𝑝) and confining pressure (𝑃𝑐), referred to herein as ‘simple effective stress’, was kept constant by changing confining pressure by the same amount as pore pressure. Linear fits of the Klinkenberg plot data were extrapolated to infinite pore pressure in order to determine slip-free permeability, 𝐾∞, and Klinkenberg’s slippage factor, 𝑏 (Klinkenberg, 1941) 𝐾𝑎 = 𝐾∞(1 +𝑏𝑃𝑝)          (Eq. 5.1) Determining 𝑏 allows calculation of dominant pore size, 𝑑 (Klinkenberg, 1941), which represents an average diameter of the smallest pore throats along those flow paths responsible for the bulk of the fluid flux through the rock, calculated using a simplified model that represents pores as cylindrical capillary tubes (Chapter 3, subsection 3.2.2) 𝑑 =8𝑐𝜆𝑃𝑝𝑏          (Eq. 5.2) where 𝑐 is Adzumi’s constant (0.9; Adzumi, 1937), 𝑃𝑝 is pore pressure, and 𝜆 is mean free path  𝜆 =  𝑅𝑇√2𝜋𝑑𝑘𝑖𝑛2𝑁𝐴𝑃           (Eq. 5.3) where 𝑅 is the gas constant, 𝑇 is temperature, 𝑑𝑘𝑖𝑛 is kinetic diameter, 𝑁𝐴 is Avogadro’s constant, and  𝑃 is pressure (Loeb, 2004). 113   Figure 5.3 Pore pressure schedule for the suite of helium (blue) and ethane (red) permeability measurements made on each sample. The ethane saturated vapour pressure at the temperature of the permeability measurements (18 °C) is 3.59 MPa.  Pore size estimates from gas slippage measurements could also be made assuming pores have slot-shaped cross sectional geometries (Randolph et al., 1984). Although some studies argue based on photomicrographs that pores of some gas shales are more slot shaped than circular in cross section (e.g. Soeder, 1988), recent analysis of a large compiled data set that includes many different gas shales suggests pores in fine-grained sedimentary rocks are more accurately modelled as circular in cross sectional geometry (Chapter 3). We therefore use the circular model (Equation 5.2) to estimate pore size from gas slippage measurements in this study.  Ethane permeability measurements were made at a range of mean pore pressures increasing from 0.69 to 3.45 MPa (Figure 5.3). Pressure was allowed to equilibrate for a minimum of 12 hours at each pressure step. Pressure was monitored during the equilibration period and sufficient time was allowed for pressure to stabilize prior to measurement. Pressure pulses of 0.24 MPa were 114  generated by bleeding off pressure from the downstream reservoir. Pressure pulses need to be small enough that flow is laminar. While early studies of the pulse decay technique suggest a maximum pressure gradient of 0.03 MPa/cm (Jones, 1997), much smaller than the gradients imparted in the present study (0.08 MPa/cm), recent work shows that pressure gradients much greater than 0.03 MPa/cm can be used and flow still be in the laminar flow regime in low permeability samples (Feng et al., 2017a). Differential pressure was allowed to decay until reaching 0.01 MPa or, for lower permeability samples where differential pressure decays very slowly, until enough data was collected for an accurate permeability measurement.  5.2.3 Permeability Calculation Permeability was calculated from the pulse decay data using the equations presented in Cui et al. (2009). An isothermal pressure-viscosity curve (Lemmon et al., 2017) was used to determine the correct ethane viscosity for each permeability calculation. Adsorption/desorption was not accounted for in any of the permeability calculations. Ethane is a strongly sorptive gas and therefore adsorption/desorption could impact the accuracy of calculated ethane permeabilities (Cui et al., 2009). However, the upstream and downstream reservoirs of the permeameters used in this study are much larger than pore volumes of the samples (~10 times), so error in our permeability measurements resulting from not accounting for adsorption/desorption is less than 10% (Cui et al., 2009).  Feng et al. (2017a, 2017b) presented an experimental setup that would eliminate the need to account for adsorption when calculating permeability from pulse decay data. In their setup the pressure pulse is generated by both increasing the upstream gas pressure and decreasing the 115  downstream gas pressure by equal amounts from equal volume reservoirs. Pressure would therefore come to equilibrium at the same pressure the sample was soaked at prior to measurement, and the quantity of ethane adsorbed would be the same before and after the pressure decay. We did not adopt the experimental setup of Feng et al. (2017a and 2017b) because, in addition to quantifying permeability at each pressure step, we wanted to quantify the rate of change of ethane gas molecules in the adsorbed state with change in pore pressure, herein referred to as the ethane desorption rate. Having an experimental setup where pressure comes to equilibrium after the pressure pulse decay at a different pressure than the initial soak period pressure allowed calculation of the ethane desorption rate.  5.2.4 Ethane Desorption Rate Calculations Ethane desorption rates were calculated using the pressure data collected during permeability measurements. The desorption rate is equivalent to the slope of the desorption isotherm at the pore pressure of the permeability measurement. Desorption rate is quantified because it represents how much liquid/semi-liquid ethane saturation changes between successive ethane permeability measurements at different mean pore pressures. Because gas pressure is bled from the system when generating differential pressure across the sample, gas pressure in the upstream reservoir, downstream reservoir, and sample pore volume come to equilibrium at a lower pressure than the initial soak period equilibrium pressure. The permeameter is a closed system once the differential pressure is created, so gas molecules that desorb from the sample due to the drop in pore pressure result in higher post-decay equilibrium gas pressure than if no molecules were to desorb (Figure 5.4). Using the pressure data, the known reservoir volumes, and estimates of pore volume, Boyle’s Law was used to calculate the amount of gas molecules desorbed per 116  MPa drop in total system pressure at the mean pore pressure of each permeability measurement (the ethane desorption rate). Pore volumes were estimated from the dimensions of the samples and porosities determined using a combination of unconfined helium pycnometry and mercury immersion. For the ethane desorption rate calculations, compressibility factors were calculated using the Peng-Robinson equation of state (Peng and Robinson, 1976).   Figure 5.4 Comparison of helium (A) and ethane (B) pulse decay data for sample B5FD2A2. Mean pore pressure increases throughout the helium permeability measurement because the upstream reservoir of the permeameter is larger than the downstream reservoir, and gas was bled from the downstream reservoir to generate the differential pressure. Mean pore pressure increases by more in the ethane measurement than the helium measurement because liquid/semi-liquid ethane desorbs to reach equilibrium at the final gas pressure, which is lower than the equilibrium gas pressure during the soak period (time < 0 on graphs). Differential pressure in the ethane permeability measurement takes more than twice as long to decay than the helium differential pressure because adsorbed liquid/semi-liquid ethane constricts flow paths resulting in lower ethane gas permeability.  5.2.5 Ethane Gas Slippage Estimates Mean free path decreases with increasing gas pressure resulting in gas slippage being less significant and therefore lower apparent permeability (Klinkenberg, 1941). When ethane is used as a probing gas, effective permeability to ethane gas varies with pore pressure not only due to 117  gas slippage, but also due to changes in liquid/semi-liquid ethane saturation resulting from adsorption and capillary condensation. In order to distinguish these two components of permeability variation, Klinkenberg’s slippage factor for ethane was calculated by multiplying the slippage factor measured for helium by the ratio of ethane mean free path to helium mean free path at a given pressure and temperature. This ratio is equal to the kinetic diameter of helium (0.26 nm, Breck, 1974) squared over the kinetic diameter of ethane (0.44 nm, Aguado et al., 2012) squared (Equation 5.3). Expected ethane permeability variation due to gas slippage was then calculated using the lowest pressure ethane permeability measurement and the calculated ethane slippage factor.  5.3 Results 5.3.1 Helium Klinkenberg Plots 𝐾∞ from extrapolation of Klinkenberg plots ranges from 3 x 10-5 to 2 x 10-2 mD (Figure 5.5). 𝐾∞ of sample TEFB9 parallel to bedding decreases with increasing simple effective stress from 2 x 10-2 mD at 7 MPa simple effective stress to 8 x 10-4 mD at 48 MPa simple effective stress. 𝐾∞ of sample TEFB9 perpendicular to bedding at 7 MPa simple effective stress is two orders of magnitude lower than to flow parallel to bedding at the same stress state. 118   Figure 5.5 Klinkenberg plot for each sample generated using the helium permeability data.  Dominant pore diameter, 𝑑, ranges from 42 nm to 473 nm (Figure 5.6). Rocks with lower permeability have smaller pores (Figure 5.6). This however is only a general trend with a high degree of scatter. For example, samples MONT and TEF21 have the same dominant pore size 119  but permeabilities that differ by an order of magnitude (yellow triangle and red star in Figure 5.6).  Figure 5.6 Dominant pore size calculated using Equation 5.2 and the helium Klinkenberg plot data. Pores are generally smaller in samples with lower 𝑲∞, but with lots of variability around the trend. Dashed line delineates the macropore-mesopore boundary below which capillary condensation is expected at ethane gas pore pressures close to the saturated vapour pressure (Gregg, 1982).  120  5.3.2 Ethane Permeability Ethane permeability of all samples under all stress states and flow orientations decreases with increasing pore pressure (Figure 5.7). At mean pore pressures less than 2.3 MPa, the decreases in permeability closely match the predicted permeability decreases due to gas slippage (Figure 5.7). Ethane permeability deviates from what is predicted due to gas slippage (becomes lower) at higher pore pressures (Figure 5.7). The pore pressure where deviation takes place varies within the sample suite. The lowest pressure permeability deviation is for sample MONT at ~2.3 MPa (Figure 5.7), whereas for other samples deviation does not occur until pressures above 3 MPa.      121     Figure 5.7 Ethane permeability measurements (coloured markers) and predicted permeability variation due to gas slippage (black markers) calculated using the ethane slippage factor, which was derived from the helium permeability data.  122  At higher pore pressures, all samples have lower permeability than what is predicted due to gas slippage, but the magnitude of the difference between measured permeability and predicted permeability varies within the sample suite; at 3.45 MPa mean pore pressure (the highest mean pore pressure measured), the difference between measured permeability and permeability predicted due to gas slippage ranges from 28 to 84 percent (Table 5.1, column 6). Table 5.1 Experimental conditions and tabulated data for each sample. Sample Simple effective stress (MPa) K∞ (mD) bhelium bethane % permeability drop at 3.45 MPa* TEFB9Pll 7 1.59E-02 0.20 0.07 33 TEFB9Pll 21 8.82E-04 0.38 0.13 39 TEFB9Pll 48 8.04E-05 0.67 0.23 45 TEFB9Pd 7 1.33E-04 1.45 0.51 28 MONT 21 3.54E-04 2.29 0.80 84 TEF21 21 2.91E-05 2.37 0.83 76 B5FD1B6 7 4.08E-04 1.27 0.44 29 B5FD2A2 21 6.26E-03 0.97 0.34 44 * Percent difference between predicted permeability and measured ethane gas  permeability at 3.45 MPa mean pore pressure   5.3.3 Ethane Desorption Rates Ethane desorption rates over the pressure range of the permeability measurements are only presented for the five highest permeability samples (Figure 5.8). Desorption rates for the three lowest permeability samples (TEF21, TEFB9Pd at 7 MPa SES, and TEFB9Pll at 48 MPa SES) are not presented because accurate calculations could not be made from the pressure data. Differential pressure was only allowed to partially decay for the low permeability measurements due to the excessive time required for full pressure decay (multiple days to weeks). Pressure data would therefore have to be extrapolated to estimate the final equilibrium pressure after decay. Additionally, because pressure data series for the lowest permeability samples are collected over 123  a longer time scale (days) than higher permeability samples (minutes to hours), small temperature fluctuations can significantly influence mean pore pressure (Figure 5.9). The data were collected in a temperature controlled environment, but at the time scale of days, temperature drifts on the order of a quarter of a degree Celsius take place (e.g. Figure 5.9D). 124   Figure 5.8 Ethane desorption rates calculated from the pulse decay pressure data. Desorption rate is elevated at high pressures for sample TEFB9Pll at 7 and 21 MPa simple effective stress and sample B5FD2A2. Desorption rate is not elevated above resolution limit for samples MONT and B5FD1B6.  125   Figure 5.9 Examples of good quality ethane pulse decay pressure data for a high permeability sample from which accurate desorption rates could be calculated (A and C) and poor quality data for a low permeability sample from which accurate desorption rates could not be calculated (B and D). Temperature fluctuation over the duration of the measurement has a significant impact on mean pore pressure for the low permeability sample (D). Data collection for the low permeability sample was stopped at ~0.07 MPa differential pressure because of the excessive time required for full decay (B).  Sample TEFB9Pll at 7 and 21 MPa simple effective stress and sample B5FD2A2 all show measurably elevated ethane desorption rates at high pore pressure (Figure 5.8). Samples MONT and B5FD1B6 do not show measurably elevated desorption rates at any mean pore pressure. Calculated desorption rates are slightly negative at some mean pore pressures for samples MONT and B5FD1B6.  126  5.4 Discussion 5.4.1 Desorption Rate Data Quality Measured desorption rates are valuable data in this study because they represent by how much liquid/semi-liquid ethane saturation is changing between successive permeability measurements, and the goal of the study is to make matrix permeability measurements at different fluid saturations. However, the apparatuses used to measure desorption rates in this study (pulse decay permeameters) are not optimized for measuring this rate. A discussion of the resulting data quality is therefore required.  The negative desorption rates calculated at some pore pressures for samples MONT and B5FD1B6 counterintuitively suggest desorption of ethane is occurring when pressure is increasing. These negative rates are not real but are an artifact resulting from the resolution limit of the experimental setup combined with temperature fluctuation (Figure 5.9). Because the upstream and downstream reservoirs of the permeameters used in this study where chosen to be much larger than the pore volumes of the samples (in order to minimize permeability measurement error due to adsorption), ethane desorbing during permeability measurements only resulted in small mean pore pressure increases (0.06 MPa at most). Small temperature fluctuations therefore result in significant absolute errors in measured desorption rates and large relative errors for small desorption rates. Because of the errors, it is possible that when the calculated desorption rates are near-zero, liquid/semi-liquid ethane saturation could be increasing by small amounts between successive permeability measurements.  127  To illustrate how near zero desorption rates that cannot be accurately quantified from the pulse decay pressure data could in fact represent small but significant changes in liquid/semi-liquid ethane saturation, desorption rate data for sample TEFB9Pll at 7 MPa simple effective stress is converted to a liquid/semi-liquid ethane saturation curve (Figure 5.10). This sample is the highest permeability sample of the suite and therefore has the highest quality pressure data for calculating desorption rates (it was the least susceptible to temperature fluctuation). An adsorbed liquid/semi-liquid fluid density of 11553 moles per m3, the density of liquid ethane at the ethane saturated vapour pressure at 18 °C (Lemmon et al., 2017), is assumed for the calculations. Liquid/semi-liquid ethane saturation in the sample increases by a small but not insignificant ~5% between 0.7 and 2.2 MPa mean pore pressure (Figure 5.10) even though desorption rate is only slightly above zero over this pressure range (Figure 5.8). Due to the resolution limit of the desorption rate measurements, sources of error (temperature fluctuation), and the assumed density, the ethane saturation curve in Figure 5.10 is a crude estimate and saturation curves for lesser quality data from lower permeability samples are not presented.  Figure 5.10 Liquid/semi-liquid ethane saturation curve for sample TEFB9Pll at 7 MPa simple effective stress.  128  5.4.2 Ethane Permeability at Low Pore Pressures At low pore pressures (< 2 MPa) the close match between apparent ethane permeability and permeability variation predicted due to gas slippage (Figure 5.7) indicates that adsorption of liquid/semi-liquid ethane is not significantly impacting ethane gas permeability. If liquid/semi-liquid ethane was blocking fluid flow, measured apparent permeability would be lower than predicted permeability. Ethane desorption rates for all samples with desorption data are near zero at pore pressures less than 2.2 MPa (Figure 5.8), indicating no changes in liquid/semi-liquid ethane saturation between successive permeability measurements, or changes so small they are not measurable with our experimental setup. No, or very small, liquid/semi-liquid ethane saturation increases with increasing pore pressure explains the lack of deviation from permeability variation predicted due to gas slippage.  5.4.3 Ethane Permeability at High Pore Pressures At high ethane pore pressures, ethane adsorption ± capillary condensation results in constrictions ± restrictions of fluid flow pathways and hence lower ethane gas permeability than is predicted due to gas slippage (Figure 5.7). Permeability decreases are substantial with up to an 84% difference between measured and predicted apparent ethane permeability (Table 5.1). The large permeability decreases indicate that permeability of fine grained rocks with nanometer-scale pore systems can be very sensitive to change in fluid saturation.   Significant increases in liquid/semi-liquid ethane saturation with increasing pore pressure are inferred from elevated desorption rates for three of the five samples with desorption data (Figure 5.8). These desorption rates however do not correlate well with the measured apparent 129  permeability decreases; desorption rates are below measurement resolution over the entire pore pressure range of the ethane permeability measurements for sample MONT (Figure 5.8) even though that sample has the largest ethane gas permeability decrease relative to predicted permeability (84%).   Difference between predicted ethane gas permeability and measured permeability has a weak negative correlation with 𝐾∞ (Figure 5.11A). This correlation is expected because lower permeability rocks generally have smaller pores, and the flow capacity of smaller flow conduits should be more sensitive to liquid/semi-liquid ethane adsorbed to conduit (pore) walls. A better segregation of the data however is found when difference between predicted ethane gas permeability and measured permeability is plotted against 𝑑 (Figure 5.11B). The two samples with markedly larger permeability drops (75% and 84%) have dominant pore sizes in the mesopore range compared to all other samples with smaller permeability drops (< 45%) and dominant pore sizes in the macropore range. 𝑑 represents an average size of the smallest pore throats along those flow paths responsible for the bulk of the fluid flux through the rock. If 𝑑 is in the mesopore range, the main fluid flow conduits can be completely blocked due to capillary condensation at pore pressures near to the saturated vapour pressure (Gregg, 1982). Complete blockage of these conduits explains the very large permeability decreases for the samples with 𝑑 less than 50 nm (MONT and TEF21, Figures 5.6 and 5.7). 130   Figure 5.11 Correlation of difference between predicted ethane gas permeability and measured permeability at 3.45 MPa pore pressure (y axes) and 𝑲∞ (A) and 𝒅 (B).  Schematic pore network models illustrate the impact of pore structure on the sensitivity of ethane gas permeability to pore pressure (Figure 5.12). The pore network models are modifications of the network models in Sakhaee-Pour and Bryant (2015). Sakhaee-Pour and Bryant showed that acyclic pore models could reproduce the non-plateau like trends of capillary pressure versus saturation that are common for gas shales. A key attribute of acyclic pore models is that narrower 131  pore throats do not limit accessibility to the wider pore throats (Sakhaee-Pour and Bryant, 2015). The pore network model depicted in Figure 5.12 A and B is for a sample with pores in the portion of the pore structure most responsible for limiting fluid flow in the macropore range. This pore network could be used to represent sample TEFB9Pll at 7 MPa simple effective stress. Ethane gas permeability is not very sensitive to pore pressure in this sample (Figure 5.7), but significant quantities of ethane gas are adsorbed at high pore pressure (Figures 5.8 and 5.10). Such data indicate that the main flow conduits have minimum pore throat sizes in the macropore size range that are not completely blocked off at ethane pressures near the saturated vapour pressure (Figure 5.12B). The sample, however, can store significant quantities of ethane in the liquid/semi-liquid state in sub-macropore scale pores whose throats do not limit accessibility to the macropores (Figure 5.12B). 132   Figure 5.12 Acyclic pore network model for a pore structure where the pores most responsible for limiting fluid flow are in the macropore range (A and B) and mesopore range (C and D). Solid red lines indicate pores filled with liquid/semi-liquid ethane. Ethane capillary condensation at high pore pressures completely blocks off the highest flow capacity flow paths in the mesoporous pore network (D) but has a limited impact on flow capacity in the macroporous pore network (B). Pore network models are modified from Sakhaee-Pour and Bryant, 2015.  In contrast to TEFB9Pll and the schematic pore network in Figure 5.12 A and B, the main flow conduits of the pore network model presented in Figure 5.12 C and D are in the mesopore size 133  range. The mesopores are completely blocked off due to capillary condensation at ethane pressures near the saturated vapour pressure (Figure 5.12 D), resulting in ethane gas permeability that is very sensitive to pore pressure. This pore network model would explain the experimental observations of sample MONT; 𝑑 was 41 nm for MONT and permeability of this sample was most sensitive to pore pressure.  5.4.4 Stress Sensitivity of Ethane Permeability Increasing stress narrows the pores and pore throats in sample TEFB9Pll resulting in decreasing 𝐾∞ (Figure 5.5). 𝑑 decreases from 473 to 259 to 145 nm at 7, 21, and 48 MPa simple effective stress respectively (Table 5.1). Concomitantly, the difference between predicted permeability and measured ethane gas permeability at 3.45 MPa mean pore pressure increases from 33% to 39% to 45% with increasing stress (Table 5.1) because liquid/semi-liquid ethane adsorbed to pore walls has a greater impact on the flow capacity of smaller flow conduits than larger flow conduits. Stress sensitivity of permeability could only be evaluated for sample TEFB9 for flow parallel to bedding because TEFB9Pll was the only plug in this study measured at multiple stress states.   5.4.5 Anisotropy of Ethane Permeability Permeability of sample TEFB9 is highly anisotropic; at 7 MPa simple effective stress 𝐾∞ to helium parallel to bedding is two orders of magnitude higher than to flow perpendicular to bedding. Calculated 𝑑 of subsample TEFB9Pd at 7 MPa simple effective stress (67 nm) is seven times smaller than 𝑑 of sample TEFB9Pll at 7 MPa simple effective stress (473 nm). The large difference in dominant pore size explains the permeability anisotropy. The large difference in 134  pore size however does not explain why ethane permeability is less sensitive to pore pressure at high pore pressures for flow perpendicular to bedding than for flow parallel to bedding; permeability to ethane gas should be more sensitive to adsorption and capillary condensation when flow is in the direction with smaller flow paths. The calculated dominant pore sizes, however, only represent an average size of the smallest pore throats along those flow paths responsible for the bulk of the fluid flux through the rock. In reality, the pore structure of sample TEFB9 is likely composed of a distribution of different sized pores. It is therefore possible that significant portions of fluid flow parallel to bedding take place along flow paths where the smallest pores are much smaller than the average, as well as in flow paths where the smallest pores are much larger than the average. For example, a continuous network of organic matter with connected, small pores (micro to mesopores) could contribute significant flow parallel to bedding. This flow would be sensitive to adsorption and capillary condensation at high ethane gas pressures. Perpendicular to bedding the organic matter might not be continuous and therefore would not contribute to flow. Perpendicular to bedding the pores most responsible for limiting fluid flow could be on average smaller than parallel to bedding, but have a narrower distribution of pore sizes that excludes the very small pores in the organic matter. Ethane permeability perpendicular to bedding could therefore be less sensitive than ethane permeability parallel to bedding to adsorption and capillary condensation at high ethane gas pressure.  5.4.6 Permeability Hysteresis  The measurements of effective permeability to ethane gas in this study were made over a range of increasing pore pressures. Permeability measurements over a range of decreasing pore pressures could also have been made to investigate the path dependence of permeability, but 135  were not made in this study due to time constraints. Capillary condensation is hysteretic (Gregg, 1982). In fine-grained reservoir rocks with mesopores, it is therefore expected that liquid/semi-liquid ethane saturation would be path dependent. Path dependent saturation should result in permeability hysteresis. Future studies should measure effective permeability to ethane gas over a range of both increasing and decreasing pore pressures to investigate permeability hysteresis. Valuable insights about pore structure could potentially be derived from analysis of permeability hysteresis loops.  5.5 Conclusion Ethane gas permeability measurements on stressed samples can be used as a tool to explore multiphase flow phenomena in fine grained sedimentary rocks at in situ reservoir stress conditions. Because pores in fine grained sedimentary rocks are generally of sub-micrometer length scale, significant liquid/semi-liquid ethane saturation exists due to adsorption and capillary condensation at ethane gas pressures near the saturated vapour pressure. The liquid/semi-liquid ethane causes restrictions and blockages of flow pathways and therefore reductions to the flow capacity of fine-grained sedimentary rocks.  The sensitivity of ethane gas permeability to adsorption and capillary condensation at high pore pressures varies within the suite of stressed samples analyzed in this study. The variability is controlled by the size of the pores most responsible for limiting fluid flow, which can be accurately determined using gas slippage measurements (Chapter 3). Dominant pore size ranges from tens to hundreds of nanometers. Samples with dominant pore sizes in the mesopore range are far more sensitive to ethane gas pore pressure because capillary condensation causes 136  complete blockage of the main flow conduits at pore pressures near the saturated vapour pressure.  The multiphase flow characteristics of sample TEFB9Pll vary with stress state; ethane permeability of sample TEFB9Pll is more sensitive to pore pressure at higher stress states. At higher stress states the pores are smaller and therefore the presence of liquid/semi-liquid ethane has a larger impact on flow capacity. The stress sensitivity of multiphase flow characteristics revealed here highlights the importance of measuring effective permeability at in situ reservoir stress conditions in order to obtain accurate reservoir simulation inputs.  137  Chapter 6: Conclusion  6.1 Overview  This thesis develops techniques for measuring key petrophysical parameters of fine-grained sedimentary reservoir rocks in the laboratory at reservoir stress states, and anisotropy of these parameters. The research is motivated by the highly stress-sensitive, anisotropic nature typical of the pore structures of fine-grained rocks, coupled with a lack of techniques available for reliable petrophysical measurements of these important, but difficult to analyze, materials. To illustrate this stress sensitivity and anisotropy, six permeability measurements of a single Montney Formation sample are presented in Figure 6.1. Measurements are made at three stress states on two subsamples, one oriented for fluid flow parallel to bedding and one for fluid flow perpendicular to bedding. For both subsamples, permeability decreases by more than two orders of magnitude between 7 and 35 MPa simple effective stress. Permeability parallel to bedding is ~2 orders of magnitude higher than perpendicular to bedding at any given stress state. The large permeability decreases with stress are a direct reflection of changes to pore geometry, and pore geometry needs to be characterized in order to predict the many geometry-dependent phenomena that dictate the economic utility of a reservoir. Of interest is pore structure characterizations that quantify the pore structure geometry of the reservoir at the stress states experienced during production. Because of how sensitive these reservoirs are to stress, it is not anticipated that pore geometry characterizations made at ambient stress states will yield adequate information to predict in situ reservoir phenomena. The techniques developed in this thesis can all be employed to make measurements on fine-grained reservoir rocks at in situ stress states, and to quantify anisotropy of the different petrophysical parameters determined using the techniques. The 138  techniques therefore yield petrophysical measurements to be used as inputs for reservoir simulators to generate accurate predictions of reservoir behaviour.  Figure 6.1 Permeability-stress relationships for a single Montney Formation sample. Helium pulse decay permeability measurements were made at multiple pore pressures at each simple effective stress to generate Klinkenberg plots. Klinkenberg plots were extrapolated to infinite pore pressure to determine K∞. Permeability decreases with increasing stress and is lower to flow perpendicular to bedding than parallel to bedding.  6.2 Technique Developments This thesis shows that the size of the smallest pore throats along those flow paths responsible for the bulk of the fluid flux through a fine-grained reservoir rock (dominant pore size) can be calculated from laboratory measurements of gas slippage. The good agreement between pore sizes calculated from gas slippage measurements and pore sizes calculated from bundle of 139  capillary tubes models in Chapter 3 shows that the pore size calculations are quantitatively accurate, and can therefore be used to predict geometry-dependent properties such as effective permeability (as in Chapter 5).   Gas slippage measurements can be made on core plugs stressed to the stress states experienced during production from fine-grained reservoir rocks. Pore size can therefore be calculated at these stresses, and anisotropy of pore size can be determined by measuring gas slippage on plugs oriented for fluid flow in different directions relative to bedding planes.  The importance of being able to measure pore size at various stress states and flow orientations is illustrated by plotting dominant pore size calculated from gas slippage measurements against 𝐾∞ in Figure 6.2. These data are for the same Montney Formation sample discussed above (Figure 6.1). For this single sample, dominant pore size varies between 10 and 600 nm depending on flow orientation and stress state. Dominant pore size is strongly positively correlated with permeability, and results in variation of permeability between 10-6 and 10-1 mD. Pore structure characterization techniques that yield pore size measurements at ambient stress conditions, such as scanning electron microscopy and low pressure gas adsorption, clearly do not yield results applicable to predicting in situ reservoir phenomena. Pore size would be overestimated by both techniques, and therefore would require correction for compression with stress. Low pressure gas adsorption would not yield any information about anisotropy. Anisotropy could be quantified using scanning electron microscopy, but in addition to not making measurements at stress, scanning electron microscopy is limited by its resolution limit and scalability (Chalmers et al., 2012a). Mercury intrusion porosimetry is another technique that can be used to measure pore 140  size, but this technique is also hampered by its resolution limit, inability to quantify anisotropy, and inability to control stress state. The gas slippage technique, which can be applied on stressed samples at different flow orientations (Figure 6.2), provides important petrophysical information that cannot be acquired using other commonly applied techniques.  Figure 6.2 Dominant pore size calculated from gas slippage measurements at different stress states and flow orientations for the same Montney Formation sample as in Figure 6.1. Pll indicates flow parallel to bedding and Pd perpendicular to bedding.  Matrix permeability is a key petrophysical parameter that can control production of hydrocarbons from fine-grained sedimentary reservoir rocks (Bustin and Bustin, 2012). Quantifying matrix permeability at the range of stress states experienced during production is therefore required for accurate reservoir simulations and production forecasts. However, 141  measuring matrix permeability in the laboratory at the exact confining pressures and pore pressures of the subsurface reservoir is challenging. Producing shale oil and shale gas reservoirs are located multiple kilometers below surface where fluid pressures are high enough to drive economic flow rates. The high confining pressures and pore pressures at these depths are not easily replicated in the laboratory. It is thus desirable to make measurements at lower pressures, such as those in Figure 6.1, and extrapolate to reservoir pressures. Extrapolation requires an effective stress law for permeability (Robin, 1973), but determining effective stress laws for fine-grained sedimentary reservoir rocks is complicated by the increased influence of gas slippage on flow rate in rocks with nanometer-scale pore systems. Prior to the research presented in Chapter 4, the severity of these complications was not recognized, and permeability effective stress laws for fine-grained reservoir rocks were published that are in error due to gas slippage (e.g. Heller et al., 2014; Letham and Bustin, 2016).  This thesis shows that gas slippage naturally drives apparent permeability effective stress law coefficients to values less than one at lower pore pressures and higher confining pressures. Distinguishing gas slippage from stress effects is challenging because they simultaneously result in apparent permeability changes with changing pressure. The two can be semi-quantitatively distinguished by analyzing Klinkenberg plots generated with measurements made at constant simple effective stress and pore pressures low enough that gas slippage is easily recognizable. If the Klinkenberg plots are linear, pore structure remains constant and apparent permeability variation is solely due to gas slippage. The permeability effective stress law coefficient is equal to 1. If Klinkenberg plots are concave up, the effective stress law coefficient is greater than 1. If Klinkenberg plots are concave down, the effective stress law coefficient is less than 1. 142  In addition to varying with stress, matrix permeability of fine-grained sedimentary reservoir rocks varies with fluid saturation (effective permeability). Being able to measure effective permeability at different fluid saturations is important because fluid saturations can change during production from a reservoir. Accurate measurements of effective permeability are therefore required for accurate production forecasts, but are difficult to acquire in the laboratory. Measurements of effective permeability at different fluid saturations can be acquired using the technique developed in Chapter 5 of this thesis, where permeability to ethane gas at a range of pore pressures up to the saturated vapour pressure of ethane is measured at laboratory temperature. Liquid/semi-liquid ethane saturation increases with increasing pore pressure due to adsorption and capillary condensation. The liquid/semi-liquid ethane restricts flow paths resulting in decreased effective permeability to ethane gas. How sensitive effective permeability is to liquid/semi-liquid ethane saturation depends on pore size. Pores that are smaller than 50 nm (mesopores and micropores) will be completely blocked due to capillary condensation at pore pressures near the saturated vapour pressure (Gregg, 1982). Therefore, if the main fluid flow conduits in a fine-grained reservoir rock are through pores smaller than 50 nm, effective permeability will be very sensitive to liquid/semi-liquid ethane saturation. Dominant pore size calculations from gas slippage measurements are especially well suited for predicting sensitivity of effective permeability to liquid/semi-liquid ethane saturation because gas slippage measurements, which are derived from permeability measurements, are weighted to the pores most responsible for limiting fluid flow.  143  6.3 Insights Gained About Fine-Grained Reservoir Rocks Gas slippage and matrix permeability are measured on a large suite of different fine-grained reservoir rocks in this thesis. The resulting data set yields general conclusions about fine-grained reservoir rocks.   Chapter 2 illustrates how variable the pore structures of these rocks can be. The average size of the pores most responsible for limiting fluid flow in two rocks with the same matrix permeability can be an order of magnitude different. Two rocks with the same sized pores can have matrix permeabilities separated by three orders of magnitude. For individual samples, permeability is always lower perpendicular to bedding than parallel to bedding, but in some strata the average size of the pores most responsible for limiting fluid flow is independent of flow direction, and in other strata pores are much smaller for flow perpendicular to bedding than parallel to bedding. The observed variability of fine-grained reservoir rock pore structures highlights the necessity of having techniques capable of accurate characterizations at reservoir stress; in order to predict reservoir phenomena that dictate the economic utility of a reservoir, where the pore structure of the reservoir fits within this spectrum of variability must be known.  The permeability effective stress laws determined in Chapter 4 give an indication of how fine-grained reservoir rocks respond to stress changes. Four of the five samples analyzed have permeability effectives stress law coefficients (Equation 4.4) of α = 1. One sample has a permeability effective stress law coefficient less than one. These data indicate that an incremental increase in pore pressure has the same effect on permeability as an incremental decrease in confining pressure, or for the case of the sample with α < 1, an incremental increase 144  in pore pressure provides a smaller permeability increase than an incremental decrease in confining pressure. No fine-grained reservoir rocks measured to date have pore structures where α > 1 and therefore an incremental increase in pore pressure provides a larger permeability increase than an incremental decrease in confining pressure.  The ethane permeability measurements coupled with gas slippage measurements in Chapter 5 yield insights about multiphase flow phenomena in fine-grained reservoir rocks. The pores in these rocks are small enough that ethane adsorbed on pore walls causes significant permeability reduction. Pores in fine-grained reservoir rocks span a wide range of sizes from more than 1000 nm to less than 10 nm. This size range straddles the macropore-mesopore boundary. If the average size of the pores most responsible for limiting fluid flow is in the macropore range, matrix permeability will be far less sensitive to adsorbed fluid than if in the sub-mesopore size range.  6.4 Future Direction This thesis is a step forward in understanding pore systems in fine-grained reservoir rocks; techniques are developed that address deficiencies in our ability to measure important petrophysical parameters of fine-grained reservoir rocks in the laboratory at reservoir stress states, and to quantify anisotropy of these parameters. However, this thesis in many respects exposes several deficiencies. Some of these deficiencies are highlighted below.  The gas slippage technique applied to fine-grained reservoir rocks can be used to measure pore size in stressed samples, which is imperative given how stress sensitive these rocks can be 145  (Figure 6.2). The gas slippage technique measures an average size of the smallest pore throats along those flow paths responsible for the bulk of the fluid flux through the rock, which is ideal when used to predict transport properties such as effective permeability. However, the storage properties of a reservoir rock are as important as transport properties, and gas slippage measurements might do a very poor job of measuring the size of the pores that store most of the fluid. For example, a gas shale could have a high proportion of poorly-connected organic matter that hosts a large quantity of very small pores. These very small pores could be capable of storing large fluid volumes. This organic porosity however might contribute very little to flow relative to a network of larger, well-connected, interparticle pores. These interparticle pores are what would be characterized by gas slippage measurements. More work is needed to develop techniques capable of pore geometry measurements, at reservoir stress, of the pores responsible for fluid storage.  How likely it is that matrix permeability measurements of fine-grained reservoir rocks made at low pressures can be extrapolated to reservoir confining and pore fluid pressures is poorly constrained. This would be better constrained if semi-quantitative determinations of permeability effective stress law coefficients for a large suite of fine-grained reservoir rocks were determined by analyzing Klinkenberg plot linearity, as in Chapter 4. The semi-quantitative permeability effective stress law coefficient determinations, and therefore the constraint, would be more robust if Klinkenberg plots are generated with data collected at the same simple effective stress but over a larger range of confining and pore fluid pressures than in previous works (e.g. Heller et al., 2014; Letham and Bustin, 2016). Klinkenberg plots of this type should be generated for a large suite of fine-grained reservoir rocks. 146  The effective permeability measurements made at different liquid/semi-liquid ethane saturations in Chapter 5 are insightful and are a step forward to understanding and investigating multiphase flow phenomena in fine-grained reservoir rocks. However, the data could be even more insightful if collected using an apparatus that has better temperature control than the apparatuses used in Chapter 5. More accurate desorption rates and therefore ethane saturation curves could then be determined. 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AAPG Bulletin, 59(1), 154–158.   155  Appendix Appendix A  Compiled Gas Slippage Data Set Study 𝑲∞ (mD) 𝒃 for helium at 18°C (MPa) Klinkenberg, 1941 2.37E+01 2.32E-02 Klinkenberg, 1941 2.44E+01 1.92E-02 Klinkenberg, 1941 3.21E+01 3.47E-02 Klinkenberg, 1941 9.23E+01 1.34E-02 Klinkenberg, 1941 1.05E+02 1.34E-02 Klinkenberg, 1941 1.70E+02 1.38E-02 Klinkenberg, 1941 1.05E+03 6.99E-03 Klinkenberg, 1941 1.35E+03 4.39E-03 Heid et al., 1950 4.95E+01 5.76E-02 Heid et al., 1950 1.87E+02 5.11E-02 Heid et al., 1950 8.60E+00 4.80E-02 Heid et al., 1950 1.10E+01 6.25E-02 Heid et al., 1950 1.35E-01 6.70E-01 Heid et al., 1950 2.30E+00 1.20E-01 Heid et al., 1950 2.70E+01 6.19E-02 Heid et al., 1950 3.30E+01 7.74E-02 Heid et al., 1950 2.10E+00 1.68E-01 Heid et al., 1950 4.85E+00 1.52E-01 Heid et al., 1950 2.60E+01 4.91E-02 Heid et al., 1950 6.20E+00 6.66E-02 Heid et al., 1950 1.09E+01 7.94E-02 Heid et al., 1950 9.60E+00 6.15E-02 Heid et al., 1950 6.10E-01 1.21E-01 Heid et al., 1950 7.20E-01 2.21E-01 Heid et al., 1950 1.18E+01 7.69E-02 Heid et al., 1950 1.65E+01 5.96E-02 Heid et al., 1950 1.25E+01 5.98E-02 Heid et al., 1950 2.40E+01 8.20E-02 Heid et al., 1950 2.10E+01 4.87E-02 Heid et al., 1950 4.40E+00 8.26E-02 Heid et al., 1950 4.60E+00 5.96E-02 Heid et al., 1950 1.74E+02 4.03E-02 Heid et al., 1950 1.20E+01 6.23E-02 Heid et al., 1950 1.16E+00 1.61E-01 156  Study 𝑲∞ (mD) 𝒃 for helium at 18°C (MPa) Heid et al., 1950 4.10E+00 6.96E-02 Heid et al., 1950 7.50E-02 5.24E-01 Heid et al., 1950 2.37E+01 3.18E-02 Heid et al., 1950 2.40E+01 1.02E-01 Heid et al., 1950 7.20E-03 2.08E-01 Heid et al., 1950 8.27E+01 3.16E-02 Heid et al., 1950 2.16E+02 2.28E-02 Heid et al., 1950 6.75E+00 8.00E-02 Heid et al., 1950 4.87E+01 3.75E-02 Heid et al., 1950 4.15E+01 9.00E-02 Heid et al., 1950 1.82E+01 2.59E-02 Heid et al., 1950 8.00E+01 1.03E-01 Heid et al., 1950 2.15E+01 7.76E-02 Heid et al., 1950 6.75E+01 7.06E-02 Heid et al., 1950 8.55E+01 2.06E-02 Heid et al., 1950 5.32E+02 3.52E-03 Heid et al., 1950 3.00E+02 9.18E-03 Heid et al., 1950 8.75E+01 1.91E-02 Heid et al., 1950 2.70E+00 1.13E-01 Heid et al., 1950 1.57E+01 2.01E-02 Heid et al., 1950 5.80E+01 3.22E-02 Heid et al., 1950 2.00E+01 8.55E-02 Heid et al., 1950 6.12E+02 1.03E-02 Heid et al., 1950 3.23E+02 1.52E-02 Heid et al., 1950 5.00E+02 2.75E-03 Heid et al., 1950 1.00E+02 4.23E-02 Heid et al., 1950 9.40E-01 1.15E-01 Heid et al., 1950 1.71E+02 3.91E-02 Heid et al., 1950 1.74E+02 1.75E-02 Heid et al., 1950 3.66E+02 2.69E-03 Heid et al., 1950 5.10E+01 3.46E-02 Heid et al., 1950 3.00E+01 3.26E-02 Heid et al., 1950 2.49E+02 1.10E-02 Heid et al., 1950 2.56E+02 9.20E-03 Heid et al., 1950 1.17E+01 5.21E-02 Heid et al., 1950 4.40E-01 3.16E-01 Heid et al., 1950 2.20E-01 6.33E-01 Heid et al., 1950 1.67E+02 1.92E-02 Heid et al., 1950 1.29E+03 5.35E-03 157  Study 𝑲∞ (mD) 𝒃 for helium at 18°C (MPa) Heid et al., 1950 1.77E+01 4.13E-02 Heid et al., 1950 1.48E+03 1.34E-03 Heid et al., 1950 4.22E+01 3.91E-02 Heid et al., 1950 8.60E+02 1.14E-02 Heid et al., 1950 9.65E+02 2.75E-02 Heid et al., 1950 4.25E+02 1.45E-01 Heid et al., 1950 2.58E+00 8.85E-02 Heid et al., 1950 4.35E-01 2.91E-01 Heid et al., 1950 3.00E-01 3.30E-01 Heid et al., 1950 9.45E+00 5.72E-02 Heid et al., 1950 1.00E+01 9.44E-02 Heid et al., 1950 1.05E+02 2.44E-02 Heid et al., 1950 2.24E+02 3.93E-02 Heid et al., 1950 3.02E+02 1.50E-02 Heid et al., 1950 9.50E-03 3.20E-01 Heid et al., 1950 2.15E-01 3.97E-01 Heid et al., 1950 5.40E-01 1.65E-01 Heid et al., 1950 1.50E-03 9.97E-01 Heid et al., 1950 1.70E-03 6.88E-01 Heid et al., 1950 4.40E-04 1.04E+00 Heid et al., 1950 9.10E+01 1.43E-01 Heid et al., 1950 1.03E+01 6.41E-02 Heid et al., 1950 1.57E+01 9.51E-02 Heid et al., 1950 7.95E+00 9.73E-02 Heid et al., 1950 7.20E+00 8.20E-02 Heid et al., 1950 9.60E+00 9.63E-02 Heid et al., 1950 1.35E+00 1.83E-01 Heid et al., 1950 2.80E+00 1.02E-01 Heid et al., 1950 1.60E+00 2.58E-01 Heid et al., 1950 4.00E+00 1.62E-01 Heid et al., 1950 2.10E+01 8.88E-02 Heid et al., 1950 8.50E+01 8.49E-02 Heid et al., 1950 2.72E+02 1.06E-01 Heid et al., 1950 2.01E+02 3.81E-02 Heid et al., 1950 2.87E+01 4.66E-02 Heid et al., 1950 2.55E+01 4.82E-02 Heid et al., 1950 1.45E+02 3.52E-02 Heid et al., 1950 1.12E+02 5.96E-02 Heid et al., 1950 2.30E+00 1.71E-01 158  Study 𝑲∞ (mD) 𝒃 for helium at 18°C (MPa) Heid et al., 1950 4.20E+00 1.01E-01 Heid et al., 1950 7.60E+00 7.23E-02 Heid et al., 1950 6.00E-01 1.64E-01 Heid et al., 1950 2.50E-01 1.73E+00 Heid et al., 1950 1.01E+01 9.95E-02 Heid et al., 1950 9.20E+01 1.71E-02 Heid et al., 1950 3.55E+01 7.19E-02 Heid et al., 1950 1.52E+02 2.59E-02 Heid et al., 1950 2.71E+02 7.63E-03 Heid et al., 1950 2.75E+01 2.79E-02 Heid et al., 1950 4.75E+01 2.28E-02 Heid et al., 1950 6.50E+01 2.71E-02 Heid et al., 1950 1.19E+02 3.79E-02 Heid et al., 1950 9.25E+01 3.93E-02 Heid et al., 1950 8.80E+01 1.11E-02 Heid et al., 1950 8.60E+01 4.56E-03 Heid et al., 1950 5.60E+01 4.74E-02 Heid et al., 1950 5.25E+01 3.36E-02 Heid et al., 1950 1.81E+02 3.48E-03 Heid et al., 1950 1.89E+02 1.31E-02 Heid et al., 1950 2.14E+02 8.28E-03 Heid et al., 1950 1.19E+02 1.32E-02 Heid et al., 1950 1.24E+02 7.92E-03 Heid et al., 1950 1.28E+02 3.54E-02 Heid et al., 1950 1.15E+02 1.54E-02 Heid et al., 1950 9.70E+00 3.05E-02 Heid et al., 1950 1.18E+01 4.17E-02 Heid et al., 1950 4.90E+00 6.60E-02 Heid et al., 1950 9.95E+01 2.67E-02 Heid et al., 1950 6.70E+01 2.63E-02 Heid et al., 1950 8.40E+01 4.68E-02 Heid et al., 1950 1.19E+02 2.97E-02 Heid et al., 1950 5.80E+01 8.45E-03 Heid et al., 1950 1.30E+00 9.67E-02 Heid et al., 1950 2.35E+00 1.38E-01 Heid et al., 1950 2.20E+01 1.03E-01 Heid et al., 1950 1.45E-01 1.91E-01 Heid et al., 1950 1.70E-01 2.58E-01 Heid et al., 1950 1.50E-02 3.54E-01 159  Study 𝑲∞ (mD) 𝒃 for helium at 18°C (MPa) Heid et al., 1950 9.30E+00 6.35E-02 Heid et al., 1950 8.50E-01 5.43E-01 Heid et al., 1950 1.07E+00 1.04E-01 Heid et al., 1950 9.30E+00 7.82E-02 Heid et al., 1950 1.58E+01 3.73E-02 Heid et al., 1950 8.00E+00 5.29E-02 Heid et al., 1950 3.70E+00 1.12E-01 Heid et al., 1950 3.05E+00 9.00E-02 Heid et al., 1950 4.20E+00 7.49E-02 Heid et al., 1950 2.11E+03 4.52E-03 Heid et al., 1950 2.76E+02 1.14E-02 Heid et al., 1950 3.30E+02 1.79E-02 Heid et al., 1950 3.50E+02 3.93E-02 Heid et al., 1950 1.26E+02 3.42E-02 Heid et al., 1950 1.55E+00 1.59E-01 Heid et al., 1950 1.04E+03 2.18E-02 Heid et al., 1950 4.80E+00 1.02E-01 Heid et al., 1950 5.90E+00 1.07E-01 Heid et al., 1950 4.60E+00 1.07E-01 Jones, 1972 1.30E-02 2.51E-02 Jones, 1972 7.34E-02 1.42E-01 Jones, 1972 1.13E-01 2.19E-01 Jones, 1972 1.20E-01 2.33E-01 Jones, 1972 1.54E-01 2.98E-01 Jones, 1972 3.13E-01 6.07E-01 Jones, 1972 3.91E-01 7.58E-01 Jones, 1972 4.12E-01 8.00E-01 Jones, 1972 4.74E-01 9.20E-01 Jones, 1972 4.99E-01 9.67E-01 Jones, 1972 7.00E-01 1.36E+00 Jones, 1972 1.47E+00 2.86E+00 Jones, 1972 2.41E+00 4.68E+00 Jones, 1972 2.66E+00 5.16E+00 Jones, 1972 2.71E+00 5.26E+00 Jones, 1972 4.95E+00 9.60E+00 Jones, 1972 8.56E+00 1.66E+01 Jones, 1972 9.96E+00 1.93E+01 Jones, 1972 1.03E+01 2.00E+01 Jones, 1972 1.33E+01 2.58E+01 160  Study 𝑲∞ (mD) 𝒃 for helium at 18°C (MPa) Jones, 1972 1.71E+01 3.33E+01 Jones, 1972 1.89E+01 3.67E+01 Jones, 1972 2.34E+01 4.55E+01 Jones, 1972 2.26E+01 4.39E+01 Jones, 1972 2.33E+01 4.51E+01 Jones, 1972 3.58E+01 6.95E+01 Jones, 1972 9.49E+01 1.84E+02 Jones, 1972 7.54E+01 1.46E+02 Jones, 1972 4.37E+01 8.48E+01 Jones, 1972 5.56E+01 1.08E+02 Jones, 1972 1.17E+02 2.27E+02 Jones, 1972 8.62E+01 1.67E+02 Jones, 1972 6.61E+01 1.28E+02 Jones, 1972 5.75E+01 1.12E+02 Jones, 1972 5.14E+01 9.97E+01 Jones, 1972 4.42E+01 8.57E+01 Jones, 1972 3.21E+02 6.24E+02 Jones, 1972 2.89E+02 5.60E+02 Jones, 1972 3.08E+02 5.97E+02 Jones, 1972 1.64E+02 3.18E+02 Jones, 1972 6.02E+02 1.17E+03 Jones, 1972 4.61E+02 8.95E+02 Jones, 1972 4.50E+02 8.74E+02 Jones, 1972 5.67E+02 1.10E+03 Jones, 1972 3.09E+02 5.99E+02 Jones, 1972 2.37E+02 4.60E+02 Jones, 1972 1.96E+02 3.81E+02 Jones, 1972 8.51E+02 1.65E+03 Jones, 1972 9.76E+02 1.89E+03 Jones, 1972 3.86E+02 7.48E+02 Jones, 1972 3.37E+02 6.53E+02 Jones, 1972 2.68E+02 5.21E+02 Jones, 1972 2.45E+02 4.76E+02 Jones, 1972 2.44E+02 4.73E+02 Jones, 1972 1.94E+02 3.77E+02 Jones, 1972 2.13E+02 4.13E+02 Jones, 1972 1.74E+02 3.38E+02 Jones, 1972 1.61E+02 3.12E+02 Jones, 1972 1.51E+02 2.93E+02 161  Study 𝑲∞ (mD) 𝒃 for helium at 18°C (MPa) Jones, 1972 1.55E+02 3.01E+02 Jones, 1972 1.32E+02 2.55E+02 Jones, 1972 1.46E+02 2.83E+02 Jones, 1972 1.74E+02 3.38E+02 Jones, 1972 1.64E+02 3.18E+02 Jones, 1972 1.52E+02 2.95E+02 Jones, 1972 1.60E+02 3.10E+02 Jones, 1972 2.04E+02 3.95E+02 Jones, 1972 1.78E+02 3.45E+02 Jones, 1972 1.55E+02 3.00E+02 Jones, 1972 1.60E+02 3.11E+02 Jones, 1972 1.10E+02 2.12E+02 Jones, 1972 2.76E+01 5.35E+01 Jones, 1972 3.05E+01 5.92E+01 Jones, 1972 3.30E+01 6.40E+01 Jones, 1972 3.69E+01 7.16E+01 Jones, 1972 3.84E+01 7.45E+01 Jones, 1972 4.19E+01 8.13E+01 Jones, 1972 4.24E+01 8.23E+01 Jones, 1972 4.63E+01 8.98E+01 Jones, 1972 4.47E+01 8.67E+01 Jones, 1972 4.56E+01 8.84E+01 Jones, 1972 4.80E+01 9.31E+01 Jones, 1972 6.81E+01 1.32E+02 Jones, 1972 1.05E+02 2.03E+02 Jones, 1972 9.60E+01 1.86E+02 Jones, 1972 1.05E+02 2.04E+02 Jones, 1972 8.78E+01 1.70E+02 Jones, 1972 9.02E+01 1.75E+02 Jones, 1972 8.25E+01 1.60E+02 Jones, 1972 7.64E+01 1.48E+02 Jones, 1972 6.74E+01 1.31E+02 Jones, 1972 7.19E+01 1.39E+02 Jones, 1972 8.06E+01 1.56E+02 Jones, 1972 7.11E+01 1.38E+02 Jones, 1972 6.10E+01 1.18E+02 Jones, 1972 5.94E+01 1.15E+02 Jones, 1972 1.41E+02 2.74E+02 Jones, 1972 1.47E+02 2.84E+02 162  Study 𝑲∞ (mD) 𝒃 for helium at 18°C (MPa) Jones, 1972 1.26E+02 2.44E+02 Sampath and Keighin, 1982 8.10E-03 1.57E-02 Sampath and Keighin, 1982 1.38E-02 2.68E-02 Sampath and Keighin, 1982 4.21E-02 8.17E-02 Sampath and Keighin, 1982 5.36E-02 1.04E-01 Sampath and Keighin, 1982 2.20E-02 4.27E-02 Sampath and Keighin, 1982 1.16E-01 2.25E-01 Sampath and Keighin, 1982 5.60E-02 1.09E-01 Sampath and Keighin, 1982 1.34E-01 2.60E-01 Sampath and Keighin, 1982 2.52E-02 4.89E-02 Sampath and Keighin, 1982 1.43E-02 2.77E-02 Sampath and Keighin, 1982 5.00E-04 9.70E-04 Sampath and Keighin, 1982 1.40E-03 2.72E-03 Sampath and Keighin, 1982 4.80E-03 9.31E-03 Sampath and Keighin, 1982 2.80E-03 5.43E-03 Sampath and Keighin, 1982 1.24E-02 2.41E-02 Sampath and Keighin, 1982 8.70E-03 1.69E-02 Sampath and Keighin, 1982 2.38E-02 4.62E-02 Sampath and Keighin, 1982 4.00E-04 7.76E-04 Sampath and Keighin, 1982 1.70E-03 3.30E-03 Randolph et al., 1984 4.24E-03 6.83E-03 Randolph et al., 1984 1.40E-03 2.25E-03 Randolph et al., 1984 3.02E-03 4.86E-03 Randolph et al., 1984 5.47E-03 8.81E-03 Randolph et al., 1984 1.87E-03 3.01E-03 Randolph et al., 1984 7.70E-04 1.24E-03 Randolph et al., 1984 5.02E-03 8.08E-03 Randolph et al., 1984 3.58E-03 5.76E-03 Randolph et al., 1984 2.33E-03 3.75E-03 Randolph et al., 1984 4.26E-03 6.86E-03 Randolph et al., 1984 2.58E-03 4.15E-03 Randolph et al., 1984 1.70E-04 2.74E-04 Randolph et al., 1984 7.50E-04 1.21E-03 Randolph et al., 1984 1.01E-03 1.63E-03 Randolph et al., 1984 7.20E-04 1.16E-03 Randolph et al., 1984 1.57E-03 2.53E-03 Randolph et al., 1984 2.19E-03 3.53E-03 Randolph et al., 1984 1.57E-03 2.53E-03 Randolph et al., 1984 3.58E-03 5.76E-03 163  Study 𝑲∞ (mD) 𝒃 for helium at 18°C (MPa) Randolph et al., 1984 9.23E-03 1.49E-02 Randolph et al., 1984 1.71E-03 2.75E-03 Randolph et al., 1984 1.54E-03 2.48E-03 Randolph et al., 1984 6.60E-04 1.06E-03 Randolph et al., 1984 8.50E-04 1.37E-03 Randolph et al., 1984 1.04E-03 1.67E-03 Randolph et al., 1984 1.47E-03 2.37E-03 Randolph et al., 1984 5.40E-04 8.69E-04 Tanikawa and Shimamoto, 2009 5.91E-06 2.95E+01 Tanikawa and Shimamoto, 2009 9.01E-06 3.32E+01 Tanikawa and Shimamoto, 2009 2.05E-05 3.37E+01 Tanikawa and Shimamoto, 2009 1.48E-05 2.44E+01 Tanikawa and Shimamoto, 2009 4.67E-05 1.42E+01 Tanikawa and Shimamoto, 2009 5.63E-05 1.29E+01 Tanikawa and Shimamoto, 2009 4.35E-05 1.18E+01 Tanikawa and Shimamoto, 2009 2.60E-05 8.29E+00 Tanikawa and Shimamoto, 2009 1.41E-05 6.95E+00 Tanikawa and Shimamoto, 2009 5.63E-05 8.30E+00 Tanikawa and Shimamoto, 2009 8.59E-05 8.80E+00 Tanikawa and Shimamoto, 2009 6.64E-05 7.27E+00 Tanikawa and Shimamoto, 2009 8.02E-05 5.84E+00 Tanikawa and Shimamoto, 2009 1.70E-04 6.29E+00 Tanikawa and Shimamoto, 2009 1.25E-04 5.13E+00 Tanikawa and Shimamoto, 2009 1.22E-04 5.60E+00 Tanikawa and Shimamoto, 2009 1.55E-04 4.70E+00 Tanikawa and Shimamoto, 2009 4.79E-05 4.62E+00 Tanikawa and Shimamoto, 2009 6.34E-05 4.62E+00 Tanikawa and Shimamoto, 2009 5.64E-05 3.94E+00 Tanikawa and Shimamoto, 2009 5.14E-05 3.26E+00 Tanikawa and Shimamoto, 2009 3.62E-05 2.77E+00 Tanikawa and Shimamoto, 2009 6.98E-05 2.47E+00 Tanikawa and Shimamoto, 2009 7.84E-05 2.77E+00 Tanikawa and Shimamoto, 2009 6.65E-05 3.07E+00 Tanikawa and Shimamoto, 2009 8.03E-05 3.50E+00 Tanikawa and Shimamoto, 2009 8.21E-05 3.88E+00 Tanikawa and Shimamoto, 2009 1.11E-04 3.61E+00 Tanikawa and Shimamoto, 2009 1.58E-04 3.61E+00 Tanikawa and Shimamoto, 2009 2.25E-04 3.66E+00 Tanikawa and Shimamoto, 2009 3.06E-04 3.41E+00 164  Study 𝑲∞ (mD) 𝒃 for helium at 18°C (MPa) Tanikawa and Shimamoto, 2009 1.96E-04 2.99E+00 Tanikawa and Shimamoto, 2009 1.51E-04 2.90E+00 Tanikawa and Shimamoto, 2009 9.92E-05 2.94E+00 Tanikawa and Shimamoto, 2009 1.12E-04 1.66E+00 Tanikawa and Shimamoto, 2009 1.20E-04 1.76E+00 Tanikawa and Shimamoto, 2009 1.70E-04 1.84E+00 Tanikawa and Shimamoto, 2009 1.59E-04 1.62E+00 Tanikawa and Shimamoto, 2009 3.78E-04 2.37E+00 Tanikawa and Shimamoto, 2009 4.89E-04 2.17E+00 Tanikawa and Shimamoto, 2009 4.89E-04 1.99E+00 Tanikawa and Shimamoto, 2009 4.67E-04 1.57E+00 Tanikawa and Shimamoto, 2009 4.56E-04 1.26E+00 Tanikawa and Shimamoto, 2009 5.25E-04 1.24E+00 Tanikawa and Shimamoto, 2009 3.96E-04 1.30E+00 Tanikawa and Shimamoto, 2009 3.37E-04 1.04E+00 Tanikawa and Shimamoto, 2009 3.87E-04 9.99E-01 Tanikawa and Shimamoto, 2009 5.64E-04 1.09E+00 Tanikawa and Shimamoto, 2009 5.51E-04 9.43E-01 Tanikawa and Shimamoto, 2009 2.15E-04 1.17E+00 Tanikawa and Shimamoto, 2009 1.14E-04 9.14E-01 Tanikawa and Shimamoto, 2009 1.71E-04 6.07E-01 Tanikawa and Shimamoto, 2009 7.51E-05 5.09E-01 Tanikawa and Shimamoto, 2009 5.14E-04 4.47E-01 Tanikawa and Shimamoto, 2009 5.64E-04 7.25E-01 Tanikawa and Shimamoto, 2009 1.17E-03 2.47E+00 Tanikawa and Shimamoto, 2009 1.06E-03 1.90E+00 Tanikawa and Shimamoto, 2009 8.40E-04 1.62E+00 Tanikawa and Shimamoto, 2009 7.64E-04 1.62E+00 Tanikawa and Shimamoto, 2009 9.01E-04 1.28E+00 Tanikawa and Shimamoto, 2009 1.14E-03 1.30E+00 Tanikawa and Shimamoto, 2009 1.28E-03 1.21E+00 Tanikawa and Shimamoto, 2009 1.38E-03 1.06E+00 Tanikawa and Shimamoto, 2009 8.60E-04 1.00E+00 Tanikawa and Shimamoto, 2009 7.84E-04 8.27E-01 Tanikawa and Shimamoto, 2009 9.46E-04 7.36E-01 Tanikawa and Shimamoto, 2009 1.04E-03 9.03E-01 Tanikawa and Shimamoto, 2009 1.20E-03 8.27E-01 Tanikawa and Shimamoto, 2009 1.44E-03 9.16E-01 Tanikawa and Shimamoto, 2009 1.87E-03 8.77E-01 165  Study 𝑲∞ (mD) 𝒃 for helium at 18°C (MPa) Tanikawa and Shimamoto, 2009 1.55E-03 7.80E-01 Tanikawa and Shimamoto, 2009 1.91E-03 7.05E-01 Tanikawa and Shimamoto, 2009 2.36E-03 7.81E-01 Tanikawa and Shimamoto, 2009 2.92E-03 7.81E-01 Tanikawa and Shimamoto, 2009 2.92E-03 8.77E-01 Tanikawa and Shimamoto, 2009 2.42E-03 1.12E+00 Tanikawa and Shimamoto, 2009 2.72E-03 1.19E+00 Tanikawa and Shimamoto, 2009 3.86E-03 1.21E+00 Tanikawa and Shimamoto, 2009 4.34E-03 1.30E+00 Tanikawa and Shimamoto, 2009 4.44E-03 1.44E+00 Tanikawa and Shimamoto, 2009 4.14E-03 1.53E+00 Tanikawa and Shimamoto, 2009 4.65E-03 2.02E+00 Tanikawa and Shimamoto, 2009 1.03E-02 1.51E+00 Tanikawa and Shimamoto, 2009 7.62E-03 1.51E+00 Tanikawa and Shimamoto, 2009 6.32E-03 1.32E+00 Tanikawa and Shimamoto, 2009 9.20E-03 1.27E+00 Tanikawa and Shimamoto, 2009 8.37E-03 1.25E+00 Tanikawa and Shimamoto, 2009 6.47E-03 1.08E+00 Tanikawa and Shimamoto, 2009 5.62E-03 1.08E+00 Tanikawa and Shimamoto, 2009 5.76E-03 9.17E-01 Tanikawa and Shimamoto, 2009 4.55E-03 9.17E-01 Tanikawa and Shimamoto, 2009 3.86E-03 8.28E-01 Tanikawa and Shimamoto, 2009 3.60E-03 1.00E+00 Tanikawa and Shimamoto, 2009 1.35E-03 5.74E-01 Tanikawa and Shimamoto, 2009 2.85E-03 5.83E-01 Tanikawa and Shimamoto, 2009 6.03E-03 8.40E-01 Tanikawa and Shimamoto, 2009 5.12E-03 7.70E-01 Tanikawa and Shimamoto, 2009 5.62E-03 7.37E-01 Tanikawa and Shimamoto, 2009 4.66E-03 7.05E-01 Tanikawa and Shimamoto, 2009 4.66E-03 5.58E-01 Tanikawa and Shimamoto, 2009 3.96E-03 4.35E-01 Tanikawa and Shimamoto, 2009 3.69E-03 3.99E-01 Tanikawa and Shimamoto, 2009 2.05E-03 3.76E-01 Tanikawa and Shimamoto, 2009 1.51E-03 2.97E-01 Tanikawa and Shimamoto, 2009 2.48E-03 2.69E-01 Tanikawa and Shimamoto, 2009 4.46E-03 2.69E-01 Tanikawa and Shimamoto, 2009 5.64E-03 2.73E-01 Tanikawa and Shimamoto, 2009 6.05E-03 3.16E-01 Tanikawa and Shimamoto, 2009 5.01E-03 3.25E-01 166  Study 𝑲∞ (mD) 𝒃 for helium at 18°C (MPa) Tanikawa and Shimamoto, 2009 6.79E-03 4.36E-01 Tanikawa and Shimamoto, 2009 7.12E-03 4.69E-01 Tanikawa and Shimamoto, 2009 8.20E-03 4.05E-01 Tanikawa and Shimamoto, 2009 9.22E-03 4.55E-01 Tanikawa and Shimamoto, 2009 1.17E-02 4.17E-01 Tanikawa and Shimamoto, 2009 9.44E-03 3.07E-01 Tanikawa and Shimamoto, 2009 1.51E-02 4.42E-01 Tanikawa and Shimamoto, 2009 1.70E-02 5.35E-01 Tanikawa and Shimamoto, 2009 1.58E-02 6.28E-01 Tanikawa and Shimamoto, 2009 1.73E-02 7.71E-01 Tanikawa and Shimamoto, 2009 1.14E-02 9.05E-01 Tanikawa and Shimamoto, 2009 9.20E-03 8.53E-01 Tanikawa and Shimamoto, 2009 8.00E-03 6.10E-01 Tanikawa and Shimamoto, 2009 9.21E-03 5.50E-01 Tanikawa and Shimamoto, 2009 1.11E-02 5.12E-01 Tanikawa and Shimamoto, 2009 1.16E-02 1.05E+00 Tanikawa and Shimamoto, 2009 1.31E-02 1.13E+00 Tanikawa and Shimamoto, 2009 1.47E-02 1.08E+00 Tanikawa and Shimamoto, 2009 2.04E-02 8.41E-01 Tanikawa and Shimamoto, 2009 2.41E-02 8.79E-01 Tanikawa and Shimamoto, 2009 1.82E-02 3.50E-01 Tanikawa and Shimamoto, 2009 2.25E-02 2.26E-01 Tanikawa and Shimamoto, 2009 2.00E-02 2.33E-01 Tanikawa and Shimamoto, 2009 1.91E-02 2.50E-01 Tanikawa and Shimamoto, 2009 1.86E-02 2.61E-01 Tanikawa and Shimamoto, 2009 1.55E-02 2.39E-01 Tanikawa and Shimamoto, 2009 2.36E-02 2.58E-01 Tanikawa and Shimamoto, 2009 2.78E-02 2.65E-01 Tanikawa and Shimamoto, 2009 5.91E-03 1.95E-01 Tanikawa and Shimamoto, 2009 7.83E-03 2.01E-01 Tanikawa and Shimamoto, 2009 9.67E-03 2.10E-01 Tanikawa and Shimamoto, 2009 1.55E-03 1.68E-01 Tanikawa and Shimamoto, 2009 3.29E-03 8.35E-02 Tanikawa and Shimamoto, 2009 2.79E-02 6.82E-02 Tanikawa and Shimamoto, 2009 3.36E-02 8.87E-02 Tanikawa and Shimamoto, 2009 3.44E-02 1.12E-01 Tanikawa and Shimamoto, 2009 2.85E-02 1.12E-01 Tanikawa and Shimamoto, 2009 2.36E-02 1.15E-01 Tanikawa and Shimamoto, 2009 1.62E-02 1.06E-01 167  Study 𝑲∞ (mD) 𝒃 for helium at 18°C (MPa) Tanikawa and Shimamoto, 2009 1.17E-02 1.61E-01 Tanikawa and Shimamoto, 2009 1.48E-02 1.41E-01 Tanikawa and Shimamoto, 2009 1.78E-02 1.79E-01 Tanikawa and Shimamoto, 2009 2.10E-02 1.64E-01 Tanikawa and Shimamoto, 2009 2.10E-02 1.46E-01 Tanikawa and Shimamoto, 2009 2.47E-02 1.57E-01 Tanikawa and Shimamoto, 2009 2.85E-02 1.40E-01 Tanikawa and Shimamoto, 2009 3.06E-02 1.74E-01 Tanikawa and Shimamoto, 2009 3.86E-02 1.66E-01 Tanikawa and Shimamoto, 2009 5.62E-02 1.82E-01 Tanikawa and Shimamoto, 2009 9.91E-03 1.48E-01 Tanikawa and Shimamoto, 2009 1.01E-02 1.71E-01 Tanikawa and Shimamoto, 2009 1.20E-02 1.71E-01 Tanikawa and Shimamoto, 2009 1.28E-02 1.81E-01 Tanikawa and Shimamoto, 2009 1.20E-02 1.46E-01 Tanikawa and Shimamoto, 2009 1.91E-02 1.33E-01 Tanikawa and Shimamoto, 2009 4.88E-02 2.62E-01 Tanikawa and Shimamoto, 2009 7.80E-02 2.23E-01 Tanikawa and Shimamoto, 2009 6.94E-02 2.58E-01 Tanikawa and Shimamoto, 2009 6.62E-02 3.12E-01 Tanikawa and Shimamoto, 2009 9.41E-02 2.62E-01 Tanikawa and Shimamoto, 2009 1.01E-01 3.12E-01 Tanikawa and Shimamoto, 2009 8.77E-02 3.26E-01 Tanikawa and Shimamoto, 2009 9.19E-02 3.66E-01 Tanikawa and Shimamoto, 2009 7.27E-02 3.61E-01 Tanikawa and Shimamoto, 2009 7.44E-02 4.24E-01 Tanikawa and Shimamoto, 2009 7.98E-02 4.83E-01 Tanikawa and Shimamoto, 2009 1.13E-01 4.12E-01 Tanikawa and Shimamoto, 2009 5.61E-02 4.18E-01 Tanikawa and Shimamoto, 2009 4.65E-02 4.36E-01 Tanikawa and Shimamoto, 2009 3.05E-02 3.26E-01 Tanikawa and Shimamoto, 2009 3.12E-02 3.61E-01 Tanikawa and Shimamoto, 2009 2.98E-02 3.99E-01 Tanikawa and Shimamoto, 2009 3.51E-02 4.30E-01 Tanikawa and Shimamoto, 2009 3.12E-02 4.69E-01 Tanikawa and Shimamoto, 2009 4.04E-02 5.12E-01 Tanikawa and Shimamoto, 2009 4.87E-02 6.02E-01 Tanikawa and Shimamoto, 2009 5.88E-02 5.76E-01 Tanikawa and Shimamoto, 2009 6.31E-02 5.93E-01 168  Study 𝑲∞ (mD) 𝒃 for helium at 18°C (MPa) Tanikawa and Shimamoto, 2009 4.13E-02 6.76E-01 Tanikawa and Shimamoto, 2009 3.59E-02 7.49E-01 Tanikawa and Shimamoto, 2009 3.27E-02 7.27E-01 Tanikawa and Shimamoto, 2009 2.65E-02 7.17E-01 Tanikawa and Shimamoto, 2009 2.30E-02 7.16E-01 Tanikawa and Shimamoto, 2009 2.04E-02 6.10E-01 Tanikawa and Shimamoto, 2009 3.68E-02 5.76E-01 Tanikawa and Shimamoto, 2009 3.19E-02 5.67E-01 Tanikawa and Shimamoto, 2009 2.71E-02 5.67E-01 Tanikawa and Shimamoto, 2009 2.25E-02 5.05E-01 Tanikawa and Shimamoto, 2009 2.53E-02 6.38E-01 Tanikawa and Shimamoto, 2009 3.27E-02 6.28E-01 Tanikawa and Shimamoto, 2009 1.34E-01 2.47E-01 Tanikawa and Shimamoto, 2009 1.51E-01 2.47E-01 Tanikawa and Shimamoto, 2009 2.04E-01 2.74E-01 Tanikawa and Shimamoto, 2009 2.77E-01 2.74E-01 Tanikawa and Shimamoto, 2009 2.14E-01 2.40E-01 Tanikawa and Shimamoto, 2009 1.86E-01 2.10E-01 Tanikawa and Shimamoto, 2009 2.71E-01 2.14E-01 Tanikawa and Shimamoto, 2009 2.77E-01 1.85E-01 Tanikawa and Shimamoto, 2009 3.12E-01 1.53E-01 Tanikawa and Shimamoto, 2009 1.19E-01 1.90E-01 Tanikawa and Shimamoto, 2009 1.16E-01 1.52E-01 Tanikawa and Shimamoto, 2009 1.73E-01 1.36E-01 Tanikawa and Shimamoto, 2009 1.62E-01 1.12E-01 Tanikawa and Shimamoto, 2009 1.91E-01 1.17E-01 Tanikawa and Shimamoto, 2009 2.00E-01 1.04E-01 Tanikawa and Shimamoto, 2009 2.05E-01 1.03E-01 Tanikawa and Shimamoto, 2009 2.41E-01 1.19E-01 Tanikawa and Shimamoto, 2009 1.37E-01 8.63E-02 Tanikawa and Shimamoto, 2009 1.37E-01 7.45E-02 Tanikawa and Shimamoto, 2009 2.71E-01 6.26E-02 Tanikawa and Shimamoto, 2009 4.44E-01 4.96E-02 Tanikawa and Shimamoto, 2009 4.14E-01 6.74E-02 Tanikawa and Shimamoto, 2009 4.88E-01 6.64E-02 Tanikawa and Shimamoto, 2009 2.65E-01 8.02E-02 Tanikawa and Shimamoto, 2009 2.78E-01 8.51E-02 Tanikawa and Shimamoto, 2009 6.45E-01 2.55E-01 Tanikawa and Shimamoto, 2009 6.01E-01 2.30E-01 169  Study 𝑲∞ (mD) 𝒃 for helium at 18°C (MPa) Tanikawa and Shimamoto, 2009 6.92E-01 1.72E-01 Tanikawa and Shimamoto, 2009 1.37E+00 1.44E-01 Tanikawa and Shimamoto, 2009 1.40E+00 1.99E-01 Tanikawa and Shimamoto, 2009 1.81E+00 2.55E-01 Tanikawa and Shimamoto, 2009 2.19E+00 1.36E-01 Tanikawa and Shimamoto, 2009 3.84E+00 1.58E-01 Tanikawa and Shimamoto, 2009 5.21E+00 1.14E-01 Tanikawa and Shimamoto, 2009 7.23E+00 1.47E-01 Tanikawa and Shimamoto, 2009 1.00E+01 1.27E-01 Tanikawa and Shimamoto, 2009 8.73E+00 9.18E-02 Tanikawa and Shimamoto, 2009 7.10E+00 1.59E-02 Tanikawa and Shimamoto, 2009 6.16E+00 2.39E-02 Tanikawa and Shimamoto, 2009 5.60E+00 3.25E-02 Tanikawa and Shimamoto, 2009 6.01E+00 3.25E-02 Tanikawa and Shimamoto, 2009 7.78E+00 3.25E-02 Tanikawa and Shimamoto, 2009 8.15E+00 3.77E-02 Tanikawa and Shimamoto, 2009 7.25E+00 4.17E-02 Tanikawa and Shimamoto, 2009 6.30E+00 3.60E-02 Tanikawa and Shimamoto, 2009 4.77E-01 2.46E-02 Tanikawa and Shimamoto, 2009 1.28E+00 3.99E-02 Tanikawa and Shimamoto, 2009 1.82E+00 5.11E-02 Tanikawa and Shimamoto, 2009 8.98E-01 5.74E-02 Tanikawa and Shimamoto, 2009 1.22E+00 8.40E-02 Tanikawa and Shimamoto, 2009 9.85E-01 8.52E-02 Tanikawa and Shimamoto, 2009 8.56E-01 7.58E-02 Tanikawa and Shimamoto, 2009 5.48E-01 7.80E-02 Tanikawa and Shimamoto, 2009 4.44E-01 8.15E-02 Tanikawa and Shimamoto, 2009 6.31E-01 8.15E-02 Tanikawa and Shimamoto, 2009 6.61E-01 1.01E-01 Tanikawa and Shimamoto, 2009 4.76E-01 9.71E-02 Tanikawa and Shimamoto, 2009 3.77E-01 9.02E-02 Tanikawa and Shimamoto, 2009 3.05E-01 1.09E-01 Tanikawa and Shimamoto, 2009 3.35E-01 1.21E-01 Tanikawa and Shimamoto, 2009 4.04E-01 1.12E-01 Tanikawa and Shimamoto, 2009 6.45E-01 1.30E-01 Tanikawa and Shimamoto, 2009 6.93E-01 1.09E-01 Tanikawa and Shimamoto, 2009 5.35E-01 1.12E-01 Tanikawa and Shimamoto, 2009 5.61E-01 1.34E-01 Tanikawa and Shimamoto, 2009 4.33E-01 1.23E-01 170  Study 𝑲∞ (mD) 𝒃 for helium at 18°C (MPa) Tanikawa and Shimamoto, 2009 4.98E-01 1.57E-01 Tanikawa and Shimamoto, 2009 4.23E-01 1.69E-01 Tanikawa and Shimamoto, 2009 5.47E+00 5.58E-02 Tanikawa and Shimamoto, 2009 5.47E+00 6.01E-02 Tanikawa and Shimamoto, 2009 5.09E+00 6.95E-02 Tanikawa and Shimamoto, 2009 4.03E+00 6.95E-02 Tanikawa and Shimamoto, 2009 3.50E+00 6.65E-02 Tanikawa and Shimamoto, 2009 2.90E+00 7.70E-02 Tanikawa and Shimamoto, 2009 2.58E+00 7.92E-02 Tanikawa and Shimamoto, 2009 2.04E+00 8.16E-02 Tanikawa and Shimamoto, 2009 2.46E+00 8.52E-02 Tanikawa and Shimamoto, 2009 2.46E+00 1.02E-01 Tanikawa and Shimamoto, 2009 2.70E+00 1.11E-01 Tanikawa and Shimamoto, 2009 4.12E+00 9.59E-02 Tanikawa and Shimamoto, 2009 3.84E+00 1.03E-01 Tanikawa and Shimamoto, 2009 4.12E+00 8.16E-02 Tanikawa and Shimamoto, 2009 3.26E+00 9.17E-02 Tanikawa and Shimamoto, 2009 3.34E+00 1.08E-01 Tanikawa and Shimamoto, 2009 4.65E-02 5.20E-01 Tanikawa and Shimamoto, 2009 4.54E-02 5.76E-01 Tanikawa and Shimamoto, 2009 4.54E-02 6.19E-01 Tanikawa and Shimamoto, 2009 4.23E-02 7.06E-01 Tanikawa and Shimamoto, 2009 2.84E-02 7.82E-01 Tanikawa and Shimamoto, 2009 1.95E-02 4.90E-01 Tanikawa and Shimamoto, 2009 2.14E-02 5.51E-01 Tanikawa and Shimamoto, 2009 1.62E-02 1.52E-01 Tanikawa and Shimamoto, 2009 1.66E-02 1.69E-01 Tanikawa and Shimamoto, 2009 1.44E-02 1.69E-01 Tanikawa and Shimamoto, 2009 1.31E-02 1.54E-01 Tanikawa and Shimamoto, 2009 1.58E-02 1.48E-01 Tanikawa and Shimamoto, 2009 3.04E+00 9.72E-02 Tanikawa and Shimamoto, 2009 2.83E+00 8.53E-02 Tanikawa and Shimamoto, 2009 3.34E+00 8.28E-02 Tanikawa and Shimamoto, 2009 3.75E+00 9.04E-02 Tanikawa and Shimamoto, 2009 2.64E+00 7.37E-02 Tanikawa and Shimamoto, 2009 3.11E+00 1.11E-01 Tanikawa and Shimamoto, 2009 4.87E-01 1.09E-01 Tanikawa and Shimamoto, 2009 5.88E-01 1.01E-01 Tanikawa and Shimamoto, 2009 5.35E-01 8.76E-02 171  Study 𝑲∞ (mD) 𝒃 for helium at 18°C (MPa) Tanikawa and Shimamoto, 2009 3.95E-01 9.71E-02 Tanikawa and Shimamoto, 2009 6.02E-01 1.19E-01 Tanikawa and Shimamoto, 2009 3.51E-01 9.99E-02 Tanikawa and Shimamoto, 2009 7.44E-02 4.36E-01 Tanikawa and Shimamoto, 2009 3.35E-02 4.00E-01 Tanikawa and Shimamoto, 2009 2.65E-02 5.20E-01 Tanikawa and Shimamoto, 2009 3.43E-02 5.20E-01 Tanikawa and Shimamoto, 2009 4.14E-02 4.56E-01 Tanikawa and Shimamoto, 2009 4.99E-02 6.20E-01 Tanikawa and Shimamoto, 2009 4.54E-02 6.57E-01 Tanikawa and Shimamoto, 2009 3.76E-02 6.66E-01 Tanikawa and Shimamoto, 2009 2.98E-02 5.84E-01 Tanikawa and Shimamoto, 2009 2.65E-02 5.35E-01 Tanikawa and Shimamoto, 2009 2.25E-02 5.05E-01 Tanikawa and Shimamoto, 2009 3.60E-03 7.69E-01 Tanikawa and Shimamoto, 2009 3.69E-03 8.91E-01 Tanikawa and Shimamoto, 2009 2.42E-03 8.65E-01 Tanikawa and Shimamoto, 2009 1.87E-03 7.80E-01 Tanikawa and Shimamoto, 2009 4.33E-01 1.04E-01 Tanikawa and Shimamoto, 2009 3.94E-01 1.77E-01 Tanikawa and Shimamoto, 2009 6.16E-02 4.43E-01 Tanikawa and Shimamoto, 2009 1.66E-03 9.16E-01 Tanikawa and Shimamoto, 2009 2.10E-03 8.77E-01 Tanikawa and Shimamoto, 2009 3.05E-02 6.76E-01 Tanikawa and Shimamoto, 2009 1.09E-02 4.76E-01 Tanikawa and Shimamoto, 2009 3.28E-03 8.16E-01 Al Ismail and Zoback, 2016 7.06E-04 2.05E+00 Al Ismail and Zoback, 2016 4.97E-04 1.85E+00 Al Ismail and Zoback, 2016 3.06E-04 2.46E+00 Al Ismail and Zoback, 2016 1.83E-04 3.62E+00 Al Ismail and Zoback, 2016 1.07E-04 3.49E+00 Al Ismail and Zoback, 2016 8.60E-05 3.53E+00 Al Ismail and Zoback, 2016 5.30E-05 4.29E+00 Al Ismail and Zoback, 2016 3.80E-05 5.15E+00 Al Ismail and Zoback, 2016 9.37E-04 2.93E+00 Al Ismail and Zoback, 2016 7.76E-04 2.41E+00 Al Ismail and Zoback, 2016 3.75E-04 3.03E+00 Al Ismail and Zoback, 2016 2.57E-04 3.35E+00 Al Ismail and Zoback, 2016 2.47E-04 1.56E+00 172  Study 𝑲∞ (mD) 𝒃 for helium at 18°C (MPa) Al Ismail and Zoback, 2016 1.26E-04 1.14E+00 Al Ismail and Zoback, 2016 9.40E-05 1.39E+00 Al Ismail and Zoback, 2016 8.50E-05 1.41E+00 Al Ismail and Zoback, 2016 7.40E-05 1.49E+00 Al Ismail and Zoback, 2016 4.70E-03 8.21E-01 Al Ismail and Zoback, 2016 4.19E-03 1.14E+00 Al Ismail and Zoback, 2016 4.13E-03 1.12E+00 Al Ismail and Zoback, 2016 4.03E-03 1.11E+00 Al Ismail and Zoback, 2016 3.87E-03 1.15E+00 Al Ismail and Zoback, 2016 1.01E-03 8.26E-01 Al Ismail and Zoback, 2016 5.41E-04 2.25E+00 Al Ismail and Zoback, 2016 2.96E-04 3.00E+00 Al Ismail and Zoback, 2016 1.74E-04 3.54E+00 Al Ismail and Zoback, 2016 1.73E-04 2.09E+00 this study 4.84E-02 1.07E-01 this study 2.34E-02 8.27E-02 this study 1.54E-02 8.58E-02 this study 6.77E-05 3.29E+00 this study 7.12E-06 5.59E+00 this study 8.77E-07 1.64E+01 this study 2.34E-03 4.37E-01 this study 2.08E-04 5.81E-01 this study 6.78E-05 1.72E+00 this study 4.08E-05 1.31E+00 this study 7.50E-04 1.83E+00 this study 1.12E-04 1.48E+00 this study 3.10E-05 2.00E+00 this study 1.53E-04 1.73E+00 this study 2.24E-05 1.11E+00 this study 3.46E-06 1.79E+00 this study 2.28E-05 3.54E+00 this study 3.55E-06 5.84E+00 this study 8.36E-04 1.60E+00 this study 4.24E-04 2.39E+00 this study 2.94E-04 2.50E+00 this study 2.21E-04 2.61E+00 this study 3.57E-05 2.19E+00 this study 1.01E-04 1.80E+00 this study 1.29E-04 1.79E+00 173  Study 𝑲∞ (mD) 𝒃 for helium at 18°C (MPa) this study 6.01E-04 1.36E+00 this study 9.56E-04 5.72E-01 this study 6.29E-04 1.94E+00 this study 1.01E-03 5.95E-01 this study 6.13E-05 2.27E+00 this study 2.50E-06 8.14E+00 this study 1.01E-01 1.71E-01 this study 1.32E-02 3.05E-01 this study 8.71E-04 1.39E+00 this study 2.77E-04 1.01E+00 this study 2.73E-05 1.76E+00 this study 8.85E-06 3.20E+00 this study 1.59E-02 2.05E-01 this study 1.33E-04 1.45E+00 this study 2.91E-05 2.37E+00 this study 3.45E-04 2.29E+00 this study 8.82E-04 3.75E-01 this study 9.62E-06 2.50E+00 this study 4.74E-06 5.68E+00 this study 8.56E-05 1.78E+00 this study 8.04E-05 6.69E-01 this study 4.11E-06 2.49E+00  

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