MODELLING SUB-CORE SCALE PERMEABILITY IN SANDSTONE FOR USE IN STUDYING MULTIPHASE FLOW OF CO 2 AND BRINE IN CORE FLOODING EXPERIMENTS

Transcription

1 MODELLING SUB-CORE SCALE PERMEABILITY IN SANDSTONE FOR USE IN STUDYING MULTIPHASE FLOW OF CO 2 AND BRINE IN CORE FLOODING EXPERIMENTS A REPORT SUBMITTED TO THE DEPARTMENT OF ENERGY RESOURCES ENGINEERING OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE By Michael H. Krause June 11, 2009

2 Copyright by Michael H. Krause 2009 All Rights Reserved ii

3 I certify that I have read this report and that in my opinion it is fully adequate, in scope and in quality, as partial fulfillment of the degree of Master of Science in Petroleum Engineering. Prof. Sally M. Benson (Principal Advisor) iii

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5 Abstract As CO 2 capture and storage moves closer to commercialization, the ability to make accurate predictions regarding storage capacity in saline aquifers becomes more important. Improving storage capacity estimates can be done by conducting detailed regional studies on saline aquifers, something which the Department of Energy regional Carbon Sequestration Partnerships have been aggressively pursuing. Improving estimates also requires experimental and theoretical work, to develop a better understanding of the impacts of heterogeneity on multi-phase systems with unfavorable mobility ratios. To study these systems, core flood experiments are conducted by injecting CO 2 into a brine saturated sandstone core at reservoir conditions, simulating injection conditions for CO 2 storage in a saline aquifer. Using an X-ray CT scanner, sub-core scale porosity is mapped in the core prior to experiments, and sub-core scale CO 2 saturation is mapped during the experiments. The results of these experiments reveal that small variations in porosity can lead to large spatial variations in CO 2 distribution, with a high degree of small scale spatial contrast in CO 2 saturation. To understand how such variable CO 2 distribution occurs, simulations of the experiment can be conducted to test the sensitivity of CO 2 saturation to different fluid parameters from multiphase-flow theory. To perform such sensitivity studies, permeability must first be calculated at the same sub-core scale as porosity and saturation are measured. However, permeability cannot be directly measured at the sub-core scale, therefore it must be calculated using other measured data, which has traditionally been porosity. Methods for calculating permeability from porosity are common, (Nelson, 1994), and straightforward to apply in sub-core scale studies because sub-core scale porosity is measured as part of the experiment. In this study, a specific subset of these porositypermeability relationships have been systematically tested using numerical simulations of the core flood experiment. Comparing the results of the predicted saturation in the v

6 simulations to the measured values in the experiment consistently indicate that while these methods are very accurate for estimating core-scale properties, they do not accurately represent the sub-core scale permeability, and the simulations do not replicate the experimental measurements. To improve the estimate of sub-core scale permeability, a new method was developed to take advantage of additional data measured as part of the experiments. The capillary pressure curve for the sandstone core is measured experimentally for use as input in simulations. This capillary pressure data can also be integrated with the core flood experiment saturation measurements to calculate permeability. Using a modified version of the Leverett J-Function, sub-core scale permeability was calculated using the capillary pressure, and sub-core scale saturation and porosity measurements. The results of simulations using this permeability method show a much improved quantitative match to experimental saturation measurements over the porosity-only based permeability models. This new method for calculating permeability shows the potential to greatly advance the study of sub-core scale phenomena in CO 2 -brine systems by providing an accurate sub-core scale permeability representation. Using this method to calculate permeability, sensitivity studies of other multi-phase flow parameters can be conducted to determine their effect on CO 2 saturation in the presence of heterogeneity. vi

7 you are only two questions away from the frontier of knowledge Dr. Steven Losh vii

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9 Acknowledgments I would first and foremost like to thank my advisor, Dr. Sally Benson for the many insightful conversations, for her deep insight, and endless patience in seeking answers to many difficult questions. The things I have gained from her are endless and beyond description, and they will come with me to many places, through many journeys, and in the end, hopefully to make a difference in the world. I would also like to thank Jean-Christophe Perrin for conducting many excellent experiments over the course of the last two years, without which, none of this and much other work would not be possible. I would like to thank Ethan Chabora for his many stimulating conversations and for attentively proofreading so many abstracts and posters for me. I also need to thank Obi Isebor for getting me through so many late nights of studying, Louis-Marie Jacquelin for keeping me company in the lab and Thanapong Boontaeng for helping me understand multi-phase flow. I would also like to thank Jonathan Ennis-King and Lincoln Paterson for making my summer in Australia possible. I would especially like to thank Jonathan for taking the time to teach me how to use and understand TOUGH2, without which I would no doubt still be error checking my input files. I would like to thank Lynn Orr and Hamdi Tchelepi for various suggestions and contributions that helped this work along and for the insight into future directions for this and other work. I am very grateful to Karsten Pruess for developing TOUGH2, without which, many scientists would be doing much different work. I am also very grateful to Dmitry Silin for providing the capillary pressure function upon which most of this work is based and for taking the time to provide feedback for this work. I am indebted to the generous financial support of the Global Climate and Energy Project (GCEP) and to its sponsors for funding this work and many others in our research group and around the world. I am also grateful to Supri-C for their support, both financial and intellectual over the past two years. ix

10 Lastly and mostly, I am most grateful to my father, a man I have admired my whole life, who never gave up in the face of adversity, a man who would sell the shirt off his back to see his son succeed, without whom I would definitely not be who I am nor where I am today. x

11 Contents Abstract... v Acknowledgments... ix Contents... xi List of Tables... xv List of Figures... xvii Chapter Introduction Statement of the Problem Outline of the Research Approach Organization of the Report... 5 Chapter Literature Review Kozeny-Carman Models Models Based on Surface Area and Saturation Models Based on Pore Dimension Fractal Models Chapter Experimental and Simulation Methods Multi-Phase Flow Experiments Multi-Phase Flow Experimental Facility Measuring Relative Permeability and Saturation Experimental Results Capillary Pressure Measurements Simulation Method Description of TOUGH2 MP Additional Simulation Comments Chapter Evaluation of Existing Methods for Calculating Permeability Experimental Data Preparation xi

12 4.1.1 CT Image processing Upscaling Relative Permeability Capillary Pressure Model Validation Saturation Results using Kozeny-Carman Models Permeability Maps of the Kozeny-Carman Models Comparison of Kozeny-Carman Model Results with Experiment Saturation Results of Fractal Models Discussion of Porosity Based Model Results Examination of Core Scale Results Experiment Porosity-Saturation Relationship Conclusions Chapter A Proposed Method for Calculating Sub-Core Scale Permeability Using the Leverett J-Function for Calculating Permeability Previous Investigations Extension of Calhoun et al. Permeability Equation Capillary Pressure Curve Fits Permeability Maps Saturation Results of Modified Leverett J-Function Models Residual Brine Saturation Simulations Zero Residual Brine Saturation Results Comparison of Core Average Results Statistical Comparison of Permeability Methods Conclusions Chapter Conclusions Summary of Findings Recommendations for Future Work xii

13 6.3 Concluding Remarks Appendix A: A Method for Estimating Specific Surface Area Nomenclature References xiii

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15 List of Tables Table 2.1 Coefficients and data range for Huet et al. (2005)...13 Table 3.1 Experimental conditions...21 Table 3.2 Coefficients of Kumagai and Yokoyama viscosity relationship...22 Table 3.3 Permeability calculation data...23 Table 3.4 Experimental data for calculating relative permeability...23 Table 4.1 Simulation initial conditions...42 Table 4.2 Simulation grid data...43 Table 4.3 Simulation 1-4 permeability parameters for Kozeny-Carman models...46 Table 4.4 Simulation 5-6 parameters for fractal models...53 Table 4.5 Core average results using traditional permeability models...58 Table 5.1 J-Function fitting parameters used to calculate permeability...64 Table 5.2 Core average results using modified Leverett J-Function method...74 Table 5.3 Linear trend line data for slice 29 average saturation comparisons...74 Table 5.4 Linear trend line data for slice average saturation comparisons...76 xv

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17 List of Figures Figure 1.1 (a) Porosity map and (b) CO 2 saturation map of a Berea sandstone core...3 Figure 2.1 Comparison of piecewise terms and Eq Figure 3.1 Relative permeability experiment diagram (Perrin et al., 2009)...19 Figure 3.2 Experimental relative permeability measurement...24 Figure 3.3 Porosity map of experiment Berea core...24 Figure 3.4 Saturation map of experiment Berea core at 100 percent CO 2 injection...25 Figure 3.5 Slice average porosity and saturation...26 Figure 3.6 Measured capillary pressure curves for the CO 2 -brine system...29 Figure 4.1 CT data visualization software CT-view...34 Figure 4.2 Image processing software CT-daqs...35 Figure 4.3 (a) Experiment porosity map, (b) Upscaled simulation porosity map...36 Figure 4.4 (a) Experiment saturation map (b) Upscaled saturation map...37 Figure 4.5 Relative permeability curve fit...38 Figure 4.6 Capillary pressure curve fit for medium sized Berea sample (ICP1)...40 Figure 4.7 Capillary pressure curve fit for small sized Berea sample (ICP2)...41 Figure 4.8 Grid used for simulations...43 Figure 4.9 Test case results after 4 PVI (a) case 1, (b) case 2, (c) case 3, (d) case Figure 4.10 Plot of specific perimeter vs. porosity of homogeneous Berea sample...47 Figure 4.11 Permeability maps using Kozeny-Carman models for (a) Simulation 1 (b) Simulation 2 (c) Simulation 3 (d) Simulation Figure 4.12 CO 2 saturation in slice 29 (a) Experiment (b) Sim. 1 (c) Sim. 2 (d) Sim. 3 (e) Sim xvii

18 Figure 4.13 Simulation vs. experiment saturation in slice 29 for Kozney-Carman models (in order of saturation contrast)...52 Figure 4.14 Permeability maps of fractal models (a) Simulation 5 (b) Simulation Figure 4.15 CO 2 saturation in slice 29 (a) Experiment (b) Sim. 5 (c) Sim Figure 4.16 Simulation vs. experimental saturation in slice 29 for fractal models (in order of saturation contrast)...56 Figure 4.17 Comparison of simulation saturation vs. porosity in Slice 29 (in order of saturation contrast)...58 Figure 4.18 Comparison of experimentally measured saturation vs. porosity...59 Figure 5.1 Flow chart for calculating permeability using capillary pressure data...63 Figure 5.2 Capillary pressure fits for Simulation 9 (ICP3) and Simulation 10 (ICP4) from medium Berea data...65 Figure 5.3 Capillary pressure fit for Simulation 11 (ICP5) from small Berea data...66 Figure 5.4 Relative permeability curve fit for S lr = Figure 5.5 Permeability maps using modified Leverett J-Function for (a) Simulation 7 & 9 (b) Simulation 8 (c) Simulation 10 and (d) Simulation Figure 5.6 Comparison of (a) fractal permeability map (Simulation 5) and (b) modified Leverett J-Function permeability map (Simulation 7 & 9)...68 Figure 5.7 CO 2 Saturation in slice 29 (a) Experiment (b) Sim. 7 (c) Sim Figure 5.8 Simulation vs. experiment saturation in slice 29 for J-Function method...70 Figure 5.9 CO 2 Saturation in slice 29 (a) Experiment (b) Sim. 9 (c) Sim. 10 (d) Sim Figure 5.10 Simulation vs. experiment saturation in slice 29 for J-Function method.73 Figure 5.11 Comparison of slice average saturation of simulations Figure 5.12 Comparison of slice average saturation of simulations xviii

19 Chapter 1 1 Introduction In recent years, CO 2 capture and storage (CCS) has gained a great deal of attention as a strategy with significant potential to greatly reduce anthropogenic carbon dioxide emissions. CCS works by capturing carbon dioxide or any other greenhouse gas (CH 4, HFC s, etc) from a point source, transporting the gas to an underground storage site, and injecting it underground for permanent storage. These storage sites typically fall into three categories, saline aquifers, depleted oil and gas reservoirs and unmineable coal seams. Some of the benefits of using saline aquifers for greenhouse gas storage are: worldwide distribution (Bradshaw and Dance, 2006), good correlation between emissions sources and storage locations (NETL, 2008), and very large storage capacities. The Intergovernmental Panel on Climate Change (IPCC) estimates worldwide storage capacity in saline formations to be 1,000-10,000 Gt of CO 2 (IPCC, 2005), but they note that the limits on capacity are highly uncertain due to the limited data available on saline formations. According to the National Energy Technology Laboratory (NETL) (2008), the US and Canada alone have an estimated capacity of 3,300 12,000 Gt of CO 2 storage capacity, enough to sequester at least 400 years of US CO 2 equivalent emissions at 2006 emissions rates. The large range of these estimates illustrates the degree of uncertainty which exists in making capacity estimates without detailed regional studies. Reasons for this uncertainty include: unknown aquifer extent, thickness, porosity and permeability, unknown seal quality, limited knowledge of geological features such as faults and fractures, large scale heterogeneities, and unknown storage efficiency. NETL (2008) defines storage efficiency, E s, as the fraction of a basin s or region s total pore volume that the CO 2 is 1

20 2 CHAPTER 1. INTRODUCTION expected to actually contact, but is more simply defined as the fraction of the total available pore volume that actually stores CO 2. Many of these uncertainties can be addressed by doing comprehensive regional studies to more precisely estimate total aquifer size and storage potential, something which the regional CCS partnerships in the US have been aggressively pursuing. Storage efficiency is difficult to estimate however as it contains three correction factors related to the total reservoir pore volume available for CO 2, and four correction factors related to displacement efficiency, each of which have a large range of uncertainty (NETL, 2008). By assigning distributions to the uncertainty in each of the parameters, a storage efficiency range of 0.01 to 0.04 is determined for confidence intervals of 15 to 85 percent respectively (NETL, 2008). The two displacement efficiency correction factors for the vertical and horizontal displacement efficiency, which are generally functions of porosity and permeability variations (NETL, 2008), one for the influence of gravity, and one arising from the fundamental principles which govern fluid flow behavior when more than one fluid is present, called multi-phase flow. In this system, CO 2 is displacing brine, however, CO 2 is a lighter, less viscous fluid, which results in an unfavorable mobility ratio, meaning that CO 2 will not efficiently displace brine. In systems with an unfavorable mobility, the displacing fluid will flow through the zone of highest permeability, possibly bypassing large portions of the reservoir, leading to inefficient displacement. While this is true in all systems, as permeability is defined as the ability for porous media to transport fluid, it is especially true in systems with unfavorable mobility. Understanding how these four factors affect and influence CO 2 storage efficiency is critical to our ability to predict storage capacity. If the displacement efficiency can be more precisely characterized, then it might be possible to optimize injection and storage strategies to increase the storage efficiency, E s, through specific knowledge of the effect of features such as porosity and permeability contrast on CO 2 storage.

21 3 CHAPTER 1. INTRODUCTION 1.1 Statement of the Problem To study the effect of heterogeneity on displacement efficiency, core flooding experiments can be used in conjunction with X-ray CT scanning to measure the CO 2 saturation in a rock core. In these experiments, a core is saturated with brine and CO 2 is injected into it, simulating the injection conditions in a saline storage reservoir, then the sub-core scale CO 2 saturation measurement provided by the CT scan can be used to provide insight into the role of heterogeneity in determining the resulting CO 2 distribution. To illustrate the problem, a porosity map of a relatively homogeneous Berea core imaged in a CT scanner is shown below in Figure 1.1 (a), and the resulting CO 2 saturation map after injecting CO 2 into the brine saturated core is shown in Figure 1.1 (b). The figure shows that while porosity varies only slightly in the core, the saturation distribution varies all the way from zero to 100 percent CO 2. The next step then is to determine what geological properties of the core in (a) give rise to the saturation distribution in (b). Figure 1.1 (a) Porosity map and (b) CO 2 saturation map of a Berea sandstone core To study the role of heterogeneity, simulations of the experiment were conducted using the ECO2N module of the TOUGH2-MP reservoir simulator, which was designed for the CO 2 -brine system. The goals of the simulations were to study different factors which affect the CO 2 distribution and to replicate and explain the large spatial saturation variations measured in the CO 2 injection experiment (Benson et al., 2008). Numerous attempts to model the experiment could not quantitatively reproduce or explain the spatial location of CO 2 or the large spatial CO 2 saturation contrast measured during the experiment.

22 4 CHAPTER 1. INTRODUCTION The goals of this research are to study the behavior of the CO 2 -brine system by conducting core flooding experiments and to validate our understanding of the results using numerical simulation. Based on the initial work in Benson et al. (2008), several factors were identified for further study: absolute permeability, relative permeability and capillary pressure. These three factors control how fluid moves through and is distributed in the core, but absolute permeability in particular is of interest in this study. Permeability cannot be directly measured at the sub-core scale as porosity and saturation can be, therefore, it must be calculated indirectly from other properties. Permeability is a unique fundamental input for these simulations, and an accurate representation will provide subsequent relative permeability and capillary pressure studies with more confident quantitative results. Once we can understand the role of these very fine scale heterogeneities, we can extrapolate our knowledge up to understand more about reservoir scale heterogeneities, and improve our understanding of fluid interaction and storage capacity estimates in CO 2 storage aquifers. 1.2 Outline of the Research Approach The goal of this effort is to determine an accurate method for calculating sub-core scale permeability. There are many methods which have been derived and developed for calculating permeability in a variety of applications. Methods which are appropriate for use at the sub-core scale were tested in this study by using them to create sub-core scale permeability maps similar to Figure 1.1, and using those permeability maps as input for numerical simulations. The results of each simulation were then quantitatively compared to experimental measurements to determine which methods for calculating permeability provided the most accurate results. The research approach presented here is to start with core flooding experiments which are conducted at reservoir conditions. To conduct the experiment, we saturate a rock core with brine, inject CO 2 and measure the pressure difference to calculate relative permeability. A CT scanner is then used to measure the sub-core scale CO 2 saturation during the experiment, and is also used prior to the experiment to calculate the sub-core scale porosity. Capillary pressure is then measured on a rock sample from the same core.

23 5 CHAPTER 1. INTRODUCTION The challenge then is to determine an accurate method for calculating permeability within the core at the same scale as the porosity and saturation are measured. To calculate sub-core scale permeability, porosity, saturation and capillary pressure data are available as previously stated. The study starts by examining methods for calculating permeability based on porosity data, these are the oldest and most common methods for predicting permeability (Nelson, 1994). Next, methods based on residual water saturation and capillary pressure data are examined. The last method to be tested was recently developed and is based on using fractal geometry to represent pore structure. After these established methods have been tested, a new relationship using the capillary pressure, saturation and porosity data to calculate permeability is proposed and the results are analyzed. 1.3 Organization of the Report Chapter 2 provides a literature survey of common methods used for calculating permeability. Permeability is important in fields such as groundwater flow, contaminant remediation, ceramics, powders, membranes, oil and gas recovery, and wastewater filtration, among others; therefore, there are dozens of methods for calculating permeability, and so a small, practical subset of suitable methods will be covered. Chapter 3 outlines the core flooding and capillary pressure experiments which were conducted. The chapter describes the experimental setups, and then explains how porosity, permeability, relative permeability and capillary pressure are measured, and presents the actual data. The chapter also includes information about the simulator used for conducting the core flood simulations. Chapter 4 describes the basic inputs of the numerical simulations, and presents the saturation results of a selected subset of permeability relationships used for numerical simulation of the core flood experiment. These results are then examined both qualitatively, and quantitatively. Chapter 5 derives a new method for calculating permeability based on the experimentally measured capillary pressure data and integrating this information with saturation and porosity measurement. The results of numerical simulations using this

24 6 CHAPTER 1. INTRODUCTION method are also shown and analyzed qualitatively and quantitatively. A statistical analysis of every simulation and some discussion follows, outlining the strengths and potential drawbacks of the existing permeability models in chapter 4, and the new permeability model in chapter 5. Chapter 6 contains a summary of the work presented in this study, and also a summary of the research findings. Lastly, recommendations for future work to improve the new permeability model are included.

25 Chapter 2 2 Literature Review Permeability was first deduced by Henri Darcy as being proportional to the length and flow rate and inversely proportional to the pressure drop, given by Darcy s Law in Eq. 2.1 (D Arcy, 1856). Darcy did not recognize the permeability is also inversely proportional to viscosity. Permeability is a macroscopic empirical parameter which describes the ability of a fluid to move through porous media such as soil, granular beds and porous rocks. u = K P L 2.1 where u is the flow rate through the medium, K is Darcy s description of permeability, and ΔP is the pressure drop across a medium of length L. In Eq. 2.1, K = k/μ, which gives the current form of Darcy s law. k is typically written with units of darcies or millidarcies, where 1 darcy is μm 2. There are many equations for calculating permeability based on different sets of information, such as grain size and sorting, residual water saturations, porosity, cementation, etc. There are also many different factors which affect permeability, such as diagenesis, clay inclusions, cementation, etc, which will not be discussed here, but which Nelson (1994) provides a good overview. Nelson organizes permeability models into several categories: Carman-Kozeny models, models based on grain size and mineralogy, models based on surface area and water saturation, well log models and models based on pore dimension. Models based on grain size and mineralogy and well log models will not be included in this study. Grain size and mineralogy models require either destructive grain size and sorting analysis, or microtomographic measurements of the grain and pore structure, neither of which is practical for our research. Well log 7

26 8 CHAPTER 2. LITERATURE REVIEW models require well log data, and are not appropriate for use in fundamental core scale studies. Not included in Nelson s paper are fractal models, which are also examined in this study. The models which are included in this study all incorporate data which is already measured as a part of our experiments, and are thus readily applicable for calculating permeability. 2.1 Kozeny-Carman Models Kozeny-Carman based models are the most common and oldest models used for estimating permeability. These models treat porous media as a bundle of capillary tubes of equal length and constant cross section. Kozeny derived Eq. 2.2 for permeability by solving the Navier-Stokes equation for all tubes passing through a point (Bear, 1972). The equation contains the terms k, which is permeability in millidarcies (md), c o, which is Kozeny s constant, M is the specific surface area per unit volume, and ϕ is the rock porosity. The constant c o has values dependent on the flow channel shape, where 0.5 corresponds to a circle, for a square, for an equilateral triangle and for a strip. Permeability (k) will be in millidarcies from here on unless noted. φ 3 k = c 2.2 o M 2 Carman (1937) extended Kozeny s equation to the widely recognized Carman- Kozeny equation by writing the specific surface area in units of surface area per grain volume (a v ) rather than bulk volume (M). φ 3 k = S 2.3 a2 v 1 φ 2 where a v is the specific surface area per grain volume in a unit volume, recognizing that the bottom term of Eq. 2.3 gives M 2 from Eq S is called the shape factor, but serves the same function as Kozeny s constant for predicting permeability. Where data is available, S is a calibration parameter used to match predicted permeability values to experimental measurements.

27 CHAPTER 2. LITERATURE REVIEW 9 By assuming spherical grain shape, the parameter a v in Eq. 2.3 can be derived in terms of average grain diameter. A slightly modified form of Eq. 2.3 to account for the length of a tortuous capillary tube is shown in Eq. 2.4 as given by Panda and Lake (1994), where D p is the average grain diameter and τ is the tortuosity. Tortuosity is defined as the ratio of the length of a flow tube to the length of the core, generally taken to be around 2 (Carman, 1937). D 2 p φ 3 k = S Τ 1 φ 2 A simple version of Eq. 2.3 can be used as a first estimate on permeability by including the surface area term in the constant S, given by Eq. 2.5 (Benson et al., 2009). In this equation, S is calibrated to the measured core average permeability and porosity. k = S φ 3 1 φ Mavko and Nur (1997) provide an additional modification of the Carman-Kozeny equation by suggesting that one must account for a known lower bound on porosity at which the pores become disconnected and flow is no longer possible, called the percolation threshold. The form is given in Eq. 2.6 where ϕ c is the percolation porosity constant and can be measured experimentally, but is generally between 2 and 5 percent (Mavko and Nur, 1997). One can see from the equation that when porosity is equal to ϕ c, permeability is equal to zero. The authors provide evidence that this becomes more important at low porosities, where the standard Carman-Kozeny fails to accurately predict permeability. φ φ 3 c k = S 2.6 a2 v 1 φ + φ 2 c An empirical model based on the Kozeny-Carman form in Eq. 2.3 is shown in Eq. 2.7 below. This model uses a variable power of porosity in the numerator to provide a better match between experimentally measured and predicted permeability (Mavko and Nur, 1997).

28 10 CHAPTER 2. LITERATURE REVIEW φ n k = S 2.7 a2 v 1 φ Models Based on Surface Area and Saturation A commonly used model in the oil and gas industry was proposed by Timur (1968) to include information about residual water saturation, S wr. He proposed a general functional form based on previous work, suggesting the general empirical model in Eq. 2.8, based loosely on the semi-theoretical bundle of capillary tubes model of Kozeny (Eq. 2.2) by empirically replacing the surface area term in the denominator with residual water saturation. k = a φb 2.8 c S wr where the coefficients a, b and c are determined statistically and S wr is the residual water saturation. By comparing residual water saturation, permeability and porosity measurement on 155 samples from three US oil fields, Timur determined values of 0.136, 4.4 and 2 for a, b and c respectively. This equation was extended by Coates by multiplying the numerator by (1-S wr ) to ensure that permeability goes to zero as the residual liquid phase goes to unity (Nelson, 1994). This model has proved to be popular in industry as both residual water saturation and porosity can be easily estimated using well logging techniques, therefore, incorporating more information to theoretically improve the estimate of permeability. 2.3 Models Based on Pore Dimension Nelson (1994) explains that it is the pore dimensions which control permeability, not porosity or residual water saturation, thereby making the claim that all previous methods are indirect measurements of permeability. Direct information about the connectivity and dimensions of the pore network will yield the most direct relationship with permeability. One straight forward manner of doing this is by using capillary pressure data, which

29 CHAPTER 2. LITERATURE REVIEW 11 relates the pore radius to the capillary pressure through the Washburn (1921) equation, shown below. P c = 2σ cos θ R 2.9 where σ is the interfacial tension and θ is the contact angle between two fluids, R is the tube radius and P c is the capillary pressure. Purcell (1949) was the first investigator to derive the fundamental relationship between permeability and capillary pressure using a bundle of capillary tubes by recognizing that permeability is the sum of the permeance of each individual tube in the bundle. By using Poiseuille s law for fluid flow in tubes and Darcy s law for fluid though through porous media, he showed the relationship in Eq can be used to estimate permeability. k = α σ Hg air cos θ 2 φ 1 1 P2 ds w 0 c 2.10 where α is a fitting factor called the Purcell Lithology Factor, which also includes unit conversions. In this manner, Purcell showed that permeability could be calculated by integrating the capillary pressure curve with respect to the wetting phase saturation, S w. In order to simplify this correlation, Calhoun et al. (1949) sought to relate permeability to the displacement pressure, which is the minimum pressure required for a non-wetting fluid phase to invade a saturated porous media, and to the value of the Leverett J-Function, J(S w ), as defined by Leverett (1940, 1942) and shown in Eq below. The J-Function is evaluated at wetting phase saturation of 1.0 to be consistent with defining the permeability in terms of displacement pressure, which is also defined at wetting phase saturation of 1.0. J S w = P c σ cos θ k φ 2.11

30 12 CHAPTER 2. LITERATURE REVIEW k = 1 p d 2 J S w Sw =1.0 2 σ cos θ 2 φ 2.12 where σ and θ are the interfacial tension and contact angle between the wetting and non wetting fluids, and p d is the displacement pressure. The J-Function will be explained in more detail in Chapters 5, but it is a dimensionless function which was shown by Leverett (1940, 1942) to reduce to the same dimensionless curve for rocks of different permeability and porosity, but of similar geological character. From this property of the J-Function, the curve for J(S w ) may be calculated using one set of capillary pressure measurements, and can then be applied to calculate the capillary pressure curve for other rocks of similar geologic character using only porosity and permeability data. Therefore once J(S w ) 1.0 is known, it can be used in Eq to predict permeability in similar rocks. Many other authors have used various forms like this, Nelson (1994) and Huet et al provide summaries of additional methods. In their paper, paper, Nakornthap and Evan (1986) derive a new form of Eq by substituting Corey s (1966) capillary pressure curve solution for Pc in the equation, shown in Eq. 2.13, then integrating to get the solution, which Huet et al. (2005) write into the form of Eq P c = P d S w S wr 1 S wr 1 λ 2.13 k = 10.66α σ Hg air cos θ 2 1 S wr 4 φ 2 1 p d 2 λ λ where λ is called the pore geometry factor by Brooks and Corey, and is for unit conversion, which can change depending on what units for interfacial tension and displacement pressure are preferred. Huet et al. (2005), then rewrite Eq into a general power law relationship, shown in Eq. 2.15, grouping all the scalar constants into a 1. k = a 1 1 p d a 2 λ λ + 2 a 3 1 S wr a 4φ a

31 CHAPTER 2. LITERATURE REVIEW 13 This final form was fitted to 89 data sets of varying properties from low to relatively high porosity, permeability and residual wetting phase saturation. Using regression analysis, the empirical solutions for the coefficients a 1, a 2, a 3, a 4 and a 5 are given in Table 2.1. The authors also sought to find conformance with other models, specifically, a general form of Timur s equation, shown in Eq This showed that general solutions such as Eq. 2.8 work just as well as Eq. 2.14, however, they must be calibrated to every data set, while Eq is considered by Huet et al. (2005) to be a general solution applicable to rocks with characteristics falling within the range of those given in the table. Table 2.1 Coefficients and data range for Huet et al. (2005) Coefficients of Eq Data Range for Calc. Coefficients a Parameter Min. Max a k (md) a ϕ (%) a S wr (%) a P d (psia) Fractal Models The last group of models to be considered is the so-called fractal models. Fractal shapes can be used to describe porous media by using characteristic radii to model features of different scale. The size and geometry of these features can be calculated using specific surface area measurements, such as using the Brunnauer-Emmett-Teller (BET) method of measuring nitrogen adsorption onto a grain surface (Pape et al., 2000). This method is explained in more detail as it is relatively new compared to previously discussed models and may not be familiar to most readers. Several authors have derived different methods of incorporating fractal shapes into permeability models, Xu and Yu (2007) derive a fractal model for calculating Kozeny- Carman constant in Eq. 2.3, while Civan (2001) derives a power law correlation which uses fractals to describe the pore volume to solids ratio. The most practical model for this work however is the model by Pape et al. (2000), who derives a power law

32 14 CHAPTER 2. LITERATURE REVIEW relationship to porosity for different sandstones using fractal geometry to describe the pore structure. Pape et al. (1999) start with a modified version of the Kozeny-Carman equations, shown in Eq. 2.16, where T is tortuosity and r eff is a characteristic effective pore throat radius. They then use fractal geometry to derive formulas for tortuosity, porosity, and effective pore throat radii in terms of a characteristic grain radius. These fractal equations are then combined and reduced to give tortuosity and effective pore throat radius as in Eq and Eq respectively (Pape et al., 1999). k = r 2 eff 8T φ 2.16 T = 0.67 φ r eff 2 = r grain 2φ By combining Eq and 2.18 into Eq. 2.16, and selecting a characteristic grain radius, the general formula in Eq can be derived, where β is determined from the characteristic grain radius. Pape et al. (1999) show that Eq is only accurate for rocks with porosity greater than 10 percent. For low porosity samples, the mean effective radius is calculated from permeability measurements using Eq and Eq Then, Eq and 2.17 are used to derive Eq. 2.20, where γ is determined from the measured mean effective radius. This equation is given by Pape et al. (1999) to be valid for rocks with porosity of 1 to 10 percent. For rocks with very low porosity, less than 1 percent, the formulation is rederived using an absolute minimum effective radius, which they show reduces to Eq. 2.21, and is valid for rocks with porosity less than 1 percent. k = β 10φ

33 CHAPTER 2. LITERATURE REVIEW 15 k = γφ k = δφ 2.21 Rather than using three different equations for calculating permeability, Pape et al. (1999) make the case that a simple linear combination of all three may be used instead since the contribution of each equation outside its given range is negligible. By averaging over many sandstone samples, an average grain diameter of 200,000 nm, an effective pore throat radius of 200 nm, an absolute minimum pore throat radius of 50 nm are used to determine the constants in Eqs respectively, to derive Eq for an average sandstone. To show that assuming a simple linear combination of all three equations to get Eq is valid, Figure 2.1 shows the permeability calculated from Eqs in their respective porosity ranges, and also shows the permeability calculated from Eq across all porosity ranges from 0 to 25 percent. The figure shows Eq deviates most from the piecewise construction at very low porosity, but matches very well in the range of 1 to 25 percent porosity, which is within the range of interest for this study. Eq is derived in the same way as Eq. 2.22, but is for Berea sandstone. k = 31φ φ φ k = 6.2φ φ φ

34 16 CHAPTER 2. LITERATURE REVIEW Figure 2.1 Comparison of piecewise terms and Eq. 2.22

35 Chapter 3 3 Experimental and Simulation Methods 3.1 Multi-Phase Flow Experiments To study sub-core scale multi-phase flow phenomena, a facility has been developed to co-inject brine and CO 2 into a rock core at reservoir conditions. During the coinjection experiment, X-ray computed tomography (CT) scanning is used to measure the saturation of the fluids in the core. This saturation data can then be combined with concurrent measurements of, pressure, temperature and flow rate to study the sub-core scale multi-phase flow behavior Multi-Phase Flow Experimental Facility First, a 2-inch diameter rock core is placed in an oven for at least 12 hours at 600 F to stabilize any clays which might be present in the core. Then the core is set inside a Teflon sleeve, placing the two ends of the core against inlet and outlet plates. The Teflon sleeve is used to seal the ends of the core against these plates, and the core with the end plates is placed inside an aluminum core holder. Then the end plates are bolted to the core holder, sealing the core inside, and water is allowed to surround the Teflon sleeve, and is pressurized to simulate reservoir confining pressure. Next, the core holder is placed inside a CT scanner and connected to a pair of dual syringe injection pumps and a fluid separator, then the core is evacuated with a vacuum pump. At this time, the core is aligned in the CT scanner and the scanner resolution is set. Then an initial scan of the core is taken before any fluids have been injected, called the dry scan. A schematic of the system is shown in Figure 3.1. After the dry scan has been conducted, CO 2 is pumped into the entire system using one of the dual syringe pumps and brought up to reservoir temperature and pressure. A 17

36 18 CHAPTER 3. EXPERIMENTAL AND SIMULATION METHODS pump is used to maintain backpressure, which is the reservoir pore pressure. To maintain reservoir temperature, the CO 2 passes through a heat exchanger before entering the core, and two electric heaters maintain reservoir temperature in the core. After the system reaches reservoir conditions, a second CT scan of the core is taken, called the CO 2 saturated scan. Next, the CO 2 is evacuated from the whole system, the other dual syringe pump is used to flood the whole system with brine. The system is again brought to reservoir conditions in the same manner as previously described, and a third CT scan is taken, called the brine saturated scan. After these scans are complete, the brine saturated core is disconnected from the system and CO 2 and brine are flowed simultaneously through a closed loop connecting the two dual syringe pumps and the separator. The two fluids are gravity separated in the separator, from which, the CO 2 dual syringe pump refills by drawing CO 2 from the top of the separator, and the brine dual syringe pump refills by drawing brine from the bottom of the separator. This closed loop circulation is conducted until the CO 2 and brine are in equilibrium with each other, that is, until the CO 2 is saturated with brine, and the brine is saturated with CO 2. Once the two fluid components are saturated with each other, they are referred to as phases, where the brine is an aqueous, or liquid phase composed of liquid brine and aqueous CO 2, and the CO 2 is a supercritical, or gas phase, composed of gaseous CO 2 and dissolved brine. Brine and CO 2 hereafter refer only to phases, not components. If the phases are not in equilibrium with each other, dryout could occur near the inlet of the core, which is caused by brine evaporating into the CO 2 phase. The undersaturated aqueous phase could also dissolve CO 2 from the core, giving erroneous saturation results. Once the phases are in equilibrium, the lines are flushed of CO 2 and reattached to the core holder, bringing the system back up to reservoir pressure.

37 CHAPTER 3. EXPERIMENTAL AND SIMULATION METHODS 19 Figure 3.1 Relative permeability experiment diagram (Perrin et al., 2009) Measuring Relative Permeability and Saturation The experiment starts by injecting 100 percent brine (brine fractional flow of 1) at a set flow rate to calculate the absolute permeability of the whole core. This is easily calculated by measuring the pressure drop across the core and using Darcy s law in Eq k = Q μ A ΔP 3.1 where Q is the flow rate, μ is the viscosity of brine, L is the length of the core, A is the cross sectional area of the core, and Δp is the pressure drop across the core. After the absolute permeability of the core has been measured, the fractional flow of CO 2 being injected is increased in stepwise increments, waiting at each step until steady state is reached to take a CT scan of the core, until 100 percent CO 2 is being injected. Steady state is defined as the time after which saturation is no longer changing in the core. At

38 20 CHAPTER 3. EXPERIMENTAL AND SIMULATION METHODS each step, the pressure drop across the core is measured at steady state, and Darcy s law for multi-phase flow in Eq. 3.2 is used to calculate the relative permeability. k r,j = kl A ΔP Q μ j 3.2 where j denotes phase j for CO 2 and brine, and k r, j is the relative permeability of the given phase. The brine saturated and dry scans of the core taken before the experiment started are used to generate the porosity map of the core using Eq The CO 2 saturated scan and the scan taken at each fractional flow after steady state was reach are used to generate a saturation map corresponding to that specific fractional flow of CO 2 using Eq. 3.4 (Akin and Kovscek, 2003). In the equations, CT i refers to the absolute CT number of voxel i in the core. Water by definition has a CT number of 0 and air has a CT number of φ i = CT i brine CT i dry CT brine CT air 3.3 where CT Brine i is the CT number measured in voxel i when the core is saturated with brine, CT dry i is the CT number measured in voxel i before injecting any fluids, and CT Brine and CT air are the previously defined CT values of water and air respectively (brine taken same as water). S CO2,i = CT exp brine i CT i CO CT 2 brine i CT i 3.4 where CT exp i is the CT number measured in voxel i during the experiment and CT CO2 i is the CT number measured in voxel i in the CO 2 saturated core. In this manner CO 2 saturation maps of the core are calculated for each fractional flow rate in the experiment. We can also use this information to integrate over the whole core to calculate the average saturation.

39 CHAPTER 3. EXPERIMENTAL AND SIMULATION METHODS Experimental Results Experimental Conditions For this investigation, the most homogeneous Berea sandstone core available was selected. The experimental conditions, core properties and CT scanning information for this experiment are shown below in Table 3.1. The experimental conditions have been selected to replicate a saline aquifer storage reservoir where the CO 2 would be in supercritical state. The salt mass fraction is below the US minimum total dissolved solids of 10,000 ppm which defines a saline aquifer because previous experiments were conducted to replicate aquifer conditions at the Otway Basin Pilot Project in Australia, where there is no defined salinity limit for saline aquifers. In that project, CO 2 is being injected into an aquifer with salinity of 6500 ppm. Table 3.1 Experimental conditions Experimental Conditions Core Description CT Scanning Information P (MPa) Diameter (cm) 5.08 Voxel Length (mm) 1 T ( C) 50 Length (cm) 20.2 Voxel Width (mm) x NaCl (ppm) 6500 Permeability (md) 85 Slice Gap (mm) 0.5 Q t (ml/min) 3 Porosity (%) 18.5 Number of Sices 132 The scanning resolution is fixed by selecting a field of view at the time of the scan, the field of view has fixed resolution of 512 by 512 pixels. The voxel length is the thickness of a single scan slice, and was selected at the beginning of the experiment from a choice of 1, 3 or 5 mm scan length. A voxel length of 1 mm was selected, with gap of 0.5 mm between slices. A gap is not desirable, but due to cooling constraints, it was necessary in order to limit the number of slices so the entire core could be measured at one time Absolute Permeability To calculate permeability, a value of viscosity must be selected, as seen from Eq. 3.1, for consistency, the viscosity relationship used by the simulator module ECO2N is used to calculate brine viscosity. It should be noted however, that introducing small amounts of NaCl to the system has an almost negligible effect on viscosity. Philips et al. (1981) give the relationship in Eq. 3.5 for viscosity, which is calibrated to experimental data sets

40 22 CHAPTER 3. EXPERIMENTAL AND SIMULATION METHODS and considered accurate within ± 2 percent at temperatures of 10 C to 350 C and pressures of 0.1 MPa to 50 MPa. μ Brine μ H2 O = m m m T 1 e.7m 3.5 where m is the molal concentration of NaCl in g-moles NaCl per kg H 2 O, T is the system temperature in C and μ is viscosity in centipoises (cp). Because CO 2 and brine are allowed to circulate in the experiment so that they are in equilibrium with each other, the brine is also saturated with CO 2, which has a small effect on viscosity. Kumagai and Yokoyama (1999) present a correlation shown in Eq. 3.6 for calculating the viscosity of brine saturated with CO 2 for pressure in the range of 0.1 to 30 MPa and temperature in the range of 0 to 5 C. This correlation has not been verified experimentally at the temperatures of our study, therefore it will not be used for viscosity calculations. Furthermore, the effect of CO 2 on viscosity is negligible, amounting to at most a few percent decrease. 1 2 μ Brine = a + bt M NaCl + c + dt M NaCl + e + ft M CO g + T M CO2 + i P μ H2 O (T,P=0.1) Table 3.2 Coefficients of Kumagai and Yokoyama viscosity relationship a d g b e h c f i where μ is viscosity in mpa s, T is temperature in K, P is pressure in MPa, M i is the molarities of CO 2 and NaCl in moles per kilogram H 2 O. The permeability was calculated by averaging the calculation over four different flow rates to verify the accuracy of the measurements, the data is shown Table 3.3 below, with an average permeability of 84.7 millidarcies determined for the core.

41 CHAPTER 3. EXPERIMENTAL AND SIMULATION METHODS Relative Permeability Table 3.3 Permeability calculation data After absolute permeability has been measured, relative permeability was measured as previously described. The fractional flows at which data were recorded are shown in Table 3.4 below. The experiment was conducted using a total flow rate of 3 ml/min into the core, starting with 100 percent brine, and gradually increasing the CO 2 fractional flow in the steps shown in the table. At each step, the relative permeability of each phase was calculated using Eq. 3.2 using the measured pressure drop across the core after steady state was achieved. The viscosity for brine was calculated using Eq. 3.5 and the viscosity of CO 2 comes from the National Institute for Standards and Technology (NIST) webbook. Flow Rate (ml/min) Relative permeability is typically shown as a function of saturation, which in this case is the average saturation in the entire core. CO 2 saturation is calculated for each voxel in the core at each fractional flow using Eq. 3.4, and then averaged over the whole core. The resulting core average saturation at each fractional flow is also included in Table 3.4. The relative permeability data plotted as a function of their corresponding saturation are shown in Figure 3.2. ΔP (psi) Flow Rate (m 3 /s) Permebility (m 2 ) E E E E E E E E Average ΔP (Pa) Table 3.4 Experimental data for calculating relative permeability Permeability (md) CO 2 Flow Rate Brine Flow Rate (ml/min) (ml/min) f CO2 f Brine S CO2 S Brine ΔP ss (Pa) k rco2 k rbrine

42 Relative Permeability 24 CHAPTER 3. EXPERIMENTAL AND SIMULATION METHODS Brine Relative Permeability CO2 Relative Permeability Chart Title Brine Saturation Figure 3.2 Experimental relative permeability measurement Porosity and Saturation Maps Using Eq. 3.3 and Eq. 3.4, porosity and saturation maps of the core are created, as shown in Figure 3.3 and Figure 3.4. For expediency, only the saturation map of CO 2 measured at 100 percent CO 2 injection is shown. The porosity map in Figure 3.3 shows that there is no apparent structured heterogeneity. There does appear to be a slight porosity gradient along the core however. Figure 3.3 Porosity map of experiment Berea core

43 CHAPTER 3. EXPERIMENTAL AND SIMULATION METHODS 25 The injection direction and gravity vector shown in Figure 3.4 are consistent with all of the following core images in this report. The figure shows a slight saturation gradient along the core, with higher average saturation near the inlet and lower average saturation near the outlet. The figure also shows that while the spatial contrast in porosity is relatively low, on the order of 10 percent porosity, the spatial saturation contrast is extremely high, from near zero up to 100 percent CO 2. Injection g Figure 3.4 Saturation map of experiment Berea core at 100 percent CO 2 injection The slice average values of porosity and saturation for the core are shown in Figure 3.5. The figure confirms from the saturation map that a small saturation gradient does exist along the core. This saturation gradient however, is not due to the influence of gravity as gravity override can be easily distinguished in saturation maps (see Perrin et al., 2009).

44 Average Porosity 26 CHAPTER 3. EXPERIMENTAL AND SIMULATION METHODS 0.25 Slice Average Values Average Saturation Porosity Saturation Slice Number Figure 3.5 Slice average porosity and saturation 3.2 Capillary Pressure Measurements Capillary forces exist when two or more fluids are present in a system due to the interfacial tension that exists between them. The interface is curved, creating a pressure difference between them, this pressure difference is termed the capillary pressure. Capillary pressure can be measured dynamically or statically, however, Brown (1951) showed that these two methods yield identical results. The dynamic method uses a centrifuge to simulate large gravitational forces on fluid in saturated rock samples (Hassler and Brunner, 1945). During the experiment, the centrifuge does not stop and readings of the amount of fluid displaced from a rock sample are taken electronically by measuring fluid levels in a collection chamber attached to the outside of the rock sample. Static methods consist of the restored state method and mercury intrusion (Hassler and Brunner, 1945, Purcell, 1949). In the restored state method, a saturated sample is placed on top of a membrane to which that liquid is wetting and permeable. The saturated sample is then surrounded by another liquid to which the membrane is not wetting. The pressure in the liquid surrounding the sample is then incremented

45 CHAPTER 3. EXPERIMENTAL AND SIMULATION METHODS 27 successively, while the liquid in the sample is forced out through the membrane at the bottom. Each time the pressure in the non wetting liquid is incremented, the system is allowed to come into equilibrium and the saturation in the sample is measured by mass balance (Hassler and Brunner, 1945). One of the limitations of this method is that true capillary equilibrium can take a long time to achieve, and a full capillary pressure curve can take weeks to attain. Another limitation of this method is the maximum pressure that can be imposed before the non wetting fluid can enter the membrane is typically much lower than pressures attainable by mercury intrusion. Mercury intrusion, or mercury injection capillary pressure (MICP), is the most direct way to measure capillary pressure. A clean and dry rock sample of any shape or geometry is placed in a sample holder, the sample is evacuated to very low absolute pressure and mercury is allowed to surround the sample. The data is obtained by successively increasing the mercury pressure and measuring the amount of mercury intruded into the sample at each pressure interval (Purcell, 1949). At each step a short interval of time is required for pressure to reach equilibrium, typically less than a minute. The entire test is usually finished in less than two hours and the maximum pressure attainable with this method is 60,000 psi or higher. Mercury intrusion is used to measure capillary pressure on rock samples in this study. We use a Micromeritics Autopore IV which can measure capillary pressure up to 30,000 psi. When the test is conducted, the measured capillary pressure is for the mercury-air system, where mercury is the non wetting phase and air is the wetting phase. The following conversion must be applied to convert the pressure readings to their CO 2 -brine system equivalents. P c,co2 brine P c,hg air = σ CO 2 brine cos θ CO2 brine σ Hg air cos θ Hg air 3.7 where σ i and θ i are the interfacial tension and contact angle of fluid system i, and P c,hg-air is the set of measured capillary pressure data points. The interfacial tension and contact angle of mercury-air are relatively well established, although with a range of uncertainty, but were taken to be 485 dynes/cm and 130 respectively. The contact angle between brine and CO 2 was not well established at the initiation of this study and was taken to be

46 28 CHAPTER 3. EXPERIMENTAL AND SIMULATION METHODS 180. The interfacial tension between CO 2 and brine is also not well established as a function of pressure, temperature and salinity, however, Chalbaud et al. (2008) report values, from which an interpolation of 28.5 dynes/cm was made. Two capillary pressure curves were generated using two samples of different size, the small sample weight was 1.870g and the medium sample weight was 3.317g, the sizes were intentionally different to determine the resolution of the test. The amount of mercury intruded into the sample is determined by automatically measuring the amount of mercury in a penstock attached to the sample holder, which is filled at the time that the sample is surrounded by mercury at the beginning of the test. Acceptable precision requires that at least 20 percent of the penstock volume is used up by the end of the test, otherwise the resolution between data points can have excessive experimental error. The minimum recommended intrusion volume is 20 percent, (Micromeritics, 2008). The medium sample used 25 percent of a medium sized penetrometer, and the small sample used 44 percent of a smaller sized penetrometer stem volumes, so both tests are considered acceptable. The capillary pressure results have already been converted to the CO 2 -brine system using Eq. 3.7 and are shown for both samples in Figure 3.6. The two tests have the same characteristic shape and are very close to one another, however, there is some difference, which must be accounted for by doing a sensitivity analysis on the capillary pressure curve fit in the following series of numerical simulations.

47 CHAPTER 3. EXPERIMENTAL AND SIMULATION METHODS 29 Figure 3.6 Measured capillary pressure curves for the CO 2 -brine system 3.3 Simulation Method The compositional simulator TOUGH2 MP was used to conduct the simulations of the core flooding experiment. The simulator was originally developed at Lawrence Berkeley National Laboratory (Pruess et al., 1999). The simulator has undergone significant extension and modification by including new fluid flow modules for different systems, such as geothermal, hydrology, condensable and non-condensable gas flows, and a variety of additional fluid systems (Pruess et al. 1999). The code has also been extended to a parallel version for use on clusters or servers, called TOUGH2 MP (Zhang et al., 2008) Description of TOUGH2 MP TOUGH2 MP is the massively parallel version of TOUGH2 V2.0 and works by subdividing up the main domain into a series of smaller domains and solving a local flow problem on each subdomain (Zhang et al, 2008). In each subdomain, the accumulation and source/sink terms (i.e. injection blocks) are solved locally, then the flow terms are solved, such that the boundary cells of a given subdomain communicate with the

48 30 CHAPTER 3. EXPERIMENTAL AND SIMULATION METHODS boundary cells of the adjacent subdomains, ensuring conservation of mass for each subdomain Mass and Energy Balance Equations In each grid block TOUGH2 is solving a mass flow balance such that all mass flows and accumulation are conservative. This is solved simultaneously for each grid block, shown in residual form below (Zhang et al., 2008). k R i x n+1 k = M i x n+1 k M i x n Δt V i j A ij F ij k x n+1 + V i q i k,n+1 = where the vectors x n, n+1 correspond to the primary variables at the current time step n and the next time step n+1, i corresponds to block i and k corresponds to component k. M corresponds to accumulation of component k in block i, F corresponds to mass or energy flows from blocks j into block i across the interface area A between the two blocks. V corresponds to the block i volume and q is the injection or production rate in block i, Δt is the current time step. After the residual vector is calculated for each component and each block, Newton s method is used as shown below in Eq. 3.9 to drive the residual to zero, indicating convergence (Zhang et al. 2008). m R i k,n+1 x m p x m,p+1 x m,p = R i i,n+1 x m,p 3.9 where m indicates the m th primary variable, p is the current iteration, and the solution is for iteration p+1 by solving the system for x m, p Thermodynamic Variables The primary variables in TOUGH2 are dependent upon the flow module being used. The ECO2N module has been developed for the brine-co 2 -NaCl system and uses the following four primary variables for isothermal systems: pressure, NaCl mass fraction, CO 2 gas (or supercritical fluid) saturation and temperature in C.

49 CHAPTER 3. EXPERIMENTAL AND SIMULATION METHODS 31 The Redlich-Kwong cubic equation of state (EOS) is used to determine component partitioning into the two phase gas-aqueous mixture and is correlated to experimental data as reported by Pruess (2005). Thermodynamic limits of the ECO2N module include system temperatures of 12 C T 110 C, and it cannot represent two or three phase mixtures of CO 2 since the simulator does not make any distinction between liquid, gas or supercritical CO 2 (Pruess, 2005). Using the cubic EOS, the mole fraction of each component is calculated in each phase, from which the molality of CO 2, n, is calculated and used to calculate the mass fraction of each component, k, in each phase. The procedure is illustrated in the equations below using the mole fraction of CO 2 in the aqueous phase, x 2 and mole fraction of H 2 O in the gas phase, y 1, provided from the cubic EOS (Pruess, 2005). n = x 2 2m M H2 O 1 x X 2 = nm CO mm NaCl + nm CO Y 1 = y 1 M H2 O y 1 M H2 O + 1 y 1 M CO where m is the molality of NaCl in the brine, which is required as an input, M k is the molecular weight of component k, and X 2 and Y 1 are the mass fractions of CO 2 in the aqueous phase and H 2 O in the gas phase respectively Thermophysical Data The thermophysical data required for simulation are density, viscosity and specific enthalpy. These properties for CO 2 are calculated using experimental data over a range of pressure and temperature and provided with the ECO2N module as part of the regular input files. The CO 2 density and viscosity calculations assume that the gas phase is pure CO 2.

50 32 CHAPTER 3. EXPERIMENTAL AND SIMULATION METHODS Brine density is correlated to experimental data for a range of pressure, temperature and salinity, and is calculated using the additive densities of brine and dissolved CO 2 as given in Eq (Pruess, 2005). 1 ρ aq = 1 X 2 ρ b + X 2 ρ CO where ρ CO2 is calculated as given by Garcia (2001). This formulation neglects the pressure dependence of partial density of dissolved CO 2 since it amounts to only a few percent of the total aqueous density (Pruess, 2005). The brine viscosity uses Eq. 3.5 developed by Philips et al. (1981) and is considered valid for salinities up to 5 molal (230,000 ppm) and assumes that brine viscosity is independent of dissolved CO 2 (Pruess, 2005) Additional Simulation Comments The version of TOUGH2 MP used by our research group has additional custom modifications. A keyword has been inserted into the mesh file which tells the simulator to keep the outlet slice of the core out of capillary contact with the rest of the core by setting the capillary gradient between the last two slices to zero (Benson et al., 2009). This has been shown by trial and error to best represent the experimental conditions measured in the lab. In addition to this, a modification to the code has been made to include an additional capillary pressure function developed by Silin et al. (2009), which is not available in the commercial release TOUGH2 MP.

51 Chapter 4 4 Evaluation of Existing Methods for Calculating Permeability Using the experimental data that has been obtained, simulations have been conducted using two types of permeability relationships discussed in Chapter 2. First, the most common relationship, that of Kozeny-Carman and its various forms are used to calculate permeability for a series of simulations. Second, the fractal models are used to calculate permeability in a series of simulations. Finally, some analysis and discussion of the results of those models is presented. 4.1 Experimental Data Preparation CT Image processing To prepare the experimental data, several graphically interactive programs have been developed to help process and evaluate large volumes of data in an efficient manner. The CT scanner takes the image of the core in slices, which are reconstructed using these programs to create 3-dimensional images. To reconstruct the composite image, first, the center of the core in the CT field of view is determined by visually examining the CT image of any slice in the core. The radius of the core in pixels is also selected visually after the center has been determined. To facilitate this, the program CT-view, shown in Figure 4.1, displays the absolute CT values of a single slice in any CT dataset and updates the view automatically as the image center and radius are adjusted. 33

52 34 CHAPTER 4. EXISTING PERMEABILITY METHOD RESULTS Figure 4.1 CT data visualization software CT-view After the core radius and center are determined, the program CT-Daqs is used to process the CT files to create different types of output files. The program, with a screenshot of the input shown in Figure 4.2, is used to load the dry core CT image, brine saturated CT image, CO 2 saturated CT image and experimental CT images discussed in chapter 3, and uses the image center and radius information from the previous step. The program CT-daqs assumes that all the CT images have the same center and radius, and that every slice in the composite image of the core also shares the same center and radius. Then using Eq. 3.3 and Eq. 3.4, CT-daqs creates a map of the porosity and saturation data by selecting the Tecplot or Upscaled Tecplot checkbox. To calculate permeability, one of the options is selected from the dropdown menu shown in Figure 4.2, the permeability methods discussed in this study have already been installed in the program. To assign porosity and permeability to a mesh file for the simulations, the Save Slice Porosity and Permeability Files checkbox is selected; these output files are used by the program varmesh, developed at LBNL, to assign the geologic

53 CHAPTER 4. EXISTING PERMEABILITY METHOD RESULTS 35 properties to a generic mesh file. The varmesh program also creates an initial conditions file, an injection conditions file, and assigns boundary conditions to the mesh file. Figure 4.2 Image processing software CT-daqs Upscaling To keep the simulations tractable, the original CT grid must be averaged, or upscaled into a courser grid. In addition to this, it is also important to keep the mesh refined enough to retain the same order of contrast measured at the experimental scale to study the effect of heterogeneity. For this study, a transverse (in a single slice) upscaling factor of 5 was selected, meaning that the properties in 25 cells in a slice are averaged into one. A longitudinal upscaling factor of two was selected, meaning the properties of two slices were averaged into one. Transverse upscaling is done arithmetically so that for a factor of five, 25 cells are arithmetically averaged together. Longitudinal upscaling also uses arithmetic averaging for porosity and saturation, but harmonically for permeability.

54 36 CHAPTER 4. EXISTING PERMEABILITY METHOD RESULTS Saturation and porosity are scalar values which are directly measured, and arithmetic averaging is appropriate for upscaling these properties. Permeability however, represents the ability for porous media to transport fluid, and averaging in the direction perpendicular to flow is done harmonically. This can be shown by solving Eq. 3.1 for the pressure drop across each layer perpendicular to flow in a multi-layer system, and then determining the effective permeability required to transmit a constant amount of fluid, q. The result of this upscaling procedure on the porosity and 100 percent CO 2 injection saturation maps are shown below in Figure 4.3 and Figure 4.4. Note that the original experimental maps are also upscaled by a factor of 2 to 1 in the slice plane because of the very large size of the data files. The figures show that the upscaling procedure has reduced the spatial contrast in the porosity and saturation maps. The porosity map in Figure 4.3 (a) has a relatively narrow range of values, and while the spatial distribution of porosity in (b) has been smoothed out, the level of contrast in the core is comparable to the original image. Figure 4.3 (a) Experiment porosity map, (b) Upscaled simulation porosity map In contrast to the porosity map, the saturation map in Figure 4.4 (a) shows values in the range of zero to 100 percent CO 2 saturation, and the upscaled image in (b) shows a significant amount of smoothing compared to the original image. It is therefore possible that when averaging adjacent cells with very large differences in CO 2 saturation, the upscaled cell may have a very different value than any of the original cells. The sensitivity of the simulations to this upscaling procedure has not been evaluated in this study, but numerical effects of the upscaling will be discussed with the simulation results.

55 CHAPTER 4. EXISTING PERMEABILITY METHOD RESULTS 37 Figure 4.4 (a) Experiment saturation map (b) Upscaled saturation map Relative Permeability TOUGH2 offers a number of built in functions for fitting relative permeability curves, however, there is no option to use a table of data points, therefore, a built in function which best matches the data must be used. Our version of TOUGH2 has some custom features, one of which is an additional relative permeability function which is not available in the general release. The functions we use are shown below in Eq. 4.1 for brine relative permeability and Eq. 4.2 for CO 2 relative permeability. k r,brine = S Brine S lr 1 S lr n Brine 4.1 k r,co2 = S CO2 S gr 1 S lrn n CO where n Brine and n CO2 are exponential fitting parameters, S lr and S gr are the residual liquid and gas (CO 2 ) saturation respectively and S lrn is an adjustable parameter which allows the endpoint gas relative permeability to be less than 1, an option not available in other curve fits.. The brine and CO 2 relative permeability fits using the above equations with the corresponding parameters are shown below in Figure 4.5. The data is shown as a function of normalized brine saturation, given by Eq. 4.3, which is the standard way of displaying relative permeability and capillary pressure data. The fit does not correspond exactly with the data because the form of the functions above do not allow a perfect fit, however, the functions do provide a better fit than other options in TOUGH2. The

56 38 CHAPTER 4. EXISTING PERMEABILITY METHOD RESULTS parameters were also adjusted slightly from the R 2 value closest to one after several simple history matching tests on a homogeneous core to get a better match to the average saturation value. S Brine,Normalized = S = S Brine S lr 1 S lr 4.3 Figure 4.5 Relative permeability curve fit Relative Permeability Fit Parameters n Brine n CO2 S lr S gr S lrn The value of residual liquid phase saturation, S lr, was selected based on previous work by Kuo et al. (2009), who found that the residual liquid phase saturation measured in the relative permeability experiment does not necessarily represent the true value. In Figure 4.5, the residual liquid saturation is the data point at the lowest brine saturation on the brine relative permeability curve (blue) because at this point, injecting 100 percent CO 2 does not further reduce the brine saturation. However, work by Perrin et al. (2009) has shown that increasing the flow rate can reduce this residual liquid saturation, therefore, its true value should be determined by other methods such as history matching, S lr of 0.20 was selected based on the several history matching simulations previously

57 CHAPTER 4. EXISTING PERMEABILITY METHOD RESULTS 39 mentioned and should be the same for both the capillary pressure and relative permeability functions Capillary Pressure Capillary Pressure Curve Fits Leverett (1941) showed that scaling capillary pressure data by the dimensional group k φ /σ, on the left side of Eq. 4.4, and plotting the results vs. wetting phase saturation, resulted in a curve, denoted by J(S w ) and called the J-Function. Furthermore, he showed that the capillary pressure data for cores of different permeability and porosity collapsed to a single J(S w ) curve when plotted in this non-dimensional form. From this, we can assume that if we know the functional form of the J-Function, we can infer the specific capillary pressure curve for rocks of different properties, given by Eq. 4.5, which includes an additional cos θ term not in Leverett s original definition to account for the contact angle between the two fluids. P c σ k φ = J S w 4.4 P c,i = σ cos θ φ i k i J S w 4.5 A number of investigators have developed functional forms for J(S w ) (Brooks and Corey, 1966, Van Genuchten, 1980), and some of these are available in TOUGH2. A new form which provides a better curve fit to typical sandstone capillary pressure curves was developed by Silin et al. (2009) and is available in the version of TOUGH2 used for this study, it is shown below in Eq The parameters A, B and λ i are empirical fitting parameters determined by the user to provide the best curve fit. The curve fits of the two capillary pressure shown in Figure 3.6 using this functional form for J(S w ) are shown in Figure 4.6 and Figure 4.7. J S w = A 1 S λ B 1 S λ 2 1 λ2 4.6

58 40 CHAPTER 4. EXISTING PERMEABILITY METHOD RESULTS Figure 4.6 Capillary pressure curve fit for medium sized Berea sample (ICP1) ICP1 Capillary Pressure Fit Parameters A B S lr λ 1 λ One can see from the figures that the curve fits still do not match all of the data points, particularly at the ends. The capillary pressure curve shape is typical of most rocks and has proven difficult to precisely fit using functional forms for the J-Function because of its distinctive shape. Due to the curve shape and the number of fitting parameters in Eq. 4.6, using a simple fitting procedure to vary the parameters and set the coefficient of variation (R 2 ) to 1 does not work, therefore the parameters are manually adjusted to get a subjective curve fit. The curve fits in these figures were selected to best match the middle range of the capillary pressure data, where most of the saturation values in the simulations are expected to be. The important difference between the two curve fits is that ICP1 in Figure 4.6 has a much flatter plateau region than ICP2.

59 CHAPTER 4. EXISTING PERMEABILITY METHOD RESULTS 41 Figure 4.7 Capillary pressure curve fit for small sized Berea sample (ICP2) ICP2 Capillary Pressure Fit Parameters A B S lr λ 1 λ Unique Capillary Pressure Curves One can see from Eq. 4.4, that the subscript i has been added, this signifies that every voxel in the simulation mesh has a unique capillary pressure curve, calculated by the function J(S w ) and scaled to that voxels unique porosity and permeability values. Recall that it was stated that J(S w ) is dimensionless and was shown by Leverett (1941) to be the same for cores of different properties. Therefore, we assume that each voxel also has the same J(S w ) function. Therefore, once we have determined the J-Function fitting parameters for ICP1 and ICP2, it is possible to directly calculate each voxels unique capillary pressure curve. This implies that permeability has the additional function of scaling the original capillary pressure data to determine each voxels unique capillary pressure curve. From this relationship, we can see that each permeability relationship also carries with it, a new and unique set of capillary pressure curves. It is this concept which we hypothesize is partly responsible for the large variations in saturation distribution as a result of only small variations in geologic parameters. This is also the reason that finding an accurate

60 42 CHAPTER 4. EXISTING PERMEABILITY METHOD RESULTS method to calculate permeability is the most important step in studying these sub-core scale saturation phenomena Model Validation Base Case Model To ensure that the model is running appropriately, a series of four simulations have been conducted to validate the model using Eq. 2.5 as the test case for permeability. These simulations were also conducted to determine a base case for the actual study. As stated in section 4.1.3, several quick simulations were conducted on a homogeneous core to establish a good relative permeability fit based on the core average saturation in the simulation and correlated to the experimentally measured average for the 100 percent CO 2 injection case, therefore, RP1 will not be examined again here. Next, the base case was set up using the parameters shown in Table 4.1. In the table, most of the information is given in Chapter 2, but the average core permeability, which was given as 85 md in Table 3.3 and is shown as 89 md below, this is due to a minor discrepancy made in the initial calculation, however, the difference of 4 md should not affect the outcome. To determine the amount of CO 2 that dissolves into the brine at phase equilibrium, trial and error was used to determine the dissolved CO 2 mass fraction where CO 2 began to evolve out of the brine before any injection occurred. The interfacial tension (σ) is calculated by interpolating the data of Chalbaud et al. (2009). The injection fractional flow for all of the simulations was selected to be 100 percent CO 2. The injection of CO 2 does not account for brine dissolved in the CO 2 because the amount is very insignificant and dry out near the core inlet should not be a problem for such low injection volumes. Table 4.1 Simulation initial conditions Simulation Conditions Thermophysical Data Injection Conditions T ( C) 50 Dissolved CO init 2 (mf) Q CO2-Gas (kg/s) 3.04E-05 P (MPa) ρ CO2 (kg/m 3 ) Q CO2-Aq (kg/s) 0.00E+00 x NaCl (ppm) 6500 ρ H2O (kg/m 3 ) Q H2O-Gas (kg/s) 0.00E+00 φ ave σ CO2-Brine (N/m) Q H2O-Aq (kg/s) 0.00E+00 k ave (md) 89 Injection Rate (ml/m) 3 Q NaCl (kg/s) 0.00E+00

61 CHAPTER 4. EXISTING PERMEABILITY METHOD RESULTS 43 The base mesh is shown in Figure 4.8 with the summary of the grid shown in Table 4.2. The cell dimensions are not exactly cubic because that would have required excessive upscaling in the planar direction. Also notice that the number of cells in each slice is the number of cells in the circular slice, not the number of cells in a square in which the circle is inscribed, in this sense, the entire mesh is actually a cylinder. The number of slices in the CT data set given in Table 3.1 was 132, the upscaling factor in the longitudinal direction is 2:1, which gives 66 active simulation slices, however there is also an inlet and outlet slice added to the core, giving 68 total slices. The inlet slice is where the injection occurs. Fluids are injected into these grid elements and allowed to intrude the inlet end of the core in whichever flow paths the simulator finds. The outlet is maintained out of capillary contact with the core by setting the capillary gradient between the outlet slice and the last slice of the core to zero. The outlet grid elements have very large volumes so that the initial pressure is maintained during the simulation, which is the same outlet pressure condition as the experiment. Injection g Figure 4.8 Grid used for simulations Table 4.2 Simulation grid data Y-Z Upscaling 5:1 Grid Dimensions 36x36x68 Cells/Slice 936 X Upscaling 2:1 Cell Size (mm 3 ) 1.27 x 1.27 x 3 Total Cells Base Case Results The model validation consisted of four simulations, test case one uses capillary pressure curve ICP2 on a heterogeneous core, test case two uses capillary pressure curve ICP2 on a homogeneous core, test case three uses capillary pressure curve ICP2 on a

62 44 CHAPTER 4. EXISTING PERMEABILITY METHOD RESULTS heterogeneous core without gravity and test case four uses capillary pressure curve ICP1 on a heterogeneous core. The main concern in the method we use to conduct the experiments is gravity override, since the core is horizontal, the system is unstable, if injection were to stop, the fluids would redistribute themselves due to buoyancy forces caused by the density difference between brine and CO 2. Often core flood experiments are done vertically, with the lighter fluid injected in the top, this leads to a stable system because if injection stops, the fluids, theoretically, will not redistribute themselves, however, experimentally, this is much harder to conduct due to the constraints of the CT scanner position. The results of the four simulations are shown below in Figure 4.9 for injection time of 6000s, or dimensionless time of 4 pore volumes injected (PVI). Injection time tau (τ), is typically reported in terms of PVI since it has more physical meaning than the actual injection time in seconds does. The results are shown in the cross section of the plane passing vertically down the length of the core to highlight any gravity effect. Test case two in (b) is the homogeneous core, which would show the strongest gravity effect, however, there is no obvious separation of phases present in the simulation, thereby confirming that the flow rate selected should be free of gravity effects. Comparing image (a) and (c) also confirms that gravity does not have an effect on the simulation results because the results of the no gravity case in (c) are the same as when gravity is present in (a). Lastly, comparing image (a) with image (d), whose only difference is the capillary pressure relationship, we can see that using ICP1 in (d) results in greater saturation contrast than using ICP2 in (a).

63 CHAPTER 4. EXISTING PERMEABILITY METHOD RESULTS 45 Figure 4.9 Test case results after 4 PVI (a) case 1, (b) case 2, (c) case 3, (d) case 4 The results show that the simulations may not have reached steady state because there is a saturation gradient across the core in all of the images in the figure. The homogeneous core should not have a saturation gradient at steady state, and therefore has not yet reached it. CO 2 dissolves on the order of 0.3 percent brine by mass (Pruess, 2005), which amounts to slightly less than 0.7g of brine at 4τ, or about 11 percent of the brine in the inlet slice, which means dryout may be important at early times near the inlet, and will certainly have an effect at times longer than 4τ. Additional simulations on the homogeneous core showed that at 8τ, the change in average saturation in the core is only 3 percent, therefore, the effect is not large. The heterogeneous cases in Figure 4.9 also appear to have a saturation gradient; however, this is not necessarily due only to dry out. Recall Figure 3.5, which showed that the experimental results had a saturation gradient; in addition the figure shows that the core has lower average porosity near the outlet. It will be shown later that saturation is closely related to porosity in these simulations, therefore, some saturation gradient should be expected. To reduce the chance that steady state will not be reached however, simulation time was increased to 5.3τ for the remaining simulations. Lastly, the results in Figure 4.9 show that using ICP1 resulted in more saturation contrast than ICP2 by comparing image (d) with image (a) respectively. It was stated in Chapter 1 that the saturation contrast in the experiment was very high, therefore, this is desirable if we want to match the experiment as best as possible, therefore, ICP1 will be used for the cases presented in this chapter.

64 46 CHAPTER 4. EXISTING PERMEABILITY METHOD RESULTS 4.2 Saturation Results using Kozeny-Carman Models Permeability Maps of the Kozeny-Carman Models Four of the Kozeny-Carman type models were selected for the first set of simulations. The four permeability models are shown in Eq , with accompanying parameters shown in Table 4.3. The table shows that the value of S, the empirical shape factor, varies greatly from one relationship to another, this is in some part caused by the upscaling, since it is harmonic, small permeability values have a disproportionate effect on the upscaled permeability, resulting in a large range of S depending on the permeability relationship selected. The value of ϕ c for Eq in the table was selected from a range of typical values given by Mavko and Nur (1997) and was not determined exactly for this Berea core. k i = S φ i 3 1 φ i k i = S a v 2 φ i 3 1 φ i k i = S φ i 5 1 φ i k i = S φ i φ c 3 1 φ i + φ c Table 4.3 Simulation 1-4 permeability parameters for Kozeny-Carman models Simulation Equation Parameter(s) Value S S a v Eq S S ϕ c 0.04

65 CHAPTER 4. EXISTING PERMEABILITY METHOD RESULTS Description of Calculating Specific Surface Area To calculate the parameter a v, a custom program was written using Matlab s image processing toolbox. The details of the method are explained in Appendix A for the interested reader, but a quick explanation is given here. Thin sections were acquired of the Berea core used in the experiment. Using Matlab, the image is converted to binary format (black and white) and the pore space is analyzed to determine the porosity and perimeter in a small sample area of the thin section called a region of interest (ROI). This process is repeated for many ROI s, in each one, producing a data point of perimeter per grain area (specific perimeter) vs. porosity, with the composite data set of the thin section shown in Figure Figure 4.10 Plot of specific perimeter vs. porosity of homogeneous Berea sample By fitting a curve to the data in Figure 4.10, we can get an equation for specific perimeter as a function of porosity. If we assume that the amount of perimeter per unit grain area is directly proportional to the amount of surface area per unit grain volume, then a v can be written as the curve fit in the figure, where the constant multiplier from the proportionality assumption can be combined with S in Eq This process is explained in more detail in Appendix A. By combining the Eq for a v with Eq. 4.8, and reducing, the result in Eq is an equation only in terms of porosity, which was

66 48 CHAPTER 4. EXISTING PERMEABILITY METHOD RESULTS previously stated to be the easiest form to calculate permeability because the porosity of the core is easily measured by the CT scanner. a v,i =.033φ i k i = S φ i φ i Permeability Maps Using the equations for permeability given above and using the experiment porosity map from Figure 4.3(a), the permeability was calculated for these four simulations and upscaled in the manner previously described, the results are shown below in Figure The figure shows that the main difference between the permeability maps is the level of contrast between the high and low permeability values, which is expected since each relationship is only a function of porosity. Based on the figure, (b) is the low heterogeneity case and (c) is the high heterogeneity case, with the maps in (a) and (d) showing relatively little qualitative difference, although (d) does appear to have more contrast than (a).

67 CHAPTER 4. EXISTING PERMEABILITY METHOD RESULTS 49 Figure 4.11 Permeability maps using Kozeny-Carman models for (a) Simulation 1 (b) Simulation 2 (c) Simulation 3 (d) Simulation Comparison of Kozeny-Carman Model Results with Experiment To better compare the simulation and experimental results, a single slice, from the experiment and each simulation is shown, but the qualitative and quantitative match to the experiment can be shown to be the same in all slices in the core in every simulation in this report. Slice 29 was selected because it has the same average saturation as the whole core and is far away from any end effects present in the experiment and simulations. The results are shown in Figure 4.12 for simulations 1-4. The results show that none of the Kozeny-Carman models matches the experimentally measured saturation values very well. Qualitatively, the match is poor, both in absolute value and in the spatial distribution of CO 2. The experiment clearly has the largest amount of spatial contrast in CO 2 saturation, while simulation 3, the model with the highest level of CO 2 saturation contrast, does not approach the level of contrast in the experiment. Moreover, the model which includes information about the specific perimeter actually does the worst job of predicting saturation distribution, it appears almost homogeneous using the selected saturation scale.

68 50 CHAPTER 4. EXISTING PERMEABILITY METHOD RESULTS

69 CHAPTER 4. EXISTING PERMEABILITY METHOD RESULTS 51 Figure 4.12 CO 2 saturation in slice 29 (a) Experiment (b) Sim. 1 (c) Sim. 2 (d) Sim. 3 (e) Sim. 4

70 52 CHAPTER 4. EXISTING PERMEABILITY METHOD RESULTS To have a quantitative understanding of how well each simulation is able to predict the experimentally measured saturation at the sub-core scale, we can plot the simulation saturation results vs. the experimentally measured results in this slice, where a perfect correlation results in a 45 degree line across the graph. The results of this are shown in Figure 4.12, plotted in order of CO 2 contrast, with the perfect correlation line shown in purple. The figure shows that there is no observable correlation in the spatial value of saturation for any of the simulations. The figure also shows that the range of saturations measured in the experiment is much larger than the range of saturations predicted in the simulations, even in the highest contrast case. Additional discussion is found in section 4.4 and at the end of chapter 5. Figure 4.13 Simulation vs. experiment saturation in slice 29 for Kozney-Carman models (in order of saturation contrast) 4.3 Saturation Results of Fractal Models The previous simulations showed that the Kozeny-Carman models do not adequately predict permeability at the sub-core scale, therefore, a different type of model built on the same principle of predicting permeability from porosity was selected for investigation. These models are described in detail in Chapter 2, but two models are taken directly from Pape et al. (2000). The models are shown below, where Eq is used to calculate

71 CHAPTER 4. EXISTING PERMEABILITY METHOD RESULTS 53 permeability in an average sandstone and Eq is used to calculate permeability specifically for a Berea sandstone (see Pape et al., 2000), k i is given in nm 2 rather than md for the fractal models. The fitting parameters S, for the equations are shown in Table 4.4. k i = 31φ i φ i φ i k i = 6.2φ i φ i φ i Table 4.4 Simulation 5-6 parameters for fractal models Calculating permeability in the same manner as described in section 4.2 gives the resulting permeability maps in Figure 4.14 using Eq and Eq for simulations 5 and 6 respectively. The figure shows that the two relationships are nearly identical for this core, however, the amount of contrast in permeability is much larger than the Kozeny-Carman permeability maps in Figure This is due to the nonlinearity of the dependence of permeability on porosity in the fractal models, which is much greater than the Kozeny-Carman models simply by observation of the powers to which porosity is raised in the fractal models. Simulation Equation Parameter Value S S

72 54 CHAPTER 4. EXISTING PERMEABILITY METHOD RESULTS Figure 4.14 Permeability maps of fractal models (a) Simulation 5 (b) Simulation 6 The simulation results for slice 29 using the permeability maps in Figure 4.14 are shown in Figure The two models show nearly identical saturation results, which is expected because of the very similar permeability maps. From the figure, The level of spatial saturation contrast in these models is much greater than the Kozeny-Carman models and is much closer to the contrast measured in the actual experiment.

73 CHAPTER 4. EXISTING PERMEABILITY METHOD RESULTS 55 Figure 4.15 CO 2 saturation in slice 29 (a) Experiment (b) Sim. 5 (c) Sim. 6 Plotting the simulation results for slice 29 vs. the experimental measurements can again show the level of accuracy of the spatial saturation prediction from the simulation. Plotting the simulation 5 and 6 results in Figure 4.16 in order of saturation contrast, it is apparent that the spatial saturation prediction from these simulations still does not match the experimental measurements; simulation 1 is also shown on the figure for reference. The figure shows that the range of saturations in simulations 5 and 6 is greater than in simulation 1-4 (Figure 4.13), which quantitatively confirms the qualitative contrast seen in Figure 4.15.

74 56 CHAPTER 4. EXISTING PERMEABILITY METHOD RESULTS Figure 4.16 Simulation vs. experimental saturation in slice 29 for fractal models (in order of saturation contrast) 4.4 Discussion of Porosity Based Model Results Examination of Core Scale Results The results presented in the previous sections consistently indicate that traditional porosity based permeability methods for simulation of sub-core scale phenomena do not accurately reproduce experimentally measured saturation. While this study is not completely exhaustive, there is an indication that extrapolating permeability from porosity using a power law relationship is not accurate enough for sub-core scale permeability prediction, and that another approach may be required. There are many equations for predicting permeability using information in addition to porosity, some of the general forms are presented in Chapter 2. If we consider the general form derived by Huet et al. (2005), shown below in Eq. 4.15, we see that it includes three additional parameters, displacement pressure at 100 percent wetting phase saturation, p d, index of pore size distribution, λ, and residual liquid saturation, S wr.

75 CHAPTER 4. EXISTING PERMEABILITY METHOD RESULTS 57 k = a 1 1 p d a 2 λ λ + 2 a 3 1 S wr a 4φ a If we consider the equation however, it is apparent that once values of p d, λ and S wr are determined for a rock type, these are simply constants which are raised to some respective power a i, and multiplied by porosity raised to some power a 5, which reduces to a general power low for permeability as a function of porosity. The same can also be said about all of the equations presented in Chapter 2, therefore, these permeability relationships are unlikely to significantly improve the correlation between simulation predicted saturation and experimentally measured saturation. These permeability equations are usually calibrated to core scale measurements of permeability and other parameters to determine accurate correlations, not to sub-core scale studies. The equations have often been shown to be quite accurate at predicting core scale and larger permeability (see Nelson, 1994), but it has never been shown that they are accurate at predicting sub-core scale permeability and it may be inappropriate use them to extrapolate down to this scale. The experimentally measured core average values of CO 2 saturation and pressure drop across the core are 50.26% and 7059 Pa respectively. Comparing these experimental values with the simulation values in Table 4.5, we can see that the simulations to an excellent job of predicting the average CO 2 saturation, and an acceptable of predicting the pressure drop. The simulations predict a larger pressure drop than was measured, this could indicate that the relative permeability relationship is incorrect, or that the core average permeability calculation is incorrect, each of which will have a strong influence on the pressure drop, therefore, a more thorough history match for the base case could further improve the results.

76 58 CHAPTER 4. EXISTING PERMEABILITY METHOD RESULTS Table 4.5 Core average results using traditional permeability models Experiment Porosity-Saturation Relationship To understand why these relationships fail to accurately predict the spatial saturation distributions, we can examine the relationship between porosity and saturation in the simulation and the experiment. Plotting the simulation saturation values vs. their corresponding porosity values in Figure 4.17 for slice 29 using selected permeability relationships, we can see that there is a clear relationship suggested between saturation and porosity. Simulation Average CO 2 Saturation Error (%) Saturation The results in the figure also show that there is a very important relationship between the degree of permeability contrast and corresponding CO 2 contrast when using porosity based permeability models. Average ΔP (Pa) Pressure Error (%) Figure 4.17 Comparison of simulation saturation vs. porosity in Slice 29 (in order of saturation contrast)

77 CHAPTER 4. EXISTING PERMEABILITY METHOD RESULTS 59 Making the same plot of the experimentally measured saturation for three different slices near the inlet, middle and outlet end of the core, we can see in Figure 4.18 that there is no discernable relationship between CO 2 saturation and porosity. This relationship has been shown to be consistent for several different cores tested under very similar conditions, with only a very heterogeneous core with structured heterogeneity showing only a very weak relationship between saturation and porosity (Perrin et al., 2009). From the figure, it is apparent that there is not necessarily a direct relationship between saturation and porosity, but using porosity-based permeability predictions causes such a relationship to exist in numerical simulations. Therefore, while porosity-based permeability estimation may be very useful for core scale predictions, these relationships do not appear to be appropriate for calculating sub-core scale permeability. Figure 4.18 Comparison of experimentally measured saturation vs. porosity Conclusions Based on this analysis, we conclude that if porosity based permeability models force saturation to be a function of porosity, and experimental results show no distinguishable relationship between saturation and porosity, another approach is required. While there are many different permeability relationships in the literature that were not discussed in

78 60 CHAPTER 4. EXISTING PERMEABILITY METHOD RESULTS Chapter 2, many of them require complex grain analysis and many reduce down to a function of porosity. Additionally, there are other models which take advantage of capillary pressure data to predict permeability because, as Nelson (1994) explains, it is the pore throats, not the pores themselves which control permeability. Since capillary pressure data is a direct measurement of pore throat structure, it is possible to use the data to improve permeability predictions. This approach is the subject of the next chapter.

79 Chapter 5 5 A Proposed Method for Calculating Sub-Core Scale Permeability As Nelson (1994) explains, it is the pore throats, not the pores themselves which control how fluid moves through porous media. Chapter 2 refers to several investigators (Purcell, 1949, Calhoun et al., 1949, Huet et al., 2005) who have used capillary pressure data to derive information about the pore throats to create permeability relationships. In this chapter, a new method is proposed for calculating permeability using capillary pressure data, which builds on the work of previous investigators. 5.1 Using the Leverett J-Function for Calculating Permeability Previous Investigations Purcell s Permeability Equation From chapter 2, Purcell (1949) proposed that permeability could be directly calculated using a capillary bundle model and integrating over the inverse of the capillary pressure curve squared. For reference, Purcell s equation is shown again below in Eq Using the capillary pressure curves in chapter 3, it is possible to do this integration numerically for the whole core. However, since capillary pressure is only measured on a representative sample of the whole core, no information is available for the unique capillary pressure curve for each voxel. Therefore, the integration is not unique to a voxel and a different approach is required. k = α σ Hg air cos θ 2 φ 1 1 P2 ds w 0 c

80 62 CHAPTER 5. MODIFIED LEVERETT J-FUNCTION METHOD Calhoun et al. Permeability Equation Calhoun et al. extended Purcell s work by determining a theoretical form for the Lithology Factor constant, shown as α in Eq. 5.1, introduced by Purcell. In the course of this, Calhoun showed that permeability can be calculated as a function of the capillary pressure by solving Leverett s J-Function (Eq. 5.2) at 100 percent wetting phase saturation, as shown below in Eq J S w = P c σ cos θ k φ 5.2 J S w Sw =1.0 = P D σ cos θ k φ 5.3 k = 1 p D 2 φ J S w σ cos θ Extension of Calhoun et al. Permeability Equation Equation 5.4 has the same problem as Purcell s equation in Eq. 5.1 in that J(S w ) 1.0 is the same for each voxel by the definition of the J-Function, and unless displacement pressure, p D, is known for each voxel, the core average p D must be used to solve Eq. 5.4 for permeability, once again, resulting in permeability as a linear function of porosity. In the course of the derivation, Calhoun et al., did not specify a theoretical reason for selecting p D as the value at which to solve J(S w ), it was just convenient for their derivation. However, Leverett (1941) showed that Eq. 5.2 is true at all saturations, therefore, Eq. 5.3 may be solved for any value of S w and its corresponding capillary pressure. This introduces saturation as a second sub-core scale measured parameter for use in calculating sub-core scale permeability. Since we do not require that saturation have any specific value, Eq. 5.4 may be rewritten in the general form shown in Eq k i = 1 p c 2 φ i J S w 2 σ cos θ 2 5.5

81 CHAPTER 5. MODIFIED LEVERETT J-FUNCTION METHOD 63 where the J-Function is given by Eq Substituting Eq. 4.6 into Eq. 5.5 gives Eq. 5.6 for permeability of element i as a function of porosity and saturation of element i, measured in the experiment. k i = S φ i A B 1 S λ 1,i S,i λ 2 1 λ2 2 σ cos θ P c The empirical factor S has again been added to the function to ensure that the core average permeability value is in agreement with the experimentally measured value. The value of capillary pressure used in Eq. 5.6 is calculated using the given capillary pressure curve fit, evaluated at the core average saturation as measured in the experiment. This average value is used because at steady state, capillary pressure must be the same everywhere in the core except near the ends where there is a minor end effect (Richardson et al., 1952). If capillary pressure is not the same, pressure gradients would be induced in the core, and the fluids will redistribute themselves unless a capillary barrier exists to prevent this. To visualize how all of the experimental data is used to calculate a permeability map using this method, a flowchart is provided in Figure 5.1. Figure 5.1 Flow chart for calculating permeability using capillary pressure data

82 64 CHAPTER 5. MODIFIED LEVERETT J-FUNCTION METHOD Capillary Pressure Curve Fits In Section 4.1.4, it was shown that the parameters A, B, λ 1 and λ 2 are not unique and therefore, the continuous function used to describe capillary pressure is subjective. This creates a uniqueness problem with Eq. 5.6 because there can be many realizations of permeability from the same dataset depending on how the user chooses to fit the data, therefore, three different curve fits were selected to test the validity of this method. As an alternative method, it would be possible to create an interpolation procedure which uses the experimental data to calculate permeability directly from the measured data. In order to maintain consistency between the capillary pressure curves used in the simulation and the permeability calculation, the function in Eq. 5.6, rather than the measured data was used to calculate permeability. The selected fitting parameters for these simulations are shown in Table 5.1. Experimentally measured brine saturation was almost as low as zero in some portions of the core, therefore, using residual values greater than zero to calculate normalized saturation (Eq. 4.3) in Eq. 5.6 resulted in nonphysical permeability values, therefore, a residual brine saturation of zero was used to calculate normalized brine saturation, S *, in Eq The different values for residual liquid used to calculate permeability and used in the actual simulation to calculate capillary pressure are designated by S lr k and S lr s respectively in the table. Table 5.1 J-Function fitting parameters used to calculate permeability Simulation P c Curve # A B λ 1 λ 2 S lr k 7 ICP ICP ICP ICP ICP The fitting parameters in simulation 7 and 8 were selected to be the same as ICP1 and ICP2 in the previous simulations in Chapter 4. The curve fit for simulation 9 and 10 correspond to the capillary pressure data in Figure 4.6 (ICP1), where simulation 9 uses the same parameters as simulation 7 but with different residual brine saturation in the simulation. Simulation 10 uses fitting parameters designed to better match the data at S lr s S

83 CHAPTER 5. MODIFIED LEVERETT J-FUNCTION METHOD 65 low and high values of brine saturation. Simulation 11 uses a curve fit to match the capillary pressure data in Figure 4.7 (ICP2) at low and high values of brine saturation. The curve fits for simulations 9 and 10 are shown in Figure 5.2 and the curve fit for simulation 11 is shown in Figure 5.3. For simulations 9-11, a new relative permeability fit was required because the normalized brine saturation has been changed. The new relative permeability function is shown in Figure 5.4 and qualitatively looks the same as Figure 4.5 where S lr = 0.20 cases. Figure 5.2 Capillary pressure fits for Simulation 9 (ICP3) and Simulation 10 (ICP4) from medium Berea data

84 66 CHAPTER 5. MODIFIED LEVERETT J-FUNCTION METHOD Figure 5.3 Capillary pressure fit for Simulation 11 (ICP5) from small Berea data Figure 5.4 Relative permeability curve fit for S lr = 0 Relative Permeability Fit Parameters n Brine n CO2 S lr S gr S lrn

85 CHAPTER 5. MODIFIED LEVERETT J-FUNCTION METHOD Permeability Maps Permeability maps were generated for Simulations 7-11 by using Eq. 5.6 and the steady state saturation map for the 100 percent CO 2 injection case, shown in Figure 4.4. The J-Function has very large values as brine saturation goes to zero, which Eq. 5.5 and 5.6 show leads to very high permeability values. To keep permeability bounded within a reasonable upper limit, a maximum of permeability of 2000 md was allowed for the simulation grids. The resulting permeability maps using the parameters in Table 5.1 are shown below in Figure 5.5, note that the scale is different than in Chapter 4. It is clear from the figure that using different Eq. 5.6 to calculate permeability dramatically changes the resulting permeability profile. It is also clear that these models have a very high level of permeability contrast compared to those presented in Chapter 4. For comparison, the model with the highest contrast in Chapter 4, which was the fractal model used for simulation 5, is shown compared on the same scale to the permeability map for simulations 7 and 9 in Figure 5.6(a). Figure 5.5 Permeability maps using modified Leverett J-Function for (a) Simulation 7 & 9 (b) Simulation 8 (c) Simulation 10 and (d) Simulation 11

86 68 CHAPTER 5. MODIFIED LEVERETT J-FUNCTION METHOD Figure 5.6 Comparison of (a) fractal permeability map (Simulation 5) and (b) modified Leverett J-Function permeability map (Simulation 7 & 9) 5.2 Saturation Results of Modified Leverett J-Function Models Residual Brine Saturation Simulations Selecting slice 29 to make a qualitative comparison again, the results of simulations 7 and 8 are shown with the experimental results in Figure 5.7. The figure shows a very good match for both cases in terms of saturation contrast, with simulation 8 appearing to have higher average saturation than simulation 7, but about the same factor of contrast between the high and low CO 2 saturation values in both simulations.

87 CHAPTER 5. MODIFIED LEVERETT J-FUNCTION METHOD 69 Figure 5.7 CO 2 Saturation in slice 29 (a) Experiment (b) Sim. 7 (c) Sim. 8 To see the accuracy of the saturation prediction, we can plot the simulation saturation values in the slice vs. the experimentally measured values, as in chapter 4. The comparison is shown in Figure 5.8, and shows that the correlation between the simulations and experiment is much improved over the porosity based methods in chapter 4. The figure shows that on average the high and low saturations appear to be underpredicted as most of the values in this region fall below the perfect correlation line given by the purple line. However, the middle range of saturation values are being

88 70 CHAPTER 5. MODIFIED LEVERETT J-FUNCTION METHOD relatively well predicted, falling along the general trend of the perfect correlation line for both simulation results. Figure 5.8 Simulation vs. experiment saturation in slice 29 for J-Function method One of the reasons that the simulations underpredict saturation at high saturation values is because of the capillary pressure fitting parameters used for ICP1 and ICP2. These relationships have poor fits at low brine saturations (see Figure 4.6 and Figure 4.7, respectively) overpredicting capillary pressure by almost an order of magnitude at very low brine saturations, this is artificially forcing CO 2 saturation to be lower. The other reason is that an artificially imposed residual liquid saturation of 20 percent was used in these simulations. In order for any cells to have more than 80 percent CO 2 saturation, the residual liquid saturation must be reduced. These two reasons explain the capillary pressure fitting parameters selected for simulations 9-11, shown in Table 5.1. In order to determine the effect of residual brine saturation in simulations 7 and 8, the same J-Function fitting parameters with zero residual liquid saturation were used for capillary pressure in simulation 9 as in simulation 7. In order to test the importance of an accurate curve fit at low brine saturations, the J- Function fitting parameters in simulation 10 and 11 were selected to better match the data at very low brine saturations.

89 CHAPTER 5. MODIFIED LEVERETT J-FUNCTION METHOD Zero Residual Brine Saturation Results The resulting CO 2 saturation in slice 29 for simulations 9-11 is plotted along with the experimental results in Figure 5.9. The saturation contrast is higher in these results, so the scale has been changed from previous comparisons to better highlight the contrast. The figure shows that simulation 9 actually results in higher contrast than the experimental measurements using the scale shown in the figure. Simulation 10 has slightly lower contrast than simulation 9, but still appears to have more than the experimental data. Simulation 11 shows the best qualitative match to the experiment, but all three simulations show very good overall qualitative matches to the experimental data. To determine how well the results compare quantitatively, the simulation results have been plotted vs. their corresponding experimental results for slice 29 in Figure The figure shows that the maximum CO 2 saturation in these simulations is higher than in the previous cases, which should be expected because the residual liquid saturation has been set to zero. The data trends of simulations 9-11 in the figure also follow the diagonal perfect correlation line more consistently than simulations 7 and 8 in Figure 5.8. Despite the improved correlation, it is apparent that there is still a maximum threshold on CO 2 saturation in the simulation near 75 percent. This maximum saturation threshold is likely due to the very low relative permeability of brine at low saturations, shown in Figure 5.4. The figure shows that at and below 25 percent brine saturation, the brine relative permeability is essentially negligible, therefore, even though the residual value is zero, the brine is essentially immobile at such low brine saturations. The upper bound on permeability of 2000 md might also have an impact on the maximum saturation. A history match testing the sensitivity of relative permeability and maximum permeability limit on these models has the potential to improve the correlation in Figure 5.9 and Figure 5.10.

90 72 CHAPTER 5. MODIFIED LEVERETT J-FUNCTION METHOD Simulation Key Figure k Params Simulation Figure k Params Simulation a Experiment c ICP4 10 b ICP3 9 d ICP5 11 Figure 5.9 CO 2 Saturation in slice 29 (a) Experiment (b) Sim. 9 (c) Sim. 10 (d) Sim. 11

91 CHAPTER 5. MODIFIED LEVERETT J-FUNCTION METHOD 73 Figure 5.10 Simulation vs. experiment saturation in slice 29 for J-Function method Comparison of Core Average Results These results have shown qualitatively that this modified Leverett method is more accurate at predicting sub-core scale CO 2 saturation, however, the previous chapter showed in Table 4.5 that the traditional permeability methods were very accurate in predicting the core average CO 2 saturation and relatively accurate at predicting the average pressure drop across the core. The average saturation and pressure drop simulations 7-11 are shown below in Table 5.2. The results indicate that this new method does a much poorer job of predicting core average values of saturation and pressure drop than the porosity based permeability models did. The saturation is still relatively good, within 9 percent in all cases, however, this is a factor of five greater error than the porosity based methods. The match in pressure drop is also relatively poor, off by an average of about 100 percent using these methods. The pressure drop however, is strongly correlated to relative permeability, and a more accurate history match could improve the prediction.

92 74 CHAPTER 5. MODIFIED LEVERETT J-FUNCTION METHOD Table 5.2 Core average results using modified Leverett J-Function method Simulation Average CO 2 Saturation Error (%) Average ΔP (Pa) 5.3 Statistical Comparison of Permeability Methods Pressure Error (%) Saturation As a representative dataset for the core, the coefficient of determination (R 2 ) values of slice 29 data are shown in Table 5.3. The R 2 values are calculated by forcing a linear trendline through the origin of the data in Figure 4.13, Figure 4.16, Figure 5.8 and Figure This fit was selected because the 45 degree perfect correlation line for these plots passes through the origin and has a fit slope of 1. The table shows that most of the R 2 values are actually negative, indicating that assigning the average saturation value to all the simulation data points would actually perform better than the curve fit. Simulations 9 and 10 are the only models with significant positive R 2 values, indicating that these models best match the experimental values. Table 5.3 Linear trend line data for slice 29 average saturation comparisons Simulation Fit Slope Fit R 2 Fig. Ref. Simulation Fit Slope Fit R 2 Fig. Ref These qualitative results from using this modified Leverett J-Function method show a greatly improved visual match to the experimental results, both in contrast (Figure 5.7 and Figure 5.9) and in absolute value (Figure 5.8 and Figure 5.10). However, the results in Table 5.3 show that even the best sub-core scale saturation match requires improvement. Therefore, it may be more statistically significant to compare slice average data because of the incomplete history match. There is also a certain amount of

93 CHAPTER 5. MODIFIED LEVERETT J-FUNCTION METHOD 75 experimental error in measuring saturation at such small scales, however, the slice average value is very precise, (Perrin and Krevor, personal communication, 2009). The curve fits for the three models which best match the experiment are shown Figure The figure shows that the R 2 of the fits using the average values is very good, over.95 for simulation 11. In addition to this, the slope of each curve fit is nearly 1, which would be a perfect correlation. The simulations do appear to overpredict the average saturation of the slices with low experimentally measured saturation, but on average, the matches to the experimental results are very good. Figure 5.11 Comparison of slice average saturation of simulations 9-11 The same plot of the slice average values of the porosity based permeability models is shown in Figure The curve fits are again linear and forced to go through the origin, the corresponding curve fit data is shown in Table 5.4. The figure shows that the low saturation values are consistently over predicted and the high saturation values are consistently underpredicted, resulting in poor slice average saturation prediction. Recall that simulations 5 and 6 had permeability models with the most contrast, however, the figure shows that these models are the worst at predicting slice average CO 2 saturation. Recall also that simulation 2 had the permeability model with the least

94 76 CHAPTER 5. MODIFIED LEVERETT J-FUNCTION METHOD contrast, however, this model actually does the best at predicting slice average saturation among these porosity based permeability models. Figure 5.12 Comparison of slice average saturation of simulations 1-6 Table 5.4 Linear trend line data for slice average saturation comparisons 5.4 Conclusions Simulation Fit Slope Fit R 2 Simulation Fit Slope Fit R The CO 2 saturation prediction has been much improved by the use of this modified Leverett method of predicting permeability. The qualitative comparisons showed much improved matches for this method over the porosity-based permeability models discussed in chapter 4. The quantitative analysis also showed that the results are much better than the porosity based methods, although the core average saturation and pressure results are less accurate.

95 CHAPTER 5. MODIFIED LEVERETT J-FUNCTION METHOD 77 It has been discussed that a more systematic and complete sensitivity study to history match average core saturation and pressure drop will likely improve the results from using this proposed permeability method. A more thorough sensitivity study may also improve the simulation match of the porosity-based permeability methods, although not to the extent to which this new permeability method improves the match. In addition to a sensitivity study, a finer simulation grid could also be used to improve the results if the simulator could be found to run faster. Simulations where such large contrast in properties exist at such a small spatial distance take days to run, if a simulator could be optimized for these sub-core scale simulations, any effect due to upscaling could be reduced. There is also some experimental error in measuring saturation at the sub-mm scale, however, there are techniques to greatly reduce this error, such as longer scan time, higher scanning power, and multiple scans (Perrin and Krevor, personal communication, 2009) which will be utilized in future studies to improve the accuracy of experimental results.

96 78 CHAPTER 5. MODIFIED LEVERETT J-FUNCTION METHOD

97 Chapter 6 6 Conclusions 6.1 Summary of Findings Chapter 3 showed that the saturation of CO 2 measured at the sub-core scale in a core flooding experiment can vary dramatically over very small spatial scales. Subsequent analysis of the porosity map of the core revealed no obvious geological explanation for this, such as large spatial contrast in porosity, or structured heterogeneity. This gave rise to the problem of determining what controls the distribution of CO 2 at the sub-core scale. In the absence of gravity and compositional changes, capillary pressure and relative permeability control the movement of fluid in a multiphase system once the geological parameters have been determined. Subsequent simulations (Benson et al., 2008) showed that these two parameters could not accurately replicate the spatial distribution of CO 2, leading to the conclusion that the permeability predictions at the sub-core scale needed further investigation. Simulations in chapter 4 showed that using Kozeny-Carman and fractal models for calculating permeability did not give accurate saturation results at the sub-core scale. The methods did show good agreement with measured core average saturation and pressure drop however. It was then discussed how most of the permeability models in general use reduce down to a function of porosity once other parameters, such as average grain diameter or residual liquid saturation have been determined. A new approach was then derived from previous work by Calhoun et al. (1949) to calculate permeability using Leverett s J-Function scaling relationship for calculating capillary pressure. This approach works by combining capillary pressure, saturation and porosity data into one formula for predicting sub-core scale permeability. Simulations 79

98 80 CHAPTER 6. CONCLUSIONS AND FUTURE WORK using this method for calculating permeability showed greatly improved results over traditional methods in terms of spatial distribution, contrast, and absolute value of CO 2 saturation, both at the sub-core scale, and at the slice average scale. These methods do not do as well as the traditional methods at predicting core average saturation and pressure drop however. The findings presented here do not invalidate the traditional permeability models, but show that they should be used with care at such small scales. Most of the data used to calibrate and validate these traditional models was collected at the core scale, and comparing the core average simulation saturation and pressure drop to the experimental results showed a very good match, while the new modified Leverett method showed a poorer match to the experiment average. 6.2 Recommendations for Future Work The findings of this report indicate that a substantial improvement has been made in predicting sub-core scale permeability in this relatively homogeneous Berea core. With this new approach proposed, the method should be thoroughly tested under a variety of conditions to determine the best implementation. After the model has been thoroughly tested and validated, the permeability map can be used as input to begin testing the effect of relative permeability and capillary pressure on saturation distribution. The first recommendation is to test the uniqueness of the permeability map. In this study, the 100 percent CO 2 injection saturation map was used as input to evaluate the Leverett-J Function and capillary pressure. Saturation maps from different fractional flow rates are also available from the experiment and should be used as input to determine how unique the permeability map is. This should only be done on a dataset which has very high confidence in spatially mapped saturation values. It was stated previously that there is some error in measuring saturation at such small spatial scales, it is possible to reduce this error using increased CT voltage, amperage and scan time (Perrin and Krevor, 2009), and therefore, a highly precise series of experiments should be conducted to test the validity of this method with respect to uniqueness.

99 CHAPTER 6. CONCLUSIONS AND FUTURE WORK 81 The second recommendation is that the ability to predict saturation at different fractional flow rates should be determined. This can be done by simply conducting simulations using the permeability map generated from the 100 percent CO 2 injection saturation map. Then, simulations at all of the measured fractional flows should be conducted using permeability maps created from experimental saturation maps measured at that respective fractional flow, this will further validate any results from the previous recommendation. The third recommendation is to investigate the effect of structured heterogeneity on this method. If one considers a core with a high level of structured heterogeneity, such that the CO 2 is forced to circumvent a portion of the core by sub-core scale geological features, the CO 2 could be artificially forced to bypass a region of the core with high permeability, however, using this method, due to the low saturation values, it would appear that the region has low permeability. There is no obvious method to correct for this effect at this time, however, it might be possible to use some type of mixed prediction-correction method that could be used with inverse modeling. Once the first two recommendations have been completed, I believe this method will be very useful for sub-core scale studies in relatively homogeneous cores. However, I do believe that some work remains to be done to extend this methods to cores with high levels of heterogeneity, particularly any structured heterogeneity which might force CO 2 to bypass certain portions of the core. 6.3 Concluding Remarks With this, I would again like to thank everyone for their valuable input in this work, especially my advisor, Sally Benson, and to the post doc who performed all of the relative permeability experiments, Jean-Christophe Perrin. I hope that future investigators find this method useful and that additional investigations following these recommendations can further validate this theory and these results.

100 82 CHAPTER 6. CONCLUSIONS AND FUTURE WORK

101 Appendix A: A Method for Estimating Specific Surface Area Specific surface area, a v, is the amount of surface area per unit of grain volume, and is traditionally measured by doing destructive grain size analysis of a rock sample (Panda and Lake, 1994), using a scanning electron microscope (Berryman and Blair, 1986) or using the method of nitrogen surface adsorption (Pape et al., 2000). Here a new method is proposed using thin sections and image analysis techniques to estimate the specific surface area. First a very thin, epoxy impregnated sample of the rock, called a thin section, is digitally scanned at a desired resolution. The rock grains in a thin section are transparent and easily distinguishable from the blue epoxy. The color image is converted to gray scale, and using the measured core average porosity as a threshold, the grayscale image is converted to binary. This process is illustrated for the thin section shown below in Figure A.1. Figure A.1 Thin section conversion to binary image Berryman and Blair (1986) use a statistical approach using thin section analysis for calculating specific surface area, but a simpler approach is taken here. The Kozeny- 83

102 84 APPENDIX A: SPECIFIC SURFACE AREA Carman permeability equation is derived assuming a bundle of capillary tubes transports fluid through the porous media, for which specific surface area is linearly proportional to specific perimeter, or the amount of perimeter per unit grain area. This linearly proportionality can be assumed to be true for general porous media (Ross, personal communication, 2008), therefore, measuring the pore perimeter and grain area can provide an estimate for specific surface area. To analyze the thin section, a small sample area, called a region of interest (ROI), is analyzed in Matlab using the image processing toolbox. The size of this ROI is determined by the user, but is generally taken to be on the same scale as the CT measurements, or the upscaled voxel area. A sample ROI is shown below in Figure A.2 (a) where the porosity is shown in white and the grain area is shown in black, the image has dimensions of 1.05 mm on each side. The trace of the pore perimeter is shown in Figure A.2 (b), which is used to calculate the total perimeter contained in the image area. Figure A.2 (c) is used to calculate the total pore area in the ROI, which is shown outlined in red, the feature outlined in green is an embedded grain and its area is subtracted from the total pore area. Figure A.2 (a) Binary ROI (b) pore perimeter trace of ROI (c) pore identification (red) and embedded feature (green) identification of ROI (each image is 1.05 mm across) The white perimeter along the boundary of the ROI in Figure A.2 (a) or (b) is not included in the total perimeter calculation because does not represent true perimeter, but is instead the internal portion of a larger pore which has been truncated by the sample area. The perimeter in each ROI is then converted to specific perimeter by dividing by