Modelling Sub-Core Scale Permeability in Sandstone …...MODELLING SUB-CORE SCALE PERMEABILITY IN...

MODELLING SUB-CORE SCALE

PERMEABILITY IN SANDSTONE FOR USE IN

STUDYING MULTIPHASE FLOW OF CO2 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

ii

© Copyright by Michael H. Krause 2009

All Rights Reserved

iii

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)

v

Abstract

As CO2 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 CO2 into a

brine saturated sandstone core at reservoir conditions, simulating injection conditions for

CO2 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 CO2 saturation is mapped

during the experiments. The results of these experiments reveal that small variations in

porosity can lead to large spatial variations in CO2 distribution, with a high degree of

small scale spatial contrast in CO2 saturation.

To understand how such variable CO2 distribution occurs, simulations of the

experiment can be conducted to test the sensitivity of CO2 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 porosity-

permeability relationships have been systematically tested using numerical simulations of

the core flood experiment. Comparing the results of the predicted saturation in the

vi

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 CO2-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 CO2 saturation in the presence of heterogeneity.

vii

“… you are only two questions away from the frontier of knowledge”

Dr. Steven Losh

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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.

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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.

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Contents

Abstract ............................................................................................................................... v

Acknowledgments.............................................................................................................. ix

Contents ............................................................................................................................. xi

List of Tables .................................................................................................................... xv

List of Figures ................................................................................................................. xvii

Chapter 1 ............................................................................................................................. 1

1 Introduction ................................................................................................................... 1

1.1 Statement of the Problem .................................................................................... 3

1.2 Outline of the Research Approach ...................................................................... 4

1.3 Organization of the Report ................................................................................. 5

Chapter 2 ............................................................................................................................. 7

2 Literature Review.......................................................................................................... 7

2.1 Kozeny-Carman Models ..................................................................................... 8

2.2 Models Based on Surface Area and Saturation ................................................ 10

2.3 Models Based on Pore Dimension .................................................................... 10

2.4 Fractal Models .................................................................................................. 13

Chapter 3 ........................................................................................................................... 17

3 Experimental and Simulation Methods ....................................................................... 17

3.1 Multi-Phase Flow Experiments ........................................................................ 17

3.1.1 Multi-Phase Flow Experimental Facility .................................................. 17

3.1.2 Measuring Relative Permeability and Saturation ..................................... 19

3.1.3 Experimental Results ................................................................................ 21

3.2 Capillary Pressure Measurements ..................................................................... 26

3.3 Simulation Method ........................................................................................... 29

3.3.1 Description of TOUGH2 MP .................................................................... 29

3.3.2 Additional Simulation Comments............................................................. 32

Chapter 4 ........................................................................................................................... 33

4 Evaluation of Existing Methods for Calculating Permeability ................................... 33

4.1 Experimental Data Preparation ......................................................................... 33

xii

4.1.1 CT Image processing ................................................................................ 33

4.1.2 Upscaling .................................................................................................. 35

4.1.3 Relative Permeability ................................................................................ 37

4.1.4 Capillary Pressure ..................................................................................... 39

4.1.5 Model Validation ...................................................................................... 42

4.2 Saturation Results using Kozeny-Carman Models ........................................... 46

4.2.1 Permeability Maps of the Kozeny-Carman Models.................................. 46

4.2.2 Comparison of Kozeny-Carman Model Results with Experiment ........... 49

4.3 Saturation Results of Fractal Models ................................................................ 52

4.4 Discussion of Porosity Based Model Results ................................................... 56

4.4.1 Examination of Core Scale Results........................................................... 56

4.4.2 Experiment Porosity-Saturation Relationship........................................... 58

4.4.3 Conclusions ............................................................................................... 59

Chapter 5 ........................................................................................................................... 61

5 A Proposed Method for Calculating Sub-Core Scale Permeability ............................ 61

5.1 Using the Leverett J-Function for Calculating Permeability ............................ 61

5.1.1 Previous Investigations ............................................................................. 61

5.1.2 Extension of Calhoun et al. Permeability Equation .................................. 62

5.1.3 Capillary Pressure Curve Fits ................................................................... 64

5.1.4 Permeability Maps .................................................................................... 67

5.2 Saturation Results of Modified Leverett J-Function Models ............................ 68

5.2.1 Residual Brine Saturation Simulations ..................................................... 68

5.2.2 Zero Residual Brine Saturation Results .................................................... 71

5.2.3 Comparison of Core Average Results....................................................... 73

5.3 Statistical Comparison of Permeability Methods ............................................. 74

5.4 Conclusions ....................................................................................................... 76

Chapter 6 ........................................................................................................................... 79

6 Conclusions ................................................................................................................. 79

6.1 Summary of Findings ........................................................................................ 79

6.2 Recommendations for Future Work ................................................................. 80

xiii

6.3 Concluding Remarks ......................................................................................... 81

Appendix A: A Method for Estimating Specific Surface Area ........................................ 83

Nomenclature .................................................................................................................... 87

References ......................................................................................................................... 91

xv

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

xvii

List of Figures

Figure 1.1 (a) Porosity map and (b) CO2 saturation map of a Berea sandstone core ....3

Figure 2.1 Comparison of piecewise terms and Eq. 2.22 ............................................16

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 CO2 injection ....25

Figure 3.5 Slice average porosity and saturation .........................................................26

Figure 3.6 Measured capillary pressure curves for the CO2-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 4 ..45

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 4 .........................................49

Figure 4.12 CO2 saturation in slice 29 (a) Experiment (b) Sim. 1 (c) Sim. 2 (d) Sim. 3

(e) Sim. 4 ....................................................................................................51

xviii

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 6 ..54

Figure 4.15 CO2 saturation in slice 29 (a) Experiment (b) Sim. 5 (c) Sim. 6 ..............55

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 Slr = 0 ..................................................66

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 .................67

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 CO2 Saturation in slice 29 (a) Experiment (b) Sim. 7 (c) Sim. 8 ...............69

Figure 5.8 Simulation vs. experiment saturation in slice 29 for J-Function method ...70

Figure 5.9 CO2 Saturation in slice 29 (a) Experiment (b) Sim. 9 (c) Sim. 10 (d) Sim.

11................................................................................................................72

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 9-11....................75

Figure 5.12 Comparison of slice average saturation of simulations 1-6......................76

1

Chapter 1

1 Introduction

In recent years, CO2 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 (CH4,

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 CO2 (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 CO2 storage

capacity, enough to sequester at least 400 years of US CO2 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, Es, as the fraction of a basin’s or region’s total pore volume that the CO2 is

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 CO2.

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 CO2, 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, CO2 is displacing brine, however, CO2

is a lighter, less viscous fluid, which results in an unfavorable mobility ratio, meaning

that CO2 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 CO2 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, Es, through specific knowledge of the effect

of features such as porosity and permeability contrast on CO2 storage.


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 CO2

saturation in a rock core. In these experiments, a core is saturated with brine and CO2 is

injected into it, simulating the injection conditions in a saline storage reservoir, then the

sub-core scale CO2 saturation measurement provided by the CT scan can be used to

provide insight into the role of heterogeneity in determining the resulting CO2

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 CO2

saturation map after injecting CO2 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 CO2. 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) CO2 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 CO2-brine system. The goals of the simulations were to study different factors

which affect the CO2 distribution and to replicate and explain the large spatial saturation

variations measured in the CO2 injection experiment (Benson et al., 2008). Numerous

attempts to model the experiment could not quantitatively reproduce or explain the spatial

location of CO2 or the large spatial CO2 saturation contrast measured during the

experiment.


The goals of this research are to study the behavior of the CO2-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 CO2

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 CO2 and measure the pressure difference to calculate relative

permeability. A CT scanner is then used to measure the sub-core scale CO2 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.


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


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.

7

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 0.98692 μm2.

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

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), co, which

is Kozeny’s constant, M is the specific surface area per unit volume, and ϕ is the rock

porosity. The constant co has values dependent on the flow channel shape, where 0.5

corresponds to a circle, 0.562 for a square, 0.597 for an equilateral triangle and 0.667 for

a strip. Permeability (k) will be in millidarcies from here on unless noted.

𝑘 = 𝑐𝑜𝜙3

𝑀2 2.2

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 (av) rather than bulk volume (M).

𝑘 = 𝑆𝜙3

𝑎𝑣2 1 − 𝜙 2

2.3

where av is the specific surface area per grain volume in a unit volume, recognizing that

the bottom term of Eq. 2.3 gives M2 from Eq. 2.2. 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.

CHAPTER 2. LITERATURE REVIEW 9

By assuming spherical grain shape, the parameter av 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 Dp 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).

𝑘 = 𝑆𝐷𝑝

2𝜙3

72𝛵 1 − 𝜙 2 2.4

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 = 𝑆𝜙3

1 − 𝜙 2 2.5

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 ϕc 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 ϕc,

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.

𝑘 = 𝑆 𝜙 − 𝜙𝑐

3

𝑎𝑣2 1 − 𝜙 + 𝜙𝑐

2 2.6

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).


𝑘 = 𝑆𝜙𝑛

𝑎𝑣2 1 − 𝜙 2

2.7

2.2 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, Swr. 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 = 𝑎𝜙𝑏

𝑆𝑤𝑟𝑐 2.8

where the coefficients a, b and c are determined statistically and Swr 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-Swr) 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


relates the pore radius to the capillary pressure through the Washburn (1921) equation,

shown below.

𝑃𝑐 = 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 Pc 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. 2.10 can be used to

estimate permeability.

𝑘 = 𝛼 𝜎𝐻𝑔−𝑎𝑖𝑟 𝑐𝑜𝑠 𝜃 2𝜙

1

𝑃𝑐2𝑑𝑆𝑤

1

0

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, Sw.

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(Sw), as defined by Leverett (1940, 1942) and shown in Eq. 2.11

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 𝑆𝑤 =𝑃𝑐

𝜎 𝑐𝑜𝑠 𝜃

𝑘

𝜙 2.11


𝑘 = 1

𝑝𝑑2 𝐽 𝑆𝑤 𝑆𝑤 =1.0

2 𝜎 𝑐𝑜𝑠 𝜃 2𝜙 2.12

where σ and θ are the interfacial tension and contact angle between the wetting and non

wetting fluids, and pd 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(Sw) 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(Sw)|1.0 is known, it can be used in Eq. 2.12 to predict permeability in similar rocks.

Many other authors have used various forms like this, Nelson (1994) and Huet et al.

2005 provide summaries of additional methods. In their paper, paper, Nakornthap and

Evan (1986) derive a new form of Eq. 2.10 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. 2.14.

𝑃𝑐 = 𝑃𝑑 𝑆𝑤 − 𝑆𝑤𝑟

1 − 𝑆𝑤𝑟 −

1𝜆

2.13

𝑘 = 10.66𝛼 𝜎𝐻𝑔−𝑎𝑖𝑟 𝑐𝑜𝑠 𝜃 2 1 − 𝑆𝑤𝑟

4𝜙21

𝑝𝑑2

𝜆

𝜆 + 2 2.14

where λ is called the pore geometry factor by Brooks and Corey, and 10.66 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. 2.14 into a

general power law relationship, shown in Eq. 2.15, grouping all the scalar constants into

a1.

𝑘 = 𝑎1

1

𝑝𝑑 𝑎2

𝜆

𝜆 + 2 𝑎3

1 − 𝑆𝑤𝑟 𝑎4𝜙𝑎5 2.15


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 a1, a2, a3, a4 and a5 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. 2.8. 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. 2.14 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)

2.4 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

a1 1017003.24 Parameter Min. Max

a2 1.7846 k (md) 0.0041 8340

a3 1.6575 ϕ (%) 0.3 34

a4 0.5475 Swr (%) 0.7 33

a5 1.6498 Pd (psia) 2.32 2176

Coefficients of Eq. 2.15 Data Range for Calc. Coefficients


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 reff 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. 2.17 and Eq. 2.18 respectively (Pape et al., 1999).

𝑘 = 𝑟𝑒𝑓𝑓

2

8𝑇𝜙 2.16

𝑇 = 0.67

𝜙 2.17

𝑟𝑒𝑓𝑓2 = 𝑟𝑔𝑟𝑎𝑖𝑛

2 2𝜙 8 2.18

By combining Eq. 2.17 and 2.18 into Eq. 2.16, and selecting a characteristic grain

radius, the general formula in Eq. 2.19 can be derived, where β is determined from the

characteristic grain radius. Pape et al. (1999) show that Eq. 2.16 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. 2.16 and Eq. 2.17. Then,

Eq. 2.16 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 10 2.19


𝑘 = 𝛾𝜙2 2.20

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. 2.19 – 2.21 respectively, to derive Eq. 2.22 for

an average sandstone. To show that assuming a simple linear combination of all three

equations to get Eq. 2.22 is valid, Figure 2.1 shows the permeability calculated from Eqs.

2.19-2.21 in their respective porosity ranges, and also shows the permeability calculated

from Eq. 2.22 across all porosity ranges from 0 to 25 percent. The figure shows Eq. 2.22

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. 2.23 is derived in the same way as Eq. 2.22, but is for Berea sandstone.

𝑘 = 31𝜙 + 7463𝜙2 + 191 10𝜙 10 2.22

𝑘 = 6.2𝜙 + 1493𝜙2 + 58 10𝜙 10 2.23


Figure 2.1 Comparison of piecewise terms and Eq. 2.22

17

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 CO2 into a rock core at reservoir conditions. During the co-

injection 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.

3.1.1 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, CO2 is pumped into the entire system using

one of the dual syringe pumps and brought up to reservoir temperature and pressure. A

18 CHAPTER 3. EXPERIMENTAL AND SIMULATION METHODS

pump is used to maintain backpressure, which is the reservoir pore pressure. To maintain

reservoir temperature, the CO2 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 CO2

saturated scan. Next, the CO2 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 CO2 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 CO2 dual syringe pump refills by drawing CO2 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 CO2 and brine are in

equilibrium with each other, that is, until the CO2 is saturated with brine, and the brine is

saturated with CO2. 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 CO2, and the CO2 is a supercritical, or gas phase, composed of

gaseous CO2 and dissolved brine. Brine and CO2 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 CO2 phase. The under-

saturated aqueous phase could also dissolve CO2 from the core, giving erroneous

saturation results. Once the phases are in equilibrium, the lines are flushed of CO2 and

reattached to the core holder, bringing the system back up to reservoir pressure.

CHAPTER 3. EXPERIMENTAL AND SIMULATION METHODS 19

Figure 3.1 Relative permeability experiment diagram (Perrin et al., 2009)

3.1.2 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.

3.1.

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 CO2 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 CO2 is being injected. Steady

state is defined as the time after which saturation is no longer changing in the core. At


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.

𝑘𝑟 ,𝑗 = 𝑘𝐿

𝐴 ∙ 𝛥𝑃 𝑄 ∙ 𝜇 𝑗 3.2

where j denotes phase j for CO2 and brine, and kr, 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. 3.3. The CO2 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 CO2 using Eq. 3.4 (Akin

and Kovscek, 2003). In the equations, CTi 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 -1000.

𝜙𝑖 =𝐶𝑇𝑖

𝑏𝑟𝑖𝑛𝑒 − 𝐶𝑇𝑖𝑑𝑟𝑦

𝐶𝑇𝑏𝑟𝑖𝑛𝑒 − 𝐶𝑇𝑎𝑖𝑟 3.3

where CTiBrine

is the CT number measured in voxel i when the core is saturated with

brine, CTidry

is the CT number measured in voxel i before injecting any fluids, and CTBrine

and CTair are the previously defined CT values of water and air respectively (brine taken

same as water).

𝑆𝐶𝑂2 ,𝑖 =𝐶𝑇𝑖

𝑒𝑥𝑝 − 𝐶𝑇𝑖𝑏𝑟𝑖𝑛𝑒

𝐶𝑇𝑖𝐶𝑂2 − 𝐶𝑇𝑖

𝑏𝑟𝑖𝑛𝑒 3.4

where CTiexp

is the CT number measured in voxel i during the experiment and CTiCO2

is

the CT number measured in voxel i in the CO2 saturated core. In this manner CO2

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.


3.1.3 Experimental Results

3.1.3.1 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 CO2 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, CO2 is being

injected into an aquifer with salinity of 6500 ppm.

Table 3.1 Experimental conditions

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.

3.1.3.2 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

P (MPa) 12.41 Diameter (cm) 5.08 Voxel Length (mm) 1

T (˚C) 50 Length (cm) 20.2 Voxel Width (mm) 0.254

xNaCl (ppm) 6500 Permeability (md) 85 Slice Gap (mm) 0.5

Qt (ml/min) 3 Porosity (%) 18.5 Number of Sices 132

Experimental Conditions Core Description CT Scanning Information


and considered accurate within ± 2 percent at temperatures of 10˚C to 350˚C and

pressures of 0.1 MPa to 50 MPa.

𝜇𝐵𝑟𝑖𝑛𝑒

𝜇𝐻2𝑂= 1 + .0816𝑚 + .0122𝑚2 + .000128𝑚3

+ .000629𝑇 1 − 𝑒−.7𝑚

3.5

where m is the molal concentration of NaCl in g-moles NaCl per kg H2O, T is the system

temperature in ˚C and μ is viscosity in centipoises (cp). Because CO2 and brine are

allowed to circulate in the experiment so that they are in equilibrium with each other, the

brine is also saturated with CO2, 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 CO2 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 CO2 on viscosity is negligible, amounting to at most a few

percent decrease.

𝜇𝐵𝑟𝑖𝑛𝑒 = 𝑎 + 𝑏𝑇 𝑀𝑁𝑎𝐶𝑙 + 𝑐 + 𝑑𝑇 𝑀𝑁𝑎𝐶𝑙

12 + 𝑒 + 𝑓𝑇 𝑀𝐶𝑂2

+ 𝑔 + 𝑕𝑇 𝑀𝐶𝑂2

2 + 𝑖 𝑃 − 0.1 + 𝜇𝐻2𝑂 (𝑇,𝑃=0.1)

3.6

Table 3.2 Coefficients of Kumagai and Yokoyama viscosity relationship

where μ is viscosity in mPa·s, T is temperature in K, P is pressure in MPa, Mi is the

molarities of CO2 and NaCl in moles per kilogram H2O. 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.

a 3.8597100 d 0.0190621 g -7.2276900

b -0.0132561 e 8.7955200 h 0.0264498

c -5.3753900 f -0.0317229 i -0.0016996


Table 3.3 Permeability calculation data

3.1.3.3 Relative Permeability

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 CO2 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 CO2 comes from the National Institute for Standards and Technology (NIST)

webbook.

Relative permeability is typically shown as a function of saturation, which in this case

is the average saturation in the entire core. CO2 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.

Table 3.4 Experimental data for calculating relative permeability

1.00 1.52 1.67E-08 10480 8.761E-14 88.77

2.00 3.15 3.33E-08 21718 8.455E-14 85.67

3.00 4.83 5.00E-08 33302 8.271E-14 83.81

4.00 6.70 6.67E-08 46195 7.950E-14 80.56

84.70

Permebility

(m2)

Permeability

(md)

Average

Flow Rate

(ml/min)ΔP (psi)

Flow Rate

(m3/s)

ΔP (Pa)

0.00 3.00 0.00 1.00 0.00 1.00 32833 0.0000 1.0000

0.15 2.85 0.05 0.95 0.04 0.96 36103 0.0038 0.8640

0.45 2.55 0.15 0.85 0.07 0.93 39230 0.0105 0.7114

0.75 2.25 0.25 0.75 0.09 0.91 38377 0.0179 0.6417

1.20 1.80 0.40 0.60 0.12 0.88 42499 0.0258 0.4636

1.50 1.50 0.50 0.50 0.14 0.86 49748 0.0275 0.3300

1.95 1.05 0.65 0.35 0.16 0.84 41220 0.0432 0.2788

2.35 0.00 1.00 0.00 0.51 0.49 7107 0.3021 0.0000

krCO2 krBrineBrine Flow Rate

(ml/min)fCO2 fBrine SCO2 SBrine ΔPss (Pa)

CO2 Flow Rate

(ml/min)


Figure 3.2 Experimental relative permeability measurement

3.1.3.4 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 CO2

measured at 100 percent CO2 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

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

Re

lati

ve P

erm

eab

ility

Brine Saturation

Chart Title

Brine Relative Permeability

CO2 Relative Permeability


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 CO2.

Figure 3.4 Saturation map of experiment Berea core at 100 percent CO2 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).

𝒈

Injection


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

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.10

0.13

0.16

0.19

0.22

0.25

0 20 40 60 80 100 120 140

Ave

rage Satu

rationA

vera

ge P

oro

sity

Slice Number

Slice Average Values

Porosity

Saturation


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 CO2-brine

system equivalents.

𝑃𝑐 ,𝐶𝑂2−𝑏𝑟𝑖𝑛𝑒

𝑃𝑐 ,𝐻𝑔−𝑎𝑖𝑟=

𝜎𝐶𝑂2−𝑏𝑟𝑖𝑛𝑒 𝑐𝑜𝑠 𝜃𝐶𝑂2−𝑏𝑟𝑖𝑛𝑒

𝜎𝐻𝑔−𝑎𝑖𝑟 𝑐𝑜𝑠 𝜃𝐻𝑔−𝑎𝑖𝑟 3.7

where σi and θi are the interfacial tension and contact angle of fluid system i, and Pc,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 CO2 was not well established at the initiation of this study and was taken to be


180˚. The interfacial tension between CO2 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 CO2-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.


Figure 3.6 Measured capillary pressure curves for the CO2-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).

3.3.1 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


boundary cells of the adjacent subdomains, ensuring conservation of mass for each

subdomain.

3.3.1.1 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).

𝑅𝑖𝑘 𝑥 𝑛+1 = 𝑀𝑖

𝑘 𝑥 𝑛+1 − 𝑀𝑖𝑘 𝑥 𝑛

− 𝛥𝑡

𝑉𝑖 𝐴𝑖𝑗𝐹𝑖𝑗

𝑘 𝑥 𝑛+1 + 𝑉𝑖𝑞𝑖𝑘 ,𝑛+1

𝑗

= 0 3.8

where the vectors xn, 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).

− 𝜕𝑅𝑖

𝑘 ,𝑛+1

𝜕𝑥𝑚𝑚

𝑝

𝑥𝑚 ,𝑝+1 − 𝑥𝑚 ,𝑝 = 𝑅𝑖𝑖 ,𝑛+1 𝑥𝑚 ,𝑝 3.9

where m indicates the mth

primary variable, p is the current iteration, and the solution is

for iteration p+1 by solving the system for xm, p+1.

3.3.1.2 Thermodynamic Variables

The primary variables in TOUGH2 are dependent upon the flow module being used.

The ECO2N module has been developed for the brine-CO2-NaCl system and uses the

following four primary variables for isothermal systems: pressure, NaCl mass fraction,

CO2 gas (or supercritical fluid) saturation and temperature in ˚C.


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 CO2 since the simulator does not make any distinction between liquid, gas or

supercritical CO2 (Pruess, 2005).

Using the cubic EOS, the mole fraction of each component is calculated in each

phase, from which the molality of CO2, 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 CO2 in the aqueous phase, x2 and mole

fraction of H2O in the gas phase, y1, provided from the cubic EOS (Pruess, 2005).

𝑛 = 𝑥2 2𝑚 + 1000

𝑀𝐻2𝑂

1 − 𝑥2

3.10

𝑋2 = 𝑛𝑀𝐶𝑂2

1000 + 𝑚𝑀𝑁𝑎𝐶𝑙 + 𝑛𝑀𝐶𝑂2

3.11

𝑌1 = 𝑦1 ∙ 𝑀𝐻2𝑂

𝑦1 ∙ 𝑀𝐻2𝑂 + 1 − 𝑦1 𝑀𝐶𝑂2

3.12

where m is the molality of NaCl in the brine, which is required as an input, Mk is the

molecular weight of component k, and X2 and Y1 are the mass fractions of CO2 in the

aqueous phase and H2O in the gas phase respectively.

3.3.1.3 Thermophysical Data

The thermophysical data required for simulation are density, viscosity and specific

enthalpy. These properties for CO2 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 CO2 density and viscosity calculations assume that the gas phase is pure

CO2.


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 CO2 as

given in Eq. 3.13 (Pruess, 2005).

1

𝜌𝑎𝑞=

1 − 𝑋2

𝜌𝑏+

𝑋2

𝜌𝐶𝑂2

3.13

where ρCO2 is calculated as given by Garcia (2001). This formulation neglects the

pressure dependence of partial density of dissolved CO2 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 CO2 (Pruess, 2005).

3.3.2 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.

33

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

4.1.1 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.

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, CO2 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

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

4.1.2 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.


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 CO2 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 CO2 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 CO2 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.


Figure 4.4 (a) Experiment saturation map (b) Upscaled saturation map

4.1.3 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 CO2 relative permeability.

𝑘𝑟 ,𝐵𝑟𝑖𝑛𝑒 = 𝑆𝐵𝑟𝑖𝑛𝑒 − 𝑆𝑙𝑟

1 − 𝑆𝑙𝑟 𝑛𝐵𝑟𝑖𝑛𝑒

4.1

𝑘𝑟 ,𝐶𝑂2=

𝑆𝐶𝑂2− 𝑆𝑔𝑟

1 − 𝑆𝑙𝑟𝑛 𝑛𝐶𝑂 2

4.2

where nBrine and nCO2 are exponential fitting parameters, Slr and Sgr are the residual liquid

and gas (CO2) saturation respectively and Slrn 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 CO2 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


parameters were also adjusted slightly from the R2 value closest to one after several

simple history matching tests on a homogeneous core to get a better match to the average

saturation value.

𝑆𝐵𝑟𝑖𝑛𝑒 ,𝑁𝑜𝑟𝑚𝑎𝑙𝑖𝑧𝑒𝑑 = 𝑆∗ =𝑆𝐵𝑟𝑖𝑛𝑒 − 𝑆𝑙𝑟

1 − 𝑆𝑙𝑟 4.3

Figure 4.5 Relative permeability curve fit

Relative Permeability Fit Parameters

nBrine nCO2 Slr Sgr Slrn

4.20 2.00 0.20 0.0 0.0

The value of residual liquid phase saturation, Slr, 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

CO2 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,

Slr of 0.20 was selected based on the several history matching simulations previously


mentioned and should be the same for both the capillary pressure and relative

permeability functions.

4.1.4 Capillary Pressure

4.1.4.1 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(Sw) 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(Sw) 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.

𝑃𝑐𝜎

𝑘

𝜙= 𝐽 𝑆𝑤 4.4

𝑃𝑐 ,𝑖 = 𝜎 𝑐𝑜𝑠 𝜃 𝜙𝑖

𝑘𝑖𝐽 𝑆𝑤 4.5

A number of investigators have developed functional forms for J(Sw) (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. 4.6. The parameters A, B and λi 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(Sw) are shown in

Figure 4.6 and Figure 4.7.

𝐽 𝑆𝑤 = 𝐴 1

𝑆∗𝜆1

− 1 + 𝐵 1 − 𝑆∗𝜆2

1𝜆2

4.6


Figure 4.6 Capillary pressure curve fit for medium sized Berea sample (ICP1)

ICP1 Capillary Pressure Fit Parameters

A B Slr λ1 λ2

.004 0.2 0.20 3.4 2.8

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 (R2) 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.


Figure 4.7 Capillary pressure curve fit for small sized Berea sample (ICP2)

ICP2 Capillary Pressure Fit Parameters

A B Slr λ1 λ2

.01 0.2 0.20 2.92 2.0

4.1.4.2 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(Sw) and scaled to that voxels unique porosity and permeability values. Recall

that it was stated that J(Sw) 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(Sw) 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


method to calculate permeability is the most important step in studying these sub-core

scale saturation phenomena.

4.1.5 Model Validation

4.1.5.1 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

CO2 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 CO2 that dissolves into the brine at

phase equilibrium, trial and error was used to determine the dissolved CO2 mass fraction

where CO2 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 CO2. The

injection of CO2 does not account for brine dissolved in the CO2 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

T (˚C) 50 Dissolved CO2init

(mf) 0.04873 QCO2-Gas (kg/s) 3.04E-05

P (MPa) 12.41 ρCO2 (kg/m3) 608.38 QCO2-Aq (kg/s) 0.00E+00

xNaCl (ppm) 6500 ρH2O (kg/m3) 993.33 QH2O-Gas (kg/s) 0.00E+00

φave 0.185 σCO2-Brine (N/m) 0.0285 QH2O-Aq (kg/s) 0.00E+00

kave (md) 89 Injection Rate (ml/m) 3 QNaCl (kg/s) 0.00E+00

Simulation Conditions Injection ConditionsThermophysical Data


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.

Figure 4.8 Grid used for simulations

Table 4.2 Simulation grid data

4.1.5.2 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

Y-Z Upscaling 5:1 Grid Dimensions 36x36x68 Cells/Slice 936

X Upscaling 2:1 Cell Size (mm3) 1.27 x 1.27 x 3 Total Cells 63648

𝒈

Injection


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 CO2. 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).


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. CO2 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.


4.2 Saturation Results using Kozeny-Carman Models

4.2.1 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. 4.7 – 4.10, 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 ϕc for Eq. 4.10 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.

𝑘𝑖 = 𝑆𝜙𝑖

3

1 − 𝜙𝑖 2 4.7

𝑘𝑖 =𝑆

𝑎𝑣2

𝜙𝑖3

1 − 𝜙𝑖 2 4.8


5

1 − 𝜙𝑖 2 4.9

𝑘𝑖 = 𝑆 𝜙𝑖 − 𝜙𝑐

3

1 − 𝜙𝑖 + 𝜙𝑐 2 4.10

Table 4.3 Simulation 1-4 permeability parameters for Kozeny-Carman models

Simulation Equation Parameter(s) Value

1 4.7 S 9253.5

S 650.4

av Eq. 4.11

3 4.9 S 260846

S 28431ϕc 0.04

4 4.10

2 4.8


4.2.1.1 Description of Calculating Specific Surface Area

To calculate the parameter av, 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 4.10.

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 av 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. 4.8. This process is explained

in more detail in Appendix A. By combining the Eq. 4.11 for av with Eq. 4.8, and

reducing, the result in Eq. 4.12 is an equation only in terms of porosity, which was


previously stated to be the easiest form to calculate permeability because the porosity of

the core is easily measured by the CT scanner.

𝑎𝑣,𝑖 = .033𝜙𝑖.79 4.11


1.42

1 − 𝜙𝑖 2 4.12

4.2.1.2 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 4.11.

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).


Figure 4.11 Permeability maps using Kozeny-Carman models for (a) Simulation 1 (b)

Simulation 2 (c) Simulation 3 (d) Simulation 4

4.2.2 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 CO2. The experiment clearly has the largest amount of spatial contrast in

CO2 saturation, while simulation 3, the model with the highest level of CO2 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.


Figure 4.12 CO2 saturation in slice 29 (a) Experiment (b) Sim. 1 (c) Sim. 2 (d) Sim. 3 (e) Sim.

4


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 CO2 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. 4.13 is used to calculate


permeability in an average sandstone and Eq. 4.14 is used to calculate permeability

specifically for a Berea sandstone (see Pape et al., 2000), ki is given in nm2 rather than

md for the fractal models. The fitting parameters S, for the equations are shown in Table

4.4.

𝑘𝑖 = 31𝜙𝑖 + 7463𝜙𝑖2 + 191 10𝜙𝑖

10 4.13

𝑘𝑖 = 6.2𝜙𝑖 + 1493𝜙𝑖2 + 58 10𝜙𝑖

10 4.14

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. 4.13 and Eq. 4.14 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 4.11. 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

5 4.13 S 0.000799

6 4.14 S 0.002923


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 4.15. 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.


Figure 4.15 CO2 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.


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

4.4.1 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, pd, index of pore size distribution, λ, and residual liquid saturation, Swr.


𝑘 = 𝑎1

1

𝑝𝑑 𝑎2

𝜆

𝜆 + 2 𝑎3

1 − 𝑆𝑤𝑟 𝑎4𝜙𝑎5 4.15

If we consider the equation however, it is apparent that once values of pd, λ and Swr

are determined for a rock type, these are simply constants which are raised to some

respective power ai, and multiplied by porosity raised to some power a5, 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 CO2 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 CO2 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.


Table 4.5 Core average results using traditional permeability models

4.4.2 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. The results in the figure also show that there is a very important

relationship between the degree of permeability contrast and corresponding CO2 contrast

when using porosity based permeability models.

Figure 4.17 Comparison of simulation saturation vs. porosity in Slice 29 (in order of

saturation contrast)

1 0.5007 0.3780 9222 30.64

2 0.4957 1.3729 9372 32.77

3 0.5053 0.5372 9155 29.69

4 0.5027 0.0199 9157 29.72

5 0.5036 0.1990 9638 36.53

6 0.4945 1.6116 9100 28.91

Average CO2

Saturation

Average ΔP

(Pa)Simulation

Saturation

Error (%)

Pressure

Error (%)


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 CO2 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

4.4.3 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


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.

61

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

5.1.1 Previous Investigations

5.1.1.1 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.

5.1. 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 𝜎𝐻𝑔−𝑎𝑖𝑟 𝑐𝑜𝑠 𝜃 2𝜙

1

𝑃𝑐2𝑑𝑆𝑤

1

0

5.1

62 CHAPTER 5. MODIFIED LEVERETT J-FUNCTION METHOD

5.1.1.2 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. 5.4.

𝐽 𝑆𝑤 = 𝑃𝑐


𝑘

𝜙 5.2

𝐽 𝑆𝑤 𝑆𝑤 =1.0 = 𝑃𝐷


𝑘

𝜙 5.3

𝑘 = 1

𝑝𝐷2 𝜙 𝐽 𝑆𝑤 1.0

2 𝜎 𝑐𝑜𝑠 𝜃 2 5.4

5.1.2 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(Sw)1.0 is

the same for each voxel by the definition of the J-Function, and unless displacement

pressure, pD, is known for each voxel, the core average pD 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 pD as the value at which to solve J(Sw), 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 Sw 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. 5.5.

𝑘𝑖 = 1

𝑝𝑐2𝜙𝑖 𝐽 𝑆𝑤 2 𝜎 𝑐𝑜𝑠 𝜃 2 5.5

CHAPTER 5. MODIFIED LEVERETT J-FUNCTION METHOD 63

where the J-Function is given by Eq. 4.6. 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.

𝑘𝑖 = 𝑆 ∙ 𝜙𝑖 𝐴 1

𝑆∗,𝑖𝜆1

− 1 + 𝐵 1 − 𝑆∗,𝑖𝜆2

1𝜆2

2


𝑃𝑐

2

5.6

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


5.1.3 Capillary Pressure Curve Fits

In Section 4.1.4, it was shown that the parameters A, B, λ1 and λ2 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. 5.6. The different values for residual liquid used to calculate permeability and used

in the actual simulation to calculate capillary pressure are designated by Slrk and Slr

s

respectively in the table.

Table 5.1 J-Function fitting parameters used to calculate permeability

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

Simulation Pc Curve # A B λ1 λ2 Slrk

Slrs

S

7 ICP1 0.004 0.2 3.4 2.8 0 0.2 0.502

8 ICP2 0.01 0.2 2.9 2 0 0.2 0.349

9 ICP3 0.004 0.2 3.4 2.8 0 0 0.502

10 ICP4 0.016 0.17 2 2.8 0 0 0.589

11 ICP5 0.025 0.25 2.2 1.4 0 0 0.535


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 Slr = 0.20 cases.

Figure 5.2 Capillary pressure fits for Simulation 9 (ICP3) and Simulation 10 (ICP4) from

medium Berea data


Figure 5.3 Capillary pressure fit for Simulation 11 (ICP5) from small Berea data

Figure 5.4 Relative permeability curve fit for Slr = 0

Relative Permeability Fit Parameters

nBrine nCO2 Slr Sgr Slrn

5.50 1.90 0.0 0.0 0.0


5.1.4 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 CO2 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


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

5.2.1 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 CO2 saturation values in both simulations.


Figure 5.7 CO2 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


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 CO2 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 CO2 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.


5.2.2 Zero Residual Brine Saturation Results

The resulting CO2 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 5.10. The

figure shows that the maximum CO2 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

CO2 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.


Figure 5.9 CO2 Saturation in slice 29 (a) Experiment (b) Sim. 9 (c) Sim. 10 (d) Sim. 11

Figure k Params Simulation Figure k Params Simulation

a c ICP4 10

b ICP3 9 d ICP5 11

Experiment

Simulation Key


Figure 5.10 Simulation vs. experiment saturation in slice 29 for J-Function method

5.2.3 Comparison of Core Average Results

These results have shown qualitatively that this modified Leverett method is more

accurate at predicting sub-core scale CO2 saturation, however, the previous chapter

showed in Table 4.5 that the traditional permeability methods were very accurate in

predicting the core average CO2 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.


Table 5.2 Core average results using modified Leverett J-Function method

5.3 Statistical Comparison of Permeability Methods

As a representative dataset for the core, the coefficient of determination (R2) values of

slice 29 data are shown in Table 5.3. The R2 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

5.10. 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 R2

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 R2 values, indicating that these

models best match the experimental values.

Table 5.3 Linear trend line data for slice 29 average saturation comparisons

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

7 0.4819 4.12 12016 70.22

8 0.4592 8.64 14242 101.76

9 0.5066 0.80 13618 92.92

10 0.5102 1.51 13229 87.41

11 0.4915 2.21 15567 120.53

SimulationAverage CO2

Saturation

Saturation

Error (%)

Average ΔP

(Pa)

Pressure

Error (%)

Simulation Fit Slope Fit R2

Fig. Ref. Simulation Fit Slope Fit R2

Fig. Ref.

1 0.8742 -96.05 4.13 7 0.8809 -0.628 5.7

2 0.8685 -722.4 4.13 8 0.8504 0.0015 5.7

3 0.8793 -27.7 4.13 9 0.9441 0.1757 5.9

4 0.8764 -65.26 4.13 10 0.9534 0.2162 5.9

5 0.8743 -5.457 4.16 11 0.916 -0.012 5.9

6 0.8589 -5.417 4.16


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

5.11. The figure shows that the R2 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 5.12. 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 CO2

saturation. Recall also that simulation 2 had the permeability model with the least


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

The CO2 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.



1 0.9973 -0.531 7 0.9605 0.9161

2 0.9878 0.3326 8 0.9154 0.9051

3 1.0061 -2.393 9 1.0082 0.791

4 1.0011 -1.145 10 1.015 0.8164

5 1.0025 -3.61 11 0.9788 0.9554

6 0.9841 -6.78


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.

79

Chapter 6

6 Conclusions

6.1 Summary of Findings

Chapter 3 showed that the saturation of CO2 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 CO2 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 CO2,

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

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 CO2

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 CO2 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.

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 CO2 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 CO2 is forced to circumvent a portion of the core by sub-core scale geological

features, the CO2 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 CO2

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.

82 CHAPTER 6. CONCLUSIONS AND FUTURE WORK

83

Appendix A: A Method for Estimating Specific Surface

Area

Specific surface area, av, 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.

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-

Figure A.1 Thin section conversion to binary image

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.

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

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)

APPENDIX A: SPECIFIC SURFACE AREA 85

the fraction of the unit area of the ROI which is grain, giving perimeter per unit grain

area. The procedure for calculating these values for the sample shown in Figure A.2 is

given below.

𝑃𝑇 = 𝑃𝑖

27

𝑖=1

+ 𝑃𝑖

33

𝑖=33

A.1

𝜙 = 𝐴𝑖 −

27𝑖=1 𝐴𝑖

33𝑖=33

𝐴𝑇 A.2

𝐴𝐺 = 𝐴𝑇 ∙ 1 − 𝜙 A.3

𝑃𝑣 = 𝑃𝑣𝐴𝑔

A.4

𝑎𝑣 = 𝛼𝑃𝑣 A.5

where PT is the total perimeter in the ROI, Pi and Ai are the perimeter and area of each

outlined feature in Figure A.2 (b), where the subscripts, i are given for each feature in

Figure A.2 (c), ϕ is the porosity of the ROI, AG is the total grain area in the ROI, Pv is the

specific perimeter in the ROI, av is the specific surface area, and α is the linear

proportionality factor between av and Pv. Plotting Pv vs. ϕ for each ROI gives the plot in

Figure A.3, which can be used to fit a power law correlation between specific perimeter

and porosity.

86 APPENDIX A: SPECIFIC SURFACE AREA

Eq. 4.8 used in the study is given below in Eq. A.6.

𝑘 = 𝑆

𝑎𝑣2

𝜙3

1 − 𝜙 2 A.6

𝑘 = 𝑆

0.033𝛼𝜙0.79 2

𝜙3

1 − 𝜙 2 A.7

Since α is a linear proportionality constant, it can be combined with the constant S,

which reduces Eq. A.7 to Eq. A.8, which is given as Eq. 4.12 in the chapter 4

permeability study.

𝑘 = 𝑆𝜙1.42

1 − 𝜙 2 A.8

Figure A.3 Specific perimeter for thin section in Figure A.1

87

Nomenclature

Abbreviations

CCS - Carbon capture and storage

CT - Computed Tomography

EOS - Equation of State

ICPj - Capillary pressure curve fit j

IPCC - International Panel on Climate Change

LBNL - Lawrence Berkeley National Laboratory

NETL - National Energy Technology Laboratory

NIST - National Institute for Standards and Technology

PVI - Pore Volumes Injected

ROI - Region of Interest

Symbols

a - Fitting parameter in Timur (1968) permeability model (md)

ai - Huet et al. (2005) fitting parameter, where i is from 1 to 5 (md or none)

av - Specific surface area per unit grain volume (m2/m

3)

A - Area (m2)

A - Fitting parameter in Silin et al. (2009) J-Function parameterization (-)

b - Porosity fitting in Timur (1968) (-)

B - Fitting parameter in Silin et al. (2009) J-Function parameterization (-)

c - Residual water saturation fitting exponent in Timur (1968) (-)

88 NOMENCLATURE

co - Kozeny’s constant (-)

Dp - Average grain diameter (m2)

Es - Storage efficiency (Fraction of void volume)

fj - Fractional flow of phase j (Fraction of total volumetric flow rate)

F - Mass or energy flux in TOUGH2 solver 𝑘𝑔

𝑚2𝑠 or

𝐽

𝑚2𝑠

ḡ - Gravitational vector (m/s2)

J - Leverett J-Function (-)

k - Permeability (md, nm2)

K - Darcy’s original definition of permeability 𝑚𝑑

𝑃𝑎∙𝑠

m - Molal concentration of NaCl in brine in Philips et al. (1981) 𝑔−𝑚𝑜𝑙𝑒𝑠 𝑁𝑎𝐶𝑙

𝑘𝑔−𝐻2𝑂

M - Specific surface area per unit bulk volume (m2/m

3)

Mi - Molarities of CO2 and NaCl in Kumagai and Yokoyama (1999) 𝑚𝑜𝑙𝑒𝑠


Mk - Molecular weight of component k in ECO2N (g/mole)

n - Molality of CO2 in brine 𝑔−𝑚𝑜𝑙𝑒𝑠 𝐶𝑂2


nj - Fitting exponent of phase j for relative permeability curve (-)

P - Fluid pressure (Pa)

Pc - Capillary pressure (Pa)

Pd - Displacement capillary pressure (entry pressure) (Pa)

Q - Injection rate (kg/s)

R - Residual vector of mass and energy balance in TOUGH2 (kg or J)

S - Shape factor used in Permeability models (varies)

Sj - Saturation of phase j (Percent or Fraction)

NOMENCLATURE 89

Slrn - Fitting parameter for relative permeability curves (-)

T - Temperature (˚C)

u - Darcy flow velocity (m/s)

V - Volume (m3)

x - Vector of primary variables in TOUGH2 solver (varies)

x1 - Mole fraction of CO2 in the aqueous phase (mol/mol)

y2 - Mole fraction of H2O in the gas phase (mol/mol)

X1 - Mass fraction of CO2 in the aqueous phase (kg/kg)

Y2 - Mass fraction of H2O in the gas phase (kg/kg)

Greek Symbols

Δ - Change in Value (-)

λ - Brooks and Corey pore geometry factor (-)

λ1 - Fitting parameter in Silin et al. (2009) J-Function parameterization (-)

λ2 - Fitting parameter in Silin et al. (2009) J-Function parameterization (-)

μ - Viscosity (Pa·s)

ϕ - Porosity 𝑉𝑜𝑖𝑑 𝑉𝑜𝑙𝑢𝑚𝑒

𝑇𝑜𝑡𝑎𝑙 𝑉𝑜𝑙𝑢𝑚𝑒

ϕc - Percolation porosity constant 𝑉𝑜𝑖𝑑 𝑉𝑜𝑙𝑢𝑚𝑒

𝑇𝑜𝑡𝑎𝑙 𝑉𝑜𝑙𝑢𝑚𝑒

ρj - Density of phase or component j (kg/m3)

σ - Interfacial Tension 𝐷𝑦𝑛𝑒𝑠

𝑐𝑚

Τ - Tortuosity in capillary tube model (m/m)

τ - Dimensionless injection time (Pore Volumes Injected)

θ - Contact angle between wetting and non wetting fluids (Degrees)

Subscripts

90 NOMENCLATURE

brine - Pertaining to aqueous brine wetting phase

CO2 - Pertaining to CO2 non-wetting gas phase

CO2-brine Pertaining to the CO2 brine system in capillary pressure conversions

eff - Effective value

gr - Pertaining to residual gas phase saturation

grain - Rock or mineral grain

hg-air - Pertaining to the Mercury-air system in capillary pressure measurements

H2O - Pertaining to water

i - Index referring to grid element or CT voxel i

j - Index referring to grid element j, or pertaining to phase j

lr - Pertaining to residual liquid or aqueous phase saturation

m - Indicating the mth

primary variable in the TOUGH2 Newton solver

ss - Indicates a measurement taken at steady state

NaCl - Pertaining to sodium chloride

p - Iteration counter in TOUGH2 Newton iteration solver

w - Pertaining to wetting phase

wr - Pertaining to residual wetting phase saturation

Superscripts

* - Pertaining to normalized saturation

brine - Pertaining to the aqueous brine wetting phase

k - Indicates a component (Brine, CO2, NaCl)

n - Indicates time step n solution in TOUGH2 simulation

91

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