The Pennsylvania State University

The Graduate School

College of Earth and Mineral Sciences

PORE-SCALE IMAGING AND LATTICE BOLTZMANN MODELING OF SINGLE-

AND MULTI-PHASE FLOW IN FRACTURED AND MIXED-WET PERMEABLE

MEDIA

A Dissertation in

Energy and Mineral Engineering

by

Christopher James Landry

© 2013 Christopher James Landry

Submitted in Partial Fulfillment

of the Requirements

for the Degree of

Doctor of Philosophy

May 2013

ii

The dissertation of Christopher James Landry was reviewed and approved* by the following:

Zuleima T. Karpyn

Associate Professor of Petroleum and Natural Gas Engineering

Dissertation Adviser

Chair of Committee

Li Li

Assistant Professor of Energy and Mineral Engineering

Russell T. Johns

Professor of Petroleum and Natural Gas Engineering

Maria Lopez de Murphy

Associate Professor of Civil Engineering

Luis Ayala

Associate Professor of Petroleum and Natural Gas Engineering

Associate Department Head for Graduate Education

*Signatures are on file in the Graduate School

iii

ABSTRACT

Three investigations of pore-scale single-phase and multiphase flow in fractured porous media

and mixed-wet porous media are presented here. With an emphasis on validating and utilizing

lattice Boltzmann models in conjunction with x-ray computed microtomography.

The objective of the first study is to investigate fracture flow characteristics at the pore-scale,

and evaluate the influence of the adjacent permeable matrix on the fracture’s permeability.

We use X-ray computed microtomography to produce three-dimensional images of a fracture in

a permeable medium. These images are processed and directly translated into lattices for

single-phase lattice Boltzmann simulations. Three flow simulations are presented for the imaged

volume, a simulation of the pore space, the fracture alone and the matrix alone. We show that

the fracture permeability increases by a factor of 15.1 due to bypassing of fracture choke points

through the matrix pore space. In addition, pore-scale matrix velocities were found to follow a

logarithmic function of the distance from the fracture. Finally, our results are compared

against previously proposed methods of estimating fracture permeability from fracture

roughness, tortuosity, aperture distribution and matrix permeability.

In the second study we present a pore-scale study of relative permeability dependence on the

strength of wettability of homogenous-wet porous media, as well as the dependence of relative

permeability on the distribution and severity of wettability alteration of porous media to a mixed-

wet state. A Shan-Chen type multicomponent multiphase lattice Boltzmann model is employed

to determine pore-scale fluid distributions and relative permeability. Mixed-wet states are

iv

created – after pre-simulation of homogeneous-wet porous medium – by altering the wettability

of solid surfaces in contact with the non-wetting phase. To ensure accurate representation of

fluid-solid interfacial areas we compare LB simulation results to experimental measurements of

interfacial fluid-fluid and fluid-solid areas determined by x-ray computed microtomography

imaging of water and oil distributions in bead packs (Landry et al. 2011). The LB simulations

are found to match experimental trends observed for fluid-fluid and fluid-solid interfacial area-

saturation relationships. The relative permeability of both fluids in the homogenous-wet porous

media is found to decrease with a decreasing contact angle. This is attributed to the increasing

disconnection of the non-wetting phase and increased fluid-solid interfacial area of the wetting

phase. The relative permeability of both fluids in the altered mixed-wet porous media is found to

decrease. However the significance of the decrease is dependent on the connectivity of the

unaltered solid surfaces, with less dependence on the severity of alteration.

In the third study we present sequential x-ray computed microtomography (CMT) images of

matrix drainage in a fractured sintered glass granule pack. Sequential imaging captured the

capillary-dominated migration of the non-wetting phase front from the fracture to the matrix in a

brine-surfactant-Decane system. The sintered glass granule pack was designed to have minimal

pore space beyond the resolution of CMT imaging, so that the pore space of the matrix

connected to the fracture could be captured in its entirety. The segmented image of the pore

space was then directly translated to a lattice to simulate the transfer of fluids between the

fracture and the matrix using lattice Boltzmann (LB) modeling. This provided us an opportunity

to validate the modeling technique against experimental images at the pore-scale. Although the

surfactant was found to alter the wettability of the originally weakly oil-wet glass to water-wet,

v

the fracture-matrix fluid transfer is found to be a drainage process, showing little to no counter-

current migration of the oil-phase. The LB simulations were found to closely match

experimental rates of fracture-matrix fluid transfer, equilibrium saturation, irreducible wetting

phase saturation and fluid distributions.

vi

TABLE OF CONTENTS

List of Tables ………………………………………………………………….. ix

List of Figures …………………………………………………………………. ix

Acknowledgements ……………………………………………………………

xv

CHAPTER 1: INTORDUCTION TO PORE-SCALE STUDIES ……………. 1

CHAPTER 2: SINGLE-PHASE LATTICE BOLTZMANN …………………

SIMULATIONS OF PORE-SCALE FLOW IN FRACTURED

PERMEABLE MEDIA

9

2.1 Summary .………………………………………………………………. 9

2.2 Introduction ……………………………………………………………. 9

2.3 Materials and Methods ………………………………………………… 17

2.3.1 X-ray Computed Microtomography Scanner …..……………….. 17

2.3.2 Fractured Permeable Media ….…...……………………………… 18

2.3.3 Image Processing ….…………………………………………….. 20

2.3.4 Fracture Pore Space Identification ...…………………………….. 22

2.3.5 Surface Area and Volume ….…………………………………….. 24

2.3.6 Aperture and Fracture Roughness .………………………………. 24

2.3.7 Lattice Boltzmann Simulations .…………………………………. 28

2.4 Results ..……………………………………………………………….. 31

2.4.1 Fracture and Matrix Pore Space Geometry .……………………… 31

2.4.2 LB Simulation Results ……..…………………………………….. 33

2.4.3 Fracture Permeability Estimation ……………………………….. 36

2.5 Conclusions ….……………………………………………………….. 43

vii

2.6 References …………..………………………………………………… 46

CHAPTER 3: RELATIVE PERMEABILITY OF HOMOGENOUS-WET ….

AND MIXED-WET POROUS MEDIA AS DETERMINED BY

PORE-SCALE LATTICE BOLTZMANN MODELING

55

3.1 Summary ..……………………………………………………………… 55

3.2 Background ….………………………………………………………….. 56

3.3 Materials and Methods ….………………………………………………. 64

3.3.1 Experimental Measurements …………………………………….... 64

3.3.2 Single-Phase BGK Lattice Boltzmann Model …………………….. 65

3.3.3 Shan-Chen Multicomponent Lattice Boltzmann Model …………. 67

3.3.4 LB Model Implementation ………………………………………… 69

3.4 Results …………………………………………………………………… 72

3.4.1 Fluid-Fluid and Fluid-Solid Interfacial Areas …………………….. 72

3.4.2 Pore Aperture and Fluid Distribution ……………………………… 77

3.4.3 Two-Phase Flow in a Circular Pore ……………………………….. 79

3.4.4 Relative Permeability of Homogenous-Wet States ………………. 81

3.4.5 Relative Permeability of Mixed-Wet States ………………………. 84

3.5 Conclusions …………………………………………………………….. 92

3.6 References ……..…………………………………………………………

95

CHAPTER 4: LATTICE BOLTZMANN MODELING AND ………………....

4D X-RAY COMPUTED MICROTOMOGRAPHIC IMAGING OF

FRACTURE-MATRIX FLUID TRANSFER

106

4.1 Summary ………………………………………………………………. 106

4.2 Introduction ……………………………………………………………... 107

4.3 Materials and Methods ………………………………………………….. 110

viii

4.3.1. Fractured Porous Medium ………………………………………… 110

4.3.2 Procedure ………………………………………………………….. 111

4.3.3 CMT Imaging …………………………………………………….... 113

4.3.4 Image Processing …………………………………………………... 113

4.4 Results ………………………………………………………………….... 115

4.4.1 Pore Space Analysis ………………………………………………... 115

4.4.2 Immiscible Displacement …………………………………….……. 121

4.4.3 LB Simulation of Fracture-Matrix Fluid Transfer ………….……... 127

4.4.4 Pore-Scale Experimental and Model Results ………………..…….. 130

4.5 Conclusions ……….……………………………………………………... 137

4.6 References …….…………………………………………………………. 140

CHAPTER 5: SUMMARY OF FINDINGS AND RECOMMENDATIONS ….

FOR FUTURE WORK

148

ix

LIST OF TABLES

Table 2.1. Pore space geometry and fracture characteristics.

…………………33

Table 2.2. Summary of LB permeability measurements and fracture

permeability estimates.

…………………41

Table 4.1. Temperature cycles for sintering of glass granules.

…………………110

LIST OF FIGURES

Figure 1.1 An example segmented CMT image showing the pore

space voxels in white (A). Lattice Boltzmann models

simulate fluids as swarming particles (B) on a discrete

lattice, where each pore voxel is represented by a LB

fluid node (C). The complex solid boundaries are also

retained by translating the CMT image 1-to-1 voxel-to-

node with the solid boundaries being translated to

bounce-back nodes, shown in yellow (D).

…………………4

Figure 2.1. Illustration of the double-sided ‘log-splitting’ approach

used to propagate a rough fracture in the porous

polyethylene rod.

…………………19

Figure 2.2. Demonstration of simple thresholding image

segmentation. At the left is a raw CMT image slice, the

raw data is cropped (shown as dashed lines) to remove

portions of fracture that were in contact with the splitting

tool. The middle plot shows the frequency of CT

registration numbers, and Gaussian fits to the solid and

pore phases in the image, the intercept of these Gaussian

fits is used to segment the image. At the right is a

segmented image, in which gray represents the pore

space and white represents the solid or grain.

…………………21

x

Figure 2.3. Example slices from the fracture space identification

procedure. The left image shows the pore space (light

gray) after 8 cycles of erosion, at which point all of the

matrix pore space is removed, this locates the fracture

(dark gray). The middle image shows the pore space in

contact with the fracture space after 2 cycles of reversed-

out erosions, the erosion cycle one short of the fracture

pore space becoming connected with the matrix pore

space; this identifies the starting point for expanding the

fracture pore space out. The right image shows the

fracture pore space after 6 cycles of expansion, the

expansion cycle one short of the fracture pore space

making contact with the grains or solid, this defines the

fracture.

…………………23

Figure 2.4. Illustration of the measure of the vertical aperture of the

fracture ( ), the perpendicular aperture ( ), the fracture

profile distances from edge of image ( ) and the

bandwidth window ( ). The light gray is the matrix pore

space, and the dark gray is the fracture.

…………………24

Figure 2.5. Fracture aperture distribution. The small initial peak is a

result of the matrix pore aperture distribution impacting

the aperture distribution of the fracture at fracture

boundaries. The distribution is overall log-normal,

consistent with fractures in natural permeable media.

…………………26

Figure 2.6. Poiseuille flow through a channel with a radius

velocity profile, comparison of LB results

and analytical solution. These results show the LB

simulation is closely approximating flow in the small

channel.

…………………31

xi

Figure 2.7. Determination of Hurst coefficient using the variable

bandwidth method. Data points are shown for four

slices, the linear function shown ( ) was

determined from 16 equidistant slices spanning the range

of the simulated pore volume. The Hurst exponent is the

slope of the fitted linear function shown.

…………………32

Figure 2.8. Normalized x-velocity glyph visualization of the

fracture-alone (above) and the all-pore (below)

simulations; the glyphs are larger and farther to the red

side of the spectrum for greater velocities.

Th28maximum normalized velocity shown in the gly29s

is .

…………………35

Figure 2.9. Mean normalized velocities ( ) as a function of

distance from the fracture ( ), and logarithmic function

fits.

…………………27

Figure 3.1. Segmented CMT image of the bead pack (A) with solid

voxels colored white, and the corresponding lattice of the

LB model (B) with the bounce-back nodes in blue.

…………………71

Figure 3.2. Specific fluid-fluid interfacial areas as a function of

wetting phase saturation for the LB simulations and

experimental CMT images.

…………………76

Figure 3.3. Fractional fluid-solid interfacial areas as a function of

wetting phase saturation for the LB simulations and

experimental CMT images, the wetting fluid-solid

interfacial areas are shown in black and the non-wetting

fluid-solid interfacial areas are shown in gray.

…………………76

Figure 3.4. Pore aperture and fluid distribution of initial …………………78

xii

homogenous-wet states.

Figure 3.5. Relative permeability in a circular pore with radius = 11

; LB simulation results and semi-analytical solution

curves.

…………………81

Figure 3.6. Relative permeability determined by LB simulations for

the initial homogenous-wettability states. The relative

permeability of the wetting and non-wetting phase are

shown in black and gray, respectively.

…………………82

Figure 3.7. Images of the LB lattice with initial homogenous-

wettability (A), the fluid distribution of the non-wetting

phase at the end of the simulation of the initial

homogenous-wettability with and (B),

and the alteration of the bounce-back densities of the

nodes in contact with the non-wetting fluid (C). This

image also summarizes the three parameters determining

the mixed-wet state, initial wettability (A), saturation of

alteration (B), and severity of alteration (C).

…………………86

Figure 3.8. Relative permeability determined by LB simulations for

the mixed-wet states of the originally homogenous-wet

states, (A) , (B) , (C)

, and (D) , with

increasing severity of alteration. The relative

permeability of the wetting and non-wetting phase are

shown in black and gray, respectively.

…………………88

Figure 4.1. Image of the fracture with the ROI highlighted (A), and

the corresponding mean fracture aperture and contact

area profiles with the ROI indicated by arrows.

…………………116

xiii

Figure 4.2. Image of the matrix pore space showing the fracture

saturated with water phase and the matrix saturated with

the oil phase, the ROI volume is highlighted (A). Also

the corresponding porosity (B) and mean aperture (C)

profiles with the ROI indicated with arrows.

…………………118

Figure 4.3. Example 0.1183 cm

3 image of sphere-packing method

used to measure pore aperture distribution. The

individual pore apertures are measured as the radius of

the spheres seen in this image. The red, purple, teal,

green, yellow, and light green represent pore aperture

measurements between ,

, ,

, , and

mm, respectively. The

background shows the solid in black and the pore space

in white.

…………………120

Figure 4.4. The fracture aperture distribution (left) and the pore

aperture distribution (right) for the portion of the core to

be imaged sequentially during water invasion of the

matrix.

…………………121

Figure 4.5. Image of oil phase (white), water phase (black), glass

(gray) contact in the fracture after initial water phase

injection. The contact angle appears in this image to be

large, near neutral wetting conditions.

…………………122

Figure 4.6. CMT images from the sequential scanning of the

surfactant water phase injection showing a volume

rendering (left) and a slice from a height of 1.0704 cm

(right). In the volume renderings the water phase is dark

orange, the solid is light orange and the oil phase is

white. In the slices the water phase is light gray, the

glass is dark gray, and the oil phase is white.

…………………124

xiv

Figure 4.7. Non-wetting phase saturation of the right and left matrix

as determined from sequential CMT imaging. Early time

refers to displacement before the interfacial tension

reduction has stabilized. The linear fits shown represent

a displacement following the rate function,

⁄

.

…………………126

Figure 4.8. Volume images of the 270x150x150 voxel3 (0.4256 x

0.2365 x 0.2365 cm3) sample volumes of the pore space

for the left and right matrix to be translated 1-to-1, voxel

to lattice node, for lattice Boltzmann simulation (A). LB

simulation setup showing the 5 lattice layer thick

pressure boundaries, with the fracture represented as the

non-wetting phase occupied pressure boundary (B).

…………………128

Figure 4.9. Non-wetting phase saturation of the right and left matrix

as determined from sequential CMT imaging and LB

drainage simulations. The only variable between the LB

simulations for the right and left matrix is the pore space

geometry.

…………………131

Figure 4.10. Comparison of experimental (left) and LB simulation

(right) vertical saturation profiles for the right (A, B) and

left (C, D) matrix, and horizontal saturation profiles for

the right (E, F) and left (G, H) matrix.

…………………133

Figure 4.11. Sample images of the LB simulations of fracture-matrix

transfer. The non-wetting phase saturation front and

simulation pore space are shown, the wetting phase is

occupies the pore space not occupied by the non-wetting

phase.

…………………135

xv

ACKNOWLEDGEMENTS

This material is based upon work supported by the National Science Foundation under

Grant No. 0747585 and the Marathon Alumni Centennial Graduate Fellowship. I would

like to thank Dr. Orlando Ayala of the University of Delaware for providing computational

resources, and for offering his expertise and time, many of the simulations shown here

would not have been possible without his help. I would also like to thank Soheil Saraji

of Piri Research Group at the University of Wyoming for providing interfacial tension

measurements. I would also like to thank my doctoral committee for their thoughtful

comments, and insightful discussion. Last I would like to thank my advisor, Dr. Zuleima

Karpyn, who had the unenviable task of keeping me on task.

1

CHAPTER 1: INTRODUCTION TO PORE-SCALE STUDIES

The goal of studying multiphase fluid flow at the pore-scale in porous media and fractured

porous media is to improve fluid transport prediction at the field-scale. Generally prediction of

fluid transport at the field scale is determined by models that discretize the formation of interest

– attributing macroscale flow properties to individual gridblocks – and simulating flow under

varying scenarios. Macroscale flow properties are conventionally determined by measuring fluid

flow in cores take from the formation of interest. Pore-scale studies attempt to elucidate the

connection between pore space characteristics and fluid mobility, with the intention of

determining macroscale flow properties to be utilized in field-scale fluid transport prediction.

Accurate determination of macroscale flow properties such as permeability, relative permeability

– saturation relationships and capillary pressure – saturation relationships are fundamental to

accurate field-scale fluid transport prediction. There are numerous techniques used to measure

macroscale flow properties of porous media and fractured porous media, all of which determine

macroscale flow properties by measuring fluid flux through cores taken from the formation of

interest under varying applied boundary conditions. Although direct measurement of macroscale

flow properties is straight forward there are downsides to these methods. For one they require

cores that can be expensive to procure, and often these cores only offer a small sampling of a

vast heterogeneous formation. The measurements themselves are also expensive, and in the case

of fractured porous media, measuring flow in the fracture and porous matrix separately can be

very difficult, if not impossible. Most importantly, these methods provide measurements, but no

2

insight into the cause for resulting macroscale flow measurements. The porous media is treated

as essentially a “black-box”.

Pore-scale studies peer into this black-box to elucidate the relationship between pore space

characteristics and resulting macroscale fluid flow. Although the physics of multiphase fluid

flow is fairly well understood, the complexity of the solid boundary conditions encountered in

pore spaces render any possibility of analytical solutions of fluid flow to be beyond practicality.

Complex non-invasive imaging techniques, such as x-ray computed microtomography (CMT)

allow us to not only image pore spaces at resolutions great enough to capture the shape of

individual pores, but also the distribution of fluids within these pores. Being able to image pore

spaces and fluid distributions at the pore-scale provides insight into multiphase fluid flow, as in

we are able to peer into the black-box, but this does not alleviate some of the issues we

encountered with conventional measurements of multiphase fluid flow. It is still expensive, and

requires cores.

Pore-scale models offer us a powerful tool to not only gain insight into the connection between

pore space characteristics and resulting macroscale fluid flow. Pore-scale models use either

imaged pore spaces or synthesized pore spaces to simulate fluid flow at the pore-scale. There are

many different types of pore-scale models in existence, including classic computational fluid

dynamics (discrete solutions of Navier-Stokes equation), pore-network models, smoothed

particle hydrodynamics, the level-set method and lattice Boltzmann (LB) models. The best

choice of model to use when investigating fluid flow at the pore-scale is dependent on the

objective of the investigation, here we employ lattice Boltzmann models. The complexity of the

3

solid boundaries simulated in LB models is only limited by computational power, or in the case

of imaged pore spaces, the resolution of the image. This is an improvement over more

commonly used pore-network models which represent the pore space as a network of discrete

geometric shapes (i.e. cylinders, triangular tubes, square tubes, spheres etc.). Briefly, LB models

can be described as simulating fluids as swarming particles on a discrete lattice. A detailed

description of the model used in the investigations presented here can be found in Chapter 3.

The complex solid boundaries of pore spaces imaged by CMT can be translated 1-to-1 voxel-to-

node onto LB lattices To better visualize what is meant by maintaining the complexity of the

solid boundaries of pore space in LB models please refer to figure 1.1. In this figure we show a

CMT image of a pore space and the corollary LB lattice. LB models simulate fluid flow in a

purely local manner, no long range information is required during the evolution of the model,

each node only requires knowledge of adjacent nodes. This means LB models have an inherent

parallelism, and simulations can carried out across many processors using parallel computing

techniques. One of the greatest challenges encountered in the implementation of LB models is

the computational expense. Although LB models are inherently parallel and simulations can be

spread across hundreds, if not thousands of processor cores, the computational demand is so

great that multiphase simulations on 1003 node

3 lattices can require upwards of a week on 200

cores to reach convergence. However, it should be noted the only expense is computational. It

is possible to run numerous scenarios at only the cost of computer time.

4

Swarming Particles Discrete LBE

A)

B) C)

D)

Figure 1.1. An example segmented CMT image showing the pore space voxels in white (A). Lattice

Boltzmann models simulate fluids as swarming particles (B) on a discrete lattice, where

each pore voxel is represented by a LB fluid node (C). The complex solid boundaries are

also retained by translating the CMT image 1-to-1 voxel-to-node with the solid boundaries

being translated to bounce-back nodes, shown in yellow (D).

There are two main objectives of the three investigations presented here in chapters 2, 3, and 4.

The first is to validate multi-phase pore-scale fluid distributions predicted by LB models against

those imaged experimentally. One of the main goals of pore-scale studies is to better understand

the connection between pore space characteristics and resulting macroscale flow, considering

5

this, it is imperative that we validate LB prediction of pore-scale fluid distribution. LB models

have in many previous investigations been used to determine macroscale flow properties in the

same way as is done conventionally, by imposing pressure boundaries and monitoring fluid flux.

In these studies LB models have been shown to accurately predict macroscale flow properties.

For further information regarding previous investigations into prediction of macroscale flow

properties by LB please refer to subsection 3.2 Background. There are however only a handful

of studies (subsection 3.2 Background) that have attempted to compare LB model pore-scale

fluid distributions against experimentally imaged pore-scale fluid distributions. The second main

objective is to take advantage of the intimate knowledge of fluid velocity and distribution offered

by the LB model to perform numerical investigations into single-phase and two-phase fluid flow

that would be difficult if not impossible to perform in a physical laboratory setting.

With these objectives in mind we present three investigations. In chapter 2 we take advantage of

the intimate knowledge of velocity (velocity is known at every LB fluid node) offered by the LB

model to investigate single-phase flow in fractures bounded by a porous matrix. We simulate

flow in a fracture bounded by porous walls and the same fracture bounded by impermeable

walls, to investigate the effect porous walls have on fracture permeability. Because we have

intimate knowledge of velocity we simulate the flow in our fracture-matrix coupled system, and

measure the flow in the fracture exclusively. This would be nearly impossible in a conventional,

physical experimental system, because it is extremely difficult to separate the measurement of

fluid flow in the fracture from that of the matrix. In this investigation we also compare the our

results to existing fracture permeability estimations based on fracture characteristics, such as the

mean aperture, standard deviation of the aperture distribution, wall roughness, tortuosity and

6

contact area. Since many formations of interest are fractured, and fluid flow is often dominated

by flow in fractures in these formations, accurate estimations of fracture permeability are

imperative to the success of field-scale fluid transport prediction.

In the third chapter we engage in validation of LB model prediction of fluid distribution by

comparison to experimental results, and investigate fluid flow in mixed-wet states using the LB

model. Experimental pore-scale fluid distribution data sets are sparse, which is one of the

reasons so few investigations have attempted to compare model and experimental pore-scale

fluid distributions. We use an experimental data set we previously published (Landry et al.

2011) to compare to our LB model results. And after presenting an argument for validation of

our methods through comparison to experimental measurements, we engage in an investigation

of wettability alteration to mixed-wet states and the effect this has on the relative permeability of

the porous medium. We take advantage of the control of solid surface wettability (the adhesion

force of every LB bounce-back node is tunable) offered by the LB model to investigate two-

phase flow in mixed-wet porous media. Although numerical studies into the relative

permeability of mixed-wet porous media using pore-network models has existed for over twenty

years (subsection 3.2 Background), this is the first thorough investigation of two-phase flow in

mixed-wet porous media using LB to the author’s knowledge. Also much like in the first

investigation, this investigation would be very difficult to perform in a conventional, physical

experimental system, due to the difficulty in engineering the wettability of physical porous

media.

7

In the fourth chapter we again engage in validation of LB model prediction of pore-scale fluid

distribution by comparison to experimental results. In this investigation we image and simulate

fracture-matrix fluid transfer at the pore-scale in a surfactant-brine-oil fluid system. Many

formations of interest are fractured and the transfer of fluids from the high-permeability, low-

storage fractures to the low-permeability, high-storage matrix is of great interest. Predicting this

migration requires a rigorous investigation into the mechanics that determine the transfer of

fluids to and from the matrix. Understanding the physics of this process can be greatly enhanced

by investigating fluid transfer at the interface of the fracture and matrix at the pore-scale. Many

previous investigations at the pore-scale have used two-dimensional etched-glass micromodels to

investigate fracture-matrix fluid transfer (see subsection 4.2 Introduction). We present a first of

its kind three-dimensional study of fracture-matrix fluid transfer at the pore-scale. We

sequentially image fracture-matrix fluid transfer at the pore-scale in a three-dimensional

fractured porous medium using CMT. This is a novel use of three-dimensional CMT imaging,

and the first time to the author’s knowledge that fracture-matrix fluid transfer has been captured

in four dimensions at the pore-scale. We compare pore-scale fluid distributions from this

experiment to LB model results to both validate the method of simulation and gain insight into

the physics of fracture-matrix fluid transfer in a surfactant-brine-oil system.

The porous media used in these investigations is all synthetic, there are two reasons we do not

use natural porous media in our investigations. First, the CMT imaging system has limited

resolution. To compare LB and experimental results it is imperative that the model system is

representative of the porous medium imaged, if there are pore spaces beyond the resolution of

the CMT imaging system, they may affect the fluid flow in the experimental system in a way that

8

cannot be represented in the simulation. Second, natural porous media can contain

heterogeneous surfaces that we cannot differentiate using the imaging techniques at our disposal.

Small differences in the wettability of natural porous media due to heterogeneous composition or

surface precipitation may affect fluid flow in the experimental system in a way we cannot

translate to the simulation. Investigations like those presented here, help to build a foundation of

validation of pore-scale modeling techniques, and highlight the strengths and limits of the pore-

scale models used. The ultimate goal is to apply these techniques to natural porous media, and

free the determination of macroscale flow properties from the “black-box” of conventional

measurement techniques.

9

CHAPTER 2: SINGLE-PHASE LATTICE BOLTZMANN SIMULATIONS OF PORE-

SCALE FLOW IN FRACTURED PERMEABLE MEDIA

2.1 SUMMARY

The objective of this work is to investigate fracture flow characteristics at the pore-scale, and

evaluate the influence of the adjacent permeable matrix on the fracture’s permeability. We use

X-ray computed microtomography to produce three-dimensional images of a fracture in a

permeable medium. These images are processed and directly translated into lattices for

single-phase lattice Boltzmann simulations. Three flow simulations are presented for the imaged

volume, a simulation of the pore space, the fracture alone and the matrix alone. We show that

the fracture permeability increases by a factor of 15.1 due to bypassing of fracture choke points

through the matrix pore space. In addition, pore-scale matrix velocities were found to follow a

logarithmic function of the distance from the fracture. Finally, our results are compared

against previously proposed methods of estimating fracture permeability from fracture

roughness, tortuosity, aperture distribution and matrix permeability.

2.2 INTRODUCTION

Current understanding of the physical phenomena concerning fluid flow through fractures is

limited. Fluid flow in fractured formations depends on both the fracture system and the

adjacent permeable rock, also called the rock matrix. Fractures control the overall conductivity

10

of the rock while the permeable matrix provides fluid storage capacity. Studying fracture

structures, flow through fractures and fracture-matrix interactions presents significant

challenges with important applications in areas such as geo-environmental remediation, geo-

hazard mitigation, geothermal exploitation, and hydrocarbon production. In particular,

development of unconventional hydrocarbon resources, such as coalbed methane, gas shales and

tight sands, depends on our ability to represent fracture flow and fracture-matrix interactions for

the design of recovery strategies and performance predictions. The deliverability of these

unconventional hydrocarbon resources is highly controlled by the connectivity of natural

fracture network, the accessibility of fractures to the matrix storing hydrocarbons and the

fracture permeability. Of fundamental interest to the study of flow in fractured permeable media

is the determination of fracture permeability from fracture and matrix pore space geometry.

The study of flow in fractures is, for the most part, limited to fractures in rock with

impermeable fracture walls. The simplest approximation of single phase fluid flow in

fractures with impermeable walls is given by the cubic law, the analytical solution to the Navier-

Stokes equation for viscous flow through wide smooth parallel plates (no-slip walls). Given

the wall is wide enough (Width >> aperture) and the length across which a pressure head is

applied is long enough (Length > Width), the cubic law is defined as:

(

)

where is the flow rate through the fracture, w is the fracture width, is the fracture

aperture, is the dynamic viscosity, is the pressure gradient, and in the context of Darcy’s

11

equation fracture permeability ( ), is defined as, ⁄ . The simplest modification

to the cubic law is to replace the aperture, with the mean aperture ( ), this is referred to as the

alternate cubic law (ACL). This simple definition of flow through fractures does not include

important factors influencing fluid flow in natural fracture geometries, such as, fracture

roughness, asperities, anisotropies and aperture distribution. These factors have been shown to

greatly influence flow in fractures (Auradou et al. 2005, Keller et al. 1995, Konzuk and

Kueper 2004, Nolte et al. 1989, Tsang and Witherspoon 1981, Tsang and Witherspoon

1983, Witherspoon et al. 1980). As the effect of these factors became apparent, empirical and

semi-analytical modifications of the cubic law were made in the hopes of developing an

estimation of fracture permeability from fracture geometry statistics. Witherspoon et al. 1980

studied flow across smooth and rough non-contacting surfaces and introduced a modification

to the fracture permeability that included the fracture roughness,

where is a surface roughness factor ranging from 1.04 to 1.65. After the introduction of fractal

geometry, it became apparent fractures possess self-affine fractal properties regardless of the

host rock (Brown and Scholz 1985, Crandall et al. 2010, Poon et al. 1992, Schmittbuhl et al.

1995). The fractal property of fractures is often described by its Hurst exponent ( ), related to

the fractal dimension ( ) by . With this knowledge, investigators began to study

flow in synthetically developed fractures that followed fractal geometries. The study of flow

through rough fractures can be difficult to quantify and observe experimentally, thus

investigators turned to recent developments in pore-scale fluid flow models, and computational

12

fluid dynamics (CFD) to numerically simulate flows in complex pore spaces. A variety of pore

scale models have been developed, here we focus on lattice Boltzmann (LB) methods

(McNamara and Zanetti 1988), and to a lesser extent its forbearer, lattice gas cellular automata

(LGCA) (Frisch et al. 1986) methods.

Lattice techniques hold great promise in their application to the study of flow in permeable

media; they are capable of simulating flow in any complex media – given a lattice of adequate

resolution is used – as well as being easily used in parallel processing. LB and LGCA methods

have been used to study single-phase flow in both permeable media (e.g., Boek and Venturoli

2010, Ferreol and Rothman 1995, Genabeek and Rothman 1996, Koponen et al. 1997, Maier et

al. 1998, Manz et al. 1999, Pan et al. 2004) and fractures (e.g., Auradou et al. 2005, Boutt et

al. 2006, Drazer and Koplik 2002, Eker and Akin 2006, Kim et al. 2003, Madadi and Sahimi

2003, Nazridoust et al. 2006, Stockman et al. 1997, Zhang and Kang 2004, Zhang et al. 1996).

Zhang et al. 1996 studied single-phase flow through synthesised self-affine fractal fracture

using LCGA; they proposed an empirical estimation of fracture permeability as a

function of mean aperture:

with a value of of 2.67, 3.15 and >4 for fractures with a Hurst exponent of 0.8, 0.3 and -0.5.

A value of 2 would be expected for flow between parallel plates, the value of is expected to

increase monotonically with roughness. Eker and Akin 2006 also investigated single phase flow

in synthesized self-affine fractal fractures with Hurst exponents ranging from -0.21 to -0.50

13

using LB methods, and following the work of Zhang et al. 1996 proposed to be a linear

function of fractal dimension,

These empirical predictions only take the fracture roughness into consideration, other factors

governing fracture geometry such as asperities and aperture distribution must also be included in

a complete prediction for the estimation of fracture permeability. Zimmerman and Bodvarsson

1996 attempted to relate the effective hydraulic fracture aperture ( ) to the statistics of aperture

distribution. The effective hydraulic fracture aperture is a measured property of flow through a

fracture, it is the equivalent aperture for flow between smooth plates for a measured flow rate

(Q),

(

)

The relation between ( ) and the statistics of aperture distribution was derived from the

application of Reynold’s lubrication equation. As they noted, asperities can be viewed as a part

of the aperture distribution, however, it is convenient to treat the contact areas or asperities

separately from the aperture distribution. The is proportional to the fracture permeability

(

⁄ ), thus their prediction for the relation of the to fracture statistics is also a

definition of fracture permeability:

(

)

14

is the standard deviation of the fracture aperture distribution and C is the fractional

contact between fracture walls. The applicability of numerical results from synthesised

fractures has merit, but validation of these methods requires studying real fracture geometries.

Imaging of fractures has become simplified by the employment of sophisticated three

dimensional imaging techniques, such as X-ray computer microtomography (CMT).

The use of X-ray CMT in the study of flow in permeable media and fractures has become

widespread, offering a non-invasive method to image complex pore geometries and fractures in

three dimensions. High resolution images of pore space are often used in conjunction with pore-

scale models to investigate pore scale flow mechanisms that are difficult to measure and

observe. A large variety of permeable media has been imaged using CMT, including macro

permeable clays (Heijs et al. 1995) soils (Perret et al. 1999), bead packs (Culligan et al.

2006), unconsolidated sands (Brusseau et al. 2006, Wildenschild et al. 2005), sandstones

(Auzerais et al. 1996, Nakashima et al. 2004) and limestones (Okabe and Blunt 2004). Many

fractures have also been studied using CMT, including single brittle fractures in Berea

sandstone (Karpyn et al. 2007), single planar artificial fractures of increasing aperture in

limestone (Ketcham et al. 2010), lab-scale hydraulic fractures in limestone (Renard et al.

2009), natural fractures in limestone (Wennberg et al. 2009), and single brittle fractures in

granite (Johns et al. 1993, Keller et al. 1999), to name a few.

Nazridoust et al. 2006 studied flow in a single tensile fracture in Berea sandstone imaged by

Karpyn et al. 2007. They proposed studying flow in fractures in a manner analogous to the

15

study of pressure losses due to friction in pipes and ducts. The resistance to flow is measured as

a friction factor, analogous to fracture permeability. For flow between smooth parallel plates

the friction factor (f) is defined as,

They introduced a new friction factor dependant on fracture tortuosity and aperture distribution,

and fitted to CFD results for flows of varying velocity,

( )

where ⁄

is the Reynolds fracture number and is the kinematic viscosity; this can be

related to the fracture permeability by the pressure drop,

(

)

where is the corrected mean fracture aperture, defined as, , is the dynamic

viscosity, and is the tortuosity,

16

where is the length of the tortuous actual path of flow and is the length of the fracture,

arriving at a semi-analytical estimation of fracture permeability,

( )

Crandall et al. 2010 also studied flow in a single fracture in Berea sandstone imaged by CMT

using CFD. They used the same image of a fracture as Nazridoust et al. 2006, and attempted to

include matrix effects on flow by making the CFD grid adjacent to the fracture permeable.

Following the work of Nazridoust et al. 2006 they redefined the friction factor for the fracture to

include the flow through the matrix,

( )

⁄

and equivalently the fracture permeability,

( (

) ⁄ )

( )

where is the matrix permeability and is the width of the matrix. Overall they found the

friction factor decreased with increasing matrix permeability, and the fracture permeability is

directly proportional to the matrix permeability.

17

As of yet, there is no agreed upon approach for estimation of fracture permeability from

fracture geometry and the flow behavior of the rock matrix. This is a fundamental issue, and a

key building block for adequate description and simulation of fluid flow through fractured

reservoirs. To the author’s knowledge, this is the first pore-scale study of flow in fractured

permeable media in which a three-dimensional rough fracture is explicitly interconnected with

the pore network of the adjacent permeable rock. We use CMT to image the fractured

permeable media at resolutions high enough to capture the solid surface curvature of the

grains; hence we are able image the entirety of the pore space including that of the matrix.

This image is translated directly into a lattice to be used by LB simulations of single phase

flow. To study the effect of the matrix on flow in the fracture, and vice versa, three simulations

are presented. One in which the entirety of the pore space is simulated (fracture in permeable

rock), one in which the matrix pore space is removed and the flow is simulated in the fracture

alone (analogous to a fracture in impermeable rock), and lastly five simulations are run on

portions of the pore space that do not contain any part of the fracture and averaged to determine

the permeability of the matrix alone. The results of these simulations are compared to the

empirical and semi-analytical estimations of fracture permeability discussed above.

2.3 MATERIALS AND METHODS

2.3.1 X-ray Microtomography Scanner

The pore space of the fractured permeable medium was imaged using X-ray (CMT).

Scanning took place at Penn State’s Center for Quantitative X-ray Imaging (CQI). X-ray CMT

imaging is a non-invasive imaging technique that produces a three-dimensional grid of voxels

18

– much like pixels for a two-dimensional image – each of which contain an integer value CMT

registration number. The scan of the core in the scan presented here is composed of 800 slices,

each of these slices being 1024 x 1024 voxels. Each slice of the three-dimensional grid of CMT

registration numbers is produced by a mathematical reconstruction of 1440 two-dimensional

images taken perpendicular to the axis of the core over the course of a full 360° core rotation.

CMT registration numbers produced by the mathematical reconstruction are dependent on the

relative X-ray attenuation, or X-ray opaqueness, of the material occupying each voxel. X-ray

attenuation by a material is a function of the density and apparent atomic number of the

material. These images are segmented to identify the material occupying each voxel, to be

further elaborated on in the image processing section. The half inch diameter cores in this

investigation result in CMT scans with a voxel resolution of 13.606 x 13.606 x 12.405 μm3, the

latter represents the thickness of the slice. However, to directly translate the CMT image of the

pore space to a LB simulation requires a cubic lattice, thus the images are treated as having

cubic voxels with side lengths of 13.606 μm.

2.3.2 Fractured Permeable Media

Porous polyethylene rods manufactured by Small Parts Inc., were used as the permeable

medium. These rods are 0.127 cm in diameter and formed by sintering medium/coarse,

angular/subangular polyethylene grains into rod form. This creates a permeable structure similar

to a medium/coarse, well-sorted unconsolidated sand or sandstone. A synthetic permeable

medium is used in lieu of a natural permeable medium to ensure the pore space of the matrix

is resolvable. Polyethylene is a malleable material, thus a double- sided ‘log-splitting’

19

approach was used to create a fracture in the porous polyethylene rods (Figure 2.1), instead of

more conventional techniques (i.e., modified Brazilian test) used to induce fractures on natural

rock. Two splitting tools are driven along the desired fracture plane, not down the center of the

rod, but along the sides. This results in a rough fracture propagating between the splitting tools.

The outer portions of the fracture that were in contact with the spitting tool are cropped from

the image and not used in the LB simulation. This crop results in a total pore space lattice

volume of 520 x 520 x 800 voxel3.

Figure 2.1. Illustration of the double-sided ‘log-splitting’ approach used to propagate a rough fracture

in the porous polyethylene rod.

20

2.3.3 Image processing

The raw, cropped CMT images are processed and segmented for analysis. The raw images

are first filtered using a 3 x 3 x 3 kernel median filter to remove salt-and-pepper type noise. To

quantify measurements of the pore space, the filtered images are segmented into the phases

present, pore and solid. The contrast between the solid phase and pore space is sufficient for

application of simple thresholding. Simple thresholding segments the image by declaring all

voxels below a threshold value as belonging to one population and all those above to the other.

The threshold value is determined by fitting Gaussian curves to a frequency plot of the CT

registration numbers (Figure 2.2) (Mees et al. 2003). The frequency plot contains three peaks,

the first is associated with the pore space, and the second two are associated with the solid,

with respect to increasing CT registration number. The intercept of the two Gaussian curves

for the respective populations is used as the threshold value, here at a CT registration number

of 17323. The two peaks of the solid represent the two different polyethylene particulates used

in the construction of the permeable rod.

21

Figure 2.2. Demonstration of simple thresholding image segmentation, at the left is a raw CMT image

slice, the raw data is cropped (shown as dashed lines) to remove portions of fracture that

were in contact with the splitting tool. The middle plot shows the frequency of CT

registration numbers, and Gaussian fits to the solid and pore phases in the image, the

intercept of these Gaussian fits is used to segment the image. At the right is a segmented

image, in which grey represents the pore space and white represents the solid or grain.

The segmented images contain some sporadic solid voxels misidentified by the segmentation

procedure in the pore space. To remove these sporadic voxels, the segmented 3D image of the

pore space undergoes a percolation. A percolation finds all like voxels in contact. First a

percolation is conducted on the solid voxels. The contact points between individual grains are

beyond the resolution of the CMT images, thus this percolation finds all the connected solid

voxels or ‘true’ solid voxels. The ‘negative’ of the connected solid voxels is then considered

the pore space. To remove any isolated pores from the image of the pore space a second

percolation is carried out on the negative of the connected solid voxels. The second percolation

finds all pore space voxels in contact, ensuring the pore space to be simulated does not

contain any unnecessary, non-interacting (isolated pores) pore space. This prepares the pore

space for direct interpretation into a lattice for use by LB simulations.

22

2.3.4 Fracture Pore Space Identification

The identification of the fracture is non-trivial; unlike previous investigations of fractures, the

distinction between the fracture and matrix is not readily apparent. To identify the pore space

associated with the fracture pore space – also simply referred to as the fracture – a sequence of

erosions, expansions and percolations were performed on the segmented images of the pore

space. A cycle of erosion substitutes all the pore space edge voxels with solid voxels, thus

removing a single layer of voxels from the pore space volume. Edge voxels are defined as those

with voxel face contact between the two phases present, namely solid and pore. Expansion is the

opposite of erosion.

The first step in the identification of the fracture pore space is the isolation of the locality of

the fracture. If the fracture aperture is generally greater than the greatest distance between

grains in the pores of the matrix, the fracture can be located by eroding the pore space until a

single pore is remaining. This point was reached after eight cycles of erosion for the pore space

presented here. This single pore only captures the core of the fracture pore space, to identify

the entirety of the fracture this pore must be dilated out. Each cycle of erosion results in a loss

of detail, to minimize this effect, the core of the fracture pore space is reversed-out by two

cycles of erosion. It is important to note that this is not two cycles of expansion. We have the

images preceding each erosion; we can retain loss of detail to these two cycles of erosions. The

dilation cannot continue in this reversing-out manner. The next reversed-out erosion opens up

the pore throats connecting to the matrix pore space, to continue in this manner would

erroneously begin to identify the matrix pore space as fracture pore space. Instead, at this point

23

we perform cycles of expansion until the fracture pore space contacts the solid surface. After

seven cycles of expansion the dilated fracture pore space overlaps solid voxels, thus six cycles

of expansion defines the fracture. This sequence is summarized in Figure 2.3.

Figure 2.3. Example slices from the fracture space identification procedure. The left image shows the pore space (light grey) after eight cycles of erosion, at which point all of the matrix pore space is removed, this locates the fracture (dark grey). The middle image shows the pore space in contact with the fracture space after two cycles of reversed-out erosions, the erosion cycle one short of the fracture pore space becoming connected with the matrix pore space; this identifies the starting point for expanding the fracture pore space out. The right image shows the fracture pore space after six cycles of expansion, the expansion cycle one short of the fracture pore space making contact with the grains or solid, this defines the fracture.

The flow in the matrix adjacent to the fracture is influenced by the high rate of flow in the

fracture, and differs significantly from the flow in the rest of the matrix. To investigate the

extent of this effect, the flow in the pore space adjacent to the fracture pore space is analyzed.

To measure this flow, the voxels adjacent to the fracture must be identified as a function of

distance from the fracture. To do so, the fracture pore space is expanded to identify voxels at

equal spatial distance from the fracture throughout the entirety of the pore space. These

expansions identify voxels as a function of distance from the fracture, and provide a measure of

flow velocity as a function of distance from the fracture.

24

2.3.5 Surface Area and Volume

The segmented images are used to measure surface areas and volumes of grains and pore space.

Volumes are calculated as the sum of all solid and pore voxels ( ). The porosity of the

matrix ( ) is defined as, ( )⁄ and the fracture porosity ( ) is defined as,

⁄ , where is the facture volume. The surface areas are determined using the image

processing software Avizo Fire 6.3. This software uses a modified marching cubes algorithm to

“wrap” a tetrahedral surface around the segmented image data for the solid phase. This surface

quantifies the solid surface area ( ). This is normalized to the bulk permeable volume of the

matrix to find the specific solid surface area ( ) defined as, ⁄ , where is the

bulk volume of the matrix defined as, .

2.3.6 Aperture and Fracture Roughness

It is assumed that the slices of the CT scan run orthogonal to the vertical plane of the

fracture; the measurement of apertures and fracture roughness are taken per slice of the scan.

There are two manners in which aperture can be measured, the perpendicular aperture (hp)

and the vertical aperture (h) (Figure 2.4). The apertures measured here are vertical apertures,

albeit the perpendicular aperture is not subject to fracture orientation, as the vertical aperture

is, considering the velocities reported here follow a Cartesian coordinate system defined by

the images, the vertical aperture was considered a more appropriate method of measurement.

The aperture distribution is shown in figure 2.5; the small initial peak is a result of the matrix

pore aperture distribution impacting the aperture distribution at fracture boundaries. The

25

fracture boundaries are defined by the matrix pore space, this will define the lower end of the

fracture aperture distribution. This effect on the aperture distribution becomes negligible as the

fracture aperture to matrix pore aperture ratio increases, but for our fracture the effect is

readily apparent. The distribution is overall log-normal, consistent with fractures in natural

permeable media (Karpyn et al. 2007, Keller et al. 1999, Ketcham et al. 2010). In this study,

fracture apertures below the maximum pore aperture found in the matrix (~0.33 mm) are

considered fracture contact areas. Pore apertures are measured in the same manner as fracture

apertures. The definition of contact area used here is,

where is the number of fracture apertures measured that are less than the maximum vertical

aperture of the matrix pores and is the total number of measured fracture apertures.

26

Figure 2.4. Illustration of the measure of the vertical aperture of the fracture (h), the perpendicular

aperture (hp), the fracture profile distances from edge of image (Bj, Bj+s) and the

bandwidth window (s). Note, The light gray is the matrix pore space, and the dark gray is

the fracture.

Figure 2.5. The fracture aperture distribution, the small initial peak is a result of the matrix pore

aperture distribution impacting the aperture distribution of the fracture at fracture

boundaries. The distribution is overall log-normal, consistent with fractures in natural

permeable media.

0

0.005

0.01

0.015

0.02

0.025

0.03

0 0.5 1 1.5 2 2.5

Fre

qu

ency

Fracture Aperture (h) (mm)

27

The fracture profile is defined by the fracture pore space in individual two-dimensional slices of

the image. The roughness of the fracture profiles of individual slices is characterized by the

Hurst exponent ( ) also referred to as the roughness exponent or self-affine exponent. We use

the variable bandwidth method to determine the Hurst exponent ( ) of the fracture profile for

individual slices. We assume our fracture is well described by fractional Brownian motion, such

that a scaling relationship exists between the size of the sampling window ( ) and the standard

deviation of the fracture profile ( ):

The Hurst exponent can be determined from the slope of a log-log plot of standard deviation

( ) vs. sampling window size ( ). The standard deviation is calculated according to a

modified form of (Dougan et al. 2000):

√

∑ ( )

where is the number of datapoints contained in the dataset, j is the index of the fracture profile

height ( ) (Figure 2.4), and is a count of the number of samples that contain an asperity,

wherein a measurement of is not possible. The assumption that a fracture is well described by

fractional Brownian motion and the variable bandwidth method is an appropriate method for

the measurement of the Hurst exponent has been found to be generally true for fractures in

natural rocks by previous investigators (Crandall et al. 2010, Renard et al. 2009). The

tortuosity ( ), previously defined, requires a measurement of the mean length of a streamline

flowing through the fracture ( ), this is found using the programme Paraview. The stream

tracer function seeds a particle at the inlet and progress the particle in the velocity field. The

28

mean length of a streamline is determined from 50 streamlines seeded at regular intervals along

the fracture inlet.

2.3.7 Lattice Boltzmann Simulations

Most petroleum engineers are accustomed to the ‘top-down’ methods of simulating fluid flow,

in which partial differential equations defining fluid flow are discretized using finite-

difference, finite-element and finite-volume techniques. The LB method has its roots in

LCGA methods, which simulate flow using a ‘bottom-up’ approach, in which, fluids are

simulated by a swarm of particles moving along discrete directions on a lattice. The streaming

and collision of these particles are governed by a set of rules such that the time-averaged motion

of the particles is consistent with the Navier-Stokes equations. The LCGA method is subject to

numerical instability, to avoid instability the LB method replaces these particles with averages

referred to as particle distribution functions. For a review of LB methods, see Succi 2001 and

Sukop and Thorne 2006. A detailed description of the LB model is given in chapter 3.

The lattice of LB methods is well suited for direct pore space lattice construction from CMT

images. Each voxel is directly translated to a single node in the Boltzmann lattice. The pore

voxels are represented by nodes on which ‘particles’, represented by particle distribution

functions, can freely move upon, the solid voxels are represented by non-interacting nodes, and

the solid boundary voxels are represented by bounce-back nodes. Bounce-back nodes are

essentially algorithm devices that return any exiting ‘particles’ back into the pore lattice domain

with the opposite momentum. Here, we use the half-way bounce-back condition on the nodes

at the solid surface, this ensures a microscopic, and hence, a macroscopic no-slip condition

29

halfway between the pore nodes and the bounce-back nodes (He et al., 1997). A density is

associated with each node, this gives the system mass, and as a result pressure. The LB

simulation presented here simulates a quasi-incompressible fluid, wherein the lattice node

density ( ) is directly proportional to the lattice pressure ( ), following from the ideal

gas equation of state. The ratio of the determining forces involved in flow, namely the

inertial and viscous forces, are conventionally defined by Reynolds numbers. The physical and

lattice Reynolds numbers ( ) are defined as,

and,

where is the reference velocity, taken as the average velocity of the flow to be

simulated, is the reference length taken as the length of the simulated volume, is the

physical dynamic viscosity, is the maximum lattice velocity, is the lattice resolution,

is the lattice kinematic viscosity and is the lattice density. Here, a water flow with a

is simulated, a stokes flow. The lattice kinematic viscosity was set to 1/6

(generally accepted to be the most numerically stable lattice kinematic viscosity), and the

resolution was set equal to 520, corresponding to CMT image resolution. To impose a flow on

the lattice the pressure at the inlet and outlet are set to constant values. To set the pressure at

the inlet and outlet, the densities of the nodes at these boundaries are maintained throughout the

simulation. Since the density is related to the pressure by the equation of state, this ensures the

pressures will remain constant at these boundaries (Zou and He 1997). All other boundaries are

treated as periodic, as in, when a particle exits a boundary it directly enters the boundary exactly

30

opposite of where it exited. The ‘particles’, driven by the pressure (lattice density) gradient

between the inlet and outlet, undergo a single stream and collision step during each iteration of

the LB simulation until steady state flow is reached. Each lattice site has an associated

macroscopic velocity vector with velocity components reported in the three Cartesian

directions ( , where ). Steady-state flow is reached when the mean

lattice velocity magnitude converges.

The open source LB code Palabos v0.7r3 was used to perform the LB simulations. Simulations

were carried out on Penn State’s HPC lion-X cluster. The LB simulations used a D3Q19

lattice with Bhatnagar-Gross-Krook (BGK) dynamics. Pan et al. (2006) investigated the

accuracy of LB simulations of flow in sphere packs using multiple-relaxation-time (MRT),

single-relaxation-time (SRT) and BGK dynamics. They found for a lattice viscosity of 1/6

(used here) the BGK model was as accurate as the more computationally expensive MRT

model. To validate the simulation, a Poiseuille flow was simulated through a narrow

cylindrical channel with a radius ( ) of 0.33 mm. The analytical solution to the velocity profile

for Poiseuille flow is,

The velocity profiles for the analytical solution and LB simulation results for Poiseuille flow

are shown in figure 2.6.

31

Figure 2.6. Poiseuille flow through a channel with a radius a = 0.33(mm) velocity profile, comparison

of LB results and analytical solution. We observe that the LB simulation is closely

approximating flow in the small channel.

2.4 RESULTS

2.4.1 Fracture and Matrix Pore-Space Geometry

A summary of all measured properties of the pore space are summarised in Table 2.1. The

matrix porosity (φm) was found to be ~0.32, significantly lower than the values of 0.40–0.50

reported by Prodanovic et al. 2006, which suggests inconsistency in the manufacturing

process of these rods. The fracture porosity (φf) of the pore space is 0.243; the volume of the

fracture is significant, and the fracture is expected to dominate flow. The roughness of the

fracture was measured for 16 slices evenly distributed along the vertical extent of the fracture

(every 0.652 mm). The Hurst exponents were determined for each of these slices; figure 2.7

shows a log-log plot of deviation vs. window size for some of these slices. The Hurst exponent

ranges from 0.51 to 0.69, with a mean Hurst exponent (ε) of 0.612; this is a nominal value of

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

-0.33 -0.27 -0.21 -0.15 -0.09 -0.03 0.03 0.09 0.15 0.21 0.27 0.33

u (

m/s

)

x (m)

LB sim

Analytical

32

roughness for permeable media. The mean Hurst exponent of 0.612 is similar to Hurst exponents

measured for tensile fractures in Berea sandstone (0.38 to 0.45) (Crandall et al. 2010), and single

hydraulically induced fractures in annular cores of Fountainbleu sandstone (0.4 to 0.5) (Renard

et al. 2009). The mean tortuosity was determined from the tortuosity measurement of 50

streamlines, and found to be 0.52. This is a relatively high tortuosity, the tortuosities measured

for a single tensile fractures in Berea sandstone ranged from 0.29 to 0.30 (Nazridoust et al.

2006) and 0.034 to 0.064 (Crandall et al. 2010). The differences in tortuosity measured by

Nazridoust et al. 2006 and Crandall et al. 2010 are a result of different methods for determining

actual length of streamlines. Crandall et al. 2010 determined the actual length from seeded

streamlines in the same manner used here, Nazridoust et al. 2006 assumed the fracture profile

was representative of the tortuosity.

Figure 2.7. Determination of Hurst coefficient using the variable bandwidth method, the data points

are shown for four slices, the linear function shown (ε = 0.612) was determined from 16

equidistant slices spanning the range of the simulated pore volume. The Hurst exponent

is the slope of the fitted linear function shown.

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2 2.5

log

(σfp

)

log (s)

Slice 200

33

2.4.2 LB Simulation Results

The objective of running the three LB simulations is to study flow in a fractured permeable

medium, a fractured impermeable medium and an unfractured permeable medium, three LB

simulation results are presented respectively, one of the entire pore space (fractured permeable

media), one of the fracture alone (fractured impermeable media), and an average of five

simulations of the matrix alone (permeable media). The velocities are normalized to the mean x-

velocity ( ), also the Darcy velocity, for each voxel with index ,

| |

| |

Table 2.1 Pore Space Geometry and Fracture Characteristics

34

A three-dimensional velocity glyph visualization is presented in figure 2.8, this visualization

illustrates the channeling of flow in the fractures for both simulations. In general, the flow

through the fracture is significantly increased by the permeable matrix. Also, the flow in the

fracture-alone simulation shows a single readily apparent high velocity choke point, while the

flow in the fracture – aided by the permeable matrix – is more widely dispersed over the fracture.

The matrix only LB simulations were run on 200 x 200 x 400 voxel portions of the pore space

that did not contain any part of the fracture; the average of five of these simulations was used to

determine matrix flow properties. The permeability was determined from the mean x-, or Darcy

velocities,

The flow in the fracture for the all-pore space simulation dominates the flow, with 88% of the

flow occurring in the fracture with a velocity 4.24 times the mean x-velocity. The permeability

of the fracture with a permeable matrix is 15.1 times greater than the fracture alone. The

permeability of the matrix in the all-pore space simulation is also increased by the proximity of

the fracture by 2.14 times. However, if the simulation included a greater extent of the matrix –

as in a greater distance between fractures – this effect would dissipate. As in, if a greater extent

of the matrix was given this apparent increase in permeability would drop to unity. Keep in

mind the periodic boundaries imply that the flow simulated is the equivalent of imposing a

distance between ‘fractures’ equal to the dimension of the simulated volume orthogonal to flow.

35

Figure 2.8. Normalized x-velocity glyph visualization of the fracture-alone (above) and the all-pore

(below) simulations; the glyphs are larger and farther to the red side of the spectrum for

greater velocities. The maximum normalized velocity shown in the glyphs is .

To explore the flow adjacent to the fracture, the velocities are plotted as a function of distance

36

from the fracture in figure 2.9. The velocities appear to decrease logarithmically as a function of

distance from the fracture following the form,

( ) ( )

where is the velocity in the x-, y- or z-direction and is the distance from the fracture. This

is a purely empirical fitting of data. The high velocities in the matrix immediately adjacent to

the fracture persist for only a short distance, a distance on the order of a single grain diameter.

The previously mentioned apparent increase in the matrix permeability in the all-pore simulation

is a result of these high velocities.

37

Figure 2.9. Mean normalized velocities ( ) as a function of distance from the fracture ( ),

and logarithmic function fits.

2.4.3 Fracture Permeability Estimation

A summary of the LB simulation results and fracture permeability estimates from previously

mentioned investigations are presented in Table 2.2. The fracture permeability determined from

LB simulation results for the all-pore simulation and the fracture-alone simulation are 31.9e-9

and 2.11e-10 m2, respectively. The ACL determines an estimate of fracture permeability from

the mean aperture, ignoring all other aspects of the fracture geometry and matrix permeability.

38

The ACL overestimates the fracture permeability in the all-pore simulation and the fracture-

alone simulation by 11.1 and 168 times, respectively. The ACL is not expected to give a good

estimate of permeability for our fracture. The ACL estimate decreases in accuracy with the

greater influence of unaccounted geometry determining flow in the fracture, such as the aperture

distribution, tortuosity and fracture roughness. If the fracture wall separation was significantly

increased, the tortuosity would decrease and the roughness would interfere with a smaller

percentage of the flow. Witherspoon et al. 1980 fit an empirical prediction of fracture

permeability to measurements from flow experiments. They determined the fracture permeability

from the mean aperture and a surface roughness factor, here set to the highest value

( ) of the range of values suggested. Witherspoon et al. 1980 overestimates the

fracture permeability in the all-pore simulation and the fracture-alone simulation by 6.71 and

102 times, respectively. Witherspoon et al. 1980 was determined empirically from flow

experiments, thus it is subject to the nature of the fractures used to fit it. If these fractures were

not as rough and tortuous as ours we would expect these characteristics to have less of an

effect on their flow experiments, and more closely approximated by the ACL. Zhang et al.

1996 determines the fracture permeability from the mean aperture and a power factor, here set to

a value ( ) linearly interpolated from the values determined for fractures with a

Hurst exponent of 0.80 ( ) and 0.30 ( ). This estimate, like the estimate of

Witherspoon et al. 1980, was determined empirically, and unlike Witherspoon et al. 1980,

from LCGA simulations on synthetic fractures, instead of flow measurements. Zhang et al. 1996

underestimates the fracture permeability in the all-pore simulation by 3.85 times, and

overestimates fracture permeability in the fracture-alone simulation by 3.94 times. The

simulated flow in Zhang et al. 1996 took place in a fracture with impermeable walls, thus their

39

estimate of fracture permeability is most appropriately tested by the fracture-alone LB

simulation, and the overestimate is a result of the differences in the fractures simulated by them

and the fracture imaged here. The fractures simulated by Zhang et al. 1996 had no induced

tortuosity, nor varying aperture. Tortuosity and varying aperture within a fracture will increase

the resistance to flow, without consideration of these fracture characteristics we would expect an

overestimation of permeability. However, considering the simplicity of their proposed estimation

of fracture permeability it is a vast improvement over ACL and Witherspoon et al. 1980. Eker

and Akin 2006 proposed the β value of the Zhang et al. 1996 is a linear function of the fractal

dimension, as we see in table 2.2 their estimate is by far the most inaccurate. This is not

surprising, considering this linear function was fit to numerical flow simulations on

synthesised fractures in impermeable rocks with abnormally high fractal dimensions

( ). Their linear function for does not fit the results of Zhang et al. 1996,

nor ours, suggesting a linear function of determined from the fractal dimension alone will

not accurately estimate fracture permeability outside from those for which it was fit.

The fracture permeability estimate of Zimmerman and Bodvarsson 1996, unlike the above

estimations of fracture permeability, is semi-analytically derived. Their estimate is only

applicable to fractures in impermeable rock, and underestimates the fracture permeability for

the fracture-alone LB simulation by approximately 48%. The underestimation indicates that

characterizing the fracture by the mean and standard deviation of the fracture aperture alone does

not provide enough information to accurately predict the fracture permeability.

Nazridoust et al. 2006 derived a semi-analytical prediction for fracture permeability that assumes

40

the flow in the fracture can be treated in much the same manner as flow in pipes.

Conventionally in pipe and duct flow, pressure losses are a function of friction factors, which

are dependent on flow rate. The cubic law, which was the basis for previously evaluated

fracture permeability estimations, assumes Stoke’s or laminar flow. The smooth, parallel plate

solution used by Nazridoust et al. 2006 includes inertial effects, thus the resistance to flow is

dependent on flow rate. The fracture permeability estimation of Nazridoust et al. 2006 is

dependent on the flow rate, fracture aperture mean, standard deviation and tortuosity. They

fitted their function for the friction factor to CFD simulations performed on two-dimensional

CMT images of fractures in Berea sandstone. These two-dimensional fracture profiles

contained no asperities. Their derivation does not account for permeable fracture walls, thus

only the fracture-alone LB simulation is comparable. Nazridoust et al. 2006 overestimate the

permeability in the fracture-alone LB simulation by 1.33 times, but it is the best estimate of those

tested. This overestimation can be attributed to the significant amount of asperities in our

fracture, which are not accounted for by Nazridoust et al. 2006, and in particular the effect these

asperities have on a three-dimensional fracture.

41

Table 2.2. Summary of LB Permeability Measurements and Fracture Permeability Estimates

Notes: Summary of LB results ( ) and fracture permeability

estimates, where - Alternate Cubic Law, - (Witherspoon et al., 1980), -

(X. Zhang et al., 1996), -(Eker and Akin, 2006), - (Zimmerman and Bodvarsson,

1996), - (Nazridoust et al., 2006) and - (Crandall et al., 2010); is the only

estimate that considers matrix permeability, and the only estimate appropriate for the all-

pore LB simulation.

Crandall et al. 2010 built upon the work of Nazridoust et al. 2006 by adding the matrix

permeability to the prediction of fracture permeability. They used the same CMT images of a

fracture in Berea sandstone as Nazridoust et al. 2006 and performed two-dimensional CFD

simulations with a homogenous, isotropic fracture wall with permeability ranging from 2E-16 to

2E-12 mm2 (0.2 to 2000 mD). Since their derivation accounts for a permeable fracture wall,

42

only the all-pore LB simulation is comparable. Crandall et al. 2010 underestimates the fracture

permeability in the all-pore simulation by 6.72 times. Although the prediction for the LB

simulation of the fracture alone (Nazridoust et al. 2006) is off by a relatively small percentage

(33%), the prediction for the all-pore space fracture permeability (Crandall et al. 2010) is off by

672%. Here are some explanations for this large difference. First, Crandall et al. 2010

validated their derivation of fracture permeability from CFD results for matrix permeability well

below our matrix permeability of 4.92E-11mm2 (49.7 D). Also, there is a fundamental

difference in the nature of the matrix for our simulations and theirs. Our simulation directly

simulates an actual matrix pore space, this allows for fluid to channel in and out of the fracture

through the matrix, helping to bypass fracture choke points. This will greatly increase the

flow in the fracture, as we observe from our results. The matrix in Crandall et al. 2010 does

not allow for this mechanism of bypass. In CFD the rock matrix is divided up into a fine-

grid of finite-volumes, and each of these volumes are homogenous. This form of simulation

assumes the finite-volumes are on the order of the representative elementary volumes (REV) of

flow for the permeable medium being simulated. The fracture pore space does not have a REV,

but the matrix does, and it’s REV is dependent on the grain size of the permeable media. The

REVs of sandstone, and the matrix simulated in the all-pore LB simulation presented here, are

many orders of magnitude larger than the finite volumes used by Crandall et al. 2010.

Crandall et al. 2010 use of CFD is arguably a fair representation of flow in fractured

permeable media with much smaller REVs.

43

2.5 CONCLUSIONS

A single rough fracture was propagated along the axis of a porous polyethylene rod; this

fractured permeable medium was imaged using CMT. The fracture pore space identification

process, consisting of a succession of erosions, percolations and expansions on the segmented

processed CMT images of the pore space, presented here appears to capture the fracture well.

The fracture was found to have a log-normal aperture distribution and a Hurst exponent of

0.612, similar to tensile fractures in Berea and Fountainbleu sandstone. Three LB simulations

were performed on the imaged pore space, one of all-pore space imaged (fracture in permeable

media), one of the fracture alone (fracture with impermeable walls), and an average of five

simulations on portions of the pore space that did not contain any part of the fracture to be

representative of flow in the matrix only. The fracture permeability was increased by a factor of

15.1 by the presence of the highly permeable matrix, and the matrix permeability was increased

by a factor of 2.14. However, this increase in the matrix permeability is a result of the presence

of the fracture, and is dependent on fracture spacing. The x-, y-, and z-velocities were found to

be a logarithmic function of the distance from the fracture. Although developing a generalised

form of this equation was beyond the scope of this paper.

Fracture permeability estimates from a variety of previous investigations were compared to our

LB simulation results; all of these estimates assume an impermeable fracture wall, except for

Crandall et al. 2010. The ACL does not account for fracture roughness, tortuosity, asperities

and aperture distribution, and has been widely accepted to grossly overestimate the fracture

permeability. The ACL overestimated the fracture permeability by two orders of magnitude.

44

Zhang et al. 1996 suggest permeability is a power function of mean aperture and a fitted power

factor β. Although this will provide a good estimate of permeability if a value of β is found

that fits the fracture flow measurements, it has no power of prediction without a function

for β dependent on fracture characteristics. Eker and Akin 2006 proposed a linear function for β

dependent on the fractal dimension of the fracture wall roughness; however their prediction for β

did not fit the results of Zhang et al. 1996, nor ours. A linear function for β dependent only

on the fracture wall fractal dimension is inadequate. Zimmerman and Bodvarsson 1996 derived

a semi-analytical prediction of fracture permeability dependent on the mean fracture aperture,

fracture aperture deviation and fractional contact area from the Reynold’s lubrication

equation. The permeability predicted by this function underestimated fracture permeability

approximately 48%, indicating accurate prediction of fracture permeability requires more

information than the mean and standard deviation of the fracture aperture provide. Nazridoust et

al. 2006 derived a semi-analytical function of fracture permeability dependent on mean

fracture aperture, fracture aperture standard deviation and flow rate from the parallel-plate flow

solution for flow with significant inertial effects. Their prediction of fracture permeability

provided the closest match to LB simulations, overestimating the fracture permeability by a mere

33%. The slight overestimation can be attributed to neglecting the effect of asperities.

Crandall et al. 2010 built upon the work of Nazridoust et al. 2006 to include the effect of matrix

permeability on flow in fractured permeable media. Their fracture permeability prediction

underestimated the fracture permeability in the LB simulation of the fractured permeable

medium by a factor of 6.72 times. As discussed, the simulations used in Crandall et al. 2010

assumed the flow in the matrix can be represented by homogenous and isotropic finite

volume elements. However, they used REVs that are much smaller than the REVs of

45

sandstones and the permeable media used in our simulations. Our permeable medium is

simulated at the pore-scale without any assumptions made regarding the matrix pore space.

This allows for bypassing of fracture choke points through the pore spaces of the matrix adjacent

to the fracture. This dynamic is not possible in CFD simulations. It is for these reasons flow in

the fracture was so much greater in our simulations.

LB methods, and other pore-scale models are tools that give us insight into fluid flow through

permeable media at scales that are very difficult to observe; particularly in the study of

fractured permeable media, for which measuring the flow in the fracture alone would prove

very difficult. It is for this reason validating our results would require more indirect methods,

such as measuring the mean permeability of the entire matrix-fracture system and comparing

this permeability to simulation results. However, this investigation was not meant to be an

evaluation of the accuracy of LB methods, the evaluation of the accuracy of single-phase LB

models has been addressed by many of the previous researchers mentioned. A comprehensive

prediction of fracture permeability from fracture and matrix characteristics for fractures in

permeable rock that includes all determining factors, including the flow behavior of the matrix,

remains elusive. However, pore-scale studies such as this one provide a tool to help minimize

unrealistic assumptions made in more simplistic representations of flow in fractured

permeable media.

46

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55

CHAPTER 3: RELATIVE PERMEABILITY OF HOMOGENOUS-WET AND MIXED-

WET POROUS MEDIA AS DETERMINED BY PORE-SCALE LATTICE

BOLTZMANN MODELING

3.1 SUMMARY

We present a pore-scale study of the dependence of relative permeability dependence on the

strength of wettability of homogenous-wet porous media, as well as the dependence of relative

permeability on the distribution and severity of wettability alteration of porous media altered to a

mixed-wet state. A Shan-Chen type multicomponent multiphase lattice Boltzmann model is

employed to determine pore-scale fluid distributions and relative permeability. Mixed-wet states

are created – after pre-simulation of homogeneous-wet porous medium – by altering the

wettability of solid surfaces in contact with the non-wetting phase. To ensure accurate

representation of fluid-solid interfacial areas we compare LB simulation results to experimental

measurements of interfacial fluid-fluid and fluid-solid areas determined by x-ray computed

microtomography imaging of water and oil distributions in bead packs (Landry et al. 2011). The

LB simulations are found to match experimental trends observed for fluid-fluid and fluid-solid

interfacial area-saturation relationships. The relative permeability of both fluids in the

homogenous-wet porous media is found to decrease with a decreasing contact angle. This is

attributed to the increasing disconnection of the non-wetting phase and increased fluid-solid

interfacial area of the wetting phase. The relative permeability of both fluids in the altered

mixed-wet porous media is found to decrease. However the significance of the decrease is

56

dependent on the connectivity of the unaltered solid surfaces, with less dependence on the

severity of alteration.

3.2 BACKGROUND

Macroscale multiphase flow properties in porous media (i.e. relative permeability and capillary

pressure) are strongly dependent on wettability. The wettability of a porous medium is initially

determined by its mineral composition. However, brines and oils can alter the wettability of

porous media, such that the multiphase flow properties of a rock can be significantly dependent

on saturation history. Subsequently, the majority of consolidated and unconsolidated porous

media have a mixed or fractional wettability. The process occurs in three steps:

1) Rocks initially develop wettability as a result of mineral-water interactions.

2) Migration of non-aqueous phases into the rock alters the wettability of the rock.

3) Subsequent cycles of displacement further alter the wettability of the rock.

Wettability is conventionally measured using the Amott-Harvey method or the USBM method.

Both measure an index of wettability from displacement experiments. Indices range from -1-to-1

for perfectly oil-wet to perfectly water-wet conditions. At the pore-scale, wettability is described

by contact angles. The contact angle is measured through the aqueous phase and ranges from 0°

to 180°, for perfectly water-wet to perfectly oil-wet, respectively. Common reservoir porous

media includes siliclastic rocks (i.e. sandstone) primarily composed of quartz, but also

containing significant amounts of feldspars, calcite, dolomite and clays; and carbonate rocks (i.e.

limestone) composed primarily of calcite, dolomite, clays and organics.

57

Conventional measurement of oil recovery and macroscale flow properties from mixed-wet

reservoir rock cores demonstrate some of the important trends observed in field-scale variations

of saturation history and wettability. Janhunandan and Morrow 1995 investigated changes in

wettability and oil recovery from Berea cores aged with two different crude oils and varying

initial water saturations (Swi), aging temperatures and brine compositions. Regardless of oil type

and brine composition the Amott index of the rock increased monotonically with Swi. Jerauld

and Rathmell 1997 observed the same trend in oil-wetness and connate water saturation for

Prudhoe Bay sandstone cores composed primarily of quartz. Core wettability trended more oil-

wet with the decrease in connate water saturation above the oil-water contact line of the

reservoir. With more oil present in the rock, a greater portion of the pore space will be in contact

with the oil, resulting in larger surface area of the pore space subject to wettability alteration.

Both studies also demonstrated the dependence of wettability alteration on the brine

composition, oil composition and aging temperature. Schembre et al. 2006 measured the

wettability and oil recovery of cores from a diamtomite (carbonate) reservoir. The Amott indices

were found to increase significantly from 0.2-0.55 to 0.55-0.85 with temperature increases from

45 to 230 C, and correlate with the percent content of water-wet clays. The increase in the

Amott indices were also found to increase oil recovery. The temperature dependence indicates

variation in the temperature of a reservoir will affect the wettability of the rock.

Once a fluid is in contact with a mineral surface in the porous media the alteration in wettability

is controlled by fluid-mineral surface chemistry. As summarized by Buckley and Liu 1998 the

mechanisms involved in crude oil/brine/solid interaction, include polar interactions between the

polar functional groups in the oil and polar surface sites on the mineral, surface precipitation,

58

acid/base reactions and ion binding. One of the most common minerals found in siliclastic rocks

is quartz. The surface interactions between quartz and brine depend on both the pH and ionic

strength of the brine. The affinity of brine to quartz in the presence of an oil phase is

proportional to the amount of ion adsorption. Ion adsorption increases with increasing ionic

strength and decreases for pHs in the proximity of the pH of zero-point surface charge (pHzpc =

1.5-4.0) (Barranco et al. 1997). The same behavior is observed in coal, with the strongest water

wettability found at alkaline and acidic conditions. Chaturvedi et al. 2009 demonstrated the

effect pH has on the wettability, and subsequently, the relative permeability of coal. At brine

pHs of 10, and 7, the intersection point of relative permeability curves were Sw = 0.65 (water-

wet) and Sw = 0.43 (weakly oil-wet), respectively. Carbonate wettability is also sensitive to brine

pH and composition, as demonstrated in Yu et al. 2009 chalk cores experienced very little

spontaneous imbibition of reservoir brine, while seawater altered the wettability to water-wet and

spontaneously imbibed the core. Evje and Hiorth 2011 modeled these results with the inclusion

water-rock chemistry, specifically the dissolution and precipitation of calcite, magnesite and

anhydrite in the presence of Mg2+

and SO42-

. They attributed the increase in wettability to the

presence of SO42-

in the seawater brine, which increased the dissolution of calcite, exposing new

water-wet surfaces. One of the most important mechanisms of wettability alteration by crude

oils is the adsorption and deposition of asphaltenes on mineral surfaces. Asphaltenes can render

initially water-wet mineral surfaces oil-wet. Saraji et al. 2010 measured the adsorption of

asphaltene to calcite, dolomite and quartz by asphaltene containing oils under flow conditions.

The asphaltene was found to adsorb the most to calcite, and to lesser extent quartz and dolomite.

Wettability was not measured, but it was concluded that the amount of asphaltene adsorbed was

sufficient to form a monolayer on all minerals. The alteration of mineral wettability is controlled

59

by mineral-fluid interactions. Knowledge of individual mineral-fluid surface chemistry provides

insight into which minerals in a porous medium are being altered. Tabriziky et al. 2011

measured changes to contact angles and wettability indices of quartz, kaolinite and calcite after

aging in presence of oil with and without asphaltene content, and brines containing either MgCl2

or Na2SO4. Wettability was measured using BET H2O surface adsorption, and contact angles

measured on flat surfaces of calcite and quartz to investigate the dependence of wettability

alteration on ionic species and the presence of asphaltene. Quartz was neutral wet for oil without

asphaltenes and distilled water, and weakly oil wet for oil with asphaltenes and distilled water.

SO42-

had a negligible effect; however, Mg- significantly altered quartz to a water wet state in the

presence of asphaltenes, and an oil-wet state without the presence of asphaltenes. Kaolinite was

neutral wet for oil with and without asphaltenes and distilled water. SO42-

, as with quartz, had a

negligible effect. Mg- significantly altered kaolinite to a water wet state in the presence of oil

with and without asphaltene. Calcite was water wet in presence of oil with and without

asphaltene and distilled water. SO42-

altered the wettability to a weakly water-wet state, and Mg-

altered the wettability further to a weakly oil-wet state. Studies such as these demonstrate the

complexity involved in fluid/mineral interactions and the resulting alteration of wettability.

There are two factors to consider in the study of wettability alteration, the chemistry of alteration

and the distribution of fluids in the pore geometry with respect to mineral constituents of the

porous media. There are few studies that have attempted to measure contact angles at the pore

scale within natural porous media, where mineral heterogeneity exists in individual pores. Robin

et al. 1995 imaged the distribution of oil and water phases in a reservoir sandstone and carbonate

before and after wettability alteration using cryo-SEM. The images provided qualitative

60

evidence of individual mineral wettability alteration within individual pores. Many of the trends

in mineral wettability alteration discussed above are found in these images. Although these

types of images confirm the assumptions of bulk studies, they do not provide much insight into

the 3D pore geometry that governs the movement of fluids. Non-destructive 3D imaging of pore

geometry and fluid distributions using x-ray computed microtomography (CMT) provides us

with a powerful tool to image and analyze pore geometries and fluid distributions in these

geometries. Many investigators have used CMT to image a variety of fluid distributions in

unconsolidated materials (Culligan et al. 2006, Brusseau et al. 2006, Brusseau et al. 2008,

Constanza-Robinson et al. 2008, Al-Raoush et al. 2009, Lebedeva and Fogden 2011) and rock

(Coles et al. 1998, Turner et al. 2004, Prodanovic et al. 2007, Iglauer et al. 2011,, Silin et al.

2011, Kumar et al. 2012). Armed with knowledge of the fluid/mineral wettability alteration and

pore geometry, pore scale models can be employed to investigate the macroscale multiphase

flow properties of mixed-wet porous media.

The focus of this paper is to further elucidate the relationship between mixed-wet states and

relative permeability. In the past two decades pore network models have been successfully used

to model wettability alteration, multiphase fluid flow of mixed-wet states, and saturation history

dependent relative permeability and capillary pressure hysteresis (Blunt 1997a, Blunt 1998a,

Blunt et al. 2002, Dixit 1996, Dixit 1998, Al-Futaisi and Patzek 2004, Hui and Blunt 2000,

Jackson et al. 2003). Kovscek et al. 1993 introduced a physically based model for wettability

alteration within individual pores; each pore is represented by a capillary tube with a star-shape

cross-section. The tube is initially occupied by the wetting phase, during drainage the wetting

phase is displaced by a piston-like mechanism by the non-wetting phase. The non-wetting phase

61

will now occupy the center of the pore, leaving the corners occupied by the wetting phase. For

the surface of the pore to make contact with the non-wetting phase the capillary pressure must

overcome a critical capillary pressure at which the wetting film destabilizes to molecular

thickness. Once contact is made the surface of the pore in contact with the oil is altered in its

strength of wettability. This concept was further extended to squares (Blunt 1997a, Blunt

1997b), triangles and lenses (Oren et al. 1998). As is noted in Blunt 1997, the disjoining

capillary pressure is not known for each individual pore, pore network models cope with this by

assigning a wettability alteration to a fraction of the pores occupied by oil. These fractions can

then be varied to observe resulting multiphase flow behavior. The extraction of representative

pore networks from 3D images was introduced by Oren et al. 1998, and has led to numerous

studies of multiphase flow in a variety of porous media imaged by CMT and modeled by pore

network models. In mixed-wet media there are three general possible modes of assigning

fractional wettability to pores - the largest pores, the smallest pores, or at random (Oren and

Bakke 2003, Hoiland et al. 2007). The choice of contact angles representing the wettability can

also be uniform or distributed at random over a range. With these free parameters almost any

wetting state can be modeled (Fenwick and Blunt 1998, Hui and Blunt 2000, Jackson et al. 2003,

Al-Futaisi and Patzek 2004, Speiteri et al. 2008, Svirsky et al. 2007, Zhao et al. 2010). There

are concerns regarding the use of pore network models for the study of wettability alteration and

its effect on multiphase flows. The simplification of the pore space into capillary tubes of

varying shape is not predisposed to the inclusion of knowledge of the mineral composition of the

pores. The surface chemistry is highly dependent on accurate representation of fluid/mineral

interfacial areas, which may be difficult to ascertain without proper inclusion of the mineral

composition of pores.

62

In this paper we use a more rigorous pore scale modeling approach. Lattice Boltzmann (LB)

methods can directly use 3D images of pore space by a one-to-one correlation of image voxels to

lattice sites. The accurate representation of any pore geometry is then only a question of lattice

resolution. This makes the model very appealing over pore network models. However, it is far

more computationally expensive, requiring on the order of 104-5

more cpu time than an

equivalent pore network model (Vogel et al. 2005). The computational needs of LB are

compensated by its inherent parallelism – in most models each node is only aware of its

neighboring nodes. Multiphase LB models have been validated against experimental

measurements of capillary pressure in bead packs (Pan et al. 2004, Schaap et al. 2007, Porter et

al. 2009) and sandstones (Ramstad et al. 2010, Ramstad et al. 2012), as well as experimental

measurements of relative permeability in sphere packs (Ghassemi and Pak 2009, Hao and Cheng

2010) and sandstones (Boek and Venturoli 2010, Ramstad et al. 2010). Investigations of mixed

wettability using LB are limited to the work of Hazlett et al. 1998. In this investigation a CMT

image of a reservoir sandstone was translated directly into a lattice for LB multiphase

displacement simulations for a strongly water-wet sandstone and a mixed-wet sandstone. The

mixed wet state was determined by simulating a drainage to irreducible water saturation. At the

end of the drainage, the pore walls in contact with the non-wetting phase are altered in

wettability. The measurements of capillary pressure and relative permeability matched

experimentally observed trends. Although there has been a fair amount of validation of LB

methods via matching experimental measurements of macroscale flow properties, few have

compared fluid distributions imaged using CMT with LB model results. Sukop et al. 2008

compared two phase fluid distributions in a sand pack imaged using CMT to distributions

63

resulting from LB simulations. The saturation per slice was found to match experimental results

well, but there was no analysis presented of interfacial area measurements. Porter et al. 2009

compared interfacial fluid-fluid areas measured by CMT to LB displacement simulations and

found a good match for drainage simulations.

Our objective is to investigate the evolution of wettability alteration as a result of saturation

history and the initial state of wettability, as well as the sensitivity of relative permeability to this

evolution. Proper pore-scale simulations of wettability alteration require accurate predictions of

fluid-solid interfacial areas. To test the ability of the LB model to predict these interfacial areas

we will first compare CMT images of oil/brine fluid distributions in water-wet and weakly oil-

wet bead packs from Landry et al. 2011 to the results of LB simulations. The bead packs can be

viewed as simple analogs of water-wet siliclastic porous media and weakly oil-wet carbonate

porous media. There are three variables of interest in our investigation of relative permeability

– saturation relationships and their dependence on wettability, the initial wetting state of the

porous media, the saturation at which wettability alteration takes place, and the degree of

alteration. As was done in the work of Hazlett et al. 1998 solid surfaces in contact with the non-

wetting fluid, after fluid distribution is established by homogenous-wet LB simulations, will be

altered to create a mixed-wet porous medium. Relative permeability measurements on the

initial and mixed-wet states will then be carried out using the LB model. We will be able to

evaluate the extent to which the LB model can recreate pore scale distributions of immiscible

fluids, and compare relative permeability measurements of mixed-wet states. In this simplified

scenario, the initial wettability, saturation of alteration and severity of alteration control the

mixed-wet state of the porous media. The only variable in our simulations is wettability, as a

64

result of these three parameters. Natural porous media is subject to a far more complex

wettability alteration process, dependent on numerous variables related to the mineral-fluid

interactions. However, the resulting evolution of relative permeability – saturation relationships

can be investigated without such considerations.

3.3 MATERIALS AND METHODS

3.3.1 Experimental Measurements

The validation of pore-scale models at the pore-scale is limited in the literature. Here we

compare LB simulation results to CMT images of brine/oil distributions in strongly water-wet

(glass) and weakly oil-wet (polyethylene) bead packs from Landry et al. 2011. Both bead types

were spherical with a size range between 0.425 and 0.600 mm, and packed in a column 25.4 mm

in diameter and ~80-90 mm in length. The fluids used were a 4-8% NaI brine and kerosene, with

dynamic viscosities of 1.00 and 2.43 cP, respectively. The bead packs were initially fully

saturated with brine, followed by capillary-dominated displacement by kerosene, then brine. At

the end of each displacement CMT images were taken with a voxel resolution of 0.0260 x 0.0260

x 0.0292 mm3 and 0.0259 x 0.0259 x 0.0274 mm

3, for the glass and polyethylene bead packs,

respectively. Due to the relatively large pore aperture of these porous media gravity plays a

significant role in the distribution of fluids in these bead packs. This results in a classic

transition-zone saturation profile in both bead packs at the end of drainage, from which fluid

distributions can be observed at varying fluid saturations. From these images specific interfacial

fluid areas were measured, as well as blob size and surface area distributions. These

65

measurements of volume and area will be compared in this paper to the results of LB

simulations.

3.3.2 Single-Phase BGK Lattice Boltzmann Model

The LB method is a mesoscopic method based on microscopic particle dynamics that provides

numerical solutions to macroscopic hydrodynamics. In short, fluids are modeled as swarms of

streaming and colliding particles. Each node on the lattice contains a particle distribution

function, , where is the index of each discrete velocity, is the location of the node on

the lattice, and is time; here we use a D3Q19 lattice (Qian et al 1992). The D3Q19 particle

distribution function has 19 discrete velocities, , including a zero velocity and 18 velocities

pointing to neighboring nodes. The particle distribution function is also referred to as the density

function – the macroscopic density at each node is represented by a distribution of densities of

particles moving along each discrete velocity. The particle distribution function evolves in time

according to the lattice Boltzmann equation,

[

]

where is the relaxation parameter, is a discrete time step ( , for one time iteration, in

lattice units 1 t.u.) and is the equilibrium particle distribution function. The left hand

side of the equation represents the streaming of particles by passing the particle distribution at

each discrete velocity to the respective neighboring node. The right-hand side of the equation

represents the collision of particles as a partial relaxation to the equilibrium particle distribution.

The relaxation parameter (also known as the collision interval) is representative of the rate of

particle collisions, and is related to the kinematic viscosity, , of the lattice by, ,

66

where is the speed of sound of the lattice, or propagation speed, equal to √ . The discrete

lattice speed unit is the ratio of the lattice spacing – equal to 1 lattice length unit –

and the time step. Here we use Bhatnar-Gross-Krook (BGK) dynamics, meaning the relaxation

parameter is defined by a single value. Although it has been shown that BGK dynamics results

in viscosity-dependent permeability measurements (Pan et al. 2004), as opposed to more

computationally demanding dynamics (e.g. Multiple-Relaxation Time dynamics), for our

purposes BGK dynamics will suffice. The D3Q19 equilibrium particle distribution function

can be calculated as

[

]

where is the weight of each discrete velocity, is the density of , and is the

macroscopic velocity moment of . The discrete velocities are defined as follows,

[ ]

[

]

with weights ⁄ , ⁄ , ⁄ . The

density is obtained by summing the particle densities, ∑ , and the macroscopic velocity

is obtained by summing the particle momentum and dividing by density , ∑ ⁄ . To

impose external forces, , on the lattice (i.e. gravity), momentum is added to the macroscopic

velocity in the following way,

where is the macroscopic velocity calculated from the particle distribution function prior to

collision. Each time iteration of the LB equation proceeds in three steps. First, the particle

67

distributions are streamed to respective nodes. Second, the macroscopic density and velocity are

determined from these new particle distributions, and when present, with the inclusion of an

exterior force as stated above. Third, the equilibrium particle distribution is obtained and the

particle distribution undergoes collision. There are three types of lattice nodes, fluid nodes,

bounce-back nodes and solid nodes. The nodes on which fluids move are fluid nodes, the nodes

on the surface of solids are bounce-back nodes, and nodes on which no fluids exist are solid

nodes. The bounce-back nodes act as algorithmic devices that return incoming streamed particle

densities with the opposite momentum, producing a no-slip condition halfway between the fluid

node and the bounce-back node.

3.3.3 Shan-Chen Multicomponent Lattice Boltzmann Model

We employ the Shan-Chen Multicomponent (MC) LB model to simulate two-phase immiscible

fluid flow (Shan and Doolen 1995, Martys and Chen 1996). The Shan-Chen MC LB model is

one of the least computationally demanding multiphase LB models, and for this reason one of the

most commonly used multiphase LB models. Two immiscible fluids are simulated on the lattice

by representing each fluid phase with its own particle distribution function, , where

is the index for each component particle distribution function. The component particle

distributions interact via a pseudo-potential interparticle force ( ) defined as,

∑

where is the parameter controlling the strength of the interparticle force, when is positive

the force is repulsive. Only the nearest-neighbor nodes are considered in the calculation of the

68

interparticle force. The adhesion force between components and solid surfaces is created by

imposing fictitious component densities, , on bounce-back nodes,

∑

where is an indicator function that denotes the existence of a bounce-back node, as in,

when is occupied by a bounce-back node. Conventionally, negative values

of are used for wetting fluids and positive values of are used for the non-wetting

fluids, with . The fictitious bounce-back densities are not densities in the

same manner as fluid nodes, they are simply values that control the strength of adhesion. These

forces are included in the lattice Boltzmann equation in the same manner as the exterior force,

The total momentum of all particles at each lattice node most be conserved by the collision

operator, thus the particle distribution functions for both components must be included in

calculations,

∑

∑

The surface densities of bounce back nodes ( ) in a Shan-Chen D3Q19 two

component system (wetting and non-wetting) can be estimated for a given contact angle by the

following equation proposed by Huang et al. 2007,

69

where is the density of the wetting fluid and is the dissolved density of the non-

wetting fluid in the wetting fluid (for values of sufficient to segregate component fluids the

dissolved density is very low). The lattice pressure at each node is determined by the D3Q19

Shan-Chen MC LB model equation of state,

[ ]

[ ]

All LB simulations presented here were executed using the open source code Palabos.

3.3.4 LB Model Implementation

There are a collection of parameters that are chosen by the user to define a LB model simulation

of fluid flow. The resolution of the lattice is an important factor when designing a LB

simulation, optimally the resolution should be as fine as computing resources allow. However,

due to the immense computational needs of LB simulations it is often the case that computing

resource limits and time constrictions determine the resolutions of LB fluid flow simulations in

porous media. Also, the necessary resolution to adequately represent flow in a porous media is

dependent on the pore structure, thus determining an adequate resolution for simulations merits

further investigation. Given our computing resources we chose to translate our CMT images

with a 1-to-1 correlation, voxel-to-node. This ensures we are using a resolution great enough to

capture the fluid-fluid and fluid-solid interfacial areas measured by CMT, and results in lattice

sizes within the limits of our computing resources. The imaged voxel resolution of the glass

bead pack is 0.0260 x 0.0260 x 0.0292 mm3, to impose a 1-to-1 correlation of voxel-to-node the

raw image of the pore geometry is resampled to 0.0260 x 0.0260 x 0.0260 mm3 and segmented

into pore and solid voxels. The pore voxels of the segmented image volume are translated as

70

lattice nodes, and the solid voxels that are in contact with pore voxels are translated as bounce-

back nodes, with all other solid voxels being ignored in the lattice (Figure 3.1). The resolution of

the imaged voxels determines the physical size of a lattice length unit, , here being the side

length of a CMT voxel, and the physical mass of lattice mass unit, , being the volume of a

CMT voxel times the density of the physical fluid being simulated.

Other parameters control the density and viscosity ratio of the fluids and the wettability of the

solid surfaces. Throughout the LB simulations presented here the sum of the lattice density of

the components, , is set to 1.0 . The time relaxation parameter , is set to

1 for both fluids; the purpose of this being two-fold. The relaxation parameter determines the

viscosity of each component, and thus the viscosity ratio of the fluids. Our experimental fluids

are kerosene and water with a viscosity ratio, ⁄ , of 2.3 and 0.43 for the glass bead

pack and polyethylene bead pack (Note: in the glass bead pack the kerosene is the non-wetting

phase, in the polyethylene bead pack the water is the non-wetting phase) displacements,

respectively. The capillary number, , is a dimensionless measure of the ratio of viscous to

capillary forces, and is defined as,

where is the dynamic velocity, is the fluid velocity in the direction of flow, and is the

interfacial tension. The experimental displacements were very low, on the order of

. Given the interfacial tension forces are so much greater than the viscous forces; we

assume simulating fluids with a viscosity ratio of 1 will suffice for our needs. Also the use of

BGK dynamics has been shown to result in viscosity-dependent permeability outside time-

relaxation parameters of 1 (Pan et al. 2006). A time relaxation parameter of 1 is both

71

numerically stable and representative of the physical system to be modeled. The interfacial

tension of the lattice is determined by the value of , we use a value of 1.8, as suggested by

Huang et al. 2007, giving a lattice interfacial tension of 0.100. In the SC MC LB model, values

of lower than 1.8 decrease the contrast of the component fluids, while higher values result in

an undesirable compression of fluid components (Schaap et al. 2007, Huang et al. 2007). As

was previously mentioned, the desired contact angle can be determined by equation proposed by

Huang et al 2007 for given bounce-back node densities and . We do not know

exactly what the contact angles are in our experimental system, we are only aware of the fact that

the glass bead pack is water-wet and the polyethylene bead pack is weakly oil-wet. To compare

LB results to experimental results we will simulate a range of contact angles.

Figure 3.1: Segmented CMT image of the bead pack (A) with solid voxels colored white, and the

corresponding lattice of the LB model (B) with the bounce-back nodes in blue.

72

3.4 RESULTS

3.4.1 Fluid-Fluid and Fluid-Solid Interfacial Areas

The objective of this investigation is to study flow in mixed-wet porous media, it is important

that the model can adequately capture the interfacial areas between solids and fluids. To validate

our method we compare the fluid surface and interfacial areas determined by LB simulations to

experimental CMT measurements. As previously mentioned and illustrated in figure 3.1 a 1-to-1

correlation of CMT voxel to lattice node is used. The image chosen was 1003 voxel

3, or ~5

beads to a side. Larger lattices up to 2003 voxel

3 were also used in initial simulations, but the

results were similar to results using the smaller 1003 voxel

3 lattices at an 1/8 of the computational

cost. In our steady-state simulations the initial density of each fluid component is uniformly set

at a desired saturation with a sum of 1, , and an exterior force of

⁄ is applied to both component fluids to initialize concurrent flow

resulting in capillary-dominated flow ( ). This setup allows the fluids to segregate

without out further input within the pore space. The disadvantage of this setup is it allows fluids

access to pores that would not be accessible in displacement processes, and also lead to a more

scattered non-wetting fluid distribution than what would be observed experimentally. All other

parameters of simulation are the same as those used in the simulation of the circular pore, with

the exception of the bounce-back densities ( ).

To compare the fluid distributions of the LB simulation to experimental images we measure the

specific fluid surface areas and specific interfacial areas. For details on the experimental

measurement of these areas refer to Landry et al. 2011. To measure these areas and volumes of

73

component fluids from the LB results the lattice must be first segmented to identify the fluid

occupying each node, here the fluid occupying the node, , is defined as the component

fluid with the greatest density,

The volume of the fluid component is,

∑

where the . The volume of the solid is the product of the sum of all the

bounce-back and non-interacting nodes and . The wetting-phase, non-wetting phase, and

solid surface areas, , are measured using a modified marching cubes algorithm,

this algorithm wraps a triangular-mesh around the segmented data (Dalla et al. 2002). The

specific fluid and solid surface areas, , are defined as, ⁄ , where is the

volume of the whole lattice. The specific solid surface area of the lattice, .

One of the drawbacks of using the LB model to simulate strongly-wetting porous media is the

occurrence of a monolayer or 1-node thick wetting phase film surrounding all bounce-back

nodes. This film is a consequence of using a mesoscopic model founded on particle dynamics;

the pseudo-potential affinity of the wetting phase on nodes adjacent to the bounce-back nodes is

too strong to be dislodged by the non-wetting phase. This monolayer can exaggerate the

measurement of the wetting-phase surface areas, which will have an effect on interfacial areas.

Very little flow occurs in these monolayers, so although they may exaggerate wetting phase

surface areas, they should not have a significant effect on flow.

74

We are interested in comparing three specific interfacial areas, the specific fluid-fluid interfacial

area, the non-wetting fluid-solid interfacial area and the wetting fluid-solid interfacial area. The

specific fluid-fluid interfacial area is the total area of the fluid-fluid interfaces (meniscus)

normalized to bulk volume and is defined as (Dalla et al. 2002),

The specific fluid-fluid interfacial area is important to the study of multiphase flows; it is here

that energy and mass are transferred between fluid phases. The specific wetting phase-solid

interfacial area, , and the specific non-wetting phase-solid interfacial area, are defined

as, . The solid surface area varies in the images, therefore we normalize the

specific fluid-solid interfacial areas to the specific solid surface area, giving us the fractional

wetting fluid-solid interfacial area, , and the fractional non-wetting fluid-solid interfacial

area, , defined as

⁄ .

The wettability alteration that will be investigated here will occur on bounce-back nodes in

contact with the non-wetting phase, and the fraction of the surface that will be altered is equal to

the fractional non-wetting fluid-solid interfacial area. Four contact angles are considered

, resulting from setting ,

respectively. A comparison of LB and experimental measurements of specific fluid-fluid

interfacial areas can be found in figure 3.2. Qualitatively, the LB simulations show a distinctive

curve trending towards a maximum at wetting phase saturations between 0.3 and 0.5. This same

curve is observed in the experimental measurements from the glass and polyethylene bead packs,

and has also been reported by both previous experimental and numerical investigations (Reeves

and Celia 1996, Brusseau et al. 2006, Culligan et al. 2006, Schaap et al. 2007, Joekar-Niaser et

75

al. 2009, Porter et al. 2009, Landry et al. 2011). We also observe an increase in specific fluid-

fluid interfacial area with decreasing contact angles, as would be expected, and is also reflected

in the experimental measurements of the weakly oil-wet polyethylene bead pack and the

moderately water-wet glass bead pack. A rough estimation by visual inspection from these

curves would suggest that the contact angles of the glass and polyethylene bead packs are near

and , respectively. However, we cannot make a quantitative comparison of

model and experimental results without knowledge of the experimental contact angles. In

general, the LB simulations capture the trends of the relationship between saturation and

interfacial fluid-fluid areas. The LB simulations also agree with the trends of the experimental

measurements of the fractional fluid-solid interfacial areas, as shown in figure 3.3. The

fractional non-wetting phase interfacial area decreases with decreasing contact angles, this is a

result of the wetting phase dominating occupation of the smaller pore spaces and forcing the

non-wetting phase into the largest pores where its fractional interface with the solid decreases.

This relationship affects the fraction of a solid surface that is in contact with a wettability altering

fluid, and also the location of the fluid in the pore space (i.e. the small pores where clay minerals

or organics may be found, or the large pores where multiple mineral constituents may be found.)

76

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

[mm

-1]

Sw

0.1

0.2

0.3

0.4

P

G

[

]

Poly beads

Glass beads

Figure 3.2: Specific fluid-fluid interfacial areas as a function of wetting phase saturation for the LB

simulations and experimental CMT images.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Xfs

[]

Sw

0.10.20.30.4PG

Poly beads

Glass beads

Figure 3.3: Fractional fluid-solid interfacial areas as a function of wetting phase saturation for the LB

simulations and experimental CMT images, the wetting fluid-solid interfacial areas are

shown in black and the non-wetting fluid-solid interfacial areas are shown in gray.

3.4.2 Pore Aperture and Fluid Distribution

77

To measure the pore aperture distribution of the pore space, first a skeleton or medial axis, of the

pore space must be constructed. The skeleton is constructed by thinning algorithms (Lee et al.

1994) using the software Avizo Fire 6.3. The skeleton of the pore space is a connected network

of voxels that represent the midpoint between solid walls in the pore space. Each voxel in the

skeleton can be thought of as the center of a sphere that makes contact with the wall in at least

two places. There is a significant amount of literature regarding the subject of pore space

topology description (Thovert et al. 1993, Lindquist et al. 1996, Bakke and Oren 1997,

Prodanovic et al. 2006). The pore aperture is the maximum radius of a sphere that can occupy a

pore. To determine the pore aperture distribution, a “sphere packing” of sequentially smaller

spheres are strung along the skeleton. First, the voxels of skeleton are labeled with their distance

to the pore wall. The labeled skeleton is then searched for voxels that fall within a user-defined

range. These voxels then become the center of a sphere – with a radius equal to the distance to

the closest pore wall – placed in the pore space, and all voxels of the pore space that fall within

this sphere are labeled. The pore aperture distribution can then be determined by simply

summing the labeled voxels. The fluid distribution can also be easily measured by masking the

segmented fluid occupation ( ) with the pore aperture-labeled pore space. The pore

aperture distribution and fluid distribution are presented in figure 3.4. We can see from the fluid

distribution that decreasing the contact angle pushes the non-wetting phase into the larger pores.

This is a result of the increasing affinity of the wetting phase for the solid surfaces, resulting in

the wetting phase dominance of smaller pore spaces. This can have a substantial effect on the

dependence of fluid mobility on wettability alteration. If only the larger pores that would be

78

occupied by the non-wetting phase in a homogenous-wet state are altered the effect on the

relative permeability of the fluids will be minimal.

0

0.05

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Volu

me

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hte

d f

raction

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Non-w

ett

ing p

hase f

luid

fra

ction

Pore aperture [mm]

0.1

0.2

0.3

0.4

Figure 3.4: Pore aperture and fluid distribution of initial homogenous-wet states.

79

3.4.3 Two-phase Flow in a Circular Pore

To test our model against a known semi-analytical solution for two-phase immiscible flow we

simulate flow in a circular pore. Two-phase immiscible flow is commonly described using an

empirical extension of Darcy’s law,

where the mean velocity of each fluid parallel to the direction of flow, , is a function of the

dynamic viscosity, , pressure gradient, , absolute or single-phase permeability, , and the

relative permeability defined as , where is the effective permeability. In a

circular pore where the wetting fluid is distributed as an annulus along the walls of the pore and

the viscosity ratio is one, the wetting phase relative permeability is a function of the wetting

phase saturation, , and the non-wetting phase relative permeability is a function of

non-wetting phase saturation, ⁄ (Ramstad et al. 2010). To simulate two-

phase flow in a circular pore at varying saturations the wetting phase fluid component is initially

distributed as an annulus along the pore walls with the non-wetting phase fluid component

occupying the nodes in the center in correlation with the desired saturation. The walls of the

pore are strongly-wetting to maintain the annular distribution of the wetting phase. Otherwise,

under weakly-wetting conditions, the wetting phase will separate and lose its annular shape, the

annular shape being an assumption of the semi-analytical solution of immiscible two-phase flow

in a circular pore. To determine the relative permeability of component fluids in the LB

simulation the momentum in the direction of flow of each fluid is summed over all pore nodes at

the given wetting phase saturations,

80

∑

and normalized to the fully saturated momentum,

To impose concurrent flow in a circular pore with a radius of 11 nodes an exterior force of

⁄ is applied to both component fluids, and periodic boundaries

are imposed at the inlet and outlet. The system is considered to have reached steady-state when

the convergence criteria were met,

A choice of was chosen, based on numerical experiments. In figure 3.5 are the results

of the LB simulation and the semi-analytical solutions for two-phase flow in a circular pore.

The difference between the semi-analytical solution and the LB simulations is due to the discrete

nature of the rendering of the circular pore and the resolution. Previous investigations have

shown multicomponent LB models are increasingly more accurate with finer resolutions

(Ramstad et al. 2010).

81

0

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0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

kr

S(w)

Figure 3.5: Relative permeability in a circular pore with radius = 11 ; LB simulation results and

semi-analytical solution curves.

3.4.4 Relative Permeability

The relative permeability of the porous media is determined by LB simulations in the same

manner as described in the section Two-phase flow in a circular pore. With the simulation

volume being bounded by walls perpendicular to flow, and as was used in the circular pore

simulations, periodic boundaries are applied in the direction flow (i.e. the inlet and outlet).

However, as was described in Martys and Chen 1996, the total momentum of the component

lattices did not display a linear relationship with exterior forcing below . To

determine the relative permeability exterior forces between

⁄ were used. The relative permeability curves for the initial homogenous-wettability

82

porous media are shown in figure 3.6. At low wetting phase saturations the non-wetting phase

relative permeability exceeds one and the wetting phase relative permeability is slightly negative.

This has also been reported in the preliminary results of Boek and Ventourilli 2010. These

saturations are at or below experimentally observed irreducible wetting phase saturations of

and for the glass and polyethylene beads respectively, and are unlikely to

occur in displacement processes. We are able to measure the relative permeability at these

saturations using the LB simulations due to the employment of a steady-state setup which allows

for any initial saturation. However, some of these simulations may not have a physical corollary.

-0.1

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1

1.1

1.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

kr

Sw

0.10.20.30.4

Figure 3.6: Relative permeability determined by LB simulations for the initial homogenous-wettability

states. The relative permeability of the wetting and non-wetting phase are shown in black

and gray, respectively.

83

Generally, figure 3.6 shows decreasing the contact angle also decreases the relative permeability

of both phases at all saturations. The variation is small due to the high porosity and connectivity

of the pore space. Also, as would be expected, the crossover point for the relative permeability

curves occurs at . The relative permeability of the fluids is governed by two competing

mechanisms that act to inhibit and promote fluid flow, namely, fluid-solid interfacial area and

connectivity, respectively. Greater fluid-solid interfacial area leads to greater resistance to flow,

while increased connectivity leads to greater mobility. The dominant factor being the

connectivity of the phase, which results in the general trend of increasing phase relative

permeability with increasing phase saturation. Unlike the fluid-solid interfacial area, we do not

have a simple method of measuring the connectivity of a fluid phase, but given that we have

measurements of the solid-fluid interfacial areas we can infer differences in relative permeability

that cannot be attributed to trends in the solid-fluid interfacial areas are a result of connectivity.

Also the connectivity correlates with the fluid distribution, with the more connected wetting

phase occupying smaller pores, as was shown in figure 3.4.

How the wetting phase and non-wetting phase respond to these competing mechanisms can be

very different. In figure 3.6 we can see the net effect on the non-wetting phase relative

permeability is an overall decrease as the fluids become more disconnected (occupy larger

pores). Although there is less fluid-solid interfacial area this effect is dominated by the decrease

in connectivity. Unlike the non-wetting phase, the wetting phase relative permeability shows

very little dependence on the wetting strength of the porous medium for . We know that

there will not be a decrease in the connectivity of the wetting phase as the contact angle is

decreased, thus the small decrease in mobility between and the other homogenous-wet

84

states can only be attributed to an increase in fluid-solid interfacial areas. It is not surprising that

the non-wetting phase relative permeability is more greatly influenced by connectivity, being the

less connected phase, and the wetting phase relative permeability is more greatly influenced by

the fluid-solid interfacial area, being the generally more connected phase. However, regarding

the non-wetting phase relative permeability, a similar previous investigation reported the

opposite effect from what is reported here. Li et al. 2005 simulating two-phase flow in a

homogenous sphere pack using a multiple-relaxation-time Shan-Chen type multicomponent LB

model reported an increase in the non-wetting phase relative permeability for neutral wet states

( ) over strongly wet states ( ). This could be attributed to differences in the pore

space geometry resulting in the prevalence of increased non-wetting phase mobility due to the

decrease in solid-fluid interfacial area, over the decrease in mobility associated with decreased

connectivity as was observed here. The homogenous sphere pack used by Li et al. 2005 may

have a generally narrower distribution of pore apertures, meaning fewer large pore spaces for the

non-wetting phase to occupy, increasing the dependence of the non-wetting phase relative

permeability on the solid-fluid interfacial area. The interaction of these competing mechanisms

is sensitive to small differences in pore space geometry. In the mixed-wet states individual pores

can have mixed-wettability resulting in the competition of these mechanisms within each pore.

3.4.5 Relative Permeability of Mixed-Wet States

As was previously stated, the bounce-back density of the nodes in contact with the non-wetting

phase after fluid distribution is established in the homogenous-wetting simulations, are altered to

create a mixed-wet porous medium (figure 3.7). The mixed-wet state is the result of three

85

parameters, the initial homogenous-wettability, the saturation at which alteration takes place, and

the severity of alteration. Here we will alter the wettability of each of four homogenous-wetting

states at a wetting phase saturation near 0.5, with four increasingly severe alterations, resulting in

sixteen mixed-wet states. The fictional bounce-back density of nodes in contact with the non-

wetting phase are altered to preferentially wet the non-wetting phase with contact angles

( ). To avoid confusion, the initial

wetting phase is from here on referred to as the wetting phase of the mixed-wet states, and used

as the wetting phase saturation in figures. Also, the mixed-wet states will be referred to

according to the following indexing, , where is the for the initial wettability, and

is the for the altered wettability (i.e. would refer to a state that was initially

wet, then altered as previously stated to ).

The initial homogenous-wet states will use the same indexing as the mixed-wet states absent a

superscript. There are numerous mixed-wet states that could be considered using this

framework, we limit this investigation to studying the mixed-wet state resulting at .

These mixed-wet states are the result of wettability alteration to the homogenous-wet states at

, and therefore the mixed-wet states are dependent on the fluid distribution of the

phases at . The wetting phase at this saturation is described as occupying the smaller

pore spaces (figure 3.4), and is the more well connected phase. Thus the surfaces altered by the

non-wetting phase are disconnected, with an increasing fraction of the surface altered (increasing

fractional solid-fluid interfacial area) with increasing contact angle.

86

Figure 3.7: Images of the LB lattice with initial homogenous-wettability (A), the fluid distribution of the

non-wetting phase at the end of the simulation of the initial homogenous-wettability with

and (B), and the alteration of the bounce-back densities of the nodes in

contact with the non-wetting fluid (C). This image also summarizes the three parameters

determining the mixed-wet state, initial wettability (A), saturation of alteration (B), and

severity of alteration (C).

The relative permeability curves and the fractional solid-fluid interfacial areas for the mixed-wet

states are shown in figure 3.8. Recapping the observations of the homogenous-wet states, the

non-wetting phase was found to decrease with decreasing contact angle due to decreasing

connectivity, while the wetting phase was found to decrease with decreasing contact angle due to

increasing solid-fluid interfacial area. In general, the wettability alteration has little effect on the

wetting phase relative permeability. The fractional wetting phase solid-fluid interfacial areas of

the mixed-wet states are generally lower than those of the mixed-wet states. Unlike in the

homogenous-wet states this decrease does not correlate with an increase of the wetting phase

relative permeability. We can conclude that any differences in the wetting phase relative

permeability of mixed-wet states cannot be attributed to differences in the solid-fluid interfacial

area. Instead the insensitivity of the wetting phase relative permeability to wettability alteration

can be attributed to the fluid distribution of the wetting phase at the saturation of alteration. At

87

the wetting phase has developed a well-connected pathway (occupies smaller pores,

has a high fractional solid-fluid interfacial area). The solid in contact with the wetting phase

remains unaltered, thus, this connected path remains in the mixed-wet states. For the

mixed-wet states there is some decrease in the wetting phase relative permeability observed for

the mixed-wet state. This decrease is limited to the

mixed-wet state as a result of its

poorer connectivity of the established unaltered wetting phase pathways in comparison to those

of the ,

, and mixed-wet states. But even this small decrease only exists for a

mixed-wet state suffering a severe alteration. Overall, the wetting phase relative permeability is

unaffected by the wettability alteration, the established unaltered wetting phase pathways that

encourage the wetting phase to occupy the smaller pores dominates the mobility of the wetting

phase.

88

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0.1-0.00.10.20.3

A)

C)

E)

G) H)

F)

D)

B)

89

Figure 3.8: Relative permeability and fractional fluid-solid interfacial areas determined by LB

simulations for the mixed-wet states of the originally homogenous-wet states, (A, B)

, (C, D) , (E, F) , and (G, H) ,

with increasing severity of alteration. The relative permeability and fractional fluid-solid

interfacial areas of the wetting and non-wetting phase are shown in black and gray,

respectively.

The non-wetting phase relative permeability is significantly altered in the mixed-wet states, and

as in the case of the wetting phase, the most significant effect is seen for the mixed-wet

states. For the ,

, and mixed-wet states the severity of the wettability alteration

has little effect on the mobility of the non-wetting phase, but the existence of a wettability

alteration does. This can be understood as a result of solid-fluid interaction on altered solid

surfaces. These surfaces are no longer dominated by the initial wetting phase, resulting in an

increase of the non-wetting phase fluid-solid interfacial area and a decrease in the non-wetting

phase mobility. The increase in the non-wetting phase relative permeability correlates with an

increase in the non-wetting phase fluid-solid interfacial area. In general, the non-wetting phase

becomes pinned by its affinity to these altered solid surfaces, the lubricating effect the wetting-

phase has on the non-wetting phase mobility is lost. The non-wetting phase responds to the

strength of alteration much as the wetting phase responded in the homogenous-wet states to

increasing strength of wettability. Unlike the ,

, and mixed-wet states, for the

mixed-wet states the severity of alteration appears to be proportional to the decrease in the

non-wetting phase relative permeability. This also can be considered in the context of the fluid

distribution of the homogenous-wet , being close to neutrally-wet the non-wetting phase

occupies smaller pore spaces and has a greater fluid-solid interfacial area than the more strongly

90

wetting scenarios. Thus, the alteration of the surface occurs not only on a larger fraction of the

solid surface, but also in the smaller pore spaces – much like the wetting phase in the

homogenous-wet states, the non-wetting phase relative permeability decreases due to the greater

fluid-solid interfacial areas. This increase in fluid-solid interfacial areas is reflected in the

measurements of non-wetting fluid-solid interfacial areas (figure 3.8). Incremental increases in

the non-wetting fluid-solid interfacial area for the mixed-wet states resulted in

incrementally decreasing non-wetting phase permeability. The non-wetting fluid-solid interfacial

area of the ,

, and mixed-wet states increases significantly with the existence of

an alteration, but the increase does not respond to increasing severity of alteration, which is

reflected in the relative permeability curves. Overall the non-wetting phase responded to the

wettability alteration with a decrease in relative permeability as a result of increasing non-

wetting fluid-solid interfacial areas.

Had the wettability alteration occurred at high wetting phase saturations we would expect the

same result to a lesser extent, simply due to less of the solid surface being altered. Wettability

alteration at lower wetting phase saturations would have a somewhat increasing effect, however,

not as much as one may at first think. The extreme example of altering the entire surface when

fully saturated with the non-wetting phase would result in a simple role reversal of the wetting

and non-wetting phases. This would suggest that at lower wetting phase saturations of

wettability alteration we could expect the wetting phase relative permeability to “rebound” and

begin to respond by increasing with decreasing fluid-solid interfacial area. However, saturations

below irreducible wetting phase saturation are unlikely to occur. At saturations near the

irreducible wetting phase saturation the wetting phase will dominate the smaller pore spaces.

91

And as occurred in our simulations, wettability alteration will not occur in these smaller pore

spaces. The wetting phase will continue to be pinned to these smaller pore spaces, and generally

will not experience any increase in mobility, as one might intuitively expect when considering

the extreme example of the entire surface being altered. At these saturations the non-wetting

phase will show a slightly greater effect than what we see here at , however the

differences will not be great. In our simulations of mixed-wet states the non-wetting phase

relative permeability has been significantly decreased by the wettability alteration. There is

very little mobility left to be lost.

Comparable studies to this one are limited for the most part to numerical experiments using pore

network modeling (Blunt et al. 2002, Valvatne and Blunt et al. 2004, Zhao et al. 2010, Gharbi

and Blunt 2012). There are a few notable differences between PNMs and LB models. As was

described in the background section, pore network models use networks of capillary tube

elements to represent the pore space topology, while LB models use direct translations of CMT

images to construct the lattice. This also means that wetting films are simulated by very

different means. In PNMs the wetting phase is simulated as thin films, allowing for very low

wetting phase saturations to maintain connectivity. Consequently in mixed-wet scenarios it is

possible for a pore to contain a thin film of one phase sandwiched between the other phase

occupying the center and corners of the pore (Blunt et al. 2002). The wetting phase films in a

lattice Boltzmann simulation are limited in their thickness to the resolution of the lattice, and

may or may not appear depending on the resolution of the lattice and the strength of the bounce-

back density ( ) used. Also PNMs simulate displacements, unlike here, where our

simulations are steady state. The PNM study of mixed-wettability in six different types of

92

limestone of Gharbi and Blunt 2012 offers a fair comparison to this work. This work

investigated the mixed-wet states of PNMs constructed from CMT images of limestone cores.

They also found a significant reduction in the non-wetting phase (oil) relative permeability for

wettability alteration fractions of 0.25 and 0.5 – similar to solid surface fractions altered at

here. However, they also report a significant reduction in the wetting phase relative

permeability, not seen here. This can be attributed to not only differences in the pore space

geometry, but also the method in which fluid films are simulated. At our wetting phase

remains well connected for homogenous-wet states , , and , and as was

previously stated there remains well-connected unaltered solid-surface for the wetting-phase to

be established, ensuring an insignificant effect on the wetting phase relative permeability for the

mixed-wet states. In the mixed-wet states of Gharbi and Blunt 2012 much of the wetting phase

existed as thin films, resulting in very low wetting phase permeability. The inability to simulate

fluid films beyond the resolution of the lattice is one of the weaknesses of the LB method.

However the ability of the LB method to simulate fluids in realistic pore geometries makes it a

very attractive technique.

3.5 CONCLUSIONS

We have presented a study of the dependence of relative permeability on wettability for both

homogenous-wet and mixed-wet states using LB modeling. Comparison between LB simulation

and experimental results show the LB simulations capture the trends of fluid-fluid and fluid-solid

interfacial areas well. Since our mixed-wet states are created by altering the wettability of the

93

solid in contact with the non-wetting phase it is important that these interfacial areas are

accurately captured to create realistic mixed-wet states.

The homogenous-wet states display a decrease in relative permeability for both phases with

decreasing contact angles. This is attributed here to the competition of two mechanisms that

increase and decrease fluid flow. With decreasing contact angle the non-wetting phase becomes

more disconnected and occupies larger pore spaces. This occupation results in less non-wetting

fluid-solid interfacial areas which acts to increase the mobility of the non-wetting phase,

however, the decreasing connectivity dominates this effect resulting in lower non-wetting phase

relative permeability. The decrease in the wetting phase relative permeability with decreasing

contact angle is a result of greater fluid-solid interfacial area which acts to decrease the mobility

of the wetting phase; any increases in connectivity of the wetting phase cannot overcome this

effect, due to the pinning of the wetting phase to the smaller pore spaces.

The pores of the mixed-wet state are themselves mixed, the interaction between the two

competing mechanisms controlling flow will also compete within individual pores. This can

lead to a rich assortment of fluid mobility behavior as a result of varying the wettability alone.

In this investigation we find that the connectivity of the wetting phase at the saturation of

wettability alteration is critical to the significance of the effect the alteration to a mixed-wet state

will have on the wetting phase mobility. In general, the severity of alteration is inconsequential

if the unaltered solid surface is well connected. As is the case for initial homogenous-wet states

with a at The non-wetting phase relative permeability demonstrates a

significant decrease in mobility proportional the severity of alteration when the solid surfaces

94

being altered includes a significant portion of the smaller pore spaces. As is the case for initial

homogenous-wet state with a at For initial homogenous-wet states with a

at the resulting mixed-wet states all show a significant decrease in the non-

wetting phase mobility, but no dependence on the severity of the alteration, even when the solid

surface is altered to neutrally wet. Overall the most important factor governing non-wetting

phase mobility in mixed-wet states is the fluid-solid interfacial areas, greater interfacial fluid-

solid surface areas lead to decreases in mobility. The responses to a wettability alteration at a

wetting phase saturation of 0.5 shown here, are comparable to what we would expect for

wettability alteration at lower wetting phase saturations due to the pinning of the wetting phase

to the smaller unaltered pore spaces.

The framework presented here to study the evolution of wettability alteration and its effect on the

relative permeability of two-phase flow in a simple porous medium could easily be applied to

natural consolidated and unconsolidated porous media. Our initial wettability states are

homogenous, for natural porous media the initial wettability will be determined by the mineral

constituents and their locations in the pore space geometry. We present simulations of varying

alteration severity, for natural porous media the level of wettability alteration is a result of

complex fluid-mineral interactions and would likely require bulk studies of wettability alteration

for individual mineral constituents. Given this knowledge can be determined using imaging and

laboratory techniques, LB simulations can be easily included to estimate fluid flow properties,

such as relative permeability, for a wide assortment of scenarios.

95

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CHAPTER 4: PORE-SCALE LATTICE BOLTZMANN MODELING AND 4D X-RAY

COMPUTED MICROTOMOGRAPHY IMAGING OF FRACTURE-MATRIX FLUID

TRANSFER

4.1 SUMMARY

We present sequential x-ray computed microtomography (CMT) images of matrix drainage in a

fractured sintered glass granule pack. Sequential imaging captured the capillary-dominated

migration of the non-wetting phase front from the fracture to the matrix in a brine-surfactant-

Decane system. The sintered glass granule pack was designed to have minimal pore space

beyond the resolution of CMT imaging, so that the pore space of the matrix connected to the

fracture could be captured in its entirety. The segmented image of the pore space was then

directly translated to a lattice to simulate the transfer of fluids between the fracture and the

matrix using lattice Boltzmann (LB) modeling. This provided us an opportunity to validate the

modeling technique against experimental images at the pore-scale. Although the surfactant was

found to alter the wettability of the originally weakly oil-wet glass to water-wet, the fracture-

matrix fluid transfer is found to be a drainage process, showing little to no counter-current

migration of the oil-phase. The LB simulations were found to closely match experimental rates

of fracture-matrix fluid transfer, equilibrium saturation, irreducible wetting phase saturation and

fluid distributions.

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4.2 INTRODUCTION

The displacement of fluids in fractured rock by transfer of fluids between the high-permeability

low-storage fractures and low-permeability high-storage surrounding rock is a common process

that occurs during hydrocarbon recovery, carbon dioxide injection, and groundwater

contamination. During an injection into, or flooding of, a fractured rock, fluid migration is

determined by the complex interaction of capillary and viscous forces that control flow in the

fractures, as well as by the transfer of fluids to and from the matrix. Predicting this migration

requires a rigorous investigation into the mechanics that determine the transfer of fluids to and

from the matrix. Understanding the physics of this process can be greatly enhanced by

investigating fluid transfer at the interface of the fracture and matrix at the pore-scale.

Experimental studies of fracture-matrix fluid transfer at the pore-scale have been limited, for the

most part, to etched-glass micromodels and two-dimensional glass bead models. Micromodels

are transparent, two-dimensional representations of pore networks (Wan et al. 1996,

Karadimitriou and Hassanizadeh 2012). Using these models, a wide variety of multiphase flow

processes involving fracture-matrix transfer have been investigated in recent years including

spontaneous imbibition into water-wet media (Rangel-German and Kovscek 2006, Hatiboglu and

Babdagli 2008, Hatiboglu and Babdagli 2011), CO2 flooding of naturally fractured oil-wet

reservoirs at immiscible, near-miscible, and miscible pressure (Er et al. 2010), continuous gas,

water and alternating gas-water injection in oil-wet naturally fractured reservoirs (Dehghan et al.

2012), and multiphase flow in multiple intersecting fractures (Buchgraber et al. 2012). These

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studies have provided invaluable insight into the physics involved in multiphase fracture-matrix

fluid transfer; however, their limitation to two-dimensions assumes flow in the missing third

dimension will have minimal impact on fluid flow. In the investigation presented here, the third

dimension is retained by utilizing X-ray computed microtomography (CMT), which is a non-

invasive three-dimensional imaging technique. This allows us to image fracture-matrix fluid

transfer of a three-dimensional porous media at the pore-scale. Also, images obtained of the

pore space can be directly inputted into pore-scale lattice Boltzmann models for further

investigation of the mechanics controlling flow.

X-ray computed microtomography (CMT) has become a commonly used method of imaging

pore spaces and fluid distributions of unconsolidated sands and beadpacks (Culligan et al. 2006,

Brusseau et al. 2006, Brusseau et al. 2008, Constanza-Robinson et al. 2008, Al-Raoush et al.

2009, Lebedeva and Fogden 2011), rock (Coles et al. 1998, Turner et al. 2004, Prodanovic et al.

2007, Iglauer et al. 2011, Silin et al. 2011, Kumar et al. 2012) and fractures (Karpyn et al. 2007)

at the pore-scale in three dimensions. Additionally, numerous investigations using CMT have

imaged artificially induced (Ketcham et al. 2010) and naturally occurring fractures in rocks

(Keller et al. 1999, Renard et al. 2009, Wennberg et al. 2009). Due to advances in CMT imaging

technology within the past two decades, it is now possible to image pore spaces and fluid

distributions at the pore scale in three dimensions. However, carrying out experiments using

CMT is time-consuming and costly. Robust, pore-scale models offer the possibility of simulating

a wide variety of scenarios in order to improve our predictive capabilities regarding fracture-

matrix fluid transfer, at the cost of only computational time. Here, we focus on the use of lattice

Boltzmann (LB) models, although numerous pore-scale modeling techniques currently exist

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(Meakin and Tartakovsky 2009). Multiphase LB models have been validated against

experimental measurements of capillary pressure in bead packs (Pan et al. 2004, Schaap et al.

2007, Porter et al. 2009) and sandstones (Ramstad et al. 2010, Ramstad et al. 2012), as well as

experimental measurements of relative permeability in sphere packs (Ghassemi and Pak 2009,

Hao and Cheng 2010) and sandstones (Boek and Venturoli 2010, Ramstad et al. 2010).

Although there has been a fair amount of validation of LB methods via matching of experimental

measurements of macroscale flow properties, few have compared fluid distributions imaged

using CMT with LB model results. Sukop et al. 2008 compared two-phase fluid distributions in

a sand pack imaged using CMT to distributions resulting from LB simulations. The saturation

per slice was found to match experimental results well. Porter et al. 2009 compared interfacial

fluid-fluid areas measured by CMT to LB displacement simulations, and found a good match for

drainage simulations.

We are interested in investigating fracture-matrix fluid transfer at the pore-scale using

experimental, sequential, three-dimensional CMT imaging of fluid transfer and lattice Boltzmann

modeling. A synthetic core was manufactured in order to contain a minimal amount of pore

space unresolvable by CMT imaging, and was subsequently fractured to serve as our fractured

porous medium. This fractured core is more closely analogous to a real rock than two-

dimensional micromodels, and it also has the previously stated advantage of allowing fluids to

move in three directions. One of the advantages of using LB models in conjunction with CMT is

the ease of translating images of the pore space into a simulation lattice. Here, we simply

translate the CMT image of the pore space 1-to-1, image voxel to lattice node, thus retaining the

complex solid boundary conditions of the pore space in the model. Also, by manufacturing the

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core to have minimal pore space unresolvable by CMT, we can be confident that the modeled

pore space is a close representation of the physical pore space observed during the experiment.

4.3 MATERIALS AND METHODS

4.3.1 Fractured Porous Medium

The fractured porous medium is a sintered glass granule pack, fractured using a modified

Brazilian test. Crushed glass was sieved for granules ranging from 115 to 210 μm in diameter.

These granules were placed in a 7.62 cm diameter, 10.2 cm long mullite tube and sintered

following the temperature cycle summarized in table 4.1. Cores with a diameter of 1.27 cm were

drilled from these sintered glass granule packs. A single, tensile fracture was induced along the

axis of the core, and the two resulting halves of the core were glued along the edges of the

fracture to ensure the two halves are immobile. The resulting fractured core is 3.12 cm in length

and ~1.5 cm in diameter.

Table 4.1: Temperature cycles for sintering of glass granules.

Step Time [h] Cumulative Time

[h] Temperature [C]

Increase 8 8 25 → 600

Hold 1.5 9.5 600

Increase 1 10.5 600 → 645

Hold 1 11.5 645

Decrease 1 12.5 645 → 600

Hold 1.5 14 600

Decrease 16 30 600 → 25

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4.3.2 Procedure

The core was placed between the inlet and outlet of the core holder and encased in fuel-resistant

Teflon heat-shrink tubing to create an airtight seal at both the inlet and outlet of the core holder.

The core abuts the face of the inlet and outlet ports. This face was designed with crisscrossing

channels in order to uniformly spread the injected/received fluid across the face of the inlet/outlet

ports for even fluid delivery/recovery to the core. The core and core holder assembly was placed

vertically in the CMT scanner under vacuum. The core was then scanned in its entirety to

acquire a dry image of the pore space. Following the dry scan, the core was presaturated with

the oil phase, 40% wt iodododecane in decane. The decane was doped with iodododecane in

order to create contrast between phases in the CMT images. The fully oil-saturated core was

then scanned in its entirety to acquire an image of the connected pore space. After scanning, the

oil phase was displaced by injection of the water phase, 5%wt NaCl DI water. The interfacial

tension between the water and oil phase was measured using the sessile drop method, and found

to be 42.86 mN/m. The flow rates throughout the experiment were maintained at a rate of 0.1 to

1.0 ml/min. The balance of viscous and capillary forces are commonly measured by the

capillary number, ⁄ , a dimensionless measure of viscous and capillary forces. The

displacements here have capillary numbers between 10-5

and 10-7

. All displacements in the

experiment were considered to be capillary-dominated.

The CMT scanner is equipped to provide real-time radiographs (2D side-view images) of a 0.532

cm length of the core during water injection. During the water injection no water was observed

entering the matrix. At the end of the initial water phase injection, the core was scanned in its

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entirety, confirming that the water phase had only succeeded in displacing the oil phase from the

fracture. Although glass is often considered a water-wet material, the presaturation with the oil

phase and exposure to x-rays rendered the core weakly oil-wet. The water phase, being the non-

wetting phase, occupies only the largest portions of the pore space necessary to connect the inlet

with the outlet, i.e. the fracture. Due to the vertical orientation of the core and the difference in

densities of the fluids, there is a difference in the pressure ( ) of the heavier water-occupied

fracture and lighter oil-occupied matrix, equal to , where and are the

densities of water and oil, is the gravitational constant, and is the distance from the top or

inlet to the core. Because there are no viscous forces driving the water phase into the matrix,

only capillary and gravitational forces are acting in the transfer of fluid to and from the matrix.

The capillary pressure ( ) is the difference in pressure between the non-wetting ( ) and

wetting phase ( ). In our system the non-wetting phase occupies the fracture, while the wetting

phase occupies the matrix, thus . For the non-wetting phase to penetrate the matrix, the

difference in pressure due to gravity forces must overcome the capillary entry pressure ( ).

The capillary entry pressure is the capillary pressure necessary for the non-wetting phase to

invade a pore occupied by the wetting phase. The capillary entry pressure is directly

proportional to the interfacial tension, and can be lowered by decreasing the interfacial tension.

In order to lower the interfacial tension of our fluid pair, 2 ml/l of dish soap was added to the

water phase, lowering the interfacial tension to 0.30 mN/m. The surfactant-water phase is

injected at the same rates as the water-phase injection. Once the surfactant-water phase

injection is commenced the surfactant-water phase mixes with the water phase already occupying

the fracture. Once the surfactant becomes established in the fractured porous medium, the

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interfacial tension of the system will drop and the surfactant water-phase will begin to displace

the oil phase (wetting phase) from the matrix. During the injection of the surfactant water phase,

a 0.506 cm section of the core was sequentially scanned, producing 14 3D images of the water

phase entering the matrix. This volume of the core is referred to as the region of interest (ROI)

from this point on. Each scan takes approximately 10 minutes, with time in between scans

ranging from 1 to 20 minutes. The injection is not stopped during the scans. The movement of

the water phase into the matrix was slow enough that the moving interface did not appreciably

blur our images. The sequential scanning of the water phase invasion was ended after 220

minutes, at which time the core was scanned in its entirety.

4.3.3 CMT Imaging

X-ray computed microtomography has become a popular technique to non-invasively image the

pore spaces of porous media in 3D. A CMT “scan” involves taking a sequence of 2-D

radiographs of a rotating sample and mathematically reconstructing them into a 3-D image

composed of voxels. Each voxel is much like a pixel in a 2D image, and contains a CMT

registration number, which is a relative measure of the x-ray attenuation of the material

occupying a particular voxel. Images reported here were taken at Pennsylvania State

University’s Center for Quantitative x-ray Imaging (CQI). Scans of the core in its entirety

resulted in images 1024x1024x1980 voxel3, while sequential scans during the water phase

invasion of the matrix were 1024x1024x321 voxel3. Only a section of the core could be

monitored during transient imaging because imaging of a larger section would have come at a

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loss to the overall number of images that could be acquired. All scans have a voxel resolution of

0.015762 x 0.015762 x 0.015767 mm3, with the latter representing height, i.e. slice thickness.

4.3.4 Image Processing

The CMT images must be further processed in order to determine meaningful information. First,

all images are filtered using a 3x3x3 kernel median filter to remove salt and pepper type noise.

The CMT registration numbers of the three phases present, i.e. glass, oil phase, and water phase,

are significantly different, allowing for identification of the phase occupying each voxel.

Segmentation here was accomplished through simple thresholding. Other, more complex

thresholding techniques, such as bi-level thresholding with indicator kriging, were attempted, but

did not appear to result in any visually apparent improvement in segmentation. In simple

thresholding, two phases are segmented according to a threshold CMT value. Voxels with CMT

registration numbers below the threshold are labeled as one phase and those with CMT

registration numbers above the threshold are labeled as the other phase. To segment our three

phase images, a two-step segmentation process was used. First, the glass and pore spaces were

segmented in the presaturation image using a threshold determined by fitting a Gaussian curve to

the peak in the CMT registration number frequency distribution corresponding to the glass. The

threshold was taken as the CMT registration number corresponding to where the Gaussian curve

has a frequency value below 1% of the peak value; all voxels with CMT registration numbers

above this value are considered to belong to the pore space. This segmentation was then used to

mask the three phase images. By subtracting out the glass, only two phases, the oil phase and

water phase, are left. These two phases were then segmented in the same manner as the

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segmentation of the glass from the pore space. With the images subsequently segmented into

glass, oil phase, and water phase, measurements of the pore space and fluid distribution could be

made.

However, before measurements associated with the matrix and fracture could be taken, the pore

space had to be labeled as belonging to either the fracture or the matrix. The fracture and matrix

pore spaces are connected, and the distinction between them is not obvious. To identify the pore

space associated with the fracture and the matrix, a process of image erosion, isolation, and

dilation, detailed in a previous study of ours (Landry and Karpyn 2012), was used. The matrix

measurements are taken from the portion of the matrix parallel to the fracture and within 0.5 cm

of the fracture face. After segmentation of the images and labeling of the pore space, volume

measurements could be made from the image.

4.4 RESULTS

4.4.1 Pore Space Analysis

During the surfactant fluid penetration of the matrix, it became clear that the matrix on either

side of the fracture was appreciably different, and that these differences resulted in significantly

different rates of water phase invasion into the matrix. To measure the characteristics of the

matrix separately, for what will be referred to from here on as the right and left sides, a line of

division was determined from the center point of the fracture aperture measurements. In our

images, the fracture is oriented perpendicular to image voxels, therefore we found it appropriate

to measure the so-called “vertical aperture” ( ), i.e. the distance between the fracture walls. An

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image of the fracture, and a corresponding mean fracture aperture profile, are shown in figure

4.1. The mean fracture aperture was measured for each slice. The ROI is also highlighted in

figure 4.1.

Figure 4.1: Image of the fracture with the ROI highlighted (A), and the corresponding mean fracture

aperture and contact area profiles with the ROI indicated by arrows.

Overall, the fracture profile shows two wavelengths of decreasing and increasing mean aperture,

with the two minimums corresponding to significant contact areas between the fracture walls.

The portion of the core to be imaged sequentially during the surfactant water phase injection of

the matrix contains, at its center, a large “asperity”, meaning that the aperture is smaller than the

average pore aperture and is thus indistinguishable from the matrix using the labeling method

A) B)

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described here. This portion of the core was chosen because it is located near the center and

contains a portion of the fracture that has a significantly varying aperture and the occurrence of

an asperity. One of the objectives of this study is to investigate the dependence of fracture-

matrix fluid transfer on the fracture aperture and the existence of asperities.

The porosity of the right and left matrix was calculated by summing the voxels of the

segmented, labeled pore space images from the presaturation scan. Calculating the porosity for

each slice, a profile of the porosity is presented in figure 4.2. The porosity of the core varies

from ~0.1 to ~0.35, with the lowest porosities being found at the bottom and top of the core.

Also, the porosity of the left matrix is generally slightly greater than that of the right matrix as a

result of the sintering process. The top, bottom, and sides, with the left matrix being closer to the

center, of the sintered sample reached greater temperatures, resulting in greater compaction. The

porosity of the left and right matrix of the portion of the core to be sequentially imaged is 0.3030

and 0.2698, respectively. Although the porosity gives a bulk measure of the volume of pore

space present, and provides information in regards to the level of sintering compaction, the

displacement of the wetting phase in the matrix by the non-wetting phase in the fracture is not

dependent on the porosity. Instead, this displacement is strongly dependent on the pore aperture

of the matrix. The porosity and the pore aperture are not necessarily related, it is both possible to

have a high porosity, small pore aperture porous medium and a low porosity, large pore aperture

porous medium. Another objective of this study was to design a synthetic 3D porous medium

that could be imaged with minimum unresolvable pore space, thereby allowing very accurate and

complete measurements of pore space geometry, such as the matrix pore aperture distribution.

118

Figure 4.2: Image of the matrix pore space showing the fracture saturated with water phase and the

matrix saturated with the oil phase, the ROI volume is highlighted (A). Also the

corresponding porosity (B) and mean aperture (C) profiles with the ROI indicated with

arrows.

A significant amount of literature regarding the description of pore space topology exists

(Thovert et al. 1993, Lindquist et al. 1996, Oren and Bakke 1997, Prodanovic et al. 2006).

Although there are numerous measurements that can be made regarding the pore space topology,

we focused on the pore aperture distribution. In order to measure the pore aperture distribution

of the pore space, first a skeleton, or medial axis, of the pore space must be constructed by

thinning algorithms (Lee et al. 1994) using the software Avizo Fire 6.3. The skeleton of the pore

space is a connected network of voxels that represents the midpoint between solid walls in the

pore space. Each voxel in the skeleton can be thought of as the center of a sphere that makes

contact with the wall in at least two places. The pore aperture is the maximum radius of a

A) B) C)

119

sphere that can occupy a pore. To determine the pore aperture distribution, a “sphere packing”

of sequentially smaller spheres are strung along the skeleton. First, the voxels of skeleton are

labeled with their distance to the pore wall. Next, the labeled skeleton is searched for voxels that

fall within a user-defined range. These voxels then become the center of a sphere – with a radius

equal to the distance to the closest pore wall – placed in the pore space. The algorithm treats all

voxels that fall within this sphere as solids in order to avoid overlapping measurements of pore

aperture between spheres from different aperture ranges. An example image of the pore

apertures is shown in figure 4.3. The mean pore aperture is defined here as the volume-weighted

mean pore aperture,

∑

where represents all pore apertures measured, is the pore aperture, and is the volume

weight of each pore aperture measured,

∑

The weight of each pore aperture is equal to the volume ( ) of the associated sphere divided by

the sum of all pore aperture sphere volumes. The mean pore aperture profile generally follows

the trends seen in the porosity profile, with the largest pore apertures occurring in the middle of

the core for both of the matrixes, and the left matrix pore apertures being generally greater than

the right (figure 4.2). The fracture and pore aperture distributions for the ROI are given in figure

4.4. The fracture and pore aperture distributions are lognormal, similar to what is observed in

CMT images of Berea sandstone (Prodanovic et al. 2007) and fractures in Berea sandstone

(Karpyn et al. 2007). The mean pore aperture of the left and right matrix is 0.0770 and 0.0721

(mm), respectively.

120

Figure 4.3. Example 0.1183 cm

3 image of sphere-packing method used to measure pore aperture

distribution. The individual pore apertures are measured as the radius of the spheres

seen in this image. The red, purple, teal, green, yellow, and light green represent pore

aperture measurements between , ,

, , , and mm,

respectively. The background shows the solid in black and the pore space in white.

121

Figure 4.4. The fracture aperture distribution and the pore aperture distribution for the portion of the

core to be imaged sequentially during water invasion of the matrix.

4.4.2 Immiscible Displacement

After the core is presaturated with the oil phase, the water phase without the addition of a

surfactant is injected. The oil phase is nearly completely displaced from the fracture, with water

phase saturations in the fracture, , while the matrix has yet to be invaded by the

water phase. Only a small amount of the oil phase is retained in the fracture near the top of the

core, and as is evident in the image presented in figure 4.5, the contact angle through the oil

phase is quite large, close to neutral wetting conditions. The thorough displacement of the oil

phase from the fracture by the water phase can be attributed to the weakly oil-wet condition. The

affinity of the wetting phase for the solid is too weak for the wetting phase to be retained in the

smaller apertures of the fracture. Also, as was previously mentioned, the non-wetting water

phase does not invade the matrix during the initial water phase injection due to capillary forces,

which can be decreased by the addition of a surfactant.

122

Figure 4.5. Image of oil phase (white), water phase (black), glass (gray) contact in the fracture after

initial water phase injection. The contact angle appears in this image to be large, near

neutral wetting conditions.

At the beginning of the surfactant water phase injection, the non-wetting water phase is already

established in the fracture. Thus, the fracture-matrix fluid transfer occurs in what has been

previously referred to as the “instantly-filled” regime (Rangel-German and Kovscek, 2006), i.e.

fracture-matrix fluid transfer does not occur until after the displacing fluid has been established

in the fracture, as opposed to the “filling” regime in which fracture-matrix transfer occurs

concurrently with the advancing displacement in the fracture. The regime of fracture-matrix

fluid transfer is often determined by the velocity at which the displacing front advances in the

fracture, with a faster moving front resulting in an “instantly-filled” fracture. The flow presented

here is very slow; however, it represents an instantly-filled fracture-matrix transfer due to the

delay in the injection of surfactants required for fluids to penetrate the matrix.

123

A few example images from the sequential scanning of the surfactant water phase injection of

the matrix are shown in figure 4.6. During the sequential scanning there are some small amounts

of the oil phase that appear in the fracture (figure 4.6). However, the total volume of the oil

phase in the fracture is much smaller than the amount of the oil phase being displaced from the

matrix suggesting there is very little, if any, movement of the oil phase from the matrix to the

fracture. The displaced oil phase from the matrix migrates in the direction of flow through the

matrix to the outlet. The small amount of oil found in the fracture appears to result from the

snap-off of oil already in, or near, the fracture (figure 4.6). This suggests that the surfactant is

also acting to alter the wettability of the solid to water-wet. The surfactant of the soap used is

sodium dodecyl sulfate (SDDS), an anionic surfactant, which has been shown to decrease the

water phase contact angle with quartz (Zdziennicka et al. 2009). We would expect that the

functional surface sites of quartz would be similar to the glass used here, leading to the observed

wettability alteration to water-wet.

124

Figure 4.6. CMT images from the sequential scanning of the surfactant water phase injection

showing a volume rendering (left) and a slice from a height of 1.0704 cm (right). In the

volume renderings the water phase is dark orange, the solid is light orange and the oil

phase is white. In the slices the water phase is light gray, the glass is dark gray, and the

oil phase is white.

125

The effect this wettability alteration has on the transfer of fluids is discussed in the following

pore-scale analysis. The non-wetting phase saturation of the matrix ( ) as a function of the

square root of time ( ⁄ ) is shown in figure 4.7. At the start of the surfactant water phase

injection, the surfactant must first mix with the water phase already present, and establish a

monolayer at the interface of the oil and water phase, before interfacial tension will be

significantly reduced. As the interfacial tension is reduced, we observe an increasing rate of

wetting-phase displacement, referred to in figure 4.7 as “early time”, which stabilizes at a near

constant rate, , with rate being a linear function of ⁄ , ⁄ , where

⁄ is

the time the displacement is initiated. This stable displacement of brine-surfactant-Decane is the

focus of this study, considering the majority of the displacement takes place after the surfactant

has been established, and the interfacial tension significantly reduced. The ⁄

for the

displacement is not a set number due to the slowly decreasing interfacial tension at early time.

Instead, we can determine an apparent start time for the stable displacement as determined by

fitting the rate function by linear regression. Fitting experimental measurements for ⁄

resulted in correlation coefficients of 0.9893 and 0.9801 for the right and left matrix, respectively

(figure 4.7). Including earlier times resulted in lower correlation coefficients, and excluding

⁄

did not result in significantly greater correlation coefficients. Therefore we can be

fairly confident that the stable displacement occurs in this time frame. The apparent ⁄

is 4.48

and 4.32 (min1/2

) for the right and left matrix, respectively. Both sides are in agreement on the

apparent ⁄

. The is 0.0787 and 0.0551 (min-1/2

) for the right and left matrix, respectively.

There is a significant difference in rate observed between the right and left matrix. Given that

the only variable between the right and left matrix is the pore space geometry, the significance of

126

the effect matrix pore space geometry has on fracture-matrix fluid transfer is highlighted. To

further understand the physics involved in this displacement, we compared the experimental

pore-scale fluid distributions to the fluid distributions of a pore-scale LB model simulating a

capillary-dominated drainage of the right and left matrix – separately but equally – in which the

only variable in the simulation of the right and left matrices will be pore space geometry.

Figure 4.7. Non-wetting phase saturation of the right and left matrix as determined from sequential

CMT imaging. Early time refers to displacement before the interfacial tension reduction

has stabilized. The linear fits shown represent a displacement following the rate function,

⁄

.

Early

Time

Stable

Displacement

127

4.4.3 LB Simulation of Fracture-Matrix Fluid Transfer

For details of the LB model used refer to chapter 3. To model the surfactant water phase

movement from the fracture to the matrix a 270x150x150 voxel3 (0.4256 x 0.2365 x 0.2365 cm

3)

sample volume (figure 4.8) of the pore space from each matrix is translated 1-to-1, voxel to

lattice node, for lattice Boltzmann simulation. This translation maintains the complexity of the

pore space solid boundary conditions – the image is directly translated into fluid and bounceback

nodes. The size of the image volume to be simulated is limited by the prohibitive computational

requirements of the LB model. Simulating the entire matrix volume scanned would be optimal,

although, it should be noted, that the sample volume used is larger than the representative

elementary volume determined by porosity and specific solid surface area measurements not

shown here. The simulation lattices are bound by solid walls orthogonal to the direction of flow,

and pressure boundaries 5 lattice layers thick at the ends. These pressure boundaries represent

the non-wetting phase occupied fracture, and the pushback of the capillary force of the wetting

phase occupied matrix. Images of the sample pore space used in LB simulations, and an

illustration of the LB simulation set up is given figure 4.8.

128

Figure 4.8: Volume images of the 270 x 150 x 150 voxel3 (0.4256 x 0.2365 x 0.2365 cm

3) sample

volumes of the pore space for the left and right matrix to be translated 1-to-1, voxel to

lattice node, for lattice Boltzmann simulation (A). LB simulation setup showing the 5

lattice layer thick pressure boundaries, with the fracture represented as the non-wetting

phase occupied pressure boundary (B).

The wettability of the core was observed previously to be very weak with a contact angle in the

range of 75° to 90°. Using the equation of Huang et al. 2007 given above, a , and a

total component density, , this contact angle can be approximated by using

. Setting the pressure boundaries is less exact; experience and

numerical experiments were performed to determine the pressure difference ( ) required for

the non-wetting phase to traverse the simulation volume, resulting in a general estimation of

0.033 (mulu-1

tu-2

). Typically, the capillary pressure of the simulation is scaled ( ⁄ )

using the physical and lattice interfacial tensions in accordance with Laplace’s law,

Non-wetting

phase Pressure

Boundary

(Fracture)

Wetting phase

Pressure Boundary

(Matrix)

Matrix:

Initially occupied by

wetting phase

Direction of DisplacementB)A)

129

where is the length scaling equal to the resolution of the image, and, and are the

physical and lattice interfacial tensions respectively. A lattice pressure difference of 0.033

(mulu-1

tu-2

) represents a physical pressure difference of 6.33 Pa, and is less than the physical

pressure difference experienced by the fluids in the portion of the core sequentially scanned.

This suggests the process of fracture-matrix fluid transfer is indeed a drainage process and not

the result of wettability alteration followed by imbibition. Had the simulation required a

pressure difference significantly greater than the pressure difference experienced in the

experiment, it would suggest the drop in the interfacial tension alone could not account for the

initialization of fracture-matrix fluid transfer. The simulation runs until the convergence

criterion ( ), defined here as,

is met. Based on numerical experiments not shown here, a convergence criteria of 0.001 was

chosen. The LB simulations each required near iterations (

) to

reach equilibrium, completing the displacement.

Time scaling of LB simulations ( ⁄ ) is vague, typically dimensionless

measurements of the forces involved (i.e. capillary number, Reynolds number) are matched to

those used in the simulation to determine a reasonable value of the time scaling (Sukop and

Thorne 2006) or chosen, as described by Hatiboglu and Babadagli 2008, “carefully” to match

scaling of the interfacial tension, viscosity and density between the LB and experimental system.

Time-scaling in the manner described by Hatiboglu and Babadagli 2008 resulted here in a gross

underestimation of the time-scaling on the order of 103 - 10

4. The displacements described here

130

occurred over a period of a few hours, whereas those described by Hatiboglu and Babadagli

2008 occurred in a matter of seconds. The LB models employed are similar, and thus the time-

scaling is of a similar magnitude, which was shown to be appropriate for the displacements

described in Hatiboglu and Babadagli 2008, but not those presented here. We determined the

time-scaling by matching the physical time of stable displacement in the experimental system to

the lattice time required for the simulation to reach equilibrium. Thus, the time-scaling is

determined by the measured time of displacement in the experiment, and not derived from the

simulation alone. This highlights one of the weaknesses of these types of LB simulations; alone

they cannot predict the rate of fracture-matrix fluid transfer for slow displacements. Unlike the

physical fracture-matrix transfer, there is no delay in the start of the displacement in the LB

simulations. To compensate for this, the LB simulation results are shifted by an amount equal to

the mean

of the right and left matrix. The stable physical displacement required is roughly

to reach equilibrium, giving a time-scaling of

⁄ .

4.4.4 Pore-Scale Experimental and Model Results

One of the main objectives of this study was to evaluate the accuracy with which the LB

simulation captures fluid distribution at the pore-scale. As was mentioned in the Introduction,

there are very few investigations that have compared LB simulation results with experimental

results at the pore-scale. Also, due to the employment of a surfactant that appears to alter the

wettability of the solid surface, we were interested in determining if the fracture-matrix fluid

transfer can be described by a drainage process, and if the differences observed in the fracture-

131

matrix transfer rate ( ) of the right and left matrix can be attributed to differences in the pore

space geometry. A comparison of as a function of for the experimental and LB results

is presented in figure 4.9.

Figure 4.9. Non-wetting phase saturation of the right and left matrix as determined from sequential

CMT imaging and LB drainage simulations. The only variable between the LB

simulations for the right and left matrix is the pore space geometry.

The LB simulation of the right matrix fits the experimental results very well; however, the

simulation of the left matrix underestimates the rate of fracture-matrix fluid transfer and,

ultimately, the equilibrium non-wetting phase saturation. This could be attributed to simulating

too low of a pressure difference; at higher capillary pressures the non-wetting phase saturation is

expected to be greater. However, the right matrix simulation correctly reproduced the

132

equilibrium saturation. It is also possible that the size of the pore space translated into lattices

for LB simulation was too small. Returning to figure 4.6, we observe the existence of elongated

pore spaces with lengths on the order of 0.10 – 0.15 cm or 63 - 95 voxels ( ). It is possible that

our simulations, which have side lengths of 150 voxels ( are too small to effectively access

these larger pore spaces during the simulated displacement, we further comment on this in the

next section. Determination of appropriate lattice sizes and resolutions for LB simulations

continue to be subjects of discussion, and with increasing computational accessibility, these

issues can be addressed. Although the LB model underestimates the fracture–matrix fluid

transfer for the left matrix, it does correctly predict the higher of the left matrix over the right

matrix for the first half of the displacement. This suggests that the stable displacement of the

wetting phase from the matrix by the non-wetting phase occupied fracture as a result of the

introduction of an interfacial tension reducing and wettability altering surfactant can be described

by a drainage process, without the consideration of wettability alteration. This is not surprising

considering a surface cannot be altered until the non-wetting phase fluid makes contact with the

surface, and this surface contact is controlled by a drainage-like displacement.

133

A) B)

C) D)

E) F)

G) H)

Experimental Model

134

Figure 4.10. Comparison of experimental and simulated vertical saturation profiles in the matrix (left

and right) adjacent to the fracture for the right (A, B) and left (C, D) matrix, and horizontal

saturation profiles for the right (E, F) and left (G, H) matrix.

To further investigate how effective the LB simulations were at replicating the experimentally

observed pore-scale fluid distributions we compare vertical and horizontal saturation profiles.

The vertical (saturation as a function of height) and the horizontal (saturation as a function of

distance from fracture) for the experimental and LB model are presented in figure 4.10.

Experimental and LB results are reported in dimensionless height ( ⁄ ) and length

( ⁄ ) units, where the reference height and length are the respective height and lengths

of the analyzed CMT image ( ) and LB simulation volume

( ). We also show some snapshots from the LB simulation

for the right and left matrix in figure 4.11.

135

Figure 4.11: Sample images of the LB simulations of fracture-matrix transfer. The non-wetting phase

saturation front and simulation pore space are shown, the wetting phase is occupies the

pore space not occupied by the non-wetting phase.

Left Matrix Right Matrix

136

The experimental vertical saturation profiles (figure 4.10A, C) do not display any dependence on

the fracture aperture, even with the presence of a sizable asperity. However, if the asperity were

larger we would expect the matrix adjacent to the asperity to experience some delay in the arrival

of the saturation front from nearby open portions of the fracture. Similar uniformity of the

vertical saturation profiles is seen in the LB simulations (figure 4.10B, D), but due to small size

of the LB simulation volumes the saturation front is less uniform, a result of piston-like

displacement occurring in individual pores. We can be fairly confident the capillary-dominated

fracture-matrix fluid transfer presented here has very little dependence on the fracture aperture,

given the fracture is occupied by the displacing phase.

The LB simulation for the left matrix (figure 4.10D) has low non-wetting phase saturations near

the bottom of the simulation volume. The blockage of the saturation front in the lower portion of

the simulation volume is a result of its small size. This becomes more apparent in the images of

the LB simulations shown in figure 4.11. As was previously stated, LB simulations are

computationally demanding, limiting the size of the simulation volume. The blockage of the

saturation front indicates that the volume simulated may be too small, and could contribute to the

underestimation of the fracture-matrix fluid transfer. Furthermore, the higher rate of

displacement in the left matrix was captured by the LB simulation at first, until the lack of

accessibility to the lower portion of the simulation volume slowed the displacement rate. The

experimental horizontal saturation profiles (figure 4.10E, G) show a saturation front that is 0.30

– 0.40 cm in length, with an irreducible wetting phase saturation near 0.90-0.95. The LB model

horizontal saturation profiles (figure 4.10F, H) roughly match the saturation front lengths of the

experimental measurements, but more importantly they accurately predict an irreducible wetting

137

phase saturation near 0.90-0.95. Also we see here that the non-wetting phase saturation front for

the left matrix reached the edge of the sample. In the LB simulation of the left matrix only the

top part of saturation front transited the simulation volume, causing the overall vertical saturation

profiles to underestimate experimental results. Again the mismatch can be attributed to

simulating to small of a volume. Considering this, we do not believe any mismatches between

the model and experimental results should be attributed to shortcomings of the LB method, but

rather are simply a result of a rough estimation of boundary conditions. Also issues involving

the size of the sample volume used to simulate fracture-matrix fluid transfer are a result of our

limited computational capacity.

4.5 CONCLUSIONS

Here we have presented a pore-scale investigation of fracture-matrix fluid transfer. A surfactant

water phase displacement of an oil-saturated oil-wet fractured synthetic porous medium was

sequentially imaged using 3D CMT imaging. The synthetic porous medium was designed to

have minimal pore space beyond the resolution of the CMT imaging used, allowing us to observe

the motion of the displacement front at the pore-scale. The displacement occurred over a period

of approximately 3 hours allowing us to image a 0.5043 cm length of the core 14 times to

produce 4D data of fracture-matrix fluid transfer. Although the displacement was not stopped

during sequential imaging, the displacement was slow enough that there was no appreciable

blurring of the 3D images. A surfactant was introduced to lower the interfacial tension between

the water and oil phase, and consequently the capillary entry pressure, to initiate fracture-matrix

fluid transfer. Images showed the surfactant altered the wettability of the porous medium

138

causing small amounts of the wetting phase in and near the fracture to snap-off and be

transported in the fracture. But, almost all of the non-wetting phase was displaced out the top of

core. There was very little, if any, counter-current migration of the wetting phase. The

surfactant was introduced into a fracture with the non-wetting water phase already established,

thus there was a delay in the stabilization of the rate of fracture-matrix fluid transfer as the

surfactant established itself at the oil-water interface. After this delay the displacement stabilized

and the non-wetting phase saturation of the right and left matrix correlated linearly with the

square-root of time. Although the differences in the porosity and pore aperture distribution of

the right and left matrix were small, the fracture-matrix transfer rate of the left matrix was

significantly greater than the right matrix. This emphasizes the importance of pore space

geometry, as this is the only variable that exists between the right and left matrix. Although the

matrix pore space geometry is an important factor determining the migration of the displacing

front, the movement of the displacing front showed very little dependence on the fracture

aperture, and the existence of an “asperity”.

One of the objectives of this investigation was to evaluate the use of a Shan-Chen type LB

models by making pore-scale comparisons of fluid distributions imaged using CMT and model

results, which are lacking in the literature. We present LB drainage simulations of a sample

volume of the left and right matrix to simulate the experimentally observed fracture-matrix fluid

transfer. Typical methods of scaling time in accordance with the interfacial tension results in a

3-4 order of magnitude underestimate of the time scaling. Here we scale time to match

experimentally observed physical time. This highlights one of the weaknesses of these types of

LB simulations; alone they cannot predict the rate of fracture-matrix fluid transfer for slow

139

displacements. The LB drainage simulation of the right matrix closely matched the

experimentally observed equilibrium saturation, displacement rate, irreducible wetting phase

saturation, as well as the vertical and horizontal saturation profiles. The LB drainage simulation

of the left matrix closely matched the experimentally observed displacement rate for the first half

of the displacement and the irreducible wetting phase saturation. However, the simulation LB

simulation of the left matrix underestimated the equilibrium saturation, which was reflected in an

underestimation of the displacement rate for the second half of the displacement and a pore-scale

fluid distribution match that was not as convincing as the simulation of the right matrix. This

underestimation is likely a result of using too small of a simulation volume, which limited access

to some of the larger elongated pores of the left matrix. We do not believe any mismatches

between the model and experimental results should be attributed to shortcomings of the LB

method. Due to the prohibitive computational demands of multicomponent LB models the

sample volumes of the image used in the LB model was significantly smaller than the ROI,

considering the simulations only contained on the order of 100 grains they replicated

experimental results fairly well. This successful replication also implies that although the

surfactant altered the wettability of the porous medium to water-wet the fracture-matrix fluid

transfer can be described as a drainage process.

140

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CHAPTER 5: SUMMARY OF FINDINGS AND RECOMMENDATIONS FOR FUTURE

WORK

In these investigations we have highlighted some of the strengths of the LB model, and its ability

to elucidate the relationship between pore space characteristics and macroscale two-phase fluid

flow. Our comparisons of pore-scale fluid distributions between the model and experiment are

promising. There are however many questions that can be addressed by future investigations.

In the first investigation we found that the presence of porous walls can greatly enhance the

permeability of the fracture. We also found that fracture permeability estimation using

previously proposed formulations dependent on fracture characteristics did a generally poor job

of predicting the permeability of the fracture simulated. The determination of fracture

permeability is greatly simplified when the existence of the porous walls has an insignificant

effect. We can expect that as the ratio of the fracture aperture to pore aperture increases the

presence of the porous walls will have less of an effect, until at a critical ratio it will have no

effect. The question then arises what is this critical ratio? One should also consider that there

are many different types of fractures that exist across vast length scales (10-6

to 104 m). At what

length scales and what type of fractures does the presence of a porous wall significantly affect

flow? It would be optimal if a formulation could be determined that estimated fracture

permeability from the fracture characteristics, but the results seen here are not promising.

However considering there are many types of fractures, does there exist a “class” of fractures for

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which the estimation of fracture permeability from the fracture characteristics is practical. We

can speculate that fractures with a large fracture aperture to pore aperture ratio would fall into

this “class”. Future investigations that collected a variety of fracture images could simulate flow

using LB and begin to find the limits of fracture permeability estimation from fracture

characteristics. Although it should be noted that it is unlikely that a robust fracture permeability

estimation from fracture characteristics could be found for fractures like the one presented here

that a small fracture aperture to pore aperture ratio – the solid boundaries of porous matrix-

fracture system a too complex to be distilled into a simple formulation.

In the second investigation we found that the relative permeability of both phases in

homogenous-wet porous media decreased with increasing wetting strength. It would be

interesting to compare this to experimental measurements of relative permeability. There are

many techniques that can be employed to change the wettability of beads, performing

conventional relative permeability measurements of homogenous-wet bead packs would help to

further connect the results of the second investigation to experimental measurements.

In the third investigation we found that LB simulation matched the experimental results for the

right matrix well, but not the left matrix. This was attributed to size of sample image translated

into a LB lattice being likely too small. However, it is also possible that the issue was the

underestimation of the pressure difference imposed in the simulations. It would be easy to

determine which of these were true by simulating the fracture-matrix fluid transfer on a larger

sample image. But what would be of greater interest to the community is a generalized study of

lattice sizes and resolutions used in LB simulations. As in, what is the representative elementary

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volume required to properly simulate flow? One way to answer this question would be to use a

pore space image of a semi-homogenous medium, such as a bead pack, and simulate increasingly

larger volumes until the macroscale flow results stabilize.

VITA: Christopher James Landry

EDUCATION:

Doctor of Philosophy, Energy and Mineral Engineering

The Pennsylvania State University, University Park, Pennsylvania May 2013

Dissertation Title: Pore-scale imaging and lattice Boltzmann modeling of single- and multi-

phase flow in fractured and mixed-wet permeable media.

Advisor: Dr. Zuleima Karpyn

Master of Science, Petroleum and Natural Gas Engineering

The Pennsylvania State University, University Park, Pennsylvania

December 2009

Thesis Title: Experimental pore-scale analysis of fluid interfacial areas in oil-wet and water-wet

bead packs.

Advisor: Dr. Zuleima Karpyn

Bachelor of Science, Geology & Physics

Western Michigan University, Kalamazoo, Michigan

December 2006

PEER-REVIEWED PUBLICATIONS:

1) Landry C.J. and Karpyn Z. (In preparation) “Lattice Boltzmann modeling and 4D x-ray

computed microtomogrpahy imaging of fracture-matrix fluid transfer”.

2) Landry C.J. and Karpyn Z. (In preparation) “Relative permeability of homogenous-wet and

mixed-wet porous media as determined by pore-scale lattice Boltzmann modeling”.

3) Landry C.J. and Karpyn Z. (2012) “Single-phase lattice Boltzmann simulations of pore-scale flow in fractured permeable media” International Journal of Oil, Gas and Coal

Technology, 5(2/3), 182-206.

4) Landry C.J., Karpyn Z. & Piri M. (2011) “Pore-scale analysis of trapped immiscible fluid

structures and fluid interfacial areas in oil-wet and water-wet bead packs” Geofluids,

11(2), 209-227.

5) Landry C.J., Koretsky C., Das S. & Lund T. (2009) “Surface complexation modeling of Co(II)

on mixtures of hydrous ferric oxide, quartz and kaolinite” Geochimica et Cosmochica

Acta, 73 , 3723-3737

AWARDS:

1) “Second Place Society of Petroleum Engineers International Student Paper Contest – Master’s

Division” Society of Petroleum Engineers Annual Technical Conference, September

2010, Florence, Italy.

2) “First Place Society of Petroleum Engineers Regional Student Paper Contest – Master’s

Division” Eastern/Mid-Continent/Rockies SPE Regional Paper Contest, March 2010,

Pennsylvania State University, State College PA, U.S.A.

3) “Second Runner-up Graduate Student Paper Contest” Canadian International Petroleum

Conference, June 2009, Calgary Canada.

4) “2008 Marathon Alumni Centennial Graduate Fellowship” College of Earth and Mineral

Sciences, October 2008, Pennsylvania State University, State College PA, U.S.A.

5) “Outstanding Student Paper in the Division of Geochemistry” American Chemistry Society,

2006 National Meeting, Adsorption of metals to geomedia symposium, Atlanta GA U.S.A.

6) “2005-2006 Arts & Sciences Research and Creative Activities Award” College of Arts &

Sciences, Western Michigan University, Kalamazoo MI U.S.A.