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Characterizing the Evolution of Porosity and Permeability in Porous Media

Undergoing Pressure Solution Creep

Submitted to the Departments of Earth, Atmospheric, and Planetary Sciences and

Chemical Engineering

in Partial Fulfillment of the Requirements for the Degree of

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Bachelor of Science in Earth, Atmospheric, and Planetary Sciences

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and Bachelor of Science in Engineering

as Recommended by the Department of Chemical Engineering

at the Massachusetts Institute of Technology

May18, 2012

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Author

Depc ent of Earft, Atmospheric, and Planetary Sciences Department of Chemical Engineering

Certified by

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Dr. Christopher Leonardi Thesis Supervisor

Professor Jesse Kroll

Thesis Supervisor

Professor Brian Evans Thesis Supervisor

Accepted by

Accepted by -MASSAC14SET INTITUTE OF TECHOLG

OCT 2 42017

LIBRARIES

Samuel Bowring

Chair, Committee on Undergraduate Program

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Barry Johnston Undergraduate Officer, Department of Chemical Engineering

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Table of Contents

Table of Figures... ii

A b stra ct ... iii

Acknow ledgem ents... iv

1. Introduction ... 1

2. Pressure Solution in Porous M edia ... 2

3. Calculating the Perm eability of Porous M edia ... 4

4. Lattice Boltzm ann M ethod for Porous M edia Flow s ... 7

4.1 Underlying Equations...8

4.2 Lattice Arrangem ent ... 9

4.3 Boundary Conditions...10

5. Generating a Pore Netw ork M odel...11

5.1 M ed ial Axis M ethod ... 11

5.2 M axim al Inscribed Spheres M ethod ... 13

6. Creation of the Netw ork M odel...14

6.1 Distance Transform ... 14

6.2 Filling and Skeletonization ... 15

6.3 Hierarchy: M aster and Slave Assignm ents... 17

7. Porosity of Consolidating M edia ... 19

7.1 Analytical Solution for Porosity...19

7.2 Calculating Porosity Using Voxelated Im ages ... 20

7.3 Im age Resolution Study ... 21

8. Sphere Pack Properties ... 23

9. Perm eability of Consolidating M edia Using Lattice Boltzm ann M odels... 24

10. Netw ork M odels of Consolidating M edia ... 30

11. Conclusions ... 36

W orks Cited... 38

Appendix A. Voxelization Process ... 41

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Appendix D. Codes ... 52

D.1 Code to Determ ine Analytical Porosity ... 52

D.2 Code to Determ ine Num erical Porosity ... 56

D.3 Code to Create the Network M odel ... 57

lmgz Function: ... 57 Voxelize-2 Function: ... 59 DistTransform Function ... 61 Fillin-2 Function ... 62 M asterSlavel Function ... 67 SphereCS Function ... 70

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Table of Figures

Figure 1. Pressure dissolution along grain interfaces and resulting transport and re-deposition of grain

m aterial at free grain surfaces... 2

Figure 2. Fluid behavior represented at different spatial scales. ... 8

Figure 3 D3Q15 three dimensional lattice arrangem ent ... 10

Figure 4. 2D skeletonization of a circle pack... 12

Figure 5. Distance transform of the pore space. ... 15

Figure 6. Filling the space w ith m axim al spheres (2-D). ... 15

Figure 7. Skeleton circles remaining from sphere packing ... 16

Figure 8. Resulting skeletal structure of the im age ... 17

Figure 9. M aster-slave hierarchy procedure... 18

Figure 10. Two-dimensional visualization of the intersection of three spheres. ... 20

Figure 11. Numerical porosity calculations for sphere packs with varying image resolution...21

Figure 12. Run tim e for porosity calculations versus resolution ... 22

Figure 13. Velocity field for sphere pack 1c_101y ... 24

Figure 14. Log-log plot of Lattice-Boltzmann permeability as a function of porosity. ... 25

Figure 15. Log-log plot of elastic compression permeability as a function of porosity...26

Figure 16. Elastic model permeability as a function of confining pressure... 27

Figure 17. Log-log scale of permeability as a function of porosity ... 28

Figure 18. Log-log plot of permeability as a function of time for sphere packs subject to pressure solution creep at vario us co nditions... 29

Figure 19. (Left) Inscribed skeleton spheres in a simple cubic sphere packing using a leniency factor of 0.75. (Right) Resulting network image of a simple cubic sphere packing. ... 31

Figure 20. Netw ork im age of data set 4c_62278y ... 32

Figure 21. Pore size and coordination number distributions for the network model of data set 4c62278y ... 3 3 Figure 22. Netw ork im age of data set 4c_20y ... 33

Figure 23. Pore size and coordination number distributions for the network model of data set 1c_341y. ... 3 4 Figure 24. Pore coordination number as a function of porosity... 35

Figure 25. Average pore size as a function of porosity... 35

Figure 26. Number of pores connecting to at least 1 other pore in the sphere pack as a function of p o ro sity . ... 3 6 Figure 27. Sample original sphere pack (left) and the resulting image after voxelization...41

Figure 28. (a) Three-dimensional image of a two-sphere intersection. (b) Two-dimensional representation of a tw o-sphere intersection. ... 44

Figure 29. Two-dimensional representation of a three-sphere intersection. ... 45

Figure 30. (a) Three-dimensional sphere-plane intersection. (b) Simplified two dimensional model of the intersection of a plane and a sphere ... 45

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Characterizing the Evolution of Porosity and Permeability in Porous Media Undergoing Pressure Solution Creep

By

Cassandra Swanberg Submitted to the

Departments of Earth, Atmospheric and Planetary Sciences And Chemical Engineering

May 17, 2012

In Partial Fulfillment of the Requirements for the Degree of Bachelor of Science in Earth, Atmospheric, and Planetary Sciences

and Bachelor of Science in Engineering

as Recommended by the Department of Chemical Engineering

Abstract

This work looks at the change in pore-scale morphological properties such as porosity and permeability using modeled sphere packs. The effects of varying pressure, temperature, and stress upon these properties are evaluated in numerically derived sphere packs undergoing creep and elastic compaction processes. This thesis will utilize the abilities of the lattice Boltzmann method and the

network model method to determine various morphological properties of these sets of packed spheres. The results from these two methods can be combined to further analyze the relationship between pore space morphology and fluid flow parameters in porous media that can be used to develop correlations to predict permeability based upon the physical structure of the pore space.

Thesis Supervisor: Christopher Ross Leonardi Title: Postdoctoral Associate

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Acknowledgements

Firstly, I'd like to thank my thesis advisor, Dr. Chris Leonardi for his continued support during my research. Without his patience and guidance, this thesis would not have been possible. I'd also like to thank Dr. Yves Bernabe for his moral support, careful explanations, and of course, his data sets upon which this work is based.

I'd like to thank the MIT Departments of Chemical Engineering and Earth, Atmospheric, and

Planetary Sciences for preparing me for my new job after graduation. With the knowledge and skills I have gained here these past years, I feel like I'm ready to take on the world.

These past four years have quite a journey and I am grateful for both my family and friends for acting as an excellent source of support. I'd like to thank my parents for believing in me and for their constant love and encouragement. Without them, I wouldn't be where I am today.

I'd also like to thank all the wonderful friends I have met here at MIT. In particular, I'd like to

acknowledge Camille McAvoy, Jennifer Hope, and E.J. Hester. You have become some of the best friends I have ever had. Thank you for your support and friendship. I don't think I could have made it through these past four years without you.

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1. Introduction

Fluid flow in porous media is an important aspect within many fields of scientific study. Two examples are chemical reactor engineering and petroleum reservoir engineering, where the ability to quantify fluid flow within complex porous media plays a pivotal role in understanding and carrying out routine production operations (1) (2). The ability to accurately measure and quantify certain flow parameters, most notably the permeability, is vital to quantifying flow behavior in porous media. Traditionally, these parameters are evaluated by laboratory experiments of cored rock, but the cost and time-savings benefits of numerically determining these parameters can be great (3) (4). However, the solution of the fluid flow equations that are used to determine these parameters can become complex. Tortuous and irregular boundaries within these systems complicate the task of solving macroscopic transport equations. Furthermore, the model size and limits on computing power constrain the

applicability of smaller scale molecular dynamic simulations (5).

A compromise must be found to keep within the limits of computational feasibility while still

maintaining sufficient complexity to accurately capture the complex behavior of the system. To accomplish this task, a number of fluid dynamic methods have been created to capture the pore-scale behavior of these systems. Two of the most common techniques are pore network models (PNM) and computational fluid dynamics techniques such as the lattice Boltzmann method (LBM). The aim of this project is to investigate the correlation between permeability predictions made using a pore network model and the LBM for realizing fluid-structure interaction in porous media undergoing both elastic compaction and pressure solution creep.

This work looks at the change in pore-scale morphological properties such as porosity and permeability using modeled sphere packs. The effects of varying pressure, temperature, and stress upon these properties are evaluated in numerically derived sphere packs undergoing creep and elastic compaction processes. This thesis will utilize the abilities of the lattice Boltzmann method and the network model method to determine various morphological properties of these sets of packed spheres. The results from these two methods can be combined to further analyze the relationship between pore space morphology and fluid flow parameters in porous media to develop a correlation to predict permeability based upon the physical structure of the pore space.

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2. Pressure Solution in Porous Media

Pressure dissolution plays an important role in a variety of different geological processes, especially in grain compaction and cementation in basins. As sediments collect in these areas, unconsolidated granular materials at lower depths becomes compacted due to overburden pressure, and the porosity, or empty space between these grains, diminishes. Initially, porosity reduction in these materials is effected through mechanical redistribution of grains by slippage, rotation, bending, or brittle fracture, but at depths greater than 1-2 kilometers, compaction is accomplished via a chemical process known as pressure solution (6).

Pressure solution is a result of an increase in grain solubility at grain interfaces due to large stresses between grains as a result of compaction (7). The increase in solubility allows the pore fluid to dissolve the grain material at these interfaces. The dissolved material is then transported through diffusive or advective processes and redistributed on a free grain surface (grain surfaces not subject to stress) or transported to another location in the medium by the pore fluid as shown in Figure 1 (6).

Transport of Material Away from Grain

Interface

Re-Deposition at Free Grain Surfaces

Figure 1. Pressure dissolution along grain interfaces and resulting transport and re-deposition of grain material at free grain surfaces. Grain compression causes an increase in solubility along the

compression interface which allows pore fluid to dissolve and transport grain material away from the grain interface and re-deposit it upon free grain surfaces.

Pressure Solution at Grain Interfaces

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Many studies have been conducted to better understand the mechanics that underlie this process, but laboratory experiments cannot easily capture the behavior observed in natural basins because of the large time scales involved (8). Therefore, much work has gone into designing numerical models that

can capture the mechanics of the process (9) (10).

Models have been created that simulate an idealized sphere pack undergoing elastic compression and pressure solution creep. These models are capable of capturing the response of the sphere packs to various thermodynamic conditions, as described in detail in Section 8. Sphere Pack Properties. The sphere packs are evaluated at a variety of different conditions to determine changes in morphological structure that influence the transport processes of fluids within these different granular consolidations. In particular, the framework of this work is designed to understand permeability, porosity, pore coordination number, and pore size and their relation to changes introduced within these various modeled sphere packs.

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3. Calculating the Permeability of Porous Media

The calculation of fluid flow through porous media can be complex, especially when considering the behavior at the pore scale where the irregular boundaries of the structure are most apparent. Over the years, various methods have been created in an attempt to handle these conditions and build equations which can generalize the structure and behavior of fluids in rocks at this level. Some of the most promising methods known to date are the lattice Boltzmann method and the pore network model. Both of these methods vary in how they calculate physical properties of porous media, but both are capable of providing effective and valid approximations these parameters at the pore scale. The lattice Boltzmann is a computational fluid dynamics method capable of simulating fluid flow within complex boundaries. Conversely, network models create a simplified skeletal structure of a sample's pore space which accurately captures the most important pore shape, connectivity and size distributions. Fluid flow in the skeleton is then solved as a series of interconnected pipes.

One of the most important problems in simulating fluid flow through porous media is simply to understand how this highly irregular geometry relates to macroscopic flow behavior. The ability of fluid to flow through a given pore geometry is typically characterized by a property known as permeability. Permeability can be thought of as a measure of the connectivity of a pore space and is an important parameter in Darcy's Law,

Q=-kA

(1)

-P,

where Q is the volumetric flow rate through the medium, A is the cross sectional area, p is the kinematic viscosity, and VP is the pressure gradient across the medium. This equation governing porous media flow has been verified both experimentally and numerically and provides the framework that governs most porous media models.

The outputs of the LBM namely drag force and fluid flux, can be used to calculate permeability using Darcy's law,

k

-- (2)

APz 9.869233*10 16* A

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the direction of fluid flow and A is the area. The constant is a conversion factor from m2 to millidarcy.

These permeability measurements, evaluated over discrete lattice volumes in the Lattice-Boltzmann method can be averaged at each lattice node and used to calculate permeability on much larger macroscopic scales (11).

If fluid flow parameters are not available, another method to determine permeability is to relate

it to the underlying morphological structure of the medium itself using structural characteristics obtained from a pore network model. Permeability correlations of this type often stem from a modified version of the Kozeny-Carman equation (12),

Or2

k

= (3)

fi

which relates permeability to the hydraulic radius (rH), the porosity of the medium (c5), and various experimentally derived constants (fl). Many variations of this equation exist in the literature and are constantly being changed and updated with the ultimate goal of understanding the general relationship between permeability and porosity in porous media (13) (14) (15) (16). In general, the Kozeny-Carman equation is used in pore network models that are able to capture details about the structure of the porous medium. These structural distributions such as the pore coordination number and pore throat radius become part of the denominator term and are fitted to experimental data to achieve accurate correlations. Many variations on this equation exist and the scientific community has yet to reach a consensus as to which porous media parameters are most important when characterizing permeability.

Keehm et al. (17) and Singh and Mohatany (18) tried various methods to modify the Kozeny-Carman equation for permeability determination. Singh and Mohatany generated exponentially correlated 3D porous media and utilized a modification of the Kozeny-Camran equation that includes porosity and pore-pore correlation length to calculate permeability (17). They used a Lattice Boltzmann model to calculate permeability directly and then fit the parameters of the pore space to a modified Kozeny-Carman equation. Their equation is only applicable if the pore space can be expressed as a linear superposition of exponentially correlated porous media.

Keehm et al. reconstructed 3D porous media using a sequential indicator simulation that uses the statistical parameters obtained from thin sections and geostatistical methods to stack them into a fully 3D image with the same properties as seen in the individual thin sections (18). Absolute permeability was determined using Lattice Boltzmann simulations and the results were then used to fit a

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version of the Kozeny-Carman equation as a function of porosity and effective surface area estimated from the thin sections.

Many definitions of the Kozeny-Carman equation exist to characterize permeability of a porous substance as a function of its underlying morphological properties. Many of these definitions are limited

by the assumptions used in creating the underlying models that determined the input properties of the

model. Since there is no single definition of a "pore" or a "pore throat", the shapes and characteristics of these complex three dimensional pore spaces are difficult to characterize. Network models provide a structure by which this can be accomplished, but even these models must be built upon certain assumptions, not all of which are valid. A paper by Bryant, King, and Mellor (15) examined the various assumptions that are often used in the construction of network models such as pore and pore throat size distributions, fixed pore shapes, fixed coordination numbers, and pore arrangement (whether or not they adhere to a specific lattice distribution) on a controlled packing of disordered spheres in a bladder (15). Their network model worked by examining each sphere in the packing and connecting it to its nearest four neighbors, thus creating a uniformly coordinated, completely disordered system. Using an effective bond length and hydraulic conductivity calculated by smoothing and segmenting the flow path between these spheres, this model was able to provide permeability measurements that agreed well with other random sphere pack models. The overall finding of this work showed that assumptions such as constant bond length between pores or that the pores are arranged in a regular lattice are invalid. Furthermore, their study revealed that closely positioned pores are correlated thus causing the actual permeability to be less than would be seen with randomly assigned bond sizes. Each of these findings limits the assumptions that can be incorporated into future models and thus affect the way parameters can be fit into modified Kozeny-Carman equations.

In this thesis, permeability will be evaluated using a network model built in Matlab that extracts the properties of a pore network directly from a set of packed spheres subject to various simulated evolutionary processes. The calculated results of this network model will include pore coordination number distributions, pore size, pore throat size, and porosity and will be used to describe permeability of a given sphere pack using a modified version of the Kozeny-Carman equation. The results of the permeability measurements of these models will be directly compared with that of the Lattice-Boltzmann predictions for accuracy and differences in computational expense associated with these two methods.

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4. Lattice Boltzmann Method for Porous Media Flows

In recent years, the Lattice Boltzmann method has become an increasingly popular method to computationally derive fluid flow behavior and physics (5). This method works not by discretizing differential equations that govern macroscopic fluid flow, but by combining averaged microscopic behavior with certain conditions such that the overall macroscopic equations are obeyed. These conditions, which include particle collision distributions as well as specific boundary conditions, are applied to particle distributions to govern fluid flow, and if appropriately specified, follow the behavior of the macroscopic Navier-Stokes equations at the nearly incompressible limit (19). By discretizing a flow geometry into a uniform grid and using averaged particle distribution functions at each point in this grid instead of single particles, the Lattice Boltzmann Method (LBM) reduces the amount of statistical noise (5) and reduces the computational expense of simulating a completely microscopic particle regime (as is typically done in lattice gas models). These models can also be used efficiently in geometries that are difficult to simulate macroscopically due to irregular and complex boundaries. Furthermore, the particle distribution interactions in the Lattice Boltzmann method depend only on particles in the grid nodes nearest them which makes this method ideal for parallelization and thus can be easily run on multi-processor computers to reduce computational time (20).

It's for these many reasons that the Lattice Boltzmann method is an ideal candidate for simulating flow behavior in porous media. The highly complex boundaries within and between pores have made other methods of modeling difficult, if not impossible to implement, thus the LBM has been used often in the literature as a tool to characterize important aspects of porous media flow. There is a large variety of literature available that presents different utilizations of the LBM in porous media including methods that utilize 3D x-ray microtomographic images of rocks (11) (21) (22) (23) and those that use 3D images reconstructed from 2-D thin sections (18). In particular, these methods differ in the implementation of the conditions under which the LBM simulations are run, such as the way in which the no-slip boundary conditions are maintained at the pore walls, the type of lattice by which the model is discretized, and the inclusion or exclusion of multiple fluid interactions within the media.

Indeed, the Lattice Boltzmann method can be modified, adapted, and combined with various other methods to handle a wide range of physical phenomena, and as the popularity of the Lattice Boltzmann method has grown, so has its range of applications. Apart from being an ideal method for handling the complex boundaries found in pore scale studies of fluid flow, the lattice Boltzmann method

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has been applied to other complex fluid conditions such as multiphase flow, multicomponent flow, and suspended particles (5). This is of particular use in oil reservoir systems where multiphase gas, oil, and water systems are prevalent. The behavior of these systems adds further complexity to the simulations because of more complex interface conditions such as the wettability of surfaces and interfacial tension between different phases. Most of the more recent work in the application of the lattice-Boltzmann method to porous media has been around expanding its applications to more complex conditions so that it can accurately capture real-world behaviors.

4.1 Underlying Equations

Fluid behavior is characterized differently at different spatial scales (Figure 2). At the microscopic level, fluid behavior can be analyzed as a set of molecular collisions. At the macroscopic level, fluid interactions are the result of pressure gradients and fluid properties such as viscosity and density (24). The intersection of these two extremes is not as clear, but can be reasonably described statistically using the Boltzmann transport equation,

af

+

af

.

P

f

.

F

=

dxdpdt

(4)

at

ax

m Op

at

collision

where F is the force field acting on the particles in the system, m is the mass of the particles and f is the distribution that defines the number of particles occupying a given volume element at a given time.

Molecular Dynamics Lattice-Boltzmann Finite Difference/Transport Equations

(Microscopic) (Mesoscopic) (MacroscoPic)

Figure 2. Fluid behavior represented at different spatial scales. At microscopic scales (left), fluids behave as a series of discrete particle collisions. At macroscopic scales (right), fluids behave and are driven by pressure differences. The Lattice-Boltzmann method uses a mesoscopic approach (center) which bridges the gap between these two behaviors and analyzes average particle

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The Lattice Boltzmann method builds upon this "mesoscopic" approach by discretizing the system and analyzing the average behavior of a distribution of particles at a discrete time within a discrete volume element. It allows only discrete particle velocities and specifies only a few possible directions for these particles to travel along a regular grid. The Lattice Boltzmann equation itself can be written as

fi(x

+

c

1

At, t + At) = fi(x, t) + fl

1(f(x, t)), (5)

where f is again the particle distribution function, At is the time step, and Ax is the lattice spacing, ci = Ax/At is the velocity, and fi is the particle collision function (5). Collision (also known as

relaxation) and streaming (also known as convection) rules at each lattice point define how these particle distributions can interact with each other and with surfaces as well as how they move to the next lattice node in the grid. Since these rules are evaluated independently at each lattice point, the Lattice Boltzmann method can be easily parallelized, thus reducing its computational demand (20). The combination of these rules within the system specify the temporal redistribution and movement of the nodal particle distributions and averages these behaviors to produce overall fluid flow behavior.

4.2 Lattice Arrangement

The Lattice Boltzmann Method requires that the simulated domain be discretized into a lattice that specifies the different movement options available to particle distributions as they stream between nodes. The general form for specifying the lattice structure in Lattice Boltzmann simulations is to use DnQv where n represents the dimensionality of the problem

(1-D, 2-(1-D, 3-D) and v represents the number of different velocity vectors a particle can follow at

any given lattice point (24). For three dimensional flow problems, two lattice definitions are generally used-- the D3Q15 and D3Q19 (25). Both these method contains a rest function as well as a number of directional velocity vectors as seen in Figure 3 for the case of D3Q15. The addition of the rest function allows the system to have a velocity of zero for a given lattice point which allows pressure to be evaluated independently from velocity at the lattice nodes (19).

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Y Z .. ... ... ... . ... CO C Cq4 X

Figure 3 D3Q15 three dimensional lattice arrangement.

For 3D systems, the D3Q19 lattice is slightly more accurate than D3Q15 due to the higher number of lattice velocities, but the computational expense associated with a more complicated system in 3D must be considered. In this work, the D3Q15 is chosen over the

D3Q19 because the lower number of velocity vectors reduce computing time and provide a

sufficient representation of the lattice velocity distribution to evaluate permeability.

4.3 Boundary Conditions

Two kinds of boundary conditions are used within the Lattice-Boltzmann simulations undertaken in this thesis. At the inlet and outlet faces of the system, periodic boundary conditions are applied. This condition uses the outlet particle distribution profile of the fluid as the receiving inlet conditions. Using this boundary condition, the fluid domain can be extended infinitely, allowing the system to reach a steady state at sufficiently high time steps (25). Immersed moving boundary wall conditions are used at the other four faces and at the fluid-solid interface along each of the solid spherical grains. This boundary condition utilizes an additional collision operator to maintain the no-slip condition along these interfaces. C13 C, C14 CIOI

...

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5. Generating a Pore Network Model

In addition to Lattice-Boltzmann models, the other commonly used method for permeability determination within porous media is network models. Since fluid can only be transported through the pore space in a rock, network models first use discretization algorithms to break the volume into pore and grain material. Only the pore material is considered in creating the model. Several methods have been previously defined to describe the flow properties through the pore space such as the medial axis method and the maximal spheres method (16). These methods reduce the pore spaces into a series of pipes of finite radius that connect subsequent pores. These methods simplify the flow structure and allow for easier calculation of parameters such as the coordination number, pore and pore-throat size distributions, and pore-throat and pore-pore body aspect ratios. Software packages have been developed to decompose 3D images into their underlying skeletal structure (26) and provide easy access to these difficult to characterize parameters, but none are available publicly. It was therefore necessary to develop a code to define network models of our data in order to obtain the correct parameters needed to define permeability within the system.

5.1 Medial Axis Method

Network modeling has been studied using a variety of different approaches. Some approaches break the pore structure down using a medial axis transform (13) (27). Others fit spheres within the pore space and connect the spheres to easily represent pores and pore space structure (I 6). The medial axis method works by defining the topological skeleton of a given geometry and then removing all the parts of the skeleton that do not contribute to overall connectivity. The skeleton of the image can be obtained by selectively eroding parts of the image down until further erosion would change the connectivity of the image or by first calculating the distance transform of the image to be considered and following the local maxima through the pore space. The distance transform is simply the calculation of the distance from any given voxel to the nearest grain space boundary. This transform will be described in greater depth in Section 0. In two dimensions, the skeleton of the image is represented by a series of lines that follow the local maxima of the distance transformed image and therefore follow the centers between successive circles. The skeletonization of the void spaces between simple set of circles in 2D is shown in Figure 4. 2D skeletonization of a circle pack

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Figure 4. 2D skeletonization of a circle pack

Once the skeleton of a given image is obtained, the small branches of the skeleton that do not contribute to the overall connectivity of the image must be removed. This can be accomplished in many ways such as removing elements that are smaller than a certain length or more robustly, by eroding branches down using an erosion algorithm while maintaining overall topographic connectivity (27).

Bernabe, Li and Maineult BLM used network simulations to study the effect of pore connectivity. The resulting permeability, porosity, pore shape and spacing characteristics calculated using this underlying pore structure skeleton defined a series of power laws that are accurate for two and three dimensional models. However, this method required an accurate measurement of the coordination number of the pores. This method was later modified to use the electrical formation factor as a way to relate formation factor, permeability, and coordination number and verified using a variety of granular materials (14).

For the purposes of this project, the medial axis or erosion method is difficult to implement on 3-dimensional images because of the difficulty in specifying the erosion algorithm in complex morphologies without losing information on the underlying structure of the material. Thus, the maximal inscribed sphere method was chosen to calculate the pore space properties of the sphere packs analyzed in this work.

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5.2 Maximal Inscribed Spheres Method

The maximal inscribe spheres method works by fitting a series of spheres within the pore space of the system in order to specify the pore space morphology of a given sample. The model created for this project uses the basic methodology outlined by Silin et al. (16) and Al-Kharusi and Blunt (28). A given grain space image is first converted into a three dimensional binary matrix using a method known as voxelization. Within this matrix, each pore space element, represented as a "0", undergoes a distance transformation which calculates the minimal distance from this location to the nearest grain space voxel in the matrix.

Once this transformation occurs, the space is filled with maximal spheres, beginning from the largest number in the distance transformed matrix and working down. As each of these spheres is created in the pore space, the distance number assigned to voxels included within each of the sphere volumes is set to zero so they are not calculated. This procedure continues until the pore space is completely filled with maximal spheres. The spheres that touch grains at two or more points are recorded and the rest are discarded.

Then, a separate hierarchy procedure determines if any of these existing maximal pore spheres touch each other or overlap, and if so, compares the radii of each of these pairings. The larger sphere becomes the master to the smaller sphere. This process continues for all touching and overlapping pore space spheres. The ultimate masters of the system (the spheres that have no other larger spheres touching them) are the pores of the system and all touching spheres describe the connectivity of these pores. The ultimate master spheres inherit the slaves of all their connecting spheres and the slaves of the slaves and so on.

The connectivity between each of the resulting ultimate master pores is analyzed by determining the common slaves between the masters, and the pore connectivity is assigned as the number of other ultimate pores connected to each of the ultimate masters. A detailed procedure for each of these processes is reviewed in Section 6. Creation of the Network Model.

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6. Creation of the Network Model

The pore network model chosen for this work was that of the maximal inscribed spheres method. This method describes the pore space of a given porous media by fitting spheres into the underlying pore space of the material, and the connections of these spheres is used to determine pore parameters such as pore size, pore throat size, and pore connectivity. The method that follows is based upon the methodology laid out by Silin et. AI(16) with some minor deviations for the use of sphere packs rather than microtomographic rock images.

6.1 Distance Transform

The distance transform takes the voxelized image and calculates the distance to the nearest sphere for each voxel in the matrix as portrayed in Figure 5. Distance transform of the pore space Other methods of doing this include manually expanding a voxelized sphere surface centered at a given pore voxel until it contacts a grain, but this is only accurate down to the size of the voxel. A fast scanning algorithm for Euclidean distance transforms that uses a similar approach is available on the Mathworks exchange (29) and is used to reduce run time for three dimensional simulations. Both of these transforms do, however, produce voxels that contain equal values because the accuracy of this approach is limited by the voxel size. This behavior can be a problem when creating the hierarchy procedure described in Section 6.3 Hierarchy: Master and Slave Assignments. Since the exact locations and radii of the spheres in each of the sphere packs is known, the exact distance from pore voxel centers to the nearest sphere surface can be calculated to minimize this occurrence. The distance from the voxel to the boundary of a nearby grain is given as

Distance = (Xvoxei - Xsphere)2 + (Yvoxei - Ysphere)2 + (Zvoxei - Zsphere)2 - Rsphere. (6)

This calculation is quite computationally intensive, especially when the matrix size becomes large, because multiple calculations must be done for each voxel. To minimize computational time, this calculation is only performed on spheres/circles that touch grain boundaries at two or more points. These circles/spheres are part of the skeletal structure of the space and are the only shapes used to determine the pore space properties and skeleton.

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(a) (b) (c)

Figure 5. Distance transform of the pore space including (a) how the distance to nearest spheres is compared for a single voxel (the yellow line is the distance to the nearest sphere assigned to that particular voxel), (b) a color coded representation of the distance transform (blue represents short distances and red represents large distance values), and (c) the compilation of the distance transform in three dimensions (pore spaces are the lighter areas between the blue circular objects in the image).

6.2 Filling and Skeletonization

The next step in the process fills the pore space with maximal spheres and then determines whether these spheres are part of the underlying skeleton of the pore structure. Each of the pore space voxels is assigned a number corresponding to the distance to the nearest grain space voxel. The filling procedure uses the largest distance voxel to inscribe the maximal sphere that can fit in the pore space at this point all other distance voxels in this space are set to zero and the procedure repeats for the next largest distance voxel until the entire pore space is filled with maximal spheres as seen in Figure 6. Filling the space with maximal spheres (2-D)

300]

0E

W

0 100 A 3M 4J 5W0 1 0 1 1

Figure 6. Filling the space with maximal spheres (2-D). Pore spaces are filled incrementally according to radii calculated in the distance transform. The above images show pore space filling at three times read from left to right: beginning, intermediate, and complete filling.

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Not all of these spheres are necessary to describe the pore space, however. Only the maximal spheres that describe the pore space skeleton are needed in the following procedures. The maximal sphere is part of the pore space skeleton if it contacts a grain space at more than two points. The result of applying this condition upon all the maximal spheres in the system is shown in Figure 7. If a sphere is deemed to be part of the skeletonized pore structure, the exact distance to each of the pore spaces it is touching is calculated and the result is stored to avoid future discrepancies due to equal spheres being in contact with one another. Connecting the centers of these spheres together gives us the skeletal structure of the pore space pictured in Figure 8. Note that this structure also contains unneeded dead end paths that do not contribute to the overall pore space connectivity.

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Figure 8. Resulting skeletal structure of the image

6.3 Hierarchy: Master and Slave Assignments

The next step in the process is to sort the skeleton circles into a hierarchy based upon size. If two skeleton spheres intersect, then the radii of these two circles are compared, and the larger sphere becomes master to the smaller sphere (Figure 9). This procedure is repeated for all skeleton sphere intersections in the pore space. The skeleton spheres that have no direct masters are deemed to be "ultimate masters" and represent the pores of the structure. The smaller slave spheres provide the connectivity between each of these pores.

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Ultimate Master Ultimate Master Master to 1 and 3 Ultimate Slave Master to 6

Slave to 4 an

2 5

Master to 5,

Slave to 2

Master to 4, Slave to

3

Master to 5,

Slave to 2 Slave to 7

Figure 9. Master-slave hierarchy procedure. Hierarchy is decided according to radius of intersecting or touching maximal spheres.

The ultimate masters then begin to consolidate slaves by inheriting not only those smaller slave spheres that directly intersect this larger sphere, but also the slaves of each of the slaves. This procedure continues following the principle that "the slave of my slave is also my slave" until the "ultimate slaves" are found. These spheres have no direct slaves, and are intersected only by spheres that are larger than them. These "ultimate slaves" can be either dead end spheres or pore throats

depending upon whether or not they are part of the path that connects "ultimate masters".

Once this inheritance procedure is completed for all the "ultimate masters", the morphological properties of the pore space can be determined. The number of pores in the sample is the number of "ultimate masters" and the size of these pores are simply the radius of these maximal spheres. The connectivity of the pores is found by analyzing how many other pores are connected to a given "ultimate master". This is accomplished by looking for common slaves. If two pores are connected, they should share a common "ultimate slave" at the smallest point in the junction. This junction is the pore throat connecting the two pore spaces and the radius of the common "ultimate slave", is recorded as the pore throat radius for this connection. The connectivity or coordination number of each of the "ultimate masters" is the number of other "ultimate masters" that it is connected to. These properties vary between different pores in the sample space, and the distribution of these properties is displayed in the form of a histogram. The averages of these properties can be used to calculate permeability for each of the sphere packs under consideration using Kozeny-Carman equations as described in Section 3.

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7. Porosity of Consolidating Media

Porosity of a porous medium is a measure of the amount of empty space within a given material defined as the ratio of the volume of the void space to the total volume of the medium. Since the medium in which flow parameters are being calculated consists of spheres, an analytical solution for the porosity of many of the sphere packs under consideration (assuming intersection volumes are created

by no more than two spheres) is easily obtainable. Porosity tells us about the amount of space in the

material, and serves as an important variable in calculating permeability.

7.1 Analytical Solution for Porosity

In the case of a sphere pack, porosity is the amount of space within the confining cube that is not occupied by a spherical grain. Therefore, the analytical definition of porosity that will be used in this project is

Porosity = Vconfining cube -(vspieres- Z vsphere-sppere intersect-Z vsphere-cube intersect)

Vconfining cube

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Appendix B. Analytical Solution for Porosity. The volume of each individual sphere can be completely described as well as the volume of intersection of spheres with the overall cubical boundaries of the sample. The intersection of two spheres can be described analytically by calculating the "lenses" formed from the intersection between the two surfaces. However, this solution is only valid for the intersection of two spheres. When the intersection volume is occupied by more than two spheres (Figure 10), a closed form analytical solution is very difficult to obtain. Analytical porosity determinations are therefore limited to sphere packs with small intersection volumes that are unlikely to contain intersections of more than two spheres. The results of these calculations will be used to verify the image resolution necessary to obtain voxelized images that yield similar calculated porosities.

R2

R1

R3

Figure 10. Two-dimensional visualization of the intersection of three spheres. An exact analytical calculation of the volume included in all three spheres is difficult to calculate.

7.2 Calculating Porosity Using Voxelated Images

As shown in Figure 27, the voxelization procedure used to calculate the network model of the system translates the original sphere pack image into a three-dimensional binary matrix, where ones represent grain space and zeros represent unoccupied pore space. It is therefore simple to calculate porosity of the resulting matrix based upon the number of ones and zeros present as

Numerical Porosity = Z Pore Voxels (7)

Total Voxels

This method of calculation is limited by the resolution of the matrix. At low matrix sizes, numerical porosity calculations oscillate due to poor representation of the spheres in the voxelization process.

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Since the voIxelization process attempts to represent the geometry of the system as a matrix, the spheres in the pack must be represented by smaller, cubical objects. Unless the resolution of the matrix is sufficiently high to accurately represent these surfaces, the overall geometry of the system will be inaccurately represented.

7.3 Image Resolution Study

Running the voxelization procedure to calculate the numerical porosity of the same image many times with different image resolutions (matrix sizes) provides a way to analyze the matrix size necessary to numerically obtain the directly calculated analytical porosity. At sufficiently high resolutions, the porosity of the image will stabilize to a single value. This value will closely correspond to the analytically derived result if sphere intersections are limited to mainly two spheres. However, if compaction is too great, the numerical value and the analytical value will not correspond. Figure 11 shows the comparison between these two calculated porosities. The numerical porosity at various matrix sizes is normalized against the analytically derived porosity for a given sphere pack.

0 0 0 0 Z 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0 50 100 15 0 200 250

3D Matrix Side Length

300 350 400

Figure 11. Numerical porosity calculations for sphere packs with varying image resolution.

21 * dat1c13y_973K80_rxyz . * dat1c1686y_873K40_rxyz A dat1e_100_rxyz.xs - dat4c62278y_873K40_rxyz t* to

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-Note that the normalized porosity for the data set 4c62278_873K40_rxyz does not normalize to 1. This is because this data set is an example of a well consolidated sphere pack in which more than two spheres intersect at any given time. The analytical solution for these data sets is incorrect and porosity must be evaluated using the numerical solution.

The degree of resolution in the numerical solution must be considered for porosity determinations. At larger image resolutions (matrix sizes), the numerical calculation of porosity will be better represented because the voxelization process is better able to represent the geometry of the sphere pack. However, the amount of time required to calculate this value also increases. Figure 12. Run time for porosity calculations versus resolution. Matrix size length3 gives the total number of voxels in the image shows the average run time for calculation of numerical porosity overall the data sets considered in this study. Data sets above 300 became impractical to run. Therefore, the resolution was limited to a matrix side length of 150 because the numerical porosity calculations in Figure 11 show very little variation at these resolutions, and the run time pictured in Figure 12 is low relative to higher length scales.

200

180

mAverage

Run Time

160 1140 v 120 u E P 100 -- 80 . 60 OW" E UO 40 8 Samu 202 0 50 100 ISO 200 250 300 350 400

3D Matrix Side Length

Figure 12. Run time for porosity calculations versus resolution. Matrix size length3 gives the total number of voxels in the image.

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8. Sphere Pack Properties

Sphere packs evaluated in this work were created using a mechanical stress simulation program designed by Dr. Yves Bernabe (13) from the MIT Department of Earth, Atmospheric and Planetary Sciences. This study evaluates two different sets of spherical grains undergoing both pressure solution creep and elastic compression at various temperatures, pressures, and time scales. Grains are initially randomly distributed within a confining box and consolidated to force grains to come into contact with one another. The grains are then relaxed and the process is repeated until the grains reach a stable configuration. Since friction and gravity are neglected within the system, the grains tend to dynamically rearrange during this process. Eventually, a stable configuration of grains is reached, and pressure dissolution and elastic compaction processes can be evaluated by changing the thermodynamic conditions of the system (volume of confining box, pressure, temperature, compaction method).

Sphere pack 1 contains 205 spherical grains, each with a radius of 40 +/- .3 microns with an initial box size of 460 microns. Sphere pack 2 contains 240 spheres, with radii of 40 +/- 2.7 microns and an initial box size of 490 microns. Each pack has an initial porosity of 43% corresponding to the expected porosity of a randomly distributed sphere pack. Pressure solution creep experiments are conducted at pressures between 0 and 120 MPa with temperatures of 500, 600, and 700 degrees Celsius over various time scales between roughly 40 and 60,000 years. Sphere packs subjected to elastic compaction (without pressure solution deformation) are also considered using confining pressures between 0 and 120 MPa.

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9. Permeability of Consolidating Media Using Lattice Boltzmann Models

All LBM simulations undertaken in this study used the Geonumerics code developed by Dr.

Christopher Leonardi from the MIT Department of Civil and Environmental Engineering. Sphere pack locations and radii were analyzed to determine permeability using the results of Lattice Boltzmann simulations. Each simulation used a single fluid phase with a kinematic viscosity of le-6 m2/s and a density of 1000 kg/m3 (i.e. water). Lattice was set set at 2e-6 m. The average and maximum velocity as well as the average drag force were calculated and used along with porosity, sample volume, and various fluid properties to calculate overall permeability of each sample according to equation 2 (see

page 4).

Initial simulations were undertaken with full cube side lengths provided by Yves Bernabe. Further investigation showed that the velocity among the edges was much higher than in the bulk of the sphere pack, which significantly changes the permeability of the overall sample. Sample dimensions were reduced by 10% on each side to remove these edge effects. A rescaled image of the sphere pack velocity field for pack

1c_101y

is shown in Figure 13. The velocity field around the edges is mostly uniform, allowing the velocity field within the bulk to dominate.

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The resulting calculated permeabilities resemble power laws as a function of porosity on a log-log scale (Figure 14. Log-log-log plot of Lattice-Boltzmann permeability as a function of porosity) suggesting that porosity strongly affects the transport properties of a given sphere pack. The power law relationship of the system at small porosities appears to vary from that at larger porosities.

10000.00 1000.00 100.00 10.00 y = 127547x3.8769 R2 = 0.9933 * Pack 4 Elastic * Pack 1 Elastic x Pack 1 Creep m Pack 4 Creep 1.00 0.0100 U x 0.1000 Porosity 1.0000

Figure 14. Log-log plot of Lattice-Boltzmann permeability as a function of porosity using models derived from both elastic and creep compressions on sphere packs 1 and 4.

25

E M.

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In sphere packs subjected to elastic compression, both sphere packs appears to follow a power law trend in a log-log relationship as shown in Figure 15. Log-log plot of elastic compression permeability as a function of porosity. 5000.00 4500.00 4 y = 116814x37.15 3 4000.00 a' 3500.00 3000.00 E * 2500.00 2000.00

+ Pack 4 Elastic A Pack 1 Elastic

1500.00 1000.00 1

0.1000

Porosity

Figure 15. Log-log plot of elastic compression permeability as a function of porosity

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Confining pressure as a function of porosity for both data sets appears to follow a linear trend as represented in the plot as seen in Figure 16. Elastic model permeability as a function of confining pressure.. Packs 1 and 4 appear to have similar slopes. Pack 4 is shifted down from pack 1 most likely due to the larger number of spheres in the system.

40 00 .00 , I I I I I I , , I I I , I I , -I I I , I I , I 3000.00 E 2000.00 .0 E W. 1000.00 60 80

Confining Pressure (Mpa)

100 120

Figure 16. Elastic model permeability as a function of confining pressure.

27 y = -11.232x + 4014.4 y = -11.36x + 3778. * Pack 4 Elastic A Pack 1 Elastic 0.00 0 20 40 140

-

- -

- -

- - -

-

-

-

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-For sphere packs subject to pressure solution creep, similar behavior is observed. Porosity and permeability on a log-log scale follow a power law relationship, but the behavior changes somewhat at low porosities. This behavior could be due to pore throats being cut off or due to passing the percolation threshold of the sphere pack. As Figure 17 shows, the various thermodynamic variables evaluated in these models do not seem to have as much of an effect upon permeability as porosity.

10000 1 1 1 1 . I I . . I I I I I I I I I y = 127215X3.8747 R2 = 0.9926 + 1.0000 0.1000 Porosity

Figure 17. Log-log scale of permeability as a function of porosity for sphere pack subject to pressure solution creep. 1000 E (U 0-100 10 * Pack 1 873K 40 MPa * Pack 1773 K 80 MPa * Pack 1873 K 80 MPa + Pack 1873 K 120 MPa A Pack 1973 K 80 MPa x Pack 4 873 K 40 MPa 0.0100 1 L

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However, the thermodynamic considerations do change the time scale at which this process occurs. Figure 18 shows how these various models set at different pressures and temperatures change permeability over time.

1000

E

Z 100 + Pack

1

Creep 873K 40 MPa

u -un+- Pack 1 Creep 773 K 80 MPa

E

a Pack 1 Creep 873 K 80 MPa

-+- Pack 1 Creep 873 K 120 MPa

10 -a- Pack 1 Creep 973 K 80 MPa

+ Pack 4.Creep 873K 40 MPa

1 10 100 1000

Time (yrs.)

Figure 18. Log-log plot of permeability as a function of time for sphere packs subject to pressure solution creep at various conditions

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10. Network Models of Consolidating Media

Analysis of the sphere packs using Lattice-Boltzmann provided a means to understand how absolute permeability varies under different conditions for the models. The pore network model provides us with a way to understand how the morphological properties of the sphere packs change and understand how the connectivity of the pores in the pack vary with different conditions. The combination of the LBM and the PNM provide a way to relate the changing morphology porous material to the permeability. These models were constructed using the procedure outlined in Section 6 and provide a way to interpret the resulting physical image of the sphere packs provided the centers and radii of the spheres are known for each iteration of the model.

Because the network model is essentially an image analysis model, it must be evaluated at sufficient resolutions to accurately capture the behavior of a given geometry. This model is also subject to a small amount of inaccuracy in the way it handles skeletonization because sphere centers are confined by the matrix which defines the voxel locations. To account for this behavior, the condition for skeletonization (that the inscribed sphere must touch a grain at two or more places) must be modified slightly. In other words, spheres can be considered part of the topological skeleton if they are less than a certain small distance away from a grain surface in more than two locations. Analyses run upon sphere packs with a known geometry show that the appropriate number to use in this case is .75 times the voxel size. The resulting sphere network for a simple cubic sphere packing using this parameter is shown in Figure 19.

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50 50 40 40 20 20 110 01 100 50 6060 so60

Figure 19. (Left) Inscribed skeleton spheres in a simple cubic sphere packing using a leniency factor of 0.75. (Right) Resulting network image of a simple cubic sphere packing.

Network models were created for all sphere packs for both elastic and pressure solution creep models. The models used a matrix size of 150 x 150 x 150 and a leniency factor of .75. Each model run produces a network image of the model as well as histograms of the coordination number and pore size distributions of the model sphere pack.

Models with more compaction exhibit very sparse network images and the connectivity of the pores (via the blue lines) portrays a general image of the permeability (Figure 20).

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150 100

4r

S

50 Ji 0 *a

I~

0 100 %nn 150 J1J

0

Figure 20. Network image of data set 4c_62278y. Data set modeled by sphere pack 4 and has undergone pressure solution creep for 62278 years. The sparse pores and low number of

interconnection (blue lines) are indicative of low porosity and permeability.

As expected, both the pore size and pore coordination number distributions reflect the small number of connected pores within the matrix (Figure 21).

Pore Coordination Number Distribution

2.5 3 35 4 45 5 Pore Size 70 60 50 0 40 30 10 160 -100 -'8

I

0 L 0 60 5 6

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Figure 21. Pore size and coordination number distributions for the network model of data set 4c_62278y.

This data set is modeled by sphere pack 4 and has experienced a simulated pressure solution creep over

62278 years.

Images experiencing smaller amounts of pressure solution or compaction show highly connected pore spaces with large pores in the network models (Figure 22).

150 100 0\

0

50 150 0 50 100

Figure 22. Network image of data set 4c_20y. Data set modeled by sphere pack 4 and has undergone pressure solution creep for 20 years. The dense pore spheres and high number of interconnection (blue lines) are indicative of high porosity and permeability.

The pore size and coordination number histograms of these models reflect the connectivity and large pore sizes reflected in the network images (Figure 23).

33

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450 260 400 -350 - 200 150 250

-~100

0 E 100 50 50 0 0 0 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12

Coordination Number PoreSize (s m)

Figure 23. Pore size and coordination number distributions for the network model of data set ic_341y. This data set is modeled by sphere pack 1 and has experienced a simulated pressure solution creep over 341 years.

In general, the data sets produced several distinct trends. As expected, pore coordination number as well as pore size closely correlates with porosity. The pore coordination number varies almost linearly with porosity for all sphere packs until porosities of about 0.3. Then, the coordination number drastically increases as the porosity approaches 0.4 as shown in Figure 24. At higher porosities, the pore throats which connect individual pores are large and the sphere pack is not well consolidated which allows for many pathways between the grains.

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8 7 .E 4 0 C 5 0 a 4 03 1 0 0 0.1 0.2 0.3 0.4 Porosity

Figure 24. Pore coordination number as a function of porosity.

The average pore size shows a linear correlation with porosity (Figure 25). This is expected since the pores are the main storage units within the pore space and thus control much of the porosity.

E 0 a, 10 9 8 7 6 5 4 3 2 1 0 0 0.1 0.2 Porosity 0.3 0.4

Figure 25. Average pore size as a function of porosity.

35 * Pack 1 Creep 873K 4 * Pack 1 Creep 773 K 8 + Pack 1 Creep 873 K 1 A Pack 1 Creep 973 K 8 * Pack 4 Creep 873K 4 - Pack 1 Elastic -Pack 4 Elastic ) MPa 0 MPa 20 MPa 0 MPa ) MPa -+0 0 .~~

~ ~

.*

o Pack 1 Creep 873K 40 MPa

* Pack 1 Creep 773 K 80 MPa * Pack 1 Creep 873 K 120 MPa

* Pack 4 Creep 873K 40 MPa

- Pack 1 Elastic

Figure

Figure 1.  Pressure dissolution along grain interfaces and  resulting transport and  re-deposition of grain material at free grain surfaces
Figure 2.  Fluid behavior  represented  at  different spatial scales. At microscopic scales  (left), fluids behave  as  a  series of discrete  particle collisions
Figure 3  D3Q15 three  dimensional lattice arrangement.
Figure  4.  2D  skeletonization  of a  circle pack
+7

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