impact of single fracture roughness on the flow, …...ii ii impact of single fracture roughness on...

Impact of Single Fracture Roughness on the Flow, Transport and Development of Biofilms

by

Scott A Briggs

A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy

Graduate Department of Civil Engineering University of Toronto

© Copyright by Scott A Briggs 2014

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Impact of Single Fracture Roughness on the Flow, Transport and

Development of Biofilms

Scott A Briggs

Doctor of Philosophy

Graduate Department of Civil Engineering

University of Toronto

2014

Abstract

This study examined the impact of systematically increasing roughness in a single fracture and

the effects on the hydraulic properties, solute transport and biofilm development in those

fractures. Biofilms were modeled using a newly developed two-dimensional Lattice-Boltzmann

Method (LBM) fluid flow model with the additional capability to simulate substrate transport

using a discrete Random Walk (RW) algorithm. The discrete modeling methods, including

LBM, RW and Cellular Automata (CA) for biofilm modeling, were able to capture small scale

effects that emerged into large scale behaviour of biofilm growth and structure development.

The two-dimensional fluid flow model using LBM was developed and validated against

analytical solutions for simple cases of parallel plate flow and a backward facing step. Fracture

flow results showed a pronounced deviation from predicted cubic law flow rates as expected and

previously reported in the literature. Two-dimensional fracture cross-sections were synthetically

produced to control the fracture roughness and results from the LBM model extended the three-

zone non-linear model of hydraulic behaviour from porous media to include fractured media.

Simulations with the LBM model showed that secondary flows, or flows not contributing to bulk

flow, could occur at Reynolds numbers lower than previously reported in the literature.

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A numerical solute transport method was added to the LBM flow simulations to model solute

transport in fractures of increasing roughness. Here the delay of breakthrough curves, including

initial, peak and tail were associated with the development and growth of eddies.

A biofilm growth model, implemented with a discrete CA method, was used to capture the local

small scale effects of fracture roughness, hydraulics and substrate transport on biofilm

development in terms of biomass and bio-structure. In two dimensions, biofilm growth was

controlled by clogging, which occurs earlier at lower biomass levels in rougher fractures.

Sensitivity analysis of biofilm development assuming a variation in bacteria shear strength was

completed. Lower biofilm shear strength does not allow for any biofilm development within the

fracture, however, when the shear strength is increased above a threshold, dependent on

Reynolds numbers and fracture roughness, biofilms begin to develop. Above the shear strength

threshold the same general trends of biofilm development hold compared to the results when no

sloughing due to shear is considered.

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Acknowledgments

I would like to thank my supervisors, Dr. Brent Sleep and Dr. Bryan Karney who have gone

above and beyond to ensure I have always had the support and encouragement needed during my

graduate work. Their ongoing feedback and guidance has been invaluable and is sincerely

appreciated.

I would also like to thank Dr. Giovanni Grasselli for his feedback and encouragement over the

years. Particularly, his insight and guidance with this document has made it stronger and more

complete. In addition I would like to thank him for letting me share offices with his students

who have also been a tremendous source of help and support.

My appreciation and thanks go out to Dr. Jennifer Drake and Dr. Sarah Dickson who provided

advice and feedback for my thesis and as a result have helped to make this document a well-

rounded and comprehensive body of work.

I would never have been able to begin or complete this journey without the support of my family,

including parents, parents-in-law and of course my wife, whom over the course of my studies

have always supported and encouraged me during successful times and difficult times and

without question allowed me to find myself along the way. I am truly fortunate to have their

genuine support allowing me to persevere in my studies.

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Table of Contents

CHAPTER 1 INTRODUCTION ................................................................................................. 1

1.1 Problem Statement ................................................................................................................................................ 1

1.2 Approach ................................................................................................................................................................ 3

1.2.1 Lattice Boltzmann Methods .......................................................................................................................... 3

1.2.2 Particle Tracking Methods ............................................................................................................................ 4

1.2.3 Biofilm Modeling ............................................................................................................................................ 6

1.3 Research Objectives .............................................................................................................................................. 7

1.4 Thesis Overview ..................................................................................................................................................... 8

CHAPTER 2 VALIDATION OF A NEWLY DEVELOPED MODEL FOR FLOW IN A

SINGLE ROCK FRACTURE ................................................................................................... 10

2.1 Introduction ......................................................................................................................................................... 10

2.2 Model Implementation ........................................................................................................................................ 14

2.2.1 Lattice Boltzmann Method .......................................................................................................................... 14

2.2.2 Boundary Conditions ................................................................................................................................... 16

2.2.3 Fracture Generation .................................................................................................................................... 17

2.3 Results and Discussion ........................................................................................................................................ 18

2.3.1 Flow Between Parallel Plates ...................................................................................................................... 18

2.3.2 Backward Facing Step ................................................................................................................................. 19

2.3.3 Flow in a Single Fracture............................................................................................................................. 20

2.4 Model Performance ............................................................................................................................................. 25

2.5 Conclusions .......................................................................................................................................................... 25

CHAPTER 3 QUANTIFICATION OF THE EFFECTS OF EDDY FORMATION ON

THE EFFECTIVE HYDRAULIC APERTURES IN ROCK FRACTURE FLOW ............ 26

3.1 Introduction ......................................................................................................................................................... 26

3.2 Methods ................................................................................................................................................................ 28

3.2.1 Flow modeling .............................................................................................................................................. 28

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3.2.2 Flow between Parallel Plates ....................................................................................................................... 30



3.3.1 Fracture Flow ............................................................................................................................................... 34

3.3.2 Tortuosity ...................................................................................................................................................... 42

3.3.3 Directionality ................................................................................................................................................ 44

3.4 Summary and Conclusions ................................................................................................................................. 44

CHAPTER 4 SOLUTE TRANSPORT IN SINGLE FRACTURES WITH INCREASING

ROUGHNESS ............................................................................................................................. 46

4.1 Introduction ......................................................................................................................................................... 46

4.2 Methods and Validation ...................................................................................................................................... 48

4.2.1 Fluid Flow Modeling in Fractures .............................................................................................................. 48

4.2.2 Solute Transport ........................................................................................................................................... 49

4.2.3 Model Validation .......................................................................................................................................... 51

4.3 Results .................................................................................................................................................................. 53

4.4 Sensitivity Analysis .............................................................................................................................................. 57

4.5 Conclusions .......................................................................................................................................................... 60

CHAPTER 5 EFFECTS OF ROUGHNESS AND SHEAR ON BIOFILM POPULATIONS

AND STRUCTURE IN A SINGLE ROCK FRACTURE....................................................... 61

5.1 Introduction ......................................................................................................................................................... 61

5.2 Model Implementation ........................................................................................................................................ 64

5.2.1 Biofilm ........................................................................................................................................................... 64

5.2.2 Substrate ....................................................................................................................................................... 66

5.2.3 Bulk Fluid Flow ............................................................................................................................................ 67


5.3 Biofilm Growth Model ........................................................................................................................................ 70

5.4 Timescales ............................................................................................................................................................ 72


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5.5.1 Biofilm with No Sloughing ........................................................................................................................... 74

5.5.2 Biofilm with Sloughing ................................................................................................................................ 80

5.5.3 Sensitivity Analysis ....................................................................................................................................... 84

5.6 Conclusions .......................................................................................................................................................... 93

CHAPTER 6 CONCLUSIONS AND RECOMMENDATIONS ............................................ 94

6.1 Overall Conclusions ............................................................................................................................................. 94

6.2 Contributions ....................................................................................................................................................... 97

6.3 Critical Appraisal ................................................................................................................................................ 98

6.4 Future Work ...................................................................................................................................................... 100

CHAPTER 7 BIBLIOGRAPHY ............................................................................................. 103

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List of Figures

Figure 2.1: Horizontal velocity profile comparing the analytical results of a Poisseuille

profile and the model results. ..................................................................................... 19

Figure 2.2: Flow, from left to right, over a backward facing step. Shown as red vertical

lines, the reattachment lengths are approximately 3, 4, 5 step heights for Re =

100, 150 and 200 respectively. The step height is half the downstream width.

Velocity is plotted with red representing the fastest velocities and blue the

slowest. The velocity profile is parabolic immediately upstream and far

downstream of the step while the zones outside of this region are omitted for

clarity. ........................................................................................................................ 20

Figure 2.3: The ratio of hydraulic aperture to mechanical aperture is plotted against

statistical roughness of the fracture as described by Renshaw (1995). The

model fits well with theoretical data. The model predictions are plotted from

a single fracture by increasing the mechanical aperture or dm. .................................. 22

Figure 2.4: The streamlines are plotted as the Re increases from 0.6 to 60. Secondary

flows develop in the form of eddies and grown to fill a larger cross-section of

the aperture. Each node is represents approximately 2 µm. ..................................... 24

Figure 2.5: Left hand side: Flow through a fracture. Right hand side: Flow through

parallel plates with the mechanical aperture equivalent to the fracture aperture

on the left. Relative velocity is plotted with yellow representing the fastest

velocities and dark blue the slowest. .......................................................................... 24

Figure 3.1: Fracture profiles b through i generated using a synthetic fracture generator

called SynFrac. Total fracture length is 100 mm and each fracture has a mean

aperture of 1.7 mm, only the fractal dimension (FD) variable is changed in

SynFrac. Fracture profile a represents a parallel plate system with an

equivalent 1.7 mm aperture. Fracture profile j represents a 16 mm long strip

from a dolomite fracture with mean aperture 0.1 mm. .............................................. 34

Figure 3.2: Flow streamlines in a fracture over a range of Reynolds number from 0.01 to

500. The fracture is a 2D slice of a 3D fracture generated in SynFrac with a

fractal dimension of 2.35. The segment shown has an overall dimension of

approximatly 1 mm2. .................................................................................................. 35

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Figure 3.3: Simulated flow streamlines in a fracture at a Reynolds number of 500. The

fracture is a 2D slice of a 3D fracture generated in SynFrac with a fractal

dimension of 2.35. The segment shown has an overall length of approximately

5 mm taken from the 100 mm long fracture simulated. ............................................ 36

Figure 3.4: Relative effective hydraulic apertures (ratio of effective to mean apertures for

each fracture respectively) for the dolomite and synthetic fractures with

varying roughness. ..................................................................................................... 38

Figure 3.5: Slope of effective aperture plots (Figure 3.4) for the dolomite and synthetic

fractures with varying roughness. .............................................................................. 39

Figure 3.6: Eddy volume for the dolomite and synthetic fractures with varying roughness. ...... 40

Figure 3.7: Flow streamlines (black lines with arrows) and the eddy volume that does not

contribute to bulk flow (thick red line). Cross-section shown represents

approximately 1.8mm2 from a segment of a SynFrac cross section with an

original fractal dimension of 2.35. ............................................................................. 41

Figure 3.8: Statistically similar synthetic fractures generated with SynFrac. Only the seed

of the pseudo random number generators is changed. ............................................... 42

Figure 3.9: Tortuosity for the dolomite and synthetic fractures with varying roughness. ........... 43

Figure 3.10: Tortuosity of statistically similar synthetic fractures generated with SynFrac.

Only the seed of the pseudo random number generators is changed. ........................ 44

Figure 4.1: Point source diffusion in 2D and the relative concentrations at a given radius

from the source. Results for time t = 1000, t = 2000 and t = 10000 are shown

with their respective analytical solutions. .................................................................. 51

Figure 4.2: Effective dispersion for the values: 0.0038 / and

0.0013 2/ after (Sukop and Thorne, 2005). The input values are given in

terms of lattice units (lu) and time steps (ts), typical for LBM applications. ............ 52

Figure 4.3: Breakthrough curves for 7 10 10 2 at Reynolds numbers 1

through 100 for synthetic fractures generated from a 2D slice of a 3D surface

with fractal dimensions (FD) 2.00 through 2.35. The ‘Slit’ represents a

parallel plate system modeled in the same way as all FD results; finally the

analytical solution for each case is shown for comparison. Concentration

profiles (C) are plotted relative to the total number of particles (M) and

normalized.................................................................................................................. 54

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Figure 4.4: Effective dispersion coefficients using data from the LBM and RW model

using the method of moments except for the analytical solutions with is

calculated from Equation 4.8. Data shown for 7 10 10 2 at

Reynolds numbers 1 through 100 for synthetic fractures generated from a 2D

slice of a 3D surface with fractal dimensions (FD) 2.00 through 2.35. ..................... 56

Figure 4.5: Breakthrough curves for 3.5 10 10 2 , 7 10 10 2

and 14 10 10 2 respectively at a Reynolds number of 50 for

synthetic fractures generated from a 2D slice of a 3D surface with fractal

dimensions (FD) 2.00 through 2.35. The ‘Slit’ represents a parallel plate

system modeled in the same way as all FD results, finally the analytical

solution for each case is shown for comparison. ....................................................... 57

Figure 4.6: Data shown is for Re = 50 for synthetic fractures generated from a 2D slice of

a 3D surface with fractal dimensions (FD) 2.00 through 2.35. Case 1 and 2 do

not meet the constraint for minimizing numerical dispersion. .................................. 58

Figure 4.7: For a set bin size when calculating the histogram, a larger number of particles

gives a more accurate description of the dispersion of particles through the

fracture without changing the overall behaviour. Data shown is for Re = 50

for synthetic fractures generated from a 2D slice of a 3D surface with fractal

dimensions (FD) 2.00 through 2.35. .......................................................................... 59

Figure 5.1: Fracture profiles b through i generated using a synthetic fracture generator

called SynFrac. Total fracture length is 100 mm and each fracture has a mean

aperture of 1.7 mm, only the fractal dimension (FD) input parameter is

adjusted in SynFrac. Fracture profile a represents a parallel plate system with

an equivalent mean 1.7 mm aperture. ........................................................................ 70

Figure 5.2: Main program loop which includes the processes of fluid dynamics, substrate

transport and biofilm growth. .................................................................................... 72

Figure 5.3: A representative sample of biofilm structure in a fracture. For the fracture

shown, Re = 50, FD = 2.35, Biofilm shear strength is 0.045 Pa. The plotted

segment is approximately 1mm of the total 100mm fracture. Blue represents

flow with streamlines plotted on top, green represent a biofilm cell and pink

represent locations where biofilms are permitted to develop. ................................... 74

Figure 5.4: Biofilm characteristics expressed by two different quantitative measurements:

relative FD on the left and relative biomass on the right. Values are relative to

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the initial FD and biomass of each respective fracture. Results are shown for

Re 1 through 100 and normalized time. ..................................................................... 76

Figure 5.5: Total biomass is plotted at time of a clogging event for fractures with FD

2.00, 2.15 and 2.35. Results for parallel plates are shown for reference. ................. 78

Figure 5.6: Biomass growth plotted against the relative hydraulic aperture as a measure of

hydraulic behavior in fractures with increasing roughness. Results shown for

Re = 1, 50 and 100. Each model is run until a clogging event negates the

usefulness of further hydraulic measurements. .......................................................... 79

Figure 5.7: Streamlines are plotted along a segment of the total fracture representing

approximately 1 mm of the model at Re = 50. Shear strength from left to

right, from top to bottom: 0.030, 0.035, 0.040, 0.045, 0.050, 40 Pa (similar to

no shear enabled in the model). Biofilm is shown in green while pink

represent locations where biofilms are permitted to develop. The plotted

results are shown at the time of a clogging event, or late-time for those shear

strengths that do not clog. .......................................................................................... 81

Figure 5.8: Results for relative change in FD for fractures with FD 2.15 and 2.35 shown

with the parallel plate case for comparison. For the case of Re = 1 all shear

strength values exhibit similar behaviour and follow the same trend. Biofilm

shear strength varies from 0.01 to 40 Pa with the case of no sloughing also

shown for comparison. Various biofilm shear strength values are highlighted

to emphasise the shift in the threshold growth values over increasingly rough

fractures. ..................................................................................................................... 83

Figure 5.9: Results for relative change in biomass for fractures with FD 2.15 and 2.35

shown with the parallel plate case for comparison. Biofilm shear strength

varies from 0.01 to 40 Pa with the case of no sloughing also shown for

comparison. Various biofilm shear strength values are highlighted to

emphasise the shift in the threshold growth values over increasingly rough

fractures. ..................................................................................................................... 84

Figure 5.10: Sensitivity of the biofilm growth behaviour to whether particles are re-

injected after being consumed by a bacteria cell. Re = 50 and FD = 2.35. ............. 85

Figure 5.11: Timescale sensitivity analysis for the case of Re = 50 and FD = 2.35. Results

are shown using for various Time Step (TS) ratios between successive steps. ......... 86

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Figure 5.12: Visualization of sensitivity of the timescale used between successive biofilm

iterations. Shown from top to bottom represent Time Step (TS) ratio of 100;

1,000; 10,000; 100,000 and 1,000,000. The segment of fractures shown

represents approximately 2 mm of the total 100 mm fracture. All five cases

are for Re = 50 and FD 2.35. ..................................................................................... 87

Figure 5.13: Biofilm FD and biomass results for the case of a TS ratio of 1,000,000 for

fractures with increasing roughness. .......................................................................... 89

Figure 5.14: Biomass growth as a percent increase plotted against the relative effective,

or hydraulic, aperture for the case of a TS ratio of 1,000,000. Presented

hydraulic apertures are normalized to unity for a relative comparison between

fractures of varying roughness. The bottom figure enables particle re-

injection relative to the top figure. ............................................................................. 90

Figure 5.15: Sensitivity analysis for diffusion coefficients in fracture with Re = 50 and

FD = 2.35. ................................................................................................................. 91

Figure 5.16: Sensitivity analysis of initial substrate concentrations in fracture with Re =

50 and FD = 2.35. ...................................................................................................... 91

Figure 5.17: Sensitivity analysis of reproducibility of the model in fracture with Re = 50

and FD = 2.35. ........................................................................................................... 92

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Chapter 1

Introduction

1.1 Problem Statement

Groundwater refers to water that is found throughout soils and in rock cracks, or fractures, is

usually found within 100 meters of the ground surface and plays a critical role in the

hydrological cycle. Groundwater is a crucial natural resource across the world, used as a source

of drinking water and for many residential, commercial and industrial processes. Specifically, in

Canada, 25% of the population rely on groundwater as a source of drinking water (Statistics

Canada, 2010) and overall groundwater accounts for 30% of the global fresh water supply

(Gleick, 1996). Undesirable substances, natural or anthropogenic in origin, are considered to be

contaminants and typically do not stay stationary but can travel significant distances and pollute

fresh drinking water sources. It then becomes necessary to address the sources of contamination

by removing polluting sources and rehabilitating or remediating contaminated sites.

In the field of contaminant hydrogeology various treatment techniques and technologies are used

to rehabilitate targeted sites around the world. Depending on the soil types, contaminant

properties and project objectives, solutions may include excavation, pump-and-treat, soil-vapour

extraction, thermal technologies or bioremediation. Without describing each method, it can be

generally said that each have their respective advantages and disadvantages and the end goal is to

remove as much pollution as possible. Commonly, contaminants have very low solubility in

water meaning that after some initial remediation, if trace amounts of the substance remain they

will slowly dissolve into the groundwater. Over time the substances continue to contaminate a

site and more significantly are likely to be transported downstream to a source of drinking water.

To address this long term contamination source, a similarly long term solution is needed and can

take the form of bioremediation.

Bioremediation takes advantage of the local bacterial populations in the soil to transform

contaminants to more inert forms. Bioremediation methods can include bioaugmentation, the

injection of new bacterial populations and biostimulation, stimulation of the growth of natural

populations by injecting food, or substrate. Bioremediation can be cost-effective over long

periods of time as the bacterial populations are self-sustaining or may require only minimal

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intervention to maintain. Conventionally, zones of high levels of bacteria are established and

take the form of a bio-barrier that acts to effect treatment of contaminants as the groundwater

moves through the bio-barrier.

Bacteria are often associated with and are divided into two forms, or phenotypes: planktonic and

biofilms. Planktonic bacterial refer to a free floating cell or group of cells in a fluid and biofilms

refer to bacteria that are attached to a surface. When bacteria form biofilms they are more

resilient to anti-microbial attack, they are more productive and can grow much faster. The

success of bacterial populations in the biofilm phenotype is such that the biofilm is considered to

be the predominant and preferred form of most bacterial species (Costerton, 2007). Therefore,

when trying to understand bioremediation at the scale of individual or groups of bacteria it

becomes a study of biofilms.

Bioremediation can be used in various soil types including fractured rock where water,

contaminants and biofilms are predominantly found within the rock fractures amongst the host

rock. To study the performance of bioremediation therefore requires the understanding of flow

and transport in fractures along with the behaviour of biofilms attached to the fracture walls.

Overall understanding of biofilm behaviour and groundwater environments is improved through

a three tiered approach: numerical modeling, lab scale testing and in-situ pilot studies. This work

concentrates on numerical simulations or modeling of biofilm in rock fractures to advance the

knowledge in the field.

Flow rates in rock fractures can become significant relative to typical porous media flow rates as

fractured media may develop zones of large apertures or high gradients or both. Gradients can

also be augmented artificially, for example during pumping at a well, and would significantly

alter the hydraulic behaviour in the surrounding media. It is not always known when flow may

become significant and when conventional fracture flow models like the cubic law breakdown

and a different approach is required. Similarly, when preferential flow paths open in fractures,

transport of solutes can be moved over significant distances but are still affected by fracture

geometry. Both the changing hydraulic and solute transport behaviours at increasing hydraulic

gradients can significantly affect the expected outcome of biofilm growth in terms of biomass,

structure and location.

The behaviour and development of the biofilm, which takes place at the micron to mm scale

(bacteria sizes are measured in microns) is expected to be strongly affected by the fracture

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aperture geometry. Significant changes in apertures or variations in roughness may considerably

alter the distribution of substrate within a fracture and therefore control the biofilm morphology.

To capture variation in aperture and roughness, numerical methods suitably able to resolve small

scale features in a single rock fracture are needed. Simulating biofilms within these fractures at

micron resolutions requires significant computational capability and practically limits the

modeling domain to a scale of millimeters or centimeters.

1.2 Approach

The intentions of this thesis are to examine the effects of fracture aperture geometry and its

associated roughness on three key aspects of biofilm development including bulk fluid

movement, substrate transport and biofilm development. Fluid flow within rock fractures is

modeled using a Lattice-Boltzmann Method (LBM) developed to resolve flows around the

unique geometry of fractures and biofilm colonies. Substrate transport is also expected to be

directly affected by aperture variations and a discrete Random Walk (RW) particle tracking

method is used capture these effects. Finally, a Cellular Automata (CA) approaches is used for

modeling biofilm development.

1.2.1 Lattice Boltzmann Methods Overall understanding of biofilm behaviour groundwater environments is improved through a

three tiered approach: numerical modeling, lab scale testing and in-situ pilot studies. Within the

scope of numerical modeling, understanding fluid flow in rock fractures remains an open

research question in the areas of contaminant hydrogeology, petroleum engineering and the long-

term disposal of nuclear waste. Conventionally, bulk flow rates in fractures have been modeled

as flow through an equivalent system of smooth parallel plates using the cubic law (see

Witherspoon et al. (1980) for a review of the early development of this law in fractures) where

flow is proportional to pressure gradient, with a proportionality constant, or transmissivity,

related to the cube of the aperture. However the cubic law approximation to flow in fractures is

not sufficient for all applications. For example, small scale numerical modeling of fluid

interactions within fractures may require the use of more comprehensive approaches. The small

scale fluid-fracture interactions are expected to play a significant role in diffusion and advection

of solutes and therefore affect biofilm substrate availability, populations and structure. Discrete

numerical methods such as the LBM can be effectively used to model rough fracture geometries

and capture small scale effects of fracture surfaces.

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LBM originates from the Lattice Gas Automata (LGA) methods. LGA methods are discrete in

space, time and particle velocity. Frisch et al. (1986) developed the first hexagonal grid, with

seven particle velocities that consisted of a lattice for which each node has six vertices connected

to other nodes. The seventh particle velocity came from the stationary case with zero velocity.

In Frisch’s model, there could be either 0 or 1 particle at any given node moving, or streaming,

in any direction. The collision step occurred when more than one particle occupied the same

node and the rules governing the collisions conserved mass and momentum before and after each

collision.

LBM evolved from Lattice-Gas Automata (LGA) to address some of its short-comings, the

primary being the Boolean treatment of particles at a node. Instead, LBM use a probability

distribution function to describe the nodal velocities and fluid momentum (Martys and

Hagedorn, 2002). In LBM the microscopic interaction of particles on a grid and the averaging of

those interactions emerge into the macroscopic continuum of a fluid.

LBM are essentially explicit finite difference approximations of the Boltzmann equation used to

describe flow (Eker and Akin, 2006) where the Boltzmann equation is a relationship that

describes the kinetics, or changes, of a thermodynamic system. The LBM are typically first

order accurate in time and 2nd order accurate in space depending on the implementation (Tolke,

2010). A popular approach to CFD modeling includes the use of the Navier-Stokes equations

which govern the motion of fluid by conserving mass and energy. Similarly to the LBM, the

Navier-Stokes equations are derived from the Boltzmann equations.

Other types of CFD start with the Navier-Stokes equations, which govern the macroscopic

movement of fluids, then discretize to get a solution to a system of partial differential equations

(Eker and Akin, 2006). The LBM, however, models the interaction of particles on a grid and

their emergent interactions which include two main steps: streaming and collision. The

streaming step is a translation of particles from one node on the grid to the next. The collision

step conserves momentum by redirection of particles which ‘collide’ or occupy the same node.

1.2.2 Particle Tracking Methods Particles within the systems are displaced via the processes of advection and diffusion.

Advection is calculated using the local velocity at the known particle coordinates while diffusion

is calculated using a discrete RW method. A random walk in space refers to the random step, or

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path, of a particle over time. With proper treatment of the particle paths and a sufficient number

of particles the resultant behaviour is similar to Brownian motion. Brownian motion refers to the

transport phenomenon describing the random movement of molecules in a fluid. The sum of

molecular collisions emerges as the physical process of diffusion at the large scale. The RW

group of methods have been developed and used extensively for the purpose of solute transport

in porous and fractured media (Ahlstrom et al., 1977; Tompson and Gelhar, 1990; Wels et al.,

1997; James and Chrysikopoulos, 1999; Delay et al., 2005; Nowamooz et al., 2013).

The discrete particles modelled in the RW process are used for simulating the transport of

dissolved substrate. Several simplifying assumptions are used for the modeling of dissolved

substrate in this work which also simplifies the numerical implementation and computational

cost. They are assumed to be neutrally buoyant and exhibit no decay or matrix diffusion.

Particle-particle interactions are not modeled nor do particles affect the flow solution. The only

forces acting on the particles are advective from the local fluid velocity and a diffusive process,

both of which are described by the following Fokker-Planck equation (James and

Chrysikopoulos, 2011):

∆ ∙ ∆ 0,1 ∙ 2 ∙ ∙ ∆ (1.1)

where Dm is the molecular diffusion coefficient, is the local velocity at the location x of the

particle at time t, 0,1 is normally distributed random number for each dimension i with mean

zero and a standard deviation of unity. For a more detailed development of the Fokker-Planck

approach the reader is referred to Delay et al. (2005) among other literature.

Numerical dispersion is minimized by ensuring ∙ ∆ ≪ ∆ (Tompson and Gelhar, 1990;

Hassan and Mohamed, 2003). A more strict limit is described by Maier et al. (2000) and further

constrains to ensure a particle moves a maximum of one half ∆ per time step.

∆ 6 ∆∆

(1.2)

With a sufficient number of particles a RW method can accurately model the process of

Brownian motion used to model diffusion. In general more than 100,000 particles are required

to sufficiently model diffusion using RW (Hassan and Mohamed, 2003).

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1.2.3 Biofilm Modeling Experimental and numerical studies of biofilms have been beneficial in developing and testing

theories of fundamental biofilm behaviours. Using micro-scale discrete numerical algorithms, the

current study examines the behaviour expressed by a biofilm developing in a fracture and

improves the understanding of the role of fracture geometry and flow rates in a single rock

fracture on biofilm growth patterns.

A CA approach to biofilm modeling is taken as CA algorithms can exhibit complex and chaotic

behaviour from simple evolutionary rules. They are based on local relationships and interactions

and have been shown to proficiently model local phenomena such as spikes or discontinuities in

population distributions. The state of a CA cell which is discrete in time and space describes

weather a bacterium is present or not at that location.

CA have been used to successfully model the emergent behaviours of biofilms including 2D

models (Wimpenny and Colasanti, 1997; Hermanowicz, 1998; Krawczyk et al., 2003; Indekeu

and Giuraniuc, 2004; Luna et al., 2004) along with 3D models (Picioreanu et al., 1998; Hunt,

2004; Eberhard et al., 2005). Continuous approaches can also be used for 3D biofilm modeling

(Alpkvist et al., 2006) and can simulate complex biofilm structures but fall short of the

heterogeneity shown by Hunt (2004) or biofilms observed experimentally (Lewandowski et al.,

1999). When modeling at the scale of a bacterium (~1 µm) a 3D cell of one cubic millimeter

will consist of a billion cells. At this scale computation costs quickly become prohibitive and

simplifications have to be made depending on the constraints and objectives of the project. For

example, 2D modeling reduces computation requirements while still being able to capture

biofilm behaviour however, it would not capture 3D channeling around a biofilm cluster for

example. As with any model, one must be cognisant of its capabilities and its limitations to

judge of the validity and applicability of the results.

The bacteria in the model are governed by simple life-cycle rules, these rules emerge into

complex behaviour suitable for modeling real systems. The bacteria or CA cells start their life-

cycle as dormant, inactive bacteria along the fracture wall; they are activated if substrate

becomes available locally and finally they consume the substrate and divide after a threshold

mass has been reached. Through modeling of biofilms we can strive to predict their behaviour

with the ultimate goal of understanding the factors that dominate their actions. Using

experimental work as a starting point, numerical biofilm models can be developed to mimic

7

these systems. Biofilms are an example of a system whose behaviour is described as emergent

where each bacterium has limited knowledge of the global state but as the system grows at a

local level it does so in a way that benefits the whole. CA can be programed under similar

conditions and is why CA models have effectively duplicated the mushroom shaped biofilm

clusters observed experimentally (Picioreanu et al., 1998; Hunt et al., 2003).

Biofilm modeling in this thesis is limited to a few key elements of a bacterium life-cycle

including substrate consumption for growth, randomization of the direction of growth and effects

of biofilm removal, or sloughing.

Sloughing of biofilm due to shear from the bulk fluid can be optionally applied to the biofilm

model and the effects of enabling this feature are studied. Biofilms are considered to behave as a

visco-elastic material however for the purposes of this study, they are assumed to be rigid with a

maximum shear strength above which sloughing occurs. When the maximum shear strength is

reached for a given cell in the model, the cell is removed which assumes the bacteria is flushed

from the system with no re-attachment downstream.

1.3 Research Objectives

In this thesis the impact of single fracture roughness on flow, transport and biofilm growth is

studied. The following research objectives were examined to develop an understanding of these

factors:

1. To investigate the effect of fracture geometry and roughness on flow in a single rock

fracture. To address this objective, a CFD code was needed capable of resolving small

scale variations in velocity throughout the fracture improving on traditional cubic law

approaches (Chapter 2).

2. To determine, quantitatively, the effects of fracture roughness and eddy formation on the

effective hydraulic aperture in rock fracture flow. To quantify these effects, synthetically

generated fractures allowing independent parameter adjustments, including roughness are

needed (Chapter 3).

3. To determine the effects of fracture geometry and roughness on solute transport in a

single rock fracture. Similar to the treatment of flow in fractures a particle transport

model which enables small scale local interactions is required to capture these effects.

8

Using the same synthetic fractures from Chapter 2 the effect of changing fracture

roughness can be analyzed (Chapter 4).

4. To investigate the effects of fracture roughness and biofilm shear strength on biofilm

population dynamics. A discrete modeling approach is required to capture local

interactions between bacteria, substrate and fluids. Analysis of biomass and fractal

dimensions of biofilm colonies were used to quantify the effects of roughness, hydraulic

behaviour and biofilm shear strength (Chapter 5).

1.4 Thesis Overview

This thesis consists of four central chapters that are intended to stand alone for the purpose of

submission to academic journals. Each chapter builds on the computational model of the

previous chapters and therefore contains some repeated background material on the numerical

methods used in earlier chapters.

Chapter 2 describes the model developed for 2D flow through rock fractures using the LBM

including background and theory leading to the development of the method. Validation of the

model is completed using simple case studies including flow between parallel plates and flow

over a backward facing step. Performance of the model is discussed and the importance of the

parallel implementation on General Purpose Graphics Processing Units (GPGPU) is established.

Cross sections from two real fractures are modeled and compared using a statistical measure of

roughness defined by the ratio of hydraulic and mechanical aperture.

Chapter 3 classifies the effects of eddy formation and growth in a single rock fracture under

increasing flow rates. Using synthetically generated fracture surfaces the appearance and growth

of eddies is described and a three-zone model of fracture flow is extended from porous media to

include fractured media. Tortuosity is also calculated and shown to follow a similar three-zone

model.

Chapter 4 adds a solute transport module to the LBM flow code. A discrete RW approach is

used to describe the process of diffusion while fluid velocities from the fracture are used to

advect the discrete particles. The RW method is validated against 2D analytical solutions for

diffusion while the combined LBM and RW model is validated using Taylor-Aris dispersion

between parallel plates. Breakthrough curves are presented and validated for parallel plate flow

9

using analytical solutions as comparison. Furthermore, breakthrough curves are reported for

varying flow rates and fracture roughnesses to characterize their respective effects.

Chapter 5 adds a biofilm growth module to the numerical code. A discrete CA approach is used

to describe the population dynamics of a biofilm. The effects of variations in flow rates, fracture

roughness, diffusion coefficients and biofilm shear strength are addressed. Sensitivity analysis

of the biofilm growth model is reported and discussed.

Chapter 6 details overall conclusions, contributions and possible future work.

10

Chapter 2

Validation of a Newly Developed Model for Flow in a Single

Rock Fracture

Abstract

Simulation of flow through rough walled rock fractures is investigated using the Lattice

Boltzmann Method (LBM) implemented on General Purpose Graphic Processing Units

(GPGPUs). The LBM model developed is an order of magnitude faster than published results for

LBM simulations run on modern CPUs. Hydraulic parameters and velocity profiles of an actual

rock fracture were calculated and compared to a smooth fracture of equivalent aperture as

predicted by the cubic law. Results showed that the applicability of the cubic law depends

highly on the fracture geometry with LBM model predictions deviating from cubic law

predictions from 10% to 50%. In particular, LBM models confirm that as the ratio of the mean

fracture aperture to the standard deviation of the aperture decreases, cubic law predictions

become increasingly inaccurate.

2.1 Introduction

Capturing the small scale behaviour in physical systems with grid spacing down to the micron

scale quickly leads to impractical computational requirements. Some modern computer

architectures have been developed, however, which are suitable for high performance parallel

computing. The challenge then becomes the implementation of conventional numerical

algorithms and methods on the new architecture. Once such architecture involves Graphics

Processing Units (GPUs) which have been used for decades for graphics within the PC

ecosystem. GPUs have evolved over time with more complex computing capabilities and are

now able to compile and run C based code and are more commonly referred to as General

Purpose GPUs or GPGPUs. Similarly, some algorithms are better suited for implementation on

parallel computing then others, factors include the method and scale of intercommunication

between nodes. As more intercommunication is required, particularly if global knowledge of the

system is required, a larger penalty is incurred because of the limited bandwidth between

computational units. Rather it is beneficial if the calculations at each node is as self-contained as

11

possible, minimizing inter-node communication. GPGPUs behave similar to Single Instruction

Multiple Data (SIMD) systems which represents a computing system that distributes the same

instruction set to many different processors. Considering this physical layout of GPGPU systems

requires the formulation of numerical methods where similar rules and equations are applied at

all nodes to effectively utilize the parallel nature of the hardware. Intuitively this encompasses

many physical processes where for example, the same equations of fluid movement is applied at

each node requiring only local knowledge of the system.

The Lattice Boltzmann Methods (LBM) are increasingly used for simulation of fluid flows in

complex geometries however their application to real engineering cases has been limited by the

required computing power. The local nature of LBM, where only next-neighbour cell

communication is required, lends to the methods suitability to parallelization. Previous work has

shown that an increase of an order of magnitude in performance can be expected when

implementing LBM on a Graphics Processing Unit (GPU) (Bailey et al., 2009; Tolke, 2010).

However such work did not show applicability and validation for flows in rock fracture of

interest to hydrogeologists.

LBM are types of numerical methods for solving Computational Fluid Dynamics (CFD)

problems. Other types of CFD start with the Navier-Stokes equations, which govern the

macroscopic movement of fluids, then discretize to get a solution to a system of partial

differential equations (Eker and Akin, 2006). In the LBM model the microscopic interaction of

particles on a grid and the averaging of those interactions emerge into the macroscopic

continuum of a fluid. These interactions include two main steps: streaming and collision. The

streaming step is a translation of particles from one node on the grid to the next. The collision

step conserves momentum by redirection of particles which ‘collide’ or occupy the same node.

LBM originates from the Lattice Gas Automata (LGA) methods. LGA methods are discrete in

space, time and particle velocity. Frisch et al. (1986) developed the first hexagonal grid, with

seven particle velocities that consisted of a lattice for which each node has six vertices connected

to other nodes. The seventh particle velocity came from the stationary case with zero velocity.

In Frisch’s model, there could be either 0 or 1 particle at any given node moving, or streaming,

in any direction. The collision step occurred when more than one particle occupied the same

node and the rules governing the collisions conserved mass and momentum before and after each

collision.

12

LBM evolved from Lattice-Gas Automata (LGA) to address some of its short-comings, the

primary being the Boolean treatment of particles at a node. Instead, LBM use a probability

distribution function to describe the nodal velocities (Martys and Hagedorn, 2002). In two

dimensions, nine velocity directions ei where i = 0,1,2...8 are sufficient to describe a continuum

fluid. Each node has 8 vertices and eo represents a particle at rest. The naming convention used

for LBM is DdQq, where d represents the dimension and q represents the number of vertices

(Sukop and Thorne, 2005). In this case the model would be D2Q9 for a two-dimensional grid, or

lattice, using nine vertices.

The velocity distribution function, f, represents the frequency of a particle occurring in any of the

nine discrete velocities. The frequencies correspond to the density of fluid in any given

direction. Therefore one can derive the macroscopic fluid density (ρ) to be the sum of all the

velocity distribution functions (Sukop and Thorne, 2005) as shown in Equation 2.1.

8

0ii

f (2.1)

Similarly, Equation 2.2 shows the macroscopic velocity u is an average of all the discrete

velocities weighted by the velocity distribution function, f.

8

0

1

ii

ei

fu

(2.2)

Using Equation 2.1 and 2.2 the microscopic quantities can be related to the desired macroscopic

velocity. The streaming is achieved in a similar method to the LGA, namely a translation of

particles however the collision rules are replaced with a continuous function. A popular collision

function is the Bhatnagar-Gross-Krook (BGK) model with a relaxation term (τ) used in Equation

2.3 (Succi, 2001). The velocity distribution function tends towards the equilibrium distribution

according by the BGK collision term Ω (Wagner, 2008).

)(1 eq

iff

i

(2.3)

where fieq is the local equilibrium value for the velocity distribution function in the direction of

link ei and varies depending on the lattice used. In the BGK model, the fluid tends towards

equilibrium at a rate governed by the relaxation term, τ (Latt, 2007). The BGK collision operator

13

expressed above along with the streaming step, which is a discretization of the Boltzmann

equation, is one of the simplest forms of the LBM and is summarized by Equation 2.4:

i

txi

fttti

exi

f ),(),( (2.4)

where x represents the position and t represents time. The function fi(x,t) is the original

distribution function at time t and fi(x+eiΔt,t+Δt) is the distribution function at time t+Δt. Over

that time, a LBM particle has moved a distance of eiΔt or to the next node in the direction of ei

(Brewster, 2007). The lattice velocity along each vertex varies such that each pseudo-particle

shall travel one node, or lattice unit, each time step. The local equilibrium distribution function

used in the BGK collision term is described by Equation 2.5.

22

23

4

2)(9

231)()(

c

u

c

ui

e

c

ui

ex

iwx

eqi

f (2.5)

where wi are weights (4/9 for i=0, 1/9 for i =1,2,3,4 and 1/36 for i =5,6,7,8) and c is the lattice

speed (Sukop and Thorne, 2005).

This study demonstrates a validated GPGPU code for simulating 2D laminar flow through rock

fractures using a D2Q9 LBM with a BGK collision model. For further development of LBM the

reader is directed to Succi (2001) and Sukop and Thorne (2005). LBM are essentially explicit

finite difference approximations of the Boltzmann equation and using a Chapman-Enskog

expansion, the Navier-Stokes equations for incompressible flow can be recovered (Eker and

Akin, 2006). The LBM are typically 1st order accurate in time and 2nd order accurate in space

depending on the implementation of the collision term (Tolke, 2010).

LBM methods, which originate from a CA structure, are efficiently parallelized in computer

programming due to the locality of the discretization. Each node is only concerned with its

direct neighbours and therefore when the lattice is distributed to parallel processors the only

required communication is at the sub-lattice boundaries (Martys and Hagedorn, 2002).

14

2.2 Model Implementation

2.2.1 Lattice Boltzmann Method

The model created is a 2D LBM using a BGK collision operator as previously discussed and

summarized in Equation 2.6:

)),(),((1

),(),( txeq

iftxftx

ifttt

iex

if

(2.6)

where the left hand side of the equation represents the streaming step and the right hand side

represents the collision step and τ is the relaxation parameter which governs the rate at which the

fluid tends towards equilibrium. For the LBM model presented τ takes the form:

213 L (2.7)

where νL is the numerical viscosity defined by the discretization of the system in lattice units.

The model runs on a GPGPU using a proprietary programming model developed by NVIDIA

called CUDA. Traditionally GPUs have specialized in graphics programming but the CUDA

model allows general purpose programs to run in parallel and become GPGPUs.

One of the drawbacks of GPGPU implementations is the discrepancy between 32-bit and 64-bit

floating point precision as current hardware has limited support for 64-bit, or double precision

calculations which are often a third or a quarter of single precision performance. Without double

precision calculations the likelihood of numerical error or the complexity required to compensate

for error increases. Error in CFD models is conventionally referred to as numerical dissipation

and describes the artificial dissipation of momentum in the fluid due to numerical error. Since

the LBM is essentially a finite difference approximation to the Boltzmann equation, it is subject

to the same numerical truncations as other finite difference methods. The numerical error can

cause dissipation of the advection term which by definition should be free of dissipation (Zhu et

al., 2006). The advection term in the LBM is represented by the streaming step or uniform

translation of data. Since the convection term is also treated in the same streaming step by LBM

(Yu et al., 2003), the LBM model presented along with other LBM models can run into

numerical difficulties.

15

To minimize the potential for numerical instabilities and maintain the second order accuracy of

the LBM, the model parameters are defined using the method laid out by Latt and Krause as part

of the OpenLB User Guide (2008). The process involves selecting physical units then

converting to lattice units to finally obtain the relaxation parameter τ. The relaxation parameter

plays an important role in the collision term of the LBM. It controls the tendency of the system

to move towards local equilibrium. In the literature, the relaxation parameter has been found to

cause numerical instabilities at values approaching 0.5 from the right hand side (τ must be

greater than 0.5 for physical viscosities). Stable values of τ close to unity are preferred for

simple implementation of the LBM and can be found using the method outlined below (Sukop

and Thorne, 2005; Sukop et al., 2013).

In this research water is the physical fluid being simulated with a kinematic viscosity, ν in a

fracture of aperture 2a and with physical velocity u. This leads to an expression for the Reynolds

number:

ua

2Re (2.8)

The dimensionless expression for Reynolds number is then used to convert from the physical

units of the system to lattice units. The fracture width is discretized into lattice nodes of length

δx with discrete time δt. In order to minimize the slightly compressible nature of the LBM and

second order accuracy the following constraints are used when determining system discretization

respectively:

3x

t

(2.9)

2~ xt (2.10)

The lattice viscosity (νL), is calculated based on the discretization of the system and the

dimensionless Reynolds number. Finally, the relaxation parameter is calculated according to

Equation 2.7 and is kept as close to unity as possible by adjusting the mesh size and maximum

lattice velocity. Lattice velocity is limited to a maximum 0.1 lattice units per time step which

arises from the approximations used in the LBM and its partial compressibility (Sukop and

Thorne, 2005)

16

2.2.2 Boundary Conditions

One of the distinct advantages of the LBM comes from its discrete nature. It is efficient for

modeling complex geometries (Chen et al., 1994; Eker and Akin, 2006; Lammers et al., 2006;

Brewster, 2007) which arises in the analysis of rock fractures. An array is stored that sets the

value of any point in the LBM grid to represent either a fluid cell or a solid boundary. At the

solid boundaries, a no-slip condition is used to create a zero velocity boundary along the surface.

A different set of collision equations are used at the solid boundary and are referred to as mid-

plane bounce-back boundary conditions (Succi, 2001). The name arises from the applied

boundary rules where particles entering a boundary at time t are sent back out with equal velocity

magnitude and opposite direction at time t+Δt this effectively puts the boundary at a distance

midway between a fluid and solid node.

Gravity driven boundary conditions are used to drive the fluid through the fracture. Solid, no-

slip boundaries are used along the fracture surfaces while periodic boundary conditions are used

on the left and right hand side of the model allowing the fluid being simulated to continually

wrap around the domain. Periodic boundary conditions simplify the simulation by creating an

infinite domain which removes any entry or exit effects and, in the simplest case of parallel plate

flow, allows the analytical Poiseuille velocity profile to develop in a much smaller domain

further reducing the computational load of the model.

For the present work fractures are considered to be vertical, thus, the force of gravity is added to

the velocity component parallel to the fracture, resulting in gravity acting along the primary

fracture axis. Gravity driven conditions are used according to the method described by Sukop

and Thorne (2005). Gravity driven flow acts on each cell of the LBM independently therefore it

is unnecessary to use conventional pressure or velocity boundary conditions as this would only

add an artificial constraint into the system, possibly creating entry or exit effects. The

acceleration due to gravity is converted to a velocity term:

dt

dummaF

(2.11)

where F is the external force added into the LBM calculations in the form of a local velocity.

The mass (m) is proportional to the density (ρ) and the relaxation parameter (τ) can be substituted

for time arriving at:

17

F

u (2.12)

where Δu represents a discrete velocity increment and is added to the velocity component

parallel to the fracture plane used to calculate the equilibrium distribution function.

2.2.3 Fracture Generation

Flow through a single rock fracture is modeled for two cases. The first, a one sided fracture

aperture collected by Boutt et al. (2006) and the second consists of a 2D slice through a fracture

generated in the laboratory.

The second data set was obtained from a dolomite block approximately 350 mm long, 250 mm

wide and 70 mm thick. The rock sample contained stylolites, which are planes of weakness,

parallel to the length of the rock. A fracture was introduced in the rock block using the method

described in Reitsma and Kueper (1994) resulting in final dimensions of 280x210x70 mm. A 3D

stereo-topometric measurement system, the Advanced Topometric Sensor (ATOS) II

manufactured by GOM mbH, determined the surface profile of the fracture walls and its aperture

distribution. For more details on the preparation of the sample and the ATOS II system see

Mondal and Sleep (2012) and Tatone and Grasselli (2009) respectively. A 16 mm 2D slice

through the 3D surface created by the ATOS II was used in the LBM. Using a 2D

approximation of the fracture to represent the 3D surface saves significant computational

resources. A 2D system cannot capture or quantify the effects of contact points in a fracture and

the impact of reducing effective apertures and increasing tortuosity (Zimmerman and

Bodvarsson, 1996). Tortuosity in fractures refers to the circuitous path a fluid particle will travel

due to the small and large scale roughness of a rock fracture. Despite this, 2D modeling is an

effective means of providing insight into the hydraulic behaviour of rough fractures.

The cubic law, which is conventionally used to describe flow through rock fractures, assumes the

fractures can be modelled as parallel plates and is used for comparison with the LBM. An

equivalent aperture, 2a, is required by cubic law calculations for flow and for the purposes of

comparison, the mechanical aperture of the two fractures being modeled is used. The flow

through parallel plates as describe by the cubic law is as follows:

L

hWa

gQ

3)2(

12 (2.13)

18

where Q is the flow rate, ν is the kinematic viscosity and 2a is the aperture. W is the width of the

fracture and in all simulations is taken as unity for the 2D models. L is the length of the fracture

and Δh represents the change in head over the length of the fracture. In the case of gravity driven

flow where gravity acts along the length of the fracture Δh = L and Equation 2.13 becomes:

3)2(12

ag

Q

(2.14)

The flow rates calculated by the cubic law are compared to those calculated by the LBM model.

2.3 Results and Discussion

2.3.1 Flow Between Parallel Plates

LBM model results of flow between parallel plates using incompressible fluids have been

compared with available analytical solutions. For laminar flow conditions, the Hagen-Poiseuille

equation can be used to describe the horizontal velocity through a cross-section:

)(

2)( 22 xa

Gxu

(2.15)

This analytical solution yields a parabolic velocity profile where 2a is the width between parallel

plates, x is the distance from the centerline and G is the driving force. For the case of gravity

driven flow G=ρg. The maximum velocity occurs at the centreline where x=0 and the average

velocity is 2/3 of the maximum velocity. Substituting for these changes gives:

2

3

a

ug avg

(2.16)

Using the non-dimensional Reynolds expression physical parameters can be converted to

equivalent lattice parameters. Lattice spacing is determined by the geometry and discretization

of the physical system. Lattice velocity is limited to a maximum 0.1 lattice units per time step

which arises from the approximations used in the LBM (Sukop and Thorne, 2005). Viscosity is

determined by constraining the relaxation parameter in Equation 2.7 to unity (τ = 1) to ensure

numerical stability.

The force of gravity in Equation 2.16 is used to drive flow in the model. For the case of parallel

plates, when the numerical model reaches steady state it compares well to the analytical solution

19

as shown in Figure 2.1. Figure 2.1 shows a horizontal velocity profile plotting the ratio of

velocity (u) to maximum velocity (Umax) at all nodes across the model domain. The parallel plate

boundaries are configured for a 256 node spacing. The only location where the Poisseuille

profile and model profile differ are at the two closest nodes to the boundary and this can be

attributed to the implementation of the bounce-back rules and is typical of the LBM BGK

approach.

Figure 2.1: Horizontal velocity profile comparing the analytical results of a Poisseuille profile and the model results.

2.3.2 Backward Facing Step

Geometry of a backward facing step consists of parallel plates with an upstream step which is

half the plate separation and a length at least 20 times the height to minimize interference from

outflow boundary conditions. Flow is gravity driven and the boundary conditions are periodic.

Upstream of the step, the flow field is given sufficient distance to develop a parabolic velocity

profile and similarly, far downstream of the step the flow becomes parabolic. Finally the flow is

gradually ramped back to the step height for the periodic boundary. All other boundaries use

standard bounce-back rules.

Backward facing step geometry is well studied but conventionally at higher Reynolds numbers

than is necessary for the study of flow in fractures. Results presented by Armaly et al. (1983) for

Re ≤ 200 are used with Re = 100 being the slowest of the available data. The relationship of the

dimensionless Reynolds number ( 2 ∙ ⁄ ) defines the flow where 2a is the characteristic

0

64

128

192

256

0 0.2 0.4 0.6 0.8 1

Node

u/Umax

Model

Poiseuille

20

length and the upstream height or step height is used. Next, u is the average inlet velocity and

finally, ν is the kinematic viscosity of water.

Generally, flow over a backward facing step at Reynolds numbers under 200 consists of three

segments: flow separation directly after the step consisting of an area recirculation followed by a

bulk flow reattachment point and finally development of a parabolic velocity profile downstream

of the step. The reattachment point refers to the end of the recirculation zone. Figure 2.2

illustrates three cases, Re = 100, Re = 150 and Re = 200 where flow is from left to right and

relative colouration from blue to red representing slow to fast fluid velocities. Reynolds

numbers are varied by adjusting lattice parameters, specifically the force of gravity for the

system. Using graphical means of measurement, the reattachment points of the model results are

in good agreement with those reported by Armaly et al. (1983) of 3, 4 and 5 times the step height

for the Reynolds numbers of 100, 150 and 200 respectively. In each case the velocity profile

becomes parabolic again far downstream.

Re = 100

Re = 150

Re = 200

Figure 2.2: Flow, from left to right, over a backward facing step. Shown as red vertical lines, the reattachment lengths are approximately 3, 4, 5 step heights for Re = 100, 150 and 200 respectively. The step height is half the downstream width. Velocity is plotted with red representing the fastest velocities and blue the slowest. The velocity profile is parabolic immediately upstream and far downstream of the step while the zones outside of this region are omitted for clarity.

2.3.3 Flow in a Single Fracture

Flow through a single rock fracture is modeled for two cases. The first, a one sided fracture

aperture collected by Boutt et al. (2006) and the second consists of a 2D slice through a fracture

generated in the laboratory.

For the first fracture data set, the cubic law deviates 8.4% from the actual flow rates determined

by the LBM model at Re = 6. Flow rates and therefore Re on the order of 6 and powers of ten

thereof are chosen based on the discretization of the fracture and the desire to maintain a

relaxation parameter close to unity. The fracture has an arithmetic mean of 359 µm which

21

translates into a fracture velocity of 1.6710-2 m/s for an equivalent parallel plate system. The

same 8.4% deviation from the cubic law is found at Reynolds numbers 0.06, 0.6, 6 and 60.

These results are in line with observations reported by other researchers where Brush and

Thompson (2003) found the cubic law to be within 10% of their Stokes Law simulations for

Reynolds numbers less than unity.

The second data set analyzed in our study has an equivalent aperture of 100 µm and at Re = 6,

the velocity through an equivalent parallel plate system is 5.9710-2 m/s. In this case the

deviation from the cubic law at Re = 6 is 50% and the same approximate deviation holds for Re

from 0.06 through 6. Again, variations in the literature can be found for example Brown (1987)

who used the Reynolds equation, which describes flow between slightly rough non-planar

surfaces, found the cubic law to hold within a factor of 2, while Tsang (1984) showed an order of

magnitude variation from the cubic law if tortuosity was ignored.

The two fracture data sets show different deviations from the cubic law which could be due to

their different physical attributes. The first data set consists of the lower profile of a fracture,

while the top profile is a smooth plate leading to a closer approximation of parallel plates then

that of the second data set where both sides are represented by a fracture profile. The equivalent

aperture is also much larger in the first data set, 359 µm versus 100 µm in the second data set

potentially affecting hydraulic properties and deviations from the cubic law.

The two data sets highlight the difficulty of using the cubic law for fracture flow as not all

fractures are within its approximations and shows the advantages of using the LBM model which

for example accounts for roughness or tortuosity intrinsically.

A measure of fracture roughness can be described in statistical terms by differentiating between

hydraulic and mechanical apertures. Conventionally they are considered equivalent when used

in the cubic law however, as discussed by Renshaw (1995), if a fracture aperture is described by

a lognormal distributed with mean B and variance σB2, then the respective calculations for

hydraulic and mechanical apertures vary. The expression relating these two quantities is as

follows:

2exp

2B

m

h

d

d (2.17)

22

where dh is the hydraulic aperture defined as the geometric mean and dm is the mechanical

aperture defined as the arithmetic mean. Zimmerman et al. (1991) and Renshaw (1995) defined

a roughness parameter as the non-dimensional ratio of mechanical aperture to standard deviation

σB2 of the fracture data:

21

1expexp

2exp

22

2

BB

B

B

md

(2.18)

Since both Equation 2.17 and 2.18 depend only on the variance of the lognormal aperture

distribution, they can be combined and are shown in Figure 2.3. The ratio of hydraulic aperture

to mechanical aperture tends towards unity as either the fracture aperture increases or the

standard deviation decreases as the walls become smoother. Experimental data by Zimmerman

and Main (2003) not shown on the graph fit well with the theoretical data. Other numerical work

by Patir and Cheng (1979) and Brown (1987) also compare similarly with the theoretical data.

The model predictions plotted in Figure 2.3 were calculated using a single fracture by increasing

the mechanical aperture (dm) and shows that, as the separation between the rock fracture walls

increases, the lognormal variance also changes.

Figure 2.3: The ratio of hydraulic aperture to mechanical aperture is plotted against statistical roughness of the fracture as described by Renshaw (1995). The model fits well with theoretical data. The model predictions are plotted from a single fracture by increasing the mechanical aperture or dm.

The main advantage of LBM models compared to other approaches is its ability to resolve small

scale effects such as an abrupt change in aperture where there is a significant change in velocity

streamlines and potential secondary flows. Figure 2.4 shows the flow streamlines calculated at

two different locations along the same fracture at three different Reynolds numbers. The first

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6 7 8 9 10

(dh/

d m)3

dm/σh

Theoretical

Model

23

location on the left is an area of a large change in aperture while the second location is of a small

depression in the fracture. Even at low Reynolds numbers, Re=0.6, the flow has zones of

recirculation, creating areas of the fluid that do not actively contribute to bulk flow. As the

Reynolds numbers are increased (6 then 60), the recirculation zones become larger and appear in

more places. These results demonstrate how the roughness of a fracture can affect fluid flow

within a fracture even at low Reynolds numbers and provide some hints to explain the

discrepancy between flow rates expected from the cubic law and results from the LBM.

Figure 2.5 compares the results from flow in the first data set compared to flow between parallel

plates with an equivalent mechanical aperture. The left hand side of Figure 2.5 consists of a rock

fracture along the base of the model with a no-slip smooth top boundary, constant gradient outlet

and parabolic inlet boundaries. The right hand side of Figure 2.5 models flow through parallel

plates spaced at an equivalent aperture calculated using arithmetic mean. It can be seen that the

actual rock fracture compresses the velocity profile much more than that of the equivalent

fracture. It is the peaks of the rock fracture that significantly change the velocity distribution,

leading to an apparently smaller equivalent aperture. The flow distribution is clearly different

from that predicted by simple parallel plates and although it cannot be seen in Figure 2.5, there

are areas of recirculation downstream of each fracture constriction (see Figure 2.4). Since this is

a complex phenomenon, it would be difficult to create a single variable that could be adjusted for

such effects. Rather, it is important that a given system be simulated with a model such as the

presented LBM model.

24

Figure 2.4: The streamlines are plotted as the Re increases from 0.6 to 60. Secondary flows develop in the form of eddies and grown to fill a larger cross-section of the aperture. Each node is represents approximately 2 µm.

Figure 2.5: Left hand side: Flow through a fracture. Right hand side: Flow through parallel plates with the mechanical aperture equivalent to the fracture aperture on the left. Relative velocity is plotted with yellow representing the fastest velocities and dark blue the slowest.

Re = 0.6

Re = 6

Re = 60

25

2.4 Model Performance

Performance of the presented LBM on the GPU is approximately an order of magnitude faster

than a comparable LBM model running on a CPU and is consistent with the findings of Tolke

(2010). Typically, performance in LBM codes in measured in Million Lattice Updates Per

Second or MLUPS, single CPU codes typically perform around 6.2 MLUPS (Bailey et al., 2009)

and more recently 88 MLUPS (Habich et al., 2013) while the GPU model in this study achieves

over 1000 MLUPS for a grid size of 2048 by 128 nodes using double precision calculations. The

comparison should be taken as a rough estimate as this is not intended to compare directly

between models which would require the equivalence of grid size, LBM implementation,

optimizations or other factors affecting the performance computer code.

2.5 Conclusions

The LBM model is well suited for simulating laminar flows through systems where complex

flow patterns are produced by the irregular boundaries found in rock fractures. Even under

laminar flow condition, tortuous flow paths and surface roughness create unique flow conditions

that the model in this study can effectively capture. The discussed model allows for the efficient

simulation and real-time rendering of fracture flow and is capable of simulating 2D systems at

the micron to millimetre scale. The GPU implementation of LBM can simulate systems in a

fraction of the time compared to CPU based codes, allowing for faster analysis and efficient

parametric studies. The LBM model presented in this study agree with other modeling of flow in

rock fractures (Tsang, 1984; Brown, 1987) and also fit well with the statistical roughness model

described by Zimmerman et al. (1991) and Renshaw (1995).

26

Chapter 3

Quantification of the Effects of Eddy Formation on the

Effective Hydraulic Apertures in Rock Fracture Flow

Abstract

The effect of eddy formation on flow in fractures of varying surface roughness was investigated

using Lattice Boltzmann simulations. Simulations were conducted for both statistically

generated hypothetical fractures and a real dolomite fracture. Simulation of flow in synthetic

fractures systematically investigated the effect of eddy formation on hydraulic conductivity with

increasingly rough fractures and Reynolds (Re) numbers ranging from 0.01 to 500. Complex

flow features, such as eddies, arising near the fracture surface were directly associated with

changes in effective hydraulic aperture. Eddies were identified in some fracture geometries at a

Re of 0.01, a value below the lowest previously reported as the minimum for eddy formation in

fractures. Rapid eddy growth above Re values of 1, followed by less rapid growth at higher Re

values, suggested a three-zone non-linear model for flow in rough fractures, similar to that found

for porous media by Chaudhary et al. (2011). This three-zone model, relating effective hydraulic

conductivity to Re, was also found to be appropriate for the simulation of water flow in the real

dolomite fracture. Not surprisingly, increasing fracture roughness led to greater eddy volumes

and lower effective hydraulic conductivities for the same Re values.

3.1 Introduction

Understanding fluid flow in rock fractures remains an open research question in the areas of

contaminant hydrogeology, petroleum engineering and the long-term disposal of nuclear waste.

Conventionally, bulk flow rates in fractures have been modeled as flow through an equivalent

system of smooth parallel plates using the cubic law (see Witherspoon et al. (1980) for a review

of the early development of this law in fractures) where flow is proportional to pressure gradient,

with a proportionality constant, or transmissivity, related to the cube of the aperture. Brown

(1987), using 2-dimensional simulations of the Reynolds equation, concluded that (i) the cubic

law, with various measures of aperture, could approximate flow through synthetically-generated

fractures to within a factor of 2, (ii) that the arithmetic average aperture gave better results than

27

the other averages they considered, and (iii) that corrections to the cubic law accounting for

tortuosity and contact area provided a better match to the Reynolds equation simulations. The

review article by Zimmerman and Bodvarsson (1996) discussed the use of various

simplifications to the Navier-Stokes equation including the lubrication equations and showed

that at low Reynolds (Re) numbers (< 1) the effective cubic law aperture was lower than the

actual aperture by a factor related to the ratio of the mean aperture and the aperture standard

deviation. They concluded that this ratio, or the geometric mean of the aperture, in combination

with a tortuosity correction factor could effectively predict hydraulic conductivities.

Brush and Thomson (2003) simulated the 3-dimensional Navier-Stokes equations in rough-

walled fractures and showed that for Re less than 1, deviation from the local cubic law (LCL)

were less than 10%. In experimental studies using magnetic resonance imaging Dijk et al.

(1999a, 1999b) found that the accuracy of the local cubic law depended strongly on the wall

roughness, with sharp discontinuities in wall profile producing complex flow patterns. Recent

experimental studies (Qian et al., 2011a) demonstrate a substantial deviation from the local cubic

law in rough fractures at Re below 150. Both Boutt et al. (2006) and Cardenas et al. (2009)

showed with simulations based on the Lattice Boltzmann Method (LBM) that fracture roughness

had a significant effect on transport, including a directional anisotropy. Experimental work by

Plouraboué et al. (2000) in self-affine rough fractures with various translations of the opposing

fracture surfaces indicated that heterogeneity in the flow field caused deviations from the parallel

plate model for fracture flow. The importance of fracture roughness, mean aperture, and

translation of fracture surfaces was also demonstrated by a perturbation analysis of tracer

dispersion by Roux et al. (1998).

At higher Reynolds numbers the pressure drop in fractures is nonlinearly related to the flow rate.

In this higher Reynolds number regime the Forcheimer quadratic equation can be used to

describe the relationship between pressure drop and flow rate (Yaarubi et al. in Faybishenko et

al., 2005). For such flows, eddy formation in rough walled fractures may become significant. In

fractured media, eddies have been deduced in fractures by nuclear magnetic resonance imaging

(Dijk et al., 1999; Dijk and Berkowitz, 1999), numerically in simplified single fracture

geometries (Qian et al., 2012) and in pore-and-throat geometries (Cao and Kitanidis, 1998;

Bouquain et al., 2012). Other numerical work by Yan and Koplik (2008) showed eddies in

fractures for non-Newtonian fluids. Cardenas et al. (2009) compared the predictions of eddy

formation between full Navier-Stokes and Stokes flow simulations. The simplification of Stokes

28

flow leads to eddy formation as a function of geometry but excludes inertial flows. Based on the

observations that the complex aperture geometry creates tortuous flow paths and increases the

length required for fluid to travel through the system, Zimmerman et al. (2004), while not

specifically addressing eddy formation, proposed a two-zone model of fracture hydraulics with

increasing Re. Chaudhary et al. (2011, 2013) investigated, through two-dimensional modelling

of the Navier-Stokes equations, the role of eddies in porous media and the reduction of hydraulic

conductivity as eddies grow with increasing Reynolds numbers. They asserted that eddies may

exist in porous media at all scales of flow causing deviations from Darcy flow and the

importance of modeling the unique pore geometry of different porous media to ensure accurate

results. Chaudhary et al. (2011) showed that bulk flow in porous media over a wide range of

Reynolds numbers is best described by a relationship characterized by three zones of differing

response of hydraulic conductivity to changes in Reynolds numbers. The first of these zones is

where traditional low Reynolds simulations hold and can predict flow in a fracture; the second

zone is where linearity breaks down and coincides with a rapid increase in eddy formation and

growth; the third zone is associated with a decrease in eddy growth rate, most likely associated

with geometrical constraints.

In fractured media, a rock fracture may be seen as a system of abruptly changing channel widths,

creating many regions of secondary flows such as eddies. Thus the results of Chaudhary et al.

(2013) suggest that fracture geometry, particularly roughness, significantly influences flow in

fractures. The current study systematically investigates the effect of fracture roughness on the

hydraulic conductivity of fractures over a wide range of Reynolds numbers. The investigation is

conducted using a 2-dimensional general purpose graphical processing unit (GPGPU) based

LBM to simulate water flow in synthetic and real fracture samples.

3.2 Methods

3.2.1 Flow modeling

When complex geometries and varying Reynolds numbers are present, a Computational Fluid

Dynamics (CFD) approach is required to capture the velocity distributions within a fracture (Dijk

et al., 1999). Only recently the advances in computational efficiency have allowed the execution

of CFD models on standard desktop computers and a number of commercially available finite

volume software packages are available to address fluid flow problems in hydrogeology

(Cardenas et al., 2007; Cardenas et al., 2009). Another CFD approach is the LBM (Boutt et al.,

29

2006; Eker and Akin, 2006; Yan and Koplik, 2008) which is a group of methods for simulating

fluid flow. LBM are based on the discrete Boltzmann equation from which the Navier-Stokes

equations can be recovered using a Chapman-Enskog expansion (Succi, 2001). LBM

intrinsically considers the unique boundaries of any given fluid regime and is used in this

implementation at Reynolds numbers approaching 500. LBM simulations do not require

computationally expensive meshing due to the local interactions and simple rule-based evolution.

When LBM is implemented on GPGPUs, the computational speed is usually increased by several

orders of magnitude with respect to the same code running on CPU (Bailey et al., 2009; Tolke,

2010). GPGPUs are widely-available components which are essentially powerful parallel

computers. The improved speed allows efficient completion of broad parametric studies while

modeling various fluid phenomena.

Comprehensive documentation of the development of LBM can be found in the literature (Succi,

2001; Sukop and Thorne, 2005; Latt, 2007). For the purpose of describing laminar flow in a

rock fracture a 2D LBM was developed using nine velocity components and the BGK collision

approximation. The two main components of LBM are the streaming step and collision step:

)),(),((1

),(),( txeq

iftxftx

ifttt

iex

if

(3.1)

where the left-hand-side of the equation represents the streaming step and the right-hand-side

represents the collision step. The velocity distribution functions, f, represent the statistical

movement of a fluid bundle along the nine velocity components, i. Direction links ei ensures

each fluid bundle moves a unit distance x each time step t. The relaxation parameter, τ, governs

the rate at which the fluid tends towards equilibrium defined by feq. For the LBM model

presented τ also represents the kinematic viscosity of the simulated fluid and must be larger than

0.5 to represent physical fluids:

(3.2)

where νL is the kinematic viscosity in lattice units.

The model runs on a Graphics Processing Unit (GPU) using a proprietary programming model

developed by NVIDIA called CUDA. Traditionally GPUs been used for graphics programming

but the CUDA model allows general-purpose programs to run in parallel on the GPU.

213 L

30

Periodic boundary conditions are used on the left and right hand side of the model, connecting

the left edge to the right, allowing the fluid being simulated to continually wrap around the

domain. This simplifies the simulation by creating an effectively infinite domain and removes

entry or exit effects associated with the development of a Poiseuille velocity profile. While

periodic boundary conditions may not be representative of fracture inflow conditions in the field,

they are intended to represent an elemental fracture segment that is part of a larger fracture

network. In the simplest case of parallel plate flow, this approach allows the analytical

Poiseuille velocity profile to develop in a much smaller domain further reducing the

computational requirements. Moreover, fluid in the fractures is acted on only by gravity which is

added (after Sukop and Thorne, 2005) to the velocity component parallel to the mean fracture

axis. Gravity driven flow acts on each cell of the LBM independently and pressure or velocity

boundary conditions are not used. The acceleration due to gravity (a) is converted to a velocity

term (u):

dt

dummaF (3.3)

where F is the external force added into the LBM calculations in the form of a local velocity. In

LBM, the mass (m) is proportional to the density (ρ) and the relaxation parameter (τ) can be

substituted for time (t) arriving at:

F

u (3.4)

where Δu represents a discrete velocity increment and is added to the velocity component

parallel to the fracture plane used to calculate the equilibrium distribution function in Equation 1.

3.2.2 Flow between Parallel Plates

The LBM model results of simulating flow between parallel plates using incompressible fluids

have been compared with analytical solutions. For laminar flow conditions, the Hagen-

Poiseuille equation can be used to describe the horizontal velocity (u) through a cross-section:

)(2

)( 22 xaG

xu

(3.5)

31

This analytical solution yields a parabolic velocity profile where 2a is the slot width, ν is the

kinematic viscosity, x is the distance from the centerline and G is the driving force. For the case

of gravity driven flow G=ρg. The maximum velocity occurs at the centreline where x=0 and the

average velocity is 2/3 of the maximum velocity. Substituting for these changes gives the driving

force for the LBM:

2

3

a

ug avg

(3.6)

Using the non-dimensional Reynolds expression physical parameters are converted to equivalent

lattice parameters:

avge

uaR

2 (3.7)

Lattice spacing is determined by the geometry and discretization of the physical system. Lattice

velocity is limited to a maximum 0.1 lattice units per time step which arises from the

approximations used in the LBM formulation to minimize compressibility effects (Sukop and

Thorne, 2005). To ensure numerical stability, the relaxation parameter, τ, typically has a value

of unity however, it can be reduced to maintain the limit on lattice velocities for higher Reynolds

numbers. Values of τ approaching 0.5 do introduce numerical error into to the simulation.

However, as shown by Sukop et al. (2013), the numerical error is relatively small compared to

the overall behaviour of hydrogeological systems. The force of gravity in Equation 3.6 is used in

Equation 3.4 to drive flow in the model. For the case of parallel plates, when the numerical

model reaches steady state it compares well to the analytical solution with a relative average

velocity error much less than 1% for Reynolds numbers up to 500 and lattice spacing down to 5

units wide (normal to the bulk flow direction).


A data set of fracture apertures was obtained for a dolomite rock sample approximately 350 mm

long, 250 mm wide and 70 mm thick. The rock sample contained stylolites, which are planes of

weakness, parallel to the length of the rock. A fracture was introduced in the rock block using

the method described in Reitsma and Kueper (1994) resulting in final dimensions of 280x210x70

mm. A 3D stereo-topometric measurement system, the Advanced Topometric Sensor (ATOS) II

manufactured by GOM mbH, determined the surface profile of the fracture walls and its aperture

32

distribution. For more details on the preparation of the sample and the ATOS II system see

Mondal and Sleep (2012, 2013) and Tatone and Grasselli (2009) respectively. A 16 mm 2D grid

slice through the 3D surface created by the ATOS II was used in the LBM model and is shown in

Figure 3.1. Using a 2D approximation of the fracture to represent the 3D surface saves

significant computational resources. A 2D system cannot capture or quantify the effects of

contact points in a fracture and the impact of reducing effective apertures and increasing

tortuosity (Zimmerman and Bodvarsson, 1996). Despite this, 2D modeling is an effective means

of providing insight into the hydraulic behaviour of rough fractures.

In addition to modeling flow in a segment of the dolomite rock fracture, flow was modeled in

synthetic fractures. Synthetic fracture generation creates systems with controlled surface

properties. To quantify the effects of surface roughness, a series of similar fractures with

increasing roughness was generated with the software package SynFrac developed by Ogilvie et

al. (2006). Ogilvie expands on earlier work (Brown et al., 1995; Glover et al., 1998a; Glover et

al., 1998b) to capture the complex nature of natural fractures with synthetic approximations.

Glover and Hayashi (1997) demonstrated that modeling a synthetic fracture at the centimeter

scale applied directly to field flow measurements at the 100 meter scale. An important

consideration when generating synthetic fractures is capturing the fracture properties at all

wavelengths. The top and bottom of a single fracture will have correlated geometry and surface

properties at long wavelengths but are mostly independent at short wavelengths. The threshold

separating long and short wavelength is called the mismatch length. SynFrac has multiple

methods for determining the mismatch length, for the purpose of this study the SynFrac

implementation of the Brown et al. (1995) mismatch length is set to 15 mm.

Using SynFrac for 2D fracture generation, two studies were conducted. First a series of fractures

were generated with increasing roughness determined by the fractal dimension input parameter.

Second, to analyse random variations that may occur in the fracture surfaces generated by

SynFrac, multiple fractures with identical characteristic parameters were created by only varying

the seeds of the Park and Miller pseudo-random number generator (SRNG) in SynFrac. A 100

mm 2D profile is manually extracted from the data and selected such that it has no contact point.

Since each 3D fracture generated with SynFrac is adjusted so the relative separation of the top

and bottom surface creates a single contact point, an equivalent adjustment was needed in 2D for

consistency between fracture studies. Therefore the arithmetic mean aperture of each fracture

was kept constant for each study by manually adjusting the profile separation. Attention is also

33

paid to the entry and exit of the fracture profile to ensure no interference with the periodic

boundary conditions for fluid flow. Other SynFrac settings are kept constant including the

resolution (1024x1024), standard deviation (1 mm) and anisotropy factor (1.0). The fracture

length of 1024 elements is expanded to a grid length with 2048 elements using interpolation,

resulting in a 48.8 micron element resolution.

Fractures were generated by specifying a fractal dimension for the 3D surface in SynFrac

ranging from 2.00 to 2.35. However, the use of a fractal dimension for defining roughness is

incomplete as fractal dimensions are not unique to an object, two similar but unique objects may

have the same fractal dimension. It has also been shown that the direction of flow in a fracture

yields varying results (Boutt et al., 2006) whereas the fractal dimension of a surface is

independent of the direction of measure. Some recent work (Tatone and Grasselli, 2009; Tatone

and Grasselli, 2010) has developed a roughness parameter used for measuring shear resistance in

rock fractures based on angular thresholding of fracture surfaces. The concept of a shear based

roughness translates well in fluid mechanics as wall shear stress compounded by the roughness

of a fracture results in drag against the bulk flow. The roughness is calculated for each surface

according to Tatone and Grasselli (2010) then an average taken to represent both fractures with a

single parameter. Larger values represent a larger roughness and were calculated for each

direction and flow modeled. Fractures with roughness between fractal dimensions of 2.00 and

2.35 were used and compared with an equivalent parallel plate system and a real dolomite

fracture (Figure 3.1). Ogilvie et al. (2006), developers of SynFrac, used complementary

software, ParaFrac, to analyse real rock fractures and found that sandstone and granodiorite

samples had fractal dimensions approaching 2.35.

34

Figure 3.1: Fracture profiles b through i generated using a synthetic fracture generator called SynFrac. Total fracture length is 100 mm and each fracture has a mean aperture of 1.7 mm, only the fractal dimension (FD) variable is changed in SynFrac. Fracture profile a represents a parallel plate system with an equivalent 1.7 mm aperture. Fracture profile j represents a 16 mm long strip from a dolomite fracture with mean aperture 0.1 mm.


3.3.1 Fracture Flow

The LCL uses the mean aperture to represent the hydraulic aperture of a fracture; however, the

roughness of the fracture causes deflections and separation of streamlines and bulk fluid

movement resulting in regions of secondary flow which could be described as eddies or as

resulting from eddies. Depending on the geometry, eddies will appear at all scales of the fracture

and all scales of flow as they are inherent to the complex geometries of rock fractures as shown

in Figure 3.2. An effective hydraulic aperture can be defined as an aperture that would result for

given flow rates and a system of parallel plates. The effective hydraulic aperture represents a

fraction of the aperture contributing to bulk flow, the remaining aperture is associated with

secondary flows. The secondary flows can be seen in Figure 3.3 to take the form of eddies or the

resulting detached streamlines downstream resulting from eddies. Secondary flow systems are

found to exist up to the mismatch length set in SynFrac to 15 mm for the models shown.

a) parallel plate

b) FD 2.00

c) FD 2.05

d) FD 2.10

e) FD 2.15

f) FD 2.20

g) FD 2.25

h) FD 2.30

i) FD 2.35

j) dolomite

fracture

35

Figure 3.2 shows a small 1 mm segment of a 100 mm fracture with fractal dimension 2.35 using

the LBM model developed for this work. At low Reynolds numbers (Re = 0.01), some regions

of the fracture show eddies that will grow to occupy a significant portion of the system Figure

3.3 represents a 5 mm segment of the fracture where secondary flows are evident and take the

form of eddies or disconnected streamlines caused by eddies.

Figure 3.2: Flow streamlines in a fracture over a range of Reynolds number from 0.01 to 500. The fracture is a 2D slice of a 3D fracture generated in SynFrac with a fractal dimension of 2.35. The segment shown has an overall dimension of approximatly 1 mm2.

36

Figure 3.3: Simulated flow streamlines in a fracture at a Reynolds number of 500. The fracture is a 2D slice of a 3D fracture generated in SynFrac with a fractal dimension of 2.35. The segment shown has an overall length of approximately 5 mm taken from the 100 mm long fracture simulated.

Eddies affect the local area where streamlines detach from the bulk flow then re-attached at a

location downstream depending on geometry and the Reynolds number (Armaly et al., 1983).

The areas of detach flow reduce the effective hydraulic aperture as they do not contribute to bulk

flow. In this work, instead of describing eddy locations, the local velocities from the LBM

model are used define an effective hydraulic aperture. Using the local velocity information and

exact aperture distribution, flow conditions are calculated using the LBM. First a gravity driven

flow cubic law is derived. Starting with Equation 3.6 rearranged for aperture:

g

ua avg

6

2 2 (3.8)

where 2a is the effective hydraulic aperture, ν is the kinematic viscosity, g is gravity and uavg is

the average velocity. The velocity at any given cross section is given by:

aW

Quavg 2

(3.9)

37

where Q is the flow rate and W is the width of the fracture (kept at unity for the 2D case studied).

Substituting Equation 3.9 into Equation 3.8 gives and equation for effective hydraulic aperture,

2a.

3/1

122

gW

Qa

(3.10)

Using the cubic law in this way an effective hydraulic aperture is calculated using the known

flow rates from the LBM model.

For this study SynFrac was used to create fractures of various roughnesses but with statistically

similar properties including an equivalent mean. The fractal dimension was varied between 2.00

and 2.35, a slice was taken from the 3D fracture create by SynFrac and the mean was adjusted to

match all other samples facture. Finally an equivalent roughness is calculated based on work by

Tatone and Grasselli (2010) and summarized in Table 3.1.

Table 3.1: Comparison of Roughness Parameters Fracture Type 3D Fractal Dimension Angular Threshold

Synthetic Fracture(using SynFrac)

2.00 9.62 2.05 10.72 2.10 12.41 2.15 14.43 2.20 16.73 2.25 19.79 2.30 23.15 2.35 27.29

Dolomite Fracture Not Applicable 8.09

Each fracture is modeled using the LBM at Reynolds numbers between 0.01 and 500. The

results show that all fractures exhibit approximately constant effective aperture at Re < 1. At Re

> 1 the effective aperture begins to decrease. The rougher synthetic fractures, while having the

same mean, have smaller effective hydraulic apertures than that of a smooth fracture, indicating a

larger fraction of the aperture contributing to secondary flows such as eddies. As the Reynolds

number increases, the rougher fractures show an increased rate of reduction of effective aperture

compared to smoother fractures. Figure 3.4 illustrates the first two zones of flow in fractures: in

Zone I at Re < 1 effective aperture is constant but dependent on initial fracture geometry; Zone II

begins at Re approaching 1 where conventional fracture modeling will break down.

38

Previous studies in porous media (Chaudhary et al., 2011) and in fractured media (Zimmerman et

al., 2004) have also shown a multi zone behaviour of hydraulic properties. Chaudhary et al.

(2011) reported the transition beginning at Reynolds numbers around 1 for porous media while

Zimmerman et al. (2004) reported a transition zone beginning at Re = 1 and becoming

significant around Re = 10.

Figure 3.4: Relative effective hydraulic apertures (ratio of effective to mean apertures for each fracture respectively) for the dolomite and synthetic fractures with varying roughness.

As Reynolds numbers continue to increase eddy growth is constrained by the increasing flow

rates being driven through the fracture. Figure 3.5 shows the rate of change of effective aperture

with change in Re (slopes of lines in Figure 3.4). The reduction in eddy growth rate represents

the boundary between Zone II and Zone III at approximately Re = 30 for the fractures generated

by SynFrac and shown in Figure 3.5. The dolomite fracture shows eddy growth at large

Reynolds numbers however the dolomite fracture still shows a three zone non-linear effective

hydraulic aperture relationship with Reynolds number.

10-2

10-1

100

101

102

103

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Reynolds Number

Rel

ativ

e E

ffec

tive

Ape

rtur

e

FD 2.00

FD 2.05FD 2.10

FD 2.15

FD 2.20

FD 2.25

FD 2.30FD 2.35

Dolomite Fracture

39

Figure 3.5: Slope of effective aperture plots (Figure 3.4) for the dolomite and synthetic fractures with varying roughness.

Absolute results between SynFrac and the dolomite fracture show some variation possibly due to

differences in geometry and surface characteristics. Each synthetic fracture represents 100 mm

of total length while the lab generated fracture is approximately 16 mm. Mean of the synthetic

fracture aperture is 1.7 mm while the real fracture is 0.1 mm. The surface characteristics were

used in SynFrac with no attempt to duplicate those of the dolomite fracture. The dolomite

fracture data shows a rebound in eddy growth (Figure 3.5) beginning at a Reynolds number of

100. The eddy growth rate would still be limited by fracture geometry and growth rates would

slow down until the fluid transitions to a more turbulent regime.

The effective hydraulic aperture, calculated from the LBM results, represents the region of the

fracture contributing to the bulk flow and is some fraction of the mean aperture. In 2D this

fraction also represents the effective volume of the fracture contributing to bulk flow. The

remainder of the fracture, defined as the eddy volume, contains complex flows, regions of

streamline separation and eddy formation (Figure 6). The eddy volume is similar to the ratio of

eddy to total volume in Chaudhary et al. (2011). Chaudhary et al. (2011) define eddy growth

from frictional drag calculations arising from CFD simulations whereas in this work the bulk

flow and eddy regions are explicitly defined by calculating an effective aperture (Equation 3.10).

10-2

10-1

100

101

102

103

-0.018

-0.016

-0.014

-0.012

-0.01

-0.008

-0.006

-0.004

-0.002

0

Reynolds Number

Slo

pe o

f E

ffec

tive

Ape

rtur

e P

lots

FD 2.00FD 2.05

FD 2.10

FD 2.15

FD 2.20FD 2.25

FD 2.30

FD 2.35Dolomite Fracture

40

Figure 3.6: Eddy volume for the dolomite and synthetic fractures with varying roughness.

The flow in the region associated with eddies is considered negligible relative to bulk flow;

however, its proximity to the boundary is important to many engineering problems. This region

would be expected to contribute to the extended breakthrough curves in the field of solute

transport. It is the region where biofilms will develop and biodegrade contaminants, and it is

also a region from which matrix diffusion would occur.

Figure 3.7 illustrates a small fracture segment showing the local effective hydraulic aperture and

how it decreases with increasing Reynolds numbers. To estimate the location of the velocity

streamline that separates bulk flow and secondary flows, a threshold is determined from the

velocity streamline data in the fracture. First, the average velocity for the entire fracture is

calculated, then any node with local velocities less than the product of the average velocity and

the eddy volume ratio (Figure 3.6) is considered to be below the velocity threshold contributing

to bulk flow; the boundary between secondary flows and the bulk flow is highlighted by a thick

red line. The local eddy volume estimate is found to accurately place the threshold streamline

within one or two LB nodes.

10-2

10-1

100

101

102

103

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Reynolds Number

Edd

y V

olum

e

FD 2.00FD 2.05

FD 2.10

FD 2.15

FD 2.20FD 2.25

FD 2.30

FD 2.35Dolomite Fracture

41

Figure 3.7: Flow streamlines (black lines with arrows) and the eddy volume that does not contribute to bulk flow (thick red line). Cross-section shown represents approximately 1.8mm2 from a segment of a SynFrac cross section with an original fractal dimension of 2.35.

The effective hydraulic aperture in Figure 3.7 illustrates the separation between the bulk flow

and secondary flow regions and the changing behaviour with changing Reynolds numbers. The

fracture surfaces act as a series of backward facing steps with many overlapping detachment and

re-attachment locations. The centre regions of the streamline plots in Figure 3.7 graphically

represent the effective hydraulic aperture presented in this work. The effective hydraulic aperture

is directly constrained by the appearance and growth of eddies. Even at the lowest Reynolds

42

numbers, complex fracture surfaces introduce boundary layer effects which would cause

deviations relative to flow calculated for a parallel plate system.

To examine the variation of results for fractures with the same statistical characteristics, five

different SynFrac fractures were created with SRNG with the same statistical properties but with

different seeds. Results showed some variation as expected, however the overall behaviour is

consistent with the three-zone model presented earlier.

Figure 3.8: Statistically similar synthetic fractures generated with SynFrac. Only the seed of the pseudo random number generators is changed.

3.3.2 Tortuosity

Using the local velocity modeled by the LBM a tortuosity value was calculated using the actual

flow path in the fracture. Brown et al. (1998) defined tortuosity as the ratio of actual fluid path

to the total projected length of the fracture. Crandall et al. (2010) used advective particle

tracking to average the fluid path of over a hundred simulated particles. When the entire velocity

profile of the fracture is known a more detailed approach can be used that traces the actual path

of the fastest moving fluid streamline and determines its length which is divided by the actual

fracture length to calculate tortuosity (Skjetne et al., 1999). The fastest streamline is assumed to

represent the natural tortuosity of the bulk flow. The path of fluid streamlines changes as seen in

Figures 3.2 and 3.6 due to both roughness and Reynolds numbers. The results in Figure 3.9

10-2

10-1

100

101

102

103

0.76

0.78

0.8

0.82

0.84

0.86

0.88

Reynolds Number

Rel

ativ

e E

ffec

tive

Ape

rtur

e m

Replicate 1

Replicate 2

Replicate 3Replicate 4

Replicate 5

43

show a trend of increased tortuosity resulting from increased roughness similar to that reported in

previous work (Tsang, 1984; Brown, 1989; Crandall et al., 2010).

Tortuosity’s dependence on the Reynolds numbers is more complex and CFD approaches are

required to make this determination. Initially flow paths are determined by the geometry, or

roughness, of the system. Then, as Reynolds number increases, eddies form and grow in regions

of abrupt aperture change perturbing existing streamlines and in turn increasing tortuosity. The

complex interaction leads to a non-linear relationship between Reynolds number and tortuosity

(Figure 3.9). General behaviour follows a three-zone trend similar to the effective hydraulic

aperture. Zone I is constant at Re < 1 with a transition to a non-linear relationship in Zone II and

III. The replicate synthetic fractures (Figure 3.10) show a more pronounced shift from Zone II to

Zone III as tortuosity increases significantly. Although the eddy growth rate in Zone III is

reduced, eddies are at their largest and could explain the significant increase at the highest

Reynolds numbers.

Figure 3.9: Tortuosity for the dolomite and synthetic fractures with varying roughness.

10-2

10-1

100

101

102

103

1.03

1.035

1.04

1.045

1.05

1.055

1.06

1.065

1.07

1.075

Reynolds Number

Tor

tuos

ity

FD 2.00

FD 2.05FD 2.10

FD 2.15

FD 2.20

FD 2.25

FD 2.30FD 2.35

Dolomite Fracture

44

Figure 3.10: Tortuosity of statistically similar synthetic fractures generated with SynFrac. Only the seed of the pseudo random number generators is changed.

3.3.3 Directionality

A measure of roughness was chosen that could account for any directional anisotropy in the

fracture. For Reynolds numbers less than 100, the vast majority of groundwater flows, flow rates

in the fractures generated by SynFrac showed less than 1% variation when reversing flow

direction. The random creation of fracture surfaces in SynFrac does not seem to create

directionally dependent fractures with the default program settings as roughness also varied by

approximately 1% (for the same SynFrac fracture between the forward and reverse directions).

Directional dependence becomes a factor when large scale fracture features are present causing

differentiation in flow streamlines. Large backward facing steps would be an example of a

geometry creating directionally sensitive results.

3.4 Summary and Conclusions

The following assertions arise from this study:

1. Eddies may be present at all scales of flow in fractures, extending below previously

reported Reynolds numbers in the literature for eddy formation (Crandall et al., 2010).

Eddies at the lowest Reynolds numbers may be only present in fractures of a minimum

roughness or in areas of rapid aperture changes.

10-2

10-1

100

101

102

103

1.03

1.035

1.04

1.045

1.05

1.055

1.06

1.065

1.07

1.075

Reynolds Number

Tor

tuos

ity

Replicate 1

Replicate 2

Replicate 3Replicate 4

Replicate 5

45

2. Existing eddies experience significant growth and new eddies form beginning at a

Reynolds number around unity. This is a similar range to previous work (Zimmerman et

al., 2004), however it is the complex flow arising at the boundaries, such as eddies, that

are directly associated with the change in effective hydraulic aperture.

3. This eddy growth behaviour suggests a three-zone non-linear model of fracture flow

similar to that found for porous media by Chaudhary et al. (2011). This work expands

the application of the three-zone model to rough fractures. In Zone I at Re < 1 effective

aperture is constant but dependent on initial fracture geometry; Zone II begins at Re

approaching 1 where conventional fracture modeling breaks down as a result of the

significant increase in eddy growth rates. The reduction in eddy growth rate represents

the boundary of Zone II and Zone III and can vary for the fracture system being modeled.

4. The three-zone model of fracture flow also applies to tortuosity as the growth of eddies in

a fracture are directly linked to a non-linear change in measured tortuosity.

5. Directionality only plays a role when large scale features are present within a fracture that

would significantly change flow characteristics. For example a large scale aperture

variation or step and such variation were not present with the default SynFrac settings.

6. Using GPGPUs computing allows for rapid analysis of a variety of parameters and their

effects on the fracture hydraulics at high resolution with the potential to scale to large

systems at a relative low cost of entry.

46

Chapter 4

Solute Transport in Single Fractures with Increasing

Roughness

Abstract

A parametric study of roughness on transport in fractures was performed using random walk

simulations at varying Reynolds (Re) numbers. Simulations were conducted for statistically

generated, hypothetical fractures where only the fracture roughness, in terms of a Fractal

Dimension, was altered. Complex flow features, such as eddies, arising near the fracture surface

were directly associated with changes in the behaviour of solute resident time. Initially, at Re

less than 10, little if any difference is apparent comparing the analytical solutions of solute

breakthrough curves with fractures of different roughnesses. At larger Re, especially Re > 20 a

significant change in behaviour is observed with increasing roughness possibly explained by the

emergence and growth of eddies. The deviations of the breakthrough curves from Fickian

behaviour are occurring at the same range of Re and FD shown to be associated with the onset

and growth of secondary flows. At the highest roughness and Re modeled, it is clear that the

fluid flow interacting with unique fracture geometries create a non-linear response to solute

transport and eddy formation is a key factor in the behaviour.

4.1 Introduction

The mechanisms governing solute transport in a single fracture remains an important and open

research question in the field of contaminant hydrogeology, carbon capture and storage, nuclear

waste storage, and oil and gas recovery. Developing a comprehensive understanding of solute

transport in fractures is underpinned by the need for accuracy in the simulation of fluid flow. To

account for the effects of tortuosity (Tsang, 1984) and Re above unity, a computational fluid

dynamic (CFD) approach is needed. The Lattice-Boltzmann method (LBM) is a CFD approach

that approximates the full Navier-Stokes (NS) equation for fluid flow and has been used in

fractured media to capture inertial forces, eddies and directional effects (Boutt et al., 2006; Eker

and Akin, 2006) and shown in Chapter 3. The LBM, which solves for local velocities

throughout the model domain, lends itself well to solute transport methods such as discrete

47

Random Walk (RW) that can effectively use that knowledge to simulate transport of solutes in a

fracture. Work by Cardenas et al. (2007) showed tailings in the breakthrough behaviour based

on the existence of eddies in a fracture. Also work by Nowamooz et al. (2013) found that real

fractures with different aperture profiles would result in non-Fickian transport behaviour with

early arrival times and late tailings. To further quantify and understand and the effect of

roughness and eddies on solute transport a systematic study of a single fracture with increasing

roughness is completed using computer modeling.

The advection-diffusion equation (ADE) is commonly used to model solute transport in

fractures. However, due to the complex interactions of the fracture geometry the ADE does not

always capture the velocity deviations from the normal distribution that have been observed

experimentally (Neretnieks et al., 1982; Jiménez-Hornero et al., 2005; Qian et al., 2011b). For

example, Qian et al. (2011b) showed experimentally that tailing was evident in fractures to

varying degrees for Re between 12.2 and 86.9 with aperture between 4 and 9 mm. Their fractures

were artificially created in the laboratory from two glass plates where roughness, or aperture

variations, are simulated by inserting small glass segments with different thicknesses along the

plates. They used a mobile-immobile model, which was developed in several earlier works

(Coats and Smith, 1964; Piquemal, 1992; Piquemal, 1993), to quantify the non-Fickian

behaviour of the breakthrough curves. Other approaches to fit the observed behaviour more

accurately than the ADE include the boundary layer dispersion (Koch and Brady, 1985; Koch

and Brady, 1987), equivalent-stratified medium (Fourar and Radilla, 2009) and macro dispersion

(Detwiler et al., 2000) models and RW methods (Ahlstrom et al., 1977; Berkowitz et al., 2006).

While there are several solute transport methods, they are often paired with fluid flow models

that are unable to adequately model inertial flows in fractures.

The transport of solutes is heavily dependent on the fluid structures that form within a fracture.

The local velocity knowledge from the LBM is more accurate than using the cubic law or Stokes

flow based models as it enables the use of high resolution velocity profiles as input into solute

transport models. Stokes flow does not account for the inertial effects of the flow regime and

any eddies present are only a function of geometry (Cardenas et al., 2009). Other system

interactions that affect solute transport include channeling, matrix diffusion, sorption and the

variation in relative advection versus diffusion. These interactions may play a role in the

observed power law tailing in fractured media (Becker and Shapiro, 2000; Knapp et al., 2000;

Kosakowski, 2004; Cardenas et al., 2009).

48

Cardenas et al. (2007) quantified the effect of eddies on fracture solute transport. A 2D NS flow

and transport model was used from the commercially available software package COMSOL to

model a 15cm long x-ray mapped fracture. The original geometry of the fracture was modeled

along with modified aperture distributions to emphasize or deemphasize various features that

would lead to an eddy forming in a particular location. Power law tailing resulted for the

fracture containing an eddy at Re less than unity and extended to Re < 5 (Cardenas et al., 2009).

Lacking, however, is a broader more systematic approach to variations in fracture roughness and

geometry and their effects on flow and solute transport at high Reynolds numbers.

4.2 Methods and Validation

4.2.1 Fluid Flow Modeling in Fractures

The Lattice Boltzmann Methods have been used in engineering applications, specifically in the

field of porous and fractured media flow (Sukop et al., 2013). Comprehensive development of

LBM theory can be found in the literature (Succi, 2001; Sukop and Thorne, 2005; Latt, 2007).

For the purpose of modeling flow in fractures a 2D LBM was developed using nine velocity

directions ei, also known as D2Q9, which models the cross-section of a given fracture profile.

LBM can be summarized with the following equation:

∆ , ∆ , , , (4.1)

where the distribution function f of a fluid packet moves according to the streaming step on the

left hand side of the equation for a given position x and time t. The right hand side of the

equation represents the collision step. Collision of the fluid packets moves the system towards

equilibrium controlled by τ and the equilibrium distribution function . For the purposes of

fluid flow in a fracture boundary conditions include the no-slip condition along the fracture

surface and periodic boundaries along the primary direction of flow. Flow velocities are

controlled by a gravitational force acting on all elements and can be controlled to develop a

desired Reynolds number ( 2 ∙ ⁄ ) where the 2a is the aperture, u is the average velocity

and ν is the kinematic viscosity. A complete development of the LBM model and validation is

presented in Chapter 2.

Fracture profiles are synthetically generated using the software package SynFrac (Ogilvie et al.,

2006). Fractures are generated that have strongly correlated top and bottom surfaces at long

49

wavelengths but are independent at short wavelengths. This approach has been effective for

modeling fractures at the centimeter scale that directly relate to field scale systems at the 100

meter scale (Glover and Hayashi, 1997). Synthetic fractures were chosen to allow for systematic

changes in roughness without changing other fracture properties. Input parameters are chosen

which represent a 100mm fracture and an average aperture of 1.7 mm. The fractal dimension

(FD) is adjusted from 2.00 to 2.35 at 0.05 increments to generate eight 3D fractures surfaces.

The 2D fracture profiles are the same cross-sections used in Chapter 3 for the quantification of

eddies.

4.2.2 Solute Transport

Solute transport is simulated by modelling the discrete movement of particles through the

fracture. Water velocities are known from the LBM simulation and the local velocity

information is used to displace particles each time step. Diffusion follows a RW process to

model discrete particle movement. The RW group of methods have been developed and used

extensively for the purpose of solute transport in porous and fractured media (Ahlstrom et al.,

1977; Tompson and Gelhar, 1990; Wels et al., 1997; James and Chrysikopoulos, 1999; Delay et

al., 2005; Nowamooz et al., 2013).

For the purposes of studying the effects of roughness in a single fracture using the discrete RW

process, particles are assumed to be neutrally buoyant and exhibit no decay or matrix diffusion.

Particle-particle interactions are not modeled nor do particles affect flow. The only forces acting

on the particles are advective from the local fluid velocity and a diffusion process, both of which

are described by the following Fokker-Planck equation (James and Chrysikopoulos, 2011):

∆ ∙ ∆ 0,1 ∙ 2 ∙ . ∙ ∆ (4.2)

where is the local velocity at the location x of the particle at time t, 0,1 is normally

distributed random number for each dimension i with mean zero and a standard deviation of

unity. For a more detailed development of the Fokker-Planck approach the reader is referred to

Delay et al. (2005).

Numerical dispersion is minimized by ensuring ∙ ∆ ≪ ∆ (Tompson and Gelhar, 1990;

Hassan and Mohamed, 2003). A more strict limit is described by Maier et al. (2000) and further

constrains to ensure a particle moves a maximum of one half ∆ per time step.

50

∆ 6 ∆∆

(4.3)

With a sufficient number of particles a RW method can accurately model the process of

Brownian motion used to model diffusion. In general, more than 100,000 particles are required

to sufficiently model diffusion using RW (Hassan and Mohamed, 2003).

Solute transport is modeled using the LBM for fluid flow and RW for particle transport. The

particles are inserted upstream in the fracture as an instantaneous injection. The residence time,

t, for each particle is tracked and plotted as a histogram representing concentration against time.

To generalize the presented data, resident times are non-dimensionalized using the relative

fracture properties and is expressed by the term pore volume or PV:

∙

∙ (4.4)

where 2a is the mechanical aperture, Q is the flow in two dimensions through the fracture

calculated from the LBM velocity data and L is the total length of the fracture in which the

particle travels.

The method of temporal moments is used to calculate an effective dispersion coefficients ( )

for comparison between the synthetic fractures of increasing roughness (Levenspiel, 1972;

Govindaraju and Das, 2007).

(4.5)

(4.6)

2 (4.7)

∙ (4.8)

where M is total mass recovered, Q the average flow, the mean travel time, is the velocity

derived by the method of moments (Equation 4.7), specifically using and L where L is the

length of the fracture.

51

4.2.3 Model Validation

For a point-source in 2D space the analytical solution for diffusion as developed by Crank (1975)

in the form shown by Sukop and Thorne (2005) is:

(4.9)

where C is the concentration, Co is the initial concentration, Dm is the molecular diffusion

coefficient, Mo the initial mass, t is time and r is the spatial coordinate. Figure 4.1 shows the

results from the RW algorithm at three different time increments and their respective analytical

solutions from Equation 4.9. The fit between RW and the analytical solution is excellent with

some variation from the analytical solution due to the random nature of the RW method.

Figure 4.1: Point source diffusion in 2D and the relative concentrations at a given radius from the source. Results for time t = 1000, t = 2000 and t = 10000 are shown with their respective analytical solutions.

Taylor-Aris dispersion between parallel plates is defined as follows (Stockman, 1997):

∙

∙ (4.10)

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.40

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

r

C/C

o

t=1000

t=2000

t=10000

respectiveanalyticalsolutions

52

where Deff is the effective dispersion coefficient, Dm is molecular diffusion coefficient, 2a is the

plate separation and uavg is the average velocity. The above equation holds for over a range of

Peclet (Pe) numbers:

√210 ≪ ≪ / (4.11)

where L is the length of the system and the Peclet number is defined as:

(4.12)

Figure 4.2 indicates a good fit between the discrete RW and the analytical dispersion equation

for parallel plate applications. Values were chosen to be similar to those found in Sukop and

Thorne (2005) as an additional measure of comparison and validation.

Figure 4.2: Effective dispersion for the values: 0.0038 / and 0.0013 / after (Sukop and Thorne, 2005). The input values are given in terms of lattice units (lu) and time steps (ts), typical for LBM applications.

Finally, using the effective dispersion coefficients calculated in 4.10, the concentration at any

point can be determined analytically for the case of parallel plate flow. If the downstream exit is

taken as the reference location and the system is allowed to evolve over time the resultant

breakthrough of solute can be plotted. The analytical solution for the concentration of a solute

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

0 5 10 15 20 25 30

Deff[lu2/ts]

Aperture [lu]

Analytical

Model

53

subject to uniform flow u at a location x at time t for a 1D instantaneous injection is (Hunt,

1978):

, (4.13)

where M is the initial concentration of particles in the system, is the porosity and taken at unity

and is the effective dispersion from Equation 4.10.

4.3 Results

Breakthrough data from the model results are compared with the analytical solution for a parallel

plate system with an equivalent mechanical (arithmetic) aperture using Equation 4.13 and

labeled as the analytical solution in Figure 4.3. Figure 4.3 shows the breakthrough curves for

7 10 ⁄ at Reynolds number 1 through 100 for the synthetic fractures generated

with original 3D fractal dimension (FD) on the range of 2.00 to 2.35. Also plotted are the results

for a parallel plate system with the same mechanical aperture as the rougher fractures, modeled

using the same LBM and RW methods (labeled as a slit). From Figure 4.3, the modeled slit

results are very close to the analytical solutions as expected. However, Non-Fickian behaviour

was apparent if the inequality of Equation 4.11 was not maintained as required and reported in

the literature (Cardenas et al., 2009; Qian et al., 2011b).

At low Re, below 10, roughness is not a large factor with only a slight impact on initial

breakthrough for rougher fractures. As Re increase we see a larger effect of roughness as

significant deviation from the analytical solution occurs. As reported in Chapter 3, the eddy

growth rate in the same group of synthetic fractures peaks at Re = 30 while eddy volume as a

ratio of total volume continues to increase at least through Re = 500. Therefore, eddy growth is

occurring at the same range of Re as is the deviation from analytical results in Figure 4.3. It

follows that some correlation between eddy growth and deviation from Fickian behaviour is

likely. The deviation from normal Fickian distributions becomes more apparent at the higher Re

and roughness reported.

54

Figure 4.3: Breakthrough curves for 7 10 ⁄ at Reynolds numbers 1 through 100 for synthetic fractures generated from a 2D slice of a 3D surface with fractal dimensions (FD) 2.00 through 2.35. The ‘Slit’ represents a parallel plate system modeled in the same way as all FD results; finally the analytical solution for each case is shown for comparison. Concentration profiles (C) are plotted relative to the total number of particles (M) and normalized.

The breakthrough curves reported in Figure 4.3 show a trend of moving to later PV including

initial, peak and late breakthrough of solutes relative to the analytical solution. Using PV non-

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1Re 1

C/M

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1Re 10

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1Re 20

C/M

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1Re 40

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1Re 50

C/M

PV0 0.5 1 1.5 2

0

0.2

0.4

0.6

0.8

1Re 100

PV

Slit

2.002.05

2.10

2.15

2.20

2.25

2.302.35

Analytical

55

dimensionalizes time as a function of the ratio of flow to fracture volume and so later PV results

will be a result of either later absolute time for solute transport, larger flow rates or smaller

fracture volumes. Although an attempt is made to ensure the mechanical aperture is the same

between all fracture samples, due to the discrete nature of the grid, this is difficult to achieve.

The fracture volume differs by 2.2% between all profiles modeled, which does not account for

the increasing residence time shown at large Re and FD. Regarding the flow rates, fractures with

rougher cross-sections experience relatively slower flow rates and would therefore tend to move

the residence times in the opposite direction than reported in Figure 4.3. Since the breakthrough

curves, including initial, peak and tail, move towards large PV it can be concluded that the solute

residence time in the fractures are larger than expected.

From the real-time display of solute transport in the model (not shown), it is evident that the

solute is entering secondary flow zones, or areas with eddies, and being retarded from earlier

breakthrough. The diffusion in and out of eddies is occurring at a rate that maintains a

breakthrough similar to the normal distribution of the analytical case for low Re and FD.

However at large Re and FD approaching 100 and 2.35 respectively, the normal distribution

breaks down and becomes non-Fickian. Non-Fickian transport behaviour is characterized by

Nowamooz et al. (2013) as having early breakthrough and late-time tailing. However, the results

of Figure 4.3 do not indicate early breakthrough but late breakthrough combined with a shift in

the peak and tailing of particles. While their experiment used transparent fracture replicas with

no matrix diffusion, their experimental fractures were driven by constant flow pumps while our

work uses a gravity, or pressure, driven boundary. Other numerical work, by Cardenas et al.

(2007), showed late-time tailing and attributed the results to eddies that formed in the fracture.

Their work resulted from finite-element analysis using the commercially available COMSOL

software package. However, the variation in late-time tailing was analysed by arbitrarily altering

the fracture profile while the current study takes a more systematic approach to evaluating the

effects of fracture roughness.

Figure 4.4 shows the effective dispersion coefficient as calculated from Equation 4.8. Similarly

to Figure 4.3, data from the slit comes from the LBM and RW model while the analytical

solution is calculated from Equation 4.9, both are plotted for comparison. Again we see little

change below Re = 10 and increasing rate of change for higher Reynolds numbers likely

associated with the emergence and growth of eddies. At larger Re, secondary flows are

contributing to an increased dispersion of solutes within the fracture.

56

The combined differentiation of factors between previously published results, including

experimental and numerical work, indicates that while the results from this study do not

duplicate results, similar trends are occurring for the conditions being modeled. Significantly

however, the deviations of the breakthrough curves from Fickian behaviour are occurring at the

same range of Re and FD shown in Chapter 3 to be associated with the onset and growth of

secondary flows.

Figure 4.4: Effective dispersion coefficients using data from the LBM and RW model using the method of moments except for the analytical solutions with is calculated from Equation 4.8. Data shown for

7 10 ⁄ at Reynolds numbers 1 through 100 for synthetic fractures generated from a 2D slice of a 3D surface with fractal dimensions (FD) 2.00 through 2.35.

Adjustments in the diffusion coefficient are expected to change the breakthrough curve

distributions. Figure 4.5 shows three cases with increasing molecular diffusion coefficient

values of 3.5 10 ⁄ , 7 10 ⁄ and 14 10 ⁄ . As

the diffusion coefficient is increased the effects due to roughness are decreased and breakthrough

curves align with the analytical solution. When advective forces dominate, either with high

Reynolds numbers or low diffusion coefficients, non-Fickian behaviour emerges at high Re and

FD likely due to the complexities of secondary flow paths. When the diffusive forces dominate,

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0 20 40 60 80 100

Dispersion Coefficien

t [m

2/s]

Re

Analytical

Slit

2.00

2.05

2.10

2.15

2.20

2.25

2.30

2.35

57

either with low Reynolds numbers or high diffusion coefficients, fracture roughness is less

controlling due to solute being able to move more freely between the bulk flow and secondary

flow regimes.

Figure 4.5: Breakthrough curves for 3.5 10 ⁄ , 7 10 ⁄ and 1410 ⁄ respectively at a Reynolds number of 50 for synthetic fractures generated from a 2D slice of a 3D surface with fractal dimensions (FD) 2.00 through 2.35. The ‘Slit’ represents a parallel plate system modeled in the same way as all FD results, finally the analytical solution for each case is shown for comparison.

4.4 Sensitivity Analysis

Based on the constraint on how far a particle is allowed to travel each time step, ideally

constrained by Equation 4.3 and restated for convenience below. To maintain the inequality, the

discrete time step can be reduced while the discretization of space is fixed for a given model. For

a given Reynolds number the maximum time step will change and the example of Re = 50 is

shown in Figure 4.6. A total of four models are run, two above and two below the empirical

limit expressed by:

∆ 6 ∆∆

(4.14)

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Dm

=3.510 m2/sC

/M

PV0 0.5 1 1.5 2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Dm

=710 m2/s

PV0 0.5 1 1.5 2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Dm

=1410 m2/s

PV

Slit 2.00 2.05 2.10 2.15 2.20 2.25 2.30 2.35 Analytical

58

Figure 4.6: Data shown is for Re = 50 for synthetic fractures generated from a 2D slice of a 3D surface with fractal dimensions (FD) 2.00 through 2.35. Case 1 and 2 do not meet the constraint for minimizing numerical dispersion.

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1Case 1

C/M

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1Case 2

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1Case 3

C/M

PV0 0.5 1 1.5 2

0

0.2

0.4

0.6

0.8

1Case 4

PV

Slit2.00

2.05

2.10

2.152.20

2.25

2.30

2.35Analytical

59

Figure 4.7: For a set bin size when calculating the histogram, a larger number of particles gives a more accurate description of the dispersion of particles through the fracture without changing the overall behaviour. Data shown is for Re = 50 for synthetic fractures generated from a 2D slice of a 3D surface with fractal dimensions (FD) 2.00 through 2.35.

Sensitivity analysis shows from Figure 4.6 that considerations of the minimum time step must be

taken into account to reduce numerical dispersion which is most prevalent in Case 1. Case 3 and

Case 4 begin to reach convergence and for the purpose of accuracy balanced with computation

limits; most models were run with parameters similar to Case 3. In terms of particle count, 215

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1215 Particles

C/M

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1218 Particles

C/M

PV

Slit2.00

2.05

2.10

2.152.20

2.25

2.30

2.35Analytical

60

particles are sufficient for most models and reduces computational requirements versus 218

particles (Figure 4.7). A smaller bin size is used to compensate for the reduced number of

particles for calculations using the method of moments and delivers sufficient accuracy.

4.5 Conclusions

1. Solute transport is affected similarly to fluid flow using the same set of fractures with

increasing roughness. Initially, at Reynolds numbers less than 10, little if any difference

is apparent comparing the analytical solutions with fractures of different roughnesses. At

larger Re, especially Re > 20 a significant change in behaviour is observed with

increasing roughness possibly explained by the emergence and growth of eddies as

reported in Chapter 3.

2. The deviations of the breakthrough curves from Fickian behaviour are occurring at the

same range of Re and FD shown in Chapter 3 to be associated with the onset and growth

of secondary flows.

3. At the highest roughness and Reynolds numbers modeled, it is clear that the flow

interacting with unique fracture geometries create a non-linear response to solute

transport and eddy formation is a key factor in the behaviour.

61

Chapter 5

Effects of Roughness and Shear on Biofilm Populations and

Structure in a Single Rock Fracture

Abstract

A parametric study of roughness on biofilm development in fractures was investigated using

numerical simulations at varying Reynolds (Re) numbers. Discrete modeling approaches to the

motion of fluid, and substrate transport in a single fracture were taken using Lattice Boltzmann

Methods (LBM), Random Walk (RW) methods respectively. Simulations were conducted for

statistically generated, hypothetical fractures where only the fracture roughness, in terms of a

Fractal Dimension, was altered. Biofilm development was modelled using a discrete Cellular

Automata (CA) approach where each node represents a group of bacteria and their evolution is

controlled by local rules based consumption of substrate. The effects of fracture roughness are

associated with non-linear changes to hydraulic behaviours in the fracture. Also studied were the

effects of changing Re, diffusion coefficients, substrate concentrations and biofilm shear

strength.

5.1 Introduction

A biofilm is the phenotypic expression of a cluster of bacteria when attached to a surface. The

biofilm phenotype is the predominant and preferred form of most bacterial species (Costerton,

2007). In fractured media stimulation of the formation of biofilms can be used as a remediation

technique to degrade undesired contaminants or to act as bio-barriers impeding transport of

contaminants. The natural background level of a bacterial community can be augmented to

encourage growth and development of biofilms that are able to degrade a given contaminant.

Experimental and numerical studies of biofilms have helped to develop and test theories of

fundamental biofilm behaviours. Using micro-scale discrete numerical algorithms, the current

study examines the behaviour expressed by a biofilm developing in a fracture and improves the

understanding of the role of fracture geometry and flow rates in a single rock fracture.

62

A brief overview of biofilm modeling follows, for more depth the reader is directed to other

more detailed reviews (Characklis et al., 1990; Purevdorj-Gage et al., 2004; de Beer et al., 2006;

Wang and Zhang, 2010; Stewart, 2012). Initial biofilm models considered the growth of the

bacterial colony with varying levels of complexity but no bulk fluid flow. For example,

Hermanowicz (2001) developed a 2D model using a discrete cellular automata (CA) grid where

biofilm growth was determined by local evolution rules. It was found that when limitations were

imposed on the external mass transfer of nutrients mushroom shaped structures resulted. These

rough structures have also been experimentally found by de Beer et al. (1994). Three-

dimensional models of biofilm growth have improved on biofilm modeling capability and, for

example, considered various attachment and detachment factors in biofilms including nutrient

starvation, chemical signaling and antimicrobial attack (Hunt et al., 2003; Hunt et al., 2004; Hunt

et al., 2005). These models also resulted in structured biofilms typically in towering or

mushroom shapes.

Some early models by Picioreanu et al. (1999) incorporated hydrodynamic considerations with

biofilm modeling. Their model qualitatively described smooth biofilms under substrate rich

conditions and Re = 10 for a single sided smooth plane or surface. A substrate rich condition is

created by these high Reynolds numbers when the fast flow compresses the mass transfer

boundary layer creating a shorter diffusion path for substrate. Conversely, under low Reynolds

numbers the boundary layer is thicker creating longer substrate diffusion paths resulting in a

substrate limited condition. Under the substrate limited conditions (Picioreanu et al., 1999)

found structures became heterogonous and mushroom shaped. Further work with this model

resulted in a shear induced detachment model (Picioreanu et al., 2001) both of which simulated

biofilm growth using a CA approach while the bulk fluid was modeled using LBM.

Further biofilm modeling algorithms were based on a continuous-continuous approach to biofilm

modeling (Kreft et al., 1998; Kreft et al., 2001). The approach treats each bacterium as a sphere

along a continuous coordinate system. As the spheres grow and divide they push outwards as a

shifting algorithm adjusts their location. This model produces similar biomass results to

comparable CA algorithms but the shape differed, the biofilms created are more rounded and

confluent compared to the rough and disjointed CA models. Under similar continuous

diffusion-reaction conditions as the CA models, the individual-based-model (Ibm) develops

towering or mushroom shaped biofilm structures.

63

A discrete approach to substrate transport was taken using the Random Walk (RW) technique to

simulate substrate advection and diffusion. The micro-scale approach allows for a unique view

on the biofilm growth process where local substrate-to-bacteria reactions drive growth. This

discrete model can capture local effects as a biofilm will only grow when a substrate particle is

physically adjacent to a bacteria cell in the numerical grid.

RW methods have been used for decades to describe transport in porous media with work dating

back to Ahlstrom et al. (1977) which built on work by Bear (1972) and Csanady (1973), since

then work has continued and further developments can be found in the literature (James and

Chrysikopoulos, 1999; Delay et al., 2005; Jiménez-Hornero et al., 2005; James and

Chrysikopoulos, 2011). RW methods use the process of Brownian motion to step particle

motion randomly in time. Each time step a particle is shifted based on a normally distributed

random number. The small movements result from smaller molecular and atomic vibrations and

collisions. When a sufficient number of particles are used, depending on the implementation and

size of the model, a normal distribution emerges from the random motion of an individual

particle and the diffusion process can be simulated (Valocchi and Quinodoz, 1989). It is also

important to maintain a sufficiently small RW time step such that a particle will not move more

than one grid cell distance per time step to reduce numerical diffusion (Tompson and Gelhar,

1990). Using the local velocity at each grid location from the bulk fluid modeling, advection of

the particles are tracked along with diffusion. The complex geometries of rock fractures create

locations of recirculation, or eddies, areas of stagnation and separation all of which add

complexity to the trajectory of each substrate particle.

Flow through a single rock fracture is analysed using a Lattice Boltzmann Method (LBM)

numerical model. LBM is discrete in time and space where many cells are used to allow the

emergent behaviours of fluids to be observed. Comprehensive development of LBM can be

found in Succi (2001), Sukop and Thorne (2005) and Latt (2008a). For the purpose of describing

laminar flow in a rock fracture a 2D LBM using a BGK collision operator is sufficient and is

summarized in Equation 1.

)),(),((1

),(),( txeq

iftxftx

ifttt

iex

if

(5.1)

where the left hand side of the equation represents the streaming step and the right hand side

represents the collision step.

64

5.2 Model Implementation

The model includes three components that work together to simulate development of biofilms in

rock fractures under flowing conditions. The biofilm growth is based on local evolutionary CA

rules while the substrate is simulated using RW and flow is modeled using a LBM approach.

5.2.1 Biofilm

A CA approach to biofilm modeling is used as CA exhibit complex and chaotic behaviour from

simple evolutionary rules. They are based on local relationships and interactions and have been

shown to proficiently model local phenomenon such as spikes or discontinuities in population

distribution. The state of a CA cell which is discrete in time and space describes whether a

bacterium is present or not at that location and its mass.

The bacteria in the model are given simple life-cycle rules. When substrate becomes available

local to a bacterial cell, the cell can consume the substrate and divide after a threshold mass has

been reached. When a substrate is present, the growth model behaves according to discrete rules.

Each bacterial cell consumes substrate when a particle is adjacent to one of its eight neighbours.

The mass of the particle is converted to biofilm mass via a yield coefficient and conservation of

mass is used to validate growth of the biofilm throughout the simulation.

Once a biofilm cell has increased in mass to twice its original mass a division process occurs. A

search is made for free spaces surrounding the nearest four neighbours and one of the free spaces

is chosen randomly for the new cell produced by the division. If no free spaces are available, one

of the four occupied neighbouring spaces is chosen randomly and the existing cell is displaced

by the newly created daughter cell. This displaced cell is then subjected to the same process,

beginning with a search for free space then displacing another cell if needed. Finally, this

continues until all new cells and displaced cells have been moved into free spaces.

EPS (extracellular polymeric substance) is excreted by the bacteria during the biofilm process

the behaviour of which is associated with the phenotypic change that occurs in bacteria when

they move from the planktonic to a biofilm state. For the purposes of this model it is assumed

that EPS and bacteria are acting together in each cell and in the growth of a biofilm cluster.

This is not entirely accurate as some bacteria have been seen to form EPS structures and then

leave and move to more nutrient rich areas (Costerton, 2007).

65

Sloughing of biofilm due to shear from the bulk fluid can be optionally applied in the biofilm

model. Biofilms observed experimentally are considered to behave as a visco-elastic material,

however for the purposes of the numerical simulations in this study, they are assumed to be rigid

with a maximum specified shear strength above which sloughing occurs. When the maximum

shear strength is reached for a given cell in the model, the cell is removed and the bacterium is

flushed from the system with no re-attachment downstream. The shear stress experienced by a

given bacterium is calculated from the velocity profile of the cross-section of the model in which

the bacterium resides. The velocity gradient or time rate of strain, dV/dy (Crowe et al., 2001)

where V is the fluid velocity and y is the distance from the wall is used to defined the shear

stress.

(5.2)

where τ is the shear stress and μ is the dynamic viscosity of the fluid. To simplify the calculation

of shear it is assumed that the velocity gradient is constant requiring only the knowledge of two

points on the velocity profile. The velocity at the wall is zero and the velocity at the maximum

streamline is a value Vmax and is found from the LBM output. Vmax is measured perpendicular to

the global horizontal axis at every cross-section along the fracture and its respective value used

for biofilm shear calculations along the wall. Therefore the shear becomes:

(5.3)

where Δy is the distance between any given biofilm cell and the maximum velocity streamline at

the required cross-section.

Since no specific physical species or mixed species biofilms are modeled where a specific shear

strength may be known, instead a range of biofilm shear strengths are used to demonstrate the

relative change in biofilm behaviour at increasing Re and fracture roughness. The shear strength

of specific biofilms species and mix culture biofilms have been measured in the literature using a

variety of methods and measurements (Möhle et al., 2007). The range of reported tensile,

compressive and shear strengths is significant and difficult to compare based on the varying

methods. Some early worked focusing on shear strength reported values between 5 to 50 N/m2

using a centrifuge to apply a tensile stress (Ohashi and Harada, 1994). Stoodley et al. (1999)

observed in-situ mixed-culture biofilms under an applied fluid shear with time lapse microscopy

66

and reported an elastic modulus E of 40 N/m2. While the shear modulus G of mixed culture

biofilms was measured in the range of 0.2 to 24 N/m2 (Towler et al., 2003). More recent work

has measure the shear strength of biofilms over similarly large ranges between 0.12 and 7 N/m2

(Chen et al., 2005; Möhle et al., 2007), All three mechanical properties of the biofilm structure,

E, G and shear strength play an important role in the biofilm behaviour and rheology, however,

for the purposes of comparison at varying Re and fracture roughnesses the shear strength is

varied over several orders of magnitude in this study.

5.2.2 Substrate

Particles within the systems are displaced via the processes of advection and diffusion.

Advection is calculated using the local velocity at the known particle coordinates while diffusion

is calculated using a discrete RW method. The displacement due to both of these factors is

summarized by Equation 5.4:

∆ ∙ ∆ 0,1 ∙ 2 ∙ ∙ ∆ (5.4)

where is the local velocity at the location of the particle in the dimension i, 0,1 is a

normally distributed random number for each dimension i with mean zero and a standard

deviation of unity and ∆ is the fundamental time step in the simulation and is taken as unity

since the calculation is performed once per time step. The molecular diffusion coefficient Dm

used is 7 10 ⁄ , similar to glucose in water.

A substrate particle exhibits two other behaviours. First, when a substrate particle hits a wall it is

reflected back into the modeling domain. Second, when a particle encounters a biofilm it can be

consumed or it may diffuse into the biofilm. Diffusion coefficients within the biofilm are

assumed to be the same as in the bulk flow. It is assumed that the substrate particles have no

interactions with other particles. There is no matrix diffusion of the substrate into the rock

fracture, no buoyancy effects and gravity is not considered to be acting on the particles. Finally

the particles do not affect the bulk fluid flow in the system.

Substrate concentrations are determined by converting a discrete number of particles into a

global concentration in terms of grams per liter. To achieve a desired 10 g/L substrate

concentration 205,312 particles are required considering the physical and numerical conversion

factors. To simplify unit conversions and increase computation speed, it is assumed that each

substrate particle, when consumed, is completely consumed and that it is of sufficient mass to

67

allow for bacterial growth leading to division. The calculations for these assumptions can be

done using representative 3D cell within the 2D slice of the fracture assuming the third

dimension has the same depth as the height and width of the individual LBM cells. Each LBM

cell in the synthetic fracture has the same discretization at 12.2 microns for a total volume of

1819 microns3. Taking a representative cylindrical shaped bacterium with mass 1 10 ,

length 4 microns and a 1 micron diameter and assuming no specific packing orientation

approximately 455 bacteria can fit in each discrete LBM cell. Next, using a biofilm yield

coefficient of 0.45 grams of biofilm produced for each gram of substrate consumed 1.01

10 of substrate must be consumed to allow growth of a discrete cell which contains

approximately 455 bacteria. The number of particles is then finally calculated from the desired

global concentration of 10 g/L knowing the fracture dimensions (100 mm x 1.7 mm x 12.2

microns).

5.2.3 Bulk Fluid Flow

Flow in the fractures is gravity driven which is implemented in the LBM, requiring only bounce

back boundaries at the walls and periodic boundaries at each end of the system in the direction of

flow. For all flow simulations shown, gravity driven flow drives the fluid from left to right, as if

gravity was acting to the right, with solid boundaries across the top and bottom of the model.

The hydrostatic pressure gradient applied by gravity drives flow in the system and results in

similar behaviour to pressure boundaries. Periodic boundary conditions are used at the ends of

the modeling domain effectively connecting the left side to the right and allowing the fluid being

simulated to continually wrap around the domain. The system is simplified with periodic

boundaries; particularly there are no entry or exit effects (Sukop and Thorne, 2005). Conversely,

velocity boundaries would impose a predetermined velocity profile at the entry and exit of the

fracture and would require a significantly longer domain to account for these effects. Other

complications arise with the implementation of velocity boundaries with LBM and must be

considered carefully to conserve mass (Zou and He, 1997).

One of the distinct advantages of the LBM comes from its discrete nature. It is efficient for

modeling complex geometries (Chen et al., 1994; Eker and Akin, 2006; Lammers et al., 2006;

Brewster, 2007) which arises in the analysis of rock fractures. An array is stored to set the value

of any point in the LBM grid to represent either a fluid particle or a solid boundary (rock surface

or bacteria). At the solid boundaries, a no-slip condition is used to create zero velocity at the

68

boundary surface. At the boundaries the LBM use a different set of collision equations as

described by Succi (2001) and are referred to as mid-plane bounce back boundary conditions.

The name arises from the applied boundary rules. Fluid particles entering a boundary at time t

are sent back out with equal magnitude and opposite direction at time t+Δt this effectively puts

the boundary at a distance midway between a fluid and solid node.

Gravity driven conditions are used according to the method described by Sukop and Thorne

(2005). The force of gravity is added to the horizontal velocity component resulting in gravity

acting along the horizontal axis, this is done for convenience to line up with the primary axis of

flow. The acceleration due to gravity is converted to a velocity term as shown in Equation 5.5.

dt

dummaF (5.5)

where F is the external force added into the LBM calculations in the form of a local velocity. In

LBM, the mass (m) is proportional to the density (ρ) and relaxation parameter (τ) can be

substituted for time arriving at Equation 5.6.

F

u (5.6)

where Δu represents a discrete velocity increment and is added to the horizontal velocity

component used to calculate the equilibrium distribution function in Equation 5.1.

To minimize the potential for numerical instabilities and maintain the second order accuracy of

the LBM, the model parameters are defined using the method laid out by Latt and Krause (2008)

as part of the OpenLB User Guide. The process involves selecting physical units then converting

to lattice units to finally obtain the relaxation parameter τ. The relaxation parameter plays an

important role in the collision term of the LBM. It controls the tendency of the system to move

towards local equilibrium. In the literature, the relaxation parameter has been found to cause

numerical instabilities at values approaching 0.5 from the right hand side (τ must be greater than

0.5 for physical viscosities) (Sukop and Thorne, 2005).


Fractures profiles used for growing biofilm are the same profiles used in Chapter 3. The profiles

are generated using SynFrac, a synthetic fracture generation software developed by Ogilvie et al.

69

(2006). Synfrac can generate 3D fractures surfaces based on several input parameters including:

fractal dimension (FD), resolution, standard deviation, anisotropy and mismatch length. The

mismatch length refers to the correlation between the top and bottom surfaces. Below the

mismatch length the fractures surfaces will be mostly independent while above it they will be

correlated. SynFrac has multiple methods for determining the mismatch length, building on

previous research (Brown et al., 1995; Glover et al., 1998a; Glover et al., 1998b). For the

purposes of this work the SynFrac implementation of the (Brown et al., 1995) mismatch length is

set to 15 mm. The total fracture is 100 mm square with a resolution set to 1024 by 1024,

standard deviation equal to 1 mm and anisotropy factor of 1.0. The 2D LBM model domain is

2048 elements in length and uses linear interpolation to expand the 1024 elements from SynFrac

resulting in a 48.8 micron grid spacing. The FD is set over a range of 2.00 through 2.35 for this

study (Figure 5.1), where the upper limit is based on work by Ogilvie et al. (2006) who found

that sandstone and granodiorite samples had fractal dimensions approaching 2.35. Given all the

parameters Synfrac generates two 3D surfaces and positions them with a separation that creates a

single contact point. For our purposes a 2D slice is taken at the same location for each FD such

that no contact point is intersected. The separation of each 2D slice is adjusted to obtain an

average mechanical aperture of 1.7 mm. To ensure no interference with the periodic boundary

conditions for fluid flow the aperture and alignment are held constant at the entry and exit of the

fracture profile.

70

Figure 5.1: Fracture profiles b through i generated using a synthetic fracture generator called SynFrac. Total fracture length is 100 mm and each fracture has a mean aperture of 1.7 mm, only the fractal dimension (FD) input parameter is adjusted in SynFrac. Fracture profile a represents a parallel plate system with an equivalent mean 1.7 mm aperture.

5.3 Biofilm Growth Model

The biofilm growth model incorporates fluid dynamics, substrate transport, biofilm population

dynamics and biofilm sloughing due to shear. Initially the model consists of fracture walls, fluid

at rest and no biofilm. The modeling starts by allowing the fluid to begin moving at a desired

Reynolds number and is given time to reach equilibrium. Substrate transport is initialized next

with particles being randomly distributed across the entire system to minimize any preferential

locations of growth. Finally the system is seeded with bacterial cells added as continuous

monolayer along the upper and lower fracture surfaces. As biofilm is assumed to be

impermeable to flow, when a continuous biofilm structure bridges the top and bottom fracture

surfaces flow will be stopped in the fracture. All biofilm models are run until a clogging event

stops flow moving through the fracture. A clogging event is defined as the threshold where the

flow rate is 1/100th of the flow rate before the biofilm started developing. The main program

loop begins from this point and proceeds as follows (also shown in Figure 5.2):

1. A fluid step is completed using the LBM incorporating any new geometry changes from

the previous biofilm step (if applicable).

a) parallel plate

b) FD 2.00

c) FD 2.05

d) FD 2.10

e) FD 2.15

f) FD 2.20

g) FD 2.25

h) FD 2.30

i) FD 2.35

71

2. A substrate transport step is completed using RW particle transport to account for

diffusion while advection is accounted for from the known fluid velocities calculated by

the LBM step.

3. If a biofilm step is called for based on the timescale analysis (discussed below) the

algorithm will proceed to modeling biofilm growth. Otherwise the algorithm loops back

to step 1.

4. Within the biofilm population growth step several sub steps are required and described

here:

a. For each biofilm cell a search is made for available substrate in any of its 8

neighbouring cells.

b. If substrate is present the substrate is consumed by the biomass cell and the cell

mass is increased according to the yield coefficient and the particle mass (only

one particle can be consumed by one biofilm cell per time step).

c. Any biofilm cell that is now above the division threshold will be divided and a

new daughter cell will be created.

d. First priority for locating a new daughter cell is given to choosing at random a

free space among the 4 neighbours sharing an edge. If no free space in available

one of the 4 occupied neighbours is displaced and replaced with the new daughter

cell. The search step is repeated for the displaced cell until all cells have found

empty space in which to move (Picioreanu et al., 1998).

e. When applied, shear is calculated for any biofilm cells adjacent to fluid cells and

any cells above the threshold strength are removed from the system, assumed to

be sloughed off and not to re-attach downstream.

f. The system is now in a new state which has accounted for biofilm growth and

shearing. The algorithm now returns to step 1.

72

LBM

Substrate TransportNo

Substrate available at any of the 8 neighbours?

Biofilm Population Dynamics

Yes

Consume substrate: Divide?

Update grid

Shear

Yes

No

No

Update grid

Run biofilm population dynamics?

Yes

Initialize grid:No Flow

No SubstrateNo Biofilm

Run LBM until equilibrium is reached

Enable substrate transport: Initialize

particles with random locations

Enable biofilm growth along fracture wall

Figure 5.2: Main program loop which includes the processes of fluid dynamics, substrate transport and biofilm growth.

5.4 Timescales

Simulated time scales of fluid flow, substrate transport, and biofilm growth in a single rock

fracture occur over several orders of magnitude. To deal with the discrepancy of the time scales,

73

the physical processes are split into two groups: fast and slow time. Fast processes include fluid

flow and substrate transport while the slow processes include biofilm consumption, division and

detachment due to shear. In general the following holds:

∆ ≪ 1 ≪ ∆ (5.7)

where fast time scales are much less than one second and slow time scales are much larger.

Specifically, time scales for the LBM and substrate transport are on the order of 10-5 seconds or

less while a typical biofilm specific growth rate (µ) is of the order 0.3 per hour or a division

every 3.3 hours. As biofilm development can take several weeks it becomes impractical to

simulate the entire domain using the smallest time scales. Using a similar approach found in the

literature (Picioreanu et al., 1999; Picioreanu et al., 2000) the domain is split into two time

scales. For the purposes of balancing computational capacity and the time required to grow

biofilms, the LBM and RW are run for 0.02 seconds between successive biofilm steps. Given

the small incremental changes that may occur if a bacterium divides it is assumed that the fluid

reaches equilibrium within the 0.02 seconds between each biofilm step.

The substrate transport time scale needed for stable and accurate results is of a similar order as

the LBM. Once a substrate particle is consumed it is converted to biomass which marginally

decreases the global concentration by approximately 10-5 g/L per particle. The discrete locations

of substrate are driving biofilm growth and development and whether a bacterium may consume

a particle is determined from the local availability of that substrate particle. When the substrate

time (fast process) is disconnected from biofilm population dynamics (slow processes) a new

field of particles is presented to the biofilm algorithm each time step and depending on the ratio

of LBM and RW time steps to biofilm growth time steps, substrate particles move varying

distance between time steps. Moreover, as the biofilm growth is heavily dependent on substrate

availability this will affect the growth rates of the biofilm. With the use of varying timescales,

the length of a biofilm time step becomes arbitrary and would require validation against known

systems to calculate the physical time elapsed for the biofilm. Sensitivity analysis of the time

scale ratio is shown later to demonstrate the global effect on biofilm populations.

74


5.5.1 Biofilm with No Sloughing

Biofilms are modeled for Re over two orders of magnitude, between 1 and 100, to capture

potential effects of secondary flows that develop within the fractures. Secondary flows are

associated with increasing roughness and Re (Chapter 3) and have a non-linear effect on the

hydraulics and solute transport in the model. The development of biofilm structures also plays a

role in both of these aspects leading to complex flow streamlines (Figure 5.3). Figure 5.3 is a

representative example of biofilm growth shown along a 1 mm segment of the 100 mm fracture,

biofilms develop large towering structures. The biofilm colonies tend to grow at locations of

local aperture constrictions, where the fracture surface is closest to the bulk flow, minimizing

substrate diffusion distances. Biofilms from the top and bottom fracture surfaces have a

tendency to grow in tandem meeting near the middle of the fracture aperture.

Figure 5.3: A representative sample of biofilm structure in a fracture. For the fracture shown, Re = 50, FD = 2.35, Biofilm shear strength is 0.045 Pa. The plotted segment is approximately 1mm of the total 100mm fracture. Blue represents flow with streamlines plotted on top, green represent a biofilm cell and pink represent locations where biofilms are permitted to develop.

Figure 5.4 represents the biofilm population over time for three different Reynolds numbers: 1,

5 and 100. Two quantities are used to describe biofilm populations. First the biomass is

reported in terms of percent increase from initial inoculation. Relative results are shown since

each fracture has a different roughness and therefore a different perimeter and will begin with a

different number of nodes allowing growth of bacteria. Secondly the FD is reported, similarly to

biomass, as a percent increase over time. On the left hand column of Figure 5.4 the FD change

over time and shown and on the right hand side the biomass change over time. The biomass

75

grows at an almost constant rate for most cases suggesting a zero-order Monod growth rate.

However the FD of the biofilm initially grows quickly then slows at late time. A larger FD is

associated with towering biofilm structures indicating that biofilms need to grow into the bulk

flow, away from the wall, to best capture substrate. As Re increases this effect is augmented

possibly due to the decrease in effective hydraulic aperture contributing to bulk flow (as

discussed in Chapter 3). At higher Re, advection becomes a more prominent transport

mechanism and therefore the highest probability of finding substrate is in the bulk flow zone,

which is decreasing at higher Re, as a result biofilm, should it grow, will need to find substrate in

this zone. Consequently biofilms must grow farther away from the fracture wall resulting in

larger FD at earlier time for increasing Re.

As a proof-of-principle the qualitative shapes formed by the biofilm are generally consistent with

the literature as discussed in the introduction of Chapter 5 which indicated that under substrate

poor conditions biofilms have been shown to develop rough towering structures (de Beer et al.,

1994; Picioreanu et al., 1999; Hermanowicz, 2001; Hunt et al., 2004). More specifically, in the

field of contaminant hydrogeology, Arnon et al. (2005) found biofilms clogging and affecting

preferential flow paths within naturally fractured chalk. While not being able to compare results

directly between the 3D experimental setup and the 2D modeled results, clogging clearly

controls the hydraulics of the fractures of both systems. The flow rates presented in this work

relative to the diffusion coefficients are such that substrate transport is limited by diffusion and

therefore in zones between the bulk flow and the wall, where diffusion is the primary mechanism

for moving between streamlines, substrate limiting conditions arise. Conversely, when the

diffusion coefficient or substrate concentrations are increased, the system develops biofilms of

more uniform thickness.

76

Figure 5.4: Biofilm characteristics expressed by two different quantitative measurements: relative FD on the left and relative biomass on the right. Values are relative to the initial FD and biomass of each respective fracture. Results are shown for Re 1 through 100 and normalized time.

For the three Re in Figure 5.4, the growth rates are faster at higher Reynolds numbers where

advection dominates the transport of substrate. The resultant biofilm development grows into the

fracture aperture, away from the surface, increasing FD and resulting in clog events sooner than

at lower Re. At higher Re, clogging occurs sooner which may be due to increasing flows alone

or, in addition, the role of secondary flow may become significant for delivering substrate to

existing colonies as eddies develop around new biofilm colonies.

Re = 1

Re = 50

Re = 100

0%

10%

20%

30%

40%

0 100 200 300

FD Change

0%

10%

20%

30%

40%

0 50 100

FD Change

0%

10%

20%

30%

40%

0 50 100

FD Change

Time

PP 2.00 2.05 2.10 2.15 2.20 2.25 2.30 2.35

0%

100%

200%

300%

400%

500%

0 100 200 300

Biomass Change

0%

100%

200%

300%

400%

500%

0 50 100Biomass Change

0%

100%

200%

300%

400%

500%

0 50 100

Biomass Change

Time

77

The effects of secondary flow, consisting primarily of eddies along the rough fracture play a non-

linear role in terms of substrate transport to the biofilm. A few factors lead to this non-linear

relationship. First, where the fractures are parallel plates, as the Re increases, the boundary layer

is compressed due to hydraulic forces. A compressed boundary layer reduces the diffusion

distance a substrate particle must travel from the bulk flow to the biofilm. However, since the

fractures are complex, rough surfaces, their effect on the boundary layer is non-linear. It was

shown (Chapter 3) that as the Re values increase, the bulk flow zone decreases resulting in an

increased thickness of boundary layer. A larger boundary layer would increase the diffusion

distance for substrate. However, the boundary layer is composed of secondary flows, typically

eddies, which contribute their own non-linearity of advection and diffusion resulting in a

complex feedback loop between fluid flow, transport behaviour and their direct effects on

biofilm development. Substrate particles that are captured in eddies and delayed in the fracture

as shown in Chapter 4 may have an increased likelihood of coming into contact with a biofilm

cell. Finally, as biofilm colonies grow and constrict flow, they will compress the boundary layer

and reduce the diffusion distance for substrate transport to the biofilm (Picioreanu et al., 2001).

While two-way biofilm-fluid interactions are not being modeled as they are in Taherzadeh et al.

(2010), the hydrodynamic conditions of the model impact biofilm growth and development. The

biofilm can only grow where substrate is available and its highest availability is through

advection (for the given diffusion coefficient) which becomes more dominant at higher Re.

The data in Figure 5.4 highlights the similarity in growth rates and shape over time regardless of

roughness but it is the clogging event that differentiates the various roughnesses. Figure 5.5

summarizes the total biofilm mass when clogging occurs which is significantly lower for the

rougher fractures. Again, at higher Re, advection dominates and the biofilm will tend to grow

into these substrate rich advection dominated zones which will cause clogging of the fracture

more rapidly and with lower biomass.

78

Figure 5.5: Total biomass is plotted at time of a clogging event for fractures with FD 2.00, 2.15 and 2.35. Results for parallel plates are shown for reference.

Plotting the biomass verses relative hydraulic aperture, calculated from Chapter 3 illustrates the

effect on hydraulic conditions and how they are affected by bio-accumulation (Figure 5.6).

Results suggest less biomass is required to reduce the effective hydraulic apertures with

increasing Re and increasing fracture roughness. Some variation is shown due to the random

nature of both the RW substrate transport and CA based biofilm model. Overall Figure 5.6

shows a linear decrease in hydraulic aperture as biomass accumulates with a departure from

linearity just before a clogging event (when the model is stopped).

Given the geometric properties of a fracture along with the flow conditions and substrate

availability it is feasible to use the model to determine the likelihood that clogging will occur.

Clogging of a fracture results in a no-flow condition, or in 3D indicates a change in preferential

flow paths. In both cases, the flow rates will be reduced as flow no longer moves according to

its original, lowest energy flow pattern. In the model, a clogged fracture can no longer remediate

fluids as contaminants are no longer being transported over the biofilm. In three-dimensional

clogging at a larger scale fluid would be forces to flow to new zones which may not be rich in

bacteria or favourable for bioremediation. Ideally an engineered bioremediation implementation

would be designed to encourage biofilm development while discouraging clogging possibly by

controlling induced flow rates, substrate concentrations, or the choice of augmented bacterial

populations.

0%

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1 10 100

Biomass at Clog Even

t

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PP 2.00 2.15 2.35

79

Figure 5.6: Biomass growth plotted against the relative hydraulic aperture as a measure of hydraulic behavior in fractures with increasing roughness. Results shown for Re = 1, 50 and 100. Each model is run until a clogging event negates the usefulness of further hydraulic measurements.

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80

5.5.2 Biofilm with Sloughing

Sloughing events due to the shear strength of the biofilm being exceeded are modeled using the

know velocity distribution in the fracture which develops according to Equation 5.2. Sloughing

occurs at the per-element level and once sloughed it is assumed the bacterium travels far

downstream (out of the domain) without re-attaching. The effect of increasing the shear strength

of the biofilm is shown graphically in Figure 5.7 for Re = 50. From left to right, top to bottom

the shear strength of the biofilm in Figure 5.7 are: 0.030, 0.035, 0.040, 0.045, 0.050 and 40 Pa.

Depending on roughness and Re the biofilm is strong enough to resist sloughing and can develop

above a threshold shear strength. Even at low strengths a biofilm can still develop, although at a

slower rate. As a biofilm builds mass and structure it will increase drag and reduce flow rates

and therefore the resulting shear stress, allowing the biofilm to grow into new areas. In addition,

advection dominates the substrate transport process and biofilm will therefore grow into the

substrate rich zones instead of in low shear zones. These two behaviours govern at all shear

strengths for a given diffusion coefficient.

While the model only simulates rigid biofilm structures, the heterogeneous, towering biofilm

colonies with narrow support structures would be expected to deform in the downstream

direction and develop into biofilm streamers as previously reported in the literature (Taherzadeh

et al., 2010). From the model results, although limited to 2D and rigid structures, it can be seen

that substrate transport still governs development and the biofilm will grow into the bulk flow if

at all possible.

81

Figure 5.7: Streamlines are plotted along a segment of the total fracture representing approximately 1 mm of the model at Re = 50. Shear strength from left to right, from top to bottom: 0.030, 0.035, 0.040, 0.045, 0.050, 40 Pa (similar to no shear enabled in the model). Biofilm is shown in green while pink represent locations where biofilms are permitted to develop. The plotted results are shown at the time of a clogging event, or late-time for those shear strengths that do not clog.

Figures 5.8 and 5.9 show changes in FD and biomass respectively for shear strengths between

0.01 and 40 Pa, including, for comparison the case where sloughing due to shear is not enabled.

. ⁄ . ⁄

. ⁄ . ⁄

. ⁄ . ⁄

. ⁄

82

For the case of smooth parallel plates there is some threshold shear strength, depending on Re,

below which no biofilm development occurs. For increasingly rougher fractures there will be

zones of local aperture change resulting in varying distances between the fracture surface and

maximum velocity stream line (Δy in Equation 5.2) which is used to predict shear forces. These

changes allow for biofilm to form in more diverse locations leading to a transition zone between

low shear strength, no growth scenario to high shear strength, high growth scenarios where some

biofilm may form at this transitional strength but not sufficiently to develop a clog. For a fixed

diffusion coefficient this threshold is dependent on roughness and Re. Again, advection is still

the dominant form of substrate transport in these models and biofilm will grow into the bulk flow

and eventually clog the fracture for shear strength values above the threshold. For shear strength

values below the threshold, the biomass cannot grow sufficiently to increase drag and reduce

flow in the fracture and therefore allow for more growth through an overall reduction of shear

and the associated sloughing. For shear strength values at the threshold, for example τ = 0.04 Pa

for the case of Re = 50 and FD = 2.15, it is not clear whether growth will continue until clogging

or if growth will plateau and reach a steady value. A key, qualitative indicator that the biofilm

shear strength is in a threshold range is to determine where the biofilm is developing. For all

shear strengths that eventually lead to clogging, biofilms tend to develop along local peaks in the

fracture surface, or aperture constrictions. However for the threshold shear strengths, biofilms

only grow in zones where apertures are relatively large, or areas where the sloughing forces

acting on the biofilm are minimized.

83

Figure 5.8: Results for relative change in FD for fractures with FD 2.15 and 2.35 shown with the parallel plate case for comparison. For the case of Re = 1 all shear strength values exhibit similar behaviour and follow the same trend. Biofilm shear strength varies from 0.01 to 40 Pa with the case of no sloughing also shown for comparison. Various biofilm shear strength values are highlighted to emphasise the shift in the threshold growth values over increasingly rough fractures.

Parallel Plates

FD = 2.15

FD = 2.35

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0.01 0.02 0.03 0.04 0.050.06 0.07 0.08 0.09 0.10.2 0.3 0.4 0.5 0.60.7 0.8 0.9 1 25 10 20 40 No Shear

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84

Figure 5.9: Results for relative change in biomass for fractures with FD 2.15 and 2.35 shown with the parallel plate case for comparison. Biofilm shear strength varies from 0.01 to 40 Pa with the case of no sloughing also shown for comparison. Various biofilm shear strength values are highlighted to emphasise the shift in the threshold growth values over increasingly rough fractures.

5.5.3 Sensitivity Analysis

Sensitivity analysis takes on two forms in this study. The first analysis covers the effects of

physical parameters including substrate concentrations and molecular diffusion coefficients. The

second analysis covers numerical assumptions made for the model including the treatment of

Parallel Plates

FD = 2.15

FD = 2.35

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Biomass Change

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0.01 0.02 0.03 0.04 0.05

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85

substrate when it is consumed, and whether the discrete substrate particles are re-introduced into

the system to maintain the same absolute concentration and how the time scales are treated

between flow, substrate transport and biofilm population dynamics. Finally the variation from

running the same model multiple times is analysed as the algorithm includes two calculations

which use pseudo random number generators: RW and choice of daughter cell locations.

The assumption that substrate particles are removed from the system after consumption is

analysed in Figure 5.10, the discrete particles for the results presented have been removed

completely from the system after being consumed by bacteria. A few alternatives were

considered including re-introducing particles along the cross-section of the fracture. However, if

particles are re-introduced too close to the fracture wall they will artificially affect the location of

biofilm growth. Instead particles are re-injected half way between the fracture surfaces at the

entry of the fracture. Since particles are re-injected along the centre of the fracture in is expected

that it will take some time and distance for those new particles to diffuse to the fracture surfaces

to allow for more uniform biofilm growth. In the case of no shear conditions, the biofilm will

clog before any effect from particle behaviour can be observed (Figure 5.10). Whereas, in the

case when sloughing due to shear is enabled, the additional particles eventually reach the fracture

surfaces and the biofilm begins a new phase of increased growth that leads to a clogging event.

Figure 5.10: Sensitivity of the biofilm growth behaviour to whether particles are re-injected after being consumed by a bacteria cell. Re = 50 and FD = 2.35.

Another important numerical modeling factor is the various timescales of the system including

flow, substrate transport and biofilm growth. Flow and substrate transport are dealt with at the

0%

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FD Increase

TimeNo Shear No Shear & Re‐Inject

Shear = 0.04 N/m² Shear = 0.04 N/m² & Re‐Inject

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Biomass Change

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86

same relative timescale, approximately 10-5 seconds, which is much faster than that of biofilm

growth. For the results shown in Figure 5.1 through 5.10, the disconnect of timescales is as

follows: flow and transport are allowed to simulate 0.02 seconds or approximately 1000 time

steps between each biofilm growth step. The assumption holds well for fluid flow since 0.02

seconds is enough time for the flow to reach equilibrium after any incremental changes in

biofilm shape. The effect of various timescales or Time Step (TS) ratios which represent the

number of LBM/RW steps relative to biofilm growth steps are reported (Figure 5.11) and results

are shown indicating the TS ratio between each biofilm growth step. The timescale variations

introduce an unknown discrepancy in the physical representation of time. Therefore, the results

are shown with relative timescales in place of a physical time. Increasing the TS ratio allows

more RW iterations between biofilm steps and substrate particles diffuse farther and are more

likely to move towards the biofilm growing on the fracture walls. This increases substrate

availability and has an effect similar to increasing the diffusion coefficient or substrate

concentration. Also shown in Figure 5.11 in the black dotted lines is the result of re-injecting

substrate particles after consumption by a biofilm node. For the case of re-injecting particles at

high TS ratios the assumption that particle re-injection does not affect results breaks down at

earlier and earlier time. From the results it is evident that at larger TS ratios the increased

number of LBM/RW time steps allows greater time for substrate diffusion between substrate

consumption/biofilm growth steps and all particles are consumed in relatively shorter times

(Figure 5.12). As particles are re-injected growth is allowed to continue.

Figure 5.11: Timescale sensitivity analysis for the case of Re = 50 and FD = 2.35. Results are shown using for various Time Step (TS) ratios between successive steps.

0%

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TimeTS Ratio = 1 TS Ratio = 10 TS Ratio = 100

TS Ratio = 1,000 TS Ratio = 10,000 TS Ratio = 100,000

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87

Figure 5.12: Visualization of sensitivity of the timescale used between successive biofilm iterations. Shown from top to bottom represent Time Step (TS) ratio of 100; 1,000; 10,000; 100,000 and 1,000,000. The segment of fractures shown represents approximately 2 mm of the total 100 mm fracture. All five cases are for Re = 50 and FD 2.35.

A graphical representation of Figure 5.11 is shown in Figure 5.12; a segment of the fracture is

captured and displayed relative to the same cross-section for values with an increasing TS ratio.

TS ratio = 1,000,000

TS ratio = 100,000

TS ratio = 10,000

TS ratio = 1,000

TS ratio = 100

88

For the lowest TS ratios, biofilm structures are narrow and biomass accumulation is minimal. As

the timescale separation increases, more substrate transport time steps are simulated such that

when a biofilm step is calculated, the likelihood of a substrate being adjacent to the biofilm in

increased. The net result is similar to increasing the diffusion coefficient or initial substrate

concentrations. By the largest TS ratios all substrate is consumed and a more homogenous

biofilm develops.

The effects of fracture roughness for the case of a TS ratio of 1,000,000 (Figure 5.13) are similar

to the trends for a TS ratio of 1,000 (Figure 5.4) with exception that at largest values, all particles

are quickly consumed, slowing growth. By simulating particle re-injection in the large TS ratio

cases, biofilm growth in not limited and is able to grow until a clogging event. Some differences

are still evident for large TS ratio models for example at late time, for the case when substrate

particles are re-injected into the fractures, the biofilm eventually stops growing into the bulk

fluid, evident from a reduction of FD growth rates. At this time, before a clog develops, the

biofilm will fill in gaps between the initial towering biofilm structures resulting in an overall

decrease in the FD and smoother biofilm (Figure 5.13). Smooth and homogeneous biofilm

structures are expected in substrate rich environments. At the stage when the rate of FD growth

is decreasing, biomass growth rates accelerate and can be explained, in part, by two factors. First,

the amount of biofilm surface area capable of consuming substrate increases as evident from the

FD results. Secondly, the global substrate concentrations are increasing as the volume of fluid in

the fracture constantly decreases with increasing biofilm volume while the discrete number of

particles remains constant.

89

Figure 5.13: Biofilm FD and biomass results for the case of a TS ratio of 1,000,000 for fractures with increasing roughness.

Hydraulic behaviour for large TS ratios values versus the results for TS ratios of 1000 follow

comparable trends, again with variations near the end of the simulation. For a TS ratio of 1,000

hydraulic apertures decreased linearly with increasing biomass with the controlling effects of

roughness and Re determining the rate of decrease. At some point of biomass accumulation,

depending on roughness and Re, the hydraulic aperture decreased rapidly until a clog formed.

For the case of a TS ratio of 1,000,000 hydraulic apertures remain linear throughout biomass

accumulation for both treatments of particle re-injection. Only a single Re is studied for this

case while all fracture roughnesses are compared in Figure 5.14. Increasing fracture roughness

leads to a more rapid decrease in hydraulic aperture as biomass accumulates which is the same

general behavior as the TS ratio of 1,000.

TS 1,000,000 Re‐inject

TS 1,000,000

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90

Figure 5.14: Biomass growth as a percent increase plotted against the relative effective, or hydraulic, aperture for the case of a TS ratio of 1,000,000. Presented hydraulic apertures are normalized to unity for a relative comparison between fractures of varying roughness. The bottom figure enables particle re-injection relative to the top figure.

Varying the diffusion coefficient in the numerical model has the same qualitative effect as

changing the TS ratio as shown by graphically in Figure 5.12. Diffusion coefficients varying

over three orders of magnitude are shown in Figure 5.15. As diffusion coefficients increase,

particles are diffusing much faster leading to high substrate availability and significantly

increased biofilm growth rates. Growth is limited when all particles are consumed which occurs

for these simulations at diffusion coefficients above 7 10 ⁄ . It would be

expected that the growth rates for the highest diffusion coefficients would not stop growing in

the time shown should particles be re-injected.

For the threshold shear strength value of 0.04 Pa the increase in diffusion coefficient results in a

more subtle change in biofilm behaviour however the same general trends hold.

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91

Figure 5.15: Sensitivity analysis for diffusion coefficients in fracture with Re = 50 and FD = 2.35.

Figure 5.16: Sensitivity analysis of initial substrate concentrations in fracture with Re = 50 and FD = 2.35.

The analysis of substrate concentration is conducted by changing the discrete number of particles

in the systems as described in Section 5.2.2. By increasing the number of discrete particles, the

0%

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TimeNo Shear, Dm = 3.5E‐10 m²/sNo Shear, Dm = 7E‐10 m²/sNo Shear, Dm = 1.4E‐9 m²/sNo Shear, Dm = 7E‐9 m²/sNo Shear, Dm = 7E‐8 m²/sNo Shear, Dm = 7E‐7 m²/sShear = 0.04 Pa, Dm = 3.5E‐10 m²/sShear = 0.04 Pa, Dm = 7E‐10 m²/sShear = 0.04 Pa, Dm = 1.4E‐9 m²/s

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No Shear, Cs = 5 g/LNo Shear, Cs = 10 g/LNo Shear, Cs = 20 g/LShear = 0.04 Pa, Cs = 5 g/LShear = 0.04 Pa, Cs = 10 g/LShear = 0.04 Pa, Cs = 20 g/L

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92

likelihood that a particle is near a biofilm cell is increased and growth rates increase with

increasing substrate (Figure 5.16).

Finally, analysis of the reproducibility of the model is shown in Figure 5.17 for the single Re =

50. Three different models runs are conducted with the same input parameters and allowed to

run through until a clogging event occurs. The final values for FD, biomass and the time to a

clogging event show a standard error between one and 6 percent. The variation between model

runs had no relation to initial fracture roughness.

Figure 5.17: Sensitivity analysis of reproducibility of the model in fracture with Re = 50 and FD = 2.35.

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93

5.6 Conclusions

Using a discrete-discrete small scale model has led to complex simulations of biofilm growth in

rock fractures using simple rules and algorithms. Based on the presented work the following

conclusions were made:

1. 2D flow is ultimately controlled by clogging events that occur at local aperture

constrictions. Aperture constrictions tend to be offset between the top and bottom

fracture surfaces since fracture surfaces are not correlated below 15 mm. Biofilm

colonies that develop on a local peak along the surface will tend to grow towards a

biofilm colony on the opposite surface. There is a slight preference for growing in the

upstream direction driven by substrate availability.

2. While biomass typically grows at a constant rate when substrate is available, FD initially

grows quickly then slows at later time. A larger FD is associated with towering biofilm

structures indicating that biofilms need to grow into the bulk flow, away from the wall, to

best capture substrate.

3. Relative growth rates are faster at higher Re indicating the dominance of advection

relative to diffusion for substrate transport for diffusion coefficients similar to sugar.

4. Roughness does not affect relative growth rates for any given Re but rougher fractures

will clog sooner with less biomass.

5. For increasing Re and FD, less biomass is required for the same reduction in the effective

hydraulic aperture. Rougher fractures may have smaller aperture constrictions and when

biomass accumulates at these locations, flow is reduced and clogging is likely to occur

sooner, at lower biomass levels. Similarly, for increasing Re, growth rates are faster,

reducing effective hydraulic aperture more quickly.

6. Shearing of biofilm significantly controls the biomass growth rates. The threshold at

which biofilm can form is dependent on shear strength, Re and roughness.

94

Chapter 6

Conclusions and Recommendations

6.1 Overall Conclusions

Using a newly developed high performance 2D model, the effects of roughness on fluid flow,

substrate transport and biofilm development in a single rock fracture are examined. A lattice

Boltzmann model (LBM) is used to simulate fluid flow through fractures and has been used to

investigate the initiation and growth of secondary flow, such as eddies, within irregular fractures.

Also the model simulates discrete particle transport throughout the fracture as well as discrete

biofilm growth using a cellular automata (CA) approach. All three physical phenomena are

strongly influenced by the roughness of the fracture which can be described by a fractal

dimension (FD). In addition, the Reynolds number (Re), substrate diffusion coefficient, biofilm

shear strength play an important role in the behaviour of biofilm in a fracture. The following

conclusions summarize the overall results and observations:

1. The LBM model simulations presented in this study are consistent with other modeling

studies of flow in rock fractures (Brown, 1987; Tsang, 1984) and also fit well with the

statistical roughness model described by Zimmerman et al. (1991) and Renshaw (1995)

(Chapter 1 and 2).

2. The LBM model is well suited for simulating laminar flows through systems where

complex flow patterns are produced by the combination of pressure gradients on the fluid

as it interacts with the kind or irregular boundaries found in rock fractures. Even under

laminar flow conditions, tortuous flow paths and surface roughness create unique flow

conditions that the current model can effectively capture. The model efficiently

simulates 2D fracture systems at the micron to millimeter scale and allows real-time

rendering of the resulting flow. The general purpose graphics processing unit (GPGPU)

implementation of LBM can simulate systems faster, by an order of magnitude, compared

to CPU based codes, allowing for faster analysis and efficient parametric studies.

(Chapter 2).

95

3. Significantly, this work shows that eddies may be present at virtually all scales of flow in

fractures, and that their first formation extends below previously reported Reynolds

numbers reported in the literature (Crandall et al., 2010). Eddies at the lowest Reynolds

numbers may require a minimum roughness, or a zone of rapid aperture change, to be

formed in fracture (Chapter 3).

4. An important threshold for eddy development occurs near a Reynolds number of around

unity: beyond this value, any existing eddies experience more rapid growth and new

eddies form more readily; this threshold is consistent with earlier work (Zimmerman et

al., 2004). However, unlike previous work, the growth is herein attributed to the complex

flow arising at the boundaries, such as eddies, that are directly associated with the change

in effective hydraulic aperture.

5. This eddy growth behaviour suggests a three-zone non-linear model of fracture flow

similar to that found for porous media by Chaudhary et al. (2011). This work expands

the application of the three-zone model to rough fractures. In Zone I at Re < 1 effective


approaching 1 where conventional fracture modeling breaks down as a result of the


the boundary of Zone II and Zone III and can vary for the fracture system being modeled

(Chapter 3).

6. The three-zone model of fracture flow also applies to tortuosity as the growth of eddies in

a fracture are directly linked to a non-linear change in measured tortuosity (Chapter 3).

7. Solute transport is affected similarly to fluid flow as is clearly demonstrated through the

flow studies that use the same set of fractures as in the hydraulic study with increasing

roughness. Initially, at Reynolds numbers less than 10, little if any difference is apparent

comparing the analytical solutions with fractures of different roughnesses. At larger Re,

especially for Reynolds numbers exceeding twenty, a significant change in behaviour is

observed with increasing roughness possibly explained by the emergence and growth of

eddies associated with the three-zone model of fracture flow (Chapter 4).

8. For low flow rates, a Fickian diffusion model is shown to accurately represent transport

phenomena. However, deviations of the breakthrough curves from Fickian behaviour

96

are progressively noted with higher flows; significantly, such deviations begin to appear

at the same range of Re and FD shown in Chapter 3 to be associated with the onset and

growth of secondary flows (Chapter 4).

9. At the highest roughness and Reynolds numbers modeled, it is clear that the fluid flow

interacting with unique fracture geometries create a non-linear response to solute

transport and as shown by the three-zone model of fracture flow, eddy formation is a key

factor in that behaviour (Chapter 4).

10. Using a discrete-discrete small scale model permits complex simulations of biofilm

growth in rock fractures using simple rules and algorithms. As expected 2D flow is

controlled by clogging events. This can occur anywhere along the fracture aperture and

act to stop fluid flow in the model, however the tendency is for biofilm to develop in

areas of aperture constriction (Chapter 5).

11. While biomass typically grows at a constant rate when substrate is available, the biomass

FD initially grows quickly then slows at later time. A larger FD is associated with

“towering biofilm structures” (structure which stretch widely across the flow aperture)

indicating, not surprisingly, that biofilms need to grow into the bulk flow, away from the

wall, to best capture substrate (Chapter 5).

12. Relative growth rates of biofilms are faster at higher Re suggesting the dominance of

advection to substrate transport for diffusion coefficients similar to sugar (Chapter 5).

13. Roughness does not affect relative growth rates for any given Re however rougher

fractures will clog sooner with less biomass. (Chapter 5).

14. For increasing Re and FD, less biomass is required to reduce the effective hydraulic

aperture. Rougher fracture, may have smaller aperture constrictions and when biomass

accumulate at these locations, flow is reduces and clogging is likely to occur sooner, with

less biomass present. Similarly, for increasing Re, growth rates are faster, reducing

effective hydraulic aperture more quickly (Chapter 5).

15. Introducing biofilm shear strength significantly controls the biomass growth rates. The

threshold at which biofilm can form is dependent on shear strength, Re an FD (Chapter

5).

97

6.2 Contributions

The significant contributions of this thesis are as follows:

1. The 2D biofilm model developed herein is the first to systematically analyse the effects

of roughness and secondary flows on the hydraulics, transport and biofilm behaviour in

fractured media. The model is shown to be capable of efficiently simulating three primary

phenomena: gravity driven flow through a single rock fracture using the LBM, solute

transport using RW and biofilm population dynamics using CA.

2. The 2D biofilm model uses high performance GPGPU programming to significantly

improve performance of the LBM and RW algorithms decreasing analysis times and

increasing the number of models feasible to run for any given project.

3. The presence of eddies was detected at Reynolds numbers below that previously reported

in the literature. The model can be used to determine when flow become complex and

require CFD tools, such as the presented model, to describe the hydraulic properties of a

fracture.

4. This work extends the three-zone non-linear model of flow in porous media to include

fractured media. The transition between these three zones is shown to be correlated with

the onset and growth of secondary flows in a fracture. In Zone I at Re < 1 effective


approaching 1 where conventional fracture modeling will break down as a result of the


the boundary of Zone II and Zone III and can vary for the fracture system being modeled

5. The emergence and growth of eddies in fractures are shown to correlate with the

transition to non-Fickian behaviour of breakthrough curves in a single rock fracture.

6. Based on the presented 2D model, fracture roughness does not affect relative growth rates

of biofilm in a single rock fracture. Instead, rougher fractures when modeled in 2D are

clogged more quickly than smoother fractures. However fracture roughness does affect

the reduction in effective hydraulic aperture as flow in rougher fractures will drop more

quickly than smooth fractures.

7. The 2D biofilm model can be used to determine biofilm growth rates and time-to-

clogging given as input: biofilm shear strength, Re and FD. Ideally an engineered

bioremediation site would encourage biofilm development while discouraging clogging

98

possibly by controlling induced flow rates or the choice of augmented bacterial

populations.

6.3 Critical Appraisal

Experience with the methods and methodologies used here offer insight into their respective

strengths and weaknesses. To this end, it is desirable to evaluate the choices made and discuss

their ramifications for this work with the benefit of hindsight.

The biofilm model is inherently deterministic and it does not consider the stochastic nature of the

hydrologic cycle, for example the uncertainty of rain events that drive most groundwater flows at

some time scale. The geologic conditions are also transient, undergoing changes in confinement

pressures over time leading to changing flow rates and aperture dilations and contractions. As

fracture networks within bedrocks change as new fractures propagate, old pathways close or

become clogged with rock fragments and debris. Within a host rock, rock properties will vary

spatially, the nature of the roughness, anisotropy or relative matrix permeability may change

over time and space. Bacterial communities are also always changing to best suit the

surrounding environment without even considering the anthropogenic components of

contamination or other engineering activities. A system that attempts to consider some or all of

these factors would be required to move towards a stochastic modeling regime, for example

using Monte Carlo methods. A broader parametric analysis significantly increases computation

cost and would benefit from using emerging parallel computer systems such as the GPGPUs

used in the work. Even so, the scalability of the presented model is unknown and

implementation would impose its own challenges. Ideally to minimize the parametric space, 2D

modeling of various system inputs and properties would give a better understanding of their

relative impacts and whether to they need to be included in Monte Carlo analysis.

The use of a 2D model limits specific results of the thesis however it is expected that the overall

results would still hold. Specifically, in 3D, biofilm structure and growth would be able to

expand in a further dimension and reduce the impact of clogging in 2D. Preferential flow paths

develop in 3D around clogging sites however flows would still be reduced as it is in 2D. Growth

outside the preferential flow paths would slow as they would be limited by substrate diffusion

while growth in the advective dominant zones would be faster but balanced with sloughing.

Similar behaviour was shown for 2D systems with maximum shear strengths applied to the

biofilm, it might have been expected that biofilm could not grow in high shear zones, and they

99

did have had delayed initial growth times, but as biomass and structure increase, the pressure

driven flow runs into more resistance. Eventually, flows diminish to an extent where biofilms

can develop and similar clogging events occur even with relatively low biofilm shear strengths.

Developing a 3D model is still a primary consideration for the extension of this work however

the complexity of implementing a 3D system in addition to the computation cost maintains the

applicability and usefulness of 2D work

The use of high performance computing will always be required when modeling systems with

large grid sizes and small time steps. With modern computers this leads to the use of parallel

computing systems of which GPGPUs are and one of the least expensive options. They offer a

pre-packaged high performance computer with little to no additional hardware requirements. One

of the big costs with GPGPUs comes with the implementation at the software level. GPGPUs

evolved in a graphics based software ecosystem where the types of problems being solved

suggest an optimal hardware design. When extending GPUs to general purpose programing a

few key issues quickly control the performance efficiency. First, GPUs have limited bandwidth

between host and GPU. Ideally any given problem should be solved entirely on the GPU to

minimize the impact of the interconnect bottleneck. This is a reasonable and achievable goal in

high performance computing as it is always beneficial to minimize external communications

whether it is between multiple nodes of a cluster or between GPUs. Secondly, this generation of

GPGPUs consist of many small processors with relatively simple logic implementations in

comparison to modern CPUs which are composed of a few large processors. Therefore it can be

expected that each type of hardware architecture is most suitable for different tasks. For

GPGPUs, code should be generated that takes advantage of hundreds of processors doing the

same task or what is known as Single Instruction Multiple Data (SIMD). In this way, the same

instruction is sent to every processor to work on its given slice of data. This works well for both

the LBM and RW implementations as each node of the grid works on similar instructions and

applies those rules to the local available data, taking information from the individual node or

immediate neighbour. However the biofilm growth model requires significant neighbour

communication at levels above immediate neighbours. During a bacteria growth step, the code

must choose at random where to grow and then check the validity of that choice. If all

immediate neighbours are occupied, the algorithm becomes recursive as it begins to shift cells

and make room for daughter cells. Both of these steps require conditional branching and

recursive functions that are not traditional strengths of GPGPUs because they break the single

100

instruction component of SIMD, during a conditional statement, there are now two instructions

for each data component. Some GPGPU implementations will calculate both branches of a

conditional statement and simply discard the unused branch as it is faster than first calculating

one, then checking the results and deciding if the other also needs calculating. Regardless, a

significant performance penalty is taken for branching and recursion. Therefore the goal would

be to create a biofilm growth model that requires little if no branching or recursion, such a

method is best suited for research in the field of computer science and collaboration with said

group would be beneficial.

6.4 Future Work

The results of this research indicate that more work is needed to refine the model and improve its

predictive capacity. To this end, several recommendations are listed below and organized into

three groups: Extending the types of fractures being modeled, numerical modeling improvements

and lab scale validation.

The existing model has several features which have not been completely explored. To this end

the following future work is recommend:

1. Compare the presented gravity based boundary conditions with other boundary

conditions, for example constant flux. This would improve the scope of problems that

could be simulated with the presented model.

2. Analyze fracture flow, transport and biofilm development results controlling for other

fracture properties including mismatch length and anisotropy. Results would extend the

conclusions already discussed to include a larger variation in fracture topologies.

3. Compare synthetic fractures and real fractures with respect to biofilm modeling and the

effects of fracture roughness.

4. Extend the single fracture results to include simple fracture networks. Moving from a

single fracture to fractured networks would allow for more general conclusions regarding

flow, transport and biofilm population dynamics in fractured media.

It is a constant requirement that numerical models be improved in terms of accuracy, ability and

performance while simultaneously reducing trade-offs. Accordingly, the following future work

is recommended to improve the numerical model developed in this thesis:

101

1. Extend the biofilm, substrate transport and fluid dynamics models to three-dimensions to

capture the expected channeling effect of fluid moving through fracture surfaces and

around biofilm colonies.

2. While moving to three-dimensional modeling is beneficial, the computational costs and

complexities of implementation are significant. Instead, developing methods for

simulating 3D phenomena in 2D could be beneficial. For example the impact of 2D

clogging could be reduced by introducing a variable biofilm permeability factor

dependent on time and space that would allow fluid to move through biofilm colonies to

simulate the effects of 3D tortuosity. The factor could be calibrated using real 3D

experiments.

3. Implement a larger variety of biofilm states. The current CA model considers bacteria to

be either active or non-existent whereas real bacteria exhibit more complex traits

including: Attached, detached, active and dormant roles and their respective

transformations. Furthermore differentiating between EPS structure and biomass would

allow for more realistic comparison with biofilms found in the lab and in natural

environments.

4. Extend the biofilm model to include a more complex model biofilm structure including

treating it as viscoelastic material and separating the biofilm processes of sloughing and

erosion.

5. Improve the parallel implementation of biofilm population dynamics to bring it up to par

with the fluid and substrate transport implementations. This would minimize the effects

of the broad timescales required and discussed in section 5.4.

Finally, it is important to continue improving the biofilm model with respect to its ability to

model biofilms actually seen in the lab and natural environment. It would be recommended to

conduct experiments to compare the behaviour of fluid, transport and biofilm growth to

numerical results of the model. Fracture profiles and surfaces can be scanned and modeled

numerically while also run in the lab at controlled flow rates. Implementing the improved

numerical steps as list above would enable running lab scale comparison tests to calibrate the

timescale effects discussed in section 5.4. Model comparison with lab scale work would

significantly strengthen the presented work and potentially lead to new avenues of research and

contribute to engineering recommendations in the field. Specifically, the model results to date

suggest that if the fractured media could be characterized for its geometry and roughness, biofilm

102

growth could be qualitatively predicted along with the likelihood of clogging and the sensitivity

to a range of parameters for example: Re, shear strength and diffusion coefficients.

103

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