Impact of Biofilm Formation in Microbial Enhanced Oil ...

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Impact of Biofilm Formation in Microbial Enhanced Oil

Recovery Performance

Cao, Jiayi

Cao, J. (2018). Impact of Biofilm Formation in Microbial Enhanced Oil Recovery Performance

(Unpublished master's thesis). University of Calgary, Calgary, AB.

http://hdl.handle.net/1880/109413

master thesis

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UNIVERSITY OF CALGARY

Impact of Biofilm Formation in Microbial Enhanced Oil Recovery Performance

by

Jiayi Cao

A THESIS

SUBMITTED TO THE FACULTY OF GRADUATE STUDIES

IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE

DEGREE OF MASTER OF SCIENCE

GRADUATE PROGRAM IN CHEMICAL ENGINEERING

CALGARY, ALBERTA

DECEMBER, 2018

© Jiayi Cao 2018

ii

Abstract

The complex trade-off between the effects of biosurfactant generation and biofilm growth has

been a challenge for successful simulation and field-scale implementation of microbial-enhanced

oil recovery (MEOR). In this work, a two-phase, two-dimensional MEOR model is developed,

including effects from interfacial tension, porosity and permeability reduction. Empirical models

are validated against experimental data. The model is discretized through two-point flux-

approximation, and the numerical solution is validated against the two-phase Buckley-Leverett

equation. The model shows a good match with previous MEOR simulation results. Monte Carlo

simulation-based sensitivity analyses of various operational parameters in a homogeneous

reservoir highlight the importance of bacteria injection concentration, which can result in 16%

difference in oil recovery by minimizing biofilm formation and optimizing biosurfactant

production. Variation in biosurfactant critical micelle concentration and biofilm density is found

to increase oil recovery by up to 15%, indicating that both strain selection and injection

concentration should accommodate reservoir rock and fluid properties.

iii

Acknowledgements

I would like to express my sincere gratitude to my supervisor, Dr. John Chen, and co-supervisor,

Dr. Hector De la Hoz Siegler, for their support, encouragement and guidance.

Thanks also to the Reservoir Simulation Group and its sponsors.

iv

Dedication

To my parents, brother and fiancé.

v

Table of Contents

Abstract ........................................................................................................................................... ii Acknowledgements ........................................................................................................................ iii

Dedication ...................................................................................................................................... iv Table of Contents .............................................................................................................................v List of Tables ................................................................................................................................ vii List of Figures and Illustrations ................................................................................................... viii List of Symbols, Abbreviations and Nomenclature ....................................................................... xi

1 CHAPTER 1: INTRODUCTION ...........................................................................................1 1.1 General Introduction ............................................................................................................1 1.2 Objectives ............................................................................................................................4

2 CHAPTER 2: LITERATURE REVIEW ................................................................................6 2.1 MEOR Overview .................................................................................................................6

2.1.1 Advantages and Disadvantages of MEOR .....................................................................8

2.2 Impact of Reservoir Conditions on Microbial Communities ...............................................9 2.2.1 Parameters Controlling Bacterial Reactions ..................................................................9

2.3 MEOR Methods .................................................................................................................12 2.4 Field Applications ..............................................................................................................13 2.5 MEOR Mechanisms ...........................................................................................................15

2.5.1 Interfacial Tension Reduction ......................................................................................15 2.5.2 Biofilm Growth ............................................................................................................16

2.6 MEOR Mathematical Models ............................................................................................19 2.6.1 Summary of MEOR Numerical Models ......................................................................22

2.7 Numerical Modelling of Porous Media Transport .............................................................26 2.7.1 Finite Difference Method .............................................................................................26

2.7.2 Finite Element Method ................................................................................................27 2.7.3 Finite Volume Method .................................................................................................28

3 CHAPTER 3: MEOR MODEL ............................................................................................29

3.1 Reservoir Model.................................................................................................................29 3.1.1 Phase Pressure and Saturation for Incompressible Fluids ...........................................34

3.2 MEOR Model.....................................................................................................................35 3.2.1 Microbial Growth and Surfactant Production Kinetics ...............................................35

3.2.2 Biofilm Model ..............................................................................................................38 3.2.3 Biosurfactant effect ......................................................................................................42

3.3 Summary of Equations and Mechanisms ...........................................................................46

3.4 Assumptions .......................................................................................................................49

4 CHAPTER 4: NUMERICAL MODEL ................................................................................51 4.1 Matlab Reservoir Simulation Toolbox ...............................................................................51 4.2 Finite Volume Discretization .............................................................................................52

4.2.1 Gridding .......................................................................................................................52 4.2.2 Two-Point Flux Approximation ...................................................................................53 4.2.3 Time discretization.......................................................................................................59

vi

4.3 IMPES Method ..................................................................................................................61

4.3.1 Boundary conditions ....................................................................................................65 4.3.1.1 Dirichlet Boundary Condition........................................................................65 4.3.1.2 Neumann Boundary Condition ......................................................................65

4.3.2 Solution Algorithm and MRST Implementation .........................................................66 4.4 Two-Phase Flow Analytical Solution ................................................................................67 4.5 Comparison to MEOR Simulations ...................................................................................70 4.6 Methodology ......................................................................................................................72

5 CHAPTER 5: MEOR SIMULATION AND SENSITIVITY ANALYSIS ..........................74

5.1 Simulation Results .............................................................................................................74 5.1.1 Initial Conditions .........................................................................................................74

5.1.1.1 Microbial input parameters ............................................................................76 5.1.2 Surfactant effect in a one-dimensional model .............................................................77

5.1.3 Surfactant effect in a two-dimensional model .............................................................82 5.1.4 Biofilm effect in a one-dimensional model .................................................................85

5.2 Biosurfactant Parameters Sensitivity Analysis ..................................................................91 5.2.1 Initial conditions ..........................................................................................................91

5.2.2 Biosurfactant parameters for IFT reduction .................................................................92 5.2.3 Biosurfactant type versus production rate ....................................................................94

5.3 Growth Rate Sensitivity Analysis ......................................................................................97

5.3.1 Initial conditions ..........................................................................................................97 5.3.2 Impact of maximum specific biomass growth rate ......................................................98

5.3.3 Impact of maximum specific biosurfactant growth rate ............................................100 5.3.4 Impact of critical nutrient concentration ....................................................................102

5.4 Biofilm and Biosurfactant Sensitivity Analysis ...............................................................104

5.4.1 High reservoir porosity and permeability ..................................................................104

5.4.1.1 Impact of microbe injection concentration and maximum specific growth

rate .....................................................................................................................105 5.4.1.2 Combined impact of microbe injection concentration and maximum

specific growth rate ...........................................................................................107 5.4.2 Low reservoir porosity and permeability ...................................................................110

5.4.2.1 Impact of microbe injection concentration and maximum specific growth

rate .....................................................................................................................110

5.5 Summary of Sensitivity Analysis Results ........................................................................112

6 CHAPTER 6: CONCLUSION AND RECOMMENDATIONS ........................................114 6.1 Conclusions ......................................................................................................................114 6.2 Recommendations for Future Work.................................................................................116

7 REFERENCES ...................................................................................................................118

vii

List of Tables

Table 2-1: Microbial Products and Associated Recovery Mechanisms ......................................... 7

Table 2-2: Summary of Key Numerical Models for MEOR to Date ............................................ 25

Table 3-1: Calibrated Values for IFT Reduction Model for Three Different Biosurfactants ....... 44

Table 4-1: Rock and Fluid properties for Model Validation ........................................................ 67

Table 4-2: Summary of Grid Size, Time Step Size, and Computational Time Required for 9

Simulation Validation Tests .................................................................................................. 68

Table 5-1: Summary of Microbial Input Parameters for MEOR Simulation ............................... 76

Table 5-2: Summary of Simulation Conditions for Biosurfactant ................................................ 77

Table 5-3: Summary of Simulation Conditions for Biofilm ......................................................... 86

Table 5-4: Summary of Input Parameters for Biosurfactant Sensitivity Analysis ........................ 92

Table 5-5: Summary of Input Parameters for Growth Rate Sensitivity Analysis ......................... 97

Table 5-6: Summary of Oil Recovery Sensitivity Analysis........................................................ 112

viii

List of Figures and Illustrations

Figure 3-1: Concentration of biofilm bacteria versus free bacteria under two sets of Langmuir

coefficients, (a) α1= 0.001, α2= 0.001 and (b) α1= 0.001, α2= 0.0017 .............................. 40

Figure 3-2: Validation of Kozeny grain-coating model against experimental data (Abbasi et

al., 2015) ............................................................................................................................... 42

Figure 3-3 Validation of the Nielsen IFT reduction model to experimental data from Daoshan

et al. (2014); Pereira et al. (2013); and McInernery et al. (2004) ......................................... 44

Figure 3-4: Summary of MEOR mechanisms .............................................................................. 47

Figure 4-1: Cell-centered two-dimensional grid ........................................................................... 53

Figure 4-2: Time discretization ..................................................................................................... 59

Figure 4-3: Relative permeability curve used for numerical model validation against

Buckley-Leverett analytical solution .................................................................................... 68

Figure 4-4: Effect of decreasing grid size on numerical simulation convergence to analytical

solution for two-phase, one-dimensional incompressible flow ............................................ 69

Figure 4-5: Effect of decreasing time step size on numerical simulation convergence to

analytical solution for two-phase, one-dimensional incompressible flow ............................ 70

Figure 4-6: Comparison of microbial transport to one-dimensional simulational results from

Sivasankar (2014) ................................................................................................................. 71

Figure 4-7: Comparison of microbial transport under sorption, or biofilm formation effects,

to simulation results from Sivasankar (2014) ....................................................................... 72

Figure 5-1: Oil-water relative permeability for a water-wet rock ................................................. 75

Figure 5-2: Effect of biosurfactant concentration on interfacial tension ...................................... 77

Figure 5-3: (a) Reservoir saturation and (b) pressure profile for a 1D waterflood simulation ..... 78

Figure 5-4: (a) Saturation, (b) metabolite concentration, and (c) bacteria/nutrient profile for a

low injection concentration MEOR case .............................................................................. 79

Figure 5-5: (a) Water saturation and (b) metabolite concentration profiles for a high bacteria

concentration MEOR case .................................................................................................... 80

Figure 5-6: (a) Metabolite concentration and (b) bacteria and nutrient concentration profiles

for a high bacteria specific growth rate MEOR case ............................................................ 81

Figure 5-7: Nutrient concentration in a 2D MEOR simulation after (a) 0.01, (b) 0.9, (c) 1.9,

and (d) 3.7 years. ................................................................................................................... 83

ix

Figure 5-8: Microbe concentration in a 2D MEOR simulation after (a) 0.01, (b) 0.9, (c) 1.9,

and (d) 3.7 years. ................................................................................................................... 83

Figure 5-9: Surfactant concentration in a 2D MEOR simulation after (a) 0.01, (b) 0.9, (c) 1.9,

and (d) 3.7 years. ................................................................................................................... 84

Figure 5-10: Water saturation distribution in a 2D MEOR simulation after (a) 0.01, (b) 0.9,

(c) 1.9, and (d) 3.7 years. ...................................................................................................... 85

Figure 5-11: Oil recovery for a low bacteria and nutrient injection concentration, and normal

biofilm density MEOR case .................................................................................................. 87

Figure 5-12: Permeability and porosity profiles across the 1D reservoir after 1000 days of

MEOR with biofilm formation ............................................................................................. 88

Figure 5-13: Nutrient, microbe and surfactant profiles across the 1D reservoir after 1000 days

of MEOR (a) with biofilm formation (b) without biofilm formation ................................... 89

Figure 5-14: (a) Oil recovery for high specific maximum microbe growth rate MEOR after

1500 days and (b) permeability concentration profiles across the 1D reservoir after 500,

1000 and 1500 days of MEOR utilizing high growth rate microbes at higher injection

concentration ......................................................................................................................... 90

Figure 5-15: (a) Oil recovery for a low biofilm density MEOR case and (b) Permeability and

microbe concentration profiles across the 1D reservoir after 500, 1000 and 1500 days ...... 91

Figure 5-16: Surface plot of (a) Oil recovery and (b) Minimum oil saturation at 20 days to

surfactant parameters, l1 and l3 ............................................................................................ 93

Figure 5-17: Contour plot of oil recovery at 20 days to surfactant parameters, l1 and l3 ............ 93


surfactant parameters, μm,met and l1 .................................................................................. 95

Figure 5-19: Contour plot of Oil recovery at 20 days to surfactant parameters, l1 and

μm,met ................................................................................................................................. 95

Figure 5-20: (a) Minimum oil saturation and (b) Oil recovery as a function of time for

different μm,met and l1 values ............................................................................................ 96

Figure 5-21: Oil recovery versus time for varying maximum specific microbe growth rates ...... 99

Figure 5-22: Minimum oil saturation versus time for varying maximum specific microbe

growth rates ........................................................................................................................... 99

Figure 5-23: Effect of varying maximum specific biosurfactant growth rate on oil recovery

and minimum oil saturation ................................................................................................ 100

x

Figure 5-24: Minimum oil saturation versus time for varying maximum specific surfactant

growth rates ......................................................................................................................... 101

Figure 5-25: Interaction effect of maximum specific microbe and surfactant growth rate on

oil recovery ......................................................................................................................... 102

Figure 5-26: Oil recovery for different values of maximum specific microbe and surfactant

growth rate, with critical nutrient limitation (red) and without critical nutrient limitation

(blue) ................................................................................................................................... 104

Figure 5-27: Effect of microbe injection concentration on (a) oil recovery and (b) minimum

oil saturation at 500, 1000, 1500, and 2000 days under moderate porosity and

permeability conditions ....................................................................................................... 106

Figure 5-28: Effect of maximum specific microbe growth rate on (a) oil recovery and (b)

minimum oil saturation at 500, 1000, 1500, and 2000 days under moderate porosity and

permeability conditions ....................................................................................................... 107

Figure 5-29: Impact of combined variation in microbe injection concentration and maximum

specific growth rate on oil rate at (a) 2000 days and (b) 1000 and 2000 days ................... 109

Figure 5-30: Oil recovery at 500, 1000, 1500, and 2000 days under low reservoir porosity

and permeability conditions for (a) varied microbe injection concentration and (b) varied

maximum specific microbe growth rate ............................................................................. 111

xi

List of Symbols, Abbreviations and Nomenclature

Symbol Definition

𝐴𝑟 Ratio of rock surface area to volume, m2/m

3

𝑎 Coat’s interpolation constant

𝐵 Concentration, total bacteria, kg/m3

𝐵𝑏 Concentration, bacteria adsorbed to biofilm, kg/m3

𝐵𝑓 Concentration, free bacteria, kg/m3

𝐵𝑖𝑛𝑗 Injection concentration, bacteria, kg/m3

𝐵𝑜 Formation volume factor, oil, m3/std m

3

𝐵𝑤 Formation volume factor, water, m3/std m

3

𝐶 Grain-coating model constant

𝑓𝑜 Fractional flow, oil

𝑓𝑤 Fractional flow, water

𝑔 Gravitational constant, m/s2

�⃗⃗� Absolute permeability tensor, mD

𝐾𝑏 Half-saturation concentration, bacteria, kg/m3

𝐾𝑚 Half-saturation concentration, metabolite, kg/m3

𝑘𝑟𝑜 Relative permeability, oil

𝑘𝑟𝑤 Relative permeability, water

𝑘𝑟𝑜𝑤𝑖 Relative permeability value, oil at initial water saturation

𝑘𝑟𝑤𝑜𝑟 Relative permeability value, water at residual oil saturation

𝑘𝑟𝑖,𝑖𝑚𝑚 Initial relative permeability for Coats' interpolation

𝑘𝑟𝑖,𝑚𝑖𝑠𝑐 Miscible relative permeability for Coats' interpolation

𝑙{1,2,3} Biosurfactant IFT reduction parameters

𝑀 Concentration, metabolite (surfactant), kg/m3

𝑀𝑖𝑛𝑗 Injection concentration, metabolite (surfactant), kg/m3

�⃗� Unit normal

𝑁 Concentration, nutrient, kg/m3

𝑁𝑖𝑛𝑗 Injection concentration, nutrient, kg/m3

𝑁𝑐𝑟𝑖𝑡 Critical nutrient concentration for surfactant production, kg/m3

𝑁𝑐𝑎 Capillary number

𝑛 Constant, Corey relative permeability

𝑝 Pressure, kPa

𝑝𝑐 Pressure, capillary, kPa

𝑝𝑜 Pressure, oil, kPa

𝑝𝑤 Pressure, water, kPa

𝑞𝑏 Source term, bacteria concentration, kg/m3

𝑞𝑛 Source term, nutrient concentration, kg/m3

𝑞𝑜 Source term, oil, m3/d

𝑞𝑤 Source term, water, m3/d

𝑅𝑏 Reaction term, bacteria, kg/m3 d

𝑅𝑛 Reaction term, nutrient, kg/m3 d

𝑅𝑚 Reaction term, metabolite (surfactant), kg/m3 d

𝑆 Surface of a volume

xii

𝑆𝑜 Saturation, oil

𝑆𝑤 Saturation, water

𝑆𝑜𝑟 Residual saturation, oil

𝑆𝑤𝑖 Initial saturation, water

𝑆𝑤𝑚𝑎𝑥 Saturation, water at residual oil saturation

𝑡 Time, d

𝑇𝑖 Transmissibility in 𝑥𝑖, m-d

�⃗� Darcy velocity, total, m/d

�⃗� 𝑜 Darcy velocity, oil, m/d

�⃗� 𝑤 Darcy velocity, water, m/d

𝑌𝑏 Yield term, bacteria

𝑌𝑚 Yield term, metabolite (surfactant)

z Depth, m

𝛼{1,2} Langmuir model parameters

Θ Mass of bacteria adsorbed per unit area, kg/m2

𝜂 Fluid viscosity, cp

𝜂𝑜 Viscosity, oil, cp

𝜂𝑤 Viscosity, water, cp

𝜃 Contact angle

𝜆 Phase mobility, total, m d/kg

𝜆𝑜 Phase mobility, oil, m d/kg

𝜆𝑤 Phase mobility, water, m d/kg

𝜇𝑏 Specific growth rate, bacteria, 1/hr

𝜇𝑚 Specific growth rate, metabolite (surfactant), 1/d

𝜇𝑚,𝑚𝑖𝑐 Maximum growth rate, microbe (bacteria), 1/d

𝜇𝑚,𝑚𝑒𝑡𝑎 Maximum growth rate, metabolite (surfactant), 1/d

𝜌 Density, kg/m3

𝜌𝑜 Density, oil, kg/m3

𝜌𝑠,𝑜 Density at standard conditions, oil, kg/m3

𝜌𝑤 Density, water, kg/m3

𝜌𝑠,𝑤 Density at standard conditions, water, kg/m3

𝜎 Interfacial tension, mN/m

𝜙 Porosity

Ω Fixed region

Abbreviation Definition

CMC Critical micelle concentration

FDM Finite difference method

FEM Finite element method

FVM Finite volume method

xiii

EOR Enhanced Oil Recovery

IFT Interfacial tension

IMPES Implicit Pressure Explicit Saturation

IOR Improved Oil Recovery

MEOR Microbial Enhanced Oil Recovery

MFD Mimetic Finite Difference

MRST MATLAB Reservoir Simulation Toolbox

MPFA Multi-Point Flux Approximation

MsMFE Multi-scale Mixed Finite Elements

MsTPFA Multi-scale Two-Point Finite-Approximation

OOIP Original Oil In Place

SINTEF Stiftelsen for industriell og teknisk forskning (Norwegian)

TPFA Two-Point Flux-Approximation

1

1 CHAPTER 1: INTRODUCTION

1.1 General Introduction

Primary oil recovery typically harvests 20% of the original oil in place (OOIP), whereas

secondary oil recovery will recover an additional 20–40% of the OOIP. Combined, primary and

secondary recovery can leave as much as half of the OOIP within the reservoir. Improved oil

recovery (IOR) or enhanced oil recovery (EOR) techniques, such as gas injection, surfactant

flooding, polymer flooding, CO2 flooding, and thermal recovery, are therefore necessary to

maximize the yield of the remaining crude oil. However, these methods are often associated with

various economic or environmental risks. For example, the use of certain chemical surfactants

may not only be detrimental to the environment in and of itself, but also have secondary impacts

due to the toxic chemicals required for its generation (Banat, 1995). To mitigate these risks,

renewed attention has been directed towards greener and more sustainable, yet economic

alternatives. Microbial enhanced oil recovery (MEOR) is one such alternative. In this

technology, residual oil mobilization is mediated through the injection of microbes, nutrients, or

a combination thereof into the reservoir.

Since Beckman first proposed utilizing microbes for mobilizing oil from porous media in 1926

(Beckman, 1926), MEOR through bacteria injection has been applied in both sandstone and

carbonate reservoirs with varying permeabilities. Recently, it has been successfully implemented

in field trials for heavy oils with API gravity as low as 16o (Patel et al., 2015). By exploiting

microbial reproduction and production of secondary metabolites, such as biopolymers,

biosurfactants, gases, and acids, MEOR can enable the selectively plugging of high-permeability

2

zones, reduce interfacial tension, reduce oil viscosity, alter rock wettability, and increase

permeability.

Both interfacial tension reduction via surfactant production and selective plugging via biofilm

formation have been identified as dominant mechanisms in MEOR field trials (Gao & Zekri,

2011). Generally, interfacial tension of 1 mN/m or below is required to facilitate the mobilization

of residual oil and significantly improve oil recovery (Gray et al., 2008). However, due to the

subsurface dilution of biosurfactant during microbial waterflooding and the difficulty of

transporting both bacteria and nutrients to the target zone (Rashedi et al., 2012), this requirement

cannot be easily met in most biosurfactant-based MEOR field applications.

In selective plugging, particulate bacteria in the size range of 1 micron or flocculated bacteria

colonies can become trapped in comparatively sized pore throats (Nielsen, 2010), restricting

fluid flow. Alternatively, biofilm formation and coating of rock surfaces can reduce reservoir

permeability. Fluid flow can then be shifted to lower permeability zones. The successful

alteration of the injection profile and mobilization of oil from unswept reservoirs has been

reported in multiple field trials in the USA (Brown & Vadie, 2002) and China (Hou et al., 2008;

Zhao et al., 2005). Thus, improving oil recovery is substantially more easily achieved through

selective plugging than through interfacial tension reduction.

Despite being more environmentally friendly and cost-effective than alternative EOR

technologies, MEOR is currently not commonly employed in industry. In 2015, Statoil claimed

to be the only company implementing MEOR in an off-shore field (Amundsen, 2016). The

3

complexity of the MEOR process and the number of variables required for optimizing

stimulation of appropriate metabolic pathways pose a significant challenge to its implementation

(Patel et al., 2015). This is compounded by the lack of reservoir simulators built for MEOR

modelling. To date, many commercial simulators are unable to model MEOR-specific reactions,

such as competition between indigenous and exogenous microbial species or influence of pH on

biomass growth (Sen, 2008).

Another challenge to MEOR implementation is that biomass formation, responsible for selective

plugging, also causes well inlet microbial clogging (“bioclogging”) and loss of injectivity, which

is one of the main limiting factors associated with MEOR (Lazar et al., 2007). Similarly, while

enhancement of microbial growth is necessary to generate sufficient surfactant concentration for

interfacial tension reduction, over-stimulation may result in wellbore clogging or impede fluid

flow. Numerous treatments have been proposed to circumvent bioclogging, including prevention

of biopolymer formation, and utilization of spores or ultra-microbacteria (Banat, 1995). Still,

optimization of MEOR is difficult in practice due to the large number of uncertain parameters,

including microbial kinetics, biofilm and biosurfactant properties. To date, a quantitative analysis

simultaneously considering reservoir, bacteria, and surfactant parameters has not been

undertaken in the context of biosurfactant production and biofilm formation.

One of the main objectives of this work is to develop and implement a MEOR model which

couples both surfactant generation and biofilm formation to oil recovery. The process will be

simulated using the MATLAB Reservoir Simulation Toolbox (MRST). MRST is an open-source

toolbox developed for rapid prototyping of new reservoir models currently under development

4

by SINTEF Applied Mathematics in Oslo, Norway. It employs a number of finite volume and

finite element solver modules, including Two-Point Flux Approximation (TPFA), Multipoint

Flux Approximation (MPFA) and Multiscale Mixed Finite Elements. The toolbox has been used

numerous times (Akindipe, 2016; Amundsen, 2016; Nielsen, 2010) for the simulation of MEOR,

which often require the coupling of transport equations to various combinations of reaction

equations.

Using the MEOR simulation model, a series of sensitivity analysis will be performed to

determine the impact of surfactant type, bacteria type, and bacteria concentration on oil recovery

under different permeability and porosity conditions. Both biological and reservoir parameters

must be considered, so that bacteria growth, and subsequently surfactant and biofilm generation,

can be coupled to bacterial transport.

1.2 Objectives

This study aims to investigate the effect of various operating parameters on biofilm formation

and surfactant production, and consequently MEOR performance. The outline of the thesis is as

follows:

1. Review MEOR mechanisms and limitations, field applications, and key mathematical

models;

2. Derive a two-dimensional, two-phase mathematical model of MEOR, considering both

interfacial tension reduction and biofilm formation, and validate the model predictions

with experimental data obtained from the open literature;

5

3. Derive the numerical model through TPFA discretization, and validate the two-phase

incompressible flow model using the Buckley-Leverett analytical solution;

4. Investigate the effect of surfactant type, bacterial injection concentration, bacteria

kinetics parameters, and biofilm density; and

5. Perform a sensitivity analysis of key design parameters in two different geological

models.

The results of the sensitivity analysis can identify gaps in mathematical models for MEOR and

enable further understanding of process optimization and operating envelopment. This study

evaluates for the first time the impact of injection parameters under different reservoir

conditions, through permeability, porosity and interfacial tension reduction effects.

6

2 CHAPTER 2: LITERATURE REVIEW

This chapter overviews key microbial enhanced oil recovery (MEOR) methods and reservoir

parameters controlling microbial survival. A survey of past MEOR field trials is presented. Two

physical mechanisms governing MEOR performance and motivating model development,

interfacial tension reduction and selective plugging, are reviewed. Lastly, literature pertinent to

the mathematical and numerical modelling of MEOR models is summarized.

2.1 MEOR Overview

MEOR is mediated by one of two mechanisms: the injection of exogenous microbes and

nutrients, or the stimulation of indigenous microbial communities via nutrient injection. These

microbes have historically been bacterial and can be further categorized based on their strain,

metabolic process or action mechanism. Depending on the strain of bacteria, its growth and

metabolism can result in the production of surfactants, acids, solvents, gases, polymers or

enzymes. Excluding enzymes, the compounds active in microbial-mediated oil mobilization are

similar to those utilized in conventional EOR processes.

Microorganisms and their metabolites can improve the interactions between fluids and the

porous media through various combinations of mechanisms. Generation of biosurfactants

decreases oil-water interfacial tension and alters wettability; generation of biogas decreases oil

viscosity and increases pressure driving force; generation of bio-acids encourages mineral

dissolution and increases absolute permeability. On the other hand, biomass growth or

biopolymer generation results in selective plugging, and the microbes themselves may enable oil

7

viscosity reduction by hydrocarbon degradation. A summary of microbial products and

associated mechanisms is presented in Table 2-1. Interfacial tension reduction and wettability

alteration, in combination with selective plugging, have been generally identified as the most

important mechanisms in improving oil recovery (Rashedi et al., 2012).

Table 2-1: Microbial Products and Associated Recovery Mechanisms

Microbial Product Mechanism

Biosurfactant Reduce interfacial tension

Cause emulsification

Improve pore-scale displacement

Alter wettability

Biomass Reverse wettability

Plug high-permeability zones

Selective degradation of higher molecular weight

hydrocarbons

Alteration of oil viscosity

Biopolymers Improve water viscosity

Control water mobility

Plug high-permeability zones

Acids Improve effective permeability by dissolving carbonates

CO2 gas production from acid and carbonate reactions

Gases Reduce oil viscosity

Solvents Reduce oil viscosity

Remove heavy hydrocarbons from pore throat

Lowers interfacial tension

Promote emulsification

8

2.1.1 Advantages and Disadvantages of MEOR

Marshall (2008) noted that the use of microorganisms in MEOR presents a key advantage over

alternative EOR technologies, as enzyme-mediated bio-catalysis allows enhanced reaction

kinetics by enabling higher specificity reactions to proceed under more less severe conditions.

A list of other important MEOR advantages is shown below (Lazar et al., 2007; Patel et al.,

2015).

1. The technology can be consistently low cost. Microbes and nutrients are inexpensive to

obtain at the required concentrations and are priced independently of global crude oil.

2. Fewer and less expensive modifications to existing waterflooding facilities are required

to implement MEOR in the field, compared to other EOR technologies.

3. Due to attachment and self-replication processes, microbes remaining after microbial

flooding can survive within the reservoir and sustain recovery over time.

4. Both microorganisms and nutrients are non-toxic and environmentally friendly.

5. Multiple microbial mechanisms can act simultaneously, such as a combination of

surfactant and biofilm generation, to enhance oil recovery.

6. The wide range of bacterial strains, including extremophiles, applicable in MEOR

enables effective application in a variety of reservoir conditions (i.e. both carbonate and

sandstone, over a range of oil viscosities).

Disadvantages of MEOR, compared to other EOR technologies, include the following:

1. Requirement for large quantities of microbial nutrient (i.e. sugar, molasses) may limit

applicability for off-shore implementation.

9

2. MEOR involving injection of exogenous species leads to increased equipment for

microbial cultivation.

3. Modelling and optimization of MEOR are difficult due to complex reactions in situ.

4. Precise control of microbial behavior in the reservoir is a challenge, due to potential

competition from indigenous microbes and difficulty predicting nutrient and microbe

transport.

2.2 Impact of Reservoir Conditions on Microbial Communities

Microorganisms are an extremely diverse set of microscopic organisms. A large subset of

microorganisms, including fungi, algae, and protozoa, is unable to grow under harsh reservoir

conditions. However, other microorganisms, such as heterotrophic bacteria and archaea, have

been found to exist up to five kilometers subsurface (Magnabosco et al., 2018). Heterotrophic

bacteria, which require organic carbon as an energy source, have been found in sandstone and

shale reservoirs. It has been suggested that heterotrophes can survive on organic material co-

deposited with sediments, which diffuse from across sandstone-shale interfaces and enable

survival in the sandstone (Krumholz, 2000). In general, bacteria are utilized in MEOR and thus

the following review will target parameters pertinent to bacterial survival.

2.2.1 Parameters Controlling Bacterial Reactions

Most bacteria strains tend to exist in the aqueous phase, but if heterotrophic they must line the

oil-water interface and metabolize organic substrates from the oil phase. Thus, during nutrient

injection in MEOR, microbial growth concentrates away from the oil-water interface. Oxygen-

deficient conditions, attributed either to prior microbial consumption or iron- and sulfur-

10

mediated depletion, require subsurface microbial communities to be generally anaerobic. Other

reservoir parameters which can limit microbial growth and metabolism within reservoir

conditions include the following (Varjani & Gnansounou, 2017):

1. Pressure, temperature, salinity, and pH

2. Porosity, permeability and dissolved solids content

3. Nutrient composition, electrolyte composition, and redox potential

Of the parameters which may impact microbial survival subsurface, temperature and salinity

have been identified as key in controlling the growth rate, death rate, and metabolic activity of

microbial communities in the reservoir. Increase in temperature generally results in enzyme

denaturation, disrupting active sites required for reaction catalysis (Marshall, 2008). However,

temperatures at which cellular function becomes sub-optimal are different for psychrophilic,

mesophilic, or thermophilic bacteria, which are associated with optimum temperatures of below

25oC, 25 – 45

oC, and 45 – 60

oC respectively. While extremophilic bacteria have been found to

exist at temperatures up to 120oC, a review by Gao and Zekri (2011) of MEOR applications in

the past decade found that all applications were conducted at reservoir temperatures below

100oC. Furthermore, 70% of the reservoir temperatures were below 55

oC, and 95% below 85

oC.

Because both temperature and pressure increase with depth, effects of pressure are coupled to

those of temperature. Increased pressure may exert indirect effects on cellular function, including

increased H2, O2 gas solubility and subsequent alteration of oxidation potential. Pressure can also

11

directly affect cellular function, such as the decreasing fluidity and water-permeability of cell

walls and increasing compaction of the DNA double helix. Additionally, with increased pressure,

chance of bacteria survival at elevated temperature is increased. (Youssef et al., 2009)

Difference in ionic strength, due to the high concentration of electrolytes in the extracellular

fluid, is an important driving force for transport of water into or out of the scells. Because NaCl

may compose up to 90 wt% of the dissolved solids in reservoirs (Donaldson et al., 1989),

matching reservoir characteristics with microbial halophilicity is important to a successful

MEOR operation. Though Gao and Zekri established that reservoir salinity below 10 wt% is a

general benchmark for MEOR treatment, various moderate and extreme halophiles have been

isolated from salt lakes and briny water with 10 – 30 wt% NaCl (Kushner, 1993). Additionally, a

correlation exists between possible reservoir pH and salinity combinations for microbial survival.

For example, halophiles surviving at 7 – 15 wt% NaCl and pH 10, may also grow at 1 – 5 wt%

NaCl and pH 7 (Gary E Jenneman, 1989). Moreover, it has been noted in numerous reports that

species with higher optimal growth salinity also have higher optimal growth temperature,

matching reservoir conditions. Lastly, moderate halophiles have been found to compete

favorably with extreme halophiles under high salinity and low nutrient conditions, which

indicates favorable outcomes for the ex situ MEOR method (Patel et al., 2015; Rodriguez-Valera,

1991)

Based on the above constraints, Yakimov et al. (1997) and Marshall (2008) identified Bacillus as

the preferred bacteria for MEOR application. Various strains of Bacillus have been found to

12

grow and produce favorable metabolites (i.e. surfactant with anti-microbial properties) under

temperatures up to 60 oC and 12 wt% NaCl.

2.3 MEOR Methods

MEOR methods are conventionally divided into two categories: those operating by ex situ or by

in situ mechanisms (Safdel et al., 2017). In in situ MEOR, generation of surfactants, acids, gases,

and polymers occurs via the in situ stimulation of injected or indigenous bacteria through

nutrient injection. In general, the in situ stimulation of indigenous bacteria is rarely observed in

field trials (Patel et al., 2015), as selective stimulation of favorable species within the subsurface

microbial community is challenging.

The delivery of exogenous bacteria in in situ MEOR may occur via huff and puff or bacterial

flooding. In the case of microbial huff and puff, microbe and nutrient injection occurs at the

production well. A cyclic shut-in and production cycle then allows for optimized production via

permeability alteration (i.e. selective plugging). Conversely, in the case of bacterial flooding,

microbe and nutrient injection occurs at the injector, followed by water flooding. Dominant

mechanisms in bacterial flooding include interfacial tension reduction and selective plugging. In

general, microbial huff and puff have been more frequently observed than bacterial flooding

(Gao & Zekri, 2011).

In contrast, in ex situ MEOR, microorganisms or their metabolites are produced and separated in

batch at surface and subsequently injected subsurface, in a mechanism similar to chemical EOR

or surfactant flooding. However, directly injecting exogenous microorganisms is associated with

13

higher risk, as the strains would need to compete favorably with those indigenous to the reservoir

and already acclimated to its conditions. Injecting biosurfactant directly would reduce the

complexity and mitigate risks associated with controlling metabolic activity, growth and

competition of microorganisms subsurface. However, the process is associated with higher costs

than conventional chemical or surfactant EOR and is less frequently reported in field trials than

in situ MEOR.

2.4 Field Applications

In the first commercial-scale MEOR project in 1954, Socony Mobil Research laboratory injected

a mix of C. acetobutylicum and molasses in Lisbon field, Arkansas, USA. The operation

resulted in marginal success, as observed from slight increase in incremental recovery.

Knowledge in MEOR processes has since advanced from bench-scale investigations to field

trials, particularly in the 1990s (Ramkrishna, 2008). After the 1990s, MEOR field trials have

become increasingly focused on microbial waterflooding due to reduced cost (Lazar et al., 2007),

in comparison to alternatives strategies such as huff and puff or the in situ stimulation of

indigenous microbiota.

Most literature reviews of MEOR field trials have concluded that the technology generally

produces favorable outcomes (Bailey et al., 2001; Patel et al., 2015; Rashedi et al., 2012). While

a review by Safdel et al. (2017) indicated that 90% of all global MEOR field trials have achieved

a positive effect on oil production, the effect of MEOR can vary and is difficult to predict

precisely. Youssef et al. (2009) reported variable residual oil recovery increase, ranging from

14

10% to as high as 340% for 2 – 8 years following MEOR. Similarly, Dietrich et al. (1996)

described production rate increasing from 10% to 500% for dolomite and sandstone reservoirs.

Although control experiments for MEOR are not feasible given the alteration of reservoir

properties following oil production, a higher number of field trials would eventually enable the

generation of comparative cases. A 1995 survey conducted on the economic effectiveness of 322

commercial projects, encompassing 2,000 wells over a variety of reservoir conditions, allowed

the preliminary evaluation of treatment outcomes given reservoir parameters (Portwood, 1995).

In 2007, Maudgalya et al. conducted a broader classification of 407 MEOR field trials by

lithology; recovery mechanism; reservoir properties (salinity, temperature, permeability);

microbial species; nutrient type; and success relative to incremental recovery. 77% of the 407

trials were carried out in sandstone and 22% in carbonate reservoirs, with Clostridium spp. and

molasses as the dominant bacteria and nutrient used in all tests respectively. Similarly, Lazar

(2007) presented both quantitative and qualitative effects of various MEOR processes in field

trials worldwide, categorizing by well injection protocol.

More recently, Safdel et al. (2017) reviewed 47 biosurfactant-based MEOR field trials across 21

countries. The classification was performed via surfactant type, in order to understand the

macroscopic impact of different biosurfactants on interfacial tension and residual oil recovery.

The dominant cultures, in addition to Clostridium, included Pseudomonas and Bacillus spp.

Currently, the use of ultra microbacteria remains a key area of research for MEOR. First

developed in Australia, ultra microbacteria is generated via selective starvation, or nutrient

15

manipulation of indigenous reservoir microbiota (Sheehy, 1991). The ultra microbacteria would

have altered surface-active properties, subsequently altered transport and attachment properties,

and can potentially enable higher control in selective plugging. This MEOR variant has thus far

been successfully implemented in Alton field, Australia (Lazar, 2007).

2.5 MEOR Mechanisms

2.5.1 Interfacial Tension Reduction

In MEOR, interfacial tension (IFT) reduction and micelle formation are typically mediated via

surfactant production. The amphiphilicity of surfactants, such as glycolipids and lipopeptides

(Banat, 1995), allows them to absorb at the oil-water interface, lowering the hydrostatic pressure

requirement for the water phase to overcome the capillary effect. At high enough concentration,

surfactants can form membrane-like structures, and allow oil mobilization and oil-in-water

emulsions flow. Because bacteria tend to exist in the aqueous phase, secretion of biosurfactants

is important for enabling the transport of hydrophobic materials into the hydrocarbon-oxidizing

cells (McInerney et al., 2005). Furthermore, because surfactants can cause wetting of

hydrophobic surfaces, they may also allow cell adhesion and enable biofilm formation.

Various results from surfactant-driven MEOR trials have been observed, from cases with major

IFT reduction and material incremental recoveries, i.e. 2 to 20% OOIP, to cases with poor or no

improvement in recovery. Due to multiple possible mechanisms involved in the MEOR process,

the success or failure observed in these experimental cases cannot be attributed solely to

surfactant production.

16

In surfactant-based MEOR, a certain threshold surfactant concentration (critical micelle

concentration) must be achieved before oil mobilization. While the feasibility of meeting this

threshold has been a contentious point in literature (Bryant & Lockhart, 2002), evidence from

field trials is generally favorable. IFT in oil-water systems tends to be 30 mN/m. (Green &

Whillhite, 2018) recommended that IFT after surfactant addition should be from 10-2

and 10-3

mN/m to observe significant oil recovery improvement, whereas Gray et al. (2008) defined the

successful biochemical surfactant IFT as below 0.4 mN/m. Youssef et al. (2007) suggested that

surfactant concentration should be 10 – 20 mg/L to achieve the required IFT reduction. At the

bench-scale, lipopeptides and rhamnolipids have been found to lower the IFT multiple orders of

magnitude, to below 0.1 mN/m (Patel et al., 2015). Also, there have been some field trials

reporting surfactant concentrations in the production fluids to be 90 mg/L to 350 mg/L (Yossef et

al., 2007; McInerney et al., 2005). However, on average biosurfactants are unable to reach this

level of reduction in the reservoir due to various factors, including dilution, sub-optimal

microbial kinetics (Gray et al., 2008), and loss of surfactant via pore wall adsorption (Zekri et al.,

2001).

2.5.2 Biofilm Growth

During biofilm formation, higher permeability channels are increasingly coated with microbial

colonies. As a result, water is forced to sweep lower permeability zones, in a mechanism referred

to as selective plugging. Variables which influence bacteria transport and attachment include the

size, hydrophobicity, flocculation tendency and surface charge of the bacterial cell (Murphy &

Ginn, 2000). Surface charge and pore throat size of the medium are also important factors.

Bacteria transport and attachment can be classified into two mechanisms: particulate plugging in

17

reservoirs with small pore throat size, and biofilm formation in reservoirs with larger pore throat

size.

Biofilm formation results from biomass accumulation on the rock surface. Biofilm growth can be

divided into three stages: cell attachment, sessile colony formation, and dispersal. The initial

cell–rock interaction is mediated by Lewis acid-base and electrostatic forces (Battin et al., 2003).

Under sufficient nutrient concentration, cells attached to the rock surface will undergo

replication and secrete biopolymers (“exopolymers”), which eventually form an extracellular

polymeric matrix. The exopolymers are bound by crosslinking of individual chains and are

embedded with daughter cells. On the other hand, bacterial dispersal may occur either actively or

passively. Passive dispersal, such as erosion or sloughing, reduces or eliminates the biofilm

(Bryers, 1987), whereas active dispersal includes mechanisms initiated by the microbial cell

(Kaplan, 2010).

Sloughing, or the sudden detachment of cell from the rock surface, occurs as a result of shear

stress from the fluid flowing parallel to the attachment surface. Sloughing is a function of

thickness, density, and strength of the biofilm (Toole, Kaplan, & Kolter, 2000). A first-order

equation describing biofilm detachment was given by Rittman (1982):

𝜇𝑠𝑙𝑜𝑢𝑔ℎ = 𝛽𝜌𝑏𝑓𝑟𝑏𝑓, 𝜏 > 𝜏𝑐𝑟𝑖𝑡

𝜇𝑠𝑙𝑜𝑢𝑔ℎ = 0, 𝜏 > 𝜏𝑐𝑟𝑖𝑡. (2.1)

Here 𝜌𝑏𝑓 is the density of the biofilm, 𝑟𝑏𝑓 is the thickness of the biofilm, 𝜏 the shear stress, 𝜏𝑐𝑟𝑖𝑡

the critical stress at which detachment occurs, and 𝛽 the rate coefficient. 𝛽 was empirically

determined to be 0.16 h-1

.

18

Several studies have shown that nutrient availability is important for biofilm growth and

dispersal. Low-nutrient conditions have been shown to induce biofilm detachment, and high-

nutrient conditions biofilm growth (Delaquiset al., 1989; James et al., 1995). The degree of

permeability reduction then depends on the relative magnitudes of growth rate and dispersal rate,

which in turn depend on nutrient concentration and fluid flowrates (Peyton, 1996). Reduced

permeability can lead to alteration in nutrient flow pathways and bacterial transport. Therefore,

for biofilm-generating microbial species, control of nutrient loading is critical to MEOR

performance.

Bacterial attachment for selective plugging has generally been modeled through reversible

equilibrium adsorption or deep bed filtration (Nielsen et al., 2014). Filtration models have been

used in cases where pore sizes are similar to cell size, and are based on a coefficient for

probability of cell capture:

𝜇𝑓𝑖𝑙𝑡𝑒𝑟 = 𝜙𝑝𝑐𝑎𝑝𝑢𝑆𝑤𝐵𝑓, 𝜏 > 𝜏𝑐𝑟𝑖𝑡. (2.2)

Here 𝜙 is the rock porosity, 𝑝𝑐𝑎𝑝 is the probability that a cell will be filtered over a distance (m-

1), 𝑆𝑤 is water saturation, 𝑢 is the fluid flowrate, and 𝐵𝑓 is the free bacteria concentration.

Reversible equilibrium adsorption is a Langmuir-type equation which relates the mass of

adsorbed bacteria to the concentration of bacteria to the concentration of free bacteria in the

water phase. Pore throat size is not limited. In this case, the concentration of bacteria in the

biofilm is calculated as a function of the surface area available for adsorption (𝐴) and the mass

of bacteria which can be adsorbed per unit area (Θ), defined by specific adsorption parameters.

19

The surface area is defined as a function of porosity, water saturation, and pore surface area, so

that

𝐵𝑏 = Θ ∗ (𝑆𝑤𝐴𝑟𝜙), (2.3)

where the mass of bacteria adsorbed per unit area, Θ, can be defined, via the Langmuir model, as

a function of 𝛼1, a parameter controlling the maximum mass of biofilm per unit area of pore

space, and 𝛼2, a parameter controlling the speed of adsorption (Amundsen, 2016),

Θ =𝛼1 ∗ 𝛼2 ∗ 𝐵𝑓

1 + 𝛼2 ∗ 𝐵𝑓. (2.4)

2.6 MEOR Mathematical Models

The multitude of complex relationships between multiple physical, chemical and biological

factors has been a persistent challenge in mathematically modelling MEOR. Yet, formulation of

a reservoir simulator is essential to the successful development of all MEOR deployment

strategies, as it allows the prediction of the transport, in situ production and action of both

biomass and metabolites.

Numerous approaches have been taken to mathematically model the MEOR process. Apart from

simplified analytical descriptions (Bryant & Lockhart, 2002), models describing the behavior of

microorganisms in porous media are mainly composed of combinations of the following three

inter-relating components:

20

1. Transport Properties: the transport properties of the bacteria are composed of the

diffusivity of the bacteria, and the longitudinal and transverse dispersivity of the porous

medium. The diffusivity component is obtained via the Stokes-Einstein equation, where

the bacteria exhibit Brownian motion due to significant fluid advection (Marshall, 2008).

2. Conservation law: the concentration of bacteria in the fluid phase is described as a

combination of microbial growth, advection and diffusion, and solid adsorption.

3. Bioclogging: permeability reduction is described as a combination of biopolymer

entrainment, or biomass attachment to pore walls, or chemical precipitation of dissolved

electrolytes such as CaCO3 (Nemati et al., 2004). Biofilm growth is described as a

combination of bacterial growth and ablation as a result of increase flow velocities

(Rittmann, 1982). Pore-network models, compared to independent capillary tube models,

can significantly improve the capability to account for changes in hydraulic conductivity

resulting from microbial growth (Thullner et al., 2004).

Mathematical models for MEOR are typically one-, two-, or three-dimensional representations of

two or, less often three phases (oil-water, or oil-water-gas). Typically, five to seven components

are included in in situ models, including oil, water, bacteria, nutrients, and active metabolite (e.g.

surfactant). While the mechanism varies, all models are based on mass balance and Darcy’s law.

Nutrients and metabolites are miscible only in the aqueous phase, and biomass partitions to the

aqueous phase, so that the oil phase consists of oil only.

21

Of the potential MEOR mechanisms observed experimentally, bioclogging, viscosity reduction

and interfacial tension reduction are of the most interest in literature. Biomass adsorption kinetics

to pore walls drive biofilm formation, and subsequently porosity and permeability reduction.

This can lead to negative effects, such as loss of injectivity at the injector, or benefits, such as

selective plugging. On the other hand, surfactant generation through microbial growth and

metabolism results in interfacial tension reduction. Several methods have been used to model the

changes in relative permeability end-points and curves resulting from reduction in interfacial

intension and residual oil saturation. These include the capillary number method, Coats’

interpolation method, and the Corey interpolation method (Nielson et al., 2010). Lastly, while

nutrients are largely assumed not to adsorb to the pore walls, surfactant adsorption is sometimes

considered (Amundsen, 2016).

Bacterial attachment and biofilm modelling has been an active area of research in MEOR. It has

been mathematically approached in two ways (Patel et al., 2015). Assuming equilibrium

partitioning of bacteria between water and oil phases is fast compared to convection, the

Langmuir-type reversible equilibrium adsorption equation can be used. Alternatively, the

attachment and detachment rates can be derived from deep bed filtration theory (Nielsen, 2015).

Finally, the resulting porosity reduction is coupled with the permeability reduction via relations

such as the Carman-Kozeny equation.

22

2.6.1 Summary of MEOR Numerical Models

Jenneman et al. (1984) were the first to model microbial transport and plugging phenomena in

Berea sandstone. However, development of increasingly comprehensive MEOR models started

in the early 1990s. Until the early 2000s, selective plugging and biomass-mediated permeability

reduction were the key focus areas for MEOR mathematical development. However, in the past

ten years, simulation of surfactant effects has been of increasing interest. Similarly, until recent

years, finite difference numerical solutions were the most widely used. However, with increased

prevalence of simulators such as MRST, UTCHEM and COMSOL, MEOR models have

nowadays been solved through other discretization methods.

Islam (1990) was the first to develop a comprehensive reservoir simulator for predicting bacterial

transport and reaction. The three-dimensional bacterial transport and growth model was coupled

to three-phase flow equations and was applied to a Canadian heavy oil reservoir. Mechanisms

included in the model are bacterial plugging through biomass, IFT reduction through surfactant

generation, viscosity reduction through bacteria and pressurization via CO2 generation. The

model was solved through the Implicit Pressure Explicit Saturation (IMPES) method using finite

difference. Dispersion and diffusion terms were neglected; bacterial growth was also neglected.

In 1991, Chang et al. proposed a three-dimensional, three-phase model describing microbial

transport phenomena in porous media. The model accounted for diffusion, adsorption,

chemotaxis, growth and decay of micro-organisms. Both porosity and permeability reduction

were considered through pore clogging and de-clogging. It was found that microbial adsorption

23

to rock service was critical to transport in reservoir. The model was solved using the IMPES

method, and Crank-Nicolson was used to formulate the finite difference transport equations.

Zhang et al. (1993) then developed a one-dimensional, three-phase model to simulate transport,

growth and metabolism of indigenous bacteria. Only reduction in permeability was considered.

Permeability alteration was assumed to be caused by biomass pore surface retention and pore-

throat plugging. Convection-dispersion was considered as part of the transport equations.

Desouky et al. (1996) developed a one-dimensional, two-phase model for simulation of

adsorption, diffusion, chemotaxis, and bacterial growth and decay. Permeability damage and

porosity effects were considered. The deposition of microbial components (cells, nutrients and

metabolites) due to sedimentation and straining were considered. The model was fairly similar to

Islam (1990), however, incorporated the interaction between indigenous and exogenous bacteria.

Behesht et al. (2008) developed a three-dimensional, two-phase model simulating the combined

surfactant and polymer effects on MEOR. Permeability reduction through polymer retention,

water phase viscosity increase through polymer and salinity/hardness interactions, and polymer

adsorption were considered. Interfacial tension alteration through biosurfactant addition, and

biosurfactant adsorption were also considered. The model was one of the first to simulate

physico-chemical effects, including flooding time schedules, washing water flow rate, substrate

concentration and permeability on recovery. The model was solved numerically through finite

differences, using the IMPES method.

24

Nielsen et al. (2009) developed a one-dimensional isothermal model for simulating the

partitioning of surfactants between the oil and water phases via a distribution coefficient. The

key mechanism modeled was the reduction of interfacial retention. Interpolation methods for

relating changes in interfacial tension to relative permeability were compared; it was found that

the capillary number method, Coat’s interpolation, and Corey method produced comparable

results in terms of incremental recovery. An implicit finite difference technique was applied to

solve the mathematical model.

Li et al. (2012) developed a coupled isothermal, biological and hydrogeological model. The

model was the first to explicitly relate residual oil saturation and capillary number. The flow and

transport equations were coupled and solved simultaneously assuming incompressible and

immiscible flow. As part of the transport equations, dispersion effects were considered.

Bioclogging effects on permeability and porosity, as well as surfactant impact on IFT were

considered. The model was solved through Comsol Multiphysics, employing a finite element

discretization.

Sivasankar (2014) was the first to develop a non-isothermal model of MEOR. The one-

dimensional, two-phase numerical model simulated temperature, flood velocity and mean fluid

velocity effects on the reservoir. Change in relative permeability as a function of surfactant

production was investigated. A similar model was built to investigate the effect of pH on

surfactant production and IFT reduction (Sivasankar, 2017).

25

Delshad et al. (2002) were the first to simulate MEOR using UTCHEM. Most recently,

Hosseininoosheri (2016) presented three-dimensional, two-phase model of MEOR using

UTCHEM, accounting for the impact of temperature, salinity, and pH on microbial growth rate

and biosurfactant production. The model also accounted for biosurfactant adsorption effect. The

simulation showed a 10 - 15% incremental oil recovery using in situ biosurfactants compared to

waterflooding.

The numerical solution of most MEOR mathematical models follows the IMPES procedure.

Most recently, Nielsen (2010) implemented the first full MEOR model in a compositional

streamline simulator, based on the implicit pressure, explicit composition (IMPEC), to decouple

flow and reactive transport. The isothermal model accounted for the injection of exogeneous

microbes and neglected adsorption, capillary pressure, diffusion and chemotaxis. The effect of

surfactant on interfacial tension was modeled. The reaction and transport processes are solved

simultaneously along the streamline.

A summary of key MEOR models is provided in Table 1. Generally, except for Li (2011), all

models developed to date have been implemented through finite difference or, seldomly, finite

volume method.

Table 2-2: Summary of Key Numerical Models for MEOR to Date

Model Dimension Phase Mechanisms

Islam, 1990 3 3 Plugging through biomass, IFT reduction through

surfactant generation, viscosity reduction through

bacteria and pressurization via CO2 generation

Chang, 1991 3 3 Transport (diffusion, adsorption), chemotaxis,

26

growth and decay of microorganisms; porosity

and permeability reduction

Zhang et al., 1992 1 3 Transport (convection-dispersion), growth and

metabolism of indigenous bacteria; permeability

reduction and pore-throat plugging

Desouky et al.,

1996

1 2 Simulation of adsorption, diffusion, chemotaxis,

and bacterial growth and decay; permeability

damage and porosity effects

Beheshet, 2008 3 2 Surfactant and polymer effect on viscosity,

permeability and interfacial tension

Nielsen et al., 2009 1 2 Surfactant effect

Li et al., 2011 2 2 Dispersion effects, bioclogging effects on

permeability and porosity, surfactant impact on

IFT

Sivasankar and

Kumar, 2014

1 2 Non-isothermal surfactant production

Hosseininoosheri,

2016

3 2 Temperature, salinity, pH effects on biosurfactant

production; biosurfactant adsoprtion

Nielsen et al., 2010 1 2 Surfactant generation modeled via compositional

streamline simulator

2.7 Numerical Modelling of Porous Media Transport

Porous media flow has been investigated extensively during the last century. However,

numerical simulation of the multicomponent reactive transport in porous media is still a

challenge, due to the large numbers of chemical equations which must be coupled. The difficulty

arises from the propagation of singular inaccuracies in unknowns, resulting in false predictions.

In general, numerical modelling of porous media flow fall broadly under three categories: finite

difference, finite volume, and finite element.

2.7.1 Finite Difference Method

Due to its robustness and ease of use, the finite difference method (FDM) has been one of the

most frequently used discretization methods in reservoir analysis. However, one of the major

27

challenges for reservoir simulation has been retaining the level of detail in geological models in

the reservoir simulation. FDM typically employs structured meshes, and therefore are unable to

represent the complex geometry of geological objects within petroleum reservoirs (Hurtado et

al., 2007). On the other hand, alternative approaches employing unstructured meshes have been

increasingly favored for this reason. These include finite volume method (FVM) (Barth, 1992),

and finite element method (FEM) (Zienckievicz & Morgan, 1983), which furthermore have

attractive local and global conservation properties.

2.7.2 Finite Element Method

In FEM, the mesh is composed of geometrically simple elements which are not necessarily

orthogonal (e.g. triangle, quadrilateral, or tetrahedral shaped elements). Governing equations in

FEM are integrated over each element, and contributions from all elements are subsequently

assembled over the domain in order to generate a large sparse matrix equation. Standard FEMs

have generally been designed for problems with smooth solutions (i.e. diffusion processes). Due

to convection-dominated steep fronts from the saturation equation, and the importance of fluids

velocity for solution coupling, standard FEMs may lead to numerical diffusion. Thus, in the

cases where FEM was applied for reservoir simulation (Ewing, 2002) velocities were calculated

using mixed FEM for the pressure equation. For the approximation of sharp moving fronts,

upwind schemes based on a discontinuous finite element approximation are typically used. This

would lead to a more accurate scheme than a standard first order finite difference scheme, and

thus reduces numerical diffusion.

28

FEMs differ from FDMs in two aspects: FEM uses piecewise polynomial approximations while

FDMs use only piecewise constant approximations. FEM scheme accuracy may be improved by

increasing the degree of the polynomials. Unlike FDM, FEMs can also be formulated for

irregular meshes, which allow local refinement of the mesh and enable better representativeness

of domain boundaries.

2.7.3 Finite Volume Method

FVM depends on the conservation of physical quantities over cell volumes. Due to its locally

conservative properties, FVM is particularly suited for the numerical simulation of various types

of conservation laws (e.g. elliptic, parabolic). Like FEM, FVM can be used on both structured

and unstructured meshes containing elements with arbitrary geometries. In FVM, a balance is

written for each mesh cell, referred to as a control volume, through integration. Through the

divergence theorem, an integrated form of the fluxes over the boundary of each control volume

can be obtained and then discretized. FVM can be divided into cell-centered or vertex-centered

methods, in which the discrete variables are located either at the centroids or vertex of each cell.

Compared to FEM, the use of unstructured meshes in FVM for discretization is still relatively

recent. FVM in unstructured meshes has become particularly relevant to reservoir simulation,

due to the geometric complexity of reservoir models (Prevost et al., 2002).

29

3 CHAPTER 3: MEOR MODEL

This chapter presents the equations used to model the MEOR operation. A two-phase,

incompressible flow model is derived in terms of pressure and saturation equations for IMPES

solution. Equations for microbial rate, interfacial tension reduction, and biofilm formation are

summarized.

3.1 Reservoir Model

The governing equations for fluid flow in porous media include the law of conservation of mass

and Darcy’s law. For a single fluid phase, the mass conservation equation in a fixed region Ω

may be written as

𝜕

𝜕𝑡∫ 𝜙𝜌𝑑𝑉Ω

+∫ 𝜌�⃗� ∙ �⃗� 𝑑𝑆𝜕Ω

= ∫ 𝑞𝑑𝑉Ω

, (3.1)

where 𝜙 is the porosity, 𝜌 the density, �⃗� the fluid velocity, 𝑑𝑉 the volume element, 𝑑𝑆 the

surface of a side of 𝑑𝑉, �⃗� the unit vector normal and outwards to the surface, and 𝑞 is a source or

sink term (mass per unit volume per unit time). Depending on whether it is greater or lesser than

zero, 𝑞 may represent either the injection or production well, respectively.

Then rewriting the surface integral in equation (3.1) as the volume integral by the divergence

theorem

∫ (𝜕

𝜕𝑡𝜙𝜌 + ∇ ∙ (𝜌�⃗� ))𝑑𝑉

Ω

= ∫ 𝑞𝑑𝑉Ω

,

30

which yields the equation of continuity

𝜕

𝜕𝑡𝜙𝜌 + ∇ ∙ (𝜌�⃗� ) = 𝑞. (3.2)

In a reservoir, fluid flow is generally driven by a pressure gradient and further controlled by

properties of the rock and the fluid, as well as gravity. The macroscopic velocity vector, �⃗� ,

denoting the effective speed and direction of flow within the pore space of a representative

volume element, can be described through Darcy’s Law. The one-dimensional Darcy’s law

(Darcy, 1856) relates the total volumetric flow rate of a fluid through a cross-sectional area of a

porous medium to the pressure gradient, medium permeability, and fluid viscosity. Incorporating

gravitational force, the differential form of the three-dimensional Darcy’s law becomes

�⃗� = −1

𝜂𝑲(∇𝑝 − 𝜌𝑔∇z), (3.3)

where 𝜂 is the fluid viscosity, 𝑲 is the absolute permeability tensor, 𝑝 is the pressure, 𝑔 is the

gravitational constant, and z is the depth. Darcy’s Law can also be derived from the Navier-

Stokes equation, which simplifies to Stokes equation assuming incompressible flow.

To model the simultaneous two-phase flow of oil and water, first the phases must be assumed to

be immiscible (i.e. that there is no mass transfer). By convention, water is assumed to be the

wetting phase relative to oil, which is then the non-wetting phase. Then, the saturation, capillary

pressure, and relative permeability relations between the two phases may be determined.

Saturation is defined as the volumetric fraction of void space within the porous medium which is

filled by fluid. Assuming that oil and water completely fill the void space, the saturation

relationship may be written as

31

𝑆𝑜 + 𝑆𝑤 = 1. (3.4)

Here, 𝑆𝑜 is the saturation of the oil phase and 𝑆𝑤 is the saturation of the water phase. Then the

conservation of mass may be written for each phase separately

𝜕

𝜕𝑡𝜙𝜌𝑜𝑆𝑜 + ∇ ∙ (𝜌𝑜�⃗� 𝑜) = 𝑞𝑜

𝜕

𝜕𝑡𝜙𝜌𝑤𝑆𝑤 + ∇ ∙ (𝜌𝑤�⃗� 𝑤) = 𝑞𝑤.

(3.5)

The fraction of a phase (oil/non-wetting or water/wetting) which is trapped, or irreducible, in the

void space following displacement from the immiscible phase is referred to as the residual

saturation. For the oil phase, the saturation at which residual oil becomes trapped by capillary

forces is denoted 𝑆𝑜𝑟. 𝑆𝑜𝑟 depends typically on rock permeability and wettability; in a water-wet

sandstone, for example, it may vary from 0.2 to 0.35 (Chen, 2007). Analogously, the connate

water saturation is denoted 𝑆𝑤𝑐.

The other two functions, capillary pressure and relative permeability, are strongly dependent on

saturations. Due to the immiscibility of the two phases, the curvature of and the interfacial

tension at their interface causes a difference in pressure, known as capillary pressure. For a two-

phase immiscible system, capillary pressure is then defined as

𝑝𝑐 = 𝑝𝑜 − 𝑝𝑤 = 𝑓(𝑆𝑤), (3.6)

where 𝑝𝑜 is the pressure of the oil phase, 𝑝𝑤 is the pressure of the water phase, and 𝑆𝑤is the

water phase saturation. Capillary pressure can be related to surface tension, porosity,

permeability, wetting angle, which is a function of temperature and fluid composition, via the J-

function. It is empirically a function of water phase saturation and saturation history (drainage or

imbibition) and can be used to infer relative permeability (Derahman and Zahoor, 2008).

32

Relative permeability is a measure of the impairment to flow one phase exerts to another. Similar

to capillary pressure, relative permeability also a function of saturation. While relative

permeability depends on the saturation history of the oil phase, it does not depend on that of the

water phase. Relative permeabilities of water and oil phases, 𝑘𝑟𝑜 and 𝑘𝑟𝑤 respectively, give rise

to mobility, the ratio of relative permeability to viscosity, and subsequently fractional volumetric

flow rate, the ratio of phase mobility to total mobility.

While relative permeability curves are typically experimentally determined, numerous empirical

methods correlating relative permeability to saturation exist. The Corey type relative

permeability curves (Lake, 1989) are

𝑘𝑟𝑜(𝑆𝑜) = 𝑘𝑟𝑜𝑤𝑐 (1 − 𝑆𝑤 − 𝑆𝑜𝑟1 − 𝑆𝑤𝑐𝑟 − 𝑆𝑜𝑟

)𝑛

(3.7)

𝑘𝑟𝑤(𝑆𝑤) = 𝑘𝑟𝑤𝑜𝑟 (𝑆𝑤 − 𝑆𝑤𝑐𝑟

1 − 𝑆𝑤𝑐𝑟 − 𝑆𝑜𝑟)𝑛

, (3.8)

where 𝑛 is the Corey exponent (assumed the same for oil and water), 𝑆𝑤𝑐𝑟 is the critical water

saturation, 𝑆𝑜𝑟 the residual oil saturation, 𝑘𝑟𝑜𝑤𝑐 the end-point relative permeability of oil at

𝑆𝑤 = 𝑆𝑤𝑐, and 𝑘𝑟𝑤𝑜𝑟 is the end-point relative permeability of water at 𝑆𝑜 = 𝑆𝑜𝑟. Then

incorporating the relative permeability into the two-phase Darcy’s law,

�⃗� 𝑤 = −𝑘𝑟𝑤𝑲

𝜂𝑤(∇𝑝 − 𝜌𝑤𝑔∇z) (3.9)

�⃗� 𝑜 = −𝑘𝑟𝑜𝑲

𝜂𝑜(∇𝑝 − 𝜌𝑜𝑔∇z). (3.10)

33

At residual oil saturation, relative permeability of oil is zero, and thus the volumetric flowrate of

the oil phase is zero by Darcy’s Law. Broadly, the value of 𝑆𝑜𝑟 is impacted by capillary trapping

forces, which is product of interfacial tension. As an extension, reduction of interfacial tension

via surfactant addition may directly impact relative permeability. The result of surfactant-

mediated MEOR then is the reduction of IFT and 𝑆𝑜𝑟, and the consequent increase of 𝑘𝑟𝑜 and

ultimately the effective permeability.

Formulation volume factor is the ratio of a volume of a phase under reservoir conditions to its

volume at standard conditions. It can be written in terms of density, which differs in the reservoir

and at the surface due to the compressibility of the fluid under pressure and temperature. This

yields

𝜌𝑜 =𝜌𝑠,𝑜𝐵𝑜

𝜌𝑤 =𝜌𝑠,𝑤𝐵𝑤,

(3.11)

where 𝜌𝑠,𝑤 and 𝜌𝑠,𝑜 are the densities of water and oil, respectively, at standard conditions. Then

equation (3.5) can be generalized

𝜙𝜕

𝜕𝑡(𝑆𝑤𝐵𝑤) + ∇ ∙ (

�⃗� 𝑤𝐵𝑤) −

𝑞𝑤𝜌𝑠,𝑤

= 0 (3.12)

𝜕

𝜕𝑡(𝑆𝑜𝜙

𝐵𝑜) + ∇ ∙ (

�⃗� 𝑜𝐵𝑜) −

𝑞𝑜𝜌𝑠,𝑜

= 0. (3.13)

Due to the existence of bacteria, substrates and surfactants in the water phase, MEOR is modeled

through a multicomponent, multiphase approach. Then the mass conservation for each

34

component is based on its concentration, respectively B, N, and M for bacteria, nutrients

(substrate), and metabolite (surfactant):

𝜕

𝜕𝑡(𝑆𝑤𝜙𝐵

𝐵𝑤) + ∇ ∙ (

�⃗� 𝑤𝐵

𝐵𝑤) =


𝐵𝑖𝑛𝑗 (3.14)

𝜕

𝜕𝑡(𝑆𝑤𝜙𝑁

𝐵𝑤) + ∇ ∙ (

�⃗� 𝑤𝑁

𝐵𝑤) =


𝑁𝑖𝑛𝑗 (3.15)

𝜕

𝜕𝑡(𝑆𝑤𝜙𝑀

𝐵𝑤) + ∇ ∙ (

�⃗� 𝑤𝑀

𝐵𝑤) =


𝑀𝑖𝑛𝑗 . (3.16)

3.1.1 Phase Pressure and Saturation for Incompressible Fluids

Assuming that the water and oil are incompressible,

𝐵𝑤 = 𝐵𝑜 = 1. (3.17)

Additionally, defining the phase mobilities as the ratio of phase relative permeability to phase

viscosity, and total mobility as the sum of phase mobilities,

𝜆𝑤 =𝑘𝑟𝑤𝜂𝑤

𝜆𝑜 =𝑘𝑟𝑜𝜂𝑜

𝜆𝑤 + 𝜆𝑜 = 𝜆.

(3.18)

Similarly, fractional flow is defined as the ratio of the phase mobility to total mobility,

𝑓𝑤 =𝜆𝑤𝜆

𝑓𝑜 =𝜆𝑜𝜆.

(3.19)

Lastly, total velocity is defined as the sum of phase velocities,

35

�⃗� = �⃗� 𝑤 + �⃗� 𝑜 . (3.20)

Then for incompressible fluids, the pressure equation may be derived (Chen, 2002),

−∇ ∙ (�⃗⃗� (𝜆∇𝑝𝑜 − 𝜆𝑤∇𝑝𝑐 − (𝜆𝑤𝜌𝑤 + 𝜆𝑜𝜌𝑜)𝑔∇𝑧)) =𝑞𝑜𝜌𝑜+𝑞𝑤𝜌𝑤. (3.21)

A similar saturation equation may be derived as

𝜙𝜕𝑆𝑤𝜕𝑡

+ ∇ ∙ (�⃗⃗� 𝑓𝑤𝜆𝑜 (𝑑𝑝𝑐𝑑𝑆𝑤

∇𝑆𝑤 − (𝜌𝑜 − 𝜌𝑤)𝑔∇𝑧) + 𝑓𝑤�⃗� ) =𝑞𝑤𝜌𝑤. (3.22)

3.2 MEOR Model

Two critical reactions driving microbial action in the reservoir in any type of MEOR application

include the consumption of substrate to enable microbial growth, and product generation (e.g.

biosurfactants).

𝑆𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒𝐵𝑎𝑐𝑡𝑒𝑟𝑖𝑎→ 𝐵𝑎𝑐𝑡𝑒𝑟𝑖𝑎

𝑆𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒𝐵𝑎𝑐𝑡𝑒𝑟𝑖𝑎→ 𝑃𝑟𝑜𝑑𝑢𝑐𝑡

3.2.1 Microbial Growth and Surfactant Production Kinetics

The Monod equation has typically been used to model bacterial growth and is an empirical

equation analogous to Michaelis-Menten kinetics and Langmuir heterogenous catalysis. Though

a few multi-substrate or non-isothermal forms exist, the single-substrate, non-inhibited (i.e.

neither substrate, product or cell-inhibited), isothermal form of the Monod equation is the most

pre-dominant in the literature. The Monod equation is written as

𝜇𝑏 = 𝜇𝑚,𝑚𝑖𝑐𝑁

𝐾 + 𝑁, (3.23)

36

where the specific growth rate of the bacteria, 𝜇, is expressed in terms of the substrate or nutrient

concentration, N, maximum growth grate, 𝜇𝑚,𝑚𝑖𝑐, and the half-saturation constant, K, the

nutrient concentration at which specific growth rate is half of its maximum value. At low N, the

reaction is first order and has been found to take on a similar form as the semi-empirical logistic

equation (Kargi, 2009).

There have been other growth kinetics models which take into consideration cell death, which

may be caused by mechanisms such as toxic metabolite accumulation (Whiting, 1992). Because

bacterial decay is not modeled separately, 𝜇𝑚,𝑚𝑖𝑐 is assumed to be the maximum growth rate net

of death rate. Notably, 𝜇𝑏 is zero when and only when the environment is depleted of nutrients,

otherwise known as the limiting substrate condition. For microbial flooding, because nutrient

concentration decreases with distance from the injector, depending on the injected bacteria to

nutrient ratio, there would exist a zone surrounding the injection well outside of which the

microbial growth is zero and the system is in equilibrium.

A similar kinetic model (Zhang, 1993) can be used to describe the rate of surfactant biosynthesis

𝜇𝑚 = 𝜇𝑚,𝑚𝑒𝑡𝑎𝑁 − 𝑁𝑐𝑟𝑖𝑡

𝐾𝑚 + 𝑁 −𝑁𝑐𝑟𝑖𝑡, (3.24)

where 𝜇𝑚 is the synthesis rate of metabolites, 𝜇𝑚,𝑚𝑒𝑡𝑎 the maximum surfactant synthesis rate,

and 𝑁𝑐𝑟𝑖𝑡 the critical concentration of nutrients below which surfactant production does not

occur. The equation models bacteria’s preferential utilization of nutrients for survival instead of

non-critical metabolite synthesis in limiting conditions.

37

As previously described, bacteria consume nutrients for both reproduction and surfactant

synthesis. The proportion of nutrients utilised for each activity is then defined through the yield

coefficient, 𝑌𝑖, where i can be either bacteria or metabolites. Assuming metabolite production

and reproduction are the only two reactions the bacteria undergo,

𝑌𝑏 + 𝑌𝑚 = 1. (3.25)

Then the reaction terms for bacteria and metabolites are functions of the bacterial concentration,

the yield coefficient and the growth or production rates,

𝑅𝑏 = 𝜇𝑏𝐵𝑌𝑏 (3.26)

𝑅𝑚 = 𝜇𝑚𝐵𝑌𝑚 (3.27)

𝑅𝑛 = −𝑅𝑏 − 𝑅𝑚. (3.28)

Similar to nutrients and biosurfactants, bacteria are assumed to be a component of the water

phase. Bacteria, surfactants and nutrients are assumed not to partition into the oil phase or adsorb

into the rock surface. Surfactant is assumed to be produced in situ only. Lastly assuming that

biofilms are not formed within the reservoir (i.e. porosity and permeability are constant), the

basic system of conservation equations for bacteria, surfactants, and substrates can be written as

𝜕

𝜕𝑡(𝑆𝑤𝜙𝐵

𝐵𝑤) + ∇ ∙ (

�⃗� 𝑤𝐵

𝐵𝑤) −

𝑞𝑏𝜌𝑤− 𝑅𝑏 = 0 (3.29)

𝜕

𝜕𝑡(𝑆𝑤𝜙𝑀

𝐵𝑤) + ∇ ∙ (

�⃗� 𝑤𝑀

𝐵𝑤) − 𝑅𝑚 = 0 (3.30)

𝜕

𝜕𝑡(𝑆𝑤𝜙𝑁

𝐵𝑤) + ∇ ∙ (

�⃗� 𝑤𝑁

𝐵𝑤) −

𝑞𝑛𝜌𝑤− 𝑅𝑛 = 0. (3.31)

38

3.2.2 Biofilm Model

Microbial species in MEOR have surface-active properties which enable them to adsorb to rock

surfaces. The consequence of this is two-fold: (1) porosity and permeability reduction via pore

bio-clogging and (2) concentration of metabolic activity towards injection sites. This

phenomenon may be modelled through the conversion of a portion of the free, or water-soluble,

bacteria into a biofilm (Thullner, 2010). The biofilm may be modelled as a homogeneous or

heterogeneous structure, and is generally assumed to coat the pore wall uniformly. Alternatively,

biomass has been modeled through a mechanism similar to biopolymer agglomeration, wherein

the bacteria flocculate into dense colonies in the water phase and obstruct flow through pore-

throats (Li, 2011).

In this study, the reversible equilibrium adsorption model is used. It is assumed that some portion

of the free-flowing bacteria is adsorbed onto the grain surfaces in a homogenous, uniform

biofilm layer. The growth of this layer would then reduce porosity and permeability. The biofilm

growth is assumed to come solely from the free-flowing bacteria (i.e., biomass produced by the

biofilm is not assumed to adsorb to the rock). Adsorbed bacteria are assumed, however, to

consume nutrients and produce surfactants at the same rate as free bacteria. Furthermore, cells

are assumed not to desorb from the biofilm surface following adsorption. Then, the reaction

terms for bacteria and metabolites can be modified by redefining the bacteria concentration

𝐵 = 𝐵𝑓 + 𝐵𝑏. (3.32)

And the mass conservation for bacteria is re-defined in terms of free bacteria:

39

𝜕

𝜕𝑡(𝑆𝑤𝜙𝐵𝑓

𝐵𝑤) + ∇ ∙ (

�⃗� 𝑤𝐵𝑓

𝐵𝑤) −

𝑞𝑏𝜌𝑤− 𝑅𝑏 = 0 (3.33)

The concentration of bacteria in the biofilm is calculated as a function of the surface area

available for adsorption (𝐴) and the mass of bacteria which can be adsorbed per unit area (Θ),

defined by specific adsorption parameters,

𝐵𝑏 = Θ ∗ 𝐴. (3.34)

The adsorption surface area, 𝐴, is defined as a function of porosity, water saturation, and pore

surface area, 𝐴𝑟. Water saturation must be considered as bacteria exists only in the water phase,

and does not contact oil through the oil phase.

𝐴 =𝑆𝑤𝐴𝑟𝜙

(3.35)

Then the mass of bacteria adsorbed per unit area, Θ, can be defined, via the Langmuir model

Θ =𝛼1 ∗ 𝛼2 ∗ 𝐵𝑓

1 + 𝛼2 ∗ 𝐵𝑓, (3.36)

where 𝛼1 is a parameter controlling the maximum mass of biofilm per unit area of pore space for

a particular microbial species and 𝛼2 is used to determine the speed of adsorption (Amundsen,

2016). In the above, the kinetics of adsorption is neglected, a consequence of which is the

retardation of bacterial transport in comparison to, for example, nutrient transport. The

relationship between free bacteria versus adsorbed bacteria is shown below in Figure 3-1 for two

different sets of Langmuir coefficients. The initial value of the coefficient is especially

important, as for exceedingly large values of 𝛼2 under high water saturation, all free bacteria will

be adsorbed to the biofilm phase. This is exemplified in Figure 3-1b, where for Sw = 0.8,

concentration of bacteria in biofilm will exceed concentration of free bacteria, and thus more

bacteria are assumed to be sequestered to the biofilm phase than in the water phase.

40

Figure 3-1: Concentration of biofilm bacteria versus free bacteria under two sets of

Langmuir coefficients, (a) 𝜶𝟏= 0.001, 𝜶𝟐= 0.001 and (b) 𝜶𝟏= 0.001, 𝜶𝟐= 0.0017

Based on the concentration of biofilm bacteria in the pore space, the density of the biofilm, and

the initial pore space, the porosity may be expressed as (Amundsen, 2016):

ϕ∗ = ϕ(1 −𝐵𝑏𝜌𝑏) (3.37)

While numerous studies have demonstrated the potential of microbial growth to lead to reduction

in porosity, permeability and dispersivity (Li, 2011), the relationship between porosity reduction

and permeability reduction under bioclogging is unclear. In 2002, Thullner proposed the use of

pore network modelling to simulate both biofilm and colony formation and found that the

simulation of colonies fit best with experimental data. In this model, growth was assumed to

occur in the smallest pores first. Pores either contain growing biomass, or do not. For a

homogenous media, the threshold value for effective porosity, defined as the ratio of reduced

porosity to original porosity at which permeability becomes zero, can be as small as 0.4. For a

media with increasing heterogeneity, the threshold value for effective porosity may be as high as

0

0.01

0.02

0.03

0 0.01 0.02 0.03

Bb

(k

g/m

3)

Bf (kg/m3)

Sw = 0.3

Sw = 0.5

Sw = 0.8

0

0.01

0.02

0.03

0 0.01 0.02 0.03

Bb (

kg

/m3)

Bf (kg/m3)

Sw = 0.3

Sw = 0.5

Sw = 0.8

41

1. Commercial simulators, such as UTCHEM, (Desouky et al., 1996) use the Carman-Kozeny to

relate porosity and permeability, giving

𝐾 =ϕ

𝑘𝑧 ∗ 𝑠𝑝𝑣2. (3.38)

In addition to this, Abbasi et al. (2018) have identified numerous other models to relate

permeability to biofilm saturation. Of these, the Kozeny grain model, where the biofilm coats the

grain, and the Kozeny pore model, where the biofilm grows vertically into the pore space, are

most well-known. The grain-coating model is the simplest (Kleinberg et al. 2003), and is written

in a form similar to the parallel capillary coating model,

𝐾∗ = 𝐾 (ϕ∗

ϕ)𝐶

, (3.39)

where permeability is related to porosity via a constant, 𝐶. Based on the maximum limit for

biomass-affected pore radius, Clement et al. (1996) proposed a value of 19/6 for 𝐶. On the other

hand, the grain pore-filling model is written,

𝐾∗ = 𝐾(1 −

𝐵𝑏𝜌𝑏)𝐶+2

(1 + √𝐵𝑏𝜌𝑏)

2 .

The Kozeny grain-coating model is selected for this work. The model is compared against the

pore-filling model and experimental data from Abbasi et al. in Figure 3-2 below, using an

empirical value of 17.9 and 6.8 for 𝐶 respectively. The R2 for the grain-coating model is 0.98,

and for the pore-filling model is 0.95.

42

Figure 3-2: Validation of Kozeny grain-coating model against experimental data (Abbasi et

al., 2015)

3.2.3 Biosurfactant effect

One of the key effects of surfactant production is oil-water IFT reduction and consequent

alteration of capillary forces, mediated by the enrichment of the amphiphilic molecules at the

interface. This is critical to mobilizing residual oil, the flow of which is controlled by viscous

and capillary forces. The displacement efficiency can then be expressed through the capillary

number

𝑁𝑐𝑎 =𝜂𝑤𝑢

𝜎 cos 𝜃, (3.40)

where 𝜂𝑤 is the viscosity of the displacing fluid (nominally, the water phase), 𝑢 is the velocity of

the displacing fluid, and 𝜃 is the contact angle. Depending on surfactant type and concentration,

IFT can be reduced up to four orders of magnitude (Fulcher et al., 1985). Then, due to limited

range of possible values for the first two variables, significant changes to the capillary number,

0

0.2

0.4

0.6

0.8

1

0 0.02 0.04 0.06 0.08 0.1

k*

/k

Bb/ρb

Experimental (Abbasi, 2015)

Kozeny pore-filling

Kozeny grain-coating

43

and as an extension significant mobilization of immobile oil, is only possible with changes in the

IFT.

The effect of surfactant on IFT can be described as a function of the surfactant concentration,

and a set of properties of the surfactant, 𝑙1, 𝑙2, and 𝑙3 (Nielsen, 2010),

𝜎∗(𝑀) = 𝜎−𝑡𝑎𝑛ℎ(𝑙3𝑀− 𝑙2) + 1 + 𝑙1−𝑡𝑎𝑛ℎ(−𝑙2) + 1 + 𝑙1

. (3.41)

Notably, there is a threshold value for 𝑀 which must be exceeded, before a significant change in

interfacial intension may be observed, and as well as a critical value after which surfactant fails

to alter interfacial tension. This value relates to the critical micelle concentration, above which

surfactant molecules will spontaneously form micellar structures. It has been observed that for

conventional chemical EOR, surfactant concentrations should be higher than the CMC, as the

presence of a significant concentration of micelles is necessary to maximize interfacial tension

reduction (Schramm & Marangoni, 2000). Another consideration is the behavior of the surfactant

at the water-oil interface; the partitioning of surfactant into the oil phase, and thus loss of

activity, has been modeled in a number of studies. We assume here that the surfactant does not

partition significantly into the oil phase.

To validate equation (3.41), three biosurfactants and their experimental effect on interfacial

tension reduction were extracted from Daoshan et al. (2004), McInernery et al. (2004), and

Pereira et al. (2013). 𝑙1, 𝑙2, and 𝑙3 were adjusted to fit the model to the data, and the parameter

and R2 values are summarized below in Table 3-1 below. With an average R

2 of 0.978 and a

44

relatively random residuals distribution, the model can be validated. A graph comparing the

predicted to the actual value is shown below in Figure 3-3.

Table 3-1: Calibrated Values for IFT Reduction Model for Three Different Biosurfactants

Rhamnolipid

(Daoshan et al., 2004)

Glycolipid

(Pereira et al., 2013)

Glycolipid

(McInernery et al., 2014)

𝑙1 0.35 0.35 0.05

𝑙2 0.2 0.2 0.2

𝑙3 1.1 0.12 400

R2 0.998 0.962 0.975

Figure 3-3 Validation of the Nielsen IFT reduction model to experimental data from

Daoshan et al. (2014); Pereira et al. (2013); and McInernery et al. (2004)

45

While the exact effect of IFT on relative permeability curves is unclear (Al-Wahaibi et al., 2006;

Shen et al., 2006), a lower IFT can reduce the curvature and residual saturation and increase end-

point relative permeabilities of relative permeability curves. Some studies have previously

compared different methods of calculating the impact of IFT reduction on relative permeability

curves (Nielsen, 2010; Amundsen, 2016), including the capillary number method, the Coats

interpolation method, and interpolation of parameters in Corey equations.

While the most complex out of the three methods procedurally, Coats’ correlation can be used to

describe changes in relative permeability curves by reduction in IFT via an interpolation

function. The interpolation function 𝑔(𝜎) is then defined

𝑔(𝜎) = (𝜎∗

𝜎)

1𝑎, (3.42)

where the exponent 𝑎 typically ranges between 4 and 10 from experimental permeability curves,

𝑔(𝜎) may range from one to an infinitesimal value, and the index (*) refers to the new or

modified value. Then 𝑔(𝜎) can be used to update the residual saturation values, to a value

between zero and initial saturation,

𝑆𝑤𝑖∗ = 𝑔(𝜎)𝑆𝑤𝑖

𝑆𝑜𝑟∗ = 𝑔(𝜎)𝑆𝑜𝑟.

(3.43)

Here 𝑔(𝜎) is used also to update relative permeability curves, which are composed of two parts,

𝑘𝑟𝑖,𝑖𝑚𝑚 and 𝑘𝑟𝑖,𝑚𝑖𝑠𝑐. 𝑘𝑟𝑖,𝑖𝑚𝑚 is the immiscible relative permeabilities of the unaltered IFT value,

𝜎. 𝑘𝑟𝑖,𝑚𝑖𝑠𝑐 is the relative permeability curve of fully miscible phases. While it is modeled here as

a straight line between the endpoints of 𝑘𝑟𝑖,𝑖𝑚𝑚, other studies have proposed using IFT-based

residual saturation curves. Note that larger values of 𝑎 reduce the impact of IFT on relative

46

permeability curves, wherein relative permeabilities change less towards the full miscibility

curves. The following relation is then sued to determine the final relative permeability curve:

𝑘𝑟𝑤∗ = 𝑔(𝜎) ∗ 𝑘𝑟𝑤,𝑖𝑚𝑚 + (1 − 𝑔(𝜎)) ∗ 𝑘𝑟𝑤,𝑚𝑖𝑠𝑐

𝑘𝑟𝑜∗ = 𝑔(𝜎) ∗ 𝑘𝑟𝑜,𝑖𝑚𝑚 + (1 − 𝑔(𝜎)) ∗ 𝑘𝑟𝑜,𝑚𝑖𝑠𝑐.

(3.44)

The Coat’s interpolation method alone does not result in the alteration of maximum relative

permeability values, and consequently at some saturation, relative permeability following IFT

reduction is less than before. Shen et al. (2006) and Nielsen (2015) proposed the additional

interpolation of 𝑘𝑟𝑜𝑤𝑖 and for 𝑘𝑟𝑤𝑜𝑟, and the Corey exponent, 𝑛, using a similar methodology,

𝑘𝑟𝑜𝑤𝑖∗ = (

𝜎∗

𝜎)1/𝑎

∗ 𝑘𝑟𝑜𝑤𝑖 + (1 − (𝜎∗

𝜎)1/𝑎

)

𝑘𝑟𝑤𝑜𝑟∗ = (

𝜎∗

𝜎)1/𝑎

∗ 𝑘𝑟𝑤𝑜𝑟 + (1 − (𝜎∗

𝜎)1/𝑎

)

𝑛∗ = (𝜎∗

𝜎)1/𝑎

∗ 𝑛 + (1 − (𝜎∗

𝜎)

1𝑎).

(3.45)

The additional interpolation step allows the end-point values at residual saturations to be

increased and the Corey exponent to be decreased. Thus, relative permeability following IFT

reduction can span a broader range of water saturation, and provides a better match with

experimental results.

3.3 Summary of Equations and Mechanisms

Key mechanisms this MEOR model simulates are summarized below in Figure 3-4. With

microbial growth and biofilm generation, porosity and subsequently permeability will be altered.

47

At the same time, microbial growth enables biosurfactant production, resulting in interfacial

tension reduction and alteration of oil and water relative permeabilities, lowering residual oil

saturation and enhancing oil recovery.

Microbial KineticsTwo-Phase

Immiscible Flow

Reservoir Rock

Properties

Porosity,

permeability

IFT, relative permeability

Sor, Swi, etc.

Figure 3-4: Summary of MEOR mechanisms

For convenience, the equations used for modelling MEOR, built within the MRST blackoil

module, are summarized below.

1. Two-phase incompressible flow

𝑆𝑜 + 𝑆𝑤 = 1

𝑝𝑐 = 𝑝𝑜 − 𝑝𝑤 = 𝑓(𝑆𝑤)

𝑘𝑟𝑤(𝑆𝑤) = 𝑘𝑟𝑤𝑜𝑟 (𝑆𝑤 − 𝑆𝑤𝑐

1 − 𝑆𝑤𝑐 − 𝑆𝑜𝑟)𝑛

𝑘𝑟𝑜(𝑆𝑜) = 𝑘𝑟𝑜𝑤𝑐 (𝑆𝑜 − 𝑆𝑜𝑟

1 − 𝑆𝑤𝑐 − 𝑆𝑜𝑟)𝑛

�⃗� 𝑤 = −𝑘𝑟𝑤𝑲

𝜂𝑤(∇𝑝 − 𝜌𝑤𝑔∇z)

�⃗� 𝑜 = −𝑘𝑟𝑜𝑲

𝜂𝑜(∇𝑝 − 𝜌𝑜𝑔∇z)

48

−∇ ∙ (�⃗⃗� (𝜆∇𝑝𝑜 − 𝜆𝑤∇𝑝𝑐 − (𝜆𝑤𝜌𝑤 + 𝜆𝑜𝜌𝑜)𝑔∇𝑧)) =𝑞𝑜𝜌𝑜+𝑞𝑤𝜌𝑤



∇𝑆𝑤 − (𝜌𝑜 − 𝜌𝑤)𝑔∇𝑧) + 𝑓𝑤�⃗� ) =𝑞𝑤𝜌𝑤

2. Microbial growth and metabolism kinetics

𝜇𝑏 = 𝜇𝑚,𝑚𝑖𝑐𝑁

𝐾 + 𝑁

𝜇𝑚 = 𝜇𝑚,𝑚𝑒𝑡𝑎𝑁 − 𝑁𝑐𝑟𝑖𝑡

𝐾𝑚 + 𝑁 − 𝑁𝑐𝑟𝑖𝑡

𝑌𝑏 + 𝑌𝑚 = 1

𝑅𝑏 = 𝜇𝑏𝐵𝑌𝑏

𝑅𝑚 = 𝜇𝑚𝐵𝑌𝑚

𝑅𝑛 = −𝑅𝑏 − 𝑅𝑚

3. Transport of microbial components within the water phase

𝜕

𝜕𝑡(𝑆𝑤𝜙𝑀) + ∇ ∙ (�⃗� 𝑤𝑀) − 𝑅𝑚 = 0

𝜕

𝜕𝑡(𝑆𝑤𝜙𝑁) + ∇ ∙ (�⃗� 𝑤𝑁) −

𝑞𝑛𝜌𝑤− 𝑅𝑛 = 0

𝜕

𝜕𝑡(𝑆𝑤𝜙𝐵) + ∇ ∙ (�⃗� 𝑤𝐵) −

𝑞𝑏𝜌𝑤− 𝑅𝑏 = 0

𝜕

𝜕𝑡(𝑆𝑤𝜙𝐵𝑓) + ∇ ∙ (�⃗� 𝑤𝐵𝑓) −

𝑞𝑏𝜌𝑤− 𝑅𝑏 = 0

4. Biofilm reduction of porosity and permeability

𝐵 = 𝐵𝑓 + 𝐵𝑏

𝐵𝑏 =𝛼1 ∗ 𝛼2 ∗ 𝐵𝑓

1 + 𝛼2 ∗ 𝐵𝑓∗𝑆𝑤𝐴𝑟𝜙

ϕ∗ = ϕ(1 −𝐵𝑏𝜌𝑏)

𝐾∗ = 𝐾 (ϕ∗

ϕ)𝐶

49

5. Biosurfactant reduction of interfacial tension and alteration of relative permeability

𝜎∗(𝑀) = 𝜎−𝑡𝑎𝑛ℎ(𝑙3𝑀− 𝑙2) + 1 + 𝑙1−𝑡𝑎𝑛ℎ(−𝑙2) + 1 + 𝑙1

𝑔(𝜎) = (𝜎∗

𝜎)

1𝑎

𝑆𝑤𝑖∗ = 𝑔(𝜎)𝑆𝑤𝑖

𝑆𝑜𝑟∗ = 𝑔(𝜎)𝑆𝑜𝑟

𝑘𝑟𝑜𝑤𝑖∗ = (

𝜎∗

𝜎)1/𝑎

∗ 𝑘𝑟𝑜𝑤𝑖 + (1 − (𝜎∗

𝜎)1/𝑎

)

𝑘𝑟𝑤𝑜𝑟∗ = (

𝜎∗

𝜎)1/𝑎

∗ 𝑘𝑟𝑤𝑜𝑟 + (1 − (𝜎∗

𝜎)1/𝑎

)

𝑛∗ = (𝜎∗

𝜎)1/𝑎

∗ 𝑛 + (1 − (𝜎∗

𝜎)

1𝑎)

3.4 Assumptions

Some of the key assumptions made in this model are summarized below:

Injected bacteria type is tolerant to reservoir conditions;

Monod equation can sufficiently model bacterial growth and surfactant production;

Indigenous bacteria are of negligible mass within the reservoir and do not consume

injected nutrients; indigenous bacteria do not impact exogenous bacteria survival;

Injected bacterial growth rate is only a factor of nutrient concentration;

Reservoir can be modeled under isothermal and constant pH and salinity conditions;

Effective bacterial growth rate also sufficiently describes bacteria death rate;

Dispersion and advection transport effects are not considered;

Chemotaxis effect on bacteria transport is negligible;

50

Nutrients are only converted for cell growth and replication or surfactant production;

The reversible equilibrium model is valid for biofilm growth;

Biosurfactant, bacteria, and nutrient are not adsorbed, apart from the loss of bacteria to

biofilm formation;

The biofilm is homogeneous, and bacteria within the biofilm have the same metabolic

reactions as free bacteria;

The biofilm does not contain a significant mass fraction of exopolymers

Biosurfactant and nutrient loss to the oil phase is negligible;

Density and viscosity of the fluid is independent of bacteria, nutrient and surfactant

concentration; and

Dispersion of bacteria and nutrient in the water phase is complete and immediate.

51

4 CHAPTER 4: NUMERICAL MODEL

4.1 Matlab Reservoir Simulation Toolbox

MATLAB Reservoir Simulation Toolbox (MRST) is used in this study to solve the microbial

and reservoir equations described in Chapter 3. MRST is an open source toolbox currently under

development by SINTEF in Oslo, Norway. It is designed for the rapid prototyping and

demonstration of new modelling concepts. Naturally, it has been frequently adopted for MEOR

modelling (Akindipe, 2016; Amundsen, 2016). This is due to the number and complexity of

MEOR mechanisms, many of which commercial simulators cannot accommodate.

The default discretization scheme in MRST is the Two-Point Flux-Approximation (TPFA)

method, a cell-centered finite volume method. In addition to TPFA, MRST contains a variety of

add-on modules for different finite volume and finite element solvers, including Multipoint Flux

Approximation (MPFA), Multiscale Mixed Finite Elements (MsMFE), Mimetic Finite

Difference (MFD), and Multiscale TPFA (MsTPFA). In contrast to finite difference codes

conventionally used to numerically model MEOR, finite element and finite volume methods

offer greater flexibility and the ability to solve a broader class of problems. This is particularly

applicable to reservoir simulation, where the complex geometry of geological objects within

petroleum reservoirs cannot be easily described by structured meshes in FDM.

52

4.2 Finite Volume Discretization

4.2.1 Gridding

Grids in two-dimensional models are composed of planar cells which may be defined by vertices

and edges. Unstructured grids may consist of a variety of polyhedral cells. The most common

cell type is triangular, such as in the Delaunay tessellation. The flexibility of unstructured grids

allows easy adaptation to change in resolution. However, in an unstructured grid, geometry of

each cell needs to be stored explicitly, and cannot be referenced using a structured multi-index.

On the other hand, structured grids often require large assumptions of the reservoir geology and

cannot easily describe complex structures at high resolution. However, cells in structured grids

are of uniform shapes and can be referenced through a multi-index (Lie, 2016), significantly

simplifying computational and memory storage costs. Choice of grid, therefore, depends on the

trade-off between memory and efficiency, and representativeness.

Furthermore, two basic formulations exist for FVM: cell- or vertex- centered. In the cell-centered

method, the grid cells themselves are defined as the control volumes. In the vertex-centered

method, the control volumes are defined separately from, and constructed around, the

computational grid cells. Dobes et al. (2006) and Haegland et al. (2009) have made an extensive

comparison of the two methods for simulation of flow in porous media. In this study, a simple 2-

dimensional, cell-centered Cartesian grid is implemented, as shown in Figure 4-1. It is composed

of rectangles with uniform spacing, where the edges are aligned with the 𝑥- and 𝑦 - axis. The

multi-index reference for cells is (𝑖1, 𝑖2, … ). These indices can be mapped uniformly to the

vertex coordinates, (𝑖1∆𝑥1, 𝑖2∆𝑥2, … ).

53

Figure 4-1: Cell-centered two-dimensional grid

4.2.2 Two-Point Flux Approximation

Finite volume discretization is implemented through the Two-Point Flux-Approximation (TPFA)

scheme. TPFA, or the cell-centered finite volume method, is used extensively in industry, as it is

relatively robust and easy to implement. In comparison to finite difference methods, finite

volume methods such as TPFA are derived from conservation of quantities over cell volumes

and thus more physically motivated. However, convergence for TPFA schemes is limited to

orthogonal grids, as different grid orientations will impact convergence.

Without loss of generality, implementation of TPFA for a flow equation requires its integration

over a control volume. Consider the simplified single-phase flow equation

∇ ∙ �⃗� = 𝑞, �⃗� = −�⃗⃗� ∇𝑝, in Ω ⊂ ℝ𝑑 . (4.1)

54

Let Ω𝑖 be a single cell with a normal vector, �⃗� 𝑖𝑗. Integrating the above over Ω𝑖 and assuming

sufficient smoothness, invoking the divergence theorem, the mass conservation equation is

derived as

∫ �⃗� ∙ �⃗� 𝑖𝑗𝑑𝑠

∂Ω𝑖

= ∫ 𝑞𝑑𝑥

∂Ω𝑖

, (4.2)

and substituting Darcy’s law,

∫ −�⃗⃗� ∇𝑝 ∙ �⃗� 𝑖𝑗𝑑𝑠

∂Ω𝑖

= ∫ 𝑞𝑑𝑥

∂Ω𝑖

. (4.3)

Two-point flux approximation derives from combination of separate one-sided flux

approximation for each neighbouring cell to the shared interface. Let the flux across the center of

a grid cell, Ω𝑖 to its interface with a neighboring cell, Ω𝑗, be denoted by

𝑢𝑖,𝑗 = ∫ −�⃗⃗� ∇𝑝 ∙ �⃗� 𝑖,𝑗𝑑𝑠

Γ𝑖,𝑗

, (4.4)

where Γ𝑖,𝑗 is the interface belonging to Ω𝑖 with a normal �⃗� 𝑖𝑗 and area 𝐴𝑖𝑗. For the same interface

belonging to the neighboring cell, then

𝑢𝑗,𝑖 = ∫ −�⃗⃗� ∇𝑝 ∙ �⃗� 𝑗,𝑖𝑑𝑠

Γ𝑗,𝑖

= ∫ �⃗⃗� ∇𝑝 ∙ �⃗� 𝑖,𝑗𝑑𝑠

Γ𝑗,𝑖

. (4.5)

The pressure gradient must be approximated in both cells. Between the center and face of each

control volume, the gradient term can be calculated with a central finite difference approximation

to give a one-sided transmissibility term, assuming that pressure is linear within each cell,

55

𝑢𝑖,𝑗 = ∫ −�⃗⃗� 𝑖∇𝑝 ∙ �⃗� 𝑖,𝑗𝑑𝑠

Γ𝑖,𝑗

≈ −𝐴𝑖𝑗�⃗⃗� 𝑖(𝑝𝑖 − 𝜋𝑖,𝑗)𝑑𝑖

|𝑑𝑖|2∙ �⃗� 𝑖,𝑗

𝑢𝑗,𝑖 = ∫ −�⃗⃗� 𝑗∇𝑝 ∙ �⃗� 𝑗,𝑖𝑑𝑠

Γ𝑗,𝑖

≈ 𝐴𝑗𝑖 �⃗⃗� 𝑗(𝑝𝑗 − 𝜋𝑗,𝑖)𝑑𝑗

|𝑑𝑗|2 ∙ �⃗� 𝑖,𝑗 .

(4.6)

Then imposing the continuity of fluxes across all faces, 𝑢𝑖,𝑗 = −𝑢𝑗,𝑖 = 𝑢𝑖𝑗, and continuity of

pressure at the interface 𝜋𝑖,𝑗 = 𝜋𝑗,𝑖 = 𝜋𝑖𝑗 , the TPFA discretization of the flux is given by

𝑢𝑖𝑗 ≈ 𝐴𝑖𝑗 (𝑑𝑖𝐾𝑖+𝑑𝑗

𝐾𝑗)

−1

(𝑝𝑖 − 𝑝𝑗). (4.7)

Note that the discretization result is similar to that from a classical finite difference scheme.

Indeed, while FVM and FDM are fundamentally different in terms of derivation and

interpretation, for low-order methods, cell-centered mass-conservative FDM gives the same

output as that from cell-centered FVM.

Next, the discretization of flow is presented. The pressure equation from Chapter 3 can be

discretized as

−∇ ∙ (�⃗⃗� (𝜆∇𝑝𝑜 − 𝜆𝑤∇𝑝𝑐 − (𝜆𝑤𝜌𝑤 + 𝜆𝑜𝜌𝑜)𝑔∇𝑧)) =𝑞𝑜𝜌𝑜+𝑞𝑤𝜌𝑤. (4.8)

Then let the fluid gravity be defined as

𝛾𝛼 = 𝜆𝛼𝜌𝛼

�̅� = 𝜆𝑤𝜌𝑤 + 𝜆𝑜𝜌𝑜 ,

(4.9)

56

and assume that the permeability tensor can be written as

�⃗⃗� = 𝑑𝑖𝑎𝑔(𝑘11, 𝑘22, 𝑘33). (4.10)

Then the pressure equation can be written in two dimensions regarding each coordinate, 𝑥1and

𝑥2, as follows,

−𝜕

𝜕𝑥1(𝑘11𝜆

𝜕𝑝𝑜𝜕𝑥1) −

𝜕

𝜕𝑥2(𝑘22𝜆

𝜕𝑝𝑜𝜕𝑥2

)

=𝑞𝑜𝜌𝑜+𝑞𝑤𝜌𝑤−𝜕

𝜕𝑥1(𝑘11 (𝜆𝑤

𝜕𝑝𝑐𝜕𝑥1

+ �̅�𝜕𝑧

𝜕𝑥1))

−𝜕

𝜕𝑥2(𝑘22 (𝜆𝑤


+ �̅�𝜕𝑧

𝜕𝑥2))

(4.11)

Applying the finite volume method, the pressure equation is integrated over the control volume

∬ −𝜕

𝜕𝑥1(𝑘11𝜆

𝜕𝑝𝑜𝜕𝑥1) −

𝜕

𝜕𝑥2(𝑘22𝜆

𝜕𝑝𝑜𝜕𝑥2

)𝑑𝑥1𝑑𝑥2Ω

=∬𝑞𝑜𝜌𝑜+𝑞𝑤𝜌𝑤−𝜕

𝜕𝑥1(𝑘11 (𝜆𝑤


+ �̅�𝜕𝑧

𝜕𝑥1))

Ω

−𝜕

𝜕𝑥2(𝑘22 (𝜆𝑤


+ �̅�𝜕𝑧

𝜕𝑥2))𝑑𝑥1𝑑𝑥2

(4.12)

Then,

57

−[∫(𝑘11𝜆𝜕𝑝𝑜𝜕𝑥1) 𝑑𝑥2]

𝑖−12,𝑗

𝑖+12,𝑗

− [∫(𝑘22𝜆𝜕𝑝𝑜𝜕𝑥2

) 𝑑𝑥1]𝑖,𝑗−

12

𝑖,𝑗+12

=∬(𝑞𝑜𝜌𝑜+𝑞𝑤𝜌𝑤) 𝑑𝑥1𝑑𝑥2

Ω

− [∫(𝑘11 (𝜆𝑤𝜕𝑝𝑐𝜕𝑥1

+ �̅�𝜕𝑧

𝜕𝑥1)) 𝑑𝑥2]

𝑖−12,𝑗

𝑖+12,𝑗

− [∫(𝑘22 (𝜆𝑤𝜕𝑝𝑐𝜕𝑥2

+ �̅�𝜕𝑧

𝜕𝑥2)) 𝑑𝑥1]

𝑖−12,𝑗

𝑖+12,𝑗

.

(4.13)

For a Taylor series expansion of a function 𝑓 around some point 𝜉,

−∫𝑓(𝑠)𝑑𝑠 = ∫(𝑓(𝜉) + (𝑠 − 𝜉)𝑓′(𝜉) +(𝑠 − 𝜉)2

2!𝑓′′(𝜉) + ⋯)𝑑𝑠.

(4.14)

Then integration with a change of variable, the approximation can be shown below, with a

second order of accuracy,

∫𝑓(𝑠)𝑑𝑠 = 𝑓(𝜉)∫𝑑𝑠 + 𝑓′(𝜉)∫(𝑠 − 𝜉)𝑑𝑠 + ⋯

∫𝑓(𝑠)𝑑𝑠 = 𝑓(𝜉)Δ𝑠 + 𝑂((Δ𝑠)2).

(4.15)

Using the mid-point rule, the source term can be approximated as

∬ (𝑞𝑜𝜌𝑜+𝑞𝑤𝜌𝑤) 𝑑𝑥1𝑑𝑥2

Ω

≈ (𝑞𝑜𝜌𝑜+𝑞𝑤𝜌𝑤) ∆𝑥1∆𝑥2

(4.16)

Then the cell-centered finite volume discretization of the pressure equation, evaluating the𝜕𝑝𝑐

𝜕𝑥1 ,

𝜕𝑝𝑐

𝜕𝑥2, 𝜕𝑝𝑜

𝜕𝑥1, and

𝜕𝑝𝑜

𝜕𝑥2 terms with a central difference, can be written as

58

−(∆𝑥2𝑘11𝜆

ℎ1)𝑖+12,𝑗

(𝑝𝑜,𝑖+1,𝑗 − 𝑝𝑜,𝑖,𝑗) + (∆𝑥2𝑘11𝜆

ℎ1)𝑖−12,𝑗

(𝑝𝑜,𝑖,𝑗 − 𝑝𝑜,𝑖−1,𝑗)

− (∆𝑥1𝑘22𝜆

ℎ2)𝑖,𝑗+

12

(𝑝𝑜,𝑖,𝑗+1 − 𝑝𝑜,𝑖,𝑗) + (∆𝑥1𝑘22𝜆

ℎ2)𝑖,𝑗−

12

(𝑝𝑜,𝑖,𝑗 − 𝑝𝑜,𝑖,𝑗−1)

= −(∆𝑥2𝑘11𝜆𝑤

ℎ1)𝑖+12,𝑗

(𝑝𝑐,𝑖+1,𝑗 − 𝑝𝑐,𝑖,𝑗)

+ (∆𝑥2𝑘11𝜆𝑤

ℎ1)𝑖−12,𝑗

(𝑝𝑐,𝑖,𝑗 − 𝑝𝑐,𝑖−1,𝑗)

− (∆𝑥1𝑘22𝜆𝑤ℎ2

)𝑖,𝑗+

12

(𝑝𝑐,𝑖,𝑗+1 − 𝑝𝑐,𝑖,𝑗)

+ (∆𝑥1𝑘22𝜆𝑤ℎ2

)𝑖,𝑗−

12

(𝑝𝑐,𝑖,𝑗 − 𝑝𝑐,𝑖,𝑗−1) − (∆𝑥2𝑘11�̅�

ℎ1)𝑖+12,𝑗

(𝑧𝑖+1,𝑗 − 𝑧𝑖,𝑗)

+ (∆𝑥2𝑘11�̅�

ℎ1)𝑖−12,𝑗

(𝑧𝑖,𝑗 − 𝑧𝑖−1,𝑗) − (∆𝑥1𝑘22�̅�

ℎ2)𝑖,𝑗+

12

(𝑧𝑖,𝑗+1 − 𝑧𝑖,𝑗)

+ (∆𝑥1𝑘22�̅�

ℎ2)𝑖,𝑗−

12

(𝑧𝑖,𝑗 − 𝑧𝑖,𝑗−1) + (𝑞𝑜𝜌𝑜+𝑞𝑤𝜌𝑤)𝑖,𝑗

∆𝑥1∆𝑥2

(4.17)

And the transport equation,



∇𝑆𝑤 − (𝜌𝑜 − 𝜌𝑤)𝑔∇𝑧) + 𝑓𝑤�⃗� ) =𝑞𝑤𝜌𝑤

(4.18)


= ∇ ∙ (�⃗⃗� 𝜆𝑤(∇𝑝𝑤 − 𝛾𝑤∇𝑧)) +𝑞𝑤𝜌𝑤

(4.19)

In two dimensions,

59


=𝜕

𝜕𝑥1(𝜆𝑤𝑘11 (

𝜕𝑝𝑤𝜕𝑥1

− 𝛾𝑤𝜕𝑧

𝜕𝑥1)) +

𝜕

𝜕𝑥2(𝜆𝑤𝑘22 (

𝜕𝑝𝑤𝜕𝑥2

− 𝛾𝑤𝜕𝑧

𝜕𝑥2)) +

𝑞𝑤𝜌𝑤,

(4.20)

and can be discretized similarly through the finite volume method in two dimensions,

(𝜙𝜕𝑆𝑤𝜕𝑡)𝑖,𝑗∆𝑥1∆𝑥2

= (∆𝑥2𝑘11𝜆𝑤

ℎ1)𝑖+12,𝑗

(𝑝𝑤,𝑖+1,𝑗 − 𝑝𝑤,𝑖,𝑗)

− (∆𝑥2𝑘11𝜆𝑤

ℎ1)𝑖−12,𝑗

(𝑝𝑤,𝑖,𝑗 − 𝑝𝑤,𝑖−1,𝑗)

+ (∆𝑥1𝑘22𝜆𝑤ℎ2

)𝑖,𝑗+

12

(𝑝𝑤,𝑖,𝑗+1 − 𝑝𝑤,𝑖,𝑗)

− (∆𝑥1𝑘22𝜆𝑤ℎ2

)𝑖,𝑗−

12

(𝑝𝑤,𝑖,𝑗 − 𝑝𝑤,𝑖,𝑗−1)

− (∆𝑥2𝜆𝑤𝛾𝑤𝑘11

ℎ1)𝑖+12,𝑗

(𝑧𝑖+1,𝑗 − 𝑧𝑖,𝑗)

+ (∆𝑥2𝜆𝑤𝛾𝑤𝑘11

ℎ1)𝑖−12,𝑗

(𝑧𝑖,𝑗 − 𝑧𝑖−1,𝑗)

− (∆𝑥1𝜆𝑤𝛾𝑤𝑘22

ℎ2)𝑖,𝑗+

12

(𝑧𝑖,𝑗+1 − 𝑧𝑖,𝑗)

+ (∆𝑥1𝜆𝑤𝛾𝑤𝑘22

ℎ2)𝑖,𝑗−

12

(𝑧𝑖,𝑗 − 𝑧𝑖,𝑗−1) + (𝑞𝑤𝜌𝑤)𝑖,𝑗

∆𝑥1∆𝑥2.

(4.21)

4.2.3 Time discretization

The discretization of time is implemented through the Euler method. This involves the uniform

partitioning of the time interval from initial time 𝑡𝑜 = 0 to the final time 𝑡𝑁 = 𝑇, through 𝑛 time

steps of size Δ𝑡. This is shown in Figure 4-2 below.

Figure 4-2: Time discretization

60

There are three time-stepping schemes, Forward Euler, Backward Euler, and Improved Euler

(Crank-Nicolson). The computational cost of the three schemes differ considerably, and an

optimal choice depends on whether the intermediate evolution details of the flow problem need

to be captured. While explicit schemes are easy to implement and have lower cost per time step,

smaller time steps are required for stability reasons, especially if velocity or mesh size are non-

uniform; on the other hand, implicit schemes are stable over a wider range of time steps, but

have higher cost per time step, particularly for non-linear problems, and fail to converge at larger

Δ𝑡.

The general layout of forward and backward Euler, and Crank-Nicolson discretization scheme

may be written as:

𝑢𝑛+1 − 𝑢𝑛

Δ𝑡+ 𝜃𝑓(𝑢𝑛+1, 𝑡𝑛+1) + (1 − 𝜃)𝑓(𝑢𝑛, 𝑡𝑛) = 0, (4.22)

where 𝜃 is the implicitness parameter,

𝜃 = 0 Forward Euler

𝜃 = 1 Backward Euler

𝜃 = 1/2 Crack Nicholson

(4.23)

61

4.3 IMPES Method

The elliptic pressure equation is time-dependent through the mobility terms, which are functions

of time-dependent saturation. With fixed saturation, the pressure equation then becomes time-

independent.

In this work, the implicit time discretization scheme is used to solve for pressure. This is per the

implicit pressure, explicit saturation (IMPES) method (Sheldon et al., 1959), which computes

pressure separately from saturation. At a certain time step, the pressure equation is solved first

through implicit time approximation, for 𝑝𝑛 at each grid block. Then the saturation at the next

time step, 𝑆𝑛+1 is solved through explicit time approximation, by substituting pressure, 𝑝𝑛, and

saturation, 𝑆𝑛,value. By decoupling the pressure and saturation equations, the sequential solver

IMPES is more efficient and requires less memory than other approaches, such as Simultaneous

Solution. Thus, it is more broadly used in industry for two-phase incompressible and slightly

compressible flows.

In order to derive the two-point flux-approximation discretization of equations from Chapter 3,

first, transmissibility corresponding to the mobility of phase 𝛼 in the 𝑥1 coordinate is defined,

𝑇𝛼,1,𝑖±

12,𝑗= (∆𝑥2𝑘11𝜆𝛼ℎ1

)𝑖±12,𝑗

, (4.24)

and transmissibility corresponding to the total mobility is defined,

𝑇1,𝑖±

12,𝑗= (∆𝑥2𝑘11𝜆

ℎ1)𝑖±12,𝑗

. (4.25)

Further, permeability is calculated through upstream weighting, where

62

𝑘𝑟𝑤|𝑖−1/2 = 𝑘𝑟𝑤|𝑖. (4.26)

Then the implicit pressure equation can be written as

−𝑇1,𝑖+

12,𝑗

𝑛 (𝑝𝑜,𝑖+1,𝑗𝑛 − 𝑝𝑜,𝑖,𝑗

𝑛 ) + 𝑇1,𝑖−

12,𝑗

𝑛 (𝑝𝑜,𝑖,𝑗𝑛 − 𝑝𝑜,𝑖−1,𝑗

𝑛 ) − 𝑇2,𝑖,𝑗+

12

𝑛 (𝑝𝑜,𝑖,𝑗+1𝑛 − 𝑝𝑜,𝑖,𝑗

𝑛 )

+ 𝑇2,𝑖,𝑗−

12

𝑛 (𝑝𝑜,𝑖,𝑗𝑛 − 𝑝𝑜,𝑖,𝑗−1

𝑛 )

= −𝑇𝑤,1,𝑖+

12,𝑗

𝑛 (𝑝𝑐,𝑖+1,𝑗𝑛 − 𝑝𝑐,𝑖,𝑗

𝑛 ) + 𝑇𝑤,1,𝑖−

12,𝑗

𝑛 (𝑝𝑐,𝑖,𝑗𝑛 − 𝑝𝑐,𝑖−1,𝑗

𝑛 )

− 𝑇𝑤,2,𝑖,𝑗+

12

𝑛 (𝑝𝑐,𝑖,𝑗+1𝑛 − 𝑝𝑐,𝑖,𝑗

𝑛 ) + 𝑇𝑤,2,𝑖,𝑗−

12

𝑛 (𝑝𝑐,𝑖,𝑗𝑛 − 𝑝𝑐,𝑖,𝑗−1

𝑛 )

− (𝑇�̅�

𝜆)1,𝑖+

12,𝑗

𝑛

(𝑧𝑖+1,𝑗 − 𝑧𝑖,𝑗) + (𝑇�̅�

𝜆)1,𝑖−

12,𝑗

𝑛


− (𝑇�̅�

𝜆)2,𝑖,𝑗+

12

𝑛

(𝑧𝑖,𝑗 − 𝑧𝑖,𝑗+1) + (𝑇�̅�

𝜆)2,𝑖,𝑗−

12

𝑛

(𝑧𝑖,𝑗 − 𝑧𝑖,𝑗−1)

+ (𝑞𝑜𝜌𝑜+𝑞𝑤𝜌𝑤)𝑖,𝑗

𝑛

∆𝑥1∆𝑥2.

(4.27)

The explicit saturation formulation is given by

63

(𝜙𝑆𝑤𝑛+1 − 𝑆𝑤

𝑛

Δ𝑡)𝑖,𝑗

∆𝑥1∆𝑥2

= 𝑇𝑤,1,𝑖+

12,𝑗

𝑛 (𝑝𝑤,𝑖+1,𝑗𝑛 − 𝑝𝑤,𝑖,𝑗

𝑛 ) − 𝑇𝑤,1,𝑖−

12,𝑗

𝑛 (𝑝𝑤,𝑖,𝑗𝑛 − 𝑝𝑤,𝑖−1,𝑗

𝑛 )

+ 𝑇𝑤,2,𝑖,𝑗+

12

𝑛 (𝑝𝑤,𝑖,𝑗+1𝑛 − 𝑝𝑤,𝑖,𝑗

𝑛 ) − 𝑇𝑤,2,𝑖,𝑗−

12

𝑛 (𝑝𝑤,𝑖,𝑗𝑛 − 𝑝𝑤,𝑖,𝑗−1

𝑛 )

− (𝑇𝑤𝛾𝑤)1,𝑖+

12,𝑗

𝑛 (𝑧𝑖+1,𝑗 − 𝑧𝑖,𝑗) + (𝑇𝑤𝛾𝑤)1,𝑖−

12,𝑗

𝑛 (𝑧𝑖,𝑗 − 𝑧𝑖−1,𝑗)

− (𝑇𝑤𝛾𝑤)2,𝑖,𝑗+

12

𝑛 (𝑧𝑖,𝑗+1 − 𝑧𝑖,𝑗) + (𝑇𝑤𝛾𝑤)2,𝑖,𝑗−

12

𝑛 (𝑧𝑖,𝑗 − 𝑧𝑖,𝑗−1)

+ (𝑞𝑤𝜌𝑤)𝑖,𝑗

𝑛

∆𝑥1∆𝑥2.

(4.28)

The transport equations for bacteria, nutrients and surfactants can similarly be discretized using

the TPFA in the IMPES method. For the free bacteria phase, the discretization is written as

(𝜙(𝑆𝑤𝐵𝑓)

𝑛+1− (𝑆𝑤𝐵𝑓)

𝑛

Δ𝑡)

𝑖,𝑗

∆𝑥1∆𝑥2 =

= 𝐵𝑓𝑇𝑤,1,𝑖+

12,𝑗


𝑛 ) − 𝐵𝑓𝑇𝑤,1,𝑖−

12,𝑗


𝑛 )

+ 𝐵𝑓𝑇𝑤,2,𝑖,𝑗+

12


𝑛 ) − 𝐵𝑓𝑇𝑤,2,𝑖,𝑗−

12


𝑛 )

− (𝐵𝑓𝑇𝑤𝛾𝑤)1,𝑖+12,𝑗𝑛

(𝑧𝑖+1,𝑗 − 𝑧𝑖,𝑗) + (𝐵𝑓𝑇𝑤𝛾𝑤)1,𝑖−12,𝑗𝑛


− (𝐵𝑓𝛾𝑤)2,𝑖,𝑗+12

𝑛(𝑧𝑖,𝑗+1 − 𝑧𝑖,𝑗) + (𝐵𝑓𝑇𝑤𝛾𝑤)2,𝑖,𝑗−12

𝑛(𝑧𝑖,𝑗 − 𝑧𝑖,𝑗−1)

+ (𝑞𝑏𝜌𝑤+ 𝑅𝑏)

𝑖,𝑗

𝑛

∆𝑥1∆𝑥2.

(4.29)

For nutrients and metabolites, the IMPES discretization is written similarly; for nutrients,

64

(𝜙(𝑆𝑤𝑁)

𝑛+1 − (𝑆𝑤𝑁)𝑛

Δ𝑡)𝑖,𝑗

∆𝑥1∆𝑥2 =

= 𝑁𝑇𝑤,1,𝑖+

12,𝑗


𝑛 ) − 𝑁𝑇𝑤,1,𝑖−

12,𝑗


𝑛 )

+ 𝑁𝑇𝑤,2,𝑖,𝑗+

12


𝑛 ) − 𝑁𝑇𝑤,2,𝑖,𝑗−

12


𝑛 )

− (𝑁𝑇𝑤𝛾𝑤)1,𝑖+

12,𝑗

𝑛 (𝑧𝑖+1,𝑗 − 𝑧𝑖,𝑗) + (𝑁𝑇𝑤𝛾𝑤)1,𝑖−

12,𝑗


− (𝑁𝑇𝑤𝛾𝑤)2,𝑖,𝑗+

12

𝑛 (𝑧𝑖,𝑗+1 − 𝑧𝑖,𝑗) + (𝑁𝑇𝑤𝛾𝑤)2,𝑖,𝑗−

12


+ (𝑞𝑛𝜌𝑤+ 𝑅𝑛)

𝑖,𝑗

𝑛

∆𝑥1∆𝑥2,

(4.30)

and for metabolites (surfactants),

(𝜙(𝑆𝑤𝑀)

𝑛+1 − (𝑆𝑤𝑀)𝑛

Δ𝑡)𝑖,𝑗

∆𝑥1∆𝑥2 =

= 𝑀𝑇𝑤,1,𝑖+

12,𝑗


𝑛 ) − 𝑀𝑇𝑤,1,𝑖−

12,𝑗


𝑛 )

+ 𝑀𝑇𝑤,2,𝑖,𝑗+

12


𝑛 ) − 𝑀𝑇𝑤,2,𝑖,𝑗−

12


𝑛 )

− (𝑀𝑇𝑤𝛾𝑤)1,𝑖+

12,𝑗

𝑛 (𝑧𝑖+1,𝑗 − 𝑧𝑖,𝑗) + (𝑀𝑇𝑤𝛾𝑤)1,𝑖−

12,𝑗


− (𝑀𝑇𝑤𝛾𝑤)2,𝑖,𝑗+

12

𝑛 (𝑧𝑖,𝑗+1 − 𝑧𝑖,𝑗) + (𝑀𝑇𝑤𝛾𝑤)2,𝑖,𝑗−

12


+ (𝑅𝑚)𝑖,𝑗𝑛 ∆𝑥1∆𝑥2.

(4.31)

65

4.3.1 Boundary conditions

4.3.1.1 Dirichlet Boundary Condition

The Dirichlet boundary condition, or the Type I boundary condition, prescribes a value to the

variable at the boundary of the domain. In this case, pressure at the boundary at both sides of the

interval is specified:

𝑝(0, 𝑥2, 𝑡) = 𝑔0(𝑥2, 𝑡), 𝑥1 = 0

𝑝(𝐿, 𝑥2, 𝑡) = 𝑔𝐿(𝑥2, 𝑡), 𝑥𝑁 = 𝐿

(4.32)

For a cell-centered grid, the value of the boundary cell is extrapolated, such that

𝑝1𝑗 = 𝑔1𝑗

𝑝𝑁𝑗 = 𝑔𝑁𝑗 . (4.33)

The Dirichlet boundary condition can be approximated to second order either by the use of

shifted grid, or by averaging the boundary cell.

4.3.1.2 Neumann Boundary Condition

The Neuman boundary condition defines the gradient of the variable at the boundary of the

problem domain, so that

𝜕𝑝

𝜕𝑥(0, 𝑥2, 𝑡) = 𝑞0(𝑥2, 𝑡), 𝑥1 = 0

𝜕𝑝

𝜕𝑥(𝐿, 𝑥2, 𝑡) = 𝑞𝐿(𝑥2, 𝑡), 𝑥1 = 𝐿

(4.34)

One of the most commonly used Neumann boundary conditions specifies no flow at the

boundary.

66

4.3.2 Solution Algorithm and MRST Implementation

The algorithm utilised for solving the model equations is summarized below:

1. Initialize domain, initial conditions and boundary conditions

2. Solve the pressure equation using the water saturation value from the preceding time-step

3. Solve the saturation equation using the oil pressure value from the previous time step

4. Calculate error

a. If errors are within tolerance, move to step 5

b. If errors exceed specified tolerance, iterate steps 2-3

c. Reduce time step if problem fails to converge

5. Solve the concentration equation iteratively

a. If error is within tolerance, move to step 6

b. Reduce time step if the problem fails to converge

6. Calculate IFT and update relevant values through interpolation of Corey equations

7. Repeat until 𝑡𝑁 = 𝑇

Amundsen (2016) developed a MEOR module to MRST for modelling effects of biopolymer for

viscosity alteration, surfactants for interfacial tension reduction, adsorption of bacteria as well as

partitioning of surfactant into the oil phase. The model was implemented into MRST under the

TwoPhaseOilWaterModel.m, a subclass of ThreePhaseBlackOilModel.m, and was composed of

two subclasses, equationsMEORa.m and equationsMEORbiofilm.m. Amundsen’s original

biofilm did not model the alteration of porosity or permeability as a result of biomass growth. In

this work, the biofilm model is modified, and a new script is generated under

67

equationsMEORbioperm.m, The rock properties are then updated appropriately with outputs

from equationsMEORbioperm.m at each time step.

4.4 Two-Phase Flow Analytical Solution

The TPFA discretized numerical model is validated against the two-phase, one-dimensional

incompressible Buckley-Leverett (Buckley & Leverett, 1942) analytical solution, in order to

ensure convergence. The rock and fluid properties as summarized in Table 4-1 are used to

generate analytical and numerical results for comparison.

Table 4-1: Rock and Fluid properties for Model Validation

Parameter Value Units

Reservoir dimensions (length x width x height) 5000 x 1 x 1 m ∗ m ∗ m

Volume dimension (length x width x height) 1 x 1 x 1 m ∗ m ∗ m

Total time 8.6 day

Oil viscosity 4 Cp

Water viscosity 1 Cp

Permeability 100 mD

Porosity 0.2

Water injection rate 100 m3/day

The following relative permeability curve is constructed using Corey’s correlation with an oil

and water exponent of 2, and is used in the numerical and analytical solution.

68

Figure 4-3: Relative permeability curve used for numerical model validation against

Buckley-Leverett analytical solution

Both the effect of numerical grid size and time step are investigated in order to determine

optimal grid block size and time step combinations for MEOR simulation in MRST. Processor

and computer configuration for these simulations are: Windows 7 PC, 64-bit Operating System,

Intel(R) Core(TM) i5-300U CPU @ 2.40GHz. The summary of cases simulated is presented

below in Table 4-2.

Table 4-2: Summary of Grid Size, Time Step Size, and Computational Time Required for 9

Simulation Validation Tests

Grid block size (Δx) Time step size (Δx) Computational time

1 m 0.01 d 37.8 s

0.5 m 0.01 d 38.9 s

0.1 m 0.01 d 54.7 s

0.05 m 0.01 d 67.3 s

0.025 m 0.01 d 85.2 s

0.1 m 0.02 d 27.8 s

0.1 m 0.005 d 100.8 s

0.1 m 0.0001 d 503.8 s

69

0.1 m 0.00001 d 6504.7 s

Figure 4-4 shows the comparison of numerical saturation profile to the Buckley-Leverett

analytical solution under varying grid step sizes. With increasing number of steps, N, and

consequently decreasing step size, Δx, the numerical solution converges to the analytical model.

In addition, the flood front is sharper, indicating lower numerical dispersion (Cho, Augustine, &

Zerpa, 2015). However, for N > 500 or Δx < 10 m, the improvement in accuracy becomes

increasingly immaterial compared to the cost in computational time.

Figure 4-4: Effect of decreasing grid size on numerical simulation convergence to analytical

solution for two-phase, one-dimensional incompressible flow

Similarly, Figure 4-5 shows a closer comparison of numerical saturation profile to the Buckley-

Leverett analytical solution with varying time step sizes. With decreasing time step size, Δt, the

numerical solution increasingly converges to the analytical model. However, in comparison to

the effect of decreasing grid size, the computational time cost is greater with lesser improvement

70

in model accuracy. Notably, at Δt > 0.1, the model no longer converges, while for Δt < 0.01,

simulation result does not differ significantly compared to analytical solution. Thus, the choice

of time step is significantly more limited by failure to converge, as well as computational time.

Figure 4-5: Effect of decreasing time step size on numerical simulation convergence to

analytical solution for two-phase, one-dimensional incompressible flow

4.5 Comparison to MEOR Simulations

The model output is compared to previous numerical simulation results from Sivasankar (2014)

and experimental results from Hossain et al. (2008), matching initial conditions and grid

parameters for a one-dimensional homogeneous reservoir. In Figure 4-6, the transport of

microbes into the reservoir is shown under low nutrient conditions. Because additional advective

and dispersive transport processes are accounted for within the Sivasankar model, whereas the

current model assumes that microbes are a well-mixed component of the water phase, the

microbial front shows more significant tails in the Sivasankar simulation.

71

Figure 4-6: Comparison of microbial transport to one-dimensional simulational results

from Sivasankar (2014)

In Figure 4-7, microbial transport considering biofilm effects is compared to results from the

alternate Sivasankar model, which includes sorption as adherence of the cells to the pore wall.

Sorption in the model is described as a first order function of free bacteria concentration in the

water phase, which is similar to the biofilm model presented in this work. Similarly to Figure

4-6, because of the lack of advective and dispersive models in the microbial transport term, the

microbe concentration lags behind that simulated in the Sivasankar model, and the experimental

core flooding results from Hossain et al. (2008).

72

Figure 4-7: Comparison of microbial transport under sorption, or biofilm formation

effects, to simulation results from Sivasankar (2014)

4.6 Methodology

To investigate the effect of surfactant and biofilm formation, the effect of input related

parameters (e.g. surface injection concentration of bacteria, microbial growth coefficients,

biosurfactant parameters, biofilm parameters) and performance variable (oil recovery) will be

studied. Specifically, the following parametric and sensitivity tests will be performed:

1. Waterflooding, compared to MEOR (no biofilm generation) saturation profiles under

different surfactant concentrations

2. Effect of altering biosurfactant parameters on oil recovery

3. MEOR (no biofilm generation) biosurfactant effect under limiting nutrient, versus

limiting bacteria conditions

4. Effect of altering critical nutrient concentration and biofilm density on oil recovery

73

5. MEOR impact on permeability, porosity and oil recovery at different specific maximum

microbe growth rates and microbe injection concentrations, under two different

conditions

74

5 CHAPTER 5: MEOR SIMULATION AND SENSITIVITY ANALYSIS

In this section, both one-dimensional and two-dimensional simulation outputs are presented in

order to examine the effects of biosurfactant and biofilm parameters. The MEOR operation is

assumed to occur through continuous microbial flooding, where bacteria and nutrients are

injected through the injector well and flow through a homogeneous porous medium. Sensitivity

analysis will be limited to the one-dimensional model, in order to reduce computational time and

facilitate interpretation.

5.1 Simulation Results

5.1.1 Initial Conditions

The reservoir is assumed to contain no bacteria initially. During microbial flooding, nutrients and

a single strain of exogenous bacteria (“microbe”) producing one active biosurfactant

(“metabolite”) are injected continuously at a fixed concentration and rate. It is assumed that the

bacteria strain produces only biosurfactant, and neither biodegrades hydrocarbons nor produce

significant quantities of other metabolites affecting oil recovery (i.e. gases, acids).

For the one-dimensional simulation, a porous medium length of 10 meters is considered. The

one-dimensional model is solved numerically in a 500 x 1 x 1 grid, consisting of 500 volumes of

dimensions 0.02m x 1m x 1m. In this case, a single injector and producer are located respectively

at (1, 1, 1) and (500, 1, 1). For each simulation, the simulation time-step is 0.01 day, and the total

simulation time 12 days. For the two-dimensional simulation, both the length and width are 400

meters. The two-dimensional model is solved numerically in a 32 x 32 x 1 grid consisting of

1024 volumes of dimensions 12.5m x 12.5m x 100m. The time step for the two-dimensional case

75

is 10 days, and the total simulation time 2000 days. The grid dimensions are increased in order to

allow solver convergence. In this case, a single injector and producer are located respectively at

(1,1,1) and (32, 32, 1), representing a quarter of a symmetric five-spot well pattern.

The density of oil is defined to be 800 kg/m3, and water 1000 kg/m

3. Both phases are assumed to

be incompressible, with viscosities of 3 cp and 1 cp respectively. IFT is assumed to be 35 mN/m.

The reservoir porosity is initially assumed to be 0.3, and permeability 100 mD. Initial water and

oil saturation are respectively 0.2 and 0.8. The producer well bottom-hole pressure is 104 kPa.

Gravity effects are ignored in both one- and two-dimensional simulations. A relative

permeability graph was generated using MRST initSimpleFluid constructor for a simplified

Corey-type two-phase fluid, as shown below in Figure 5-1. Based on Treiber et al. (1972),

because krw at Sor is less than 50% of kro at Swi, the rock can be deemed to be water-wet.

Figure 5-1: Oil-water relative permeability for a water-wet rock

76

5.1.1.1 Microbial input parameters

In previous MEOR models, typical injection concentration has ranged from 10-5

to 10-1

kg

substrate/m3, and 10

-5 to 10

-1 kg microbe/m

3 (Landa-Marbán, Radu, & Nordbotten, 2017; S. M.

Nielsen et al., 2014). Default input parameters used for microbial growth, permeability and

interfacial tension reduction are based on those previously used by Larceda et al. (2012), Nielsen

et al. (2010), and Amundsen (2016). They are summarized in the following table.

Table 5-1: Summary of Microbial Input Parameters for MEOR Simulation

Parameter Description Value Units

𝜇𝑚,𝑚𝑖𝑐 Maximum specific microbe growth rate 0.2 1/day

𝜇𝑚,𝑚𝑒𝑡𝑎 Maximum specific metabolite growth rate 0.2 1/day

𝐾𝑏 Half saturation constant, bacterial growth 1 kg/m3

𝐾𝑚 Half saturation constant, surfactant production 1 kg/m3

𝑌𝑏 Yield coefficient, bacteria 0.78

𝑌𝑚 Yield coefficient, surfactant 0.22

𝛼1 Langmuir constant 0.001

𝛼2 Langmuir constant 0.0017

𝜌𝑏 Biofilm density 1000 kg/m3

𝐶 Parametric porosity-permeability constant for

biomass pore radius

19/6

𝑙1 Surfactant property 10-4

𝑙2 Surfactant property 0.2

𝑙3 Surfactant property 1.5 *104

𝑎 Coat’s interpolation constant 6

𝐴𝑟 Surface area per volume rock 3 *105 m2/m3

77

The effect of biosurfactant concentration on IFT is shown below in Figure 5-2, given an initial

oil-water IFT of 35 mN/m. The CMC of the biosurfactant is approximately 4E–4 kg/m3, and the

ultimate IFT achievable is 2.92E–3 mN/m.

Figure 5-2: Effect of biosurfactant concentration on interfacial tension

5.1.2 Surfactant effect in a one-dimensional model

In order to independently investigate the effect of biosurfactant production on oil recovery, a

series of preliminary one-dimensional simulations were run without biomass formation,

eliminating alteration in porosity and permeability. The initial conditions for these runs are

summarized in the table below.

Table 5-2: Summary of Simulation Conditions for Biosurfactant

Case Bacteria injection

concentration (kg/m3)

Nutrient injection


Maximum Specific

Growth Rate (hr-1

)

1 0 0 --

2 0.01 0.01 0.2

3 1 1 0.2

4 1 1 1 (microbe)

0.2 (metabolite)

78

Case 1 is a simulation of waterflooding only, without injected nutrients or bacteria or

biosurfactant production. The saturation profile and pressure profiles are presented below for

waterflooding at an injection rate of 0.1 m3/day after 6 days. Stabilization of reservoir pressure is

shown in Figure 5-3. The pressure at the injector well is maintained within 1% deviation from

initial pressure (1040 kPa to 1034 kPa) throughout the waterflood, at a constant injection rate.

The saturation curve illustrates that the simulation is consistent with frontal advance. In this case,

the flood front reaches the producer well and leads to breakthrough at 12 days. The oil recovery

after 12 days, in this case, is 46.6%.

Figure 5-3: (a) Reservoir saturation and (b) pressure profile for a 1D waterflood simulation

Following Case 1, two additional cases are simulated to investigate the effect of increasing

bacteria concentration. In Case 2, low concentrations of nutrient and bacteria (0.01 kg/m3) are

injected simultaneously. Nutrient concentration and maximum specific bacteria growth rate are

arbitrarily selected to be sufficiently large and small respectively to prevent nutrient depletion.

The maximum specific growth rate of bacteria and maximum specific surfactant growth rate are

79

maintained at 0.2 hr-1

. The saturation profile for this case (Figure 5-4a) does not show

significantly reduced SOR compared to that of Case 1. As shown in Figure 5-4b, a low

concentration of biosurfactant is generated at the nutrient front, reaching 1.9E-4 kg/m3 at 3 m

and matching a slight alteration in the saturation curve at the same location. The oil recovery is

also 46.6%, unimproved from Case 1. Because biofilm is not modeled at this stage, no injectivity

loss from permeability or porosity reduction is shown.

Figure 5-4: (a) Saturation, (b) metabolite concentration, and (c) bacteria/nutrient profile

for a low injection concentration MEOR case

80

In Case 3, a high concentration of both nutrient and bacteria are injected (1 kg/m3), with the

same maximum bacteria and surfactant growth rates, 0.2 hr-1

. The higher injection concentration

ensures significant biosurfactant production for maintaining critical nutrient concentration to

enable IFT reduction. The saturation profile clearly exhibits a secondary displacement front

compared to Case 2, as shown in Figure 5-5. The secondary displacement front coincides with

the length at which maximum surfactant is realized, at 2.5 m. After 12 days, the oil recovery in

Case 3 is 49.0%. Case 3 shows an 2.4% increase in oil recovery compared to Case 2 and Case 1.

Comparatively, the peak surfactant concentration in Case 3 is 0.249 kg/m3 higher than in Case 2,

and 0.250 kg/m3 than in Case 1. The minimum SOR achieved in this case is 0.08, a 60%

reduction to that from Cases 1 and 2. This illustrates the dependence of interfacial tension

reduction and consequently oil recovery, on surfactant concentration. Meeting the CMC

requirement, in this case 4E–4 kg/m3, is necessary to improve surfactant efficacy.

Figure 5-5: (a) Water saturation and (b) metabolite concentration profiles for a high

bacteria concentration MEOR case

81

In the current model, both microbial growth rate and surfactant production rate are related to

nutrient concentration; however, surfactant production rate is not directly related to microbial

growth rate. Instead, surfactant production rate is a function of microbial concentration, such

that, without generating sufficient biomass, CMC cannot be reached. On the other hand, once

microbial growth and surfactant growth deplete reservoir nutrient concentration below critical

nutrient concentration, surfactant production ceases. Therefore, the injected nutrient condition

must be sufficient to not only allow sufficient microbial concentration, but also maintain

secondary metabolite production. Then in Case 4, to investigate the effect of limiting nutrient on

bacteria growth and oil recovery, the maximum specific growth rate of bacteria is increased to 1

hr-1

.

Figure 5-6: (a) Metabolite concentration and (b) bacteria and nutrient concentration

profiles for a high bacteria specific growth rate MEOR case

For Case 4, bacteria and nutrient injection concentration are both 1 kg/m3. As shown in Figure

5-6, under high growth rates, nutrient concentration depletes and approaches zero towards the

producer well. Due to the substrate-limiting Monod growth rate equation, both microbial and

surfactant growth rate become zero under high concentration of injected microbes and high

82

microbial growth rate. This then leads to a plateau of microbial and surfactant concentrations.

Assuming microbes do not decay in the absence of nutrients, this leads to a slug of non-

replicating microbes and constant concentration surfactants advancing with the waterfront. Thus,

as a result of increasing bacteria specific growth rate, the concentration of in situ surfactant has

decreased from 0.25 in Case 3 to 0.08 in the current case. The same phenomenon is not observed

in Cases 2 and 3, where nutrient concentration is in excess of microbial concentration, such

microbial growth rate is non-zero. However, in this case, the oil recovery is still 49.0%. This can

be attributed to the fact that the CMC requirement is 4E–4 kg/m3, which Case 4 is achieves

despite nutrient-limiting conditions.

5.1.3 Surfactant effect in a two-dimensional model

In the two-dimensional case, water is injected at 864 m3/day in a five-spot pattern. The nutrient

injection concentration is 0.01 kg/m3, and a bacteria injection concentration is 0.005 kg/m

3. Both

the water and oil phases are assumed to be incompressible; the oil viscosity is 20 cp and water

viscosity 1 cp. Initial water and oil saturation are respectively 0.3 and 0.7. The same relative

permeability graph from the one-dimensional model is used. Similar microbial kinetics

parameters are used as the one-dimensional model and are summarized in Table 5-1.

The water saturation, microbe, nutrient, and metabolite profiles for 0.01, 0.9, 1.9 and 3.7 years

are shown in Figure 5-8 to Figure 5-10. Similarly to the 1D simulation cases, bacteria and

metabolite are limited to the nutrient front, or the secondary displacement front, at approximately

180, 250, and 320 m away from the injector well for 0.9, 1.9 and 3.7 years respectively.

83

Figure 5-7: Nutrient concentration in a 2D MEOR simulation after (a) 0.01, (b) 0.9, (c) 1.9,

and (d) 3.7 years.

Figure 5-8: Microbe concentration in a 2D MEOR simulation after (a) 0.01, (b) 0.9, (c) 1.9,

and (d) 3.7 years.

(a) (b)

(c) (d)

(a) (b)

(c) (d)

84

Figure 5-9: Surfactant concentration in a 2D MEOR simulation after (a) 0.01, (b) 0.9, (c)

1.9, and (d) 3.7 years.

As shown in Figure 5-8 to Figure 5-9, peak bacteria and metabolite concentration lag 20 m

behind the secondary displacement front. The nutrient concentration is the highest at the

injection well and decreases towards the production well due to microbial metabolism. Notably,

the surfactant concentration profile shows a sharper peak than the bacteria concentration profile,

and additionally lags 10 m behind. This can be attributed to the dependence of in situ surfactant

growth rate on microbial concentration, wherein metabolite production is insignificant at some

small bacteria concentration value. From Figure 5-9, it can be seen that the CMC for surfactant is

not reached until 1.9 years after flooding begins, and that the nutrient-limited surfactant did not

break-through even after 3.7 years of microbial flooding.

(a) (b)

(c) (d)

85

Figure 5-10: Water saturation distribution in a 2D MEOR simulation after (a) 0.01, (b) 0.9,

(c) 1.9, and (d) 3.7 years.

From Figure 5-10c, breakthrough occurs at 1.91 years. At this point, a secondary displacement

front can be observed at approximately 150 - 200 m, where a substantial decrease in water

saturation can be observed over an incremental area, matching with the surfactant peak zone as

shown in Figure 5-9c. As seen from both Figure 5-10c and Figure 5-10d, a small difference in

the water saturation between microbial swept and unswept regions does exist, confirming the

effect of interfacial tension reduction on reducing SOR.

5.1.4 Biofilm effect in a one-dimensional model

While biofilm density is generally assumed to be 1000 kg/m3 in most MEOR models (Tsezos

and Benedek, 1980), experimental measurements of biofilm density can vary as much as 5 – 10

times even through a cross-section (Zhang and Bishop, 1994). This can thus induce a source of

(a) (b)

(c) (d)

86

uncertainty in biofilm modelling. To compare the impact of microbe growth rate and biofilm

density on surfactant-mediated MEOR performance, three additional cases are run under the

conditions summarized in Table 5-3.

Table 5-3: Summary of Simulation Conditions for Biofilm

Case Bacteria injection


Nutrient injection


Biofilm Density

(kg/m3)

Max Growth

Rate (hr-1

)

5 0.1 0.1 1000 0.2

6 0.2 0.2 1000 1

7 0.2 0.2 500 0.2

As shown below in Figure 5-11, water breakthrough occurs at approximately 400 days. At 1000

days, prior to secondary surfactant breakthrough, oil recovery is 2% higher in Case 5 compared

to similar biosurfactant-only conditions. This increase can be attributed to biofilm formation

within the pore space, which leads to reduced porosity and physical displacement of the oil

phase. The minimum porosity and permeability reached are respectively 0.2994 and 99.94mD

respectively. As shown in Figure 5-12 and Figure 5-13, the permeability and porosity profiles

across the one-dimensional reservoir reach minimum values at maximum biofilm mass

concentration.

87

Figure 5-11: Oil recovery for a low bacteria and nutrient injection concentration, and

normal biofilm density MEOR case

In Cases 2 and 3, where biofilm does not exist, the free bacteria moves parallel to the nutrient

front. In contrast, in Case 5, the biofilm growth lags behind the nutrient and metabolite fronts,

which can be seen from the local nutrient concentration maximum at 350 m and the

corresponding decrease in metabolite concentration at 350 m. Notably, because biofilm bacteria

are assumed to behave similarly to free bacteria in terms of nutrient metabolism, nutrient is

depleted closer to the injection well in the biofilm case compared to the biosurfactant-only case,

due to the combined concentration of both injected bacteria and biofilm bacteria. Additionally,

biofilm concentration is higher towards the injection well than the free bacteria concentration in

the equivalent surfactant-only case, as shown in Figure 5-13a and Figure 5-13b. This is because

the concentration of bacteria grows continuously with time and is not assumed to decay.

Furthermore, concentration of biofilm increases towards the producer well. This is a result of the

assumption that daughter cells from biofilm growth partition spontaneously into the water phase

as free bacteria, so that the daughter cells from biofilm growth are released to the water phase

and increase local free bacteria concentration. Because biofilm bacteria concentration is a

88

function of both water saturation and free bacteria concentration, the biofilm grows thicker

deeper into the reservoir. This mechanism is further demonstrated in Figure 5-14.

Thus in Case 5, the dominant mechanisms for oil recovery are physical displacement by the

biofilm layer, and alteration of relative permeability by biosurfactant production. Still, the oil

recovery increase compared to the case without biofilm formation is incremental, which

indicates that, for heterogeneous reservoirs, the large disadvantage associated with biofilm-

driven injectivity loss may eventually outweigh the small gain in oil recovery from selective

plugging or physical displacement.

Figure 5-12: Permeability and porosity profiles across the 1D reservoir after 1000 days of

MEOR with biofilm formation

89

Figure 5-13: Nutrient, microbe and surfactant profiles across the 1D reservoir after 1000

days of MEOR (a) with biofilm formation (b) without biofilm formation

Subsequently, in Cases 6 and 7, the effect of increased maximum specific microbial growth rate

and injection concentration, and decreased biofilm density are independently simulated. As

shown in Figure 5-14 and Figure 5-15, although oil recovery is improved prior to surfactant

breakthrough in the case of low biofilm density, the subsequent drop in rate of oil recovery is

also more significant for this case. Furthermore, by reducing biofilm density by 50% the fraction

of permeability reduction is increased from 0.06 mD (Case 5) to 0.32 mD (Case 7) after 1000

days. Comparatively, even by increasing maximum specific microbe growth rate by 500% and

90

doubling injection concentration, permeability reduction is increased from 0.06 mD (Case 5) to

0.18 mD (Case 8). The reduction in reservoir permeability hinders the flow of liquid through

Darcy’s law, thus hindering oil recovery.

These results indicate that early-onset reservoir plugging is much more likely to occur when

using bacteria with lower biofilm density, instead of higher injection concentration of, or higher

maximum specific growth rate of, high biofilm density bacteria. Thus, if biofilm-prone bacteria

strains are utilized for MEOR, these results demonstrate the importance of selecting favorable

strains for forming denser biofilms, as this allows for operation under a broader range of

injection concentrations. Operating at higher bacteria concentrations, without limiting nutrients,

may then lead to increased biosurfactant concentration and deeper penetration of microbes into

the reservoir, without loss of injectivity.

Figure 5-14: (a) Oil recovery for high specific maximum microbe growth rate MEOR after

1500 days and (b) permeability concentration profiles across the 1D reservoir after 500,

1000 and 1500 days of MEOR utilizing high growth rate microbes at higher injection

concentration

91

Figure 5-15: (a) Oil recovery for a low biofilm density MEOR case and (b) Permeability

and microbe concentration profiles across the 1D reservoir after 500, 1000 and 1500 days

5.2 Biosurfactant Parameters Sensitivity Analysis

A variety of surfactants can be generated from different bacteria strains for MEOR, such as

rhamnolipids, lipopeptides and glycolipids. Rhamnolipids have been identified as the most

potent class of biosurfactant (Patel et al., 2015), but can be produced by only select strains

(Lazar, Petrisor, & Yen, 2007). Furthermore, bacteria strains do not necessarily produce

surfactants at the same rate (Geetha, Banat, & Joshi, 2018). In order to investigate the relative

importance of biosurfactant quality versus biosurfactant growth rate in MEOR operation, two

sets of sensitivity analyses are performed.

5.2.1 Initial conditions

A summary of the initial conditions used during the sensitivity analysis of biosurfactant

parameters, 𝑙1, 𝑙3, and 𝜇𝑚,𝑚𝑒𝑡𝑎, is provided in the following table. Reservoir properties used are

similar to those for one-dimensional MEOR simulation previously. The rate of water injection

for sensitivity analyses cases is 800 m3/day.

92

Table 5-4: Summary of Input Parameters for Biosurfactant Sensitivity Analysis


𝑑𝑡 Time step 1 𝑑𝑎𝑦

𝑇𝑁 Total time 2000 𝑑𝑎𝑦

Volume dimension 1 x 100 x 100 𝑚 𝑥 𝑚 𝑥 𝑚

Grid dimension 400 x 100 x 100 𝑚 𝑥 𝑚 𝑥 𝑚

𝑆𝑤𝑖 Water saturation, initial 0.3

𝑆𝑜𝑖 Oil saturation, initial 0.7

𝐾𝑏 Half saturation constant, bacterial growth 1 kg/m3

𝐾𝑚 Half saturation constant, surfactant growth 1 𝑘𝑔/𝑚3



𝑎 Coat’s interpolation constant 6

𝜇𝑚,𝑚𝑖𝑐 Maximum specific microbe growth rate 0.2 1/𝑑𝑎𝑦

𝑞𝑛 Nutrient injection concentration 5 *10-2

𝑘𝑔/𝑚3

𝑞𝑏 Bacteria injection concentration 5 *10-3

𝑘𝑔/𝑚3

5.2.2 Biosurfactant parameters for IFT reduction

First, a sensitivity analysis is performed to compare the impact of the two biosurfactant

interfacial reduction factors, 𝑙1 and 𝑙3, from Equation (3.41). 𝑙1 has been shown in Chapter 3 to

drive the ultimate reduced interfacial tension value, whereas 𝑙3 is important for determining the

CMC of the surfactant. In the following section, the results of a Monte Carlo simulation-based

sensitivity analysis utilising random sampling is presented. The upper and lower parameter

boundaries are selected based on literature values. For 𝑙1, the sampling range is between 0.05 and

2, corresponding to an ultimate IFT value of 1.4 or 21.9 mN/m respectively. For 𝑙3, the sampling

range is between 2,000 and 10,000, corresponding to a CMC of 3E–4, or 1E–3 respectively.

93


surfactant parameters, 𝒍𝟏 and 𝒍𝟑

Figure 5-17: Contour plot of oil recovery at 20 days to surfactant parameters, 𝒍𝟏 and 𝒍𝟑

Figure 5-16 shows the effect of both 𝑙1 and 𝑙3 on oil recovery and minimum oil saturation, where

minimum oil saturation is defined as the lowest oil saturation achieved during the 20 day MEOR

operation. Clearly, 𝑙1 has a more significant effect on both oil recovery and minimum oil

l3

l1

90007500600045003000

1.8

1.5

1.2

0.9

0.6

0.3

>

–

–

–

–

–

< 0.450

0.450 0.475

0.475 0.500

0.500 0.525

0.525 0.550

0.550 0.575

0.575

Oil Recovery

Contour Plot of Oil Recovery vs l1, l3

94

saturation, as the slope of the surface in the 𝑙1 direction is steeper. A contour plot of oil recovery

to different combinations of 𝑙1 and 𝑙3 is shown in Figure 5-17, where the increase in oil recovery

correlates strongly and non-linearly to a decrease in 𝑙1. An interaction effect exists between 𝑙1

and 𝑙3, where for 𝑙1 < 0.6, a less significant correlation between oil recovery and 𝑙3 can be

observed. This indicates that only at a low enough ultimate interfacial tension value, will the

CMC of the surfactant significantly impact residual oil recovery, confirming results from section

5.1.

5.2.3 Biosurfactant type versus production rate

To date, a correlation between a microbe’s maximum specific biosurfactant production rate and

the chemical properties of the biosurfactant concerning surface tension reduction has not been

described in literature. In this study, a sensitivity analysis is performed by varying the highest

impact IFT parameter, 𝑙1, and the maximum specific surfactant growth rate, 𝜇𝑚,𝑚𝑒𝑡𝑎. The upper

and lower parameter boundaries are again selected based on literature values. For 𝑙1, the

sampling range is again between 0.05 and 2, whereas for 𝜇𝑚,𝑚𝑒𝑡𝑎, the sampling range is between

0.05 and 0.35.

95


surfactant parameters, 𝝁𝒎,𝒎𝒆𝒕 and 𝒍𝟏

Figure 5-19: Contour plot of Oil recovery at 20 days to surfactant parameters, 𝒍𝟏 and

𝝁𝒎,𝒎𝒆𝒕

From Figure 5-18 and Figure 5-19, the impact of bacteria surfactant type and surfactant growth

rate are not equal. The magnitude of change in the direction of surfactant type (𝑙1) significantly

outweighs that of effective production rate. Approximately 90% of the highest recovery cases are

µ (m,meta)

l1

0.300.250.200.150.10

1.8

1.5

1.2

0.9

0.6

0.3

>

–

–

–

–

–

< 0.450

0.450 0.475

0.475 0.500

0.500 0.525

0.525 0.550

0.550 0.575

0.575

Oil Recovery

Contour Plot of Oil Recovery vs l1, mu_surf

96

achieved with the entire range of 𝜇𝑚,𝑚𝑒𝑡, 0.05 to 0.035. This is because, given relatively low

biosurfactant CMC, an effective concentration for IFT reduction can be easily met at typical

growth rates. The highest RF is achieved at (𝑙1, 𝜇𝑚,𝑚𝑒𝑡) = (0.05, 0.28).

The oil recovery and minimum oil saturation as a function of time and 𝑙1, for 600 simulations of

different 𝜇𝑚,𝑚𝑒𝑡 and 𝑙1 combinations, is shown below in Figure 5-20.

Figure 5-20: (a) Minimum oil saturation and (b) Oil recovery as a function of time for

different 𝛍𝐦,𝐦𝐞𝐭 and 𝐥𝟏 values

With decreasing 𝑙1, the inflection point in the minimum oil saturation corresponding to the

secondary displacement front occurs earlier on in time and reaches a lower value. The time lag to

minimum oil saturation inflection corresponds to time at which biosurfactant concentration

exceeds CMC. For 𝑙1 = 2, the minimum oil saturation achieved is 0.37, whereas for 𝑙1 = 0.05,

the minimum oil saturation achieved is 0.23. This is due to the improved reduction in IFT from

21.9 mN/m to 1.4 mN/m. Using Corey’s interpolation function with 𝑛 = 6, the difference in

modified IFT would result in a 40% difference in modified Sor, corresponding to the 0.37 and

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0.23 values respectively. Then as shown in Figure 5-20b, the reduction in minimum oil

saturation achieved corresponds to a difference of 17% oil recovery. Due to the lack of variation

in the 3D line-plot, it can be inferred that change in 𝑙1 is the dominant effect on the alteration of

saturation dependence on time.

Therefore, to optimize MEOR operations, the microbial species should be selected for surfactant

efficacy, under non-limiting reservoir nutrient conditions for surfactant production. However, in

many field applications, selection of bacteria strain for MEOR is limited by other considerations,

including suitability to reservoir conditions, such as temperature, pressure, salinity, pH, etc.

5.3 Growth Rate Sensitivity Analysis

5.3.1 Initial conditions

A summary of the initial conditions used during the sensitivity analysis of maximum specific

growth rate parameters, 𝜇𝑚,𝑚𝑖𝑐 and 𝜇𝑚,𝑚𝑒𝑡𝑎, is provided in the following table. Reservoir

properties used are similar to those for one-dimensional MEOR simulation previously. The rate

of water injection for sensitivity analyses cases is 0.1 m3/day. In order to independently evaluate

the effect of biosurfactant production under various conditions, biomass is not modeled in this

section. Maximum specific microbe and surfactant growth rate are each varied between 0.001 hr-

1 and 1 hr

-1, while holding the other constant at 0.002 hr

-1.

Table 5-5: Summary of Input Parameters for Growth Rate Sensitivity Analysis


𝑑𝑡 Time step 0.01 𝑑𝑎𝑦

𝑇𝑁 Total time 20 𝑑𝑎𝑦

98

Volume dimension 0.02 x 1 x 1 𝑚 𝑥 𝑚 𝑥 𝑚

Grid dimension 10 x 1 x 1 𝑚 𝑥 𝑚 𝑥 𝑚

𝑆𝑤𝑖 Water saturation, initial 0.2

𝑆𝑜𝑖 Oil saturation, initial 0.8

𝐾𝑏 Half saturation constant, bacterial growth 0.5 kg/m3

𝐾𝑚 Half saturation constant, surfactant growth 0.5 𝑘𝑔/𝑚3



𝑞𝑛 Nutrient injection concentration 1 𝑘𝑔/𝑚3

𝑞𝑏 Bacteria injection concentration 1 𝑘𝑔/𝑚3

𝑙1 Surfactant property 10-4

𝑙2 Surfactant property 0.2

𝑙3 Surfactant property 1.5 *104

5.3.2 Impact of maximum specific biomass growth rate

In addition to biosurfactant type and specific maximum growth rate of biosurfactant, the impact

of maximum specific microbe growth rate on minimum oil saturation and oil recovery is

investigated. In the context of interfacial tension reduction, microbial concentration plays two

roles. With increasing microbial growth rate, nutrient is increasingly depleted, thus decreasing

rate of biosurfactant synthesis due to substrate-limiting Monod kinetics. However, with

increasing microbial growth rate, local microbial concentration is increased, leading to an

increase in biosurfactant synthesis rate. As shown in Figure 5-21, with increasing maximum

specific growth rate of bacteria, there is a decrease in oil recovery. Furthermore, there is an

inflection point at 0.7 hr-1

leading to more substantial rate of ultimate oil recovery decrease per

increase in microbial growth rate. From Figure 5-22, the Monte Carlo simulation shows that as

maximum specific microbe growth rate decreases, minimum oil saturation from days 4 - 7

generally increases. Additionally, the time to realize the IFT reduction effect also increases by up

to 1.2 days. This can be attributed to the role of microbial concentration in surfactant growth

99

rate, where increased microbial concentration allows CMC of biosurfactant to be reached earlier

on. Ultimately, minimum SOR tend towards similar values, thus indicating that the CMC value

is achievable for a range of microbial growth rates, granted sufficient microbial residence time

within the reservoir.

Figure 5-21: Oil recovery versus time for varying maximum specific microbe growth rates

Figure 5-22: Minimum oil saturation versus time for varying maximum specific microbe

growth rates

100

5.3.3 Impact of maximum specific biosurfactant growth rate

Similarly, the effect of independently altering biosurfactant growth rate is studied. Figure 5-23

shows an asymptotic relationship between biosurfactant growth rate and oil recovery at a fixed

maximum specific microbe growth rate of 0.002 hr-1

. This again confirms that sufficiently large

biosurfactant concentration for meeting CMC is a critical requirement for MEOR success.

However, below a maximum specific biosurfactant growth rate of 0.03 hr-1

and 0.3 hr-1

, there is a

critical impact on minimum oil saturation and oil recovery achieved after 20 days, respectively.

This indicates that, at any rate above 0.03 hr-1

, the CMC value can be reached within 20 days. On

the other hand, at any rate below 0.3 hr-1

, the time required to generate sufficient surfactant to

realize CMC can negatively impact on oil recovery.

Figure 5-23: Effect of varying maximum specific biosurfactant growth rate on oil recovery

and minimum oil saturation

Below, the impact of maximum specific surfactant growth rate on minimum oil saturation is

shown as a function of time, for 600 Monte Carlo simulations. Clearly, oil saturation is reduced

before day 1 in this case, where maximum specific microbe growth rate is held at 0.002 hr-1

. In

the case shown in Figure 5-22, maximum surfactant microbe growth rate is held at 0.002 hr-1

and

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the reduction of IFT is not observed until day 3. This can be justified by the first order effect of

𝜇𝑚,𝑚𝑒𝑡 on surfactant growth rate, compared to the non-linear effect of 𝜇𝑚,𝑚𝑖𝑐.

Figure 5-24: Minimum oil saturation versus time for varying maximum specific surfactant

growth rates

Finally, the interaction effect between maximum specific microbe and surfactant growth rate on

oil recovery is investigated. As shown in Figure 5-25, maximum specific surfactant growth rate

plays the dominant effect on oil recovery, where the change in oil recovery to an incremental

change in maximum specific surfactant growth rate is much more significant than to an

incremental change in maximum specific microbe growth rate. However, at maximum specific

surfactant growth rate above 0.2 hr-1

, there is a stronger interaction effect from maximum

specific microbe growth rate, such that oil recovery increases for lower maximum specific

microbe growth rate and higher maximum specific surfactant growth rate. This may be due to the

interplay between nutrient depletion arising from higher maximum specific microbe growth rate,

which is associated with a higher yield constant, and meeting CMC with higher maximum

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specific surfactant growth rate. Note that the lowest oil recovery achieved at this system

corresponds to (𝜇𝑚,𝑚𝑖𝑐, 𝜇𝑚,𝑚𝑒𝑡𝑎) = (0.98, 0.03). Additionally, the highest oil recovery occurs

after maximizing maximum specific surfactant growth rate, and minimizing maximum specific

microbe growth rate, so that nutrient depletion is minimized while sufficient biosurfactant

production rate is maintained.

Figure 5-25: Interaction effect of maximum specific microbe and surfactant growth rate on

oil recovery

5.3.4 Impact of critical nutrient concentration

In previous sensitivity analyses, critical nutrient concentration 𝑁𝑐𝑟𝑖𝑡 was assumed to be 0 kg/m3

such that substrate production occurs even under limited nutrient conditions. However, under

reservoir conditions, nutrients tend to be diluted during transport. Given possibly unfavorable

growth conditions, Larceda et al. (2012) and Zhang et al. (1992) proposed using the empirical

103

metabolite growth model from Bajpai and Reuss (1982), wherein below a certain nutrient

concentration, surfactant production ceases. The original critical nutrient concentration for the

production of ethanol from glucose from Bajpai and Reuss was 0.03 kg/m3. In MEOR modelling

literature, values ranging from 0.01 to 0.05 kg/m3

have been used. In this section, the effect of

critical nutrient concentration for surfactant-mediated MEOR is investigated. Values of 𝑁𝑐𝑟𝑖𝑡 are

sampled through a uniform distribution between 0.0005 to 0.05 kg/m3; 𝜇𝑚,𝑚𝑒𝑡 and 𝜇𝑚,𝑚𝑖𝑐 are

sampled with the same upper and lower bounds as the previous sensitivity analysis. 600

simulations are run, comparing the effect of 𝑁𝑐𝑟𝑖𝑡 on oil recovery for different combinations of

𝜇𝑚,𝑚𝑒𝑡 and 𝜇𝑚,𝑚𝑖𝑐. As shown in Figure 5-26, the oil recovery is similar for combinations of

𝜇𝑚,𝑚𝑒𝑡 and 𝜇𝑚,𝑚𝑖𝑐 with 𝑁𝑐𝑟𝑖𝑡 = 0 and with 0.0005 < 𝑁𝑐𝑟𝑖𝑡 < 0.05. Thus, for moderate nutrient

injection concentration, some small critical nutrient condition below 0.05 kg/m3 has relatively

insignificant effect on oil recovery.

104

Figure 5-26: Oil recovery for different values of maximum specific microbe and surfactant

growth rate, with critical nutrient limitation (red) and without critical nutrient limitation

(blue)

5.4 Biofilm and Biosurfactant Sensitivity Analysis

5.4.1 High reservoir porosity and permeability

In a review from Maudgalya et al. (2007) on global MEOR field trials, it was found that 77% of

407 all trials were carried out in sandstone and 22% in carbonate reservoirs. Conventionally,

MEOR has been applied to sandstone due to the lowered risk of bioclogging, given higher

porosity and permeability conditions. However, to date, a quantitative sensitivity analysis of

bacteria strain and injection concentration impact on oil recovery and residual oil saturation has

not been performed in sandstone reservoirs. In the following section, the impact of bacteria

injection concentration and maximum specific bacteria growth rate on both oil recovery and

minimum oil saturation over 2000 days will be assessed, considering both biosurfactant

105

production and biofilm formation effects. A homogeneous one-dimensional tube with 0.3

porosity and 100 mD permeability will be used to replicate a sandstone reservoir.

5.4.1.1 Impact of microbe injection concentration and maximum specific growth rate

First, a sensitivity analysis is performed to evaluate the impact of bacteria injection concentration

on oil recovery and minimum residual oil saturation. As shown below in Figure 5-27a, at the

beginning of MEOR operation, the correlation between oil recovery and microbe injection

concentration is nearly linear, and substantially less favorable for injection concentration below

0.05 kg/m3. However, as simulation time increases, lower bacteria injection concentration

becomes increasingly favorable, and after 1500 days, injection concentration below 0.05 kg/m3 is

optimal for recovery. The optimum injection concentration for oil recovery for 500, 1000, 1500,

and 2000 days are respectively >1 kg/m3, 0.27 kg/m

3, 0.01 kg/m

3 and 0.01 kg/m

3. From Section

5.1.4, this can be explained by the point of surfactant front breakthrough. Prior to surfactant front

breakthrough, increasing injection concentration and biofilm thickness in the reservoir promotes

oil recovery, which matches the result from 500 days. Subsequently, increased biofilm thickness

following surfactant breakthrough results in decreased oil recovery, matching results from 1000

– 2000 days.

Additionally, Figure 5-27b shows that above an injection concentration of 0.15 kg/m3, minimum

residual oil saturation achieved is independent of time after 1000 days. However, at injection

concentration below 0.025 kg/m3, there is a significant time requirement for achieving MEOR

effect. For example, when injection concentration is 0.005 kg/m3, 2000 days is required prior to

reaching SOR of 0.09, or producing sufficient biosurfactant to reach CMC. The time required for

106

reaching minimum SOR is reduced as injection concentration increases, so that in the range of

0.005 kg/m3 to 0.15 kg/m

3, sufficient injection concentration required to reach biosurfactant

CMC value is achieved prior to 1000 days. For all simulation cases, minimum SOR plateaus

with injection concentration above 0.05 kg/m3. This can be attributed to the increased

concentration of retained microbe in the reservoir from biofilm formation, thus leading to

nutrient depletion and reduced maximum rate of surfactant production. Another possible

mechanism, matching with the results from Figure 5-27a, is that the CMC value is achieved

between 500 and 1000 days for injection concentrations above 0.2 kg/m3. Consequently, for

increasing microbe injection concentration above a critical value of 0.05 kg/m3, rate of oil

recovery increase is increasingly hindered by bioclogging, and not by improved by further

biosurfactant production.

Figure 5-27: Effect of microbe injection concentration on (a) oil recovery and (b) minimum

oil saturation at 500, 1000, 1500, and 2000 days under moderate porosity and permeability

conditions

107

Next, the impact of maximum specific microbe growth rate assuming fixed injection

concentration is assessed. Contrary to the effect of increasing microbial injection concentration,

with increased maximum specific microbe growth rate, ultimate SOR is not achieved within

2000 days. Due to the relatively low injection concentration of 0.001 kg/m3, varying maximum

microbial specific growth rate between 0.1 and 1 hr-1

does not allow sufficient biosurfactant to

reach CMC within 2000 days. On the other hand, oil recovery is clearly dominated by time

instead of microbial growth rate, and increases slightly with increase in maximum specific

growth rate, indicating that permeability has not become limiting in any of the cases. Thus, both

the oil recovery and minimum SOR graphs respectively increase and decrease monotonically

with maximum specific microbial growth rate.

Figure 5-28: Effect of maximum specific microbe growth rate on (a) oil recovery and (b)

minimum oil saturation at 500, 1000, 1500, and 2000 days under moderate porosity and

permeability conditions

5.4.1.2 Combined impact of microbe injection concentration and maximum specific growth rate

In this section, interaction effect between injection concentration and maximum specific microbe

growth rate is assessed. 600 Monte Carlo simulations were run, with the lowest randomly

108

selected injection concentration at 0.001522kg/m3, and the highest at 0.99908kg/m

3. The

minimum microbial growth rate simulated was 0.10104, and highest 0.99994 hr-1

. As previously

seen, injection concentration plays a far greater effect on oil recovery when the parameters are

varied between 0.001 to 1 kg/m3 and 0.001 to 1 hr

-1 respectively. After 2000 days, from Figure

5-29a, oil recovery is maximized at 76% at a microbe injection concentration of 0.017966 kg/m3,

and a microbial growth rate of 0.29877 hr-1

. The combination of a relatively low microbe

injection concentration and relatively high microbial growth rate enables optimized biosurfactant

production while mitigating unfavorable biofilm generation and bioclogging. A minimum oil

recovery of 60% occurred at an injection concentration and maximum specific growth rate of

0.99539kg/m3 and 0.39888hr

-1 respectively, matching with the highest permeability reduction

and limited biosurfactant generation. Notably, interaction effect between maximum specific

microbe growth rate and microbe injection concentration is insignificant, as the variation in oil

recovery for different growth rates at constant injection concentration is small.

Figure 5-29b shows the increased effect of biofilm on oil recovery for different microbe injection

concentrations. At 1500 days, injection concentration of 0.38 kg/m3 is optimal due to increased

biosurfactant production and moderate biofilm effects. At 2000 days, because biosurfactant

CMC value has been satisfied even at lower injection concentrations, bioclogging limits the

ultimate recovery from higher injection concentrations.

109

Figure 5-29: Impact of combined variation in microbe injection concentration and

maximum specific growth rate on oil rate at (a) 2000 days and (b) 1000 and 2000 days

110

5.4.2 Low reservoir porosity and permeability

Historically, MEOR has been applied in both sandstone and carbonate reservoirs, which have

been known to differ in terms of porosity and permeability. However, studies reviewing

guidelines for MEOR bacteria injection concentration and bacteria growth kinetics for different

reservoir properties have not been found in literature. In this section, the sensitivity analysis of

maximum specific microbe growth rate and microbe injection concentration is repeated in a low

porosity and permeability reservoir. 600 Monte Carlo simulation cases are run bracketing

maximum specific microbe growth rates of 0.1 to 1 hr-1

and injection concentrations of 0.001 to

0.8 kg/m3, assuming reservoir porosity and permeability of 0.1 and 5 mD respectively.

5.4.2.1 Impact of microbe injection concentration and maximum specific growth rate

Under low porosity and permeability conditions, compared to the high porosity case, minimum

oil saturation does not change as significantly as a function of time, between 500 to 2000 days,

above 0.001 kg/m3 injection concentration. This indicates that sufficient microbe concentration

to produce the interfacial tension reduction via biosurfactant is achieved prior to water

breakthrough in the MEOR operation.

On the other hand, the alteration in recovery factor as a function of injection concentration is

similar at 2000 days, with a decrease of 12%/kg/m3 in this case, compared to 17%/kg/m

3 in the

higher porosity case. This is shown in Figure 5-30a. Furthermore, greater oil recovery is

achieved after 2000 days in this case than in the higher porosity case, respectively 82% and 75%.

Additionally, at injection concentration of approximately 0.8 kg/m3, recovery factor does not

improve after 500 days, whereas under higher porosity conditions, the recovery factor improves

111

by 18% from 500 days to 2000 days. Finally, the optimum injection concentration is fixed as a

function of time in the lower porosity and permeability case, at approximately 0.02 kg/m3,

whereas the value varies as a function of time in the higher porosity and permeability case. These

observations are likely due the increase in pore velocity with decreased porosity and constant

injection rate (800 m3/day). The 500 day case at lower reservoir porosity is comparable to the

1500 or 2000 day case in the higher porosity case in terms of relative time to water breakthrough.

Additionally, recovery at 0.8 kg/m3 injection concentration in the lower porosity case, 72.2%, is

higher than the recovery at the same injection concentration in the higher porosity case, 62.5%.

This may be explained by the increased effect of physical displacement of oil from the pore with

bioclogging in low permeability conditions. However, the effect of biofilm growth in limiting

fluid flow and thus oil recovery is possibly less pronounced in this case due to the relative

increase in pore velocity.

Figure 5-30: Oil recovery at 500, 1000, 1500, and 2000 days under low reservoir porosity

and permeability conditions for (a) varied microbe injection concentration and (b) varied

maximum specific microbe growth rate

112

Varying the maximum specific microbe growth rate by the same range results in a much

narrower range of oil recovery under the low porosity case compared to the high porosity case.

As shown in Figure 5-30b, recovery varies less than 20% within 2000 days, whereas in Figure

5-29, recovery varies by over 40%. This again can be explained by the reduction in pore volume,

which allows further movement of surfactant front along the reservoir. Ultimately, no loss of

injectivity has been observed on in the simulated time-scales despite maintaining microbe

kinetics parameters and injection concentration. This may be explained by the exclusion of pore

size distribution in the model, which implies that biofilm growth would only affect the maximum

pore radius.

5.5 Summary of Sensitivity Analysis Results

Table 5-6 summarizes the impact of each of the parameters on oil recovery at 2000 days

independently and in combination. As seen below, the most important parameters for MEOR

performance is bacteria injection concentration, and subsequently biosurfactant type and

biosurfactant production rate. While biofilm density was not included in the sensitivity analysis,

as can be seen from Case 7, it may also play a large effect on oil recovery. Therefore, selecting

the correct bacteria strain based on biofilm formation propensity, biofilm properties, and

biosurfactant CMC value, and subsequently controlling rate of bacteria injection, are pertinent to

MEOR implementation success.

Table 5-6: Summary of Oil Recovery Sensitivity Analysis

Parameters Range Low Value High Value Variation

l1 (0.056, 2.00) 0.47 0.58 0.11

113

l3 (2004.2, 9996) 0.72 0.75 0.03

l1, l3 (0.050, 2.00);

(2002, 9972)

0.43 0.58 0.15

µm,met (0.009, 0.998) 0.7 0.72 0.02

µm,met, l1 (0.0507, 0.3482);

(0.0549, 2.000)

0.430 0.585 0.155

µm,met, l1, l3 (0.0506, 0.3498)

(0.0534, 2.000);

(2008, 9990)

0.426 0.580 0.154

µm,mic (0.003, 0.997) 0.62 0.69 0.07

µm,met, µm,mic (0.0013, 0.999);

(0.0015, 0.999)

0.64 0.72 0.08

Ncrit (0.0005, 0.0050) 0.68 0.68 0

Ncrit, µm,met (0.0005, 0.0050);

(0.0015, 0.9991)

0.68 0.72 0.04

Ncrit, µm,met, µm,mic (0.0005, 0.0050);

(0.0015, 0.9999);

(0.0014, 0.9993)

0.65 0.72 0.07

qb (high φ) (0.0088, 0.997) 0.60 0.76 0.16

µm,mic (high φ) (0.104, 0.997) 0.743 0.746 0.003

qb, µm,mic (high φ) (0.0015, 0.9991);

(0.101, 1.000)

0.60 0.76 0.16

qb (low φ) (0.0022, 0.999) 0.70 0.82 0.12

µm,mic (low φ) (0.104, 0.997) 0.77 0.79 0.02

qb, µm,mic (low φ) (0.0015, 0.9991);

(0.1010, 1.0000)

0.70 0.82 0.12

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6 CHAPTER 6: CONCLUSION AND RECOMMENDATIONS

6.1 Conclusions

In this work, a two-phase, two-dimensional finite volume model is built on MRST to

simultaneously model microbial kinetics, surfactant generation, and biofilm generation in

MEOR. Biological and geological variables are coupled through equations relating microbial

growth to both relative permeability alteration via biosurfactant production, and porosity and

permeability reduction via biofilm formation. In particular, permeability and porosity reduction

as a function of free bacteria concentration, and subsequently biofilm concentration, is modeled.

Each of the critical MEOR models, including biosurfactant effect on interfacial tension, biofilm-

mediated porosity reduction on permeability, and microbial growth, are validated against and

showed reasonable match to experimental data from literature. The numerical two-phase

compressible flow model is validated against the two-phase Buckley-Leverett analytical solution

in order to investigate impact of step and time step size on solver convergence, and obtain

optimal simulation given computational time. Results of the MEOR model are then compared to

previous simulation results from literature, matching directionally while highlighting the

importance of advective and dispersive transport. Finally, parametric studies and Monte Carlo

simulation-based sensitivity analysis are performed by bracketing empirical parameters from

literature, resulting in the following observations:

Under excessively high maximum specific microbe growth rate, nutrient depletion occurs

and biosurfactant production rate plateaus, limiting maximum biosurfactant

concentration.

In the case of nutrient-limited biosurfactant production, oil recovery depends on whether

biosurfactant concentration exceeds CMC.

115

In this model, critical nutrient concentration does significantly impact MEOR outcome.

After exceeding the CMC value as determined by the l1 parameter, biosurfactant

production no longer affects oil recovery or residual oil saturation; instead, biofilm

formation plays a dominant effect in decreasing porosity and permeability, and

decreasing oil recovery.

Under current model assumptions, biofilm grows thicker towards the producer, indicating

that injectivity loss is more likely to occur further away from the wellbore in

homogeneous geology.

Biofilm formation has been shown to both slightly improve oil recovery through physical

displacement prior to water breakthrough, but depending on local concentration, may

negatively impact oil recovery following secondary, or surfactant front, breakthrough.

Under higher porosity conditions, oil recovery is initially higher for increased microbe

injection concentration due to improved biosurfactant growth rate, but eventually

plateaus due to bioclogging effects. In contrast, oil recovery is initially lower for low

microbe injection concentration due to limited biosurfactant growth rate, but ultimately

surpasses recovery under higher injection concentration due to less significant

bioclogging effects.

Without a pore size distribution model, under homogeneous reservoir conditions, loss of

injectivity is not predicted within 2000 days under a range of different microbial and

operating parameters, even for low porosity reservoirs.

Assuming the Langmuir model for bacteria partitioning and Kozeny grain-coating model

for permeability reduction are applicable, biofilm density plays a greater role in

permeability reduction than increasing bacteria injection concentration or maximum

116

specific growth rate. Thus, MEOR performance is highly dependent on the selection of

high density biofilm strains, as this allows for operation under a broader range of

injection concentrations, resulting in increased biosurfactant concentration and deeper

penetration of microbes into the reservoir.

Critical parameters for MEOR performance under biofilm and biosurfactant generation

conditions include bacteria injection concentration, biofilm density, biosurfactant type

and biosurfactant production rate. Thus, in the design of an MEOR operation, first the

appropriate strain must be selected based on biofilm formation propensity, biofilm

properties, and biosurfactant CMC value. Subsequently, bacteria injection concentration

and waterflooding rate must be adjusted to prevent biofilm formation. Finally, sufficient

nutrient injection concentration should be maintained to prevent pre-mature nutrient

depletion and insufficient biosurfactant production.

6.2 Recommendations for Future Work

It is recommended that in future work, improved models for microbial decay and biosurfactant

partitioning to the oil phase are added in order to more closely simulate biological behavior.

Depending on the bacteria metabolic pathways, i.e. degradation of hydrocarbons, product-

inhibited growth may be considered. Diffusion and dispersion terms should be considered in

order to better understand microbial flow in the reservoir, which can alter biofilm and

biosurfactant profiles. Chemotaxis effect, and the effect of bioclogging on microbial transport

should be coupled to a pore size distribution model, under a three-dimensional reservoir with

heterogeneus geology.

117

In this work, the dominant MEOR mechanism and shortcoming, biosurfactant generation and

bioclogging respectively, have been simulated. However, other effects including visocisty

reduction through biodegradation and biogas production, as well as wettability alteration are

opportunities for further simulation studies. In addition, while the impacts of temperature,

salinity and pH effects on microbial growth are well known, they have not been quantified in

terms of biofilm formation and biosurfactant efficacy, and would be useful for more detailed

process design optimization.

118

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