Stanley Babatunde (main adviser, together with J.M. Nordbotten), 2014.

Stanley Babatunde (main adviser, together with J.M. Nordbotten), 2014.
Modeling of oil reservoirs with
focus on microbial induced effects
Master’s thesis in Applied and Computational Mathematics
Stanley Owulabi Babatunde
Department of Mathematics
University of Bergen
November 2014
Dedication
This piece of work is dedicated to the entire Babatunde family in Ghana, Australia
and Norway for their numerous support in bringing me up this far.
i
Acknowledgement
All thanks and praise go to the almighty God for the courage and determination
granted me to be able to put this piece together. It is solely by His grace, wisdom
and knowledge which has guided me in conducting this research studies in spite
of all difficulties. I am eternally grateful to Him.
My thanks also goes to my supervisors; Associate professor Florin Adrian Radu
and Professor Jan Martin Nordbotten whose wonderful comments and advice have
made this work a success. As a matter of fact, you made me believe that I have
so much strength and courage to persevere in the midst of staggering oppositions.
What kept my head up the sea was your great motivation. You understood me
and were very tolerant and determined to see me through and you indeed achieved
your aim. Honestly, I aspire to emulate you if I ever have the chance to step into
your shoes in the future. The exposures you both gave me to attend NUPUS
workshop and conference in Germany gave me much insight into research. I will
never forget those memories. To all the wonderful people at the University of
Stuttgart, I thank you for your kindness.
I cannot rule out the contributions of the various bibliographies and scientific
papers which fed me in putting this piece together. To the various authors, I
express my profound gratitude to you all.
I want to specially thank Mr. Emmanuel Babatunde and Veronica Babatunde for
the immense financial support they offered me which has given me the sound mind
to undertake this research work. I thank all others who assisted, encouraged and
supported me during this research, not forgetting my colleague master students
and my brother Kingsley Bawfeh for their support.
Last but not least, I appreciate my best friend Elorm Agoba for her words of
encouragement.
It is my wish that the good Lord bless you all for the contributions you made.
ii
Abstract
As an abstract to this thesis, we review some literatures in EOR and discussed
the processes, strength and weakness of Microbial Enhanced Oil Recovery techniques. A two phase flow model comprising water and oil via the concept of mean
pressure has been formulated using mass conservation equations, Darcy’s law and
constitutive relations. This resulted in a set of coupled nonlinear parabolic partial
differential equation with primary variables being the mean pressure and water
saturation. We discretized these equations in one dimension using a control volume discretization scheme in space and implicit Euler in time. We employed the
IMPES approach which decoupled the primary variables. A model validation test
was made by comparison with an analytical solution and with the Couplex-Gas
benchmark. The model was used to investigate two major mechanisms by which
the activities of bacterial helps in enhancing the recovery of the residual oil.
iii
Contents
Dedication
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Acknowledgement
ii
Abstract
iii
Contents
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List of Figures
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List of Tables
viii
Abbreviations
ix
Symbols
x
1 General introduction
1
2 Microbial Enhanced Oil Recovery
2.1 Phases of oil recovery . . . . . . . . . . . . . . . . . . .
2.2 Some conventional EOR methods and mechanisms . . .
2.2.1 Gas Injection . . . . . . . . . . . . . . . . . . .
2.2.2 Chemical flooding . . . . . . . . . . . . . . . . .
2.2.3 Thermal recovery . . . . . . . . . . . . . . . . .
2.2.4 Polymer flooding . . . . . . . . . . . . . . . . .
2.3 The MEOR method . . . . . . . . . . . . . . . . . . . .
2.3.1 Types of MEOR . . . . . . . . . . . . . . . . .
2.3.2 General processes of MEOR . . . . . . . . . . .
2.3.3 By-Products of microbial and their effect on the
2.3.4 Advantages of MEOR method . . . . . . . . . .
2.3.5 Problems of MEOR method . . . . . . . . . . .
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3 Mathematical model
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3.1 Two-phase model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.1.1 The governing equations . . . . . . . . . . . . . . . . . . . . 13
iv
3.2
3.1.2 The mean value formulation . . . . . . . .
Microbial transport mechanisms in porous media
3.2.1 Transport equations . . . . . . . . . . . .
3.2.2 Adsorption . . . . . . . . . . . . . . . . .
3.2.3 Growth and decay . . . . . . . . . . . . .
3.2.4 Exponential growth . . . . . . . . . . . . .
4 Numerical modeling
4.1 Grid . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Spatial discretization . . . . . . . . . . . . . . .
4.2.1 The finite difference methods . . . . . .
4.2.2 The two point flux approximation . . . .
4.2.3 Dirichlet boundary conditions . . . . . .
4.2.4 Neumann boundary conditions . . . . . .
4.3 The fully discrete scheme for two-phase flow . .
4.3.1 Discretization of the transport equation .
4.3.2 The diffusion term . . . . . . . . . . . .
4.3.3 The source term . . . . . . . . . . . . . .
4.3.4 The convection term . . . . . . . . . . .
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5 Numerical results and analysis
5.1 Model validation with the couplex-gas benchmark . .
5.1.1 Benchmark simulation one . . . . . . . . . . .
5.1.2 Benchmark simulation two . . . . . . . . . . .
5.2 Model validation with an analytical solution . . . . .
5.2.1 The set of equations and the parameters used
5.2.2 Comparison of results . . . . . . . . . . . . . .
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6 Modeling of MEOR activities
6.1 A case study . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Effects of introducing microbial into the model . . . . . . . . . .
6.2.1 Inter-facial tension reduction with bacteria concentration
6.2.2 Viscosity reduction with bacteria concentration . . . . .
6.3 Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . .
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7 Summary and conclusion
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Bibliography
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v
List of Figures
2.1
Processes of MEOR . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1
4.2
4.3
4.4
4.5
4.6
An example of a cell-centered grid in 2-D . . . . . . . . . . . . . .
Equal distant time discretization . . . . . . . . . . . . . . . . . .
Space interval divided into equal sized cells . . . . . . . . . . . . .
Cell-centered space discretization. . . . . . . . . . . . . . . . . . .
Cell centered grid with points xi and the cell walls xi+ 1 . . . . . .
2
Dirichlet boundary conditions for cell-centered grid by adding ghost
cells at the ends of the interval. . . . . . . . . . . . . . . . . . . .
4.7 Dirichlet boundary conditions for cell-centered grid by adding half
cells at the ends of the interval. . . . . . . . . . . . . . . . . . . .
4.8 Cell centered grid with points xi and the cell walls xi+ 1 . . . . . .
2
4.9 Flow of fluxes across discontinuity points in 1D . . . . . . . . . .
4.10 Standard nomenclature for control-volume discretization in 1D . .
n
n
4.11 Linear interpolation to obtain interface values Ci−
for
1 and C
i+ 12
2
the central difference approximation . . . . . . . . . . . . . . . . .
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5.1
5.2
. 48
5.3
5.4
6.1
6.2
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6.4
6.5
6.6
6.7
6.8
van Genuchten’s parameters . . . . . . .
Benchmark simulation one. Pressure and
at time T = 45 days. . . . . . . . . . . .
Benchmark simulation two. Pressure and
at time T = 45 days. . . . . . . . . . . .
Error plots for pressure and saturation .
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water saturation profiles
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water saturation profiles
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A case study. Pressure and water saturation profiles at time T =
45 days. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
An oil reservoir . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Inter-facial tension correlation with bacteria concentration . . . .
Saturation profile for van Genuchten parameter with different concentrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fraction of oil remaining in the reservoire after 45 days using van
Genuchten parameters with focus on inter-facial tension reduction
Saturation profile for Brooks Corey parameter with different concentrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fraction of oil remaining in the reservoire after 45 days using Brooks
Corey parameters . . . . . . . . . . . . . . . . . . . . . . . . . . .
Viscosity correlation with bacteria concentration . . . . . . . . . .
vi
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6.9
Saturation profile for van Genuchten parameter with focus on viscosity reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.10 Fraction of oil remaining in the reservoire after 45 days with focus
on viscosity reduction . . . . . . . . . . . . . . . . . . . . . . . . . .
6.11 Water saturation profile for the sensitivity analysis of van Genuchten’s
parameter n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.12 Fraction of oil remaining in the reservoir after 100 days with different van Genuchten’s parameters . . . . . . . . . . . . . . . . . . . .
vii
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List of Tables
2.1
By-Products effects on rock and oil . . . . . . . . . . . . . . . . . .
5.1
5.2
5.3
Benchmark fluid properties and other parameters . . . . . . . . . . 48
Error analysis for saturation . . . . . . . . . . . . . . . . . . . . . . 55
Error analysis for pressure . . . . . . . . . . . . . . . . . . . . . . . 55
6.1
General parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
viii
9
Abbreviations
EOR
Enhanced Oil Recovery
MEOR
Microbial Enhanced Oil Recovery
REV
Representative Elementary Volume
ix
Symbols
φ
Porosity of the porous medium
m
K̂
→
−
U
permeability of the porous medium
MilliDarc [mD]
Darcy’s velocity
[m/s]
Kro
relative permeability of oil
MilliDarc [mD]
Krw
relative permeability of water
MilliDarc [mD]
K̂o
effective permeability of oil
MilliDarc [mD]
K̂w
effective permeability of water
MilliDarc [mD]
g
acceleration due to gravity
[m/s2 ]
o
oil
w
water
t
time coordinate
seconds (s)
x
space coordinate
meters (m)
τ
time step
seconds (s)
h
spatial step
meters (m)
S
saturation
So
saturation of oil
Sw
saturation of water
P
mean pressure
pascal [M P a]
Pe
entry pressure
pascal [M P a]
Pc
capillary pressure
pascal [M P a]
Po
pressure of oil
pascal [M P a]
Pw
pressure of water
pascal [M P a]
ρo
density of oil
[kg/m3 ]
x
ρw
density of water
[kg/m3 ]
µo
viscosity of oil
[kgm−1 s−1 ]
µw
viscosity of water
[kgm− 1s− 1]
Fvo
source/sink term of oil
Fvw
source/sink term of water
FvT
source/sink term
λT
Total mobility of the displacing fluid
λd
Total mobility of the displaced fluid
C
bacterial concentration
D
dispersion coefficient
ν
velocity of bacterial in water
λ
Brooks-Corey parameter for the porous medium
[m/s]
xi
Chapter 1
General introduction
Energy is an essential force which the world cannot survive without. Despite the
huge investments in other sources of energy such as biofuels, solar energy and
wind energy; fossil fuels will still remain the key supply of energy source for many
years to come (Graus et al. 2011). The current concern the global oil industry
faces is the increase rate of unproductive and ageing wells. An oil well can be said
to be unproductive when approximately about 30 % of the oil in place has been
recovered; thus, a substantial quantity of oil is left in them after the application
of conventional oil extraction methods. Moreover, there is a dire need to produce
more crude oil to meet the worldwide rising energy demand which illustrates the
necessity of progressing Enhanced Oil Recovery (EOR) processes. These methods
try to overcome the main obstacles in the way of efficient oil recovery such as the
low permeability of some reservoirs, the high viscosity of the crude oil, and high
oil-water inter-facial tensions that may result in high capillary forces retaining
the excess oil in the reservoir rock (Bubela, 1987). With the global campaign on
protecting the environment through reducing the green house effect by probably
Carbon Capture and Storage (CCS) [11] and other possible means, a big challenge is thrown to the oil industry to come out with efficient but environmentally
friendly EOR techniques.
During oil production, primary oil recovery can account for between 30-40 % oil
productions, while additional 15-25 % can be recovered by secondary methods
such as water injection, leaving behind about 35-55 % of oil as residual oil in the
reservoirs (Cosse, 1993). This residual oil is usually the target of many enhanced
1
oil recovery technologies. Recovery of this residual oil is at present a big challenge
for many oil companies and there is a continuous search for a cheap and efficient
technology that can help in its recovery. Additional recovery from residual oil can
lead to increase in global oil production as well as prolonging the productive life
of many oilfields. The techniques employed for recovery of this residual oil are
generally termed Enhanced Oil Recovery (EOR) methods.
Thesis outline
In chapter 2 of this work, we shall review some relevant literatures available on
MEOR so as to be abreast with current trends and development in this field. A
formulation of a mathematical model consisting of a two-phase flow of oil and
water will be given in chapter 3. All mathematical equations encountered in this
work will be discretized in one dimension using the finite volume discretization
scheme and this will be organized in chapter 4. For our model to be trusted and
dependable for further simulations, we will seek to either construct an analytical
solution and perform some convergence analysis or run some simulations of a know
benchmark to see if the results using our model will be consistent with that of the
benchmark. This will be done in chapter 5. The major goal of this thesis is to
analyze numerically how the introduction of bacteria into the reservoir can help
to recover the residual oil. In the last two chapters 6 and 7 of this work, we shall
investigate some mechanisms by which microbial models are incorporated into the
reservoir for the enhancement of the residual oil. Conclusions and outlook will be
given in the last chapter 7.
2
Chapter 2
Microbial Enhanced Oil Recovery
The notion of using Microbial Enhance Oil Recovery (MEOR) stems as far back in
the 1920s where ZoBell [3] studied and observed a gradual separation of oil caused
by sulfate-reducing bacteria. Ever since, there has been numerous research carried
in this area which has been reported in the literature. This has been summarized
by Bryant and Burchfield [4].
2.1
Phases of oil recovery
Enhanced Oil Recovery (EOR) is generally considered as the third, or last, phase
of useful oil production. It is mostly referred to as the tertiary oil production.
The first or primary phase of oil production begins with the discovery of an oil
field using the natural stored energy to move the oil wells by expansion of volatile
components or pumping of individual wells to assist the natural drive. When this
oil is depleted, production declines.
A secondary phase of oil production begins when supplemental energy is added to
the reservoir by injection of water. As the oil-to-water production ration of the
field approaches an economic limit of operation, the net profit diminishes because
the difference between the value of the produced oil and the cost of the water
treatment and injection becomes too narrow, the tertiary period of production
begins where the water injection phenomenon in the secondary stage is changed.
3
The residual oil is retained mainly by viscous and capillary forces. These forces are
influenced by several parameters such as; surface/inter-facial tension, wettability,
permeability, viscosity just to mention but a few. The aim of EOR is to alter these
parameters in several beneficial ways so as to extract the residual oil.
2.2
Some conventional EOR methods and mechanisms
There are several EOR techniques known to recover the residual oil. The type
used for a specific reservoir mainly depends on the properties of the reservoir.
This aspect brings to light some commonly used EOR techniques.
2.2.1
Gas Injection
It is called miscible flooding [5] and is the most commonly used method. It is a
general term for injection processes that introduce miscible gases into the reservoir.
A miscible displacement process maintains reservoir pressure and improves oil
displacement because the inter-facial tension between oil and water is reduced.
This refers to removing the interface between the two interacting fluids. This
allows for total displacement efficiency.
Gases used in this process include carbon-dioxide (CO2 ) and nitrogen (N2 ). The
fluid most commonly used for miscible displacement is carbon-dioxide since it
has the tendency to reduce the oil viscosity and is less expensive than liquefied
petroleum gas. Oil displacement by carbon-dioxide injection relies on the phase
behaviors of the mixtures of these two gas and the crude. These behaviors are
strongly dependent on reservoir temperature, pressure and crude oil composition.
As oil and gas have a cognate symbiosis in the same structural trap, their physical
and chemical properties are similar. As such, the gas drive oil method has the
potential to deliver better displacement process efficiency and higher recovery rates
than other techniques. However, this theory is relevant only under specific reservoir
conditions. If these specific conditions are present, then the volume expansion of
4
the injected gas, which acts to move the oil, takes precedent over the smaller
chemical reactions from the gas drive process at the oil and gas interface.
2.2.2
Chemical flooding
In a chemical flood, chemicals are injected with the water flood to improve the
displacement efficiency [5]. A chemical solvent is specially developed for adaptation to the specific structural characteristics and physiochemical properties of a
reservoir.
After injecting with water, chemical reactions form new chemical sediment, which
can reduce the contradiction between layers, increase volume and amount of water
injected. This can improve the degree to which reserves can be recovered, while
improving production efficiency.
However, this type of chemical reaction would take place in a poor reservoir so it
will also produce oil pollution and the capacity for water absorption would be damaged. Most wells cannot achieve a satisfactory result using this method, making
it counterproductive, with the negative effects outweighing the benefits.
2.2.3
Thermal recovery
Thermal method raises the temperature of regions of the reservoir to heat the
crude oil in the formation and reduces its viscosity or vaporise part of the oil
thereby decreasing the mobility ratio. It includes the injection of hot water, steam
and other gases or by conducting combustion in situ of oil or in gas.
The increase in heat reduces the surface tension, increases the permeability of the
oil and improves the reservoir seepage conditions [5]. The heated oil may also
vaporise and then condense, forming improved oil.
This approach however, requires substantial investment in special equipment. Due
to the heat effect, this method causes severe damage to the underground well
structure, as well as poses safety risks in the larger production process. For these
reasons, the method is not generally used.
5
2.2.4
Polymer flooding
Polymer flooding is also widely used as gas flooding. It is used to retrieve oil left
behind after conventional recovery processes [5]. It is an augmented water flooding
technique introduced in the 1960’s, mainly used for heterogeneous reservoirs, to
retrieve oil after areas in the reservoir with high permeability have been highly
water flooded.
It is a method in which high-molecular-weight polyacrylamides are injected into
the water, so as to increase the viscosity of fluid, improve volumetric sweep efficiency and thereby further increasing the oil recovery factor.
When oil is displaced by water, the oil/water mobility ratio is so high that the
injected water fingers through the reservoirs. By injecting polymer solution into
reservoirs, the oil/water mobility ratio can be much reduced, and the displacement
front advances evenly to sweep a larger volume. The viscoelasticity of polymer
solution can help displace oil remaining in micro pores that cannot be otherwise
displaced by water flooding.”
Most of the EOR methods discussed above are known to be costly and task challenging since most of them involves a significant change in the reservoir condition
and others even cause a permanent damage to the reservoir. Until recently, the
use of microbial to enhance the recovery of the residual oil is emerging as a cheap
and effective EOR method [6]. They are cheap in that these microbial are found
everywhere on earth. Research has shown that they are among the first life forms
to appear on earth and are present in most of its’ habitats [7]. They inhabits the
soil, water, acidic hot springs, radioactive waves and deep portions of the earth’s
crust. It is worth noting that about 40 millions of microbes can be found in a
gram of soil and a million cells of microbes are contained in a milliliter of fresh
water. With approximately 5 x 1030 bacteria living on earth, forming a biomass
which exceeds that of all plants and animals [8], it is undeniable that they are
found everywhere on planet earth.
6
2.3
The MEOR method
This is a biological method which involves injecting bacteria into the oil reservoir to improve the recovery efficiency [5]. Experimental results using a particular
species in a reservoir have shown that through the metabolism of large population
bacteria, large amounts of organic acids can be produced. These organic acids
may act to restore vitality to an ageing well, increase its productivity and thereby
acting to induce a substantial increase in oil recovery.
Compared with most other EOR methods, MEOR is emerging as the cheapest
way to recover the residual oil due to its’ low economical cost, low risk for the
personnel and a little destroyer (if any) to the environment.
2.3.1
Types of MEOR
There are three types of MEOR processes known to recover oil from reservoirs [9].
The first type is similar to conventional chemical flooding. Metabolites with favorable oil-displacing properties are generated as bacteria are grown ex situ on suitable nutrients. Fermentation broth with (or without) cells removed is injected into
candidate reservoirs to displace oil. The second type makes use of the in situ fermentation of nutrients(e.g Molasses) injected into the reservoir. Potential bacteria
species are inoculated to carry out the desired biological conversions. Metabolites
and biogas are generated in place, allowing for the release of oil trapped in the
pores of the reservoir. The third type is similar to the second type but no nutrient
is supplied ex situ [9]. The bacteria species injected into the reservoir must be
capable of utilizing hydrocarbons in the reservoir to generate metabolites in situ.
7
2.3.2
General processes of MEOR
The processes generally involves:
• Injection of microbial along with nutrients into the well and closing for approximately 20 days.
• The microbial then propagates and produce polymers, gases, surfactants and
organic acid (their metabolites)
• These metabolites then aid in propagating oil by changing both the physical
properties of the rock and crude oil itself.
• The gas (carbon-dioxide and methane) restore the gas drive phenomenon
of the oil pushing it to the mouth of the well.
Figure 2.1: Processes of MEOR
8
2.3.3
By-Products of microbial and their effect on the rock
and oil
Below is table 2.1 showing details of the metabolites of microbial and their numerous effects on the crude oil and the reservoir rock.
By-Products
Effects
Acids
• Modification of reservoir rock.
• Improvement of porosity and permeability.
• Reaction with calcareous rocks and CO2 production.
Biomass
• Selective or non selective plugging.
• Emulsification through adherence to hydrocarbons.
• Modification of the solid surfaces.
• Degradation and alteration of oilfields.
• Reduction of oil viscosity and soil pour point.
• Desulfurization of oil.
Gasses (CO2 , CH4 , H2 )
• Reservoir repressurization.
• Oil swelling.
• Viscosity reduction.
• Increase of permeability due to solubilization of carbonate rocks by CO2 .
• Reduction of oil viscosity and soil pour point.
• Desulfurization of oil.
Surface active agents
• Lowering of inter-facial tension.
• Emulsification.
Table 2.1: By-Products effects on rock and oil
9
2.3.4
Advantages of MEOR method
The merits of using MEOR are numerous and this aspect of the thesis brings to
light some of the major advantages. Firstly, It is a fact that microbes live almost
everywhere and the fact that they feed on themselves and other nutrients such as
molasses, makes the use of MEOR relatively cheap as compared to other enhanced
methods. Secondly, no capital expenditure is required for its’ treatment, making
it economically attractive for marginal producing wells. Also, during the implementation of MEOR activities, only minor modifications to the field facilities are
required and results are mostly realized within two to three weeks after the treatment. Again, the operations can be implemented on small pilot areas of about 5 10 wells. Furthermore, MEOR has been proven to be a fast and simple application
commercially. Finally, it is environmentally friendly in that no harsh chemicals
or additives are used as in the case of most other EOR methods since it utilizes
indigenous micro-organisms. In General, this method increases oil recovery factor
and thus life of the field is extended for years and thereby reducing production
decline. These amongst others makes this method phenomenal.
2.3.5
Problems of MEOR method
In spite of its’ numerous advantages, MEOR technique faces some problems which
are outlined in (Lazar, 2007). To begin with, microbes produce H2 S and SO2
causing bio-corrosion of the equipment and contamination of ground water. To
add to that, in the transportation process, injectivity of microbes is lost due to
microbial plugging of the well-bore; thus, dispersion or transportation of all necessary components to the target zone is of much concern and if this is not properly
addressed, the whole notion of using microbes to enhance the recovery of the residual oil will never materialize.
Another major concern is the optimization of the desired in-situ metabolic activity due to the effect of variables such as PH, temperature, salinity, and pressure
for any in-situ MEOR operation. This leads to another problem being isolation
of microbial strains, adaptable to the extreme reservoir conditions of PH, temperatures, pressure and salinity (Sen, 2008).
Finally, the low in-situ concentration of bacteria metabolites is a headache.
10
In spite of the various advantages of MEOR over other EOR methods, MEOR
has not gained credibility in the oil industry because the value of MEOR can only
be determined by the results of field trials. Again, MEOR’s literature is mainly
based on laboratory data and a shortage of field trials can be seen in this field.
Also, because of reservoir heterogeneity, it is so difficult to extrapolate laboratory
results into what is to be expected in the field or predict what will happen in a
new field based on the results obtained from another field. Furthermore, few of
the tests explain the mechanisms of oil recovery or offer a reasonable analysis of
the application outcome. In addition, as (Moses, 1991) pointed out, the follow-up
time of most field trials is not long enough to determine the long-term effects of
the process. Finally, the precise mechanisms of in-situ MEOR operations are still
unclear; thus more research is required in this field (Xu and Lu, 2011).
11
Chapter 3
Mathematical model
3.1
Two-phase model
This section presents the models and solution algorithms for the saturation and
pressure distributions in a two-phase flow through a porous media. The mathematical model, which was developed from the popular Darcy’s equation and the continuity equation, contains a substance source/sink term. Using the finite volume
discretization scheme, the implicit pressure explicit saturation (IMPES) method
was used to solve for pressure and saturation. The implementation was carried
out in Matlab.
We note that the relative permeabilities and capillary pressure are functions of the
fluid saturation.
The two main dependent variables of interest in two- phase flow in porous medium
are saturation and pressure. Saturation is the ratio of the volume that a fluid occupies to the pore volume of the porous medium. The relative amounts of oil, gas
or water that will flow when more than one phase is present in a porous medium
are dependent on the individual phase saturation. On the other hand, reservoir
pressure is used for characterizing a reservoir, estimating its oil capacity and predicting its future behavior. The production of oil and water in a well is a function
of the reservoir pressure, which depends on the amount of oil and water in the
reservoir, which on its part, is described by the saturation (Craft and Hawkins,
1991).
This thesis therefore, models a two-phase flow in a porous media with a source/sink
12
term, for the determination of the saturation and pressure distributions. The governing equations are formulated in one-dimension for the saturation and pressure
respectively. Although many reservoirs are modeled as two-dimensional space coordinates (Blunt, 2001) due to the fact that petroleum reservoirs are usually more
permeable in the horizontal direction than in the vertical direction, this research
is limited to just one-dimension.
3.1.1
The governing equations
In the absence of hydrodynamic dispersion, the Darcy’s equation (3.1) is written
to relate the superficial velocity of the simultaneous flow of each phase to the
pressure gradient of the phase:
→
−
Krα K̂
−
Uα =−
(∇Pα + ρα →
g)
µrα
(3.1)
for
α = {Oil(o), W ater(w)}
→
−
where U [m/s] is the production by cross sectional area of the flow; P [Pa] is the
fluid pressure; g [m/s2 ] is the acceleration due to gravity; K̂[m2 ] , Krα , µ[pas] and
ρ[kg/m3 ] are the single phase permeability, relative permeability, viscosity and
density respectively.
Due to the surface tension and curvature of the inter-phase between the two phases,
one phase referred to as the wetting phase being water tends to wet the porous
medium more than the other phase, referred to as the non-wetting phase being oil
(Xue, 2004). Since the void volume is completely occupied by the two fluid, the
following equation applies for the saturations:
Sw + So = 1
(3.2)
The pressures of the two phases are related to each other through the capillary
pressure Pc (Sw ), (Xue, 2004) given by;
Po − Pw = Pc (Sw ).
13
(3.3)
By relating the effective permeabilities K̂o and Kˆw to the single phase permeability
K̂, the relative permeabilities Kro and Krw can be defined as:
Kˆw = Krw K̂
K̂o = Kro K̂.
The relative permeabilities are empirically taken to be functions of saturation and
are assumed to be independent of direction (Aziz and Settari, 1979). From the
continuity equation, we have:
→
−
∂ (ρα Sα φ)
+ ∇ · (ρα U α ) = Fv,α , α = w, n
∂t
(3.4)
Where φ[-] is the porosity or void fraction of the porous media and Fv,α [kg/m3 s]
represent the sink or source capacities for each respective phases and t[s] is the
production time.
Equations (3.4) represent the mathematical model for the flow of two immiscible phases in porous media. In order to solve it for the transient saturation and
pressure of each phase, the following additional information is to be provided:
• Capillary pressure and relative permeabilities as functions of saturation.
• Appropriate boundary and initial conditions.
• The porosity and fluid properties (densities and viscosities).
In this model, we assume that densities and porosity are constant and also we
neglect gravity since a transverse one dimensional flow is considered. With these
assumptions, substituting the Darcy law equation (3.1) into the continuity equation for each phase results in;
φ∂ (Sw )
Krw K̂
Fv,w
−∇·
(∇Pw ) =
∂t
µw
ρw
(3.5)
φ∂ (So )
Kro K̂
Fv,o
−∇·
(∇Po ) =
.
∂t
µo
ρo
(3.6)
It is worth noting that equations (3.2), (3.3), (3.5) and (3.6) give the fully coupled
equations for the two phase model of oil and water. In order to solve them, these
14
equations have to be either coupled or decoupled into pressure and saturation
equations depending on the formulation and the numerical method used.
Making So the subject in equation (3.2) and substituting into equation (3.6) gives;
Kro K̂
Fv,o
φ∂ (1 − Sw )
−∇·
(∇Po ) =
∂t
µo
ρo
(3.7)
To eliminate the saturations, we add equations (3.5) and (3.7). This results in:
− ∇ · K̂[λw ∇Pw + λo ∇Po ] = Fv,T
(3.8)
Where,
λα =
Krα
,
µα
α = w, o
(3.9)
and
Fv,T =
Fv,o Fv,w
+
.
ρo
ρw
(3.10)
Equation (3.9) is called the phase mobility whilst equation (3.10) is the total source
or sink term
Various formulations such as the mean value formulation, the global pressure formulation and the fractional flow formulation can be used. In this research, we
make use of only the mean value formulation.
15
3.1.2
The mean value formulation
P =
Po + P w
.
2
(3.11)
From equations (3.3) and (3.11), we have
1
Po = P + Pc (Sw )
2
(3.12)
1
Pw = P − Pc (Sw ).
2
(3.13)
Substituting (3.12) and (3.13) into (3.8) gives,
1
1
− ∇ · K̂ λw ∇ P − Pc (Sw ) + λo ∇ P + Pc (Sw )
= Fv,T .
2
2
(3.14)
Now, using P and Sw as primary variables, we solve equations (3.14) and (3.5)
using the finite volume discretization scheme. We note that these two equations
are nonlinear partial differential equations. Although Newton’s fix point iterative
solver can be used to solve it nonlinearly, this thesis employed the IMPES method
which takes care of the nonlinearity. For an alternative robust implicit discretization approach to these equations, we refer to [32]. For our system of equation
under consideration, we have:

1
1

−∇
·
K̂
λ
∇
P
−
P
(S
)
+
λ
∇
P
+
P
(S
)
= Fv,T

w
c
w
o
c
w
2
2








φ∂(Sw )


− ∇ · K̂ λw ∇(P − 12 Pc (Sw )) = Fρv,w

∂t
w



















+ appropriate boundary conditions
+ appropriate initial conditions
16
(3.15)
Simplifying equation (3.15) and applying boundary and initial conditions, we have;


−∇ · K̂ λT ∇(P ) + 12 λd ∇ (Pc (Sw )) = Fv,T









φ∂(Sw )


− ∇ · K̂ λw ∇(P − 12 Pc (Sw )) = Fρv,w

∂t
w



















+ appropriate boundary conditions
Sw (x, 0) = Sw0 (x)
0
P (x, 0) = P (x)
Where, λT = λw + λo being the total mobility and
λd = λo − λw .
17
(3.16)
3.2
Microbial transport mechanisms in porous
media
The main goal of MEOR is to recover the residual oil by the use of microbial. As
earlier noted, the residual oil left after the application of primary and secondary
means of oil production is about 35-55%. The thesis at this stage brings to bare the
transport mechanisms that may describe some basic mechanisms regulating the
dynamics and interaction between microbial organisms and the porous medium
with divers compositions of fluids [21]. We recall that one of the major challenges
facing MEOR processes is transport: thus, getting the microbial to hit the target
zones. Anaerobic microbial organisms such as Clostridium, Bacillus, Pseudomonas
just to mention but a few are normally found in subsurface porous medium. If
we however inject oxygen into the reservoir, aerobic organisms may also survive.
Thus, we often find a large variation of microbial types in such environment. The
microbial organisms of interest considered in this work are bacteria. Therefore we
use this term for the organisms even if the more general term microbial organisms
could have been used.
3.2.1
Transport equations
With the assumption that the porous medium contains some amount of water,
let C be the concentration of the bacteria in water. From [22] and [23], a simple
way to model spatial and temporal variation of C at macro scale is by the simple
advection-dispersion equation (3.17)
∂C
∂C
∂ 2C
=D 2 −ν
∂t
∂x
∂x
(3.17)
in which D is the hydrodynamic dispersion coefficient, and ν stands for the velocity
of the water driven bacteria.
3.2.2
Adsorption
During transport, a fraction of the bacteria will be adsorbed by the solid surface.
Denoting the attached bacteria concentration by ψ, a general governing equation
18
for transport of bacteria in water-saturated porous media is [[22], [23]]:
∂C ρb ∂ψ
∂ 2C
∂C
+
=D 2 −ν
∂t
φ ∂t
∂x
∂x
(3.18)
where φ is the porosity and ρb is the (dimensionless) dry bulk density.
This thesis does not take into consideration certain factors such as equilibrium
formulations in which retardation growth will be taken care of and also adsorptions involving kinetics.
3.2.3
Growth and decay
The growth and decay of the bacteria are not taken into account in the transport
equations (3.17) and (3.18). To make way for them, we include a source term q in
the transport equation:
∂ 2C
∂C
∂C ρb ∂ψ
+
=D 2 −ν
+q
∂t
φ ∂t
∂x
∂x
(3.19)
where q represents the rate of growth or decay, depending on its sign. For simple
systems, a common choice for the source term q is the Monod equation [26] which
is given by
q = ϕmax
C
Kc + C
(3.20)
where:
q is the specific growth rate of the microorganisms ϕmax is the maximum specific growth rate of the microorganisms, C is the concentration of the limiting
substrate for growth and Kc is the ”half-velocity constant”(the value of C when
q
ϕmax
= 0.5). ϕmax and Kc are empirical coefficients to the Monod equation.
For multicomponent systems, the source term qi for the component i will depend
on the concentrations of one or more of the other components, and on chemical
reactions. Denoting the chemical reaction by R and assuming that the bacteria
are living in the water, the conservation based equations for the concentration of
a component i in water (Ciw ) are then given as
19
∂ 2 Ciw
∂Ciw
∂(Sw φCiw ) ρb ∂ψ
+
−D
= qiw + R(Ciw )
+
ν
i
2
∂t
φ ∂t
∂x
∂x
(3.21)
We note that equation (3.21) forms part of our model problem equation (3.16)
3.2.4
Exponential growth
We consider a stationary system, and assume that the concentrations of the essential nutrients for a specific bacteria, remain sufficiently high. The number n(t)
of bacteria in the system will then increase exponentially: n(t) = n0 bt/T where
n0 = n(0) is the initial number of bacteria, b is the growth factor, and T is the
time needed for the number n(t) to increase by a factor b.
The discretization of the main mathematical models equations (3.16) and (3.21)
are discussed in chapter 4 of this thesis.
20
Chapter 4
Numerical modeling
The term numerical modeling often refers to solving a partial or an ordinary differential equation. With regards to our set of coupled nonlinear partial differential
equation we need to solve it numerically. The finite difference method described in
[12] and the two point flux approximation [24] which is a control volume method
are used. This aspect of the thesis brings to light the theoretical background for
the discretization, sets up the discretization schemes for the equations and show
how they are decoupled.
4.1
Grid
Most continuous partial differential equations have their solutions in the infinite
dimensional space hence a solution should be sought for in the finite dimensional
space. This brings to bear the concept of discretization. Thus, our problem has
to be solved numerically by discretizing it. The same accounts for if you want
to represent a mathematical function f (x) numerically, you must first define the
points xi where the function is evaluated. Defining these points simply implies to
define a suitable grid on the domain of the function.
The way the grids are defined can be thought of as dividing the interval into cells
with walls separating the cells. From a two - dimensional perspective, the grids
are generated by putting out several points in the domain, and then connecting
the points by straight lines not intersecting each other. When we look at the grids
as containing cells, the lines are the cell walls whilst the grid points are the corners
of the cells. We then can choose the points xi to be the same as the grid points, or
21
we can let the points xi be the middle points of the cells, giving us a cell-centered
grid [10], see fig. 4.1.
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
Figure 4.1: An example of a cell-centered grid in 2-D
Control-volume method mostly uses a cell-centered grid. Usually, to define the
points xi for a cell-centered grid, one needs to divide the interval into cells first,
then find the middle point of the cells and let them be the points xi [10]. But
for an equidistant one-dimensional grid, it does not matter if one defines the discretization points xi or the walls of the cells first.
0
•|
k
•
t1
•
•
tj
•
tj+1
•
•
tm−1
T
•|
tm
t
Figure 4.2: Equal distant time discretization
Our problem has to be discretized in both space and time. The time interval is
[0 , T], thus starting from initial time to = 0 to final time tm = T . The size of
the time steps are given by 4tj = tj − tj−1 . We have equal distance time interval,
where time step size is constant. Thus, 4 = τ for j = 0, ..., m. Hence, τ is found
from τ =
T
.
m
0
|
x1
2
|
x3
2
|
|
xi− 1
2
|
xi+ 1
2
|
|
xn− 1
2
Figure 4.3: Space interval divided into equal sized cells
22
L
|
xn+ 1
2
x
We again use the control-volume method in space. Thus, we discretize the space
interval [0 , L] into a cell-centered grid. For convenience regarding the spacing,
we consider equal distant grid. We then start by dividing the interval [0 , L] into
n equal cells. The walls of the cells are then given by xi+ 1 = ih for i = 0, ..., n
2
where h = x
i+ 12
−x
i− 21
=
L
.
n
Then x = 0 and x
1
2
i+ 12
= L gives the boundaries of
the domain, see fig. 4.3.
0 x1
| • | •
x1
x3
2
2
|
•
xi
| • | •
xi+ 1
xi− 1
2
2
|
•
xn L
| • |
xn− 1 xn+ 1
2
x
2
Figure 4.4: Cell-centered space discretization.
After dividing the space intervals into cells, we take the middle points of the cells
to be the grid points x1 , x2 , ..., xn ; see figure 4.4. The reason being that, since
we have an equal distant grid, the distance between the neighboring grid points
will be the same as the size of the cells, 4xi = xi+1 − xi = h. Then x1 =
xi = ih −
4.2
h
2
for i = 0, ..., n. The last grid point is xn = nh −
h
2
=L−
h
2
and
h
.
2
Spatial discretization
This aspect looks at the spatial discretization techniques since the only discretization technique used for the time is the backward Euler method. Several discretization techniques exist for discretizing the spatial component of a partial differential
equation but the method used mostly in this thesis is the two point flux approximation, being a control volume-method. Since this work is limited to one-dimension,
this two point flux approximation is equivalent to the finite difference method for
a cell centered grid.
4.2.1
The finite difference methods
This method has its base from the Taylor series expansion for u(x + h) = u(xi+1 )
and u(x−h) = u(xi−1 ) and is the most elementary spatial discretization technique.
Although not widely used today due to its inability to contain the conservative
property required for most problems, it is widely described in books on numerical
analysis, such as [10, 16–18]. The Taylor series and finite difference method are
23
also the bases for the time discretization equation.
From the Taylor series
h2 00
h3 000
u(xi+1 ) = u(xi ) + hu (xi ) + u (xi ) + u (xi ) + ...,
2
3!
0
(4.1)
we get the forward difference approximation for the first derivative,
u0 (xi ) =
u(xi+1 ) − u(xi )
+ O(h)
h
(4.2)
The order of the method is given by the term O(h). It represents the terms which
are truncated from the Taylor series and it is that which determines the order of
convergence.
In the same vein, we get the backward difference approximation
u0 (xi ) =
u(xi ) − u(xi−1 )
+ O(h),
h
from
u(xi−1 ) = u(xi ) − hu0 (xi ) +
h2 00
h3
u (xi ) − u000 (xi ) + ...,
2
3!
(4.3)
(4.4)
From equations (4.2) and (4.3), it is seen that both forward and backward difference approximations are of order O(h), making them not to be the best.
Now, subtracting equation (4.4) from equation (4.1) gives the central difference
approximation scheme
u0 (xi ) =
u(xi+1 ) − u(xi−1 )
+ O(h2 ),
2h
(4.5)
which is a better approximation.
Finding an approximation for the second derivative, we use
u(xi+1 ) − u(xi )
h
h2
h3
= u0 (xi ) + u00 (xi ) + u000 (xi ) + u0000 (xi )...
h
2!
3!
4!
(4.6)
u(xi ) − u(xi−1 )
h
h2
h3
= u0 (xi ) − u00 (xi ) + u000 (xi ) − u0000 (xi )...
h
2
3!
4!
(4.7)
Subtracting equation (4.7) from equation (4.6) results in
u00 (xi ) =
u(xi+1 ) − 2u(xi ) + u(xi−1 )
+ O(h2 )
h2
24
(4.8)
which gives the central difference approximation scheme for the second derivative
u00 (xi ).
Although this method is very simple to implement, its major challenge is seen
when applied to very complex geometry. Also, due to where it stems from, makes
it not to have the conservative property.
25
4.2.2
The two point flux approximation
We consider the ordinary stationary differential equation
− (K(x)Px )x = Q(x)
(4.9)
where
K(x) denote the permeability, Q(x) the source term and Px =
∂p
.
∂x
Assume
q = −K(x)Px .
(4.10)
Then from equations (4.9) and (4.10), we have
− qx = Q(x).
(4.11)
We want to solve equation (4.10) for q. We first begin to discretize it using fig.4.5
below
|
xi− 3
2
∆xi−1
•
xi−1
∆xi
•
xi
|
xi− 1
2
∆xi+1
•
xi+1
|
xi+ 1
2
|
xi+ 3
2
∆xi+2
•
xi+2
|
xi+ 5
2
Figure 4.5: Cell centered grid with points xi and the cell walls xi+ 1
2
.
The grid points xi for i = 0, 1, 2, ..., n are the middle points of the cells. The walls
are of cell with mid point xi given by xi− 1 and xi+ 1 . Integrating equation (4.11)
2
2
over each cell, we get;
Z
−
xi+ 1
2
Z
xi− 1
2
Q(x)dx
xi− 1
2
2
which results in
Z
qi+ 1 − qi− 1 =
2
xi+ 1
qx dx =
2
xi+ 1
2
Q(x)dx.
(4.12)
xi− 1
2
We now seek to find an expression for qi+ 1 by K and P . This is done by re-writing
2
equation (4.11) as
Px = −
q
.
K(x)
26
(4.13)
If we now integrate equation (4.13) from the middle point xi to the middle point
xi+1 , we get
Z
xi+1
xi+1
Z
Px dx = −
xi
xi
=⇒
Z
q
dx
K(x)
xi+1
Pxi+1 − P xi = −qi+ 1
2
xi
1
dx
K(x)
hence
Pxi+1 − P xi
qi+ 1 = − R xi+1
.
1
2
dx
K(x)
xi
We now need an approximation for the integral
Z
xi+1
xi
1
dx
K(x)
from grid point xi to grid point xi+1 . These are the middle points in two neighboring cells, thus, we integrate over two cells. We assume that K(x) is constant
on each cell, denoted by the values at the grid point, Ki ≈ K(xi ). We let ∆xi
denote the distance between the walls of the cell. Thus, ∆xi = xi+ 1 − xi− 1 . We
2
2
approximate the integral by taking the average over the two cells involving xi and
xi+1 .
=⇒
Z
xi+1
xi
1
1
dx =
K(x)
2
∆xi+1 ∆xi
+
Ki+1
Ki
.
Thus we get the following expression
qi+ 1 = − 2
1
2
Pi+1 − Pi
∆xi+1
Ki+1
+
∆xi
Ki
which becomes
qi+ 1 = −ai+1 (Pi+1 − Pi )
2
if we let
27
(4.14)
ai =
1
1
2
∆xi
Ki
+
∆xi−1
Ki−1
.
This holds for a non-equidistant cell centered grid. For equidistant cell-centered
grid, we have ∆xi = ∆xi−1 = h for i = 0, ..., n. This gives
ai =
1
h
2
1
Ki
+
1
(4.15)
Ki−1
In this thesis, only equidistant cell-centered grids are considered. Now, substituting for q, equation (4.12) becomes
P − Pi−1
i
−
h
1
1
+
2 Ki
Ki−1
P − Pi
i+1
=
h
1
1
+
2 Ki+1
Ki
Z
xi+ 1
2
Q(x)dx
(4.16)
xi− 1
2
which can be written as
ai (Pi − Pi−1 ) − ai+1 (Pi+1 − Pi ) = bi
(4.17)
where bi is defined as
Z
xi+ 1
2
bi =
Q(x)dx .
(4.18)
xi− 1
2
Transposing equation (4.17), we have
− ai Pi−1 + (ai + ai+1 )Pi − ai+1 Pi+1 = bi
(4.19)
for i = 1, ..., n. Thus, we have a system of n equations. The unknown are
[Po , P1 , ..., Pn , Pn+1 ]. That is, there are n + 2 unknown. In order to obtain a
unique solution, we need an additional boundary conditions. In this thesis, two
commonly boundary conditions are discussed.
28
4.2.3
Dirichlet boundary conditions
Under this system of boundary condition, the value of the unknown function
is specified at end point of the domain. With regards to our stationary onedimensional problem (4.9) on the domain [0 , L] with the unknown pressure
p = p(x). The Dirichlet boundary conditions can be given as
p(0) = p0
(4.20)
p(L) = pL .
(4.21)
and
Since we are using the cell-centered grid in figure 4.5, when the problem is discretized,
p 1 = p0 ,
2
and
pn+ 1 = pL .
2
It is however not a straight forward issue in handling Dirichlet boundary conditions for a cell-centered gird and we need to find a trick in doing it. It is quite
straight forward in the case of vertex centered grid since the boundary points will
coincide with the grid points at the end.
x0
x1
| • | • |
x1
x3
2
2
•
xi
| • | •
xi− 1
xi+ 1
2
2
|
•
xn L
| • |
xn− 1 xn+ 1
2
x
2
Figure 4.6: Dirichlet boundary conditions for cell-centered grid by adding
ghost cells at the ends of the interval.
In the book by J. W. Thomas [25], two ways of handling Dirichlet boundary
conditions are presented for cell-centered grids. One way is simply by adding
ghost cells at the ends of the interval as shown in figure 4.6. Then we assume
that the Dirichlet boundary conditions are prescribed at the ghost cells and we
discretize in the first and last cells as done in the figure above. This will result in
additional grid points x0 and xn+1 and we assume
p0 = po ,
29
and
pn+1 = pL .
In [25], it is noted to be a first order approximation making it not be sufficient
if the solution largely depends on the boundary conditions. The second simplest
way is to add half cells at the ends of the intervals as shown in figure 4.7. This
thesis however makes
only.
x0 usex1of the first approach
xi
| • | • | • | • | • |
x1
xi+ 1
x3
xi− 1
2
2
2
2
xn L
|
•
• |
xn− 1 xn+ 1
2
x
2
Figure 4.7: Dirichlet boundary conditions for cell-centered grid by adding half
cells at the ends of the interval.
Using the first approach of adding ghost cells normally handle the Dirichlet boundary conditions the same as we would have done for a vertex centered grid. The
difference lies in how the grid points xi are defined and also in the points in which
we evaluate the p values. An illustration of how to manage the boundary condition
is given below for
− (Kpx )x = Q.
(4.22)
Where we make use of the first approach by adding ghost cells in the ends of
the interval. From the discretization of the problem as shown in equation (4.19)
results in the system of equations
−a1 p0 + [a1 + a2 ]p1 − a2 p2 = b1
−a2 p1 + [a2 + a3 ]p2 − a3 p3 = b2
...
...
...
−an−1 pn−2 + [an−1 + an ]pn−1 − an pn = bn−1
−an pn−1 + [an + an+1 ]pn − an+1 pn+1 = bn
With Dirichlet boundary conditions, we require two boundary conditions p0 = po
and pn+1 = pL for the pressure. Moving the boundary terms to the right hand
30
side of the system of equations results in
[a1 + a2 ]p1 − a2 p2 = b1 + a1 po
−a2 p1 + [a2 + a3 ]p2 − a3 p3 = b2
...
...
...
−an−1 pn−2 + [an−1 + an ]pn−1 − an pn = bn−1
−an pn−1 + [an + an+1 ]pn = bn + an+1 pL .
Writing this in matrix form implies






A=





a1 + a2
−a2
0
.
.
−a2
0
.
.
.
a2 + a3 −a3 .
.
.
...
... ...
...
...
...
... ...
...
...
...
−an−1 an−1 + an
0
−an
0
0
0

b 1 + a1 P o

b2








.







0

.





.





.
.
an
an + an+1
and








b=







b3
.
.
.
bn−1
bn + an+1 pL
which gives the linear system
Ap = b
for p = [p1 , p2 , p3 , ..., pn , pn+1 ]T which can then be solved by the use of a numerical
linear solver.
31
4.2.4
Neumann boundary conditions
In the case where we have Neumann boundary conditions, the values of the first
derivative of the function is specified at the boundaries. Thus, in a one-dimensional
case, this boundary condition is given as
px (0) = pα
(4.23)
px (L) = pβ .
(4.24)
and
From figures [4.4 and 4.6], we again realize that x 1 = 0 and xn+ 1 = L. Since
2
2
qi+ 1 is the expression for the derivative of the pressure p at the cell wall xi+ 1 , the
2
2
boundary condition can then be written as
q1 = − 2
1
2
pα
1
K0
+
1
K1
and
qn+ 1 = − 2
1
2
pβ
1
Kn
+
1
Kn+1
With K0 and Kn+1 unknown, lets assume that K0 = K1 and Kn = Kn+1 . Assuming also that we have homogeneous Neumann boundary conditions, that is
pα = pβ = 0. This then results in
q1 = 0
(4.25)
qn+ 1 = 0
(4.26)
2
and
2
making the values for K0 and Kn+1 to be irrelevant.
Showing also how to deal with Neumann boundary conditions for the discretization
of equation (4.9)
−(K(x)Px )x = Q(x).
Using again our previous discretization, from equations (4.12) and (4.18), we have
32
qi+ 1 − qi− 1 = bi ,
2
2
for i = 1, ..., n. For the first equation, when i = 1, we get
q 3 − q 1 = bi .
2
2
Here we can insert the boundary condition at x = 0 (4.25) which gives
q 3 = b1 .
2
Similarly, the last equation for i = n, equation (4.26) also becomes
−qn− 1 = bn .
2
Now from the fact that
q1− 1 = −ai + 1(pi+1 − pi ),
2
the first and the last equation become
a2 p 1 − a2 p 2 = b 1
−an pn−1 + an pn = bn
The rest of the equations follows as before. Again, representing the systems of
equations in matrix form
Ap = b,
where this time

a2
−a2
0
.
.
.
0

 −a2 a2 + a3 −a3 .
.
.
.


...
... ...
...
...
 0
.

A=
...
... ...
...
...
.
 .

...

−an−1 an−1 + an −an
 .
0
0
0
0
−an
an
33






.





and

b1

 b2


 .

b=
 .

 .

 b
 n−1
bn







.






We will again solve the linear system for p = [p1 , p2 , p3 , ..., pn , pn+1 ]T . When we
have homogeneous Neumann boundary conditions, pα = pβ = 0, and b simplifies
to b = [b1 , b2 , ..., bn−1 , bn , ]T .
34
4.3
The fully discrete scheme for two-phase flow
We now introduce the discrete form of the continuous problem equation (3.16)
by discretizing it using the two point flux approximation. Applying the IMPES
results in the decoupled system of equations;

h
i
n+1
1
n
n
n


−∇ · K̂ λT (Sw ) ∇(P
) + 2 λd (Sw ) ∇ (Pc (Sw )) = Fv,T








n+1
n

n+1
−Sw
Sw

1
n
n

φ τ
− ∇ · K̂λw (Sw ) ∇(P
− 2 Pc (Sw )) = Fρv,w

w



+ appropriate boundary conditions

















Sw (x, 0) = Sw0 (x)
0
P (x, 0) = P (x)
in which the mean pressure is obtained implicitly whilst the water saturation is
obtained explicitly.
=⇒
Swn+1 = f (Swn , P
n+1
)
and
P
n+1
= f (P n+1 , Swn )
The pressure equation above in one dimension is,
n+1
"
−
d
dP
λT (Swn )
dx
dx
1
+ λd (Swn )
2
dPc (Swn )
#
= Fv,T
dx
(4.27)
Discretizing equation (4.27) using the two point flux approximation described
above gives


n+1
1
dPc (Swn ) 

n dP
+ λd (Swn )
−
λT (Sw )
 dx =
Ωi |
{z dx } |2
{z dx }
Z
A
B
Recal that from figure 4.5
35
Z
Fv,T ds
Ωi
(4.28)
x
|
i− 32
∆xi−1
•
xi−1
x
∆xi
•
xi
|
i− 21
x
|
i+ 12
∆xi+1
•
xi+1
x
|
i+ 32
∆xi+2
•
xi+2
|
xi+ 5
2
Figure 4.8: Cell centered grid with points xi and the cell walls xi+ 1
2
.
with ∆xi = |Ωi | = |xi+ 1 − xi− 1 |.
2
2
From equation (4.28), A and B are fluxes and defined by
n+1
Ai− 1 = λT (Swn ) dPdx |i− 1
2
2
n+1
Ai+ 1 = λT (Swn ) dPdx |i+ 1
2
2
n
(Sw )
Bi− 1 = 12 λd (Swn ) dPcdx
|i− 1
2
2
n
(Sw )
|i+ 1
Bi+ 1 = 12 λd (Swn ) dPcdx
2
2
i−
1
2
i+
1
2
•
i
Figure 4.9: Flow of fluxes across discontinuity points in 1D
We assume that the properties of the medium are constant on each control volume.
Simplifying the fluxes results in
−[Ai+ 1 − Ai− 1 + Bi+ 1 − Bi− 1 ] = Fv,T · dx
2
2
2
2
=⇒
Ai− 1 − Ai+ 1 + Bi− 1 − Bi+ 1 = Fv,T · dx
2
2
2
Expanding, we have
36
2
(4.29)
P n+1 − P n+1 i
i−1
n
λT Si−
1
2
4xi
P n+1 − P n+1 i+1
i
n
− λT Si+
1
2
4xi
1 n n
+ λd Si+ 1 Pc (Si+1
− Pc (Sin ))
2
2
P (S n ) − P (S n )
1
c
c
i
i−1
n
− λd Si− 1
= Fv,T 4 xi
2
2
4xi
(4.30)
n+1
Seeking Pi+1
from the above equation (4.30), we have
n+1
Pi+1
= Pin+1 +
1
n
λT Si+
1
h n+1
n
λT Si−
Pin+1 − Pi−1
1
2
2
1 n 1 n n
n
Pc (Sin ) − Pc (Si−1
) − λd Si+
Pc (Si+1
− Pc (Sin ))
+ λd Si−
1
1
2
2
2
2
i
2
−Fv,T (i, n) 4 xi
(4.31)
where we make use of the forward difference approximation of the derivatives.
Rearranging equation (4.31) gives
− λT
n
Si−
1
2
n+1
Pi−1
h
+ λT
n
Si−
1
2
+ λT
n
Si+
1
2
i
Pin+1
− λT
n
Si+
1
2
n+1
Pi+1
= bn+1
i
(4.32)
where
i
1 n 1h n n
n
S
S
λ
λ
S
bn+1
=
P
(S
)
+
+
λ
Pc (Sin )
1
1
1
d
c
d
d
i−1
i
i− 2
i− 2
i+ 2
2
2
1 n n
2
Pc (Si+1
) + FTn+1
+ λd Si+
1
i 4 xi
2
2
(4.33)
n
n
If we now let am = λT Si−
and
a
=
λ
S
, then equation (4.33) can be
1
i+1
T
i+ 1
2
2
written in the form of equation (4.19) as
n+1
n+1
− am Pm−1
+ [am + am+1 ]Pmn+1 − am+1 Pm+1
= bn+1
m
37
(4.34)
From equation (4.34), the systems of equations we get looks like
−a1 P0 + [a1 + a2 ]P1 − a2 P2 = b1
−a2 P1 + [a2 + a3 ]P2 − a3 P3 = b2
...
...
...
−am−1 Pm−2 + [am−1 + am ]Pm−1 − am Pm = bm−1
−am Pm−1 + [am + am+1 ]Pm − am+1 Pm+1 = bm
With Dirichlet boundary conditions, we require two boundary conditions P (0) =
P0 and P (L) = PL for the pressure. We note that it is the pressure difference
between these two boundary conditions which actually drive the flow. Since
this thesis makes use of only the cell centered grid, it implies, P1/2 = P0 whilst
Pn+1/2 = PL . In order to make the boundary conditions coincide with P (0) and
P (L) respectively, we add ghost cells at the end of the interval and assume that
the boundary conditions are specified at the ghost cells. Discretizing these ghost
cells will lead us to get additional grid points x0 and xn+1 . We then assume finally
that P0 = P0 and Pn+1 = PL .
We move the boundary terms to the right hand side of the system of equations to
give
(a1 + a2 )P1 − a2 P2 = b1 + a1 P0
−a2 P1 + (a2 + a3 )P2 − a3 P3 = b2
...
...
...
−am−1 Pm−2 + (am−1 + am )Pm−1 − am Pm = bm−1
−am Pm−1 + (am + am+1 )Pm = bm + am+1 PL
Writing this in matrix form implies
38

An+1







=






a1 + a2
−a2
0
.
.
.
0

−a2
a2 + a3
−a3
.
.
.
.
0
−a3
...
...







.






0
.
a3 + a4 −a4
.
.
...
...
...
...
...
...
...
...
...
−ai−1 am−1 + am
.
0
0
0
−am
0
and

b
n+1






=






b 1 + a1 P 0

b2






.






b3
.
.
bm−1
bm + am+1 PL
which gives the linear system
AP = b
for
P = [P1 , P2 , P3 , ..., Pm−1 , Pm ]T
which can then be solved by the use of a numerical linear solver.
39
.
.
.
am
am + am+1
We note from figure 4.10
i − 21
δxi− 1
i−1
i
2
w
W
i + 21
δxi+ 1
i+1
2
P
e
E
∆x
Figure 4.10: Standard nomenclature for control-volume discretization in 1D
that
n
=
λT Si+
1
n )+λ (S n )
λT (Si+1
T
i
2
and
n
λT Si−
=
1
n )
λT (Sin )+λT (Si−1
2
2
2
In this thesis, the upwind approximation of mobilities is employed.
From the saturation equation in equation (3.16), we have
φ∂ (Sw )
1
d
Fv,w
n+1
n d
n
P
− Pc (Sw )
−
λw (Sw )
=
∂t
dx
dx
2
ρw
(4.35)
Integrating equation (4.35) using the two point flux approximation results in
Sin+1 − Sin X
Fv,w
φ
−
F =
4x
4t
ρw
where
P
F = F1 + F2 + F3 + F4
with
n
F1 = −λw (Si−
1) ·
2
dP
| 1
dx i− 2
n
F2 = − 12 λw (Si−
1) ·
2
n
F3 = 12 λw (Si+
1) ·
2
dPc
| 1
dx i− 2
dPc
| 1
dx i+ 2
40
(4.36)
n
F4 = λw (Si+
1) ·
2
dP
| 1
dx i+ 2
Substituting for F1 , F2 , F3 and F4 into equation (4.36) and using the backward
Euler discretization scheme for the time derivative gives,
dp 1 dp (S n )
Sin+1 − Sin
c
w
n
n
+ λw Si−
− λw Si−
1
1
2
2
4t
dx 2
dx
dp 1 dp (S n )
Fv,w
c
w
n
n
− λw Si+
+ λw Si+
=
4 xi .
1
1
2
2
dx 2
dx
ρw
φ
(4.37)
Seeking Sin+1 from equation (4.37), we have
4t
1
n+1
n+1
n
n
n
n
[λw (Si−
− Pi−1
) − λw (Si−
1 )(Pc (Si ) − Pc (Si−1 ))+
1 )(Pi
2
2
2
φ4x
2
1
Fv,w
n+1
n+1
n
n
n
n
λw (Si+
) − λw (Si+
(4x2i )].
1 )(Pi+1 − Pi
1 )(Pc (Si+1 ) − Pc (Si )) +
2
2
2
ρw
(4.38)
Sin+1 = Sin +
n+1
We note that, from the pressure equation (4.31) we first obtain Pi+1
which we
then use to find Sin+1 , the saturation equation (4.38).
41
4.3.1
Discretization of the transport equation
We now discretize the conservative transport equation (3.21) in chapter 3 using the
control volume method. Omitting the adsorption term and re-arranging results in
∂(θC)
∂C
∂ 2C
+ν
− D 2 − qw = 0
∂t
∂x
∂x
(4.39)
where θ = Sw φ is the moisture content.
It is clear from the equation that C can be obtained explicitly. Integrating equation (4.39) on the standard nomenclature for control-volume methods figure 4.10
and applying the backward Euler discretization method for the time component,
we have
Z
i+ 21
φ
i− 12
Z i+ 1
Z i+ 1
Z i+ 1
n+1 n+1
n
2
2
2
(Swi
Ci ) − (Swi
Cin )
∂Cin
∂ 2 Cin
dx+
ν
dx−
D
dx−
qin dx = 0
2
1
1
1
∆t
∂x
∂x
i− 2
i− 2
i− 2
(4.40)
Now, Sw is already known from equation (4.38). This implies that equation (4.40)
reduces to
Z
i+ 21
φ
i− 12
Cin+1 − Cin
dx+
∆t
Z
i+ 12
i− 12
∂C n
ν i dx−
∂x
Z
i+ 12
i− 12
∂ 2 Cin
dx−
D
∂x2
Z
i+ 12
i− 12
qin dx = 0 (4.41)
This aspect of the thesis explains into details the discretization. We first consider
the spatial components of equation (4.41) being
Z
i+ 12
i− 21
dC n
ν i dx −
dx
Z
i+ 12
i− 12
d2 Cin
D
dx −
dx2
Z
i+ 12
i− 12
qin dx = 0
(4.42)
Each term in this equation is evaluated and simplified separately using the centraldifference finite volume model. The parts are then reassembled into a discrete
equation relating C at node i to the C values at nodes (i + 1) and (i − 1).
42
4.3.2
The diffusion term
The second term in equation (4.42), being the diffusion term, shows the balance
of transport by diffusion into the control volume . This integral can be evaluated
exactly as
Z
i+ 21
i− 12
d2 Cin
D
dx =
dx2
dC
D
dx
i+ 12
dC
− D
dx
(4.43)
i− 12
Replacing the two diffusive fluxes by finite-difference approximations, we have
n
− Cin
Ci+1
dC
n
≈ Di+ 1
D
− Cin )
= Di+ 1 (Ci+1
2
2
dx i+ 1
δxi+ 1
2
2
dC
D
dx
Di+ 1
where Di+ 1 =
2
δxi+ 1
2
2
n
Cin − Ci−1
n
)
= Di− 1 (Cin − Ci−1
2
2
δxi− 1
i− 21
≈ Di− 1
2
Di− 1
and Di− 1 =
2
δxi− 1
2
2
From figure 4.10, δxi+ 1 = (i + 1) − i and δxi− 1 = i − (i − 1).
2
2
n
n
We bear in mind that Cin , Ci+1
, and Ci−1
are the values of C n at the nodes
i, (i + 1), and (i − 1) of figure 4.10. These are the discrete unknown that are obtained by solution of the finite volume model equations. Since this thesis considers
only the case of uniform D, it implies Di+ 1 = Di− 1 = D. Equation (4.43) now
2
2
becomes
Z
i+ 21
D
i− 12
4.3.3
d2 Cin
n
n
dx = Di+ 1 (Ci+1
− Cin ) − Di− 1 (Cin − Ci−1
)
2
2
dx2
(4.44)
The source term
The discrete contribution of the source term is obtained by assuming that (q n +
R(C n )) has the uniform value of (q n )i throughout the control volume. Thus,
Z
i+ 12
i− 12
q n dx ≈ qin ∆x
43
(4.45)
The distribution of qin ∆x will be supplied as an input to the model.
4.3.4
The convection term
The convective term in equation (4.42) can be integrated once exactly as
i+ 12
Z
ν
i− 21
dCin
dx = (νC n )i+ 1 − (νC n )i− 1
2
2
dx
(4.46)
Ci−1
Ci
Ci+1
i−1
i−
1
2
i
i+
1
2
i+1
n
n
Figure 4.11: Linear interpolation to obtain interface values Ci−
1 and C
i+ 1
for the central difference approximation
2
2
.
n
In evaluating the right hand side of the above expression, the values of Ci+
1 and
2
n
n
Ci−
1 need to be estimated. In the finite volume method, the values of Ci are
2
stored only at the nodes i, (i + 1), and (i − 1). The method for determining
n
n
n
an interface value such as Ci+
1 from the nodal values such as Ci and Ci+1 has
2
important consequences for the accuracy of the numerical model. Various methods
n
n
n
exist for estimating Ci+
1 in terms of the nodal values Ci+1 and Ci . In this thesis,
2
the common linear interpolation method as depicted in figure 4.11 is used. The
linear interpolation formula can be written as:
n
n
n
Ci+
1 = βi+ 1 Ci+1 + (1 − βi+ 1 )Ci
2
2
in which
βi+ 1
2
2
i + 12 − i
=
(i + 1) − i
44
(4.47)
(4.48)
Thus, equations (4.48) and (4.47) constitutes the central difference scheme for
approximating the derivatives
d(νC n )
(νC n )i+1 − (νC n )i
|i+ 1 ≈
2
dx
(i + 1) − i
n
n
n
Again, using linear interpolation to estimate Ci−
1 in terms of Ci−1 and Ci gives
2
n
n
n
Ci−
1 = βi− 1 Ci−1 + (1 − βi− 1 )Ci
2
2
where
βi− 1
2
(4.49)
2
i − i − 12
=
i − (i − 1)
(4.50)
Since this thesis is limited to only uniform mesh and also due to the fact that the
nodes are located midway between the cell faces, it implies βi− 1 = βi+ 1 = 12 . We
2
2
now substitute equations (4.47) and (4.49) into (4.46) and rearranging, results in
Z
i+ 12
ν
i− 12
dCin
n
n
− Cin ) − νi− 1 βi− 1 (Ci−1
− Cin ) + νi+ 1 Cin − νi− 1 Cin
dx = νi+ 1 βi+ 1 (Ci+1
2
2
2
2
2
2
dx
(4.51)
Since ν is a uniform parameter, we have νi+ 1 = νi− 1 = ν. Hence the last two
2
2
terms in the preceding equation cancels. This reduces equation (4.51) to
Z
i+ 21
ν
i− 12
dCin
n
n
dx = νβi+ 1 (Ci+1
− Cin ) − νβi− 1 (Ci−1
− Cin )
2
2
dx
(4.52)
If we now substitute equations (4.44) , (4.45) and (4.52) into equation (4.42) and
simplify, it results in the discrete form of the spatial components. This gives
n
n
− ai+1 Ci+1
+ ai Cin − ai−1 Ci−1
= bni
(4.53)
where
ai+1 =
1
(D 1 − νβi+ 1 )
2
∆x i+ 2
45
(4.54)
ai−1 =
1
(D 1 − νβi− 1 )
2
∆x i− 2
(4.55)
ai = ai+1 + ai−1
(4.56)
bni = qin
(4.57)
For the discretization of the time component in (4.41), we have
Z
i+ 21
φ
i− 12
Cin+1 − Cin
dx = φ
∆t
Cin+1 − Cin
∆t
∆x
(4.58)
which is considered to be a constant averaging of the concentration Cin on each cell.
We now add equation (4.58) and (4.53) to give us the fully discretized equation for
our transport model (4.39). Hence, for our discretized transport model, we have
φ
Cin+1 − Cin
∆t
n
n
∆x − ai+1 Ci+1
+ ai Cin − ai−1 Ci−1
= bni
(4.59)
Rearranging equation (4.59), Cin+1 can be derived explicitly from the equation
Cin+1 = Cin +
∆t n
n
ai+1 Ci+1
− ai Cin + ai−1 Ci−1
+ bni
φ∆x
46
(4.60)
Chapter 5
Numerical results and analysis
The numerical results of the simulations are presented in this chapter and they are
based on the discretization discussed in chapter 4. In order to test the accuracy of
the model, we run our model on some simulations problem designed in an article
published by Brahim Amaziane, Mladen Jurak and Ana Keko [27]. Their work
focused on modeling and simulations of immiscible compressible two-phase flow
in porous media by the concept of global pressure. Their model was tested with
the Couplex-Gas benchmark (2006) which has currently been proposed to improve
the simulation of the migration of hydrogen produced by the corrosion of nuclear
waste packages in an underground storage. The results of their simulations are
presented for the differences in phase pressures and water saturation.
5.1
Model validation with the couplex-gas benchmark
For the numerical analysis, a set of data was taken from the couplex-Gas benchmark with incompressible wetting phase (water) and the ideal gas law ρg (pg ) = cg pg
for the nonwetting phase (hydrogen) and a set of van Genuchten’s parameters.
The fluid properties and other parameters used are given in table 5.1 below, in
which n and entry pressure Pe are the van Genuchten’s parameters.
47
Parameter
Values
K
1 mD
φ
0.1
n
2
L
100 m
Pe
2 MP a
ρw
996.5 kg/m3
ρo
2 kg/m3
µw
0.869 x 10−3 P a s
µg
9 x 10−6 P a s
Table 5.1: Benchmark fluid properties and other parameters
The van Genuchten’s capillary pressure and relative permeabilities are given by
the relations:
1/n −1/m
Sw
−1
Pc (Sw ) = Pe
,
m i2
√ h
1/m
Krw (Sw ) = Sw 1 − 1 − Sw
2m √
1/m
Kro (Sw ) = 1 − Sw 1 − Sw
In the above formulas, we have m = 1 − 1/n and assume that water and gas
residual saturations are equal to zero. Figure 5.1 shows how they look.
50
1
40
0.8
Krw
Relative permeability
30
c
P [MPa]
Kro
20
0.4
0.2
10
0
0
0.6
0.2
0.4
0.6
0.8
0
0
1
0.2
0.4
0.6
0.8
1
X [m]
X [m]
(a) Capillary pressure
(b) Relative permeabilities
Figure 5.1: van Genuchten’s parameters
Three different simulations corresponding to gas injection, imbibition and gas
source terms were presented in this papers. This thesis showcases just two of
them.
48
5.1.1
Benchmark simulation one
In this simulation, the gas is injected on the left end of the porous medium initially saturated with water. As in other simulations, they use Dirichlet boundary
conditions for the global pressure and a Dirichlet boundary condition for the water
saturation on injection boundary completed by the Neumann condition on the output end of the domain. Their governing equations were solved using the following
boundary conditions:
Sw (0, 4) = 0.4,
p(0, t) = 2.0,
p(L, t) = 0.1,
∂
Sw (L, t) = 0
∂x
and initial conditions:
Sw (x, 0) = 1.0,
p(x, 0) = 0.1.
The source terms are equal to zero. Thus Fw = Fg = 0. With the assumptions
that our nonwetting phase is gas, we solved our governing equation (3.16) with
the same boundary conditions. We defined their global pressure in terms of our
water pressure using equation (3.13) and specify boundary conditions for the water phase pressure. This yielded the following boundary conditions:
Sw (0, 4) = 0.4,
pw (0, t) = 4.09 − Pc (0.4)/2,
pw (L, t) = 0.1,
∂
Sw (L, t) = 0.
∂x
and initial conditions:
Sw (x, 0) = 1.0,
pw (x, 0) = 0.1.
The same fluid properties were ascribed to the oil and the simulation time T is
45 days. The results of simulations one are found in figure 5.2. The results are in
correspondence with the ones in [27], please see Fig. 5 in [27].
49
1
po
pw
pmean
4
0.9
3
0.8
2
Sw
phase pressure [MPa]
5
0.7
1
0.6
0
0.5
−1
0
20
40
60
80
0.4
0
100
20
40
60
80
100
x[m]
x[m]
(a) Pressure phases
(b) Saturation
Figure 5.2: Benchmark simulation one. Pressure and water saturation profiles
at time T = 45 days.
5.1.2
Benchmark simulation two
Simulation two concerns an imbibition process in which pure water is injected in
the porous domain filled with 30% of gas. Their governing equations were solved
using the following boundary conditions:
Sw (0, 4) = 1.0,
p(0, t) = 4.0,
p(L, t) = 0.5,
∂
Sw (L, t) = 0
∂x
and initial conditions:
Sw (x, 0) = 0.7,
p(x, 0) = 0.5
whilst ours was solved using
Sw (0, 4) = 1.0,
pw (0, t) = 4.0,
pw (L, t) = 2.3 − Pc (0.7)/2,
and initial conditions:
Sw (x, 0) = 1.0,
pw (x, 0) = 2.3 − Pc (0.7)/2.
Again, the source terms are zero. Thus Fw = Fg = 0.
50
∂
Sw (L, t) = 0
∂x
The above simulation two yielded the following results in figure 5.3 which are
consistent with Fig. 6 Simulation 2 in [27].
Po
0.95
Pw
3
0.9
Pmean
0.85
Sw
phase pressure [MPa]
4
2
0.8
0.75
1
0.7
0.65
0
0
20
40
60
80
0
100
20
40
60
80
100
x[m]
x[m]
(a) Pressure phases
(b) Saturation
Figure 5.3: Benchmark simulation two. Pressure and water saturation profiles
at time T = 45 days.
The results of both simulations are very much consistent with the physics. We see
in simulation one that a higher pressure was imposed at the injection boundary so
as to be able to drive the less dense phase being gas into the much denser phase
being water. Only small regions of the domain is occupied by the gas and the
remaining part of the domain is fully saturated with water causing all pressures to
coincide after over 60 meters into the domain. This must be so because capillary
effect is absent at regions which are fully saturated with one phase. In simulation
two, after 45 days, 20 meters into the domain is fully saturated with water. This
caused the phase pressures to coincide 20 meters into the domain. Both results
are trivial in that it is much easier for a less dense fluid to be displaced by a high
dense fluid.
The results from both simulations are enough to attest to the fact that our model
problem equation (3.16) can be trusted for further simulations. We again prove
this point by constructing an analytical solution for the mean pressure and water
saturation and show some convergence analysis.
51
5.2
Model validation with an analytical solution
The main aim of this section is to show some convergence analysis for our model.
As spelled out in the article of Radu et. al. [19], we first construct an analytical
solution which fits the system and perform several numerical simulations with
different sizes of the time and space step. Secondly, we compare our numerical
solutions with the constructed analytical solutions by comparing the errors in the
2-norm and the L∞ -norm. A constructed numerical solution only implies that
we choose a simple solution which we know satisfies the initial and boundary
conditions and then we adjust the equations such that the analytical solutions will
be the exact solutions of the set of equations. This procedure is quite common
when comparing an analytical solution with its’ numerical solution.
5.2.1
The set of equations and the parameters used
We now solve numerically our equation (3.16) within a dimensionless domains of
time t [0, T ] and space x [0, L]. With regards to parameters to describe the
capillary pressure and the relative permeabilities, the Brooks-Corey parameters of
the type;




−1
Pc (Sw ) = Sw λ
Krw (Sw ) = SwM


 K (S ) = (1 − S )2 (1 − S )N
ro
w
w
w
with λ = 2, being the pore size distribution index which ranges between
0.2 ≤ λ ≤ 3, N =
2
λ
+ 1 and M = N + 2 is used.
e and Se . These equations will
We then construct an analytical solutions for P
w
then super impose initial and boundary conditions for our numerical solution.
Sew (x, t) = 0.2 + 0.8xt, Sw [0, 1]
(5.1)
e (x, t) = xt(1 − x) + 2t.
P
(5.2)
We now use our analytical solutions to find the respective sources and sink terms
for our mean pressure and water saturation. For simplicity, we let Pc = P cap ,
52
Sw = S, Fv,T = F T , Fv,w = F w , λT = λT , λd = λd , µo = µo , µw = µw and ρw = ρw
Then equation (3.16) in one dimension gives
− λT (S)P x + 12 λd (S)(P cap (S))x
= FT
w
φSt − ( λw (S)P x − 21 λw (S)(P cap (S))x x = Fρw
x
=⇒
1 d
cap
d
cap
− (λ (S)x P x + λ (S)P xx + (λ (S)x (P (S))x + λ (S)(P (S))xx ) = F T
2
(5.3)
T
T
and
1 w
Fw
cap
w
w
φSt − (λ (S))S Sx P x + λ (S)P xx −
(λ (S))S Sx (P (S))xx
= w (5.4)
2
ρ
Now from the brooks-Corey parameters, we have
(λT (S))S =
mS m−1 2(1 − S)(1 − S n ) + n(1 − S)2 S n−1
−
µw
µo
2(1 − S)(1 − S n ) + n(1 − S)2 S n−1 mS m−1
(λ (S))S = −
+
µo
µw
d
(5.5)
(5.6)
1
(P
cap
S −( λ +1)
(S))S = −
λ
((P cap (S))SS =
where
(1 + λ) −( 1 +2)
S λ
λ
λT (S) =
S m (1 − S)2 (1 − S n )
+
µw
µo
λd (S) =
(1 − S)2 (1 − S n ) S m
− w
µo
µ
(5.7)
(5.8)
If we now differentiate (5.5) to (5.8) with respect to x, taking note of the chain rule
and substituting the terms respectively into equations (5.3) and (5.4); simplifying
gives
53
Ffw
mS m−1
S m mS m−1
= ρ φSt −
S
P
+
P
(Sx )2
x x
xx w −
µw
µ
2µw
1 S m 1 + λ −( 1 +2)
Sx
+
S λ
2 µw
λ2
w
1 −( 1 +1)
S λ
λ
(5.9)
and
1
FfT = (λT (S))S Sx Px + λT (S)Pxx + (λd (S))S (Sx )2
2
1
1 + λ −( 1 +2)
− λd (S)
Sx
S λ
2
λ2
1 −( 1 +1)
S λ
λ
(5.10)
in which we have from (5.1) and (5.2)
5.2.2
e =t−2 x t
P
x
(5.11)
e = −2t
P
xx
(5.12)
e x = 0.8t
(S)
(5.13)
e xx = 0
(S)
(5.14)
e t = 0.8x
(S)
(5.15)
Comparison of results
To compare the numerical solution with the analytical one, we require some form
of measure for the difference between them, in the form of a norm. From Florin
Radus’ lecture notes on numerical analysis [20], we will make use of both the
L2 -norm and L∞ -norm which are defined as
Z
||u||2 =
21
|u|dx
and
||u||∞ = max |u|
x→[0,1]
for a function u. We will then compute the error for P and Sw as
E = ||uanal (x, T ) − unum (x, T )||2
54
where uanal (x, T ) and unum (x, T ) denote the analytical and numerical solutions
respectively at t = T , being the final time. In this test, we let t [0, 1], making
T = 1. The squared error is then given by
2
E = ||uanal (x, T ) −
unum (x, T )||22
Z
=
1
|uanal (x, T ) − unum (x, T )|2 dx,
0
from x [0, 1]. We recall that when we solve the equations numerically, we do
divide the interval [0,L] into subintervals which we call cells with the midpoints
x1 , x2 , ..., xn , and integrate over each cell as seen in chapter 4. A similar thing is
done here where we arrive at
2
E =
n Z
X
i=1
xi+ 1
2
|uanal (x, T ) − unum (x, T )|2 dx,
xi− 1
2
A simple percentage error analysis was made for the 2-norm and the ∞-norm
respectively and the results shown in tables 5.2 and 5.3.
t
h
% 2-norm
Reduction
% ∞-norm
Reduction
0.2
0.2
9.0877
-
13.3333
-
0.1
0.1
4.8353
1.8794490518
7.2727
1.833335625
0.9103
0.05
0.05
2.4956
1.9375300529
3.8095
1.9090956818
0.9542
Convergence rate (α)
0.025
0.025
1.2543
1.9896356534
1.9512
1.9523882739
0.9925
0.0125
0.0125
0.62668
2.0014999681
0.98765
1.9755986432
1.0011
Table 5.2: Error analysis for saturation
t(day)
h(m)
% 2-norm
Reduction
% ∞-norm
Reduction
0.2
0.2
2.757
-
6.4231
-
0.1
0.1
1.7341
1.5898737097
3.9349
1.6323413556
0.6689
0.05
0.05
0.93812
1.8484842024
2.1078
1.8668279723
0.8863
0.025
0.025
0.48041
1.9527486938
1.0808
1.9502220577
0.9655
0.0125
0.0125
0.23699
2.0271319465
0.50733
2.1303687935
1.0194
Convergence rate (α)
Table 5.3: Error analysis for pressure
Figure 5.4 below shows the error plot. It is candid from the figure that as the time
step decreases, the errors become smaller and converge. The convergence is of
order 2 as seen from the error reduction columns in table 5.2 and 5.3 respectively.
55
0.2
0.18
0.16
0.14
pressure
time
0.12
saturation
0.1
0.08
0.06
0.04
0.02
0
0
1
2
3
4
5
Error
6
7
8
9
10
Figure 5.4: Error plots for pressure and saturation
Theoretically, we know that
E 2 = ||uanal (x, T ) − unum (x, T )||22
But
E 2 = κ2 (4t)2p
where p is the convergence rate and κ, a constant.
=⇒
Ei2 = κ2 (4t)2p
2p
4t
2
2
Ei+1 = κ
2
=⇒
Ei
Ei+1
2
=
κ2 (4t)2p
2p
κ2 4t
2
Hence, we have the rate of convergence p given by
p = log2
Ei
Ei+1
The last columns of tables 5.2 and 5.3 reveal that the model has a linear rate of
convergence. It is clear from the tables that our model converges.
56
Chapter 6
Modeling of MEOR activities
6.1
A case study
Based on our benchmark in chapter 5, we carefully formulated a base case study
on which we simulate the activities of microbial to see their effects on the model.
We considered a case in which a reservoir of length L = 100m having an initial oil
saturation of 30%. An imbibition process was performed by first injecting water
into the left side of the porous domain. For the numerical test of this case, a set
of data was taken from a paper published by Sidsel M. Nielsen on 1D Simulation
for Microbial Enhanced Oil Recovery with Metabolite Partitioning [28]. Table 6.1
summarizes the fluid properties and other parameters used for this case study and
other simulations. We again employed van Genuchten’s correlation functions with
the same assumptions that the water and oil residual saturations are equal to zero.
In this simulation, we specify Dirichlet boundary conditions for the water pressure
and the Dirichlet condition for water saturation on injection boundary completed
by a Dirichlet condition on the output end of the domain. Thus, we solve our
governing equation (3.16) using the following boundary conditions:
Sw (0, t) = 1,
p(0, t) = 40.0,
p(L, t) = 1 − Pc (0.7)/2,
and initial conditions:
Sw (x, 0) = 0.7,
p(x, 0) = 1 − Pc (0.7)/2
57
Sw (L, t) = 0.7
The source terms are equal to zero. Thus Fw = Fo = 0. This implies that we
have imposed a very high water injection pressure and lowered it at the output
boundary. The results of this simulation are shown in figure 6.1 for the time
instance of 45 days. We notice from the figure below that water saturation is one
only on the boundary. This demands that both phase pressures must start at the
same point on the boundary just as seen.
po
0.95
pw
40
0.9
pmean
w
0.85
20
S
phase pressure [MPa]
60
0.8
0.75
0.7
0
0.65
−20
0
20
40
60
80
0
100
20
40
60
80
100
x[m]
x[m]
(a) Pressure phases
(b) Saturation
Figure 6.1: A case study. Pressure and water saturation profiles at time T =
45 days.
58
6.2
Effects of introducing microbial into the model
This section focuses on how we incorporate microbial models into the two-phase
flow model for enhancement of the oil recovery. Most of the concepts employed
in this section are basically the same as in the papers by Islam (1990) [29] and
Nielsen et al. [28]. As seen in table 2.1 of chapter 2, enhancement of the oil
recovery through microbial action can be performed through several mechanisms
such as;
• Reduction of oil-water inter-facial tension and alteration of wettability by
surfactant production and bacterial presence.
• Selective plugging by bacteria and their metabolites.
• Viscosity reduction by gas production or degradation of long-chain saturated
hydrocarbons.
• Generation of acids that dissolve rock improving absolute permeability.
Out of these four mechanisms, the first two are believed to have the greatest impact
on recovery, see please (Jenneman et al., 1984 and Bryant et al., 1989.)
Islam (1990) came out with a mathematical model for MEOR describing growth
which leads to plugging, reduction of viscosity and inter-facial tension and the
production of gas. Thus, he investigated four different mechanisms by which
bacteria helps in recovery of the residual oil. This research investigated two of the
mechanism presented by Islam (1990).
Bacteria produce surfactants by consumption of substrates. Surfactant decreases
oil/water inter-facial tension and can be distributed between both phases. Several
methods are used to model relative permeability changes as a function of interfacial tension. A correlation between surfactant concentration and inter-facial
tension σ was employed. Actually, a reduction of inter-facial tension decreases
residual oil saturation and straightens the relative permeability curves approaching full miscibility (Coats, 1980; Al-Wahaibi et al., 2006).
The two mechanisms investigated in the paper are:
• Inter-facial tension-reducing surfactant generation
59
• Oil viscosity-reducing surfactant generation
Based on the model described in [28], we modify our reactive transport model
equation (3.21) to describe convection-diffusion, bacterial growth and metabolite
production, where metabolite is surfactant. Thus, our water phase now consist
of water, bacteria and metabolite. The oil phase consists primarily of oil, but
contains also metabolite. Figure 6.2 illustrates components and phases of the flow
system. Surfactant can lower the oil/water inter-facial tension that has an effect
on relative permeability curves. We assume the following for our model:
• Bacterial growth rate can be described by Monod kinetics being independent of
temperature, pressure, pH and salinity.
• Metabolite is surfactant and can be distributed between both phases according
to a distribution constant Ki and masses of water and oil.
• Neglecting adsorption of bacterial and thus plugging porous medium by bacteria.
• No substrate and metabolite adsorption to pore walls.
• Negligible chemotaxis.
• Isothermal system with incompressible flow.
• No volume change on mixing.
−→ Oil
Metabolite
↑↓
−→
Water
Metabolite
Bacteria
Substrate −−−−−→ Bacteria
Figure 6.2: An oil reservoir
6.2.1
Inter-facial tension reduction with bacteria concentration
In modeling bacteria-generated surfactant flood, it is assumed that inter-facial
tension is a function of bacteria concentration. In many cases, the relationship
between these two parameters looks like figure 6.3. This figure has been extrapolated from data taken from the graph. Water-flooded systems have inter-facial
60
tensions between oil and water that are around (20 − 30mN/m). In order to increase recovery significantly, a good surfactant should decrease inter-facial tension
three or four orders of magnitude (Fulcher et al., 1985; Shen et al., 2006).
2
10
0
σ[N/m]
10
−2
10
−4
10 −5
10
0
10
Cm
Figure 6.3: Inter-facial tension correlation with bacteria concentration
We also make use of the correlation between inter-facial tension and relative permeability curves as given in Islam (1990). They followed the work by Bang and
Caudle [30] who predicted that the curves were related to inter-facial tension in
the following manner:
Kro
Krw
σmax − σ(Cb )
= Kro (So ) + (So − Kro )
σmax
σmax − σ(Cb )
= Krw (Sw ) + (Sw − Krw )
σmax
This formulation assumes that the relative permeabilities to water and oil are
straight lines extending from zero to one. Thus, in the limit as σ → 0, these
relative permeability curves approaches straight line forms. The van Genuchten’s
relative permeability curves will also be used and a comparison analysis will be
made for the water saturations and their corresponding fractions of oil left in the
reservoir per time. Also, the following capillary pressure curve was used in order
to incorporate dependence of the capillary pressure on σ(Cb ). As specified in [31]
which is used by the popular Eclipse simulator we have
σ(Cb )
pc (σ(Cb ), Sw ) = Pe Pc (Sw )
σmax
61
(6.1)
where pc becomes 0 if σ is 0.
Initial reservoir conditions for this simulation are found in table 6.1 at the end
of this chapter. We compare our results with the base case study. We inject a
mixture of bacteria together with substrate into the reservoir after water flooding
and we compare results for the case of van Genuchten and Brooks-Corey parameters. The result of this simulations are found in figures 6.4 and 6.6 respectively.
Water flooding
0.95
C = 10−1
0.9
−3
C = 10
Sw
0.85
C = 10−5
0.8
0.75
0.7
0.65
0
20
40
60
80
100
x[m]
Figure 6.4: Saturation profile for van Genuchten parameter with different
concentrations
62
Total oil remaining
0.26
MEOR
Water flooding
0.24
0.22
0.2
0
1
2
3
4
Total production time of 45 days 6
x 10
Figure 6.5: Fraction of oil remaining in the reservoire after 45 days using van
Genuchten parameters with focus on inter-facial tension reduction
With a simulation time of just 45 days, the total oil left in the reservoir with the
case of water flooding (C = 0kg/m3 ) and (C = 10−5 kg/m3 ) gave a mass of 22.56%
and 22.51% respectively. As we increase the concentration of the bacteria together
with substrates (C = 10−3 kg/m3 ), the remaining mass of oil left is 21.72%. Finally
with a high concentration of (C = 10−1 kg/m3 ), the remaining mass of oil left is
20.79%. We note here that any further increase in bacteria concentration beyond
(C = 10−1 kg/m3 ) does no longer have much significant effect on the Inter-facial
tension since it approaches the minimum. See figure 6.4. It is trivial to note that,
both the case of water flooding and that of (C = 10−5 kg/m3 ) gave quite close
percentages of mass recovered. This is expected since bacteria concentration has
to increase substantially away from the injection well so as to mobilize oil near
the production well bore. However, as the concentration of the bacteria increases
oil production increases. As the production time increases, more oil will be mobilized with the case of bacteria than that of water flooding because bacteria will
grow and degrade more hydrocarbons which will in turn create an oil bank which
will increase the recovery factor. Figure 6.5 displays the fraction of oil left in the
reservoir for a highly concentrated MEOR flooding as against water flooding.
With the same simulation time of 45 days, similar arguments as in the case of van
Genuchten’s parameters hold. But comparatively, van Genuchten’s parameters
had a better significant recovery than Brooks-Corey parameters. This can be seen
63
in the two figures. A simple reason could be due to the choice of the parameter λ
being used. In this thesis, we choose λ = 2 which is commonly used. With regards
to MEOR modeling, various parameters ought to be experimented in order to find
a best fit for effective recovery. The total oil remaining in the reservoir with the
case of water flooding (C = 0kg/m3 ) and (C = 10−5 kg/m3 ) is 24.93% and 24.87%
respectively. As we increase the concentration of bacteria to (C = 10−3 kg/m3 ),
the remaining mass of oil left is 24.63%. Finally with a high concentration of
(C = 10−1 kg/m3 ), the remaining mass of oil left is 24.42%. Indeed with our
choice of λ, the Brooks-Corey’s parameters has performed poorly against the van
Genuchten’s parameters. See figure 6.7 which displays the fraction of oil left in
the reservoir for a highly concentrated MEOR flooding as against water flooding
using Brooks Corey’s parameters.
Water flooding
0.95
C = 10−1
C = 10−3
C = 10−5
0.9
Sw
0.85
0.8
0.75
0.7
0.65
0
20
40
60
80
100
x[m]
Figure 6.6: Saturation profile for Brooks Corey parameter with different concentrations
64
Total oil remaining
0.265
MEOR
Water flooding
0.26
0.255
0.25
0.245
0.24
0
1
2
3
Production time of 45 days
4
6
x 10
Figure 6.7: Fraction of oil remaining in the reservoire after 45 days using
Brooks Corey parameters
65
6.2.2
Viscosity reduction with bacteria concentration
The viscosity of oil can be decreased drastically in the presence of bacteria. However, no conclusive experiment has been made to investigate how oil viscosity
correlates with bacteria concentration Islam (1990). This thesis employs the correlation curve used in the paper being an analogy to solvent flood. We use an
interpolation formula to derive the relation with some data points taken from the
figure. Figure 6.8 displays the linear correlation curve between oil viscosity and
bacteria concentration used in this thesis.
12
muo[mN/m]
10
8
6
4
0
0.002
0.004 0.006
Cm
0.008
0.01
Figure 6.8: Viscosity correlation with bacteria concentration
In this simulation, an oil viscosity of 10 mPa.s was used together with the same
reservoir conditions and parameters. Figure 6.10 compares recovery results of
viscosity-reducing bacteria with that of water flooding after 45 days of simulation.
66
1
water flooding
c = 10−1
c = 10−4
−3
c = 10
c = 10−5
c = 10−2
0.95
0.9
0.85
S
w
0.8
0.75
0.7
0.65
0.6
0.55
0.5
0
20
40
60
80
100
x[m]
Figure 6.9: Saturation profile for van Genuchten parameter with focus on
viscosity reduction
Total oil remaining
0.258
MEOR
Water flooding
0.256
0.254
0.252
0.25
0.248
0
1
2
3
Production time of 45 days
4
6
x 10
Figure 6.10: Fraction of oil remaining in the reservoire after 45 days with
focus on viscosity reduction
The figure show that oil recovery declines in a similar way as water flooding. The
only difference is the delay in the case of MEOR. We note that as the concentration of bacteria increases from (C = 10−5 kg/m3 ) to (C = 10−3 kg/m3 ), the
recoveries are just similar to that of water flooding. The amount of oil recovered
67
with this range of bacteria solution are between 24.98% and 24.81%. Concentrations of (C = 10−2 kg/m3 ) and (10−1 kg/m3 ) gave quite a significant recovery of
24.69% and 24.48%. This indicates a sharp recovery as can be seen in the figure
above. Although the fact still remains that bacteria have a positive effect on the
residual oil in both mechanisms investigated, the percentage recovery from viscosity reduction bacteria are much less than bacteria generated surfactant flooding.
Comparing concentration of (C = 10−1 kg/m3 ) for both cases, bacteria generated
surfactant flooding gave a final recovery of 20.79% as against 24.48% for the case
of viscosity reduction bacteria. Figure 6.10 displays the fraction of oil left in the
reservoir for a highly concentrated MEOR flooding as against water flooding with
focus on viscosity reduction. Inter-facial tension reduction has always proven to
be the best way to recover the residual oil. However, no candid conclusion can be
made from this mechanism. Two factors of uncertainty are involved here. The first
is that the nature of inter-facial tension reduction of oil viscosity are still not clear
since this field lacks data. The second factor is that the presence of moderately
viscous oil limits the benefit of oil viscosity reduction (Islam 1990).
6.3
Sensitivity analysis
We perform a sensitivity analysis using the van Genuchten’s parameter n to see
its effect on the sweep efficiency. With this simulation, we considered only the
case of water flooding and a total production time of 100 days. Figure 6.11 and
6.12 displays the water profile curves and the fraction of oil remaining after 100
days. For n = 2.5 the total oil left is 15.6% whilst that of n = 2 gave 12.48%.
With n as low as 1.5, a greater fraction of the oil has been recovered leaving just
6.28%. This reveals that as the parameter n decreases, the recovery increases.
Similar arguments also hold for the Brooks Corey parameter λ in which as λ
decreases, the recovery increases. The fact of the matter is that, these parameters
are properties of the reservoir and cannot be altered. But a good knowledge of
them will help to know which mechanism to employ for MEOR technology to be
efficient.
68
1
Sw
0.9
0.8
0.7
0
n =1.5
n =2.5
n =2
20
40
60
80
100
x[m]
Figure 6.11: Water saturation profile for the sensitivity analysis of van
Genuchten’s parameter n
Total oil remaining
0.3
n=1.5
n=2
n=2.5
0.25
0.2
0.15
0.1
0.05
0
2
4
6
8
10
Total production time of 100 days 6
x 10
Figure 6.12: Fraction of oil remaining in the reservoir after 100 days with
different van Genuchten’s parameters
69
Reservoir model parameters and fluid properties
Parameter
Values
K
2 mD
φ
0.3
n
2
λ
2
L
100 m
T
45 days
Pe
2 MP a
ρw
1000 kg/m3
ρo
800 kg/m3
µw
0.869 x 10−3 P a s
µo
9 x 10−4 P a s
µo (base)
10 x 10−3 P a s
σbase
29 mN/m
D
1.92e − 5 m2 /s
Swi
0.7
Sor
0.3
Pwi
2 MP a
Pe
2 MP a
Cbi
0 kg/m3
Table 6.1: General parameters
70
Chapter 7
Summary and conclusion
This thesis presented a mathematical model for MEOR activities consisting of a
two-phase flow and a transport equation. The model was discretized using the
finite volume method and the IMPES approach. We validated our model by comparing it with the Couplex-Gas benchmark and with an analytical solution of
which both proved the validity of the model. We assumed that the bacteria are
affecting the inter-facial tension and the viscosity of the oil. Although several
mechanisms exists for modeling MEOR activities, we investigated two of them.
Indeed our valid model has been able to showcase numerically that the activities of microbes does help in the recovery of the residual oil. This was seen in both
mechanisms investigated. We first saw the case where inter-facial force reducing
bacteria were introduced with various concentrations and their corresponding effect. The recovery curve in figure 6.5 of page 63 clearly shows that MEOR flooded
surfactants had much oil recovered than conventional water flooding. We also saw
in the second mechanism that bacteria affect the viscosity of the oil in a beneficial
way. This was also demonstrated in figure 6.10, page 67 where viscosity reducing
bacteria were seen to affect the residual oil. Comparatively, a reduction in the
inter-facial force proves to be the best way to recover the residual oil as against
viscosity reduction.
In this thesis, we have shown that the parameters of the relative permeability
curves are very sensitive to the recovery. This was demonstrated in the case for
the van Genuchten’s parameter n where we discovered that the as n decreases, the
71
recovery increases. See figure 6.12, page 69. As reported by the paper Nielsen et
al [28], it is indeed true that several parameters must be experimented so as to
find a best fit with experimental data. Every reservoir has its own physio-chemical
characteristics which ought to be well studied. This of course will provide a good
clue on which species of bacteria to employ and which kind of mechanism to be
used. If this is not factored, the notion of using MEOR, though will produce result
but may not be optimum. In Nielsen et al (2003), they investigated a less efficient
bacteria as against an efficient one in which the latter gave a recovery of 44% as
against 9% by the former. Although comparatively, it is much lower, a 9% oil
recovery is very significant.
It must be noted here that this thesis only provided a simple advection-diffusion
equation which takes care of flow of bacteria in the water phase. The performance
of MEOR would have been far better if growth of bacteria and substrate distribution has been factored. On the whole, MEOR is efficient and if well implemented
can help recover the residual oil without causing much harm to the environment
and will cut down high cost of oil production. Statoil has proven this fact and
they are the only oil company so far in the world, using MEOR technology.
Outlook
• We hope to extend our advection-diffusion equation to a more general one
which will address some challenges such as pore clogging, adhesion and adsorption etc facing MEOR transport.
• We look forward to consider other cases such as growth rates, substrate
distribution coefficient where we can see the effects of bacteria leaving in
both phases.
• To investigate all other possible mechanisms by which bacteria actions can
be incorporated into the model in order to enhance the sweep efficiency.
• To come out with a new mechanism of modeling bacteria activities for the
enhancement of the sweep efficiency.
• To consider higher dimensional modeling of MEOR activities.
72
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