Modeling a well stimulation process using the meor technique

IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308
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Volume: 03 Issue: 03 | Mar-2014, Available @ http://www.ijret.org 153
MODELING A WELL STIMULATION PROCESS USING THE MEOR
TECHNIQUE
Nmegbu C.G.J.1
, Pepple D.D2
1, 2
Department of Petroleum Engineering, Rivers State University of Science and Technology, Rivers State, Nigeria
Abstract
Microbial enhanced oil recovery remains the most environmental friendly, cost effective recovery technique in oil production,
particularly for wellbore stimulation. This research investigates the effects of microbial growth rate, microbial and nutrient
concentrations for well stimulation purposes. A representative model incorporating microbial concentration, its growth rate and skin
factor is developed, validated and discussed. An explicit formulation which poses a solution to the equation for the model is used to
describe the reservoir pressure responses. It is observed through plots of reservoir pressure against reference distances that flow and
production rates improved as a result of an improved BHP when the microbial parameters were incorporated to the fluid transport
equation at same injection rates and same reservoir parameters. The trend followed by the pressure profile plots correlates with that
expected of a well stimulation pressure profile.
Keywords: Well Stimulation, MEOR, Permeability, MEOR stimulation
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1. INTRODUCTION
The current need for maximizing oil recovery from the
reservoir has prompted the evaluation of improved oil
recovery methods and various EOR techniques. Microbial
EOR is an of aspect biotechnology, utilizing the potentials of
microbes to significantly influence oil flow and its recovery.
However, a sound and reliable engineering technique in
optimizing microbial formulations are required to maximize
these potentials.
The use of microbes for hydrocarbon recovery has been
credible, and loss of crude during the process can be
considered insignificant compared to the amount of increased
recovery. Pressure, salinity, pore structure and mainly
temperature often limit the functionality of microbes during
any MEOR application.
The patented process described by ZoBell showed that
bacterial products such as gases, acids, solvents, surfactants
and cell biomass released oil from sand packed columns in a
laboratory test [1]. Subsequent studies have shown that [2];
1. Viable bacteria and various nutrients required for
growth can be transported through cores.
2. Insitu growth of bacteria results in significant
reduction in formation permeability.
3. Permeability reduction is selective for high
permeability cores and improves sweep even under
conditions where cross flow of fluids between regions
occur.
Taylor et al conducted a theoretical and experimental
investigation to effectively quantify reduction in permeability
as a result of enhanced microbial growth in a porous media
[3]. They observed that enhanced biological activities in sand
column reactors can significantly reduce permeability. An
analytical relationship was then established between the
biofilm thickness and resulting permeability reduction.
A one-dimensional, two-phase, compositional numerical
simulator to model the transport and growth of bacteria and oil
recovery in MEOR process was developed by Sarkar et al. [4].
In their model, permeability reduction was modeled using the
effective medium theory an implicit-pressure, explicit-
concentration algorithm was used to solve pressure and mass
concentration equations.
Islam presented a mathematical formulation to describe and
explain microbial transport in a multiphase multi directional
flow through a porous media [5]. In his formulation, physical
dispersion terms were neglected in the component transport
equation, since metabolic product actions were not included in
the model, considerations which relate biomass to metabolic
and their activities were defined.
Nielson et al used a correlation between IFT and surfactant
concentration. Usually, a reduction in IFT causes a decrease in
residual oil saturations, therefore affecting the permeability
curve end points [2]. They investigated the following methods
[2, 6, 7];
1. Capillary number and normalized residual oil
saturation correlations.
2. Coats interpolated between 𝐾𝑟 and the interpolation
of factors of core types relative to permeability
curves.

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They recommended the latter, in which more parameters can
be estimated in order to obtain a better fit with experimental
data.
Knapp et al also developed a 1 –Dimensional mathematical
model to effectively describe the microbial plugging process
[8]. The impact of cellular growth and microbial retention on
temporal reduction in permeability of porous media were the
main objectives investigated by this model. They assumed the
development of stationary phase is solely due to the biomass
retention therefore convective transport is the dominant
mechanism for microbial mobilization. Their governing
equation included a convection dispersion equation for
bacteria and nutrient transport, and a mass conversion
equation for stationary phase development.
Zhang et al presented a three-phase, multiple species, one-
dimensional mathematical model to simulate biomass growth,
bioproduct formation, and substrate consumption during in-
situ microbial growth, and to predict permeability reduction as
a result of in-situ growth and metabolism in porous media [9].
All the model parameters considered by respective authors are
ideal for successful MEOR implementation. They are
considered the most relevant as a result of multiple microbial
oil recovery studies and mainly include microbial transport,
microbial concentration, Interfacial Tension (IFT) reduction
parameters, microbial kinetics, mobility control, viscosity
reduction etc. This study basically aims at examining
microbial formulations that can be applied as well stimulation
alternatives and permeability alteration agents.
2. MATERIALS AND METHOD
The fundamental theories of Fluid flow and Monod growth
kinetics would serve as a basis for modeling well stimulation
processes by MEOR application as a result of permeability
alteration caused by metabolite production by the choice
microbe (clostridium sp).
Fig -1: Effects of well bore vicinity damage on the pressure
profile and BHP levels [10]
Region A is the damaged region, while region B is the
undamaged region.
2.1 Choice of Microbe
Having the ability to effectively withstand reservoir with the
most challenging conditions, particularly temperature and
salinity, Clostridium sporonges proves the best stimulation
microbe, prior to its metabolite production (butanol-
CH3(CH2)3OH and acetones(2𝐶𝐻3 𝐶𝑂𝑂𝐻) that alters the
absolute permeability of reservoir rock after reaction to
produce calcium acetate, carbon dioxide and water.
Being a thermophile, with a temperature tolerance range of
about 50 – 700
C (122 – 1580
F), this microbe can thrive in
relatively high reservoir temperature condition, averaging
about 600
C (1400
F)
Pores must be twice the diameter of the microbe for effective
transportation to occur. Ideally, clostridium sp records about
4.0𝜇𝑚 length and 0.6𝜇𝑚 thick. This proves convenient
enough to be transported in a carbonate pore throat averaging
1.16𝜇𝑚 minimum. An optimum pH for microbial existence
and transport in the porous media lies between 4.0 – 9.0, and
clostridium sp lies between this limit (4.5 - 4.7).
3. MATHEMATICAL MODELING
3.1 Microbial Growth Rate
The growth rate expression applied for bacteria are often the
Monod expression based on the Michelis-Menten enzyme
kinetics [2, 5, 11]. The Monod expression with one limiting
substrate is widely used, but it is empirical in the context of
microbial growth.
The Monod growth rate for one limiting substrate without any
inhibition will be used in this work:
𝐺 = 𝐺 𝑚𝑎𝑥
𝐶 𝑛
𝑘 𝑠+ 𝐶 𝑛
(1)
Where
Gmax is the maximum growth rate obtained in excess nutrient
(hr-1
)
ks is the substrate concentration to half Gmax (mg/l)
Cn is the nutrient concentration (mg/l)
3.2 Fluid Transport Equation
The most general form of a single phase fluid flow equation in
a porous media is presented below in equation (2), making no
assumptions regarding fluid type or pressure dependency on
rock and fluid properties [12];

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𝜕
𝜕𝑥
𝑈𝑥
1
𝐵
∆𝑥 +
𝜕
𝜕𝑦
𝑈𝑦
1
𝐵
∆𝑦 +
𝜕
𝜕𝑧
𝑈𝑧
1
𝐵
∆𝑧 + 𝑞 =
𝑣 𝑏 𝜙𝑆 𝑖 𝐶𝑡
𝛼 𝑐 𝐵 𝑜
𝜕𝑃
𝜕𝑡
𝜕
𝜕𝑥
𝑈𝑥
1
𝐵
∆𝑥 +
𝜕
𝜕𝑦
𝑈𝑦
1
𝐵
∆𝑦 +
𝜕
𝜕𝑧
𝑈𝑧
1
𝐵
∆𝑧 + 𝑞 =
𝑣 𝑏 𝜙𝑆 𝑖 𝐶𝑡
𝛼 𝑐 𝐵 𝑜
𝜕𝑃
𝜕𝑡
(2)
3.3 Assumptions for Model Development
1. Fluid flow is one-dimensional single phase, and takes
place in a uniform porous medium.
2. Metabolite production mostly bioacids [2]
3. Isothermal system as reservoir fluctuations in
temperature is regarded minimal [4]
4. A change in temperature will alter the individual
values of 𝐶𝑡, 𝜇 𝑜, 𝐵 𝑎𝑛𝑑 𝑃
5. No break in injection rates of nutrient and bacteria
during the process
6. Microbial decay not considered.
7. No indigenous bacteria present.
8. Flow in the reservoir is in the direction of the
wellbore.
9. Chemotaxis (movement of microbes towards an
increasing concentration of substrate) not considered.
10. No substrate and metabolite adsorption on the pore
walls, so Langmuir equilibrium isotherm not
considered.
11. Flow is laminar (reservoir contains only oil).
12. Unsteady state flow conditions.
13. Other factors affecting growth rates such as salinity
and pH remain constant.
With these assumptions imposed on equation (2) the flow of
fluid in the reservoir was
Representing a 1-dimensional, single phase flow system as:
𝜕
𝜕𝑥
𝑈𝑥
1
𝐵
∆𝑥 + 𝑞 =
𝑣 𝑏 𝜙𝐶𝑡
𝛼 𝑐 𝐵 𝑜
𝜕𝑃
𝜕𝑡
(3)
From Darcy’s law for a 1-dimensional flow system:
𝑈𝑥 = − 𝛽𝑐
𝐴 𝑥 𝑘 𝑥
𝜇 𝑜
𝑑𝑃
𝑑𝑥
(4)
Substituting equation (4) into (3), we have:
𝜕
𝜕𝑥
− 𝛽𝑐
𝐴 𝑥 𝑘 𝑥
𝜇 𝑜
𝑑𝑃
𝑑𝑥
1
𝐵
∆𝑥 + 𝑞 =
𝛼 𝑐 𝐵 𝑜
𝜕𝑃
𝜕𝑡
(5)
Accounting for microbial concentration[2]:
𝜕
𝜕𝑥
−𝛽𝑐
𝐴 𝑥 𝑘 𝑥 𝐶 𝑏
𝜇 𝑜
𝑑𝑃
𝑑𝑥
1
𝐵
∆𝑥 + 𝑞𝐶𝑏 =
𝑣 𝑏 𝜙𝐶𝑡 𝐶 𝑏
𝛼 𝑐 𝐵 𝑜
𝜕𝑃
𝜕𝑡
(6)
Equation (6) is known as the component transport equation for
microbes.
For slightly compressible fluids such as oil,
Formation volume factor, B =
𝐵 𝑜
1+𝑐 𝑃−𝑃𝑜
For initial boundary conditions,
P = Po, therefore B = Bo.
Neglecting the negative sign on the LHS of equation (6), the
equation is reduced to:
𝜕
𝜕𝑥
𝛽𝑐
𝜇 𝑜 𝐵
𝑑𝑃
𝑑𝑥
∆𝑥 + 𝑞𝐶𝑏 =
𝛼 𝑐 𝐵
𝜕𝑃
𝜕𝑡
(7)
Incorporating the Monod equation to account for microbial
growth rate, we have;
𝜕
𝜕𝑥
𝛽𝑐
𝜇 𝑜 𝐵
𝑑𝑃
𝑑𝑥
∆𝑥 + 𝑞𝐶𝑏 + 𝐺 =
𝛼 𝑐 𝐵
𝜕𝑃
𝜕𝑡
(8)
For stimulation, skin factor must be considered (showing the
relativity of permeability and radii of investigation ), and is
represented as thus [10];
𝑆 =
𝑘
𝑘 𝑠𝑘𝑖𝑛
− 1 𝑙𝑛
𝑟 𝑠𝑘𝑖𝑛
𝑟 𝑤
(9)
Incorporating skin factor into equation (8), we have
∂
∂x
βc
Ax kx Cb
μo B
dP
dx
∆x + qCb + G =
vb ϕCtCb S
αc B
∂P
∂t
(10)
Equation (10) can be used to predict pressure in the reservoir
after microbial injection.
4. SOLUTION TO MATHEMATICAL
FORMULATION
Rewriting equation (10) as a second order derivative, we have:
𝜕2 𝑃
𝜕𝑥2 𝛽𝑐
𝜇 𝑜 𝐵
∆𝑥 + 𝑞𝐶𝑏 + 𝐺 =
𝑣 𝑏 𝜙𝐶𝑡 𝐶 𝑏 𝑆
𝛼 𝑐 𝐵
𝜕𝑃
𝜕𝑡
(11)
Applying central difference approximation in space (x) and
forward difference approximation in time (t), we have [12];
Fig -2: Discrete points representation (grid positions)
𝜕2 𝑃
𝜕𝑥2 =
𝑃𝑖+1
𝑛
−2𝑃𝑖
𝑛
+𝑃𝑖−1
𝑛
∆𝑥2 and
I +1ii - 1
∆𝑥

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𝜕𝑃
𝜕𝑡
=
𝑃𝑖
𝑛+1
−𝑃𝑖
𝑛
∆𝑡
Where ‘i’ is position and ‘n’ is the time step. Applying the
approximations to Equation (11) gives:
𝑃𝑖+1
𝑛
−2𝑃𝑖
𝑛
+𝑃𝑖−1
𝑛
∆𝑥2 𝛽𝑐
𝜇 𝑜 𝐵
∆𝑥 + 𝑞𝐶𝑏 + 𝐺 =
𝛼 𝑐 𝐵
𝑃𝑖
𝑛+1
−𝑃𝑖
𝑛
∆𝑡
(12)
Rearranging equation (12) we have:
𝛽𝑐
𝜇 𝑜 𝐵
∆𝑥
𝑃𝑖+1
𝑛
∆𝑥2 − 2 𝛽𝑐
𝜇 𝑜 𝐵
∆𝑥
𝑃𝑖
𝑛
∆𝑥2 +
𝛽𝑐
𝜇 𝑜 𝐵
∆𝑥
𝑃𝑖−1
𝑛
∆𝑥2 + 𝑞𝐶𝑏 + 𝐺 =
𝛼 𝑐 𝐵
𝑃𝑖
𝑛+1
−𝑃𝑖
𝑛
∆𝑡
(13)
The above can now be written as:
𝛽𝑐
∆𝑥𝜇 𝑜 𝐵
𝑃𝑖+1
𝑛
− 2 𝛽𝑐
𝑃𝑖
𝑛
+
𝛽𝑐
𝑃𝑖−1
𝑛
+ 𝑞𝐶𝑏 + 𝐺 =
𝛼 𝑐 𝐵
𝑃𝑖
𝑛+1
−𝑃𝑖
𝑛
∆𝑡
(14)
Taking 𝛽𝑐
to be M, rewriting equation (14) gives:
𝑀𝑃𝑖+1
𝑛
− 2𝑀𝑃𝑖
𝑛
+ 𝑀𝑃𝑖−1
𝑛
+ 𝑞𝐶𝑏 + 𝐺 =
𝛼 𝑐 𝐵
𝑃𝑖
𝑛+1
− 𝑃𝑖
𝑛
(15)
For initial boundary conditions, all pressure values at any
position ‘i’ at present time step ‘n’ are the same, so the values
of 𝑃𝑖
𝑛
𝑎𝑛𝑑 𝑃𝑖−1
𝑛
are all equal and known. The only
unknown is the pressure value at position ‘i’ at a new time
step n+1.
In order to make 𝑃𝑖
𝑛+1
the subject, we first multiply through
equation (15) by the inverse of
𝛼 𝑐 𝐵∆𝑡
, we have;
𝑀𝑃𝑖+1
𝑛
− 2𝑀𝑃𝑖
𝑛
+ 𝑀𝑃𝑖−1
𝑛
+
𝑞𝐶𝑏 + 𝐺 = 𝑃𝑖
𝑛+1
− 𝑃𝑖
𝑛
(16)
Let
𝑣 𝑏 𝜙 𝐶𝑡 𝐶 𝑏 𝑆
= C, we can write;
𝐶 𝑀𝑃𝑖+1
𝑛
− 2𝑀𝑃𝑖
𝑛
+ 𝑀𝑃𝑖−1
𝑛
+ 𝐶 𝑞𝐶𝑏 + 𝐺 = 𝑃𝑖
𝑛+1
−
𝑃𝑖
𝑛
(17)
Equation (17) can now be written as thus;
𝑃𝑖
𝑛+1
= 𝑃𝑖
𝑛
+ 𝐶 𝑀𝑃𝑖+1
𝑛
− 2𝑀𝑃𝑖
𝑛
+ 𝑀𝑃𝑖−1
𝑛
+
𝐶 𝑞𝐶𝑏 + 𝐺 (18)
5. MODEL VALIDATION
Fig -3: Discretization of reservoir showing
dimensions,production and injection points
Table -1: Field parameters
Parameters Values
Depleted reservoir pressure 1500 psi
Permeability of damaged
zone( 𝐾𝑠𝑘𝑖𝑛 )
30 ft
Total compressibility, 𝐶𝑡
10*10-6
psi-1
Transmissibility coefficient 1.127
Formation porosity 20%
Wellbore radius, 𝑟𝑤
0.25 ft
Damaged radius, 𝑟𝑠𝑘𝑖𝑛
2ft
Formation permeability, k 160md
Volume conversion factor, 𝛼 𝑐
5.615
∆𝑥 1000 ft
∆𝑦 50 ft
∆𝑧 200 ft
Oil formation volume factor,
𝐵𝑜
1.00 rb/stb
Oil viscosity, 𝜇 𝑜
10 cp
Permeability of damaged zone,
𝐾𝑠𝑘𝑖𝑛
30md
∆𝑡 30days

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Table -2: Nutrient and microbial parameters [11]
Parameters Values
Max microbial growth
rate, 𝐺 𝑚𝑎𝑥
0.343hr-1
The substrate concentration at
half Gmax, ks
12.8 (𝑚𝑔/𝑙)
Nutrient concentration, 𝐶𝑛 45 𝑚𝑔/𝑙
Microbial concentration, 𝐶𝑏 3.5 × 10−4 𝑐𝑒𝑙𝑙𝑠
𝑚𝑙
=
10𝑐𝑒𝑙𝑙𝑠/𝑓𝑡3
5.1 Calculation of Constants
Skin factor, S =
𝐾
𝐾 𝑠𝑘𝑖𝑛
− 1 𝑙𝑛
𝑟 𝑠𝑘𝑖𝑛
𝑟 𝑤
Microbial growth rate,𝐺 = 𝐺 𝑚𝑎𝑥
𝐶 𝑛
𝑘 𝑠+ 𝐶 𝑛
Constants, 𝐶 =
𝛼 𝑐 𝐵 𝑜∆𝑡
𝑣 𝑏 𝜙 𝐶𝑡 𝐶 𝑏 𝑆
Transmissibility term, 𝑀 =
𝛽 𝐶 𝐴 𝑥
𝐾 𝑎𝑣𝑔
1000 𝑥
𝐶 𝑏
𝜇 𝐵 𝑜∆𝑥
We have;
S=9.01
𝐺 =6.4 day-1
C = 0.093
M= 1.071
5.2 Calculation of Pressure Responses at Different
Grid Blocks
For time step 1, ∆𝑡 = 30𝑑𝑎𝑦𝑠
Setting initial boundary conditions, Pi=1500psi
Recalling,
𝑃𝑖
𝑛+1
= 𝑃𝑖
𝑛
+ 𝐶 𝑀𝑃𝑖+1
𝑛
− 2𝑀𝑃𝑖
𝑛
+ 𝑀𝑃𝑖−1
𝑛
+ 𝐶 𝑞𝐶𝑏 + 𝐺
For grid block 1, i=1
𝑃1
𝑛+1
= 1500 + 0.093 1.071(1500) − (2 × 1.071)(1500)
+ 1.071(1500) + 0.093 40 10 + 6.4
𝑃1
𝑛+1
= 1537.79𝑝𝑠𝑖
𝑃2
𝑛+1
= 1500 + 0.093 1.071(1500) − (2 × 1.071)(1500)
+ 1.071(1500)
+ 0.093 −150 10 + 6.4
𝑃2
𝑛+1
= 1360.39𝑝𝑠𝑖
𝑃3
𝑛+1
= 1500 + 0.093 1.071(1500) − (2 × 1.071)(1500)
+ 1.071(1500) + 0.093 0 10 + 6.4
𝑃3
𝑛+1
= 1500.60𝑝𝑠𝑖
𝑃4
𝑛+1
= 1500 + 0.093 1.071(1500) − (2 × 1.071)(1500)
+ 1.071(1500) + 0.093 0 10 + 6.4
𝑃4
𝑛+1
= 1500.60𝑝𝑠𝑖
𝑃5
𝑛+1
= 1500 + 0.093 1.071(1500) − (2 × 1.071)(1500)
+ 1.071(1500) + 0.093 0 10 + 6.4
𝑃5
𝑛+1
= 1500.60𝑝𝑠𝑖
𝑃6
𝑛+1
= 1500 + 0.093 1.071(1500) − (2 × 1.071)(1500)
+ 1.071(1500) + 0.093 100 10 + 6.4
𝑃6
𝑛+1
= 1594.07
6. RESULTS AND DISCUSSION
Fig -4: Pressure profile with MEOR model
The plot above shows a BHP of 1360.4psi at the production
point at grid block 2
6.1 Comparison with Fluid Flow Equation Excluding
Microbial Parameters and Skin Factor
General fluid flow equation is given as:
𝜕
𝜕𝑥
𝛽𝑐
𝐴 𝑥 𝑘 𝑥
𝜇 𝑜 𝐵 𝑜
𝑑𝑃
𝑑𝑥
∆𝑥 + 𝑞 =
𝛼 𝑐 𝐵 𝑜
𝜕𝑃
𝜕𝑡
(6)
1300
1350
1400
1450
1500
1550
1600
1650
0 2 4 6 8
pressure,psi
Grid point

__________________________________________________________________________________________
Applying finite difference approximation to the above, we
have:
𝑃𝑖
𝑛+1
= 𝑃𝑖
𝑛
+
𝑞 +
𝑀 𝑥 𝑖+1
2
𝑃𝑖+1
𝑛
−
𝑀𝑥𝑖+12+𝑀𝑥𝑖−12𝑃𝑖𝑛+𝑀𝑥𝑖−12𝑃𝑖−1𝑛 (7)
Where:
is the injection or production rate factor,
𝑀 𝑥 𝑖±1
2
fluid transmissibility term, 𝑀 =
𝛽 𝐶 𝐴 𝑥 𝐾 𝑥
𝜇 𝐵 𝑜∆𝑥
Solving for constants,
C=8.4225
𝑀 =0.107
Fig -5: Pressure profile (with water) without MEOR
Field result Model prediction Deviation
230psi 240 psi 10 psi
It is shown from figure 5 that a BHP of 240psi exists at the
production point in grid block 2, with same injection and
production rates, field parameters etc. used in the MEOR
model. it is observed that there is a significant increase of
pressure at the boundaries and injection points, but minimal
pressure response at the production point. This implies that
there is a resistance to fluid flow around the wellbore region,
possibly skin effect which is responsible for the existence of a
low pressure response at the BHP at the production point
Fig -6: Pressure profiles comparison
7. CONCLUSIONS
The BHP levels in 1, 2 and 3 in fig 1.1 represents bottom hole
pressure level at improved, ideal and reduced permeability
respectively. The comparison of both models as shown in fig
(1.6) correlates with the pressure profile for an improved
wellbore vicinity as shown in fig (1.1). It is observed that the
BHP records about 240psi before the Meor formulation. A
bottom hole pressure of 1360.39psi is established after MEOR
treatment, this pressure increase of about 1120.39psi implies
that there is an improved oil flow towards the wellbore, prior
to the improvement in permeability and damage reduction
around the wellbore vicinity. It is recommended that an
optimum microbial concentration must be investigated so as to
ascertain a concentration limit to prevent microbial plugging
or clogging.
ACKNOWLEDGEMENTS
The authors would love to appreciate the department of
petroleum engineering RSUST for her relentless guidance and
support in course of this research. The contributions of Dr.
Jackson Akpa and Ohazuruike Lotanna, amongst others, are
highly recognized.
REFERENCES
[1]. Lazar, I. and Constantinescu, P. (1997): Field trials results
of MEOR. Microbes and Oil Recovery, 1: pp. 122-131.
[2]. Nielson, S.M., Shapiro, A.A., Michelsen, M.L. and
Stenby, E.H. (2010). 1D simulations for microbial enhanced
oil recovery with metabolite partitioning. Transp. Porous
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[3]. Taylor SW, Milly PCD, Jaffe PR. (1990). Biofilm growth
and the related changes in the physical-properties of a porous-
medium. 2. Permeability. Water Resour Res 26:pp. 2161–
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0
500
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2500
0 2 4 6 8
pressure,psi
0
500
1000
1500
2000
2500
0 2 4 6 8
pressure,psi
model
normal
Grid point
Grid point

__________________________________________________________________________________________
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[10]. Ahmed, T (2010) Reservoir Engineering Handbook , 4th
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Applied Reservoir Simulation .Henry Doherty memorial fund
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Texas.

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