The Capillary number hypothesis is very empirical in Surfactant flooding Enhanced Oil Recovery (EOR) method. which is of modest experience on the North Sea and many other offshore platforms. The capillary number drives the force wetting processes, which is controlled by the balance between capillary and viscous forces. The mobilization of oil trapped in pores of water-wet rock is steered by capillary number that is typically within specific ranges (〖10〗^(-5) to〖 10〗^(-4)). There is high uncertainty and confusion in the parameterization of capillary number formula, as every quantity is given on a macroscale level. As demonstrated herein, a new microscopic capillary number parameterization was proposed. This paper is written to improve the numerical formulation of capillary number in surfactant flooding model. The new formula for capillary number was derived based on existing equations as a function of residual oil saturation and tested. Thus, the proposed mobility mechanism easily accounts for a broader critical range of capillary number (〖10〗^(-6) to〖 10〗^(-4)) in comparison with available models with a critical capillary number (〖10〗^(-5) to〖 10〗^(-4)). We used an existing model to quantify the effect of capillary number on a miscible and immiscible relative permeability curves by computing the interpolation parameter F_kr as a tabulated function of the Logarithm (base 10) of the capillary number using a new capillary number formulation.

1. The New Capillary Number Parameterization for Simulation in Surfactant Flooding
JOGCE
The New Capillary Number Parameterization for Simulation in
Surfactant Flooding
Mustapha Abdullahi
School of Chemical Engineering and Analytical Science, University of Manchester, Manchester M13 9PL, UK
Email: mustapha.abdullahi@manchester.ac.uk, Tel: +447388119656
The Capillary number hypothesis is very empirical in Surfactant flooding Enhanced Oil Recovery
(EOR) method. which is of modest experience on the North Sea and many other offshore
platforms. The capillary number drives the force wetting processes, which is controlled by the
balance between capillary and viscous forces. The mobilization of oil trapped in pores of water-
wet rock is steered by capillary number that is typically within specific ranges (𝟏𝟎−𝟓
to 𝟏𝟎−𝟒
).
There is high uncertainty and confusion in the parameterization of capillary number formula, as
every quantity is given on a macroscale level. As demonstrated herein, a new microscopic
capillary number parameterization was proposed. This paper is written to improve the numerical
formulation of capillary number in surfactant flooding model. The new formula for capillary
number was derived based on existing equations as a function of residual oil saturation and
tested. Thus, the proposed mobility mechanism easily accounts for a broader critical range of
capillary number (𝟏𝟎−𝟔
to 𝟏𝟎−𝟒
) in comparison with available models with a critical capillary
number (𝟏𝟎−𝟓
to 𝟏𝟎−𝟒
). We used an existing model to quantify the effect of capillary number on
a miscible and immiscible relative permeability curves by computing the interpolation parameter
𝑭 𝒌𝒓 as a tabulated function of the Logarithm (base 10) of the capillary number using a new
capillary number formulation.
Keywords: Critical capillary number; Capillary number; capillary desaturation; miscibility; Interpolation parameter.
INTRODUCTION
Up to two-thirds of the crude oil remains trapped in the
reservoirs after primary and secondary recovery in an
average oil reservoir (Yan and Qiu, 2016). EOR is then
required to optimise the depletion, as the remaining oil is
trapped in the pore structure of the reservoir.
The fluid flow in porous media is mainly governed by three
forces: gravitational force, viscous force and capillary
forces. These three forces are integrated through the
Capillary number (𝑁𝑐) and the bond number (𝐵𝑜) defined
by:
𝑁𝑐 =
𝑢𝜇
𝜎
(1)
𝐵𝑜 =
∆𝜌𝑔𝑑2
𝜎 (2)
𝑢 is Darcy’s velocity, µ is the viscosity of the displacing
fluid, 𝜎 is the interfacial tension between oil and the
surfactant solution, ∆p is the density difference between
the phases, g is acceleration due to gravity and d is a
characteristic dimension taken here as the pore diameter.
The role of interfacial tension and capillary number in a
dynamic air-liquid interface during an immiscible fluid
displacement flow driven by pressure was investigated by
Yan & Qiu (2016). The critical capillary number was
studied analytically, and they derived the theory regarding
the critical capillary number with the wettability effect in a
pressure-driven capillary tube. Their results show that the
critical capillary number is associated with the contact
angle, slip length and capillary radius. Basante (2010)
conducted a laboratory experimental work using two
sandstone cores to demonstrate the effect of capillary
Journal of Oil, Gas and Coal Engineering
Vol. 3(1), pp. 023-033, August, 2018. © www.premierpublishers.org. ISSN: 0767-0974
Research Article

2. The New Capillary Number Parameterization for Simulation in Surfactant Flooding
Abdullahi M. 024
number on mobility control. The result of the experiment
shows that raising the flow rate by a factor of 10
(0.03ml/min to 0.3 ml/min) will decrease the oil recovery
factor considerably. The water-oil relative permeability
characterises two-phase flow, and it’s difficult to determine
the functional form in a reservoir study. The addition of
various chemical agents (For example, Surfactant) during
chemical flooding in oil production will significantly change
the interfacial tension and increase the degree of difficulty
in measurement. Shen et al. developed an improved
method of measuring water-oil relative permeability
curves, shown that the relative permeabilities of both water
and oil phases will increase with decreasing interfacial
tension (Shen et al., 2006).
A logarithm relationship exists between water-oil two-
phase relative permeability and interfacial tension (Shen et
al., 2006).
Base on several experiments, relative permeability is
considered as a function of saturation, interfacial tension,
and properties of core pore only (Shen et al., 2006).
𝐾𝑟𝑤 = 𝑚 𝑤(𝑆 𝑤
∗ ) 𝑛 𝑤
(3)
𝐾𝑟𝑜 = 𝑚 𝑜(1 − 𝑆 𝑤
∗ ) 𝑛 𝑜 (4)
where 𝑚 𝑤 and 𝑚 𝑜 are coefficients of water and oil relative
permeability functions respectively
Note that 𝑚 𝑤 is the water relative permeability at 𝑆 𝑤
∗
=
1 and 𝑚 𝑜 is the oil relative permeability at 𝑆 𝑤
∗
= 0,
respectively.
By definition above,
𝑚 𝑤 = (𝐾𝑟𝑤
∗ ) 𝑆 𝑜𝑟
(5)
𝑚 𝑜 = (𝐾𝑟𝑜
∗ ) 𝑆 𝑤𝑐
(6)
Therefore the normalise formula can be derived from the
two relative permeability equations above:
𝐾𝑟𝑤
∗
= (𝑆 𝑤
∗ ) 𝑛 𝑤
(7)
𝐾𝑟𝑜
∗
= (1 − 𝑆 𝑤
∗ ) 𝑛 𝑜 (8)
The exponential indexes, 𝑛 𝑤 𝑎𝑛𝑑 𝑛 𝑜 are found to be related
to interfacial tension and the pore size distribution
parameters, 𝜆 𝑤 and 𝜆 𝑜 (Brooks and Corey, 1964), as
𝑛 𝑜 = 𝑛 𝑜(𝜎 𝑤𝑜, 𝜆 𝑜) (9)
𝑛 𝑤 = 𝑛 𝑤(𝜎 𝑤𝑜, 𝜆 𝑤) (10)
A particular relation between exponential constants and
interfacial tension was derived, and the two-phase relative
permeability model from the relative permeability
equations above has the form(Shen et al., 2006):
𝐾𝑟𝑤
∗
= (𝑆 𝑤
∗ )[𝑜.9371.𝑙𝑜𝑔(𝜎 𝑤𝑜)+𝜆 𝑤] (11)
𝐾𝑟𝑜
∗
= (1 − 𝑆 𝑤
∗ )[0.1960.𝑙𝑜𝑔(𝜎 𝑤𝑜)+𝜆 𝑜] (12)
where 𝜆 𝑤 and 𝜆 𝑜are constants for water and oil relative
permeability, respectively, for the same rock type. With the
sandstone cores associated with the fluids in Shen’s
experiment, the curve fitting of the experimental data leads
to 𝜆 𝑜=2.006 and 𝜆 𝑤 =3.807 (Shen et al., 2006).
Xu et al. (2011) conducted experiments to determine the
surfactant’s performance, such as the relationship
between surfactant concentration and oil/water interfacial
tension and the relationship between the surfactant
concentration and the water viscosity. Their results show
that oil/water interfacial tension will decrease as the
surfactant concentration increase, surfactant flooding has
the capacity of enhancing oil recovery. Their results show
that the optimum surfactant concentration is 2%, which
can improve oil recovery by the percentage of 0.22. Ren
et al. (2018) used both numerical and analytical method to
characterise the migration, trapping and accumulation of
𝐶𝑂2 in a saline aquifer during geological sequestration.
They used a 1D two-phased-flow model and solved the
model equations using the method of characteristics. Their
results demonstrated that the 𝐶𝑂2accumulated by
permeability hindrance is greater than that accumulated by
capillary trapping.
Alquaimi et al. (2018) proposed a new capillary number
definition for fractures that depends on force balance and
incorporates geometrical characteristics of the fracture
model. They conducted an experimental desaturation
procedure to test their capillary number definition and
quantify the relationship between the pressure and
trapped ganglions. Bryan & Kantzas (2009) performed
core flooding experiments to investigate how Alkali-
Surfactant flooding can lead to improved heavy oil
recovery. It was determined from their results that the
performance of surfactant alone was not sufficient to
emulsify oil, but can only increase the water-wetting of the
glass, but the combination of alkali and surfactant can
reduce the oil-water interfacial tension and oil/water
emulsions will be produced. Furthermore, their results
show that the mechanism of emulsification and
entrainment, which occurs during high rate flow in lower
permeability cores is not as efficient in recovering
additional oil.
Laforce et al. (2008) used analytical solutions to study the
development of multi-contact miscibility in simultaneous
water and gas (SWAG) injection into a reservoir, they
considered the application of 𝐶𝑂2 storage in enhanced oil
recovery using a fully compositional one-dimensional and
three-phase flow through porous media. Their results
demonstrate that miscibility does not develop when the
fraction of water in the injection mixture is sufficiently high
and define the minimum gas fraction necessary to achieve
miscibility and highlights the importance of improved
relative permeability models.
Lohne et al. (2012) investigated the influence of capillary
forces on segregated flow behind the displacement front

3. The New Capillary Number Parameterization for Simulation in Surfactant Flooding
J. Oil, Gas Coal Engin. 025
by numerical simulations of homogeneous and
heterogeneous models. Their results show that the
positive effect of gravity segregation is that the oil floats
up, accumulates under low permeable rocks and thereby
increases the effective horizontal oil mobility. They found
the magnitude of the incremental oil production to increase
with increasing curvature of oil relative permeability.
Hence, the positive effect of decreasing IFT is larger in
mixed-wet formation than in water-wet formations.
Keshtkar et al. (2016) developed an explicit composition
and explicit-saturation method to study surfactant flooding
sensitivity analysis on an oil production reservoir. Their
results show that the addition of surfactant causes a
reduction in IFT between water and oil phases and
subsequently will trigger the mobility of the trapped oil and
increase the oil production level.
Felix et al. (2015) carried out various experiments to
implement surfactant polymer flooding. Different slugs
were injected after water flooding, and their results showed
different displacement efficiencies based on the
mechanism chosen for the implementation of the
surfactant polymer flooding. These experiments revealed
the importance of selecting the right tool for the surfactant
flooding as to optimise recovery. Xavier. (2011)
experimentally determined the influence of surfactant
concentration on hydrocarbon recovery. The interfacial
tension between brine and kerosene was studied with the
use of sodium dodecyl sulphate (SDS) as a means of
reducing the interfacial tension. His findings were that the
IFT decreases as the surfactant concentration increases
and reaches a point of critical micelle concentration
(CMC). Thus was able to find a critical surfactant
concentration at 0.3 wt% of the surfactant.
Cheng et al. (2005) presented results on developed
miscibility by gas injection in petroleum reservoir with the
aim to define a minimum miscibility pressure (MMP) for the
fluid system. He proposed a method for the optimum
number of grid cells and time step size for numerical
simulation models and verified 1D first-contact miscibility
displacement. Khanamiri et al. (2015) conducted
laboratory surfactant flooding experiments with aged
sandstone cores, surfactant sodium dodecylbenzene
sulfonate was used at a concentration of 0.05wt% and
0.2wt% to enhance oil recovery. The result shows the
effect of surfactant concentration on Interfacial tension i.e.
decrease in IFT with an increase in surfactant
concentration. They discovered that the low salinity
surfactant flooding with 0.2wt% surfactant concentration
did not result in higher oil recovery that the flooding with
0.05wt% surfactant concentration in tertiary low salinity
surfactant injection. This is because the tertiary low salinity
surfactant injection after secondary low salinity water
injection is more efficient than the tertiary surfactant
injection where the surfactant is injected after high salinity
(secondary) and low salinity water (tertiary) injection.
Whitson et al. (1997) proposed a mathematical formulation
for relative permeability and the interpolation function for a
Gas-oil system as a function of the capillary number:
𝐹𝑘𝑟 = 1 − [(𝛼𝑁𝑐) 𝑛
+ 1]−1
(13)
Where 𝑛 ≃ −0.75 seems to fit data they conducted by
laboratory measurements. The scaling parameter α is
used to fit the measured data.
Thus, by referring to the latest previously published
literature on the miscibility development in surfactant
flooding, it can be concluded that the capillary number and
the interfacial tension plays a vital role in accurately
predicting miscibility in Surfactant flooding to enhance oil
recovery. The previous literature has not critically
evaluated the mobility components as a function of the
miscibility level between surfactant, water, and oil, most of
the research and discussion conducted in the literature are
focused on miscibility in gas and oil system. Therefore is
need to come up with a numerical model that will
accurately give a clear picture of the transport components
as a function of miscibility and various formulations in
surfactant-water and oil system.
METHODOLOGY AND MODEL SETUP
Reservoir simulation is a useful tool for estimating the
future behaviour of petroleum fields. In some cases, it can
also be used for identifying particular phenomena in a
specific task. In this study, the investigation was performed
using numerical simulation experiments. New formulation
for the capillary number was derived. And Eclipse Black
Oil model with surfactant option was used for simulating
the displacement process to see the effect of off surfactant
and to test the transport mechanism which was derived as
shown in Figure 1. Another simulation was conducted to
validate the implementation of surfactant flooding using
Matlab Reservoir Simulation Tool (MRST).
Figure 1: Simulation process

4. The New Capillary Number Parameterization for Simulation in Surfactant Flooding
Abdullahi M. 026
1. Proposed new capillary number equation
Capillary pressure relationship is expressed by a single
dimensionless function known as the Leverett Number
(Leverett, 1941):
𝑁𝐿𝑒 =
𝑃𝑐
𝜎𝑇(𝜃)
√
𝐾
𝜙 (14)
by defining 𝑇(𝜃) as:
𝑇(𝜃) =
𝑃𝑐(𝑆)𝑟𝑛
2𝜎
(15)
In a cylindrical oil mass of average radius a, the average
length as proposed by (Stegemeier, 1974) is:
𝐿 =
2𝜎
𝑎𝛻𝜙
(16)
In the case of wetting phase trapping, residual fluid is held
in rings, interconnected with only thin water layers as
described by Basante (2010). The geometrical factor for
the contact angle interface passing through the pore is:
𝑇(𝜃) =
𝑐𝑜𝑠(𝜃 − 𝜂)
1 + (𝑟𝑡/𝑟 𝑛
)(1 − 𝑐𝑜𝑠𝜂)
(16)
𝑇(𝜃) is the pore shape wetting factor, The maximum
interface curvature exists at 𝑛 𝑚 Basante (2010) is given as:
𝑛 𝑚 = 𝜃 − 𝑎𝑟𝑐 𝑠𝑖𝑛 [
𝑠𝑖𝑛 𝜃
1 + (𝑟𝑛/𝑟𝑡)
] (17)
Figure 2: MultiPore Model
For a nonwetting immobile (Oil) phase such as that shown
in figure 2 above, a force balance demonstrating the oil will
be displaced if the applied pressure exceeds the net
restraining capillary pressure as proposed by (Stegemeier,
1974):
∆𝑃𝐴 = 𝛻𝛷 . ∆𝐿 ˃
2𝜎
𝑟𝑛
. 𝜓 = ∆𝑃𝑐
(18)
The applied potential gradient is defined as:
𝛻𝛷 =
𝑑𝑝
𝑑𝐿
+ ∆𝜌𝑔(1 + 𝐺) (19)
The first term is the applied pressure gradient and the
second contains gravity and other acceleration terms.
The LHS of inequality is comprised of the pressure
gradient and length of the alternative flow path around the
trap and can be defined as a dimensionless length
regarding the pore entry radius:
𝑚 = ∆𝐿/𝑟𝑛 (20)
The geometrical factor ψ for the interface front with an
advancing angle α and receding angle β is similar to that
given by Melrose and Brandner (Melrose, 1974):
𝜓 = 𝑇(ά) −
𝑇(𝛽)
𝑟𝑏
𝑟𝑛
⁄
(21)
𝑟𝑏
𝑟𝑛
⁄ is the pore body/ pore neck radii,
𝑟𝑏
𝑟𝑛
⁄ is comparible
to the Difficulty index by (Dullien et al., 1972) and its
defined as:
𝐷 = [
1
𝑟𝑛
−
1
𝑟𝑏
] ∫ ∫
𝛼{𝑟𝑏, 𝑟𝑛}. 𝑑𝑟𝑛 . 𝑑𝑟𝑏
0
𝑟 𝑏
0
∞
(22)
The difficulty index D is an index measuring the difficulty of
recovering waterflood residuals in tertiary surfactant
flooding.
For special cases of 𝛼 = 𝛽 = 0, 𝜓 = 𝑟𝑛[(1/𝑟𝑛) − (1/𝑟𝑏)].
The fluid-rock only partially describes the structure of the
rock k; the pore inlet size distribution can be express by
combining Eq (14) and eq (15):
𝑟𝑛(𝑆) =
2√𝐾/𝜙
𝑁 𝐿𝑒(𝑆) (23)
Substituting equation Equations (21), (22) and (24) into
equation (19) gives
[
1
𝑁𝐿𝑒
2
(𝑆)
] . [
1
𝜓(𝑆)
] . [2𝑚(𝑆)] . [
𝐾
𝜙
𝛻𝛷
𝜎
] ≥ 1 (24)
m= dimensionless alternative part length (a dimensionless
length regarding the pore entry radius).
We can define the dimensionless length of the entire multi-
pore trapped oil mass as:
𝑓 =
∆𝐿
𝑎 (25)
where a is the average radius of trapped oil mass.
When equation (26) is combined with equation (16) an
expression for the alternative flow path regarding
interfacial tension and the pressure gradient is:
∆𝐿 = √
2𝜎𝑓
𝛻𝛷 (26)
By using equation (21), (24) and (27), The dimensionless
flow path will be given as:
𝑚(𝑆) = [𝑁𝐿𝑒(𝑆)] . [
𝑓
2
]
1/2
. [
𝜙𝜎
𝐾𝛻𝛷
]
1/2
(27)

5. The New Capillary Number Parameterization for Simulation in Surfactant Flooding
Trapping at a higher ratio of
𝛻𝛷
𝜎
will result from the
separation at both original and at weaker filament
locations. The new pattern will have smaller, but still
closely fitting pieces, so that in the low viscous/capillary
J. Oil, Gas Coal Engin. 027
ratio region, residual saturation may be practically
unchanged.
By combining equations (25) and (28), the ratio of the
viscous to capillary forces originally proposed by Brooks
and Corey (1964) is equated to three properties of the
fluid-rock system:
𝑁𝑐1(𝑆) =
𝐾𝛻𝛷
𝜎
≥ [𝜙𝑁𝐿𝑒
2
(𝑆)] . [𝜓2
(𝑆)]. [1/2𝑓] (28)
The first right-hand side of the inequality defines the
geometric of the rock pore network, the second term
defines pore body/pore neck radii and its connection with
a contact angle, and the third is a constant fluid geometric
property.
Another dimensionless number 𝑁𝑉𝐶 is obtained by
substituting Darcy’s law into Equation (29) and placing
relative permeability in the rock-fluid property term, these
results in the equation below:
𝑁𝑐(𝑆) =
𝑢𝜇
𝜎
≥ [𝜙𝑁𝐿𝑒
2
(𝑆)] . [𝐾𝑟𝑤(𝑆)𝜓2
(𝑆)]. [1/2𝑓] (29)
This number, which differs from 𝑁𝑐1 by a factor of 𝐾𝑟𝑤,
segregates all rock properties to the Right-hand Side and
thereby provides a good measure for comparative ease of
recovery from different rocks.
Although 𝑁𝐿𝑒(S) in equation (30) was derived as a function
of initial saturations, it can be expressed in terms of
normalized residual oil,𝑆 𝑅 = 𝑆 𝑜𝑟𝑐 𝑆 𝑜𝑟⁄ , because 𝑆 𝑜𝑟𝑐 itself is
a single function of initial saturation (S). 𝑆 𝑜𝑟 is defined as
the maximum trapped saturation or the residual saturation
at a small 𝑁𝑐.
𝑆 𝑜𝑟 is a rock property only, the relative permeability to
water, 𝐾𝑟𝑤, can be expressed in terms of normalized
saturation , when we substitute this function into Equation
(30) relates normalized residual oil and capillary number:
𝑁𝐶(𝑆 𝑅) =
𝑢𝜇
𝜎
≥ [𝜙𝑁𝐿𝑒
2
(𝑆 𝑅)] . [𝐾𝑟𝑤(𝑆 𝑅)𝜓2
(𝑆 𝑅)] . [1/2𝑓] (30)
The above is the Capillary number equation derived as a
function of the residual saturation or initial saturation
depending on availability of experimental data.
The parameters in the proposed capillary number equation
are given as follows:
𝑁𝐿𝑒 Leverett Number
 ϕ Porosity
 𝜓 Geometrical factor for curvature
 K Permeability
 𝑆 𝑤 Water saturation
 𝑁𝑐 Capillary number
 f Dimensionless length of the entire pore

The Leverett Number is:
𝑁𝐿𝑒 =
𝑃𝑐
𝜎
√
𝐾
𝜙
(31)
The geometrical factor for curvature is:
𝜓 = 𝑇(𝛼) −
𝑇(𝛽)
𝑟𝑏 𝑟𝑛⁄ (32)
where 𝛼 is the advancing contact angle of oil/water
interface, 𝛽 is the receding contact angle of oil/water
interface, 𝑟𝑏 is pore body radii, 𝑟𝑛 is pore neck radii
And the Dimensionless length of the entire pore is given
as:
𝑓 =
𝐿
𝑎 (33
where 𝐿 is Length of multi pore oil mass, 𝑎 is the average
radius of a multi pore oil mass.
The geometrical factor and the dimensionless length (f)
values can only be obtained experimentally. To test the
proposed equation, various parameters may be used as
proposed by Melrose and Brandner (1975). Thereby given
the value of 𝜓 = 0.35 and 𝑓 = 2.7, the capillary number
and the interpolation parameters of various sample
computed and given in
Table 1: Calculations of Capillary number and the interpolation parameter using the proposed formulation
Calculations of Capillary number and the interpolation parameter 𝑭 𝒌𝒓
ϕ NLe Krw(Sw) ψ f
NC =
ϕNLe
2
Krw((Sw))ψ2
2f
log 10 (Nc) Nc interpolant

6. The New Capillary Number Parameterization for Simulation in Surfactant Flooding
0.3 0.44 0.000002 0.35 18 4 X 10−8
-7.4 0
0.48 0.0049 1 X 10−6
-5.3 0
0.51 0.12 1.2X 10−4
-4.3 0.5
0.62 0.4 1.6 X 10−3
-3.7 1
0.77 0.8 2 X 10−2
-1.6 1
Abdullahi M. 028
2. Numerical Simulation model for surfactant
flooding
The purpose of the numerical simulation is to verify the
adaptability of the new formulation and the reliability of the
parameters used. Also to identify miscibility from capillary
number dynamics and the effect of relative permeability
interpolation to numerically replicate miscibility. To
demonstrate the development effect of surfactant flooding,
analyse the model-based calculated capillary number by
studying the dynamic changes in Velocity, interfacial
tension, and local adsorption. Also to examine the various
effect of transport component as a function of miscibility.
The surfactant flooding simulation was implemented on
MRST. The surfactant model in Eclipse assumes black-oil
fluid representation. However, in MRST the surfactant is
assumed to be only dissolved in the aqueous phase and is
added to the injected water as a mass per volume
concentration (Kg/Sm3
). The geological model used in the
simulation is a modified 1-D surfactant data file from the
MRST. The surfactant specific data consisting of tabulated
values are obtained from experimental work done by Xu et
al. (2011) as tabulated in Table 2. The reservoir physical
properties and fluid properties used in the simulation are
given in Table 3 and Table 4 respectively.
Table 2: Surfactant properties used [5]
Csurf
(Kg sm3
)⁄
IFT(N m⁄ ) 𝜇 𝑤 (cP) Adsorption
(Kg Kg)⁄
0 0.05 0.61 0
30 1E-05 0.8 0.0005
100 1E-06 1 0.0005
Table 3: Reservoir physical parameters (Jørgensen,
2013)
Porosity Φ 0.3
Permeability X 𝐾𝑥𝑥 100 mD
Permeability Y 𝐾𝑦𝑦 100 mD
Permeability Z 𝐾𝑧𝑧 20 mD
Initial pressure 𝑃𝑖 300 bar
Top depth 1000 m
Table 4: Reservoir Fluid parameters (Jørgensen, 2013)
Initial water saturation 𝑆 𝑤,𝑖 0.2
Surface Density of water 𝜌 𝜔,𝑠𝑐 1080 Kg sm3⁄
Surface Density of oil 𝜌 𝑜,𝑠𝑐 800 Kg sm3⁄
Reference viscosity of oil 𝜇 𝑜,𝑟𝑒𝑓 0.61 mPa s
Reference viscosity of water 𝜇 𝑤,𝑟𝑒𝑓 5.0 mPa s
Reference pressure 𝑃𝑟𝑒𝑓 300 bar
SIMULATION RESULTS AND DISCUSSION
The simulation is carried out using MRST by comparing
various parameters including the capillary number
between cases of zero surfactants (𝐶 = 0𝑘𝑔/𝑠𝑚3
) and fifty
surfactant concentration (𝐶 = 50𝑘𝑔/𝑠𝑚3
) along a reservoir
grid cell with size (x). The position of the reservoir grid is
donated with x in metres and the reservoir grid used range
from 0 to 100 metres.

7. The New Capillary Number Parameterization for Simulation in Surfactant Flooding
Figure 3: Capillary number profile within the grid with
surfactant concentration of 0𝒌𝒈/𝒔𝒎 𝟑
and 50𝒌𝒈/𝒔𝒎 𝟑
J. Oil, Gas Coal Engin. 029
Figure 4: Interfacial tension profile within the grid with surfactant concentration of 0𝒌𝒈/𝒔𝒎 𝟑
and 50𝒌𝒈/𝒔𝒎 𝟑
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0 50 100
IFT
Position x (m)
0 surfactant concentration
time= 1day
time= 240
day
time= 480
day
0
0.01
0.02
0.03
0.04
0.05
0.06
0 50 100
IFT
Position x(m)
50 surfactant concentration
time= 1day
time=
240days
time=
480days

8. The New Capillary Number Parameterization for Simulation in Surfactant Flooding
Figure 5: Absolute velocity profile within the grid with surfactant concentration of 0𝒌𝒈/𝒔𝒎 𝟑
and 50𝒌𝒈/𝒔𝒎 𝟑
Figure 6: Oil saturation distribution map after flooding with surfactant concentration of 0𝒌𝒈/𝒔𝒎 𝟑
and 50𝒌𝒈/𝒔𝒎 𝟑
The capillary numbers in each grid cell were calculated
using the equation 𝑁𝑐 =
𝑢𝜇
𝜎
in MRST. The flow regime
pattern in each grid cells is defined, and each cell has its
unique 𝑁𝑐 value. From Figure 3, The capillary number can
be seen increasing from 10−8
to 10−2
. The interfacial
tension decreases along the grid cells from 0.05𝑁/𝑚 to
10−6
𝑁/𝑚 as seen in Figure 4. Figure 4 shows the velocity
profiles and it can be deduced that the velocity remain
constant within the grids cell and does not changes. This
clearly indicates that the velocity of the phases does not
affect the miscibility development. It can be observed that
both the capillary number and interfacial tension profiles
shows a sharp front and smeared front as the simulation
time increases at a certain point along the grid block which
is caused by the presence of surfactant in the injecting fluid
added from the injector well located in the first grid. There
is no transition zone on the profile because there is no
diffusion in the block and the permeability is constant. The
Abdullahi M. 030
sharp front on the figures indicates the miscible zone, and
the smeared front is the immiscible zone.
Figure 6 show different distribution maps after flooding
with a surfactant concentration of 0𝑘𝑔/𝑠𝑚3
and 50𝑘𝑔/𝑠𝑚3
.
The figure clearly show the influence of surfactant in EOR
flooding as the model with surfactant concentration of
50𝑘𝑔/𝑠𝑚3
produces larger oil recovery after flooding.
5.00E-10
5.00E-09
0 50 100
Vel
Position x(m)
0 surfactant concentration
time= 1 day
time= 240
day
time= 480
day
5.00E-10
5.00E-09
0 50 100
Vel
Position x(m)
50 surfactant concentration
time= 1 day
time= 240
day
time= 480
day
0 01 1

9. The New Capillary Number Parameterization for Simulation in Surfactant Flooding
Figure 7: Interpolation parameter against the Log(Nc) to
identify miscibility from capillary number dynamics
Miscibility is achieved by interpolation between the
immiscible relative permeability curves and the miscible
relative permeability curves. The Interpolation parameter
𝐹𝑘𝑟 values must span between the range value of [0, 1].
Miscibility is the function of the interpolation parameter.
The interpolation parameter is described by a function
(log10 𝑁𝑐). This logarithmic function is defined with an
equation:
𝐹𝑘𝑟 =
log10 𝑁𝑐 − log10 𝑁𝑐
𝑁𝑜𝑠𝑢𝑟𝑓
log10 𝑁𝑐
𝑠𝑢𝑟𝑓
− log10 𝑁𝑐
𝑁𝑜𝑠𝑢𝑟𝑓
(34)
where 𝑁𝑐 is the model-based capillary number, 𝑁𝑐
𝑁𝑜𝑠𝑢𝑟𝑓
is the minimal values of the capillary numbers, 𝑁𝑐
𝑠𝑢𝑟𝑓
is the
maximal values of the capillary numbers.
The interpolation parameter 𝐹𝑘𝑟 value of 0 implies
immiscible conditions and a value of 1 implies miscible
conditions. The interpolation parameter 𝐹𝑘𝑟 is computed to
identify miscibility from capillary number dynamics and
show the effect of relative permeability interpolation
function to numerically replicate miscibility. From Figure 7
is can be clearly seen that the model is immiscible at the
first timesteps in the grid cells during the surfactant
flooding, partially miscible at median timesteps and fully
miscible at the end of the grid. The gradual miscibility
development of the model at every timestep is caused by
the increasing level of the surfactant concentration during
the flooding. This means miscibility develops gradually as
we flood.
Figure 8: Water and Oil miscible/immiscible relative
permeability curves
Figure 8 represents the two sets of relative permeability
curves for water and oil, one curve for immiscible
conditions (𝐹𝑘𝑟=0) and one curve for fully conditions
(𝐹𝑘𝑟=1). Once the value for 𝐹𝑘𝑟 is determined, the two
relative permeability are scaled and averaged according to
a specific method. The lower and upper end-points on the
two relative permeability curves are used to calculate new
end-point saturations by a weighted average with 𝐹𝑘𝑟=1.
As 𝐹𝑘𝑟 vary between 0 and 1, the relative permeability
calculations vary. Since the relative permeability defined in
the input deck consist of some discrete points and is not
defined by a continuous function. Instead, new saturation
variables will be created, and these two saturation values
are used to calculate the relative permeability for both
miscible and immiscible conditions at the target saturation,
𝑆 𝑤.
Using these new saturation values, the relative
permeability is interpolated in the miscible and immiscible
table.
J. Oil, Gas Coal Engin. 031

10. The New Capillary Number Parameterization for Simulation in Surfactant Flooding
The effective relative permeability in the grid cell with
saturation 𝑆 𝑤 and miscibility factor 𝐹𝑘𝑟 is then the weighted
average of the two curves (Equation below).
𝐾𝑟𝑤(𝑆 𝑤) = 𝐹𝑘𝑟. 𝐾𝑟𝑤,𝑚𝑖𝑠𝑐 + (1 − 𝐹𝑘𝑟 )𝐾𝑟𝑤,𝑖𝑚𝑚𝑖𝑠𝑐
(35)
𝐾𝑟𝑜(𝑆 𝑤) = 𝐹𝑘𝑟. 𝐾𝑟𝑜,𝑚𝑖𝑠𝑐 + (1 − 𝐹𝑘𝑟 )𝐾𝑟𝑜,𝑖𝑚𝑚𝑖𝑠𝑐 (36)
Figure 9: Comparison between the influences of
Interfacial tension on Capillary number computed using
standard formulation in MRST and the newly proposed
formulation for capillary number
Figure 9 shows a comparison between the newly derived
formulation proposed and the standard formulation. The
capillary number was plotted against interfacial tension
computed from the new formulation proposed and the
standard formulation. At first, the interfacial tension σ is
introduced at a pore level, the effect of surfactant is to
modify the interfacial tension to make σ a function of the
surfactant concentration, C. At an upscale level, the
change in relative permeability 𝐾𝑟 will depend on the
capillary number 𝑁𝑐 which measures the ratio between the
viscous and capillary forces and is defined as
𝑁𝑐 =
𝑢𝜇
𝜎
(37)
As the surfactant concentration is increased during the
flooding, the interfacial tension reduces, thus the residual
oil or trapped oil is forced to move. From the Proposed
formulation and standard formulation lines, the maximum
capillary number reached is greater using the newly
proposed formulation.
Figure 10 Comparison between the new and standard
capillary desaturation curve
Figure 10 shows the capillary desaturation curve, this
shows the relation between the capillary number 𝑁𝑐 and
the residual oil saturation of the model. The residual oil
saturation is deduced from the relation 𝑆 𝑜𝑟 =
1
1+(
𝑁 𝑐
𝜆
)
𝛽
(Jørgensen, 2013). Given average values to be used in
sandstone as λ = 0.0012 and 𝛽 = 1.25. From figure 9, the
oil mobility is observed to have begun at 𝑁𝑐 = 10−6
which
is known as the critical value of 𝑁𝑐 when the newly
proposed formulation is used. Using the standard
formulation or model-based calculated capillary number,
the mobility is seen to have begun at 𝑁𝑐 = 10−5
. Literature
study has shown that critical value of 𝑁𝑐 in most capillary
desaturation curves is between 10−5
and 10−4
. Thus the
new 𝑁𝑐 formulation provides a broader critical value range
than the standard formulation.
The recovery factor of the simulations using MRST and
Eclipse@ is computed as:
𝑅𝑒𝑐𝑜𝑣𝑒𝑟𝑦 𝑓𝑎𝑐𝑡𝑜𝑟 =
𝑟𝑒𝑐𝑜𝑣𝑒𝑟𝑎𝑏𝑙𝑒 𝑜𝑖𝑙
𝑂𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑖𝑛 − 𝑝𝑙𝑎𝑐𝑒 𝑜𝑖𝑙
Table 5: Comparison between the recovery factor of
Eclipse@ and MRST
Eclipse® MRST
OIL(recovery) 107.39 sm3
112.67 sm3
Recovery factor 0.191 0.2
By studying the total oil recovered in the two models
above, the production data for oil is well matched. The
small discrepancy may relate to errors by the simulator as
the eclipse@ indicated few warnings during the simulation
process, among was convergence issue, and also
numerical dispersion issue may cause the small difference
in the recovery factor. The oil bank seems to moves faster
and has a higher oil recovery factor in MRST than in
Eclipse@ as it gave a higher recovery factor as seen in .
MRST is thus a vital and optimised tool for enhanced oil
recovery simulation.
Abdullahi M. 032

11. The New Capillary Number Parameterization for Simulation in Surfactant Flooding
CONCLUSIONS AND RECOMMENDATIONS
CONCLUSIONS
 A new equation for calculation of capillary number as a
major transport component and mechanism for oil
mobilisation was derived. The equation was tested and
proven reasonable. The equation can be accepted as a
working hypothesis since the result of the capillary
numbers obtained all fell within the range of 𝑁𝑐 as
observed experimentally in various papers and models
available. Thus, the proposed mechanism easily
accounts for a broader critical range of capillary number
(10−6
to 10−4
) as compare with the standard models
with less broad critical capillary number ranges
(10−5
to 10−4
).
 Miscibility was identified clearly from capillary number
dynamics, and the effect of relative permeability
interpolation parameter was used to replicate
miscibility.
 Absolute velocity of the phases was seen not to have
any influence on the capillary number distribution
during the flooding. Therefore, the addition of surfactant
in flooding doesn’t change the velocities of the phases
in the grid block.
 An empirical performance of miscibility was certainly
seen in the two relative permeability curves plotted,
also seen in the plot of capillary number against the
interpolation parameter which shows how miscibility
develops and finally seen from the capillary
desaturation curve. These certainly illustrated the effect
of miscibility in the surfactant flooding in enhance
recovery compared to the water flooding.
 The surfactant model in MRST produced significantly
higher oil (recovery) and a higher recovery factor in the
surfactant model implemented in MRST than Eclipse®.
RECOMMENDATIONS
 To investigate other methods that capture miscibility
effects on relative permeability curves
 Experimental works should be carried out to describe
the phase behaviour of oil/water system containing
surfactant and published data validation.
 Full-scale modeling of a 3-phase system (Oil-Water-
Gas) to study the effect of miscibility and hysteresis
simultaneously.

NOMENCLATURE
𝑁𝐿𝑒 Leverett Number
𝑁𝑉𝐶 Capillary number
𝑃𝑐 Capillary pressure
P Pressure
q Flow rate
r radius
R Ratio of maximum pore radius/average pore radius
determined by resistivity
S Saturation of nonwetting phase
𝑆 𝑅 Normalized residual oil saturation
T(θ) geometric factor for contact angle of interface passing
through toroidal pore
u Darcy’s velocity
α̂ Dullien pore volume distribution function
α Advancing contact angle of oil/ water interface
β Receding contact angle of oil/water interface
∇Φ Potential gradient
ϕ Porosity
ᵑ pore angle interface
∆ρ density difference
σ Interfacial tension
θ Advancing contact angle of fluid used to determine rock
property
ψ The geometrical factor for curvature
b pore body
c cylinder
d drainage
m maximum
n pore neck
or Immobile nonwetting phase at maximum trapping
orc immobile nonwetting phase below maximum trapping
wr immobile wetting phase at maximum trapping
wrc immobile wetting phase below maximum trapping
𝑉𝑏 Bulk volume
f Dimensionless length of the entire pore
K Permeability
μ Viscosity
L Core length
A area
𝜆 Mobility of phase
𝐾𝑟 Relative permeability
𝑆 𝑛 Normalized saturation
𝑆𝑔𝑟 Residual gas saturation
𝑆𝑔𝑟 Initial gas saturation
C Trapping characteristic of the porous media
M Mobility ratio
ω Todd-Longstaff mixing parameter
𝑁𝑐 Capillary number
X Fractional distance of current saturation between
drainage curve end point and hysteresis saturation
J. Oil, Gas Coal Engin. 033

12. The New Capillary Number Parameterization for Simulation in Surfactant Flooding
REFERENCES
AlQuaimi BI, Rossen WR (2018). Capillary Desaturation
Curve for Residual Nonwetting Phase in Natural
Fractures, SPE Journal, Preprint(September 2017), pp.
1–15. doi: 10.2118/189448-PA.
Basante VA (2010). Effect of Capillary Number on the Oil
Recovery using Oil-Water Emulsion Injection in
Coreflooding, in 13th Brazilian Congress of Thermal
Sciences and Engineering. Uberlândia, MG, Brazil.
Brooks RH, Corey AT (1964). Hydraulic properties of
porous media, Hydrol Pap Fort Collins CO Colo State
Univ, 3(3), p. 27 pgs. doi: citeulike-article-id:711012.
Bryan J, Kantzas A (2009). Potential for alkali-surfactant
flooding in heavy oil reservoirs through oil-in-water
emulsification, Journal of Canadian Petroleum
Technology, 48(2), pp. 37–46. doi: 10.2118/09-02-37.
Cheng N (2005). Special Topics On Developed Miscibility
Special Topics On Developed Miscibility, October,
https://core.ac.uk/download/pdf/52100024.pdf.
Dullien FAL, Dhawan G K, Gurak N, Babjak L (1972). A
relationship between pore structure and residual oil
saturation in tertiary surfactant floods, Society of
Petroleum Engineers Journal, 12(04), pp. 289–296.
Felix U, Ayodele TO, Olalekan O, (2015). Surfactant-
Polymer Flooding Schemes (A Comparative Analysis),
Nigeria Annual International Conference and Exhibition
held in Lagos, Nigiria.
Jørgensen K, (2013). Implementation of a Surfactant
Model in MRST With Basis in Schlumberger ’ s Eclipse,
(June).
Keshtkar S, Sabeti M, Mohammadi AH (2016). Numerical
approach for enhanced oil recovery with surfactant
flooding, Petroleum. Elsevier Ltd, 2(1), pp. 98–107. doi:
10.1016/j.petlm.2015.11.002.
Khanamiri HH, Torsæter O, Stensen JÅ, (2015).
Experimental Study of Low Salinity and Optimal Salinity
Surfactant Injection, Europec 2015. doi:
10.2118/174367-MS.
Laforce T, London IC, Orr FM (2008). SPE 116119
Development of Gas/Oil Miscibility in Water and Gas
Injection, in, pp. 21–24.
Leverett MC (1941) Capillary Behavior in Porous Solids,
Transactions of the AIME, 142(01), pp. 152–169. doi:
10.2118/941152-G.
Lohne A, Purwanto EY, Fjelde I (2012). SPE 154495
Gravity Segregated Flow in Surfactant Flooding,
(June), pp. 4–7.
Melrose JC (1974). Role of Capillary Forces In
Detennining Microscopic Displacement Efficiency For
Oil Recovery By Waterflooding, Journal of Canadian
Petroleum Technology, 13(04), p. 9. doi:
dx.doi.org/10.2118/74-04-05.
Ren B, Delaney JM, Lake LW, Bryant SL (2018). Interplay
Between Permeability Retardation and Capillary
Trapping of Rising Carbon Dioxide in Storage
Reservoirs, in SPE Annual Technical Conference and
Exhibition. Texas, pp. 9–11.
Shen P, Zhu B, Li X, Wu Y, (2006). The Influence of
Interfacial Tension on Water / Oil Two-Phase Relative
Permeability, SPE/DOE Symposium on Improved Oil
Recovery, (1), pp. 1–8.
Stegemeier GL (1974). Relationship of Trapped Oil
Saturation to Petrophysical Properties of Porous Media,
SPE Improved Oil Recovery Symposium. doi:
10.2118/4754-MS.
Whitson CH, Fevang O (1997). Generalized
Pseudopressure Well Treatment in Reservoir
Simulation, in.
Xavier JA (2011). Effect of Varying Surfactant
Concentration on Interfacial Tension in Tertiary
Recovery of Crude Oil.
Xu F, Guo X, Wang W, Zhang N, Jia S, Wang X, Key S
(2011). SPE 145036 Case Study : Numerical
Simulation of Surfactant Flooding in Low Permeability
Oil Field’. doi:10.2118/145036-MS.
Yan C, Qiu H (2016) Critical Capillary Number of Interfacial
Film Displacement in a Capillary Tube, pp. 1–17.
1610.07749.pdf
Accepted 26 July 2018
Citation: Abdullahi M (2018). The New Capillary Number
Parameterization for Simulation in Surfactant Flooding.
Journal of Oil, Gas and Coal Engineering. 3(1): 023-033.
Copyright: © 2018 Abdullahi M. This is an open-access
article distributed under the terms of the Creative
Commons Attribution License, which permits unrestricted
use, distribution, and reproduction in any medium,
provided the original author and source are cited.