The solution of immiscible fluid flow by means of
optimal homotopy analysis method
Dipak J. Prajapati
N. B. Desai
Government Engineering
College,Sector-28,
Gandhinagar, Gujarat, India
Head, Department of
Mathematics
A. D. Patel Institute of
Technology, New V. V. Nagar,
Gujarat, India
[email protected]
[email protected]
ABSTRACT
In this paper the mathematical model of immiscible fluid flow
in homogeneous porous media with capillary pressure effects
known as fingering phenomenon is discussed. The resulting
nonlinear partial differential equation is reduced to the
nonlinear ordinary differential equation by similarity
transformation and its approximate solution is obtained using
basic optimal homotopy analysis method (OHAM) under
appropriate conditions. The numerical values and graphical
presentation are given using Mathematica
Keywords
FLUID FLOW THROUGH POROUS MEDIA, FINGERING
PHENOMENON, SIMILARITY TRANSFORMATION,
BASIC OPTIMAL HOMOTOPY ANALYSIS METHOD.
I. INTRODUCTION
In primary oil recovery process, oil is pushed to the surface of
earth by natural pressure of the reservoir. It allows about
5% 𝑡𝑜 10% of the oil in the reservoir to be extracted. In
secondary recovery process water or gas is injected to drive
the residual oil remaining after the primary recovery process
to the surface wells. This allows 25% 𝑡𝑜 30% of the oil in
the reservoir to be extracted.
In this paper, we consider the fingering phenomenon occurred
during water injection in secondary oil recovery process as
shown in figure 1.
When a fluid contained in a porous medium is displaced by
another of lesser viscosity, instead of regular displacement of
the whole front, protuberances may occur which shoot
through the porous medium at relatively great speed. This
phenomenon is called fingering and the protuberances which
represent instabilities in the displacement problem are called
fingers. The displacement problems of this type (particularly,
the stabilization of fingers) have much current importance in
the secondary recovery processes of petroleum technology.
This instability phenomenon is discussed by many researchers
from different points of view. Saffman and Taylor (1985)
derived a classical result for the shape of finger in the absence
of capillary. Recently many researchers have discussed the
shape, size and velocity prediction of fingers under different
situation with different view ( Lange et al.,1998; Brailovsky et
al.,2006; Zhan and Yortsos , 2000; Wang and Feyen ,1998).
Joshi and Mehta (2009) have discussed the solution by the
group invariant method of the instability phenomenon arising
in fluid flow through porous media. Kinjal (2010) has
obtained the power series solution of fingering phenomenon.
II. MATHEMATICAL MODELLING
In this problem, it is considered that there is a uniform water
injection into an oil saturated porous medium of homogeneous
physical characteristics having length 𝐿.The protuberances
may occur due to the viscosity of oil and water. Since the
entire oil at the initial boundary (being measured in the
direction of displacement), is displaced through a small
distance due to water injection, therefore, it is assumed that
complete saturation exists at the initial boundary.
For the mathematical formulation, we consider the governing
law which is Darcy’s law, here, as valid for the investigated
oil-water flow system and assume further that the
macroscopic behaviour of fingers is governed by statistical
treatment. In the statistical treatment of fingers only the
average behaviour of the two fluids involved is taken into
consideration.
The saturation of injected fluid, 𝑆𝑖 (𝑥, 𝑡) is then defined as the
average cross-sectional area occupied by the injected fluid at
level 𝑥 and time t, i.e. 𝑆𝑖 (𝑥, 𝑡). Thus the saturation of injected
fluid in porous medium represents the average cross-sectional
area occupied by fingers.
During water injection in secondary oil recovery process, the
seepage velocity Vi of injected fluid (water) and the seepage
velocity Vn of native fluid (oil) may be written (by Darcy’s
law) as
𝑉𝑖 =
−
𝑘𝑖
𝜇𝑖
𝐾
𝜕𝑝𝑖
𝜕𝑥
(1)
𝑉𝑛 =
−
𝑘𝑛
𝜇𝑛
𝐾
𝜕𝑝𝑛
𝜕𝑥
(2)
where K is the permeability of the homogenous porous
medium, 𝑘𝑛 and 𝑘𝑖 are relative permabilities of native fluid
(oil) and injected fluid (water) which are functions of
saturations 𝑆𝑛 and 𝑆𝑖 of oil and water respectively. 𝑝𝑖 and 𝑝𝑛
denote the pressures of water and oil respectively while 𝜇𝑖 and
𝜇𝑛 are the constant kinematic viscosities of the water and oil
respectively.
Regarding phase densities as constant, the equations of
continuity are
𝑃
𝜕𝑆𝑖
𝜕𝑡
𝜕𝑉𝑖
+
𝜕𝑥
where 𝑝̅ =
𝑃
𝜕𝑆𝑛
𝜕𝑡
+
𝜕𝑉𝑛
𝜕𝑥
=
0
(4)
where 𝑃 is the porosity of the medium which is considered as
constant.
The porous medium is considered to be fully saturated. From
the definition of phase saturation ( Scheidegger (1960)), 𝑆𝑖 +
𝑆𝑛 = 1.
(5)
If the substances are immiscible, then there is a surface
tension occurring at their contact line which creates a
capillary pressure
𝑝𝑐 (𝑆𝑖 ) = 𝑝𝑛 − 𝑝𝑖
(6)
It is well known that relative permeability is the function of
displacing fluid saturation. Then at this stage for definiteness
of the mathematical analysis, we assume standard forms of
Scheidegger and Johnson (1961) for the analytical
relationship between the relative permeability, phase
saturation and capillary pressure as
𝑘𝑖 = 𝑆𝑖 ,
𝑘𝑛 = 𝑆𝑛 = 1 −
𝑆𝑖
(7)
𝑝𝑐 = −𝛽𝑆𝑖
(8)
where β is constant (negative sign shows the direction of
saturation of water opposite to capillary pressure)
Substituting values of 𝑉𝑖 and 𝑉𝑛 from equations (1) and (2)
in equations (3) and (4) respectively,
𝑃
𝜕𝑝𝑖
𝜕𝑥 𝜇𝑖
𝜕𝑥
( 𝐾
)
(
𝑘𝑛
𝐾
𝜕𝑥 𝜇𝑛
𝜕𝑝𝑖
Eliminating
𝑃
𝜕𝑝𝑐
𝜕𝑥
𝜕𝑆𝑖
𝜕𝑡
𝜕𝑥
𝜕
=
𝜕𝑝𝑛
𝜕𝑥
𝜕𝑆𝑖
𝜕𝑡
=
(9)
𝑃
𝜕
)
𝜕𝑆𝑛
𝜕𝑡
=
(10)
from (6) and (9),
𝑘
[ 𝑖𝐾(
𝜕𝑥 𝜇𝑖
𝜕𝑝𝑛
𝜕𝑥
−
)]
𝜕𝑥
+
𝑝𝑛 −𝑝𝑖
From (16),
2
= 𝑝̅ +
(16)
𝑝𝑛 +𝑝𝑖
2
𝜕𝑝𝑛
𝜕𝑥
is the constant mean pressure.
=
1 𝜕𝑝𝑐
(17)
2 𝜕𝑥
Using (17) in (13), we get ,
𝐾 𝑘
𝑀= ( 𝑖−
2 𝜇𝑖
𝑘𝑛 𝜕𝑝𝑐
𝜇𝑛
)
(18)
𝜕𝑥
Substituting value of 𝑀 in Equation (15), we obtain
𝜕𝑆𝑖 1 𝜕
𝑘𝑖 𝑑𝑝𝑐 𝜕𝑆𝑖
+
[𝐾
]=0
(19)
𝜕𝑡 2 𝜕𝑥 𝜇𝑖 𝑑𝑆𝑖 𝜕𝑥
Since 𝑘𝑖 = 𝑆𝑖 & 𝑝𝑐 = −𝛽𝑆𝑖 , we have
𝜕𝑆𝑖 𝛽 𝐾 𝜕
𝜕𝑆𝑖
𝑃
−
(𝑆
)=0
(20)
𝜕𝑡 2 𝜇𝑖 𝜕𝑥 𝑖 𝜕𝑥
The appropriate conditions to solve nonlinear equation (20)
are:
𝑆𝑖 (0, 𝑡) = 𝑆𝑖𝑜 , 𝑓𝑜𝑟 𝑡 >
0
(21)
𝑆𝑖 (𝐿, 𝑡) = 𝑆𝑖1 𝑓𝑜𝑟 𝑡 >
(22)
0
and
(23)
𝑆𝑖 (𝑥, 0) = 𝑔(𝑥) 𝑓𝑜𝑟 𝑥 > 0 , 0 ≤ 𝑔(𝑥) < 𝑆𝑖𝑜
The small radiation in the saturation of the injected fluid will
be
𝜕𝑆𝑖 (0, 𝑡)
= 𝜖 𝑓𝑜𝑟 𝑡 > 0
(24)
𝜕𝑥
The equation (20) is a nonlinear partial differential equation
which describes the immiscible oil-water fluid flow through
porous media.
Using dimensionless variables
𝑥
𝐾𝛽𝑡
𝑋= ,
𝑇=
,
(25)
𝐿
2𝜇𝑖 𝐿2 𝑃
equations (20), (21), (23)𝑎𝑛𝑑 (24) reduce to
𝜕𝑆𝑖
𝜕
𝜕𝑆𝑖
=
(𝑆
)
(26)
𝜕𝑇 𝜕𝑋 𝑖 𝜕𝑋
𝑆𝑖 (0, 𝑇) = 𝑆𝑖𝑜
𝑓𝑜𝑟 𝑇 >
0
(27)
𝑃
(11)
Using (10) 𝑎𝑛𝑑 (11), we get
𝜕
2
𝑝
2 𝑐
=
(3)
𝑘𝑖
𝑝𝑛 +𝑝𝑖
𝑝𝑛 =
1
0
𝜕
𝑘𝑛 𝜕𝑝𝑐
𝜕𝑆𝑖
𝜕 𝐾 𝜇𝑛 𝜕𝑥
𝑀
𝑃
+
[
+
]=0
(15)
𝑘𝑛 𝜇𝑖
𝑘 𝜇
𝜕𝑡 𝜕𝑥
1+
1+ 𝑛 𝑖
𝑘𝑖 𝜇𝑛
𝑘𝑖 𝜇𝑛
The value of the pressure of native fluid can be written in the
form
𝑘
𝑘𝑛 𝜕𝑝𝑛
𝜇𝑖
𝜇𝑛
[𝐾 ( 𝑖 +
)
𝜕𝑥
−𝐾
𝑘𝑖 𝜕𝑝𝑐
𝜇𝑖 𝜕𝑥
]=0
(12)
Integrating both sides with respect to 𝑥, we have
𝑘
𝑘𝑛 𝜕𝑝𝑛
𝜇𝑖
𝜇𝑛
𝐾( 𝑖 +
−𝑀
where 𝑀 is the integrating constant.
(13) can be rewritten as
𝜕𝑝𝑐
𝜕𝑝𝑛
−𝑀
𝜕𝑥
=
+
𝑘𝑖 𝑘𝑛
𝑘 𝜇
𝜕𝑥
𝐾( + ) 1+ 𝑛 𝑖
𝜇𝑖 𝜇𝑛
𝑘𝑖 𝜇𝑛
Using (11) and (14), we obtain,
)
𝜕𝑥
−𝐾
𝑘𝑖 𝜕𝑝𝑐
𝜇𝑖 𝜕𝑥
=
𝑆𝑖 (𝑋, 0) = 𝑔(𝑋) 𝑓𝑜𝑟 𝑋 > 0, 0 ≤ 𝑔(𝑋) <
𝑆𝑖𝑜
(28)
𝜕𝑆𝑖 (0, 𝑇)
= 𝜖 𝑓𝑜𝑟 𝑇 > 0
𝜕𝑋
Choosing similarity transformation,
𝑋
(29)
(13)
𝑆𝑖 (𝑋, 𝑇) = 𝑓(𝑛) 𝑤ℎ𝑒𝑟𝑒 𝑛 =
(14)
the governing equation (26) reduces to the ordinary
differential equation
𝑓(𝑛)𝑓 ′′ (𝑛) + [𝑓 ′ (𝑛)]2 + 2 𝑛 𝑓 ′ (𝑛) = 0
(31)
together with the conditions
𝑓(0) = 𝑆𝑖𝑜
(32)
𝑓 ′ (0) = 𝜔 ≠ 0
(33)
2√𝑇
(Mehta, 1977)
(30)
We solve equation (31) using basic optimal HAM.
III.
SOLUTION USING THE BASIC
OPTIMAL HOMOTOPY ANALYSIS
METHOD (OHAM)
Due to the initial conditions (32) and (33), we choose
𝑓0 (𝑛) = 𝑆𝑖𝑜 + 𝜔𝑛
(34)
as the initial approximation of 𝑓(𝑛).
Besides we choose the auxiliary linear operator as
𝜕 2 𝜑(𝑛; 𝑞)
ℒ[𝜑(𝑛; 𝑞)] =
(35)
𝜕𝑛2
with the property ℒ[𝑓] = 0 𝑤ℎ𝑒𝑛 𝑓 = 0
(36)
Furthermore, based on governing equation (31), we define
such a nonlinear operator
2
𝜕 2 𝜑(𝑛; 𝑞)
𝜕𝜑(𝑛; 𝑞)
+(
)
𝜕𝑛2
𝜕𝑛
𝜕𝜑(𝑛; 𝑞)
+ 2𝑛
(37)
𝜕𝑛
Let 𝑐0 denote a nonzero auxiliary parameter .Then we
construct the zero-order deformation equation
(1 − 𝑞)ℒ[𝜑(𝑛; 𝑞) − 𝑓0 (𝑛)]
= 𝑞𝑐0 𝐻(𝑛)𝒩(𝜑(𝑛; 𝑞))
(38)
subject to the initial conditions
𝜑(0; 𝑞) = 𝑆𝑖𝑜
(39)
𝜕𝜑(𝑛; 𝑞)
|
=𝜔
(40)
𝜕𝑛
𝑛=0
𝒩(𝜑(𝑛; 𝑞)) = 𝜑(𝑛; 𝑞)
where 𝑞 ∈ [0,1] is the embedding parameter, 𝐻(𝑛) is
nonzero auxiliary function and 𝜑(𝑛; 𝑞) is an unknown
function.
When 𝑞 = 0 , we have from (36) and (38),
𝜑(𝑛; 0) = 𝑓0 (𝑛)
(41)
When 𝑞 = 1, the equations (38),(39),(40) are equivalent to the
equations (31),(32),(33) provided
𝜑(𝑛; 1) =
𝑓(𝑛)
(42)
Therefore, according to equations (41) and (42), the solution
𝜑(𝑛; 𝑞) varies from the initial guess 𝑓0 (𝑛) to the solution
𝑓(𝑛) of the equation (31) as the embedding parameter 𝑞
increases from 0 to 1.
Obviously, 𝜑(𝑛; 𝑞) is determined by the auxiliary linear
operator ℒ, the initial guess 𝑓0 (𝑛) and the convergencecontrol parameter 𝑐0 . We have great freedom to select all of
them. Assuming that all of them are so properly chosen that
the Taylor series
𝜑(𝑛; 𝑞) = 𝑓0 (𝑛) +
∑∞
(𝑛)
(43)
𝑓
𝑞𝑚
𝑚=1 𝑚
exists and besides converges at 𝑞 = 1, we have using (42)
the homotopy-series solution
dividing them by 𝑚! and finally setting 𝑞 = 0, we have the so
called high order deformation equation
ℒ[𝑓𝑚 (𝑛) − 𝜒𝑚 𝑓𝑚−1 (𝑛)] =
𝑐0 𝐻(𝑛)ℛ𝑚 (𝑛)
(46)
subject to the conditions
𝑓𝑚 (0) = 0, 𝑓′𝑚 (0) = 0, 𝑚 ≥ 1
1
where ℛ𝑚 (𝑛) = (𝑚−1)!
(44)
𝑚=1
where
1 𝜕 𝑚 𝜑(𝑛; 𝑞)
𝑓𝑚 (𝑛) =
|
𝑚! 𝜕𝑞𝑚
𝑞=0
(45)
Differentiating the zero order deformation equation (38) 𝑚
times with respect to the embedding parameter 𝑞 and then
𝜕𝑚−1 𝒩(𝜑(𝑛;𝑞))
|
𝜕𝑞 𝑚−1
(48)
𝑞=0
0 , 𝑤ℎ𝑒𝑛 𝑚 ≤ 1
and 𝜒𝑚 = {
(49)
1 , 𝑤ℎ𝑒𝑛 𝑚 > 1
For simplicity, assume 𝐻(𝑛) = 1; hence the solution of (46)
can be expressed in the form
𝑓𝑚 (𝑛) = 𝜒𝑚 𝑓𝑚−1 (𝑛) + 𝑐0 ℒ −1 [ℛ𝑚 (𝑛)] + 𝐶0
+ 𝐶1 𝑛
(50)
where the constants 𝐶0 𝑎𝑛𝑑 𝐶1 are determined by the
substitution of (50) into (47).
Taking 𝑚 = 1 𝑎𝑛𝑑 𝑚 = 2 𝑖𝑛 (50), 𝑤𝑒 ℎ𝑎𝑣𝑒 ,
𝑓1 (𝑛) = 𝑐0 {
𝜔2 𝑛2
2
𝜔2 𝑛2
𝑓2 (𝑛) = 𝑐0 [
2
+
+
𝜔𝑛3
3
𝜔𝑛3
3
}
+ 𝑐0 {
and
𝑆𝑖𝑜 𝜔2 𝑛2
2
+
𝑆𝑖𝑜 𝜔𝑛3
3
𝜔2 𝑛4
2
+
+
𝜔3 𝑛3
2
𝜔𝑛5
10
+
}]
So
𝜔2 𝑛2 𝜔𝑛3
𝑓(𝑛) = 𝑆𝑖𝑜 + 𝜔𝑛 + 𝑐0 {
+
}
2
3
𝜔2 𝑛2 𝜔𝑛3
+ 𝑐0 [
+
2
3
𝑆𝑖𝑜 𝜔2 𝑛2 𝑆𝑖𝑜 𝜔𝑛3 𝜔3 𝑛3
+ 𝑐0 {
+
+
2
3
2
2
4
5
𝜔 𝑛
𝜔𝑛
+
+
}]
2
10
+⋯
Therefore the saturation of injected fluid is
𝑆𝑖 (𝑋, 𝑇)
= 𝑆𝑖𝑜 +
+ 𝑐0 [
+ 𝑐0 {
𝜔𝑋
2√𝑇
+
+ 𝑐0 {
𝜔2 𝑋 2
𝜔𝑋 3
+
3}
8𝑇
24(√𝑇)
𝜔2 𝑋 2
𝜔𝑋 3
+
3
8𝑇
24(√𝑇)
𝑆𝑖𝑜 𝜔2 𝑋 2
𝑆𝑖𝑜 𝜔𝑋 3
𝜔3 𝑋 3
𝜔2 𝑋 4
+
+
+
3
3
8𝑇
32𝑇 2
24(√𝑇)
16(√𝑇)
𝜔𝑋 5
∞
𝑓(𝑛) = 𝑓0 (𝑛) + ∑ 𝑓𝑚 (𝑛)
(47)
5 }]
+⋯
320(√𝑇)
Determining the optimal value of 𝒄𝟎 :
As given by Liao, the discrete squared residual error at the
𝑚𝑡ℎ order of approximation is
𝐸𝑚 =
𝐾
𝑚
𝑗=0
𝑘=0
1
∑ [𝑁 (∑ 𝑓𝑘 (𝑗∆𝑥))]
𝐾
2
1
where ∆𝑥 = . Since the squared residual 𝐸𝑚 is dependent
𝐾
upon 𝑐0 , the optimal homotopy approximation is gained by
𝑑𝐸𝑚 (𝑐0 )
=0
𝑑𝑐0
Here the optimal value of 𝑐0 is determined by the minimum of
𝐸10 corresponding to the nonlinear algebraic equation
𝐸′10 = 0 and it is c0 = −3.353767389897125. For this
value of 𝑐0 , we get the convergent homotopy-series solution.
IV.
NUMERICAL AND GRAPHICAL
REPRESENTATION
Figure 2 represents the graph of 𝑆𝑖 (𝑋, 𝑇) verses distance 𝑋 for
time 𝑇 = 0.1, 0.2, 0.3, 0.4, 0.5 𝑎𝑛𝑑 0.6. Numerical values
of Table 1 are used for Figure 2.
V. CONCLUSION
The basic optimal HAM is applied to find the solution of the
nonlinear differential equation (31) .The value of the
convergence-control parameter is obtained by the minimum of
the discrete squared residual. It is found that the saturation of
the injected fluid 𝑆𝑖 (𝑋, 𝑇) increases smoothly with increase in
distance X.
The BVPh, a Mathematica package, is used to obtain
numerical presentation. Table 1 indicates the numerical values
of saturation of injected water for different distance 𝑋 and
time 𝑇
Figure 1: Secondary oil recovery process in oil reservoir
X=0
X=0.1
X=0.2
X=0.3
X=0.4
X=0.5
X=0.6
X=0.7
X=0.8
X=0.9
X=1
Table 1: Numerical values of the Saturation 𝑺𝒊 (𝑿, 𝑻) of injected water
T=0.1
T=0.2
T=0.3
T=0.4
T=0.5
0.2
0.2
0.2
0.2
0.2
0.2015120
0.2010920
0.2008980 0.2007810
0.20070
0.202690
0.2020530
0.2017220 0.2015120
0.2013640
0.2034140
0.2028020
0.2024190 0.2021560
0.2019630
0.2037640
0.203320
0.2029620
0.202690
0.2024790
0.2038970
0.2036380
0.2033530 0.2031070
0.2029020
0.2039380
0.203810
0.2036130 0.2034140
0.2032320
0.203947
0.2038940
0.2037730 0.2036260
0.2034770
0.203949
0.2039290
0.2038630 0.2037640
0.2036510
0.2039540
0.2039430
0.203910
0.2038490
0.2037690
0.204306
0.203947
0.203933
0.203897
0.203845
T=0.6
0.2
0.200640
0.2012520
0.2018140
0.202310
0.2027290
0.203070
0.2033370
0.2035370
0.2036810
0.203781
OHAM Solution
0.205
0.204
0.203
T=0.1
0.202
T=0.2
Si(X,T) 0.201
T=0.3
0.2
T=0.4
0.199
0.198
T=0.5
0.197
T=0.6
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1
X
Figure 2: Saturation of injected water at distance 𝑿 for different time 𝐓
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