The present paper analytically discusses the phenomenon of fingering in double phase flow through homogenous porous media by using variational iteration method. Fingering phenomenon is a physical phenomenon which occurs when a fluid contained in a porous medium is displaced by another of lesser viscosity which frequently occurred in problems of petroleum technology. In the current investigation a mathematical model is presented for the fingering phenomenon under certain simplified assumptions. An approximate analytical solution of the governing nonlinear partial differential equation is obtained using variational iteration method with the use of Mathematica software.
Analytical and numerical simulation of the problems arising in oilwater displacement has become a predictive tool in oil industry. In oil recovery process, oil is produced by simple natural decompression without any pumping effort at the wells. This stage is referred to as primary recovery, and it ends when a pressure equilibrium between the oil field and the atmosphere occurs. Primary recovery usually leaves 70%–85% of oil in the reservoir. To recover part of the remaining oil, a fluid (usually water) is injected into some wells (injection wells) while oil is produced through other wells (production wells). This process serves to maintain high reservoir pressure and flow rates. It also displaces some of the oil and pushes it toward the production wells. This stage of oil recovery is called secondary recovery process.
It is a very wellknown physical fact that when a fluid having greater viscosity flowing through a porous medium is displaced by another fluid of lesser viscosity then, instead of regular displacement of whole front, protuberance takes place which shoot through the porous medium at a relatively very high speed, and fingers have been developed during this process as shown in Figure
Representation of fingers in a cylindrical piece of homogeneous porous media.
As shown in Figure
Schematic presentation of fingering (instability) phenomenon.
The seepage velocity of water (injected fluid) (
The equations of continuity of two phases are given as [
From the definition of phase saturation [
The capillary pressure
For definiteness we assume capillary pressure
The relative permeability of water and oil is considered from the standard relationship due to Scheidegger and Johnson [
The equations of motion for saturation are obtained by substituting the values of (
Eliminating
Combining (
Integrating (
On simplifying,
Substituting the value of (
Expressing
Thus from (
Substituting the value of
Using (
or
Considering the dimensionless variables,
in (
In order to solve (
Following the variational iteration method [
Define the operator
Define the components
Here the initial approximation
Using (
Let
If the first finite
The numerical values of the saturation of water
Numerical values of saturation of water at different values of time and distance.

 









0  0.136275  0.273109  0.410519  0.548521  0.687134 

0.0004  0.136679  0.273519  0.410934  0.548941  0.687560 

0.0016  0.137894  0.274748  0.412178  0.550202  0.688836 

0.0036  0.139918  0.276797  0.414253  0.552303  0.690964 

0.0064  0.142752  0.279666  0.417158  0.555244  0.693943 

0.01  0.146396  0.283355  0.420892  0.559025  0.697772 
The plot of time (
The plot of distance (
In the present investigation the phenomenon of fingering has been analytically discussed by considering the mass flow rate of injected water to determine the saturation of injected water for different values of time and distance. It is concluded that by considering the mass flow rate of oil and water, the saturation of injected water advances faster in comparison with the saturation of injected water neglecting the mass flow rate. The values of parameters used in present investigation are shown in Table
Values of different parameters.
Parameter  Value 




0.01 kg/ 

20 k Pa 

1000 kg/ 



1 m 

0.6800 
In the present study the mass conservation equation and Darcy’s law are considered for isothermal flows where the effect of temperature to the system is neglected; however, the mathematical model can be developed for nonisothermal flows. Analytical methods are the most widely used classical reservoir engineering methods in the petroleum industry in predicting petroleum reservoir performance. We concluded that the present variational iteration method used for finding the approximate analytical solution was found to be easy, accurate, and efficient in comparison with other analytical methods.
Seepage velocity of injected fluid (meter/second)
Seepage velocity of native fluid (meter/second)
Permeability of homogeneous porous medium (meter^{2})
Relative permeability of injected fluid (dimensionless)
Relative permeability of native fluid (dimensionless)
Viscosity of injected fluid (pascal second)
Viscosity of native fluid (pascal second)
Density of injected fluid (kg/meter^{3})
Density of native fluid (kg/meter^{3})
Mass flow rate of water (kg/(second·meter^{3}))
Mass flow rate of oil (kg/(second·meter^{3}))
Pressure of injected fluid (pascal)
Pressure of native fluid (pascal)
Porosity of homogeneous porous medium (dimensionless)
Capillary pressure coefficient (pascal)
Saturation of water (dimensionless)
Linear coordinate for distance (meter)
Linear coordinate for time (second)
Linear coordinate for distance (dimensionless)
Linear coordinate for time (dimensionless)
Length of porous medium (meter)
Capillary pressure (pascal).