Mathematical Model and Solution for Fingering Phenomenon in Double Phase Flow through Homogeneous Porous Media

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.


Introduction
Analytical and numerical simulation of the problems arising in oil-water 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 well-known 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 1. This phenomenon is called fingering or instability phenomenon. In the statistical treatment of the fingers only average cross-sectional area occupied by the fingers is considered while the size and shape of the individual fingers are neglected [1]. Many researchers have discussed this phenomenon from various view points. Sheideger and Johnson have discussed the statistical behavior of fingering in homogeneous porous media without capillary pressure [1]. Verma has examined the behavior of fingering in a displacement process through heterogeneous porous media with capillary pressure and pressure dependent phase densities [2]. Mehta has used special relation with capillary pressure and he used singular perturbation technique to find its solution [3]. Verma [7]. All the above researches have neglected the external sources and sink in the mass conservation equations. In the present study the mathematical model has been presented by considering the mass flow rates of oil and water in the equations of continuity, and the governing nonlinear partial differential equation has been obtained for saturation of injected water.

Statement of Problem
As shown in Figure 1, a welldeveloped fingers flow is furnished on account of uniform water injection into the oil saturated isotropic, homogeneous porous medium. The schematic presentation of fingers is expressed in Figure 2.
Our particular interest in the present investigation is to develop a mathematical model by considering the mass flow rate of oil and water and discuss the fingering phenomenon analytically by using variational iteration method.

Mathematical Formulation
The seepage velocity of water (injected fluid) ( ) and oil (native fluid) ( ) is given by Darcy's law [8] = − ( ) , where is the permeability of the isotropic, homogeneous porous medium, and are the respective relative permeability of water and oil, and and are the respective pressure of water and oil, and are the respective viscosity of water and oil.
The equations of continuity of two phases are given as [8] ( where and are the constant mass flow rate of water and oil, and are density of water and oil, and are the saturation of water and oil, respectively, and is the porosity of the medium.
From the definition of phase saturation [1], The capillary pressure , defined as the pressure discontinuity of the flowing phases across their common interface, is given by For definiteness we assume capillary pressure as a linear function of the saturation of water ( ) as where is a positive constant. The relative permeability of water and oil is considered from the standard relationship due to Scheidegger and Johnson [1] given by = , The equations of motion for saturation are obtained by substituting the values of (1) in (2), respectively, as Eliminating / from (4) and (8) we get Combining (9) and (10) and using (3) we get Integrating (11) with respect to , where 1 is a constant of integration.
Expressing as = + (1/2) , where = ( + )/2 is a constant mean pressure, we have Thus from (15) and (12) we get Substituting the value of 1 in (14) and on simplification we have Using (6) and (5) in (17) and after some simplification, we get where porosity and permeability are treated as constant for isotropic, homogeneous porous medium. Considering the dimensionless variables, in (19), we get where = 2 2 / . In order to solve (21) completely the following specific initial and boundary conditions are considered:
Here the initial approximation V 0 is assumed from the initial condition where the function ( ) is considered to be in parabolic nature. Pradhan et al. [5] have discussed the fingering phenomenon numerically by assuming ( ) to be a linear function of space variable.

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The Scientific World Journal Using (24) and (25) we get the following iterations with the help of Mathematica software: ) , Further approximations can be similarly obtained. Considering the first four approximations, the resulting approximate analytical solution is given by

Convergent Analysis
be an operator from Hilbert space H to H. The series solution

(29)
Remark 2. If the first finite , = 1, 2, . . . , , are not less than one and ≤ 1 for > , then, of course, the series solution ∑ ∞ =0 V ( ) of problem converges. In other words, the finite terms do not affect the convergence of the series solution [10]: Based on the above theorem the approximate analytical solution given by (27) is convergent.

Numerical and Graphical Presentation of Solution
The numerical values of the saturation of water ( , ) are shown in Table 1 for different values of time and distance. The graphical representation of the same has been shown in Figures 3 and 4. From Figures 3 and 4, it is observed that saturation of injected water increases with the space variable and time variable . This resembles well with the physical phenomenon of the problem.

Conclusion
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 2; however the parameters and can be assumed as the function of space variable in the case of anisotropic, heterogeneous porous medium, and the relative permeabilities and are assumed as function of saturation under the equilibrium condition. These relative 6 The Scientific World Journal permeabilities can also be assumed as a function of effective saturation under the nonequilibrium effects. The capillary pressure has been assumed to depend only on the saturation of the wetting phase (water); this capillary pressure can also depend on the surface tension, porosity, permeability, and the contact angle with the rock surface of the wetting phase which in turn depends on the temperature and fluid composition; with such assumption on capillary pressure the parameters can also be included to study its effect in future. Darcy law is considered in two-phase system without the gravitational forces; the differential form of the Darcy law can be extended to three-phase system with and without gravitational forces. The present mathematical model for onedimensional flow can also be extended to two-dimensional, three-dimensional flows for isotropic, homogeneous and anisotropic, heterogeneous porous media.
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.