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.
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 . 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 . Verma has examined the behavior of fingering in a displacement process through heterogeneous porous media with capillary pressure and pressure dependent phase densities . Mehta has used special relation with capillary pressure and he used singular perturbation technique to find its solution . Verma and Mishra have discussed similarity solution for instability phenomenon in double phase flow through porous media . Pradhan et al. have discussed the solution of instability phenomenon by finite element method . Meher et al. discussed the solution of instability phenomenon arising in double phase flow through porous medium with capillary pressure using Exponential self similar solutions technique . Patel et al. have discussed the power series solution of fingering phenomena in homogeneous porous media . 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.
2. 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.
3. Mathematical Formulation
The seepage velocity of water (injected fluid) () and oil (native fluid) () is given by Darcy’s law  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  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 ,
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  given by
Integrating (11) with respect to , where is a constant of integration.
Expressing as , where is a constant mean pressure, we have
Substituting the value of in (14) and on simplification we have
or where porosity and permeability are treated as constant for isotropic, homogeneous porous medium.
Considering the dimensionless variables,
in (19), we get where .
In order to solve (21) completely the following specific initial and boundary conditions are considered:
4. Solution of Problem
Define the operator as
Define the components , , as
Here the initial approximation is assumed from the initial condition where the function is considered to be in parabolic nature. Pradhan et al.  have discussed the fingering phenomenon numerically by assuming to be a linear function of space variable.
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
4.1. Convergent Analysis
Theorem 1. Let be an operator from Hilbert space H to H. The series solution converges if such that , where
Remark 2. If the first finite , , are not less than one and for , then, of course, the series solution of problem converges. In other words, the finite terms do not affect the convergence of the series solution : Based on the above theorem the approximate analytical solution given by (27) is convergent.
5. 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.
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 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 one-dimensional 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.
|:||Seepage velocity of injected fluid (meter/second)|
|:||Seepage velocity of native fluid (meter/second)|
|:||Permeability of homogeneous porous medium (meter2)|
|:||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/meter3)|
|:||Density of native fluid (kg/meter3)|
|:||Mass flow rate of water (kg/(second·meter3))|
|:||Mass flow rate of oil (kg/(second·meter3))|
|:||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).|
A. E. Schedegger and Johnson, “The stastical behavior of instabilities in displacement process in porous media,” Canadian Journal of Physics, vol. 39, no. 2, pp. 326–334, 1961.View at: Google Scholar
A. P. Verma, “Statistical behavior of fingering in a displacement in heterogeneous porous media with capillary pressure,” Canadian Journal of Physics, vol. 47, no. 3, pp. 319–324, 1969.View at: Google Scholar
M. N. Mehta, An asymptotic expansion in fluid flow through porous media [Ph.D. thesis], Veer Narmad South Gujarat University, Surat, India, 1978.
V. H. Pradhan, M. N. Mehta, and T. Patel, “Numerical solution of nonlinear equation representing one dimensional instability phenomena in porous media by finite element technique,” International Journal of Advanced Engineering Technology, vol. 2, no. 1, pp. 221–227, 2011.View at: Google Scholar
R. Meher, M. N. Mehta, and S. K. Meher, “Exponential self similar solutions technique for instability phenomenon arising in double phase flow through porous medium with capillary pressure,” Applied Mathematical Sciences, vol. 4, no. 25-28, pp. 1329–1335, 2010.View at: Google Scholar
K. R. Patel, M. N. Mehta, and T. R. Patel, “The power series solution of fingering phenomenon arising in fluid flow through homogeneous porous media,” International Journal on Applications and Applied Mathematics, vol. 6, no. 2, pp. 497–509, 2011.View at: Google Scholar
Z. Chen, Reservoir Simulation-Mathematical Techniques in Oil Recovery, Society for Industrial and Applied Mathematics, 2007.
J.-H. He, “Variational iteration method—a kind of non-linear analytical technique: some examples,” International Journal of Non-Linear Mechanics, vol. 34, no. 4, pp. 699–708, 1999.View at: Google Scholar