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

Abstract

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.

1. 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 and Mishra have discussed similarity solution for instability phenomenon in double phase flow through porous media [4]. Pradhan et al. have discussed the solution of instability phenomenon by finite element method [5]. 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 [6]. Patel et al. have discussed the power series solution of fingering phenomena in homogeneous porous media [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.

Figure 1 

Representation of fingers in a cylindrical piece of homogeneous porous media.

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.

Figure 2 

Schematic presentation of fingering (instability) phenomenon.

3. Mathematical Formulation

The seepage velocity of water (injected fluid) (𝑉_𝑖) and oil (native fluid) (𝑉_𝑛) is given by Darcy’s law [8] 𝑉_𝑖𝑘=−^𝑖𝜇_𝑖𝐾𝜕𝑃_𝑖,𝑉𝜕𝑥_𝑛𝑘=−^𝑛𝜇_𝑛𝐾𝜕𝑃_𝑛,𝜕𝑥 (1) where 𝐾 is the permeability of the isotropic, homogeneous porous medium, 𝑘_𝑖 and 𝑘_𝑛 are the respective relative permeability of water and oil, and 𝑃_𝑖 and 𝑃_𝑛 are the respective pressure of water and oil, 𝜇_𝑖 and 𝜇_𝑛 are the respective viscosity of water and oil.

The equations of continuity of two phases are given as [8] 𝜕𝑚𝜌_𝑖𝑆_𝑖+𝜕𝜌𝜕𝑡^𝑖𝑉_𝑖𝜕𝑥=𝑞_𝑖,𝜕𝑚𝜌_𝑛𝑆_𝑛+𝜕𝜌𝜕𝑡^𝑛𝑉_𝑛𝜕𝑥=𝑞_𝑛, (2) where 𝑞_𝑖 and 𝑞_𝑛 are the constant mass flow rate of water and oil, 𝜌_𝑖 and 𝜌_𝑛 are density of water and oil, 𝑆_𝑖 and 𝑆_𝑛 are the saturation of water and oil, respectively, and 𝑚 is the porosity of the medium.

From the definition of phase saturation [1], 𝑆_𝑖+𝑆_𝑛=1. (3)

The capillary pressure 𝑃_𝐶, defined as the pressure discontinuity of the flowing phases across their common interface, is given by 𝑃_𝐶=𝑃_𝑛−𝑃_𝑖. (4)

For definiteness we assume capillary pressure 𝑃_𝐶 as a linear function of the saturation of water (𝑆_𝑖) as 𝑃_𝐶=−𝛽𝑆_𝑖, (5) where 𝛽 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 𝑘_𝑖=𝑆_𝑖,𝑘_𝑛=𝑆_𝑛=1−𝑆_𝑖. (6)(7)

The equations of motion for saturation are obtained by substituting the values of (1) in (2), respectively, as 𝜕𝑚𝜌_𝑖𝑆_𝑖𝜕𝑡=𝑞_𝑖+𝜕𝜕𝑥𝐾𝜌_𝑖𝑘_𝑖𝜇_𝑖𝜕𝑃_𝑖,𝜕𝜕𝑥𝑚𝜌_𝑛𝑆_𝑛𝜕𝑡=𝑞_𝑛+𝜕𝜕𝑥𝐾𝜌_𝑛𝑘_𝑛𝜇_𝑛𝜕𝑃_𝑛.𝜕𝑥 (8)(9)

Eliminating 𝜕𝑃_𝑖/𝜕𝑥 from (4) and (8) we get 𝜕𝑚𝜌_𝑖𝑆_𝑖𝜕𝑡=𝑞_𝑖+𝜕𝜕𝑥𝐾𝜌_𝑖𝑘_𝑖𝜇_𝑖𝜕𝑃_𝑛−𝜕𝑥𝜕𝑃_𝐶.𝜕𝑥 (10)

Combining (9) and (10) and using (3) we get𝑞0=^𝑖𝜌_𝑖+𝑞_𝑛𝜌_𝑛+𝜕𝐾𝑘𝜕𝑥𝜇^𝑖_𝑖𝑘+𝐾^𝑛𝜇_𝑛𝜕𝑃_𝑛𝑘𝜕𝑥−𝐾^𝑖𝜇_𝑖𝜕𝑃_𝐶.𝜕𝑥 (11)

Integrating (11) with respect to 𝑥, 𝐶₁=𝑞_𝑖𝜌_𝑖+𝑞_𝑛𝜌_𝑛𝑥+𝐾𝑘_𝑖𝜇_𝑖𝑘+𝐾^𝑛𝜇_𝑛𝜕𝑃_𝑛𝑘𝜕𝑥−𝐾^𝑖𝜇_𝑖𝜕𝑃_𝐶,𝜕𝑥 (12) where 𝐶₁ is a constant of integration.

On simplifying, 𝜕𝑃_𝑛=𝐶𝜕𝑥𝐾𝑘_𝑖/𝜇_𝑖+¹𝐾𝑘_𝑛/𝜇_𝑛+𝐾𝑘_𝑖/𝜇_𝑖𝐾𝑘_𝑖/𝜇_𝑖+𝐾𝑘_𝑛/𝜇_𝑛𝜕𝑃_𝐶−𝑞𝜕𝑥_𝑖/𝜌_𝑖+𝑞_𝑛/𝜌_𝑛𝑥𝐾𝑘_𝑖/𝜇_𝑖+𝐾𝑘_𝑛/𝜇_𝑛. (13)

Substituting the value of (13) in (10), 𝜕𝑚𝑆_𝑖=𝑞𝜕𝑡_𝑖𝜌_𝑖+𝜕𝐾𝑘𝜕𝑥𝜇^𝑖_𝑖𝐶₁𝐾𝑘_𝑖/𝜇_𝑖+𝐾𝑘_𝑛/𝜇_𝑛+𝐾𝑘_𝑖/𝜇_𝑖𝐾𝑘_𝑖/𝜇_𝑖+𝐾𝑘_𝑛/𝜇_𝑛𝜕𝑃_𝐶−𝑞𝜕𝑥_𝑖/𝜌_𝑖+𝑞_𝑛/𝜌_𝑛𝑥𝐾𝑘_𝑖/𝜇_𝑖+𝐾𝑘_𝑛/𝜇_𝑛−𝜕𝑃_𝐶.𝜕𝑥 (14)

Expressing 𝑃_𝑛 as 𝑃_𝑛=𝑃+(1/2)𝑃_𝐶, where 𝑃=(𝑃_𝑖+𝑃_𝑛)/2 is a constant mean pressure, we have 𝜕𝑃_𝑛=1𝜕𝑥2𝜕𝑃_𝐶.𝜕𝑥 (15)

Thus from (15) and (12) we get 𝐶₁=𝑞_𝑖𝜌_𝑖+𝑞_𝑛𝜌_𝑛𝑥+𝐾𝑘_𝑖𝜇_𝑖𝑘+𝐾^𝑛𝜇_𝑛12𝜕𝑃_𝐶𝑘𝜕𝑥−𝐾^𝑖𝜇_𝑖𝜕𝑃_𝐶.𝜕𝑥 (16)

Substituting the value of 𝐶₁ in (14) and on simplification we have 𝜕𝑚𝑆_𝑖=𝑞𝜕𝑡𝜌^𝑖_𝑖+𝜕−𝜕𝑥𝐾𝑘_𝑖2𝜇_𝑖𝜕𝑃_𝐶.𝜕𝑥 (17)

Using (6) and (5) in (17) and after some simplification, we get 𝑚𝜕𝑆_𝑖=𝑞𝜕𝑡𝜌^𝑖_𝑖+𝐾𝛽2𝜇_𝑖𝜕𝑆𝜕𝑥^𝑖𝜕𝑆_𝑖𝜕𝑥 (18)

or 𝜕𝑆_𝑖=𝑞𝜕𝑡𝑚𝜌^𝑖_𝑖+𝐾𝛽2𝑚𝜇_𝑖𝜕𝑆𝜕𝑥^𝑖𝜕𝑆_𝑖,𝜕𝑥 (19) where porosity 𝑚 and permeability 𝐾 are treated as constant for isotropic, homogeneous porous medium.

Considering the dimensionless variables, 𝑥𝑋=𝐿,𝑇=𝐾𝛽2𝜇_𝑖𝑚𝐿²𝑡, (20)

in (19), we get 𝜕𝑆_𝑖𝜕𝜕𝑇=𝐴+𝑆𝜕𝑋^𝑖𝜕𝑆_𝑖,𝜕𝑋 (21) where 𝐴=2𝜇_𝑖𝐿²𝑞_𝑖/𝐾𝛽𝜌_𝑖.

In order to solve (21) completely the following specific initial and boundary conditions are considered: 𝑆_𝑖()(𝑋),𝑆𝑋,0=𝑓_𝑖(0,𝑇)=𝑓₁(𝑇),𝑆_𝑖(𝐿,𝑇)=𝑓₂(𝑇). (22)

4. Solution of Problem

Following the variational iteration method [9–11], we obtain the following iteration formula for (21): 𝑆_𝑖𝑘+1(𝑋,𝑇)=𝑆_𝑖𝑘(𝑋,𝑇)+^𝑇₀(−1)𝜕𝑆_𝑖𝑘−𝜕𝜕𝜏𝑆𝜕𝑋^𝑖𝑘𝜕𝑆_𝑖𝑘𝜕𝑋−𝐴𝑑𝜏. (23)

Define the operator 𝑂[𝑆_𝑖𝑘] as𝑂𝑆_𝑖𝑘=(−1)^𝑇₀𝜕𝑆_𝑖𝑘−𝜕𝜕𝜏𝑆𝜕𝑋^𝑖𝑘𝜕𝑆_𝑖𝑘𝜕𝑋−𝐴𝑑𝜏. (24)

Define the components 𝑣_𝑘, 𝑘=0,1,2,…, as 𝑣₀=𝑓(𝑋)=0.01𝑋²,𝑣₁=𝑂𝑣₀,𝑣₂=𝑂𝑣₀+𝑣₁,⋮𝑣_𝑘+1=𝑂𝑣₀+𝑣₁+⋯+𝑣_𝑘. (25)

Here the initial approximation 𝑣₀ is assumed from the initial condition where the function 𝑓(𝑋) is considered to be in parabolic nature. Pradhan et al. [5] have discussed the fingering phenomenon numerically by assuming 𝑓(𝑋) to be a linear function of space variable.

Using (24) and (25) we get the following iterations with the help of Mathematica software: 𝑣₁=−^𝑇₀𝜕𝑣₀𝜕𝜕𝜏−𝐴−𝑣𝜕𝑋⁰𝜕𝑣₀𝑣𝜕𝑋𝑑𝜏,₁=𝑇0.68+0.0006𝑋²,𝑣₂=−^𝑇₀𝜕𝑣₀+𝑣₁−𝜕𝜕𝜏−𝐴𝑣𝜕𝑋⁰+𝑣₁𝜕𝑣₀+𝑣₁𝑣𝜕𝑋𝑑𝜏,₂=−2.71051×10⁻²⁰𝑇𝑋²+𝑇³0.000272+7.2×10⁻⁷𝑋²+𝑇²0.0068+0.000036𝑋²,𝑣₃=−^𝑇₀𝜕𝑣₀+𝑣₁+𝑣₂−𝜕𝜕𝜏−𝐴𝑣𝜕𝑋⁰+𝑣₁+𝑣₂𝜕𝑣₀+𝑣₁+𝑣₂𝑣𝜕𝑋𝑑𝜏,₃=𝑇⁷5.59543×10⁻¹¹+4.44343×10⁻¹³𝑋²+𝑇⁶4.896×10⁻⁹+5.184×10⁻¹¹𝑋²+𝑇⁵3.5904×10⁻⁷+2.592×10⁻⁹𝑋²+𝑇⁴0.00001564+8.64×10⁻⁸𝑋²+𝑇³0.000045333+0.00000144𝑋²,𝑣₄=−^𝑇₀𝜕𝑣₀+𝑣₁+𝑣₂+𝑣₃−𝜕𝜕𝜏−𝐴𝑣𝜕𝑋⁰+𝑣₁+𝑣₂+𝑣₃𝜕𝑣₀+𝑣₁+𝑣₂+𝑣₃𝑣𝜕𝑋𝑑𝜏,₄=𝑇¹⁵3.31505×10⁻²⁴+7.89762×10⁻²⁶𝑋²+𝑇¹⁴7.25168×10⁻²²+1.97441×10⁻²³𝑋²+𝑇¹³8.59045×10⁻²⁰+2.30347×10⁻²¹𝑋²+𝑇¹²7.18117×10⁻¹⁸+1.72761×10⁻¹⁹𝑋²+𝑇¹¹4.41144×10⁻¹⁶+9.59781×10⁻¹⁸𝑋²+𝑇¹⁰2.07243×10⁻¹⁴+4.22303×10⁻¹⁶𝑋²+𝑇⁹8.47521×10⁻¹³+1.52854×10⁻¹⁴𝑋²+𝑇⁸3.26248×10⁻¹¹+4.73225×10⁻¹³𝑋²+𝑇⁷1.047744×10⁻⁹+1.24416×10⁻¹¹𝑋²+𝑇⁶2.77168×10⁻⁸+2.592×10⁻¹⁰𝑋²+𝑇⁵4.6512×10⁻⁷+4.1472×10⁻⁹𝑋²+𝑇⁴2.2667×10⁻⁷+4.32×10⁻⁸𝑋². (26) Further approximations can be similarly obtained. Considering the first four approximations, the resulting approximate analytical solution is given by 𝑆_𝑖(𝑋,𝑇)=𝑣₀+𝑣₁+𝑣₂+𝑣₃+𝑣₄=0.01𝑋²+𝑇¹⁵3.31505×10⁻²⁴+7.89762×10⁻²⁶𝑋²+𝑇¹⁴7.25168×10⁻²²+1.974405746938776×10⁻²³𝑋²+𝑇¹³8.590445×10⁻²⁰+2.30347×10⁻²¹𝑋²+𝑇¹²7.18117×10⁻¹⁸+1.72761×10⁻¹⁹𝑋²+𝑇¹¹4.41144×10⁻¹⁶+9.59781×10⁻¹⁸𝑋²+𝑇¹⁰2.07243×10⁻¹⁴+4.22303×10⁻¹⁶𝑋²+𝑇⁹8.47521×10⁻¹³+1.52854×10⁻¹⁴𝑋²+𝑇⁸3.26248×10⁻¹¹+4.73225×10⁻¹³𝑋²+𝑇⁷1.1037×10⁻⁹+1.28859×10⁻¹¹𝑋²+𝑇⁶3.26128×10⁻⁸+3.1104×10⁻¹⁰𝑋²+𝑇⁵8.2416×10⁻⁷+6.7392×10⁻⁹𝑋²+𝑇⁴0.00001587+1.296×10⁻⁷𝑋²+𝑇³0.00031733+0.00000216𝑋²+𝑇²0.0068+0.000036𝑋²+𝑇0.68+0.0006𝑋². (27)

4.1. Convergent Analysis

Theorem 1. Let 𝐴[𝑢]=^𝑡₀(−1)^𝑚()!()𝑚−1𝜏−𝑡^𝑚−1𝐿𝑢_𝑘(𝜏)+𝑁𝑢_𝑘(𝜏)−𝑔(𝜏)𝑑𝜏 (28) be an operator from Hilbert space H to H. The series solution 𝑢(𝑡)=∑^∞_𝑘=0𝑣_𝑘 converges if ∃0<𝛾<1 such that ‖𝑣_𝑘+1‖≤𝛾‖𝑣_𝑘‖∀𝑘∈𝑁∪{0} [10], where 𝑣₀=𝑢₀,𝑣₁=𝐴𝑣₀,𝑣₂=𝐴𝑣₀+𝑣₁,⋮𝑣_𝑘+1=𝐴𝑣₀+𝑣₁+⋯+𝑣_𝑘. (29)

Remark 2. If the first finite 𝛽_𝑖, 𝑖=1,2,…,𝑙, are not less than one and 𝛽_𝑖≤1 for 𝑖>𝑙, then, of course, the series solution ∑^∞_𝑘=0𝑣_𝑘(𝑡) of problem converges. In other words, the finite terms do not affect the convergence of the series solution [10]: 𝛽₀=𝑣₁𝑣₀𝛽=68.06,₁=𝑣₂𝑣₁𝛽=0.0104<1,₂=𝑣₃𝑣₂𝛽=0.0088<1,₃=𝑣₄𝑣₃𝛽=0.0122<1,₄=𝑣₅𝑣₄⋮=0.0190<1 (30) 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 𝑆_𝑖(𝑋,𝑇) 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 𝑋 and time variable 𝑇. This resembles well with the physical phenomenon of the problem.

Table 1 

Numerical values of saturation of water at different values of time and distance.

Figure 3 

The plot of time (𝑇) versus saturation of water (𝑆𝑖) for different values of distance (𝑋).

Figure 4 

The plot of distance (𝑋) versus saturation of water (𝑆𝑖) for different values of time (𝑇).

6. 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 𝑚 and 𝐾 can be assumed as the function of space variable in the case of anisotropic, heterogeneous porous medium, and the relative permeabilities 𝑘_𝑖 and 𝑘_𝑛 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 𝑃_𝐶 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.

Table 2 

Values of different parameters.

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.

Nomenclature

𝑉𝑖:	Seepage velocity of injected fluid (meter/second)
𝑉𝑛:	Seepage velocity of native fluid (meter/second)
𝐾:	Permeability of homogeneous porous medium (meter2)
𝑘𝑖:	Relative permeability of injected fluid (dimensionless)
𝑘𝑛:	Relative permeability of native fluid (dimensionless)
𝜇𝑖:	Viscosity of injected fluid (pascal second)
𝜇𝑛:	Viscosity of native fluid (pascal second)
𝜌𝑖:	Density of injected fluid (kg/meter3)
𝜌𝑛:	Density of native fluid (kg/meter3)
𝑞𝑖:	Mass flow rate of water (kg/(second·meter3))
𝑞𝑛:	Mass flow rate of oil (kg/(second·meter3))
𝑃𝑖:	Pressure of injected fluid (pascal)
𝑃𝑛:	Pressure of native fluid (pascal)
𝑚:	Porosity of homogeneous porous medium (dimensionless)
𝛽:	Capillary pressure coefficient (pascal)
𝑆𝑖:	Saturation of water (dimensionless)
𝑥:	Linear coordinate for distance (meter)
𝑡:	Linear coordinate for time (second)
𝑋:	Linear coordinate for distance (dimensionless)
𝑇:	Linear coordinate for time (dimensionless)
𝐿:	Length of porous medium (meter)
𝑃𝐶:	Capillary pressure (pascal).