An Analytic Soluion of Fingering Phoenomenon Arising in Fluid Flow through Porous Media by Using Techniques of Calculus of Variation and Similarity Theory

The present paper represents an analytical solution of fingering phonomenon arising in double phase flow through homogeneous media under certain initial & boundary condition using techniques of calculus of variation and similarity theory. The numerical and graphical representation of solution has been given the graph of saturatin F(η) of injected liquid, is increasing after η = 0.5 for t > 0, which indicates that when injected liquid entries into native liquid at common-interface, then suddenly the native liquid enters into injected liquid due to difference in wettability. Hence initial saturation will decrease and then after η > 0.5 the saturation uniformly increases parabolically which is physically consistent with the available theory.

Saturation of Oil η(x, t) The interface curve K Intrinsic permeability

Introduction
It is a very well-known physical fact that when a fluid, contained in a porous media, is displaced by another of lesser viscosity, instead of regular displacement of whole front, perturbations (fingers) occur which shoot through the porous medium at relatively great speed.This phenomenon of occurrence of instabilities is called fingering.
Immiscible flow of heavy oil in a porous formation by high temperature pressurized water has been numerically studied.
The physical region is a square domain in the horizontal plane with low and high pressure points at the opposite corners along one of the diagonals.Water, the invading fluid, when introduced at high pressure displaces the in situ oil towards the low-pressure production zone.The extent of displacement of oil by water through the porous medium in a given amount of time and the appearance of preferential flow paths (fingers) is the subject of the present investigation.
The resistance to water-oil movement arises from the viscous forces in the fluid phases and the capillary force at their interface.Based on their relative magnitudes, various forms of displacement mechanisms can be realized.As the viscosity ratio of heavy oil to water is large, viscous forces in the oil phase become dominant and constitute the major factor for controlling the flow distortions in the porous formation.A mathematical model that can treat the individual fluid pressures, capillary effects and heat transfer has been employed in the present work.A fully implicit, two-dimensional numerical model has been used to compute the pressure and temperature fields.The domain decomposition technique has been adopted in the numerical solution since the problem is computationally intensive.Naturally occurring oil-rich reservoirs to which the present study is applicable are inhomogeneous and layered.A qualitative study has been carried out to explore the effect of permeability variations on the flow patterns.Numerical calculations show that non-isothermal effects as well as layering, promote the formation of viscous fingers and consequently the sweep efficiency of the high-pressure waterfront.
In the statistical treatment of fingering (Scheidegger, A.E., 1961) only average cross sectional area occupied by the fingers, is taken into account, the size and shape of the individual fingers are disregarded.Scheideger and Johnson (1961) introduce the idea of discussing the statistical behaviour of instabilities in homogeneous porous media and considered the phenomenon without the effecct of capillary pressure.Verma (1964) has examined the behaviour of fingering phenomenon in a displacement process through heterogeneous porous medium from statistical point of view.It has been shown that fingers may be stabilized in homogeneous media statistical view point by (Scheidegger, A.E., 1960) many authors, for example, Chouke (1959), Jecquard (1940), Verma (1924), have investigated this phenomenon with different aspects.

Formulation of Problem
Let water be injected with constant velocity into a dipping oil saturated porous medium of homogeneous physical characteristic.The displacement of oil by water gives rise to a well-developed finger flow as shown in Figure 1(a), 1(b).
From Darcy's law the seepage velocity of water V w and oil V 0 can be written as where ρ w and ρ 0 are constant densities of water and oil respectively, α is the inclination of the bed, g is acceleration due to gravity.
The equation of continuity of the phases is given by From the definition of phase saturation we have The capillary pressure, which is defined as the pressure discontinuity of the following phases across the common interface is written as The equation of motion for saturation can be obtained by substitution of the values of V w and V 0 from equations ( 1) and (2) in equation ( 3) and ( 4) respectively.Thus we have Substituting the value of ∂P w ∂X from Eq. ( 7), Eq. ( 8) reduces to ¢ www.ccsenet.org/jmrISSN: 1916-9795 Now considering Eq. ( 9) and ( 10) Integrating both sides w.r.t."x" we have where q is the constant of integration or, Substituting the value of ∂P 0 ∂X from equation ( 13), equation ( 10) becomes (Appendix A).
The value of the pressure of oil (P 0 ) can be written as Where P is the mean pressure.Since P, the mean pressure is constant hence we have Substituting the value of ∂P 0 ∂X in equation ( 12) we get (Appendix B) using the above value of "q" equation ( 14) reduces to (Appendix C)

A Special case study
For definitions of the mathematical analysis, we assume a standard form for the relationship between capillary pressure, permeability of water & permeability of oil with phase saturation as where B and C are constant.
Using the above values equation ( 18) reduces as follow : This is the equation of motion for saturation, with the boundary condition when x = 0 then S w (0, t) = S w0 , Since an exact solution of equation ( 20) is difficult to obtain due to non-linear forms there in hence we have obtain the approximate solution of the above problem by Rayleigh-Ritz method :

Solution procedure
Since capillary pressure is very small in porous media, hence assuming that the capillary pressure P c as zero or B as zero, equation ( 20) reduces to where Using Birkhof's technique of one parameter group transformation, defined as where parameter a 0, and q, u, v are real numbers to be determined.21) using above value becomes

Hence we have ∂S
equation ( 24) is absolute conformed invaraint under T 1 provided and choosing an arbitrary constant "A" as follows Thus the invariants of group T 1 is given by η = x l (28) and where dash represent differentiations w.r.t.'η' Substituting the above values in equation ( 21) we have This is an ordinary differential equation of first order.
Let trial solutionis : substituting the value of F, F we have Hence from equation ( 29) that is S w = t 2 F(η) we have Case 2.
(2)Reducing this variational problem to a minimizing problem by assuming an approximate solution in the form Where the trial functions φ i (x) satisfy the boundary conditions and φ i (x) = 0 on the C of its region R.
Let the integral to be extremised be

Journal of Mathematics Research
September, 2009 Such that y(a) = A, and y(b) = B.
Substituting (31) in ( 32) by replacing y by y in I, giving I as a function of the unknown's c i .Then c's become parameters, which are so determiend as to extremise I.This requires Solving these equations, we get the values of c i , which when substituted in (31) give the desired solution.It's solution is equivalent to extremising the following integral : The functional of the above problem is as follows : Since the Euler's equation gives Eq. ( 35).
Now assuming the trail function as Differentiating with respect to η we have Now replacing F by F, and substituting the values of F and F in equation ( 34) we have It's stationary value is given by dI dβ = 0. Differentiating equation ( 38) with respect to β we have Since the trail solutin of the second order functional equation will be of degree at most two hence β = 2.So the approximate solution is Hence from equation ( 29) that is S w = t 2 F(η) we have

Conclusion
The equation represents analytical solution of fingering phenomenon arising double phase flow through homogeneous media under initial & boundary condition.
The graphical presentation has been given by figure -3 that is the graph of F(η) verses η.The graph of saturation F(η) of injeted liquid is increasing after η = 0.5 for t > 0, which indicates that when injected liquid entries into native liquid at common-interface, then suddenly the native liquid enters into injected liquid due to difference in wettability.
Hence initial saturation will decrease and then after η > 0.5 the saturation uniformly increases parabolically which is physically consistent with the available theory.