Analytical Treatment and Convergence of the Adomian Decomposition Method for Instability Phenomena Arising during Oil Recovery Process

An abstract result is proved for the convergence of Adomian decomposition method for partial differential equations that model porous medium equation. Moreover, we prove that this decomposition scheme applied to a porous medium equation arising in instability phenomena in double phase flow through porous media is convergent in a suitable Hilbert space. Furthermore, this technique is utilized to find closed-form solutions for the problem under consideration. 1. Introduction The oil-water movement in a porous medium is an important problem of petroleum technology and water hydrology (Scheidegger [1]). The motion of two immiscible fluids in a homogenous porous medium was obtained by Buckley and Leveret without considering capillary pressure. The basic assumption underlying in the present investigation is that the oil and water form two immiscible liquid phases and water represents preferentially wetting phase. During secondary oil recovery when a fluid (oil) contained in a porous matrix of an oil formatted region in an oil reservoir is displaced by another fluid of lesser viscosity, that is, water, instead of regular displacement of the whole front, perturberance (fingers) may occur which shoot thorough the porous medium at relatively great speed. These protuberances are called fingers. This phenomenon is called instability. Scheidegger [1] analyzed the statistical behavior of instability in a displacement process through a homogeneous porous medium with capillary pressure and pressure-dependent phase densities. This problem has great importance for oil production in petroleum technology. Displacement of oil from a porous matrix by an external force which gives rise to a pressure gradient is known as forced instability. Instability can occur in cocurrent and counter-current flow modes. The stability of a water flood depends on the mobility ratio between oil, water, heterogeneity of the porous medium, segregation of the fluids in the reservoir, and dissipation of fluid fronts caused by capillary pressure. Instabilities may occur in both miscible and immiscible processes and originate on the interface between oil and water. These frontal instabilities are often characterized by a number of penetrating fingers of displacing fluid. Many of the oil fields around the world contain high viscous oil which gives a mobility ratio greater than unity ( ). Such mobility ratios are unfavorable and may cause the occurrence of frontal instabilities. If , the injected water may channel through the oil in an unstable manner. This viscous fingering leads to premature