Adomian Decomposition of Buckley-Leverett Equation with Capillary Effects

Abstract

Petroleum reservoir engineering problems are known to be inherently nonlinear. Consequently, solutions to the complete multiphase flow equations have been principally attempted with numerical methods. However, simplified forms of the problem have been solved some 60 years ago, when the Buckley-Leverett formulation was introduced. Ever since that pioneer work, which neglected the capillary term, this formulation has been widely accepted in the petroleum industry. By using the method of characteristic, the multiphase one-dimensional fluid flow was solved. However, the resulting solution was a triple-valued one for a significant region. For decades, the existence of multiple solutions was considered to be the result of nonlinearity. Buckley and Leverett introduced shock utilizing the concept of material balance, and, two decades later, when numerical solutions were possible, it was discovered that the triple-value problem disappeared if the complete flow equation, including the capillary pressure form, is solved. Numerical methods, however, are not free from linearization. In fact, every numerical solution imposes linearization at some point of the solution scheme. Therefore, a numerical technique cannot be used to definitively state the origin of multiple solutions. In this article, a semi-analytical technique, the Adomian decomposition method (ADM), capable of solving nonlinear partial differential equations without any linearizing assumptions, is used to unravel the true nature of the one-dimensional, two-phase flow. Results show that the Buckley-Leverett shock is neither necessary nor accurately portrayed in the displacement process. By using the ADM, the solution profile observed through numerous experimental studies was rediscovered. This article opens up an opportunity to seek approximate but close to exact solutions to the multiphase flow problems in porous media.