Numerical Solution of Nonlinear Partial Differential Equations Arising In Fluid Flow through Porous Medium

Abstract

In this research work, we discuss mathematical solutions of Co-current and Counter-current in Homogeneous porous medium. Imbibition involves the movement of one immiscible fluid through one more. In the porous material or medium, imbibition phenomenon is considered by increasing saturating phase. This phenomenon can be divided into two different cases: Free (spontaneous or force Free) and Forced. We shall restrict our study to the free imbibition phenomenon. If the saturating part is drained into a porous material or media through resources of duct heaviness then this imbibition phenomenon is known as Spontaneous imbibition. It is a method of curved edges between the saturating and non-saturating phase lacking of any exterior force. Spontaneous imbibition divided into two different types due to the flow of phases: (1) Co-current imbibition (2) counter-current imbibition. newlineIf saturating in addition non-saturating parts passage in the similar path, then this type of Spontaneous imbibition known as Co-current imbibition while both are not moving in the same direction is known as counter-current. newlineWe referred many research papers for the understanding of the basic concepts of both the phenomenon s mathematical model. Many authors and research scholar has been investigated this phenomenon in the different point of view. Many researchers have been obtained numerical explanation of these phenomenons in Homogeneous, Heterogeneous and Cracked porous material or medium. Brownscombe, Dyes, Scheideggar, Mehta and Verma, Graham and Richardson did a remarkable work and played a very important role in this research field. The mathematical formulation of Counter-current and Co-current imbibition phenomenon is lead to the nonlinear partial differential equations. We are converting these governing equations into system of ODEs by using polynomial based differential quadrature process with different types of grid points (Uniform and Chebyshev-Gauss-Lobatto). The Differential Quadrature technique newlinePage X newlineis very powerful and efficient