A Study of Free Boundary Problems of Fluid Flow in Porous Media by Mixed Methods

Abstract

In this article, the ow of uids in porous media is studied as a free or moving boundary problem by the mixed method. In particular, a new weak formulation for the problem of seepage of uids through a porous media is discussed and analyzed mathematically and numerically. The new formulation is in a mixed form and is suitable for the use of mixed nite element methods in the numerical approximation. It is proved that the weak formulation and its nite element discretization has a solution which can be approximated by a sequence of regularized problems. 1. Introduction In this paper, we are concerned with the free or moving boundary value problem in the study of the ow of uids through a heterogeneous porous media. Such problems are important in many branches of science and engineering. For example, in the areas of soil science, agricultural engineering , and groundwater hydrology, the movement of uids and their dissolved components in both saturated and unsaturated soils is an important environmental consideration. In petroleum engineering, improved recovery of oil and gas is based on simulation of multiphase and multicomponent uid transport in deep rocks. In both application areas, mass transfer across phase boundaries is an important consideration which can be discussed in the context of free or moving boundary problems. Another important application of the free or moving boundary problem is water seepage through a dam, or rain water creeping through an unsatu-rated zone. The underlying physics of the petroleum and seepage problems are very similar. For comparison, assume that there are two uids owing simultaneously in the porous medium. In unsaturated ow, these uids are water and air, while in the petroleum problem, the uids are assumed to be water and oil. Relevant material properties, including the capillary pressure and relative permeability are assumed to be known. Free boundary problems are also seen in other areas of petroleum industry such as basin simulation. The research of basin simulation is important