Local Mesh Refinement for Degenerate Two-phase Incompressible Flow Problems

In this paper we consider nite element methods for a degenerate elliptic-parabolic partial diierential system which describes the ow of two incompressible, immiscible uids in porous media. Local mesh reenement techniques for solving this degenerate system are discussed. 1. TWO-PHASE FLOW EQUATIONS In this paper we consider nite element methods for the ow of two incompressible, immiscible uids in porous media < d , d 3: (1:1) ?r f((s)rp + 0 1 (s))g = q(s); @ t s ? r f(r + w (s)rp + 0 2 (s))g = q w (s); where and are the porosity and absolute permeability of the porous system, s is the (reduced) saturation of the wetting phase (e.g. water), is the total mobility, p and are the so-called global pressure and complementary pressure respectively, w is the wetting phase mobility, 0 1 and 0 2 are the given functions of s, and q and q w are the sink/source terms. s is related to through (1:2) s = S(); where S(x;) is a strictly increasing function of. For the derivation of (1.1) from the usual two-phase ow equations, refer to 6, 10]. The pressure equation is given by the rst equation of (1.1), while the saturation equation is described by the second equation. They