 
Summary: THE EXISTENCE OF WEAK SOLUTIONS TO
SINGLE POROSITY AND SIMPLE DUALPOROSITY MODELS
OF TWOPHASE INCOMPRESSIBLE FLOW*
Todd Arbogast
Department of Mathematical Sciences,
Rice University, Houston, Texas 772511892
Key words and phrases: porous medium, dualporosity, twophase, ellipticparabolic,
integrodifferential, degenerate
Abstract. It is shown that there exists a weak solution to a degenerate and singu
lar ellipticparabolic partial integrodifferential system of equations. These equa
tions model twophase incompressible flow of immiscible fluids in either an ordinary
porous medium or in a naturally fractured porous medium. The full model is of
dualporosity type, though the single porosity case is covered by setting the matrix
tofracture flow terms to zero. This matrixtofracture flow is modeled simply in
terms of fracture quantities; that is, no distinct matrix equations arise. The equa
tions are considered in a global pressure formulation that is justified by appealing
to a physical relation between the degeneracy of the wetting fluid's mobility and the
singularity of the capillary pressure function. In this formulation, the elliptic and
parabolic parts of the system are separated; hence, it is natural to consider various
boundary conditions, including mixed nonhomogeneous, saturation dependent ones
