The Existence of Weak Solutions to Single Porosity and Simple Dual-porosity Models of Two-phase Incompressible Flow*†

Abstract. It is shown that there exists a weak solution to a degenerate and singular elliptic-parabolic partial integro-differential system of equations. These equations model two-phase incompressible flow of immiscible fluids in either an ordinary porous medium or in a naturally fractured porous medium. The full model is of dual-porosity type, though the single porosity case is covered by setting the matrixto-fracture flow terms to zero. This matrix-to-fracture flow is modeled simply in terms of fracture quantities; that is, no distinct matrix equations arise. The equations 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 of the first three types. A weak solution is obtained as a limit of solutions to discrete time problems. The proof makes no use of the corresponding regularized system. The hypotheses required for various earlier results on single-porosity systems are weakened so that only physically relevant assumptions are made. In particular, the results cover the cases of a singular capillary pressure function, a pure Neumann boundary condition, and an arbitrary initial condition.