Mathematical model and numerical solution of filtration problem for two immiscible fluids

In this paper, a one-dimensional nonlinear time-dependent problem is studied, the problem being a mathematical model of a flow of two immiscible fluids in porous medium. Fluid flow is described by two nonlinear parabolic degenerated differential equations, which act in subdomains separated by a priori unknown moving boundary. The main feature of this problem is that both functions and fluxes have jumps through unknown boundary. On this boundary some additional conditions are imposed to connect jumps of functions with jumps of fluxes.
The studied problem is formulated in a fixed domain. Scalar relations between values of solution functions on a moving boundary give additional conditions for finding jumps of these functions.
The implicit finite-differences approximation of the problem is constructed and algorithm for its implementation is suggested. The numerical experiments demonstrate the adequateness of constructed model to physical process and good efficiency of proposed algorithm.