Analysis of one-dimensional horizontal two-phase flow in geothermal reservoirs

In the absence of capillarity the single-component two-phase porous medium equations have the structure of a nonlinear parabolic pressure (equivalently, temperature) diffusion equation, with derivative coupling to a nonlinear hyperbolic saturation wave equation. The mixed parabolic-hyperbolic system is capable of substaining saturation shock waves. The Rankine-Hugoniot equations show that the volume flux is continuous across such a shock. In this paper we focus on the horizontal one-dimensional flow of water and steam through a block of porous material within a geothermal reservoir. Starting from a state of steady flow we study the reaction of the system to simple changes in boundary conditions. Exact results are obtainable only numerically, but in some cases analytic approximations can be derived. When pressure diffusion occurs much faster than saturation convection, the numerical results can be described satisfactorily in terms of either saturation expansion fans, or isolated saturation shocks. At early times, pressure and saturation profiles are functionally related. At intermediate times, boundary effects become apparent. At late times, saturation convection dominates and eventually a steady-state is established. When both pressure diffusion and saturation convection occur on the same timescale, initial simple shock profiles evolve into multiple shocks, for which no theory is currently available. Finally, a parameter-free system of equations is obtained which satisfactorily represents a particular case of the exact equations.