The existence of weak solutions to immiscible compressible two-phase flow in porous media: The case offields with different rock-types
Brahim Amaziane, Leonid Pankratov, Andrey Lvovich Piatnitski · Discrete and Continuous Dynamical Systems - B · 2013
We study a model describing immiscible, compressible two-phase flow, such as water-gas, through heterogeneousporous mediataking into account capillary and gravity effects. We will consider a domain made up of severalzones with different characteristics: porosity, absolute permeability, relative permeabilitiesand capillary pressure curves. This process can be formulated as a coupled system of partial differentialequations which includes a nonlinear parabolic pressure equation and a nonlinear degenerate diffusion-convectionsaturation equation. Moreover the transmission conditions are nonlinear and the saturation is discontinuous atinterfaces separating different media. There are two kinds of degeneracy in the studied system: the first oneis the degeneracy of the capillary diffusion term in the saturation equation, and the second one appears in theevolution term of the pressure equation.Under some realistic assumptions on the data, we show the existence of weak solutions with the help of an appropriateregularization and a time discretization. We use suitable test functions to obtain a priori estimates.We prove a new compactness result in order to pass to the limit in nonlinear terms. This passage to the limitis nontrivial due to the degeneracy of the system.