We consider the flow of two-phases, say oil and water, in a porous medium. The classical model of this flow involves a elliptic-hyperbolic system, based on the Darcy's law. The saturation is governed by a hyperbolic conservation law and pressure obeys an elliptic equation. The problem of existence of global weak solutions for this model is still open. The main difficulty being the lack of regularity of the velocity field.
We propose two ways of modify the Darcy's law. The resulting models are still hyperbolic-elliptic system but they have a more regular velocity field. We show the existence of global-in-time solutions for those models and compare them.