### Numerical Study for Two-Phase Flow with Gravity in Homogeneous and Piecewise-Homogeneous Porous Media

#### Abstract

In this work we study two-phase flow with gravity either in 1-rock homogeneous media or 2-rocks composed media, this phenomenon can be modeled by a non-linear scalar conservation law with continuous flux function or discontinuous flux function, respectively. Our study is essentially from a numerical point of view, we apply the new Lagrangian-Eulerian finite difference method developed by Abreu and Pérez and the Lax-Friedrichs classic method to obtain numerical entropic solutions. Comparisons between numerical and analytical solutions show the efficiency of the methods even for discontinuous flux function.

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DOI: https://doi.org/10.5540/tema.2020.021.01.21

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