Existence and Asymptotic Behavior of Measure-Valued Solutions for Three-Phase Flow in Porous Media

We establish the existence of physically admissible measure-valued solutions for the initial boundary value problem modelling the Bow of water, gas, and oil in a one-dimensional reservoir under water flooding, and we prove the convergence of the time averages of the expected values for the saturations of water and gas to the state which represents pure water, at a rate which depends only on the shape of the flux functions near this state. We also present some useful general results about the weak convergence of probability measures to Dirac measures.