# American Institute of Mathematical Sciences

April  2014, 7(2): 317-346. doi: 10.3934/dcdss.2014.7.317

## Numerical analysis of a non equilibrium two-component two-compressible flow in porous media

 1 King Abdullah University of Science and Technology (KAUST), Computer, Electrical and Mathematical Sciences & Engineering, 23955-6900, Thuwal 2 Ecole Centrale de nantes, Laboratoire de Mathématiques Jean Leray, UMR CNRS 6629, 1, rue de la Noé, 44321 Nantes

Received  May 2013 Revised  June 2013 Published  September 2013

We propose and analyze a finite volume scheme to simulate a non equilibrium two components (water and hydrogen) two phase flow (liquid and gas) model. In this model, the assumption of local mass non equilibrium is ensured and thus the velocity of the mass exchange between dissolved hydrogen and hydrogen in the gas phase is supposed finite.
The proposed finite volume scheme is fully implicit in time together with a phase-by-phase upwind approach in space and it is discretize the equations in their general form with gravity and capillary terms We show that the proposed scheme satisfies the maximum principle for the saturation and the concentration of the dissolved hydrogen. We establish stability results on the velocity of each phase and on the discrete gradient of the concentration. We show the convergence of a subsequence to a weak solution of the continuous equations as the size of the discretization tends to zero. At our knowledge, this is the first convergence result of finite volume scheme in the case of two component two phase compressible flow in several space dimensions.
Citation: Bilal Saad, Mazen Saad. Numerical analysis of a non equilibrium two-component two-compressible flow in porous media. Discrete & Continuous Dynamical Systems - S, 2014, 7 (2) : 317-346. doi: 10.3934/dcdss.2014.7.317
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