# American Institute of Mathematical Sciences

March  2008, 9(2): 281-308. doi: 10.3934/dcdsb.2008.9.281

## A nonlinear degenerate system modelling water-gas flows in porous media

 1 Université du Sud Toulon Var, MC2 (Inria Futurs) and CPT/Imath, Av. de l'université, 83957 La Garde, France 2 Ecole Centrale de nantes, Laboratoire de Mathématiques Jean Leray, UMR CNRS 6629, 1, rue de la Noé, 44321 Nantes, France

Received  February 2007 Revised  October 2007 Published  December 2007

The aim of this paper is to study a system modelling the flow of an incompressible phase (water) and a compressible phase (gas) in porous media. Two kinds of degeneracy appear for this problem: a dissipative term and an evolution term degenerate with respect to the saturation. Global weak solutions are established for the system by introducing several approximate models. The first one consists in obtaining a non-degenerate dissipative system. The second one is a time discretization method in order to overcome the degeneracy in the evolution term. At this step, the subproblem is a non- degenerate elliptic system which is strongly coupled and highly nonlinear. Then the Leray-Schauder fixed point theorem instead of a classical Schauder fixed point theorem is the key point to solve such a problem.
Citation: Cédric Galusinski, Mazen Saad. A nonlinear degenerate system modelling water-gas flows in porous media. Discrete & Continuous Dynamical Systems - B, 2008, 9 (2) : 281-308. doi: 10.3934/dcdsb.2008.9.281
 [1] Shifeng Geng, Zhen Wang. Best asymptotic profile for the system of compressible adiabatic flow through porous media on quadrant. Communications on Pure & Applied Analysis, 2012, 11 (2) : 475-500. doi: 10.3934/cpaa.2012.11.475 [2] Shifeng Geng, Lina Zhang. Large-time behavior of solutions for the system of compressible adiabatic flow through porous media with nonlinear damping. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2211-2228. doi: 10.3934/cpaa.2014.13.2211 [3] Brahim Amaziane, Leonid Pankratov, Andrey Piatnitski. An improved homogenization result for immiscible compressible two-phase flow in porous media. Networks & Heterogeneous Media, 2017, 12 (1) : 147-171. doi: 10.3934/nhm.2017006 [4] 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 [5] Yangyang Qiao, Huanyao Wen, Steinar Evje. Compressible and viscous two-phase flow in porous media based on mixture theory formulation. Networks & Heterogeneous Media, 2019, 14 (3) : 489-536. doi: 10.3934/nhm.2019020 [6] Brahim Amaziane, Leonid Pankratov, Andrey Piatnitski. The existence of weak solutions to immiscible compressible two-phase flow in porous media: The case of fields with different rock-types. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1217-1251. doi: 10.3934/dcdsb.2013.18.1217 [7] Mario Ohlberger, Ben Schweizer. Modelling of interfaces in unsaturated porous media. Conference Publications, 2007, 2007 (Special) : 794-803. doi: 10.3934/proc.2007.2007.794 [8] Radosław Kurek, Paweł Lubowiecki, Henryk Żołądek. The Hess-Appelrot system. Ⅲ. Splitting of separatrices and chaos. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 1955-1981. doi: 10.3934/dcds.2018079 [9] Laurent Lévi, Julien Jimenez. Coupling of scalar conservation laws in stratified porous media. Conference Publications, 2007, 2007 (Special) : 644-654. doi: 10.3934/proc.2007.2007.644 [10] Leda Bucciantini, Angiolo Farina, Antonio Fasano. Flows in porous media with erosion of the solid matrix. Networks & Heterogeneous Media, 2010, 5 (1) : 63-95. doi: 10.3934/nhm.2010.5.63 [11] Cedric Galusinski, Mazen Saad. Water-gas flow in porous media. Conference Publications, 2005, 2005 (Special) : 307-316. doi: 10.3934/proc.2005.2005.307 [12] Ioana Ciotir. Stochastic porous media equations with divergence Itô noise. Evolution Equations & Control Theory, 2019, 0 (0) : 0-0. doi: 10.3934/eect.2020010 [13] Sébastien Court. Stabilization of a fluid-solid system, by the deformation of the self-propelled solid. Part II: The nonlinear system.. Evolution Equations & Control Theory, 2014, 3 (1) : 83-118. doi: 10.3934/eect.2014.3.83 [14] Sébastien Court. Stabilization of a fluid-solid system, by the deformation of the self-propelled solid. Part I: The linearized system.. Evolution Equations & Control Theory, 2014, 3 (1) : 59-82. doi: 10.3934/eect.2014.3.59 [15] S. Bonafede, G. R. Cirmi, A.F. Tedeev. Finite speed of propagation for the porous media equation with lower order terms. Discrete & Continuous Dynamical Systems - A, 2000, 6 (2) : 305-314. doi: 10.3934/dcds.2000.6.305 [16] Youcef Amirat, Laurent Chupin, Rachid Touzani. Weak solutions to the equations of stationary magnetohydrodynamic flows in porous media. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2445-2464. doi: 10.3934/cpaa.2014.13.2445 [17] Ting Zhang. The modeling error of well treatment for unsteady flow in porous media. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2171-2185. doi: 10.3934/dcdsb.2015.20.2171 [18] T. L. van Noorden, I. S. Pop, M. Röger. Crystal dissolution and precipitation in porous media: L$^1$-contraction and uniqueness. Conference Publications, 2007, 2007 (Special) : 1013-1020. doi: 10.3934/proc.2007.2007.1013 [19] Olivier Pironneau, Alexei Lozinski, Alain Perronnet, Frédéric Hecht. Numerical zoom for multiscale problems with an application to flows through porous media. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 265-280. doi: 10.3934/dcds.2009.23.265 [20] D. G. Aronson. Self-similar focusing in porous media: An explicit calculation. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1685-1691. doi: 10.3934/dcdsb.2012.17.1685

2018 Impact Factor: 1.008