Convergence of iterative coupling for coupled flow and geomechanics

In this paper we study solving iteratively the coupling of flow and mechanics. We demonstrate the stability and convergence of two widely used schemes: the undrained split method and the fixed stress split method. To our knowledge this is the first time that such results have been rigorously obtained and published in the scientific literature. In addition, we propose a new stress split method, with faster convergence rate than known schemes. These results are specially important today due to the interest in hydraulic fracturing ([1], [3], [4] and [5]), in oil and gas shale reservoirs. This paper is dedicated to the 60th anniversary of C.J. van Duijn, because of his impact of applying rigorous mathematics to real world problems. The research of A.M. was partially supported by the GNR MOMAS (Modélisation Mathématique et Simulations numériques liées aux problèmes de gestion des déchets nucléaires) (PACEN/CNRS, ANDRA, BRGM, CEA, EDF, IRSN). He would like to thank Institute for Computational Engineering and Science (ICES), UT Austin for hospitality in April 2009, 2010 and 2011. The research by M. F. Wheeler was partially supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences through DOE Energy Frontier Research Center: The Center for Frontiers of Subsurface Energy Security (CFSES) under Contract No. DE-SC0001114. Andro Mikelić Université de Lyon, CNRS UMR 5208, Université Lyon 1, Institut Camille Jordan, 43, blvd. du 11 novembre 1918, 69622 Villeurbanne Cedex, FRANCE Tel.: +33-426234548 Fax: +33-956109885 E-mail: [email protected] Mary F. Wheeler The Center for Subsurface Modeling, Institute for Computational Engineering and Sciences The University of Texas at Austin, 201 East 24th Street Austin, TX 78712, U. S. A. Tel.: +1-512-475-8625 Fax: +1-512-232-2445 E-mail: [email protected]