On the interface law between a deformable porous medium containing a viscous fluid and an elastic body∗

Coupled multiphase flow and geomechanics models are computationally costly and complex to implement. In order to take advantage of petascale and future exascale computing power, parallel domain decomposition offers an opportunity for decoupling realistic subsurface problems. The basic idea is to reduce the complexity of these multiphysics problems by applying the coupling only in those domains where it is needed. Thus, in classical poroelastic modeling, one needs to take into account both the flow and geomechanics effects in the reservoir (pay zone). When the flow-geomechanics-interactive pay zone and the geomechanics-only non-pay zone are in contact, the natural question is what to set at their interface. In this paper we undertake a rigorous derivation of the interface conditions between a poroelastic medium (the pay zone) and an elastic body (the non-pay zone). We suppose that the poroelastic medium contains a pore structure of the characteristic size ε and that the fluid/structure interaction regime corresponds to diphasic Biot’s law. The question is challenging because the Biot’s equations for the poroelastic part contain one unknown more than the Navier equations for the non-pay zone. The solid part of the pay zone (the matrix) is elastic and the pores contain a viscous fluid. The fluid is supposed viscous and slightly compressible. We study the case when the contrast of property is of order ε2, i.e. the normal stress of the elastic matrix is of the same order as the fluid pressure. We suppose a periodic matrix and obtain the a priori estimates. Then we let the characteristic size of the inhomogeneities tend to zero and pass to the limit in the sense of the two-scale convergence. The obtained effective equations represent a two-scale system for 3 displacements and 2 pressures, coupled through the interface conditions with the Navier equations for the elastic displacement in the non-pay zone. We prove uniqueness for the homogenized 2-scale system. Then we introduce several auxiliary problems and obtain a problem without the fast scale. This new system is diphasic quasistatic and corresponds to the diphasic effective behavior already observed in papers by M. Biot on the ∗The research of Andro Mikelić is 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 The Center for Subsurface Modeling, ICES, UT Austin for hospitality in April 2009, 2010 and 2011. Mary F. Wheeler is partially supported by the DOE grant DE-FG0204ER25617, and the Center for Frontiers of Subsurface Energy Security under Contract No. DE-SC0001114. †E-mail:[email protected]