Block-preconditioned Krylov Methods for Coupled Multiphase Reservoir Flow and Geomechanics

The present work focuses on numerical solution of partial differential equations coupling multiphase flow and mechanical processes in geological formations. The balance equations are discretized using finite volume and finite element techniques. A backward implicit time integration scheme is selected. Linearization of the system of nonlinear algebraic equations produces a Jacobian matrix characterized by block structure due to the coupled nature of mass and momentum balance equations. Based upon approximate block-factorization of the Jacobian, we propose a two-stage preconditioner for fully implicit simulation. Generalized Constrained Pressure Residual approach is used to construct a pressure-displacement system, involving the unknowns characterized by long range error components. Specifically, in the first stage elliptic components of the coupled problem are tackled, with the reduced system being solved by the fixed-stress block-partitioned algorithm recently advanced in the context of single-phase poromechanics. Once pressure and displacement degrees of freedom have been updated, a second stage preconditioner is applied to deal with the other unknowns, namely saturations. Note that, from an algebraic standpoint, both CPR and fixed-stress strategies are built on particular choices of sparse Schur complement approximations, which combine algebraic and physically-based arguments. Numerical results are presented to illustrate performance and robustness of the proposed preconditioner.