Multiscale and block-partitioned solvers for coupled reservoir flow and geomechanics

Understanding the hydro-mechanical behavior of subsurface formations is key in many subsurface engineering applications, such as hydrocarbon recovery, subsurface hydrology, and geologic carbon storage. Although one-way coupled or staggered in-time approaches may be successfully used in practice, it is unfortunately not in general known a priori when these weaker approximation assumptions are sufficiently accurate. Thus, the simultaneous solution of the governing equations becomes an essential requirement for making predictions, quantifying the uncertainty, and assessing risk. This talk focuses on efficient algorithms for the fully-implicit solution of the matrix systems that result from discretization and linearization of poromechanics equations. First, a multiscale finite element-finite volume (MSFE-FV) method for single-phase flow in elastic media is presented. Based on a two-grid approach, the method allows for the construction of accurate coarse-scale systems using locally computed basis functions. In contrast to upscaling methods, once the coarse-scale solution is obtained, it is mapped onto the original fine-scale resolution, again using the basis functions. Beyond providing approximate solutions at low computational cost, the multiscale method, in conjunction with a smoothing step, can also be applied as an effective two-level preconditioner for the fine-scale problem within a Krylov method. Second, a novel two-stage preconditioner for accelerating the iterative solution of coupled multiphase poromechanics equations is proposed based upon an approximate block-factorization of the Jacobian matrix. In the first stage, a generalized Constrained Pressure Residual (CPR) approach is used to construct a pressure-displacement system that involves the unknowns characterized by long range error components. Specifically, the reduced pressure-displacement system is 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. Numerical results are presented to illustrate accuracy, robustness, and efficiency of the proposed algorithms.

Speaker: Nicola Castelletto

Monday, 04/24/17

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Green Earth Sciences Building

367 Panama St, Room 104
Stanford University
Stanford, CA 94305

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