A multiscale preconditioner for microscale deformation of fractured porous media

The direct numerical simulation (DNS) of elastic deformation on digitized images of porous materials at the micron (or pore) scale requires the solution of large systems of linear(ized) equations. Krylov solvers are instrumental but suffer from slow convergence without a preconditioner. We present a multiscale preconditioner that significantly accelerates DNS, is scalable on parallel machines, and can be non-intrusively applied within existing codes. The preconditioner is an algebraic reinterpretation of a recent pore-level multiscale method (PLMM) proposed by the authors. It combines a global preconditioner with a local smoother to simultaneously attenuate low- and high-frequency errors, respectively. Like PLMM, a single application of the global preconditioner yields an approximate solution that is sufficiently accurate in a wide range of applications (e.g., geologic CO₂/H₂ storage). The combination with a smoother enables improving the approximation even further. While all cracks are assumed to be static, we propose an adaptive strategy to update the preconditioner efficiently for evolving-crack problems without affecting the convergence rate of the Krylov solver. We validate the preconditioner against DNS and test its convergence on various 2D/3D microstructures and crack patterns. The agreement and performance are favorable.