Efficiency Comparison of Direct and Krylov Subspace Iterative Parallel Solvers for Hydraulic Fracture Propagation Simulations in Shale Reservoirs

ABSTRACT: Hydraulic fracturing in unconventional shale reservoirs poses significant challenges due to complex geological features and the computational intensity of simulating fracture propagation. In this study, we introduce a fast parallel hydraulic fracture simulator that integrates the Displacement Discontinuity Method (DDM) for modeling rock deformation with the Newton-Raphson and Picard iterative methods for solving fluid flow equations. To address the evolving computational demands - arising from dynamically increasing matrix dimensions as fractures propagate - we implement two OpenMP-based parallel solvers: a direct method solver based on LU-decomposition and a Krylov subspace iterative solver employing the GMRES algorithm with incomplete LU preconditioning. Two test cases, representative of real field scenarios, are used to evaluate the accuracy and computational efficiency of these solvers. The simulation results demonstrate that all solvers accurately predict fracture geometries, while significant reductions in computational time are achieved through parallelization. Specifically, the Krylov subspace iterative solver exhibits superior scalability for larger, more complex simulations, achieving speedups up to 34, compared to a maximum speedup of 29 for the direct method solver. These findings provide critical insights into optimizing hydraulic fracture simulations, thereby facilitating more effective design and calibration of hydraulic fracturing treatments in unconventional reservoirs.