Numerical investigation on a block preconditioning strategy to improve the computational efficiency of DFN models

Simulation of fluid flow dynamics in fractured porous media is an important issue in several subsurface models. The intricate network generated by hundreds of fractures produces complex multi-scale geometries that can be modelled in different ways. In contrast to homogenization-based techniques, discrete fracture network (DFN) models explicitly represent the fracture planes and their properties, prescribing continuity constraints for the fluid flow along the linear intersections. We focus on the formulation of the DFN model as a PDE-constrained optimization problem as originally proposed in [1].  This approach uses a non-conforming mesh and decouples the global problem in local ones, thus being suitable for an effective parallel implementation [2]. Imposing the flow continuity by a Lagrange-multiplier technique gives rise to a linearized algebraic problem where the global matrix K has a symmetric saddle-point structure with a rank-deficient leading block. In this work, we focus on accelerating the iterative solution of the system with matrix K by introducing effective block preconditioning techniques. First, an appropriate permutation of K is performed, in order to avoid a singular leading block though losing the global symmetry. Then, we restrict K to the coarse space of the fracture traces and solve inexactly the projected matrix by either an explicit or a matrix-free approach. The granular properties and the structure of K blocks are properly exploited in order to guarantee an efficient parallel implementation. The proposed algorithm is tested in applications of increasing size to verify its robustness and effectiveness in the acceleration of the iterative linear solver.[1]      S. Berrone, S. Pieraccini, S. Scialò. A PDE-constrained optimization formulation for Discrete Fracture Network flows, SIAM J. Sci. Comput., Vol. 35, pp. B487-B510, (2013) [2]      S. Berrone, S. Scialò, F. Vicini. Parallel meshing, discretization and computation of flow in massive Discrete Fracture Networks, SIAM J. Sci. Comput., Vol. 41, pp. C317-C338, (2019)