A novel and efficient preconditioner for solving lagrange multipliers-based discretization schemes for reservoir simulations

We present a novel and efficient preconditioning technique to solve the non-symmetric system of equations associated with Lagrange multipliers-based discretization schemes, such as Mixed Hybrid Finite Element method (MHFE) and Mimetic Finite Difference method (MFD). These types of discretization have been gaining popularity lately and here we develop a fully dedicated preconditioner for them. Preconditioners are key to improve the efficiency of Krylov subspace methods, that provide a solution to the sequence of large-size, and often ill-conditioned, systems of equations originating from reservoir numerical simulations. The mathematical model of flow in porous media is governed by a set of two coupled nonlinear equations: The momentum and mass balance equations, discretized using either the MHFE or the MFD, and the Finite Volume method (FV), respectively. Unknowns are located on elements (element pressure and saturation) and faces (face pressure and phase capillary pressure), the latter behaving as Lagrange multipliers. The problem is solved by adopting a fully implicit approach and linearization is provided by a Newton-Raphson method, which leads to a block-structured Jacobian matrix. An original numerical formulation of the mass balance equation, where the continuity of fluxes is strongly imposed with the aim of increasing the efficiency of the nonlinear iteration, has been investigated. The resulting block Jacobian is not symmetric, thus requiring special preconditioning tools for its efficient solution. The preconditioning approach exploits the Jacobian block structure to develop a multi-stage strategy that addresses separately the problem unknowns. A crucial point is the approximation of the resulting Schur complements, which is carried out at an algebraic level by applying proper restriction operators to the full matrix blocks. The selection of such restrictors is carried out with the aid of a domain decomposition technique algebraically enhanced by a dynamic minimal residual strategy. The proposed block preconditioner has been tested through an extensive experimentation on unstructured and highly heterogeneous reservoir systems, pointing out its robustness and computational efficiency.