Tareq M. Al-Shaalan

Studies of Robust Two Stage Preconditioners for the Solution of Fully Implicit Multiphase Flow Problems

The solution of the linear system of equations for a large scale reservoir simulation has several challenges. Preconditioners are used to speed up the convergence rate of the solution of such systems. In theory, a preconditioner defines a matrix M that can be inexpensively inverted and represents a good approximation of a given matrix A. In this work, two-stage preconditioners consisting of the approximated inverses M1 and M2 are investigated for multiphase flow in porous media. The first-stage preconditioner, M1, is approximated from A using four different solution methods: (1) constrained pressure residuals (CPR), (2) lower block Gauss-Seidel, (3) upper block Gauss-Seidel, and (4) one iteration of block Gauss-Seidel. The pressure block solution in each of these different schemes is calculated using the Algebraic Multi Grid (AMG) method. The inverse of the saturation (or more generally, the nonpressure) blocks are approximated using Line Successive Over Relaxation (LSOR). The second stage preconditioner, M2, is a global preconditioner based on LSOR iterations for the matrix A that captures part of the original interaction of different coefficient blocks. Several techniques are also employed to weaken the coupling between the pressure block and the nonpressure blocks. Effective decoupling is achieved by: (1) an IMPES-like approach designed to preserve the integrity of pressure coefficients, (2) Householder transformations, (3) the alternate block factorization (ABF), and (4) the balanced decoupling strategy (BDS) based on least squares. The fourth method is a new technique developed in this work. The aforementioned preconditioning techniques were implemented in a parallel reservoir simulation environment, and tested for large-scale two-phase and three-phase black oil simulation models. This study demonstrates that a two-stage preconditioner based on balanced decoupling strategy (BDS) or ABF combined with Gauss-Seidel sweeps, that also incorporate nonpressure solutions for M, delivers both the fastest convergence rate and the most robust option overall without compromising parallel scalability.

Interior boundary-aligned unstructured grid generation and cell-centered versus vertex-centered CVD-MPFA performance

Grid generation for reservoir simulation must honor classical key constraints and be boundary aligned such that control-volume boundaries are aligned with geological features such as layers, shale barriers, fractures, faults, pinch-outs, and multilateral wells. An unstructured grid generation procedure is proposed that automates control-volume and/or control point boundary alignment and yields a PEBI-mesh both with respect to primal and dual (essentially PEBI) cells. In order to honor geological features in the primal configuration, we introduce the idea of protection circles, and to generate a dual-cell feature based grid, we construct halos around key geological features. The grids generated are employed to study comparative performance of cell-centred versus cell-vertex control-volume distributed multi-point flux approximation (CVD-MPFA) finite-volume formulations using equivalent degrees of freedom. The formulation of CVD-MPFA schemes in cell-centred and cell-vertex modes is analogous and requires switching control volume from primal to dual or vice versa together with appropriate data structures and boundary conditions. The relative benefits of both types of approximation, i.e., cell-centred versus vertex-centred, are made clear in terms of flow resolution and degrees of freedom required.

Correction to: Interior boundary-aligned unstructured grid generation and cell-centered versus vertex-centered CVD-MPFA performance

Due to an oversight, some author’s corrections were not carried out during Performing proof corrections stage. The Publisher apologizes for these mistakes. The original article was corrected.