Efficient reordered nonlinear Gauss–Seidel solvers with higher order for black-oil models

Abstract

The fully implicit method is the most commonly used approach to solve black-oil problems in reservoir simulation. The method requires repeated linearization of large nonlinear systems and produces ill-conditioned linear systems. We present a strategy to reduce computational time that relies on two key ideas: (i ) a sequential formulation that decouples flow and transport into separate subproblems, and (ii ) a highly efficient Gauss-Seidel solver for the transport problems. This solver uses intercell fluxes to reorder the grid cells according to their upstream neighbors, and groups cells that are mutually dependent because of counter-current flow into local clusters. The cells and local clusters can then be solved in sequence, starting from the inflow and moving gradually downstream, since each new cell or local cluster will only depend on upstream neighbors that have already been computed. Altogether, this gives optimal localization and control of the nonlinear solution process.This method has been successfully applied to realfield problems using the standard first-order finite vol-Ø.S. Klemetsdal ume discretization. Here, we extend the idea to firstorder dG methods on fully unstructured grids. We also demonstrate proof of concept for the reordering idea by applying it to the full simulation model of the Norne oil field, using a prototype variant of the open-source OPM Flow simulator.

Comparison of nonlinear field-split preconditioners for two-phase flow in heterogeneous porous media

“…This technique is commonly used in the sequential fully implicit method in which the total velocity is also fixed during the resolution of the transport problem. In both options, we use Newton's method with damping to solve the discrete transport problem, although more efficient nonlinear solvers are available [20][21][22][23][24].…”

“…Homotopy methods are robust for large time steps and can achieve, in some cases, unconditional nonlinear convergence. Using a different approach, ordering-based methods [20][21][22][23][24] accelerate nonlinear convergence thanks to a reordering technique based on the flow direction. The reordered systems have block-triangular structure and can therefore be efficiently solved with backward substitution.…”

Smooth Implicit Hybrid Upwinding for Compositional Multiphase Flow in Porous Media

“…We mention here that other nonlinear strategies can be combined with our approach to further improve convergence. Recent developments for the fully implicit simulation of flow in porous media include-but are not limited to-physics-based damping strategies [5][6][7]9], ordering-based nonlinear solvers [33][34][35], nonlinear preconditioning [36][37][38][39], nonlinear multigrid [40][41][42], continuation methods [43,44], and Anderson acceleration [45,46].…”

Heat Flow Transport Model by Gauss-Seidel Type Iteration Methods for Gas and Solid Materials

“…The previously stated that, the numerical method by the Gauss-Seidel equation is recurring calculation that can solve algebraically on unknown variables. It can solve a set of linear or nonlinear algebra which suitable for energy transfer problems (Pu & Yuan, 2019;Klametsdal, Rasmussen, Meyner, & Lie, 2020;He & Wang, 2019;Paradezhenko, Melnikov, & Reser, 2019). The grid of the column's sectional area for Gauss-Seidel calculation is shown in Figure 2.…”