Ordering-based Nonlinear Solver for Fully-implicit Simulation

Abstract

The Fully-Implicit method (FIM) is often the method of choice for the temporal discretization of the partial differential equations governing multiphase flow in porous media, especially when nonlinearity is severe. The FIM offers unconditional stability, but requires the solution of large coupled systems of nonlinear algebraic equations. Newton-based methods – often with damping heuristics - are employed to solve the nonlinear systems. However, Newton-based solvers can suffer from convergence problems, especially for large time steps in the presence of highly nonlinear flow physics. To overcome such convergence problems, the timestep is reduced, and the Newton steps are restarted from the solution of the previous (converged) time step. Recently, potential ordering and the reduced-Newton method were used to solve immiscible three-phase flow in the presence of buoyancy and capillary effects (e.g., Kwok & Tchelepi, JCP, 2007). Here, we extend the potential-ordering method to interphase mass transfer. Specifically, we deal with the black-oil model with variable bubble-point pressure. The convergence properties and the computational efficiency of the potential-ordering nonlinear solver are superior to existing damped Newton methods. The nonlinear iteration process starts with the latest pressure field. Here, we use the Algebraic MultiGrid (AMG) from Fraunhofer (Stueben, International Multigrid Conference, 1983). Based on the latest pressure (potential) distribution, a directed graph is formed, in which nodes represent grid cells (control volumes) and edges represent phase fluxes between cells. As proposed by Natvig & Lie (ECMOR, 2008), and Shahvali & Tchelepi (SPE RSS, 2013), Tarjan's strongly connected components algorithm is used to order the nonlinear discrete system into a block triangular form. For the transport step, the potential ordering is used to update the saturations/compositions, one cell (control volume) at a time – from highest to lowest phase potential. The algorithm deals effectively with mass transfer between the liquid and gas phases, including phase disappearance (e.g., gas going back in solution) and reappearance (e.g., gas leaving solution), as a function of pressure and composition. The new nonlinear ordering-based approach was implemented in Stanford’s general-purpose research simulator (AD-GPRS), and was applied to challenging Black-Oil problems using highly heterogeneous permeability fields (e.g., layers of the SPE10 test case). Detailed robustness and performance comparisons of the potential based solver with state-of-the-art nonlinear/linear solvers (e.g. damped Newton with CPR-based AMG) are presented for variable bubble-point black-oil problems using highly detailed 3D heterogeneous models. The results show that for large time steps (e.g., corresponding to fluid throughput – CFL – numbers on the order of hundreds to thousands), our nonlinear ordering-based solver reduces the number of nonlinear iterations significantly, which also leads to gains in the overall computational cost.