Incorporating Algebraic Multigrid when Solving Reservoir Simulation Equations

Abstract

Adaptive-implicit petroleum reservoir simulations often result in huge, very ill-conditioned linear systems of equations. The convergence of single-level solvers, for these systems tends to deteriorate considerably as these systems become larger. The primary difficulty is the coupled nature of the model equations that exhibit characteristics of both hyperbolic and nearly elliptic systems. Multilevel methods provide a technique to efficiently solve nearly elliptic systems independent of size whereas preconditioned Krylov solvers are more robust for both. Therefore, multi-stage preconditioning methods are commonly used in practice to solve coupled systems. In this thesis a two-stage combinative method, the constrained pressure residual (CPR) method, has been implemented in a commercial black-oil reservoir simulator. A decoupling preconditioner has been developed to approximately decouple and extract the nearly elliptic pressure sub-system. An algebraic multigrid (AMG) method is used to efficiently solve the pressure system followed by an incomplete LU (ILU) technique as a robust second stage solver. The primary objective is to generate systems that are suitable for an efficient AMG solution while simultaneously ensuring a fast overall convergence of the solver. The combinative solver and the dynamic row-sum (DRS) preconditioner have been developed to include several optional variants. A study of these parameters has been performed to determine their importance and select reasonable default values. The accuracy of the new combinative method has been verified. It is generally able to produce small residual vectors with fewer solver iterations. Simulator efficiency has been improved for an assortment of test cases. Several simulations had clock times between 1.5 and 3 times faster with the combinative solver. Parameter choices leading to a small number of fast AMG cycles for the pressure solution at the sacrifice of some accuracy generally led to the best performance.