A Parallel Linear Solver Algorithm for Solving Difficult Large Scale Thermal Models

摘要

One of the best methods to solve the large sparse system of linear equations that arise from reservoir flow equations is the Flexible Generalized Minimum Residual(FGMRES)1 method with the two stage Constrained Pressure Residual(CPR)preconditioner2,3.In the CPR method,a pressure-like equation is formed from the fully implicit matrix algebraically.For a commercial reservoir simulator,this first stage pressure equation is solved by the Algebraic Multi-Grid(AMG)method4,5.The second stage precondi- tioner uses Incomplete LU(K)factorization3 for the fully implicit system of equations,where K denotes the level of in-fill of the factorization. The linear solver with CPR-AMG-ILUK preconditioner was shown to be scalable and very efficient for solving large scale isothermal problems.However,this is not always the case for thermal problems. Certain large scale thermal models have encountered significant difficulties in linear and nonlinear solver convergence.Linear solver failures result in bad solution vector updates,which in turn result in nonlinear solver convergence problems.The end result is exceedingly long run times or runs failing to finish altogether. A new solver algorithm that we call SWIFT was developed to address this problem.The central idea of this algorithm is to exploit the sparsity within the neq x neq sub-matrices and use the magnitude of the in-fill terms to adaptively modify the factorization sparsity pattern to improve accuracy.It combines block diagonal scaling,equilibration,reordering and threshold-base incomplete LU factorization to form a robust linear solver.Several parallel variations of the SWIFT algorithm were evaluated and implemented. The new solver algorithm was implemented in the reservoir simulator and applied to several large scale thermal models with up to 34 million cells.For these models it reduced linear iterations by 80%to 93% and gave simulator run time speed-up factors of 2 to 5.5.