Development of A Fast Auxiliary Subspace Pre-conditioner for Numerical Reservoir Simulators

Abstract

Abstract The Fast Auxiliary Space Preconditioning (FASP) method is a general framework for constructing effective preconditioners for solving discretized partial differential equations (PDEs). By constructing appropriate auxiliary spaces, FASP transforms a complicated problem into a sequence of simpler solver-friendly systems. In this paper, we present a new FASP preconditioner for petroleum reservoir simulations. According to the analytic characteristics for the pressure and saturation variables in the black oil model, we choose appropriate auxiliary spaces for different parts of the Jacobian systems arising from the fully implicit method (FIM) with coupled implicit wells. By combining the new preconditioner with Krylov subspace methods (such as the GMRes method), we construct an efficient and robust solver, which can be easily generalized to more complicated models like the modified black oil model for simulating polymer flooding. Preliminary numerical experiments demonstrate the advantages of this new preconditioner. Introduction The solution of the linear system of equations arising from fully implicit method for a large scale reservoir simulation has many challenges. Very often more than 80% of CPU time in fully implicit reservoir simulation is usually spent on solving Jacobian systems resulting from the Newton linearization. These linear systems are large, sparse, highly nonsymmetric and ill-conditioned. The Krylov subspace methods1, such as BiCGstab and GMRes, are efficient iterative methods for these Jacobian systems. Since these iterative methods do not converge very well by themselves, many preconditioners have been proposed over the years. They mainly fall in two categories: (i) purely algebraic preconditioners and (ii) preconditioners based on the different properties of the variables. In the category (i), there are, for example, point-wise incomplete lowerupper factorization (ILU) methods2, 3, 4, 5, block ILU methods6, 7, nest factorization methods8, 9, and SVD-reduction methods10. In the category (ii), on the other hand, the methods are based on the understanding that pressure variables and saturation variables have different PDE properties; the representative examples are the Combinative method11, Constrained Pressure Residual (CPR) method12, 13, and multi-stage methods14, 15, 16.