Multigrid pressure solver for 2D displacement problems in drilling, cementing, fracturing and EOR

Abstract

Numerical simulators for the design of oilfield services technologies are required to be fast and robust, hence the implementation of underlying mathematical models is expected to be highly efficient. We present an optimized numerical algorithm based on the multigrid approach for solving the pressure equation in 2D problems of fluid-fluid displacement. This type of problems arise in oilfield services applications, such as drilling, cementing, hydraulic fracturing and EOR. The displacement of fluids with different properties in a narrow annular gap (drilling, cementing) or a slot (hydraulic fracturing, production) is governed by the 2D width-averaged model in the lubrication approximation. In numerical implementation, the most time-consuming is the elliptic Poisson pressure equation, which has coefficients with discontinuities (jumps) in the case of displacement of fluids with viscosity contrast. Specifically for this case of discontinuous coefficients, we designed and implemented numerically the black-box multigrid algorithm, which resolves the divergence issue typical of standard realization of multigrid algorithm using bilinear prolongation operators. The algorithm is tested on a number of matrices corresponding to discretization of elliptic equation at rectangular staggered meshes with different resolution. The performance of the algorithm is compared against several other sparse solvers available for commercial use: smoothed aggregated algebraic multigrid (PyAMG software package), direct sparse solver from Intel MKL library (DSS), Bi-Conjugate Gradient-Stabilized solver preconditioned by ILUT factorization implemented into BLAS package, and our in-house Conjugated Gradient algorithm preconditioned by ILU(2) factorization. It was found that the performance of the black-box multigrid solver is significantly higher (from 7x to 30x in terms of CPU time depending on the mesh resolution) as compared to the closest solver under consideration, which allows us to recommend this solver for implementation into existing 2D drilling simulators, fracturing design tools, cement placement and production simulators.