Scalable Multi–stage Linear Solver for Coupled Systems of Multi–segment Wells and Complex Reservoir Models

Abstract

AbstractWellbore flow and interactions between wells and the reservoir can be complex. Accurate modeling of these behaviors is especially important for advanced (multi-lateral) wells. This paper describes a new scalable linear solver for flow simulation of detailed unstructured models with coupled multi-lateral multi-segment wells. The linear solver accommodates systems associated with both the fully implicit approach and the adaptive implicit method (AIM).A general-purpose research simulator (GPRS) serves as the computational platform, in which reservoir flow is described with a thermal-compositional AIM formulation, and a multi-segmented well (MsWell) model is used to describe wellbore flow. In the MsWell model, the wellbore is discretized into segments; pressure, velocity, and phase fractions are computed for each segment. Multi-lateral multi-segment wells add a large number of equations and unknowns, which are coupled to the reservoir model.The new linear solver is a generalized two-stage constraint pressure residual (CPR) preconditioner. The first stage is a global pressure system of equations, where the pressure coupling between the reservoir and the complex wells is represented accurately. This coupled pressure system is obtained algebraically, where the multi-segment wells are reduced to a form similar to that of standard wells. The two-stage scheme serves as the inner loop of a generalized minimum residual (GMRES) iterative solver. Algebraic multigrid (AMG) is used to compute the first-stage pressure solution; a special block-based incomplete lower-upper (ILU) preconditioner with arbitrary levels of fill-in is used for the second stage. This multi-stage solution algorithm was implemented using multi-level, sparse, block-based (MLSB) data structures specially designed for AIM computations on unstructured grids. We demonstrate the superior performance of this new solver compared with state-of-the-art methods using a variety of compositional problems for highly detailed reservoir models with unstructured grids and complex multi-segment wells.