Linear solver framework for coupled systems of multi-segmented wells and unstructured reservoir models

Yuanlin Jiang and Hamdi Tchelepi

Multi-segmented well (MsWell) model can describe the multi-phase flow behavior inside the wellbore. However, it has a large number of governing equations and variables. The properties of those equations are very different from those of reservoir equations and conventional well equations. This requires an efficient solution strategy to solve the system.

A robust MsWell model has been developed and integrated into GPRS (General Purpose Research Simulator developed at Stanford). This talk will focus on the development of efficient solution strategy for a system with highly heterogeneous reservoir and a large number of MsWell models.

We  developed a new matrix format which avoids tedious and expensive traditional matrix build-up and fill-in process. This matrix format is essentially a wrapper and it reuses some basic data in the simulator. The new format is fully compatible with adaptive implicit simulation and un-structured grid. An efficient matrix-vector operation is provided so as Krylov solvers can solve the matrix.

Constraint pressure residual (CPR) method has proven to be one of the most successfu  preconditioning techniques for reservoir simulation. We extend the idea to the system with MsWell model. In the first stage, MsWell matrices are reduced to those of conventional well model, and then the reduced matrices and reservoir matrix go through standard true IMPES reduction to generate pressure system.  AMG solver is employed as the first stage solver and Block ILU(k) solver is the one for the second stage. This two-stage CPR preconditioner significantly speeded up the linear convergence of the systems with MsWell model.