A Multi-Pass Procedure for Redistributing Linearized Mass Balance Errors to Improve Simulator Performance

Abstract

Abstract In a generalized reservoir simulator for compositional and black oil models, the primary governing equations are mass conservation equations and volume balance equations with pressure and component masses as the primary unknowns. The phase equilibrium is honored in the calculation of partial volume derivatives. The Newton-Raphson method is widely chosen to solve this non-linear system: the equations are first linearized then solved using direct or iterative linear solvers. Direct solvers are not practical for large systems; therefore, iterative solvers are used. Iterative solvers are not exact and are usually converged to a specified criterion. The stricter the criterion, the higher the computational effort required by the linear solver. On the other hand, a loose convergence criterion can lead to more Newton iterations or false physical solutions. In addition to the convergence tolerances for linear solvers, certain criteria must be applied to the non-linear Newton iterations. One of the main convergence criteria is that the volume error at any gridblock is less than a small fraction of its pore volume. The linear solver tolerance should be somehow related to this non-linear tolerance, or the linear solutions should be adjusted to help the convergence of the non-linear iterations. Because of the limited accuracy of iterative linear solvers, a two-pass procedure is proposed to improve the linear solutions by redistributing linearized mass balance errors. In the first step, all the gridblocks are divided into different groups according to whether the mass change is positive or negative and by the magnitude of contribution to local fluid volume change. Then a constrained optimization problem is defined and solved to adjust the linear solutions, satisfying global mass conservation. With these modified solutions, the linearized volume errors for all the gridblocks are calculated. In the second step, a criterion similar to the aforementioned volume convergence criterion for Newton iteration is imposed on the linearized volume error for each gridblock. With reasonable linear solutions, most of the gridblocks should satisfy this criterion. For gridblocks that do not satisfy this condition, another constrained optimization problem is defined to ensure local volume balance with local mass solutions further adjusted. With this two-pass procedure, global mass conservation and local volume balance are achieved for the implicit formulation of the governing equations, with little or no degradation of the Newton convergence, and usually without needing to tighten the linear solver convergence tolerances.