Learning to Solve Parameterized Single-Cell Problems Offline to Expedite Reservoir Simulation

Abstract

The reservoir simulation system of residual equations is composed by applying a single parameterized nonlinear function to each cell in a mesh. This function depends on the unknown state variables in that cell as well as on those in the neighboring cells. Anecdotally, the solution of these systems relies on both the level of nonlinearity of this single-cell function as well as on how tightly the cell equations are coupled. This work reformulates this system of equations in an equivalent that is only mildly nonlinear. In an amortized offline regression stage, the single-cell equation is solved over a sampling of possible neighboring states and parameters. A neural network is regressed to this data. An equivalent residual system is formed by replacing the single-cell residual function with the neural network, and we propose three alternative algorithms to solve these preconditioned systems. The first method applies a Picard iteration that does not require Jacobian matrix evaluations or linear solution. The second applies a modified Seidel iteration that additionally infers locality automatically. The third algorithm applies Newton's method to the preconditioned system. The solvers are applied to a one-dimensional incompressible two-phase displacement problem with capillarity and a general two-dimensional two-phase flow model. We investigate the impacts of neural network regression accuracy on the performance of all methods. Reported performance metrics include the number of residual/network evaluations, linear solution iterations, and scalability with time step size. In all cases, the proposed methods significantly improve computational performance relative to the use of standard Newton-based solution methods.