Explicit Singly Diagonally Implicit Runge-Kutta Methods and Adaptive Stepsize Control for Reservoir Simulation

Simulation of fluid flow in petroleum reservoirs is an essential tool in understanding, predicting and controlling advanced oil recovery methods. The major computational effort in reservoir simulation comes from solving a very large system of differential equations describing the fluid flow and the complex behaviour of advanced oil recovery methods. Choosing an appropriate method in the numerical solution of a large system of differential equations involves deciding on factors such as the order of the integration scheme, stability properties and concern for computational efficiency. Current simulators normally uses first order integration schemes applied with heuristically guided strategies for controlling time-step sizes. In the solution process of complex recovery methods this can lead to unnecessary computations and inappropriately small time-steps. We establish a fully implicit integrator of high order applied with an adaptive time-step selection supported by error estimates. We describe the explicit singly diagonally implicit Runge-Kutta (ESDIRK) methods with an embedded scheme for error estimation. This class of methods is computationally efficient, A- and L-stable as well as stiffly accurate. The embedded method providing the error estimate is of different order than the method used for advancing the solution. Based on this error estimate, the time-step is computed by a predictive control law. The predicitive control law is designed based on a model of the numerical method (ESDIRK) itself. Implicit integration involves the solution of a system of coupled nonlinear residual equations which need to be solved iteratively. Fast convergence of the iterative solver is crucial and may be controlled by the time-step size. We present a strategy for adaptive stepsize selection that mitigates the trade off between the convergence rate and time-step size. Consequently, the stepsize selection rule keeps the error estimate bounded (i.e., close to a user-specified tolerance) and at the same time maintains a good convergence rate of the equation solver. In addition, the controller has the ability to combine the above stepsize selection rule with classical time-step control that limits maximum change in key variables.