Exponential Time Integrators for 3D Reservoir Simulation

We investigate exponential time integrators which we use, in conjunction with an upwind weighted finite volume discretization in space, for the efficient and accurate simulation of advection-diffusion processes including non-linear chemical reactions in highly heterogeneous 3D oil reservoirs with more than 528k unknowns. The exponential integrators are based on the variation of constants solution and solve the linear system exactly. While this is at the expense of computing the exponential of the stiff matrix representing the finite volume discretization, the use of real Leja points or the Krylov subspace technique to approximate the exponential makes these methods competitive compared to standard finite difference-based time integrators. We investigate two exponential time integrators, the second-order accurate Exponential Euler Midpoint (EEM) scheme and Exponential Time Differencing of order one (ETD1). All our numerical examples, which include advection-diffusion-reaction simulations performed on the classical SPE10 test case, demonstrate that our methods are highly competitive compared to standard semi-implicit time integrators. This competitiveness comprises two components: efficiency and accuracy. Our results suggest that exponential time integrators such as the ETD1 and EEM schemes can readily be applied to large-scale 3D reservoir simulations with several million of unknowns.