Accurate and efficient simulation of multiphase flow in heterogeneous porous media motivates the development of space-time multiscale strategies for the coupled nonlinear flow (pressure) and saturation transport equations. The flow equation entails heterogeneous high-resolution (fine-scale) coefficients and is global (elliptic or parabolic). The time-dependent saturation profile, on the other hand, may exhibit sharp local gradients or discontinuities (fronts) where the solution accuracy is highly sensitive to the time-step size. Therefore, accurate flow solvers need to address the multiscale spatial scales, while advanced transport solvers need to also tackle multiple time scales. This paper presents the first multirate multiscale method for space-time conservative multiscale simulation of sequentially coupled flow and transport equations. The method computes the pressure equation at the coarse spatial scale with a multiscale finite volume technique, while the transport equation is solved by taking variable time-step sizes at different locations of the domain. At each coarse time step, the developed local time-stepping technique employs an adaptive recursive time step refinement to capture the fronts accurately. The applicability (accuracy and efficiency) of the method is investigated for a wide range of two-phase flow simulations in heterogeneous porous media. For the studied cases, the proposed method is found to provide 3 to 4 times faster simulations. Therefore, it provides a promising strategy to minimize the tradeoff between accuracy and efficiency for field-scale applications.

We present ADM-LTS, an adaptive multilevel space-time-stepping scheme for transport in heterogeneous porous media. At each time step, firstly, the flow (pressure) solution is obtained. Then, the transport equation is solved using the ADM-LTS method, which consists of two stages. In the first stage, an initial solution is obtained by imposing the coarsest space-time grid. This initial solution is then improved, in the second stage, by imposing a space-time adaptive grid on the cells where the solution does not satisfy the desired quality. The quality control is based on error estimators with user-defined threshold values. The time-integration procedure, in which the coarsest-scale solution provides local flux boundary conditions for sub-domains with local time refinement, is strictly mass conservative. In addition, the method employs space-time fine grid cells only at the moving saturation fronts. In order to ensure local mass conservation at all levels, finite-volume restriction operators and unity prolongation operators are developed. Several numerical experiments have been performed to analyze the efficiency and accuracy of the proposed ADM-LTS method for both homogeneous and heterogeneous permeability fields on two and three dimensional domains. The results show that the method provides accurate solutions, at the same time it maintains the computational efficiency. The ADM-LTS implementation is publicly available at https://gitlab.com/darsim2simulator.