Application of space–time conservation element and solution element method in streamline simulation

Abstract

Streamline simulation has been developed as an alternative to conventional Eulerian methods for the simulation of multiphase fluid flow in oil and gas reservoirs. In streamline simulation the saturation equations are solved over streamlines as a function of time-of-flight (TOF).We present a parallel implementation of this method. Furthermore, to increase the accuracy and speed of simulation, a new method called space–time conservation element and solution element (CE/SE) is implemented to solve the saturation equations along streamlines. CE/SE has many non-traditional features, including a unified treatment of space and time and stable numerical behavior with no need to introduce total variation diminishing schemes. As flux is a time–space property, in CE/SE, both time and space are discretized and treated on the same footing. In CE/SE, parameters and their derivatives are considered as independent variables and are computed simultaneously at each time-step that leads to local and global flux conservation. In addition by introduction of solution element and conservation element, cell fluxes can be calculated without extrapolation. To show the strength of method, the Buckley–Leverett equation is solved using CE/SE and compared to the finite volume method (FVM). Then, the method is employed in a streamline simulator to simulate water injection in a heterogeneous oil reservoir. CE/SE can capture the correct solution better than the FVM especially near sharp fronts. Though CE/SE is a second order method, its accuracy is higher than the third order Leonard’s scheme and its order of convergence is higher than current methods in the literature. In addition, the simulation time of CE/SE is about 10 percent lower than the other second and third order methods we test.