Accurate numerical modeling of fluid transport is essential in reservoir management. Higher-order methods help to improve accuracy by reducing the numerical diffusion, which is common for all first order methods. In this paper, we present an implementation of a MUSCL-type second-order finite volume method and demonstrate its capabilities on 2D and 3D unstructured grids. This includes corner point grids that are typically used in reservoir modeling.
A second order finite volume method is compared to standard first order method in terms of accuracy, performance and an ability to handle nonlinearities. There are several ways to build a second order finite volume method. In this paper we choose an optimization-based strategy to compute the steepest possible linear reconstruction. At the same time, a steepness-limiting procedure is included in the optimization as constraint. This ensures that the steepest possible reconstruction that does not lead to oscillations is computed. As a result, sharper fronts compared to standard schemes are obtained.
The paper demonstrates the described method on several benchmark cases with emphasis on relevant for practical reservoir simulation test cases. In particular, we use Norne field open data set, which enables cross validation with other implementations. We test the method on the transport case, where an analytical solution is known, to verify convergence behavior and to isolate the errors. Furthermore, the performance of first- and second-order methods is compared on multiphase flow problems typical for improved oil recovery: solvent and CO2 injection. The second order method shows superior performance in terms of accuracy.
This paper verifies the desirable properties of higher order method for reservoir simulation. Moreover, all the described implementations are available in an open source reservoir simulator Open Porous Media (OPM). As a result, these methods are accessible for reservoir engineers and can be used with industry standard modeling setups.