Multilevel Multiscale Mimetic Method for Two-Phase Flows in Porous Media

Flow simulations in porous media formations involve a wide range of strongly coupled scales. For example, the length scale of short and narrow channels is on the order of micrometers, while the size of a simulation domain may be several kilometers. In modeling two-phase flow the strongest multiscale influence arises from the heterogeneous structure of the subsurface environment. The permeability of rock formations is highly heterogeneous and may span several orders of magnitude, from nearly impermeable barriers to high-permeable flow channels. For such complex systems fully resolved simulations become computationally intractable. It is also well understood that employing simple averages of the fine-scale parameters in a model of the same form has significant limitations. The goal of multiscale modeling is to developed methods which upscale fine-scale model, not just its parameters. Many different approaches that take into account the multiscale nature of the problem have been proposed, such as the Multiscale Finite Element methods, the Multiscale Finite Volumes method, the Multilevel Upscaling method (MLUPS). All of these methods, except MLUPS, consider a two-level structure: coarse and fine scale partitions. Using a two-level structure most multiscale methods achive a coarsening factor of approximately 10 in each coordinate direction, while the trends in fine-scale realizations of large reservoirs requires a coarsening factor of 100 or more. Multilevel framework was realized in MLUPS method but this approach does not produce conservative velocity fields, which is desirable for two-phase flow simulation. In our work we propose the new Multilevel Multiscale Mimetic method (M3). This approach brings together a novel subgrid modeling algorithm for developing mimetic discretizations on coarse scales with algebraic multigrid for estimating moments of the flux on the edges. Multilevel hierarchy of coarse scale discretizations makes it very flexible and computationally efficient. Due to the algebraic nature of the method it can be naturally adopted to the different types of fine scale discretization, such as: Mixed Finite Element method, Finite Volume method, Mimetic Finite Difference method. Moreover the method can handle full permeability tensor and general types of fine and coarse scale partitions. The idea of the method consists in an algebraic transformation of a fine-scale linear system to a coarse scale. The coarsening procedure is based on approximated values of the flux moments on the edges. To estimate these moments we apply a small number of algebraic multigrid cycles to the global flow problem. The stencil of a coarse-scale discretization as well as the structure of the linear system are similar to the fine scale ones hence the same transformation may be performed recursively. The algebraic transformation is defined in such a way that velocity fields on all levels are conservative. This is the advantage over MLUPS method. The method is applied to two test cases. The first test case consists of two highly heterogeneous quarter five- spot problems in 2D. The permeability fields are generated using GSLIB library with different anisotropy angles. The second test case is 2D version of a 3D upscaling benchmark taken from 10th SPE Comparative Solution project. To preform 2D simulations we consider one of the fluvial layers of 3D reservoir model defined in SPE project which is the challenge test for the most of multiscale methods.