The Multiscale Finite Volume Method on Unstructured Grids

Finding a pressure solution for large-scale reservoirs that takes into account fine-scale heterogeneities can be very computationally intensive. One way of reducing the workload is to employ multiscale methods that capture local geological variations using a set of reusable basis functions. One of these methods, the multiscale finite-volume (MsFV) method is well studied for 2D Cartesian grids, but has not been implemented for stratigraphic and unstructured grids with faults in 3D. With reservoirs and other geological structures spanning several kilometers, running simulations on the meter scale can be prohibitively expensive in terms of time and hardware requirements. Multiscale methods are a possible solution to this problem, and extending the MsFV method to realistic grids is a step on the way towards fast and accurate solutions for large-scale reservoirs. We present a MsFV solver along with a coarse partitioning algorithm that can handle stratigraphic grids with faults and wells. The solver is an alternative to traditional upscaling methods, but can also be used for accelerating fine-scale simulations. Approximate solutions computed by the new MsFV solver are compared with fine-scale solutions computed by a standard twopoint scheme for grids with realistic permeability and geometries. The results show that the MsFV method is suitable for solving realistic permeabilities, but can fail for highly anisotropic systems. The implementation is a suitable framework for further experimentation with partitioning algorithms and MsFV variants. To achieve better precision, the implementation can use the MsFV method as a preconditioner for Arnoldi iterations using GMRES, or for smoothing cycles using Dirichlet Multiplicative Schwarz (DMS). Introduction Multiscale methods have been proposed as a way of bridging the gap in resolution between geological models (cell sizes: centimeters to decimeters in the vertical direction, meters to tens of meters in horizontal direction) and dynamic simulation models (cell sizes: meters to tens of meters in vertical direction, tens of meters to hundred of meters in horizontal direction). As an alternative to traditional upscaling techniques, multiscale methods (Efendiev and Hou 2009) can resolve fine-scale qualities with reduced computational complexity (Kippe et al. 2008) on highly detailed reservoir models by creating basis functions that relate the fine scale (geological model) to a coarser scale (dynamic model). Two methods have received particular interest in industry, the multiscale finite-volume (MsFV) methods (Jenny et al. 2003) and the multiscale mixed finite-element (MsMFE) method (Hou and Wu 1997). Both methods compute flow solutions using degrees-of-freedom associated with a coarse grid, but differ in the way they construct the multiscale basis functions that are used to account for fine-scale effects in the coarse-scale system and reconstruct approximate solutions on grids with fine or intermediate resolution. In the MsMFE method the basis functions are constructed by unit flow across faces in the coarse grid, whereas the basis functions in the MsFV method are computed using a dual-grid formulation with unitary pressure values at each vertex of the coarse blocks. The MsFV method has been extended to a wide variety of problems in subsurface flow, including density-driven flow (Lunati and Jenny 2008) and compressible multiphase flow (Lunati and Jenny 2006; Zhou and Tchelepi 2008), by adding an extra set of correction functions. The method can also be employed in an iterative framework on its own (Hajibeygi and Jenny 2011) or as a preconditioner (Lunati et al. 2011), having a close relationship to domain-decomposition methods (Nordbotten and Bjørstad 2008). A particular advantage of the MsFV method, compared with multigrid and domain-decomposition methods, is that the multiscale method can reconstruct a conservative flux field at any iteration stage, which is crucial if the flux field is used to solve transport problems. Application to unstructured grids is a remaining challenge for the MsFV method. Although the modeling approaches used by the industry today are predominantly structured, they typically lead to irregular and unstructured simulation grids. Very complex grids having unstructured connections and degenerate cell geometries arise naturally when representing structural framework like faults, joints, and deformation bands, and/or stratigraphic architectural characteristics like channels, lobes, clinoforms, and shale shale/mud drapes. Similarly, unstructured connections are induced when local grid refinement, structured or unstructured, is used to improve the modeling of (deviated) wells. With the exception of a highly idealized model of faults using 2D structured grids (Hajibeygi et al. 2011), the MsFV method has so far only been formulated and applied to regular Cartesian grids. Extending the MsFV formulation from uniform Cartesian to industry-standard grids with complex geometries/topologies and high aspect/anisotropy ratios is therefore a key step on the road to widespread adoption in industry. In particular, it is desirable to develop automated coarsening strategies that perform well for complex methods and preferably adapt to geological features and