Multiscale Finite-Volume CVD-MPFA Formulations on Structured and Unstructured Grids (original) (raw)
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The Multiscale Finite Volume Method on Unstructured Grids
SPE Reservoir Simulation Symposium, 2013
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.
Computational Geosciences, 2014
A Dirichlet-Neumann representation method was recently proposed for upscaling and simulating flow in reservoirs. The DNR method expresses coarse fluxes as linear functions of multiple pressure values along the boundary and at the center of each coarse block. The number of flux and pressure values at the boundary can be adjusted to improve the accuracy of simulation results, and in particular to resolve important fine-scale details. Improvement over existing approaches is substantial especially for reservoirs that contain high permeability streaks or channels. As an alternative, the multiscale mixed finite-element (MsMFE) method was designed to obtain fine-scale fluxes at the cost of solv-
Defect and Diffusion Forum, 2010
This paper addresses the key issue of calculating fluxes at the control-volume interfaces when finite-volumes are employed for the solution of partial differential equations. This calculation becomes even more significant when unstructured grids are used, since the flux approximation involving only two grid points is no longer correct. Two finite volume methods with the ability in dealing with unstructured grids, the EbFVM-Element-based Finite Volume Method and the MPFA-Multi-Point Flux Approximation are presented, pointing out the way the fluxes are numerically evaluated. The methods are applied to a porous media flow with full permeability tensors and non-orthogonal grids and the results are compared with analytical solutions. The results can be extended to any diffusion operator, like heat and mass diffusion, in anisotropic media.
Multiscale Modeling & Simulation, 2006
We analyse and further develop a hierarchical multiscale method for the numerical simulation of two-phase flow in highly heterogeneous porous media. The method is based upon a mixed finite-element formulation, where fine-scale features are incorporated into a set of coarse-grid basis functions for the flow velocities. By using the multiscale basis functions, we can retain the efficiency of an upscaling method by solving the pressure equation on a (moderate-sized) coarse grid, while at the same time produce a detailed and conservative velocity field on the underlying fine grid.
The Multiscale Finite-Volume Method on Stratigraphic Grids
SPE Journal, 2014
Finding a pressure solution for large and highly detailed reservoir models with fine-scale heterogeneities modeled on a meter scale is computationally demanding. One way of making such simulations less compute intensive is to employ multiscale methods that solve coarsened flow problems using a set of reusable basis functions to capture flow effects induced by local geological variations. One such method, 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. We present an open-source implementation of the MsFV method in 3D along with a coarse partitioning algorithm that can handle stratigraphic grids with faults and wells. The resulting solver is an alternative to traditional upscaling methods, but can also be used for accelerating fine-scale simulations. To achieve better precision, the implementation can use the MsFV method as a preconditioner for Arnoldi iterations using GMRES, or as a preconditioner in combination with a standard inexpensive smoother.
Advances in Water Resources, 2013
In the present work, the multiscale finite volume (MsFV) method is implemented on a new coarse grids arrangement. Like grids used in the MsFV methods, the new grid arrangement consists of both coarse and dual coarse grids but here each coarse block in the MsFV method is a dual coarse block and vice versa. Due to using the altered coarse grids, implementation, computational cost, and the reconstruction step differ from the original version of MsFV method. Two reconstruction procedures are proposed and their performances are compared with each other. For a wide range of 2-D and 3-D problem sizes and coarsening ratios, the computational costs of the MsFV methods are investigated. Furthermore, a matrix (operator) formulation is presented. Several 2-D test cases, including homogeneous and heterogeneous permeability fields extracted from different layers of the tenth SPE comparative study problem are solved. The results are compared with the fine-scale reference and basic MsFV solutions.
Geofluids, 2004
The permeability of the Earth's crust commonly varies over many orders of magnitude. Flow velocity can range over several orders of magnitude in structures of interest that vary in scale from centimeters to kilometers. To accurately and efficiently model multiphase flow in geologic media, we introduce a fully conservative node-centered finite volume method coupled with a Galerkin finite element method on an unstructured triangular grid with a complementary finite volume subgrid. The effectiveness of this approach is demonstrated by comparison with traditional solution methods and by multiphase flow simulations for heterogeneous permeability fields including complex geometries that produce transport parameters and lengths scales varying over four orders of magnitude.