Computational Geosciences

, Volume 12, Issue 3, pp 337–350

# Multiscale finite-volume method for density-driven flow in porous media

Original paper

## Abstract

The multiscale finite-volume (MSFV) method has been developed to solve multiphase flow problems on large and highly heterogeneous domains efficiently. It employs an auxiliary coarse grid, together with its dual, to define and solve a coarse-scale pressure problem. A set of basis functions, which are local solutions on dual cells, is used to interpolate the coarse-grid pressure and obtain an approximate fine-scale pressure distribution. However, if flow takes place in presence of gravity (or capillarity), the basis functions are not good interpolators. To treat this case correctly, a correction function is added to the basis function interpolated pressure. This function, which is similar to a supplementary basis function independent of the coarse-scale pressure, allows for a very accurate fine-scale approximation. In the coarse-scale pressure equation, it appears as an additional source term and can be regarded as a local correction to the coarse-scale operator: It modifies the fluxes across the coarse-cell interfaces defined by the basis functions. Given the closure assumption that localizes the pressure problem in a dual cell, the derivation of the local problem that defines the correction function is exact, and no additional hypothesis is needed. Therefore, as in the original MSFV method, the only closure approximation is the localization assumption. The numerical experiments performed for density-driven flow problems (counter-current flow and lock exchange) demonstrate excellent agreement between the MSFV solutions and the corresponding fine-scale reference solutions.

## Keywords

Gravity Counter-current flow Lock-exchange problem Multiscale methods Finite-volume methods Multiphase flow in porous media Reservoir simulation

## References

1. 1.
Aarnes, J.: On the use of a mixed multiscale finite elements method for greater flexibility and increased speed or improved accuracy in reservoir simulation. Multiscale Model. Simul. 2(3), 421–439 (2004)
2. 2.
Aarnes, J., Kippe, V., Lie, K.: Mixed multiscale finite elements and streamline methods for reservoir simulation of large geomodel. Adv. Water Resour. 28, 257–271 (2005)
3. 3.
Arbogast, T.: Numerical subgrid upscaling of two phase flow in porous media. Tech. Rep., Texas Institute for Computational and Applied Mathematics, The University of Texas at Austin (1999)Google Scholar
4. 4.
Arbogast, T.: Implementation of a locally conservative numerical subgrid upscaling scheme for two phase darcy flow. Comput. Geosci. 6, 453–481 (2002)
5. 5.
Arbogast, T., Bryant, S.L.: Numerical subgrid upscaling for waterflood simulations. SPE 66375, presented at the SPE Symp on Reservoir Simulation, Houston, 11–14 February (2001)Google Scholar
6. 6.
Chen, Z., Hou, T.Y.: A mixed finite element method for elliptic problems with rapidly oscillating coefficients. Math. Comput. 72(242), 541–576 (2003)
7. 7.
Christie, M.A., Blunt, M.J.: Tenth SPE comparative solution project: a comparision of upscaling techniques. SPE 66599, presented at the SPE Symp on Reservoir Simulation, Houston, 11–14 February (2001)Google Scholar
8. 8.
Hou, T.Y., Wu, X.H.: A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comp. Phys. 134(1), 169–189 (1997)
9. 9.
Jenny, P., Lee, S.H., Tchelepi, H.: Multi-scale finite-volume method for elliptic problems in subsurface flow simulation. J. Comp. Phys. 187(1), 47–67 (2003)
10. 10.
Jenny, P., Lee, S.H., Tchelepi, H.: Adaptive multiscale finite-volume method for multi-phase flow and transport in porous media. Multiscale Model. Simul. 3(1), 50–64 (2004)
11. 11.
Jenny, P., Lee, S.H., Tchelepi, H.: Adaptive fully implicit multi-scale finite-volume method for multi-phase flow and transport in heterogeneous porous media. J. Comp. Phys. 217, 627–641 (2006)
12. 12.
Lunati, I., Jenny, P.: Multi-scale finite-volume method for compressible flow in porous media. J. Comp. Phys. 216, 616–636 (2006)
13. 13.
Lunati, I., Jenny, P.: Treating highly anisotropic subsurface flow with the multiscale finite-volume method. Multiscale Model. Simul. 6(1), 208–218 (2007)
14. 14.
Renard, P., de Marsiliy, G.: Calculating equivalent permeability: a review. Water Resour. Res. 20(5-6), 253–278 (1997)Google Scholar
15. 15.
Wolfsteiner, C., Lee, S.H., Tchelepi, H.A.: Well modeling in the multiscale finite volume method for subsurface flow simulation. Multiscale Model. Simul. 5(3), 616–636 (2006)