##
**Multiscale finite-volume method for compressible multiphase flow in porous media.**
*(English)*
Zbl 1220.76049

Summary: The Multiscale Finite-Volume (MSFV) method has been recently developed and tested for multiphase-flow problems with simplified physics (i.e. incompressible flow without gravity and capillary effects) and proved robust, accurate and efficient. However, applications to practical problems necessitate extensions that enable the method to deal with more complex processes. In this paper we present a modified version of the MSFV algorithm that provides a suitable and natural framework to include additional physics. The algorithm consists of four main steps: computation of the local basis functions, which are used to extract the coarse-scale effective transmissibilities; solution of the coarse-scale pressure equation; reconstruction of conservative fine-scale fluxes; and solution of the transport equations. Within this framework, we develop a MSFV method for compressible multiphase flow. The basic idea is to compute the basis functions as in the case of incompressible flow such that they remain independent of the pressure. The effects of compressibility are taken into account in the solution of the coarse-scale pressure equation and, if necessary, in the reconstruction of the fine-scale fluxes. We consider three models with an increasing level of complexity in the flux reconstruction and test them for highly compressible flows (tracer transport in gas flow, imbibition and drainage of partially saturated reservoirs, depletion of gas–water reservoirs, and flooding of oil–gas reservoirs). We demonstrate that the MSFV method provides accurate solutions for compressible multiphase flow problems. Whereas slightly compressible flows can be treated with a very simple model, a more sophisticate flux reconstruction is needed to obtain accurate fine-scale saturation fields in highly compressible flows.

### MSC:

76M12 | Finite volume methods applied to problems in fluid mechanics |

76N99 | Compressible fluids and gas dynamics |

76S05 | Flows in porous media; filtration; seepage |

76T99 | Multiphase and multicomponent flows |

### Keywords:

compressibility; multiscale methods; finite-volume methods; multiphase flow in porous media; reservoir simulation
PDF
BibTeX
XML
Cite

\textit{I. Lunati} and \textit{P. Jenny}, J. Comput. Phys. 216, No. 2, 616--636 (2006; Zbl 1220.76049)

Full Text:
DOI

### References:

[1] | Aarnes, J.E.; Kippe, V.; Lie, K.A., Mixed multiscale finite elements and streamline methods for reservoir simulation of large geomodel, Adv. water res., 28, 257-271, (2005) |

[2] | Arbogast, T., Implementation of a locally conservative numerical subgrid upscaling scheme for two phase Darcy flow, Comput. geosci., 6, 453-481, (2002) · Zbl 1094.76532 |

[3] | Arbogast, T.; Bryant, S.L., A two-scale numerical subgrid technique for waterflood simulations, Soc. petrol. eng. J., 446-457, (2002) |

[4] | Chen, Z.M.; Hou, T.Y., A mixed multiscale finite element method for elliptic problems with oscillating coefficients, Math. comput., 72, 242, 541-576, (2003) · Zbl 1017.65088 |

[5] | M.A. Christie, M.J. Blunt, Tenth SPE comparative solution project: A comparison of upscaling techniques, SPE 66599, presented at the SPE Symposium on Reservoir Simulation, Houston, February 11-14, 2001. |

[6] | Dagan, G., Flow and transport in porous formations, (1989), Springer-Verlag New York |

[7] | Gautier, Y.; Blunt, M.J.; Christie, M.A., Nested girding and streamline-based simulation for fast reservoir performance prediction, Comput. geosci., 3, 295-320, (1999) · Zbl 0968.76059 |

[8] | Hoeksema, R.J.; Kitanidis, P.K., Analysis of the spatial structure properties of selected aquifers, Water resour. res., 21, 4, 563-572, (1985) |

[9] | Hou, T.Y.; Wu, X.H., A multiscale finite element method for elliptic problems in composite materials and porous media, J. comput. phys., 134, 1, 169-189, (1997) · Zbl 0880.73065 |

[10] | Jenny, P.; Lee, S.H.; Tchelepi, H., Multi-scale finite-volume method for elliptic problems in subsurface flow simulation, J. comput. phys., 187, 1, 47-67, (2003) · Zbl 1047.76538 |

[11] | Jenny, P.; Lee, S.H.; Tchelepi, H., Adaptive multiscale finite-volume method for multi-phase flow and transport in porous media, Multiscale model. simulat., 3, 1, 50-64, (2004) · Zbl 1160.76372 |

[12] | P. Jenny, S.H. Lee, H. Tchelepi, Fully implicit adaptive multi-scale finite-volume algorithm for multi-phase flow in porous media, J. Comput. Phys., 2006 (submitted for publication). · Zbl 1160.76373 |

[13] | Renard, Ph.; de Marsiliy, G., Calculating equivalent permeability: a review, Water resour. res., 20, 5-6, 253-278, (1997) |

[14] | Wen, X.H.; Gómez-Hernández, J.J., Upscaling hydraulic conductivities in heterogeneous media: an overview, J. hydrol., 183, 1-2, R9-R32, (1996) |

[15] | C. Wolfsteiner, S.H. Lee, H.A. Tchelepi, Well modeling in the multiscale finite volume method for subsurface flow simulation. Multiscale Model. Simulat., 2006 (submitted for publication). · Zbl 1205.76175 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.