An efficient multi-point flux approximation method for discrete fracture-matrix simulations.

*(English)*Zbl 1402.76131Summary: We consider a control volume discretization with a multi-point flux approximation to model Discrete Fracture-Matrix systems for anisotropic and fractured porous media in two and three spatial dimensions. Inspired by a recently introduced approach based on a two-point flux approximation, we explicitly account for the fractures by representing them as hybrid cells between the matrix cells. As well as simplifying the grid generation, our hybrid approach excludes small cells in the intersection of the fractures and hence avoids severe time-step restrictions associated with small cells. Excluding the small cells also reduces the condition number of the discretization matrix. For examples involving realistic anisotropy ratios in the permeability, numerical results show significant improvement compared to existing methods based on two-point flux approximations. We also investigate the hybrid method by studying the convergence rates for different apertures and fracture/matrix permeability ratios. Finally, the effect of removing the cells in the intersections of the fractures are studied. Together, these examples demonstrate the efficiency, flexibility and robustness of our new approach.

##### MSC:

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

65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |

76M20 | Finite difference methods applied to problems in fluid mechanics |

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\textit{T. H. Sandve} et al., J. Comput. Phys. 231, No. 9, 3784--3800 (2012; Zbl 1402.76131)

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[1] | Barenblatt, G.E.; Zheltov, I.; Kochina, I., Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks, J. appl. math. USSR, 24, 1286-1303, (1960) · Zbl 0104.21702 |

[2] | Warren, J.; Root, P., The behavior of naturally fractured reservoirs, SPE softw. pract. exp., 3, 45-255, (1963) |

[3] | Lim, K.T.; Aziz, K., Matrix – fracture transfer shape factors for dual-porosity simulators, J. petrol. sci. eng., 13, 169-178, (1995) |

[4] | Duguid, J.; Lee, P., Flow in fractured porous media, Water resour. res., 13, 558-566, (1977) |

[5] | K. Pruess, Y. Wu, A new semi-analytical method for numerical simulation of fluid and heat flow in fractured reservoirs, SPE Adv. Tech. Ser., Earth Sciences Division, Lawrence Berkeley Laboratory, HydroGeoLogic Inc., no. 1, 1993, pp. 63-72. |

[6] | Pruess, K.; Narasimhan, T., On fluid reserves and the production of superheated steam from fractured, vapor-dominated geothermal reservoirs, J. geophys. res., 87, 9329-9339, (1982) |

[7] | Pruess, K.; Narasimhan, T., A practical method for modeling fluid and heat flow in fractured porous media, Spe j., vol. 25, (1985), Lawrence Berkeley Laboratory, pp. 14-26 |

[8] | Dietrich, P.; Helmig, R.; Sauter, M.; Hötzl, H.; Köngeter, J.; Teutsch, G., Flow and transport in fractured porous media, (2005), Springer |

[9] | Reichenberger, V.; Jakobs, H.; Bastian, P.; Helmig, R., A mixed-dimensional finite volume method for two-phase flow in fractured porous media, Adv. water resour., 29, 1020-1036, (2006) |

[10] | Martin, V.; Jaffré, J.; Roberts, J.E., Modeling fractures and barriers as interfaces for flow in porous media, SIAM J. sci. comput., 26, 1667-1691, (2005) · Zbl 1083.76058 |

[11] | Gebauer, S.; Neunhäuserer, L.; Kornhuber, R.; Ochs, S.; Hinkelmann, R.; Helmig, R., A equidimensional modelling of flow and transport processes in fractured porous systems I, (), 335-342 |

[12] | Neunhäuserer, L.; Gebauer, S.; Ochs, S.; Hinkelmann, R.; Kornhuber, R.; Helmig, R., A equidimensional modelling of flow and transport processes in fractured porous systems II, (), 343-350 |

[13] | Hægland, H.; Assteerawatt, A.; Dahle, H.; Eigestad, G.; Helmig, R., Comparison of cell- and vertex-centered discretization methods for flow in a two-dimensional discrete-fracture – matrix system, Adv. water resour., 32, 1740-1755, (2009) |

[14] | B. Mallison, Practical gridding algorithms for discrete-fracture modeling workflows, in: Proceedings of the 12th European Conference On the Mathematics of Oil Recovery, Oxford, UK, 2010. |

[15] | Lee, S.H.; Lough, M.F.; Jensen, C.L., Hierarchical modeling of flow in naturally fractured formations with multiple length scales, Water resour. res., 37, 443-455, (2001) |

[16] | A. Moinfar, W. Narr, M.-H. Hui, B. Mallison, S.H. Lee, Comparison of discrete-fracture and dual-permeability models for multiphase flow in naturally fractured reservoirs, in: Proceeding of the SPE Reservoir Simulation Symposium, 2011. |

[17] | Karimi-Fard, M.; Gong, B.; Durlofsky, L.J., Generation of coarse-scale continuum flow models from detailed fracture characterizations, Water resour. res., 42, (2006), (W10423) |

[18] | Berkowitz, B., Characterizing flow and transport in fractured geological media: a review, Adv. water resour., 25, 861-884, (2002) |

[19] | Karimi-Fard, M.; Durlofsky, L.; Aziz, K., An efficient discrete-fracture model applicable for general-purpose reservoir simulators, Spe j., 9, 227-236, (2004) |

[20] | Matthäi, S.; Mezentsev, A.; Belayneh, M., Finite element-node-centered finite-volume two-phase-flow experiments with fractured rock represented by unstructured hybrid-element meshes, SPE reserv. eval. eng., 10, 740-756, (2007) |

[21] | Hoteit, H.; Firoozabadi, A., Multicomponent fluid flow by discontinuous Galerkin and mixed methods in unfractured and fractured media, Water resour. res., 41, W11412, (2005) |

[22] | Aavatsmark, I., An introduction to multipoint flux approximation for quadrilateral grids, Comput. geosci., 6, 405-432, (2002) · Zbl 1094.76550 |

[23] | Edwards, M.G.; Rogers, C.F., Finite volume discretization with imposed flux continuity for the general tensor pressure equation, Comput. geosci., 2, 259-290, (1998) · Zbl 0945.76049 |

[24] | Aziz, K.; Aziz, K.; Settari, A., Petroleum reservoir simulation, (1979), Elsevier Applied Science Publishers London and New York |

[25] | K.-A. Lie, S. Krogstad, I.S. Ligaarden, J.R. Natvig, H.M. Nilsen, B. Skaflestad, Open source matlab implementation of consistent discretisations on complex grids, Comput. Geosci., 2011, doi:10.1007/s10596-011-9244-4. · Zbl 1348.86002 |

[26] | T. Russell, Stability analysis and switching criteria for adaptive implicit methods based on the cfl condition, in: SPE Symposium on Reservoir Simulation, 1989, pp. 97-107. |

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