Abstract
Fully implicit time-space discretizations applied to the two-phase Darcy flow problem leads to the systems of nonlinear equations, which are traditionally solved by some variant of Newton’s method. The efficiency of the resulting algorithms heavily depends on the choice of the primary unknowns since Newton’s method is not invariant with respect to a nonlinear change of variable. In this regard, the role of capillary pressure/saturation relation is paramount because the choice of primary unknowns is restricted by its shape. We propose an elegant mathematical framework for two-phase flow in heterogeneous porous media resulting in a family of formulations, which apply to general monotone capillary pressure/saturation relations and handle the saturation jumps at rocktype interfaces. The presented approach is applied to the hybrid dimensional model of two-phase water-gas Darcy flow in fractured porous media for which the fractures are modelled as interfaces of co-dimension one. The problem is discretized using an extension of vertex approximate gradient scheme. As for the phase pressure formulation, the discrete model requires only two unknowns by degree of freedom.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Ahmed, R., Edwards, M., Lamine, S., Huisman, B.: Control-volume distributed multi-point flux approximation coupled with a lower-dimensional fracture model. J. Comput. Phys. 284, 462–489 (2015)
Alboin, C., Jaffre, J., Roberts, J., Serres, C.: Modeling fractures as interfaces for flow and transport in porous media. Fluid Flow Transp. Porous Media 295, 13–24 (2002)
Angot, P., Boyer, F., Hubert, F.: Asymptotic and numerical modelling of flows in fractured porous media. Math. Model. Numer. Anal. 43, 239–275 (2009)
Brenner, K., Groza, M., Guichard, C., Lebeau, G., Masson, R.: Gradient discretization of hybrid dimensional Darcy flows in fractured porous media. Numer. Math. 134(3), 569–609 (2016). doi:10.1007/s00211-015-0782-x
Brenner, K., Groza, M., Guichard, C., Masson, R.: Vertex approximate gradient scheme for hybrid dimensional two-phase Darcy flows in fractured porous media. Math. Model. Numer. Anal. 49, 303–330 (2015)
Brenner, K., Hennicker, J., Masson, R., Samier, P.: Gradient discretization of hybrid-dimensional Darcy flow in fractured porous media with discontinuous pressures at matrix-fracture interfaces. IMA J. Numer. Anal. (2016). doi:10.1093/imanum/drw044
Brenner, K., Hennicker, J., Masson, R., Samier, P.: Hybrid dimensional modelling and discretization of two phase Darcy flow through DFN in porous media. In: ECMOR XV- 15th European Conference on the Mathematics of Oil Recovery, 29 August–1 September 2016. Amsterdam (2016)
Cancès, C., Pierre, M.: An existence result for multidimensional immiscible two-phase flows with discontinuous capillary pressure field. SIAM. J. Math. Anal. 44, 966–992 (2012)
Ding, D., Langouet, H., Jeannin, L.: Simulation of fracturing induced formation damage and gas production from fractured wells in tight gas reservoirs. SPE 153255, 246–258 (2012). doi:10.2118/153255-PA
Eymard, R., Guichard, C., Herbin, R.: Small-stencil 3D schemes for diffusive flows in porous media. Math. Model. Numer. Anal. 46, 265–290 (2010)
Eymard, R., Guichard, C., Herbin, R., Masson, R.: Gradient schemes for two-phase flow in heterogeneous porous media and Richards equation. ZAMM - J. Appl. Math. Mech. 94, 560–585 (2014)
Flauraud, E., Nataf, F., Faille, I., Masson, R.: Domain decomposition for an asymptotic geological fault modeling. C. R. Acad. Bulg. Sci. Méc. 331, 849–855 (2003)
Hoteit, J., Firoozabadi, A.: An efficient numerical model for incompressible two-phase flow in fracture media. Adv. Water Resour. 31, 891–905 (2008)
Karimi-Fard, M., Durlofsky, L.J., Aziz, K.: An efficient discrete-fracture model applicable for general-purpose reservoir simulators. SPE J. 9(2), 227–236 (2004)
Martin, V., Jaffré, J., Roberts, J.: Modeling fractures and barriers as interfaces for flow in porous media. SIAM J. Sci. Comput. 26, 1667–1691 (2005)
Monteagudu, J., Firoozabadi, A.: Control-volume model for simulation of water injection in fractured media: incorporating matrix heterogeneity and reservoir wettability effects. SPE J. 12, 355– 366 (2007)
Reichenberger, V., Jakobs, H., Bastian, P., Helmig, R.: A mixed-dimensional finite volume method for multiphase flow in fractured porous media. Adv. Water Resour. 29, 1020– 1036 (2006)
Sandve, T., Berre, I., Nordbotten, J.: An efficient multi-point flux approximation method for discrete fracture-matrix simulations. J. Comput. Phys. 231, 3784–3800 (2012)
Si, H.: TetGen, TetGen, a delaunay-based quality tetrahedral mesh generator. ACM Trans. Math. Softw. 41(2), Article 11 (2015). doi:10.1145/2629697
Tunc, X., Faille, I., Gallouët, T., Cacas, M., Havé, P.: A model for conductive faults with non matching grids. Comput. Geosci. 16, 277–296 (2012)
Acknowledgements
The authors would like to thank ENGIE EP and Storengy for supporting this work and allowing its publication. This work was also supported by the GeoPor project funded by the French National Research Agency (ANR) with the grant ANR-13-JS01-0007-01 (project GEOPOR).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Brenner, K., Groza, M., Jeannin, L. et al. Immiscible two-phase Darcy flow model accounting for vanishing and discontinuous capillary pressures: application to the flow in fractured porous media. Comput Geosci 21, 1075–1094 (2017). https://doi.org/10.1007/s10596-017-9675-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10596-017-9675-7