Mathematical modeling of the two-phase fluid flow in inhomogeneous fractured porous media using the double porosity model and finite element method

V. I. Vasilyev ; M. V. Vasilyeva ; A. V. Grigorev ; G. A. Prokopiev

Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 160 (2018) no. 1, pp. 165-182

Résumé

Numerical simulation of the two-phase fluid flow in a fractured porous media using the double porosity model with a highly inhomogeneous permeability coefficient has been studied. A system of equations has been presented for the case of two-phase filtration without capillary and gravitational effects, which is a connected system of equations for pressure and saturation in a porous medium that contains a system of cracks. Different variants of specifying the flow functions between the porous medium and cracks have been considered. The numerical implementation for velocity and pressure approximation is based on the finite element method. To discretize the saturation equation, the classical Galerkin method with counter-flow approximation has been used. The results of numerical calculations for the model problem using various interflow functions have been presented.

Détail

Keywords: two-phase filtration, inhomogeneous media, fractured-porous media, double porosity model, interflow functions, finite element method, numerical stabilization.

Bibliographie

[1] Barenblatt G. I., Zheltov Iu. P., Kochina I. N., “Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks [strata]”, J. Appl. Math. Mech., 24:5 (1960), 1286–1303 | Zbl

[2] Bourgeat A., “Homogenized behavior of two-phase flows in naturally fractured reservoirs with uniform fractures distribution”, Comp. Methods Appl. Mech. Eng., 47:1 (1984), 205–216 | DOI | MR | Zbl

[3] Arbogast T., Douglas J. (Jr.), Hornung U., “Derivation of the double porosity model of single phase flow via homogenization theory”, SIAM J. Math. Anal., 21:4 (1990), 823–836 | DOI | MR | Zbl

[4] Gwo J. P., Jardine P. M., Wilson G. V., Yeh G. T., “A multiple-pore-region concept to modeling mass transfer in subsurface media”, J. Hydrol., 164:1 (1995), 217–237 | DOI

[5] Kalinkin A. A., Laevsky Y. M., “Mathematical model of water-oil displacement in fractured porous medium”, Sib. Electron. Math. Rep., 12 (2015), 743–751 | MR | Zbl

[6] Vasilyeva M. V., Vasilyev V. I., Timofeeva T. S., “Numerical solution of the convective and diffusive transport problems in a heterogeneous porous medium using finite element method”, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 158, no. 2, 2016, 243–261 (In Russian) | MR

[7] Vasilyeva M. V., Vasilyev V. I., Krasnikov A. A., Nikiforov D. Ya., “Numerical simulation of single-phase fluid flow in fractured porous media”, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 159, no. 1, 2017, 100–115 (In Russian) | MR

[8] Ginting V., Pereira F., Presho M., Wo S., “Application of the two-stage Markov chain Monte Carlo method for characterization of fractured reservoirs using a surrogate flow model”, Comput. Geosci., 15:4 (2011), 691–707 | DOI | MR | Zbl

[9] Salimi H., Bruining J., “Improved prediction of oil recovery from waterflooded fractured reservoirs using homogenization”, SPE Reservoir Eval. Eng., 13:1 (2010), 44–55 | DOI

[10] Vabishchevich P. N., Grigoriev A. V., “Numerical modeling of fluid flow in anisotropic fractured porous media”, Numer. Analys. Appl., 9:1 (2016), 45–56 | DOI | DOI | MR | Zbl

[11] Warren J. E., Root P. J., “The behavior of naturally fractured reservoirs”, Soc. Pet. Eng. J., 3:3 (1963), 245–255 | DOI

[12] Kazemi H., Merrill L. S. (Jr.), Porterfield K. L., Zeman P. R., “Numerical simulation of water-oil flow in naturally fractured reservoirs”, Soc. Pet. Eng. J., 16:6 (1976), 317–326 | DOI

[13] Douglas J. (Jr.), Arbogast T., Paes-Leme P. J., Hensley J. L., Nunes N. P., “Immiscible displacement in vertically fractured reservoirs”, Transp. Porous Media, 12:1 (1993), 73–106 | DOI

[14] Karimi-Fard M., Durlofsky L. J., Aziz K., “An efficient discrete fracture model applicable for general purpose reservoir simulators”, SPE Reservoir Simul. Symp., Soc. Petr. Eng., 2003, SPE-79699-MS, 11 pp. | DOI

[15] Efendiev Y., Lee S., Li G., Yao J., Zhang N., “Hierarchical multiscale modeling for flows in fractured media using generalized multiscale finite element method”, Int. J. Geomath., 6:2 (2015), 141–162 | DOI | MR | Zbl

[16] Akkutlu I. Y., Efendiev Y., Vasilyeva M. V., “Multiscale model reduction for shale gas transport in fractured media”, Comput. Geosci., 20:5 (2016), 953–973 | DOI | MR | Zbl

[17] Karimi-Fard M., Gong B., Durlofsky L. J., “Generation of coarse-scale continuum flow models from detailed fracture characterizations”, Water Resour. Res., 42:10 (2006), Art. W10423, 13 pp. | DOI

[18] Gong B., Karimi-Fard M., Durlofsky L. J., “Upscaling discrete fracture characterizations to dual-porosity, dual-permeability models for efficient simulation of flow with strong gravitational effects”, SPE J., 13:1 (2008), 58–67 | DOI | MR

[19] Lee S. H., Jensen C. L., Lough M. F., “Efficient finite-difference model for flow in a reservoir with multiple length-scale fractures”, SPE J., 5:3 (2000), 1–11 | DOI

[20] Li L., Lee S. H., “Efficient field-scale simulation of black oil in a naturally fractured reservoir through discrete fracture networks and homogenized media”, SPE Reservoir Eval. Eng., 11:4 (2008), 750–758 | DOI

[21] Donea J., Huerta A., Finite Element Methods for Flow Problems, John Wiley Sons, 2003, xii+350 pp. | DOI