Abstract
A macroscopic model of two-phase flow of compressible liquids in a compressible double porosity medium is developed and used to analyse various qualitative mechanisms of the occurrence of memory (delay). The two main mechanisms are non-instantaneous capillary redistribution of liquids, and non-instantaneous relaxation of pressure. In addition, cross effects of memory arise, caused by asymmetric extrusion of liquids from pores due to phase expansion and pore compaction, as well as nonlinear overlap of compressibility and capillarity (non-linear extrusion). To construct the model, the asymptotic method of two-scale averaging in the variational formulation is applied. Complete averaging has been achieved due to the separation of nonlocality and nonlinearity at different levels of the asymptotic expansion. All delay times are explicitly defined as functions of saturation and pressure.
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REFERENCES
Barenblatt, G.I., Zheltov, Yu.P., and Kochina, I.N., Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks [strata], J. Appl. Math. Mech. (Engl. Transl.), 1960, vol. 24, no. 5, pp. 1286–1303.
Arbogast, T., Douglas, J., and Hornung, U., Derivation of the double porosity model of single-phase flow via homogenization theory, SIAM J. Math. Anal., 1990, vol. 21, no. 4, pp. 823–836.
Panfilov, M., Macroscale Models of Flow through Highly Heterogeneous Porous Media, Dordrecht: Kluwer Academic Publishers, 2000.
Amaziane, B., Jurak, M., Pankratov, L., and Vrbaski, A., Some remarks on the homogenization of immiscible incompressible two-phase flow in double porosity media, Discrete Contin. Dyn. Syst.- Ser. B, 2018, vol. 23, no. 2, pp. 629–665.
Amaziane, B., Milisic, J.P., Panfilov, M., and Pankratov, L., Generalized nonequilibrium capillary relations for two-phase flow through heterogeneous media, Phys. Rev. E, 2012, vol. 85, p. 016304.
Amaziane, B., Pankratov, L., Jurak, M., and Vrbaski, A., A fully homogenized model for incompressible two-phase flow in double porosity media, Appl. Anal., 2015, vol. 95, no. 10, pp. 2280–2299.
Bourgeat, A., Luckhaus, S., and Mikelic, A., Convergence of the homogenization process for a double-porosity model of immiscible two-phase flow, SIAM J. Math. Anal., 1966, vol. 27, no. 6, pp. 1520–1543.
Bourgeat, A. and Panfilov, M., Effective two-phase flow through highly heterogeneous porous media, Comput. Geosci., 1998, vol. 2, pp. 191–215.
Yeh, L.M., Homogenization of two-phase flow in fractured media, Math. Models Methods Appl. Sci., 2006, vol. 16, pp. 1627–1651.
Arbogast, T., A simplified dual porosity model for two-phase flow, in Computational Methods in Water Resources IX, vol. 2: Mathematical Modeling in Water Resources, Russell, T.F., et al., Eds., Southampton: Computational Mechanics Publ., 1992.
Ait Mahiout, L., Amaziane, B., Mokrane, A., and Pankratov, L., Homogenization of immiscible compressible two-phase flow in double porosity media, Electron. J. Differ. Equations, 2016, vol. 52, pp. 1–28.
Amaziane, B. and Pankratov, L., Homogenization of a model for water-gas flow through double-porosity media, Math. Methods Appl. Sci., 2016, vol. 39, pp. 425–451.
Jafari, I., Masihi, M., and Nasiri Zarandi, M., Experimental study on imbibition displacement mechanisms of two-phase fluid using micromodel: Fracture network, distribution of pore size, and matrix construction, Phys. Fluids, 2017, vol. 29, no. 11, p. 122004.
Khoshkalam, Y., Khosravi, M., and Rostami, B., Visual investigation of viscous cross-flow during foam injection in a matrix-fracture system, Phys. Fluids, 2019, vol. 31, p. 023102.
Yao, C.C. and Yan, P.Y., A diffuse interface approach to injection-driven flow of different miscibility in heterogeneous porous media, Phys. Fluids, 2015, vol. 27, no. 8, p. 083101.
Li, H., Guo, H., Yang, Z. et al., Evaluation of oil production potential in fractured porous media, Phys. Fluids, 2019, vol. 31, p. 052104.
Jafari, I., Masihi, M., and Nasiri Zarandi, M., Numerical simulation of counter-current spontaneous imbibition in water-wet fractured porous media: Influences of water injection velocity, fracture aperture, and 826 grains geometry, Phys. Fluids, 2017, vol. 29, no. 11, p. 113305.
Rokhforouz, M.R. and Akhlaghi Amiri H.A., Phase-field simulation of counter-current spontaneous imbibition in a fractured heterogeneous porous medium, Phys. Fluids. 2017. V. 29. P. 062104.
Hussein, M., Multiphase flow simulations in heterogeneous fractured media through hybrid grid method, AIP Conf. Proc., 2013, no. 1558, p. 2048.
Saedi, B., Ayatollahi, S., and Masihi, M., Free fall and controlled gravity drainage processes in fractured porous media: Laboratory and modelling investigation, Can. J. Chem. Eng., 2015, vol. 93, p. 2286.
Allaire, G., Homogenization and two-scale convergence, SIAM J. Math. Anal., 1992, vol. 28, pp. 1482–1518.
Meirmanov, A.M., Mathematical Models for Poroelastic Flows, vol. 1 of Atlantis Studies in Differential Equations, Beijing: Atlantis, 2014.
Meirmanov, A.M., Reiterated homogenization in the problems on filtration of underground liquids, Nauchn. Vedomosti Belgorod. Gos. Univ. Ser.: Mat. Fiz., 2012, no. 17 (136), pp. 178–193.
Nikolaevskii, V.N., Mekhanika poristykh i treshchinovatykh sred (The Mechanics of Porous and Fractured Media), Moscow: Nedra, 1984.
Scheidegger, A.E., The Physics of Flow through Porous Media, Toronto: Univ. Press, 1974.
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This work was supported by the Scientific Committee of the Ministry of Education and Science of the Republic of Kazakhstan, grant no. AR05132680.
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Panfilov, M.B., Baishemirov, Z.D. & Berdyshev, A.S. Macroscopic Model of Two-Phase Compressible Flow in Double Porosity Media. Fluid Dyn 55, 936–951 (2020). https://doi.org/10.1134/S001546282007006X
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DOI: https://doi.org/10.1134/S001546282007006X