Multiscale modeling filtration processes in porous media

Article

Published: 23.03.2018

Authors: Dimitrienko Yu.I., Bogdanov I.O.

Published in issue: #3(75)/2018

DOI: 10.18698/2308-6033-2018-3-1738

Category: Mechanics | Chapter: Mechanics of Liquid, Gas, and Plasma

The article considers a mathematical model for the multiscale process of filtration of weakly compressible liquids in periodic porous media. Computational filtration models based on the Darcy law present rule of thumb test for estimating the parameters of porous medium with a complex internal geometry. These models often use experimental data on the permeability of porous medium or approximate empirical relationships for the parameters of local flows in pores. That is why three-dimensional local problems using the method of asymptotic averaging are formulated and presented here. These are stationary problems about the flow of some fictitious linearly viscous incompressible medium. The dependence of their results and solutions on the internal pore geometry is shown. Averaging of the local equations is performed, being the basis for obtaining the global problem of unsteady filtration of weakly compressible liquids

References
[1] Coutelieris F.A., Delgado J.M.P.Q. Transport Processes in Porous Media. Berlin, Springer-Verlag, 2012, 236 p.
[2] Chen X., Zhang Y., Yan S. Two-dimensional simulations of resin flow in dual-scale fibrous porous medium under constant pressure. Journal of Reinforced Plastics and Composites, 2013, vol. 32 (22), pp. 1757–1766.
[3] Gantois R., Cantarel A., Dusserre G., Felices J.N., Schmidt F. Mold filling simulation of resin transfer molding combining BEM and level set method. Applied Mechanics and Materials, 2011, vol. 62, pp. 57–65.
[4] Loudad R., Saouab A., Beauchene P., Agogue R., Desjoyeaux B. Numerical modeling of vacuum-assisted resin transfer molding using multilayer approach. Journal of Composite Materials, 2017, vol. 51 (24), pp. 3441–3452.
[5] Song Y.S. Mathematical analysis of resin flow through fibrous porous media. Applied Composite Materials, 2006, vol. 13 (5), pp. 335–343.
[6] Yang B., Tang Q., Wang S., Jin T., Bi F. Three-dimensional numerical simulation of the filling stage in resin infusion process. Journal of Composite Materials, 2016, vol. 50 (29), pp. 4171–4186.
[7] Barenblatt G.I., Entov V.M., Ryzhik V.M. Dvizgenie zhidkostey i gazov v prirodnykh plastakh [Movement of liquids and gases in natural seams]. Moscow, Nedra Publ., 1984, 211 p.
[8] Masket M. The flow of homogeneous fluids through porous media. New York, London, McGraw-Hill Publ., 1937, 763 p. [In Russ.: Masket M. Techenie odnorodnykh zhidkostey v poristoy srede. Moscow, Izhevsk, Institut kompyuternykh issledovaniy Publ., 2004, 628 p.].
[9] Dil D.O., Bubenchikov A.M. Vestnik Tomskogo gosudarstvennogo universiteta — Tomsk State University Journal, 2013, no. 6 (26), pp. 70–78.
[10] Evstigneev D.S., Savchenko A.V. Interexpo Geo-Sibir — Interexpo GEO-Siberia, 2015, vol. 2, no. 3, pp. 64–69.
[11] Bodunov N.M., Brekhovskikh P.V. Filtratsiya vyazkoy zhidkosti cherez poristuyu sredy s ispolzovaniem zakona Brinkmana [Filtration of a viscous liquid through a porous medium using the Brinkman law]. Sbornik dokladov Vserossiyskoy nauchno-prakticheskoy konferentsii s mezhdunarodnym uchastiem “Novye tekhnologii. Materialy i oborudovanie rossiyskoy aviakosmicheskoy ot-rasli”. Kazan, 10–12 avgusta 2016 g. V dvukh tomakh. Tom 1 [Proceedings of the “All-Russian scientific conference with international participation “New technologies. Materials and equipment for the Russian aviation and aerospace industry”. Kazan, August 10–12, 2016. In 2 volumes. Vol. 1]. Kazan, Akade-miya nauk Respubliki Tatarstan Publ., 2016, pp. 643–649.
[12] Dimitrienko Yu.I., Bogdanov I.O. Matematicheskoe modelirovanie i chislennye metody — Mathematical Modeling and Computational Methods, 2017, no. 2, pp. 3–27.
[13] Dimitrienko Yu.I., Zakharova Yu.V., Bogdanov I.O. Universitetskiy nauchnyy zhurnal — Humanities and Science University Journal, 2016, no. 19, pp. 33–43.
[14] Dimitrienko Yu.I., Levina A.I., Galitsyn A.A. Vestnik MGTU im. N.E. Baumana. Seria Estestvennye nauki — Herald of the Bauman Moscow State Technical University. Series: Natural Sciences, 2011, special issue “Mathematical Mode-ling”, pp. 50–65.
[15] Dimitrienko Yu.I., Shpakova Yu.V., Bogdanov I.О., Sborshchikov S.V. Ma-tematicheskoe i chislennoe modelirovanie gazodinamicheskikh protsessov v kompozitsionnom materiael pri otverzhdenii [Mathematical and numerical modeling of gas-dynamic processes in a composite material during curing]. Trudy Mezhdunarodnoy konferentsii, posvyashchennoy 90-letiyu so dnya rozhdeniya akademika G.I. Marchuka “Aktualnye problemy vychislitelnoy i prikladnoy matematiki”. Novosibirsk, 19-23 oktyabrya 2015 g. [Proceedings of the International conference dedicated to the 90th anniversary of academician G.I. Marchuk “Morden problems of computational mathematics and mathematical modeling”. Novosibirsk, October 19–23, 2015]
[16] Dimitrienko Yu.I., Shpakova Yu.V., Bogdanov I.О., Sborshchikov S.V. Inzhenernyy zhurnal: nauka i innovatsii — Engineering Journal: Science and Innovation, 2015, no. 12 (48). DOI: 10.18698/2308-6033-2015-12-1454
[17] Dimitrienko Yu.I. Mekhanika sploshnoi sredy. V 4 tomakh. Tom 2. Universalnye zakony mekhaniki i elektrodinamiki sploshnykh sred [Continuum mechanics. In 4 volumes. Vol. 2. Universal laws of continuous medium mechanics and electrodynamics]. Moscow, BMSTU Publ., 2011, 559 p.
[18] Belyaev A.Yu. Usrednenie v zadachakh teorii filtratsii [Averaging in problems of the theory of filtration]. Moscow, Nauka Publ., 2004, 200 p.
[19] Dimitrienko Yu.I. Mekhanika kompositsionnykh materialov pri vysokikh temperaturakh [Mechanics of composite materials at high temperatures]. Moscow, Mashinostroenie Publ., 1997, 368 p.
[20] Dimitrienko Yu.I., Sokolov A.P. Metod konechnykh elementov dlya resheniya lokalnykh zadach mekhaniki kompozitsionnykh materialov [The finite element method for solving local problems in the mechanics of composite materials]. Moscow, BMSTU Publ., 2010, 66 p.
[21] Zienkiewicz O.C., Taylor R.L., Zhu J.Z. The Finite Element Method. Its Basis and Fundamentals. Seventh Edition. Oxford, Butterworth-Heinemann Publisher, 2013, 756 p.