AbstractAbout AuthorsReferences
Filtration of suspended solid particles in porous material simulates the processes of strengthening foundations, creating waterproof walls in rock, construction and reconstruction of roads, colmatation (deposition of particles) in the bottom-hole zone of the well by components of drilling fluid during oil production, the operation of filter elements of treatment facilities, and much more. The purpose of this work was to study the filtration of a monodisperse suspension of high concentration in a homogeneous porous medium having pores of various sizes and configurations. A suspension was pumped into the porous medium under pressure, displacing pure liquid containing no particles from the pores. It is assumed that the main reason for particle retention is a size mechanism: particles pass freely through large pores and become stuck in narrow pores, the diameter of which is smaller than the particle size. The nonlinear dependence of the sediment growth rate on the concentration of suspended particles, characteristic of a highly concentrated suspension, is modeled. With the slow movement of the suspension in the porous material, the deposited particles remain motionless. They cannot be torn away from the framework of the porous medium by the carrier fluid and impacts of suspended particles. A mathematical model describes the transformation of suspended particles into sediment and sets the rate of sediment growth. A solution to the filtration problem in implicit integral form and a simple algebraic relation (Riemann invariant) relating the concentrations of suspended and deposited particles are obtained. The problem is solved for a linear filtration function and a general nonlinear concentration function. An asymptotic solution was constructed near the concentration front of suspended and deposited particles, specifying an approximate solution in the form of explicit algebraic formulas. It is shown that the asymptotic behavior is close to the exact solution; the error decreases with increasing order of the asymptotic expansion.
Keywords: filtration, porous material, suspension, dimensional particle retention mechanism, exact solution, asymptotics.
Keywords: filtration, porous material, suspension, dimensional particle retention mechanism, exact solution, asymptotics.
L.I. KUZMINA1, Candidate of Sciences (Physics and Mathematics), Associate Professor (This email address is being protected from spambots. You need JavaScript enabled to view it.);
Yu.V. OSIPOV2, Candidate of Sciences (Physics and Mathematics), Professor
Yu.V. OSIPOV2, Candidate of Sciences (Physics and Mathematics), Professor
1 National Research University Higher School of Economics (20, Myasnitskaya Street, Moscow, 101000, Russian Federation)
2 National Research Moscow State University of Civil Engineering (26, Yaroslavskoe Highway, Moscow, 129337, Russian Federation)
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9. Galaguz Yu.P., Kuzmina L.I., Osipov Yu.V. The problem of filtering a suspension in a porous medium with sediment. Mekhanika zhidkosti i gaza. Izvestiya Rossiiskoi akademii nauk (Izv. RAN. MZhG). 2019. Vol. 54(1), pp. 85–97. (In Russian). DOI: https://doi.org/10.1134/S0015462819010063
10. Kuzmina L.I., Osipov Yu.V. Determining the Lengmur coefficient of the filtration problem. International Journal for Computational Civil and Structural Engineering. 2020. Vol. 16(4), pp. 48–54. DOI: 10.22337/2587-9618-2020-16-4-48-54
11. Сафина Г.Л. Моделирование фильтрации двухчастичной суспензии в пористой среде // Промышленное и гражданское строительство. 2022. № 2. С. 31–35. DOI: https://doi.org/10.33622/0869-7019.2022.02.31-35
11. Safina G.L. Modelling of filtration of a two-particle suspension in a porous medium. Promyshlennoe i grazhdanskoe stroitel’stvo. 2022. No. 2, pp. 31–35. (In Russian). DOI: https://doi.org/10.33622/0869-7019.2022.02.31-35
12. Сафина Г.Л. Расчет профилей осадка двухчастичной суспензии в пористой среде // Промышленное и гражданское строительство. 2020. № 11. C. 110–114. DOI: https://doi.org/10.33622/0869-7019.2020.11.110-114
12. Safina G.L. Calculation of deposit profiles of a two-particle suspension in a porous medium. Promyshlennoe i grazhdanskoe stroitel’stvo. 2020. № 11. С. 110–114. (In Russian). DOI: https://doi.org/10.33622/0869-7019.2020.11.110-114
13. Zhang H., Malgaresi G.V.C., Bedrikovetsky P. Exact solutions for suspension colloidal transport with multiple capture mechanisms. International Journal of Non-Linear Mechanics. 2018. Vol. 105, pp. 27–42. DOI: https://doi.org/10.1016/j.ijnonlinmec.2018.07.007
14. Kuzmina L.I., Nazaikinskii V.E., Osipov Y.V. On a deep bed filtration problem with finite blocking time. Russian Journal of Mathematical Physics. 2019. Vol. 26 (1), pp. 130–134. DOI: 10.1134/S1061920819010138
15. Vyazmina E.A., Bedrikovetskii P.G., Polyanin A.D. New classes of exact solutions to nonlinear sets of equations in the theory of filtration and convective mass transfer. Theoretical Foundations of Chemical Engineering. 2007. Vol. 41(5), pp. 556–564. DOI: 10.1134/S0040579507050168
16. Осипов Ю.В., Жеглова Ю.Г. Моделирование переноса и захвата частиц в пористой среде // Промышленное и гражданское строительство. 2019. № 11. С. 31–35. DOI: 10.33622/0869-7019.2019.11.56-60
16. Osipov Yu.V., Zheglova Yu.G. Modelling of transport and retention of particles in porous media. Promyshlennoe i grazhdanskoe stroitel’stvo. 2019. No. 11, pp. 56–60. (In Russian). DOI: 10.33622/0869-7019.2019.11.56-60
17. Malgaresi G., Collins B., Alvaro P., Bedrikovetsky P. Explaining non-monotonic retention profiles during flow of size-distributed colloids. Chemical Engineering Journal. 2019. Vol. 375. ID 121984. DOI: 10.1016/j.cej.2019.121984
18. Vaz A, Maffra D, Carageorgos T, Bedrikovetsky P. Characterisation of formation damage during reactive flows in porous media. Journal of Natural Gas Science and Engineering. 2016. Vol. 34, pp. 1422–1433. DOI: 10.1016/j.jngse.2016.08.016
19. Кузьмина Л.И., Осипов Ю.В., Царева В.И. Обратная задача для линейной функции фильтрации // Промышленное и гражданское строительство. 2020. № 6. С. 64–68. DOI: 10.33622/0869-7019.2020.06.64-68
19. Kuzmina L.I., Osipov Yu.V., Tsareva V.I. Inverse problem for a linear filtration function. Promyshlennoe i grazhdanskoe stroitel’stvo. 2020. No. 6, pp. 64–68. (In Russian). DOI: 10.33622/0869-7019.2020.06.64-68
20. Alvarez A.C., Hime G., Marchesin D., Bedrikovetsky P.G. The inverse problem of determining the filtration function and permeability reduction in flow of water with particles in porous media. Transport in Porous Media. 2007. Vol. 70 (1), pp. 43–62. DOI: https://doi.org/10.1007/s11242-006-9082-3
2. Ибрагимов М.Н., Семкин В.В., Шапошников А.В. Цементация грунтов инъекцией растворов в строительстве. М.: АСВ, 2017. 266 с.
2. Ibragimov M.N., Semkin V.V., Shaposhnikov A.V. Tsementatsiya gruntov inektsiei rastvorov v stroitel’stve [Cementation of soils by injection of solutions in construction]. Moscow: ASV. 2017. 266 p.
3. Christodoulou D., Lokkas P., Droudakis A., Spiliotis X., Kasiteropoulou D., Alamanis N. The development of practice in permeation grouting by using fine-grained cement suspensions. Asian Journal of Engineering and Technology. 2021. Vol. 9 (6), pp. 92–101. DOI: https://doi.org/10.24203/ajet.v9i6.6846
4. Мамедов Г.Н., Сулейманова И.Г., Тагиров Б.М. Высокоэффективный легкий заполнитель из стек-лосодержащих отходов // Строительные материалы. 2020. № 12. С. 66–71. DOI: https://doi.org/10.31659/0585-430X-2020-787-12-66-71
4. Mammadov H.N., Suleimanova I.H., Tahirov B.M. High-effective lightweight aggregate obtained from glass-containing waste. Stroitel’nye Materialy [Construction Materials]. 2020. No. 12, pp. 66–71. (In Russian). DOI: https://doi.org/10.31659/0585-430X-2020-787-12-66-71
5. Федорова Г.Д., Александров Г.Н., Скрябин А.П. Активация структурообразующих свойств оксида графена в цементных композитах // Строительные материалы. 2020. № 1–2. С. 17–23. DOI: https://doi.org/10.31659/0585-430X-2020-778-1-2-17-23
5. Fedorova G.D., Aleksandrov G.N., Scryabin A.P. Activation of structure-forming properties of graphene oxide in cement composites. Stroitel’nye Materialy [Construction Materials]. 2020. No. 1–2, pp. 17–23. (In Russian). DOI: https://doi.org/10.31659/0585-430X-2020-778-1-2-17-23
6. Федорова Г.Д., Скрябин А.П., Александров Г.Н. Исследование влияния оксида графена на прочность цементного раствора // Строительные материалы. 2019. № 1–2. С. 16–22. DOI: https://doi.org/10.31659/0585-430X-2019-767-1-2-16-22
6. Fedorova G.D., Skriabin A.P., Aleksandrov G.N. The study of the influence of graphene oxide on the strength of cement stone using river sand. Stroitel’nye Materialy [Construction Materials]. 2019. No. 1–2, pp. 16–22. DOI: https://doi.org/10.31659/0585-430X-2019-767-1-2-16-22 (In Russian).
7. Santos A., Bedrikovetsky P., Fontoura S. Analytical micro model for size exclusion: Pore blocking and permeability reduction. Journal of Membrane Science. 2008. Vol. 308, pp. 115–127. DOI: https://doi.org/10.1016/j.memsci.2007.09.054
8. Bashtani F., Ayatollahi S., Habibi A., Masihi M. Permeability reduction of membranes during particulate suspension flow; analytical micro model of size exclusion mechanism. Journal of Membrane Science. 2013. Vol. 435, pp. 155–164. DOI: 10.1016/j.memsci.2013.01.043
9. Галагуз Ю.П., Кузьмина Л.И., Осипов Ю.В. Задача фильтрации суспензии в пористой среде с осадком // Механика жидкости и газа. Известия Российской академии наук (Изв. РАН. МЖГ). 2019. № 1. С. 86–98. DOI: 10.1134/S0568528119010067
9. Galaguz Yu.P., Kuzmina L.I., Osipov Yu.V. The problem of filtering a suspension in a porous medium with sediment. Mekhanika zhidkosti i gaza. Izvestiya Rossiiskoi akademii nauk (Izv. RAN. MZhG). 2019. Vol. 54(1), pp. 85–97. (In Russian). DOI: https://doi.org/10.1134/S0015462819010063
10. Kuzmina L.I., Osipov Yu.V. Determining the Lengmur coefficient of the filtration problem. International Journal for Computational Civil and Structural Engineering. 2020. Vol. 16(4), pp. 48–54. DOI: 10.22337/2587-9618-2020-16-4-48-54
11. Сафина Г.Л. Моделирование фильтрации двухчастичной суспензии в пористой среде // Промышленное и гражданское строительство. 2022. № 2. С. 31–35. DOI: https://doi.org/10.33622/0869-7019.2022.02.31-35
11. Safina G.L. Modelling of filtration of a two-particle suspension in a porous medium. Promyshlennoe i grazhdanskoe stroitel’stvo. 2022. No. 2, pp. 31–35. (In Russian). DOI: https://doi.org/10.33622/0869-7019.2022.02.31-35
12. Сафина Г.Л. Расчет профилей осадка двухчастичной суспензии в пористой среде // Промышленное и гражданское строительство. 2020. № 11. C. 110–114. DOI: https://doi.org/10.33622/0869-7019.2020.11.110-114
12. Safina G.L. Calculation of deposit profiles of a two-particle suspension in a porous medium. Promyshlennoe i grazhdanskoe stroitel’stvo. 2020. № 11. С. 110–114. (In Russian). DOI: https://doi.org/10.33622/0869-7019.2020.11.110-114
13. Zhang H., Malgaresi G.V.C., Bedrikovetsky P. Exact solutions for suspension colloidal transport with multiple capture mechanisms. International Journal of Non-Linear Mechanics. 2018. Vol. 105, pp. 27–42. DOI: https://doi.org/10.1016/j.ijnonlinmec.2018.07.007
14. Kuzmina L.I., Nazaikinskii V.E., Osipov Y.V. On a deep bed filtration problem with finite blocking time. Russian Journal of Mathematical Physics. 2019. Vol. 26 (1), pp. 130–134. DOI: 10.1134/S1061920819010138
15. Vyazmina E.A., Bedrikovetskii P.G., Polyanin A.D. New classes of exact solutions to nonlinear sets of equations in the theory of filtration and convective mass transfer. Theoretical Foundations of Chemical Engineering. 2007. Vol. 41(5), pp. 556–564. DOI: 10.1134/S0040579507050168
16. Осипов Ю.В., Жеглова Ю.Г. Моделирование переноса и захвата частиц в пористой среде // Промышленное и гражданское строительство. 2019. № 11. С. 31–35. DOI: 10.33622/0869-7019.2019.11.56-60
16. Osipov Yu.V., Zheglova Yu.G. Modelling of transport and retention of particles in porous media. Promyshlennoe i grazhdanskoe stroitel’stvo. 2019. No. 11, pp. 56–60. (In Russian). DOI: 10.33622/0869-7019.2019.11.56-60
17. Malgaresi G., Collins B., Alvaro P., Bedrikovetsky P. Explaining non-monotonic retention profiles during flow of size-distributed colloids. Chemical Engineering Journal. 2019. Vol. 375. ID 121984. DOI: 10.1016/j.cej.2019.121984
18. Vaz A, Maffra D, Carageorgos T, Bedrikovetsky P. Characterisation of formation damage during reactive flows in porous media. Journal of Natural Gas Science and Engineering. 2016. Vol. 34, pp. 1422–1433. DOI: 10.1016/j.jngse.2016.08.016
19. Кузьмина Л.И., Осипов Ю.В., Царева В.И. Обратная задача для линейной функции фильтрации // Промышленное и гражданское строительство. 2020. № 6. С. 64–68. DOI: 10.33622/0869-7019.2020.06.64-68
19. Kuzmina L.I., Osipov Yu.V., Tsareva V.I. Inverse problem for a linear filtration function. Promyshlennoe i grazhdanskoe stroitel’stvo. 2020. No. 6, pp. 64–68. (In Russian). DOI: 10.33622/0869-7019.2020.06.64-68
20. Alvarez A.C., Hime G., Marchesin D., Bedrikovetsky P.G. The inverse problem of determining the filtration function and permeability reduction in flow of water with particles in porous media. Transport in Porous Media. 2007. Vol. 70 (1), pp. 43–62. DOI: https://doi.org/10.1007/s11242-006-9082-3
For citation: Kuzmina L.I., Osipov Yu.V. Filtration of suspension in a porous material. Stroitel’nye Materialy [Construction Materials]. 2023. No. 9, pp. 89–93. (In Russian). DOI: https://doi.org/10.31659/0585-430X-2023-817-9-89-93