AbstractAbout AuthorsReferences
Filtration of suspensions and colloids in porous materials occurs during the construction and operation of hydraulic structures, tunnels and underground storage facilities. Filtration models are used to calculate the penetration of grout into loose soil, when treating drinking water and industrial wastewater. During the filtration process, suspended particles pass through large pores and get stuck at the entrance of small-diameter pores. The trapped particles form a stationary deposit. A model of filtration of a polydisperse suspension in a porous material is considered. The purpose of the work is to study sediment profiles – the dependence of the concentration of deposited particles on the distance to the porous material inlet at a fixed time. An exact solution to the model was constructed using the method of characteristics. It has been shown that when filtering a polydisperse suspension, the distribution of sediment differs for different types of particles. The sediment profile of the largest particles always decreases monotonically, but the sediment profile of the smallest particles is not monotonic. It decreases at short times, then a maximum point appears on the graph, moving along the porous medium as time increases. After the maximum point reaches the porous material exit, the sediment profile becomes monotonically increasing. The sediment profiles of intermediate size particles and the total sediment profile are either monotonic or non-monotonic depending on the model parameters. The behavior of maximum points of non-monotonic profiles has been studied.
L.I. KUZMINA1, Candidate of Sciences (Physics and Mathematics), Associate Professor;
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 Moscow State University of Civil Engineering (26, Yaroslavskoe Highway, Moscow, 129337, Russian Federation)
1. Zhu G., Zhang Q., Liu R., Bai J., Li W., Xiao Feng X. Experimental and numerical study on the permeation grouting diffusion mechanism considering filtration effects. Geofluids, 2021. https://doi.org/10.1155/2021/6613990
2. Ибрагимов М.Н., Семкин В.В., Шапошников А.В. Цементация грунтов инъекцией растворов в строи-тельстве. 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 Enginee-ring and Technology. 2021. Vol. 9 (6), pp. 92–101.https://doi.org/10.24203/ajet.v9i6.6846
4. Мамедов Г.Н., Сулейманова И.Г., Тагиров Б.М. Высокоэффективный легкий заполнитель из стеклосодержащих отходов // Строительные материалы. 2020. № 12. С. 66–71. 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). 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. 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. 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. (In Russian). https://doi.org/10.31659/0585-430X-2019-767-1-2-16-22
7. Guedes R.G., Al-Abduwani F., Bedrikovetsky P., Currie P.K. Deep bed filtration under multiple particle-capture mechanisms. SPE Journal. 2009. Vol. 14. No. 3, pp. 477–487. https://doi.org/10.2118/98623-PA
8. Кузьмина Л.И., Осипов Ю.В., Шайдуллина А.М. Динамика частиц в пористой среде // Промышленное и гражданское строительство. 2021. № 10. C. 72–77. https://doi.org/10.33622/0869-7019.2021.10.72-77
8. Kuzmina L.I., Osipov Yu.V., Shaydullina A.M. Particle dynamics in a porous medium. Promyshlennoe i grazhdanskoe stroitel’stvo. 2021. No. 10, pp. 72–77. (In Russian). https://doi.org/10.33622/0869-7019.2021.10.72-77
9. Осипов Ю.В., Жеглова Ю.Г. Моделирование переноса и захвата частиц в пористой среде // Промышленное и гражданское строительство. 2019. № 11. С. 56–60. https://doi.org/10.33622/0869-7019.2019.11.56-60
9. 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).
https://doi.org/10.33622/0869-7019.2019.11.56-60
10. 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.
https://doi.org/10.1016/j.memsci.2007.09.054
11. 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.
https://doi.org/10.1016/j.memsci.2013.01.043
12. Gitis V., Rubinstein I., Livshits M., Ziskind G. Deep-bed filtration model with multistage deposition kinetics. Chemical Engineering Journal. 2010. Vol. 163, pp. 78–85. https://doi.org/10.1016/j.cej.2010.07.044
13. Сафина Г.Л. Моделирование фильтрации двухчастичной суспензии в пористой среде // Промышленное и гражданское строительство. 2022. № 2. С. 31–35. https://doi.org/10.33622/0869-7019.2022.02.31-35
13. 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). https://doi.org/10.33622/0869-7019.2022.02.31-35
14. Sun N.Z. Mathematical modeling of groundwater pollution. New York: Springer. 2014. 377 p.
15. Herzig J.P., Leclerc D.M., le Goff P. Flow of suspensions through porous media–application to deep filtration. Industrial & Engineering Chemistry Research. 1970. Vol. 62. No. 5, pp. 8–35.
https://pubs.acs.org/doi/abs/10.1021/ie50725a003
16. Bedrikovetsky P. Upscaling of stochastic micro model for suspension transport in porous media. Transport in Porous Media. 2008. Vol. 75. No. 3, pp. 335–369. https://doi.org/10.1007/s11242-008-9228-6
17. 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. No. 4, pp. 48–54. https://doi.org/10.22337/2587-9618-2020-16-4-48-54
18. Кузьмина Л.И., Осипов Ю.В. Фильтрация частиц в пористом материале // Строительные материалы. 2023. № 9. C. 89–93. https://doi.org/10.31659/0585-430X-2023-817-9-89-93
18. 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). https://doi.org/10.31659/0585-430X-2023-817-9-89-93
19. 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. No. 5, pp. 556–564.
https://doi.org/10.1134/S0040579507050168
20. 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. https://doi.org/10.1016/j.ijnonlinmec.2018.07.007
21. Полянин А.Д. Точные решения дифференциальных, интегральных, функциональных и других математических уравнений. М.: ИПМех РАН, 2023. 600 с.
21. Polyanin A.D. Tochnye resheniya differentsial’nykh, integral’nykh, funktsional’nykh i drugikh matematicheskikh uravnenii [Exact solutions to differential, integral, functional and other mathematical equations]. Moscow: IPMech RAN. 2023. 600 p.
22. Полянин А.Д., Журов А.И. Методы разделения переменных и точные решения нелинейных уравнений математической физики. М.: ИПМех РАН, 2021. 383 с.
22. Polyanin A.D., Zhurov A.I. Metody razdeleniya peremennykh i tochnye resheniya nelineinykh uravnenii matematicheskoi fiziki [Methods of separation of variables and exact solutions to nonlinear equations of mathematical physics]. Moscow: IPMech RAN. 2021. 383 p.
23. Кузьмина Л.И., Осипов Ю.В., Царева В.И. Обратная задача для линейной функции фильтрации // Промышленное и гражданское строительство. 2020. № 6. С. 64–68.
https://doi.org/10.33622/0869-7019.2020.06.64-68
23. 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). https://doi.org/10.33622/0869-7019.2020.06.64-68
24. 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. No. 1, pp. 43–62.
https://doi.org/10.1007/s11242-006-9082-3
25. Сафина Г.Л. Расчет профилей осадка двухчастичной суспензии в пористой среде // Промышленное и гражданское строительство. 2020. № 11. С. 110–114.
25. Safina G.L. Calculation of deposit profiles of a two-particle suspension in a porous medium // Promyshlennoe i grazhdanskoe stroitel’stvo. 2020. No. 11, pp. 110–114. (In Russian).
26. 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. 121984.
https://doi.org/10.1016/j.cej.2019.121984
27. Осипов Ю.В., Астахов М.Д. Расчет фильтрации бидисперсной суспензии в пористой среде // Инженерно-строительный вестник Прикаспия. 2020. Т. 31. № 1. С. 69–72.
27. Osipov Yu.V., Astakhov M.D. Calculation of filtration of bidisperse suspension in a porous medium. Inzhenerno-stroitel’nyj vestnik Prikaspiya. 2020. Vol. 31. No. 1, pp. 69–72. (In Russian).
2. Ибрагимов М.Н., Семкин В.В., Шапошников А.В. Цементация грунтов инъекцией растворов в строи-тельстве. 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 Enginee-ring and Technology. 2021. Vol. 9 (6), pp. 92–101.https://doi.org/10.24203/ajet.v9i6.6846
4. Мамедов Г.Н., Сулейманова И.Г., Тагиров Б.М. Высокоэффективный легкий заполнитель из стеклосодержащих отходов // Строительные материалы. 2020. № 12. С. 66–71. 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). 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. 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. 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. (In Russian). https://doi.org/10.31659/0585-430X-2019-767-1-2-16-22
7. Guedes R.G., Al-Abduwani F., Bedrikovetsky P., Currie P.K. Deep bed filtration under multiple particle-capture mechanisms. SPE Journal. 2009. Vol. 14. No. 3, pp. 477–487. https://doi.org/10.2118/98623-PA
8. Кузьмина Л.И., Осипов Ю.В., Шайдуллина А.М. Динамика частиц в пористой среде // Промышленное и гражданское строительство. 2021. № 10. C. 72–77. https://doi.org/10.33622/0869-7019.2021.10.72-77
8. Kuzmina L.I., Osipov Yu.V., Shaydullina A.M. Particle dynamics in a porous medium. Promyshlennoe i grazhdanskoe stroitel’stvo. 2021. No. 10, pp. 72–77. (In Russian). https://doi.org/10.33622/0869-7019.2021.10.72-77
9. Осипов Ю.В., Жеглова Ю.Г. Моделирование переноса и захвата частиц в пористой среде // Промышленное и гражданское строительство. 2019. № 11. С. 56–60. https://doi.org/10.33622/0869-7019.2019.11.56-60
9. 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).
https://doi.org/10.33622/0869-7019.2019.11.56-60
10. 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.
https://doi.org/10.1016/j.memsci.2007.09.054
11. 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.
https://doi.org/10.1016/j.memsci.2013.01.043
12. Gitis V., Rubinstein I., Livshits M., Ziskind G. Deep-bed filtration model with multistage deposition kinetics. Chemical Engineering Journal. 2010. Vol. 163, pp. 78–85. https://doi.org/10.1016/j.cej.2010.07.044
13. Сафина Г.Л. Моделирование фильтрации двухчастичной суспензии в пористой среде // Промышленное и гражданское строительство. 2022. № 2. С. 31–35. https://doi.org/10.33622/0869-7019.2022.02.31-35
13. 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). https://doi.org/10.33622/0869-7019.2022.02.31-35
14. Sun N.Z. Mathematical modeling of groundwater pollution. New York: Springer. 2014. 377 p.
15. Herzig J.P., Leclerc D.M., le Goff P. Flow of suspensions through porous media–application to deep filtration. Industrial & Engineering Chemistry Research. 1970. Vol. 62. No. 5, pp. 8–35.
https://pubs.acs.org/doi/abs/10.1021/ie50725a003
16. Bedrikovetsky P. Upscaling of stochastic micro model for suspension transport in porous media. Transport in Porous Media. 2008. Vol. 75. No. 3, pp. 335–369. https://doi.org/10.1007/s11242-008-9228-6
17. 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. No. 4, pp. 48–54. https://doi.org/10.22337/2587-9618-2020-16-4-48-54
18. Кузьмина Л.И., Осипов Ю.В. Фильтрация частиц в пористом материале // Строительные материалы. 2023. № 9. C. 89–93. https://doi.org/10.31659/0585-430X-2023-817-9-89-93
18. 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). https://doi.org/10.31659/0585-430X-2023-817-9-89-93
19. 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. No. 5, pp. 556–564.
https://doi.org/10.1134/S0040579507050168
20. 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. https://doi.org/10.1016/j.ijnonlinmec.2018.07.007
21. Полянин А.Д. Точные решения дифференциальных, интегральных, функциональных и других математических уравнений. М.: ИПМех РАН, 2023. 600 с.
21. Polyanin A.D. Tochnye resheniya differentsial’nykh, integral’nykh, funktsional’nykh i drugikh matematicheskikh uravnenii [Exact solutions to differential, integral, functional and other mathematical equations]. Moscow: IPMech RAN. 2023. 600 p.
22. Полянин А.Д., Журов А.И. Методы разделения переменных и точные решения нелинейных уравнений математической физики. М.: ИПМех РАН, 2021. 383 с.
22. Polyanin A.D., Zhurov A.I. Metody razdeleniya peremennykh i tochnye resheniya nelineinykh uravnenii matematicheskoi fiziki [Methods of separation of variables and exact solutions to nonlinear equations of mathematical physics]. Moscow: IPMech RAN. 2021. 383 p.
23. Кузьмина Л.И., Осипов Ю.В., Царева В.И. Обратная задача для линейной функции фильтрации // Промышленное и гражданское строительство. 2020. № 6. С. 64–68.
https://doi.org/10.33622/0869-7019.2020.06.64-68
23. 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). https://doi.org/10.33622/0869-7019.2020.06.64-68
24. 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. No. 1, pp. 43–62.
https://doi.org/10.1007/s11242-006-9082-3
25. Сафина Г.Л. Расчет профилей осадка двухчастичной суспензии в пористой среде // Промышленное и гражданское строительство. 2020. № 11. С. 110–114.
25. Safina G.L. Calculation of deposit profiles of a two-particle suspension in a porous medium // Promyshlennoe i grazhdanskoe stroitel’stvo. 2020. No. 11, pp. 110–114. (In Russian).
26. 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. 121984.
https://doi.org/10.1016/j.cej.2019.121984
27. Осипов Ю.В., Астахов М.Д. Расчет фильтрации бидисперсной суспензии в пористой среде // Инженерно-строительный вестник Прикаспия. 2020. Т. 31. № 1. С. 69–72.
27. Osipov Yu.V., Astakhov M.D. Calculation of filtration of bidisperse suspension in a porous medium. Inzhenerno-stroitel’nyj vestnik Prikaspiya. 2020. Vol. 31. No. 1, pp. 69–72. (In Russian).
For citation: Kuzmina L.I., Osipov Yu.V. Sedimentation of heterogeneous particles in a porous material. Stroitel'nye Materialy [Construction Materials]. 2024. No. 8, pp. 63–68. (In Russian). https://doi.org/10.31659/0585-430X-2024-827-8-63-68