FILTRATION PROBLEM WITH NONLINEAR FILTRATION AND CONCENTRATION FUNCTIONS
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Abstract
ncient architectural buildings are of great value for all modern humanity. Over time, under the influence of vibrations, water and other man-made and natural factors, the foundations of such buildings are destroyed, the soil structure changes. Currently, one of the most popular methods of strengthening soils and strengthening foundations is the jet grouting technology. When the liquid grout passes through the porous rock, the suspended particles of the grout form a deposit. In this paper, we study a one-dimensional model of suspension deep bed filtration in a porous medium with different particle capture mechanisms. The considered filtration model consists of the balance equation for the masses of suspended and retained particles and the kinetic equation for deposit growth. In this case, the deposit growth rate is determined by the concentration function of suspended particles, which, in turn, depends on the properties of the suspension and the geometry of the porous medium. The solution to the problem is obtained for linear and non-linear concentration functions. An asymptotic solution of the problem is constructed for both types of functions near the concentration front of suspended and retained particles. It is shown that the asymptotic and numerical solutions are close over a long time interval.
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References
Li B., Han J. Conservation and regeneration of historical buildings // IOP Conference Series: Earth and Environmental Science, 2021, vol. 787, 012179. DOI: https://doi.org/10.1088/1755-1315/787/1/012179
Makowski A., Polanska B. Application of jet grouting technology for the renovation of historic buildings // IOP Conference Series: Journal of Physics, 2020, vol. 1425, 012201. DOI: https://doi.org/10.1088/1742-6596/1425/1/012201
Redina T. Reconstuction of the historical building foksal 13 and 15 in Warsaw // Acta Polytechnica CTU Proceedings, 2019, vol. 23, pp. 44-48. DOI: https://doi.org/10.14311/APP.2019.23.0044
Sengupta R. Grouting for strengthening the Qutb Minar, Delhi // Mortars, Cements and Grouts used in the Conservation of Historic Buildings, 1981, pp. 147-164.
Wanik K. Use of jet grouting technique to realize substructures of historic buildings: The example of an apartment building in Warsaw // Geotechnical Engineering for the Preservation of Monuments and Historic Sites, 2013, pp. 769-777. DOI: https://doi.org/10.1201/b14895-89
Chernyakov A.V. Application of jet grouting during foundation reinforcement and reconstruction of historic buildings on land occupied by the state museum and reservation «Tsarutsyno» // Soil Mechanics and Foundation Engineering, 2011, vol. 48(5), pp. 769-777. DOI: https://doi.org/10.1007/s11204-011-9145-5
Hofmann H. Unterfangungen mit einpressung von zementsuspension unter hohem Druck // Beton- und Stahlbetonbau, 1989, vol. 84 (8), pp. 199-202. DOI: https://doi.org/10.1002/best.198900290
Morey J.J. Les domaines d'application du jet-grouting // Revue Francaise de Geotechnique, 1992, vol. 61, 17-30. DOI: https://doi.org/10.1051/geotech/1992061017
Drooff E.R., Furth A.J., Scarborough J. A. Jet grouting to support historic buildings // Geotechnical Special Publication, 1995, vol. 50, pp. 42-55.
Modoni G., Croce P., Mongiovi L. Theoretical modelling of jet grouting // Geotechnique, 2006, vol. 56 (5), pp. 335-347. DOI: https://doi.org/10.1680/geot.2006.56.5.335
Tien C. Principles of filtration // Elsevier: Amsterdam, 2012.
Tien C. Introduction to cake filtration // Elsevier: Amsterdam, 2006. DOI: https://doi.org/10.1016/B978-044452156-9/50010-2
Sacramento R.N., Yang Y., You Z., Waldmann A., Martins A.L., Vaz A.S.L., Zitha P.L.J., Bedrikovetsky P. Deep bed and cake filtration of two-size particle suspension in porous media // Journal of Petroleum Science and Engineering, 2015, vol. 126, pp. 201-210. DOI: https://doi.org/10.1016/j.petrol.2014.12.001
Fallah H., Barzegar Fathi H., Mohammadi H. The mathematical model for particle suspension flow through porous medium // Geomaterials, 2012, vol. 2, pp. 57-62. DOI: https://doi.org/10.4236/gm.2012.23009
Fallah H., Fallah A., Rahmani A., Afkhami M., Ahmadi A. Size exclusion mechanism, suspension flow through porous medium // International Journal of Modern Nonlinear Theory and Application, 2012, vol. 1, pp. 113-117. DOI: https://doi.org/10.4236/ijmnta.2012.14017
Santos A., Bedrikovetsky P. Size exclusion during particle suspension transport in porous media: stochastic and averaged equations // Computational and Applied Mathematics, 2004, vol. 23(2-3), pp. 259-284. DOI: https://doi.org/10.1590/S0101-82052004000200009
You Z., Bedrikovetsky P., Kuzmina L. Exact solution for long-term size exclusion suspension-colloidal transport in porous media // Hindawi Publishing Corporation Abstract and Applied Analysis, 2013, vol. 2013, 680693. DOI: https://doi.org/10.1155/2013/680693
Kuzmina L.I., Osipov Yu.V. Exact solution for 1D deep bed filtration with particle capture by advection and dispersion // International Journal of Non-Linear Mechanics, 2021, vol. 137, 103830. DOI: https://doi.org/10.1016/j.ijnonlinmec.2021.103830
Nazaikinskii V.E., Bedrikovetsky P.G., Kuzmina L.I., Osipov Y.V. Exact solution for deep bed filtration with finite blocking time. // SIAM Journal on Applied Mathematics, 2020, vol. 80 (5), pp. 2120-2143. DOI: https://doi.org/10.1137/19M1309195
Kuzmina L.I., Osipov Y.V. and Zheglova Y.G. Analytical model for deep bed filtration with multiple mechanisms of particle capture // International Journal of Non-linear Mechanics, 2018, vol. 105, pp. 242-248. DOI: https://doi.org/10.1016/j.ijnonlinmec.2018.05.015
Kuzmina L.I., Osipov Y.V., Gorbunova T.N. Asymptotics for filtration of polydisperse suspension with small impurities. // Applied Mathematics and Mechanics (English Edition), 2021, vol. 42(1), pp. 109-126. DOI: https://doi.org/10.1007/s10483-021-2690-6
Osipov Y.V. Calculation of filtration of polydisperse suspension in a porous medium // MATEC Web of Conferences, 2017, vol. 117, 00131. DOI: https://doi.org/10.1051/matecconf/201711700131
Kuzmina L.I., Osipov Yu.V. Deep bed filtration asymptotics at the filter inlet // Procedia Engineering, 2016, vol. 153, pp. 366-370. DOI: https://doi.org/10.1016/j.proeng.2016.08.129
Safina G.L. Calculation of retention profiles in porous medium // Lecture Notes in Civil Engineering, 2021, vol. 170, pp. 21-28. DOI: https://doi.org/10.1007/978-3-030-79983-0_3
Safina G.L. Numerical solution of filtration in porous rock // E3S Web of Conferences, 2019, vol. 97, 05016. DOI: https://doi.org/10.1051/e3sconf/20199705016
Osipov Yu., Safina G., Galaguz Yu. Filtration model with multiple particle capture // Journal of Physics Conference Series, 2019, vol. 1425(1), 012110. DOI: https://doi.org/10.1088/1742-6596/1425/1/012110
Yang S., Russell T., Badalyan A., Schacht U., Woolley M., Bedrikovetsky P. Characterisation of fines migration system using laboratory pressure measurements // Journal of Natural Gas Science and Engineering, 2019, vol. 65, pp. 108-124. DOI: https://doi.org/10.1016/j.jngse.2019.02.005
Osipov Yu., Safina G., Galaguz Yu. Calculation of the filtration problem by finite differences methods // MATEC Web of Conferences, 2018, vol. 251, 04021. DOI: https://doi.org/10.1051/matecconf/201825104021
You Z., Osipov Y., Bedrikovetsky P., Kuzmina L. Asymptotic model for deep bed filtration // Chemical Engineering Journal, 2014, vol. 258, pp. 374-385. DOI: https://doi.org/10.1016/j.cej.2014.07.051
Dev Burman G.K., Das P.K. Groundwater exploration in hard rock terrain: An experience from eastern India // The Hydrological Basis for Water Resources Management, 1990, vol. 197, pp. 19-30.