ASYMPTOTICS OF a PARTICLES TRANSPORT PROBLEM

Vestnik MGSU 11/2017 Volume 12
  • Kuzmina Ludmila Ivanovna - National Research University Higher School of Economics Candidate of Physical and Mathematical Sciences, Associate Professor, Department of Applied Mathematics, National Research University Higher School of Economics, 20 Myasnitskaya st., Moscow, 101000, Russian Federation.
  • Osipov Yuri Viktorovich - Moscow State University of Civil Engineering (National Research University) (MGSU) Candidate of Physical and Mathematical Sciences, Associate Professor, Department of Applied Mathematics, Moscow State University of Civil Engineering (National Research University) (MGSU), 26 Yaroslavskoe shosse, Moscow, 129337, Russian Federation.

Pages 1278-1283

Subject: a groundwater filtration affects the strength and stability of underground and hydro-technical constructions. Research objectives: the study of one-dimensional problem of displacement of suspension by the flow of pure water in a porous medium. Materials and methods: when filtering a suspension some particles pass through the porous medium, and some of them are stuck in the pores. It is assumed that size distributions of the solid particles and the pores overlap. In this case, the main mechanism of particle retention is a size-exclusion: the particles pass freely through the large pores and get stuck at the inlet of the tiny pores that are smaller than the particle diameter. The concentrations of suspended and retained particles satisfy two quasi-linear differential equations of the first order. To solve the filtration problem, methods of nonlinear asymptotic analysis are used. Results: in a mathematical model of filtration of suspensions, which takes into account the dependence of the porosity and permeability of the porous medium on concentration of retained particles, the boundary between two phases is moving with variable velocity. The asymptotic solution to the problem is constructed for a small filtration coefficient. The theorem of existence of the asymptotics is proved. Analytical expressions for the principal asymptotic terms are presented for the case of linear coefficients and initial conditions. The asymptotics of the boundary of two phases is given in explicit form. Conclusions: the filtration problem under study can be solved analytically.

DOI: 10.22227/1997-0935.2017.11.1278-1283

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FINELY DISPERSED COMPOSITE BINDER FOR REINFORCING SOILS BY INJECTION METHOD

Vestnik MGSU 11/2017 Volume 12
  • Grishin Andrey Nikolaevich - Moscow State University of Civil Engineering (National Research University) (MGSU) Post-graduate student, Department of Cementing Substances and Concrete Technology, Moscow State University of Civil Engineering (National Research University) (MGSU), 26 Yaroslavskoe shosse, Moscow, 129337, Russian Federation.
  • Panchenko Igor Yakovlevich - Moscow State University of Civil Engineering (National Research University) (MGSU) Doktor-ingenieur Habilitatus, Head of the Department of the Scientific Research Institute of Expertise and Engineering, Moscow State University of Civil Engineering (National Research University) (MGSU), 26 Yaroslavskoe shosse, Moscow, 129337, Russian Federation.
  • Kharchenko Igor Yakovlevich - Moscow State University of Civil Engineering (National Research University) (MGSU) , Moscow State University of Civil Engineering (National Research University) (MGSU), 26 Yaroslavskoe shosse, Moscow, 129337, Russian Federation.
  • Bazhenov Marat Il’darovich - Moscow State University of Civil Engineering (National Research University) (MGSU) Candidate of Technical Sciences, Head of the Composites and Concrete Structures Sector of the Scientific Research Institute of Expertise and Engineering, Moscow State University of Civil Engineering (National Research University) (MGSU), 26 Yaroslavskoe shosse, Moscow, 129337, Russian Federation.

Pages 1289-1298

Subject: we consider the problem of supplying the construction industry, in particular underground construction, with mineral binder for diluted aqueous suspensions that meet the requirements for reinforcement of low-strength sand and clastic soils by injections into the reinforced soil mass. Research objectives: substantiating possibility of using amorphous biosilica in combination with carbide sludge, whose particles size does not exceed 10 mm on average, as a binder for aqueous suspensions being injected. Materials and methods: as raw materials we used: common construction hydrated lime from “Stroimaterialy” JSC, Belgorod, hydrated lime in the form of carbide sludge from the dumps of Protvino plant (carbide sludge, hereafter), active mineral admixture biosilica from the group of companies “DIAMIX” and a plasticizer Sika viscocrete 5 new. Test methods are in accordance with applicable standards. To obtain samples of impregnated soils, a specially developed technique was used in the form of a unidirectional model. Results: properties of the composite binder prepared with different compositions are presented. The optimal component ratios are determined. The following properties of aqueous suspensions are studied: conditional viscosity, sedimentation and penetrating ability. Conditional viscosity is no more than 40 sec on average. Sedimentation does not exceed 1.2 %. Soil-concrete obtained by injection of a dilute aqueous suspension based on this composite binder has a compressive strength in the range from 4.44 to 12.5 MPa. Conclusions: utilization of finely dispersed composite mineral binder, which is based on interaction of amorphous silica with calcium hydroxide, as a binder for high penetration aqueous suspensions has been substantiated. This binder is not inferior to foreign analogues in terms of its strength and technological parameters and can be used for reinforcement of loose and low-strength soils. In case of using carbide sludge, the ecological and environment protection problems are being solved since it is a waste product in production of acetylene.

DOI: 10.22227/1997-0935.2017.11.1289-1298

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Asymptotics of the filtration problem for suspension in porous media

Vestnik MGSU 1/2015
  • Kuzmina Ludmila Ivanovna - Higher School of Economics Department of Applied Mathematics, Moscow Institute of Electronics and Mathematics, Higher School of Economics, 20 Myasnitskaya str., Moscow, 101000, Russian Federation; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .
  • Osipov Yuri Viktorovich - Moscow State University of Civil Engineering (National Research University) (MGSU) Candidate of Physical and Mathematical Sciences, Associate Professor, Department of Computer Science and Applied Mathematics, Moscow State University of Civil Engineering (National Research University) (MGSU), 26 Yaroslavskoe Shosse, Moscow, 129337, Russian Federation; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .

Pages 54-62

The mechanical-geometric model of the suspension filtering in the porous media is considered. Suspended solid particles of the same size move with suspension flow through the porous media - a solid body with pores - channels of constant cross section. It is assumed that the particles pass freely through the pores of large diameter and are stuck at the inlet of pores that are smaller than the particle size. It is considered that one particle can clog only one small pore and vice versa. The particles stuck in the pores remain motionless and form a deposit. The concentrations of suspended and retained particles satisfy a quasilinear hyperbolic system of partial differential equations of the first order, obtained as a result of macro-averaging of micro-stochastic diffusion equations. Initially the porous media contains no particles and both concentrations are equal to zero; the suspension supplied to the porous media inlet has a constant concentration of suspended particles. The flow of particles moves in the porous media with a constant speed, before the wave front the concentrations of suspended and retained particles are zero. Assuming that the filtration coefficient is small we construct an asymptotic solution of the filtration problem over the concentration front. The terms of the asymptotic expansions satisfy linear partial differential equations of the first order and are determined successively in an explicit form. It is shown that in the simplest case the asymptotics found matches the known asymptotic expansion of the solution near the concentration front.

DOI: 10.22227/1997-0935.2015.1.54-62

References
  1. Barenblatt G.I., Entov V.M., Ry- zhik V.M. Theory of Fluid Flows Through Natural Rocks. Dordrecht, Kluwer Academic Publishers, 1990, 395 p.
  2. Bedrikovetsky P. Mathematical Theory of Oil and Gas Recovery with Applications to Ex-USSR Oil and Gas Fields. Dordrecht, Kluwer Academic, 1993, 576 p.
  3. Khilar K.C., Fogler H.S. Migrations of Fines in Porous Media. Dordrecht, Kluwer Academic Publishers, 1998, 173 p.
  4. Tien C., Ramarao B.V. Granular Filtration of Aerosols and Hydrosols. 2nd ed. Amsterdam, Elsevier, 2007, 512 p.
  5. Tufenkji N. Colloid and Microbe Migration in Granular Environments: A Discussion of Modeling Methods. Colloidal Transport in Porous Media. 2007, pp. 119—142. DOI: http://dx.doi.org/10.1007/978-3-540-71339-5_5.
  6. Baveye P., Vandevivere P., Hoyle B.L., DeLeo P.C., De Lozada D.S. Environmental Impact and Mechanisms of the Biological Clogging of Saturated Soils and Aquifer Materials. Critical Reviews in Environmental Science and Technology. 1998, vol. 28, pp. 123—191. DOI: http://dx.doi.org/10.1080/10643389891254197.
  7. Vidali M. Bioremediation. An Overview. Pure and Applied Chemistry. 2001, vol. 73, no. 7, pp. 1163—1172. DOI: http://dx.doi.org/10.1351/pac200173071163.
  8. Gitis V., Dlugy C., Ziskind G., Sladkevich S., Lev O. Fluorescent Clays — Similar Transfer with Sensitive Detection. Chemical Engineering Journal. 2011, vol. 174, no. 1, pp. 482—488. DOI: http://dx.doi.org/10.1016/j.cej.2011.08.063.
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  10. You Z., Badalyan A., Bedrikovetsky P. Size-Exclusion Colloidal Transport in Porous Media-Stochastic Modeling and Experimental Study. SPE Journal. 2013, vol. 18, no. 4, pp. 620—633. DOI: http://dx.doi.org/10.2118/162941-PA.
  11. 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. DOI: http://dx.doi.org/10.1007/s11242-008-9228-6.
  12. Chalk P., Gooding N., Hutten S., You Z., Bedrikovetsky P. Pore Size Distribution from Challenge Coreflood Testing by Colloidal Flow. Chemical Engineering Research and Design. 2012, vol. 90, no. 1, pp. 63—77. DOI: http://dx.doi.org/10.1016/j.cherd.2011.08.018.
  13. Mays D.C., Hunt J.R. Hydrodynamic and Chemical Factors in Clogging by Montmorillonite in Porous Media. Environmental Science and Technology. 2007, vol. 41, no. 16, pp. 5666—5671. DOI: http://dx.doi.org/10.1021/es062009s.
  14. Civan F. Reservoir Formation Damage : Fundamentals, Modeling, Assessment, and Mitigation. 2nd ed. Amsterdam, Gulf Professional Pub., 2007.
  15. Gitis V., Rubinstein I., Livshits M., Ziskind G. Deep-Bed Filtration Model with Multistage Deposition Kinetics. Chemical Engineering Journal. 2010, vol. 163, no. 1—2, pp. 78—85. DOI: http://dx.doi.org/10.1016/j.cej.2010.07.044.
  16. Noubactep C., Care S. Dimensioning Metallic Iron Beds for Efficient Contaminant Removal. Chemical Engineering Journal. 2010, vol. 163, no. 3, pp. 454—460.
  17. Yuan H., Shapiro A.A. A Mathematical Model for Non-Monotonic Deposition Profiles in Deep Bed Filtration Systems. Chemical Engineering Journal. 2011, vol. 166, no. 1, pp. 105—115. DOI: http://dx.doi.org/10.1016/j.cej.2010.10.036.
  18. Santos A., Bedrikovetsky P. A Stochastic Model for Particulate Suspension Flow in Porous Media. Transport in Porous Media. 2006, vol. 62, pp. 23—53. DOI: http://dx.doi.org/10.1007/s11242-005-5175-7.
  19. You Z., Bedrikovetsky P., Kuzmina L. Exact Solution for Long-Term Size Exclusion Suspension-Colloidal Transport in Porous Media. Abstract and Applied Analysis. 2013, vol. 2013, 9 p. DOI: http://dx.doi.org/10.1155/2013/680693.
  20. Herzig J.P., Leclerc D.M., Goff P. Le. Flow of Suspensions Through Porous Media — Application to Deep Filtration. Industrial and Engineering Chemistry. 1970, vol. 62 (5), pp. 8—35. DOI: http://dx.doi.org/10.1021/ie50725a003.
  21. Alvarez A.C., Bedrikovetsky P.G., Hime G., Marchesin D., Rodrigues J.R. A Fast Inverse Solver for the Filtration Function for Flow of Water with Particles in Porous Media. J. of Inverse Problems. 2006, vol. 22, pp. 69—88. DOI: http://dx.doi.org/10.1088/ 0266-5611/22/1/005.
  22. 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. DOI: http://dx.doi.org/10.1134/S0040579507050168.
  23. You Z., Osipov Y., Bedrikovetsky P., Kuzmina L. Asymptotic Model for Deep Bed Filtration. Chemical Engineering Journal. 2014, vol. 258, pp. 374—385. DOI: http://dx.doi.org/10.1016/j.cej.2014.07.051.
  24. Kuzmina L.I., Osipov Yu.V. Particle Transportation at the Filter Inlet. International Journal for Computational Civil and Structural Engineering. 2014, vol. 10, iss. 3, pp. 17—22.
  25. Kuzmina L.I., Osipov Yu.V. Matematicheskaya model' dvizheniya chastits v fil'tre [Mathematical Model of Particle Motion in the Filter]. Voprosy prikladnoy matematiki i vychislitel'noy mekhaniki : sbornik nauchnykh trudov [Problems of Applied Mathematics and Computational Mechanics]. Moscow, MGSU Publ., 2014, vol. 17, pp. 295—304. (in Russian)

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HIGH-QUALITY ORNAMENTAL FINE CONCRETES MODIFIED BY NANOPARTICLES OF TITANIUM DIOXIDE

Vestnik MGSU 6/2012
  • Bazhenov Yuriy Mikhaylovich - Moscow State University of Civil Engineering (National Research University) (MGSU) Doctor of Technical Sciences, Professor, Head of the Department of Technologies of Cohesive Materials and Concretes, Moscow State University of Civil Engineering (National Research University) (MGSU), 26 Yaroslavskoe shosse, 129337, Russian Federation.
  • Korolev Evgeniy Valer'evich - Moscow State University of Civil Engineering (MSUCE) Doctor of Technical Sciences, Professor, Moscow State University of Civil Engineering (MSUCE), 26 Yaroslavskoe shosse, Moscow, 129337, Russian Federation.
  • Lukuttsova Natal'ya Petrovna - Bryansk State Academy of Engineering and Technology (BSAET) Doctor of Technical Sciences, Professor +7 (4832) 74-60-08, +7 (4832) 74-05-13, Bryansk State Academy of Engineering and Technology (BSAET), 3 pr. St.-Dimitrova, Bryansk, 241037, Russian Federation; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .
  • Zavalishin Sergey Iosifovich - Moscow State University of Civil Engineering (MSUCE) Candidate of Technical Sciences, Professor, Moscow State University of Civil Engineering (MSUCE), 26 Yaroslavskoe shosse, Moscow, 129337, Russian Federation.
  • Chudakova Ol'ga Andreevna - Bryansk State Academy of Engineering and Technology (BSAET) postgraduate student, +7 (4832) 74-60-08, +7 (4832) 59-56-39, Bryansk State Academy of Engineering and Technology (BSAET), 3 pr. St.-Dimitrova, Bryansk, 241037, Russian Federation; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .

Pages 73 - 78

Ultrasonic method of generation of a stable suspension of nano-particles of titanium dioxide and the strengthening properties of the ornamental fine concrete that contains cement binders with a nano-dispersed additive constitute the subject of the research covered by the authors. Nanoparticles react with the basic chemical elements that compose the concrete and act as crystallization centres. Therefore, the concrete porosity is reduced, while physical and technology-related properties of the ornamental fine concrete are improved.
The authors have proven that the application of the nano-dispersed additive that contains titanium dioxide influences the processes of the structure formation in respect of fine ornamental concretes and improves the strength, as well as the water and cold resistance of fine concretes. The improvement is attributed to the dense concrete structure and strong adhesion between cement grains and between the cement and the aggregate. This conclusion is based on the data obtained through the employment of an electronic microscope used to identify the porosity of fine concretes.

DOI: 10.22227/1997-0935.2012.6.73 - 78

References
  1. Drinberg A.S., Kalinskaya T.V., Itsko E.F. Neorganicheskie pigmenty, proizvodstvo i perspektivy [Inorganic Pigments, Production and Prospects]. Lakokrasochnye materialy i ikh primenenie [Paint-and-Lacquer Materials and Application]. 2007, no. 12, pp. 20—28.
  2. Latyshev Yu.V., Lenev L.M. Tseny na TiO2 — stabil’ny!? Chego mogut zhdat’ potrebiteli etogo syr’ya? [Prices for TiO2: are They Stable!? What Can Consumers Expect of This Material?] Lakokrasochnye materialy i ikh primenenie [Paint-and-Lacquer Materials and Application]. 2007, no. 12, pp. 12—19.
  3. Lukuttsova N.P., Chudakova O.A., Khotchenkov P.V. Dekorativno-otdelochnye izdeliya na osnove nanomodifitsiruyushchey dobavki [Ornamental Finishing Products That Contain Nano-Modifiers]. Vestnik BGTU im. V.G. Shukhova [Proceedings of Voronezh State University of Technology]. 2011, pp. 67—72.

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Asymptotic solution of the filtration equation

Vestnik MGSU 2/2016
  • Kuzmina Ludmila Ivanovna - Higher School of Economics Department of Applied Mathematics, Moscow Institute of Electronics and Mathematics, Higher School of Economics, 20 Myasnitskaya str., Moscow, 101000, Russian Federation; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .
  • Osipov Yuri Viktorovich - Moscow State University of Civil Engineering (National Research University) (MGSU) Candidate of Physical and Mathematical Sciences, Associate Professor, Department of Computer Science and Applied Mathematics, Moscow State University of Civil Engineering (National Research University) (MGSU), 26 Yaroslavskoe Shosse, Moscow, 129337, Russian Federation; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .

Pages 49-61

The problem of filtering a suspension of tiny solid particles in a porous medium is considered. The suspension with constant concentration of suspended particles at the filter inlet moves through the empty filter at a constant speed. There are no particles ahead of the front; behind the front of the fluid flow solid particles interact with the porous medium. The geometric model of filtration without effects caused by viscosity and electrostatic forces is considered. Solid particles in the suspension pass freely through large pores together with the fluid flow and are stuck in the pores that are smaller than the size of the particles. It is considered that one particle can clog only one small pore and vice versa. The precipitated particles form a fixed deposit increasing over time. The filtration problem is formed by the system of two quasi-linear differential equations in partial derivatives with respect to the concentrations of suspended and retained particles. The boundary conditions are set at the filter inlet and at the initial moment. At the concentration front the solution of the problem is discontinuous. By the method of potential the system of equations of the filtration problem is reduced to one equation with respect to the concentration of deposit with a boundary condition in integral form. An asymptotic solution of the filtration equation is constructed near the concentration front. The terms of the asymptotic expansions satisfy linear ordinary differential equations of the first order and are determined successively in an explicit form. For verification of the asymptotics the comparison with the known exact solutions is performed.

DOI: 10.22227/1997-0935.2016.2.49-61

References
  1. Barenblatt G.I., Entov V.M., Ryzhik V.M. Theory of Fluid Flows through Natural Rocks. Dordrecht, Kluwer Academic Publishers, 1990, 396 p.
  2. Bedrikovetsky P. Mathematical Theory of Oil and Gas Recovery with Applications to Ex-USSR Oil and Gas Fields. Dordrecht, Kluwer Academic, 1993, 576 p. DOI: http://www.doi.org/10.1007/978-94-017-2205-6.
  3. Khilar K.C., Fogler H.S. Migrations of Fines in Porous Media. Dordrecht, Kluwer Academic Publishers, 1998, 173 p. DOI: http://www.doi.org/10.1007/978-94-015-9074-7.
  4. Tien C., Ramarao B.V. Granular Filtration of Aerosols and Hydrosols. 2nd ed. Amsterdam, Elsevier, 2007, 512 p.
  5. Baveye P., Vandevivere P., Hoyle B.L., DeLeo P.C., Sanchez De Lozada D. Environmental Impact and Mechanisms of the Biological Clogging of Saturated Soils and Aquifer Materials. Critical Reviews in Environmental Science and Technology. 1998, vol. 28, pp. 123—191. DOI: http://www.doi.org/10.1080/10643389891254197.
  6. Jeong S., Vigneswaran S. Assessment of Biological Activity in Contact Flocculation Filtration Used as a Pretreatment in Seawater Desalination. Chemical Engineering Journal. 2013, vol. 228, pp. 976—983. DOI: http://www.doi.org/10.1016/j.cej.2013.05.085.
  7. Khare P., Talreja N., Deva D., Sharma A., Verma N. Carbon Nanofibers Containing Metal-Doped Porous Carbon Beads for Environmental Remediation Applications. Chemical Engineering Journal. 2013, vol. 229, pp. 72—81. DOI: http://www.doi.org/10.1016/j.cej.2013.04.113.
  8. Inyang M., Gao B., Wu L., Yao Y., Zhang M., Liu L. Filtration of Engineered Nanoparticles in Carbon-Based Fixed Bed Columns. Chemical Engineering Journal. 2013, vol. 220, pp. 221—227. DOI: http://www.doi.org/10.1016/j.cej.2013.01.054.
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  11. Chalk P., Gooding N., Hutten S., You Z., Bedrikovetsky P. Pore Size Distribution from Challenge Coreflood Testing by Colloidal Flow. Chemical Engineering Research and Design. 2012, vol. 90. Pp. 63—77.
  12. Santos A., Bedrikovetsky P. A Stochastic Model for Particulate Suspension Flow in Porous Media. Transport in Porous Media. 2006, vol. 62, pp. 23—53.
  13. Vollebregt H.M., Van der Sman R.G.M., Boom R.M. Model for Particle Migration in Bidisperse Suspensions by Use of Effective Temperature. Faraday Discussions. 2012, vol. 158, pp. 89—103. DOI: http://dx.doi.org/10.1039/C2FD20035J.
  14. Sund N., Bolster D., Mattis S., Dawson C. Pre-asymptotic Transport Upscaling in Inertial and Unsteady Flows Through Porous Media. Transport in Porous Media. 2015, vol. 109, issue 2, pp. 411—432.
  15. Mathieu-Potvin F., Gosselin L. Impact of Non-uniform Properties on Governing Equations for Fluid Flows in Porous Media. Transport in Porous Media. 2014, vol. 105, issue 2, pp. 277—314. DOI: http://dx.doi.org/10.1007/s11242-014-0370-z.
  16. Hönig O., Doster F., Hilfer R. Traveling Wave Solutions in a Generalized Theory for Macroscopic Capillarity. Transport in Porous Media. 2013, vol. 99, no. 3, pp. 467—491. DOI: http://dx.doi.org/10.1007/s11242-013-0196-0.
  17. Yuan H., You Z., Shapiro A., Bedrikovetsky P. Improved Population Balance Model for Straining-Dominant Deep Bed Filtration Using Network Calculations. Chemical Engineering Journal. 2013, vol. 226, pp. 227—237. DOI: http://dx.doi.org/10.1016/j.cej.2013.04.031.
  18. Gitis V., Rubinstein I., Livshits M., Ziskind G. Deep-bed Filtration Model with Multistage Deposition Kinetics. Chemical Engineering Journal. 2010, vol. 163, no. 1—2, pp. 78—85. DOI: http://dx.doi.org/10.1016/j.cej.2010.07.044.
  19. You Z., Osipov Y., Bedrikovetsky P., Kuzmina L. Asymptotic Model for Deep Bed Filtration. Chemical Engineering Journal. 2014, vol. 258, pp. 374—385. DOI: http://dx.doi.org/10.1016/j.cej.2014.07.051.
  20. Yuan H., Shapiro A., You Z., Badalyan A. Estimating Filtration Coefficients for Straining from Percolation and Random Walk Theories. Chemical Engineering Journal. 2012, vol. 210, pp. 63—73. DOI: http://dx.doi.org/10.1016/j.cej.2012.08.029.
  21. Kuzmina L.I., Osipov Yu.V. Inverse Problem of Filtering the Suspension in Porous Media. International Journal for Computational Civil and Structural Engineering. 2015, vol. 11, no. 1, pp. 34—41.
  22. 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. DOI: http://dx.doi.org/10.1007/s11242-008-9228-6.
  23. Kuzmina L.I., Osipov Yu.V. Particle Transportation at the Filter Inlet. International Journal for Computational Civil and Structural Engineering. 2014, vol. 10, no. 3, pp. 17—22.
  24. Herzig J.P., Leclerc D.M., Legoff P. Flow of Suspensions Through Porous Media — Application to Deep Filtration. Industrial and Engineering Chemistry. 1970, vol. 62 (5), pp. 8—35. DOI: http://dx.doi.org/10.1021/ie50725a003.
  25. 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. DOI: http://dx.doi.org/10.1134/S0040579507050168.
  26. Bedrikovetsky P.G., Marchesin D., Checaira F., Serra A.L., Resende E. Characterization of Deep Bed Filtration System from Laboratory Pressure Drop Measurements. Journal of Petroleum Science and Engineering. 2001, vol. 32, no. 3, pp. 167—177. DOI: http://dx.doi.org/10.1016/S0920-4105(01)00159-0.
  27. Yuan H., Shapiro A., You Z., Badalyan A. Estimating Filtration Coefficients for Straining from Percolation and Random Walk Theories. Chemical Engineering Journal. 2012, vol. 210, pp. 63—73. DOI: http://dx.doi.org/10.1016/j.cej.2012.08.029.
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