RESEARCH OF BUILDING MATERIALS

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

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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
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  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.
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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.
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  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.
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