Abstract
A model for deep bed filtration of a polydisperse suspension with small impurities in a porous medium is considered. Different suspended particles move with the same velocity as the carrier water and get blocked in the pore throats due to the size-exclusion mechanism of particle retention. A solution of the model in the form of a traveling wave is obtained. The global exact solution for a multiparticle filtration with one high concentration and several low concentrations of suspended particles is obtained in an explicit form. The analytic solutions for a bidisperse suspension with large and small particles are constructed. The profiles of the retained small particles change monotony with time. The global asymptotics for the filtration of a polydisperse suspension with small kinetic rates is constructed in the whole filtration zone.
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BEDRIKOVETSKY, P. Mathematical Theory of Oil and Gas Recovery with Applications to Ex-USSR Oil and Gas Fields, Springer Science and Business Media, Des Moines (2013)
CIVAN, F. Reservoir Formation Damage (Fundamentals, Modeling, Assessment, and Mitigation), Gulf Professional Publishing, Houston (2007)
LU, B., LIU, X., DONG, P., TICK, G. R., ZHENG, C., ZHANG, Y., MAHMOOD-UL-HASSAN, M., BAI, H., and LAMY, E. Quantifying fate and transport of nitrate in saturated soil systems using fractional derivative model. Applied Mathematical Modelling, 81, 279–295 (2020)
ELIMELECH, M., GREGORY, J., JIA, X., and WILLIAMS, R. Particle Deposition and Aggregation: Measurement, Modelling, and Simulation, Butterworth-Heinemann, New York (2013)
DOMGA, R., TCHATCHUENG, J. B., TOGUE-KAMGA, F., and NOUBACTEP, C. Discussing porosity loss of FeO packed water filters at ground level. Chemical Engineering Journal, 263, 127–134 (2014)
ZEMAN, L. J. and ZYDNEY, A. L. Microfiltration and Ultrafiltration: Principles and Applications, Marcel Dekker, New York (1996)
BRADFORD, S. A., YATES, S. R., BETTAHAR, M., and SIMUNEK, J. Physical factors affecting the transport and fate of colloids in saturated porous media. Water Resources Research, 38, 1327–1334 (2002)
GITIS, V., RUBINSTEIN, I., LIVSHTS, M., and ZISKIND, G. Deep-bed filtration model with multistage deposition kinetics. Chemical Engineering Journal, 163, 78–85 (2010)
BEDRIKOVETSKY, P. Upscaling of stochastic micro model for suspension transport in porous media. Transport in Porous Media, 75, 335–369 (2008)
TIEN, C. Principles of Filtration, Elsevier, Oxford (2012)
BEDRIKOVETSKY, P., SIQUEIRA, F. D., FURTADO, C., and SOUZA, A. L. S. Modified particle detachment model for colloidal transport in porous media. Transport in Porous Media, 86, 353–383 (2011)
HERZIG, J. P., LECLERC, D. M., and GOFF, P. L. Flow of suspensions through porous media—application to deep filtration. Industrial and Engineering Chemistry Research, 62, 8–35 (1970)
POLYAKOV, Y. S. and ZYDNEY, A. L. Ultrafiltration membrane performance: effects of pore blockage/constriction. Journal of Membrane Science, 434, 106–120 (2013)
TUFENKJI, N. Colloid and microbe migration in granular environments: a discussion of modelling methods. Colloidal Transport in Porous Media, Springer, Berlin (2007)
TUFENKJI, N. and ELIMELECH, M. Correlation equation for predicting single-collector efficiency in physicochemical filtration in saturated porous media. Environmental Science and Technology, 38, 529–536 (2004)
POLYANIN, A. D. and MANZHIROV, A. V. Handbook of Mathematics for Engineers and Scientists, Chapman and Hall/CRC Press, Boca Raton (2006)
SHARMA, M. and YORTSOS, Y. Transport of particulate suspensions in porous media: model formulation. AIChE Journal, 33, 1636–1643 (1987)
SHAPIRO, A. Elliptic equation for random walks: application to transport in microporous media. Physica A: Statistical Mechanics and Its Applications, 375, 81–96 (2007)
YUAN, H. and SHAPIRO, A. Colloid transport and retention: recent advances in colloids filtration theory. Colloids: Classification, Properties and Applications, Nova Science Publishers, New York (2013)
MACKIE, R. I. and ZHAO, Q. A framework for modeling removal in the filtration of polydisperse suspensions. Water Research, 33, 794–806 (1999)
HARRIS, T. C., HOGG, A. J., and HUPPERT, H. E. Polydisperse particle-driven gravity currents. Journal of Fluid Mechanics, 472, 333–371 (2002)
BENNACER, L., AHFIR, N. D., BOUANANI, A., ALEM, A., and WANG, H. Suspended particles transport and deposition in saturated granular porous medium: particle size effects. Transport in Porous Media, 100, 377–392 (2013)
TRZASKUS, K., ELSHOF, M., KEMPERMAN, A., and NIJMEIJER, K. Understanding the role of nanoparticle size and polydispersity in fouling development during dead-end microfiltration. Journal of Membrane Science, 516, 152–161 (2016)
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, 105, 242–248 (2018)
ZHANG, H., MALGARESI, G. V. C. P., and BEDRIKOVETSKY, P. Exact solutions for suspension-colloidal transport with multiple capture mechanisms. International Journal of Non-Linear Mechanics, 105, 27–42 (2018)
POLYANIN, A. and ZAITSEV, V. Handbook of Nonlinear Partial Differential Equations, Chapman and Hall/CRC Press, Boca Raton (2012)
VYAZMINA, E. A., BEDRIKOVETSKII, P. G., and 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, 41, 556–564 (2007)
ALVAREZ, A. C., HIME, G., MARCHESIN, D., and 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, 70, 43–62 (2007)
ALVAREZ, A. C., HIME, G., SILVA, J. D., and MARCHESIN, D. Analytic regularization of an inverse filtration problem in porous media. Inverse Problems, 29, 025006 (2013)
CHALK, P., GOODING, N., HUTTEN, S., YOU, Z., and BEDRIKOVETSKY, P. Pore size distribution from challenge coreflood testing by colloidal flow. Chemical Engineering Research and Design, 90, 63–77 (2012)
KUZMINA, L. I., NAZAIKINSKII, V. E., and OSIPOV, Y. V. On a deep bed filtration problem with finite blocking time. Russian Journal of Mathematical Physics, 26, 130–134 (2019)
KUZMINA, L. I. and OSIPOV, Y. V. Deep bed filtration asymptotics at the filter inlet. Procedia Engineering, 153, 366–370 (2016)
ANDREUCCI, D. and TEDEEV, A. F. Asymptotic behavior for the filtration equation in domains with noncompact boundary. Communications in Partial Differential Equations, 42, 347–365 (2017)
KUZMINA, L. I., OSIPOV, Y. V., and GALAGUZ, Y. P. A model of two-velocity particles transport in a porous medium. International Journal of Non-Linear Mechanics, 93, 1–6 (2017)
MALGARESI, G., COLLINS, B., ALVARO, P., and BEDRIKOVETSKY, P. Explaining nonmonotonic retention profiles during flow of size-distributed colloids. Chemical Engineering Journal, 375, 121984 (2019)
POLYANIN, A. and DILMAN, V. V. Methods of Modeling Equations and Analogies in Chemical Engineering, CRC Press/Begell House, Boca Raton (1994)
TANG, Y. D., JIN, T., and FLESCH, R. C. C. Effect of mass transfer and diffusion of nanofluid on the thermal ablation of malignant cells during magnetic hyperthermia. Applied Mathematical Modelling, 83, 122–135 (2020)
GERBER, G., WEITZ, D. A., and COUSSOT, P. Propagation and adsorption of nanoparticles in porous medium as traveling waves. Physical Review Research, 2, 033074 (2020)
CHALK, P., GOODING, N., HUTTEN, S., YOU, Z., and BEDRIKOVETSKY, P. Pore size distribution from challenge coreflood testing by colloidal flow. Chemical Engineering Research and Design, 90, 63–77 (2012)
GALAGUZ, Y. P., KUZMINA, L. I., and OSIPOV, Y. V. Problem of deep bed filtration in a porous medium with the initial deposit. Fluid Dynamics, 54, 85–97 (2019)
YOU, Z., BEDRIKOVETSKY, P., BADALYAN, A., and HAND, M. Particle mobilization in porous media: temperature effects on competing electrostatic and drag forces. Geophysical Research Letters, 42, 2852–2860 (2015)
BEDRIKOVETSKY, P., ZEINIJAHROMI, A., SIQUEIRA, F. D., FURTADO, C., and DE SOUZA, A. L. S. Particle detachment under velocity alternation during suspension transport in porous media. Transport in Porous Media, 91, 173–197 (2012)
YOU, Z., BADALYAN, A., YANG, Y., and BEDRIKOVETSKY, P. Formation damage challenges in geothermal reservoirs: laboratory and mathematical modeling. Formation Damage During Improved Oil Recovery: Fundamentals and Applications, Gulf Professional Publishing, Oxford (2018)
DEL VIGO, A., ZUBELZU, S., and JUANA, L. Numerical routine for soil water dynamics from trickle irrigation. Applied Mathematical Modelling, 83, 371–385 (2020)
BORAZJANI, S., ROBERTS, A. J., and BEDRIKOVETSKY, P. Splitting in systems of PDEs for two-phase multicomponent flow in porous media. Applied Mathematical Letters, 53, 25–32 (2015)
BORAZJANI, S. and BEDRIKOVETSKY, P. Exact solutions for two-phase colloidal-suspension transport in porous media. Applied Mathematical Modelling, 44, 296–320 (2017)
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Kuzmina, L.I., Osipov, Y.V. & Gorbunova, T.N. Asymptotics for filtration of polydisperse suspension with small impurities. Appl. Math. Mech.-Engl. Ed. 42, 109–126 (2021). https://doi.org/10.1007/s10483-021-2690-6
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DOI: https://doi.org/10.1007/s10483-021-2690-6