A NONLINEAR MATHEMATICAL MODEL OF FILTRATION PROCESSES IN POROUS MEDIA WITH TIME-DEPENDENT PERMEABILITY DUE TO COLMATATION
Keywords:
filtration, porous media, Darcy law, nonlinear diffusion, colmatation, permeability, degenerate parabolic equation.Abstract
Filtration of fluids through porous media underlies numerous applications in hydrogeology, petroleum engineering, and water treatment. Classical models based on the linear Darcy law assume constant medium permeability, which is inadequate for long-term processes in which suspended particles progressively deposit within the pore space and reduce its transmissivity — a phenomenon known as colmatation (clogging). This work proposes a new nonlinear mathematical model of non-stationary filtration that couples the pressure field with the dynamics of pore clogging through a permeability-degradation function depending on the accumulated deposit concentration. The resulting initial-boundary value problem is a quasilinear degenerate parabolic equation. A finite-difference scheme with an implicit time-stepping procedure is constructed to solve the problem numerically. Qualitative analysis and preliminary computations indicate that the proposed model captures the experimentally observed decline of filtration rate over time substantially better than the classical linear model, and it reduces to the latter as a limiting case. The model is applicable to the design of groundwater intake systems, sand and slow filters in water treatment, and injection wells in oil-field water-flooding operations.Downloads
References
[1] Bear, J. Dynamics of Fluids in Porous Media. American Elsevier, New York, 1972.
[2] Polubarinova-Kochina, P.Ya. Theory of Ground Water Movement. Princeton University Press, 1962.
[3] Barenblatt, G.I., Entov, V.M., Ryzhik, V.M. Theory of Fluid Flows Through Natural Rocks. Kluwer Academic Publishers, 1990.
[4] Herzig, J.P., Leclerc, D.M., Le Goff, P. Flow of suspensions through porous media — application to deep filtration. Industrial & Engineering Chemistry, 62(5), 1970, pp. 8–35.
[5] Samarskii, A.A. The Theory of Difference Schemes. Marcel Dekker, New York, 2001.
[6] Civan, F. Reservoir Formation Damage:Fundamentals, Modeling, Assessment, and Mitigation. 3rd Edition, Gulf Professional Publishing, Elsevier, 2016.
[7] Khilar, K.C., Fogler, H.S. Migration of Fines in Porous Media. Springer, Dordrecht, 2013.
[8] Raissi, M., Perdikaris, P., Karniadakis, G.E. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics, 378, 686–707, 2019.
[9] Karniadakis, G.E., Kevrekidis, I.G., Lu, L., Perdikaris, P., Wang, S., Yang, L. Physics-informed machine learning. Nature Reviews Physics, 3, 422–440, 2021.
[10] Zhang, K., Chen, Y., Li, S.Nonlinear filtration models for heterogeneous porous media with dynamic permeability. Computers & Geosciences, 2022.
[11] Hassan, A., Jo, H., Park, J. Porous media transport va modeling. Recent advances in pore-scale and continuum-scale modeling of transport processes in porous media. Advances in Water Resources, 2023.
[12] Wang, S., Perdikaris, P. Physics-informed neural networks for scientific computing in porous media flow and transport problems. Computer Methods in Applied Mechanics and Engineering, 2024.
