Abstract
A new model for unsaturated flow in porous media, including capillary hysteresis and dynamic capillary effects, is analyzed. Existence and uniqueness of solutions are established and qualitative and quantitative properties of (particular) solutions are analyzed. Some results of numerical computations are given. The model under consideration incorporates simple ‘play’-type hysteresis and a dynamic term (time-derivative with respect to water content) in the capillary relation. Given an initial water content distribution, the model determines which parts of the flow domain are in drainage and which parts are in imbibition. The governing equations can be recast into an elliptic problem for fluid pressure and an evolution equation for water content. Standard methods are used to obtain numerical results. A comparison is given between J.R. Philip's semi-explicit similarity solution for horizontal redistribution in an infinite one-dimensional domain and solutions of the new model.
Similar content being viewed by others
References
H.W. Alt, S. Luckhaus and A. Visintin, On non-stationary flow through porous media, Ann. Matem. Pura Appl. 136 (1984) 303–316.
S.N. Antontsev, A.V. Kazhikhov and V.N. Monakhov, Boundary Value Problems in Mechanics of Non-homogeneous Fluids (North-Holland, Amsterdam, 1990).
J. Bear, D. Zaslavsky and S. Irmay, Physical Principles of Water Percolation and Seepage (Z.uitg. Paris, 1968).
A. Beliaev, Unsaturated porous flows with play-type capillary hysteresis, Russian J. Math. Phys. 8(1) (2001) 1–13.
A. Beliaev and S.M. Hassanizadeh, A theoretical model of hysteresis and dynamic effects in capillary relation for two-phase flow in porous media, Transport Porous Media 43(3) (2001) 487–510.
A. Bourgeat and M. Panfilov, Effective two-phase flow through highly heterogeneous porous media: Capillary non-equilibrium effects, Comput. Geosci. 2(3) (1998) 191–215.
E. DiBenedetto and M. Pierre, On the maximum principle for pseudoparabolic equations, Indiana Univ. Math. J. 30(6) (1981) 821–854.
S. Flügge, C. Truesdell and W. Noll, Handbuch der Physik, Bd.3/3, Die nicht-linearen Feldtheorien der Mechanik (Springer, Berlin, 1965).
S.M. Hassanizadeh, Dynamic effects in the capillary pressure-saturation relationship, in: Proc. of the 4th Internat. Conf. on Civil Engineering, 4–6 May 1997, Sharif University of Technology, Teheran, Iran, Vol. 4, pp. 141–149.
S.M. Hassanizadeh and W.G. Gray, Thermodynamic basis of capillary pressure in porous media, Water Resources Res. 29(10) (1993) 3389–3405.
A.S. Kalashnikov, Some problems of the qualitative theory of nonlinear degenerate second order parabolic equations, Russian Math. Surveys 42(2) (1987) 169–222.
M.A. Krasnosel'skii and A.V. Pokrovskii, Systems with Hysteresis (Springer, Berlin, 1989).
Y. Mualem, Modified approach to capillary hysteresis based on a similarity hypothesis, Water Resourses Res. 9 (1973) 1324–1331.
Y. Mualem, A conceptual model of hysteresis, Water Resourses Res. 10 (1974) 514–520.
M. Panfilov, Upscaling two-phase flow in double porosity media: nonuniform homogenization, in: Recent Advances in Problems of Flow and Transport in Porous Media, eds. J.M. Crolet and M.E. Hatri, (Kluwer Academic, Dordrecht, 1998) pp. 195–215.
J.R. Philip, Horizontal redistribution with capillary hysteresis, Water Resources Res. 27 (1991) 1459–1469.
P.A.C. Raats and C.J. Van Duijn, A note on horizontal redistribution with capillary hyteresis, Water Resources Res. 31(1) (1995) 231–232.
D.E. Smiles, G. Vachaud and M.A. Vauclin, Test of the uniqueness of the soil moisture characteristic during transient, non-hysteretic flow of water in a rigid soil, Soil Sci. Soc. Amer. Proc. 35 (1971) 535–539.
F. Stauffer, Time dependence of the relations between capillary pressure, water content and conductivity during drainage of porous media, in: IAHR Sympos. on Scale Effects in Porous Media, Thessaloniki, Greece, August 29–1 September 1978.
G.C. Topp, A. Klute and D.B. Peters, Comparison of water content-pressure head data obtained by equilibrium, steady-state, and unsteady-state methods, Soil Sci. Soc. Amer. Proc. 31 (1967) 312–314.
G. Vachaud, M. Vauclin and M.A. Wakil, Study of the uniqueness of the soil moisture characteristic during desorption by vertical drainage, Soil Sci. Soc. Amer. Proc. 36 (1972) 531–532.
C.J. Van Duijn and L.A. Peletier, Nonstationary filtration in partially saturated porous media, Arch. Rational Mech. Anal. 78(2) (1982) 173–198.
A. Visintin, Differential Models of Hysteresis (Springer, Berlin, 1994).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Beliaev, A., Schotting, R. Analysis of A New Model for Unsaturated Flow in Porous Media Including Hysteresis and Dynamic Effects. Computational Geosciences 5, 345–368 (2001). https://doi.org/10.1023/A:1014547019782
Issue Date:
DOI: https://doi.org/10.1023/A:1014547019782