Abstract
Saturation overshoot and hysteresis for two phase flow in porous media are briefly reviewed. Old and new challenges are discussed. It is widely accepted that the traditional Richards model for twophase flow in porous media does not support non-monotone travelling wave solutions for the saturation profile. As a concequence various extensions and generalizations have been recently discussed. The review highlights different limits within the traditional theory. It emphasizes the relevance of hysteresis in the Buckley–Leverett limit with jump-type hysteresis in the relative permeabilities. Reviewing the situation it emerges that the traditional theory may have been abandoned prematurely because of its inability to predict saturation overshoot in the Richards limit.
Similar content being viewed by others
References
R. Hilfer, Adv. Chem. Phys. XCII, 299 (1996)
M. Muskat, The Flow of Homogeneous Fluids Through Porous Media (McGraw Hill, New York, 1937)
A.E. Scheidegger, The Physics of Flow Through Porous Media (University of Toronto Press, Canada, 1957)
R.E. Collins, Flow of Fluids Through Porous Materials (Reinhold Publishing Co., New York, 1961)
J. Bear, Dynamics of Fluids in Porous Media (Elsevier Publ. Co., New York, 1972)
G.D. Marsily, Quantitative Hydrogeology – Groundwater Hydrology for Engineers (Academic Press, San Diego, 1986)
G.P. Willhite, Waterflooding, series SPE Textbook Series, Vol. 3 (Society of Petroleum Engineers, USA, 1986)
L.W. Lake, Enhanced Oil Recovery (Prentice Hall, Englewood Cliffs, 1989)
F.A.L. Dullien, Porous Media – Fluid Transport and Pore Structure (Academic Press, San Diego, 1992)
M. Sahimi, Flow and Transport in Porous Media and Fractured Rock (VCH Verlagsgesellschaft mbH, Weinheim, 1995)
M. Eliassi, R.J. Water Resour. Res. 37, 2019 (2001)
M.I.J. van Dijke, K. Sorbie, Phys. Rev. E 66, 046302 (2002)
I. Fatt, AIME Petrol. Trans. 207, 144 (1956)
M.M. Dias, A.C. Payatakes, J. Fluid Mech. 164, 305 (1986)
M. Blunt, P. King, Phys. Rev. A 42, 4780 (1990)
M. Blunt, M.J. King, H. Scher, Phys. Rev. A 46, 7680 (1992)
M. Ferer, G.S. Bromhal, D.H. Smith, Phys. Rev. E 67, 051601 (2003)
R. Hilfer, H. Besserer, “Old problems and new solutions for multiphase flow in porous media”, in Porous Media: Physics, Models, Simulation, edited by A. Dmitrievsky, M. Panfilov (World Scientific Publ. Co., Singapore, 2000), p. 133
E. Buckingham, “Studies on the movement of soil moisture,” Tech. Rep. (U.S. Department of Agriculture, Bureau of Soils, 1907)
L.A. Richards, Physics 1, 318 (1931)
R.D. Wyckoff, H. Botset, Physics 7, 325 (1936)
M. Muskat, M. Meres, Physics, 7, 346 (1936)
M. Leverett, Trans. AIME 142, 152 (1941)
R. Hilfer, Phys. Rev. E 73, 016307 (2006)
R. Hilfer, R. Helmig, Adv. Water Res. 27, 1033 (2004)
W.O. Smith, P.D. Foote, P.F. Busang, Physics 1, 18 (1931)
S. Buckley, M. Leverett, Trans. AIME 146, 107 (1942)
J. Challis, Phil. Mag. Ser. 3, 32, 494 (1848)
G. Stokes, Phil. Mag. Ser. 3, 33, 349 (1848)
P. Lax, Comm. Pure Appl. Math. 7, 159 (1954)
P. LeFloch, Hyperbolic Systems of Conservation Laws (Birkhäuser, 2002)
H. Alt, S. Luckhaus, A. Visintin, Ann. Mat. Pura Appl. 136, 303–316 (1984)
K. Aziz, A. Settari, Petroleum Reservoir Simulation (Applied Science Publishers Ltd., London, 1979)
R. Helmig, Multiphase Flow and Transport Processes in the Subsurface (Springer, Berlin, 1997)
Z. Chen, G. Huan, Y. Ma, Computational Methods for Multiphase Flow in Porous Media (SIAM, 2006)
B. Biswal, P.E. Øren, R.J. Held, S. Bakke, R. Hilfer, Image Anal. Stereol. 28, 23 (2009)
L. Anton, R. Hilfer, Phys. Rev. E 59, 6819 (1999)
R. Hilfer, H. Besserer, Physica B 279, 125 (2000)
R. Hilfer, Physica A 359, 119 (2006)
E. Gerolymatou, I. Vardoulakis, R. Hilfer, J. Phys. D 39, 4104 (2006)
R. Hilfer, Physica A 371, 209 (2006)
S. Manthey, M. Hassanizadeh, R. Helmig, R. Hilfer, Adv. Water Res. 31, 1137 (2008)
R. Hilfer, Modeling and simulation of macrocapillarity, in CP1091, Modeling and Simulation of Materials, edited by P. Garrido, P. Hurtado, J. Marro (American Institute of Physics, New York, 2009), p. 141
R. Hilfer, F. Doster, Transp. Porous Med. 82, 507 (2010)
F. Doster, P. Zegeling, R. Hilfer, Phys. Rev. E 81, 036307 (2010)
F. Doster, R. Hilfer, New J. Phys. 13, 123030 (2011)
F. Doster, O. Hönig, R. Hilfer, Phys. Rev. E 86, 016317 (2012)
R. Hilfer, F. Doster, P. Zegeling, Vadose Zone J. 11, vzj2012.0021 (2012)
O. Hönig, F. Doster, R. Hilfer, Transp. Porous Med., 196 (2013)
F. Doster, R. Hilfer, Water Resour. Res. 50, 1 (2014)
E.G. Youngs, Soil Sci. 86, 202–207 (1958)
J. Briggs, D. Katz, “Drainage of water from sand in developing aquifer storage (1966), paper SPE1501 presented 1966 at the 41st Annual Fall Meeting of the SPE, Dallas, USA
R. Glass, T. Steenhuis ad, J. Parlange, Soil Sci. 148, 60 (1989)
S. Shiozawa, H. Fujimaki, Water Resour. Res. 40, W07404 (2004)
D. DiCarlo, Water Resour. Res. 40, W04215 (2004)
J.L. Nieber, Geoderma 70, 207 (1996)
F. Otto, J. Diff. Eq. 131, 20 (1996)
F. Otto, Adv. Math. Sci. Appl. 7, 537 (1997)
S. Geiger, D. Durnford, Soil Sci. Soc. Am. J. 64, 460 (2000)
M. Eliassi, R.J. Glass, Water Resour. Res. 38, 1234 (2002)
A. Egorov, R. Dautov, J. Nieber, A. Sheshukov, Water Resour. Res. 39, 1266 (2003)
D. DiCarlo, Adv. Water Res. 28, 1021 (2005)
C. van Duijn, L. Peletier, I. Pop, SIAM J. Math. Anal. 39, 507 (2007)
L. Cueto-Felgueroso, R. Juanes, Phys. Rev. Lett. 101, 244504 (2008)
L. Cueto-Felgueroso, R. Juanes, Phys. Rev. E 79, 036301 (2009)
D. diCarlo, Water Resour. Res. 49, 4531 (2013)
H. Alt, S. Luckhaus, Math. Z. 183, 311 (1983)
T. Fürst, R. Vodak, M. Sir, M. Bil, Water Resour. Res. 45, W03408 (2009)
D. DiCarlo, M. Mirzaei, B. Aminzadeh, H. Dehghanpur, Transp. Porous Med. 91, 955 (2012)
Y. Xiong, J. Hydrology 510, 353 (2014)
J. Nieber, R. Dautov, E. Egorov, A. Sheshukov, Transp. Porous Med. 58, 147 (2005)
C. van Duijn, Y. Fan, L. Peletier, I. Pop, Nonlinear Anal. Real World Appl. 14, 1361 (2013)
T. Bauters, D. DiCarlo, T. Steenhuis, Y. Parlange, J. Hydrology 231-232, 244 (2000)
M. Deinert, J. Parlange, K. Cady, T. Steenhuis, J. Selker, Water Resour. Res. 39, 1263 (2003)
S. Bottero, M. Hassanizadeh, P. Kleingeld, T. Heimorvaara, Water Resour. Res. 47, W10505 (2011)
R. Hilfer, P.E. Øren, Transp. Porous Med. 22, 53 (1996)
Y. Mualem, Water Resour. Res. 10, 514 (1974)
J.C. Parker, R.J. Lenhard, Water Resour. Res. 23, 2187 (1987)
R. Lenhard, J. Parker, Water Resour. Res. 23, 2197 (1987)
M.T.v. Genuchten, Soil Sci. Soc. Am. J. 44, 892 (1980)
L. Luckner, M.T.v. Genuchten, D. Nielsen, Water Resour. Res. 25, 2187 (1989)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Hilfer, R., Steinle, R. Saturation overshoot and hysteresis for twophase flow in porous media. Eur. Phys. J. Spec. Top. 223, 2323–2338 (2014). https://doi.org/10.1140/epjst/e2014-02267-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1140/epjst/e2014-02267-x