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1R. Hilfer, “Transport and relaxation phenomena in porous media,” Advances in Chemical Physics XCII, 299 (1996).
2M. Muskat, The Flow of Homogeneous Fluids through Porous Media (McGraw Hill, New York, 1937).
3A.E. Scheidegger, The Physics of Flow Through Porous Media (University of Toronto Press, Canada, 1957).
4R.E. Collins, Flow of Fluids through Porous Materials (Reinhold Publishing Co., New York, 1961).
5J. Bear, Dynamics of Fluids in Porous Media (Elsevier Publ. Co., New York, 1972).
6G.d. Marsily, Quantitative Hydrogeology – Groundwater Hydrology for Engineers (Academic Press, San Diego, 1986).
7G.P. Willhite, Waterflooding, SPE Textbook Series, Vol. 3 (Society of Petroleum Engineers, USA, 1986).
8L.W. Lake, Enhanced Oil Recovery (Prentice Hall, Englewood Cliffs, 1989).
9F.A.L. Dullien, Porous Media - Fluid Transport and Pore Structure (Academic Press, San Diego, 1992).
10M. Sahimi, Flow and Transport in Porous Media and Fractured Rock (VCH Verlagsgesellschaft mbH, Weinheim, 1995).
11M. Eliassi and R.J. Glass, “On the continuum-scale modeling of gravity-driven fingers in unsaturated porous media: The inadequacy of the Richards equation with standard monotonic constitutive relations and hysteretic equations of state,” Water Resour. Res. 37, 2019 (2001).
12M.I.J.van Dijke and K. Sorbie, “Pore-scale network model for three-phase flow in mixed-wet porous media,” Phys.Rev.E 66, 046302 (2002).
13I. Fatt, “The network model of porous media I. capillary pressure characteristics,” AIME Petroleum Transactions 207, 144 (1956).
14M.M. Dias and A.C. Payatakes, “Network models for two-phase flow in porous media I: Immiscible microdisplacement of non-wetting fluids,” J. Fluid Mech. 164, 305 (1986).
15M. Blunt and P. King, “Macroscopic parameters from simulations of pore scale flow,” Phys. Rev. A 42, 4780 (1990).
16M. Blunt, M.J. King, and H. Scher, “Simulation and theory of two-phase flow in porous media,” Phys. Rev. A 46, 7680 (1992).
17M. Ferer, G.S. Bromhal, and D.H. Smith, “Pore-level modeling of drainage: Crossover from invasion percolation fingering to compact flow,” Phys.Rev.E 67, 051601 (2003).
18R. Hilfer and H. Besserer, “Old problems and new solutions for multiphase flow in porous media,” in Porous Media: Physics, Models, Simulation, edited by A. Dmitrievsky and M. Panfilov (World Scientific Publ. Co., Singapore, 2000) p. 133.
19E. Buckingham, “Studies on the movement of soil moisture,” Tech. Rep. (U.S.Department of Agriculture, Bureau of Soils, 1907).
20L.A. Richards, “Capillary conduction of liquids through porous mediums,” Physics 1, 318 (1931).
21R.D. Wyckoff and H. Botset, “Flow of gas-liquid mixtures through unconsolidated sands,” Physics 7, 325 (1936).
22M. Muskat and M. Meres, “Flow of heterogeneous fluids through porous media,” Physics 7, 346 (1936).
23M. Leverett, “Capillary behaviour in porous solids,” Trans. AIME 142, 152 (1941).
24R. Hilfer, “Macroscopic capillarity and hysteresis for flow in porous media,” Phys.Rev.E 73, 016307 (2006).
25R. Hilfer and R. Helmig, “Dimensional analysis and upscaling of two-phase flow in porous media with piecewise constant heterogeneities,” Adv. Water Res. 27, 1033 (2004).
26W.O. Smith, P.D. Foote, and P.F. Busang, “Capillary rise in sands of uniform spherical grains,” Physics 1, 18 (1931).
27S. Buckley and M. Leverett, “Mechanism of fluid displacement in sands,” Trans. AIME 146, 107 (1942).
28J. Challis, “On the velocity of sound, in reply to the remarks of the astronomer royal,” Phil.Mag.Ser.3 32, 494 (1848).
29G. Stokes, ‘‘On a difficulty in the theory of sound,” Phil.Mag.Ser.3 33, 349 (1848).
30P. Lax, “Weak solutions of nonlinear hyperbolic equations and their numerical computation,” Comm. Pure Appl.Math. 7, 159 (1954).
31P. LeFloch, Hyperbolic Systems of Conservation Laws (Birkhäuser, 2002).
32H. Alt, S. Luckhaus, and A. Visintin, “On nonstationary flow through porous media,” Annali di Matematica Pura ed Applicata 136, 303–316 (1984).
33K. Aziz and A. Settari, Petroleum Reservoir Simulation (Applied Science Publishers Ltd., London, 1979).
34R. Helmig, Multiphase Flow and Transport Processes in the Subsurface (Springer, Berlin, 1997).
35Z. Chen, G. Huan, and Y. Ma, Computational Methods for Multiphase Flow in Porous Media (SIAM, 2006).
36B. Biswal, P.E. Øren, R.J. Held, S. Bakke, and R. Hilfer, “Modeling of multiscale porous media,” Image Analysis and Stereology 28, 23–34 (2009).
37L. Anton and R. Hilfer, “Trapping and mobilization of residual fluid during capillary desaturation in porous media,” Physical Review E 59, 6819 (1999).
38R. Hilfer and H. Besserer, “Macroscopic two phase flow in porous media,” Physica B 279, 125 (2000).
39R. Hilfer, ‘‘Capillary pressure, hysteresis and residual saturation in porous media,” Physica A 359, 119 (2006).
40E. Gerolymatou, I. Vardoulakis, and R. Hilfer, “Modeling infiltration by means of a nonlinear fractional diffusion model,” J.Phys.D 39, 4104 (2006).
41R. Hilfer, ‘‘Macroscopic capillarity without a constitutive capillary pressure function,” Physica A 371, 209 (2006).
42S. Manthey, M. Hassanizadeh, R. Helmig, and R. Hilfer, “Dimensional analysis of two-phase flow including a rate-dependent capillary pressure-saturation relationship,” Adv. Water Res. 31, 1137 (2008).
43R. Hilfer, “Modeling and simulation of macrocapillarity,” in CP1091, Modeling and Simulation of Materials, edited by P. Garrido, P. Hurtado, and J. Marro (American Institute of Physics, New York, 2009) p. 141.
44R. Hilfer and F. Doster, “Percolation as a basic concept for macroscopic capillarity,” Transport in Porous Media 82, 507 (2010).
45F. Doster, P. Zegeling, and R. Hilfer, ‘‘Numerical solutions of a generalized theory for macroscopic capillarity,” Physical Review E 81, 036307 (2010).
46F. Doster and R. Hilfer, “Generalized Buckley-Leverett theory for two phase flow in porous media,” New Journal of Physics 13, 123030 (2011).
47F. Doster, O. Hönig, and R. Hilfer, ‘‘Horizontal flows and capillarity driven redistribution,” Phys.Rev.E 86, 016317 (2012).
48R. Hilfer, F. Doster, and P. Zegeling, “Nonmonotone saturation profiles for hydrostatic equilibrium in homogeneous media,” Vadose Zone Journal 11, vzj2012.0021 (2012).
49O. Hönig, F. Doster, and R. Hilfer, “Travelling wave solutions in a generalized theory for macroscopic capillarity,” Transport in Porous Media DOI, 196 (2013).
50F. Doster and R. Hilfer, “A comparison between simulation and expriment for hysteretic phenomena during two phase immiscible displacement,” Water Resources Research 50, 1 (2014).
51E. G. Youngs, ‘‘Redistribution of moisture in porous materials after infiltration: 2,” Soil Sci. 86, 202–207 (1958).
52J. Briggs and 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.
53R. Glass and T. Steenhuis ad J. Parlange, “Mechanism for finger persistence in homogeneous unsaturated, porous media: theory and verification,” Soil Science 148, 60 (1989).
54S. Shiozawa and H. Fujimaki, “Unexpected water content profiles und flux-limited one-dimensional downward infiltration in initially dry granular media,” Water Resources Research 40, W07404 (2004).
55D. DiCarlo, “Experimental measurements of saturation overshoot on infiltration,” Water Resources Research 40, W04215 (2004).
56J.L. Nieber, “Modeling of finger development and persistence in initially dry porous media,” Geoderma 70, 207 (1996).
57F. Otto, “{L}^{1}-contraction and uniqueness for quasilinear elliptic-parabolic equations,” Journal of Differential Equations 131, 20 (1996).
58F. Otto, “{L}^{1}-contraction and uniqueness for unstationary saturated-unsaturated porous media flow,” Adv.Math.Sci.Appl. 7, 537 (1997).
59S. Geiger and D. Durnford, “Infiltration in homogeneous sands and a mechanistic model of unstable flow,” Soil Sci.Soc.Am.J. 64, 460 (2000).
60M. Eliassi and R.J. Glass, ‘‘On the porous-continuum modeling of gravity-driven fingers in unsaturated materials: Extension of standard theory with a hold-back-pile-up effect,” Water Resour. Res. 38, 1234 (2002).
61A. Egorov, R. Dautov, J. Nieber, and A. Sheshukov, “Stability analysis of gravity-driven infiltrating flow,” Water Resources Research 39, 1266 (2003).
62D. DiCarlo, “Modeling observed saturation overshoot with continuum additions to standard unsatured theory,” Adv. Water Res. 28, 1021 (2005).
63C.van Duijn, L. Peletier, and I. Pop, “A new class of entropy solutions of the Buckley-Leverett equation,” SIAM J.Math.Anal. 39, 507–536 (2007).
64L. Cueto-Felgueroso and R. Juanes, “Nonlocal interface dynamics and pattern formation in gravity-driven unsaturated flow through porous media,” Phys.Rev.Lett. 101, 244504 (2008).
65L. Cueto-Felgueroso and R. Juanes, “Stability analysis of a phase-field model of gravity-driven unsaturated flow through porous media,” Phys.Rev.E 79, 036301 (2009).
66D. diCarlo, ‘‘Stability of gravity driven multiphase flow in porous media: 40 years of advancements,” Water Resources Research 49, 4531 (2013).
67H. Alt and S. Luckhaus, “Quasilinear elliptic-parabolic differential equations,” Math. Z. 183, 311–341 (1983).
68T. Fürst, R. Vodak, M. Sir, and M. Bil, ‘‘On the incompatibility of Richards’ equation and finger-like infiltration in unsaturated homogeneous porous media,” Water Resources Research 45, W03408 (2009).
69D. DiCarlo, M. Mirzaei, B. Aminzadeh, and H. Dehghanpur, “Fractional flow approach to saturation overshoot,” Transport in Porous Media 91, 955 (2012).
70Y. Xiong, ‘‘Flow of water in porous medi with saturation overshoot: A review,” J. Hydrology 510, 353–362 (2014).
71J. Nieber, R. Dautov, E. Egorov, and A. Sheshukov, “Dynamic capillary pressure mechanism for instability in gravity-driven flows; review and extension of very dry conditions,” Transp.Porous Med. 58, 147–172 (2005).
72C.van Duijn, Y. Fan, L. Peletier, and I. Pop, “Travelling wave solutions for degenerate pseudo-parabolic equations modelling two-phase flow in porous media,” Nonlinear Analysis: Real World Applications 14, 1361–1383 (2013).
73T. Bauters, D. DiCarlo, T. Steenhuis, and Y. Parlange, “Soil water content dependent wetting front characteristics in sands,” J. Hydrology 231-232, 244–254 (2000).
74M. Deinert, J. Parlange, K. Cady, T. Steenhuis, and J. Selker, “Comment on ”on the continuum-scale modeling of gravity-driven fingers in unsaturated porous media: The inadequacy of the Richards equation with standard monotonic constitutive relations and hysteretic equations of state” by Mehdi Eliassi and Robert J. Glass,” Water Resources Research 39, 1263 (2003).
75S. Bottero, M. Hassanizadeh, P. Kleingeld, and T. Heimorvaara, ‘‘Nonequilibrium capillarity effects in two-phase flow through porous media at different scales,” Water Resources Research 47, W10505 (2011).
76R. Hilfer and P.E. Øren, “Dimensional analysis of pore scale and field scale immiscible displacement,” Transport in Porous Media 22, 53 (1996).
77Y. Mualem, “A conceptual model for hysteresis,” Water Resour.Res. 10, 514 (1974).
78J.C. Parker and R.J. Lenhard, “A model for hysteresic constitutive relations governing multiphase flow. 1. saturation-pressure relations,” Water Resour.Res. 23, 2187 (1987).
79R. Lenhard and J. Parker, “A model for hysteretic constitutive relations governing multiphase flow 2. permeability-saturation relations,” Water Resources Research 23, 2197 (1987).
80M.T.v. Genuchten, “A closed form equation for predicting the hydraulic conductivity of unsaturated soils,” Soil Sci.Soc.Am.J. 44, 892–898 (1980).
81L. Luckner, M.T.v. Genuchten, and D. Nielsen, “A consistent set of parametric models for the two-phase flow of immiscible fluids in the subsurface,” Water Resources Research 25, 2187–2193 (1989).