| 1 | R. Hilfer, “Transport and relaxation phenomena in porous
media,” Advances in Chemical Physics XCII, 299 (1996). |
| 2 | M. Muskat, The Flow of Homogeneous Fluids through Porous
Media (McGraw Hill, New
York, 1937). |
| 3 | A.E. Scheidegger, The Physics of Flow Through Porous Media (University of Toronto Press, Canada, 1957). |
| 4 | R.E. Collins, Flow of Fluids through Porous Materials (Reinhold Publishing Co., New York, 1961). |
| 5 | J. Bear, Dynamics of Fluids in Porous Media (Elsevier Publ. Co., New York, 1972). |
| 6 | G.d. Marsily, Quantitative Hydrogeology – Groundwater Hydrology
for Engineers (Academic Press, San Diego, 1986). |
| 7 | G.P. Willhite, Waterflooding, SPE Textbook
Series, Vol. 3 (Society of
Petroleum Engineers, USA, 1986). |
| 8 | L.W. Lake, Enhanced Oil Recovery (Prentice Hall, Englewood Cliffs, 1989). |
| 9 | F.A.L. Dullien, Porous Media - Fluid Transport and Pore
Structure (Academic Press, San
Diego, 1992). |
| 10 | M. Sahimi, Flow and Transport in Porous Media and Fractured
Rock (VCH Verlagsgesellschaft mbH, Weinheim, 1995). |
| 11 | M. 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). |
| 12 | M.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). |
| 13 | I. Fatt, “The network model of porous media I. capillary
pressure characteristics,” AIME Petroleum Transactions 207, 144 (1956). |
| 14 | M.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). |
| 15 | M. Blunt and P. King, “Macroscopic parameters from simulations of pore
scale flow,” Phys. Rev. A 42, 4780 (1990). |
| 16 | M. Blunt, M.J. King, and H. Scher, “Simulation and theory of two-phase flow in porous
media,” Phys. Rev. A 46, 7680 (1992). |
| 17 | M. 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). |
| 18 | R. 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. |
| 19 | E. Buckingham, “Studies on the movement of soil moisture,” Tech. Rep. (U.S.Department of
Agriculture, Bureau of Soils, 1907). |
| 20 | L.A. Richards, “Capillary conduction of liquids through porous
mediums,” Physics 1, 318 (1931). |
| 21 | R.D. Wyckoff and H. Botset, “Flow of gas-liquid mixtures through unconsolidated
sands,” Physics 7, 325 (1936). |
| 22 | M. Muskat and M. Meres, “Flow of heterogeneous fluids through porous
media,” Physics 7, 346 (1936). |
| 23 | M. Leverett, “Capillary behaviour in porous solids,” Trans. AIME 142, 152 (1941). |
| 24 | R. Hilfer, “Macroscopic capillarity and hysteresis for flow in
porous media,” Phys.Rev.E 73, 016307 (2006). |
| 25 | R. 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). |
| 26 | W.O. Smith, P.D. Foote, and P.F. Busang, “Capillary rise in sands of uniform spherical
grains,” Physics 1, 18 (1931). |
| 27 | S. Buckley and M. Leverett, “Mechanism of fluid displacement in sands,” Trans. AIME 146, 107 (1942). |
| 28 | J. Challis, “On the velocity of sound, in reply to the remarks
of the astronomer royal,” Phil.Mag.Ser.3 32, 494 (1848). |
| 29 | G. Stokes, ‘‘On a difficulty in the theory of sound,” Phil.Mag.Ser.3 33, 349 (1848). |
| 30 | P. Lax, “Weak solutions of nonlinear hyperbolic equations
and their numerical computation,” Comm. Pure Appl.Math. 7, 159 (1954). |
| 31 | P. LeFloch, Hyperbolic Systems of Conservation Laws (Birkhäuser, 2002). |
| 32 | H. Alt, S. Luckhaus, and A. Visintin, “On nonstationary flow through porous media,” Annali di Matematica Pura ed Applicata 136, 303–316 (1984). |
| 33 | K. Aziz and A. Settari, Petroleum Reservoir Simulation (Applied Science Publishers Ltd., London, 1979). |
| 34 | R. Helmig, Multiphase Flow and Transport Processes in the
Subsurface (Springer, Berlin, 1997). |
| 35 | Z. Chen, G. Huan, and Y. Ma, Computational Methods for Multiphase Flow in Porous
Media (SIAM, 2006). |
| 36 | B. 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). |
| 37 | L. Anton and R. Hilfer, “Trapping and mobilization of residual fluid during
capillary desaturation in porous media,” Physical Review E 59, 6819 (1999). |
| 38 | R. Hilfer and H. Besserer, “Macroscopic two phase flow in porous media,” Physica B 279, 125 (2000). |
| 39 | R. Hilfer, ‘‘Capillary pressure, hysteresis and residual
saturation in porous media,” Physica A 359, 119 (2006). |
| 40 | E. Gerolymatou, I. Vardoulakis, and R. Hilfer, “Modeling infiltration by means of a nonlinear
fractional diffusion model,” J.Phys.D 39, 4104 (2006). |
| 41 | R. Hilfer, ‘‘Macroscopic capillarity without a constitutive
capillary pressure function,” Physica A 371, 209 (2006). |
| 42 | S. 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). |
| 43 | R. 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. |
| 44 | R. Hilfer and F. Doster, “Percolation as a basic concept for macroscopic
capillarity,” Transport in Porous Media 82, 507 (2010). |
| 45 | F. Doster, P. Zegeling, and R. Hilfer, ‘‘Numerical solutions of a generalized theory for
macroscopic capillarity,” Physical Review E 81, 036307 (2010). |
| 46 | F. Doster and R. Hilfer, “Generalized Buckley-Leverett theory for two
phase flow in porous media,” New Journal of Physics 13, 123030 (2011). |
| 47 | F. Doster, O. Hönig, and R. Hilfer, ‘‘Horizontal flows and capillarity driven
redistribution,” Phys.Rev.E 86, 016317 (2012). |
| 48 | R. Hilfer, F. Doster, and P. Zegeling, “Nonmonotone saturation profiles for hydrostatic
equilibrium in homogeneous media,” Vadose Zone Journal 11, vzj2012.0021 (2012). |
| 49 | O. Hönig, F. Doster, and R. Hilfer, “Travelling wave solutions in a generalized theory
for macroscopic capillarity,” Transport in Porous Media DOI, 196 (2013). |
| 50 | F. Doster and R. Hilfer, “A comparison between simulation and expriment for
hysteretic phenomena during two phase immiscible displacement,” Water Resources Research 50, 1 (2014). |
| 51 | E. G. Youngs, ‘‘Redistribution of moisture in porous materials
after infiltration: 2,” Soil Sci. 86, 202–207 (1958). |
| 52 | J. 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. |
| 53 | R. Glass and T. Steenhuis ad J. Parlange, “Mechanism for finger persistence in homogeneous
unsaturated, porous media: theory and verification,” Soil Science 148, 60 (1989). |
| 54 | S. 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). |
| 55 | D. DiCarlo, “Experimental measurements of saturation overshoot
on infiltration,” Water Resources Research 40, W04215 (2004). |
| 56 | J.L. Nieber, “Modeling of finger development and persistence in
initially dry porous media,” Geoderma 70, 207 (1996). |
| 57 | F. Otto, “ -contraction and uniqueness for quasilinear
elliptic-parabolic equations,” Journal of Differential Equations 131, 20 (1996). |
| 58 | F. Otto, “-contraction and uniqueness for
unstationary saturated-unsaturated porous media flow,” Adv.Math.Sci.Appl. 7, 537 (1997). |
| 59 | S. Geiger and D. Durnford, “Infiltration in homogeneous sands and a
mechanistic model of unstable flow,” Soil Sci.Soc.Am.J. 64, 460 (2000). |
| 60 | M. 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). |
| 61 | A. Egorov, R. Dautov, J. Nieber, and A. Sheshukov, “Stability analysis of gravity-driven infiltrating
flow,” Water Resources Research 39, 1266 (2003). |
| 62 | D. DiCarlo, “Modeling observed saturation overshoot with
continuum additions to standard unsatured theory,” Adv. Water Res. 28, 1021 (2005). |
| 63 | C.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). |
| 64 | L. 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). |
| 65 | L. 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). |
| 66 | D. diCarlo, ‘‘Stability of gravity driven multiphase flow in
porous media: 40 years of advancements,” Water Resources Research 49, 4531 (2013). |
| 67 | H. Alt and S. Luckhaus, “Quasilinear elliptic-parabolic differential
equations,” Math. Z. 183, 311–341 (1983). |
| 68 | T. 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). |
| 69 | D. DiCarlo, M. Mirzaei, B. Aminzadeh, and H. Dehghanpur, “Fractional flow approach to saturation
overshoot,” Transport in Porous Media 91, 955 (2012). |
| 70 | Y. Xiong, ‘‘Flow of water in porous medi with saturation
overshoot: A review,” J. Hydrology 510, 353–362 (2014). |
| 71 | J. 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). |
| 72 | C.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). |
| 73 | T. Bauters, D. DiCarlo, T. Steenhuis, and Y. Parlange, “Soil water content dependent wetting front
characteristics in sands,” J. Hydrology 231-232, 244–254 (2000). |
| 74 | M. 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). |
| 75 | S. 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). |
| 76 | R. Hilfer and P.E. Øren, “Dimensional analysis of pore scale and field scale
immiscible displacement,” Transport in Porous Media 22, 53 (1996). |
| 77 | Y. Mualem, “A conceptual model for hysteresis,” Water Resour.Res. 10, 514 (1974). |
| 78 | J.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). |
| 79 | R. Lenhard and J. Parker, “A model for hysteretic constitutive relations
governing multiphase flow 2. permeability-saturation relations,” Water Resources Research 23, 2197 (1987). |
| 80 | M.T.v. Genuchten, “A closed form equation for predicting the
hydraulic conductivity of unsaturated soils,” Soil Sci.Soc.Am.J. 44, 892–898 (1980). |
| 81 | L. 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). |
| 82 | http://www.openfoam.com/. |
