J. Jpn. Soc. Soil Phys. No. +*0, p..10*,**1 Y. Mualem M. Th. van Genuchten * Reviewing classical studies in soil physics A new model for predicting the hydraulic conductivity of unsaturated porous media By Y. Mualem Water Resources Research, Vol. +,, No.-, /+-/,, (+310) and A closed-form equation for predicting the hydraulic conductivity of unsaturated soils By M. Th. van Genuchten Soil Science Society of America Journal, Vol..., No./, 23,232 (+32*) Ken ichirou KOSUGI* * Graduate School of Agriculture, Kyoto Univ., Kyoto, 0*02/*,, Japan Key words : Water retention curve, unsaturated hydraulic conductivity, pore size distribution, Mualem-van Genuchten model + q Y K s q Y K Y q K Richards, Hendrayanto et al., +333 * 0*02/*, : Mualem-van Genuchten
48 +*0,**1 inverse method pedotransfer function van Genuchten et al., +333 - van Genuchten +32* Mualem +310 Mualem-van Genuchten Mualem +310 van Genuchten +32* Koutsoyiannis and Kundzewicz,**1 Institute for Scientific Information, Scopus, Google Scholar Mualem +310 +,**- 11-1.1 van Genuchten +32*,,0*.,,,10 +,*/. Penman +3.2 +,//0 112 103 Mualem-van Genuchten Mualem +310 van Genuchten +32* Mualem +310 Childs and Collis-George, +3/* ; Millington and Quirk, +30+ Burdine, +3/- ; Wyllie and Gardner, +3/2 Mualem van Genuchten +32* van Genuchten +32* SSSAJ van Genuchten, +312 Rien Mualem-van Genuchten Mualem +310 van Genuchten +32* van Genuchten Mualem-van Genuchten, Mualem,+ Mualem +310 r f r frdqdr rrdr f r dr R min R A+2 ; R r qr frdrq r q r
: Y. Mualem 49 q r... q q rq s q *+ S e S e qqr q sq r r Y YAr A,* *.+/ cm,,, Mualem +310 X, X X rrdr rr dr frfrdrdrx rr Mualem a r, r f R. l + r. L f R. l, r. L f r l + l,. R. r. L + R r, r A/ pr, Lpr, l +pr, l, L R 0 r, l +r, l, r. l,r. l + r. r. + l +l,rr A1 R, rr Hagen-Poiseuille, + rr R K K s K r arrgrrrfrfrdrdr K r K K s G R, r, r r r Mualem r r + R Hagen-Poiseuille R, r, r A. pr. g 2v f L pr. g 2v f + pr. g l + 2v f, l, v L, l +,l, R, r, r f, f +, f, R, r, r f, f +, f, A, ff +f, T R, r, r T G Mualem Burdine +3/- Millington and Quirk +30+ qq r power function K qq r r q sq r, S e
50 +*0,**1 + rfrdr Y + Y dq dq dy dy + Y dse + Fig. + Mualem +310, Figure + A combination of cylindrical tubes used to evaluate the hydraulic conductivity of a pair of capillary elements. (Mualem +310, Figure +) K rs e, Mualem,- Mualem + r r Mualem +310 Childs and Collis- George +3/* + r, A+. TRrrGRrrr, frfrdrdr TRrrGRrrr, frfrdrdr K r K rs e Mualem +310 rr r, r, rr Mualem rr + Ll +l, Mualem +310./ *./ Childs and Collis-George +3/* * Millington and Quirk +30+.- Kunze et al. +302 +.- *./ Hagen-Poiseuille, Burdine +3/- Wyllie and Gardner +3/2 K rs, e Kozeny Averjanov +3/* K rs e -././. *./
: Y. Mualem 51 Mualem, +310 *././ empirical S e, -././ Mualem +310./ Mualem Mualem, Y. : A catalogue of the hydraulic properties of unsaturated soils, Proj...,, Technion- Israel Inst. of Technol., Haifa, Israel, +310 UNSODA Leij et al., +330 +32/ ; Kosugi, +33. - Mualem-van Genuchten -+ Mualem +310 S e Y K r S e Y K r closed-form Mualem +310 Brooks and Corey +30. S eyy c l YY c S e+ YY c K rs,,l e Y c l pore size distribution index, Fig., Mualem-van Genuchten Mualem-Brooks and Corey a b a*.**/ cm + n, m*./ Y c,** cm l+ *./ van Genuchten +32*, Figure. (a) Water retention and (b) relative hydraulic conductivity functions based on Mualem-van Genuchten (indicated by solid lines) and Mualem-Brooks and Corey (indicated by broken lines) models, with parameter values of a*.**/ cm +, n,, m*./, Y c,** cm, l+, and *./. (modified from van Genuchten +32*, Figure.),a Y c,** cm l+ Brooks and Corey Y Y c q q s J Y c Mualem +310,b ; Mualem +310 *./ YY c K r + Y c van Genuchten +32* Richards
52 +*0,**1 YY c dq dy * -, van Genuchten van Genuchten +32* S e+ay n m a Y m n,a a*.**/ cm + n, m*./ S van Genuchten +32* m+ Ahuja and Swartzendruber +31, Endelman et al. +31. Haverkamp et al. +311 van Genuchten m+ Brutsaert +300 -- m+ Brutsaert +300 van Genuchten +32* m n K rs e f ns e f n+, x S e x ay f ns e x +m +n dx +x +m xy m f ns em y m++n +y +n dy m++n* m++n f ns e + ay dx f ns e++s e +m m K rs e ++S e +m m, m++n Mualem-van Genuchten K r Y,b Mualem-Brooks and Corey van Genuchten +32* Mualem +310 *./,b K ry *./ - Mualem-van Genuchten m+ +n a, n, m *./ K ry K r - van Genuchten +32* - - Mualem-van Genuchten + K r. Mualem +310./ *./ Wösten and van Genuchten +322 Schuh and Cline +33* Kosugi +333
: Y. Mualem 53 Corey Y S eay mn a+y c mnl - Fig. - Hygiene Sandstone Silt Loam G.E. - van Genuchten +32*, Figure / 1 Observed (open circles) and calculated (solid lines) curves for soil water retention and relative hydraulic conductivity, for Hygiene Sandstone (upper panels) and Silt Loam G.E. - (lower panels). (modified from van Genuchten +32*, Figures / and 1) K r van Genuchten +32* m+,n Burdine-van Genuchten Mualemvan Genuchten -. Brooks and Corey van Genuchten +32* Brooks and a,a m+ +n, a Y,** cm van Genuchten S van Genuchten +32* a van Genuchten and Nielsen +32/ m++n m n l n Brooks and Corey a. Y+a S e, m m n n m Y+a +..+ van Genuchten Mualem-van Genuchten m++n m S e+ay n +n+
54 +*0,**1 S e n n n + n,.02 n m++n +n *m+ m *+ n m++n n S e+ay ++m m /a a *.*+ cm + +a+** cm m *.+ * +. Fig.. m n *.. n van Genuchten and Nielsen +32/, Figure, Water retention functions based on Eq. (,/), when n was increased and m was simultaneously decreased such that the product mn remains *... (modified from van Genuchten and Nielsen +32/, Figure,) a n K s q s van Genuchten +32* K s q s Mualemvan Genuchten a n q r a n n S eay n m* Y S e + heavy clay S e m+ ++m n Y+a S e+ Y+a S e* /a glass beads S e /a m *.+ m* m+ van Genuchten m a a,a van Genuchten S S Y * Y * dqdy C Y Y * Y,
: Y. Mualem 55 / a *.*+ cm + m a S ey b ds edy ds edyy ds edy Fig. / (a) Water retention functions based on Eq. (-0) and (b) their derivatives, when a was fixed at *.*+ cm + and m was changed. Solid circle on each ds e/dy Y curve represents maximal ds e/dy value. 0 Y * +** cm m a S ey b ds edyy ds edyy ds edy Fig. 0 (a) Water retention functions based on Eq. (-2) and (b) their derivatives, when Y * was fixed at +** cm and m was changed. Solid circle on each ds e/dy Y curve represents maximal ds e/dy value. Y *m +m a van Genuchten van Genuchten, +312 a Kosugi, +33. S e+myy * ++m m 0a Y * +** cm m a Y+** cm Y * 0b 0a ds edyy ds edy q sq r C ds edy m Y+** cm / a Y+** cm ds edyy
56 +*0,**1 n a m Y * van Genuchten Mualem., van Genuchten +32* Gardner, +3/0 ; Brutsaert, +300 f r ln r ln r m s Kosugi, +33. +330 fr exp lnrrm,,s, r m r m S e *./ s * S eqlnyy ms Y m r m Y Y mar m S e*./ Q x x Qx,p +, expu,,du Q x Microsoft Excel NORMSDIST QxNORMSDISTx Q x erfcx, expu, du Qxerfcx, Mualem K rs e QQ + S es, Kosugi, +330 Q + Q S eq x xq + S e Q + S e S e x Microsoft Excel NORMSINV Q + S enormsinvs e Mualem-van Genuchten - Kosugi, +330 r may m s Kosugi, +331.- Childs and Collis-George +3/* Mualem +310 Burdine +3/- K rs e h Raats, +33, ; Kosugi, +333 Mualem *./ b + h, Burdine, b, h+ Raats, +33, K rs e mbn+bn h I x p, q
: Y. Mualem 57 I xpq u p+ +u q+ du u p+ +u q+ du m++n b+ h, K rs e QQ + S ebs h Kosugi +333,** b h K rs e QQ + S ebs.. Y q q r Luckner et al. +323 q r Nimmo +33+ q r Luckner et al. q r SÀír et al. +32/ q b q bq r q r Y qq r Y q q r q r Mualem +310 Nimmo +33+ Mualem S e Y q* Rossi and Nimmo +33. Fayer and Simmons +33/ +33/ multimodal model Othmer et al., +33+ ; Durner, +33. ; Ross and Smettem, +33- Y q* Mualem +310 Kosugi et al.,,**, ;,**. / Mualem +310 Childs and Collis-George +3/* Burdine +3/- Mualem +310 Mualem Childs and Collis-George Burdine van Genuchten van Genuchten Mualem
58 +*0,**1 Mualem +310 S ey Mualem-Brooks and Corey Richards Mualem-van Genuchten Mualem van Genuchten Mualem +310 van Genuchten +32* Mualem-van Genuchten +3-2**21 Ahuja, L.R. and Swartzendruber, D. (+31,) : Anim- proved form of soil-water di#usivity function. Soil Sci. Soc. Am. Proc. -0 : 3+.. Averjanov, S.F. (+3/*) : About permeability of subsurface soils in case of incomplete saturation. Eng. Collect., 1. Brooks, R.H. and Corey, A.T. (+30.) : Hydraulic properties of porous media. Hydrol. Pap. no. -. Colorado State Univ., Ft. Collins, USA. Brutsaert, W. (+300) : Probability laws for pore-size distributions. Soil Sci. +*+ : 2/3,. Burdine, N.T. (+3/-) : Relative permeability calculations from pore-size distribution data. Trans. Am. Inst. Min. Metall. Pet. Eng., +32 : 1+11. Childs, E.C. and Collis-George, N. (+3/*) : The permeability of porous materials. Proc. Roy. Soc. London, Ser. A,,*+ : -3,.*/. Durner, W. (+33.) : Hydraulic conductivity estimation for soils with heterogeneous pore structure. Water Resour. Res., -* :,++,,-. Endelman, F. J., Box, G.E.P., Boyle, J.R., Hughes, R.R., Keeney, D.R., Northrup, M.L. and Sa$gna, P.G. (+31.) : The mathematical modeling of soil-waternitrogen phenomena. EDFB-IBP-1.-2. Oak Ridge National Laboratory, Oak Ridge, Tenn., USA. Fayer, M. J. and Simmons, C.S. (+33/) : Modified soil water retention functions for all matric suctions. Water Resour. Res., -+ : +,--+,-2. Gardner, W.R. (+3/0) : Representation of soil aggregate-size distribution by a logarithmic-normal distribution. Soil Sci. Soc. Am. Proc.,,* : +/++/-. Haverkamp, R., Vauclin, M., Touma, J., Wierenga, P. J. and Vachaud, G. (+311) : A comparison of numerical simulation models for one-dimensional infiltration. Soil Sci. Soc. Am. J..+ :,2/,3.. Hendrayanto, Kosugi, K., Uchida, T., Matsuda, S. and Mizuyama, T. (+333) : Spatial variability of soil hydraulic properties in a forested hillslope. J. For. Res.,. : +*1++.. Kosugi, K. (+33.) : Three-parameter lognormal distribution model for soil water retention. Water Resour. Res., -* : 23+3*+. Kosugi, K. (+330) : Lognormal distribution model for unsaturated soil hydraulic properties. Water Resour. Res., -, :,031,1*-. Kosugi, K. (+331) : A new model to analyze water retention characteristics of forest soils based on soil pore radius distribution. J. For. Res.,, : +2. Kosugi, K. (+333) : General model for unsaturated hydraulic conductivity for soils with lognormal pore-size distribution. Soil Sci. Soc. Am. J., 0- :,1*,11.,**. pp. -/./ Kosugi, K., Hopmans, J.W. and Dane J.H. (,**,) : Parametric models. In : Methods of Soil Analysis, Part., Physical Methods. (eds J.H. Dane and G.C. Topp), pp. 1-31/1. Soil Sci. Soc. Am., Madison, WI, USA. Koutsoyiannis, D. and Kundzewicz, Z.W. (,**1) : Quantifying the impact of hydrological studies. Hydrological Sciences-Journal-des Sciences Hy-
: Y. Mualem 59 drologiques, /, (+) :-+1. Kunze, R. J., Uehara, G. and Graham, K. (+302) :Factors important in the calculation of hydraulic conductivity. Soil Sci. Soc. Am. Proc., -, : 10* 10/. Leij, F. J., Alves, W. J., van Genuchten, M. Th. and Williams, J.R. (+330) : The UNSODA unsaturated soil hydraulic database. Res. Rep. 0** R-30 *3/. +*- pp. US Environmental Protection Agency, Cincinnati, Ohio, USA. Luckner, L., van Genuchten, M.Th. and Nielsen, D.R. (+323) : A consistent set of parametric models for the two-phase flow of immiscible fluids in the subsurface. Water Resour. Res.,,/ :,+21,+3-. Millington, R. J. and Quirk, J.P. (+30+) : Permeability of porous solids, Trans. Faraday Soc., /1 : +,** +,*0. Mualem, Y. (+310) : A new model for predicting the hydraulic conductivity of unsaturated porous media. Water Resour. Res., +, : /+-/,,. +33/ : 2 :./-.0+. Nimmo, J.R. (+33+) : Comment on the treatment of residual water content in A consistent set of parametric models for the two-phase flow of immiscible fluids in the subsurface by L. Luckner et al. Water Resour. Res.,,1 : 00+00,. +32/ : +,* : 031.. Othmer, H., Diekkrüger, B. and Kutílek, M. (+33+) : Bimodal porosity and unsaturated hydraulic conductivity. Soil Sci., +/, : +-3+/*. Penman, H.L. (+3.2) : Natural evaporation from open water, bare soil and grass. Proc. Roy. Soc. Lond. Ser. A, Math. and Phys. Sci., +3- (+*-,) :+,*+./. Raats, P.A.C. (+33,) : A superclass of soils. In : Proc. Int. Worksh. Indirect methods for estimating the hydraulic properties of unsaturated soils, Riverside, CA, +++- Oct. +323. (eds M.Th. van Genuchten et al.), pp..//+. Univ. Calif., Riverside, CA, USA. Rossi, C. and Nimmo, J.R. (+33.) : Modeling of soil water retention from saturation to oven dryness. Water Resour. Res., -* : 1*+1*2. Ross, P. J. and Smettem, K.R. J. (+33-) : Describing soil hydraulic properties with sums of simple functions. Soil Sci. Soc. Am. J., /1 :,0,3. Schuh, W.M. and Cline, R.L. (+33*) : E#ect of soil properties on unsaturated hydraulic conductivity pore-interaction factors. Soil Sci. Soc. Am. J., /. : +/*3+/+3. SÀír, M., Vogel, T. and Císlerová, M. (+32/) : The analytical expression of the retention curves and hydraulic conductivities (in Czech). Vodohosp. CÀas., -- : 1.2/. van Genuchten, R. (+312) : Calculating the unsaturated hydraulic conductivity with a new closedform analytical model. Res. Rep. 12-WR-*2. Princeton Univ., Princeton, USA. van Genuchten, M.Th. (+32*) : A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci. Soc. Am. J.,.. : 23, 232. van Genuchten, M.Th., Leij, F. J. and Wu, L. (+333) : Characterization and measurement of the hydraulic properties of unsaturated porous media, Proc. Int. Worksh., Riverside, CA,,,,. Oct. +331. U.S. Salinity Lab. and Univ. Calif., Riverside, CA, USA. van Genuchten, M.Th. and Nielsen, D.R. (+32/) : On describing and predicting the hydraulic properties of unsaturated soils. Ann. Geophys. - : 0+/0,2. Wösten, J.H.M. and van Genuchten, M.Th. (+322) : Using texture and other soil properties to predict the unsaturated soil hydraulic functions. Soil Sci. Soc. Am. J., /, : +10,+11*. Wyllie, M.R. J. and Gardner, G.H.F. (+3/2) : The generalized Kozeney-Carman equation. World Oil, +.0 :,+*,,2.
60 +*0,**1 Mualem-van Genuchten Mualem +310 van Genuchten +32* van Genuchten n a m Y * :,**1 - +2 :,**1 0 0