Y. Mualem. M. Th. van Genuchten

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1 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

2 48 +*0,**1 inverse method pedotransfer function van Genuchten et al., 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 ,** van Genuchten +32*,,0*.,,,10 +,*/. Penman ,// 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

3 : 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

4 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 Kunze et al *./ Hagen-Poiseuille, Burdine +3/- Wyllie and Gardner +3/2 K rs, e Kozeny Averjanov +3/* K rs e -././. *./

5 : 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., / ; Kosugi, 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

6 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 Haverkamp et al van Genuchten m+ Brutsaert 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

7 : 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+

8 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,

9 : 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

10 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, 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, 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

11 : 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 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

12 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 : 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 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-

13 : 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/ : +,* : 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 (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 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.

14 60 +*0,**1 Mualem-van Genuchten Mualem +310 van Genuchten +32* van Genuchten n a m Y * :,** :,**1 0 0