www.VadoseZoneJournal.org | 618 2011, Vol. 10 Alternatve Analy tcal Expressions for the General van Genuchten– Mualem and van Genuchten– Burdine Hydraulic Conductvity Models The van Genuchten expressions for the unsaturated soil hydraulic proper tes, first pub- lished in 1980, are used frequently in various vadose zone flow and transport applicatons assuming a specific relatonship between the m and n soil hydraulic parameters. By com- parison, probably because of the complexity of the hydraulic conductvity equatons, the more general solutons with independent m and n values are rarely used. We expressed the general van Genuchten–Mualem and van Genuchten–Burdine hydraulic conductvity equa- tons in terms of hypergeometric functons, which can be approximated by infinite series that converge rapidly for relatvely large values of the van Genuchten–Mualem parameter n but only very slowly when n is close to one. Alternatve equatons were derived that provide very close approximatons of the analytcal results. The newly proposed equatons allow the use of independent values of the parameters m and n in the soil water retenton model of van Genuchten for subsequent predicton of the van Genuchten–Mualem and van Genuchten–Burdine hydraulic conductvity models, thus providing more flexibility in fiƫng experimental pressure-head-dependent water content, θ (h), and hydraulic conduc- tvity, K(h), or K( θ) data. The unsaturated hydraulic conductvity is a key property for modeling water fow in the vadose zone. Tis property can be measured directly (Klute and Dirksen, 1986) or indirectly (van Genuchten and Leij, 1992) or estimated implicitly using inverse meth- ods (Hopmans and Šimůnek, 1999; Butters and Duchateau, 2002). Based on pore size distribution theories proposed by Childs and Collis-George (1950), Burdine (1953), and Mualem (1976), among others, several approaches for predicting the unsaturated hydraulic conductivity from measured soil water retention data have been presented in the literature. Tese theories produced predictive expressions for the relative hydraulic conductivity, K r , which is the ratio between the hydraulic conductivity, K, at a certain water content, θ , or pressure head, h, and the hydraulic conductivity at saturation, K s : ( ) ( ) r s , , K h K h K θ θ = [1] A frequently used equation for describing the soil water retention curve is (van Genuchten, 1980) ( ) e s r 1 m n r S h − θ−θ ⎡ ⎤ = = + α ⎢ ⎥ ⎣ ⎦ θ −θ [2] where S e is the efective saturation, θ s and θ r are the saturated and residual soil water con- tents, respectively, and α , m, and n are empirical parameters. As shown by van Genuchten (1980) and van Genuchten and Nielsen (1985), Eq. [2] may be combined with the pore size distribution models of Burdine (1953) or Mualem (1976) to lead to predictive expres- sions for the relative hydraulic conductivity of unsaturated soils. Te Burdine (1953) and Mualem (1976) models difer in the manner in which efective pore radii are estimated (Vereecken, 1995). Te model of Mualem (1976) is given by ( ) ( ) { } ( ) { } ⎡ ⎤ ⎣ ⎦ = ⎡ ⎤ ⎣ ⎦ ∫ ∫ e 2 e e e 0 r e 2 1 e e 0 1 d 1 d S l S hS S K S hS S [3] Alternatves for the van Genuchten hydraulic conductvity equatons are presented using approximations of infinite series of hypergeometric func- tons. These approximatons match very closely to the analytcal results and allow the use of independent parameter values in the soil water retenton model for subsequent pre- dicton of the hydraulic conductvity. D. Dourado Neto, Q. de Jong van Lier, and K. Reichardt, Univ. of São Paulo, Piraci- caba (SP), Brazil; M.Th. van Genuchten, Dep. of Mechanical Engineering, Federal Univ. of Rio de Janeiro, Rio de Janeiro (RJ), Brazil; K. Metselaar, Dep. of Environmental Sciences, Wageningen Univ., Wageningen, the Nether- lands; and D.R. Nielsen, Dep. of Land, Air and Water Resources, Univ. of California, Davis, CA 95616. *Corresponding author (qdjvlier@ esalq.usp.br). Vadose Zone J. 10:618–623 doi:10.2136/vzj2009.0191. Received 22 Dec. 2009. Posted online 26 Apr. 2011. © Soil Science Society of America 5585 Guilford Rd., Madison, WI 53711 USA. All rights reserved. No part of this periodical may be reproduced or transmiƩed in any form or by any means, electronic or mechanical, including photo- copying, recording, or any informaton storage and retrieval system, without permission in writng from the publisher. Original Research D. Dourado Neto Q. de Jong van Lier* M.Th. van Genuchten K. Reichardt K. Metselaar D.R. Nielsen