Theory and equations

Soil hydraulic property models

The unsaturated soil hydraulic properties θ(h) and K(h) are described with closed-form equations Šimůnek et al. (2009). In all models the effective saturation is

S_e = (θ − θ_r) / (θ_s − θ_r)

with the residual and saturated water contents θ_r and θ_s.

van Genuchten – Mualem

van Genuchten (1980) used the statistical pore-size distribution model of Mualem (1976) to obtain a predictive equation for the unsaturated conductivity from the retention function:

θ(h) = θ_r + (θ_s − θ_r) / [1 + |αh|ⁿ]^m for h < 0; θ(h) = θ_s for h ≥ 0

K(h) = K_s S_e^l [1 − (1 − S_e^1/m)^m]², m = 1 − 1/n, n > 1

α [1/L], n and l are empirical parameters; l is the pore-connectivity parameter, estimated as 0.5 by Mualem (1976) (the parameter l).

van Genuchten with an air-entry value of −2 cm

For fine-textured soils with n close to 1 the van Genuchten–Mualem conductivity drops steeply just below saturation, which makes the numerical solution unstable and the predictions poor. Vogel et al. (2001) and Schaap and van Genuchten (2006) introduced a small minimum capillary height h_s = −2 cm: the retention curve is the van Genuchten function with the fictitious saturated water content θ_m chosen so that θ(h_s) = θ_s,

θ_m = θ_r + (θ_s − θ_r) (1 + |α h_s|ⁿ)^m

and the conductivity follows the modified model below with θ_a = θ_r, θ_k = θ_s, K_k = K_s.

Modified van Genuchten (Vogel and Císlerová)

Vogel and Císlerová (1988) added flexibility near saturation with a non-zero air-entry value h_s:

θ(h) = θ_a + (θ_m − θ_a) / (1 + |αh|ⁿ)^m for h < h_s; θ(h) = θ_s for h ≥ h_s

K(h) = K_s K_r(h) for h ≤ h_k; K(h) = K_k + (h − h_k)(K_s − K_k)/(h_s − h_k) for h_k < h < h_s; K(h) = K_s for h ≥ h_s

K_r = (K_k/K_s) (S_e/S_ek)^l [ (F(θ_r) − F(θ)) / (F(θ_r) − F(θ_k)) ]²

F(θ) = [1 − ((θ − θ_a)/(θ_m − θ_a))^1/m]^m, S_ek = (θ_k − θ_r)/(θ_s − θ_r)

θ_m ≥ θ_s and θ_a ≤ θ_r are fictitious extrapolations of the retention curve; K_k is a measured conductivity at the water content θ_k (h_k = h(θ_k)). With θ_a = θ_r, θ_m = θ_k = θ_s and K_k = K_s the original van Genuchten–Mualem model results.

Brooks and Corey

S_e = |αh|⁻ⁿ for h < −1/α; S_e = 1 for h ≥ −1/α

K = K_s S_e^{2/n + l + 2}

Brooks and Corey (1964); here n is the pore-size distribution index (λ) and 1/α the air-entry (bubbling) pressure head.

Kosugi (log-normal)

Kosugi (1996) assumed a log-normal pore-size distribution; with Mualem's model:

S_e = ½ erfc[ ln(h/h₀) / (√2 σ) ] for h < 0

K = K_s S_e^l { ½ erfc[ ln(h/h₀) / (√2 σ) + σ/√2 ] }²

In the parameter table α holds h₀ (the median pressure head) and n holds σ (the standard deviation of ln |h|), as in HYDRUS.

Durner (dual porosity)

Durner (1994) divided the pore space into two overlapping regions, each with a van Genuchten–Mualem function; their linear superposition describes multimodal pore systems:

S_e = w₁ [1 + (α₁ h)ⁿ₁]^−m₁ + w₂ [1 + (α₂ h)ⁿ₂]^−m₂

K(S_e) = K_s (w₁ S_e1 + w₂ S_e2)^l ( w₁ α₁ [1 − (1 − S_e1^1/m₁)^m₁] + w₂ α₂ [1 − (1 − S_e2^1/m₂)^m₂] )² / (w₁ α₁ + w₂ α₂)²

with the weights w₁ = 1 − w₂ and m_i = 1 − 1/n_i.

Look-up tables

Instead of a model, θ(h), K(h) and C(h) can be given as tables (file Mater.in of HYDRUS-1D); values between the tabulated heads are interpolated.

Temperature dependence

With temperature-dependent hydraulic properties, the pressure head is scaled with the surface tension σ and the conductivity with the dynamic viscosity μ and the density ρ of water at the soil temperature T, relative to a reference temperature Philip and de Vries (1957); Constantz (1982):

h(T) = h(T_ref) σ(T) / σ(T_ref)

K(T) = K(T_ref) [μ(T_ref) ρ(T)] / [μ(T) ρ(T_ref)]

The conductivity of water-saturated soil increases by about 2 % per °C, mostly through the viscosity.

Typical values

The soil catalog of Carsel and Parrish (1988) gives average parameters of the textural classes; Rosetta predicts them from the texture Schaap et al. (2001); Zhang and Schaap (2017).

Literature cited in this topic

Brooks, R. H., and A. T. Corey (1964). Hydraulic Properties of Porous Media. Hydrology Paper No. 3, Colorado State University, Fort Collins, CO, 27 pp.
Carsel, R. F., and R. S. Parrish (1988). Developing joint probability distributions of soil water retention characteristics. Water Resources Research 24(5): 755–769. doi:10.1029/WR024i005p00755
Constantz, J. (1982). Temperature dependence of unsaturated hydraulic conductivity of two soils. Soil Science Society of America Journal 46(3): 466–470. doi:10.2136/sssaj1982.03615995004600030005x
Durner, W. (1994). Hydraulic conductivity estimation for soils with heterogeneous pore structure. Water Resources Research 30(2): 211–223. doi:10.1029/93WR02676
Kosugi, K. (1996). Lognormal distribution model for unsaturated soil hydraulic properties. Water Resources Research 32(9): 2697–2703. doi:10.1029/96WR01776
Mualem, Y. (1976). A new model for predicting the hydraulic conductivity of unsaturated porous media. Water Resources Research 12(3): 513–522. doi:10.1029/WR012i003p00513
Philip, J. R., and D. A. de Vries (1957). Moisture movement in porous materials under temperature gradients. Transactions, American Geophysical Union 38(2): 222–232. doi:10.1029/TR038i002p00222
Schaap, M. G., and M. Th. van Genuchten (2006). A modified Mualem–van Genuchten formulation for improved description of the hydraulic conductivity near saturation. Vadose Zone Journal 5(1): 27–34. doi:10.2136/vzj2005.0005
Schaap, M. G., F. J. Leij, and M. Th. van Genuchten (2001). Rosetta: a computer program for estimating soil hydraulic parameters with hierarchical pedotransfer functions. Journal of Hydrology 251(3–4): 163–176. doi:10.1016/S0022-1694(01)00466-8
van Genuchten, M. Th. (1980). A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Science Society of America Journal 44(5): 892–898. doi:10.2136/sssaj1980.03615995004400050002x
Vogel, T., and M. Císlerová (1988). On the reliability of unsaturated hydraulic conductivity calculated from the moisture retention curve. Transport in Porous Media 3: 1–15. doi:10.1007/BF00222683
Vogel, T., M. Th. van Genuchten, and M. Císlerová (2001). Effect of the shape of the soil hydraulic functions near saturation on variably-saturated flow predictions. Advances in Water Resources 24(2): 133–144. doi:10.1016/S0309-1708(00)00037-3
Zhang, Y., and M. G. Schaap (2017). Weighted recalibration of the Rosetta pedotransfer model with improved estimates of hydraulic parameter distributions and summary statistics (Rosetta3). Journal of Hydrology 547: 39–53. doi:10.1016/j.jhydrol.2017.01.004
Šimůnek, J., M. Šejna, H. Saito, M. Sakai, and M. Th. van Genuchten (2009). The HYDRUS-1D Software Package for Simulating the One-Dimensional Movement of Water, Heat, and Multiple Solutes in Variably-Saturated Media, Version 4.08. HYDRUS Software Series 3, Department of Environmental Sciences, University of California Riverside, Riverside, CA, USA, 330 pp. https://www.pc-progress.com/en/Default.aspx?h1d-description

All sources: References.