Theory and equations

Hysteresis

The relation between water content and pressure head depends on whether the soil is wetting or drying. The water content on a drying branch is higher than on a wetting branch at the same head (ink-bottle effect, contact angle, entrapped air).

Kool and Parker model

HYDRUS incorporates hysteresis with the empirical model of Scott et al. (1983), as modified by Kool and Parker (1987) to account for air entrapment. The main drying curve has the parameters θ_r, θ_s, α, n (and θ_m); the main wetting curve has the same θ_r and n and its own θ_s^w, α^w (often α^w ≈ 2α). With hysteresis in the conductivity, the main wetting curve also has K_s^w.

Scanning curves start at a reversal point (h_Δ, θ_Δ), where the direction of the change of the water content changes. They are scaled from the main curves so that they pass through the reversal point:

A drying scanning curve is the main drying curve scaled towards θ_r:

θ(h) = θ_r + a^d [θ^d(h) − θ_r], a^d = (θ_Δ − θ_r) / (θ^d(h_Δ) − θ_r)

A wetting scanning curve is the main wetting curve scaled so that it passes through the reversal point and reaches, at saturation, the water content θ_s that remains after air entrapment:

θ(h) = θ_r + a^w [θ^w(h) − θ_r], θ_r = θ_s − a^w (θ_s^w − θ_r)

θ_s = θ_s − (θ_s − θ_Δ) / [1 + R (θ_s − θ_Δ)], R = 1/(θ_s − θ_s^w) − 1/(θ_s − θ_r)

The trapped-air relation for θ_s follows Land (1968), as in Kool and Parker (1987): wetting from a dry state ends at θ_s^w, wetting from saturation ends at θ_s. With hysteresis in the conductivity, the scanning curves of K are scaled in the same way between K at the reversal point and K_s.

A node changes its branch when its water content changes in the opposite direction by more than the water content tolerance. The initial condition lies on the main drying or the main wetting curve (Initially drying / wetting curve), or on the scanning curves of a previous run (restart).

The pumping effect and the model of Lenhard et al.

While simple to implement, the Kool–Parker model can suffer from a pumping effect: under repeated small wetting and drying cycles the hysteresis loops can drift to physically unrealistic parts of the retention function. The model of Lenhard et al. (1991) and Lenhard and Parker (1992) eliminates pumping by keeping track of the historical reversal points: a scanning curve that returns to an earlier reversal point continues on the curve of that earlier reversal (closed scanning loops). Hysteresis in the retention curve (no pumping, Lenhard et al.) selects it.

Parameters

The hysteresis parameters are entered on the page Materials & parameters: θ_m (drying), θ_s, α and K_s of the wetting branch. For the parameter estimation the constraints θ_m = θ_s, α^w = 2α and K_s^w = K_s are available.

Literature cited in this topic

Kool, J. B., and J. C. Parker (1987). Development and evaluation of closed-form expressions for hysteretic soil hydraulic properties. Water Resources Research 23(1): 105–114. doi:10.1029/WR023i001p00105
Land, C. S. (1968). Calculation of imbibition relative permeability for two- and three-phase flow from rock properties. Society of Petroleum Engineers Journal 8(2): 149–156. doi:10.2118/1942-PA
Lenhard, R. J., and J. C. Parker (1992). Modeling multiphase fluid hysteresis and comparing results to laboratory investigations. In: M. Th. van Genuchten, F. J. Leij, and L. J. Lund (eds.), Proc. Intl. Workshop on Indirect Methods for Estimating the Hydraulic Properties of Unsaturated Soils, University of California, Riverside, CA, pp. 233–248.
Lenhard, R. J., J. C. Parker, and J. J. Kaluarachchi (1991). Comparing simulated and experimental hysteretic two-phase transient fluid flow phenomena. Water Resources Research 27(8): 2113–2124. doi:10.1029/91WR01272
Scott, P. S., G. J. Farquhar, and N. Kouwen (1983). Hysteretic effects on net infiltration. In: Advances in Infiltration, Publ. 11-83, American Society of Agricultural Engineers, St. Joseph, MI, pp. 163–170.

All sources: References.