Theory and equations

Dual-porosity models

Dual-porosity models assume that water flows only in one part of the pore space (inter-aggregate pores, fractures, the mobile region), while water in the other part (intra-aggregate pores, the rock matrix, the immobile region) does not move but exchanges water and solutes with the mobile region Šimůnek et al. (2003); Šimůnek and van Genuchten (2008). They describe non-equilibrium (preferential) flow in structured soils.

Governing equations

∂θ_mo/∂t = ∂/∂x [ K(h) (∂h/∂x + cos α) ] − S_mo − Γ_w

∂θ_im/∂t = −S_im + Γ_w

θ = θ_mo + θ_im

θ_mo and θ_im are the water contents of the mobile and immobile regions, S_mo and S_im their sink terms and Γ_w the transfer rate of water from the mobile to the immobile region [1/T]. The hydraulic functions of the mobile region are van Genuchten–Mualem functions; the immobile region has its own retention parameters θ_r,im, θ_s,im (and α_im, n_im).

Mass transfer driven by the water content

The transfer rate is proportional to the difference of the effective saturations of the two regions Šimůnek et al. (2001); Šimůnek et al. (2003):

Γ_w = ω [S_e^mo − S_e^im]

with the first-order rate coefficient ω [1/T]. The transfer within a time step is limited so that the immobile region does not exceed saturation or fall below its residual water content.

Mass transfer driven by the pressure head

The transfer is driven by the difference of the pressure heads of the two regions, with a conductivity of the interface evaluated with the van Genuchten parameters of the immobile region (α_im, n_im; ω takes the role of the saturated conductivity of the interface):

Γ_w = ½ [K_a(h_mo) + K_a(h_im)] (h_mo − h_im)

where h_im is the head of the immobile region at its current water content.

Durner's model

The model of Durner (1994) also has two pore regions, but in equilibrium: one Richards equation with a bimodal retention function (Durner).

Solute transport

With a dual-porosity flow model, solute transport uses the mobile–immobile formulation with an exchange proportional to the concentration difference plus the solute carried by the water transfer (non-equilibrium transport).

Literature cited in this topic

Durner, W. (1994). Hydraulic conductivity estimation for soils with heterogeneous pore structure. Water Resources Research 30(2): 211–223. doi:10.1029/93WR02676
Šimůnek, J., and M. Th. van Genuchten (2008). Modeling nonequilibrium flow and transport processes using HYDRUS. Vadose Zone Journal 7(2): 782–797. doi:10.2136/vzj2007.0074
Šimůnek, J., N. J. Jarvis, M. Th. van Genuchten, and A. Gärdenäs (2003). Review and comparison of models for describing non-equilibrium and preferential flow and transport in the vadose zone. Journal of Hydrology 272(1–4): 14–35. doi:10.1016/S0022-1694(02)00252-4
Šimůnek, J., O. Wendroth, N. Wypler, and M. Th. van Genuchten (2001). Non-equilibrium water flow characterized by means of upward infiltration experiments. European Journal of Soil Science 52(1): 13–24. doi:10.1046/j.1365-2389.2001.00361.x

All sources: References.