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Theory and equations

Non-equilibrium solute transport

Non-equilibrium transport describes early breakthrough and long tailing that the equilibrium advection–dispersion equation cannot reproduce Šimůnek and van Genuchten (2008). Rusteau offers the models of HYDRUS-1D.

Two-site sorption (chemical non-equilibrium)

The sorption sites are divided into two fractions Selim et al. (1976); van Genuchten and Wagenet (1989): on type-1 sites (fraction f) sorption is instantaneous, on type-2 sites it is kinetic (first order):

se = f Ks cβ / (1 + η cβ)
∂sk/∂t = α [ (1 − f) Ks cβ/(1 + η cβ) − sk ] − φk

with the first-order rate coefficient α [1/T] and the degradation φk on the kinetic sites. It is selected with f < 1 (transport parameters). f = 0 gives a one-site kinetic sorption model.

Mobile–immobile water (physical non-equilibrium)

The liquid phase is divided into a mobile (flowing) region θmo and an immobile (stagnant) region θim van Genuchten and Wierenga (1976). Solutes are exchanged by diffusion at the rate α [1/T]:

∂(θim cim)/∂t + ∂((1 − f) ρ sim)/∂t = α (c − cim) − φim

where f is the fraction of sorption sites in contact with the mobile water. It is selected with θim > 0. With the dual-porosity flow models the immobile water content is transient and the water transfer carries solute as well.

Dual non-equilibrium

Physical and chemical non-equilibrium combined: mobile–immobile water and, in the mobile region, equilibrium (fraction fem) and kinetic sorption sites with the rate ωS Šimůnek and van Genuchten (2008). In the reaction table, μg holds fem and γg holds ωS, as in HYDRUS-1D.

Dual permeability

Advection–dispersion in both the fracture and the matrix domain, coupled by the solute transfer Γs = αs(1 − w)θm(cf − cm) + Γwc (dual permeability) Gerke and van Genuchten (1993a).

Initial condition

With Non-equilibrium phase initially in equilibrium the initial kinetic or immobile concentrations are set in equilibrium with the liquid concentrations; otherwise they are given on the page Soil profile (Non-eq. conc.).

Typical behaviour

  • small α: the non-equilibrium phase hardly takes part (early breakthrough, as with less sorption);
  • large α: approaches equilibrium transport;
  • intermediate α: asymmetric breakthrough curves with tailing.

Literature cited in this topic

  • Gerke, H. H., and M. Th. van Genuchten (1993a). A dual-porosity model for simulating the preferential movement of water and solutes in structured porous media. Water Resources Research 29(2): 305–319. doi:10.1029/92WR02339
  • Selim, H. M., J. M. Davidson, and R. S. Mansell (1976). Evaluation of a two-site adsorption-desorption model for describing solute transport in soils. In: Proceedings of the Summer Computer Simulation Conference, Washington, D.C., pp. 444–448.
  • van Genuchten, M. Th., and P. J. Wierenga (1976). Mass transfer studies in sorbing porous media I. Analytical solutions. Soil Science Society of America Journal 40(4): 473–480. doi:10.2136/sssaj1976.03615995004000040011x
  • van Genuchten, M. Th., and R. J. Wagenet (1989). Two-site/two-region models for pesticide transport and degradation: Theoretical development and analytical solutions. Soil Science Society of America Journal 53(5): 1303–1310. doi:10.2136/sssaj1989.03615995005300050001x
  • 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

All sources: References.