Theory and equations

Colloid, virus and bacteria transport

Colloids, viruses and bacteria are transported like solutes but are retained by kinetic attachment to the solid phase rather than by sorption. HYDRUS-1D (and Rusteau) uses two kinetic sites with attachment and detachment Schijven and Šimůnek (2002); Bradford et al. (2003):

∂(θ c)/∂t + ρ ∂s_e/∂t + ρ ∂s₁/∂t + ρ ∂s₂/∂t = ∂/∂x (θ D ∂c/∂x) − ∂(q c)/∂x − R

ρ ∂s_i/∂t = θ ψ_i k_att,i c − k_det,i ρ s_i − (decay)

with the concentration c of the colloids in the liquid, the concentrations s₁, s₂ attached to the two sites, the attachment and detachment coefficients k_att,i, k_det,i [1/T] and the dimensionless functions ψ_i that describe blocking, ripening or straining.

Blocking and straining functions

iPsi	Function
0	no blocking, ψ = 1
1	Langmuir blocking, ψ = 1 − s/s_max
2	ripening: attachment increases with the attached mass
3	random sequential adsorption (RSA) Adamczyk et al. (1994); Johnson and Elimelech (1995)
4	depth-dependent straining, ψ = ((d_c + z − z₀)/d_c)^−β Bradford et al. (2003)
5	straining combined with Langmuir blocking

s_max (SMax) is the maximum solid-phase concentration of a site and d_c the diameter of the sand grains.

Attachment from filtration theory

With attachment rates from filtration theory, the attachment coefficient follows from clean-bed filtration theory Logan et al. (1995):

k_att = 3(1 − n) / (2 d_c) · η · α · v

with the porosity n, the collector (grain) diameter d_c, the single-collector efficiency η Rajagopalan and Tien (1976), the sticking efficiency α (entered in the column of the attachment coefficient) and the pore-water velocity v. The particle diameter d_p enters η.

Parameters

The parameters are entered per solute (colloid) and material on the page T / θ dependence, colloids: rKa1, rKd1, SMax1, rKa2, rKd2, SMax2 and the blocking types or the diameters Dc, Dp.

Literature cited in this topic

Adamczyk, Z., B. Siwek, M. Zembala, and P. Belouschek (1994). Kinetics of localized adsorption of colloid particles. Advances in Colloid and Interface Science 48: 151–280. doi:10.1016/0001-8686(94)80008-1
Bradford, S. A., J. Šimůnek, M. Bettahar, M. Th. van Genuchten, and S. R. Yates (2003). Modeling colloid attachment, straining, and exclusion in saturated porous media. Environmental Science & Technology 37(10): 2242–2250. doi:10.1021/es025899u
Johnson, P. R., and M. Elimelech (1995). Dynamics of colloid deposition in porous media: blocking based on random sequential adsorption. Langmuir 11(3): 801–812. doi:10.1021/la00003a023
Logan, B. E., D. G. Jewett, R. G. Arnold, E. J. Bouwer, and C. R. O'Melia (1995). Clarification of clean-bed filtration models. Journal of Environmental Engineering 121(12): 869–873. doi:10.1061/(ASCE)0733-9372(1995)121:12(869)
Rajagopalan, R., and C. Tien (1976). Trajectory analysis of deep-bed filtration with the sphere-in-cell porous media model. AIChE Journal 22(3): 523–533. doi:10.1002/aic.690220316
Schijven, J. F., and J. Šimůnek (2002). Kinetic modeling of virus transport at the field scale. Journal of Contaminant Hydrology 55(1–2): 113–135. doi:10.1016/S0169-7722(01)00188-7

All sources: References.