Theory and equations

Non-equilibrium water flow, centrifuge and density-dependent flow

These options of the HYDRUS code are not offered by the HYDRUS-1D interface (further options).

Non-equilibrium water flow

Following Ross and Smettem (2000), the water content does not follow the equilibrium retention curve θ_eq(h) instantly but approaches it with the time constant τ_w:

∂θ/∂t = (θ_eq(h) − θ) / τ_w

which over a time step gives

θ − θ_old = (θ_eq(h) − θ_old) (1 − e^−Δt/τ_w)

τ_w = 0 is equilibrium flow. The convergence test of the iteration uses the change of the equilibrium water content. The iteration gets harder: a smaller time step no longer increases the storage term, so strong infiltration may not converge.

Centrifuge

In a centrifuge the gravity term cos α of the Richards equation is replaced by the centrifugal acceleration relative to gravity:

ω² (R + d) / g

with the angular velocity ω = 2π rpm/60, the distance R of the soil surface from the axis of rotation, the depth d below the surface and the gravitational acceleration g. Rusteau writes the centrifugal term in conservative form (the flux of each element with the acceleration at its midpoint). A column draining to a bottom at h = 0 reaches the analytical equilibrium

h(d) = −ω²/(2g) [ (R + L)² − (R + d)² ]

(L the length of the column).

Density-dependent flow

The density ρ and the dynamic viscosity μ of the solution depend on the liquid concentration c of solute 1:

ρ/ρ₀ = a₀ + a₁ c + a₂ c² + a₃ c³, μ/μ₀ = b₀ + b₁ c + b₂ c² + b₃ c³

The conductivity scales with ρ/μ, the gravity term and the storage with ρ; HYDRUS solves ρ ∂θ/∂t = ∂/∂z[K(∂h/∂z + ρ)] − S while its mass balance sums ρθ, so the balance error equals the solution mass gained by changes of the density.

Literature cited in this topic

Ross, P. J., and K. R. J. Smettem (2000). A simple treatment of physical nonequilibrium water flow in soils. Soil Science Society of America Journal 64(6): 1926–1930. doi:10.2136/sssaj2000.6461926x

All sources: References.