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Theory and equations

Boundary conditions of water flow

The Richards equation needs a condition at the soil surface (x = L) and at the bottom (x = 0) Šimůnek et al. (2009).

Pressure head and flux conditions

h(x, t) = h0(t) at x = 0 or x = L (first type, Dirichlet)
−K (∂h/∂x + 1) = q0(t) at x = 0 or x = L (second type, Neumann)

with the prescribed head h0 and flux q0 (positive upward).

Atmospheric boundary condition

The potential fluid flux across the soil surface is determined by the external conditions (precipitation, potential evaporation), but the actual flux depends on the moisture conditions in the soil. The atmospheric boundary condition of Neuman et al. (1974) limits the absolute flux and the pressure head at the surface:

|−K (∂h/∂x + 1)| ≤ E at x = L
hA ≤ h ≤ hS at x = L

E is the maximum potential rate of infiltration or evaporation, hA the minimum pressure head allowed at the surface (−hCritA, the dry limit) and hS the maximum (hCritS, the ponding depth; 0 without a surface layer). When one of the head limits is reached, the boundary switches from a flux to a head condition: during evaporation from a dry surface the surface stays at hA and the actual evaporation falls below the potential one; during rain on a saturated surface the water ponds up to hS and the excess runs off. When the actual flux again falls within the limits, the boundary switches back.

With a surface layer, ponded water accumulates and its height h changes with the balance of precipitation, infiltration and evaporation:

−K (∂h/∂x + 1) = q0(t) − dh/dt at x = L

Free drainage

A unit vertical hydraulic gradient at the bottom (gravity flow):

∂h/∂x = 0, q = −K(h) at x = 0

suitable where the water table is far below the domain.

Deep drainage

The bottom flux depends on the position of the groundwater level Hopmans and Stricker (1989):

q(h) = −Aqh exp(Bqh |h − GWL0L|)

with the empirical parameters Aqh [L/T] and Bqh [1/L] and the reference position of the groundwater level GWL0L.

Seepage face

The bottom of a lysimeter or a column with a free outlet: as long as the bottom is unsaturated the flux is zero; once the pressure head reaches hSeep (usually 0) the head is fixed at hSeep and water flows out.

Horizontal drains

The bottom flux equals the discharge to parallel drains or ditches, computed from the groundwater level with the formulas of Hooghoudt (1940) (homogeneous profiles, equivalent depth) and Ernst (1962) (layered profiles with vertical, horizontal and radial resistances), as implemented in SWAP van Dam et al. (1997), with the drain spacing, the entrance resistance and the conductivities of the layers.

Time-variable conditions

Heads and fluxes at both boundaries, precipitation, evaporation and the dry limit can change with time (time-variable data). Time steps end at the times of the records.

Literature cited in this topic

  • Ernst, L. F. (1962). Grondwaterstromingen in de verzadigde zone en hun berekening bij aanwezigheid van horizontale evenwijdige open leidingen. Verslagen van Landbouwkundige Onderzoekingen 67(15), Pudoc, Wageningen, 189 pp.
  • Hooghoudt, S. B. (1940). Bijdragen tot de kennis van eenige natuurkundige grootheden van den grond, 7. Algemeene beschouwing van het probleem van de detail ontwatering en de infiltratie door middel van parallel loopende drains, greppels, slooten en kanalen. Verslagen van Landbouwkundige Onderzoekingen 46(14)B: 515–707.
  • Hopmans, J. W., and J. N. M. Stricker (1989). Stochastic analysis of soil water regime in a watershed. Journal of Hydrology 105(1–2): 57–84. doi:10.1016/0022-1694(89)90096-6
  • Neuman, S. P., R. A. Feddes, and E. Bresler (1974). Finite Element Simulation of Flow in Saturated-Unsaturated Soils Considering Water Uptake by Plants. Third Annual Report, Project No. ALO-SWC-77, Hydraulic Engineering Laboratory, Technion, Haifa, Israel.
  • van Dam, J. C., J. Huygen, J. G. Wesseling, R. A. Feddes, P. Kabat, P. E. V. van Walsum, P. Groenendijk, and C. A. van Diepen (1997). Theory of SWAP Version 2.0. Report 71, Department of Water Resources, Wageningen Agricultural University, Wageningen, 167 pp.
  • Šimůnek, J., M. Šejna, H. Saito, M. Sakai, and M. Th. van Genuchten (2009). The HYDRUS-1D Software Package for Simulating the One-Dimensional Movement of Water, Heat, and Multiple Solutes in Variably-Saturated Media, Version 4.08. HYDRUS Software Series 3, Department of Environmental Sciences, University of California Riverside, Riverside, CA, USA, 330 pp. https://www.pc-progress.com/en/Default.aspx?h1d-description

All sources: References.