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Theory and equations

Root water uptake

Sink term

The sink term S of the Richards equation is the volume of water removed per unit time from a unit volume of soil by plant roots. Feddes et al. (1978) defined it as

S(h) = α(h) Sp

where the water stress response function α(h) (0 ≤ α ≤ 1) reduces the potential uptake Sp. The potential uptake is the potential transpiration Tp distributed over the root zone with the normalized root distribution b(x):

Sp = b(x) Tp, b(x) = b'(x) / ∫LR b'(x) dx
S(h, hφ, x) = α(h, hφ, x) b(x) Tp

b'(x) is the root distribution of the soil profile (only relative values matter) and LR the root zone. The actual transpiration is the integral of S over the root zone, Ta = ∫ S dx.

Water stress response functions

Feddes et al. (1978) Feddes et al. (1978): piecewise linear. Water uptake is zero close to saturation (h > h1 = P0, oxygen deficiency) and below the wilting point (h < h4 = P3); it is optimal between h2 = POpt and h3, and varies linearly in between. h3 depends on the potential transpiration: h3 = P2H for Tp ≥ r2H, P2L for Tp ≤ r2L, and is interpolated linearly in between.

S-shape (van Genuchten 1987) van Genuchten (1987):

α(h) = 1 / [1 + (h/h50)p]

with the pressure head h50 at which the uptake is halved and the exponent p (≈ 3). It does not account for oxygen stress.

Osmotic (salinity) stress

The osmotic head of the soil solution is hφ = Σ aj cj with the osmotic coefficients aj of the solutes. Following van Genuchten (1987):

  • additive: the osmotic head adds to the pressure head, α = α(h + hφ);
  • multiplicative: α = α(h) αφ(hφ), with an S-shaped αφ = 1/[1 + (hφ/hφ50)p] or the threshold–slope function of Maas and Hoffman (1977); Maas (1990): no reduction up to a threshold, then a linear decrease with the slope.

Compensated root water uptake

Roots may take up more water from less stressed parts of the root zone to compensate for stressed parts. With the water stress index ω (the ratio of actual to potential transpiration without compensation) and the critical stress index ωc Jarvis (1989); Šimůnek and Hopmans (2009):

ω = ∫LR α(h, hφ, x) b(x) dx
S = α(h, hφ, x) b(x) Tp / max(ω, ωc)

For ω ≥ ωc the uptake is fully compensated (Ta = Tp); below, Ta = Tp ω/ωc. ωc = 1 means no compensation.

Root distribution

The root distribution b'(x) is set on the page Soil profile, constant, linear, or with the functions of Hoffman and van Genuchten (1983) (constant in the upper 20 % of the rooting depth Lr, then linear to zero at Lr) and Vrugt et al. (2001):

b(z) = (1 − z/zm) exp[ −(pz/zm) |z − z| ]

with the maximum rooting depth zm, the depth of maximum root density z and the shape parameter pz (above z; p = 1 below).

Root growth

With root growth the root distribution follows the Hoffman–van Genuchten function of the current rooting depth LR(t), which is read from the time-variable data, interpolated from a table, or given by the Verhulst–Pearl logistic growth function Šimůnek and Suarez (1993):

LR(t) = Lm L0 / [L0 + (Lm − L0) e−r t]
r = −1/(tm − t0) · ln[ L0(Lm − Lmed) / (Lmed(Lm − L0)) ]

with the initial rooting depth L0 at the sowing time t0, the maximum rooting depth Lm and the known rooting depth Lmed at the time tm; t is the time since sowing. Before sowing and after harvest there are no roots; a period repeats the cycle.

See also: root solute uptake · page Root water and solute uptake

Literature cited in this topic

  • Feddes, R. A., P. J. Kowalik, and H. Zaradny (1978). Simulation of Field Water Use and Crop Yield. Simulation Monographs, Pudoc, Wageningen, 189 pp.
  • Hoffman, G. J., and M. Th. van Genuchten (1983). Soil properties for efficient use of water and salinity control. In: H. M. Taylor, W. R. Jordan, and T. R. Sinclair (eds.), Limitations to Efficient Water Use in Crop Production, ASA, CSSA, SSSA, Madison, WI, pp. 73–85.
  • Jarvis, N. J. (1989). A simple empirical model of root water uptake. Journal of Hydrology 107(1–4): 57–72.
  • Maas, E. V. (1990). Crop salt tolerance. In: K. K. Tanji (ed.), Agricultural Salinity Assessment and Management, ASCE Manuals and Reports on Engineering Practice No. 71, American Society of Civil Engineers, New York, pp. 262–304.
  • Maas, E. V., and G. J. Hoffman (1977). Crop salt tolerance – current assessment. Journal of the Irrigation and Drainage Division, ASCE 103(IR2): 115–134.
  • van Genuchten, M. Th. (1987). A Numerical Model for Water and Solute Movement in and below the Root Zone. Research Report No. 121, U.S. Salinity Laboratory, USDA-ARS, Riverside, CA.
  • Vrugt, J. A., M. T. van Wijk, J. W. Hopmans, and J. Šimůnek (2001). One-, two-, and three-dimensional root water uptake functions for transient modeling. Water Resources Research 37(10): 2457–2470. doi:10.1029/2000WR000027
  • Šimůnek, J., and D. L. Suarez (1993). Modeling of carbon dioxide transport and production in soil: 1. Model development. Water Resources Research 29(2): 487–497. doi:10.1029/92WR02225
  • Šimůnek, J., and J. W. Hopmans (2009). Modeling compensated root water and nutrient uptake. Ecological Modelling 220(4): 505–521. doi:10.1016/j.ecolmodel.2008.11.004

All sources: References.