H(z,t) = soil matric potential
sient flow of water in a stable and uniform zone
S(z,t) = soil osmotic potential
of soil. The root system is considered as a diffuse
sink for water that permeates each depth of soil
layer uniformly, though not necessarily at the
Kw = soil hydraulic conductivity at
same root length density through the root zone.
Macroscopic models are derived from Darcy's
∆x = distance between roots at position
law, expressed in hydraulic head terms
where H (z,t) and S(z,t) are mea-
Jw = KwdH/dx
∆z = soil depth increment.
where Jw = flux of water (L3 L2 t1)
The major drawback of these models is that they
utilize a gross spatial average of matric and os-
dH = hydraulic head difference (L)
motic potentials and neglect the decrease in water
dx = distance along the flow path (L).
potential and change in the salt concentration at
the soil/root interface as well as the rhizosphere.
Incorporating the soil water diffusivity term,
Dw = Kw(dΨ/dθ), in eq 1
Microscopic models consider the diffusion of
Jw = Dw θ/ x.
water towards a single root (Gardner 1960). The
The vertical flow through a thin layer of soil
is dependent on root geometry, rooting length,
where water content changes with time and dis-
tance can be solved with Richards (1931) continu-
steady-state conditions, the rate of water uptake
per unit root length, qr, from the soil at a uniform
θ/ t = / z Kw H/ z + Kw/ z
qr = 2ΠKw[Hp Hs]/ln (rsoil/rroot)
= / z D θ/ z + Kw/ z.
where Kw = soil hydraulic conductivity
Equation 3 is further modified for water ex-
Hp = matric head in root epidermis or to-
traction by the plant roots
tal leaf water potential
Hs = matric head in soil surrounding root
θ/ t = θ/ z (Kw H/ z) + A(z,t)
rsoil = radius of soil cylinder surrounding
where A(z,t) is the root water extraction in refer-
rroot = root radius.
ence to soil depth, z, and time, t.
Several workers assumed that A(z,t) is a func-
Root water uptake of a specific soil volume can
tion of root activity. None, however, has assumed
be estimated by multiplying the qr with root
that root activity is a function of water potential
length density, Lv. The transpiration rate, T, is as-
difference between plant and soil, distance be-
sumed to be equal to uptake rate and can be cal-
tween the uniformly spaced roots, and some
measure of conductivity in the rootsoil system.
The root water uptake model of Nimah and
T = ∑ qri Lvi .
Hanks (1973) can be written as
A(z,t) = Hroots + [RRES(z)] H(z,t)
Taylor and Klepper (1975) proposed the follow-
θfinal = θinitial (qr) (Lv) (Hp Hs)
where Hroots = effective water potential in root at
where θfinal and θinitial are the volumetric water
contents at the end and beginning of a measuring
Rc = coefficient to account for longitu-
dinal resistance in xylem
Water uptake rates of species differ even when