Soil Water Flow - Transport in the Atmosphere-Vegetation-Soil Continuum - page 152

Question 4.10: A layered vertical column consists of 25 cm of a loam soil with k s =
5 cm d -1 overlain by 75 cm of a sandy soil with k s = 25 cm d -1 . On top of the column a
water layer of 10 cm is maintained, and the bottom is open to the atmosphere.
a) How large is the effective hydraulic conductivity of this column?
b) How large is the lux density q through the column?
c) How large is the pressure head at the loam-sand interface?
d) Draw a head diagram of the column, including H, h , and z .

4.5.2 Unsaturated Soil
In unsaturated soils air volumes are present and the water low channels are smaller
than those in saturated soil. The water phase is bounded partially by solid surfaces
and partially by an interface with the air phase. In contrast to the positive water pres-
sure found in saturated soils, the water pressure within the liquid phase is caused by
water elevation, attraction to solid surfaces, and the surface tension of the air-water
interface and is lower than zero. As the water content decreases, the liquid pressure
decreases and the water phase is constrained to narrower and more tortuous channels
(Jury et al., 1991 ).
In 1907, Edgar Buckingham proposed a modiication of Darcy's law (Eq. ( 4.13 ))
to describe low through unsaturated soil. This modiication rested primarily on two
assumptions:

1. The driving force for water low in isothermal, rigid, unsaturated soil containing no
solute membranes and zero air pressure potential is the sum of matrix and gravitational
potential.
2. The hydraulic conductivity of unsaturated soil is a function of the water content or matrix
potential.

In head units, the Buckingham-Darcy lux law may be expressed for vertical
low as:

∂ ( )
∂

hz
z

∂
∂ =−

H
z

∂
∂ +

h
z


 


 

qk =− () θ k () θ =− k () θ 1 (4.16)

Similar to saturated low, the lux density q (m d -1 ) is the water low per unit cross-
sectional area per unit time. Several points should be stressed about Eq. ( 4.16 ). First,
it is a differential equation that is written across an ininitely small thin layer of soil
over which h and k ( θ ) are constant. It may not be written across a inite layer of
soil unless the water content and matrix head of the layer are uniform. Second, the
derivative in Eq. ( 4.16 ) is a partial derivative, because in unsaturated soil h may be a
function of both z and t . The partial derivative ∂ h /∂ z implies that the derivative with
respect to z is taken at constant t ; it is the instantaneous value of the slope of h ( z ):