Page 213 - Soil and water contamination, 2nd edition

200 Soil and Water Contamination

Box 11.III (continued) Physics of groundwater and soil water flow

In the case of saturated groundwater flow, the right-hand side of Equation (11.III.4) can
be rewritten as:
1 n h
w
S s (11.IIIf)
t t
w
-1
where S = the specific storage [L ], which is a proportionality constant relating the
s
changes in water volume per unit change in hydraulic head. Thus, a change in water flow
in space results in a change in the hydraulic head. In the case of an unconfined aquifer,
the specific storage is equal to the part of the pore volume that can yet be occupied by
water. This is often somewhat less than the porosity, because the soil matrix is rarely
completely dry. In the case of a confined aquifer, the specific storage is much smaller and
depends on the changes in pore volume as result of changes in hydraulic head, and thus
on the compressibility of the rock matrix.
In the case of unsaturated soil water flow, the right-hand side of Equation (11.III.4)
equals:
1 w n
(11.IIIg)
t t
w
which implies that a change in water flow in space results in a change in moisture content
of the soil in time.

The momentum equation is represented by Darcy’s law, which relates the flow rate to the
hydraulic gradient. The one-dimensional form of Darcy’s law for flow in direction x reads:
h
q K (11.IIIh)
x x
x
-1
where K = the hydraulic conductivity [L T ], which depends on the grain size distribution
x
of the soil matrix and varies over several orders of magnitude. The saturated hydraulic
-9
-1
-2
-5
-1
-1
conductivity is in the order of 10 m s for gravel, 10 m s for sand, and 10 m s for
clay. In unsaturated conditions, the air in the soil greatly reduces the connectivity of the
pores and hence also greatly reduces the hydraulic conductivity. Accordingly, the hydraulic
conductivity is a function of soil moisture (and thus of the pressure head).
The combination of Darcy’s law (Equation 11.IIIh) and the continuity equation
(Equation 11.IIId) yields:
h h h 1 n
K K K w (11.IIIi)
x x x y y y z z z w t
This equation has numerous analytical solutions for various simplified situations for both
groundwater and unsaturated flow. Because unsaturated flow largely consists of vertical
drainage driven by gravity forces, unsaturated models are usually one-dimensional. For
more complex distributed models, the equations are mostly solved numerically using
available model codes, such as MODFLOW (McDonald and Harbaugh, 1988), a widely
used three-dimensional finite-difference groundwater flow model.

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