Environmetal Soil Properties and Behaviour

117Soil\u2013Water Systems
The total soil\u2013water potential \u3c8 is generally considered to be given as 
\u3c8 = \u3c8m + \u3c8\u3c0 + \u3c8g in analyses of moisture transfer in soils, where \u3c8m is the 
matric potential, \u3c8\u3c0 is the osmotic potential or solute potential, and \u3c8g is 
the gravitational potential that is ignored in the case of horizontal flow, as 
shown in Figure 3.21. A detailed discussion of the Darcy coefficient will 
be given in Chapter 7. When one expresses each potential as the energy 
per unit weight of soil\u2013water, that is, hydraulic head, the commonly used 
one-dimensional equation with the gravitational term for analyses of water 
transfer is described as follows:
 
v k
x
kw= \u2212
\u2202
\u2202 \u2212( ) ( )\u3b8
\u3c8 \u3b8 (3.24)
where \u3c8w is the water potential, that is, \u3c8w = \u3c8m + \u3c8\u3c0. In the case of flow in 
granular soils, the osmotic potential can be ignored, and the water potential 
is expressed only in terms of the matric potential, that is, \u3c8w = \u3c8m .
There are at least three types of unsaturated fluid flow in soils. These relate 
directly to what happens to the soil during or as a result of the unsaturated 
Distance x from water source
Vo
lu
m
et
ric
 w
at
er
 co
nt
en
t \u3b8
0
\u3b8ini
\u3b8sat
Soil column
Transmission zone
Wetting zone
Wetting front
Wetting front
profile
Quasi-saturated
water content
\u3b8sat = Saturated
Double mariotte flask
Ai
r e
nt
ry
 at
 el
bo
w 
in
let
 
FIguRE 3.21
Characteristics of a wetting front profile. The shaded area represents the volumetric water 
content in the zone behind the wetting front. The double Mariotte flask provides a source of 
water with a constant head defined by the position of the elbow inlet. The flask can be raised 
or lowered to provide different constant heads, from negative to positive.
118 Environmental Soil Properties and Behaviour
flow, and can be classified by tracking the status of the fabric of the soil sub-
ject to the unsaturated flow. They are
\u2022	 No change in soil volume and no change in soil fabric during and as 
a result of unsaturated flow, analogous to a rigid porous block.
\u2022	 No change in soil volume but change in soil fabric (i.e., change in 
pore geometry) during and as a result of unsaturated flow. This is 
the most likely case for nonswelling soils. It can also be the case for 
swelling soils under confinement.
\u2022	 Change in soil volume and change in soil fabric, that is, change in 
pore geometry and porosity during and as a result of unsaturated 
flow. This case will be considered in detail in Chapter 4.
For the case of no change in soil volume and little-to-no change in pore 
geometry and porosity, unsaturated fluid flow in such soils can be generally 
determined in terms of changes in the volumetric water content \u3b8 at a point 
by the mass conservation law. The equation of continuity, which states that 
the flow of water into and out of a unit volume of soil is equal to the rate of 
change of the volumetric water content, is given as:
 
\u2202
\u2202 = \u2212
\u2202
\u2202
\u3b8
t
v
x 
 (3.25)
where t = time. Combining Equations (3.24) and (3.25), one obtains the equa-
tion with the gravitational term for water movement in water uptake (includ-
ing the film flow) in the absence of external forces such as overburden forces, 
as follows:
 
\u2202
\u2202 =
\u2202
\u2202
\u2202
\u2202
\uf8eb
\uf8ed\uf8ec
\uf8f6
\uf8f8\uf8f7 +
\u2202
\u2202
\u3b8 \u3b8 \u3c8 \u3b8
t x
k
x
k
x
w( )
( )
 (3.26)
or
 
C
t x
k
x
k
xw
w
w
w w\u3c8 \u3c8 \u3c8 \u3c8 \u3c8( ) \u2202\u2202 = \u2202\u2202 \u2202\u2202\uf8eb\uf8ed\uf8ec
\uf8f6
\uf8f8\uf8f7 +
\u2202
\u2202( )
( )
 (3.27)
where C(\u3c8w) refers to water capacity, that is, C(\u3c8w) = d\u3b8/d\u3c8w.
In the case of fully saturated flow, since the potential \u3c8 responsible for flow 
consists primarily of the pressure potential \u3c8p and gravitational potential \u3c8g, 
that is, \u3c8 = \u3c8p + \u3c8g, one obtains:
 
\u2202
\u2202 =
\u2202
\u2202
\u2202
\u2202
\uf8eb
\uf8ed\uf8ec
\uf8f6
\uf8f8\uf8f7 +
\u2202
\u2202
n
t x
k n
x
k n
x
p( )
( )\u3c8
 (3.28)
119Soil\u2013Water Systems
where n refers to porosity. When porosity n and permeability coefficient k 
(n) are constant, Equation (3.28) takes the form of a Laplace equation with 
constant permeability coefficient k:
 
\u2202
\u2202
\u2202
\u2202
\uf8eb
\uf8ed\uf8ec
\uf8f6
\uf8f8\uf8f7 =x k x
p\u3c8 0
 
Introducing the water diffusivity coefficient D(\u3b8) = k(\u3b8)(d\u3c8w /d\u3b8) into Equation 
(3.26), one obtains the commonly cited equation for partly saturated flow as 
follows:
 
\u2202
\u2202 =
\u2202
\u2202
\u2202
\u2202
\uf8eb
\uf8ed\uf8ec
\uf8f6
\uf8f8\uf8f7 +
\u2202
\u2202
\u3b8 \u3b8 \u3b8 \u3b8
t x
D
x
k
x
( )
( )
 (3.29)
Equation (3.29) is used because water content gradients are sometime easier 
to measure and also because water flow is more easily solved with diffusiv-
ity rather than conductivity.
The gravitational term is ignored for horizontal flow (Figure 3.21), and also 
in the case of flow in clays because the osmotic and matric potentials are 
dominant in comparison to the gravitational potential. In this case, the fol-
lowing relationship is commonly used for analysis of water transport:
 
\u2202
\u2202 =
\u2202
\u2202
\u2202
\u2202
\uf8eb
\uf8ed\uf8ec
\uf8f6
\uf8f8\uf8f7
\u3b8 \u3b8 \u3b8
t x
D
x
( ) (3.30)
The theoretical moisture profiles predicted from an evaluation of the relation-
ship given in Equation (3.30) can be seen in Figure 3.22, with initial and bound-
ary conditions: \u3b8 = \u3b8i (constant) at x > 0, t = 0 and \u3b8 = \u3b8sat. at x = 0, t > 0, where \u3b8i is 
the initial water content and \u3b8sat is the saturated water content. In reality, there 
will always be some air voids trapped in the void spaces in \u201csaturated\u201d soils at 
soil surface and hence, the quasi-saturated soil condition is generally assumed 
to be the fully saturated condition for most practical applications.
Time-wetting studies show that a linear relationship exists between the 
wetting front distance x from the source and the square root of time (\u221at ) 
required for the wet front to reach x, as shown, for example, by tests on a 
kaolinite (kaolin clay) at three different densities under the no-volume change 
condition (Figure 3.23). Determination of water entry and the characteristics 
of water transfer in soils is facilitated by using a system that allows one to 
observe and record the rate of advance of the wetting front into the soil such 
as the system shown in Figure  3.21. As we have indicated previously, the 
advantage of using the movable double Mariotte water source system shown 
120 Environmental Soil Properties and Behaviour
0 10 20 30 40 50
4
8
12
16
20
24
t (minutes)
Di
sta
nc
e o
f W
et
tin
g F
ro
nt
 x 
fro
m
 W
at
er
 S
ou
rc
e 1.2 g/cm3
1.3 g/cm3
1.4 g/cm3
FIguRE 3.23
Wetting front advance, x, in relation to square root of time t for a kaolinite clay at three differ-
ent densities, 1.2, 1.3, and 1.4 g/cm3.
D decreases as \u3b8
increases; D = be\u2013c\u3b8
Distance from Water Source, x
Vo
lu
m
et
ric
 W
at
er
 C
on
te
nt
, \u3b8 D increases faster than \u3b8
increases; D = bec\u3b8
D increases linearly 
with \u3b8; D = a\u3b8
D increases slower
than \u3b8 increases; D = b(1\u2013ec\u3b8)
D = Constant
FIguRE 3.22
Theoretical moisture profiles predicted from evaluation of the diffusion equation (Equation 
3.30), showing the nature of the D relationships relative to the profiles obtained.
121Soil\u2013Water Systems
in the left-hand side of Figure 3.21 lies in its ability to impose both positive 
and negative hydraulic heads to the sample. The tops of the tubes are sealed. 
This means that exposure to atmosphere is only at the elbow junction at the 
bottom end of the tubes. By this means, a constant head can be maintained at 
the level of the elbow atmospheric outlets. The zero hydraulic head situation 
shown in the illustration depicted in Figure 3.21 is best used to test the water 
uptake capability of the test sample, that is, the capability of the soil\u2013water 
potential \u3c8w (in soil engineering terminology, this would be soil suction S) to 
draw water into the