Theory of Water Flow through Unsaturated Soils - Unsaturated Soil Mechanics in Engineering Practice - page 346

classification is based on the pattern of permeability vari-
ation. A soil is called heterogeneous, isotropic if the coeffi-
cient of permeability in the x -direction, k x , is equal to the
coefficient of permeability in the y -direction at any point
within the soil mass (i.e., k x =

function for a typical soil. There is strong nonlinearity to
the water storage function at the point of inflection on the
SWCC. The nonlinearity of the water storage soil property
can give rise to numerical instability and errors in computing
water balances if not properly taken into account during
the solution of the seepage partial differential equation.
There is also an independent water storage curve for the
drying and wetting processes since there is hysteresis in
the SWCC.

k y at B )
(Fig. 7.21a). However, the magnitude of the coefficient of
permeability in either direction can vary from point A to
point B since its magnitude depends on matric suction. The
variation in the coefficient of permeability with respect to
matric suction is often assumed to follow a single-valued
functional relationship even though it is known that the
soil is hysteretic with respect to the wetting and drying
processes.

k y at A and k x =

7.4.5 One-Dimensional Flow in Unsaturated Soil
There are numerous situations where the water flow is
predominantly in one direction. Let us consider a covered
ground surface with the water table located at a specified
depth, as shown in Fig. 7.22. The surface cover prevents
the vertical flow of water from the ground surface.
The pore-water pressures are negative with respect to the
water table under static equilibrium conditions. The negative
pore-water pressure head has a linear distribution with depth
(i.e., line 1). Its magnitude is equal to the gravitational head
(i.e., elevation head) measured relative to the water table. In
other words, the hydraulic head (i.e., the gravitational head
plus the pore-water pressure head) is zero throughout the
soil profile. This means that the change in head with respect
to depth, and likewise the hydraulic gradient, is equal to
zero. Therefore, there can be no flow of water in the vertical
direction (i.e., q wy =

7.4.3 Heterogeneous, Anisotropic Steady-State Seepage
Figure 7.21b illustrates the heterogeneous, anisotropic case.
The ratio of the coefficient of permeability in the x -direction,
k x , to the coefficient of permeability in the y -direction, k y ,
is a constant at any point (i.e., k x /k y at A

=
a constant not equal to unity). The magnitude of the
coefficients of permeability k x and k y can also vary with
matric suction from one location to another, but their ratio
is assumed to remain constant. Anisotropic conditions can
also be oriented in any two perpendicular directions.
The third case of coefficients of permeability occurs when
there is a continuous variation in the coefficient of perme-
ability (Fig. 7.21c). The permeability ratio k x /k y may not
be a constant from one location to another (i.e., k x /k y at
A

= k x /k y at B

0).
The soil surface would be exposed to the environment if
the covering were removed from the ground surface. Envi-
ronmental changes could produce flow in a vertical direction
and subsequently alter the negative pore-water pressure head
profile. Steady-state evaporation would cause the pore-water
pressures to become more negative, as illustrated by line 2
in Fig. 7.22. Hydraulic head changes to a negative value
since the gravitational head remains constant. The hydraulic
head has a nonlinear distribution from a zero value at the
water table to a more negative value at ground surface. An
assumption is made that the water table remains at a constant
elevation for this example. The nonlinearity of the hydraulic
head profile is caused by the spatial variation in the coefficient
of permeability. Water flows in the direction of the decreasing
hydraulic head, which means that water flows from the water
table upward to the ground surface. The upward constant
flux of water is designated as being positive for steady-state
evaporation.
Steady-state infiltration causes a downward water flow.
The negative pore-water pressure increases from the static
equilibrium condition. This condition is indicated by line 3
in Fig. 7.22. The hydraulic head profile starts with a pos-
itive value at ground surface and decreases to zero at the
water table. Therefore, water flows downward with a con-
stant negative flux for steady-state infiltration.
The above one-dimensional illustration of flow involves the
application of moisture flux boundary conditions. A steady
rate of evaporation or infiltration is used as the boundary

k x /k y at B ), and different directions may have different
permeability functions.
The following steady-state seepage formulations deal with
heterogeneous, isotropic and heterogeneous, anisotropic
cases. The casewhere there is a continuous variation in perme-
ability with space is not considered. All steady-state seepage
analyses assume that the pore-air pressure has reached a
constant equilibrium value. When the equilibrium pore-air
pressure is atmospheric, the water coefficient-of-permeability
function with respect

=

u w ,
has the same absolute value as the permeability function with

to matric suction, k w u a −

respect to pore-water pressure, k w − u w .

7.4.4 Ability of Unsaturated Soil to Store
and Release Water
The simulation of transient flow processes requires a char-
acterization of a water storage property that changes with
the degree of saturation (or volumetric water content) of
the soil. The water storage property is part of the partial
differential equation describing a transient process.
The water storage soil property associated with water flow
through an unsaturated soil is given the symbol m 2 and is
equal to the arithmetic slope of the (volumetric water content)
SWCC. The differentiation of the mathematical equation
for the SWCC serves as a measure of the water storage soil
property. Figure 7.14 showed the form of the water storage