Solving Saturated/Unsaturated Water Flow Problems - Unsaturated Soil Mechanics in Engineering Practice - page 401

in turn is a function of pore-water pressure, which is a
component of hydraulic head h w . The nonlinear equations
require several iterations to produce convergence. The water
coefficient of permeability at each node is revised to a new
constant value following each solution for the hydraulic
heads.
For the first iteration, the k wy values at all points can be set
equal to the saturated coefficient of permeability k s .The n

varying coefficient-of-permeability values throughout the
unsaturated soil column. The above analysis can also be
applied to the steady-state downward flow of water through
an unsaturated soil. Once again, the hydraulic head bound-
ary conditions at two points along the soil column must
be known.

−
2 linear equations can then be solved simultaneously using a
procedure such as the Gaussian elimination technique. The
computed hydraulic heads are used to calculate new values
for the water coefficient of permeability. The coefficient-
of-permeability values at each point must be in agreement
with the coefficient of permeability versus soil suction func-
tion. The revised coefficient-of-permeability values k wy are
then used for the second iteration. New hydraulic heads
are then computed for all depths. The iterative procedure is
repeated until there is no longer a significant change in the
computed hydraulic heads and the computed coefficients of
permeability.
Figure 8.34 illustrates typical distributions for the pore-
water pressure and the hydraulic head along the unsaturated
soil column. Flow is occurring under steady-state evap-
oration conditions. The nonlinearity of the flow equation
results in a nonlinear distribution of the hydraulic head and
the pore-water pressure head. The equipotential lines are not
equally spaced along the column. This is different from the
uniformly spaced equipotential lines for a homogeneous,
saturated soil column. The difference is the result of the

8.3.3 Flux Boundary Conditions Applied
Infiltration into an unsaturated soil column is another
example which can be used to illustrate the solution of the
nonlinear differential seepage equation (Fig. 8.35). Steady-
state infiltration can be established as a result of sprinkling
irrigation. Let us assume a constant downward water flux
q wy which is less than the saturated coefficient of perme-
ability of the soil. Steady-state flow can be described using
Eq. 7.30. The hydraulic head distribution can be determined
by solving the finite difference form of the steady-state
flow equation (i.e., Eq. 8.37). The hydraulic head boundary
condition at the ground surface is a computed value.
However, the water flux q wy is known and is constant
throughout the soil column for steady-state conditions.
The soil column can be discretized into n nodal points
with an equal spacing y (Fig. 8.35). The water flux at
point i can be expressed in terms of the hydraulic heads at
points i + 1 and i − 1 using Darcy's law:

h w (i + 1 ) − h w (i − 1 )

2 y

q wy =− k wy (i) A (8.38)

Figure 8.35 One-dimensional steady-state water flow through an unsaturated soil with a flux
boundary condition.