Theory of Water Flow through Unsaturated Soils - Unsaturated Soil Mechanics in Engineering Practice - page 351

7.5 PARTIAL DIFFERENTIAL EQUATIONS
FOR TRANSIENT SEEPAGE

However, practical engineering problems are generally
solved with respect to the x-, y-, and z -Cartesian coordinate
system. As a result it is necessary to perform a transfor-
mation of the major and minor coefficients of permeability
into the Cartesian coordinate system.

The steady-state water flow partial differential equations can
be expanded to include transient or unsteady-state formula-
tions. It is necessary to take changes in water storage into
account when considering a transient analysis. A transient
analysis allows the geotechnical engineer to give considera-
tion to a wider range of possible questions that can be asked.
The computational power of the computer also allows the
engineer to perform parametric-type analyses where many
of the questions involve possible changes that might occur
over time.
Many situations are encountered in engineering practice
where the ground surface is a moisture flux boundary. In
other words, the climatic conditions at a site give rise to a
varying moisture flux at the ground surface. The conversion
of thermal (and other variable) boundary conditions into an
actual evaporative flux becomes of significant interest in
solving geotechnical engineering problems. Moisture flux
boundary conditions open the way for the consideration of a
variety of new design possibilities (e.g., design of soil cover
systems). This topic has devoted one chapter to the assess-
ment of the moisture flux conditions at the ground surface.
Unsteady-state or transient formulations are required when
solving problems where there is a net moisture flux imposed
at the ground surface as a boundary condition changing
with time.

7.5.2 Unsteady-State Seepage in Anisotropic Soil
The term anisotropy is used to refer to soil conditions where
the coefficient of permeability varies with respect to direction.
The coefficients of permeability in the x - and y -directions
are assumed to be different at any point in the soil mass.
The conditions associated with anisotropy will be first dis-
cussed followed by the derivation of the transient saturated-
unsaturated soil seepage. The coefficient of permeability also
varies with respect to space (i.e., heterogeneity) due to vari-
ations in matric suction throughout the soil mass.
Unsteady-state water flow through an anisotropic soil is
analyzed by considering the continuity of the water phase.
The pore-air pressure is assumed to remain constant with
time (i.e., ∂u a /∂t 0). According to Freeze and Cherry

(1979), “The single-phase approach to unsaturated flow
leads to techniques of analysis that are accurate enough for
almost all practical purposes, but there are some unsaturated
flow problems where the multiphase flow of air and water
must be considered.” The water phase partial differential
equation can be obtained in a similar manner to that used
for the formulation of water flow through an isotropic soil.

=

7.5.2.1 Anisotropic Permeability
Let us consider the general case of a variation in the coef-
ficient of permeability with respect to space (heterogeneity)
and direction (anisotropy) in an unsaturated soil, as illus-
trated in Fig. 7.26. At a particular point, the largest or major
coefficient of permeability, k w 1 , occurs in the direction s 1 ,
which is inclined at an angle α to the x -axis (i.e., horizontal).
The smallest coefficient of permeability is in a perpendicular
direction to the largest permeability (i.e., in the direction s 2 )

7.5.1 Uncoupled Two-Dimensional, Unsteady-State
Formulations
Water flow through an earth dam during the filling of its
reservoir is an example of two-dimensional, unsteady-state
or transient flow. Eventually, the flow of water through the
dam will reach a steady-state condition. Subsequent fluctua-
tions of water level in the reservoir will again initiate further
unsteady-state water flow conditions. Infiltration and evapo-
ration cause a continuously changing flow regime. Transient
analyses of water flow are strongly influenced by conditions
in the unsaturated zone (Freeze, 1971).
The following uncoupled formulation satisfies the conti-
nuity equation for the water phase. Analyses that take the
interaction between the fluid flows (e.g., water and air) and the
soil structure equilibrium into consideration are part of com-
bined analyses and are given consideration in Chapter 16.
The derivation of the partial differential equations for two-
dimensional, unsteady-state water flow in two directions (i.e.,
the x - and y -directions) is first considered in this chapter. Fluid
flow in the third direction (i.e., the z -direction) is assumed to
be negligible.
Consideration is first given to the handling of anisotropic
permeability properties before deriving the equations for
transient water flow through saturated-unsaturated soil
systems. Water flow properties are generally defined with
respect

k w 2

k w 1

α

B

Heterogeneity:

k w 2

k w 1

α

k wi at A = k wi at B
Where i = 1, 2, x or y

α

Direction of minor
permeability

A

Anisotropy:

s 2

y

v wy k w 1

k w 2

k w 1
k w 2

v w 2

A = B = constant

1

≠

s 1

α

Direction of major
permeability

x

v wx

Figure 7.26 Consideration of the orientation of the principal
coefficients of permeability in heterogeneous and anisotropic unsat-
urated soil.

to major and minor coefficients of permeability.