Ground Surface Moisture Flux Boundary Conditions - Unsaturated Soil Mechanics in Engineering Practice - page 325

6.4.5 Convergence Difficulties
The convergence of the highly nonlinear partial differential
moisture flow equation is the most pressing challenge facing
soil cover modelers (Shackelford, 2005). However, signif-
icant advances have been made in resolving problems of
nonconvergence. There are two conditions that need to be
satisfied in order to ensure that the correct modeling solu-
tion is attained. First, the solution of the PDE must converge
for every time step of the lengthy design modeling period
(e.g., 10 years). Second, the modeler needs assurance that
the solution has converged to the correct solution. There are
mathematical criteria that can be used to ensure that both of
the mentioned criteria are met while solving the PDEs.
It is possible to plot tornado diagrams of the profiles of
minimum and maximum values of soil suction or saturation
levels on an annual basis. The tornado diagrams will be
quite consistent from one year to the next once steady-state
conditions have been reached.
The most successful solution to date has involved the use
of the AGR technique (Oden, 1989; Yeh, 2000). When using
this technique, the finite element mesh is continually refined,
as necessary, in order to meet the conditions that promote
convergence to the correct solution.
A simple finite difference formulation can be used to
illustrate the adaptive time-and-space technique for solv-
ing nonlinear PDEs. Figure 6.58 illustrates the finite differ-
ence solution of the one-dimensional consolidation equation.
When solving the finite difference equation, it is possible to
define a variable β to ensure convergence to the correct solu-
tion (i.e., similar to the coefficient of consolidation times the
time step and divided by the square of the spatial distance
between nodes). In order for the finite difference solution to
be correct, the defined variable β needs to be kept less than
a value of 0.5. If β becomes greater than 0.5, errors can be
incurred, as described in Fig. 6.58.
The coefficient of consolidation of a soil can change as
the coefficient of permeability of the soil changes. Changes
can be substantial and the β variable can readily go outside

acceptable limits and cause the solution to be in error. The
problem should be solved by changing either the time inter-
val or the spatial interval in order to bring the β vari-
able into an acceptable range. It is easy to see how this
can be accomplished when using a finite difference solu-
tion. When performing a finite element solution, the spatial
changes need to be accomplished by changing the size of
the finite elements. Changing element sizes while the solu-
tion is underway is a challenge, but this technique has been
incorporated into the solvers of all finite element solutions
used by SoilVision Systems (Fredlund and Thode, 2010).
The AGR technique appears to generally meet the condi-
tions necessary for convergence to the correct solution when
solving highly nonlinear PDEs.

6.4.6 Verification of Cover Designs
Field monitoring is an essential part of quality assurance
(QA) for cover systems. Measurements need to be taken to
ensure that the cover system is performing as anticipated
from the design. Field monitoring can provide verification
that the cover performance is satisfactory.
A typical field monitoring system for a cover system may
involve (i) the installation of a lysimeter, (ii) the installation
of a weather station, (iii) soil suction measurements within
the soil cover, and (iv) water content measurements within
the soil cover (Fig. 6.59). Equipment is available commer-
cially for each of the mentioned measurements. All types
of instrumentation do not necessarily need to be installed in
all cover systems. Rather, there is often a hierarchy on the
instrumentation monitoring items. For example, a lysime-
ter might be considered to be the most important item of
instrumentation.
Design details for the lysimeter are important in order to
obtain accurate measurements of infiltration. The lysimeter
may need to be on the order of 10-20m in length, of proper
depth, and filled with appropriate material in order to yield
accurate results (Benson, 2002).
Weather stations are reasonably priced and can be placed
at the site of a proposed cover system. While the cost of the
weather station is modest, its maintenance and servicing may
entail considerable expenditure. It must also be remembered
that visual field inspections and reporting need to be an
ongoing part of the monitoring process.

U 0, k +1 − U 0, k

U 2, k + U 4, k − 2 U 0, k

= =

C v

Δ z 2 Δ t

Δ t C v

β =

r

z 2 Δ

U 2, k

+ For stability β must be less than 0.5

6.4.7 Uniqueness of Cover Design Procedure
There are many assumptions that need to be made as part of
the design procedure for a cover system. The soil conditions
can change with time due to the effects of weathering and
freeze-thaw conditions with the result that the soil properties
become far from the initially measured or assumed values.
The changes can prove to differ by orders of magnitude
from initial compacted or placement conditions. This does
not make a realistic design impossible but simply shows that
much greater care and detail must be given to the assessment
of the unsaturated soil properties.

=

Crandell (1956) noted

1-2 r l

1. truncation errors and
2. round-off errors

U 0, k U 0, k +1

r

- that led to convergence to the
wrong solution

r

U 4, k

The space t increments

affect both of these errors

Δ z and time Δ

Δ t

Time

Schematic form of equation

Figure 6.58 Controlling convergence through use of adjustment
of the time-and-space steps.