Environmental Engineering Reference
In-Depth Information
mechanics processes. Consideration of the long-term
storage of radioactive waste products has drawn con-
siderable attention to the solution of TMF models.
In this situation, it is possible that the total stresses
change with time, moisture flow occurs (i.e., liquid
and vapor), and heat is generated with time.
7. There are multiple processes associated with waste
rock management and the operation of heap leach
deposits. Heat may be generated with time as water
or an acid is allowed to move through the waste rock
material. Moisture flow may be in a liquid and/or vapor
form. Heat generation can also initiate significant
advective air flow through the rock material. There may
also be chemical reaction processes involved that result
in a change of mass with respect to time.
8. The management of municipal solid waste materials
has similarities to the processes mentioned for waste
rock materials. Physical processes involve air flow (and
gases such as methane), vapor water flow, liquid water
flow, and heat flow as well as chemical reactions that
result in a loss of mass from the system with time.
Each of the problems described above can be viewed as
the solution of a series of partial differential equations. The
PDE equations can be solved in a “coupled” or “uncoupled”
manner. The equations might also be solved in terms of one
process being dominant or several solutions may be man-
ually solved by iterating between independent processes.
While a coupled solution would appear to be the best type
of solution, the rigor associated with fully coupled solutions
might not be warranted.
the modeler. The solving process is illustrated in Fig. 15.1.
In this example, pore-water pressure changes are calculated
for the next time step. The solution of the water PDE may
involve a series of iterations to accommodate nonlineari-
ties associated with unsaturated hydraulic conductivity and
water storage soil parameters. The new pore-water pressures
are then imported into the stress analysis and subsequent
deformations (i.e., x- and y- coordinate strain values) are
computed. The solution process then reverts back to the
water flow PDE where consideration is given to moving
forward to the next time step.
When the solution of the water flow and stress-
deformation PDEs are uncoupled it is possible to proceed
with the entire water flow solution for the entire time span
of interest and then solve the stress-deformation analysis
using the pore-water pressures previously calculated. The
uncoupled analysis assumes that the pore-water pressure
calculation influences the stress-deformation analysis but
the reverse is not true (i.e., the stress-deformation results
do not affect the pore-water pressure analysis).
It is also possible to consider the solution to the swelling
soil problem in an uncoupled, iterative manner where there
is an iterative process between the water flow and stress-
deformation PDEs as shown in Fig. 15.2. This procedure
moves the solution toward a coupled type of solution but
may not meet all the requirements of a fully coupled solu-
tion. The iterative-type solution will likely provide a more
accurate representation of the nonlinear soil properties,
and as a result, the solution should be more accurate. The
additional modeling effort may or may not be warranted
in solving a particular problem at hand.
A coupled solution means that two or more PDEs are
being solved simultaneously, as shown in Fig. 15.3. A so-
called coupling matrix is placed between the PDEs being
solved. The coupling matrix ensures that the limiting condi-
tions associated with each PDE involved are satisfied before
moving on to the next time increment. Necessary iterations
are performed through the coupling matrix to ensure that
nonlinear soil property conditions are satisfied as well as
other conditions related to the effect of one PDE solution
on the solution of other PDEs.
There are other solution procedures that are sometimes
used when solving multiple PDEs. For example, the assump-
tion might be made that certain components of the coupling
15.2 COUPLED AND UNCOUPLED SOLUTIONS
The terms coupled and uncoupled have particular meaning
with respect to geotechnical engineering applications. The
swelling of a soil can be used to illustrate the meaning of
uncoupled and coupled processes. Let us consider a two-
dimensional cross section through a soil mass. There are two
stress-deformation PDEs and a water flow PDE involved.
Hydraulic head drives water flow and there are stress and
deformation state variables in stress-deformation analysis.
An uncoupled solution means that each PDE is solved in
an independent manner and information is transferred from
one PDE to the other in a hierarchical manner designated by
Uncoupled
Water partial differential equation
- A function of time t
- State variable, hydraulic head,
h w
Stress partial differential equations
- Stress state and deformation
variables
u w / (
ρ w g )
Y
=
+
(
s x - u a ), (
s y - u a ), ( u a - u w )
Figure 15.1 Modeling swelling of soil as uncoupled solution.
 
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