Environmental Engineering Reference
In-Depth Information
F
⎩
⎨
⎧
⎭
⎬
⎫
u
F
+
[5.5]
∆
t
w
j
+
θ
where θ is the temporal discretization factor in the range between zero and 1, ∆
t
the
time increment and
j
the time station index. It is evident from [5.4] that the coupling
between the equations disappears in steady state conditions.
For the different procedures that can be used to solve the final system [5.5]
(displacement or pressure only formulations, uncoupled, monolithic and staggered
ones) and their numerical properties, we refer to [LEW 98].
5.1.3
. Possible numerical problems
Applications of numerical methods to real subsidence problems caused by water
withdrawal are sometimes limited by the size of the calculations and difficulty in
correctly modeling the domain in which the phenomenon takes place. These
problems are defined in three dimensions at a regional scale. Traditional techniques
involve simplified solutions: axially symmetric models, plane stress or plane strain
conditions are those usually used.
These assumptions are, however, too drastic. The assumption of rotational
symmetry is questionable, for instance, when analyzing an aquifer of variable
thickness or when pumping wells are irregularly distributed over the surface. In
similar cases a homogenized mathematical model is very useful: governing
equations are integrated (homogenized) over the thickness, i.e. the smallest
dimension of the domain. For the related numerical solution, we refer to [SIM 89a].
By using this procedure it is possible to analyze a three-dimensional situation, by
applying a two-dimensional model.
A second problem quite common in the numerical simulation of subsidence
concerns the way of analyzing zones a long way from the pumping sites. Subsidence
phenomena are almost always defined in a domain that can be assumed to be
unbounded, due to its lateral extent. The most frequently used technique within the
finite element method consists of locating the external boundary at a limited
distance from the center.
Boundary conditions that apply to infinity are then fixed at this fictitious
boundary. It is evident that the accuracy of the solution depends on the position of
this assumed boundary. Obviously, the solution improves with increasing lateral
extension of the modeled domain, but the number of unknowns in the discretized
model also increase. Further, because the problems can involve very long time
