Environmental Engineering Reference
In-Depth Information
This type of equation can be solved by using the method of characteristics. The
characteristic lines are current lines that are perfectly followed by the pollutant.
The resolution of analytical equations is often impossible, usually because
Darcy's velocity field is itself of a certain complexity. It is therefore necessary to
proceed numerically, often by using the finite differences method, which appears to
be an effective tool for this type of problem [BEA 87]. The motion of a particle is
simply given by the equation:
∆
x
=
vx
()
i
∆
t
[13.31]
So the particle trajectory (
particle tracking
) can be followed step by step. The
choice of the coordinate
x
i
is essential in this type of method. If we adopt the
departure coordinate
x
0
we use Euler's method, whose convergence is conditional
[ZIE 89, ZIE 95]. It is then necessary to proceed by using a sufficient number of
small steps, especially in areas with strong gradients (wells, springs, etc.). Midpoint
methods or higher-order methods are generally much more efficient, but they
require iterating. This method has been introduced in several commercial codes for
calculating flows in porous media, often based on the finite differences method.
13.6. Finite elements modeling of the problem with advection and diffusion
13.6.1.
Galerkin's method
Consider now a finite element discretization of the full problem (advection plus
dispersion), but under steady state condition (without storage). The equations to
solve are very similar to those of flow in porous media. If the advection velocity
becomes zero, the equations of the two problems are similar. It is therefore
interesting to develop elements equivalent to those that are used for the flows. Let us
build up isoparametric elements:
x =
Nx
C =
NC
L
L
[13.32]
L
L
where
x
are the coordinates,
L
is an index indicating the node number and
C
L
the nodal
concentrations. Let us adopt Galerkin's method; weighting and interpolation functions
are identical:
WN
=
[13.33]
L
L
