MULTILAYERED SYSTEMS
General Considerations
Earth scientists and geotechnical engineers are increasingly
challenged to solve environmental problems related to waste
disposal facilities and remediations of polluted sites by
pollutants containment systems.
The design of such structures typically involves
some form of "barrier" that separate the contaminants from
the general sub-soil and groundwater systems.
The primary objective of a barrier is to minimize as much
as possible the advective and dispersive transport rate of
the pollutants in solution.
The knowledge of the parameters that characterize the different
types of mineral barriers versus different types of pollutants is of outstanding importance.
While their assessment in laboratory tests doesn't pose any
particular problem from a practical point of view, the interpretation
of the results of the experiments is difficult, because the model
that describes correctly the transport properties of the contaminant
in porous media can be solved analytically only in
very special cases. The validity of these analytical solutions is
usually extrapolated to explain the results of the experiments,
very often with poor agreement.
The most important factor causing these discrepancies is that the
analytical solutions available in the literature are obtained
by applying right at the top of the sample two kinds of boundary
conditions, given by continuity of concentration (so called first type) and
continuity of flux (so called third type). In addition, a finite or a
semi-infinite system can be considered, resulting in four different
situations. In fact,
as suggested in (1) and (2), the appropriate model
should be chosen by taking into account the presence of all
other elements in the system, including those in direct contact with
the soil sample, such as very thin and long connecting lines.
Results
In papers (1) and (2), we propose a numerical solution of the contaminant
transport equations in porous media
for a multi-layer system with general geometry, where every layer
represents an element of the apparatus used for the assessment of
the seepage velocity, the diffusion-dispersion coefficient and
the sorption capacity. For this multi-layer system it is
possible, by using simple mathematical expressions, to introduce a
transformation that reduces the general and variable system geometry
to an equivalent and more simple geometry where the seepage velocity
becomes constant in each layer. If the seepage velocity vis constant,
the advection-diffusion equation can by simplified into a purely
diffusive equation by considering a Galilean transformation into a
reference system moving with velocity v with respect to the
laboratory system. In addition, it is
simpler to take into account the boundary conditions imposed on the
system in an appropriate manner. For the solution of the diffusion
equation and the definition of the concentration profile
in the equivalent system, we propose
an algorithm based on a simple implicit finite difference (IFD)
scheme in which each layer is discretized separately and
the boundary conditions are
derived from the continuity of flux and concentration.
We calculate the concentration profiles, as
functions of space and time, for some
particular systems and we investigate the validity of the
various analytical solutions used in the literature and of the
boundary conditions adopted for their derivation. To this purpose, we
perform a parametric study by varying the diffusion coefficients of
the layers of a chosen equivalent system and study the behaviour of
the numerical solution in some relevant cases.
-
L. Demeio, D. Sani and M. Manassero, A numerical method for the solution
of the diffusion equation in multilayered systems, Internal Report
N. 5/1998, Dipartimento di Matematica "V. Volterra", Universita' degli
Studi di Ancona.
-
L. Demeio, M. Manassero e D. Sani, Modelli matematici degli esperimenti
di trasporto di sostanze inquinanti: studio numerico dell'equazione di
diffusione, Proc. IV Congresso Nazionale della SIMAI, Giardini
Naxos (ME), 1-5 Giugno 1998, vol. 2, p. 320.