IDENTIFICATION OF AQUIFER TRANSMISSIVITY WITH MULTIPLE SETS OF DATA USING THE DIFFERENTIAL SYSTEM METHOD
M. Giudici,^ G. A. Meles/ G. Parravicini,^ G. Ponzini/ and C. Vassena^
Universita degli Studi di Milano, Dipartimento di Scienze della Terra, Sezione di Geofisica, viaCicognara?, Milano, Italy, \Mauro.Giudici, Giansilvio.Ponzini,Chiara.Vassena}@unimi.it Universita degli Studi di Milano, Dipartimento di Fisica, via Celoria 16,120133 Milano, Italy, Guido.Parravicini@unimi.it
Abstract
The mass balance equation for stationary flow in a confined aquifer and the phenomenological Darcy's law lead to a classical elliptic PDE, whose phenomenological coefficient is transmissivity, T, whereas the unknown function is the piezometric head. The differential system method (DSM) allows the computation of
T
when two independent data sets are available, i.e., a couple of piezometric heads and the related source or sink terms corresponding to different flow situations such that the hydraulic gradients are not parallel at any point. The value of T at only one point of the domain, xo, is required. The T field is obtained at any point by integrating a first order partial differential system in normal form along an arbitrary path starting from
XQ.
In this presentation the advantages of this method with respect to the classical integration along characteristic lines are discussed and the DSiVI is modified in order to cope with multiple sets of data. Numerical tests show that the proposed procedure is effective and reduces some drawbacks for the application of the DSM.
keywords: Inverse problems, porous media, multiple data sets
1. Problem definition and classical methods of solution
We consider ground water flow in a confined aquifer, i.e.
a
permeable porous geological formation with upper and lower impermeable boundaries. The mass balance equation for stationary flow (which means that the fluid density is constant and the porous medium is not deforming), can be written as
d^{Td,h)^dy{Tdyh)^f,
(1) where
T
is the aquifer transmissivity
[L?/T], h
is the piezometric head
[L]
and / is the source term, i.e. the well discharge rate of abstracted water per unit area of the aquifer
[L/T].
The development of
a
forecasting model requires the solution to (1) with respect to
h,
so that
T
and / must be known.
Please use the following format when citing this chapter:
Giudici, M., Meles, G.A., Parravicini, G., Ponzini, G., and Vassena, C, 2006, in IFIP International Federation for Information Processing, Volume 202, Systems, Control, Modeling and Optimization, eds. Ceragioli, F., Dontchev, A., Furuta, H., Marti, K., Pandolfi, L., (Boston: Springer), pp.
175181.
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Data on
T
are usually obtained from the interpretation and processing of well
tests,
which are very much influenced by the well characteristics (head losses due to screen and drain effect, pump position, etc.) and provide a value which can be representative of a region with a limited radius around the well, say of the same order of magnitude of the screened intervals, which could be of the order of tens of meters. As a consequence, these values are not representative of the flow processes at a regional scale, where flow is modelled in aquifers whose lateral extensions could be as great as tens or hundreds of kilometers and for which the spacing of the numerical grid could be hundreds of meters. The T field has to be estimated, for example with the solution of an inverse problem for equation (1). This requires the computation of T, given
h
and / and the least prior knowledge of
T.
In the mathematical and geophysical literature this inverse problem has been classically posed as a Cauchy problem, and the solution is found by integration along the flow lines (see, e.g., [8], [9], [10], [2], [3], [13]). For this it is necessary to assign T at a point for each flow line. The application of such an approach to real cases is very difficult, practically impossible. In fact it is difficult to measure
T
along the inflow or outflow boundary of the domain or wherever at a point along each flow line. It is also difficult to determine the flow lines with enough precision from head data which are available at a limited number of irregularly scattered points. Moreover, since the
T
field depends upon the hydraulic gradient, grad/i, the integration of (1) with respect to
T
along a flow line is intrinsically unstable. Since the integration along each flow path is independent from the integration along the neighbouring flow lines, the instability could lead to results which do not respect any regularity of the
T
field among nearby flow lines (see [3] for a discussion about practical aspects). Other approaches, related to non linear leastsquares techniques, possibly with regularization, or in the framework of maximum likelihood estimation, assume some knowledge of the unknown parameter [18], [1], e.g., the fact that it is piecewise constant so that the domain can be partitioned into a number of subdomains where
T
is constant. This approach is known as zonation. Instead of using additional prior information on T, which always poses problems of data effectiveness, other methods can reduce the above mentioned problem for inversion through the use of data measured at different times and therefore related to different flow situations. See [15], [14], [12], [1], [4], [16], [19], [7] among the others. The next section describes one of these methods, the Differential System Method (DSM).
DSM with multiple data
sets
177
2.
The Differential System Method
The simplest version of the DSM, see [5] and [11], allows for a solution of the inverse problem when two independent sets of data,
I
(/i^ \
Z^''
) , I = 1,2>,
and the value of
T
at only one point
XQ
of the domain are available. In this case equation (1) can be written for both data sets and leads to a system of first order partial differential equations for T, which can be written in the normal form gradr=ra + b, (2) if the following
independence condition
holds: detAT^O, (3) where the elements of
A
are given by the relations
Ai, = dM ^.
(4) The
T
field is obtained at any point x by integration of the differential equation (2) in the unknown function
T
along any line connecting x to xo, where the value,
TQ,
of T at
XQ
is the initial value for the integration. The integration path, 7, can be chosen according to a
stability condition
that requires that the line integral /
\ai\dl
be small in order that the error propagation along the integration line 7
he
small. The DSM has been tested with stationary [5] and transient [17] synthetic data. A discussion on the discrete stability of the method is given in [5] and numerical experiments are shown in [6]. The numerical tests so far performed show that the stability condition is important also for the choice of the starting point
XQ.
In fact if
XQ
is chosen in an area where a is great, numerical errors prevent the computation of T with a good confidence but for a small neighborhood of
XQ.
Unfortunately the data on
T
are usually available where well tests can be performed; as a consequence, Xo should correspond to the location of an existing well where tests have been performed and nobody can guarantee that a is small there. Another difficulty for the application of the DSM to real cases is the fact that data sets independent on the whole domain can be obtained for a variation of the physical boundary conditions, which is nevertheless quite rare and above all cannot be controlled. In fact boundary conditions vary as a response to climate change, modification of land use, and so on. On the other hand, a variation of the pumping schedule modifies the flow field in limited regions surrounding the pumping wells only and not throughout the whole aquifer (see, e.g., [16]).
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3.
The Differential System Method with multiple sets of data
The difficulties discussed at the end of the previous section might
be
mitigated if the
DSM is
modified
in
order
to
deal with several sets
of
data,
i.e. M
pairs
/'/i(0_
/(0\ / == 1,...,M,
with
M > 2.
Equation
(1) can be
written
for all
the available data sets.
The
standard version
of
the
DSM can be
applied
if we
locally choose
the
best pair
of
sets
of
data
to
build
the
matrix
A, as
defined by (4),
and
compute
the
vectors
a and b to be
used
in (2).
In particular multiple data sets
can be
used pairwise
to
compute
the
vectors a
and b in the
following way.
The
domain
is
subdivided
in
subregions, where a pair
of
data sets
can be
found that best satisfies
the
following conditions:
1.
the
independence condition;
2.
the
stability condition;
3.
the
smallness
of
fi{A),
the
condition number
of A.
In particular,
;^{A)
is
computed
as
follows
ii A
II111 ii.4 ^ii
 ^ '^ '^ rs^
M^)^
\\A
II i^^^^i , (5)
where the Frobenius norm
is
used. Once the vectors
a
and
b
have been computed with the best pair
of
sets,
the DSM can be applied with the standard procedure in each subregion.
4.
Numerical tests
In this section some results
of
simple numerical tests
are
shown. More complex cases have also been analysed, but the results are qualitatively very similar, so that this simple case could
be
more easy
to be
analysed
and
interpreted. The reference Log(T) field
is
represented
in
figure
1,
together with
the po
sition
of
the abstraction wells that
are
used
to
generate
the
synthetic head data. In particular
one set of
data
(set 0) is
obtained with
no
pumping wells
and
nine data sets (sets
1 to 9)
correspond
to the
cases when
one
well
at
time
is
pumping, with
the
discharge rates
(in
L/s) plotted
in
figure
1. For
each data
set
the noisefree head data are obtained with
a
finite difference solution
of
the
discrete balance equation;
the
assigned Dirichlet boundary conditions
are
linearly varying from left (100
m) to
right
(80
m). Then
the
data
are
corrupted with
an
uncorrelated noise. Here we show the results when the noise
is
introduced with a truncation
of
the piezometric heads
at the
third decimal digit. The results
of
the standard
DSM
applied
to the
data sets
0 and 4 are
shown in figure
2
when
the
starting point
is at
well no.
or
9.
The
differences between the two cases
are
apparent.
In
particular when
the
starting point corresponds
to
DSM with multiple data sets
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Figure
1.
Reference Log(T) field (T in m^/s). Circled numbers show the positions of the abstraction wells; numbers above the labels show discharge rates in L/s.
well no. 9 negative transmissivities have been identified in a large region (the black area of the bottom plot of figure 2).
Figure
2.
Log(T) field identified with the standard
DSM.
The crosses denote the positions of the starting point. Gray scale is the same as for
figure
1.
The results obtained when all the sets of data are used simultaneously, with the technique described in the previous section, are shown in figure 3, again for the starting point at well no. 1 or 9. These results show that this approach is very useful to reduce the dependence of the final solution on the starting point, which can be chosen almost everywhere without worsening the results of the DSM.