Identification of aquifer transmissivity with multiple sets of data using the Differential System Method

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Identification of aquifer transmissivity with multiple sets of data using the Differential System Method
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  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 16,1-20133 Milano, Italy, Guido.Parravicini@unimi.it Abstract  The mass balance equation for stationary flow in a confined aquifer and the phe-nomenological Darcy's law lead to a classical elliptic PDE, whose phenomeno-logical coefficient is transmissivity, T, whereas the unknown function is the piezometric head. The differential system method (DSM) allows the computation 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.  175-181.  176  PROCEEDINGS, IFIP-TC7,  TURIN 2005 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 least-squares 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 problems of data effectiveness, other methods can reduce the above mentioned problem for inversion through the use of data measured at different times and therefore 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 equation (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 integration 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]).  178  PROCEEDINGS, IFIP-TC7,  TURIN 2005 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 II--111 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 complex 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 noise-free head data are obtained with  a  finite difference solution  of  the  discrete 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 179 •W jom.) 1000- -  t  .' .> _) ; U fiii D S :;i; il <l2 rn> ft u: ® IKDDN @ 4IK(<) ••-  -5 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.
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