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Identification of aquifer point sources and partial boundary condition from partial overspecified boundary data

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In this work, we present a new mathematical method that allows recovering of the wells fluxes and hydraulic heads on a part of the boundary where they are not known, for an aquifer domain having overspecified boundary data on another part of its
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  Surface Geosciences (Hydrology – Hydrogeology) Identification of aquifer point sources and partial boundarycondition from partial overspecified boundary data Najla Tlatli Hariga a , Rachida Bouhlila b, *, Amel Ben Abda c a  LAMSIN-ENIT & INAT, B.P. 37, 1002 Tunis-Belvedere, Tunisia b  LMHE-ENIT, B.P. 37, 1002 Tunis-Belvédère, Tunisia c  LAMSIN-ENIT, B.P. 37, 1002 Tunis-Belvedere, Tunisia Received 21 June 2007; accepted after revision 26 November 2007Available online 18 January 2008 Presented by Ghislain de Marsily Abstract In this work, we present a new mathematical method that allows recovering of the wells fluxes and hydraulic heads on a part of the boundary where they are not known, for an aquifer domain having overspecified boundary data on another part of its boundary.The method is based on the minimisation of an energy-like error functional of Andrieux and Ben Abda [Inverse Probl. 22 (2006)115 – 133] for the missing-data recovering step and on the Reciprocity Gap principle of the same authors [Inverse Probl. 12 (1996)553 – 564] for point sources identification.  To cite this article: N.T. Hariga et al., C. R. Geoscience 340 (2008). # 2007 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved. RésuméIdentification des points sources et d’une condition aux limites sur une partie de la frontière d’un aquifère à partir deconditions auxlimites partielles surabondantes. Nousnous proposonsdanscetravailde résoudreun problèmede Cauchy,afinderetrouver les débits des puits et la condition de charges imposées sur une partie de la frontière d’un aquifère, à partir de donnéespartielles surabondantes sur une autre partie de la frontière. La résolution se fait en deux étapes : on commence par minimiser lafonctionnelle d’erreur d’énergie développée par Andrieux et Ben Abda [Inverse Probl. 22 (2006) 115 – 133] pour compléter lesdonnées, puis on applique le principe de réciprocité introduit par les mêmes auteurs [Inverse Probl. 12 (1996) 553 – 564] pourdéterminer les débits.  Pour citer cet article : N.T. Hariga et al., C. R. Geoscience 340 (2008). # 2007 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved. Keywords:  Hydrogeology; Inverse problem; Cauchy problem; Data completion; Point sources; Reciprocity gap principle  Mots clés :  Hydrogéologie ; Problème inverse ; Problème de Cauchy ; Complétion de données ; Points sources ; Principe d’écart à la réciprocité 1. Introduction The leading inverse problem in hydrogeology dealswith estimating the hydrodynamical parameters of theaquifer: permeabilities, transmissivities, and storagecoefficients. The most frequent problem and also the http://france.elsevier.com/direct/CRAS2A/  C. R. Geoscience 340 (2008) 245 – 250* Corresponding author. E-mail address:  bouhlila.rachida@enit.rnu.tn (R. Bouhlila).1631-0713/$  –  see front matter # 2007 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.doi:10.1016/j.crte.2007.11.006  most studied in the literature is indeed that of thedetermination of transmissivities from piezometric datain a steady-state flow regime [9,10]. However, in manyrealsituations,uncertaintiescanberelatedtotheaquiferdomain itself, its boundary conditions and also, usually,on the external constraints, such as withdrawal rates inwells, drillings, and recharge.The knowledge of the aquifer withdrawal rates canrepresent a largely unknown factor in real problems of groundwater resources modelling. The acquisition of these data is expensive and prone to great uncertainties.The usual techniques of specific measurements orestimation, using, e.g., surveys of power consumptionor crop surfaces, type and degree of growth of theirrigated crops and also satellite images, provide onlyorders of magnitude. Indeed, usually, this term of external forcing constitutes data to be refinedin a globalmodel calibration procedure.The problem under consideration can be statedmathematically as the recovery of unknown heads on apart of the domain boundary and of the prescribed fluxfor known point sources from Cauchy data. Theidentification procedure relies on two steps:   the recovery of the missing boundary data, which isreduced to solving the Cauchy problem;   the identification of the well fluxes.The first step is the most difficult one, because, sinceHadamard [8], the Cauchy problem is known to beseverely ill posed. The most popular method known inthe hydrogeology community, as reported in the reviewpapers [9,10], consists, roughly speaking, in drawingthe flow streamlines. Eventhough this method is easy toimplement, it is very sensitive to measurement errors. Itis therefore usually exploited together with a regulariz-ing procedure [4].Our approach in solving the Cauchy problem isborrowed tothesolidmechanicscommunity:itreliesona method of minimizing a misfit energy-like function.We adapt this energy-like error to the Darcyframework. 2. The model We will focus, in this note, on extending the datacompletion algorithm based on an energy errorfunctional, initially introduced for the Laplace equation[3] to the Darcy framework  [5]. The potential application concerns the identification of well extrac-tion in an aquifer with known transmissivities andprescribed piezometric levels on a part of its boundary,in steady-state conditions.Using the notations shown in Fig. 1, a simplifiedmathematical model is given by:  Div ð Tgrad ð h ÞÞ ¼ X k  Q k  d sk   in V h  ¼  H   on G m T   @ h @ n ¼  f on  G m 8>>><>>>: (1) where T  isthetransmissivity,whichcanbeascalarifthemedium is isotropic and homogenous, or a function incase of heterogeneity, or a second-order tensor for theanisotropic case.The hydraulic head is denoted by  h ,  Q k  is the well abstraction, corresponding to a point sourcelocated at  S  k   with coordinates (  x k   ,y k  ), and  d  is the Diracdistribution.  G  m  is the portion of the boundary of   V where both  h  and its normal derivative are known (i.e.an overspecified condition) and  G  u  is the part of theboundary where all information is lacking. This situa-tion can correspond to an arbitrary boundary for anaquifer known to extend over G  u , but where no informa-tion is available. It is for example the case of a deepaquifer continuing beyond the shoreline below the seabottom. Indeed, from a practical viewpoint, as theknown piezometric levels correspond to measuredvalues in boreholes and are presented as interpolatedisolines, it is easy to calculate their normal derivative,even on an internal part of the domain, but close to theboundary isopiezometric line, in order to obtain theoverspecified data. We propose, in this paper, to recon-struct the missing data using an energy-like error func-tional introduced in [3] and then to determine the pointsource fluxes via the reciprocity gap principle [1,2,6]. 3. Mathematical identification process 3.1. Data completion Let us consider the above Cauchy problem(equation 1). Provided the data  H   are compatible with  N.T. Hariga et al./C. R. Geoscience 340 (2008) 245 – 250 246Fig. 1. The mathematical domain with wells.Fig. 1. Domaine du modèle mathématique avec les différents puits.  the flux  f , the data completion step is achieved in aneighbourhood of the boundary of the domain wherethere is no well. A fictitious inner boundary  G  F  istherefore introduced. Then the energy-like errorfunctional is constructed on the fictitious domain  V F with  @ V F  =  G  m  [  G  u  [  G  F  and  G  H =  G  u  [  G  F . Extendingthe data means finding ( f ,  f  ) such that we obtainequation. (2):  Div ð Tgrad ð h ÞÞ ¼  0 in V F h  ¼  H  ;  T   @ h @ n ¼  f  on G m h  ¼  f  ;  T   @ h @ n ¼  ’  on G H 8>>>><>>>>: (2)The approach in the energy-like error functionalmethod developed in [3] follows two steps. First, weconsider, for a given pair ( h , t  ), the two following mixedwell-posed problems (equations (3a) and (3b)):  Div ð Tgrad ð h 1 ÞÞ ¼  0  in  V F h 1  ¼  H   on  G m T   @ h 1 @ n ¼  h  on  G H 8>><>>: (3a)  Div ð Tgrad ð h 2 ÞÞ ¼  0  in  V F h 2  ¼  t  on G H T   @ h 2 @ n  ¼  f  on  G m 8>><>>: (3b)The second step is to build an energy-like errorfunctional on the pair ( h , t  ), using an energy norm,denoted E. Indeed, these fields are obviously equal onlywhen the pair ( h , t  ) meets the real data ( f ,  f  ) on theboundary  G  H  =  G  u  [  G  F . We propose then to solve thedata completion problem via the following minimiza-tion algorithm (equation (4)): ð ’ ;  f  Þ ¼  argmin E  ð h ; t  Þ  (4) E  ð h ; t  Þ ¼  12 Z  V F  ð Tgrad ð h 1 Þ Tgrad ð h 2 ÞÞ 2 d V  (5) 3.2. The reciprocity gap principle After the first step of reconstructing the missingboundary data, we deal with recovering the unknownwell fluxes via the Reciprocity Gap functional. Thisfunctional has been introduced in [2] in the context of planar cracks identification from overspecified bound-ary data. It relies on the well-known Mawell – Bettireciprocity principle. As introduced in [2], it relates theoverdetermined boundary data to the unknown quan-tities (here the well fluxes).We multiply the first equation of the system (1) by avirtual field  v  satisfying Div(grad m ) = 0, then weintegrate it over the whole domain  V ; we use Green’ssecond formula, we find:  R ð v Þ ¼ X k  Q k  v ð S  k  Þ  (6) where  R ð v Þ ¼ Z  @ V T   @ h @ nv  @ v @ nh  d G  (7) Notice that the left-hand side of the equality (6) or (7) istotally known: ( f ,  H  ) are the overdetermined boundarydata and we can select  v . More precisely, for  k   wells with unknown fluxes, weevaluate  R (  z  j ),where  z isthecomplexvariable,suchthatthe real and imaginary parts of   z  j are harmonic:  R ð  z  j Þ ¼ X k  Q k   z  jk   (8) where  z k   =  x k   + i  y k   is the affix of the point source  S  k  . Then we exploit the reciprocity gap functional withvarious particular fields  m .Therefore, the determination of the fluxes of acollection of wells amounts to solving a linear system:  AQ  ¼  b  (9) where  A  ¼ ð  z i j Þ  is the Wandermonde matrix,  Q  = ( Q  j )and  b  = (  R (  z i )). To clarify this step, we carry out an example in thecase of two wells located at the points M 1 (  x 1 ,  y 1 ) andM 2 (  x 2 ,  y 2 ) with fluxes  Q 1  and  Q 2 . The particularharmonic  v  chosen are: v 1  ¼  1  r v 1 n  ¼  0 v 2  ¼  x þ iy  r v 2 n  ¼  1 þ i Then we apply equations (6) and (7) for each  v :  R ð v 1 Þ ¼ X k  Q k  v 1 ð S  k  Þ ¼  Q 1  þ Q 2  ¼ Z  @ V T   @ h @ n d G  R ð v 2 Þ ¼ X k  Q k  v 2 ð S  k  Þ ¼  Q 1 ð  x 1  þ iy 1 Þþ Q 2 ð  x 2  þ iy 2 Þ  R ð v 2 Þ ¼ Z  @ V T   @ h @ n  ð  x þ iy Þ Th ð n  x  þ in  y Þ d G  N.T. Hariga et al./C. R. Geoscience 340 (2008) 245 – 250  247  Finally, we have to solve one of the equivalent linearsystems, corresponding to the real and imaginary partsof system (8), with  k   = 2: Q 1  Q 2 ½  ¼ 1  x 1 1  x 2   ¼ Z  @ V T   @ h @ n d G Z  @ V T   @ h @ n x  hn  x   d G " # or Q 1  Q 2 ½   1  y 1 1  y 2   ¼ Z  @ V T   @ h @ n d G Z  @ V T   @ h @ n y  hn  y   d G " # so that we obtain directly  Q 1  and  Q 2. 4. Numerical trials The resolution of the problem given by equation (2)was carried out using the finite-element method (FEM).First, we complete the boundary data via an optimisa-tion problem, then we determine thewells fluxes via theresolution of a linear problem.For minimizing the energy functional  E   (equation(4)), we have to compute the gradient of   E   with respecttoall the components of the unknown field  h  (here equaltothenumberofnodesofthemesh).Therefore,wehavechosen the adjoint state method because it does notdepend on the number of components, and it makes itpossible to evaluate the gradient in any direction usingonly the determination of two adjoint fields.All the computations have been run on FemLab [7].The numerical tests are performed on the followinggeometry (Fig. 2), corresponding to a rectangulardomain of 20 km  10 km, with a homogeneoustransmissivity  T   = 0.001 m 2 s  1 with overspecified dataontheuppersideandmissingdataonthelowerone.Theoverspecified data are extracted from the exact solution h (  x ,  y ) of the direct problem given by equation (1) withpoint sources coordinates (  x k  ,  y k  ) and intensity  Q k  : h ð  x ;  y Þ ¼ X k  Q k  2 p T   log  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð  x   x k  Þ 2 þð  y   y k  Þ 2 q  (10) The domain is meshed with a regular mesh of triangularelements with linear interpolation, characterized by1100 nodes and 348 elements. To test the efficiencyof the proposed reconstruction processes, differentcases have been studied. 4.1. The case of multiple wells We consider in this case a collection of five wellswith positive and negative fluxes. Well locations andfluxes are shown in Table 1, whereas Table 2 sums up the calculated fluxes and the relative errors compared tothe exact values. These last values correspond to theexactsolution of the forward problem givenby equation(10). On Tables 3 and 4, we consider the same fivewells with fluxes values 10 and 100 times less than those of Table 2. As the piezometric levels are linear withrespect to the fluxes (as shown in equation (10)), theerror remains almost identical for the three cases.Fig. 3a shows the reconstructed data (the hydraulichead anditsnormalderivative)overtheboundarywheredata are missing, the lower one, in the case of these fivewells. Fig. 3b represents the hydraulic head over the  N.T. Hariga et al./C. R. Geoscience 340 (2008) 245 – 250 248Fig. 2. The studied domain with the five wells.Fig. 2. Domaine étudié avec cinq puits.Table 1Position (  X  , Y  ) and rates ( Q k  ) of the five tested wellsTableau 1Position (  X  , Y  ) et débit  Q k   des cinq puits testés  X   (km) 2 5 10 15 18 Y   (km) 3 7 5 6 4 Q k   (l s  1 )   50   70 150   30 80Table 2Exactvaluesoffluxes( Q k   exact )andcomputedones( Q k   calc )inthecaseof five wellsTableau 2Valeurs exactes des debits ( Q k   exact ) et valeurs calculées ( Q k   calc ) dansle cas de cinq puits Q k   exact  (l s  1 )   50   70 150   30 80 Q k   calc.  (l s  1 )   57   68.1 152.1   33.3 85.1Relative error (%) 14 2.7 1.4 11 6.25  entire domain in the case of five wells. We note that thecompletion results match very well the exact ones. 4.2. Sensitivity to noisy data To take into account the sensitivity of the recoveredwell fluxes to noisy data, a uniform white noise (withzero mean), is applied to the Dirichlet data on  G  m  in thecase of a single well located at  x  = 2 km and  y  = 5 km,with a rate  Q  =  100 l s  1 . Table 5 shows the errors fora noise level up to 8%. One can see that, despite thenoise, the data recovering error remains acceptable, fora noise level less then 6%. 4.3. Sensitivity to the relative position In the third case, we test the sensitivity to the relativeposition of two wells. We consider two wells withfluxes:  Q 1  =  – 100 l s  1 and  Q 2  =  50 l s  1 , separatedby a variable distance  d  , and we compute the relativeerroronrecoveredfluxesforeachdistance(seeTable6).One can observe that if the wells are well separated (farfromeachother),thefluxesarewellidentified,butwhenthe distance between the wells decreases, the identifica-tion procedure is less accurate.  N.T. Hariga et al./C. R. Geoscience 340 (2008) 245 – 250  249Table 3Exactvaluesoffluxes( Q k   exact ) andcomputedones( Q k   calc )inthecaseof five wells, where flux are a tenth of those of  Table 2Tableau 3Valeurs exactes des débits ( Q k   exact ) et valeurs calculées ( Q k   calc ) dansle cas de cinq puits, avec des flux 10 fois plus petits que ceux duTableau 2 Q k   exact  (l s  1 )   5   7 15   3 8 Q k   calc.  (l s  1 )   5.7   6.8 15.2   3.3 8.5Relative error (%) 14 2.8 1.3 10 6.25Table 4Exactvaluesoffluxes( Q k   exact ) andcomputedones( Q k   calc )inthecaseof five wells, where flux are a hundredth of those of  Table 2Tableau 4Valeurs exactes des débits ( Q k   exact ) et valeurs calculées ( Q k   calc ) dansle cas de cinq puits, avec des flux 100 fois plus petits que ceux duTableau 2 Q k   exact  (l s  1 )   0.5   0.7 1.5   0.3 0.8 Q k   calc.  (l s  1 )   0.57   0.68 1.52   0.33 0.85Relative error (%) 14 2.8 1.3 10 6.25Fig. 3. ( a ) Hydraulic head and its normal derivative over  G  u ; ( b ) isovalue of the hydraulic head in the case of five wells.Fig. 3. ( a ) Charge hydraulique et sa dérivée normale sur  G  u  ; ( b ) isovaleur de la charge hydraulique, dans le cas de cinq puits.Table 5Various noise levels: single well located at  x 0  = 2 km and  y 0  = 5 kmwith a flux  Q 0  =  100 l s  1 Tableau 5Différents bruits dans le cas d’un puits situé à  x 0  = 2 km et  y 0  = 5 km,avec un débit  Q 0  =  100 l s  1 Noise level (%) 0 2 4 8Relative error (%) 0.3 3 4 6Table 6Sensitivity to the relative location of two wells with  Q 1  =  100 l s  1 and  Q 2  =  50 l s  1 Tableau 6Influence de la distance relative entre les puits, dans le cas de deuxpuits avec  Q 1  =  100 l s  1 et  Q 2  =  50 l s  1 Distance  d   (km) 16 10 6 3 2 1.5Relative error (%) 2 6 8 9 15 17
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