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Journal of Hydrology 509 (2014) 115–131
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Journal of Hydrology
journal homepage: www.elsevier .com/ locate / jhydrol
Analytical solutions for analysing pumping tests in a sub-verticaland anisotropic fault zone draining shallow aquifers
0022-1694/$ - see front matter � 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.jhydrol.2013.11.014
⇑ Corresponding author. Address: 1039 Rue de Pinville, 34000 Montpellier,France. Tel.: +33 (0) 4 67 15 79 82; fax: +33 (0) 4 67 15 79 90.
E-mail addresses: [email protected] (B. Dewandel), [email protected](B. Aunay), [email protected] (J.C. Maréchal), [email protected](C. Roques), [email protected] (O. Bour), [email protected] (B. Mougin),[email protected] (L. Aquilina).
B. Dewandel a,⇑, B. Aunay b, J.C. Maréchal a, C. Roques c, O. Bour c, B. Mougin d, L. Aquilina c
a BRGM, Water Dept., New Water Resource & Economy Unit, 1039 Rue de Pinville, 34000 Montpellier, Franceb BRGM, French Regional Geological Survey of Réunion Island, 5 Rue Sainte Anne, CS 51016, 97404 Saint-Denis Cedex, Francec OSUR Research Federation – Géosciences, UMR 6118, University of Rennes 1, CNRS, Av. du Général Leclercq, 35042 Rennes, Franced BRGM, French Regional Geological Survey of Brittany, Rennes Atalante Beaulieu, 2 Rue de Jouanet, 35700 Rennes, France
a r t i c l e i n f o s u m m a r y
Article history:Received 19 July 2013Received in revised form 4 November 2013Accepted 8 November 2013Available online 22 November 2013This manuscript was handled by P.Kitanidis, Editor-in-Chief, with theassistance of Christophe Darnault, AssociateEditor
Keywords:Pumping testCompartmented aquiferFaultCrystalline aquiferGroundwater dating
We present new analytical solutions for examining the influence, during a pumping test in a well, of aninfinite linear and anisotropic strip-aquifer that drains shallow aquifers of different diffusivity and thick-ness. The whole system is confined and the aquifer geometry can be represented by a ‘T’, an aquifergeometry resembling a sub-vertical fault or a sub-vertical vein cross-cutting shallower aquifers. The pro-posed solutions are based upon an unconventional application of well-image theory, without limitationof the diffusivity contrast between the three domains. Solutions for drawdown were developed for thethree domains, i.e. the strip-aquifer and the two shallow compartments, and flow signatures are dis-cussed in detail and compared to numerical modelling. The proposed solutions are not shown to be exactsolutions to the appropriate partial differential equation, but very good and useful approximations. Thesolutions were applied to a 63-day pumping test in a steep fault zone in crystalline aquifer rock of Brit-tany, France. After that, the flow contributions of the fault zone and of the shallow aquifers deduced fromgroundwater dating were compared to analytical solutions. The solutions and theoretical type-curveexamples can help in understanding flow processes from tests conducted in settings that are similar tosuch a conceptual model.
� 2013 Elsevier B.V. All rights reserved.
1. Introduction
In hydrogeology, the oil-and-gas industry, and geothermalactivities, well testing is a critical element for assessing thehydrogeological properties of the rock. The evaluation of theseproperties, as well as their variability in space, is essential forimproving the management of the resource (e.g., Raghavan,2004; Dewandel et al., 2012), or because detailed groundwater-flow data are required for sophisticated pressure and transfermodelling, as for instance when studying nuclear waste deposits,geothermal energy, or contaminant transport.
In discontinuous aquifers, determination of an adequate concep-tual model prior to quantification of the hydrogeologic parametersby modelling of the tests, is a particular challenge because thehydraulic signature can be non-unique (Rafini and Larocque,2012). Well-test data are extensively used in numerical models forevaluating hydraulic conductivity and storativity fields. Though
different approaches have been used, the problem is particularly dif-ficult to tackle in fractured media because of system heterogeneity(Neuman, 2005). For instance, numerical models can be determinis-tic (e.g. Walton, 1987; Rathod and Rusthon, 1991), based on a sto-chastic continuum (e.g. Neuman, 1987; Molz et al., 2004; Illmanand Hughson, 2005, etc.) or based on discrete fracture networks thatexplicitly consider the complex geometry of fracture networks (e.g.Long et al., 1982; Cacas et al., 1990a,b; de Dreuzy et al., 2002, etc.).Various approaches are also used for developing inverse methodsin fractured media (Zimmerman et al., 1998; Lavenue and deMarsily, 2001; Franssen and Gomez-Hernandez, 2002; Illmanet al., 2009), or for testing on numerical models our ability to imagethe properties of heterogeneous media from well tests (Tiedemanet al., 1995) or hydraulic-head data (Le Goc et al., 2010). Among thesemethods, hydraulic tomography has shown very promising results(e.g. Gottlieb and Dietrich, 1995; Butler et al., 1999; Yeh and Liu,2000; Huang et al., 2011; Illman et al., 2007, 2008, 2009; Yin andIllman, 2009). Although these approaches may provide a detailedsite characterisation (Meier et al., 1998; Sánchez-Vila et al., 1999),such high-resolution techniques required intensive field work, suchas several hydraulic tests, collection of numerous pressure-headdata, and detailed geological knowledge. Such data are generallyonly available for experimental sites, but not where most of tests
Nomenclature
List of symbolsa distance to discontinuity in mex exponential functionk hydraulic conductivity in m/sL half-width of the pumped compartment (e.g. Domain
1; strip aquifer), in mQ pumping flow-rate, in m3 s�1
r distance to pumping well, in mrw well radius, in mS1, S2, S3 storage coefficients of the strip aquifer, right- and
left side compartments respectively, dimensionless(–)
Ss1, Ss2, Ss3 specific storage coefficients of the strip aquifer, right-and left-side compartments respectively, m�1
s1, s2, s3 drawdown of the strip aquifer, right- and left-sidecompartments respectively, in m
sD dimensionless drawdowntDL dimensionless time according to the distance to the
boundary, isotropic casetDLx dimensionless time according to the distance to the
boundary, anisotropic caset time in secondsT1, T2, T3 transmissivity of the strip aquifer, right- and left-side
compartments respectively, in m2/sTxx and Tyy principal axes of transmissivity anisotropy in the hor-
izontal plane of the strip aquifer, in m2/sx, y coordinates of a Cartesian system, in mg1, g2, g3 diffusivity of the strip aquifer, right- and left-side
compartments respectively, in m2/sWðurÞ ¼
R1u
1y expð�yÞÞdy with y variable of integration. Wellfunction
Fig. 1. The ‘T’ aquifer. Conceptual sketch of the infinite anisotropic linear-stripaquifer, Domain 1 (D1; width 2L), separating two semi-infinite half-spaces ofdissimilar properties (domains 2 and 3).
116 B. Dewandel et al. / Journal of Hydrology 509 (2014) 115–131
are conducted. For example, for water-supply purposes generallyjust data from one well and one test are available. Analytical solu-tions for analysing pumping-test data, such as the ones developedhere, are thus still of wide interest. They are useful for evaluatingaveraged properties of the media (e.g. hydrogeological properties,flow behaviour, geometry), and can help in constraining the numer-ical models particularly in fractured aquifers where field data maynot be sufficient, if they exist, to justify the use of a detailed numer-ical model.
We present new analytical solutions for examining the influ-ence in a well test of an infinite linear and anisotropic strip-aquiferthat drains shallower aquifers of differing diffusivity and thicknesswith a geometry that resembles a ‘T’ (the ‘T-aquifer’; Fig. 1). How-ever, the thickness of the shallow aquifers on the sides of the stripaquifer can be the same as that of the strip one. The flow behav-iour, i.e. flow signature, has also been examined in detail. Thisconceptual model may represent the main features of several geo-logical conditions, such as a fault zone cross-cutting shallow aqui-fers, or dykes and veins intruding igneous rock, which are favouritediscontinuities for borehole siting in crystalline aquifers (Sander,2007; Dewandel et al., 2011). It may also represent a buried chan-nel embedded in other materials, two faults separating a centralstrip with dissimilar hydraulic properties, cut and filled paleo-val-leys of lava flows, etc.
The research reported in this article is based on a large body ofearlier works in hydrogeology and petroleum literature. Numerousworks (e.g. Maximov, 1962; Bixel et al., 1963; Nind, 1965; Fenske,1984; Raghavan, 2010) have examined the effects of well tests per-formed near a single linear discontinuity separating materials ofdissimilar properties, or the case of two arbitrarily intersectingboundaries (van Poollen, 1965). Others focused on the behaviourof a semi-permeable fault separating aquifers with similar proper-ties (e.g. Yaxley, 1987; Abbaszadeh and Cinco-Ley, 1995; Shanet al., 1995) or with dissimilar properties (e.g. Ambastha et al.,1989; Rahman et al., 2003; Charles et al., 2005; Anderson, 2006),the behaviour of two intersecting leaky faults (Abdelaziz and Tiab,2004), or the behaviour of pumping well located in an infinite con-ductive vertical or horizontal fracture (Gringarten and Ramey,1974; Gringarten et al., 1974). Much work was also done on thebehaviour of pumping in a strip aquifer of permeable material lim-ited by at least two impermeable or constant-head boundaries (e.g.Muskat, 1937; Ferris et al., 1962; Earlougher et al., 1968; Lennoxand Vandenberg, 1970; Chan et al., 1976; Strelsova and McKinley,1984; Ehlig-Economides and Economides, 1985; Larsen and
Hovdan, 1987; Yeung and Chakrabarty, 1993; Kuo et al., 1994;Onur et al., 2005), or the behaviour of composite radial systems(e.g. Bixel and van Poollen, 1967; Butler, 1988; Abbaszadeh andKamal, 1989; Bratvold and Horne, 1990; Butler and Liu, 1993;Acosta and Ambastha, 1994), or that of composite linear systems(Bourgeois et al., 1996).
However, few works have investigated the influence on apumping test of a strip aquifer bounded laterally by aquifers of dif-ferent diffusivity and thickness, such as in a ‘T-aquifer’. Boonstraand Boehmer (1986); Boonstra and Boehmer, 1987 proposed asolution that assumed a high contrast in transmissivity betweenthe inner and the two outer regions, while Butler and Liu (1991)proposed a solution without limitations of aquifer diffusivitybetween the three regions. Others have investigated troughnumerical modelling the flow induced by a pumping test carried
B. Dewandel et al. / Journal of Hydrology 509 (2014) 115–131 117
out in a fault embedded in low-permeable matrix with variousinclinations of the fault plane (Rafini and Larocque, 2012; Lerayet al., 2013). Nonetheless, these works do not consider anisotropyin hydraulic conductivity of the linear strip aquifer, which can bevery high for a fractured system for instance, or the effect of shal-low aquifers.
The solutions proposed here are based upon unconventionalapplication of the well-image theory without limitation of the diffu-sivity contrast between the three domains. The method of images is aclassic technique for solving boundary-value problems, such thoseencountered in heat conduction, elastostatics, electrostatics and,particularly, in groundwater flow, to solve the problem of no-flowor constant boundaries, or leaky boundaries (Ferris et al., 1962;Kruseman et al., 1990; Anderson, 2000; Anderson, 2006). Solutionswith this technique also have the advantage of providing convenientforms compared to solutions from differential equations.
Solutions for drawdown are developed for the isotropic andanisotropic cases and for the three domains, and are compared tonumerical modelling. The hydrodynamic signatures are discussed.Then, the proposed solutions are applied to a 63-day pumping testcarried out in a steep fault zone in a crystalline aquifer of Brittany(western France; Roques et al., 2013).
2. Mathematical model
The problem of interest consists in evaluating the drawdown, asa function of coordinates x, y and time, produced by pumping with-in an infinite linear strip aquifer of one material (Domain 1; D1;Fig. 1) limited on both sides by aquifers of differing properties (Do-mains 2 and 3). The width of the strip aquifer is 2L and lateral aqui-fers, Domains 2 and 3, have an infinite lateral extension outside ofthe layer boundary. The pumping well, located in the inner region(D1), is at the origin of a Cartesian coordinate system (x = 0, y = 0)and at a distance a from the closest discontinuity. The y-axis is par-allel to the two discontinuities. The well fully penetrates the aqui-fer and produces at a constant rate Q. The overall domain (i.e., D1,D2, D3) is assumed to be under confined condition, and Domain 1is anisotropic in hydraulic conductivity onto the horizontal planewhile the other two domains are isotropic. The aquifer thicknessof each domain can be different (Fig. 1).
The proposed solutions are based upon an unconventionalapplication of the well-image theory. This technique, originallyproposed by Fenske (1984) for the case of one discontinuity sepa-rating aquifers with dissimilar diffusivities, has been developed forthe case presented above. The proposed solutions are not shown tobe exact solutions to the appropriate partial differential equation,but very good and useful approximations. It is, however, believed,that drawdown solutions could be obtained by numerical inversionof Laplace transform, although this is not the method used here.
In a general manner and using the well-image theory for thecase presented above, one can consider the drawdown in Domain1 to consist of two components: direct drawdown caused by thepumping well, and reflected drawdown caused by the reflectionof each image-well drawdown from the two discontinuities intoDomain 1. The total drawdown can thus be represented by thesum of drawdown values from the pumping well and the image-wells located across the boundaries, each image-well beingstrengthened to take into account the contrast in properties be-tween the three domains. Fig. 2a presents graphically the solutionfor drawdown in Domain 1, each dot corresponding to an imagewell with its distance from the pumping well and its strength. Notethat this technique is widely used for solving the particular cases ofno-flow and constant-head-boundary conditions (e.g. Ferris et al.,1962; Kruseman et al., 1990); in these cases image strength is 1or �1. More information on the well-image theory can be foundfor example in Anderson (2000) and Anderson (2006).
The drawdown in domains 2 and 3 is the transmitted draw-down across the boundaries. The solutions of drawdown dependon the properties of the domain of interest and are the analoguesof the solution in Domain 1. Consequently, drawdown can be rep-resented by the sum of drawdowns caused by the producing welland two infinite well-image series lying outside the domain ofinterest, each well (pumping and image wells) being strengthenedto consider the aquifer properties of each domain (Fig. 2b and c).
Along the boundaries, conditions vary from discontinuity L1(between D1 and D2) to discontinuity L2 (between D1 and D3)and the following criteria must be satisfied:
1. The drawdown must be equal on both sides of thediscontinuities:– then s1(a,y, t) = s2(a,y, t) along discontinuity L1,– and s1(�[2L � a],y, t) = s3(�[2L � a],y, t) along discontinuity
L2;2. The specific discharges orthogonal to discontinuities must be
equal on both sides of the boundaries:– then T1
@s1ða;y;tÞ@x ¼ T2
@s2ða;y;tÞ@x along discontinuity L1,
– and T1@s1ð�½2L�a�;y;tÞ
@x ¼ T3@s3ð�½2L�a�;y;tÞ
@x along discontinuity L2;
where si(x,y, t) are drawdowns in each domain (i = 1, 2, 3) and Ti
are the transmissivities of each domain, and L the half-width of thestrip aquifer.
2.1. Isotropic case
We first consider the case of a pumping well located within anisotropic infinite linear strip aquifer (D1) separated on both sidesby domains 2 and 3, each domain characterised by dissimilar diffu-sivities (g1 – g2 – g3) and thicknesses.
Using the well-image theory described above, the general solu-tion for drawdown valid for Domain 1 can be expressed by thedrawdowns of the producing well and four infinite image-wellseries lying outside D1, such as (Fig. 2a):
s1ðx;y;tÞ¼Q
4pT1
W S14T1t ½x2þy2�� �
þX1
n¼0;2;4;...
AnW S14T1 t ½ð2nLþ2a�xÞ2þy2�� �
þX1
n¼2;4;...
BnW S14T1t ½ð2nL�xÞ2þy2�� �
þX1
n¼2;4;...
CnW S14T1t ð�ð2nL�2aÞ�xÞ2þy2h i� �
þX1
n¼2;4;...
DnW S14T1t ð�2nL�xÞ2þy2h i� �
8>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>:
9>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>;
ð1aÞ
where W (u) is the well-function (or exponential integral functionalso noted �Ei [�u]), S1 and T1 (m2/s) are the storage and the trans-missivity of the strip aquifer (D1), respectively, a is the distancefrom the pumping well to the closest discontinuity (m), L is thehalf-width of the strip aquifer (m) and Q is the pumping rate (m3
s�1). An, Bn, Cn and Dn are the image-well strengths that accountfor the properties of neighbour aquifers, domains 2 and 3.
For domains 2 and 3, the drawdown solutions are the analoguesof Eq. (1a), and correspond to the drawdown caused by the pump-ing well and two well-image series lying outside the domain(Fig. 2b and c). For a well lying in D2, it is:
s2ðx;y;tÞ¼Q
4pT2
b0W S24T2t ½x2þy2�� �
þX1
n¼2;4;...
EnW S24T2t ½ð�2nL�xÞ2þy2�� �
þX1
n¼2;4;6;...
FnW S24T2 t ð�ð2nL�2aÞ�xÞ2þy2h i� �
8>>>><>>>>:
9>>>>=>>>>;ð1bÞ
and for a well lying in D3:
Fig. 2. Graphical display of the image-well series. (a) For a well located in Domain 1; (b) for a well located in Domain 2; and (c) for a well located in Domain 3. Dots representthe location of the pumping well and its images across the two discontinuities with corresponding strength (An to Hn).
118 B. Dewandel et al. / Journal of Hydrology 509 (2014) 115–131
s3ðx;y; tÞ¼Q
4pT3
b00W S34T3 t ½x2þy2�� �
þX1
n¼0;2;4;...
GnW S34T3t ½ð2nLþ2a�xÞ2þy2�� �
þX1
n¼2;4;6;...
HnW S34T3 t ð2nL�xÞ2þy2h i� �
8>>>><>>>>:
9>>>>=>>>>;ð1cÞ
where S2 and S3 are the storage of Domain 2 and Domain 3 respec-tively (–), T2 and T3 (m2/s) are the corresponding transmissivities.En, Fn, Gn, Hn, b0 and b00 are the image-well strengths that also ac-count for properties of neighbour aquifers.
Now, the problem lies in evaluating the strength of each image-well.
2.1.1. Drawdown solutions for the three domainsWe first consider the first image according to L1 (discontinuity
between D1 and D2; Fig. 3a). This case refers to the influence of apartial hydrologic barrier that separates two domains with con-trasting diffusivities, i.e. g1 – g2, gi = Ti/Si; i = 1, 2 (Maximov,1962; Bixel et al., 1963; Nind, 1965; Fenske, 1984; Raghavan,2010).
The result of applying the two boundary conditions at the dis-continuity L1, i.e. s1(a,y, t) = s2(a,y, t) and T1
@s1ða;y;tÞ@x ¼ T2
@s2ða;y;tÞ@x , is
that the first strength coefficients A0 and b0 depend upon the diffu-
sivity contrast, the distance to the limit, and time. Appendix Agives the detail of their solutions (Eqs. (A3a) and (A3b)).
For this case, the drawdown solutions for domains 1 and 2 cor-respond to the ones proposed by Fenske (1984) (Appendix A; Eqs.(A1a) and (A1b)). Furthermore, where there is no-diffusivity con-trast (i.e. g1 = g2), the solutions for drawdown in the two domainsare exact solutions of the appropriate partial differential equation(Nind, 1965; Raghavan, 2010).
Considering the case of two parallel discontinuities and thus thecase of a strip aquifer, the pumping well has to be imaged aboutthe second discontinuity (L2) –Fig. 3b, and then imaged about L1(Fig. 3c). After that, the first image-well according to L1 is imagedabout L2 (Fig. 3d), and so on, which gives an infinite number of im-age-wells (Figs. 2 and 3b, c, d). Therefore, each iteration results in anew image-well whose strength has to be evaluated according toboundary conditions at the discontinuities. Appendix A presentsthe calculation of the strengths for the first image-wells (Eqs.(A4)–(A7)).
Looking at Fig. 3a, b, c and d, the strength of the nth image-well depends on the product of all previous image-strengths,therefore the strengths of each image-well series can be ex-pressed as geometrical series such as, for drawdown in Domain1 (in Eq. (1a)):
Fig. 3. Graphical display of the first image-wells and calculation of the corresponding strength coefficients. (a) First image according to L1; (b) first image according to L2; (c)second image according to L1; (d) second image according to L2; and (e) third image according to L1.
B. Dewandel et al. / Journal of Hydrology 509 (2014) 115–131 119
120 B. Dewandel et al. / Journal of Hydrology 509 (2014) 115–131
An ¼Xn
i¼0;2;4
a1i
Xn
i¼2;4;6
a10i Bn ¼
Xn
i¼2;4;6
a2i
Xn
i¼2;4;6
a20i Cn
¼Xn
i¼4;6;8
a3i
Xn
i¼2;4;6
a30i and Dn ¼
Xn
i¼2;4;6
a4i
Xn
i¼2;4;6
a40i ð2aÞ
where
a1!4i ¼ s1!4
i T1 � c1!4i T2
s1!4i T1 þ c1!4
i T2
s1!4i ¼ e
�u1!4ið Þ
e�v1!4
ið Þ u1!4i ¼ S1
4T1t R1!4i
� �2
c1!4i ¼ W �u1!4
ið ÞW �v1!4
ið Þ v1!4i ¼ S2
4T2t R1!4i
� �2
8>><>>:
and
a10!40i ¼ s10!40
iT1�c10!40
iT3
s10!40i
T1þc10!40i
T3
s10!40i ¼ e
�u10!40ið Þ
e�v10!40
ið Þ u10!40i ¼ S1
4T1t R10!40
i
� �2
c10!40i ¼ W �u10!40
ið ÞW �v10!40
ið Þ v10!40i ¼ S3
4T3t R10!40
i
� �2
8>><>>:
where e is the exponential function and Ri the distance from image-well to discontinuities, such as:
R1i ¼ ½ð2iLþaÞ2þy2 �
1=2; R2
i ¼ ½ð2iL�aÞ2þy2�1=2; R3
i ¼ ½ð2L½i�2��aÞ2þy2�1=2;
R4i ¼ ½ð2L½i�2�þaÞ2þy2�
1=2
R10
i ¼ ½ð2L½i�1�þaÞ2þy2 �1=2; R20
i ¼ ½ð2L½i�1��aÞ2þy2�1=2; R30
i ¼ ½ð2L½i�1��aÞ2þy2 �1=2;
R40
i ¼ ½ð2L½i�1�þaÞ2þy2 �1=2
For the solution of drawdown in Domain 2 (Eq. (1b)), the image-well strengths are:
b0ðas defined in Eq:A3bÞ; Fn ¼ CnbIIn and En ¼ Dnb
In ð2bÞ
where
bI;IIn ¼
2sI;IIn cI;II
n T2
sI;IIn T1þcI;II
n T2sI;II
n ¼e �uI;II
nð Þ
e �v I;IInð Þ cIII
n ¼W �uI;II
n
� �W �v I;II
n
� � uI;IIn ¼
S14T1t RI;II
n
� �2
v I;IIn ¼
S24T2t RI;II
n
� �2
8><>:
and RIn ¼ ½ð2L½n� 2� þ aÞ2 þ y2�
1=2; RII
n ¼ ½ð2nL� aÞ2 þ y2�1=2
Finally, for the solution of drawdown in Domain 3 (Eq. (1c)) theimage-well strengths are:
b00;Gn ¼ AnbIIIn and Hn ¼ Bnb
IVn ð2cÞ
where b00 is the reciprocal of b0 according to L2 (see Eqs. (A4) inAppendix A) and
bIII;IVn ¼ 2sIII;IV
n cIII;IVn T2
sIII;IVn T1þcIII;IV
n T2sIII;IV
n ¼ e �uIII;IVnð Þ
e �v III;IVnð Þ cIII;IV
n ¼W �uIII;IV
n
� �W �v III;IV
n
� � uIII;IVn ¼ S1
4T1t RIII;IVn
� �2
v III;IVn ¼ S2
4T2t RIII;IVn
� �2
8><>:
and
RIIIn ¼ ½ð2L½nþ 1� � aÞ2 þ y2�
1=2; RIV
n ¼ ½ð2L½n� 1� � aÞ2 þ y2�1=2
As a result, the strength coefficients depend on geometrical ser-ies that are functions of the diffusivities of each domain, time, dis-tance from pumping well to the first discontinuity, and width ofthe strip aquifer. Note that for the case of time and space-constantstrength coefficients, i.e. equal-diffusivity ratio between the com-partments, the solutions are exact and that for the limiting casesof no-flow and constant-head boundaries – strength coefficientsare thus equal to 1 or � 1, respectively – drawdown solutions inDomain 1 are exact and identical to the ones given in, for example,Ferris et al. (1962) and Kruseman et al. (1990).
2.1.2. Drawdown solution at the pumping wellFor a producing well of radius rwðxrw ¼ yrw
¼ffiffiffi2p
rwÞ and for theparticular case where the well is at the centre of the strip aquifer(i.e. a = L and rw/L ? 0) and properties of domains 2 and 3 areidentical (i.e. T2 = T3; S2 = S3; then g2 = g3), Eqs. (1a) and (2a) aresimplified. The resulting drawdown solution at the pumping wellis then:
spwðrw;tÞ¼Q
4pT1W
S1
4T1tr2
w
� �þ2
X1n¼1;2;3;...
Yn
i¼1;2;3;...
aiWS1
4T1tð2nLÞ2
� �( )
ð3Þ
where
ai ¼siT1 � ciT2
siT1 þ ciT2si ¼
e�u0i
e�v 0i
ki ¼W u0i� �
W v 0i� � u0i ¼
S14T1t ð½ð2i� 1ÞL�2Þ
v 0i ¼S2
4T2t ð½ð2i� 1ÞL�2Þ
8<:
Drawdown solutions at any point in the strip aquifer and in theright- and left-side compartments are given in Appendix B.
Fig. 4 presents the result of Eq. (3) using dimensionless time
tDL ¼ T1tS1L2
� �and dimensionless drawdown sDðtDL; rw=LÞð
¼ 2pT1Q spwðrw; tÞÞ for the case where T1 = 10xT2 and diffusivity ratios
g1/g2 vary from 0.025 to 30.A convenient starting point for the analysis of (3) is the log–log
diagnostic plot developed by Bourdet et al. (1983) for constantpumping rates. In this plot, both drawdown (sD) and its derivative(s0D) with respect to the natural logarithm of time (i.e. @s/@ ln t) arerepresented. Among others, this log–log plot is commonly used foridentifying flow regimes during a test (e.g. Ehlig-Economides,1988; Bourdet et al., 1989; Spane and Wurstner, 1993; Renardet al., 2009). For all presented derivative curves in this work, thederivatives were computed numerically according to the algorithmproposed by Bourdet et al. (1989).
At the start of pumping, when the cone of depression has notyet reached the discontinuity (tDL < 0.04 on Fig. 4), (3) is identicalto the solution of an infinite homogeneous aquifer (Theis, 1935).For large enough tDL, a condition when Theis’ equation reduces toCooper-Jacob’s equation, a straight line on a semi-logarithmic plotwith a slope 1/2 characterises dimensionless drawdown. Its loga-rithmic derivative, @sD/@ ln tDL, is a constant equal to 1/2 reflectingthe first infinite-radial flow to the well. Therefore, for short times,the logarithmic derivative of drawdown will converge to@s/@ ln t = Q/4pT1.
Once the cone of depression starts to reach the discontinuities(tDL > 0.04), it is the storage ratio between the strip-aquifer andthe adjacent aquifers that changes the character of the drawdowncurves (Strelsova, 1988). If, for a constant transmissivity ratio be-tween the three compartments—as in Fig. 4—the storage of theun-pumped domain increases (i.e. S1/S2 < 1 and g1/g2 > 1), the finalradial flow will occur later compared to the equal-diffusivity case(curve D, Fig. 4), after a transitional period characterised by a flat-ter drawdown curve. This is exactly what is shown by curves E, Fand G on Fig. 4. This lower decrease of drawdown reflects thepressure support provided by the storage of the outside compart-ments, and thus the flatter signature of derivative curves, whichcauses the discontinuities to appear temporarily as rechargingboundaries.
Conversely, if the storage of outside compartments is lower(S1/S2 > 1 and g1/g2 < 1; curves A, B and C on Fig. 4), the final radialflow will also occur later than for the equal-diffusivity case, andlater than for the previous cases (i.e. g1/g2 > 1). The transitionalperiod will be sharper with an increase in the derivative slopeduring the intermediate stage of pumping compared to theequal-diffusivity case (curve D), causing the discontinuities toappear temporarily as no-flow boundaries.
Fig. 4. Type-curves of dimensionless drawdown (sD) and derivatives s0D� �
, and dimensionless time (tDL) for a pumping well centred in a strip aquifer (L = a) with variousdiffusivity ratios and T1=10 � T2. Corresponds to (3).
B. Dewandel et al. / Journal of Hydrology 509 (2014) 115–131 121
For very late stages of pumping (i.e. t ?1 or tDL ?1), ki and si
in Eq. (3) tend to 1 and, therefore, ai tends to the contrast in trans-missivity between the domains: (T1 � T2)/(T1 + T2). The evaluationof the logarithmic derivative of the dimensionless drawdown forlate-pumping stages leads to:
limtDL!1
@sD
@ ln tDL� 1
21þ 2
X1n¼1;2;...
T1 � T2
T1 þ T2
� �n !
For the case where the ratio (T1 � T2)/(T1 + T2) is either higherthan � 1 (T2 > T1) or lower than 1 (T2 < T1), this last equation canbe regarded as a Taylor series that tends, for n ?1, to a constantvalue that depends on the transmissivity of the strip aquifer (D1)and those of the adjacent domains. The value tends to @sD/@ lntDL � T1/2T2. As a result, the logarithmic derivative of drawdown(@s/@ ln t) will converge to a constant value inversely proportionalto the transmissivity of the adjacent domains (@s/@ ln t � Q/4p T2).By extension, where the properties of the two adjacent domainsare different (g2 – g3) the logarithmic derivative of drawdown,@s/@ ln t, will converge to a constant value inversely proportionalto the arithmetic average of transmissivity from domains 2 and3: @s/@ ln t � Q/[2p (T2 + T3)]. This shows that, for late-stage pump-ing, the drawdown slope does not depend on the transmissivity ofthe pumped aquifer (T1), but on those of the adjacent compart-ments even if they are superficial (h1� h2 and h3, Fig. 1). Thisbehaviour agrees with previous work that examined drawdownaround a well within a central strip or a vertical fracture embeddedin a matrix with differing diffusivity (Gringarten et al., 1974; Butlerand Liu, 1991; Rafini and Larocque, 2012).
This analysis is also valid for the general solution (a – L; Eq.(1a)). We repeated it for the drawdown solutions in domains 2and 3 (Eqs. (1b) and (1c)) and found that for late-stage pumping@s/@ ln t tends to the same limit (see also Numerical evaluation ofsolutions, hereafter). This result indicates that the semi-log methodfor uniform aquifers (Cooper and Jacob, 1946) is applicable to awide variety of non-uniform aquifers (e.g., van Poollen, 1965;Butler and Liu, 1991; Meier et al., 1998; Strelsova, 1988;Sánchez-Vila et al., 1999). However, tests done in compartmentedaquifers have to be long enough, e.g. 5–30 days, to allow a properestimation of the transmissivity of the external compartmentswith the use of Cooper-Jacob method.
2.2. Anisotropic case
In an ideal anisotropic aquifer, the hydraulic conductivity in thehorizontal plane (x,y) can be represented by a tensor with twoorthogonal major and minor axes, Txx and Tyy (Hantush, 1966; Han-tush and Thomas, 1966; Ramey, 1975; Neuman et al., 1984; Heilw-
eil and Hsieh, 2006). Note that for a fully penetrating pumpingwell, flow lines are parallel as well as being parallel to the bottomof the aquifer (or orthogonal to the z-axis). Consequently, we donot consider anisotropy of hydraulic conductivity in the verticalplane. Assuming that the main axes are oriented along the x � andy-directions—kyy or Tyy are thus parallel to the discontinuities—, theanisotropic domain, i.e. Domain 1, can be transformed into anequivalent isotropic domain by defining a new set of coordinatessuch as: _x ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiTyy=T1
px and _y ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiTxx=T1
py with T1 ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiTxxTyy
p.
This re-scaling, valid for Domain 1, cannot be applied to do-mains 2 and 3 because they have to be isotropic. For the adjacentdomains and following Bear and Dagan (1965) and Anderson(2006), we found that the re-scaling transformation can be written
as €x ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiTyy=Txx
px0 and ÿ = y for Domain 2, and as x
v¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiTyy=Txx
px00
and yv¼ y for Domain 3, where x0 and x00 are functions of the real-
coordinate system that assumed at discontinuity L1 (i.e. betweenD1 and D2): x = x0 = a, and at the second discontinuity L2 (i.e. be-tween D1 and D3): x = x00 = �(2L � a). These re-scaling relation-ships show that domains 2 and 3 must be shifted upward alongthe x-axis to account for the transformed width and coordinatesof Domain 1. Also, re-scaling must satisfy the long time that draw-down solutions have to be characterised by straight lines on asemi-log plot whose slope depends on the average of transmissiv-ity of the two external compartments. However, at this stage of thedevelopment, x0 and x00 are unknown functions of x, but known atthe discontinuities, which allows developing a solution in D1 only.x0 and x00 will be evaluated numerically to find drawdown solutionsfor domains 2 and 3 in the real-coordinates system (see Numericalevaluation of the solutions, hereafter).
The boundary conditions along the two discontinuities as ex-pressed at the beginning can be applied with the new sets of trans-formed coordinates, and the strength of each image-well can be re-evaluated according to the technique presented previously(Appendix A).
2.2.1. Drawdown solutions for the three domainsFor simplicity, we consider the particular case where the pump-
ing well is located at the centre of the strip aquifer (i.e. a = L) andthe properties of domains 2 and 3 are identical (i.e. T2 = T3 andS2 = S3); the drawdown solutions for the three domains wherea – L and g1 – g2 – g3 are given in Appendix C.
The drawdown solution for a well in Domain 1 is:
s1ðx;y;tÞ¼Q
4pffiffiffiffiffiffiffiffiffiffiffiffiffiTxxTyy
pW S1
4T21 t
Tyyx2þTxxy2 � �
þX1
n¼1;2;3;...
Yn
i¼1;2;3;...
axyi W S1
4T21 t½Tyyð2nL�xÞ2þTxxy2 �
� �
þX1
n¼1;2;3;...
Yn
i¼1;2;3;...
axyi W S1
4T21 t½Tyyð�2nL�xÞ2þTxxy2 �
� �8>>>><>>>>:
9>>>>=>>>>;
ð4aÞ
122 B. Dewandel et al. / Journal of Hydrology 509 (2014) 115–131
where
axyi ¼
sxyi T1�cxy
i T2
sxyi T1þcxy
i T2sxy
i ¼e�uxy0
i
e�vxy0i
kxyi ¼
W uxy0
i
� �W vxy0
i
� � uxy0
i ¼S1
4T21 tð½Tyyð2i�1ÞL�2þTxxy2Þ
vxy0i ¼
S24T2 t
Tyy
Txx½ð2i�1ÞL�2þy2
� �8<:
For a well located in the right-side compartment (x0 > 0), thedrawdown solution is:
s2ðx0 ;y;tÞ¼Q
4pT2bxy
0 WS2
4T2tTyy
Txxx02þy2
� �� �þ
X1n¼1;2;3;...
bxyn
Yn
i¼1;2;3;...
axyi W
S2
4T2tTyy
Txxð�2nL�x0 Þ2þy2
� �� �( )ð4bÞ
and for a well in the left-side compartment (for this case x00 = x0 withx0 < 0), the drawdown solution is:
s3ðx0 ;y;tÞ¼Q
4pT2bxy
0 WS2
4T2tTyy
Txxx02þy2
� �� �þ
X1n¼1;2;3;...
bxyn
Yn
i¼1;2;3;...
axyi W
S2
4T2tTyy
Txxð2nL�x0 Þ2þy2
� �� �( )ð4cÞ
where
bxy0 ¼
2sxy0 cxy
0 T2
sxy0 T1þcxy
0 T2sxy
0 ¼e�uxy0
1
e�vxy01
cxy0 ¼
W uxy0
1
� �W vxy0
1
� � uxy0
1 ¼S1
4T21 t½TyyL2þTxxy2�
vxy0
1 ¼S2
4T2 tTyy
TxxL2þy2
h i8<:
and
bxyn ¼
2sxyn cxy
n T2
sxyn T1 þ cxy
n T2
Thus, the drawdown solutions are mathematically similar tothose developed for the isotropic case (Appendix B).
2.2.2. Drawdown solution at the pumping wellConsidering the drawdown solution at the pumping well
(rw= radius of the well), Eq. 4a simplifies and becomes:
spw rw ;t;ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiTyy=Txx
q� �¼ Q
4pffiffiffiffiffiffiffiffiffiffiffiffiffiTxxTyy
p Wr2
wS1
4T21tðTxxþTyyÞ
!þ2
X1n¼1;2;3;...
Yn
i¼1;2;3;...
axyi W
S1
4T21t½Tyyð2nLÞ2 �
!( )
ð5Þ
where axyi has been defined in Eq. 4a, but with
uxy0
i ¼S1
4T21tð½Tyyð2i� 1ÞL�2Þ and vxy0
i ¼S2
4T2tTyy
Txx½ð2i� 1ÞL�2
� �.
Fig. 5a presents the result of Eq. (5) using dimensionless time,tDL, and drawdown, sD, for the case of T1 = 10xT2, diffusivity ratiog1/g2 = 10.0, and the anisotropy ratio in Domain 1, Tyy/Txx, varyingfrom 0.1 to 30.
As discussed previously, Eq. (5) is similar to (3) and thus leadsto a similar behaviour of the derivative curves. During the earlypumping stage @sD/@ ln tDL = 1/2(or @s/@ ln t = Q/4pT1), depictingthe first infinite-acting radial flow when the discontinuities havenot been reached by pumping, whereas for late-stage pumping@sD/@ ln tDL tends to T1/2T2 (or @s/@ ln t � Q/4pT2). By extension,where T2 – T3, @s/@ ln t will also tend to Q/[2p(T2 + T3)] for late-stages of pumping. However, compared to the isotropic case(Txx = Tyy, curve D on Fig. 5a), the duration of the first radial flowto the well is reduced when decreasing the ratio Tyy/Txx (curves Eand F), or increased with a higher ratio (curves A, B and C). Notethat in case of very low ratio values, the first radial flow may notappear. In the same manner, the second radial flow occurs morerapidly for low anisotropy ratios, or shifts to longer times for highratio values. This second flow regime may also appear very late, ormay even never happen, when the strip aquifer is the most perme-able compartment with a very high anisotropy ratio. This differ-ence in behaviour shows that when the major axis of thetransmissivity tensor is parallel to the discontinuity sides (i.e. par-allel to the y-axis; Tyy/Txx > 1), drawdown progresses more quicklyalong the y � axis meaning that, compared to the isotropic case,discontinuities are perceived later in time. Conversely, whenTyy/Txx < 1 drawdown will progress more rapidly toward the limits,leading to a faster reaction of the compartments embedding thestrip aquifer. The time at which the limits are reached by pumpingdepends thus on the transmissivity perpendicular to the
discontinuities, i.e. along the x-axis, and can be expressed bytdisc. = L2S/4Txx. This time corresponds on Fig. 5a to the time atwhich derivative curves leave the first radial flow. Note that in caseof isotropic media, Txx has to be replaced by T1 in Eqs. (4) and (5).
Furthermore, increasing the anisotropy ratio (curves A, B and Con Fig. 5a) induces an increase of the diffusivity contrast betweenthe y-axis and the adjacent compartments that, similar to theisotropic aquifer-strip case with g1/g2 > 1 (Fig. 4), gives a flattersignature to the derivative curves (or a smaller decrease of draw-down). This causes the discontinuities to appear temporarily asrecharging boundaries, even if the adjacent compartments are lesstransmissive. This is particularly well demonstrated on Fig. 5b(curves A, B and C), where the dimensionless time is computedaccording to transmissivity along the x-axis tDLx ¼ Txxt
S1L2
� �. With this
new dimensionless time, limits are perceived at the same timewhatever the Tyy/Txx ratio.
Conversely, decreasing the anisotropy ratio will induce (curvesE and F on Fig. 5b), similar to the isotropic case (g1/g2 < 1 Fig. 4), adecrease in the diffusivity contrast between the y-axis and theadjacent compartments, which increases the sharpness of the tran-sitional period between the two radial flow regimes and will causethe discontinuities to appear, temporarily, as no-flow boundaries.
3. Discussion and application to field data
This section focuses on the numerical evaluation of the pro-posed solutions and their application on a pumping test performedin crystalline rocks of Brittany (western France).
3.1. Numerical evaluation of solutions
The performance of Eqs. (2) and (4a), and those given in Appen-dices, has been compared to numerical modelling to evaluate theiraccuracy and to evaluate the functions x0 and x00 required for thedrawdown solutions in domains 2 and 3 when the pumped com-partment (Domain 1) is anisotropic (see Eqs. (4b) and (4c) andsolutions in Appendix C).
The numerical modelling was done with the 3D ‘MARTHE’hydrodynamic modelling code (Modelling Aquifers with Rectangu-lar grids in Transitory regime for Hydrodynamic flow calculation;Thiéry, 1993; Thiéry, 1994; Thiéry, 2010) developed at BRGM(French Geological Survey). This code allows 3D hydrodynamicand hydrodispersive modelling of groundwater flow in porousmedia. The hydrodynamic calculations use the integrated finite-difference method with an implicit scheme; here, we used themethod of conjugate gradients with Cholesky decomposition. Con-vergence of the calculations is checked with several criteria, suchas the difference of hydraulic head between two successive itera-tions, and residual error flow.
The grid geometry used in the numerical model is a square40 � 40 km grid with a constant head boundary condition on eachside. In the x, y plane, cell size varies from 1 � 1 m at and near thepumping well to 500 � 500 m near the model boundaries. Verti-cally, the pumped aquifer is a 100-m-thick layer subdivided into25 4-m-thick layers. The aquifer is capped by an impermeablelayer (no-flow condition) making the pumped layer under confinedcondition.
Several diffusivity ratios, thickness ratios between the three do-mains, and widths of the strip aquifer were used for testing the iso-tropic ((2)) and anisotropic (Eq. (4), Eqs. C1–C3) solutions atvarious observation-well locations, i.e. within the strip aquiferand in the right- and left-side compartments. Here, we only showa few of these tests with two examples (Fig. 6a and b) done for test-ing the anisotropic solutions with g1 – g2 – g3 (Eqs. C1–C3,Appendix C). In the model, the pumping well fully penetrates the
(a)
(b)
Fig. 5. Type-curves of dimensionless drawdown (sD) and derivatives s0D� �
for a pumping well centred in the strip (L = a) with various anisotropy ratios Tyy/Txx; T1 = 10 � T2.Corresponds to Eq. (5). (a) As a function of dimensionless time, tDL and (b) as a function of anisotropy normalised dimensionless time, tDLx.
B. Dewandel et al. / Journal of Hydrology 509 (2014) 115–131 123
aquifer and is located at the centre of the grid (x = 0; y = 0) and atthe centre of 100-m-wide strip aquifer (2L = 100 m). The geometryof the aquifer is similar to the one in Fig. 1, the strip aquifer (D1)has a thickness, h1, of 100 m while the adjacent compartmentsare 48 and 24 m thick for the right- (h2) and left-side (h3) compart-ments, respectively. In both cases, strip transmissivity T1 is2 � 10�2 m2/s (hydraulic conductivity, k1 = 2 � 10�4 m/s), storagecoefficient S1 2.0 � 10�3 (specific storage, Ss1 = 2 � 10�5 m�1),right-side transmissivity T2 is 2.4 � 10�3 m2/s (k2 = 5 � 10�5 m/s)and S2 is 9.6 � 10�4 (Ss2 = Ss1), and left-side transmissivity T3
1.2 � 10�2 m2/s (k3 = 5 � 10�4 m/s) and S3 is 4.8 � 10�4 (Ss3 = Ss1);thus, g1 = 10.0, g2 = 2.5 (g1/g2 = 0.4) and g3 = 25.0 (g1/g3 = 4.0).Only the anisotropy ratio Tyy/Txx differs between the two examples,being 10.0 for model 1 (Fig. 6a) and 20.0 for model 2 (Fig. 6b). Dur-ing the modelling of up to 48 days of pumping (about 70,000 min)the flow rate was maintained constant (Q = 2.8 � 10�4 m3/s), themodel boundaries where never reached and the pumped aquiferwas never de-saturated.
First, we evaluated the performance of Eq. (4a) and then weused the drawdown computed with several numerical models inthe outer regions (right-side: Domain 2 and left-side: Domain 3)for evaluating and generalising the functions x0 and x00 used for ana-lytical computation of drawdown in the two adjacent domains.
3.1.1. Evaluation of the solution in the pumped compartment (stripaquifer, Domain 1)
Fig. 6a and b compare the results of numerical modelling withanalytically computed drawdown values. As shown in the figures,drawdown values and derivatives computed with Eq. (4a) forobservation wells in the pumped compartment (Domain 1) per-fectly match the values from numerical modelling (curves A, B
and C on the figures). The root mean square error (RMSE) was cal-culated for each numerical-analytical computation of drawdown(see inserted tables on figures). Values range between 1.1 � 10�4
and 3.1 � 10�4 m, showing that Eq. (4a) agrees with the numericalmodelling, which is a good indication that it is accurate.
3.1.2. Evaluation of the solutions in the un-pumped compartments(domains 2 and 3)
As assumed previously, drawdown solutions in domains 2 and 3depend on x0 and x00 that are functions of x. These two functionswere evaluated empirically while prescribing various values of x0
and x00 in the solutions for domains 2 and 3 (solutions are givenAppendix C) until the lowest RMSE values were obtained betweenthe drawdown computed numerically and that computed with thetwo solutions. Then, values for x0 and x00 were compared to the real-coordinates x and y (i.e. the coordinates of the observation wells)for evaluating the functions. Fig. 7 presents the values found forx0 and x00 for various x coordinates and for various anisotropy ratios(Tyy/Txx = 0.05, 0.2, 5, 10 and 20). This graph summarises the anal-yses made with various sets of diffusivity ratios between the com-partments and various compartment thicknesses used in thenumerical model (h1 = 100.0 m; h2 and h3 vary from 24.0 to100.0 m). Only the aquifer-strip width was kept constant(2L = 200.0 m); nonetheless, tests were also done with otherstrip-aquifer widths; see for instance Fig. 6, where 2L = 100.0 m.Note that, for convenience, the figure represents absolute valueswhereas in the real-coordinates system x > 0 and x0 > 0 in Domain2, and that x < 0 and x00 < 0 in Domain 3.
The graph shows that, to account for the anisotropy in Domain1, the transformed-scale in domains 2 and 3 results in a shift alongthe x-axis. The evaluated relationships are linear, which agrees
Fig. 6. Comparison of drawdown, s, and derivative, s0 , computed with MARTHE software (plain dots: s; open dots: s0) and the analytical solutions for the anisotropic case (Eqs.C1–C3; plain curves: s; dotted curves: s0). See the text for model parameters. The inserted tables show RMSE values for various observation-well location (x,y coordinates). (a)Tyy/Txx = 10.0 and (b) Tyy/Txx = 20.0.
124 B. Dewandel et al. / Journal of Hydrology 509 (2014) 115–131
with earlier work with such a configuration (Bear and Dagan,1965). They can be expressed as a function of the anisotropy ratioin Domain 1, Tyy/Txx, and the distances from the pumping well tothe discontinuities. From this, it results that for Domain 2 (x > a):
x0 ¼ xffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiTxx=Tyy
qþ a 1�
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiTxx=Tyy
q� �ð6aÞ
which gives for x > a:
€x ¼ ðx� aÞ þ affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiTyy=Txx
qand for Domain 3 (x < �[2L � a]):
x00 ¼ xffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiTxx=Tyy
qþ ð2L� aÞ 1�
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiTxx=Tyy
q� �ð6bÞ
which gives for x < �½2L� a� : xv¼ ðx� ½2L� a�Þ þ ½2L� a�
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiTyy=Txx
p.
Note that when the pumping well is located at the centre of thestrip aquifer, i.e. a = L, Eqs. (6a) and (6b) become identical.
The analysis was also done for various sets of y values (y = 0,150, 300, 500, 1100 m), but this did not show any significant differ-ences in terms of prescribed x0 and x00 values; which agrees with theproposed re-scaling technique (€y ¼ y and y
v¼ y, see Section 2.2
above.).These empirical relationships were used for computing the
drawdown in domains 2 and 3 shown on Fig. 6 (curves D, E, Fand G), where the strip-aquifer width is 100.0 m. RMSE valuesfor each numerical–analytical drawdown computation are as low
as those given for the solution in Domain 1, between 1.4 � 10�4
and 4.0 � 10�4 m, showing that the solutions—Eq. (4b) and (4c),and the ones in Appendix C with the use of Eqs. (6a) and(6b)—agree with the numerical modelling, which, again, is a goodindication that they are accurate.
3.2. Application to field data: drawdown and fluxes modelling
The example is a recent (October 2011) 63-day pumping testperformed near the City of Rennes in a fault zone that cross-cutsa shallow aquifer system and is a few tens of metres wide(Fig. 8a Saint-Brice-en-Coglés site, Brittany; Roques et al., 2012;Roques et al., 2013). The geology of the area is dominated by horn-fels schists that have been exposed to weathering processes, lead-ing to a 10 m thick regolith layer and to a 40–50-m thick denselyfractured rock probably issued from weathering processes(Maréchal et al., 2004; Wyns et al., 2004; Dewandel et al., 2006;Lachassagne et al., 2011). Together, the regolith layer and the frac-tured zone form a shallow aquifer. The pumping test took place ina 216-m-deep well intersecting the steeply inclined fault zonefrom 110 m below ground surface. During the test, a packer wasplaced at 80 m depth in the well to ensure that pumping was car-ried out within the fault zone. During the pumping test, water leveland flow rate were continuously monitored (automatic recorderswith measurements every minute, cross checked by manualmeasurements). Flow rate varied between 42.8 and 47.8 m3/h.
Fig. 7. Evaluation of the transformed scale along the x-axis for establishing drawdown solutions in Domain 2 (x0) and in Domain 3 (x00).
B. Dewandel et al. / Journal of Hydrology 509 (2014) 115–131 125
The initial water level was 5.3 m below soil surface and at the endof pumping drawdown was 22.1 m. The test was not influenced byrecharge from rainfall.
3.2.1. Drawdown modellingThe derivative curve of measured drawdown shows three main
characteristic flow regimes (Fig. 8b): (A) the early response of thefault zone, (B) a period of transition controlled by the geometricalproperties of the fault zone, and (C) the overall hydraulic responseof the system. This sequence of flow regimes agrees with the Tiab(2005) and Rafini and Larocque (2012) models for a well located intransmissive zone embedded in a low-permeability matrix. Ini-tially, during the first 1000 min of pumping, the response is typicalof a fault zone where water is mainly extracted from local fracturestorage, potentially fed by proximal matrix storage and/or localleakage from overlying aquifers. Then, during a transitory periodbetween 1000 and 50,000 min (35 days) of pumping, the derivativevariations were interpreted as a classic linear-flow response (deriv-ative slope around 0.5) until 10,000 min (7 days) of pumping, fol-lowed by pseudo-steady-state flow conditions (derivative slopenear to 1) that can be related to the total finite size of the perme-able fault system. Finally, the system reached a radial flow regimeuntil the end of pumping, showing that the whole system wasaffected.
The almost constant derivative value implies a global trans-missivity for the system of about 2–7 � 105 m2/s, which reflectsthe average transmissivity of the surrounding compartments. Asthe diagnostic showed the response of a well pumping an infi-nite strip aquifer bounded by aquifers of various properties, thefield data were interpreted with the anisotropic solution (Eq.(5)). In this modelling, the flow-rate variations were consideredas applying the principle of superposition (e.g. Kruseman et al.,1990).
Because the diagnostic from derivative curve shows a global re-sponse of the outer compartments, no evidence can be found thatone of the limits is reached before the other, or that a transmissiv-ity contrast exists between the left- and right compartments. Con-sequently, we assumed that the pumping well is located at thecentre of the strip aquifer and that the properties of the left- andright-side domains are identical. The hydrogeologic parametersof Eq. (5) that best fit both observed drawdown and derivativeare, for the strip aquifer: T1 = 4.9 � 10�3 m2/s and S1 = 4.5 � 10�2
(–), and for the external compartments: T2 = 2.5 � 10�5 m2/s,S2 = 1.5 � 10�4 (–). The width of the pumped compartment (2L)was estimated at 27 m, which seems to be realistic consideringthe available geological data. Anisotropy in transmissivity in the
horizontal plane Tyy/Txx was also necessary to fit the data set, andwas evaluated at 30.0, meaning that the strip aquifer is 30 timesmore transmissive parallel to the fault. A quite important wellborestorage effect—corresponding to a borehole radius of 2.5 m—wasnecessary as well, particularly to fit the first two hundred minutesof pumping. This important value can be due to large fractureopenings close to the well and/or may reveal a dual-porosity func-tioning of the strip aquifer. Quadratic head losses(2.5 � 104 m�5 s2) were also integrated in the modelling. Resultsare presented on Fig. 8b. The model correctly fits observed data,even if the behaviour corresponding to the total finite size of thepermeable fault system cannot be reproduced by the model (deriv-ative slope of 1 on observed data); differences between observedand model data are however small, not more than a fewcentimetres.
3.2.2. Fluxes modelling and comparison to geochemical dataIn addition to water-level and flow-rate measurements, the
pumped water was regularly sampled for groundwater dating(CFC12 concentrations) and other geochemical analyses, for evalu-ating the contributions of the fault zone and of the shallow com-partments (Roques et al., 2012, 2013a, b). Based on sophisticatedgeochemical modelling (CFC, anions, cations, trace elements, iso-topes), Roques et al. (2013) identified three poles, two of whichbeing related to deep compartments with water of a high apparentage (fault zone and low-permeability matrix of the fault zone). Forsimplification, as our solutions consider two main aquifer systems,the relative geochemical contributions (Fig. 8c) of the fault zoneand the shallow compartments were estimated by considering asimple binary mixing model (Plummer and Busenberg, 2000;Ayraud et al., 2008). This geochemical model assumes that mixingoccurred between two poles, one characterised by water of a highapparent age (55 ± 5 years; CFC12 concentrations of 0–50 pptv atthe pumping well under ambient flow conditions) representingthe signature of the entire fault-zone water, and the other by waterfrom the shallow weathered and fractured aquifer (mean apparentage of 10 ± 10 years; CFC12 concentrations of 500–600 pptv mea-sured in shallow boreholes in the weathered zone before thepumping). After this, the contributions of the fault zone and ofthe shallow compartments deduced from the geochemical modelwere compared to those computed with the proposed analyticalsolutions.
Using the solutions described in this work, the fluxes from theadjacent compartments entering the strip aquifer can be estimatedfrom the solutions while integrating one of the two flux conditionsat the discontinuities (i.e. x = L) for �1 < y <1, such as:
(a)(a)
Fig. 8. Application to a field example in crystalline aquifers of Brittany (western France). (a) Site location (map is modified from Chantraine et al. (2001)). (b) Observed andmodelled drawdown and derivative, and measured flow rate; use of solutions of Eq. (5). (c) Flow contributions deduced from groundwater dating (CFC12; (Roques et al.,2012; Roques et al., 2013)) and computed with Eq. (7).
126 B. Dewandel et al. / Journal of Hydrology 509 (2014) 115–131
Q adjacent comp: ¼ 2Z þ1
�1T1@s1ðL; y; tÞ
@xdy
¼ 2Z þ1
�1T2@s2ðL; y; tÞ
@xdy ð7Þ
with, and taking the solution of the right-side compartment at thelimit (x = x0 = L):
T2@s2ðL;y;tÞ
@x¼� Q
4p
bxy0
�2LTyyTyyTxx
L2þy2 Exp � S2
4T2 tTyy
TxxL2þy2
h i� �
þX1
n¼1;2;3;...
bxyn
Yn
i¼1;2;3;...
axyi
�2L 1�2nð ÞTyyTyyTxx
Lð1�2nÞ½ �2þy2 Exp � S2
4T2 tTyy
Txxð�2nL�LÞ2þy2
h i� �8>>><>>>:
9>>>=>>>;
where Qadjacent comp. is the flux entering the strip aquifer, andQ = Qstrip�aquifer + Qadjacent comp. (Q = pumping flow-rate; andQstrip�aquifer: flux from the strip aquifer). Note that the integrals inEq. 7 are multiplied by two to account for the two discontinuitiesL1 and L2.
Fluxes were computed, and the relative contributions of thefluxes from the strip aquifer (Qstrip�aquifer/Q; or contributionfrom the fault zone) and from the adjacent compartments(Qadjacent comp./Q; or contribution from the shallow compartments)were compared to those deduced from geochemical analyses(Fig. 8c). Both hydrodynamical and geochemical models agree,which clearly confirms the influence of the recent water solicitedduring pumping. They show that this sub-vertical fault zone is a thin,but efficient, aquifer allowing a rapid transfer of pressure able to
capture water from a shallow aquifer. Such a result agrees with Lerayet al. (2013) and Roques et al. (2013). At the end of the pumping,about 15–20% of the pumped water came from the shallow fracturedaquifer.
4. Conclusion
Our study shows that the unconventional application ofwell-image theory, originally proposed by Fenske (1984) forestablishing drawdown solutions for a medium composed of twocompartments with varying diffusivity, can be extended to the caseof an infinite linear and anisotropic strip-aquifer that drains shal-low aquifers of differing diffusivities and thicknesses. The use ofthis application of the image-well theory is assumed to be applica-ble to other well functions. However, functions should be derivableon x to get mathematical expressions of strength coefficients. Forexample, solutions could be derived for a partially penetrating welllocated in a 3-D anisotropic strip aquifer using the Hantush andHantush-leaky well functions (Hantush, 1964; Hantush, 1966).
The proposed solutions can be applied to an aquifer structurewith a ‘T’ geometry, i.e. with a thickness of the strip aquifer beingvery high compared to those of the adjacent compartments (h1 > h2
and h1 > h3). This aquifer geometry resembles a sub-vertical faultor vein cross-cutting shallow aquifers, which are appreciatedhydrogeological targets for well siting in crystalline-rock aquifers.
(b)
(c)
Fig. 8 (continued)
B. Dewandel et al. / Journal of Hydrology 509 (2014) 115–131 127
An interesting point of the proposed solutions is the possibilityto compute fluxes from adjacent compartments entering the stripaquifer (or fault zone) and to compare these to other estimates(geochemical, flow-metre, etc.). This is particularly well-demon-strated by the pumping test on the Saint Brice-en-Coglès site(Fig. 8), where the contribution of shallower aquifers is confirmedby geochemical data. Although this aspect should be demon-strated further by other case studies. Such computations can bevery useful in terms of groundwater resource management,assessing the impact of pumping on shallow aquifers and/orevaluating the contribution of water from shallow aquifers inthe pumping well.
Acknowledgements
The authors are grateful for a research-sponsorship from BRGM(France) and from the CASPAR Research Project co-funded by theFrench Water Agency of Loire-Brittany (AELB), the Regional ofCouncil of Brittany, the Department of Ille-et-Vilaine, the City ofRennes and the French Ministry for Education and Research. C.Darnault, Associate Editor of the Journal, and the two anonymousJournal referees are thanked for their useful remarks and com-ments that improved the quality of the paper. We are grateful toDr. H.M. Kluijver for revising the English text.
Appendix A
Calculation of strength coefficient for the first image-wellsaccording to discontinuities L1 and L2 (Fig. 3a–d).
We first consider the first image according to L1 (discontinuitybetween D1 and D2; Fig. 3a). In this case, the problem refers to theinfluence of a partial hydrologic barrier on a well test (Bixel et al.,1963; Fenske, 1984; Maximov, 1962; Nind, 1965; Raghavan, 2010).
Applying the first boundary condition, it results that drawdownin Domain 1 is:
s1ðx;y;tÞ¼Q
4pT1W
S1
4T1t½x2þy2�
� �þA0W
S1
4T1t½ð2a�xÞ2þy2�
� � �ðA1aÞ
and that drawdown in Domain 2 is:
s2ðx; y; tÞ ¼Q
4pT2b0 W
S2
4T2t½x2 þ y2�
� � �ðA1bÞ
Equating Eqs. (A1a) and (A1b), and for the condition x = a (i.e. s1
(a,y, t) = s2(a,y, t) at the discontinuity), we obtain:
1þ A0 ¼T1
T2b0
W S24T2t ½a2 þ y2�� �
W S14T1t ½a2 þ y2�� � ¼ T1
T2
b0
c0ðA1cÞ
Then, applying the second boundary condition, discharge on the leftside of the discontinuity is:
T1@s1
@x¼�Q
4p
Exp � S14T1 t ½x2þy2 �
� �S1
4T1 t ½x2þy2�2x4T1 t
S1
þA0
Exp � S14T1 t ½ð2a�xÞ2þy2 �
� �S1
4T1 t ½ð2a�xÞ2þy2 �2x�4a
4T1 tS1
8<:
9=; ðA2aÞ
where Exp is the exponential function and on the right side, is
T2@s2
@x¼ �Q
4pb0
Exp � S24T2t ½x2 þ y2�
� �S2
4T2t ½x2 þ y2�2x4T2t
S2
ðA2bÞ
128 B. Dewandel et al. / Journal of Hydrology 509 (2014) 115–131
To meet the requirement of the second boundary condition,that is the discharge on both sides must be equal (i.e.T1
@s1ða;y;tÞ@x ¼ T2
@s2ða;y;tÞ@x ), we obtain:
1� A0 ¼ b0
Exp � S24T2t ½a2 þ y2�
� �Exp � S1
4T1t ½a2 þ y2�� � ¼ b0
s0ðA2cÞ
Eqs. (A1c) and (A2c) are solved simultaneously to find A0 andb0:
A0 ¼s0T1 � c0T2
s0T1 þ c0T2: ðA3aÞ
and
b0 ¼2s0c0T2
s0T1 þ c0T2ðA3bÞ
where
c0 ¼W S1
4T1t ½a2 þ y2�� �
W S24T2t ½a2 þ y2�� � and s0 ¼
Exp � S14T1t ½a2 þ y2�
� �Exp � S2
4T2t ½a2 þ y2�� �
This case that considers the system composed of only two do-mains (D1 and D2) is exactly the solution proposed in Fenske(1984) for various diffusivity contrasts (i.e. g1 – g2). Furthermore,where there is no-diffusivity contrast (i.e. g1 = g2) the solutionsfor drawdowns in the two domain are identical to Nind (1965)and Raghavan (2010).
Reciprocally, the first image-well strengths according to thesecond discontinuity (L2) are obtained such as (Fig. 3b):
b00 ¼2s00c00T3
s00T1 þ c00T3and C2 ¼
s00T1 � c00T2
s00T1 þ c00T2ðA4Þ
where
c00 ¼W S1
4T1t ½ð2L� aÞ2 þ y2�� �
W S34T3t ½ð2L� aÞ2 þ y2�� � and
s00 ¼Exp � S1
4T1t ½ð2L� aÞ2 þ y2�� �
Exp � S34T3t ½ð2L� aÞ2 þ y2�
� �Considering the case of two discontinuities these first image-wellshave to be imaged about the discontinuities and strengths of thesenew images evaluated according to boundary conditions, and thesame process applied iteratively for each new image-wells.
For the second image according to L1 (discontinuity betweenD1 and D2; Fig. 3c), we obtain:
B2 ¼ C2s1T1 � c1T2
s1T1 þ c1T2F2 ¼ C2
2s1c1T3
s1T1 þ c1T3ðA5Þ
where
c1 ¼W S1
4T1t 4L� að Þ2 þ y2h i� �
W S24T2t ½ð4L� aÞ2 þ y2�� � and
s1 ¼Exp � S1
4T1t ½ð4L� aÞ2 þ y2�� �
Exp � S24T2t ½ð4L� aÞ2 þ y2�
� �For the second image according to L2 (discontinuity between
D1 and D3; Fig. 3d), we obtain:
D2 ¼ A0s01T1 � c01T3
s01T1 þ c01T3G0 ¼ A0
2s01c01T3
s01T1 þ c01T3ðA6Þ
where
c01 ¼W S1
4T1t 2Lþ að Þ2 þ y2h i� �
W S34T3t ½ð2Lþ aÞ2 þ y2�� � and
s01 ¼Exp � S1
4T1t ð2Lþ aÞ2 þ y2h i� �
Exp � S34T3t ½ð2Lþ aÞ2 þ y2�
� �For the third image according to L1 (discontinuity between D1 andD2; Fig. 3e), we obtain:
A2 ¼ B2s2T1 � c2T2
s2T1 þ c2T2E2 ¼ F2
2s2c2T2
s2T1 þ c2T2ðA7Þ
where
c2 ¼W S1
4T1t ½ð4Lþ aÞ2 þ y2�� �
W S24T2t ð4Lþ aÞ2 þ y2h i� � and
s2 ¼Exp � S1
4T1t ½ð4Lþ aÞ2 þ y2�� �
Exp � S24T2t ½ð4Lþ aÞ2 þ y2�
� �and so on.
Appendix B
Complement of Eq. (3), the pumping well is located at the cen-tre of the strip aquifer and properties of Domains 2 and 3 areidentical.
Solution of drawdown for a well in the strip aquifer (Domain 1):
s1ðx;y;tÞ¼Q
4pT1
W S14T1 t ½x2þy2�� �
þX1
n¼1;2;3;...
Yn
i¼1;2;3;...
aiW S14T1t ½ð2nL�xÞ2þy2�� �
þX1
n¼1;2;3;...
Yn
i¼1;2;3;...
aiW S14T1 t ½ð�2nL�xÞ2þy2�� �
8>>>><>>>>:
9>>>>=>>>>;ðB1Þ
where
ai¼siT1�ciT2
siT1þciT2si¼
e�u0i
e�v 0i
ki¼W u0i� �
W v 0i� � u0i¼
S14T1t 2i�1ð ÞL½ �2þy2� �
v 0i ¼S2
4T2t ð½ð2i�1ÞL�2þy2Þ
8<:
Drawdown solution in the right-side compartment (Domain 2) is:
s2ðx;y;tÞ¼Q
4pT2b0W
S2
4T2t½x2þy2 �
� �þ
X1n¼1;2;3;...
bn
Yn
i¼1;2;3;...
aiWS2
4T2t½ð�2nL�xÞ2þy2 �
� �( )
ðB2Þ
and drawdown solution in the left-side compartment (Domain 3) is:
s3ðx;y;tÞ¼Q
4pT2b0W
S2
4T2t½x2þy2 �
� �þ
X1n¼1;2;3;...
bn
Yn
i¼1;2;3;...
aiWS2
4T2t½ð2nL�xÞ2þy2 �
� �( )
ðB3Þ
where
bn ¼2sncnT2
snT1þcnT2sn ¼
e�u0n
e�v 0nkn ¼
W u0n� �
W v 0n� � u0n ¼
S14T1 t ð½ð2n�1ÞL�2þy2Þ
v 0n ¼S2
4T2 t ð½ð2n�1ÞL�2þy2Þ
8<:
Appendix C
Complement of Eq. (4), the pumping well is located in an aniso-tropic strip aquifer (a – L) and properties of Domains 2 and 3 aredifferent. The solution of drawdown for a well in the strip aquifer(Domain 1) is:
B. Dewandel et al. / Journal of Hydrology 509 (2014) 115–131 129
s1ðx;y;tÞ¼Q
4pffiffiffiffiffiffiffiffiffiffiffiffiffiTxxTyy
p
W S14T2
1 t½Tyyx2þTxxy2 �
� �þ
X1n¼0;2;4;...
Axyn W S1
4T21 t
Tyyð2nLþ2a�xÞ2þTxxy2h i� �
þX1
n¼2;4;...
Bxyn W S1
4T21 t½Tyyð2nL�xÞ2þTxxy2 �
� �
þX1
n¼2;4;...
Cxyn W S1
4T21 t½Tyyð�ð2nL�2aÞ�xÞ2þTxxy2 �
� �
þX1
n¼2;4;...
Dxyn W S1
4T21 t½Tyyð�2nL�xÞ2þTxxy2 �
� �
8>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>:
9>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>;ðC1Þ
where
Axyn ¼
Xn
i¼0;2;4
a1xyi
Xn
i¼2;4;6
a1xy0
i
a1xyi with
u1xyi ¼
S1
4T21t½Tyyð2iLþaÞ2þTxxy2�
v1xyi ¼
S24T2 t
Tyy
Txxð2iLþaÞ2þy2
h i8<:
a1xy0
i withu1xy0
i ¼ S1
4T21 t½Tyyð2L½i�1�þaÞ2þTxxy2�
v1xy0
i ¼ S34T3t
Tyy
Txxð2L½i�1�þaÞ2þy2
h i8<:
8>>>>>>>><>>>>>>>>:
Bxyn ¼
Xn
i¼2;4;6
a2xyi
Xn
i¼2;4;6
a2xy0
i
a2xyi with
u2xyi ¼ S1
4T21 t½Tyyð2iL� aÞ2 þ Txxy2�
v2xyi ¼ S2
4T2tTyy
Txxð2iL� aÞ2 þ y2
h i8<:
a2xy0
i withu2xy0
i ¼ S1
4T21t½Tyyð2L½i� 1� � aÞ2 þ Txxy2�
v2xy0
i ¼ S34T3t
Tyy
Txxð2L½i� 1� � aÞ2 þ y2
h i8<:
8>>>>>>>><>>>>>>>>:
Cxyn ¼
Xn
i¼4;6;8
a3xyi
Xn
i¼2;4;6
a3xy0
i
a3xyi with
u3xyi ¼ S1
4T21 t
Tyy 2L i� 2½ � � að Þ2 þ Txxy2h i
v3xyi ¼ S2
4T2tTyy
Txx2L i� 2½ � � að Þ2 þ y2
h i8><>:
a3xy0
i withu1xy0
i ¼ S1
4T21t
Tyy 2L i� 1½ � � að Þ2 þ Txxy2h i
v1xy0
i ¼ S34T3t
Tyy
Txx2L i� 1½ � � að Þ2 þ y2
h i8><>:
8>>>>>>>>><>>>>>>>>>:
and
Dxyn ¼
Xn
i¼2;4;6
a4xyi
Xn
i¼2;4;6
a4xy0
i
a4xyi with
u4xyi ¼ S1
4T21t
Tyy 2L i� 2½ � � að Þ2 þ Txxy2h i
v4xyi ¼ S2
4T2tTyy
Txx2L i� 2½ � � að Þ2 þ y2
h i8><>:
a4xy0
i withu4xy0
i ¼ S1
4T21t
Tyy 2L i� 1½ � � að Þ2 þ Txxy2h i
v4xy0
i ¼ S34T3 t
Tyy
Txx2L i� 1½ � � að Þ2 þ y2
h i8><>:
8>>>>>>>>><>>>>>>>>>:
with
a1xy0!4xy0
i ¼ s1xy0!4xy0
i T1 � c1xy0!4xy0
i T3
s1xy0!4xy0
i T1 þ c1xy0!4xy0
i T3
s1xy0!4xy0
i ¼ e�u1xy0!4xy0
i
� �e�v1xy0!4xy0
i
� �c1xy0!4xy0
i ¼ W �u1xy0!4xy0i
� �W �v1xy0!4xy0
i
� �
8>>><>>>:
and
a1xy!4xyi ¼ s1xy!4xy
i T1 � c1xy!4xyi T2
s1xy!4xyi T1 þ c1xy!4xy
i T2
c1xy!4xyi ¼ W �u1xy!4xy
ið ÞW �v1xy!4xy
ið Þ
s1xy!4xyi ¼ e
�u1xy!4xyið Þ
e�v1xy!4xy
ið Þ
8>><>>:
Drawdown solution in the right-side compartment (Domain 2)is:
s2ðx;y;tÞ¼Q
4pT2
bxy0 W S2
4T2 tTyy
Txxx02þy2
h i� �þ
X1n¼2;4;6;...
Exyn W S2
4T2 tTyy
Txxð�2nL�x0Þ2þy2
h i� �
þX1
n¼2;4;6;...
Fxyn W S2
4T2 tTyy
Txxð�ð2nL�2aÞ�x0Þ2þy2
h i� �8>>>><>>>>:
9>>>>=>>>>;ðC2Þ
with x0 ¼ xffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiTxx=Tyy
pþ a 1�
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiTxx=Tyy
p� �where
bxy0 with
uxy1 ¼
S14T2
1t½Tyya2 þ Txxy2�
vxy1 ¼
S24T2
2tTyy
Txxa2 þ Txxy2
h i8<:
Fxyn ¼ Cxy
n b3xyn b3xy
0 withuIIIxy
n ¼ S14T2
1tTyyð2nL� aÞ2 þ Txxy2h i
v IIIxyn ¼ S2
4T2tTyy
Txxð2nL� aÞ2 þ y2
h i8><>:
Exyn ¼ Dxy
n b4xyn b4xy
0 withuIVxy
n ¼ S14T2
1tTyyð2L½n� 2� þ aÞ2 þ Txxy2h i
v IVxyn ¼ S2
4T2tTyy
Txxð2L½n� 2� þ aÞ2 þ y2
h i8><>:
and
b3xy;4xyn ¼ 2s3xy;4xy
n c3xy;4xyn T2
s3xy;4xyn T1 þ c3xy;4xy
n T2
Finally, drawdown solution in the left-side compartment (Do-main 3) is:
s3ðx;y;tÞ¼Q
4pT3
bxy0
0 W S34T3 t
Tyy
Txxx002þy2
h i� �þ
X1n¼0;2;4;...
Gxyn W S3
4T3 tTyy
Txx2nLþ2a�x00ð Þ2þy2
h i� �
þX1
n¼2;4;6;...
Hxyn W S3
4T3 tTyy
Txx2nL�x00ð Þ2þy2
h i� �8>>>><>>>>:
9>>>>=>>>>;
ðC3Þ
with
x00 ¼ xffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiTxx=Tyy
qþ 2L� að Þ 1�
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiTxx=Tyy
q� �where
bxy0
0 withuxy0
1 ¼S1
4T21t
Tyy 2L� að Þ2 þ Txxy2h i
vxy0
1 ¼S3
4T3tTyy
Txx2L� að Þ2 þ y2
h i8><>:
Gxyn ¼ Axy
n b2xyn b2xy
n withuIxy
n ¼ S1
4T21t
Tyy 2L n� 1½ � þ að Þ2 þ Txxy2h i
v Ixyn ¼ S3
4T3tTyy
Txx2L n� 1½ � þ að Þ2 þ y2
h i8><>:
Hxyn ¼ Bxy
n b1xyn b1xy
n withuIIxy
n ¼ S14T2
1tTyy 2L nþ 1½ � � að Þ2 þ Txxy2h i
v IIxyn ¼ S3
4T3tTyy
Txx2L nþ 1½ � � að Þ2 þ y2
h i8><>:
and
b1xy;2xyn ¼ 2s1xy;2xy
n c1xy;2xyn T3
s1xy;2xyn T1 þ c1xy;2xy
n T3
Appendix D. Supplementary data
Supplementary data associated with this article can be found, inthe online version, at http://dx.doi.org/10.1016/j.jhydrol.2013.11.014.
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