Predictive modelling of hydraulic head responses to dipole flow experiments in a fractured/karstified limestone aquifer: Insights from a comparison of five modelling approaches to

$Page 1: Predictive modelling of hydraulic head responses to dipole flow experiments in a fractured/karstified limestone aquifer: Insights from a comparison of five modelling approaches to$
Journal of Hydrology 454–455 (2012) 82–100

Contents lists available at SciVerse ScienceDirect

Journal of Hydrology

journal homepage: www.elsevier .com/locate / jhydrol

Predictive modelling of hydraulic head responses to dipole flow experimentsin a fractured/karstified limestone aquifer: Insights from a comparisonof five modelling approaches to real-field experiments

Jacques Bodin a,⇑, Philippe Ackerer b, Alexandre Boisson c,1, Bernard Bourbiaux d, Dominique Bruel e,Jean-Raynald de Dreuzy c, Frederick Delay b, Gilles Porel a, Hamid Pourpak d,2

a Université de Poitiers, CNRS IC2MP UMR 7285, 40 avenue du Recteur Pineau, 86022 Poitiers Cedex, Franceb Laboratoire d’Hydrologie et de Géochimie de Strasbourg, Université de Strasbourg/EOST-CNRS, 1 rue Blessig, 67084 Strasbourg Cedex, Francec Géosciences Rennes, Université de Rennes-CNRS, Campus de Beaulieu, 35042 Rennes Cedex, Franced IFP Energies nouvelles (IFPEN), 1-4 avenue de Bois-Préau, 92852 Rueil-Malmaison Cedex, Francee Centre de Géosciences/Mines-ParisTech, 35 rue St Honoré, 77300 Fontainebleau, France

a r t i c l e i n f o

Article history:Received 12 March 2012Received in revised form 26 May 2012Accepted 31 May 2012Available online 16 June 2012This manuscript was handled by CorradoCorradini, Editor-in-Chief, with theassistance of Fritz Stauffer, Associate Editor

Keywords:Carbonate aquiferComparison of modelling approachesPredictive modellingDipole flow experiments

0022-1694/$ - see front matter � 2012 Elsevier B.V. Ahttp://dx.doi.org/10.1016/j.jhydrol.2012.05.069

⇑ Corresponding author.E-mail address: [email protected] (J.

1 Present address: Indo-French Centre for GroundwRoad, Hyderabad 500 606, India.

2 Present address: TOTAL E&P, Centre Scientifique etLarribau, 64000 Pau, France.

s u m m a r y

Five modelling approaches were tested against a well-studied limestone aquifer constituting theHydrogeological Experimental Site (HES) of Poitiers, France. The modelling exercise consisted of predict-ing the drawdown responses of a series of observation wells for two dipole (pumping–injection) flowexperiments involving two distinct well pairs at the HES. The differences between model predictionsappear to be mainly related to the hydraulic datasets used for model parameterisation and, to a muchlesser extent, to the conceptual modelling approach (equivalent porous medium vs. discrete fracturenetworks), the model dimensionality (two-dimensional vs. three-dimensional), and/or the parameterisa-tion approach (forward vs. inverse). Despite the abundance and diversity of calibration/parameterisationdata, all of the models failed to predict the drawdowns with a reasonable degree of accuracy. Only theorder of magnitude of the drawdowns was correctly predicted by three of the five models, whereas allmodels failed to predict the drawdown behaviour at both intermediate and late times. The primarysource of error is attributed to a lack of information in the single-well pumping test data, which didnot capture the multi-permeability structure of the aquifer as revealed a posteriori by the responses todipole flow experiments. This highlights the need to develop novel (or improved) approaches to charac-terise the hydraulic properties of limestone aquifers.

� 2012 Elsevier B.V. All rights reserved.

1. Introduction

1.1. General

A critical issue in modelling groundwater flow and solutetransport in heterogeneous aquifers is determining an optimal bal-ance between model complexity and the available calibration/parameterisation data. Oversimplified models may fail to captureimportant aspects of the flow structure and dynamics, whereasundersimplification may result in costly models that are difficult,if not impossible, to implement due to the scarcity of field data(Gomez-Hernandez, 2006; Hill, 2006; Hunt et al., 2007; Doherty,

ll rights reserved.

Bodin).ater Research (IFCGR), Uppal

Technique Jean Féger, avenue

2011). This issue is particularly relevant for fractured/karstifiedlimestone aquifers, which are known to be highly heterogeneousbecause they contain low-resistance pathways sometimesenlarged by dissolution, in which pressure-head perturbationsand/or solutes may propagate much more rapidly than withinthe intergranular permeability of the rock matrix (White, 2002).As a contribution to this topic, the aim of the present paper is tocompare five modelling approaches that have been applied to awell-characterised limestone aquifer constituting the Hydrogeo-logical Experimental Site (HES) of Poitiers, France.

1.2. Modelling limestone aquifers: literature overview/background

We present here an overview of modelling approaches that canbe applied to fractured/karstified limestone aquifers, which high-lights the need for our comparative study. Further details andreferences on this specific topic can be found in the review by Kov-acs and Sauter (2007), and more general thoughts on the modelling

http://dx.doi.org/10.1016/j.jhydrol.2012.05.069

mailto:[email protected]

http://dx.doi.org/10.1016/j.jhydrol.2012.05.069

http://www.sciencedirect.com/science/journal/00221694

http://www.elsevier.com/locate/jhydrol

J. Bodin et al. / Journal of Hydrology 454–455 (2012) 82–100 83

of fractured aquifers can be found in the review articles by Berko-witz (2002) and Neuman (2005) and in the books by Bear et al.(1993), National Research Council (1996), Adler and Thovert(1999) and Faybishenko et al. (2000).

As noted by Kovacs and Sauter (2007), a primary distinctionmust be drawn between lumped parameter models and distrib-uted models. Lumped parameter (also known as black-box orgrey-box) models subsume the flow and transport processes intoglobal transfer functions based either on time series analysis meth-ods (Rimmer and Salingar, 2006; Bailly-Comte et al., 2008) ormulti-reservoir systems (Geyer et al., 2008; Fleury et al., 2009).Such approaches are generally both appropriate and sufficientwhen the modelling objective is merely to relate the global re-sponse of the aquifer (e.g., spring outflow) to a succession of inputevents (e.g., rainfall recharge). However, distributed parametermodels are required to address spatial variability (e.g., multi-wellwater levels and/or concentrations). The present study focuses onthis latter issue.

The distributed modelling approaches that can be applied tofractured/karstified limestone aquifers can be distinguishedaccording to three main categories: discrete network simulations(Jeannin, 2001; Jourde et al., 2002; Jaquet et al., 2004), continuumapproaches based on either Equivalent-Porous-Medium (EPM)assumptions (Larocque et al., 1999; Scanlon et al., 2003; Dafnyet al., 2010; Peleg and Gvirtzman, 2010) or dual/triple-continuumapproximations (Sauter, 1993; Garfias et al., 1998; Wu et al.,2004), and hybrid models that combine elements of both discretenetwork simulations and continuum approaches (Spiessl et al.,2007; Reimann and Hill, 2009; Wu et al., 2009). In all cases, thedescription of spatial variability can be deterministic or stochastic(Neuman, 2005). The discrete framework is conceptually appealingbecause preferential flow paths are explicitly accounted for; unfor-tunately, this framework is also the most difficult to work with interms of parameterisation and calibration because field informa-tion concerning the spatial distribution and individual geometryof fractures and karst conduits is often limited. In contrast, contin-uum approaches may seem less suited to accurately depicting theinfluence of preferential flow paths over a wide range of scales butcan be coupled with inverse methods to optimise the modelcalibration (Larocque et al., 1999; Scanlon et al., 2003).

1.3. Aim and scope of the present comparison

The comparison of alternative modelling approaches againstreal-world field experiments (Marschall and Elert, 2003; Alonsoet al., 2005; Hodgkinson et al., 2009; Tsang et al., 2009) has proven

Table 1The five modelling approaches compared.

Modelling approach Main dataset used for model calibration

2D continuum/slug-test K-pattern Transmissivity values interpreted from(Audouin and Bodin, 2008)

2D continuum/automatic downscalinginverse method

2004 pumping-test drawdowns

2D continuum/automatic downscalinginverse method

2005 pumping-test drawdowns

3D continuum/sequential inverse method Petrophysical well-log and core data (Bpumping-test drawdowns

3D hybrid DFN–EPM continuum/directparameterisation with no recalibrationof model parameters

Fracture and karst network statistics; trinterpreted from cross-borehole (Audousingle-borehole (Audouin and Bodin, 20values interpreted from the 2005 pumpDelay, 2007b).

Purely discrete (3D) fracture-pipenetworks

Fracture and karst network statistics; 20

to be fruitful for advancing the state of the art in the modelling offractured media. However, all of the above-cited studies wereexclusively related to crystalline rock environments because theywere primarily motivated by problems associated with radioactivewaste disposal. To the best of the authors’ knowledge, this type ofinter-model comparison has never been carried out in carbonaterocks, with a partial exception being the work by Scanlon et al.(2003), who compared a two-dimensional EPM model with alumped parameter model for simulating regional groundwaterflow in a karst aquifer in the US. In the present study, we comparefive distributed modelling approaches that were tested against awell-characterised limestone aquifer (Hydrogeological Experimen-tal Site of Poitiers, HES, France). The applied models are listed inTable 1 and further described below in the Section 2.3. For thepurpose of comparison, a predictive (blind) modelling exercisewas setup on the basis of the H+/P database (De Dreuzy et al.,2006; Audouin et al., 2008), which houses all of the data collectedat the HES (see Section 2.1). The modelling exercise consisted ofpredicting the drawdown responses of a series of observation wellsfor two hypothetical dipole (pumping–injection) hydraulic testsinvolving two distinct well pairs at the HES. These experimentswere performed ‘‘in the real world’’ once the predictive simulationswere fully completed. An important distinction between thecalibration/parameterisation data and the predictive modellingscenario is the difference in boundary conditions that change fromsingle-well (pumping or slug-test) experiments to dipole flowhydraulic-tests.

Owing to its specificities, the present study does not pretend to beexhaustive because (i) it focuses only on flow dynamics and (ii) theinter-model comparison does not include any multi-continuummodelling approach. In fact, two multi-continuum models wereimplemented in parallel to the approaches listed in Table 1. TheIFPEN team tested a dual-porosity model as an alternative to theiradopted single-porosity approach, but no significant benefit wasnoted in terms of calibration accuracy (Pourpak et al., 2009). Onlythe single-porosity model was retained for their predictive simula-tions. A distributed triple-porosity model was also originally imple-mented in parallel to the other approaches by an additionalmodelling team, but its calibration was, unfortunately, not achieved.

2. Methodology

2.1. Description of the site and available dataset

The Hydrogeological Experimental Site (HES) of Poitiers, France(Fig. 1), investigates a confined limestone aquifer approximately

/parameterisation Modellingteam

Forward(F)/Inverse(I)

Stochastic (S)/Deterministic(D)

Code

cross-borehole slug tests Geosc.Rennes

F D STK

LHYGES I S DSI2004

LHYGES I S DSI2005

ourbiaux et al., 2007); 2004 IFPEN I D SQI

ansmissivity valuesin and Bodin, 2008) and07) slug tests; storativitying tests (Kaczmaryk and

IC2MP F S CPM

04 pumping-test drawdowns MinesParis

F S DFN

Fig. 1. The location of the Hydrogeological Experimental Site (HES) in France.

84 J. Bodin et al. / Journal of Hydrology 454–455 (2012) 82–100

100 m thick. The HES includes 32 fully penetrating wells that weredrilled in two separate phases, in 2002–2003 and 2004, down tothe aquifer basement (Toarcian marls) located at a depth ofapproximately 125 m. Most of the wells were arranged accordingto a regular grid in a square area of 210 � 210 m (Fig. 2). All HESwells are either fully screened or open over the entire aquiferthickness.

Extensive hydrogeophysical investigations have been per-formed to infer the permeability structure of the HES aquifer(Bourbiaux et al., 2007; Audouin et al., 2008). The couplingbetween flowmeter and borehole imaging logs indicates that flow-paths in the aquifer are strongly constrained within subhorizontalkarst conduits and subvertical fractures. As frequently observed inlimestone aquifers (Filipponi et al., 2009) and illustrated in Fig. 3,the karst conduits develop only in specific limestone layers locatedin this area at depths of 50 m, 85 m and 115 m. The karst-conduitopenings observed in borehole video images range in size from0.01 m to 3 m. The relative importance of subvertical fracture flow-paths vs. subhorizontal karst-conduit flowpaths is difficult toassess since the subvertical fractures are undersampled by thevertical wells. However, despite both the horizontal and clusterednature of karst-conduit networks, the three karstic levels are, atleast since 2005, indubitably interconnected since the pressure-head perturbations induced by pumping tests or slug tests propa-gate very rapidly between the wells intercepting distinct levels(Kaczmaryk and Delay, 2007a; Audouin and Bodin, 2008). The ver-tical interconnectivity between the three karstic levels may be sup-ported both by the limestone fracturing and by the boreholesthemselves.

Two series of pumping-test experiments were performed in2004 and 2005, following the completion of each drilling cam-paign. For each test, a constant pumping flow rate ranging between30 and 70 m3 h�1 was prescribed over a period ranging from 24 to167 h (Table 2). As shown in Fig. 4, the drawdown responses of thetwo pumping-test campaigns differ notably. Whereas the 2004drawdown curves are clearly separated in time and amplitudeaccording to the lag distance between observed and pumped wells,most drawdown curves recorded in 2005 are merged together inboth amplitude and time (i.e., exhibiting the same drawdownvalue at the same time), regardless of the lag distance. The differ-ences between the 2004 and 2005 drawdown curves were inter-preted in previous studies (Delay et al., 2007; Kaczmaryk andDelay, 2007a, 2007b; Riva et al., 2009) as a possible enhancement

in inter-well connectivity induced by the 2004 pumping testexperiments (resulting from the unclogging of clay-filledflowpaths) and/or the drilling of the 2005 wells. As is apparent ina comparison between Figs. 4 and 3, the similar drawdownresponses observed during the 2005 pumping tests correspondsystematically to wells intercepting high-K flowpaths (which areassumed to provide a high inter-well connectivity), whereas thewells intercepting neither karst-conduit nor fracture flowpaths ex-hibit different hydraulic responses. Note that a similar well-fieldbehaviour has been previously reported by Day-Lewis et al.(2000) in the fractured crystalline rock aquifer of Mirror Lake(USGS Hydrogeology Research Site, New Hampshire, USA).

Various deterministic and stochastic approaches have beenused to interpret the drawdown data from the 2004 and 2005pumping-test campaigns (Delay et al., 2004; Bernard et al., 2006;Delay et al., 2007; Kaczmaryk and Delay, 2007a, 2007b; Rivaet al., 2009). A synthetic presentation of the physical and mathe-matical concepts underlying each of these approaches can be foundin the article by Riva et al. (2009). The ranges of T- and S-values ob-tained with the different interpretation methods are summarisedin Table 3.

A series of single- and cross-borehole slug-test experimentswere carried out in support of the 2005 pumping-test campaign.As the comparison between Table 3 and Table 4 shows, the T-val-ues interpreted from slug tests exhibit a higher variability thanthose interpreted from pumping tests. According to Audouin andBodin (2008), this finding reflects a finer sensibility of slug teststo heterogeneity. Whereas pumping tests involve the entire frac-ture/karst-conduit/matrix system because of the limited storagecapacity of karst conduits and fracture voids, the head perturbationinduced by a slug test propagates only through the flowpaths withthe lowest hydraulic resistance. Cross-borehole slug-test interpre-tations are therefore expected to yield hydrodynamic parameterscharacterising preferential flowpaths corresponding to fracturesand/or karst conduits, whereas pumping tests performed over amuch larger time scale are more likely to define the bulk propertiesof the entire fracture/karst-conduit/matrix system. As a valuablecomplement to cross-borehole slug tests, single-borehole slug testsin the wells intercepting neither karst conduits nor fracture flow-paths (e.g., wells M08, M09, and M10) are expected to yield repre-sentative T-values of the limestone rock matrix.

Complete details about pumping tests and slug tests, includingraw drawdown data, as well as the interpreted hydrodynamicparameters were made available to the modelling teams throughthe H+/P database (De Dreuzy et al., 2006). Other available datasetsincluded borehole flowmeter measurements, borehole images, adetailed lithologic/sedimentary description of two cored boreholes,fracture statistics from outcrop studies (Bourbiaux et al., 2007),and karst-network statistics computed from cave maps in thevicinity of the HES (Bodin and Razack, 1997).

2.2. Overview of the modelling exercise and shared assumptions

Two test problems were considered. Both were based on dipoleexperiments involving one pumping well and one injecting welloperated simultaneously. The experiments were marked as (i)M06–M22 and (ii) M12–M15 with the first mark for the pumpingwell and the second mark for the injecting well. Pumping/injectingflow rates were set at 30 m3 h�1 and the duration of each experi-ment was fixed at 48 h. The modelling exercise consisted ofpredicting the drawdown responses in ‘‘target’’ observation wells(Table 5) that were presumed to have different levels of connectiv-ity with the pumping and injecting wells.

To minimise the boundary effects on simulated drawdowns, themodelled domain was extended away from the HES, assuming con-stant head conditions located at least 2 km away from the wells.

Fig. 2. The location of the wells at the HES. The wells are grouped by letters: (a) the 11 pumping wells in the 2004 campaign, (b) the 19 observation wells in the 2004campaign, (c) the 8 pumping wells in the 2005 campaign and (d) the 30 observation wells in the 2005 campaign.

Fig. 3. Flowing structures identified in HES wells from borehole imaging and heat-pulse flowmeter data. Modified from Audouin et al. (2008).


The influence of the local piezometric gradient (0.003 towards theNNW) on drawdown responses was assumed to be negligible. Theeffects of turbulent flow that might prevail in karst conduits duringthe two dipole experiments were also assumed to be negligible.Actually, this assumption was dictated by the inability of the testedmodels to account for turbulent flow conditions. The potential lossof accuracy in predicted drawdowns resulting from this assump-tion was not evaluated beforehand. In a study addressing theimpact of turbulent flow in the simulation of hydraulic heads with-in a karstic aquifer in southern Florida, USA, Shoemaker et al.(2008) reported head differences from laminar elevations rangingfrom �18 to +27 cm. Transposing these results to other aquifersis, of course, hardly conceivable.

Mathematically the problem can be formulated by the followinggoverning differential equation and boundary conditions:

Basic equation of groundwater flow:

Ss@H@t¼ rðKrHÞ þ qs ð1Þ

Table 2The pumping rates and duration of pumping tests in the 2004 and 2005 campaigns.

Startingday

Pumpingwell

Q(m3 h�1)

Period of pumpingtest (h)

2004Campaign

07/03/2003

M02 63.8 24

25/03/2003

M03 30.6 24

19/03/2003

M04 58.4 24

13/03/2003

M05 57.0 24

24/11/2003

M06 67.4 72

02/12/2003

M07 63.5 57

15/12/2003

M11 59.0 79

30/01/2004

MP4 41.0 90

20/02/2004

MP5 54.8 167

23/02/2004

MP6 62.8 70

04/03/2004

MP7 60.7 125

2005Campaign

05/01/2005

M22 65.0 120

18/01/2005

M19 64.5 144

10/02/2005

M21 62.3 96

22/02/2005

M16 64.5 138

22/03/2005

M12 68.3 96

05/04/2005

M13 61.4 99

12/05/2005

M15 31.3 76

30/05/2005

M20 62.9 93

Fig. 4. Representative drawdown responses during the 2004 and 2005 pumpingtests. Modified from Kaczmaryk and Delay (2007b).


where Ss is the specific storage coefficient (L�1); K is hydraulic con-ductivity (LT�1); H is hydraulic head (L), t is time (T) and qs is thesink/source term (T�1). The value of qs is zero except at the locationsof the pumping and injection wells:

for the dipole M06–M22:

qsðM06Þ ¼ �30 h�1 ð2aÞ

qsðM22Þ ¼ þ30 h�1 ð2bÞ

for the dipole M12–M15:

qsðM12Þ ¼ þ30 h�1 ð3aÞ

qsðM15Þ ¼ þ30 h�1 ð3bÞ

Boundary conditions:

@Hðx P LÞ@t

¼ @Hðx 6 �LÞ@t

¼ @Hðy P LÞ@t

¼ @Hðy 6 �LÞ@t

¼ 0;

L P 2000 m ð4Þ

where x and y are the horizontal coordinate axes, aligned with theprincipal components of the hydraulic conductivity tensor, and L isthe distance from the centre of the HES beyond which the influenceof the dipole experiments is assumed to be negligible. The model-ling exercise consists in numerically solving the set of Eqs. 1, 2a,2b, 3a, 3b, 4 for the head-time variations in the ‘‘target’’ observationwells (Table 5).

2.3. The tested models

The applied models are listed in Table 1. Below we provide abrief summary of each modelling approach, with references tothe major publications where they were presented and applied.For clarity and brevity, we will refer to the models by theirthree-letter symbols.

2.3.1. The 2D continuum/slug-test K pattern (STK)The STK model has been generated with the sole use of the large

body of hydraulic test data available on the HES site. To combinethe information from the numerous well tests and slug tests, wedevelop a methodology intermediary between the point-wiseinterpolation of transmissivities and the hydraulic tomography(Gottlieb and Dietrich, 1995; Yeh and Liu, 2000). The basic princi-ples of the composition are that well tests sample larger zones thanslug tests and that cross-borehole hydraulic tests are most sensi-tive to the zone between the pumped well and the observationwell (Oliver, 1993). The composition method operates a spatialre-organisation of the information contained in the hydraulic tests.Compared to the classical inverse method using point-wise perme-abilities for regularisation, the composition method uses the addi-tional information of the spatial localisation of the sensitivity.Compared to hydraulic tomography, composition is a direct meth-od and not an inverse method; it does not begin from the tran-sient-state head chronicles but from their interpretation.However, this method does not require the intensive computationsrequired by transient-state simulations and parameter estimation.

First, pumping tests characterise larger zones than slug tests. Amore precise indication of the area sampled by a pumping test isgiven by the radius of influence of the pumped well:R ¼ 1:5

ffiffiffiffiffiffiffiffiffiffiffiffiðT=SÞ

pt where T/S is the aquifer diffusivity (de Marsily,

Table 3Ranges of the transmissivity and storativity values interpreted from the 2004 and 2005 pumping-test data. TG, k, r2: the geometric mean, integral scale and variance, respectively,of the local log-transmissivities; Tf and Sf: the transmissivity and storativity, respectively, of the fracture continuum; Sm: the storativity of the matrix continuum.

Pumping testcampaign

Interpretation method References T (m2 s�1) S (–)

2004 Theis method Unpublished 1.6 � 10�3 to 4.4 � 10�2 1.4 � 10�5 to7.3 � 10�2

Cooper–Jacob method Unpublished 2.4 � 10�3 to 4.4 � 10�3 3.6 � 10�4 to9.3 � 10�2

Fractal single-mediumapproach

Delay et al. (2004) and Bernard et al.(2006)

2.6 � 10�3 to 4.8 � 10�3 2.0 � 10�5 to5.6 � 10�3

Homogeneous dual-mediumapproach

Delay et al. (2007) and Kaczmaryk andDelay (2007b)

Tf = 1.1 � 10�3 to 2.1 � 10�3 Sm = 1.4 � 10�4 to3.9 � 10�3

Sf = 2.4 � 10�5 to4.8 � 10�4

Fractal dual-mediumapproach


Tf = 1.3 � 10�3 to 4.6 � 10�3 Sm = 3.0 � 10�4 to1.5 � 10�3

Sf = 2.4 � 10�5 to2.8 � 10�4

Stochastic type-curvesapproach

Riva et al. (2009) TG = 6.5 � 10�4 to 1.3 � 10�3; k = 100 m;r2 = 1.5

–

2005 Theis method Unpublished 4.2 � 10�3 to 3.0 � 10�2 4.3 � 10�5 to2.9 � 10�2

Cooper–Jacob method Unpublished 2.2 � 10�3 to 3.7 � 10�3 8.2 � 10�4 to2.8 � 10�1

Fractal single-mediumapproach

Delay et al. (2004) and Bernard et al.(2006)

2.0 � 10�3 to 7.3 � 10�3 2.2 � 10�5 to4.2 � 10�3

Homogeneous dual-mediumapproach


Tf = 1.3 � 10�3 to 1.8 � 10�3 Sm = 4.5 � 10�4 to2.4 � 10�2

Sf = 3.2 � 10�5 to1.5 � 10�3

Fractal dual-mediumapproach


Tf = 1.1 � 10�3 to 7.9 � 10�3 Sm = 6.2 � 10�4 to4.1 � 10�3

Sf = 4.3 � 10�5 to8.5 � 10�4

Stochastic type-curvesapproach

Riva et al. (2009) TG = 6.5 � 10�4 to 1.4 � 10�3; k = 130 m;r2 = 1.0–1.5

–

Table 4Ranges of the transmissivity values interpreted from single- and cross-borehole slug-test experiments.

Slug-test experiments References T (m2 s�1)

Single-borehole slug tests (allinterpreted T-values)

Audouin andBodin (2007)

7.2 � 10�5 to6.4 � 10�2

Single-borehole slug tests in wellsM08, M09, and M10

Audouin andBodin (2007)

7.2 � 10�5 to4.1 � 10�4

Cross-borehole slug tests Audouin andBodin (2008)

4.4 � 10�3 to7.7 � 10�2


1986). Two-dimensional characteristic diffusivity values (T/S) arebest estimated from the Theis solution that is sensitive to the earlytime drawdown response and range in the interval [1–100 m2 s�1](Table 3). The characteristic time at which the entire drilled zone(R � 100 m) has been influenced by pumping ranges between45 s and 1 h. The drawdown data show that the characteristic timeis, on average, 2 min. After only 20 min, the zone effectivelysampled by the pumping test is at least three times larger thanthe well-location area. Because the pumping tests lasted approxi-mately 3 days, they characterise the mean hydraulic properties ofboth the drilled zone and a larger concentric zone beyond the wells(outer zone). The division of the site into two concentric zones isnot only descriptive but also hydraulically meaningful. The charac-teristic transmissivity of the outer zone derived from late-timepumping-test drawdowns with the Cooper–Jacob method(3.5 � 10�3 m2 s�1, see Table 3) is approximately one order ofmagnitude smaller than the mean transmissivity of the drilledzone obtained from slug tests (2 � 10�2 m2 s�1, Audouin and Bod-in, 2008; see Table 4). The storage coefficient S is assumed to behomogeneously equal to 2 � 10�3 (Bernard and Delay, 2008). For

the STK model, heterogeneity comes exclusively fromtransmissivity.

Transmissivity heterogeneity can also be modelled in the drilledzone based on the transmissivity dataset derived from the slug-testinterpretations (Audouin and Bodin, 2007, 2008). Because a cross-borehole test is most sensitive to the area directly around theinjecting and pumping wells (Oliver, 1993), we intend to preferen-tially affect the interpreted transmissivity from this test betweenthe boreholes. As a simple rule, only the nearest cross-boreholetest neighbour will be considered. Because the influence of thetransmissivity of a given point is thought to be maximal to thenearest cross-borehole test, we decide conversely to affect thetransmissivity derived from the cross-hole to the closest pointsof the segment between the holes. When two cross-borehole testshave been performed from either well, the derived transmissivitiesare not necessarily equal because of the surrounding heterogene-ity. In this case, we affect the transmissivity obtained at the pie-zometer only to the closest points to the part of the segmentfrom the well to the piezometer.

This set of rules results in the decomposition of the drilled zonearea into triangles derived from the Voronoï tessellation based onthe well locations. Each triangle of the tessellation originates froma well WO and is directed towards another well WT. The trianglecovers the area made up of the closest point to the part of thesegment from WO to WT. If a slug test has been performed withinjection in WO and observation in WT, the triangle is given thetransmissivity obtained by the cross-hole test interpretation inWT. For the HES site, cross-borehole slug-test data were directlyable to fill 35% of the triangles. We complete the transmissivityfield with the following additional rules. If no test has beenperformed from WO to WT, the transmissivity of the triangle orig-inating from WO is set equal to the transmissivity derived from the

Table 5Overview of the two test problems. The ‘‘target’’ observation wells are those in which the drawdown responses had to be predicted.

Dipole experiment Pumping well Injecting well Pumping/injecting flow rate (m3 h�1) Target observation wells

M06–M22 M06 M22 30 M07, M09, M16, M20, M21, MP6M12–M15 M12 M15 30 M04, M07, M09, M14, M21, MP6

Fig. 5. Log10 T map of the STK model derived from the composition of nearest-neighbour cross-hole slug and pumping test interpretations. The transmissivity of the dark bluezone, corresponding to a very low transmissivity zone, is set to 10�6 m2 s�1.


test from WT to WO, if available. With this additional rule, up to70% of the transmissivity field has been covered. The remainingunfilled triangles were given the transmissivity of one of the twoother neighbouring triangles. We reiterate the application of thisrule up to the completion of the transmissivity field. The low-transmissivity zone around M09, M10 and M18 is given a muchlower transmissivity than the rest of the model 10�6 m2 s�1.Because its value is much smaller than that of the other transmis-sivities, this transmissivity does not influence the predictions.

The obtained transmissivity model is heterogeneous (Fig. 5).Even if the transmissivity field resolution scale is the inter-welldistance, it may display larger-scale transmissivity structures. Out-side of the low transmissivity zone, transmissivity within thedrilled zone ranges over two orders of magnitude. The interest ofthis model is its simplicity and the use it makes of the availablehydraulic data. Its limitations are its somewhat artificial sharptransmissivity interfaces and the absence of fitting to the transienthead data (a direct rather than inverse model).

2.3.2. The 2D continuum/automatic downscaling inverse method (DSI)Although DSI is versatile and can be applied to various three-

dimensional spatially distributed problems, it is only discussedbelow with regard to its application to the inversion of a two-dimensional diffusive (Darcian) flow in a confined aquifer. As iswell known, flow is averaged over the wetted thickness of the res-ervoir, with the storage capacity S (–) and the transmissivity T(L2 T�1) as hydraulic parameters at a given planar location x = (x1,x2) but averaged or assumed constant along the vertical directionx3. The flow equation is solved by using a Galerkin-conform fi-nite-element method (e.g., Kinzelbach (1986)) on a non-structuredtriangular meshing. In the present case dealing with a modelled

domain of 1.5 � 1.5 km2, including the HES plus a wide surround-ing area to push away the boundary conditions far from the area ofinterest, the mean size of the meshes is about 3–5 m on a side.

Inversion of interference testing is conducted within the frame-work of the so-called gradient methods, minimising a simple least-square objective function:

Jðh;pÞ ¼ nT � n; nðpÞ ¼ hm � hðpÞ ð5Þ

n is the vector of errors between observed heads (or drawdowns) hm

and simulated heads h(p) at the same locations and times, and p isthe vector of sought parameters conditioning the direct flow prob-lem. It was deemed useless to add a regularisation term in (5) as a‘‘distance’’ between the sought parameters p and any prior estima-tion or measurement p0 because the values of storage capacity andtransmissivity previously identified at the HES (Table 3) were, inessence, representative of wide portions of the aquifer and not ofpoint values suited to discrete calculation involving small finite ele-ments. Notably, the drawdown values concealed in the objectivefunction were sampled from raw data according to a logarithmictime step, which is assumed (at least for a homogeneous medium)to give even weight to data recorded at short and long pumpingtimes.

Because the parameterisation technique (see below) is able tohandle a large vector p, the Quasi-Newton method associated witha BFGS algorithm (see, e.g., Byrd et al. (1994)) is chosen as the opti-misation technique seeking p in the parameter space. This tech-nique rests on the calculation of the gradient of the objectivefunction, rJ = oJ/op, calculated here with the adjoint-state method(for details, see Ackerer and Delay (2010)).

The basic ideas pertaining to the parameterisation technique inthe DSI method are as follows: (i) some kind of parsimony princi-


ple stating that a good solution to a spatially distributed problem isthe one with the minimum number of parameters required todepict the fields, here of S and T, yielding the minimal objectivefunction; (ii) the ability to discover the spatial structure of theparameter fields as the consequence of inversion and not as a fea-ture conditioned by a prior guess (e.g., a prescribed type of covari-ance function). The parameters S and T are located at the nodes of aparameter grid made of triangular meshes superimposed onto thefinite element computation grid of the direct problem. Both gridsare independent, and the transfer of parameter values betweengrids is made by linear interpolation on the parameter grid and di-rect projection onto the calculation grid.

For a given design of the parameter grid, the optimiser handlesand modifies the parameters until a minimal value of J is found. Ifthis value is not considered small enough, the parameter grid is re-fined by splitting each triangle of the entire grid or of portions ofthe grid into four smaller triangles. The four smaller triangles aredelineated between the vertices of the initial larger triangle andthe mid-points of each side of the initial triangle. When resumingthe optimisation with the refined grid, the initial parameter valuesat the vertices of the sub-triangles are assigned by interpolation ofthe values at the three vertices of the previous larger triangle.Multiple ‘‘stochastic’’ solutions to the same inverse problem canbe obtained either by changing the shape of the first parametergrid defined before the inversion run or by adding random noiseto the initial parameter values at each refinement step. The abilityto refine only selected portions of any parameter grid avoids theneed to seek a higher resolution in areas of the parameter fieldwhere a higher resolution would be useless (e.g., far from locationswhere observed data are available). Criteria such as the local valueof the objective function in a triangle G of the parameter grid,JG � 0, or the local component of the gradient, oJ/opG � 0, aremapped to delineate subareas that will not be refined.

For the specific case of interference data from the HES, twotypes of inversion were performed for both 2004 and 2005 pump-ing-test campaigns. The first type focused on seeking S and T fieldsinverting at once all of the pressure transients at all availableobserved points of a test pumping in a single well. All tests wereinverted, and 100 runs per test were launched. The second typeof inversion used all of the drawdown data from a given year(2004 or 2005) merged in the same set. This set was designed asthe responses to a fictitious pumping scenario stressing eachpumping well sequentially in time, each stress period being fol-lowed by a relaxation period allowing the drawdowns to returnto zero. The detailed results are reported in Ackerer and Delay(2010). The findings can be summarised briefly as follows:

� There are no significant differences in the sought S and T fieldsamong different inversions of different single pumping tests of agiven year.� There are no significant differences between inverse solutions

to a single pumping test of a given year and solutions to the fic-titious pumping scenario of the same year.� Sought fields of T values significantly differ between years 2004

and 2005. The range of T values does not change, remainingbetween 10�6 and 10�2 m2 s�1, but the statistical distributionof ln(T) evolves from being roughly Gaussian in 2004 to a distri-bution skewed towards high values in 2005. This change alsoimpacts the spatial distribution by delineating a compact patchof high transmissivity values centred on the HES. In addition,the storage capacity does not evolve significantly between2004 and 2005, with values within 10�3 – 5 � 10�3 at the HES.� All T fields show a well-structured spatial correlation, although

this was not prescribed by the DSI method (see above). Theinference of this structure by means of variogram calculationsresults in exponential variograms for 2004 ln(T) fields with an

effective correlation length of 100 m. In contrast, 2005variograms become linear or nested bilinear shaped with nocorrelation length for the range of distances investigated bythe inversion exercises.

The predictions of head response to dipole experiments dis-cussed in this paper were calculated using 20 stochastic (S, T) fieldsinverted from the fictitious pumping scenarios discussed above.Calculations were conducted using the same (direct) code as thatused for the interference inversions. The variability across simula-tions is illustrated in Fig. 6. The drawdown curves reported in Figs.7a,b and 8a,b are composed of the mean drawdown values com-puted over the 20 simulations for each time step.

2.3.3. The 3D continuum/sequential inverse method (SQI)The SeQuential Inverse (SQI) method adopted here was part of

an integrated methodology beginning with the combined analysisof all available static/geological and dynamic/flow data from wellb-ores and other surveys (outcrop observations and surface geophys-ics) to generate a geostatistical facies model of the HES (Bourbiauxet al., 2007). The flow model was directly derived from this geosta-tistical model. Then the petrophysical parameters and the spatialdistribution of facies were sequentially calibrated to well-pumpingtests (SQI method). The modelling and calibration procedure issummarised below together with specific HES application features.

2.3.3.1. Geostatistical modelling. The geostatistical model was builtusing a proprietary software of IFPEN formerly developed in part-nership with the Centre of Geostatistics at the Paris School ofMines. The background of this facies modelling workflow andmethod can be found in Matheron et al. (1987), Galli et al.(1993), Doligez et al. (1999), and Ravenne et al. (2002). First, thedistribution of eight facies along wellbores and between wellswas characterised from drilling reports, production logs, wellborewall images, and related data. Then a stationary truncated mono-Gaussian algorithm was used to simulate the 3D distribution of fa-cies on a Cartesian grid, with cell dimensions equal to 10 m hori-zontally and 1 m vertically. That geostatistical realisationhonoured the facies observations at the multiple well locations.Considering the estimated contrast between facies flow properties,the initial eight-facies model was simplified into an equivalentlumped-facies model with two contrasted facies, a tight matrix fa-cies and a conductive ‘‘water’’ facies.

2.3.3.2. Flow simulation and calibration methods/tools. Flow simula-tion. A proprietary well-test numerical simulation program (Blanc,1995; Rahon et al., 1996) was used to simulate pumping and inter-ference tests. A single-porosity modelling approach was adoptedfor this study, although a dual-porosity approach was tested else-where (Pourpak et al., 2009). Both the petrophysical properties offacies and the spatial distribution of facies can be optimised.

The inversion of petrophysical properties is performed throughthe iterative optimisation of a pressure objective function express-ing the mean square differences between the field-measured pres-sure data and the simulated pressures predicted by the flow model.The optimisation technique (Rahon et al., 1996) is based on theflow model gradients with respect to the petrophysical parametersto be calibrated, which included facies porosity and permeability inthe present study.

The spatial distribution of facies is optimised by the so calledgradual deformation method. This geostatistical parameterisationtechnique was introduced and implemented by Hu (2000) to con-tinuously modify a Gaussian random function while honouring theunderlying geostatistical parameters. It is a multi-step processinvolving the optimisation of a few independent gradual deforma-tion parameters at each step. Optimisation is again driven by the

Fig. 6. Examples of stochastic simulations of drawdowns for the dipole M12–M15 with the DSI 2005 (20 simulations) and CPM models (50 simulations). Thin curves: singlerealisations; bold curves: mean drawdowns; circles: HES experimental data.


pressure objective function calculated from flow simulation resultson the gradually deformed model. The gradual deformationtechnique was implemented within a proprietary, constrainedreservoir-modelling platform, CondorFlow™, coupled with thewell-test simulator described before.

2.3.3.3. Implementation: a sequential calibration methodol-ogy. Designing a methodology of calibration was the main initialpurpose of the study, and it turned out that, after proper initialisa-tion of the flow model, the following sequence, i.e., petrophysicalcalibration followed by facies deformation, provided the most sat-isfactory calibration.

Selected well data for the model calibration. The flow modelwas calibrated to the field pressures measured during the wellM07 pumping test, including well M07 pressure drawdown dataand pressure measurements in eight observation wells showingsignificant interference with well M07: C1, M02, M04, M05, M06,M09, MP4 and MP6. Other pumping-test data were not taken intoaccount, whereas facies information from all wells was integratedinto the underlying geostatistical model.

Regarding model parameterisation/initialisation, the flowboundary conditions, consisting of the water recharge of the stud-ied domain, were simulated by large peripheral buffer cells. The

porosity and permeability of those cells were initially calibratedfrom the inversion of the entire testing period (57 h) of the selectedwell (M07) and were kept unchanged in the subsequent steps ofcalibration focusing on the HES domain. Because the storativityproperty was calibrated via the porosity parameter in the petro-physical inversion step, the compressibility parameter had to beset at a fixed value (0.001 bar�1, corresponding to a specific storagevalue of 4 � 10�6 m�1 for the calibrated buffer-cell porosity valueof 0.04) estimated from previous HES studies (Delay et al., 2004;Bernard, 2005) and from preliminary simulations and inversiontests.

As regard petrophysical inversion, the conductive-facies poros-ity and permeability, and, to a lesser extent, the tight-facies perme-ability, were calibrated. That inversion led to a very highpermeability value of the conductive (‘‘water’’) facies, close to9 � 10�3 m s�1. At the end of that calibration, a few ‘‘problematic’’wells (M05 for instance) remained unmatched. Then, using thepreviously calibrated facies’ petrophysical parameters, the modelwas tuned further by deforming the distribution of facies, leadingto a considerable reduction in the discrepancies between thesimulated and measured pressures of ‘‘problematic’’ wells.

It is worth noting that (i) the HES model was calibrated to wellM07 pumping-test data alone and not to other pumping tests and

Fig. 7a. Predicted and measured drawdowns for the dipole M06–M22. The DSI, CPM and DFN curves are composed of mean drawdown values computed at each time stepfrom Monte Carlo simulations involving 20, 50 and 10 simulations, respectively (Fig. 6).


that and (ii) only the first 24 h of well testing were taken into ac-count for calibration. Considering the karstic nature of the HESaquifer, that limited set of field data was largely insufficient toconstrain the location and connectivity of flowpaths, which was apriori expected based on geological studies (Bourbiaux et al., 2007)and a posteriori revealed by the dipole experiments discussed inthat paper.

2.3.4. The 3D hybrid DFN–EPM continuum approach (CPM)The purpose here was to build a three-dimensional continuum

model incorporating as much prior knowledge as possible aboutaquifer properties and geometry in the parameterisation processbased on previously published studies, and to use this modeldirectly to simulate the two dipole experiments without any recal-ibration of the model parameters against single-well pumping-testdata. Owing to the (assumed) key role played by the fractures andthe karst conduits, preferential flowpaths were explicitly incorpo-rated into the model following the heterogeneous continuumporous-medium (CPM) approach initially proposed by Jacksonet al. (2000) and further developed by Svensson (2001a,b) and

Botros et al. (2008). The method basically consists of two steps:first, a discrete fracture network (DFN) representing the three-dimensional network of fractures and karst conduits is simulated;second, the model grid is superimposed to the fracture network,and an effective permeability tensor is computed within each gridblock for subsequent use in a continuum numerical framework.

A stochastic Monte Carlo approach was used to account foruncertainty resulting from the partial knowledge of the geometri-cal and hydraulic characteristics of the actual fracture/karstnetwork. The commercially available software FRACMAN-7, fromGolder Associates, was used to generate a set of 50 DFNs withstatistical parameters inferred from the analysis of (i) fractures ob-served on analogue outcrops (Bourbiaux et al., 2007), (ii) maps ofcaves located near the experimental site (Bodin and Razack,1997), and (iii) borehole logging data (Audouin et al., 2008). Theborehole logging data were only used in a probabilistic framework,i.e., the simulated DFNs were not designed to explicitly honour thelocation of actual fractures and/or karst conduits identified in thedifferent HES boreholes. Each fracture was assigned a randomtransmissivity value drawn from the lognormal distribution

Fig. 7b. Predicted and measured drawdowns for the dipole M06–M22. The DSI, CPM and DFN curves are composed of mean drawdown values computed at each time stepfrom Monte Carlo simulations involving 20, 50 and 10 simulations, respectively (Fig. 6).


obtained from the interpretation of cross-borehole slug tests,which were expected to yield reliable estimates of the hydraulicproperties of preferential flowpaths (see the discussion in Section2.1). The chosen CPM model dimensions were 300 m � 300 m inthe horizontal plane (the area covered by the regular array ofHES wells; see Fig. 2) and 100 m in the vertical plane (the aquiferthickness). To minimise boundary (geometry) effects, fracture net-works were originally simulated within a domain of 1500 � 1500� 100 m and then truncated to the CPM domain size.

The CPM model grid superimposed to the DFNs was a three-dimensional finite-difference grid consisting of 30 layers, 60 rowsand 60 columns. Layers were either 5 m or 2 m thick, representingthe thickness of sedimentary layers observed in borehole images,and a uniform mesh size of 5 � 5 m was used horizontally. Effec-tive CPM properties were derived for each grid block containingone or more fractures according to the Oda tensor method (Oda,1985), which accounts for the number, area, orientation, and trans-missivity of fractures. Specific storage (Ss) values were jointlyassigned according to the following empirical relationship, derived

from the pairs of permeability and storativity values interpretedfrom cross-borehole slug tests:

Ss ¼ 10�11�logffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiKx�Ky�Kz

3pð Þ ð6Þ

where Kx, Ky, and Kz are the principal values of the effective perme-ability tensor. The resulting field of jointed K and Ss values was as-sumed to realistically capture the correlation and anisotropystructure of preferential flowpaths. Grid blocks containing no frac-ture were assigned a constant permeability value Km = 2.5 �10�6 m s�1, corresponding to the geometric mean of permeabilityvalues interpreted from single-borehole slug tests in wells inter-cepting neither karst-conduit nor fracture flowpaths, and a constantspecific storage value Ssm = 1.1 � 10�5 m�1 based on the dual-med-ium interpretation of the 2005 pumping-test data by Delay et al.(2007) and Kaczmaryk and Delay (2007b). To minimise boundaryeffects in the flow simulations, the heterogeneous CPM domainwas surrounded by a homogeneous buffer extending to 4 km in hor-izontal directions, which was discretised into grid elements of

Fig. 8a. Predicted and measured drawdowns for the dipole M12–M15. The DSI, CPM and DFN curves are composed of mean drawdown values computed at each time stepfrom Monte Carlo simulations involving 20, 50 and 10 simulations, respectively (Fig. 6).


increasing size. The permeability and storativity values of theseelements were set equal to the geometric and arithmetic mean,respectively, of the CPM domain properties. Constant head condi-tions were applied to the external boundaries of the full domain.

The numerical code MODFLOW-2000 (Harbaugh et al., 2000;Hill et al., 2000), using the interface Groundwater Vistas (Rumb-augh and Rumbaugh, 2004), was used to simulate the two dipolescenarios in the 50 CPM models. The variability across simulationsis illustrated in Fig. 6. The drawdown curves reported in Figs. 7a,band 8a,b are composed of the mean drawdown values computedover the 50 models for each time step.

2.3.5. Purely discrete (3D) fracture/pipe networks (DFNs)The discrete fracture network modelling strategy followed dur-

ing the proposed two-step exercise at the HES site is, in essence,similar to the one already followed within a particular task (Rejeband Bruel, 2001) of the DECOVALEX research program, a 15-year-long international cooperative project for code validation againstin situ experiments (Tsang, 2009). Among the lessons learned, weconcluded that incorporating as much geological structural infor-mation as possible was of great importance to the accuracy of

predictions. The primary assumptions for the 3D stochastic DFNapproach and recent modelling capabilities for groundwater man-agement issues in shallow fractured systems can be found in Bruelet al. (2008).

Our main objective at the HES site is to capture some of thehydraulic behaviours of a fractured and karstified limestone blockthat is 200 m � 200 m � 100 m in size, observed through a set ofvertical boreholes with a typical separation of approximately50 m. The objective is not to separately match a collection of localobservations made within this particular block. With regard to theconceptualisation of the DFN geometrical model, rather roughinformation is considered. Three horizontal, hydraulically activestructures (karstified stratigraphic horizons) are reported.However, Fig. 3 shows that there is no horizontal persistency: few-er than 30% of the wells connect to the karstic level at 50 m,approximately 50% of the wells are connected to the karstic levelat 85 m, and approximately 40% of them show some hydraulic linkwith the deeper level at 115 m. None of the wells connect all threelevels. Therefore, these horizontal structures are represented in themodel by horizontal 2D sub-networks of fractures with a bimodalaperture distribution, open regions and sealed regions and with a

Fig. 8b. Predicted and measured drawdowns for the dipole M12–M15. The DSI, CPM and DFN curves are composed of mean drawdown values computed at each time stepfrom Monte Carlo simulations involving 20, 50 and 10 simulations, respectively (Fig. 6).


correlation length (isotropic) for the sealed patches that is approx-imately equal to the inter-well distance (50 m). Subverticalfractures also exist in the block and will be superimposed to thethree structures, purely at random, with similar statistics (threemain sets of fractures grouped around azimuth angles of 45�,170� and 120�, respectively), as observed along outcrops nearby.Based on these vertical fractures, connections between the hori-zontal flowpaths are established. The calibrated total fracture den-sity is approximately 1 � 10�3 m�3. Random fracture radii arearbitrarily distributed according to a power law distribution withan exponent of 3 and a minimum radius set to 5 m. The globalmodelled areal fracture intensity index d32 (Xu et al., 2006) isapproximately 8 m2/m3, and we check that within the connected(thus flowing) part of the network of fractures, the number ofintersections per well is appropriate. Dealing with flow simulation,hydraulic boundaries (prescribed hydrostatic condition) were as-signed at some distance from the edges of the block (±2 km inthe x and y directions), but only the horizontal structures weregenerated at this scale. Using the reported results of well-testanalysis (see Table 3) is inappropriate in this approach becausethey result from conceptual approaches including equivalent

porous medium and do not translate into local properties at the‘fracture object’ scale. Thus, hydraulic properties of fractures werecalibrated so that the model could qualitatively reproduce the vari-ety of 2004 pumping-test responses. We assume that fluid flowalong any flowing path follows Darcy’s law (transmissivity is linearwith fracture aperture) and that the so-called cubic law (transmis-sivity scales with the cube of the fracture aperture) is valid atlocations where the identified karst conduits were likely developedduring 2005 test phases. Calibration tests suggest transmissivitiesin the order of 10�4–10�3 m2 s�1 in fractures and 10�3–10�2

m2 s�1 in karstified horizons. The impact of local heterogeneitiesup to 2 orders of magnitude greater in the immediate vicinity ofperturbed wells, with the superposition of karst pipes centimetricin diameter (M04, M05, M19, M20 and M21), was accounted forbut not really calibrated. The storativity of fracture elements isintroduced, with larger values in horizontal layers (S = 5 � 10�4)than in vertical fractures (S = 2 � 10�4). A set of ten equiconsistentstatistical realisations of the fracture network has been con-structed. Each of these realisations has been used to perform bothdipole scenarios, with global fluxes of ±30 m3/h prescribed atoperated wells over a period of 2 days. Average drawdowns are


then calculated at the selected wells for comparisons. The variabil-ity of the responses (e.g., standard deviation), in space and overtime, were also made available. Hydraulic heads at injection andproduction wells are also obtained as well as the temporal evolu-tion of the spreading of the injected mass of fluid into the variousfractures connected to the wells, which resembles to a dynamicflow log. Taking advantage of this added value of DFN approacheswould be of primary importance for any further analysis involvedin solute transport studies. It is also interesting to note that thelocal flow distribution in a well stabilises towards a steady statelater than the hydraulic head obtained at the well head.

2.3.6. About the Representative Elementary Volume (REV)With the exception of the DFN model, all the other models are

based on continuum approaches that implicitly assume the exis-tence of a Representative Elementary Volume (REV) (Bear, 1972),i.e. a volume of very small size as compared to the size of theHES domain and in which the hydraulic properties of the aquiferare statistically homogeneous. In practice however, the existenceof such REV is hardly provable and might even be opposed to thewell-known scaling behaviour of fracture and karst systems (Neu-man, 2005). Assessing the size (or the existence) of the REV of theHES aquifer would necessitate robust estimates of the hydraulicproperties of the aquifer over a range of support scales. Unfortu-nately, the difficulty in obtaining such robust estimates at the scaleof inter-well distances (as illustrated in Table 3) and the lack ofhydraulic data at support scales smaller than the minimum inter-well distance cannot support any assessment of the REV concept.Despite possible inconsistency, we assume that the use of contin-uum approaches is defendable from a pragmatic point of view inorder to address the spatial heterogeneity of the HES aquifer.

2.4. The dipole field experiments

The two pumping/injection dipole experiments were conductedduring Spring 2009. Due to technical constraints in the field exper-imental setup, some delay was unavoidable between the pumpingand the initiation of the (re-)injection signals. This delay was lim-ited to less than 10 s for both dipole experiments. Drawdownresponses were monitored in both the ‘‘target’’ observation wellslisted in Table 5 and in all of the other available boreholes of theHES experimental site. The collected data are publicly availablethrough the H+/P database (http://hplus.ore.fr). Because of thesmall amplitude of water-level variations (typically less than20 cm), drawdowns had to be corrected for barometric pressurechanges during the experiments (1000–1013 mbar). The correctionwas performed using a constant barometric efficiency calculatedaccording to the Clark method, see e.g. Davis and Rasmussen(1993). Implicit in the use of this method is the assumption thatchanges in barometric pressure are transmitted instantaneouslyto the well/aquifer system, which is commonly accepted for con-fined aquifers. Note that Rasmussen and Crawford (1997) and Tolland Rasmussen (2007) propose a more elaborate correction meth-od based on regression deconvolution, which enables delayedresponses due to wellbore storage effects to be accounted for.However, this method requires at least 2 weeks of hourly data col-lected at constant time intervals, which were not available in thepresent study.

3. Results and discussion

Figs. 7a,b and 8a,b show the drawdown responses predicted bythe five modelling teams for the two dipole scenarios as well as the‘‘real’’ drawdowns monitored in the target observation wells dur-ing the two field experiments. The analysis will be developed in

two stages. In the first stage, we focus only on the inter-compari-son of model outputs, i.e., the real drawdown responses aredisregarded. The comparison between predicted and observeddrawdowns is discussed in the second stage. The following issuesare examined: (i) the magnitude of the drawdowns, (ii) the dynam-ics of the drawdown responses at early times, and (iii) the late-time dynamics.

3.1. Inter-comparison of model outputs

The magnitude of simulated drawdowns is significantly greaterfor the DSI 2004, SQI, and DFN models than for the other models.This difference is very likely due to the use of 2004 pumping-testdata for calibrating these models. Although the 2004 data were as-sumed to be partially biased because of changes in the hydrody-namic behaviour of the HES between the first and secondpumping-test campaigns (see the discussion in the Section 2.1),three modelling teams used these data because of the relativelycentral location of the 2004 pumping wells compared to the loca-tion of the 2005 pumping wells.

The analysis of model dynamics at early times was based on thenormalised times required for the models to reach 10% of theirmaximum (positive or negative) drawdowns at the observationwells. Normalisation was conducted by dividing the times requiredto reach 10% of the maximum drawdown by the squared distancesfrom the observation wells to the most influential dipole well(depending on the sign of drawdowns). The results of these calcu-lations, as well as the basic statistics involved, are summarised inTable 6. Among these statistics, median values are considered tobe more relevant than means for analysing the relative (bulk)propagation velocity of pressure-head perturbations within themodels because the medians are less influenced by delayedresponses in ‘‘poorly connected’’ wells (e.g., M14 in models DSIand STK; see the discussion below). On this basis, the models wereordered in Table 6 from the lowest to highest median responsetimes, which range gradually over two orders of magnitude. TheDSI 2005 and CPM models are found to be the most reactive atearly times, i.e., those in which the pressure-head perturbationspropagate the most rapidly. In contrast, the SQI and DFN modelshave the most delayed responses. Here again, the influence of cal-ibration/parameterisation datasets seems to be significant becausethese two sets of models were calibrated with 2005 and 2004 data,respectively. In contrast, the cross-analyses of Tables 6 and 1 donot show any obvious relationship between the model characteris-tics (2D/3D, discrete/continuum, forward/inverse modelling) andtheir median response times. The variation coefficient values inTable 6 provide further information because these values can beregarded as rough measurements of the hydraulic heterogeneitywithin each model. Interestingly, the two models based on discretefracture network simulations (CPM and DFN) have the lowest val-ues, which suggests high fracture-network connectivity and/ormay reflect a homogenisation artefact of the CPM method.

The comparison of drawdown behaviours at late times revealsfurther differences between the models. Due to the balanced dipoleconfiguration considered in both test problems, all simulateddrawdowns are expected to reach a steady state at a given time.This time may be, of course, either within or beyond the simulationperiod (48 h). Only the CPM and SQI models systematically reach asteady state within the simulation period, at pumping times on theorder of 8 h and 14 h, respectively. Whereas the behaviour of theCPM model at late times may seem consistent with its early-timereactivity, both indicating a rapid transfer of pressure interferencesthrough the aquifer, the late-time behaviour of the SQI model isless straightforward in view of the responses of the other modelswith comparable early-time responses. The faster achievement ofsteady state in the SQI model suggests a rapid transfer of pressure

http://hplus.ore.fr

Table 6Normalizised times (h m�2) required to reach 10% of the maximum (positive or negative) drawdown values in the ‘‘target’’ observation wells during the two dipole experiments.Normalisation is carried out by dividing by squared distance from the observation well to the main dipole well of influence (depending on the sign of drawdowns).

Dipole test Obs. well DSI 2005 CPM DSI 2004 STK SQI DFN HES

M06–M22 M07 3.2 � 10�7 2.8 � 10�7 3.3 � 10�6 1.3 � 10�5 1.3 � 10�5 5.3 � 10�5 4.9 � 10�7

M09 7.6 � 10�8 4.4 � 10�7 1.3 � 10�6 9.1 � 10�6 1.6 � 10�5 5.0 � 10�5 2.8 � 10�5

M16 3.7 � 10�6 1.2 � 10�6 2.3 � 10�6 5.6 � 10�6 4.7 � 10�6 3.9 � 10�5 3.5 � 10�7

M20 1.9 � 10�6 1.2 � 10�6 1.2 � 10�4 8.9 � 10�6 6.4 � 10�5 3.0 � 10�5 2.0 � 10�7

M21 1.8 � 10�5 2.4 � 10�7 3.2 � 10�6 1.3 � 10�5 4.6 � 10�6 6.6 � 10�5 1.3 � 10�6

MP6 6.5 � 10�8 5.5 � 10�7 1.2 � 10�6 6.8 � 10�6 7.4 � 10�6 2.0 � 10�5 5.1 � 10�7

M12–M15 M04 7.8 � 10�8 6.7 � 10�7 2.5 � 10�6 6.2 � 10�6 9.6 � 10�6 8.7 � 10�5 2.9 � 10�7

M07 7.0 � 10�8 4.8 � 10�7 6.3 � 10�7 6.0 � 10�6 8.4 � 10�6 1.1 � 10�4 1.3 � 10�7

M09 2.1 � 10�5 4.2 � 10�7 4.6 � 10�7 1.1 � 10�5 2.5 � 10�4 6.9 � 10�5 1.9 � 10�5

M14 3.0 � 10�4 4.5 � 10�7 3.2 � 10�4 9.8 � 10�5 1.8 � 10�5 6.7 � 10�5 4.1 � 10�6

M21 1.1 � 10�7 1.2 � 10�6 1.8 � 10�5 1.0 � 10�5 1.0 � 10�5 1.1 � 10�4 1.3 � 10�7

MP6 2.9 � 10�7 6.5 � 10�7 1.2 � 10�6 1.9 � 10�5 1.7 � 10�5 1.4 � 10�4 7.6 � 10�7

Mean 2.8 � 10�5 6.5 � 10�7 3.9 � 10�5 1.7 � 10�5 3.5 � 10�5 6.9 � 10�5 4.6 � 10�6

Median 3.1 � 10�7 5.1 � 10�7 2.4 � 10�6 9.6 � 10�6 1.2 � 10�5 6.7 � 10�5 5.0 � 10�7

Standard deviation 8.4 � 10�5 3.6 � 10�7 9.4 � 10�5 2.6 � 10�5 6.9 � 10�5 3.5 � 10�5 9.1 � 10�6

Variation coefficient 3.0 0.6 2.4 1.5 2.0 0.5 2.0


interferences through the aquifer but with a delayed response atearly times. This behaviour is thought to result from differenthydraulic properties in the inner region (HES domain) and in theouter region of the model, which may have perturbed the propaga-tion of pressure interferences within the model; in fact, the param-eters of the outer region were fixed initially and kept unchangedthroughout the calibration process, which only included theparameters of the inner region.

Three of the models (DSI 2005, DFN, and STK) show a distinctbehaviour at late times depending on the considered dipolescenario: although they systematically reached a steady state atintermediate times (12–24 h) for the dipole M06–M22 (exceptfor observation well M09 explicitly located in a low permeabilityzone, see discussion below), the drawdowns simulated for the di-pole M12–M15 remain transient at late times. Only the STK modelseems to approach a steady state at the end of the simulation per-iod. The dissimilarity between the M06–M22 and M12–M15 sce-nario outputs likely originates from an elongated low-permeability zone extending perpendicularly to the M12–M15axis, which was included in the three models on the basis of pump-ing-test and borehole data from wells M08, M09, M10 and M14(see Figs. 3 and 4). The DSI 2004 and SQI models accounted for asimilar low-permeability zone but with less contrasting perme-ability values in comparison to the other models, which had a low-er impact on simulated drawdowns. Only the CPM model did notexplicitly account for such a low-permeability zone because of itsstochastic parameterisation based on unconditional simulations.

Based on the above inter-comparison of model outputs, one candraw the following two conclusions: (i) the most notable differ-ences between the predicted drawdowns are related to the hydrau-lic datasets used for model parameterisation; and (ii) neither theconceptual modelling approach (EPM vs. DFN), nor the modeldimension (2D vs. 3D), nor the parameterisation approach(forward vs. inverse) show any significant difference in terms ofdrawdown magnitude and dynamics at early and late times.

3.2. Field experiments

Fig. 9 shows drawdown curves in addition to those presented inFigs. 7a,b and 8a,b. As can be seen in this figure, some observationwells show oscillatory responses in the first 100 s. This oscillatorycharacter of pumping-induced responses is consistent to that ofthe slug-induced responses observed by Audouin and Bodin(2008) in the same wells and are likely due to inertial effects atthe pumping and/or injection and/or observation wells, see e.g.

Butler Jr. and Zhan (2004). Such oscillations were never observedduring the 2004 and 2005 pumping-test experiments because ofthe limited time-resolution (60 s) of the data loggers used for theautomatic measurement of pressure/head drawdowns. Anothertypical feature well visible in Fig. 9 is the pseudo-steady statereached by some observation wells at intermediate times (M05,M17, M13, and M19) before returning to a transient state. Theduration of this pseudo-steady-state phase was approximately2000 s during the M06–M22 experiment and was one order ofmagnitude shorter (�200 s) during the M12–M15 experiment.

The late-time behaviour of drawdowns for both dipole experi-ments was somewhat ambiguous: whereas a stabilisation of draw-downs was hardly discernible in any of the observation wells at theend of the M06–M22 experiment (Figs. 7a and 7b), the drawdownbehaviour of some wells during the second 24-h period of theM12–M15 experiment might be interpreted as a steady state(e.g., Figs. 8a and 8b: M04, M07, M21, and MP6). An overall attain-ment of a steady state during both dipole experiments might yet beexpected because of (i) the dipole configuration; (ii) the shortdistance between pumping and injection wells; and (iii) the veryrapid transfer of pressure interferences through the aquifer, whichwas highlighted by previous pumping and/or cross-borehole slug-test experiments (Kaczmaryk and Delay, 2007a; Audouin and Bod-in, 2008). To assess both the interplay of the pumping/injectioninterferences on drawdown responses and the origin of the pseu-do-steady state at intermediate times, a further experiment wascarried out by re-pumping well M12 at the same rate over a periodof 45 min but without reinjecting the withdrawn water in wellM15. As illustrated in Fig. 10, the monopole drawdown responsessuperpose remarkably well to the dipole drawdowns at early timesand then diverge systematically, which marks the asynchronoussuperposition of pumping and reinjection signals. The time atwhich this divergence occurs ranges from 10 to 1000 s, dependingboth on the distance and connectivity of observation wells with thepumping/injection wells. It is worthwhile to note that the apparentpropagation velocity of the reinjection signal to the highest con-nected observation wells ranged up to 20 m s�1. The secondremarkable feature appearing in Fig. 10 is the disappearance ofthe pseudo-steady state at intermediate times (see, e.g., MP5),when only the pumping well was active. This and the aboveobservations suggest that the dipole drawdown behaviours atintermediate and late times were strongly controlled by the mul-ti-permeability structure of the limestone aquifer, in which a frac-tion of the pumping/reinjection signals is transmitted rapidlythrough high-diffusivity flowpaths, while the remaining fractions

Fig. 9. The oscillatory drawdowns at early times.

Fig. 10. A comparison of drawdowns induced by the monopole M12 and the dipoleM12–M15 in the same observation wells (M16, MP5 and M08).


of signals propagate more slowly in flowpaths of lower diffusivity.The pseudo-steady state would therefore reflect the rapid conver-gence (and annihilation) of opposite pressure perturbations ofequal amplitude propagated through high-diffusivity flowpaths.This pseudo-steady state would be maintained until some delayedpart of the pumping or reinjection signals propagating in lower-diffusivity flowpaths reached the observation well under consider-ation and broke the previous pseudo-equilibrium. The ‘‘true’’steady state (not observed during the field experiments) isexpected to occur later, when the pressure disturbances propagat-ing in the lowest-diffusivity flowpaths have had enough time todiffuse across the inter-well dipole area.

3.3. Comparison between predicted and observed drawdowns

As shown in Figs. 7a, 7b and 8a, 8b, all the models failed to pre-dict the drawdowns with a reasonable degree of accuracy. Table 7presents the sum of squared differences between simulated andobserved drawdowns during the two 48-h periods of dipole exper-iments. According to this (imperfect/debatable) criterion, the threemost reliable models were, in order of relative accuracy, the STKmodel, the DSI-2005 model, and the CPM model. In fact, only theorder of magnitude of the drawdowns was correctly predicted bythese three models, and the small differences between their‘‘scores’’ in Table 7 can be considered as not significant. The generaloverestimation of drawdowns by models DSI 2004, SQI, and DFN,which were parameterised with 2004 pumping-test data, is verylikely due to changes in the hydrodynamic behaviour of the HESbetween the first and second pumping-test campaigns. Such

changes could not be rigorously certified from the 2004 and2005 pumping-test datasets only because different pumping wellswere used in the two test campaigns, but a new (monopole) pump-ing experiment conducted recently in well M06 and monitored inall of the HES wells undoubtedly confirms the modification of thehydrodynamic behaviour of the HES since the first pumping-testcampaign.

Not surprisingly, none of the models predicted the oscillatorybehaviour of drawdowns at early times because the flow equationssolved by the different models did not account for inertial effects.More disappointing was the general failure of the models to fore-cast the drawdown behaviours at both intermediate times (pseu-do-steady state) and late times (transient state after 48 h ofpumping/reinjection). As shown in Figs. 7a and 7b, the upward/downward evolution of the simulated head levels was evensystematically inverted for some observation wells (e.g., M07 andM20). Whereas a fine comparison between simulated and observeddrawdowns at either intermediate or late times would be uselessbecause of evident behaviour discrepancies, a comparison of re-sponse-time values as defined in Table 6 may be useful to assessthe relative ability of the models to capture the high-velocitypropagation of pumping/reinjection signals through preferentialflowpaths. The normalised response times of the two ‘‘real’’ dipoleexperiments were thus determined following the same conventionused for the simulated drawdowns and are reported in the last col-umn of Table 6. The comparison between simulations and observa-tions shows that the CPM model was the most reliable in terms ofmedian response time, but this model clearly underestimated theflowpath heterogeneity, as indicated by its lower variation coeffi-cient value. The same remark holds for the DFN model, whichhas a similar variation coefficient value. Because both the CPMand DFN models were based on discrete fracture network simula-tions, this failing may reflect an over-connectivity of simulatedfracture networks and/or may result from a homogenisationartefact of the CPM method. Unlike above, the variation coefficientvalue of the SQI model outputs was exact, but the median responsetime was underestimated by two orders of magnitude. This resultis likely due to the use of the biased 2004 pumping-test dataset(see discussion above), and the agreement between variation coef-ficient values is believed to be non-significant. One could counter-argue that only the average transmissivity (and/or storativity)value(s) might have been corrupted by the use of 2004 pumping-test data but that the spatial structure of inverted T- and S-fieldsmay be reliable if the spatial structure of flowpaths remained un-changed between the two pumping-test campaigns. However, thisassumption is contradicted by the results of the variogram analysisconducted by Ackerer and Delay (2010) through their inversions ofthe 2004 and 2005 datasets (see Section 2.3.2). Only the DSI-2005

Table 7Sum of squared differences between simulated and observed drawdowns during the two 48-h periods of dipole experiments.

Dipole test Obs. well (DSI 2004 – HES)2 (DSI 2005 – HES)2 (SQI – HES)2 (STK – HES)2 (CPM – HES)2 (DFN – HES)2

M06–M22 M07 1.8 � 101 5.0 � 10�1 1.4 � 101 2.8 � 10�1 8.2 � 10�1 7.1M09 1.3 � 101 6.2 � 10�2 5.0 2.3 � 10�2 5.3 � 10�2 1.9 � 10�2

M16 4.5 1.3 � 10�1 5.1 3.5 � 10�2 3.6 � 10�2 2.4 � 10�1

M20 4.0 � 10�1 2.3 � 10�1 9.8 � 10�1 3.9 � 10�1 4.5 � 10�1 1.1M21 2.0 1.6 � 10�1 7.5 9.9 � 10�2 4.3 � 10�2 2.1MP6 9.1 1.4 � 10�1 7.2 3.2 � 10�2 3.7 � 10�2 3.3 � 10�1

Sum 4.7 � 101 1.2 4.0 � 101 8.6 � 10�1 1.4 1.1 � 101

M12–M15 M04 8.9 1.6 � 10�2 2.7 1.1 � 10�2 4.4 � 10�3 6.4 � 10�1

M07 9.0 4.5 � 10�2 2.7 4.7 � 10�3 1.5 � 10�2 1.8 � 10�1

M09 6.7 2.6 � 10�2 3.1 � 10�1 7.0 � 10�2 1.7 � 10�1 2.7M14 8.4 � 10�2 9.6 � 10�3 6.0 � 10�3 1.0 � 10�3 2.4 � 10�2 5.1 � 10�1

M21 3.5 2.0 � 10�2 1.5 1.8 � 10�1 4.6 � 10�3 3.4 � 10�1

MP6 9.1 9.9 � 10�2 5.4 5.8 � 10�2 6.8 � 10�2 2.5Sum 3.7 � 101 2.2 � 10�1 1.3 � 101 3.3 � 10�1 2.9 � 10�1 6.9

Sum (both dipoles) 8.5 � 101 1.4 5.2 � 101 1.2 1.7 1.8 � 101


and STK model outputs were consistent, to some extent, with boththe median and variation coefficient of observed response times.Here again, however, it is difficult to draw clear insights from thisresult, except that models based on a continuum approach appearat least as suitable as models based on DFN simulations to capturethe propagation of pressure-head perturbations in preferentialflowpaths.

4. Summary and conclusion

Five modelling approaches have been applied to a well-studiedlimestone aquifer. The modelling exercise consisted of predictingthe drawdown responses of a series of observation wells for twodistinct dipole experiments. The main findings of this work canbe summarised as follows:

1. The differences between model predictions appear to bemainly related to the hydraulic datasets used for modelparameterisation and, to a much lesser extent, to the concep-tual modelling approach (EPM vs. DFN), the model dimen-sion (2D vs. 3D), and/or the parameterisation approach(forward vs. inverse). This finding counterbalances, to someextent, the outcomes of recent studies suggesting that theerrors resulting from an inadequate choice of conceptualmodel and parameterisation methods might be greater thanthose resulting from uncertainties in parameter values (Neu-man and Wierenga, 2003; Rojas et al., 2010b; Ye et al., 2010).While acknowledging the relevance of multi-model aggrega-tion methods advocated by these authors to account for con-ceptual uncertainty in groundwater modelling, the presentwork emphasises the need to better discriminate among cal-ibration/parameterisation data to enhance the predictivecapabilities of models. Approaches such those recently pro-posed by Rojas et al. (2010a) and Neuman et al. (2012)may be useful in such an analysis. Intuitively, one mayexpect that the impact of model parameterisation/calibra-tion from flow data tends to override and hide the impactsof conceptual model choice as the heterogeneity or complex-ity of the aquifer increases.

2. In the present case, despite both the abundance and diversityof calibration/parameterisation data, all of the models failedto predict the drawdowns with a reasonable degree of accu-racy. Only the order of magnitude of the drawdowns wascorrectly predicted by three of the five models, whereas allmodels failed to predict the drawdown behaviour at both

the intermediate and late times. Rather than reflecting anygeneral inadequacy of the modelling approaches, the differ-ences between simulated and observed drawdowns arelikely attributable to an insufficient/inadequate hydrauliccharacterisation of the aquifer. A first example is the generaloverestimation of drawdown magnitude by the DSI 2004,SQI, and DFN models, which were corrupted by the use ofbiased datasets from the early pumping-test campaign. Sec-ond and more meaningful is the basic nature of discrepanciesbetween the form of simulated and observed drawdownresponses, the latters being characterised by a typicalpseudo-steady-state at intermediate times that was not fore-casted by any of the models. Actually, the dipole drawdownswere strongly controlled by the multi-permeability structureof the limestone aquifer, in which a fraction of the pumping/reinjection signals was transmitted rapidly through high-dif-fusivity flowpaths, while the remaining fractions of signalspropagated more slowly in flowpaths of lower diffusivity.Each of the tested models was theoretically capable of han-dling – through a proper parameterisation and spatial distri-bution of T- and S-values – such a superposition of differentflowpaths with contrasted hydraulic properties. But thehydraulic dataset available for model calibration/parame-terisation did not capture the multi-permeability structureof the aquifer as revealed a posteriori by the responses todipole flow experiments.

3. The present study highlights the need to develop novel (orimproved) approaches to characterise the hydraulic proper-ties of limestone aquifers. Conventional (single-well) pump-ing tests seem insufficiently sensitive to discriminatebetween the multiple ranges of flowpath diffusivities control-ling the propagation of pressure-head perturbations withinthe aquifer. In contrast, dipole configuration emphasises thehydraulic contrast between flowpaths, which may lead to aclearer signature of multi-permeability flow systems (e.g., inthe present case study, pseudo-steady-state drawdowns atintermediate times). In fact, the dipole configuration providesa good signature of the (solicited) well-connected flowpathnetwork between the two wells for all interference wellslocated in the flow-investigated area. In contrast, the flow sig-nature of a monopole configuration rapidly integrates thecontribution of the low-diffusivity flowpaths and/or mediumas the distance from the tested well increases because of the(acknowledged) fractal nature of the main (percolating) flow-path network. As suggested by Fig. 10, the comparison of


monopole drawdowns to dipole drawdowns may allow forthe separation in time of the responses of high- and lower-dif-fusivity flowpaths, which should enable more precise assess-ments of the ranges of T- and S-values of the subflow systems.

4. A possible source of inaccuracy that was anticipated prior tothe simulations was the inability of the tested models toaccount for turbulent flow conditions that might haveprevailed in karst conduits during the dipole experiments.Actually, this issue may be regarded as being of secondaryimportance with regard to the nature of model errors, whichwere mainly dominated by discrepancies between the formof simulated and observed drawdown responses. Non-Darcyflow effects might have had consequences on the magnitudebut not on the form of drawdown responses, which areexpected to be relatively similar under either turbulent orlaminar flow conditions, see e.g. Elsworth and Doe (1986).

Acknowledgments

This work was financially supported by the French CNRS-INSUthrough the research programs EC2CO-CYTRIX/MACH, GDR/HTHS,and SO/H+ and by the ‘‘Poitou-Charentes Soils-Water ResearchProgram’’. The assistance of Benoit Nauleau and Denis Paquet inconducting the two dipole field experiments is gratefullyacknowledged.

References

Ackerer, P., Delay, F., 2010. Inversion of a set of well-test interferences in a fracturedlimestone aquifer by using an automatic downscaling parameterizationtechnique. J. Hydrol. 389 (1–2), 42–56.

Adler, P.M., Thovert, J.-F., 1999. Fractures and Fracture Networks. Theory andApplications of Transport in Porous Media. Kluwer Academic Publishers,Dordrecht, The Netherlands, 429 pp.

Alonso, E.E., Alcoverro, J., Coste, F., Malinsky, L., Merrien-Soukatchoff, V., Kadiri, I.,Nowak, T., Shao, H., Nguyen, T.S., Selvadurai, A.P.S., Armand, G., Sobolik, S.R.,Itamura, M., Stone, C.M., Webb, S.W., Rejeb, A., Tijani, M., Maouche, Z.,Kobayashi, A., Kurikami, H., Ito, A., Sugita, Y., Chijimatsu, M., Börgesson, L.,Hernelind, J., Rutqvist, J., Tsang, C.-F., Jussilao, P., 2005. The FEBEX benchmarktest: case definition and comparison of modelling approaches. Int. J. Rock Mech.Min. 42, 611–638.

Audouin, O., Bodin, J., 2007. Analysis of slug-tests with high-frequency oscillations.J. Hydrol. 334 (1–2), 282–289.

Audouin, O., Bodin, J., 2008. Cross-borehole slug test analysis in a fracturedlimestone aquifer. J. Hydrol. 348 (3–4), 510–523.

Audouin, O., Bodin, J., Porel, G., Bourbiaux, B., 2008. Flowpath structure in alimestone aquifer: multi-borehole logging investigations at the hydrogeologicalexperimental site of Poitiers, France. Hydrogeol. J. 16 (5), 939–950.

Bailly-Comte, V., Jourde, H., Roesch, A., Pistre, S., Batiot-Guilhe, C., 2008. Time seriesanalyses for Karst/River interactions assessment: case of the Coulazou river(southern France). J. Hydrol. 349 (1–2), 98–114.

Bear, J., 1972. Dynamics of Fluids in Porous Media. Elsevier, New York, 764 pp.Bear, J., Tsang, C.H., de Marsily, G., 1993. Flow and Contaminant Transport in

Fractured Rock. Academic Press, San Diego, CA, USA, 560 pp.Berkowitz, B., 2002. Characterizing flow and transport in fractured geological

media: a review. Adv. Water Resour. 25, 861–884.Bernard, S., 2005. Caractérisation hydrodynamique des réservoirs carbonatés

fracturés: Application au Site Expérimental Hydrogéologique (SEH) del’Université de Poitiers. Ph.D. Thesis, Univ. Poitiers, 230 pp.

Bernard, S., Delay, F., 2008. Determination of porosity and storage capacity of acalcareous aquifer (France) by correlation and spectral analyses of time series.Hydrogeol. J. 16 (7), 1299–1309.

Bernard, S., Delay, F., Porel, G., 2006. A new method of data inversion for theidentification of fractal characteristics and homogenization scale fromhydraulic pumping tests in fractured aquifers. J. Hydrol. 328, 647–658.

Blanc, G., 1995. Numerical well test simulations in heterogeneous reservoirs. In:AAPG Conference, Nice.

Bodin, J., Razack, M., 1997. Application du concept de Surface ElémentaireReprésentative (S.E.R.) à l’étude comparée entre karstification et tectoniquedans le département de la Vienne (France). In: 6th Conference on limestonehydrology and fissured aquifers, La Chaux de Fonds (Switzerland), pp. 259–262.

Botros, F.E., Hassan, A.E., Reeves, D.M., Pohll, G., 2008. On mapping fracturenetworks onto continuum. Water Resour. Res. 44 (8), W08435. http://dx.doi.org/10.1029/2007WR006092.

Bourbiaux, B., Callot, J.P., Doligez, B., Fleury, M., Gaumet, F., Guiton, M., Lenormand,R., Mari, J.L., Pourpak, H., 2007. Multi-scale characterization of an

heterogeneous aquifer through the integration of geological, geophysical andflow data: a case study. Oil Gas Sci. Technol. 62 (3), 347–373.

Bruel, D., Faisal, K., Engerrand, C., 2008. Upscaling of slug test hydraulic conductivityusing a Discrete Fracture Network modelling in granitic aquifers. In: Ahmed, S.,Jayakumar, R., Salih, A. (Eds.), Groundwater Dynamics in Hard Rock Aquifers,Sustainable Management and Optimal Monitoring Network Design. CapitalPublishing Company, New Delhi, pp. 123–133.

Butler Jr., J.J., Zhan, X., 2004. Hydraulic tests in highly permeable aquifers. WaterResour. Res. 40 (12), 1–12.

Byrd, R.H., Lu, P., Nocedal, J., Zhu, C., 1994. A limited memory Algorithm for BoundConstraint Optimization. Northwest Univ., Dpt. of Electrical Engineering andComputer Science, Tech. Rep., NAM-08, 24 pp.

Dafny, E., Burg, A., Gvirtzman, H., 2010. Effects of Karst and geological structure ongroundwater flow: the case of Yarqon-Taninim Aquifer, Israel. J. Hydrol. 389 (3–4), 260–275.

Davis, D.R., Rasmussen, T.C., 1993. A comparison of linear regression with Clark’smethod for estimating barometric efficiency of confined aquifers. Water Resour.Res. 29 (6), 1849–1854.

Day-Lewis, F.D., Hsieh, P.A., Gorelick, S.M., 2000. Identifying fracture-zone geometryusing simulated annealing and hydraulic-connection data. Water Resour. Res.36 (7), 1707–1721.

De Dreuzy, J.R., Bodin, J., Le Grand, H., Davy, P., Boulanger, D., Battais, A., Bour, O.,Gouze, P., Porel, G., 2006. General database for ground water site information.Ground Water 44 (5), 743–748.

de Marsily, G., 1986. Quantitative Hydrogeology. Academic Press, Inc., 440 pp.Delay, F., Porel, G., Bernard, S., 2004. Analytical 2D model to invert hydraulic

pumping tests in fractured rocks with fractal behavior. Geophys. Res. Lett. 31,L16501. http://dx.doi.org/10.1029/2004GL020500.

Delay, F., Kaczmaryk, A., Ackerer, P., 2007. Inversion of interference hydraulicpumping tests in both homogeneous and fractal dual media. Adv. Water Resour.30 (3), 314–334.

Doherty, J., 2011. Modelling: picture perfect or abstract art? Ground Water 49 (4),455.

Doligez, B., Beucher, H., Geffroy, F., Eschard, R., 1999. Integrated reservoircharacterization: improvement in heterogeneous stochastic modelling byintegration of additional external constraints, In: Schatzinger, R., Jordan, J.(Eds.), Reservoir Characterization Recent Advances, AAPG Memoir, pp. 333–342.

Elsworth, D., Doe, T.W., 1986. Application of non-linear flow laws in determiningrock fissure geometry from single borehole pumping tests. Int. J. Rock Mech.Min. Sci. Geomech. Abstr. 23 (3), 245–254.

Faybishenko, B., Witherspoon, P.A., Benson, S.M. (Eds.), 2000. Dynamics of Fluids inFractured Rock, vol. 122. Geophysical Monograph Series, Washington, DC, 400pp.

Filipponi, M., Jeannin, P.Y., Tacher, L., 2009. Evidence of inception horizons in karstconduit networks. Geomorphology 106 (1–2), 86–99.

Fleury, P., Ladouche, B., Conroux, Y., Jourde, H., Dörfliger, N., 2009. Modelling thehydrologic functions of a karst aquifer under active water management – theLez spring. J. Hydrol. 365 (3–4), 235–243.

Galli, A., Beucher, H., Le Loc’h, G., Doligez, B., Heresim group, 1993. The Pros andCons of the truncated Gaussian method. In: Armstrong, M., Dowd, P.A. (Eds.),Geostatistical Simulations. Kluwer Academic Publishers, pp. 217–233.

Garfias, J., Andre, C., Llanos, H., Herrera, I., 1998. A dual-porosity approach tosimulate groundwater flow in fractured porous media: application in the Itxinakarstic aquifer, Spain. IAHS-AISH Publ. 253, 367–377.

Geyer, T., Birk, S., Liedl, R., Sauter, M., 2008. Quantification of temporal distributionof recharge in karst systems from spring hydrographs. J. Hydrol. 348 (3–4),452–463.

Gomez-Hernandez, J.J., 2006. Complexity. Ground Water 44 (6), 782–785.Gottlieb, J., Dietrich, P., 1995. Identification of the permeability distribution in soil

by hydraulic tomography. Inverse Probl. 11 (2), 353–360.Harbaugh, A.W., Banta, E.R., Hill, M.C., McDonald, M.G., 2000. Modflow-2000, the

U.S. Geological Survey Modular Ground-water Model – User Guide toModularization Concepts and the Ground-water Flow Process. Open-FileReport 00-92, U.S. Geological Survey.

Hill, M.C., 2006. The practical use of simplicity in developing ground water models.Ground Water 44 (6), 775–781.

Hill, M.C., Banta, E.R., Harbaugh, A.W., Anderman, E.R., 2000. MODFLOW-2000, theU.S. Geological Survey Modular Ground-water Model – User Guide to theObservation, Sensitivity, and Parameter-Estimation Processes and Three Post-processing Programs. U.S. Geological Survey Open-File Report 00-184,210 pp.

Hodgkinson, D., Benabderrahmane, H., Elert, M., HautojÃrvi, A., Selroos, J.O., Tanaka,Y., Uchida, M., 2009. An overview of Task 6 of the Aspö Task Force: modellinggroundwater and solute transport: improved understanding of radionuclidetransport in fractured rock. Hydrogeol. J. 17 (5), 1035–1049.

Hu, L.Y., 2000. Gradual deformation and iterative calibration of Gaussian-relatedstochastic models. Math. Geol. 32 (1), 87–108.

Hunt, R.J., Doherty, J., Tonkin, M.J., 2007. Are models too simple? Arguments forincreased parameterization. Ground Water 45 (3), 254–262.

Jackson, C.P., Hoch, A.R., Todman, S., 2000. Self-consistency of a heterogeneouscontinuum porous medium representation of a fractured medium. WaterResour. Res. 36 (1), 189–202.

Jaquet, O., Siegel, P., Klubertanz, G., Benabderrhamane, H., 2004. Stochastic discretemodel of karstic networks. Adv. Water Resour. 27 (7), 751–760.

Jeannin, P.Y., 2001. Modelling flow in phreatic and epiphreatic karst conduits in theHölloch cave (Muotatal, Switzerland). Water Resour. Res. 37 (2), 191–200.

http://dx.doi.org/10.1029/2007WR006092

http://dx.doi.org/10.1029/2004GL020500


Jourde, H., Cornaton, F., Pistre, S., Bidaux, P., 2002. Flow behavior in a dual fracturenetwork. J. Hydrol. 266, 99–119.

Kaczmaryk, A., Delay, F., 2007a. Improving dual-porosity-medium approaches toaccount for karstic flow in a fractured limestone: application to the automaticinversion of hydraulic interference tests (Hydrogeological Experimental Site,HES – Poitiers – France). J. Hydrol. 347 (3–4), 391–403.

Kaczmaryk, A., Delay, F., 2007b. Interference pumping tests in a fractured limestone(Poitiers – France): inversion of data by means of dual-medium approaches. J.Hydrol. 337 (1–2), 133–146.

Kinzelbach, W., 1986. Groundwater Modelling: an Introduction with SamplePrograms in Basic. Elsevier, Amsterdam, 333 pp.

Kovacs, A., Sauter, M., 2007. Modelling karst hydrodynamics. In: Goldscheider, N.,Drew, D. (Eds.), Methods in Karst Hydrogeology. Taylor & Francis/Balkema,Leiden, The Netherlands, pp. 201–222.

Larocque, M., Banton, O., Ackerer, P., Razack, M., 1999. Determining karsttransmissivities with inverse modelling and an equivalent porous media.Ground Water 37 (6), 897–903.

Marschall, P., Elert, M., 2003. Overall Evaluation of the Modelling of the TRUE-1Tracer Tests – Task 4. SKB TR 03-12, Swedish Nuclear Fuel and WasteManagement CO, Stockholm, Sweden.

Matheron, G., Beucher, H., de Fouquet, C., Galli, A., Guerillot, D., Ravenne, C., 1987.Conditional Simulation of the Geometry of Fluvio-deltaic Reservoirs, SPE 62ndAnn. Tech. Conf. & Exh., Dallas, Texas, pp. 591–599.

National Research Council, 1996. Rock Fractures and Fluid Flow: ContemporaryUnderstanding and Applications. National Academy Press, Washington, DC, 551pp.

Neuman, S.P., 2005. Trends, prospects and challenges in quantifying flow andtransport through fractured rocks. Hydrogeol. J. 13 (1), 124–147.

Neuman, S.P., Wierenga, P.J., 2003. A Comprehensive Strategy of HydrogeologicModelling and Uncertainty Analysis for Nuclear Facilities and Sites. ReportNUREG/CR-6805, US Nuclear Regulatory Commission, Washington, USA.

Neuman, S.P., Xue, L., Ye, M., Lu, D., 2012. Bayesian analysis of data-worthconsidering model and parameter uncertainties. Adv. Water Resour. 36, 75–85.

Oda, M., 1985. Permeability tensor for discontinuous rock masses. Geotechnique 35(4), 483–495.

Oliver, D.S., 1993. The influence of nonuniform transmissivity and storativity ondrawdown. Water Resour. Res. 29 (1), 169–178.

Peleg, N., Gvirtzman, H., 2010. Groundwater flow modelling of two-levels perchedkarstic leaking aquifers as a tool for estimating recharge and hydraulicparameters. J. Hydrol. 388 (1–2), 13–27.

Pourpak, H., Bourbiaux, B., Roggero, F., Delay, F., 2009. An integrated method forcalibrating a heterogeneous/fractured reservoir model from wellbore flowmeasurements: case study. SPE Reserv. Eval. Eng. 12 (3), 433–445.

Rahon, D., Blanc, G., Guérillot, D., 1996. Gradients method constrained by geologicalbodies for history matching. In: Proceedings of the 5th European Conference onthe Mathematics of Oil Recovery, pp. 283–293.

Rasmussen, T.C., Crawford, L.A., 1997. Identifying and removing barometricpressure effects in confined and unconfined aquifers. Ground Water 35 (3),502–511.

Ravenne, C., Galli, A., Doligez, B., Beucher, H., Eschard, R., 2002. Quantification offacies relationships via proportion curves. In: Armstrong, M. et al. (Eds.),Geostatistics Rio 2000, 31st International Geological Congress, Rio de Janeiro, 6–17 August 2000, Proceedings 2002. Kluwer Academic Publishers, pp. 19–39.

Reimann, T., Hill, M.E., 2009. MODFLOW-CFP: a new conduit flow process forMODFLOW-2005. Ground Water 47 (3), 321–325.

Rejeb, A., Bruel, D., 2001. Hydromechanical effects of shaft sinking at the Sellafieldsite. Int. J. Rock Mech. Min. 38 (1), 17–29.

Rimmer, A., Salingar, Y., 2006. Modelling precipitation-streamflow processes inkarst basin: the case of the Jordan River sources, Israel. J. Hydrol. 331 (3–4),524–542.

Riva, M., Guadagnini, A., Bodin, J., Delay, F., 2009. Characterization of theHydrogeological Experimental Site of Poitiers (France) by stochastic welltesting analysis. J. Hydrol. 369 (1–2), 154–164.

Rojas, R., Feyen, L., Batelaan, O., Dassargues, A., 2010a. On the value of conditioningdata to reduce conceptual model uncertainty in groundwater modeling. WaterResour. Res. 46, W08520. http://dx.doi.org/10.1029/2009WR008822.

Rojas, R., Kahunde, S., Peeters, L., Batelaan, O., Feyen, L., Dassargues, A., 2010b.Application of a multimodel approach to account for conceptual model andscenario uncertainties in groundwater modelling. J. Hydrol. 394 (3–4), 416–435.

Rumbaugh, J.O., Rumbaugh, D.B., 2004. Guide to Using Groundwater Vistas.Environmental Simulations Inc., Reinholds, Pennsylvania, USA.

Sauter, M., 1993. Double porosity models in karstified limestone aquifers: fieldvalidation and data provision. Hydrogeological processes in karst terranes. In:Proc. International Symposium & Field Seminar, Antalya, Turkey, 1990, pp. 261–279.

Scanlon, B.R., Mace, R.E., Barrett, M.E., Smith, B., 2003. Can we simulate regionalgroundwater flow in a karst system using equivalent porous media models?Case study, Barton Springs Edwards aquifer, USA. J. Hydrol. 276 (1–4), 137–158.

Shoemaker, W.B., Cunningham, K.J., Kuniansky, E.L., Dixon, J., 2008. Effects ofturbulence on hydraulic heads and parameter sensitivities in preferentialgroundwater flow layers. Water Resour. Res. 44, W03501. http://dx.doi.org/10.1029/2007WR006601.

Spiessl, S.M., Prommer, H., Licha, T., Sauter, M., Zheng, C., 2007. A process-basedreactive hybrid transport model for coupled discrete conduit-continuumsystems. J. Hydrol. 347 (1–2), 23–34.

Svensson, U., 2001a. A continuum representation of fracture networks. Part I:Method and basic test cases. J. Hydrol. 250, 170–186.

Svensson, U., 2001b. A continuum representation of fracture networks. Part II:Application to the Aspo Hard Rock laboratory. J. Hydrol. 250, 187–205.

Toll, N.J., Rasmussen, T.C., 2007. Removal of barometric pressure effects and earthtides from observed water levels. Ground Water 45 (1), 101–105.

Tsang, C.F., 2009. Introductory editorial to the special issue on the DECOVALEX-THMC project. Environ. Geol. 57 (6), 1217–1219.

Tsang, C.F., Stephansson, O., Jing, L., Kautsky, F., 2009. DECOVALEX project: from1992 to 2007. Environ. Geol. 57 (6), 1221–1237.

White, W.B., 2002. Karst hydrology: recent developments and open questions. Eng.Geol. 65 (2–3), 85–105.

Wu, Y.S., Liu, H.H., Bodvarsson, G.S., 2004. A triple-continuum approach formodelling flow and transport processes in fractured rock. J. Contam. Hydrol.73 (1–4), 145–179.

Wu, Q., Zhou, W., Pan, G., Ye, S., 2009. Application of a discrete-continuum model tokarst aquifers in North China. Ground Water 47 (3), 453–461.

Xu, C., Dowd, P.A., Mardia, K.V., Fowell, R.J., 2006. A connectivity index for discretefracture networks. Math. Geol. 38 (5), 611–634.

Ye, M., Pohlmann, K.F., Chapman, J.B., Pohll, G.M., Reeves, D.M., 2010. A model-averaging method for assessing groundwater conceptual model uncertainty.Ground Water 48 (5), 716–728.

Yeh, T.C.J., Liu, S., 2000. Hydraulic tomography: development of a new aquifer testmethod. Water Resour. Res. 36 (8), 2095–2105.

http://dx.doi.org/10.1029/2009WR008822

http://dx.doi.org/10.1029/2007WR006601

Documents

Predictive modelling of hydraulic head responses to dipole flow experiments in a fractured/karstified limestone aquifer: Insights from a comparison of five modelling approaches to