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Geostatistical optimization of water reservoircharacterization case of the “Jeffra de Medenine”aquifer system (SE Tunisia)Hayet Chihi a , Nicolas Jeannee b , Houcine Yahyaoui c , Habib Belayouni d & Mourad Bedir aa Georesources Laboratory, Centre for Water Researches and Technologies , University ofCarthage , Borj Cedria Ecopark, BP 273, Soliman , Tunisiab Geovariances , 49 bis Av Franklin Roosevelt, 77215 Avon Cedex , Paris , Francec Regional Commissary for Agricultural Development of Medenine , Medenine , Tunisiad Faculty of Sciences of Tunis, Department of Geology , University Tunis El Manar , 1068 ,Tunis , TunisiaPublished online: 17 Jul 2013.

To cite this article: Hayet Chihi , Nicolas Jeannee , Houcine Yahyaoui , Habib Belayouni & Mourad Bedir , Desalination andWater Treatment (2013): Geostatistical optimization of water reservoir characterization case of the “Jeffra de Medenine”aquifer system (SE Tunisia), Desalination and Water Treatment, DOI: 10.1080/19443994.2013.812988

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Geostatistical optimization of water reservoir characterization caseof the “Jeffra de Medenine” aquifer system (SE Tunisia)

Hayet Chihia,*, Nicolas Jeanneeb, Houcine Yahyaouic, Habib Belayounid, Mourad Bedira

aGeoresources Laboratory, Centre for Water Researches and Technologies, University of Carthage, Borj CedriaEcopark, BP 273, Soliman, TunisiaEmail: hayet_chihi@yahoo.frbGeovariances, 49 bis Av Franklin Roosevelt, 77215 Avon Cedex, Paris, FrancecRegional Commissary for Agricultural Development of Medenine, Medenine, TunisiadFaculty of Sciences of Tunis, Department of Geology, University Tunis El Manar, 1068 Tunis, Tunisia

Received 25 December 2012; Accepted 24 May 2013

ABSTRACT

This study attempts to characterize the organization, geometry and continuity of aquifersystems in a faulted setting, by geostatistical methods. It concerns the “Jeffara de Medenine”aquifers, in South-Eastern Tunisia. The quality of architectural reservoir modelling dependsheavily on available data and on the fault network at the origin of its compartmentalization.In our case study, the available data consist mainly of boreholes: (i) usually sparse: the datadistribution and density are very uneven within the study area, depending on the aquifersand the river network; (ii) they do not, usually penetrate the entire aquifer formation. There-fore, aquifers situated at a great depth remain unattainable for many drillings, leaving largeareas under-informed and (iii) they are supplemented by seismic data which, although ofvariable quality, provide useful information for building the fault network at a large scale.To deal with this lack of data, an original geostatistical approach is applied in order to makethe best use of the available data: (i) borehole data corresponding to the geological interfaces:these are exact data (equal to) and (ii) information provided by the end of drilling; these areuncertain data using inequalities (less than, greater than, between). The estimation of theTuronian reservoir top (taken as an example in this study) may indeed be constrained by theexact and inequality well values, thus avoiding some inconsistencies during interpolation bykriging under inequality constraints. Fault parameters are also explicitly incorporated in theinterpolation procedure. This geostatistical approach is used for depth estimation within the“Jeffara de Medenine” aquifer system and is compared to classical kriging and evaluatedthrough the quality of estimation, the adopted assumptions and method limitations. Thus,estimation procedures can be improved to build geometric models that describe as well aspossible the geological reality.

*Corresponding author.

Presented at the 6th International Conference on Water Resources in Mediterranean Basin (WATMED6), 10–12October 2012, Sousse, Tunisia

1944-3994/1944-3986 � 2013 Balaban Desalination Publications. All rights reserved.

Desalination and Water Treatmentwww.deswater.com

doi: 10.1080/19443994.2013.812988

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Keywords: Architectural Model; Geostatistics; Kriging with inequality; Aquifer system;Borehole; Sparse data; Faults

1. Introduction

In southern Tunisia, good-quality water resourcesare relatively rare and are not easily renewed due to thesemi-arid climate in this area. The lack of resources isthe result mainly of the significant demographic growth,which has caused an increase in the demand for waterboth for irrigation and for drinking. In addition, the areanow faces a serious problem of water-quality deteriora-tion caused by considerable economic growth coupledwith overexploitation of water resources.

The aquifers supply the area with water of a salin-ity varying from 1 to 7 g/l. The reasons for this rela-tively high mineralization are marine intrusion,brackish water intrusions drained from adjacent saltysystems and the proximity of hyper-saline watersystems like Sebkhas and Chotts.

The study area (Fig. 1), the “Jeffara de Medenine”,includes a multi-aquifer system called the Zeuss Kou-tine aquifer which is the main water resource in thearea. It is a vulnerable system and hence needs to befurther documented and monitored in order to pre-serve it and improve its water quality.

One of the most important features that character-ize this water system is its geological complexity, inparticular due to the intense tectonics affectingthe various aquifers. In general, fault systems influencecommunication between reservoirs. For a better under-standing of the water flow paths within the faulted

reservoir units, it is imperative to build the reservoirarchitecture of the site where the geometry of the unitsand their spatial extensions are established.

Geometric modelling of each water reservoir unitimplies that the spatial distribution of the depthvariable within each surface boundary is estimated.However, high spatial variability of “depth” within afaulted surface boundary and sparse data network areknown to be major causes of uncertainty.

Our major goal in this study is to investigatewhether geostatistical techniques can accurately recon-struct the unknown surfaces on the basis of valuesobserved at a small number of points in the studyarea. Classical kriging was first applied, to account forthe spatial continuity of the target horizons throughthe variogram analysis. Kriging also allows explicitintegration of the fault system knowledge in theestimation process. Then, an original kriging variant,called kriging with inequality constraints was testedto avoid losing the information from wells which hadnot reached the target horizon. Although not new, toour knowledge, this methodology has not beenapplied previously to the modelling of aquifers.

2. Materials and methods

The construction of a geologically consistent reser-voir model involved the following major steps: (i)

Fig. 1. Map of the studied domain and data location.

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creation of a structural framework, (ii) delineation ofthe reservoir and (iii) integration of different types ofinformation through geostatistical techniques to modelthe surfaces bounding the reservoir units using theISATIS software [1].

2.1. Data base

Data and information were obtained from differentsources and studies [2–6]: 16 seismic cross-sections,four petroleum wells as well as 49 water wells andboreholes (Fig. 1) were used to provide direct infor-mation on the surface and subsurface geology neededfor the spatial modelling of the different boundingsurfaces.

Each data type has its specific features, whichinfluence how it is integrated into the modellingprocess and affect the quality of the model. However,these raw data alone would be useless without carefuland methodical processing and interpretation. Geore-ferencing is a crucial step in the modelling process,whereby all available information is combined andorganized in a common coordinate system: it has tocover the entire studied zone and to be preciseenough not to lose or distort information.

2.2. Geologic exploration: building a structural framework

The proposed modelling methodology starts with acareful geological analysis of (i) the lithostratigraphicunits that compose the aquifer system within the studyarea and (ii) fault structures that affect the differentreservoir units. The objective is to improve the reser-voir characterization of the aquifer system in order todefine the geometric parameters required for thegeometric modelling steps.

The multi-aquifer system includes all the layersfrom the Jurassic to the Mio-plio-quaternary. Thecarbonate formations of the Jurassic, the Albo-Aptian,the Turonian and the lower Senonian constitute amultilayer hydrogeological unit [3,4]. The lateralextent of the different aquifers at various depths iscontrolled by the structural evolution of the area andits vicinity during Jurassic to present times [2,4]. Theconnections between aquifers are possible eitherthrough faults or by vertical leakage.

Two main fault classes were identified: (i) thelarge-scale faults inferred from seismic data [5,6] and(ii) the small-scale faults observed on geologicalcross-sections reconstructed through lithostratigraphiccorrelation using boreholes (Fig. 2) or documented ongeological maps [7].

Figs. 2 and 3 show the most prominent faultstructures identified and correlated in the studiedarea, they display a NW-SE striking trend with throwstowards the NE. These faults run from SW to NE: theTebaga fault, the Medenine fault, the Mareth fault, theZarat fault, the Lella Gamoudia fault and the OumZassar fault. Several minor, unevenly spaced faults inthe studied field were mapped on the geologic cross-sections. The main ones were correlated and they arethe Ksar Chrarif fault, the Zeuss fault and the Koutinefault. They display a NE-SW striking trend withthrows towards the NW for the Ksar Chrarif andKoutine faults and to the SE for the Zeuss faults. Allthese faults created the compartmentalization of thearea and built up a system of horst and graben struc-tures, within a globally down-tilted domain towardsthe NE.

This fault classification has several importantimplications from a modelling point of view, as itallows a definition of the fault hierarchy that has to beincluded in the modelling procedures.

2.3. Geostatistical modelling

A consistent architectural model is constituted notonly of the surfaces that fit the observation data, but alsoof correct relationships between the geological inter-faces. For this purpose, classical kriging and krigingwith inequality constraint estimates are compared andevaluated in order to build geometric models that reflectas well as possible the geological reality.

These kriging estimators are but variants of thebasic linear regression estimator. They are based onthe regionalized variable theory where the value of avariable z(x) at a point with coordinates vector x isconsidered a realization of a random variable Z(x).The collection of spatially correlated random variables{Z(x), x 2 R}, where R denotes the study region, istermed a random function [8].

2.3.1. Ordinary kriging

We consider the problem of estimating the valueof the target variable “depth”, at any unsampled loca-tion x, of a bounding surface using a given neighbour-hood fZðXaÞa ¼ 1; . . . ; ng, at the same boundingsurface. The kriging estimator is a linear regressionestimator Z�ðxÞ defined as:

Z�ðxÞ ¼Xn

a¼1

kaZðxaÞ ð1Þ

where ka is the weight assigned to ZðxaÞ [9].

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The weights are chosen so as to minimize the esti-mation error variance:

r2j ¼ Var½Z�ðxÞ � ZðxÞ� ð2Þ

under the constraint of unbiasedness of the estimator.These weights are obtained by solving the linear equa-tions of the kriging system. Kriging requires the useand hence the estimation and modelling of a vario-gram function that describes the spatial variability ofthe random variable ZðxÞ:

cðhÞ ¼ 1

2Var½ZðxÞ � Zðxþ hÞ� ð3Þ

where h is the distance separating data Z(x) andZ(x + h).

2.3.2. Kriging with inequality constraints

To enhance the estimation of the target variable“depth”, kriging with inequality constraint makes itpossible to use the information obtained from wellsnot intercepting the target horizon.

Accordingly as shown on Fig. 4, kriging withinequality constraints can take into account “harddata” consisting of exact values: ZðxaÞ ¼ za; a ¼ 1;. . . ;m; and complementary “inequality data”ZðxaÞ 2 Aa ¼ ½aa;�1½; a ¼ mþ 1; . . . ; n, where the

Fig. 3. Fault network correlated from seismic and geological data.

Fig. 2. Geologic cross-section within the study area.

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datum value za is only known to lie within aninterval Aa.

The method is based on a two-step process.First, inequality data have to be replaced by a new

set of exact data consistent with the original exactpoints. The way to replace the intervals is to calculatethe conditional expectation of the target variable ateach inequality data location ZCE

a . To calculate theconditional expectation, a Gibbs Sampler [10] tech-nique is used to simulate, for each inequality, a givennumber of realizations of the target variable accordingto its variogram model and conditioned by the inter-vals and the exact data [8]. Then, the average value ofthe realizations at each inequality data point is calcu-lated. These average values are an approximation tothe conditional expectation.

The simulation of ZCEa ; a ¼ mþ 1; . . . ; n conditionally

on: Zb ¼ zb; 1 � b � m and Zb 2 Ab;mþ 1 � b � nincluding b ¼ a is implemented by repeating the follow-ing sequence:

(1) Assign a random value za within Aa ¼ ½aa;�1½for each site xa; a ¼ mþ 1; . . . ; n

(2) Select an index a0 at random in the set ofinequality data {a ¼ mþ 1; . . . ; n}.

(3) Ignore the value at this site and estimate it bykriging from the current values zb at all othersites; also compute the corresponding kriging

variance r�2a0.(4) Replace the value at this site by the kriged

value plus a simulation of the error, condition-ally on the inequality data at xa0: the new

za0 ¼ ZKa0 þ r�a0 U where U is a standard normal

random variable chosen so that za0 honours theinequality.

(5) Go back to 2, and loop many times.(6) Calculate the conditional expectation at this site

by averaging the set of simulations za0 ða ¼ mþ1; . . . ; nÞ. The conditional expectation ZCE

a0 is in

fact the most probable value of the variable at theinequality data locations.

ZCEa0 ¼ E½ZðxÞnZa 2 Aa08a� ¼

X

b

kaE½ZbnZa 2 Aa08a� ð4Þ

Then, the second step is to estimate the target vari-able using ordinary kriging with both the exact dataand the conditional expectation values that replace theinequalities. It is also possible to consider the condi-tional variance r�2a derived from the simulations (step6) as a variance of measurement error. This has theadvantage of giving less confidence to the conditionalexpectation values than to the exact data. However, itshould be noted that, by doing that, the resulting vari-ance may still be too optimistic; indeed, we do notaccount for the fact that the errors made when replac-ing inequalities by the conditional expectation are notindependent.

3. Results

The results shown are for the estimation of theTuronian upper boundary. Remember that our casestudy is characterized by two main features that makethe modelling task relatively difficult: (i) the spatialdistribution of values is random, in that some obser-vations are close to each other and some others arescattered. (ii) Geometric fault parameters compartmen-talize the reservoir surface into subdomains withdifferent data density. These two features are twomajor causes of uncertainties. An important contribu-tion of geostatistics is the assessment of the uncer-tainty on unsampled values that usually helps tohandle compartments with different data density andthus to choose the relevant kriging method.

A key step before prediction is the modelling ofthe spatial distribution of Turonian depth variable.The experimental variograms were calculated in twospecific directions (Fig. 5): along the normal NW-SEfaults, the major continuity direction; and across thefaults that is along the SW-NE subsidence direction.This variogram shows an extremely high variabilityowing to the fact that it may include couples where xand x + h are taken in two different compartments.

The directional variograms show an anisotropicbehaviour: (i) A stationary structure expressed in theNW-SE direction. This locally stationary structure is

Fig. 4. Geological cross-section showing the available datafor the Top Turonian reservoir:exact data: borehole data corresponding to the geologicalinterfaces: Z(x)A. Mjirda =�409m,inequality data: information provided by the end ofdrilling: Z(x)Zeuss 1 <�200m.

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related to the depth variability, at a small scale, withineach compartment. (ii) A drift structure expressed inthe NE-SW direction. The drift structure reflects thecontinuous increase in the depth within the overalldown-tilted study area towards the NE direction.

For each target location, the neighbouring samplesmust be located within the considered compartment,between the fault boundaries. Therefore, the modelparameters must be inferred from variogram valuesfrom only a few distance classes. Accordingly, thevariogram has to be fitted on the basis of their behav-iour towards small and medium distances, only thestationary behaviour is taken into account for depth

interpolation. The variogram was fitted with ananisotropic stationary spherical structure with a rangeof 10,000m along the NW-SE direction and 5,000malong the NE-SW direction and a sill of 2000m:cðhÞ ¼ 2000 spherical ð10000N120; 5000N30Þ.

Ordinary kriging was applied using only exactdata. Fig. 6(A) shows the kriged depth map of the topTuronian reservoir. It shows that ordinary krigingwith local search neighbourhoods provides goodresults in the central compartment where the datadensity is high. This is confirmed by the low krigingvariance values (Fig. 6(B)). On the contrary, in the NEcompartments, ordinary kriging is inappropriate: thereare inconsistencies resulting from two special proper-ties of the estimating procedures: (i) the first concernsthe neighbourhood information which has to berestricted inside the compartment and (ii) the secondis the violation of the constraints at wells where thehorizon has not been reached by the boreholes.

The method of kriging with inequality constraintsmakes it possible to take into account the constraintinformation obtained from wells which have not inter-cepted the target horizon. It makes the best use of theavailable data and thus increases neighbourhoodinformation within each compartment. Fig. 7(A)shows the depth map of the top Turonian reservoir. Itdemonstrates that kriging, with a local search neigh-bourhood, is successful in all the compartments of thestudy area. In the NE compartments, the inequalitiesare now adequately taken into account in the model-ling process. Associated kriging variance values aresmall even around the inequalities because, in thepresent case, most inequalities correspond to wellswhere the horizon has not been reached (depthP cer-tain value). Consequently, the probable value for thesewells is close to the end of the borehole and the asso-ciated kriging variance is small. As discussed in thesection above, obtained kriging variances might be

Fig. 5. Directional variograms of the Turonian Horizon.N30: NE-SW directional variogram, N120: NW-SEdirectional variogram.

Fig. 6. (A): Depth map of the Top Turonian reservoir calculated by ordinary kriging. Location of exact data (black crosses +)and inequality data (pink circles ). (B): Kriging variance map.

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slightly underestimated as the correlation of errors isignored.

Fig. 8 presents a NE-SW cross-section intersectingseven boreholes. Four of them are shown in black andrepresent the exact data and three boreholes areshown in pink and represent inequality data (thedepth values of the Top Turonian reservoir are indi-cated for each borehole). This figure presents, as well,in blue continuous line the interpolated surface usingordinary kriging and in pink dashed line the interpo-lated surface with inequality constraints. Large differ-ences between the two estimators arise particularly atA, B and C locations:

(1) In location A, ordinary kriging does not respectthe dip direction of the Zarat Fault. In fact, theZarat fault dips to the NE direction and not to theSW and this is a physically inconsistent result.

(2) In location B, ordinary kriging cannot directlyhandle the inequality constraints imposed byHenchir Jdidi well, this measurement is ignoredand the surface is overestimated with a largekriging variance.

(3) In Location C, ordinary kriging violates theconstraints at Smar well where the horizon has

not been reached by the borehole, the surface isextrapolated with a large kriging variance.

Contrary to ordinary kriging, the constrainedkriging method incorporates both the exact and theinequality data. It respects the dip direction at locationA and improves the estimation in B and C. Also,kriging with inequality constraints provides a smallerkriging variance in areas only informed with inequali-ties (Fig. 6(A) and (B)).

4. Conclusions and perspectives

Characterizing aquifer systems in faulted settingswith scarce data is a challenging task. The “Jeffarade Medenine” aquifer, in South-Eastern Tunisia, is acase in point. The complex geological settingrequired first a thorough review of the available datain order to build a detailed structural framework thatconstituted the fundamental part of the geometricmodelling. Afterwards, it was shown how appropri-ate geostatistical modelling can provide suitableestimates for the target faulted surface. The appliedkriging approach accurately integrated both exact

Fig. 7. (A): Depth map of the Top Turonian reservoir calculated by kriging with the inequalities method. Location ofexact data (black crosses +) and inequality data (pink circles ). B: Kriging variance map.

Fig. 8. Blue continuous line: Interpolated surface based on exact data only; Pink dashed line: Interpolated surface basedon exact and inequalities. Boreholes designed in black are exact data and boreholes designed in pink are inequalities, thedepth values are indicated for each borehole.

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data and inequality constraints, improving predic-tions of each horizon.

To enhance the estimation of a given horizon, itmight be possible to incorporate the correlationbetween successive horizons within a cokrigingapproach. Although classical, this method is compli-cated and requires further work in the present case asthe relationships between the different horizons mayvary from one geological compartment to another.

Improved procedures for estimating geologic inter-faces make it possible to build geometric models thatreflect as well as possible the geological reality, whichresults in a better assessment of water resources andsubsequently a better management of the aquifersystem.

Symbols

ZðxaÞ — Regionalized variable (depth), m

Z�ðxÞ — Z(x) estimator, m

cðhÞ — A variogram function that describes the spatialvariability of the regionalized variable ZðxÞ, m2

ka — Kriging weights

r2j — Kriging variance, m2

ZCEa — Conditional expectation of ZðxaÞ, m

Aa — Inequality interval Aa ¼ ½aa;�1�

References

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[3] B. Ben Baccar, Contribution a l’etude hydrogeologique del’aquifere multicouche de Gabes Sud [Contribution to thehydrogeological study of multilayer aquifer South of Gabes],Doctoral thesis, University of Paris Sud, Orsay, 1982.

[4] A. Mammou, Caracteristiques et evaluation des ressources eneau du Sud Tunisien [Characteristics and evaluation of waterresources in Southern Tunisia], PhD thesis, University ofParis-Sud, Orsay, 1990.

[5] H. Chihi, Fault system analysis of the Zeuss-Koutine aquifer(Southern Tunisia): New insights into reservoir modelling,Fifth international Symposium of IGCP 506 on: Marine andNon-Marine Jurassic Global Correlation and Major GeologicalEvents, Hammamet, Tunisia, March 28–31, 2008.

[6] H. Chihi, M. Bedir, H. Belayouni, Variogram identificationaided by a structural framework for improved geometricmodeling of faulted reservoirs: Jeffara Basin, SoutheasternTunisia, Natural Resources Research 22(2) (2013) 139–161.doi: 10.1007/s11053-013-9201-0.

[7] S. Bouaziz, Geologic map of Medenine (1/100.000), 1990.[8] J.P. Chiles, P. Definer, Geostatistics: Modelling Spatial

Uncertainty, John Wiley & Sons, New York, 2012.[9] G. Matheron, Pour une analyse krigeante des donnees regio-

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[10] S. Gemanand, D. Geman, Stochastic relaxation, Gibbs distri-bution and the Bayesian restoration of images, IEEE Trans.Pattern Anal. Mach. Intell. 6 (1984) 721–741.

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