Multi-point Modeling of Clay Lensesand its Impact on Aquifer Vulnerability

Whitney Trainor and Jef CaersEarth, Energy, and Environmental SciencesStanford Center for Reservoir Forecasting

April 30 & May 1, 2009

Abstract

Effective groundwater management requires hydrogeologic models builtfrom various data sources. While multi-point geostatistical algorithms havebeen widely applied in petroleum reservoir characterization, applications tohydrogeology are still few. In this paper we show how the multi-point algo-rithm snesim is employed to characterize facies distributions, using a datasetof a groundwater system from Arhus, Denmark. Geological heterogeneity ofthis area is the result of glacial depositional and erosional processes. This geo-logic scenario has created some clay lens features on the surface and within theburied valleys. The clay coverage and lens features determine whether a flowpath exists between a surface contaminant and an extraction well. Bayesianclassification of lithology from resistivity was performed such that the moreextensive resistivity soundings could be used as soft data (representing prob-ability of clay) in the snesim simulations. These facies realizations are thenused for flow simulations with tracers placed exhaustively at the surface. Thegeobody idea was recycled to define the aquifer vulnerability measure: num-ber of tracer concentration bodies intersecting with extraction wells. How torepresent the changing support volume of the resistivity soundings is one ofseveral outstanding issues. Preliminary results suggest another data set withmore near-surface information will be more consequential to vulnerability re-sults.

1 Introduction & Motivation

Much of the world’s drinking water is supplied from groundwater sources. Over thepast several decades, many aquifers have been compromised by surface-born con-taminants due to urban growth and farming activities. Further contamination willcontinue to be a threat until critical surface recharge locations are zoned as ground-water protection areas. This can only be successfully achieved if the hydraulicallycomplex connections between the contaminant sources at the surface and the un-derlying aquifers are understood.

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Denmark is one example of this type of scenario. Since 1995, in an effort toidentify crucial recharge zones, extensive geophysical datasets were collected over37% of the Danish countryside—the areas designated as particularly valuable dueto their high water extraction. The work presented in this report utilizes the geo-physical data and flow information available in an area north of the city of Arhus.As with the rest of the country, the data was collected with the intention of makingmore informed decisions regarding the designation of recharge proctection zones.The magnitude of these decisions is considerable, as it could involve the relocationof farms and large compensations for these explusions. Consequently, incorrectlyidentifying an area as hydraulically connected to important aquifers or vice versa,can lead to a costly error.

Although spatial information has an important role and long tradition in decisionmaking for petroleum applications, this has not been the case for the groundwaterfield. The long-term goals of this research are to develop a methodology that canquantify the value of information for spatial decision-making for groundwater man-agement. An important distinction should be made clear here: although the ArhusNorth data and geology are used in this study, these are mainly used to create arealistic synthetic case. Therefore, no conclusions should be drawn to a particularDanish case from the results in this study.

This report will describe the uncertainty in the subsurface heterogeneity, thegeophysical information and how these uncertainties were taken into account duringthe subsurface modeling. More importantly, flow simulation is performed on thesemodels. These results were used to define a measure of vulnerability—the criticalmeasure for the Danish groundwater decision.

1.1 Geological Scenario: Buried Valleys

Buried Valleys are considered the informal term for Pleistocene (Quaternary) sub-glacial channels. They have also been described as the result of “waxing and wanningof Pleistocene ice sheets” [BurVal Working Group, 2006]. The primary method val-leys are formed is by subglacial meltwater erosion (sudden “outbursts” of meltwaterreleased by glacial lakes). Thus, the valley formation is directly related to the mor-phology and erodability of the geological strata. The secondary method is throughdirect glacial erosion by ice sheets [BurVal Working Group, 2006].

Several of the processes that create and fill buried valleys are important for un-derstanding the complexity of the Danish aquifer systems and their vulnerability tosurface-born pollutants. In Denmark, the superposition of 3 different generations ofglaciation have been observed. Thus, multi-generation glacial valleys cross-cut eachother and can also appear to abruptly end (as seen in Figure 1). The existence andlocation of these glacial valleys can be thought of as the primary level of Denmark’saquifer system structure. If largely filled with sand, the buried valley has potentialfor being a high volume aquifer (reservoir).

However, these buried valleys can be “re-used” as revealed by the observed cut-and-fill structures. This describes the secondary level of uncertainty of heterogeneityin the Danish aquifer systems.

Most cut-and-fill structures are narrower than the overall buried val-

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Figure 1: Network of Buried Valleys; darker to lighter representing older to youngergenerations

ley, but in some places very wide structures that span the entire valleywidth can be seen. The complex internal structure can be observed inseismic surveys, electromagnetic surveys and occasionally in boreholedata. [Sandersen & Jørgensen, 2003]

The possible combinations of heterogeneity are indeterminable, on account of theerosion and deposition processes that create both the valleys themselves and thestructures inside them. Figure 2 shows a few different possible internal hetero-geneities and varying extent of overlying strata, which deems the valley as actually“buried.”

Figure 2: Possible Eroded Substrate & Internal Structures of Buried Valleys modi-fied from [Sandersen & Jørgensen, 2003]

1.2 Hydrogeology: Aquifer Vulnerability

Due to the generally complex internal structure of the valleys, poten-tially protective clay layers above the aquifers are likely to be discon-tinuous....The aquifers inside the valley will thus have a varying degreeof natural protection. Even if laterally extensive clay layers are present,the protective effect will only have local importance if the surround-ing sediments are sand-dominated. ...The valleys may therefore createshort-circuits between the aquifers in the valley and the aquifers in thesurrounding strata. [BurVal Working Group, 2006]

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Experience has led Danish hydrologists to believe that aquifers with a clay capthickness of less than 15m are more vulnerable to anthropogenic contamination,specifically the leaching of nitrate [Thomsen, et. al, 2004]. Thus, these clay fea-tures, whether they are interior structures (as in scenario H in Figure 2) or thecontinuous cap (as in scenario I in Figure 2) are the determining factor whethersurface contaminants reach the underlying or neighboring aquifers.

2 Geophysical Dataset:

Time-Domain Electromagnetic Soundings

Favorable electrical conductivity contrasts exist between the flow-barrier clay faciesand the high permeability sand facies; clay generally has an electrical resistivity lessthan 30 ohm-meters, whereas sand is usually greater than 80 ohm-meters. Hence,most of the geophysical surveys in Denmark have been either electrical or electro-magnetic (EM). The Time-domain EM method (TDEM or TEM) works with atransmitter loop that turns on and off a direct current to induce currents and fieldsinto the subsurface [Christiansen, 2003].

How the TEM method works and measures the earth’s resistivity structure canbe summarized into six steps with the appropriate Maxwell equation1:

1. The constant current Iwire in the transmitter produces a primary magneticfield Hp:

∇×Hp = Iwire

2. Current suddenly terminates; the changing magnetic field induces a secondaryelectric field Es in the earth that attempts to oppose the change (where µ ismagnetic permeability of the earth):

∇× Es =−∂(µHp)

∂t

3. The induced electric field produces an image current, which in turn producesa secondary magnetic field Hs (where σ is electrical conductivity of earth):

Is = σEs

∇×Hs = Is

4. Current diffuses outward and downward over time (t). The diffusion rate:

• is proportional to 1/sqrt(t)

• depends on earth conductivity (σ)

5. The diffusing, time varying current produces time varying secondary magneticfield.

∂(∇×Hs)

∂t=∂Isdt

1modified from lecture notes of D. Alumbaugh, University of California, Berkeley

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6. This decaying magnetic field Hs produces time-varying voltage V in the re-ceiver coil s (where n is the unit normal vector through the receiver coil):

∇V = −∫∂(µHs)

∂t· nds

As depicted in Figure 3, the induced fields are the result of the current being turnedoff in the smaller transmitter coil (Tx) which is inside the larger receiver loop (Rx).The receiver loop measures the changing magnetic field (H) from the induced cur-rents (It). In short, one can think of the TEM method as sampling the earth asdescribed in step 4 – outward and downward. At later time gates (measurementtimes – which range from 10−6 to 10−3 seconds), the measurement swath is wider,but the signal is weaker and perhaps less reliable. The TEM measurement presentsa challenge, as this changing volume support is not easily represented on regulargrids, commonly used to create the lithological models. This is discussed further inthe Section 2.1.

As the Maxwell Equations suggest, the geophysical inversion to obtain resistivitymodels from TEM data is computationally demanding. The state of the science isdeveloping feasible 3D inversions. In view of that fact, it is understandable that themore than 3,000 TEM soundings in the Arhus North area were inverted using 1Dinversion codes. The resulting 1D electrical resistivity models, representing the X-and Y-locations of the transmitter loop, are parameterized into vertical layers, eachwith an electrical resistivity (ρ) and thickness value (h). The geophysical inversionalso provides an uncertainty measure on both of these parameters (also described inSection 2.1).

Figure 3: Left: Tx & Rx loops on surface of conductive, layered earth; Center:Induced, diffusing currents It; Right: Secondary H field

2.1 Calibration of TEM to lithology

The real goal is to relate these TEM models of electrical resistivity to the property ofconsequence: lithological facies that have different flow properties and affect aquifervulnerability. Several challenges exist in relating or calibrating the TEM modelsto lithology. The first is silt. Silt has the electrical properties of sand, and theflow properties of clay. Second, again as seen in Figure 3, is the changing TEMvolume of support. The geophysical inversion partially accounts for this throughthe simulation of the physics. However, the simulations assume homogeneous half-space (where resistivity can only change vertically). And lastly, the lithological

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information available (as what is common in the hydrogeologic community) aredrillers logs. These data are subjective, lithological observations made while well-cuttings are being excavated. The drillers log classifications for the Arhus Northarea were reduced from more than 20 to two: either clay or non-clay (gravel, sand,silt, limestone, etc).

Two general approaches are common when calibrating the larger support data(usually with more exhaustive coverage) to the property of interest at a smaller scale.The first technique is to calibrate the larger support data to represent a probabilityof the high resolution property existing at a certain location. This is useful foruse with multiple-point algorithms which use Bayes law to combine probabilities(such as a probability of a repeating geologic pattern). The second is to utilize thevolume support data as a proportion measurement. This approach was consideredin attempt to downscale TEM models to clay proportions. However, results wereunworkable, limited by the poor point-scale variogram from the driller’s logs.

Along with the ≈3,000 TEM models, Arhus North has ≈750 drillers logs. How-ever, only 7 TEM-drillers log pairs are within 25 meters map distance (XY location).Perfect collocation most likely doesn’t exist since the TEM measurement can be ru-ined by metal in the well head or casing. On average, the TEM models reach 300meters depth while the drillers logs are usually 100 meters deep. Both the TEMmodels and the drillers logs were re-sampled at 1 meter vertical spacing, such thatsome TEM layers’ resistivities were repeated to account for layer thickness >> 1m(this may account for the high modes seen in Figure 4). Generally Figure 4 showsa favorable resistivity separation between sand and clay. However, it is importantto note that the effects of re-sampling, the different support size of the two mea-surements, and the subjective nature of the drillers logs have not been taken intoaccount.

Also included in Figure 4 is the error estimates from the TEM inversion. Theconfidence interval for each parameter is based on a linear approximation to thecovariance of the estimation error Cest:

Cest = (GT CG)−1 (1)

where C is the covariance matrix of the estimated data error andG is the Jacobian, iethe partial derivatives of the data vector to the model parameters of layer resisitivity(ρ) and thickness (h) [Tarantola & Valette, 1982]. Standard deviations on modelparameters are calculated as the square root of the diagonal elements in Cest. Theanalysis gives a standard deviation factor (STDF) on the parameter ps (representingρ or h) [Christiansen, 2003].

STDF (ps) = exp(√Cest,ss) (2)

The inversion is executed in the logarithmic space. Therefore, the quotient andproduct with the model parameter ps and the resulting STDF factor can be used todefine a confidence interval of two standard deviations (2σ):

2σ =

[ps

STDF (ps), ps ∗ STDF (ps)

](3)

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Figure 4: Resistivity (Ohm-m) and Lithology of 7 collocated TEM models andDrillers Logs

These confidence intervals give a measure of the inaccuracy of the TEM models,both the resistivity value and its vertical location, due to data error. This can leadto misclassification of lithology at the 7 “collocations” with the drillers logs. Toaccount for this, the variability is modeled using realizations of each of the original7 collocated TEM models. Each realization is modified according to the confidenceintervals of the original. Thirty realizations of the 1D TEM model were made bydrawing from a multivariate normal distribution. This distribution’s mean is definedby the original TEM model parameters, and the diagonal of the covariance is definedby the uncertainty values from the TEM parameters. Since resistivity and thicknessare negatively correlated in the TEM response, the off-diagonal terms were assigneda -0.6 value. This value is typical for statistical rock physics simulations involvingattributes that are known to be physically negatively correlated.

The calibration provides the likelihood of the electrical resistivity ρ given thelithology being clay or non-clay. Using Bayes rule (Equation 4), the “posteriors”or “soft probabilities” can be calculated: given the TEM model resisitivities whatis the probability of the lithology being clay or non-clay? The marginal (p(ρ)) andprior (p(litho)) are scaling factors and were assigned a value of 1.

p(litho|ρ) =p(litho)p(ρ|litho)

p(ρ)(4)

As is apparent in Figure 4, the resistivity values discriminate well between clay andnon-clay. Figure 5, shows the 3D view of the locations of TEM models, with theiroriginal posterior values.

3 Geostatistical Simulation

As mentioned in Section 1.1, two levels of subsurface heterogeneity exist: the loca-tion of the buried valleys (the reservoir) and the interior structure of the valleys.Although the locations of the valleys themselves could be represented stochasticallyusing the patterns of Figure 1, current work focuses on the interior structures. Thus,

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Figure 5: Clay posterior; 3D view from above

Figure 6: Deterministic Model of Buried Valley (Red) and Non-valley (Blue) Loca-tions. Vertical Exaggeration x25. View from South

valley and non-valley are modeled deterministically from the TEM models (shownin Figure 6).

Very little information exists in the geological literature on the dimension of theseinternal structures. To represent the uncertainty on clay lens dimension and spatialpatterns, several binary clay lens training images (TI–representation of repeatingpatterns of geologic system) were created (seen in Figure 7) using the GSLIB Ellipsimprogram [Deutsch, et. al, 1998]. Blue represents the clay (flow-barrier) facies, andred is the non-clay (permeable) facies.

These TI’s are used with the algorithm snesim within Stanford’s GeostatisticalEarth Modeling Software (SGeMS) [Remy, et. al, 2002]. Single Normal EquationSimulation (snesim) is a multiple-point geostatistical algorithm, which generates astochastic facies realization using the TI [Strebelle, 2002]. How snesim accomplishesthis can be roughly summarized into seven steps

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Figure 7: 5 different Clay Lens Training Images

On the Training Image:

1. snesim scans the TI for all independent patterns using a template

2. The template captures different data patterns

3. The “Search tree” is used to organize and store different “data events” orunique patterns from the TI

On the simulation grid:

4. Randomly select a location

5. Define a “data event” within the template

6. Use the search tree to find similar patterns and define the probabilities on thepossible discrete values for the center of the template

7. Draw from these probabilities to determine the center value.

The region feature was used to only simulate clay features within the buried valleylocations.

Additionally, to test the influence of the posterior on both the snesim realizationand eventually the flow simulations, a synthetic, less discriminating data set ofresistivity and lithology was created (Figure 8) as an alternative to the one providedin Figures 4 and 5.

However, to make either the data-calibrated posterior or the less-discriminatingposterior compatible with the TI’s of Figure 7 and the snesim algorithm, smoothedcubes of the posteriors were created. In order to avoid artifacts due to the verti-cal support of the TEM models (ie Figure 5), the vertical sampling of the TEM-posteriors was decimated such that only every 3rd cell (or 12m) was retained. Using

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this “thinned” set of posteriors as hard data in an SGSIM realization, the final pos-teriors were obtained. Figure 9 shows the same cross-section location for both thesynthetic and original posteriors.

4 Flow Simulation: Defining a Vulnerability Mea-

sure

All the binary snesim realizations were then utilized in flow simulation. The clayfacies was assigned a permeability of 0.1 milli-Darcy (mD) and non-clay 1000 mD.Specifically of interest is how surface-born contaminants will reach extraction (pump-ing) wells. To observe how different flow property heterogeneity will influence thesurface-to-well hydraulic connections, a conservative tracer was placed exhaustivelyon the surface. The sources (boundary recharge and rainfall) and sinks (extractionwells) are simulated for 20 years.

A conventional metric to compare surface-to-well connectivity is the volume oftracer pumped out. However, we propose a different method extending the geobodyidea to concentration bodies. The geobody concept was developed to identify con-nected bodies (cluster identification) in binary facies models [Hoshen & Kopelman, 1976],and later used to modified the snesim algorithm to condition to tracer tests (hy-draulic connection data) [Renard & Caers, 2008].

To utilize the geobody idea, a concentration threshold is defined for the sim-ulation grid at the last time step. For this study, a threshold of 10−10 lbs/stb(pounds per standard barrel – field units in an oil reservoir simulator). Defininga threshold allows for the continuous concentration cube to be transformed to abinary representation of connected concentration bodies (Figure 10). A connectionevaluation can then be made about how many independent concentration bodiesintersect the 61 pumping wells, or the total length of these concentration body-wellintersections can be evaluated. This discrete measure is advantageous compared tothe more abstract metric of the volume of tracer pumped out. The concentrationbodies metric provides a more concrete notion of the number of connections from

Figure 8: Resistivity (Ohm-meter) vs Lithology: Example of less discriminatingTEM

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Figure 9: Cross-Section at X=567375. Top: X-Section of Synthetic Posterior. Bot-tom: X-section of Original Posterior. Red color means higher probability of clay atthat location. Grey indicates non-valley location.

Figure 10: Top: Log Concentrations. Bottom: Binary representation withthreshold=10−10. Black lines represent locations of pumping wells.

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the surface to extraction wells, and is therefore considered a suitable indicator ofaquifer vulnerability.

Figure 11: Number of concentration body intersects from flow simulations on real-izations from original posterior; Clay percentage for SNESIM realizations: Left plot15%, right plot 25%. Shapes depict different random seeds

In Figures 11 and 12 the x-axis represents the area of the largest clay lens featuredin the TI. Above each string of models is the TI number that was used to generatethem (shown in Figure 7). The plot of the left represents the realizations with 15%clay, and on the right, those with 25% clay are plotted. Each group has three modelsgenerated with different random seeds.

Specifically, the plots of Figure 11 show two trends. First, as the size of thelargest clay lens increases, the number of concentration bodies-to-well connectionsdecreases. And second, the realizations with 25% clay (right) show fewer connectionsthan those with 15% (left). Figure 12 displays the total length of all the concentra-tion body-well connections. Again, the two trends (decrease in length with increasein clay lens area and clay percentage) are generally seen except for the suite of mod-els from TI 5. The length of the intersections is longer for TI 5. Unlike the othertwo training images, TI 5 has more clay features of different sizes and shapes. Thesecomplex patterns may restrict the possible preferential flow paths from the surfaceto well, which may explain why the intersection lengths are longer.

The cross-sections in Figures 13 and 14 give insight into how the clay influencesthe tracer. Figure 13 demonstrates the log tracer concentration and permeabilityfor the small clay lenses of TI 1, while Figure 14 is from a model created with TI5, with distribution of smaller and larger clay lenses. Figures 15 and 16 show tworesults from models created with TI 3 and TI 4 respectively.

The smaller clay features of Figure 13 result in more “fingering” of the concen-tration bodies. In Figure 14, the left side demonstrates an absence of clay (whichis consistent with the clay posterior), and consequently the average tracer depth is≈ 10 cells. The right side has more clay and more limited tracer penetration (av-erage penetration ≈ 5 cells). Therefore, information with more near-surface detailswill have increased consequences for the vulnerability measurement.

Figures 17 and 18 are the results from the realizations that used the soft prob-ability from the synthetic, less-discriminating posterior. These figures demonstrate

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Figure 12: Length of concentration body-well intersection from flow simulations onrealizations from original posterior

Figure 13: Cross-Section at X=567375 formodel from TI1 15% clay. Top: LogTracer Concentration. Bottom: Perme-ability (Pink=Clay)

Figure 14: Cross-Section at X=567375 formodel from TI3 15% clay. Top: LogTracer Concentration. Bottom: Perme-ability (Pink=Clay)

the same trends of Figures 11 and 12. The flow simulations on the models createdfrom TI 5 (with ≈ 3 different clay lens sizes) show the same tendency to have longerconcentration body connections with the wells. Also similar to the simulations withthe original posterior, the models with the greatest clay lens area and 15% claydemonstrate greater concentration body-well connections. This could be becausethe marginal (clay percentage) and the length of the clay lens of TI given to thesnesim algorithm are incompatible, causing the actual clay lenses in the realizationsto be not as large.

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Figure 15: Cross-Section at X=567375 formodel from TI2 15% clay. Top: LogTracer Concentration. Bottom: Perme-ability (Pink=Clay)

Figure 16: Cross-Section at X=567375 formodel from TI2 15% clay Scale×2. Top:Log Tracer Concentration. Bottom: Per-meability (Pink=Clay)

Figure 17: Number of Intersections from realizations using synthetic posteriors. Leftplot: models with 15% Clay. Right plot: models with 25%

5 Summary

The work presented here represents the first developments of a workflow whichincludes geological information, geostatistical modeling and flow simulation. Themodeling is performed with the objective of resolving the groundwater decision:which locations are vulnerable to surface contaminants and therefore should be pro-tected? Currently, modeling of the buried valleys is done deterministically usingthe ≈3,000 TEM models. It was these resistivity models that were also used todetermine the likelihood of the presence of interior clay structures. After generatingsnesim realizations conditioned to these soft probabilities and different clay lenstraining images, flow simulation was performed. A measure of vulnerability was es-tablished recycling the geobody concept for tracer concentration bodies. However, itwas observed from the cross-sections of tracer concentrations that the heterogeneityin the upper 5 cells (20m) really determines the preferential flow paths of the tracer.Therefore, a different data source should be considered to extract more near surfaceinformation.

Electrical profiling, where current and potential electrodes are inserted into the

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Figure 18: Intersection Lengths from realizations using synthetic posteriors. Leftplot: models with 15% Clay. Right plot: models with 25%

ground at different spacings, gives good resolution of near-surface conductive fea-tures such as clay. However, because of the need for galvanic contact with the sub-surface (insertion electrodes) and the particular spacing between electrodes neededfor the inversion algorithms (moving of electrodes), this method was too labor in-tensive and hence expensive for extensive coverage. The method was improved bymounting the instruments on a vehicle that could be pulled; it is now known as thePACES method (pulled array continuous electrical sounding) [Sørensen, K. I., 1996].Secondly, the design provided more electrode separations. Having more currentelectrodes, the new method provides better possibilities for interpreting multiplenear-surface layers, and, hence, the presence or absence of protective clay covers.

Besides utilizing PACES, other future work may include analyzing: the uncer-tainty of drillers logs, how decimation may have affected the near-surface informationin the posterior, how to speed up flow simulations and more simulations (results) formore resolute conclusions. We saw that the critical depths for aquifer vulnerabilitywere in the top five grid cells. However, in order to make a viable soft probabilitycube, only every 3rd TEM-posterior was retained. Also, the changing (diffusing)TEM volume support was not truly accounted for. Each one of the 20 year sim-ulation, requires 1 hour on 8 nodes using the parallel Eclipse flow simulator. Themass conservation computation (necessary for the tracer) significantly increases theCPU time. For future work, it maybe worthwhile to consider streamline simulationin place of flow simulations. Further simulations with varying clay lens sizes areneeded to define in which situations, higher quality data is more useful in detectingthe consequential subsurface heterogeneity.

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[Deutsch, et. al, 1998] Deutsch, C., and Journel, A., 1998. GSLIB GeostatisticalSoftware Library and User’s Guide, Oxford University Press: New York.

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[Thomsen, et. al, 2004] Thomsen, R., Søndergaard, V.H., and Sørensen, K.I., 2004.Hydrogeological Mapping as a Basis for Establishing Site-specific GroundwaterProtection Zones in Denmark. Hydrogeology Journal, V. 12, 550-562.

[Remy, et. al, 2002] Remy, N., and Journel, A., 2002. J.GsTL: A GeostatisticalTemplate Library in C++, Computers & Geosciences; Proceedings of theIAMG2001, Annual Conference of the International Association for Mathe-matical Geology, 28, 8, 971-979.

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