Numerical simulation of subsurface
characterization methods: application to a natural
aquifer analogue
Janet Whittaker
*& Georg Teutsch
Applied Geology, Geological Institute, University of Tubingen, Sigwartstr. 10, D-72076 Tubingen, Germany
(Received 18 June 1997; revised 8 April 1998; accepted 20 November 1998)
Information from an outcrop is used as an analogue of a natural heterogeneous aquifer in order to provide an exhaustive data set of hydraulic properties. Based on this data, two commonly used borehole based investigation methods are simulated numerically. For a scenario of sparse sampling of the aquifer, the process of regionalization of the borehole hydraulic conductivity values is simu-lated by application of a deterministic interpolation approach and conditioned stochastic simulations. Comparison of the cumulative distributions of particle arrival times illustrates the eects of the sparse sampling, the properties of the individual investigation methods and the regionalization methods on the ability to predict ¯ow and transport behaviour in the real system (i.e. the exhaustive data set). Ó 1999 Elsevier Science Ltd. All rights reserved
1 INTRODUCTION
The reliability of model predictions of groundwater ¯ow and contaminant transport is heavily dependent on an accurate representation of the hydraulic properties of the subsurface. However, cost-eciency and practicali-ties limit the number of boreholes that may be drilled for investigation, leading to uncertainty in the identi®cation of the aquifer characteristics. In addition, the investi-gation methods have individual properties according to, for example, their scale of measurement and
dimen-sionality26. In general, a detailed model parameter set
must be generated from the sparse borehole measure-ments. For a heterogeneous medium the method of parameter regionalization between the borehole posi-tions also has signi®cant in¯uence on the predicposi-tions of ¯ow and transport.
2 APPROACH
The objective of this study is to assess the information provided by two commonly used borehole based sub-surface investigation methods. For an aquifer whose hydraulic properties are known exhaustively, numerical algorithms to simulate the determination of hydraulic conductivities from (a) sieve analyses of cores and (b) ¯owmeter measurements are applied at dierent bore-hole positions. Based on this borebore-hole information, both a deterministic approximation approach and a stochas-tic simulation approach are then used to generate pa-rameter ®elds for the complete aquifer. Thus the process of characterization of the aquifer using sparse data is simulated for a hypothetical aquifer. The eect of the incomplete knowledge of the hydraulic properties is examined by the simulation of a groundwater ¯ow and solute transport scenario for both the aquifer data sets based on the limited borehole data and the exhaustive data set (`reality').
Random ®eld generators, which are used for re-gionalization of borehole data, are based on univariate and bivariate statistical descriptors. Thus they incor-porate various statistical assumptions. The resulting
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*Corresponding author. Present address: Department of
Civil and Structural Engineering, University of Sheeld, Sheeld S1 3JD. Tel.: +44-114 222 5728; fax: +44-114 222 5700; e-mail: j.j.whittaker@sheeld.ac.uk
realizations appear unable to capture the complexity of real sedimentary structures. In particular, the low en-tropy of extreme values (i.e. the continuity and inter-connectedness of the high conductivity regions which are critical factors determining ¯uid and solute
trans-port9) is not reproduced by most statistical methods,
which tend to generate high entropy realizations12.
Gaussian methods are also unable to handle abrupt transitions in hydraulic conductivities, or represent large-scale trends characteristic of natural geological features, where the assumption of statistical stationarity
is inappropriate16,2.
Therefore, in order to be as close as possible to geological reality, an alternative approach is taken, such that the geometric and hydraulic data are obtained from an accessible natural system. Detailed information from an outcrop is taken as a natural analogue of an aquifer system deposited in a similar sedimentary environment. The outcrop is interpreted and mapped in terms of lithofacies elements, whereby it is assumed that each category of element is formed as a result of a speci®c type of sedimentary event and possesses characteristic hydraulic properties. The procedure of mapping
archi-tectural elements19 has been applied for many years in
oil exploration (see e.g. Ref.8) and recently used by
Huggenberger et al.11, Aiken1, Bierkens and Weerts3,
Jussel et al.14,15 and Lapperre et al.17 in the context of
generating hydraulic data representative of an aquifer. Other approaches using realistic hydrogeological infor-mation as a basis for numerical investigations of ¯ow and transport have involved the simulation of
deposi-tional processes25,27or the application of transmissivity
or hydraulic conductivity measurements from test
sites5,7. Further methods for representing heterogeneous
sedimentary deposits are discussed by Koltermann and
Gorelick16and Anderson2.
3 THE EXHAUSTIVE DATA SET: `REALITY'
The outcrop used as a basis for the exhaustive data set was situated in a gravel pit on the Swiss/German border. Believed to have been deposited under a braided river environment, the sediments in this area comprise layers of poorly sorted sands and gravels, sand lenses and lenses of gravels with practically no sand matrix (Open Framework Gravel). Such heterogeneous deposits are a typical feature of alluvial aquifers of Southern Germany
(e.g. the Neckar Valley24,10).
From a high resolution photographic image, an in-terpretation of the lithofacies elements comprising an
area of 25´ 5 m2 of the outcrop was digitized (see
Fig. 1). Hydraulic properties were assigned to each ele-ment, based on published data of measurements also
carried out in the same sedimentary deposits11,14,15. The
elements consist of eight classes, falling into three main categories: Brown/Grey Gravels, Sand and Open
Framework Gravels (in order of increasing hydraulic conductivity). The corresponding hydraulic
conductivi-ties range over 5 orders of magnitude: from 1.8´10ÿ5
to 1.1´10ÿ1 m sÿ1. The hydraulic conductivity is
as-sumed to be isotropic at small scale and the variability within a lithofacies element is not considered. In order to reduce the non-normality of the hydraulics conduc-tivity distribution, all geostatistical analyses were per-formed on the log normal transformations of hydraulic conductivities. The variance of the ln-transformed
hy-draulic conductivities, r2
lnK, is 4.7 and the geometric
mean conductivity, KG, is 1.19´ 10ÿ4 m sÿ1. The
his-togram of the lnK values still shows a skew towards
high conductivities. The porosity of the lithofacies ranges typically from 17% for the poorly sorted Brown
and Grey Gravels to 43% for the sands14. For the
out-crop in question, the mean porosity is 20%. Recognizing that the variability in porosity is not very signi®cant in comparison to the heterogeneity of the hydraulic con-ductivity, the mean value of 20% was assigned as the porosity and eective porosity of each lithofacies cate-gory. Thus an exhaustive data set of the hydraulic properties of the outcrop was formed, and is used in this study as a representation of `reality'.
As an example ¯ow scenario, steady-state con®ned ¯ow governed by a ®xed head gradient of 0.001 over the length of the aquifer is simulated using a standard 3D
®nite dierence model18. Since the system does not
consist of layered, homogeneous formations, both hor-izontal and vertical inter-cell conductivities were deter-mined from the geometric means of the cell hydraulic conductivities. To illustrate the transport properties of a contaminant in such a ¯ow regime, particles are uni-formly distributed along the left hand boundary and tracked in the direction of ¯ow using piecewise
analyt-ical solutions for each model block22. Fig. 1(c) shows
how the paths of the particles are strongly in¯uenced by the structures of high conductivity. Local dispersion is neglected, so that the variation in arrival times of the particles at the right hand boundary occurs solely as a result of the heterogeneity in the hydraulic conductivity ®eld. Although the particles were initially distributed uniformly over the depth of the left hand boundary, due to the eective homogeneity in the vicinity of the boundary, this case is equivalent to that of particles distributed according to the ¯ux across the boundary.
Whilst the average time of arrival of the particles at the right hand boundary is 0.912 years, the median arrival time is 0.607 years, re¯ecting the skew in the hydraulic conductivities to higher values. These arrival times correspond to eective hydraulic conductivities of
1.74´ 10ÿ4 and 2.61´10ÿ4 m sÿ1, respectively.
Com-parison with the conductivities of the individual litho-facies elements shows that these eective conductivities are only an order of magnitude greater than the con-ductivities of the poorly sorted gravels (typically
conductivities (typically 1.0´10ÿ1 m sÿ1 for the Open Framework Gravels). Thus the lower conductivity gravels show a dominant in¯uence on the eective hy-draulic conductivity and average/median arrival times. For an isotropic two dimensional region having a conductivity distribution described by a random log normal function, the eective conductivity is given by
the geometric mean,KG4. The prediction of the arrival
time usingKGcalculated from the exhaustive data set is
1.33 years. The much earlier average arrival time cal-culated for the exhaustive data set may be due to the non-lognormality of the data set, the extent and con-nectivity of the high conductivity regions (leading to statistical anisotropy) and/or the in¯uence of the boundary conditions. The variance of the particle ar-rival times is 0.663, and, as an arbitrary measure of the early arrival times, the 5% quantile of the particle ar-rival times is 0.42 years.
The simulated ¯ow and transport scenario for the exhaustive data set is viewed as the representation of
reality. The remainder of this paper considers the situ-ation when knowledge of the aquifer is reduced through sparse sampling of the aquifer. The situation when only four boreholes are available for sampling this section of aquifer is investigated. Two subsurface investigation methods are simulated: regionalization of the simulated borehole based conductivity values is carried out using both deterministic approaches and a geostatistical technique in order to generate hydraulic conductivity ®elds representing the aquifer section. Their ability to reproduce the `real' ¯ow and transport behaviour is used in the assessment of each approach.
4 NUMERICAL SIMULATION OF SUBSURFACE CHARACTERIZATION METHODS
In practice an aquifer must be characterized on the basis of a limited number of boreholes. In an investigation of the role of data in the characterization of heterogeneous Fig. 1.(a) Sedimentary structures of a sand and gravel outcrop. (b) The mapping of the lithofacies. (c) Simulated pathlines for
sand and gravel aquifers using geostatistical simulations,
Egglestonet al.7observed a threshold number of data of
approximately three measurements per integral volume. Above the threshold number an increase in data brought more bene®t through increased conditioning than
through improved variogram de®nition. Egglestonet al.7
concluded that, whilst some test sites have a large amount of redundant data, most real world situations are not sampled densely enough to characterize the subsurface heterogeneities. In the remainder of this pa-per we consider a realistic situation of sparse sampling of the exhaustive data set to examine the consequences in terms of unreliability in the prediction of groundwater ¯ow and solute transport.
In this example a cluster of four boreholes is posi-tioned in the aquifer with spacings of 2, 1 and 4 m be-tween boreholes, from left to right (see Fig. 1(c)). The leftmost borehole is initially positioned at the arbitrary distance of 5 m from the left hand boundary; other positions are considered later. In the following two sections algorithms are described for the numerical simulation of two commonly used subsurface investi-gation methods. In both cases, the exhaustive data set is used to simulate, as far as possible, the hydraulic con-ductivities that would be derived from the methods if they were carried out in the hypothetical aquifer.
4.1 Sieve analysis
As an indirect method of obtaining the K values of core samples, sieve analysis uses the correlation between grain size and hydraulic conductivity. Correlations of grain size with conductivity sometimes appear to be poor, probably attributable to textural grain packing
eects at a smaller scale23. However,Kvalues derived by
Jussel et al.14 from sieve analyses in these deposits
compared well with permeameter measurements in horizontal and vertical directions for undisturbed ma-terial from individual lithofacies elements. In other de-posits showing a similar degree of heterogeneity good agreement was found between the average conductivities
derived from sieve analyses and pumping tests10.
In simulating sieve analyses of a core for the ex-haustive data set, the relationship between grain size and K value is assumed to be valid. The classi®cation of the lithofacies elements situated at the position of the borehole are examined and it is assumed that it would be possible to carry out sieve analysis on each `distin-guishable' segment of the core, where distinguishable is de®ned as a segment of one class of lithofacies ele-ment with a length of at least 5 cm. All errors in the experimental process and interpretation are ignored.
Thus a complete analysis of the cores produces pointK
values representative of the lithofacies elements present at the borehole positions. Gaps in the vertical
distribu-tion ofKvalues occur where elements are too thin to be
analysed.
4.2 Flowmeter measurements
By measuring the vertical ¯ux at speci®ed intervals throughout the depth of a well, ¯owmeters provide the vertical distribution of ¯uxes into the well. Assuming that the aquifer is layered and that each layer is homo-geneous and of uniform thickness (equal to the
¯ow-meter intervaldz) an eective hydraulic conductivity for
each layer may be obtained from the ¯ux distribution.
The eective conductivities are representative ofKin the
vicinity of the well. In a ®rst algorithm28the eectiveK
values were approximated as means of the hydraulic conductivities in a region assumed to represent the area of in¯uence of the well. A second algorithm is described here. The pumping test is simulated using the exhaustive conductivity ®eld in the ®nite dierence model. Cells within the borehole are assigned a very large hydraulic conductivity and water is withdrawn at a constant rate in the uppermost layer. Thus the distribution of the ¯ux into the well is regulated by the hydraulic conductivities in the vicinity of the well. It is assumed that the ¯ow is not aected by the borehole and ¯ow within the well.
The ¯ux,Qi, into the well over each ¯owmeter interval
dzcan be calculated from cell-by-cell mass balances. The
total ¯uxQwithdrawn from the well is
QX
i
Qi: 1
The distribution of conductivities may be related to the ¯ux distribution following several approaches. Those based on the expansion of analytical solutions for radial ¯ow are not suitable for this two dimensional example,
however the commonly applied approach13,20of
steady-state analysis of a strati®ed aquifer can also be applied to this system. Assuming that the ¯ux through each layer is proportional to the transmissivity of the layer leads to the formula
Ki
K
Qi=dz
Q=B ; 2
whereBis the total depth of the aquifer. The borehole
conductivity distribution is dimensionalized using K, a
conductivity representative of the complete aquifer depth. This may be obtained, for example, from a pumping test carried out in a well fully penetrating the aquifer. Simulating steady-state pumping from a well situated at the centre of the aquifer, two dimensional analysis of the head dierence generated in the aquifer
leads to a value of K1.9654´10ÿ4 m sÿ1. Other
al-ternatives, better re¯ecting the local eective
conduc-tivities, could beKvalues determined from slug tests or
short-term pumping tests, however analyses of these approaches are based on radial ¯ow, therefore not applicable to this example. Whilst eqn (2) holds for steady-state conditions, it may also be applied under
quasi-steady-state conditions, which are typically
The hydraulic conductivities obtained from the ¯owmeter measurements are eective values represent-ing the area de®ned by a depth equal to the ¯owmeter interval and the extent of the in¯uence of the pumping.
Therefore in this two dimensional example, theKvalues
are representative of an area with depthdzeither side of
the borehole. The horizontal extent of the area of sup-port depends on the ¯ow rate and the length of time since the start of pumping. Earlier times better re¯ect the hydraulic conductivity distributions in the near vicinity
of the borehole. Whilst the mean of lnKis dependent on
the scaling conductivity K, it can be shown that the
variance, r2
lnK, is independent of K. This two
dimen-sional simulation and analysis of pumping tests is un-able to take into account the eects of the radial ¯ow to
the well, which should lead to theKvalues further from
the well having less in¯uence21. However, we believe this
algorithm is able to capture the essential characteristics of ¯owmeter measurements, and is certainly an im-provement on means over assumed regions of in¯uence.
4.3 A borehole pro®le
As an example of the hydraulic conductivities simulated for the two techniques, the borehole situated at a dis-tance of 12 m from the left hand side is considered. The
Kvalues of the sieve analysis, which coincide with the
realK values in the core, are shown over the depth of
the borehole in Fig. 2. In addition, the K values
ob-tained from the simulation of ¯owmeter measurements is displayed for analyses carried out for intervals of 5, 15
and 40 cm length. As clearly observed, the variance from
the mean lnK decreases with increasing interval. The
mean lnKitself increases with increasing interval. The
maximum K values derived from the ¯owmeter
mea-surements are much lower than those of the Open Framework Gravels, however the eects of all high conductivity regions are seen, even where the lenses were to too narrow to be distinguished in the sieve analyses (e.g. at the height of approx. 4 m). Fig. 3 illustrates the conductivities in the vicinity of the well and the ¯owlines to the well. The ¯ux is uniform between each ¯owline, showing that the geometry of the high conductivity lenses strongly in¯uence the ¯ux distributions in the well. The lenses can be so situated that, even very close to the well, ¯ow is nearly vertical, thus the assumption of horizontal ¯ow towards a well does not appear to be valid for heterogeneous deposits such as these. Fig. 4 displays histograms of the hydraulic conductivities of the exhaustive data set at the borehole, and the ¯ow-meter measurement (15 cm interval).
5 GENERATION OF AQUIFER PARAMETERS FROM BOREHOLE DATA
The `reduced' information about the aquifer may be used in a variety of ways to construct a conductivity ®eld
for the aquifer. At the most basic level, a meanKfrom
the simulated investigation methods can be used as an eective conductivity for the aquifer, from which a sin-gle particle arrival time can be calculated, given the head
gradient and a constant eective porosity. Thus the geometric mean of the conductivities gained from the borehole investigation methods predict arrival times of 1.80 and 1.57 years for the sieve analyses and the ¯ow-meter measurements, respectively, much later than the mean of the `real' arrival times.
5.1 A deterministic interpolation approach
In order to gain an indication of the variability in arrival times, a simple block interpolation of the borehole in-formation is performed, such that the borehole con-ductivities are applied in regions extending sideways up to a boundary or the midpoint between two boreholes (i.e. the one dimensional equivalent of polygon inter-polation) (see Fig. 5). Fig. 6 illustrates the cumulative distributions of the arrival times for both investigation methods, based on the initial positions of the boreholes, together with the `real' arrival times plotted in bold. As an indication of the accuracy of the predictions of the cumulative distributions of the arrival times, the mean square errors of the quantiles of the predicted arrival times compared with the quantiles of the real arrival times are calculated. The results are 9.1 for sieve analysis and 0.16 for ¯owmeter (interval 15 cm). The block
in-terpolation of the pointKvalues from the sieve analysis
Fig. 5.lnKdistribution for block interpolation of borehole data derived from simulations of sieve analysis.
Fig. 4.Histograms of the real lnKvalues at the borehole (left) and those derived by the simulation of ¯owmeter measurements in the borehole (right).
creates both very fast and very slow pathways, therefore exaggerating early and late arrival times. For the ¯ow-meter measurements, the sideways extension of values representative of a larger scale of support provide a more reliable prediction of the arrival times. However as the ¯owmeter interval increases, the variability in the arrival times decreases.
5.2 A stochastic simulation approach
The block interpolation does not use any information about the statistical nature of the architectural elements shown by the correlation between borehole conductivity
values. Variogram analysis of the boreholeKvalues in
the vertical and horizontal directions can provide esti-mates of correlation lengths and the degree of statistical anisotropy. Stochastic simulation is chosen in preference to kriging, which has the tendency to smooth data, due to favouring regional accuracy over point accuracy. Whilst it might be desirable to characterize the dierent lithofacies structures (e.g. through the indicator ap-proach), the level of sampling in this example is too low.
Therefore the realizations of K ®elds are generated by
sequential Gaussian simulation6, conditioned by the
borehole data.
A normal score transformation of the ln Kvalues was
performed, so that the transformed values have a nor-mal distribution. Variograms were obtained for the
normalized lnK values. In the vertical direction only
data pairs corresponding to the same borehole were used. Fig. 7 shows the variogram in the vertical
direc-tion for the normalized lnK values derived from sieve
analysis, together with a ®tted variogram model. Due to the density of data in the vertical direction, the model can be ®tted with some certainty. Consistent with the assumption that the hydraulic conductivities are con-stant within a lithofacies element, the nugget is set to zero. Although it is also possible to consider the sill of the variogram as an unknown, it is here taken to be equal to the variance of the unweighted sampled data. The principal parameter of uncertainty lies in the degree of statistical anisotropy of the structures, due to the lack
of information for the determination of the range in the horizontal variogram (Fig. 8). Travel times resulting from the ®rst ten realizations of conductivity ®elds using the maximum horizontal range are illustrated in Fig. 9a; results for the minimum range are illustrated in Fig. 9(b). Both show the tendency to overestimate the mean arrival times. There is wider spreading of the re-sults in the case of the maximum horizontal range, which is probably due to the increasing in¯uence of the non-ergodicity of the system.
For the simulations of the ¯owmeter measurements, the variogram in the vertical direction shows the eect of
changing the scale of support of theKvalues. In
addi-tion to a decrease in variance as shown in the sill, in-creasing the ¯owmeter intervals results in a loss in resolution and an increase in range (Fig. 10). In all cases the horizontal variogram remains similar. The increase in ¯owmeter interval also leads to an increase in the
geometric meanKGof the individual realizations. As for
the sieve analysis simulations, the ln K values were
normalized and the variograms ®tted (an intermediate Fig. 7.Vertical variogram of the normalized lnKsieve analysis
data with ®tted variogram (dashed line).
Fig. 8. Horizontal variogram of the normalized lnK sieve analysis data with ®tted variograms (dashed lines).
value of the horizontal range was used). Travel times corresponding to the ®rst ten arrival times are illustrated for a small interval (5 cm) and a large interval (40 cm) in Fig. 11. Whilst small intervals for ¯owmeter measure-ments give a slightly higher variability, in this case it appears to be at the expense of a slightly lower mean
ln K. The mean square error between the `true' and the
computed arrival time quantiles in each realization are 0.26 (40 cm interval) and 0.43 (5 cm interval), both av-eraged over 50 realizations.
6 VARIABILITY OF RESULTS OVER THE AQUIFER
In order to examine the variability of the results ac-cording to the position of the boreholes, 17 dierent positions of the borehole con®guration were considered (retaining the initial spacing of 2, 1 and 4 m between the boreholes). The ®rst borehole was positioned in turn at 1 m intervals at distances of between 1 and 17 m from the left hand boundary. The mean and variance of the sampled data vary greatly between positions, showing that the level of sampling is not large enough to guar-antee accurate representation of the aquifer statistics.
For the sieve analysis simulations KG ranges from
8.74´ 10ÿ5to 1.24´ 10ÿ4m sÿ1, whilstr2
lnKranges from
1.7 to 7.0. In the case of ¯owmeter measurements with
the intermediate interval of 15 cm, KG ranges from
4.05´ 10ÿ5 to 1.29 ´10ÿ4 m sÿ1 andr2
lnK ranges from
0.74 to 4.2. Thus, in this sparsely sampled example, the ¯owmeter measurements fail to re¯ect the variation of
the real aquifer (r2
lnK 4:7), whilst the sieve analyses
can both over- and underestimate the variance.
Whilst the earlier example of the block interpolation provided a fairly good approximation of the arrival times, the variability according to position is large (see Table 1: case a for the combined statistics of all 17 po-sitions), providing considerable uncertainty in the
pre-dictions. The block interpolation of ¯owmeter
Fig. 11.Cumulative distribution of particle arrival times for the exhaustive data set and ten realizations for ¯owmeter data
with interval: (a)dz5 cm; (b)dz40 cm.
Fig. 9.Cumulative distribution of particle arrival times for the exhaustive data set and 10 realizations for sieve analysis data using: (a) the minimum ®tted horizontal range; (b) the
maxi-mum ®tted horizontal range.
measurements provides a more stable and accurate prediction of arrival times (see Table 1: case b), however in positions where the predictions were the worst, the arrival times could be better predicted using Gaussian simulations. On an average, block interpolations of borehole data from both investigation methods tend to underestimate the early arrival times, as indicated by the means of the 5% quantile of particle arrival times.
Variogram analysis shows that, according to posi-tion, it is not always possible to interpret the horizontal variograms. Considering the positions at which the
variance in the sampled lnKvalues is the least and the
greatest for sieve analysis (cases d and f, respectively) and for ¯owmeter measurements (cases h and j, re-spectively), 50 realizations of sequential Gaussian sim-ulations of the conductivities were performed. The statistics of the arrival times are shown in Table 1. The results of the block interpolations corresponding to the cases d, f, h and j are displayed as cases c, e, g and i in Table 1 for comparison. Prediction of mean arrival time is most accurate using the simulated ¯owmeter measurements having low sample variance, irrespective of regionalization method, although arrival times are not as variable as in reality. In general, the Gaussian simulations tend to overestimate the early arrival times, whilst block interpolation leads to some very early ar-rival times, suggesting that the higher conductivity re-gions created by the former regionalization method are less connective than in `reality', whereas the latter method overemphasises connectivity.
7 DISCUSSION AND CONCLUSIONS
In this paper information gained from an outcrop was used as a representation of a real aquifer deposited un-der similar conditions to create a detailed data set ex-hibiting natural heterogeneity. The exhaustive data set has been used to simulate the application of two com-monly used borehole based investigation methods and
to illustrate the characteristics of the K values they
generate in such a system. The accuracy of the aquifer characterization was assessed using a scenario of con-®ned ¯ow and advective solute transport.
The mean hydraulic conductivity is predicted fairly accurately by both investigation methods, however since the exhaustive data set is skewed towards higher con-ductivities, the simulations in general had the tendency to overestimate the mean arrival times. The spread in ar-rival times is observed to be dependent on the variance of the conductivity and the spatial connectivity of high and low conductivity regions. The cumulative distributions of the arrival times varied according to: (a) the properties of the investigation method (e.g. scale of integration, dimensionality), (b) the position of the boreholes and (c) the method of generating the data sets (here illustrated with interpolation and conditional simulation). The low
level of sampling led to signi®cant uncertainty in the values and a dependency on the position of the bore-holes. The particular inaccuracy in estimating the early and late arrival times highlights the diculties in creating reliable model data sets based only on a few ®eld mea-surements for the prediction of contaminant migration and the planning of aquifer remediation.
Whilst the Gaussian simulations based on sieve ana-lyses were better able to represent high permeability lenses and therefore better reproduced the variability of the exhaustive data set, this did not lead to a better prediction of the arrival times at the right hand boundary. Fig. 12
shows the `real' lnKdistribution together with those of
two conditioned realizations generated from the borehole information. The simulations based on the simulations of ¯owmeter measurements were consistently more accu-rate, despite their failure to generate any regions of
per-meability higher than 1.6´10ÿ3 m sÿ1. This can be
explained by the observation that the high permeability lenses are not very signi®cant in the determination of the mean arrival times at the right hand boundary. Thus the
¯owmeter Kvalues, representing a much larger area of
support are applicable e.g. for the estimation of the mean contaminant ¯ux across a control plane (vertically aver-aged). Furthermore, it becomes obvious from Fig. 12 that
the larger structures of the K®eld are quite well
repro-duced in the ¯owmeter realizations.
However, as is observed in Fig. 1, the high perme-ability lenses play a very signi®cant role in the distri-bution of particle location on the right hand boundaries.
Thus the accuracy of local predictions of concentrations is heavily dependent on the ability to reproduce the con®guration and connectivity of the high conductivity lenses. At the chosen low level of sampling of the aquifer both deterministically and stochastically generated
hy-draulic conductivity ®elds were unable to generate K
®elds capable of predicting the real vertical distribution of the particles over the right hand boundary, i.e. the local concentrations.
Although this study was carried out on 2D data sets, the principles behind the simulations are also applicable to three dimensions. Quantitatively, the statistics of particle arrival times would be dierent if ¯ow and transport in the third dimension were taken into account: due to the increased possibilities for connective regions of higher conductivity and opportunities for bypassing of low conductivity regions the average arrival time is expected to be earlier (based on stochastic analysis of eective hydraulic conductivity in two and three
di-mensions4), and the variance of the arrival times would
be lower. However, the question of whether investigation methods and regionalization techniques are capable of reproducing the features of heterogeneous hydraulic conductivity distributions that determine groundwater ¯ow and solute transport is common to numerical sim-ulations in both two and three dimensions.
This illustrative example shows the importance of considering real structures and geometry of naturally heterogeneous porous media to gain further under-standing of our ability or inability to predict the
gration and fate of contaminants under practical con-ditions.
ACKNOWLEDGEMENTS
This study was supported by the European Com-mission through the European Groundwater Research Programme (Human Capital and Mobility Programme) and the Environment and Climate Programme (Con-tract Number ENV4-CT96-5035).
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