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Advances in Water Resources Vol. 22, No. 8, pp 819±829, 1999
Ó 1999 Elsevier Science Ltd
Printed in Great Britain. All rights reserved
0309-1708/99/$ ± see front matter
Numerical simulation of subsurface
characterization methods: application to a natural
aquifer analogue
Janet Whittaker* & Georg Teutsch
Applied Geology, Geological Institute, University of T
ubingen, Sigwartstr. 10, D-72076 T
ubingen, 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 simulated 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
2 APPROACH
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 practicalities 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 investigation methods have individual properties according to,
for example, their scale of measurement and dimensionality26. In general, a detailed model parameter set
must be generated from the sparse borehole measurements. For a heterogeneous medium the method of
parameter regionalization between the borehole positions also has signi®cant in¯uence on the predictions of
¯ow and transport.
The objective of this study is to assess the information
provided by two commonly used borehole based subsurface 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 borehole positions. Based on this borehole information, both
a deterministic approximation approach and a stochastic simulation approach are then used to generate parameter ®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 regionalization of borehole data, are based on univariate
and bivariate statistical descriptors. Thus they incorporate various statistical assumptions. The resulting
*
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
819
820
J. Whittaker, G. Teutsch
realizations appear unable to capture the complexity of
real sedimentary structures. In particular, the low entropy of extreme values (i.e. the continuity and interconnectedness of the high conductivity regions which
are critical factors determining ¯uid and solute transport9) 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 architectural 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 information as a basis for numerical investigations of ¯ow
and transport have involved the simulation of depositional processes25,27 or the application of transmissivity
or hydraulic conductivity measurements from test
sites5,7. Further methods for representing heterogeneous
sedimentary deposits are discussed by Koltermann and
Gorelick16 and 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 interpretation 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 element, 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 conductivities range over 5 orders of magnitude: from 1.8 ´ 10ÿ5
to 1.1 ´ 10ÿ1 m sÿ1 . The hydraulic conductivity is assumed 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 conductivity distribution, all geostatistical analyses were performed on the log normal transformations of hydraulic
conductivities. The variance of the ln-transformed hydraulic conductivities, r2ln K , is 4.7 and the geometric
mean conductivity, KG , is 1.19 ´ 10ÿ4 m sÿ1 . The histogram of the ln K 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 outcrop 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 conductivity, the mean value of 20% was assigned as the
porosity and eective porosity of each lithofacies category. 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 horizontal and vertical inter-cell conductivities were determined from the geometric means of the cell hydraulic
conductivities. To illustrate the transport properties of a
contaminant in such a ¯ow regime, particles are uniformly distributed along the left hand boundary and
tracked in the direction of ¯ow using piecewise analytical 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. Comparison with the conductivities of the individual lithofacies elements shows that these eective conductivities
are only an order of magnitude greater than the conductivities of the poorly sorted gravels (typically
2.0 ´ 10ÿ5 m sÿ1 ), and much lower than the highest
Numerical simulation of subsurface characterization methods
821
Fig. 1. (a) Sedimentary structures of a sand and gravel outcrop. (b) The mapping of the lithofacies. (c) Simulated pathlines for
advective transport in a con®ned aquifer.
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 hydraulic 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, KG 4. The prediction of the arrival
time using KG calculated from the exhaustive data set is
1.33 years. The much earlier average arrival time calculated for the exhaustive data set may be due to the
non-lognormality of the data set, the extent and connectivity of the high conductivity regions (leading to
statistical anisotropy) and/or the in¯uence of the
boundary conditions. The variance of the particle arrival times is 0.663, and, as an arbitrary measure of the
early arrival times, the 5% quantile of the particle arrival 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 situation 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
822
J. Whittaker, G. Teutsch
sand and gravel aquifers using geostatistical simulations,
Eggleston et al.7 observed 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. Eggleston et 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 paper 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 positioned in the aquifer with spacings of 2, 1 and 4 m between 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 investigation methods. In both cases, the exhaustive data set is
used to simulate, as far as possible, the hydraulic conductivities 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, K values derived by
Jussel et al.14 from sieve analyses in these deposits
compared well with permeameter measurements in
horizontal and vertical directions for undisturbed material from individual lithofacies elements. In other deposits 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 exhaustive 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 `distinguishable' segment of the core, where distinguishable
is de®ned as a segment of one class of lithofacies element 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 point K
values representative of the lithofacies elements present
at the borehole positions. Gaps in the vertical distribution of K values 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 homogeneous and of uniform thickness (equal to the ¯owmeter interval dz ) an eective hydraulic conductivity for
each layer may be obtained from the ¯ux distribution.
The eective conductivities are representative of K in the
vicinity of the well. In a ®rst algorithm28 the eective K
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
dz can be calculated from cell-by-cell mass balances. The
total ¯ux Q withdrawn from the well is
X
Qi :
1
Q
i
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 approach13,20 of steadystate 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 Qi =dz
;
Q=B
K
2
where B is 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 K 1.9654 ´ 10ÿ4 m sÿ1 . Other alternatives, better re¯ecting the local eective conductivities, could be K values 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
reached after a short period of pumping.
Numerical simulation of subsurface characterization methods
The hydraulic conductivities obtained from the
¯owmeter measurements are eective values representing 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, the K values
are representative of an area with depth dz either side of
the borehole. The horizontal extent of the area of support 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 ln K is dependent on
it can be shown that the
the scaling conductivity K,
2
This two dimenvariance, rln K , is independent of K.
sional simulation and analysis of pumping tests is unable to take into account the eects of the radial ¯ow to
the well, which should lead to the K values 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 improvement on means over assumed regions of in¯uence.
823
and 40 cm length. As clearly observed, the variance from
the mean ln K decreases with increasing interval. The
mean ln K itself increases with increasing interval. The
maximum K values derived from the ¯owmeter measurements 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 ¯owmeter measurement (15 cm interval).
4.3 A borehole pro®le
As an example of the hydraulic conductivities simulated
for the two techniques, the borehole situated at a distance of 12 m from the left hand side is considered. The
K values of the sieve analysis, which coincide with the
real K values in the core, are shown over the depth of
the borehole in Fig. 2. In addition, the K values obtained from the simulation of ¯owmeter measurements
is displayed for analyses carried out for intervals of 5, 15
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 mean K from
the simulated investigation methods can be used as an
eective conductivity for the aquifer, from which a single particle arrival time can be calculated, given the head
Fig. 2. Vertical distribution of K values derived from simulations of sieve analysis (segment length 5 cm) and ¯owmeter measurements (with intervals of 5, 15 and 40 cm) for a borehole situated 12 m from the left hand boundary.
824
J. Whittaker, G. Teutsch
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 ¯owmeter measurements, respectively, much later than the
mean of the `real' arrival times.
5.1 A deterministic interpolation approach
Fig. 3. (a) ln K distributution in the vicinity of a borehole.
(b) Flowlines for steady-state ¯ow to the well.
In order to gain an indication of the variability in arrival
times, a simple block interpolation of the borehole information is performed, such that the borehole conductivities are applied in regions extending sideways up
to a boundary or the midpoint between two boreholes
(i.e. the one dimensional equivalent of polygon interpolation) (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 interpolation of the point K values from the sieve analysis
Fig. 4. Histograms of the real ln K values at the borehole (left) and those derived by the simulation of ¯owmeter measurements in
the borehole (right).
Fig. 5. ln K distribution for block interpolation of borehole data derived from simulations of sieve analysis.
Numerical simulation of subsurface characterization methods
825
Fig. 6. Cumulative distribution of particle arrival times for the
exhaustive data set and the deterministic parameter ®elds.
creates both very fast and very slow pathways, therefore
exaggerating early and late arrival times. For the ¯owmeter 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 borehole K values in
the vertical and horizontal directions can provide estimates 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 approach), 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 K values was
performed, so that the transformed values have a normal distribution. Variograms were obtained for the
normalized ln K values. In the vertical direction only
data pairs corresponding to the same borehole were
used. Fig. 7 shows the variogram in the vertical direction for the normalized ln K 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 constant 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
Fig. 7. Vertical variogram of the normalized ln K sieve analysis
data with ®tted variogram (dashed line).
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 results 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 the K values. In addition to a decrease in variance as shown in the sill, increasing 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 mean KG of the individual realizations. As for
the sieve analysis simulations, the ln K values were
normalized and the variograms ®tted (an intermediate
Fig. 8. Horizontal variogram of the normalized ln K sieve
analysis data with ®tted variograms (dashed lines).
826
J. Whittaker, G. Teutsch
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 maximum ®tted horizontal range.
Fig. 11. Cumulative distribution of particle arrival times for
the exhaustive data set and ten realizations for ¯owmeter data
with interval: (a) dz 5 cm; (b) dz 40 cm.
6 VARIABILITY OF RESULTS OVER THE
AQUIFER
Fig. 10. Vertical variograms of the ln K ¯owmeter data for
intervals: (a) dz 5 cm; (b) dz 15 cm; (c) dz 40 cm.
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 measurements 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 averaged over 50 realizations.
In order to examine the variability of the results according 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 guarantee accurate representation of the aquifer statistics.
For the sieve analysis simulations KG ranges from
8.74 ´ 10ÿ5 to 1.24 ´ 10ÿ4 m sÿ1 , whilst r2ln K ranges 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 and r2ln K 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 (r2ln K 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 positions), providing considerable uncertainty in the predictions. The block interpolation of ¯owmeter
r.
1
r.
1
r.
1
r.
1
±
Block
Block
Block
s.G.s.
Block
s.G.s.
Block
s.G.s.
Block
s.G.s.
4.7 (complete data set)
±
±
1.7
1.7
7.0
7.0
0.74
0.74
4.2
4.2
±
s.a.
f.m.
s.a.
s.a.
s.a.
s.a.
f.m.
f.m.
f.m.
f.m.
Reality
a
b
c
d
e
f
g
h
i
j
(s.a. sieve analysis; f.m. ¯owmeter measurement; Block deterministic block interpolation; s.G.s. sequential Gaussian simulations; b.p. borehole positions; r. realizations)
±
960.0
5.4
1.8
0.50
160.0
58.0
0.13
0.32
10.0
14.0
0.66
730.0
4.5
3.2
0.28
160.0
50.0
0.56
0.10
8.7
14.0
0.42
0.11
0.27
0.09
0.84
0.02
0.33
0.21
0.62
0.31
0.60
0.91
9.2
2.1
0.44
1.4
4.7
3.2
0.85
0.99
3.2
2.7
±
17 b.p.
17 b.p.
1 b.p.
b.p.; 50
1 b.p.
b.p.; 50
1 b.p.
b.p.; 50
1 b.p.
b.p.; 50
Mean 5% quantile of
arrival times (years)
Mean arrival time
(years)
Averaged
statistics
Regionalization
method
Mean variance of ln K
(borehole data)
Investigation
method
Case
Table 1. Statistics of particle arrival times for data sets generated by sampling the aquifer using a four borehole con®guration
Mean variance of
arrival times
Mean square error of
arrival times
Numerical simulation of subsurface characterization methods
827
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 position, it is not always possible to interpret the horizontal
variograms. Considering the positions at which the
variance in the sampled ln K values is the least and the
greatest for sieve analysis (cases d and f, respectively)
and for ¯owmeter measurements (cases h and j, respectively), 50 realizations of sequential Gaussian simulations 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 arrival times, suggesting that the higher conductivity regions 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 under similar conditions to create a detailed data set exhibiting natural heterogeneity. The exhaustive data set
has been used to simulate the application of two commonly 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 conductivities, the simulations in general had the tendency to
overestimate the mean arrival times. The spread in arrival 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
828
J. Whittaker, G. Teutsch
level of sampling led to signi®cant uncertainty in the
values and a dependency on the position of the boreholes. 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 measurements for the prediction of contaminant migration
and the planning of aquifer remediation.
Whilst the Gaussian simulations based on sieve analyses 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' ln K distribution together with those of
two conditioned realizations generated from the borehole
information. The simulations based on the simulations of
¯owmeter measurements were consistently more accurate, despite their failure to generate any regions of permeability 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 K values, representing a much larger area of
support are applicable e.g. for the estimation of the mean
contaminant ¯ux across a control plane (vertically averaged). Furthermore, it becomes obvious from Fig. 12 that
the larger structures of the K ®eld are quite well reproduced in the ¯owmeter realizations.
However, as is observed in Fig. 1, the high permeability lenses play a very signi®cant role in the distribution 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 hydraulic 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 dimensions4), 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 simulations 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 understanding of our ability or inability to predict the mi-
Fig. 12. ln K distributions of: (a) the interpretation of the real sedimentary deposits; (b) a Gaussian realization conditioned by
simulated sieve analysis borehole data; (c) a Gaussian realization conditioned by simulated ¯owmeter borehole data.
Numerical simulation of subsurface characterization methods
gration and fate of contaminants under practical conditions.
ACKNOWLEDGEMENTS
This study was supported by the European Commission through the European Groundwater Research
Programme (Human Capital and Mobility Programme)
and the Environment and Climate Programme (Contract Number ENV4-CT96-5035).
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Advances in Water Resources Vol. 22, No. 8, pp 819±829, 1999
Ó 1999 Elsevier Science Ltd
Printed in Great Britain. All rights reserved
0309-1708/99/$ ± see front matter
Numerical simulation of subsurface
characterization methods: application to a natural
aquifer analogue
Janet Whittaker* & Georg Teutsch
Applied Geology, Geological Institute, University of T
ubingen, Sigwartstr. 10, D-72076 T
ubingen, 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 simulated 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
2 APPROACH
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 practicalities 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 investigation methods have individual properties according to,
for example, their scale of measurement and dimensionality26. In general, a detailed model parameter set
must be generated from the sparse borehole measurements. For a heterogeneous medium the method of
parameter regionalization between the borehole positions also has signi®cant in¯uence on the predictions of
¯ow and transport.
The objective of this study is to assess the information
provided by two commonly used borehole based subsurface 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 borehole positions. Based on this borehole information, both
a deterministic approximation approach and a stochastic simulation approach are then used to generate parameter ®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 regionalization of borehole data, are based on univariate
and bivariate statistical descriptors. Thus they incorporate various statistical assumptions. The resulting
*
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
819
820
J. Whittaker, G. Teutsch
realizations appear unable to capture the complexity of
real sedimentary structures. In particular, the low entropy of extreme values (i.e. the continuity and interconnectedness of the high conductivity regions which
are critical factors determining ¯uid and solute transport9) 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 architectural 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 information as a basis for numerical investigations of ¯ow
and transport have involved the simulation of depositional processes25,27 or the application of transmissivity
or hydraulic conductivity measurements from test
sites5,7. Further methods for representing heterogeneous
sedimentary deposits are discussed by Koltermann and
Gorelick16 and 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 interpretation 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 element, 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 conductivities range over 5 orders of magnitude: from 1.8 ´ 10ÿ5
to 1.1 ´ 10ÿ1 m sÿ1 . The hydraulic conductivity is assumed 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 conductivity distribution, all geostatistical analyses were performed on the log normal transformations of hydraulic
conductivities. The variance of the ln-transformed hydraulic conductivities, r2ln K , is 4.7 and the geometric
mean conductivity, KG , is 1.19 ´ 10ÿ4 m sÿ1 . The histogram of the ln K 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 outcrop 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 conductivity, the mean value of 20% was assigned as the
porosity and eective porosity of each lithofacies category. 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 horizontal and vertical inter-cell conductivities were determined from the geometric means of the cell hydraulic
conductivities. To illustrate the transport properties of a
contaminant in such a ¯ow regime, particles are uniformly distributed along the left hand boundary and
tracked in the direction of ¯ow using piecewise analytical 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. Comparison with the conductivities of the individual lithofacies elements shows that these eective conductivities
are only an order of magnitude greater than the conductivities of the poorly sorted gravels (typically
2.0 ´ 10ÿ5 m sÿ1 ), and much lower than the highest
Numerical simulation of subsurface characterization methods
821
Fig. 1. (a) Sedimentary structures of a sand and gravel outcrop. (b) The mapping of the lithofacies. (c) Simulated pathlines for
advective transport in a con®ned aquifer.
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 hydraulic 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, KG 4. The prediction of the arrival
time using KG calculated from the exhaustive data set is
1.33 years. The much earlier average arrival time calculated for the exhaustive data set may be due to the
non-lognormality of the data set, the extent and connectivity of the high conductivity regions (leading to
statistical anisotropy) and/or the in¯uence of the
boundary conditions. The variance of the particle arrival times is 0.663, and, as an arbitrary measure of the
early arrival times, the 5% quantile of the particle arrival 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 situation 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
822
J. Whittaker, G. Teutsch
sand and gravel aquifers using geostatistical simulations,
Eggleston et al.7 observed 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. Eggleston et 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 paper 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 positioned in the aquifer with spacings of 2, 1 and 4 m between 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 investigation methods. In both cases, the exhaustive data set is
used to simulate, as far as possible, the hydraulic conductivities 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, K values derived by
Jussel et al.14 from sieve analyses in these deposits
compared well with permeameter measurements in
horizontal and vertical directions for undisturbed material from individual lithofacies elements. In other deposits 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 exhaustive 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 `distinguishable' segment of the core, where distinguishable
is de®ned as a segment of one class of lithofacies element 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 point K
values representative of the lithofacies elements present
at the borehole positions. Gaps in the vertical distribution of K values 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 homogeneous and of uniform thickness (equal to the ¯owmeter interval dz ) an eective hydraulic conductivity for
each layer may be obtained from the ¯ux distribution.
The eective conductivities are representative of K in the
vicinity of the well. In a ®rst algorithm28 the eective K
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
dz can be calculated from cell-by-cell mass balances. The
total ¯ux Q withdrawn from the well is
X
Qi :
1
Q
i
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 approach13,20 of steadystate 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 Qi =dz
;
Q=B
K
2
where B is 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 K 1.9654 ´ 10ÿ4 m sÿ1 . Other alternatives, better re¯ecting the local eective conductivities, could be K values 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
reached after a short period of pumping.
Numerical simulation of subsurface characterization methods
The hydraulic conductivities obtained from the
¯owmeter measurements are eective values representing 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, the K values
are representative of an area with depth dz either side of
the borehole. The horizontal extent of the area of support 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 ln K is dependent on
it can be shown that the
the scaling conductivity K,
2
This two dimenvariance, rln K , is independent of K.
sional simulation and analysis of pumping tests is unable to take into account the eects of the radial ¯ow to
the well, which should lead to the K values 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 improvement on means over assumed regions of in¯uence.
823
and 40 cm length. As clearly observed, the variance from
the mean ln K decreases with increasing interval. The
mean ln K itself increases with increasing interval. The
maximum K values derived from the ¯owmeter measurements 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 ¯owmeter measurement (15 cm interval).
4.3 A borehole pro®le
As an example of the hydraulic conductivities simulated
for the two techniques, the borehole situated at a distance of 12 m from the left hand side is considered. The
K values of the sieve analysis, which coincide with the
real K values in the core, are shown over the depth of
the borehole in Fig. 2. In addition, the K values obtained from the simulation of ¯owmeter measurements
is displayed for analyses carried out for intervals of 5, 15
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 mean K from
the simulated investigation methods can be used as an
eective conductivity for the aquifer, from which a single particle arrival time can be calculated, given the head
Fig. 2. Vertical distribution of K values derived from simulations of sieve analysis (segment length 5 cm) and ¯owmeter measurements (with intervals of 5, 15 and 40 cm) for a borehole situated 12 m from the left hand boundary.
824
J. Whittaker, G. Teutsch
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 ¯owmeter measurements, respectively, much later than the
mean of the `real' arrival times.
5.1 A deterministic interpolation approach
Fig. 3. (a) ln K distributution in the vicinity of a borehole.
(b) Flowlines for steady-state ¯ow to the well.
In order to gain an indication of the variability in arrival
times, a simple block interpolation of the borehole information is performed, such that the borehole conductivities are applied in regions extending sideways up
to a boundary or the midpoint between two boreholes
(i.e. the one dimensional equivalent of polygon interpolation) (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 interpolation of the point K values from the sieve analysis
Fig. 4. Histograms of the real ln K values at the borehole (left) and those derived by the simulation of ¯owmeter measurements in
the borehole (right).
Fig. 5. ln K distribution for block interpolation of borehole data derived from simulations of sieve analysis.
Numerical simulation of subsurface characterization methods
825
Fig. 6. Cumulative distribution of particle arrival times for the
exhaustive data set and the deterministic parameter ®elds.
creates both very fast and very slow pathways, therefore
exaggerating early and late arrival times. For the ¯owmeter 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 borehole K values in
the vertical and horizontal directions can provide estimates 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 approach), 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 K values was
performed, so that the transformed values have a normal distribution. Variograms were obtained for the
normalized ln K values. In the vertical direction only
data pairs corresponding to the same borehole were
used. Fig. 7 shows the variogram in the vertical direction for the normalized ln K 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 constant 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
Fig. 7. Vertical variogram of the normalized ln K sieve analysis
data with ®tted variogram (dashed line).
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 results 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 the K values. In addition to a decrease in variance as shown in the sill, increasing 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 mean KG of the individual realizations. As for
the sieve analysis simulations, the ln K values were
normalized and the variograms ®tted (an intermediate
Fig. 8. Horizontal variogram of the normalized ln K sieve
analysis data with ®tted variograms (dashed lines).
826
J. Whittaker, G. Teutsch
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 maximum ®tted horizontal range.
Fig. 11. Cumulative distribution of particle arrival times for
the exhaustive data set and ten realizations for ¯owmeter data
with interval: (a) dz 5 cm; (b) dz 40 cm.
6 VARIABILITY OF RESULTS OVER THE
AQUIFER
Fig. 10. Vertical variograms of the ln K ¯owmeter data for
intervals: (a) dz 5 cm; (b) dz 15 cm; (c) dz 40 cm.
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 measurements 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 averaged over 50 realizations.
In order to examine the variability of the results according 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 guarantee accurate representation of the aquifer statistics.
For the sieve analysis simulations KG ranges from
8.74 ´ 10ÿ5 to 1.24 ´ 10ÿ4 m sÿ1 , whilst r2ln K ranges 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 and r2ln K 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 (r2ln K 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 positions), providing considerable uncertainty in the predictions. The block interpolation of ¯owmeter
r.
1
r.
1
r.
1
r.
1
±
Block
Block
Block
s.G.s.
Block
s.G.s.
Block
s.G.s.
Block
s.G.s.
4.7 (complete data set)
±
±
1.7
1.7
7.0
7.0
0.74
0.74
4.2
4.2
±
s.a.
f.m.
s.a.
s.a.
s.a.
s.a.
f.m.
f.m.
f.m.
f.m.
Reality
a
b
c
d
e
f
g
h
i
j
(s.a. sieve analysis; f.m. ¯owmeter measurement; Block deterministic block interpolation; s.G.s. sequential Gaussian simulations; b.p. borehole positions; r. realizations)
±
960.0
5.4
1.8
0.50
160.0
58.0
0.13
0.32
10.0
14.0
0.66
730.0
4.5
3.2
0.28
160.0
50.0
0.56
0.10
8.7
14.0
0.42
0.11
0.27
0.09
0.84
0.02
0.33
0.21
0.62
0.31
0.60
0.91
9.2
2.1
0.44
1.4
4.7
3.2
0.85
0.99
3.2
2.7
±
17 b.p.
17 b.p.
1 b.p.
b.p.; 50
1 b.p.
b.p.; 50
1 b.p.
b.p.; 50
1 b.p.
b.p.; 50
Mean 5% quantile of
arrival times (years)
Mean arrival time
(years)
Averaged
statistics
Regionalization
method
Mean variance of ln K
(borehole data)
Investigation
method
Case
Table 1. Statistics of particle arrival times for data sets generated by sampling the aquifer using a four borehole con®guration
Mean variance of
arrival times
Mean square error of
arrival times
Numerical simulation of subsurface characterization methods
827
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 position, it is not always possible to interpret the horizontal
variograms. Considering the positions at which the
variance in the sampled ln K values is the least and the
greatest for sieve analysis (cases d and f, respectively)
and for ¯owmeter measurements (cases h and j, respectively), 50 realizations of sequential Gaussian simulations 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 arrival times, suggesting that the higher conductivity regions 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 under similar conditions to create a detailed data set exhibiting natural heterogeneity. The exhaustive data set
has been used to simulate the application of two commonly 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 conductivities, the simulations in general had the tendency to
overestimate the mean arrival times. The spread in arrival 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
828
J. Whittaker, G. Teutsch
level of sampling led to signi®cant uncertainty in the
values and a dependency on the position of the boreholes. 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 measurements for the prediction of contaminant migration
and the planning of aquifer remediation.
Whilst the Gaussian simulations based on sieve analyses 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' ln K distribution together with those of
two conditioned realizations generated from the borehole
information. The simulations based on the simulations of
¯owmeter measurements were consistently more accurate, despite their failure to generate any regions of permeability 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 K values, representing a much larger area of
support are applicable e.g. for the estimation of the mean
contaminant ¯ux across a control plane (vertically averaged). Furthermore, it becomes obvious from Fig. 12 that
the larger structures of the K ®eld are quite well reproduced in the ¯owmeter realizations.
However, as is observed in Fig. 1, the high permeability lenses play a very signi®cant role in the distribution 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 hydraulic 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 dimensions4), 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 simulations 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 understanding of our ability or inability to predict the mi-
Fig. 12. ln K distributions of: (a) the interpretation of the real sedimentary deposits; (b) a Gaussian realization conditioned by
simulated sieve analysis borehole data; (c) a Gaussian realization conditioned by simulated ¯owmeter borehole data.
Numerical simulation of subsurface characterization methods
gration and fate of contaminants under practical conditions.
ACKNOWLEDGEMENTS
This study was supported by the European Commission through the European Groundwater Research
Programme (Human Capital and Mobility Programme)
and the Environment and Climate Programme (Contract Number ENV4-CT96-5035).
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