pressure transient data was reduced to a binary set: 1 Ž
. yes if an observation zone responds to an injection,
Ž .
and 0 no otherwise, and they successfully visual- ized connections between wells. The result strongly
supports the existence of two separated conductive zones, one at a depth of approximately 30 m and
other at 60 m.
From the large number of interference data de- scribed above, three sets of injection data in series
were selected for each zone: injection into wells 0-0, SE-1 and SE-3 for the upper zone, and injection into
wells 0-0, SW-1 and SW-3 for the lower zone. These injection intervals are located approximately between
depths of 20 to 30 m and between depths of 50 to 60 m. Intervals with interference response are also lo-
cated in these depths. We believe these intervals are confined in the upper and lower conductive zones
shown in Fig. 1b. Injection flow rates were, on the average, 6.4 lrmin for well 0-0, 5.9 lrmin for well
SE-1 and 6.5 lrmin for well SE-3 in the upper zone. In the lower zone, injection flow rates were 4.5
lrmin for well 0-0, 5.3 lrmin for well SW-1 and 1.4 lrmin for well SW-3. It should be noted that for a
pair of wells, injecting at the first and observing at the second does not always give the same pressure
Ž response when their roles are reversed i.e. injecting
. at the second and observing at the first . For exam-
ple, injection into well 0-0 gives about 1.5 m change of head in well SE-3; on the other hand, injection
into well SE-3 gives 0.2 m change of head in well
Ž .
0-0 for the upper zone see Fig. 6 . This is because the packer configurations were changed before doing
the reciprocal tests. Ž
. The template model of the site consists of a
32 = 32 regular grid with 2.5 m fracture elements which covers an 80 = 80 m area both for the upper
and lower zones. The total number of nodes and elements are 1089 and 2112, respectively. Wells
SW-4 and SE-4 are not included in the model, because the interference response is not observed
during a 600-s injection. In all the calculations dis- cussed below, we impose an initial head value of 0
m for all nodes, so that the relative head changes of wells are examined. We also assign constant pressure
Ž .
conditions 0 m on the outer boundary. As an initial configuration, all fracture elements have transmissiv-
y5
Ž
2
.
y4
ity 10 m rs , aperture 2.30 = 10
m. This ini- tial transmissivity is the average value of the aquifer
estimated from the drawdown analyses of the three Ž
. wells, 0-0, SW-1 and SE-1 Cohen, 1993 . Note that
the transmissivity of the outer region beyond a radius of 37.5 m from the origin is held fixed at 10
y5
Ž
2
. m rs throughout the inversion, because the pres-
sure transients are not so sensitive to the transmissiv- ity distribution of the outer region. In other words, it
is difficult to resolve the regions away from the wells and in particular, the regions near the bound-
aries.
The equivalent specific storage of the fracture elements are set to be 0.1 and 0.01 m
y1
for the upper and lower zones, respectively, and are held
constant throughout the inversion. These values are selected by trial and error. It is generally necessary
to use a higher-than-usual specific storage in discrete fracture network models where the conductivity of
an element is coupled to its volume through the cubic law. Because it is impossible to model all the
fractures and pore spaces in the rock enumeratively due to the computational limitations, the specific
storage value in the model is an effective one that represents the storage of the volume of rock that
Ž .
surrounds the fracture element. Mauldon et al. 1993 used a similar approach for their simulation of the
fracture flow experiments conducted at Stripa Mine in Sweden. Our focus, too, is to estimate the distribu-
tion of the transmissivity contrast. Therefore, we feel it is justified to use the effective value for the
specific storage.
In our forward simulation, we set the boundary condition at the well as the constant rate using the
average flow rate over the injection period. Except for the very early time, when the compliance of the
injection plumbing affected the apparent flow rate, the flow did not change greatly. Replacement aper-
tures specified in the inversion are 1.07 = 10
y3
, 4.96 = 10
y4
, 2.30 = 10
y4
and 1.07 = 10
y4
m, which correspond to transmissivities of 10
y3
, 10
y4
, 10
y5 y6
Ž
2
. and 10
m rs , respectively. The upper and lower zones were modeled separately.
5. Results
5.1. The upper zone Four cluster sizes, 1, 10, 20 and 40 were used for
the inversion to investigate spatial characteristics of
the transmissivity distribution within the upper con- ductive zone. For each cluster size, inversions were
run seven times starting from seven different random seeds. Inversions using cluster size of 10 with
anisotropic clusters, with strikes at N30W and N60E, were also conducted. These major fracture sets were
identified from the measurements of outcrops and
Ž .
the acoustic televiewer logs. Annealing cooling
schedules for four cluster sizes are shown in Fig. 4. In these schedules, the total number of the element
change in an inversion is set to be same for all cluster size cases. Namely, in the case of cluster size
40, the control parameter T is lowered after 20 configuration changes are accepted, whereas the con-
trol parameter T is lowered after 800 changes are accepted in the case of cluster size 1. We set all
inversions will be stopped when the annealing sched- ule is exhausted. This means that the inversions
using larger cluster sizes need less number of itera- tion. For example, the inversions using cluster sizes
1 and 40 requires approximately 22,460 and 980 total iterations, respectively.
Fig. 5 summarizes the inversion results. The low- Ž
. est minimum objective function
energy , 4.7, is reached in the seed 4 of the cluster size 10, starting
Ž .
from the initial objective function energy of 3.33 =
3
Ž .
10 . The minimum objective function energy varies to some extent depending on a random seed for a
given cluster size. The averaged minimum objective Ž
. function energy is smallest for a cluster size of 1.
As the cluster size decreases, the averaged minimum Ž
. objective function energy also decreases. Compari-
son of pressure transients between observed and calculated data for the minimizing configuration ob-
Ž .
tained using the cluster size 10 seed 4 is illustrated in Fig. 6. A relatively good match is achieved.
In order to represent average properties of the annealing solutions obtained using the same cluster
size, seven annealing solutions are combined to pro- duce one configuration by means of median filters,
which are commonly used as velocity filters in data
Ž processing of vertical seismic profiling e.g. Hardage,
. 1985 . That is, the median transmissivity value
Ž .
transmissivity value itself, not its logarithm among seven values is selected for each fracture element so
as to produce one median configuration. Five me- dian-filtered configurations are shown in Figs. 7–9.
The first result is found using a cluster size of 1. The
Ž .
Fig. 4. Annealing cooling schedules used in the inversion.
Ž .
Fig. 5. Inversion results for the upper zone: minimum objective function energy vs. cluster sizes.
second, third and fourth results are found using cluster sizes of 10, 20 and 40 with isotropic clusters.
The fifth result is found using a cluster size of 10 with anisotropic clusters, with strikes at N30W and
Ž .
Fig. 6. Pressure match for minimizing configuration obtained using cluster size of 10 seed 4 for the upper zone. Symbols and lines represent the observed and calculated data, respectively.
Ž .
Ž .
Fig. 7. Median-filtered inversion result with cluster size 1 U1 and cluster size 10 U2 for the upper zone. The injection test data into well Ž
. 0-0, SE-1 and SE-3 in series filled circles was used in the inversion.
Ž .
Ž .
Fig. 8. Median-filtered inversion result with cluster size 20 U3 and cluster size 40 U4 for the upper zone.
Ž .
Fig. 9. Median-filtered inversion result with cluster size 10 and anisotropic shapes for the upper zone U5 .
N60E. We refer to these as cases U1 through U5. It is difficult to observe the major feature common to
all the results from these figures. However, the an- nealing results consistently show high transmissivi-
ties around well 0-0. SW-3 is relatively isolated in U3 and U5 results. This is because SW-3 had a very
low interference response from all of three well
Ž .
injections see Fig. 6 . Aside from these two fea- Ž
. tures, U1–U4 isotropic clusters do not seem to
show a particular spatial pattern. Since the inversion result is based on limited
information and the hydraulic inversion is inherently non-unique, it is difficult to say how ArealB such
results are. Rather, we believe that assessing the validity of the result can best be done by testing how
Ž .
the result model predicts hydraulic behavior of the data, which is not used to build the model. Thus, to
evaluate the inversion results, a cross validation test was conducted. We modeled the SW-2 injection test,
which was not included in the inversion as injection
Ž .
data. TRINET Karasaki, 1987 was again used to simulate injection into SW-2. Fig. 10 shows predic-
tion errors for injecting at SW-2 for each cluster size Ž
. results U1–U5 . The prediction error, which is the
sum of the squared difference between the calculated Ž .
and observed data given by Eq. 3 , is approximately 4 for cases U1 and U2, while it is more than 15 for
U3, U4 and U5. This result indicates that cases U1 and U2 are the good models among the results of
five cluster types. From these results, we observe that cluster sizes of 1 and 10 are suitable to obtain
good results.
Ž The omnidirectional semi-variogram for U2 clus-
. ter size of 10 is given in Fig. 11. The directional
semi-variograms were also calculated to check the existence of geometric anisotropies. However, no
significant anisotropies are observed in Fig. 11. A practical range of the spatial correlation is approxi-
Ž mately 4.0 m since both the element length and
. spacing are 2.5 m, we refer to this value as 5.0 m .
This practical range is a little smaller than that obtained from each annealing solution because of the
smoothing effect of the median filters. However, it is
Ž .
nearly consistent with the range 5–7.5 m estimated from the optimal cluster size, in that the optimal
cluster size corresponds to 20–40 of the number of fractures within the practical range of spatial correla-
Ž .
tion Nakao et al., 1999 . 5.2. The lower zone
We repeated the same procedure for the lower conductive zone. Fig. 12 summarizes inversion re-
Fig. 10. Predicted errors of injection into well SW-2 for five annealing solutions for the upper zone.
Ž sults. The lowest minimum objective function en-
. ergy , 140.0, is reached in the seed 7 of the cluster
size 1, starting from the initial objective function Ž
.
3
energy of 4.82 = 10 . The averaged minimum ob- Ž
. jective function energy is smallest for a cluster size
of 1. As the cluster size decreases, the minimum Ž
. objective function energy also decreases. For clus-
ter size of 10, the anisotropic-shapes case reaches smaller energy than that of isotropic-shapes case.
Comparison of pressure transients between observed
Ž .
Fig. 11. Omnidirectional and directional semi-variograms of transmissivity for the inversion results with cluster size s 10 U2 for the upper zone.
Ž .
Fig. 12. Inversion results for the lower zone: minimum objective function energy vs. cluster sizes.
and calculated data for the minimizing configuration Ž
. obtained using the cluster size 1 seed 7 is illus-
trated in Fig. 13. Although a good match between observed and calculated data was obtained in the
Ž .
Fig. 13. Pressure match for minimizing configuration obtained using cluster size of 1 seed 7 for the lower zone. Symbols and lines represent the observed and calculated data, respectively.
Ž .
Ž .
Fig. 14. Median-filtered inversion result with cluster size 1 L1 and cluster size 10 L2 for the lower zone. The injection test data into well Ž
. 0-0, SW-1 and SW-3 in series filled circles was used in the inversion.
Ž .
Ž .
Fig. 15. Median-filtered inversion result with cluster size 20 L3 and cluster size 40 L4 for the lower zone.
Ž .
Fig. 16. Median-filtered inversion result with cluster size 10 and anisotropic shapes for the lower zone L5 .
lower zone, the absolute head value of the injection into SW-3 is quite large. This causes the relatively
Ž .
high minimum objective functions energies rather than those of the upper zone cases.
Five median-filtered configurations are shown in Figs. 14–16. The first result is found using a cluster
size of 1. The second, third and fourth inversion solutions are found using cluster sizes of 10, 20 and
Fig. 17. Predicted errors of injection into well SW-2 and SE-1 for five annealing solutions for the lower zone.
Ž .
Fig. 18. Omnidirectional and directional semi-variograms of transmissivity for the inversion results with cluster size s 1 L1 for the lower zone.
40 with isotropic clusters. The fifth result is found using a cluster size of 10 with anisotropic clusters,
with strikes at N30W and N60E. We refer to these as cases L1 through L5. A major feature common to all
the results is that well SW-3 is clearly isolated. This is because SW-3 had a very high head change while
Ž .
injection see Fig. 13 . Annealing results also show a relatively high transmissivity area between wells
SW-2 and SE-2. In order to evaluate the inversion results, a cross
validation test was again conducted. We modeled the SW-2 and the SE-1 injection tests, which were not
included in the inversion as injection data. Fig. 17 shows predicted errors for injecting at SW-2 and
Ž .
SE-1 for each cluster size results L1–L5 . For clus- Ž
. ter size of 10, the anisotropic-shapes case L5 ob-
tains smaller error than that of isotropic-shapes case Ž
. L2 . This finding will be discussed in the next
section. The prediction errors are smallest for L1 in Ž
both injections. This result indicates that L1 cluster .
size of 1 is the most suitable among the results of five cluster types.
The directional and omnidirectional semi-vario- Ž
. grams for L1 cluster size of 1 are given in Fig. 18.
A practical range of the spatial correlation is less Ž
than 2.5 m since both of the element length and .
spacing are 2.5 m, we refer to this value as 2.5 m . This practical range is consistent with the range
Ž .
around 2.5 m estimated from the optimal cluster Ž
. size Nakao et al., 1999 . It is impossible to observe
whether geometric anisotropies of the correlation structure exist within the range of 2.5 m, because of
the resolvable scale used in the configuration. Note that this is the first attempt to estimate the horizontal
correlation length of transmissivity for both the up- per and lower conductive zones at Raymond field
site.
6. Discussion
