vvEPA
United States
Environmental Protection
Agency
Robert S. Kerr Environmental
Research Laboratory
Ada, OK 74820
Research and Development
EPA/600//M-91/005 Jan. 1991
ENVIRONMENTAL
RESEARCH BRIEF
Macrodispersion and Spatial Variability of Hydraulic Conductivity in a Sand
and Gravel Aquifer, Cape Cod, Massachusetts
Kathryn M. Hessa, Steven H. Wolfbc, Michael A. Celiabd and Stephen P. Garabedian*
Introduction
Macrodispersion is the field-scale hydrodynamic spreading of
solutes in an aquifer caused by local variations in ground-water
velocity. These variations are caused, in large part, by small-
scale variations in the hydraulic properties within the aquifer.
Several stochastic transport equations have been developed
that relate macrodispersion to aquifer heterogeneity, particularly
spatial variations of hydraulic conductivity (K) (Dagan, 1982;
Gelhar and Axness, 1983; Neuman et al., 1987). These equations
have been field-tested to a limited degree by comparing the
macrodispersion observed in field tracer tests to that predicted
using a statistical analysis of the variability in K. The tracer test
conducted in a sandy aquifer at Base Borden, Ontario (Sudicky,
1986; Freyberg, 1986) was a pioneering effort in such studies.
The link between macrodispersion and aquifer heterogeneity
has been further investigated in experiments conducted by the
U.S. Geological Survey (USGS) on Cape Cod, Massachusetts.
As part of the USGS studies, a natural-gradient tracer test was
conducted in which the transport of bromide, a nonreactive
tracer, was monitored in the sand and gravel aquifer for two
years (LeBlanc et al., in press). From analyses of spatial
moments of the distribution of bromide at 16 times during the
test, a dispersivity tensor was calculated for the field transport
experiment (Garabedian et al., in press). As a complementary
effort, detailed investigations of the variability of K in the aquifer
were made at the tracer-test site, and the stochastic transport
'U.S. Geological Survey, Marlborough, MA 01752; bDept. of Civil
Eng., Massachusetts Institute of Technology, Cambridge, MA 02139;
°Now at ENSR Consulting and Eng., Acton, MA 01720; "Now at Dept.
of Civil Eng., Princeton University, Princeton, NJ 08540.
equations developed by Gelhar and Axness (1983) were used in
conjunction with the statistical analysis of the K data to estimate
macrodispersion forthe aquifer. This Research Brief summarizes
these investigations of the variability in K at the Cape Cod site and
the relation of that variability to macrodispersion. This research
was supported by the USGS Toxic-Substances Hydrology
Program and the R. S. Kerr Environmental Research Laboratory
of the U.S. Environmental Protection Agency.
Two methods of measuring K were evaluated: permeameter
analyses of cores and f lowmeter tests in wells. More than 1900
estimates of K were obtained from the permeameter and f lowmeter
tests. Geostatistical analyses of these data yielded estimates of
the mean, variance, and correlations scales forthe K distribution
in the aquifer.
Estimates of macrodispersivities based on the statistical analysis
of the K distribution agreed well with the macrodispersivities
calculated fromthetracertest. The range in asymptotic longitudinal
dispersivities that was estimated from the statistical properties of
the f lowmeter K data, using the equations of Gelhar and Axness
(1983) and assuming horizontal isotropy, was 0.23 to 1.2 meters;
this range brackets the value of 0.96 meters calculated from the
tracer test. The theory also correctly predicted the tracer-test
result that longitudinal dispersion greatly exceeds transverse
dispersion. The components of transverse dispersivity were
underestimated by the transport theory, however, probably
because temporal variations in the direction of flow caused
additional dispersion of the tracer during the field experiment that
was not accounted for in the theory of Gelhar and Axness (1983).
Estimation of the transverse horizontal dispersivity was greatly
improved after the effects of transient flow were incorporated
using Rehfeldt's (1988) modification of the transport theory.
W£Q Printed on Recycled Paper
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Objectives
The objectives of this research were to:
1. Evaluate methods for determining the small-scale
variability of K for a sand and gravel aquifer.
2. Statistically characterize the spatial distribution of
K at the Cape Cod tracer-test site.
3. Assess the applicability of the stochastic transport
equationsforestimatingmacrodispersion from the
variance and correlation scales of the spatial
distribution of K.
Methods of Measuring Hydraulic Conductivity
Two methods were used to determine K over small intervals of
the aquifer: laboratory multi-port-permeameter tests of intact
cores (Wolf, 1988; Wolf et al., in press) and in-situ borehole
flowmeter tests (Hess, 1989). For the permeameter
measurements, relatively undisturbed, 4.8-cm (centimeter) -
diameter cores of the medium to coarse sand and gravel aquifer
were obtained using the method of Zapico et al. (1987). Cores
were taken at 16 locations (Figure 1) from a zone that was 6 to
7.5 meters thick and located immediately below the water table.
Lateral distances between coring locations ranged from 1 to 24
meters. X-rays were taken of the cores and the resulting
radiographs were used to identify intervals within the cores that
had similar stratigraphy or inferred grain size. These were the
intervals over which permeameter measurements were made.
A schematic of the constant-head, multi-port permeameter is
shown in Figure 2. The liner in which the core was collected
served as the permeameter body, and manometer probes were
inserted through the liner into the core at the chosen intervals. K
was calculated from permeameter tests on 825 vertical core
intervals which averaged 7.3 cm in length. The K values of
intervals which contained large gravel were not included in the
final data set because the large gravels obstruct flow in the
permeameter, and the resulting K values were thought to be non-
representative of the aquifer (Wolf et al., in press).
Boreholeflowmetertestswereconducted in sixteen5-cmdiameter
wells screened over a 12-m interval just below the water table.
The wells were installed near the coring locations and were
separated by distances of 1 to 24 meters (Figure 1). The borehole
flowmeter test is based on a method developed by Hufschmied
(1986) and modified by Rehfeldt et al. (1989). This procedure
involves measuring the incremental increase in discharge up the
well with a highly sensitive, impeller flowmeter, while maintaining
a constant drawdown in the well by pumping near the water table
at a steady rate (Figure 3). The method is analogous to a
standard aquifer test, except that discharge is measured at short
intervals along the screen, instead of only at the well head. This
method allows calculation of K for each interval.
About 70 estimates of K over vertical intervals of 15 cm were
obtained in each of the 16 wells using the flowmeter method, for
a total of 1109 measurements. Only the 668 K values from the
zone above an altitude of 6 meters were used in the final analysis
of the spatial distribution of K. This is the vertical interval over
which permeameter measurements were made and through
which the bromide tracer traveled in the natural-gradient test.
The array of wells and coring locations is offset about 25 meters
from the tracer-test site (Figure 1). The statistical characterization
of the K distribution at this location should be representative of
the tracer-test zone as well because the depositional environment
of the fluvially-derived glacial-outwash sediments is consistent
across the test area.
Results of Flowmeter and Permeameter Tests
Both methods produced detailed profiles of the variability of K
with altitude within the aquifer. Figure 4 presents four profiles
obtained by each method from the central cluster (Figure 1)
where wells and coring locations are separated by only one meter
from their nearest neighbor. A high degree of variability is
observed in the vertical direction for both types of K measurements.
Greater spatial continuity is observed in the horizontal direction;
several zones of similar K are evident in Figure 4. Some of these
zones are horizontally continuous across the entire area of
investigation. From these qualitative observations, the horizontal
correlation scale of the K distribution was expected to be much
greater than the vertical correlation scale.
The borehole-f lowmeter method yielded a geometric mean K of
0.11 cm/s (centimeters per second) (Table 1), which is similar to
the horizontal mean of 0.13 cm/s estimated from a nearby aquifer
test (LeBlanc et al., 1988) and from the tracer test (LeBlanc et al.,
in press). In contrast, the geometric mean of the permeameter
K values was 0.035 cm/s. This value represents a vertical mean
because the permeameter tests were conducted on vertical
cores. The ratio of the flowmeter to permeameter means (3:1) is
similarto the horizontal-to-verticalanisotropy (2:1 to 5:1) previously
reported for this aquifer (LeBlanc et al., 1988), which suggests
that measurement direction may cause the difference in means
obtained from the two methods. However, if K within the thin,
uniform intervals over which permeameter measurements were
made (5 to 10 cm) is isotropic, then the permeameter tests should
have yielded reasonable estimates of horizontal K. Acomparison
of measurements on intact and homogenized, repacked cores
(Wolf, 1988) seems to support the assumption of isotropy at the
scale of the permeameter measurements. Wolf et al. (in press)
address other possible causes of the underestimation of K by the
permeameter method, including non-representative sampling
and loss of porosity in the samples by compaction.
The variances of the natural logarithm of K (InK) for the two
measurement techniques also differed (Table 1). The flowmeter
variance of 0.24 was significantly higher than the permeameter
variance of 0.14, even though the averaging interval of 7.3 cm for
the permeameter measurements was half of the 15-cm averaging
interval for the flowmeter measurements.
The permeametertests were more laborious and time consuming
than the flowmeter tests. A flowmeter profile could be obtained
in half a day in the field, whereas an equivalent permeameter
profile required a half-day of field sampling and several days of
laboratory analyses. The flowmeter method was a more efficient
method for obtaining detailed profiles of K and yielded a mean
value similarto those estimated by other field tests conducted in
this aquifer. The flowmeter method has also been used
successfully in other aquifers (Hufschmied, 1986; Rehfeldt et al.,
1989; Molz et al., 1989). In the flowmeter-test analysis, however,
induced flow into the well and aquifer stratification are both
assumed to be horizontal. Significant deviation from these
assumptions in some situations could limit the usefulness of the
f lowmetertest for determining the vertical distribution of horizontal
K.
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EXPLANATION
AREA OF TRACER CLOUD N WHOM BROMDE
CONCENTRATIONS EXCEEDED 1 MLUQRAM
PER LITER
WATER-TABLE CONTOUR. AUGUST 2. 1985 —
Shows altitude of water table. Contour
Interval .1 meters. Datum ia aea level
PREDICTED PATH OF TRACER CLOUD
OBSERVATION WELL, AND IDENTFIER
WELL USED FOR IN-STTU FLOWMETER TEST
ANDDENTFIER
LOCATION OF CORES USED FOR PERMEAMETER
ANALYSIS. AND DENTFIER
Figure 1. Tracer-test site in abandoned gravel pit, showing water-table contours, observed location of nonreactive bromide tracer cloud at various
times after injection, location of long-screened wells used for borehole-flowmeter tests, and location of coring sites used for multi-port-
permeameter tests. Water-table map and bromide-cloud areas (from LeBlanc et al., in press).
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manometers
Figure 2. Schematic of multi-port, constant-head permeameter used to measure hydraulic conductivity on 5-cm-diameter cores (from
Wolf et al., in press).
DRAWDOWN
WATER TABLE
PUMP INTAKE
SCREENED
INTERVAL
.INFLOW
IMPELLER FLOWMETER
V
Figure 3. Schematic of hydraulic test used to measure hydraulic conductivity in 5-cm-diameter, long-screened wells using an impeller flowmeter.
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a
N
0.01
F447
F446
F445
0.1 0.01 0.1 0.01 0.1
HYDRAULIC CONDUCTIVITY, IN CENTIMETERS PER SECOND
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§
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TT1
TT2
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TT4
0.01
0.1 00)1 0.1 0.01 0.1 0.01
HYDRAULIC CONDUCTIVITY, IN CENTIMETERS PER SECOND
0.1
14
12
Figure 4. Hydraulic-conductivity profiles from (a) borehole-flowmeter tests and (b) permeameter tests for the central cluster where sampling
locations are separated by approximately one meter. Dotted lines indicate the respective geometric mean hydraulic-conductivity
values. Cross-hatched shading indicates zones of hydraulic conductivity greater than the geometric mean. Slanted-line shading
indicates zones less than the geometric mean.
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Table 1. Results of geostatistical analysis of hydraulic-
conductivity (K) data from permeameter and borehole-
flowmeter tests.
is consistent with the predominant horizontal layering observed
in the stratigraphy at the site.
Flowmeter Permeameter Macrodispersion Estimates
Number of K values
Mean vertical spacing (centimeters)
Geometric mean (centimeters
per second)
Variance of natural logarithm of K
Vertical correlation scale (meters)
Minimum
Best-fit
Maximum
Horizontal correlation scale (meters)
Minimum
Best-fit
Maximum
668
15
.11
.24
.08
.19
.46
1.4
2.6
5.2
825
7.3
.035
.14
.13
.18
.39
.9
1.2
2.6
Geostatistical Analysis
Ageostatistical analysis of the spatial variability of K was performed
on the borehole-flowmeter and permeameter data to obtain
estimates of the correlation structure. A variogram (Journel and
Huijbregts, 1978) in the vertical direction for the permeameter
data is shown in Figure 5. A negative-exponential model fit this
experimental variogram well (Figure 5) and was applied to all
other variograms during the analysis. Estimatesofthecorrelation
scales were obtai ned by fitting this model to the variogram values
using a sill value that was similar to the sample variance. Models
were initially fit by minimizing the sum of squares of the difference
between the model and variogram values. A visual check was
also conducted and, in some cases, the correlation scale and sill
value were adjusted slightly to provide a better fit of the rising limb
of the model to the variogram values.
The estimated vertical correlation scales were similar for the two
data sets (Table 1). Best-fit vertical correlation scales were 0.19
and 0.18 meters for the flowmeter and permeameter data,
respectively. By incorporating uncertainty associated with the
sample variance using a method developed by Rehfeldt (1988),
a range of estimates around the best-fit value was found to be
0.08 to 0.46 meters for the flowmeter measurements and 0.13 to
0.39 meters for the permeameter measurements.
The horizontal variogram values (Figure 6) showed moredeviation
about the fitted model than did the vertical variogram values
(Figure 5). This may be due, in part, to the limited number of
sampling locations in the horizontal direction; only 16 locations,
spaced 1 to 24 meters apart, were used for each type of
hydraulic-conductivity test. The horizontal correlation scales
ranged from 1.4 to 5.2 and 0.9 to 2.6 meters, with best-fit values
of 2.6 and 1.2 meters, for the flowmeter and permeameter data,
respectively (Table 1). Because of the large range in values, the
differences between the permeameter and flowmeter horizontal
correlation scales are probably not statistically significant. In
addition, no statistically significant horizontal anisotropy was
observed during the analysis. Collection of data from additional
sampling locations may be necessary to detect any horizontal
anisotropy in the K distribution. If horizontal anisotropy is
present, it is probably small. The large ratio of horizontal to
vertical correlation scales (5:1 to 25:1) obtained for both data sets
Dispersivity values were estimated using the stochastic transport
theory of Gelhar and Axness (1983) and the variance and
correlation scales determined in the geostatistical analysis (Table
2). Isotropy in the plane of stratification was assumed in the
analysis (Case 1, Gelhar and Axness, 1983) because there was
no evidence of horizontal anisotropy. The range in asymptotic
longitudinal dispersivity estimated from the flowmeter data was
0.23 to 1.2 meters, with a best-fit value of 0.5 meters; this range
encompasses the value of 0.96 meters observed in thetracertest
(Garabedian et al., in press). The dispersivities calculated
from the permeameter data were consistently tower than those
from the flowmeter data (Table 2) because the estimates of
variance and horizontal correlation scale from the permeameter
data are tower, in general, than those from the flowmeter
measurements.
Application of the stochastic equations to the results of the
geostatistical analysis also indicated that longitudinal dispersivity
exceeds the transverse components, as was observed in the
tracer test. The stochastic analysis, however, underestimated
the magnitude of the transverse components (Table 2). Shifts in
the hydraulic-gradient direction during the two-year-long tracer
test (Garabedian et al., in press) may have caused additional
lateral mixing and, thus, enhanced the transverse horizontal
dispersion observed in the tracer test. An estimate of 0.025
meters for transverse horizontal dispersivity (Table 2) was
calculated using the method of Rehfeldt (1988) that incorporates
the effects of the transient flow on dispersion. This estimated
value agrees well with the transverse horizontal dispersivity of
0.018 meters observed in the tracer test (Garabedian et al, in
press).
Effective Hydraulic Conductivity
The correlation scales from the variogram analyses were also
used in the stochastic transport equations to estimate the
anisotropy in the effective K tensor. The horizontal-to-vertical
anisotropy ratio of K, estimated by the method of Gelhar and
Axness (1983) using either the flowmeter or permeameter
correlation scales, is about 1.2:1, which is slightly smaller than
the anisotropy of 2:1 to5:1 obtained from the nearby aquifer test
(LeBlanc et al., 1988). The anisotropy of the effective K tensor
is also smaller than the ratio of mean K values obtained from the
flowmeterand permeameter measurements (3:1). Thissuggests
that the difference between the horizontal-K values from the
flowmeter and the vertical-K values from the permeameter may
be only partly explained by anisotropy and different measurement
directions.
Conclusions
The following conclusions can be made on the basis of results of
this research:
1. Flowmeter tests in long-screened wells provided a
relatively fast and easy method for assessing the
small-scale variability in K (hydraulic conductivity)
at the Cape Cod site. The resulting geometric
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Table 2 Component* of macrodisperslvlty estimated from hydraulic-conductivity data using the stochastic theory
of Gelhar and Axness (1983) and the modifications of the theory of Rehfeldt (1988).
Con-elation Scales
(meters)
Variance
Flowmeter
Minimum
Best-fit
Maximum
Permeameter
Minimum
Best-fit
Maximum
onni\
Horizontal
0.20 1.4
.24 2.6
.29 5.2
.12 .9
.13 1.2
.16 2.6
Vertical
0.08
.19
.46
.13
.18
.39
Macrodispersivity (meters)
Longitudinal
0.23
.50
1.2
.09
.12
.33
Transverse
Horizontal Vertical
10 • ia»
10 7 ia7
10 7 10 7
10 7 ia7
10 7 10*
10« 10*
Tracer Test
Garabedian et al.
(in press)
.96
.018
.0015
Flowmeter, accounting
for transient flow
.96
.025
iu
O
cc
0.3
0.2
0.1
0.0
0.0
0.5
»
X
= 0.13
= 0.18 meters
i
1.0
1.5
2.0
3.0
MEAN SEPARATION DISTANCE OF LAG CLASS, h, IN METERS
Figure 5. Vertical variogram of natural-logarithm of hydraulic conductivity from the permeameter data. Exponential model of the form
YA = oj (1 - exp(-/yX)) was fit to the experimental variogram, where Yft = the semivariance,
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0.3
0.2
UJ
O
0.1
0.0
0.0 5 10 15 20
MEAN SEPARATION DISTANCE OF LAG CLASS, h, IN METERS
Figure 6. Horizontal variogram of natural-logarithm of hydraulic conductivity from the borehole-flowmeter data. Exponential model of the form
Y/, = o? 0- exp(-/yX) was fit to the experimental variogram, where yft = the semivariance, tf, = the sill value, and X = the correlation
scale. Vertical bar indicates 95% confidence interval about the sample variance. Horizontal bar indicates the corresponding horizontal
correlation-scale range.
mean of K was similar to the mean calculated from
tracer and aquifer tests.
2. The permeameter method of measuring K was
more time consuming and produced a mean which
was significantly less than that determined by other
methods. This lower mean may be the result of
non-representative sampling, loss of porosity in
the samples, and anisotropy in the sediments.
3. The permeameter results provided an estimate of
the vertical correlation scale which is similar to that
from the flowmeter tests. However, estimates of
the variance of InK and the horizontal correlation
scale from the permeameter tests are lower, in
general, than those from the flowmeter
measurements.
4. Horizontal anisotropy in the K field was not
observed, and isotropy in the plane of stratification
was assumed in the macrodispersion calculations.
A large ratio of horizontal to vertical correlation
scales was observed and ranged from 5:1 to 25:1.
5. The ratio of horizontal to vertical K indicated by the
stochastic analysis (1.2:1) was less than the range
in ratios calculated from a nearby aquifer test (2:1
to 5:1).
6. Stochastic transport equations predicted a range
of asymptotic long itudinal-macrodispersivities from
the flowmeter measures of K variability (0.23 to 1.2
meters) which brackets the value observed in the
tracer test (0.96 meters). Estimates of longitudinal
macrodispersivity using statistical parametersfrom
the permeameter measurements were consistently
lower than the tracer-test value.
7. Transverse horizontal and transverse vertical
dispersivities were underestimated, probably
because transient flow effects are not taken into
account in the stochastic transport equations of
Gelhar and Axness (1983).
8. The transverse horizontal dispersivity (0.025
meters) estimated by the modified stochastic
equations of Rehfeldt (1988), which incorporate
the effects of lateral shifts in the hydraulic-gradient
direction, agrees well with the value observed in
the tracer test (0.018 meters).
Literature Cited
Dagan, G. (1982) Stochastic modeling of groundwater flow by
unconditional and conditional probabilities: 2. The solute transport,
Water Resources Research. 18, 835-848.
Freyberg, D.L. (1986) A natural gradient experiment on solute
transport in a sand aquifer: Spatial moments and the advection
and dispersion of nonreactive tracers. Water Resources Research.
22,2031-2046.
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Garabedian, S.P., LeBlanc, D.R., Gelhar, L.W., and Celia, M.A.
(in press) Large-scale natural-gradient tracer test in sand and
gravel, Cape Cod, Massachusetts: 2. Analysis of spatial
moments for a nonreactive tracer, Water Resources Research.
Gelhar, L.W., and Axness, C.L (1983) Three-dimensional
stochastic analysis of macrodispersion in aquifers, Water
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Hess, K.M. (1989) Use of a borehole flowmeter to determine
spatial heterogeneity of hydraulic conductivity and
macrodispersion in a sand and gravel aquifer, Cape Cod,
Massachusetts, Proceedings. NWWA Conference on New Field
Techniques for Quantify ing the Physical and Chemical Properties
of Heterogeneous Aquifers, Dallas, TX, 497-508.
Hufschmied, P. (1986) Estimation of three-dimensional statistically
anisotropic hydraulic conductivity field by means of single well
pumping tests combined with flowmeter measurements,
Hvdrogeoloqie. 2,163-174.
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Teasdale, W.E., and Paillet, F.L. (1988) Hydrogeologic controls
on solute transport in a plume of sewage-contaminated ground
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Rehfeldt, K.R., Hufschmied, P., Gelhar, L.W., and Schaefer,
M.E. (19891 Measuring hydraulic conductivity with the borehole
flowmeter. EPRI Topical Report EN-6511, Electric Power
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sand and gravel aquifer, Engineer's thesis, Dept. of Civil Eng.,
Mass. Inst. of Tech.
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hydraulic conductivities calculated from multi-port permeameter
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Disclaimer
The information in this document has been funded wholly or in
part by the United States Environmental Protection Agency
under interagency agreement number DW14933317 with the
U.S. Geological Survey. This document has been subjected to
the Agency's peer and administrative review and has been
approved for publication as an EPA document.
Quality Assurance Statement
All research projects making conclusions or recommendations
based on environmentally related measurements and funded by
the Environmental Protection Agency are required to participate
in the Agency Quality Assurance Program. This project was
conducted under an approved Quality Assurance Program Plan.
The procedures specified in this plan were used without exception.
Information on the plan and documentation of the quality assurance
activities and results are available from the Principal Investigator.
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