1 Interpretation of Transmissivity Estimates from 2 Single-Well, Pumping Aquifer Tests 3 By Keith J. Halford1, Willis D. Weight2, and Robert P. Schreiber3
4 Abstract 5 Interpretation of single-well tests with the Cooper-Jacob method remains more
6 reasonable than most alternatives. Drawdowns from 628 simulated single-well tests
7 were interpreted with the Cooper-Jacob straight-line method. Error and bias as a
8 function of vertical anisotropy, partial penetration, specific yield, and interpretive
9 technique were investigated for transmissivities that ranged from 10 to 10,000 m²/d.
10 Cooper-Jacob transmissivity estimates in confined aquifers were affected minimally by
11 partial penetration, vertical anisotropy, or analyst. Cooper-Jacob transmissivity
12 estimates of simulated unconfined aquifers averaged twice known values. One percent
13 of these estimates were more than 10 times known values. Transmissivity estimates
14 were not improved by interpreting results with an unconfined solution. An empirical
15 corrector reduced the bias and magnitude of error in Cooper-Jacob transmissivity
16 estimates for unconfined aquifers. The empirical corrector is available as a Visual Basic
17 for Applications function in a single-well analysis spreadsheet.
1 USGS, 333 W Nye Ln, Room 203, Carson City, NV 89706; e-mail:[email protected]; Tel:775-887-7613 2 Montana Tech of The University of Montana, 1300 West Park St., Butte, MT 59701; e-mail:[email protected]; Tel:406-496-4329 3 CDM, One Cambridge Place, 50 Hampshire Street, Cambridge, MA 02139; e-mail:[email protected]; Tel:617-452-6251
1 1 Introduction 2 Single-well aquifer tests provide data in many situations in which aquifer
3 pumping-test data from multiple wells would be a luxury for the ground-water analyst.
4 The primary factor limiting the performance of multi-well aquifer tests is cost, particularly
5 in areas with large depth to water (Belcher et al, 2001). Limiting analysis to existing
6 wells predominately occurs in the arid West, but can occur in many areas. For
7 example, irrigation wells are installed far apart to minimize interference. This separation
8 causes adjacent wells to be poor observation wells.
9 The ground-water analyst faces many problems and challenges where hydraulic
10 testing is limited to an isolated well. These problems include well-known and
11 quantifiable effects such as borehole head loss during pumping, as well as much more
12 difficult-to-quantify factors such as lateral and vertical anisotropy. Many effects could
13 only be quantified with reasonable accuracy using multiple observation wells or flow
14 logs (Hanson and Nishikawa, 1996). An analyst can still derive significant useful
15 information from single-well tests.
16 Single-well aquifer tests frequently are analyzed with the Cooper-Jacob (1946)
17 method because of its simplicity. Transmissivity is estimated by fitting a straight line to
18 drawdowns on an arithmetic axis versus time on a logarithmic axis in a semilog plot.
19 Drawdowns in confined and unconfined aquifers have been analyzed with the Cooper-
20 Jacob method regardless of differences between field conditions and theory.
2 1 Single-well tests from partially penetrating wells in unconfined aquifers depart
2 greatly from the Theis (1935) model. The Cooper-Jacob method is a simplification of
3 the Theis solution so the production well should fully penetrate a confined,
4 homogeneous, and isotropic aquifer. Unconfined aquifer tests are affected by vertical
5 anisotropy and specific yield in addition to transmissivity and storage coefficient. These
6 additional parameters control vertical gradients that are created by partial penetration
7 and drainage from the water table. Likewise, leakage from adjacent confining beds also
8 could affect the Cooper-Jacob method. Transmissivity likely will be overestimated
9 where leakage occurs.
10 Transmissivity estimates from single-well tests in unconfined aquifers also are
11 affected by discharge rate, test duration, and interpretive technique. The transition from
12 the release of water from storage owing to the compressibility of the medium and fluid to
13 drainage of pores is less likely to be observed during a test of relatively small discharge
14 or short duration. Interpretation is hampered because a final drawdown limb, which has
15 a slope predicted by the Cooper-Jacob method, is absent.
3 1 Purpose and Scope 2 This article documents how interpreting single-well, pumping aquifer tests with
3 the Cooper-Jacob method affects transmissivity estimates. Drawdowns in 800 single-
4 well tests were simulated with radial, MODFLOW models that were defined with
5 transmissivities between 10 and 10,000 m²/d. Drawdowns from 628 of these simulated
6 tests would maintain water levels above pump intakes and were interpreted with the
7 Cooper-Jacob, straight-line method. Transmissivity error and bias as a function of
8 vertical anisotropy, partial penetration, specific yield, and interpretive technique were
9 investigated. Transmissivity estimates were improved with informed interpretation or
10 with an empirical, corrector algorithm.
11 Approach 12 Effects of unmet assumptions on the reliability of Cooper-Jacob transmissivity
13 estimates were investigated by interpreting 628 simulated drawdowns. The
14 transmissivity of each aquifer was estimated with a Cooper-Jacob analysis of simulated
15 drawdowns. Effects of unmet assumptions were quantified by comparing Cooper-Jacob
16 transmissivity estimates to specified values, herein referred to as known transmissivity.
17 Hydraulic properties were limited to plausible ranges. Transmissivities ranged
18 from 10 to 10,000 m²/d. Transmissivities of less than 10 m²/d were excluded because
19 slug tests are more practical than pumping tests. A single specific storage of 5 x 10-6 m-
20 1 was assigned to all aquifers because transmissivity estimates were insensitive to
21 specific storage. Specific yields ranged from 0.05 to 0.3. Vertical anisotropies ranged
22 between 0.02 and 0.2 and were assumed to represent a sedimentary system.
4 1 Pumping wells that penetrated between 10 and 100 percent of aquifers were
2 analyzed. All partially penetrating wells were open at the top of the aquifer or water
3 table (Figure 1). Wellbore storage and skin effects of the pumping wells were simulated
4 which decreased maximum potential pumping rates. Simulation results were
5 considered physically impossible and rejected where simulated water levels in the
6 pumping well were less than 3 m above the bottom of the well. Wells in unconfined
7 aquifers were simulated to penetrate 25 or more percent of the saturated thickness so
8 water levels would remain above the pump intakes.
9 All single-well tests were simulated over 2-day periods to balance testing
10 effectiveness and operational constraints. Transmissivity estimates from tests of longer
11 duration are less ambiguous because drainage from the water table will be observed
12 and drawdowns will follow a late-time response. Actual single-well tests typically range
13 between 1 and 3 days in duration because of operational constraints.
5 1 Simulated Aquifer Tests 2 All single-well aquifer tests were simulated with two-dimensional MODFLOW
3 models (McDonald and Harbaugh, 1988; Harbaugh and McDonald, 1996). The
4 production well in each model was simulated as a high conductivity zone where water
5 was removed from the uppermost cell and flow was apportioned within MODFLOW. All
6 models extended 100,000 m from the production well along a row that was discretized
7 into 99 columns (Figure 1). Pumping wells were simulated in column 1 and aquifer
8 material was simulated with columns 2 through 99. Column 2 was 0.02 m wide and the
9 remaining columns were 1.15 times the width of the previous column. The 100-m thick
10 aquifers were uniformly subdivided into 100 rows. All models had no flow lateral
11 boundaries, initial heads of 0 m, and a single stress period of 50 time steps. The first
12 time step was 0.1 second in duration. Each successive time step was 1.3 times greater
13 than the previous one.
14 Hydraulic properties were modified consistently to simulate axisymmetric, radial
15 flow. Simulated pumpage was withdrawn from column 1 so that distance from the left
16 edge of column 1 to a column node would be equivalent to the radial distance from the
17 pumping well. Hydraulic conductivity and storage of the ith column were multiplied by
18 2πri to simulate radial flow where ri is the distance from the outer edge of the first
19 column to the center of the ith column. Axisymmetric, radial flow has been solved with
20 MODFLOW by using many layers and a single row (Reilly and Harbaugh, 1993; Clemo,
21 2002). A single MODFLOW layer is more convenient because input is defined easily,
22 all conductances are computed within the BCF package, and output is checked quickly.
6 1 Pumping rates were simulated near the maximum rate that could be pumped
2 practically from an aquifer of a given transmissivity. Pumping rates were greater in
3 more transmissive aquifers (Table 1). Realistic wellbore-storage effects were simulated
4 by increasing well diameters as pumping rates increased. A storage coefficient of 1
5 was assigned in row 1 of column 1 to simulate wellbore storage. Conductances
6 between columns 1 and 2 were reduced tenfold to simulate skin where a well was
7 present.
8 Interpretation 9 Cooper-Jacob transmissivity was estimated for each single-well test by a
10 mechanistic approach and separately by six analysts, including the three authors and
11 three volunteers. Transmissivity was estimated automatically with the mechanistic
12 approach. A semilog slope was defined by the drawdowns at 0.5 and 2 d after pumping
13 started. A minimum time of 0.5 d was selected as a compromise between avoiding the
14 early-time complications due to wellbore storage, partial penetration, and water-table
15 effects and assumed measurement sensitivity for actual aquifer tests. Experience
16 guided analysts’ best fit of semilog slopes. All simulated drawdowns and analyses can
17 be retrieved from http://nevada.usgs.gov/tech/groundwater_05.htm.
7 1 Confined Analysis
2 Cooper-Jacob transmissivity estimates in confined aquifers were affected
3 minimally by partial penetration, vertical anisotropy, and interpretative technique
4 (Figure 2). Transmissivities of 100 m²/d or greater were estimated within 10 percent of
5 their value in all but one case. A steady additional drawdown from partial penetration
6 and vertical anisotropy was established before 12 hours of pumping had elapsed. This
7 additional drawdown minimally affected estimates of slope and transmissivity. Confined
8 aquifer test results were unambiguous and transmissivity estimates varied little among
9 analysts.
10 Unconfined Analysis
11 Transmissivities of unconfined aquifers were overestimated with a mechanistic
12 application of Cooper-Jacob (Figure 3). More than 75 percent of known transmissivity
13 values between 10 and 1000 m²/d were overestimated. Estimates averaged twice
14 known transmissivity values in this range. One percent of estimates were more than 10
15 times known transmissivity values. About 80 percent of known transmissivity values
16 between 1000 and 10,000 m²/d were estimated within a factor of two.
17 Estimates by analysts were more accurate than mechanistic estimates of
18 transmissivity (Figure 4). Analysts improved transmissivity estimates most where
19 known transmissivity values ranged between 250 and 5000 m²/d. More than 90 percent
20 of these transmissivity estimates were within a factor of two of the known values.
21 Interpretation did not significantly improve transmissivity estimates or remove bias
22 where known transmissivity values ranged between 10 and 100 m²/d.
8 1 Transmissivity estimates were not improved by interpreting results with an
2 unconfined analytical solution instead of Cooper-Jacob. Results from an unconfined
3 aquifer with a transmissivity of 1000 m²/d were interpreted with the Moench unconfined
4 solution (Barlow and Moench, 1999). Transmissivity, specific storage, vertical
5 anisotropy, specific yield, and skin were estimated simultaneously to minimize an
6 unweighted, sum-of-squares objective function. Transmissivity estimates ranged
7 between 800 and 1300 m²/d while matching all aspects of the drawdown curve (Figure
8 5). A similar transmissivity of 900 m²/d was estimated by applying Cooper-Jacob to the
9 final drawdown limb.
10 Specific storage, vertical anisotropy, and specific yield could not be estimated
11 uniquely with an unconfined Moench solution (Barlow and Moench, 1999). Estimates of
12 these hydraulic properties with an unconfined Moench solution ranged over more than
13 2-orders of magnitude and several estimates were physically unreasonable (Table 2).
14 Initial estimates strongly affected final estimates. Initial estimates were limited to
15 plausible values for all six solutions.
16 Effects of Individual Parameters
17 Specific yield, vertical anisotropy, and partial penetration systematically affected
18 Cooper-Jacob transmissivity estimates in unconfined aquifers (Figure 6). Effects of
19 individual parameters on transmissivity were more pronounced on mechanistic
20 estimates. These effects largely were eliminated by analysts where known
21 transmissivity values ranged between 500 and 10,000 m²/d. Cooper-Jacob
22 transmissivity estimates exceeded known transmissivity values more as specific yield
23 increased (Figure 6a).
9 1 Decreases in vertical anisotropy proportionally displace the maximum difference
2 between Cooper-Jacob estimates and known transmissivity (Figure 6b). Transmissivity
3 estimates likely will be worst where vertical anisotropy ranges between 0.05 and 0.1.
4 Estimates differ most from known transmissivity between 50 and 100 m²/d, which is the
5 range where errors could not be compensated with interpretation.
6 Partial penetration affected transmissivity estimates more than specific yield or
7 vertical anisotropy (Figure 6c). Overestimation of transmissivity was greatest for fully
8 penetrating wells in unconfined aquifers. Transmissivity estimates were least biased
9 where partial penetration was 25 percent.
10 Empirical Corrector
11 An empirical corrector was developed to improve transmissivity estimates from
12 single-well tests on unconfined aquifers. Cooper-Jacob transmissivity, vertical
13 anisotropy, partial penetration, and specific yield define a multiplier that corrects the
14 Cooper-Jacob transmissivity estimate. The common log of the empirical corrector (C) is
10 0.1 0.1 -0.5 v (0.3-0.89Kxz ) 1 log(C) = 0.63Kxz (tanh(v) + PP - 2.237) e ...... (1)
log(T ) - Kxz0.2 - PP - log(S ) - 1 2 v = CJ Y ...... (2) log(4PP) 5.03 Kxz0.110 10 3 where, Kxz is vertical anisotropy (dimensionless), PP is partial penetration
4 (dimensionless), SY is specific yield (dimensionless), and TCJ is a Cooper-Jacob
5 transmissivity estimate (L²/T). Coefficients and exponents in equations (1) and (2) were
6 estimated by minimizing sum-of-squares differences between known transmissivity and
7 mechanistic-transmissivity estimates. The empirical corrector is applied as a Visual
8 Basic for Applications function in the spreadsheet Pumping_Cooper-
9 Jacob_wCORRECTOR.xls and can be retrieved from
10 http://nevada.usgs.gov/tech/groundwater_05.htm. This workbook is a modification of
11 the previously published Cooper-Jacob.xls (Halford and Kuniansky, 2002).
12 More than 65 percent of known transmissivity values between 10 and 250 m²/d
13 can be corrected within a factor of two (Figure 7), assuming that 95 percent of specific
14 yield and vertical anisotropy estimates are within a factor of two of their actual values.
15 The average of all estimates for a given transmissivity was less biased and was within 5
16 percent of the known value. Equation (1) is of limited value where known transmissivity
17 exceeds 500 m²/d because limitations of the Cooper-Jacob method can be better
18 compensated with interpretation.
11 1 Hydraulic Conductivity 2 Hydraulic conductivity of confined aquifers was unambiguously determined as
3 the transmissivity estimate divided by aquifer thickness, rather than the screen length,
4 whenever transmissivity was 100 m²/d or greater (Figure 2). Hydraulic conductivity was
5 not defined clearly by either aquifer thickness or screen length where transmissivity was
6 less than 50 m²/d. Screen length might be appropriate for estimating hydraulic
7 conductivity from transmissivity but only where transmissivity is appreciably less than 10
8 m²/d.
9 Hydraulic conductivities of unconfined aquifers could not be interpreted
10 unambiguously from transmissivity estimates with either aquifer thickness or screen
11 length. Aquifer thickness was better than screen length for interpreting hydraulic
12 conductivity with estimates that ranged from 0.5 to 4 times known hydraulic
13 conductivities (Figure 6c). Hydraulic conductivity estimates ranged from 1.6 to 8 times
14 known values where transmissivity estimates were divided by screen length.
15 Design Effects 16 Well and aquifer head losses can be better differentiated with improved
17 transmissivity estimates from single-well aquifer tests. Head losses, from poor well
18 completion practice (skin) and partial penetration, will be overestimated where
19 transmissivity has been overestimated. An analyst could conclude falsely that the well
20 was completed poorly and that completion of future wells could be improved. This
21 scenario is likely to occur where transmissivity ranges between 10 and 250 m²/d.
12 1 Interpretive techniques from this paper can be used to improve the design of
2 isolated production wells. Wells will more likely produce at designed rates with
3 improved skin and transmissivity estimates because screen lengths and well diameters
4 can be sized appropriately. The expense of poor design can be significant where
5 deepening a well or drilling an additional well likely is a result.
6 Conclusions 7 Cooper-Jacob analysis of single well tests in unconfined aquifers overestimates
8 transmissivity. Transmissivity estimates averaged twice known values between 10 and
9 1000 m²/d and exceeded known values by more than an order of magnitude in a few
10 cases. Transmissivity estimates also were not improved by interpreting results with an
11 unconfined solution. Cooper-Jacob transmissivity estimates in confined aquifers
12 differed little from known values regardless of partial penetration or vertical anisotropy.
13 Hydraulic conductivity of confined aquifers was estimated unambiguously using the
14 transmissivity estimate divided by aquifer thickness where transmissivity exceeded 100
15 m²/d.
13 1 Transmissivity estimates from single-well tests can be improved with informed
2 interpretation or empirical corrections. The final limb of an unconfined response can be
3 identified by an analyst where transmissivity exceeds 500 m²/d. All transmissivity
4 estimates are improved qualitatively because analysts are aware of likely biases. The
5 empirical corrector that was developed in this paper can reduce systematic bias where
6 transmissivity ranges between 10 and 250 m²/d. Improved transmissivity estimates
7 should improve skin estimates that would allow for better well design. This simple
8 approach provides a useful tool to improve transmissivity estimates from a single-well
9 test.
10 Acknowledgements 11 The authors express their appreciation to Eve L. Kuniansky, Randell J. Laczniak,
12 and David E. Prudic of the U.S. Geological Survey who generously assisted this study
13 by analyzing many simulated aquifer tests. The authors thank Randy Hanson, Rhett
14 Everett, and Ed Weeks of the U.S. Geological Survey and anonymous reviewers that
15 helped improve the paper.
14 1 References 2 Barlow, P.M., and A.F. Moench. 1999. WTAQ—A computer program for calculating drawdowns and
3 estimating hydraulic properties for confined and water-table aquifers. USGS Water-Resources
4 Investigations Report 99-4225.
5 Belcher, W.R., P.E. Elliott, and A.L. Geldon. 2001. Hydraulic-property estimates for use with a transient
6 ground-water flow model of the Death Valley Regional Ground-Water Flow System, Nevada and
7 California. USGS Water-Resources Investigations Report 01-4210.
8 Clemo, T., 2002. MODFLOW-2000 for cylindrical geometry with internal flow observations and improved
9 water table simulation. Technical Report BSU CGISS 02-01, Boise State University, Boise, Idaho.
10 Cooper, H.H., and C.E. Jacob. 1946. A generalized graphical method for evaluating formation constants
11 and summarizing well field history. American Geophysical Union Transactions 27: 526–534.
12 Halford, K.J., and E.L. Kuniansky. 2002. Documentation of spreadsheets for the analysis of aquifer
13 pumping and slug test data. USGS Open-File Report 02-197.
14 Hanson, R. T., and Tracy Nishikawa. 1996. Combined use of flowmeter and time-drawdown data to
15 estimate hydraulic conductivities in layered aquifer systems. Ground Water 34, no. 1: 84-94.
16 Harbaugh, A.W., and M.G. McDonald. 1996. Programmer's documentation for MODFLOW-96, an update
17 to the U.S. Geological Survey modular finite-difference ground-water flow model: USGS Open-File
18 Report 96-486.
19 McDonald, M.G., and A.W. Harbaugh. 1988. A modular three-dimensional finite-difference ground-water
20 flow model. USGS Techniques of Water-Resources Investigations, book 6, A1.
21 Reilly, T.E., and A. W. Harbaugh. 1993. Computer Note: Simulation of cylindrical flow to a well using the
22 U.S. Geological Survey modular finite-difference ground-water flow model. Ground Water 31, no. 3:
23 489-494.
15 1 Theis, C.V. 1935. The relation between the lowering of the piezometric surface and the rate and duration
2 of discharge of a well using ground water storage. Transaction of American Geophysical Union 16:
3 519–524.
4
16 1 Tables 2 Table 1.—Well diameters and discharges assigned to simulated aquifers...... 18 3 Table 2.—Six unconfined Moench solutions for a simulated single-well test in an aquifer with a 4 transmissivity of 1000 m²/d, specific yield of 0.1, vertical anisotropy of 0.2, and a partial 5 penetration of 50 percent...... 18
6
17 1 2 Table 1.—Well diameters and discharges assigned to simulated aquifers.
Transmissivity, Well Discharge, m²/d Diameter, m in m³/d 10 0.05 20 25 0.05 25 50 0.1 50 100 0.1 100 250 0.15 250 500 0.15 500 1000 0.2 1000 2500 0.3 2500 5000 0.3 5000 10,000 0.4 10,000 3 4 Table 2.—Six unconfined Moench solutions for a simulated single-well test in an aquifer with a transmissivity of 5 1000 m²/d, specific yield of 0.1, vertical anisotropy of 0.2, and a partial penetration of 50 percent. RMS Transmissivity, Specific Vertical Specific Solution Skin error, in in m²/d Storage, in m-1 Anisotropy Yield m 1 360 0.000003 0.44 0.51‡ 0.0 0.012 2 550 0.000005 0.35 0.33 3.0 0.008 3 790 0.000026 0.80▲ 0.55‡ 7.2 0.005 4 1000 0.000008 0.15 0.09 9.7 0.003 5 1300 0.0000004† 0.005▲ 0.002‡ 12.0 0.004 6 1400 0.000170† 1.40▲ 0.52‡ 17.0 0.006 †Specific-storage estimate is not in likely range of 0.000003 to 0.00003 m-1. ▲Vertical anisotropy is not in likely range of 0.02 to 0.5. ‡Specific yield is not in likely range of 0.01 to 0.35. 6 7
18 1 FIGURES 2 Figure 1.—Definition sketch of hypothetical well and aquifer system with grid for numerical 3 solution...... 20 4 Figure 2.—Cooper-Jacob estimates of transmissivity in confined aquifers for all combinations of 5 partial penetration and vertical anisotropy with slope defined by drawdowns at 12 and 48 6 hours...... 20 7 Figure 3. —Cooper-Jacob estimates of transmissivity for all combinations of storage coefficient, 8 partial penetration, and vertical anisotropy with slope defined by drawdowns at 12 and 48 9 hours...... 21 10 Figure 4. —Cooper-Jacob estimates of transmissivity for all combinations of storage coefficient, 11 partial penetration, and vertical anisotropy with slope defined by analyst...... 22 12 Figure 5.—Example fits of unconfined Moench and Cooper-Jacob solutions to MODFLOW 13 solution for an aquifer with a transmissivity of 1000 m²/d, specific yield of 0.1, vertical 14 anisotropy of 0.2, and a partial penetration of 50 percent...... 23 15 Figure 6. —Selected Cooper-Jacob estimates of transmissivity with specific yields between 0.05 16 and 0.3, partial penetrations of 25 to 100 percent, and vertical anisotropies of 0.02 to 0.2 17 with slope defined by drawdowns at 12 and 48 hours...... 23 18 Figure 7. —Corrected Cooper-Jacob estimates of transmissivity for all combinations of storage 19 coefficient, partial penetration, and vertical anisotropy with slope defined by drawdowns at 20 12 and 48 hours...... 24
19 1
2 Figure 1.—Definition sketch of hypothetical well and aquifer system with grid for numerical solution.
100,000 D /
M² 10,000 N I , Y T VI
SSI 1,000 SMI AN
R m
T 100 u
B im ax O M AC
J m
- u im Average - 75% 10 in
PER M 25% - Average O
O KNOWN C
1 10 100 1,000 10,000 KNOWN TRANSMISSIVITY, IN M²/D 3 4 Figure 2.—Cooper-Jacob estimates of transmissivity in confined aquifers for all combinations of partial 5 penetration and vertical anisotropy with slope defined by drawdowns at 12 and 48 hours.
20 100,000 D /
M² 10,000 N
I m ,
Y mu
T xi
VI Ma
SI 1,000 S SMI
N m
A u m R i 100 n T Mi B O C A J - 10 ER Average - 75% P
O 25% - Average O C KNOWN 1 10 100 1,000 10,000 KNOWN TRANSMISSIVITY, IN M²/D 1 2 Figure 3. —Cooper-Jacob estimates of transmissivity for all combinations of storage coefficient, partial 3 penetration, and vertical anisotropy with slope defined by drawdowns at 12 and 48 hours.
21 100,000 D /
M² 10,000 N
I m , u Y im T x
VI Ma
SI 1,000 S SMI
N m A u
R im 100 n T Mi B O C A J - 10 ER Average - 75% P
O 25% - Average O C KNOWN 1 10 100 1,000 10,000 KNOWN TRANSMISSIVITY, IN M²/D 1 2 Figure 4. —Cooper-Jacob estimates of transmissivity for all combinations of storage coefficient, partial 3 penetration, and vertical anisotropy with slope defined by analyst.
22 5.4 Numerical, T = 1000 m²/d Unconfined Moench, T = 1100 m²/d
RS Cooper-Jacob, T = 900 m²/d E
T 5.2 E M N I WN, O
D 5.0 AW DR
4.8 0.001 0.01 0.1 1 10 TIME, IN DAYS 1 2 Figure 5.—Example fits of unconfined Moench and Cooper-Jacob solutions to MODFLOW solution for an 3 aquifer with a transmissivity of 1000 m²/d, specific yield of 0.1, vertical anisotropy of 0.2, and a partial 4 penetration of 50 percent.
5 6 Figure 6. —Selected Cooper-Jacob estimates of transmissivity with specific yields between 0.05 and 0.3, 7 partial penetrations of 25 to 100 percent, and vertical anisotropies of 0.02 to 0.2 with slope defined by 8 drawdowns at 12 and 48 hours.
23 100,000 D / ² M
N 10,000 I , Y T VI m SSI I 1,000 mu
M xi S Ma m
N u
A im n R Mi T 100 B O C A J - Average - 75% 10
PER 25% - Average O
O KNOWN C 1 10 100 1,000 10,000 KNOWN TRANSMISSIVITY, IN M²/D 1 2 Figure 7. —Corrected Cooper-Jacob estimates of transmissivity for all combinations of storage coefficient, 3 partial penetration, and vertical anisotropy with slope defined by drawdowns at 12 and 48 hours.
24