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ARCSWESTRemedial Activities atSelected UncontrolledHazardous Waste Sites inthe Zone of Regions IX and X

GROUNDWATER CAPTUREZONE ANALYSIS AND

MODELING SIMULATIONS

AMERICAN RIVER STUDY AREARANCHO CORDOVA, CALIFORNIA

Environmental Protection AgencyContract No. 68-W9-0031

CKMHILL

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LGROUNDWATER CAPTURE

ZONE ANALYSIS ANDMODELING SIMULATiONS

AMERICAN RIVER STUDY AREARANCHO CORDOVA, CALIFORNIA

IEPA CONTRACT NO. 68-W9-0031

WORK ASSIGNMENT NO. 31-31-9L16CH2M HILL PROJECT NO. BAE69204.02.02

Prepared for:

| U.S. Environmental Protection Agency>- Region EX

75 Hawthorne Streetj San Francisco, California 94105

, . Prepared by:

: ^ • • ; . ' • ' • ' , : ' V ; : • • • • • ' ; ' , ' " ' CH2MHILL ; ' • • ' / ' ', Redding and Oakland, California4 ' ' . - . • : , • . • - • . - - . • ' . • • • ' • • ' .

June 17,1994f , - , • . • • ' • ' . - ' , . ' , ' ; - "

CONTENTS

Page

Purpose 1Introduction 1Conclusions and Recommendations 1Aquifer Test Evaluation 3Groundwater Model Construction and Calibration 6Groundwater Capture Evaluation 11Works Cited 23

TABLES

1 Comparison of Aquifer Properties Used in Groundwater ModelingSimulations With Estimates Presented in the EE/CA 5

2 Assumed Extraction Rates for the GroundwaterModeling Simulations 12

FIGURESFollows Page

1 Well Location Map 32 Simulated Versus Observed Drawdown in the Upper Aquifer After 10 Days

of Pumping 63 Simulated Versus Observed Drawdown in the Middle Aquifer After 10 Days

of Pumping 64 Simulated Versus Observed Drawdown in the Lower Aquifer After 10 Days

of Pumping 65 Simulated Flowlines for Wellfield 1 Upper Aquifer 126 Simulated Flowlines for Wellfield 1 Middle Aquifer 127 Simulated Flowlines for Wellfield 1 Lower Aquifer 128 Town and Chicago Well Flowlines Deep Reinjection for Wellfield 1 169 Simulated Flowlines for Wellfield 2 Upper Aquifer 1810 Simulated Flowlines for Wellfield 2 Middle Aquifer 1811 Simulated Flowlines for Wellfield 2 Lower Aquifer 1812 Town and Chicago Well Flowlines Deep Reinjection for Wellfield 2 18

Appendix A: Theoretical Basis for the MLU Program Code

mtf:US10/007B.WSl 11

Groundwater Capture ZoneAnalysis and Modeling Simulations

'Purpose

The purpose of this report is to present the results of groundwater modeling simulationsperformed to estimate me extent of capture zones created by several proposed groundwaterextraction wells hi the vicinity of the American River near Rancho Cordova, California.The interpretations of aquifer tests performed at the site are also described as they relate tothe refinement of the hydraulic properties assumed for input to the groundwater model.

CH2M HILL performed this effort in an oversight role to investigate the validity of theassumptions and conclusions presented in the May 1993 Engineering Evaluation/CostAnalysis (EE/CA) for the American River Study Area produced by Aerojet. It is intendedas a management tool to focus future remedial activities in the American River Study Areaand is not intended to be an exhaustive analysis of the hydrogeologic conditions that cur-rently exist, or will exist hi the future as a result of response actions.

A response action is necessary in the American River Study Area because groundwatercontamination presents a potential threat to the long-term beneficial use of groundwaterresources in the area.

Introduction

A plume of contaminated groundwater has been detected offsite from the Aerojet Facilityand it is continuing to move northwesterly toward and beneath the American River. Thisplume is composed predominantly of trichlorethene (TCE) and it is the target of a plannedresponse action to remediate the contaminated aquifers in the area. Current data shows thatthe three uppermost aquifers in the vicinity of the American River contain levels of TCEthat exceed the State of California Maximum Contaminant Levels (MCLs). A descriptionof the hydrogeologic and water quality data is available in the document titled EngineeringEvaluation and Cost Analysis far the American River Study Area, produced by AerojetGeneral Corporation, May 1993. Additional aquifer testing data collected since May 1993were also used in this analysis.

Conclusions and Recommendations

The main conclusions reached from the groundwater capture zone analysis and modelingsimulations, and the recommended course of action, are as follows.

1. The continued migration of the American River Study Area plume is a potentialthreat to municipal wells of the Fair Oaks Water District (FOWD).

mtf:US10/D07A.W51

2. The implementation of the removal action recommended in Aerojet's May 1993EE/CA is an environmentally sound first step in an iterative approach to control themigration of the contaminant plume in the American River Study Area.

3. A significantly higher extraction rate than discussed in the EE/CA, on the order of2,000 gallons per minute (gpm), may be required to completely capture the TCEtarget area in the upper, middle, and lower aquifers. The estimated pumping rate ishigher than had been contemplated in the EE/CA because the aquifers have beentested and found to be more transmissive. The model used in this analysis assumesthat the aquifer properties disclosed in the testing at Well Cluster 4300-02 extendthroughout the modeled area. The assumption was made because there are no otherquantitative data on the properties of the upper or middle aquifers. The travel timesshown on the figures, as indicated by the tic marks, do not reflect actualgroundwater contaminant migration observed at the site. This discrepancy should becontinually evaluated through implementation of this project

4. An extraction well network, designed to contain the observed contamination, willrequire additional upper, middle, and lower aquifer wells on the north and southsides of the American River.

5. On the basis of groundwater monitoring data collected, including data from thenewly installed Greenvale (Aerojet Well No. 1559-61) and Oak Glenn (Aerojet WellNo. 1556-8) wells, the modeling shows that the recharge of groundwater in theproposed locations of the May 1993 EE/CA will not accelerate the identified con-taminant target area toward the FOWD water supply wells.

6. Recent groundwater quality data collected from Aerojet Well No. 1407 by Aerojetand the Central Valley Regional Water Quality Control Board (RWQCB) indicatethat TCE concentrations above MCLs (at 7.7 ug/1) have been detected (RWQCBInspection Report dated April 27, 1994). -These data were not received in time forinclusion hi this analysis. Previous sampling rounds for Well No. 1407 have consis-tently produced results below MCLs. Additionally, data collected from Aerojet WellNo. 1526, located between the TCE plume and Well No. 1407, have yielded samplesthat have been consistently "nondetect" for TCE in previous sampling rounds. Untilthis situation is fully evaluated, we recommend that recharge in the vicinity of WellNo. 1407 be in the deeper aquifer.

7. Therefore, Aerojet should initiate action to produce future phases of groundwaterextraction in the northern and southern areas of the American River.

We recommend that Aerojet implement the following course of action in the Ameri-can River Study Area:

• Installation of the proposed system as identified in the May 1993 EE/CA,with the exception of an increased, design flow rate.

mtf:US10A»7A.W51

r

• Evaluation and implementation of extraction, treatment, and disposal ofgroundwater in the southern portion of the American River Study Area.

• Continue monitoring and studying the actual response of the aquifer andinfluence on the contaminant plume in the American River Study Area fromgroundwater extraction. This study should be the basis for using a phasedapproach to address the groundwater contamination in the vicinity. It shouldinclude the collection of actual data, as well as the refining of the model withnewly collected data.

All groundwater models arc simplified versions of real world systems that approximatelysimulate the relevant reactions of a real world system. Models are only as precise as thedata and assumptions upon which they are based. Because real world systems are verycomplex, as is the case with the American River Study Area, the modeling results presentedin this report should be viewed as an exercise and as a useful tool to help facilitate removaland response action decisions.

In general, the results of this modeling exercise, when compared to the known real worldmovement of the contaminant plume in the American River Study Area, are not in completeagreement Of specific concern is that the travel times and resultant flow rates are ex-tremely high values. In reality, the historically monitored groundwater movement in theAmerican River Study Area has not proved to be this rapid. Therefore, the actual captureof groundwater contaminants estimated in this report may differ from the actual capturezones realized in real world operations. If the actual transmissivity of the aquifers is lessthan what was used in the model, then less pumping from potentially more wells may berequired.

Aquifer Test Evaluation

Aquifer Test Description

Several aquifer tests have been performed in Extraction Wells 4300, 4301, and 4302 whichwere screened specifically in the upper, middle, and lower aquifers, respectively. This wellcluster, located north of the American River, is shown on Figure 1. Initially, three separateaquifer tests were conducted at the well cluster during November 1993. Each test consistedof pumping one well in the cluster for approximately 4 days, and monitoring the water levelresponse in the adjacent extraction wells and in surrounding monitoring wells. An addi-tional aquifer test was conducted in December 1993, consisting of pumping all three extrac-tion wells concurrently for a period of 10 days, and monitoring the water level response insurrounding monitoring wells.

mtf:US10/007A.W51

<**&+

1S36-SB*

(10*7)

1OMIItfOu

except 1190-2 Felt. 1991

FIGURE 1WELL LOCATION MAPRANCHO CORDOVA, CALIFORNIAAEROJET - AMERICAN RIVER AREA

Aquifer Test Analysis

The method used to interpret the data from the two sets of aquifer tests varied depending onthe test design. The single-well pumping tests were evaluated using the computer programMLU developed by CJ. Hemker, 1993. The results from the multiple well test wereevaluated using the groundwater model MicroFem, also written by Hemker, et al, 1987 to1994. A more detailed description of these analysis methods is presented below.

Single-Well Test Analysis

The computer program MLU was used to calculate hydraulic parameter estimates from thesingle-well aquifer tests. MLU is a transient, multi-aquifer simulation that uses a leastsquares, curve-fitting algorithm to calculate aquifer and aquitard parameters (aquifer trans-missivity [T], aquifer storage coefficient [S], aquitard resistance [R], and aquitard storagecoefficient [S']) on the basis of time-drawdown data collected during aquifer tests. Thesolution technique accounts for both leakage between aquifers and storage of water in theaquitards and aquifers. Any number of parameters can be fixed based on prior knowledgeof their approximate values; the program then estimates the values of the remainingparameters.

A more complete description of this program, including the governing equations and thesolution technique is presented in Appendix A. In Table 1, the results of the MLU analysisare summarized and compared to the estimates of aquifer parameters presented in theEE/CA. The EE/CA estimates were based on an aquifer test of a long-screened watersupply well that was located approximately 1 mile to the southeast of short-screened Wells4300, 4301, and 4302.

Table 1Comparison of Aquifer Properties Used In Groundwater Modeling

Simulations With Estimates Presented in the EE/CA

EE/CA Data Single-Well Tests Multiple-Well Tests

Aquifer Transmissivity (fiVd)

Upper Aquifer

Middle Aquifer

Lower Aquifer

2,000

550

1,730

22,000*

1.8301

90*

16,000*

4,000*

125*

Aquitard Vertical Resistance (days)

Upper/Middle

Middle/Lower

N/A

N/A

80*

900*

80*

900*

•Parameter Values in the Vicinity of Wells 4300, 4301, and 4302N/A * Not available

mtf:USl<V007A.W51

Multiple-Well Test Analysis

Because the MLU program can evaluate only one pumped aquifer at a time, a three-dimensional groundwater flow model was used to evaluate the multiple-well test results. Afinite element MicroFem groundwater model was developed by CH2M HILL to estimate thecapture zones that would be produced by operating the proposed extraction wells locatednear the American River. This model was based on the results of the single-well tests andthe stratigraphic data presented in the American River Study Area EE/CA. The results ofthe 10-day multiple-well aquifer test provided an additional opportunity to test the validityof the existing groundwater model, and provided another data set with which the modelcould be further calibrated. A more complete discussion of the groundwater modelconstruction and development is provided in the following section.

The 10-day multiple-well aquifer test was simulated using the finite element groundwatermodel and, following small adjustments to the assumed aquifer transmissivities, produced agood match between observed and simulated drawdown in the vicinity of the extractionwells. The observed water levels at the end of the 10-day test were increased by 0.5 foot toapproximately account for a regional rising water level trend of about 0.5 foot observedduring the course of the test. Comparisons of observed and simulated drawdown at variousdistances from the pumping wells are shown on Figures 2 through 4. The agreementbetween simulated and observed drawdown improves as radial distance from the pumpingwell increases. Because the objective of this modeling effort is to define the extent of thecapture zone created by extraction wells, it is more important to match the observeddrawdown at the distant edges of the cone of depression. The aquifer parametersdetermined from the multiple-well aquifer test were similar to those obtained from thesingle-well tests. Table 1 presents the aquifer parameter estimates that are based on themultiple-well test.

Groundwater Model Construction and Calibration

The groundwater model selected to evaluate the extent of capture produced by the proposedextraction wells in the American River Study Area is MicroFem. This finite element modelwas selected because it is capable of simulating transient groundwater flow in multipleaquifer systems, and includes leakage between adjacent aquifers at the site. It is capable ofgenerating three dimensional flowlines that allow for the evaluation of capture zones inthree dimensions. The capability of incorporating the leakage between aquifers, and theinfluence of leakage on the capture of contaminated groundwater by extraction wells, wasneeded for the evaluation of the leaky aquifer system that exists in the study area.

Groundwater Model Assumptions and Limitations

The conclusions presented in this report are based on a groundwater model that was con-structed using many assumptions regarding the distribution of aquifer properties across theAmerican River Study Area. The uncertainty associated with our current understanding of

mtf:US10/007A.W51

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Wen 1395

100

SIMULATED-DRAWDOWN

1,000 10,000

DISTANCE (ft)

ROO14M.03

RGURE 2SIMULATED VERSUS OBSERVEDDRAWDOWN IN THE UPPER AQUIFERAFTER 10 DAYS OF PUMPINGAEROJET-AMERICAN RIVER AREARANCO CORDOVA, CALIFORNIA

QBHHIU.

& 3

!Q

100

Well 1396

\ SIMULATEDDRAWDOWN

Well 1539"

Well 1526

Well 1532

1,000

DISTANCE (ft)

Well 1407

10,000

RDO1494JK

RGURE 3SIMULATED VERSUS OBSERVEDDRAWDOWN UN THE MIDDLEAQUIFER AFTER 10 DAYS OFPUMPINGAEROJET-AMERICAN RIVER AREARANCHO CORDOVA, CAUFORNIA

I— 1— )— H-C—.!H/Ii. I—

o

8

7

6

5

4

3

2

1

Well 1397

.SIMULATED .DRAWDOWN

Well 1540

100 1,000 10,000

DISTANCE (ft)

RDO14M_01

FIGURE 4SIMULATED VERSUS OBSERVEDDRAWDOWN IN THE LOWERAQUIFER AFTER 10 DAYS OF PUMPINGAEROJET-AMERICAN RIVER AREARANCHO CORDOVA. CALIFORNIA

\fmtrn fi IUm

the actual aquifer properties at the site is still large. As such, these predictions of aquiferresponse to extraction well pumping also contain significant uncertainty. The constructionof any response action at the site must be associated with sufficient field information, col-lected during operation, to confirm that the desired extent of capture is being achieved bythe system in place.

The capture zone simulations presented in this analysis were performed assuming steady-state conditions. This neglects the inclusion of transient influences on the groundwatersystem such as seasonal fluctuations in groundwater levels and recharge rates. However,the use of a steady-state model is appropriate because the objective of the groundwatermodeling effort is to evaluate the long-term performance of an extraction system at contain-ing and extracting contaminated groundwater.

Groundwater Model Construction

The principal objective of the model prepared for this analysis was to test the assumptionsand conclusions presented in the EE/CA. Thus, CH2M HILL performed this modelingeffort primarily in an oversight role. The model should not be considered to be an exhaus-tive analysis of the hydrogeologic conditions in the American River area. However, themodel is useful for evaluation of hydrogeologic conditions in the study area to determinethe need and scope of response action to control the movement of contaminated ground-water.

The groundwater model used in this analysis has evolved since September 1993, as addi-tional field data have been collected. The original groundwater model was a five-layermodel and was based entirely on the aquifer and aquitard geometries, and the aquifer prop-erties presented in the May 1993 EE/CA Report. The upper three model layers representthe upper, middle, and lower aquifers at the site. The fourth and fifth layers combine torepresent the regional deeper aquifer, which was subdivided into two layers to allow greatervertical resolution of deep flow paths. The original aquifer properties obtained from infor-mation presented in the EE/CA are summarized in Table 1.

The lateral boundary conditions of the model were defined as constant heads in all layers,and are based on water level contour maps presented in the Aerojet EE/CA (Figures 2-10through 2-12). The upper and lower boundaries were defined as no flow boundaries.

Groundwater Model Calibration

The original calibration of the groundwater model was based on comparing the simulatedgroundwater levels across the study area with the observed water levels measured in moni-toring wells. The calibration procedure consisted of setting the constant heads, located atthe model perimeter, to produce groundwater flow directions and gradients to match theobserved values. This procedure resulted in a close agreement between the simulated andthe observed water levels.

mtf:US10/007A.W51 10

r

Revisions to Groundwater Model

After completion of the single-well tests at Wells 4300, 4301, and 4302, the model wasrevised to reflect the findings from these tests. These tests suggested mat the transmissivityof the upper aquifer was significantly higher than the values estimated for the EE/CA, andthat the transmissivity of the lower aquifer was significantly lower. The thickness of eachof the aquifer units was presented in the EE/CA (Figures 2-7 through 2-9), and the influ-ence of aquifer thickness on transmissivity was retained in the refined version of thegroundwater model. The aquifer thickness was accounted for by adjusting all transmissivityvalues in a single layer by a common factor. This approach is equivalent to adjusting thehydraulic conductivity of the aquifer materials while retaining the original estimates ofaquifer thickness. The leakance values assigned between model layers were based on theresults of the MLU analysis described above (see Table 1).

The material properties assigned to the groundwater model were further refined on the basisof the 10-day aquifer test conducted at the site. As previously discussed, the 10-daymultiple- well test was simulated using the MicroFem groundwater model. Following smalladjustments to the assumed aquifer transmissivities, a good match between the observed andthe simulated drawdown hi the vicinity of the extraction wells was obtained (see Figures 2through 4). The results of this calibration effort were critical because the purpose of con-structing the groundwater model was to evaluate the response of the aquifer system to pro-posed extraction well networks. These results suggest that the groundwater model producessimulated water levels in response to pumping that are close to those that will be observedin the vicinity of Wells 4300, 4301, and 4302.

However, in areas remote from the Well 4300 series, where little field data are available,model predictions are less certain. As the capture wellfield is constructed, a program ofaquifer testing and model refinement should be performed to enhance the predictive capabil-ity of the model.

Groundwater Capture Evaluation

The following groundwater capture evaluation considers two potential wellfields designed toremediate contaminated groundwater in the vicinity of the American River. Wellfield 1 wasproposed by Aerojet in the EE/CA, and consisted of two extraction well clusters pumping atthe rates presented in Table 2. The Aerojet wellfield was based on the results of the two-dimensional model, Flowpath, which relied on aquifer parameters from a well locatedapproximately 1 mile to the southeast of Wells 4300-4302. The extraction rates in Table 2were based on the pumping tests in November and December 1993, and not on the assumedextraction rates in the EE/CA.

Wellfield 2 was designed by CH2M HILL to capture the entire plume of contaminatedgroundwater north of the American River. The extraction rates for this wellfield are alsolisted in Table 2.

mtf:US10/D07A.W51 11

Table 2Assumed Extraction Rates for the Groondwater

Modeling Simulations

Well Name Upper Aquifer Middle Aquifer Lower AquiferEE/CA WeDfleld 1

Wells 4300 to 4302Cluster 1

224224

197197

Total Wellfield Pumpage

3434

910Wellfield 2

Wells 4300 to 4302Cluster 1Cluster 2

Cluster3Upper Well 4

285240240235285

195200105105

0Total Wellfield Pumpage

353535350

2,030

Notes: All Values in gpmWell Locations Shown on Figures 5 through 7 and 9 through 11

Wellfield 1 (Wellfield Proposed by Aerojet in EE/CA)

Configuration and Extraction Rates

The original extraction network proposed by Aerojet consisted of two extraction wellclusters located on the downgradient edge of the TCE plumes. The locations of these wellclusters are shown on Figure 5. Wells 4300 through 4302 have been constructed and wereused to define aquifer parameters. The pumping rates for the second well cluster wereassumed to be equal to the rates of Wells 4300 through 4302 during the 10-day aquifertest. Because all three wells will be pumping when the final extraction network is in opera-tion, these are appropriate pumping rates for use in the capture zone simulation. The pump-ing rate is a combined 455 gpm from the well cluster as follows: 224 gpm from the upperaquifer, 197 gpm from the middle aquifer, and 34 gpm from the lower aquifer. Theassigned pumping rates used in the capture zone simulations are summarized in Table 2.

Capture Zone Extent

The results of these simulations suggest that the extraction well configuration proposed byAerojet will need to be augmented by additional extraction wells to capture the TCE plumenorth of the American River hi all three contaminated aquifers. The extraction well loca-tions, and the groundwater pathlines indicating the extent of capture from these wells for theupper, middle, and lower aquifers, are shown on Figures 5 through 7. The significant dif-ference between the capture zones presented in the EE/CA Report and those presented hereis mainly a result of the differences in the assumed transmissivities of the aquifer units.The transmissivity of the upper aquifer used in the EE/CA analysis is eight to ten timeslower than the value used in this analysis, and the transmissivity of the middle aquifer used

mtf:US10/D07A.WSl 12

COM)

1190-2

T ™* RocnCRQO WoH

Upper Aqutfsr

.Mkfcfla

— --- -Uoww Aquifer

-- - Oaep Aqutfar

NOTES:

Ticfc mono rvprBflont 1 y9or trovn tfino*

Scal« In FeetBOO 1000 2000 3000

FIGURE 5_S1MULAIED-FL.QWLINES_FOR WELLFIELD 1UPPER AQUIFERRANCHO CORDOVA. CAUTORNIAAEROJET - AMERICAN RIVER AREA

(to«)

1999-88*

1190-2

Wflfl

Uppw Aquffcr

Aquffer

NOTES:TICK nHnliv repfvsflnt 1 yoor

Seals In FeetQ 80Q 1000 ±OQP 3000

FIGURE 6-SIMULAIED-ELOVWJNES-FOR WELLFIELD 1MIDDLE AQUIFERRANCHO CORDOVA. CALIFORNIAAEROJET - AMERICAN RIVER AREA

CIM«)

1190-2 Ft6.

-Recharge Wsfl

Upper Aquffer

MMdta Aquffer

Lower Aquffer

FIGURE 7SIMULAIEQ-aOWUNES-FOR WELLFIELD 1LOWER AQUIFERRAMCHO CORDOVA. CAUFORMAAEROJET - AMERICAN BVtR AREA

r

in the EE/CA analysis is almost four times lower than the value used here (See Table 1).The lower assumed transmissivities used in the EE/CA resulted in much larger capturezones being predicted with similar assumed extraction rates.

Treated Groundwater Recharge

For the purposes of this analysis, the recharge of treated groundwater was assumed to occurto the deeper regional aquifer beneath the contaminated aquifers at the site. The results ofthe simulations suggest that the recharge of treated groundwater has no significant impacton the extent of capture achieved by the extraction wells. In all of the simulations per-formed, the quantity of recharge water was equal to the total extraction rate from the extrac-tion wells; 910 gpm for Wellfield 1, and 2,030 gpm for Wellfield 2. The locations of therecharge wells are shown on Figures 5 through 12.

Impacts on Town and Chicago Wells

Another objective of mis analysis was to investigate the potential impacts of the plannedresponse action on nearby groundwater production wells. Two FOWD wells are locateddowngradient of the TCE target area in the American River Study Area. These wells areknown as the Town Well and the Chicago Well, and their locations are shown on Figure 1.Well logs obtained for both the Town and Chicago Wells indicate that the total depth ofeach well is 600 feet. These wells produce water from deeper regional aquifers, below thecontaminated portions of the upper, middle, and lower aquifers.

FOWD records indicate that these wells are currently operated only in peak demand periodsduring the summer months. Total pumpage from the Town Well was 0, 3.1 and 1.0 acre-feet (ac-ft) in 1990, 1991, and 1992, respectively. Total pumpage from the Chicago Wellwas 9.0, 3.7, and 1.2 ac-ft in 1990, 1991, and 1992 respectively. These productionquantities correspond to average pumping rates of between 0 and 5.6 gpm assumingcontinuous pumping throughout the year. In actuality, the wells were pumped atsignificantly higher rates for limited durations of time. Actual 1990 through 1992 produc-tion rates for the Town and Chicago Wells are not available. Because this analysis wasperformed assuming steady state conditions, a constant pumping rate had to be assigned tothe production wells. The pumping rates assigned to these wells was 600 gpm for theChicago Well and 1,050 gpm for the Town Well. These values were obtained from welltesting data collected at the time of well construction.

We concluded that for the purposes of this analysis it was more appropriate to simulate thehydraulic conditions that will actually result from the operation of these wells than use anartificially low average annual pumping rate. The groundwater pathlines that provide waterto both the Town Well and the Chicago Well are shown in Figure 8. It should be notedthat this figure is based on conservative assumptions and most of the time the Town andChicago Wells will be shut down.

mtf:US10/007A.W51 16

LEGEND13*1-4

• ^EXTENT OF TCE TARGET AREAIN THE LOWER AQUIFER

oocttre beloweofrtomfhated area)1190-2 Fib. 1091

T ™ RSCnOTOjS WOT

Upper Aquifer• MMcfle Aquifer

Lower Aqutfer

Deap Aqutfer

Scale h Fmt1000 2000

FIGURE 8JQWhL^ND_CHICAGO-WELLFLOWUNES - DEEP RECHARGEFOR WELLFIELD 1RANCHO CORDOVA. CALIFORNIAAEROJET - AMERICAN RIVER AREA

Wellfield 2

Configuration and Extraction Rates

Because Wellfield 1 proposed by Aerojet in the EE/CA will not likely capture the TCEplume, the model was used to develop a groundwater extraction network that was capableof capturing the TCE plume that exists in the aquifers north of the American River. Noneof the extraction networks evaluated here are intended to address contamination that residesin aquifers south of the American River. These zones of contamination will be addressed infuture response actions.

The configuration of the extraction network required to capture the TCE plume is shown onFigures 9 through 11. This extraction network consists of four well clusters along with oneadditional upper aquifer well, and pumps a total of 2,030 gpm; including 1,285 gpm fromthe upper aquifer (240 to 285 gpm per well), 605 gpm from the middle aquifer (100 to 200gpm per well), and 140 gpm from the lower aquifer (35 gpm per well). The assumedpumping rates in this capture zone simulation are summarized in Table 2. Because of thelow transmissivity of the lower aquifer, much of the groundwater flowing into the extractionwells screened in the lower aquifer originates from deeper units (see Figure 11). However,the analysis indicates that the majority of the lower aquifer contamination is pulled toward,and eventually would be removed by the proposed well network.

Small gaps in the modeled capture zones still persist but are not deemed significant withinthe context of this report. In the model such gaps could be filled by increased pumpingfrom the area between Well Cluster 4300-02 and Cluster 2. Such fine tuning of the capturezone was not performed for this evaluation because there are still large uncertainties in theassumed aquifer properties, and the precision of the existing groundwater model cannotdefinitively determine whether these small gaps in the capture zones will actually exist oncethe system is installed. Held measurements should be performed during the constructionand operation of the extraction system to confirm that the extraction system is actuallycapturing the desired target area.

Treated Groundwater Recharge

Associated recharge of the total extraction rate from Wellfield 2 (2,030 gpm) was simulatedat the reinjection location proposed by Aerojet in the western portion of Sailor Bar Park.The locations of these recharge wells are shown on Figure 12.

Impacts on Town and Chicago Wells

The impacts of Wellfield 2 on downgradient production wells were evaluated using thegroundwater model. The flowlines associated with the Town Well and Chicago Well under

mtf:US10/007A.W51 18

Torn »(1047)

(1MH)

15S6-58*

SAILOR BARPARK BOUNDARY

except 1190-2 Fe£l991

4- -Rechong* Ifeft

Upper Aqutfw

. MkMte Aqutftr

Lower Aquifer

Deep Aquifer

NOTES:Tick marks lepiescnt 1 yoor traval times.

FIGURE 9SIMULATED FLOWLINESFOR WELLFIELD 2UPPER AQUIFERRANCHO CORDOVA. CALIFORNIAAEROJET - AMERICAN RIVER AREA

Upper Aquffcr

MWdla Aqutfar

NOTES:Tick murk* lepieaeiit 1 year travel times.

FIGURE 10SIMUUTED FLOWLINESFOR WELLFIELD 2MIDDLE AQUIFERRANCHO CORDOVA. CALIFORNIAAEROJET - AMERICAN RIVER AREA

(1048)

1556-58*

,1559-81

•xnpt 1190-2 Fed. 1091

•* RochoTQ®

Upper Aquifer

MWdto A<julfer

Loww Aquffer

Deep AquHor

NOTES:Tide morics represent 1 ywor travel times.

FIGURE 11SIMULATED ̂ LOWUNESFOR WELLFIELD 2LOWER AQUIFERRANCHO CORDOVA. CAUFORNUAEROJET - AMERICAN RIVER AREA

TCE TARGET AREAM THE UJWER

^ W0R flbonoofwo«raspt 1190-2 F«b. 1991

Upp«r Aquffer

-UMdte Aquffsr

• Lxiwer Aoutfer

— Deep Aquifer

NOTES:Tick morks rvprovont 5 ywr travof tfmos.

300Scole In Feat

1000 IPX 3000

FIGURE 12TOWN AND CHICAGO WELLFLOWLINES - DEEP RECHARGEFOR WELLFIELD 2RANCHO CORDOVA, CALIFORNIAAEROJET - AMERICAN RIVER AREA

ri

Wellfield 2 conditions are presented on Figure 12. As in the previous recharge simulation,most of the water pumped from the Town Well appears to originate from the reinjectionwellfield.

Work Cited

Hemker, C. J. 1992. MLU-A Microcomputer Program for Aquifer Test Analysis for Un-steady-State Flow in Multiple-Aquifer Systems. Amsterdam, The Netherlands.

Hemker, C. J., and H. Van Elburg. 1987-1994. MicroFem: Microcomputer MultilayerSteady-State Finite Element Groundwater Modeling. Elangsgracht 83-1016 TR-Amsterdam,The Netherlands.

Aerojet General Corporation. 1993. Engineering Evaluation and Cost Analysis for theAmerican River Study Area. May.

ri._ mtf:USl(VD07A.W51 23

p

rir.

r

p.I

P.

Appendix ATheoretical Basis for the MLU Program Code

III1

i

Journal of Hydrology. 90 (1987) 231-248 231Elsevier Science Publishers B.V.. Amsterdam — Printed in The Netherlands

.[5]

UNSTEADY FLOW TO WELLS IN LAYERED AND FISSUREDAQUIFER SYSTEMS

iC.J. HEMKER and C. MAAS

Institute of Earth Sciences, Free University. P.O. Box 71S1 1007 MC Amsterdam. (TheNetherlands)The Netherlands Waterworks' Testing and Research Institute (KIWA). P.O. Box 1072, 3430 BBNieuwegein (The Netherlands)

(Received June 30, 1986; revised and accepted October 27, 1986)

ABSTRACT

Hemker, C.J. and Maas, C., 1987. Unsteady flow to weils in lay-wed and fissured aquifer systems.J. Hydrol.. 90: 231-249.

A solution has been developed for the calculation of drawdowns in leaky and confined multi-aquifer systems, pumped by a well of constant discharge penetrating one or more of the aquifers.In contrast to earlier solution* the effect* of elastic storage in separating and bounding aquitardshave now completely been accounted for.

The computing technique is based on the numerical inversion of the Laplace transform. Twodifferent methods are used and reaulta are compared with fiit unaSytical solution. Both StehfestValgorithm and Schapery V least squares method yield accurate results in a fraction of the computa-tion time required for the analytical evaluation.

Selected sets of time-drawdown and distance-drawdown curves are plotted to illustrate multi-ple-aquifer well flow and to compare new solutions with results which were previously published.The analogy with flow ie unconiined and Assured Aquifers in demonstrated by multilayer models,representing multiple-porooity formatioas with linear and diffusive crossflow.

I INTRODUCTION

j Practical groundwater investigations often show that an entire system of! interconnected, more or less permeable layers responds to the withdrawal ofj water from one of its components. Some hydrogeological environments may bei really complex indeed, but it is frequently found that these-systems can beI characterized as a sequence of alternating aquifers and aquitards of relativelyI large horizontal extent. In order to analyse the drawdown distribution causedI by pumping wells in such leaky multiple-aquifer systems, several analyticalj solutions have been recently developed (Hemker, 1984,1985; Hunt, 1985; Maas,j 1986), The resulting analytical expressions may appear simple in matrix nota-I tion, but especially transient flow solutions require much computational effort| when drawdowns are evaluated with some precision.

j 'Stehfest, 1970.j "Schapery, 1962.

j 0022-1694/87/S03.50 © 1987 Elsevier Science Publishers B.V.

232

NOTATION

A6,«:,-d{D,, Uf

KKgK,, K*

RS,. S'S,,, S.s,, s;"s,,s'iSB

system matrix as defined by eqn. (6)ailjf - <pS/c,)1/l

hydraulic resistance of tth aquitard (DUKfi

thickness of tth aquifer, ith aquitard(b, coth b,)lc,Tj&,./(<:, 7} sinh A,)diagonal matrix with elements K<>(>\/ui~)zero-order modified Bessel function of the second kindhydraulic conductivity of ith aquifer, tth aquitardtotal number of aquifers in a multiple-aquifer systemparameter of Laplace transformdischarge vectorpumping rate from ith aquiferradial distance from pumping welleigenvector matrixstorage coefficient of tth aquifer (S.-A). ith aquitard (Sspecific storage of ith aquifer, ith aquitarddrawdown in ith aquifer, ith aquitardLaplace transform of s,. sjdimensionless drawdown (4xTts!Qt)time since pumping starteddimensionless time (TttlStr*)diagonal matrix with elements Tt

transmissivity of tth aquifer {KjD,}.;•• - .5 T^ I

eigenvalues of system matrixvertic&l coordinate

Computer programs for the estimation of hydraulic characteristics fromaquifer-test data generally take hundreds or thousands of function evalua-tions, which renders the application of analytical solutions for transient multi-ple-aquifer flow less suitable. Confronted with this practical problem., bothauthors found a solution by turning to the technique of numerical inverseLaplace transformation. Since different inversion methods were used, viz.Schapery's least squares (1962) and Stehfest (1970), this provided a good oppor-tunity for comparing their performances, in particular the accuracy of results.

Another important advantage of the numerical inversion technique is thatthe effects of elastic storage of the aquitards are now no longer neglected butcan be fully incorporated in the mathematical formulation. In this respect thepresented solution can be regarded as the n-aquifer generalization of the singleand two-aquifer solutions given by Hantush (1960) and Neuman and Wither-spoon (1969a) respectively. The applicability of the presented n-iayer transientflow solution is further enhanced by demonstrating how well flow problems ofdouble and multiple-porosity formations can be treated in a likewise manner..

233

NO DRAWDOWN NO FLOW

NO DRAWDOWN 2 t NO FLOW* r-.ji--i.iM> l"

Fig. 1. Schematic diagram c-f a well in a leaky or confined multipSe-aquifer system.

STATEMENT OF THE PROBLEM

The objective can be stated as finding a relatively fast and sufficientlyaccurate computing method to determine the drawdown s in all the aquifers ofa leaky multiple-aquifer system as a function of time t and distance to a pumpedwell r. The aquifer system consists of n aquifers and n 4- 1 aquitards, as shownschematically in Fig. 1. Actually this diagram represents different systems, asboth no-drawdown and impervious (no-flow) boundaries are considered at thetop and (or) base of the system. All aquifers and aqmtards are horizontal andof infinite extent, homogeneous, isotropic and individually compressible. Thewell is of infinitesimal radius, fully penetrates one or more of the aquifers anddischarges at a constant rate Q. It is further assumed that the system layersshow sufficiently contrasting conductivities to neglect horizontal flow in theaquitards and resistance to vertical flow in the aquifers (Hemker, 1985),

The multi-aquifer well flow problem can now be formulated by a system of2n simultaneous partial-differential equations with initial and .boundary con-ditions for the unknown drawdown. When the radial How component in theaquifers is considered, s(r, i) must satisfy the equation:

(fSj I d$;

~dr* + ~r Jr

i = 1,2,

sf(r, 0) = 0

s,{oo, t) = 0

Tf dz

n

T-. dz

(1)

234

while vertical flow in the aquitards is governed by:

a? ~ K ° d t * l ~ 1 ' 2 - - - - - n

a?(r.«,0) = 0

S;(r> 2i-l> ^) = S j -S ('"»*)

where ^ = hydraulic conductivity; S, = specific storage; T = aquifer trans-missivity: S = storage coefficient, and indices are used to indicate the suc-cession of layers, while primes refer to aquitards (Figs. 1 and 5).

When a leaky system is considered s0 = 0 and sa+l = G, but if top and (or)base of the system are impervious, no-flow system boundary conditions must beused instead:

— s'(r, z,,, t} =

and (or):

£<.*I (3)

? 0 —

SOLUTION IN THE LAPLACE DOMAIN

With the aid of Laplace transformation, the systems of partial differentialeqns. (1) and (2) can be reduced to systems of ordinary differential equations,which may be treated by appropriate analytical techniques to find a solutionin the Laplace domain. Each author independently developed an answer to thesame problem. Although the methods are essentially the same, they differ inapproach, notation and final formulation. In terms of matrix functions thesolution may be written (Appendix A):

which is equivalent to the result of eigenvalue analysis (Appendix B):

«(r, P) = 7T- VKV^q (5)ZTCJ3

The calculation of the Laplace transformation of the drawdown vector s isbased on the construction of the tridiagonal system matrix A, which is dennedby:

235

A -

"/*0

i)

-/"»

/I

If*-I

Q -ff, «« + e«.i* •*• ^«

(6)

1

I

\

t

Ij

II

iI

where: dt = pSJT- «C; = (6,- coth &.-)/<:,.T,t /<,- - b../^ smh fa,); 6, = a,D,'; ando, = (pSrf/^i)"2- Diagonal matrix T contains th«: transmissivities, while q issimply the discharge vector (Q,, Q*, . , <?W)T. To calculate the square matricesV and K, the system matrix is first transformed into a symmetric matrix D andthen decomposed into its eigenvalues u>, and normalized eigenvectors r,-:T l f tAT~ in = D = JRWJir1. Diagonal matrix A' is given by its non-zero ele-ments K0(r^/Hui) and matrix V is denned by V ~ T""K R. More details are givenwith the derivation of eqris. (4) and (5) in Appendices A and B.

NUMERICAL INVERSION OF THE LAPLACE TRANSFORMED SOLUTION

As it is the purpose of the authors to obtain 8 formula which is easy toevaluate, analytical inversion of the Laplace transformed solution has notbeen attempted. A large number of different numerical inversion techniquesare available at present, suggesting that no single method suits all purposes.A bibliography listing over 260 titles on the subject has been compiled byPiessens (1975). Systematic comparative studies; were conducted by Cost (1964)and by Davies and Martin (1979). The latter paper has the widest scope, but theformer should be of interest to the hydrologist too, as the type of probleminvestigated by Cost shows similarities to the hydraulic head problem ofgroundwater flow.

Davies and Martin suggest that more than one method should be used on anyunknown function, because every method breaks down on some functions.

Undoubtedly, the easiest method of numerical Laplace inversion is due toTer Haar (1951) and Schapery (1962). It states that.:

/(O * (7)

Ter Haar chooses « = i, while Schapery proposes o = 0.5 (more properly:a = exp(-y), y = Euler's number = 0.57721. . ,). Is is readily verified that TerHaar's method is exact when:

= at + b (8)

where o and b are arbitrary constants, while Schapary's method is exact when:

= a In t + b (9)

236

For the methods to be applicable it turns out that the conditions (8) or (9) haveto be satisfied only approximately in a certain interval around t.

Sternberg (1969), Brutsaert and Corapcioglu (1976) and Barents (1982) usedSchapery's method to solve problems that are closely related to the problem ofthe present paper. The method indeed yields good results in many practicalcases. For the problem at hand, however, deviations were found for smallvalues of time, which were considered to be unacceptable by the authors.

Two other numerical inversion methods are selected, viz. a second methodby Schapery (1962) (known as Schapery's least squares), and the algorithmpresented by Stehfest (1970). In both cases evaluation of the transformedfunction Is required in the real p-domain only. Stehfest's method is found byDavies and Martin (1979) to be accurate on a fairly wide range of functions.Schapery's least squares method, on the other hand, is stated to be rarelyaccurate. Nevertheless Cost (1964) finds the latter method to be very promising,at least for viscoelastic stress analysis.

In this paragraph both methods are briefly described. Their performancesare shown for an arbitrarily chosen testcase and the results are compared witha recently developed analytical solution (Maas, 1987).

The numerical Inversion method by Stehfest requires a fixed number N ofJ(p) evaluations for each value of I. The approximate value /*(£) at t of thefunction fit) is obtained by:

/(O * (10)

where N must be an even number. The weighting coefficients u, depend on Nonly and need to be calculated once:

- i)!) (11;

An optimal value for N should be chosen, as it depends on the type of functionto be evaluated and the precision with which this is done (eigenvalues, hyperbolic and Bessel functions were computed with a relative accuracy of at leasttwelve digits). Greater values for N improve the result theoretically, butrounding errors limit this value in practice. Calculations with different valuesfor N therefore form an essential part of testing this method. .,

Stehfest's method has also been successfully applied to problems in ground-water hydrology by Moench and Ogata (1981, 1984), Moench (1984) and Strelt-sova (1982, 1984).

Schapery's least squares method assumes the original function /(i) to be wellapproximated by m exponential terms:

(12)

where gs and a; are parameters. After choosing the ot; judiciously the gf aredetermined by minimizing the total square error E* between f(t) and/*(!), giversby:

237

so:

00

- f

ao

« j

or.

In other words; in order to minimize E1, f(p) and f*(p) should coincide in mpoints. In view of eqn. (12):

CURVE A r»«jm. A,

Fig. 2. Time-drawdown curves for two-aquifer systems with different top and base conditions.

238

TABLE 1

Stehfest's algorithm tested against an analytical solution of the problem shown in Fig. 2, model D

Time(days)

StehfescN *> &

Stehfestff - 10

StehfestN - 12

s, (m) s,(m) 8s(m) s (m)

Analytical solutionfMaas, 1987, eqn. (26)1

s, (m) 3j (m)

i.io-s2£

i.io-'25•i.io-325i.io-1•*?£,

5i.io-1251.10°25

0«000

-a-0-0000.00270.03610.18240.34030.46740.53150.53650.53630.5363

0.01640.09620.35910.65841.00431.48711.85402.21S72.68903.04173.39213.83834.09904.24864.31314.31804.31784.3178

0e€>0G3

— 0

300.00300.03580.18220.34040.46790.53150.53620.5362Q.5363

0.01650.09570.35900.65851.00441.48721.85412.21682.68913.04193.39213.83824.09974. 24884.31304.31784.31784.3178

0000

-0-00

-c00.00300.03570.18220.34040.46810.53140.53620.53630.5363

0.01650.09560.35910.65861.00451.48721.85412.21682.68913.04193.39213.83834.09984.24874.31294.31774.31784.3178

000

000000.00000.00300.03560.18220.34040.46810.53140.53620.53630.5363

0.01640.09560.3591C.6585i.0044

1.4372i.85412.21682.689C3.04193.39213.83834.09984.24S74.31294.31774.31784.3178

K1)-W l/o*'(I = 1, m)

Choosing m values a. and equating:

- = I\ui/

(» = m) (13)

a system of m linear equations is obtained, which can be solved for g..From eqn. (12) it appears that m evaluations of J(p) suffice to calculate /*(*)

for any value of t. Schapery makes plausible (and it is confirmed by ourexperiments) that /*(*) tends to /(*) as m tends to infinity.

From the form of eqn. (12) it follows that Schapery's method, when appliedto well hydraulics, is better suited for the recovery period than for the draw-down period. For this reason it is profitable to subtract the available steady-state solution from eqn. (3) before inversion.

For the geohydrological scheme depicted in Fig. 2, model D, results ofStehfest's and Schapery's algorithms are presented for a selected number oftime values (Tables 1 and 2). The tables also display the results obtained withan analytical formula that was derived using a generalized Fourier transformtechnique (Maasr 1987). It appears from the tables that the number of function-

239

TABLE 2

Schapery's method tested against an analytical solution of the problem shown in Fig. 2, model D

Time(days)

1.1Q-*25i.io-4

25i.icr'25i.ur1

251.10"'-251.10°25

Schaperym ~ 25

s,(m)

0.08140.08070.07880.07S70.069?0.05300.0291

-0.005?-0.6463-0.0278

0.05360.20890.33220,45130.53580.53990.53630.5355

•SjOn)

1.39851.40291.41571.43691.47841.69641.77312.0639Z61083.03583.41943.87554.08664.22574.31S24.32264.31784.3168

Schaperym - 60

s,(m)

C00040.0003

-0.00040.000200002

-0,00020.0004

-000040.00040.00270.0360018190.34080.4677O.S315O.S362O.S363O.t.363

s,(m)

0.01960.09740.35700.65971,00521.48631.85572.21552.69033.04173.39143.83864.09924.2492•4.31274.31794.31784.3179

Schapeiym « IOC'

s,(m)

000

-0-0-0

000.00000.00310.03560.18220.34040.4681C.S3140.5362Q.53630.5363

Analytical solution(Maas, 1987, eqn. (26)!

ts(m)

001640.09560.35910.65861.00441.48721.S154S2.JU682.6S913.1*193.3S213.83834.C998<!.2487<;.31294.31774.31784.3 !76

s,(m)

000000000.00000.00300.03560.18220.34040.46810.53140.53620.53630.5363

Sj(m)

0.01640.09560.35910.55851.00441.48721.85412.21682.68903.04193.39213.83834.09964.24874.31294.3i774.31784.3178

al evaluations (N and m, respectively) are decisive for the accuracy of theresults. The authors suggest that for practical purposes Stehfest's method beused with Ar = 10 and that of Schapery with m » 100. Which of the twoalgorithms is to be preferred depends primarily on the number of time valuesfor which the original function f(t) is required. Roughly speaking the twoalgorithms are competitive when this number equals ten, It is safe, however, touse both methods together and compare the outcome. From a computationalpoint of view the analytical solution shown m the last columns of Tables 1 and2 is far inferior to the approximate ones, especially for small values of time. Itshould not be employed unless high accuracy is necessary for theoreticalreasons.

EXAMPLES OF WELL FLOW IN MULTIPLE AQUIFERS

The computing technique described in the previous sections can be used toanalyse the drawdown behaviour of simple or rather complex multiple-aquifersystems with leaky or confined boundaries. However, anconfined and heteroge-neous aquifer problems can also be treated, as will be discussed in the nextsection. Calculations for all examples given are carried out by the same in-teractive program, written in double precision GW-BASIC, compiled and runon an Olivetti M24 microcomputer. Processing speed appeared to be no serious

240

»<mj

/

'• • • ' ' 'Jin> I ! \H I /I \i •• ; i ' . - / . ' «-'' I / / !/••" /

' ' If JTO'5 10-S

x>C.-« ^Ul(<My)^» 10- jib" ;o-̂ -£-. ,^tu-

pl_./ u /- ./I

2000 r(m) 60OO ,o no3 •"Mm) *O4

Fig. 3. Development of the piezometric depressions in the pumped and top aquifer of a leakyfour-aquifer system.

limitation since, for example, the computation of 400 drawdowns in a four-aquifer system (Fig. 3) only required 8 min.

As a first example the two-aquifer and interjacent aquitard system, used byNeuman and Witherspoon (I969b) to show the effects of unpumped aquifers andaquitard storage, is modified to include aquitards at the top and base of bothaquifers. The original time-drawdown curves were obtained in two differentways: a leaky system with high hydraulic resistances of the upper and loweraquitard and & confined system with zero-storage of these aquitards producedthe same results (Fig. 2, model Al and A2). Alternative pairs of drawdowncurves were computed for three cases, all with identical characteristics forboth aquifers and aquitards, but differing with respect to the applied boundaryconditions (Fig. 2, model B, C and D). Upper and lower aquifer drawdowns arefound to be considerably reduced by these alterations, showing the effect ofeach individual adjustment to the model.

For a better comparison with published results and to. allow any set ofconsistent units, the use of dimensionless parameters may be preferred. Sincethe present solution only assumes horizontal flow in aquifers and vertical flowin aquitards, a selected set of dimensionless parameters should be based on therelated hydraulic characteristics: T, S and c, S', respectively. In a multiple-aquifer S3/stem comprising n aquifers and (n + 1) aquitards dimensionlessdrawdown SD can be expressed as a function of (4n -f I) dimensionless par-ameters: /.D = dimensionless time for the pumped kth aquifer (T^ijS/,^), 2ndimensionless leakage parameters (^/Tlc,)"2 and (r2/T|c^i)l/2 and 2ra dimension-less storage parameters SJSk {> ^ k} and S//Sft, where SD Is defined bySD = 4jiTts/QA. Apart from these parameters the top and base conditions of the

241

system should be given as no-drawdown or no-flow boundaries. Although thedrawdown curves in Fig. 2 are given on a dimensionless scale, actual computa-tion has been carried out using the parameters: indicated in the lower part ofthis figure.

A second example to illustrate the application of the presented solutiontechnique concerns the visualization of a developing cone of depression arounda pumped well in the second aquifer of a leaky four-aquifer system (Fig. 3). Itrepresents the fresh and brackish groundwater system in the polder areaaround a well field at Lexmond in The Netherlands (Hemker, 1984). The usedset of hydraulic parameters is only partly based on aquifer test results; especi-ally storativity values are estimates only. The induced potentiometric depres-sion in the upper aquifer is shown in the same figure, clearly demonstrating itsdelayed development

EXTENSION TO FISSURED AQUIFER SYSTEMS

As explained by Streltsova-Adams (1978) and Streltsova (1982, 1984) there isan obvious similarity between well flow solutions for layered systems andfissured formations. Hence it is of interest to determine to what extent theapplicability of the presented multiple-aquifer technique can be expanded toinclude the related solutions for heterogeneous systems. Most solutions forgroundwater flow in fissured formations are based on the double-porosityconcept and assume either a linear (also referred OT as : pseudo-steady state,lumped) or a diffusive (transient, capacitive, distributed) type of block-to-fis-sure flow behaviour (Streltsova, 1984; Barker, 1985), Linear crossflow impliesthat the rate of flow is proportional to the average difference in heads between

1 ] fissures and blocks, while diffusive crossflow is based on the development of ahead distribution in the matrix material of the blocks.

The influence of doubie-porosity with linear crossflow can be simulated by

I add ing a zero-transmissivity aquifer and an interjacent zero-storativity aquit-ard to the system. Unconfined conditions of multiple-aquifer systems can bemodelled in exactly the same way (Hemker, 1985). When, for example, an

. unconfirmed fissured aquifer is considered, drawdown behaviour can be re-p produced by a confined three-aquifer model. A quantitative confirmation was; sought in the type curves presented by Boulton and Streltsova-Adams (1978).

However, although five parameters are given for each set of curves, while fourft are sufficient, not enough information is provided for a reconstruction. With

the aid of published drawdown values for the same carves (Streltsova-Adams,1978) it was found that the unknown aquifer transmissivity is probably

f l m*day"1.. Table 3 shows the results of calculations obtained from the analyti-cal solution for unconfined flow in a fissured formation, the analytical solutionfor confined flow in a three-aquifer system and the corresponding Stehfest

I numerical inversion approximation.m When the block geometry of a double-porosity formation can be idealized as

infinite slabs, solutions for diffusive block-to-fissure flow are identical to an

I

I

1

V I

242

TABLE 3

Drawdown values for comparable triple-porosity models

J (day)

5 HTa

1.25 1C-6

2.5 HT*5 l<r5

1.25 lO"8

2,5 10-*5 HP4

1.25 IC"S

2.5 10 -s

5 10-"'1.25 10-£

0.0250.050.1250.250.51.252.5

4 Ti t

rS4

0.20.5V

25

102060

10020C500

100020005000

10*2 10*S 10"

10s

Dimensionless

Streltsova1

0.00100.04460.18820.42550.73340.88761.03401.37351.80132.35923.16503.74224.23704.67724.82004.85954.88864.9353

drawdown

Hemker2

0.00110.04443.18640.42180.73080.88601.03161.36831.79372.35013.15603.73484.23294.68134.83434.89004.96275.0752

Hemker and Mamr

0.90100.04460.18540.42170.73070.88601.03181.36831.793?2.34SS3.15593.73484.23294.68124.83434.89024.9626a.0751

'Streltaova-Adams, 1978: table IV, rs Hemker, 1985: Eqn. (11),3 Using Stehfeat(JV

0.1. r/(riCj)l/I - 1. 5,/S,

10): eee Fig. 4, model A, r - 1m.100, S./S, - 10.

aquifer-aquitard system with no-flow boundaries. Similarly, triple-porositymedia with diffusive flow and slab-shaped blocks correspond with confinedlayered systems of two aquitards and one aquifer. To illustrate these analogies,the multilayer counterparts of: (A) an unconfmed double-porosity formationwith linear crossflow; (B) an unconfined double-porosity formation with dif-fusive crossflow; and (C) a confined triple-porosity formation with diffusivecrossfiow are shown in Fig. 4. For each model dimensionless drawdown curvesare plotted to demonstrate their different hydraulic behaviour.

Thus far only double- and triple-porosity single formations have been dis-cussed. There is, however, no restriction to also include multiple-porositylayered formations. The solution is found in the composition of the propersystem matrix. To include diffusive crossflow in some layer, the correspondingdiagonal element of the system matrix is extended with a source term toaccount for contribution from the matrix blocks: (bj tanh b^CjT^ Two extrablock parameters are introduced (<v, S/) when one such term is added; in easeof multiple-porosity the procedure of adding terms can be repeated,

Linear crossflow can be included in a multilayer model in a similar way byadding the term fr?/[c,-T,(l -i- bj}} to the appropriate diagonal element. E thematrix block drawdown is of interest an alternative solution is found byincreasing the number of aquifers considered. The block storativity value is

243

\

^^ fa:$/m. •

1C)"' I K> to* 1O3 1O" 'lo 10°

Fig. 4. Time-drawdown curves for triple-porosity formations with linear and/or diffusive croasflow.

assigned to an aquifer of (nearly) Eero-transmissivity, while the additionalzero-storativity aquitard accounts for tha block resistance (Fig. 4, model A). Ifthe fissured formation is no top or base aquifer, however, the simple structureof aquifer succession is broken. As a result the tridiagonal property of thesystem matrix will be lost, but this doesn't impede the application of thepresented matrix solution method.

CONCLUSIONS

The main object of this paper has been to develop an efficient solution to thetransient multiple-aquifer well flow problem, accounting for elastic storage inaquifers as well as in aquitards. Following essentially the same techniques, theauthors independently arrived at comparable results, using the Laplace trans-form and numerical Laplace inversion. The two approaches prove to be com-petitive from an economical point of view. Being essentially approximate, themethods are recommended to be used together in order to check each other. Acomparison of the results with those obtained by a fully analytical solutionshows that high accuracy is attainable.

It has been shown by examples that the solutions are well suited to theoretic-ally investigate the dynamical behaviour of multiple-aquifer systems. Featuressuch as double or multiple porosity are easily incorporated in the models.

The primary stimulus of the research reported in this paper has been itsapplication to computerized evaluation of pumping tests. Multiple-aquifer testshave been successfully interpreted by both authors. The subject is felt to be ofsufficient practical interest to justify publication as &. separate paper.

244

APPENDIX A. DERIVATION OF EQN. (4) IN TERMS OF MATRIX FUNCTIONS

The reader who is not familiar with the use of matrix differential calculus is advised to consul;.Maas (1986). A stratified porous medium is considered, containing an infinite number of layers. Thefollowing (infinite) matrices are defined:

(a) column matrices:

* {«,} = %

*" {<}' = si

1 {<?<} - ft

(b) diagonal matrices:

K

S {S,} - &

S'

C {CaJ « c;

r {TZ} - r.(c) fluperdiagonal matrix:

* {#U*U ~ !-

The matrix H, being introduced for operational purposes, is defined to have the following property:

where If1 ie the transpose of H and I is the infinite unit matrix.The matrix differential equation describing the hydraulic head in the aquitards reads:

jrjl«'(r.2,0 - S" |- f{r. 2, i) (A!)OZ* OJ

Putting £ - '[Z| — z)/I/J for each aquitard, eqn. (Al) can be written:

. . . .

(notice that 4" runs from 0 to 1). The boundary conditions with respect to C are given by:

f(r. I, t) - Hf(r, 6, i) - s(r, <) - <A2a)

C- ' ^^ r .CtLO - HC-l4-f(r.S I Q-, 0 (A2b)3C 5£

When the initial condition is taken to be:

•'(r.e.O) - 0 (A2c)

the Laplace transform of eqn. (Ai" is given by:

245

IsI

Equations (A2a, b) remain unaltered under Laplace traiufonnation, except for a bar appearingabove «' and •. By inspection the solution of eqn. (Al"), using eqn. (A2a). is found to be:

«"(»•, C. p) - •inh-|{>/pCS;H«nh{tt - Ov/p5S'}HT •>• tina.{^pCS'}]i(r. p)

Using Darcy's law the downward flux i through the aquiijs-os ia found from eqn. (A3):

(A3)

[-cosh 1(1 - OjpCS'}lf + cosh {(^pCS'JJafr.p) <A4)

With regard to the aquifers the matrix differential equation describing the hydraulic head is'.

4«<r.<>TV,'»(r, £) - Hi(r, 0. £> •(- i(r. 1, £)C/i

where:

V? - HL+L —

The boundary conditions are:

lim - 2nr T — «(r. «) -= o,_« dr

*(oo. I) " 0

With the initial condition:

a(r, 0) - 0

the Laplace transform of eqna. (AS) and (A6a, b) reads:

r. p) - Hi(r, 0, p) + i(r. 1, p) « pSi(r, 0

1

P ^

«(oo. p) « 0

In view of eqn. (A4), eqn. (AS') can be written:

TV?i(r, p) *• B(p)i(r, p) •» |»Si(r, p)

where:

B - HC-1

(AS)

(A6a)

(A6b)

(A6c)

lim — 2itr T — i(r, p),~t or '

(A6b')

(A5">

(A7)

The matrix B is seen to be symmetrically tridiagonai. In scalar notation:

I ic, tanhypc, S;" c;., tanhvjsc.-,, S/,,

(A7')

M

246

The general solution of eqn. (A5") is given by:

where:

c, and Cj are column matrices containing constants of integration, c, and c, are solved by using eqn.(AS) and the boundary conditions (A6aQ and (A6bO. It is found that:

c2 - 0

so the final solution reads:

e(r,p) «= -^- K^{rjA(p)\T'lq2xp

A finite system of n aquifers and n + I aquitards can be obtained by setting:

I0(r, p) « C

Alternatively, an impervious base may be introduced by setting:

and an impervious tap iayer is obtained likewise by:

(A9a)

(A9b)

<A10bj

(Alie)

(Allb)

Numerical evaluation of the matrix function Kojrx/j4} is performed as described by Maas (1986).

APPENDIX B. DERIVATION OF EQN. (5) BY EIGENVALUE ANALYSIS

The application of the Laplace transform to the boundary value problem given by eqna. (i) and(2) yields a system of 2n ordinary differential equations.

Vertical flow in the aquitards IE described by ra equations:

ff'i"1,2,.... ,r. (Bl)

whiles,, = H a i l = 0 when a ieaky system is considered. By substituting of - pS^/tf/and 6, - <2,£fthe solution obtained can be written:

einhtei* ~ a.- .̂-) 8inh(a,.r,_, - a^z)

from which it follows that:

247

Fig. 6. Definition sketch.

sinh 6, *inh 6, (B2)

If, however, the base of aquiturd i is impervious, the second boundary condition of eqn. (Bl) shouldbe replaced by:

—oz 0

which leads to a solution of the form:

cosh 6,

and hence:

dS"i a,cosh 6, (B3)

When, aubsequently, the system of differentia) equations for horizantal flow is considered, eqn. (2).Laplace transformation will yield:

<B4)

«.(oo.p) - 0

ft-

Substituting «qn. (B2) into the third and fourth term of eqtt. (B4) and rearranging leads to thefollowing expression:

d'Sf I dst t>i "

/6. cotti b^ 6jt, coth6itl pS\ . ^ bitl _\ CiT, n^.Ti TjSi eu,Ti9inh6,ti*

1>l

At this stage the system oi n equations is readily expressed IR matrix notation:

Li - Am (B5)

248

where L is the Laplace operator 8*ldr* + Ifrd/dr. i is the vector of transformed drawdowns and Ais the tridiagonal n y. n system matrix, as defined by eqn. (6). From eqns. (B3) and (B4) it followsthat systems with no-flow boundaries only require that «a and (or) «„. ,.„ of matrix A are replacedby (6, tanh 6,)/c,T, and (or) (6.., tanh 6..,)/c..,T..

Solving eqn. (B5) for • is essentially similar to the steady well flow multiple-aquifer solution(Hemker, 1984). A slightly different approach will be used, however, to fully benefit from the nearlysymmetric property of A. We define a symmetric tridiagonal matrix D and then calculate its p.eigenvalues and eigenvectors:

ri/2AT-M „ D = RWR-1 ' (36)

where T is a diagonal matrix with Tt along the main diagonal, Wisthen x n diagonal matrix withthe eigenvalues ID,-, and R is the n x n matrix containing the corresponding eigenvectors in itscolumns. Since D is symmetric, the eigenvectors can be normalized to achieve an orthoncrma!matrix R, thus R'1 = RT.Let matrix V be defined by V «= T""2 R, then:

V-- -= R~1T^ = fiT1" - FTr (37)

Substituting eqns. (B6) and (B7) into (B5) leads to:

.Li - yWV-'a (38)

This system of differential equations can be uncoupled and solved for the boundary conditions, justlike the steady-flow problem, to obtain:

s = i VKV- lg i'89)P

where K is the n x n diagonal matrix with /^(rN/ui~) as non-zero elements and g is the vector givenby QilZnT/, i == 1. 2, ... * a. By further simplification the final solution may be written:

where 9 is simply the discharge vector (<?,. Q2 ..... Q.)T. This last equation shows that itsnumerical evaluation only implies the eigenvalue decomposition of a symmetric tridiagonal matrixand the calculation of hyperbolic and Bessel functions.

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