(XANF-921082--2

EstimatingThe Water Table Under The RadioactiveWasteManagement Site in Area 5 of the Nevada Test Site D_2 0.].3506

The Dupuit- Forcheimer Approximation

byF. Tom Lindstrom, Lawrence E. Barker, David E. Cawlfield,

Dale D. Daffern, Brian L. Dozier, Dudley F. Emer, Warren R. Strong

Reynolds Electrical & EngineeringCo., Inc. _ _'_ !'_;,_i_'_PostOffice Box 98521 _" ,_,_,_'_'_:_ ,

Las Vegas, NV 89193-8521 'i_, /,/i/_'(i t _9_

ABSTRACT

To adequatelymanage the low level nuclearwaste (LLW) repositoryin Area 5 of theNevada Test Site (NTS), a knowledgeof the water table under the site is paramount.The estimatedthicknessof the add intermountainbasinalluvium is roughly900 fc_et.Very littlereliablewater table data for Area 5 currentlyexists. The Special ProjectsSectionof the ReynoldsElectrical& EngineeringCo., Inc. Waste ManagementDepartment is currentlyformulatinga long-rangedrillingand samplingplan in supportof a ResourceConservation Recovery ACt (RCRA) Part B permit waiver forgroundwater monitoring and liner systems. An estimate of the water table under theLLW repository, called the Radioactive Waste Management Site (RWMS) in Area 5, isneeded for the drilling and sampling plan. Very old water table elevation estimates atabout a dozen widely scrJtteredtest drill holes, as well as water wells, are availablefrom declassified U.S. Geological Survey, Lawrence Uvermore National Laboratory,and Los Alamos National Laboratory drilling logs. A three-dimensional steady-statewater-flow equation for estimating the water table elevation under a thick, very dryvadose zone is developed using the Dupuit assumption. A prescribed positive verticaldownward infiltration/evaporation condition is assumed at the atmosphere/soilinterface. An approximation to the square of the elevation head, based uponmultivariate cubic interpolation methods, is introduced. The approximate is forced tosatisfy the governing elliptic (Poisson) partial differential equation over the domain ofdefinition. The remaining coefficients are determined by interpolating the water tableat eight "boundary points." Several realistic scenarios approximating the water tableunder the RWMS in Area 5 of the NTS are discussed.

ACKNOWLEDGEMENT

The authors wish to thank Dr. Lynn Ebeling of Reynolds Electrical & Engineadng Co.,Inc. (REECo) and Dr. Britt Jacobson of the Reno, Nevada branch of the DesertResearch Institute for their helpful reviews of this manuscript. We also wish to thankPatricia Herrin and Yvonne Lewis of the REECo Information Products Section for theirinvaluable help in typing, proofreading, and assembling this manuscript.

1_ISTRIBUTION OF THIS DOCUMENT IS UNL|_41TEO= i m _%-rn

f lVl b i trr

1.0 INTRODUCTION

The Dupuit (1863) approximation has been used by _roundwater modelers andhydrologists in a variety of situations (Hantush, 1967; Bear and Verruijt, 1987). ltworks well when the slope of the water table in a phreatic aquifer is very small.Slopes varying from a one-foot (ft) dse in a 1,000-foot run to a ten-foot rise in 1,000-foot run are quite common. Besides the assumption of shallow slope over the domainof definition (i.e., that portion in the x-y plane over which the hydraulic head H isdefined) D, the assumption of a thin capillary fringe relative to the thickness of theentire vadose zone is made. For example, a capillary fringesixfeet thick relativeto atotalvadose zone thicknessof 900 feet, qualifies. Also,withthe assumptionofshallowslopes alongthe water table, the assumptionof horizontalequ!ipotentialsismade. Bear and Verruijt (1987) give a good introductorydiscussionof the Dupuit orDupuit-Forcheimer(D-F) conceptson pages45-52 of their book.

The phreatic surface or water table elevation head H, neglecting the capillary fringepressure (suction or tension), is assumed to be a surface of constant atmosphericpressure. Using the above mentioned concepts, Lindstrom et al. (199;2)derived atwo-dimensional Dupuit-Forcheimer approximation to the water table under theRadioactive Waste Management Site (RWMS) in Area 5 at the Nevada Test Site(NTS). Figure 1 shows the location of NTS in Nevada.

2.0 GOVERNING EQUATION

2.1 Assumptions

lt is assumed that:

1. The slope of the phreatic surface is much smaller than 0.1 ft/ft.2. The bottom of the aquifer is flat and Impermeable.3. The recharge (infiltration)-exfiltration (evaporation) rate qrchg(ft/day) can be

represented by a bilinear expression.4. The aquifer is homogenous and isotropic with respect to hydraulic

conductivity K (ft/day).

The governing equationcan bewritten as "

o_2H2 o_2H2 (1)

+ _2 =-2q=hg/K, ,I o_X2

which is a Poisson (elliptic) partial differential equation (pde)in the square of the watertable elevation head H (ft) relative to mean sea level as the reference datum.

L i

Figure 1. Map of Nevada showingthe locationof the Nevada Test Site.

3

The domainof definitionD togetherwith its boundary1"is sketched in Figure2. Thepoint(xo, Y0) in D is calledthe expansionpointfor the approximatesolution. The fourouter heavy dots'representthe locationsof the four calibrationpointsfor representingthe infiltration/evaporationfunctionqrchgas a bilinearform.

aH aH components)Since we know neither H (x,y) nor the normal derivative ( _ andat every point along 1",we cannot solve Equation 1 in the classical boundary valuesense via eigenfunctlon expansion methods. We introduce an approximate solution toH(x,y) that will require knowing H at eight calibration points together with representingqrchgas a bilinear form. The eight calibration well locations are also shown In Figure2.

3.0 APPROXIMATE SOLUTION

Lindstromet al. (1992) give the detailsof the approximatesolutionmethod. A briefsummary of their work follows.

The superpositionprincipleallowswriting

H2 (x,y) = H_.(x,y)+ H_)(x,y) , (2)

whereH_ satisflestheLaplaceproblem

a2 H_a2 H_. + = o,for ali (x,y)in O (3)

ax2 ay2

and

7

H_ = ,T_,AI,(Z)i (x,y) . (4)I=0

Span {d>o, 4>1, ... 4>7}forms a basis for a real linear vector sub space of dimensioneight of the real linear vector space (dimension 16) of ali multivariate polynomials intwo variables of degree three or less. The eight d:)'sare defined in Appendix B. Theeights Al's are found as the elements of the solution vector of the linear system

X A = H_ (5)" ' DISCLAIMER

This report was prepared as an account of work sponsored by an agency of the UnitM StatesGovernment. Neither the United States Government nor any agency thereof, nor any of their

employees, makes any warranty, express or implied, or assumes any legal liability o,' responsi-bility for the accuracy, completeness, or usefulness of any information, apparatus, product, orprocess disclose,d, or rcprc_nts that its use would not infringe privately owned rights. Refer-once heroin to any specific commercial product, process, or service by trade r_amc, trademark,manufacturer, or otherwise does not necessarily constitute or imply its ¢ndo,rr,cmcnt, recom-mendation, or favoring by the United States Govcrnrnent or any agency there,of. The views

and opinions of authors exprcssc, d heroin do not necessarily state or reflect those of theUnited Sta_cs Liovernmcm or any ag©hey i.hc_=ui'.

i,

5

wherethe 8x8 dense"designmatrix"X hasthe structure

• o(X_,y_)_(x_,y_)._,2(x_,y_)... '_7(×_,y_)

x = , (6)

L_'o(Xs,ys)_',(xs,ys) .... d>7(xs,Ys)J

the "known"vectorH_ hasthe structure

H2 = (H2(x,,yl),H2(x2,Y2),...H2(xs,ys))T , (7)

and the "unknown"vectorA of coefficientshas the structure

A = (AO,A1,A2,...A 7)T . (8)

H;_satisfiesthe Poissonproblem',

a2 H2 -2 qrchg (x,y)°_2H2 + = (9)ohX2 o_y2 K

with

q_,Q(x,y)= I_o+13,(x-xo)+ 1_2(y-yo)133(x-xo)(y-yo), (_o)

(Box,Hunter,and Hunter,1978;Prenter,1975)and

7

H_ = __,gi _i (x,y) + y (x,y) , (11)1=0

where

7(x,Y) = -(Y-Yo Po+_1 (X-Xo)+ (Y-Yo) + (X-Xo)(Y-Yo) . (12)K

t

Since the eight calibrationwell elevations have already been used in estimating A, wefollow the classical Nle approach and set H_ = 0 at each of the eight calibration welllocations. We then have the linear system

XB = -y , (13)

where X is defined in Equation 6 and y is defined as

y = ,, , , , , , , Lytxl,y,j,ytx2,y2j,yLx3,y3j,...yLxs,ys))T .(14)

With ali the Ai and Bi coefficients thusly known, Equation 2 is used to find H2 (x,y) forany (x,y) in D. Actually, the positive square root is chosen, since it is conventional todiscuss H rather than H2.

A word of warning is in order. X can be singular for certain well patterns (geometries).The choice of well patterns is thus an important issue. We are currently researchingthe "optimaldesign"well patternproblembased on the above outlinedD-F model.The D'Arcy specificdischargecomponentsqxand qy are calculatedat each point(x,y)in D by usingthe formulas

_H_.+o_H_

qx =_K;)I'I =_K-_ - a---x-"_)x 2 H (ft/day) (15)

and

qy = -K ;)14 = _K _)y ony (ft/day) (16)ony 2 H "

4.0 APPLICATION OF THE D-F MODEL

A summary of the currently believed basin-wide geology and hydrology of the 15-milediameter closed basin called Frenchman Flat is given in Lindstrom et al. (1992). Theyalso give a summary of the precipitation pattern as documented in Winograd andThordarson (1975). Suffice it to say that the region is very arid (<5 inches totalprecipitation per year). The vadose zone is about 240 meters (780 ft) thick and iscomprised of gravelly, sandy alluvium. The alluvium is largely derived from tuffaceousrock in the surrounding mountains.

il

4.10rialna! Calibration Drill Holes/Water Wells

The primarymissionof the NTS over the past 40 years has been the testingofnuclearweapons. Since the end of abovegroundtesting in 1962, nucleardeviceshave been detonatedwell belowthe groundsurface. This has requiredthe drillingoflarge diameter,deep boreholes.Unfortunately, thoughthere have been five (Neagle,1992) undergroundnucleardevicesdetonatedat or near the northend of FrenchmanFlat (and aboutten other mixedwaterwell and test boreholesscatteredthroughoutFrenchmanFlat), very scantygeologyand hydrologydata existfor ali these drillholes.

Accuratemeasurementsof the water table depth are absent from most records.Some water wells, as well as post-testdrillbackholes,exist in FrenchmanFlat. Figure3 showsa sketchof the locationof severalof the drill holes/waterwells closestto theRWMS. Water table elevationsare also shown,but these are more than likelyestimated. A crescent-shapedregioncan be envisionedforD (Figure 3). The designmatrixX was found to be nonsingularand mildlyill-conditionedusingthe geometriclocationsof the eightcalibrationdrill holes/wellsgiven in Table 1. Our own linearsystemsolverusingGauss eliminationwith full pivotingand iterativeimprovement(Young and Gregory, 1973), was used to find the _ and B Vectorsof coefficients.Usingthese coefficientsand assumingqrcho= 0 over D, the estimatedwater tableunderRWMS is shown as a contourplot in-Figure4. Figure4 clearlyshowsthe waterat the water table to be flowingin an east-southeasterlydirectionwith specificdischargecomponentsqx= 0.075 _day and qv = -0.026 Wt/day,based on a K value of30.0 Wt/day(Daffern et al., 1990). This places the water flowtowardsthe center of theplaya in Frenchman Flat.

4.2 Monte Carlo Estimated Water Table Under the RWMS

The eightcalibration1drill hole/waterwell elevationsused ingeneratingthe estimatedwater table under RWMS (simulation2, above) were gathered from variousU.S.GeologicalSurvey and LawrenceLivermoreNationalLaboratorymaps and drillinglogsover abouta 20-year time span. Sincethe primaryconcernof the drillerswasgenerallynot in loggingthe depthto the water table, someof the water tableelevationsmay be in considerableerror. Also, the regionof calibrationis an arearoughly25,000 feet in diameter, lt is highlyunlikelythat there are no interveninggeologicstructuresin the water table region. The D-F model used here assumesthatthe water table region has homogeneousand areallyisotropic,hydrologicproperties.This is a very big assumptionand one which is probablynottrue on such a grandscale as 25,000 feet. Therefore,the followinguncertaintyanalysisis in order.

A rather exhaustiveliteratureand personaltelephone interviewsurvey yieldsthecurrentlybelievedwater table elevationsinthe originaleight drillholes/waterwellsused for calibration. Table 2 summarizesthis data. Figure5 showsthe estimatedwater table under RWMS based uponthese latest findings.

i

2°4_778,0OO Uel 1==

776,000 20412'

_4,000

772.000 2,407,_r ,,, 2,383'

7"/0.000 I _2'386'T

RY_tS

Area 5I_ ;_1.000 I . ,r" 2,403.4' 2,360.6'

,,,,...

1::: ?S4,0000 •Z 2,_7,3'

762.000

UeS¢

_0,000 I=12,410'

_I.000

756,000

U_n

7S4,0002,401'

lm TestWell2,3e_ I=!

7SO,O00 2,396'

7_1.000 WegSDB

2,403'

740.000

Easting

Figure 3. Location of eight calibrationdrillholes/waterwellsusedinthe D-F model.9

LI

I0

t,

11

12

i i , !I

' ° "_II....1I ,,_i M ;_Jl_-

o_ -_| ' __v_ _ B Q

Z Z Z Z Z Z ;Z

_ ° o _ _

_z.o9 "-'_- ' ._g

L_

'" ._---- 2408 w '_

-! |o

w u_ _

_- _- 24 O? ..... -

_............. _ n_

,., - 2406 .__ \ I_

Z Z Z Z Z Z Or_

r_

13

In comparingTables 1 and 2, we observe a mean elevation increase of 27 feet in drillhole UE5F, which is a majorelevationchange. The other changesare less than orequal to two feel Observethe major uncertaintyvalue associatedwith Well 1. Verypoordrillinglogsexist for thiswater weil, hence, the majoruncertaintyin its meanelevation.

In comparingTables 1 and 2, we observe the substitutionof drill hole RNM2S forUE5N. The measured water table elevationat UE5N differeddramaticallyfromthoseof nearbywells and drill holes. This suggeststhat the measurement mightbesubstantiallyin error or _ffectedby notablydifferen*_geology. Additionally,the watertable elevationat drillhole UE5F increasedby 27 feet, whichis a majorchange. Noother differenceexceeded two feet.

A Monte Carlo analysiswas used to assessthe impactof water table elevationuncertainties. Probabilitydistributions,detailedbelow,were placed on each watertable elevation. Usingthe random numbergenerator from the statisticspackageMINITAB release 7.1, 3,000 realizationsof the simulatedelevationswere run. Foreach realization,water table elevationswithinRWMS were estimatedusingtheapproximationdescribedhere with constantrechargeof 0.0 ft/day.

Random water table elevations were assumed statisticallyindependent. Ali elevationsexcept Well 5b were assumed normallydistributed,with the mean equal to theobservedwater table elevationand standarddeviationof one-half the statederror.Well 1 was drilledfor reasons;otherthan water table characterization,and hence hasa large associateduncertainty. Due to asymmetricuncertainty,the water tableelevationat Well 5b was assumedto be Iognormallydistributed.The 2.5th and 97.5thpercentilesof the Iognorrnaldistributionwere set equal, respectively,to the lowerandupper limits.

At each point in the RWMS the sample mean, 2.5th percentile,and 97.5th percentilewere calculatedfromthe realizations. These are displayed in Figures6, 7, and 8,respectively. Figure6, in whilchthe samplemean is displayed,is similarto Figure 5,in which no uncertaintyis accountedfor. Hey,ce,despite model nonlinearitywhichmighthave lead to complexerror propagation,uncertaintiesof the magnitudesconsideredhere have little Impact on mean estimatedelevation. Figures7 and 8displaypercentilesof the samplingdistribution.Unlikethe piezometrichead itself,there is no reasonthese sud:acesshouldobey Equa*.ion1.

The roughnessof Figures7 and 8 can be attributedto surface "flatness."That is, thecurvatureof the surface is so small that, comparatively,Monte Carlo samplingerror isnonnegligible. This would not hold if the surfacecurved more sharply,asdemonstratedby Monte Carlo studies (not reported here) usingelevations resulting inless flat surfaces. Standarddeviationof samplingerror is inverselyproportionalto thesquare root of the numberof realizations. Hence, the numberof realizationsrequiredfor negligiblesamplingerror is impracticallylarge.

i4

rl, III' _I' irq111rl' , rlll Ii'q,ll, ,iIi_i ellqllV '_ 'ii pl llep 11, "IIIII' l,I_ll r""lill "'II" ' tlqlllS'Ir' ' il_l""' ,', ........ qf " ',q,' 'qql,rl , ' fql....

=

- 16

i

- Jl_.-= ,_ _

= _H _=..

_ - oa |o _.=, >,

=. =. a., I '_o . ! |

k,.,,,z z z z ," z z

°_" _ ..,.

uJ N

°- -_p..o - 0

.-/ Q

F./I- o

//

_- _ E// -o8

o.i::

o

:3

o LE

§_ , \ , , , , r_Z Z Z Z Z Z 0

p_ r_

17

=

Figures7 and 8, taken together, providepoint-by-pointapproximate 95 percentconfidenceintervalsfor estimatedelevation. These are flat enoughto renderidentificationof gradient directionimpossible. However,they do show the magnitudeof the gradient is small. Hence, any flow would be slow,thoughof unknowndirection.Figure8 somewhatcountedntuitivelyhas a localminimum. To understandwhy, letE(x,y) denote the estimatedelevationat a point (x,y) and let S(x,y) be its estimatedstandard deviation. Treatingestimatesas normallydistributed,though not strictlymathematicallycorrect, facilitatesexposition.

Under this assumption,an approximate 95 percent confidence Interval for E(x,y), thetrue mean is E(x,y) + 2 S(x,y). In general, E(x,y) Increasesin X (water tableelevationsare greater to the east) but S(x,y) decreases in X (uncertaintyconcerningwater table elevationsis greaterto the west, where there are fewer nearbywells/drillholes). Hence, the upperconfidencelimit [E(x,y) + 2 S(x,y)] is the sum of anincreasingand decreasing function, lt is not surprisingsuch a sum has a localminimum. The lowerconfidence limit[E(x,y) - 2 S(x,y)] is the slim of two functionsIncreasingin X and exhibitsno localextrema.

5.0 INFILTRATION/EVAPORATION

5.1 Infiltration (direct surface recharge}

So far qrchghas been set to zero in ali simulations. The reason for this is that wehave been unable to find any publishedwork on directrechargeestimationconductednear the RWMS in Area 5 of the NTS. Chfoddemass balance methodsfor estimatingqrchgdo existand have been used by Stone (1985) in estimatingthe effect of varioussurfacecoveringsover calcariousalluviumin New Mexicoand Western Australia.They have also been used by Scanlon (1991)in estimatingrecharge in westernTexasdesert soils. However, the assumptionson whichthe chloridemass balance andisotopicchlorine methods rest may not be valid for the extremely dry high desertconditions of Frenchman Flat, Nevada (Scanlon, 1991).

In the interest of seeing the effects of a nonzero direct recharge, two additional watertable simulations have been made. In the first case qrchgwas set at 0.003 ft/day.This is equivalent to slightly over one foot of water per year. Figure 9 shows the

result of setting qrchgto +0.003 ft/day and estimating the water table using thecalibration data listed in Table 2.

5.2 Evaporation (direct surface exflltratlon)

In the second case qrchgwas set at -0.003 ft/day which, again, is equivalentto slightlyoverone foot of water evaporatedper year. Figure 10 showsthis simulation. Inthiscase the current calibrationwell data (Table 2) were used.

18

O_

;n

° c0z . z z z z z z

' C

I I I I I

__.._.___"_''-'-----'_ 2408 _"'

_ W

o _ _",..,_,

o _'_*_'_ 2407 _ o

_ w_m

_ W

0

,., - 2406 _ '" iT

Z Z Z Z Z Z 0

_ ,

19

i , i, ,

V_6

.. . ][I c:)-

I _ _ _

"-__';, , J 8o, =_z_|,,] ®• .>_

Z Z Z Z Z Z Z,- CO CO L4

_Z_og --" (,_

0

"' _ 2408 "'

°$

COCO

"-, ..........

'"""

-o -_'""'_'_' 2407 _ o

\\ w :3

8-,_ I I I I t lr_z' z z z z zp,,,,, _ p,,

o o o _ _ ._ !_ _ _ _ _C- _ CO COp_ I_ •

2O

Comparisonof Rgure 5, where the estimatedwater table has been computedusing

qrchg.=0, withFigure 9 (qrcho= +0.003 ft/day) and Figure10 (qrchg= -0.003 ft/day)clearly showsthat infiltration-causesa slightelevationincrease,whereas evaporationcauses a slightelevationdecrease inthe water table surface. Since qrchg IS realisticbut small,the effect is to producea small perturbation;hence Figures9 rind 10 arevery similar. Since the scale used here is of the orderof 25,000 feet and since theRWMS is off to the northwesternside of the regionof model calibration,these figuresshowthat very littleeffect on the water table elevationsis expected under either directrechargeor directevaporation. Indeed, usingqrchg= 0.003 ft/day, obtainedfrom theWinograd and Thordarson(1985) precipitationmap, Is probablythree to four ordersofmagnitudetoo high. Muchsmallervalues, eitherpositiveor negative,of qrchgwillhave a much smaller effect on an already smalleffect. Thus, lt is our opinionthatdirect rechargeand/orevaporationmay be neglected5nany further use of the two-dimensionalwater table estimatormodel DUP3DSP (Lindstromet al. 1992).

6.0 CONCLUSIONS AND FUTURE WORK

The feasibility of usingthe Dupuit-Forcheimermodelingassumptionsfor estimationofthe water table underthe RWMS in Area 5 of the NTS has been clearly establishedinthis work. However, it remainsto be seen via directvalidationboreholesaroundtheRWMS borderthat the modelspredictionsare accurate, The belief in the hypothesisthat the water table under the RWMS is "flatterthan a pancake"is stillstrong(Cox,1991). Unfortunatelythe state of the art in deep boreholedrillingIs suchthat the U.S.Department of Energy rightfullyrefusesto allow any surface-to-watertable drillingwithinthe boundariesof the RWMS itself. The potentialfor creatingdirect leaks ordirecttransportpaths from the surface to the water table if a "swisscheese"typedrillingoperationwere to be allowed is too great. Therefore, lt is doubtfulthat a directvalidationof the accuracyof the D-F model discussedhere in for the RWMS will beachieved. The D-F modeldiscussedhereincan, however,be validatedusingsurface-to-water table boreholedata aroundthe boundariesof the RWMS. The three initialRWMS boundary characterizationsurface-to-watertable boreholesare currentlybeingdrilled. These three water table elevationswillbe used as a directcheck on the"degree of goodness"of the estimatedwater table under the RWMS.

As was stated earlier inthe warningaboutthe potentiallysingular natureof designmatrixX if a *bad" geometricwell patternis chosen,work is currentlyunderway todiscoverthe propertiesof X if 1:hewell pattern is perturbedaway from thesedesigns.

Effortsare currentlyunderway,to conduct a jointwell patterndesign researchprojectwith Desert Research Institutestaff hydrologistswith an eye towardusingthe D-Fmodel to aid in choosingan "optimal"design. For example, the classicaldesignsbased upon D-optimality(dete_rminant)yieldingminimumcovarlance matrixcomponents,or K-optimality(spectralconditionnumber)yieldingwell-conditionednessof the design matrixor some weightedcombinationof both D and K optimalitycriteriawill be considered. Simpleexlensionof the D-F model from usingeightcalibration

=

21

" " ' " !lr IIq '11j ' I, ' "11 ,,, Ii , ,_ qlr I_I,"_IIIIr ' Hl '__JI" ..... Ilrl

well data sets (the PUrelyInterpolatorycase) to nine or morecalibrationwell data setsand a variational(least squares)conceptalso gives riseto the same type of optimalitycriterion.

In ali the above cases the D-F model DUP3D8P is used essentially in the "inversemode." That is, we are solvingan inverse or parameterestimationproblem. Anotherinverse mode use of the D-F model is that of locatinghiddengeologicstructures. Forexample, suppose that _ and B have been determined from eightcalibrationwell logsand that qrchghas been defined at the four outer calibrationpoints. Then, if anabsurdlylarge moundor depressionin the water table is predicted,lt probablyIndicatesthe assumptionsof the modelsare beingviolated. This in turn meanshiddengeologicor hydrologicstructures(e,g., faults, fractures,springs,drains)arepresent. Virtuallyno use of the D-F model has been made to date with respectto thislast concept. However, as moredata becomeavailablewe plan to use the model to"prospect"for hiddenstruct_Jres.

22

J

7.0 LITERATURE CITED

Bear, J. and A. Verruljt (1987)Modelinggroundwater flow and pollution:theoryand appilcatlonsof transportInporousmedia. Reldel, Dordrecht,414 pp.

Box, G.E.P,, W. G. Hunter, and J. S. Hunter(1978)Statlstlcsfor Experimenters: An introductionto deslgn, data analysis,and modelbuilding. Wiley, New York, 653 pp,

Carslaw, H. S. and J. C. Jaeger (1957)Conductionof heat in solids. OxfordPress, 510 pp.

Case, C. J. Davis, R. French,and S. Raker (1984)Site characterizationinconnectionwith the low level RadioactiveWasteManagementSite in Area 5 of the Nevada Test Site, Nye County,Nevada - FinalReport,USDOE/NV/10163-13. ReynoldsElectrical& EngineeringCo., Inc., P.O.Box 98521, Las Vegas, NV 89193-8521.

Cox, Warren (1991)Pdvatecommunication

Daffem, D. D., L L Ebeling,and W. B. Cox (1990)Unsaturatedzone characterizationof the Area 5 Radioe.ctiveWaste ManagementSite: Phase I. Preliminarylaboratorystudiesforthe determinationof soilmoisturecharacteristiccurves and unsaturatedhydraulicconductivitles.U.S.Departmentof Energy Report, USDOE/NV/10630-3. ReynoldsElectrical&EngineeringCo., Inc., P.O. Box98521, Las Vegas, NV 89193-8521.

Dozier, B. L., and S. E. Rawlinson(1991)Conceptualmodel for the geologyin Area 5. U.S. Departmentof Energy Report,USDOE/NV/10630-14. ReynoldsElectrical & EngineeringCo., Inc., P.O. Box98521, Las Vegas, NV 89193-8521.

Dupuit,J. (1863)Etudestheoriqueset pratiquessur le movementdes eaux dane les canauxdecouvertset _ travers les terrainsperm _ables_ Dunod, Pads.

Hantush,M.S. (1967)Growthand decay of groundwatermoundsIn responseto uniformpercolation.Water Res. Res 3:227-234.

23

Lindstrom,F. T., L E. Barker, D. E. Cawlfleld, D. D. Daffern, B. L. Dozier, D. F. Emer,and W. R. Strong(1992)

Estimatingthe water table underthe RadioactiveWaste ManagementSite inArea 5 of the Nevada Test Site: the Dupu!t-Forcheimerapproximation, tj.S.Department of Energy Report,_USDOE/NV/10630-24.ReynoldsElectrical&EngineeringCo., Inc., P.O. Box98t;21, Las Vegas, NV 89193-8521.

Neagle, C. C. (1992)Personalcommunication.

Prenter, P. M. (1975)Splines and variational methods. Wiley, New York, 3_3 pp.

Wlnograd, !. J. andW. Thordar_on(1975)Hydrogeologicand hyd;'ochemicalframework, South-CentralGreat Basin,Nevada-Califorr_ia,with special referenceto the Nevada Test Site: hydrologyofnucleartest sites. USGS prof. paper #712-C, U.S. Govt. PrintingOffice,Washington,DC.

Young, D. M., and R. T. Gregory (1973)A Survey of NumericalMathematics: Volume I1. Addlson-Wesley,Reading, MA.

24

,6I,

8.0 NOMENCLATURE

i. DependentVariables._.ymbols Meanlna

H(x,y) Watertableelevationabovemean (ft)asthedatum

HL(X,y) Laplacewatertableelevatlon (ft)Hp(x,y) Polssonwatertableelevatlon (ft)

II. IndependentVariablesSymbols. Meanina Units,

x EastingNevadaCoordinate (ft)y NorthingNevadaCoordinate (ft)

III. LiquidFluxesSymbols Meanina Units

qx EastingDarcyfluxif positiveand (ft/day)westingflux if negative

qy NorthlngDarcyflux if positiveand (ft/day)southingflux If negative

, ,

IV. Fluidan_,PorousMediumPropertiesSymbols, Meaning _,Unlts

K : Saturatedhydraulicconductivity(20°C) (ft/day)i

V. Infiltrationof EvaporationRate.Symbols Meanina Unit__.._s

qrchg Infiltration(+)or evaporation(-) rate (ft3/ft2-day)reflectiveof verticalhydraulicconductivitypropertyof porousmedium

25

,,, ,, .... ql=, ,=' ,,rll_I_ , rll ' _;1_

9.0 APPENDIX A

The eight basic functions,ali of whichsatisfy the Laplace partialdifferential equation,

_----_+ _ = 0, and used In the D-F modelDUP3DSP, are defined as:o_x2 o_y2

Oo (x,y) = 1.0 (A.1)

_1 (x,y) = x- x0 _A.2)

• _(x,y)=(x-Xo)2-(y-ro)_ (A,3)

_3(x,y ) = (X-X0)((X-X0)2-3(y-y0) 2) (A,4)

_4(x,Y) = Y-Y0 (A.5)

,_,(x,y)=(X-Xo)(y-yo) lA,61

•,/,,,/:/,-,olI/x-xo/'' /-_(Y-Yo) 2 (A.7)

•,(x,y)=(X-Xo)(y-yo)((X-Xo)=-(y-yo)_) lA.8/

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