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VOL. 80, NO. 35 JOURNAL OF GEOPHYSICAL RESEARCH DECEMBER 10, 1975

A Theoretical Study of Heat Extraction From AquifersWith Uniform Regional Flow

A. C. GRING^RTEN ^ND J.P. S^UTBureaude RecherchesGdologiquest MiniSres,Sert;iceGdologique ational, Orleans,France

A mathematicalmodel is presented or investigatinghe non-steadystate emperaturebehaviorof apumpedaquiferduring reinjection f a fluid at a temperature ifferent rom that of the nativewater.Resultsare presentedn terms of dimensionlessarameters nd shouldbe helpful in the designofgeothermal pace-heatingrojects.Applicationso practicalcases re also ncluded.

INTRODUCTION MATHEMATICAL MODEL

The use of geothermal energy for space heating and theproduction of electricity s currently imited to a small numberof regionsaround the world with exceptionallyhigh geother-mal gradients Krugerand Otte, 1973].Geothermal space eat-ing or air conditioningcan also be used, however, n regionswith normal temperature radientsf it is possible nd econom-ically feasible o drill a well deep enough o reach an aquiferwith adequate water temperature.Aquifers at depths of theorder of 2000 m have already been used for space heating[Maugis, 1971]and were ound suitablewith the currentspace-heating echnology.Other heatingprocesseshouldpermit heuseof much shallower formations. Free surface aquiferscanalso be used for air cooling, or even for spaceheating, bymeans of heat pumps. With the energy crisisand increasingfuel prices,aquifersmay thus becomea very important sourceof heat energy.Disposalof the heat-depleted ater, however,may often bea problem because f the mineralcontents, he temperature,orthe volumes nvolved, all of which would prevent the waterfrom being wasted into sewage ines. One solution consistsofreinjecting he water back into the aquifer. This proceduremaintains he reservoirpressure, reventssubsidence, nd in-sures n indefinitesupplyof water. It alsopermits he recoveryof the heat contained in the rock, but as a result it creates azone of injectedwater around each njectionwell at a differenttemperature rom that of the native water. These zoneswillgrow with time and will eventually each he productionwells.After breakthroughoccurs, he water temperature s no longerconstantat the production wells, and this may reducedras-tically the efficiency f the whole operation.It is thus important to design such a system n order toprevent injected water breakthrough before a specified imeand to maintain the temperaturevariations at the productionwells after breakthroughwithin reasonableimits. Although afew authors have considereda similar problem for a singlerecharging-discharging ell pair [Houpeurt et al., 1965; La-gardeand Maugis, 1966],no general heoryhasbeenpublishedto date.

It is the purposeof this paper to developa mathematicalmodel for investigating he non-steady state temperaturebe-havior of production wells during the reinjectionof heat-de-pleted water into aquiferswith uniform regional low. Resultsare presented n terms of dimensionless arametersand shouldbe helpful for the designof such systems.Copyright 1975 by the AmericanGeophysicalUnion.

Our discussionwill be based on an analytical model.A rectilinearsystem s placedsuch hat the x, y plane coin-cides with the midplane of the aquifer. A few simplifyingassumptionshat are usual n this type of problemare made.These are listed below and will be discussed later.1. The aquifer s assumedo be horizontaland of uniformthickness . The cap rock and the bedrock,aboveand belowthe aquifer, are impermeableo flow and of infiniteextent nthe vertical direction. The system s thus symmetricalwithrespect o the midplane of the aquifer.2. Flow is assumed o be steady,since he duration of thetransient low period s short n comparisonwith the lengthoftime required o reach hermalequilibrium.The total injectionrate Q is constantand equal o the total production ate. Allwells fully penetrate he aquifer. The flow field due to therecharging nd discharging ells s superimposedn a naturalsystem f arealparallel low of Darcy velocityV0,whoseorien-tation is at an angle a with the x axis, as measured ounter-

clockwise from that axis.3. Initially, the water and rock in the aquifer and the cap

rock and bedrock are at the same emperatureTo. (Actually,the cap rock and bedrock emperatures re neither denticalnor uniform nitially, because f the geothermal radient,butthis fact can be neglected,sincewe only consider emperatureperturbations.) t time t = 0 the temperature f the injectedwater s setequal o Tt and is maintained onstant hereafter.Thermalequilibrium s supposedo takeplace nstantaneouslybetween he water and the rock in the aquifer, so that any-where n the aquifer he rockhas he same emperature s hesurrounding luid.4. In the aquifer the effectof the thermal conductivitysneglectedn the horizontaldirection high Pecletnumber).Furthermore,he aquifer s assumedo be thin enough hat hetemperatures alwaysuniform n theverticaldirectioninfinitevertical thermal conductivity).5. In the cap rock and the bedrockhe effectof the hori-zontal hermalconductivitys alsoneglected,nd the verticalthermal conductivity s finite. The temperatureremains con-stant and equal to the initial temperature t infinity in thevertical direction.

6. There is no heat transfer coefficientbetween he aquiferand the cap rock and bedrock, nd the aquifer emperaturesassumedo be equal o the caprock temperature t the contactbetween he cap rock and the aquifer (z = h/2).7. The product of the densityand the heat capacity orboth the water and the rock, and he caprock vertical hermalconductivity,are constant. Differences n viscositybetween4956

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GRINGARTENNDAUTY:EATXTRACTIONROMQUIFERSSTREAMLINES

+2 YINJECTION

Fig. . Visualizationf streamhannel.

4957

injectedaterndativeaterreeglected.no-mixing'conditionndistonlikeisplacementressumednhequi-fer.Thebovessumptionsllowachtreamhanneleavingparticularnjectionello ereatedndependently.heem-peratureorrespondingogiventreamhannel,oundedytwo treamlinesb ndb db,an e escribedy one-dimensionalunction(S,) n he quifernd y wo-dimensionalunctiona(S,, ) n heapockrhe ed-rock,beinghetreamhannelrearomheorrespondinginjectionellFigure).The ifferentialquationoverninghe ateremperatureT?withinstreamhannelsobtainedywritingheatbalancen n lementf treamhanneletweenhereasand+dSromhenjectionellFigure). heerivationis imilaro hatublishedy auwerier1955]nd arslawandaeger1959]or ne-dimensionalasslow.heesults

OTto'(S, )hpC3T(S')_-- at asor(S,, t) (1)= K 0-'--------

cD -

0.4.: 0.3

0.2

INJECTIONF OLDATER ;NJECTIONF(SPACEEATING) ---''C-'.

BOO0o NORMALNDNVERTEDEVEN, ISOLATEDNVERTEDIVESPO'A FZVE SDOffi STAGGEREDINEDRIVE DIRECTLINE DRIVE SKEWEDOURPOT

2 3 4 6 8 , 0.3 0.4 0.6 0.8 1VISCOSITYATIO,o/i

Fig.. Effectf iscosityation reakthroughimeor ariouswellatternsafterataeportedyCraig1971]).wheresheatef lowithinhetreamhannel,oC:sthe atereatapacity,ACA- PpwCw3-1 tp)pmCms heaquifereatapacitypmC,,einghatf he atrixnd>beinghequiferorosity),ndas heapockhermalconductivity.The apockemperaturesgovernedy he eaton-ductionequation , t) _ p__C_T(S,t) h(s,z _ , '- (2)Off -- K Ot

HEATROMAP OCKYCONDUCTIONTRhdS'KR-'-'zz -

HEATRANSPORTED HEALEAVING HEELEMENT hdSdtwCw,S+dS)dthdWTwS+dS)--

ADIABATICHEATROMEDROCKYCONDUCTIONTRdS.KR -- h,Z Z = -

HEATRANSPORTEDBY THE WATERENTERINGHEELEMENT,, owCwuS) t dwTS

Fig.. Heatalancen streamhannellement.

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4958 GRtNGARTENND AUTY:EAT XTRACTIONROM QUIFERSwhere DRCR s the cap rock heat capacity.The temperaturesmust also satisfy the following conditions:

TR(S, , t) = T?(S, t) = To t < hS/q (3)ahead of the hydrodynamic ront,

T?(0, t) = To t < 0T?(0, t) = T t > 0

(4)

T?(S, t) = T?(S, h/2, t) ' t andS (5)!irnRi(S,, t) = To (6)

Thesimultaneousolutionf (1) and 2), subjecto condi-tions (3)-(6), is easily obtained from Lauwerier's one-dimen-sionalsolution.The result or the water emperaturewithin thestream channel is obtained asTo -- T,.'(S, t)

To -- Ti= erfc PwCw)22 -1/2KRpRCR t (7)

In the case n which here s no heatexchange y conductionbetween he aquifer and the cap rock and bedrock Kn = 0),(7) reduces o a step function'ToTo(S,t)- hSTo T = u t =0-To -- To(S, )

To ri

t < (pC/pwC)(hS/q)(8)u t p.4C.4p, o = 1

t >

whichcorrespondso a pistonlikehermal ront displacementwith a velocity qual o pC/pACA imes hat of the hydro-dynamic ront. The water temperatures Toaheadof the ther-mal front and T, behind t. On the other hand, f Kn is differentfrom zero, there is a transition zone behind the thermal frontin which the temperaturevaries from To o Tt.If the streamlineseaving recharge ellreach dischargewell, the water temperature at that dischargewell within anelementarystreamchannel s obtainedby substituting he totalstream channel area between he two wells, Smx, or S in (7).(Except in a few simple cases,Smxmust be computednumer-ically.)The resultcan be written in termsof dimensionlessarame-ters as

(9)where

X = (poCopACa/KapnCR)(Qh/D') (10)to = (pwCo/paCa)(Qt/D'h) (11)

D is some haracteristicength, ndd(Smax/D)/d(p/Q)s thedimensionlessderivative of the total stream channel area withrespecto the stream unction.Heat transfer etweenhe aqui-fer and the cap rock or bedrock s negligibleor X > 10 .The water emperature t therecharge ellat a given ime ois thenobtained y integratinghe right-hand ideof (9) withrespect to p for all stream channels that have reached thedischargewell at to:

x

'-: 0 9

- 0.8

-'" 0.7

Zl,I

'----0 '

o 0.5

:.=r 0.4

x , 0.3o

0.2

x. 0.1

18S

135

90 0 ,--ANGLE OF AREAL

0 I I I i II I I I i I { i I i I I0 10 z 10QD Q/bOY0

Fig. 4. Maximumwater emperaturet theproduction ellof a doublet sa function f Q/hDVoand heangle f areaflow.

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GRINGARTENAND $AUTY: HEAT EXTRACTIONFROM AQUIFERS 4959

1,0

: %CPACOh'R C.

1503OO

10 1 10 10 10 10 (PwCwJ0 )2 PACA2hDII'IENSIONLESSIfIE,X(td-':nC T (t- ----)c rFig. 5. Dimensionlessemperatureversusdimensionlessime at the productionwell of a doublet n a deep confinedaquifer (Q/ > 104) for various values of X 'from 0 to 300.

To- To(t) ;d(Sma,/D)-- e fc,- (--tffi-

= To[/(), X, to] (12)where OP)depends n the stream unction.This result s perfectlygeneral.Tw can then be evaluated fthe flow field is known.

DISCUSSIONEquation 12) will also yield satisfactoryesults f not all ofthe rather estrictive ssumptionsf the mathematical odelare satisfied.The total injection rate may change f the duration of thechange s short in comparisonwith the total time length of

interest and if the change does not modify drastically thestreamchannel: he injection rate to be used n (10) and (11) isthen equal to the total volume of injectedwater dividedby thetotal injection time.On the other hand, a variable injection temperaturecan betaken into account exactly by superposition. f, for instance,the injection emperature s Tt for time 0 to time t, Tt. or t tot.,etc., the temperatureof the water at the productionwell attime t is given byTo - Tw(t) = (To - rt,)rwo[f(P), X, tD]

+ (Ttx - Tt.)TwD[fOP),, tD -- tin]+ (Tt.- Tt3)TwD[fOP),, tD -- tD.]+ "' (13)

and for a continuous njection temperature function Tt(t),' dr(r)oo[/(P),D- D]r(14)o(t) To dr 'The effectof the contrastof viscosity etween n-placewaterand injected water is more difficult to evaluate. Information,however, may be obtained from studieson water flooding inthe oil industry. Water flooding s a secondary ecoveryproc-ess n which oil is displacedby injectedwater. Calculations orviscosity atios of unity were comparedwith laboratoryexperi-ments at viscosity atios other than unity for a variety of wellarrangements. esultswere usuallygiven n terms of patternareal sweepefficiencyor fraction of pattern area contactedbywater at breakthrough. These values can be transformed toyield the to value (from (11)) at which breakthroughof thethermal ront occursor eachwell pattern n the case n whichthere is no heat exchangewith the cap rock or the bedrock (In order to evaluate the influenceof the viscosity atio onthe advancementof the thermal fron. for variouswell patternsthe ratio of the breakthrough to value at a viscosityratio ofunity to that at a viscosity atio of other than unity has beenp!otted n Figure3 versushe ratio of the viscosity f nativewater to that of injectedwater. The data used are those pub-lished, y Craig [1971]. t is apparent rom Figure 3 that break-through occurs ater for viscosity atios less than unity andearlier for viscosity atios greater han unity. In otherwords, fcold water is injected nto a warm aquifer (as occurs n geother-mal space eating),he temperature ill remainconstant tthe production ells or a period qnger han hat computedwith our method. On the other hand, if the reinjectedwater is

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4960 GRINGARTEN ND $AUTY:HEAT EXTRACTION ROM AQUIFERS

- 00

t

ii

*

OF. 0,5

C::) ,3

'-'

KRPRCR

Fig.6. Dimensionlessemperaturet theproduction ellof a doubletn a deep onfinedquiferQo > 10) for variousvalues of 3, from 0.3 to infinity.

warmer han the aquifer,as occurs n air cooling, he temper-ature at the productionwells will rise earlier than would bepredicted.RECHARGING-DISCHARGING WELL PAIR

The previous esultshavebeenapplied o the caseof a singlerecharging-dischargingell pair (called a 'doublet' hereafter)in uniformlow.Therecharge ell s on thex axisat a pointwith coordinates +a, 0), and the dischargewell on the x axisat a point with coordinates-a, 0). (The recharge ell s 0.9downstream hena = 0.) The characteristicengthD in (10)and 11) is now hedistance etweenhe wo wells D - 2a). 'Information on the flow field created by a recharging-dis- , 0.7charging ellpair n a homogeneousnd sotropicquifer 0.6with niformlowsavailablen heiteratureDaCostandBennett, 1960; Grove et al., 1970]. The stream function can be , 0.sexpressedn termsof two parameters nly: the angleof arealflowa anda dimensionlessarameter , 0.4Qt> = Q/hDVo (15)

which characterizes he importance of the well recharge-dis-charge rate compared o that of the areal flow.The dimensionlessater temperature t the productionwellcan thus be expressedrom (12) as a function of four dimen-sionlessarametersonly:To- ro(t) = rwz>(ot,Qz>, X, tz>) (16)To-- Ti

Equation 16) hasbeenevaluatedby computeras a functionof to for various valuesof a, Qt>,and 3,. As was mentioned

before,,accountsor heheatransferyconductionith hecap rock and the bedrock,whereasa and QD set he maximuminterflowbetweenwells, .e., the total amountof injectedwaterin the water producedat the productionwell after an infinitetime (to = co). The maximum interflow is independentof 3,and is also equal o the maximum emperature hangeTwt>m,x

= 0

QD Q/hDVo6x %CwPACQh... 10 K oRCR D

0.3 0.2

0.1O_ I I i t ill I I I I lilt_1 10 10 2

owC Qtt D - OACA hFig. 7. Dimensionlessemperature t the productionwell versus i-mensionlessime for a = 0 , Qo = 6, and 3, = 104

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GRINGARTENND AUTY:EAT XTRACTIONROM QUIFERS2O

15

11C- 10

29C ..NJ_ECT.ONT[$PERAI'ORE

!/" ........ MEANNJE!I5C INJECTION EMPERATURE

o ,,, 30 _ 29oc 20 = MEAN NJECTI TEMPERATURE= 10 5oc .....

0 i I0 1 2 .......3_C_.__I.....3 4TIME,IN YEARS

Fig. . Temperaturefwatert he roductionell ndereriodichangen njectionemperature.

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0 PRODUCTIONWELLIFINJECTION WELLowCwPACQhx: -- 104K oR CRQD:Q/hDVo 6 m 0

! .

AREAL FLOtl

Fig. 9. Propagation of thermal front.

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4962 GRINGARTENNDSAUTY:HEATEXTRACTIONROMAQUIFERSof the water at the productionwell. TwDmas plottedversusQoin Figure 4 for various values of a from 0 to 180. Twolimitingcases re of interest. 1) At Qo valuesgreater han 104,interflow s maximum and independent f a: Toomax l,and Too s only a function of 3,and to. This is the case n mostdeep confined aquifers. (2) On the other hand, at low Qovalues, when areal flow dominates (as in most free surfaceaquifers), it is possible o have no interflow between wells,except f a = 180.It is not practical o presentn this papera setof curves hatwill coverall practical ases. hese re provided nly or deepaquifers Qo > 104). Two s plotted n Figure 5 versus o forvariousvaluesof 3, rom 0.3 to infinity and n Figure6 versus ,(to - 1) for 3,values anging rom 0 to 300. Figures5 and 6 canbe used o calculate 1) the spacingbetween he two wells nordernot to haveany change n temperature t the productionwell during a specified eriodand (2) the temperature hangeat the productionwell after breakthrough.

EXAMPLE OF CALCULATIONThermal front breakthroughoccursat to = 1.04 if 3, > 10(Figure 5) and at 3, (to - 1) = 0.5 if 0 < 3, < 10 (Figure 6).

Actually, his last formula can be usedwith a good approx-imation for all 3, values.Hence f there must be no change nthe producedwater emperatureor a periodof lengthAt (thatis, if At is the useful ife of the doublet), he spacing,n cgsunits, must at least be equal to

D= 2.Q./Xt/I(4-{-(1-40 CoPRC)

+2(pC)2t] J (17)Equation 17) hasbeenused o evaluate he spacing etweenwellsof isolated oublets hat will be drilled for space eatingin the 1800-m-deepDogger aquifer near Paris, France. For h= 50 m, Q = 100 ma/h, at = 30 years,pwCw = 1 cal/cm*/s,pC 0.5 cal/cma/C, and K = 6 X 10 a cal/cm/s/C,equation 17) yieldsD = 63 m. This value ncludes safetyfactor, sincecold water is injected nto a warm aquifer. Theresults rom (17) would be D = 917 m ifK were aken aszero.One suchschemehas beenoperating n Melun, France, since

1969.In free surhce aquifers,whereareal flow is usually mpor-tant, it is theoreticallypossible o locate he wellsso hat therewill be no interflow at all. In that case he temperaturewillalways emainconstant t the productionwell. Unfortunately,this may not alwaysbe practical,as s llustratedby the follow-

ing example.A 70-m-thick reesurface quifer s to be used orspaceheating for months and for air cooling for 4 months,by meansof heat pumps.The water emperaturen the aquiferhasbeen ound o be constantndequal o llC all yeararound, a finding which indicates that there is no heat ex-change hrough the 3-m-thick unsaturated one. The rate ofwater to be withdrawn and injected is 900 ma/h, and theinjection temperature s 5C during the space-heating tageand 29C during the air-coolingperiod. Aquifer porosity s10%,and the areal low hasa real velocitymagnitude f 1.7m/d. How should he doubletbe designedn order to keep hetemperature t the productionwell as constantas possible?

Figure 4 indicates hat no interflow will take place betweenthe.wells f QD s ess han 1.8. D must thereforebe greater hanQ/(1.8h. Vo), or D > 965 m. However, in this particular situa-tion the two wells must be drilled on a propertywithin aheavilybuilt area, and the maximum practicalspacingbetweenthe wells s only 300 m in the direction of areal flow. QD s thusequal to 6. Figure 4 then indicates hat interflow will be theleast and that it will be equal to 38% if the rechargewell isplaceddirectly downgradeof the dischargewell (a = 0).The time variation of the water temperatureat the produc-tion well due to the seasonal hange n injection emperature sobtainedby first computing he TwD unction rom (16) for theappropriatevaluesof a, QD, and 3,and then substitutingt into(13). The calculationswere made on the assumptionhat theaquifer s boundedabove and below by two adiabaticbound-aries (3, = co), in order to obtain an upper limit for the pro-ducedwater temperature.TwD(O, , co, D) is plotted n Figure7 versus D, and the temperaturevariation in time at the pro-duction well obtained from (13)is shown n Figure 8. Propaga-tion of the thermal front is presented n Figure 9.With a 300-m spacing, he temperatureat the productionwell will thus be a periodic function of time having the sameperiod as the injection temperature function and oscillatingbetween10C and 15C. This was found to be inadequate,andit was decided o use wells with partial penetration o takeadvantageof the high anisotropybetween he horizontal andthe vertical directions.Water is pumped out from the top ofthe aquifer and reinjectedat the bottom. The systemhas beenoperating or about 1 year and has been ound to be satisfac-tory so far.

Acknowledgments.he investigationseportedhereinhave beencarriedout at the Bureaude Recherches fologiques t Minifires(BRGM) in Orl6ans, rance. he authors ish o expressheirappre-ciation o the managementf BRGM for permissiono publishhispaper.Thanksare extendedo J. Goguel,BRGM vice-president,ormany stimulatingdiscussionsn the paper opic. The assistance fP. A. Landel n setting p thecomputer rogramsgratefully cknowl-edged.

REFERENCESCarslaw,H. S., andJ. C. Jaeger,Conductionf Heat n Solids, nded.,p. 396, Clarendon, Oxford, 1959.Craig, F. F., The ReservoirEngineering spects f Waterflooding,Monogr.Ser., vol. 3, Societyof PetroleumEngineers, allas, Tex.,1971.Da Costa,J. A., and R. R. Bennett, he patternof flow n the vicinityof a recharging nd dischargingair of wells n an aquiferhavingareal parallel flow, Int. Ass.Sci. Hydrol. Publ.52, 524-536, 1960.Grove, D. B., W. A. Beetem, and F. B. Sower, Fluid travel timebetween rechargingnd dischargingellpair n an aquiferhavinga uniform egional low ield, WaterResour. es., (5), 1404,1970.Houpeurt, A., J. DeLouvrier, and R. lffiy, Fonctionnementd'undoublet hydraulique de refroidissement,Houille Blanche, no. 3,239-246, 1965.Kruger,P., and C. Otte, Geothermal nergy,StanfordUniversityPress,Palo Alto, Calif., 1973.Lagarde,A., and P. Maugis, M6thodesd'6tudedes possibilit6sd'exploitationesnappes 'eausouterrainesour e chauffager-bain, Rdf 14072, nst. Fr. du P6trole,Rueil-Malmaison, rance,Dec. 1966.Lauwerier, . A., The transport f heat n an oil layercaused y theinjectionof hot fluid, Appl. Sci. Res.,Sect.A, 5, 145, 1955.Maugis,P., Exploitation 'unenapped'eauchaude outerraineourle chauffage rbain dans a r6gionparisienne, nn. Mines, no. 5,135-142, 1971.

(ReceivedJanuary 27, 1975;revisedAugust 18, 1975;acceptedAugust 18, 1975.)