SUSTAINABLE HEAT FARMING OF GEOTHERMAL SYSTEMS: A CASE STUDY OF HEAT EXTRACTION AND THERMAL RECOVERY IN A MODEL EGS FRACTURED RESERVOIR

Transcription

1 PROCEEDINGS, Thiy-Sih Wokshop o Geoheal Resevoi Egieeig Safod Uivesiy, Safod, Califoia, Jauay 3 - Febuay, SGP-TR-9 SUSTAINABLE HEAT FARMING OF GEOTHERMAL SYSTEMS: A CASE STUDY OF HEAT EXTRACTION AND THERMAL RECOVERY IN A MODEL EGS FRACTURED RESERVOIR Daiel Sue,, Do B. Fo, Bia J. Adeso 3, Doald L. Koch 4, Philipp Rudolf vo Roh, ad Jeffeso W. Tese,* Akiso Cee fo a Susaiable Fuue ad he Coell Eegy Isiue, Coell Uivesiy, Ihaca, NY 4853, USA Isiue of Pocess Egieeig, ETH Zuich, Soeggsasse 3, 89 Zuich, Swizelad 3 College of Egieeig ad Mieal Resouces, Wes Vigiia Uivesiy, Mogaow, WV 656, USA 4 School of Cheical ad Bioolecula Egieeig, Coell Uivesiy, Ihaca, NY 4853, USA * Coespodig auho: jw54@coell.edu ABSTRACT To addess he quesio of eewabiliy of Ehaced Geoheal Syses (EGS) a coducio-doiaed, odel EGS esevoi was evaluaed as a epeseaive wos case o esiae hea eacio duig poducio ad heal ecovey followig shu dow. I he odel syse wae is ijeced a specified aes ad epeaues io a sigle ecagula facue suouded by a ifiie aou of ipeeable ho ock. Duig he eacio phase, wae oves alog he facue eacig hea fo he adjace ock ai leadig o local coolig ad heal dawdow of he esevoi. Whe he wae ijecio is sopped, coducive hea asfe fo he suoudig hoe ock egios leads o heal ecovey of he coole zoes i he esevoi. The ae of ecovey is coolled locally by he epeaue gadie ha is iduced duig he heal dawdow. A wodiesioal aheaical odel was developed o descibe hea asfe fo boh eacio ad ecovey. Regadig he ecovey, a advaced aalyical appoach was developed ha is capable of descibig he epeaue duig ecovey a evey posiio alog he facue. Ou appoach leads o he sae esul fo he epeaue a he ile posiio, as peseed i ealie eseach usig a diffee appoach. I addiio, ueical siulaios wee caied ou usig he TOUGH code o sudy he ipoace of he assupios eployed i he aalyical descipio ad o eed he applicabiliy of he odel by eablig siulaio of opeaig cycles wih aleaig eacio ad ecovey ies. The effec of eglecig hea coducio i he ock i he diecio paallel o he flow i he facue was aalyzed by copaiso of he aalyical odel o he TOUGH siulaios. Fo a fied facue aea, low flow aes ca esul i heal dawdow localized aoud he fluid ile wih hea coducio i he paallel diecio becoig sigifica. BACKGROUND AND MOTIVATION Oe ipoa feaue of ay opeaig geoheal esevoi syse has o do wih is aicipaed poducio susaiabiliy ove he log e. Alhough geoheal esevois ca be depleed duig poducio if echage aes ae isufficie o ovecoe local coolig of ock ad losses of fluid pessue, wih pope aagee hydoheal esevois have bee show o be poducive fo log peiods of ie. I Egieeed Geoheal Syses he siuaio is diffee wih o ecod of log e field esig ad as a esul he eewabiliy of EGS i geeal is ofe quesioed. Aelsso e al. () defie he eewabiliy of a geoheal souce as he abiliy o aiai he isalled capaciy idefiiely. Theally, he esevoi would be i seady-sae codiio, i.e. he ae of hea eacio by he wokig fluid ad he echage ae fo he bulk ock ae equal. Howeve, he eewable capaciy is fequely oo sall fo coecial develope due o ecooy of scale i ifasucue develope ad opeaio coss (Sayal 5). Theefoe, cosideaios abou he susaiabiliy of geoheal syses us also iclude he ecovey effec afe a sop of hea eacio (Megel ad Rybach ). Sayal (5) defies he susaiable capaciy of a geoheal esevoi as he capaciy, ha ca be ecooically aiaied ove he aoized life ie of a powe pla. Accodig o his eview of opeaig hydoheal plas aoud he wold, he susaiable capaciy is 5 o 45 ies he eewable capaciy wih he faco beig os likely aoud. Alhough, eploiig he susaiable capaciy eveually esuls i sigifica coolig of he esevoi ad ecovey ies i he ode of hudeds of yeas, hee ae good easos o defie such geoheal esouce opeaio as susaiable. Coplee ecovey of he heal eegy is eveually guaaeed ad eve wih ecovey ie

2 scales of hudeds of yeas he esouce base is lage eough o allow fo log-e eegy poducio (Tese, Adeso e al. 6). The possible displacee of fossil fuel cosupio would educe evioeal polluio oday ad help o peseve hese fuels fo fuue geeaios as aw aeial ad fo possibly cleae ad oe efficie powe applicaios (Sayal 5). Geoheal esouces ca be cosideed eewable o ie-scales of echological/socieal syses, wheeas fossil fuel eseves eew i geologic ie scales oly (Rybach, Megel e al. ). Modelig he asie hea asfe i geoheal esevois allows sudyig diffee opeaig saegies wih he oveall goal o icease he susaiable capaciy ad ake he bes use of he hea flow fo he suoudigs ha is iduced by he eacio of hea. The objecive of his wok is o odel boh opeaioal odes of geoheal esevois: eacio ad ecovey phase. Duig he eacio phase, liquid wae is ciculaed hough he esevoi ude pessue o eac hea. Duig he ecovey phase, he poducio flow is shu dow ad he hea flow fo he suoudig ock echages he esevoi. Boh, a ueical ad a aalyical odel o descibe hea ad ass asfe i he esevoi ae developed. The esuls of boh odels ae copaed o each ohe fo validaio ad o bee udesad he effec of soe of he assupios ade i he aalyical odel. THEORY Aalyical Model The aheaical appoach o he hea asfe poble is based o he odel esevoi geoey give i Figue. The odel syse used fo a disceely facued EGS esevoi was adaped fo ealie woks by Gigae e al. (975) ad Wude ad Muphy (978). Alhough ealisic esevois cosis of a ewok of facues, a sigle facue odel adequaely capues heal ecovey hough hea coducio fo he ho ock suoudig he esevoi. A sigle ecagula, veical facue of cosa widh b sepaaes wo blocks of hoogeeous, isoopic, ipeeable ock. A Caesia coodiae syse has bee placed such ha he plae coicides wih he ock/facue ieface. I his epo, we assue wae o be he geoheal wokig fluid. Figue : Maheaical odel fo sigle facue i ho ock. (a) shows he hee-diesioal odel (3D), (b) shows he sae odel educed o wo diesios (D). The dashed lie i he iddle of he facue idicaes he syey i he -diecio. The ock is assued o eed hoizoally () o ifiiy. Rock ad facue eed o ifiiy i he y- diecio. Iiially, ( ) he whole syse is a uifo epeaue T,. Fo he hea eacio phase, wae is ijeced a,z a a cosa epeaue T w, ad a a cosa ass flow ae &. I is flowig upwads hough he facue o he oule a ad zl. Fuheoe, he followig assupios ae ade: The epeaue vaiaio i he wae i - diecio is isigifica, as he apeue of he facue is vey sall copaed o he facue legh (b/l<<), ad is egleced. Fo all he wae epeaue is equal o he ock epeaue a. I ohe wods, he hea asfe esisace a he ock/wae ieface is egleced, which is a easoable assupio fo ay pacical case (Wude ad Muphy 978; Ogio, Yaaua e al. 999). Coducio i he veical diecio (z) i boh he facue ad he ock foaio is egleced, as

3 well as adiaive hea asfe i he facue. Because he apeue (b) of he facue i - diecio is sall copaed o he facue ee i y-diecio (H), such ha b/h<<, we ca ± ad eglec hea asfe eed he y-ais o i y. This effecively educes he aheaical hea asfe poble o wo spaial diesios. Hea asfe occus oly by coducio i he ock i -diecio ad foced covecio alog he z-diecio i he facue (Gigae, Wihespoo e al. 975). No hea flu occus a he ock/fluid ieface duig ecovey. The assupio is jusified by he sall volue, ad hece, sall hea capaciy of he facue copaed o he volue/hea capaciy of he ock ha has bee cooled duig he pecedig eacio phase. The desiy ad specific hea capaciy of boh ock ad wae, ad he hea coduciviy of he ock ae cosa. The saic fluid pessue i he facue is se o eceed he vapo pessue of he wae by a lage eough agi o keep he facue i sigle phase flow. Fis eacio phase The followig deivaio of a soluio fo he epeaue field i a esevoi duig he fis eacio phase is based o he epo of Wude ad Muphy (978). A oe-diesioal diffeeial eegy balace wihi he ock yields T α () whee α is he ock heal diffusiviy, i.e. he aio of heal coduciviy (k ) ad he poduc of desiy (ρ ) ad specific hea capaciy (c ) k α () ρ c ad TT(, is he epeaue. Followig he assupios of egligible hea asfe esisace a he ock/fluid ieface ad of egligible epeaue vaiaios i -diecio i he wae, he epeaue T(, descibes he ock epeaue a he ieface ad also he wae epeaue ove he eie facue widh a he especive z-posiio. The epeaue depedece o z is ioduced i he bouday codiio fo he ock/fluid ieface ρ wc wb + ρ wc wub k (3) z wih U epeseig he flow velociy of he wae, b he half-widh of he facue, ad ρ w ad c w he desiy ad specific hea capaciy of he wae. Wih he iiial ad bulk epeaue of he ock specified as T, ad he cosa wae epeaue a he ijecio posiio ake as T w, he addiioal bouday codiios ae T, z, Tw, (4) (, > T (, T, (5) ad he iiial codiio is T, ) T (6) (, The aalyical soluio is give i es of he diesioless, oalized epeaue Θ ad is based o he soluio of a classical asie hea asfe poble as peseed fo eaple by Caslaw ad Jaege (959) ad Apaci (966) T(, Tw, + βz Θ(, ef (7) T, Tw, α whee k β (8) ρ c Ub w w Fis ecovey phase A advaced appoach was developed o descibe he epeaue i he sigle facue odel aalyically. We coside a liea hea sik alog he facue, i.e. i z-diecio a. Because of ou ealie siplificaio o wo diesios, he sik appeas a all y fo y o yh. The lie sik epeses he hea eacio by he flowig fluid i he facue duig he eacio phase. The eegy balace icludig he hea sik becoes T q( δ ( ) α ρ c (9) whee q( is he hea flu a he ock/wae ieface ad δ () is he Diac dela fucio. The discepacy o Eq. () i he igh had side of he equaio is esolved by oig ha he bouday codiio i Eq. (3) hadles he hea sik i he pevious foulaio. As eioed befoe, he epeaue T(, is valid fo boh ock ad wae a he ieface. The coespodig iiial ad bouday codiios of Eq. (9) ae agai Eq. (5) ad Eq. (6). The hea flu q( akes accou of he secod bouday codiio. Duig he eacio phase, hea flows fo he ho egios of ock io he ovig fluid i he facue. The flu q( ca be deeied based o he epeaue pofile fo he eacio i Eq. (7). No hea flu occus a he ock/wae ieface duig he ecovey phase, accodig o he assupio lised above, ad q( is he zeo. Wih e beig he ie a he ed of he eacio phase ad u T w, T,, we ge q( k u ( z) k β ep, 4α, < e πα () > e

4 Eq. (9) was solved usig a vaiable asfoaio ad he followig Gee s fucio (Duffy ) H ( ) ( ) G(, ) ep () 4πα( ) 4α ( ) whee H(-') is he Heaviside sep fucio, which is equal o fo posiive agues ad equal o fo egaive agues. Fo deails o he deivaio, see Appedi B. The soluio i iegal fo is e H ( ) z z ( β ) Θ(,, ) + d ep π 4α () Eq. () ca be used fo boh eacio ad ecovey phase, i.e. fo [, ), if oe eacio peiod wih subseque ecovey is cosideed. Fo < e, he Heaviside sep fucio H( ') effecively educes he iegaio liis fo [, e ] o [,]. The epeaue a ie e ca be defied as he lii of Eq. () fo e. The iegal i Eq. () has a closed fo soluio whe,z. The ecovey soluio ( > e ) a,z i es of Θ is Θ(, z, a (3) π e The sae soluio was epoed by Wude ad Muphy (978). They did o lieally esic he soluio o he ijecio poi a,z bu assued a cosa dawdow epeaue i hei deivaio. Noeheless, i ay pacical siuaio he epeaue duig dawdow is cosa oly a he ile posiio, whee he fluid ca be assued o ee he syse a a cosa epeaue. Usig Taylo seies epasio ad e wise iegaio, he iegal ca be solved aalyically fo ad z [,L], i.e. fo all posiios alog he facue. Fo he deivaio see Appedi B. The fially esulig ifiie seies is Θ(, π ξ k + + ( ) ( + ) e + + π efc ( )!! ( ξ ) + ep( ξ ) k k + k ξ, > ( + )( )...( k + ) e, (4) whee βz ξ ad is a bioial αe coefficie. Eq. (4) oly applies fo he ecovey phase ad alog he facue, i.e. fo > e ad. Θ is iial a he ile ad aial a he oule ad he values ca be used o obai a age fo he ee of he ecovey of he geoheal esevoi. Fo eaple, whe Θ a he oule, wae could agai be eaced a he iiial ock epeaue. If fuhe ie fo ecovey is specified, he z-posiio a which Θ is eached, will eea fuhe owads he ile, i.e. owads (, z). Whe he ecovey ie becoes lage copaed o he eacio ie, >> e, he hea sik due o he eacio fluid esebles a ipulsive poi souce a ad. The fis e of he ifiie seies i Eq. (4) is [ ( ξ ) ξ π efc( ξ )] e Θ ep (5) π The sae e ca be foud fo a ipulsive poi souce of agiude e Q( z) a ad. q( d 4k e u ep πα [ ( ξ ) ξ π efc( ξ )] (6) Nueical Model The TOUGH ueical siulao (Puess, Oldebug e al. 999) was used fo ueical odelig of eacio ad ecovey peiods i he sigle-facue syse. TOUGH is a geeal pupose ueical siulaio poga fo uli-diesioal o-isoheal flows of uli-phase, ulicopoe fluid iues i facued ad poous edia. The code was developed a he Lawece Bekeley Naioal Laboaoy ad is wie i sadad FORTRAN77. I eploys a iegal fiie diffeece ehod (IFDM) i disceizig he ediu, which has he advaage of iegula disceizaio i uli-diesios. Tie is disceized fully iplicily as a fis-ode backwad fiie diffeece ad flues ae copued usig upsea weighig (Puess, Oldebug e al. 999). PeaSi (Thudehead Egieeig, Mahaa, KS) was used as pe- ad pos-pocessig sofwae a he fo ed. PeaSi allows o ieacively defie he esh ad paaees fo he odel ad he ceaes a ipu file fo he TOUGH code. Afe uig he TOUGH code PeaSi ca be used o pocess he siulaio esuls fo gaphical epeseaio. Figue shows a scheaic view of he esh applied fo he ueical siulaio i TOUGH. Because of he syey alog he y,z-plae i he iddle of he facue, oly half of he basic odel i Figue was ipleeed. The siulaed doai eeds 3 i, i y, ad 6 i z. Wih a siple esiae fo he peeaio deph of he heal coolig i he -diecio, α, we fid ha 3 will be eached afe abou 7 yeas oly. The fis colu of gidblocks alog he z-ais ad a epeses he facue. The colu of facue cells ad he fis colu of ock cells e o i ae boh.3 wide. Fo hee o, he esh is logaihically spaced io 8 oe gidblocks i he -diecio. I he z- diecio, he esh is equally paiioed io.5

5 Figue : Mesh fo TOUGH ueical siulaio of a sigle facue i ho ock. (b) shows he coplee gid wih diesios ad he posiios of fluid ijecio ad oule. The odel ca be picued as he igh-had half of he wo diesioal syse i Figue, Gaph (a), icludig a wo diesioal block of ock ad half he facue o is lef-had side. The lies idicaig he esh cause he odel o appea hoogeeously dak i his oveall view. The deailed view i (a) shows he sucue of he esh aoud he ijecio posiio. The wae flow occus wihi he fis colu of cells which icopoae he.3 wide facue as scheaically idicaed i (a). Noe ha he ea cell wih he souce of ass does o appea i he figue. blocks ecep ea he ile whee he z-spacig was educed o.5 fo he fis e ees fo he ijecio poi a, z. The flow was assued o be plug flow hough he facue; heefoe, he esh was o efied i he y-diecio ad cosiss of oe laye of gidblocks i he,z-plae. No hea ad ass flow occus a he boudaies a y ad y, ad hece, o hea o ass flow ca occu i he y- diecio, he sae as i he aalyical odel. The facue cell a z6 is se o fied sae ad epeses a ass sik fo he wae flow alog he facue duig he eacio phase. The fied sae specificaio keeps he cell's popeies cosa a hei iiial value, iespecive of occuig hea ad ass echage. Howeve, he cell ha is cosideed as he poducio well posiio i he aalysis, is he facue cell a zl5. The disace bewee he wo cells was ioduced o avoid udesiable effecs fo he fied sae cell defiiio o he spaial doai of iees of z [,L]. A ea cell was added below he facue cell a,z fo he ijecio of fluid io he facue. The ea cell icludes a cosa souce of ass se o & epeseig he wae ijecio. Pessue ad ehalpy of he ijeced wae have bee specified o be MPa ad kj/kg, especively, which coespods o a epeaue of T w, C. Fo he ecovey phase, he ass souce i he ile cell is se o zeo. The fied sae cell a he opposie ed of he facue a z6 is ow eabled ad se o he sae iiial epeaue as he suoudig cells ( global iiial codiios ). I ode o lii he hea asfe fo he suoudig ock o coducio, he poosiy ad peeabiliy of he ock ae se o. TOUGH icludes oly Dacy s Law flow odels fo poous edia, so o odel fluid flow i a ope facue, he poosiy of he facue aeial is se o., ad is peeabiliy o -9. While his is o a igoous odel of fluid flow i a ope facue ad will o epoduce pessue dop accuaely, he copaiso wih he heoy which elies oly o a specificaio of he flow ae i he cack will o be affeced. A colu of. wide fied sae cells was added a he 3 bouday o achieve a cosa epeaue i he bulk ock. Pessue ad epeaue of his s colu of cells i ae fied a he iiial codiios. All ohe boudaies of he esh ae iacive, i.e. o hea o ass flow occus a he boudaies. As give by (Asead ad Tese 987) he dawdow ae of sigle facue geoheal

6 esevois is deeied by he paaee & /A, he wae ass flow ae & divided by he ock/waeieface aea, AHL. All flow aes i he discussio ae give as aea-oalized ass flows & /A, which is equivale o he heal eacio ae pe ui facue aea whe uliplied by he hea capaciy ad epeaue of he wae. The agiude of & /A scales diecly wih he ae of heal dawdow. RESULTS AND DISCUSSION Figue 3 copaes he esuls of he aalyical eacio soluio i Eq. (7) ad he ueical siulaio fo he diesioless, oalized epeaue Θ a fou diffee locaios alog he facue. Aalyical ad ueical siulaio esuls ach closely fo all posiios. Geeally, Θ deceases fo is oigial value of as hea is eaced fo he syse. A he ile posiio (,z) we see a shap declie o Θ, as equied by he bouday codiio i Eq. (4). Fo he esevoi ad flow codiios specified (fo a suay, see Appedi A), he epeaue a he oule (, zl5 ) says close o he iiial epeaue fo he fis 5 yeas. The peeaio of he heal coolig fo alog he facue ad io he ock i -diecio is discussed below. ecovey is oalized by he eacio ie e ad ages fo, he sa of he ecovey, o 6, whee he ie fo ecovey would be five ies he duaio of he eacio phase. The ueical iegaio soluio will be used o copae he esuls of he aalyical odel ad he TOUGH ueical odel, because i allows o iclude posiios i he ock ai whee. Figue 4: Ted of oalized epeaue Θ a he posiio, z wih ie duig he ecovey phase. The ie ais is oalized by he eacio ie e. The coiuous black lie shows he soluio obaied fo he ueical iegal soluio, he dashed lies show he esul of he aalyical epessio i Eq. (4) wih a iceasig ube of es of he ifiie seies beig cosideed. Figue 5 cobies he esuls of he ueical iegaio soluio ad he TOUGH ueical siulaio fo he ecovey phase a he fou posiios alog he facue ioduced i Figue 3. The epeaue i he TOUGH siulaio eceeds Figue 3: Copaiso of he aalyical soluio fo he eacio give i Eq. (7) ad he TOUGH siulaio esul. The gaphs show he oalized epeaue Θ fo a 3 yeas eacio peiod wih a aeaoalized wae ass flow ae of 6 & A 8 kg/( 4 s) ( U.33 /s) a fou diffee locaios alog he facue. I ode o veify he ifiie seies soluio fo he ecovey phase i Eq. (4) i is copaed o a ueical soluio of he iegal i Eq. () i Figue 4. The wo soluios ach well, whe eough es of he ifiie seies ae cosideed. The ie ais i Figue 4 ad i he followig illusaios of Figue 5: Resuls of he TOUGH siulaio ad he ueical iegaio fo he ecovey phase a fou diffee posiios alog he facue, afe 3 yeas of hea eacio a a aea-oalized wae 6 ass flow ae of & A 8 kg/( s) 4 ( U.33 /s).

7 he ueical iegaio soluio fo posiios elaively close o he ile, as ca be see fo, z i Figue 5. The absolue diffeece is Θ.44 a e 6. A possible easo fo he divegece is he diffee eae of coducio hea asfe i he z- diecio i he wo odels. The aalyical soluio is based o a eegy balace ha eglecs coducio i z i he eie syse, wheeas he TOUGH odel does coside coducio i z. Ieesigly, he TOUGH soluio esuls i fase heal ecovey. I ode o ivesigae he ipoace of hea coducio i we copaed aalyical ad TOUGH siulaio esuls fo a siuaio whee he epeaue gadies i he z-diecio becoe egligible. Theefoe, he aea-oalized fluid ass flow ae fo he eacio phase was iceased o a 3 aificially high value of & A kg/( s), esulig i a epeaue dawdow alos uifo i as show i Figue 6. Figue 7: Resuls of he TOUGH siulaio ad he aalyical soluio fo he ecovey phase afe a 3 yea hea eacio phase a a aea-oalized wae ass flow ae of 3 & A kg/( s) ( U 3.33 /s) fo he posiio, z. The sae qualiaive esul was obaied fo posiios i he ock ai a diffee disaces fo he facue i -diecio. Figue 8 gives a eaple fo he locaio 5 io he ock ad a he sae disace o he ile posiio i z-diecio of. The applicabiliy of he assupio of egligible hea coducio i he diecio of he flow cooly ade fo he deivaio of aalyical soluios (Bodvasso 969; Gigae, Wihespoo e al. 975; Wude ad Muphy 978; Bodvasso ad Tsag 98; Cheg, Ghassei e al. ) sees o be depede o he flow ae of he eacio phase ad he esulig epeaue pofile. 6 & A 8 kg/( s) Figue 6: Tepeaue field afe 3 yea eacio peiod wih a aificially high aeaoalized wae ass flow ae of 3 & A kg/( s) ( U 3.33 /s). Ideed, he esuls of TOUGH ad of he ueical iegaio soluio ach uch close fo he heal ecovey fo he epeaue field wih egligible gadies i he z-diecio, as ca be see i Figue 7. 3 & A kg/( s) Figue 8: Resuls of he TOUGH siulaio ad he aalyical soluio fo he ecovey phase afe a 3 yea hea eacio phase a aea-oalized wae ass 6 flow aes of & A 8 kg/( s) ad 3 & A kg/( s) fo he posiio 5, z.

8 Coveioal scalig aalysis ca be used o copae he ode of agiude of hea coducio i he wo diecios heoeically. The es i he goveig equaio scale as T T ~ (7) δ T T ~ (8) z δ z whee T, δ, ad δ z ae he appopiae chaaceisic quaiies fo epeaue ad legh i ad especively. Hece, we ca defie a diesioless quaiy hea coducio i z δ Λ ~ (9) hea coducio i δ z I ode o deeie he elaive ipoace of hea coducio alog each ais, he chaaceisic legh scale i boh he - ad z-diecio us be deeied. δ ad δ z ca be cosideed o be he peeaio deph of he heal coolig fo alog he - ad z-ais, especively. Defiig he posiio of he heal coolig fo o be he value of o z a which Θ equals a abiay value ϑ, i ca be descibed as δ Θ( δ,, ϑ ef () α βδ z Θ(, δ z, ϑ ef () α wih β as defied i Eq. (8). Hece, ou diesioless quaiy scales as follows Λ ~ β () U & The ueical values fo he wo flow aes eioed i his aicle ae β 3 fo he low 7 flow ae ad β 5 fo he high flow ae. I ohe wods, fo he high flow ae hea coducio i eceeds hea coducio i z by si odes of agiude, while fo he low flow ae he wo becoe oe copaable. Fo ϑ. 95, he heal peeaio dephs afe he 3 yea eacio phase a he lowe flow ae coespodig o β ae δ 86 ad δ z 497, wheeas fo he high flow ae he heoeical peeaio deph i z would be δ z 4' 333. If we iegae heal beakhough o he fluid oule posiio wih is associaed egaive effecs fo he suface isallaio io ou cosideaios, ad hus esic δ z o L, we ca coclude ha he -D coducio odel should be applied oly fo log facues, whee δ L <<. / CONCLUSIONS AND OUTLOOK The heal behavio of a odel facued geoheal esevoi duig hea eacio ad heal ecovey was cosideed i his wok. Fisly, a aheaical odel was developed ad solved aalyically; secodly, ueical siulaios wee caied ou o ivesigae he ipoace of he assupios ade i he aalyical odel ad o eed he applicabiliy of he sigle-facue odel o eable he ueical siulaio of opeaio i eacio/ecovey cycles. The ueical odel i cobiaio wih he aalyical soluios was applied o ivesigae he ipoace of hea coducio i he ock ai paallel o he facue. The obseved depedece o he aea-oalized fluid flow ae could be cofied quaiaively by a diesioless paaee. Low aea-oalized flow aes ca esul i heal dawdow localized aoud he ile posiio, ad hus, ceae heal gadies i he diecio paallel o he facue of copaable agiude o he gadies i he diecio ohogoal o he facue. The os ipoa specific fidigs ae suaized below. A advaced appoach was developed o descibe he syse aheaically wih a liea hea sik alog he facue legh beig used o ealize he hea eacio fo he ho ock by he wae flow hough he facue. A closed fo aalyical soluio descibig he epeaue a he wae ile posiio duig he ecovey (ad eacio) phase was deived. Alhough based o a diffee aheaical appoach ha used i a ealie sudy by Wude ad Muphy (978), ou soluio fo he ecovey epeaue a he ile was cosise wih ha epoed ealie. I addiio, we developed a aalyical soluio fo heal ecovey descibig he epeaue a all posiios alog he facue. This oe geeal ifiie seies soluio allows o deeie he epeaue duig he ecovey phase a he posiio of he wae oule, fo eaple. The oule epeaue is cucial as i coespods diecly o he poducio epeaue of he geoheal fluid ha would be uilized a he suface fo geeaig powe ad/o povidig hea. Usig ueical iegaio, ou aalyical appoach was successfully used o descibe he asie heal behavio of he eie spaial doai. Nueical siulaio esuls fo he developed TOUGH odel ad he aalyical soluio coespod closely fo he esed age of diffee opeaig codiios ad posiios i he esevoi. This fidig gives us cofidece i odelig

9 dawdow ad ecovey sceaios fo which aalyical soluios ae iacable. The developed diesioless paaee ca be applied o esiae he sigificace of wodiesioal coducio of hea. The oediesioal siplificaio epeses a lowe boud fo he epeaue ecovey i geoheal syses, as he addiioal coducio i he secod diesio leads o fase ecovey. The odel developed i he sudy ca be used o evaluae saegies fo he opeaio of geoheal esevois. Such saegies should ai a akig he bes use of he susaiable capaciy of a give esevoi while guaaeeig he epeaue equiees of he ifasucue a he suface. Aog he opeaioal sceaios o be esed ae: Resevoi opeaio wih seveal cycles, each cosisig of oe eacio ad oe ecovey phase. Megel ad Rybach () copaed opeaio sceaios wih vayig cycle duaios while he oveall eacio ie ad he ass flow ae duig eacio wee kep cosa. They foud ha he oveall eaced heal eegy iceases wih shoe cycle peiods. Ou odel shows qualiaively cosise esuls. These basic esuls should be ivesigaed i deail icludig diffee wae ass flow aes ad esevoi popeies such as heal coduciviy of he ock. Diffee o Megel ad Rybach () we wa o copae he oveall eegy of he eaced ho fluid isead of is ehalpy. Gaphical epeseaios of he epeaue i he odeled esevoi iuiively visualize how he esevoi iplees a coue-cue hea echage behavio whe heal beakhough is eached. The epeaue alog he facue gadually iceases fo T w, a he fluid ijecio poi o T, a he oule. Befoe beakhough is eached, he egio close o he oule posiio is iacive wih espec o hea asfe fo ock o wae, because he wae is aleady heaed up o T, befoe i eaches he egio. Coay o he coo oio ha heal beakhough deeies he eiaio of he esevoi usage, we wa o eploe ways how o use esevois ha have see beakhough fo peheaig he geoheal fluid befoe i is asfeed o a secod esevoi. By his eas oe could ake bee use of he heal eegy i he fis esevoi ad siulaeously pospoe beakhough i he secod esevoi, ad hece, guaaee a cosa poducio epeaue of T, fo uch loge ie. I ode o icease he ock/wae ieface aea, ha is acively asfeig hea o he fluid quickly, a highe flow ae a he begiig of he eacio phase could be advaageous. The flow ae should decease lae o o peve ealy beakhough. The effec o he oal eegy ha ca be eaced fo a esevoi eeds o be ivesigaed, alog wih saegies how o deal wih he vayig ass flow ae wih espec o he suface isallaio. REFERENCES Asead, H. C. H. ad J. W. Tese (987). Hea Miig. Lodo, New Yok, E. & F.N. Spo. Apaci, V. S. (966). Coducio Hea Tasfe. Readig, Addiso-Wesley. Aelsso, G., A. Gududsso, e al. (). "Susaiable Poducio of Geoheal Eegy: Suggesed Defiiio." IGA-News 43(Jauay- Mach): -. Bodvasso, G. S. (969). "O he Tepeaue of Wae Flowig Though Facues." Joual of Geophysical Reseach 74(8): Bodvasso, G. S. ad C. F. Tsag (98). "Ijecio ad Theal Beakhough i Facued Geoheal Resevois." Joual of Geophysical Reseach 87(B): Caslaw, H. S. ad J. C. Jaege (959). Coducio of Hea i Solids. Ofod, Ofod Uivesiy Pess. Cheg, A. H. D., A. Ghassei, e al. (). "Iegal Equaio Soluio of Hea Eacio fo a Facue i Ho Dy Rock." Ieaioal Joual fo Nueical ad Aalyical Mehods i Geoechaics 5(3): Duffy, D. G. (). Gee's Fucios wih Applicaios. Washigo, D.C., Chapa & Hall/CRC. Gadshey, I. S. ad I. M. Ryzhik (7). Table of Iegals, Seies, ad Poducs. New Yok, Acadeic Pess. Gigae, A. C., P. A. Wihespoo, e al. (975). "Theoy of Hea Eacio fo Facued Ho Dy Rock." Joual of Geophysical Reseach 8(8): -4. Megel, T. ad L. Rybach (). Poducio Capaciy ad Susaiabiliy of Geoheal Doubles. Poc. of he Wold Geoheal Cogess, Kyushu - Tohoku, Japa. Ogio, F., M. Yaaua, e al. (999). "Hea asfe fo ho dy ock o wae flowig hough a cicula facue." Geoheics 8(): -44. Puess, K., C. Oldebug, e al. (999). TOUGH Use's Guide, Vesio.. Bekeley, CA, Lawece Bekeley Naioal Laboaoy.

10 Rybach, L., T. Megel, e al. (). A Wha Tie Scale ae Geoheal Resouces Reewable? Poc. of he Wold Geoheal Cogess, Kyushu-Tohoku, Japa. Sayal, S. K. (5). Susaiabiliy ad Reewabiliy of Geoheal Powe Capaciy. Poc. of he Wold Geoheal Cogess, Aalya, Tukey. Tese, J. W., B. J. Adeso, e al. (6). The Fuue of Geoheal Eegy, Massachuses Isiue of Techology: 358. Wude, R. ad H. Muphy (978). Theal Dawdow ad Recovey of Sigly ad Muliply Facued Ho Dy Rock Resevois. Los Alaos, New Meico, Los Alaos Scieific Laboaoy of he Uivesiy of Califoia. APPENDIX A. Noeclaue ad Applied Nueic Values A HL 5 ock/facue ieface 6 α.3 /s heal diffusiviy ock b.3 half-widh of facue k β (-) diesioless paaee ρwc wub c 5 J/(kgK) specific hea capaciy ock c w 484 J/(kgK) specific hea capaciy wae δ, δ z () heal peeaio deph H legh of he odel i y (TOUGH) L 5 legh of facue i z Λ ~ β (-) di.less paa. copaig coducio i ad z & (kg/s) wae ass flow ae & A (kg/( s)) aea-oalized wae ass flow ae k.9 W/(K) heal coduciviy ock k w.6 W/(K) heal coduciviy wae (TOUGH) ρ 7 kg/ 3 desiy ock ρ kg/ 3 desiy wae w T w, C wae ijecio epeaue T, C iiial ock epeaue T (, T Θ(, T T U (/s) e, w, w, diesioless epeaue wae flow velociy βz ξ siilaiy vaiable fo, α e B. Deivaio fo he Ifiie Seies Recovey Soluio i Eq. (4) We defie a ew vaiable which esuls i hoogeeous bouday codiios ad siplifies poble solvig u T(, T, (3) Followig he defiiio of Θ i Eq. (7) he elaio of Θ ad u is u Θ (4) u wih u defied as u(,z,. The asfoaios of Eq. (9) ad is bouday ad iiial codiios i Eq. (5) ad Eq. (6) ae u u q( δ ( ) α ρ c (5) u (, ) (6) u (, (7) The appopiae Gee s fucio fo he doai of + R ad R is show i Eq. (). Followig he coo appoach fo applyig Gee s fucios he soluio o he PDE i Eq. (5) is he iegal q( ) δ ( ) u(, G(, ) d d (8) ρ c The d iegal ca be evaluaed applyig he sifig popey of he dela fucio. The uppe iegaio bouday fo ca be educed fo o e, he ie a he ed of he eacio phase, because he hea flu q( is assued o be fo he ecovey phase ( > e ). The hea flu q( duig he eacio phase ca be deeied applyig he deivaive of he epeaue pofile fo he eacio i Eq. (7) as show i Eq. (). The esul is ku z q z β (, ) ep, < e (9) πα 4α Subsiuig he Gee s fucio show i Eq. () ad he hea flu show i Eq. (9) io Eq. (8) we ge he geeal iegal soluio show above i Eq. (). Fo > e, i.e. fo he ecovey peiod, Eq. () ca be ewie o be u(, e β z ep + d u π 4α (3) Usig he Taylo seies epasio fo he e Eq. (3) ca be siplified o ( ) ( e ep 4 u z,, ) + u π z β + d α (3)

11 Fo, he iegal i each e of he suaio i Eq. (3) ca be epessed as a icoplee gaa fucio + Γ + +, ),, ( e u z u ξ ξ π (3) wih ξ as defied below Eq. (4) ad he icoplee gaa fucio defied as Γ s a d e s a τ τ τ ), ( (33) Applyig he subsiuio τ he icoplee gaa fucio ca be evaluaed usig he followig iegal foud i Gadshey ad Ryzhik (7) ( )+ pv efc p d e v p )!! ( ) ( π, ) 3)...( )( ( ) ( ) ( > + p k pv v e k k k k v p (34) ad subsiued i Eq. (3) o yield he ifiie seies soluio give above i Eq. (4).