Thermal Breakthrough Calculations to Optimize Design of a Multiple

GRC Transactions, Vol. 38, 2014

Thermal Breakthrough Calculations to Optimize Design of a Multiple-Stage Enhanced Geothermal System Tianyu Li, Sogo Shiozawa, and Mark McClure The University of Texas at Austin Department of Petroleum and Geosystems Engineering, Austin, TX [email protected]

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Keywords EGS, multiple stages, thermal breakthrough, optimization, horizontal wells

1. Introduction

ABSTRACT

1.1 Premise

:HSHUIRUPHGDVLPSOHRSWLPL]DWLRQSURFHGXUHDQGVHQVLWLYLW\ DQDO\VLVWRH[DPLQHGHVLJQVIRU(QKDQFHG*HRWKHUPDO6\VWHPV (*6  WKDW LQYROYH KRUL]RQWDO ZHOOV DQG PXOWLSOH IUDFWXULQJ VWDJHV7KHVHQVLWLYLW\DQDO\VLVLQFOXGHGFDOFXODWLRQVRIWKHUPDO EUHDNWKURXJKDQGWKHPD[LPXPÀRZUDWHWKDWFRXOGEHDFKLHYHG WKURXJK WKH V\VWHP FRQVLGHULQJ SUHVVXUH GURS LQ WKH ZHOO DQG LQ WKH UHVHUYRLU  &RQYHQWLRQDOO\ (*6 ZHOOV KDYH EHHQ QHDUO\ vertical and stimulated in openhole sections in a single stage. We investigated a design involving two parallel horizontal wells. 7KH¿UVWZHOOZRXOGEHGULOOHGDQGFRPSOHWHGZLWKFDVLQJDQG WKHQVWLPXODWHGVHTXHQWLDOO\LQVWDJHVXVLQJFDVHGKROHSDFNHUV rated to high temperature). The second well would be drilled WKURXJKWKHVWLPXODWHGUHJLRQFUHDWHGE\WKH¿UVWDQGFRPSOHWHG openhole. For different combinations of well spacing, fracture WUDQVPLVVLYLW\DQGQXPEHURIVWDJHVWKHRSWLPDOÀRZUDWHZDV GHWHUPLQHGWRPD[LPL]HSUHVHQWYDOXH39 RIUHYHQXHFRVWZDV not considered). The calculations showed that stimulating with PXOWLSOHVWDJHVZRXOGUDGLFDOO\LPSURYHHFRQRPLFSHUIRUPDQFH GHOD\LQJ WKHUPDO EUHDNWKURXJK DQG DOORZLQJ D KLJKHU RYHUDOO ÀRZUDWHWREHFLUFXODWHGWKURXJKWKHV\VWHP$WORZZHOOVSDFLQJDQGORZQXPEHURIVWDJHVLWLVRSWLPDOWRFLUFXODWHÀXLGPRUH VORZO\WKDQWKHPD[LPXPSRVVLEOHUDWHLQRUGHUWRGHOD\WKHUPDO breakthrough. With greater well spacing and with more stages, WKHUPDOEUHDNWKURXJKZLOOEHUHODWLYHO\GHOD\HGDQGLWLVRSWLPDO WRFLUFXODWHDWWKHKLJKHVWSRVVLEOHÀRZUDWH,WLVRSWLPDOWRXVHWKH ORZHVWZHOOVSDFLQJZKHUH139LVPD[LPL]HGE\FLUFXODWLQJDWWKH PD[LPXPSRVVLEOHUDWH:KHQLWLVRSWLPDOWRFLUFXODWHDWWKHPD[LPXPSRVVLEOHUDWH139LVVHQVLWLYHWRUHVHUYRLUWUDQVPLVVLYLW\ :KHQLWLVRSWLPDOWRFLUFXODWHDWOHVVWKDQWKHPD[LPXPSRVVLEOH UDWH139LVXQDIIHFWHGE\UHVHUYRLUWUDQVPLVVLYLW\2SWLPL]LQJ WRPD[LPL]HDJJUHJDWHKHDWH[WUDFWLRQLQVWHDGRI139KDVRQO\D limited effect on NPV. Most thermal drawdown calculations were

,Q(QKDQFHG*HRWKHUPDO6\VWHPV(*6 K\GUDXOLFVWLPXODWLRQLVXVHGWRLQFUHDVHWKHÀRZUDWHWKDWFDQEHDFKLHYHGWKURXJK DKLJKWHPSHUDWXUHORZSHUPHDELOLW\IRUPDWLRQ)OXLGLVFLUFXODWHG between injection and production wells. Cold water is injected and KHDWVXSDVLWÀRZVWKURXJKWKHIRUPDWLRQWRZDUGWKHSURGXFWLRQ ZHOO7KHKRWZDWHURUVWHDPSURGXFHGWRWKHVXUIDFHPD\EHDSSOLHGIRUGLUHFWXVHRUIRUHOHFWULFLW\JHQHUDWLRQ Economic success for EGS requires that the time-discounted UHYHQXHIURPHOHFWULFLW\RUKHDWSURGXFWLRQLVVXI¿FLHQWWRRIIVHW H[SHQVHVSOXVSURYLGHDQDGGLWLRQUDWHRIUHWXUQ$QRSWLPDO(*6 GHVLJQ ZRXOG EDODQFH PDUJLQDO FRVW DQG UHYHQXH WR PD[LPL]H WLPHGLVFRXQWHGUHYHQXHPLQXVH[SHQVHV 3URMHFW UHYHQXH LV VWURQJO\ DIIHFWHG E\ WKH WHPSHUDWXUH RI SURGXFHGÀXLGDQGWKHSURGXFWLRQUDWHERWKRIZKLFKDUHOLNHO\WR change over time. Over the long term, the temperature of produced ÀXLGZLOOGHFOLQHDVWKHIRUPDWLRQFRROVGXHWRWKHORVVRIKHDWWR WKHFLUFXODWLQJÀXLG,WKDVEHHQVKRZQWKDWDQH[FHVVLYHO\KLJK FLUFXODWLRQUDWHPD\OHDGWRSRRUKHDWVZHHSHI¿FLHQF\*ULQJDUWHQ'RHHWDO 2QWKHRWKHUKDQGWKHWLPHYDOXHRI PRQH\VXJJHVWVWKDWKLJKFLUFXODWLRQUDWHPD\EHGHVLUDEOHEHFDXVH LWZLOOFDXVHUHYHQXHWREHHDUQHGDVHDUO\DVSRVVLEOH ,QWKLVVWXG\ZHXVHGDVLPSOHFRQFHSWXDOPRGHODQGUHODWLYHO\ simple calculations to calculate optimal injection rates and well VSDFLQJ GLVWDQFH EHWZHHQ LQMHFWRU DQG SURGXFHU  IRU GLIIHUHQW ZHOOERUHFRQ¿JXUDWLRQV0DQ\VLPSOL¿FDWLRQVDUHPDGHEXWWKH VWXG\ SURYLGHV D VHQVLWLYLW\ DQDO\VLV RQ WKH HIIHFW RI GLIIHUHQW parameters and gives insight into the behavior of the optimization problem. 7KHSULPDU\REMHFWLYHRIWKHVWXG\ZDVWRLQYHVWLJDWHWKHHIIHFW of multiple stage simulation on the economic performance of an EGS doublet. To date, most EGS projects have been performed 313

Li, et al.

LQQHDUO\YHUWLFDOZHOOVLQDVLQJOHVWDJHLQWRRSHQKROHVHFWLRQV RIZHOOERUH7ZRH[FHSWLRQVDUHWKH6FK|QHEHFNSURMHFWZKLFK XVHGSDFNHUVLQVHYHUDOVWLPXODWLRQVWDJHV=LPPHUPDQQHWDO   DQG WKH 1HZEHUU\ SURMHFW ZKLFK XVHG GLYHUWLQJ DJHQWV WRDWWHPSWDQHIIHFWLYHO\PXOWLSOHVWDJHVWLPXODWLRQ3HWW\HWDO  :HDUHLQWHUHVWHGLQLQYHVWLJDWLQJWKHSHUIRUPDQFHRIDQ (*6GRXEOHWLQYROYLQJÀRZEHWZHHQSDUDOOHOKRUL]RQWDOZHOOV as shown in Figure 1. In this design, a zonal isolation technolRJ\VXFKDVSDFNHUV LVXVHGWRSXPSVHTXHQWLDOO\LQWRVHSDUDWH VHFWLRQV RI RQH RI WKH ZHOOERUHV UHIHUUHG WR DV PXOWLSOH VWDJH stimulation). The laterals are oriented so that the stimulated region DWHDFKVWDJHIRUPVURXJKO\WUDQVYHUVHWRWKHKRUL]RQWDOODWHUDO 6LPLODUGHVLJQVKDYHEHHQGLVFXVVHGE\*ULQJDUWHQHWDO  *UHHQDQG3DUNHU 1DOODDQG6KRRN -XQJ  DQG6KLR]DZDDQG0F&OXUH 

VPDOOQXPEHURIZLGHO\VSDFHIUDFWXUHVDWWKHZHOOERUH5LFKDUGV HWDO,WRDQG.DLHGDSDJHRI%URZQHWDO 0L\DLULDQG6RULPDFKL:\ERUQHWDO%DULDHWDO 'H]D\HVHWDO $FDUHIXOLQYHVWLJDWLRQDWWKH6RXOW] SURMHFW VXJJHVWHG WKDW RQO\ D KDQGIXO RI WKH FULWLFDOO\ VWUHVVHG IUDFWXUHVDWWKHZHOOERUHDFWXDOO\DSSHDUHGWREHFRPHFRQGXFWLYLW\ ÀXLGSDWKZD\V(YDQV  Numerical simulations based on continuum models of the resHUYRLUVXFKDVGXDOSRURVLW\PRGHOVWHQGWREHRYHUO\RSWLPLVWLF DQGYDVWO\XQGHUHVWLPDWHWKHQDWXUDOWHQGHQF\IRUÀRZORFDOL]DWLRQ LQIUDFWXUHQHWZRUNV'LVFUHWHIUDFWXUHQHWZRUNVLPXODWLRQVWKDW FRXSOHÀXLGÀRZZLWKWKHVWUHVVHVLQGXFHGE\IUDFWXUHGHIRUPDWLRQ GHPRQVWUDWH KRZ LQWHUDFWLRQ RI PHFKDQLFDO DQG K\GUDXOLF SURFHVVHVZLWKWKHYDJDULHVRIÀRZLQIUDFWXUHQHWZRUNVFUHDWHV ORFDOL]HGGHIRUPDWLRQDQGVSDUVHO\SRSXODWHGIUDFWXUHQHWZRUNV 0F&OXUHDQG+RUQH0F&OXUH (YHQLIDODUJHVWLPXODWHGQHWZRUNLVFUHDWHGRQO\DVPDOOPLQRULW\RIWKDWQHWZRUN PD\DFWXDOO\EHSDUWRIDVWURQJK\GUDXOLFFRQQHFWLRQEHWZHHQ DQLQMHFWLRQDQGSURGXFWLRQZHOO6HFWLRQIURP0F&OXUH  :KHQÀRZLQJIUDFWXUHVDUHZLGHO\VSDFHGWKHKHDWVZHHS HI¿FLHQF\LVXQOLNHO\WREHRSWLPDO5DSLGWKHUPDOEUHDNWKURXJK KDVEHHQREVHUYHGDWVRPH(*6SURMHFWV7HQPDHWDO  $VHFRQGSUREOHPKDVEHHQHVWDEOLVKLQJVXI¿FLHQWK\GUDXOLF FRQQHFWLRQEHWZHHQWKHLQMHFWLRQDQGSURGXFWLRQZHOOV)RUH[DPSOHWKLVSUREOHPZDVGRFXPHQWHGDW)HQWRQ+LOO%URZQHW DO 5RVHPDQRZHV3DUNHU DQGWKHZHOO*3.DW 6RXOW]*HQWHUHWDO  The Soultz EGS project has achieved the highest circulating ÀRZUDWHRIDQ\(*6SURMHFW%XWHYHQDWWKH6RXOW]SURMHFWWKH ÀRZUDWHLVQRWDVKLJKDVQHHGHG)XUWKHUPRUHWKH6RXOW]SURMHFW LVLQDORFDWLRQZKHUHWKHUHLVDOUHDG\FRQVLGHUDEOHQDWXUDOSHUPHDELOLW\IURPSUHH[LVWLQJODUJHIDXOW]RQHV'H]D\HVHWDO  8QOHVVSURSSDQWLVXVHGZHDUHXQOLNHO\WRDFKLHYHJUHDWHUÀRZ UDWHRQDSHUIUDFWXUHEDVLVDQGWKHHI¿FDF\DQGFRVWHIIHFWLYHQHVV RI SURSSDQW LV XQFHUWDLQ  7KHUHIRUH WKH SULPDU\ UHDVRQ ÀRZ UDWHVKDYHEHHQLQVXI¿FLHQWKDVEHHQWKDWQRWHQRXJKIUDFWXUHV KDYHEHHQVWLPXODWHGDQGSDUWLFLSDWHGLQDK\GUDXOLFFRQQHFWLRQ EHWZHHQWKHLQMHFWLRQDQGSURGXFWLRQZHOOV$Q\SURSRVHGVWUDWHJ\ WRLPSURYHHFRQRPLFSHUIRUPDQFHIRU(*6PXVWDWWHPSWWR¿QG DZD\WRGUDPDWLFDOO\LQFUHDVHWKHQXPEHURIÀRZLQJSDWKZD\V that can be created in the reservoir. 0XOWLSOHVWDJHK\GUDXOLFIUDFWXULQJLVDQREYLRXVFDQGLGDWHIRU LPSURYLQJWKHQXPEHURIÀRZLQJSDWKZD\V:KHQÀXLGLVSXPSHG LQWRDOHQJWK\ZHOOERUHVHFWLRQÀXLGWHQGVWRÀRZLQWRWKH¿UVW few fractures that form or that are stimulated at the wellbore. As DUHVXOWVWLPXODWLRQLVQRWHYHQO\GLVWULEXWHGDORQJWKHZHOO,IWKH stimulation is performed with multiple stages, fractures are forced WRIRUPDWHDFKVWDJHFUHDWLQJDPXFKPRUHHYHQO\VSDFHGIUDFWXUH QHWZRUN,QWKHRLODQGJDVLQGXVWU\WKHXVHRIPXOWLSOHK\GUDXOLF fracture stages has enabled a huge amount of new production from SUHYLRXVO\VXEHFRQRPLFVKDOHIRUPDWLRQV.LQJ  An objection to the use of multiple stages in EGS has been that openhole packers have not been proven to work at high temperaWXUH+RZHYHU6KLR]DZDDQG0F&OXUH SRLQWHGRXWWKDW LQUHFHQW\HDUVFDVHGKROHSDFNHUVKDYHEHFRPHDYDLODEOHWKDW DUHUDWHGWRDWOHDVWƒ&(*6SURMHFWVKDYHQHDUO\DOZD\VEHHQ FRPSOHWHGLQRSHQKROHERWKWRVDYHFRVWDQGWRPD[LPL]HFRQWDFW ZLWK QDWXUDO IUDFWXUHV EHFDXVH WKH JRDO KDV EHHQ WR VWLPXODWH

Figure 1. An EGS doublet of horizontal wells connected by vertical fracture stages (normal or strike-slip faulting regime). The wells are oriented toe to heel in order to encourage equal flow rates between stages.

In prior work, we discussed how using multiple stages would LQFUHDVHWKHPD[LPXPÀRZUDWHDFKLHYDEOHWKURXJKWKHV\VWHP DQGVKRZHGKRZDGGLQJVWDJHVFRXOGGUDPDWLFDOO\LQFUHDVHÀRZ UDWH6KLR]DZDDQG0F&OXUH 7KHREMHFWLYHRIWKLVSDSHU LVWRH[WHQGWKDWZRUNE\LQYHVWLJDWLQJWKHHIIHFWRIKDYLQJPXOtiple stages on thermal breakthrough. Thermal breakthrough is a FRPSOH[IXQFWLRQRIPDQ\IDFWRUVLQFOXGLQJWKHRYHUDOOÀRZUDWH WKURXJKWKHV\VWHPWKHQXPEHURIÀRZLQJIUDFWXUHVWKHVSDFLQJ EHWZHHQWKHODWHUDOVDQGWKHVSDFLQJEHWZHHQWKHÀRZLQJIUDFWXUHV DORQJWKHODWHUDOV:HSHUIRUPHGDVLPSOHRSWLPL]DWLRQWRLGHQWLI\ WKHLGHDOODWHUDOVSDFLQJDQGÀRZUDWHGHSHQGLQJRQWKHQXPEHU of stages along the wells. We calculated the present value of the UHYHQXHGHULYHGIURPWKHV\VWHPIRUGLIIHUHQWQXPEHUVRIVWDJHV DVVXPLQJRSWLPDOZHOOVSDFLQJDQGÀRZUDWH

1.2 Conceptual Models for an EGS Reservoir The classical EGS design has been based on the concept of VKHDUVWLPXODWLRQWKDWDQLQFUHDVHLQÀXLGSUHVVXUHLQGXFHVVOLS RQSUHH[LVWLQJIUDFWXUHVZKLFKH[SHULHQFHDUHVXOWLQJLQFUHDVHLQ WUDQVPLVVLYLW\7KHKRSHKDVEHHQWKDWWKLVSURFHVVZRXOGFUHDWHD ODUJHFORVHO\VSDFHGZHOOFRQQHFWHGQHWZRUNRIQDWXUDOIUDFWXUHV ZKLFKZRXOGDFFHVVVXI¿FLHQWYROXPHRIURFNWRHQDEOHHFRQRPLFDOO\YLDEOHSURGXFWLRQ ([SHULHQFHKDVUHYHDOHGOLPLWDWLRQVWRWKHFRQYHQWLRQDO(*6 GHVLJQ:HOOORJVKDYHVKRZQWKDWÀRZWHQGVWRORFDOL]HLQWRD 314

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QDWXUDOIUDFWXUHV +RZHYHULWLVSRVVLEOHWRSHUIRUPVWLPXODWLRQ RXWRISHUIRUDWHGFDVHGZHOOV%HQGDOOHWDO UHSRUWHGRQD ZHOOWKDWZDVUHFHQWO\VWLPXODWHGLQJUDQLWHDWNPGHSWKRXW of a perforated, cased well at 190°C. Furthermore, we do not believe that shear stimulation is necHVVDULO\WKHEHVWVWUDWHJ\IRUFUHDWLQJDQHIIHFWLYH(*6UHVHUYRLU $QRSWLPDO(*6GHVLJQZRXOGEHKLJKO\UHSHDWDEOHEXWVKHDU stimulation is dependent on the local details of the natural fracture QHWZRUN,WPD\EHDGYDQWDJHRXVLQ(*6WRDWWHPSWFRQYHQWLRQDO K\GUDXOLFIUDFWXULQJLQZKLFKÀXLGLVSXPSHGRXWRIWKHZHOOZLWK WKHLQWHQWLRQRIFUHDWLQJDQGSURSDJDWLQJQHZO\IRUPLQJIUDFWXUHV )RUDYDULHW\RIUHDVRQVLQYHVWLJDWRUVKDYHSUHGRPLQDQWO\FODLPHG WKDWWKHSULPDU\PHFKDQLVPRIVWLPXODWLRQLQ(*6LVVKHDUVWLPXODWLRQVXPPDUL]HGE\0F&OXUHDQG+RUQH 0F&OXUHDQG +RUQH DUJXHGWKDWLQIDFWQHZIUDFWXUHVSUREDEO\KDYH IRUPHGDQGSURSDJDWHGDWPDQ\(*6VLWHV There are several operational decisions that could be made WRLQFUHDVHWKHSUREDELOLW\RIQHZK\GUDXOLFIUDFWXUHVIRUPLQJ,I injection is performed into a perforated, cased hole, new fractures ZRXOGEHPRUHOLNHO\WRIRUPWKDQGXULQJSXPSLQJLQWRRSHQKROH EHFDXVHWKHSHUIRUDWLRQVZRXOGEHXQOLNHO\WRLQWHUVHFWQDWXUDO IUDFWXUHV$OVRLWKDVEHHQREVHUYHGLQWKHRLODQGJDVLQGXVWU\ WKDWWKHSXPSLQJRIDPRUHYLVFRXVÀXLGLVPRUHOLNHO\WRFUHDWHD³VLPSOH´IUDFWXUHJHRPHWU\UDWKHUWKDQDFRPSOH[IUDFWXUH QHWZRUN&LSROODHWDO  A “simple” fracture at each stage might appear to be undeVLUDEOH EHFDXVH LW ZRXOG QRW FUHDWH D ODUJH QXPEHU RI ÀRZLQJ IUDFWXUHVZKLFKLVUHTXLUHGIRUDTXDOLW\(*6UHVHUYRLU%XWWKHXVH RIPXOWLSOHIUDFWXULQJVWDJHVZRXOGQDWXUDOO\FUHDWHDODUJHQXPEHU RIÀRZLQJIHDWXUHV7KXVZHSURSRVHWRFUHDWHDODUJHQXPEHURI “simple” fractures using multiple stages, rather than attempting to create a single massive fracture network from a single stage. $QLPSRUWDQWXQNQRZQLVZKHWKHUQHZO\FUHDWHGK\GUDXOLF fractures would be capable of self-propping. The concept of shear stimulation relies on the mismatch of asperities to hold open QDWXUDO IUDFWXUHV$VSHULW\ PLVPDWFK PD\ QRW RFFXU RQ QHZO\ IRUPHGIUDFWXUHV2QWKHRWKHUKDQGQHZO\IRUPLQJIUDFWXUHVPD\ EHVXI¿FLHQWO\URXJKWKDWWKH\DUHDEOHWRVHOISURSHVSHFLDOO\LQ KDUGURFNVXFKDVJUDQLWH-XQJ  Regardless of stimulation mechanism, either new fractures or shear stimulation of natural fractures, it is clear that having additional fracture stages would permit the formation of additional ÀRZSDWKZD\VE\SUHYHQWLQJLQMHFWLRQÀXLGIURPGLYHUWLQJLQWRD small number of features at the wellbore.

GURS LQ WKH ZHOOV DQG LQ WKH UHVHUYRLU 'LIIHUHQW FRPELQDWLRQV RISDUDPHWHUVZHUHH[DPLQHGWR¿QGWKHRSWLPDOUDWHVSDFLQJ QXPEHURIVWDJHVDQGWUDQVPLVVLYLW\)RUFRPSDULVRQZHUHSHDWHG WKH RSWLPL]DWLRQ PD[LPL]LQJ WKH FXPXODWLYH KHDW SURGXFWLRQ instead of NPV.

2.1 Flow Rate Calculation )RUHDFKFRPELQDWLRQRISDUDPHWHUVWKHPD[LPXPÀRZUDWH SRVVLEOHWKURXJKWKHV\VWHPZDVFDOFXODWHG7KHFDOFXODWLRQVLQcluded pressure gradient in the injection well, the reservoir, and WKHSURGXFWLRQZHOO'RZQWRWKHWDUJHWGHSWKWKHJHRPHWU\RI the injection and production wells was based on the deep wells *3.DQG*3.DWWKH6RXOW](*6SURMHFW+RZHYHUXQOLNHDW Soultz, we assumed that at the target depth, the wells deviated to become horizontal. The wells were oriented toe-to-heel, as shown LQ)LJXUH)RUVLPSOLFLW\WKHWHPSHUDWXUHVLQWKHLQMHFWRUZHOO production well, and the reservoir were assumed to be constant EXWGLIIHUHQWLQHDFKRIWKHWKUHH  The wellhead pressure of the injector, WHPinj, and the wellhead pressure of the producer, WHPprod ZHUH VSHFL¿HG WR EH  03D DQG03DUHVSHFWLYHO\IROORZLQJWKH6RXOW]FLUFXODWLRQWHVW GHVFULEHGE\7LVFKQHUHWDO )ORZZDVDOVRGULYHQE\WKH GLIIHUHQFHLQGHQVLW\EHWZHHQWKHÀXLGLQWKHLQMHFWLRQZHOODQGWKH SURGXFWLRQZHOO)ORZUDWHFRXOGKDYHEHHQLQFUHDVHGE\XVLQJD KLJKHULQMHFWLRQSUHVVXUHEXWSUDFWLFDOO\H[FHVVLYHO\KLJKLQMHFWLRQSUHVVXUHPD\OHDGWRH[FHVVLYHÀXLGORVV)ORZUDWHFRXOGKDYH EHHQLQFUHDVHGE\SXPSLQJWKHSURGXFWLRQZHOOEXWWKLVZRXOG involve additional cost and parasitic power loss. In practice, it ZRXOGSUREDEO\EHHFRQRPLFWRSXPSWKHSURGXFWLRQZHOODQGLW PD\EHIHDVLEOHWRXVHDKLJKHULQMHFWLRQSUHVVXUHWKDQZHDVVXPHG but we chose these values in order to give conservative estimates. For the calculation of pressure drop in the wells, it was asVXPHGWKDWDOOÀXLGHQWHUHGRUH[LWHGWKHZHOOERUHVDWDVLQJOH ORFDWLRQWKHPLGGOHRIHDFKRIWKHWZRODWHUDOVHYHQWKRXJKZH ZHUHPRGHOLQJZHOOVZLWKPXOWLSOHVWDJHV 7KHV\VWHPZDVPRGeled as four nodes connected in series: WHPinj, BHPinj, BHPprod, and WHPprod, where BHPinj and BHPprod are the bottomhole presVXUHRIWKHLQMHFWRUDQGWKHSURGXFHUZHOOVUHVSHFWLYHO\:HOOERUH SUHVVXUHGURSFDOFXODWLRQVZHUHXVHGWRFDOFXODWHǻPinj, equal to WHPinj - BHPinjDQGǻPprod, equal to WHPprod - BHPprod 'DUF\¶V ODZZDVXVHGWRFDOFXODWHǻPres, equal to BHPinj - BHPprod. All ÀXLGLQWKHV\VWHPZDVDVVXPHGWREHVLQJOHSKDVHOLTXLGZDWHU )RU D JLYHQ YDOXH RI UHVHUYRLU WUDQVPLVVLYLW\ WKH ÀRZ UDWH WKURXJK WKH V\VWHP ZDV FDOFXODWHG E\ QXPHULFDOO\ VROYLQJ WKH IROORZLQJQRQOLQHDUHTXDWLRQZLWKÀRZUDWHq, as the unknown:

2. Methodology

WHPprod = WHPinj +
Pinj (q) +
Pres (q) +
Pprod (q) 

7RPD[LPL]HWKHSUHVHQWYDOXH39 RIDQ(*6GRXEOHWZH DQDO\]HGWKHHIIHFWRIWKHIRXUNH\HOHPHQWVLQWKHIUDFWXUHGHVLJQ  ÀRZUDWH WKHQXPEHURIIUDFWXUHVWDJHV WKHVSDFLQJ EHWZHHQWKHWZRKRUL]RQWDOZHOOVDQG IUDFWXUHWUDQVPLVVLYLW\ )RUHDFKFRPELQDWLRQRISDUDPHWHUVDQDQDO\WLFDOH[SUHVVLRQZDV used to calculate the temperature of produced water over time. These values were converted into thermal production and electricLW\SURGXFWLRQDQG¿QDOO\SUHVHQWYDOXH)RUHDFKFRPELQDWLRQ RIZHOOVSDFLQJIUDFWXUHWUDQVPLVVLYLW\DQGQXPEHURIVWDJHVD YDULHW\RIFLUFXODWLRQUDWHVZHUHFRQVLGHUHGXSWRWKHPD[LPXP SRVVLEOHUDWHZKLFKZDVFDOFXODWHGE\FRQVLGHULQJWKHSUHVVXUH

2.1.1 Pressure Change Calculation in the Injection Well and the Production Well 7KLVVHFWLRQH[SODLQVKRZ
Pinj q and
Pprod q were FDOFXODWHG7KHWRWDOSUHVVXUHJUDGLHQWdp/dz) can be calculated as WKHVXPRIWKHIULFWLRQDOJUDGLHQWdp/dz)FWKHK\GURVWDWLFJUDGLHQWdp/dz)HDQGWKHDFFHOHUDWLRQDOJUDGLHQWdp/dz)A+DVDQDQG .DELU DQGLVJLYHQE\

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(dp / dz) = (dp / dz) F + (dp / dz) H + (dp / dz) A  315

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7KH K\GURVWDWLF DQG WKH DFFHOHUDWLRQDO JUDGLHQWV DUH UHSUHVHQWHGE\ (dp / dz) H =
g  sin




(dp / dz) A =
(w / A)dv / dz =

vdv / dz 



Table 2. Geometry and well head pressures of the injection well and production well.

Wellbore

where ȡLVWKHGHQVLW\RIÀXLGș is the wellbore angle from horizontal line, wLVWKHÀXLGPDVVRIÀRZUDWHA is cross-sectional area of casing, and vLVLWVYHORFLW\ The frictional pressure gradient is: (dp / dz) F =
fv
/ 2d 

Injector

Injector possible extension (cased hole)



2

where d is well or pipe diameter and f is friction factor, which deSHQGVRQWKHWXUEXOHQFHRIWKHÀXLGDQGDOVRRQWKHSLSHURXJKQHVV &KHQ SURSRVHGWKHIROORZLQJHTXDWLRQWRFDOFXODWHWKH Fanning friction factor: f =

1

 2 log(
/ d
5.0452 log )  3.7065 Re  

2



Producer Producer possible extension (openhole)



whereİ is pipe roughness, and A is the dimensionless parameter JLYHQE\
=

(
/ d )1.1098 7.149 + 2.8257 Re





Fluid properties were chosen for fresh water and are given in Table 1. The temperature in the injection well was assumed to be 60°C, and the temperature in the production well was assumed to EHƒ&7KHVXUIDFHURXJKQHVVRIFDVLQJZDVDVVXPHGWREH PLFURQVDVHVWLPDWHGIRUWKHZHOOERUHFDVLQJRI*3.DW6RXOW] E\0pJHOHWDO EDVHGRQPHDVXUHPHQWVRIWKHZHOOKHDG and bottomhole pressure during injection.

Injector

Reservoir

Producer

60

190

180

Fluid density (kg/m^3)

983.2

873.9

885

Fluid viscosity (cp)

0.466

0.142

0.151

 

3.07E-13

 

Temperature (°C)

Transmissivity (m^3)

9Ǫ

4610 – [4760-6110]

9Ǫ

0-534 534-4084 4084-4638 4638 – [4788-6138]

WHP (MPa)

83.7 70.9 70.9 90 75

4

0

13ǫ

90

7

79.2

8½

0.75

0

2.1.2 Pressure Change Calculation in the Reservoir 7KLVVHFWLRQH[SODLQVKRZ
Pres q was calculated. The presVXUHFKDQJHWKURXJKWKHUHVHUYRLUZDVFDOFXODWHGIURP'DUF\¶V ODZDVVXPLQJVWHDG\VWDWHOLQHDUÀRZWKURXJKDIUDFWXUHZLWK height hP DQGWUDQVPLVVLYLW\TP): qµ D  
Pres = Th


()

Table 1. Properties used in the flow rate calculations. The reservoir transmissivity shown is the “baseline” transmissivity. Properties

0-2373 2373-3449 3449-4282 4282-4550 4550-4610

Wellbore Angle From Horizontal (°)

&RQVLGHULQJ WKH ÀRZ JHRPHWU\ VKRZQ LQ )LJXUH  HDFK molecule of water will pass through the same length of wellbore ODWHUDODVLWSDVVHVWKURXJKWKHV\VWHPUHJDUGOHVVRIZKLFKVWDJH LW ÀRZHG WKURXJK  DQG WKDW OHQJWK LV HTXDO WR KDOI RI WKH WRWDO OHQJWKRIERWKWKHODWHUDOV7RDFFRXQWIRUWKLVHIIHFWHYHU\WLPH a stage was added, the length of the horizontal lateral in both the LQMHFWRUDQGWKHSURGXFHUZDVLQFUHDVHGE\RQHKDOIWKHVSDFLQJ EHWZHHQVWDJHVP

0.8981

( )

MD (m)

Pipe Diameter (in)

where ǻ3res is BHPinj - BHPprod03D qLVÀRZUDWHRIWKHUHVHUYRLUNJV ȝLVWKHYLVFRVLW\RIÀXLGFSRUIRUXQLWFRQVLVWHQF\ LQ(TXDWLRQ03DV DQGD is the distance between the injector DQGSURGXFHUP  The Soultz project was used to make a baseline estimate for WKHWUDQVPLVVLYLW\RIDQ(*6UHVHUYRLU'XULQJWKHFLUFXODWLRQWHVWDW6RXOW]DNJVÀRZUDWHZDVVXVWDLQHGEHWZHHQ *3.DQG*3.7KHRSHQKROHVHFWLRQRIWKHZHOOVZDVURXJKO\ PDQGWKHZHOOVHSDUDWLRQZDVURXJKO\P8VLQJWKRVH parameters, and using the reservoir and wellbore properties given LQ7DEOHDQG7DEOHZKLFKZHUHEDVHGRQWKH6RXOW]FLUFXODWLRQWHVW ZHHVWLPDWHGWKHRYHUDOOUHVHUYRLUWUDQVPLVVLYLW\WREH îm:HXVHGWKLVYDOXHDVRXUEDVHOLQHWUDQVPLVVLYLW\ for a single stage. 7KH H[DFW QDWXUH DQG JHRPHWU\ RI WKH IUDFWXUH QHWZRUN DW 6RXOW]RUWKHQHWZRUNFUHDWHGLQRXUK\SRWKHWLFDO(*6V\VWHP  is not important for the calculation. We are using a single number, UHVHUYRLUWUDQVPLVVLYLW\WRDFFRXQWIRUWKHDJJUHJDWHÀRZFDSDFLW\ of the created fracture network, whether it was a single, planar

7DEOHGHVFULEHVWKHJHRPHWU\RIWKHLQMHFWLRQDQGSURGXFWLRQZHOOVEDVHGRQWKH6RXOW]ZHOOV*3.DQG*3. LQFOXGLQJ GHSWK FDVLQJ GLDPHWHU DQG LQFOLQDWLRQ 7LVFKQHU HW DO   7DEOH  FRQWDLQV WKH H[WHQGHG ODWHUDOV RQ WKH KRUL]RQWDO ZHOOV ZKLFK ZHUH QRW SUHVHQW LQ *3. DQG *3.  7KH ZHOOERUH OHQJWKVLQFUHDVHGE\PIRUHYHU\VWDJHWKDWZDVDGGHG7KH H[WHQGHGKRUL]RQWDOSDUWRIWKHLQMHFWLRQZHOOZDVDVVXPHGWREH FRPSOHWHGZLWKFDVLQJDQGSHUIRUDWLRQVFDVHGKROHFRPSOHWLRQ  The horizontal lateral of the production well was assumed to be openhole. The roughness of the formation in the openhole was assumed to be 2000 microns. Pressure drop in the perforations was not included in the calculation. 316

Li, et al.

,QVRPHFDOFXODWLRQVGHVFULEHGLQ6HFWLRQ ZHXVHGWKH IXOOVROXWLRQRI*ULQJDUWHQHWDO ZKLFKFDOFXODWHVWKHUPDO GUDZGRZQIRUSDUDOOHOIUDFWXUHVRI¿QLWHVSDFLQJ7KHVROXWLRQDVVXPHVDSHULRGLFERXQGDU\FRQGLWLRQLQ¿QLWHVHULHVRIIUDFWXUHV  7KHVROXWLRQUHTXLUHVDQXPHULFDOLQYHUVH/DSODFHWUDQVIRUP7R SHUIRUPWKHFDOFXODWLRQZHXVHGDFRGHWKDWZDVZULWWHQE\'RH HWDO  $YHU\LPSRUWDQWLQSXWSDUDPHWHUZDVIUDFWXUHKHLJKWVHWWR P7KLVSDUDPHWHUVWURQJO\DIIHFWVWKHUHVXOWVEHFDXVHLWLV SURSRUWLRQDOWRWKHYROXPHRIÀXLGDFFHVVHGGXULQJFLUFXODWLRQ 7KLVLVDGLI¿FXOWYDOXHWRFKDUDFWHUL]HDQGPZDVFKRVHQDV DIDLUO\FRQVHUYDWLYHYDOXH

K\GUDXOLFIUDFWXUHDGHQVHQHWZRUNRIVWLPXODWHGQDWXUDOIUDFWXUH a large, thick, shear stimulated fault zone, or network of both new DQGSUHH[LVWLQJIUDFWXUHV ,QRXUK\SRWKHWLFDO(*6GRXEOHWZHDVVXPHGh to be 200 m. The wellbore spacing is a design parameter. When there were PXOWLSOH VWDJHV ZH DVVXPHG WKDW ÀRZ ZDV HYHQO\ GLVWULEXWHG EHWZHHQHDFKVWDJH7KHUHIRUHWKHÀRZUDWHSHUVWDJHZDVq/S, where SLVWKHQXPEHURIVWDJHV)RUDSSOLFDWLRQLQ(TXDWLRQ WKHWRWDOV\VWHPWUDQVPLVVLYLW\HTXDOWRTS7DEOHVKRZVWKHÀXLG properties used in the calculations.

2.2 Heat Extraction from the Fractured Rock :HXVHGWKHDQDO\WLFDOVROXWLRQGHYHORSHGE\*ULQJDUWHQHW DO WRFDOFXODWHWKHÀXLGWHPSHUDWXUHDWWKHSURGXFHUDVD function of time. This solution assumes a series of parallel planar IUDFWXUHVLQWKHUHVHUYRLUZLWKKHLJKWRIKDQGVSDFLQJRI/7KH distance between the fractures is twice the half fracture spacing, ;e  DQG WKH WRWDO QXPEHU RI IUDFWXUHV HTXDO WR WKH QXPEHU RI VWDJHV LV17KHZDWHUIURPWKHLQMHFWLRQZHOOÀRZVWKURXJKWKH IUDFWXUHVJHWVKHDWHGXSE\WKHIRUPDWLRQDQGHQWHUVWKHSURGXFtion well. Figure 2 shows a schematic of the problem setup.

Table 3. Properties used in heat extraction calculations. Properties

Hot water production q h

Q

Q

L

q Xe

Values

Units

Production time

t

946080000

s

Fracture height

Comments Equivalent to 30 years

h

200

m

-

Formation temperature

TRO

190

°C

-

Injecting temperature

TWO

60

°C

-

Thermal conductivity of rock

KR

3.01248

J/(m*s*°C)

For granite

Heat capacity of rock

cR

1000

J/(kg*°C)

-

Heat capacity of water

cW

4420

J/(kg*°C)

-

Rock density

ȡR

2500

kg/m3

-

Water density

ȡW

873.9

kg/m

3

Number of stages

N

Varies

-

-

Well spacing

L

Varies

m

-

Injection rate

q

Varies

kg/s

-

In formation

Cold water injection

Figure 2. Illustration of the Gringarten et al. (1975) analytical solution.

2.3 Conversion to Electricity Production 7KHUDWHRIWKHUPDOHQHUJ\SURGXFWLRQIURPWKH(*6SURMHFW was calculated with the following equation:

In most of the calculations in this paper, we assumed that the half fracture spacing for our model was large enough that the spacLQJFRXOGEHFRQVLGHUHGLQ¿QLWH$IWHUPDQLSXODWLQJWKHHTXDWLRQ JLYHQE\*ULQJDUWHQHWDO IRUVLQJOHIUDFWXUHZHJRWDQ idealized equation to calculate the water temperature at producer IRUPXOWLSOHIUDFWXUHVZLWKLQ¿QLWHIUDFWXUHKDOIVSDFLQJ 'H¿QLQJTwo for injecting temperature, Tro for original formation temperature, Kr and Kw IRUWKHUPDOFRQGXFWLYLW\RIURFNDQG ZDWHUUHVSHFWLYHO\cr and cw IRUWKHKHDWFRQGXFWLYLW\RIURFNDQG water, Q for the water injection rate per fracture per unit height, ȡr and ȡw IRUGHQVLW\RIURFNDQGZDWHUDQG t for production time, the following equation gives the water outlet temperature, Tw , DVVXPLQJLQ¿QLWHIUDFWXUHVSDFLQJ

 L* Tw = TWO + (TRO
TWO ) *erf  

Q = qCW
T ,

where the ǻ7 LV FKDQJLQJ ZLWK WLPH DQG T LV PDVV ÀRZ UDWH NJV Cw LVWKHKHDWFDSDFLW\RIZDWHU-NJƒ& DQGǻ7 is the WHPSHUDWXUHGLIIHUHQFHLQWKHV\VWHPƒ& :KHQWKHSURGXFHG ÀXLGUHDFKHGƒ&LWZDVDVVXPHGWKDWQRIXUWKHUHOHFWULFLW\ could be produced. &XPXODWLYHHOHFWULFLW\SURGXFWLRQZDVFDOFXODWHGE\DGGLQJ XSWKHKHDWSURGXFWLRQIRUHDFKVSHFL¿FWLPHSHULRGGXULQJWKH SURGXFWLRQXQWLOWKHWRWDOSURGXFWLRQWLPHWG\HDUV DQGPXOWLSO\LQJE\HI¿FLHQF\eff: n

* cR *
R ) / (t )  ,  cW *
W *Q 

(K

E ( td ) =

R

(Q *
t * eff ) , i

i=0

ZKHUHQLVWKHQXPEHURIWLPHSHULRGV:HDVVXPHGWKHHQHUJ\ FRQYHUVLRQ HI¿FLHQF\ WR EH  IURP (TXDWLRQ  IURP 7HVWHU 

ZKHUH4LVFDOFXODWHGE\XVLQJWKHWRWDOPDVVLQMHFWLRQUDWHT

Q=

Symbols

q . N *
R * h

2.4 Present Value (PV) Calculation 7KH 139 RI WKH SURMHFW ZDV FDOFXODWHG E\ WKH IROORZLQJ equation:

The values of the properties we put in the previous equations DUHVKRZQLQ7DEOH 317

Li, et al.

td=T

NPV ( i,n ) =


td=0

P * E ( td ) , td (1+ i )

where i is the discount rate, PLVWKHSULFHRIHOHFWULFLW\ td is the SURGXFWLRQWLPH\HDUV DQGT is the total production time for the SURMHFW\HDUV :HDVVXPHGWKHSULFHRIHOHFWULFLW\WREHFHQWV per kWh and the discount rate to be 16%.

2.5 Present Value (PV) Optimization We calculated the NPV for different combinations of parameters. The optimization algorithm was simple. Values were XQLIRUPO\VDPSOHGWKURXJKRXWWKHVROXWLRQVSDFHDQGWKHFRPbination of parameters that gave the highest NPV was selected. %HFDXVHWKHWUXO\RSWLPDOSRLQWVOD\EHWZHHQWKHVDPSOHSRLQWV WKLVZDVDQDSSUR[LPDWHRSWLPL]DWLRQSURFHGXUH7KHGLVWULEXWLRQV of parameters are given in Table 4. Several separate optimization calculations were performed. )RUWKH¿UVWVHWRIFDOFXODWLRQVWUDQVPLVVLYLW\ZDVVHWWRWKHEDVHOLQHYDOXHRIîm:HFDOFXODWHGWKHPD[LPXPSRVVLEOH ÀRZUDWHIRUHDFKFRPELQDWLRQRIZHOOVSDFLQJDQGVWDJHV7KHQ ZHIRXQGWKHRSWLPDOÀRZUDWHIRUHDFKFRPELQDWLRQRISDUDPHWHUV ZKLFKZDVVRPHWLPHVHTXDOWRWKHPD[LPXPSRVVLEOHÀRZUDWH  1H[WPD[LPXP139XVLQJWKHRSWLPDOUDWH YHUVXVWKHQXPEHU RIVWDJHVZDVFDOFXODWHGIRUHDFKYDOXHRIZHOOVSDFLQJ)LQDOO\ WKH PD[LPXP 139 ZDV FDOFXODWHG IRU HDFK QXPEHU RI VWDJHV using the optimal rate and spacing. ,QWKHQH[WVHWRIDQDO\VHVWKHRSWLPL]DWLRQSURFHGXUHZDV SHUIRUPHGXVLQJWKUHHGLIIHUHQWYDOXHVRIUHVHUYRLUWUDQVPLVVLYLW\ 7KHDQDO\VLVZDVSHUIRUPHGIRUDQGPHWHUZHOOVSDFLQJ In the third set of calculations, the optimization was performed ZLWK WKH REMHFWLYH RI PD[LPL]LQJ FXPXODWLYH KHDW SURGXFWLRQ instead of NPV. These calculations used the baseline value for WUDQVPLVVLYLW\ )LQDOO\ WKH RSWLPL]DWLRQ SURFHGXUH ZDV UHSHDWHG XVLQJ WKH EDVHOLQHVHWWLQJVDQG¿QLWHIUDFWXUHVSDFLQJZLWKDODWHUDOOHQJWK of 1000 m.

Figure 3. The maximum flow rate for wells with different spacing and stages.

Figure 4. Typical temperature and cumulative production profile.

Table 4. The ranges of parameters in the optimization process.

Parameters Flow rate Well spacing Stages Transmissivity

Minimum Value

Maximum Value

Total Number of Values Distribution

qmax*log(-4)

qmax

40

log

100 m

800m

8

linear

1

30

30

linear

1.02333E-13

9.21E-13

3

linear

Figure 5. Net present value versus flow rate for 400 meter well spacing and 5, 15, and 25 stages.

3.3 Optimal NPV and Flow Rate for Different Well Spacing

3. Results

$VDQH[DPSOHRIWKHRSWLPL]DWLRQSHUIRUPHGE\WKHFRGH Figure 5 shows NPV versus circulation rate for 400 meter spacLQJDQGDQGVWDJHV)LJXUHVKRZVWKHÀXLGÀRZUDWH to achieve the optimal NPV for different values of wells spacing and number of stages. The optimal NPV for multiple well spacing VHWWLQJVYHUVXVWKHQXPEHURIVWDJHVLVJLYHQLQ)LJXUH

3.1 Maximum Flow Rate )LJXUH  VKRZV WKH PD[LPXP SRVVLEOH ÀRZ UDWH IRU ZHOOV ZLWKVSHFL¿FQXPEHURIVWDJHVDQGZHOOVSDFLQJDVVXPLQJWKH WUDQVPLVVLYLW\LV 3.07*10
13 m 3 .

3.2 Temperature and Cumulative Electricity Production Profile

3.4 Optimal NPV for Different Transmissivity )LJXUHDQG)LJXUHVKRZVWKHRSWLPDO139XQGHUGLIIHUHQW WUDQVPLVVLYLW\VHWWLQJVIRUZHOOVZLWKVSDFLQJRIPDQG PUHVSHFWLYHO\

)LJXUH  VKRZV WKH WHPSHUDWXUH DQG FXPXODWLYH HOHFWULFLW\ SURGXFWLRQYHUVXVSURGXFWLRQWLPHIRUDW\SLFDOZHOOZLWKVWDJHV and 400 m spacing. 318

Li, et al.

Figure 10. Maximum NPV with optimal combinations of flow rate and well spacing.

Figure 6. The rates to achieve optimal NPV for different spacing settings.

Figure 11. Optimal rates for each number of stages. Figure 7. Optimal NPV for different spacing setting.

Figure 12. Comparison of the NPV between optimizing NPV and maximizing heat production.

Figure 8. The optimal NPV for wells with 200 m spacing and different transmissivity.

3.5 Maximum NPV and Optimal Combination of Flow Rate and Well Spacing )LJXUHVKRZVWKHPD[LPXP139ZLWKWKHRSWLPDOFRPELQDWLRQVRIÀRZUDWHDQGVSDFLQJIRUHDFKQXPEHURIVWDJHV7KH RSWLPDOUDWHWKDWJHQHUDWHVWKHPD[LPXP139IRUWKHRSWLPDO combination of parameters is given in Figure 11.

3.6 Comparison Between Optimizing NPV and Maximizing Heat Production )LJXUH)LJXUH)LJXUHVKRZ139KHDWSURGXFWLRQ ÀRZUDWHDQGRSWLPDOZHOOVSDFLQJGHSHQGLQJRQZKHWKHUWKH SURFHGXUHZDVGHVLJQHGWRPD[LPL]H139RUWRWDOKHDWSURGXFtion.

Figure 9. The optimal NPV for wells with 800 m spacing and different transmissivity.

319

Li, et al.

Figure 16. Optimal flow rate versus number of stages with different values of fracture spacing using the finite fracture spacing Gringarten solution.

Figure 13. Comparison of the heat production between optimizing NPV and maximizing heat production.

4. Discussion 4.1 Maximum Flow Rate )LJXUHVKRZVWKHPD[LPXPÀRZUDWHWKURXJKWKHV\VWHP as a function of the number of stages and for different values of IUDFWXUHVSDFLQJ7KHPD[LPXPUDWHLVOLPLWHGE\SUHVVXUHGURS in the wellbore and in the reservoir. At low numbers of stages, WKHSUHVVXUHGURSLVDOPRVWHQWLUHO\LQWKHUHVHUYRLU7KHSUHVVXUH GURS LQ WKH UHVHUYRLU VFDOHV LQYHUVHO\ ZLWK VSDFLQJ VR JUHDWHU rates are possible with lower spacing. The pressure drop in the UHVHUYRLUVFDOHVOLQHDUO\ZLWKWKHQXPEHURIVWDJHVDQGVRDVWKH number of stages grows larger, the proportion of pressure drop in WKHZHOOERUHVJURZV$VDUHVXOWÀRZUDWHWKURXJKWKHV\VWHPV JURZV URXJKO\ OLQHDUO\ ZLWK QXPEHU RI VWDJHV ZKHQ WKHUH DUH fewer stages and greater well separation. As the overall pressure drop in the reservoir increases, the marginal effect of adding stages increases, as the pressure drop in the wellbore limits further LQFUHDVHLQUDWH7KLVDQDO\VLVZDVGHVFULEHGLQJUHDWHUGHWDLOE\ 6KLR]DZDDQG0F&OXUH 

Figure 14. Comparison of the well spacing between optimizing NPV and maximizing heat production.

4.2 Thermal Drawdown )LJXUHVKRZVDQH[DPSOHRISURGXFHGWHPSHUDWXUHYHUVXV time for a particular combination of parameters. In this case, WKHUPDOGUDZGRZQEHJLQVDIWHURQO\DIHZ\HDUVEXWSURJUHVVHV VORZO\HQRXJKWKDWWKHFXWRIIWHPSHUDWXUHRIƒ&LVQRWUHDFKHG XQWLOQHDUO\\HDUV)RUVORZHUUDWHVWKHWHPSHUDWXUHGHFOLQH is more gradual, but for higher rates, thermal decline can be rather abrupt.

4.3 Optimal Flow Rate Figure 5 shows NPV as a function of circulation rate for 400 meter well spacing and 5, 15, and 25 stages. If circulation rate is too high, thermal breakthrough is too rapid and NPV is not PD[LPL]HG,IFLUFXODWLRQUDWHLVWRRORZWKHUPDOEUHDNWKURXJK LV OLPLWHG EXW WKH SURGXFWLYH FDSDFLW\ RI WKH UHVHUYRLU LV QRW PD[LPL]HG7KHFXUYHVGRQRWH[WHQGEH\RQGWKHPD[LPXPUDWH SRVVLEOH ,Q )LJXUH  WKH RSWLPDO UDWH LV DOZD\V OHVV WKDQ WKH PD[LPXPSRVVLEOHUDWH%XWIRUVRPHFRPELQDWLRQVRISDUDPHWHUV WKH RSWLPDO UDWH LV HTXDO WR WKH PD[LPXP UDWH DQG WKH FXUYHV PRQRWRQLFDOO\LQFUHDVH Figure 6 shows the NPV-optimizing circulation rate as a funcWLRQRIWKHQXPEHURIVWDJHVDQGIRUDYDULHW\RIYDOXHVIRUZHOO spacing. Two distinct trends are apparent.

Figure 15. NPV versus number of stages with different values of fracture spacing using the finite fracture spacing Gringarten solution.

3.7 Effect on Finite Fracture Spacing In the previous sections, spacing between the fracture stages ZDVDVVXPHGLQ¿QLWH7KLVZRXOGRQO\EHDQDGHTXDWHDVVXPSWLRQ LIWKHVWDJHVZHUHDGHTXDWHO\VSDFHG7RWHVWWKHHIIHFWRIWKLVDVsumption, we repeated the optimization using the full Gringarten HW DO   VROXWLRQ ZKLFK DVVXPHV ¿QLWH IUDFWXUH VSDFLQJ The total lateral length was assumed to be 1000 m. The spacing between stages was assumed to be the total lateral length, 1000 PGLYLGHGE\WKHQXPEHURIVWDJHVPLQXVRQH 320

Li, et al.

cases, increasing the number of stages increases NPV. At lower circulation rate, increasing the number of stages results in a linear increase in NPV. Some diminishing return is seen, but even IRUVWDJHVWKHUHLVFRQVLGHUDEOHPDUJLQDO139IURPDGGLQJ VWDJHV,QFUHDVLQJIURPRQHWRWKLUW\VWDJHVUHVXOWVLQDURXJKO\ [LQFUHDVHLQ139 )LJXUH  VKRZV WKH RSWLPDO ÀRZ UDWH IRU HDFK QXPEHU RI stages, assuming the optimal well spacing. The optimal rate increases, as the addition of further stages increases the volume of URFNDFFHVVHGE\WKHV\VWHPDOORZLQJKHDWWREHH[WUDFWHGPRUH UDSLGO\ZLWKDVLPLODUWKHUPDOGUDZGRZQ

In cases with smaller number of stages and shorter well VSDFLQJWKHRSWLPDOUDWHLVOLPLWHGE\WKHSRWHQWLDOIRUWKHUPDO EUHDNWKURXJK )OXLG LV FLUFXODWHG PRUH VORZO\ WKDQ LV SRVVLEOH WKURXJKWKHV\VWHP)RUH[DPSOH)LJXUHVKRZVWKDWIRUWKHRQH VWDJHPVSDFLQJFDVHWKHPD[LPXPFLUFXODWLRQUDWHLVDURXQG 14 kg/s, but Figure 6 shows that the optimal rate is around to 2 kg/s. This is not an optimal design, because if the wells were spaced IXUWKHUDSDUWWKHUPDOEUHDNWKURXJKZRXOGEHIXUWKHUGHOD\HGDQG NPV would be increased. Increasing the spacing would lower the PD[LPXPFLUFXODWLRQUDWHEXWWKHUHZRXOGEHQRSHQDOW\EHFDXVH WKHRSWLPDOUDWHLVORZHUWKDQWKHPD[LPXPUDWH,QFUHDVLQJWKH QXPEHURIVWDJHVLQFUHDVHV139)LJXUH EHFDXVHLWHQDEOHVD higher total rate to be used without accelerating thermal breakWKURXJKEHFDXVHLWVSUHDGVÀRZRYHUDJUHDWHUQXPEHURIIUDFWXUHV In cases with a greater number of stages and greater well VSDFLQJLWLVRSWLPDOWRFLUFXODWHÀXLGDWWKHPD[LPXPSRVVLEOH rate. In these cases, increasing the number of stages increases 139)LJXUH SULPDULO\EHFDXVHLWLQFUHDVHVWKHWRWDOÀRZUDWH WKURXJK WKH V\VWHP ,Q WKLV FDVH LQFUHDVLQJ WKH ZHOO VSDFLQJ further would not be advantageous because it would lower the WRWDOÀRZUDWHZLWKRXWKDYLQJDVXI¿FLHQWO\EHQH¿FLDOHIIHFWLQ GHOD\LQJ WKHUPDO GUDZGRZQ7KH RSWLPDO GHVLJQ RFFXUV DW WKH PLQLPXPZHOOVSDFLQJZKHUHLWLVVWLOORSWLPDOWRFLUFXODWHÀXLG DWWKHPD[LPXPSRVVLEOHUDWH )LJXUHVKRZV139DVVXPLQJFLUFXODWLRQDWWKHRSWLPDOUDWH  DVDIXQFWLRQRIZHOOVSDFLQJDQGQXPEHURIVWDJHV([FHSWIRU the largest number of stages, the highest NPV is reached with the 600 meter well spacing. For 25 stages and larger, the 400 meter well spacing has higher NPV. With such a large number of stages, 139 LV FRQWUROOHG E\ PD[LPXP ÀRZ UDWH UDWKHU WKDQ WKHUPDO breakthrough, and lowering the spacing allows rate to be increased.

4.6 Optimizing NPV Versus Optimizing Cumulative Heat Extraction of 30 Years 'HSHQGLQJ RQ SURMHFW REMHFWLYHV FLUFXODWLRQ UDWH PD\ EH FKRVHQ WR RSWLPL]H HLWKHU 139 RU FXPXODWLYH KHDW H[WUDFWLRQ %HFDXVH ZH DVVXPHG D FRQVWDQW HI¿FLHQF\ IRU FRQYHUWLQJ KHDW WRHOHFWULFLW\WKLVZDVHTXLYDOHQWWRPD[LPL]LQJWRWDOHOHFWULFLW\ production. An optimization based on NPV will favor higher rates EHFDXVHLWLVPRUHYDOXDEOHWRSURGXFHHOHFWULFLW\VRRQHU)LJXUH 12 compares the NPV depending on if the rate and spacing are VHOHFWHG WR PD[LPL]H 139 RU WRWDO HOHFWULFLW\ SURGXFWLRQ 7KH RSWLPL]DWLRQIRUWRWDOHOHFWULFLW\SURGXFWLRQUHVXOWVLQDQ139WKDW LVVXUSULVLQJO\FORVHWRWKHRSWLPL]DWLRQIRU1397KHQRQVPRRWKQHVVRIWKHFXUYHLVFDXVHGE\WKHHUURULQKHUHQWWRRXULQH[DFW RSWLPL]DWLRQDOJRULWKP)LJXUHVKRZVDJJUHJDWHKHDWSURGXFWLRQIRUWZRFDVHVPD[LPL]LQJHLWKHU139RUKHDWSURGXFWLRQ,Q VRPHFDVHVWKHKHDWSURGXFWLRQLVQHDUO\HTXDOLQWKHWZRFDVHV DQGLQRWKHUFDVHVWKH\DUHIDLUO\GLIIHUHQW)LJXUHVKRZVWKDW WKHPD[LPL]DWLRQRIWRWDOSURGXFWLRQFDVHWHQGVWR¿QGDVOLJKWO\ greater well spacing to be optimal. The abrupt difference in total KHDW SURGXFWLRQ EHWZHHQ  DQG  VWDJHV )LJXUH   RFFXUV because the optimization algorithm chooses to use a lower well VSDFLQJVWDUWLQJDWVWDJHV)LJXUH 7KH139EHWZHHQWKHVH WZRFDVHVVKRZVRQO\DVOLJKWGHYLDWLRQ)LJXUH VXJJHVWLQJ WKDWLQWKLVFDVHGHFUHDVLQJZHOOVSDFLQJUHVXOWVLQRQO\DVOLJKW increase in NPV, but a substantial drop in total heat production.

4.4 Effect of Reservoir Transmissivity )LJXUHDQG)LJXUHVKRZWKH139DWRSWLPDOUDWHYHUVXV number of stages for three different values of reservoir transmisVLYLW\DWZHOOVSDFLQJRIPDQGPUHVSHFWLYHO\$WORZ ZHOOVSDFLQJWKH139LVFRPSOHWHO\LQVHQVLWLYHWRUHVHUYRLUWUDQVPLVVLYLW\7KLVLVEHFDXVHLWLVRSWLPDOWRFLUFXODWHÀ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ÀRZUDWHPXFKE\LQFUHDVLQJWKHQXPEHU RIVWDJHV7KHUHIRUHIRUWKHKLJKHVWWUDQVPLVVLYLW\FDVHWKH m spacing, and the largest number of stages, the marginal NPV for adding stages is limited.

4.7 Finite Fracture Spacing Figures 15 and 16 show the NPV versus number of stages and optimal rate versus number of stages for four values of well spacLQJFDOFXODWHGXVLQJWKH*ULQJDUWHQVROXWLRQZLWK¿QLWHIUDFWXUH VSDFLQJDVVXPLQJDWRWDOODWHUDOOHQJWKRIP 7KHVH¿JXUHV FDQEHFRPSDUHGWR)LJXUHVDQGZKLFKZHUHFDOFXODWHGZLWK WKH*ULQJDUWHQVROXWLRQDVVXPLQJLQ¿QLWHIUDFWXUHVSDFLQJ7KHHIIHFWRIWKH¿QLWHIUDFWXUHVSDFLQJLVJUHDWHVWIRUWKHODUJHVWQXPEHU of stages, when the spacing between each stage is smallest. The optimal fracture spacing is a bit greater. In this case, the 600 m VSDFLQJRXWSHUIRUPVWKHPVSDFLQJIRUVWDJHVXQOLNHIRUWKH LQ¿QLWHVSDFLQJFDVHZKHQWKHPVSDFLQJKDVDKLJKHU139 7KHRSWLPDOUDWHDWVWDJHVDQGRUPVSDFLQJ WXUQVRXWWREHKLJKHUWKDQIRUWKHLQ¿QLWHVSDFLQJFDVH,QWKHVH FDVHV139LVPD[LPL]HGE\ÀRZLQJDWWKHPD[LPXPSRVVLEOH ÀRZUDWH,QWKHFDVHVZKHUHWKHLQ¿QLWHIUDFWXUHVSDFLQJVROXtion was used, the calculations of pressure loss in the wellbore DVVXPHGDVSDFLQJRIP)RUVWDJHVWKLVZDVDQHIIHFWLYH ODWHUDOOHQJWKRIP7KHJUHDWHUODWHUDOOHQJWKUHGXFHGWKH PD[LPXPSRVVLEOHÀRZUDWHUHODWLYHWRWKH¿QLWHVSDFLQJFDVH

4.5 Optimal Parameters Figure 10 shows the optimal NPV as a function of the numEHURIVWDJHVDVVXPLQJWKHRSWLPDOVSDFLQJDQGÀRZUDWH,QDOO 321

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7KHWRWDORSWLPDOV\VWHP139IRUVWDJHVLVRQO\PRGHVWO\ ORZHUIRUWKH¿QLWHIUDFWXUHVSDFLQJFDVHFRPSDUHGWRWKHLQ¿QLWH fracture spacing calculation. This indicates that a lateral length RIPDFFHVVHVVXI¿FLHQWURFNYROXPHIRUHFRQRPLFSURGXFWLRQ2EYLRXVO\LIWKHODWHUDOOHQJWKDQGVWDJHVSDFLQJZHUHWRR ORZWKHUPDOGUDZGRZQZRXOGRFFXUUDSLGO\DVDOOWKHKHDWVWRUHG LQWKHURFNYROXPHZDVGHSOHWHG%XWIXUWKHULQFUHDVHVLQODWHUDO OHQJWKEH\RQGPPD\QRWUHVXOWLQPXFKDGGLWLRQDO139 A back of the envelope calculation can be performed to check WKLV UHVXOW7KH YROXPH RI URFN LQ D FXEH [[ P LV 1.2×10 m$VVXPLQJDWKHUPDOGUDZGRZQRIƒ&IURPƒ& WRƒ& WKLVYROXPHFRQWDLQVî9 kWhr of heat. With a GUDZGRZQRIƒ&IURPƒ&WRWKHLQMHFWLRQWHPSHUDWXUHRI ƒ& WKHKHDWFRQWHQWLVî10 kWhr. The latter value repreVHQWVDQXSSHUERXQGRQWKHUPDOUHFRYHU\DFKLHYHGZLWKSHUIHFW KHDWVZHHSHI¿FLHQF\DQGLIWKHHQWLUHUHVHUYRLUFDQEHFRROHG WR WKH LQMHFWLRQ ÀXLG WHPSHUDWXUH EHIRUH WKHUPDO EUHDNWKURXJK RFFXUV +RZHYHU LW QHJOHFWV KHDW FRQGXFWLRQ IURP RXWVLGH RI WKHYROXPHRIURFNIURPWKHWRSERWWRPDQGVLGHV ,QWKH stage, 600 m spacing case shown in Figures 15 and 16, the total KHDWSURGXFWLRQIURPWKHPRGHOZDVURXJKO\î9 kWhr. This SUHGLFWHGUHFRYHU\LVWKHRUHWLFDOO\SRVVLEOHWKRXJKFOHDUO\UHSUHVHQWVDQDJJUHVVLYHDVVXPSWLRQDERXWKHDWVZHHSHI¿FLHQF\2Q the other hand, the assumption of a “formation” height of 200 m PD\EHFRQVHUYDWLYHIRUDV\VWHPZLWKDVSDFLQJRIXSWRP $FKLHYLQJKLJKKHDWVZHHSHI¿FLHQF\ZRXOGEHVLJQL¿FDQWO\ DLGHGE\WKHXVHRIPXOWLSOHVWDJHIUDFWXULQJZKLFKZRXOGHQDEOH a large number of stimulated fractures to be formed. If a cased hole completion were used in one of the wells, packers could be used to isolate sections of the wellbore and pump diverting PDWHULDOVLIVRPHIUDFWXUHSDWKZD\VZHUHH[FHVVLYHO\FRQGXFWLYH Flow through perforations from the cased hole would serve as DWKURWWOHWKDWZRXOGSUHYHQWH[FHVVLYHUDWHIURPDQ\SDUWLFXODU VWDJH%HFDXVHKHDWVZHHSLVXQOLNHO\WREHFORVHWRLGHDOLWZRXOG SUDFWLFDOO\EHGHVLUDEOHWRRYHUGHVLJQWKHV\VWHPZLWKDORQJHU ODWHUDODQGPRUHVWDJHVWKDQZRXOGEHQHFHVVDU\EDVHGRQLGHDO VZHHSHI¿FLHQF\

FLUFXODWLRQ UDWH E\ DOORZLQJ KLJKHU UDWHV DQG GHOD\LQJ WKHUPDO breakthrough. The optimal wellbore spacing occurs at the lowest possible VSDFLQJZKHUH139LVPD[LPL]HGE\ÀRZLQJDWWKHPD[LPXP SRVVLEOH ÀRZ UDWH WKURXJK WKH V\VWHP 3UDFWLFDOO\ VSHDNLQJ LW PD\EHDGYDQWDJHRXVWRXVHDVPDOOHUVSDFLQJEHFDXVHWKLVPD\ UHGXFHIUDFWXULQJFRVWVDQGLQFUHDVHWKHSUREDELOLW\RIHVWDEOLVKLQJ a good connection between the wellbores. $VORQJDVWKH139LVRSWLPL]HGE\ÀRZLQJDWWKHPD[LPXP SRVVLEOHUDWHWKURXJKWKHV\VWHPLWLVDGYDQWDJHRXVWRLQFUHDVH IUDFWXUHWUDQVPLVVLYLW\IRUH[DPSOHE\XVLQJSURSSDQW 7KLVZLOO RFFXUXQOHVVWKHIUDFWXUHWUDQVPLVVLYLW\LVDOUHDG\YHU\KLJKRU the well spacing is low. 2SWLPL]LQJIRUPD[LPXPKHDWH[WUDFWLRQUDWKHUWKDQ139KDG DVXUSULVLQJO\VPDOOHIIHFWRQWKHUHVXOWV,WZDVIRXQGWKDWZLWKLQ WKHHUURURIRXUDSSUR[LPDWHRSWLPL]DWLRQVFKHPHWKHUHVXOWVZHUH not too different. In some cases, it appeared that decreasing wellERUHVSDFLQJPD\UHVXOWLQVXEVWDQWLDOORVVRIWRWDOKHDWH[WUDFWLRQ with minimal improvement of NPV. For design of an actual multiple stage EGS doublet, a more VRSKLVWLFDWHGSK\VLFDOPRGHODQGRSWLPL]DWLRQVFKHPHZRXOGEH needed. In addition, optimal parameters would need to be calFXODWHGFRQVLGHULQJPDUJLQDOFRVWV)RUH[DPSOHDGGLQJVWDJHV ZRXOGLQFUHDVHFRVWE\UHTXLULQJPRUHFDVLQJPRUHFHPHQWDQG a greater volume of water during stimulation. Increasing well spacing would increase water consumption needed per stage. In addition to marginal costs, a go/no go decision on a commercial EGS project would require considering of all costs associated ZLWKWKHSURMHFWRYHUWKHOLIHWLPHRIWKHV\VWHP%HFDXVHPDNHXS ZHOOVFRXOGEHGULOOHG(*6ZHOOVFRXOGEHSHUPLWWHGWRWKHUPDOO\ decline at a timescale shorter than the life span of the power plant. The calculations show that a lateral length of 1000 m is VXI¿FLHQWWRDFFHVVDQDGHTXDWHYROXPHRIURFN7KLVOHQJWKLV FRPSDUDEOHWRPDQ\RLODQGJDVZHOOVGULOOHGLQXQFRQYHQWLRQDO resources. 7KLV VHQVLWLYLW\ DQDO\VLV GHPRQVWUDWHV WKDW PXOWLSOH VWDJH IUDFWXULQJGHVLJQVUDGLFDOO\LPSURYH(*6HFRQRPLFSHUIRUPDQFH 'HYHORSLQJ DQG WHVWLQJ WHFKQRORJ\ WKDW HQDEOHV PXOWLSOH VWDJH fracturing and horizontal drilling for EGS needs to be a high SULRULW\IRUIXWXUHGHYHORSPHQW

4.8 Practical Issues Affecting Well Spacing 3UDFWLFDOO\WKHUHPD\EHVHYHUDOGLVDGYDQWDJHVWRDWWHPSWLQJ ODUJHZHOOVSDFLQJEH\RQGPHWHUV)LUVWDVWKHVL]HRI the stimulated region increases, the volume of water required for LQMHFWLRQZLOOLQFUHDVHQRQOLQHDUO\GXHWRLQFUHDVHGOHDNRIIDQG increased height growth. The greater the distance between the LQMHFWLRQDQGSURGXFWLRQZHOOVWKHPRUHGLI¿FXOWLWZLOOEHFRPH to establish a good connection between the wells. On the other KDQGODUJHVWLPXODWLRQGHVLJQHGIRUJUHDWHUVSDFLQJPD\UHVXOW in a greater effective fracture height than the 200 m assumed in our calculations.

Acknowledgements 7KDQN\RXWRWKH&RFNUHOO6FKRRORI(QJLQHHULQJDWWKH8QLYHUVLW\ RI 7H[DV DW$XVWLQ IRU VXSSRUWLQJ WKLV UHVHDUFK 7KDQN \RXYHU\PXFKWR7KRPDV'RHIRUKLVWKRXJKWIXOFRPPHQWVRQ this subject and also for sharing his code for implementing the *ULQJDUWHQHWDO ¿QLWHIUDFWXUHVSDFLQJFDOFXODWLRQ

References

5. Conclusions

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