Effective solid-to-fluid heat transfer coefficient in EGS reservoirs

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1 Engineering Conerence International ECI Digital Archive 5th International Conerence on Porou Meia an Their Alication in Science, Engineering an Inutry Reeree Proceeing Summer Eective oli-to-lui heat traner coeicient in EGS reervoir Xiao-Long Ouyang Tinghua Univerity Rui-Na Xu Tinghua Univerity Pei-Xue Jiang Tinghua Univerity Follow thi an aitional work at: htt://c.engconintl.org/orou_meia_v Part o the Material Science an Engineering Common Recommene Citation Xiao-Long Ouyang, Rui-Na Xu, an Pei-Xue Jiang, "Eective oli-to-lui heat traner coeicient in EGS reervoir" in "5th International Conerence on Porou Meia an Their Alication in Science, Engineering an Inutry", Pro. Kambiz Vaai, Univerity o Caliornia, Riverie; Pro. Arian Bejan, Duke Univerity; Pro. Akira Nakayama, Shizuoka Univerity; Pro. Oronzio Manca, Secona Univerità egli Stui Naoli E, ECI Symoium Serie, (014). htt://c.engconintl.org/orou_meia_v/13 Thi Conerence Proceeing i brought to you or ree an oen acce by the Reeree Proceeing at ECI Digital Archive. It ha been accete or incluion in 5th International Conerence on Porou Meia an Their Alication in Science, Engineering an Inutry by an authorize aminitrator o ECI Digital Archive. For more inormation, leae contact ranco@bere.com.

2 Proceeing o the 5th International Conerence on Porou Meia an it Alication in Science an Engineering ICPM5 June -7, 014, Kona, Hawaii EFFECTIVE SOLID-TO-FLUID HEAT TRANSFER COEFFICIENT IN EGS RESERVOIRS Xiao-Long Ouyang, Rui-Na Xu, Pei-Xue Jiang* Key Laboratory or Thermal Science an Power Engineering o Minitry o Eucation Key Laboratory o CO Utilization an Reuction Technology, Deartment o Thermal Engineering, Tinghua Univerity, Beijing, , China ABSTRACT The reent work eveloe a three-equation local thermal non-equilibrium moel to reict the eective oli-to-lui heat traner coeicient in the enhance geothermal ytem reervoir bae on the volume averaging metho. Due to the high rock-to-racture ize ratio, the oli thermal reitance eect in the internal rock cannot be neglecte in the eective oli-to-lui heat traner coeicient. The reent three-equation local thermal non-equilibrium moel can conier the ynamic variation o the oli thermal reitance in tranient heat traner by introucing the enetration temerature ierence. The moel wa valiate by comarion with ore-cale numerical imulation an macro-cale LTNE moel numerical imulation. The reult how that the three-equation local thermal non-equilibrium moel ha a high accuracy. INTRODUCTION The enhance geothermal ytem (EGS) extract the thermal energy rom ee unergroun ry hot rock to rouce electric ower, which i a otential alternative energy in the near uture. To achieve thi, a coolant lui will be injecte into the reervoir with ry hot rock an heate to a higher temerature. Thi heat traner roce can be imulate by the local thermal non-equilibrium (LTNE) moel with coniering the oli-to-lui heat traner in orou meia. In the LTNE moel, the olito-lui heat traner i calculate by the hae volume average temerature ierence. However, the claic LTNE moel uually neglect the oli thermal reitance in the oli-to-lui heat traner coeicient, while the oli thermal reitance lay an imortant role in the oli-to-lui heat traner in EGS. Thi i becaue the EGS reervoir are conit o racture rock, where the ize o each iece o rock i much larger than that o the racture. A a reult, the rock center temerature can be much higher than the olilui interacial temerature. The correcte oli-to-lui heat traner coeicient with the oli thermal reitance eect i calle the eective oli-to-lui heat traner coeicient. There are two main aroache to conier the oli thermal reitance eect in orou meia. The irt aroach i aing a contant correction in the oli-tolui heat traner coeicient: β k = h, w + h where β=6, 8, 10 or lab, cyliner, an here, reectively, by Stuke [1], an β=(1-ε) by Gelet et al. []. Thi aroach i imle but neglect ome ynamic inormation in the tranient oli thermal conuction, which will be hown in the ollowing ection. The econ aroach i couling the raial heat conuction equation in the article with the macro-cale energy equation, a by Hanley an Hegg [3]. Thi aroach can conier more ynamic inormation in the tranient oli thermal conuction, but require one more imenion to calculate the raial heat conuction which conume more calculation time. With combining thee two aroache, the reent work will ut orwar a more recie an eicient moel or the eective oli-to-lui heat traner coeicient to reict the temerature iel in EGS reervoir bae on volume averaging metho. NOMENCLATURE = thickne o the rock lab h = Eective oli-to-lui heat traner coeicient h = Soli-urace-to-lui heat traner, w coeicient N = racture number r = Raial coorinate in each iece o rock T = Temerature T = Penetration temerature ierence x = x coorinate in macro-cale y = y coorinate in macro-cale z = z coorinate in macro-cale Greek Symbol β = Coeicient o the oli thermal reitance, eine by Eq. (1) (1)

3 γ = Contant coeicient o the oli thermal reitance, eine by Eq. (8) Subcrit e = Eective = Flui hae = Soli hae w = wall urace o the oli hae Other = Intrinic hae average = Macro-cale graient 1 Eective oli-to-lui heat traner coeicient moel z y Flui T x Soli T Figure 1: Schematic iagram o uniormly racture EGS reervoir The LTNE moel in the reervoir can be exree a eρ c + ρ c u T = = k, e T + ah ( T T ) ( 1 e) ρc k, e T ah ( T T ) where h i the eective oli-to-lui heat traner coeicient. Here we ue the notation,, to itinguih the macro-cale an the ore-cale variable. E.g., T i the macro-cale intrinic hae average o the ore-cale temerature, T. Aume the rock in orou meia are in hae o uniorm arallel large lab with a ame thickne o, a hown in Figure 1. In a rereentative element volume, the local thermal conuction occur mainly in the raial irection o each iece o rock. Then the tranient oli thermal conuction in the internal rock can be coniere bae on the econ aroach mentione above: r c T = k r r () (3) Thu, thi equation can be relate to Eq. () with the ollowing bounary conition: ( ), ( w ) k = h T T = h T T w r r = T r = r = T r = 0 w = 0 where r i the raial coorinate o each iece o rock, an T w i the average wall urace temerature. In aition, T i the intrinic hae average o T, thu T 0 (4) = T r (5) A a reult, the eective oli-to-lui heat traner coeicient, h, can be erive reciely rom Eq. ()- (5). However, the ore-cale coorinate, r, bring in one more imenion in the geometry, an Eq. (5) introuce an integral between two variable. Thee equation are not convenient to olve. To erive more eicient moel or h, Eq. (3)-(5) nee to be imliie by moeling h bae on Eq. (1). Note that β in Eq. (1) will be coniere a a variable in the ollowing analyi. 1.1 Penetration temerature ierence equation Deine the enetration temerature ierence: 3 T = ( T T ) 0 r r (6) Here, T i the volume average o temerature ierence, T T, weighte by (r/ ). Since (r/ ) =0 at the center o the rock an (r/ ) =1 at the urace o the rock, T exree the ivergence between the near-wall temerature an the oli average temerature. It alo exree the enetration egree o urace temerature o the oli hae. Thereore, T i an imortant variable to ecribe the tranient oli thermal conuction in macro-cale. Thu, ucale Eq. (3) with volume averaging an Eq. (4)-(6) to eliminate the ore-cale coorinate. Then the enetration temerature ierence equation can be obtaine: ( ) 4 h T T 8k ρc = T T 3 ( w) (7)

4 1. Cloure moeling an the three-equation LTNE moel To cloure the moeling, h can be moele bae on Eq. (1). Since β i coniere a variable or a more recie moel, we rooe an aumtion that β i recirocal to the relative enetration temerature ierence: T T w β = γ (8) T where γ i a contant relate to the hae o the rock. The contant γ can be erive rom the exact olution o one-imenional tranient thermal conuction in a large lab. That i length irection, whoe with were b. The working lui, water, lowe through the racture along the length irection uniormly. The our outer urace vertical to the low irection were covere by inulate wall. The reent calculation comare the cae with ierent racture number N. For all the cae, thee arameter kee the ame: W=L=1m, b=1mm, the total low rate through the ytem wa 0.165kg/, the initial temerature o the ytem wa 145, the inlet lui temerature wa 75. The rock inie the ytem are aume imermeable. L b π γ = 4 (9) 3 Thu, the three-equation LTNE moel i eveloe bae on Eq. (1), (), (7) an (8) to conier the tranient oli thermal conuction in the internal rock: eρ c + ρ c u T = + = 3 k, e T ah ( T T ) ( 1 e) ρc = k, e T ah ( T T ) 4 h ( T T ) 8k ρc ( T T w) (10) where the eective oli-to-lui heat traner coeicient i moele a 1 1 γ k T T w h = h, w + T 1 the wall urace temerature can be obtaine by ( ), ( ) w w (11) h T T = h T T (1) Note that the three-equation LTNE moel can be ue in calculation o one, two or three imenion in macrocale. T, T an T are the three main variable or the equation. Valiation by ore-cale numerical imulation Figure how a mall enhance geothermal ytem with a ew artiicial racture in the rock, whoe croection area wa W W an whoe length wa L. There were N traight racture oene uniormly arallel to the W W N=5 Figure : Schematic iagram o the calculation cae Inlet L Rock Fracture b Outlet Figure 3: Geometry in the ore-cale numerical imulation W N.1 Pore-cale numerical imulation moel Uner thee conition, the ore-cale numerical imulation can be imlemente in only a twoimenional element hown in Figure. The geometry o each element i eicte in Figure 3. Hal o a rock an hal o a racture with ymmetry bounarie were coniere in thi imulation. Since the lui velocity in the racture i low with the maximum Re=165, the laminar low momentum equation wa occuie to olve the ore-cale low iel, an the tranient energy equation to olve the ore-cale temerature iel.. Macro-cale numerical imulation moel For the macro-cale moel, one-imenional low in the length irection were coniere in the governing equation. Becaue o the narrow racture, the oliurace-to-lui heat traner coeicient h,w i a large a more than 000W/(m K), while the oli thermal

5 reitance i larger than 1 h, w. Thu, 1 h, w i negligible in thi calculation. With thi aumtion, the oneimenional three-equation LTNE moel can be written a: ερ c + ρ c u T γ k T T = ε k + a ( T T ) T ( 1 ε) ρc = T γ T T 4k ρc = T T t 3 T T γ k T T ( 1 ε ) k ( ) a T T ( ) (13) ερ c + ρ c u T β k = ε k + a ( T T ) (17) ( 1 ε) ρc T β k ( 1 ε ) k a ( T T = ) where β =6 or large lab. The eective thermal conuctivitie in Eq. (16) an (17) are et or the ame reaon with the three-equation LTNE moel..3 Comarion o reult where the eective thermal conuctivity o each hae can be written a the material thermal conuctivity multilie by it volume raction ince the oli an lui hae are arrange arallel. The thermal ierion in the eective thermal conuctivity o the lui hae can be neglecte ue to the racture are traight. The initial an bounary conition: x = 0: T = T0, = 0 x = L: = = 0 t = 0: T = T = T, T = 0 i (14) where the initial conition T = 0 will lea to ivergence, in act T wa et to be a very mall value in the numerical calculation. The oli rock in thi ytem can be regare a large lab, thu N Nb a =, ε = (15) W W The local thermal equilibrium (LTE) moel an the twoequation LTNE moel with Stuke correlation were alo calculate to comare with the three-equation LTNE moel. Thu, The LTE moel: ερ c + ( 1 ε ) ρc + ρ c u t x (16) T εk ( 1 ε) k = + The two-equation LTNE moel with Stuke correlation: x Figure 4: Comarion between the ore-cale an macrocale numerical imulation Figure 4 how the reult o the lui temerature at the outlet. The racture number enity lay an imortant role on the heat traner erormance o the reent ytem. A hown by the ore-cale numerical imulation reult, or the conition with the contant ma low rate, the lui temerature at the outlet kee the initial temerature or a erio an then ecreae to the inlet temerature with N=100. However, the lui temerature at the outlet ecreae at the very beginning to a relative low temerature with N=5 or N= ue to the larger velocity in each racture. Although the oli-urace-tolui heat traner coeicient h,w i large enough, the oli-to-lui heat traner ability i till not goo enough with a lower racture number enity ue to the larger oli thermal reitance in each iece o rock. It i not eicient to extract the thermal energy rom the rock without a large enough racture number enity. The reult o the three-equation LTNE moel agree very well with thoe o the ore-cale numerical imulation. It imlie that the three-equation LTNE moel ha a high accuracy in reicting the tranient oli thermal conuction eect in orou meia. However, the reult o the LTE moel can only aroach to that o the orecale numerical imulation or the cae with N=100. It mean the cae with more racture uch a N=100 aroximately atiie the local thermal equilibrium

6 conition, while the other two cae o not. It houl be notice that the reult o the LTNE moel without coniering the oli thermal reitance eect in the internal rock will be the ame a thoe o the LTE moel becaue h,w i aume large enough. Hence, the oli thermal reitance eect in the internal rock ominate the heat traner erormance with a maller racture number enity. Thereore, the LTNE moel coniering the oli thermal reitance eect in the internal rock mut be ue or thee two cae. Figure 4 alo comare the three-equation LTNE moel with the two-equation LTNE moel with Stuke correlation. Thee two moel give largely ierent reiction o the outlet lui temerature at the beginning tage with N=5 or N=. At the later tage, thee two reiction are cloe while the latter moel over-reict a little. It can be inerre that the contant β in Stuke correlation cannot reict the ynamic variation o the oli thermal reitance in the beginning tage o tranient thermal conuction. But the threeequation LTNE moel can overcome thi roblem by introucing the enetration temerature ierence. CONCLUSIONS Due to the high rock-to-racture ize ratio in EGS reervoir, the oli thermal reitance eect in the internal rock cannot be neglecte in the eective olito-lui heat traner coeicient. The reent threeequation local thermal non-equilibrium moel can conier the ynamic variation o the oli thermal reitance in tranient heat traner by introucing the enetration temerature ierence. The valiation how that the three-equation local thermal non-equilibrium moel ha a higher accuracy than the two-equation LTNE moel with Stuke correlation, eecially at the beginning tage. The reult alo how that it i not eicient to extract the thermal energy rom the rock without a large enough racture number enity becaue o the larger oli thermal reitance in the internal rock. ACKNOWLEDGEMENT The author woul like to acknowlege uort rom the National Natural Science Founation o China (No ), the National Science Fun or Creative Reearch Grou o China (No ) an the Inutrial Technology Develoment Program (No. B ). REFERENCES [1]Stuke B (1948) Berechnung De Wärmeautauche in Regeneratoren Mit Zylinrichem Un Kugelörmigem Füllmaterial. Angewante Chemie 0: 6-68 []Gelet R, Loret B, Khalili N (013) Thermal Recovery From a Fracture Meium in Local Thermal Non- Equilibrium. Int J Numer Anal Met 15: [3]Hanley D, Hegg PJ (1969) The Eect o Thermal Conuctivity o the Packing Material On Tranient Heat Traner in a Fixe Be. Int J Heat Ma Tran 1: