A THREE-DIMENSIONAL SOLUTION FOR HEAT EXTRACTION FROM A FRACTURE IN HOT DRY ROCK USING THE BOUNDARY ELEMENT METHOD

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1 PROCEEDINGS, Twenty-Seventh Wokshop on Geothemal Resevoi Engineeing Stanfod Univesity, Stanfod, Califonia, Januay 28-30, 2002 SGP-TR-7 A THREE-DIMENSIONAL SOLUTION FOR HEAT EXTRACTION FROM A FRACTURE IN HOT DRY ROCK USING THE BOUNDARY ELEMENT METHOD A. Ghassemi,A.H.-D.Cheng,S.Taasovs Depatment of Geology & Geological Engineeing Univesity of Noth Dakota Gand Foks, ND Depatment of Civil Engineeing Univesity of Mississippi Univesity, MS ABSTRACT Heat extaction fom a factue in a geothemal esevoi is typically modeled based on the assumption that heat conduction is onedimensional and pependicula to the factue. In this pape an integal euation fomulation is used to model the thee-dimensional heat ßow in the esevoi. This method esults in a numeical pocedue in which the discetization of the esevoi geomety is entiely eliminated, leading to a much moe ef- Þcient scheme. In addition to poviding the tempeatue distibution in the esevoi, the thee-dimensional bounday integal euation fomulation povides an efficient means fo calculating the induced themal stesses on the factue suface and in the esevoi. INTRODUCTION The Hot Dy Rock (HDR) concept of geothemal exploitation involves dilling two o moe wells to suitable depths to connect pemeable factues of natual o man-made oigin, injecting cold wate into one well, and ecoveing hot wate fom the othe. A numbe of analytical and numeical solutions exist fo modeling heat extaction fom a factue. See, fo example, Willis-Richads and Walloth (995) fo a compehensive eview. With few exceptionssuchastheþnite element solution by Kolditz (995), GOECRACK, and the bounday element model by Cheng et al., 200; the heat conduction in the esevoi is modeled Figue : Heat extaction fom a plana factue. as one-dimensional and pependicula to the factue suface. The pimay eason fo such simpliþcation is due to the difficulty in modeling an unbounded domain by numeical discetization. In this wok, the govening euations fo the thee-dimensional heat extaction fom a single factue embedded in an inþnite geothemal esevoi ae tansfomed into an integal euation. The pocedues fo solving the esulting integal euation ae descibed and some numeical esults ae pesented. MATHEMATICAL MODEL Figue illustates, schematically, a view of heat extaction fom a hot dy ock system by ciculating wate though a natually existing o man-made factue. The factue is assumed to be ßat and of Þnite size. The geothemal esevoi containing the factue

2 is assumed to be of inþnite extent. Othe assumptions ae simila to these postulated in Cheng et al., 200. SpeciÞcally, it is assumed that (i) all popeties such as factue thickness, pemeability, and esevoi heat capacity ae constant; (ii) the injection ate of cold wate is steady; the esevoi is impemeable to wate and the factue has no stoage capacity; hence the poduction ate of hot wate is eual to the injection ate; and (iii) the heat stoage and dispesion effects in the factue ßuid ßow ae negligible. FRACTURE FLOW It is assumed that the factue width is small such that the ßow in the factue is lamina and govened by the lubication ßow euation: 2 p(x, y) = π2 µ w 3 (x, y); (x, y) x, y A () whee ( 2 ) is the gadient opeato in two spatial dimensions, p is the ßuid pessue, µ the ßuid viscosity, w the factue width, and A is the factue suface (see Figue ). Note that: = w v (2) is the dischage pe unit width, and v is the aveage ßow velocity given by: R w(x,y) v(x, y) = w(x, y) 0 v(x, y, z) dz (3) whee v is the ßow velocity. Assuming that (i) theßuid is incompessible, (ii) the factue wall is impemeable to ßuid ßow, and (iii) the factue width does not change with time, the ßuidcontinuityeuationcanbewittenas: 2 (x, y) =0 (4) whee ( 2 ) is the two-dimensional divegence opeato. Solving euation () fo (x, y) and substituting into euation (4) yields a second ode patial diffeential euation: h i 2 w 3 (x, y) 2 p(x, y) = 0 (5) The above euation can be used to solve fo the pessue distibution within the factue using appopiate bounday conditions and the knownfactuewidth,. Thedischageandaveage velocity can then be obtained fom () and (2). Fo the cuent poblem, the bounday condition is: p = 0 on A (6) n In addition, thee exist ßow singulaities in the factue plane due to the injection and extaction wells. Hence euation (4) must be modiþed to include these singulaities, i.e., 2 (x, y) =Q [δ(x e ) δ(x i )] (7) whee Q is the injection and extaction ate (must be eual), δ is the Diac delta function, X e =[(x x e ), (y y e )], X i =[(x x i ), (y y i )], (x i,y i ) and (x e,y e ) ae the coodinates of the injection and extaction wells, espectively. Similaly, (5) needs to be modiþed to: h i 2 w 3 (x, y) 2 p(x, y) = c [δ(x i ) δ(x e )] (8) whee c = π 2 µq. HEAT TRANSPORT IN FRACTURE The heat tanspot euation fo the factue can be witten as: 2 [(x, y)t(x, y, 0, t)] = 2K ρ w c w T(x, y, z, t) ;fox,y A (9) z whee the factue ßow Þeld, (x, y), isknown by solving the ßuid ßow euation; T is the wate tempeatue; ρ w is the wate density; c w is the speciþc heat of wate; and K is the ock themal conductivity. Note that heat stoage and dispesion tems have been neglected basedonealieþndings (Cheng et al., 200). 2

3 The heat conduction in the ock can be modeled by a diffusion euation in thee dimensions: 2 3 T(x, y, z, t) = ρ c T(x, y, z, t) ; K t fo x,y,z Ω (0) whee ρ is the ock density, c is the speciþc heat of ock, and 2 3 is the thee-dimensional Laplacian opeato. Note that the same notation, T, is used fo the tempeatue of the esevoi ock and wate in the factue, because tempeatue must be continuous between the two media. The govening euations, (9) and (0), ae subject to initial and bounday conditions. The initial tempeatue of the ock and the wateinfactueisassumedtobeaconstant: T(x, y, z, 0) =T o () At the injection point (x i,y i,0), thetempeatue is eual to the injection wate tempeatue: T(x i,y i,0,t)=t wo (2) It is clea that thee exist singulaities at the injection and extaction points; these must be included in the govening euation (9) as souce and sink tems: 2 [(x, y)t(x, y, 0, t)] (3) = 2K T(x, y, z, t) + ρ w c w z Q [T(x e,y e,0,t) δ(x e ) T wo δ(x i )] O, by expanding the uantities unde the divegence opeato and noting the ßow euation (7), the above euation can be witten as: (x, y) 2 T(x, y, 0, t) 2K T(x, y, z, t) ρ w c w z = 0 (4) LAPLACE TRANSFORM In ode to facilitate the use of the Laplace tansfom techniue, it is moe desiable to have a zeo value fo the initial condition. Hence a dimensionless tempeatue deþcit is deþned: T d = T o T (5) T o T wo Euations (4) and (0) peseve the same fom in the new vaiable, i.e., (x, y) 2 T d (x, y, 0, t) = 2K T d (x, y, z, t) ρ w c w z (6) K 2 T d (x, y, z, t) 3 T d (x, y, z, t) =ρ c t The initial condition then becomes: (7) T d (x, y, z, 0) =0 (8) At the injection point, the dimensionless tempeatue deþciency is: T d (x i,y i,0,t)= (9) Fom the ßow singulaity in (7), we can wite (6) in this fom: ρ w c w 2 [(x, y)t d (x, y, 0, t)] T 2K d (x, y, z, t) = (20) z ρ w c w Q [T d (x e,y e,0,t) δ(x e ) δ(x i )] Pefoming Laplace tansfom on (7)-(20), esults in: h ρ w c w 2 (x, y) T e i d (x, y, 0, s) 2K e T d (x, y, z, s) z = (2) ρ w c w Q et d (x e,y e,0,s) δ(x e ) s δ(x i) K 2 3 e T d (x, y, z, s) =sρ c e T d (x, y, z, s) (22) et d (x i,y i,0,s)= s (23) whee s is the Laplace tansfom paamete. 3

4 INTEGRAL EQUATION Euations (2) and (22) ae deþned inthee spatial dimensions. Solving tansient, theedimensional poblems in an inþnite domain still poses a challenge fo moden day computes. Howeve, by utilizing Geen s function, the system can be conveted into a twodimensional integal euation deþnedonthe factue suface, thus signiþcantlyeducing the computational effot. To model the tempeatue in the esevoi due to a continuous point heat souce with stength eu, weintoducethe following euation: K 2 e 3 T d (x, y, z, s) sρ c T e d (x, y, z, s) = eu δ x x 0 (24) whee δ is the Diac delta function, and x 0 is the souce location. The solution of (24) is given by the Geen s function: eg = eu µ 4πK R exp ρ c s R (25) whee: R = (x x 0 ) 2 +(y y 0 ) 2 +(z z 0 ) 2 (26) Then it can be shown that the tempeatue in the esevoi due to a continuous distibution of souces on the factue suface A is given by: ρ w c w 4πK K et d (x, y, z, s) = (27) R h A 2 (x 0,y 0 ) T e i d (x 0,y 0,0,s) f dx 0 dy 0 s f i ρwc w Q T e d (x e,y e,0,s) f e 4πK whee β = ρ c s K, f = R exp ( β R ), f i = R i exp ( β R i ), f e = R e exp ( β R e ),and: R e = R i = R = (x x e ) 2 +(y y e ) 2 + z 2 (28) (x x i ) 2 +(y y i ) 2 + z 2 (29) (x x 0 ) 2 +(y y 0 ) 2 + z 2 (30) Because the ßow Þeld contains singulaities inthefactuedomaina due to injection and extaction, euation (27) cannot be eadily integated. To emove the singulaity, a egulaization techniue can be applied to obtain: et d (x, y, z, s) = (3) ρ wc w Q R 4πK A 2 (x 0,y 0 ) et dª fi dx 0 dy 0 + ρ w c w T ee 4πK d f e s T e d (x, y, 0, s) f i whee T e h d = et d (x 0,y 0,0,s) T e i d (x, y, 0, s) and h et e d = et d (x e,y e,0,s) T e i d (x, y, 0, s). The integand is no longe singula at the injection andextactionpoints,(x i,y i,0) and (x e,y e,0). To avoid dealing with deivatives in the integand, the integal in (3) can be tansfomed based on the divegence theoem and noting that the integal pefomed ove the factue bounday, A, vanishes due to the no ßux condition n = 0 on A (see (6)): et d (x, y, z, s) = (32) ρ w c w Q ~T e 4πK d f e s T e ª d (x, y, 0, s) f i ρ w c w R 4πK ~ A T d g exp ( β R ) dx 0 dy 0 whee: = x (x 0,y 0 ) R x 0 + y (x 0,y 0 ) R y 0 (33) g = R 2 ( + βr ) (34) Applying the above euation on the factue suface, (x, y) A and z = 0, esults in: 4

5 Figue 2: Computational mesh. et d (x, y, 0, s) = (35) ρ w c w Q ~T e 4πK d F e s T e ª d (x, y, 0, s) F i ρ w c w R 4πK ~ A T d G dx 0 dy 0 whee F e = e exp ( β e ), F i = i exp ( β i ), G = 2 ( + β) exp ( β ), and: = e = i = (x x 0 ) 2 +(y y 0 ) 2 (36) (x x e ) 2 +(y y e ) 2 (37) (x x i ) 2 +(y y i ) 2 (38) Note that euation (35) is now deþnedina two-dimensional space, x, y A and can be utilized fo an integal euation solution. NUMERICAL IMPLEMENTATION To solve the system epesented by (35), the factue suface, A, is discitized into a numbeofelementsandisdeþned by a total of n nodes (see Figue 2) Then, an unknown tempeatue deþcit T ej d is assigned to each node, except fo the node at the injection point whee et d = /s is the imposed bounday condition. Hence thee exist n unknown discete tempeatues. Euation (35) is applied to the n nodes (excluding the injection point) by selecting the nodal locations as the base points: ~T j d = (39) ρ w c w Q {h ~T e 4πK d ~ i T j d F e s ~ T j d F i } ρ N wc w P el R h ~T(η 0 4πK S el, ξ 0 ) ~ i T j d G n= x (η 0, ξ 0 ) η 0 + y(η 0, ξ 0 ) ξ 0 dη 0 dξ 0 whee ~ T j d is the cuent nodal point, ~ T e d is tempeatue at the extaction point, η 0 and ξ 0 ae the element local coodinates, and T(η 0, ξ 0 ) is the tempeatue at some point inside element. Note that x = 0 η fo suae elements. Afte pefoming numeical intega- 0 tion ove the elements, each euation becomes a linea euation involving T ej d, j =,...,n, as unknowns. The linea system can then be solved fo the discete tempeatues T ej d. In the pesent implementation of the numeical scheme and pogam development the tempeatue distibution in a cicula cack is found by using suae, fou-node linea elements. In addition, the dipole solution is used to model ßuid ßow due to the injection and extaction pocess in the factue. The noßow bounday condition fo the cicula cack is imposed by using the method of images (Stack, 989) and supeposition to Þnd the appopiate potential function. The potential is then used to calculate the ßuid dischage fo evey factue element (Figue 3). When the extaction well falls inside an element, it is moved to the neaest nodal point. The tempeatue inside an element is intepolated using standad shape functions fo the element, N i : N i = 4 ( ± x0 )( ± y 0 ) All othe function ae calculated exactly at each nodal point. The double integals in euation (39) ae integated numeically ove each element using the Gaussian integation pocedue. Fo 5

6 each element both the fou-point and ninepoint integation schemes ae used. If the absolute value of the elative eo, value 4 value 9 value 9 is less than then esult of the ninepoint integation outine is used. Othewise, the element is futhe divided into fou pats and the nine-point scheme is applied ove each sub-element. If the eo is still too lage, each sub-element is again subdivided into smalle elements until the eo is small enough o a speciþed maximum numbe of iteations is eached. A diect solve is used to solve the linea system of euations. An iteative solve can also be used, howeve, in this case it yields simila esults but convegence is vey slow. Once the nodal tempeatues ae obtained, the extaction point tempeatue can be calculated by (i) using the nodal values (ii) o using a contou integation aound the oiginal extaction point. In the latte case, the integation path consists of eight nodes neaest to the extaction point, and the tempeatue is then given by: ~T e d = R Q Γεe n(x 0,y 0 ) ~ T d (x 0,y 0 ) ds It is necessay to tansfom the solution back into the time domain. This can be achieved by using an appoximate Laplace invesion method, e.g., the Stehfest (970) method (Cheng et al., 994). EXAMPLE The solution of the thee-dimensional heat conduction is examined below though a numeical example using the following data set: 3 m3 R = 300 m; Q = 5x0 sec ρ w =.0 g/cm 3 ; ρ = 2.65 g/cm 3 ; K = 2.58 Wm K ; c w = 4.05x0 3 Jkg K ; c =.x0 3 Jkg K, Figue 4 pesents the nomalized tempeatue deþcit as a function of time. It can be obseved that a Þne gid yields a highe value of tempeatue. Howeve, the cuves do convege as a Þne gid is used. Also, the values of T d obtained fom the nodal values is highe than the values obtained using the aveaging scheme descibed above. Figue 5 illustates the nomalized extaction tempeatue fo the same poblem. Figue 6 shows that a lage factue esults in a highe extaction tempeatue, as expected. The 3D model pediction is compaed with the one-dimensional heat conduction model of Gingaten and Sauty (975), as epoted by Kolditz (995), in Figue 7. It can be obseved that ou solution pedicts a highe extaction tempeatue, this agees with the obsevation of Kolditz (995). The same numeical appoach was used to model the onedimensional heat extaction fom a cicula factue using the data of Rodemann (982). The esult is shown in Figue 8. The black contou lines epesent the one-dimensional analytic solution of Rodemann (986). Theefoe, the Numeical esults pesented heein ae in ualitative ageement with othe published esults. SUMMARY AND CONCLUSION The thee-dimensional bounday integal euation fo heat conduction in a geothemal esevoi has been solved numeically. The integal euation solution eliminates the need fo discetization of the geothemal esevoi. In addition to igoous testing of the model, futue activities include modeling of esevoi elasticity, themo-elasticity and pooelasticity. These can cause the factue width to depend on the factue ßuid pessue, tempeatue, and esevoi compliance. The pesent fomulation lends nicely itself to these developments. Effots in these diections ae undeway. 6

7 Tempeatue, C -T Factue Radius=300m; Injection Rate: 5 lites/sec T nodal (50x50) T avg (50x50) T nodal (30x30) T avg (30x30) D injection well: (-50m,0) extaction well: (50m,0) 2-step Laplace Invesion Time, yeas Figue 3: Fluid ßow in a cicula factue fom an injection/extaction pai (dipole solution). Figue 5: Nomalized extaction tempeatue fo example one Factue Radius=300m; Injection Rate: 5 lites/sec T nodal (50x50) T avg (50x50) T nodal (30x30) T avg (30x30) Nodal Tempeatue; Injection Rate: 5 lites/sec; (70x70) gid R=300 m R=20 m T D injection well: (-50m,0) extaction well: (50m,0) 2-step Laplace Invesion injection well: (-90m,0) extaction well: (90m,0) 6-step Laplace invesion Time, yeas Time, yeas Figue 4: Tempeatue deþcit as a function of time fo a 300 m factue. Figue 6: Extaction tempeatue fo two diffeent factue adii. 7

8 Tempeatue, 0 C Factue Radius=300m; Injection Rate: 5 lites/sec; (70x70) gid Analytical solution is fom Kolditz (995). Gingaten (D), unbounded cack 3D BEM solution, 6 steps Time, yeas Figue 7: Compaison of 3D numeical and analytical D solutions. ACKNOWLEDGMENT The Þnancial suppot of the US Depatment of Enegy (DE-FG07-99ID3855) is gatefully acknowledged. Cheng, A.-D., Ghassemi, A., and Detounay, E., 200. A two-dimensional solution fo heat extaction fom a factue in hot dy ock. Int. J. Numeical & Analytical Methods in Geomech., 25, Kolditz, O., 995. Modelling ßow and heat tansfe in factued ocks: Dimensional effect of matix heat diffusion, Geothemics, 24, Rodemann, H., 982. Analytical model calculations on heat exchange in a factue, in Haenel, R. (ed.), Uach Geothemal Poject, pp Stuttgat. Stehfest, H., 970. Numeical invesion of Laplace tansfoms, Comm. ACM, 3, and 624. Sack, O. D.L., 989. Goundwate Mechanics, Pentice Hall, New Jesy, 732 p. Willis-Richads, J. and Walloth, T., 995. Appoaches to the modeling of HDR esevois: a eview, Geothemics, 24, E E E E E E E E E E E E E E E E+02 Figue 8. Fluid tempeatue fo the poblem of Rodemann (982). REFERENCES Cheng, A.H.-D., Sidauuk, P. and Abousleiman, Y., 994. Appoximate invesion of the Laplace tansfom, Mathematica J., 4,