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2 Geothermal Resources Council Transactions, Vol. 26, September 22-25, 2002 Analysis of Thermal Effectiveness of Geothermal Multi-Borehole Circulating System S. Fomin, K. Yoshida and T. Hashida Fracture Research Institute, School of Engineering, Tohoku University, Sendai , Japan Keywords Geothermal energy, heat transfer, injection and production wells, ma thema tical model, multi- borehole s ys tem. ABSTRACT A three-dimensional model (FRACSIM-3D) developed in Tohoku University for numerical simulation of heat and fluid flow within the fractured media is used in the analysis of geothermal reservoir performance. The model effectively simulates two main stages of the geothermal reservoir exploitation, namely, (i) hydraulic stimulation of the existing natural fractures within the Hot Dry Rock (HDR) reservoir and (ii) forced convection through the fractured media when the filtrating fluid extracts the heat from the hot rock and delivers it to the production well. Since the heat accumulated by the fluid within the system of injection boreholes can constitute a substantial fraction of the total thermal output of the geothermal power plant, the model of heat and mass transfer in the fractured media at the heat extraction stage should be coupled with the equations which describe the heat transport in the system of injection and production wells. Mathematical modeling of heat flow within the multiborehole circulating system is proposed in this study. On the basis of this model the major parameters that affect the thermal productivity of the geothermal power plant are analyzed. The effective regimes of the fluid circulation and optimal geometry of the multi-borehole system are proposed. lntrod uction HDR geothermal energy extraction systems have received much attention recently. The basic concept of these systems is to develop a water circulation system through the subsurface fracture network in Hot Dry Rock (HDR). Owing to a low thermal conductivity of the rock, the water circulation paths have a significant influence on heat extraction. Therefore, one of the aspects of the general problem of reservoir modeling is the proper approximation of the fracture distribution within the rock. A series of geophysical investigations has confirmed that subsurface fracture networks can commonly be described by fractal geometry. Hirata [l] has investigated several fault systems in Japan and concluded that fractal geometry was a useful tool to characterize the geometry of the fault systems. Main at al. [2] also have noted that subsurface fractures could be characterized by using a methodology in which the number of fractures is related to the fracture length. The improved three-dimensional modeling procedure FRACSIM-3D for subsurface fracture networks, developed by the research group in Tohoku University [3], is based on the relationship between the fracture length and a number of fractures, as suggested by Main, et. al, [2], Scholz, et. al, [4], and Watanabe and Takahashi [5]. It incorporates the elements of the approximate model proposed by Willis-Richards, et. al, [6] in which the fracture shear displacements and openings, variation of the shape of the stimulated rock volume, and pressure compliant fracture apertures are taken into account. The numerical model FRACSIM-3D has proven to be an appropriate approximatemodel capable of simultaneously addressing the problems associated with hydraulic stimulation, fluid circulation and heat extraction. The simulation of the heat transport through the randomly distributed fissures is based on the unsteady energy conservation equation which accounts for the molecular conduction, forced convection and thermal dispersion [7]. In the case of the multi-borehole arrangement the heat accumulated by the fluid on its way through the system of injection wells can constitute a significant supplement for the total thermal output of the geothermal power plant [8,9]. Moreover, in the regions of the high volcanic activity or high formation temperature, this multi-borehole circulating system can be solely used for heat mining without developing the fractured reservoir. Therefore the heat transport within the boreholes should be included in the model. In the present study the simple mathematical model of the heat flow within the system of the injection and production wells of differ- 2 79

3 Fomin, et. a/. ent geometry and configuration is proposed. This model is used for assessment of the multi-borehole system s thermal performance and for defining the optimal flow regimes. Using the general algorithm, which incorporates the heat extraction within the fractured media, heat accumulation in the injection boreholes, and heat dissipation during the flow within the production wells, the overall variation of the fluid temperature (after passing through the reservoir and borehole system) in the production well with time is obtained. System Model As was mentioned above, the numerical model FRACSIM- 3D developed in Tohoku University [3,6] has proven to be an appropriate approximate model capable of addressing simultaneously the problems associated with hydraulic stimulation, fluid circulation and heat extraction. The simulation of the heat transport through the randomly distributed fissures is based on the unsteady energy conservation equation, which accounts for the molecular conduction, forced convection and thermal dispersion [7]. In this model the 3D fracture network, which consists of the randomly-distributed penny-shaped subsurface fractures, is mapped on a regular cubical grid of the discretized reservoir domain. It is assumed that the flow properties of a stochastic fracture network depend on the fluid pressure. In the 3D case the penny-shaped fractures are generated stochastically within a fracture generation volume and a fractal fracture length distribution is assumed. The permeability is strongly affected by the fracture distribution and fracture apertures. It can be shown that the mean temperature of the rock may differ from the temperature of the liquid only in the initial stage of fluid injection in the vicinity of the injection well. Therefore, the conditions of the thermal equilibrium can be assumed and the 2-temperature model can be reduced to one-temperature equation with regard to the admixture temperature. Since the fluid velocity within the geothermal reservoirs created in the fractured rock can be relatively high and the characteristic sizes of the solid blocks of the rock, which constitute the fissured reservoir media, are relatively large, the effect of thermal dispersion is an important factor which has been included into the mathematical model of the heat transport within the fissured geothermal reservoir. In FRACSIM-3D the temperature of the liquid Tb at the bottom of the injection well is assumed given and the temperature at the bottom of the production well is computed. However, obviously the temperature of the liquid Tb at the bottom of the injection well differs from its value Tnj on the surface in the onset of the injection well. In reality Tb is unknown and should be found from the analysis of the heat and mass transfer processes in the borehole and surrounding rocks. The knowledge of this value is important not only for the solution of the above problem. The other factor that motivates the importance of this heat transfer analysis is the idea of constructing the heat generating plants without developing the fractured subsurface reservoirs [8, 91. In these circulating systems, which incorporate several injection and production well (see Figure l), the heat is obtained by the circulating fluid due to the heat transfer from the surrounding high-temperature-rock. To this reason, the analysis of the multi borehole system performance (with or without circulation in the subsurface reservoir) is of major importance for the further development in the area of geothermal energy utilization. The boreholes in this kind of geothermal system are normally rather long, so that the ratio of the well radius r,,, to its characteristic depth His a small parameter, r,jh<<l; the borehole can be also curved [9]. We will assume that the axial curvature radius of the borehole R, is large enough and is of the order of the borehole depth, R,-H. In this case the locally orthogonal coordinates (s,r) can be employed, where s is the coordinate measured along the central axial line of the borehole and r is the distance in the radial direction measured from the central line (transverse to s). Due to the small ratio r,jh<<l, the mass, momentum and energy equations can be presented in terms of the averaged quantities over the borehole cross-section. The averaging procedure for the mass, momentum and energy conservation equations is straight-forward and leads to the following equations for the incompressible flow 2 9 ~,,,Pw = G(T), aw ap 22 p-+-+l- gpcoscp = 0, 37 as rw where G is a flow rate through the cross-section of the borehole, w, T, p are the mean velocity, temperature and pressure of a fluid in a borehole, respectively; qw is a heat flux on the wall, cp is the angle between the coordinate s and vertical z axis. Due to its small value, the effect of the shear stress on the temperature variation in the borehole is ignored. The heat flux on the borehole s wall qw is unknown and can be defined by coupling the heat flow in the borehole with heat transfer in the rock. The initial distribution of temperature in the surrounding thermally undisturbed rock is determined by the geothermal temperature distribution within the earth, which generally is a well known function of vertical coordinate z. Normally this quantity is approximated by a linear function as Trl,,o = Tr(t) = Tro f Tz, where r = IT,(0) - Tr (H)I H is the mean geothermal gradient and Tm is the temperature on the rock at the level of fluid entrance into the borehole (at the neutral level for injection borehole and at the bottom of the production well). Assuming constant thermo-physical properties of the rock, the equation for temperature distribution T, in the surrounding rock in cylindrical coordinates (r, s) can be presented by the following non-dimensional model mathematical model: F~=O, e,=o; (1) (3) (4) (5) 280

4 Fomin, eta/. where and Z = Scoscp) and employing the method of characteristics after some simplifying manipulations solution of equation (1 1) can be presented: Equation (4) describes the temperature distribution in the rock surrounding the borehole, equation (5) is the initial condition at z=o, equation (6) represents the Newtonian law of a heat transfer on the borehole wall with heat transfer coefficient h,, equation (10) is the condition of the finite temperature of the rock in infinity. An approximate solution of the problem (4)-(7) is found in [lo] in a following form where The heat flux on the wellbore face qw yields <P(Fo) = 1/[1+ Biln(Z(Fo))], Z(Fo) = 1 +{(Bi)dFo. Function {(Si) has been obtained in [lo] by comparing the exact solution of the problem (4)-(7) with the approximate (9) and employing the non-linear regression method: e(z, FO) = e, exp[-q(fo)bz /cos cp] - (14) [l - exp( -@( Fo)BZ / cos cp] / B(D( Fo) cos cp. The dimensional temperature of the fluid flow in the borehole T, measured in "C, can be obtained from (14) accounting for the formula The other regime of interest occurs when the fluid is injected at a constant pressure and the flow rate varies with time. In this case solution (14) is no longer valid and, therefore, additional (and more complex) computations are required. Nevertheless, the above solution can be used for the analysis of the overall output of the geothermal system in coupe with mathematical models of the heat transport in the fractured media. In the particular case when the reservoir is not developed and only the multi-borehole system is utilized for accumulating the geothermal energy from the HDR [8, 91 solution (14) represents the basic equation for the assessment of the heat gained during the fluid circulation. The latter case is briefly described within the section below. {(Si) = ( Bi) / ( Bi). (10) Combining equations (3) and (9) in a non-dimensional form, yields where The initial and boundary conditions are Equations (1 1)-( 13) constitute the boundary value problem for the temperature within the borehole. Although the last term on the right-hand side of equation (1) would be more complex in the case of compressible fluid, the solution procedure for the incompressible fluid can be readily extended to compressible fluids. Assuming constant flow rate in the borehole (G and Ware constant), constant coefficient of heat transfer along the wellbore face (B is constant) and constant angle of the borehole inclination cp (Le. K,- =-coscp is constant Figure 1. The proposed configuration of the borehole system for effective heat extraction; (A)- vertical cross-section, (B)- horizontal cross-section, where the circle denotes the production well and squares - injection wells. (1 - injection wells, 2 - production well, 3 - fractured reservoir, 4 - surface). 281

5 Fomin, et. a/. Assessment of Thermal Performance of the Geothermal System The multi-borehole system, which has a number of injection wells (4 injection wells indicated by squares on projection (B) in Figure 1) and one production well (circle on projection (B) in Figure l), is considered. In order to elongate the injection wells designated for the heat extraction purpose we assumed the curved or inclined injection wells with inclination angle cp between the borehole axis and axis z. The bottoms of the injection wells are located in vicinity of the bottom of production well, so that the development of the full size reservoir is not required [8,9]. This configuration of the boreholes system is convenient for estimating the amount of thermal energy acquired in the boreholes without extra heating within the fractured reservoir. The approximate the temperature in the bottom of the injection well can be computed from the equations (14) and (15). For instance the temperature within the injection well T,# in the dimensional form in the case of N injection wells of the same radius and the same inclination angles can be presented as follows where p = 2nr,Jh,, ~(clg,~,), Tjnj is an injection water temperature, Grot is a total flow rate for N injection boreholes (Gror=GN) and parameter Y=N/coscp represents the configuration of the borehole s system. Ignoring the heat obtained by fluid from the fractured reservoir leads to the following condition that equalizes the temperature at the bottom of the injection wells at z=h and temperat re in the entry point of the production well: TI lz=h = Tb = Tj, where T,, is the temperature in the production well. If the px%tuction well is not insulated, then from (14) W Number of injection wells I I 0.0 : I I 1 I I qo, [kg/ s e cl 0.1 I [ - I 1 year Oyears years Figure 2. Variation of effectiveness qw vs flow rate in the production well. (A- the effect of the number of injection boreholes, B - sensitivity with regard to circulation time). The effectiveness of this simplified geothermal circulation system (heat extraction from the fractured area is ignored) can be characterized by the ratio which represents the real increment of the fluid s temperature in the exit of the circulationsystem relative to the absolute maximum which is equal to a geothermal step HT. Variation of the effectiveness criterion (1 8) regarding the flow rate of the circulating fluid presented in Figure 2 demonstrates the existence of the local maximum for a specified flow rate. As it can be seen, the effectiveness of the circulation system increases monotonously with increase of the flow rate and at some certain point, say G=Go, which depends on the number of boreholes (Figure 2, A) or exploitation time (Figure 2, B) or some other parameters (for instance, angle of the borehole inclination) effectiveness reach its maximum and then reduces with further increase of the flow rate. Hence, if the number of production wells and their inclinations are fixed the optimal regime of circulation can be suggested by the right choice of the flow rate. The results presented in Figure 2 (A) show that the number of the injection boreholes within the circulating system significantly affects effectiveness of the geothermal system whereas for the different periods of the system exploitation the effectiveness remains practically the same (Figure 2, B). The computed temperatures on the exit from the production well in the case when the geothermal system is simulated in whole, accounting for heat extraction from the fractured reser- 2 82

6 Fomin, eta/. n 0 U fl c, E h + El 1601 Flow Rate = 34 kg/sec 40 Flow Rate = 80 kg/sec 0 I I I I Time [years] Figure 3. Temperature in the exit of the geothermal system T~lz=o vs time. (Solid line presents computations from the complete model including multi-borehole system and fractured reservoir; dashed line - computations only for reservoir where the heating in the injection wells is ignored). voir and multi-borehole circulating system, are presented in Figure 3. In general, thermal efficiency of the geothermal reservoir is greatly affected by the reservoir geophysical properties (fracture length, orientation and density, the size of stimulated volume, initial temperature, etc.). Computations results illustrated in Figure 3 are obtained by using the numerical simulating code calibrated for the Hijiory deep reservoir typical conditions [3]. Hundreds of numerical tests were performed for the different randomly distributed fractures in order to simulate the experimentally observed Hijiori reservoir growth during the stimulation period (in the sense of growth direction and volume expansion). For instance, it was found that the fractal dimension of fracture size distribution should be equal to 2.45, the fracture distribution density 0.7m-, the simulated fracture length should be from the interval of 6 to 100 meters, etc. As can be seen, the heat gained from the fractured reservoir dominates, however with time the contribution of heat accumulated within the injection wells in the overall heat production increases. For the geothermal power plant based on the reservoir concept the temperature in the production well does not have local maximum at the specified flow rate as it happens in the case of multiborehole circulating system. The effectiveness of the reservoir monotonously reduces with increasing the flow rate. Acknowledgements The work was supported in part by the Research for the Future Program of the Japan Society for the Promotion of Science (JSPS-RFTF 97P00901) and by The Ministry of Education, Culture, Sports, Science and Technology Grant-in Aid for COE Research (No. 1 I CE2003) and Grant for Scientific Research (B) (No ). References [ll Hirata,T., PureAppl. Geophis., 131, (1989), p [2] Main, L. G., Peacock, S. and Meredith, P. G., Pure Appl. Geophys., 133, (1990), p [3] Shimizu, A., Watanabe, K., Willis-Richards, J. and Hashida, T., Proc. World Geothermal Congress., Morioka, (2000), p [4] Scholtz, C.H., Dawers, N. H., Yu, J. Z., Anders, M. N., and Cowie, P.A., J. Geophys. Res., 98, (1993), p.951. [5] Watanabe, K. and Takahashi, T., J. Geophys Res., Vol. 100, B 1, (1 993, p.521. [6] Willis-Richards, J., Watanabe, K. and Takahashi, H., J. Geophysical Res., 101, (1996), p [7] Fomin, S., Shimizu, A. and Hashida, T., Mathematical Modelling of Convection Heat Transfer in a Geothermal Reservoir of Fractal Geometry, Proc. 1 2Ih International Conference of Heat Transfer, Grenoble, August 18-23, (2002), accepted, in press. [8] Sa1amatinA.N. and Chugunov V.A., The System for the Geothermal Heat Production, Patent No , USSR, (1978), [in Russian]. [9] Green, L.H. and Shulman G., The Economic Impact of Reducing Deep Drilling Costs For Heat Mining Power Plants, Proceedings of TECEC-32, Washington D.C., ( I996), p [lo] Fomin, S., Chugunov, V., Hashida, T., Saito, S. and Suto, Y., Heat Flux on the Bore-Face and Temperature Distribution in the Formation, GRC Transactions, Vol. 26, (2002), accepted, in press. 2 83