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Optimization of the utilization of deep borehole heat exchangers

Optimization of the utilization of deep borehole heat exchangers ylkong@mail.iggcas.ac.cn Key Laboratory of Shale Deep-borehole heat exchangers (DBHE) are generally coaxial pipes installed in deep Gas and Geoengineering, boreholes and has become an alternative approach to utilize geothermal energy. Institute of Geology and Geophysics, Chinese Since the performance of the DBHE system can be affected by several parameters, it is Academy of Sciences, important to optimize the design of parameters for the DBHE. In this paper, based on Beijing 100029, China the analytical method, we carried out the sensitivity analysis of DBHE design param- Full list of author information is available at the end of the eters, including outer pipe diameter, inner pipe diameter, flow rate, outer pipe materi- article als, grout materials, and borehole depth during continuous operation for 4 months. The sensitivity analysis results indicate that the heat extraction rate can be significantly affected by outer pipe diameter, borehole depth, and flow rate. The effects of grout materials, inner pipe diameter and outer pipe materials are of second-order. Finally, an optimization method based on the lowest Average Energy Cost index was proposed to optimize these DBHE design parameters under different geological conditions. Given the cost in this study, a combination scheme of all the optimal parameters is given for different depth wells under different geological conditions. Keywords: Deep-borehole heat exchanger, Geothermal energy, Sensitivity analysis, Optimal heat extraction rate, Economic analysis Introduction Based on the differences in burial depth, utilization mode, and storage medium, geother - mal energy is usually divided into three types: shallow geothermal energy (0–200  m), medium–deep hydrothermal energy (200–3000 m), and hot dry rocks energy (> 3000 m) (Wang 2015). The China Geothermal Energy Development Report released in August 2018 shows that the shallow geothermal energy is the main method used in China for geothermal heating, which has been rapidly developed. The extent of hydrothermal heat - ing is also increasing steadily. However, the development in the utilization of geothermal resources has some difficulties due to the large areas demanding of shallow geothermal energy and the uneven distribution of hydrothermal energy (Kong et al. 2014). Moreo- ver, when utilizing the hydrothermal energy, reinjection of geothermal wastewater must be carried out for maintaining the pressure of geothermal reservoirs (Rybach 2003). However, with a low reinjection rate, reinjection could be quite difficult in sandstone reservoirs (Kong et al. 2014; Su et al. 2018; Ungemach 2003). Therefore, deep-borehole heat exchangers (DBHE) have become an alternative approach to utilize geothermal © The Author(s) 2020. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creat iveco mmons .org/licen ses/by/4.0/. Pan et al. Geotherm Energy (2020) 8:6 Page 2 of 20 energy (Alimonti et  al. 2018). The principle of DBHE is to install a coaxial pipe into a deep borehole to inject cold water into the outer pipe and extract hot water from the inner pipe, forming a closed-loop system (Rybach and Hopkirk 1995). The heat energy can be transported during this process through the heat transfer between circulating fluid and surrounding rock. There is no interaction between circulating fluid and natural groundwater; it is thus called “obtaining heat without extracting groundwater”. It could be found that the principle of DBHE is quite similar to the coaxial borehole heat exchangers (CBHE) except for the depth (Acuña et al. 2011; Holmberg et al. 2016; Kohl et al. 2002; Sapinska-Sliwa et al. 2016). However, at least to the authors’ knowledge, the definition of DBHE depth range does not appear in any literature. In this study, the borehole heat exchangers (BHE) deeper than 500  m were defined as DBHE in consid - eration of the heat exchanging efficiency. The research on BHE carried out in the last decade concerns detailed investigations of the heat transfer performance of CBHE and DBHE. For the CBHE, several studies have focused on investigating its performance and applicability. Acuña et  al. (2011) implemented six distributed thermal response tests (DTRTs) on a multi-pipe CBHE. Acuña and Palm (2013) presented results from three DTRTs carried out on two coaxial pipe-in-pipe BHE at different flow rates. They pointed out that the ground thermal conductivity and the borehole thermal resistance required for designing a CBHE system could be determined by an in  situ thermal response test on a completed borehole. Zanchini et al. (2010) investigated the effects of thermal short- circuiting and flow velocity on the thermal efficiency of CBHE. The results show that the effects of flow velocity and thermal short-circuiting are both important and that the latter can be reduced considerably by employing a low conductivity material for the inner tube. Raymond et al. (2015) evaluated the impact of changing the water flow rate, the inner and outer pipe SDR on the thermal resistance of CBHE. The results show that considerable advantages could be obtained by increasing the water flow rate, and the outer pipe SDR. Beier et al. (2013) developed a CBHE model for the vertical temperature profiles, which can be used instead of the mean temperature approximation to estimate borehole resistance. Furthermore, Luo et  al. (2019) presented an analytical model for CBHE that specifically considers geothermal gradient. Over the past few years, more and more research work has been carried out to inves- tigate the DBHE system performance. Several studies suggest that the heat transfer per- formance of the DBHE correlates significantly with the geothermal gradient and thermal conductivity of the surrounding rock (Bu et al. 2012; Chen et al. 2019; Cheng et al. 2013; Kong et al. 2017a, b; Le Lous et al. 2015; Nalla et al. 2005; Noorollahi et al. 2015; Temple- ton et al. 2014). Le Lous et al. (2015) also conducted the sensitivity analysis of the volu- metric heat capacity of the rock. They found that low volumetric heat capacity value can reduce the thermal recovery time. Except for geological conditions, parameters related to DBHE design and operational settings can also affect the heat performance of DBHE. Fang et al. (2018), Holmberg et al. (2016), and Wang et al. (2017) found that the circu- lating fluid injected through the annular space can reduce the heat loss than through the inner pipe. Moreover, the performance of different circulating fluids in such heat transfer has been evaluated, proving that water is the most suitable fluid for circulation P an et al. Geotherm Energy (2020) 8:6 Page 3 of 20 (Alimonti and Soldo 2016; Bu et al. 2012; Kujawa et al. 2004; Le Lous et al. 2015; Nalla et al. 2005; Noorollahi et al. 2015; Templeton et al. 2014; Wight and Bennett 2015). Furthermore, Nalla et al. (2005) pointed out that a key parameter affecting the perfor - mance of the DBHE is the fluid residence time, which represents the flow rate. The other DBHE design parameters have also been investigated, including the borehole diameter, the diameter of the outer pipe, the diameter of the inner pipe, the depth of the borehole, grout materials and pipe materials (Alimonti and Soldo 2016; Beier and Holloway 2015; Chen et al. 2019; Lhendup et al. 2014; Nalla et al. 2005; Wang et al. 2017). According to the results of these analyses, borehole diameter, the outer pipe diameter, grout thermal conductivity, outer pipe thermal conductivity, and borehole depth can positively influ - ence the heat transfer performance of DBHE (Alimonti and Soldo 2016; Chen et al. 2019; Nalla et al. 2005; Wang et al. 2017). In contrast, the inner pipe diameter and inner pipe wall thermal conductivity could negatively influence the heat transfer performance of DBHE (Alimonti and Soldo 2016; Beier and Holloway 2015; Chen et al. 2019; Lhendup et al. 2014; Nalla et al. 2005). Therefore, it can be concluded that all these DBHE design parameters can affect the system performance, and the optimal design of DBHE can be obtained, while it was rarely given in the literature. Herein, we simulated the effects of design parameters such as outer and inner pipe diameter, flow rate, grout and outer pipe materials, and DBHE depth on the heat extrac - tion rate. The parametric sensitive study is conducted by using the analytical method proposed by Beier (2014) and Beier et al. (2014). Importantly, our study revealed that it is useful for improving the DBHE system performance to allocate these design param- eters simultaneously. Further, we developed an optimization method to identify the opti- mal DBHE configuration in different regions. We considered 3 rock types of sandstone, limestone, and granite due to their wide distribution and utilization for geothermal energy exploitation. Since the optimal design for the DBHE is to improve the heat trans- fer capacity and reduce the cost, our optimization method is to do the economic analysis by applying the Average Energy Cost (AEC) index (Rodríguez and Díaz 2009). Methods Optimization method In this study, we used the index of Average Energy Cost to optimize the DBHE design, which includes the outer pipe diameter, inner pipe diameter, flow rate, outer pipe mate - rials, grout materials, and DBHE depth. The average energy cost represents the specific cost of heat in yuan per kilowatt-hour. It was calculated from the total well cost, which consists of the drilling cost C , g rout drill cost C , coaxial pipe cost C , and circulating water cost C , and is divided by the grout pipe water system’s thermal energy output Q: C + C + C + C drill grout pipe water AEC = . (1) The system’s thermal energy output Q is calculated with heat extraction rate, DBHE depth, and working time. In this study, we defined the heat extraction rate as the maximum rate at which the inlet temperature is near to 0  °C at the end of a heat extraction period. Fang et  al. (2018) used a similar definition for the optimal rate of Pan et al. Geotherm Energy (2020) 8:6 Page 4 of 20 heat extraction, where inlet temperature is maintained at 5 °C at the end of a period of heat extraction. In addition, the analytical method used for calculating the heat extraction rate is given in "Coaxial heat exchanger model" section. The drilling cost for different borehole diameter intervals is presented in Budget Standard of Geological Survey Projects by China Geological Survey (https ://max. book1 18.com/html/2017/0903/13152 0083.shtm), but only 4 ranges (#1–4) of bore- hole diameter are given, and the depth range is 500–1000 m. In order to calculate the AEC of DBHE with a depth range of 500–3000  m, the drilling cost of other depth is calculated by the relation between well depth and drilling cost (Daniilidis et al. 2017; Heidinger 2010; Olasolo et  al. 2016), as shown in Table  1. The borehole diameter intervals of #1–4 are 201–250 mm, 251–300 mm, 301–350 mm, 351–400 mm, respec- tively. It is, therefore, with a constant thickness of grout between the outer pipe and the surrounding soil (0.023  m), the outer pipe diameter intervals of #1–4 are 0.178– 0.227  m, 0.228–0.277  m, 0.278–0.327  m, 0.328–0.377  m, respectively. The coaxial pipe cost is calculated based on the weight per length of coaxial pipe, and a fixed cost per unit mass of steel (Nalla et  al. 2005). The normalized coaxial loop cost is 7 yuan per linear foot of loop, which consists of a 1-in. inner pipe and a 4-in. outer pipe (Liu et al. 2018). The circulating water cost is calculated based on the volume of water, and the normalized water cost is 0.0021 yuan per gallon (Liu et  al. 2018). The grout cost is calculated based on the volume of grout used for sealing the outer pipe diameter. The normalized grout cost is 2.310 yuan per gallon (Liu et al. 2018), and the thermally enhanced grout is 7 and 10.5 yuan per gallon (Liu et al. 2018). The principle of the optimization is to find the DBHE design with minimum AEC value. Since the AEC is highly related to the heat transfer performance of DBHE, the key to minimize the AEC of DBHE is to obtain the optimal heat transfer efficiency. The optimization procedures are as illustrated in Fig. 1 , and the details are as follows. Table 1 The drilling cost of different borehole diameter intervals and rock types Rock type #1 (yuan/m) #2 (yuan/m) #3 (yuan/m) #4 (yuan/m) BHE depth (m) Sandstone 1497 1599 2135 2393 500 Sandstone 2994 3198 4270 4786 1000 Sandstone 5988 6396 8540 9572 1500 Sandstone 8982 9594 12,810 14,358 2000 Sandstone 11,976 12,792 17,080 19,144 2500 Sandstone 17,964 19,188 25,620 28,716 3000 Limestone 2159 2374 3086 3450 500 Limestone 4318 4748 6172 6900 1000 Limestone 8636 9496 12,344 13,800 1500 Limestone 12,954 14,244 18,516 20,700 2000 Limestone 17,272 18,992 24,688 27,600 2500 Limestone 25,908 28,488 37,032 41,400 3000 Granite 3670 4036 5246 5865 500 Granite 7340 8072 10,492 11,730 1000 Granite 14,680 16,144 20,984 23,460 1500 Granite 22,020 24,216 31,476 35,190 2000 Granite 29,360 32,288 41,968 46,920 2500 Granite 44,040 48,432 62,952 70,380 3000 P an et al. Geotherm Energy (2020) 8:6 Page 5 of 20 The first step is to choose the simulated scenario of certain DBHE materials, DBHE depth, and rock type. The second step is varying the outer pipe diameter and choosing one. The third step is to obtain the inner pipe diameter by matching the annular space area with the area of the cross-section of the inner pipe. The fourth step is to modify the flow rate to obtain the relationship between the flow rate and heat extraction rate. The fifth step is calculating the approximate gradient of each flow rate and select the opti - mal flow rate where the gradient decreases to a minimum (in our study, this was when the gradient was less than 0.2/0.5 = 0.4). The sixth step is to calculate the optimal heat extraction rate with the obtained pipes diameter and flow rate. The seventh step is calcu - lating the AEC value of this DBHE design. The eighth step is calculating the AEC values of other DBHE designs by redoing step 2 to step 7. Finally, the ninth step is finding the optimal DBHE design with the lowest AEC values. Coaxial heat exchanger model In this study, we applied the analytical method proposed by Beier (2014) and Beier et  al. (2014) to calculate the heat extraction rate. In their method, the governing equations of heat transfer in the inner pipe, outer pipe, grout, and ground are established, respectively. The two equations in pipes are coupled together by the shunt heat transfer through the wall of the inner pipe. The fluid flowing through the annulus exchanges heat with the surrounding Fig. 1 Procedure to obtain the optimal design of DBHE with a total of n scenarios Pan et al. Geotherm Energy (2020) 8:6 Page 6 of 20 grout. Heat conduction through the grout and ground occurs radially out from the outer pipe. The circulating fluid enters the BHE through the annulus in this model. The energy conservation equation of the annulus is ∂T N H A ∂T s f D1 D1 D1 + + N (T − T ) + N T − T = 0, 12 D1 D2 g D1 Dg (2) r =1 ∂z 2 ∂t D D 0 < Z < 1, 0 < t D D where N is a dimensionless thermal conductance (reciprocal of resistance) of the ground, the symbol w represents the flow rate, and N is a dimensionless thermal con- ductance (reciprocal of resistance) of the grout. The ratio of the volumetric heat capaci - ties of the circulating fluid, c , and the ground, c , is designated as H . The parameter A f s f D1 is a ratio of the flow area. The energy conservation equation of the inner pipe is ∂T N H A ∂T s f D2 D2 D1 − + + N (T − T ) = 0, 0 < Z < 1, 0 < t . (3) 12 D2 D1 D D ∂z 2 ∂t D D The parameter A is a ratio of the inner pipe area and the circular area based on the out- D2 side pipe outer wall radius. Neglecting heat conduction in the axial direction, the heat conduction equation for the grout is ∂T H ∂T ∂T g Dg Dg Dg = + , 1 < r < r ,0 < z < 1, 0 < t . (4) D Db D D κ ∂t r ∂r ∂r D D D The heat conduction equation for the ground surrounding the borehole is ∂T 1 ∂T ∂T Ds Ds Ds = + , r < r ,0 < z < 1, 0 < t . (5) Db D D D ∂t r ∂r ∂r D D D Equations (1) and (2) require boundary conditions. When z = 0 , there is (T − T ) = N , z = 0, 0 < t . D1 D2 s D D (6) When z = 1 , there is, T = T , z = 1, 0 < t . D1 D2 D D (7) At the outer pipe/grout interface where r = 1, an energy balance sets the heat transfer from the circulating fluid to the grout equal to the heat conducted into the grout. That is, N T − T ∂T g Dg D1 Dg = , r = 1, 0 < z < 1, 0 < t . (8) D D D κN ∂r s D At the grout/ground interface, r = r , an energy balance sets the heat conduction Db rate from the grout equal to the heat conduction rate into the ground, and the grout and ground temperatures are equal at this interface, P an et al. Geotherm Energy (2020) 8:6 Page 7 of 20 ∂T ∂T Dg Ds κ = , r = r , 0 < z < 1, 0 < t , (9) D Db D D ∂r ∂r D D ∂T = ∂T , r = r , 0 < z < 1, 0 < t . Dg Ds D Db D D (10) The ground temperature approaches its undisturbed value as the distance from the borehole increases, ∂T → 0, asr →∞, 0 < z < 1, 0 < t . Ds D D D (11) At the start of heat injection ( t = 0 ), the circulating fluid, grout, and ground are all at the undisturbed ground temperature. This condition requires the dimensionless tem - peratures to be zero, T = T = 0, 0 ≤ z ≤ 1, t = 0, D1 D2 D D (12) T = 0, 1 ≤ r ≤ r , 0 ≤ z ≤ 1, t = 0, Dg D D D Db (13) T = 0, r ≤ r , 0 ≤ z ≤ 1, t = 0. (14) Ds Db D D D Detailed calculation of the heat transfer coefficient and the analytical solution can be found in references of Beier et al. (2014). Sensitivity analysis and simulated scenarios The principle of the sensitivity analysis is that when analyzing the impact of one param - eter on the heat extraction rate of DBHE, the other parameters should be kept fixed. In this study, we performed the sensitivity analysis of each design parameter (outer pipe diameter, inner pipe diameter, flow rate, outer pipe materials, grout materials, and depth) by applying the coaxial pipe heat exchanger model. The parameters in the model are given in Table 2, and the system running time is a heating cycle of 4 months. Besides, in order to investigate the applicability of DBHE in different rock types, 3 rock types (sandstone, limestone, granite) were chosen in the calculation. Furthermore, we considered the average, minimum, and maximum values of thermal conductivity for the selected rock types. Since the heat capacity of selected rock types changes little under constant temperature, we chose the mean value of rock volumetric heat capacity (Clauser 2011a, b; McKenna et  al. 1996; Thomas et  al. 1973; Cho et  al. 2009). Table  3 summarizes the details of the thermal properties of selected rock types. With these con- siderations, we established 54 scenarios to simulate the effect of each design parameter. The overview of these scenarios is listed in Table  4, and the details of each scenario are as follows. In order to analyze the effects of outer pipe diameter on the heat extraction rate, the diameter was set at 0.177, 0.180 m, and then increased incrementally in 0.020 m steps, while keeping all other parameters that act upon the DBHE constant. This was repeated until the outer pipe reached a diameter of 0.400 m. Using this method, the diameter of the borehole increases accordingly. Besides, we set the thickness of the grout between the outer pipe and the surrounding soil at 0.023  m. Recogniz- ing that the heterogeneity of geological materials strongly affects the heat transfer Pan et al. Geotherm Energy (2020) 8:6 Page 8 of 20 Table 2 Parameters used in the model Parameter Symbol Amount Unit Borehole diameter d 0.2 m Inner pipe outer diameter d 0.09 m po Inner pipe wall thickness t 0.00734 m Outer pipe outer diameter d 0.177 m eo Outer pipe wall thickness t 0.00587 m −1 Flow rate w 11.6 L s −1 −1 Pipe wall thermal conductivity k , k 0.5 W m K pp ep −1 −1 Grout thermal conductivity k 0.73 W m K 6 −3 −1 Grout volumetric heat capacity c 3.8 × 10 J m K −3 Water density ρ 1000kg m 6 −3 −1 Water volumetric heat capacity c 4.19 × 10 J m K −1 −1 Water thermal conductivity k 0.59 W m K −3 −1 −1 Water viscosity μ 1.14 × 10 kg m s Water Prandtl number Pr 8.09 – Average ground temperature T 15 °C Borehole depth D 2000 m Table 3 Values of thermal properties for the selected rock types Rock type Minimum thermal Average thermal Maximum thermal Volumetric conductivity value conductivity value conductivity value heat capacity −1 −1 −1 −1 −1 −1 −3 −1 (W m  K ) (W m  K ) (W m  K ) (J m  K ) Sandstone 2.06 3.895 5.73 2.05 × 10 Limestone 1.2 2.15 3.1 2.155 × 10 Granite 2.12 2.87 3.62 2.33 × 10 performance of DBHE, the influence of outer pipe diameter on the heat extraction rate was carried out under different geological conditions (scenario #1A–I). In order to investigate the influences of the inner pipe diameter, the parameters in the DBHE were kept constant except the change of inner pipe diameter. Since the diameter of the inner pipe cannot exceed that of the outer pipe, the range of inner pipe diameters is 0.050–0.140 m and the increment used for analysis is 0.010 m, total- ing 10 steps. Since rocks are heterogeneous materials, the impact of inner pipe diam- eter on heat extraction rate was investigated under different geological conditions (scenario #2A–I). For the analysis of the effects of flow rate on heat extraction  rate, other parame - ters in the DBHE were maintained at a constant level, and the flow rate was changed. −1 The range of the calculated flow rate is 11.6–81.6  L  s , with an incremental step of −1 5.0  L  s used for the analysis. The range of Reynolds number of the inner pipe is from 172,010 to 1,210,000, and the range of Reynolds number of the outer pipe is from 50,755 to 357,040. Scenarios #3A–I were designed to investigate the impact of flow rate on heat extraction rate under different geological conditions. For the analysis of the change in heat extraction rate with different outer pipe mate - rials, we considered three outer pipe wall thermal conductivity values (0.5, 30, 45) in the calculation. The first one represents the concrete outer pipe, and the last two P an et al. Geotherm Energy (2020) 8:6 Page 9 of 20 Table 4 Simulated scenarios for the sensitivity analysis Scenario Outer Inner Flow Outer pipe Grout DBHE Rock Description ID pipe pipe rate wall thermal thermal depth type −1 diameter diameter (L s ) conductivity conductivity (m) −1 −1 −1 −1 (m) (m) (W m  K ) (W m  K ) #1A-C Var 0.09 11.6 0.5 0.73 2000 Sand- #1A, #1D, stone #1G cor- respond to the minimum thermal conductiv- ity value #1D-F Var 0.09 11.6 0.5 0.73 2000 Lime- #1B, #1E, #1H stone correspond to the average thermal conductiv- ity value #1G-I Var 0.09 11.6 0.5 0.73 2000 Granite #1C, #1F, #1I correspond to the maximum thermal conductiv- ity value #2A-C 0.177 Var 11.6 0.5 0.73 2000 Sand- #2A, #2D, stone #2G cor- respond to the minimum thermal conductiv- ity value #2D-F 0.177 Var 11.6 0.5 0.73 2000 Lime- #2B, #2E, #2H stone correspond to the average thermal conductiv- ity value #2G-I 0.177 Var 11.6 0.5 0.73 2000 Granite #2C, #2F, #2I correspond to the maximum thermal conductiv- ity value #3A-C 0.177 0.09 Var 0.5 0.73 2000 Sand- #3A, #3D, stone #3G cor- respond to the minimum thermal conductiv- ity value #3D-F 0.177 0.09 Var 0.5 0.73 2000 Lime- #3B, #3E, #3H stone correspond to the average thermal conductiv- ity value Pan et al. Geotherm Energy (2020) 8:6 Page 10 of 20 Table 4 (continued) Scenario Outer Inner Flow Outer pipe Grout DBHE Rock Description ID pipe pipe rate wall thermal thermal depth type −1 diameter diameter (L s ) conductivity conductivity (m) −1 −1 −1 −1 (m) (m) (W m  K ) (W m  K ) #3G-I 0.177 0.09 Var 0.5 0.73 2000 Granite #3C, #3F, #3I correspond to the maximum thermal conductiv- ity value #4A-C 0.177 0.09 11.6 Var 0.73 2000 Sand- #4A, #4D, stone #4G cor- respond to the minimum thermal conductiv- ity value #4D-F 0.177 0.09 11.6 Var 0.73 2000 Lime- #4B, #4E, #4H stone correspond to the average thermal conductiv- ity value #4G-I 0.177 0.09 11.6 Var 0.73 2000 Granite #4C, #4F, #4I correspond to the maximum thermal conductiv- ity value #5A-C 0.177 0.09 11.6 0.5 Var 2000 Sand- #5A, #5D, stone #5G cor- respond to the minimum thermal conductiv- ity value #5D-F 0.177 0.09 11.6 0.5 Var 2000 Lime- #5B, #5E, #5H stone correspond to the average thermal conductiv- ity value #5G-I 0.177 0.09 11.6 0.5 Var 2000 Granite #5C, #5F, #5I correspond to the maximum thermal conductiv- ity value #6A-C 0.177 0.09 11.6 0.5 0.73 var Sand- #6A, #6D, stone #6G cor- respond to the minimum thermal conductiv- ity value P an et al. Geotherm Energy (2020) 8:6 Page 11 of 20 Table 4 (continued) Scenario Outer Inner Flow Outer pipe Grout DBHE Rock Description ID pipe pipe rate wall thermal thermal depth type −1 diameter diameter (L s ) conductivity conductivity (m) −1 −1 −1 −1 (m) (m) (W m  K ) (W m  K ) #6D-F 0.177 0.09 11.6 0.5 0.73 var Lime- #6B, #6E, #6H stone correspond to the average thermal conductiv- ity value #6G-I 0.177 0.09 11.6 0.5 0.73 var Granite #6C, #6F, #6I correspond to the maximum thermal conductiv- ity value represents pipes made of steel. Further, nine scenarios (#4A–I) were designed to investigate the impact of outer pipe wall thermal conductivity on the heat extraction rate under different geological conditions. Subsequently, we applied three types of grout materials in the simulation to analyze the influences of grout materials. The first one is bentonite–water mixtures with ther - mal conductivity value of 0.73, the second one is a mix of bentonite and silica sand with thermal conductivity value of 1.73, and the last one is a mix of bentonite and graphite with thermal conductivity value of 2.77 (Liu et al. 2018). Furthermore, we designed nine scenarios (#5A-F) to investigate the impact of outer pipe wall thermal conductivity on the heat extraction rate under different geological conditions. We also changed the DBHE depths to see what effect they had on heat transfer capac - ity. In our study, the depth range was set from 500 to 3000 m, with an incremental step of 500  m. Then, scenarios #6A–I were designed to investigate the influences of DBHE depth on the heat extraction rate under different geological conditions. Results and discussion The sensitivity analysis of DBHE design parameters The influence of outer pipe diameter In scenario #1A–I, the simulated results of the heat extraction rate with different outer pipe diameter under different geological conditions are presented in Fig.  2. Fig- ure  2 suggests that as the diameter of the outer pipe increases, the associated increase in heat extraction rate gradually decreases. The heat extraction rate is more sensitive to the increase of the outer pipe diameter when the pipe is smaller and does not lin- early increase with outer pipe diameter. For example, in scenario #1B, when we increase the diameter from 0.180 to 0.220  m, the heat extraction rate increases from 178 to −1 190  W  m (increase by 6.7%). However, when the outer pipe diameter increases from −1 0.360 m to 0.400 m, the heat extraction rate only increases from 221.7 to 229.1 W m (increase by 3.3%). The heat extraction rate is, therefore, more sensitive to increases in the outer pipe diameter while the pipe remains relatively small. Wang et al. (2017) also Pan et al. Geotherm Energy (2020) 8:6 Page 12 of 20 Fig. 2 Influence of different outer pipe diameter on heat extraction rate under different geological conditions simulated the outlet temperature and heat transfer capacity under different outer pipe diameters. The results demonstrate that the outlet temperature and heat transfer capac - ity both increase with an increase to the outer pipe diameter. Because this study is con- cerned with changes in the optimal heat extraction rate, the trend of heat extraction rate in our study is different from the trend of outlet temperature in Wang et al. (2017). Figure  2 also suggests that as the rock thermal conductivity values increase, the heat extraction rate also increases in all three rock types. Moreover, there is a more notice- able change of heat extraction rate in scenario #1A–C than in scenario #1D–F and sce- nario #1G–I. This is because sandstone has a wider range of thermal conductivity value than limestone and granite. By comparing the results of scenario #1A with scenario #1C, we could observe that when the rock thermal conductivity is increased from the mini- mum value (scenario #1A) to the average value (scenario #1B), the heat extraction rate will be improved by 50.9–51.6%. When the rock thermal conductivity is increased from the average value (scenario #1B) to maximum value (scenario #1C), the heat extrac- tion rate can only be improved by 24.5–25.4%. The heat extraction rate, with a double increase in rock thermal conductivity, will not increase exponentially. Chen et al. (2019) also pointed out that the marginal performance gain from increasing soil thermal con- ductivity is gradually decreasing. The influence of inner pipe diameter Figure  3 presents how the heat extraction rate is changing against different inner pipe diameter under different geological conditions (scenario #2A–I). Figure  3 shows that as the inner pipe diameter increases the heat extraction rate decreases, and the heat extrac- tion rate is linearly related to the inner pipe diameter. It is also clear that as the diam- eter of the inner pipe decreases, the heat extraction rate change in value is minimal. For example, in scenario #2B, when the inner pipe diameter decreases from 0.10 to 0.05 m, −1 the heat extraction rate increases from 147.7 to 150.4 W m , only 1.8% was improved. In addition, the reduction of the inner pipe diameter can be regarded as an increase to P an et al. Geotherm Energy (2020) 8:6 Page 13 of 20 Fig. 3 Influence of different inner pipe diameters on heat extraction rate under different geological conditions Fig. 4 Influence of different flow rates on heat extraction rate under different geological conditions the outer pipe diameter, the purpose of which is to increase the area of the annular space and improve the heat transfer capacity. The influence of flow rate From scenario #3A–I, we evaluated the influence of flow rate on heat extraction rate under different geological conditions, and the results are depicted in Fig.  4. Fig- ure  4 indicates that as the flow rate increases, the heat extraction rate increases rap - idly at first. However, the trend of growth in the heat extraction rate will continue to decrease and eventually terminate. Bu et  al. (2012) and Wang et  al. (2017) also found that with an increase in flow rate, the heat transfer capacity would continue to increase. This trend is similar to that illustrated in Fig.  4, with an increase that eventually becomes minimal. This implies that increasing only the flow rate may not Pan et al. Geotherm Energy (2020) 8:6 Page 14 of 20 lead to an increase in the heat extraction rate. Hence, there is an optimal flow rate for DBHE. However, Fig.  4 also shows that the optimal flow rate is not constant, but changes with geological conditions. Thus, when designing a DBHE, we should con - sider the influence of the heterogeneity of geological materials. Based on the analysis results of outer pipe diameter, inner pipe diameter, and flow rate, we could conclude that altering only one of the three parameters could lead to an increase in the rate of heat extraction. However, changing only one parameter can- not achieve optimal transfer efficiency; therefore, the three parameters need to allo - cated simultaneously. The key to achieving the optimal transfer efficiency is to ensure the smooth flow of the circulating water in the DBHE, which means to make sure that the area of the annular space matches the area of the cross-section of the inner pipe. Then, an optimal flow rate that ensures that the DBHE achieves the optimal heat extraction rate is required. The influence of outer pipe materials Figure  5 illustrates the impacts of pipe materials on heat extraction rate under dif- ferent geological conditions (scenario #4A to #4I). It is apparent that the outer pipe wall thermal conductivity value has a limited impact on the heat extraction rate. For example, in scenario #4B, when the outer pipe wall thermal conductivity increases by 590%, the heat extraction rate only be improved by 9%. Moreover, when the outer pipe wall thermal conductivity value is large, keeping an increase in the outer pipe wall thermal conductivity value will not affect the heat transfer performance. Figure  5 also shows that the heat transfer performance of DBHE could be improved by apply- ing a steel outer pipe. Nevertheless, installing the steel outer pipe instead of concrete will increase initial investment, as the DBHE is typically 2–3  km long (Chen et  al. 2019). Therefore, it is not economical to apply steel outer pipe to improve the perfor - mance of DBHE system. Fig. 5 Influence of outer pipe materials on heat extraction rate under different geological conditions P an et al. Geotherm Energy (2020) 8:6 Page 15 of 20 The influence of grout materials Figure  6 demonstrates the effects of varying grout thermal conductivity values on heat extraction rate under different geological conditions (scenario #5A to #5I). From Fig.  6, we can see that the optimization effect of applying thermally enhanced grout materi - als under a good geological condition (high rock thermal conductivity value) is better than that under a poor geological condition (low rock thermal conductivity value). Take scenario #5A and # 5C for example; when we applied thermally enhanced grout mate- rial with thermal conductivity value of 1.73 in DBHE under a poor geological condition (scenario #5A), the heat extraction rate will be improved by 4.3%, while in scenario #5C, the heat extraction rate could be increased by 8.2%. In addition, from the results of sce- nario #5C, we can also observe that when the grout thermal conductivity is increased from 0.73 to 1.73, the heat extraction rate would be improved by 8.2%. When the grout thermal conductivity is increased from 1.73 to 2.77, the heat extraction rate could only be improved by 2.3%. This means the heat extraction rate will not increase exponentially with a double increase in grout thermal conductivity. When compared with the analysis results of pipe materials, it can be concluded that it is more economical to apply ther- mally enhanced grout materials, instead of using a steel outer pipe. The influence of DBHE depth In this study, we defined the depth of 500 m as the lower limit of DBHE in consideration of its heat exchange efficiency. In order to investigate the influence of DBHE depth, the depth range was set from 500 to 3000  m, with an incremental step of 500  m. Figure  7 presents the change in heat extraction rate with DBHE depth under different geological conditions (scenario #6A–I). From Fig.  7, we could find that the heat extraction rate is sensitive to the BHE depth. For instance, in the results of scenario #6B, when we increase the BHE depth from 500 to 1000 m, the heat extraction rate will be increased by approx. 30.0%. However, increasing the DBHE depth will increase the drilling cost. It is, there- fore, the optimal design is different for different depth wells. Fig. 6 Influence of grout materials on heat extraction rate under different geological conditions Pan et al. Geotherm Energy (2020) 8:6 Page 16 of 20 Fig. 7 Influence of DBHE depth on the optimal heat extraction rate under different geological conditions We should point out that the working time of the system could also affect the heat transfer performance of DBHE. While under continuous heat extraction the difference of outlet temperature after 1  year and after 10  years of operation is minimal  (Bu et  al. 2012; Kong et  al. 2017b; Chen et  al. 2019); this phenomenon shows that the mode of continuous heat extraction is not affected by the period of operation. Therefore, the results we simulated are credible, although the long-term analysis is not conducted in this study. Optimization of DBHE As mentioned in "Optimization method" section, there is an optimal DBHE design for each depth under certain geological conditions. Therefore, we obtained the optimal DBHE designs for different depths and rock types by applying the index of AEC, as shown in Fig. 8. Figure  8 shows that the AEC of DBHE in granite and limestone is much higher than the AEC of DBHE in sandstone. Even with a minimum thermal conductivity value, the AEC of DBHE in sandstone is still lower than the AEC of DBHE in granite and lime- stone because the drilling cost in granite and limestone is much higher than in sand- stone. Figure  8 also suggests that the AEC value increases when DBHE depth increases from 500 to 1000 m and deceases when the DBHE depth increases from 1000 to 3000 m. This is because the drilling cost we used was calculated by the relationship between well cost and well depth (Lukawski et al. 2014), and the drilling cost per meter for 1000 m is higher than that for 500 m. When the depth range is 2000–3000 m, the increase of the well depth leads to little increase in drilling cost per meter. Although we employed the relationship between well cost and well depth during the simulation process, there is still a strong uncertainty between the drilling cost and well depth (Lukawski et al. 2014, 2016). In our study, the AEC value decreases when the DBHE depth increases from 1000 to 3000 m, but it does not mean the AEC value will still decrease when the DBHE depth larger than 3000 m. Actually, according to the published well costs, the drilling cost per meter increases rapidly when well depth lager than 3000  m (Gul and Aslanoglu 2018; P an et al. Geotherm Energy (2020) 8:6 Page 17 of 20 Fig. 8 The average energy cost of DBHE with different depths Fig. 9 The comparison of optimal design and non-optimization design Lukawski et  al. 2014, 2016). Therefore, in our cases, a DBHE of 3000  m should be the most economical choice when one wants to get the maximum heat with the lowest cost. In addition, Fig.  8 shows that the AEC value is similar when DBHE depth is 500 and 2000  m, which means the economic benefit of building a 2000  m DBHE is similar to build 4 DBHEs of 500 m. In this case, it is sure one 2000 DBHE is more economical than 4 DBHEs of 500 m due to the lower area demanding. In order to verify the validity of the optimization method, the non-optimization design was compared to the optimal design under the same geological conditions, as shown in Fig. 9. It is obvious that the optimization method is quite effective. Take the DBHE depth of 500  m as an example, the AEC value decreases 22.1% after optimization. Therefore, the optimization method proposed in this study is robust and can significantly improve the performance of DBHE system. Pan et al. Geotherm Energy (2020) 8:6 Page 18 of 20 In our cases, the optimal designs indicate that the most economical outer pipe diam- eter is 0.220 m because the drilling cost will increase a lot when the outer pipe diameter increases larger than 0.22  m. The most economical outer pipe and grout materials are concrete, and a mix of bentonite and graphite, respectively. The optimal flow rate is dif - −1 ferent with different well depths and rock types, and the range is 8.5–63 L s . Moreover, the steel outer pipe has not been considered in any optimal design, which means it is not economical for DBHE system. Conclusions and outlook We carried out a number of sensitivity analyses to identify the effect of design param - eters on the heat transfer performance of DBHE. Since the motivation of this work was to obtain the optimal DBHE design, we proposed a procedure to optimize these design parameters. On the basis of these studies, we have drawn the following conclusions. The heat extraction rate is very sensitive to the outer pipe diameters, well depth, and flow rate. While the grout materials, inner pipe diameter, and outer pipe materi - als have a minor effect. For fixed rock thermal properties, any increase in the contact area between the circulating water and the rock will enhance the heat extraction rate. Moreover, in order to obtain the optimal design, these parameters need to be allocated simultaneously. The optimal DBHE designs reveal that the most economical outer and inner pipe diameter was 0.220  m and 0.1544  m, respectively. For fixed pipe diameters, outer pipe materials, grout materials and the optimal flow rate vary with the change of well depths and rock thermal properties. The results also show that it is more economical to apply thermally enhanced grout materials, instead of using a steel outer pipe. Finally, it should be noted that the heat transfer characteristics of DBHE are also related to several other parameters such as the thermal conductivity of the heat transfer medium, the thermal insulation performance of the inner pipe. Nonetheless, the pro- posed optimization method can significantly improve the heat transfer performance of DBHE. List of symbols Variables −3 −1 −2 −1 d: Diameter, m; f: Friction factor; c: Volumetric heat capacity, J m K ; h: Convective film coefficient, W m K ; H: Ratio −1 −1 2 of volumetric heat capacities; k: Thermal conductivity, W m K ; A: Area, m ; N: Dimensionless thermal conductance; r: −1 −1 Radius, m; R: Thermal resistance, K m W ; Re: Reynolds number; T: Temperature, °C; V: Flow velocity, m s ; t: Time, s; w: 3 −1 −1 Flow rate, m s ; z: Vertical depth coordinate, m; Pr: Prandtl number; P: Heat extraction rate, W m ; Q: Thermal energy output, J; C: Cost, Yuan; D: Borehole depth, m. Greeks −1 −1 −3 κ: Ratio of thermal conductivities; μ: Viscosity, kg m s ; ρ: Density, kg m . Subscripts b: Borehole; pi: Inside of inner pipe; po: Outside of inner pipe; pp: Inner pipe; D: Dimensionless; eo: Outside of outer pipe; ei: Inside of outer pipe; ep: Outer pipe; f: Circulating fluid; g: Grout; s: Ground (or soil); 1: Flow path number 1; 2: Flow path number 2. Acknowledgements This research is supported by the National Key Research and Development Program of China (No. 2018YFB1501801). Authors’ contributions SP performed the simulations. SP and YK prepared the manuscript. CC provided the Matlab code. YK, CC, ZP and JW improved and revised the manuscript. All authors read and approved the final manuscript. P an et al. Geotherm Energy (2020) 8:6 Page 19 of 20 Availability of data and materials The datasets generated and analyzed during the current study are available from the corresponding author on reason- able request. Consent for publication Not applicable. Competing interests The authors declare that they have no competing interests. Author details Key Laboratory of Shale Gas and Geoengineering, Institute of Geology and Geophysics, Chinese Academy of Sciences, Beijing 100029, China. Innovation Academy for Earth Science, Chinese Academy of Sciences, Beijing 100029, China. 3 4 University of Chinese Academy of Sciences, Beijing 100049, China. Helmholtz Centre for Environmental Research-UFZ, 04318 Leipzig, Germany. Received: 26 September 2019 Accepted: 27 January 2020 References Acuña J, Mogensen P, Palm B. Distributed thermal response tests on a multi-pipe coaxial borehole heat exchanger. HVAC & R Res. 2011;17(6):1012–29. Acuña J, Palm B. Distributed thermal response tests on pipe-in-pipe borehole heat exchangers. Appl Energy. 2013;109:312–20. Alimonti C, Soldo E. Study of geothermal power generation from a very deep oil well with a wellbore heat exchanger. Renew Energy. 2016;86:292–301. Alimonti C, Soldo E, Bocchetti D, Berardi D. 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Optimization of the utilization of deep borehole heat exchangers

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Abstract

ylkong@mail.iggcas.ac.cn Key Laboratory of Shale Deep-borehole heat exchangers (DBHE) are generally coaxial pipes installed in deep Gas and Geoengineering, boreholes and has become an alternative approach to utilize geothermal energy. Institute of Geology and Geophysics, Chinese Since the performance of the DBHE system can be affected by several parameters, it is Academy of Sciences, important to optimize the design of parameters for the DBHE. In this paper, based on Beijing 100029, China the analytical method, we carried out the sensitivity analysis of DBHE design param- Full list of author information is available at the end of the eters, including outer pipe diameter, inner pipe diameter, flow rate, outer pipe materi- article als, grout materials, and borehole depth during continuous operation for 4 months. The sensitivity analysis results indicate that the heat extraction rate can be significantly affected by outer pipe diameter, borehole depth, and flow rate. The effects of grout materials, inner pipe diameter and outer pipe materials are of second-order. Finally, an optimization method based on the lowest Average Energy Cost index was proposed to optimize these DBHE design parameters under different geological conditions. Given the cost in this study, a combination scheme of all the optimal parameters is given for different depth wells under different geological conditions. Keywords: Deep-borehole heat exchanger, Geothermal energy, Sensitivity analysis, Optimal heat extraction rate, Economic analysis Introduction Based on the differences in burial depth, utilization mode, and storage medium, geother - mal energy is usually divided into three types: shallow geothermal energy (0–200  m), medium–deep hydrothermal energy (200–3000 m), and hot dry rocks energy (> 3000 m) (Wang 2015). The China Geothermal Energy Development Report released in August 2018 shows that the shallow geothermal energy is the main method used in China for geothermal heating, which has been rapidly developed. The extent of hydrothermal heat - ing is also increasing steadily. However, the development in the utilization of geothermal resources has some difficulties due to the large areas demanding of shallow geothermal energy and the uneven distribution of hydrothermal energy (Kong et al. 2014). Moreo- ver, when utilizing the hydrothermal energy, reinjection of geothermal wastewater must be carried out for maintaining the pressure of geothermal reservoirs (Rybach 2003). However, with a low reinjection rate, reinjection could be quite difficult in sandstone reservoirs (Kong et al. 2014; Su et al. 2018; Ungemach 2003). Therefore, deep-borehole heat exchangers (DBHE) have become an alternative approach to utilize geothermal © The Author(s) 2020. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creat iveco mmons .org/licen ses/by/4.0/. Pan et al. Geotherm Energy (2020) 8:6 Page 2 of 20 energy (Alimonti et  al. 2018). The principle of DBHE is to install a coaxial pipe into a deep borehole to inject cold water into the outer pipe and extract hot water from the inner pipe, forming a closed-loop system (Rybach and Hopkirk 1995). The heat energy can be transported during this process through the heat transfer between circulating fluid and surrounding rock. There is no interaction between circulating fluid and natural groundwater; it is thus called “obtaining heat without extracting groundwater”. It could be found that the principle of DBHE is quite similar to the coaxial borehole heat exchangers (CBHE) except for the depth (Acuña et al. 2011; Holmberg et al. 2016; Kohl et al. 2002; Sapinska-Sliwa et al. 2016). However, at least to the authors’ knowledge, the definition of DBHE depth range does not appear in any literature. In this study, the borehole heat exchangers (BHE) deeper than 500  m were defined as DBHE in consid - eration of the heat exchanging efficiency. The research on BHE carried out in the last decade concerns detailed investigations of the heat transfer performance of CBHE and DBHE. For the CBHE, several studies have focused on investigating its performance and applicability. Acuña et  al. (2011) implemented six distributed thermal response tests (DTRTs) on a multi-pipe CBHE. Acuña and Palm (2013) presented results from three DTRTs carried out on two coaxial pipe-in-pipe BHE at different flow rates. They pointed out that the ground thermal conductivity and the borehole thermal resistance required for designing a CBHE system could be determined by an in  situ thermal response test on a completed borehole. Zanchini et al. (2010) investigated the effects of thermal short- circuiting and flow velocity on the thermal efficiency of CBHE. The results show that the effects of flow velocity and thermal short-circuiting are both important and that the latter can be reduced considerably by employing a low conductivity material for the inner tube. Raymond et al. (2015) evaluated the impact of changing the water flow rate, the inner and outer pipe SDR on the thermal resistance of CBHE. The results show that considerable advantages could be obtained by increasing the water flow rate, and the outer pipe SDR. Beier et al. (2013) developed a CBHE model for the vertical temperature profiles, which can be used instead of the mean temperature approximation to estimate borehole resistance. Furthermore, Luo et  al. (2019) presented an analytical model for CBHE that specifically considers geothermal gradient. Over the past few years, more and more research work has been carried out to inves- tigate the DBHE system performance. Several studies suggest that the heat transfer per- formance of the DBHE correlates significantly with the geothermal gradient and thermal conductivity of the surrounding rock (Bu et al. 2012; Chen et al. 2019; Cheng et al. 2013; Kong et al. 2017a, b; Le Lous et al. 2015; Nalla et al. 2005; Noorollahi et al. 2015; Temple- ton et al. 2014). Le Lous et al. (2015) also conducted the sensitivity analysis of the volu- metric heat capacity of the rock. They found that low volumetric heat capacity value can reduce the thermal recovery time. Except for geological conditions, parameters related to DBHE design and operational settings can also affect the heat performance of DBHE. Fang et al. (2018), Holmberg et al. (2016), and Wang et al. (2017) found that the circu- lating fluid injected through the annular space can reduce the heat loss than through the inner pipe. Moreover, the performance of different circulating fluids in such heat transfer has been evaluated, proving that water is the most suitable fluid for circulation P an et al. Geotherm Energy (2020) 8:6 Page 3 of 20 (Alimonti and Soldo 2016; Bu et al. 2012; Kujawa et al. 2004; Le Lous et al. 2015; Nalla et al. 2005; Noorollahi et al. 2015; Templeton et al. 2014; Wight and Bennett 2015). Furthermore, Nalla et al. (2005) pointed out that a key parameter affecting the perfor - mance of the DBHE is the fluid residence time, which represents the flow rate. The other DBHE design parameters have also been investigated, including the borehole diameter, the diameter of the outer pipe, the diameter of the inner pipe, the depth of the borehole, grout materials and pipe materials (Alimonti and Soldo 2016; Beier and Holloway 2015; Chen et al. 2019; Lhendup et al. 2014; Nalla et al. 2005; Wang et al. 2017). According to the results of these analyses, borehole diameter, the outer pipe diameter, grout thermal conductivity, outer pipe thermal conductivity, and borehole depth can positively influ - ence the heat transfer performance of DBHE (Alimonti and Soldo 2016; Chen et al. 2019; Nalla et al. 2005; Wang et al. 2017). In contrast, the inner pipe diameter and inner pipe wall thermal conductivity could negatively influence the heat transfer performance of DBHE (Alimonti and Soldo 2016; Beier and Holloway 2015; Chen et al. 2019; Lhendup et al. 2014; Nalla et al. 2005). Therefore, it can be concluded that all these DBHE design parameters can affect the system performance, and the optimal design of DBHE can be obtained, while it was rarely given in the literature. Herein, we simulated the effects of design parameters such as outer and inner pipe diameter, flow rate, grout and outer pipe materials, and DBHE depth on the heat extrac - tion rate. The parametric sensitive study is conducted by using the analytical method proposed by Beier (2014) and Beier et al. (2014). Importantly, our study revealed that it is useful for improving the DBHE system performance to allocate these design param- eters simultaneously. Further, we developed an optimization method to identify the opti- mal DBHE configuration in different regions. We considered 3 rock types of sandstone, limestone, and granite due to their wide distribution and utilization for geothermal energy exploitation. Since the optimal design for the DBHE is to improve the heat trans- fer capacity and reduce the cost, our optimization method is to do the economic analysis by applying the Average Energy Cost (AEC) index (Rodríguez and Díaz 2009). Methods Optimization method In this study, we used the index of Average Energy Cost to optimize the DBHE design, which includes the outer pipe diameter, inner pipe diameter, flow rate, outer pipe mate - rials, grout materials, and DBHE depth. The average energy cost represents the specific cost of heat in yuan per kilowatt-hour. It was calculated from the total well cost, which consists of the drilling cost C , g rout drill cost C , coaxial pipe cost C , and circulating water cost C , and is divided by the grout pipe water system’s thermal energy output Q: C + C + C + C drill grout pipe water AEC = . (1) The system’s thermal energy output Q is calculated with heat extraction rate, DBHE depth, and working time. In this study, we defined the heat extraction rate as the maximum rate at which the inlet temperature is near to 0  °C at the end of a heat extraction period. Fang et  al. (2018) used a similar definition for the optimal rate of Pan et al. Geotherm Energy (2020) 8:6 Page 4 of 20 heat extraction, where inlet temperature is maintained at 5 °C at the end of a period of heat extraction. In addition, the analytical method used for calculating the heat extraction rate is given in "Coaxial heat exchanger model" section. The drilling cost for different borehole diameter intervals is presented in Budget Standard of Geological Survey Projects by China Geological Survey (https ://max. book1 18.com/html/2017/0903/13152 0083.shtm), but only 4 ranges (#1–4) of bore- hole diameter are given, and the depth range is 500–1000 m. In order to calculate the AEC of DBHE with a depth range of 500–3000  m, the drilling cost of other depth is calculated by the relation between well depth and drilling cost (Daniilidis et al. 2017; Heidinger 2010; Olasolo et  al. 2016), as shown in Table  1. The borehole diameter intervals of #1–4 are 201–250 mm, 251–300 mm, 301–350 mm, 351–400 mm, respec- tively. It is, therefore, with a constant thickness of grout between the outer pipe and the surrounding soil (0.023  m), the outer pipe diameter intervals of #1–4 are 0.178– 0.227  m, 0.228–0.277  m, 0.278–0.327  m, 0.328–0.377  m, respectively. The coaxial pipe cost is calculated based on the weight per length of coaxial pipe, and a fixed cost per unit mass of steel (Nalla et  al. 2005). The normalized coaxial loop cost is 7 yuan per linear foot of loop, which consists of a 1-in. inner pipe and a 4-in. outer pipe (Liu et al. 2018). The circulating water cost is calculated based on the volume of water, and the normalized water cost is 0.0021 yuan per gallon (Liu et  al. 2018). The grout cost is calculated based on the volume of grout used for sealing the outer pipe diameter. The normalized grout cost is 2.310 yuan per gallon (Liu et al. 2018), and the thermally enhanced grout is 7 and 10.5 yuan per gallon (Liu et al. 2018). The principle of the optimization is to find the DBHE design with minimum AEC value. Since the AEC is highly related to the heat transfer performance of DBHE, the key to minimize the AEC of DBHE is to obtain the optimal heat transfer efficiency. The optimization procedures are as illustrated in Fig. 1 , and the details are as follows. Table 1 The drilling cost of different borehole diameter intervals and rock types Rock type #1 (yuan/m) #2 (yuan/m) #3 (yuan/m) #4 (yuan/m) BHE depth (m) Sandstone 1497 1599 2135 2393 500 Sandstone 2994 3198 4270 4786 1000 Sandstone 5988 6396 8540 9572 1500 Sandstone 8982 9594 12,810 14,358 2000 Sandstone 11,976 12,792 17,080 19,144 2500 Sandstone 17,964 19,188 25,620 28,716 3000 Limestone 2159 2374 3086 3450 500 Limestone 4318 4748 6172 6900 1000 Limestone 8636 9496 12,344 13,800 1500 Limestone 12,954 14,244 18,516 20,700 2000 Limestone 17,272 18,992 24,688 27,600 2500 Limestone 25,908 28,488 37,032 41,400 3000 Granite 3670 4036 5246 5865 500 Granite 7340 8072 10,492 11,730 1000 Granite 14,680 16,144 20,984 23,460 1500 Granite 22,020 24,216 31,476 35,190 2000 Granite 29,360 32,288 41,968 46,920 2500 Granite 44,040 48,432 62,952 70,380 3000 P an et al. Geotherm Energy (2020) 8:6 Page 5 of 20 The first step is to choose the simulated scenario of certain DBHE materials, DBHE depth, and rock type. The second step is varying the outer pipe diameter and choosing one. The third step is to obtain the inner pipe diameter by matching the annular space area with the area of the cross-section of the inner pipe. The fourth step is to modify the flow rate to obtain the relationship between the flow rate and heat extraction rate. The fifth step is calculating the approximate gradient of each flow rate and select the opti - mal flow rate where the gradient decreases to a minimum (in our study, this was when the gradient was less than 0.2/0.5 = 0.4). The sixth step is to calculate the optimal heat extraction rate with the obtained pipes diameter and flow rate. The seventh step is calcu - lating the AEC value of this DBHE design. The eighth step is calculating the AEC values of other DBHE designs by redoing step 2 to step 7. Finally, the ninth step is finding the optimal DBHE design with the lowest AEC values. Coaxial heat exchanger model In this study, we applied the analytical method proposed by Beier (2014) and Beier et  al. (2014) to calculate the heat extraction rate. In their method, the governing equations of heat transfer in the inner pipe, outer pipe, grout, and ground are established, respectively. The two equations in pipes are coupled together by the shunt heat transfer through the wall of the inner pipe. The fluid flowing through the annulus exchanges heat with the surrounding Fig. 1 Procedure to obtain the optimal design of DBHE with a total of n scenarios Pan et al. Geotherm Energy (2020) 8:6 Page 6 of 20 grout. Heat conduction through the grout and ground occurs radially out from the outer pipe. The circulating fluid enters the BHE through the annulus in this model. The energy conservation equation of the annulus is ∂T N H A ∂T s f D1 D1 D1 + + N (T − T ) + N T − T = 0, 12 D1 D2 g D1 Dg (2) r =1 ∂z 2 ∂t D D 0 < Z < 1, 0 < t D D where N is a dimensionless thermal conductance (reciprocal of resistance) of the ground, the symbol w represents the flow rate, and N is a dimensionless thermal con- ductance (reciprocal of resistance) of the grout. The ratio of the volumetric heat capaci - ties of the circulating fluid, c , and the ground, c , is designated as H . The parameter A f s f D1 is a ratio of the flow area. The energy conservation equation of the inner pipe is ∂T N H A ∂T s f D2 D2 D1 − + + N (T − T ) = 0, 0 < Z < 1, 0 < t . (3) 12 D2 D1 D D ∂z 2 ∂t D D The parameter A is a ratio of the inner pipe area and the circular area based on the out- D2 side pipe outer wall radius. Neglecting heat conduction in the axial direction, the heat conduction equation for the grout is ∂T H ∂T ∂T g Dg Dg Dg = + , 1 < r < r ,0 < z < 1, 0 < t . (4) D Db D D κ ∂t r ∂r ∂r D D D The heat conduction equation for the ground surrounding the borehole is ∂T 1 ∂T ∂T Ds Ds Ds = + , r < r ,0 < z < 1, 0 < t . (5) Db D D D ∂t r ∂r ∂r D D D Equations (1) and (2) require boundary conditions. When z = 0 , there is (T − T ) = N , z = 0, 0 < t . D1 D2 s D D (6) When z = 1 , there is, T = T , z = 1, 0 < t . D1 D2 D D (7) At the outer pipe/grout interface where r = 1, an energy balance sets the heat transfer from the circulating fluid to the grout equal to the heat conducted into the grout. That is, N T − T ∂T g Dg D1 Dg = , r = 1, 0 < z < 1, 0 < t . (8) D D D κN ∂r s D At the grout/ground interface, r = r , an energy balance sets the heat conduction Db rate from the grout equal to the heat conduction rate into the ground, and the grout and ground temperatures are equal at this interface, P an et al. Geotherm Energy (2020) 8:6 Page 7 of 20 ∂T ∂T Dg Ds κ = , r = r , 0 < z < 1, 0 < t , (9) D Db D D ∂r ∂r D D ∂T = ∂T , r = r , 0 < z < 1, 0 < t . Dg Ds D Db D D (10) The ground temperature approaches its undisturbed value as the distance from the borehole increases, ∂T → 0, asr →∞, 0 < z < 1, 0 < t . Ds D D D (11) At the start of heat injection ( t = 0 ), the circulating fluid, grout, and ground are all at the undisturbed ground temperature. This condition requires the dimensionless tem - peratures to be zero, T = T = 0, 0 ≤ z ≤ 1, t = 0, D1 D2 D D (12) T = 0, 1 ≤ r ≤ r , 0 ≤ z ≤ 1, t = 0, Dg D D D Db (13) T = 0, r ≤ r , 0 ≤ z ≤ 1, t = 0. (14) Ds Db D D D Detailed calculation of the heat transfer coefficient and the analytical solution can be found in references of Beier et al. (2014). Sensitivity analysis and simulated scenarios The principle of the sensitivity analysis is that when analyzing the impact of one param - eter on the heat extraction rate of DBHE, the other parameters should be kept fixed. In this study, we performed the sensitivity analysis of each design parameter (outer pipe diameter, inner pipe diameter, flow rate, outer pipe materials, grout materials, and depth) by applying the coaxial pipe heat exchanger model. The parameters in the model are given in Table 2, and the system running time is a heating cycle of 4 months. Besides, in order to investigate the applicability of DBHE in different rock types, 3 rock types (sandstone, limestone, granite) were chosen in the calculation. Furthermore, we considered the average, minimum, and maximum values of thermal conductivity for the selected rock types. Since the heat capacity of selected rock types changes little under constant temperature, we chose the mean value of rock volumetric heat capacity (Clauser 2011a, b; McKenna et  al. 1996; Thomas et  al. 1973; Cho et  al. 2009). Table  3 summarizes the details of the thermal properties of selected rock types. With these con- siderations, we established 54 scenarios to simulate the effect of each design parameter. The overview of these scenarios is listed in Table  4, and the details of each scenario are as follows. In order to analyze the effects of outer pipe diameter on the heat extraction rate, the diameter was set at 0.177, 0.180 m, and then increased incrementally in 0.020 m steps, while keeping all other parameters that act upon the DBHE constant. This was repeated until the outer pipe reached a diameter of 0.400 m. Using this method, the diameter of the borehole increases accordingly. Besides, we set the thickness of the grout between the outer pipe and the surrounding soil at 0.023  m. Recogniz- ing that the heterogeneity of geological materials strongly affects the heat transfer Pan et al. Geotherm Energy (2020) 8:6 Page 8 of 20 Table 2 Parameters used in the model Parameter Symbol Amount Unit Borehole diameter d 0.2 m Inner pipe outer diameter d 0.09 m po Inner pipe wall thickness t 0.00734 m Outer pipe outer diameter d 0.177 m eo Outer pipe wall thickness t 0.00587 m −1 Flow rate w 11.6 L s −1 −1 Pipe wall thermal conductivity k , k 0.5 W m K pp ep −1 −1 Grout thermal conductivity k 0.73 W m K 6 −3 −1 Grout volumetric heat capacity c 3.8 × 10 J m K −3 Water density ρ 1000kg m 6 −3 −1 Water volumetric heat capacity c 4.19 × 10 J m K −1 −1 Water thermal conductivity k 0.59 W m K −3 −1 −1 Water viscosity μ 1.14 × 10 kg m s Water Prandtl number Pr 8.09 – Average ground temperature T 15 °C Borehole depth D 2000 m Table 3 Values of thermal properties for the selected rock types Rock type Minimum thermal Average thermal Maximum thermal Volumetric conductivity value conductivity value conductivity value heat capacity −1 −1 −1 −1 −1 −1 −3 −1 (W m  K ) (W m  K ) (W m  K ) (J m  K ) Sandstone 2.06 3.895 5.73 2.05 × 10 Limestone 1.2 2.15 3.1 2.155 × 10 Granite 2.12 2.87 3.62 2.33 × 10 performance of DBHE, the influence of outer pipe diameter on the heat extraction rate was carried out under different geological conditions (scenario #1A–I). In order to investigate the influences of the inner pipe diameter, the parameters in the DBHE were kept constant except the change of inner pipe diameter. Since the diameter of the inner pipe cannot exceed that of the outer pipe, the range of inner pipe diameters is 0.050–0.140 m and the increment used for analysis is 0.010 m, total- ing 10 steps. Since rocks are heterogeneous materials, the impact of inner pipe diam- eter on heat extraction rate was investigated under different geological conditions (scenario #2A–I). For the analysis of the effects of flow rate on heat extraction  rate, other parame - ters in the DBHE were maintained at a constant level, and the flow rate was changed. −1 The range of the calculated flow rate is 11.6–81.6  L  s , with an incremental step of −1 5.0  L  s used for the analysis. The range of Reynolds number of the inner pipe is from 172,010 to 1,210,000, and the range of Reynolds number of the outer pipe is from 50,755 to 357,040. Scenarios #3A–I were designed to investigate the impact of flow rate on heat extraction rate under different geological conditions. For the analysis of the change in heat extraction rate with different outer pipe mate - rials, we considered three outer pipe wall thermal conductivity values (0.5, 30, 45) in the calculation. The first one represents the concrete outer pipe, and the last two P an et al. Geotherm Energy (2020) 8:6 Page 9 of 20 Table 4 Simulated scenarios for the sensitivity analysis Scenario Outer Inner Flow Outer pipe Grout DBHE Rock Description ID pipe pipe rate wall thermal thermal depth type −1 diameter diameter (L s ) conductivity conductivity (m) −1 −1 −1 −1 (m) (m) (W m  K ) (W m  K ) #1A-C Var 0.09 11.6 0.5 0.73 2000 Sand- #1A, #1D, stone #1G cor- respond to the minimum thermal conductiv- ity value #1D-F Var 0.09 11.6 0.5 0.73 2000 Lime- #1B, #1E, #1H stone correspond to the average thermal conductiv- ity value #1G-I Var 0.09 11.6 0.5 0.73 2000 Granite #1C, #1F, #1I correspond to the maximum thermal conductiv- ity value #2A-C 0.177 Var 11.6 0.5 0.73 2000 Sand- #2A, #2D, stone #2G cor- respond to the minimum thermal conductiv- ity value #2D-F 0.177 Var 11.6 0.5 0.73 2000 Lime- #2B, #2E, #2H stone correspond to the average thermal conductiv- ity value #2G-I 0.177 Var 11.6 0.5 0.73 2000 Granite #2C, #2F, #2I correspond to the maximum thermal conductiv- ity value #3A-C 0.177 0.09 Var 0.5 0.73 2000 Sand- #3A, #3D, stone #3G cor- respond to the minimum thermal conductiv- ity value #3D-F 0.177 0.09 Var 0.5 0.73 2000 Lime- #3B, #3E, #3H stone correspond to the average thermal conductiv- ity value Pan et al. Geotherm Energy (2020) 8:6 Page 10 of 20 Table 4 (continued) Scenario Outer Inner Flow Outer pipe Grout DBHE Rock Description ID pipe pipe rate wall thermal thermal depth type −1 diameter diameter (L s ) conductivity conductivity (m) −1 −1 −1 −1 (m) (m) (W m  K ) (W m  K ) #3G-I 0.177 0.09 Var 0.5 0.73 2000 Granite #3C, #3F, #3I correspond to the maximum thermal conductiv- ity value #4A-C 0.177 0.09 11.6 Var 0.73 2000 Sand- #4A, #4D, stone #4G cor- respond to the minimum thermal conductiv- ity value #4D-F 0.177 0.09 11.6 Var 0.73 2000 Lime- #4B, #4E, #4H stone correspond to the average thermal conductiv- ity value #4G-I 0.177 0.09 11.6 Var 0.73 2000 Granite #4C, #4F, #4I correspond to the maximum thermal conductiv- ity value #5A-C 0.177 0.09 11.6 0.5 Var 2000 Sand- #5A, #5D, stone #5G cor- respond to the minimum thermal conductiv- ity value #5D-F 0.177 0.09 11.6 0.5 Var 2000 Lime- #5B, #5E, #5H stone correspond to the average thermal conductiv- ity value #5G-I 0.177 0.09 11.6 0.5 Var 2000 Granite #5C, #5F, #5I correspond to the maximum thermal conductiv- ity value #6A-C 0.177 0.09 11.6 0.5 0.73 var Sand- #6A, #6D, stone #6G cor- respond to the minimum thermal conductiv- ity value P an et al. Geotherm Energy (2020) 8:6 Page 11 of 20 Table 4 (continued) Scenario Outer Inner Flow Outer pipe Grout DBHE Rock Description ID pipe pipe rate wall thermal thermal depth type −1 diameter diameter (L s ) conductivity conductivity (m) −1 −1 −1 −1 (m) (m) (W m  K ) (W m  K ) #6D-F 0.177 0.09 11.6 0.5 0.73 var Lime- #6B, #6E, #6H stone correspond to the average thermal conductiv- ity value #6G-I 0.177 0.09 11.6 0.5 0.73 var Granite #6C, #6F, #6I correspond to the maximum thermal conductiv- ity value represents pipes made of steel. Further, nine scenarios (#4A–I) were designed to investigate the impact of outer pipe wall thermal conductivity on the heat extraction rate under different geological conditions. Subsequently, we applied three types of grout materials in the simulation to analyze the influences of grout materials. The first one is bentonite–water mixtures with ther - mal conductivity value of 0.73, the second one is a mix of bentonite and silica sand with thermal conductivity value of 1.73, and the last one is a mix of bentonite and graphite with thermal conductivity value of 2.77 (Liu et al. 2018). Furthermore, we designed nine scenarios (#5A-F) to investigate the impact of outer pipe wall thermal conductivity on the heat extraction rate under different geological conditions. We also changed the DBHE depths to see what effect they had on heat transfer capac - ity. In our study, the depth range was set from 500 to 3000 m, with an incremental step of 500  m. Then, scenarios #6A–I were designed to investigate the influences of DBHE depth on the heat extraction rate under different geological conditions. Results and discussion The sensitivity analysis of DBHE design parameters The influence of outer pipe diameter In scenario #1A–I, the simulated results of the heat extraction rate with different outer pipe diameter under different geological conditions are presented in Fig.  2. Fig- ure  2 suggests that as the diameter of the outer pipe increases, the associated increase in heat extraction rate gradually decreases. The heat extraction rate is more sensitive to the increase of the outer pipe diameter when the pipe is smaller and does not lin- early increase with outer pipe diameter. For example, in scenario #1B, when we increase the diameter from 0.180 to 0.220  m, the heat extraction rate increases from 178 to −1 190  W  m (increase by 6.7%). However, when the outer pipe diameter increases from −1 0.360 m to 0.400 m, the heat extraction rate only increases from 221.7 to 229.1 W m (increase by 3.3%). The heat extraction rate is, therefore, more sensitive to increases in the outer pipe diameter while the pipe remains relatively small. Wang et al. (2017) also Pan et al. Geotherm Energy (2020) 8:6 Page 12 of 20 Fig. 2 Influence of different outer pipe diameter on heat extraction rate under different geological conditions simulated the outlet temperature and heat transfer capacity under different outer pipe diameters. The results demonstrate that the outlet temperature and heat transfer capac - ity both increase with an increase to the outer pipe diameter. Because this study is con- cerned with changes in the optimal heat extraction rate, the trend of heat extraction rate in our study is different from the trend of outlet temperature in Wang et al. (2017). Figure  2 also suggests that as the rock thermal conductivity values increase, the heat extraction rate also increases in all three rock types. Moreover, there is a more notice- able change of heat extraction rate in scenario #1A–C than in scenario #1D–F and sce- nario #1G–I. This is because sandstone has a wider range of thermal conductivity value than limestone and granite. By comparing the results of scenario #1A with scenario #1C, we could observe that when the rock thermal conductivity is increased from the mini- mum value (scenario #1A) to the average value (scenario #1B), the heat extraction rate will be improved by 50.9–51.6%. When the rock thermal conductivity is increased from the average value (scenario #1B) to maximum value (scenario #1C), the heat extrac- tion rate can only be improved by 24.5–25.4%. The heat extraction rate, with a double increase in rock thermal conductivity, will not increase exponentially. Chen et al. (2019) also pointed out that the marginal performance gain from increasing soil thermal con- ductivity is gradually decreasing. The influence of inner pipe diameter Figure  3 presents how the heat extraction rate is changing against different inner pipe diameter under different geological conditions (scenario #2A–I). Figure  3 shows that as the inner pipe diameter increases the heat extraction rate decreases, and the heat extrac- tion rate is linearly related to the inner pipe diameter. It is also clear that as the diam- eter of the inner pipe decreases, the heat extraction rate change in value is minimal. For example, in scenario #2B, when the inner pipe diameter decreases from 0.10 to 0.05 m, −1 the heat extraction rate increases from 147.7 to 150.4 W m , only 1.8% was improved. In addition, the reduction of the inner pipe diameter can be regarded as an increase to P an et al. Geotherm Energy (2020) 8:6 Page 13 of 20 Fig. 3 Influence of different inner pipe diameters on heat extraction rate under different geological conditions Fig. 4 Influence of different flow rates on heat extraction rate under different geological conditions the outer pipe diameter, the purpose of which is to increase the area of the annular space and improve the heat transfer capacity. The influence of flow rate From scenario #3A–I, we evaluated the influence of flow rate on heat extraction rate under different geological conditions, and the results are depicted in Fig.  4. Fig- ure  4 indicates that as the flow rate increases, the heat extraction rate increases rap - idly at first. However, the trend of growth in the heat extraction rate will continue to decrease and eventually terminate. Bu et  al. (2012) and Wang et  al. (2017) also found that with an increase in flow rate, the heat transfer capacity would continue to increase. This trend is similar to that illustrated in Fig.  4, with an increase that eventually becomes minimal. This implies that increasing only the flow rate may not Pan et al. Geotherm Energy (2020) 8:6 Page 14 of 20 lead to an increase in the heat extraction rate. Hence, there is an optimal flow rate for DBHE. However, Fig.  4 also shows that the optimal flow rate is not constant, but changes with geological conditions. Thus, when designing a DBHE, we should con - sider the influence of the heterogeneity of geological materials. Based on the analysis results of outer pipe diameter, inner pipe diameter, and flow rate, we could conclude that altering only one of the three parameters could lead to an increase in the rate of heat extraction. However, changing only one parameter can- not achieve optimal transfer efficiency; therefore, the three parameters need to allo - cated simultaneously. The key to achieving the optimal transfer efficiency is to ensure the smooth flow of the circulating water in the DBHE, which means to make sure that the area of the annular space matches the area of the cross-section of the inner pipe. Then, an optimal flow rate that ensures that the DBHE achieves the optimal heat extraction rate is required. The influence of outer pipe materials Figure  5 illustrates the impacts of pipe materials on heat extraction rate under dif- ferent geological conditions (scenario #4A to #4I). It is apparent that the outer pipe wall thermal conductivity value has a limited impact on the heat extraction rate. For example, in scenario #4B, when the outer pipe wall thermal conductivity increases by 590%, the heat extraction rate only be improved by 9%. Moreover, when the outer pipe wall thermal conductivity value is large, keeping an increase in the outer pipe wall thermal conductivity value will not affect the heat transfer performance. Figure  5 also shows that the heat transfer performance of DBHE could be improved by apply- ing a steel outer pipe. Nevertheless, installing the steel outer pipe instead of concrete will increase initial investment, as the DBHE is typically 2–3  km long (Chen et  al. 2019). Therefore, it is not economical to apply steel outer pipe to improve the perfor - mance of DBHE system. Fig. 5 Influence of outer pipe materials on heat extraction rate under different geological conditions P an et al. Geotherm Energy (2020) 8:6 Page 15 of 20 The influence of grout materials Figure  6 demonstrates the effects of varying grout thermal conductivity values on heat extraction rate under different geological conditions (scenario #5A to #5I). From Fig.  6, we can see that the optimization effect of applying thermally enhanced grout materi - als under a good geological condition (high rock thermal conductivity value) is better than that under a poor geological condition (low rock thermal conductivity value). Take scenario #5A and # 5C for example; when we applied thermally enhanced grout mate- rial with thermal conductivity value of 1.73 in DBHE under a poor geological condition (scenario #5A), the heat extraction rate will be improved by 4.3%, while in scenario #5C, the heat extraction rate could be increased by 8.2%. In addition, from the results of sce- nario #5C, we can also observe that when the grout thermal conductivity is increased from 0.73 to 1.73, the heat extraction rate would be improved by 8.2%. When the grout thermal conductivity is increased from 1.73 to 2.77, the heat extraction rate could only be improved by 2.3%. This means the heat extraction rate will not increase exponentially with a double increase in grout thermal conductivity. When compared with the analysis results of pipe materials, it can be concluded that it is more economical to apply ther- mally enhanced grout materials, instead of using a steel outer pipe. The influence of DBHE depth In this study, we defined the depth of 500 m as the lower limit of DBHE in consideration of its heat exchange efficiency. In order to investigate the influence of DBHE depth, the depth range was set from 500 to 3000  m, with an incremental step of 500  m. Figure  7 presents the change in heat extraction rate with DBHE depth under different geological conditions (scenario #6A–I). From Fig.  7, we could find that the heat extraction rate is sensitive to the BHE depth. For instance, in the results of scenario #6B, when we increase the BHE depth from 500 to 1000 m, the heat extraction rate will be increased by approx. 30.0%. However, increasing the DBHE depth will increase the drilling cost. It is, there- fore, the optimal design is different for different depth wells. Fig. 6 Influence of grout materials on heat extraction rate under different geological conditions Pan et al. Geotherm Energy (2020) 8:6 Page 16 of 20 Fig. 7 Influence of DBHE depth on the optimal heat extraction rate under different geological conditions We should point out that the working time of the system could also affect the heat transfer performance of DBHE. While under continuous heat extraction the difference of outlet temperature after 1  year and after 10  years of operation is minimal  (Bu et  al. 2012; Kong et  al. 2017b; Chen et  al. 2019); this phenomenon shows that the mode of continuous heat extraction is not affected by the period of operation. Therefore, the results we simulated are credible, although the long-term analysis is not conducted in this study. Optimization of DBHE As mentioned in "Optimization method" section, there is an optimal DBHE design for each depth under certain geological conditions. Therefore, we obtained the optimal DBHE designs for different depths and rock types by applying the index of AEC, as shown in Fig. 8. Figure  8 shows that the AEC of DBHE in granite and limestone is much higher than the AEC of DBHE in sandstone. Even with a minimum thermal conductivity value, the AEC of DBHE in sandstone is still lower than the AEC of DBHE in granite and lime- stone because the drilling cost in granite and limestone is much higher than in sand- stone. Figure  8 also suggests that the AEC value increases when DBHE depth increases from 500 to 1000 m and deceases when the DBHE depth increases from 1000 to 3000 m. This is because the drilling cost we used was calculated by the relationship between well cost and well depth (Lukawski et al. 2014), and the drilling cost per meter for 1000 m is higher than that for 500 m. When the depth range is 2000–3000 m, the increase of the well depth leads to little increase in drilling cost per meter. Although we employed the relationship between well cost and well depth during the simulation process, there is still a strong uncertainty between the drilling cost and well depth (Lukawski et al. 2014, 2016). In our study, the AEC value decreases when the DBHE depth increases from 1000 to 3000 m, but it does not mean the AEC value will still decrease when the DBHE depth larger than 3000 m. Actually, according to the published well costs, the drilling cost per meter increases rapidly when well depth lager than 3000  m (Gul and Aslanoglu 2018; P an et al. Geotherm Energy (2020) 8:6 Page 17 of 20 Fig. 8 The average energy cost of DBHE with different depths Fig. 9 The comparison of optimal design and non-optimization design Lukawski et  al. 2014, 2016). Therefore, in our cases, a DBHE of 3000  m should be the most economical choice when one wants to get the maximum heat with the lowest cost. In addition, Fig.  8 shows that the AEC value is similar when DBHE depth is 500 and 2000  m, which means the economic benefit of building a 2000  m DBHE is similar to build 4 DBHEs of 500 m. In this case, it is sure one 2000 DBHE is more economical than 4 DBHEs of 500 m due to the lower area demanding. In order to verify the validity of the optimization method, the non-optimization design was compared to the optimal design under the same geological conditions, as shown in Fig. 9. It is obvious that the optimization method is quite effective. Take the DBHE depth of 500  m as an example, the AEC value decreases 22.1% after optimization. Therefore, the optimization method proposed in this study is robust and can significantly improve the performance of DBHE system. Pan et al. Geotherm Energy (2020) 8:6 Page 18 of 20 In our cases, the optimal designs indicate that the most economical outer pipe diam- eter is 0.220 m because the drilling cost will increase a lot when the outer pipe diameter increases larger than 0.22  m. The most economical outer pipe and grout materials are concrete, and a mix of bentonite and graphite, respectively. The optimal flow rate is dif - −1 ferent with different well depths and rock types, and the range is 8.5–63 L s . Moreover, the steel outer pipe has not been considered in any optimal design, which means it is not economical for DBHE system. Conclusions and outlook We carried out a number of sensitivity analyses to identify the effect of design param - eters on the heat transfer performance of DBHE. Since the motivation of this work was to obtain the optimal DBHE design, we proposed a procedure to optimize these design parameters. On the basis of these studies, we have drawn the following conclusions. The heat extraction rate is very sensitive to the outer pipe diameters, well depth, and flow rate. While the grout materials, inner pipe diameter, and outer pipe materi - als have a minor effect. For fixed rock thermal properties, any increase in the contact area between the circulating water and the rock will enhance the heat extraction rate. Moreover, in order to obtain the optimal design, these parameters need to be allocated simultaneously. The optimal DBHE designs reveal that the most economical outer and inner pipe diameter was 0.220  m and 0.1544  m, respectively. For fixed pipe diameters, outer pipe materials, grout materials and the optimal flow rate vary with the change of well depths and rock thermal properties. The results also show that it is more economical to apply thermally enhanced grout materials, instead of using a steel outer pipe. Finally, it should be noted that the heat transfer characteristics of DBHE are also related to several other parameters such as the thermal conductivity of the heat transfer medium, the thermal insulation performance of the inner pipe. Nonetheless, the pro- posed optimization method can significantly improve the heat transfer performance of DBHE. List of symbols Variables −3 −1 −2 −1 d: Diameter, m; f: Friction factor; c: Volumetric heat capacity, J m K ; h: Convective film coefficient, W m K ; H: Ratio −1 −1 2 of volumetric heat capacities; k: Thermal conductivity, W m K ; A: Area, m ; N: Dimensionless thermal conductance; r: −1 −1 Radius, m; R: Thermal resistance, K m W ; Re: Reynolds number; T: Temperature, °C; V: Flow velocity, m s ; t: Time, s; w: 3 −1 −1 Flow rate, m s ; z: Vertical depth coordinate, m; Pr: Prandtl number; P: Heat extraction rate, W m ; Q: Thermal energy output, J; C: Cost, Yuan; D: Borehole depth, m. Greeks −1 −1 −3 κ: Ratio of thermal conductivities; μ: Viscosity, kg m s ; ρ: Density, kg m . Subscripts b: Borehole; pi: Inside of inner pipe; po: Outside of inner pipe; pp: Inner pipe; D: Dimensionless; eo: Outside of outer pipe; ei: Inside of outer pipe; ep: Outer pipe; f: Circulating fluid; g: Grout; s: Ground (or soil); 1: Flow path number 1; 2: Flow path number 2. Acknowledgements This research is supported by the National Key Research and Development Program of China (No. 2018YFB1501801). Authors’ contributions SP performed the simulations. SP and YK prepared the manuscript. CC provided the Matlab code. YK, CC, ZP and JW improved and revised the manuscript. All authors read and approved the final manuscript. P an et al. Geotherm Energy (2020) 8:6 Page 19 of 20 Availability of data and materials The datasets generated and analyzed during the current study are available from the corresponding author on reason- able request. Consent for publication Not applicable. 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