The single well geothermal heating (SWGH) technology has attracted extensive attention. To enhance heat extraction from SWGH, a mathematical model describing heat transfer is set up, and the key influence factor and heat transfer enhancement method are discussed by thermal resistance analysis. The numerical results show that the thermal resistance of rock is far greater than that of well wall and fluid. So, reducing rock thermal resistance is the most effective method for enhancing the heat extraction power. For geothermal well planning to drill: rock thermal resistance can be reduced by increasing well diameter and rock thermal conductivity; the temperature difference between liquid and rock can be raised by increasing well depth. For already existing geothermal well: an insulator with thermal conductivity of 0.2 W/(mK) is sufficient to preserve fluid enthalpy; a decrease in injection water temperature causes the increase of heat extraction power from geothermal well and heat output from heat pump simultaneously; increasing injection velocity causes the increase of pump power consumption and heat extraction power from geothermal well as well as net heat output between them. The entrepreneurs may refer to the above data in actual project. Furthermore, filling composite materials with high thermal conductivity into leakage formation is proposed in order to reduce the thermal resistance of rocks.

In northern China, coal accounts for 92% energy use for building heating systems, which contributes significantly to winter smog air pollution. As a consequence, there is a more pressing need to utilize renewable energy for building heating in order to reduce coal consumption and air pollution. Geothermal energy has shown significant potential because of its low environmental impact, low greenhouse gas emissions and technology feasible [^{2} buildings by the end of 2017 in China [

Recently, without hydrothermal geothermal resource constrains, geothermal water reinjection, and corrosion and scaling, the single well geothermal heating (SWGH) is widely accepted. The most crucial part in SWGH is deep borehole heat exchanger (DBHE), which is also named coaxial borehole heat exchanger or wellbore heat exchanger, as shown in

Some researchers studied the utilizations of geothermal energy from abandoned oil and gas wells owing to low drilling cost by retrofitting them into DBHE [

Some researchers have carried out experimental and numerical simulation researches on DBHE. However, there is very few studies reported on the heat transfer enhancement of DBHE for building heating through thermal resistance analysis. The main purpose of the current study is to propose some effective ways to enhance the heat transfer of DBHE using thermal resistance analysis method. First, a mathematical model describing the heat transfer between surrounding rocks, well tube and insulation tube as well as the circulating fluid is developed. And then the performance of DBHE and the key influence factors on heat extraction are discussed by thermal resistance analysis. The novelty of this research is as follows: enhancing heat transfer by analyzing and reducing thermal resistance; indicating that poor thermal conductivity of rocks is a major obstacle to the performance improvement of DBHE, and thus reducing rock thermal resistance is the most effective method for enhancing the heat extraction power; studying the effect of injection water temperature on heat extraction from DBHE and heat pump performance simultaneously; analyzing the relationship between heat extraction, injection mass flow rate and power consumption of circulating pump; and proposing some effective methods for the enhancement of heat transfer of the existing DBHE and the new geothermal well planning to drill to install DBHE respectively. Furthermore, two schemes are suggested in order to reduce the thermal resistance of rocks. One is the technology of enhanced geothermal system (EGS), which creates artificial fractures in the rocks by hydraulic fracturing technology and injecting fluid into them; the other is the technology of filling composite materials with high thermal conductivity into leakage formation, which is a common technology in oil and gas industry.

SWGH mainly consists of DBHE, circulating pump and heat pump system. The structure diagram of DBHE is shown in

To simplify the mathematical model, some assumptions have been made as follows: The mathematical model of DBHE mainly includes heat conduction in rocks and heat convection in circulating fluid inside well tube and insulation tube [

Equations for extraction channel:

Equations for injection channel:

Energy equation for rocks:

The heat transfer in rocks is controlled by conduction, and the energy equation for rocks is given [

Dittus-Boelter formula is used to calculate the convective heat transfer coefficient.

For injection channel:

For extraction channel:

Boundary conditions:

The heat transfer from well wall to injection fluid is expressed as:

The heat exchange between well wall and rock is determined by:

In _{3} = 50 W/(mK).

Assuming that rock temperature keeps unchanged when

Performance of heat pump unit:

Because the fluid does not touch with rocks, the extracted fluid flows directly into the evaporator of heat pump. With supply and return water temperature of 45/40°C in the user side, the relationship of coefficient of performance (COP) and _{out} is given as follows by Liu et al. [

The output heating of the heat pump can be determined according to the following equation:

The heat extracted from the geothermal well is estimated as follows:

Finite volume method with the implicit scheme is used to discretize

The process of heat transfer from rocks to fluid includes three thermal resistances as shown below [

Heat transfer power per unit well depth is calculated as follows:

Thus, the heat extracted from the geothermal well can also be expressed by

The geothermal well was drilled specially for installing DBHE in 2016 at Qingdao, China. At the first spudding section, the casing with size of

Parameter | Value |
---|---|

_{0} |
288.15 K |

_{g} |
27 K/km |

2605 m | |

_{1} |
0.21 W/(mK) |

_{4} |
2800 kg/m^{3} |

_{4} |
920 J/(kgK) |

_{4} |
3.49 W/(mK) |

30 t/h |

The key performance indicators for geothermal single well include extraction temperature _{out}, extraction power _{E} and performance attenuation. The main factors that affect the three key indicators include well depth _{4}, rock thermal conductivity _{4}, geothermal gradient _{g}, insulation material thermal conductivity _{1}, injection temperature _{in} and injection mass flow rate _{T} concerned with _{4}, _{4} and _{in}. The second method is to increase the heat transfer temperature difference Δ_{g} and _{in}. The third method is to reduce heat loss decided by thermal conductivity of insulation material _{1}. For a new geothermal well planning to drill, _{4}, _{4}, _{g} can be selected. While for an existing geothermal well, _{in}, _{in} and _{1} can be optimized to enhance heat transfer.

The _{0}, _{4} and _{4} are kept constant in the present simulations. The well type listed in _{g}, _{1}, _{4}, _{in} and _{in} are variables, and their effects on extraction temperature _{out} and extraction power _{E} are evaluated. The basic parameters used in calculation are presented in _{0}, _{g} and

Well type | Well tube size (mm) | Insulation tube size (mm) |
---|---|---|

B | ||

M | ||

S |

Parameter | Value |
---|---|

_{0} |
288.15 K |

_{g} |
30 K/km |

3000 m | |

_{1} |
0.0 W/(mK) |

_{4} |
2700 kg/m^{3} |

_{4} |
1098 J/(kgK) |

_{4} |
3.0 W/(mK) |

_{in} |
1 m/s |

_{in} |
278.15 K |

Well type | M |

_{out} and the extracted heat power _{E} with time at the first heating season. As demonstrated in _{out} and _{E} firstly decrease sharply, and then drops slowly. The temperature of the rock is relatively high at the beginning, and the temperature difference between rock and fluid is large, thus resulting in a higher _{out} and _{E}. Nevertheless, _{4} drops as the heat extraction process proceeds, and the temperature difference between rock and fluid gradually becomes smaller, leading to a gradually decrease in _{out} and _{E}. _{out} is respectively 291.24, 290.2 and 289.23 K at the 5th, 10th and 21st weeks.

_{out} and _{E} with the heating year. _{out} and _{E} are the average value at each heating season. It is obvious that _{out} and _{E} drop apparently with time, but the reduction rate becomes small gradually. _{out} and _{E} are respectively 290.15, 288.61, 288.28 K and 755.01, 660.02, 639.42 kW at 1st, 10th and 20th heating season which will be helpful for the selection of heat pump.

The influence radius of rock is 7.24 m at the first heating season. According to _{H}, _{S}, _{R}, and _{T} is respectively 6.86 ^{–4}, 7.14 ^{–4}, 2.22 ^{–1} and 22.34 ^{–2} mK/W for M type geothermal well. It is clear that _{R} is much greater than _{H} and _{S}. According to _{R} can be reduced by improving rock thermal conductivity and enlarging well tube diameter.

The effect of rock thermal conductivity _{4} on _{out} and _{E} is illustrated in _{4} results in rises of _{out} and _{E}. The average _{out} and _{E} at each heating season are respectively 288.81, 290.50, 292.11 K and 672.71, 777.52, 877.62 kW for _{4} = 2.5, 3.0 and 3.5 W/(mK). _{E} increases by nearly 100 kW when _{4} increases by 0.5 W/(mK).

The thermal conductivity of the rock has significant influence on the heat extraction rate and heat recovery. According to _{R} is inversely proportional to _{4}. The bigger _{4}, the smaller _{R}. _{R} is 3.14 ^{–1}, 2.61 ^{–1} and 2.24 ^{–1} mK/W, respectively, for _{4} = 2.5, 3.0 and 3.5 W/(mK). The location with higher _{4} should be selected for drilling a new geothermal well in practical engineering. The _{4} is determined by natural conditions. Thus, filling composite material with high thermal conductivity into leakage formation is an effective measure to improve thermal conductivity and reduce thermal resistance of rocks.

_{out} and _{E} as a function of _{g}. As can be discovered, _{out} and _{E} depend largely on _{g}, and increasing _{g} will cause the increase of _{out} and _{E}. Δ_{g}, thus resulting in the corresponding increases in _{L} and _{E} when _{T} remains unchanged according to _{out} and _{E} at the heating season are respectively 287.06, 290.50, 293.94 K and 563.97, 777.52, 991.06 kW for _{g} = 20, 30 and 40 K/km. _{E} increases by 213.55 kW when _{g} varies from 20 to 30 K/km. _{E} has the same variation trend as _{g} varies from 30 to 40 K/km.

The effect of well diameter on _{out} and _{E} are shown in _{E} increases with increasing well diameter. Average _{out} and _{E} at the whole heating season are 295.17, 290.50, 289.07 K and 702.62, 777.52, 812.85 kW, respectively. _{E} for M and B type well increase by 74.90 and 110.23 kW compared with S type well. The cross sectional area for M and B type well is respectively 1.52 and 1.89 times larger than that of S type well, while _{E} of M and B type well is only 1.12 and 1.16 times larger than that of S type well. That is to say the smaller the well diameter, the deeper influence of well diameter on _{E}. _{R}, _{H}, _{S} and _{T} all vary with the well diameter according to _{T} of S, M and B type well is respectively 27.36 ^{–2}, 26.24 ^{–2} and 25.74 ^{–2} mK/W, and R_{T} for M and B type well decreases by 4.09% and 5.92% respectively than that of S type well. It can be concluded that an increase in well diameter leads to increasing _{E}, however drilling cost also increases with the enlargement of well diameter. Therefore, the economic performance should be considered for the selection of well diameter.

_{out} and _{E} as a function of well depth are described in _{out} and _{E}. The increasing well depth will lead to an increase in _{out} and _{E.} This is because an increased _{g}, and thus the increase of Δ_{E}. _{out} and _{E} are 284.33, 290.50, 298.66 K and 387.04, 777.52, 1285.85 kW, respectively, at _{E} increases by 390.48 and 508.33 kW, respectively, indicating that the growth rate of _{E} becomes larger. But a deeper geothermal well will result in a higher drilling cost. To sum up, the system performance, drilling cost and payback period should be considered comprehensively so as to decide a reasonable well depth during practical design.

In actual project, the insulator is easily replaced. It is particularly meaningful to analyze the effect of insulator on DBHE performance. _{1} for insulator on _{E} at the first heating season. It can be observed that, the decreasing _{1} will cause an increase in _{E}, the smaller the _{1}, the bigger the _{E}. More heat is prevented from extraction channel to injection channel with smaller _{1}, thus resulting in a higher _{E}. Average _{E} at the whole heating season is respectively 777.49, 770.32, 763.27, 756.39 and 749.66 kW for _{1} = 0, 0.05, 0.1, 0.15 and 0.2 W/(mK). _{E} decreases by 13.61 kW when _{1} rises from 0.1 to 0.2 W/(mK), which implies that the thermal conductivity of insulator has little effect on _{E}.

The fluid temperatures in injection and extraction channel _{2} and thus a drop in temperature for _{1} as well as _{out}. It is obviously the bigger the _{1}, the greater the heat loss, and hence, the lower the _{out}. _{out} is 289.40, 289.31, 289.22, 289.13 and 289.04 K, respectively, for _{1} = 0, 0.05, 0.1, 0.15 and 0.2 W/(mK). Compared to _{1} = 0 W/(mK), _{out} drops by 0.36 K, and _{E} decreases by 27.22 kW with reduction ratio of 3.58% for _{1} = 0.2 W/(mK), implying that an ordinary insulator, such as polypropylene with thermal conductivity of 0.2 W/(mK), has met the need for insulation.

_{L} and _{2} _{1} = 0. It is evident _{L} and _{2} increase rapidly with well depth. This is attributed to the fact that rock temperature increases with well depth, which leads to a rapid rise of _{2} and thus a nearly linear increasing _{L}. The increasing well depth is an effective means for heat transfer enhancement between rock and geothermal well, but the economic performance needs to be considered.

_{E} and _{T} vs. _{in} is plotted in _{in}, the higher the _{E} and _{T}. The decrease in _{in} causes the increase of Δ_{E}. _{E} is 812.40, 777.49 and 743.91 kW, respectively, for _{in} = 275.65, 278.15 and 280.65 K. COP is respectively 4.51, 4.70 and 4.91 for _{in} = 275.65, 278.15 and 280.65 K, indicating that COP increases with the increase of _{in}. _{T} is respectively 1043.89, 987.86 and 934.29 kW for _{in} = 275.65, 278.15 and 280.65 K, implying that _{T} increases with the decrease of _{in} although COP decreases. The above analysis results show that under working conditions of heat pump, the lower the _{in}, the higher the _{E} and _{T}.

_{E} as a function of _{in} is plotted in _{E}. Average _{E} at the whole heating season is respectively 715.44, 777.49 and 800.91 kW with _{in} of 0.5, 1.0 and 1.5 m/s. An increase in _{in} results in an increase in _{3} according to _{H} according to _{H} is respectively 10.64 ^{–4}, 8.46 ^{–4} and 7.07 ^{–4} mK/W at _{in} = 0.5, 1.0 and 1.5 m/s, reflecting that the influence of _{in} on _{E} becomes small with increasing _{in}. Generally, increasing velocity leads to an increase in pump power consumption. The relationship between pump power consumption, _{in} and _{E} is given as follows [

where ^{3}/h; _{P} is the head of pump, m; _{p} is the pump efficiency; _{P} is the power consumption of pump, kW.

_{P}, _{in} and _{E}. _{P} increases with the rising _{in}, and they are respectively 8.83, 10.65 and 12.31 kW at _{in} = 0.5, 1.0 and 1.5 m/s. The difference between _{E} and _{P} is respectively 706.61, 766.84 and 788.60 at _{in} = 0.5, 1.0 and 1.5 m/s, which indicates that increasing _{in} can cause the increase of net heat output from geothermal well and pump power consumption. The performance of heat pump should be considered to obtain the optimal _{in}.

Parameters | Values | ||
---|---|---|---|

_{in} (m/s) |
0.5 | 1 | 1.5 |

_{P} (m) |
65.36 | 56.02 | 41.75 |

_{p} (%) |
54.07 | 76.86 | 74.29 |

_{P} (kW) |
8.83 | 10.65 | 12.31 |

_{E} (kW) |
715.44 | 777.49 | 800.91 |

_{E}–_{P} (kW) |
706.61 | 766.84 | 788.60 |

In the present study, a mathematical model describing heat conduction in rocks and heat convection in circulating fluid inside well tube and insulation tube is developed. And then some methods of heat transfer enhancement for SWGH are proposed and discussed by thermal resistance analysis. The main following conclusions can be drawn:

The extracted water temperature and extracted heat power both decrease with time, and the decreasing rate is decreased. For the 1st, 10th and 20th heating season, the average extracted water temperature is 290.15, 288.61 and 288.28 K, respectively, and the average extracted heat power is 755.01, 660.02 and 639.42 kW, respectively. The rock temperature always falls with time. The distance between two wells should not be less than 100 m to avoid interference.

The main factor affecting heating power is the thermal resistance of the rock, which is much larger than that of well wall and fluid. Reducing thermal resistance of rock is the most effective method to boost the extracted heat power. The thermal resistance of rock can be reduced by increasing well diameter and rock thermal conductivity. The extracted heat power increases by 100 kW when the rock thermal conductivity increases by 0.5 W/(mK).

The increasing well depth and geothermal gradient are also beneficially to promote extracted heat power. The heat transfer power per unit well depth increases with increasing well depth, and it is about 40 W/m at the top of the well and reaches almost 400 W/m at the bottom of the well. The extracted heat power is proportional to the geothermal gradient. The extracted heat power increases 213.54 kW with the increase of geothermal gradient of 10 K/km.

The thermal conductivity of insulation material has little impact on the extracted power. The extracted heat power decreases by 13.61 kW as the conductivity of insulation material rises from 0.1 to 0.2 W/(mK). The reduction ratio is just 1.78%. The decreasing injection water temperature is beneficial to the extracted heat power. The extracted heat power and output power of heat pump are respectively 812.40, 777.49, 743.91 kW and 1043.89, 987.86, 934.29 kW when injection water temperature is 275.65, 278.15 and 280.65 K.

Increasing injection velocity can cause the increase of pump power consumption, heat extraction from geothermal well and net heat output between them. The performance of heat pump should be considered when choosing injection velocity so as to get the optimal system performance.

_{1}:

flow area of extraction channel, m^{2}

_{3}:

flow area of injection channel, m^{2}

_{4}:

specific heat of rock, J/(kg K)

_{P}:

specific heat of fluid, J/(kg K)

_{4}:

well depth, m

well depth, m

Prandtl number

_{E}:

heat extracted from geothermal well, kW

_{T}:

Output heating of heat pump, kW

Reynolds number

_{H}:

the convective thermal resistance for fluid in the geothermal well, mK/W

_{R}:

the conductive thermal resistance in the rock, mK/W

_{S}:

the conductive thermal resistance for well wall, mK/W

_{T}:

total thermal resistance, mK/W

_{12}:

heat exchange rate between injection and extraction channel, K/s

_{23}:

heat exchange rate between fluid and well wall, K/s

_{0}:

ground surface temperature, K

_{1}:

fluid temperature in extraction channel, K

_{2}:

fluid temperature in injection channel, K

_{3}:

well wall temperature, K

_{4}:

rock temperature, K

_{g}:

geothermal gradient, K/km

_{in}:

injection temperature, K

_{out}:

extraction temperature, K

_{in}:

injection velocity, m/s

_{1}:

flow velocity in extraction channel, m/s

_{2}:

flow velocity in injection channel, m/s

_{1}:

thickness of insulation material, m

_{4}:

thickness of well wall, m

_{e}:

hydraulic diameter, m

_{1}:

convective heat transfer coefficient for inner wall of insulation tube, W/(m^{2} K)

_{2}:

convective heat transfer coefficient for outer wall of insulation tube, W/(m^{2} K)

_{3}:

convective heat transfer coefficient for well wall, W/(m^{2} K)

_{L}:

heat conductivity coefficient for unit length, W/(m K)

mass flow rate, t/h

_{L}:

Heat transfer power per unit well depth, W/m

distance from well wall, m

_{1}:

inner radius of insulation tube, m

_{2}:

outer radius of insulation tube, m

_{3}:

inner radius of well wall, m

_{4}:

outer radius of well wall, m

_{∞}:

the influence radius of rock, m

_{1}:

thermal conductivity of insulation material, W/(m K)

_{3}:

thermal conductivity for well wall, W/(m K)

_{4}:

thermal conductivity of rock, W/(m K)

water density, kg/m^{3}

_{4}:

density of rock, kg/m^{3}

_{4}:

rock thermal diffusivity, m^{2}/s

simulation time, s

temperature difference, K

coefficient of performance

deep borehole heat exchanger

enhanced geothermal system

single well geothermal heating