Sage Journals: Discover world-class research

Abstract

Closed-loop U-shaped geothermal wells show great potential owing to their special well-depth structure, which can provide a good flow rate and heat extraction. However, no advanced process parameter optimization method is available for U-shaped geothermal wells. To effectively describe the heat transfer processes of U-shaped geothermal wells, an analytical solution that couples transient heat conduction in the surrounding soil (or rocks) with the quasisteady heat transfer process in boreholes was developed. This modelling approach depends on many common elements, such as the thermophysical properties of the working fluid and series of resistances for various elements in the wellbore. Subsequently, based on the exergy analysis method, the optimal operating flow rate was defined and a design method for the optimal flow rate was developed. Results indicate that to obtain the maximum exergy efficiency, different optimal flow rates for the U-shaped geothermal well are achieved at different stages of the heating period. This findings of this study expand the research ideas of the process parameter optimization of U-shaped geothermal wells and provide a theoretical basis for developing an optimal circulating-flow-rate design for U-shaped geothermal wells.

Keywords

Geothermal energy u-shaped geothermal well optimal flow rate exergy analysis heat transfer

Introduction

As The global warming issue becomes more prominent, the target of carbon neutral has prompted a shift in current energy systems. To reduce greenhouse gas emissions and keep the global temperature rise within 1.5 °C, a transition to sustainable energy must be made at the earliest (Blohm, 2021; Grubb, 1992). As a renewable and clean source of energy, geothermal energy offers the advantages of stability, efficiency, and continuity (Cadelano et al., 2019; Jung et al., 2015), and can be used for power generation and winter heating (Karlsdóttir et al., 2019; Liddell et al., 2016). The use of geothermal energy can reduce CO₂ from energy supply sources by reducing its emission amount as well as from the energy consumption end by injecting and storing (Wu et al., 2021). Geothermal energy projects have great practical significance in regions with high-temperature geothermal resources to improve energy systems, revitalize the local economy, promote high-quality development, and realize the goals of achieving carbon peak as well as carbon neutrality.

Pumping–irrigation geothermal energy by extracting hot water from the formation, offering an advanced technology for the development and use of geothermal energy. However, subsequently, problems such as groundwater pollution, excessive sand production at the wellhead, hydrological decline, and even surface-water-level sinking may occur (Kamila et al., 2021; Kaya et al., 2011; Yin et al., 2021), which will increase the difficulty of subsequent geothermal development. Currently, these problems can be addressed using geothermal water recharge technology, which can achieve heat extraction without water and exhibits zero pollution as well as sustainability (van Oort et al., 2021).

Recently, closed-loop geothermal systems based on geothermal water recharge technology that consist of injection, horizontal, and production wells have been proposed (Fallah et al., 2021; Li, et al., 2019a; Sun et al., 2018). These systems are efficient and sustainable. Furthermore, they can reduce risks associated with drilling and improve the economy (Grayson and Gans, 2012); in addition, they demonstrate good applicability in the field (Song, et al., 2018a). Compared with geothermal systems based on the downhole coaxial heat exchanger, which are widely used at present (Beier et al., 2013; Davis and Michaelides, 2009; Li et al., 2020; Ma et al., 2020; Song et al., 2019; Song, et al., 2018b; Zhang et al., 2019), closed-loop U-shape geothermal wells can provide higher flow rate and better heat extraction owing to their special well-depth structure, thus showing great potential in the development and utilization of geothermal energy.

However, several current studies on U-shaped geothermal wells focus on their concept and nature (Li, et al., 2019a). Existing studies have developed mathematical models for flow and heat transfer to determine the relation between the well structure, wellbore size, process parameters, and working fluid physical parameters and the heating power and outlet temperature (Li, et al., 2019a; Liao et al., 2021; Lyu et al., 2020; Sun et al., 2019). The finding of such studies have laid the foundation for developing geothermal energy technology using U-shaped geothermal wells. However, many mathematical models have not been verified because of the lack of field or indoor experimental data. Additionally, a set of optimization design methods for their process parameters is lacking. These shortcomings have affected the early commercial usage of geothermal energy technology using U-shaped geothermal wells.

Exergy analysis, a method based on the second law of thermodynamics, considers both the “quality” and “quantity” of energy, and is a very useful tool for engineering design and optimization, particularly in power and energy systems (Moran and Enrico, 1994). Currently, the low utilization rate of geothermal energy has prompted the adoption of advanced methods for achieving more optimal process parameters for geothermal wells. However, the exergy balance is instrumental in the efficient use of resources because it can determine the location, type, and true magnitude of energy resource waste (Wall, 2003).

In geothermal systems, exergy analysis has been mainly employed in power systems (Hepbasli and Akdemir, 2004; Kurşun, 2020; Ozgener et al., 2005). For geothermal heating systems, simple exergy analysis was performed (Malana et al., 2020; Ozgener et al., 2005); however, the optimization of process parameters was not studied precisely. Furthermore, exergy analysis of geothermal wells mainly focuses on groundwater source heat pumps, and closed-loop U-shaped geothermal wells, which have been proposed in recent years, are rarely reported.

In this study, a mathematical model for the flow and heat transfer of U-shaped geothermal wells was developed to analyze the relation between different flow rates and the outlet fluid temperature as well as the relation between different flow rates and heat extraction. Based on the exergy analysis method, a design method for the optimal flow rate was developed and applied to a geothermal well in Xi’an, China, as an example. The findings of this study can expand the research ideas of the process parameter optimization of U-shaped geothermal wells and provide a theoretical basis for developing an optimal circulating-flow-rate design for U-shaped geothermal wells.

Physical Problem and mathematical model

Description Of U-shaped geothermal well operation

A U-shaped geothermal well was developed by joining a vertical well and a horizontal well. The developed well comprised four sections: a descending section, a diagonal section, a horizontal section, and an ascending section. Linear and heat-conducting cement was used in the horizontal section to improve heat transfer, whereas insulation materials were used in the ascending section to preserve heat loss. A working fluid medium was used to achieve heat exchange through the descending, diagonal, horizontal, and ascending sections and then through the heat pump system to complete the closed-loop cycle. The working fluid in a wellbore exchanges heat with the casing wall via forced convection, and heat conduction occurs in solid parts, including casings, cement sheaths, and rocks. This configuration breaks through the limitation of the low working flow of the downhole coaxial heat exchangers and simultaneously increases the use of high-temperature formation heat energy in the deepest part of the well. The working principle of the developed U-shaped geothermal well is presented in Figure 1.

Figure 1.

Schematic of a U-shaped geothermal well.

To formulate the energy equations needed to describe the thermal behaviour of the entire wellbore profile and surrounding formations, the following assumptions were made:

The first type of boundary condition is the formation set at an infinite distance from the wellbore; the formation follows a simple geothermal gradient distribution. In the International Ground Source Heat Pump Association method (Ahn et al., 2005), the change in heat transfer between the formation and the wellbore over time is reflected by the change in the thermal resistance of the formation over time.

The circulating working fluid returning from the U-shaped geothermal well can fully exchange heat with the external environment through the heat exchanger, and the inlet temperature is maintained at ambient temperature.

The equations do not consider radiation heat transfer or phase change of the working fluid in the downhole.

Mathematical Model

Considering the circulating working fluid in the borehole as the research object, a microelement section (the control body) was used for heat transfer analysis (Figure 2).

Figure 2.

Schematic of the simplified heat transfer mathematical model.

Consider the heat flow inside the borehole during in period Δτ. The heat balance equation is expressed as follows:

U_{P}^{n + 1} - U_{P}^{n} = Q_{N}^{n \to n + 1} - Q_{S}^{n \to n + 1} + Q_{E}^{n \to n + 1} + S_{P}^{n \to n + 1}

(1)

For simplicity, the upper and lower labels of each physical quantity are specified as follows: the superscript represents time and the subscript represents position. Therefore, U_Pⁿ⁺¹ (J) denotes the internal energy of the control body at the (n + 1)th time step; U_Pⁿ (J) represents the internal energy of the control body at the nth time step; Q_Nⁿ^{→n + 1} (J) represents the heat entering the control body from above by the working fluid from the nth time step to the (n + 1)th time step; Q_Sⁿ^{→n + 1} (J) represents the heat exiting the control body from above by the working fluid from the nth time step to the (n + 1)th time step; Q_Eⁿ^{→n + 1} (J) represents the heat transferred to the control body from the surrounding rock and soil layers from the nth time step to the (n + 1)th time step; and S_Pⁿ^→n+1 (J) represents the heat generated by the control body.

To express the changes in various parameters with respect to the position and time, the average value is used to represent various energies. Thus, these energies can be further expressed as follows:

U_{P}^{n + 1} = (\bar{ρ c T})_{P}^{n + 1} \frac{π}{4} d_{o}^{2} Δ L,

(2)

U_{P}^{n} = (\bar{ρ c T})_{P}^{n} \frac{π}{4} d_{o}^{2} Δ L,

(3)

Q_{N}^{n \to n + 1} = (\bar{c \dot{m} T})_{N}^{n \to n + 1} Δ τ,

(4)

Q_{S}^{n \to n + 1} = (\bar{c \dot{m} T})_{S}^{n \to n + 1} Δ τ,

(5)

Q_{E}^{n \to n + 1} = \frac{{(\bar{T_{G} - T_{P}})}^{n \to n + 1}}{{\bar{R}}_{E}^{n \to n + 1}} Δ τ,

(6)

S_{P}^{n \to n + 1} = {(\bar{\frac{8 f {\dot{m}}^{3}}{π^{2} d_{i}^{5} ρ^{2}}})}_{P}^{n \to n + 1} Δ L Δ τ .

(7)

where

(\bar{M})_{P}

denotes the average value of a certain physical quantity in the control body at a certain time;

(\bar{M})_{N}^{n \to n + 1}

represents the average value of the entrance surface of a certain physical quantity from the nth to the (n + 1)th time step;

(\bar{M})_{S}^{n \to n + 1}

represents the average value of the exit surface of a certain physical quantity from the nth to the (n + 1)th time step;

(\bar{M})_{P}^{n \to n + 1}

represents the average value of a certain physical quantity in the control body from the nth to the (n + 1)th time step;

{\bar{R}}_{E}^{n \to n + 1}

(K/W) denotes the average total thermal resistance from the formation to the working fluid from the nth to the (n + 1)th time step; ρ (kg/m³) represents the density of the fluid; c (J/(kg·K)) represents the specific heat of the working fluid; T_P (K) represents the temperature of the working fluid in the control body; T_N (K) represents the temperature of the working fluid flowing into the control body; T_S (K) represents the temperature of the working fluid flowing out of the control body; T_G (K) represents the temperature of the formation; d_i (m) represents the inner diameter of the borehole;

m (\cdot)

(kg/s) represents the mass flow of the working fluid; f represents the circulating friction coefficient of the working fluid; ΔL (m) represents the length of the control body; and Δτ (s) represents the time step.

Hence, Eq. (1) can be expressed as

\begin{aligned} [(\bar{ρ c T})_{P}^{n + 1} - (\bar{ρ c T})_{P}^{n}] \frac{π}{4} d_{i}^{2} Δ L \\ = (\bar{c \dot{m} T})_{N}^{n \to n + 1} Δ τ - (\bar{c \dot{m} T})_{S}^{n \to n + 1} Δ τ + \frac{{(\bar{T_{G} - T_{P}})}^{n \to n + 1}}{{\bar{R}}_{E}^{n \to n + 1}} Δ τ + {(\bar{\frac{8 f {\dot{m}}^{3}}{π^{2} d_{i}^{5} ρ^{2}}})}_{P}^{n \to n + 1} Δ L Δ τ . \end{aligned}

(8)

If the control body is sufficiently small, i.e., when ΔL tends to 0, the change in the physical and flow parameters with respect to the position can be ignored and considered as a constant. Moreover, the parameter value of the centre point of the control body can be used to replace the average value. Then, Eq. (8) can be converted to

\begin{aligned} [(ρ c T)_{P}^{n + 1} - (ρ c T)_{P}^{n}] \frac{π}{4} d_{i}^{2} Δ L \\ = (c \dot{m} T)_{N}^{n \to n + 1} Δ τ - (c \dot{m} T)_{S}^{n \to n + 1} Δ τ + \frac{{(T_{G} - T_{P})}^{n \to n + 1}}{R_{E}^{n \to n + 1}} Δ τ + {(\frac{8 f {\dot{m}}^{3}}{π^{2} d_{i}^{5} ρ^{2}})}_{P}^{n \to n + 1} Δ L Δ τ . \end{aligned}

(9)

If the time step is sufficiently small, that is, when Δτ tends to 0, the parameter value at the previous time can be used to replace the parameter average value over time, and Eq. (9) can be further simplified as

\begin{aligned} (ρ c)_{P}^{n} [T_{P}^{n + 1} - T_{P}^{n}] \frac{π}{4} d_{i}^{2} Δ L \\ = (c \dot{m} T)_{N}^{n} Δ τ - (c \dot{m} T)_{S}^{n} Δ τ + \frac{{(T_{G} - T_{P})}^{n}}{R_{E}^{n}} Δ τ + {(\frac{8 f {\dot{m}}^{3}}{π^{2} d_{i}^{5} ρ^{2}})}_{P}^{n} Δ L Δ τ . \end{aligned}

(10)

Dividing both sides by Δτ and further organizing Eq. (10), we achieve

\begin{aligned} \frac{(ρ c)_{P}^{n} \frac{π}{4} d_{i}^{2} Δ L}{Δ τ} T_{P}^{n + 1} \\ = (c \dot{m})_{N}^{n} T_{N}^{n} - (c \dot{m})_{S}^{n} T_{S}^{n} + [\frac{(ρ c)_{P}^{n} \frac{π}{4} d_{i}^{2} Δ L}{Δ τ} - \frac{1}{R_{E}^{n}}] T_{P}^{n} + \frac{T_{G}^{n}}{R_{E}^{n}} + {(\frac{8 f {\dot{m}}^{3}}{π^{2} d_{i}^{5} ρ^{2}})}_{P}^{n} Δ L . \end{aligned}

(11)

The final discrete equation is expressed as

T_{P}^{n + 1} = \frac{a_{N}^{n}}{a_{P}^{n + 1}} T_{N}^{n} + \frac{a_{S}^{n}}{a_{P}^{n + 1}} T_{S}^{n} + \frac{a_{P}^{n}}{a_{P}^{n + 1}} T_{P}^{n} + \frac{a_{G}^{n}}{a_{P}^{n + 1}} T_{G}^{n} + \frac{s_{P}^{0}}{a_{P}^{n + 1}},

(12)

where

a_{P}^{n + 1} = \frac{(ρ c)_{P}^{n} \frac{π}{4} d_{i}^{2} Δ L}{Δ τ}

a_{N}^{n} = (c \dot{m})_{N}^{n}

a_{S}^{n} = (c \dot{m})_{S}^{n}

a_{P}^{n} = \frac{(ρ c)_{P}^{n} \frac{π}{4} d_{i}^{2} Δ L}{Δ τ} - \frac{1}{R_{E}^{n}}

a_{G}^{n} = \frac{1}{R_{E}^{n}}

, and

s_{P}^{0} = {(\frac{8 f {\dot{m}}^{3}}{π^{2} d_{o}^{5} ρ^{2}})}_{P}^{n} Δ L

Extending Eq. (12) to the entire borehole, we achieve

{\begin{matrix} T_{P, 1}^{n + 1} = \frac{a_{P, 0}^{n}}{a_{P, 1}^{n - 1}} T_{P, 0}^{n} + \frac{a_{P, 1}^{n}}{a_{P, 1}^{n - 1}} T_{P, 2}^{n} + \frac{a_{P, 2}^{n}}{a_{P, 1}^{n - 1}} T_{P, 1}^{n} + \frac{a_{A, 1}^{n}}{a_{P, 1}^{n - 1}} T_{G, 1}^{n} + \frac{s_{P, 1}^{n}}{a_{P, 1}^{n}} \\ T_{P, 2}^{n + 1} = \frac{a_{P, 1}^{n}}{a_{P, 2}^{n}} T_{P, 1}^{n} + \frac{a_{P, 2}^{n}}{a_{P, 2}^{n}} T_{P, 3}^{n} + \frac{a_{P, 2}^{n}}{a_{P, 2}^{n}} T_{P, 2}^{n} + \frac{a_{A, 2}^{n}}{a_{P, 2}^{n}} T_{G, 2}^{n} + \frac{s_{P, 2}^{n}}{a_{P, 2}^{n}} \\ \dots \dots \\ T_{P, i - 1}^{n + 1} = \frac{a_{P, i - 2}^{n}}{a_{P, i - 1}^{n}} T_{P, i - 2}^{n} + \frac{a_{P, i}^{n}}{a_{P, i - 1}^{n}} T_{P, i}^{n} + \frac{a_{P, i}^{n}}{a_{P, i - 1}^{n}} T_{P, i - 1}^{n} + \frac{a_{A, i - 1}^{n}}{a_{P, i - 1}^{n}} T_{G, i - 1}^{n} + \frac{s_{P, i - 1}^{n}}{a_{P, i - 1}^{n}} \end{matrix}

(13)

Further, the total heat extraction power of the working fluid is calculated using Eq. (14) (Lyu et al., 2017):

Φ = \sum_{i = 1}^{n} c_{i} (T_{P_{i}} - T_{Pi - 1}) ρ_{i} q_{V},

(14)

where Ф(W) denotes the heat extraction power of the working fluid and q_V (m³/s) represents the injection rate of the U-shaped geothermal well.

From the viewpoint of the first law of thermodynamics, the larger the Ф value, the better. However, this is not the case when considering the second law of thermodynamics. It should be considered in terms of the heat extraction availability. Heat exergy can be expressed in Eq. (15) (Ameri et al., 2010).

E_{x},_{Φ} = Φ (1 - \frac{T_{0}}{T_{out}}),

(15)

where E_x,Ф (W) denotes the heat exergy power, T₀ (K) represents the environmental temperature, and T_out (K) represents the outlet temperature of the working fluid.

Using the proposed mathematical model, the heat extraction and outlet temperature can be calculated, and these values can be substituted in Eq. (15), yielding the heat exergy power of the U-shaped geothermal well.

Definite Solution conditions

The following definite solution conditions are required to achieve equation a closed equation.

Determination Of various thermal resistances

If the contact thermal resistance is not considered, the heat transfer process from the formation to the working fluid is described as follows. The formation first transfers heat to the cement ring by conduction (the ascending section has an insulating layer). Then, the cement ring transfers heat to the casing by conduction and the casing transfers heat to the working fluid by convection. According to the thermoelectric analogy for the theoretical calculation of heat transfer, each heat transfer link can be regarded as the heat transfer resistance. Thus, the sum of these thermal resistances R_E in Eq. (6) and Eqs. (8)–(11) can be expressed as

R_{E} = R_{1} + R_{2} + R_{3} + R_{4} + R_{5},

(16)

where R₁ (K/W) denotes the thermal convection resistance between the casing and the working fluid; R₂ (K/W) represents the thermal conduction resistance of the casing; R₃ (K/W) represents the thermal conduction resistance of the cement ring; R₄ (K/W) represents the thermal conduction resistance of the insulation; and R₅ (K/W) represents the thermal conduction resistance of the rock formation.

In Eq. (16), the calculation of thermal resistance proceeds as follows:

The thermal convection resistance (R₁) can be calculated as

R_{1} = \frac{1}{π d_{i} h Δ L},

(17)

where h (W/(m²·K)) denotes the convective heat transfer coefficient, which is defined as

h = \frac{λ_{f} N u_{f}}{d_{i}},

(18)

where λ_f (W/(m·K)) represents the heat conductivity coefficient of the working fluid. Nu_f represents the Nusselt number (a dimensionless characteristic number), which is obtained using the Gnielinski equation (Gnielinski, 2013):

N u_{f} = \frac{(\frac{f}{8}) (R e - 1000) P r_{f}}{1 + 12.7 \sqrt{\frac{f}{8}} (P r_{f}^{\frac{2}{3}} - 1)} [1 + {(\frac{d}{l})}^{\frac{2}{3}}] c_{t},

(19)

where Re denotes the Reynolds number, Pr_f denotes the Prandtl number, and l (m) denotes the pipe length. f denotes the Darcy friction factor of turbulent flows in a tube and is calculated using the Filonenko equation (Gnielinski, 2013):

f = (1.82 \lg R e - 1.64)^{- 2} .

(20)

In Eqs. (19) and (20), the range of application is Re = 3000–6,000,000 and Pr = 0.5–2000; moreover, c_t denotes the correction coefficient of the physical property. For liquids, c_t is determined by using Eq. (21):

c_{t} = {(\frac{P r_{f}}{P r_{w}})}^{0.01}, \frac{P r_{f}}{P r_{w}} = 0.05 \sim 20.

(21)

The thermal conduction resistance (R₂, R₃, and R₄) is calculated using Eq. (22):

R_{i} = \frac{1}{2 π λ_{i} Δ L} \ln (\frac{d_{outer}}{d_{inner}}),

(22)

where λ_i (W/(m·K)) denotes the thermal conductivity of materials in the ith layer and d_inner (m) and d_outer (m) represent the inner and outer diameters of materials in the ith layer, respectively.

The calculation of the thermal conduction resistance of the rock formation (R₅) is based on Eqs. (23)–(24) (Ferreira et al., 2017):

R_{5} = \frac{I (X_{rb})}{2 π λ_{g} Δ L},

(23)

X_{rb} = \frac{r_{b}}{2 \sqrt{a τ}},

(24)

where I(X_rb) denotes the dimensionless temperature; X_rb represents the dimensionless time; λ_g (W/(m·K)) denotes the thermal conductivity of the rock formation; a (m²/s) represents the thermal diffusion coefficient of the formation; τ (s) represents the cumulative contact time between the working fluid and the formation; and r_b (m) denotes the concentric circle radius of the wellbore structure and stratigraphic interface.

I(X_rb) in Eq. (23) can be determined using Eqs. (25)–(28) (Ahn et al., 2005):

I (X_{rb}) = 0.5 (\begin{matrix} - \ln X_{rb}^{2} - 0.57721566 + 0.9999193 X_{rb}^{2} - 0.24991055 X_{rb}^{4} \\ + 0.05519968 X_{rb}^{6} - 0.00976004 X_{rb}^{8} + 0.00107857 X_{rb}^{10} \end{matrix}) 0 \leq X_{rb} \leq 1,

(25)

I (X_{rb}) = 0.5 (\frac{1}{2 X_{rb}^{2} e^{X_{rb}^{2}}}) (\frac{A}{B}) X_{rb} > 1,

(26)

where

A = X_{rb}^{8} + 8.5733287 X_{rb}^{6} + 18.059017 X_{rb}^{4} + 8.637609 X_{rb}^{2} + 0.2677737,

(27)

B = X_{rb}^{8} + 9.5733223 X_{rb}^{6} + 25.632956 X_{rb}^{4} + 21.099653 X_{rb}^{2} + 3.9684969.

(28)

Thermophysical Properties of the working fluid

Water is characterized by good thermodynamic properties and nontoxicity and is easy to obtain and odourless; thus, it is frequently used as a heat transfer working medium in the production of geothermal energy. The density, viscosity, thermal conductivity, and specific heat of water can be determined from data in the steam tables in SI units (Bacon and Parrish, 1973).

Initial Conditions

In the heat transfer process in the wellbores, the formation temperature can generally be set as the initial temperature of the wellbore:

T_{P, i}^{0} = T_{G, i} .

(29)

Boundary Conditions

Based on the conditions for heat transfer between the U-shaped geothermal well and the rock formation, the circulating water inlet is set as a mass flow inlet boundary condition, which can be expressed as

T_{P, 1}^{0} = T_{P, 1}^{1} = \dots T_{P, 1}^{n} = T_{i n l e t} .

(30)

The outer boundary of the porous geothermal reservoir is retained at a constant temperature, and the bottom and top boundaries are treated as adiabatic surfaces, which can be expressed as

T_{G, i}^{0} = T_{G, i}^{1} = T_{G, i}^{2} = \dots \dots T_{G, i}^{n} = T_{G 0} + G \cdot H

(31)

where T_G⁰ (K) denotes the ground temperature; G (K/m) represents the geothermal gradient; and H (m) denotes the vertical depth.

Model Solution

The temperature distribution in the well can be calculated (Figure 2). The solution steps are as follows:

Step 1: The basic parameter values, including the well structure, borehole size, geothermal gradient, the inlet temperature of the working fluid, and injection rate are provided as inputs.

Step 2: The time step is meshed and estimated.

Step 3: The initial wellbore temperature distribution is assumed.

Step 4: The thermophysical properties of the working fluid are determine.

Step 5: Various thermal resistances are calculated using Eqs. (16)–(28).

Step 6: The wellbore temperature distribution is calculated using Eq. (13).

Step 7: The temperature difference between steps 6 and 3 is compared. If the difference is greater than the required accuracy, the previous temperature in step 3 is replaced by substituting the temperature calculated in step 6; then the temperature distributions are recalculated by repeating steps 4–6. If the temperature difference is less than the required accuracy, step 8 is performed.

Step 8: The heat extraction power of the working fluid is calculated using Eq. (14).

Step 9: The heat exergy power of the working fluid is calculated using Eq. (15).

Step 10: The temperature distribution of the entire wellbore, the heat extraction power, and the heat exergy power of the working fluid at different times are output.

Figure 3 depicts a flowchart to help understand the solution for the mathematical model.

Results and discussion

Experimental Validation of the mathematical model

The mathematical model must be validated using experimental data or other calculated results to determine whether the mathematical model can accurately describe the temperature distribution and heat extraction using a U-shaped geothermal well. This work uses the test data of in situ experimental platform a U-shaped geothermal well to verify the mathematical model.

Figure 3.

Flowchart of the mathematical model solution.

Experimental System and basic parameters

To accelerate the process of middle deep geothermal energy usage, Yanchang Petroleum International Exploration and Development Co. Ltd, set up an in situ experimental platform of a U-shaped geothermal well in a community in Xi’an. The basic parameters of the U-shaped geothermal well are presented in Table 1.

Table 1.

Basic parameters of the U-shaped geothermal well.

Parameter	Value
Inner diameter of the first casing	0.2245 m
Outer diameter of the first casing	0.2445 m
Outer diameter of first cementing	0.3111 m
Depth diameter of the first casing	750 m
Inner diameter of the second casing	0.1273 m
Outer diameter of the second casing	0.1397 m
Outer diameter of the second cementing	0.2159 m
Depth of deflection point	2340 m
Vertical well depth	2500 m
Horizontal well length	490 m m
Insulating layer depth	750 m
Insulating layer thickness	0.0286 m
Thermal conductivity of casing	40 W/(m·K)
Thermal conductivity of cementing	0.8 W/(m·K)
Thermal conductivity of insulating layer	0.05 W/(m·K)
Ground temperature	285.8 K
Geothermal gradient	0.0258 K/m

The experimental detection system and test equipment are presented in Figures 4 and 5 (Li et al., 2018; Li, et al., 2019b).

Figure 4.

Experimental tube system and detection system.

Figure 5.

Experimental site and test equipment; A) temperature sensor of the outlet well, B) temperature sensor of the inlet well, and C) electromagnetic flowmeter.

Comparison Between simulation and experimental results

A series of experiments were conducted under seven sets of condition, and the duration of each experiment performed under each set of conditions exceeded 72 h. Table 2 presents the experimental parameters of each set of experimental conditions.

Table 2.

Experimental duration and parameters of the working conditions.

Condition	Experiment time	Flow rate (m³/h)	Inlet temperature (K)
1	From 9 am on January 19, 2020, to 1 pm on January 22, 2020	50	285
2	From 1 pm on March 3, 2020, to 4 pm on March 6, 2020	55	285
3	From 9 am on March 29, 2020, to 1 pm on April 1, 2020	60	285
4	From 1 pm on April 17, 2020, to 4 am on April 20, 2020	24	285
5	From 12 am on June 2, 2020, to 6 pm on June 5, 2020	37	285
6	From 6 pm on August 25, 2020, to 8 pm on August 28, 2020	37	290
7	From 10 am on September 11, 2020, to 8 pm on September 14, 2020	37	280

The inlet and outlet temperatures (T_in and T_out, respectively) are the average of two temperature sensors recorded at the same time. The flow rate is the average of the instantaneous flow rates of two flow sensors. Figure 6 presents the measured and theoretically calculated values of the outlet temperature under each condition.

Figure 6.

Comparison of experimental and calculated results of outlet temperature under each condition.

The comparison results indicate a considerable difference between the calculated and experimental results in the initial stage of the experiment. After approximately 9 h, the experimental results are almost the same as the calculated results. This is because the well was filled with water before the start of the experiment; thus, the water that initially exited the well was in full contact with the formation. Thus, the experimentally obtained outlet temperature is considerably higher than the calculated one. The experiment did not enter a normal cycle until all the original water in the well was changed, and the experimentally obtained outlet temperature corresponds to the calculated one.

Table 3 presents the experimental and calculates results obtained under each working condition after 72 h of circulation, as well as the relative error between the two results.

Table 3.

Comparison of the experimental and calculated results after 72 h.

Condition	Flow rate (m³/h)	Inlet temperature (K)	Experimental outlet temperature (K)	Calculated outlet temperature (K)	Relative error (%)
1	50	285	303.35	305.14	0.59
2	55	285	302.29	303.72	0.47
3	60	285	300.83	302.56	0.58
4	24	285	316.69	315.65	0.33
5	37	285	308.42	309.60	0.38
6	37	290	311.60	311.61	0.10
7	37	280	305.54	307.44	0.62

Figure 6 and Table 3 show that a certain difference between the calculated and experimental results. The difference in the results arises from the mathematical model that was developed herein under some assumptions to reduce the difficulty of the solution. However, considering the change trend, the calculated results agree well with the experimental results. Especially, under conditions 2, 4, 5, and 6, the relative error and maximum temperature difference between the calculated and experimental results are <0.5% and <1.5 K, respectively. In contrast, the relative error under conditions 1 and 7 is larger and the maximum temperature difference is approximately 2 K. The reason for these results is closely related to the experimental process. During the experiments, the inlet temperature under conditions 1 and 7 was unstable compared with the other conditions (Figure 6), causing a considerable change in the working fluid temperature, thus resulting in a large relative error. However, objectively speaking, the developed mathematical model achieves high calculation accuracy, accurately predicting the temperature of the circulating working fluid in the U-shaped geothermal well.

Sensitive Factor analysis

The most important parameters required for evaluating the performance of U-shaped geothermal wells are the outlet temperature and heat extraction power. The outlet temperature determines the heat energy availability, and the heat extraction power determines the heating capacity of geothermal wells.

After the U-shaped geothermal well was developed and put into operation, its well structure was fixed, while the relevant parameters of the working fluid, pipe string, and formation were also maintained constant. Additionally, for the simplicity of operation, the inlet temperature was generally the same as the ambient temperature and was considered to remain constant. Therefore, the circulating flow rate of the working fluid in the U-shaped geothermal well is the key process parameter that can be designed optimally.

Therefore, a geothermal well in Xi’an, China, was considered as an example to study the variation in the outlet temperature and heat extraction power with respect to the flow rate during the heating period.

Basic Parameters

The heating period in Xi’an generally starts on November 15 and lasts until March 15 of the following year, ∼120 days. The temperature data from November 15, 2019, to March 15, 2020, were obtained from the Xi’an Meteorological Bureau and are shown in Figure 7.

Figure 7.

Temperature during the heating period in xi’an, China.

From November to March of the following year, one working condition is set every month, a total of five working conditions. The average temperature in each month is set as the inlet temperature of the working fluid, and the flow rate is set at 1–60 m³/h. The thermal resistance of the rock formation is calculated using Eqs. (22)–(27), which is set as the monthly average value. The calculation parameters are shown in Table 4, and the other variables are the same as those listed in Table 1.

Table 4.

Calculation parameters under each working condition.

Condition	Month	Inlet temperature (°C)	Inlet temperature (K)	Formation resistance (K/W)	Flow rate (m³/h)
1	November	5.688	278.838	0.1288	1–60
2	December	3.468	276.618	0.1806	1–60
3	January	1.810	274.960	0.2051	1–60
4	February	5.690	278.840	0.2188	1–60
5	March	9.267	282.417	0.2263	1–60

Outlet Temperature changes with the flow rate

Figure 8 depicts the relationship between the outlet temperature of the working fluid and the flow rate of the U-shaped geothermal well from November to March of the following year.

Figure 8.

Outlet temperature changes with the flow rate.

As shown in Figure 8, each curve shows the same temperature change trend; as the flow rate increases, the outlet temperature of the working fluid first increases and then decreases. However, the outlet temperature becomes increasingly less sensitive to changes in the flow rate.

This is because when the flow rate is smaller, the heat exchange between the working fluid and the formation is more sufficient. In the endothermic zone (descending section, diagonal section, horizontal section, and the lower part of the ascending section), the temperature of the working fluid increases more rapidly. Similarly, in the heat-release zone (upper part of the ascending section), the temperature of the working fluid decreases more dramatically. Therefore, if the flow rate does not exceed a certain value and the insulation depth of the ascending section is insufficient, the temperature of the working fluid will decrease rapidly during the upward flow. Hence, in the first half of the curve (Figure 8), the outlet temperature increases with the flow rate. When the flow rate exceeds a certain value, a greater flow rate, yields a shorter heat transfer time between the working fluid and the formation, resulting in insufficient heat transfer between them. Hence, in the latter half of the curve, a greater flow rate affords a lower outlet temperature. Therefore, there is a flow rate value for each working condition to maximize the outlet temperature of the working fluid. For example, in the case of working condition 1in November, when the flow rate is set as 9 m³/h, the maximum outlet temperature achieved, i.e., 321.03 K (Figure 8). There are two reasons for this result. First, November is the initial heating period for the U-shaped geothermal well, which shows great heat-release potential. Second, compared with December and January, the inlet temperature is higher in November. Subsequently, with an increase in the thermal resistance, the heating capacity decreases. Therefore, at the end of the heating period (March), the maximum inlet temperature is 282.4 K; however, the outlet temperature is only 219.9 K, which does not reach the maximum temperature in November.

Table 5 shows the calculation results of the corresponding flow rate and maximum outlet temperature for each month.

Table 5.

Calculation results of the maximum outlet temperature.

Condition	Month	Inlet temperature (K)	Outlet temperature (K)	Corresponding flow rate m³/h
1	November	278.838	321.03	9
2	December	276.618	320.18	8
3	January	274.960	319.83	6
4	February	278.840	319.84	6
5	March	282.417	319.93	6

Owing to the difference in the boundary conditions for each working condition, the maximum value of the outlet temperature and the corresponding flow rate differed (Table 5). Of course, this does not mean that the flow rate at which the highest outlet temperature is achieved is the optimal value. It is also necessary to continue to investigate the change in heat extraction power with respect to the flow rate.

Heat Extraction power changes with the flow rate

Figure 9 shows the relationship between the heat extraction power and flow rate of the U-shaped geothermal well from November to March of the following year.

Figure 9.

Heat extraction power changes with flow rate.

Each curve in Figure 9 shows the same change trend; as the flow rate increases, the heat extraction power increases. However, the flow rate becomes increasingly less sensitive to changes in the flow rate.

This is because if the flow rate is increased, the mass of the fluid that exchanges heat with the formation over unit time increases. Figure 9 also depicts that the heat extraction power tends to plateau at the later part of the curves. Of course, a high flow rate is not beneficial because a higher flow rate, yields a lower the outlet temperature. An excessively high flow rate will reduce the heat energy availability. Therefore, the exergy analysis method is effective analysis approach.

Optimal Flow rate based on exergy analysis

The exergy of each working condition can be calculated using Eq. (15). The greater the exergy, the more available thermal energy extracted from the geothermal well. According to the conclusion obtained in Section 3.2., when the flow rate exceeds a certain value, a higher flow rate, yields a lower the outlet temperature. However, when the flow rate is higher, the heat extraction is also greater. Under the premise that the ambient temperature is set as a fixed value, it is impossible to determine whether exergy increases or decreases with the flow rate; hence, further analysis is necessary. The flow rate obtained at the maximum exergy is set as the optimal flow rate. The main task of this section is to determine the optimal flow rate based on exergy analysis.

Fitting Of the function of outlet temperature and heat extraction power with flow rate

Based on the analysis in Section 3.2., both the outlet temperature and heat extraction will change with changes in the flow rate, and the change trend observed under different working conditions is the same. Hence, both the outlet temperature and heat extraction power are functions of the flow rate.

Thus, Eq. (15) can be expressed as

E_{x},_{Φ} = Φ (q_{V}) [1 - \frac{T_{0}}{T_{out} (q_{V})}] .

(32)

Using calculus to determine the extreme value, the following two equations can be satisfied when the maximum exergy value is achieved:

\frac{d E_{x},_{Φ}}{d q_{V}} = 0,

(33)

\frac{d^{2} E_{x},_{Φ}}{d q_{V}^{2}} < 0.

(34)

At this time, the flow rate that satisfies Eq. (33) is the optimal flow rate.

Therefore, determining how to fit the function of the outlet temperature and heat extraction power changing with the flow rate using curves shown in Figures 8 and 9 has become the key to achieve the optimal flow rate design.

Analyzing the curves shown in Figure 8, the function should meet the following relations:

q_{V} \in (0, + \infty),

(35)

q_{V} \to 0, T_{out} \to T_{0},

(36)

q_{V} \to + \infty, T_{out} \to T_{0},

(37)

q_{V} = q_{V 0}, T_{out} = T_{\max},

(38)

q_{V} \to 0, \frac{d T_{out}}{d q_{V}} \to + \infty,

(39)

q_{V} \to + \infty, \frac{d T_{out}}{d q_{V}} \to 0,

(40)

q_{V} = q_{V 0}, \frac{d T_{out}}{d q_{V}} = 0,

(41)

where q_V0 (m³/h) represents the flow rate at the maximum outlet temperature.

Based on the aforementioned relations and using the function approximation principle (Denœux, 1997), the function form can be achieved as

T_{out} = T_{0} + \exp (a + b q_{V} + c q_{V}^{- 1} + d q_{V}^{- 2}),

(42)

where, a, b, c, and d represent undetermined coefficients obtained using the least-squares fitting method.

Similarly, based on the shape of the curve shown in Figure 9, a function can be obtained for the heat extraction power,

Φ = \exp (e + f q_{V}^{- 1} + g q_{V}^{- 2} + h q_{V}^{- 3}),

(43)

where e, f, g, and h represent undetermined coefficients obtained using the least-squares fitting method.

According to the least-squares principle (Verma et al., 2006; Wong-Loya et al., 2015), the deviation function is defined as

ϕ (a, b, c) = \sum_{i = 1}^{n} {(T_{out, i} - a - b q_{V, i} - c q_{V, i}^{- 1} - d q_{V, i}^{- 2})}^{2}

(44)

To minimize φ (a, b, c, d), we must determine the partial differential of Eq. (44) and make it 0; then,

{\begin{matrix} \frac{\partial φ}{\partial a} = - 2 \sum_{i = 1}^{n} (T_{out, i} - a - b q_{v, i} - \frac{c}{q_{v, i}} - \frac{d}{q_{v, i}^{2}}) = 0 \\ \frac{\partial φ}{\partial b} = - 2 \sum_{i = 1}^{n} (T_{out, i} - a - b q_{v, i} - \frac{c}{q_{v, i}} - \frac{d}{q_{v, i}^{2}}) q_{v, i} = 0 \\ \frac{\partial φ}{\partial c} = - 2 \sum_{i = 1}^{n} \frac{(T_{out, i} - a - b q_{v, i} - \frac{c}{q_{v, i}} - \frac{d}{q_{v, i}^{2}})}{q_{v, i}} = 0 \\ \frac{\partial φ}{\partial d} = - 2 \sum_{i = 1}^{n} \frac{(T_{out, i} - a - b q_{v, i} - \frac{c}{q_{v, i}} - \frac{d}{q_{v, i}^{2}})}{q_{v, i}^{2}} = 0 \\ \sum_{i = 1}^{n} T_{out, i} = n a + \sum_{i = 1}^{n} q_{v, i} b + \sum_{i = 1}^{n} \frac{c}{q_{v, i}} + \sum_{i = 1}^{n} \frac{d}{q_{v, i}^{2}} \\ \sum_{i = 1}^{n} q_{v, i} T_{out, i} = \sum_{i = 1}^{n} a q_{v, i} + \sum_{i = 1}^{n} q_{v, i}^{2} b + n c + \sum_{i = 1}^{n} \frac{d}{q_{v, i}} \\ \sum_{i = 1}^{n} \frac{T_{out, i}}{q_{v, i}} = \sum_{i = 1}^{n} \frac{a}{q_{v, i}} + n b + \sum_{i = 1}^{n} \frac{c}{q_{v, i}^{2}} + \sum_{i = 1}^{n} \frac{d}{q_{v, i}^{3}} \\ \sum_{i = 1}^{n} \frac{T_{out, i}}{q_{v, i}^{2}} = \sum_{i = 1}^{n} \frac{a}{q_{v, i}} + \sum_{i = 1}^{n} \frac{b}{q_{v, i}^{2}} + \sum_{i = 1}^{n} \frac{c}{q_{v, i}^{3}} + \sum_{i = 1}^{n} \frac{d}{q_{v, i}^{4}} \end{matrix} .

(45)

They can be simply expressed as

{\begin{matrix} l_{12} b + l_{13} c + l_{14} d = l_{1, T} \\ l_{22} b + l_{23} c + l_{24} d = l_{2, T} \\ l_{32} b + l_{33} c + l_{34} d = l_{3, T} \end{matrix},

(46)

a = \bar{T} - b \bar{q_{v, i}} - \frac{c}{\bar{q_{v, i}}},

(47)

where

{\begin{matrix} l_{i j} = l_{j i} = \sum_{i = 1}^{n} q_{v, i} \times (\frac{1}{q_{v, i}}) - \frac{1}{n} (\sum_{i = 1}^{n} q_{v, i}) (\sum_{i = 1}^{n} \frac{1}{q_{v, i}}) \\ l_{i, T} = \sum_{i = 1}^{n} T_{out, i} q_{v, i} - \frac{1}{n} (\sum_{i = 1}^{n} q_{v, i}) (\sum_{i = 1}^{n} T_{out, i}) \end{matrix},

(48)

\bar{T_{out, i}} = \frac{1}{n} \sum_{i = 1}^{n} T_{out, i}, \bar{q_{v, i}} = \frac{1}{n} \sum_{i = 1}^{n} q_{v, i} .

(49)

Solving Eqs. (46) and (47) can yield the three undetermined coefficients a, b, and c, thereby fitting the specific form of Eq. (42).

According to the aforementioned ideas, considering the calculation result obtained in the case of November as an example, the fitted function between the outlet temperature and flow rate and that between the heat extraction power and flow rate are presented in Eqs. (50) and (51), respectively.

T_{out} = 278.8375 + \exp (4.042171 - 0.02062 q_{V} - 1.56713 q_{V}^{- 1} + 0.542234 q_{V}^{- 2}),

(50)

Φ = \exp (14.19166 - 12.4088 q_{V}^{- 1} + 16.48168 q_{V}^{- 2} - 8.19646 q_{V}^{- 3}),

(51)

Figure 10 shows a comparison of the function image depicted using Eqs. (50) and (51) and the calculated value obtained using the mathematical model.

Figure 10.

Comparison of the fitting function and calculated value.

Figure 10 also shows that the fitting function is consistent with the calculated value obtained using the mathematical model. Moreover, the fitting function can be used to perform exergy analysis.

Exergy Analysis based on the fitting function

Substituting the fitting function into Eq. (15), the heat exergy absorbed by the working fluid from the formation can be expressed as

E_{x},_{Φ} = \exp (e + f q_{V}^{- 1} + g q_{V}^{- 2} + h q_{V}^{- 3}) [1 - \frac{T_{0}}{T_{0} + \exp (a q_{V} + b q_{V}^{- 1} + c q_{V}^{- 2} + d)}]

(52)

Considering the data obtained in the case of November as an example, the curve depicting changes in the heat exergy with respect to the flow rate is shown in Figure 11.

Figure 11.

Heat exergy change with the flow rate for November data.

As the flow rate increases, the heat exergy first increases and then decreases; the maximum exergy value is obtained at point E on the curve (Figure 11). The flow rate corresponding to point E is the optimal flow rate. The working fluid can obtain the maximum heat exergy value for the U-shaped geothermal well using this flow rate operation.

Similarly, the curve depicting changes in the heat exergy with respect to the flow rate in each month during the heating period can also be obtained (Figure 12). The designed operating flow rate and the corresponding maximum exergy power calculation results are shown in Table 6.

Figure 12.

Heat exergy change with the flow rate during the heating period.

Table 6.

Design results for optimal flow rate.

Condition	Month	Designed operating flow rate (m³/h)	Maximum exergy power (kW)
1	November	26.0	93.674
2	December	21.0	73.672
3	January	19.0	69.505
4	February	18.0	55.020
5	March	17.5	44.725

Figure 12 and Table 6 show that owing to different boundary conditions, the optimal flow rate for each month during the heating period differs. In the initial stage of the heating period, the exergy power is greater, and the optimal flow rate is higher. Thus, the U-shaped geothermal well must be operated at a higher flow rate. At the end of the heating period, the exergy power becomes smaller and the optimal flow rate decreases;, hence, the well must be operated at a smaller flow rate. The operating flow rate must be adjusted in time as the heating process progresses to obtain the maximum exergy power.

Optimal Flow rate design process

The subsequent analysis demonstrates that the proposed mathematical model can be used to calculate the exergy power of U-shaped geothermal wells developed under different flow rates to obtain the optimal flow rate during operation. The steps for designing the optimal flow rate of the U-shaped geothermal wells are listed below:

Step 1: The basic parameters such as the well structure, formation, working fluid, and pipe string physical parameters are provided as input.

Step 2: The weather in the heating area during the heating period is investigated to determine the inlet temperature of the U-shaped geothermal well.

Step 3: The wellbore heat transfer model (established in Section 2) is employed at each stage of the heating period to calculate the outlet temperature and heat extraction power under a certain range of flow rates.

Step 4: Sufficient sample data are obtained and used to fit the function of the outlet temperature and heat extraction power changing with the flow rate.

Step 5: Eq. (32) is used to determine the change in exergy efficiency with respect to the flow rate. The flow rate corresponding to the maximum exergy efficiency is the optimal flow rate of the U-shaped geothermal well.

The flowchart shown in Figure 13 helps understand the process of determining the optimal flow rate of the U-shaped geothermal well.

Conclusions

Taking Xi’an area as an example, this work developed an evaluation (mathematical) model for obtaining the optimal flow rate of a U-shaped geothermal well based on exergy analysis. Some significant findings are summarized below:

The proposed heat extraction mathematical model for the U-shaped geothermal well is highly accurate and yields results that are consistent with the experimental results. The maximum relative error between the calculated and experimental results is only 0.62%.

As the flow rate of the U-shaped geothermal well increases, the outlet temperature of the working fluid first increases and then decreases; however, the outlet temperature becomes increasingly less sensitive to changes in the flow rate. In the initial month of heating (November), the highest outlet temperature can be reached at 321.03 K, and the corresponding flow rate is 9 m³/h.

As the flow rate increases, the heat extraction power increases; however, it becomes increasingly less sensitive to changes in the flow rate. A higher flow rate does not imply better performance because it reduces the heat energy availability.

As the flow rate increases, the heat exergy first increases and then decreases, and the flow rate obtained at the maximum exergy value is the optimal flow rate. During the heating period from November to March of the following year, the monthly maximum heat exergy power is 93.7, 73.7, 69.5, 55.0, and 44.7 kW, and the corresponding optimal flow is 26, 21, 19, 18, and 17.5 m³/h, respectively. Therefore, the operating flow rate can be adjusted in time using the method for determining the best flow developed in this work to obtain the maximum exergy power.

Figure 13.

Flowchart for determining the optimal flow rate.

The object of the exergy analysis in this study is the benefits obtained from the rock formation, excluding the cost of equipment. In the future, we will continue to investigate the optimal flow rate from the following two aspects:

From exergy efficiency, study the flow rate design with the largest ratio of benefit to cost will be studied.

Coupling process parameters with economic parameters, the flow rate design with the most economical flow will be investigated.

Footnotes

Acknowledgements

The authors would like to express appreciation for the support by the National Natural Science Foundation of China [No.51874252], PetroChina Innovation Foundation [No.2020D-5007-0312], Sichuan Province Science and Technology Planning Project [No.2019JDTD0026], PetroChina-Southwest Petroleum University Innovation Consortium Project [No.2020CX040103], and Yanchang Petroleum Group geothermal project (No.GJM18TLH0005).

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China [grant number 51874252], PetroChina Innovation Foundation [grant number 2020D-5007-0312], Sichuan Province Science and Technology Planning Project [grant number 2019JDTD0026], PetroChina-Southwest Petroleum University Innovation Consortium Project [grant number 2020CX040103], and Yanchang Petroleum Group geothermal project (grant number GJM18TLH0005).

ORCID iD

Dong Xiao

References

Ahn

Kim

S-Y

Song

(2005) Cost-benefit analysis of the geothermal energy resources on the private and public respects. The Korean Society for Geosystem Engineering 42: 318–329.

Ameri

Kiaahmadi

Khanaki

, et al. (2010) Energy and exergy analyses of a spark-ignition engine. International Journal of Exergy 7(5): 547–563.

Bacon

Parrish

(1973) STEAM TABLES IN SI UNITS. Mechanical Engineer's Reference Book 22(4): 2–86.

Beier

Acuña

Mogensen

, et al. (2013) Borehole resistance and vertical temperature profiles in coaxial borehole heat exchangers. Applied Energy 102: 665–675.

Blohm

(2021) An enabling framework to support the sustainable energy transition at the national level. Sustainability 13(7): 3834. https://www.mdpi.com/2071-1050/13/7/3834.

Cadelano

Cicolin

Emmi

, et al. (2019) Improving the energy efficiency, limiting costs and reducing CO₂ emissions of a museum using geothermal energy and energy management policies. Energies 12(16): 3192. https://www.mdpi.com/1996-1073/12/16/3192.

Davis

Michaelides

(2009) Geothermal power production from abandoned oil wells. Energy 34(7): 866–872.

Denœux

(1997) Function approximation in the framework of evidence theory: a connectionist approach. Vol. 1.

Fallah

Chen

, et al. (2021) Globally scalable geothermal energy production through managed pressure operation control of deep closed-loop well systems. Energy Conversion and Management 236: 114056.

10.

Ferreira

Hafemann

Barbosa Jr.

, et al. (2017) A numerical study on the thermal behaviour of wellbores. SPE Production & Operations 32(04): 564–574.

11.

Gnielinski

(2013) On heat transfer in tubes. International Journal of Heat and Mass Transfer 63: 134–140.

12.

Grayson

Gans

(2012) Closed Loop Circulating Systems Enhance Well Control and Efficiency With Precise Wellbore Monitoring and Management Capabilities. In: SPE/IADC Managed Pressure Drilling and Underbalanced Operations Conference and Exhibition.

13.

Grubb

(1992) Climate change. Nature 358(6386): 448–448.

14.

Hepbasli

Akdemir

(2004) Energy and exergy analysis of a ground source (geothermal) heat pump system. Energy Conversion and Management 45(5): 737–753.

15.

Jung

Y-J

Kim

Lee

Y-H

, et al. (2015) A study on the energy efficiency of a geothermal heat pump system in use the outdoor reset control application. Journal of the Korean Solar Energy Society 35: 45–52.

16.

Kamila

Kaya

Zarrouk

(2021) Reinjection in geothermal fields: An updated worldwide review 2020. Geothermics 89: 101970. https://doi.org/10.1016/j.geothermics.2020.101970.

17.

Karlsdóttir

Heinonen

Pálsson

, et al. (2019) Life cycle assessment of a geothermal combined heat and power plant based on high temperature utilization. Geothermics 82: 101727. https://doi.org/10.1016/j.geothermics.2019.101727.

18.

Kaya

Zarrouk

O ', et al. (2011) Reinjection in geothermal fields: A review of worldwide experience. Renewable & Sustainable Energy Reviews 15(1): 47–68.

19.

Kurşun

(2020) Energy and exergy analysis of a concentrated photovoltaic recuperator design for a geothermal based multi-generation system. Applied Thermal Engineering 181: 115932.

20.

Liu

Xiao

, et al. (2019a) Concepts and characteristics of a U-shaped docking well geothermal heat extraction system. Energy Exploration & Exploitation 37: 865–883.

21.

Guan

Liu

, et al. (2020) Heat transfer performance of a deep ground heat exchanger for building heating in long-term service. Renewable Energy 166: 20–34.

22.

Guan

Wang

, et al. (2018) Experimental and numerical studies on heat transfer characteristics of vertical deep-buried U-bend pipe to supply heat in buildings with geothermal energy. Energy 142: 689–701.

23.

Guan

Wang

, et al. (2019b) Experimental and numerical studies on heat transfer characteristics of vertical deep-buried U-bend pipe in intermittent heating mode. Geothermics 79: 14–25.

24.

Liao

Sun

, et al. (2021) Geothermal exploitation and electricity generation from multibranch U-shaped well–enhanced geothermal system. Renewable Energy 163: 2178–2189.

25.

Liddell

Unsworth

Pek

(2016) Magnetotelluric imaging of anisotropic crust near fort McMurray, alberta: Implications for engineered geothermal system development. Geophysical Journal International 205(3): 1365–1381.

26.

Lyu

Chen

(2020) Thermal performance of an energy pile group with a deeply penetrating U-shaped heat exchanger. Energies 13: 5822.

27.

Lyu

Song

, et al. (2017) Numerical analysis of characteristics of a single U-tube downhole heat exchanger in the borehole for geothermal wells. Energy 125: 186–196.

28.

Zhao

Yin

, et al. (2020) A coupled heat transfer model of medium-depth downhole coaxial heat exchanger based on the piecewise analytical solution. Energy Conversion and Management 204: 112308.

29.

Malana

Geraldinne

Malana

, et al. (2020) Energy and Exergy Analysis of BN-06 Wellhead Geothermal Power Plant in Province of Biliran, Philippines.

30.

Moran

Enrico

(1994) Exergy analysis: Principles and practice. Journal of Engineering for Gas Turbines and Power-transactions of The Asme - J ENG GAS TURB POWER-T ASME 116: 285–290.

31.

Ozgener

Hepbasli

Dincer

(2005) Energy and exergy analysis of geothermal district heating systems: An application. Building and Environment 40(10): 1309–1322.

32.

Song

Zheng

, et al. (2018a) Heat extraction performance of a downhole coaxial heat exchanger geothermal system by considering fluid flow in the reservoir. Geothermics 76: 190–200.

33.

Song

Shi

, et al. (2018b) Numerical analysis of the heat production performance of a closed loop geothermal system. Renewable Energy 120: 365–378.

34.

Song

Zheng

, et al. (2019) Study on thermal conductivity of cement with thermal conductive materials in geothermal well. Geothermics 81: 1–11.

35.

Sun

Yao

(2019) Literature review on a U-shaped closed loop geothermal energy development system. Energy Sources, Part A: Recovery, Utilization, and Environmental Effects 42: 1–13.

36.

Sun

Yao

, et al. (2018) Geothermal energy development by circulating CO₂ in a U-shaped closed loop geothermal system. Energy Conversion and Management 174: 971–982.

37.

van Oort

Chen

Ashok

, et al. (2021) Constructing Deep Closed-Loop Geothermal Wells for Globally Scalable Energy Production by Leveraging Oil and Gas ERD and HPHT Well Construction Expertise. In: SPE/IADC International Drilling Conference and Exhibition.

38.

Verma

Andaverde

Santoyo

(2006) Statistical evaluation of methods for the calculation of static formation temperatures in geothermal and oil wells using an extension of the error propagation theory. Journal of Geochemical Exploration 89(1): 398–404.

39.

Wall

(2003) Exergy tools. Proceedings of The Institution of Mechanical Engineers Part A-journal of Power and Energy - PROC INST MECH ENG A-J POWER 217: 125–136.

40.

Wong-Loya

Andaverde

del Rio

(2015) Improved method for estimating static formation temperatures in geothermal and petroleum wells. Geothermics 57: 73–83.

41.

Hao

, et al. (2021) Experimental research on carbon storage in a CO₂-based enhanced geothermal system. Renewable Energy 175: 68–79.

42.

Yin

Song

, et al. (2021) Analysis of flow and thermal breakthrough in leaky downhole coaxial open loop geothermal system. Applied Thermal Engineering 194: 117098.

43.

Zhang

, et al. (2019) Performance analysis of a downhole coaxial heat exchanger geothermal system with various working fluids. Applied Thermal Engineering 163: 114317.

Research on Design Method for Optimal Flow Rate of U-shaped Geothermal Well Based on Exergy Analysis

Abstract

Keywords

Introduction

Physical Problem and mathematical model

Description Of U-shaped geothermal well operation

Mathematical Model

Definite Solution conditions

Determination Of various thermal resistances

Thermophysical Properties of the working fluid

Initial Conditions

Boundary Conditions

Model Solution

Results and discussion

Experimental Validation of the mathematical model

Experimental System and basic parameters

Comparison Between simulation and experimental results

Sensitive Factor analysis

Basic Parameters

Outlet Temperature changes with the flow rate

Heat Extraction power changes with the flow rate

Optimal Flow rate based on exergy analysis

Fitting Of the function of outlet temperature and heat extraction power with flow rate

Exergy Analysis based on the fitting function

Optimal Flow rate design process

Conclusions

Footnotes

Acknowledgements

Declaration of conflicting interests

Funding

ORCID iD

References