Optimized geothermal energy extraction from hot dry rocks using a horizontal well with different exploitation schemes

Abstract

In the foreseeable future, the geothermal exploitation from hot dry rocks (HDR) using a horizontal well will bear potential. Thus, in-depth studies should be conducted on the selection of injection-production scheme (IPS) and working fluid, design of reinjection parameters, optimization of wellbore structure and materials, and analysis of geological settings. This paper proposed a fully coupled model to study the above scientific questions. For Model A, the working fluid was injected into the annulus and then flowed out of the thermal insulation pipe (TIP). Its temperature passes through two stages of temperature rise and two stages of temperature decline. But for model B, the working fluid was injected into the TIP and then flowed out of the annulus. Its temperature undergoes five stages, four stages of temperature rise and one stage of temperature decline. The results show that the Model A is the best IPS owing to its high outlet temperature, stable thermal recovery, and low fluid injection volume. In Model A, when the working fluid was supercritical carbon dioxide and the liquid injection volume was 135.73 m³/d, the heat recovery ratio (HRR) was as high as 85.40%, which was 17.85% higher than that of the Model B whose working medium was water, and its liquid injection volume was only 25% of that. Meanwhile, over ten years of continuous production, the outlet temperature decreased by 7.5 °C and 18.38 °C in the latter. The optimal working fluid has a low volume heat capacity and thermal conductivity for any IPS. Sensitivity studies showed that for the area that met the HDR standard, the effect of reinjection temperature on the outlet temperature can be ignored. As for Model A, HRR drops sharply by 6.74–9.32% when TIP goes from completely adiabatic to nonzero thermal conductivity. Meanwhile, the horizontal segment length of the TIP is shorter when Model A obtains the optimal outlet temperature compared with Model B. In addition, the correlation between the outlet temperature and different formations of thermophysical properties was seriously affected by the IPS and exploitation period, which was summarized in detail.

Introduction

Geothermal energy is one of the promising supplements to fossil fuels (Sayigh 1999; Gupta and Roy 2006); it is characterized as reproducible and nonpolluting (Esen et al. 2007a, 2007b; Balbay and Esen 2013; Gharibi et al. 2018; Nian and Cheng 2018). In addition, geothermal resources are almost inexhaustible and barely affected by weather conditions (Adams et al. 2014; Aliyu and Chen 2017; Arat and Arslan 2017; Liu et al. 2018), different from weather-dependent energy sources (tidal, wind, and solar energies). Thus, geothermal resources, especially for hot dry rocks (HDR), have received considerable research attention because of their huge potential for electricity generation and space heating (Lee 2014; Li and Zhang 2017; Willems et al. 2017; Zeng et al. 2017; Shi et al. 2018, 2019). Since the Fenton Hill project in the 1970s (Lu 2018), enhanced geothermal system (EGS) technology has developed over the past 50 years in several countries. Although EGS was not first applied in the EU, France's Soultz plant and Germany's Landau plant were the first two commercial-scale EGS power plants. EGS development has also been ambitious in Australia. The world's largest EGS power plant was schedule to be operational in Habanero, Australia, a 1 MWe demonstration EGS power plant began operating at the beginning of May 2013 (Asai et al. 2018; Lu 2018; Gong et al. 2020).

Given HDR's extremely low porosity and permeability, large-scale hydro-fracturing is usually required before geothermal exploitations to provide channels for flow and heat exchange (None 2009; Wang et al. 2012a, 2012b; Li et al. 2022). However, the fractures created by hydro-fracturing can be blocked under geo-stress and water–rock interactions, rendering geothermal resources unexploitable (Xu and Pruess 2004; Pruess 2006). On the other hand, cap rocks can be easily fractured because of their high brittleness, which results in a huge loss of water (Hendron 1987; Brown et al. 1999; Brown and Duchane 1999). Large-scale hydro-fracturing is not only expensive but can also cause environmental problems or induce earthquakes (Majer et al. 2007; Kraft and Deichmann 2014; Anyim and Gan 2020). Therefore, an economical, environmentally friendly, safe, and reliable method must be developed for the enhanced geothermal exploitation of HDR.

One commonly adopted method for geothermal exploitation of HDR is circulating the working fluids through wellbores (Brown 2000; Pruess 2006; Zhang et al. 2013; Wang et al. 2018), in which the costly and complex hydro-fracturing can be avoided. In addition, oil and gas wells have been increasingly abandoned worldwide when petroleum reservoirs became depleted without economic feasibility. Therefore, numerous researchers began to study abandoned oil and gas wells (AOGW) for heat extraction and power generation. If the AOGW can be retrofitted to geothermal systems for the extraction of thermal energy from the HDR, not only can the environmental risk be reduced effectively, but the geothermal utilizations will also become cheaper without high-cost drilling (Su and Sun 1996; Wang and Liu 1997; Nian and Cheng 2018).

Kujawa et al. (2004, 2005) proposed the seminal research of retrofitting AOGW for geothermal production based on a double-pipe heat exchanger and assessed the possibility and usefulness of accessing geothermal energy from existing production wells (Kujawa et al. 1998, 2003). Subsequently, significant advances (Davis and Michaelides 2009; Bu et al. 2012; Cheng et al. 2013; Angrisani et al. 2016; Mokhtari et al. 2016) have occurred in the AOGW for geothermal energy production, but previous models were focused on vertical wells, which cannot be used for simulating the unique heat and mass transfer characteristics of the flow in a horizontal well. Many works have been done on a horizontal ground-coupled heat pump systems in shallow strata (Esen et al. 2007a, 2007b; Gonzalez et al. 2012; Go et al. 2016; Bulmez et al. 2022). However, there is little research on heat extraction from horizontal wells in deep strata. Thus, Cui et al. (2017) and Sun et al. (2018) attempted to study the performance of geothermal development in horizontal wells, and the technical and economic feasibility was also assessed. In addition, based on the updated conceptual model of the field, Seyedrahimi-Niaraq et al. (2021a, 2021b) presented an unsaturated numerical model for the NW Sabalan geothermal reservoir by incorporating some new exploration data and considering the unsaturated zone. For predicting the reservoir production capacity, a 30-year production response for the various electricity generation scenarios was carried out and the optimal production zone was determined.

However, no scholars have dynamically analyzed the heat extraction of a horizontal well with different exploitation schemes from the time–space perspective. In addition, we believe that in the foreseeable future, geothermal development using a horizontal well will be a potential heat recovery method. For this purpose, in-depth studies should focus on selecting injection-production scheme (IPS) and working fluids, design of reinjection parameters, optimization of wellbore structure and materials, and analysis of thermophysical geological properties.

A fully coupled model was proposed to study the geothermal exploitation of HDR by recycling water/supercritical carbon dioxide (SCCO) in a horizontal well via a closed loop. In addition, two kinds of IPS were focused on in this study: (a) Fluid is injected into the annulus and then flows out of the thermal insulation pipe (TIP). (b) Fluid is injected into the TIP and then flows out of the annulus. To better describe the heat exchange in geothermal exploitations under transient temperature and pressure, the thermophysical parameters of the working fluids were obtained by dynamically invoking the database of the National Institute of Standards and Technology (NIST). In addition, to study the different IPS, we put forward the "injection-production switch" parameters. We first investigated each exploitation mode's optimal reinjection rate (q_best) based on the outlet temperature. In addition, we studied the mode with the highest outlet temperature and that with the most stable thermal recovery. Then, we dynamically analyzed the effects of various factors on the geothermal exploitation performance under different IPS, working fluids, and exploitation periods (t_exp). The factors were divided into three main categories: (a) reinjection parameters, (b) wellbore structure and materials, and (c) geological factors. Finally, for the four exploitation modes, the variation coefficients of the outlet temperatures corresponding to various sensitivity parameters were calculated, which can contribute to the integrated optimization of geothermal extraction schemes.

The superiority of this model over the previous models presents a comprehensive and more realistic numerical model for the geothermal exploitation of a horizontal well and predicts the performance of the different production scenarios. There are mainly three contributions of this paper to the previous research: (a) Type curve analyses were used to understand complex heat exchanging processes for different exploitation modes. (b) Key design variables were deeply discussed from the space–time perspective to obtain conceptual design guidelines. (c) The correlation between the outlet temperature and different stratum of thermophysical properties was summarized in detail.

Modeling

Figure 1 depicts the structure of the horizontal well for geothermal exploitation. The bottom of the wellbore, which is different from the wells used in oilfields, was sealed with cement or packer to prevent direct contact between the injected working fluids and the rock (Cui et al. 2017). In addition, two kinds of IPS (fluid loop) were used to extract heat:

• a) [Model A] (Fig. 1a): The fluid is injected into the annulus and then flows out of the TIP. The injected low-temperature working fluid flows through the annulus between the TIP and casing along the wellbore, being heated up simultaneously by the geothermal reservoir. At the bottom of the horizontal well, the heated fluid with the highest temperature flows into the TIP and returns to the surface for heat exchange.

• b) [Model B] (Fig. 1b): The fluid is injected into the TIP and then flows out of the annulus. The heat-transmission fluid is first injected into the TIP of the vertical segment, followed by the working fluid to the well bottom, after which flow begins and proceeds toward the horizontal segment of the TIP. When the working fluid reaches the toe point of the long horizontal tube, it begins to flow into the annulus where it extracts heat energy from the surrounding HDR.

Fig. 1

Schematic of different IPS for geothermal exploitation of a horizontal well

In addition, the heated working fluids were pumped to the ground through the vertical wellbore for residential heating or electricity generation: at present, the most common technology in geothermal power generation is the dual cycle geothermal power system, including the organic Rankine cycle and the Kalina cycle (Wu et al. 2009; Bina et al. 2018). The main principle of the system is that the high-temperature geothermal fluid is pumped into a heat exchanger to transfer its thermal energy to another working medium (usually a low-boiling working fluid), which is heated and evaporated into a steam turbine to do work to generate electricity. After completion of the heat exchange (generate electricity), the cooled working fluid was used for reinjection.

To facilitate the research, we defined four exploitation modes in which the working fluid can be water or SCCO:

• (1) [Model A, water] representing the IPS of [Model A] and with water as the working fluid;

• (2) [Model A, SCCO] representing the IPS of [Model A] and with SSCO as the working fluid;

• (3) [Model B, water] representing the IPS of [Model B] and with water as the working fluid;

• (4) [Model B, SCCO] representing the IPS of [Model B] and with SCCO as the working fluid.

In addition, to facilitate the description of the relevant thermal extraction process, we abbreviated the pipes of different segments as follows (the same below):

• (1) TIP_V represents the vertical segment of the TIP.

• (2) TIP_H represents the horizontal segment of the TIP.

• (3) ANN_H represents the horizontal segment of the annulus.

• (4) ANN_V represents the vertical segment of the annulus.

Transient temperature field governing equation of the model

In accordance with the heat flow process of the model, the temperature field governing equation can be derived. The implicit difference method was used to solve the transient temperature distribution in this research. In addition, Appendix A shows the mesh generation method of the thermal extraction system.

Heat transfer model inside the vertical pipes

Conduction and convection dominated the heat transfers of the working fluids in the vertical pipes (including the TIP_V and ANN_V) during the exploitation. In addition, in the vertical pipes of the wellbore, the vertical component of the working fluid's downward/upward seepage velocity was highly valued, and the radial component can be neglected (Huang et al. 2019; Xu 2020; Zhang et al. 2021). Thus, the heat transfer equation inside the vertical pipes can be written as follows:

$$\frac{\partial }{\partial x}\left({\lambda }_{f}\frac{\partial T}{\partial x}\right)+\frac{\partial }{\partial z}\left({\lambda }_{f}\frac{\partial T}{\partial z}\right)-\frac{\partial }{\partial z}\left({{\rho }_{f}{c}_{f}v}_{z}T\right)=\frac{\partial }{\partial t}\left({\rho }_{f}{c}_{f}T\right)$$

(1)

where T is the temperature field, v_z is the seepage velocity of the working fluid in the vertical pipes, and λ_f, ρ_f and c_f represent the thermal conductivity, density and specific heat capacity of the working fluid, respectively.

In the vertical wellbore, the flow direction of the working fluid in the annulus was opposite that of the working fluid in the TIP. Therefore, for this research, we innovatively put forward the "injection-production switch" parameters (S₁, S₂, and S₃), which can be used to define the convection terms of the governing equation with different flow directions of the working fluid:

$$\left\{\begin{array}{ll}{S}_{1}=\frac{{v}_{z}+\left|{v}_{z}\right|}{2\left|{v}_{z}\right|}\\ {S}_{2}=\frac{{v}_{z}}{\left|{v}_{z}\right|} \\ {S}_{3}=\frac{{v}_{z}-\left|{v}_{z}\right|}{2\left|{v}_{z}\right|}\end{array}\right.$$

(2)

In the vertical conduit, when the flow direction of the working fluid is downward, v_z > 0; otherwise, v_z < 0. Thus, S₁, S₂, and S₃ were all constants. In addition, if v_z > 0, then ${S}_{1}=1$, ${S}_{2}=1$, ${S}_{3}=0$; if v_z < 0, ${S}_{1}=0$, ${S}_{2}=-1$, and ${S}_{3}=-1$. In addition, the upwind difference scheme was adopted for the convection term (Ewing et al. 1994; Lazarov et al. 1996; Yuan 2010; Shi 2021):

$$\frac{\partial }{\partial z}\left({\rho }_{f}{c}_{f}{v}_{z}T\right)=\frac{{\rho }_{f}{c}_{f}{v}_{z}}{\Delta z}\left({-s}_{1}{T}_{i,j-1}^{n+1}{+s}_{2}{T}_{i,j}^{n+1}{-s}_{3}{T}_{i,j+1}^{n+1}\right)$$

(3)

where ${T}_{i,j-1}^{n+1}$, ${T}_{i,j}^{n+1}$ and ${T}_{i,j+1}^{n+1}$ represent the temperature at the next time period in discrete grid nodes (i, j−1), (i, j) and (i, j + 1), respectively.

The difference scheme of Eq. (1) can be defined as follows:

$${a}_{1}\left({T}_{i+1,j}^{n+1}-2{T}_{i,j}^{n+1}+{T}_{i-1,j}^{n+1}\right)+{a}_{2}\left({T}_{i,j+1}^{n+1}-2{T}_{i,j}^{n+1}+{T}_{i,j-1}^{n+1}\right)-{a}_{3}\left({-s}_{1}{T}_{i,j-1}^{n+1}{+s}_{2}{T}_{i,j}^{n+1}{-s}_{3}{T}_{i,j+1}^{n+1}\right)={a}_{4}\left({T}_{i,j}^{n+1}-{T}_{i,j}^{n}\right)$$

(4)

where ${T}_{i,j}^{n}$ is the current temperature in the discrete grid nodes (i, j), and ${T}_{i+1,j}^{n+1}$ and ${T}_{i-1,j}^{n+1}$ represent the temperature at the next time period in discrete grid nodes (i + 1, j) and (i−1, j), respectively.

Equation (4) can be simplified as below:

$${a}_{1}{T}_{i-1,j}^{n+1}+\left(-2{a}_{1}-2{a}_{2}-{s}_{2}{a}_{3}-{a}_{4}\right){T}_{i,j}^{n+1}+{a}_{1}{T}_{i+1,j}^{n+1}+\left({a}_{2}+{s}_{1}{a}_{3}\right){T}_{i,j-1}^{n+1}+\left({a}_{2}+{s}_{3}{a}_{3}\right){T}_{i,j+1}^{n+1}={-a}_{4}{T}_{i,j}^{n}$$

(5)

where

$${a}_{1}=\frac{{{(\lambda }_{f})}^{t-1}}{{\left(\Delta x\right)}^{2}},{a}_{2}=\frac{{{(\lambda }_{f})}^{t-1}}{{\left(\Delta z\right)}^{2}}, {a}_{3}=\frac{{\left({\rho }_{f}{c}_{f}\right)}^{t-1}{v}_{z}}{\Delta z}, {a}_{4}=\frac{{\left({\rho }_{f}{c}_{f}\right)}^{t-1}}{\Delta t}$$

(6)

where ∆t is the time step, and ∆x and ∆z are the radial and discrete longitudinal spacing, respectively.

Heat transfer model inside the horizontal pipes

Heat conduction and convection dominated the heat transfer of the working fluids in the horizontal conduits (including the TIP_H and ANN_H). Thus, the temperature field equation can be expressed as follows (Huang et al. 2019; Wang 2019; Zhang et al. 2021):

$$\frac{\partial }{\partial x}\left({\lambda }_{f}\frac{\partial T}{\partial x}\right)+\frac{\partial }{\partial z}\left({\lambda }_{f}\frac{\partial T}{\partial z}\right)-\frac{\partial }{\partial x}\left({{\rho }_{f}{c}_{f}v}_{r}T\right)=\frac{\partial }{\partial t}\left({\rho }_{f}{c}_{f}T\right)$$

(7)

where T is the temperature field, v_r is the seepage velocity of the working fluid in the horizontal pipes, and λ_f, ρ_f and c_f represent the thermal conductivity, density and specific heat capacity of the working fluid, respectively. In the same manner, the upwind difference scheme was adopted for the convection term (Ewing et al. 1994; Lazarov et al. 1996; Yuan 2010; Shi 2021):

$$\frac{\partial }{\partial x}\left({\rho }_{f}{c}_{f}{v}_{r}T\right)=\frac{{\rho }_{f}{c}_{f}{v}_{r}}{\Delta x}\left({-s}_{1}{T}_{i-1,j}^{n+1}{+s}_{2}{T}_{i,j}^{n+1}{-s}_{3}{T}_{i+1,j}^{n+1}\right)$$

(8)

where ${T}_{i-1,j}^{n+1}$, ${T}_{i,j}^{n+1}$ and ${T}_{i+1,j}^{n+1}$ represent the temperature at the next time period in discrete grid nodes (i−1, j), (i, j) and (i + 1, j), respectively.

The difference scheme of Eq. (7) can be defined as follows:

$${c}_{1}{T}_{i-1,j}^{n+1}+\left(-2{c}_{1}-2{c}_{2}-{s}_{2}{c}_{3}-{c}_{4}\right){T}_{i,j}^{n+1}+{c}_{1}{T}_{i+1,j}^{n+1}+\left({c}_{2}+{s}_{1}{c}_{3}\right){T}_{i,j-1}^{n+1}+\left({c}_{2}+{s}_{3}{c}_{3}\right){T}_{i,j+1}^{n+1}=-{c}_{4}{T}_{i,j}^{n}$$

(9)

where

$${c}_{1}=\frac{{{(\lambda }_{f})}^{t-1}}{{\left(\Delta x\right)}^{2}}, {c}_{2}=\frac{{{(\lambda }_{f})}^{t-1}}{{\left(\Delta z\right)}^{2}}, {c}_{3}=\frac{{\left({\rho }_{f}{c}_{f}\right)}^{t-1}{v}_{r}}{\Delta x}, {a}_{4}=\frac{{\left({\rho }_{f}{c}_{f}\right)}^{t-1}}{\Delta t}$$

(10)

where ∆t is the time step, ∆x is the discrete radial spacing, ∆z is the discrete longitudinal spacing, ${T}_{i,j}^{n}$ is the current temperature in the discrete grid nodes (i, j), and ${T}_{i+1,j}^{n+1}$ and ${T}_{i-1,j}^{n+1}$ represent the temperature at the next time period in discrete grid nodes (i + 1, j) and (i−1, j), respectively.

Heat transfer model for the HDR and impermeable medium

For the HDR and impermeable medium (pipes/cements), heat conduction dominates the heat transfer rather than convection. Thus, their transient temperature field governing equation can be given by the following (Fang 2018; Huang et al. 2019):

$$\frac{\partial }{\partial x}\left({\lambda }_{s}\frac{\partial T}{\partial x}\right)+\frac{\partial }{\partial z}\left({\lambda }_{s}\frac{\partial T}{\partial z}\right)=\frac{\partial }{\partial t}\left({\rho }_{s}{c}_{s}T\right)$$

(11)

where T is the temperature field, and λ_s, ρ_s and c_s represent the thermal conductivity, density and specific heat capacity of the impermeable media, respectively.

The difference scheme of Eq. (11) can be defined as follows:

$${b}_{1}{T}_{i-1,j}^{n+1}+\left(-2{b}_{1}-2{b}_{2}-{b}_{3}\right){T}_{i,j}^{n+1}+{b}_{1}{T}_{i+1,j}^{n+1}+{b}_{2}{T}_{i,j-1}^{n+1}+{b}_{2}{T}_{i,j+1}^{n+1}=-{b}_{3}{T}_{i,j }^{n}$$

(12)

where

$${b}_{1}=\frac{{\left({\uplambda }_{s}\right)}^{t-1}}{{\left(\Delta x\right)}^{2}}, {b}_{2}=\frac{{\left({\uplambda }_{s}\right)}^{t-1}}{{\left(\Delta z\right)}^{2}}, {b}_{3}=\frac{{\left({\uprho }_{s}{\mathrm{c}}_{s}\right)}^{t-1}}{\Delta t}$$

(13)

where ∆t is the time step, ∆x is the discrete radial spacing, ∆z is the discrete longitudinal spacing, ${T}_{i,j}^{n}$ is the current temperature in the discrete grid nodes (i, j), and ${T}_{i-1,j}^{n+1}$, ${T}_{i,j}^{n+1}$, ${T}_{i+1,j}^{n+1}$ ${T}_{i,j-1}^{n+1}$ and ${T}_{i,j+1}^{n+1}$ represent the temperature at the next time period in discrete grid nodes (i−1, j), (i, j), (i + 1, j), (i, j−1) and (i, j + 1), respectively.

Initial and boundary conditions of the model

The initial (undisturbed) geothermal temperature was assumed to be a known function of depth. Thus, the initial condition can be written as follows:

$${T|}_{t=0}={T}_{\mathrm{sur}}+{g}_{T}z$$

(14)

where T_sur is the initial surface temperature, and g_t is the geothermal gradient.

The inner/outer boundary of the temperature field was assumed to be equal to the (undisturbed) geothermal temperature. Therefore, the model's inner and outer boundary conditions can be defined by Eqs. (15) and (16), respectively. In the vertical pipes, the coupled boundary conditions at the interfaces between the working fluids and tube walls can be given by Eq. (17). In addition, the coupled boundary conditions at the interfaces between the working fluids in the vertical and horizontal pipes can be described as Eq. (18).

$${T|}_{r=0}={T}_{\mathrm{sur}}+{g}_{T}z$$

(15)

$${T|}_{r={r}_{e}}={T}_{\mathrm{sur}}+{g}_{T}z$$

(16)

$$\left\{\begin{array}{ll}{h}_{c}\left({T}_{\mathrm{bf}}-{T}_{\mathrm{bp}}\right)={-{\lambda }_{s}\frac{\partial T}{\partial r}|}_{r={r}_{\mathrm{fp}}}\\ {{T}_{\mathrm{bf}}|}_{r={r}_{\mathrm{fp}}}={{T}_{\mathrm{bp}}|}_{r={r}_{\mathrm{fp}}}\end{array}\right.$$

(17)

$$\left\{\begin{array}{ll}{{\lambda }_{f}\frac{\partial T}{\partial r}|}_{r={r}_{\mathrm{vh}}}+\left|{\rho }_{\mathrm{f}}{c}_{\mathrm{f}}{v}_{r}\right|{\left({T}_{\mathrm{vf}}-{T}_{\mathrm{hf}}\right)|}_{r={r}_{\mathrm{vh}}}={{\lambda }_{s}\frac{\partial T}{\partial r}|}_{r={r}_{\mathrm{vh}}} \\ {{T}_{\mathrm{vf}}|}_{r={r}_{\mathrm{vh}}}={{T}_{\mathrm{hf}}|}_{r={r}_{\mathrm{vh}}}\end{array}\right.$$

(18)

$${h}_{\mathrm{c}}=0.023\frac{{\lambda }_{\mathrm{f}}}{L}{R}_{\mathrm{ef}}^{0.8}{P}_{\mathrm{rf}}^{n}$$

(19)

where r_e is the radial distance from the left boundary to the right, r_fp is the boundary between the working fluid and the vertical pipe, r_vh is the boundary between the working fluids in the vertical pipe and working fluids in the horizontal pipe, T_bf and T_bp are the working fluid temperature and vertical pipe temperature at the working fluid/vertical, T_vf and T_hf are the working fluid temperature in the vertical pipe and the working fluid temperature in the horizontal pipe at the horizontal pipe/vertical pipe interface, λ_s represents the thermal conductivity of the impermeable media, and λ_f, ρ_f and c_f represent the thermal conductivity, density and specific heat capacity of the working fluid, respectively. In addition, h_c is the convection heat transfer coefficient, L is the characteristic length, R_ef is Reynolds number, P_rf is Prandtl number, n is the coefficient, and n = 0.3 (when the fluid is heated) or n = 0.2 (when the fluid is cooled).

In the longitudinal direction, we assumed that no heat transfer occurred at the top and bottom of the model. Thus, their boundary conditions can be described as Eqs. (20) and (21). In addition, the coupled boundary condition at the interfaces between the working fluids in the horizontal pipes and tube walls can be defined by Eq. (22).

$${{\lambda }_{\mathrm{top}}\frac{\partial T}{\partial z}|}_{z=0\left(0\le r\le {r}_{e}\right)}=0$$

(20)

$${{\lambda }_{\mathrm{bottom}}\frac{\partial T}{\partial z}|}_{z={z}_{\mathrm{max}\left(0\le r\le {r}_{e}\right)}}=0$$

(21)

$$\left\{\begin{array}{ll}{{\lambda }_{f}\frac{\partial T}{\partial z}|}_{z={z}_{b}}\left(0<r<{r}_{e}\right)={{\lambda }_{s}\frac{\partial T}{\partial z}|}_{z={z}_{b}} \left(0<r<{r}_{e} \right) \\ {{T}_{\mathrm{hbf}}|}_{z={z}_{b}}={{T}_{\mathrm{hbs}}|}_{z={z}_{b}}\end{array}\right.$$

(22)

where λ_top and λ_bottom are the medium thermal conductivity of the upper boundary and lower boundary, λ_f and λ_s are the thermal conductivity of working fluid and impermeable media, T_hbf and T_hbs are the working fluid temperature and horizontal pipe temperature at the horizontal pipe/working fluid interface, r_e is the radial distance from the left boundary to the right, and z_max is the total well depth from the surface.

Results and discussion

Based on the information provided above, in this part, the numerical solutions of the four exploitation modes are obtained and discussed in detail. In addition, based on the study of type curves, the sensitivities of the three main categories of factors were analyzed: (a) reinjection parameters, (b) wellbore structure and materials, and (c) geological factors. The coefficients of variation of the outlet temperatures corresponding to various sensitivity parameters were also calculated. Appendix B lists the basis parameters used for calculation. In addition, the transient thermophysical parameters (ρ_f, c_f, and λ_f) of the working fluids (water/SCCO) were obtained by dynamically invoking the database of NIST.

Maximum outlet temperature and type curve analysis of different exploitation modes

To study the optimal outlet temperature for the different exploitation modes, we simulated the outlet temperatures (t_exp: 1.0 year) of different q_in. To understand complex heat exchanging process for different exploitation modes more conveniently, P₁–P₆ are marked in Fig. 2, and their meanings are as follows: P₁ represents the temperature of injection fluid at the wellhead; P₂ represents the fluid temperature when it enters the horizontal segment from the vertical segment; P₃ is the fluid temperature at the bottom of the horizontal segment of the pipe; P₄ indicates the fluid temperature when it enters the vertical segment from the horizontal segment; In the Model A, P₅ represents the production fluid temperature at the wellhead, however, P₅ represents the fluid temperature at the interface between the HDR and the insulation formation when the produced fluid migrates upward in the Model B; P₆ represents the production fluid temperature at the wellhead in the Model B.

• (1) As shown in Fig. 2a–d, for the four exploitation modes, the outlet temperatures all increased first and then decreased with the increase in q_in. For any mode, the q_best maximizes the outlet temperature (T_rmax).

Fig. 2

Comprehensive comparison of the four exploitation modes. a–d Outlet temperature (t_exp = 1.0 year) at different q_in for the various exploitation modes. e–h Type curve analysis of different exploitation modes (t_exp = 1.0 year, q_in = q_best) and the temperature of the working fluid in different segments of the pipe for the various exploitation modes. i q_best and T_rmax for the different exploitation modes (t_exp = 1.0 year)

The above phenomenon can be explained as follows (Fig. 2e–h): when the q_in is extremely low, the heat loss is excessive in the P₃-P₄-P₅ ([Model A]) or P₅-P₆ ([Model B]) stage although the working fluid can be heated to a higher temperature in the P₁-P₂-P₃ ([Model A]) or P₁-P₂-P₃-P₄-P₅ ([Model B]) stage. In addition, when the q_in is extremely high, the heat loss of the working fluid is small in the P₃-P₄-P₅ ([Model A]) or P₅-P₆ ([Model B]) stage. However, the temperature rise is very limited in the P₁-P₂-P₃ ([Model A]) or P₁-P₂-P₃-P₄-P₅ ([Model B]) stage, thus, there is a certain q_best leads to T_rmax.

In addition, the four exploitation modes had different q_best, with values of 339.29 ([Model A, water]), 135.72 ([Model A, SCCO), 542.87 ([Model B, water]), and 271.43 m³/day ([Model B, SCCO]) and corresponding T_rmax of 161.36 °C, 170.80 °C, 134.10 °C, and 152.14 °C, respectively. Therefore, as shown in Fig. 2a–d, i, among all the exploitation modes, [Model A, SCCO] has the highest outlet temperature, and it requires the least amount of fluid injection (q_best = 135.72 m³/day) compared with the other exploitation modes.

• (2) During the circulation of the working medium fluid, for [Model A], the working fluid temperature passes through four stages: two stages of temperature rise and two stages of temperature decline. However, for [Model B], the working fluid temperature undergoes five stages: four stages of temperature rise and one stage of temperature decline. The above phenomenon is attributed to the difference in the heat exchange caused by the variation in IPS.

For [Model A] (Fig. 2e–f), when a cooled working fluid is injected into the ANN_V, it will be heated by the surrounding rocks. Then, the working fluid enters the ANN_H and is heated by the HDR around the casing. Therefore, in the annulus (including the vertical and horizontal segments), the working fluid undergoes two stages (P₁–P₂–P₃) of heating. Subsequently, the working fluid enters the TIP_H, because the temperature of the working fluid in the TIP_H is higher than that of the fluid in the ANN_H. Thus, the working fluid transfers heat to the annulus, resulting in its temperature decrease (P₃–P₄). Finally, the working fluid enters the TIP_V. Similarly, the high-temperature working fluid in the TIP_V is cooled by the low-temperature reinjection fluid in the ANN_V, resulting in a decrease in the fluid temperature (P₄–P₅).

For [model B] (Fig. 2g–h), when a cooled working fluid is injected into the TIP_V, the working fluid of the TIP_V will be heated by the high-temperature production fluid in the ANN_V. The working fluid then flows into the TIP_H, which is similarly heated by the fluid in the ANN_H. Thus, the working fluid involves two stages (P₁–P₂–P₃) of heating in the TIP (including the vertical and horizontal segments). Subsequently, the working fluid enters the ANN_H because the temperature of the working fluid in the ANN_H is significantly lower than the ambient temperature (T_b). Hence, the working fluid is continually heated, causing its temperature to rise rapidly (P₃–P₄). Finally, the working fluid enters the ANN_V, and given that the temperature of the working fluid at the bottom (z = 2900–3000 m) is slightly lower than the T_b of the HDR, the working fluid is also continually heated (P₄–P₅). However, with the upward migration of the working fluid in the ANN_V, its temperature will be higher than that of the surrounding formation and the fluid in the TIP_V. Therefore, the working fluid in the ANN_V transfers heat to the surrounding rock and the TIP_V, resulting in its temperature reduction (P₅–P₆). Thus, in the ANN_V, the working fluid experiences a temperature increase (P₄–P₅) before a temperature decrease (P₅–P₆), which is different from that in [model A].

• (3) Under the same conditions, the T_rmax of [model A] is greater than that of [model B] because after the working fluid is injected into the annulus, it can be fully heated by the surrounding rock and HDR. Then, the hot working fluid flows into the TIP with a particularly low thermal conductivity (TC). Thus, the working fluid loses less heat, resulting in an increased outlet temperature. However, for [Model B], when a fluid is injected into the TIP, although the working fluid is heated to a high temperature in the TIP (including the vertical and horizontal segments) and the ANN_H, given the high TC of the casing and the transition formation/caprock relative to the TIP, the working fluid in the ANN_V will release a large amount of heat to the radial direction (especially in the upper strata). Therefore, the outlet temperature is relatively low. Thus, the use of a low-TC material at the top of the vertical casing can significantly increase the outlet temperature of [Model B].

In addition, under the same IPS, when the working fluid is SCCO, its outlet temperature is greater than that when the working fluid is water because SCCO has a smaller volume heat capacity (VHC) than water and is therefore more easily heated. In addition, SCCO exhibits a relatively minimal heat loss due to its lower TC compared with water. Thus, for geothermal exploitations using a horizontal well, we should attempt to use fluids with low VHC and TC.

Stability comparison of the four geothermal exploitation modes

To compare the stability of the four geothermal exploitation modes, we simulated the outlet temperatures (q_in = q_best) of the different modes at varied t_exp. As shown in Fig. 3, for the four exploitation modes, all the outlet temperatures decrease with the t_exp, but the outlet temperature change (δT_out) in 10 years shows variation ($\updelta {T}_{\mathrm{out}}={{T}_{\mathrm{out}}|}_{t=10 \mathrm{years}}-{{T}_{\mathrm{out}}|}_{t=1 \mathrm{year}}$). In addition, we defined |δT_out| as the absolute value of δT_out (the same below). In [Model B], the |δT_out| is significantly greater than that of [Model A], and [Model A, SCCO] is the most stabilized mode with the minimum |δT_out| of 7.50 °C.

Fig. 3

Stability comparison of the four geothermal exploitation modes. a Outlet temperature at different t_exp (q_in = q_best). b δT_out of the different exploitation modes

The above phenomenon can be explained as follows: For [Model A], when a working fluid is injected into the ANN_V, it can be continuously heated by the surrounding rock and HDR. Thus, the fluid temperature is relatively high (P₂ in Fig. 2e–f) when the working fluid reaches the ANN_H. Therefore, compared with [Model B], the working fluid absorbs relatively little heat from the HDR around the horizontal pipes in [Model A]. Hence, for [Model A], the T_b of HDR around the ANN_H decreases more slowly than that in [Model B]. Thus, under the same t_exp, the T_b of [Model A] will be high. As a result, the working fluid of [Model A] can be heated to a high temperature before it enters the TIP_V (P₄ in Fig. 2e–f). Thus, the |δT_out| is small in [Model A] under the same conditions.

Sensitivity analysis of reinjection temperature (T _in)

Given that the working fluid is constantly exchanging heat with its surrounding environment during circulation, especially in the HDR, a strong heat exchange occurs. Thus, when we studied the sensitivity of T_in, the T_b of HDR around the horizontal pipes was considered. Thus, for different T_b, the heat recovery ratio (HRR) (η_s = T_out/T_b) at different T_in was simulated. In addition, the effects of T_in on the η_s were studied by linear regression analysis (Table 1), and we defined K_slope as the slope of η_s with respect to T_in.

Table 1 The fitting equations of each curve in Fig. 4

Fig. 4

Sensitivity analysis of T_in. a–d η_s (t_exp = 1.0 year) of different T_in at various T_b for the four exploitation modes. e ∆T_sense and K_slope at different T_b for the four exploitation modes (t_exp = 1.0 year)

The following cognitions can be obtained (Fig. 4):

• (1) For the four exploitation modes, under any T_b, all η_s increase with the T_in, Based on Fourier's first law, with the increase of T_in, less heat is released from the working fluid in the vertical output tube into the fluid in the injection tube. Therefore, in the vertical output tube, the temperature of the working fluid decreases relatively less (P₄-P₅ stage for [Model A] and P₅-P₆ stage for [Model B], as shown in Fig. 2e–h).

In addition, as shown in Fig. 4e, the outlet temperature variation (${\Delta T}_{\mathrm{sense}}={{T}_{\mathrm{out}}|}_{T\mathrm{in}=55\mathrm{^\circ{\rm C} }}-{{T}_{\mathrm{out}}|}_{T\mathrm{in}=15^\circ{\rm C} }$) decreases with the increase in T_b. However, the ∆T_sense is different for the various exploitation modes when T_b is in the range of 74 to 200 °C. For [Model A, water], the ∆T_sense drops from 7.85 to 1.47 °C (K_solop = 0.1832–0.0335); from 5.86 to 1.17 °C (K_solop = 0.1473–0.0259) for [Model A, SCCO]; from 11.08 to 1.72 °C (K_solop = 0.2771–0.0424) for [Model B, water]; from 6.81 to 1.30 °C (K_solop = 0.1714–0.0316) for [Model B, SCCO]. Thus, when the working fluid is water, the effect of T_in on η_s is greater than that when the working fluid is SCCO.

• (2) In this research, when K_slope $\le$ 0.05, the T_in has little influence on η_s because when K_slope = 0.05, when the T_in increases/decreases by 40 °C (the T_in ranges from 15 to 55 °C), the disturbance of the outlet temperature is at 2% (0.05% × 40 = 2%). Therefore, the degree of influence is extremely low when K_slope $\le$ 0.05. On the contrary, when K_slope > 0.05, the T_in has a relatively large influence on η_s. Thus, when T_b > 146 °C, the K_slope < 0.05 for the four exploitation modes. Therefore, in the geothermal exploitation of horizontal well, when the T_b of stratum reaches 146 °C, the T_in has almost no influence on η_s.

Sensitivity analysis of the TC (λ _TIP) of the TIP

Figure 5 shows the HRR (η_s = T_out/200 °C, the same below) at different λ_TIP. The following cognitions can be obtained:

• (1) For the four exploitation modes, their HRR all decrease with the increase in λ_TIP because as the λ_TIP increases, the working fluid in the output pipe releases more heat. As a result, the fluid temperature drops increasingly during this stage (P₄-P₅ stage for [Model A] and P₅-P₆ stage for [Model B], as shown in Fig. 2e–h).

• (2) In this section, we define the temperature variation (${\Delta T}_{\mathrm{sense}}$) as ${\Delta T}_{\mathrm{sense}}={{T}_{\mathrm{out}}|}_{{\uplambda }_{TIP} = 46\mathrm{ W}/(\mathrm{m K})}-{{T}_{\mathrm{out}}|}_{{\uplambda }_{TIP} = 0 \mathrm{W}/(\mathrm{m K})}$, the |∆T_sense| (same as below) is the absolute value of ∆T_sense. ∆T_sense represents the difference between the maximum outlet temperature and the minimum outlet temperature within the variation range of the sensitivity parameter. Thus, the influence degree of λ_TIP on the outlet temperature can be ranked from strong to weak as follows (Fig. 5b): |∆T_sense|= 40.53 °C ([Model A, water]) >|∆T_sense|= 32.23 °C (Model B, water) >|∆T_sense|= 27.85 °C ([Model A, SCCO]) >|∆T_sense|= 20.76 °C (Model B, SCCO).

Fig. 5

Sensitivity analysis of λ_TIP for the four exploitation modes. a HRR (t_exp = 1.0 year, q_in = q_best) at different λ_TIP for the four exploitation modes. b ∆T_sense (t_exp = 1.0 year, q_in = q_best) of the different exploitation modes

Thus, in [Model A], the disturbance of λ_TIP to the HRR is significantly greater than that in [Model B]. However, as shown in Fig. 5a, in [model A], a sudden drop in the HRR occurs when λ_TIP > 0. For [Model A, water], the HRR plummets from 89.63 to 80.31% and from 92.14 to 85.40% for [Model A, SCCO]. Such a phenomenon transpires because for [model A], if the TIP is fully adiabatic (λ_TIP = 0), theoretically, the working fluid will experience no heat loss in the TIP (P₃-P₄-P₅ stage in Fig. 2e–f). However, for λ_TIP > 0, a certain amount of heat will be lost. As a result, a sharp drop in the HRR will occur.

However, for [Model B], the working fluid in the ANN_V not only releases heat to the reinjection fluid in the TIP_V but also transfer a large amount of thermal energy to the transition formation/caprock. Thus, the heat released by the working fluid in the ANN_V and then into the TIP_V is only a fraction of the total heat loss (P₅-P₆ stage in Fig. 2g–h), which is notably different from [Model A] whose working fluid in the TIP_V only releases heat to the ANN_V. Therefore, in [Model B], with the increase in λ_TIP, the heat dissipation of the working fluid causes relatively little disturbance to the total loss of heat. Thus, as shown in Fig. 5a, the HRR only presents a gradually decreasing trend with the increase in λ_TIP, and no precipitous drop occurs for [Model B]. In addition, the above reasons explain why the λ_TIP has a larger effect on the HRR in [Model A].

Sensitivity analysis of the length (L _TIP) of the TIP

Figure 6 shows the HRR of different L_TIP for the four exploitation modes. The following cognitions can be obtained:

• (1) For the four exploitation modes, their outlet temperatures are positively correlated with the L_TIP. However, the outlet temperatures do not increase indefinitely but eventually reach a stable state. In addition, the outlet temperature increases quickly first and then at a lower rate with the increase in L_TIP.

• (2) Within the variation range (L_TIP = 100–2500 m) of the sensitivity parameter, if the difference $(\Delta \eta ={{\eta }_{s}|}_{{L}_{\mathrm{TIP}} = 2500\mathrm{m}}-{{\eta }_{s}|}_{{L}_{\mathrm{TIP}} = {L}_{\mathrm{best}}})$ between the maximum HRR (L_TIP = 2500 m) and the HRR at a certain length (L_best) is less than 1% (∆η < 1%), the HRR can reach the optimal level at that length (L_best). In other words, the increase in the L_TIP (when L_TIP > L_best) has little effect on the HRR. Hence, as shown in Fig. 6b, the corresponding L_best differs for the four exploitation modes, with values of 900 ([Model A, water]), 600 ([Model A, SCCO]), 1200 ([Model B, Water]), and 800 m ([Model B, SCCO]).

Fig. 6

Sensitivity analysis of the L_TIP for the four exploitation modes. a HRR (t_exp = 1.0 year, q_in = q_best) with different L_TIP for the four exploitation modes. b ∆T_sense and L_best of different exploitation modes (t_exp = 1.0 year, q_in = q_best)

Therefore, for [Model A], the L_best is significantly shorter than that of [model B] because under the same conditions, for [Model A], when the working fluid reaches the horizontal segment of the pipe, its temperature (P₂ in Fig. 2e–f) is significantly higher than that of [Model B]. Therefore, when the L_TIP is shorter, the working fluid can also be sufficiently heated to achieve the optimal outlet temperature. On the contrary, [Model B] needs a longer horizontal pipe to achieve the optimal outlet temperature. This requirement also explains why the outlet temperature of [model B] is more affected by the L_TIP compared with [model A]: ∆T_sense = 35.69 °C ([Model B, water]) > ∆T_sense = 32.39 °C ([Model B, SCCO]) > ∆T_sense = 22.25 °C ([Model A, water]) > ∆T_sense = 18.68 °C ([Model A, SCCO]). In this section, we define the temperature variation (∆T_sense) as ${\Delta T}_{\mathrm{sense}}={{T}_{\mathrm{out}}|}_{{L}_{\mathrm{TIP}}=2500\mathrm{m}}-{{T}_{\mathrm{out}}|}_{{L}_{\mathrm{TIP}}=100\mathrm{m}}$.

Sensitivity analysis of geological thermophysical parameters

The correlation between the outlet temperature and different formations of thermophysical parameters is seriously affected by the IPS and t_exp. These formations include the caprock, transition formation, and HDR, which are summarized in Table 2.

Table 2 Changes in the correlation between the outlet temperature and geological thermophysical parameters under different IPS and t_exp

[+ ,−] indicates that in the early stage of exploitation, the outlet temperature is positively correlated with the studied parameter, whereas in the later stage of exploitation, the outlet temperature is negatively correlated with the studied parameter. [ +] represents that the outlet temperature is always positively correlated with the studied parameter. [−] denotes that the outlet temperature is always negatively correlated with the studied parameter.

Sensitivity analysis of the transition formation

Figure 7 shows the HRR with different λ_tran at various t_exp for the four exploitation modes. The following cognitions can be obtained:

• (1) In [Model A], the correlation between the HRR and the λ_tran differs at various t_exp. As shown in Fig. 7a–b, e, in this study, when t_exp ≤ 3.0 years, the HRR is positively correlated with the λ_tran. However, when t_exp ≥ 5.0 years, the HRR is inversely correlated with the λ_tran. However, when this correlation changes, the t_exp may vary depending on the model parameters (or actual geological setting parameters). The reasons for the changes in the correlation between the HRR and the λ_tran are as follows.

Fig. 7

Sensitivity analysis of the λ_tran at different t_exp for the four exploitation modes. a–d HRR (q_in = q_best) with different λ_tran at various t_exp for the four exploitation modes. e ∆T_sense (q_in = q_best) at different t_exp for the different exploitation modes

In the early stage of exploitation, for [Model A], with the increase in the λ_tran, the transition formation becomes more favorable to heat the working fluid in the ANN_V (P₁-P₂ stage in Fig. 2e–f; depth range: z = 200–2800 m). Therefore, when the working fluid reaches the ANN_H, the temperature is high (P₂ in Fig. 2e–f). Thus, the outlet temperature is also high. However, given that the transition formation continuously transfers heat to the upper stratum, over time (especially in the late stage of exploitation), the greater the λ_tran, the more heat is transferred to the upper stratum from the lower stratum. This condition leads to a decrease in the T_b of the transition formation (z = 200–2800 m), which lowers the temperature (P₃ in Fig. 2e–f) of the working fluid when it enters the TIP_H, thus resulting in a decrease in the HRR. Therefore, in the early stage of exploitation, the HRR increases with the λ_tran, whereas in the later stage of exploitation, the HRR decreases.

• (2) However, for [Model B], as shown in Fig. 7c–d, e, given the large λ_tran, the fluid in the ANN_V releases more heat to the surrounding rock, especially in the upper part of the transition formation (P₅-P₆ stage in Fig. 2g–h). Therefore, as shown in Fig. 7e, the HRR consistently decreases with the λ_tran, but the |∆T_sense| gradually decreases over time. In this section, we define the temperature variation (${\Delta T}_{\mathrm{sense}}$) as ${\Delta T}_{\mathrm{sense}}={{T}_{\mathrm{out}}|}_{{\uplambda }_{\mathrm{tran}}=6 \mathrm{W}/(\mathrm{m K}) }-{{T}_{\mathrm{out}}|}_{{\uplambda }_{\mathrm{tran}}=0.9\mathrm{W}/(\mathrm{m K})}$.

Figure 8 shows the HRR with different VHC_tran for the four exploitation modes. The following cognitions can be obtained:

• (1) For [model A], the fluid in the ANN_V continuously absorbs heat from the surrounding rock. Therefore, as mentioned above, under the same conditions, the larger the VHC_tran, the more heat the surrounding rock will release. Thus, the working fluid in the ANN_V can absorb more heat. This condition allows the working fluid to have a higher temperature (P₃ in Fig. 2e–f) before entering the TIP_H. As a result, the outlet temperature increases. Similarly, in the later stage of exploitation development, the larger the VHC_tran, the more thermal energy exists in the reservoir, especially around the depth (H = 3000 m) where the horizontal pipe is located. Hence, the outlet temperature will also be higher. Thus, for [Model A], in the whole exploitation process, the HRR is always positively correlated with the VHC_tran.

Fig. 8

Sensitivity analysis of the VHC_tran for the four exploitation modes. a HRR (t_exp = 1.0 year, q_in = q_best) with different VHC_tran for the four exploitation modes. b) ∆T_sense (q_in = q_best) at different t_exp for the various exploitation modes

In addition, the above factors explain why the influence of VHC_tran on the outlet temperature becomes more significant with the t_exp over a period of 1–10 years. The temperature variation (${{{\Delta T}_{\mathrm{sense}}=T}_{\mathrm{out}}|}_{{\mathrm{VHC}}_{\mathrm{tran}}=4.5\times {10}^{6} \mathrm{J}/({\mathrm{m }}^{3} \cdot \mathrm{K})}-{{T}_{\mathrm{out}}|}_{{\mathrm{VHC}}_{\mathrm{tran}}=0.6\times {10}^{6} \mathrm{J}/({\mathrm{m }}^{3} \cdot \mathrm{K})}$) increases from 2.28 to 7.88 °C for [Model A, water] and increases from 3.70 to 10.06 °C for [Model A, SCCO].

• (2) However, for [Model B], a large VHC_tran will lead to the increased heat release of the working fluid in the ANN_V (P₅-P₆ stage in Fig. 2g–h). Under the same conditions, if the VHC_tran is large, then the temperature of the surrounding rock increases by 1 °C, and the surrounding rock needs to absorb more heat from the working fluid in the ANN_V. Therefore, the temperature of the working fluid decreases more during this stage (P₅-P₆ in Fig. 2g–h). Thus, the HRR is always inversely correlated with the VHC_tran. In addition, as shown in Fig. 8b, the |∆T_sense| also decreases with the t_exp, the principle is similar to that of the VHC_cap above.

Sensitivity analysis of the HDR

As shown in Fig. 9a–d, for the four exploitation modes, the correlation between the HRR and λ_HDR differs at various t_exp. In this study, when t_exp ≤ 3.0 years, the HRR is positively correlated with the λ_HDR. However, when t_exp ≥ 5.0 years, the HRR is inversely correlated with the λ_HDR. Still, when this correlation changes, the t_exp may vary depending on the model parameters. The reasons for the changes in the correlation between the HRR and λ_HDR are as follows.

Fig. 9

Sensitivity analysis of the λ_HDR at different t_exp for the four exploitation modes. a–d HRR (q_in = q_best) with different λ_HDR at various t_exp for the four exploitation modes. e ∆T_sense (q_in = q_best) at different t_exp for the various exploitation mode

In the early stage of exploitation, with the increase in the λ_HDR, the HDR becomes more favorable to heat the working fluid. Therefore, for [Model A], when the working fluid reaches the TIP_H, the temperature increases (P₃ in Fig. 2e–f). In the same manner, for [Model B], when the working fluid in the ANN_V enters the section of insulation formation (z = 2800–2900 m), the temperature increases (P₅ in Fig. 2g–h). Thus, the HRR increases with the λ_HDR in the early stage of exploitation.

However, given that the HDR also continuously transfers heat to the upper stratum and the working fluids, over time (especially in the late stage of exploitation), the greater the λ_HDR is, the more heat is lost. This condition leads to a decrease in the T_b of the HDR, which lowers the temperature of the working fluid (for [Model A], the temperature node is P₃ in Fig. 2e–f; for [Model B], the temperature node is P₅ in Fig. 2g–h), thus resulting in a decrease in the HRR. Therefore, in the early stage of exploitation, the HRR increases with the λ_HDR, whereas in the later stage of exploitation, the HRR decreases. The above factor also explains why the |∆T_sense| increases with t_exp in the late stage of exploitation. In this section, we define the temperature variation (${\Delta T}_{\mathrm{sense}}$) as:${\Delta T}_{\mathrm{sense}}={{T}_{\mathrm{out}}|}_{{\uplambda }_{\mathrm{HDR}}=6 \mathrm{W}/(\mathrm{m K}) }-{{T}_{\mathrm{out}}|}_{{\uplambda }_{\mathrm{HDR}}=0.9\mathrm{W}/(\mathrm{m K})}$.

Figure 10 shows the HRR with different VHC_HDR for the four exploitation modes. The following cognitions can be obtained:

Fig. 10

Sensitivity analysis of the VHC_HDR at different t_exp for the four exploitation modes. a HRR (t_exp = 1.0 year, q_in = q_best) with different VHC_HDR for the four exploitation modes. b ∆T_sense (q_in = q_best) at different t_exp for the various exploitation modes

In the early stage of exploitation, the working fluid in the horizontal segment continuously absorbs heat from the HDR. Therefore, as mentioned above, under the same conditions, the larger the VHC_HDR, the more heat the surrounding rock will release, and the more heat the working fluid in the horizontal segment can absorb. Thus, for [Model A], this condition increases the working fluid temperature (P₃ in Fig. 2e–f) before entering the TIP_H. In the same manner, for [Model B], when the working fluid in the ANN_V enters the section of insulation formation (z = 2900 m), the temperature also increases (P₅ in Fig. 2g–h). As a result, the HRR increases with the λ_HDR. Similarly, in the later stage of exploitation development, the larger the VHC_HDR is, the more thermal energy exists in the HDR, especially around the depth (H = 3000 m) where the horizontal pipe is located. Hence, the working fluid can be heated to a high temperature. Thus, for the four exploitation modes, in the whole exploitation process, the HRR is always positively correlated with the VHC_HRD.

In this section, we define the temperature variation (${\Delta T}_{\mathrm{sense}}$) as ${{T}_{\mathrm{out}}|}_{{\mathrm{VHC}}_{\mathrm{HDR}}=4.5\times {10}^{6} \mathrm{J}/({\mathrm{m }}^{3} \cdot \mathrm{K})}-{{T}_{\mathrm{out}}|}_{{\mathrm{VHC}}_{\mathrm{HDR}}=0.6\times {10}^{6} \mathrm{J}/({\mathrm{m }}^{3} \cdot \mathrm{K})}$. As shown in Fig. 10b, for the different exploitation modes, the all ∆T_sense increase with the t_exp because with the increase in the t_exp, the differences in the VHC_HRD lead to greater differences in the heat energy of HDR. As a result, the outlet temperature experiences a greater disturbance.

Comprehensive comparison of sensitivity analysis

To comprehensively compare the influence degree of various parameters on the HRR in this study, based on the principle of mathematical statistics, we adopted the coefficient of variation as the index of sensitivity evaluation (Abdi 2010; McAuliffe 2015):

$${C}_{\mathrm{v}}=\frac{s}{\left|\overline{y }\right|}$$

(23)

$$s=\sqrt{\frac{1}{N}\sum_{i=1}^{N}{\left({y}_{i}-\overline{y }\right)}^{2}}$$

(24)

where C_v is the coefficient of variation, s is the standard deviation of the sample data, $\left|\overline{y }\right|$ is the average of sample data, y_i represents a data sample, and N represents the number of sample data.

The value of C_v reflects the sensitivity of HRR to various influencing factors. The greater the C_v value, the greater the influence of this factor on the HRRs. Figure 11 summarizes the results of the sensitivity comparison of the studied parameters. For the four exploitation modes, although the sensitivity of each mode to various parameters varies, in the studied parameter ranges, λ_TIP and L_TIP have the greatest degree of sensitivity. Thus, the design of the TIP must be optimized. The effect of the T_in on the HRR decreases with the increase in T_b. Hence, for the area that meets the HDR standard (T_b > 150 °C), the effect of the T_in on the HRR can be ignored. In addition, the caprock with a high TC damages the geothermal exploitation performance. However, the correlations between the HRR and thermophysical properties of different formations are seriously affected by the IPS and t_exp.

Fig. 11

Coefficient of variation of the outlet temperature corresponding to various sensitivity parameters for the four exploitation modes

Conclusions

In this study, a fully coupled model was established to synthetically analyze the geothermal exploitation using a horizontal well under different IPS and working fluids. Dynamic optimization analysis was carried out from the space–time perspective. The following conclusions have been drawn:

• (1) Under the same conditions, [Model A] is the best IPS because of its high outlet temperature, stable thermal recovery, and low fluid injection volume. If [Model B] must be adopted as the exploitation scheme, the casing pipe in the upper part of the formation should be made of materials with a low TC. Moreover, for any IPS, when a fluid (e.g., SCCO) with low VHC and TC is used as the working fluid, its HRR will be notably high.

• (2) Type curve analysis shows that: (a) For [Model A], the working fluid temperature passes through four stages (two stages of temperature rise and two stages of temperature decline); (b) For [Model B], the working fluid temperature undergoes five stages (four stages of temperature rise and one stage of temperature decline).

• (3) When the T_b exceeds 146 °C, the T_in has almost no influence on the outlet temperature (with the increase in the T_in, the outlet temperature disturbance remains below 1%). Therefore, for the area that meets the HDR standard (T_b > 150 °C), the effect of the T_in on the outlet temperature can be ignored. In addition, each mode has an optimal q_best (a maximum outlet temperature exists in the relation curve between the outlet temperature and q_in), which varies depending on the IPS and the working fluid.

• (4) The TIP properties seriously affect the HRR: (a) The study of variation coefficient showed that among all the sensitivity parameters studied in this research, λ_TIP and L_TIP have the greatest influence on the HRR. Therefore, the materials of TIP must be optimized. (b) For [model A], when λ_TIP = 0, the HRR is the highest; otherwise, the HRR sharply declines. However, in [Model B], regardless of whether the TIP is completely adiabatic, the outlet temperature decreases gradually with λ_TIP. Thus, the TIP of completely adiabatic materials (λ_TIP = 0) is a great addition to [Model A]. (c) Given economic considerations, a long horizontal pipe is not optimal. The research showed that the L_TIP becomes shorter when [model A] obtains the optimal outlet temperature compared with [model B].

• (5) The influence of IPS and t_exp should be fully considered when analyzing the influence of different formations of thermophysical parameters on the outlet temperature (Table 1).

• (6) The model in this paper does not consider the influence of chemical reaction between the working medium and casing on geothermal development. Therefore, in further research, a fully coupled thermo-hydro-mechanical-chemical model should be constructed to study the dynamic evolution characteristics of thermal extraction during the geothermal development of the horizontal wells. In addition, considering the economic factors of geothermal exploitation will be the key direction of geothermal development numerical simulation optimization research.

Appendices

Appendix A

In this model, the finite difference method was used to discretize the transient temperature field governing equation. In addition, the mesh at the wellbore location was refined. In the area where the vertical or horizontal well exists, the spacing was set as 0.01 m to describe the temperature precisely. However, the grid with equal spacing of 2.0 m was used for the formations. Figure 12 shows the details of mesh generation.

Fig. 12

Schematic of the discrete grid of the model

Appendix B

Tables 3 and 4 list the parameters of the base model using the horizontal well technology for the geothermal exploitation of HDR. Figure 13 shows the initial T_b of the model. When the working fluid is water or SCCO, the reinjection temperature is 20 °C and 32 °C, respectively. In the sensitivity analysis, the volume reinjection rate of each model was set to the optimal values): 339.39 ([Model A, water]), 135.72 ([Model A, SCCO]), 542.87 ([Model B water]), and 271.43 m³/day ([Model B, SCCO]).

Table 3 Geometry parameters of the wellbore

Table 4 Thermo-physical properties of various media

Fig. 13

Initial T_b of the model