Introduction

The application of fossil fuels has brought about the Industrial Revolution and has provided humanity with tremendous developmental impetus over the past few centuries. In the current global environment, energy demand is steadily on the rise. Energy diversification not only offers a broader range of energy choices to cater to the specific needs of different regions and countries but also enhances the stability of energy supply. By reducing reliance on a single energy source, it mitigates the risk of supply disruptions. Geothermal energy, as a stable, widely distributed, and pollution-free clean energy source, has garnered attention from many nations. Hot dry rock refers to high-temperature rock formations buried deep underground with low porosity and permeability (Li et al. 2020). The heat stored within them requires artificial modification through the creation of enhanced geothermal systems (EGS) for extraction (Baujard et al. 2021). The transformation of the reservoir is an irreversible process, necessitating the simulation of the overall production dynamics of the reservoir. This allows for the exploration of the impact of key production parameters on the reservoir’s dynamic behavior, thereby providing a basis for the formulation of production schemes.

EGS research has been conducted for more than 30 years, but in the past it was limited to countries such as the US, the UK, Japan, France, Germany, Switzerland, and Australia (Genter et al. 2003; Zhu et al. 2019; Leon and Kumar 2008; Karastathis et al. 2011). The abundance of dry heat rock resources has attracted the attention of many scholars. Modeling and experimental studies on EGS have developed rapidly. The issue of the EGS presents a classic example of a multi-domain, multiphysics coupling challenge. Within the framework of the EGS consisting of the injection wells, production wells, porous reservoirs, and surrounding impermeable rock formations, different domains adhere to distinct governing equations. Nonetheless, physically valid connectivity conditions must be met at the interfaces of these domains. Concurrently, in the simulation of EGS, the temperature field and permeation field necessitate coupled solutions, occasionally entailing joint solutions involving solid deformation and component transport. Gringarten et al. (1975) presented a theory for heat extraction from a fractured hot dry rock based on the assumption of the infinite and uniform apertures of parallel vertical fractures. Cold water enters through the bottom of each fracture, and the heat conductivity transfers from the hot rock to the fractured rock via the water flow. Their solution could be used in determining the water temperature and the ratio of the amount of heat extracted by means of the water flow within the fractures. Ashena et al. (2023) investigate the effect of a change in the thermal conductivity on the net heat energy, the net power, and the coefficient of performance (COP), at different circulation rates. Zhou and Hou (2013) proposed a numerical model describing hydraulic fracturing process for simulating the dynamic fracture expansion process and the fracture fluid momentum and mass exchange between the fracture fluid and t pore fluid.

In dry-thermal rock reservoirs, the interactions between temperature, pressure, and percolating bodies are very complex. Bower and Zyvoloski (1997) developed a numerical model of heat-water coupling in fractured rock. Shaik et al. (2011) determined the effective permeability tensor in geothermal reservoir simulations by investigating the fluid flow process, and they followed with a two-dimensional model to simulate the geothermal reservoirs. Nowak et al. (2011) proposed a multi-physical field porous medium THM coupled microscopic continuous medium method based on mass conservation, momentum conservation, and energy conservation equations with two field system engineering experiments and established a three-dimensional model while implementing parallel computation for the model. Jiang et al. (2013, 2014), as well as Chen and Jiang (2015), conducted numerical simulations of the permeation and heat transfer processes in EGS based on the Brinkman equation. They considered the impact of permeability and thermal imbalances between the surrounding rock and pore fluids on the lifespan of EGS. Subsequently, Cao et al. (2016) explored various geothermal well layouts and the relationship between injected water flow rate, heat extraction efficiency, EGS lifespan under the influence of structural stresses. Saeid et al. (2015) performed coupled simulations of low-temperature geothermal systems in one and two dimensions, considering different regimes of laminar and turbulent thermal conductivity. By comparing various parameters, they derived a quantitative relationship indicating that the lifespan of a geothermal field depends on porosity, flow rate, well spacing, reservoir temperature, and injected water temperature. However, the systematic study of how parameters related to injection wells and fracture width impact EGS heat production remains unexplored. Furthermore, the alterations in fluid properties occurring due to temperature changes within the geothermal rock reservoir have not been thoroughly considered. Therefore, it is imperative to comprehend the influence of parameters like injection well characteristics, fracture width, and fluid properties on the reservoir and production aspects. This understanding is crucial for optimizing the heat production capacity of EGS.

The output response of an EGS is jointly governed by the stress field, temperature field, and fracture permeation field within the matrix rock blocks. In this study, we integrated the coupling relationships between the fluid medium in the EGS, the permeability of fractures, and the reservoir rock. This integration led to the establishment of mathematical models for the stress field, temperature field, and permeation field within the EGS. Initially, we analyzed the dynamic characteristics of the temperature field during prolonged exploitation of the EGS. Subsequently, we conducted a comprehensive study and discussion on the impact of reservoir properties such as injection well temperature, injection well pressure, primary fracture width, and secondary fracture density, as well as other production parameters. The results of these efforts provide a deeper understanding of the dynamic nature of the reservoir temperature field and how the properties of the reservoir and production parameters collectively influence heat production in the EGS.

Coupled control equations for dry heat rock mining

Basic assumptions

An EGS is complex and can be simplified into a model composed of matrix rock mass and fracture system. The phase change of water caused by temperature is ignored, and the reservoir is a single-phase water-saturated reservoir. In the context of heat transfer in the model, only the heat transfer between the matrix rock and the fluid, as well as the variation in the temperature field within the matrix rock, has been considered. Temperature changes resulting from other factors have not been taken into account. In this study, the water flow in the fracture system is assumed to be laminar, in accordance with Darcy’s law, and the deformation of both the matrix rock mass and the fracture system is small (Pruess and Narasimhan 1985).

Control equations of the coupled model

For the EGS, the equations for the displacement and stress fields of the rock can be expressed as follows (Sun et al. 2016):

$$ \mu u_{i,jj} + \left( {\mu + \lambda } \right)u_{j,ji} - \alpha_{B} p_{,j} - \beta_{T} T_{s,i} + F_{i} = 0 $$
(1)
$$ \lambda = \frac{E\nu }{{\left( {1 + \nu } \right)\left( {1 - 2\nu } \right)}},{ }\mu = E/2\left( {1 + \nu } \right) $$
(2)
$$ \beta_{T} = \alpha_{T} E/\left( {1 - 2\nu } \right) $$
(3)

where u is the displacement [m]; \(\lambda\) and \(\nu \) are the Lame constants [Pa], respectively; \(\alpha_{T}\) is the thermal expansion coefficient of the bedrock [K−1]; \(\alpha_{B}\) is the Biot coupling coefficient; \(\sigma_{ij}\) is the tensor of the stress in the bedrock [Pa]; \(T_{s}\) is the temperature field of the bedrock [K]; and \(F_{i}\) is the physical force acting in the bedrock [N].

Considering the deformation of cracks, the controlling equation for the deformation generated by the interaction between the cracks and the rock mass can be expressed as follows:

$$ u_{n} = \frac{{\sigma_{n}{\prime} }}{{K_{n} }}, u_{s} = \sigma_{s}{\prime} /K_{s} $$
(4)
$$ \sigma_{n}{\prime} = \sigma_{n} - p,\,\,\sigma_{s}{\prime} = \sigma_{s} , $$
(5)

where \( u_{n}\) is the normal displacement of the crack [m]; \(u_{s}\) is the tangential displacement of the crack [m]; \(K_{n}\) and \(K_{s}\) are the normal and tangential stiffness of the crack [Pa/m], respectively; and\(\sigma_{n}{\prime}\) and \(\sigma_{s}{\prime}\) are the normal and tangential effective stresses at the boundary of the crack [Pa/m].

Considering the porous seepage flow of matrix rocks with low permeability, the governing equation of seepage flow can be expressed as follows (Tenma et al. 2008):

$$ S\frac{\partial p}{{\partial t}} + \nabla \cdot u = - \frac{\partial e}{{\partial t}} + Q $$
(6)
$$ u = - \frac{\kappa }{\eta }\left( {\nabla p + \rho g\nabla z} \right) $$
(7)

where S is the water storage coefficient of the matrix [Pa−1]; t is time, s; \(\kappa\) is the permeability of the matrix [m2]; e is the volume strain of the matrix mass; \(\eta\) is the dynamic viscosity of the seepage medium [Pa·s]; and Q is the source-sink term of the seepage [kg/(m2·s)].

Consider the temperature field of the matrix of the EGS, the control equation can be expressed as follows:

$$ c_{s} \rho_{s} \frac{{\partial T_{s} }}{\partial t} = \lambda_{s} \nabla^{2} T_{s} + W $$
(8)

where \(c_{s}\) is the specific heat capacity of the matrix rock mass [J/(kg·K)]; \(\rho_{s}\) is the density of the matrix rock mass [kg/m3]; W is the external heat source term [W]; and \(\lambda_{s}\) is the heat transfer coefficient of the matrix rock mass [W/(m·K)].

Considering the fluid seepage within the fracture, the controlling equation can be expressed as follows (Ma et al. 2023):

$$ d_{f} s_{f} \frac{\partial p}{{\partial t}} + \nabla_{\tau } \cdot\left( { - d_{f} \frac{{\kappa_{f} }}{\eta }\nabla p} \right) = - d_{f} \frac{{\partial e_{f} }}{\partial t} + Q_{f} $$
(9)

where \(\kappa_{f}\) is the permeability of the fracture in the enhanced geothermal system [m2]; \(Q_{f}\) is the flow exchange between the fracture face and the matrix rock [kg/(m2·s)]; \(d_{f}\) is the width of the fracture [m]; \(s_{f}\) is the water storage coefficient of the fracture [Pa−1]; and n is the normal direction of the fracture boundary.

Considering the heat field of the fluid itself in the fracture, the governing equation can be expressed as follows (Sun et al. 2017):

$$ d_{f} \rho_{f} c_{f} \frac{{\partial T_{f} }}{\partial t} + d_{f} \rho_{f} c_{f} u_{f} \nabla_{\tau } T_{f} = d_{f} \nabla_{\tau } \cdot\left( {\lambda_{f} \nabla_{\tau } T_{f} } \right) + W_{f} $$
(10)

where \(c_{f}\) is the specific heat capacity of the fluid medium [J/(kg·K)]; \(u_{f}\) is the flow velocity of the fluid flowing along the fracture [m/s]; \(\rho_{f}\) is the density of the fluid flowing in the fracture [kg/m3]; \(T_{f}\) is the temperature of the fluid in the fracture [K]; \(\lambda_{f}\) is the coefficient of heat conduction of the fluid in the fracture [W/(m·K)]; and \(W_{f}\) is the heat conducted at the fracture boundary through the matrix rock boundary to the fluid flowing in the fracture [W].

For the \(W_{f}\) term on the right side of the equation, it is assumed that the heat exchange between the fluid in the fracture and the matrix rock boundary obeys Newton’s law of cooling, and the heat exchange between the fluid medium in the fracture and the matrix rock boundary can be calculated by the convective heat transfer coefficient. When the temperature of the fluid medium in the fracture differs from the temperature between the matrix rock boundary, the heat absorbed by the fluid medium in the fracture from the matrix rock on a unit area can be expressed as follows (Li et al. 2015):

$$ W_{f} = h\left( {T_{s} - T_{f} } \right) $$
(11)

where h is the heat transfer coefficient between media [W/(m2·K)].

Coupling relationship between physical fields

Coupling between seepage and stress at the boundary surface of fracture and matrix rocks

Geothermal exploitation system is an organic system, and there are joint effects of temperature field, seepage field, and stress field among rock mass and cracks and the action mechanism among temperature field, seepage field, and stress field (Zhou et al. 1998; Li et al. 2023). In an EGS, the heat exchange at the boundaries of fractures affects the temperature distribution within the reservoir rock. The coupling relationship between the temperature field and the stress field can alter the properties of the fracture network. As the primary flow pathways in the fluid medium, changes in the fracture network can impact its permeability, which is crucial for the coupled behavior between stress and flow fields in enhanced geothermal systems. To study the coupling characteristics of this system, Louis (1974) conducted experiments investigating the relationship between flow and stress in a rock mass containing a single fracture. He proposed an empirical exponential formula to explain the behavior of this coupled system:

$$ k_{f} = k_{0} {\text{exp}}\left( { - \alpha \sigma_{n}{\prime} } \right) $$
(12)

where \(k_{0}\) is the permeability coefficient of the fracture at \(\sigma_{n}{\prime} = 0\) [m2]; \(k_{f}\) is the permeability coefficient of the fracture in the matrix rock [m2]; \(\alpha\) is the influence coefficient [1], and the magnitude of this coefficient is related to the state of existence of the fracture in the matrix rock.

Changes in fluid media properties

In EGS, the properties of the fluid also undergo changes as the fluid temperature varies within the fractures. The variation in fluid temperature has a significant impact on the viscosity of the fluid. The relationships between water viscosity, water density, specific heat capacity, and thermal conductivity of water with respect to temperature are expressed by the following formulas (Qu et al. 2017; Shen et al. 2022):

$$ \mu_{w} \left( T \right) = \left\{ {\begin{array}{*{20}c} {1.38 - 0.028T + 1.36 \times 10^{ - 4} T^{2} - 4.64 \times 10^{ - 7} T^{3} + 8.90 \times 10^{ - 10} T^{4} } \\ { - 9.08 \times 10^{ - 13} T^{5} T^{3} + 3.84 \times 10^{ - 16} T^{6} 273 \le T < 413} \\ {0.004 - 2.11 \times 10^{ - 5} T + 3.86 \times 10^{ - 8} T^{2} - 2.40 \times 10^{ - 11} T^{3} 413 \le T < 533} \\ \end{array} } \right. $$
(13)
$$ \rho_{w} \left( T \right) = 838.47 + 1.40T - 0.003T^{2} + 3.72 \times 10^{ - 7} T^{3} $$
(14)
$$ C_{pw} \left( T \right) = 12010.15 - 80.41T + 0.31T^{2} - 5.38 \times 10^{ - 4} T^{3} + 3.62 \times 10^{ - 7} T^{4} $$
(15)
$$ \lambda_{w} \left( T \right) = - 0.869 + 0.009T - 1.58 \times 10^{ - 5} T^{2} + 7.98 \times 10^{ - 9} T^{3} $$
(16)

Numerical model solving and validation

Computational models

Figure 1 illustrates the EGS wellbore and fracture network model. In the diagram, "I" is the injection well, "P" is the production well, and the two horizontal fractures connected with the injection well and the production well are the main fractures of the reservoir. Secondary cracks are a group of vertical cracks that connect two horizontal main cracks. The physical parameters of the dry rock model are presented in Table 1. (Xiao et al. 2019).

Fig. 1
figure 1

Geothermal system reservoir model

Full size image
Table 1 The calculation parameters of the model
Full size table

Model solving methods

Based on the aforementioned coupled model, along with the initial and boundary conditions of the EGS rock mass, it is possible to solve for the temperature field, flow field, and stress field of the fractures and matrix rock mass. In this study, the numerical solution of the coupled model for the EGS is achieved using COMSOL Multiphysics. The governing equations of the various physical fields are combined and solved together, allowing for the computation of relevant numerical values for each field. For simulating fractures in the enhanced geothermal system, it is common to employ thin-layer elements (Leung and Zimmerman 2012), which have a certain thickness. However, in the case of EGS, the calculation domain is large and the fracture network within it is extensive and complex. Each thin-layer fracture would generate a large number of grids, leading to an excessively large grid count that hinders the computation of final results. Another approach is to transform the thin-layer fracture elements typically used in simulations into one-dimensional line elements without thickness for computation (Chen et al. 2014) The difference in results is not significant under the same parameters (Qu et al. 2019; Li et al. 2015), and the use of line elements simplifies the grid when performing finite element simulation and computation. The model shown in Fig. 1 is simplified to a two-dimensional model as depicted in Fig. 2, for simulation and analysis. For the two-dimensional model of the EGS under the condition of one-dimensional line elements, the stress field, temperature field, and flow field of the system are imported into COMSOL Multiphysics for solution.

Fig. 2
figure 2

Two-dimensional computational model

Full size image

Model coupling verification

This study employs COMSOL Multiphysics to solve the THM coupling model. In order to validate the effectiveness and computational accuracy of the model, two case studies are chosen for validation in this work.

Ghassemi and Zhang (2004) investigated the variations in temperature, pressure, and stress near a circular wellbore after a sudden temperature drop. Utilizing identical physical parameters, a comparative analysis between their findings and the results obtained from modeling and computation in COMSOL Multiphysics is presented in Fig. 3, for reference. Lauwerier (1955) gave an analytical solution for the variation of fracture water temperature with time in the study of heat transfer in oil reservoirs. After a certain time t, the temperature of the water in the fracture is distributed along the x-axis as follows:

$$ T\left( {x,t} \right) = T_{0} + \left( {T_{in} - T_{0} } \right){\text{erfc}}\left( {\frac{{(\lambda_{s} x)/\left( {\rho_{w} C_{w} d_{f} } \right)}}{{2\sqrt {\lambda_{s} /(\rho_{s} C_{s} )u_{f} \left( {u_{f} t - x} \right)} }}} \right)U\left( {t - x/u_{f} } \right) $$
(17)

where erfc is the residual error function; U is the unit step function; and \(u_{f}\) is the fluid flow rate in the fracture, m/s.

Fig. 3
figure 3

Distribution and variation of temperature and pore pressure

Full size image

The THM coupling model of a single fracture was solved using COMSOL Multiphysics, and the results were compared with the analytical solution, as depicted in Fig. 4. The numerical solution closely aligns with the analytical solution. By contrasting the numerical simulation with the analytical solution, it becomes evident that the mathematical model and solution approach presented in this paper are in concordance with theoretical analytical expectations. This underscores the feasibility of the mathematical model and computational methodology proposed in this study.

Fig. 4
figure 4

Variation of fluid temperature in fracture

Full size image

Results and discussion

Evolutionary characteristics of the temperature field

Currently, the geothermal recovery rate γ of a geothermal reservoir is commonly used to evaluate the geothermal recovery performance of an enhanced geothermal system, and γ is calculated as follows:

$$ \gamma = \frac{{(T_{{{\text{avg}}\left( {{\text{init}}} \right)}} - T_{{{\text{avg}}\left( t \right)}} )C_{s} \rho_{s} V_{s} }}{{C_{s} \rho_{s} V_{s} }} \times 100{\text{\% }} $$
(18)

where \(T_{{{\text{avg}}\left( {{\text{init}}} \right)}}\) is the initial average temperature of the rock [K]; \(T_{{{\text{avg}}\left( t \right)}}\) is the average temperature of the rock at moment t [K]; and \(V_{s}\) is the volume of the rock [m3].

The variation of the temperature field of dry thermal rocks under different time conditions obtained by COMSOL Multiphysics is shown in Fig. 5. From the result of Fig. 5, it can be seen that the temperature field of the rock mass near the production wells maintains a stable value for a long time during the first years of the mining process. After a long time of mining, the low-temperature region of the matrix rock expands, the temperature increase of the fluid during the flow of the fracture network slows down, and the low-temperature region gradually spreads along the fluid flow side. After about 50 years of mining, the low-temperature region spreads near the production wells, thermal breakthrough occurs in the production wells, the temperature of the production wells decreases rapidly, and the thermal power of the production wells decreases sharply.

Fig. 5
figure 5

Temperature distribution in reservoirs at different times

Full size image

Effect of the injection well temperature

To study the effect of the inlet temperature on the production well output, the temperature of the production well and the variation of the thermal power of the production well were obtained by taking increments of 15 °C at a time at a pressure of 20 MPa (2900 psi) and an initial inlet temperature of 20 °C, and the variation trends are shown in Figs. 6 and 7. From the results in Figs. 6 and 7, it can be seen that the variation trends at different temperatures do not vary much over the 30-year stable mining period. The effect of temperature on the variation of thermal power varies roughly linearly, which is consistent with the analysis under Eq. (17) for the variation of fracture water temperature. High-temperature fluids from producing wells can be injected directly into injection wells without cooling treatment after power generation or other reuse.

Fig. 6
figure 6

Temperature change of production well in the cases with different injection well temperature

Full size image
Fig. 7
figure 7

Variation of thermal power of production well in the cases with different injection well temperature

Full size image

Effect of the injection well pressure

The injection well pressure significantly affects the fluid flow rate in the fracture network, and Fan et al. (2018) established an analytical model considering the heat-fluid–solid coupling dry-thermal rock reservoir based on the discrete fracture network model to analyze the effect of injection volume on the temperature of the discrete fracture network. With the increase of injection volume, the temperature at the fracture exit end changes rapidly at the early stage and slows down at the later stage. Meanwhile, the thermal breakthrough at the exit end of the production well becomes earlier and earlier with the increase of injection volume. In order to study the effect of injection well pressure on production well output, the variation of temperature and thermal power of the outlet well were obtained by increasing the inlet pressure by 4 MPa (5800 psi) for each calculation starting from 10 MPa (1450 psi) under the condition that the injection well temperature was 60 °C and the production well pressure was 6 MPa (8700 psi). The variation of temperature and thermal heat extraction rate at different pressures are shown in Figs. 8 and 9. From the results in Figs. 8 and 9, it can be seen that as the pressure increases, the temperature decreases at an accelerated rate after about 15 years of stable extraction, and the higher the pressure, the faster the rate of decrease. The effect of increasing pressure on the production heat power of the producing wells is mainly concentrated in the first 25 years of exploitation, and increasing the pressure affects the flow rate of the fluid in the fracture and the change of the heat field. Compared to the relatively stable heat power variation at 22 MPa (3191 psi), further increasing the pressure leads to an accelerated decline in heat power after 25 years of exploitation. Raising the injection well pressure results in a sharp initial increase in heat power from the production well, yet boosting the pressure at the injection well predominantly impacts the initial phase of heat production. Moreover, increasing the pressure causes the injected well fluid to permeate into more distant zones, concurrently expediting the thermal breakthrough process.

Fig. 8
figure 8

Temperature change of production well in the cases with different injection well pressure

Full size image
Fig. 9
figure 9

Variation of thermal power of production well in the cases with different injection well pressure

Full size image

Effect of the width of the main crack

The main fracture is connected to the injection well and the outlet well. As the main circulation channel for fluid media, the width of the main fracture has a large impact on the temperature field distribution of the entire enhanced geothermal system. A smaller fracture width will cause the fluid medium percolation process in the main fracture to slow down, which will inject insufficient thermal energy extraction from the matrix rock at the end of the well. A larger fracture width will make the fluid flow faster and flow too quickly from the injection well to the end of the main fracture, the fluid medium does not fully exchange heat with the matrix rock body near the injection well area, the temperature of the matrix rock body near the production well drops faster, and the thermal breakthrough of the production well is advanced (Hu et al 2014). In order to study the effect of the width of the main fracture, the width of the secondary fracture was taken to be 0.0001 m, and the change in the width of the main fracture was calculated by taking increments of 0.0002 m each time starting from 0.0001 m, and the temperature field at each width and the change in the thermal power of the production well were obtained as shown in Figs. 10 and 11. From the results in Figs. 10 and 11, it can be seen that the variation of the width of the main fracture has a similar effect on the variation of the production thermal power as the variation of the pressure on the thermal power. As the width of the fracture increases, the flow rate of fluid medium along the main fracture becomes faster and the thermal power of the production well shows an upward trend. After the stable exploitation period of 20–25 years, the curve of the production thermal power is staggered compared to the fracture with smaller width due to thermal breakthrough, at which time the temperature of the output well of the fracture with larger width decreases faster and the production thermal power is less than that of the fracture with smaller width.

Fig. 10
figure 10

Temperature change of production well

Full size image
Fig. 11
figure 11

Variation of thermal power of production well

Full size image

Effect of the paracrack density

In the process of dry-thermal rock extraction, two main fractures established by hand are then artificially hydraulically fractured to form a secondary fracture channel connecting the two main fractures, which together with the secondary fractures form the fracture network of the enhanced geothermal system. The degree of controlled fracturing affects the density of the secondary fractures, and the change in the density of the secondary fractures affects the connectivity between the injection wells and the output wells on the one hand, and on the other hand, the increase in the density of the secondary fractures leads to a more uniform distribution of the secondary fractures in the matrix rock, which results in a more adequate heat exchange between the fluid and the matrix rock and reduces the rate of thermal breakthrough in the production wells to a certain extent. The variation of the temperature of the water outlet well, the thermal power of the production well, and the geothermal extraction rate in the dry heat rock area obtained by varying the number of secondary fractures in the extraction area with a primary fracture width of 0.0005 m and a secondary fracture width of 0.0001 m are shown in Figs. 12 and 13. From the results in Figs. 12 and 13, it can be seen that there is not much fluctuation in the mining power when the analysis of fracture density is performed at a pressure of 14 MPa (2030 psi). The density of fractures primarily impacts the connectivity between the injection and production wells within the geothermal rock, resulting in a larger contact area between the fluid medium and the geothermal reservoir. Starting from six secondary fractures, further increasing the density of secondary fractures diminishes its effect on the reservoir’s heat power. In the context of geothermal rock reservoir, there exists a marginal impact with increasing fracture density, and a higher number of fractures lead to greater hydraulic fracturing difficulty. When devising extraction strategies, it is essential to consider the combined influence of hydraulic fracturing complexity and fracture density, aiming for an optimal balance.

Fig. 12
figure 12

Temperature change of production well

Full size image
Fig. 13
figure 13

Variation of thermal power of production well

Full size image

Summary and conclusions

In this study, a mathematical model for the stress field, temperature field, and permeability field of an enhanced geothermal system (EGS) was established based on the coupling relationship between fluid medium, fracture permeability, and reservoir rocks. Full coupling simulations were conducted using COMSOL Multiphysics to investigate the impact of injection well temperature, injection well pressure, main fracture width, and secondary fracture density on the distribution of reservoir temperature field and the thermal–electric output of production wells. The key conclusions drawn from these studies are as follows:

  1. (1)

    In an enhanced geothermal system, as the production process progresses, the low-temperature area starts to expand from the injection well and causes thermal breakthrough, and the temperature of the production well drops significantly. The rate of heat breakthrough in producing well is related to the distance between injection well and producing well.

  2. (2)

    The impact of temperature decrease on the temperature and thermal power output of production wells generally follows a linear trend. As for the injection well pressure, its effect on the output temperature and thermal power of production wells is substantial. With an increase in injection well pressure, the overall thermal breakthrough speed of the system experiences significant changes. Around the threshold of 22 MPa, higher injection well pressures lead to a notable acceleration in the thermal breakthrough speed.

  3. (3)

    The influence of fracture width on this system and pressure appears to follow similar patterns. Excessive fracture width accelerates thermal breakthrough. As for the variation in output from the geothermal rock reservoir under different secondary fracture densities, increasing fracture density has a substantial impact on output wells when the fracture density itself is relatively low. However, as the number of fractures increases, the impact of increasing fracture density on the thermal power output of production wells stabilizes.