#### Abstract

The surface morphology of rough fractures significantly affects the fluid flow and heat transfer characteristics in the fractures. A thermal-flow coupling model with specific geometric fractures was established to investigate the influence of surface morphology on the heat transfer characteristics of a single fracture. The effect of temperature on the physical properties of rocks and fluids was included in the study to reflect the actual situation more realistically. The research results show that the temperature of the fluid in the rough fracture is nonlinearly distributed along the flow direction and the higher the flow velocity, the higher the heat transfer efficiency. The fracture surface morphology has a significant impact on the heat transfer characteristics, and the surface fluctuation will greatly affect the flow velocity, causing the fluid temperature to change abruptly at the fracture surface. Under the same flow rate, with the increase of the fluctuation degree of the fracture surface and the fluctuation frequency, the larger the heat exchange area of the fracture surface, the stronger the heat exchange performance. The heat transfer efficiency of the fracture is directly related to the heat transfer area of the fracture, so even with the same permeability, the heat transfer performance of fractures with different surface topography is different.

#### 1. Introduction

Continuous exploration and development of deep earth extended resources has become a common phenomenon [1–3]. The development and utilization of hot dry rock geothermal resources with huge reserves, clean and unaffected by climate within 3-10 km below the surface, have attracted much attention at present [4]. Enhanced geothermal system (EGS) is used to form open and connected fractures in deep reservoir by hydraulic fracturing. Through connected fractures, the working fluid can be heated cyclically by contacting rock and returning to the surface to extract heat from dry-hot reservoir, which can effectively relieve energy demand [5]. Therefore, the study of fluid flow and heat transfer in fractures of rock reservoirs is crucial for the effective development of geothermal resources [6]. However, in underground engineering, the fractures formed by rock mass rupture usually have very complex surface morphology and void structure [7, 8], directly affecting the flow and heat transfer characteristics of the fluid [9].

In previous researches on convection heat transfer in rock fractures, the shape of the fracture is dominated by a single fracture; the fracture surface is divided into two types: smooth and rough, most of which are smooth [10]. Recently, studies have shown that evaluating the heat transfer efficiency of fracture flow without considering the change of fracture surface topography can make a big difference [11], Compared with smooth fractures, the roughness of rock fractures improves the overall heat transfer intensity to a certain extent [12]. However, for different surface topographies, the heat transfer performance in rough fractures may be lower than that in smooth fractures due to the dominant channeling effect and curved flow path [13, 14]. Both the pore size and roughness distribution in the fracture will affect the distribution of the flow field and the thermal breakthrough curve in the fracture to a certain extent [15]. Research shows that the rougher the crack, the stronger the convection heat exchange capacity [16]. Although the flow heat transfer process in rough fractures has been studied, there is still only a preliminary qualitative conclusion on the role of the rough topography of the fractures in the heat transfer process.

Numerous studies have attempted to quantify the roughness of the fracture surface by using JRC, fractal dimension, roughness, etc. [17, 18], to quantitatively explore the effect of surface topography on flow and migration. For example, heat transfer studies using 3D printed fractures with surface planar of different JRC have shown that heat transfer efficiency is positively related to volume flow and roughness [19]. Or use the fractal dimension (*D*) and planar waviness (*Ra*) to characterize the surface roughness of the fracture, and obtain that the heat transfer efficiency of the fracture is directly related to the surface roughness of the fracture [20]. There is also a quantitative description of the morphological characteristics of the rough surface of the fracture through the root mean square of the surface slope and the planar length. It is found that the temperature of the fracture surface will fluctuate violently with the change of the topography, etc. [10]. It is worth noting that whether JRC or fractal dimension is used to quantify the topographic characteristics of the whole fracture area, it cannot accurately reflect the surface structure characteristics of the local fracture area. Therefore, the influence of the complex surface morphology of the fracture on the flow and heat transfer characteristics remains to be studied.

Meanwhile, due to the technical difficulties of boundary condition control and monitoring in the fracture, it is difficult to study the fluid flow and heat transfer characteristics in the fracture in field and indoor experiments. Numerical simulation can clearly identify the fluid flow and heat transfer process in fractures with complex surfaces [21, 22]. It has been widely used in flow heat transfer analysis of rough fractures [23, 24].

Therefore, in order to further clarify the relationship between rough fracture surface morphology and convective heat transfer and flow, in this study, a variety of fracture flow heat transfer models with different void structures are established to analyze the influence of surface morphology on the flow and heat transfer process. Through numerical simulation results, the influence of surface fluctuation and surface area on flow and heat transfer characteristics of fractures were compared and analyzed. The main purposes of this study are as follows: (1) to promote the understanding of heat transfer in inhomogeneous pore fractures by clarifying the heat transfer characteristics of different fracture geometries and (2) the method to determine the heat transfer characteristics under the influence of section morphology and flow rate is also introduced.

#### 2. Theory Background

Consider that in the fracture of geothermal reservoir, the process of fluid flowing through the fracture for heat transfer is in a long-term stable operation cycle, and in the experiment, the flow and heat transfer behavior of the fracture is usually stable after a certain period of time [25].

In this study, we consider a process in which the hydrothermal coupling between water and rock in the fracture is in a stable state. The basic assumptions are as follows: (1)Fluid flow and heat transfer are steady state(2)The rock sample is considered to be impermeable(3)The water flow is laminar(4)Viscous and incompressible fluid(5)Heat transfer by radiation is not considered(6)There is no dissolution or precipitation between water and rocks(7)No additional heat source and heat loss

A single-phase fluid flow in fractures is governed by the mass continuity equation (Equation (1)) and the Navier-Stokes equation (Equation (2)) [21], as given by
where (kg/m^{3}) is the fluid density, (m/s) is the velocity, (Pa•s) is the fluid dynamic viscosity, and (Pa) is the reduced pressure.

In the process of fluid flowing through the fractures of geothermal reservoirs, the heat transfer is mainly determined by the convection and diffusion between the heat transfer fluids and the heat conduction in the rock. According to Fourier’s law [21], the energy conservation equation of rock in the steady-state is expressed as where (°C) and [W/(m•°C)] are the rock temperature and the heat conduction coefficient of the rock.

Heat transport in the fluid is predominantly governed by convection and conduction, and the energy conservation equation of fluid in the fracture in the steady state is expressed as where is the specific heat capacity of the fluid at constant pressure and (°C) and [W/(m•°C)] are the fluid temperature and heat conduction coefficients of the fluid.

#### 3. Methods

##### 3.1. Fracture Models

In the past, the research on the flow and heat transfer characteristics of fracture fluid is mostly based on the single fracture cylinder model [10, 26]1. In this paper, the physical model of flow and heat transfer is a two-dimensional plane model with length (L) of 100 mm and width (2R) of 51 mm, which is formed from the plane intercepted from a cylindrical sample with a single fracture. The model is composed of the lower impermeable rock area and the middle fracture (as shown in Figure 1).

Fractures with different void structures are designed, and the influence of fracture surface morphology on fluid flow and heat transfer characteristics is analyzed (see Figure 2). The fractured surfaces are generated by periodic curves. For rough fractures, we design three different amplitudes (mm) and wavelength (mm) to characterize the roughness (see Figure 2). Therefore, in this study, 28 fractures with different morphological characteristics were obtained. The pore size of fractures in EGS reservoirs generally ranges from micrometers to centimeters. In laboratory experiments, the pore size of real rock sample fractures is also between 0 and 2 mm [27, 28], so we fixed the mechanical pore size bs of all fractures to 1 mm.

Once the geometric structure of the model is completed, the mesh can be generated directly, and the triangular mesh is used to divide the fracture model. In order to ensure the correctness of the simulation results, we are working to verify the independence of the grid and ensure that the number of grids does not affect the final results. The meshing of fractures is shown in Figure 3.

##### 3.2. Simulation Setup

In order to more accurately understand the fluid flow and heat transfer characteristics in the fracture, the effect of temperature on the physical properties of the fluid is studied. For the effect of temperature on the physical properties of granite, a general empirical relation for thermal conductivity as a function of temperature is written as follows [29]: where the is thermal conductivity of rock and (°C) is temperature of rock.

The empirical relation for and is represented below [30]: where is density of rock and is the specific heat capacity of rock.

Water is the typically used as a heat exchange medium in geothermal mining. The existing literature gives the physical properties of water under the influence of different temperatures when it remains liquid. The relationship between the dynamic viscosity of water and temperature is an important factor controlling fluid flow. The dynamic viscosity of water can be calculated using Equations (7)-(8) [31].

When temperature at 273-413 K: When temperature at 413-553 K:

The density of a fluid at 100 k to 300 k is

Specific heat capacity of water between 273.15 K and 553 K is written by

Thermal conductivity of water at 273.15 K to 553 K is

Considering the extremely low permeability of matrix, we ignore the fluid flow in matrix and assumed that the viscosity flow. The inner wall of the fracture is considered to be an impermeable and nonslip boundary. For the initial and boundary conditions of the flow field, the volume flow rate was applied at the inflow boundary with a fixed pressure pout for the outflow boundary. To determine flow conditions, the Reynolds number is where (kg/s) is the mass flow rate, (m) is the width of fracture, and (m) is aperture of fracture.

In this simulation test, the calculated *Re* for all fractures is within the range of 0.1-100. It means that mass flow rate is between 0.001 and 0.1 kg/s to analyze the influence of flow rate on heat transfer characteristics. The relationship between Reynolds number and mass flow rate is as follows (Figure 4).

For temperature field, the temperatures of the upper and lower boundaries of the rock are set to a constant temperature . In the previous study, the heat conduction along the axial direction of the cylinder is usually neglected [32], so both the left and right boundaries of the rock are set as thermal adiabatic boundaries. The temperature of the fluid at the fracture inlet is set to , and the outlet of the fracture is set as adiabatic boundaries. The parameters used in the numerical simulation are shown in Table 1.

#### 4. Results and Discussion

##### 4.1. Numerical Results

Fluid flow and heat transfer results in a single fracture were obtained using numerical calculations. In order to evaluate the effect of fracture morphology on flow and heat transfer characteristics, the cubic law was used to calculate the equivalent hydraulic aperture of the rough fractures by
where (mm) is the mechanical fracture aperture, is the fracture length, and *p* is the pressure gradient.

It should be noted that the calculation results of hydraulic aperture obtained by us are obtained under the assumption that the temperature is 25 °C. Under the influence of wall roughness, heat transfer and solute migration behaviors are closely related to flow behavior [9, 33]. Therefore, flow characteristics can reflect heat transfer behavior to a certain extent. This paper quantifies the roughness of fractures by using equivalent hydraulic aperture.

For heat transfer in fractures, the heat () absorbed by the fluid from the rock in the whole fracture by convection is represented by

where and is mean temperature of the fluid at the fracture outlet and inlet. (kg/m^{3}) is the fluid density. (T), is the specific heat capacity of the fluid related to temperature. (W/m) is the absorbed heat of water in fracture.

Only the simulation results when the fluid flow Reynolds number is 100 are listed for the large amount of calculation data, and other results will be reflected in the subsequent analysis. The flow and heat transfer results of 28 curves with different morphologies (type, wavelength, and amplitude) are shown in Table 2.

According to Table 2, fractures with similar surface morphology numbered 20, 21, and 22 are analyzed. When the wavelength is fixed at 0.2 mm and the amplitude is increased from 0.1 mm to 0.15 mm, the equivalent hydraulic aperture of the fracture is reduced from 0.806 mm to 0.706 mm, and the osmotic pressure is increased from 173.1 Pa to 273.4 Pa. But at the same time, its heat transfer efficiency will also increase, which is reflected in that the heat absorbed by the water in fracture 2 ranges from 4397 (W/m), while the heat absorbed is 5009 (W/m). Therefore, the increase of fluctuation amplitude on the fracture surface will increase the flow resistance so that there needs to be a large pressure difference between the fracture inlet and outlet to maintain the same flow rate, while the heat transfer efficiency of the fracture will be improved. On the contrary, with the increase of its wavelength , the flow resistance caused by amplitude decreases. The fractures with surface amplitude of 0.1 mm and of 0.2 mm, 0.4 mm, and 0.8 mm numbered 20, 25, and 28 are analyzed. It is obvious that the pressure difference between inlet and outlet is affected by the fracture shape and the change of pressure difference is more significant when the flow rate increases. Therefore, this study uses the equivalent hydraulic aperture to evaluate the roughness between fractures. In addition, the length of the fracture surface is also included in the flow heat transfer coefficient of the fracture.

##### 4.2. Effect of Flow Velocity on Heat Transfer

For fractures with different morphologies, the variation law of temperature distribution with flow velocity is similar. The study found that the fluid velocity in the fracture is the main factor affecting the heat transfer efficiency in the fracture in Figure 5. The (a), (b), (c), and (d) are the temperature distributions when the mass flow rate is 0.001 kg/s, 0.01 kg/s, 0.05 kg/s, and 0.1 kg/s, respectively. Under the condition of fluid flow and constant temperature, the center temperature of the fractured rock sample increases to both sides. For lower flow rates, the temperature distribution shows a cold front-like shape in the middle of the rock sample. At high flow rates, the temperature planar shows a smooth pattern on both sides of the sample.

**(a)**

**(b)**

**(c)**

**(d)**

Understanding the temperature distribution of fluid flow in fractures is important for accurately understanding fluid flow and heat transfer processes in individual rock fractures. The fluid temperature along the mid-section of the fracture was chosen to explore the fluid temperature distribution in Figure 6. It can be seen from the figure that the fluid exhibits nonlinear distribution characteristics in the fracture, which is consistent with the previous literature conclusions [34]. The temperature of the fluid rises faster near the fracture inlet, especially when the flow is 0.001 kg/s; the temperature change is very obvious; and the fluid temperature rises rapidly from 25 °C to more than 240 °C. This is because the temperature difference between the fluid near the fracture inlet and the rock surface is large and the heat transfer efficiency is significantly improved. As the fluid flows through the fracture wall, the fluid temperature increases, the temperature difference between the fluid and the fracture rock wall decreases, the heat transfer efficiency decreases, and the fluid temperature rise curve tends to be flat and finally tends to a stable value. For different fracture forms, the distribution of fluid temperature along the flow direction is consistent.

Figure 7 shows the relationship between outlet temperature and flow velocity of different types of fractures. Consistent with the previously reported experiments, the temperature of steady-state flowing fluid at the fracture outlet decreases with the increase of flow velocity. As the fluid flow rate increases from 0.001 kg/s to 0.1 kg/s, the temperature of the fluid at the fracture outlet decreases from about 249 °C to about 35 °C, with an obvious decrease.

For fluids with higher flow rates, the time for heat exchange through the fracture is shorter, which makes the temperature of the fluid with high flow rate in the fracture lower than that of the low flow rate at the same flow position in the fracture. It also increases the mean temperature difference between the fluid and the rock wall, as shown in Figure 8. When the flow velocity increases from 0.001 kg/s to 0.01 kg/s, the difference between the average temperature of fractured rock surface and fluid rapidly increases from about 1 °C to nearly 4 °C. The increased temperature difference increases the heat transfer efficiency between the fluid and the rock. In the thermal boundary layer theory, the higher flow velocity makes the heat transfer boundary layer thinner, resulting in a decrease in the convective thermal resistance and an increase in the convective heat transfer intensity in the fracture.

Although the increase of flow rate leads to the decrease of outlet fluid temperature, higher flow rate will increase the heat exchange intensity between fluid and rock and increase the total heat transfer between rock and fluid in the fracture as shown Figure 9. With the increase of flow rate, the total heat absorbed by the fluid (W/m) is also increasing, but the increasing trend is slowing down. As the flow rate increases from 0.001 kg/s to 0.01 kg/s, the absorbed heat increases from about 1000 to about 3500. However, when the flow rate increases from 0.01 kg/s to 0.1 kg/s, the heat only increases to about 4000, and the increasing range decreases. On the one hand, due to the restriction of the inlet fluid temperature, the temperature difference between the fluid and the fracture wall will not increase with the flow rate. On the other hand, the heat transfer in the rock is mainly through heat conduction, which is affected by the fixed temperature boundary of the rock, and the heat conduction in the boundary between the fracture wall and the fixed temperature is also limited. In reservoir heat exploitation, the total heat absorbed by the fluid in the fracture of geothermal reservoir can be increased by increasing the flow rate at the fracture inlet or reducing the fluid temperature at the fracture inlet; meanwhile, it is also necessary to consider obtaining a reasonable outlet temperature.

##### 4.3. Influence of Fracture Morphology

Studying the effect of fracture roughness on fluid flow and heat transfer is of great significance for understanding the fluid flow and heat transfer mechanism in fractures. For the transformation of heat storage in geothermal development, a problem that must be faced is how to create artificial fractures to maximize the amount of heat production. The roughness of the fracture surface plays a dual role in the fluid flow process and the heat transfer process [11]. Due to the protrusions or grooves on the rough fracture surface, the fluid will generate eddy currents in local areas, which can cause the water temperature in some areas of the fracture to be significantly higher than others (see Figure 10). In this simulation, we found that very little of the heat absorbed by this part of the fluid is carried out of the fracture by the fluid.

The fluid velocity and temperature in the fracture are compared, as shown in Figure 11. In the convective heat transfer simulation, the temperature distribution of rough fracture is related to the fracture surface morphology. The protrusions and grooves on the rough fracture surface will hinder the flow of the fluid so that the temperature cold front of the fluid will appear serrated. As we all know, due to the viscous effect, the fluid has a velocity boundary layer in the flow process. The velocity is zero at the boundary of the fracture wall and reaches the maximum near the middle of the fracture. For the fluid in the fracture, the temperature distribution is opposite to the velocity distribution. In the same fracture section, the fluid temperature in the middle of the fracture is lower than that on the wall. The reason for this phenomenon is that due to different flow rates, the temperature of the fluid with low flow rate is mainly controlled by thermal conduction. With the increase of flow rate, the intensity of thermal convection increases. Therefore, the difference of velocity will lead to different convective heat transfer efficiency, resulting in the difference of fluid temperature distribution. In general, the temperature distribution is closely related to the roughness, and the fracture shape is the main factor affecting the temperature distribution after the fluid is fully developed.

Even at the same flow rate, the fracture morphology also affects the flow heat transfer between the fractures. Figure 12 shows the outlet temperature and total heat transfer of fractures with different surface morphology at a flow rate of 0.1 kg/s. It can be seen that when the equivalent hydraulic aperture in the fracture is the same, the outlet temperature and total heat transfer of the three kinds of rough fractures are higher than those of smooth parallel plate fractures. Therefore, when the hydraulic properties are the same, increasing the roughness of fractures is conducive to improve the heat transfer efficiency. Compared with the same fracture, the outlet temperature and heat transfer efficiency of the fracture decrease with the increase of the equivalent hydraulic aperture at a given velocity. Under the action of confining pressure, the aperture of the fracture decreases, which improves the flow and heat transfer efficiency. The application of confining pressure is helpful to the flow and heat transfer of the fracture [32].

The equivalent hydraulic aperture of fractures is related to the surface morphology. Figure 13 shows the relationship between the equivalent hydraulic aperture and the total amount of absorbed heat of water in three kinds of rough fractures and the wavelength of the amplitude A of the change on the fracture surface. (The dotted line represents the fracture opening, and the solid line represents the total heat transfer.) The upper part of the area where the solid line in the figure is located is the heat exchange between water and rock in the fracture, and the dotted line and the lower part represent the hydraulic aperture in the fracture. When the fixed flow velocity is 0.1 kg/s and the wavelength is 0.8 mm, the increase of surface protrusion amplitude will significantly increase the roughness of the fracture and increase the flow resistance in the fracture. The equivalent hydraulic aperture increases with the increase of roughness, thereby increasing the heat transfer intensity, although it will hinder the fluid flow [12]. On the contrary, the increase of the amplitude and cycle length of the protrusion on the fracture surface reduces the roughness of the fracture, increases the permeability of the fracture, but reduces the heat transfer capacity.

In convective heat transfer theory, the rough fracture surface increases the contact area between the fluid and the fractured rock. The larger the contact area between fluid and rock, the more sufficient the convective heat transfer. In Figure 14 (the dotted line represents the fracture surface length, and the solid line represents the total heat transfer), with the increase of the amplitude , the length of the fracture surface increases and the total heat transfer increases. With the increase of wavelength , the fracture length decreases and the total heat transfer decreases.

In summary, even with the same permeability, the heat transfer of fractures with different shapes is different, as shown in Figure 12. The local part of the rough fracture has different shapes of protrusions and grooves, which have different influence mechanisms on the fluid retention and flow behavior. Because the surface morphology of the fracture determines the convective heat transfer area in the fracture, the local morphology characteristics of the fracture need to be considered in the seepage heat transfer analysis. In general, fractures with larger heat exchange area have higher heat exchange efficiency.

#### 5. Conclusions

The surface topography of the fracture affects the fluid flow in the geothermal reservoir, and this numerical analysis studies the effect of the fracture shape on the heat transfer characteristics of the water flow in a single fracture. By designing rough fractures of different types and shapes, the influence of the surface morphology of the fractures is analyzed, and the following conclusions are obtained: (1)Whether in smooth or rough fractures, the fluid temperature distribution along the fracture flow direction and vertical flow direction is nonlinear. If the average water temperature in the fracture is simply defined as the average of the inlet and outlet water temperatures, the temperature of the whole fluid will be underestimated(2)The protrusions and grooves on the rough surface will greatly affect the fluid velocity on the fracture surface, lead to the sudden change of surface fluid temperature, and make the change of fluid temperature in the fracture more complex. With the increase of the roughness, the heat transfer efficiency of the fracture improves(3)The increase of fracture roughness reduces the flow capacity, but increases the heat transfer capacity in the fracture. The fracture morphology has a significant impact on the heat transfer capacity of the fracture. The fluctuation degree and shape of the surface affect the heat exchange area of the fracture surface and then change the heat transfer efficiency. Even under the same permeation conditions, the heat transfer capacity of different surface morphologies varies considerably. In general, the fractures with larger surface area, the greater the heat transfer efficiency

#### Data Availability

Some data used to support the results of this study are included in this paper, and the rest data used to support the results of this study can be obtained from the corresponding authors.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

This work was supported by the Program for Guangdong Introducing Innovative and Enterpreneurial Teams (No. 2019ZT08G315), the Natural Science Foundation of China (Nos. 52174083 and U2013603), the Natural Resources Science and Technology Project of Hunan Province (No. 2020-12), and the Shenzhen Basic Research (general project) (JCYJ20190808153416970).