Hot Dry Rock (HDR) is the most potential renewable geothermal energy in the future. Enhanced Geothermal System (EGS) is the most effective method for the development and utilization of HDR resources, and fractures are the main flow channels and one of the most important conditions for studying heat transfer process of EGS. Therefore, the heat transfer process and the heat transfer mechanism in fractures of EGS have been the hot spots of research. Due to the particularity of the mathematical models of heat transfer, research in this field has been at an exploratory stage, and its methods are mainly experimental tests and numerical simulations. This paper introduces the progress on heat transfer in fractures of Hot Dry Rock EGS in detail, provides a comparative analysis of the research results and prospects for future research directions: It is suggested that relevant scholars should further study the mathematical equations which are applicable to engineering construction of seepage heat transfer in irregular fractures of the rock mass, the unsteady heat transfer process between multiple fractures of the rock mass and the heat transfer mechanism of the complex three-dimensional models of EGS.

In recent years, primary energy demand is largely provided by non-renewable energy, which has led to global warming. Therefore, energy shortage has always been the focus of sustainable development. Many countries have gradually improved their research on renewable energy to reduce dependence on non-renewable energy sources such as, fossil fuels, coal, etc. [

The thermal energy of HDR which is a potential geothermal energy source is stored in a variety of metamorphic or crystalline rock bodies. Hydraulic pumps, therefore, are often required to stimulate the rock formations, causing fractures in the rock, which create a network of fractures as flow paths in the geothermal reservoir between the injection and production wells, resulting in the formation of EGS for heat extraction [

It is predicted that if only 2% of the EGS resource is recovered, underground thermal reservoirs have the potential to provide enough energy for more than 5,000 years in China [

After reading a lot of literature and theoretical analysis, the author believes that the mechanism of heat transfer in fractures of Hot Dry Rock EGS is mainly divided into two directions, one is convective heat transfer mechanism, and the other is seepage heat transfer mechanism. The convective heat transfer mechanism mostly refers to the theories and mathematical equations that are continuously optimized based on Newton’s law of cooling, and the seepage heat transfer mechanism mostly refers to the theories and mathematical equations that are continuously optimized based on Darcy’s Law.

At present, the vast majority of scholars at home and abroad think that the heat transfer process between fluids and rocks in Hot Dry Rock EGS is mainly convective heat transfer supplemented by heat conduction. Therefore, the research work is mainly based on the mechanism of “convection & heat conduction”.

Convective heat exchange in Hot Dry Rock EGS refers to the heat exchange between the fluid and the rock mass when the fluid flows over the surface of the rock mass or the fractured surface of the rock mass [

where

Heat conduction in Hot Dry Rock EGS refers to the spontaneous heat transfer process from high temperature to low temperature caused by the temperature gradient in the hot dry rock mass. When the heat transfer medium is injected into Hot Dry Rock EGS, there is a temperature gradient between the fractured surface of the rock mass and the rock interior, which will inevitably cause heat conduction, because the temperature of the heat transfer medium is always lower than that of the rock mass. The heat conduction in Hot Dry Rock EGS is the main heat supplement way in which the heat transfer medium can continuously extract heat, and the amount of heat conduction depends on the internal energy of the rock mass at that position [

In fact, for a certain large-scale Hot Dry Rock EGS, its heat conduction is an unsteady state process. When the heat extraction rate is greater than the heat recovery rate, the temperature of the rock mass will gradually decrease with time going by. When the temperature of the fractured surface of the rock mass is reduced to a certain extent, HDR geothermal well will lose the ability to extract heat, or the outlet temperature of the heat transfer medium will not reach the design value. At this time, HDR wells need to be sealed until the geothermal temperature meets the requirements to open the well to get heat [

The rock is in pressure equilibrium prior to hydraulic fracturing, and a pressure difference is formed after hydraulic fracturing between the upper and lower thermal reservoirs and the fracturing fluid, which leads to deformation of the rock, resulting in the formation of small fractures similar to porous media [

The research models of fracture structure of Hot Dry Rock EGS, so far, are divided into two main types, two-dimensional model and three-dimensional model respectively. Since two-dimensional models are easier to model, scholars have studied fracture structures that are closer to the real situation, with the main directions being single fracture and complex fracture, where the research on the effect of a single fracture structure on heat transfer has focused on the effect of fracture surface roughness on heat transfer. However, scholars usually build 3D single-fracture models, 3D parallel-fracture models or 3D simple cross-fracture models for numerical simulation calculations due to the complexity of three-dimensional modeling.

Tsang [

In recent years, more detailed studies on the influence of the fracture surface roughness on the heat transfer process have been carried out.

Neuville et al. [

Li et al. [

Luo et al. [

He et al. [

where

The numerical simulation results of He and Rong et al showed that the roughness of the fracture morphology of the fractured surface of the rock mass not only promoted the increase of heat transfer intensity but also promoted the effect with the increase of the roughness under the same flow rate and this promotion was weakened when the flow rate increased. Similarly, under the certain flow rate, the temperature of the fluid at the outlet in the rough fracture was higher than that in the smooth fracture and the temperature of the fluid at the outlet increased with the increase of the roughness of the fractured surface. Also, the pressure difference between inlet and outlet increased with the increase of roughness of the fractured surface of the rock mass. The two-dimensional model used in this numerical simulation cannot represent the anisotropy of rock mass between the rough fractures, and the author did not make numerical simulation on the relationship between parameter B and the fracture width, so the effect of MCF on heat transfer process should be further studied.

He et al. [

Wu et al. [

Zhou et al. [

Gao et al. [

Ma et al. [

Vasilyeva et al. [

Similar to the complex network studied by Vasilyeva et al. [

Huang et al. [

Xin et al. [

Jiang et al. [

Similar to the geometric model studied by Jiang et al. [_{a}) related to EGS heat extraction(The schematic diagram of the three-dimensional model and its grid system is shown in _{a} reduced the heat exchange between rock and fluid and the effective stress, and expanded the porosity and permeability of the rock thermal reservoir to a limited extent. Moreover, the model developed by Cao et al. [

Sun et al. [

Similar to the geometric model established by Sun et al., Pandey et al. [

Li et al. [

Ma et al. [

In the 1950s, the former Soviet Union and some Western scholars studied the water flow characteristics of single-fractured rock and summarized the mathematical expression:

Lomize proposed the equations [

Laminar flow:

Turbulent flow:

Reynolds number:

among them,

Louis proposed the equations [

Laminar flow:

Turbulent flow:

The Reynolds number curve is shown in

Both scholars strictly divided the water flow state of the single-fractured rock mass, but this may not be consistent with the water flow state in the actual fractured rock mass. So Su et al. [

where

Instead of dividing the states of water flow,

Asai et al. [

where

Kluge et al. [

Huang et al. [

Chapman [

In recent years, more and more scholars successively studied in depth the heat transfer coefficient of heat transfer process of fractured rock mass.

Zhao [

Heninze et al. [

When

where

However, this method was biased when calculating the rock temperature of 140°C and almost all calculations were still greater than the estimated value. And the authors believed that the possible cause was that experiment under the condition of unknown pressure affected the fluid flow. The simulation experiment had inconsistency in the data set and the model was too simple, so the reliability of the conclusion should be further studied.

Bai et al. [

For OHTC, they believed that the reason for the abnormal values of OHTC may be that the denominator contained subtraction or more variables, and for this reason, a new mathematical expression of OHTC (h) was proposed (The mathematical expression is as

For LHTC, a numerical model based on COMSOL model was developed by Bai et al. The results showed that [

LHTC had relationship with the roughness of the fractured surface of the rock mass, and in a certain range of fracture width, the narrower the fracture and the stronger the local heat transfer capacity.

The increase of water flow rate can significantly improve the local heat transfer ability of the rough fractured surface of the rock mass within the fixed fracture width of the rock mass.

The polynomial established by the experimental data showed that LHTC had a negative correlation with the fractured surface of the rock mass, and LHTC had a larger value at the groove of the rough crack surface.

Not only that, but the team compared LHTC with OHTC, and the comparison showed that:

The OHTC was always larger than LHTC at fixed fracture width and same flow.

As the water flow increased, the OHTC value increased, but the arithmetic mean of LHTC increased firstly and then decreased.

Although the results of the experiment were more accurate and the results of the experiment were more reliable, there were also shortcomings. First of all, the experiment only studied the single-fractured rock mass, and did not consider whether unsteady heat conduction process among fractures in multiple fractures would affect the heat transfer coefficient.

Ma et al. [

When joint roughness coefficient (JRC) of the fracture was kept constant, the total heat transfer coefficient was positively correlated with the volume flow rate. When the volume flow rate is kept constant, the total heat transfer coefficient increased with the increase of JRC.

The results of the distribution of the local heat transfer coefficient showed that the local heat transfer coefficient increased to a maximum value at the inlet and then decreased to a relatively constant value along the flow direction.

The volume flow rate did not affect the distribution of the local heat transfer coefficient. When the curvature of the fracture surface was kept constant, the local heat transfer coefficient increased with the increase in volume flow rate.

Although the distribution of local heat transfer coefficients was analyzed by Ma et al., the results can be better verified if they are compared and analyzed with the local heat transfer coefficient equations derived by other authors.

After the analysis and summary of the research status of the mathematical models of heat transfer of Hot Dry Rock EGS at home and abroad, the following conclusions and suggestions can be drawn:

At present, the experimental tests and numerical simulations of the mathematical models of heat transfer of Hot Dry Rock EGS in the domestic and foreign countries are mostly studied for the horizontal or vertical single fracture or intersecting fracture of the rock mass. This single-fractured rock model provides more convenient conditions for experimental tests and numerical simulations. However, there is a great deviation from the actual heat transfer process of Hot Dry Rock EGS, and the results of these numerical simulations are difficult to guide the actual situation. Although there are many researches on irregular fractures, most of them are only at the level of academic research and the methods used are too complicated for engineering construction. Therefore, it is suggested that the relevant scholars should further study the mathematical equations, which are applicable to engineering construction, of seepage heat transfer in irregular fractures of the rock mass to improve the efficiency of construction.

At present, the experimental tests and numerical simulations of the mathematical models of heat transfer of Hot Dry Rock EGS in the domestic and foreign countries are mostly studied for steady heat transfer process. The steady heat transfer process cannot test or simulate the temperature change trend of the rocks around the HDR geothermal wells over time, so it is difficult to estimate the service life of Hot Dry Rock EGS. Whether the heating by themselves and the heating from surrounding HDR during the non-heating period can maintain the continuous and stable operation of EGS during the heating period is a worth problem for further study. Therefore, it is suggested that the unsteady heat transfer process and the mathematical models of heat transfer among the multiple fractures of HDR should be studied in depth by scholars, which has great practical significance to guide the practical application of the project.

At present, limited by the experimental test conditions, domestic and foreign scholars have a wide range of research on computer programming of heat transfer numerical simulation of Hot Dry Rock EGS, and many numerical simulation methods and procedures provide convenient conditions for scholars to study HDR in the future. However, the current numerical simulation is only fits the two-dimensional model deeply, which usually ignores the problem of anisotropy of the rock mass. Although some domestic and foreign scholars have also found this defect and carried out some numerical simulation studies for the three-dimensional model, these three-dimensional models are usually ideal simple single-fractured rock models which still have some bias compared with the actual situation. Therefore, considering the anisotropy of rock masses, the heat transfer process and mathematical models of heat transfer on Hot Dry Rock EGS in the model of complex three-dimensional fractures should be studied deeply with a view to getting closer to the real situation.