Geothermal development requires an understanding of changes in pore permeability caused by repeated thermal shock fatigue damage in hot dry rock. Research on the subject can aid the evaluation of the longevity and mining value of enhanced geothermal systems (EGS). However, few relevant studies are currently available. In this study, the change characteristics of pore permeability in granite following different heating–cooling cycle temperatures (250°C, 350°C, 450°C, 550°C, and 650°C) and numbers of cycles (1, 5, 10, 15, and 20 cycles) were analyzed. Results show that with increasing temperature, the uneven thermal expansion and thermal shock effect of minerals promote crack development, leading to increases in the porosity and permeability of granite, particularly at temperatures above 450°C. When the heat treatment temperature was below 450°C, the number of cycles only slightly affected the porosity and permeability; meanwhile, when the temperature exceeded 450°C, the porosity and permeability increased significantly with an increase in the number of cycles. Moreover, three-dimensional nonlinear fitted relationships among porosity (or permeability), cycle temperature, and number of cycles have been established for the first time with correlation coefficients (R2) above 0.9, which reveals the change rules of pore permeability after quenching in hot dry rock. The results can be used to evaluate the efficiency of geothermal reservoir energy extraction and aid in geothermal reservoir design.

The depletion of traditional fossil fuels and the deterioration of the environment due to the combustion of fossil fuels render necessary the search for new alternative green and sustainable energy sources [1]. One such renewable energy source, geothermal energy, is favored by many countries because of its wide distribution, rich reserves [2, 3], strong stability [4], low environmental impact [5], low greenhouse gas emissions [6], low unit cost, and the relatively easy and quick construction of geothermal plants. In the past decade, geothermal energy has been widely used for activities such as heating, bathing [7], medical treatment [8], construction of crop greenhouses, food drying, and power generation [914].

Hot dry rock (HDR) is a high-temperature rock mass that is generally buried 3–10 km below the ground surface in the absence of water or steam at temperatures typically ranging from 150°C to 650°C [12]. HDR at temperatures exceeding 200°C is more advantageous for geothermal use, and the presence of such geothermal reserves has gradually become a key variable influencing the future direction of a country’s energy structure [13]. Currently, enhanced geothermal systems (EGS) provide an effective way to exploit HDR [14]. The commercial development of EGS technology can produce thousands of megabytes of electricity [15]. The United States has established Frontier Observatory for Research in Geothermal Energy (FORGE) for the promotion of EGS technology [16], which is mainly used to inject cold water into injection wells [17]. After a sufficient heat exchange between the cold water and the HDR, heat energy is extracted from production wells [18]. The economic value and mining life of EGS depend to a large extent on the permeability of its underground fracture network. If the permeability decreases, the production temperature and related power generation of the HDR decrease as well [19]. The thermal shock fatigue damage of HDR after repeated injections of cold water significantly affects the development of the fracture network of the reservoir. Therefore, it is helpful for geothermal energy development to exploring the change characteristics of the pores and permeability of the HDR following multiple thermal shocks.

Porosity and permeability as important physical parameters of rocks are widely studied [20, 21]. Owing to the high temperature in the deep underground, variations in the crack and porosity of rocks are either related to changes in the microcrack network induced by thermal expansion or structural damage [22, 23]. Rock permeability is largely affected by fracture morphology and pore structure (such as tortuosity, connectivity, and volume). Numerous experimental studies have attempted to quantify the relationship between the permeability of different rocks and its effects, and temperature plays a crucial role among various effects on porosity and permeability [24, 25]. By establishing a crack porosity model, Etienne and Poupert explored the change characteristics of granite crack porosity after heat treatment and found that crack porosity increased significantly when the temperature exceeded 400°C [26]. Ravalec et al. heated fine-grained granitic mylonite from La Bresse (France) at temperatures ranging from 20°C to 700°C and tested its permeability. The permeability of granite increased slowly up to 400°C and rapidly up to 700°C [25].

Zhao et al. explored the influence of high temperature and high pressure on the permeability of large-sized granite and observed that peak permeabilities occurred at temperatures in the 75°C–150°C and 300°C–450°C [27]. Previous studies have also found that when the heat treatment temperature is lower than the critical point of phase transition in quartz minerals, the porosity and permeability of rocks are controlled by thermal stress and internal water vapor caused by thermal stress [27, 28]. When the temperature is high, the porosity and permeability of rocks change significantly, particularly the phase transformation of quartz minerals [29]. Table 1summarizes additional published studies regarding the effects of high temperatures on the permeability of granite [3, 2933].

These research results have laid an important foundation for the development and utilization of underground engineering. However, natural cooling often fails to meet the actual conditions of some underground projects, such as the development of HDR and the ventilation of mines at deep depths [34, 35]. During the development of HDR, multiple injections of cold water induce the dual effects of thermal shock and fatigue damage within the rock. The cold state of the water relative to the surrounding hot rocks can change the density and mineral structures by shrinking the rock and changing the thermal stress field, thereby changing the associated porosity and permeability. Nonetheless, relationships describing the changes in the porosity and permeability of HDR after thermal shock fatigue damage have thus far been rarely determined.

In this study, a preliminary relationship describing the change in the pore permeability of HDR under the effect of quenching heating–cooling cycles is presented. Experimental materials and methods, together with an established permeability model, are introduced in Section 2. The variations in porosity and permeability with cycle temperature and the number of quenching cycles are introduced in Section 3. Section 4 discusses the three-dimensional nonlinear fitted relationships between porosity (or permeability) and cycle temperature and the number of cycles. Section 5 summarizes the conclusions of this research.

2.1. Sample Preparation and Experimental Procedures

The test granite used in this study originated from Shandong Province, China. All rock samples were cut from a large granite block in an identical direction to maintain sample consistency. All samples were of a standard size in accordance with the recommendations of the International Society for Rock Mechanics. X-ray diffraction results indicate that the granite consisted of the following minerals: quartz (32.0%), feldspar (58.4%), hornblende (6.1%), and black mica (3.4%). Table 2 lists basic information describing sample density, sample size, heating and testing equipment, temperature, and range of the number of cycles.

Figure 1 shows the heating and cooling phases of an experimental thermal cycle at different specified temperatures. The samples were heated to a specific temperature by using a KSL-1700X-A2 sintering furnace at 10°C/min. The technical parameters of the sintering furnace are listed in Table 3. Each specific cycle temperature was kept constant for about 1 h. After the specified heat treatment period was completed, the samples were taken out of the sintering furnace and then immersed in cold water, cooled to room temperature. They were subsequently taken out of the water and placed in a drying oven at 50°C for drying before being placed back into the sintering furnace for the next cycle. To compare the evolutionary characteristics of the pore permeability of granite under different thermal shock times, the samples were subjected to 1, 5, 10, 15, and 20 cycles.

The porosity of each specimen was measured at the end of the 1st, 5th, 10th, 15th, and 20th cycles. Two samples were subjected to each experimental condition, and the average values of each experimental parameter were determined. The porosity of the samples after heat treatment was evaluated using a PoreMaster 60 GT mercury pressure instrument. The main technical parameters of the instrument were as follows: pressure sensor range, 0–50 psia; measurement aperture distribution, 1080–0.003 μm; accuracy rate; 0.11%; resolution, 0.000763 psia; and sample volume, 3.2 cc. A B007 type “Super eyes” electron microscope was also used to observe the surface condition of the samples after heat treatment.

2.2. Model of Permeability Change Rate

Rock permeability plays an important role in underground engineering [36] but is difficult to accurately measure. A permeability model is typically used to derive permeability from other more available parameters, and porosity is one such parameter used in many previous studies [3744].

In the present study, the Carman–Kozeny model, which is widely used in engineering geology and hydrogeology, was used to estimate the permeability of samples after a heating–cooling cycle. The form of this model [45, 46] is as follows:
(1)k=cϕ31ϕ2,
where k is the permeability coefficient, ϕ is the porosity, and c is the Carman–Kozeny model coefficient. Equation (1) shows that the permeability coefficient exhibits a positive correlation with porosity. Permeability increases as the porosity increases. Porosity is affected by temperature; thus, permeability is also affected by temperature.
The parameter k was used to better describe changes in permeability—that is,
(2)ΔkT=kLTkL250kL250,(3)ΔkL=kLTk1Tk1T,
where ΔkT is the effect of temperature on the permeability change rate; ΔkL is the effect of the number of cycles on permeability change rate; and the index letters T and L denote the temperature series and cycle number series, respectively; that is, T=250,350,450,550,650 and L=1,5,10,15,20. Thus, kL250 and k1T are the permeability coefficients after L cycles at 250°C and one cycle at T °C, respectively.
By substituting the T and L series into Equations (2) and (3), Equations (4) and (5) are derived, as follows:
(4)ΔkT=1ϕL2502ϕLT31ϕLT2ϕL25031(5)ΔkL=1ϕ1T2ϕLT31ϕLT2ϕ1T31.

The permeability change rate of granite under different temperatures and numbers of cycles were calculated using Equations (1)–(5) and are discussed in the following sections.

3.1. Variations in the Porosity of Granite

Figure 2 presents the variations in the porosity of granite with cycle temperature (T) and the number of cycles (L). Two distinct stages of variations in porosity with temperature were identified. Stage 1 was stable and occurred from 250°C to 450°C. During this stage, the porosity remained unchanged or hardly changed with an increase in temperature, suggesting that only slight structural modifications in the pore were induced below 450°C. Stage 2 was a rapid growth stage occurring between 450°C and 650°C. In this study, the porosity increased rapidly with an increase in temperature, with the maximum porosity exceeding 7% (Figure 2(a)). This quick increase in porosity was attributed to the increase in the density of internal cracks in the granite and the significant change in the pore structure.

The change in porosity with the number of cycles was caused by temperature (Figure 2(b)). At temperatures below 550°C, the change in porosity was less dependent on the number of cycles. However, at 550°C or higher, the porosity of granite increased as the cycle number increased. For instance, when the heat treatment temperature was 650°C, the porosities after 1 and 20 cycles were 3.51% and 7.65%, respectively. It can be inferred that with an increase in the number of cycles, cracks in the granite were further developed, increasing the density and width of cracks; consequently, the connectivity between cracks increased.

3.2. Variations in the Permeability of Granite

The relationship between the rate of change in the permeability coefficient of granite after heating–cooling cycles was determined using Equations (4) and (5), and the results are presented in Figure 3.

The rate of change in permeability with temperature also had two obvious stages (Figure 3(a)). At temperatures ranging from 250°C to 450°C, the permeability coefficient did not change significantly. This shows that when the temperature is lower than 450°C, pores and fractures in the granite were poorly developed, and the connectivity between pores was weak. However, at temperatures between 450°C to 650°C, the porosity of the granite increased with increasing temperature as the connectivity between pores increased [47]; thus, the permeability coefficient increased rapidly as the temperature increased.

The number of cycles also played an important role in the change in permeability, particularly above 450°C (Figure 3(b)). When the temperature was lower than 450°C, the permeability fluctuated slightly as the number of cycles increased. However, when the temperature exceeded 450°C, the expansion of minerals increased the number of granite fractures. As the number of cycles increased, the pore connectivity of the granite was strengthened, increasing the permeability.

4.1. Damage Mechanism of Granite under Thermal Shock

The development of pores and fractures in granite, which affects its permeability, is mainly influenced by the thermal expansion of minerals. The thermal dilatation of quartz and feldspar is depicted in Figure 4 (data from Winkler) [48]. The thermal dilatation of these minerals increases as temperature increases, particularly above 450°C. For instance, quartz undergoes a phase transformation when the temperature reaches 573°C at which point volume increases by about 5% [49, 50]. The difference between the expansion coefficients of different minerals leads to the generation of cracks and even connects different pores.

Thermal shock is another main factor for the enhancement of porosity and permeability of granite, as determined in this study. The mechanical model of the internal temperature stress of a rock sample during heating and cooling can be divided into thermal expansion stress and secondary temperature stress [51]. The uneven expansion between minerals causes crack initiation in the internal pores of the samples. However, the secondary temperature stress caused by thermal shock causes the cracks in the samples to further expand, leading to increases in the porosity and permeability of the samples [52].

In the current study, cracks were observed on the samples by using the B008 type “Super eyes” electron microscope, which magnified the surface of the samples up to 500 times. Cracks on the surface of the samples were considerably affected by the temperature and the number of thermal cycles (Figure 5). Figure 5(a) shows how surface cracks change with temperature after one cycle only. At low temperatures (250°C or 350°C), no visible cracks appeared on the sample surface. However, when the temperature reached 450°C or higher, cracks were considerably noticeable. Furthermore, the crack width at 650°C was markedly larger than that at 450°C. Figure 5(b) shows the surface crack characteristics of the samples after 5, 10, 15, and 20 cycles at 450°C. As the number of cycles increased, not only did the width of the cracks increase, the connectivity between the cracks increased as well. This finding confirms that cyclic thermal shock aggravated the thermal damage of the granite.

Previous studies have found similar effects of temperature on the permeability of granite, as shown in Figure 6 [29, 49]. From 105°C to 450°C, the permeability of granite increases slowly. By contrast, from 450°C to 600°C, the permeability of granite increases rapidly.

4.2. Three-Dimensional Nonlinear Relationship between Porosity, Cycle Temperature, and Number of Cycles

On the basis of the aforementioned results, granite porosity is not only affected by temperature but also by the number of heating–cooling cycles the rock has been exposed to. However, whether a functional relationship exists between porosity, temperature, and the number of cycles has yet to be determined. Thus, a three-dimensional nonlinear model of porosity was fitted to the data, as follows:
(6)φ=φL,T=110.9+1.12L+1.13T0.008LT0.004T2+1.84×105LT2+6.3×106T31.17×108LT33.5×109T4,R2=0.986,
where T is the thermal cycle maximum temperature and L is the number of cycles. The three-dimensional nonlinear fitting coefficient R2 of Equation (6) was 0.986, indicating that the correlation among the three parameters was considerably high (Figure 7). The fluctuation of the three-dimensional fitted surface is relatively large as the porosity changed greatly at varying temperatures and numbers of cycles.

4.3. Three-Dimensional Nonlinear Relationship between Permeability, Temperature, and Number of Cycles

The change in permeability is divided into the temperature effect ΔkT and the effect of the number of cycles ΔkL. Similar to porosity, three-dimensional nonlinear fitted relationships were obtained between the changes in permeability and temperature and the number of cycles, as follows:
(7)ΔkT=802.6+131.2L+5.135T3.83L2+0.01L2T+5.7×104LT2+6.64×106T3,R2=0.972,(8)ΔkL=2183.07L2.16T+0.19L2+0.019LT+0.008T20.001L2T3.26×105LT21.2×105T3+1.44×106L2T2+1.39×108LT3+6.5×109T4,R2=0.937.

Equation (7) represents the three-dimensional nonlinear fitted relationship of ΔkT, T, and L that describes the effect of temperature. The associated correlation coefficient R2 was 0.972, indicating that the correlation between ΔkT, T, and L was high. Moreover, when the values of the temperature and the number of cycles were large, ΔkT (Figure 8(a)) and ΔkL varied greatly (Figure 8(b)). When the temperature was 650°C and the number of cycles was 5, ΔkL was about 0.82; however, when the temperature was 650°C and the number of cycles was 15, ΔkL was about 9.48. Thus, permeability is significantly influenced by temperature and number of cycles.

To explore the effect of changes in the cyclical temperature on the permeability of granite, its porosity after heat treatment was measured, and the results were used to evaluate the changes in permeability. The conclusions drawn are as follows:

  • (1)

    When granite was subjected to a 250°C–350°C thermal shock cycle, no obvious cracks appeared on its surface. However, when the cycle temperature exceeded 450°C, the number of surface cracks increased with an increase in the number of heating–cooling cycles

  • (2)

    Thermal cycling temperature significantly affected the porosity and permeability of granite. Porosity and permeability increased with cycle temperature, particularly above 450°C. Therefore, 450°C can potentially be a critical value of permeability change in granite

  • (3)

    The number of heating–cooling cycles increased the damage to the pore structure of granite. The change in permeability was not apparent in granite subjected to few (<15) cycles; by contrast, a significant change was observed when the number of cycles was high (>15). In future studies, more heating–cooling cycles should be conducted to elucidate the permeability evolution of geothermal reservoirs in granite

  • (4)

    The three-dimensional nonlinear fitted relationships among porosity (or permeability), cycle temperature, and number of cycles had correlation coefficients (R2) greater than 0.9

The set of three-dimensional nonlinear fitted models presented in this study can be used to elucidate the influence of temperature and the number of heating–cooling cycles on the porosity or permeability of granite, as well as to provide an important reference for the future development of geothermal energy.

The data used to support the findings of this study are available from the corresponding author upon request.

No conflict of interest exists in the submission of this manuscript, and the manuscript is approved by all authors for publication.

This research was supported by the open foundation of Key Laboratory of Deep Earth Science and Engineering (Sichuan University), Ministry of Education (No. DESE202102), the Department of Science & Technology of Guangdong Province (No. 2019ZT08G315), the National Natural Science Foundation of China (Grant No. 41672279 and No. 52004167), and Shenzhen City Clean Energy Research Institute.

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