1 Introduction

Geothermal energy is stable, environmental protection and highly efficient and does not affect air quality, and therefore serves as an effective alternative source of energy to traditional fossil energy (Zhang et al. 2018; Hu et al. 2019a, b; Sun and Hu 2019; Feng et al. 2020; Xie et al. 2020; Hu et al. 2021; Long et al. 2021; Li et al. 2022). Geothermal energy within the first 10 km of Earth’s upper crust is about 1.3 × 1027 J and this could supply global energy for approximately 217 million years (Lu 2018). Hot dry rock (HDR) has been widely studied as an important geothermal resource. On one hand, many studies have addressed HDR reservoir reconstruction technology, including hydraulic fracturing (Sharma et al. 2004; Zimmermann and Reinicke 2010), thermally induced fracturing and chemical stimulation (Hardin et al. 2003; Nami et al. 2008). On the other hand, the physical and mechanical properties of HDR have been explored through laboratory tests and simulation (e.g., wave velocity, resistivity, thermal conductivity, porosity, permeability, compressive strength and tensile strength) (Akdag et al. 2018; Feng et al. 2017, 2018, 2020; Hu et al. 2018; Jia et al. 2019; Jin et al. 2019; Qin et al. 2019; Yang et al. 2019; Wu et al. 2019a, b, c; Zhang et al. 2021a, b). Moreover, the anisotropic characteristics of rocks also affect their mechanical properties and failure modes (Kong et al. 2023; Xi et al. 2023). In addition, the influencing factors in the interaction process between low-temperature fluids and high-temperature rocks have also been widely studied, such as the range of ore temperature, number of cycles, heating and cooling rates, and fluid type (Hu et al. 2022; Wu et al. 2019a, b, c; Wang and Konietzky 2019; Xi et al. 2023; Zhang et al. 2021a, b). And these studies have greatly promoted the development and utilization of HDR.

Previous studies mainly focused on the granite geothermal reservoir, while the sandstone thermal reservoir has rarely been evaluated. The study of sandstone geothermal reservoir is of great significance to the development and utilization of geothermal energy. Zimmermann and Reinicke (2010) used laboratory and field experiments to analyze hydraulic fracturing of deep sandstone reservoirs to develop enhanced geothermal systems. Müller et al. (2010) evaluated the representative elementary volume (REV) of fractured geothermal sandstone reservoirs. Haffen et al. (2013) used thermal conductivity and temperature logs to determine fluid-flow zones in geothermal sandstone reservoirs. Schmidt et al. (2017) studied the interaction between hydrothermal NaCl solutions with fracture surfaces of the geothermal reservoir sandstone of the Upper Rhine Graben and Su et al. (2018) reviewed research on cold water reinjection in sandstone geothermal reservoirs in China. These research results facilitate the understanding of the significance of sandstone geothermal reservoir development and utilization. Nevertheless, the physical and mechanical properties of sandstone after cyclical thermal shock treatment in water are rarely studied, which can provide important theoretical support for sandstone geothermal reservoir development.

In this study, the characteristics of the variation of P-wave velocity of sandstone after the application of multiple thermal shock treatments have been studied. Previous studies showed that the thermal damage test temperature of HDR reached approximately 600 °C because the occurrence temperature of HDR could reach 500 °C or higher (Tomac and Sauter 2018; Sha et al. 2020). In the investigation, the test temperature was set from 100 to 800 °C accordingly. Sections 3.1 and 3.2 analyzed the variation characteristics of P-wave velocity in red sandstone under two high-temperature actions. In the first experiment, the P- wave velocity of sandstone after thermal shock at 600 °C was lower than 800 °C. This result is contrary to previous test results that have shown that the P-wave velocity of sandstone decreased with increasing temperature (Sun et al. 2016; Hu et al. 2019a, b). The second thermal shock experiment of sandstone was conducted to verify the abnormal phenomenon of P-wave velocity in sandstone. Interestingly, the P-wave velocity of sandstone was still found to be abnormal at some temperatures and in a number of cycles. In Sect. 4.1, the relationship between the damage factor of sandstone and temperature, and the number of cycles has also been studied. Section 4.2 focuses on the three-dimensional nonlinear relationship between P-wave velocity, temperature, and number of cycles. Sections 4.3 and 4.4 analyze the possible factors for abnormal changes in P-wave velocity of red sandstone. This systematic study of the P-wave velocity characteristics of sandstone after cyclical thermal shock aims to provide a reference for the development and utilization of geothermal energy and other underground projects.

2 Materials and methods

The red sandstone samples used in the two experiments were taken from Linyi city, Shandong Province, China. The mineral composition of red sandstone was analyzed by XRD, mainly including quartz (77%), albite (10%), laumontite (9%), calcite (3%), and goethite (1%). Red sandstone has a relatively dense mass structure and is composed of standard Ф50 × 100 mm cylinders. Red sandstone samples were heated in a KSL-1700 ×  muffle furnace at a rate of 10 °C/min. Samples were heated in the muffle furnace for two hours once a specified temperature was reached. Samples were subsequently rapidly cooled in water at room temperature (approximately 25 °C). When the samples were cooled to room temperature in water, removed them from the water. Subsequently, the samples were dried in a drying oven at 40 °C. It is worth noting that the temperature setting of the drying oven should not be too high, as higher temperatures can cause some damage to the sandstone, thereby affecting the experimental results. The complete process of “heating-quick cooling” of red sandstone was repeated once. Table 1 shows the temperatures and cycle times applied in the two experiments. The number of cycles (0) in Table 1 indicates that the samples were not rapidly cooled in water; these were cooled naturally in the KSL-1700X muffle furnace. In the first experiment, red sandstone samples were heated to 100 °C, 200 °C, 300 °C, 400 °C, 500 °C, 600 °C and 800 °C respectively and subsequently rapidly cooled 0, 2, 4, 6 and 8 times at each specified temperature. In the second experiment, red sandstone samples were heated to 300 °C, 400 °C, 500 °C, 600 °C, 650 °C and 700 °C respectively, and rapidly cooled 0, 1, 2, 3 and 4 times at each given temperature. The RSM-SY6 acoustic wave detector was used to measure P-wave velocity of red sandstone after the heat treatment. We applied Vaseline to the probe of the wave velocity meter to improve the accuracy of the wave velocity measurement and this facilitated the efficient coupling of the rock sample and probe.

Table 1 Temperature and quick cooling number of two experiments
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3 Experimental result

3.1 The variation of P-wave velocity with temperature

The change in wave velocity after application of the cyclical thermal shock to red sandstone in the first experiment is shown in Fig. 1a. Wave velocity varied according to three approximate temperature ranges. In the range of 100–400 °C, P-wave velocity decreased slowly and approximately linearly. The rates of change of wave velocity were 8.5%, 12.4%, 14.0%, 14.1% and 16.6% respectively, under five different thermal shock conditions. When the heat treatment temperature was in the range of 400–600 °C, the P-wave velocity of the samples decreased rapidly. The change rates of the P-wave velocity were 38.0%, 49.8%, 50.4%, 51.3% and 52.6% respectively, under the number of cycles of 0, 2, 4, 6 and 8. However, when the temperature was between 600 and 800 °C, the wave velocity of the samples increased. To verify the temperature range of the abnormal P-wave velocity of the red sandstone after thermal shock, we set specific temperatures of 300 °C, 400 °C, 500 °C, 600 °C, 650 °C and 700 °C in the second test. In the second group of experiments, the change of P-wave velocity of the samples in the range of 300–600 °C was similar to that of the first group. Wave velocity decreased with increasing temperature (Fig. 1b). Nevertheless, P-wave velocity fluctuated greatly under different cycles in the heat treatment temperature range of 600–700 °C. Interestingly, the P-wave velocity showed an upward trend at 600 °C or 650 °C under different cycles, except for where there were 0 cycles.

Fig. 1
figure 1

The Variation of P-wave velocity with temperature. a First experiment; b second experiment

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3.2 The variation of P-wave velocity with number of cycles

The number of cycles also played a role in the change of P-wave velocity of the red sandstone. Temperature and number of cycles differed between the two groups of experiments but the change in common characteristics of P-wave velocity of red sandstone with the number of cycles in the two groups remained apparent (Fig. 2). The common feature was that P-wave velocity decreased slowly with the increase in the number of cycles when the temperature was below 400 °C. P-wave velocity decreased significantly with the increase in the number of cycles when the temperature was above 400 °C. The dual interaction of temperature and cycle number had a significant effect on the P-wave velocity of red sandstone.

Fig. 2
figure 2

The variation of P-wave velocity with number of cycles. a First experiment; b second experiment

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4 Experimental discussion

4.1 Damage factor

Many researchers use damage factor D to evaluate the degree of damage to rock under high temperatures (Liu and Xu 2015; Hu et al. 2019a, b). In this study, we analyzed the degree of thermal damage to red sandstone caused by the temperature and number of cycles. Therefore, the temperature damage factor DT and the number of cycles damage factor DL are defined in formulae (1) and (2) below:

$$D_{T} = 1 - \left( {\frac{{V_{PT} }}{{V_{P0} }}} \right)^{2}$$
(1)
$$D_{L} = 1 - \left( {\frac{{V_{PL} }}{{V_{P0} }}} \right)^{2}$$
(2)

where VPT and VPL are longitudinal wave velocity dominated by temperature effect and cycle effect respectively; and VP0 is initial longitudinal wave velocity. Additionally, the temperatures T of the two sets of experiments are T1 = [100, 200, 300, 400, 500, 600, 800]−1 and T2 = [300, 400, 500, 600, 650, 700]−1. The number of cycles L are L1 = [0, 2, 4, 6, 8]−1 and L2 = [0, 1, 2, 3, 4]−1. The damage factor D of the two groups of experiments is shown in Fig. 3. When the temperature was below 600 °C, damage factor D increased in the two groups of red sandstone. Results showed that damage to red sandstone increased with an increase in temperature. However, when the temperature was higher than 600 °C, damage factor D of the two groups of experimental sandstone decreased or tended to be gentle. The influence of cycle times on the damage factor of red sandstone is shown in Fig. 3c, d. Interestingly, the damage factor did not always increase with the number of cycles. The same sample could fluctuate under a different number of cycles, e.g. 300 °C and 400 °C in Fig. 3d. This also demonstrated the complexity of the damage to red sandstone under the coupling effect of temperature and number of cycles.

Fig. 3
figure 3

Change of damage factor of red sandstone with temperature and number of cycles. a First experiment-temperature; b second experiment-temperature; c first experiment-number of cycles; d second experiment-number of cycles

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In fact, the damage to red sandstone mainly comes from three aspects: ① during the heating process, the uneven thermal expansion of minerals inside the sandstone leads to the generation of intergranular cracks (Feng et al. 2017; Xi et al. 2023); ② When high-temperature sandstone interacts with low-temperature water, the temperature gradient inside the sandstone rapidly changes, leading to thermal shock damage, which promotes crack propagation and connectivity; ③ Fatigue damage caused by periodic heating and cold water cooling (Hu et al. 2021, 2022). The degree of damage to red sandstone varies under different temperatures and thermal shock times. With the increase of temperature and thermal shock frequency, the damage of red sandstone gradually increases, and the damage factor also shows a significant upward trend.

4.2 Nonlinear fitting relationship of P-wave velocity with temperature and number of cycles

Experimental results showed that P-wave velocity was significantly affected by temperature and number of cycles, yet exploring the functional relationship between P-wave velocity, temperature and number of cycles remained important. Figure 4 shows the nonlinear fitting relationship between P-wave velocity and temperature, and the number of cycles in the two tests. Data points in the two tests were evenly distributed on the surface. The nonlinear fitting formulae between the two test parameters are shown in Eqs. (3) and (4) below.

$$\begin{array}{*{20}c} \begin{gathered} V_{P1} = 2.406 + 0.126N - 4 \times 10^{ - 4} T - 1.15 \times 10^{ - 4} NT + 4.06 \times 10^{ - 6} T^{2} \\ - 2.8 \times 10^{ - 8} NT^{2} - 1.05 \times 10^{ - 8} T^{3} \\ \end{gathered} & {R^{2} = \, 0.974} \\ \end{array}$$
(3)
$$\begin{array}{*{20}c} \begin{gathered} V_{P2} = - 1.73 + 0.552N + 0.03T - 2.6 \times 10^{ - 3} NT - 7.42 \times 10^{ - 5} T^{2} \\ + 2.43 \times 10^{ - 6} NT^{2} + 5.16 \times 10^{ - 8} T^{3} \\ \end{gathered} & {R^{2} = \, 0.926} \\ \end{array}$$
(4)

where VP is P-wave velocity, T is treatment temperature and N is the number of cycles. The correlation coefficient R2 of the two red sandstone high-temperature tests were 0.974 and 0.926 respectively. This indicated that P-wave velocity had a strong correlation with temperature and number of cycles. The VP (T, N) function relation can be written as the general formula (5) below:

$$V_{P} = a + bN + cT + dNT + eT^{2} + fNT^{2} + gT^{3}$$
(5)
Fig. 4
figure 4

Nonlinear fitting relationship of P-wave velocity with temperature and number of cycles. a First experiment; b second experiment

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The P-wave velocity variation characteristics of red sandstone could be evaluated more effectively through the nonlinear fitting relationship of P-wave velocity, temperature and number of cycles given above. This can also support the physical exploration and inversion analysis of HDR reservoirs in the development of deep geothermal resources. During the long-term exploitation of geothermal resources, with changes in reservoir temperature and injection production cycle, the model established above can be used to evaluate the trend of P-wave velocity changes under different geological conditions, thereby helping to understand the evolution of geothermal reservoir fractures.

4.3 Effect of sandstone expansion on P-wave velocity of red sandstone

The thermal expansion of quartz has a significant effect on red sandstone since the main mineral of red sandstone is quartz. Figure 5 shows the thermal expansion characteristics of quartz and different types of sandstone. Somerton and Selim (1961) found that the volume of thermal expansion of quartz and three types of sandstones increased with increasing temperature and this was especially evident at temperatures above 400 °C (Fig. 5a). However, when the temperature was above 573 °C (transformation of α quartz to β quartz), the thermal expansion of quartz and three types of sandstone was gentle. Hartlieb et al. (2016) tested the relationship between the thermal expansion of sandstone and temperature and found similar results to those of Somerton and Selim (1961) (Fig. 5b).

Fig. 5
figure 5

Variation characteristics of thermal expansion of quartz and sandstone with temperature. a The volume expansion of quartz and three kinds of sandstone (Somerton and Selim 1961); b Thermal expansion of sandstone heat-treated twice (Hartlieb et al. 2016)

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Thermal expansion of sandstone caused by high temperatures leads to damage and the development of sandstone fractures is accelerated during the quenching process. Iwasaki and Torikai (1993) explored the change of surface crack density (N) after quartz lascar quenching and divided the quenching temperature (Tq) region (Fig. 6). The quenching temperature region is as follows:

$$T_{q} < \sim 200\;^\circ {\text{C }}\left( {{\text{region}}\;{\text{I}}} \right)$$
$$\sim \, 200\;^\circ {\text{C }} < T_{q} < \sim 600\;^\circ {\text{C }}\left( {{\text{region}}\;{\text{II}}} \right)$$
$$\sim 600\;^\circ {\text{C }} < T_{q} \left( {{\text{region}}\;{\text{III}}} \right)$$
Fig. 6
figure 6

The variation characteristics of surface crack density of the quartz lasca with quenching temperature (Iwasaki and Torikai 1993)

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To further analyze the relationship between quenching temperature and crack density, Tq and N are linearly fitted as follows in different regions as follows:

$$\begin{array}{*{20}c} {{\text{Region}}\;{\text{ I:}}} & {N_{1} = \, - 0.982 + 0.0075T} & {R^{2} = \, 0.94} \\ \end{array}$$
(6)
$$\begin{array}{*{20}c} {{\text{Region}}\;{\text{II:}}} & {N_{2} = \, 0.546 + 0.0018T} & {R^{2} = \, 0.95} \\ \end{array}$$
(7)
$$\begin{array}{*{20}c} {{\text{Region}}\;{\text{III:}}} & {N_{3} = \, 0.625 + 0.0021T} & {R^{2} = \, 0.96} \\ \end{array}$$
(8)

Crack density N seemed to increase with an increase in temperature T, indicating a strong positive correlation between crack density and temperature. This was demonstrated in that the thermal shock damage interaction between quartz and other minerals led to changes in the physical and mechanical properties of red sandstone. In this study, multiple cycle thermal shock was significant to crack development and damage of red sandstone and this also explains why P-wave velocity declined.

4.4 Suggestions for why P-wave velocity to rose abnormally

The crystal state of quartz in red sandstone changes under high temperatures. This is a phase transformation and the transformation forms of quartz differ at different temperatures. Seven crystalline states of quartz occur in nature, apart from the melting state, as shown in Fig. 7a (Kingery 1976; Wu et al. 2019a, b, c). These phase transition states can be divided into two types: reconstruction transitions and displacement transitions (Kingery 1976; Kihara 1990).

Fig. 7
figure 7

Changes of unit cell of silicon dioxide in displacement transition and reconstruction transition. a Phase transition of silica at different temperatures; b atomic structure of quartz; c displacement transition of quartz; d displacement transition of quartz; e reconstruction transition of quartz; f α quartz to β quartz (Kingery 1976; Wu et al. 2019a, b, c)

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Reconstruction transitions of silica break the original atomic connection and form a new atomic structure (Fig. 7b–e). This phase transition mode is relatively slow and compared with reconstruction transitions, the displacement transitions occur easily since the chemical bonds between atoms do not need to be destroyed nor do changes the basic structure need to occur. The displacement transitions can be accomplished by twisting the atomic structure (Kingery 1976) and this process is relatively rapid (Fig. 7b–d). The temperatures of the two tests in this study were below 800 °C. The main process of high-temperature thermal shock was the transformation of α quartz to β quartz, as shown in Fig. 7f.

Previous studies showed that when α quartz transformed into β quartz, cracking occurred due to the rapid transition speed or thermal expansion between different minerals could cause rock cracking damage. This led to a decrease in P-wave velocity (Hu et al. 2019a, b; Wu et al. 2019a, b, c). P-wave velocity increased abnormally in the range of 600–700 °C in two high-temperature thermal shock experiments of red sandstone (Fig. 1). We propose three hypotheses to explain the abnormal rise of P-wave velocity. When the thermal shock temperature was higher than the phase transition temperature (T > 573 °C), morphology of the quartz crystal changed from granular to needle-like and subsequently the needle-like crystal morphology could be retained by the thermal shock effect (Iwasaki and Torikai 1993; Hosaka et al. 1995). The granular and acicular crystal morphology of the quartz was evident in the study of Iwasaki and Torikai (1993). This process might decrease fractures between minerals and abnormally increase P-wave velocity.

The decrease in P-wave velocity was mainly caused by the formation of a continuous fracture network. The effect of a microstructural closure could explain the abnormal increase of P-wave velocity after the minimum point. This occurred due to the formation of a small quantity of melt that filled some cracks, as shown Fig. 8a (Lebedev and Zharikov 2000). Furthermore, the increase in fractures after multiple thermal shocks made the internal framework of sandstone relatively loose. However, local melting reconnected the loose skeleton and consolidated rapidly under the action of cold water and this caused the wave velocity of sandstone to increase abnormally.

Fig. 8
figure 8

Changes of internal cracks width in red sandstone before and after thermal shock. a Small quantities of melt filling the cracks; b crack width before putting in water; c rack width after being placed in water

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The uneven expansion of minerals at a high temperature caused cracks in rocks. Simultaneously, crack density increased with an increase in temperature (Fig. 8b). However, during thermal shock, the rock rapidly heated the surrounding water and even boiled for a short time (Van Otterloo et al. 2015). A pressurized vapor film formed between the water and the rock. The higher the rock temperature, the greater the pressure generated in the water. We suggest that the transient pressure during a thermal shock may compress the rock fissures, as shown Fig. 8c. This means that temperature has a competitive and synergistic effect on the density and width of cracks in rock before and after thermal shock.

Our three hypotheses each have limitations. Indeed, not all samples have abnormal P-wave velocity in the range of 600–700 °C. These hypotheses could also be influenced by other factors, such as the cooling time of samples taken from the muffle furnace to the water. The P-wave velocity anomaly in this experiment remains an important research topic for future investigation and may make geophysical exploration more accurate and reliable.

5 Conclusions

In this study, P-wave velocity characteristics were investigated through two red sandstone cyclical thermal shock tests and the influence of temperature and number of cycles on P-wave velocity were analyzed. The main conclusions are as follows:

  1. 1.

    P-wave velocity of red sandstone decreases overall with an increase in temperature and number of cycles, and this is particularly evident at temperatures above 400℃.

  2. 2.

    The relationship of damage factor D was established by P-wave velocity. Evidence showed that damage factor D increased with an increase in temperature and number of cycles.

  3. 3.

    In the two tests, the nonlinear fitting correlation coefficient R2 of P-wave velocity with temperature and number of cycles was above 0.9, which indicates that the fitting function can better evaluate the variation characteristics of P-wave velocity.

  4. 4.

    After repeated thermal shocks in water, P-wave velocity of sandstone could rise abnormally in the range of 600–700 °C and this could be caused by quartz phase transformation or partial melting filling fractures.