Introduction

Grotto temples are cave-style Buddhist relics carved into cliffs, integrating architecture, murals and sculptures, and carrying rich historical, cultural and artistic value1,2. Based on the lithology of the excavated rocks, the grotto temples can be classified into five types, including sandstone, conglomerate, mudstone, limestone and crystalline rock, with sandstone grotto temples being the most widely distributed3. In China, sandstone grotto temples account for more than 80% of all grotto temples4. As a sedimentary rock, sandstone is characterized by high porosity, weak cementation and strong water absorption, making it susceptible to weathering under the influence of the external environment5.

Studies have confirmed that water is a key factor leading to the weathering of sandstone, and its erosion methods include dry-wet cycles, freeze-thaw cycles, and others4,6. Among them, dry-wet cycles are the most common with significant deterioration effects, which have been widely studied by scholars7,8,9,10. The dry-wet cycles of grotto sandstone often occur in the presence of salts since natural water bodies generally contain salts11,12. In general, the soluble salts in water mainly include sulfates, chlorides, carbonates and nitrates13,14, with sodium sulfate (Na₂SO₄) being the most destructive15,16. Under the influence of dry-wet cycles, soluble salts undergo crystallization and dissolution processes. As water evaporates, soluble salts crystallize gradually, generating crystallization pressure17,18. When environmental humidity rises, salt crystals dissolve rapidly, and some salts (e.g., Na₂SO₄) combine with water molecules to form crystalline hydrates (e.g., Na₂SO₄·10H₂O), thereby generating hydration pressure19. Under the combined effects of repeated crystallization and dissolution of salts, as well as water erosion (including softening, dissolution, and hydrolysis), the sandstone undergoes progressive damage and deterioration. Such deterioration not only compromises the mechanical properties and structural integrity of the sandstone, but also exerts a pronounced influence on its thermal properties20,21,22.

Thermal property parameters, primarily including thermal conductivity, thermal diffusivity, specific heat capacity and thermal effusivity, are critical indicators for characterizing the physical properties of rocks23. They affect the temperature distribution of the rock mass and are crucial for heat transfer, thermal stress calculation and thermal damage analysis2,6. Previous studies have demonstrated that the thermal properties of rocks are jointly influenced by internal structures (e.g., pore systems) and external conditions (e.g., pressure)24,25,26. During dry-wet cycles involving soluble salts, processes such as crystallization, hydration, softening, and dissolution alter the mineral composition, pore characteristics, and crack density of rocks, thereby inducing dynamic changes in thermal parameters18. In recent years, scholars have gradually revealed the evolution laws and controlling factors of the thermal properties of sandstone during the dry-wet cycling process through indoor simulation experiments. For example, Sun and Zhang27 found that dry-wet cycles can reduce the thermal conductivity of sandstone, with 30 cycles being the critical point for parameter changes. Bu et al.28 further demonstrated that pores and microcracks are the primary factors contributing to the deterioration of thermal conductivity and the enhancement of thermal heterogeneity. These findings have significantly advanced our understanding of the thermal behavior of rocks. However, most existing studies have focused on conventional parameters such as thermal conductivity and thermal diffusivity, while thermal effusivity has received far less attention.

Thermal effusivity determines the heat transfer intensity of materials in unsteady temperature fields, with its physical meaning being the square root of the product of density, specific heat capacity and thermal conductivity23,29,30. As a comprehensive thermal property parameter, thermal effusivity is closely associated with the preservation state of rocks and can be employed to assess their deterioration31,32,33. Meanwhile, thermal effusivity can be measured in a non-contact manner by means of active infrared thermography34,35, thereby preventing damage to the rock mass during parameter testing. Therefore, thermal effusivity has significant application potential in geotechnical engineering and cultural relic protection projects. However, current studies have primarily focused on testing methods and theoretical calculations30,36, and a systematic understanding of the evolution of sandstone thermal effusivity during dry-wet cycles in soluble salt solutions is still lacking. Additionally, the effects of factors such as mineral composition, pores, and cracks on the thermal effusivity under cyclic conditions are still not fully clarified and warrant further investigation.

This study aims to investigate the stage-dependent deterioration characteristics of the thermal effusivity of sandstones during dry-wet cycles involving soluble salts, clarify its evolution mechanisms and main influencing factors at different stages, and establish a corresponding predictive model. To achieve this objective, cyclic deterioration tests were performed on sandstones from the Upper Jurassic Suining Formation using sodium sulfate solution, and the evolution process of thermal effusivity was analyzed using active infrared thermography. On this basis, degradation parameters of the sandstone were obtained using testing techniques such as scanning electron microscopy, nuclear magnetic resonance, and X-ray diffraction, and the relationship between these parameters and thermal effusivity was investigated through regression analysis. The findings are expected to advance the application of thermal effusivity in the diagnosis of deterioration and damage assessment of grotto temples, thereby providing a scientific basis for preventive conservation.

Methods

Sample preparation

The sandstone samples used in the experiment were collected from the Yuanjue Cave stone-carving area in Anyue County, Sichuan Province, China (105°20'39.48” E, 30°05'14.96” N). The site was originally constructed during the Song Dynasty (1091 AD) and is now a National Key Cultural Relics Protection Unit. Geologically, the Yuanjue Cave stone-carving area lies in the central part of the gentle fold belt of central Sichuan, at the junction of the Longnüsi semi-annular structure and the Weiyuan radial structure. The geological structure of the area is relatively simple, with exposed strata comprising sandstones and mudstones of the Upper Jurassic Suining Formation, along with Quaternary unconsolidated deposits. The grotto sculptures are situated on the sandstone cliff at the top of the hill. The Yuanjue Cave stone-carving area has a subtropical monsoon humid climate, with an average annual rainfall of 1025.8 mm and an average of 147 rainy days per year. The humid and rainy climate subjects the sandstone to prolonged dry-wet cycles.

Previous studies have shown that the sandstone is cemented with calcite and sericite, exhibiting a loose structure and poor cementation36,37. According to the recommendations of the International Society of Rock Mechanics (ISRM) for determining thermal properties of rocks38, the rock blocks were processed into cylindrical samples 50 mm in diameter and 25 mm in height, with polished sides and ends. The deviations in sample height and diameter were within 0.3 mm, and the flatness error of both end faces was controlled within 0.05 mm. Twenty-four sandstone samples with similar wave velocities (1.74–1.79 km/s) were selected for testing. Details regarding Yuanjue Cave and the rock samples are provided in Fig. 1.

Fig. 1: Location and overview of the Yuanjue Cave and sample preparation.
Fig. 1: Location and overview of the Yuanjue Cave and sample preparation.
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a Geographic information. b Stone carvings in the scenic area. c Preparation of sandstone samples using fallen rocks from the scenic area.

Experimental design

Following previous studies4,39, an anhydrous sodium sulfate was used to prepare a soluble salt solution with a concentration of 10% to reasonably induce the deterioration of sandstone within the experimental period. The 24 sandstone samples were divided into three parallel test groups (denoted as I, II, and III), each consisting of 8 samples. The eight samples in each group underwent 0–7 dry-wet cycles, respectively, to obtain sandstone samples with varying degrees of deterioration. It should be noted that the three parallel groups were subjected to identical experimental conditions, procedures, and testing methods, with the aim of ensuring the reliability of the results. For ease of identification, the samples were labeled in the format “group–cycle number”. The group was denoted using Roman numerals (I, II, III), while the cycle number was indicated using Arabic numerals (0–7). For example, I–1 denotes the sample that underwent one cycle in the first parallel test group.

Prior to the experiments, the samples were dried in an oven at 105 °C for 24 h to ensure consistency in their initial state. It should be noted that at 105 °C, the thermal stress within sandstone is relatively low and does not induce significant changes in its macro and micro structure40,41. The oven used in this study has a maximum temperature of 300 °C with a precision of ±0.1 °C. Each deterioration cycle lasted 24 h and consisted of three stages. In the drying stage, the samples were heated to 105 °C at a rate of 5 °C/min and maintained at this temperature for 10 h42,43. Afterward, in the cooling stage, the samples were removed and allowed to naturally cool at room temperature (25 °C) for 4 h. Finally, in the immersion stage, the cooled samples were immersed in a sodium sulfate solution at room temperature for 10 h.

It should be noted that the purpose of this study is to analyze the evolution law of thermal effusivity and its influencing factors through accelerated deterioration tests, rather than to simulate the salt weathering process of grotto sandstone under natural conditions. After the cyclic tests, the samples were measured for thermal effusivity and basic properties, followed by further analyses using scanning electron microscopy (SEM), nuclear magnetic resonance (NMR), and X-ray diffraction (XRD). The testing workflow is illustrated in Fig. 2, and the detailed testing procedures are provided below.

Fig. 2: Sample parameter testing.
Fig. 2: Sample parameter testing.
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a Measurement process. b Some of the equipment used.

Thermal effusivity test

Currently, thermophysical properties of rocks are primarily measured using instruments like the Hot Disk TPS 2500S and ISOMET 2114. Although these instruments can provide accurate results, the contact-based measurement methods they employ impose strict requirements on sample surface flatness and minimum dimension. This technical characteristic makes them unsuitable for samples with rough surfaces or complex shapes44,45.

In this study, infrared thermography was employed for non-contact measurement of thermal effusivity. Infrared thermography can be classified as either passive or active. Passive infrared thermography uses the infrared radiation of an object itself to carry out testing, whereas active infrared thermography applies an external heat source to modify the temperature field of the target, subsequently capturing and analyzing its infrared radiation. Compared with the passive approach, active thermography significantly enhances thermal contrast and the signal-to-noise ratio, yielding superior measurement performance.

The experimental system consisted of an infrared thermal imager, a 2000 W far-infrared quartz heating plate, a heat flux sensor, a computer, and a multifunctional tripod (Fig. 2b). Prior to the experiment, the heating plate was preheated for 5 min to achieve thermal equilibrium. Subsequently, the distance between the heating plate and the sample was precisely set to 0.35 m using a fixed bracket, and the sample was subjected to constant-power heating for 360 s. Measurements with the JT2025R heat flux sensor (accuracy ±3%, Beijing Jiantong Technology Co., Ltd.) indicate that the heat flux density at the top of the sample was qw = 1850 W·m−2. During the heating process, temperature images of the sample surface were captured using a Fluke Ti480 Pro infrared thermal imager. The infrared camera provides a spatial resolution of 640 × 480 pixels, a thermal sensitivity of ≤0.05 °C, and a temperature measurement accuracy of ±2%. To minimize the influence of sample surface roughness on temperature measurement accuracy, the emissivity of the thermal imager was set to 0.93. In related studies, this value has been regarded as the average emissivity of sandstone46,47.

After the tests were completed, the heat transfer process of the sandstone samples was analyzed using heat transfer theory. Since the direction of thermal radiation is always perpendicular to the sample surface, the heat within the sandstone primarily transfers along the depth direction. According to heat transfer theory, if heat is transmitted predominantly in one direction, or if conduction in that direction is dominant, the heat transfer process can be described using a one-dimensional equation. For a sandstone sample with an initial temperature of t₀, subjected to a constant heat flux qw at time zero, the one-dimensional heat conduction equation can be expressed as follows36:

$$\frac{\lambda }{\rho c}\frac{{\partial }^{2}t(x,\tau )}{\partial {x}^{2}}=\frac{\partial t(x,\tau )}{\partial \tau },(0 < x < +\infty )$$
(1)

Boundary condition:

$$x=0,q={q}_{w}$$
(2)

Initial condition:

$$\tau =0,t\left(x,0\right)={t}_{0}$$
(3)

Where t is the temperature (°C); λ is the thermal conductivity (W·m−1·K−1); ρ is the material density (kg·m−3); c is the specific heat capacity (J·kg−1·K−1); x is the distance from the sample surface (m); and τ is the time (s).

The calculation formula for the thermal effusivity can be obtained by using the separation of variables method.

$$\vartheta =\sqrt{\lambda \rho c}=\frac{1.12{q}_{{\rm{w}}}\sqrt{\tau }}{t(0,\tau )-{t}_{0}}$$
(4)

Where \(t(0,\tau )\) represents the temperature of the surface of the sample at time τ.

According to the heating time τ, heat flux density qw and temperature t, the thermal effusivity ϑ of the sandstone sample can be calculated by using formula (4).

Testing of basic properties of sandstone

The sample appearance and dry mass Ms were recorded using a Canon RP digital camera and an electronic balance with 0.1 g accuracy, respectively. Subsequently, the sandstone samples were subjected to a 24 h vacuum saturation treatment, and the saturated mass Msat was measured. Based on the principle of the water displacement method, the volume V of the saturated samples was measured using a beaker and graduated cylinder (accuracy ±0.1 ml), ensuring that the eye level was aligned with the liquid surface during readings. The volume and mass of each sample were measured three times, and the average values were taken to reduce random errors. Finally, the porosity p of each sample was calculated using Eq. (5).

$$p=\frac{{V}_{{\rm{v}}}}{V}\times 100 \% =\frac{{M}_{{\rm{sat}}}-{M}_{{\rm{s}}}}{{\rho }_{w}\times V}\times 100 \%$$
(5)

Where Vv represents the volume of the pores.

Nuclear magnetic resonance (NMR) test

NMR utilizes the magnetic resonance phenomenon of atomic nuclei to study the pore structure of materials, offering the advantages of being rapid and non-destructive48. The saturated samples were sequentially placed into the nuclear magnetic resonance (NMR) analyzer to obtain T2 spectra and NMR images. The test was conducted using a MacroMR12-150H-I nuclear magnetic resonance imaging analyzer produced by Suzhou Niumag Analytical Instrument Corporation, Suzhou, China. The instrument operates at a main frequency of 12.8 MHz, with a magnetic field strength of 0.3 ± 0.05 T and a magnetic field homogeneity of ≤50 ppm.

Scanning electron microscopy (SEM) analysis

SEM is a high-resolution, high-contrast imaging technique used to analyze the pores, microcracks, and particle contact relationships in rocks49. Prior to SEM analysis, a 2 mm cubic sample was extracted from each sample. Surface dust and loose particles were removed, followed by drying, gold coating, and vacuum treatment. The samples were observed using a JSM-7610F Scanning Electron Microscope at 1000× magnification to obtain clear micrographs.

X-ray diffraction (XRD) measurements

To elucidate the influence mechanism of dry-wet cycles on the thermal effusivity of sandstone, XRD was further employed to analyze the mineral composition of the samples. The dried samples were ground into powder finer than 400 mesh, and 15 g was selected for XRD analysis. An XRD-6100 X-ray powder diffractometer was used, with a goniometer radius of 185 mm and an angular accuracy of ±0.001. During the test, the scanning rate was set to 10°/min, with a diffraction angle range of 5°–80° and a step size of 0.02°. After the test, Jade software was used for phase identification and semi-quantitative analysis of the diffraction patterns.

Results

Evolution characteristics of appearance, volume and mass

Appearance, volume and mass can intuitively reflect the deterioration state and material loss of sandstone samples12. According to the experimental scheme, the image, volume, and mass data of the samples under different numbers of cycles were collected, as shown in Fig. 3. Figure 3a shows that at 0 cycle, the sandstone surface is flat and smooth, retaining its original appearance. As the number of cycles increases, phenomena such as cracking, hollowing and powdering appear on the sandstone surface. As cracks continue to propagate and extend, severe spalling occurs in the samples. By the end of 7 cycles, the sandstone sample exhibits a rough, porous surface with severely compromised structural integrity, showing a 36.87% volume reduction (Fig. 3b). In addition, as shown in Fig. 3c, after 7 cycles the sample mass drops from 103.43 g to 65.7 g, representing a loss of 36.48%. It can be observed that the sample mass exhibits an accelerating decline with increasing cycle numbers, indicating that spalling primarily occurs in the later stages. The mass loss reflects the gradual change process of appearance and volume, while the alterations in appearance and volume serve as strong evidence of mass loss. Together, these changes reveal the evolutionary law of sandstone from minor damage to severe deterioration during dry-wet cycles.

Fig. 3: Macroscopic characteristics of sandstone samples.
Fig. 3: Macroscopic characteristics of sandstone samples.
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a Surface morphology of samples at different cycles (side and top views). b Volume variation with cycle numbers. c Mass variation with cycle numbers.

Evolution law of thermal effusivity

Thermal effusivity comprehensively characterizes the thermal properties of rocks under different deterioration states. In general, higher thermal effusivity indicates less rock deterioration and a faster rate of heat transfer33,50. In this study, the temperature field of the samples was collected at the end of thermal excitation (Fig. 4a), and the average temperature of the top surface was extracted using SmartView software (Fig. 4b). Based on this, the thermal effusivity of samples with different cycle numbers was calculated using Eq. (4), as shown in Fig. 4c. It should be noted that although the thermal effusivity of the samples was not measured at exactly the same temperature, the influence of temperature on the calculated values can be regarded as negligible. Thermal effusivity is defined as the square root of the product of density, specific heat capacity and thermal conductivity, among which the specific heat capacity and thermal conductivity exhibit a strong correlation with temperature. When the temperature increases from 63.03 °C to 78.65 °C, the thermal conductivity shows a slight decrease, whereas the specific heat capacity exhibits a slight increase51,52. Therefore, the changes in these two factors offset each other to a certain extent, resulting in an insignificant influence of temperature on the calculation results of thermal effusivity.

Fig. 4: Infrared thermography test results of sandstone samples.
Fig. 4: Infrared thermography test results of sandstone samples.
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a Temperature field of the samples. b Average temperature of the top surface of the samples in (a). c Thermal effusivity calculated from temperature data in (b).

In Fig. 4a, there is a significant difference in the temperature fields of the samples. At lower cycle numbers, the top surfaces exhibit lower temperatures, appearing as yellow-green regions in the thermal images. As the number of cycles increases, the deterioration phenomena such as cracking, powdering and hollowing lead to a rise of the top surface temperature and cause localized heat accumulation effects (shown as red or orange-yellow high-temperature zones). Data in Fig. 4b show that the average temperature on the top surface rises from 63.03 °C at 0 cycles to 78.65 °C at 7 cycles. This indicates that dry-wet cycles significantly reduce the heat transfer efficiency of the samples, causing more heat to be retained on the surface layer. Consequently, thermal effusivity in Fig. 4c drops from 1402.38 to 900.65 W·s¹/²·m−2·K−1 after 7 cycles. In summary, dry-wet cycles in sodium sulfate solution significantly affected the thermal transfer properties of sandstone, leading to a progressive decline in thermal effusivity.

To further explore the evolution characteristics of the thermal effusivity during the cycling process, this section introduces the loss rate δn of thermal effusivity, which is used to quantitatively describe its degradation degree relative to the initial state. The calculation is defined by the following formula:

$${\delta }_{n}=\frac{{\vartheta }_{0}-{\vartheta }_{n}}{{\vartheta }_{0}}\times 100 \%$$
(6)

Where ϑ0 and ϑn represent the thermal effusivity after undergoing 0 and n (n = 0, 1, ⋯, 7) dry-wet cycles, respectively.

Figure 5 presents the calculated results of the thermal effusivity loss rate. Although some data points show slightly higher dispersion, the deterioration process of thermal effusivity can still be divided into two stages based on the overall evolution trend: the rapid period (0–3 cycles) and the fluctuating period (4–7 cycles). During the rapid stage, the loss rate of thermal effusivity increases from 0 to 24.43%. The large increase in the loss rate during this stage indicates that the degree of sandstone deterioration is continuously intensifying, and the heat transfer efficiency is significantly reduced. In contrast, during the fluctuating stage, the loss rate of the thermal effusivity no longer follows a simple rule of increase or decrease. The loss rate shows a fluctuating trend of rising first, then falling, and then rising again, with values of 26.71%, 29.86%, 27.02%, and 35.68%, respectively. Therefore, the thermal effusivity exhibits different deterioration rates and evolution characteristics at different stages.

Fig. 5: Loss rate of thermal effusivity and its degradation stages.
Fig. 5: Loss rate of thermal effusivity and its degradation stages.
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During the fluctuation stage, extensive spalling occurred on the surface of the samples.

At present, studies on the evolution stages of thermal parameters during dry-wet cycles remain relatively limited. Most studies have regarded the variation of thermal parameters during dry-wet cycles as a single increasing or decreasing process27. For example, Bu et al.28 found in the dry-wet cycles tests that the thermal conductivity of sandstone showed a continuous downward trend of 12.97%. However, different from the above research results, this study found that the thermal effusivity fluctuated significantly in the later stages of the cycle. This phenomenon is attributed to the high concentration of the salt solution and the low degree of sandstone cementation, which resulted in large-area spalling of the samples in the later stages of the cycling (Fig. 3). As is well known, rock weathering typically progresses from the surface inward. Surface spalling exposes the less weathered interior regions of the rock, thereby enhancing heat transfer performance in the short term. As the dry-wet cycles continue, the newly exposed areas gradually deteriorate, creating a “spalling-deterioration” cycle that leads to fluctuations in the thermal effusivity. Consequently, the evolution of the thermal effusivity exhibits a distinct two-stage pattern, with spalling being the primary driver of its fluctuations.

Analysis of pore characteristics

NMR can accurately characterize the pore structure of rocks by detecting the relaxation times of hydrogen nuclei in pore fluids. The technique primarily measures two relaxation parameters: longitudinal relaxation time (T1) and transverse relaxation time (T2). Due to its faster measurement process, T2 has become the primary indicator for analyzing the characteristics of pore structure. Previous studies show that larger pores in rocks allow greater molecular mobility of fluids, resulting in longer T248,53. Therefore, the T2 is positively correlated with pore radius r, as shown in Eq. (7).

$$r={\rho }_{2}{F}_{s}{T}_{2}$$
(7)

Where ρ2 is the surface relaxation intensity, in μm/ms；Fs is the geometric factor. For a given pore shape, Fs is a constant.

Figure 6 reveals the evolution of the T2 spectrum of sandstone samples under dry-wet cycles. The T2 peak time shifts rightward with increasing cycle numbers, indicating a progressive increase in pore size in the sandstone samples. The T2 spectrum morphology transitions from an initial bimodal distribution (0–1 cycles) to a unimodal pattern (2–7 cycles). Simultaneously, the peak intensities of the T2 spectra show an increasing trend, indicating a gradual rise in pore quantity within the samples. Notably, the peak attenuation observed at 6–7 cycles is primarily due to the decrease of water content, which results from the reduced sample volume. Therefore, the evolution of the T2 spectrum reflects the increase in both pore size and pore quantity in the sandstone.

Fig. 6: Variation patterns of T2 spectrum (with partial enlarged views) and water content in samples under different numbers of cycles.
Fig. 6: Variation patterns of T2 spectrum (with partial enlarged views) and water content in samples under different numbers of cycles.
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The figure illustrates the variations in pore size and pore quantity of the sandstone sample.

Magnetic resonance images (Fig. 7) more clearly demonstrate the dynamic evolution of pore structures in the samples. In the magnetic resonance images, the color gradient (red > yellow > green > blue) reflects regional differences in water content. Red areas indicate water-rich zones with high porosity, while blue areas correspond to dense structures with low porosity. In the initial state, the magnetic resonance images display a uniform blue background, indicating a dense structure and low porosity in the samples. As the dry-wet cycles progress, yellow-green areas gradually extend from both ends of the samples toward the center, indicating a continuous increase in porosity. After seven cycles, the yellow-green areas became widely distributed, indicating that the samples had transformed into loose porous media. The similar evolution laws of the T2 spectrum and magnetic resonance images indicate that the dry-wet cycles altered the pore characteristics of the sandstone samples, resulting in a significant increase in their porosity.

Fig. 7: Magnetic resonance images of sandstone samples.
Fig. 7: Magnetic resonance images of sandstone samples.
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The color gradient (red > yellow > green > blue) reflects the differences in water content across different regions.

Analysis of morphological features and mineral composition

SEM is widely used to investigate microstructural alterations in rocks. Figure 8 displays the SEM images of sandstone samples at varying deterioration levels under 1000× magnification. At zero cycles, the sandstone sample exhibits a compact structure with no visible cracks and only minor debris adhering to the surface. As dry-wet cycles progress, weathering-prone minerals such as feldspar and mica are gradually eroded and decomposed. This leads to weakened cementation between grains, accompanied by an increase in both the number and size of pores and microcracks. In the later stages of the dry-wet cycles, the sandstone microstructure underwent significant changes. Erosion of the cementing material intensified, and the connectivity of pores and cracks increased significantly, resulting in sandstone samples becoming loose media with structural damage and pore development.

Fig. 8: Microstructural evolution characteristics of samples under different numbers of cycles.
Fig. 8: Microstructural evolution characteristics of samples under different numbers of cycles.
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All images were captured at 1000× magnification.

To investigate the variation patterns of mineral composition in sandstone samples under dry-wet cycles, whole-rock analysis was conducted using XRD, with the results shown in Fig. 9a. The results indicate that fresh sandstone is primarily composed of quartz, albite, K-feldspar, muscovite, calcite and hematite. Except for quartz, the diffraction peak intensities of albite, K-feldspar, muscovite, calcite, and hematite are relatively low. The mineral contents were quantified using Jade software based on the diffraction patterns of sandstone powder (Fig. 9b). It can be seen from Fig. 9 that the content of various minerals shows a fluctuating trend during the dry-wet cycle, which is similar to findings reported in some studies12,54.

Fig. 9: Test results of sandstone minerals.
Fig. 9: Test results of sandstone minerals.
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a X-ray diffraction patterns. b Mineral content.

Discussion

Thermal effusivity depends on the square root of the product of density, specific heat capacity, and thermal conductivity. Therefore, the deterioration mechanism of thermal effusivity can be revealed by analyzing the effects of pores, cracks and mineral content on these three parameters. All sandstone samples are kept dry during the measurement of thermal effusivity. Dry sandstone samples can be regarded as solid-gas biphasic media, where the thermophysical properties of the gas phase are significantly lower than those of the solid phase27,55. Dry-wet cycles promote the development of internal pores and cracks in the rock, increasing the gas-phase proportion. These structural changes weaken the continuity of heat transfer paths and the heat storage capacity per unit volume, thereby reducing thermal conductivity and specific heat capacity26. Simultaneously, the expansion of the pore-crack network also reduces the volume proportion of solid minerals, leading to a decrease in rock density. It is noteworthy that although the mineral content exhibits minor fluctuations during the dry-wet cycles (Fig. 9), the magnitude of these changes is limited and has a relatively small impact on the overall thermal properties6. Therefore, pores and cracks are the main factors contributing to the deterioration of thermal effusivity.

To further analyze the dominant roles of pores and cracks at different stages, the variations in porosity and number of cracks of the sample are presented (Fig. 10). It should be noted that the number of cracks was manually counted from the fused visible-light and infrared images of the samples, and this parameter originates from one of our previous studies36. Although this method may have difficulty capturing small cracks deep within the samples, given that weathering typically progresses from the surface inward, the vast majority of weathering cracks appear in the shallow surface layer of the samples. Therefore, this method can effectively ensure the reliability of the crack count results. As shown in Fig. 10, during the rapid period, the porosity increases sharply from 21.83% to 28.95%, while the number of cracks rises from 0 to 3. In this stage, the development of pores and cracks increased the resistance to heat transfer, with the effect of pores playing a dominant role, resulting in an average loss rate of thermal effusivity of 11.58% across different cycle numbers. Upon entering the fluctuation period, the average loss rate of thermal effusivity surged to 29.82%. At this stage, the porosity remained around 33%, whereas the number of cracks increased sharply. The initiation and propagation of cracks caused large-scale spalling of the samples (Fig. 3), leading to fluctuations in thermal effusivity. In this stage, the influence of cracks on thermal effusivity became significantly stronger. It can be observed that the influence of pores and cracks on thermal effusivity evolves dynamically across different stages. During the rapid period, pores play the dominant role while the influence of cracks gradually emerges. In the fluctuation period the effect of cracks significantly intensifies, whereas the effect of pores tends to stabilize.

Fig. 10: Variations in the porosity and the number of cracks of sandstone samples.
Fig. 10: Variations in the porosity and the number of cracks of sandstone samples.
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This figure shows the differentiated variations of porosity and crack number across different degradation stages.

Figure 11 presents the relationships between thermal effusivity and the number of cracks, as well as between thermal effusivity and porosity. It can be seen that both relationships are well fitted by negative exponential functions, with correlation coefficients exceeding 0.9. Therefore, we attempted to construct an evolution model of thermal effusivity based on a negative exponential function (Eq. (8)), which predicts the thermal effusivity from porosity and the number of cracks.

$$\vartheta ={a{\rm{e}}}^{-b}+c$$
(8)

Where a, b, and c are the coefficients of the model.

Fig. 11: Relationships between the thermal effusivity and porosity, and between the thermal effusivity and the number of cracks.
Fig. 11: Relationships between the thermal effusivity and porosity, and between the thermal effusivity and the number of cracks.
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Both parameters exhibit a strong correlation with thermal effusivity.

Referring to relevant studies56,57, this section constructs the mathematical expression of coefficient b using porosity (p) and the number of cracks (d), while coefficients a and c are treated as constants. To facilitate model solving, the raw data were nondimensionalized using Eq. (9). A multivariate optimization analysis was performed using the nonlinear least-squares method in MATLAB, and the expressions for coefficients a, b, and c were determined, as shown in Eq. (10). Subsequently, substituting Eq. (10) into Eq. (8) yielded the evolution model of thermal effusivity, as shown in Eq. (11). Table 1 lists the relevant model parameters.

$$\left\{\begin{array}{l}\bar{d}=\frac{d}{{\sum }_{n=0}^{7}{d}_{n}}\\ \bar{p}=\,{\log }_{10}\,(p\times 100)/{\log }_{10}({p}_{0}\times 100)\end{array}\right.$$
(9)
$$\begin{array}{l}{\rm{a}}=3748\\ b=-1.42\bar{p}-2.5\bar{d}\\ {\rm{c}}=500\end{array}$$
(10)
$$\overline{\vartheta }=3748\cdot {e}^{-1.42\overline{p}-2.5\overline{d}}+500$$
(11)

Where p and d represent the porosity (measured in %) and the number of cracks (measured in strips), respectively; p0 is the initial porosity of the sample; \(\overline{p}\) and \(\overline{d}\) represent the nondimensionalized values of p and d, respectively; \(\overline{\vartheta }\) represents the predicted values of thermal effusivity, in W ∙ s1/2 ∙ m−2 ∙ K−1.

Table 1 Relevant parameters of the evolution model of thermal effusivity
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Based on the data in Table 1, a 1:1 scatter plot of the predicted and measured thermal effusivity values was plotted (Fig. 12). In Fig. 12, the data points cluster near the ideal matching line y = x, with all points falling within the ±15% error margin, indicating the high accuracy of the prediction model. In terms of modeling approach, the thermal effusivity prediction model developed in this study adopts a negative exponential form, with two structural parameters, porosity and crack number, as the core variables, and it belongs to an empirical model32. This modeling idea also has certain reference value for the prediction of physical and mechanical parameters such as wave velocity and strength during the rock weathering process. In addition, the model has a simple form and its parameters are easy to obtain, which can well describe the nonlinear variation characteristics of thermal effusivity during the sandstone deterioration process6. Compared to the thermal parameter models (primarily thermal conductivity) in Table 2, this model introduces the crack parameter, which improves the response ability to structural deterioration. Especially in the later stage when cracks expand rapidly and local spalling is significantly intensified, the model capability to predict the evolution of thermal performance becomes more prominent. Considering that grotto temples have undergone hundreds to thousands of years of natural weathering and small-scale weathering cracks are commonly developed in the grotto sandstone, it is of great practical significance to incorporate the crack index into the thermal parameter prediction model37.

Fig. 12: A 1:1 scatter plot of the measured versus predicted values of the thermal effusivity.
Fig. 12: A 1:1 scatter plot of the measured versus predicted values of the thermal effusivity.
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The closer a data point is to the red solid line, the smaller the prediction error.

Table 2 Some thermal parameter prediction models
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It should be noted that research on predictive models of thermal effusivity remains relatively limited, and the model proposed in this study is still a preliminary exploration, and its applicability needs to be further validated in a broader range of scenarios58. Compared with theoretical models or mixing models, this model has not yet introduced parameters such as mineral composition, pore size and saturation, leaving room for improvement in representing the influence of microstructural features and moisture conditions on thermal effusivity. Despite these limitations, the model developed in this study still provides a valuable tool for predicting thermal effusivity. To further quantify the influence of pores and cracks on thermal effusivity, we performed a sensitivity analysis of the model parameters. Referring to a previous study59, sensitivity is defined as the ratio of the relative error of the output parameter to the relative error of the input parameter. Taking porosity as an example, its sensitivity \(S(\bar{p})\) can be calculated using Eq. (12).

$$S(\bar{p})=\frac{\left(\frac{|\Delta \bar{\vartheta }|}{\bar{\vartheta }}\right)}{\left(\frac{|\Delta \bar{p}|}{\bar{p}}\right)}=\frac{|\Delta \bar{\vartheta }|}{|\Delta \bar{p}|}\cdot \frac{\bar{p}}{\bar{\vartheta }}$$
(12)

Where \(|\Delta \bar{\vartheta } |/\bar{\vartheta }\) denotes the relative error of the thermal effusivity, and \(|\Delta \bar{p}|/\bar{p}\) denotes the relative error of porosity.

When \(|\Delta \bar{p}|/\bar{p}\) approaches 0, Eq. (12) can be rewritten as follows.

$$S(\bar{p})=\left|\frac{\partial \bar{\vartheta }(\bar{p},\bar{d})}{\partial \bar{p}}\right|\cdot \frac{\bar{p}}{\bar{\vartheta }}$$
(13)

By substituting Eq. (11) into Eq. (13), the sensitivity of porosity can be obtained. Similarly, the sensitivity of cracks can be calculated, and the results are shown in Fig. 13. The analysis reveals that the sensitivity of pores consistently exceeds that of cracks throughout the cycling process. This is because the degree of crack development is relatively low in the early stage of the cycle, failing to significantly change the continuity of the heat transfer path. Meanwhile, most cracks remain closed during the cycles, resulting in low air content within the cracks, thereby reducing their ability to impede heat transfer. In terms of changing trends, the sensitivity of porosity decreases with the number of cycles, whereas that of cracks shows the opposite pattern. These results indicate that pores play a dominant role in the evolution of thermal effusivity, though this dominance rapidly attenuates with crack development.

Fig. 13: Variation of the sensitivity of pores and cracks with the number of cycles.
Fig. 13: Variation of the sensitivity of pores and cracks with the number of cycles.
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The sensitivities of porosity and cracks exhibit opposite trends.

In this paper, dry-wet cycle tests were conducted on sandstone samples from the Upper Jurassic Suining Formation (grotto sandstone of the Yuanjue Cave, Sichuan, China) exposed to sodium sulfate solution, and the evolution law and deterioration mechanism of thermal effusivity were investigated. On this basis, an evolution model of thermal effusivity was established using laboratory test data. The main conclusions are summarized as follows:

  1. (1)

    The dry-wet cycles in sodium sulfate solution significantly affected the heat transfer performance of the sandstone samples. As the number of cycles increased, deterioration phenomena such as hollowing, cracking, and powdering appeared in the sandstone samples, leading to reduced thermal effusivity and causing heat accumulation effects.

  2. (2)

    The evolution process of thermal effusivity can be divided into a rapid period and a fluctuation period. The rapid period is primarily characterized by a continuous decline, while the fluctuation period shows a short-term recovery in thermal effusivity due to the peeling of weathered surface layers of the samples. The differences between the two stages reveal the dynamic and complex nature of the thermal performance degradation process of rocks under dry-wet cycles.

  3. (3)

    The deterioration of thermal effusivity was primarily caused by pore growth and the initiation and expansion of cracks. These structural changes not only reduced rock density but also weakened the continuity of heat transfer pathways and the heat storage capacity per unit volume. During the rapid period, pores play the dominant role while the influence of cracks gradually emerges. In the fluctuation period the effect of cracks significantly intensifies, whereas the effect of pores tends to stabilize.

  4. (4)

    An evolution model of thermal effusivity was established using a negative exponential function, which predicts the thermal effusivity through porosity and the number of cracks. The sensitivity analysis based on the model showed that pores played a dominant role in the evolution of the thermal effusivity, but their dominance rapidly decayed with the development of cracks, confirming a competitive mechanism of pore attenuation versus crack enhancement.