Quantitative evaluation of weathering degree through Fuzzy-AHP method and petrophysics analysis for sandstone carvings

The assessment of weathering degree of sandstone carvings plays an important role in scientific conservation. However, the current state of weathering assessment research cannot meet the needs of scientific evaluation. The Jurassic sandstones used in this study were selected, and soluble salt and acid simulation experiments were performed on standard samples. Non-destructive methods are used to determine the weight, surface hardness, ultrasonic velocity, and water absorption. Furthermore, mineral composition and petrophysical properties are analyzed in the laboratory on small specimens. It is revealed that the main mineral, such as quartz and feldspar, is shown by oblate and prolate in shape. The moisture content of rock samples has a significant impact on their compressive strength. The deterioration process increases the porosity of the rock. As the number of cycles increases, so does the quality, surface hardness, and ultrasonic velocity. Water absorption coefficient, on the other hand, increases over cycles. In addition, the Fuzzy-AHP method is adopted to further evaluate degree of weathering on experiment samples. The macroscopic results show that the weathering degree is 0.3271 for sodium sulfate solution, indicating moderate weathering; 0.1951 for pH = 1 sulfuric acid and 0.1060 for pH = 2 sulfuric acid, showed low weathering; and 0.0516 for pH = 3 sulfuric acid and 0.0421 for pH = 4 sulfuric acid, inferring very low weathering.


Introduction
Damage to cultural heritage is frequently caused by salt, rainwash, and ice crystallization. The granular disintegration (Fig. 1a), tafoni (Fig. 1b), salt efflorescence (Fig. 1c), losing (Fig. 1d), and scaling ( Fig. 1e) are representative examples of a deterioration that may be influenced significantly by rock weathering. Generally, Jurassic sandstone, being entirely crystalline, originates from geological periods between the Cretaceous and the Triassic. The petrophysical properties of sandstone are controlled by the rock fabric, which includes grain size, sorting, and roundness. Furthermore, the swelling and shrinking of clay minerals, as well as alteration, may cause mic-cracking within the rock mass.
Rock weathering caused by freeze-thaw (Piantelli et al. 2020;Tianbiao et al. 2016), salt crystallization (Jingke et al. 2021;Zhongjian et al. 2015), thermal shock (Murru et al. 2018) and dissolution (Lanqin et al. 2018) has become a popular focus of research. However, little works have been conducted to determine the weathering degree, as it takes into account numerous weathering factors. Incipiently, typical cavernous weathering (i.e., 'Tafoni,' 'honeycomb,' 'alveolar,' etc) commonly occurs in sandstone and granitic outcrops, which have been described by Mustoe (1983) and analyzed by Mellor (1997), who stated that this weathering formation is initiated at weak points, then, elevated humidity  (Qingyang, Northwest China); d loss damage on Jurassic sandstone in Chongqing and e scaling of strongly layered in Chengde may promote core softening, which in turn leads to further cavern enlargement (Alexandrowicz 1989). Afterward, several researchers used chemical indices to determine weathering degree (Chiu and Ng 2014;Dong et al. 2015). It should be noted that chemical indices may change as a result of mineral transformation or replacement, they are not credible indicators to evaluate weathering degree. In addition, mechanical and physical properties of rocks, such as Schmidt hammer rebound value, P-wave velocity, porosity, compressive and tensile strength, govern their physical behavior (Martinho et al. 2017;Wilhelm et al. 2016;Yun et al. 2021;Yılmaz 2012). As a result, physical indexes acquired from laboratory accelerated deterioration tests were mostly used in quantitative evaluation. Because cultural relics are a finite resource, most researchers recommend using non-destructive methods to determine the degree of weathering in situ.
Most deterioration experiments have previously been studied under the influence of various environmental factors. Common characterization indices include weight, hardness, ultrasonic velocity and so on. However, there has been little research on a comprehensive analysis of these indices. Conducting a comprehensive assessment of sandstone carvings before carrying out detailed scientific conservation work has become a critical step. As a result, exploring a suitable approach to assessing the weathering degree of carving remains to be discussed. Originally, Saaty advocated the analytical hierarchy process (AHP) for the first time in 1979 (Saaty 1979;Wind and Satty 1980). It is widely used for risk prediction and geo-environmental assessment due to its efficiency in dealing with qualitative data (Alshehri et al. 2015;Ercanoglu et al. 2008;Nefeslioglu et al. 2013;Ramkar and Yadav 2021). The AHP technique, on the other hand, is unable to reflect actual human thinking styles, and assessment outcomes always become subjective and imprecise. Hence, Zadeh (1973) further proposed fuzzy theory combined with AHP to improve the flexibility in judgment and decision. Furthermore, recent studies reveal that the Fuzzy-AHP method is successfully applied to natural hazards assessment (Hategekimana et al., 2018;Rezaei et al. 2019;Xiaoling et al. 2013).
In this study, the Fuzzy-AHP method was adopted to determine the weathering degree. A simulation experiment was carried out using sodium sulfate solution and sulfuric acid. There are four pH gradients (pH = 1,2,3 and 4). Furthermore, Jurassic sandstone specimens were subjected to a non-destructive test, which included surface hardness, ultrasonic velocity, penetration coefficient, photomicrograph, and quality. Both of these deterioration indicators were collected. Afterward, AHP was used to determine the weights in the hierarchical framework of the weathering degree assessment. Finally, FAHP was adopted to calculate weathering degree assessment outcomes. Furthermore, this method for calculating weathering indices will further promote the scientific conservation of future geotechnical heritage.

Petrophysical properties
Danzishi rock carving was located in Nanan District, Chongqing, and was excavated in Jurassic quartz arkose. Furthermore, plenty of statues were carved into the Jurassic stratum sandstone cliff. The analysis of samples collected from the Danzishi rock carving (Fig. 2) 1 3 revealed that the texture was predominantly sandy and/or massive, with poor sorting and roundness. Table 1 detailed the sampling of petrophysical properties.
In addition, Jurassic sandstone (as shown in Fig. 3) is primarily composed of medium-sized sand, quartz, and feldspar, which differ in that regard depending on direction. The feldspar inclusions frequently exhibit serictic alteration. Quartz is commonly observed in idiomorphic to hypidiomorphic single-crystal shapes, but it can also be found in polycrystalline quartz aggregates. Chlorite and epidote are frequently formed from partial feldspar. Plagioclase and quartz predominate in this matrix, with subordinated sericite and chlorite. Furthermore, quartz crystal has a preferred grain orientation in an oblate shape, whereas feldspar has a prolate fabric. The coefficient of expansion of

Experiment schedule of sodium sulfate solution
According to a testing report of the study area, sodium sulfate with 5 percentage concentrations was selected to conduct a simulation experiment on sample cubes (10 × 10 × 10 cm). Figure 4 shows the non-destructive methods used in this experiment study. A Proceq Equotip 550 Leeb with impact body D was used due to the good correlation between surface hardness and rock surface strength. Twelve sample points were evenly set on sample cubes. The single impact method was employed to minimize damage to the sample's surface. Similarly, ultrasonic velocity was measured by Pundit PL-200 using directly transmit method, as presented in Fig. 4b. As for water absorption coefficient, the Karsten tube measurement was conducted on sample cubes. First, the bottom end of the glass was wellconnected to sample cube with an elastic film material. The glass tube was water-filled to zero readings at time t = 0. Subsequently, the readings provide values of absorbed water volume for 2 h, taking readings at different time intervals. The water absorption coefficient was calculated using the equation ω = W∕ √ t , where W represents water absorption per unit area, and t represents time. Figure 5 depicts a rough sketch of the experiment schedule. The samples were completely immersed in a salt solution for about 12 h. Samples were then dried for 12 h in an air oven set to 80℃. After that, the specimens were removed from the oven and allowed to cool to air temperature for 12 h. Finally, the surface salinity of the specimens was dissolved by immersing them in distilled water. It should be noted that the whole process is one cycle, with each cycle lasting three days.

Experiment schedule of sulfuric acid solution
Similarly, dilute sulfuric acid was chosen based on the hydrogen ion content in the testing report. A dilute sulfuric acid solution was subjected to four pH gradients (pH = 1,2,3 and 4). Twelve cube samples were used and divided into four groups, as shown in Fig. 6. Each group's cubes were immersed in a sulfuric acid solution containing varying concentrations of hydrogen ions for 12 h, dried in an air oven for 12 h, and then allowed to cool at room temperature for another 12 h. Following that, samples will be returned to their containers to soak for the next cycle. After 9 cycles, hardness, ultrasonic, and absorption tests were conducted by non-destructive methods.

Result of sodium sulfate solution
As presented in Fig. 7, specimens subjected to salt damage are characterized by granular disintegration and scaling. It is worth noting that there is a strong correlation between grain size and water absorption. Water absorption is greater in the sample with a larger grain size. In addition, the change laws of ultrasonic velocity, weight, and hardness are shown in Fig. 7d, e, and f. Both of these indicators are observed to decrease over repeated cycles. Granular disintegration and scaling are the primary causes of weight and hardness loss.  It is clear that the samples exhibit a greater degree of decrease in hardness and ultrasonic velocity. These samples' damage levels range from strong to weak in the following order: pH = 1, pH = 2, pH = 3 and pH = 4. The results, nevertheless, do not show a good law for specimen weight and water absorption.

Determine the hierarchical model
For weathering assessment degree, the judgment of the importance degree of different conditions is the first step to assess weathering degree through weight calculation. In this study, this hierarchical model includes three layers (as shown in Fig. 9): The first layer is called as goal layer (A), i.e., specimens weathering degree in this study; the second one is the criterion layer which indicates sodium sulfate solution (B); the third layer, as well as index layer, contains the measurement indicators, including surface hardness (C 1 ), ultrasonic velocity (C 2 ) and water absorption coefficient (C 3 ).

Determine the judgment matrix (A) and the weight calculation
Firstly, the weight judgment matrix can be established according to the 1 to 9-degree compare method proposed by Saaty (2006). Afterward, the weight of each layer, as well as where = 1 , 2 , ..., n is the weight vector, max is the largest eigenvalue of matrix A, which can be calculated by using the following equation: Furthermore, the element of the weight vector should be normalized via the Equation: Actually, it should be noted that consistency check plays an important role in the whole process. The purpose of consistency check is to avoid the interference of other factors on the reliability and accuracy of matrix ranking. The specific calculation formula is as follow: where CI is the consistency indicator, and the value of CI is calculated by Eq. (6). RI is called random indicator, and its value for different scales is presented in Table 2.
Therefore, we first rank the criteria level index in order of importance. The importance degree of different conditions varies from strong to weak in the following order: sodium sulfate solution, pH = 1 sulfuric acid, pH = 2 sulfuric acid, pH = 3 sulfuric acid and pH = 4 sulfuric acid. Furthermore, the results of the weight judgment matrix are shown in Tables 3, 4 and 5. (

Determine the assessment set and membership degree
Evaluation set refers to the set composed of the evaluation grade of the possibility of evaluation target as the element, V= v 1 , v 2 , v 3 , ..., v n . Generally, weathering degree is divided into five levels, i.e., no weathering, low weathering, moderate weathering, high weathering, and very high weathering geologically. Moreover, the risk is generally scaled from 0 (0%) to 1 (100%) mathematically. Combined with the expert opinion and the classification standard of weathering degree in engineering geology, the assessment set is illustrated as follows in this study. V = [v 1 , v 2 , v 3 , v 4 , v 5 ] (7) where In addition, the membership function, including the intuitive method, inferential method, F statistical method, and Gaussian distribution, is established according to the characteristics of the index system. Furthermore, the triangular membership function is selected in this article because the membership function is equivalent to the evaluation result, which has been successfully applied in the risk assessment of earthen sites in China by Yumin (2019) and Zhiqian (2016). In addition, the fuzzy set is generated using triangular membership function (Fig. 10), as shown in Eq. (9):

Calculate the comprehensive assessment
To assess weathering degree, establishing a decision matrix (D) is necessary. The decision matrix and its normalized result (F) are as followed.
Likewise, the value of other conditions is 0.1951 for pH = 1 sulfuric acid, 0.1060 for pH = 2 sulfuric acid, 0.0516 for pH = 3 sulfuric acid, and 0.0421 for pH = 4 sulfuric acid, as shown in Fig. 11.

Discussion
Rock weathering caused by a complex interplay of physical and chemical processes is still poorly understood (Turkington and Paradise, 2005). The petrophysical properties such as grain fabric, grain size distribution, and compressive strength are a more active role in these processes (Ruedrich and Siegesmund, 2006).

Grain fabric
As presented in Fig. 12, petrographic analyses (in cross-polarized light) and SEM tests on standard thin sections were carried out to provide a qualitative description of the fresh rock fabric at various magnifications. 618, 2629, 1.90, 2531 591, 2446, 2.05, 2525 564, 2187, 1.30, 2528 483, 2037, 1.50, 2525 468, 1895, 1.20, 2512 425, 1817, 1.40, 2501 385, 1738, 1.05, 2499 175, 0.178, 0.183, 0.144 0.167, 0.166, 0.197, 0.143 0.160, 0.148, 0.125, 0.143 0.137, 0.138, 0.144, 0.143 0.132, 0.128, 0.115, 0.143 0.120, 0.123, 0.135, 0.142 0.109, 0.118, 0.101, 0.142 According to Fig. 12, cross-polarized light observations demonstrate that grains have a preferred orientation in an oblate or prolate shape, and little gypsum cement occurs. While the microstructure is characterized by layered arrangement and tight particle cementation. Slight erosion has been performed at the boundary of each layer. In addition, the separate effects of erosion, hydration action, and salt crystallization of acidrich/salt-rich water will further lead to the deterioration within the rock mass. It is easily noticeable in Fig. 13 at various magnifications. For example, samples soaked in sodium sulfate (Fig. 13a, b, and c) exhibited large dissolving pores, which provide the channel for salt-rich fluid to enter and conduct the hydration and crystallization action. Samples immersed in dilute sulfuric acid solution, on the other hand, showed micro-cracking, as shown in Fig. 13f. Moreover, the white calcium sulfate crystals can be seen on the mineral surface (Fig. 13h to l). It is interesting to note that micro-cracking and small-scale cavities (Fig. 13l) have been developed on the mineral surface, which is likely to result in mineral crystal fracture and aqueous solution loss.

Pore size distribution
As shown in Fig. 14, mercury porosimetry (Brakel et al.1981) is used to measure the pore size distribution of samples. Experiment results reveal that the medium porosity of the Jurassic sandstone varies between 8.01 percent for the fresh sandstone and 14.20 percent for the sandstone undergoing the deterioration process. The porosity of specimens varies from high to low is in the following order: samples immersed in sodium sulfate solution, samples soaked in sulfuric acid solution with pH = 2, samples soaked in sulfuric acid solution with pH = 1, samples immersed in sulfuric acid solution with pH = 3, fresh sandstone and samples immersed in sulfuric acid solution with pH = 4. Although the cycle of salt solution is shorter than the cycle of acid solution, it is obvious that samples subjected to sodium sulfate solution degradation have largest porosity. Sulfuric acid-soaked samples showed a good correlation between porosity and weathering degree. In general, the porosity of samples with pH = 1 and pH = 2 hydrogen ions concentrations is greater than that of pH = 3 and pH = 4. As the concentration of hydrogen ions increases, so does the weathering degree of samples. The value of weathering degree is 0.1951 for samples soaked in pH = 1 sulfuric acid solution and, 0.1060, 0.0516, or 0.0421 for sampled soaked in pH = 2, pH = 3, pH = 4 sulfuric acid solution, respectively.
Damage commonly occurs in the surface layer of samples during sulfuric acid deterioration experiments, which is characterized by disintegration macroscopically and mic-cracking on particle surface at the microlevel. Non-destructive testing results show that surface hardness and ultrasonic velocity decrease over a testing cycle. The decay rate increases as the concentration of hydrogen ion increases. However, sodium sulfate solution appears to be able to be capable of penetrating the specimen's interior and causing damage, as seen by delamination and disintegration. This has been attributed to pore expansion and connection, which can be confirmed by the change of porosity. Stress development due to crystal growth, thermal expansion, and hydration, on the other hand, all play key roles in the deteriorating process (Miyazaki et al.1992;Schmelzer et al. 2006;Winkler and Wilhelm, 1970). Therefore, the weathering degree performed worse than samples damaged by sulfuric acid.
In addition, much work has been conducted on the relationship between compressive strength and porosity (Ruedrich et al. 2010). This study shows a positive correlation between porosity and weathering degree (Fig. 15). There is no doubt that porosity is a critical rock mechanics metric. It is also a powerful tool for connecting rock physics, hydraulics, and mechanical indicators. In future research, we will strive to consider more weathering factors based on rock porosity to develop more detailed weathering models.

Conclusions
According to an analysis of the AHP-Fuzzy assessment method and petrophysical properties, some major conclusions can be drawn: (1) The weathering degree of specimens varies from high to low in the following order: samples immersed in sodium sulfate solution, soaked in sulfuric acid solution with pH = 1, soaked in sulfuric acid solution with pH = 2, samples immersed in sulfuric acid solution with pH = 3, and samples immersed in sulfuric acid solution with pH = 4. (2) Based on the Fuzzy-AHP method, salt-simulate deterioration samples exhibit moderate weathering (0.3271), low weathering for pH = 1 sulfuric acid (0.1951) and pH = 2 sulfuric acid (0.1060), very low weathering for pH = 3 (0.0516) and/or 4 sulfuric acids (0.0421). (3) According to petrophysical analysis, comprehensive strength and porosity perform a clear correlation to weathering resistance, with increasing moisture content exhibiting lower comprehensive strength. In addition, samples with higher porosity have worse weathering resistance. (4) The microstructure is characterized by layer arrangement and moderate erosion at the boundary of each layer, as observed by SEM under different magnifications. (5) The quantitative evaluation of weathering degree is critical in preventive conservation. It should also be emphasized that continuous environmental monitoring at sites is essential.