Suggestion of advanced regression model on friction angle of fault gouge in South Korea

Seong-Woo Moon, Hyun-Seok Yun, Yong-Seok Seo

Department of Earth and Environmental Sciences, Chungbuk National University,1 Chungdae-ro, Cheongju-si, Chungcheongbuk-do, 28644, South Korea

Keywords:Fault gouge Friction angle Simple regression analysis Structural equation model analysis Multiple regression analysis

ABSTRACT Although friction characteristics of fault gouge are important to understand reactivation of fault,behavior of earthquake,and mechanism of slope failure,analysis results of fault gouge have low accuracy mostly than those of soils or rocks due to its high heterogeneity and low strength.Therefore,to improve the accuracy of analysis results, we conducted simple regression analysis and structural equation model analysis and selected major influential factors of friction characteristics among many factors,and then we deduced advanced regression model with the highest coefficient of determination (R2) via multiple regression analysis. Whereas most coefficients of determination in simple regression analysis are below 0.3-0.4, coefficient of determination in multiple regression analysis is remarkably large as 0.657.

1. Introduction

Friction characteristics of fault gouge are major factor that has to be analyzed for understanding mechanism of slope failure or estimating fault reactivation potential (Lupini et al., 1981; Marone,1998; Ikari et al., 2011; Ji et al., 2013). However, fault gouge is mixed irregularly with fine-grained clay and coarse-grained breccia, and fault gouge has high heterogeneity and low shear strength. Thus, its friction characteristics can be affected by many factors such as particle size distribution, presence of foliation, and difference of composite minerals (Erickson and Wiltschko, 1991;Billi,2005;Loew et al.,2010;Haines et al.,2013;Alder et al.,2016).For this reason, many researches have been carried out to identify the influential factors on friction characteristics of fault gouge.

Looking for studies related to breccia content, Woodcock et al.(2006) reported that crackle and mosaic breccias, which contain a large amount of breccia, are classified as particle-supported fault rock with friction between particles, whereas chaotic breccia and fault gouge,which contain a lower breccia content,are classified as matrix-supported breccia, with the friction characteristics depending on the matrix materials.Henderson et al.(2010)divided the fault rock in the basal shear zone into particle- and matrixsupported breccias based on a particle size analysis, and conducted ring shear tests to show that the shear strength of the particlesupported breccias was higher than that of the matrix-supported breccias. Moon et al. (2014) and Yun et al. (2015, 2019) presented regression models between the breccia content, clay content,normal stress and shear strength, and proposed that the shear strength was positively correlated with the breccia content and negatively correlated with the clay content.

It is also known that the breccia clast shape affects the friction characteristics of the fault core, in addition to the breccia content.Anthony and Marone (2005) conducted a double-direct shear test to identify the effects of the size, shape and roughness of the granular particles in fault gouge on the friction characteristics.Their results indicated that the shear strength tends to increase when the particles are more angular or rough,whereas it tends to decrease when the particles are smaller due to the stress being concentrated on a thin layer in the fault gouge.Abe and Mair(2009)simulated direct shear tests on fault gouge using a threedimensional discrete element method to analyze the effect of breccia clast shape on the friction characteristics, and demonstrated that the smoothing of the breccia shape caused the friction characteristics to decrease as follows:angular(highest)>pseudoangular > spherical (lowest). Kim et al. (2016) performed computerized tomography scans on fault cores, and analyzed the sphericity, elongation, flatness and slenderness of breccia clasts larger than 4.75 mm. The breccia clast shape was compared with the results of direct shear test, and it was noted that the friction angle decreased as the clasts became more spherical.

The types of clay minerals that comprise fault gouge are also known to affect the shear strength of the gouge.Clay minerals are divided into kaolin, illite and smectite groups in geological engineering, and exhibit the following degree of swelling: smectite(high)>illite>kaolin(low)(González de Vallejo and Ferrer,2011).These notable swelling characteristics have led many researchers to search for correlations between the clay mineral content, clay mineral type,swelling and shear strength of fault gouge.Saffer and Marone(2003) conducted a double shear test on both natural and synthetic clay-rich gouges, and confirmed that a specimen containing a higher illite content (illite > smectite) has a higher coefficient of friction,reflecting the difference in swelling between the two minerals. Ikari et al. (2007,2009) found that the coefficient of friction in a fault core decreases as the clay content increases, and that the coefficient of friction in a fault core with a high montmorillonite(smectite)content is lower than that in a fault core with a high illite and/or chlorite content.

It is known that the texture,such as foliation and slickensides,in fault gouge affects the friction characteristics. Holdsworth (2004)and Carpenter et al. (2012) explained that parts of the San Andreas Fault have a very weak shear strength that is dependent on the alignment of phyllosilicate and clay minerals, with this fabric observed as a foliation in specimens. Tesei et al. (2012) also noted that a specimen containing an alignment of silicate minerals has a lower coefficient of friction than a specimen without foliation,with pore fluid pressure and foliation playing a key role in reducing the shear strength. These studies highlight that the friction characteristics of fault gouge are influenced by a variety of factors,including the breccia content, breccia clast shape, clay content and composition, and fabric.

However, above researches are confined to a certain fault or a small area, and have disadvantage of considering only effect of specific factors such as particle size distribution, and shape of breccia. Because fault gouge is affected by many factors, it is necessary to consider many factors for identifying overall characteristics of fault gouge. In this study, we carried out sampling and laboratory tests on many fault zones, and conducted simple regression analysis and structural equation model(SEM)analysis to select major influential factors using results of laboratory tests.Influential factors that were determined from those analyses are unit weight, porosity, water content, breccia content and silt/clay content.We suggested advanced regression model with the highest coefficient of determination (R2) via multiple regression analysis using by categorizing and setting the number of independent variables on influential factors.It is expected that this analysis method can be useful to improve reliability of results while analyzing heterogeneous materials as well as fault gouge.

2. Sampling and laboratory tests for fault gouge

2.1. Sampling locations and outcrops of fault gouge

A total of 224 specimens were collected and tested in 62 fault zones across South Korea to measure the physico-mechanical properties of fault gouge (Fig.1a). The width of the fault gouge in each location was greater than 10 cm to ensure that each specimen consisted entirely of fault gouge, since a 10-cm-diameter ring sampler with a 5-cm height was used. Most of the sampling locations were in Gyeongsang-do,southeastern Korea,since large-scale fault zones, such as the Yangsan and Ulsan fault zones, are mainly distributed in this region (Fig. 1b). The host rock types of the collected fault gouge specimens consist of granite (42%), sedimentary rocks (34%), and volcanic rocks (24%). The sedimentary rocks are clastic in nature, and are mostly mudstones and shales.The volcanic rocks are mainly andesites and tuffs. The outcrops of each rock type are described in two representative locations.

Fig.1. (a)Simplified map of fault distribution showing the locations of sampling sites in South Korea;and(b)Geological map of Gyeongsang-do,southeastern Korea,where many large-scale fault zones are distributed and most of the sampling locations in this study are concentrated.

Twenty-six types of specimens were collected from fault gouge in granite. The breccia content was low, and a distinct boundary between the fault gouge and damage zone was observed in many of the outcrops.Two typical fault gouge outcrops in granite are shown in Fig.2a and b.The fault gouge in the Ulsan Fault Zone consists of a reddish and greenish gouge and is 100 cm wide (Fig. 2a). Some microbreccias (<1 mm particle size) are presented in the fault gouge zone, and slickensides and slickenlines (striations) are clearly observed.The fault gouge zone along the Yangsan Fault Zone contains large amounts of whitish gouge and is as narrow as 50 cm(Fig. 2b). Well-developed slickensides and slickenlines are presented in both fault gouges,and they consist mainly of quartz and feldspar are observed in the slickensides.

Twenty-one types of specimens were collected from fault gouge in sedimentary rock.The breccia content is higher than that in the fault gouge in granite, and many sections contain weathered damage zone with soil and clay.Two typical outcrops of fault gouge in sedimentary rock are shown in Fig.2c and d.The fault gouge in the Ulsan Fault Zone,which is one of the major fault zones in South Korea, is composed mainly of greenish gouge that strikes NS and dips 60°E, and contains breccia (clast size of 1 cm) and cataclasite(Fig.2c).The fault gouge along the Sinnyeong Fault Zone possesses clear boundaries that are characterized by grayish gouge and greenish damage zone (Fig. 2d).

Fig.2. Representative outcrop photographs of fault gouge in each rock type:(a,b)Ulsan and Yangsan Fault Zones,respectively.They consist of granitic gouge with slickensides and slickenlines;(c,d)Ulsan and Sinnyeong Fault Zones,respectively;(e,f)Yeonil Tectonic Line and Beomgok Fault Zone,respectively.Outcrops of sedimentary and volcanic fault zones have relatively high breccia contents than those of granitic fault zone.

Fifteen types of specimens were collected from fault gouge in volcanic rock.The breccia content was high,similar to that of fault gouge in sedimentary rock, with soil and clay observed in the damage zone in many sections.Two typical fault gouge outcrops in volcanic rock are shown in Fig. 2e and f. The fault zone along the Yeonil Tectonic Line comprises andesitic tuff host rock and 50-cmwide whitish gouge and breccia intermixed with grayish gouge(Fig. 2e). The damage zone on the left side of the fault comprises severely crushed andesitic tuff.The fault gouge along the Beomgok Fault consists of intermixed reddish and thin greenish gouge, and the fault gouge exhibits slickensides and slickenlines(Fig.2f).Felsic porphyry occurs in the upper right section of the fault core, and andesitic tuff is presented in the lower left section of the fault.

2.2. Laboratory tests on fault gouge

Water content test,specific gravity test,particle size analysis,Xray diffraction(XRD)analysis,and direct shear test were conducted to measure the physico-mechanical properties of the fault gouge.The size of the fault gouge specimens(10 cm in diameter,and 5 cm in height)was large enough to provide a representative elementary volume (REV) for analyzing the physical properties of heterogeneous fault gouge(Yun et al., 2018).

The dry unit weights were measured according to ASTM D2216-10 (2010). Drying was performed at <60°C to minimize chemical alteration of clay minerals,which can occur at higher temperatures.The specific gravity was measured using a pycnometer according to ASTM D854-10 (2010). The porosity and void ratio were then calculated using the relationship between the dry unit weight and specific gravity. The water content of the natural fault gouge specimens is not an intrinsic characteristic of the samples, since it can vary with climate and region. Therefore, the water contents were calculated based on the relationship between the degree of saturation, void ratio and specific gravity under the assumption that the voids in the fault gouge were water-saturated.

The breccia and silt/clay contents were measured via the sieve test method according to ASTM D422-63 (2007), and the soilwashing test method was applied according to ASTM D1140-17(2017) to prevent clay aggregation during drying. The fault breccia clast size has numerous classification criteria across the 0.1-5 mm size range (Higgins, 1971; Twiss and Moores, 1992;Snoke et al., 1998; Clark and James, 2003; Woodcock and Mort,2008). Here, the unified soil classification system (USCS), which is commonly used in civil engineering studies and internationally benchmarked by ASTM D2487-17 (2017), was used to classify the grains larger than 4.75 mm as breccia and smaller than 0.075 mm as silt/clay.

The composite mineral contents can be quantitatively calculated via XRD analysis based on the diffraction lines of the different minerals, which are obtained by bombarding the specimens with X-rays.The specimens were dried at room temperature(30°C)and powdered prior to XRD analysis.The XRD patterns of the specimens were measured on a X’Pert-Pro/MRD (Philips, Japan) using Nifiltering CuKα radiation (1.54056 ?) with a 40-kV acceleration voltage,30-mA current,and scanning at 0.02°per second from 5°to 65°.The mineral contents were calculated using the Siroquant v.3.0 program. The mineral contents of quartz, feldspar, illite, kaolin,smectite, etc. were obtained from XRD analysis. The quartz and feldspar were grouped into quartz+feldspar content and used for analysis because they have similar effect on the friction angle due to their strong resistance to weathering compared to other composite minerals. In addition, because each content of illite, kaolin and smectite has a large deviation to be insignificant for statistical analysis, illite, kaolin and smectite were also grouped into clay mineral content and used for analysis.

The friction characteristics of fault gouge, including the friction angle and cohesion,were measured via direct shear tests.The fault gouge specimens for these tests were collected using ring samplers that were the same size as the shear box (80 mm in diameter and 30 mm in height).The specimens were saturated for more 48 h to consider the worst condition of fault gouge and then consolidated to avoid the supersaturation.The direct shear tests were conducted at shear rates of 0.4-1.6 mm/min (0.5%-2% of the sample diameter), in accordance with ASTM D3080-98 (1998), and at normal stresses of 54 kPa, 108 kPa, and 162 kPa. The stresses and displacements during shearing were automatically measured by the load cells and a linear variable differential transformer (LVDT).

Physico-mechanical properties gained from laboratory tests are dry and saturated unit weights, specific gravity, porosity, water content, breccia content, sand content, silt/clay content,quartz + feldspar content, clay mineral content (containing illite,kaolin and smectite), friction angle and cohesion.

3. Procedure and methods of analyses

The procedure of analyses is shown as the flowchart in Fig. 3.Among the physico-mechanical properties from laboratory tests,the influential factors for friction angle are first selected from simple regression and SEM analyses, and then some influential factors, which have a multicollinearity, are eliminated to avoid duplication effect during the data analysis. After that, the most advanced equation is determined for estimating the friction angle of fault gouge via multiple regression analysis.

3.1. Simple regression analysis

A simple regression analysis is a statistical analysis method that is used to identify the effect of one independent variable on one dependent variable, as follows:

where Y is the dependent variable,X is the independent variable,α is the coefficient of regression, and β is the intercept.

The simple regression analysis results are assessed using the coefficient of regression (α) and the coefficient of determination(R2) with the reliability of the simple regression model increasing as R2approaches one, which means that the correlation between the independent and dependent variables is high. Here, R2< 0.3 represents a low correlation,0.3 ≤R2<0.6 represents a moderate correlation, and R2≥0.6 represents a high correlation (Zikmund,2000; Moore et al., 2013; Sanchez, 2013). These R2criteria are generally accepted as key indices for evaluating the reliability of statistical models,such as the implementation of R2in assessing the simple regression, multiple regression, or goodness of fit index(GFI)in structural model analyses.The coefficient of regression(α)means that the independent variables affect the dependent variable,with this being a one-to-one comparison for simple regression analysis, such that it is possible to use non-standardized coefficients.However,if simple regression analysis is applied to many factors or multiple regression analysis is performed,then it should be compared using standardized coefficients. For example, the coefficient of regression between the friction angle and porosity cannot be directly compared with the coefficient of regression between the friction angle and breccia content.This is because the porosity and breccia content units are % and wt% (percent by weight),respectively,which means that the regression coefficients must first be standardized to effectively evaluate the influence of these two factors. Multiple regression analysis can also be applied when analyzing many factors.

Fig. 3. Flowchart on the procedure of the analyses.

3.2. SEM analysis

SEM analysis is a statistical analysis proposed by Wright(1921)that can express the correlation between the various variables affecting the dependent variables.The advantages of this analytical approach are the ability to display the correlations in diagram formand intuitively understand the correlations between different variables, and set many dependent variables (Hox and Bechger,1999; Yung, 2008; Ullman and Bentler, 2013). However, the disadvantages of SEM analysis are that researchers should have a specialized knowledge of the analysis object to obtain reliable results via SEM analysis and that estimating equation (regression equation) is not provided.

SEM analysis consists of observed variables, latent variables(unobserved variables), and an error term (Fig. 4). The observed variables are directly identifiable variables that correspond to the data being analyzed. The latent variables are variables that are virtually determined to better understand the correlation between variables,such that the researcher’s knowledge is reflected when a latent variable is determined.The model that contains the observed and latent variables is called the external model, and the model between the latent variables is called the internal model. Furthermore, the variables that describe other variables are called exogenous variables, and the variables that are explained by other variables are called endogenous variables. The error term plays a role in better explaining the endogenous variable since the endogenous variable is not fully explained by an exogenous variable.The error term attached to the observed variable is called the measurement error, and the error term attached to the latent variable is called structural error. The overall model includes the external model, internal model and error terms.

3.3. Multiple regression analysis

Multiple regression analysis is a type of regression analysis that identifies the relationship between one dependent variable and several independent variables, such that:

where Xnis the n-th independent variable, and αnis the n-th coefficient of regression.

Eq. (2) is simply an extension of Eq. (1) to include multiple independent variables. Similar to simple regression analysis, the reliability of multiple regression analysis is based on R2and αn.However, multiple regression analysis should meet significance level for the detailed model as well as R2for the entire model.Furthermore, αncan be calculated for each independent variable and compared with the other independent variable to determine the degree of influence on the dependent variables.

4. Selection of influential factors on friction characteristics

4.1. Results of simple regression analysis

A simple regression analysis was performed to analyze the correlation between the friction angle and various physical characteristics of fault gouge, which included the unit weight, specific gravity,porosity,water content,breccia content,sand content,silt/clay content, quartz + feldspar content and clay mineral content that were measured from the laboratory tests. The simple regression analysis results are shown in Fig. 5 and Table 1 for granite,sedimentary rock, volcanic rock, and all of the rocks (granite,sedimentary rock and volcanic rock).The overall R2is less than 0.5 for each of the factors and rock types(Table 1),but the factors show distinct trends. Here, the presence or absence of a distinct trend was assessed based on the distribution patterns in Fig. 5, with a distinct trend being identified when R2≥0.1.

Fig.4. Schematic diagram of SEM analysis.The entire model for SEM analysis is composed of internal and external models.The external model shows the relationships between the observed variables and an unobserved variable. The internal model consists of the unobserved variables (latent variables).

Fig.5. Relationships between the friction angle and(a)unit weight(results of dry and saturated unit weight are same),(b)specific gravity,(c)porosity,(d)water content,(e)breccia content, (f) sand content, (g) silt/clay content, (h) quartz + feldspar content, and (i) clay mineral content in granite (▲), sedimentary rock (○), and volcanic rock (■).

Table 1 Summary of the simple regression analysis results for granite,sedimentary rock,volcanic rock,and all of the rocks.Triangle(▲)means a positive correlation whereas inverted triangle (▼)means a negative correlation. φ is the friction angle.

Both the dry and saturated unit weights tend to be proportional to the friction angle in all of the rock types (granite, sedimentary rock, volcanic rock, and granite + sedimentary rock + volcanic rock) (Fig. 5a). This is because the constituent particles become more concentrated as the unit weight increases, resulting in an increase in material strength.The relationship between the dry and saturated unit weights and the friction angle of fault gouge is consistent with previous soil studies (NAVFAC, 1986; Edil and Benson, 2007; Ersoy et al., 2013). Furthermore, the specific gravity does not exhibit a distinct trend in all rock types due to its sporadic distribution(Fig.5b).This is because the specific gravity is the unit weight of the particles themselves, such that there is no significant difference between the rock types. The porosity and water content tend to be inversely proportional to the friction angle, such that a higher porosity will lower the friction angle(Fig. 5c and d). Furthermore, a high-porosity specimen is likely to have internal fractures or microcracks (Vallejo and Mawby, 2000;Sulem et al. 2004), and the porosity is known to be inversely proportional to the unit weight, friction angle, uniaxial compressive strength (UCS), P-wave velocity, and other factors (Bjerrum et al.,1961; Sch?pfer et al., 2009; Kanji, 2014). It has also been reported that the friction angle decreases with increasing natural water content. Morrow et al. (2000) explained that this inverse correlation likely occurs because clay absorbs a large amount of water,with the additional water content decreasing the friction angle.

The particle size distribution results show that the friction angle tends to increase as the breccia content increases in all of the rock types (Fig. 5e). Gutierrez and Muftah (2011) explained that this proportional relationship between the breccia content and friction angle is due to the rolling resistance of the breccia.In other words,if there are breccias on the shear surface,then the stress transmitted to the shear surface is converted to an additional moment(rotation)for the breccias, resulting in a stress loss. This means that more stress is required for slip along the shear plane, resulting in an increase in the friction angle. The friction angle is also known to be affected by breccias fracturing,as well as the shape of the crushed breccias under high normal stress (1 MPa or higher) (Xia et al.,2011). However, the effect of breccia fracturing the breccia is considered small in this study since the normal stress applied during the direct shear tests is less than 200 kPa. The relationship between the sand content and friction angle appears to exhibit a positive correlation between sedimentary rock and all of the rocks,but no apparent trend is observed in granite and volcanic rock(Fig.5f).The friction angle tends to decrease as the silt/clay content increases, with the same trend observed in studies that analyzed fine-grained soils and clay materials(Fig.5g)(Adunoye,2014;Park and Jeong, 2018).

No correlations between the quartz + feldspar content and friction angle are observed in any of the rock types (Fig. 5h).Furthermore, the clay mineral content tends to be proportional to the friction angle in granite and sedimentary rock,but it tends to be inversely proportional to the friction angle in volcanic rock,with no trend observed in all of the rocks due to the sporadic distribution of the clay mineral content data (Fig. 5i). Therefore, it is considered that this is not the general tendency among all of the fault gouge specimens due to this random tendency to appear in granite, and sedimentary and volcanic rocks. The reasons for the little effect of clay mineral content will be explained in Discussion.

In conclusion, the simple regression analysis results indicate that the dry unit weight, saturated unit weight, porosity, water content, breccia content and silt/clay content are the factors that influence the friction angle of fault gouge.

4.2. Results of SEM analysis

In the SEM analysis, the observed variables are the physicomechanical properties that were obtained via the laboratory tests,and the latent variables are set as the following physical properties:particle size distribution, mineral composition and friction characteristics, which are based on the characteristics of the observed variables. The observed variables corresponding to the physical properties are the dry unit weight, saturated unit weight, specific gravity, porosity and water content. The observed variables corresponding to the particle size distribution include the breccia, sand and silt/clay contents.The observed variables corresponding to the mineral composition include the quartz+feldspar and clay mineral contents. The observed variables corresponding to the friction characteristics include the friction angle and cohesion. The SEM design for this study is shown in Fig. 6.

The SEM analysis results are evaluated using the GFI for the entire model and CR (critical ratio) or Pr (p value) for the detailed model,which includes the internal and external models.The GFI is in the 0-1 range and can be evaluated using the R2criterion outlined in Section 3.1. The CR and Pr criteria are based on the significance levels,such that the significance is 95%when the absolute CR value is greater than 1.96,99%when the absolute CR value is greater than 2.58,and 99.9%when the absolute CR value is greater than 3.3.At that time, 0.05, 0.01 and 0.001 of Pr values correspond to 1.96(95%significance level),2.58(99%significance level),and 3.3(99.9%significance level) of CR criterion. It means that the study hypothesis (alternative hypothesis) has a 5%,1% or 0.01% chance of being rejected and a 95%, 99% or 99.9% chance of being accepted.

Fig.6. SEM analysis results.Because the mineral composition →clay mineral content in external model and the mineral composition →friction characteristics in internal model have CR values of-0.78 and 0.7 respectively,they are not statistically significant.Here,absolute CR values of 1.96,2.58 and 3.3 correspond to Pr values of 0.05(95%significance level),0.01 (99% significance level), and 0.001 (99.9% significance level), respectively. ***: Pr < 0.001, **: Pr < 0.01, *: Pr < 0.05.

Table 2 Correlation analysis results for the internal model, which include the physical properties,particle size distribution, mineral composition and friction characteristics.

The SEM results for the analysis to determine the factors affecting the friction angle in fault gouge are shown in Fig. 6 and Tables 2 and 3. The GFI for the entire model is 0.666, which corresponds to a high degree of reliability based on the R2criterion,indicating that the relationship between the observed and latent variables is well-organized in the model. Furthermore, the CR values for each detailed model show that effects of the mineral composition on clay mineral content in external model and friction characteristics in internal model (hereafter, effect of A on B is abbreviated as A →B by using arrow)are not considered influential factors because they do not meet the CR criterion (Fig. 6). The dry unit weight, silt/clay content, quartz + feldspar content and cohesion are marked with a bar(-)in Fig.6 to indicate that these latent variables are assessed in a relative sense based on a certain variable when describing the observed variable.Therefore,the CR values of the factors marked with a bar(-)are always considered significant.In conclusion, the presented SEM analysis highlights that the dry unit weight,saturated unit weight,specific gravity,porosity,water content,breccia content,sand content and silt/clay content are the factors that influence the friction angle of fault gouge.

Table 3 Correlation analysis results for the external model.The mineral composition →clay mineral content external model is not valid due to its low absolute CR value,which is not statistically significant.

5. Suggestion of advanced regression model via multiple regression analysis

5.1. Multicollinearity analysis and applied data for multiple regression analysis

Here, three methods are employed for the selection of the influential factors, multicollinearity analysis, and selection of the applied data to minimize the analysis process and maximize the reliability of the multiple regression analysis results. Multiple regression analysis results can differ greatly with the number of applied independent variables, with more independent variables leading to a greater number of multiple regression models that need to be considered. Therefore, it is necessary to determine the influential factors that are primarily correlated with the dependent variable to select the most appropriate model. This was reviewed by performing simple regression and SEM analyses, with both analyses identifying the dry unit weight, saturated unit weight,porosity,water content,breccia content and silt/clay content as the influential factors.

The highly multicollinear variables were removed to ensure that there was no duplication during the data analysis.Multicollinearity is a phenomenon in which one independent variable is highly correlated with another variable.The relationship between the dry and saturated unit weights is one example. The dry unit weight is calculated using the void ratio and specific gravity(Eq.(3)),and the saturated unit weight is calculated using the dry unit weight and void ratio (Eq. (4)) as follows:

where γdryis the dry unit weight,γsatis the saturated unit weight,Gsis the specific gravity, and e is the void ratio.

This leads to a high correlation between these two independent variables. If both independent variables are applied to multiple regression analysis, they will introduce a duplicate effect that reduces the efficiency of the analysis and potentially biases the results.

Therefore, multicollinearity analysis is performed on the selected influential factors, with the results shown in Table 4. The criterion for the presence or absence of multicollinearity is determined via the variability injection factor (VIF), where a VIF value greater than 10 indicates a multicollinear variable. The VIF values for the dry unit weight, saturated unit weight and porosity are all much greater than 10,which highlight their multicollinearity with each other. However, the VIF value for the water content is 9.15,which is close to 10.This VIF value reflects the influence of porosity on the water content,since the water content was calculated under the assumption of full saturation.

The applied data can largely be divided into raw data and categorized data. The raw data use the measured values of the independent variables, such that the reliability of the regression analysis decreases as the deviation in the raw data increases.These deviations can be compensated by first categorizing the data, and then applying the categorized data to the analysis. The categorization is subdivided into equal and unequal interval categorizations(Fig. 7). Equal interval categorization divides the difference between the maximum and minimum values into equal intervals,whereas unequal interval categorization classifies the data as a percentage of the data volume. The equal and unequal interval categorizations are both divided into five classes in this study.

5.2. Results of multiple regression analysis

A schematic diagram of the models that are used in the multiple regression analysis is shown in Fig. 8. The dry unit weight (or saturated unit weight, or porosity), water content, breccia content and silt/clay content are determined to be the independent variables via simple linear regression analysis, SEM analysis, and multicollinearity analysis. Twenty-five different models are constructed by grouping two, three or four of the independent variables into a given model.Furthermore,three applied datasets(raw data, equal interval categorization, and unequal interval categorization)are considered,yielding a total of 75 analytical models that are reviewed to analyze the friction angle in fault gouge.

The representative models,which yielded high R2values among different multiple regression analytical models,are listed in Table 5.The reliability of each model is determined by evaluating whether both the R2value and reliability of each independent variable are significant. The model with more independent variables is considered more descriptive when the two aforementioned conditions are the same. These criteria identify the No. 4 regression model as a high-reliability model that is the most appropriate model in Table 5, since it has the highest R2value and all of its influential factors possessed high significance levels.

Although the R2values for the correlation between the friction angle and each influential factor were generally low via simple regression analysis (0.123-0.463 in Table 1), multiple regression analysis effectively increased the explanatory power of the influential factors, yielding R2values of up to 0.657, which indicated significant results in terms of both their applicability and reliability.The most reliable regression model for the friction angle is

where BC is the breccia content(wt%)and SC is the silt/clay content(wt%).

This model is derived from the No.4 regression model in Table 5,and is the proposed regression equation for estimating the friction angle in fault gouge.

Table 4 Multicollinearity analysis results for the influential factors.The dry unit weight,saturated unit weight and porosity are all multicollinear with each other(i.e.their VIF values are much greater than 10).

Fig.7. Categorization of the data applied for multiple regression analysis:(a)Equal interval categorization,where the data are equally divided into bins that are determined based on the difference between the maximum and minimum values;and(b)Unequal interval categorization,where the data are unequally divided into bins that are determined based on the percentage of the data volume.

Fig.8. Breakdown of the 75 analytical models that were reviewed to analyze the friction angle in fault gouge.Twenty-five different cases involving the independent variables were considered, with each applied to three cases involving different datasets.

Table 5 Representative multiple regression analysis results.Model No.4 appears to be the most reliable model,as the R2 value for the entire model is high,and the reliability is also significant for each of the influential factors.

6. Discussion

6.1. The reason for combined simple regression and SEM analyses

In simple regression analysis,coefficient of determination(R2)is the only factor that determines the significance of results.However,in highly distributed data,such as analysis results of heterogeneous materials,distinct tendency can be shown even if R2is low.In this case, there is no statistical criterion that determines the significance of results because researchers judge visually and subjectively the tendency and/or distribution of data.Thus,we conducted SEM analysis to use objective criterion and verify the significance for overcoming these problems. Although in this study it may seem unmeaningful to conduct respectively two analyses because the result of simple regression analysis is similar to the result of SEM analysis and we just consider influential factors selected commonly from both results of analyses,we think that this analysis procedures can be useful for carrying out future research of heterogeneous materials containing fault gouge.

6.2. The reason for little effect of clay mineral content on friction characteristics

As mentioned in Introduction, it is known that clay minerals consisting fault gouge affect friction characteristics. However, the reasons why clay minerals have little effect on friction characteristics may be thought as follows.Fault gouge means fault materials with less than 30%of the breccia content,but the fine-grain content of specimens sampled in this study varies from 30% to 90%. Even though the clay minerals content,which are obtained from XRD,is high, if fine-grain content in the fault gouge, which are measured from sieve tests, is low, friction characteristics of fault gouge are believed to be more affected by coarse-grain content. These problems are originated from that the clay mineral contents of XRD do not match well with the silt/clay content of sieve tests because specimen size of XRD is much smaller than that for sieve test.Here,particle sizes of fault gouge and fault breccia are classified in some groups, but if specimens are separately subdivided according to fine-grain and/or coarse-grain contents and more specimens are sampled,it is expected to find out effect of clay minerals on friction characteristics.

7. Conclusions

Here we conducted series of laboratory tests and statistical analyses on heterogeneous fault gouge in order to select influential factors on friction characteristics and to suggest advanced regression model. The procedures and results of analyses are as follows.

A total of 224 undisturbed specimens were collected and tested from 62 fault zones in South Korea for the unit weight, specific gravity, porosity, water content, particle distribution, composite mineral content,friction angle and cohesion.Simple regression and SEM analyses were performed to identify the influential factors that affect the friction angle, and simple regression and SEM analyses revealed that the dry unit weight, saturated unit weight, porosity,water content, breccia content and silt/clay content are the main factors affecting the friction angle of fault gouge. Multiple regression analysis was used to improve the reliability of the relationship between the influential factors and friction angle since simple regression analysis yields low R2values(0.123-0.463).The result of multicollinearity analysis, which is performed before multiple regression analyses, indicates that only one factor among dry unit weight,saturated unit weight and porosity should be considered in multiple regression analysis. When the saturated unit weight,breccia content and silt/clay content were set as the independent variables,R2value of regression model increased up to 0.657 while each independent variable is statistically significant. We expect that these analysis procedures can be useful methods to increase reliability of results in analyzing heterogeneous materials as well as fault gouge.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work was supported by Postdoctoral Fellowship Program funded by the Ministry of Education of the Republic of Korea through the Chungbuk National University in 2020.