Correlation of Geo-Mechanics Parameters with Uniaxial Compressive Strength and P-Wave Velocity on Dolomitic Limestone Using a Statistical Method

Abstract

This paper estimates the correlation between the physical and mechanical characteristics of dolomitic limestone. Forty samples were measured and the results were used to derive the empirical equations for the relations between the uniaxial compressive strength (UCS), elastic modulus (E), P-wave velocity ($v_{p}$), Poisson’s ratio (μ), point load strength, and density (ρ). The correlation results for the intact dolomitic limestone samples showed high correlation between UCS and the other parameters, and between $v_{p}$ and the other parameters. Determination of physical and mechanical characteristics in situ or in the laboratory is always a costly process. In this paper, correlation relationships were proposed using $v_{p}$ and UCS to evaluate the other parameters where the results were in good agreement. To validate the empirical equations, a Student’s t test was conducted on the test data; the calculated t-test values were much higher than the tabulated values, indicating a generally good, linear relationship among the physical and mechanical properties of the intact dolomitic limestone. The results suggest that the $v_{p}$ and UCS values of dolomitic limestone samples can be used to estimate the ρ, μ, and E of the intact rock.

1 Introduction

The geo-mechanical properties of a rock mass are fundamental in mining and civil construction design. However, the natural discontinuities and inhomogeneity of any given rock mass can create internal and external uncertainties when measuring such properties (Ozcelik et al. 2012; Mikaeil et al. 2013; Azadan and Ahangari 2014). Moreover, determining the physical and mechanical properties of rocks from deep mines or from highly fractured rocks in a laboratory or in the field can be expensive and time-consuming. These difficulties may be overcome if reliable empirical relationships could be determined among at least the most critical properties needed in the design process. Statistical analysis and regression analysis were used in previous studies to derive the empirical correlation among various geo-mechanical properties of rocks. Sharma et al. (2011) examined the relationships between the Schmidt hammer rebound numbers and the impact strength index, slake durability index, and P-wave velocity using empirical equations. Diamantis et al. (2011) investigated the relationships between the P-wave velocity and the physical and mechanical properties and petrographic characteristics of peridotites from central Greece using simple regression analyses and confidence intervals. These empirical relationships were used to estimate the geo-mechanics properties of a rock mass.

Sonic and ultrasonic measurements are also useful tools for assessing rock properties and have been used in previous studies. Ultrasonic methods are nondestructive and relatively easy to use, both in the field and in the laboratory. Ultrasonic methods are being increasingly employed in mining, geological, and underground engineering to determine the dynamic properties of rocks for use in in situ and laboratory experiments. The P-wave velocity of a rock is closely related to the intact rock structure and mineral compositions. Zhou et al. (2014) examined a slope damaged by excavation at a hydropower station site using acoustic testing technology. Many investigators demonstrated that a rock’s P-wave velocity is strongly correlated with its geo-mechanical properties. Some of these empirical relationships are listed in Table 1. In this study we investigate the correlation between the physical and mechanical properties of dolomitic limestone samples using P-wave velocities and uniaxial compressive strength tests.

Table 1 Empirical relation reported between rock mechanical parameters and P-wave velocity

2 Study Area and Samples

2.1 Study Area

The study site was the Huize lead and zinc mine, located between the Yunnan and Guizhou provinces in southwest China (Fig. 1). The landform in the study area is complex and includes alpine morphologies, gorges, and riverine landscapes. The Niu-Lan river flows through the mining area at an elevation of 1561 m. The peak elevation is 2668.9 m and the elevation of the mine entrance is 2538 m, a relative elevation difference of close to 1000 meters. Multiple faults run through the mining area, leading to the formation of cracks in the rock mass. In addition to the faults, the high stress in the deep parts (1500 m) of the rock mass (42.34–45.95 MPa, based on three tests) causes fragmentation of the rock mass. Therefore, it is very difficult to obtain intact cores from depths over 2000 m. Due to high in situ stress of the study site the drilled cores were fragmented and discs were cut from the cores (Fig. 2). The main aim of this study is to predict the physical and mechanical parameters of the rock in a deep mine using empirical relationships based on selected derived parameters. This can reduce test costs and save time when only a small number of samples can be obtained.

Fig. 1

Maps showing the study area

Fig. 2

Discs from drilled cores obtained from 1500 m depth

The geological structure of the study area (Fig. 3) includes the north-east to south-west fold and fault, forming an anticline. The Kuang-shan Chang fault, the Qi-lin Chang fault and the Yin-chang Po fault form three overlapping tile structures. The north–south Dong Tou fault also affects the study area. The sampling was carried out between the Qi-lin Chang fault and the Yin-chang Po fault, closer to the Qi-lin Chang fault. In this area, the upper strata of the Devonian system, the upper Paleozoic Carboniferous system, and the Permian strata are widely distributed. Mount E-mei basalt along the Qi-lin Chang reverse fault is exposed in the southern and western areas of the mining area, and there are a few weathering residues in the middl of the area.

Fig. 3

Geological structure; a plane map of the geological structure; b A–A’ geologic sections

2.2 Rock Samples

Forty samples were obtained from the mine at a depth of 1500 m below ground level. The rock samples were mainly dolomitic limestone and were fresh to slightly weathered. To preserve the in situ conditions, the rock specimens were saturated prior to measuring the rock properties. The specimens were classified as fine to medium dolomitic limestone of the carboniferous system in the Permian strata. An image of the dolomitic limestone taken in situ is shown in Fig. 4. Several macroscopic structures and visible features were observed in the rock mass. Signs of medium-grade dynamic metamorphism associated with tectonic and ore-forming fluid activities were found in the rock mass; this included evidence of pyritization (oxidized into limonite), development of silicide, chloritization, epidotization, and calcite and hydrothermal alteration. The most obvious indication of such metamorphism were calcite veins observed on the fresh surface of the rock mass, as seen in Fig. 4. The rock samples are light in color, composed mainly of gray and light gray, medium to thick lamellar to cryptic limestone, and Tiger porphyritic dolomitic limestone with thick layered dolomite. The samples are mainly limestone, with Tiger porphyry intercalated with dolomitic limestone. The rock contains large amounts of calcite and siliceous nodules or lenses, and a small amount of shale and breccia limestone. Dolomite veins, calcite and pyrite can be observed clearly in the joint planes of the rock mass, as shown in Fig. 4.

Fig. 4

Photos of dolomite limestone taken in the study site; a supported area; b non-supported area

The rock testing was performed on cylindrical drill core samples of dimensions (length × diameter) 100 × 50 mm or 130 × 63 mm, as seen in Fig. 5. To reduce the influence of the sample size on the strength testing process, the diameters of the cylindrical rock specimens were prepared with a length-to-diameter ratio of 2.0–2.1. The two ends of each specimen were ground and lapped parallel, to an accuracy of ± 0.3 mm. The core surfaces must be parallel to prevent surface irregularities; the surface roughness of the two ends of the specimen deviated by only ± 0.05 mm.

Fig. 5

Rock Samples taken from 1000 m depth; original samples (left) and prepared samples (right)

3 Determining the Physical and Mechanical Parameters

Of the physical and mechanical parameters of a rock, the uniaxial compressive strength (UCS) and P-wave velocity ($v_{p}$) are the two most representative parameters. To understand the characteristics of the rock and its behavior under various engineering conditions, a mechanical strength test of the rock must be performed first. The most important strength test is the uniaxial compressive strength test. The UCS of a rock is an important indicator of the rock’s bearing capacity. The elastic modulus ($E$) and Poisson’s ratio ($\mu$) of a rock sample can be derived from the results of the uniaxial compressive strength test (Eqs. 1 and 2). The P-wave velocity of an intact rock block mainly depends on $E$, $\mu$, and the density ($\rho$), as shown in Eq. (3). In addition, the microstructure and water inside the rock can also greatly influence the UCS and $v_{p}$. Thus, it is necessary to study the empirical relationship between $v_{p}$ and UCS and other parameters. Using reliable empirical relationships to evaluate the physical and mechanical parameters of rock samples can effectively reduce experimental costs and time. Thus, this study can have broad engineering applications.

$E$ is the elastic modulus, $\varepsilon_{radial}$ is the radial strain, $\varepsilon_{axial}$ is the ax

$$E = \frac{\text{UCS}}{{\varepsilon_{axial} }}$$

(1)

$$\mu = \left| { - \frac{{\varepsilon_{radial} }}{{\varepsilon_{axial} }}} \right|$$

(2)

$$v_{p} = \sqrt {\frac{{E\left( {1 - \mu } \right)}}{{\rho \left( {1 + \mu } \right)(1 - 2\mu }}}$$

(3)

where $E$ is elastic modulus, $\varepsilon_{radial}$ is radial strain, $\varepsilon_{axial}$ is axial strain, $\mu$ is Poisson’s ratio and $\rho$ is density.

3.1 Testing Instruments and Procedures

Various standard tests were performed on the prepared samples to determine the properties of the rocks from the study area. The density and porosity were measured according to the International Society for Rock Mechanics (ISRM) standards (1981). The density of the dolomitic limestone was calculated as the ratio of the specimen mass divided by its volume. The $v_{p}$ value was determined using an acoustic emission testing system (ADLINK, USA) according to ISRM suggested testing methods (2007). The deformation was recorded by strain gauges attached to the prepared core samples. The elastic modulus was obtained from the gradient of the stress–strain curves using the tangent method. To eliminate the friction effect at the top and bottom ends of the sample, the top and bottom surfaces of the core specimens were ground and covered with petroleum jelly. The $v_{p}$ value was obtained as shown in Fig. 6a. To ensure the accuracy of the test results, three tests were carried out on each sample and the mean value of the three measurements was obtained. The uniaxial compressive strength was determined using a hydraulic servo mechanical testing machine (INSTRON-1346, INSTRON, Melbourne, Australia) following ASTM (1986) standards, as shown in Fig. 6.

Fig. 6

Testing setup; a the ADLINK acoustic emission testing system; b, c the hydraulic servo mechanical testing machine (INSTRON-1346))

3.2 Physical and Mechanical Parameters

The physical and mechanical parameters of the sedimentary rock samples determined by the laboratory tests are shown in Table 2. The values of $\rho$ and $\mu$ were found to be within a relatively limited range for the saturated samples, but the values of E, $v_{p}$, UCS and the point load index ($I_{s}$) extended over a broader range. $\rho$ was within 2.683–2.841 g/cm³, with a mean value of 2.740 g/cm³.

Table 2 Geotechnical properties of studied samples

The $I_{s}$ values, measured on irregular blocks using a digital point load test system based on ISRM guidelines (1985), were within 0.84–6.09 MPa, with a mean value of 3.10 MPa. The $v_{p}$ values were measured according to the American Standard Testing Method (ASTM) standards (1983) for measuring the travel time of ultrasound pulses in a rock specimen. The $v_{p}$ values fluctuated between 3161 and 6148 m/s, with a mean value of 4293 m/s. The $E$ values varied considerably, from 6.405 to 47.936 GPa, with a mean value of 25.789 GPa. The UCS were within a very broad range (15.765–124.74 MPa), with a mean value of 60.07 MPa. The significant fluctuations of UCS, $v_{p}$, and $E$ can be attributed to the degree of weathering of the samples, their moisture content, and their inherent mineral compositions.

4 Regression Analyses

The physical and mechanical parameters of rocks are strongly related to UCS and $v_{p}$. Many researchers have also described the empirical relationship among them. However, the physical and mechanical properties of rocks are influenced by many factors such as the rock type, composition, porosity, joints, and water content. In this study, linear and nonlinear regression analyses were used to quantify the relationships among the various physical and mechanical properties ($v_{p}$, $I_{s}$, UCS, $\rho$, $E$, and $\mu$) of the sedimentary rock samples. For convenience and practical testing considerations the empirical relations between $v_{p}$ and the physical and mechanical parameters UCS, $\rho$, $E$, and $\mu$ were derived, then the empirical relations between UCS and the physical and mechanical parameters $I_{s}$, $\rho$, $E$, and $\mu$ were determined. The confidence intervals were set at 95%, and the correlation coefficient $R^{2}$ for each regression was calculated when determining the equation of the line of best fit for each regression.

4.1 Relationships Between $v_{p}$ and the Other Rock Parameters

It is generally accepted that the mechanical properties of a rock sample increase linearly with increasing $v_{p}$ values. Our tests were designed to find the correlation between $v_{p}$ and $\rho$, $E$, $\mu$, and UCS. All the parameters showed a reasonable linear relationship with a high $R^{2}$ value, except for $\mu$, where a high coefficient of determination was obtained for the logistic function. The best-fit relationships are shown in Figs. 7, 8, 9 and 10 and are quantified in Eqs. (4)–(7).

$$\rho = 4.545 \times 10^{ - 5} v_{p} + 2.54\quad R^{2} = 0.67$$

(4)

$$E = 0.013v_{p} - 30.71\quad R^{2} = 0.83$$

(5)

$${\text{UCS}} = 0.034v_{p} - 86.36\quad R^{2} = 0.80$$

(6)

$$\mu = 0.52 - \frac{91}{{30.33 + \left( {v_{p} } \right)^{0.69} }}\quad R^{2} = 0.97$$

(7)

where UCS is the uniaxial compressive strength, in MPa, determined from the unconfined compression test; $E$ is in GPa; $v_{p}$ is in m/s; $\rho$ is in g/cm³; and $\mu$ is dimensionless. $R^{2}$ is the correlation ratio for the linear/nonlinear regression analysis results.

Fig. 7

Variation of ρ with increasing $v_{p}$

Fig. 8

Variation of E with increasing $v_{p}$

Fig. 9

Variation of UCS with increasing $v_{p}$

Fig. 10

Variation of μ with increasing $v_{p}$

The relations between $v_{p}$ and $E$, UCS, and $\mu$ are best expressed by Eqs. (5), (6), and (7) (R² = 0.83, $R^{2} = 0.8$, and $R^{2} = 0.97$, respectively), compared with Eq. (4) ($R^{2} = 0.67$) which expresses the best fit between $v_{p}$ and $\rho$. Many researchers found a close correlation between $v_{p}$ and the geo-mechanical properties; however, these empirical equations are related to specific rocks of certain origins and may not apply to other rock types.

As shown above, Eqs. (1)–(3) indicate that $\rho$, $E$ and UCS have a good positive linear relation with $v_{p}$, although the $R^{2}$ value in Eq. (1) is relatively low. Equation (4) shows that $\mu$ has a good positive nonlinear relation with $v_{p}$. These results indicate that $\rho$, $E$, $\mu$ and UCS increase with increasing $v_{p}$. However, these empirical relations need to be validated by the Student’s t test (see Sect. 5).

4.2 Relationships Between the UCS Value and the Other Rock Parameters

Previous research (Gunsallus and Kulhawy 1984; Panek and Fannon 1992; Singh and Singh 1993; Kahraman 2001) provided empirical relations between UCS and $I_{s}$. Similar to Sect. 4.1, we determined the relationships between UCS, the most fundamental rock strength parameter, and the rock’s other physical and mechanical properties (Figs. 11, 12, 13 and 14). The regression lines for $E$, $\rho$, and $\mu$ estimated from the UCS value using the empirical equations derived in this research can be expressed as

$$\rho = 0.0013 \times {\text{UCS}} + 2.657\quad R^{2} = 0.871$$

(8)

$$E = 0.347 \times {\text{UCS}} + 4.95\quad R^{2} = 0.825$$

(9)

$$\mu = 8.767 \times 10^{ - 4} \times {\text{UCS}} + 0.199\quad R^{2} = 0.743$$

(10)

$${\text{UCS}} = 20.91 \times I_{s} - 4.79\quad R^{2} = 0.956$$

(11)

where $I_{s}$ represents the point load index, in MPa.

Fig. 11

Variation of UCS with increasing $I_{s}$

Fig. 12

Variation of E with increasing UCS

Fig. 13

Variation of ρ with increasing UCS

Fig. 14

Variation of μ with increasing UCS

Figures 11, 12, 13 and 14 show the best results of the linear regression analyses, where all the raw data are within the confidence intervals. The physical and mechanical parameters ($E$, $\rho$, $I_{s}$ and $\mu$) increase linearly with increasing UCS. In these estimates, the determination coefficients ($R^{2}$) of all the formulas (Eqs. 8–11) are very good. The highest determination coefficient was found for Eq. (11), suggesting that UCS can be predicted accurately using $I_{s}$ and Eq. (11). The coefficients $\rho$, $E$, and $\mu$ from Eqs. (8), (9), and (10), respectively, were also determined with reasonable accuracy, suggesting that properties such as $\rho$, $E$, and $\mu$ are closely related to the UCS value. Before applying these empirical relationships, their validity must be checked by the Student’s t test, as presented in Sect. 5.

The 1:1 slope line is used to evaluate the fit between the measured value and predicted value. The fitting line and 1:1 slope line are shown in Figs. 15 and 16 where the dataset located on the 1:1 slope line indicates an exact correlation. The larger the deviation from the slope line, the lower the accuracy. The R² values in Figs. 15 and 16 provide a good index for validating the reliability of the empirical relationship, indicating that $v_{p}$ and UCS can be used reliably to predict the other parameters.

Fig. 15

Cross-correlation of measured and predicted values from $v_{p}$

Fig. 16

Cross-correlation of measured and predicted values from UCS

5 Student’s t Test

The high correlation between the rock mechanical parameters can be verified by the Student’s t test, assuming that the observations are chosen randomly and are normally distributed. Our test results were compared with the computed $t$ value and the tabulated $t$ value using the null hypothesis. The t-test compares the difference between two averages for the variation in the data. In Eq. (12), the numerator represents the absolute value of the difference between the two averages, and the denominator represents data dispersion. The expression for the t-test is given by:

$$t = \frac{{\left| {\overline{{x_{1} }} - \overline{{x_{2} }} } \right| }}{{\sqrt {\frac{{s_{1}^{2} }}{{n_{1} }} + \frac{{s_{2}^{2} }}{{n_{2} }}} }}$$

(12)

where $\overline{{x_{1} }}$ represents the tabulated mean, $\overline{{x_{2} }}$ represents the computed mean, $s_{1}^{2}$ represents the tabulated variance, $s_{2}^{2}$ represents the computed variance, and n represents the number of samples.

Once the $t$ value is calculated, it is compared with the tabulated value. If the calculated value is higher than the tabulated one, this indicates good correlation. In Sect. 4, the confidence interval was set at 95% and the significance level was 0.05, obtaining a critical $t$ value of 1.684. The tabulated and calculated values are listed in Table 3. The calculated values of the t-test outperform the tabulated values, indicating that all the parameters have strong correlation among themselves. Thus, our results can be used to predict these parameters using the UCS value and the $v_{p}$ value.

Table 3 Tabulated and calculated t values of the Student’s t test

6 Conclusions

In this paper, the physical and mechanical properties ($E$, UCS, $v_{p}$, $\mu$, and $\rho$) of dolomitic limestone were tested on 40 specimens. The results indicate significant correlations between the parameters $E$, $\mu$, and $\rho$ and the parameters $v_{p}$ and UCS. These empirical relations were expressed as equations which can be used to obtain important index properties of the dolomitic limestone and are accurate for general applications. The Student’s t test results showed higher values than the tabulated values, confirming that the physical and mechanical properties ($\rho$, $\mu$, and $E$) of dolomitic limestone in the Huize lead and zinc mine can be predicted using the proposed correlation equations. These equations may also apply to dolomitic limestone in other locations. Thus, the results of evaluating the physical and mechanical parameters of rocks presented in this paper may have broad applications in underground geotechnical engineering. Hence, in future research this method should be further investigated and applied to different rock types.