Introduction

Seismic techniques are well understood, nondestructive, and low-cost methods that can be easily performed in the laboratory and in-situ. These techniques are widely used in civil, geological, mining, and rock engineering applications to characterize the dynamic properties of rocks. The seismic properties of rocks are influenced by numerous factors, such as the rock type, texture, mineralogical composition, grain shape and size, density, porewater, porosity, and water content1,2,3,4,5,6.

Characterization of rock materials in the field or laboratory requires the measurement of the mechanical and physical properties of rocks, which are critical tasks in many geotechnical and rock engineering projects. Moreover, such experimental studies are extremely expensive, tedious, and time consuming. Reliable modeling of empirical relationships among the mechanical and physical properties of rock materials can eliminate the need for such an expensive and tedious work and facilitate the estimation of the necessary critical design parameters.

Numerous researchers have already derived possible relationships between P-wave velocity (Vp) and the mechanical and physical features of limestone using statistical methods like regression analyses (Table 1). Tugrul and Zarif1 examined the relationships between total porosity (nt), uniaxial compressive strength (UCS), and Vp using linear empirical equations and found strong negative and positive relationships among the correlated parameters. Using simple linear relations, Yasar and Erdogan2 reported strong relationships among UCS, modulus of elasticity (E), density (ρ), and Vp. Kahraman and Yeken3 conducted various tests, for instance, absorption of water by weight (WA), ρ, porosity (n), and Vp, on 14 different carbonate rocks and reported strong negative and positive correlations between Vp and all studied physical parameters. Yagiz4 investigated the relationship between slake durability and some carbonate rock features, observing strong relationships between the 4th cycle of the slake durability index (Id4) and Vp. Yagiz7 assessed the geotechnical properties of carbonate rocks using Vp tests and found high to low positive and negative correlation coefficients in the relationships among UCS, E, effective porosity (n′), WA, Schmidt rebound value (SHV), 2nd cycle of the slake durability index (Id2), dry density (ρd), saturated density (ρs), and Vp using statistical analyses, including Student’s t test. Arman et al.5 reported moderate correlations among indirect tensile strength (ITS), point load index (PLI), and Vp. Using 45 limestone core specimens, Najibi et al.8 revealed relatively powerful relationships between E and UCS with Vp. Stan-Kieczek9 studied the elastic properties of carbonate rocks via laboratory and in-situ tests and reported a strong relationship between E and Vp. Pappalardo et al.10 performed a detailed laboratory characterization of the limestones of heritage Baroque monuments and discovered strong positive and negative correlations between limestone Vp and E, UCS, and n. Jamshidi et al.11 developed empirical equations to estimate the mechanical properties of travertine building stones from Vp and Schmidt hardness. Jamshidi et al.12 examined the effects of n and ρ on the relationship between UCS and Vp and noted a strong correlation between Vp and UCS. Ferriodooni and Khajevad13 investigated the relationships between engineering properties, slake durability index (SDI) of some travertine samples under the wetting–drying cycle, and observed a strong relationship between Id2 and Vp. Using an indirect method of comparative evaluation to approximate the compressive strength of limestone, Ali et al.14 reported a strong correlation between Vp and UCS. Kurtulus et al.15 estimated the UCS using SHV and Vp, which proved to be strongly correlated. Wen et al.6 examined the correlation between geomechanics parameters and UCS and Vp on 40 dolomitic limestone specimens and reported significant correlations among the parameters of E, UCS, ρ, and Poisson’s ratio (ν).

Table 1 Data of the correlation between some mechanical and physical properties of either carbonate rocks or limestone from published literature and this study.
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Herein, the relationship between the mechanical and physical properties of limestone and Vp were investigated. Further, the result of carbonate rocks or limestone obtained in previous studies were compared with those obtained in this study in terms of UCS, PLI(Is(50)), Id2, γn, Gs(c), WA, and n. Moreover, the reliability of the empirical relationship was validated using Student’s t test and the predicted and measured cross-correlation values from Vp.

Study area and geological settings

The study area, located along the Hafit Mountain, has been geologically well documented by numerous previous studies16,17,18,19,20 (Fig. 1). The carbonate rocks of the Hafit Mountain—an asymmetric and doubly plunging anticline—were dissected using numerous sets of faults. Tertiary carbonate rocks provide unique outcrops for three core rock units in the study area.

Figure 1
figure 1

Location map of the sample sites with the geology of the Hafit Mountain (generated with ArcGIS 10.821).

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In the Hafit Mountain, the oldest rock unit is the Early Eocene Rus Formation, aged 55–49 Myr. It is thick bedded, massive, and generally appears grayish white in color. At some levels, brownish color chert nodules are dominated with dolomite layers. The Middle to Late Eocene Dammam Formation, aged 49–34 Myr, overlies the Rus Formation and exhibits some cavernous and fractured limestone layers, locally converting to chalky and dolomitic limestone with soft marl beds. Nummulitic limestone with marl beds is also available in different localized outcrops. The Early Oligocene Asmari Formation, aged 34–29 Myr, comprising mainly gypsiferous mudstone, nummulitic marly limestone, chalky and dolomitic limestone, and marl, is the youngest rock unit in the study area22.

The Early Miocene Lower Fars Formation, aged 23–16 Myr, comprises gypsite evaporates that are interbedded with friable marls and mudstones with gypsum veins, topped with a calcrete layer. The upper part comprises conglomeratic sandstone, rich in reworked chert and ophiolitic rock fragments23. The Miocene to Pliocene Barzaman Formation, aged 23–2 Myr, comprises a pebble–cobble conglomerate interbedded with sandstones and mudstones. The formation yields evidence for cycles of sedimentation from pluvial (wet) to arid, and the sediments of the arid intervals exhibit a white or pink color owing to dolomite alterations24.

Rock sampling and laboratory studies

Over 100 limestone rock blocks were collected along the seven selected sampling locations (Figs. 1 and 2A, B). Each rock block was carefully inspected for laboratory testing and analysis. The rock blocks were appropriately represented and provided standard testing specimens without visible defects, such as alteration zones and fractures. According to the American Society for Testing and Materials25 standards, core samples were acquired from 94 selected rock blocks for physical and mechanical tests (Fig. 2C–E]). Table 2 lists the number of samples used for each test (from seven sampling locations) with their average (x) and standard deviation values. The tests were conducted on intact and natural rock samples. If the tests did not meet the suggested standards—owing to either core sample features or rock failing along the existing weakness plane—the results were not considered for examination.

Figure 2
figure 2

(A) General view of the Rus Formation and limestone outcrops, (B) a group of rock blocks collected from the field and used for coring and testing, (C) a cored rock block sample with core samples, (D) core samples prepared for the UCS test, and (E) test sample for the SDI test.

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Table 2 Number of samples used for each test and the range within one standard deviation of the average.
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With reference to the ASTM26 standards, for the Vp test, a portable pulse generator unit control, Pundit Lab, and two transducers with 25.4-mm diameter and 250-kHz frequency were used to measure the Vp of the core samples (Fig. 3A). Further, 321 Vp tests were conducted on limestone core specimens. According to Anon27 classifications, the Vp of limestone indicates highly scattered ranges from 2.08 to 7.62 and can be classified as very low to very high.

Figure 3
figure 3

Laboratory equipment used for (A) Vp measurements, Pundit Lab, (B) UCS test, servo plus evolution press (C) PLI test, and (D) SDI test.

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The UCS, PLI, Id2, γn, Gs(c), WA, and n values of the limestone samples were determined following the ASTM and ISRM standards. Table 2 lists the descriptive statistical distribution of the test results. The UCS tests were conducted on 159 NX-sized core samples, which were prepared based on the ASTM28 standard, with approximately 2:1 length to diameter ratio. Further, smooth sample end surfaces were prepared, and a 0.5–1-MPa constant loading rate was axially applied to the core specimens (Fig. 3B). The PLI test was conducted on 167 regular NX-sized core samples following the ASTM29 standard (Fig. 3C). If any sample failed either tests along existing cracks, weathered surfaces, or other weakness planes, the test results for such a sample was excluded. Moreover, 94 test samples were arranged for the SDI test from each rock block, and the SDI tests were conducted based on the ASTM30 standard (Fig. 3D). Based on a study by Franklin and Chandra31, Id2 was evaluated as a very high to extremely high SDI. The γn values of regular limestone core samples were calculated for approximately 500 test samples of UCS, PLI, and ITS test samples following the suggested method of the ISRM32. Based on the recommended methods of the ISRM32, 321 core samples were used to calculate the Gs(c). Further, the WA and n values were determined for each of the 94 limestone rock blocks using representative samples.

Mineralogical and textural studies of rock units

Mineralogical and textural evaluation was conducted on 27 selected carbonate rock samples covering the entire study area. The representative and detailed evaluation were performed on two selected carbonate rock samples, L2 and L11, discussed in here. As shown in Fig. 4 [L2–A1 and 2], clear dolomite rhombohedra crystals were found within a calcite matrix and cement, which can be described as dolostones. The X-ray diffraction (XRD) studies also show dolomite as the dominant mineral (Fig. 4 [L2–A3]). The chemical composition of the dolostones was approximately 0.8 wt% SiO2, 12 wt% MgO, 41 wt% CaO, and 0.3 wt% Fe2O3 (Table 3). The limestones, classified as fossiliferous limestone and dominated by calcite mineral, contained foraminifera, calcareous algae, and coral reef fragments. They were the most common materials in the study area (Fig. 4 [L11–B1, 2, and 3]). The chemical composition of the fossiliferous limestone was approximately 0.5 wt% SiO2, 0.2 wt% MgO, 55 wt% CaO, and 0.1 wt% Fe2O3 (Table 3). The wackestone facies—particularly those found in fossiliferous limestone—indicate sedimentary deposition in a shallow marine environment.

Figure 4
figure 4

(L2–A1) Images of cross-polarized light showing relatively large dolomite crystals, (L2–A2) and (L2–A3) showing texture and mineralogy of limestone: calcite matrix and cement enlarged using SEM in (L2–A2), minerals identified using XRD in (L2–A3). (L11–B1) Images of cross-polarized light showing relatively large fossils embedded in a dominantly calcite matrix (L11–B2) and (L11–B3) showing texture and mineralogy of limestone: calcite matrix enlarged using SEM in (L11–B2) and minerals identified using XRD in (L11–B3).

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Table 3 Chemical analyses of the selected limestone.
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Statistical analyses and discussions

Linear and nonlinear regression analyses are the commonly used and accepted methods for investigating empirical relationships between the mechanical and physical properties of rocks. Numerous researchers have already suggested the empirical relations between the mechanical and physical properties for specific rock types; however, studies only on carbonate/limestone were considered herein (Table 1). The rock type, composition, porosity, water content, and joints have a significant impact on the mechanical and physical properties of rocks. In this study, linear and nonlinear regression models were used to investigate the relationships between some mechanical and physical properties of limestone with respect to Vp. 95% confidences interval for the parameters were constructed. The correlation coefficient R which measures the strength and direction of the relationship between two variables for each regression mode was computed using the best line fit equation (Figs. 5, 6, 7, 8, 9, 10, 11). Table 4 lists the R-values. Regression analysis revealed a strong relationship between the Vp and the PLI, Id2, γn, Gs(c), WA, and n and a moderate relation between Vp and UCS. Furthermore, as far as independence of the residuals is concerned, the residual plots for those models; UCS, PLI(Is(50)) Id2, γn, Gs(C), WA, n and Vp are presented in Fig. 12 (a–g). 95% confidence intervals for the true parameters are also given in Table 4.

Figure 5
figure 5

Relationship between the Vp and UCS of limestone.

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Figure 6
figure 6

Relationship between the Vp and PLI (Is(50)) of limestone.

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Figure 7
figure 7

Relationship between the Vp and Id2 of limestone.

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Figure 8
figure 8

Relationship between the Vp and γn of limestone.

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Figure 9
figure 9

Relationship between the Vp and Gs(C) of limestone.

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Figure 10
figure 10

Relationship between the Vp and WA of limestone.

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Figure 11
figure 11

Relationship between the Vp and n of limestone.

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Table 4 Empirical equations between Vp and some tested properties of limestone.
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Figure 12
figure 12

The residual plots of (a) UCS versus Vp, (b) PLI(Is(50)) versus Vp, (c) Id2 versus Vp, (d) γn versus Vp, (e) Gs(C), Vp, (f) WA versus Vp and (g) n versus Vp.

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The validity of the models were tested using Student’s t test, and the confidence levels were set at 95% and 0.05 (α = 0.05), respectively (Table 4).

The derived equations for limestone were compared with the available equations for the same rock types in the literature (Table 1). The relationships between Vp and UCS, PLI, Id2, γn, Gs(c), WA, and n obtained from this study is compared with those of obtained in previous studies (Figs. 13, 14, 15, 16, 17, 18). The R–values, which are considered a good indicator for the strength and direction of the relationship between two variables for each regression model, were computed using the best line fit equations like linear, exponential, power and logarithmic. Those models were chosen based on the empirical distribution of obtained data. As shown in Table 1 and Figs. 13, 14, 15, 16, 17 and 18, the computed equations varied and the regression coefficient (R) ranged from 0.46 to − 0.98. Such variations can be attributed to the origin and other features of rocks, including composition, porosity, and water content.

Figure 13
figure 13

Comparison of results obtained in this study with those obtained in previous studies; Vp versus UCS of limestone.

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Figure 14
figure 14

Comparison of results obtained in this study with those obtained in previous studies; Vp versus PLI (Is(50)) of limestone.

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Figure 15
figure 15

Comparison of results obtained in this study with those obtained in previous studies; Vp versus Id2 of limestone.

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Figure 16
figure 16

Comparison of results obtained in this study with those obtained in previous studies; Vp versus γn of limestone.

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Figure 17
figure 17

Comparison of results obtained in this study with those obtained in previous studies; Vp versus WA of limestone.

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Figure 18
figure 18

Comparison of results obtained in this study with those obtained in previous studies; Vp versus n of limestone.

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As shown in Figs. 19, 20, 21, 22, 23, 24 and 25, the predicted and measured values, which are acquired from Vp, were cross-correlated. The best fit between the measured and predicted values can be evaluated with the 1:1 slope line that indicates a perfect correlation; the accuracy level of the measured values declined with an increased deviation from the 1:1 slope line. This study proves the reliability of estimating the mechanical and physical properties of limestone from Vp values.

Figure 19
figure 19

Cross-correlation of the measured and predicted values estimated from Vp in terms of UCS values of limestone.

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Figure 20
figure 20

Cross-correlation of the measured and predicted values estimated from Vp in terms of PLI (Is(50)) values of limestone.

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Figure 21
figure 21

Cross-correlation of the measured and predicted values estimated from Vp in terms of Id2 values of limestone.

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Figure 22
figure 22

Cross-correlation of the measured and predicted values estimated from Vp in terms of γn values of limestone.

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Figure 23
figure 23

Cross-correlation of the measured and predicted values estimated from Vp in terms of Gs(C) values of limestone.

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Figure 24
figure 24

Cross-correlation of the measured and predicted values estimated from Vp in terms of WA values of limestone.

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Figure 25
figure 25

Cross-correlation of the measured and predicted values estimated from Vp in terms of n values of limestone.

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Conclusions

An intensive experimental program—over 1750 tests on limestone samples collected from seven different locations along the study—were conducted using several standard testing methods. Regression analyses, both linear and nonlinear, were used to generate empirical correlations between Vp and the mechanical and physical features of limestone, i.e., UCS, PLI, Id2, γn, Gs(c), WA, and n. Based on the test results, the following conclusions were accomplished:

  1. 1.

    The statistical analyses indicate significant correlations between Vp and UCS, PLI, Id2, γn, Gs(c), WA, and n. Thus, a Vp test—a simple, fast, economical, and nondestructive method for characterizing rock—can be used to predict the UCS, PLI, Id2, γn, Gs(c), WA, and n of limestone.

  2. 2.

    The correlation of coefficient, R, between UCS, PLI, Id2 and Vp range 0.67–0.72.

  3. 3.

    The WA, n, γn, and Gs(c) of limestone provide the best correlation with Vp (R =  − 0.87, − 0.86, 0.76, and − 0.76, respectively).

  4. 4.

    For all cases, the calculated t-test statistics were highly significant which confirm the UCS, PLI, Id2, γn, Gs(c), WA, and n of limestone in the study area can be reliably estimated using the proposed correlation equations.

While the results of this study may have wide common usage in engineering applications, the provided equations apply only for the specified rock. Further research is necessary to apply these results to other rock types.

Statistical analysis

Descriptive statistics of the data is presented in Table 2. Since the sample size is large (n = 94), the intervals within one standard deviation of the means are calculated using the empirical rule to check the data spread. The Statistical package, Minitab, is used to investigate the empirical correlations between various parameters through linear and nonlinear regression analyses. To test the validities of the regression models, the Student’s t tests statistics, critical values, and the p-values of the variable relationships, as well as the 95% confidence intervals of the true parameters are summarized in Table 3. According to the central limit theorem, there is no need to verify normality for large samples (n >  = 30) since the sample mean is approximately normally distributed.