Introduction

Uniaxial Compressive Strength (UCS), tensile strength (TS) and the elasticity modulus (E) are important geotechnical parameters widely used in rock engineering projects [1, 2]. A large number of geotechnical hazards such as landslides [3], pillar instability [4], fault activation [5] and ground subsidence [6] are related to water-rock interaction effects, as the presence of water significantly reduces rock strength and stiffness. Therefore, a comprehensive and in-depth knowledge of water-rock interaction effects on rock properties is crucial for preventing and solving problems related to rock mechanics applications.

Many works have studied the relations between dry and saturated uniaxial compressive strength for different rocks [7,8,9,10,11,12,13,14,15,16,17]. The influence of water content on the tri-axial compressive strength, elastic modulus, tensile strength, shear strength and fracture toughness has been reported by several researchers [18,19,20,21]. All of them revealed that presence of water reduces strength and deformations of different rocks. This reduction is due to variation in mineralogy, texture and lithology in different rock types.

Determining strength and deformation of rocks by direct and destructive tests requires core specimens with high quality, partly expensive, difficult operation and considerable time, while in indirect methods, these specifications are not necessary [22,23,24,25]. Different studies proposed empirical equations and developed experimental models for indirect estimation of strength and deformation parameters of rocks in dry and saturated conditions. Although many studies have attempted to describe the effect of water on the mechanical properties, no one has been focused on peridotites. But, due to the fact that peridotites have become important earth crust components in some areas like south-eastern Europe, especially in the countries of former Yugoslavia, Albania, Greece, Turkey etc., many studies should be focused on them. For this reason, within the framework of the present research, thirty five samples taken from central part of Greece are tested to determine the dynamic (Compressional, VP and shear wave velocity VS) and the mechanical (uniaxial compressive strength, UCS, elasticity modulus, E and Brazilian tensile strength, BTS) properties in saturated states. The same dynamic and mechanical characteristics from the same locations with dry conditions as well as physical properties in dry (dry unit weight, γdry) and saturated states (saturated unit weight, γsat) were performed by Diamantis [26, 27]. Both dry and saturated conditions were based on the recommended methods of International Society of Rock Mechanics (ISRM) and American Society for Testing and Materials (ASTM). The objective of this research is to assess the effect of saturation on strength and deformation of the peridotite rocks. In other words, understanding the effect of the water presence of on the quality of engineering materials is the most important goal pursued by this research. Thus, simple regression analyses were used for indirect estimating the strength and deformation of peridotite rocks through the physical and dynamic properties and some empirical equations were proposed between dry and saturated characteristics. The determination coefficients (R2) and the equations of the fitted lines were calculated by the “least squares” method.

Materials and methods

The peridotites used for the experiments were retrieved from two regions of the central Greece. The first area is located at the eastern-southeastern parts of Kallidromo Mt. (near the villages of Kallidromo, Reginio and Modio, Fig. 1), while the second area is the western parts of Othrys Mt. (near the villages of Moschokarya, Makryrrachi and Domokos, Fig. 1). Basing on the field investigations, the peridotites are mainly medium-grained, homogeneous and isotropic belonging to the ophiolitic complex of Orthrys. According to Diamantis [26, 27], they present a granular or porphyritic structure, compact texture without preferred mineral orientation and they are dominantly formed by olivine, orthopyroxene (mainly enstatite) and chromite minerals belonging to the parent rocks. In fact these Peridotites have been influenced by serpentinization (i.e., low-temperature metamorphic process, [17, 28]) and the primary magnetic minerals transformed into secondary minerals. Serpentinization degree of the tested samples varies between 3 and 27% [26, 27].

Fig. 1
figure 1

a Study areas and ophiolitic complexes in Greece and neighboring countries. b Lithological map of the study area at Othrys Mt. c Lithological map of the study area at Kallidromo Mt. The exact locations of the samples collected are also displayed on both lithological maps (coordinates belong to the Greek Geodetic Reference System 1987: GGRS87)

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The data pertains to peridotites samples from thirty-five sites in the above-mentioned study areas (from the surface). The test samples were cored from fresh intact rock blocks and were cut in cylindrical specimens (6 to 10 per sample). A diamond cutter with an acute blade and a grinder were used for the cutting and polishing operations respectively. The size, shape and ends of samples followed the ISRM [29] testing specifications. The height-to-diameter ratio of the test specimens was recommended to be between 2.0 and 2.5 and the diameter of the prepared cylindrical rock specimens ranged between 53 and 55 mm with the aim of reducing the uncertainty of the sample size on the measured properties and especially on strength. Firstly, the specimens were inspected macroscopically and only the homogeneous, isotropic, un-weathered (or slightly weathered), and free of visible joints, cracks, fissures, or other discontinuities, peridotites were considered.

Results and discussion

Physical properties

The physical properties including effective porosity (ne), water absorption (Wa), dry unit weight (γdry) and saturated unit weight (γsat) were obtained by Diamantis [26, 27] for 35 samples of peridotite rocks according to ISRM [29] methods. The results of physical properties tests were calculated using the following equations.

$$n\, = \,\frac{{V_{v} }}{{V_{t} }}\, = \,\frac{{(M_{{{\text{sat}}}} \, - \,M_{s} )/\rho_{w} }}{{(M_{{{\text{sat}}}} \, - \,M_{{{\text{sub}}}} )/\rho_{w} }}\, \times \,100$$
(1)
$$\gamma_{{{\text{dry}}}} \, = \,\frac{{W_{{{\text{dry}}}} }}{{V_{t} }}\, = \,\frac{{M_{s} \, \times \,g}}{{(M_{{{\text{sat}}}} \, - \,M_{{{\text{sub}}}} )/\rho_{w} }}$$
(2)
$$\gamma_{{{\text{sat}}}} \, = \,\frac{{W_{{{\text{sat}}}} }}{{V_{t} }}\, = \,\frac{{M_{{{\text{sat}}}} \, \times \,g}}{{(M_{{{\text{sat}}}} \, - \,M_{{{\text{sub}}}} )/\rho_{w} }}$$
(3)
$$W_{a} \, = \,\frac{{M_{{{\text{sat}}}} \, - \,M_{s} }}{{M_{s} }}\, \times \,100$$
(4)

where, Vv is the volume of the voids (m3), Vt is the total volume of the specimen (m3), Msat is the saturated mass of the specimen (dry on the surface, g), Msis the dry mass of the specimen (g), Msub is the submerged mass of the specimen (g),Wdry is the dry weight of the specimen (kN/m3), g is the gravitational acceleration (9.807 m/s2), and ρw, is the density of water (1 g/cm3).

The minimum, the maximum, the average value and the standard deviation of the physical properties are listed in Table 1. The effective porosity values range between 0.06 and 0.26%. The porosity of the studied rocks classified as vary low porosity based on [30] classifications. Average value of water absorption is equal to 0.05%. Average value of dry and saturated unit weight are obtained in range of 30.64–33.33 kN/m3 and 30.66–33.34 kN/m3, respectively. By unit weight point of view the studied samples are classified as very high according to IAEG [31] classification.

Table 1 Basic descriptive statistical analysis of the physical properties for the peridotite rocks [26, 27]
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Correlations between physical properties were performed and graphs of regression analyses are shown in Fig. 2. Based on the results direct linear correlations (y = ax + b) with strong coefficient of determination (R2 = 1 and 0.99) were obtained between dry and saturated unit weight and porosity and water absorption, respectively (Fig. 2a and b). Good reverse linear correlations with high coefficient of determination equal to R2 = 0.94 and R2 = 0.95 was found between dry unit weight and porosity, and saturated unit weight and water absorption, respectively (Fig. 2c and d). Therefore, when porosity and water absorption decrease the unit weight increases.

Fig. 2
figure 2

Correlations between a dry unit weight and saturated unit weight, b porosity and water absorption, c dry unit weight and porosity and d saturated unit weight and water absorption

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Ultrasonic wave velocity

Several works [23, 32,33,34] studied the relation between ultrasonic wave velocity and different characteristics of rocks and showed that rock properties are closely related to wave velocity. Especially, Kurtulus et al. [33] studied the relation between porosity and VP in serpentinized ultrabasic rocks by simple regression analysis and presented experimental equation with coefficient of determination R2 = 0.87.

In this research, ultrasonic wave velocity including primary or compressional wave (VP) and secondary or shear wave (VS) in saturated conditions were performed based on recommend methods of [35]. The prepared core specimens with diameter of 54 mm and a length to diameter ratio in range of 2–2.5 were used for determining P and S wave velocity of studied rocks. Also, Vp and Vs values in dry states obtained by Diamantis, [26, 27] were used in order to be compared with saturated values. The P-wave velocity values ranged from 7048 to 7981 m/s and 7591 to 8174 m/s in dry and saturated conditions, respectively (Table 2). Therefore, the P-wave velocity averagely 4.61% increased after saturation. According to IAEG [31] rock classification, the P-wave velocity values classified as very high. The S-wave velocity values were in range of 3816–4560 m/s and 4111–4733 m/s, in dry and saturated conditions, respectively (Table 2). So, the S-wave velocity averagely 5.01% increased after saturation. The minimum, the maximum, the average value and the standard deviation of the ultrasonic wave velocities are listed in Table 2.

Table 2 Basic descriptive statistical analysis of the ultrasonic wave velocity for the peridotite rocks
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Correlations between porosity and dry and saturated P-wave velocities are presented in Fig. 3a. As can be seen, reverse logarithmic relations were found between the porosity and P-wave velocity with coefficient of determination R2 = 0.86 and R2 = 0.77 for dry and saturated conditions, respectively. Correlations between porosity and dry and saturated S-wave velocity as reverse logarithmic relations with coefficient of determination equal to R2 = 0.88 and R2 = 0.87 for dry and saturated conditions are shown in Fig. 3b. Therefore, when porosity of the studied rocks increases the ultrasonic wave velocity decreases. Also, the presence of water in the peridotites increases the sound velocities independently of effective porosity values.

Fig. 3
figure 3

Correlations between porosity and a dry and saturated P-wave velocity and b dry and saturated S-wave velocity

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Correlations between dry unit weight with dry and saturated P and S wave velocity are presented in Fig. 4a and b, respectively, and determination coefficient was found between 0.74 and 0.85. These relations are directly linear, therefore when unit weight of the studied rocks increases the P and S wave velocity increase.

Fig. 4
figure 4

Correlations between dry unit weight and a dry and saturated P-wave velocity and b dry and saturated S-wave velocity

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Correlation between dry VP and saturated VP is presented in Fig. 5a. This relation is direct linear with coefficient of determination R2 = 0.91.Direct linear correlation between dry and saturated VS with coefficient of determination R2 = 0.93 is shown in Fig. 5b. Other correlations between VP and VS in dry and saturated conditions are presented in Fig. 5c and d.

Fig. 5
figure 5

Correlations between a dry and saturated P-wave velocity, b dry and saturated S-wave velocity, c dry P-wave velocity and dry S-wave velocity and d dry P-wave velocity and saturated S-wave velocity

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Dynamic elastic constants

Elastic characteristics of rocks are measured from dynamic and static procedures [36]. In the static or destructive method, uniaxial stresses are applied on the core samples until failure occurs. The static elastic properties and uniaxial compressive strength of the rocks are obtained by studying the stress-deformation curves (estimated in next section, [37]). In dynamic or non-destructive method, elastic constants were calculated by using obtained results of compressional and shear ultrasonic wave velocities (VP and VS) [38]. In this research, dynamic elastic constants including bulk modulus (K), shear modulus (G), Poisson’s ratio (ν) and elasticity modulus (E) are calculated by the following equations [35]:

$$K = \rho_{b} (V_{P}^{2} - \frac{4}{3}V_{S}^{2} )$$
(5)
$$G = \rho_{b} V_{S}^{2}$$
(6)
$$\nu = \frac{{(V_{P}^{2} - 2V_{S}^{2} )}}{{2(V_{P}^{2} - 2V_{S}^{2} )}}$$
(7)
$$E = 2G(1 + \nu )$$
(8)

where ρ is the bulk density, VP and VS are the compressional and shear ultrasonic wave velocities of the samples, respectively. The minimum, the maximum, the average value and the standard deviation of the dynamic elastic constants are listed in Table 3.

Table 3 Basic descriptive statistical analysis of the dynamic elastic constants for the peridotite rocks
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In order to determine the relationships between dynamic elastic constants in dry and saturated conditions, statistical correlations were performed by simple regression analysis. Relationships between dry and saturated bulk modulus, dry and saturated shear modulus and dry and saturated elasticity modulus are illustrated in Fig. 6a, b and d, respectively. The relations are linear and the determination coefficients range between 0.67 and 0.97. While relationships between dry and saturated Poisson’s ratio is direct power relation with low determination coefficient equal to R2 = 0.55 (Fig. 6c). Because the Poisson’s ratio has small values (lower than 0.5) in comparison to other dynamic properties, the change of its determination coefficient is very sensitive, and this causes to decrease of the coefficient. Also, inhomogeneity or laboratory test errors can sometimes be led to decrease in coefficient of determination.

Fig. 6
figure 6

Correlations between a dry and saturated bulk modulus, b dry and saturated shear modulus, c dry and saturated Poisson’s ratio and d dry and saturated elasticity modulus

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Strength and deformation parameters

In this research for determining strength and deformation of the studied rocks, some mechanical properties including uniaxial compressive strength (UCS), elasticity modulus (E) and Brazilian tensile strength (BTS) were performed in saturated conditions, while the same property values in dry conditions were obtained by Diamantis [26, 27].

The UCS tests were carried out using a servo-controlled hydraulic testing machine in accordance with ASTM [39]. The elasticity modulus was derived from the slope of the stress–strain curves at the 50% of the maximum UCS. The prepared core specimens with average diameter of 54 mm (NX, range from 53 to 55 mm) and a length to diameter ratio in range of 2–2.5 [40] were used for determining UCS of the studied rocks. The UCS values were ranged from 65.21 to 241.56 MPa and 52.32 to 240.90 MPa for dry and saturated conditions, respectively, exhibiting a great fluctuation. This great difference may be due to the internal discontinuities, macroscopically undetected and/or the different degrees of serpentinization and/or petrographic variety and/or the structural complexity of peridotites.

Therefore, the uniaxial compressive strength of the studied rocks decreased averagely 8.19% after saturation. Average values of UCS were obtained equal to 142.07 and 132.53 MPa for dry and saturated conditions, respectively. Therefore, based on ISRM [29], the tested samples were classified as high to extremely high strength rocks in both dry and saturated conditions. The elasticity modulus values were 26.40 to 69.34 GPa and 21.27 to 68.75 GPa, in dry and saturated conditions, respectively. So, the elasticity modulus of the studied rocks decreased averagely 8.43% after saturation. This means that the presence of water in the rock structure reduces its modulus of elasticity.

The Brazilian tensile strength test as simple indirect method for determining tensile strength of rocks were carried out on prepared core specimens with thickness to diameter ratios 0.5 based on ISRM [29]. Brazilian tensile strength was determined by the following equation:

$${\text{BTS}}\, = \,\frac{2P}{{\pi Dt}}$$
(9)

where, P is the maximum load, D is the diameter, and t is the thickness of the specimen.

The BTS values were ranged from 11.76 to 24.93 MPa and 9.74 to 24.06 MPa in dry and saturated conditions, respectively. Average values of BTS were obtained equal to 18.67 and 16.64 MPa in dry and saturated conditions, respectively. Therefore, the Brazilian tensile strength of the studied rocks was reduced averagely 11.71% after saturation. Results of mechanical properties testing indicated that after saturation average value of BTS decrease (reduce = 11.71%) more than UCS (reduce = 8.19%). Therefore, saturation affected more tensile strength of studied rocks than compressive strength. Similar results were found by [41,42,43,44,45] for several rocks.

Obtained values of mechanical properties including uniaxial compressive strength, elasticity modulus and Brazilian tensile strength for the studied rocks in dry and saturated conditions are listed in Table 4.

Table 4 Values of the mechanical properties for the studied peridotite rocks in dry [26, 27] and saturated conditions
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Effect of porosity on strength and deformation

Some researchers attempted to estimate indirect of the UCS of different rocks by using porosity values and developed empirical equations [23, 46,47,48,49,50]. In this regard, Harthi [51] studied the porosity of igneous rock and linear regression equations were presented for predicting uniaxial compressive strength: UCS = 274–8.51n for n < 20% and UCS = 104–1.01n for n > 20%. [52] investigated the influence of apparent porosity on uniaxial compressive strength of some volcanic rocks.

In this study, correlations between porosity and uniaxial compressive strength, elasticity modulus and Brazilian tensile strength in dry and saturated conditions were found by simple regression analysis and graphs are illustrated in Fig. 7a to c, respectively. Reverse logarithmic equations with determination coefficient in range of 0.69–0.80 was found. Based on the results it’s obvious when the porosity decreases, the UCS, E and BTS increase.

Fig. 7
figure 7

Correlations between porosity and a dry and saturated uniaxial compressive strength, b dry and saturated elasticity modulus and c dry and saturated Brazilian tensile strength

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Effect of dry unit weight on strength and deformation

Some researcher assessed the effect of dry unit weight on uniaxial compressive strength of rocks (e.g. [34, 53,54,55,56,57]). Plots of correlations between dry unit weight with dry and saturated UCS are illustrated in Fig. 8a. The linear equations are obtained with medium coefficient of determination R2 = 0.68 and R2 = 0.70 for dry and saturated conditions, respectively. The linear correlations presents relationships between dry unit weight with dry and saturated elasticity modulus with coefficient of determination equal to R2 = 0.64 for dry and saturated conditions, respectively (Fig. 8b). Based on the trend lines of Fig. 8c, the relation between dry unit weight and BTS in dry conditions presents little better correlation (R2 = 0.80) than saturated stages (R2 = 0.77). As a general conclusion, when dry unit weight increases, the UCS, E and BTS increase.

Fig. 8
figure 8

Correlations between dry unit weight and a dry and saturated uniaxial compressive strength, b dry and saturated elasticity modulus and c dry and saturated Brazilian tensile strength

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Relationships between wave velocity, strength and deformation

Many researchers studied relationships between wave velocity and uniaxial compressive strength [23, 58,59,60,61], tensile strength [62, 63] and elasticity modulus [64,65,66,67] for several rocks.

The relationships between dry P-wave velocity with uniaxial compressive strength, elasticity modulus and Brazilian tensile strength in dry and saturated conditions are presented in Fig. 9a to c, respectively. Based on the results, these relations are directly linear and determination coefficient obtained in range of 0.77–0.85. In this research, beside correlations of VP, the relationships between dry S-wave velocity with UCS, E and BTS in dry and saturated conditions are analyzed by regression technique that is illustrated in Fig. 9d to f. Results are close to VP correlations and direct linear relations with coefficient of determination in range of 0.73–0.83 were obtained.

Fig. 9
figure 9

Correlations between dry P-wave velocity and a dry and saturated uniaxial compressive strength, b dry and saturated elasticity modulus, c dry and saturated Brazilian tensile strength and correlation between dry S-wave velocity and d dry and saturated uniaxial compressive strength, e dry and saturated elasticity modulus and f dry and saturated Brazilian tensile strength

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Relationships between strength and deformation parameters

In recent years, the relationships between dry and saturated UCS for various rocks were studied by many researchers and experimental equations have been concluded (e.g. [7,8,9,10,11, 68, 69]). Based on the graph of Fig. 10a, direct linear correlation is established between dry and saturated UCS and very high coefficient of determination (R2 = 0.98) was found. Many studies, attempted to correlate the Edry with Esat [42,43,44, 70,71,72]. In this research, the linear equation presents very strong correlation with coefficient of determination R2 = 0.99 between dry and saturated elasticity modulus (Fig. 10b). Relationships between Brazilian tensile strengths in dry and saturated conditions are illustrated in Fig. 10c. There is a direct linear relation with good coefficient of determination equal to R2 = 0.92.

Fig. 10
figure 10

Correlations between a dry and saturated uniaxial compressive strength, b dry and saturated elasticity modulus and c dry and saturated Brazilian tensile strength

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Many researchers have investigated the relations between strength and deformation characteristics for several rocks. In this research, for assessing the relationships between mechanical properties including UCS, E and BTS in dry and saturated conditions some correlations were performed. Correlations between dry UCS and E in dry and saturated conditions with acceptable coefficient of determination (R2 = 0.83) is shown in Fig. 11a. Direct linear correlations between dry BTS with dry and saturated values of UCS are shown in Fig. 11b. Medium coefficient of determination equal to 0.63 and 0.66 were obtained for dry and saturated conditions, respectively. Correlations between dry values of BTS with dry and saturated elasticity modulus with low coefficient of determination (0.58 and 0.59) are illustrated in Fig. 11c.

Fig. 11
figure 11

Correlations between a dry uniaxial compressive strength and dry and saturated elasticity modulus, b dry Brazilian tensile strength and dry and saturated uniaxial compressive strength and c dry Brazilian tensile strength and dry and saturated elasticity modulus

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Summary and conclusions

In this research, the effect of water saturation on the physical properties, wave velocities, strength and deformation of 35 samples of peridotite rocks were investigated. 35 samples of peridotite rocks, taken from the same places in central Greece as they reported in Diamantis [26, 27], were tested in saturated conditions and the values of static and dynamic tests were determined.

Peridotite rock samples were collected from two regions of the central Greece including eastern-southeastern part of Kallidromo Mt. and western part of Othrys Mt. The investigated samples are peridotites which are mainly medium-grained, homogeneous, isotropic rocks with granular or porphyritic structure. Results of tests on physical properties show the porosity of the studied rocks is classified as very low while the unit weight as very high. Obtained values of the ultrasonic wave velocities show P and S waves were increased after saturation. Regression analyses show reverse logarithmic and direct linear relations between porosity and unit weight with P and S wave velocities in dry and saturated conditions, respectively. According to the results, the mechanical properties namely UCS, E, and BTS were decreased after saturation. Reverse logarithmic equations was found between porosity with UCS, E, and BTS. Direct linear correlations among dry unit weight with UCS, E, and BTS indicated that when γdry increases, the mechanical properties are increased. The relationships between dry P and S wave velocities with UCS, E and BTS in dry and saturated conditions were assessed and R2 obtained in range of 0.73–0.85. Linear correlations between dry and saturated values of UCS, E and BTS were performed and regression equations with strong coefficient of determination were obtained. Results of this research in the form of some experimental equations for determining static and dynamic properties of peridotite rocks in dry and saturated conditions with coefficient of determination in range of 0.55–1 are presented. To sum up, the above-mentioned results revealed change in the physical, dynamic, and mechanical properties with the presence of water. So, the presence of water or moisture can be led to decrease of rock engineering quality when they are used in engineering projects.