The Effect of the Petrography, Mineralogy, and Physical Properties of Limestone on Mode I Fracture Toughness under Dry and Saturated Conditions

Abstract

Determining the fracture toughness of rock materials is a challenging, costly, and time-consuming task, as fabricating a sharp crack in rock specimens will lead to failure of the specimen, and preparing specimens for determining the rock fracture toughness requires special equipment. In this paper, the relationship between mode I fracture toughness (K_IC) with the rock index properties, mineralogy, and petrography of limestone is investigated using simple nonlinear and simple/multiple linear regression analyses to provide alternative methods for estimating the fracture toughness of limestones. The cracked chevron notched Brazilian disk (CCNBD) method was applied to 30 limestones with different petrographic and mineralogical characteristics under both dry and saturated conditions. Moreover, the index properties of the same rocks, including the density, porosity, electrical resistivity, P and S wave velocities, Schmidt rebound hardness, and point load index, were determined. According to the statistical analyses, a classification based on the petrography of the studied rocks was required for predicting the fracture toughness from index properties. By classifying the limestones based on petrography, reliable relationships with high correlations can be introduced for estimating the fracture toughness of different limestones using simple tests.

1. Introduction

Fracture toughness is defined as material resistance against crack growth [1]. The development and coalescence of cracks are the most important factors of rock failure with quasi-brittle behavior. Therefore, for analyzing the failure of rock materials, determining the fracture toughness is essential. This parameter is widely used in the fields of rock blasting, underground space stability, hydraulic fracturing, earthquake dynamic analysis, rock slope stability, and for evaluation of the drill-ability of rock masses.

Different studies have recently been conducted to develop some models for predicting the mechanical properties [2,3,4,5,6,7,8,9,10,11,12,13,14] and fracture toughness [15,16,17,18,19,20] of rock materials, including limestone. Akram et al. [21] reported a reverse relationship between mechanical properties and the percentages of sparite and allochems, and a direct relationship between the mechanical properties and the percentages of micrite and dolomite of limestone. Aligholi et al. [22] provided some relationship between the mechanical properties with the basic physical and dynamic characteristics of igneous rocks. Roy et al. [23] showed that the strength of sedimentary rocks decreased when the water content increased; consequently, the mechanical properties are strongly influenced by the percentage of water saturation. In addition, they estimated the rock fracture toughness using Young’s modulus and the Brazillian tensile strength. By investigating the effect of limestone texture on its mechanical behavior, Ajalloeian et al. [24] concluded that the best model for predicting Young’s modulus of carbonate rocks is multiple regression, by using some petrographic characteristics such as dolomitic percentage, the average grain size, and the percentage of allochems and carbonate. Zhixi et al. [25] reported a close relationship between K_IC and the physicomechanical properties of rock materials. According to Saeidi et al. [26], K_IC can be predicted using brittleness index. Some empirical relationships to predict K_IC from the index properties are outlined in

Table 1. Some empirical relationships between K_IC and index properties.

Empirical Relation	Rock Type	R2 (%)	Units	Reference
KIC = 0.00035 VP − 0.18	Granite and marble	64	KIC (MPa m1/2), VP (m/s)	[27]
KIC = 0.00071 VS − 0.29	Granite and marble	44	KIC (MPa m1/2), VS (m/s)	[27]
KIC = 2.45 ρ − 5.19	Granite and marble	26	KIC (MPa m1/2), ρ (gr/cm3)	[27]
KIC = −0.5 n + 1.7	Granite and marble	36	KIC (MPa m1/2), n (%)	[27]
KIC = 3.21 ρ − 6.95	Different types	91	KIC (MPa m1/2), ρ (gr/cm3)	[28]
KIC = 0.45 VP − 0.58	Sedimentary and igneous rocks	55	KIC (MPa m1/2), VP (Km/s)	[29]
KIC = 0.9 VS − 1.06	Sedimentary and igneous rocks	60	KIC (MPa m1/2), VS (Km/s)	[29]
KIC = 0.0037 e0.0022 ρ	Sedimentary and igneous rocks	54	KIC (MPa m1/2), ρ (Kg/m3)	[29]
KIC = −0.8111 VP + 1.7901	Sandstone	79	KIC (MPa m1/2), VP (m/s)	[23]
KIC = 0.000361 VP − 0.332	Sandstone	92	KIC (MPa m1/2), VP (m/s)	[25]
KIC = 0.0006147 VS − 0.5517	Sandstone	90	KIC (MPa m1/2), VS (m/s)	[25]
KIC = 0.000054074 VP + 0.3876	Shale	56	KIC (MPa m1/2), VP (m/s)	[25]
KIC = 0.0001021 VS + 0.349	Shale	64	KIC (MPa m1/2), VS (m/s)	[25]
KIC = 0.0995 IS50 + 1.11	Sedimentary rock	45	KIC (MPa m1/2), IS50 (MPa)	[30]

.

In this study, the effect of the saturation condition on the fracture toughness of limestone has been investigated, considering the types of petrography. Moreover, because of the complexity of measuring fracture toughness, the index characteristics of the limestones studied were determined, and their correlations with fracture toughness were examined. As the determination of K_IC is one of the most destructive, time consuming, and costly tests, it is very useful to predict it using non-destructive and simple test methods, which is one of the main aims of this study.

2. Materials and Methods

2.1. Materials

Thirty limestone blocks collected from different regions of Iran were studied (Figure 1). Some photomicrographs of the studied samples are represented in Figure 2. The locations, formation ages, and petrographic classification [31,32] of the studied limestones are summarized in

Table 2. Age, sampling zone, and type of rocks.

Zone	Formation	Specimen No.	Age	Rock Type
Koppeh Dagh	Chehel Kaman	R9	Paleocene	Sandy Limestone
	Pesteligh	R2	Paleocene	Sandy Limestone
		R10		Sandy Limestone
	Kalat	R11	Maastrichtian	Sandy Limestone
	Neyzar	R1	Maastrichtian	Sandy Limestone
	Aitamir	R12	Upper Aptian–Middle Cenomanian	Rudstone
	Sarcheshmeh	R13	Upper Barremian–Middle Aptian	Grainstone
	Tirgan	R6	Barremian–Aptian	Grainstone
		R5		Mudstone
		R4		Packstone
		R3		Grainstone
		R14		Grainstone
		R15		Packstone
	Shurijeh	R7	Upper Jurassic–Lower Cretaceous	Sandy Limestone
		R8		Sandy Limestone
	Mozduran	R17	Upper Jurassic	Dolomitic limestone
		R16		Dolostone
Central Iran	Qaleh Dokhtar	R18	Callovian–Kimmeridgian	Wackestone
	Esfandiar	R19	Tithonian	Floatstone
	Jamal	R20	Early Gzhelian–Asselian	Wackestone
		R21		Framestone
Zagros	Asmari	R30	Oligo–Myocene	Dolomitic limestone
		R29		Dolomitic limestone
		R28		Dolo Mudstone
		R27		Dolo Mudstone
		R26		Dolo Mudstone
		R25		Dolo Mudstone
		R24		Dolo Mudstone
		R23		Packstone
		R25		Dolomitic limestone

. Among the studied formations, Asmari is the youngest, which belongs to the Oligo–Miocene era, while Jamal is the oldest formation with the age of early Gzhelian to Asselian.

2.2. Physicomechanical Tests

Different standard tests have been employed to determine the physical, mechanical, and dynamic properties of the studied limestones. Cylindrical cores with a diameter of 54 mm (NX) were extracted from limestone blocks for such a purpose. The physical properties, including the dry density and porosity, were determined according to the ISRM suggested method [34]. The electrical resistance was measured using a portable digital device (Figure 3A). In this method, the resistivity value was measured based on the following relationship between the electrical resistance and geometrical features of the sample, including the radius, length, and cross sectional area of the rock cores (Equation (1)).

$$\mathrm{R}={\mathsf{\rho }}_{\mathrm{e}}\frac{\mathrm{L}}{\mathrm{A}}$$

(1)

The samples were fully saturated with water in a vacuum state for conducting tests under a saturated condition. Moreover, the specimens were placed in an oven at 105 °C for 24 h for conducting tests under a dry condition. Then, they were placed in a desiccator containing calcium silicate powder in a vacuum state for 3 h (Figure 3B). The ultrasonic wave velocity of the samples was determined using a portable digital device, as per the ISRM suggested method [35] (Figure 3C). The point load index was measured using a digital device, according to the ISRM suggested method [36] (Figure 3D). The point load experiments tests were carried out under both dry and saturated conditions on cylindrical cores axially with a length to diameter ratio of 0.3. The measured physical and mechanical properties of the studied rocks are presented in

Table 3. Engineering characteristics of the studied rocks.

Sample Name	VS (Km/s)	VP-Dry (Km/s)	VP-Sat (Km/s)	n (%)	ρSat (Ω.m)	ρd (gr/cm3)	Water Ab. %	RN	IS50-Dry (MPa)	IS50-Sat (MPa)	KICF-Dry (MPa m1/2)	KICF-Sat (MPa m1/2)	KICW-Dry (MPa m1/2)	KICW-Sat (MPa m1/2)	KIC-Dry (MPa m1/2)	KIC-Sat (MPa m1/2)
R1	1.392	4.704	5.007	3.5	0.881	2.597	1.33	38	1.58	1.056	1.089	0.428	1.223	0.476	1.249	0.489
R2	1.59	2.477	3.499	13.7	0.001	2.398	5.7	30	0.645	0.226	0.394	0.126	0.436	0.141	0.45	0.144
R3	2.865	6.008	6.067	1.7	11,085.196	2.691	0.64	38	1.817	2.045	1.357	1.453	1.523	1.632	1.532	1.643
R4	3.219	6.404	6.124	1.5	14,465.061	2.696	0.56	41	1.704	1.667	1.239	1.293	1.361	1.46	1.367	1.469
R5	3.413	6.235	6.342	1.7	11,434.227	2.687	0.65	41	0.888	0.953	1.072	0.906	1.197	1.007	1.205	1.012
R6	2.395	6.319	6.114	1.6	19,190.949	2.685	0.61	36	1.698	1.735	1.149	1.176	1.279	1.304	1.287	1.312
R7	3.941	6.048	6.048	1.5	11.503	2.686	0.56	40	2.145	2.075	1.549	1.547	1.727	1.727	1.773	1.772
R8	3.424	6.057	6.261	1.7	7.363	2.69	0.64	37	2.578	2.163	1.655	1.443	1.831	1.59	1.891	1.647
R9	2.212	4.704	5.184	5.4	623.915	2.556	2.13	40	1.573	0.693	1.011	0.649	1.12	0.728	1.126	0.733
R10	1.692	2.718	2.925	11.1	0.104	2.378	4.66	37	0.48	0.293	0.31	0.231	0.347	0.254	0.355	0.263
R11	2.501	3.764	3.965	7.4	193.511	2.444	3.03	41	2	1.333	1.26	0.419	1.381	0.459	1.386	0.46
R12	3.842	5.139	5.714	3.1	3878.783	2.635	1.17	35	1.68	1.653	1.241	_	1.364	_	1.369	_
R13	3.676	5.324	5.615	3.2	3435.236	2.606	1.24	44	1.547	1.093	1.017	0.996	1.133	1.097	1.14	1.102
R14	2.851	6	6.044	1.7	5409.651	2.653	0.65	42	1.28	1.547	1.011	1.02	1.12	1.11	1.126	1.157
R15	3.269	6.109	5.92	2.3	5800.758	2.649	0.86	44	1.28	1.28	0.866	1.166	0.948	1.286	0.951	1.293
R16	2.438	4.87	5.323	8.6	1066.068	2.516	3.42	40	2.08	1.627	1.099	0.971	1.217	1.074	1.224	1.079
R17	2.951	6	5.938	2.7	4319.988	2.661	1.02	44	0.667	0.4	0.832	0.803	0.913	0.91	0.916	0.916
R18	2.793	5.782	5.844	2.6	5162.635	2.65	0.98	36	1.756	1.447	1.111	1.012	1.243	1.129	1.25	1.136
R19	4.451	6.544	6.301	1.4	11,958.108	2.726	0.5	42	1.392	1.17	1.11	1.028	1.227	1.137	1.233	1.144
R20	3.527	6.277	6.385	1.5	20,879.598	2.687	0.55	37	1.488	1.622	1.061	1.193	1.187	1.327	1.195	1.335
R21	2.928	5.798	6.249	2	15,471.565	2.679	0.74	41	1.531	0.845	1.074	1.363	1.194	1.509	1.201	1.517
R22	1.815	3.984	4.081	18.4	531.007	1.98	9.31	36	0.609	0.707	0.589	0.377	0.65	0.413	0.653	0.414
R23	3.346	4.762	4.734	6.9	407.021	1.57	4.41	44	0.956	0.854	0.726	0.487	0.791	0.53	0.793	0.531
R24	3.151	5.582	5.202	9.1	158.746	2.5	3.64	51	2.218	1.602	0.963	0.867	1.05	0.945	1.052	0.947
R25	2.584	4.922	4.63	10.2	157.699	2.5	4.09	43	2.03	1.108	0.836	0.604	0.917	0.654	0.92	0.654
R26	3.425	5.167	5.003	9.2	540.381	2.51	3.67	47	2.086	1.315	1.144	0.806	1.249	0.881	1.252	0.883
R27	3.51	5.424	5.2	6.8	501.929	2.59	2.62	57	2.806	2.056	1.241	1.149	1.364	1.266	1.369	1.271
R28	2.405	5.04	4.915	8.9	257.932	2.5	3.54	46	0.353	1.037	1.124	0.887	1.224	0.972	1.227	0.975
R29	2.303	4.196	5.074	13.3	684.546	2.11	6.27	43	0.627	0.561	0.334	0.292	0.364	0.318	0.365	0.318
R30	2.787	4.591	4.631	11.9	502.143	2.43	4.89	45	1.266	1.24	0.765	0.511	0.844	0.56	0.849	0.562

.

2.3. Mode I Fracture Toughness Tests

Several methods have been proposed to measure the K_IC of quasi-brittle materials [19,37,38,39,40,41,42,43]. In this study, CCNBD specimens were used. The K_IC of the studied specimens was determined under dry and saturated conditions according to the ISRM suggested method [35]. Based on this method, the results of the fracture toughness tests are acceptable for samples that are located in the valid range, according to Figure 4. To determine the position of the samples in the valid range, the geometric characteristics of the samples shown in Figure 5 and the following formulas (Equations (2) to (5)) were used. The valid ranges are the space enclosed by the graphs of Equations (6) to (11), as shown in Figure 4.

$${\mathsf{\alpha }}_0=\frac{{\mathrm{a}}_0}{\mathrm{R}}$$

(2)

$${\mathsf{\alpha }}_1=\frac{{\mathrm{a}}_1}{\mathrm{R}}$$

(3)

$${\mathsf{\alpha }}_{\mathrm{B}}=\frac{\mathrm{B}}{\mathrm{R}}$$

(4)

$${\mathsf{\alpha }}_{\mathrm{s}}=\frac{{\mathrm{R}}_{\mathrm{s}}}{\mathrm{R}}$$

(5)

$$\mathrm{Line}0\to {\mathsf{\alpha }}_1\ge 0.4$$

(6)

$$\mathrm{Line}1\to {\mathsf{\alpha }}_1\ge \frac{{\mathsf{\alpha }}_{\mathrm{B}}}{2}$$

(7)

$$\mathrm{Line}2\to {\mathsf{\alpha }}_{\mathrm{B}}\le 1.04$$

(8)

$$\mathrm{Line}3\to {\mathsf{\alpha }}_1\le 0.8$$

(9)

$$\mathrm{Line}4\to {\mathsf{\alpha }}_{\mathrm{B}}\le 1.1729\times {\mathsf{\alpha }}_1^{1.6666}$$

(10)

$$\mathrm{Line}5\to {\mathsf{\alpha }}_{\mathrm{B}}\ge 0.44$$

(11)

The notch opening width of the specimens was less than 1 mm (Figure 6). The chevron notch crack cross-section under loading mode I are represented in Figure 6C,D before and after failure, respectively. For all of the samples, the parameters of ${\mathsf{\alpha }}_0=0.2428$ and R = 26.5 mm were constant. The value of αs was 0.7679. The values of α₁ and α_B were different, and were determined based on the position of each sample in the given valid range (Figure 5). The value of α1 is calculated using Equation (12).

$${\mathsf{\alpha }}_1=\sqrt{{\mathsf{\alpha }}_{\mathrm{s}}^2-{\left(\sqrt{{\mathsf{\alpha }}_{\mathrm{s}}^2-{\mathsf{\alpha }}_0^2}-\frac{{\mathsf{\alpha }}_{\mathrm{B}}}{2}\right)}^2}$$

(12)

To determine K_IC, the methods presented by Atkinson et al. [44], Fowell [45], and Wang et al. [46] were used (Equations (13) and (14)). The resulting K_IC values are presented in Table 3. A_i is determined using Equations (15) to (19), and T_i is determined in terms of the a/R ratio in Atkinson’s proposed method.

$${\mathrm{K}}_{\mathrm{IC}}=\frac{{\mathrm{P}}_{\mathrm{Max}}}{\mathrm{RB}}\sqrt{\frac{\mathsf{\alpha }}{\mathsf{\pi }}}\sqrt{\frac{{\mathsf{\alpha }}_1-{\mathsf{\alpha }}_0}{\mathsf{\alpha }-{\mathsf{\alpha }}_0}}{\mathrm{N}}_{\mathrm{I}}$$

(13)

$${\mathrm{N}}_{\mathrm{I}}={{\displaystyle \sum }}_{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{T}}_{\mathrm{i}}{\left(\frac{\mathrm{a}}{\mathrm{R}}\right)}^{2\mathrm{i}-2}{\mathrm{A}}_{\mathrm{i}}\left(\mathsf{\theta }\right)$$

(14)

$${\mathrm{A}}_1=1-4{\mathrm{Sin}}^2\mathsf{\theta }$$

(15)

$${\mathrm{A}}_2=8{\mathrm{Sin}}^2\mathsf{\theta }\left(1-4{\mathrm{Cos}}^2\mathsf{\theta }\right)$$

(16)

$${\mathrm{A}}_3=-4{\mathrm{Sin}}^2\mathsf{\theta }\left(3-36{\mathrm{Cos}}^2\mathsf{\theta }+48{\mathrm{Cos}}^2\mathsf{\theta }\right)$$

(17)

$${\mathrm{A}}_4=-16{\mathrm{Sin}}^2\mathsf{\theta }\left(-1+24{\mathrm{Cos}}^2\mathsf{\theta }-80{\mathrm{Cos}}^4\mathsf{\theta }+64{\mathrm{Cos}}^6\mathsf{\theta }\right)$$

(18)

$${\mathrm{A}}_5=-20{\mathrm{Sin}}^2\mathsf{\theta }\left(1-4{\mathrm{Cos}}^2\mathsf{\theta }+240{\mathrm{Cos}}^4\mathsf{\theta }-448{\mathrm{Cos}}^6\mathsf{\theta }+256{\mathrm{Cos}}^8\mathsf{\theta }\right)$$

(19)

The value of u and ν in equations provided by Fowell (Equations (20)–(21)) are determined using α₀ and α_S. Although the u and ν values are different in the Wang method, Equations (20)–(21) are similarly used [27,39,47].

$${\mathrm{K}}_{\mathrm{IC}}=\frac{{\mathrm{P}}_{\mathrm{Max}}}{\mathrm{B}\sqrt{\mathrm{D}}}{\mathsf{\gamma }}_{\min }^{*}$$

(20)

$${\mathsf{\gamma }}_{\min }^{*}=\mathrm{u}·{\mathrm{e}}^{\mathrm{v}·{\mathsf{\alpha }}_1}$$

(21)

3. Results and Discussion

3.1. Analyzing the Behavior of Samples under Dry and Saturated States

After conducting the fracture toughness test on the selected limestones, it was observed that in most of the samples, including sandy limestones, dolomite limestones, floatstones, and mudstones, the fracture toughness was lower in the saturated state than in the dry condition. However, the toughness values of the wackeston, packstone, grainstone, and framestone samples did not show any dependency on the saturation state. To investigate this, factors such as the porosity, petrography of thin sections, and mineralagy of the samples were examined.

Samples R20 and R18 were two wackeston samples and both contained kaolinite. If the presence of this clay mineral had an effect on the toughness value in a saturated or dry state, we would have seen the same behavior in these samples as well, i.e., less fracture toughness under a saturation condition. Thus, it can be somehow concluded that the presence of kaolinite had no effect on the variations of toughness as a function of the saturation state. The porosity of the R20 and R18 samples was equal to 1.5 and 2.6, respectively. In general, it was observed that porosity was the most dominant factor affecting the variations of fracture toughness for wakestone, packstone, grainstone, and framestone, which did not show any particular trend considering the saturation state. In the samples with a porosity of more than 2.5%, the fracture toughness decreased under the saturated state, while the samples with a porosity less than 2.5% showed a higher fracture toughness under a saturated state.

Samples R15, R4, and R23 were placed in packston category in terms of petrography. According to the XRD and EDS analyses, the R15 sample contained the montmorillonite clay mineral (Figure 7). It is notable that, in this sample, the fracture toughness was higher under a saturated state. The porosity of the R15, R4, and R23 samples was 1.5, 2.3, and 6.9, respectively. Again, it was observed that, irrespective of mineralogy, porosity is the main factor controlling the fracture toughness as a function of the saturation state. In the studied packstone samples, if the porosity was higher than 2.5%, the fracture toughness was higher under a dry state, but if the porosity was less than 2.5%, the fracture toughness was higher under a saturated state.

Samples R14, R3, R6, and R13 were categorized as grainstone according to their petrography, and their porosity values were equal to 1.6, 1.7, 1.7, and 3.2 respectively. R14 contained broken particles (Figure 8), R3 contained glauconite clay mineral, and R6 contained both glauconite clay mineral and broken particles, as well as microcracks. All of these samples showed higher fracture toughness values under the saturated state, and the only common parameter in these samples was a lower porosity.

Therefore, the presence of water-sensitive clay minerals in the samples of wackeston, packston, and grainstone had no effect on the variations of fracture toughness as a function of the saturation state, and the only factor affecting it was the porosity of the sample. It should also be mentioned that the R1, R2, and R10 rock samples contained glauconite; the R11, R29, and R30 rock samples contained montmorillonite; R18 contained kaolinite; and sample R9 contained palygorcite. According to our analyses, because of the high porosity values of these rock samples, which were more than 2.5%, their fracture toughness values were lower under a saturated state.

3.2. Normality Test of the Data

Normality of errors was considered by default, and because the dependent variables were associated with errors, data normality should be examined. Initially, the histograms, normal curves, and box diagrams for the dependent variable were plotted (Figure 9). Then, the skewness and kurtosis of the data were determined. Skewness represented symmetry or asymmetry of the data, and kurtosis indicate if the data have heavy/light tails in comparison with the normal distribution. In general, if the skewness and kurtosis were between −2 to 2, the data will have normal distribution. In addition, by using the box diagram, the scattered data were detected. In a usual plotting box diagram, if data have a significant difference compared with the rest, it are identified as scattered data and are treated in different ways. Generally, that data are deleted and replaced with the mean or the first or third quartile. Regarding the skewness and kurtosis values, dependent variables are normal.

3.3. Simple Regression

Linear/nonlinear simple regression with a 95% confidence level is used to determine the correlation between the fracture toughness and index properties. According to the results of the Pearson correlation, there is a high correlation among the K_IC calculated using the Atkinson et al. [44], Fowell [45], and Wang et al. [46] methods for the dry (

Table 4. Correlation among K_IC determined from three different methods under a dry condition.

		KICW-Dry (MPa m1/2)	KIC-Dry (MPa m1/2)
KICF-Dry (MPa m1/2)	RMSE	0.0108	0.0181
	R2 (%)	99.91	99.75
	Adj. R2 (%)	99.91	99.74
	p-Value	0.000	0.000
Equation		KICW-Dry = 1.11203 KICF-Dry − 0.00633	KIC-Dry = 1.1329 KICF-Dry − 0.018

) and saturated states (

Table 5. Correlation among K_IC determined from three different methods under a saturated state.

		KICW-Sat (MPa m1/2)	KIC-Sat (MPa m1/2)
KICF-Sat (MPa m1/2)	RMSE	0.0109	0.01492
	R2 (%)	99.94	99.89
	Adj. R2 (%)	99.94	99.89
	p-Value	0.000	0.000
Equation		KICW-Sat = 1.11653 KICF-Sat − 0.00839	KIC-Sat = 1.13465 KICF-Sat − 0.01444

). Therefore, only the results of the Fowell method are used to determine the relationship between the K_IC and index properties.

Firstly, the correlation between the fracture toughness of the dry and saturated states with the index properties was investigated for all of the samples. The simple linear regression showed poor results. This is because limestone has a wide range of textural and petrological features. Therefore, according to the textural diversity of limestone, petrographic studies of the samples were conducted using optical microscopy and the samples were classified in five classes accordingly (

Table 6. Rock classes based on petrographic study.

Rock Code	Class	Petrography	Rock Code	Class	Petrography
R5	1A	Mudstone	R10	2	Sandy limestone
R18	1A	Wackestone	R8	2	Sandy limestone
R20	1A	Wackestone	R7	2	Sandy limestone
R21	1A	Framestone	R2	2	Sandy limestone
R19	1B	Floatstone	R1	2	Sandy limestone
R4	1B	Packstone	R22	3A	Dolomitic limestone
R23	1B	Packstone	R16	3A	Dolostone
R15	1B	Packstone	R29	3A	Dolomitic limestone
R6	1B	Grainstone	R17	3A	Dolomitic limestone
R3	1B	Grainstone	R30	3A	Dolomitic limestone
R13	1B	Grainstone	R24	3A	Dolostone
R14	1B	Grainstone	R25	3B	Dolo mudstone
R12	1B	Rudstone	R28	3B	Dolo mudstone
R9	2	Sandy Limestone	R26	3B	Dolo mudstone
R11	2	Sandy Limestone	R27	3B	Dolo mudstone

). Mud supported limestones were placed in class 1A, grain supported samples were categorized as class 1B, sandy limestones were placed in class 2, coarse crystal dolomite limestones were placed in class 3A, and microcrystalline dolomite limestones were placed in class 3B.

By applying this classification, more meaningful correlations were obtained between the fracture toughness and physical characteristics of the studied limestones (

Table 7. Linear correlation between the index properties and dry and saturated fracture toughness for all of the limestones and individual classes.

		n	VS	VPDry	VPSat	ρSat	ρd	WA	RN	IS50Dry	IS50Sat
All Data	KICFDry	46.92	29.27	44.22	41.34	5.40	35.16	48.86	0.96	52.27	68.56
	KICFSat	58.68	44.09	73.27	71.23	28.30	41.22	57.91	1.82	29.48	58.98
Class 1A	KICFDry	94.28	65.43	51.56	97.81	83.07	95.45	95.84	23.65	27.52	2.36
	KICFSat	1.66	4.12	11.58	5.02	34.64	3.49	3.43	2.18	23.83	1.64
Class 1B	KICFDry	46.24	0.62	18.49	44.49	35.26	48.58	47.47	54.02	87.42	72.93
	KICFSat	73.79	8.65	55.35	67.90	43.97	73.10	73.96	34.50	76.39	74.30
Class 2	KICFDry	83.36	64.00	85.38	79.39	0.25	75.86	83.17	35.52	98.41	91.78
	KICFSat	69.89	87.24	85.19	80.28	0.64	80.10	68.56	15.76	65.77	82.81
Class 3A	KICFDry	30.41	16.66	30.05	14.33	8.70	55.93	35.42	0.00	63.35	45.03
	KICFSat	56.08	24.81	51.65	42.39	26.22	69.65	58.76	0.95	49.00	23.58
Class 3B	KICFDry	68.80	27.68	9.00	38.35	69.34	44.88	68.53	41.50	0.12	25.05
	KICFSat	95.84	30.80	40.67	67.06	29.94	67.12	95.06	84.82	6.23	64.02

). The relationship between the K_IC and index properties for each class were determined under dry (Figure 10) and saturated conditions (Figure 11). The best relationship was determined for each class as well as for all of the rock samples based on simple linear/nonlinear regression analyses (

Table 8. Statistical parameters of the simple linear/nonlinear regression between K_IC and the index properties.

	Dependent Variable	Independent Variables	Predictive Model	R2 (%)	p-Value
All Data	KICF-Dry	IS50-Sat	y = 0.891 × e0.596	69.5	0.000
	KICF-Sat	VP-Sat	y = 0.008 × e2.758	75.2	0.000
Class 1A	KICF-Dry	VP-Sat	y = −0.087x + 1.62	97.9	0.011
		n	y = 0.044x + 0.994	94.3	0.029
		ρSat	y = 1.4586 × e−0.032	93.1	0.035
		ρd	y = −3.229ln(x) + 4.257	95.5	0.023
		W.A.	y = 0.999 × e0.107x	96.0	0.020
Class 1B	KICF-Dry	IS50-Sat	y = 0.616 ln(x) + 0.871	73.1	0.003
		IS50-Dry	y = 0.756 × e0.902	88.4	0.000
	KICF-Sat	VP-Sat	y = 0.003 × e3.24	80.3	0.003
		n	y = 1.567 × e−0.164x	84.6	0.001
		ρSat	y = 0.122 × 0.246	83.5	0.001
		ρd	y = 0.141 × e0.786x	86.7	0.001
		W.A.	y = 1.362 × e−0.233x	86.9	0.001
		IS50-Sat	y = 0.887ln(x) + 0.795	79.6	0.003
Class 2	KICF-Dry	VP-Dry	y = 1.359ln(x) − 0.889	86.8	0.002
		VP-Sat	y = 0.038 × 2.08	80.5	0.006
		n	y = −0.102x + 1.683	83.4	0.004
		ρd	y = 8.823ln(x) − 7.161	76.1	0.010
		W.A.	y = −0.239x + 1.654	83.2	0.004
		IS50-Sat	y = 0.573ln(x) + 1.14	97.1	0.000
		IS50-Dry	y = 0.647 × e1.036	99.4	0.000
	KICF-Sat	Vs	y = 0.553x − 0.631	87.3	0.002
		VP-Sat	y = 0.022 × e0.66x	84.3	0.004
		n	y = 2.338 × e−0.994	89.2	0.001
		W.A.	y = 0.898 × e−0.939	88.7	0.002
		IS50-Sat	y = 0.663x − 0.051	82.8	0.004
Class 3A	KICF-Dry	IS50-Dry	y = 0.36ln(x) + 0.74	64.9	0.050
	KICF-Sat	ρd	y = 0.025 × e3.697	74.8	0.026
Class 3B	KICF-Dry	VP-Sat	y = 3.524ln(x) − 4.537	95.3	0.024
	KICF-Sat	VP-Sat	y = 0.0004 × e1.068x	90.7	0.048
		n	y = −0.156x + 2.229	97.5	0.013
		W.A.	y = −0.359x + 2.111	97.2	0.014
		RN	y = 1.767ln(x) − 5.969	89.2	0.055

). Classification of the samples according to petrographic studies showed acceptable relationships obtained from simple linear/nonlinear regression. Fracture toughness is generally inversely related to porosity and water absorption percentage, and is directly related to the ultrasonic wave velocity, density, and point load index. It is clear from Table 8 that by applying petrographic classification to the data, better correlations between fracture toughness and index properties were obtained.

3.4. Multiple Linear Regression

Multiple linear regression examines the simultaneous effect of several independent variables on a dependent variable. Various combinations of rock index properties were used to predict the K_IC of the studied rocks. In addition, for verifying the accuracy of the results, statistical accuracy controllers, including root mean squared error (RMSE), VIF, R², R² Adjust, and p-Value, were used. Initially, the p-value index was used to control the overall validity of the proposed models. A proposed model was considered to be statistically correct if its p-value was less than 0.05. The closer to zero, the more efficient the model. Then, the validity of each of the dependent variables was determined using the VIF criterion and the p-value. The value of the VIF criterion for each variable used in the model should be less than 10. The closer the RMSE indicator is to zero, the more functional the model. The most reliable models according to the statistical controllers are presented in

Table 9. Statistical parameters of the multiple linear regression between the K_IC and index properties.

Class	Dependent Variable	Independent Variable	VIF	p-Value	Predictive Model	RMSE	R2 (%)	Adj. R2	p-Value
All Data	KICF-Sat	Constant		0.000	KICF−Sat = 0.2646 VP−Sat + 0.3123 IS50−Sat − 0.924	0.1557	85.22	84.08	0.000
		VP-Sat (Km/s)	1.43	0.000
		IS50-Sat (MPa)	1.43	0.000
		Constant		0.000	KICF−Sat = −0.04204 n + 0.3769 IS50−Sat + 0.656	0.1879	78.46	76.81	0.000
		n (%)	1.31	0.000
		IS50-Sat (MPa)	1.31	0.000
		Constant		0.000	KICF−Sat = −0.0873 W.A. + 0.3765 IS50−Sat + 0.627	0.1931	77.27	75.52	0.000
		W.A. (%)	1.33	0.000
		IS50-Sat (MPa)	1.33	0.000
		Constant		0.002	KICF−Sat = 0.1824 VS + 0.523 ρd + 0.3246 IS50−Sat − 1.367	0.1934	78.07	75.43	0.000
		VS (Km/s)	1.36	0.005
		ρd (gr/cm3)	1.26	0.004
		IS50-Sat (MPa)	1.54	0.001
		Constant		0.018	KICF−Sat = 0.1540 VS + 0.000014 ρSat + 0.377 ρd + 0.3367 IS50Sat − 0.997	0.1786	82.03	79.04	0.000
		VS (Km/s)	1.43	0.011
		ρSat (Ω.m)	1.32	0.030
		ρd (gr/cm3)	1.48	0.030
		IS50-Sat (MPa)	1.54	0.000
		VS (Km/s)	9.06	0.008	KICF-Sat = 0.1059 VS + 0.000020 ρSat + 0.3928 IS50Sat	0.1936	96.30	95.87	0.000
		ρSat (Ω.m)	1.69	0.002
		IS50-Sat (MPa)	8.69	0.000
		ρSat (Ω.m)	1.61	0.001	KICF-Sat = 0.000024 ρSat + 0.5984 IS50Sat	0.2182	95.11	94.75	0.000
		IS50-Sat (MPa)	1.61	0.000
	KICF-Dry	VPDry (Km/s)	1.18	0.001	KICF-Dry = 0.116301 VP-Dry + 0.270473 IS50-Dry	0.1778	70.00	69.72	0.000
		IS50-Dry (MPa)	1.18	0.000
		Constant		0.000	KICF−Dry = − 0.03318 n + 0.2757 IS50−Dry + 0.789	0.1677	73.56	71.60	0.000
		n (%)	1.14	0.000
		IS50-Dry (MPa)	1.14	0.000
		Constant		0.000	KICF−Dry = −0.0701 W.A. + 0.2646 IS50Dry + 0.786	0.1711	72.48	70.45	0.000
		W.A. (%)	1.19	0.000
		IS50-Dry (MPa)	1.19	0.000
Class 1A	KICF-Sat	Constant		0.032	KICF−Sat = −0.6096 VS + 0.000040 ρSat + 2.518	0.0210	99.64	98.91	0.060
		VS (Km/s)	1.82	0.047
		ρSat (Ω.m)	1.82	0.039
	KICF-Dry	Constant		0.001	KICF−Dry = 0.038435 VP−Dry + 0.062731 n + 0.72566	0.0000	100	100	0.001
		VP-Dry (Km/s)	3.95	0.004
		n %	3.95	0.001
		Constant		0.003	KICF−Dry = −0.000001 ρSat − 0.85596 ρd + 3.3853	0.0001	100	100	0.003
		ρSat (Ω.m)	2.82	0.010
		ρd (gr/cm3)	2.82	0.005
Class 1B	KICF-Sat	Constant		0.034	KICF−Sat = −0.1115 W.A. + 0.379 IS50−Sat + 0.67	0.1077	89.75	85.66	0.003
		W.A. (%)	1.74	0.040
		IS50-Sat (MPa)	1.74	0.039
Class 2	KICF-Sat	Constant		0.004	KICF−Sat = 0.3552 VS + 0.2052 VP−Sat − 1.122	0.1244	96.86	95.30	0.001
		VS (Km/s)	2.16	0.01
		VP-Sat (Km/s)	2.16	0.025
		Constant		0.005	KICF−Sat = 0.3570 VS + 2.052 ρd − 5.366	0.0919	98.29	97.43	0.000
		VS (Km/s)	2.00	0.003
		ρd (gr/cm3)	2.00	0.007
Class 3A	KICF-Sat	ρSat (Ω.m)	1.19	0.003	KICF-Sat = 0.000140 ρSat + 0.4767 IS50-Sat	0.0903	98.87	98.30	0.000
		IS50-Sat (MPa)	1.19	0.000
	KICF-Dry	Constant		0.043	KICF−Dry = 0.679 VS − 0.0687 RN + 0.3097 IS50−Dry + 1.597	0.0711	97.28	93.20	0.041
		VS (Km/s)	4.82	0.042
		RN	4.97	0.041
		IS50-Dry (MPa)	1.36	0.025

. From this table, it can be clearly seen that by taking into account the petrographic characteristics of the studied limestones, their fracture toughness can be successfully estimated by means of their index properties.

4. Conclusions

In this study, the effect of the petrogeraphical, mineralogical, and physical properties of different limestones on fracture toughness is investigated under both dry and saturated conditions. Some statistical relationships are stablished to determine the correlation between fracture toughness and these properties, which can be determined using simple methods. The main conclusions of the conducted statistical analyses are listed as follows:

• The fracture toughness of the studied limestones decreased under a saturated state, except for the samples that were be categorized as wacketon, packstone, grainstone, and framestone, which indicates a complex behavior. In these limestone classes, porosity is the main factor affecting the ratio of dry to saturated fracture toughness. In the samples with a porosity of less than 2.5%, fracture toughness is higher under saturated condition, while in samples with a porosity more than 2.5%, fracture toughness is higher under dry condition.
• No meaningful relationship between clay content with the ratio of saturated to dry fracture toughness is found.
• The value of K_IC calculated from the different methods proposed by Atkinson, Fowell, and Wang were similar and showed perfect correlations.
• K_ICsat and K_ICdry of the limestone could be predicted with a high accuracy from index tests if careful classification based on their petrography was utilized.