Mechanical Behavior of Carbonate Rocks at Crack Damage Stress Equal to Uniaxial Compressive Strength

1 Introduction

A number of researchers such as Brace et al. (1966), Bieniawski (1967), Brady and Brown (1993), Martin and Chandler (1994), Hatzor and Palchik (1997), Pettitt et al. (1998), Eberhardt et al. (1999), Katz and Reches (2004) and Cai et al. (2004) have investigated different stages of stress–strain behavior of brittle rocks during uniaxial compression. Two stress–strain diagrams in Fig. 1 obtained by the author for carbonate rocks, exhibit three main characteristic stress levels (crack initiation stress σ_ci, crack damage stress σ_cd and uniaxial compressive strength σ_c) which are represented by points A, B and C, respectively, over the total volumetric strain curve. The crack initiation stress (σ_ci, point A) is the stress level at which microfracturing begins. The point A is the end of the elastic stage (linear portion) of stress–strain curve. The crack damage stress (σ_cd, point B) is the stress at the onset of dilation: when σ_cd is attained, the rock volume begins to increase (Schock et al. 1973; Brace 1978; Paterson 1978; Palchik and Hatzor 2002). The crack damage stress (σ_cd, point B) is the stress level at which maximum total volumetric strain (ε_cd) is attained. The uniaxial compressive strength (σ_c, point C) is the maximum axial stress (at failure). In Fig. 1, strain ε_cd at σ_cd (point B) is the maximum total volumetric strain, and strain ε_a max at σ_c is the maximum axial strain.

Fig. 1

Stress–strain behavior of brittle rocks during uniaxial compression. Crack initiation stress (σ_ci), crack damage stress (σ_cd) and uniaxial compressive strength (σ_c) are represented by points A, B and C, respectively. The total volumetric strain (ε_v) is calculated as a sum of the component strains: ε_v = ε_a + ε_R1 + ε_R2, where ε_a is axial strain, and ε_R1 and ε_R2 are radial strains measured in orthogonal directions (Palchik and Hatzor 2002): a type 1—total volumetric strain curve has a reversal point (B) and σ_cd < σ_c; b type 2—there is no reversal point in total volumetric strain curve, and σ_cd = σ_c

Martin and Chandler (1994), Eberhardt et al. (1999) and Palchik and Hatzor (2002) have shown that the crack damage stress (σ_cd) is defined as the point (see point B in Fig. 1a) where a total volumetric strain reversal occurs and unstable crack growth begins. In this case (see Fig. 1a), crack damage stress σ_cd is lower than the uniaxial compressive strength (σ_c). Indeed, Brace et al. (1966), Bieniawski (1967), Martin (1993), Pettitt et al. (1998), Eberhardt et al. (1999), Heo et al. (2001) and Katz and Reches (2004) have found that the crack damage stresses σ_cd of granites, sandstones and quartzite vary from 0.71σ_c to 0.84σ_c. They have also shown that the ratios σ_ci/σ_c and σ_ci/σ_cd for above-mentioned rocks range from 0.39 to 0.6 and from 0.52 to 0.82, respectively.

Hatzor and Palchik (1997) and Palchik and Hatzor (2002) have shown that there exist total volumetric strain curves (in heterogeneous carbonate rocks) which do not have any point of reversal, with the maximum total volumetric strain (ε_cd) attained at the uniaxial compressive strength (σ_c). In this case, crack damage stress is equal to uniaxial compressive strength (i.e. σ_cd = σ_c, B = C in Fig. 1b). Thus, there are two types of total volumetric strain curves in brittle rocks: type 1 (see Fig. 1a), with a point of reversal (B) in the total volumetric strain curve, and σ_cd < σ_c; and type 2 (see Fig. 1b), where the total volumetric strain curve has no reversal point and, therefore, σ_cd = σ_c.

Type 1 (Fig. 1a) curves have been studied by Brace et al. (1966), Bieniawski (1967), Pettitt et al. (1998), Eberhardt et al. (1999), Heo et al. (2001), Katz and Reches (2004), etc., whereas little attention has been paid to type 2 mechanical behavior of brittle rocks (Fig. 1b). In this paper, we intend to study relations between characteristic compressive stress levels, strains and mechanical properties of heterogeneous carbonate rocks exhibiting type 2 behavior of the total volumetric strain curve (i.e. at σ_cd = σ_c).

2 Testing and test results

Mechanical properties of carbonate rock samples exhibiting type 2 behavior of the total volumetric strain curve are summarized in Table 1. These samples were collected from Adulam chalk, Aminadav dolomite, Bina limestone, Yarka limestone, Yagur dolomite, and Nekorot limestone formations. The NX (d = 54 mm) sized cylindrical rock samples having the ratio L/d = 2 (here L and d are the length and diameter of a sample, respectively) were prepared. The samples were ground to the planeness of 0.01 mm and cylinder perpendicularity within 0.05 radians. Prior to testing, rock samples were oven dried at the temperature of 110°C for 24 h.

Table 1 Observed values of compressive stress levels, elastic modulus, dry bulk density, Poisson’s ratio, porosity, maximum axial strain and maximum total volumetric strain

Uniaxial compressive tests were performed at the Rock Mechanics Laboratory of the Ben-Gurion University. The tests were conducted using a load frame (TerraTek system, model FX-S-33090) at a constant strain rate of 10⁻⁵/s. The load frame operates under a hydraulic closed-loop servo-control. The load frame stiffness and maximum axial force are 5 × 10⁹ N/m and 1.4 MN, respectively. The axial strain cantilever set has a 10% strain range, and the radial strain cantilevers have a strain range limit of 7%, with the linearity of 1% over the full scale in both sets. The servo-controlled press description, physical and mechanical properties of the studied rock formations are presented in detail elsewhere (Hatzor and Palchik 1997; Palchik and Hatzor 2002, 2004).

The results of uniaxial tests are given in Table 1. The latter presents the values of crack initiation stress (σ_ci), σ_cd = σ_c (here σ_cd is the crack damage stress and σ_c is the uniaxial compressive strength), elastic modulus (E), dry bulk density (ρ), Poisson’s ratio (ν), porosity (n), maximum axial strain ε_a max (at σ_c) and maximum total volumetric strain ε_cd (at σ_cd) for each of 24 studied rock samples.

Porosity (n, %) was calculated from the measured values of dry bulk density (ρ in Table 1) and specific gravity of the solids (G _s = 2.7–2.8 g/cm³): n = [1 − (ρ/G _s)] × 100%. The precision of the porosity estimation is 0.1%. The elastic modulus (E) and Poisson’s ratio (ν) were calculated using linear regressions along the linear portion (elastic stage) of the stress–strain curve. The total volumetric strains (ε_v) calculated as a sum of the component strains (ε_a, ε_R1 and ε_R2) were plotted versus axial stresses for each of 24 studied rock samples. Figure 2 demonstrates examples of axial stress–total volumetric strain curves for five rock samples (ad 80, ca3541, gn3-2a, gn2-1b and rc9) exhibiting ε_cd = 0.134% at σ_cd = σ_c = 174 MPa, ε_cd = 0.149% at σ_cd = σ_c = 174 MPa, ε_cd = 0.161% at σ_cd = σ_c = 150 MPa, ε_cd = 0.236% at σ_cd = σ_c = 177 MPa and ε_cd = 0.134% at σ_cd = σ_c = 63.1 MPa, respectively, in point the B = C.

Fig. 2

Examples of observed axial stress–total volumetric strain curves (type 2) for five studied rock samples

In Table 1, the observed values of crack initiation stress (σ_ci) and crack damage stress or uniaxial compressive strength (σ_cd = σ_c) vary from 15 to 145 MPa with the mean of 61.7 MPa and from 31.9 to 273.9 MPa with the mean of 101.4 MPa, respectively. The difference (D) between σ_c and σ_ci and the ratio k = σ_ci/σ_c for each of the studied rock samples were calculated. Values of D and k range from 11.2 to 128.9 MPa (D _mean = 39.7 MPa) and from 0.45 to 0.78 (k _mean = 0.62), respectively, for all studied samples. Here, standard deviations (Δ) of the mean D and k values are significant—29.2 MPa and 0.09, respectively. Standard deviation of the mean has been calculated as follows:

$$ \Updelta = \sqrt {\frac{{\sum\nolimits_{i = 1}^{n} {(q_{i} - q_{m} )^{2} } }}{n - 1}} $$

(1)

where i = 1, 2,…, n is the number of observed sample (n = 24), q _i is the value of the observed parameter (D or k) in the ith sample, q _m is the arithmetic mean of parameters observed in n samples.

Note also that the values of D and k are not constant even for a single set of samples within the same rock formation: 11.9 MPa < D < 29.4 MPa (0.47 < k < 0.78), 72.2 MPa < D < 128.9 MPa (0.53 < k < 0.59), 12.8 MPa < D < 49.8 MPa (0.47 < k < 0.63), 20.5 MPa < D < 63.9 MPa (0.6 < k < 0.66) and 40.5 MPa < D < 97 MPa (0.45 < k < 0.75) for Adulam chalk, Aminadav dolomite, Bina limestone, Yagur dolomite and Nekorot limestone, respectively. The standard deviation (Δ) of the mean k within the same rock formation ranges from 0.03 to 0.12. For example, Nekorot limestone has a large Δk = 0.12, whereas Yagur dolomite exhibits relatively a small Δk = 0.03.

Table 1 demonstrates the values of elastic modulus (E), porosity (n) and Poisson’s ratio (ν) for each of 24 studied rock samples. Here, 6,100 MPa < E < 64,000 MPa, 5.4% < n < 28.5% and 0.13 < ν < 0.31. Mean values of E, n and ν are 31,600 MPa, 16.5% and 0.23, respectively, and standard deviations (Δ) of mean E, n and ν values are 17,300 MPa, 6.5% and 0.04, respectively. The values of E, n and ν in heterogeneous carbonate rocks are also non-constant even for samples within the same rock formation. For example, Bina limestone exhibits 21,000 MPa < E < 42,830 MPa, 7.5% < n < 26.4% and 0.2 < ν < 0.31.

In Table 1, maximum total volumetric strain (ε_cd) and maximum axial strain (ε_a max) range from 0.084 to 0.9% with the mean of 0.2% and from 0.19 to 1.36% with the mean of 0.39%, respectively. Thus, the maximum axial strain (ε_a max) is 1.5–2.5 times larger than the maximum total volumetric strain (ε_cd) in case where crack damage stress (σ_cd) is equal to uniaxial compressive strength (σ_c). For example, rock sample bina 7 exhibiting ε_cd = 0.132% at σ_cd = 64.2 MPa and ε_a max = 0.29% at σ_c = 64.2 MPa (see Fig. 1b), has the ratio ε_a max/ε_cd = 0.29/0.132 = 2.2.

3 Relations between mechanical properties and compressive stresses

3.1 Effect of elastic modulus, porosity and E/n ratio on σ_ci and σ_cd = σ_c

Figure 3a shows how the elastic modulus (E) influences the values of σ_ci and σ_cd = σ_c. Relations σ_ci − E and σ_c − E best follow a polynomial law with good squared regression coefficients R ² = 0.86 and 0.91 for σ_ci − E and σ_c − E, respectively. An increase in the elastic modulus (E) from 6,100 to 64,000 MPa leads to an increase in σ_ci and σ_c = σ_c values from 15 to 145 MPa and from 31.9 to 273.9 MPa, respectively. Hence, an increase in the elastic modulus by a factor of 10.5 leads to an increase in the values of σ_ci and σ_cd = σ_c by a factor of 9.7 and 8.6, respectively.

Fig. 3

Relations between compressive stress levels and a elastic modulus (E), and b porosity (n)

On the other hand, values of σ_ci decrease from 145 to 15 MPa and from 273.9 to 31.9 MPa, respectively, with porosity increase (n, Fig. 3b) from 5.4 to 28.5%. In Fig. 3b, polynomial correlations (R ² = 0.76–0.79) between the porosity (n) and σ_ci and σ_cd = σ_ci values are obtained.

An increase in the values of stresses σ_ci and σ_cd = σ_c with increasing ratio E/n is shown in Fig. 4a. The latter demonstrates that σ_ci and σ_cd = σ_c are well correlated (R ² = 0.82–0.86) with the ratio E/n. It is not surprising, since the compressive stresses σ_ci and σ_cd = σ_c depend on the elastic modulus E (Fig. 3a) and porosity n (Fig. 3b).

Fig. 4

Relations between compressive stress levels and ratios a E/n and b E/λ

3.2 Effect of E/λ ratio on σ_ci and σ_cd = σ_c

Porosity n is a measure of void space (pores and open cracks) and represents a ratio between the void space (V _p) and bulk volume (V).

$$ n = \frac{{V_{\text{p}} }}{V} $$

(2)

The bulk volume (V) is the initial volume of a sample before loading, and therefore, it does not reflect the change in the volume due to compression. Palchik and Hatzor (2002) have proposed to use the ratio between the volume of voids and change in the bulk volume due to compression, since such ratio reflects the mechanical behavior of the rock matrix. This ratio can be represented as a ratio between the volume of voids (V _p) and the maximum compaction (V _c) of a rock sample:

$$ \lambda = \frac{{V_{\text{p}} }}{{V_{\text{c}} }} $$

(3)

where V _c is the maximum decrease in a sample volume (maximum compaction of a sample), which is attained at the maximum total volumetric strain ε_cd (at crack damage stress σ_cd).

The parameter λ can be presented as

$$ \lambda = \frac{{V_{\text{p}} }}{{V_{\text{c}} }} = \frac{{V_{\text{p}} }}{V}/\frac{{V_{\text{c}} }}{V} = \frac{n}{{\varepsilon_{\text{cd}} }} $$

(4)

where V _c/V = ε_cd, ε_cd is the maximum total volumetric strain at the crack damage stress σ_cd.

When we use λ = V _p/V _c instead of n = V _p/V, the ratio E/n can be rewritten as E/λ. Relations between E/λ and compressive stress levels are presented in Fig. 4b. From Fig. 4b it is clear that the values of σ_cd = σ_c and D = σ_c − σ_ci are well correlated with E/λ. Here, power dependences between σ_cd = σ_c (R ² = 0.96), D (R ² = 0.87) and E/λ with good squared regressions coefficients R ² are obtained:

$$ \sigma_{\text{cd}} = \sigma_{\text{c}} = a\left(\frac{E}{\lambda }\right)^{b} $$

(5)

$$ D = c\left(\frac{E}{\lambda }\right)^{d} $$

(6)

where coefficients a = 2.75, b = 0.6, c = 0.82 and d = 0.64.

Note that the use of E/λ ratio (Fig. 4b) instead of E/n (Fig. 4a) versus the compressive stress σ_cd = σ_c allows us to increase the value of R ² from 0.86 to 0.96. Hence, the effect of the ratio E/λ on uniaxial compressive strength (σ_c) is more pronounced than the effect of E/n.

4 Conclusions

The mechanical behavior of heterogeneous carbonate rocks exhibiting total volumetric strain curves of type 2 was studied. Studied rock samples exhibiting a wide range of mechanical properties (31. 9 MPa < σ_cd = σ_c < 273.9 MPa, 6,100 MPa < E < 64,000 MPa and 5.4% < n < 28.5%) were collected from different geological settings of Israel. From the results of this study it can be concluded that:

• Crack initiation (at the crack initiation stress σ_ci) for studied heterogeneous carbonate rocks occurs at 0.45σ_c/0.78σ_c (at significant standard deviations ΔD = 29.2 MPa and Δk = 0.09 for all studied samples).

• Values of the difference (D) between the uniaxial compressive strength and crack initiation stress, and the ratio (k) between the crack initiation stress and uniaxial compressive strength are not constant even for samples within the same rock formation. The standard deviation (Δ) of the mean k within the same rock formation varies between 0.03 and 0.12.

• Values of the maximum axial strain (ε_a max) and maximum volumetric strain (ε_cd) at σ_cd = σ_c are 0.19/1.36% and 0.084/0.9%, respectively. At σ_cd = σ_c, the maximum axial strain (ε_a max) is 1.5–2.5 times the maximum volumetric strain (ε_cd).

• The ratio between the elastic modulus (E) and parameter λ strongly influences the values of σ_cd = σ_c and D = σ_c − σ_ci. Power dependencies between σ_cd = σ_c, σ_c − σ_ci and E/λ are obtained. Parameter λ is a ratio between the volume of voids (V _p) and the maximum compaction (V _c) of rock sample. Parameter λ is calculated as n/ε_cd, where ε_cd is the maximum total volumetric strain.