Summary

The damage of a porous rock (Euville oolitic limestone) was studied through uniaxial stress-cycling tests. An experimental device, allowing the simultaneous and continuous measurement of strains (in two perpendicular directions) and five elastic wave hree P waves and two S waves) velocities in two different directions under fully controlled hydrous conditions, was developed for the work presented in this paper. Hence, the damage was monitored in a really precise and continuous way through the evolution of dynamic and static elastic moduli. The evolutions of wave velocities and elastic moduli, which reproduce very remarkably the shape of the stress–strains curves, showed that the limestone, initially isotropic, became progressively anisotropic during uniaxial loading due to microcrack damage. Indeed, even if the creation of microcracks is probably scattered and isotropic before the coalescence of microcracks, as shown in past studies, pre-existing microcracks and penny-shaped pores which are perpendicular (or almost perpendicular) to the uniaxial stress direction closed, whereas axially-oriented microcracks opened. The anisotropy of the damage is completely reversible but some of the damage is irreversible. VP(90°), which cannot record the opening of these microcracks, started to decrease just before the macroscopic failure of the sample and can detect, therefore, very precisely the macroscopic failure of the material. The influence of water on the strength and deformation of the Euville limestone was analysed by considering both the hydromechanical and physio-chemical (‘Rehbinder effect’, intergranular pressure solution and subcritical cracking) effects of water.

Introduction

The mechanism of crack initiation and propagation in the brittle regime (low temperature and confining pressure) of rocks has long been a subject of intensive research and much progress has been made in both the understanding and modelling of stress-induced cracking. In one of the first comprehensive reviews on this subject, Bieniawski (1967) shed light on a number of crack evolution mechanisms activated by a compressive stress. The observations of Bieniawski (1967) were performed, from microscopy on thin sections, on fully unloaded rock samples that had been subjected to uniaxial compression. Similar techniques have been used by many other authors (e.g. Hadley 1976; Wong 1985; Homand et al. 2000) in their investigations carried out on the cracks of various rocks, which were subjected to compression loading. These studies confirmed a preferential orientation of induced cracks in the direction of the maximal compressive stress. As these authors have pointed out, the use of these techniques only permits the observation of cracks above a certain thickness, which depends on the observation resolution. Some of these researchers put the absence of the shear cracks in their observations down to a possible closure of these cracks once the sample was unloaded (Wong 1985; Homand et al. 2000).

Another observation technique for the cracking evolution of rock was reported by Zhao (1998), who studied the development of cracks on the surface of rock specimens under uniaxial compression tests with monitoring under direct scanning electron microscope (SEM) observation. This study found that the extending microcracks first initiate in a scattered and random manner and then the coalescence into a macrofracture occurs. To be more precise, all defects, at different angles in respect to the stress axis, can extend and there is no definite preferential orientation when the stress is low. When the ultimate rupture is impending, however, the coalescence of cracks occurs in the direction of maximal stress. This statement (e.g. cracking is first scattered and then localized) has also been demonstrated by acoustic emission (AE) analysis (e.g. Lockner et al. 1991; Lockner & Byerlee 1992). It should also be noted that a great amount of data on rocks in natural conditions (e.g. Kern et al. 1999; Hoxha et al. 2005) show that such rocks are naturally affected by imperfections and a natural cracking which is often preferentially oriented (e.g. rocks in mine pillars).

A stress–strain curve of quasi-brittle rocks is generally described using a four-region model including (Paterson & Wong 2005): (1) The closure of existing crack region. (2) Linear, elastic behaviour region. (3) Stable crack growth region beginning at the onset of crack initiation defined as the stress level when the plot of stress versus lateral strain is no longer linear. (4) Unstable crack growth region beginning from the damage threshold, where the volumetric strain reversal occurs. The evolutions in the state of damage are accompanied by the characteristic evolutions of different physical parameters: elastic stiffness, permeability, electric resistance, ultrasonic (US) wave velocity and acoustic emissions (Paterson & Wong 2005). The evolution of one of these characteristics could be used to describe the cracking evolution.

Some authors (e.g. Martin & Chandler 1994; Lau & Chandler 2004; Heap & Faulkner 2008; Heap et al. 2009a) have performed increasing-amplitude cyclic loading experiments to investigate and quantify the contribution of microcracking to the static elastic response of rocks. Their results have also shown that generally, for crystalline rocks, the level of microcracking damage greatly influences static elastic moduli such as decreases in Young's modulus and the increase in Poisson's ratio with the increasing number of cycles indicated the accumulation of damage. However research, both in the laboratory and in situ, has more generally used the measurements of US waves velocities and their attenuation to show the presence of open microcracks and fractures and to monitor the progressive damage of rocks (Sayers & Kachanov 1995). Experimental studies of US wave anisotropy during a mechanical loading include those of Sayers et al. (1990), Scott et al. (1993), Ayling et al. (1995), Fortin et al. (2005), Dodds et al. (2007) for sandstone, Schubnel et al. (2005) for calcite rocks (limestone and marble) and Dewhurst & Siggins (2006), Dodds et al. (2007), Sarout (2006) and Sarout et al. (2007) for shale. It resulted from these studies that the anisotropy of rocks develops during loading due to the opening of microcracks and no discussion was made on how static and dynamic derived moduli differ during the loading. However, many discussions may be found in the literature on how static and dynamic (determined on fully unloaded samples) derived moduli differ. The difference is mainly attributed to the liability of static measurements to be affected more by the presence of cracks and non-linear response of rock than the dynamic measurements (King 1983; Eissa & Kazi 1988). Generally, the dynamic modulus is higher than the static even though the opposite is sometimes true (Ciccotti & Mulargia 2004; Ciccotti et al. 2004); it depends on the nature of the rock: the higher the modulus of elasticity is the greater is the agreement between the static and dynamic value (Eissa & Kazi 1988). The compilation and analysis of many data on static and seismically determined Young's moduli of different kinds of rocks made by Eissa & Kazi (1988) has shown that the correspondence between the two moduli is rather low. In addition, Ciccotti et al. (2004) and Ciccotti & Mulargia (2004) found that there is no evidence of loading frequency dependence (over nine decades) in the elastic moduli.

Furthermore, as rocks always contain some porosity, many authors have shown that their short-term (e.g. Colback & Wiid 1965; Rutter 1972a, 1974; West 1994; Baud et al. 2000a; Grgic et al. 2005; Jeong et al. 2007) and long-term (e.g. Sammonds et al. 1995; Baud & Meredith 1997; Grgic & Amitrano 2009; Heap et al. 2009b, c) behaviours are influenced by the presence of fluids (usually water) through hydromechanical coupling and physio-chemical effects (e.g. ‘Rehbinder effect’, intergranular pressure solution (IPS), subcritical cracking).

The work presented here is part of a series of studies (e.g. Grgic et al. 2005, 2006; Hoxha et al. 2005; Grgic & Amitrano 2009) aimed at increasing the understanding and at quantifying the mechanisms behind the short- and long-term behaviour of rocks in relation to the problem of stability of underground mines and quarries in France. In these shallow environments, rocks constituting pillars (e.g. iron ore, limestone, gypsum rock) exist in partially saturated conditions because of the artificial ventilation of rooms, which induces a decrease in the surrounding atmosphere's relative humidity. As noted earlier, this may impact on both the short- and long-term behaviour of pillars rocks. For the purpose of this study, we will only examine short-term behaviour because Grgic & Amitrano (2009) have already looked at long-term stability.

Our study focused on the continuous characterization of the crack damage (through the study of the evolution of elastic moduli) of a porous oolitic limestone (from Euville, France) during uniaxial mechanical loading and under different hydrous conditions. The use of uniaxial compression tests is justified because, after excavation, the stress state in mine pillars is uniaxial at the periphery while horizontal stress is not significant (1–3 MPa) at the centre. In this brittle regime (low and constant temperature and low confining pressure), deformation mechanisms such as cataclastic flow, pore collapse and intracrystalline plasticity due to the motion of dislocations, which are widely described for porous carbonate rocks in the specialized literature (e.g. Rutter 1972a,b; Baud et al. 2000b; Vajdova et al. 2004; Schubnel et al. 2005) are generally excluded.

An experimental device was developed and used for the purpose of this study, which permits the simultaneous and continuous measurements of strains (in two perpendicular directions) and five elastic wave velocities with different directions and polarisations (horizontal, vertical and 45°-to-horizontal VP; horizontally and vertically polarized VS in the horizontal direction) during uniaxial compression tests and under fully controlled hydrous conditions. To date, we know of no specialized literature relating similar experiments (i.e. simultaneous and continuous measurements of five elastic wave velocities) and the results and the experimental techniques presented in the paper are transposable to other rocks. The influence of water content on the mechanical behaviour (i.e. strength and deformation) of the Euville limestone will be analysed by considering both the hydromechanical (i.e. variations in effective stress due to variations in capillary pressure) and physio-chemical (i.e. ‘Rehbinder effect’, IPS, subcritical cracking) effects.

Experimental Methods and Samples

Description and petrophysical properties of the studied rock: Euville oolitic limestone

The ‘Euville rock’ is a well-known Malm (Upper Jurassic) oolitic limestone from Euville (Northeastern France). It is formed by oolites marked by the presence of residues of echinoderms (crinoid ossicles). SEM observations are presented in Fig. 1. Oolites are cemented by different types of calcites: large syntaxial crystals, several hundreds of micrometers in size; small equant calcite (with an average size around 10 μm) and microcalcites (around 3 μm). Oolites are made of concentric layers of microcalcite.

Figure 1

Low and high magnification SEM (scanning electron microscopy) images of oolitic Euville limestone.

Figure 1

Low and high magnification SEM (scanning electron microscopy) images of oolitic Euville limestone.

The porosity of Euville limestone was studied using different porosimetry methods: water porosity under vacuum, total porosity and mercury intrusion porosimetry (Purcell test). The water porosity under vacuum technique is the best for the determination of the connected porosity. Indeed, if the sample is saturated under vacuum, all the connected porosity is accessible, even the smallest pores, thanks to the high wetting properties of water. The water porosity under vacuum is equal to 22 per cent. The porous spectrum, obtained from the first mercury injection of the Purcell test, represents a bimodal distribution of the entrance radius of pores (throats) which thereby defines two kinds of pores: intra-oolite and inter-oolite pores. Intra-oolite pores are the smaller of the two and correspond to the 0.01 μm < r < 1.5 μm range, whereas inter-oolite pores are in the 1.5 μm < r < 80 μm range. The former corresponds to ∼60 per cent of the total (bulk) mercury porosity whereas the latter corresponds to ∼40 per cent. The total (bulk) mercury porosity, determined from the first injection, is equal to 22 per cent. The free mercury porosity, determined from the second injection, is equal to 7.5 per cent and the trapped mercury porosity, which is not significant in a rock with a well-connected porous network, is equal to 14.5 per cent.

The free porosity (=7.5 per cent), which is the porosity at which fluids circulate, is small for Euville limestone. Usually for this type of rock, high values of free porosity imply high permeability. However, in this case, the small value of free porosity is balanced by a large pore radius with a resulting relatively high intrinsic permeability (k= 4 × 10−15 m2). The steady-state intrinsic permeability k of Euville limestone was determined in a triaxial cell under 1 MPa confining pressure using the Darcy law.

We chose to set, for all tests, the same constant deformation rate (12 μm min−1) sufficiently low to ensure drained conditions during the uniaxial compression tests. This is justified by the value of the intrinsic permeability (k= 4 × 10−15 m2).

Description of the hydromechanical experiments

The hydromechanical device used in this study (Fig. 2) is a hermetically sealed uniaxial compression cell in which samples can be subjected to saturated or partially saturated moisture conditions. Four cylindrical samples (of height h = 100 mm and diameter ϕ = 50 mm), cored in the direction perpendicular to the horizontal bedding of the material, were tested in this cell and their deformation was measured with strain gages (two in the axial direction and two in the lateral direction, for each sample).

Figure 2

Hydromechanical device: a cylindrical sample (h= 100 mm, ϕ= 50 mm) of Euville limestone with five piezoceramic transducers is put in a hermetic uniaxial compressive cell with water or salt solution.

Figure 2

Hydromechanical device: a cylindrical sample (h= 100 mm, ϕ= 50 mm) of Euville limestone with five piezoceramic transducers is put in a hermetic uniaxial compressive cell with water or salt solution.

Samples were initially saturated in carbonate water (C(Ca2+) ∼ 20 mg L−1; C(HCO3) ∼ 70 mg L−1) that is chemically equilibrated with the Euville limestone to inhibit purely chemical reactions (e.g. dissolution) that could occur between rock forming minerals (essentially calcite) and pore fluid, as shown by many authors (e.g. Feng & Ding 2007; Feng et al. 2009). The partial pressure of water vapour, and therefore the capillary pressure, was imposed in the porous medium using the saturated salt solutions method (ISO standard 483, 2005) (the partial pressure of water vapour depends on the chemical composition of the salt solution). Indeed, the isothermal equilibrium between liquid water and its vapour in a porous medium is given by the Kelvin law:  

1
formula
where hr is the relative humidity, pv is the partial pressure of water vapour, p0v is the vapour pressure under thermodynamic equilibrium, pc is the capillary pressure, ρl is the density of liquid water, R is the gas constant, Mv is the molar mass of water vapour and T is the absolute temperature.Three hydrous conditions were imposed on the Euville limestone in this study:

Sample A: hr= 100 per cent, saturated condition (immersion in water); pc= 0 MPa.

Sample B: hr= 85 per cent (hr imposed by salt solution KCl); pc= 22 MPa.

Sample C: hr= 66 per cent (hr imposed by salt solution NaNO2); pc= 56.2 MPa.

The saturated condition is the condition before mining (i.e. the rocks are below the water-table). The partially saturated conditions are imposed on pillars rock during mining because the water-table is lowered (mine drainage is pumped from the mine) and the ventilation of rooms induces drying of the rocks. hr= 85 per cent, 66 per cent are values measured in underground mines during mining works.

After an equilibrium was obtained with the hydrous environment, a uniaxial compression test was performed on each sample. The same constant deformation rate (12 μm min−1) was applied to all samples. The loading path corresponds to a uniaxial compression with increasing unloading-reloading cycles and the peak stress for the first cycle was about σ33∼−2.5 MPa. A relaxation phase of 15 min was applied before each unloading to dissipate viscous effects (e.g. viscoelasticity). For the saturated sample A and for the partially saturated sample C (hr= 66 per cent), the stress was half unloaded for each cycle whereas the sample B was completely unloaded at each cycle to look at the irreversibility of strain and wave velocities. However, this change in the unloading condition is not likely to affect the results and the interpretation of the deformation mechanisms.

Experimental methodology for the measurements of the velocities of US waves

During our compression tests, the US wave velocities were measured using the standard Ultrasonic Pulse Transmission technique (Birch 1960), which measures the traveltime of a solitary elastic pulse through two piezoelectric detectors held in contact with the rock sample of a known length.

The piezoelectric transducers used in our study have a bandwidth frequency range between 100 and 230 kHz (resonant frequency = 150 kHz). These specific transducers have three layers (Fig. 3)—the first layer is sensitive to compression waves (P waves), whereas the second and third layers are sensitive to shear waves (S waves) with two perpendicular (transversal) directions of polarization. The advantage of these transducers is that with a pair of ceramics, three wave velocities can be measured simultaneously at the same point. For the measurement of US wave velocities in the radial direction, a pair of diametrically opposed piezoceramics were glued onto the middle part of the sample surface (Fig. 2). For the measurement of US wave velocities in the axial direction, a pair of piezoceramics was located at both ends of the cylindrical sample, at the centre of each of the circular surfaces (Fig. 2). Axial stress was applied to the sample via perforated pistons at both ends.

Figure 3

Schematic representation of the three-layer piezoceramic transducer and location of the five transducers on the cylindrical sample surface (the directions of wave polarization and propagation are indicated).

Figure 3

Schematic representation of the three-layer piezoceramic transducer and location of the five transducers on the cylindrical sample surface (the directions of wave polarization and propagation are indicated).

The experimental device for the measurement of US wave velocities consisted of

  • 1

    a generator of sinusoidal pulses

  • 2

    a multiplexer allowing the automatic measurement of velocities on eight channels

The acquisition time for eight signals is almost instantaneous (i.e. a few microseconds). This experimental device permits the continuous measurement of elastic wave velocities which is different from similar experiments to be found in specialized literature (e.g. Fortin et al. 2005; Dewhurst & Siggins 2006; Sarout 2006; Dodds et al. 2007; Sarout et al. 2007).

Because the samples have a cylindrical symmetry, one would expect that the anisotropy due to the microcracks induced during uniaxial loading to be isotropic transverse. For an isotropic transverse material, characterized by five independent elastic constants, the study of anisotropy evolution during mechanical loading requires the measurement of five independent US wave velocities. Fig. 3 shows how the piezoelectric transducers were located on the sample surface.

Before proceeding, we shall first define the reference frame (x1, x2, x3), with (x1, x2) being the horizontal bedding plane, in Fig. 3. Measurements of the velocities of three compression waves (P) and two shear waves (S), in three different directions, were given by the five piezoceramic transducers. These velocities were referenced with respect to the horizontal bedding plane (Fig. 3), i.e. VP(0°) for the horizontal–parallel (i.e. θ= 0°) P-wave velocity, VP(45°) for the 45°-to-horizontal (i.e. θ= 45°) P-wave velocity, VP(90°) for the horizontal–perpendicular (i.e. θ= 90°) P-wave velocity, graphic for the horizontal–parallel velocity of horizontally polarized S wave, and graphic for the horizontal–parallel velocity of vertically polarized S wave. For an isotropic transverse material, graphic is equal to the horizontal–perpendicular S-wave velocity VS(90°).

Prior to the uniaxial compression tests, free-stress US wave velocities were measured on each sample. These confirmed that the material was initially isotropic. Indeed, for the saturated sample A: VP(0°)= 3050 m s−1, VP(90°)= 3040 m s−1, VP(45°)= 3040 m s−1, graphic m s−1, graphic m s−1.

Arrival time pick was based on the first break in the signal noise and estimated from approximately 5 per cent of the first peak amplitude. Data were recorded at a sampling rate of 100 MHz, corresponding to ±0.01 μs for P-wave and S-wave arrival time uncertainties.

Once the velocities were known, the coefficients of the elastic stiffness tensor were calculated using the well-known Christoffels equations for elastic wave propagation (Dieulesaint & Royer 1974; Liao et al. 1997; Lo et al. 1986; Hornby 1998; Sarout 2006):  

2
formula
 
3
formula
We used the convention of negative compressions for these equations in which Cij represents the coefficients of the elastic stiffness tensor in Voigt notation, and θ= 45° in our case. From these five independent elastic moduli, the values of Young's moduli E11 and E33, and three Poisson's ratios ν12, ν13 and ν31 could hence be calculated as (Podio et al. 1968; Sarout 2006)  
4
formula
 
5
formula
where D is the following determinant:  
6
formula

Results

Fig. 4 represents the evolution of the axial, lateral and volumetric strains and the evolution of P- and S-wave velocities during the increasing-amplitude uniaxial stress-cycling test on sample A (saturated condition). The evolution of wave velocities reproduce very remarkably the shape of the stress–strains curves.

Figure 4

Evolution of the axial, lateral, and volumetric strains (a) and evolution of P waves (b) and S waves (c) velocities—uniaxial compression test on sample A (saturated condition).

Figure 4

Evolution of the axial, lateral, and volumetric strains (a) and evolution of P waves (b) and S waves (c) velocities—uniaxial compression test on sample A (saturated condition).

The linearity threshold of the lateral strain curve (Bieniawski 1967) corresponds to σ33∼−5 MPa. In the field of rock mechanics, this threshold indicates the onset of the initiation (and opening) of stable axial microcracks (Bieniawski 1967). The dilatancy threshold, which is defined from the volumetric strain curve, indicates the onset of the dilatant volumetric deformation of the material (Wong et al. 1997). This corresponds to σ33∼−8.5 MPa. In the field of rock mechanics, this dilatancy threshold is often considered as representing the onset of unstable growth (or coalescence) of microcracks (e.g. Bieniawski 1967; Martin & Chandler 1994).

Figs 5 and 6 represent the evolution of the axial, lateral and volumetric strains and the evolution of P- and S-wave velocities during the uniaxial compression tests on the partially saturated samples B (hr= 85 per cent) and C (hr= 66 per cent), respectively. For the sample B, the stress was completely unloaded at each cycle to study the irreversibility of strain and wave velocities.

Figure 5

Evolution of the axial, lateral, and volumetric strains (a) and evolution of P waves (b) and S waves (c) velocities—uniaxial compression test on sample B (hr = 85 per cent).

Figure 5

Evolution of the axial, lateral, and volumetric strains (a) and evolution of P waves (b) and S waves (c) velocities—uniaxial compression test on sample B (hr = 85 per cent).

Figure 6

Evolution of the axial, lateral and volumetric strains (a) and evolution of P waves (b) and S waves (c) velocities—uniaxial compression test on sample C (hr= 66 per cent).

Figure 6

Evolution of the axial, lateral and volumetric strains (a) and evolution of P waves (b) and S waves (c) velocities—uniaxial compression test on sample C (hr= 66 per cent).

The evolution of wave velocities and stress–strain curves for these partially saturated samples was found to be qualitatively similar to those in the first experiment on the saturated sample A (Fig. 4).

Fig. 7 represents the evolution of the dynamic stiffness coefficients Cij and the dynamic engineering parameters, that is Young's moduli E11 and E33, and Poisson's ratios υ12, υ13 and υ31 during the uniaxial compression test on sample B (hr= 85 per cent). Because the wave velocities were measured continuously, the continuous evolution of the dynamic engineering parameters during the uniaxial loading is observed providing evidence of the progressive damage of the material.

Figure 7

Evolution of dynamic stiffness coefficients and dynamic engineering parameters (Young's moduli and Poisson's ratios)—uniaxial compression test on sample B (hr = 85 per cent).

Figure 7

Evolution of dynamic stiffness coefficients and dynamic engineering parameters (Young's moduli and Poisson's ratios)—uniaxial compression test on sample B (hr = 85 per cent).

To study the evolution of the damage from the beginning of loading until the macroscopic failure, we have represented in Figs 8(a) and (b) the evolution of static Young's modulus E33 and Poisson's ratio υ31 as a function of axial stress σ33 for all samples. These coefficients were classically calculated from the unloading curve of each cycle with the secant method (Deere & Miller 1966; Eissa & Kazi 1988), from the slope of the line joining a specified stress level (i.e. the stress level at the beginning of an unloading) and the half of this stress level on the stress–strain curve.

Figure 8

Evolution of static Young's modulus E33 (a) and Poisson's ratio υ31 (b) as a function of axial stress σ33 (coefficients were determined from the unloading curve of each cycle with the secant method)—all samples. (c) Evolution of static Young's modulus E33 and dynamic Young's moduli E33 and E11 as a function of unloading–reloading cycle number—uniaxial compression test on sample B (hr= 85 per cent). Coefficients were determined at the very beginning of each loading (σ33∼ 0 MPa) and the cycle number 0 corresponds to the initial loading.

Figure 8

Evolution of static Young's modulus E33 (a) and Poisson's ratio υ31 (b) as a function of axial stress σ33 (coefficients were determined from the unloading curve of each cycle with the secant method)—all samples. (c) Evolution of static Young's modulus E33 and dynamic Young's moduli E33 and E11 as a function of unloading–reloading cycle number—uniaxial compression test on sample B (hr= 85 per cent). Coefficients were determined at the very beginning of each loading (σ33∼ 0 MPa) and the cycle number 0 corresponds to the initial loading.

Discussion

Evolution of the damage during the uniaxial loading

In Fig. 4 (Sample A), the increase in all wave velocities, at the beginning of the loading, indicates the closure of pre-existing microcracks and penny-shaped pores in the rock. VP(90°) increases the most, followed by VSV(0°), VP(45°), VP(0°) and VSh(0°), indicating the preferential closure of pre-existing microcracks and penny-shaped pores (which can be considered as microcracks) which are perpendicular, or almost perpendicular, to the uniaxial stress direction. It is worth emphasizing that the maximum compaction of Sample A (calculated from the volumetric strain curve) is about 0.025 per cent, which is negligible compared to the initial bulk porosity (∼22 per cent). Therefore, the reduction in the bulk porosity during uniaxial loading is negligible, which is always the case with hard rocks such as the Euville limestone, and cannot explain the increase in wave velocities.

The decrease in wave velocities, from the onset of stable microcracking (σ33∼−5 MPa) up until the macroscopic failure of the sample, indicates the progressive formation and opening of microcracks. Because wave velocities started to decrease slowly in the following order: VSh(0°), VP(0°), VP(45°), VSV(0°) and VP(90°) (from σ33∼−5, −5, −8, −8 and −11.5 MPa, respectively), this indicates the preferential opening of microcracks parallel to the direction of the applied uniaxial stress. Both the total number and the total summarized length of cracks probably increased with the stress applied. In contrast, the mean crack length is supposed to be quasi-constant even though some authors (Hadley 1976; Wong 1985) have reported that the mean crack length increases with the stress deviator. Homand et al. (2000), in agreement with Zhao (1998), suggested that cracking evolution is mostly due to the initiation of new cracks than to the growth of pre-existing cracks. However, Homand et al. (2000) found that the coalescence of cracks occur more rapidly in the direction of stress.

From the dilatancy threshold, which represents the onset of unstable growth (or coalescence) of microcracks, VSh(0°) and VP(0°) decreased strongly. Generally speaking, the change from increasing to decreasing velocities at higher uniaxial stress marks the change from perpendicular microcracks closure-dominated to dilatancy (i.e. opening of axial microcracks)-dominated deformation. It is worth emphasizing that AE (Acoustic Emission) analysis (e.g. Ohnaka & Mogi 1982; Lockner 1993; Grgic & Amitrano 2009; Eslami 2010) has already demonstrated on different kinds of rocks that AE activity increases strongly at the onset of dilatancy. However, VP(90°), whose direction corresponds to the direction of applied uniaxial stress, continued to increase after the dilatancy threshold because of the closure of pre-existing microcracks and penny-shaped pores which are perpendicular, or almost perpendicular, to the uniaxial stress direction, and because VP(90°) cannot record the opening of microcracks parallel to the direction of the applied uniaxial stress. VP(90°) started to decrease just before the macroscopic failure of the sample; this velocity can detect therefore very precisely the macroscopic failure of the material.

In all cases (Figs 4–6), the initial values of P-wave velocities were identical (i.e. isotropic) whatever the direction of propagation and the initial values of S-wave velocities were also identical (i.e. isotropic) whatever the direction of polarization. This confirms that the material was initially isotropic. During loading and before the coalescence of microcracks (and macroscopic failure), the creation of microcracks is probably scattered and isotropic, as shown in past studies (Lockner et al. 1991; Lockner & Byerlee 1992; Zhao 1998). The anisotropy of the wave velocities, and therefore of the material, develops progressively during the loading due to the preferential closure of pre-existing microcracks and penny-shaped pores which are perpendicular (or almost perpendicular) to the direction of the uniaxial stress and the preferential opening of microcracks parallel to the direction of the applied uniaxial stress. This result is in good agreement with past studies on sandstone (Sayers et al. 1990; Scott et al. 1993; Fortin et al. 2005; Dodds et al. 2007), calcite rocks (Schubnel et al. 2005) and shale (Dewhurst & Siggins 2006; Sarout 2006; Dodds et al. 2007; Sarout et al. 2007).

It is worth emphasizing that this anisotropy of the wave velocities, and therefore of the damage, is reversible (i.e. the anisotropy decreases during unloading). Indeed, as observed in Fig. 5 (sample B), when the uniaxial stress is completely reduced (i.e. at each unloading), there is no more difference (except for the last cycle) between the different P-wave velocities (i.e. wave velocities become again isotropic: VP(90°)VP(45°)VP(0°)), as it was the case for the initial values (i.e. before the uniaxial compression test). This is also observed for the S-wave velocities. Moreover, after a complete unloading, all waves velocities become smaller than the initial values (i.e. before the uniaxial compression test), indicating that some of the damage is obviously irreversible.

Figs 8(a) and (b) show the evolution of static Young's modulus E33 and Poisson's ratio υ31 (secant moduli) as a function of axial stress σ33 for all samples. Because the secant modulus is calculated on the upper half of each unloading cycle, as explained in Section 3, it gives a better representation of the state of damage for a specified stress level. Indeed, as said above, the damage diminishes with decreasing stress (i.e. some of the damage is reversible), especially close to the zero. In all cases, Young's modulus was found to increase along with wave velocities at the beginning of loading due to the closure of pre-existing microcracks and penny-shaped pores which are perpendicular, or almost perpendicular, to the uniaxial stress direction. Static Young's modulus E33 started to decrease along with VP(90°) just before the macroscopic failure of the samples, together with the dynamic Young's modulus E33 (Fig. 7). Poisson's ratio increased continuously from the beginning of the loading until the failure, which indicates that the damage is anisotropic. This decrease in Young's modulus and increase in Poisson's ratio have already been observed in many past studies (e.g. Eberhardt et al. 1999; Heap & Faulkner 2008; Heap et al. 2009a).

In Fig. 8(c), both the static Young's modulus E33 and the dynamic Young's moduli E33 and E11 were determined, from the uniaxial compression test on sample B (hr= 85 per cent), at the very beginning of each loading (σ33∼ 0 MPa) with the tangent method. Unlike the static and dynamic Young's moduli presented in Figs 7 and 8(a), in Fig. 8(c) all Young's moduli decreased continuously from the beginning of loading until the failure, indicating the progressive development of damage. Indeed, because these moduli were determined for σ33∼ 0 MPa, they were not yet affected by the closure of created microcracks and pre-existing microcracks and penny-shaped pores which are perpendicular, or almost perpendicular, to the uniaxial stress direction. The values of dynamic Young's moduli E33 and E11 were found to be very close, indicating isotropic damage, as emphasized earlier thanks to the evolution of wave velocities.

Furthermore, it is worth emphasizing that the values of dynamic and static Young's moduli E33 were found to be very close (Fig. 8c) at the beginning of loading for the Euville limestone. These moduli differ with increasing stress (static modulus decreases more than dynamic modulus), illustrating that static modulus is more affected by the presence of cracks, as observed by many authors (e.g. King 1983; Eissa & Kazi 1988).

The influence of water on the deformation and strength

Fig. 9 represents the evolution of the axial, lateral and volumetric strains during uniaxial compression tests on all samples (unloading–reloading cycles are not represented). For a given uniaxial stress, the deformations are greater the more the relative humidity hr increases. In addition, Fig. 8(a) shows that the static Young's modulus E33 increases with the decrease in relative humidity hr.

Figure 9

Evolution of the axial, lateral and volumetric strains—all samples (unloading–reloading cycles are not represented).

Figure 9

Evolution of the axial, lateral and volumetric strains—all samples (unloading–reloading cycles are not represented).

The uniaxial compression strength σc and the dilatancy threshold also increase with the decrease in relative hygrometry hr: σc= 14 MPa for sample A (saturated), 16.4 MPa for sample B (hr= 85 per cent) and 20.8 MPa for sample C (hr= 66 per cent). In addition, the decrease (from the linearity threshold of the lateral strain curve until the macroscopic failure) in all wave velocities is all the smaller that the relative humidity decreases.

These effects of water on the deformation and strength of rocks have already been demonstrated by many authors (e.g. Lajtai et al. 1987; West 1994; Baud et al. 2000a; Grgic et al. 2005; Grgic & Amitrano 2009) and may be both mechanical and physio-chemical. Rutter (1972a) has investigated the effects of water on the strength and ductility of a fine-grained limestone and found that wetting reduced the uniaxial strength by 30 per cent.

The mechanical effect of water

First, pore water has a significant influence on the deformation and strength of porous rocks such as Euville limestone through hydromechanical coupling, which can be formulated using the effective stress concept (Terzaghi 1936; Bishop 1959; Nur & Byerlee 1971; Coussy 1995). The total stress tensor may be decomposed into an effective stress tensor and a pressure tensor  

7
formula
modulus and the drained Poisson's ratio.

Within the framework of partially saturated porous media, the fluid pressure tensor σp is usually represented by the π function (Coussy 1995), which corresponds to the equivalent interstitial pressure. Hence, the partitions of the elastic stress tensor and the elastic mean stress for partially saturated porous media can be written as follows:  

8
formula
where B is the Biot coefficient (B= 0.8 for Euville limestone).

The π function is written as follows (Coussy 1995):  

9
formula
where Sl and pg, respectively denote the liquid saturation and the gas (water vapour and dry air) pressure.

This function allows the generalization of the pore pressure in both saturated and partially saturated cases: π is equal to the liquid pressure in the saturated domain (π=pl when Sl= 1) and is negative in the partially saturated domain (π < 0 when Sl < 1). It is calculated from water sorption curves (liquid saturation versus capillary pressure). Fig. 10 (left) represents the water sorption curves (adsorption and desorption paths) of the Euville limestone.

Figure 10

(a) Water sorption curves obtained on the Euville limestone at 293.15 °K: liquid saturation Sl versus relative humidity hr and capillary pressure pc. (b) Loading paths in the space of effective stress invariants and failure surface of Euville limestone in the saturated domain.

Figure 10

(a) Water sorption curves obtained on the Euville limestone at 293.15 °K: liquid saturation Sl versus relative humidity hr and capillary pressure pc. (b) Loading paths in the space of effective stress invariants and failure surface of Euville limestone in the saturated domain.

Variations in relative humidity imply variations in capillary pressure (eq. 1) and, depending on the porosity and on the shape of the porous network, variations in the effective stress (eqs 8–9). A desorption path was imposed on our Euville limestone samples, corresponding to a drying from hr= 100 to hr= 85 per cent and 66 per cent. The corresponding values of the π function are: 0, −0.1 and −0.26 MPa. These values obtained from the drying of our Euville limestone, which is a macroporous rock (cf.Section 2.1), are not significant compared to those obtained from the drying of a rock with much smaller pores, for example iron ore (Grgic et al. 2005) or argillite (Grgic et al. 2006). Indeed, because the Euville limestone is a macroporous rock, its suction curves (Fig. 10, left) are very flat in the range 0 per cent < hr < 97 per cent and the material only adsorbs and desorbs water in the range of high moistures or small capillary pressures (i.e. hr > 97 per cent ↔pc <4 MPa), corresponding to the filling and emptying of large size pores. The result is that the π function, calculated from eq. (9), has small values for our Euville limestone because the area under the suction curves is not significant. Generally speaking, the values of the π function are greater the more the material is microporous. The suction curves of our Euville limestone being very flat, its adsorption properties are no significant; this is confirmed by the small value (∼1 m2 g−1) of its specific surface area (SSA) measured by adsorption using the BET isotherm method. This small value of SSA is mainly due to the large size of pores. This is not due to the physio-chemical properties of the rock forming minerals because the SSA does not differ markedly depending on the substance adsorbed (water or nitrogen).

Fig. 10 (right) shows, in the space of effective stress invariants (σ′m, σeq), the failure surface of Euville limestone obtained from triaxial compression tests in saturated condition. The failure points of the uniaxial compression tests on samples A–C are also shown here.

The two first invariants of the Cauchy stress tensor, the mean stress σm and the shear stress (or Von Mises equivalent stress) σeq, are expressed as follows:  

10
formula
where s is the deviatoric stress.

Under saturated condition, positive water pressure (i.e. π=pl > 0) influences the failure and the mode of deformation of porous rocks, depending on the value of the effective mean stress σ′m (or confining pressure). Generally speaking, rocks deform as ductile material at high effective mean stresses and brittle material at low effective mean stresses. In addition, the strength is greater the more the effective mean stresses increases.

Under partially saturated condition (i.e. π < 0), the capillary attraction forces may harden porous rocks. Indeed, capillary attraction forces induce the increase in the π function and in the effective mean stress σ′m (eqs 8–9), as shown in Fig. 10 (right). From O to P and from O to Q, the hydrous paths correspond to drying at hr= 85 and 66 per cent, respectively. From O to A, the loading path corresponds to uniaxial compression in the saturated domain (sample A). From P to B and from Q to C, the loading paths correspond to uniaxial compressions in the partially saturated domain (samples B and C). Because the initial drying of samples B and C is equivalent to the application of a confining pressure, this may explain why the uniaxial compression strength and the Young's modulus increase and why the opening of microcracks decreases (i.e. the dilatancy threshold increases) with the decrease in relative hygrometry.

However, because the increase in the π function is not significant because of the large size of pores, the hardening of the Euville limestone cannot be explained by hydromechanical coupling. This is illustrated in Fig. 10 (right); the failure thresholds of the partially saturated samples B and C are above the effective failure surface of the saturated material. Therefore, other phenomena are also involved in this hardening.

The physio-chemical effects of water

Rehbinder effects (Rehbinder & Lichtman 1957) These effects have been observed in past studies for single crystals (Rehbinder & Lichtman 1957) and porous rocks (Colback & Wiid 1965; Lajtai et al. 1987; West 1994; Grgic et al. 2005; Shakoor & Barefield 2009). Rehbinder & Lichtman (1957) attributed the weakening of single crystals by the action of water to a reduction of interfacial tension and suggested that this reduces the work hardening component due to the pile-up of dislocations at the crystal–fluid interface, and increases the probability of emergence of dislocations onto the crystal surface. Westwood & Goldheim (1968) have suggested that the adsorbance of a polar fluid onto the crystal surface modify the interactive behaviour of point and line (dislocations) defects which largely determine the mobility of dislocations during the plastic deformation of ionic crystals, such as calcite. They also suggested that this is not a surface effect, like the alteration of the surface energy of the crystal, but one which penetrates up to 10 μm into the crystal. Therefore, as proposed by Rutter (1974), it would be expected that this effect would diminish with increasing grain size in proportion to the specific grain surface area. Generally speaking, the ‘Rehbinder effect’, due to the reduction of the surface energy of different kinds of walls (e.g. pores, flaw, microfissure) by the adsorbed water, promotes the propagation of these walls if the stress is high enough.

Fluids (water, brine) are always adsorbed on these surfaces in rocks existing in natural environments or in manmade structures (e.g. underground mines and quarries, road cuts and open pits) and, therefore, the ‘Rehbinder effect’ always operates. The rate of physio-chemical reactions in rocks like granite is probably too slow to be effective over the short time it takes to complete a uniaxial compression test (Lajtai et al. 1987). Therefore, for this kind of low porosity rock, the physio-chemical effects are often small compared with the hydromechanical effect. However, as noticed by Rutter (1972a, 1974), the weakening effect of water through action at grain boundaries is likely to be greatest in rocks of high porosity and small grain size in which a large surface area is exposed to the pore fluid. The Euville limestone is, indeed, a high porosity rock (∼22 per cent) but grain size is not small (100–200 μm). Rutter (1974) also suggested that, according to experimental data obtained on a fine-grained limestone, ‘Rehbinder effects’ are much more significant at low temperatures, for which they are not overridden by the increased mobility of point defects resulting in intracrystalline plasticity mechanism, and under unconfined conditions. Therefore, in our experiments on the Euville limestone performed at room temperature and without confining pressure, ‘Rehbinder effects’ are probably of great importance. They can explain the softening (i.e. increase in ductility and decrease in strength) of the material as the relative humidity increases. Indeed, the magnitude of the Rehbinder effects depends on the concentration of water at grain surfaces, which is proportional to the partial pressure of water vapour.

Intergranular pressure solution The adsorbed water also has the effect of promoting IPS in porous rocks. IPS is a physio-chemical mechanism whereby the concentration of normal stress at grain contacts causes local dissolution of the solid material, transport of the solutes (through the interstitial fluid) out of the contact and precipitation of the solid material on the less stressed faces of the grains (Rutter 1976; Zubtsov et al. 2004). The kinetics of this mechanism depends on many factors, which include the fluid chemistry (in our case, the interstitial fluid was initially chemically equilibrated with the Euville limestone), the stress state, the temperature. IPS is a well-known mechanism which is of great importance in carbonate rocks such as limestone (Rutter 1972a; Renard et al. 2005; Le Guen et al. 2007) or chalk (Pietruszczak et al. 2006) due to the high solubility of calcite in carbonate water at room temperature. The following general chemical reaction describes the water–rock interaction in limestone:  

11
formula
In a study of calcite aggregate compaction in the presence of aqueous fluids at room temperature, Zhang & Spiers (2005a,b) have shown that compaction is inhibited by adding phosphate ions (solute species that is known to slow down the calcite dissolution and precipitation) to the pore fluid suggesting, thus, that IPS is the dominant deformation mechanism. This mechanism affects negatively the mechanical properties (deformation, strength) of porous rocks on the macroscale. The degradation of the macroscopic mechanical properties (deformation, strength) can be attributed to the reduction in the effective solid contact area between solid grains (Pietruszczak et al. 2006). The IPS is a mechanism which is often invoked (e.g. Zubtsov et al. 2004; Renard et al. 2005; Le Guen et al. 2007) to explain the long-term deformation and failure of porous rocks in the presence of water, resulting in pressure solution creep (PSC). However, even in the short range of hours (∼8 hr) corresponding to uniaxial compression test, a very fast dissolution of grain contacts may occur, as noticed by Pietruszczak et al. (2006) for chalk. Obviously, IPS reactions are much more significant and fast in chalk than in limestone. Therefore, since IPS reactions occur in the presence of the pore fluid, this could also explain the softening (i.e. decrease in failure and dilatancy thresholds and increase in deformation) of the Euville limestone with the increase in the relative humidity of the testing environment. Indeed, the proportion of water adsorbed in the porous network depends on the relative humidity hr.

Subcritical cracking Finally, the subcritical cracking is another mechanism that could be invoked to explain the physio-chemical effect of water on the mechanical properties (deformation, strength) of the Euville limestone during our experiments. Indeed, this deformation mechanism is enhanced by the presence of water through physio-chemical reactions at crack tips.

The enhancement of subcritical crack growth by stress corrosion at crack tips is often considered to be the main cause of the time-dependent brittle behaviour of silicate minerals such as quartz (e.g. Scholz 1972) and silicate rocks (e.g. Scholz 1968; Atkinson 1982; Atkinson 1984; Atkinson & Meredith 1987; Jeong et al. 2007). Even though it is very often considered that this weakening effect of water is more significant in long-term experiments than in short-term ones, it has sometimes been invoked to explain the short-term brittle behaviour of silicate rocks (e.g. Jeong et al. 2007).

In the case of silicates, the stress corrosion reaction is well known; water chemically attacks through a hydrolysis reaction the strong siloxane (Si–O–Si) bonds at the crack tip thus inducing bond rupture—aided by tensile stress—and the formation of terminal silanol (Si–OH) groups with weaker bonds. Indeed, chemical reaction rates in solids are known to depend on mechanical stress levels (e.g. Lawn 1993), which modify activation energy barriers. In addition, stress corrosion cracking is facilitated at higher water vapour pressures (e.g. Meredith & Atkinson 1985) and in water-saturated rocks (Waza et al. 1980; Meredith & Atkinson 1983). Water changes the activation energy of the chemical reaction and increases the crack propagation velocity (Scholz 1972; Atkinson 1984).

In the case of carbonate rocks such as the Euville limestone, the nature of stress corrosion reactions is not well understood, as noticed by Atkinson (1984). In addition, Rutter (1974) suggested that there is no evidence that this hydrolytic weakening, as described for quartz and other silicates (Griggs 1967; Martin 1972; Scholz 1972), plays any role in the deformation of wet calcite rocks because it is not expected that hydrolytic weakening will be important in the deformation of ionic crystals such as calcite. However, in chemically reactive environments such as that of rocks exposed to atmospheric conditions, many other mechanisms may be involved in subcritical crack growth: dissolution, diffusion, ion-exchange and microplasticity (Atkinson 1984).

Dislocation activity allowing plastic flow at crack tips may be an important mechanism of crack extension, even at room temperature, and could lead to subcritical cracking of some minerals such as calcite (Atkinson 1984). In addition, dislocation mobility is thought to be strongly influenced by the presence of structurally bond water (Atkinson 1984; Grgic & Amitrano 2009). Dissolution of calcite (eq. 11)—maybe aided by tensile stress—at crack tips may also be important in crack growth, as proposed by Atkinson (1984). Indeed, there is no reason why interfacial reactions, similar to those involved in IPS mechanism, should not be important in subcritical cracking. The solubility of calcite increases with decreasing temperature, as said above while describing the IPS mechanism.

Therefore, for calcite rocks, at conditions common in mining engineering environments (low temperature, presence of water), microplasticity and dissolution are likely to be of significance in crack growth. However, in a study of time-dependant compaction of calcite aggregate in the presence of aqueous fluids, Zhang & Spiers (2005a,b) have discounted the subcritical crack growth mechanism based on the absence of acoustic emissions during the experiments. In addition, AE measurements performed during uniaxial creep tests on some samples of the Euville limestone showed no evidence for significant crack growth deformation mechanism; the AE activity remained moderate even in the presence of water (saturated condition or high vapour pressure) during the significant (axial and lateral) deformation of the samples. Hence, subcritical cracking is not the main mechanism responsible for the short and long-term deformation of our wet limestone samples.

Conclusions

In the work presented here, we studied the continuous evolution of damage of a porous rock (Euville oolitic limestone) during uniaxial stress-cycling tests and under different (fully controlled) hydrous conditions. The damage was monitored in a really precise and continuous way through the evolution of both static and dynamic elastic moduli. These moduli were calculated from strains (in two perpendicular directions) and five elastic wave velocities in two different directions (three P-waves and two S-waves) respectively which were measured continuously and simultaneously during the hydromechanical tests.

The evolutions of wave velocities and elastic moduli reproduce very remarkably the shape of the stress–strains curves. From the beginning of the uniaxial loading until the onset of stable microcracking, the increase in wave velocities (VP(90°) and graphic increase the most) indicates the preferential closure of pre-existing microcracks and penny-shaped pores which are perpendicular, or almost perpendicular, to the uniaxial stress direction. From this onset, the decrease in wave velocities (graphic and VP(0°) decrease the first) indicates the preferential opening of microcracks parallel to the direction of the applied uniaxial stress. From the dilatancy threshold, corresponding to the onset of unstable growth (or coalescence) of microcracks, graphic and VP(0°) decrease strongly. However, VP(90°), which cannot record the opening of microcracks parallel to the direction of the applied uniaxial stress, start to decrease just before the macroscopic failure of the sample.

The wave velocities, and therefore the material, were initially isotropic and became progressively anisotropic during loading. Indeed, during loading, the creation of microcracks is scattered and isotropic. The anisotropy was caused by both the preferential closure of pre-existing microcracks and penny-shaped pores, which are perpendicular (or almost perpendicular) to the direction of the uniaxial stress and the preferential opening of axially oriented microcracks. After a complete unloading, the anisotropy of the damage decreases and is completely reversible (i.e. wave velocities become again isotropic) but some of the damage is irreversible (i.e. wave velocities become smaller than the initial values).

The uniaxial compressive strength, the dilatancy threshold and the static Young's modulus all increased as the relative humidity decreased. In addition, the deformation decreased with the relative humidity. This influence of water (i.e. hardening) was analyzed in terms of effective stresses. Indeed, capillary attraction forces, which increase in proportion to the decrease in relative humidity, induce the increase in the effective mean stress and therefore may harden porous rocks. However, because the increase in the effective mean stress is not a significant because of the largeness of pores, the hardening of the Euville limestone cannot be explained by hydromechanical coupling. Therefore, other mechanisms (‘Rehbinder effect’, IPS and subcritical cracking) were proposed to explain the physio-chemical influence of water on the mechanical behaviour of the Euville oolitic limestone.

In situ monitoring of seismic wave velocities could be used in mines to infer macroscopic crack damage of pillars. Indeed, knowledge of the damage process is important for estimating the long-term stability of underground mines and quarries in France. Particularly, since VP(90°) start to decrease just before the macroscopic failure, this velocity could be used to detect very precisely the failure of mining pillars. Furthermore, the triggering of some mining collapses by water injection is likely to involve intrinsic weakening by physio-chemical mechanisms (aided by stress) as described earlier along with mechanical pore pressure effects. Finally, the results obtained on the crack damage process from this study can be used to confirm or withdraw some assumptions often made in the modelling of damage with constitutive laws.

The experimental techniques, and the associated results, presented in this paper are transposable to the study of other rocks and under pressure and temperature conditions corresponding to very deep rocks for which other deformation mechanisms are activated. Afterwards, we are going to improve these techniques by measuring continuously and simultaneously acoustic emissions and wave velocities on different rocks samples under different hydrous, temperature and pressure conditions.

Acknowledgments

This research was carried out thanks to subsidies from the Ministries for Industry and Research and the Lorraine Region within the GISOS (www.gisos.org) framework. The authors express their gratitude to these organizations. We are also grateful to Michael J. Heap and the second (anonymous) reviewer for their helpful comments that have significantly improved the quality of the manuscript.

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