PUBLICATIONS Journal of Geophysical Research: Solid Earth RESEARCH ARTICLE 10.1002/2017JB014147 Special Section: Seismic and micro-seismic signature of fluids in rocks: Bridging the scale gap
Stress-dependent permeability and wave dispersion in tight cracked rocks: Experimental validation of simple effective medium models Joel Sarout1
, Emilie Cazes1,2, Claudio Delle Piane1, Alessio Arena1,3, and Lionel Esteban1
1
Key Points: • Experiments on thermally cracked Carrara marble in dry and water-saturated conditions up to 50 MPa effective pressure • Low-amplitude–high-frequency and low-amplitude–low-frequency elastic moduli, permeability, and porosity as a function of effective pressure • Joint effective medium modeling of elastic moduli, wave dispersion, and permeability to quantify the evolution of crack network morphology
Correspondence to: J. Sarout,
[email protected]
Citation: Sarout, J., E. Cazes, C. Delle Piane, A. Arena, and L. Esteban (2017), Stressdependent permeability and wave dispersion in tight cracked rocks: Experimental validation of simple effective medium models, J. Geophys. Res. Solid Earth, 122, 6180–6201, doi:10.1002/2017JB014147. Received 1 MAR 2017 Accepted 22 JUL 2017 Accepted article online 26 JUL 2017 Published online 19 AUG 2017
CSIRO Energy, Perth, Western Australia, Australia, 2Now at Terrasol, Lyon, France, 3Now at Thermo Fisher Scientific, Canberra, ACT, Australia
We experimentally assess the impact of microstructure, pore fluid, and frequency on wave velocity, wave dispersion, and permeability in thermally cracked Carrara marble under effective pressure up to 50 MPa. The cracked rock is isotropic, and we observe that (1) P and S wave velocities at 500 kHz and the low-strain (<105) mechanical moduli at 0.01 Hz are pressure-dependent, (2) permeability decreases asymptotically toward a small value with increasing pressure, (3) wave dispersion between 0.01 Hz and 500 MHz in the water-saturated rock reaches a maximum of ~26% for S waves and ~9% for P waves at 1 MPa, and (4) wave dispersion virtually vanishes above ~30 MPa. Assuming no interactions between the cracks, effective medium theory is used to model the rock’s elastic response and its permeability. P and S wave velocity data are jointly inverted to recover the crack density and effective aspect ratio. The permeability data are inverted to recover the cracks’ effective radius. These parameters lead to a good agreement between predicted and measured wave velocities, dispersion and permeability up to 50 MPa, and up to a crack density of ~0.5. The evolution of the crack parameters suggests that three deformation regimes exist: (1) contact between cracks’ surface asperities up to ~10 MPa, (2) progressive crack closure between ~10 and 30 MPa, and (3) crack closure effectively complete above ~30 MPa. The derived crack parameters differ significantly from those obtained by analysis of 2-D electron microscope images of thin sections or 3-D X-ray microtomographic images of millimeter-size specimens.
Abstract
Plain Language Summary
In this contribution, a combination of laboratory measurements and effective medium models is used to recover quantitative information about the evolution of the morphology of the crack network in a thermally cracked Carrara marble. The rock is tested in dry and water-saturated conditions under effective pressure loading up to 50 MPa. The properties measured and modeled are elastic wave velocities at 500 kHz, elastic moduli at 0.01 Hz, porosity, and permeability. The recovered crack network parameters lead to a good agreement between the measured and the predicted elastic, wave dispersion, and transport properties. However, they differ significantly from those derived from 2-D and 3-D image analyses. The evolution of the crack parameters suggests the existence of three deformation regimes during the effective pressure loading. The wave dispersion data and the evolution of the crack network morphology can be used to upscale laboratory-derived properties to the reservoir formation scale and depth.
1. Introduction
©2017. American Geophysical Union. All Rights Reserved.
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Laboratory experimentation on rocks under stress is often necessary for a robust interpretation of geophysical data for subsurface exploration and reservoir monitoring applications, e.g., the recovery of oil, gas, geothermal energy, or the geo-sequestration of CO2. Well-controlled laboratory experiments on relatively small rock samples often contribute to a better understanding of the physical mechanisms giving rise to the observed behavior at a larger scale [e.g., Aizawa et al., 2008; Barnhoorn et al., 2010; Sarout et al., 2017]. However, the upscaling process is not straightforward. This is often due to (i) the inherent coupling between various physical mechanisms, (ii) the frequency dependence of some physical phenomena (e.g., characteristic times related to viscous flow and diffusion), and (iii) the heterogeneous nature of rocks at all scales. Theories for modeling frequency-dependent elastic properties (dispersion) reported in the literature are based either on a single physical mechanism such as the squirt-flow model [e.g., Mavko and Jizba, 1991] or the viscous flow/inertial drag transition model [e.g., Biot, 1962], or on coupled mechanisms such as the Biot/squirt (BISQ) model combining the two formers [e.g., Dvorkin and Nur, 1993]. Wave dispersion in rocks can be significant as indicated by direct laboratory measurements [e.g., Batzle et al., 2006; Adelinet et al., 2010]
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and predicted by rock physics models [e.g., Schubnel and Guéguen, 2003; Guéguen and Sarout, 2011; Muller et al., 2010, and references therein]. Yet laboratory elastic data are often used directly for modeling/interpreting field measurements without “correction” accounting for possible dispersion effects. Defining such “corrections” usually requires a background theory accounting for the plausible causes of fluid migration within the rock’s pore space during wave propagation. Models based only on Biot’s poroelasticity often fail in explaining the observed dispersion because they do not include additional fluid-induced dispersions occurring at a scale smaller than the poroelastic representative volume. Effective medium models involving the squirt-flow mechanism are more successful in reproducing the observed dispersions. However, they require explicit and quantitative microstructural information about the rock such as length scale, shape factors, and spatial/orientation distributions. For complex natural rock microstructures, the theoretician/modeler usually faces two important challenges: (i) selecting the appropriate physical mechanism(s) operating during wave propagation and (ii) defining relevant and quantitative microstructural descriptors that can be upscaled. In most cases, carefully conducted laboratory experiments, and the issuing data, are vital for our ability to interpret field measurements or validate the existence of postulated physical mechanisms for upscaling purposes. A convenient way to conduct experiments under well-controlled conditions involves the use of synthetic materials for which the microstructure is well characterized and reproducible. Several attempts have been reported in the literature for manufacturing synthetic rocks with controlled microstructures that are simpler to analyze than typical microstructures found in natural rocks [e.g., Ass’ad et al., 1992, 1996; Rathore et al., 1995; Tillotson et al., 2011, 2012; Stewart et al., 2013; De Figueiredo et al., 2012, 2013; Amalokwu et al., 2015; Ding et al., 2014]. The manufacturing process of these synthetic rocks allows for a relatively accurate control of the final microstructure. In principle, microstructural parameters such as the crack density are either already known at the manufacturing stage or can be assessed independently (e.g., 2-D electron microscope images of thin sections and 3-D images using X-ray microtomography). In this paper, we take an intermediate route in the scale of complexity, somewhere between complex natural sedimentary rocks and much simpler synthetic rocks. We start from an isotropic and homogeneous natural rock, the Carrara marble (CM), for which (i) natural porosity and permeability are negligible at room conditions and (ii) mineralogical composition is close to that of pure calcite. A cracked sample of CM is obtained by thermal quenching. The rock physics response under variable effective pressure up to 50 MPa is then measured under dry and water-saturated conditions. The following measurements are reported here: (i) elastic properties at two contrasting frequencies (0.01 Hz and 500 kHz), (ii) porosity and gas permeability, and (iii) volumetric strain under drained conditions. We focus on the impact of the microstructure, pressure (depth), pore fluid compressibility, and probing wave frequency on the elastic and hydraulic response of the rock. Varying the effective pressure allows us to modify the microstructure of the thermally cracked specimen in a deterministic way up to an equivalent depth of ~4 km. Although cracks are expected to close progressively with increasing effective pressure, an ambiguity remains about whether their aperture, their radius, or their number is mostly impacted by pressure. Because these microstructural parameters impact differently elastic wave velocity and permeability, this knowledge is valuable for modeling and interpretation of field seismic data to recover, for instance, information on permeability. Using existing effective medium models for elasticity and permeability in cracked media, we can compare the modeled and measured permeability, P and S wave velocities, and dispersion (between 500 kHz and 0.01 Hz) and infer the evolution of the microstructure with effective pressure or depth in this cracked rock.
2. Material and Methods In its natural state, the CM is an isotropic and homogeneous rock made of 98% calcite with a relatively narrow grain size (diameter) distribution centered around 150 μm, virtually no porosity (< 0.5%), and a random orientation distribution of polyhedral calcite grains [e.g., De Bresser et al., 2005; Delle Piane et al., 2015, and references therein]. Two specimens 38 mm in diameter and 76 mm in length are extracted from the same block (see experimental workflow in Figure 1). The first sample is kept intact and subjected to a petrophysical characterization. The second specimen (TC300) is heated in an oven the temperature of which is regulated at 300°C. For a nominal thermal diffusivity in the CM of d ~ 0.8 106 m2/s and a characteristic diffusion length corresponding to the specimen’s radius (l = 19 mm), the theoretical heat diffusion time within the specimen is t ~ l2/d = 7.5 min.
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Figure 1. Laboratory workflow for characterizing and testing the tight cracked Carrara marble.
To ensure a uniform temperature field within its volume, the specimen is left to equilibrate at 300°C for 2 h. It is then quenched in a tap water tank at room temperature 20°C. The uniformity of the resulting network of microcracks was indirectly assessed through repeated P wave velocity measurements at room conditions conducted at several locations in specimen, i.e., along the specimen axis and across its diameter at several heights. Within an experimental uncertainty of few percents, no significant heterogeneity and no anisotropy were observed. Furthermore, this uniformity was directly assessed through microstructural imaging as reported partly in Delle Piane et al. [2015] and more extensively in Cazes [2012] (i.e., two-dimensional electron microscopy on thin sections and three-dimensional X-ray imaging). Both specimens are subsequently subjected to a petrophysical characterization involving the following steps (see experimental workflow in Figure 1): (i) drying in a vacuumed oven at 105°C for 48 h, then slow cooling at room conditions (20°C and 50% relative humidity) for several days; (ii) measurement of helium porosity and permeability at several effective pressures up to 35 MPa in dry conditions using an automated gas porosimeter-permeameter relying on the pulse decay method (Coretest AP-608). Note that a Klinkenberg correction was implemented to account for gas slippage at the pore walls; (iii) saturation with a water pressure of 2 MPa; and (iv) nuclear magnetic resonance (NMR) spectroscopy using an Oxford Maran Ultraspectrometer operating at 2 MHz to characterize the relative size distribution of fluid-filled pores/cracks. Subsequently, the thermally cracked specimen TC300 is dried in a vacuumed oven at 105°C for 48 h, then allowed to slowly cool down at room conditions (20°C and 50% relative humidity) for several days. It is instrumented with high-sensitivity semiconductor (high gage factor) strain gages glued to its lateral surface at midheight (see Figure 2). It is finally set up in the triaxial stress vessel for further experimentation at variable effective pressure. In order to minimize possible hysteresis effects, the TC300 specimen in its dry state is first subjected to a preliminary continuous confining pressure cycle up to 35 MPa and down to 2 MPa at a rate of 0.1 MPa/min. It is then subjected to two consecutive pressure cycles up to a maximum effective pressure of 50 MPa, corresponding to a depth of about 4 km in a normally pressurized subsurface environment. During the first cycle, the specimen TC300 is dry and the confining pressure is increased stepwise up to 50 MPa at a rate of 0.1 MPa/min then decreased to and maintained at 2 MPa. The specimen is then
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Figure 2. Carrara marble specimen TC300 and laboratory equipment: (a) triaxial stress vessel (autonomous triaxial cell), (b) thermally quenched Carrara marble specimen, (c) internal diagram of the autonomous triaxial cell, and (d) schematic of the rock specimen instrumentation and pore/confining fluids and corresponding pumps. The pore fluid valves are open during the stepwise effective pressure loading (fully drained) and closed during the oscillating stress probing test conducted at each effective pressure step.
saturated in situ (within the triaxial stress vessel) by injecting water at a pressure of 1 MPa. To achieve optimal water saturation, several pore volumes of water are flushed through the specimen from its bottom end while the top end pore line is open to atmosphere (Figure 2). The top pore line is then connected to a pore fluid pump, and water pressure is regulated at 1 MPa at both ends of the specimen. During the second cycle, each stepwise increase of the confining pressure is applied at a rate of 0.1 MPa/min, while the pore fluid pumps are regulating at 1 MPa. In addition, at the end of each pressurization step, the pore pressure within the specimen is allowed to equilibrate to ensure fully drained conditions, i.e., until the displacement of the regulating pore fluid pumps vanishes (up to an hour).
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Figure 3. (a) Stress path and (b) pore pressure response during the oscillating stress probing of the water-saturated TC300 specimen. At low pressure, pore pressure oscillations are recorded during the oscillating stress probing test; above ~30 MPa effective pressure, these oscillations vanish.
At each pressurization step during the two cycles, low-amplitude–high-frequency (HF, 0.5 MHz) and low-amplitude–low-frequency (LF, 0.01 Hz) elastic moduli are estimated. The HF moduli are derived from P and S wave velocities measured along the specimen’s axis using the pulse transmission technique [Birch, 1960]. The LF moduli are derived from the measurement of axial and radial strains during a specific “oscillating stress probing” test (Figure 3a). This test consists of applying a small oscillating axial stress on the specimen at a constant confining pressure, while the pore fluid lines are closed and the fluid pressure at the specimen’s ends is monitored (Figure 3b). First, a static differential stress of 1 MPa is applied on the specimen to ensure optimal mechanical coupling between the actuator and the specimen. An additional oscillating stress is then applied at a frequency of 0.01 Hz so that the specimen experiences a total oscillating differential 5 stress of 1 ± 0.5 MPa (Figure 3). The small (εax max ≤10 ) axial and radial strains resulting from this oscillating stress probing are measured, thanks to the high-sensitivity semiconductor strain gages attached to the specimen (Figure 2).
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Figure 4. (top left) 2-D electron microscope images acquired in backscattered electron mode of a thin section extracted from the intact Carrara marble. (top right) 3-D X-ray micro-CT image of a millimetre-size sample extracted from the TC300 specimen after microcrack recognition and labeling [see Delle Piane et al., 2015, and references therein]. Microcracks in the TC300 sample are located mainly at the interface between polyhedral calcite grains. (bottom) T2 relaxation time distribution obtained from 2 MHz NMR scanning. Water has been injected into both samples with a maximum pressure of 2 MPa prior to NMR scanning. Note that 1/T2 is proportional to the surface-to-volume (S/V) ratio of water-filled pores.
3. Results 3.1. Porosity and Permeability The NMR data (graph in Figure 4) suggest that a single family of fluid-filled micropores exists in the intact rock and that a second family of larger “cavities” (larger volume-to-surface ratio associated with a larger T2 relaxation time) appears in the TC300 sample, which is confidently attributed to microcracks generated by thermal quenching. These microcracks are further evidenced by the color-coded X-ray micro–computed tomography (CT) image of the TC300 sample: they appear to occur at the junction between polyhedral calcite grains (color-coded void space in Figure 4). This suggests the conceptual model of an isotropic solid embedding randomly located and oriented, possibly intersecting microcracks. The combined analysis of the NMR and X-ray data suggests that the thermally induced cracks are commensurate with the grain size, whereas the microvoids originally present in the natural rock are smaller (V/S ratio) by at least 2 orders of magnitude as indicated by the T2 relaxation time. The TC300 sample is significantly more permeable than the intact sample. For both specimens, permeability decreases with increasing effective pressure and tends toward a similar asymptotic, quasi-null value at high effective pressure (Figure 5b). The contrast in porosity between the two samples is, however, much less pronounced, and for both, the connected porosity remains lower than 0.5% (Figure 5a). Porosity decreases with increasing effective pressure and tends toward a similar asymptotic, quasi-null value at high effective pressure. The magnitude of volumetric strain recorded upon pressure increase is consistent with the
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Figure 5. (a) Porosity and volumetric strain and (b) helium permeability of the intact and thermally cracked samples as a function of effective pressure. (c) Correlation between permeability and porosity.
concomitant decrease in porosity (Figure 5a). Figure 5c shows the evolution of the permeability as a function of the stress-sensitive porosity for the intact and the thermally cracked specimens. There is a linear dependence of permeability on the stress-sensitive porosity for both specimens, which suggests that no significant percolation effects are to be expected up to 50 MPa effective pressure. The data reported in Figure 5 are also tabulated in Table 1. Based on these experimental data, the pressure dependences of porosity and permeability of the intact and thermally cracked specimens are empirically approximated by ϕ ∼ 0:37e0:051Peff where r 2 > 0:97 k ∼ 2e0:056Peff k ∼ 4ϕ 510
2
where r 2 > 0:97
;
(1)
where r > 0:99 2
Table 1. Experimental Data: Gas Porosity and Permeability as a Function of Effective Pressure for the Intact and Thermally Cracked (TC300) Carrara Marble (see Figure 5) Intact Effective pressure (MPa) 2.0 5.5 12.4 19.3 26.2 33.1
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Thermally Cracked (TC300)
Porosity (%)
Permeability (μD)
0.38 0.29 0.17 0.12 0.10 0.08
1.60 1.20 0.69 0.46 0.40 0.26
Effective Pressure (MPa)
Porosity (%)
Permeability (μD)
2.0 5.5 12.4 22.8 33.1
0.45 0.36 0.25 0.13 0.09
4.85 3.74 2.53 1.52 0.88
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for the intact rock and ϕ ∼ 0:48e0:051Peff k ∼ 5e0:054Peff
where r 2 > 0:99 where r 2 > 0:99
2
k ∼ 10:8ϕ 610
;
(2)
where r 2 > 0:99
for the thermally cracked rock. In the above empirical correlations effective pressure is in megapascal (MPa), porosity is in percent (%), and permeability is in micro-Darcy (μD). 3.2. Elastic Moduli In the following, the letters HF and LF refer to high frequency (500 kHz) and low (0.01 Hz) frequency, respectively, and the letters SAT and DRY refer to the water-saturated and dry states of the rock, respectively. At each effective pressure stage, the HF bulk and shear moduli in dry and water-saturated conditions are obtained from the P and S wave velocities and corresponding mass density data [Cazes, 2012] K ¼ ρrock
V 2P
4 2 VS ; 3
G ¼ ρrock V 2S ;
where ρrock stands for the mass density of the dry or water-saturated rock. Note that for the calculation of the elastic moduli under varying effective pressure the dry and water-saturated mass densities determined at room conditions are corrected for the measured deformation of the specimen. At each effective pressure stage, the LF Young’s modulus E and Poisson’s ratio ν in dry and water-saturated conditions are obtained from the stress (axial) and strains (axial and radial) measured during the oscillating stress probing test. The average axial and radial strains recorded by the two pairs of strain gages are used for this calculation (see Figure 2). The equivalent bulk and shear moduli are obtained using conventional elasticity equations for isotropic elastic solids [e,g, Mavko et al., 2009] K¼
E ; 3ð1 2v Þ
G¼
E : 2ð1 þ v Þ
Figure 6 and Table 2 report the evolution of high- and low-frequency bulk and shear moduli of the thermally cracked sample TC300 in dry and water-saturated conditions. Except the HF bulk modulus of the watersaturated rock, which remains relatively insensitive to pressure, all other moduli increase with increasing effective pressure and seem to reach a plateau beyond a threshold pressure of ~30 MPa. This suggests microcrack closure as a plausible underlying mechanism. The asymptotic values reached by these moduli are consistent with the values reported for pure calcite in the literature, i.e., Kcalcite ~ 80 GPa and Gcalcite ~ 30 GPa [Mavko et al., 2009]. Comparing Figures 6a and 6b, we observe the following: 1. In dry conditions, at low effective pressures, KDRY HF ~ KDRY LF and GDRY HF ~ GDRY LF, suggesting that there is negligible wave dispersion between 0.01 Hz and 0.5 MHz in the dry cracked rock. 2. In water-saturated conditions, at low effective pressure, KSAT HF > KSAT LF and GSAT HF > GSAT LF, suggesting that fluid-induced wave dispersion exists in the water-saturated cracked rock. 3. In water-saturated conditions, at high effective pressure, GSAT HF ~ GSAT LF, suggesting that there is negligible shear wave dispersion between 0.01 Hz and 0.5 MHz, which is consistent with the expected crack closure occurring with increasing effective pressure; 4. At high effective pressure, KDRY HF < KDRY LF and KSAT HF < KSAT LF, which is an unexpected result. The LF moduli appear higher than the HF moduli. This peculiarity is possibly due to an experimental bias in the determination of the LF bulk moduli at high effective pressure; i.e., the semiconductor strain gages were probably deformed beyond their linear range limit when effective pressure reached 40 MPa or more so that the recorded strains appeared smaller than the actual deformation experienced by the rock specimen. Comparing Figures 6c and 6d, we observe the following: 1. At high frequency (500 kHz) and low effective pressure, KSAT HF > KDRY HF and GSAT HF > GDRY HF, suggesting that the fluid present in the microcracks contributes to the overall stiffness of the rock; i.e., the fluid does not have the time to be fully displaced by the propagating ultrasonic waves.
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Figure 6. Low-amplitude–high-frequency (HF, 500 kHz) elastic moduli and low-amplitude–low-frequency (LF, 0.01 Hz) elastic moduli of the thermally cracked sample TC300 in dry and water-saturated conditions. (a) Comparison of highand low-frequency moduli in dry conditions. (b) Comparison of high- and low-frequency moduli in water-saturated conditions. (c) Comparison of dry and water-saturated moduli at high frequency. (d) Comparison of dry and water-saturated moduli at low frequency.
Table 2. Experimental Data: Elastic Moduli K and G as a Function of Effective Pressure in the Thermally Cracked Sample TC300: High Frequency (HF, 500 kHz), Low Frequency (LF, 0.01 Hz), Dry Rock (DRY), and Water-Saturated Rock (SAT) (see Figure 6) Dry (TC300) Effective Pressure (MPa) 1.5 5 10 20 30 40 50
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Water-Saturated (TC300)
HF (500 kHz) K
DRY HF
DRY HF
G
LF (0.01 Hz) DRY LF
K
DRY LF
G
(GPa)
(GPa)
(GPa)
(GPa)
15.6 35.3 63.6 67.7 76.1 75.8 75.5
11.3 12.6 23.2 25.0 25.7 25.9 26.1
7.0 18.8 38.1 59.5 72.1 83.2 92.4
6.4 12.7 18.2 23.0 24.1 25.6 26.6
Effective Pressure (MPa) 1 5 10 20 30 40 50
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HF (500 kHz) K
SAT HF
G
SAT HF
LF (0.01 Hz) K
SAT LF
SAT LF
G
(GPa)
(GPa)
(GPa)
(GPa)
74.2 73.5 78.3 76.6 78.0 75.8 77.4
19.3 23.8 24.3 25.6 26.0 26.2 26.5
21.4 38.9 61.4 82.1 83.7 93.3 94.4
9.4 15.8 19.3 24.0 25.8 26.6 27.6
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Figure 7. Biot-Gassmann predictions for (a) bulk and (b) shear moduli K and G respectively for the thermally cracked sample TC300.
2. At high frequency (500 kHz) and high effective pressure, KSAT HF ~ K DRY HF and GSAT HF ~ GDRY HF, consistent with microcrack closure with increasing effective pressure beyond ~30 MPa. 3. At low frequency (0.01 Hz) and low effective pressure, KSAT LF > K DRY LF and GSAT LF > GDRY LF, suggesting that the stress probing induces an oscillating deformation of the rock specimen that is at least partially undrained. This is confirmed by the fluid pressure oscillations recorded at the specimen’s end faces and associated with the oscillating stress probing for effective pressure below ~30 MPa (see top plot in Figure 3b). Note that for a fully drained rock KSAT LF = KDRY LF. 4. At low frequency (0.01 Hz) and high effective pressures, KSAT LF → KDRY LF and GSAT LF → GDRY LF, which is consistent with the expected microcrack closure beyond ~30 MPa effective pressure. Overall, the picture that emerges from these observations is that at low effective pressure the fluid has time to be displaced, although probably not fully drained, during the oscillating stress probing at 0.01 Hz (LF moduli). The higher the effective pressure, the more microcracks close, the less the fluid has time to drain (flow) within and out of the rock because of the concomitant permeability decrease. 3.3. Biot-Gassmann Predictions and Wave Dispersion Biot-Gassmann predictions [Biot, 1941; Gassmann, 1951] for the water-saturated rock in the undrained regime are computed from the dry moduli, corresponding in the poroelastic context to the drained moduli. BiotGassmann equations predict that the shear modulus G is not affected by the presence of fluid. However, the undrained bulk K modulus of the water-saturated rock is expected to be larger than its dry (or drained) counterpart because water is stiffer than air. For the dry specimen TC300, we have observed in Figure 6a that wave dispersion is negligible; i.e., the magnitude and pressure sensitivity of the HF and LF moduli are similar in that case. Therefore, Biot-Gassmann undrained moduli stemming from these two sets of dry rock data (HF and LF) are also similar in magnitude and pressure sensitivity. At low effective pressure, Biot-Gassmann predictions for the bulk modulus K are lower than the bulk modulus under water-saturated conditions obtained at high frequency (HF, 0.5 MHz) (Figure 7a). Qualitatively similar observations can be made for the shear modulus G (Figure 7b). This suggests that Biot-Gassmann equations do not predict adequately the HF moduli derived from ultrasonic wave velocities on the water-saturated cracked rock at low effective pressure. Biot-Gassmann undrained moduli derived from dry rock data constitute the relaxed LF moduli, and at low effective pressure (open microcracks), Biot-Gassmann undrained moduli are lower than the HF moduli derived from ultrasonic wave velocities. This suggests that fluid-induced wave dispersion (e.g., squirt-flow) exists between 0.01 and 500 kHz. In contrast, at higher effective pressure, Biot-Gassmann predictions for each type of modulus (K or G) derived from HF and LF dry rock data are commensurate with the measured HF water-saturated rock data. This is again consistent
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Figure 8. Experimentally determined P and S wave dispersions between 0.01 Hz and 500 kHz in the thermally cracked sample TC300: (a) as a function of effective pressure and (b) as a function of crack porosity.
with the closure of microcracks with increasing effective pressure, i.e., reduction of the porosity to the extent that its impact on all measured moduli becomes negligible. At each effective pressure step, wave dispersion for each wave mode (P or S) is estimated from the experiments as the relative difference between the HF water-saturated rock data and the Biot-Gassmann LF predictions derived from the dry rock data. Because the two sets of dry rock moduli (HF and LF) are similar in magnitude and pressure sensitivity, their average is used to compute the Biot-Gassmann LF undrained moduli. For comparison purposes, the difference between HF and LF velocities is normalized by the corresponding wave velocity in pure calcite derived from Kcalcite ~ 80 GPa and Gcalcite ~ 30 GPa [Mavko et al., 2009] ΔV ¼
V SAT HF V SAT LF ; V calcite
(3)
where V stands for the measured P or S wave velocity in the water-saturated specimen TC300 at high (500 kHz) and low frequency (0.01 Hz). The maximum S wave dispersion between 0.01 Hz and 500 kHz occurs at the lowest effective pressure, or highest porosity, and is ~ 26% (Figure 8). It is significantly larger than the maximum P wave dispersion of ~9% in this water-saturated cracked rock (Figure 8). Both P and S wave dispersions decrease monotonically with increasing effective pressure (or crack porosity reduction) toward a negligible value above ~30 MPa. This is again consistent with the expected closure of microcracks. The observed dispersion can therefore be confidently attributed to the presence of water-filled microcracks. The remainder of this article is dedicated to the modelling of the reported experimental data, that is: 1. HF moduli for the water-saturated cracked rock are modeled using effective medium theory. The model predictions are compared to the experimentally determined HF moduli reported in Figure 6. 2. Wave dispersion predictions are computed and compared to the experimentally determined wave dispersion reported in Figure 8. 3. Permeability of the cracked rock is also modeled using effective medium theory since the observed dependency of permeability on porosity appears to be linear (see Figure 5), suggesting that no connectivity reduction occurs in this effective pressure range (far above the percolation threshold). The theoretical permeability is compared to the experimentally determined one.
4. Modeling Wave Velocity/Dispersion and Permeability 4.1. Conceptual Model of the Rock Microstructure Based on the data reported so far, the thermally cracked CM is modeled as a homogeneous and isotropic elastic solid (aggregate of randomly oriented and naturally fused polyhedral calcite grains), embedding
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Figure 9. Conceptual model of the rock microstructure of the thermally cracked sample TC300: isotropic distribution of identical cracks embedded in an isotropic elastic solid.
ϕ ¼ 1 eV C :NV ;
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randomly oriented and spaced microcracks having a finite diameter 2R and aperture 2w (Figure 9). The number of microcracks per unit volume is NV, and their aspect ratio is ξ = w/R. For sake of simplicity, the microcracks are modeled as oblate spheroids (ξ < 1). This same model is used for predicting both (i) the HF elastic moduli using an effective medium model in the framework of the non-interaction approximation [Guéguen and Sarout, 2011] and (ii) the permeability of the rock away from the percolation threshold using the concept of hydraulic radius [Sarout, 2012]. In this model, the spheroidal microcracks with ξ < 1 are allowed to overlap/intersect so that a hydraulic connectivity is possible and fluid can flow through the rock macroscopically. The porosity of such a medium reads [Garboczi et al., 1995] (4)
where NV is the number of microcracks per unit volume and VC is the volume of a single spheroidal microcrack 4 V c ¼ πξR3 ; 3
(5)
In this context, the crack density ρ = NV R3 [Walsh, 1965] reads ρ¼
3 logð1 ϕ Þ: 4πξ
(6)
The detailed modeling and inversion workflow are illustrated in Figure 10. The HF and LF dry rock moduli being similar, their average is used as input in Biot-Gassmann equations to compute the LF undrained moduli. However, if only one of these two data sets was available, it could have been used instead for this purpose. The HF water-saturated moduli are modeled using the non-interaction approximation, which is suitable for cracked media [e.g., Grechka and Kachanov, 2006a, 2006b]. This model yields the bulk and shear moduli as a function of the crack density ρ and the aspect ratio ξ, or equivalently, as a function of the porosity ϕ and the microcracks aspect ratio ξ. This is because ϕ and ρ are related once the geometry of the cracks is set in the microstructural model (oblate spheroids with ξ < 1). At each effective pressure stage, the porosity is independently known, so is the crack density, thanks to equation (6). Therefore, a value of the single unknown model parameter ξ is determined by fitting simultaneously the experimentally determined KSAT HF and GSAT HF moduli with the effective elasticity model. The uniqueness of the numerical solution ξ i of the inverse problem at each effective pressure step i is verified in the porosity interval of interest. Similarly, the permeability of the rock is modeled using the concept of hydraulic radius, which is suitable away from the percolation threshold. This model yields the permeability of the cracked rock as a function of the porosity ϕ, the cracks aspect ratio ξ, and the cracks radius R. At each effective pressure step, the corresponding values of ξ are known from the inversion of the elastic moduli, so that a value of R is determined by fitting the modeled permeability to the measured one. The uniqueness of the numerical solution Ri of the inverse problem at each effective pressure step i is also verified in the porosity interval of interest. 4.2. Elasticity Model and Data Inversion The HF elastic moduli of a medium containing fluid-filled randomly spaced and oriented microcracks with identical aspect ratio (Figure 9) are modeled in the long wavelength approximation using the non-interaction scheme originally introduced by Kachanov [1993] and Sayers and Kachanov [1995], for dry cracks and
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Figure 10. Modeling and data interpretation workflow for tight cracked rocks applied to the thermally cracked sample TC300.
extended by Shafiro and Kachanov [1997] to fluid-saturated cracks. Schubnel and Guéguen [2003] and Guéguen and Sarout [2011] used this approach to predict wave dispersion in cracked rocks with various crack orientation distributions. This approach is based on the excess elastic compliance induced by stresssensitive microcracks embedded in an elastic solid. Guéguen and Sarout [2011] have provided a convenient set of equations describing the model approximated to the first order in aspect ratio ξ for the total compliance of a transversely isotropic elastic medium in which the cracks are the sole source of anisotropy 1 þ h½α11 þ ψ s β1111 E0 1 HF þ h½α33 þ ψ s β3333 SSAT 3333 ¼ E0 v0 HF SSAT þ hψ s β1133 1133 ¼ E0 1 þ vo α11 β1111 HF SSAT ¼ þ h þ ψ s 1212 2E 0 2 3 hα þ α i 1 þ v 0 11 33 HF SSAT þh þ ψ s β1133 1313 ¼ 2E 0 4
HF SSAT 1111 ¼
(7)
where αij and βijkl are the second- and fourth-rank crack density tensors introduced by Sayers and Kachanov [1995]; E0 and ν0 are the Young’s modulus and Poisson’s ratio of the background rock, i.e., assumed to be pure calcite in the CM, and where 32 1 v 20 h¼ ; (8) 3ð2 v 0 ÞE 0 with
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v 0 δf 1 ψS 1 2 1 þ δf : πξ E0 3 ð 1 2v Þ δf ¼ 0 4 1 v 20 K f
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Note that the aspect ratio of the cracks appears in these equations only in the parameter δf. If the bulk modulus of the saturating fluid Kf goes to zero (dry cracks), then δf goes to infinity, and ψ s goes to 1. Therefore, within the first-order approximation in ξ, the effective elastic moduli of a dry cracked medium do not depend on the aspect ratio of the cracks. We specialize these general equations to the case of an isotropic elastic medium embedding randomly oriented microcracks. In this simpler case the components of the secondand fourth-rank crack density tensors, αij and βijkl, reduce to α11 ¼ α33 ¼ ρ ¼ NV R3 ρ β1111 ¼ β3333 ¼ ; 5 ρ β1133 ¼ 15
(10)
where ρ is the conventional crack density parameter originally introduced by Walsh [1965] and reported in equation (6) for spheroidal cracks. Using equations (6), (8), (9), and (10) and converting the compliances in equation (7) into stiffnesses yields 4 10πξE 0 πξE 0 þ MSAT HF ðϕ; ξ Þ ¼ 3 20πξ ð1 þ v 0 Þ hE 0 ð5 þ 2ψ S Þ logð1 ϕ Þ 4πξ ð1 2v 0 Þ hE 0 ð1 þ ψ S Þ logð1 ϕ Þ 4 ¼ K SAT HF ðϕ; ξ Þ þ GSAT HF ðϕ; ξ Þ 3 10πξE 0 GSAT HF ðϕ; ξ Þ ¼ 20πξ ð1 þ v 0 Þ hE 0 ð5 þ 2ψ S Þ logð1 ϕ Þ
(11)
where M = K + 4 G/3 is the so-called P wave modulus. Therefore, the P and S wave velocities read rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . ffi . SAT HF GSAT HF ðϕ;ξ Þ HF MSAT HF ðϕ;ξ Þ ; V ð ϕ; ξ Þ ¼ ð ϕ; ξ Þ ¼ V SAT S P ρsat
ρsat
(12)
where ρsat = (1 ϕ) ρsolid + ϕ ρfluid is the known density of the fluid-saturated rock. Note that the crack density is the product of the number density of microcracks NV by their cubed radius R3. As such, data inversion to recover separate information about the evolution of NV and R with effective pressure suffers from an inherent ambiguity: the respective contributions of NV and R to the measurable variations of elastic moduli with pressure/depth remain unknown. In addition, in the case of water-saturated cracks, the effective elastic moduli depend on the aspect ratio and, in fact, on the product of the crack density and the crack aspect ratio (see equations (7), (9), and (10)). Therefore, in the water-saturated case, data inversion to recover separate information about the evolution of R and ξ suffers a similar ambiguity. A two-step data inversion strategy is adopted so that porosity, elasticity, and permeability data are combined to better constrain the evolution of the microstructural parameters; i.e., we first recover ρ and ξ from elasticity and porosity data; then, using this information as input, we recover R from the permeability data. For each effective pressure (or porosity) step i, the effective elastic moduli (equation (11)) are compared to the experimental HF moduli derived from the measured ultrasonic wave velocities at that particular step i. For a given (and known) porosity ϕ i at that step i, the only unknown parameter of the model is the aspect ratio of the cracks ξ at that step. The CM being made of 98% calcite, we assume that the elastic properties of the solid phase are known: Kcalcite ~ 80 GPa and Gcalcite ~ 30 GPa. For each step i, we obtain numerically a unique value of ξ in the range of interest ξ < 101 by solving the inverse problem simultaneously for the bulk and shear moduli. This is practically carried out at each step i by minimizing a cost function quantifying the least squares distance between the experimental and theoretical elastic moduli K and G, i.e., qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi SAT HF SAT HF ffi SAT HF HF 2 Ji ¼ K ðϕ i ; ξ i Þ K SAT þ G ð ϕ ; ξ Þ G : (13) i i EXP EXP The resulting non-monotonic evolution of the aspect ratio ξ with effective pressure is shown in Figure 11a: ξ varies slightly in a plausible range of 103 to 4.103. This figure also shows the evolution with pressure of the corresponding crack density ρ derived using equation (6) and the already known porosity ϕ. Crack density decreases monotonically with decreasing porosity to reach an asymptotic value ρmin ~ 0.14. Note that the percolation threshold for a cracked medium is theoretically predicted to occur at a crack density ρpercolation ~ 0.13 [e.g., Guéguen and Dienes, 1989; Berkowitz and Balberg, 1993; Garboczi et al., 1995; SAROUT ET AL.
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Benson et al., 2006; Pervukhina and Kuwahara, 2008; Sarout, 2012]. The percolation threshold corresponds to the point at which the connectivity of the crack network, and therefore the permeability, vanishes, whereas the porosity is not zero. On the other hand, as expected, the maximum value of the crack density is obtained at the lowest effective pressure or highest porosity ρmax ~ 0.51. Figure 11b compares the measured and the modeled wave velocities in the watersaturated rock in the HF unrelaxed regime. The effective medium model developed with the non-interaction approximation fits reasonably well the HF laboratory data at all pressures, despite the relatively high value the crack density parameter reaches at low effective pressure (ρmax ~ 0.51 > 0.1). Figure 11c compares the experimental and modeled fluid-induced dispersion of P and S wave velocities between 500 kHz (HF unrelaxed regime) and 0.01 Hz (LF Biot-Gassmann undrained regime). 4.3. Cutoff Frequencies and Model Validation Biot’s cutoff frequency fB separates two poroelastic regimes: (i) a quasi-static regime dominated by global viscous flow (drained and undrained regimes) and (ii) a dynamic regime dominated by inertial drag between the solid and fluid phases (inertial regime)). It is a function of the porosity ϕ and permeability k of the medium, and the fluid density ρfluid and viscosity η fb ¼
ηϕ : 2πkρfluid
(14)
The drainage cutoff frequency is the frequency separating the drained from the undrained regimes in a fluid-saturated poroelastic medium. It is based on the concept of fluid diffusion and the associated characteristic time/length. Fluid drainage in the saturated cracked rock in the radial direction is inhibited by the Viton sleeve Figure 11. (a) Crack density ρ and effective aspect ratio ξ, (b) that surrounds the rock specimen. Let us therequality of fit for P and S wave velocities and dispersions fore assume that the fluid within the specimen between 0.01 Hz and 500 kHz (Figure 11b) in the thermally cracked sample TC300. drains only toward the two ends of the specimen, into the dead volumes associated with the pore fluid lines/valves (Figure 2d). In poroelasticity, the diffusion coefficient is related to the permeability of the rock k, to the undrained bulk modulus KU, to Biot and Skempton coefficients b and B, and to the fluid viscosity η D∼
k KU B : b η
(15)
Unfortunately, no information is available on the values of Biot and Skempton coefficients b and B or on their evolution with effective pressure in this cracked rock TC300. The characteristic diffusion time τ D, or its inverse,
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Figure 12. Cutoff frequencies derived for the thermally cracked sample TC300: (a) as a function of effective pressure and (b) as a function of crack porosity.
the characteristic diffusion frequency fD is related to the diffusion length, taken to be half the specimen’s length L/2, and to the diffusion coefficient D [Adelinet et al., 2010] so that fD ∼ γ
4k K U μL2
(16)
where γ = B/b is unknown. The undrained bulk modulus is calculated from the average of LF dry and HF dry moduli using Biot-Gassmann equations. Because γ is unknown for the thermally cracked CM, fD cannot be determined explicitly. However, the LF moduli measured on the water-saturated specimen exhibit a transition at ~30 MPa effective pressure from a semidrained regime (low effective pressure) to an undrained regime (high effective pressure). This is supported by the fact that beyond ~30 MPa effective pressure, the oscillating stress probing does not generate a significant fluid pressure oscillation in the (closed) pore fluid lines. Taking a pragmatic approach for estimating the evolution of the drainage frequency with effective pressure or porosity, we empirically determine a reasonable value of γ, so that the drained-toundrained transition effectively occurs at ~30 MPa effective pressure. In other words, at an effective pressure of ~30 MPa, we infer that fD ~ 0.01 Hz and find that γ ~ 5.7 for a porosity of ϕ ~ 0.1% and a permeability of k ~ 0.001 mD. The squirt-flow cutoff frequency fSQ is the frequency at which fluid-induced energy dissipation at the microcracks scale (squirt-flow between neighbor microcracks) is maximal. It is related to the Young’s modulus E0 of the background solid (fused polyhedral calcite grains), the aspect ratio of the cracks ξ, and the fluid viscosity η [Le Ravalec et al., 1996; Adelinet et al., 2011] f SQ ∼
E0 3 ξ 24η
(17)
This frequency separates the relaxed (f < fSQ) from the unrelaxed (f > fSQ) deformation regimes. Using the results of the microstructural parameters identification obtained in the previous section, these cutoff frequencies are estimated and plotted as a function of effective pressure or porosity (Figure 12). For comparison purposes, the frequency of the ultrasonic probing (~500 MHz) and that of the oscillating stress probing (0.01 Hz) are also reported in this figure. We observe that the ultrasonic probing frequency (~500 MHz) is always larger than the squirt-flow cutoff frequency in the range of effective pressure or porosity experienced by the cracked CM. This confirms that ultrasonic wave propagation occurs in the unrelaxed regime. Biot cutoff frequency is so large (~107 kHz) that it is irrelevant for the problem at hand. Finally, as discussed above, the drainage cutoff frequency is constructed by setting γ = B/b ~ 5.7 so that the oscillating stress probing of the water-saturated specimen is drained below ~30 MPa effective pressure and undrained above that value. Because of this transitional behavior, and the disturbance induced by the dead volumes in the pore fluid lines when the valves are closed, the LF moduli in the water-saturated cracked CM are disregarded from now on.
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4.4. Permeability Model and Data Inversion A simple permeability model dedicated to cracked media was reported by Sarout [2012]. This model accounts for the presence of identical randomly oriented, spaced, and possibly intersecting spheroidal microcracks (Figure 9). A simplified version of this model applies above the percolation threshold so that the crack network is fully connected. This modeling choice for the CM is supported by the observation that in this rock (i) the permeability evolves linearly with porosity and (ii) the minimum crack density obtained at the highest effective pressure (or lowest porosity) remains higher than the crack density predicted at the percolation threshold in a cracked medium (ρmin > ρpercolation). Let us therefore assume that the network of microcracks in the thermally cracked CM remains in the fully connected regime for all effective pressures tested. In this regime, the permeability of the rock is solely due to a fully connected network of spheroidal microcracks and can be modeled using the concept of hydraulic radius [Guéguen and Dienes, 1989] k ∼ αϕm2
(18)
where m = VC/SC is the hydraulic radius of the spheroidal microcracks defined as their volume-to-surface ratio and α is a dimensionless parameter derived from Poiseuille’s law and related to the geometry of the hydraulically conducting network of microcracks. It is on the order of α ~ 1/3 for a network of spheroidal microcracks [Sarout, 2012]. The permeability of this fully connected cracked medium is explicitly related to its microstructural parameters by [Sarout, 2012] ϕR2i ξ 2i 1 ξ 2i 16 k ðϕ i ; ξ i ; Ri Þ ∼ (19) pffiffiffiffiffiffiffiffi2 27 qffiffiffiffiffiffiffiffiffiffiffiffiffi2 2ξ 2i þ2 1ξ 2i 2 1 ξ i þ ξ 2i log 2 ξ i
Note that for each effective pressure (or porosity) step i, the only remaining unknown microstructural parameter is the crack’s radius Ri, since ξ i is known from the inversion of the elasticity data in the previous section. At each effective pressure, equation (19) is fitted to the corresponding porosity-permeability data point (Figure 5c), using the corresponding value of aspect ratio ξ i obtained earlier (Figure 11a). At each step i, a unique value of crack radius Ri is numerically recovered in the range of interest R = 106 to 103 m (1 μm to 1 mm). The resulting non-monotonic evolution of the radius Ri with effective pressure is shown in Figure 13a: R is found to vary slightly in a plausible range 20 to 124 μm. Figure 13b compares the measured and the modeled permeability of the cracked rock in the fully connected regime using the recovered microstructural parameters. The model fits very well the experimental permeability data.
5. Discussion The microstructural analysis of the thermally cracked CM suggests that cracking occurs at the boundary between the originally fused polyhedral calcite grains (Figure 4, top right). These cracks are due to temperature-induced strain incompatibilities at these locations associated with the anisotropic nature of the thermal expansion tensor of calcite, which is collinear with the elastic tensor of this mineral (crystallographic directions). 5.1. Impact of Pressure on Crack Geometry Figure 11a shows that the effective crack density decreases monotonically with decreasing crack porosity (or increasing effective pressure), which is a well-established result for rocks often reported in the literature [e.g., Mavko et al., 2009]. This is due to the compliant or stress-sensitive and nonspherical geometry of cracks. This figure also shows that in contrast to the crack density, the effective crack aspect ratio has a non-monotonic evolution, exhibiting a maximum value ξ max ~ 0.004 at a porosity of ϕ critical ~ 0.3% (or an effective pressure Peff ~ 10 MPa). Figure 13a shows that the effective crack radius also evolves nonmonotonically, exhibiting a minimum value Rmin ~ 20 μm at a similar porosity ϕ critical ~ 0.3% (or effective pressure Peff ~ 10 MPa). These maximum and minimum values ξ max and Rmin occur in the early stages of the effective pressure loading (Peff ~ 10 MPa). Above this critical effective pressure, the crack aspect ratio decreases, whereas the crack radius increases. Because the aspect ratio is also a function of the crack radius, and in order to clarify the evolution of the effective geometry of the cracks, we compute the corresponding evolutions of the crack aperture 2w = 2ξR. The crack diameter 2R and aperture 2w are two independent quantifiers of the crack geometry. In addition, using the definition of the crack density, the number of cracks per unit volume NV SAROUT ET AL.
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Figure 13. (a) Effective radius R and (b) quality of fit for permeability in the thermally cracked sample TC300.
can be inferred from the knowledge of the crack radius R and crack density ρ. Figures 14a and 14b show that the number of cracks per unit volume NV and their effective radius Reff evolve non-monotonically with the effective crack aperture 2w, suggesting that three deformation regimes exist in this tight cracked rock when it is subjected to an increasing effective pressure: 1. When Peff < 10 MPa the number of cracks increases, their radius decreases, their aspect ratio increases, and their aperture remains largely unaffected. 2. When Peff > 10 MPa, the number of cracks decreases, their radius increases, their aspect ratio decreases, and their aperture decreases. 3. When Peff > 30 MPa, all the cracks are effectively closed. To summarize these findings, a conceptual model of the evolution of the effective geometry of an individual crack with effective pressure is devised (see Figure 15a). A relatively small effective pressure (Peff < 10 MPa) induces a reduction in the effective crack diameter at constant aperture. In contrast, the application of a larger effective pressure (Peff > 10 MPa) induces primarily a reduction of the effective crack aperture while the crack radius increases. Finally, as noted earlier, above ~30 MPa effective pressure, from an elasticity and permeability standpoint, the rock behaves essentially as if the cracks were closed Obviously, this model is way too simplistic compared to actual cracks in natural rocks. Actual crack surfaces are not smooth as they appear in the schematic of Figure 15a. They are rather rough surfaces, with asperities as schematized in Figure 15b [e.g., Kachanov et al., 2010; Sevostianov and Kachanov, 2008; Glubokovskikh et al., 2016; Gao and Gibson, 2012; Fortin et al., 2011]. When a relatively small pressure is applied to such cracks (Peff < 10 MPa), opposing asperities are expected to come to contact first, plausibly inducing a reduction in the effective crack size and an increase in the apparent number of cracks (larger cracks are split into smaller
Figure 14. Cross-plots of crack parameters with increasing effective pressure up to 50 MPa in the thermally cracked sample TC300.
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Figure 15. Conceptual models of the progressive closure of crack in the thermally cracked sample TC300.
ones), with a minimal impact on the crack aperture. At higher pressure (10 MPa < Peff < 30 MPa), the stress resolved at the location of these asperities should increase drastically, in proportion to the inverse of their cross section. The smallest and more compliant asperities should first yield mechanically with increasing stress, leaving the largest and stiffest asperities to sustain the increasingly large compressive stress. When approaching the 30 MPa effective pressure, asperities tend to progressively yield mechanically, leading to a more systematic and pronounced closure of the crack network. Above ~30 MPa confining pressure, elastic wave velocities (or elastic moduli) are virtually unaffected by the residual void space. In contrast, permeability at this stage is still not zero. The overall evolution of the permeability with the microstructural parameters is illustrated in Figure 16. Permeability consistently decreases with decreasing crack density ρ or crack aperture 2w. This monotonic permeability reduction occurs despite the non-monotonic evolution of the number of cracks per unit volume NV or their radius R. 5.2. Comparison With 2-D and 3-D Image Analyses A quantitative microstructural analysis was also conducted on this thermally cracked sample of CM (Figure 4 and Delle Piane et al. [2015]) and used for comparison with the corresponding values inverted from laboratory measurements. Values of crack density ρ, radius R, and aspect ratio ξ at room pressure have been derived from: (i) high resolution 2-D electron microscopy images of thin sections and (ii) 3-D X-ray microtomography images of millimeter-size plugs. After image acquisition, cracks were detected from the images based on a mapping of their contrast in electron scattering and X-ray attenuation with respect to the surrounding calcite matrix. Semi-automatic routines were developed ad hoc by Arena et al. [2014] to segment and separate intersecting cracks, and quantify their geometrical attributes. Details of the imaging techniques and results are provided in Arena et al. [2014] and Delle Piane et al. [2015], respectively (overview in Figure 4, top). Let us compare the crack density ρimg, radius Rimg, and aspect ratios ξ img obtained from 2-D and 3-D quantitative image analyses to the corresponding ρeff, Reff, and ξ eff obtained from the inversion of laboratory data at the lowest effective pressure available in this data set (1 MPa). Overall, the effective microstructural
Figure 16. Evolution of the permeability with the microstructural parameters crack density ρ, effective radius R, effective aperture 2w, and number density NV.
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Table 3. Microstructural Parameters ρ, R, and ξ Derived From Laboratory Measurements at 1 MPa Effective Pressure Using Effective Elasticity/Permeability Models and From 2-D and 3-D Image Analyses of the Same Rock at Room Pressure [see Delle Piane et al., 2015] Crack density ρ () Crack radius R (μm) 3 Crack aspect ratio ξ (× 10 ) Crack aperture 2w = 2 ξ R (nm) Image pixel size (nm)
Data Inversion
2-D imaging
3-D imaging
0.51 41 2.1 168 na
0.84 8.4 ± 5.5 38 ± 27 319 85
1.10 60.4 ± 36.1 24 ± 11 1450 800
parameters derived here can differ significantly from their image-derived counterparts (see Table 3). However, compared to 2-D image analysis, 3-D image analysis yields values of ρimg3-D, Rimg3-D, and ξ img3-D closer to the corresponding effective parameters ρeff, Reff, and ξ eff. The most notable similarity is for the crack radius R: Reff ~ 41 μm versus Rimg3-D = 60 μm. In contrast, the crack density ρimg3-D is twice larger than ρeff and the aspect ratio ξ img3-D is 10 times larger than ξ eff. During the 3-D X-ray micro-CT imaging, the sample was at room pressure, whereas ρeff, Reff, and ξ eff were derived for the rock under 1 MPa effective pressure, the lowest available. This difference in stress state of the rock could explain part of the observed discrepancy in the magnitude of the microstructural parameters. At room pressure (zero effective pressure), the crack density could be higher than ρeff = 0.51 due to the presence of more open cracks; but would it reach ρimg3-D = 1.1? The answer to this question remains unknown in view of the data available. Even if such a benign difference in pressure is the cause of the observed discrepancies in ρ, R, and ξ (Table 3), larger effective pressures typically encountered at depth in the Earth’s crust will lead to an even more significant discrepancy with the image analysis results obtained on a rock at room pressure; i.e., ρeff drops to ρmin = 0.14 at 50 MPa effective pressure, contrasting even more with ρimg3-D = 1.1. In addition, at lower effective pressure, the aperture of the cracks is intuitively expected to be larger. Based on the radius and aspect ratio, the aperture 2w of the cracks is estimated (Table 3). We observe that weff < wimg2-D < wimg3-D, suggesting another possible cause of the observed discrepancies. It appears that the effective aperture derived from the laboratory data inversion is much lower than the resolution achieved in the 3-D X-ray micro-CT images analyzed: weff ~ 168 nm < 800 nm. On the other hand, even if the pixel size of the 2-D electron microscope images is in principle sufficient to resolve the crack aperture derived from the inversion of laboratory data, geometrical biases associated with imaging 2-D thin sections extracted from 3-D objects are expected. Indeed, the estimation of crack length and aperture from 2-D cross sections of 3-D cracks is subject to a truncation/orientation ambiguity; i.e., the length of the imaged crack corresponds to the actual crack diameter only if the 2-D image intersects the center of the crack. If this is not the case, the imaged crack length will appear smaller than the actual crack length, and therefore bias the derived microstructural statistics. A similar rationale holds for the crack aperture; i.e., if the 2-D image is not orthogonal to the crack under consideration, the estimation of its aperture will be biased by parallax errors. Such an ambiguity will affect the statistics of crack size and aperture derived from 2-D images and will lead to a discrepancy with the corresponding value derived from 3-D images as noted in Delle Piane et al. [2015] and with the results of the laboratory data inversion presented here. The crack aperture 2w derived from the inversion of laboratory data leads to a good agreement between the effective medium model predictions and the laboratory data in terms of (i) elastic wave velocities (moduli), wave dispersion, and permeability (see Figures 11a, 11b, and 13b). 5.3. Possible Caveats of the Modeling Approach The effective medium modeling and laboratory data inversion yield a single value of crack radius R and aspect ratios ξ. These effective medium models are only an approximation of nature and as such they are most probably too simplistic. The approximation resides not only in the simplicity of conceptual model used (Figure 9) but also in the modeling schemes used; i.e., the effective elasticity model relies on the noninteraction approximation, and the effective permeability model relies on the concept of equivalent hydraulic radius. At any given effective pressure the crack geometry parameters constitute an effective but approximate representation of the actual crack distribution in the rock. In other words, the actual crack distribution affects wave propagation, wave dispersion, and permeability in a way similar to an equivalent population of identical cracks having the geometrical attributes Reff, weff, ξ eff, and ρeff. Considering a distribution of crack SAROUT ET AL.
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radii and aspect ratios in the conceptual model similar to the grain size distribution would probably be more realistic. At any given effective pressure, the inverted crack size 2R, aspect ratios ξ, and aperture 2w can be thought of as “average” values representing the actual crack population affecting the laboratory measurements. This “averaging scheme” is, however, not trivial and results from the effective medium models used: the effective aspect ratio ξ eff results from the effective permeability model, while the crack radius Reff results form the effective elasticity model. With regard to the modeling schemes used for this cracked CM, it seems that the non-interaction approximation and the concept of hydraulic radius yield a reasonably good match with wave velocity/dispersion and permeability data, respectively. Unexpectedly, the crack density derived from the effective elasticity model and laboratory data reaches a value of ρeff = 0.51, which is significantly higher than the admitted limit of validity of the non-interaction approximation applied to effective elasticity, i.e., ρ ~ 0.1 [Grechka and Kachanov, 2006b].
6. Conclusion Based on a petrophysical characterization and a laboratory experimentation on the thermally cracked Carrara marble at variable effective pressures in dry and water-saturated conditions, we observed that (i) gas permeability in the intact and thermally cracked rock decreases toward a similar asymptotic value with increasing effective pressure; (ii) wave dispersion between 0.01 Hz and 500 MHz in water-saturated conditions reaches a maximum of ~26% for S waves and ~9% for P waves at the lowest effective pressure tested (1 MPa), and the highest crack density value (~0.5); and (iii) wave dispersion virtually vanishes above a threshold effective pressure of ~30 MPa. Using an effective elasticity model in the non-interaction approximation, the P and S wave velocity data are jointly inverted to recover the evolution of the crack density ρ and the effective aspect ratio ξ of the cracks as a function of effective pressure (or depth). Using a simple effective permeability model, and the crack parameters derived from the prior elasticity analysis, the permeability data are inverted to recover the evolution of the effective radius R of the cracks as a function of effective pressure (or depth). The recovered microstructural parameters lead to a good agreement between predictions and measurements up to 50 MPa effective pressure and a maximum crack density ρmax ~ 0.5 (at 1 MPa). The derived crack geometry parameters differ significantly from those obtained from the analysis of 2-D electron microscope images of thin sections and from 3-D X-ray microtomography images of millimeter-size specimens. The evolution of the crack geometry parameters suggests that three deformation regimes exist with increasing depth in this tight cracked rock, involving a non-monotonic evolution of the effective radius and aspect ratio of the cracks. We also showed that by combining laboratory experimentation with an effective medium modeling of the data, it is possible for a cracked rock to (i) correct high-frequency ultrasonic data obtained in the laboratory for the interpretation of field seismic data obtained at a much lower frequency and (ii) assess the evolution of the morphology of the crack network with increasing effective pressure to predict permeability evolution with depth.
Acknowledgments We acknowledge the financial support provided by CSIRO through a Strategic Research Fund and the assistance of the technical personnel of CSIRO’s Geomechanics and Geophysics Laboratory: Bruce Maney, Shane Kager, Leigh Kiewiet, Stephen Firns, and David Nguyen. The data not reported in Tables 1–3 can be made available upon request to the lead author. Finally, we would like to thank the Associate Editor and two reviewers for their insightful and constructive comments, which greatly helped improve the original manuscript.
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