Variants of the strain-amplitude dependence of elastic wave velocities

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Nov 22, 2004 - NANJING INSTITUTE OF GEOPHYSICAL PROSPECTING AND INSTITUTE OF PHYSICS PUBLISHING. JOURNAL OF ... form for solving applied problems of seismic prospecting and seismology. ...... 101 11553–64. Johnston ...
NANJING INSTITUTE OF GEOPHYSICAL PROSPECTING AND INSTITUTE OF PHYSICS PUBLISHING

JOURNAL OF GEOPHYSICS AND ENGINEERING

J. Geophys. Eng. 1 (2004) 295–306

PII: S1742-2132(04)82667-0

Variants of the strain-amplitude dependence of elastic wave velocities in rocks under pressure E I Mashinskii Institute of Geophysics, Siberian Branch of the RAS, prosp Akad Koptyuga 3, Novosibirsk, 630090, Russia

Received 29 June 2004 Accepted for publication 20 October 2004 Published 22 November 2004 Online at stacks.iop.org/JGE/1/295 doi:10.1088/1742-2132/1/4/008

Abstract The dependence of the compressional and shear wave velocities on strain amplitude in the Madra dolomites is studied. The experiments were performed using an ultrasonic pulse transmission technique (f = 1 MHz) with a uniaxial pressure of P = 1–60 MPa. An amplitude dependence of wave velocity in the range εd ∼ (1–3) × 10−6 under a pressure of 5–20 MPa is found. The compressional velocity depends on strain amplitude but the shear velocity does not depend on amplitude change. Determined by the first arrival time, the compressional velocity increases with amplitude. However, if the compressional velocity is determined by the measurement of peak time, the velocity decreases with amplitude. For the upward and downward pressure, the curve Vp (Pax ) exhibits open hysteresis, yielding a residual component of velocity. An intersection of the branches of the hysteretic loop is observed in more porous dolomites in comparison with less porous dolomite. This intersection is manifested most in the residual components of the dynamic bulk modulus and Poisson’s ratio. The dynamic parameters can be used as is supposed as additional criteria in a geological interpretation. Keywords: inelasticity, nonlinear stress–strain relation, hysteresis, nonlinear seismics, amplitude dependence of wave velocity and attenuation, inelastic seismic parameters

1. Introduction It is of great importance that wave velocities and attenuation show amplitude dependence on strain levels applied in seismic exploration. There are many works which show the perspectivity of attenuation application in geophysical prospecting (Wang 2002, 2003, 2004, Wang and Guo 2004). This parameter is the most sensitive to changes in the structure and lithology of the rock. However, the amplitude dependence of wave velocities is also of interest because this parameter is easily measured and can be widely used in an improved form for solving applied problems of seismic prospecting and seismology. The amplitude dependence of wave velocity has been known for a long time but the mechanisms of this phenomenon 1742-2132/04/040295+12$30.00

and range of deformation have not been reliably established (Johnston and Toksoz 1980, Winkler et al 1979, Johnson et al 1996, Ten Cate et al 1996a, 1996b, Zinszner et al 1997). Another important aspect of this problem noted recently relates to the very character of this dependence, leading to variablesign velocity variations. The assumption that the velocity can both decrease and increase with increasing amplitude was made earlier, when the influence of the microplasticity on the modulus–stress relation was established (Mashinskii 1989, 1994, 2003). However, this aspect of the problem remained unexplored and has not been discussed in the literature. With the appearance of the theoretical work by McCall and Guyer (1994), interest in such investigations increased. This work shows that the dynamic modulus (wave velocity) can increase with stress if there is positive curvature in the

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static stress–strain relationship. This has prompted further investigation. My investigation was additionally stimulated by an experiment on microplasticity (Mashinskii 2001) which showed the multilevel character of the stress–strain relations σ (ε), giving all possible variants of the modulus dependence on strain (an increase, a decrease or constant behaviour). It assumes that the wave velocity will increase with amplitude if the dynamic stress–strain curve has positive curvature. Now, I present a brief review of theoretical and experimental works of the amplitude dependence of wave velocities and then describe the experiments. Some data on the amplitude dependence of wave velocities are obtained by direct measurement. Indirect evidence comes mainly from the amplitude dependence of attenuation. This enables us to judge the characteristic of the dependence of velocity on strain. Studying deformations of sandstones and quartzites, Cook and Hodgson (1965) obtained σ (ε) curves, which show that a static modulus rises with increasing strain. This dependence is the curve with positive curvature as described by McCall and Guyer (1994), and Guyer et al (1995). The dynamic modulus (wave velocity) also increases with stress. Winkler et al (1979), studying wave attenuation and velocity in sandstones, found that Vp decreases with strain amplitude at ε > 10−6 . The same results were obtained by Johnston and Toksoz (1980). An investigation of velocity variation in sandstones under confining pressures of up to 50 MPa showed that velocity change is insignificant (Stewart et al 1983). The wave velocity can also be calculated from the propagation speed of tone bursts (Ten Cate et al 1996). Tutuncu et al (1998a, 1998b) made the general conclusion that Young’s modulus (wave velocity) decreases with strain amplitude for various types of sandstones. A multitude of experiments have been recently performed using a longitudinal resonant method (Johnson 1996, Ten Cate and Shankland 1996, Zinszner et al 1997). In this case the wave velocity (Vp ) is computed from the resonant frequency. The all bar resonant experiments on sedimentary rocks (sandstones, limestones, marble, chalk) also show that wave velocity always decreases with increasing strain amplitude. There are some observations that contradict the wellknown experimental data: a rare measurement of the amplitude dependence of wave velocity in a monocrystal and also in a field experiment (Mashinskii et al 1999, Mashinskii 2001). In a monocrystal of natural quartz under a confining pressure of 10 MPa, Vp (f = 1 MHz) decreases with increasing amplitude whereas Vs increases (0.5%). Both velocities display hysteresis in the strain range 10−7–10−6. In situ measurement in the ground (f = 500 Hz) showed an increase in Vp with amplitude (Mashinskii et al 1999). There is an open hysteresis that is proof of the irreversible deformation caused by the seismic wave. In spite of this, one can maintain that in the overwhelming majority of experiments, Vp decreases with increasing amplitude. The theoretical studies of Mavko (1979), Stewart and Toksoz (1983), and other authors yield a direct dependence of Vp and attenuation on strain amplitude (the velocities 296

decrease and the attenuation increases). The theoretical results obtained by Xu et al (1998) differ from the above by an exponential dependence of kinematic and dynamic wave parameters. Note that the majority of authors base their theoretical calculations on a stress–strain curve with negative curvature (Van Den Abeele et al 1997). Thus, the theoretical work by McCall and Guyer (1994) differs from other works as it proposes a model combining the possibility of both a decrease and an increase of the quasistatic and dynamic modulus with increasing strain. The model uses the Preisach–Mayergoyz (PM) space and various sets of a large number of hysteretic mesoscopic elastic units (HMEU) which simulate a medium with hysteresis. The model aims to obtain a curve σ (ε) close to the real one. In some cases a HMEU combination results in a high density of these units at small static pressures that gradually decreases with increasing pressure. In contrast, in the other case, a low HMEU density at small pressures gradually increases with pressure. Owing to this, in the first variant, the stress– strain diagram obtained from the equation of state has negative curvature, i.e., represents the σ (ε) curve with a decreasing elastic modulus. In the second variant, the stress–strain diagram has positive curvature and, accordingly, an increasing modulus. The dynamic modulus also obeys this rule and is reflected on a large quasistatic hysteretic loop in the form of small inner loops. Therefore, having the curves σ (ε) with derivatives of opposite sign, one can theoretically obtain both increasing and decreasing dependences of the wave velocity on amplitude. This paper is devoted to the experimental study of the amplitude dependence of acoustic wave velocities in consolidated sedimentary rocks with stress–strain curve of positive curvature. According to the theory of McCall and Guyer (1994) at least, one should expect an increase in the static and dynamic modulus with stress (or strain). This and other problems (explored to a lesser extent) are discussed below.

2. Experimental configuration and methodology Experiments were conducted on dry samples of the Madra and Yurubchen dolomites of Western Siberia (a depth interval of 2800–3850 m). The sample form is a 50 mm cube. The samples were carefully prepared and polished (the roughness of surfaces, Rz , was 10–20 µm). Opposite cube faces were parallel within an accuracy of 0.2 mm per edge length. The X-, Y- and Z-axes were marked on each cubic sample (the Z-axis coincided with the borehole axis). The sequence of X- and Y-axes was chosen in accordance with the corkscrew rule. The stress was uniaxial along the chosen axis. The procedure precluded any bending of samples in the process of their uniaxial compression. The experiments were performed using an ultrasonic pulse transmission technique with uniaxial pressures of P = 1–60 MPa, and by the measurement of the static strain lmax to 500 µm. A block diagram of the set-up is illustrated in figure 1. The apparatus includes acoustic transducers for the excitation and reception (the acoustic channel) of P and S pulses with

Variants of the strain-amplitude dependence of elastic wave velocities in rocks under pressure

Figure 1. (a) Block diagram of the set-up, (b) P-wave pulse, (c) S-wave pulse.

the central frequencies fp = 750 kHz and fs = 350 kHz. On one side of the cube is a source and on the opposite side the receiver is attached. The study of the amplitude dependence of wave velocities was conducted using two time measurement approaches. (1) The interval time was measured from the starting moment to the first arrival point with the propagation of P and S pulses. (2) The interval time was measured from the starting moment to the peak-amplitude point with the propagation of P and S pulses. Only one-way transmission effects are measured. Absolute accuracy of the measurement of wave speed is about 1% but it should be noted that we are not interested in the absolute accuracy of the measurement. However, the relative accuracy for the amplitude dependence (i.e., precision or repeatability) is much better. The calibration of the acoustic channel shows an accuracy of 0.2% for P-waves and 0.4% for S-waves. The static strain was measured by an electric strainmeter operating in conjunction with an amplifier and high-precision voltmeter. The sensitivity of the measurement of the absolute strain is 0.1 × 10−6 m which corresponds to the relative strain ε = 2 × 10−6 . Relative strain, εst = L/L, is calculated

concerning a cube size (about 5 cm). If the signal/noise relation is taken of the order of 5 then the minimum strain is εmin = ±1 × 10−5 . The relative accuracy of the static measurement is better. In this paper, I present only the data for stress along the Z-axis. Pressure was carried out in a stepwise (discrete) mode from 1 MPa through 2, 3, 5, 10, 20, 30, 40, 50 and 60 MPa, after which the unloading was done in the reverse order. At each pressure the static strain l, amplitude A, and interval times tp 0 , tp A and ts 0 , ts A (P and S pulses) were measured consistently. After the load was applied, the static strain, strain amplitude of the wave pulse and interval times were measured. Then the next load was applied and those measurements were conducted again. This procedure was performed during the loading–unloading and the stress–strain curve drawn using data on the static strains obtained at each pressure. The waveform is illustrated in figures 1(b) and (c). It is important to note that the first arrival time is in fact the time of the first minimum. Therefore the wavefront length can be calculated as tp = tp A − tp 0 and ts = ts A − ts 0 . At each pressure the wave speed (interval time) is measured for four amplitudes (A1.0 , A1.5 , A2.1 , A3.0 ), i.e. the Amax /Amin relation is 3. The pulse amplitude was changed in the strain range (1–3) × 10−6. 297

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3. Experimental results

K p =0.4%

Kp =2.4%

60

The principal theoretical component in static and dynamic studies is the stress–strain curve. Most rocks have a σ (ε) curve with negative curvature; the static modulus decreases with pressure. Some rocks have a σ (ε) curve with positive curvature; the static modulus increases with pressure. If such a rule applies to the dynamic curve σ (ε), then the dynamic modulus (wave velocity) will also follow this rule, i.e. the modulus will decrease or increase, depending on the characteristic of the curvature, with stress amplitude. I consider a possible physical reason for the ‘abnormal’ behaviour of the elastic modulus which can take place in a real rock. The total strain consists of three main components (Mashinskii 2003a), εi = εi−e + εv−e + εµ ,

σ tσ εi Ei tσ εi tσ = = , ηef ηef Trel

(2)

where tσ is the stress duration, ηef is an effective viscosity, Trel = ηef /Ei is the relaxation time, and Ei = σi /εi is the instantaneous Young modulus. The εµ is microplastic strain; it is residual deformation. On the one hand microplastic strain is the time-independent deformation, however, on the other hand it is the amplitudedependent deformation. The instantaneous (local) Young modulus is given by Ei =

σi , εi−e + εv−e (tσ ) + (|ε|)

(3)

where εv−e (tσ ) is the time-dependent viscoelastic component and εµ (|ε|) is the strain-dependent microplastic component (with variable sign). A study of the inelasticity of many rocks from the curves σ (ε) has shown that the behaviour of the inelastic component (residual strain) varies: a negative curvature results most often, and a positive curvature is infrequent (Mashinskii 1989, 1994, 2001). A negative curvature of σ (ε) leads to the decrease of the modulus with stress and a positive curvature leads to an increase with stress. This can be explained by considering equation (3). Usually the εi−e , εv−e (tσ ), (|ε|) components increase with increasing stress for most rocks. It leads to the modulus decrease. However in certain rocks or microstructures, the inelastic component increment (at least the microplastic component) can decrease with stress. This takes place, for example, in dolomite and argillite. The decrease εinelastic = εv−e (tσ ) + (|ε|) means an increase of the Ei modulus in (3). The decrease in εv−e is possible if 298

50 40 30 20 10 0 0

10

20

30

Axial Strain (10-4 ) Figure 2. Static stress–strain diagrams for dry Madra dolomites with porosity Kp = 2.4% and 0.4%.

(1)

where the component εi−e represents an ideally elastic Young modulus Ei−e = σi /εi−e , corresponding to deformation of the monocrystalline grains of a rock skeleton. Two latter terms in (1) represent the inelastic component. The strain εv−e corresponds to the viscoelastic behaviour of rocks dependent on the magnitude and time of stress. For example, in a Maxwell model, the viscoelastic strain (for σ = constant) is εv−e =

Axial Stress (MPa)

3.1. Theoretical precondition

the relaxation time Trel depends on the stress. Then the value εv−e (Trel ) (tσ = const) will decrease and the modulus Ei will increase with stress under the condition that an increase in Trel occurs as well. This is a hypothesis and requires more research. As to the microplastic component, it is a reality. The microplastic strain can decrease with increasing stress and, consequently, the modulus can increase due to the microplastic component. The non-standard change of the modulus is possible during unloading as well, due to microplasticity. The classic viscoelastic mechanism does not assume such behaviour. There is an explanation of the difference between the static and dynamic Young moduli. Differences between measured static and dynamic elastic moduli are caused by different inelastic contributions to stress–strain relationships which behave as a function of strain amplitude and frequency (energy and strain rate). The increase of frequency (the increase of the deformation speed) leads to the decrease of the time-dependent viscoelastic component (εv−e (tσ )). Then, according to expression (3), modulus Ei increases with increasing frequency. The microplastic component (εµ (|ε|)) is the time-independent component. Therefore the microplastic component does not contribute to deformation if the strain (stress) does not change. The microplastic component contributes to deformation when the strain changes. However, the microplastic value can increase and even decrease with strain. The decrease of the microplastic contribution with increasing strain is possible, owing to the ‘resorptional’ (diffused) effect. It leads also to the modulus increase with increasing strain (Mashinskii 1994). 3.2. Quasistatic and dynamic measurements Dolomites have a stress–strain curve with positive curvature, which is governed by the behaviour of the quasistatic modulus as a function of stress. The σ (ε) curves of the Madra dolomites with porosity Kp = 2.4% and 0.4% are presented in figure 2. The plot shows an open hysteretic loop and the residual strain, which appear after the unloading. The residual strain is εres = 7 × 10−4 for dolomite with porosity

Variants of the strain-amplitude dependence of elastic wave velocities in rocks under pressure

4000

2300

3500

2200

Vp

Vs (m/sec)

Velocity (m/sec)

0.4%

Vst

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2500

2.4% 2100

2000

Vs 2000

1900 0

10

20

30

40

50

60

Axial Stress (MPa) 1500 0

10

20

30

40

50

60

Axial Stress (MPa)

Figure 3. Compressional and shear velocities, and the ‘quasistatic’ velocity, Vst , as a function of axial stress.

3750 0.4%

Vp (m/sec)

3600

3450

2.4%

3300

3150 0

10

20

30

40

50

60

Axial Stress (MPa)

Figure 4. Vp as a function of pressure for Madra dolomites with Kp = 0.4% and 2.4%.

Kp = 2.4%. According to the curve, the static modulus increases with pressure. The Vst velocity calculated by means of the instantaneous modulus (Esti = σi /εi ) increases with pressure as well; see figure 3 (it should be emphasized that the modulus behaviour is of interest here, rather than the accuracy of its value). Vp and Vs increase with increasing pressure. With decreasing pressure this Vst , Vp and Vs decrease; forms a hysteretic loop. The behaviour of wave velocities during the unloading of the sample complies with the unloading curve of the σ (ε) in figure 2, where the static modulus at the time of unloading is higher than the loading modulus at all pressures. Vp and Vs during the unloading obey the same rule (figures 4 and 5). This rule is sometimes violated, and in this case the velocity at a given pressure during the unloading can be lower than at the same pressure during the loading. This phenomenon, observable as an intersection of the hysteretic branches and leading to interesting conclusions, is considered below in greater detail.

Figure 5. Vs as a function of pressure for Madra dolomites with Kp = 0.4% and 2.4%.

Such an effect is observed in Madra dolomites with porosity Kp = 2.4% but it is absent in dolomite with Kp = 0.4% (figure 4). The plot shows that the intersection of the curves (Vp as function Pax ) takes place at pressure Pax ∼ 10 MPa. An intersection is absent in the Vs (Pax ) for both porosity values; the corresponding hysteretic loop is normal (figure 5). In order to elucidate the origin of the hysteretic intersection, the velocity dependences on pressure were considered from a different point of view. For the same dolomites, I plotted the dependences of Poisson’s ratio νd , the dynamic bulk modulus Kd and the coefficient Vs /Vp as a function of pressure (figure 6). The Kd and νd were calculated from the known formulae,   4 2 E(1 − γ 2 ) 2 Kd = ρ V p − V s = , (4) 3 3(1 − 2γ 2 ) Vp2 − 2Vs2 1 − 2γ 2 = , νd =  2 2 2(1 − γ 2 ) 2 Vp − Vs

(5)

where ρ = 2740 kg m−3 is the density of dolomite, and E is Young’s modulus. The diagrams of dolomite with Kp = 2.4% show an increase in νd and in Kd and a decrease in Vs /Vp as a function of pressure in the range Pax = 10–20 MPa, i.e. at approximately the same pressure, from which Vp starts to decrease during the unloading (see the intersection in figure 4). The hysteretic loops of all parameters show a residual hysteresis, which is largest (15%) for the parameter νd (7% for Kd and 2% for Vs /Vp ). The dolomite with Kp = 0.4% exhibits a closed hysteresis with nonintersecting branches, and only the Kd has a small residual hysteresis (about 2%). The diagrams in figure 6 clearly demonstrate that dolomites differ in these dynamic parameters (particularly in νd ). The inadequate decrease of Vp after the intersection is apparently caused by the inelastic decompaction of the rock (a drop in its rigidity), which occurs during the unloading of the sample. The process of an anomalous decrease in the rigidity starts from a certain pressure. The effect is in the more porous sample, but this effect is absent in the less porous sample over the entire interval of pressures. 299

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3800 0.4%

0.4%

0.22 2.4%

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0.2

Vp (m/sec)

Dynamic Poisson's Ratio (νd)

0.24

0.18

2.4%

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A1=1.0

60

A2=1.5

(b)

A3=2.1

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0.4%

Amax=3.0

3200

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0

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70

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2.4%

Figure 7. Compressional velocities of Madra dolomites (Kp = 2.4% and 0.4%) for four amplitudes A1 = 1.0 × 10−6 , A2 = 1.5 × 10−6 , A3 = 2.1 × 10−6 and Amax = 3.0 × 10−6 at different pressures. The velocity is computed using the arrival time. Errors are within the symbol size.

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(c )

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0. 6 35

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0. 6 25

2 .4% Vs / Vp

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2100

A1=1.0 A2=1.5

0. 5 95

0.4 %

2050

A3=2.1 Amax=3.0

0. 5 85 0

10

20 30 40 Axial Stress (MPa)

50

60

Figure 6. (a) Dependences of the dynamic Poisson ratio, (b) dynamic bulk modulus and (c) coefficient Vs /Vp on the pressure of Madra samples with Kp = 0.4% and 2.4%.

2000 0

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30

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60

70

Axial Stress (MPa)

3.3. The amplitude dependence of wave velocities

Figure 8. Shear velocities of Madra dolomites (Kp = 0.4% and 2.4%) for four amplitudes A1 = 1.0 × 10−6 , A2 = 1.5 × 10−6 , A3 = 2.1 × 10−6 and Amax = 3.0 × 10−6 at different pressures. The velocity is computed using the arrival time. Errors are within the symbol size.

The influence of amplitude on wave velocity was studied at different pressures. Vp and Vs were measured at four strainamplitude levels. The measurement shows that Vp changes with increasing amplitude but not at all pressures. The Vs change with amplitude is small. The wave velocity as a function of amplitude (from 1 to 3 microstrain) for the Madra dolomite is presented in figures 7–10. The velocity was calculated first using the measurement of the first arrival time and, second, the peak-amplitude time. There are qualitative differences between these results. The total picture of Vp and Vs obtained using the arrival time on four amplitudes is presented in figures 7 and 8. Vp increases with amplitude for both porosity values. However, the change in the value of Vp with amplitude is

greater at low pressure than at high pressure. The wave velocities behaviour calculated from the peak time is more complicated. The greatest change in velocity takes place at low pressure, i.e. these changes exactly exceed the measurement errors. Therefore I present  the amplitude dependence of the arrival velocity Vp 0 , Vs 0 and the peak   velocity Vp A , Vs A only for pressures of 5–20 MPa. In figure 9 it is seen that the increase in Vp 0 and the decrease in Vp A with increasing amplitude occur simultaneously. The biggest change in velocity is 1.2% (at 5 MPa). Vs 0 and Vs A are not practically changed. The insignificant increase in Vs 0 with amplitude is seen only for the porosity of 0.4%; figure 10. However, as the Vs 0 change is

300

Variants of the strain-amplitude dependence of elastic wave velocities in rocks under pressure 2200

(a) 3 40 0

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5M P a 10 M P a 20 M P a 5M P a 10 M P a 20 M P a

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Vs (m/sec)

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2000

20MPa

3 10 0

3 00 0

1900 1

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3

Strain Amplitude (microstrain) 2 90 0 1

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Strain Amplitude (microstrain) (b)

Figure 10. Dependence of computed shear velocity of Madra dolomites with Kp = 0.4% at pressures of 5–20 MPa on strain amplitude. The solid line shows the velocity computed using the arrival time, and the dashed line shows the velocity computed using the peak time. Representative error bars are shown. 16.9

5MPa

Vp (m/sec)

Arrival and Peak Time (µ sec)

3500

3300

5MPa 10MPa 10MPa

15.9

20MPa 20MPa 30MPa 30MPa 40MPa

14.9

40MPa 50MPa 50MPa 60MPa

3100 1

2

3

Strain Amplitude (microstrain)

60MPa

13.9 1

2

3

Strain Amplitude (microstrain)

Figure 9. Dependence of computed compressional velocity of Madra dolomite with (a) Kp = 2.4% and (b) Kp = 0.4% at pressures of 5–20 MPa on strain amplitude. The solid line shows the velocity computed using the arrival time, and the dashed line shows the velocity computed using the peak time. Representative error bars are shown.

close to the measurement error, I consider that there is only a tendency towards the speed increasing. The influence of amplitude on the velocity can  be0  well observed by direct comparison of the arrival time tp,s and   the peak time tp,s A . The decrease in tp 0 and the increase in tp A with amplitude occur simultaneously; figure 11. It is clearly seen at low pressures only (from 5 MPa to 20 MPa). The tp A is independent of amplitude at pressures from 30 MPa to 60 MPa. The influence of amplitude on the ts0 and tsA is insignificant. The dependence of the Vs /Vp and Poisson’s ratio, νd , on strain amplitude is illustrated in figures 12 and 13. The change in νd with amplitude reaches 7–10% at a pressure of 5 MPa.

Figure 11. Variation of arrival time (solid line) and peak time (dashed line) of compressional wave as a function of strain amplitude in dry Madra dolomite with Kp = 2.4% at pressures of 5–60 MPa. Errors are within the symbol size. Arrows emphasize the 5 MPa curves.

4. Discussion The static and dynamic experiments on dolomites display hysteresis and residual phenomena. The stress–strain curve σ (ε) has positive curvature. After unloading, the residual strain (microplastic strain) has a large value (7 × 10−4 for a high-porosity dolomite). This value greater exceeds the order of the measurement errors. The dynamic hysteretic loops of Vp and Vs are open, that is, in compliance with the residual strain on the σ (ε) curve. The residual velocity component has a value of about 2%. Some authors consider that the small hysteretic loops, reflecting the small-amplitude wave propagation and taking place inside a larger loop, retain their 301

E I Mashinskii 0.636

0.632

5MPa

0.628

Vs /Vp

5MPa 10MPa 10 MPa 0.624

20 MPa 20 MPa

0.62

0.616 1

2

3

Strain Amplitude (microstrain)

Figure 12. Dependence of the coefficient Vs /Vp of dry Madra dolomite with Kp = 2.4% at pressures of 5–20 MPa on strain amplitude. The solid line shows the coefficient computed using the arrival time, and the dashed line shows the coefficient computed using the peak time.

Dynamic Poisson's Ratio

0.19

5MPa 0.18

5MPa 10MPa 10MPa

0.17

20MPa 20MPa

0.16 1

2

3

Strain Amplitude (microstrain)

Figure 13. Dependence of the dynamic Poisson ratio of dry Madra dolomite with Kp = 2.4% at pressures of 5–20 MPa on strain amplitude. The solid line shows the Poisson ratio computed using the arrival time and the dashed line shows the Poisson ratio computed using the peak time.

slopes (the elastic modulus) in the σ –ε axes at all pressure levels (Holcomb 1981, Tutuncu et al 1998). However, the result obtained here assumes that the small hysteretic loops must change their slope with pressure and/or strain. Usually, the wave velocity during unloading is higher than during loading. In some cases, this rule is violated and there is an intersection of hysteresis branches (as is the case with dolomite of Kp = 2.4%; see figure 4). I note in particular that an intersection of hysteresis branches is absent on the static stress–strain curves. The ‘abnormal’ decrease in Vp after the intersection can be attributed to decompaction of the rock and/or to the anomalous decrease in its rigidity. In our case, the intersection begins at the unloading pressure Pax ≈ 10 MPa. On the hysteretic curve, one can see that an increased rigidity of the rock obtained during the loading 302

is kept only in the pressure interval above 10 MPa. This is indicated by a higher unloading velocity compared to loading velocity. However, starting from a certain unloading pressure, the so-called ‘collapse’ of a rock occurs when the rock loses its rigidity with a higher rate. It is interesting to note that this process is only observed in more porous dolomites. In less porous dolomites, the effect of collapse does not arise, and the condition of higher rigidity during the unloading than during the loading is retained over the entire range of pressures. I consider that the consolidation processes in a low-porosity rock during loading are of irreversible nature and are related to some other deformation mechanisms. In this rock the irreversible deformation is more steady, i.e. its value remains constant during the unloading process on all pressure intervals. The consolidation processes in other rocks are not unstable. Therefore the inelastic deformation in these rocks during unloading can essentially differ. I do not firmly maintain that the reason for the intersection occurrence lies only in the porosity value. I state only a fact that in our case the intersection occurs in a dolomite with greater porosity. A decrease in Kd and in νd , which begins with the same unloading pressure (about 10 MPa) is evidence of a decrease in rock rigidity (see in figure 6). An increase in Vs /Vp takes place at the same pressure. It is caused by the greater decrease of Vp compared to Vs . It should be noted that the residual hysteresis in Kd and particularly in νd substantially exceeds the corresponding residual component for Vs /Vp . Generally, such an indicator as the residual component of any parameter requires further investigation. I want to note that, in principle, the intersection of the hysteresis branches was known long ago. For example, P- and S-wave velocities as a function of pressure in dry and saturated Bedford limestone show the same behaviour (Johnston and Toksoz 1980). The authors give the following explanation. After loading, pore closure takes place, which is the reason for the intersection. As the rock is unloaded the pores open, reducing the velocities to a value well below those obtained during the loading cycle. The confirmation lies in the static stress–strain curve. There are some differences between the rock type and the conditions of the experiment in the considered cases. The Bedford limestone has a porosity of 11.9%. Our sandstone has a porosity of 2.4% and the pressure applied to the sample is below more than an order of 2. There is another example also: the intersection of the hysteresis branches (in the range 1 MPa to 10 MPa) for frequency shift on increasing and decreasing pressure (Zinszner et al 1997). Dry Meule sandstone shows more attenuation on the unloading curve after the intersection in comparison with attenuation on the loading curve which is evidence of lower velocity on unloading. The known hypothesis of the hysteretic intersection can be improved if microplastic inelasticity is included in the deformation process. The static and dynamic moduli depend on the behaviour of the microplastic component. The indicator of inelasticity is the residual strain, which is caused by microplasticity. It is necessary to note that in our case the relaxation process gives an insignificant small deformation (viscoelastic strain) since the consolidated rock has a

Variants of the strain-amplitude dependence of elastic wave velocities in rocks under pressure 17

Arrival Time and Peak Time (µ sec)

relaxation time which is much greater than the loading time, even in statics. The accumulation of the residual strain occurs with increasing pressure. The contribution of microplasticity increases with pressure. During the unloading the opening of the cracks (pores) occurs. This process causes a decrease in wave velocity but, in spite of this, the velocity during unloading is almost always greater than during loading. It happens because the rock is irreversibly changed by the loading (at least, while the measurement of wave velocity is conducted). During loading an accumulation of microplastic strain appears. The maximal value of the microplastic component takes place on the maximal stress. After the loading is finished the rock has greater stiffness than before. This increased stiffness (the unloading modulus is greater than the loading modulus) is usually invariable through the entire unloading. After unloading, i.e. the removal of the elastic forces, we see the residual microplastic strain. In some rocks the microplastic component can vary (increase or decrease) during unloading which leads to a change in rock stiffness (modulus and wave velocity). This is the nature of microplasticity. In the given case, the rock with lower porosity appears more stable with respect to microplastic variation than the rock with greater porosity. The latter is less stable as this rock has a bigger contact area which is the source of microplasticity. I understand that it is possibly an incomplete explanation, however the inelastic hypothesis demands further study. The influence of strain amplitude on wave velocity is established. However this influence is complex. The amplitude dependence of wave velocity is considered in two ways. The first approach uses the arrival time, and the second approach uses the peak time (accordingly, Vp,s 0 , Vp,s A velocities). These results vary qualitatively. The compressional velocity responds to the amplitude change but the shear velocity weakly responds to the amplitude change. Vp 0 increases with amplitude in qualitative agreement with the behaviour of the derivative σi /εi = Ei of the quasistatic relation σ (ε). Vp A decreases with increasing amplitude. The change in Vp is less with increasing pressure. The greatest change in Vp is at the pressure range 5–20 MPa (it is 1.3% at 5 MPa). The principal question is of the reliability of the results obtained. The accuracy of the measurement is 0.2% for P-waves and 0.4% for S-waves, i.e., it is the timing error of determining the arrival time and the peak time. It can be seen in detail in table 1, where the arrival time, the peak time, the change of the time t in the amplitude interval (A1 –Amax ) and the time errors (t) are shown for dolomites with Kp = 2.4%. Note that the longitudinal pulse has a pronounced onset since the source and receiver are attached to the sample under pressure and the contacts are wetted. The time errors were determined in the calibration. In this case, the source and the receiver were pressed to each other under pressure. In that position, the arrival and peak times were measured for all amplitudes. In other words, the time errors t (in the interval tA1 –tAmax ) were determined as the maximal deviation of the tp and ts times in the amplitude interval A1 –Amax under calibration. Data from table 1 show

Z X Y Z X Y

16

15

14

13 1

2

3

Strain Amplitude (microstrain)

Figure 14. Variation of the arrival time (solid line) and peak time (dashed line) of the compressional wave (lengthways of the axis Z, X and Y) as a function of strain amplitude in dry Madra dolomite with Kp = 2.4%.

that the arrival time depends on amplitude at all pressures. However, for reliability, I take into consideration only the results at low pressure where the value of the time change exceeds significantly the error value. It is applied to the peak time as well. At high pressure I shall imply tendency only. The accuracy of the measurement of the time of the shear wave is less. The changes of arrival and peak time of the shear wave with amplitude are very small. Therefore shear data can be considered as a probability or tendency only at the single pressure. I say a tendency, implying that it is seen in some system in the time behaviour but there is no chaotic change of time with amplitude change (see table 1). The measurements for the X-axis and Y-axis performed on the same sample are evidence of the repeatability; figure 14. They show qualitatively the same result, i.e. the time aspires to change with increasing amplitude. In this work, unfortunately, I cannot consider data for S-waves as the result. Complete evidence of the amplitude dependence can be obtained by the expansion of the strain-amplitude range. The increase of wave velocity with amplitude contradicts the existing concept. The main task is to explain why my data show so much difference from the multitude of previously published data. There are two possible reasons. The first explanation is perhaps debatable and it requires investigation. The second explanation is the most likely. The first concerns the difference in measurement methodology and the second concerns the lithology and microstructure. Most previous data were obtained using a longitudinal resonant method whereas our data were obtained using the pulse transmission method measuring travel times of 303

E I Mashinskii

Table 1. Arrival and peak times, change of time t in the amplitude interval (A1 –Amax ) and time errors in the amplitude interval (A1 –Amax ) of compressional waves for Madra dolomite with Kp = 2.4% at pressures of 5–60 MPa. Compressional wave, time (µs) 0

t (error)

tp A

−0.22

0.02

16.62 16.64 16.68 16.72

+0.10

0.02

14.82 14.75 14.73 14.70

−0.12

0.03

16.19 16.21 16.23 16.26

+0.07

0.03

A1 A2 A3 Amax

14.53 14.52 14.49 14.45

−0.08

0.03

15.77 15.81 15.83 15.84

+0.07

0.02

30

A1 A2 A3 Amax

14.41 14.36 14.35 14.30

−0.09

0.02

15.52 15.53 15.54 15.54

+0.02

0.02

40

A1 A2 A3 Amax

14.28 14.22 14.19 14.16

−0.12

0.03

15.34 15.34 15.34 15.35

+0.01

0.00

50

A1 A2 A3 Amax

14.15 14.11 14.10 14.08

−0.07

0.03

15.21 15.21 15.21 15.21

0.00

0.01

60

A1 A2 A3 Amax

14.08 14.05 14.03 14.00

−0.08

0.01

15.08 15.08 15.08 15.08

0.00

0.01

P (Mpa)

A

tp

5

A1 A2 A3 Amax

15.09 15.05 14.96 14.87

10

A1 A2 A3 Amax

20

tp (Amax –A1 )

longitudinal and shear waves through the rock sample. A similar method and apparatus were applied by Timur (1977) and Stewart et al (1983). It is known that there is no principle difference between these methods. Despite this there are some nuances, which I now consider. In the resonant method the uninterrupted movement of the sample material comes from an earth shaker. In the pulse method the material is exposed to a short affect, i.e. the shake time is low in comparison to the uninterrupted movement. The shaking of the rock in the resonant method occurs for several minutes. As a result the longer dynamic affect can change the mechanic characteristics of the medium. After vibration the rock becomes less rigid, i.e. the rock softens and the elastic modulus is lower. This leads to a decrease in wave velocity and increase in attenuation. The change of the mechanic characteristic of material after a longer shake is possible for several reasons. The accumulation of the residual deformation is possible as it takes place, for example, when metal fatigue occurs. The microcracks are opened and closed under the influence of an acoustic stress wave (Morris et al 1979). Johnson et al (1996) have observed relaxation effect in rocks after a frequency sweep. It may take several minutes for a rock to return to its original ‘linear’ elastic state after the frequency 304

tp (Amax –A1 )

t (error)

sweep. After relaxation a rock can return to previous nonelastic behaviour; perfect memory is lost as a result of either exceeding the previous absolute maximum stress or extensive cycling (Holcomb 1981). The relaxation means that there is restoration of the mechanical properties after the long influence of the shake. Then it is possible to assume that the change of mechanical properties can take place during the sinusoidal vibrations. Relaxation is the evident sign of inelasticity. The nonelastic rock can be softened drastically (the decrease of Q) by the longer shake. There is one remark yet. Apparently, the wave velocity decrease and the attenuation increase received by the resonant method are connected with the peak wave characteristic but they are not connected with the arrival time characteristic. Then it is all right and there is no contradiction. According to our data the wave velocity determined by the peak time decreases with increasing amplitude; the attenuation determined by the length of the wavefront increases with increasing amplitude. As is known, the attenuation can be  determined using the length of the wavefront t f = tp A −tp 0 (Blair and Spathis 1982, Mashinskii and D’yakov 1999). Thus in our case as well as in works of other authors the attenuation increases with strain amplitude. This is because

Variants of the strain-amplitude dependence of elastic wave velocities in rocks under pressure

the t f increases with increasing amplitude. The behaviour of the peak velocity obeys the known law as well. Finally, I consider the main reason why our data show so much difference with other data. This is the lithology and microstructure difference. Most previous experiments have been performed on sandstones (Berea, Navajo, Meule, Fontainebleau, Massilon) and other rocks (limestones, shales and granite). As for dolomite, I have not found any previous experimental data. The unusual behaviour of dolomite can be explained by the inelastic character of this rock. In this case, the strain increment with increasing stress goes with increasing rate, whereas in the other case the rate of increment of strain decreases with increasing stress. Such behaviour is possible owing to the microplastic component of inelasticity, though the other inelastic mechanism is not excluded. The study of rock deformation by the method of separation of the elastic component from the inelastic component shows that the modulus can increase with increasing stress (Mashinskii 2001). The non-standard change of the modulus is possible also with decreasing stress. If it takes place, then in the theory of Guyer et al (1995), showing that the behaviour of minor hysteretic loops (dynamic effect) is qualitatively similar to the behaviour of the larger (quasistatic) loop, it is necessary to introduce the corresponding addition. The minor hysteretic loops must also change their slope with increasing stress as well as with strain amplitude. Since the effect of the non-standard behaviour of the modulus takes place in the quasistatic regime of the deformation, a natural question arises about the legitimacy of this effect in dynamics. Can such a situation occur in the dynamic regime of the deformation? There is no problem at least for the microplastic component as it is the timeindependent component. Microplasticity of some materials takes place both in statics and dynamics to frequencies of 10 MHz. To confirm this I cite the rare example of direct measurement of velocity increase with increasing amplitude in figure 15 (a plot from Mashinskii et al (1999)). The propagation of longitudinal waves (f = 500 Hz) in a sand– clay rock was studied in situ. An evident increase of wave velocity with amplitude is observed, which is 3% (an accuracy of 0.7%). There is a hysteretic intersection and residual component of wave velocity. In addition I have noticed yet another indirect fact, which was described in Johnston and Toksoz (1980b). The Q factor of Plexiglas increases with strain amplitude by 12% in the strain range 1 × 10−6 to 3 × 10−5. This means that the attenuation decrease and the wave velocity increase. Certainly, it is the slight increase of Q factor, therefore it can be considered as a tendency only. However, if the amplitude increases, then the tendency can proceed with the law. In summary, the following conclusions can be drawn. The strain-amplitude dependence of P- and S-wave velocities in dry dolomite was studied. The acoustic pulse amplitude corresponds to a strain of 10−6; the uniaxial pressure range is from 1 to 60 MPa. The change of compressional velocity with amplitude is established in the pressure range 5–20 MPa. The change of shear velocity with amplitude is absent. On the one hand, the wave velocity determined by the measurement of

Figure 15. Dependence of the arrival time (compressional velocity) on strain amplitude in a sand–clay rock in situ. L is the distance between source and receiver. The amplitude increase and decrease are distinguished by symbols.

the first arrival time increases with increasing amplitude. On the other hand, the velocity determined by the measurement of the peak-amplitude time decreases with amplitude. The unusual behaviour of the wave velocity with amplitude is presumably explained by features of the rock inelasticity (at least microplasticity), which require further research.

5. Conclusion The results of this study show that wave velocity can increase with increasing amplitude. I suppose that specific features of the stress–strain curves influence the behaviour of the elastic modulus as a function of stress. Undoubtedly, there are ‘fine’ aspects concerning the curves graphically representing the equation of state. These are multilevel curves, which can change curvature in accordance with the level of the energy applied. These investigations provide an impetus for further research into the nonlinear propagation of seismic waves. I want to report that new data about the strain-amplitude dependence were recently obtained (a paper in press). The correctness of the above conclusions is confirmed by in situ observations 305

E I Mashinskii

in dry and saturated rocks. In future, it is necessary to explore the informativeness of hysteretic parameters in relation to applied problems of differentiation of rocks in terms of their material composition, porosity and fluid saturation.

Acknowledgments These experiments were performed at the Siberian Institute of Geology and Geophysics. The author thanks G N D’yakov for this work. I wish to thank Paul Johnson for valuable remarks, which essentially improved this paper.

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