Strain amplitude-dependent attenuation of P and S waves in dry and water-saturated sandstone under confining pressure

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1 Russian Geology and Geophysics 50 (2009) Strain amplitude-dependent attenuation of P and S waves in dry and water-saturated sandstone under confining pressure E.I. Mashinskii * Trofimuk Institute of Petroleum Geology and Geophysics, Siberian Branch of the RAS, 3 prosp. Akad. Koptyuga, Novosibirsk, , Russia Received 16 March 2008; received in revised form 6 October 2008; accepted 26 December 2008 Available online xx July 2009 Abstract The dependence of P- and S-wave attenuation on strain amplitude and frequency has been studied experimentally in dry and water-saturated sandstone samples under a confining pressure of 20 MPa. Attenuation of P and S reflections was measured at a frequency of 1 MHz in a strain range of ε ( ) The measured P-wave attenuation (Q 1 P ) in dry sandstone and S-wave attenuation (Q 1 S ) in dry and saturated samples turned out to be inversely proportional to strain amplitude while Q 1 P in saturated sandstone showed no strain dependence. The frequency-dependent attenuation spectra in dry and saturated sandstone differed considerably in S waves but were generally similar for P reflections. Strain amplitude variations were found out to influence the frequency dependence of attenuation and to shift the relaxation spectra of S waves toward high frequencies relative to those of P waves. As strain increased, the S-wave attenuation peak in saturated sandstone became notably (to 40%) narrower. The unusual strain amplitude-dependent behavior of attenuation may be due to joint action of viscoelastic and microplastic mechanisms. The reported results may be useful for improving geological interpretation of acoustic and seismic data. 2009, IGM, Siberian Branch of the RAS. Published by Elsevier B.V. All rights reserved. Keywords: anelasticity; nonlinear stress-strain relation; hysteresis; anelastic seismic parameters; relaxation spectra; strain amplitude dependence of wave velocity and attenuation Introduction * Corresponding author. address: MashinskiiEI@ipgg.nsc.ru (E.I. Mashinskii) Investigation into strain amplitude effects in acoustic and seismic soundings has a long history but an unusual amplitude-dependent behavior of wave velocities and attenuation is quite a recent discovery (Mashinskii, 2004, 2005a,b, 2007; Mashinskii et al., 1999; Zaitsev et al., 1999; Zaitsev and Matveev, 2006). Laboratory and field experiments based on cyclic strain amplitude measurements showed that wave velocities may increase and attenuation may decrease with strain. This contradicts the previous reports of velocity decrease and attenuation increase with increasing strain amplitude (Johnson et al., 1996; Ostrovsky and Johnson, 2001; Stewart et al., 1983; Tutuncu et al., 1994, 1998; Winkler, 1983, 1985; Winkler and Plona, 1982; Winkler et al., 1979; Zinszner et al., 1997). The discovered unusual effects extend the views of the behavior of seismic velocities and attenuation as a function of strain amplitude bringing to light all its possible ways (increase, decrease or independence of amplitude). These results were supported in a number of theoretical studies which put forward new attenuation mechanisms (Fillinger, 2006; Mashinskii, 2006a; Zaitsev et al., 2003; Zaitsev and Matveev, 2006). The update knowledge of strain-amplitude dependence of attenuation made basis for further studies with the same method, and new discoveries followed (Mashinskii, 2005a). Laboratory measured increase in the quality factor of sandstone and other rocks associated with greater pulse energy was accompanied by nonlinear attenuation decrease. More thorough investigation led to unexpected results concerning the behavior of frequency-dependent attenuation at different strain levels (Mashinskii, 2006a), such as shift of the P and S relaxation spectra along the axes of frequency and Q 1, changes in the relaxation spectra width, etc. These effects were known earlier for some polycrystals but with no regard to strain amplitude. For instance, S-wave attenuation in meltbearing olivine was found out to change as a function of frequency, as well as temperature, mean grain size, and melt content (Jackson et al., 2004), especially at low frequencies. At ultrasonic frequencies, P-wave attenuation showed a peak which shifted with fluid saturation change (Taylor and Knight, 2003).

2 E.I. Mashinskii / Russian Geology and Geophysics 50 (2009) Effects of this kind can be used to develop new diagnostic methods for acoustic logging or seismic exploration applications, etc. (Dvorkin et al., 2003; Mavko and Dvorkin, 2005). This paper presents an experimental study of strain amplitudedependent attenuation in dry and water-saturated sandstone under confining pressure, which has important implications for dispersion mechanisms. Instruments and methods Laboratory tests were applied to sandstone samples (recovered from a depth of 2250 m) at a confining pressure of 20 MPa and at room temperature. The samples were cylinders 4 cm in diameter and 2 cm in length, with a density of 2.2 g/cm 3 and a porosity of 15%, which were dry or saturated (about 70% water). The tests were carried out with standard instruments as in (Winkler, 1983; Jones, 1995; Mashinskii, 2006b). The physical model included three layers, with the measured rock placed between two Be bronze layers that acted as a delay line and a dampfer, their composition providing identical S and P reflections from interfaces. Ultrasonic signals at a frequency of 1 MHz were excited and received using piezoceramic transducers polarized along the P and S waves, each transducer being a source-receiver couple. Attenuation is given by (Winkler, 1983) Q 1 = αv/8.686πf = αλ/8.686π, (1) where α is the absorption coefficient, in db/m, V is the phase velocity, in m/s, and f is the frequency in Hz; α is found as (Winkler and Plona, 1982) α(ω) = L ln R 23 A top (f) R 12 A bot (f) (1 R 12 2 (f)), (2) where L is the sample double length, in m, A top (f) and A bot (f) are the Fourier amplitudes of reflections from the top and bottom of the sample, respectively, and R 12 (f) and R 23 (f) are, respectively, the top and bottom reflection coefficients. With the boundaries being identical in our case, R 12 (f) = R 23 (f). The reflection coefficient is given by R(f) = ρ r V r (f) V b ρ b (f) ρ r V r (f) + ρ b V b (f), (3) where ρ r and ρ b are the densities, in kg/m 3, V r (f) and V b (f) are the wave velocities, in m/s, in rock and Be bronze, respectively. The experimental procedure was as follows. Attenuation was measured at different strain amplitudes that changed in a closed cycle. First it increased discretely from the minimum to the maximum and then decreased back to the initial (minimum) value: ε min = ε 1 ε 2 ε max = ε 6 ε 1. The relative strain values in the signal were ε 1 = , ε 2 = , ε 3 = , ε 4 = , ε 5 = , and ε 6 = (i.e., these are relative microstrains). Thus, the full cycle (ε 1 ε 6 ε 1 ) included eleven amplitudes (six increasing and five decreasing ones). Attenuation was measured at each strain level, and the spectra were calculated in the bandwidth f min max = MHz free from diffraction effects. Range of strain amplitudes and justification of measurements. The strain amplitude was found as (Adushkin et al., 1999): ε max = υ/v = 2πu/λ, where υ is the velocity of particles, V is the wave velocity, u is the displacement of particles, and λ is the wavelength. The instruments we used are not designed to measure directly displacement or strain. Therefore, strain was estimated from the known electric signal and parameters of the piezoelectric transducer (Young s modulus, permittivity, electromechanic coupling coefficient, and piezoelectric strain constant h). The value h (V/m), the size of the confined piezoelectric plate, and Young s modulus control the resonance frequency and strain of the plate (or its absolute displacement). The source voltage was from 10 to 60 V (six strain amplitudes at every 10 V). The displacement of rock particles after the signal passed through the buffer and the sample was about m at the minimum strain amplitude and m at the maximum amplitude, according to the voltage. The relative strain in the signal was (0.3 2) 10 6, which corresponded to strain amplitudes reported in (Tutuncu et al., 1994; Winkler and Plona, 1982) for similar instruments. The objective of this study was to investigate qualitatively the behavior of attenuation as a function of strain amplitude. The knowledge of the approximate amplitude of relative strain appears to be quite sufficient. The focus was on the change of attenuation at different strains relative to the initial strain ε min = ε 1, given (along the x axis) as the relative value ε/ε min, where ε is the current strain and ε min is the calculated strain. In fact, they were relative measurements within a certain range (ε max /ε min = 6) with a single variable parameter, other conditions being invariable. Relative variations of attenuation caused by changing strain amplitude were reliable within the studied range. Therefore, the error in the absolute amplitude of relative strain was of minor significance for understanding the strain amplitude dependence of attenuation and the respective relaxation spectra. The strain range of applied in the reported tests is known from many publications to be a zone of nonlinear effects. Experimental results Figure 1 shows strain amplitude dependence of P- and S-wave attenuation at the main frequency. Attenuation appears to be almost the same in dry and wet (water-saturated) samples. Attenuation of P wave does not change much with strain amplitude while the behavior of S-wave attenuation is quite different. As strain grows, Q S 1 decreases monotonically for 8%, and the absolute attenuation in wet sandstone is three times as high as that in dry samples. The relaxation spectra of P-wave attenuation (Fig. 2) have a peak, and the frequency dependences of attenuation in dry

3 728 E.I. Mashinskii / Russian Geology and Geophysics 50 (2009) Fig. 2. P-wave attenuation as a function of frequency at six strain amplitude levels in dry and saturated sandstone. 1 6, dry sandstone, amplitudes successively from one to six; 7 12, saturated sandstone, amplitudes successively from one to six. Fig. 1. P- and S-wave attenuation as a function of strain amplitude in dry (1, 3) and saturated (2, 4) sandstone at a confining pressure of 20 MPa (here and below). and saturated sandstone are different at frequencies lower and higher than the peak. Namely, the curves for dry and wet sandstone virtually coincide and displace almost parallel to one another at high frequencies but diverge at low frequencies. Water saturation destroys the attenuation peak, and the displacement at growing strain is not parallel; there is a tendency of another peak forming, with a faster attenuation, in the low-frequency domain. The S-wave attenuation spectra (Fig. 3) for dry and water-saturated sandstone likewise have a peak. Saturating sandstone with water increases attenuation (by a factor of four) and shifts the peak toward high frequencies. As in the case of P-wave attenuation, attenuation of S waves decreases as strain amplitude increases. Furthermore, the attenuation peak in wet sandstone narrows down, i.e., the rock quality factor increases, while dry sandstone shows no peak width change. The most important result of the study is that the S-wave attenuation peak changes its width as strain amplitude changes in wet sandstone (Fig. 4). The plot shows the ratio of the peak width to frequency as a function of strain amplitude at level 0.7: [ f 0.7 /f att. peak ](ε 1 6 ). As strain amplitude increases, this ratio remains invariable in dry sandstone but decreases linearly Fig. 3. S-wave attenuation as a function of frequency at six strain amplitude levels in dry and wet sandstone. 1 6, dry sandstone, amplitudes successively from one to six; 7 12, saturated sandstone, amplitudes successively from one to six. Fig. 4. Ratio of attenuation peak width to peak frequency as a function of S-wave amplitude in dry and saturated sandstone. 1, dry sandstone; 2, saturated sandstone; 3, ratio of peak widths in saturated to dry sandstone.

4 E.I. Mashinskii / Russian Geology and Geophysics 50 (2009) in wet sandstone, for 27% within the given strain range. Another curve in the same figure corresponds to relative change of the peak width for wet and dry sandstone samples, and the difference is rather high (33%). Thus, the attenuation peak width is a good indicator of fluid saturation in sandstones. Note that the reported effects are valid within the given strain amplitude and frequency ranges. Discussion The previous (Mashinskii, 2005a, 2006a) and present studies allow some inferences on the responses of P- and S-wave attenuation to strain amplitude change. Attenuation at the main frequency obeys the law Q 1 P, S kε n, (4) where k is a dimensionless value and n 1, i.e., attenuation is inversely proportional to strain amplitude. The latter is calculated with a simple empirical relationship for attenuation with hybrid relaxation-hysteresis mechanism (Arzhavitin, 2004; Mashinskii, 2006a). In our case, for the used range of strain amplitudes, n varies from to Therefore, attenuation changes quite slowly, which is evident from a small slope of curves in Fig. 1. The largest n = corresponds to a 8 10% change of attenuation as strain amplitude grows from minimum to maximum, or six times. However, the strain-amplitude dependent change of attenuation is faster in saturated than in dry sandstone. This is, to a certain extent, a diagnostic feature to bear in mind in further investigation. It is important that the rate of strain-dependent attenuation change is specific to each rock. In some rocks these rates may be twice the above values (16 18% against 8 10%) within the same strain range (Mashinskii, 2005a). According to the reported data, S-wave attenuation is more sensitive to strain amplitude change, with responses seen in displacement of the attenuation peaks along the x (y) axes and in peak width change. The effect of peak narrowing and broadening is a most promising indicator useful for the practice of discriminating rocks according to fluid saturation. Saturated and dry sandstones differ in their responses to strain change: the attenuation peak width changes significantly even within a small amplitude range in wet sandstone but remains invariable in dry samples. This effect has to be investigated in detail in the future. The influence of strain amplitude on the behavior of attenuation peak needs a theoretical background. The classical formula for the relaxation attenuation peak is (Dvorkin et al., 2003; Mavko et al., 1998) Q 1 ωτ (ω) = 1 + (ωτ) 2, (5) where ω is the angular frequency, τ = 1/ 2f att. peak is the relaxation time, and is the relaxation strength. As mentioned above, the attenuation peak width appears to be the most informative factor. In principle, the effect of its change is known in polycrystals but it may have several causes, e.g., temperature change. The Fuoss Kirkwood equation (Cordero et al., 2003) can extend equation (5) as the attenuation process in it is described by a spectrum rather than a single relaxation time τ: Q 1 1 (ω, T) = (ωτ) α +(ωτ) β, (6) where α = β < 1. In this case, α and β control the peak width in the low-temperature (where ωτ < 1) and high-temperature domains, respectively. Equation (6) can be updated as Q 1 1 (ω, ε) = (ε) (ωτ) α +(ωτ) β, (7) where ε is the changing strain amplitude, (ε) is the amplitude-dependent relaxation strength, and α and β are the strain-dependent parameters. Then, the strength (ε) is responsible for the peak attenuation Q 1 max, while α and β control the peak broadening-narrowing in response to strain amplitude change. This is yet the basic approach which requires solid experimental and theoretical support. Thus, the attenuation peak width can discriminate between wet and dry sandstones better than other strain-dependent parameters of attenuation. This fact has important implications for seismic exploration of hydrocarbon reservoirs. There are already some encouraging results in this field and some examples of using attenuation effects in ultrasonic logging. For instance, the Q P and Q S attenuation spectra were suggested to be used as diagnostic of reservoir and non-reservoir rocks (Dvorkin et al., 2003; Mavko and Dvorkin, 2005). It is reasonable to complement this approach with the data of the present study on strain-amplitude dependence of attenuation, which will provide new indicators and improve the performance of seismic exploration methods. Conclusions Laboratory tests that explored attenuation of acoustic waves in water-saturated and dry sandstone as a function of strain amplitude varying from 10 7 to 10 6 revealed unknown effects indicating a complex behavior of attenuation parameters, especially the relaxation spectra. The main results of the study can be summarized as follows. 1. S-wave relaxation spectra can provide more information on the degree of fluid saturation of sandstone than the spectra of P waves. 2. S-wave relaxation spectra differ in water-saturated and dry sandstone in such parameters as relaxation strength, peak frequency, and width of attenuation peak. 3. The S-wave attenuation peak width in water-saturated sandstone narrows down as strain amplitude increases while that in dry sandstone remains invariable. 4. The discovered sensitivity of the S-wave attenuation peak width to fluid saturation is a promising diagnostic factor useful in discriminating between water-saturated and dry sandstone.

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