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Shear wave attenuation in dry and saturated sandstone at seismic to ultrasonic frequencies
Int. J. Rock Mech. Min. Sci. & Geomech. Abso'. Vol.30, No.7, pp. 755-761, 1993 Printed in Great Britain
()148-91162/93 $6.00 + 0.00 Pergamon Press Ltd
Shear Wave Attenuation in Dry and Saturated Sandstone at Seismic to Ultrasonic Frequencies D. H. GREEN]' H. F. WANG~t B. P. BONNER*
Ultrasonic and forced-oscillation methods have been applied to measure shear attenuation in Berea sandstone and a fused glass bead sample in the frequency ranges of.03 to 100 Hz and 600 to 1000 kHz. The Qs of dry Berea sandstone at atmospheric pressure falls in the range 100 to 150for frequencies between .03 Hz and 20 Hz. The ultrasonic results show a much lower Qs (
INTRODUCTION Understanding relationships between seismic data and rock properties is essential to inferring the state of the subsurface from seismology studies. Elastic wave velocities have been proven to be indicative of lithology, porosity, and saturation [1-4]. Other rock properties such as pore microstructure, permeability to fluid flow, and pore fluid viscosity are less influential in determining the velocity of elastic waves. Rather, these properties have their primary effect on the anelastic aspects of seismic wave propagation. Rock anelasticity can be characterized by the mechanical quality factor Q, which is inversely proportional to the fractional decrease in wave energy density per cycle of harmonic loading. An alternative measure of anelasticity applicable to wave propagation is the attenuation coefficient ct, the factor controlling the exponential decrease in amplitude with distance. For Q > 10, these two measures are related by Q =
rig ~V
(i)
w h e r e f i s frequency (Hz) and V is wave velocity [5]. Measurements of attenuation in rock are typically carried out over relatively narrow bands of frequency spanning seismic (10"2-103 Hz), sonic (103-105 Hz), and ultrasonic (105-106 Hz) regions [6-8]. Since wave attenuation is most likely controlled by different processes in different frequency bands, it is not
surprising that measurements of rock anelasticity (Q-t or et) vary with the frequency of measurement [9]. U n f o r t u n a t e l y , quantitative comparisons of measurements made at different frequencies are difficult since the contrasting experiments are usually conducted using different samples. Sample variability can produce significant variation in measured Q for a particular rock type under a uniform set of experimental conditions. Few studies have examined the attenuation behavior of the same block of rock over the entire geophysical frequency range of 10.2 - 106 Hz [9-10]. Presented here are results of measurements of shear attenuation in samples of Berea sandstone under controlled conditions of pressure, saturation, and strain-amplitude. Measurements at 1 Hz to 80 Hz were made using a lowstrain torsional oscillator and ultrasonic methods were applied to obtain attenuation values at 600 kHz to 1.00 MHz. The shear mode was studied in part because there is some uncertainty as to whether or not extensional (and therefore also compressional) apparent moduli are cross-coupled with the slow-compressional modulus in fully-saturated rocks at the intermediate frequencies of resonant- bar techniques (103 Hz to < 105 Hz) [11-12]. Torsional resonant modes are not affected by this geometric influence. EXPERIMENTAL PROCEDURE
Sample Descriptions Three Berea sandstone samples were selected for this study. The Berea sandstone is a fine-to-medium grained Mississippian greywacke containing approximately 8 percent clay (predominantly kaolinite) and well-rounded and sub-angular grains ( - 150/x) with quartz overgrowths. The three samples studied here are
]'Department of Geological Sciences, Ohio University, Athens, Ohio 45701, USA :[:Department of Geology and Geophysics, University of Wisconsin, Madison, Wisconsin, 53706, USA *Lawrence Livermore National Laboratory, Livermore, California 94550, USA 755
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Fig. 1. Torsional oscillator apparatus for low-frequency Q measurements. Length of sample shown corresponds to 5 cm. The assembly shown here is placed into a chamber for controlled humidity experiments.
distinguished by their air-injection permeabilities of 100 rod, 300 rod, and 450 md and their corresponding porosities of 18.5%, 20.2%, and 21.3%. Although not measured, apparent grain size showed moderate increases with porosity. A synthetic sample of fused glass beads was also examined. The sample was made by sintering (and densifying) a well-packed assemblage of glass beads to achieve a uniform porous material having a homogenous solid component [13]. The constituent soda-lime glass beads were 95% spherical with diameters of 210 to 250 p and densities of 2460 + 20 kg m"3. The porosity of the resulting aggregate is 31.5%.
Low-Frequency Torsion Apparatus A torsional stress-strain apparatus is employed here to characterize the complex shear modulus at low frequencies (10.2 to 102 Hz) and at strains as low as lfY~. In operation, a sinusoidal torque is applied to a 9 mm by 50 mm cylindrical sample assembly by electromagnets (Figure 1). The relative rotation across the sample assembly is monitored and converted into sample shear strain. By monitoring the in-phase and out-of-phase strain response (with respect to the applied shear stress), the complex shear modulus can be determined and used to calculate the shear velocity and Qs of the sample. The torsion apparatus is designed for use within a controlled humidity chamber. The system produces values of shear velocity Vs and attenuation (Qs"l or ols) as
functions of water partial pressure, strain amplitude, and frequency. The apparatus contribution to the apparent attenuation has been determined by calibration experiments in which a 6061-T6 aluminum cylinder with Q > 105 was substituted for the Berea sample. This small correction is applied to the apparent loss to obtain the sample Q'~. These system effects become become large only at frequencies above 100 Hz, out of the range of the test frequencies used here.
High-Frequency Pulse-Echo Apparatus The ultrasonic procedure followed here is described in detail in another paper (Green and Wang [14]) and is similar to that developed by Winkler and Plona [15] for compressional waves. The pulse-echo method determines sample attenuation through comparison of two ultrasonic pulses, one of which has traveled through the rock. The other pulse serves as a reference from which to measure the amplitude of the damped wave. A schematic of the sample assembly is shown in Figure 2. During an experiment, the assembly is placed inside a pressure vessel and is subjected to a hydrostatic confining stress. The assembly consists of a cylindrical rock sample, typically 2.5 cm in length, bounded on either side by a relatively high-Q (low-attenuation) buffer material, usually steel. Opposite the sample, attached to one buffer (the coupling buffer) is a broadband piezoelectric shear transducer within a pressure-proof steel housing. The housing, buffers, and
ROCK MECHANICS IN THE 1990s
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E X P E R I M E N T A L RESULTS
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Fig. 2. Sample assembly used for high-frequency Q measurements (after Winkler and Plona [15]). Hardened-steel buffers were used in these measurements. rock sample are all 5.08 cm (2.00 in.) in diameter and are jacketed in flexible tubing to isolate the rock sample from the external pressure medium (hydraulic oil). The backing buffer contains a pore fluid port through which the pore pressure in a saturated sample may be controlled independently of the confining pressure. In operation, the transducer emits an ultrasonic shear pulse, part of which is reflected at the couplingbuffer/rock interface. This reflection, marked "A" in Figure 2, is detected by the same transducer and recorded by a digitizer. The remainder of the original pulse travels through the rock sample, is partially reflected by the rock/backing-buffer interface, and also returns to the transducer ("B" in Figure 2). Of the two pulses, only one has traveled through the rock sample but both have the same source and receiver effects. In processing the data, each waveform is Fourier transformed and corrected for diffraction and reflection/transmission effects. Shear-wave diffraction corrections have been published and were employed here where appropriate [16]. This method gives a shear attenuation coefficient accurate to within 0.5 dB/cm, equivalent to an uncertainty in Qs of approximately + 5 for the rocks considered here [14]. This method produces values of as and Qs which can be presented as functions of pressure (5-70 MPa), saturation (dry or saturated), and frequency (600-1000 kHz). Shear-wave velocities are also obtained in the course of the waveform analysis.
Ultrasonic Pulse-Echo Shear-Wave Propagation Pulse-echo shear waveforms were recorded at confining pressures of 5 MPa to 70 MPa using the dry 100 md sample, producing the shear attenuation spectra shown in Figure 3. There is a general increase in shear attenuation with frequency at all confining pressures, with the largest attenuation ( - 5 . 5 dB/cm) observed at the lowest applied pressure (5 MPa) and the lowest attenuation values (0.7 dB/cm) found at the highest pressures (Pc > 30 MPa). The pressure effect on as is most easily seen by examining a single frequency component. Figure 4 shows the variation of the attenuation coefficient with pressure for a 800 kHz shear wave (800 kHz being near the spectral maximum of the ultrasonic pulse magnitudes). For the dry i00 md Berea sample, a pronounced decrease in Us is seen, particularly at low pressures (Pc<20 MPa). At high pressures, attenuation approaches an asymptotic value near 1 dB/cm. Also shown in Figure 4 are results for the other dry samples at 20, 30, and 70 MPa as well as for watersaturated 450 mD Berea sandstone at effective pressures of 60 and 70 MPa (P~ffbeing defined here as the simple difference Pc-P~,,~). The data for the other dry samples follow the trend of the 100 md Berea but are more attenuating. The saturated 450 md Berea sample is much more lossy to the extent that the attenuated reflection could not be detected at effective pressures below 60 MPa (P~fr being defined here as the simple difference P c - P ~ ) . The saturated sample was tested at a constant confining pressure of 70 MPa, so these data (and also the dry-sample results) were obtained on an unloading path in which the pore pressure was raised.
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Fig. 3. High-frequency spectra of the shear-wave attenuation coefficient for the dry 100 md Berea sandstone at pressures from 5 to 70 MPa. The displayed frequencies represent the approximate half-maximum bandwidth of the shear pulses.
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Fig. 6. Shear-wave attenuation spectra for the dry 300 m d (solid)and the water-saturated450 m d Bcrea sample, and the synthetic sample (dashed; 20, 30, and 70 M P a top to bottom).
The shear wave velocity for these samples and pressures ranged from approximately 2200 m/s to 2600 m/s, the glass-bead sample displaying the lowest velocity due to its higher porosity. The velocity information can be combined with values o f the attenuation coefficient via (1) to give the shear quality factor Qs. Spectra o f Qs for the dry 100 md Berea sandstone are displayed in Figure 5 for pressures ranging from 5 to 70 MPa. Most noticeable is the decrease in Qs with frequency at high pressures. At low confining pressure, Qs is essentially
independent of wave frequency. In general, there is a large increase in Q with pressure below 20-30 MPa, followed by a more gradual increase in Q at higher pressures. Somewhat surprising is the absence of an asymptotic high-pressure value for Qs at 70 MPa in the dry samples. The saturated Berea sample, on the other hand, shows the same low Q ( - 20) at both 60 and 70 MPa, a value comparable to Qs o f the dry samples at 5 MPa (Figure 6). Spectra o f Ors and Qs for the dry 300 md Berea and
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ROCK MECHANICS IN THE 1990s synthetic samples as well as the saturated 450 md Berea sample are shown in Figures 6 and 7. The dry samples display similar attenuation dependence on frequency. Evident in these figures is the much stronger shear wave attenuation in the saturated Berea sample as compared to the dry samples, even at high effective pressures. The least attenuation and the least pressure-dependence of attenuation, is observed in the fused glass bead sample.
Low-Frequency Torsion Results from the torsion experiments are shown in Figure 8 for the 300 md Berea sandstone, where Qs is given as a function of frequency. This sample was argon-saturated and tested at ambient pressure. Argon saturation places the rock in a very dry state, enabling examination of non-fluid-related anelasticity in the solid frame of the sandstone. The data span nearly four decades in frequency and show Qs to be in the range 100 to 130 below 10 Hz. Error estimates give 80% confidence intervals of 5:8 Qs=100. At higher frequencies, Qs shows an overall rapid increase as frequency becomes large. This intriguing result should be viewed as preliminary, awaiting corroboration from more tests employing higher frequencies, particularly those above 50 Hz. There is evidence for a system resonance at 130 Hz and this is possibly an influence here. DISCUSSION Attenuation mechanisms can be characterized by their sensitivities to state and loading parameters including pressure, temperature, pore fluid saturation, strain amplitude, and wave frequency. For example,
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759
theories invoking viscous flow of thin films of fluid at or near grain contacts predict a dependence of attenuation on pressure, degree of saturation, and frequency [1719]. Dissipation theories based on averaged relative flow of pore fluid do not explicitly take into account fluid/crack interactions and predict that attenuation will be influenced by effective pressure primarily through its effect on permeability [20]. Scattering of wave energy off of grains, pores, or larger heterogeneities is likely to become increasingly effective as frequency increases due to shortening of the wavelength relative to the scatterer size [21]. When more than one attenuation mechanism is operating in a rock, as will usually be the case, the most efficient dissipation process will dominate the anelastic behavior of the rock. The most likely mechanisms of ultrasonic attenuation in the dry Berea sandstone samples are scattering and microcrack-based fluid-film deformation. Winkler [8] found scattering to dominate compressional wave attenuation in dry Berea and fused glass bead samples in the ultrasonic range. This conclusion was supported by the f 4 dependence of the attenuation coefficient in both materials, a relationship predicted by simple weak scattering theories [21]. Winkler and Murphy [22] suggested that the scattering in the synthetic samples was controlled by the contrast in bulk moduli of the beads (grains) and the porous composite. If a similar contrast in shear moduli controls shear wave scattering, then to a first approximation, the composite/bead shear modulus ratio is determined solely by the porosity and is independent of saturant. This suggests that if such a scattering mechanism dominates attenuation in both dry and saturated samples, then the shear wave loss in a particular sample ought to be independent of its saturation state. Because we did not measure shear attenuation on the same sample in dry and saturated states, we cannot test this possibility directly. But the very large difference between the attenuation of the saturated and dry Berea samples would indicate that a significant saturation dependence exists. Also, a power-law frequency dependence o f f f or less is seen in the attenuation spectra, much milder than the fourthpower sensitivity predicted by simple acoustic scattering. It is not clear to what extent weak acoustic scattering theory applies to shear waves, but if acoustic and shear scattering are similar then weak scattering is not likely to be dominating shear wave attenuation in these samples. An alternative mechanism for wave dissipation in "dry" rocks invokes the viscous flow of fluid trapped in minute cracks about grain-contact asperities [19,23]. Although predictions of pressure and frequency dependences for this mechanism are rather sensitive to the assumed microcrack geometry, closure of these cracks by high confining pressures should bring about a pronounced decrease in attenuation. This is observed in our shear data (Figure 4). This mechanism may still be important at full saturation, although the dissipation behavior will be modified as the fluid will flow as a
760
ROCK MECHANICS tN TttE i990s
compressible liquid between cracks and equant pores. The shear attenuation results for saturated samples of the 450 md Berea sample are for 60 and 70 MPa effective pressure. Not surprisingly, these data show little pressure sensitivity at these high pressures, consistent with scattering, frictional, or crack-flow based attenuation theories. Although attenuation results for the dry 450 md sample are not yet available, they are very likely to be in the same range (as - 2 dB/cm; Figure 4) as those for the other dry samples at these pressures. Thus, the large attenuation in the saturated sandstones represents a very large saturation influence, as has been reported elsewhere [24]. This argues against the simple scattering due to grain-to-composite rigidity contrasts discussed above as a dominant loss mechanism in the saturated rock. Also, the frequency dependence of the attenuation coefficient, while greater than that for the dry samples, is o n l y f ~2"s. Dissipation involving fluid-flow is the most likely explanation for the strong shear wave damping. Whether this is due to a global (i.e., Blot) or local (i.e., microerack-localized) flow mechanisms cannot be determined without more observations at lower pressures. Klimentos and McCann [25] found that the global-flow meehanism accounted for their observations of compressional wave attenuation in clay-free sandstones at effective pressures above 40 MPa. Similarly, Mavko and Jizba [26] assume in their velocity dispersion model that only global-flow dissipation is operative at "high pressure" (90 MPa for the Navajo sandstone, > 50 MPa for the Fontainebleau sandstone) due to complete closure of compliant microcracks. Ongoing work is being directed towards obtaining a more complete attenuation/pressure relationship for both saturated and dry samples. The low-frequency Qs data for the 300 md Berea sample are similar to those reported elsewhere for dry sandstones below 100 Hz [27,28]. Yin, et al. [28] observed extensional Q-values greater than 100 at 100 Hz and greater than 200 at 700 Hz in dry Berea sandstone. Paffenholz and Burkhardt [27] reported constant Qs values of at least 100 in dry sandstones between .01 and 100 Hz. Unlike our findings, their results show a decrease in Qs above 100 Hz in sandstones as well as in dolomites and limestones. Our torsion measurements were made using a maximum shear strain amplitude of 5 x 10"6. This is below the threshold amplitude above which nonlinear dissipation begins to dominate in this sample. Although very little fluid is likely to exist in the argon-saturated rock, enough may be present to provide modulus reduction by adsorbtion at grain contacts or on pore surfaces [27,29]. Published reports of Qs measurements in dry Berea sandstone in the kilohertz range generally fall in the range 55 to 150 [9,30,31]. Combined with the data of Figure 8, these results suggest that a slight shear loss minimum (Qs maximum) may exist between 100 and 1000 kHz, although we have no direct measurements in
this band as yet to examine this possibility. With respect to the ultrasonic data, it is evident that the lowfrequency shear attenuation is much less than that observed at high frequencies, where Qs may be less than 10 at ambient pressure in dry rocks. It is unlikely that the same loss process is dominate at these oppposite ends of the frequency band. Ongoing work is being directed towards obtaining Qs data for these samples in the intermediate kilohertz range. CONCLUSIONS Our pulse-echo ultrasonic studies show that at high frequencies, shear-wave Q is less than 100 in Berea sandstone, with full water saturation producing low Qsvalues ( - 2 0 ) even at high effective pressures. The frequency-dependence for the shear attenuation coefficient t~s is less than f2.5 for either the dry or saturated state, indicating that weak acoustic scattering cannot describe the observed shear losses. At low confining pressures, Qs is nearly independent of frequency. A global "Biot-type" fluid-flow dissipation mechanism may be responsible for the large shear attenuation observed in the saturated Berea sandstone at high effective pressures where most narrow cracks would be closed. Fluid-film deformation at graincontact gaps is the most likely cause of dissipation in the room-dry samples. The low-frequency torsion experiments show that Qs in the dry Berea sandstone ranges between 100 and 150 over nearly three decades in frequency, from .03 to 20 Hz. There are indications that shear attenuation (Qs-l) may experience a minimum at higher frequencies, between 100 and 1000 Hz. Future experiments planned in this project will examine the anelasticity c,f our samples in this resonant-bar frequency range.
Acknowledgments. We thank Pat Berge for providing the sintered glass-bead samples. Useful reviews were provided by Mike Batzle and Joel Johnston. This research was supported by the Office of Basic Energy Sciences, Department of Energy under grant DE-FG02-91ER14194, Department of Energy contract W-6405-ENG-48 (B.P. Bonner), and by The Petroleum Research Fund grant 25144-GB2 (D.H. Green). REFERENCES 1.
2. 3.
Castagna, J.P., Batzle, M.L., and Eastwood, R.L., Relationships between compressional-wave and shearwave velocities in clastic silicate rocks, Geophysics, 50, 571-581 (1985). Robertson, J.D., Carbonate porosity from $/P travettime ratios, 53rd Ann. lnternat. Mtg., Soc. Explor. Geophys., Expanded Abstracts, 356-358 (1983). Gregory, A.R., Fluid saturation effects on dynamic elastic properties of sedimentary rocks, Geophysics, 41, 895-921 (1976).
ROCK MECHANICS IN THE 1990s 4.
Murphy, W.F., Acoustic measures of partial gas saturation in tight sandstones, J. Geophys. Ices., 89, 11549-11559 (1984). 5. Johnston, D.H., and Toksrz, M.N., Defmitions and terminology, in Seismic Wave Attenuation, edited by M.N. Toksrz and D.H. Johnston, 459 pp., Society of Exploration Geophysicists, Tulsa (1981). 6. MeKavanagh, B., and Stacey, F.D., Mechanical hysteresis in rocks at low strain amplitudes and seismic frequencies, Phys. Earth. Planet. Int., 8, 246-250 (1974). 7. Murphy, W.F., Effects of partial water saturation on attenuation in Massilon sandstone and Vycor porous glass, J. Acous. Soc. Am., 71, 1458-1468 (1982). 8. Winkler, K.W., Frequency dependentultrasonic properties of high-porosity sandstones, J. Geophys. Ices., 88, 94939499 (1983). 9. Lucet, N., and Zinszner, B., Effects of heterogeneities and anisotropy on sonic and ultrasonic attenuation in rocks, Geophysics, 57, 1018-1026 (1992). 10. Blair, D.P., A direct comparison between vibrational resonance and pulse transmission data for assessment of seismic attenuation in rock, Geophysics, 55, 51-60 (1990). 11. White, J.E., Biot-Gardner theory of extensional waves in porous rods, Geophysics, 51,742-745 (1986). 12. Mrrig, R., and Burkhardt, H., Experimental evidence for Biot-Gardner theory, Geophysics, 54, 524-527 (1989). 13. Berge, P., Wang, H.F., and Bonner, B.P., Pore pressure buildup coefficient and pore compressibility in synthetic and natural sandstones, Int. J. Rock Mech. Min. Sci. (this issue), (1993). 14. Green, D.H., and Wang, H.F., Shear-wave velocity and attenuation from pulse-echo studies of Berea sandstone, manuscript in preparation, (1993). 15. Winkler, K.W., and Plona, T.J., Technique for measureing ultrasonic velocity and attenuation spectra in rocks under pressure, J. Geophys. Res., 87, 10776-10780 (1982). 16. Green, D.H., and Wang, H.F., Shear wave diffraction loss for circular, piane-polarized source and receiver, J. Acous. Soc. Am., 90, 2697-2704 (1991). 17. Mavko, G.M., and Nur, A., Wave attenuation in partially saturated rocks, Geophysics, 44, 161-178 (1979). 18. Murphy, W.F., Winkler, K.W., and Kleinberg, R.L., Contact microphysics and viscous relaxation in sandstones, in Physics and Chemistry of Porous Media, edited by D.L. Johnson and P.N. Sen, American Institute of Physics, New York (1984). 19. Murphy, W.F., Winkler, K.W., and Kleinberg, R.L., Acoustic relaxation in sedimentary rocks: Dependence on grain contacts and fluid saturation, Geophysics, 51,757766 (1986). 20. Biot, M.A., Theory of propagation of elastic waves in a fluid-saturated porous solid., J. Acous. Soc. Am., 28,168191 (1956). 21. Sayers, C.M., Ultrasonic velocity dispersion in porous materials, J. Phys. D., 14, 413-420 (1981). 22. Winkler, K.W., and Murphy, W.F., Scattering in glass beads: effects of frame and pore fluid compressibilities, J. Acous. Soc. Am., 76, 820-825 (1984). 23. Tutuncu, A.N., and Sharma, M.M., The influence of fluids on grain contact stiffness and frame moduli in sedimentary rocks, Geophysics, 57, 1571-1582 (1992).
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24. Toks6z, M.N., Johnston, D.H., and Timur, A., Attenuation of seismic waves in dry and saturated rocks: I. Laboratory measurements, Geophysics, 44, 681-690 (1979). 25. Klimentos, T., and McCann, C., Relationships between compressional wave attenuation, porosity, clay content, and pormeability of sandstones, Geophysics, 55,998-1014 (1990). 26. Mavko, G., and Jizba, D., Estimating grain-scale fluid effects on velocity dispersion in rocks, Geophysics, 56, 1940-1949 (1991). 27. PaffenhoLz, J., and Burkhardt, H., Absorption and modulus measurements in the seismic frequency and strain range on partially saturated sedimentary rocks, J. Geophys. Ices., 94, 9493-9507 (1989). 28. Yin, C.-S., Batzlc, M.L., and Smith, B.J., Effects of partial liquid/gas saturation on extensional wave attenuation in Berea sandstone, Geophys. Ices. Left., 19, 1399-1402 (1992). 29. Murphy, W.F., Winklcr, K.W., and Kleinbcrg, R.L., Frame modulus reduction in sedimcntary rocks: The effect of adsorption on grain contacts, Geophys. ICes. Lett., 11,805-808 (1984). 30. Clark, V.A., Tittmann, B.R., and Spenccr, T.W., Effect of volatiles on attenuation (Q.I) and vcloeity in sedimentary rocks, J. Geophys. Ices., 85, 5190-5198 (1980). 31. Winkler, K.W., and Nut, A., Seismic attenuation: Effects of pore fluids and frictional sliding, Geophysics, 47, 115, (1982).
()148-91162/93 $6.00 + 0.00 Pergamon Press Ltd
Shear Wave Attenuation in Dry and Saturated Sandstone at Seismic to Ultrasonic Frequencies D. H. GREEN]' H. F. WANG~t B. P. BONNER*
Ultrasonic and forced-oscillation methods have been applied to measure shear attenuation in Berea sandstone and a fused glass bead sample in the frequency ranges of.03 to 100 Hz and 600 to 1000 kHz. The Qs of dry Berea sandstone at atmospheric pressure falls in the range 100 to 150for frequencies between .03 Hz and 20 Hz. The ultrasonic results show a much lower Qs (
INTRODUCTION Understanding relationships between seismic data and rock properties is essential to inferring the state of the subsurface from seismology studies. Elastic wave velocities have been proven to be indicative of lithology, porosity, and saturation [1-4]. Other rock properties such as pore microstructure, permeability to fluid flow, and pore fluid viscosity are less influential in determining the velocity of elastic waves. Rather, these properties have their primary effect on the anelastic aspects of seismic wave propagation. Rock anelasticity can be characterized by the mechanical quality factor Q, which is inversely proportional to the fractional decrease in wave energy density per cycle of harmonic loading. An alternative measure of anelasticity applicable to wave propagation is the attenuation coefficient ct, the factor controlling the exponential decrease in amplitude with distance. For Q > 10, these two measures are related by Q =
rig ~V
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w h e r e f i s frequency (Hz) and V is wave velocity [5]. Measurements of attenuation in rock are typically carried out over relatively narrow bands of frequency spanning seismic (10"2-103 Hz), sonic (103-105 Hz), and ultrasonic (105-106 Hz) regions [6-8]. Since wave attenuation is most likely controlled by different processes in different frequency bands, it is not
surprising that measurements of rock anelasticity (Q-t or et) vary with the frequency of measurement [9]. U n f o r t u n a t e l y , quantitative comparisons of measurements made at different frequencies are difficult since the contrasting experiments are usually conducted using different samples. Sample variability can produce significant variation in measured Q for a particular rock type under a uniform set of experimental conditions. Few studies have examined the attenuation behavior of the same block of rock over the entire geophysical frequency range of 10.2 - 106 Hz [9-10]. Presented here are results of measurements of shear attenuation in samples of Berea sandstone under controlled conditions of pressure, saturation, and strain-amplitude. Measurements at 1 Hz to 80 Hz were made using a lowstrain torsional oscillator and ultrasonic methods were applied to obtain attenuation values at 600 kHz to 1.00 MHz. The shear mode was studied in part because there is some uncertainty as to whether or not extensional (and therefore also compressional) apparent moduli are cross-coupled with the slow-compressional modulus in fully-saturated rocks at the intermediate frequencies of resonant- bar techniques (103 Hz to < 105 Hz) [11-12]. Torsional resonant modes are not affected by this geometric influence. EXPERIMENTAL PROCEDURE
Sample Descriptions Three Berea sandstone samples were selected for this study. The Berea sandstone is a fine-to-medium grained Mississippian greywacke containing approximately 8 percent clay (predominantly kaolinite) and well-rounded and sub-angular grains ( - 150/x) with quartz overgrowths. The three samples studied here are
]'Department of Geological Sciences, Ohio University, Athens, Ohio 45701, USA :[:Department of Geology and Geophysics, University of Wisconsin, Madison, Wisconsin, 53706, USA *Lawrence Livermore National Laboratory, Livermore, California 94550, USA 755
756
ROCK MECHANICS IN THE 1990s
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Fig. 1. Torsional oscillator apparatus for low-frequency Q measurements. Length of sample shown corresponds to 5 cm. The assembly shown here is placed into a chamber for controlled humidity experiments.
distinguished by their air-injection permeabilities of 100 rod, 300 rod, and 450 md and their corresponding porosities of 18.5%, 20.2%, and 21.3%. Although not measured, apparent grain size showed moderate increases with porosity. A synthetic sample of fused glass beads was also examined. The sample was made by sintering (and densifying) a well-packed assemblage of glass beads to achieve a uniform porous material having a homogenous solid component [13]. The constituent soda-lime glass beads were 95% spherical with diameters of 210 to 250 p and densities of 2460 + 20 kg m"3. The porosity of the resulting aggregate is 31.5%.
Low-Frequency Torsion Apparatus A torsional stress-strain apparatus is employed here to characterize the complex shear modulus at low frequencies (10.2 to 102 Hz) and at strains as low as lfY~. In operation, a sinusoidal torque is applied to a 9 mm by 50 mm cylindrical sample assembly by electromagnets (Figure 1). The relative rotation across the sample assembly is monitored and converted into sample shear strain. By monitoring the in-phase and out-of-phase strain response (with respect to the applied shear stress), the complex shear modulus can be determined and used to calculate the shear velocity and Qs of the sample. The torsion apparatus is designed for use within a controlled humidity chamber. The system produces values of shear velocity Vs and attenuation (Qs"l or ols) as
functions of water partial pressure, strain amplitude, and frequency. The apparatus contribution to the apparent attenuation has been determined by calibration experiments in which a 6061-T6 aluminum cylinder with Q > 105 was substituted for the Berea sample. This small correction is applied to the apparent loss to obtain the sample Q'~. These system effects become become large only at frequencies above 100 Hz, out of the range of the test frequencies used here.
High-Frequency Pulse-Echo Apparatus The ultrasonic procedure followed here is described in detail in another paper (Green and Wang [14]) and is similar to that developed by Winkler and Plona [15] for compressional waves. The pulse-echo method determines sample attenuation through comparison of two ultrasonic pulses, one of which has traveled through the rock. The other pulse serves as a reference from which to measure the amplitude of the damped wave. A schematic of the sample assembly is shown in Figure 2. During an experiment, the assembly is placed inside a pressure vessel and is subjected to a hydrostatic confining stress. The assembly consists of a cylindrical rock sample, typically 2.5 cm in length, bounded on either side by a relatively high-Q (low-attenuation) buffer material, usually steel. Opposite the sample, attached to one buffer (the coupling buffer) is a broadband piezoelectric shear transducer within a pressure-proof steel housing. The housing, buffers, and
ROCK MECHANICS IN THE 1990s
757
E X P E R I M E N T A L RESULTS
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Coupling
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Rock Sample Backing Buffer Fore Ffuid Inlet
Fig. 2. Sample assembly used for high-frequency Q measurements (after Winkler and Plona [15]). Hardened-steel buffers were used in these measurements. rock sample are all 5.08 cm (2.00 in.) in diameter and are jacketed in flexible tubing to isolate the rock sample from the external pressure medium (hydraulic oil). The backing buffer contains a pore fluid port through which the pore pressure in a saturated sample may be controlled independently of the confining pressure. In operation, the transducer emits an ultrasonic shear pulse, part of which is reflected at the couplingbuffer/rock interface. This reflection, marked "A" in Figure 2, is detected by the same transducer and recorded by a digitizer. The remainder of the original pulse travels through the rock sample, is partially reflected by the rock/backing-buffer interface, and also returns to the transducer ("B" in Figure 2). Of the two pulses, only one has traveled through the rock sample but both have the same source and receiver effects. In processing the data, each waveform is Fourier transformed and corrected for diffraction and reflection/transmission effects. Shear-wave diffraction corrections have been published and were employed here where appropriate [16]. This method gives a shear attenuation coefficient accurate to within 0.5 dB/cm, equivalent to an uncertainty in Qs of approximately + 5 for the rocks considered here [14]. This method produces values of as and Qs which can be presented as functions of pressure (5-70 MPa), saturation (dry or saturated), and frequency (600-1000 kHz). Shear-wave velocities are also obtained in the course of the waveform analysis.
Ultrasonic Pulse-Echo Shear-Wave Propagation Pulse-echo shear waveforms were recorded at confining pressures of 5 MPa to 70 MPa using the dry 100 md sample, producing the shear attenuation spectra shown in Figure 3. There is a general increase in shear attenuation with frequency at all confining pressures, with the largest attenuation ( - 5 . 5 dB/cm) observed at the lowest applied pressure (5 MPa) and the lowest attenuation values (0.7 dB/cm) found at the highest pressures (Pc > 30 MPa). The pressure effect on as is most easily seen by examining a single frequency component. Figure 4 shows the variation of the attenuation coefficient with pressure for a 800 kHz shear wave (800 kHz being near the spectral maximum of the ultrasonic pulse magnitudes). For the dry i00 md Berea sample, a pronounced decrease in Us is seen, particularly at low pressures (Pc<20 MPa). At high pressures, attenuation approaches an asymptotic value near 1 dB/cm. Also shown in Figure 4 are results for the other dry samples at 20, 30, and 70 MPa as well as for watersaturated 450 mD Berea sandstone at effective pressures of 60 and 70 MPa (P~ffbeing defined here as the simple difference Pc-P~,,~). The data for the other dry samples follow the trend of the 100 md Berea but are more attenuating. The saturated 450 md Berea sample is much more lossy to the extent that the attenuated reflection could not be detected at effective pressures below 60 MPa (P~fr being defined here as the simple difference P c - P ~ ) . The saturated sample was tested at a constant confining pressure of 70 MPa, so these data (and also the dry-sample results) were obtained on an unloading path in which the pore pressure was raised.
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Fig. 3. High-frequency spectra of the shear-wave attenuation coefficient for the dry 100 md Berea sandstone at pressures from 5 to 70 MPa. The displayed frequencies represent the approximate half-maximum bandwidth of the shear pulses.
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Fig. 6. Shear-wave attenuation spectra for the dry 300 m d (solid)and the water-saturated450 m d Bcrea sample, and the synthetic sample (dashed; 20, 30, and 70 M P a top to bottom).
The shear wave velocity for these samples and pressures ranged from approximately 2200 m/s to 2600 m/s, the glass-bead sample displaying the lowest velocity due to its higher porosity. The velocity information can be combined with values o f the attenuation coefficient via (1) to give the shear quality factor Qs. Spectra o f Qs for the dry 100 md Berea sandstone are displayed in Figure 5 for pressures ranging from 5 to 70 MPa. Most noticeable is the decrease in Qs with frequency at high pressures. At low confining pressure, Qs is essentially
independent of wave frequency. In general, there is a large increase in Q with pressure below 20-30 MPa, followed by a more gradual increase in Q at higher pressures. Somewhat surprising is the absence of an asymptotic high-pressure value for Qs at 70 MPa in the dry samples. The saturated Berea sample, on the other hand, shows the same low Q ( - 20) at both 60 and 70 MPa, a value comparable to Qs o f the dry samples at 5 MPa (Figure 6). Spectra o f Ors and Qs for the dry 300 md Berea and
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ROCK MECHANICS IN THE 1990s synthetic samples as well as the saturated 450 md Berea sample are shown in Figures 6 and 7. The dry samples display similar attenuation dependence on frequency. Evident in these figures is the much stronger shear wave attenuation in the saturated Berea sample as compared to the dry samples, even at high effective pressures. The least attenuation and the least pressure-dependence of attenuation, is observed in the fused glass bead sample.
Low-Frequency Torsion Results from the torsion experiments are shown in Figure 8 for the 300 md Berea sandstone, where Qs is given as a function of frequency. This sample was argon-saturated and tested at ambient pressure. Argon saturation places the rock in a very dry state, enabling examination of non-fluid-related anelasticity in the solid frame of the sandstone. The data span nearly four decades in frequency and show Qs to be in the range 100 to 130 below 10 Hz. Error estimates give 80% confidence intervals of 5:8 Qs=100. At higher frequencies, Qs shows an overall rapid increase as frequency becomes large. This intriguing result should be viewed as preliminary, awaiting corroboration from more tests employing higher frequencies, particularly those above 50 Hz. There is evidence for a system resonance at 130 Hz and this is possibly an influence here. DISCUSSION Attenuation mechanisms can be characterized by their sensitivities to state and loading parameters including pressure, temperature, pore fluid saturation, strain amplitude, and wave frequency. For example,
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759
theories invoking viscous flow of thin films of fluid at or near grain contacts predict a dependence of attenuation on pressure, degree of saturation, and frequency [1719]. Dissipation theories based on averaged relative flow of pore fluid do not explicitly take into account fluid/crack interactions and predict that attenuation will be influenced by effective pressure primarily through its effect on permeability [20]. Scattering of wave energy off of grains, pores, or larger heterogeneities is likely to become increasingly effective as frequency increases due to shortening of the wavelength relative to the scatterer size [21]. When more than one attenuation mechanism is operating in a rock, as will usually be the case, the most efficient dissipation process will dominate the anelastic behavior of the rock. The most likely mechanisms of ultrasonic attenuation in the dry Berea sandstone samples are scattering and microcrack-based fluid-film deformation. Winkler [8] found scattering to dominate compressional wave attenuation in dry Berea and fused glass bead samples in the ultrasonic range. This conclusion was supported by the f 4 dependence of the attenuation coefficient in both materials, a relationship predicted by simple weak scattering theories [21]. Winkler and Murphy [22] suggested that the scattering in the synthetic samples was controlled by the contrast in bulk moduli of the beads (grains) and the porous composite. If a similar contrast in shear moduli controls shear wave scattering, then to a first approximation, the composite/bead shear modulus ratio is determined solely by the porosity and is independent of saturant. This suggests that if such a scattering mechanism dominates attenuation in both dry and saturated samples, then the shear wave loss in a particular sample ought to be independent of its saturation state. Because we did not measure shear attenuation on the same sample in dry and saturated states, we cannot test this possibility directly. But the very large difference between the attenuation of the saturated and dry Berea samples would indicate that a significant saturation dependence exists. Also, a power-law frequency dependence o f f f or less is seen in the attenuation spectra, much milder than the fourthpower sensitivity predicted by simple acoustic scattering. It is not clear to what extent weak acoustic scattering theory applies to shear waves, but if acoustic and shear scattering are similar then weak scattering is not likely to be dominating shear wave attenuation in these samples. An alternative mechanism for wave dissipation in "dry" rocks invokes the viscous flow of fluid trapped in minute cracks about grain-contact asperities [19,23]. Although predictions of pressure and frequency dependences for this mechanism are rather sensitive to the assumed microcrack geometry, closure of these cracks by high confining pressures should bring about a pronounced decrease in attenuation. This is observed in our shear data (Figure 4). This mechanism may still be important at full saturation, although the dissipation behavior will be modified as the fluid will flow as a
760
ROCK MECHANICS tN TttE i990s
compressible liquid between cracks and equant pores. The shear attenuation results for saturated samples of the 450 md Berea sample are for 60 and 70 MPa effective pressure. Not surprisingly, these data show little pressure sensitivity at these high pressures, consistent with scattering, frictional, or crack-flow based attenuation theories. Although attenuation results for the dry 450 md sample are not yet available, they are very likely to be in the same range (as - 2 dB/cm; Figure 4) as those for the other dry samples at these pressures. Thus, the large attenuation in the saturated sandstones represents a very large saturation influence, as has been reported elsewhere [24]. This argues against the simple scattering due to grain-to-composite rigidity contrasts discussed above as a dominant loss mechanism in the saturated rock. Also, the frequency dependence of the attenuation coefficient, while greater than that for the dry samples, is o n l y f ~2"s. Dissipation involving fluid-flow is the most likely explanation for the strong shear wave damping. Whether this is due to a global (i.e., Blot) or local (i.e., microerack-localized) flow mechanisms cannot be determined without more observations at lower pressures. Klimentos and McCann [25] found that the global-flow meehanism accounted for their observations of compressional wave attenuation in clay-free sandstones at effective pressures above 40 MPa. Similarly, Mavko and Jizba [26] assume in their velocity dispersion model that only global-flow dissipation is operative at "high pressure" (90 MPa for the Navajo sandstone, > 50 MPa for the Fontainebleau sandstone) due to complete closure of compliant microcracks. Ongoing work is being directed towards obtaining a more complete attenuation/pressure relationship for both saturated and dry samples. The low-frequency Qs data for the 300 md Berea sample are similar to those reported elsewhere for dry sandstones below 100 Hz [27,28]. Yin, et al. [28] observed extensional Q-values greater than 100 at 100 Hz and greater than 200 at 700 Hz in dry Berea sandstone. Paffenholz and Burkhardt [27] reported constant Qs values of at least 100 in dry sandstones between .01 and 100 Hz. Unlike our findings, their results show a decrease in Qs above 100 Hz in sandstones as well as in dolomites and limestones. Our torsion measurements were made using a maximum shear strain amplitude of 5 x 10"6. This is below the threshold amplitude above which nonlinear dissipation begins to dominate in this sample. Although very little fluid is likely to exist in the argon-saturated rock, enough may be present to provide modulus reduction by adsorbtion at grain contacts or on pore surfaces [27,29]. Published reports of Qs measurements in dry Berea sandstone in the kilohertz range generally fall in the range 55 to 150 [9,30,31]. Combined with the data of Figure 8, these results suggest that a slight shear loss minimum (Qs maximum) may exist between 100 and 1000 kHz, although we have no direct measurements in
this band as yet to examine this possibility. With respect to the ultrasonic data, it is evident that the lowfrequency shear attenuation is much less than that observed at high frequencies, where Qs may be less than 10 at ambient pressure in dry rocks. It is unlikely that the same loss process is dominate at these oppposite ends of the frequency band. Ongoing work is being directed towards obtaining Qs data for these samples in the intermediate kilohertz range. CONCLUSIONS Our pulse-echo ultrasonic studies show that at high frequencies, shear-wave Q is less than 100 in Berea sandstone, with full water saturation producing low Qsvalues ( - 2 0 ) even at high effective pressures. The frequency-dependence for the shear attenuation coefficient t~s is less than f2.5 for either the dry or saturated state, indicating that weak acoustic scattering cannot describe the observed shear losses. At low confining pressures, Qs is nearly independent of frequency. A global "Biot-type" fluid-flow dissipation mechanism may be responsible for the large shear attenuation observed in the saturated Berea sandstone at high effective pressures where most narrow cracks would be closed. Fluid-film deformation at graincontact gaps is the most likely cause of dissipation in the room-dry samples. The low-frequency torsion experiments show that Qs in the dry Berea sandstone ranges between 100 and 150 over nearly three decades in frequency, from .03 to 20 Hz. There are indications that shear attenuation (Qs-l) may experience a minimum at higher frequencies, between 100 and 1000 Hz. Future experiments planned in this project will examine the anelasticity c,f our samples in this resonant-bar frequency range.
Acknowledgments. We thank Pat Berge for providing the sintered glass-bead samples. Useful reviews were provided by Mike Batzle and Joel Johnston. This research was supported by the Office of Basic Energy Sciences, Department of Energy under grant DE-FG02-91ER14194, Department of Energy contract W-6405-ENG-48 (B.P. Bonner), and by The Petroleum Research Fund grant 25144-GB2 (D.H. Green). REFERENCES 1.
2. 3.
Castagna, J.P., Batzle, M.L., and Eastwood, R.L., Relationships between compressional-wave and shearwave velocities in clastic silicate rocks, Geophysics, 50, 571-581 (1985). Robertson, J.D., Carbonate porosity from $/P travettime ratios, 53rd Ann. lnternat. Mtg., Soc. Explor. Geophys., Expanded Abstracts, 356-358 (1983). Gregory, A.R., Fluid saturation effects on dynamic elastic properties of sedimentary rocks, Geophysics, 41, 895-921 (1976).
ROCK MECHANICS IN THE 1990s 4.
Murphy, W.F., Acoustic measures of partial gas saturation in tight sandstones, J. Geophys. Ices., 89, 11549-11559 (1984). 5. Johnston, D.H., and Toksrz, M.N., Defmitions and terminology, in Seismic Wave Attenuation, edited by M.N. Toksrz and D.H. Johnston, 459 pp., Society of Exploration Geophysicists, Tulsa (1981). 6. MeKavanagh, B., and Stacey, F.D., Mechanical hysteresis in rocks at low strain amplitudes and seismic frequencies, Phys. Earth. Planet. Int., 8, 246-250 (1974). 7. Murphy, W.F., Effects of partial water saturation on attenuation in Massilon sandstone and Vycor porous glass, J. Acous. Soc. Am., 71, 1458-1468 (1982). 8. Winkler, K.W., Frequency dependentultrasonic properties of high-porosity sandstones, J. Geophys. Ices., 88, 94939499 (1983). 9. Lucet, N., and Zinszner, B., Effects of heterogeneities and anisotropy on sonic and ultrasonic attenuation in rocks, Geophysics, 57, 1018-1026 (1992). 10. Blair, D.P., A direct comparison between vibrational resonance and pulse transmission data for assessment of seismic attenuation in rock, Geophysics, 55, 51-60 (1990). 11. White, J.E., Biot-Gardner theory of extensional waves in porous rods, Geophysics, 51,742-745 (1986). 12. Mrrig, R., and Burkhardt, H., Experimental evidence for Biot-Gardner theory, Geophysics, 54, 524-527 (1989). 13. Berge, P., Wang, H.F., and Bonner, B.P., Pore pressure buildup coefficient and pore compressibility in synthetic and natural sandstones, Int. J. Rock Mech. Min. Sci. (this issue), (1993). 14. Green, D.H., and Wang, H.F., Shear-wave velocity and attenuation from pulse-echo studies of Berea sandstone, manuscript in preparation, (1993). 15. Winkler, K.W., and Plona, T.J., Technique for measureing ultrasonic velocity and attenuation spectra in rocks under pressure, J. Geophys. Res., 87, 10776-10780 (1982). 16. Green, D.H., and Wang, H.F., Shear wave diffraction loss for circular, piane-polarized source and receiver, J. Acous. Soc. Am., 90, 2697-2704 (1991). 17. Mavko, G.M., and Nur, A., Wave attenuation in partially saturated rocks, Geophysics, 44, 161-178 (1979). 18. Murphy, W.F., Winkler, K.W., and Kleinberg, R.L., Contact microphysics and viscous relaxation in sandstones, in Physics and Chemistry of Porous Media, edited by D.L. Johnson and P.N. Sen, American Institute of Physics, New York (1984). 19. Murphy, W.F., Winkler, K.W., and Kleinberg, R.L., Acoustic relaxation in sedimentary rocks: Dependence on grain contacts and fluid saturation, Geophysics, 51,757766 (1986). 20. Biot, M.A., Theory of propagation of elastic waves in a fluid-saturated porous solid., J. Acous. Soc. Am., 28,168191 (1956). 21. Sayers, C.M., Ultrasonic velocity dispersion in porous materials, J. Phys. D., 14, 413-420 (1981). 22. Winkler, K.W., and Murphy, W.F., Scattering in glass beads: effects of frame and pore fluid compressibilities, J. Acous. Soc. Am., 76, 820-825 (1984). 23. Tutuncu, A.N., and Sharma, M.M., The influence of fluids on grain contact stiffness and frame moduli in sedimentary rocks, Geophysics, 57, 1571-1582 (1992).
76l
24. Toks6z, M.N., Johnston, D.H., and Timur, A., Attenuation of seismic waves in dry and saturated rocks: I. Laboratory measurements, Geophysics, 44, 681-690 (1979). 25. Klimentos, T., and McCann, C., Relationships between compressional wave attenuation, porosity, clay content, and pormeability of sandstones, Geophysics, 55,998-1014 (1990). 26. Mavko, G., and Jizba, D., Estimating grain-scale fluid effects on velocity dispersion in rocks, Geophysics, 56, 1940-1949 (1991). 27. PaffenhoLz, J., and Burkhardt, H., Absorption and modulus measurements in the seismic frequency and strain range on partially saturated sedimentary rocks, J. Geophys. Ices., 94, 9493-9507 (1989). 28. Yin, C.-S., Batzlc, M.L., and Smith, B.J., Effects of partial liquid/gas saturation on extensional wave attenuation in Berea sandstone, Geophys. Ices. Left., 19, 1399-1402 (1992). 29. Murphy, W.F., Winklcr, K.W., and Kleinbcrg, R.L., Frame modulus reduction in sedimcntary rocks: The effect of adsorption on grain contacts, Geophys. ICes. Lett., 11,805-808 (1984). 30. Clark, V.A., Tittmann, B.R., and Spenccr, T.W., Effect of volatiles on attenuation (Q.I) and vcloeity in sedimentary rocks, J. Geophys. Ices., 85, 5190-5198 (1980). 31. Winkler, K.W., and Nut, A., Seismic attenuation: Effects of pore fluids and frictional sliding, Geophysics, 47, 115, (1982).