LINK BETWEEN ATTENUATION AND VELOCITY DISPERSION

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1 LINK BETWEEN ATTENUATION AND VELOCITY DISPERSION Jack Dvorkin Stanford University and Rock Solid Images April 25, 2005 SUMMARY In a viscoelastic sample, the causality principle links the attenuation of a stress wave to the changes in its speed as the frequency varies. This rigorous theoretical link helps explain the velocity-frequency dispersion and attenuation commonly observed in rocks with fluid. Usually, the attenuation is maximum in the frequency range where the velocity change is most rapid. Experiments indicate that a link between velocity change and attenuation persists even in cases where both are functions of a different variable, not necessarily frequency. This link can be observed in a sample as well as among several samples. The variables responsible for velocity dispersion and attenuation include the pore-fluid viscosity; temperature; confining and pore pressure; strain amplitude at the wave; water saturation; methane hydrate content; scale of inhomogeneity; and porosity. Perhaps a universal principle exists, not formulated as yet, that links velocity dispersion to attenuation for a multitude of underlying physical variables. CAUSALITY IN VISCOELASTICITY In a linear viscoelastic body, the stress σ is related to strain ε as σ(t) = m(t τ)ε(τ) dτ m ε, (1) where t is time and m is a kernel. Then in the Fourier domain σ (ω) = M(ω) ε (ω), (2) 1

2 E 1 E 2 where M(ω) is the complex modulus and ω is the angular frequency. Causality of the signal requires that there be a connection between the complex modulus M(ω) and inverse quality factor Q 1 (ω), which are called the Kramers-Kronig relations: ω Q 1 (ω) = πm R (ω) M R (ω) M R (0) = ω π M R (α) M R (0) dα α α ω, Q 1 (α)m R (α) dα α α ω, (3) where M R (ω) is the real part of M(ω). One of the simplest representations of a viscoelastic body is the standard linear solid (SLS) which is a combination of elastic springs and a dashpot (Figure 1). η Figure 1. A schematic representation of the standard linear solid. For SLS M(ω) = M [M 0 + i ω ω r M M 0 ] M + i ω ω r M M 0, (4) where M 0 = E E 2 1, M = E 2, ω r = E (E + E ) 1 2 1, (5) E 2 + E 1 η and E 1, E 2, and η are the elastic moduli and viscosity associated with the two springs and dashpot as shown in Figure 1. The inverse quality factor for SLS is directly related to the complex modulusfrequency dispersion through equation 2

3 Q 1 = E 2 ω /ω r E 1 (E 2 + E 1 ) 1+ (ω /ω r ). (6) 2 Indeed, if the viscous dashpot has η = 0 then ω r = and from Equation (6) Q 1 = 0. Simultaneously, from Equation (4) the complex modulus M(ω) = M 0 and is constant. In other words, if there is no modulus-frequency dispersion there is no attenuation and vice versa. An example of modulus-frequency dispersion and the corresponding attenuation according to the SLS model is shown in Figure 2. Figure 2. An example of SLS modulus-frequency dispersion and corresponding inverse quality factor. In SLS the maximum inverse quality factor is directly linked to the modulus dispersion which once again illustrates the causality link between the two characteristics of wave propagation: 1 Q max = 1 2 M M 0 M M 0. (7) EXPERIMENTAL EVIDENCE OF VISCOELASTIC CAUSALITY Although it is difficult to conduct measurements on the same rock sample in a wide frequency range, it has been documented that the high-frequency velocity is larger than the low-frequency velocity as predicted by Gassmann s equation (e.g., Mavko and Jizba, 1991). The corresponding bell-shaped dependence of attenuation on frequency has been 3

4 documented by, e.g., Murphy (1982) on a Massillon sandstone sample (Figure 3). Lucet (1989) compares velocity and attenuation data on a limestone sample at sonic (resonant bar) and ultrasonic frequency and observes velocity-frequency dispersion and attenuation (Figure 4), although, the classical dispersion shapes are not evident in this figure simply due to a small (two) number of samples. Figure 3. The P- and S-wave inverse quality factor versus frequency in Massillon sandstone (after Murphy, 1982). The S-wave data are marked by white crosses. The upper set of curves is for 90% water saturation, the middle set of curves is for 0.1% water saturation (dry sample exposed to atmospheric humidity), the bottom set of curves is for the oven-dry sample. Figure 4. The P- and S-wave velocity and inverse quality factor versus frequency in water-saturated limestone (after Lucet, 1989). The S-wave data are marked by white crosses. OTHER EXPERIMENTAL EVIDENCE OF CAUSALITY It appears that attenuation is linked to velocity dispersion where both vary not necessarily due to frequency change but as functions of many other variables. 4

5 Velocity Change and Attenuation Due to Pore-Fluid Viscosity A classical example is due to Nur (see Bourbie et al., 1987) where the velocity and attenuation simultaneously changed, following the pattern shown in Figure 2, as the viscosity of the pore fluid increased (Figure 5). In this example, the viscosity of the saturating fluid was manipulated by changing the temperature of the sample from 80 to 100 C. The viscosity (in fact the temperature) affect the velocity and attenuation in the same way as the frequency would in a viscoelastic body. Figure 5. Influence of the viscosity of the saturating fluid (glycerol) on the S-wave velocity and attenuation in Bedford limestone. Ultrasonic experiments. The viscosity variation was induced by temperature variation. The data are by Nur as shown in Bourbie et al. (1987). Velocity Change and Attenuation Due to Strain Amplitude The experimental data discussed here are due to Winkler (1979) and Murphy (1982). Both the velocity and attenuation varied versus the strain on the wavefront in dry rock (Figure 6, 7, and 8). The attenuation appears to be linked to the velocity-strain dispersion. The threshold strain above which the velocity starts to decrease and attenuation increase is approximately The strain of a seismic wave away from the source rarely exceeds this limit. Therefore, the examples shown in this section are of fundamental 5

6 significance in spite of their limited applicability to problems of seismic exploration. Figure 6. Extensional resonant-bar velocity and attenuation data for a dry Massillon sandstone sample. Zero confining pressure. Top velocity versus strain. Bottom the inverse quality factor versus strain. The sample has 0.22 porosity and 740 md permeability. The mineralogy is 0.88 quartz; 0.05 clay; 0.03 feldspar; and 0.04 amorphous silica. Attenuation and velocity changes are attributed to frictional losses. After Winkler (1979). Figure 7. Extensional resonant-bar velocity and attenuation data for a dry Berea-350 sandstone sample. 10 MPa confining pressure. Top velocity versus strain. Bottom the inverse quality factor versus strain. The sample has 0.22 porosity and 380 md permeability. The mineralogy is 0.82 quartz; 0.06 clay; 0.02 feldspar; and 0.06 amorphous silica. Attenuation and velocity changes are attributed to frictional losses. After Winkler (1979). 6

7 Figure 8. Shear resonant-bar modulus and attenuation data for a 92%-saturated Massillon sandstone at about 70 Hz. The sample has 0.23 porosity and 737 md permeability. The mineralogy is 0.88 quartz; 0.05 clay; 0.03 feldspar; and 0.04 amorphous silica. After Murphy (1982). Velocity Change and Attenuation Due to Pressure The connection between velocity dispersion and attenuation persists in examples where the velocity variation is due to the confining and/or pore pressure. The experimental data shown here are due to Winkler (1979), Yin (1992), and Prasad (2002). Figure 9. Extensional and shear-wave resonant-bar velocity and attenuation data for a dry Berea-350 sandstone sample versus confining pressure. The S-wave data are marked by crosses. Top velocity. Bottom the inverse quality factor. The sample is the same as used in experiments shown in Figure 7. After Winkler (1979). 7

8 Attenuation accompanies velocity change no matter why this change occurs: whether it is due to confining pressure in a dry sample, pore pressure in a gas-saturated sample, or pore pressure in water-saturated sand and shale (Figures 9 to 13). Figure 10. Shear-wave resonant-bar velocity and attenuation data for a dry Berea-350 sandstone sample versus differential pressure. Top velocity. Bottom the inverse quality factor. The lower attenuation curve is for zero pore pressure and varying confining pressure. The upper attenuation curve is for constant confining pressure 350 bars and varying pore pressure of nitrogen gas. The velocity curves for these two cases are essentially the same. The sample is the same as used in experiments shown in Figure 7. After Winkler (1979). Figure 11. Same as Figure 10 but for extensional waves. After Winkler (1979). 8

9 Figure 12. Freeman silt, ultrasonic experiment at 30 MPa confining pressure and varying pore pressure. Top velocity (solid parallel to the bedding and dashed normal to the bedding). Bottom attenuation coefficient (solid parallel to the bedding and dashed normal to the bedding). After Yin (1992). Figure 13. High-porosity unconsolidated sand with brine. Top velocity (the upper curve is for the P- wave velocity while the lower curve is for the S-wave velocity, also marked by white-cross symbols). Bottom the inverse quality factor (the inverse S-wave quality factor is marked by white-cross symbols). After Prasad (2002). Velocity Change and Attenuation Due to Methane Hydrate Saturation The following empirical relations used in this example are due to Guerin and Goldberg (2002) and are based on full-waveform logging data in a Mallik methane hydrate site in 9

10 Canada: V p = S mh, V s = S mh ; Q p 1 = S mh, Q s 1 = S mh ; (8) where the velocity is in km/s and saturation in %. As the gas hydrate content in the pore space increases, so does the velocity and inverse quality factor. The S-wave inverse quality factor appears to exceed that for the P- waves (Figure 14). One explanation for this phenomenon is that it is due to a larger relative dispersion (ΔV /V ) of V s over the whole saturation range than of V p : ΔV p V p max(v p ) min(v p ) max(v p )min(v p ) = 0.566, ΔV s max(v ) min(v ) s s = (9) V s max(v s )min(v s ) Figure 14. Velocity and attenuation versus methane hydrate concentration of the pore space. The S- wave data are represented by dashed lines while the P-wave data are represented by solid lines. After Goldberg and Guerin (2002). Scale Effects on Velocity Dispersion and Attenuation In a layered system made of n individual layers of thickness d the elastic-wave velocity depends on the ratio of the wavelength ( λ) the thickness ( d). The smaller this ratio the faster the velocity. The upper, ray theory, velocity (V pray ) is calculated by assuming that the total travel time through the system is the sum of the travel times through individual layers. The lower, effective medium, velocity (V peffmed ) is calculated by assuming that 10

11 the effective elastic modulus ( M EffMed ) of the system is the harmonic average of the individual elastic moduli ( M i ). The corresponding equations are: 1 V pray n = n 1 1 V pi ; i=1 V peffmed = M EffMed /ρ peffmed, (10) 1 M EffMed n = n 1 1 M i, M i = ρ i V 2 pi, ρ peffmed = n 1 ρ i. i=1 i=1 n The transition from the low- λ /d to high- λ /d behavior occurs between λ /d = 2 and 20. The attenuation is maximum at this transition (Marion et al., 1994). A schematic behavior of the velocity and inverse quality factor versus the ratio is shown in Figure 15. d Figure 15. Velocity and attenuation versus the wavelength over layer thickness due to scale effect on wave propagation (schematic). A layered system is shown at the top. Once again, attenuation is maximum where the velocity changes the most. The behavior shown in Figure 15 is very similar to that shown for SLS in Figure 2. As the wavelength increases, the frequency decreases, and the velocity decreases. At the same time, the inverse quality factor reaches its maximum and then decreases again. 11

12 Porosity Effects on Velocity Dispersion and Attenuation Marion (1990) measured the P-wave velocity and attenuation in a suspension of glass beads in water for porosity between the critical porosity 0.4 and maximum possible porosity 1.0. The velocity gradually decreases with increasing porosity (Figure 16). Simultaneously, the amplitude of the signal increases (attenuation decreases). Maximum attenuation (minimum amplitude) is observed at the critical porosity. This is an example of a link between velocity dispersion and attenuation among several physical samples as opposed to that for a single sample subject to changing conditions. Figure 16. Suspension of glass beads in water, ultrasonic experiment. Top velocity. Bottom relative amplitude. After Marion (1990). Saturation Effects on Velocity Dispersion and Attenuation Perhaps the most compelling and practically important results are for velocity and attenuation variations versus the water saturation of a rock sample. A large number of measurements have been reported for various saturation conditions and at various scales, including core plugs, resonant bar, well logs, and seismic. Here we present only two resonant-bar examples. In the sandstone sample (Figure 17) there are two attenuation peaks for the extensional waves and one for shear waves. The low-saturation attenuation peaks for 12

13 both waves correspond to the abrupt velocity decrease as the rock s conditions change from the oven-dry state to humidified state. The high-saturation E-wave peak corresponds to the velocity increase as the saturation approaches 100%. Figure 17. Massillon sandstone, resonant bar experiment. Top velocity (E-wave and S-wave). Bottom the inverse quality factor (E-wave and S-wave). The S-wave data are marked by white crosses. After Murphy (1982). The attenuation peak in limestone (Figure 18) is at high saturation. It corresponds to the velocity increase as the saturation approaches 100%. Figure 18. Limestone, resonant bar experiment. Top E-wave velocity under two different saturation methods. Bottom the inverse quality factor. After Cadoret (1995). 13

14 Phase Transition Effects on Velocity Dispersion and Attenuation The final example is for velocity change and accompanying attenuation during steamwater transition. In this ultrasonic experiment, the temperature and confining pressure in a water-saturated sandstone sample were maintained at 150 C and 100 bar, respectively. As the pore pressure increased, the stem in the sample condensed into water. The velocity increased during this transition while the attenuation peaked. Figure 19. Velocity and amplitude changes during steam-water transition as the pore pressure varies. Steam condenses into water at about 5 bar. After DeVilbiss (1980). CONCLUSION Examples of lab data shown here indicate that the velocity dispersion and attenuation are linked to each other as the velocity varies due to various factors, including frequency, viscosity, scale, pressure, strain, porosity, saturation, and phase transition. If this link can be described by a common, as yet unknown, principle, it may lead to a better understanding and prediction of attenuation for practical remote sensing of the subsurface. REFERENCES Bourbie, T., Coussy, O., and Zinzner, B., 1987, Acoustic of porous media, Gulf 14

15 Publishing. Cadoret, T., 1993, Effet de la saturation eau/gas sur les proprietes acoustiques des roches, Ph.D. thesis, University of Paris. DeVilbiss, J.W., 1980, Wave dispersion and absorption in partially saturated rocks, Ph.D. thesis, Stanford University. Guerin, G., and Goldberg, D., 2002, Sonic waveform attenuation in gas hydrate-bearing sediments from the Mallik 2L-38 research well, Mackenzie Delta, Canada, JGR, 107, /2001JB Lucet, N., 1989, Vitesse et attenuation des ondes elastiques soniques et ultrasoniques dans les roches sous pression de confinement, Ph.D. thesis, University of Paris. Marion, D., Mukerji, T., and Mavko, G., 1994, Scale effects on velocity dispersion: From ray to effective medium theories in stratified media, Geophysics, 59, Marion, D.P., 1990, Acoustical, mechanical, and transport properties of sediments and granular materials, Ph.D. thesis, Stanford University. Mavko, G., and Jizba, D., 1991, Estimating grain-scale fluid effects on velocity dispersion in rocks, Geophysics, 56, Murphy, W.F., 1982, Effects of microstructure and pore fluids on the acoustic properties of granular sedimentary materials, Ph.D. thesis, Stanford University. Prasad, M., 2002, Acoustic measurements in unconsolidated sands at low effective pressure and overpressure detection, Geophysics, 67, Winkler, K.W., 1979, The effects of pore fluids and frictional sliding on seismic attenuation, Ph.D. thesis, Stanford University. Yin, H., 1992, Acoustic velocity and attenuation of rocks: Isotropy, intrinsic anisotropy, and stress-induced anisotropy, Ph.D. thesis, Stanford University. 15