Estimates of velocity dispersion between seismic and ultrasonic frequencies

It is generally accepted that acoustic velocities in fluid‐saturated rocks vary with frequency. Evidence comes from experimental measurements and from theoretical causality arguments. We have developed a simple analysis technique that gives estimates of total velocity dispersion between zero frequency and any measurement frequency. The technique requires compressional (P) and shear (S) wave velocity measurements on dry and fully saturated rock. Assuming that the dry velocities are independent of frequency, the Biot‐Gassmann equations are used to calculate the zero‐frequency velocities in the fully saturated rock. Any difference between the measured velocities and the calculated zero‐frequency velocities is interpreted as evidence of dispersion. Application of this analysis technique to a variety c ultrasonic data sets gives consistent results. In many rocks, dispersion between zero frequency and ultrasonic frequencies is on the order of 10 percent at low effective stress, and it decreases to only a few percent at higher stresses. Dispersion varies with degree of saturation and with fluid viscosity in the same way as do low‐frequency attenuation measurements. The results are readily interpreted in terms of the same local‐flow absorption/dispersion mechanism that has been used to explain recent laboratory attenuation measurements. This apparent dispersion places upper bounds on seismic‐to‐sonic velocity differences. It also points out possible discrepancies between seismic velocities and ultrasonic laboratory measurements.

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The relation between seismic P‐ and S‐wave velocity dispersion in saturated rocks

Geophysics ◽

10.1190/1.1443537 ◽

1994 ◽

Vol 59 (1) ◽

pp. 87-92 ◽

Cited By ~ 35

Author(s):

Gary Mavko ◽

Diane Jizba

Keyword(s):

Wave Velocity ◽

Large Scale ◽

Velocity Dispersion ◽

Seismic Velocity ◽

Ultrasonic Velocities ◽

S Wave ◽

Local Flow ◽

Wave Velocities ◽

Shear Compliance

Seismic velocity dispersionin fluid-saturated rocks appears to be dominated by tow mecahnisms: the large scale mechanism modeled by Biot, and the local flow or squirt mecahnism. The tow mechanisms can be distuinguished by the ratio of P-to S-wave dispersions, or more conbeniently, by the ratio of dynamic bulk to shear compliance dispersions derived from the wave velocities. Our formulation suggests that when local flow denominates, the dispersion of the shear compliance will be approximately 4/15 the dispersion of the compressibility. When the Biot mechanism dominates, the constant of proportionality is much smaller. Our examination of ultrasonic velocities from 40 sandstones and granites shows that most, but not all, of the samples were dominated by local flow dispersion, particularly at effective pressures below 40 MPa.

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A dynamic elastic model for squirt-flow effect and its application on fluid-viscosity-associated velocity dispersion in reservoir sandstones

Geophysics ◽

10.1190/geo2019-0312.1 ◽

2020 ◽

Vol 85 (4) ◽

pp. MR201-MR212

Author(s):

Zhi-Qiang Yang ◽

Tao He ◽

Chang-Chun Zou

Keyword(s):

Velocity Dispersion ◽

Pore Space ◽

Pore Fluid ◽

Quantitative Prediction ◽

Elastic Model ◽

Fluid Viscosity ◽

Porous Rocks ◽

S Wave ◽

Flow Effect ◽

Reservoir Sandstones

Velocity dispersion is a common phenomenon for fluid-charged porous rocks and carries important information on the pore structure and fluid in reservoir rocks. Previous ultrasonic experiments had measured more significant non-Biot velocity dispersion on saturated reservoir sandstones with increasing pore-fluid viscosity. Although wave-induced local squirt-flow effect could in theory cause most of the non-Biot velocity dispersion, its quantitative prediction remains a challenge. Several popular models were tested to predict the measured velocities under undrained conditions, but they either underestimated the squirt-flow effect or failed to simultaneously satisfy P- and S-wave velocity dispersions (especially for higher viscosity fluids). Based on the classic double-porosity theory that pore space is comprised of mainly stiff/Biot’s porosity and minor compliant porosity, an effective “wet frame” was hypothesized to account for the squirt-flow effect, whose compliant pores are filled with a hypothesized fluid with dynamic modulus. A new dynamic elastic model was then introduced by extending Biot theory to include the squirt-flow effect, after replacing the dry-frame bulk/shear moduli with their wet-frame counterparts. In addition to yielding better velocity predictions for P- and S-wave measurements of different fluid viscosities, the new model is also more applicable because its two key tuning parameters (i.e., the effective aspect ratio and porosity of compliant pores) at in situ reservoir pressure could be constrained with laboratory velocity measurements associated with pore-fluid viscosities.

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Local and global fluid-flow effects on dipole flexural waves

Geophysics ◽

10.1190/geo2018-0040.1 ◽

2020 ◽

Vol 85 (1) ◽

pp. D1-D11

Author(s):

Elliot J. H. Dahl ◽

Kyle T. Spikes

Keyword(s):

Fluid Flow ◽

Low Frequency ◽

Summation Method ◽

Sensitivity Analyses ◽

Flexural Wave ◽

Biot Theory ◽

S Wave ◽

Local Flow ◽

Flow Effects ◽

Wave Induced

Wave-induced fluid flow (WIFF) can significantly alter the effective formation velocities and cause increasing waveform dispersion and attenuation. We have used modified frame moduli from the theory of Chapman together with the classic Biot theory to improve the understanding of local- and global-flow effects on dipole flexural wave modes in boreholes. We investigate slow and fast formations with and without compliant pores, which induce local flow. The discrete wavenumber summation method generates the waveforms, which are then processed with the weighted spectral semblance method to compare with the solution of the period equation. We find compliant pores to decrease the resulting effective formation P- and S-wave velocities, that in turn decrease the low-frequency velocity limit of the flexural wave. Furthermore, depending on the frequency at which the local-flow dispersion occurs, different S-wave velocity predictions from the flexural wave become possible. This issue is investigated through changing the local-flow critical frequency. Sensitivity analyses of the flexural-wave phase velocity to small changes in WIFF parameters indicate the modeling to be mostly sensitive to compliant pores in slow and fast formations.

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Estimating seismic velocities at ultrasonic frequencies in partially saturated rocks

Geophysics ◽

10.1190/1.1443587 ◽

1994 ◽

Vol 59 (2) ◽

pp. 252-258 ◽

Cited By ~ 62

Author(s):

Gary Mavko ◽

Richard Nolen‐Hoeksema

Keyword(s):

Large Scale ◽

Scale Effects ◽

Pore Space ◽

Degree Of Saturation ◽

Velocity Versus ◽

Seismic Velocities ◽

Pressure Gradients ◽

Pore Scale ◽

S Wave ◽

Partially Saturated

Seismic velocities in rocks at ultrasonic frequencies depend not only on the degree of saturation but also on the distribution of the fluid phase at various scales within the pore space. Two scales of saturation heterogeneity are important: (1) saturation differences between thin compliant pores and larger stiffer pores, and (2) differences between saturated patches and undersaturated patches at a scale much larger than any pore. We propose a formalism for predicting the range of velocities in partially saturated rocks that avoids assuming idealized pore shapes by using measured dry rock velocity versus pressure and dry rock porosity versus pressure. The pressure dependence contains all of the necessary information about the distribution of pore compliances for estimating effects of saturation at the finest scales where small amounts of fluid in the thinnest, most compliant parts of the pore space stiffen the rock in both compression and shear (increasing both P‐ and S‐wave velocities) in approximately the same way that confining pressure stiffens the rock by closing the compliant pores. Large‐scale saturation patches tend to increase only the high‐frequency bulk modulus by amounts roughly proportional to the saturation. The pore‐scale effects will be most important at laboratory and logging frequencies when pore‐scale pore pressure gradients are unrelaxed. The patchy‐saturation effects can persist even at seismic field frequencies if the patch sizes are sufficiently large and the diffusivities are sufficiently low for the larger‐scale pressure gradients to be unrelaxed.

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Estimating grain‐scale fluid effects on velocity dispersion in rocks

Geophysics ◽

10.1190/1.1443005 ◽

1991 ◽

Vol 56 (12) ◽

pp. 1940-1949 ◽

Cited By ~ 348

Author(s):

Gary Mavko ◽

Diane Jizba

Keyword(s):

High Frequency ◽

Velocity Dispersion ◽

Rock Properties ◽

S Wave ◽

Unknown Parameters ◽

Local Flow ◽

Essential Information ◽

Flow Effect ◽

Total Dispersion ◽

Aspect Ratios

The magnitude of the grain‐scale local flow effect on velocity dispersion in saturated rocks is quantified, by estimating the high‐frequency unrelaxed shear and bulk frame moduli, which are then combined with the Biot formulation to predict total dispersion. The method is relatively independent of assumptions about idealized pore geometries and unknown parameters such as pore aspect ratios. The local flow effect depends on the heterogeneity of pore stiffness, in particular the presence of compliant cracks and grain contacts; the pressure dependence of the dry rock properties is shown to contain the essential information about the distribution of pore stiffnesses needed to estimate the high‐frequency saturated behavior. To first order, the unrelaxed wet frame compressibility at any given pressure is shown to be approximately the dry frame compressibility at very high pressure; second order corrections add the additional compressibility gained by replacing an amount of mineral equal to the compliant pore volume with fluid. The method predicts that the difference between relaxed and unrelaxed shear compliance is simply proportional to that in bulk. The results for total dispersion (local flow plus Biot) explain quite well the measured P- and S-wave dispersion for a variety of saturated rocks.

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Calculation of matrix permeability from velocity and attenuation of ultrasonic S-wave

Journal of Geophysics and Engineering ◽

10.1093/jge/gxab065 ◽

2021 ◽

Vol 18 (6) ◽

pp. 984-994

Author(s):

Guangquan Li ◽

Chaodi Xie

Keyword(s):

Average Distance ◽

Low Frequency ◽

Wave Model ◽

Berea Sandstone ◽

Frequency Range ◽

S Wave ◽

Matrix Permeability ◽

Saturated Rock ◽

Novel Method ◽

Preferential Pathways

Abstract Previously, hydrogeologists and petroleum engineers use seepage experiments to measure permeability. This paper develops a novel method to calculate matrix permeability from velocity and attenuation of an ultrasonic S-wave. At first, permeability is derived as a function of frequency when an S-wave scans a fluid-saturated rock. Substituting the permeability into a previous S-wave model gives theoretical velocity and attenuation, in which the nexus parameter is the average distance of aperture representing pores. Fitting the predicted velocity and quality factor against the measured counterparts yields permeability in the full frequency range. For Berea sandstone, the inverted permeability at low frequency (0.0376 Darcy) is comparable to Darcy permeability (0.075 Darcy), confirming that Berea sandstone is homogenous. For Boise sandstone, the inverted permeability at low frequency is 0.0457 Darcy, much lower than Darcy permeability (1 Darcy). When S-wave scans the rocks, its velocity and attenuation are dominated by matrix pore throats and the inverted permeability represents matrix permeability. Unlike Berea sandstone, Boise sandstone has fractures and widely distributed grain diameters. The fractures and the large pores (due to large grain diameter) are preferential pathways that increase Darcy permeability far more than matrix permeability.

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On The Low-Frequency Elastic Response of Pierre Shale During Temperature Cycles

Geophysical Journal International ◽

10.1093/gji/ggab384 ◽

2021 ◽

Author(s):

Stian Rørheim ◽

Andreas Bauer ◽

Rune M Holt

Keyword(s):

Wave Velocity ◽

Low Frequency ◽

P Wave ◽

Rock Properties ◽

Fluid Viscosity ◽

S Wave ◽

Temperature Cycles ◽

Wave Velocities ◽

Young’S Moduli ◽

P Wave Velocity

Summary The impact of temperature on elastic rock properties is less-studied and thus less-understood than that of pressure and stress. Thermal effects on dispersion are experimentally observed herein from seismic to ultrasonic frequencies: Young’s moduli and Poisson’s ratios plus P- and S-wave velocities are determined by forced-oscillation (FO) from 1 to 144 Hz and by pulse-transmission (PT) at 500 kHz. Despite being the dominant sedimentary rock type, shales receive less experimental attention than sandstones and carbonates. To our knowledge, no other FO studies on shale at above ambient temperatures exist. Temperature fluctuations are enforced by two temperature cycles from 20 via 40 to 60○C and vice versa. Measured rock properties are initially irreversible but become reversible with increasing number of heating and cooling segments. Rock property-sensitivity to temperature is likewise reduced. It is revealed that dispersion shifts towards higher frequencies with increasing temperature (reversible if decreased), Young’s moduli and P-wave velocity moduli and P-wave velocity maxima occur at 40○C for frequencies below 56 Hz, and S-wave velocities remain unchanged with temperature (if the first heating segment is neglected) at seismic frequencies. In comparison, ultrasonic P- and S-wave velocities are found to decrease with increasing temperatures. Behavioural differences between seismic and ultrasonic properties are attributed to decreasing fluid viscosity with temperature. We hypothesize that our ultrasonic recordings coincide with the transition-phase separating the low- and high-frequency regimes while our seismic recordings are within the low-frequency regime.

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Retrieval of source-extent parameters and the interpretation of corner frequency

Bulletin of the Seismological Society of America ◽

10.1785/bssa07306a1499 ◽

1983 ◽

Vol 73 (6A) ◽

pp. 1499-1511

Author(s):

Paul Silver

Keyword(s):

Body Wave ◽

Amplitude Spectrum ◽

Low Frequency ◽

P Wave ◽

Corner Frequency ◽

S Wave ◽

Dislocation Source ◽

Statistical Measures ◽

Dislocation Models ◽

Planar Dislocation

Abstract A method is proposed for retrieving source-extent parameters from far-field body-wave data. At low frequency, the normalized P- or S-wave displacement amplitude spectrum can be approximated by |Ω^(r^,ω)| = 1 − τ2(r^)ω2/2 where r^ specifies a point on the focal sphere. For planar dislocation sources, τ2(r^) is linearly related to statistical measures of source dimension, source duration, and directivity. τ2(r^) can be measured as the curvature of |Ω^(r^,ω)| at ω = 0 or the variance of the pulse Ω^(r^,t). The quantity ωc=2τ−1(r^) is contrasted with the traditional corner frequency ω0, defined as the frequency at the intersection of the low- and high-frequency trends of |Ω^(r^,ω)|. For dislocation models without directivity, ωc(P) ≧ ωc(S) for any r^. A mean corner frequency defined by averaging τ2(r^) over the focal sphere, ω¯c=2<τ2(r^)>−1/2, satisfies ωc(P) > ωc(S) for any dislocation source. This behavior is not shared by ω0. It is shown that ω0 is most sensitive to critical times in the rupture history of the source, whereas ωc is determined by the basic parameters of source extent. Evidence is presented that ωc is the corner frequency measured on actual seismograms. Thus, the commonly observed corner frequency shift (P-wave corner greater than the S-wave corner), now viewed as a shift in ωc is simply a result of spatial finiteness and is expected to be a property of any dislocation source. As a result, the shift cannot be used as a criterion for rejecting particular dislocation models.

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Multifrequency full-waveform sonic logging in the screened interval of a large-diameter production well

Geophysics ◽

10.1190/geo2012-0328.1 ◽

2013 ◽

Vol 78 (5) ◽

pp. B243-B257 ◽

Cited By ~ 5

Author(s):

Majed Almalki ◽

Brett Harris ◽

J. Christian Dupuis

Keyword(s):

Phase Velocity ◽

Velocity Dispersion ◽

Large Diameter ◽

Low Frequency ◽

Production Well ◽

Frequency Wave ◽

Full Waveform ◽

Phase Velocity Dispersion ◽

Production Wells ◽

Sonic Logging

A set of field experiments using multiple transmitter center frequencies was completed to test the application potential of low-frequency full-waveform sonic logging in large-diameter production wells. Wireline logs were acquired in a simple open drillhole and a high-yield large diameter production well completed with wire-wound sand screens at an aquifer storage and recovery site in Perth, Western Australia. Phase-shift transform methods were applied to obtain phase-velocity dispersion images for frequencies of up to 4 kHz. A 3D representation of phase-velocity dispersion was developed to assist in the analysis of possible connections between low-frequency wave propagation modes and the distribution of hydraulic properties. For sandstone intervals in the test well, the highest hydraulic conductivity intervals were typically correlated with the lowest phase velocities. The main characteristics of dispersion images obtained from the sand-screened well were highly comparable with those obtained at the same depth level in a nearby simple drillhole open to the formation. The sand-screened well and the open-hole displayed an expected and substantial difference between dispersion in sand- and clay-dominated intervals. It appears that for clay-dominated formations, the rate of change of phase velocity can be associated to clay content. We demonstrated that with appropriate acquisition and processing, multifrequency full-waveform sonic logging applied in existing large-diameter sand-screened wells can produce valuable results. There are few wireline logging technologies that can be applied in this setting. The techniques that we used would be highly suitable for time-lapse applications in high-volume production wells or for reassessing formation properties behind existing historical production wells.

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