Upscaling Aspects of Spatial Scaling

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Erick French
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1 Aspects of Spatial Scaling 288

2 We need to relate measurements at different scales Lab Logs Crosswell VSP Surface Seismic How does laboratory rock physics apply to the field? frequency differences sample size differences wavelength differences Seismic velocity depends not just on the rock and fluid properties, but also on the measurement scale relative to the geologic scale 289

3 0.15 Scale effects on measured velocity RT EMT J.1 Time (µs) Waves were propagated through periodic media created by stacking plastic and steel disks. At the top, the effective layer thickness is large compared with the wavelength; at the bottom it is small compared with the wavelength. The waveforms show that both the travel time and amplitude/frequency depend on the ratio wavelength to layer thickness. The velocities in the two limits are described wellby ray theory and effective medium theory, respectively. Marion et al. (1994) 290

4 J.2 Apparent velocity picked from the layered medium experiment (top) and numerical simulations of the experiment (bottom). 291

5 One dimensional scale effects in layered media Normal incidence propagation Effective medium limit (λ >> d): 1 M EMT = 1 = S i M i 2 ρ i (Backus average) 2 S EMT 2 S EMT = ρ ave M EMT = S i 2 Ray theory limit (λ << d): S RT = S i S RT < S EMT 292

6 Stanford Rock Physics Laboratory - Gary Mavko Random arrangement of high and low impedance layers: a laboratory VSP 293

7 Scale effects in a laboratory simulated VSP experimental 5% pick (propagator matrix) 20 10% pick (propagator matrix) 20% pick (propagator matrix) 40 approximate recipe Kennet-Frazer number of disks effective medium 100 α=27 α= ray theory 140 J propagation times (microseconds) In a second experiment the plastic and steel disks were stacked randomly, to create a medium with random plastic and steel interval thicknesses. Waves were propagated through the growing stack, to roughly simulate a VSP. 294

8 Scale Effects on Seismic Velocities Interval velocities in thinly layered media J.4 This slide shows the apparent interval velocity in each plastic interval in the laboratory VSP. The difference of arrival times picked from the waveforms at the top and bottom of each plastic interval were divided into the interval thickness to get the velocity. The bulk plastic velocity is ~2500 m/s. We see that this ray theory approach gives nonsense values when the interval thickness becomes small relative to the wavelength. 295

9 λ λ Short wavelength and long wavelength synthetic seismograms for plane wave propagation through a 2-D random heterogeneous medium with Gaussian spatial autocorrelation function 296

10 Velocity Dispersion in 2-D Random Heterogeneous Media ray theory 3-D Velocity ray theory 2-D ray theory 1-D 3640 effective medium λ/a J.5 Comparison of numerical wave propagation in 2-D heterogeneous medium with a Gaussian spatial autocorrelation function and ray theory predictions of Boyse (1986) 297

11 Short Wavelength Behavior The variance of the traveltime fluctuations around the mean traveltime can be related to the variance of the slowness fluctuations (Müller et al., 1992) For plane waves in a heterogeneous medium with a Gaussian spatial autocorrelation function: σ T 2 = πlaσ S 2 L:pathlength a:spatial correlation length σ 2 T :traveltime variance σ 2 S :slowness variance 298

12 Short Wavelength Behavior Ray theory in random media S = S + S 0 difference between ray theory slowness and mean slowness: 2 L S = S RT S 0 = σ S a D 0 R SS D = α ζ dζ R ss :spatial autocorrelation function a:correlation length σ S 2 :slowness variance L:pathlength (Boyse, 1986) 299

13 1.1 Sandstone Vrb/Vus Pressure Shear Extension /Q 1.1 Limestone 1.0 Vrb/Vus Pressure Shear Extension /Q Velocity dispersion versus attenuation for sandstone and limestone samples (Lucet, 1989) J.6 300

Velocity Dispersion in Heterogeneous Limestone x-ray image 5300 5250 5200 ray theory 3-D velocity 5150 5100 5050 5000 ray theory 2-D 4950 effective

14 Velocity Dispersion in Heterogeneous Limestone x-ray image ray theory 3-D velocity ray theory 2-D 4950 effective medium Comparison of velocities computed from average traveltimes in numerical simulations with theoretical predictions λ/a J.7 301

RESEARCH PROPOSAL. Effects of scales and extracting methods on quantifying quality factor Q. Yi Shen

RESEARCH PROPOSAL. Effects of scales and extracting methods on quantifying quality factor Q. Yi Shen RESEARCH PROPOSAL Effects of scales and extracting methods on quantifying quality factor Q Yi Shen 2:30 P.M., Wednesday, November 28th, 2012 Shen 2 Ph.D. Proposal ABSTRACT The attenuation values obtained