Southwest Research Institute
Chris L. Hackert and
Jorge O. Parra;
Southwest Research
Institute
FREQUENCY-DEPENDENT
ELASTIC WAVE PROPAGATION
IN DRY AND SATURATED
VUGGY CARBONATE CORES:
EXPERIMENT & SIMULATION
Introduction
Carbonate rocks often contain cavities called vugs, which may range in size
from millimeters to meters. The presence of these vugs contributes to making
carbonate rocks difficult to characterize. The porosity, permeability, and rigidity
of the rock matrix may be very different than the same properties measured in
a heterogeneous, vuggy core. In this poster we use experimental
measurements and finite-difference modeling to investigate the frequencydependent characteristics of wave propagation in saturated and dry vuggy
carbonate cores. The two cores we use in this study are from the Ocala
Formation, a Florida carbonate aquifer. Core #7 is fairly dense, with a
moderate number of vugs. Core #41 has many large and well-connected
vugs. The detailed vug structure is obtained through x-ray computed
tomography (CT) and used to drive the simulations. We find that our models
capture the experimentally observed behavior, which is that vugs attenuate the
ultrasonic waves in a frequency-dependent manner consistent with stochastic
scattering theory. Dry cores are generally more attenuating than fully
saturated cores, but this can break down as the wavelength becomes
comparable to the vug size. This study of high-frequency wave propagation in
vuggy cores is an analog to seismic-frequency wave propagation in rocks with
large karsts.
Method
Experimental compressional and shear velocities were obtained in each core
under dry and saturated conditions. the full waveform response of these
measurements was approximately 250 kHz.
The simulated full waveform compressional wave data is based on a structure
derived from x-ray computed tomography (CT). This yields a detailed threedimensional map of the approximate density of each core, with a resolution of
about 0.25 mm horizontally and 2mm vertically. The vug structure determined
fro the CT image of each core were performed in 2-D and 3-D with finitedifference programs. The simulations model wave propagation in each core
under saturated and dry conditions, and also examine a homogeneous (nonvuggy) equivalent core, with the same average density and P-velocity.
The effect of the internal core structure on the wave propagation is best
characterized by time-frequency analysis, which shows the amplitude of various
frequency components of the transmitted wave as a function of time. This is
accomplished by Fourier transforms on a short time window, which moves
across the data trace. This type of data analysis helps to characterize
frequency-dependent attenuation, which has potential as a tool for reservoir
characterization.
X – Ray CT Data
Figure 1: Comparison of a core-end photograph (from core #41) and dry density
slice derived from x-ray CT data. The CT data slice is approximately 2mm behind
the core end pictured here. Most of the many vugs visible in the photograph
continue into the core body and are also captured in the x-ray CT data. This
validates the use of the CT data as a tool for imaging vugs.
Core Descriptions
Figure 2: Range of matrix densities and vuggy porosities determined from each
slice of the x-ray CT data. The green curve gives the average and standard
deviation of densities. The red curve is the vuggy porosity. While both core #7 and
core #41 are carbonates, core #7 is a sandy grainstone and core #41 is a
wackestone. They are separated in the Ocala formation by a vertical distance of
more than 116 ft (35 m). Core #41 has many more vugs than core #7, and also has
a less dense matrix with significantly greater heterogeneity. These two factors
combine to give the core #41 an effective permeability more than 100 times higher
than that of core #7.
2-D Models
Figure 3: Density images of the finite difference simulation domain for
each 2-D core model. Water-filled vugs in each core are clearly visible
as low density cavities. Each core is bounded on top and bottom by steel
end caps, and surrounded by an oil bath. P-waves are initialized by a
planar array of point sources in the lower end cap, and propagated to a
receiver in the upper end cap.
Experimental Measured Waveforms
Figure 4: Experiments are for water-saturated and dry cores.
Simulated Waveforms
Figure 4:
Simulations are for
water-saturated
(green), dry (red),
and an equivalent
homogeneous (nonvuggy) saturated
core (black).
Comparing the
black and green
curves shows the
vugs strongly
attenuate the wave
through scattering.
Experimental Results
Figure 5: Timefrequency
representation of
experimental core
waveforms under
saturated and dry
conditions. The color
scale indicated dB of
signal energy, with a
60-dB range between
low energy (blue) and
high energy (red). In
each core, the
transition from
saturated to dry is
accompanied by a
relative increase in
energy around
400kHz, although the
overall attenuation is
higher in the dry cores.
Core #7: 2-D Model Results
Figure 6: Time frequency representation for 2-D simulated waveforms in core #7 as: (a)
equivalent non-vuggy core, (b) vuggy saturated core, and (c) vuggy dry core. Compared to the
uniform core (a), vugs in the heterogeneous core (b) preferentially attenuate the higher
frequencies. While overall attenuation is higher in the dry core (c), frequencies around 400 kHz
are comparatively less attenuated.
Core #41: 2-D Model Results
Figure 7: Time frequency representation for 2-D simulated waveforms in core #41 as: (a)
equivalent non-vuggy core, (b) vuggy saturated core, and (c) vuggy dry core. Results are similar
to core #7, in Figure 3. This core is more vuggy than core #7, and so the relative attenuation
effects are stronger.
3-D Models
Figure 8: Cutaway view of 3-D models for core #7 and core #41,
showing vug structure derived from x-ray CT data. The surrounding oil
bath is not shown. Due to computational limits, only the central portion
of the large diameter core #41 could be simulated at this time.
Simulated 3-D Wave Structure
Figure 9: Wave image showing z-displacement at a vertical slice through the core #41
center at t=0.0208 ms: (a) equivalent homogeneous core, (b) saturated vuggy core. In the
heterogeneous core the direct wave is both faster and more attenuated as it travels along
narrow high-velocity pathways in the core. The amplitude scale is reduced by a factor of
five in image (b) to accommodate the weaker wave.
Core #41:
3-D Model Results
Figure 10: Time frequency representation for simulated waveforms in core #41 as: (a)
equivalent non-vuggy core, and (b) vuggy saturated core. As in the 2-D models, the
higher frequencies are preferentially attenuated by scattering attenuation from the vugs.
Correlation Lengths
Figure 11: Autocorrelation functions and fits to the 3-D structure of core #7 and core #41.
The red curve is the x-direction autocorrelation. The green curve is the y-direction
autocorrelation. The blue curve is the z-direction autocorrelation. The dotted lines are fits to
these curves. These results show that the horizontal autocorrelation is fairly independent of
orientation, but that the vertical autocorrelation is different than the horizontal, especially in
core #41. This is due to vugs and molds which are elongated in the vertical direction.
Figure 12. See description on next slide>>
See Larger scale graph on previous slide>>
Figure 12: Scattering attenuation as transmission loss computed from 2-D finite
difference simulations (solid lines) and predicted from statistical autocorrelation
functions (dashed lines). The red curves are for water saturated conditions and the
green curves from dry cores. The transmission loss from the simulations is derived
from the difference between the equivalent homogeneous core and the model vuggy
core in the Fourier spectrum amplitudes. Core #41 has much more vuggy porosity than
core #7, so the difference between the saturated and dry conditions is more
pronounced. The correspondence between simulation and stochastic theory is good
overall, but breaks down at high frequencies in the dry cores, especially core #41. This
failure is due to the extreme contrast between the many air-filled vugs and the rock
matrix, which violates the small-perturbation assumptions of the stochastic theory.
Notice that in both dry cores, attenuation near 400 kHz is much less than expected.
This agrees well with the experiments (see Figure 5).
Conclusion
We have demonstrated that two- and three-dimensional simulation of
elastic wave propagation in vuggy cores can capture at least qualitatively
the observed experimental behavior. The vugs strongly attenuate the
ultrasonic wave, probably leading to strong dispersion between the
ultrasonic core measurement frequencies and the frequencies of sonic
logging or surface seismic. The presence of larger-scale cavities in the
centimeter to meter size (e.g. karsts) would lead to similar attenuation and
dispersion across the sonic and seismic frequency range.
The degree of attenuation measured in the simulated cores is
generally consistent with that predicted by stochastic scattering theory.
Dry cores with air-filled vugs are strongly heterogeneous, however, and
the small-perturbation assumption breaks down at high frequencies.
Time-frequency analysis shows that frequencies near 400 kHz are
relatively less attenuated in dry cores than in saturated cores. This is
especially true for core #7, in both experiment and simulation. We
attribute this to two effects. The first is random variability due to the
particular distribution of vugs and vug sizes. The second is that the wave
transitions into the geometric optics regime at higher frequencies, and
individual raypaths may go around rather than through the vugs. This
happens more in the dry cores because the impedance contrast between
dry matrix and air-filled vug is so great.
Suggested References
Cohen L., 1989, Time-frequency distribution - a review: Proc.
IEEE, 77, 941-981.
Frenje, L. and Juhlin, C., 2000, Scattering attenuation: 2-D and
3-D finite difference simulations vs. theory: J. Appl. Geophys., 44,
33-46.
Parra, J.O., Hackert, C.L., Ababou, R., and Sablik, M.J., 1999,
Dispersion and attenuation of acoustic waves in randomly
heterogeneous media: J. Appl. Geophys., 42, 99-115.
Acknowledgments
Access to data from the two cores used in this study
was provided by Michael Bennett of the South Florida
Water Management District. Support for this work was
provided by the U.S. Department of Energy, under
Contract No. DE-AC26-99BC15203. The assistance of
Mr. Purna Halder (DOE) is gratefully acknowledged.