Chris L. Hackert and Jorge O. Parra; Southwest Research Institute Southwest Research Institute FREQUENCY-DEPENDENT ELASTIC WAVE PROPAGATION IN DRY AND SATURATED VUGGY CARBONATE CORES: EXPERIMENT & SIMULATION

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 frequency- dependent 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. Introduction

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 three- dimensional 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 finite- difference programs. The simulations model wave propagation in each core under saturated and dry conditions, and also examine a homogeneous (non- vuggy) 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.

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. X – Ray CT Data

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. Core Descriptions

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. 2-D Models

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 (non- vuggy) saturated core (black). Comparing the black and green curves shows the vugs strongly attenuate the wave through scattering.

Figure 5: Time- frequency 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. Experimental 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 #7: 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. Core #41: 2-D Model Results

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. 3-D Models

Figure 9: Wave image showing z-displacement at a vertical slice through the core #41 center at t= 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. Simulated 3-D Wave Structure

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. Core #41: 3-D Model Results

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. Correlation Lengths

Figure 12. See description on next 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). See Larger scale graph on previous slide>>

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, Frenje, L. and Juhlin, C., 2000, Scattering attenuation: 2-D and 3-D finite difference simulations vs. theory: J. Appl. Geophys., 44, 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,

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-AC26-99BC The assistance of Mr. Purna Halder (DOE) is gratefully acknowledged. Acknowledgments