Читать книгу Geophysical Monitoring for Geologic Carbon Storage - Группа авторов - Страница 75

SHRB measurements determine dynamic Young's modulus E, shear modulus G, and their related attenuations (vibration damping factors) a _E and a _G respectively (given by 1/2Q _E and 1/2Q _G using the seismic quality factors). Note that for anisotropic rock, E is the longitudinal modulus along the core axis, and G is the torsion modulus around the core axis. During an experiment, fundamental‐mode resonances of both longitudinal‐mode vibration and torsional‐mode vibration are measured, using combinations of longitudinal piezoelectric source and the axial accelerometer, and of the torsional source and the tangential accelerometers, respectively. For each resonance, its peak frequency f _c and the half‐power width Δf are measured. From these values, the quality factor of the resonance is obtained by Q ^–1=Δf/f _c . In our experiments, f _c and a=1/2 Q values are measured and computed automatically by the spectrum analyzer.

Once the peak frequencies and the quality factors for both longitudinal and torsion resonances are obtained, these are used to invert numerically for the dynamic moduli and attenuations of the rock sample. The code used for the inversion consists of a one‐dimensional, frequency‐domain wave propagation model with multiple, homogeneous segments (layers) with unknown complex Young's modulus and shear modulus for the sample segment (a propagator method, e.g., Aki & Richards, 1980). The other model parameters, such as the dimension and the density of the sample, properties of the steel bars, and source and sensor mass, are measured and known.

The forward modeling code computes accelerations at one end of the model as a function of the frequency, corresponding to either a longitudinal or torsional wave input from the source end. From the ratios between the accelerations and the force (or stress) at the source, simulated frequency response functions are computed. Similar to the experiment, the central frequencies and the half‐power widths of simulated resonance peaks are measured. Once both experimentally measured and numerically computed central frequencies and half‐power widths of longitudinal and torsional resonances are obtained, the elastic moduli (from the differences in the central frequencies) and the related attenuations (from the differences in the peak widths) in the model are adjusted so that the differences becomes smaller. Using these new parameters, corrected frequency response functions are computed and updated resonance frequencies and half‐power widths are obtained. This process is repeated until the differences between measured and computed central frequencies and the peak widths become sufficiently small.

More details of the numerical modeling and the inversion procedure, including examples for synthetic materials (acrylic and polycarbonate samples), are presented previously (Nakagawa, 2011). In Figure 5.3, examples of experimentally measured frequency response functions (measured accelerometer voltage outputs normalized by the source input voltage of the spectrum analyzer) are shown, along with fitted, numerically simulated frequency response functions. The sample is an intact, dry, Carbon Tan #1 core, under a confining stress of 9.6 MPa.

Figure 5.3 Examples of experimentally measured frequency response functions (circles) for longitudinal (E mode) and torsional (G mode) resonances for dry Carbon Tan #1 core. Response functions computed for the elastic moduli and attenuations determined by the inversion are also shown in solid curves for comparison: (a) Amplitude frequency response functions and (b) phase frequency response functions.