Validation of elastic wave measurements of rock fracture compliance using numerical discrete particle simulations

We test various methods of quantifying the compliance of single and multiple rock fractures from synthetic ultrasonic data. The data are generated with a 2D discrete particle scheme which has previously been shown to treat fractures in agreement with linear-slip theory. Studying single fractures, we find that delays derived from peak amplitudes do not correspond to group delays, as might be expected. This is due to waveform distortion caused by the frequency-dependent transmission across the fracture. Instead the delays correspond to an expression for phase delays, which we derive from linear-slip theory. Phase delays are a unique function of compliance, whereas group delays are non-uniquely related to compliance. We believe that this property of group delays has hindered the wider application of deriving fracture compliances from traveltimes. We further show that transmission coefficients derived from waveform spectra yield more accurate fracture compliances than those obtained from ratios of signal peak amplitudes. We also investigate the compliance of a set of parallel fractures. Fracture compliance can only be determined from transmission coefficients if the fracture spacing is so large that the first arriving pulse is not contaminated by reverberations. In the case of contamination the direct measurement of group or phase delays is not practical. However, we demonstrate that in such cases of strong waveform distortion the coda wave interferometry method is very effective for determining relative fracture compliance. First break delays in the fracture set data are related to those observed in single fracture simulations. This means that fracture set compliance can be estimated from first break data if used together with numerical simulations.