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

Abstract

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 travel times. 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 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 2 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.3

Rock fracture compliance derived from time delays of elastic waves

“…The normal fracture compliance Z was controlled by lowering the bond stiffness between grid points on either side of the fracture interface. The elastic fracture properties in such particle based schemes are in agreement with the linear slip theory as has previously been shown by Möllhoff and Bean (2009) and Toomey, Bean and Scotti (2002).…”

“…Analyzing the time delays between main waveform peaks shows that the delays increase monotonically with increasing dynamic Z , as expected for phase delays. The delays are somewhat smaller than predicted by the analytical expression for phase delays t ph given by Möllhoff and Bean (2009). This discrepancy was not seen in numerical data (see Fig.…”

“…We derived phase and group delays for the first data set using the standard methods of picking maximum peaks and the maxima of waveform envelopes respectively. By plotting the delays against the known normal fracture compliance Z in each simulation, we can compare the results to the analytical solution for phase delays given by (Möllhoff and Bean 2009) and group delays given by (Pyrak‐Nolte, Cook and Myer 1987) where ρ is the intact rock density, v the intact rock compressional wave velocity and ω the angular frequency. Because expressions and were derived for monochromatic waves we only observe good agreement between numerical and theoretical time delays if the synthetic data have first been band‐pass filtered in a narrow band around the central frequency of 1 MHz, see Fig.…”

“…We derived phase and group delays for the first data set using the standard methods of picking maximum peaks and the maxima of waveform envelopes respectively. By plotting the delays against the known normal fracture compliance Z in each simulation, we can compare the results to the analytical solution for phase delays given by (Möllhoff and Bean 2009)…”

“…Both phase and first break delays are monotonically related to Z , whereas group delays are not. Möllhoff and Bean (2009) explore these relationships using analytical and numerical data. They pointed to a need for a clear understanding of how to extract phase, group and first break measures from recorded data in order to quantify Z .…”