Microseismic Monitoring and Geomechanical Modelling of CO2 - bris

CHAPTER 3.
INVERTING SHEAR-WAVE SPLITTING MEASUREMENTS FOR FRACTURE PROPERTIES
3.3 Synthetic testing of inversion method
Before applying this technique to a real dataset I develop synthetic examples to demonstrate and
understand the inversion process better. The first step is the construction of an initial elastic model
using the rock physics model outlined above. A range of plausible raypath arrival azimuths and
inclinations are chosen. For each case described here, 130 synthetic data points are produced, using
a grid of 13×10 (azimuth × inclination) points, the limits of this grid being defined specifically for
each case. The splitting parameters ψ and δV S for each raypath are calculated using the Christoffel
equation. I add noise to the data by assuming a random error distribution between ±10 ◦ for ψ and
±0.5% for δV S , which are typical error ranges for real splitting data (e.g., Al-Harrasi et al., 2010).
Because there will also be uncertainty regarding the event location, and therefore the angle of wave
propagation through the rock, I also add noise of ±10 ◦ to the inclinations and azimuths of the synthetic
data - this is done after the splitting operators have been computed.
This then represents the ‘observed’ dataset, which I use to invert for the initial model parameters.
The proximity of the initial parameters used to construct the elastic model and those found by the
inversion will indicate the linearity of the objective function around the point of interest. The extent
of the confidence interval within the misfit space will show how unique a solution is. Where well
constrained, unique solutions can be found, this implies a well posed problem, with features that are
resolvable with the data available. Where a wide misfit minimum, or multiple solutions that fit the
data are found, this implies an under-determined problem, and that the data are not sufficient to
resolve the structures present.
3.3.1 Sensitivity of δ and γ
My first use of synthetic data is to test the sensitivity of the inversion to γ and δ. These parameters
control the strength of a VTI fabric, so I anticipate that they will be highly sensitive to the angle of
ray propagation with respect to this structure, i.e., the angle of inclination. I perform 3 inversions,
with subhorizontal (0-30 ◦ ), subvertical (60-90 ◦ ) and oblique (30-60 ◦ ) arrivals. In each case there is
a full range of arrival azimuths from 0-180 ◦ , and the initial elastic model has γ=0.04, δ=0.1, ξ=0.04
and α=120 ◦ . In Figures 3.5 - 3.7 I plot the RMS misfit contours as a function of γ, δ and α, at the
best fit value of ξ.
The first thing that I note from Figures 3.5 - 3.7 is the apparent linearity of the solutions. For
every parameter, for every set of arrival angles, the rock physics parameters found by the synthetic
inversion matches the initial parameters used to create the model. Furthermore, in almost every case,
excepting γ and δ for subvertical arrivals, the misfit surface describes an ellipse around the best fit
model, with no alternative best fit model. The equations used to generate the inversion are complex
and nonlinear. However, it appears that, around the regions of interest they can be approximated by
a linear relationship.
However, the misfit surfaces surrounding the best fit models are not circular, showing instead a
high degree of ellipticity in some cases, implying that the there are differences in how well constrained
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