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Classification of sonic wave Chapter 9 261

discontinuities of various primary orientations, distributions, and dispersions
around the primary orientation are generated using a discrete fracture network
(DFN) model. DFN model is used to generate multiple realizations of material
containing discontinuities with specific spatial properties. Fast-marching
method (FMM) is then used to calculate the compressional wavefront travel
times from a single transmitter to 28 receivers/sensors placed along the bound-
aries of the fractured material. A compressional wavefront travel-time dataset is
built by collecting the FMM simulations of travel times for each realization of
fractured material. Each fractured material has a dimension of 150 mm by
150 mm and is discretized using 500 by 500 grids for accurate FMM simulation
of wavefront propagation. Each grid has a length and breadth of 0.3 mm. Each
realization of material containing discontinuities also has a user-assigned label
corresponding to the overall spatial characteristics of the embedded network of
discontinuities. For each user-assigned label, DFN model generates 10,000 real-
izations. FMM simulations of wavefront propagation for 10,000 realizations of
fractured material containing discontinuities of specific orientation, distribu-
tion, and dispersion (similar to that shown in Fig. 9.12) take around 2 h on Dell
workstation with 3.5 GHz Intel Xeon CUP and 32 GB RAM.
The set of travel times computed for the 28 receivers for each realization of
fractured material along with the corresponding user-assigned label of the real-
ization constitutes the labeled compressional wavefront travel-time (LCT)
dataset (Fig. 9.13B). Data-driven models are developed on the LCT dataset
to learn to relate a set of FMM-derived travel times with the user-assigned
label of the realization, which describe the spatial characteristics of the embed-
ded network of static discontinuities. When generating the multiple realiza-
tions of material containing discontinuities with a specific label (e.g.,
material containing randomly distributed discontinuities of random lengths

exhibiting þ10 degrees dispersion around the primary orientation of
60 degrees), the statistical properties of the embedded discontinuities are
the same for all the realizations having the same label but differ from the sta-
tistical properties of the embedded discontinuities for realizations correspond-
ing to other labels.
The wavefront propagation is affected by a variety of mechanical and phys-
ical properties. For purposes of generating the LCT dataset, we assume the
background material is clean sandstone. The compressional wave velocity in
clean sandstone is around 4500 m/s. Each discontinuity is assumed to be open
and filled with air. The compressional wave velocity in discontinuity is assumed
to be 340 m/s. The width of each discontinuity is 0.3 mm. The embedded net-
work of discontinuities is generated using the discrete fracture network (DFN)
method. When creating multiple realizations using DFN method, the network of
discontinuities is defined by the probability distribution of location, primary ori-
entation, dispersion, and length of the discontinuities. The location, orientation,
and length of the discontinuities are characterized using various probability dis-
tributions, similar to the work done by Fadakar Alghalandis [18]. The detailed