Characterization of mechanical discontinuities based on data-driven classification of compressional-wave travel times

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Abstract

Wave propagation and diffusive transport phenomena are influenced by the mechanical discontinuities in material. This study shows that certain bulk properties of the network of low-velocity mechanical discontinuities (e.g. air-filled cracks) in a material can be characterized by processing compressional-wave travel times using traditional data-driven classification techniques. To that end, we perform three tasks in chronological order: (1) use the discrete fracture network (DFN) method to create two-dimensional (2D) numerical models of crack-bearing material embedded with various types of low-velocity mechanical discontinuities, (2) use the fast marching method (FMM) to simulate the propagation of the wave/diffusion front from a single source through the 2D crack-bearing material to multiple receivers placed on the boundary of the material, and (3) train 9 data-driven classifiers to characterize the crack-bearing materials (i.e. bulk properties of the network of mechanical discontinuities in the crack-bearing material) by learning from the simulations of travel times detected by multiple receivers placed around the crack-bearing material. The classifiers identified the orientation, spatial distribution, and dispersion of the low-velocity mechanical discontinuities. Voting classifier performs the best among the 9 classifiers. For the characterization of bulk dispersion and distribution of discontinuities, the sensors located on the adjacent boundaries are more important; whereas for the characterization of bulk orientation of discontinuities, the sensors located on the opposite side are more important.

Introduction

A discontinuity refers to a plane exhibiting sharp change in physical or chemical characteristics. This study focusses on low-velocity mechanical discontinuity (e.g. air-filled or water-filled cracks in solid materials). The terms fracture and crack generally refer to mechanical discontinuity. Characterization of mechanical discontinuities are critical for rock mechanics, geotechnical projects, structural health monitoring, geothermal reservoir development, and hydraulic fracturing. Characterization of subsurface fractures is critical for forecasting and optimizing the production of fossil/geothermal energy. Characterization of fracture systems helps in the evaluation of the oil recovery potential in unconventional reservoirs. For purposes of extracting and producing of subsurface earth resources, the mechanical discontinuities need to be characterized at different scales. Well testing and seismic tomography is used to characterize the beddings, joints, and faults at meter to kilometer scale.1 Resistivity/dielectric imaging is a well logging technique used to quantify the beddings and fractures in the near-wellbore region at centimeter to meter scale. In the civil engineering discipline, sonic and ultra-sonic waves are utilized to measure the discontinuities in steel, concrete, and other materials at centimeter scale for purposes of structure health monitoring. Various laboratory testing techniques, such as ultrasonic, sonic, and acoustic emission measurements, are developed for the characterization of cracks at millimeter scale.

Characterizing discontinuities (e.g. fractures and cracks) in material under laboratory conditions is a challenging task. Three common laboratory-based fracture-characterization techniques are acoustic emission (AE) monitoring, ultrasonic imaging, and computer tomography (CT) scanning. AE measurements are used to characterize the fractures by locating and detecting the AE events that generate various types of acoustic/mechanical waves due to fracture/crack formation and propagation. AE is a relatively cheap and simple technique for 3D characterization, but it is a dynamic measurement that requires substantial changes in the internal structure of the material when the AE measurement is being acquired. CT scanning is based on different X-ray absorption of elements and molecules. CT scanning can reconstruct the 3D internal structure of a material and embedded discontinuities at microscale resolution. Compared to other methods, CT scanning requires a radioactive source and is relatively time consuming and expensive. Ultrasonic imaging uses transmission and reflection of mechanical waves to locate the discontinuities. To overcome the disadvantages of AE and CT scanning, we investigate the feasibility of utilizing multipoint compressional-wave travel-time measurements along with machine learning to predict the bulk characteristics of static mechanical discontinuities in two-dimensional crack-bearing material. A motivation for this study is that if the classification models can characterize the two-dimensional crack-bearing material with high accuracy based on only the multi-point compressional-wave travel times, then the proposed method can be improved further for real-world characterization of three-dimensional crack-bearing materials by jointly processing compressional and shear waveforms and traveltimes.

This study aims to characterize the network of discontinuities in crack-bearing (fractured) material by processing only the compressional-wave travel time detected at multiple locations. The characterization is achieved by using data-driven classification of the multipoint compressional travel times. To that end, we perform three tasks in chronological order: (1) use the discrete fracture network (DFN) method to create two-dimensional (2D) numerical models of crack-bearing material embedded with various types of mechanical discontinuities, (2) use the fast marching method (FMM) to simulate the propagation of the wave/diffusion front from a single source through the 2D crack-bearing material to multiple receivers placed around the crack-bearing material, and (3) train 9 data-driven classifiers to characterize the crack-bearing materials (i.e. bulk properties of the network of mechanical discontinuities in the crack-bearing material) by learning from the simulations of travel times detected by multiple receivers placed around the crack-bearing material. One of our objectives is use synthetic data to train the classifiers to identify the orientation, spatial distribution, and dispersion of mechanical discontinuities. The existence of cracks influences the propagation of the sonic waves; the data-driven classifiers will extract/learn the patterns of the sonic wave propagation across the entire sample to facilitate the characterization of crack-bearing materials.

The configuration of source/transmitter and sensors/receivers for the laboratory-based measurement of travel times is inspired by real-world laboratory experiments.2 The 2D simulation models and the position of the sonic sensors are similar to the real-word experiment. The equipment and design of the experiments are much simple than CT scanning experiments. Moreover, combining machine learning models and acoustic propagation measurements enables the fast characterization of material in a noninvasive manner. Sonic wave propagation process is simulated using the Fast-Marching Method (FMM), which enables fast simulation of the sonic propagation process, i.e. the traveltimes recorded by the sensors. Different 2D materials models with different types of crack cluster are built with a stochastic modeling of fractures. The simulation experiments simulate the sonic wave propagation process originated from a sonic source on the boundary of the 2D material, and the sonic wavefront travel time is detected by placing 28 signal receivers on the boundary of the 2D material. Fracture system (or crack cluster) with different statistical parameters are embedded in the materials to create different types of crack-bearing (fractured) material. The various types of material differ in terms of the locations, dispersions, and orientations of the cracks in the embedded crack cluster. The training and testing datasets are created by simulating the sonic wave propagation processes in the crack-bearing material. Different machine learning models are applied to explore the possibility of using sonic wavefront travel time for characterizing certain statistical attributes of the crack-bearing systems. It is to be noted that compressional wave traveltime measurement will be more sensitive to closed/cemented high-velocity cracks as compared to open low-velocity cracks. Consequently, the travel time recorded by the 28 sensors for materials containing high-velocity discontinuities will have higher information content for purposes of the characterization of the network of mechanical discontinuities.

Popular machine learning methods to solve the fracture/crack characterization problem include Artificial Neural Network (ANN), Random Forest, Support Vector Machine (SVM), and Convolutional Neural Network (CNN). Unsupervised learning methods, such as K-means clustering, have also been used to find patterns in the measurements on crack-bearing materials. Zhou et al. (2018) applied ANN and SVM to categorize fracture events by processing acoustic emission (AE) signals produced during rock fracturing.3 Liu et al. used ANN to predict rock types from AE measurements acquired during uniaxial compression based on the assumption that different rocks have different failure modes that generate distinct AE waveforms.4,5 Farhidzadeh et al. extracted several features, like amplitude, duration, and dominant frequency, from the AE signal, and used SVM to learn from the extracted features whether the AE event (i.e. the crack) is produced by tensile or shear process.5,6 Apart from laboratory data, researches have used simulated data to develop data-driven workflows for monitoring and predicting the fracture evolution process. Moore et al. applied ANN to learn from simulation data generated using finite-discrete element model for predicting whether two fractures will coalesce based on fracture parameters, such as fracture orientations, distances between two fractures, and the minimum distance from one of the fractures to the nearest boundary.6 Miller et al. applied CNN to process 2D images acquired from rock-fracturing simulation to learn the features that are most predictive of final fracture length distribution.7

Section snippets

FMM introduction

Eikonal equation characterizes the evolution of a closed surface through a material with a specified velocity function. The fast-marching method (FMM) is developed to solve the Eikonal equation, expressed as||u(x)||=1/f/(x)forxΩu(x)=0forxΩwhere u(x) represents the travel time of the wave/diffusion front (first arrival time) to reach the location x, f(x) represents the velocity function for the material, Ω is the open set with well-behaved boundary, ∂Ω is the boundary, and x is the

Travel-time measurement setup

The setup for sonic traveltime measurements, the locations of the 28 sensors and 1 source, and the default properties of matrix and discontinuity are introduced in this section. This is essential for generating travel-time dataset corresponding to various types of crack-bearing materials. The 2D numerical models of crack-bearing material implemented in this study are inspired by the laboratory experiments conducted at the Integrated Core Characterization Center (IC3) presented in Ref. 2 and

Workflow to train and test the data-driven classifiers

Three numerical experiments are conducted in this study to develop data-driven classifiers that can differentiate crack clusters embedded in materials based on crack orientation, dispersion, and spatial distribution. The 9 data-driven classifiers used in this study are k-nearest-neighbors (KNN), linear support vector machine (SVM), radial basis function (RBF) SVM, decision tree, random forest, AdaBoost, Naïve Bayes, artificial neural network (ANN), and voting classifier. Such workflow for

Description

In a crack-bearing material formed under certain stress regime, the cracks exhibit a distribution of orientations that can be described using the von-Mises distribution. The dispersion describes the randomness of the crack orientations around a dominant orientation. The probability density function of the von-Mises distribution is expressed as16:f(x|μ,k)=ekcos(xμ)2πI0(k)where x is the main orientation, μ is a measure of the mode and k (kappa) is the dispersion factor. I0 is the modified Bessel

Assumptions and limitations

The data-driven classifiers in this study were trained and tested on simulated dataset generated for simple compressional wavefront propagation through simple crack-bearing materials. Extensive study is required to understand the generalization of the proposed data-driven workflow for laboratory-based characterization of crack-bearing materials. The study is based on wavefront propagation simulations. Experimental validations are still needed to further validate the deployment of the

Conclusions

The study proposed a noninvasive material characterization method that analyzes the compressional wave traveltime detected at multiple receivers (i.e. multipoint measurements). Fast-marching simulation of compressional-wavefront propagation in a 2D numerical model of crack-bearing material generated large datasets of compressional-wave traveltimes detected by 28 sensor/receivers placed along the boundary of crack-bearing material. The traveltime datasets are used to develop 9 data-driven

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgement

The research work was made possible due to the support from the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Chemical Sciences, Geosciences, and Biosciences Division, under Award Number DE-SC0020675.

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