Predictions of macro-scale fracture geometries from acoustic emission point cloud data in a hydraulic fracturing experiment

  • Jesse Hampton
  • Marte Gutierrez
  • Luke Frash
Open Access
Original Paper - Production Geophysics


Observations of laboratory fracture testing by means of acoustic emission (AE) can provide a wealth of information regarding the fracturing process and the subsequent damage of the material or structure under load. A method for determining the structure of macro-scale fractures from a point cloud of AE events was developed and tested at the laboratory scale. An unconfined hydraulic fracturing experiment was performed on granite while monitoring acoustic emissions from six piezoelectric transducers on the surfaces of the specimen. The granite specimen dimensions were 30 × 30 × 25 cm3. The motivation of the AE analysis was to provide location information for a secondary wellbore placement which intersects the hydraulic fracture to complete a hydraulically connected binary well-hydraulic fracture system. Information gained from the 3D event source locations was used to optimize the location and orientation of secondary production well placement to intersect the induced hydraulic fracture. A direct method was developed to predict the location, orientation and shape of the hydraulic fracture from a dense cloud of AE events. The predicted fracture plane, which was assumed to be planar, was rotated in both the pitch and roll directions, while an average error of AE event distance between the assumed plane and the individual event locations was determined for each rotation iteration, resulting with a predicted fracture plane in the minimum error orientation. Post-test fracture observations were made and digitized in 3D space from coring and slabbing the granite specimen. The actual fracture observations were compared with the predicted fracture structure from AE and showed marked correlation. The simple method of determining macro-scale fracture location, orientation, and extents provided a useful tool for materials that exhibit large and disperse clouds of AE activity where coalesced fracture structure is not apparent.


Acoustic emission Hydraulic fracturing Microseismic Fracture geometry Scale-model experiments 


Hydraulic fracturing is a standard practice in oil-, gas-, and geothermal heat bearing formations to increase reservoir permeability due to the commonality of micro and nanodarcy permeability in many of these reservoirs. Introducing hydraulic fractures in oftentimes complex environments of stress and rock structure makes predictions of hydraulically connected and accessed fractures difficult. Predictions of induced and activated natural fracture geometries are important for determining stimulation effectiveness, geometry of the connected reservoir structure, drainage predictions, and provide a useful metric for stimulation procedure alteration for future treatments in similar formations or fields. Several methods exist for monitoring hydraulic fracturing treatments and obtaining information regarding the geometry of the induced subsurface changes including microseismic, microdeformation, fiber optic distributed acoustic sensing, and others. Microseismic observations provide source locations of individual fractures occurring within an observable frequency bandwidth (Shapiro et al. 2006; Kaka et al. 2017; Eisner et al. 2013; Warpinski 2009; Maxwell et al. 2002; Mayerhofer et al. 2010; Ishida et al. 2012; Ma et al. 2017). Although these observations can sometimes provide a substantial amount of fracture data, they are limited in observation to energy released in a specified wave frequency. It is estimated that much of the fracture deformation and energy release throughout a hydraulic fracturing treatment is aseismic (Maxwell et al. 2009; Warpinski et al. 2012; Zhang et al. 2018). Though microseismic observations do not provide the entire story behind a hydraulic fracturing treatment, they often provide significant numbers of located fracture events in which further analysis can be performed.

Several methodologies exist for extracting induced fracture network location information from a cloud of microseismic events, including stimulated reservoir volume (SRV) estimation, clustering geometries, discrete fracture network (DFN) estimation, collapsing, and others (Zimmer 2011; Maxwell 2011; Jones and Stewart 1997; Microseismic Inc. 2011). SRV estimation is typically performed through binning and shrink-wrapping methods, where the hydraulic connection is either assumed between the wellbore and a microseismic event if enough events occupy the space between them, or assuming a hydraulic connection within a whole cloud of events based on proximity relationships, respectively. These cloud-based techniques are valuable for qualitative analyses of stimulation effectiveness and reservoir drainage estimations, but lack in actual fracture geometry predictions. In the literature (Microseismic Inc. 2011) DFN predictions have been made from individual microseismic event moment tensor solutions and determined fault plane predictions. The fault planes for individual events are plotted as their own discrete fracture planes and the cloud of microseismic events becomes a cloud of individual fault plane solutions. These fault planes are then often assumed to be a complex group of separate but interacting fractures. Though obtaining fault plane solutions for individual events is valuable, assuming each event is part of a separate and competing macro-scale fracture is an oversimplification. Individual microseismic event observations can be attributed to the main hydraulic fracture growth, but also can be stress- or leakoff-induced natural fracture activation, secondary fracturing occurring non-planar to the main hydraulic fracture, among others. The inability to distinguish which type of fracture is observed from microseismic observations makes macro-scale DFN estimation from single fault plane solutions difficult. It is also often assumed that the microseismic events also stem not from the main hydraulic fracture growth, which in many cases is a slow aseismic tensile opening, but rather from, the induced fractures surrounding the tip of the progressing fracture in the shear stress-dominated zone along with slip at pre-existing surfaces or sliding on the newly induced hydraulic fracture behind the fracture tip. This would mean that fracture plane predictions from individual microseismic event fault plane solutions would be oriented at a wide array of angles from the progressing hydraulic fracture. Though DFNs created from single events can be difficult to interpret, macro-scale fracture estimation from groupings of individual microseismic events can provide reasonable estimations of actual fracture geometries, especially in terms of having possible identifications of the slow tensile opening main fractures that oftentimes reside within a large cloud of located events.

In this work, predictions of coalesced fractures were determined by fitting a plane to a cloud of acoustic emission data from a laboratory hydraulic fracture test. AE refers to the generation of transient elastic waves in a material caused by the sudden occurrence of fractures or frictional sliding along discontinuous surfaces (Mogi 2007), and is a higher frequency analog to microseismic monitoring. AE testing illuminates microcracking and relatively small failures in a material under loading in real-time. Although the AE method for determining the initiation and/or propagation of flaws or friction processes is informative, a disadvantage of the AE method is that a particular test or microcrack is not perfectly reproducible due to the sudden and sometimes random formation of a crack (Grosse and Ohtsu 2010). This is especially true in AE observations of geologic materials.

Just as in field microseismic monitoring, AE recordings of rock fracture tests in the laboratory can be difficult to interpret because AE only provides part of the story of rock response to loading. For instance, in brittle and heterogeneous materials it is difficult to determine the macro-scale fracture structure from individual AE events because throughout the loading and failure process, a damage zone is created where many AEs are generally observed and the scatter of locations can be moderate to quite large depending on scale of heterogeneities, boundary conditions, and mechanical properties. This microcrack damage zone eventually coalesces into a macro-scale fracture, but the observations throughout the test are often of the individual microcracks that could or could not be directly connected to the main coalesced fracture. This observation is the motivation for using cloud-based imaging techniques to determine overall coalesced fracture location, orientation and roughness. The following will present and discuss a method to determine fracture structure in a cloud of events and the benefits and shortfalls of this approach.

Test methods and equipment

An unconfined laboratory hydraulic fracture was carried out while monitoring AE with the motivation of quantifying overall fracture structure from the discrete AEs observed relating to microcracking. The following section describes the AE analysis, laboratory equipment, specimen characteristics and testing procedures.

Acoustic emission analysis

Observed AE point source locations were used to develop a straightforward method for macro-scale fracture identification. The AE fracture identification process was then compared to the actual fracture surface observed from images post-test. AE fracture identification was performed post-test and uses 3D AE event source location data to generate the predicted fracture plane. A method for identifying fracture surfaces based on unfiltered AE data was created. This method does not require previous knowledge of the fracturing directions or geometry. The requirements for the method are at a minimum a set of 3D AE event source location results. Additional data, such as AE event amplitudes, correlation coefficients, event time, and source mechanism solutions can be used for data filtering to further enhance results by providing weighting factors if an event attribute is found to be more indicative of being associated with a macro-scale fracture plane than other parameters. Though additional information regarding the individual AE microcrack observations can be used for filtering the observed dataset or determining weighting parameters, the process described here made no assumptions on the individual event contribution to macro-scale fracture, but instead used all located AE events.

The method for creating the fracture plane from a cloud of data is an error reduction process in which a planar fracture is assumed and rotated in both the pitch and roll directions about a user-specified origin and compared with AE event source location data. The comparison takes place by calculating an error term from the perpendicular distances of each AE event source location found inside a set of bounds parallel to the rotated fracture plane, to the assumed plane. The bounds are user-specified and can be optimized based on the cloud characteristics or determined a priori from knowledge of the material and/or fracturing behavior expected. For the example discussed here, the bounds were specified from expected widths of the fracture process zone observed during notched beam bending experiments (Hampton et al. 2012). Figures 1 and 2 show conceptual sketches of this rotated fracture plane and error estimation. Figure 1 shows an arbitrarily oriented plane residing within a grouping of AE events (red dots) with its center at a user-specified origin. The principal axes of the plane are shown here to rotate so that the plane can be oriented in a discretized set of rotation angles. Computations are performed at each orientation iteration to determine the quality of fit for that orientation amongst the AE source locations. Figure 2 shows a view parallel to the assumed fracture plane (purple), where the AE events may or may not reside within the specified bounds (orange region). Axes y′ and x′ are the principal coordinate system axes for a given iteration in the pitch direction (i.e., figure shows the assumed fracture plane at an iteration angle of theta degrees in the roll direction with a given pitch orientation defined by x′ and y′).

Fig. 1

Conceptual sketch of AE events (red dots) and their proximity to a rotated fracture plane

Fig. 2

Error estimation from the perpendicular distance of each individual AE event to the fracture plane within a set of bounds. The x′ and y′ axes are the principal axes associated with the lowest error pitch orientation of the assumed plane. The plane is then rotated in increments of Δθ to perform an error estimation under each orientation. εi is the perpendicular distance between AE event (i) and the assumed fracture plane

To perform the AE fracture identification process, a range of coordinates is selected for the center/origin of the assumed fracture plane. Typical origin locations were selected around the perimeter of the openhole fracturing interval as seen in the conceptual sketch in Fig. 3. The purpose of selecting multiple origin locations is to reduce the fracture surface error by alternating where the fracture can initiate. For instance, it is possible for fractures to initiate perpendicular or tangentially from the borehole wall because of near wellbore complexity and damage. This technique allows for gridding the entire specimen with possible origins if the actual fracture origin is unknown. This would be useful in a very complex specimen with many pre-existing natural fractures that could be stimulated. Once a range of values has been selected for origin locations, pitch and roll axes are chosen. The assumed fracture plane is then aligned in an arbitrary orientation in the pitch and roll axis directions. An iteration frequency must be specified to determine how coarse the fracture search will be in both the pitch and roll directions. For this work, iteration angles were spaced by no more than 2.5°.

Fig. 3

Conceptual sketch of picking multiple origin locations around an openhole wellbore. The assumed origin locations can be user-specified or gridded throughout the entire specimen if no a priori information exists

An error term is calculated for each iteration and can be represented in several ways, but for most testing, it was determined that Eq. (1) showed the highest accuracy results.
$${\text{Error}}=\frac{{\left( {\frac{{\mathop \sum \nolimits^{} {\varepsilon _i}}}{n}} \right)}}{d}$$
where εi is the perpendicular distance from individual AE events to the assumed fracture plane within the chosen bounds, d is the total bounding distance assumed for the data, and n is the total number of AE events. The numerator in Eq. (1) is the average perpendicular distance of all AE event source locations to the rotated fracture plane. The denominator is the total bounded distance between positive and negative bounds on either side of the rotated fracture plane. This term simplifies the overall error associated with the perpendicular distances from AE events to the assumed fracture plane in each iteration, but should show a relatively high accuracy if the assumption of a planar fracture is valid. A conceptual sketch of an error versus iteration angle plot is shown in Fig. 4 for three separate origin locations defined as the three colored lines. This plot shows the possibility of differing origin locations having altered reductions in error. The main drop in error is evident in all three origin locations at nearly the same iteration angle and would suggest that if these origin locations are relatively close together, a macro-scale fracture exists in this location and orientation.
Fig. 4

Conceptual sketch of an error versus iteration angle plot. The colors of the lines each represent a different origin location. The reductions in error can possibly signify the existence of a fracture plane with that iteration angle from the assumed origination orientation

Test materials, equipment and procedure

The test material used was granite that was obtained from the Liesveld Quarry in Lyons, Colorado. Granite was chosen because of its high strength, relative homogeneity, and low permeability. Hydraulic fracture testing was performed using a dual pump fluid injection system. Data monitoring for the hydraulic fracturing tests included flow rate, pump pressure, wellhead pressure, and acoustic emissions.

Test materials

A granite sample with dimensions of 30 × 25 × 30 cm3 was used for the hydraulic fracturing test. The y-direction of the sample was slightly shorter than the others because the sample contained one unfinished and rough side, which made the sample a good candidate for unconfined testing because of the inability to put an irregularly shaped block into a loading apparatus. The granite material properties are found in Table 1. The elastic properties are reported from the average of several unconfined compressive strength tests. Tensile strength was measured using the Brazilian tensile test. Fracture toughness was measured using a notched beam fracture toughness method. Using granite for this type of acoustic emission fracture prediction testing allowed for relatively large amounts of events to be recorded throughout the hydraulic fracture tests because of the brittle nature of the crystalline rock.

Table 1

Material properties of granite specimen used in the study (Frash 2012; EMI 2010; Morrell 2012)

Material property


Young’s modulus (GPa)


Poisson’s ratio




Density (g/cm3)


Permeability (µD)




Tensile strength (MPa)


Fracture toughness (MPa m0.5)



Hydraulic injection was performed through a steel casing to a target openhole interval using a dual Teledyne ISCO syringe pump system. Injection well diameter and openhole fracturing interval diameter were approximately 10 mm and 6 mm, respectively. The injection fluid was an 80-weight gear oil with an approximate viscosity of 120 cP. MISTRAS Group, Inc. acoustic emission data collection hardware and software were used throughout all acoustic material characterization and laboratory hydraulic fracture testing. A MISTRAS PCI-2-8 system was used, which contained three PCI-2 boards and a total of six possible signal inputs. High sensitivity wideband WSα piezoelectric transducers were used. Each sensor has operating frequency range of 100–1000 kHz and a resonance of 125 kHz. Custom AE data analysis software was built in MATLAB to process the large number of AE signals in a relatively short period of time for source characterization, moment tensor solutions, parametric hit data analysis and coalesced fracture location and orientation prediction (Hampton et al. 2018). This work will focus on the latter; coalesced fracture location and orientation prediction.

Test procedure

Prior to hydraulic fracture testing, sample characterization was performed using the AE system to understand the material wave velocity, attenuation, and proper transducer location. Active and passive AE testing was performed using auto-sensor tests (ASTs) and pencil lead break tests, respectively. ASTs operate each sensor in sequence as pulsers and receivers and with known sensor locations, arrival times and signal initiation times, wave velocity structure could be determined. Sample characterization was performed before and after drilling and casing the wellbore to determine if significant changes within the velocity structure occurred providing updates to the propagation velocity input in the MISTRAS source location software.

Once all pre-test sample characterization was performed and the wellbore was drilled and cased, the fracturing procedures could be initiated. Wellhead pressure was monitored using a T-junction and pressure transducer. Once trapped hydraulic system air was bled from the lines, a constant pressure injection interval was performed to check the casing for seal leaks. Upon completion of the constant pressure injection, the fracturing process was initiated at a constant flow rate of approximately 0.1 mL/min.

Test results and observations

The granite sample was hydraulically fractured under unconfined conditions through a steel casing to a target interval. The steel casing depth was approximately 100 mm, while the openhole interval length extended an additional 50 mm down into the specimen. The openhole interval was originally intended to have a length of 100 mm starting from the cased depth to an overall bottom-hole depth of 200 mm, but a tough inclusion or layer existed within the granite sample, which prevented drilling further. Hydraulic fracture orientation predictions could not be made prior to stimulation because the sample was unstressed. Although the overall fracturing direction was not estimated prior to testing, it was predicted that the hydraulic fracture would produce a larger and more extensive network in the upper half of the sample because of the possible hard inclusion or layer encountered during the openhole interval drilling procedure.

AE source locations

Nearly 4300 AE events were observed throughout the hydraulic fracturing experiment. Figure 5a shows the large cloud of observed AE events throughout the experiment. The amplitude of each event is represented proportionally to the sizes of the circles. The colors of each event represent the correlation coefficient from the multiple regression analysis used in 3D source location. Red represents the highest correlation coefficient, meaning best location (1.0/1.0), and blue for the lowest correlation (0/1.0), with a color gradient between. From Fig. 5 the apparent total depth of the majority of the fracture network generation appeared to stay above the bottom-hole of the well as predicted from encountering the stiff inclusion or layer during drilling.

Fig. 5

AE event source locations showing very dense microcracking in the top half of the specimen. Correlation coefficient, or location accuracy is shown by color where 1/1 represents no location error and 0/1 represents very large location error. The colorbar on the right shows the distribution of color gradient. AE event amplitude is represented by the relative sizes of the circles. a Raw AE events prior to filtering. b AE source locations filtered by only those containing higher than a 0.995/1.0 correlation coefficient and 40 dB amplitude. c AE source locations filtered by only those containing higher than a 0.995/1.0 correlation coefficient and 50 dB amplitude

A secondary wellbore was required to be placed in the sample to test the ability to create a simulated injector-producer well scheme commonly found in geothermal reservoir exploitation. Secondary wellbore location optimization was difficult in such a large cloud of high amplitude and correlation coefficient AE events. It was desired that the second wellbore be located in a region of dense fracturing to provide a hydraulic connection between the original injection well and the production well through the fracture network. In other words. a macro-scale hydraulic fracture location and orientation was desired. The AE events were filtered to only contain higher correlation coefficient and amplitude data in efforts to pinpoint a main fracturing plane and/or a dense region of fractures believed to be hydraulically connected to the injection well. Figure 5b, c show these reductions and one main hydraulic fracture appeared to propagate in nearly the same orientation as the yz-plane. A very dense region of AE events existed approximately 40 mm away from the injection well at a depth of 100 mm. This region of events, coupled with the apparent direction of the main fracture was used as inputs in the secondary well trajectory plan. The second wellbore was placed at an angle of 25° from the vertical in the positive x and z direction. This path was chosen to ensure if a hydraulic fracture existed in the yz-plane, the wellbore should pass through the fracture at some point as it traverses to the other side of the sample. Figure 6 shows the injection well and the angled production well in a reduced cloud of AE events. It was uncertain as to whether this wellbore location would be sufficient for a binary injector-producer scheme until flow through tests were performed.

Fig. 6

3D view of the filtered AE event source locations and injection (vertical) and production (angled) wellbores

AE fracture surface predictions

A more advanced method of determining the location of fracture faces was desired for future tests because of the uncertainty associated with choosing a secondary wellbore location based only on amplitude and correlation coefficient filtered AE events. The method needed to be robust enough to just use AE location data and provide reasonable fracture orientation, extent, and direction from very dense clouds of data, where a coalesced fracture could not reliably be inferred. The method described above is a simple planar-based approach using distances from each AE event source location to an assumed fracture plane to determine optimal orientation of a predicted fracture.

The AE fracture detection process was applied to the data collected throughout the unconfined hydraulic fracturing test. The error determined from each of the iteration angles of the roll axis is shown in Fig. 7 from the lowest error pitch orientation. From this figure, the well-defined minimum error indicated that a fracture plane was likely in the orientation of approximately 1.7 radians from the positive roll axis. Figures 8 and 9 show the predicted hydraulic fracture appeared to intersect both the injection and production wells, which was later confirmed using fluid injection and recovery tests and post-test sample coring and slicing (Frash 2012; Hampton 2012).

Fig. 7

Fracture surface fit error versus iteration angle showing one main hydraulic fracture and other possible secondary fracture orientations. The plot shows all nine origin locations that were used (nine points surrounding a circular wellbore). The well-developed error minimum at approximately 1.7 radians in the roll direction for this particular lowest error pitch direction is shown

Fig. 8

Predicted hydraulic fracture from AE data. AE source locations have been extensively filtered for viewing, but not for the fracture surface prediction procedure

Fig. 9

Predicted hydraulic fracture from AE data

The injection well was over-cored and removed to determine the orientation of the actual fracture network. After over-coring, the granite block was slabbed at approximately one-inch intervals to view the existing hydraulic fracture network within the sample. Figure 10 shows photos of the hydraulic fracture network stenciled for clearer view. The complex network within the block existed from several hydraulic fracture injection tests, some of the latter were not recorded with AE due to software buffering issues. The fracture data from the photographs were digitized and shown in Fig. 11. The dark blue points in this figure are digitized fracture locations from the photographs, while the light blue mesh represents the interpolated surface between the measured points. This complex fracture network in an unconfined sample shows how induced stress changes from the hydraulic fractures can alter re-fracture orientation and create quite a complex system of fractures. Figure 12 shows the hydraulic fracture from Fig. 11 that is most nearly aligned with the predicted hydraulic fracture orientation determined. Comparisons made between the main hydraulic fracture predicted from the AE data and the largest fracture observed in the network from the slabbed sample can be seen in Figs. 13 and 14. Upon initial observations, the two fractures appeared to match in location, orientation and moderately in roughness trend. The roughness of the predicted hydraulic fracture was determined based on the lateral variability observed at the lowest error plane orientation. Though roughness predictions from AE data are interesting, the main take-away from these images are the general agreement between the predicted and actual hydraulic fracture orientation and extent.

Fig. 10

Slabbed sample photos showing complex network of hydraulic fractures stenciled for clearer view; depth increases from top left to bottom right

Fig. 11

All observed hydraulic fractures from slab photos. Dark blue circles represent the digitization of actual fracture locations from the images in Fig. 10. The light blue mesh is just an interpolated spline surface between these points

Fig. 12

Main hydraulic fracture visible from the slabbed sample data. Darker blue circles represent the individual data points digitized from the slab images. Lighter blue surface is an interpolation between the slab data

Fig. 13

Zoomed comparison between the AE predicted hydraulic fracture (black) and the digitized fracture location from the slab photos (blue)

Fig. 14

Zoomed comparison between the AE predicted hydraulic fracture (black) and the digitized fracture location from the slab photos (blue)


A procedure for using AE source locations to identify possible macro-scale fractures existing within the block was developed. The method is a simple planar-based approach where an error is calculated from the perpendicular distance of each AE event and an assumed fracture plane. The fracture plane is then rotated in both the pitch and roll directions to determine the lowest error orientation within the cloud of AE events. The method provided a very reliable estimation of a macro-scale hydraulic fracture as seen in the comparison images of the predicted fracture and the actual fracture, as measured from photographs taken from the slabbed block post-test.

The AE predicted fracture was at an approximate 10° offset from the yz-plane. The complexity observed from the slabbed block photos show the hydraulic fracture network to be very complex and contain not only planar fractures, but also substantial fracture curvature. The assumed planar fracture matched very well in most regions until the actual fracture showed non-planar behavior, as seen closely in Figs. 13 and 14.

The applicability of this method is reliant on the ability to observe relatively large quantities of AE events associated with a few macro-scale coalesced fractures. This assumes that the observed AE cloud resulted in a coalesced fracture, and not just a highly damaged and isolated grouping of microcracks. For instance, two identical experimental setups would produce very different fracture structure if testing a densely naturally fractured material and a homogeneous and isotropic material. The same AE fracture prediction method applied to both tests would likely give rise to differing predicted fracture error.

Summary and conclusions

An AE method to predict coalesced fracture structure, orientation and location is presented. Predictions of the coalesced fracture throughout laboratory hydraulic fracture testing in a brittle granite showed marked correlation with the actual fracture structure observed from coring and slabbing the specimen post-test. The following are the summary and conclusions from this work:

  • A simple planar-based coalesced fracture structure prediction method was developed and validated against actual laboratory hydraulic fracture data and provided a reasonable correlation using only AE event source locations as input variables.

  • This method tends to break down when the fracture structure alters from the assumed planar-based approach. Assuming non-planar fractures in regions of progressive low error might help mitigate this issue and provide a complex, but low error solution.

  • The ability to iterate through multiple origin locations for the macro-scale fracture structure prediction reduces the error associated with forcing a planar fracture to contain any specific coordinates. This functionality also allows the prediction of multiple fracture sets in a sample with a single run.

Further work is necessary to modify this process to include AE parametric data and/or source mechanism data as weighting factors for the refinement of a coalesced fracture prediction tool. Using AE event time would also serve to refine this method, as an iterative predictive tool that updates based on event time and location distribution; coalesced spline type geometry changes would also be a next step for incorporating complex curving fractures without the use of nonplanar fracture assumptions. The applicability of this method to other materials, specifically brittle civil engineering materials like concrete, is evident from this study. If AE observations are made within a concrete structure and post-observation measurement of the fractures is obstructed, this method could provide reasonable estimations of the extent and number of macro-scale fractures residing within the structure. Additional research is required to determine if this or a modified routine can more accurately define coalesced fractures occurring in less brittle materials or complex loading environments which produce curved fractures.



The authors gratefully acknowledge financial support was provided by the US Department of Energy under DOE Grant No. DE-FE0002760. The opinions expressed in this paper are those of the authors and not the DOE.


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Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Colorado School of MinesGoldenUSA

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