PROCEEDINGS, Thirty-Ninth Workshop on Geothermal Reservoir Engineering
Stanford University, Stanford, California, February 24-26, 2014
SGP-TR-202
1
Strain Measurement of Geological Samples Subjected to Triaxial Stresses Experienced
During Hydraulic Loading
Yarom Polsky, Lawrence Anovitz, Ke An and Luc Dessieux
Oak Ridge National Laboratory
1 Bethel Valley Rd
Oak Ridge, TN, 37831 U.S.A
e-mail: [email protected]
Keywords: hydraulic fracture, strain measurement, rock mechanics testing, triaxial testing
ABSTRACT
Understanding stress, strain and material failure relationships, and having the ability to predict these quantities for known load
conditions, is crucial to all geomechanics and, in some instances, reservoir flow applications. The constitutive equations governing
the deformation of geological materials are typically adequate for bulk or large scale deformation and stress analyses. However,
these rules are generally less precise in their ability to make accurate predictions in physical processes where highly localized
material heterogeneity exists or where the presence of geometric irregularities such as micro-cracks may be present. This is
especially relevant to EGS where hydraulic fracture propagation models are needed to develop optimal reservoir creation strategies
and where fracture permeability is significantly influenced by regional stress states and may affect reservoir operation strategies.
The deficiencies of the models used to describe these physical processes are a practical reality necessitated by the manner in which
rock properties must be obtained. Conventional rock mechanics tests subject samples to controlled load conditions and measure
bulk deformations of the sample or more localized deformations only on exposed surfaces of the sample. They are currently unable
to comprehensively map the deformation state within the sample. For processes such as fracture, however, the state of a particular
region within the rock drives the overall failure behavior of the sample.
The authors believe that developing a means to measure strains within samples subjected to hydraulic fracture loading conditions
will provide a useful tool for understanding the localized effects not captured by conventional techniques and may serve as a
method for improving hydraulic fracture models. An ongoing effort at Oak Ridge National Laboratory endeavors to develop a
neutron diffraction based strain measurement capability to interrogate the strain state of a geological sample, at arbitrary internal
locations, subjected to a triaxial stress state. The basis of the method and initial results for simple load conditions were reported at
last year’s Stanford Geothermal Workshop. This work will report results from recent neutron diffraction strain measurement
experiments in which marble samples were subjected to load conditions more representative of hydraulic fracturing operations
within a pressure cell specially designed for the reported strain measurement technique.
1. INTRODUCTION
Understanding and modeling hydraulic fracture propagation has been an active area of research since it became a viable practice for
enhancing production in Oil & Gas applications in the early 1950’s. A variety of models have been developed over the years to
describe the hydraulic fracture propagation process including the PKN, KGD, and assorted 3-D models. They are generally based
on linear elastic fracture mechanics and posit that fracture initiation occurs when a critical stress intensity factor is exceeded. This
factor is generally related to the presence of a crack in the material which produces a region of elevated stress near the crack tip.
For the simple 2D case of a mode 1 crack developing radially from a hole in a plate, the regional stress is given as (Rice, 1968)
[
]
√
[
]
(1)
where are the stresses near the crack tip, KI is the stress intensity factor, and r and θ are cylindrical coordinates for a reference
frame with origin at the tip of the crack.
Conventional methods, such as strain gauges, are generally unable to measure the strain corresponding to this stress unless the
crack extends to an exposed surface. This is a significant limitation for bulk samples because it prevents comprehensive
characterization of the material stress state and relationship of this state to material failure. Appropriate failure criteria for rocks are
therefore arguably best determined using conventional rock mechanics testing and measurement methods by statistically comparing
the predictions of various criteria developed over the years to numerous experiments (Colmenares and Zoback, 2002). While this
empirical technique may be practical and statistically effective, it does not provide insight into the fundamental mechanisms and
structural features that drive failure.
There have also been a relatively large number of experimental efforts in recent years studying hydraulic fracture at the laboratory
scale. Much of this work has focused on using techniques such as acoustic emissions to better understand fracture event onset,
location, evolution and propagation dynamics (Stroisz et al, 2013and Damani et al, 2013). It is often accompanied by post-mortem
examination of the specimens using inspection methods such as X-Ray computed tomography or scanning electron microscopy. It
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has also often involved comparison of experimental results to simplified fracture initiation pressure models such as the Hubert and
Willis model:
(PC-Po)=T+3σh-σH
(2)
There is generally significant scatter in the data produced during these experiments indicating that a number of factors such as
material heterogeneity, grain boundary effects or the presence of stress rising features may have a significant influence on fracture
onset. The current lack of means to perform measurements internal to the sample is a significant limitation to further understanding
of this variability.
This work is part of a long term effort to adapt neutron diffraction based strain measurement techniques to geological applications.
The highly penetrating nature of the neutron permits the use of such techniques even when otherwise opaque structures such as
pressure vessels surround the measured specimen. Successful development of the method will permit strain mapping of the internal
regions of core samples subjected to triaxial stress conditions while undergoing hydraulic loading. This enabling capability can be
used to study many critical factors affecting or driving the hydraulic fracture process including, but not limited to, residual strains in
specimens prior to loading, localized strain variations within otherwise symmetrically loaded samples and the association of these
states with stress raising features or material variability, and strain redistribution within specimens during loading. This quantitative
tool can in turn be used to develop better models of the fracture initiation process and/or validate existing models.
The path to developing the methodology for implementing this capability is admittedly challenging. The sample environment and
experimental setup must be carefully designed for use with neutron diffraction instruments, the scattering characteristics of the
experimental setup and sample must be well understood in order to ensure that sufficient scattering statistics are accumulated to
enable accurate calculation of lattice parameters, and processing of the acquired neutron intensity data must be carefully performed
and related to calibration experiments in order to accurately calculate strain of the interrogated sample volumes. This paper will
describe the most recent efforts completed in this pursuit.
2. EXPERIMENTAL DESCRIPTION
A detailed description of the geothermal pressure cell was provided in a paper published at last year’s Stanford Geothermal
Workshop (Polsky et al, 2013). The cell design permits independent pressurization of the core surfaces in the axial and radial
direction to produce compressive loads representing confining stresses similar to conventional triaxial test methods. The core
specimen dimensions for the cell are 38.1 mm diameter by 152.4 mm length. Carthage Marble specimens were used in the
experiments described in this publication. A 75 mm long, 6.35 mm diameter hole was drilled through the center of one end of the
sample. Fluid pressure was to be simultaneously applied through the sample axial surface and this hole to create a fracture inducing
stress, resembling a stress on a borehole wall, with equivalent axial confining pressure.
Hydraulic fracture for this configuration is only achievable if the pressure applied to the hole exceeds the radial confining pressure.
It was discovered that this was not possible during trials performed prior to the neutron experiments because the pressures were not
adequately isolated from each other. This configuration uses an o-ring seal between the rock surface and axial flow component of
the pressure cell. Measurements performed after initial test failure indicated that the axial faces of all samples were between 5 and
10 degrees misaligned with the cylindrical axis of the sample. This sealing surface misalignment prevented the o-ring from
performing its sealing function, allowing the bypassed fluid to act on the sleeve delivering radial confining pressure. This sleeve is
unable to resist outward radial pressure and failed repeatedly during the staging of the experiment.
There was insufficient time for re-machining of the samples to meet alignment specifications prior to the neutron experiment so the
decision was made to epoxy a stainless steel tube in the hole to deliver fracture pressure. The tube was inserted to a depth of
approximately 25 mm. This sample configuration is depicted in the left of the Figure 1 below and only permits application of radial
confining pressure in addition to internal pressurization of the hole in the sample. This method was then successfully demonstrated
for initiating hydraulic fracture in the laboratory.
Neutron diffraction lattice spacing measurements were made through the pressure cell at 2 axial locations within the core sample at
each load step. The axial locations were 25 mm above the hole bottom and 30 mm above the hole bottom. Four points with gage
volume dimensions 2 mm x 2 mm x 12 mm were measured at roughly 2 mm from the hole wall for a total of 8 measurement points
per load step (see Figure 1 left for reference). The 12 mm dimension corresponds to the axial direction of the sample.
Lattice strain measurement was done at the VULCAN engineering diffractometer at the Spallation Neutron Source, Oak Ridge
National Laboratory (An et al, 2011). Lattice plane measurements were performed per the load schedule shown below in table 1.
The internal pressure level in load step 2 was meant to be sufficiently far from the expected failure load to ensure that fracture
initiation did not occur before the sample was placed in the beam. If rock failure occurs before the sample is placed in the beam
then measurements must be repeated over the entire load history for a new specimen to capture the strain evolution with load.
Future experiments will perform laboratory characterization of the fracture initiation pressure for multiple samples to identify an
appropriate peak internal pressure value that is closer to the fracture initiation pressure. Adjustment of the internal pressure
following neutron measurements produced fracture of the Carthage Marble sample at approximately 31 MPa. A photo of the
sample following fracture is shown in the right of Figure 1 below. This tensile failure in the plane perpendicular to the axial
direction is a result of the lack of compressive confining pressure in the axial direction. Future modification of the experimental
setup will ensure that the minimal principal stress is in the radial direction in order to produce fracture directions more
representative of subsurface conditions.
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Figure 1: Sample configuration and measurement points (left), Pressure cell with sample in Vulcan Neutron Instrument
(center) and Post fracture core sample (right)
Confining Pressure Internal Pressure
Baseline 0 MPa 0 MPa
Load Step 1 14.5 MPa 0 MPa
Load Step 2 14.5 MPa 17.25 MPa
Table 1: Confining and internal pressures for measurement steps.
3. RESULTS
3.1 Lattice Strain Measurements
Strains associated with lattice deformations were calculated in the standard manner by evaluating peak shifts associated with
particular crystallographic planes using the relation
(3)
where is the measured lattice plane spacing at a load measurement point and is the lattice plane spacing with no external
loading. Lattice values at the 0 MPa load state were taken for the of each measured location for the strain mapping
measurement in the pressure cell experiments. This provides a strain value relative to the baseline condition, which may contain
residual stresses, however it does not necessarily provide a strain value of the rock relative to a stress free state. More representative
stress free values are typically obtained from powder diffraction samples. Such data was not obtained in time for this
publication but is planned in the near future.
Accurate strain calculation using this technique is a meticulous process requiring extensive instrument calibration and
understanding of the scattering characteristics of the system exposed to the beam. The data reduction procedures must also be
carefully performed to ensure that the neutron intensity data collected during the experiment is sufficient and accurately related to
lattice spacing values. The first two sets of experiments performed at the Spallation Neutron Source Vulcan beam line have been
learning experiences on the path to establishing the validity of the approach. The first set of experiments performed as part of this
effort in December of 2012 focused on lattice strain measurement of geological specimens during uniaxial compression tests to
develop diffraction specific elastic constants. For relatively isotropic, texture free materials (materials without preferred grain
orientations), macroscopic stresses can then be related to lattice plane family strains using the constitutive relation (Hutchings et al,
2005):
[
] (4)
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The uniaxial load tests verified that strains can be reliably and accurately measured for geological specimens. The stress-strain
curves obtained for the Carthage Marble calcite crystal lattice planes are shown below in Figure 2. The elastic moduli for the
different planes are consistent with literature values as reported last year (Polsky et al, 2013). A cubic polynomial fit for the
measured macroscopic stress versus lattice strain is also shown in each graph.
Figure 2: Stress vs strain curves for calcite lattice planes measured for uniaxial compressive load tests
The fidelity of the data calculated in the triaxial loading experiments was in general poor. The 8 measurement points, 4 lattice
planes per point (006, 018, 104 and 113) and two load steps evaluated during the experiment result in 64 possible lattice strain
measurement points. The selection criteria typically used for a good fit of the neutron intensity data are the chi-squared statistic and
the number of raw neutron counts measured in the fitting region. The latter was particularly low for the experiments performed and
is attributed to scattering and attenuation by the titanium pressure cell, whose scattering tendencies were not adequately
characterized prior to the measurements. Improved statistics could have been achieved with increased counting time or selection of
a pressure cell material with better scattering characteristics, such as Aluminum. The poorer quality of the data is illustrated by
comparing a representative fit from the uniaxial load tests, which did not require the pressure cell, with a representative fit from the
pressure cell experiments as shown in the figure below.
Figure 3: Peak fits uniaxial load test (left) and pressure cell test (right)
A small set of the total, 18 of 64 points, were considered to have reasonable d0 and d values to justify lattice strain calculations. It is
important to note that both the initial and shifted lattice plane must be accurate in order to calculate an accurate strain value. With
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respect to threshold measurement counts for selection, the bank 1 and bank 2 detectors of the instrument showed different
sensitivities so different selection criteria were used for the data produced from each bank. Bank 1 measured peaks were fitted if
their intensities exceeded 1000 counts and bank 2 peaks were fitted if their intensities exceeded 300 counts. Peak fits were accepted
for strain calculation only with chi-squared values below 1.5. Additionally, it was noticed when initially using the peak fitting
algorithm that peak identified by the algorithm sometimes varied based on the initial values set by the user. Sensitivity analysis was
performed on all candidate peaks by evaluating the output produced by a range of initial lattice spacing and peak width values and
only peaks with less than 0.001% variation were accepted.
The lattice strains, d values and chi-squared values calculated from reasonable data points are shown are shown below in Table 1
for illustrative purposes. It is reiterated that although these values in principal seem reasonable, the data fidelity would be much
improved with better neutron counting statistics. Also for reference, the measurement point locations within the sample and strain
measurement directions of the instrument are shown from the plan view below in Figure 4. Understanding this orientation is
necessary for relating the strain measured by the instrument to a more convenient cylindrical coordinate system with origin at the
center of the sample. Bank 2, for example, measures radial strain in the sample cylindrical coordinate system while bank 1
measures circumferential strain.
Point Load # Bank Plane d0 d Lattice Strain Chi Squared
2 1 1 104 3.019956 3.019235 -238.7452003 1.36
2 1 1 006 2.830121 2.829467 -231.085526 0.90799999
5 1 1 104 3.019252 3.018724 -174.8777512 1.17
7 1 2 018 1.915553 1.915275 -145.1278038 1.12
5 2 1 104 3.019252 3.018763 -161.9606446 1.17
4 2 2 113 2.28942 2.289328 -40.18485031 1.05
4 2 2 104 3.045671 3.045507 -53.84691912 1.16
7 2 2 018 1.915553 1.915471 -42.8074817 1.12 Table 1: d0, d , lattice strain and Chi-squared values for select measurement points
Figure 4: Measured strain directions and locations of measurement points in r,θ plane of sample
3.2 Comparison to Finite Element Results
Finite element analysis (FEA) of the sample load cases was performed with COMSOL software for comparison to lattice strain
measurement results. Figures 5 and 6 below show radial and circumferential stresses versus the distance from the hole surface to
the outer diameter of the core sample. The length unit on the x-axis is inches. The strain values measured in the experiment
represent lattice strains and are not equivalent to macroscopic strains. They must be related to macroscopic stresses using
diffraction elastic constants as defined in equation (4). This is not feasible for the performed experiments because the complete
strain state was not captured due to the limited data available as explained above. All load case 1 strain values for example are
circumferential strains. Because the radial strains near the borehole wall are low for this load case it is nonetheless possible to
estimate the macroscopic stresses from the stress-strain curves shown in Figure 2.
Estimates of macroscopic stresses were made for applicable data points using this method. Only points for the 018 plane were not
estimated because there were no lattice stress-stain curves for this plane available from the uniaxial load tests. It is also noted that
the estimated stresses correspond to average stresses within the gage volume of measurement as opposed to element stresses as
typically obtained from FEA. Accurate relation of the measured stresses to simulated stresses requires averaging of the simulated
stresses over the measurement gauge volume. The average circumferential and radial stresses for load case 1 are 24 MPa and 5 MPa
respectively. The load case 2 values are 13 MPa in the circumferential direction and 16 MPa in the radial direction. Results of the
stresses as determined from the lattice strains are displayed in Table 2. Most of the stresses are in reasonable agreement
demonstrating the fundamental viability of the strain measurement technique.
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Figure 5: Radial (left) and Circumferential (right) stresses from hole surface to sample outer diameter for 14.5 MPa
confining pressure, 0 MPa internal pressure (x-axis is distance in inches from the hole surface)
Figure 6: Radial (left) and Circumferential (right) stresses from borehole wall to sample outer diameter for 14.5 MPa
confining pressure, 17.25 MPa internal pressure
Point Load # Bank Plane Lattice Strain Macroscopic Stress Stress Direction
2 1 1 104 -238.7452003 25 θ
2 1 1 006 -231.085526 22 θ
5 1 1 104 -174.8777512 17 θ
7 1 2 018 -145.1278038 N/A θ
5 2 1 104 -161.9606446 14 R
4 2 2 113 -40.18485031 3 θ
4 2 2 104 -53.84691912 5 θ
7 2 2 018 -42.8074817 N/A θ Table 2: Estimated macroscopic stress corresponding to measured lattice strain for select points and load steps
CONCLUSION
Significant progress has been made demonstrating the applicability of neutron diffraction based strain measurement methods to
rock mechanics applications. This publication reported lattice strain measurements of the interior of a Carthage Marble sample
subjected to a hydraulically-induced triaxial stress state. The evolution of this effort has been a learning experience made difficult
by the structural complexity of geological samples, which complicates acquisition and interpretation of neutron measurements, and
the interactions of the pressure cell with the neutron beam. Processing of the acquired neutron intensity data has also been a
challenge, but it is believed that a reasonable methodology has been developed for reducing data to strain values and it is believed
that these strain values can be reliably related to macroscopic stresses.
Further advancement of this technique holds great promise for geological applications, including geothermal energy extraction, and
the rock mechanics community in general. The ability to measure the strain experienced in the interior volume of a sample at
arbitrary locations in a high pressure environment is a unique and valuable tool. The authors are aware of no other method for
making measurements of the strain state within bulk samples. It is believed that these techniques, once developed, will enable
otherwise difficult or impossible insights into critical subsurface applications such as hydraulic fracturing and will serve as both a
means for improving theory and understanding of such processes as well as a means for validating simulation tools.
Future experimental efforts will focus on improving the neutron scattering characteristics of the pressure cell and honing the
procedure for determining when adequate counting statistics have been acquired. An Aluminum pressure vessel has been
constructed to address the former concern and is expected to provide a 3-4 time reduction in beam attenuation and scattering.
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Consideration will also be given to redesigning the mating interface between the pressure vessel and the axial core sample face to
eliminate the sealing issue experienced during the reported experimental work.
ACKNOWLEDGEMENT
Research supported by the Geothermal Technologies Office, Office of Energy Efficiency and Renewable Energy, U.S. Department
of Energy under contract DE-AC05-00OR22725, Oak Ridge National Laboratory, managed and operated by UT-Battelle, LLC.
The research at ORNL's Spallation Neutron Source was sponsored by the Scientific User Facilities Division, Office of Basic
Energy Sciences, U.S. Department of Energy.
The authors would also like to thank Mr. Harley Skorpenske and Mr. Dunji Yu for their support during the neutron diffraction
experiment.
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