University of Southern California Dissertations and Theses

Stress and deformation analysis on fluid-exposed reservoir rocks

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Stress and deformation analysis on fluid-exposed reservoir rocks

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Content STRESS AND DEFORMATION ANALYSIS ON FLUID-EXPOSED RESERVOIR ROCKS
by
Rayan Dabloul
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(PETROLEUM ENGINEERING)
August 2022
Copyright 2022 Rayan Dabloul
Dedication
To my beloved parents, Ahmed and Jamila, my wife Mrouj and daughter Aaliya, my sisters Ala’a, Rawan,
Ilaf, and my brother Ghassan.
ii
Acknowledgements
I would like to send my deepest thanks and gratitude to my advisor, Professor Birendra Jha. Your guidance,
patience and encouragement are what made my work as it is today. Your support through this journey is
what kept me going. I am sincerely honored to have been part of your research group. I would also like to
thank Professor BoCheng Jin for his guidance and patience. It has been an honor working with you. This
work would hardly have been completed without your support. My appreciation also goes to Professor
Iraj Ershaghi for serving in my research committee and for the support through my research.
My deepest thanks goes to Dr. Ghaithan Muntasheri for his unlimited support during my research.
Finally, I am grateful to my parents, Ahmed and Jamila, my wife Mrouj and my daughter Aaliya, my
sisters Ala’a, Rawan, and Ilaf, and my brother Ghassan. You have provided me moral and emotional support
to my life at USC even though we have been thousands of miles apart. You have given me the opportunities
and experiences that have made me who I am. This journey would not have been possible if not for you,
and I dedicate this achievement to you.
iii
TableofContents
Dedication ii
Acknowledgements iii
ListofTables vi
ListofFigures vii
Abstract xi
Chapter1: Introduction 1
Chapter2: ExperimentalMethodology 5
2.1 Single Edge Notched Beam Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Digital Image Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Uniaxial Compression Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Chapter3: ExperimentalSetup 9
3.1 Single-Edge Notched Beam Experiment and DIC . . . . . . . . . . . . . . . . . . . . . . . . 9
3.1.1 Specimen Properties and Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.1.2 Experimental Workflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2 Uniaxial Compression Test, DIC and Active Seismic Monitoring . . . . . . . . . . . . . . . 14
3.2.1 Specimen Properties and Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2.2 Brine Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2.3 Experimental Workflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Chapter4: ExperimentalResults 19
4.1 Single-Edge Notched Beam Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.1.1 Deformation and elastic moduli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.1.2 Fracture Toughness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.1.2.1 STM Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.1.2.2 Fischer-Cripps’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.1.2.3 Energy Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.1.2.4 Effect of pore fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.1.3 J-integral analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.1.4 Fracture Propagation Analysis using DIC . . . . . . . . . . . . . . . . . . . . . . . 28
4.1.5 Numerical Modeling and Simulation of SENB . . . . . . . . . . . . . . . . . . . . . 31
4.1.6 Model construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
iv
4.1.7 Model results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.1.7.1 Stress invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.1.7.2 Stress components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.1.7.3 Stress intensity factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.2 Uniaxial Compression Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.2.1 Scanning Electron Microscope (SEM) and X-Ray Diffraction . . . . . . . . . . . . . 46
4.2.2 Acoustic monitoring of fracture propagation . . . . . . . . . . . . . . . . . . . . . . 47
4.2.3 Fracture Propagation Using DIC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.2.4 Numerical Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
Chapter5: Conclusions 62
Bibliography 64
v
ListofTables
3.1 Berea Sandstone Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2 Specimen Dimension and Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.3 Fracture toughness calculated results for dry, water–exposed and scCO
2
–exposed specimens 11
3.4 Specimens and Flaws’ dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.5 Specimen Dimension and Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.6 Synthetic Brine Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.1 Mechanical properties of the fixture and the specimens. For each fluid type, median values
over all the experiments with that fluid type are used. . . . . . . . . . . . . . . . . . . . . . 21
4.2 SENB numerical simulation inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.3 Specimens’ Fluid type and exposure duration . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.4 Simulation imput of the Uniaxial Compression for Indiana Limestone specimen . . . . . . 54
vi
ListofFigures
2.1 An illustration of our setup of Single Edge Notch Beam (SENB) testing with Digital Image
Correlation (DIC). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 A schematic of our SENB specimen with dimensions, boundary conditions and load-
displacement configuration. P is the load applied by the loading pin at the middle of the
top surface, andδ is the load cell displacement. The specimen rests on two support pins.c
is the depth or length of the pre-existing notch into the specimen. Pins can be considered
rigid for practical purposes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Illustration of specimen with two parallel flaws. a. test setup. b. S wave transducers’
position. c. flaws geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.1 Specimen is rectangular in section with a length of 50 mm, width of 25 mm, and thickness
of 10 mm. The fracture develops along the notch axis and the final fracture surface area,
neglecting the surface roughness, can be approximated as 240 mm
2
. . . . . . . . . . . . . . 10
3.2 Experimental workflow with different steps taken to perform SENB tests and DIC analysis
in this study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.3 Post-mortem analysis of the fracture surface geometry in case of (a) the D1 dry specimen
and (b) the W1 water-exposed specimen. The horizontal axes represent the X and Y
axes in microns. The color scales represent the depth of the fracture surface in microns.
The post-mortem analysis shows a higher roughness of the fracture surface of the dry
specimen compared to the fracture surface of the water-exposed specimen. . . . . . . . . . 14
3.4 System setup and process of how specimens are exposed to different fluid types. . . . . . 17
3.5 S-wave transducers setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.6 Instron Uniaxial compression system (left) and National Instrument PXIe-1088 data
acquisition system (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.1 Load vs. displacement curves of dry (top), water-exposed (middle), and scCO
2
-exposed
(bottom) specimens. The insets show horizontal displacement fields ( U
x
) at the highlighted
time steps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
vii
4.2 Flexural stress and flexural strain of dry, water-exposed, and scCO
2
-exposed specimens.
Dry and scCO
2
-exposed specimens have higher values compared to the water-exposed
specimens. Black lines indicate the lowest and highest flexural stress and strain values,
lower and higher horizontal blue lines indicate the 25 percentile and 75 percentile values,
respectively. Red horizontal lines indicate the median values. . . . . . . . . . . . . . . . . . 22
4.3 Fracture toughness results for different pore fluid types. . . . . . . . . . . . . . . . . . . . . 25
4.4 Time evolution of the deformation field U
x
in millimeters for dry (left column), water-
exposed (middle column), and scCO
2
-exposed (right column) specimens based on the DIC
analysis. δ is the load cell displacement at any given time, hence evolution inδ is a proxy
for evolution in time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.5 Time evolution of the deformation field U
y
in millimeters for dry (left column), water-
exposed (middle column), and scCO
2
-exposed (right column) specimens based on the DIC
analysis. δ is the load cell displacement at any given time, hence evolution inδ is a proxy
for evolution in time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.6 Horizontal displacement (U
x
) at peak load along the horizontal direction (X) for dry and
water-exposed specimens at three different Y values from the bottom edge. Inset figure
shows the full spatial distribution of theU
x
displacement field in the two cases. . . . . . . 31
4.7 Mesh of the 3D simulation model of the SENB test. Dimensions of the specimen and the
three pins and their relative positions are identical to our experimental setup in Fig. 2. The
initial notch on the bottom edge is difficult to see because of the mesh lines. The black
box shows the zoom-in window where we analyze stress fields and crack propagation in
detail in Figures 16-17 below. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.8 Simulation load vs. displacement curve of dry specimens. . . . . . . . . . . . . . . . . . . . 33
4.9 Horizontal (U
x
) and vertical (U
y
) displacement results of a SENB test simulation. Bending
leads to tensile crack initiation at the tip of the notch as shown by the middle row of figures. 34
4.10 Change in the Von Mises stress and the volumetric stress for the dry specimen at three
successive time steps. The volumetric stress is positive under compression. Fracture
initiation at the notch can be seen at later time steps, i.e., middle and bottom rows. . . . . 37
4.11 Stress path analysis in the Von Mises vs. volumetric stress space. The increase in the von
Mises stress under compression indicates that the left and top pins evolve towards shear
failure, as the notch evolves towards tensile failure. The inset figure shows the stress
paths in the vicinity of the propagating crack, confirming the dominance of tensile failure
near the tip. Sign convention: positive values of volumetric stress indicates compression. . 38
4.12 Crack propagation in a SENB simulation, shown in a zoom-in view (black box in Figure 4.7
around the notch. The colored background is the normal stressσ xx
, which evolves with
time. The number near the top of each figure indicates the time step value. . . . . . . . . . 39
viii
4.13 Spatial distribution and temporal evolution of the normal stressσ yy
in the vicinity of the
notch. The zoom-in view window is the same as in the figure above. . . . . . . . . . . . . 40
4.14 Spatial distribution and temporal evolution of the von Mises stress in the vicinity of the
notch. The zoom-in view window is the same as in the figure above. . . . . . . . . . . . . 41
4.15 Profiles of stress components along and across the notch showing the effects of crack
propagation and specimen bending. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.16 Stress profiles from our simulation agree well with the Irwin crack tip solution functions
(scaled), which are shown in Eq. (4.6). The agreement is better closer to the crack tip
(X = 0,Y = 1 mm). Away from the tip, SENB loading conditions dominate which are
not captured in the analytical solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.17 Evolution of the stress intensity factor K
I
ahead of the crack tip as the crack length
evolves. Crack propagation events are marked by a drop inK
I
ahead of the tip. Crack
length is measured along the notch or loading axis. Crack length along the crack surface
is 1-5% longer and increases as the crack opening displacement or aperture increases with
time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.18 Evolution of the J integral ahead of the crack tip as the crack length evolves . . . . . . . . 45
4.19 Top plate load vs load cell displacement for (sc)CO
2
, brine, and mixture of brine and (sc)CO
2
46
4.20 SEM Results. The left figure is taken before scCO
2
and brine-mixture exposure. The right
figure is taken after the exposure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.21 (a) Transmitted S-waves through the specimen as the load on the specimen increases. (b)
As the load increases, the first arrival time and amplitude increase to a maximum before
dropping due to macroscopic failure. The load at which maximum is reached is different
for the two curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.22 Transmitted signal first arrival amplitude as a function of the applied load . . . . . . . . . 48
4.23 Spatial distribution and time evolution of the horizontal displacement (Ux) and the vertical
displacement (Uy) on the surface of the D4 specimen exposed to air. The displacements
are in mm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.24 Spatial distribution and time evolution of the horizontal displacement (Ux) and the vertical
displacement (Uy) on the surface of the B2 specimen exposed to brine. The displacements
are in mm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.25 Spatial distribution and time evolution of the horizontal displacement (Ux) and the vertical
displacement (Uy) on the surface of the SB3 specimen exposed to a mixture of brine and
scCO
2
. The displacements are in mm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.26 Horizontal (U
x
) and vertical (U
y
) displacement results of a uniaxial compression specimen
simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
ix
4.27 Change in the von Mises stress and volumetric stress for a dry specimen throughout a
uniaxial compression test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.28 (σ xx
) and (σ yy
) results of a uniaxial compression simulation for a dry specimen . . . . . . 57
4.29 Comparison of (σ xx
), (σ yy
) , and (σ xy
) results of a uniaxial compression dry specimen
simulation at failure for 4 sets of simulation runs . . . . . . . . . . . . . . . . . . . . . . . . 58
4.30 Comparison of Von Mises Stress and Volumetic Stress at failure for 4 sets of simulation runs 59
4.31 Stress path analysis in the Von Mises vs. volumetric stress space. The increase in the
von Mises stress under compression indicates that the top and between the two flaws
evolve towards shear failure, as the notch evolves towards tensile failure. Sign convention:
positive values of volumetric stress indicates compression . . . . . . . . . . . . . . . . . . 60
4.32 Zoomed-in Stress path analysis in the Von Mises vs. volumetric stress space Sign
convention: positive values of volumetric stress indicates compression. . . . . . . . . . . . 60
4.33 Evolution of the stress intensity factorK
I
ahead of the top crack tip as the crack length
evolves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
x
Abstract
Injection of fluids during wastewater disposal and geologic carbon sequestration causes induced stresses
and changes in rock’s elastic and failure response, for example, elastic modulus and fracture toughness.
An accurate understanding of such changes in the response requires modeling and analysis of fluid-
induced changes in rock’s stress and deformation states for which core-scale mechanical loading tests
are often employed. Using experiments and simulations of Single Edge Notched Beam (SENB) test on
water-exposed and supercritical (sc) CO
2
-exposed Berea sandstones, and Uniaxial Compression (UC) test
on brine-exposed and brine-(sc) CO
2
exposed Indiana limestone, the effect of pore fluid on tensile crack
propagation, the plane strain fracture toughness, and spatial distributions of shear and tensile stresses are
quantified. Digital Image Correlation (DIC) is used to monitor the changes in local stress and deformation
fields during crack initiation and propagation in the SENB test. In addition to DIC, active seismic in the
(UC) test is included to detect the onset of crack intiation and propagation. Representative numerical sim-
ulation models of dynamic crack propagation under SENB and UC loading are built using the extended
finite element method. Results are analyzed using the stress path analysis to identify shear and tensile
stress concentration regions, which provide precursory information for the timing and location of failure
events during the test. Stress profiles along and across the growing crack is analyzed and compared with
Irwin’s classical solutions to understand the impact of SENB loading and UC. Evolution in stress intensity
factor and crack length shows subcritical crack growth prior to Griffith-like failure at the structure scale.
xi
Chapter1
Introduction
Long-term injection of CO
2
in deep saline aquifers for carbon sequestration [8, 46, 39, 13, 74], enhanced
oil recovery [82, 81], and enhanced methane recovery [36] causes substitution of in-situ pore fluids (brine
or hydrocarbons) with the injected supercritical (sc) CO
2
and a mixture of the scCO
2
and the ambient
fluids. Brine in the saline aquifer reacts with the CO
2
to form acidic mixtures which can react with the
minerals in the grains and in the cementation material between the grains. This can alter the porome-
chanical response of the reservoir rock under hydromechanical loads of natural or anthropogenic origin,
e.g. hydraulic fracturing [15], injection-induced stresses [67, 82, 81], earthquakes and ground deforma-
tion [71, 55]. To understand and predict such changes in the poromechanical response, multiple studies
have investigated the effects of fluid substitution on properties of the rock [12, 3, 14, 30, 11, 61]. Such stud-
ies are required to understand the chemo-mechanical effects of pressurized scCO
2
on rock’s mechanical
properties for ensuring safe long-term storage. We believe that it is important to understand the effect
of individual fluid components—scCO
2
and water—separately, because scCO
2
and water have different
physical and chemical properties, and the properties of the CO
2
-water mixture does not always follow
linearly from the properties of the individual components. For example, the CO
2
-water mixture density
varies non-linearly and non-monotonically with the CO
2
mass fraction, which has implications for CO
2
storage capacity [20, 24]. Gassmann’s model, which assumes homogeneous isotropic mono-mineral rock
1
saturated with chemically inert fluid under an undrained condition [23, 10] provides a theoretical basis
for estimating the change in elastic moduli (e.g. bulk modulus) resulting from the pore fluid substitution.
Improved fluid substitution models have emerged since then, e.g. [45]. However, the question of how the
pore fluid affect the state of stress and failure properties of the rock still remains. Fluid-induced changes
in shear and/or tensile stresses is of utmost importance because it dictates the timing, magnitude and lo-
cation of near-wellbore and caprock fracturing, injection-induced seismic/aseismic events [81], and fluid
leakage events [55] within and outside the reservoir [13, 56]. At smaller scales, changes in rock failure
properties lead to changes in hydraulic conductivity and diffusivity [51, 15], fluid transport and spreading
mechanisms [77, 69], and hydraulic fracturing characteristics [19, 53, 52].
Many fracture modeling tools are based on the Irwin-Griffith brittle failure theory according to which
a crack extends when the strain energy release rate becomes equal to the surface energy required to create
the two sides of the fracture segment [26, 48]. As the energy available from the released strain energy
exceeds what is required to create the new surfaces, unstable crack growth occurs leading to the catas-
trophic failure of the specimen [48]. [40] made an important modification to the theory and proposed a
critical stress intensity factor, normally referred to as the fracture toughness, which is defined as a measure
of the ability of a material to resist an unstable fracture propagation. Strength and toughness properties
of a rock are related to each other due to the micromechanical processes and granular texture that are
common to both. To estimate the rock strength, there are mainly two kinds of measurement methods [25]:
(1) static methods, which include uniaxial compression (ASTM D 2938-86), triaxial compression, Brazilian
test (diametrical compression of disk-shaped rock specimens), bending test (ASTM E1820), torsion and
hollow cylinder test, and (2) dynamic methods, which include resonant bar method and ultrasonic pulse
method [2, 32, 1]. Using these and other methods, several experimental studies have been conducted to
quantify fluid-induced changes in elastic and failure properties of the rock. For example, one study [64]
found negligible changes in elastic properties of scCO
2
and CO
2
-water-saturated limestone specimens
2
under in-situ conditions. Another study [72], using the dynamic method, showed a drop in the elastic
modulus and strength of CO
2
-brine saturated carbonate rocks.
While extremely beneficial in improving our understanding of fluid’s role in altering rock’s failure
response, these studies have also raised some questions which require a quantitative understanding of
stress distribution and evolution during the failure process. For example, the Brazilian test method can
be used to examine fracture initiation and propagation, if the spatial distributions of principal stresses,
stress trajectories, and the ratio of minimum to maximum principal stress are known [17, 49]. However,
a measurement of the in-situ state of stress is difficult in practice due to the tensorial nature of the stress
tensor; there are six independent stress components at each point in a 3D domain. The principal stress
coordinate system also has six unknowns; three principal stresses and three principal directions, which
may evolve in time with the near-wellbore flow and deformation processes and the far-field boundary
conditions. Due to these issues, numerical modeling and simulation has emerged as the most promising
tool for quantifying the state of stress during dynamic failure. However, such a numerical model and its
stress analysis does not exist in the literature for one of the standard rock fracture toughness tests, the
Three Point Bending (3PB) or Single Edge Notched Beam (SENB) test. A SENB test is used to estimate
flexural and tensile failure properties of a rock under uniaxial compression because a tension-based failure
test for highly brittle specimens, or for specimens with an unexpected brittle response, is hard to conduct
safely in the lab.
In hydrocarbon reservoirs, P wave velocity reaction to crack initiation and propagation is different
depending on the porous media properties such as rock type, density, grain size, porosity and the type of
pure fluid, and reservoir conditions including weathering, bedding planes and cracks’ roughness and filling
material [42]. Increased spectral amplitudes of the P wave pulses are observed in both intact and fractured
specimens. Specific stiffness of a fracture is to increase in presence of a fluid with a bulk modulus greater
than air, using the displacement discontinuity theory [58]. P wave velocity is found to be higher in high
3
pore fluid saturation, while S wave velocity is found to be lower as saturation increases [43]. ""Gassmann’s
theory"" Effects on rock strength of different fluid saturation levels in different types of rocks have been
investigated in several studied [18]. In sandstone rocks, moisture content can have adverse effect on uni-
axial compression strength (UCS) if large amount of clay mineral contents are present [31]. Vasarhelyi
indicated that effective porosity is more influential to the sensitivity of water contents on strength reduc-
tion for different sandstone rocks [70]. Here, results from an integrated study that combines experiment
and simulation of SENB tests on fluid-exposed (water and scCO
2
) Berea sandstone specimens and exper-
iment and simulation of UC tests on fluid-exposed (brine and scCO
2
) Indinana limestone are presented.
In all experiments, Digital Image Correlation (DIC) is utilized to quantify the displacement and strain
fields, which are then analyzed to understand the effect of fluid type on crack propagation and tough-
ness alteration processes. Active seismic is utilized to detect and monitor the onset of crack initiation and
propagation. High-resolution finite element model of the experiment is built, the model is calibrated with
experimentally measured displacement fields, and the model-predicted stress field is used to analyze the
evolution and mode of failure (tensile and shear) at different points in the domain.
4
Chapter2
ExperimentalMethodology
2.1 SingleEdgeNotchedBeamTest
The Three Point Bending (3PB) or Single Edge Notched Beam (SENB) test is a type of mechanical loading
test designed to measure fracture toughness of solid materials, e.g. metal, ceramic, concrete and rocks. It is
especially useful to measure tensile strength and failure properties of highly brittle materials or materials
displaying predominantly brittle response under the imposed boundary conditions. This holds true for our
selected Berea sandstone specimens under uniaxial unconfined loading conditions. The specimen lays on
two rigid rollers while axial loading is applied parallel to an existing notch [41]. Axial loading is applied
to the rock specimen by a closed-loop hydraulic compressive machine (Instron machine) with a static
load cell of± 30 kilo Newton, 5 kilogram weight and 50 Newton-meter bolt torque. The machine applies
the downward load to the specimen via a rigid cylindrical pin attached to the load cell that moves at a
prescribed displacement rate. The load-displacement data from Instron is output by the Blue Hill data
collection software. An illustration of the test setup is shown in Figure 2.1. Figure 2.2 shows a schematic
of the SENB test specimen with its load-displacement configuration.
5
Figure 2.1: An illustration of our setup of Single Edge Notch Beam (SENB) testing with Digital Image
Correlation (DIC).
L P B H c Support Pins
Loading Pin
δ
S Figure 2.2: A schematic of our SENB specimen with dimensions, boundary conditions and load-
displacement configuration. P is the load applied by the loading pin at the middle of the top surface,
andδ is the load cell displacement. The specimen rests on two support pins.c is the depth or length of the
pre-existing notch into the specimen. Pins can be considered rigid for practical purposes.
6
2.2 DigitalImageCorrelation
Digital Image Correlation (DIC) is a method to quantitatively measure 2D displacement and strain fields
on flat surfaces of solid bodies under dynamic loads using high resolution pictures of the surface at a rela-
tively high frame rate [65]. In this technique, reference speckles are created on a specimen surface. Before
loading, the position field of all the speckles provides the reference or undeformed configuration of the
specimen in terms of material or Lagrangian position vectors. As the specimen deforms under the applied
load, the speckles are displaced following the deformation, which gives us the current or deformed config-
uration of the specimen. Correlating the deformed configuration to the reference configuration allows one
to quantify the displacement vector field ( U
x
andU
y
in a Cartesian 2D space) and the strain tensor field
(ε
xx
,ε
yy
, andε
xy
), which evolve in time. The speckles are created by coating the specimen surface with
a flat-white paint. The specimens are left to dry and then sprayed with a black flat paint until a speckling
pattern is achieved. Our DIC system consists of a Grasshopper camera with a resolution of 5 MegaPixel
and Nikon Micro-Nikkor 60mm f/2.8D lens. The image frames are stored by FlyCapture SDK software at
a predetermined frame rate. Post-processing is performed using the Vic-2D correlation software, which
is able to extract the correlation among images taken at successive time steps using one of the three cri-
teria [73]: Squared Difference (SD), normalized squared difference (NSD), and Zero-Normalized Squared
Difference (ZNSD). The NSD criterion is insensitive (yields the same correlation coefficient) to scaling up
or down of the brightness, however it is sensitive (yields different correlation coefficient) to linear shifts in
brightness. The Normalized Square Difference (NSD) criterion and uniform lighting conditions are used
to obtain the correlation coefficients in the experiments.
7
2.3 UniaxialCompressionTests
Uniaxial compression test is a mechanical test used to characterize the mechanical behavior of solid ma-
terials including rocks. The specimen is positioned between two parallel cylindrical cast-iron disks [27].
The load is applied by a closed-loop hydraulic compressive machine (Instron machine) at a predetermined
load-cell displacement rate. Illustration of the test setup is shown in Figure 2.3.
P
Upper aw
Lower aw
S
L
t
W
S
C
β
a b c
ROI
Figure 2.3: Illustration of specimen with two parallel flaws. a. test setup. b. S wave transducers’ position.
c. flaws geometry
8
Chapter3
ExperimentalSetup
3.1 Single-EdgeNotchedBeamExperimentandDIC
3.1.1 SpecimenPropertiesandGeometry
Berea sandstone consists of mainly quartz mineral grains bonded by silica. Berea sandstone composition
is summerized in Table 3.1. The specimens have 18-20% porosity and permeability in the range of 100–200
mD [62, 50] and a dry bulk density of 2.1 g/cm
3
. The dimensions of all specimens are 50 mm by 25 mm by
10 mm (with an error of± 1 mm). The notch is created, at least 48 hours before exposing the specimens
to water or scCO
2
, at the center of one of the longer sides across the entire thicknessH of the specimen
(Figure 2.2) using a diamond saw. The notch is 1 mm in depth (c) and extends across the entire thickness
H in all specimens (Figure 3.1). Specimen’s properties and geometry are summarized in Table 3.2. The
faces and sides of the specimens are flattened using a Buehler MetaServ 250 Grinder-Polisher such that
the opposite sides are parallel to minimize any stress non-uniformity during loading, which can alter the
failure behavior of the specimen.
9
Table 3.1: Berea Sandstone Composition
Berea Sandstone Composition
Silica SiO2 93.13%
Alumina Al2O3 3.86%
Ferric Oxide Fe2O3 0.11%
Ferrous Oxide FeO 0.54%
Magnesium Oxide MgO 0.25%
Calcium Oxide CaO 0.10%
Figure 3.1: Specimen is rectangular in section with a length of 50 mm, width of 25 mm, and thickness of 10
mm. The fracture develops along the notch axis and the final fracture surface area, neglecting the surface
roughness, can be approximated as 240 mm
2
.
10
Table 3.2: Specimen Dimension and Properties
Length WidthB ThicknessH SpanL Porosityϕ Permeabilityk Density (dry)ρ b
50 mm 25 mm 10 mm 30 18-20% 100-200 md 2.1 g/cm
3
3.1.2 ExperimentalWorkflow
The tests are conducted on specimens with different pore fluid types: air, water at room pressure and
temperature (25
◦ C), and scCO
2
. Six tests are perofrmed on dry (exposed to air) specimens, six tests on
water-exposed specimens, and three tests on scCO
2
-exposed specimens. Results from the tests on the dry
specimens are used as the base case to analyze the effects of water and scCO
2
on geomechanical properties
of our specimens. The saturation protocol for water-exposed specimens are as follows. Dry specimens are
immersed in a flask full of water at room pressure and temperature. The specimens are kept in water for
different duration ranging from 72 hours to 240 hours(Table 3.3).
Table 3.3: Fracture toughness calculated results for dry, water–exposed and scCO
2
–exposed specimens
Test Case Saturation Type Saturation Duration, hr Fracture Toughness, MPa.m
1/2
D1 Dry – 1.17
D2 Dry – 0.89
D3 Dry – 0.93
D4 Dry – 1.01
D5 Dry – 1.03
D6 Dry – 1.00
W1 Water 240 0.84
W2 Water 240 0.78
W3 Water 240 0.85
W4 Water 72 0.68
W5 Water 72 0.87
W6 Water 72 0.80
C1 scCO
2
336 0.96
C2 scCO
2
168 1.01
C3 scCO
2
168 0.91
The exposure protocol for scCO
2
specimens are as follows. An autoclave containing a stainless steel
pressure chamber cell is used. The inside diameter of the chamber cell is 1.25 inch, wide enough to contain
the specimen without touching the cell wall. The chamber is heated up to 50
◦ C to reach scCO
2
conditions
11
upon gas injection. A 99.99% pure CO
2
Coleman Grade cylinder at 830 psi is used as the CO
2
source. A
Teledyne Isco pump Model 260D is connected to the chamber cell to raise the CO
2
pressure inside the
chamber cell. In order to reach supercritical conditions inside the gas reservoir of the pump, the reservoir
is wrapped with a heating tape and allowed it to heat up to 50
◦ C for at least 1 hour before injection. CO
2
is injected into the chamber cell until the chamber pressure increases to 3000 psi (± 50 psi). The same
pressure is used for the three CO
2
-exposed specimens. 3000 psi pressure is used because it is a typical
subsurface pressure during CO
2
injection in many storage and enhanced oil recovery projects, e.g. in the
Morrow-B sandstone of the Farnsworth CO
2
storage project in Texas [82]. The exposure duration for the
three specimens are given in Table 3.3. Prior to taking the specimens out of the chamber cell, the pressure
is decreased at 100 psi/min until room pressure is reached to avoid possible cracking of the specimen due
to sudden depressurization.
The specimens are then taken to perform the SENB test along with the DIC. The SENB tests are per-
formed at a constant load-cell displacement rate of 0.07 mm/min of the loading pin, which remains in
contact with the specimen top surface. This displacement rate falls in the range of American Society for
Testing and Materials (ASTM) standards for this test. The loading cell is adjusted so that the load axis is
aligned with the notch axis. The DIC system is placed two feet away from the Instron machine and it faces
the front surface of the specimen as shown in Figure 2.1. The camera frame rate is set to be 2 frames/second,
sufficient for the Vic-2D software to post-process the images. Analysis of the post-processed images in-
dicates the path of crack propagation before the specimens fails catastrophically. Figure 3.2 shows our
experimental workflow in this study.
Post-mortem analysis of the fracture surfaces are performed by using Keyence VHX-5000 microscope [44]
that allows us to capture the roughness and topography of the ruptured surfaces of the dry and water-
exposed specimens. Roughness of the walls of a fracture is an important property of the fracture. It
controls the hydraulic permeability of flow through the fracture and the elastic stiffness of the host rock
12
Sample Preparation
• Cutting and polishing the specimens
• Saturating specimens with water and scCO
2
• Speckling the specimens
Experimental Design
• 3-Point-bending test setup
• Load cell setup
• Displacement rate
• DIC system setup and calibration
• Instron and DIC synchronization
Data Acquisition
• Obtaining load vs displacement curves
• Post-process DIC results via Vic-2D
Analyzing the results
• Total fracture energy results
• Young’s modulus calculations
• Obtaining displacement field
• Detecting crack initiation and propagation
• Numerical simulation analysis
Figure 3.2: Experimental workflow with different steps taken to perform SENB tests and DIC analysis in
this study
13
along and across the fracture plane [33, 34, 60] as external load is applied on the fracture. In our exper-
iments, it is observed that the fracture surface in a dry specimen is rougher than the fracture surface in
a wet specimen (Figure 3.3). Although the effect of fluid type (gas vs. water) on the fracture compliance
ratio (the normal compliance to the shear compliance) is known [35], it is not well-understood how the
fluid type can affect the degree of roughness of the fracture and thereby the permeability and stiffness of
the host rock.
a. b.
4170
2979
1787
596
4773
3182
1591
0
Figure 3.3: Post-mortem analysis of the fracture surface geometry in case of (a) the D1 dry specimen and
(b) the W1 water-exposed specimen. The horizontal axes represent the X and Y axes in microns. The color
scales represent the depth of the fracture surface in microns. The post-mortem analysis shows a higher
roughness of the fracture surface of the dry specimen compared to the fracture surface of the water-
exposed specimen.
3.2 UniaxialCompressionTest,DICandActiveSeismicMonitoring
3.2.1 SpecimenPropertiesandGeometry
The specimens used in the tests are Indiana limestone. The dimensions of the specimens are about 101
mm by 50 mm by 25 mm (with an error of± 1 mm). Each specimen’s dimensions are shown in Table 3.4.
Table 3.5 lists porosity and permeability of the specimens as given by the vendor [62]. Prior to the test, the
specimens are put in an VWR 1410M stainless vacuum oven at 50
◦ C for at least two days to ensure that
the specimens are free of moisture prior to their exposure to fluids used in the experiments i.e. air, brine,
14
brine and scCO
2
mixture, and scCO
2
. The specimens are also put in the oven after exposure to remove
any moisture prior to the uniaxial compression tests.
Table 3.4: Specimens and Flaws’ dimensions
Specimen#
Specimen dimensions Flaw dimensions
Length
(mm)
Width
(mm)
Thickness
(mm)
L1
(mm)
L2
(mm)
Spacing
S (mm)
Continuity
C (mm)
Flaw inclination
angleβ (
◦ )
D-1 102.00 50.82 25.86 9.12 8.59 1.64 24.44 45
D-3 102.39 51.21 25.65 8.34 9.37 1.93 24.52 45
D-4 102.86 52.29 25.80 13.63 12.6 0 14.92 45
D-5 103.70 51.15 25.61 11.42 10.17 0 15.90 45
D-6 102.59 50.43 25.87 9.30 9.44 2.80 15.65 45
S-1 103.52 50.55 25.86 9.17 9.09 0 26.10 45
S-2 102.45 50.84 25.87 9.31 9.13 0.90 24.99 45
B-1 102.90 50.85 25.95 9.65 9.00 0.80 26.06 45
B-2 102.66 51.21 25.87 10.31 9.84 0 14.63 45
SB-1 102.67 50.84 25.81 10.39 9.54 0 27.71 45
SB-2 102.67 50.84 25.31 10.01 10.28 0 24.52 45
SB-3 102.38 50.74 25.56 10.07 10.67 0 15.59 45
Table 3.5: Specimen Dimension and Properties
Length Width Thickness Porosity Permeability Density
10.1 cm 5.0 cm 2.5 cm 12-14% 20 md 2.4 g/cm
3
3.2.2 BrineComposition
The synthetic brine used is based on known chemical composition of five different salts. The brine is made
in batches. The total solution volume per patch is 500 ml. The concentration of each ion/cation are shown
in Table 3.6.
Table 3.6: Synthetic Brine Composition
Ion/cation Concentration (mg/L)
Na
+
23992
Cl
− 38123
K
+
21
Mg
2+
320
Ca
2+
1459
SO
2− 4
3260
15
3.2.3 ExperimentalWorkflow
The tests are performed in two main stages. The first stage includes creating the two through-going flaws,
drying the specimens, saturating the specimens with three different pore fluid types, and then painting and
speckling them. The second stage involves performing the uniaxial compression test while capturing the
DIC and the S waves’ signals and then processing the data. The specimen preparation starts by creating
two through-going flaws perpendicular to the xz plane as in figure 2.3. First, 1.2-mm hole is created using
a drill press with 1.15 mm drill bit, then a scroll saw is used to create the flaws, with the aim to keep the
flaw angle and length the same for all specimens so that consistent comparisons are made on the different
pore fluid specimens’ effect on rock and fracture initiation and propagation. The specimens are exposed to
four different pore fluid types, air, brine, brine and scCO
2
mixture and scCO
2
. Before that, the specimens
are put in a vacuum oven at 50
◦ C for at least 2 days to make sure they don’t keep any moisture that could
affect the quality of the results and eliminate the presence of any water molecules in the pores, especially
for the dry and scCO
2
case. Following that, the specimens are prepared for the tests with the 4 different
pore fluid types. The first test involves specimens with just air, which is the state of the specimen after
creating the flaws and drying them. To expose the specimens to scCO
2
, brine, and brine/CO
2
mixture, a
VWR 1410M stainless vacuum oven containing a stainless-steel pressure chamber cell is used. The cell is
heated-up to 50
◦ C to reach scCO
2
conditions before injecting the desired fluid. In the case of scCO
2
, the
gas is injected in into the pressure chamber cell until 3000 psi pressure is reached. See Figure 3.4. For the
case of brine, the pressure chamber cell containing the specimen with brine is filled up to 0.5 cm above
the top of the specimen and then the cell is closed and pressurized with Ultra-pure Helium gas to reach
3000 psi at 50
◦ C. For the brine and scCO
2
mixture case, the pressure chamber cell is filled with the same
brine made of the same composition, then scCO
2
is injected until 3000 psi pressure is reached. scCO
2
is
constantly injected until the pressure stabilizes at 3000 psi for the duration of exposure. In order to prepare
the specimens for DIC analysis, the specimens are sprayed with white paint and wait until it became dry.
16
transducer
Figure 3.4: System setup and process of how specimens are exposed to different fluid types.
Then the specimens are speckled with flat black paint until a speckling pattern is achieved. The region of
interest (ROI) is set to be the area containing the 2 flaws on the xz plane 2.3.
Shear wave signals are monitored using one pair of Panametrics V153RM transducers. The transducers
have central frequency of 1 MHz. The transducers are positioned between the two flaws to monitor the
initiation and propagation of cracks between the flaws, see Figure 2.3. The transducers are attached to
the specimen using steel plates with attached springs to keep the transducers attached to the specimen,
Figure 3.5. Oven-baked honey is used to couple the transducers to the specimens while adhesive tape
is placed between the specimen and the transducers to prevent the honey from penetrating through the
specimen. A constant load of 1000 N is applied to each specimen prior to the uniaxial compression test
for 40 minutes while monitoring the shear waves. The transmitted shear waveform signals are recorded
every half a second using a data acquisition system controlled by LabView software. 5077PR Square Wave
Pulser/Receiver is used to to generate the S wave signal. Figure 3.6 shows the instron, DIC and data
acquisition system.
For this study, more than ten uniaxial compression tests are conducted on Indiana limestone specimens.
The uniaxial compression test starts by determining the displacement rate of the instron machine. In this
case, the displacement rate is set at 0.01 mm/min. This rate is found to be suitable to allow data acquisition.
17
Figure 3.5: S-wave transducers setup.
Figure 3.6: Instron Uniaxial compression system (left) and National Instrument PXIe-1088 data acquisition
system (right).
18
Chapter4
ExperimentalResults
4.1 Single-EdgeNotchedBeamTest
4.1.1 Deformationandelasticmoduli
Using the SENB experiments, load vs. load-cell displacement curve is plotted for dry and fluid-exposed
specimens in Figure 4.1. The test data shows that the maximum flexural stress, which is the maximum
stress corresponding to the peak load P
o
at the outer surface during the SENB test, is in the range of
6.11 to 7.26 MPa for the dry specimens (Figure 4.2). The flexural stress is calculated using the following
equation [6]:
σ =
3PL
2HB
2
(4.1)
whereσ is the stress at the outer surface in the load span region in MPa,P is the applied force in Newton,
L is the support span in mm, B is the width of specimen in mm, and H is the thickness of specimen in
mm. The maximum flexural strain is calculated using the following equation [6]:
ε=
6δB
L
2
(4.2)
19
0.05 0.10 0.15 0.25 0.30
100
300
500
700
900
1100
D 4
D 5
D 6
0.00 0.20
Load, N
0.05 0.10 0.15 0.20 0.25 0.30
100
300
500
700
900
1100
0.00
Load, N
0.05 0.10 0.15 0.20 0.25 0.30
100
300
500
700
900
1100
0.00
Load Cell Displacement, mm
Load, N
W 4
W 5
W 6
C 1
C 2
C 3
Figure 4.1: Load vs. displacement curves of dry (top), water-exposed (middle), and scCO
2
-exposed (bottom)
specimens. The insets show horizontal displacement fields ( U
x
) at the highlighted time steps.
20
Table 4.1: Mechanical properties of the fixture and the specimens. For each fluid type, median values over
all the experiments with that fluid type are used.
Pore Fluid Type Dry Water scCO
2
Young’s Modulus of the Fixture (GPa),E
i
184
Poisson’s Ratio of the Fixture,ν i
0.3
Young’s Modulus of Specimen,E (GPa) 0.288 0.258 0.272
Poisson’s Modulus of Specimen,ν 0.28
Fracture Surface Area,A
f
(m
2
) 0.00024
Peak Load (N) 970 776 1018
where ε is the maximum strain at the outer surface, and δ is the load-cell displacement at the mid-span
L/2 corresponding to loadP . The flexural modulus is
E
f
=
L
3
4HB
3
dP
dδ
, (4.3)
where
dP
dδ
is the slope of the tangent to the straight line portion of the load-deflection curve in Figure 4.1.
The reasoning behind using the straight line portion is that it constitutes the linear elastic region of the
mechanical response of the tested specimens. This will allow us to obtain Young’s modulus E of the
specimens from their flexural modulus [59]. Table 4.1 shows the values obtained based on specimen’s pore
fluid type.
For the water-exposed specimens, the data shows that both the peak load and displacement values are
considerably lower, with the peak flexural stress values falling in the range between 3.51 and 6.07 MPa
and the peak flexural strain values falling between 0.043 and 0.044 as shown in Figure 4.2. In the case of
the specimens exposed to scCO
2
, the peak flexural stress values are similar to those for the dry specimens.
However, the peak flexural strain values for scCO
2
specimens are smaller compared to those for the dry
specimens. Post-mortem comparison of dry and water-exposed specimens shows rougher fracture surfaces
for the dry specimens (Figure 3.3), which indicates a stronger bonding across the two surfaces.
21
Flexural Stress, MPa
Pore Fluid Type
Dry Water CO
2
1
3
5
7
Flexural Strain
0.030
0.034
0.038
0.042
0.046
Pore Fluid Type
Dry Water CO
2
Figure 4.2: Flexural stress and flexural strain of dry, water-exposed, and scCO
2
-exposed specimens. Dry
and scCO
2
-exposed specimens have higher values compared to the water-exposed specimens. Black lines
indicate the lowest and highest flexural stress and strain values, lower and higher horizontal blue lines
indicate the 25 percentile and 75 percentile values, respectively. Red horizontal lines indicate the median
values.
4.1.2 FractureToughness
As per the Griffith-Irwin theory [26, 38, 40], a pre-existing crack of certain length begins to grow rapidly
when the stress intensity factorK at the crack tip exceeds the fracture toughnessK
Ic
(mode-I fracture)
of the specimen, which happens when the crack length exceeds a certain threshold marking the end of
subcritical crack growth (SCG) period and the onset of unstable growth.K depends on the load boundary
condition and the specimen geometry [48]. At the structure length scale, which could be the specimen
thickness in case of a mechanical loading test,
K =
P
H
√
L
c
f(ζ ), (4.4)
whereζ = c/L
c
is the dimensionless crack length,P is the load applied at the specimen boundary,H is
the specimen thickness,L
c
is the characteristic size of the specimen, andf(ζ ) is a dimensionless function
dependent on the geometry. [40] showed that K is related to the tip-local stress tensor σ ij
, (i = x,y,
j =x,y in 2D) as
K =
√
2πr
σ ij
g
ij
(θ )
, (4.5)
22
where (r,θ ) denotes a point in polar coordinates near the crack tip, with origin (0,0) at the tip and θ measured from the crack axis, andg
ij
are trigonometric functions. For a mode-I fracture growing along
they axis [48],








g
xx
g
yy
g
xy








=








cos(θ/ 2)[1+sin(θ/ 2)sin(3θ/ 2)]
cos(θ/ 2)[1− sin(θ/ 2)sin(3θ/ 2)]
cos(θ/ 2)sin(θ/ 2)cos(3θ/ 2)








(4.6)
K can be calculated using either of the two methods above. In case of mechanical loading experiments,
where stress data is not available, the common practice is to calculateK using the load acting at the domain
boundary, which is done in this section. In a later section on the numerical model development of SENB,
K is calculated using the tip stress which can be obtained from the simulation model.
In order to produce consistent results of fracture toughness, which are independent of specimen di-
mensions, plane strain conditions must be ensured because in the experiment the strain is measured only
on the 2D surface of the specimen [21]. There are several methods to calculate fracture toughness. Be-
low, the methods using the experimental load-displacement curve and the specimen and notch geometry
parameters of a SENB test are briefly presented.
4.1.2.1 STMMethod
This is based on the Standard Test Method for Measurement of Fracture Toughness, which uses the fol-
lowing version of Eq. (4.4):
K
Ic
=
P
o
L
(HH
N
)
1/2
B
3/2
f(c/B) (4.7)
where
f(c/B)=
3(c/B)
1/2

1.99− (c/B)(1− c/B)

2.15− 3.93(c/B)+2.7(c/B)
2

2(1+2c/B)(1− c/B)
3/2
(4.8)
23
whereH
N
is the net specimen thickness (H
N
=H if no side grooves are present). Substituting the values
for the D1 test,K
Ic
=0.43 MPa.m
1/2
is obtained. However, a limitation of this method is that it requires
specimens with an aspect ratio (length-to-width ratio) of 4:1 to avoid edge effects.
4.1.2.2 Fischer-Cripps’sMethod
The equation for the second method is as follows [21]:
K
Ic
=
P
o
S
HB
3/2

2.9

c
B

1/2
− 4.6

c
B

3/2
+21.8

c
B

5/2
− 37.6

c
B

7/2
+38.7

c
B

9/2

(4.9)
whereS is the total length. This method also requires the aspect ratio (length-to-height) of the specimen
to be at least 4:1.
4.1.2.3 EnergyMethod
The third method is the energy method suggested by the International Union of Laboratories and Experts
in Construction Materials, systems and Structures, of fracture mechanics of concrete (RILEM FMC 50) [54],
which uses the mechanical energy obtained from the load-displacement curve. The loading curve repre-
sents the derivative of the fracture energy [21]. Thus, the fracture energy can be calculated by integrating
the load-displacement curve, and to calculate the total fracture energyG
F
the following equation is used:
G
F
=
W
o
+mgδ o
A
f
, (4.10)
where W
o
is the area under the load-displacement curve shown in Figure 4.1, m is the specimen mass
between the supporting pins, which is calculated asm= Total Specimen Mass× Span/Specimen Length,
g is the acceleration due to gravity,δ o
is the load-cell displacement at failure, andA
f
is the surface area of
24
Pore Fluid Type
1.1
1.0
0.9
0.8
0.7
Fracture Toughness, MPa.m
1/2
Dry Water CO
2
Figure 4.3: Fracture toughness results for different pore fluid types.
the fracture. The fracture surface area can be estimated from the the cross-sectional area of a planar surface
passing through the notch as illustrated in Figure 3.1. Table 3.2 shows the values used in the calculations.
The Mode-I fracture toughness is then calculated for each specimen using the following equation:
K
Ic
=
p
G
F
E (4.11)
In this study, the energy method is used to compute K
Ic
from the experiment. The energy method
is robust because it accounts for the entire load-displacement behavior, not just the peak load. Also, it is
less sensitive to the specimen aspect ratio, which for our specimens is closer to 2:1, not 4:1. The flexural
modulus E
f
is used to obtain values for Young’s modulus E [59]. Young’s Modulus of the water- and
scCO
2
-exposed specimens decreased by 11% and 5.5% compared to the dry specimen modulus, respectively.
The corresponding shear modulus (G = E/[2(1+ν )]) of dry, water and scCO
2
-exposed specimens are
113 MPa, 101 MPa, and 106 MPa, respectively. Figure 4.3 and Table 3.3 show the fracture toughness values
for dry, water-exposed, and scCO
2
-exposed specimens. The values are within the range of values (0.7 to
1.3 MPa· m
1/2
) reported in the literature for sandstones.
25
4.1.2.4 Effectofporefluid
It is found that the total fracture energy for the water-exposed and scCO
2
-exposed specimens are lower
compared to that of the dry specimen. This is believed to be due to possible fines migration, mineral
dissolution and poor consolidation upon water and scCO
2
exposure. The effect of moisture in reducing
the tensile strength and fracture toughness of compacted clay [29, 76] and rocks [28, 70, 57, 84] is known
in geotechnical engineering. Barton’s classic paper [9], and references therein, suggest reduction in the
tensile and shear strengths of crack (strength corrosion) due to pore water exposure, especially at high
pore pressure which causes a reduction in the effective intergranular compression and consolidation as per
Terzaghi’s relation [75]:σ ′
=σ − αp I, whereσ ′
is the effective stress tensor (compression positive), σ is the total stress tensor from overburden, tectonic loading, and expansion/contraction of the surrounding
rock,p is the excess pore pressure, andϕ < α < 1 is the Biot-Willis parameter. The volumetric or mean
stress is computed from the trace of the stress tensor, e.g.,σ v
=(1/3)trace(σ ) and similarly forσ ′
v
. Water
molecules react with silicon-oxygen bonds in quartz to convert them into mechanically weaker silicon-
hydroxyl bonds held together by hydrogen bonds, which is one of the mechanisms of stress corrosion
cracking (SCC). Based on the Charles-Hillig theory [16], SCC refers to changes in the crack tip radius, and
thereby stress concentration [38], due to formation of the reaction byproduct. SCC leads to subcritical
crack growth in rocks [7], where a crack grows at stress intensity factor values less than the critical stress
intensity factor, i.e. dc/dt > 0 for K
Iscc
< K < K
Ic
, where K
Iscc
is the critical stress intensity for
SCC to occur. Such moisture-induced weakening, or time-dependent failure, leads to crack extension and
subsequent coalescence of several cracks to form a macroscopic failure surface, which results into a drop
in macroscopic toughness [68].
The effect of pore fluid type on elastic moduli and strengths of a rock is still an active area of research
with many open questions. For example, a recent study reported a higher dynamic shear modulus of
26
confined water-exposed Berea sandstone specimens compared to the shear modulus of dry Berea speci-
mens [61]. [5] finds a decrease in elastic moduli as well as in tensile and compressive strengths of their
scCO
2
-exposed shale specimens due to dissolution and pore pressure-induced strain. Reduction in elastic
moduli, measured under both static and dynamic (e.g., ultrasonic) loading conditions, in presence of water
and other types of fluid (e.g., kerosene, crude oil, scCO
2
) has been observed and discussed [79, 45, 78, 5,
80]. [45] finds that viscosity and molecular polarity of the saturating fluid lead to hardening and softening
of the porous specimens, which show as increase and decrease in bulk and shear moduli, especially at
low values of the effective compressive stress ( σ ′
v
) when the cracks and voids are open and provide the
additional surface area for the fluid to interact with the solid. Clay content, dry pore surface area, differ-
ential stress, loading rate, fluid compressibility, and pH in presence of background formation water are
other important parameters known to affect rock’s mechanical properties. For example, water, which has
a higher polarity than hydrocarbons, e.g., kerosene and crude oil, lead to a decrease in the shear modulus
whereas hydrocarbons lead to an increase in the shear modulus at low differential stresses.
4.1.3 J-integralanalysis
Fracture initiation and growth processes in rocks and similar porous media are often nonlinear due to yield-
ing at high stress, viscoelasticity and creep, sub-microscale plasticity, and intergranular interactions [37,
66]. The concept of the critical stress intensity factor in Griffith’s theory is limited to LEFM which leads to
crack-tip singularity and requires regularization techniques such as the idea of process zone. For nonlinear
elastic processes, which almost always precede the onset of plastic deformation [22], a popular method to
characterize the tip-scale stress field is the J-integral method [63]. J-integral is a path-independent integral
on a contour line around the crack tip and it is defined as the rate of change in the mechanical energy (sum
27
of elastic energy and cohesive energy) of the crack system per unit change in the crack length [48]. For a
crack along they-direction,
J =
Z
Γ
σ ij
ε
ij
δ yj
− σ ij
∂u
i
∂y

n
j
dΓ (4.12)
where Γ is the contour around the tip, n is the outward unit normal on Γ , and δ yj
is the Kronecker
delta function. J-integral is not limited to LEFM and can be used to study fracture processes in case of
nonlinearity near the tip region.
4.1.4 FracturePropagationAnalysisusingDIC
Using the DIC system, the displacement along the vertical and horizontal axes of the specimen are ob-
tained for different pore fluid types. The measured displacement comes from correlating the reference,
undeformed, images att = 0 with the images captured fort > 0 until failure. The DIC results show that
the specimens undergo different stages of deformation and the pore fluid type exerts some control on the
onset and duration of those stages, which confirms our observations in the load-displacement analysis.
Figures 4.4 and 4.5 show the horizontal (U
x
) and vertical (U
y
) DIC displacement fields, respectively, for
specimens with three different pore fluid types. There are three main features of the deformation be-
havior: (1) the region below the loading pin and between the two support pins is displaced downward,
(2) growth of the crack displaces the left side region of the notch leftward (negative x-displacement) and
the right side region rightward (a positive x-displacement), and (3) the two regions on the two sides of
the notch rotate around the loading pin as the notch develops into a crack, causing rightward motion on
the top left side and leftward motion on the top right side. The regions outside of the two support pins
has least amount of displacement because of the zero-displacement boundary condition of these pins. The
DIC results show the effect of notch’s presence in each specimen. In the case of U
x
, as we get closer to
the notch, which has a tip position atY = 1 mm att = 0,U
x
magnitude increases sharply due to stress
concentration near the notch [50], which is a function of the tip radius [38]. It is also noticed in Figure 4.4
28
0.05
0
-0.05
time
δ=0.065 mm
δ=0.200 mm
δ=0.280 mm δ=0.255 mm
δ=0.150 mm
δ=0.060 mm δ=0.015 mm
δ=0.105 mm
δ=0.180 mm
Figure 4.4: Time evolution of the deformation field U
x
in millimeters for dry (left column), water-exposed
(middle column), and scCO
2
-exposed (right column) specimens based on the DIC analysis. δ is the load
cell displacement at any given time, hence evolution inδ is a proxy for evolution in time.
0.00
-0.05
-0.10
δ =0.065 mm
δ =0.200 mm
δ =0.280 mm δ =0.255 mm
δ =0.150 mm
δ =0.060 mm δ =0.015 mm
δ =0.105 mm
δ =0.180 mm
time
Figure 4.5: Time evolution of the deformation field U
y
in millimeters for dry (left column), water-exposed
(middle column), and scCO
2
-exposed (right column) specimens based on the DIC analysis. δ is the load
cell displacement at any given time, hence evolution inδ is a proxy for evolution in time.
29
that the discontinuity in theU
x
field exists close to the notch tip, which indicates the potential location of
the tip of unstable crack propagation immediately before the peak load. Analysis of theU
y
displacement
field in Figure 4.5 shows that the emergence of a nascent crack decreases the stiffness and increases the
compliance of the specimen as evidenced by large values of downward displacements in the region close
to the notch.
This behavior is further analyzed in Figure 4.6 by comparingU
x
at different Y distances from the notch
for a dry specimen (D4) and a water-exposed specimen (W4) before failure.
The line closest to the notch exhibits largest displacement increment, and as we move further away
from the notch (Y = 10 mm), the displacement increment shows an increasingly elastic response. The
displacement gradient
∂Ux
∂x
|
X=notch
along theY =1.35 mm profile is approximately an order of magnitude
larger compared to the gradient along the Y = 10 mm profile. As a result, stress relaxation via crack
propagation along a path parallel to the Y-axis is anticipated.
30
-30 -20 -10
-0.05
0.00
0.05
0.10
@Y=1.35mm
@Y=5mm
@Y=10mm
0 10 20 30
Ux
0.04
-0.04
0
0.02
-0.02
,mm
-0.05
0.00
0.10
Ux
,mm
0.05
-0.05
0
X-Axis, mm
-30 -20 -10 0 10 20 30
0.05
Dry Specimen
Water-exposed Specimen
Figure 4.6: Horizontal displacement (U
x
) at peak load along the horizontal direction (X) for dry and water-
exposed specimens at three different Y values from the bottom edge. Inset figure shows the full spatial
distribution of theU
x
displacement field in the two cases.
4.1.5 NumericalModelingandSimulationofSENB
To understand the effect of fluid type on spatial distribution of the state of stress and how the state evolves
in time, high-resolution simulations of our SENB test are performed while accounting for dynamic fracture
propagation along the notch. DIC and load-displacement results discussed above cannot provide spatial
distributions of the stresses, especially in high resolution. This necessitates a numerical modeling ap-
proach. However, modeling the distribution of stresses inside our specimen is challenging because the
31
numerical model must be representative of the SENB experiment in terms of specimen dimensions in 3D,
specimen’s physical properties, and the physical processes active during the experiment.
4.1.6 Modelconstruction
The above challenge is addressed by building a high-resolution 3D finite element model of our specimen
as shown in Figure 4.7. Abaqus is used, which is a finite element simulator. The Extended Finite Element
Method (XFEM) is used to model crack initiation and propagation from the notch, with a failure criterion
based on the local maximum principal stress. The punch (loading pin) and support pins are modeled as
discrete rigid bodies and the specimen is modeled as deformable and brittle. A standard 3D stress element
type (C3D8R) is used with Hex-shaped elements in a structured meshing routine. The domain is meshed
using an average element size of 0.25 mm by 0.5 mm by 1.25 mm in the three Cartesian directions, respec-
tively. To model the load, a loading step is applied with nonlinear large deformations where the initial time
increment size is 1 sec and the minimum and maximum step sizes are 1 and 180 seconds, respectively. The
Figure 4.7: Mesh of the 3D simulation model of the SENB test. Dimensions of the specimen and the three
pins and their relative positions are identical to our experimental setup in Fig. 2. The initial notch on the
bottom edge is difficult to see because of the mesh lines. The black box shows the zoom-in window where
we analyze stress fields and crack propagation in detail in Figures 16-17 below.
32
0 0.05 0.1 0.15 0.2 0.25 0.3
Load cell displacement, mm
0
200
400
600
800
1000
Load, N
Figure 4.8: Simulation load vs. displacement curve of dry specimens.
material properties used in the 3-D numerical model are given in Table 4.2. The value of the maximum
principal stress, which is related to the tensile strength of the fluid-exposed specimen, is chosen via model
calibration. Figure 4.9 shows U
x
and U
y
incremental displacements from the SENB simulation at three
different timesteps during the test. The model is calibrated by achieving an acceptable agreement between
the experimentally measured displacements (Figures 4.4 and 4.5) and numerically simulated displacements
(Figure 4.9).
Table 4.2: SENB numerical simulation inputs
Material Properties
Elastic Parameters
Young’s Modulus 290 MPa
Poisson’s Ratio 0.28
Damage Evolution Displacement at Failure 0.28 mm
Time Step Parameters
Maximum Number of Increments 10000
Increment Size (sec)
Initial Minimum Maximum
0.01 1.00E-11 1
Meshing Parameters
Meshing Dimensions N
x
N
y
N
z
Number of Cells 200 50 8
33
Figure 4.9: Horizontal (U
x
) and vertical (U
y
) displacement results of a SENB test simulation. Bending leads
to tensile crack initiation at the tip of the notch as shown by the middle row of figures.
34
The overall displacement fields of the specimen appear consistent with the experimental results. The
simulated load vs load-cell displacement from the Instron machine agrees with the same curve obtained
from the simulation model as we can see in Figures 4.1 and 4.8. The simulated horizontal displacement
field shows blue color ( U
x
< 0) on the left and red color (U
x
> 0) on the right side of the notch near
the bottom edge of the specimen. This agrees with the experimental Fig. 4.4. Near the top edge of the
specimen, the colors switch; U
x
> 0 on the left side of the notch and U
x
< 0 on the right side. This
also agrees with the experiments; see the column for water-exposed specimen in Fig. 4.4. The simulated
vertical displacement field shows a yellow-colored region with downward displacement between the two
support pins, which agrees with the experimental Fig. 4.5. Next, the stress field is analyzed in terms of
invariants and components of the stress tensor. This provides important insights into the failure processes
during SENB, which cannot be obtained from experiments or analytical fracture mechanics models.
4.1.7 Modelresults
Spatial distribution and temporal evolution of the stress tensor control the location and timing of micro-
scale cracking events, which determine the catastrophic tensile fracturing of the specimen during SENB
loading. However, the SENB stress solution cannot be obtained from an experiment or the classical fracture
mechanics theory, e.g., Irwin’s analytical solution in Eqs. (4.5)-(4.6). Here, the high resolution simulation
model is used to obtain the stress components (σ xx
,σ yy
,σ xy
in the plane strain configuration) and analyze
it to extract features that could be characteristic of failure during SENB loading. Given that many constitu-
tive models of rock deformation are expressed in terms of the first stress invariant I
1
and the second stress
invariantJ
2
of the deviatoric stress tensor, the stress is analyzed in the invariant space. The first invariant
is defined as I
1
= 3σ v
, and the second deviatoric invariant is defined as J
2
= (1/2)trace

s
2

, where
s =σ − σ v
I is the deviatoric stress defined as the traceless part of the stress tensor. A commonly used
quantity in those constitutive models is the von Mises stressσ vm
=
√
3J
2
. The stress path or trajectory of
35
selected points within the domain is analyzed to learn about the location and timing of micro-scale crack-
ing events which often precede the catastrophic tensile fracturing of the specimen. Understanding these
precursory signals can benefit predictive modeling of rock fracturing during CO
2
injection and hydraulic
fracturing of naturally fractured carbonates or shales.
4.1.7.1 Stressinvariants
To analyze how the state of stress at different points within the domain evolves toward mechanical failure
or stability, a stress path analysis is performed in the invariant space of von Mises vs. volumetric stress
(Figure 4.11).
√
J
2
controls inelastic deformation and shear-induced failure response of the specimen. I
1
controls the volumetric expansion or contraction response of the specimen. The notch point is seen to
evolve towards tensile failure because the volumetric stress change is tensile. Points near the top and
left pin evolve towards shear failure, which is aligned with the DIC observation. The stress trajectory at
the left pin is seen to evolve farther than the trajectory at the top pin, which indicates that the left pin
is closer to the Mohr-Coulomb failure envelope. The right pin, being symmetrically positioned, follows
the same stress path as the left pin. This analysis is valuable to identify the failure onset time failure
location (relative to the pin positions), and failure mode (shear vs. tensile) at different points inside the
domain. These observations also have practical implications for hydraulic fracturing in the field. It shows
a mechanism for shear activation of natural/pre-existing fractures, especially those oriented favorably to
the propagating hydraulic (tensile) fracture.
Figure 4.10 shows the change in the von Mises stress field and the volumetric stress field, at the same
time steps as those of the displacement results shown above in Figure 4.9.
36
Figure 4.10: Change in the Von Mises stress and the volumetric stress for the dry specimen at three suc-
cessive time steps. The volumetric stress is positive under compression. Fracture initiation at the notch
can be seen at later time steps, i.e., middle and bottom rows.
4.1.7.2 Stresscomponents
The spatial distribution and time evolution of stress components in the vicinity of the notch are further
analyzed, where the deformation magnitudes are highest and most dynamic. Figures 4.12 and 4.13 show
σ xx
andσ yy
fields at 12 successive time steps. See, e.g. t = 75.89 andt = 83.5 plots; crack propagation
causes relaxation of the tip stress and an increase in stress immediately behind the tip. Seet=123.5 and
t = 138.3 plots; crack propagation causes an increase in the stress behind the tip. This high-resolution
analysis shows quantitatively how stress increase is followed by discrete crack initiation and propagation
events, which are followed by either stress relaxation or stress increase depending on the dimensionless
crack length. Given the importance of the von Mises stress in determining the onset of failure at small-
37
0 2 4 6 8
2
4
6
8
10
12
top pin
left pin
notch
Von Mises Stress, MPa
Volumetric Stress, MPa
time
shear
failure
tensile
failure
time
top pin
left pin
notch
-0.35 -0.25 -0.15 -0.05 0.05
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
below tip
base of crack
tip of crack
Von Mises Stress
Volumetric Stress
time
time
Figure 4.11: Stress path analysis in the Von Mises vs. volumetric stress space. The increase in the von
Mises stress under compression indicates that the left and top pins evolve towards shear failure, as the
notch evolves towards tensile failure. The inset figure shows the stress paths in the vicinity of the propa-
gating crack, confirming the dominance of tensile failure near the tip. Sign convention: positive values of
volumetric stress indicates compression.
scale, its distribution and evolution is analyzed in Figure 4.14 within the same zoom-in view as the normal
stresses above. A region of low von Mises stress (blue colored lobe att=75.89,83.5,123.5) is seen ahead
of the tip, which promotes tensile cracking along the notch axis. Later, att=170.4 and180, two regions
of high von Mises stress emerge at 45
◦ angles from the crack path on the two sides of the crack, where
shear failure may occur if a favorably oriented weak plane exists at those locations.
An important question is how the stresses vary along and across the crack. In Figure 4.15, the stress
profiles are plotted along the specimen length at selected distances from the base of the crack. This is
similar to the U
x
displacement profiles shown in Figure 4.6, but now it is done for stress which holds
information about deformation yet to occur. It is insightful to compare the simulated stress field with
Irwin’s analytical stress, as shown in Figure 4.16. Near the crack tip, away from the boundary effects, the
simulated and analytical solutions are expected to show good agreement. As x increases and we move
38
Figure 4.12: Crack propagation in a SENB simulation, shown in a zoom-in view (black box in Figure 4.7
around the notch. The colored background is the normal stressσ xx
, which evolves with time. The number
near the top of each figure indicates the time step value.
39
Figure 4.13: Spatial distribution and temporal evolution of the normal stressσ yy
in the vicinity of the notch.
The zoom-in view window is the same as in the figure above.
40
Figure 4.14: Spatial distribution and temporal evolution of the von Mises stress in the vicinity of the notch.
The zoom-in view window is the same as in the figure above.
41
-20 -10 0 10 20
X Axis, mm
-1
0
1
2
3
, MPa
@Y=1mm
@Y=5mm
@Y=10mm
σ
xx -20 -10 0 10 20
X Axis, mm
-1
0
1
2
, MPa
-20 -10 0 10 20
X Axis, mm
-4
-2
0
2
4
, MPa σ
y y σ
xy Figure 4.15: Profiles of stress components along and across the notch showing the effects of crack propa-
gation and specimen bending.
42
-10 010
Distance along X, mm
-1
-0.5
0
0.5
1
Stress, MPa
-10 010
Distance along X, mm
-1
-0.5
0
0.5
1
σ
y y -10 010
Distance along X, mm
-1
-0.5
0
0.5
1
g xx g xy g y y σ
xx σ
xy Figure 4.16: Stress profiles from our simulation agree well with the Irwin crack tip solution functions
(scaled), which are shown in Eq. (4.6). The agreement is better closer to the crack tip (X =0,Y =1 mm).
Away from the tip, SENB loading conditions dominate which are not captured in the analytical solution.
away from the tip, the effect of the support pins and finite dimensions of the specimen cause the simulated
solution to differ from the analytical solution.
4.1.7.3 Stressintensityfactor
The mode-I toughness (K
Ic
) calculated using Eqs. (4.7) and (4.9) provide values at the specimen length
scale. Also, these values are calculated at the peak load P
o
, which marks the time of specimen breaking
into two pieces i.e. an unstable Griffith-like failure. Before this time, the experimental data, which is
limited mostly to load vs. displacement at the top pin, cannot shed light on crack dynamics, which could
be dominated by subcritical crack growth (SCG) or static fatigue. SCG refers to pre-existing flaws and
cracks growing quasi-statically (≪ 0.1 m/s) at stress levels below the critical stress required for unstable,
instantaneous rupture [7]. SCG can be an important phenomenon during CO
2
sequestration because the
CO
2
-brine mixture is acidic/corrosive at reservoir temperature and at injection-induced stress, which can
lead to stress corrosion cracking of the rock, which in turn leads to SCG [4, 47].
Given the high resolution of our simulation model, we can investigate the early time crack dynamics
in terms of stress field and crack length available from the simulation. We can calculate stress intensity
K near the tip using the Irwin crack-tip solutions, shown in Eqs. (4.5)-(4.6), which can be plotted along
43
with the crack length to test the SCG hypothesis. The results are plotted in Figure 4.17 for a point(r,θ )=
(0.1 mm,0) on the center line of the notch whereg
ij
= 1. Based on this result, many crack propagation
events indeed seem to correspond to SCG and lead to a drop inK, which implies stress relaxation and is
expected from the fracture mechanics theory. Also, theseK values are smaller than the experimentalK
Ic
values calculated above. This is expected because the experimental calculation is at the length scale of the
specimen and based on the load boundary conditionP instead of the local stress stateσ ij
. Since J-integral
is not limited to LEFM and hence can be used to study fracture processes in case of nonlinearity near the
tip region. We are able to calcuate the J integral value at each pre-specified contour line as shown in 4.18.
020406080 100 120 140 160 180
time (sec )
0
0.02
0.04
0.06
SIF K
I
(MPa.m
1/2
)
0
1
2
3
Crack length (mm)
Figure 4.17: Evolution of the stress intensity factorK
I
ahead of the crack tip as the crack length evolves.
Crack propagation events are marked by a drop in K
I
ahead of the tip. Crack length is measured along
the notch or loading axis. Crack length along the crack surface is 1-5% longer and increases as the crack
opening displacement or aperture increases with time.
44
0 1 2 3 4
Crack length (mm)
0
0.01
0.02
0.03
0.04
0.05
J value (MPa.mm)
contour
crack tip
η
Γ
x
y
Figure 4.18: Evolution of the J integral ahead of the crack tip as the crack length evolves
4.2 UniaxialCompressionTest
Using the UC experiments, load vs. load-cell displacement curve is plotted for dry and fluid-exposed spec-
imens in Figure 4.19. The data is recorded at the Instron machine while applying the load. As the dis-
placement increases the load increases until where it achieves straight lines, before failure, for all types
of fluid-exposed specimens. This section of the curve indicates elastic deformation in all specimen and
Young’s Modulus can be calculated. The curves then arrive to the failure points where load measure-
ments fall down for the same displacement increment, indicated plastic failure due to propagation of flows
through the body of the specimens.
The displacement rate for tests performed on dry and specimens exposed to (sc)CO
2
is 0.02 mm/min.
The displacement rate for specimens exposed to brine and a mixture of (sc)CO
2
and brine is 0.01 mm/min.
45
0 0.1 0.2 0.3 0.4 0.5 0.6
Top plate displacement, mm
0
0.5
1
1.5
2
2.5
3
3.5
4
Load, kN
D6
B2
SB3
Figure 4.19: Top plate load vs load cell displacement for (sc)CO
2
, brine, and mixture of brine and (sc)CO
2
4.2.1 ScanningElectronMicroscope(SEM)andX-RayDiffraction
The scanning electron microscope analysis is performed on a specimen before and after exposure to the
brine+scCO
2
mixture, which showed evidence of displacement and accumulation of solid particles where
macroporous material fills most of the micropores. X-Ray Diffraction analysis also showed presence of
sodium chloride crystals after saturation with synthetic brine and scCO
2
mixture.
Figure 4.20: SEM Results. The left figure is taken before scCO
2
and brine-mixture exposure. The right
figure is taken after the exposure.
46
Table 4.3: Specimens’ Fluid type and exposure duration
Specimen # Fluid type Exposure duration (hrs)
D-1 Dry -
D-3 Dry -
D-4 Dry -
D-5 Dry -
D-6 Dry -
S-1 scCO
2
168
S-2 scCO2 168
B-1 Brine 168
B-2 Brine 168
SB-1 Brine+scCO
2
168
SB-2 Brine+scCO
2
168
SB-3 Brine+scCO
2
336
4.2.2 Acousticmonitoringoffracturepropagation
Transmitted shear wave signals are recorded continuously during the uniaxial compression tests are shown.
In Figure 4.21, transmitted shear wave signals of a specimen exposed to scCO
2
. The sensitivity of the trans-
mitted signals to crack initiation was examined by monitoring changes in amplitude and arrival time of
the signal with loading. The stacked signals show an increase in arrival time for the whole duration of the
test with increasing axial load until failure. The results in Figure 4.21 indicate that the transmitted wave
amplitudes of first arrival increased as they were sensitive at the beginning to the closing of the microc-
racks that existed before the start of applying the load until we reached 35kN. As the load increases, the
signal amplitude starts to decrease which is a sign that the cracks started to form and specimen is close to
failure.
Another investigation of the transmitted signal data showed there is a trend of increase in the first
arrival signal amplitude as the fluid type changes from scCO
2
alone to brine to brine and scCO
2
mixture
at the same test conditions as shown in 4.22. This mainly contributes to the increase in stiffness values
which are the slope of the straight line of the load vs displacement curves as we move from scCO
2
to brine
to mixture of brine and scCO
2
.
47
37.96 39.56 41.16 42.76
Time (us)
10 kN
13 kN
15 kN
20 kN
25 kN
31 kN
35 kN
37 kN
41 kN
45 kN
10 20 30 40 50
Load, kN
0
0.1
0.2
0.3
0.4
First arrival amplitude, V
40
40.5
41
41.5
42
42.5
First arrival time, us
Figure 4.21: (a) Transmitted S-waves through the specimen as the load on the specimen increases. (b)
As the load increases, the first arrival time and amplitude increase to a maximum before dropping due to
macroscopic failure. The load at which maximum is reached is different for the two curves.
0 5 10 15 20 25 30 35 40 45
Load, kN
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
First arrival amplitude, V
(sc)CO2
Brine specimen#1
(sc)CO2 and Brine specimen#1
Brine speciment#2
(sc)CO2 and Brine specimen#2
Figure 4.22: Transmitted signal first arrival amplitude as a function of the applied load
48
4.2.3 FracturePropagationUsingDIC
As previously mentioned, the digital images taken during the uniaxial compression tests are correlated
using Vic-2D. Figures 4.23, 4.24, 4.25 provides the horizontal and vertical displacements in the ROI at
different load stages of specimens exposed to air, brine, and a mixture of brine and scCO
2
, respectively.
Figure 4.25 DIC post processing images of one of the specimens saturated with brine and scCO
2
for 14 days
at 50(
◦ ) and 3000 psi. The figure column on the right is the horizontal displacement and the figure column
on the left is the vertical displacement at different loading stages. The DIC results here show that tensile
cracks initiated at the external tips of the flaws at 35.496kN in the figure on the top right. The DIC also
detected the shear effect at the internal tips of the flaws at 35.895 kN as shown in the second two figures.
Finally, the tensile cracks dominated as more loading was applied to the specimen until it fails.
49
22.814 kN
23.274 kN
24.623 kN
Ux Uy
0.61
0.45
0.30
-0.01
0.15
-0.16
-0.31
0.03
-0.07
-0.16
-0.25
-0.34
-0.44
-0.53
time
Figure 4.23: Spatial distribution and time evolution of the horizontal displacement (Ux) and the vertical
displacement (Uy) on the surface of the D4 specimen exposed to air. The displacements are in mm.
50
31.569 kN
31.601 kN
31.610 kN
Ux Uy
0.61
0.45
0.30
-0.01
0.15
-0.16
-0.31
0.03
-0.07
-0.16
-0.25
-0.34
-0.44
-0.53
time
Figure 4.24: Spatial distribution and time evolution of the horizontal displacement (Ux) and the vertical
displacement (Uy) on the surface of the B2 specimen exposed to brine. The displacements are in mm.
51
35.496 kN
35.895 kN
35.923 kN
Ux Uy
0.61
0.45
0.30
-0.01
0.15
-0.16
-0.31
0.03
-0.07
-0.16
-0.25
-0.34
-0.44
-0.53
time
Figure 4.25: Spatial distribution and time evolution of the horizontal displacement (Ux) and the vertical
displacement (Uy) on the surface of the SB3 specimen exposed to a mixture of brine and scCO
2
. The
displacements are in mm.
4.2.4 NumericalModeling
Using the finite element based numerical modeling, we are able to obtain consistent results for the horizon-
tal (Ux) and vertical (Uy) displacement fields. Figure 4.26 shows these fields from the base case simulation
model at five successive time steps. It also shows the evolution of the cracks that emerge from the tips
52
of the two flaws. Looking at Ux, we see that compression in the y-direction leads to expansion of the
specimen in the x-direction. Both negative and positive values across the centerline. Expansion of the
specimen in the x-direction is the result of the compression of the y-direction which is the Poisson effect.
In the Uy plot, all the values are negative which means compression in the model. Four cracks emerge
from the four tips of the two flaws. They emerge at an angle to the maximum compression axis which is
the y axis. As time evolves, they become parallel to the maximum principal compression. Moreover, the
cracks are not equal in length. The inner crack is shorter due to stress shadowing. They are in the shadow
of the longer cracks which are closer to the free boundary. Figure 4.28 shows the normal stresses at the
same five-time steps as Fig. 13. Complex high-resolution patterns emerge during axial compression, and
crack propagation, which is not possible to obtain from DIC or other experimental measurements. The two
cracks interact with each other via the stress field. Normal stress in the x-direction is increasing in magni-
tude between the two inner cracks and also rotating with time as the cracks grow which leads to more red
color in the contour plot. Figure 4.27 shows von Mises and volumetric stresses, which are two invariants
of the stress tensor, which control deformation failure and deformation pattern. Von Mises controls the
onset of shear failure and magnitude. Volumetric stress controls volumetric deformation, expansion, and
contraction. High volume stress around the tip of the flaws is responsible for the initiation and growth of
the cracks. The inner two cracks are surrounded by blue region which is aligned with their tensile nature
because the two sides of the crack are under tension and that’s why it’s growing in tension mode. There
is however some compression yellow region which is aligned with the vertical compression axis causing
compression. Both tensile and compressive regions leads to crack bending. Figure 4.29 and Figure 4.30
show the result of sensitivity analysis, the sensitivity of the simulation model results, that is stress fields
and crack trajectory to variability in Young’s modulus and Poisson ratio. The Young’s modulus and Pois-
son ratio are varied to capture the effect of different pore fluids which are brine, super-critical CO
2
and
air. As the Young’s modulus decreases, the crack length usually on average decreases and it becomes more
53
crooked. It is more deviated from a straight crack aligned with the maximum compression axis, which
is Y axis. Similarly, as the Poisson ratio increases, more crookedness in the crack path appeared. As the
brittleness index decreases, the rock becomes more ductile and less brittle and causing shorter and crooked
fractures. As Young’s Modulus decreases, von Mises and Volumetric stress also decrease.
Table 4.4: Simulation imput of the Uniaxial Compression for Indiana Limestone specimen
Material Properties
Elastic Parameters
Young’s Modulus 400 MPa
Poisson’s Ratio 0.34
Damage Evolution Displacement at Failure 0.01
Damage Initiation Maxps 0.2
Step Parameters
Maximum Number of Increments 10000
Increment Size
Initial Minimum Maximum
0.01 1.00E-4 10
Meshing Parameters
Meshing Dimensions Nx Ny Nz
Number of Cells 48 92 13
Four simulations are performed with four sets of values of the elastic propertiesE andν for the spec-
imen to investigate the effect of variation in the properties induced by the action of the pore fluids. Fig-
ures 4.29 and 4.30 show results from each of the four simulation runs.
54
Figure 4.26: Horizontal (U
x
) and vertical (U
y
) displacement results of a uniaxial compression specimen
simulation.
55
Figure 4.27: Change in the von Mises stress and volumetric stress for a dry specimen throughout a uniaxial
compression test
56
Figure 4.28: (σ xx
) and (σ yy
) results of a uniaxial compression simulation for a dry specimen
Using the high resolution UC model, we are able to establish the relationship between the Von Mises
Stress and Volumetric Stress as in Figures 4.31 and 4.32. We see multiple groups of stress paths. For
example, regions at the upper flaw to the left of the bottom tip and lower flaw to the right of the top edge
(square shape and diamond shape, respectively) are for future shear crack development. Regions at the tip
of the cracks (circle and x shapes) are at the tension crack tip with low stress magnitudes which is aligned
with stress relaxation at the tip. The region in the center of the specimen (+ shape) has intermediate stress
magnitudes between the tip and base groups because it is in a special region between the two flaws. The
region to the right of the top flaw’s bottom crack tip (star shape) shows the origin of the tension crack
and it has completely relaxed due to the growth of that crack. The region where the triangle is is where
tension crack 1 begins at time 1.2 seconds which corresponds to the stress path bending. Explain the 2nd
bending. A new tension crack can begin at the location of the triangle point.
57
Figure 4.29: Comparison of (σ xx
), (σ yy
) , and (σ xy
) results of a uniaxial compression dry specimen simula-
tion at failure for 4 sets of simulation runs
58
Figure 4.30: Comparison of Von Mises Stress and Volumetic Stress at failure for 4 sets of simulation runs
59
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Volumetric Stress, MPa
0
1
2
3
4
5
6
7
8
9
10
11
Von Mises Stress, MPa
top flaw top crack tip
top flaw top crack base R
top flaw top crack base L
top faw bottom crack base R
top flaw bottom crack base L
top flaw bottom crack tip
between flaws
bottom flaw top crack base R
Figure 4.31: Stress path analysis in the Von Mises vs. volumetric stress space. The increase in the von
Mises stress under compression indicates that the top and between the two flaws evolve towards shear
failure, as the notch evolves towards tensile failure. Sign convention: positive values of volumetric stress
indicates compression
-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Volumetric Stress, MPa
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Von Mises Stress, MPa
top flaw top crack tip
top flaw top crack base R
top flaw top crack base L
top faw bottom crack base R
top flaw bottom crack base L
top flaw bottom crack tip
between flaws
bottom flaw top crack base R
Figure 4.32: Zoomed-in Stress path analysis in the Von Mises vs. volumetric stress space Sign convention:
positive values of volumetric stress indicates compression.
60
0 2 4 6 8 10
time (sec)
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
SIF K
I
(MPa.m
1/2
)
0
5
10
15
20
25
30
35
Crack length (mm)
Figure 4.33: Evolution of the stress intensity factorK
I
ahead of the top crack tip as the crack length evolves
61
Chapter5
Conclusions
The failure processes and properties of Berea Sandstone and Indiana Limestone specimens exposed to
different types of pore fluids are examined by integrating laboratory experiment and numerical simulation
studies. Single Edge Notch Bending (SENB) experiments are conducted on Berea Sandstone and Uniaxial
Compression (UC) experiments are conducted on Indiana Limestone with Digital Image Correlation (DIC)
to quantify the impact of crack propagation on the deformation field. Active seismic monitoring are also
conduced during the UC to detect the onset and propagation of the cracks. High resolution models of
the SENB and UC experiments are built with dynamic crack propagation to obtain the stress field, and
its invariants, which have not been analyzed in the literature. The existence of, and differences between,
stress concentration regions ahead of the growing crack and along the lines joining the loading and support
pins of the SENB setup are demonstrated. The role of the stressed regions in creating tensile and shear
failure events are analyzed. This is an important result because the stress state, which determines the
timing, location and magnitude of failure events, cannot be measured directly in an experiment or from
the classical fracture mechanics theory.
62
The presence of water in the pores is found to lead to a 16-20% drop in the plane strain fracture tough-
ness value compared to scCO
2
-exposed and dry specimens, which yield similar toughness values. scCO
2
-
exposed specimens show lowest flexural strain compared to the air or water as the pore fluid. Berea Sand-
stone exhibits a decrease of flexural stress and strain, Young’s modulus, fracture toughness and hardness
with exposure to fluids.
The potential of integrating experimental and simulation methods to improve our understanding of
stress and deformation evolution during the SENB test is demonstrated. Such an integrated analysis can
be used to estimate changes in fracture toughness, which is important for modeling near-wellbore hy-
draulic fracturing [15], injection-induced caprock fracturing [83] and leakage of stored fluids [55] during
hydrocarbon recovery and CO
2
storage.
It is found that the rate of decrease of the first arrival amplitude of the transmitted wave intensified
well before the initiation of the cracks. The amplitude of transmitted wave signals increase as brine-fluid
is introduced to the specimen due to pore accumulation and presence of solid precipitation.
63
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69

Abstract (if available)

Abstract Injection of fluids during wastewater disposal and geologic carbon sequestration causes induced stresses and changes in rock’s elastic and failure response, for example, elastic modulus and fracture toughness. An accurate understanding of such changes in the response requires modeling and analysis of fluid-induced changes in rock’s stress and deformation states for which core-scale mechanical loading tests are often employed. Using experiments and simulations of Single Edge Notched Beam (SENB) test on water-exposed and supercritical (sc) CO2-exposed Berea sandstones, and Uniaxial Compression (UC) test on brine-exposed and brine-(sc) CO2 exposed Indiana limestone, the effect of pore fluid on tensile crack propagation, the plane strain fracture toughness, and spatial distributions of shear and tensile stresses are quantified. Digital Image Correlation (DIC) is used to monitor the changes in local stress and deformation fields during crack initiation and propagation in the SENB test. In addition to DIC, active seismic in the (UC) test is included to detect the onset of crack initiation and propagation. Representative numerical simulation models of dynamic crack propagation under SENB and UC loading are built using the extended finite element method. Results are analyzed using the stress path analysis to identify shear and tensile stress concentration regions, which provide precursory information for the timing and location of failure events during the test. Stress profiles along and across the growing crack is analyzed and compared with Irwin’s classical solutions to understand the impact of SENB loading and UC. Evolution in stress intensity factor and crack length shows subcritical crack growth prior to Griffith-like failure at the structure scale.

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Asset Metadata

Creator Dabloul, Rayan (author)

Core Title Stress and deformation analysis on fluid-exposed reservoir rocks

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Petroleum Engineering

Degree Conferral Date 2022-08

Publication Date 06/30/2022

Defense Date 09/29/2021

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag 3-point bending,active eismic,brine,crack initiation,Dic,EOR,fracture toughness,horizontal displacement,J-integral,OAI-PMH Harvest,propagation,SIF,simulation,supercritical CO2,transmitted signal,unaxial compression,vertical displacement,volumetric stress,Von Mises stress

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Jha, Birendra (committee chair), Ershaghi, Iraj (committee member), Jin, Bo (committee member)

Creator Email dabloul.rayan@gmail.com,dabloul@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-oUC111352076

Unique identifier UC111352076

Legacy Identifier etd-DabloulRay-10802

Document Type Dissertation

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Rights Dabloul, Rayan

Type texts

Source 20220706-usctheses-batch-950 (batch), University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

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