Chemo-Poro-Elastic Fracture Mechanics of Wellbore
Cement Liners: The Role of Eigenstress and Pore
Pressure on the Risk of Fracture
MASSAC HLISETTS INSTITUTE
OF FECHNOLOLGY
by
JUL 02 2015
Thomas Alexander Petersen
LIBRARIES
B.S., North Carolina State University (2011)
Submitted to the Department of Civil and Environmental Engineering
in partial fulfillment of the requirements for the degree of
Master of Science
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2015
@
Massachusetts Institute of Technology 2015. All rights reserved.
Signature redacted
Author ...............
Department of Civil and Environmental Engineering
May 21, 2015
Certified by......
Signature redacted
.........
Franz-Josef Ulm
Professor of Civil and Environmental Engineering
Supervisor
I IThesis
Accepted by ...
Signature red acted
Z. .. .20
(
Heidi Nepf
Donald and Martha Harleman Professor of Civil and Environmental
Engineering
Chairman, Department Committee on Graduate Theses
2
Chemo-Poro-Elastic Fracture Mechanics of Wellbore Cement
Liners: The Role of Eigenstress and Pore Pressure on the Risk
of Fracture
by
Thomas Alexander Petersen
Submitted to the Department of Civil and Environmental Engineering
on May 21, 2015, in partial fulfillment of the
requirements for the degree of
Master of Science
Abstract
Between 2001 and 2010, United States natural gas wells have been drilled at a mean
annual rate of 24,500. Moreover, an investigation in the Marcellus region revealed a
3.4% incidence rate of well-barrier leakages that were caused primarily by casing and
cementing problems. Considering the detrimental consequences even a single failed
well can have on the health of vast expanses of ecosystems, the quality of groundwater
aquifers, and the production efficiency of fossil resources, ensuring the integrity of
cement liners is of utmost importance. While much attention has been devoted to
the mechanical analysis of the cement sheath during temperature and casing pressure
cycles in the hardened state, modeling efforts of the early-age shrinkage and porepressure developments have thus far proved inadequate. This motivates us to study
the cement sheath as a poro-elastic media under growth and stiffening of its solid
structure, and connect bulk stress and pressure development to worst-case fracture
scenarios.
Specifically, a bottom-up approach is herein developed to incorporate the microscale behavior of the hydrating cement phases into a predictive risk-of-fracture
model. We incorporate recent findings of the driving mechanism of eigenstress development in CSH-gel and connect it, via Levine's theorem, to pore-pressure changes
in the sheath. Coupled to the boundary conditions of an inner steel casing and an
outer rock formation, the bulk stress in the sheath is calculated incrementally with
reference to the growing solid skeleton. The added risk due to the off-center placement of the casing is quantified in a novel Laurent series solution to the stress state.
Finally, energy release rates are derived for (i) the micro-annulus formation along
the steel-cement and rock-cement interfaces, and (ii) the occurrence of a single radial
fracture emanating from the steel-cement interface.
Thesis Supervisor: Franz-Josef Ulm
Title: Professor of Civil and Environmental Engineering
3
4
Acknowledgments
I am incredibly grateful to have found an environment in the MIT community in which
I am challenged academically and supported personally. Thus, I must firstly express
my sincere gratitude to Professor Franz Ulm for his breadth of knowledge in cement
science and poromechanics, and his continued effort to engage me in interesting and
novel research topics. He has an innate ability to communicate the broader impacts
of our work. After research meetings, I always left his office more motivated than
when I arrived.
Additionally, I must give thanks to the hard working engineers and scientists at
the Schlumberger-Doll Research Center inn Cambridge, MA and the SchlumbergerRiboud Product Center in Clamart, France. Through weekly phone conferences and
a summer internship I gained invaluable insight into the research challenges of the oil
and gas drilling industry. Their expertise and generous aid has allowed the X-CEM
project to reach targeted solutions otherwise not possible.
Funding for my graduate studies was generously supplied by the National Science
Foundation Graduate Fellowship program and Schlumberger. I thank these organizations much for their investment in science and engineering; to enabling the work of
many inspired students and researchers.
Next, I wish to thank the Civil and Environmental Engineering Department as
a whole. I have fostered numerous professional and personal relationships that will
last far into the future. Both the Parsons' Lab and Building 1 are filled with diverse,
intelligent and passionate people that contribute to my scientific and personal growth
in this collaborative community.
Both within and beyond my life in Cambridge, friends and family have made the
last three years of my life some of the most enjoyable. In particular, I wish to thank
my dear friend, Fatima, who taught me much about the way of life at MIT from her
experiences as an undergraduate. She is an invaluable confidant, and is someone I
care for deeply. My immediate family has had the greatest influence in shaping my
success. I wish to thank my sister Becky and my brother Karl for their constant
availability and for sharing their insights and perspectives on my life's conundrums.
I am most appreciative of the continued sacrifices made by my parents Kim and
Martin to maintain my wellbeing and happiness; their support is immeasurable and
their influence is recognized in all facets of my life.
5
6
Contents
1
21
Introduction
1.1
Industrial Context
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
1.2
Research Motivation and Problem Statement . . . . . . . . . . . . . .
23
1.3
The Primary Cementing Process . . . . . . . . . . . . . . . . . . . . .
26
1.4
T hesis O utline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
31
2 The Microstructure of Hardening Cement Paste
2.1
2.2
Structure of Hardened Cement Paste (HCP) . . . . . . . . . . . . . .
31
. . . . . . . . . . .
32
. . . . . . . . . . . .
33
. . . . . . . . . . . . . .
34
. . . . . . . . . . . . . . . . . . . . . . . . . .
35
2.1.1
Calcium Hydroxide Crystals (Portlandite)
2.1.2
Calcium-Silicate-Hydrate Gel (CSH gel)
2.1.3
Anhydrous Cement Grains (Clinker)
2.1.4
Pore Structure
Classification of the Characteristic Length Scales in Hardening Cement
P aste . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.37
2.3
37
2.2.1
Level '0'. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2
L evel I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
2.2.3
Level II
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
2.2.4
Level III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
Modified Powers-Brownyard Model For the Calculation of Volume Fractio ns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
. . . . . . . . . . . . . . . .
40
. . . . . . . . . . .
45
Sample Model Output: The Evolution of Volume Fractions. .
46
2.3.1
Partitioning of Volume Fractions
2.3.2
Characteristic Time of Cement Hydration
2.3.3
7
2.4
.
48
. . . . . . . . . . . . . . .
48
. . . . . . . . . . . .
49
Continuum Micromechanics: A Three-Level Cement Thought-Model
2.4.1
The Principle of Scale Separability
2.4.2
Concepts in Continuum Micromechanics
2.4.3
Homogenization schemes for elastic properties composite materia ls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
2.4.4
Three-Level Homogenization Scheme for Hardening Cement Paste 53
2.4.5
Sample Model Output: Evolution of Poroelastic Constants. . .
57
2.4.6
Chapter Summary
. . . . . . . . . . . . . . . . . . . . . . . .
61
63
Bulk Eigenstress Development
3.1
On the Origin of Cement Eigenstresses During Hydration . . . . . . .
64
3.2
Upscaling the Microscopic Driving Forces . . . . . . . . . . . . . . . .
66
. . . . . . . . . . . . . . . . . . . . . . . . .
66
3.2.1
3.3
Levin's Theorem
An Incremental State Equation for the Mass Balance of Hydrating
Cem ent Paste . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
72
3.3.1
Mass Balance of the Water . . . . . . . . . . . . . . . . . . . .
72
3.3.2
The Simplifying Assumption of Uniform Bulk Eigenstress Developm ent . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.3
3.4
4
51
76
Sample Model Output: Comparison with the Down-Hole Pressure of an Oil W ell . . . . . . . . . . . . . . . . . . . . . . . .
80
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
82
Chapter Summary
Stress Developments for Concentric and Eccentric Steel Casing Place83
ments
4.1
4.2
An Introduction to the Method of Complex Variables for Problems of
the Plane Theory of Elasticity . . . . . . . . . . . . . . . . . . . . . .
84
4.1.1
Derivation of the Airy Stress Function in Complex Variables .
84
4.1.2
The Kolosov-Muskhelishvili Equations
. . . . . . . . . . . . .
88
4.1.3
The Kolosov-Muskhelishvili Equations in Polar Coordinates
.
90
Elements of Poromechanics: A Three-Phase Poro-Composite Cylinder
under Eigenstress Loading . . . . . . . . . . . . . . . . . . . . . . . .
8
92
4.3
4.2.1
Poromechanical Constitutive Relations . . . . . . . . . . . . .
92
4.2.2
The Boundary Conditions . . . . . . . . . . . . . . . . . . . .
93
The Stress State in a Cement Sheath with a Concentrically Placed
C asin g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1
94
Sample Model Output: Stress Evolution for a Concentrically
Placed Casing . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.4
Stress State in a Cement Sheath with an Eccentrically Placed Casing.
4.4.1
102
Constructing Coordinate Systems for the Steel, Cement, and
Rock Dom ains: . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.4.2
The Bilinear transformation . . . . . . . . . . . . . . . . . . .
104
4.4.3
The Kolosov-Muskhelishvili Formulas for the Mapped System
106
4.4.4
Matching the Boundary Contours Using the Chebyshev Polynom ials
4.4.5
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
Sample Model Output: The Stress Evolution for an Eccentrically Placed Casing.
4.5
5
Chapter Summary
. . . . . . . . . . . . . . . . . . . . . . .111
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
117
119
Fracture Criteria
. . . . . . . . . . . . . . . . . .
120
5.1
Fracture Mechanics in Porous Media
5.2
Microannulus Formation . . . . . . . . . . . . . . . . . . . . . . . . . 122
5.3
5.2.1
Microannulus Along the Steel-Cement Interface (SC)
. . . . .
123
5.2.2
Microannulus along the rock-cement interface (RC) . . . . . .
125
5.2.3
Sample Model Output: Energy Release Rate due to Interfacial
D ebonding . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
126
Radial Fracture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
128
. . . . . . .
130
5.3.1
Connection to the Chemo-Poro-Mechanics Solver
5.3.2
Method of Continuation
. . . . . . . . . . . . . . . . . . . . .
132
5.3.3
Green's Function for an Edge Dislocation . . . . . . . . . . . .
134
5.3.4
Singular Integral Equation for the Crack Surface Boundary
C ondition . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
145
5..35
Calculating the Stress Intensity Factor and the Surface Displacem ent . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.6
Sample Model Output: Evolution of Energy Release Rate due
to Radial Fracture
5.3.7
6
. . . . . . . . . . . . . . . . . . . . . . . .
Chapter Summary
. . . . . . . . . . . . . . . . . . . . . . . . . .
159
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
161
Conclusions and Perspectives
6.1
6.2
156
Sample Model Output: Energy Release Rate due to the Loss of
Shear Traction
5.4
154
167
Review and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
6.1.1
Problem Synopsis . . . . . . . . . . . . . . . . . . . . . . . . .
168
6.1.2
Modeling Contributions and Findings . . . . . . . . . . . . . .
168
N ext Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
170
A Effective Stiffnesses of Steel Casing and Rock Formation
171
B Laurent Series Representation of Analytic Functions in the Complex
Plane
173
C Upscaling Poroelastic Constants
175
C.1 Level I: CSH Gel with Gelpore Pressure
C.2 Level II: CSH Gel with Macro-Pores
. . . . . . . . . . . . . . . .
175
. . . . . . . . . . . . . . . . . .
176
C.3 Level III: Reinforcement by Rigid, Slippery inclusions (Anhydrous Cement Grains and Non-Reactive Additives), CSH Gel+Macropores
D Analytic Functions with Branch Cuts
10
. .
178
181
List of Figures
1-1
Energy sources for United States electricity generation measured in
trillions of kilowatthours. The reference case for the projected energy
sources assumes a 'business as usual' scenario. (Source of figure: [3]).
1-2
An illustration of the primary cementing procedure for an oil or gas
well. The figure has been adopted from Ref. [40].
2-1
22
. . . . . . . . . . .
Scanning electron microscope images of (a) the CSH phase -
29
adapted
from Constantinides and Ulm [20]-, and (b) a portlandite (CH) crystal
precipitated in a CSH matrix -
2-2
adapted from Nelson and Guillot 1601.
33
The Voigt-Reuss-Hill bounds of the a) bulk and b) shear moduli of
CSH solid sheets as calculated by the atomistic simulations of Qomi et
al. [6 9]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2-3
Illustration of the nano- to micro-scale structure of CSH. This figure
is adapted from [871 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2-4
34
38
Depiction of the three-level upscaling scheme utilized to arrive at the
bulk poroelastic constants and the bulk eigenstress. . . . . . . . . . .
11
39
2-5
(a) A snapshot of a simulation box filled with spherical CSH particles
for a polydispersity 6
=
0.47. The polydispersity is a measure of the
standard deviation of the particle sizes used in the mesoscale colloidal
simulation.
6 is quantified in units of 0.5(rM + rm), the average of
the maximum rm and minimum rm sphere radii and the color code
signifies the particle sizes. (b) The packing density r as a function of
the polydispersity 6; the shaded region highlights the range of jamming
volume fractions 7j (adapted from Ref. [531). . . . . . . . . . . . . . .
2-6
43
(a) The time evolution of the hydration degree measured against values
calculated from calorimetry data for Class G oil well cement.
The
modeled values are calculated by Eq.(2.14), where a = 5.3 s-, b = 6.4,
c = 230, and d
=
4.3. (b) The hydration affinity plotted against the
degree of hydration. (Data provided by Schlumberger-Doll Research
C enter.)
2-7
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
The volume fractions in a hydrating cement sample are plotted as
a function of the degree of hydration
. The results are based on
our modified Powers and Brownyard model (see Table 2.1 for input
parameters). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2-8
The microstructure of 100-day old cement paste with a w/c of 0.30
(from R ef. [261). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2-9
47
55
The evolution of the bulk modulus of the cement is plotted for an REV
at the three characteristic length scales defined by Level I (CSH-gel),
Level II (CSH-gel + macropores), and Level III (CSH-matrix + rigid
inclusions) (see Table 2.1 for input parameters). All values have been
normalized by the bulk modulus of the CSH-solid. . . . . . . . . . . .
12
58
2-10 The evolution of the Biot modulus of the cement is plotted for an REV
at the three characteristic length scales defined by Level I (CSH-gel),
Level II (CSH-gel + macropores), and Level III (CSH-matrix + rigid
inclusions) (see Table 2.1 for input parameters). The modulus has been
normalized by the bulk modulus of the CSH-solid. The secondary axis
offers a comparison of the CSH-gel packing density.
. . . . . . . . . .
59
2-11 The bulk elastic properties properties of the cement, K and G, and the
biot coefficient of the cement are plotted as a functino of the degree of
hydration (see Table 2.1 for input parameters) .
3-1
. . . . . . . . . . . .
The relation between eigenstress development and packing density for
cement hydrating at constant pressures of 1 MPa and 10 MPa
3-2
60
[86J.
65
Levin's theorem is used to upscale the eigenstress in the microstructure to the macroscale (the volume phases are not drawn in proper
proportion and scale).
3-3
. . . . . . . . . . . . . . . . . . . . . . . . . .
68
(a) The pressure evolution is plotted against the evolution of the ratio
between the characteristic times of hydration and fluid mobility. The
time ratio is plotted in log scale on the secondary axis. (b) The evolution of the bulk eigenstresses, decomposed the stresses acting in the
CSH-solid (1 - b)-* and the porespace -bp.
3-4
. . . . . . . . . . . . . .
78
A comparison between the model simulated pressure and the pressure
in the pressure in an oil well. As the pressure data is proprietary, the
curve has been smoothed and the input parameters to the model have
been om itted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-1
81
A diagram of the wellbore geometry for the case of a casing placed
concentrically w.r.t the hole.
The cement sheath is bounded at its
interior by a steel casing and at its exterior by a geologic formation.
The inner, circular boundary of the steel is located at a distance RO
from the origin. The interfaces SC and RC are located at distances of
R1 and R 2 from the origin respectively.
13
. . . . . . . . . . . . . . . . .
94
4-2
(a) The effective radial stress and (b) the effective hoop stress development along the interfaces of the steel and cement (blue) and the rock
and cement (red) is plotted in function of the degree of hydration. The
input parameters have been summarized in Table 2.1, and the scenarios
of a stiff (solid lines) and soft (dashed lines) are plotted.
4-3
. . . . . . . 101
Three-dimensional plot of the effects of the fluid exchange coefficient
A and the rock Young's modulus ER on the radial stress (top row)
and the hoop stress (bottom row) at complete hydration. Stresses are
plotted for SC (left column) and RC (right column); E(
=
1) ~ 23
G Pa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-4
103
Contours in the reference coordinate system (z-plane) are mapped via
the bilinear transformation into a conformal image ((-plane); the eccentric boundaries SC and RC are mapped into the concentric boundaries
S C and R C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-5
106
Panel of (a),(b) the radial and (c),(d) the hoop stress evolution for
a cement sheath with an eccentrically placed casing
(4,
= 0.8) as a
function of the hydration degree for the material parameters provided
in Table 2.1 and the borehole dimensions provided in Table 3.1. The
plots separate stresses that evolve along (a),(c) the steel-cement interface (SC) and (b),(d) the rock-cement interface (RC). Thick (thin)
lines correspond to stresses along the thickest (thinnest) portion of the
sheath; colors represent different fluid exchange coefficients between
formation and sheath; ER
=
40 GPa. . . . . . . . . . . . . . . . . . . 112
14
4-6
Panel of (a),(b) the radial and (c),(d) the hoop stress evolution for
a cement sheath with an eccentrically placed casing ( 6e = 0.8) as a
function of the hydration degree for the material parameters provided
in Table 2.1 and the borehole dimensions provided in Table 3.1. The
plots separate stresses that evolve along (a),(c) the steel-cement interface (SC) and (b),(d) the rock-cement interface (RC). Thick (thin)
lines correspond to stresses along the thickest (thinnest) portion of the
sheath; blue (orange) lines present model output for a soft formation;
A = 1 x 10
4-7
5
s/m . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
113
(a) The effective radial stress and (b) the effective tangential stress
at complete hydration for a wellbore hole geometry with an eccentric
casing plotted as a function of the angular component 0 along the steelcement interface (red) and the rock-cement interface (blue); ER = 40
GPa. The line thickness indicates the degree of eccentricity, which has
been varied between 0.2 and 0.8 . . . . . . . . . . . . . . . . . . . . .
4-8
114
The shear stress along the SC (red) and RC (blue) interfaces plotted
(a) as a function of the degree of hydration and (b) as a function of the
angular coordinate 0 at complete hydration; A = 1 x 10-
and ER = 40
GPa. The line thickness indicates the degree of eccentricity, which has
been varied between 0.2 and 0.8 . . . . . . . . . . . . . . . . . . . . .
4-9
114
Three-dimensional plot, investigating the influence of the stiffness ER
and Newton coefficient A on the magnitude of the maximum shear
stress experienced along SC and RC. The degree of eccentricity is set
at 6 e = 0.8 and the remaining input parameters are gathered from
T able 2.1.
5-1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
115
Dominant fracture scenarios for the plane geometry of the cement sheath. 121
15
5-2
The evolution of (b) the energy release rate and (c) the stress intensity
factor for micro-annulus formation along the steel-cement (SC) and
rock-cement (RC) interfaces calculated from the results of the chemoporomechanics solver shown in panel (a). Solid lines indicate a stiff
formation (ER =40 GPa) and the dashed lines indicate a soft formation
(ER = 5 G Pa). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-3
127
The (a) the energy release rate and (b) the stress intensity factor for
micro-annulus formation along the steel-cement (SC) and rock-cement
(RC) interfaces are plotted for different ratios of the rock and cement
bulk moduli KR/KC. The bulk modulus of the of the cement, the porepressure, and the radial stress along the interfaces have been calculated
by the chemo-poromechanics solver and are evaluated at complete hydration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5-4
Regions of continuation.
5-5
Shapes for the fundamental function for the dislocation density p(t) Oc
aU under the loading of a uniform surface pressure for (a) an embed-
. . . . . . . . . . . . . . . . . . . . . . . . .
135
at
ded crack, (b) a closed crack with the left tip ending at a bi-material
interface, and (c) an edge crack. . . . . . . . . . . . . . . . . . . . . .
5-6
152
The evolution of the (a) (c) energy release rate ((b) (d) stress intensity
factor) for a radial crack emanating from the steel-cement interface.
The top panel (a)(b) corresponds to the open crack geometry, and the
bottom panel (c) (d) corresponds to the closed crack geometry. Values
are consistent with the stress state shown in 4-2b for ER = 40 GPa. .
5-7
156
The evolution of the (a)(c) energy release rate ((b)(d) stress intensity
factor) for a radial crack emanating from the steel-cement interface.
The top panel (a)(b) corresponds to the open crack geometry, and the
bottom panel (c)(d) corresponds to the closed crack geometry. Values
are consistent with the stress state given by the input parameters in
Table 2.1 and lowering the permeability to A = 1 x 10' s/m.
16
. . . .
157
5-8
Diagrams depicting (a) the fracture toughness and (b) the critical stress
intensity factor for white ordinary Portland cement with a water-tocement ratio of w/c
=
0.4. These results were obtained by the study
of Hoover and Ulm [37].
5-9
. . . . . . . . . . . . . . . . . . . . . . . . .
158
Diagram depicting the potential energy stored in the shear connection
between the steel casing and the cement sheath upon developing a
radial crack of closed shape. . . . . . . . . . . . . . . . . . . . . . . .
5-10 The surface displacement nO(r)
=
160
juo] for p, < r < P2 plotted along
the line of the crack pi < r < P2 for (a) the open geometry (the edge
crack; x = 0) and (b) the closed crack
(x
-+ oc) for a crack that has
propagated the width of the sheath. The thickness of the lines are
proportional to the penetration depth of the crack (P2 -pi)/(R 2 - R1 ).
Values correspond to complete hydration of the cement
= 1 and are
consistent with the stress state shown in 4-2b for ER = 40 GPa.
. . .
163
5-11 A comparison of the surface displacement uo(r) = 1 uo] along the crack
p,
<
r < P2 between the open crack geometry (blue) and the closed
crack geometry (red). Values correspond to complete hydration
= 1
and are consistent with the stress state shown in 4-2b for ER = 40 GPa. 164
5-12 The energy stored in the elastic shear bond between the steel and the
cement AE plotted in relation to the normalized penetration depth of
the crack (r - Pi)/ (P2 - Pi). Values correspond to complete hydration
= 1 and are consistent with the stress state shown in 4-2b for ER = 40
G P a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
5-13 The crack surface displacement bounded by steel Es = 200GPa and
rock ER = 40 GPa. The y-axis has been elongated to show the effect
on the surface displacement as the crack tips terminate at bimaterial
interfaces with varying stiffnesses. . . . . . . . . . . . . . . . . . . . . 165
17
18
List of Tables
2.1
Material input parameters (slurry density: 1.9 g/cc). The constants
used for the hydration affinity have been optimized in a non-linear
regression with calorimetry data for Class G oil well cement. . . . . .
2.2
48
Phase volume fractions and homogenized poroelastic constants for an
REV at the characteristic length scales, Level I-III. The Biot coefficients and Biot moduli at the three scales are derived in Appendix C.
57
3.1
Borehole input parameters . . . . . . . . . . . . . . . . . . . . . . . .
80
5.1
Restrictions on the parameters I, and I2 that ensure an effective hoop
stress in the tensile regime . . . . . . . . . . . . . . . . . . . . . . . .
5.2
131
The first root of the characteristic equation describing the singular
behavior of the crack tip terminating at the RC bi-material interface;
assuming v = vR = 0.27
. . . . . . . . . . . . . . . . . . . . . . . . .
19
150
20
Chapter 1
Introduction
1.1
Industrial Context
The extraction of oil and gas from underground reservoirs will continue to rise in the
upcoming decades to meet the demands of human life and prosperity, and to help industrialize emerging economies. In fact, the U.S. Energy Information Administration
estimates that the world's energy consumption will rise from 524 quadrillion (1015)
British thermal units (Btu) in 2010 to 820 quadrillion Btu in 2040, where 85% of the
increase is to occur in developing nations [751. In order to satisfy the global energy
demand, petroleum and other liquid fuels are expected to contribute the majority of
the energy supply.
On the domestic front, the largest growth in energy supply is expected from
previously inaccessible natural gas sources. The underlining prediction of the 2014
Annual Energy Outlook is the disruptive transformation of the American energy landscape by unconventional gas production. Fig. 1-1 shows the anticipated, dominant
growth of natural gas as a source of electricity generation for the United States. Two
advantages to its increased utilization are cleaner electricity generation, and a domestic stimulus to the job market [381. Additionally, because it is estimated that the U.S.
shale source rocks contain 42 trillion cubic meters of natural gas -
approximately
65 times the current annual domestic consumption-and vast supplies of 'tight' oil,
domestic natural gas production can ensure long-term energy security and a reduc21
6
History
Projections
2012
5
4
3
2
1
0
1990
2000
2010
2020
2030
2040
Figure 1-1: Energy sources for United States electricity generation measured in trillions of kilowatthours. The reference case for the projected energy sources assumes a
'business as usual' scenario. (Source of figure: [3]).
tion in the net import of petroleum. From an environmental standpoint, the recently
released report by the Intergovernmental Panel on Climate Change identifies natural
gas as a promising bridge technology [43]:
"Greenhouse gas emissions from energy supply can be reduced significantly
by replacing current world average coal-fired power plants with modern,
highly efficient natural gas conibiiied-cycle power plants ...
provided that
natural gas is available and the fugitive emissions associated with extraction and supply are low or mitigated."
Under safe and regulated practices, natural gas power plants reduce CO 2 emissions
by 50% compared to coal-fired alternatives [381.
Because the production of conventional and unconventional liquid fuel sources
is directly related to the rate of new well completions, and because unconventional
wells require a high density of drilling sites to maximize the yield of a shale formation
(up to 16 wells are drilled in a 2-hectare area [381), the practice of constructing wells
for oil and gas extraction is assured to increase for decades to come. Furthermore, to
ensure that the benefits of natural gas as a transitional energy source are maximized,
production wells must establish safe, efficient extraction methods. The engineering
22
challenge thus posed is the design of production wells that eliminate uncontrolled fluid
loss and unintended fluid migration by properly sealing the well and reducing the risk
of mechanical failure during construction and operation. Otherwise, the potential
of natural gas as a "clean" carbon emitting alternative may not be harnessed. This
thesis is concerned with the loadings and the mechanisms of failure incurred by the
wellbore cement liner as it sets during the construction process.
1.2
Research Motivation and Problem Statement
In the construction of an oil or natural gas well, the cement sheath is placed between
the geologic formation and the steel casing for the purpose of zonal isolation, including the sealing of the reservoir from the fluids of overlying and underlying strata.
Loss of the sealing function carries harmful environmental consequences, creates potential hazards in rig and oil-well operations, and assumes loss of revenue due to
decreased production capacity and expensive repair operations
[231 1321.
Nonetheless,
recent evidence suggests that current design practices are insufficient in safeguarding
against interzonal flow. For instance, studies have indicated that leakages of methane
into groundwater aquifers during unconventional gas production (i. e., gas production
by hydraulic fracturing) are the result of impaired cement casings [451 [621 [77]. In
Pennsylvania, a region noted for its high incidence of methane leakages, Ingraffea et
al. (2014) estimate that 6.2% (1.0%) of unconventional (conventional) wells drilled
between 2000-2012 have compromised casings [411. It is known that early-age shrinkage phenomena and pore pressure developments are primary contributors to sheath
failure 113], yet contemporary modeling efforts rarely and inadequately incorporate
their effects.
For a set cement specimen, pressure and temperature cycles of the well entail
stresses that can cause failure along the casing-cement and rock-cement interfaces
termed microannulus formation; and excessive tensile stresses may initiate radial
cracks in the cement that emanate from the inner casing surface [321
14].
At early
ages, before the cement is subjected to loadings of the testing and production cycles,
23
the setting cement induces pore-pressure changes and chemical shrinkage phenomena
that cause bulk volume changes. Under constraint of the steel and rock boundaries,
these volume changes can lead to stresses that impair the integrity of the sheath. The
ensuing formation of microannuli and radial cracks produce fissures that drastically
increase the permeability of the cement liner.
An accurate, predictive design tool must reproduce the physico-chemical behavior of setting cement. Current work at the molecular and meso-scales is revealing the
fundamental driving forces of the cement volume changes and their relation to the
calcium-silicate-hydrate (CSH) packing-density [53] [54] 1861, and potential improvements of cement strength and fracture toughness through fine-tuning its chemical
structure
[69] [8]. However, the integration of these findings into predictive engi-
neering models has yet to be explored. Additionally, the classification of confined
water into free, constrained, and chemically bound water molecules 182] has elucidated the details of the H 2 0 kinetics during hydration and lends opportunity for
improvements in modeling the hydrating cement phase morphology. Thus, this work
aims at connecting these recent discoveries at the engineering scale of the materials
to the structural scale of wellbore cement liners. Utilizing a bottom-up approach
to the chemo-poro-mechanical evaluation of early-age cement, we represent CSH as
a composition of gel-solid and gel-porespace and homogenize the elementary phase
properties of the cement to the macroscale. Hence, the stiffening behavior and shrinkage caused by self-balancing eigenstresses can be tracked in time and may utimately
be linked to the boundary conditions of the wellbore system.
At the engineering scale, additional challenges remain. An important design parameter, yet to be studied in connection with the early-age driving forces of cement
volume change, is the eccentricity of the production casing with respect to the wellbore hole. In an experimental setup, Albawi investigated the fraction of interfacial
debonding due to thermal cycling of specimens containing a centrically and an eccentrically located casing [4]. Using Computer Tomography, the eccentric casing (with a
degree of eccentricity 6,
=
0.5; at its thinnest portion, the sheath thickness is half the
thickness of an equivalent concentric sheath) showed a greater fraction of debonding
24
and subsequent creation of fluid channels. To the best of our knowledge, no analytic
or semi-analytic solution exists for the boundary value problem of a material domain
confined by two eccentric, elastically deforming circular contours. Here, the loss of
axisymmetry will cause amplification of the radial and hoop stresses and give rise
to shear stresses otherwise not present. In this work, we solve the boundary value
problem and analyze the severity of stress amplification for various stiffness ratios of
cement and rock formation and fluid exchange coefficients between the cement and
the rock.
Finally, the physico-chemical model of cement hydration and the bulk stress developments must be associated to the liner's risk of failure. Though the debonding
of the interfaces and the cracking of the sheath provide the most ostensible pathways
for fluid migration, failure criteria have yet to be defined in the context of fracture
criteria. Albawi offered an empirical investigation of cement fracture of set specimens
subjected to thermal cycling
[41,
whereas Bonnett and Parfitis identify one of the
primary culprits of crack initiation: the decrease of the pore-pressure below the formation pressure leads to an absolute volume reduction of the cement [131. Bois et al.
added that micro-annulus formation is critically affected by the dynamics of cement
hydration 112] -
findings that necessitates an accurate coupling between pressure,
eigenstress, and stiffening mechanisms. In the analysis of the failure criteria, industry
continues to relate the cement's yield strength rather than fracture toughness to its
resistance to fracture. For instance, Bois et al. consider a tensile criterion in designing against radial cracking 1121. Ravi et al. simulate debonding and fracture using
a finite-element-analysis, but their smeared-crack model is poorly described and it is
uncertain whether the worst-case-scenario of a single radial crack is considered
[711.
In response to this deficiency in the mechanistic response modeling of the sheath,
work in this thesis adds analytic solutions to the energy release rates (resp. stress
intensity factors) of micro-annulus formation along the steel-cement and rock-cement
interfaces and a single radial fracture emanating from the steel-cement boundary.
The integration of a chemo-poro-mechanics and a fracture model provides a
novel, holistic approach to identifying the functional design requirements of a wellbore
25
cement sheath.
1.3
The Primary Cementing Process
Since 1903 Portland cement has been used in drilling wells to separate oil and gas
horizons from each other and overlying water aquifers. In the last century, the primary
cementing operation has continued to improve the seal between the steel-cement and
rock-cement interfaces, where new cement formulations have provided more durable
options that can be tailored to the cementing job at hand. Nonetheless, the most
commonly used cement type is the Class G oil well cement, the product of grinding
ordinary Portland cement clinker with calcium sulfate additives. Primary cementing
commences after the wellbore hole has been drilled and is filled with drilling mud.
After running a string of steel casings down the hole, the two-plug method replaces
the drilling mud with the freshly mixed cement slurry (Fig. 1-2 provides a diagram
of the mechanical devices used to complete a two-plug cementing job). Guided by a
bottom and a top plug, the cement is pumped down the interior of the steel casing.
The plugs reduce the contamination of the slurry by drilling mud and remove any
residual cement along the interior surface of the steel casing. Once the bottom plug
reaches the depth of the well and makes contact with the guide shoe, an increase in
pressure punctures the plug diaphragm and allows the cement to escape around the
bottom of the casing. The continued application of pressure using displacement fluid,
pushes the cement up the annular column between the formation and the steel casing.
Centralizers are often used to reduce the offset of the steel casing with the borehole
center and allow uniform flow of the cement around casing; in Chapter 4 it will be
shown that a centered casing also reduces the risk of fracture. Once cement reaches
the entire length of the annular gap, the cement is allowed to set. In the event of
deep wells the primary cementing operation is completed in segments.
26
1.4
Thesis Outline
The completion of the cement placement, described above, is the point at which
we commence our analysis of the early-age stress and pressure developments in the
cement sheath. Hence, this thesis is organized in the following manner:
Chapter 2 gives a description of the dominant phases of the cement microstructure, and adopts the now classic Powers and Brownyard model to partition
their volume fractions in function of the degree of hydration1 . Novel to this thesis
is the incorporation of a recent finding that links the fractions of low-density and
high-density CSH to its polydisperse packing density. With this description of the
cement phase morphology, three characteristic length scales are identified and the
poro-elastic properties are upscaled using the Mori-Tanaka and self-consistent homogenization schemes.
Next, Chapter 3 offers insight into the origin of the eigenstress developments of
the CSH-solid and, by means of Levine's Theorem, offers a device to connect these
to the bulk scale. Additionally, this chapter constructs a pressure state equation that
allows the chemical demand for water to be related to the porespace changes due to
the mechanical loadings and the growth of the solid skeleton. Hence, the drained
nature of the sheath promotes a fluid exchange with the formation.
Moving on to Chapter 4, the bulk eigenstress is linked to the boundary conditions of the steel casing and the rock formation. Whereas the elastic stress state for a
steel casing placed concentric with the wellbore hole is solved by classical means, the
eccentric case invokes complex variables and the framework pioneered by Muskhelishvili [58] to resolve the loss of axisymmetry.
Finally, in Chapter 5 the macroscopic stress evolution is connected to the most
prominent in-plane fracture scenarios. We define our failure criteria in terms of linear
elastic fracture mechanics, and devise solutions to the energy release rates caused
by micro-annulus formation (MA) along the steel-cement and rock-cement interfaces
1The degree of hydration quantifies the fraction of cement clinker undergone reaction. In other
words, it defines the extent to which the cement reaction is complete.
27
and radial fracture (RAD) emanating from the steel-cement interface. In the case of
RAD, complex variable theory handles the loss of axisymmetry, and the solution to a
singular integral equation measures the crack opening displacement. With the energy
release rates of MA and RAD at hand, failure criteria can be defined in terms of the
toughness of the cement and the bonds along the interfaces.
In Chapter 6, we conclude with a review of our findings, recommendations for
more robust cement sheath designs, and an outlook onto future work to improve the
predictive power of our chemo-poro-mechanics solver.
28
Casing
Displacement Fluid
Cement
Slurry
Top Plug
Bottom
Plug
Fboat Collar
Centralizwr
Guide Shoe Job In Procms
Figure 1-2: An illustration of the primary cementing procedure for an oil or gas well.
The figure has been adopted from Ref. [40].
29
30
Chapter 2
The Microstructure of Hardening
Cement Paste
The first part of this chapter introduces the primary constituents of hardened cement
paste and provides the qualities and dimensions pertinent to the modeling of its
microstructure. We continue by describing the segmentation of the constituents as a
function of the degree of hydration, which is based on the framework of the Powers
and Brownyard model. Novel findings in the meso-scale simulations of CSH packing
density by Masoero et al. [53] are incorporated to reproduce the transition between the
formation of low-density CSH and high-density CSH at early and late stages. Where
previous endeavors have incorporated an aspect ratio of the solid CSH phase [74] [76]
[63], this work utilizes the results of the colloidal cement model to fit a power-law
relation between the packing density of CSH and the degree of hydration. Finally,
the homogenization of the elastic material properties to the macroscale is achieved in
a three-level construction using the self-consistent and the Mori-Tanaka schemes.
2.1
Structure of Hardened Cement Paste (HCP)
Cement is a heterogeneous material, with a complex structure. It is composed of
phases or zones of materials with distinctive physico-chemical properties. Here, we
employ the term "phase" not in a chemical sense associated with uniform chemical
31
composition, but as a material subdomain that behaves uniformly at the length scale
of investigation. Indeed, cement exhibits distinctive phases at multiple length scales
that serve as fundamental, intermediate, or bulk levels of structural and chemical
characterization. For instance, the calcium-silicate-hydrate gel, often described as the
"glue" in cementitious materials, behaves as a random composite at the nanoscale,
yet attains the characteristics of a homogeneous medium at the scale of micro- to
milimeters.
At the latter scale, the characteristic dimension of a heterogeneity I
is much smaller than the characteristic dimension of the representative elementary
volume (REV) Y --
< Y. For HCP, the volumetrically dominating phases are
calcium-silicate-hydrates (CSH), calcium hydroxide (CH; also referred to as portlandite), residue of unhydrated particles, and pore space (water or air filled). Additional minor compounds, such as calcium sulfates, aluminates, and ferrites also
constitute a significant portion of the solid volume, though they exhibit limited influence on the mechanical performance of hydrated cement paste.
In this section, we provide a brief description of the phases that most influence
the structure and mechanical behavior of Class-G cement, the most unibiquitous well
cement in use. The last section classifies three characteristic length scales or levels
that serve as a basis for the poromechanical modeling of the cement paste behavior.
2.1.1
Calcium Hydroxide Crystals (Portlandite)
Calcium hydroxide (CH or portlandite) is a crystalline precipitate that forms in the
water-filled pores of hydrating cement. After hardening, it constitutes up to 25%
of the solid volume fraction of cement, and adds considerably to its stiffness. At
the micrometer scale. it may be considered a stiff inclusion in the CSH matrix [18].
As seen in Fig. 2-1b, CH has a layered structure with weak interlayer forces and
negligible hydrogen bonding that give rise to cleavage patterns along well defined
planes. Within the modeling efforts of this thesis, we do not distinguish CH from
the remaining hydration products. This will be noted in the multiscale model of the
cement volume fractions to come.
32
(b)
(a)
Figure 2-1: Scanning electron microscope images of (a) the CSH phase - adapted
from Constantinides and Ulm 120]-, and (b) a portlandite (CH) crystal precipitated
in a CSH matrix - adapted from Nelson and Guillot 160].
2.1.2
Calcium-Silicate-Hydrate Gel (CSH gel)
The primary constituent of cement is the calcium-silicate-hydrate gel which occupies
approximately 50-60% of the total volume. Fig. 2-la displays the reticular texture
of CSH gel in an image taken by a scanning electron microscope 120]. Studies of
fractured surfaces of HCP show that the gel forms two distinctive products: i) the
gel that forms around anhydrous grains, termed the 'outer product', and ii) the fibers
that grow in the water-filled space between the grains, attach to the grains, and form
radiating columns, termed the 'inner product' [171 1801. Similarly, Tennis and Jennings 1811 define low density and high density CSH components. While the presence
of two distinct forms of CSH has been known since Taplin [79], the influence of the
high-density and low-density products on the bulk elastic properties have only recently been revealed 120]. Strikingly, stiffness was determined an intrinsic property of
each component at the scale of hundreds of nanometers, such that the homogenized
modulus of the gel depends only on their volumetric contribution. Recent work in
the field of computational materials science has described CSH gel as a polydisperse
assembly of nano-scale colloidal particles, effectively linking packing density to the
elastic properties of the system [541. Here, the packing density r defines the vol33
2
-30
60
S60-
35-
~65-
-
25
-
(A)
Cn
.2
1.4
1.6
Ca/Si
1
2
1.8
(B)
,I
20
,
1.2
,
1.4
1.6
CaSi
,
,
0 55
45
'
'
70
1.8
2
Figure 2-2: The Voigt-Reuss-Hill bounds of the a) bulk and b) shear moduli of CSH
solid sheets as calculated by the atomistic simulations of Qomi et al. [691.
ume fractions of the CSH solid and the gel-pores (occupied by non-structure water,
i.e. interlayer water) that make up the CSH gel. A transition between low-density
and high-density products is observed over the progression of the hydration reaction.
Hence, the bulk modulus k, and shear modulus go, intrinsic to the CSH solid, were
calculated using atomistic simulations for varying calcium-to-silica ratios (Ca/Si) 1691.
The colloidal nature of CSH lends the material a high surface area-to-volume
ratio, measured at approximately 700 m 2 /cm
3
using the Brunauer-Emmett-Teller
method [80]. In particular, Brunauer and co-workers were able to probe the porespace
of the gel using H 2 0 as a sorbate, separating adsorbed water molecules from interlayer
water molecules; this separation is critical in determining the densification of cement
paste during reaction.
Additionally, recent ultra precision data retrieved through
small-angel neutron scattering and X-ray scattering have determined the CSH gel
density at 2.604 Mg/cm 3 with a water mass fraction of 0.174 and Ca/Si equal to
1.7 [5].
2.1.3
Anhydrous Cement Grains (Clinker)
Before reacting with water, ordinary Portland cement clinker is composed of 50-70%
alite (C 3 S) 1 , 15-30% belite (-C
2 S)
2,
and 10-30% aluminate, ferrite, or other minor
'Here, we use the notation pervasive in cement chemistry; Ca 3 S = 3CaO-SiO,.
2
C 2S=2CaO-SiO 2; where the compound is wholly or largely found as the 3 polymorph.
34
compounds
[80].
Alite reacts rapidly with water, converting around 70% of its mass
into CSH gel phase after 28 days of hydrating and nearly all after 1 year. Particularly
at early ages, alite is the reactant that adds most to the strength and fracture toughness properties of cement. The belite reaction is comparably slow, having reacted
30% after 28 days and 90% after 1 year. Hence, its early-age contribution to the
strength and toughness properties is minor, though the 1-year compressive strength
has been found comparable to that of alite [801. Aluminate and ferrite may substantially modify the cement reaction rate, but are of little significance to strength and
durability.
As can be expected, the rate of reaction is strongly dependent on the particle
size distribution of the clinker grains. Diamond [261 reports typical sizes between 2
Am and 80 um in diameter, where the typical mean diameter is around 10-12 Am.
Due to the limited space in the pore-structure of cement paste, hydration products
typically attach to and coat the clinker grains.
While the water-to-cement mass
ratio (w/c) is an important control of the cement porosity and strength, where lower
ratios typically improve mechanical performance, the Powers and Brownyard model
demonstrated experimentally that a ratio of at least 0.38 is required to completely
hydrate the mix 1611.
2.1.4
Pore Structure
The void space in HCP has a complex structure and manifests itself at a range of
length scales. Though the Union of Pure and Applied Chemistry has a system to
classify pore sizes 3 , the modeling efforts in this thesis require only to distinguish
between gel and capillary pores.
The void space in CSH gel is constituted by its interlayer spacing and micro- to
fine mesopores. There have been several attempts to model the structure of CSH at
the nanoscale. For example, Taylor 1801 reports that the interlayer spacing has been
estimated between 0.5 and 2.5 nm and makes up around a third of the gel porosity. A
more recent study by McDonald et al.
3
1551
used nuclear magnetic resonance relaxation
Micropores: < 2 nm, mesopores: 2-50 nm, and macropores: > 50 nm.
35
analysis to estimate the intra and inter CSH sheet widths and relative specific areas.
They report sheet widths of 1.5nm and 4.1nm for the intra (interlayer water spacing)
and inter (nano porosity) CSH spacing, respectively, and found the ratio of the specific
areas of the two pore types to be 2.4. In this thesis, the interlayer space is assigned as
a part of the CSH solid, such that the volume fraction of the remaining gel pores are
a primary determinant of the colloidal packing density. The gel pores are responsible
for the diminishing stiffness of the CSH gel from its early formation of high density
products to its late formation of low density products.
Even more pronounced, the capillary pores have a considerable impact on the
mechanical performance of cement. They are defined as the water or air filled residue
of space between cement grains; the initial space available is controlled by the w/cratio. Hence, the fraction of capillary porosity remaining in a set specimen determines
much of the change in bulk volume, and, consequently, influences properties such as
fracture toughness and compressive strength. For low w/c-ratios and at late stages of
hydration, the capillary voids have sizes from 10-100 nm, while high w/c-ratios and
early ages of hydration produce voids at sizes of 3-5 pm [18J.
The mechanism for the transport of water through the cement structure is
strongly dependent on size and the degree of saturation. Recently, quasi-elastic neutron scattering was used to divide the water in Portland cement into free, chemically
bound, and constrained populations [821. The constrained portion was associated
primarily with water adsorbed onto the surfaces and contained in the pores (<10nm)
of the CSH solid. For ultra-confined H 2 0, located in the interlayer spacing of the
nanogranular CSH, molecular dynamics simulations calculated the diffusivity of water to be 1/
1 00 0
th
of the bulk quantity over a rang of Ca/Si ratios [68] The mobility
water was reduced largely due to the hydrophilicity of the calcium-silicate sheets.
In the mesopores (the CSH pores), Feldman and Sereda showed for the unsaturated
system that phase changes must be accounted for by differences in the chemical potentials of the gas and liquid states and the change in free energy due to surface
adsorption [31]. They concluded that the water in pores up to 10nm in diameter are
influenced by surface forces. On the other hand, fluid transport at the macroscale
36
and in a saturated medium is principally determined by gradients in pressure.
2.2
Classification of the Characteristic Length Scales
in Hardening Cement Paste
The heterogeneity of cement-based binders manifests itself at different scales. In a
bottom-up approach, one identifies the smallest length scale at which the fundamental
phases of the material do not change from one cement material to another.
By
rearranging and re-proportioning these fundamental 'ingredients' the microstructure
of any binder, of similar chemistry, can be constructed. In a step-wise approach, these
phases are upscaled to resolutions that identify a new characteristic morphology. The
levels of resolutions used in the upscaling of our oil well cement system are described
below.
2.2.1
Level '0'
Recently, Jennings provided evidence that the fundamental building block of the CSH
solid has the structure of an amorphous colloidal 'globule', containing nano-porosity
(structural water) 1461. It is illustrated in Fig. 2-3. While the structure of CSH
at Level '0' can be speculated, it has thus far evaded access by mechanical testing
equipment. Nonetheless, Ulm et al. [871 were able to backcalculate the moduli of the
CSH solid from nano indentations performed at Level I. By deducing the statistical
means of two phases (HD and LD) that became apparent in the probability density
function of the indentation modulus, and knowing their respective gel porosities, the
bulk and shear moduli were calculated to be k, = 31.8 GPa and g, = 19.1 GPa.
These values correspond well with the bounds indicated in Fig. 2-2, determined from
molecular simulations, once the nano-porosity has been accounted for.
37
LEVEL I:
m
C-S-H matrix <I rnh
Two types of C-S-H
LEVEL '0':
C-S-Hl solid
Gel porosity
'Globules':
Basic Bldg. Block
Nanoporosity
d >16.6nm
5.6 nrn-
LD C-S-Hl
37% gel porosiqt
B~asic Bluiling Block
18% nanoporosity
(structural water)
SEM image
H D C-S-H4
24% gel powosity
Figure 2-3: Illustration of the nano- to micro-scale structure of CSH. This figure is
adapted from [87]
2.2.2
Level I
At Level I (10-1 m to 10-6 m) at least two phases of CSH have been detected,
high-density CSH (HD) and low-density CSH (LD) [871 -
they are considered the
building blocks of a cement binder. In fact, indentation tests have confirmed the
presence of both phases and their inherent stiffnesses for mixes with different w/c
ratios 121 [21]. Their volumetric proportions in the CSH gel vary during hydration and
can be calculated indirectly using a power law described by the meso-scale simulations
of Masoero et al.
1531
[54] (see Section 2.3). Instead, direct calculation is made of the
gel-porosity, which defines the local density of CSH ; HD and LD differ only in their
gelpore volumes. Here, the gel porosity must be distinguished from the structural
water at Level '0', which due to its low mobility is considered to be part of the
elementary CSH solid phase [68]. For the subsurface conditions of a wellbore, the
38
Self-Consistent
Mori-Tanaka
0
*
*1
*
*
0
*
0a
0
0
0
0
0
0
0o
0
0
0
10
0
0
00
J.
0
0
0
0
00
*
go* 0 0
0 0
Level I:
C-S-H solid + gelpores
D = 10-8 - 10- 7m
Level II:
C-S-H gel + macropores
D = 10-6 - 10-m
Level III:
Hydrating matrix, C-S-H
gel, macropores, nonreactive inclusions
D =i1-
3
_ 10-IM
Figure 2-4: Depiction of the three-level upscaling scheme utilized to arrive at the bulk
poroelastic constants and the bulk eigenstress.
gelporosity and capillary porosity (at Level II) are considered saturated throughout
the hydration process.
2.2.3
Level II
Mechanical effects relevant at Level II are observed at length scales ranging between
10-6 m and 10-' m. Here, saturated capillary pores are surrounded by a CSH matrix.
The CSH matrix is composed of HD and LD described at Level I. Though real cement
systems contain secondary products, such as portlandite and aluminates, which could
be added as additional inclusions in the CSH gel, explicit representation was dismissed
to preserve model parsimony.
2.2.4
Level III
Level III (> 10-3 m) represents the composite of a porous CSH gel and unhydrated
cement inclusions. Additionally, silica fume (SiO 2 ) is often added to improve the com-
39
pressive strength, fracture toughness, and bond strength of the cement; these silica
particles are typically smaller than 1 pm in diameter with average diameters ranging
between 0.1 pm and 0.2 pm -
50 to 100 times smaller than the size of the cement
particles. Their fineness greatly improves the cement packing density, decreasing the
volume fraction of capillary porosity that degrades the cement's mechanical performance. Similarly, the permeability is reduced, a quality desired in wellbore cements
to prevent or reduce the inter-strata migration of formation fluids [60].
Commonly, cement slurries used in oil and gas well applications use blends of
silica fume and silica flour. While silica fume improves the strength and lowers the
permeability of the mix, silica flour less than 75 pm of the mix
[33].
pulverized quartz with a typical particle size
improves the particle size gradation and reduces the water demand
At temperatures up to 85'C, silica flour behaves as an inert filler [50j.
Though silica fume has a propensity to react with calcium hydroxide, our model
introduces the silica mix as inert, rigid inclusions and it is typically added by an
amount of 0-40% by weight of cement (BWOC).
2.3
Modified Powers-Brownyard Model For the Calculation of Volume Fractions
2.3.1
Partitioning of Volume Fractions
Work by Powers and Brownyard [151 [671 in the late 1940s resulted in a model of
hydrating cement that continues to be applied to the reaction process of the chemically
reactive porous media. The core of the Powers and Brownyard model is based on the
mass balance of a water-cement mixture. Careful accounting of the morphology of the
cement constituents enables a link between volume changes and texture parameters
as a function of the hydration degree [86]. The Powers and Brownyard model acts as
the foundation of our chemo-poro-elasite modeling framework.
Following their method, the constituents of a cement paste sample are described
40
by the phase volume fractions,
(2.1)
f =
where V, is the volume of phase r and V is the overall/ reference volume of the
sample. Therefore, a unit volume of our composite material is partitioned into free
(evaporable) water and unreacted cement, reaction products, and gel and capillary
porosity:
fw. + fc, = fw + fc +
fhp
+ fs
1
(2.2)
fhp = fg + fhc
where fw0 (fco) and fw (fc) represent the initial and instantaneous volume fractions
of the water (cement clinker), respectively;
fhp
stands for the volume fraction of
the hydraction product, which is further separated into gel water fg and hydrated
cement
fhc.
Finally, the chemical shrinkage is quantified by f,. Since our system
is considered to evolve under saturated conditions of the pore spaces, fw and fg
respectively quantify the capillary and the gel water porosities. It is important to
recognize that the volume fractions are provided in reference to an initial unit volume;
that is, quantities are defined in a Lagrangian manner.
The initial volumetric water fraction no may be written in terms of the watercement mass ratio w,
w
no = fwO
c
C
+ PPC
and is equal to the initial volume of water occupied before reaction.
(23)
It is a key
design parameter, influencing the strength and fracture toughness of the hardened
material. pw and pc designate the mass densities of the water and the cement clinker,
respectively.
It follows that the initial volume fraction of the cement clinker is provided by
fco
= 1 - no and the initial mass per unit volume of the system can be written as,
41
Mo = no pw+ (1n-
pO)pc.
(2.4)
The instantaneous mass evolves as a function of the degree of hydration
= (1 - mc)/nmcO
-
the mass fraction of cement clinker undergone reaction -
(2.5)
such that the phase
densities and volume fractions are related to the water influx of a drained specimen
by:
M( )
-
M
= pc (-Afc( ))
(2.6)
+ Pw (-Afw( )) + Php fhp( ) + Pw
Here, Afc (Af,) refers to the volume of cement (water) per unit volume of sample that
has reacted to form CSH gel-fhp= Afc+Afw -fs. Per definition, the volume fraction
of cement clinker evolves linearly as a function of according to
f, ( ) =
(1
-
no) (1
-).
Hence, the change in mass of the system is directly linked to the chemical shrinkage
of the cement.
Having categorized the cement phases using the framework pioneered by Powers
and Brownyard, one degree of freedom lies in bridging the intrinsic properties of
cement clinker and water with those of CSH. In this work, the composition of the
CSH is idealized as a mixture of early-stage, low-density gel and late-stage, highdensity gel [20] [461'. These dynamics indicate a nonlinearity between the packing
density of CSH and the degree of hydration, such that the ratio between the volume
of cement reactant and hydration product increases with . Here, the packing density
r, defines the fraction of solid CSH in the hydration product, and can be modeled as a
colloidal cement system based on the meso-scale simulations of Masoero et al. [53] [54].
'This deviates drastically from the original Powers Brownyward model, which assumes fhp as a
linear function of [67]
42
a)b)
0.78
-
0.75
-
0.69
-
0.72
0.66
0.63
0
0.1 0.2 0.3 0.4 0.5
Figure 2-5: (a) A snapshot of a simulation box filled with spherical CSH particles for
a polydispersity 6 = 0.47. The polydispersity is a measure of the standard deviation
of the particle sizes used in the mesoscale colloidal simulation. 6 is quantified in units
of 0.5(rM + rm), the average of the maximum rM and minimum rm sphere radii and
the color code signifies the particle sizes. (b) The packing density rj as a function
of the polydispersity 6; the shaded region highlights the range of jamming volume
fractions tlj (adapted from Ref. [53]).
The exponential relation, with fitted parameters a and 0, is given by
Vhc
(1/
Vh
=lnQ(
im=
G
Herein,
O(2.7)
p
/ln(77m );
Io '
a=-
=
7O
Tiim
o is the percolation threshold, identified as the hydration degree at which
the hardening cement first forms a continuous solid phase and begins to resist shear
deformation.
Its accompanying packing density is denoted as 7o.
Moreover, the
asymptotic behavior may be described by a maximum density of the CSH constituent
rllim, which depends on the polydispersity of the CSH particles (r7um ~ 0.64 for lowdensity CSH, rlum ~ 0.74 for high-density CSH; confirmed experimentally [20] and
through simulation [53]). Figure 2-5b portrays the relation between the limit packing
fraction rilm and the polydispersity of the colloidal CSH model.
43
Utilizing the relation in Eq.(2.7), the volume fractions for the constituents of the
hydration product are calculated via,
fhc( )
=
(1
-
fg =
(2.8)
Ahp (0)
(2.9)
T1( ))fhp()-
Indeed, we note that the packing density 77 is the additive inverse of the gel porosity.
-
Furthermore, we can write the density of the hydration products as Php = r1 PCSH+(I
T)pw.
Next, noting a mass balance of the reactants and products -
Phpfhp-
pcAfc +pAf
=
the fraction of the hydration products becomes
fhp = Pw - Pc fco Pw - Php
Pw
Pw -
(2.10)
fs
Php
and the fraction of the saturated capillary pores is obtained by
fW
=
fW "+
Pc - Php fco
Pw - Php
+
Php
s
(2.11)
Pw - Php
Finally, the chemical shrinkage drives an external water supply whose volume may
be represented as
fs
In the above,
#3
=--+(1 -
PO)( =3efK.
(2.12)
denotes the effective sink term coefficient, the second degree of
freedom of our model; it is a tool to predict the influx of water required to saturate
the H2 0 demand of the reaction and the surface adsorption and thus determine, under
zero effective stress, the densification of the constituents.
Finally, we recognize that the above volume fractions assume a statistically homogeneous mixture of hydrating cement. In many cases, however, it is of interest to
modify the mechanical properties of a cement mixture without influence on its hydration kinetics. The addition of non-reactive inclusions permit such changes. Admitting
these inclusions, whose standard of measurement is by weight of cement (BWOC),
'In other words, the packing density q and the gel porosity o
44
1 - 77 must always add to 1.
1.0
model
0.8--0C 0.6
data
.2.5
>)4-'
- 2.0
O 1.5
0.4
W
.01.,
0.2
0.5,
0.0
10
10
time - t [s]
10
0.8
0.6
0.4
0.2
degree of hydration - [-]
1.0
(b)
(a)
Figure 2-6: (a) The time evolution of the hydration degree measured against values
calculated from calorimetry data for Class G oil well cement. The modeled values
are calculated by Eq.(2.14), where a = 5.3 s--1, b = 6.4, c = 230, and d = 4.3. (b)
(Data provided by
The hydration affinity plotted against the degree of hydration.
Schlumberger-Doll Research Center.)
the components of Eq.(2.2) must be rescaled by (1 - fNR); with:
- -BWOC
fNR
where
2.3.2
PNR
(2.13)
PNR
W+
C
PC
1I+ PC BWOC)
PNR
is the density of the non-reactive inclusions.
Characteristic Time of Cement Hydration
As the mechanics and chemistry of the cement are principally affected by the growth
of the solid skeleton, it is advantageous to replace the time dependence of variables
By doing so, the stiffness and eigenstress
evolutions in the sheath will be shown to have a close to linear relation with
.
with a degree of hydration dependence.
Correspondingly, we define a characteristic time of hydration Thyd that measures the
time required to react an additional mass fraction of cement dt/d. It is governed by
an Arrhenius type kinetics law [24]:
1 -e-
dg
dt
A()e-Ea/RT = hQZT);
45
AQ )
=
a 1 -'eb
In the above, the hydration affinity,
A( ) = de E,/RT =
dt
,
(2.15)
(1
measures the change in the Helmholtz free energy T of the system as the reaction
progresses. It is an intrinsic material function of the cement and is independent of
heat flux boundary conditions. As such, it depends exclusively on the stoichiometry
of the chemical cement phases and the molar masses and chemical potentials of their
reactants. Nonetheless, the advancement of the hydration affinity A( ) may be captured by the empirical relation given in Eq.(2.14). This necessitates the non-linear
regression of four constants a, b, c, and d that depend on the type of cement
1351.
The factor Ea/RT accounts for an activation controlled reaction; here, Ea, R, and T
are the activation energy, the ideal gas constant, and the absolute temperature (i.e.,
measured in K), respectively. Fig. 2-6a displays the relationship between the curing
time and the hydration degree of a Class G oil well cement. Here, the hydration
affinity well represents the calorimetry data of a representative cement sample curing
at 85'C (see Fig. 2-6b).
2.3.3
Sample Model Output: The Evolution of Volume Fractions.
Throughout this thesis, we connect the varying components of our modeling effort by
providing the results of a comprehensive simulation. In doing so, we choose to restrict
the input parameters to typical conditions encountered during a primary cementing
operation.
Our idealized model for the evolution of the volume fractions as a function of
the hydration degree is illustrated in Figure 2-7. All material properties and mix
design parameters have been written into Table 2.1. Per definition (see Eq.(2.5)),
one observes that the clinker vanishes linearly as the reaction progresses, while the
pronounced formation of low-density CSH at the early stages and high-density CSH at
the latter stages evokes the non-linear relationship of hydration products. As defined
46
1.0.
clinker
0.5
0.0
0.2
0.4
0.6
0.8
1.0
Figure 2-7: The volume fractions in a hydrating cement sample are plotted as a
function of the degree of hydration . The results are based on our modified Powers
and Brownyard model (see Table 2.1 for input parameters).
by Eq.(2.7), the decrease in the instantaneous packing density of the CSH gel is due
to a loss in the void-to-solid ratio. Therefore, the volume of the hydrated cement
(i.e., the CSH solid) is still linearly related to . Finally, the green segment near the
top of Fig. 2-7 evidences the chemical shrinkage, measuring the densification of the
reactants for an unloaded system with incompressible fluid and incompressible solid
phases:
fs)
M
dM
(2.16)
Pw
For an unloaded, saturated, and incompressible cement paste, it measures the mass
of water absorbed into the paste. Here, the mass of water that is chemo-physically
bound into the CSH structure and the mass adsorbed onto the gelpore surfaces have
a smaller volume than the mass of water that previously occupied the macropore
space; i. e., the change in capillary pore volume due to the growth of hydrated matter
dqo/dk (see Section 3.3.1).
47
Table 2.1: Material input parameters (slurry density: 1.9 g/cc). The constants used
for the hydration affinity have been optimized in a non-linear regression with calorimetry data for Class G oil well cement.
Mix Design (Class G)
Hydration Kinetics Param (Class G)
w/c
0.45
a[1/s] = 3.231
b
NR-BOWC
0.35
c = 95.191
d
Pc [g/cc
3.15
Ea[kJ/mol]
31.472
PNR[g/cc] (SiO 2 )
2.65
Molecular Scale Prop. of CSH
PCSH [g/cc
2.43
ks[GPa]
49
gs [GPa]
23.5
-
3.930
0.402
Mesoscale Prop. of CSH
O= 0.263
fL
Bulk Water Prop.:
2.4
20.505
0.15
0.01
pw[g/cc] = 1.0
0.640.7
e*
3/1000
kfl[GPa] = 2.0
Continuum Micromechanics: A Three-Level Cement Thought-Model
This section of the chapter reviews concepts that allow a heterogeneous cement material to be treated in a continuous manner. More succinctly, we recount the most
successful techniques that predict the macroscopic constitutive behavior from a material's microscopic structure and apply them to the three-level cement model using
the relations devised by Constantinides and Ulm [20].
2.4.1
The Principle of Scale Separability
Classical homogenization techniques wish to replace the complex, heterogeneous material behavior at the microscale by a representative, homogeneous medium at the
macroscale. A key concept in the development of homogenization techniques is the
representative elementary volume (REV): A material sample that contains a statistically sufficient number of inhomogeneities that are evenly distributed within its
48
volume to construct a fictitious, and homogeneous material equivalent. In order to
ensure that the REV can be replaced by such an equivalent homogeneous material
body, we must introduce the relevant characteristic length scales and ascertain relevance of the principle of scale separability'. For an inhomogeneity of characteristic
length 1 and embedded in the matrix of an REV of characteristic length 2 , we require 1 < Y. Similarly, for the scale of the overall structure 9, it is necessary that
Y
<
-9. For instance, the anhydrous clinker grains and capillary pores of HCP are
pervasive throughout its microstructure and have a discernible influence on the bulk
material behavior, yet are not perceived by an observer at the structural level.
A second condition requires that the applied mechanical loading varies over sufficiently large distances. Thus, the fluctuation length of the loading A should be
large compared to the dimensions of the REV, Y < A. This makes accessible the
mathematical tools of differential calculus.
These conditions ensure, from a material perspective, that continuum theory can
be applied at the structural level of an REV 191].
2.4.2
Concepts in Continuum Micromechanics
In our bottom up approach, the smallest relevant scale is identified, after which we
progressively coarsen the granularity of our model, identifying all 'levels' at which new
phase can be identified. As alluded to earlier, a phase is defined as a spatial fragment
of the REV that behaves uniformly according to its on-average stress or strain state.
It need not be homogeneous in composition. Apportioning volume fractions to the
phases of a level, the Hashin boundary conditions 7 allow the macroscopic strain and
stress states to be approximated by the volume averages of the microscopic strain
6Here, we assume a static system, unaffected by percolation phenomena and in which long-range
correlation effects are not present.
7
See, for instance, the Ph.D. thesis by Georgios Constantinides for a complete description of the
Hashin conditions {191; It also contains a more in depth review of continuum micromechanics herein
utilized.
49
and stress states:
E = (E(z)) =
=
(or(z))
=
fj E(z)dV
(2.17a)
fj a(z)dV.
(2.17b)
These relations only hold if a regular stress or strain boundary conditions apply:
on
V:
u = E -n;
or
t = E -n
(2.18)
Here, V and aV denote the volume and boundary surface of the REV. E and E are
the homogenized macroscopic strain and stress, and E(z) and o-(z) are the localized
strain and stress at position z. Finally, u and t are the displacement and traction
vectors acting on a surface perpendicular to the unit normal n. Naturally, one may
wish to write explicitly the relation between the local microscopic strain and stress
states and their upscaled quantities. Hence, linear micromechanics theory has devised
strain A and stress B concentration operators, that map the macroscopic states to
the local microstates:
o-(z)
E(z) = A(z) : E;
=
B(z) : E
(2.19)
where the assumption of linear elasticity has reduced the concentration operators to
forth-order tensor fields, and we have adopted dyadic notation. Whence, it follows
that
E = (c(z))v = (A(z) : E)=
E = (a-(Z))V = (B(z) :
(A(z)) : E -+ (A(z))v
=
I
(2.20a)
(B(z)) :E -+ (B(z))v = ]
(2.20b)
=
and I is the diagonal 4th-order identity tensor. It is now possible to express the linear
elastic constitutive relations for the homogenized medium in terms of the localization
laws. Taking the material properties at the microscale, i.e. the stiffness tensors c',
and the volume fractions of the microphases r, the macroscopically effective stiffness
50
tensor C can be calculated from Eq.(2.20) and Eq.(2.17) as
(c, :r(z))v = (c, : A(z))v : E = C: E
E = (c 1 : -(Z))V = (c 1 :
(Z))V : E = C 1
(2.21a)
E
(2.21b)
which implies
C = (Cr : A(z))v
C-
= (c;1 : B(z))v.
(2.22a)
(2.22b)
The expressions for C in Eq.(2.22a) and Eq.(2.22b) become equivalent in the limit
1/Y -+ 0 -
i. e., the operation of localizing the macroscopic strain to the microstrain
via A and localizing the macroscopic stress to the microstress via B become equivalent.
2.4.3
Homogenization schemes for elastic properties composite materials
We wish to use common techniques in micromechanics to deconstruct our complex
poro-mechanical system into a mathematically accessible problem.
Many efforts in
the past have made it possible to make accurate estimates of the macroscopic behavior
of linear elastic materials by simplifying the complex morphology of its microsystem.
For example, the famous result by Eshelby
[30] proved that the strain in an ellipsoidal
inclusion, which is embedded in an infinite medium and is subjected to a uniform
eigenstress -
a self-equilibrating stress that is not caused by an external loading8
-,
is constant. Hence, by simplifying the geometry of the inhomogeneities in an REV
to ellipsoids, several upscaling schemes have been devised that relate the strains in
the inclusion phases E
cin
and the strain in the matrix Emat. For the case of a single
8
It derives from the German word Eigenspannung and refers to a residual or "self"/materiallygenerated load, such as is present in thermal expansion. For more information on eigenstresses see
e.g., the book by Mura [57].
51
inclusion one may calculate directly from Eshelby,
I"ind =
[
+ Pr : (Cinci - cat)] -1 Emat,
(2.23)
where the loading of an eigenstress has been replaced by an equivalent far-field matrix
strain. In the above, Cinc and Cmat refer to the stiffness tensors of the inclusion
and matrix, respectively, and Pr is the forth-order Hill tensor -
it depend on the
shape of the inclusions and the elastic properties of the matrix [19]. One recognizes
immediately that the strain concentration tensor field for the inclusion is uniform and
given by Ainci
-
[i + p(Cincl - Cmat)] -1.
The rapid attenuation of the stress gradient near the inclusion often allows the result to be applied directly to dilute systems. However, for materials densely populated
by particles, additional consideration must be taken to account for their interaction.
Using the strain averaging relation in Eq. (2.17b) and considering the volume fractions
Jr = V/V
of a multiphase composite, the strain in the matrix is calculated as
'r [11 + IPr : ((Cind ci
Jmt
ma)]
i)
)-1
E
(2.24)
where E is the strain applied to the REV. It follows directly from Eq. (2.23) that the
strain concentration tensor for an individual phase r is estimated as
Aes t = [I + Pr : (Cr - Cmat ) 1 :
f [J+ Pr : (Cn - Cmat)]
(2.25)
Next, we elaborate on two options of how to chose the matrix and inclusion phases
and expound upon the circumstances under which they are best utilized.
The Mori-Tanaka Scheme (MT)
The Mori-Tanaka upscaling procedure chooses a distinct phase as the reinforcing
medium [56] [10]. It is well appropriated to cases for which there is a dominant matrix morphology, and the difference in stiffness between the matrix and the inclusions
52
is great. Hence, it is often chosen to model stiff/rigid inclusions or pore space embedded in an elastic, continuous matrix phase (e.g., aggregate embedded in a matrix
of bitumen or entrained air infused into a cement mixture).
In Eq.(2.25), we assign r
=
1 as the matrix phase and r > 1 as the inclusion
phases, such that the strain concentration operator is written as:
AMI
=
[I + Pr : (C, - C 1 )]-
: (
f
[1i + Pr : (Cn
(2.26)
C)]-)
Because the stiffness properties of the matrix phase are known, the Mori-Tanaka
scheme can be solved in an explicit fashion.
The Self-Consistent Scheme (SCS)
The second upscaling procedure of note is the self-consistent scheme, where Cscs is
chosen from the overall moduli, dependent on the properties of the phases composing
the REV 1361 151]. SCS has been utilized successfully on polyerstals, where the phases
are dispersed in the medium
[911.
Geomechanics provides another prominent appli-
cation, as granular materials, such as clay or shale, often have a disordered structure
in which none of the phases mimic dominant matrix or inclusion characteristics [88].
Consequently, the implicit nature of the REV homogenization is transformed into an
as the matrix stiffness:
integral equation by choosing the composite stiffness Cc'm
ASCS
2.4.4
- [RI + IPr : (C,
-
Ccop
:
fn [1[ +n ]Pr : (Cn
-
)-1
CcornP)P1)
(2.27)
Three-Level Homogenization Scheme for Hardening Cement Paste
In the following, we utilize the upscaling schemes previously introduced to measured
the homogenized bulk and shear moduli for the REVs at the three characteristic
length scales identified in Section 2.2. It is demonstrated that the matrix-inclusion
morphology plays an important role in distributing the stresses in the microstructure.
53
Level '0': The CSH-solid 'globule'
The power-law in Eq.(2.7) fits the proportions of low-density and high-density CSH
to the packing density of the colloidal CSH system. Hence, the fraction of CSHsolid (including structural water and nanopores) in the gel is measured by q and the
fraction of porespace is given by c = 1 - 7. Though anisotropy exists in the sheet-
like orientation of the CSH layers
'globule' -
1691,
our elementary building block -
the CSH
can be modeled as an isotropic solid due to the disordered orientation of
the particles. The stiffness tensor of the CSH-solid thus simplifies to
Chc
where K =
6
ij 41
(2.28)
3ksK + 2gsJ
is the volumetric part of the fourth-order unit tensor and J = R -K
is its deviatoric part.
Level I: CSH solid + gelpores = CSH gel
Level I includes the CSH solid and gelpores (V1
=
Vhp = Vh
+ Vg). At this scale,
the CSH solid can be modeled as colloidal grains. Upon percolation - the point
at which a continuous force path through the solid is able to resist shear stresses
- the CSH gel resembles a disordered array of contacting grains intermixed with
some porosity.
A granular material of this sort was studied by Hershey [361 and
Krbner [51], who upscaled the elastic modulus and its bounds for a polycrystalline
aggregate. Their self-consistent scheme in Eq.(2.27), has been adapted to the study
of cement by Constantinides and Ulm [201 [191. Hence, the Level I stiffness tensor is
estimated by:
C1
= 77 Ch
AsCS + (1 ---)
Cg
: ASCS(2.29)
where C9 = 0 is the modulus of the gelpore space and Ch, has been given in Eq. (2.28)
above. Recognizing that IP = Sesh : C
om p
in Eq.(2.27), where Sesh is the Eshelby
tensor 9 of phase r [30], and assuming the shape of spherical inclusions, the bulk and
9 See
for instance the review article by Zaoui [91], which provides the form of Sesh for a spherical
and isotropic inclusion.
54
Figure 2-8: The microstructure of 100-day old cement paste with a w/c of 0.30 (from
Ref. 126]).
shear moduli are recovered as,
K1
k-
4rG/g,
_
(2.30)
4G/g, + 3(1 - r7)rs
and
1
2
G,
YS
5
4
3
r (2+r)
16
(2.31)
1
+
16
2
/-V1
44 (1 - r,) - 480'q + 40072 + 408rr - 120rsr + 9r2(2 +
i/) 2
where rs = ks/gs = 2(1 + vs)/3(1 - 2v,) > 0. It is apparent that the stiffness of the
CSH-gel has a direct relation to the packing density of the colloidal grains, since ks
and g, are considered intrinsic to the CSH-solid.
Level II: CSH gel + macropores = cement binder
At Level II, the REV is represented as a matrix of CSH-gel weakened by the random
17 = Vhp + Vw). Thus, the total porosity manifests
arrangement of capillary pores (V
itself at two levels; gelpores at the sub-micron scale and macropores at the micron
55
scale. The effect of the gelpores on the Level II stiffness is accounted for in the homogenized moduli of the CSH-gel, K and G 1. Furthermore, the clear distinction
between phases and the embedded nature of the capillary pores in the CSH-matrix
suggest use of the Mori-Tanaka in calculating the Level II homogenized stiffness tensor, Cn = 3K1 K + 2G 1 J:
Knl
K1
Gil
G1
In the above, r1 = K1 /G1 and Oil
4(1 -0#1) K
4+3#%r1
((2.32)
(8 + 9ri) (1 - #i)
8 + 9r1 + 605(2 +r 1 )(
=
M
(2.33)
(Vg + Vw)/(V 1 ) is the volume fraction of the
large, capillary pores at level I.
Level III: Cement binder + anhydrous clinker + non-reactive inclusions
+
The coarsest resolution of the cement paste is achieved at Level III (V, = Vl
Vc + Vnr), for which the length scale of a typical REV is measured in milimeters. A
scanning electron microscope image of cement at Level III is provided in Fig. 2-8.
Here, we see that grains of anhydrous cement clinker are surrounded by hydrated
CSH-gel. In the case that structural additives, such as silica flour, are added to the
mix, an additional phase resides in the space separating residual clinker grains. It
should be noted, that Fig. 2-8 provides a snapshot of the arrangement of phases in
well hydrated cement sample. At earlier ages, the proportions of the cement binder
and the inclusions will vary substantially; In fact, the phases playing the roles of the
matrix and the embedded inclusions is poorly defined and may switch during the
reaction. Consequently, the random arrangement, size, and multitude of phases of
the inhomogeneities at Level III suggests the self-consistent scheme as the best suited
upscaling procedure. Following the scheme, the bulk modulus is given by
3r1 + 4
Ki1
nm
3(1 - n 11)r(2
56
(2.34)
Table 2.2: Phase volume fractions and homogenized poroelastic constants for an REV
at the characteristic length scales, Level 1-111. The Biot coefficients and Biot moduli
at the three scales are derived in Appendix C.
Level III: (hydrating
Level I: (CSH solid +
Level II: (CSH gel +
matrix, CSH gel,
gelpores)
macropores)
macropores,
non-reactive inclusions)
V1 = Vp
Matrix
Vii-
i=7nn
Fraction
=ln71
Inclusion
1- 7
0
050
Biot
Coefficient1-1-
b1 =
Solid Biot
= 0
bn =
y[(1
-L-_
1-
Modulus
nm =
I/n
/VV
finc = (V + Vnr)/Vi
+ n0y#I'
= 01(1 -
- Knlk,
bi
- #")(b - #1) +
- bJN)--
-(1
ni=
= V,VI
=
- K11k,
= VI + Vc + Vnr
V
VpV
= V p/Vn
Fraction
Total Porosity
Vh + Vw
-
finc)
= bi
1
)
Reference
Volume
and the shear modulus is calculated as
8(3 - 2nrim)
(15
-
24rInm)rii
)
Gil
-
1
1
24 (2 - 3nm11
Gi,
9(8ni - 5) 2r 2 + 48(11nr 1
-
29nm
11 + 15)rl1 + 64(3
-
+
(2.35)
2n,,I)2
In the above, the ratio of the Level II moduli, r 1 = K11/G 1 , has simplified the
expressions.
2.4.5
Sample Model Output: Evolution of Poroelastic Constants.
Incorporating the input parameters previously noted in Table 2.1, the upscaling relations derived in the previous section provide access to the stiffening behavior of the
cement slurry for an REV at Level I, Level II, and Level III. Thus, we have plotted
57
u.6
0.5-
I, C-S-H gel
II, C-S-H gel+capillary pore
-- d=
III, Cement paste+NR inclusions
-d=
0.4
0.30.2
0.1
0.2
0.4
0.6
0.8
1.0
Figure 2-9: The evolution of the bulk modulus of the cement is plotted for an REV at
the three characteristic length scales defined by Level I (CSH-gel), Level II (CSH-gel
+ macropores), and Level III (CSH-matrix + rigid inclusions) (see Table 2.1 for input
parameters). All values have been normalized by the bulk modulus of the CSH-solid.
the model output for the three bulk moduli in Fig. 2-9. Per definition, the cement is
able to withstand loads once the hydration reaction has moved beyond the percolation threshold and a continuous solid matrix has been established. We note that none
of the moduli exceed the stiffness of the CSH-solid. In fact, the added gel-porespace
(at Level I) and macro-porespace (at Level II) act to reduce the stiffness due to the
drained character of the cement. However, the comparably rigid behavior of the residual clinker grains and added structural inclusions help to increase the elastic stiffness
once the bulk modulus is homogenized to Level III. Interestingly, additives such as
silica flour improve cement strength and stiffness by increasing the polydispersity of
the cement phase constituents and, consequently, decreasing the capillary porespace.
It follows that such non-reactive inclusions, which are incorporated into the homogenization scheme at Level III, indirectly enhance the stiffness of the cement at a smaller
length scale.
In the succeeding chapter, additional poromechanical constants will be of use in
describing the constitutive behavior of the hardening cement paste. In particular, it
58
1 0 1~
-
0.8
-
1.0
d =I, C-S-H gel
d=II, C-S-H gel+capillary pore
d=III, Cement paste+NR inclusions
0.8
0.6
.0.6
0.4-
-0.4
0.2
-0.2
packing density
0.0
0.2
0.6
0.4
0.8
0.0
Figure 2-10: The evolution of the Biot modulus of the cement is plotted for an REV
at the three characteristic length scales defined by Level I (CSH-gel), Level II (CSHgel + macropores), and Level III (CSH-matrix + rigid inclusions) (see Table 2.1 for
input parameters). The modulus has been normalized by the bulk modulus of the
CSH-solid. The secondary axis offers a comparison of the CSH-gel packing density.
will be required to relate eigenstresses generated in the microscale phases to each other
and the bulk, and applied displacements at the macroscale to the strain localization in
the phases. To achieve this, we calculated the relevant Biot coefficients - operators
that localize the macroscopic strain to the porespace - and Biot moduli - operators
10
that measure the porosity change due to a microphase eigenstress . Expressions for
these constants and the porespace and packing fractions are provided in Table 2.2.
Fig. 2-10 depicts the evolution of the scale dependent Biot moduli as related to the
eigenstress generation in the CSH-solid (in the graphic, their normalized inverses
are plotted); plotted beside is the packing fraction of the CSH-gel. Before solid
percolation, the Biot moduli are directly related to the gel packing density; at Level
I a 1-to-1 correlation is observed. This is easily explained by the assumption that
any induced stress in the CSH-solid induces a subsequent unrestrained volume change
thereof. In other words, an eigenstress o-* in the solid phase produces a volume change
provided in Ap"A more detailed investigation follows in Chapter 3, and the derivations are
pendix C.
59
0.6
K/k,
-
G/ks
Cn-
(1-b)
- 0.4-
0.2
o
A0.2
0.2
0.4
0.6
0.8
1.0
Figure 2-11: The bulk elastic properties properties of the cement, K and G, and the
biot coefficient of the cement are plotted as a functino of the degree of hydration (see
Table 2.1 for input parameters).
o-*/N of the CSH-solid, whose proportion of the REV is measured by r. Once the
cement paste has hydrated beyond
o and is able to resist deviatoric stresses, the
evolution of 1/Nd is determined by the competing processes of the cement stiffening
and gel densification. The effect of an eigenstress is attenuated at larger scales as the
CSH-solid composes a lessening volume fraction of the REV.
Finally, we plot the two macroscopic elastic constants K = Km and G
next to the compressibility index (1 - b) = (1 - bm11) in Fig. 2-11.
=
Gm1
We recall that
the Biot coefficient computes the macroscpic change in strain due to a change in
porosity.
For our drained specimen, b must always be bounded between b = 1 (an
incompressible solid) and b =
#
(the uniform localization of strain). For our cement
paste, it is reasoned that applied loads before percolation act to drive fluid out of
the system and only rearrange the anhydrous cement clinker. Thus, (1 - b) = 0. For
>
0, stresses are concentrated in the compressible solid matrix, reducing the Biot
coefficient as the matrix grows.
60
2.4.6
Chapter Summary
In this chapter, the relevant phases of a neat cement paste were identified and their
properties were tracked down to the nanoscale. Considering these properties as intrinsic building blocks of the cement composite, the representation of an array of
mix designs is satisfied without sacrificing model parsimony.
Given the typically
observed dimensions of the micro phases, three characteristic length scales were identified at which new, homogenized phases were revealed. Introducing the Powers and
Brownyard model for the partitioning of the volume fractions during reaction, we
incorporated a novel model of the CSH-gel by Masoero et al. [53] to characterize its
nanotexture. Finally, the elastic properties of the cement were upscaled using the
Mori-Tanaka and Self-Consistent models for the matrix-inclusions morphology.
Key results are summarized in Table 2.2, and will provide the important link
between the micro and macroscopic material responses to internal and external loadings in Chapter 3. Moreover, the homogenized moduli will be adopted in a continuum
linear elastic stress analysis of the cement sheath in Chapter 4 and further utilized
for the calculation of the energy release rates for the fracture scenarios in Chapter 5.
As a necessary component of the risk-of-failure assessment of the wellbore linear, we
have constructed an advanced material model that can assess the stiffening behavior
of the cement in function of the degree of hydration. Most importantly, the framework has been set to investigate the dynamics between pore-pressure and eigenstress
developments and their impacts on the structural response of the cement. The next
step is to relate cement chemistry to poromechanics.
61
62
Chapter 3
Bulk Eigenstress Development
Ulm et al. [871 showed that the mechanical behavior of cement is well described
by elastic poromechanics models. These models capture the couplings between the
eigenstress in the CSH-gel, the pore-pressure development, and the deformation of the
porous skeleton. In this work, we incorporate these couplings into the mass balance of
the water in the macro- and gel-pores, while considering the reactivity of the media.
In the first section of the chapter, we introduce Levin's theorem, which allows us
to relate the localized pressure and eigenstress developments in the poroelastic skeleton to the changes in the bulk porosity. We proceed, by introducing novel findings
that explain the physico-chemical driving forces of chemical shrinkage. In particular, the water consumption due to the stoichiometry of the cement reaction and the
confinement of water in the gelpore cavities are compared to the growth of the solid
skeleton. It will be shown that net-attractive interactions of colloidal gel particles
and pore-pressure changes experienced as a result of the water demand during the
hydration reaction are responsible for the bulk volume changes of the cement paste.
Moving on, it is observed that the growth, stiffening, and deformation of the
solid skeleton are intimately linked to the pore-pressure and fluid exchange with the
formation. Hence, the aim of this chapter is to derive an equation for the discretized
pore-pressure development of an REV at the bulk scale and relate it to the boundary conditions of the sheath. At fixed degree of hydration
fraction
fhp,
and fixed solid volume
the single-valued pressure in the pores and the eigenstress in the chemoe63
lastic skeleton are upscaled to the mean bulk stress of an unrestrained REV: the bulk
eigenstress. Under restraint of the steel and the rock, radial and, in the case of an
eccentrically placed casing, tangential stress gradients drive a non-uniform pressure
development in the sheath. However, it will be shown for typical hydration kinetics and flow parameters that this gradient is small compared to the overall drop in
pressure.
3.1
On the Origin of Cement Eigenstresses During
Hydration
Chemical shrinkage -
also termed Le Chatelier contraction - is best observed when
an incompressible cement paste, before solid percolation, hydrates under saturated
conditions. Here, the difference in density of the hydration products on the one hand
and the cement and water reactants on the other hand is measured by the absorption
of water into the cement paste structure. Under these conditions, it is assumed that
the cement phases are absent of deformation due to loading. Powers examined the
chemical shrinkage of 10 Berkeley cement samples and measured absorption rates
between 0.024 g and 0.05 g of H 2 0 per g of cement [65]. While direct evaluation of
the "absolute" volume changes -
that is, the volume of the additional water mass
consumed by the hydration reaction -,
sample at early cement ages
(
can be made for the incompressible bulk
< co), the onset of a continuous solid framework allows
the cement slurry to resist the volume contraction arising in the microstructure. For
unsaturated pastes, the further diminution of the bulk volume is often attributed
to capillary forces that emerge due to the formation of menisci. This phenomenon
is often termed self-desiccation [9]. However, recent experimental results for water
saturated cement hydrating under constant pressure conditions suggest that the bulk
cement specimen shrinks even in the absence of menisci formation [861. This shrinkage
is typically one order of magnitude less than that of the chemical shrinkage. In this
work, we wish to make a link between the chemo-poro-mechanical testing by Ulm et
64
60
-
p=10 MPa
50
.p=1
MPa
40
C)30
20
10
0
0.5
0.55
0.6
0.65
0.7
0.75
Packing Density, q;
Figure 3-1: The relation between eigenstress development and packing density for
cement hydrating at constant pressures of 1 MPa and 10 MPa [86].
al. 1861 and the colloidal mesoscale simulations by Masoero et al. 153]; the simulations
lent evidence that the mean interparticle distance of the colloidal CSH spheres was
greater than the equilibrium separation, leading to a net-attractive interaction. This
phenomena is in direct agreement with the observation that, though shrinkage was
recorded during the experimental test, a positive effective stress was calculated (i.e.
internal stresses rather than external boundary loads must be responsible for the
volume change). Under comparison with the model generated (positively correlated)
relation between packing density and degree of hydration, Fig. 3-1 gives evidence to
the linear relation between o* and rq. The difference in the onset of a detectable
eigenstress for cement hydrating at 1 MPa versus cement hydrating at 10 MPa is
explained by the difference in an initial prestress.
In conclusion, there is strong
evidence that the colloidal system generates eigenstresses that are directly related
to its interparticle potentials. This means that the eigenstress is determined by the
packing density of the CSH-gel and must therefore be treated as a phenomena intrinsic
to its phase.
65
3.2
Upscaling the Microscopic Driving Forces
Our cement liner undergoes internal loading that manifests itself in its microstrueture: The CSH-gel shrinks due to internal eigenstresses that grow in proportion to the
colloidal packing density, and the water demand of the hydration reaction decreases
the fluid pressure in the pores. While these forces are considered to espouse uniformly
in the phases of the microstructure, the inhomogeneities act to redistribute the loads.
Hence, we are tasked to build a framework that upscales the local eigenstresses to
the engineering scale.
In combination with the homogenized elastic constants de-
rived in Section 2.4.4, an expression for the homogenized eigenstress facilitates the
construction of the continuum level poromechanical constitutive relations.
3.2.1
Levin's Theorem
In this section of the chapter we introduce Levin's theorem, a tool that maps the
effect of localized driving forces onto the bulk specimen. More succinctly, the theory
herein reviewed allows us to connect the pressure acting at the solid-fluid interface
dp and the eigenstress developed in the CSH-gel d-* to the homogenized bulk stress
state of the REV. Imperative to the formulation of this relation is the stationarity of
the phase volumes. Since our mechanical analysis deals with reactive cement, whose
solid volume grows continuously, the homogenization must be applied at a constant
degree of hydration and for incremental changes in the boundary conditions. Thus,
we tailor the approach by Dormieux et al. (Ref. [271, pg. 156-159) to our multiphase
chemically reactive porous material.
We begin by writing the local constitutive relation,
in V : [do(z)
=
c(z) : dE(z) + doP(z)],
66
(3.1)
c(z)
00
in 1nc
Ch
in Vh"
0
in VO
d;a=
0
in V4nc
do-*1
in Vh,
-dZpl
in VO
(3.2)
were z C V is the position vector for an REV defined in the region V. Herein, the
microscopic resolution includes elements of Level I and Level II (see Section 2.2),
where the matrix volume consists of CSH-solid and the rigid inclusions -
comprised
of the anhydrous clinker grains and the non-reactive inclusions - and their volume
contributions have been segmented into Vh and
Vinc,
respectively. In addition, the
pore volume Vp consists of the capillary and gel pores, such that:
1ic + Vh
+ VO
=
(3.3)
V
Finally, Chc is the stiffness tensor of the CSH solid and doP defines the local eigenstress. Application of the internal loading at constant hydration degree is carried out
throughout the remainder of this thesis and will not be explicitly indicated in the
subsequent.
The solution to this upscaling problem resides in the linear elasticity of the
material skeleton behavior and the self-balanced nature of the eigenstresses.
The
problem is deconstructed into two sub-problems.
" dE1 : The stress state due to a regular displacement boundary condition of the
form Eq.(2.18) where the eigenstresses have been set to zero.
" dE 2 : The stress state due to the loading of the eigenstresses where the displacement of the REV boundary has been set to zero.
Thus, the principal of superposition applies and we can construct the stress of an
.
REV subjected to bulk deformations and an internal loading, dE = dE1 + dE 2
67
du= dE - z
du 1
dE - z
sub-problem 1
du2
0
sub-problem 2
Figure 3-2: Levin's theorem is used to upscale the eigenstress in the microstructure
to the macroscale (the volume phases are not drawn in proper proportion and scale).
Sub-problem 1
This first problem, in which a regular displacement boundary condition is admitted
without eigenstresses acting in the microstructure, is stated mathematically as follows:
in V: V -da, = 0
(3.4a)
in V: doi = c(z) : dEi(z)
(3.4b)
on
av : dui(z)
= dE - z.
(3.4c)
As usual, the local strain increment is related to the strain at the scale of the REV
by the strain concentration operator, dEc(z) = A(z) : dE. Substituting this relation
into Eq.(2.17b), the homogenized stress dE 1 = (doa-)v reads as
dE= (c(z) : A(z))v : dE
d,=
# 0 1 : (dc1(z))v, = #01 : (A(z))v4 : dE
where the change in porosity d
1
(3.5a)
(3.5b)
, measured in reference to the initial porosity #0
=
(V, + Vg)/V, is simply the localization of the change in mean strain onto the pore
volume. The operator that relates pore volume changes to the macroscopic change in
68
strain is known as the Biot tensor, here identified as b =
#o1
: (A(z))v,. Furthermore,
as our poroelastic skeleton is considered statistically isotropic, the stiffness tensor Ch,
has a well known form and the relations in Eq.(3.5a) and Eq.(3.5b) are expressed
more simply as
dE1 = 2GdE' + KdEml
or
d Ei
= 2G(dElj - dEm6itj) + KdEm6ij
(3.6a)
(3.6b)
do1 = b dEm.
where dE' is the strain deviator, and dEm, is the mean strain, which equals 1/3 of the
volumetric strain dEv = tr(dE). The homogenized values of the elasticity constants
K = Km and G = GI, and the biot coefficient b = bm are derived in detail in
Section 2.4.4 and Appendix C.
Sub-problem 2
The second sub-problem fixes the displacement of the REV boundaries and measures
the change in stress due to the drained, incremental evolution of the eigenstresses.
Thus, the boundary value problem is posed as,
in V : V . dO 2 = 0
in V: do 2
on
v:
=
(3.7a)
c(z) : de 2 (z) + do-
(3.7c)
du 2 (z) = 0
and the zero displacement condition implies (dE2 (z))v
(3.7b)
=
0. Noting again that the
corresponding macroscopic stress is the mean microscopic stress in V, and using the
mean strain calculated in the first sub-problem as a virtual displacement, we can
write the virtual work of the mixed system as,
dE: dE 2
=
(dE1(z): do 2 (z))v
=
(dE1(z) : c(z) : dE 2 (z))V + (dEi(z) : doa(z))v (3.8)
69
Using the Hill Lemmal and the result of the zero displacement condition, the first
term on the right-hand side vanishes:
(dE1(z) : c(z) : dK2(Z))V = (de(z))v : (c(z))v : (dE 2 (z))V
=
0,
(3.9)
Substituting the strain localization condition for the second term in Eq.(3.8), we
incorporate the relation classically known as Levin's theorem,
(3.10)
dE2 = (A(z) : doa(z))v,
which provides a means to upscale the microstress to the resolution of the REV. Next,
the localization is separated into the volume portions of the pore space and the CSH
solid -
the two phases that contain the eigenstresses. Here, it is advantageous to
begin our analysis for a system that is absent of rigid inclusions, i.e.,
finc
=
0, with
a new porosity denoted by 0o. Thus, Eq.(3.10) is calculated by:
dE2
=
(1
-
q0)1 : (A(z))vh-du*
-
: (A(z))v,dp
(3.11)
Similarly, we can seek an expression for the change in porosity. Because the
average strain is zero, the change in porosity is equal to the change in the solid
1 The Hill lemma is the remarkable result that, for a statically admissible stress field s and a
geometrically compatible strain field e, the work averaged over the microstructure is equal to the
dyadic product of the average stress and the average strain:
(s : E) = (s) : (e)
70
volume fraction. Noting again that the rigid inclusions do not deform, we can write, 2
d2=
-1
(1
o) ((do 2 (z))vh
-
(o(do
= 1 : c- 1
2
(z))V,
hc
-
ld*)
-
dE 2 + (1
o)ldu*)
-
(3.12)
1: C-1
oldp + ol : (A(z))v dp
+ (1 - So)1
(A(z))vcdu* + (1 - Oo)ldu*
Finally, the consistency condition I = qo(A(z))v, + (1
-
Oo)(A(z))vhC provides the
b = 1 - (1 - &)1 : (A(z))v,
(3.13)
necessary relation to the Biot tensor,
and we simplify Eq.(3.12) and Eq.(3.11) for the isotropic case,
d E2
d2
=
(1 -b)do*
-bdp
or
dE 2 = (1 - b)da*6,j - bdp6ij
(3.14b)
p + d*
NP
No'
In the above, 1/NP = 1
(3.14a)
(-50 1 + b) is the inverse of the Biot modulus with
c
respect to the fluid pressure and 1/N'* = 1 : c
(-150 + b) is the inverse of the
Biot modulus with respect to the CSH eigenstress. These two parameters quantify
the pore volume change caused by an increase in the pressure or eigenstress while the
macroscopic strain is held constant. As a result, and in the absence of rigid inclusions,
we have demonstrated that
1
NP|5,nc=O
1
No'*\I fco
1
N\fcO'
(3.15)
The more general case, for which finc > 0, is derived in Appendix C using the three
level homogenization scheme. Thus, under the self-consistent scheme of choice3 , the
2
The stress in the hydrated cement due to elastic strain is sought as: (1 -
o)(do2(z))Vhc
=
-
dE2 - (# (d'2W) V,
3
We selected a self-consistent scheme to upscale the properties of the Level II microphase (absent
of ridgid inclusions) to the Level III macrophase (containing rigid inclusions; see Section 2.4.4).
71
addition of rigid inclusions rescales the change in porosity and Biot modulus as follows,
d#2 = (1 - fim)d
1
(1 - fic)
=
.finc
N|f1,,>o
N|5,_o
2;
(3.16)
Therein, the pore volume change is simply proportioned to the reduction in compressible volume.
Combining the results of the two sub-problems, we arrive at the poromechanical
constitutive relations for our cement specimen:
d
= 2GdE' + KdEml + (1 - b)do-* - b dp
d= bdEm +
3.3
da-*
dp
+
N
N
(3.17a)
(3.17b)
An Incremental State Equation for the Mass Balance of Hydrating Cement Paste
By tracing the contents of evaporable and non-evaporable (structural) water over
time, the Powers-Brownyard model elucidates the transformation of cement clinker
and water into the cement binder (see Section 2.3). The deformation of the porespace
due to external and internal loadings and the mass absorption of water caused by
chemical shrinkage require the fluid content to be traced in measure of the chemical
and physical morphology of the CSH solid. In the following, we present an incremental
mass balance of water in the capillary porespace.
3.3.1
Mass Balance of the Water
The porespace of cement may broadly be categorized into macropores (capillary pores)
and gelpores (formed in the CSH gel). Using the Lagrangian porosity 0 -
the total
and incremental loading (dp, do-*, dEm)
pore volume at a given hydration degree
72
the fluid mass content is defined as
per unit initial reference volume -,
m= p=fl,
(3.18)
where pf denotes the fluid mass density 1241. Under drained and saturated conditions,
the mass content of an REV obeys,
dm
dt
= MO
-
(Omhyd +
(3.19)
fmsurf)
where the temporal change in fluid mass dm/dt is driven by the difference in the
external water supply, Mo, and the use of H 2 0 molecules in the creation of CSH
( 6 mhyd +
6
rfmsurf)-
In particular, the two source terms refer to (i) the stoichiometric
water demand of the reaction
water
6
mhyd, also termed water of constitution or structural
[681; i.e. the specific mass of H 2 0 required in satiating the chemical require-
ments of CSH, and (ii) the adsorption of water onto the gel pore surfaces
6
msurf
driven by the interaction potential between adsorbed and bulk water in the gelpores.
To elaborate:
(i) The stoichiometric sink term
6
mhyd
refers to the observation that ig of cement
requires between 0.2g and 0.25g of water to produce CSH and CH products.
Translated into volume fractions, this stoichiometric term thus reads:
PC
p=
<
d
di
'hyd
hyd plfcO dOj =t Ohyd dt(3.20)
where pc /pfl = 3.15 is the cement-to-liquid mass density ratio, fco = (1
fnr)
-
fo) (1
-
6mhyd
is the initial cement volume content; where as d</dt is the reaction rate,
which is described by a hydration kinetics law (see Section 2.3.2). For w/c = 0.45
the stoichiometric sink term is on the order of
Ahyd
= 0.27 - 0.33.
(ii) The adsorption sink term &msurf was discovered by Powers
1651 and has recently
been quantified by reactive molecular simulations that traced the state of water
in CSH [70]. The driving force of the adsorption sink term is the interaction
73
potential; that is, the interparticle potential between the water adsorbed on
the CSH gelpore surface and the bulk water in the gelpores, which in good
approximation can be viewed as constant over the hydration process. On the
other hand, given the surface nature of the adsorption phenomena, the rate of
water adsorption is scaled by the change of the surface area of the gel-porosity,
or more generally, by the specific surface area:
6
msurf
pfl
ac-wnH 2 0 M dSG <pfl < dt
(3.21)
surf
dt
where ac-w is the number of C - w bonds per surface; approximated at 2.4
bonds/nm 2 , rH 2 O is the number of water molecules per bond
(
10H 2 0), Mw =
18g/mol is the molar mass of water, and SG is the gelpore surface, which has
been shown to increase almost linearly with the hydration degree [831. A rough
estimate of this term is provided by considering that the specific surface of cement paste is SG/I~ 80 - 300m 2 /g, where P = 2g/cc is the average paste
density. Feldman and Sereda calculated that the specific surface area of a colloidal cement system is around 200m2/g [31]. It should be noted that most of
the measureable specific surface area is due to the gelpores; the macropores contribute little. Thus,
!surf
~ 0.11 -0.41, which means that the surface adsorption
term is of the same order of magnitude as the stoichiometric sink term.
From a mass balance perspective, it is advantageous to separate the chemical
and mechanical changes to the water mass. Thus, the change in the mass content
accounted for in Eq.(3.19) is likewise obtained by considering the differential variation
in the porosity , and the fluid compressibility:
dm
pfl
+ dpn q
pfl
(3.22)
(i) The first term, djp,*,Em = dqo, measures the incremental change in the porosity
due to the morphology of the REV phases -
calculated at constant pressure
p, eigenstress o*, and volumetric strain Em,. For chemically reactive materials,
74
this term is coined the chemical porosity [24] 1271. Both the changes in the
capillary and the gelpore spaces contribute to this term, such that the evolution
of
01p,,*,Em
in function of
is depicted by the dashed contour in Fig. 2-7.
(ii) The second term, dqj measures, at constant hydration degree, the change in
porosity due to an incremental loading of the pore-solid system. More specifically, it measures the deformation of the solid skeleton due to an incremental
volume strain, pressure and eigenstress loading d(Em, p, a*) [24].
(iii) The final term
d#
Pfl
quantifies the mass change due to the compressibility of
the fluid, where, under isothermal conditions,
P
=
g with 1/kf
the fluid
compressibility; i.e. the inverse of the fluid bulk modulus, kfl.
Equating the mass balances described by Eq. (3.19) and Eq. (3.22), one obtains the
state equation for the fluid mass content in the cement under isothermal hydration:
b
dEm
dt
+
1 do-* ! dp
,pfV 2 P
1
+
= k,-__3___(3.23)
N dt
M dt
7f M
Thyd
In the above equation, the first term on the left-hand-side (l.h.s.) of the equation
quantifies the effect on the porespace due to an incremental bulk volume strain dEm
of the REV. The second term on the L.h.s. models the effect of the eigenstress development in the CSH-gel, where N(s) = No* is the corresponding Biot modulus of
the solid matrix (Vh, + Vin,). The third term on the l.h.s. accounts for the change
in pore-pressure and its influence in compressing both the solid matrix and the fluid
in the macro and gel pores, 1/M( ) = 1/NP +
#/kfl.
Finally, the terms on the right)
hand-side of the equation quantify, in order of appearance, the spatial gradient (V 2
of the pore-pressure and the effective sink term. Here, kc, pf, and qfl are the cement permeability, fluid density, and fluid viscosity respectively, while
Thyd =
(~)
is the characteristic time of cement hydration, dictating the rate of water consumption by the reaction. The effective sink term quantifies the combined effect of the
physico-chemical changes to the cement system: the stoichiometric water demand,
the adsorption of water to the gelpore surfaces, and the growth of the solid skeleton.
75
Hence,
3
(3.24)
=hyd
- surf -
3.3.2
/ef
The Simplifying Assumption of Uniform Bulk Eigenstress Development
Once the pressure state equation is applied to the boundary value problem of our
cement sheath, the pore-pressure in the cement sheath varies in direct response to
the restraints and flux conditions of the steel and the rock interfaces. As the cement
undergoes volume contraction, the difference in the resistance of the steel and the rock
produces a radial gradient in volumetric strain and consequently pressure. Addition-
ally, the case of an eccentrically placed casing will guarantee a tangential gradient
perpendicular to the radial direction - due to the non-uniform distribution of hydrating matter around the production well. If, in good approximation, the characteristic
dimensions of the state equation allow the driving forces and pressure be estimated
as uniform, the coupling of Eq.(3.23) to the stress solver in Chapter 4 is vastly simplified without loss of predictive power. More succinctly, we are enabled to apply the
effective bulk eigenstress dE*( )
=
(1- b)do-*( ) - bdp( ) along the boundary contours
without modification. This is because the volume change of an unrestrained annulus
under uniform eigenstress is equivalently achieved by the uniform application of the
.
stress along the (unrestrained) boundaries 4
In the following, it will be shown that the volumetric strain in the eccentric
annulus varies only marginally, despite the significant variations in the radial and
hoop stress components across the domain. Since, the eigenstrain development of
the CSH-gel phase is assumed an intrinsic and uniform property, it remains only to
outline the conditions that render the spatial variations in the pressure negligible to
ensure d(o-*, p, Em) as approximately uniform.
The assumption of a uniform pressure development requires justification through
analysis of the governing scaling relations. The cause of the gradient in pressure is
4
1n the case that the eigenstress is non-uniformly applied to the cement sheath, another approach
must be taken to integrate the local response across the volume of the body.
76
the influx or efflux of water along the rock-cement interface (RC), which is incited
by the consumption of water by the cement. By showing that the cement sheath is
thin with respect to the parameters governing the flow of water through the porestructure of the cement and the rock, it can be shown that the characteristic length
scale of the pressure variation extends close to or beyond the domain of the sheath.
In the case of the eccentric boundaries, the sheath will assume additional variations
of pressure in the tangential direction. However, since the circumference is greater
than the thickness of the sheath (i.e., ir(R2 + R 1 ) > (R2 - R1 )) and the flow is
principally driven by the pressure differential between between sheath and formation,
radial variations generally far exceed tangential variations - 2O9r
>>
Therefore, it
I
r a0
suffices to show that the pressure front extends close to or beyond the RC boundary.
Isolating the pressure components of Eq. (3.23) and non-dimensionalizing, the
diffusion length of the pressure within the sheath is 6p( )
kc(()pflfl,
Thyd(
-
V
_hy.
) is the characteristic time of hydration, and 5
pressure to be modeled as uniform in good estimation.
Here, Ac(d) =
> 1 allows the
It must be remembered
that the permeability of the sheath k, changes with the microtexture of the cement,
and is directly related to the capillary porosity; k
characteristic pore size [891 [801 166].
_
1~F(O)
where 1p denotes the
Thus, near the beginning of the hydration
reaction Ac is large and allows for rapid equilibration of the pressure within the
sheath. At later stages of the reaction, the hydration kinetics slows (i.e., Thyd becomes
large), minimizing further developments of the pressure variations within the sheath.
These dynamics are further investigated by discretizing Darcy's law to model the flow
of water between the formation and the sheath and only considering radial flow:
U' = Afl(p(r = R2, t)
-
PF)
(3.25)
The above Newton coefficient depends on the flow characteristics of the cement and
the rock, Afi =
n7fl
(k+
\l1C
1R
).
Moreover, the discretized expression for the fluid
velocity allows us to relate the volume of the water consumed during the reaction
dQ.""
dt
~
Thyd
)
and the volume of water entering along the RC boundary
77
1.0
0.9A
- -
Pressure
Time ratio
0.8
1.0
-(1-b)o*
0.5-
n n,
/PF
((1 b)o-*- bAp)/PF,
0.2
0.8
0.6
0.4
1.0
Figure 3-3: (a) The pressure evolution is plotted against the evolution of the ratio
between the characteristic times of hydration and fluid mobility. The time ratio is
plotted in log scale on the secondary axis. (b) The evolution of the bulk eigenstresses,
decomposed the stresses acting in the CSH-solid (1 - b)u* and the porespace -bp.
dI2fiu
dt
-MR
flf
2 27r.
Comparison of these two quantities shows that the pressure varia-
tion depends on the two characteristic time scales,
dQfux
2MAlThyd
dQsink
pflR 2 (1 - R / R2)
such that the characteristic time of hydration
acteristic time of fluid influx
mf
Thyd=
Thyd
-
T.6
dt/d competes with the char-
= pnR 2 (1 - R2/R2)/(2MAn). Considering the param-
eters governing the fluid mobility in the cement and the formation and the hydration
kinetics, several regimes of the pressure development emerge. We study these by
considering the output of a sample simulation.
Sample Pressure Output of the Poro-Mechanics Model:
Table 2.1 provides the input parameters for the sample simulations presented in this
study. We chose to restrict ourselves to typical conditions encountered during primary
78
cementing operations. In doing so, several parameters (Afi, PF, and
0) were adjusted
within the range of observable values to allow model resemblance with the pressure
evolution of typical wellbore measurements. Figure 3-3a demonstrates the dependence
of the pressure changes within the sheath on the ratio between the characteristic time
of hydration and the characteristic time of mass exchange
"
(Thyd/Tfl):
At early curing times, the rate of the hydration reaction is fast compared to
the rate of recharge
pressure.
Thyd
Tfl
-
1 - 100, producing a rapid decrease in the cement
Here, kc is large and the water entering the cement sheath moves
rapidly to equilibrate the pressure in the cement,
1
c -+ oc. It is thus the rock
permeability that limits influx of water to the sheath, Afl
-
. Because
Thyd
is small, the initial, rapid decrease in pressure is the dominant mechanism of
the bulk eigenstress development (see Figure 3-3b).
" At a degree of hydration of
-~ 0.4, the water demand of the reaction and the
changes in the pore space are balanced by the Darcy flux into the annulus, such
that a pressure minimum is realized.
" At latter maturity, the rate of the hydration reaction slows, such that the
pressure changes in the cement are principally affected by the influx of wa-
ter
Thyd
rfl _ 10 _ 107. As a consequence, the decreased permeability of the
cement drives variations in pressure.
pressure drop attenuates as
Thyd ->
However, since the chemically induced
oc and the pressure profile must adhere to
the zero-flux condition along SC, it is reasonable to assume that the pressure
gradient is small and concentrated near RC. As kc <
kF, the Newton coeffi-
cient is again well approximated by A = Pfi'Q. Moreover, Fig. 3-3b shows that
the bulk effective eigenstress becomes less dependent on the pressure and more
dependent on the eigenstress developed in the CSH-gel phase.
Accepting the arguments on the uniform development of pressure and the CSH
eigenstress, the state equation in Eq.(3.23) can be simplified to the discrete form
P
Thyd
M d&*
bM (dEm)
=
+
+
d
N d<
d
-Tf
PF
79
(3.27)
PF
Table 3.1: Borehole input parameters.
Geometry
Constitutive Properties
Inner casing rad., Ro [m]
0.1
Casing modulus, ES [GPa]
200
Outer casing rad., R 1 [m]
0.11
Poisson's ratio casing, vs [-]
0.27
0.16
Rock. modulus, ER [GPa]
40(5)
Borehole rad., R2
[M]
Poisson's ratio rock, Vu
Mass Exchange Prop.
0.04
Exchange coef., AF [s/mi
8 x 10-7
PF
0.3
Cement Placement Conditions
Formation pressure, PF [GPa]
where P = P/pF;
[-]
Initial pressure,
po [GPa]
Temperature, T [C 0]
0.04
85.9
are the normalized, scalar quantities of the pressure and
eigenstress of the cement sheath.
3.3.3
Sample Model Output: Comparison with the DownHole Pressure of an Oil Well
The red curve in Figure 3-4 shows data for the pressure evolution near the bottom of
an oil well5 . As this data is proprietary, the curve has been smoothed and normalized,
such that the dimensional information cannot be recreated. The figure serves only
to demonstrate the ability of our model to reproduce down-hole pressure dynamics.
Hence, our simulated pressure curve is plotted atop the data in blue.
Firstly, we recognize distinctive pressure variations prior to the onset of the
cement solid percolation, which substantiates around t = 14 hr. Initially, the oil well
is filled with drilling mud, equaling the hydrostatic pressure of the formation. As
the slurry is pumped up the annular region, the pressure quickly increases due to the
applied pumping load, the drag forces along the annular walls, and the difference in
densities between the two fluids. Allowed to set, the incompressible cement slurry
begins exchanging water with the formation, causing a net reduction in the cement
paste mass. During this process, the height of the annular cement column drops.
'Source: Schlumberger-Doll Research Center.
80
5
1.2
1.1-
4
1.0
-3
0.9
9 0.8
2
0.7
0.6
0.5.
0
-
modeled pressure
-
measured pressure
time ratio
5
15
10
t
20
[hrs]
Figure 3-4: A comparison between the model simulated pressure and the pressure in
the pressure in an oil well. As the pressure data is proprietary, the curve has been
smoothed and the input parameters to the model have been omitted.
However, this drop in height is not enough to substantiate the fairly linear pressure
drop observed between t ~ 3 hr and t ~ 14 hr. Instead, the downward flow of
the cement paste induces vertical drag forces along the boundaries that relieve the
down-hole pressure 1601. It should be recognized that our model does not consider
the pressure changes caused by these shear forces (in Figure 3-4 a line was fit for
appearance); we are interested only in the stresses succeeding percolation. Thus, all
simulations in our work initialize the pore-pressure at the adjacent formation pressure.
As was recognized in Figure 3-3a, the temporal pressure variations within the
sheath find direct correlation to the hydration rate. The initial severe drop, gradual
level off, and eventual recovery of the pressure are attributed to the exponential decay
in the reaction rate. At a degree of hydration
= 0.57 it is expected that the water
demand by the reaction and the changes in pore space are balanced by the Darcy flux
into the annulus, such that a pressure minimum is realized.
81
3.4
Chapter Summary
The first section of Chapter 3 discussed the origin of the eigenstress development
in the CSH solid. Stated succinctly, mesoscale simulations showed that the mean
separation of the CSH spheres was greater than the equilibrium separation, causing
net-attractive interactions between the particles. Hence, tensile eigenstresses develop
in the solid phase at constant volume that are a function of the packing density r1.
After incorporating the recent insights into the nature of these self-equilibrating
stresses, we produced a mass balance of the water in an REV. Here, changes to the
H 2 0 mass were calculated for the chemical demand of the reaction, the adsorption
of water onto the gel-pore surfaces, and the change in the pore space due to the
growth in hydrated matter, and related to the physical distortion of the pore space
due to eigenstresses (dp, do-*) as well as a prescribed regular displacement along the
REV boundary. Thereafter, we placed the model of our material element into the
setting of the cement sheath, which hydrates along an inner impermeable steel barrier
and an outer permeable rock barrier.
It was discovered, by investigating several
key characteristic length and time scales, that the pressure in the sheath remains
approximately uniform. The chapter was concluded by demonstrating the ability of
the pressure state equation to reproduce the down-hole pressure evolution of early-age
wellbore field data.
In the succeeding chapter, we couple our state equation to the momentum balance
of the cement sheath under the mechanical boundary conditions of the steel and rock.
82
Chapter 4
Stress Developments for Concentric
and Eccentric Steel Casing
Placements
This chapter of the thesis presents a detailed account of the stress and displacement
fields which have been derived analytically for a cement sheath under eigenstress
development. After the placement of the cement slurry during the primary cementing
operation, net attractive eigenstresses develop in the CSH solid gel. After percolation
of the solid skeleton and upon generating compressibility in the system1 , these tensile
stresses cause the cement to shrink and produce bulk stresses that risk fracturing the
sheath.
Placed between two circular boundaries, we link the poromechanical model introduced in Chapters 2 and 3 with the stress state due to the boundary constraints.
Two scenarios are considered: i) The case of a steel casing placed concentrically with
respect to the wellbore hole, and ii) the case of a casing placed eccentrically with
respect to the wellbore. In the case of the eccentric geometry, our solution employs
spectral methods in the complex plane that require the stress and displacement states
to be approximated by truncating Laurent series.
'Before percolation, the cement slurry is incompressible, b = 1, such that the development of
eigenstresses in the CSH gel are not experienced at the bulk scale: (1 - b)do-* = 0.
83
The mechanical solutions assume a linear elastic material behavior, and they are
thus linearly related to the pressure and eigenstress driving forces.
Consequently,
we can linearly superpose the solutions of an unrestrained, reacting specimen and a
restrained, inactive specimen. With models that relate the cement stiffening behavior
and the pore-pressure changes to the stress evolution of the wellbore geometry, the
coupled pressure state equation, Eq.(3.27), is solved incrementally to track the bulk
stress as a function of the degree of hydration.
Critical stresses at risk of impairing the sealing function of the liner will serve as
input to the fracture mechanics model in Chapter 5.
4.1
An Introduction to the Method of Complex Variables for Problems of the Plane Theory of Elasticity
Herein, we will make extensive use of the method of complex variables to solve solid
mechanics problems in two-dimensional linear elasticity. The method was pioneered
by Muskhelishvili [58] and was expanded upon by England [28]. Here, we provide a
brief description of the relations relevant to the derivations of the stress states and
fracture energy release rates of the cement annulus.
4.1.1
Derivation of the Airy Stress Function in Complex Variables
In two-dimensions, an elastic material free of inertial and body forces is considered
statically admissible if the stress tensor is divergence free and symmetric. In other
words, a static stress field in a Cartesian coordinate system satisfies the equations of
84
equilibrium,
aX+ a
=0
ax
ay
(4.1a)
09EY +
""=0
Ox
(4. 1b)
ay
where E
is the stress tensor in index notation, respecting the symmetry Eg
=
Eji. These relations are necessary and sufficient to define two functions A(x, y) and
B(x, y), such that
a = -E
ax
aY
A
EX
(4.2a)
S-
aax
= Ex
(4.2b)
=0
09X1y
a2Bs
a
0
(4.3a)
Oy
ay
Eyx,
and one recognizes
a2Aa2A
EXX
2
aaaxay
2B
(9X x + ay
8E 8 ,a
aE+ ax
Oy
ax
It follows that
a2A
=
ayB.
-
axay
OXOy
(4.3b)
-
Herein, it becomes convenient to define a potential,
U(x, y), called the Airy stress function, such that A =
8yU
and B = 9xU. The
stresses are related to the Airy stress function by
2
U
a2
a2
u
= EaX
(4.4a)
=E
(4.4b)
u
aa=
--EY =-EYX
and the biharmonic property of U is revealed V 2 V 2 U
(4.4c)
=
0; here, V 2
=
X
+
OYY
is
the Laplace operator. Due to the single-valued and continuous nature of Eij up to
its second derivative, U must be single-valued and continuous up to its fourth-order
derivative.
85
Moving on, the constitutive relations between displacement and stress for plane
strain are expressed as:
(2
K--G)
3
ExX=
EYY=
K-
E2X = G (
ay
aG
+(K+--G', (4.5b)
(9X
3
ay
4
(K+-G)
3
__
j+
+
Dn
(4.5a)
,
ax
3
ay
(4.5c)
ax
where K and G are the elastic bulk and shear moduli of the body, and un
and uY are
the displacements in the x- and y-directions, respectively. The above equations can
be recast into the more convenient form,
EX = A
ax+(A+2G) ", (4.6a)
ax
ay
EYY = A
+ (A+ 2G)
ay,
(4.6b)
Exy = G (and+ an3
ay1
(4.6c)
)
ax
where we have made use of the constant of Lam6 A = (K -
2G).
Utilizing the relations
above in conjunction with Eq.(4.4a) and Eq.(4.4b), it can be shown that
2G an
ay 2
2Gany
au2
ax
ay
-
ax
V2U
(4.7a)
A V2U
2(A + G)
(4.7b)
2 (A
+ G)
Next, we wish to develop the Airy stress function in the complex plane, and show that
it can be resolved by two analytic potentials o(z) and x(z). As was proceeded by
Muskhelishvili [58], one can introduce a function P
aU
2G-
ax
an
2Ga
ay
a2U
---
ax2
+
2U
aaU
=--+AP
aV2
86
=
V 2 U into Eq. (4.7), such that
A+2G
P
2(A + G)
A+2G
(4.8b)
2(A + G)
(4.8a)
One may further define a conjugate function
conditions, 0_P = DyQ and iByP = -
Q
that observes the Cauchy-Riemann
Q.
Remark:
For a complex valued function defined by the complex coordinate z
=
x + iy, the
Cauchy-Riemann conditions are summarized by the following single equation:
=f(z)
0
(4.9)
where an overbar denotes complex conjugation (i. e. f = x - iy) 2 , and the complex partial derivative is referred to as the Wirtinger derivative.
If we define a
function in the complex plane f(z) whose real and imaginary parts are given by
f(z) = P(x, y) + iQ(x, y), the Cauchy-Riemann equations ensure its analyticity: The
function is complex differentiable and can be expanded into a power series.
Continuing, we define the integral of f(z) as follows,
(z) =p + iq = 'Jf(z)dz
(4.10)
where the 1/4 has been introduced for convenience of notion in future expressions.
The analyticity of ib(z) asserts Dp
=
i9yq = 'P and
0 yp
-Dq =
Q.
After
inserting the above relations into Eq.(4.8) and integrating, the local displacement
functions are recovered as
2G'~-2G,
=
2GuY=
U
+2G)
x + 2(A
(A+G)P
+ CX
(A + G)
ax
&U
W
Dy
+
2 (A + 2 G)q~
(A + G)
q + Cy.
(4.11a)
(4.11b)
The constants C, and C, measure rigid body displacement and are irrelevant in the
2
The following important distinctions should be pointed out in the notation of complex functions.
For a complex valued function f(z) = p(x, y) + iq(x, y):
f(z) = p(x, y) - iq(x, y)
f() = p(x, -y) + iq(x, -y)
f(z) = f( ) = p(x, -y) - iq(x, -y).
87
determination of the stress field. Since both p and q are harmonic, P can be expressed
as:
P = V 2 (px + qy) = xV2p + yV2q + 2 O+2
8x
ay
= 4
ax
(4.12)
and we find from the definition V 2 U = P,
V 2 (U -px - qy) = 0
(4.13)
U = px + qy + g(x, y)
where g(x, y) is a real-valued harmonic function. Now, as R [7ZA(z)]
=
px + qy, it is
quickly realized that the Airy stress function may be written simply as
U = R [-Mb(z) + x(z)]
11
U = 2[(z)
(4.14)
+ X(z) + z4(z) + x(z)]
where we must define R[x(z)] = g(x, y). The real parts of f4b(z) and x(z) have been
extracted by adding their complex conjugates to the expression and halving the result.
4.1.2
The Kolosov-Muskhelishvili Equations
With a simple expression of the Airy stress function at hand, we can seek representations for EiZ and ui in terms of the potentials 4)(z) and x(z). Returning to Eq.(4.11),
the relevant partial derivatives are given as
(z) + -fb'(z)
(Z)~' ~ i z'}-VXz + X'( Z) + _4(z) + z_' (z-) + ')(41)
('Ox I2
2k
ay
where axz = 1,
2
[-4I(z) + ZV'(z) + X'(z) + Db(z) - zb' (z) - x(z)]
ayz
=
(4.15a)
(4.15b)
i, a2- = 1, and O.J = -i, and where a prime denotes the
derivative' d/dz. Adding the two components above as 9xU + h8yU and noting the
3
1t should be remembered that d/dz = O/z + a/dz. However, for analytic functions, obeying
the Cauchy-Riemann equations, a/DT= 0.
88
relation to the displacement in Eq.(4.11), we find,
(DU .DU\
2G(u,, + zu.) =
jj)
A ++ G
=
=
2(A +2G)
O + i OU
Dx
Dy
'I(z)
((A +
+ 2)p(Z)
G)
-
zV(z)
-
(4.16)
q1(z)
WA(z) - z1'(z) - X1(z)
where the substitution T (z) = X'(z) has been made and K =(A+3G)/(A+G)
=
3-4v.
The result in Eq.(4.16) allows the x- and y-components of the displacement to be
treated as a single complex function and was first presented in a similar form by
Kolosov
[49].
The displacement vector along with the representation of the stress field
to follow, act as the basis for the analysis of mechanics problems in two-dimensional
elasticity using complex variables. The formulas will henceforth be referred to as the
Kolosov-Muskhelishvili equations.
Before explicitly defining the complex functions that define the stress components, we draw attention to another important result:
++
=
f1 + if,
= 1(z) + zv(z) + T(z)
(4.17)
which has a relevant mechanical interpretation. It is the resultant force due to the
stress applied normal to the arc connecting two points pi and P2:
u
) =zD
ax
In the above, n = n(x)
+
ay
2 Eijnx)
+iEijnY)ds +C.
(4.18)
P
in(y) is the unit vector defining the positive normal to the
arc. It should be understood that the resultant force above does not depend on the
path traversed from pi to
P2,
and is unique up to an arbitrary constant C. In this
thesis, Eq. (4.18) is used to ensure traction continuity between rigidly bonded material
regions4.
Taking the partial derivatives of Eq.(4.17) with respect to the Cartesian coordi4
For a more detailed derivation of Eq.(4.18) see Refs. [58] or [28].
89
nates, the complex functions for the stress components emerge as
Exx + i
EYY - iE2,
=
02U
azXy
ax2
-
02U
(4.19a)
&x~y
+
(4.19b)
2
where
EXX + iEZY = O(z) + O(z) + z'(z) + O(z)
(4.20a)
EYY - iEXY =
(4.20b)
(z) + O(z) - z#'(z) -(z
and we have set O(z) = <'(z) and O(z) = T'(z).
4.1.3
The Kolosov-Muskhelishvili
Equations in Polar Coordi-
nates
As our cement liner is bounded by two circular contours, the wall of the steel casing at
the interior and the wellbore hole at the exterior, mechanics solution are best resolved
in polar coordinates. For the displacement vector, the following well known relation
is established between the Cartesian and the polar displacement components,
(4.21)
Ux + iUy = (ur + iUO)ei,
where r and 0 are the radial and angular components, respectively.
ourselves that z =
x+ iy
We remind
= relo where ej0 = cos(O)+i sin(O). Consequently, Eq.(4.16)
can be rewritten as,
.
2G(u, + iuo) = e-'O rA(z) - zVb'(z) - 1(z)]
90
(4.22)
In order to write the stress components in polar coordinates, we recall the transformation equations for plane stress:
2
ZrrEr
=
-
" Z cos(20) - Ey sin(20) (4.23b)
Eoo =
YY +
+
2
" + EY/I
_
2
2 EY cos(20) + Exy sin(20)
2
Z
-
2
(4.23c)
2 E" sin(20) + Ex cos(20).
Er= - E
(4.23a)
2
One recognizes by manipulating Eqn.(4.23) that the following relations emerge:
Err + EOO = Exx + EYY = 2 [O(z) +
#(z)]
(4.24a)
EOO - Err + 2iErO = 2e2 0 [z#'(z) +
0(z)]
(4.24b)
By adding or subtracting the two expressions above, the Kolosov-Muskhelishvili formulas can be written in polar coordinates:
Err - ZErO
E 00 +
=
ZErO =
z
_O(z)
z
z
O(z) + #(z) + zq'(z) + -V(z)
z
O(z) + #(z) - z#'(z)
-
(4.25a)
(4.25b)
Additionally, an important derivative of the displacement vector can be calculated:
2G
-_ +
)
=
Kb(z) - #(z) + z#'(z) + -(z)
(4.26)
This final relation allows the displacement boundary conditions for a circle to be
written in terms of
#
and V rather than their integrals.
91
4.2
Elements of Poromechanics: A Three-Phase PoroComposite Cylinder under Eigenstress Loading
4.2.1
Poromechanical Constitutive Relations
As the cement annulus is governed by the physiochemical evolution of the hydrating
matter, the solution is framed within the theory of poromechanics. Linking the incremental solid and pore-pressure changes to the bulk scale, the constitutive relations of
the steel, cement and rock are sought as:
dEm(r, 0) = Ks dEm(r, 0)
dm
m(r, 0) +dp =KdEm(r, 0) +(1 - b)(du* +dp)
z CC
z ES
(4.27a)
z E R
(4.27c)
d = {S, C, R}.
(4.27d)
(4.27b)
dEm(r, 0) + dp = KRdEm(r, 0) + dp
dSi (r,0) = 2Gd(dE 7 - dEm6nij)
Here, upper case symbols denote bulk parameters and states, while lower case symbols
denote the subsystem parameters and states. Thus, Kd( ), and Gd( ) denote the bulk
modulus and shear modulus of the respective domains of the steel S, cement C, and
rock R, though we omit explicit indication of properties describing the cement due
to their prevalence in this text. b( ) is the biot modulus of the cement. The stresses
have been separated into their volumetric (dEm) and deviatoric (dSij) parts.
The bulk elastic modulus K( ), the bulk shear modulus G( ), and the Biot coefficient b( ) of the cement are obtained by considering the microtexture and morphology
of the subsystems and vary in function of the degree of hydration,
. The early-age
behavior of the relevant cement phases and the appropriate upscaling relations have
been detailed in Chapter 2.
As our solution procedure is placed into a poromechanics framework, consideration of pore pressure effects must be given not only to the cement, but also to the
rock. It is quickly realized that this consideration has a fundamental effect on the
92
traction boundary condition along the rock-cement interface.
4.2.2
The Boundary Conditions
In this chapter of the thesis we link the incremental bulk eigenstress evaluated at
a constant degree of hydration, dE*I
= (1 - b) * d-*k - bdpk , to the stresses de-
veloped due to the confinements of the rock and either (i) a concentrically and (ii)
an eccentrically placed steel casing. While the solution to the linear elastic stress
development for the concentric case can be adapted to the well-known problem of
a thick-walled pressure vessel under uniform boundary loads (e.g., see Ref.
[84]),
an
analytical solution for the eccentric case with elastic boundary conditions has yet to
be described. It should be observed that the eccentricity of the casing causes the steel
and rock to produce a tangentially varying resistance to the bulk volume changes of
the cement. For both cases, the linear elastic solution must satisfy the momentum
balance V - a = 0 with the following boundary conditions:
" Constant pressure (or no stress) applied to the inner surface of the steel casing,
t(n = -er, r = Ro) = Cn.
(4.28)
* Traction and displacement continuity along the steel-cement interface,
t(n = er, r = R1 )] = 0
(4.29a)
u(r = R 1 )j = 0.
(4.29b)
* Traction and displacement continuity along the rock-cement interface,
[t(n = er, r = R 2 +
[u(r = R 2 + Ae)j
=
e)] =
0,
where Ae is the magnitude of the eccentricity.
93
0
(4.30a)
(4.30b)
e
A zero far-field effective stress condition,
(o-(r
4.3
-+
(4.31)
oc) + p1) = 0.
The Stress State in a Cement Sheath with a Concentrically Placed Casing.
Formatina
Ce ent
Sh ath
AfG K(
RO
f
)
/
)
G(
\4
XsRR
R
Figure 4-1: A diagram of the wellbore geometry for the case of a casing placed
concentrically w.r.t the hole. The cement sheath is bounded at its interior by a steel
casing and at its exterior by a geologic formation. The inner, circular boundary of
the steel is located at a distance RO from the origin. The interfaces SC and RC are
located at distances of R1 and R2 from the origin respectively.
A cement sheath hydrating under saturated conditions, having uniform and
isotropic properties, and confined by material boundaries that are uniform and isotropic,
94
may be reduced to a one-dimensional problem. Radial symmetry reduces the stress
tensor and displacement vector fields to functions of r. In particular, the hydrating
liner, undergoing bulk volume changes, can be evaluated as a thick-walled cylinder confined at its interior and extcrior by elastic springs; the springs represent the
equivalent elastic stiffnesses of the steel casing and rock formation.
The solution
to a cylinder under uniform boundary loading is well known (see for instance the
book by Timoshenko [841) and is readily applied to the problem at hand. Hence, the
momentum balance needs to be satisfied incrementally or in rate form in order to
accommodate the growth of the material system:
V
OE
at
= 0
,)=10
(i)+()
a2E
Ot
at
1 a(M+3zz)
r
0
-0
3
= raKaGDr +Kt -GDr
(433a)
In the above, the constitutive relations of the two-dimensional effective stress tensor
can be developed in the form
8(+a2E
a
&(Zrr + P)
(K
>)4 Drr+ (K -2G
at
-
at
where Dij =
aEig/at
o
3)3)d
a(EO A
zz
d19*+Z
dt +p
K( 2 C) Drr+(K A4GDoo +d(Z +p)(43b
K - -3
K - -G
3
dt(4
Drr +
K - -G
3
Doo E*
d
dt
(4.33a)
3 b
(4.33c)
stands for the strain rate and dE*/dt is the prestress rate; it is
the rate at which the effective bulk eigenstress develops in the cement. The prestress
95
rate is written as:
d(E* + D)
dl
6E*
=
=
(1- b)
du
(1 - b) dt
dp
+ d=
dt
dt
do*
E* + 6P
(4.34b)
dt
dt
(4.34a)
In
6P = dt
(4.34c)
Noting that Eqn.(4.33) assume plane strain conditions, such that dEzz/dt = 0, the
remaining components of the strain tensor are found via:
Err
Dr
E00= U'r(r)
Ero = 0.
(4.35)
and it is well known that a velocity solution of the radial displacement can be found
by assuming an expression of the form,
aUr - (Cir + C r
2
at
(4.36)
).
The boundary conditions in Eqn.(4.28)-(4.31) are posed as follows:
9 Along the inner surface of the steel casing:
dErr(I = Ro)
(4.37)
dt
e At the interface between the steel and the cement SC:
(rI=R)
dp = dE-(r = RI)
rr
+
dt
dt
dt
du+ (r = R1 )
du-(r = R1
dt
dt
(4.38a)
)
d
96
(4.38b)
. At the interface between the rock and the cement RC:
dE,(r = R 2 )
dp
dE-(r = R2 )
dp
+
=
+
dt
dt
dt
d
du+ (r = R 2 )
du-(r = R2
dt dt(4.39b)
dt
dt
)
(4.39a)
o Finally, the change in the effective far-field stress in the rock formation,
d-g
dt
dt
oc0)
+
dp
dt
6-=0
requires the far-field displacement field to remain undeformed dUr (r
(4.40)
-
oc) /dt =
0.
In the above, the superscript + indicates the limiting value of the stress (resp. deformation) in approaching a boundary from the left side, where the left side is defined
with respect to a counter-clockwise traversal around the contour. Similarly, the superscript - indicates the limiting value of the stress (resp. displacement) in approaching
the boundary from the right side. As the pressure in the cement drops upon hydration, it must be recognized that the formation in proximity to RC experiences a
pore-pressure change similar to that of the cement. Thus, the pressure drop dp/dt
must be accounted for in Eq.(4.39a) when resolving the traction along the wellbore
hole aE- /&t. In the event of a micro annulus formation along RC or in the limiting
case that the rock stiffness tends toward zero, this condition asserts that the effective
stress along the outer interface of the cement sheath is equal to the pressure p.
The principal strains are calculated from Eq.(4.35) and Eq.(4.36), via,
2
)
(4.41a)
Doo(r) = (C1 + C 2 r- 2 )
(4.41b)
Dr,(T) = (C1 - C 2 r-
97
and insertion into the constitutive relations of Eqn.(4.33a) and (4.33b) results in:
rr =
(2K + 2G C1-2G
3
at
=
2K + --G
at
+ 6E*
(4.42a)
C1 + 2GC + 6 E*
(4.42b)
)r2
3 )r2
azz
2K - 4-G
at
C1 +6E*
(4.42c)
3)
Next, we notice that the linear response of the interfaces allows them to be
replaced with pseudo-springs of equivalent stiffnesses, as depicted in Figure 4-1.
Specifically, the effective stiffnesses measure the material resistance of the steel and
rock due to a unit expansion of the SC and RC interfaces.
They have been de-
rived in Appendix A utilizing the displacement boundary conditions in Eq.(4.39b)
and Eq.(4.38b), and are denoted by xs (steel) and xR (rock). It follows that the
constants C1 and C2 can be solved from the two traction boundary conditions in
Eq.(4.38a) and Eq.(4.39a):
+ 6E* = xs(C1R1 + C2 R-)
2K +
G
C 1 - 2G
2K +
G
C1 - 2G2 +6
*
=
-XR(C1R2
+
(4.43a)
C2R-) - bR 6 p
(4.43b)
As mentioned above, the drop in pressure along the adjacent lying formation has been
incorporated, where we will assume that the rock is incompressible with respect to
its fluid pressure (i.e., the Biot coefficient of the rock is set to bR = 1), such that
Eq.(4.43b) becomes:
2K + 2 G C1
-
2G
+6E* + 6 p = -xR(CR
2
+
C2 R- 1 )
(4.44)
To solve the system of equations, a two-step approach will be employed to separately
account for the effect of i) the bulk eigenstress development in the cement and ii)
the pressure drop in the adjacent lying portion of the formation. Hence, C1 (C2)
has corresponding contributions of Cl* (C27) and C6P (C 6P) that may be added
98
linearly:
C1 =Cg*
+ C6P
(4.45a)
C2 =C
+ C2'.
(4.45b)
Their explicit solution reads
2G(1
6E*
C 26
C 6*
2
R6E*
_ Z2) +
XS + Z2XR
6*
(4.46a)
XR - XS
_
2(4G + 3K)(XR + zS) + (1
-
3(XR + xs)
W )(3XR(XS - 2K) - 2G(3xs + XR) + 4G(3K + G))
2
(4.46b)
and
( .,;R + 2G ) -c2 c
C
R 2p
-
<XS
(2G + 6K + 3xR)
3(xR + xs)
(4.47b)
*
XR
1P
c,
(4.47a)
*
c6 = =
where u = R 2 /R 1 . This allows the stress rate solution to be recast, noting separately
D(Zrr
+ P)
at
= (2K+
G
cl*
-2G
r2
c6E*+
+
the effects of the prestress rate the and pressure drop in the formation:
1(2K
+ 2G)
6 -2G
1 sp
c6+
a(EOO + p)
at
= 12K
+
+
2
2G)
c,
+2G
c
1] 6E*+
(2K+
2G) c6p+
2G Q
)
(4.48a)
(4.48b)
at
1(2K
4G) c
+1
6E*+
L(2K
- G) c"p+ 1 1p
(4.48c)
and the mean stress rate can be calculated by,
M = (2KcbE* + 1)6E* + 2Kc'p6p
at
(4.49)
For consistency, we express the solution above in form of the Kolosov-Muskhelishvili
formulas given by Eqn.(4.25a,b). Radial symmetry enables us to simplify their ex-
99
C
+1
6P
pression as follows,
O(Zrr
+-P) =
=
2K + 2G
3
C1 -- 2G C2+
2K + 2 G
3
C1 + 2G
r2
6E*
(4.50a)
+ 6E*
(4.50b)
)
at
20(z) - ei 2 0v(z)
&(Zoo+p) = 2#(z) + ei20v)(z)
=
)
at
r2
where the absence of shear stresses makes O(z) real-valued. We recognize immediately
that
O(z) =2K +
C1 + 6E*
(4.51a)
2GC 2
(4.51b)
Z2
)(Z) =
Finally, remembering that 1(z)
G
f
=
#(z)dz,
'(z)
=
f
(z)dz, and K = 3 - 4v the
radial displacement solution is confirmed:
f
at
-___
2G
-
z'(z)
-
(z)(4.52)
.
au,(r)
+
[Cir
.r
4.3.1
Sample Model Output: Stress Evolution for a Concentrically Placed Casing
The effective radial stress E,, + p within the sheath shows particular sensitivity to
the dynamics of the pressure.
We see from Fig. 4-2 that the maximum compres-
sive stress at both the interior and exterior boundaries coincides with the pressure
minimum around
= 0.4 (for the related pressure curve see Fig. 3-3). As the pres-
sure increases, the radial stress begins to increase and eventually enters into the
tensile regime. Toward the beginning of the reaction, the effective stress is strongly
dependent on pressure changes, as the solid cement matrix has yet to gain stiffness.
Clearly portrayed in Fig. 2-11, the overall bulk modulus advances nearly linearly with
and builds much stiffness by
~ 0.5. In consequence, upon advanced hardening
100
0.25
0.15
-
0.10
0.05
stress along SC (ER =40 GPa)
--
stress along RC (ER=40 GPa)
- -
stress along SC (ER =5 GPa)
-
stress along RC (Et=5 GPa)
-
0.00
8
0.20-
A
0.10
--
r
-0.05
-0.10
-0.15
0.15
0.05
0.00
0.2
0.4
0.6
0.8
1.0
0.05
- --
0.2
0.4
0.6
0.8
1.0
(b) effective hoop stress
(a) effective radial stress
Figure 4-2: (a) The effective radial stress and (b) the effective hoop stress development
along the interfaces of the steel and cement (blue) and the rock and cement (red) is
plotted in function of the degree of hydration.
The input parameters have been
summarized in Table 2.1, and the scenarios of a stiff (solid lines) and soft (dashed
lines) are plotted.
(
> 0.5), CSH eigenstresses more drastically influence the sheath's mechanical stress
state; whence the boundary restraints induce tensile loads.
Additionally, the eventual equilibrium of the pressure to that of the formation
pressure places the system into a state of residual tension: Initially, during the period
of accelerated hydration, a dramatic pressure drop is imposed on a relatively incompressible slurry; much of the system is still composed of a fluid mixture of water and
clinker grains. As the matter hydrates, the growth of porous CSH gel increases the
compressibility of the system. As the reaction rate slows and upon re-pressurization,
the compressible cement matrix is placed into a state of residual tension. This phenomenon further increases the effective tensile stress in the solid skeleton; beyond
effects purely due to the eigenstress in the solid and pore space.
In fact, were the
system to remain incompressible in course of the pressure evolution, no additional
additional Cauchy stresses would be created. Instead, the hardening sheath is most
vulnerable to micro-annulus formation during the period of pressure recovery.
The top panels in Fig. 4-3 demonstrate the influence of the system's permeability
and the rock stiffness on the radial stress along SC and RC. Clearly, greater tensile
stresses develop for more permeable systems with a less movable rock boundaries.
101
However, it must be pointed out that the pressure for systems with low A has not yet
recovered p < PF when
=
1; it is expected that the pressure will increase even once
the cement has set.
The rock's Young's modulus, ER, has a pronounced impact on the generation of
effective hoop stresses Eoo + p within the sheath. Both for stiff and soft formations,
the pressure drop due to early hydration induces immediate tensile stresses along the
inside of the sheath; the reduction in volume constricts the sheath around the steel
casing. A short compressive regime is witnessed along RC for stiff formations, as
the exterior bond opposes the inward displacement. Most noticeably, we recognize
that robust adhesion to a stiff formation near the late hardening stages promotes
substantial increases to the sheath's final-state hoop stress at both r
=
R1 and r
= R2;
in the event of a micro-annulus formation along RC we expect the risk of radial
cracking to be minimized due to the substantial reduction in the tangential stress.
The bottom panels in Fig. 4-3 displays the influence of the rock Young's modulus
ER and the Newton coefficient A on the hoop stress generation. In general, a more
compliant formation reduces the hoop stresses in sheath.
As a final remark, we
recognize that Eo is greater along the inner circumference (true nearly everywhere
in the A - ER space), prompting radial crack initiation along SC.
Because the solid skeleton of the cement grows during the pressure evolution, the
stress development in the sheath is path dependent. This means that the incremental addition to the homogenized bulk eigenstress at a constant degree of hydration
depends on the instantaneous volume fractions of the pore-space and the CSH solid.
4.4
Stress State in a Cement Sheath with an Eccentrically Placed Casing.
In the event that the steel casing is placed eccentrically with respect to the wellbore
hole, the stress and displacement states lose their radial axis symmetry. For large
eccentricities, the maximal radial and hoop stresses encountered along the two inter102
SC: (Err + Ap)/pF
RC: (Err + Ap)/PF
0.2
0.1
1e-06
0.1
1e-06
0
0z
0
1e-07
1e-07
-0.1
-0.1
I
1e-08
10
30
70
50
SC: (Eoo + Ap)/pF
-0.2
1 e-08
10
20
30
RC: (EoO + Ap)/pPF
0.3
0.25
1 e-06
0.2
1e-07
0.1
10
20
30
0
0.15
1 e-07
0.1
0.05
1e-08
ER [GPa]
0.3
0.2
0.05
1e-08
-0.2
0.25
1 e-06
0.15
0z
0.2
0
10
ER
20
30
[GPa]
Figure 4-3: Three-dimensional plot of the effects of the fluid exchange coefficient A
and the rock Young's modulus ER on the radial stress (top row) and the hoop stress
(bottom row) at complete hydration. Stresses are plotted for SC (left column) and
23 GPa.
RC (right column); E( = 1)
103
faces is significantly amplified. Here, it is important to quantify the added risk of
impairment, such that a maximum allowable offset is defined during primary cementing. With an engineering solution in mind, this section derives analytical solutions
for the stress field in an eccentric wellbore and constructs maps that indicate the location and maximal amplification of the principal stresses along the steel-cement and
rock-cement boundaries. This is done for a range of rock-to-cement stiffness ratios,
allowing for quick access to an accurate estimate of the added risk of impairment for
on-site conditions.
4.4.1
Constructing Coordinate Systems for the Steel, Cement,
and Rock Domains:
The first step calls for the calculation of the stress field in an eccentric geometry. We
utilize the method of complex variables, introduced by Muskhelishvili [581, and map
the cement interfaces onto circular contours centered at the origin. This allows for
the complete description of the contours by their radial coordinate. In doing so, it
is advantageous to evaluate the stresses and displacements of the steel, cement, and
rock in separate coordinate systems, the z-, (-, and v-planes:
" The steel casing is placed at the center of the physical system in the z-plane.
" The eccentric boundaries of the cement annulus are conformally mapped onto
concentric circles in the (-plane.
" The geologic formation is mapped to the v-plane by translating the z-plane in
the direction opposite the eccentricity, Ae.
4.4.2
The Bilinear transformation
A conformal mapping preserves the magnitude and sense of the angle between two
linear elements during transformation from the domain to the image region and is
univalent.
The relevant conformal mapping of the cement domain is the bilinear
transformation. By appropriately specifying two parameters ao and a,, the function
104
places the origin of the reference coordinate system - the z-plane - at the center of
the steel-cement interface
(SC =S
C) and the origin of the mapped system - the
(-plane - at the center of two concentric circular contours, the mapped steel-cement
(RC
(SC = S n C) and rock-cement interfaces
= R n C). Thus, movement between
the reference and the transformed domains is defined by:
w(z)
1+z~c~)
w(()
(4.53b)
ao + z
=+
=
1
+ ce 1z
(4.53a)
=Oa
( + aoa1)((45b
=
1 + ao(
'0
where w and w are the mapping and inverse mapping functions, respectively. Within
the transformed domain, C
=
pe" allows the boundary contours SC and RC to be
defined solely by the radii p = R 1 and p =R
2,
and the angular component varies
between 0 < 19 < 27r. The radii of the transformed interfaces are calculated from the
geometry of the physical system by
Rf
4R =
- 1
V I + (2Rjc1)2
2
2R=
2Rja1
-
1,2
(4.54)
where
(Ry
a,
O=
2
-
-
R2) 2
-
2Ae(R? + R2) + A(4.55a)
(4.55b)
.
2R1
and Ae is the eccentricity.
The representation of the region of the cement sheath in the reference and
mapped systems is compared in Fig. 4-4. It is apparent that the bilinear function
maintains the shape of any circular contour during its mapping. Nonetheless, it is
critical to observe that the equally spaced rays emanating from the origin of the
concentric geometry in Figure 4-4b provide a curvature in Figure 4-4a, such that
105
(-Plane
z-Plane
RC
SC
P
1
0
Figure 4-4: Contours in the reference coordinate system (z-plane) are mapped via the
bilinear transformation into a conformal image ((-plane); the eccentric boundaries SC
and RC are mapped into the concentric boundaries SC and W.
their intersections with SC and RC cluster along the thin section of the sheath. This
will require additional attention since the boundary points along SC
RC
i-4
'-4
SC (resp.
RC) must be expressed as 6sc(0) (resp. GRc( 6 )) in order to write the condi-
tions of stress and displacement continuity.
4.4.3
The Kolosov-Muskhelishvili
Formulas for the Mapped
System
For the poromechanics boundary value problem at hand, we choose to evaluate the
stress and displacement states using the Kolosov-Muskhelishvili formulas defined in
Eq. (4.25) and Eq. (4.26). Under conformal transformation, these are rewritten in
106
incremental form as follows:
dE, -idEp)
=
#*(() + #*(() +
*(
(2
2G
w '2j
0
+
=*(() -
4,*((),
=
0*(()
4*(()
#(z)
(w(()) = #*(() and
=
(4.56c)
')(z)
though the asterisk shall be omitted in the subsequent.
commonly proceeded (see Ref.
(d
Cw(
/
#'*(() +
The complex potentials are defined such that
(w(()) =
___
-
(2 W(2)
(du,* + iduy,)
(4.56a)
(2I(*)
[58]),
we choose to seek a solution for
/d
=
As is
and
Vd
{S, C, R}) in form of Laurent series:
00
00
Os=
A
Os (z)
z
=
k=-xo
Bs Zk
(4.57a)
Bk zk
(4.57b)
B z'k
(4.57c)
k=-00
00
00
q#c =
c (z) =
A
>3
k=-o
k=-xo
-1
-2
OR
k
R(Z)
3
=
k=-oo
k=-cc
Because the far-field locations within the rock formation are assumed unaffected by the
hydration dynamics of the cement, dE(IV1
-4
oc) = 0 and duR(IVI
oc), the series
-+
in Eqn. (4.57a) and (4.57b) neglect terms with positive powers. With the above expressions at hand, the boundary equations can be simplified into power series of z, C, and
v. In doing so, it is necessary to expand the functional expressions in the transformed
Kolosov-Muskhelishvili formulas, ((2 p2 )(w'(() /w'(()) and ((2 /p 2 )(w (()/ w'(()) , into
series expansions. Utilizing the mapping function w((), it can be shown that,
2
=
-(ocl +cai)p2 + (2a0ac
and
107
+ 1)p 1 (
-
aop
2
2
(4.58)
Z=E l nQ(
(2
2p
if n=0
2
+ (n -
-2n"
4.4.4
+ (n + 1)an 2p 2 if n > 0.
2 p_
1)a
(4.59)
Matching the Boundary Contours Using the Chebyshev
Polynomials
The driving mechanisms of the early-age isothermal stress development within the
hardening cement phase are the self-balancing loads due to eigenstress development,
and pore-pressure changes. The total stress in the cement sheath is calculated by
) onto the region d = C of the
linearly superposing the real-valued stress -E*(
boundary value problem.
It should be noted that the formation adjacent to RC
undergoes the same pressure drop, such that the term -dp(
) must be superposed
unto the rock. Complete continuity of traction and displacement ensures that both
the normal and shear stresses generated by the shrinking (resp. expanding) cement
specimen are transferred to the steel and rock. Solving for a divergence-free stress
tensor, the boundary conditions read:
" Traction-free conditions along the inner surface of the steel casing (zo = Roe0 ):
#s(zo) + #s(zo)
-
zo#'s (zo)
-
=
zs(Zo)
zo
0,
r = Ro; 0 < 0 < 27r
" Traction and displacement continuity along SC (resp. SC): (zi = R1 ie
1Zien)
qs(zi) + #s(zi) -
'(1)
(4.61)
=Oc
((1) + #C ()
zi#'s(ZI)
(2 s
20
21
- r-4's(zi)
(1)
Q
O'(1
21 WO'(1
108
1
dE*
(4.60)
'-+
(I =
2Gs 5 sOs(zi) -
2C
1
Z10 (ZI-)i) -_- s(zi)
S
2Gc
=COC((l) -
C(1) -
where r = RI; 0 < 0 < 27
'-*
)2
2
(4.62)
1
( )(I bi'(1)
__
(1 bj(()
C
o'((G) 0'c((1)
7212
'G
p = R 1 ; 0 <,d < 27r.
* Traction and displacement continuity along RC (resp. RC): (V2
(2 =
R2 ei-*
7 2 e2)
O5R(V2)
+ OR (V2)
2
-(V
+ A,e
(G2 ) + Oc ((2) -Z2
2C
2GR
R R(V2) -
OR(V2) -
-
R
(
U) ((2)
(22
= R 2 ; 0 < 0 < 27r
-+
p
=
dE*Jk
2
OR(V2
W'((2)
-22
'C((2)
KCC ((2) -OC ((2) -R2
2Gc
22 W'(G)
=G
where
(4.63)
(U 2 + Aeei20)O/R(V 2 )
I_
dP
V2
'C(( 2 ) - )2
2 W'((2)
'(()
12
-
'02
-jOR(V2)
2
()
2
=c
R (5~V
2) -
R2
22
1
(4.64)
'G
R 2 ; 0 < 9 < 27.
It is then seen that the equalities along the interfaces can be simplified to systems
of equations of the form:
00
CseikO=
k=-oo
-1
k=-oo
= C
+ iC and
Dk
(4.65a)
DiekO
(4.65b)
00
Djeik =
where Ck
Cie ik7
k=-oo
k=-oo
= DR + iD
are complex valued coefficients, and
Ck, DR, and D' are expressed as a linear combination of A, and B,. The hat
over the angle above, b, indicates the argument of the v-plane. As noted above, the
phase angles of z (resp. v) and the mapped coordinate ( do not correspond. Hence,
the required conditions of stress continuity in Eqn.(4.61) and (4.63) and displacement contintuity in Eqn.(4.62) and (4.64) must be matched through an additional
functional relation.
109
The Fourier series in Eq.(4.65) may readily be decoupled into their real and
imaginary components, such that one needs to relate etO to e". Utilizing the inverse
mapping function in Eq.(4.53) one can show for z along SC:
eo = cos(O) + i sin(O)
= 9(19) + Ig(i) + 7r/2)
0 < d < 27
(4.66)
1 ao - (1 + aOai)RieR
R1
-1 + a1R1ie01
Similarly, one can show along RC:
=cos(0) + i sin(0)
e
el
0 < 0 < 27
ao - (1 + aoa1)R2e0
1R2 I
-1
+ a1 R 2 e 0
(4.67)
As a reminder, Chebyshev polynomials of the first kind Tk allow the orthogonal
functions of cos(kO) to be expressed in terms of the fundamental modes cos(O) = g(7),
where
Tk(cos(O)) = Tk(g(i9)) = cos(kO).
(4.68)
and the discrete orthogonality condition for the polynomials reads,
{
0
N
Z Ti(cos(0m))Tj(cos(0m))
=
m=1
7rN
2
)
ir(m -
J
N/2 if i = j
7 0
j
= 0
N
Om =
if i
if i =
(4.69)
m = 1, 2,3, ... , N
Hence, one can relate the reference and mapped phase angles, while maintaining the
ability to express the boundary conditions in terms of power series of orthogonal
110
components. In particular, it can be proved along the respective boundary that
00
C eik9
=
S
knCne
(4.70a)
n DnRe
(4.70b)
n=1
00
DceikO
=
n1=1
where one may truncate the above series to an order N after the desired accuracy has
been achieved. The coefficients are calculated by,
e
1 EN= T(cos(0m))Tk(g(im))
if k = 0
M 0 Tn(cos(0m))T(g(?3m))
if k > 0
=
{
=0 Tn(cos(m))Tk(g(5m))
2 EN=0Tn (cos(m))T(g(dm))
if k
=
0
if k > 0
and k < N.
Now, the boundary conditions may be written without reference to the arguments
of the systems z, (, and v. Upon convergence of the Laurent series in the holomorphic
domains S, C, and R, the stress state is approximated by truncating the series in
Truncation at the nth-mode, (" = pne i", yields a system of iOn + 5
Eq.(4.57).
equations 5 that produce the coefficients Ad and B d for the domains d =
4.4.5
{S, C, R}.
Sample Model Output: The Stress Evolution for an Eccentrically Placed Casing.
Placing the steel casing eccentrically with respect to the wellbore hole creates an
uneven distribution of cement around the casing. Hence the sheath is segmented into
thick and thin portions. Herein, we define the degree of eccentricity 6, = Ae/(R2-RI),
which measures the fractional reduction of the thinnest section of the sheath when
compared with the regular geometry. In order to test the impact of the eccentricity
5
The 10n
+
5 equations result from 5 boundary conditions, where the real components - the
cos(kt9) modes - contribute 5n + 5 equations and the imaginary components - the sin(k0) modes
- contribute the remaining 5n equations. The final boundary condition, the assumption of zero
far-field stress, eliminates the positive powers of OR and 1/)R.
111
0.1 (a)-0.
~10.0
-
0.1 (b)
+
0.+
-C
-0.1
.
-
-0.2
A
thinnest section
thickest section
0.2
0.4
-
-02
0.6
0.8
1.0
1
s/m
x10
A=8 xlI
7
s/m
r
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
1.0
0.3
0.3
(d)
(c)
30.2-
0.2
0.1
0.1
0.0
0.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0
-
9
1.0
Figure 4-5: Panel of (a),(b) the radial and (c),(d) the hoop stress evolution for a
cement sheath with an eccentrically placed casing (6 e = 0.8) as a function of the
hydration degree for the material parameters provided in Table 2.1 and the borehole
dimensions provided in Table 3.1. The plots separate stresses that evolve along (a),(c)
the steel-cement interface (SC) and (b),(d) the rock-cement interface (RC). Thick
(thin) lines correspond to stresses along the thickest (thinnest) portion of the sheath;
colors represent different fluid exchange coefficients between formation and sheath;
ER = 40 GPa.
on stress distribution, the boundary value problem was coupled to the pressure state
equation and was solved incrementally.
Figure 4-5 and Figure 4-6 show the results for the stresses in the radial and
tangential direction along SC and RC at the thinnest and thickest segments of the
sheath for 6, = 0.8. More precisely, Figure 4-5 displays the evolution of the stresses
for low and high rock permeability values; the Newton coefficient A was varied by
over an order of magnitude from 8 x 10-7 s/M to 1
x
10-5 s/M.
In the highly
permeable system, the cement only experiences a marginal pressure drop, as the
112
0.2
0.2
(a)
(b)
0.1
0.1
++
-
0.0
.
0.0
-
s=
section
thickest section
-ER
-thinnest
-
-0~-0.1
0.2
0.4
=5 GPa
8
0 GPa
ER =
0.6
0.8
0.
1.0
0.3 (c)
0.3 (d)
0.2-
0.2
2
0.2
0.
0.6-7
0.1.
0.8
1.0
0.4
0.6
0.8
1.0
0.4
0.6
0.1
-0.1
0.0
0.0--
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
Figure 4-6: Panel of (a),(b) the radial and (c),(d) the hoop stress evolution for a
6
cement sheath with an eccentrically placed casing ( e = 0.8) as a function of the
hydration degree for the material parameters provided in Table 2.1 and the borehole
dimensions provided in Table 3.1. The plots separate stresses that evolve along (a),(c)
the steel-cement interface (SC) and (b),(d) the rock-cement interface (RC). Thick
(thin) lines correspond to stresses along the thickest (thinnest) portion of the sheath;
blue (orange) lines present model output for a soft formation; A = 1 x 10-5 s/i.
formation readily supplies the water that is being consumed during reaction. As
a results, little variation in the pressure is experienced as the cement paste gains
compressiblity and the magnitude of the residual tension in the CSH is curtailed.
As a comparison, the final state of the Cauchy stress in the low permeable system
is heightened by the cycle of drastically lowering and thereafter recovering the porepressure. In both cases, the magnitude of the stresses depends on the local thickness
of the sheath and the thick segment bears a larger load.
An important dynamic between the fluid exchange coefficient and the relative
importance of the eigenstress development is discerned by comparing the simulation
113
03
0.15.
0.20.10-
T
SC:
-
0.05-
6,=0.2
0.1
SC: 6,=0.4
SC:
SC:
-
6,=0.6
6,=0.8
RC: ,=0.2
RC: 6.=0.4
RC: 6,=0.6
RC: ni,=0.8
-
-
-i
i/2
0
-r/2
-7t
7t
7r/2
0
-7r/2
7r
9
0
(b) effective tangential stress; low
permeability A = 1 x 10- 7s/m
(a) effective radial stress; high permeability
A = 8 x 10- 7s/m
Figure 4-7: (a) The effective radial stress and (b) the effective tangential stress at
complete hydration for a wellbore hole geometry with an eccentric casing plotted as
a function of the angular component 0 along the steel-cement interface (red) and the
rock-cement interface (blue); ER = 40 GPa. The line thickness indicates the degree
of eccentricity, which has been varied between 0.2 and 0.8.
0.04
-
0.03
-
SC:
RC: J,=0.2
RC: 6,=0.4
0.02
RC: 6,=0.6
RC: J,=0.8
-
SC:
SC:
6,=0.6
6 =0.8
0.00
0.02
-0.02
0.01
0.0
6,=0.2
SC: ,=0.4
0.2
0.4
0.6
0.8
1.0
-7r
7r/2
0r/2
T
(b)
(a)
Figure 4-8: The shear stress along the SC (red) and RC (blue) interfaces plotted (a)
as a function of the degree of hydration and (b) as a function of the angular coordinate
6 at complete hydration; A = 1 x 10- and ER = 40 GPa. The line thickness indicates
the degree of eccentricity, which has been varied between 0.2 and 0.8.
114
(a)
SC: max(Ero)/PF
(b)
0.2
1e-06
K: MaXk2r9)/PF
0.2
0.18
0.18
0.16
0.16
0.14
1e-06
0.14
0.12
0.12
0.1
0.1
0.08
0.08
1e-07
1 e-08, 10
1e-07
0.06
l-80 30
50
0.06
0.04
0.04
0.02
0.02
le-081
70
-
10
30
50
70
E11 [GPa
ER [GPa
Figure 4-9: Three-dimensional plot, investigating the influence of the stiffness ER
and Newton coefficient A on the magnitude of the maximum shear stress experienced
0.8 and the remaining
along SC and RC. The degree of eccentricity is set at C
2.1.
input parameters are gathered from Table
results for the high permeable systems in Figure 4-6 and the low permeable system in
Figure 4-5 (blue lines). Systems in which fluid is readily exchanged between cement
and formation are more susceptible to the eigenstress generation in the CSH solid.
Here, the pressure drop during reaction is moderated by rapid fluid recharge, such that
the bulk stresses vary according to the increase in o-* and the stiffening behavior of
cement. Consequently, the difference in the relative magnitudes of the stresses along
the thick and thin segments of the sheath increase during the hardening process.
On the other hand, low permeable systems are more heavily dependent on the
uniform pressure evolution in the sheath. This is particularly evident when comparing
the pressure curve in Figure 3-3 to the respective residual radial stresses along SC and
RC plotted in Figure 4-5a and Figure 4-5b, respectively. Here, the residual loading due
to the growth of compressible hydrating matter during the pressure recovery enacts a
uniform stress development along both thin and thick portions of the sheath. Hence,
the curves progress nearly in parallel.
In Figure 4-6, the stress solver was run for a high stiffness and a low stiffness
formation.
As seen in panels (a) and (b), a compliant formation places the radial
115
stress along SC and RC into states of compression and tension, respectively. Such an
occurrence is troublesome for drilling contractors that often choose to add expanding agents to the cement to safeguard against the build-up of tensile stresses; for a
low stiffness formation, hydrating under the development of a uniform bulk eigenstress, the cement sheath is inevitably placed in tension along one of the interfaces.
Nonetheless, the overall magnitude of the stresses is far exceeded for stiff formations.
Interestingly, the hoop stress development with a soft rock interface shows accentuated stresses along the thin section (displayed in panels (c) and (d)). Where a
formation with a high modulus is well able to resist the residual stresses incurred by
the compressible hydrating matter, a soft formation gives way. Instead the stresses
are transferred tangentially toward the thin portion of the sheath. At this location,
the event of debonding along RC is expected to significantly increase the risk of radial
fracture.
To better understand the stress profile along the interfaces, Fig. 4-7 plots the
radial (left panel) and hoop (right panel) stresses for a range of eccentricities as a
function of the angular component 0. For the parameters selected, the thick segment
is imparted more pronounced tensile loads upon increasing the eccentricity.
More
importantly, however, one should recognized that the stresses in the thin segment deviate to a greater degree from the concentric solution than those in the thick segment.
It follows that in guarding against the stress amplification due to the casing offset,
it is of benefit to seek parameters that locate the stress increase along the thicker
portion.
Finally, the loss of axisymmetry engenders interfacial shear stresses not present
in the concentric geometry. The uneven distribution of hydrating mass acts to pull the
thin section of the cement toward the thicker section. Thereby, the cement is sheared
along the interfaces. Here, Fig. 4-8 shows (a) the maximum shear stress as the cement
hydrates and (b) the angular distribution of stresses at complete hydration. We see
that the maximum shear stress is located 450 from the narrowest segment of the
sheath.
Additionally, the shear stress tends to be greatest along RC and reaches
magnitudes up to half the radial stress; this is confirmed for a large space of ER
116
and A values graphed in Fig. 4-9. This addition of shear stress, places the sheath
at risk of mixed mode fracture, and complicates the investigation of where along the
interface fracture will originate. While this thesis does not make an effort to answer
this question, it is a condition important to be aware of.
4.5
Chapter Summary
In this Chapter we coupled the boundary conditions of the cement liner to the pressure
state equation developed in the previous chapter. As poromechanics theory is only
able to reconcile the response of the system due to two mechanical loadings -
(i)
a regular displacement condition along the REV boundary, and (ii) the pressure
acting along the solid-pore interface -
the homogenization using Levine's theorem
was administered at a constant degree of hydration and the pressure state equation
(Eq. 3.27) was advanced incrementally. This implies the path dependence of the stress
developments at the bulk scale. Herein, the stresses cannot be obtained from the final
values of the constituent volume fractions and the eigenstresses, but instead depend
on the mass of the transient state solid system at the time of loading.
Once a poromechanical framework for the constitutive relations of the cement was
constructed, the stress in the sheath was resolved for the geometry of a concentrically
placed casing and an eccentrically placed casing. In general, the stress development in
the sheath showed a strong dependence on the interplay between the time of hydration
and the ability for fluid to be exchanged with the formation. Compressive stresses in
the sheath were accentuated in the case of low fluid exchange coefficients. Of greatest
concern, however, was the recovery of the pressure at the latter stages of hydration.
For eigenstresses developing under low pressure conditions, the rebound in pressure
places the sheath into a state of high radial tensile stress.
The solution to the stress state in the eccentric geometry was derived using
the Kolosov-Muskhelishvili formulas and the technique of conformal mapping. No
analytic (or semi-analytic) solution was otherwise found in literature, such that this
117
novel approach is of industrial benefit in quickly 6 estimating the magnification of
stresses along the interfaces. Additionally, the eccentricity of the casing produced
shear stresses along SC and RC.
With our chemo-poro-elastic stress solver at hand, the chapter to follow derives
the energy release rate for the most critical in-plane fracture scenarios: The debonding
of the SC and RC interfaces, and radial fracture.
6
Numerical approaches, such as finite-element analysis require much attention to the meshing of
the geometry and can be computationally burdensome.
118
Chapter 5
Fracture Criteria
Chapter 2 and Chapter 3 characterized the early-age behavior of cement from its
microphases. Therein, we measured the transient state stiffening and shrinkage phenomena and upscaled them using a three-level homogenization scheme. Moving on to
Chapter 4, our discoveries for the behavior the cement REV were integrated with the
boundary conditions for the wellbore system. It was shown that the coupled behavior
between the eigenstresses -
both in the solid (du*) and the pore-space (dp) -
and
the growth of the solid mass resulted in a path-dependent final stress state. For a
large range of input parameters, a shrinking cement specimen produced radial and
hoop tensile stresses along the interfaces, placing the sheath at risk of fracture impairment. This risk will be characterized in the present chapter by developing the most
critical fracture criteria for the in-plane geometry. Afforded the fracture toughness
of the cement and its interfacial bonds, our work derives energy release rates that
predict the advancement of fracture. In particular, we expound upon the following
fracture scenarios:
o Micro-annulus formation: MA is a small gap that forms between the liner and
the casing or the rock (see cases (a) and (b) in Fig. 5-1). Often it is the result
of pressure and temperature cycles in the production channel of the casing.
However, at early ages, shrinkage and pore-pressure changes are the culprits
of the interfacial tensile stresses. Typically, only partial debonding is observed
119
due to the irregularity of the hole geometry and potential eccentric placement
of the casing. Nonetheless, we appropriate the worst-case scenario of complete
.
debonding to our calculation of the energy release rate1
* Radial fracture: Cement shrinkage causes tensile hoop stresses that concentrate
along the SC boundary. As formation fluid accumulates in the open fracture, it
acts to propagate the crack upward, and by extension, radially outward
[92].
In
our analysis, we calculate the energy release rate for a unit depth of the sheath
due to a single crack emanating along SC and propagating radially outward.
Several other studies have solved similar problems. Bowie and Freese provided
the stress intensity factor for an edge crack in a circular ring with a uniform
tension applied along the external boundary [14]. Delale and Erdogan produced
the stress intensity factor for a radial crack in a hollow cylinder [25].
The
solution closest to the problem at hand, was given by Luo and Chen, who
solved for the energy release rate of a crack in the intermediate matrix of a threephase composite cylinder [52]. Wang and Shen expanded upon their approach
by adding a sliding interface [90]. The solution outlined in the following utilizes
elements inspired by all of the aforementioned works.
5.1
Fracture Mechanics in Porous Media
The fracture approach we herein adopt is linear elastic fracture mechanics, which
requires the evaluation of the energy release rate, defined as:
(F)
=-9
t
(5.1)
where Epot is the elastic potential energy and F is the area of the crack surface, such
that the energy release rate is measured in J/m2 . Consider then a possible fracture
process that occurs under drained conditions
(P
= 0) at a given hydration degree
'In order to account for the increased risk due to casing eccentricity, the stress along the interface
may need to be modified by the stress concentration calculated in Section 4.4
120
Ip,r
4-
(a) microannulus along the (b) microannulus along the
rock-cement interface
steel-cement interface
(c) radial fracture
Figure 5-1: Dominant fracture scenarios for the plane geometry of the cement sheath.
(.
Such a fracture process will reduce the traction vector along the line of the crack
oriented by the outward normal n -
progressed value t+(n) = pn.
from its initial value t -(n)
= E - n to its
This stress release is associated with the release of
potential energy under constant boundary conditions and is yielded by:
-
<9cr (5.2)
(E-n -pn) - dF
(o='p(r)o
2
aF
pn)
aprd<~ir
where uj stands for the jump in displacement as a consequence of the drained fracture
propagation. !cr denotes the criticalfracture energy; this threshold must be attained
in order to substantiate an advance in the crack. Moreover, in order for crack propagation to proceed stably, the derivative of the release rate must remain negative.
This requires the amount of energy released during propagation must decreases as
the crack advances. Hence, the stability criterion reads
S
a2pot < 0
(5.3)
By drawing on the law of energy conservation, one may equivalently regard g as
the amount of work by external forces required to restore the system to its original
physical state, before crack advancement. In doing so, we utilize Clapeyron's formula
W(E) =
2
[T(u) + T*(E)]
121
(5.4)
where the amount of internal energy generated W(E) equals the sum of the work due
to externally prescribed forces T*(E) and externally prescribed displacements T(u).
Approaching fracture from this perspective is well suited to the analysis of microannulus formation at the interfaces. Here, the initial and final physical states correspond to
completely bonded and debonded interfaces, such that the work necessary to restore
the system is readily at hand.
For all fracture criteria, the stress state of the sheath E and the pressure in the
macro-pores (capillary and gel) p are determined by the chemo-poro-elastic model
outlined in Chapters 3 and 4. Thus, it remains only to determine the relevant crack
opening vector.
Stress Intensity Factor
Due to the pervasive use of stress intensity factors in lieu of energy release rates to
investigate fracture problems, Irwin [44] proved a straightforward conversion:
Kp,
=
/9E'
<
(5.5)
Ccr 1
Here, direct comparison to the critical stress intensity factor,
Acr,
of a material is
achieved; an intensive property that quantifies a material's resistance to fracture.
E' = E()/(1 - 0A) is the reduced Young's modulus under the assumption of plane
strain.
5.2
Microannulus Formation
In deriving solutions to the energy release during micro-annulus formation it is useful
to define two equivalent moduli xs and XR for the casing and the rock. These moduli
were previously invoked to calculate the evolution of the stress state in a concentric
geometry (see Section 4.3). We recall that xs is the stiffness of the casing with respect
to uniform pressure acting against its outer circumference. It associates the stress
required to substantiate a unit inward displacement. Similarly, XR is the equivalent
122
stiffness of the formation, measuring the stress along RC required to expand the well
by a unit of normalized displacement. These constants are derived in Appendix A.
5.2.1
Microannulus Along the Steel-Cement Interface (SC)
To calculate the energy release of complete debonding along the interface of the
cement sheath and the steel casing Clapeyron's formula is used to relate the internal
energy to the prescribed boundary stresses. Accordingly, the energy released during
fracture is equal to the mechanical work required to close the crack and bring the
system back to its unruptured state.
The symmetry of the problem allows for a
simple expression of the energy release rate,
gsc
1
2
1
-(E-(R 1 ) + p) ur(Ri)]
2
=
(5.6)
where the dissipated energy gSC equals the loss of energy due to the release of the
prescribed stresses (1/2)(T*(E, p). In the above, E- is the radial stress state preceding debonding and 1[lrl = u- - u+ is the opening caused along SC. In light of
the intended use for engineering applications, the above calculates the energy release
for the drained poro-mechanics material. In this worst-case-scenario, pressurized water enters the crack volume after rupture and works to further extend the opening
displacement. Denoting u+ as the displacement of the steel at R 1 , we find
+
E=-("P)
R,
(5.7)
The displacement of the debonded, inner surface of the cement u; can also be calculated in a straightforward manner. Here, the constitutive equations (4.33) are used
to seek a solution in the form of u; = Cir + C2 /r. Accordingly, the boundary value
123
problem supplies two equations,
K+
=K
K+
C2
(C1 R 2
)
-XR
=K
G
C1--
+ (K
~G) (ci-1
C1 - R
3G
4c
(C2-
-2G
+ (K-
G
C2
(C1
(5.8a)
(C1+
(5.8b)
)
- (Er-,+P)
that define the constants for the radial displacement as
C2
R2
where z
=
(C2
2
6G-3xR
~2G +6K +3>rR
1
(2G + 6K + 3xR) (-(E (R1 ) + p))
2
2 (2G(G+ K) - xR(3K +G)),
-G(2G+ 3xR +6K)
)
Cl
(5.9a)
(5.9b)
R2 /R 1 . Substituting these coefficients back into the displacement relation
reveals the equivalent stiffness of the cement sheath -
the resistance of the exposed
cement surface to an applied pressure:
(R)
Ur
+ P) = ZC(Ri)
_
R,
2xR(3G + (3K + G)
(6G - 3xR)
) + 4G(3K + G)(1 - -o2
2 +2G+6K+3xR
2
(5.10)
)
rr
Hence, we are enabled to calculated the opening displacement along SC by
p)
(,4R)
1
+
N
)R
/R 1
.
iurl = (E;,
(5.11)
Finally, plugging the above into Eq.(5.6), the energy release rate due to the microannulus formation is given by
gsC
-
+ )2
(
+
I R
(5.12)
with (x) = (1/2) (x + Ixl) > 0, indicating the unilateral nature of the crack opening.
124
Microannulus along the rock-cement interface (RC)
5.2.2
Micro-annulus formation along the outer boundary releases the stress that bonds the
rock formation to the cement sheath. Mimicking the approach from the above, the
energy release rate is found by
- (R 2 ) + p) [Ur (R 2 )]
_ *
gRC
(5.13)
where E- +p is the effective radial stress along RC before debonding, and Ur(R 2 )
U is the opening displacement (71
U,-
=
(u-) denotes the displacement at R 2 as one
approaches RC from the interior (exterior)). Here, the radial displacement of the
wellbore hole after fracture is given by
+
(5.14)
(= XR P) R2
Again, the constitutive relations in Eq. (4.33) allow the radial displacement of the
relevant cement sheath to be calculated by assuming a solution of the form U=
Cr
+ C2 /r. In particular, the boundary conditions read
KS
C1R1 +
=
(K +4 G)
-= K+
(C1
-
G) (C1 -
+ (K
+
-
K -
2G)
C+
C ) (5.15a)
G
C + C2) (5.15b)
and the coefficients are evaluated as
6G+3Xs
2G+6K - 3;<s
C2
R1
_
4G 2 (I _ C2)
C2(5.16a)
R
(2G - 3 xs + 6K)(-(E- + p))
+ 2G(6K(1 - U2 ) + 3xs(1 + 3; 2 )) + 3xs (2K + xs)
125
(5.16b)
Whence, the equivalent stiffness of the cement for a pressure acting against its outer
boundary is measured by
4G2 (1
(R 2 )
=
C
-
.2) + 2G(6K(1 - 02) + 3xs +xs( + 3 2 )) + 3xs (2K + xs)
6G + 3xs + (2G + 6K - 3xs)-0 2
(5.17)
Hence, the displacement of the outer surface of the sheath, which once adjoined the
rock, is calculated by
=
(R2
xC
(5.18)
R2.
With the opening displacement at hand, we arrive at the following expression for the
energy release rate due to micro-annulus formation along RC:
gRC
5.2.3
1
2
Rr
+
(5.19)
Sample Model Output: Energy Release Rate due to Interfacial Debonding
The fracture energy release rate due to MA is calculated for the stress evolutions
given in Figure 5-2a, for the cases of a stiff (ER= 40 GPa) and a soft (ER = 5
GPa) rock formation.
The results for 9 and IC as a function of
are displayed
in Figure 5-2b and Figure 5-2c. Risk of debonding is capacitated once Err + Ap
along SC and RC enters into a state of tension at around
~ 0.6 - 0.7. Beyond
this threshold, the risk of fracture becomes a function of the competing processes of
material toughening/increase in bond strength and the stress increase along r
=
R,
and r = R 2 . From Eqn.(5.19) and (5.12), it is seen that g is linearly related to
the build-up of radial stresses and inversely related to the moduli of the separating
materials.
While 9 along SC and RC is comparable at the early stages of hardening, the
disproportionate increase in stress at the outer portion of the sheath promotes a more
considerable increase in gRC. Additionally, because 1/xs is small compared to 1/XR,
the stiffening of the cement (decrease in 1/xC) more dramatically reduces the energy
126
(a)
0.14
0.12
0.10-
- - -
stress along SC (ER =40 GPa)
stress along RC (ER =40 GPa)
stress along SC (ER =5 GPa)
stress along RC (ER =5 GPa)
0.080.06
0.04
0.02
0.00
(b)
0.6
120
--
100
-
~~~~"~~=-
'
.---
- -
0.7
0.8
1.0
0.9
along SC
micro-a nnulus
micro-a
nnulus along SC
micro-a nnulus along RC
80
6040
--
20
0.6
0.7
0.8
0.9
1. 0
0.9
1.0
21
(C)
-
micro-annulus along SC
-
micro-annulus along RC
1.5
1.
0.5
0.0c
0.6
0.7
0.8
Figure 5-2: The evolution of (b) the energy release rate and (c) the stress intensity
factor for micro-annulus formation along the steel-cement (SC) and rock-cement (RC)
interfaces calculated from the results of the chemo-poromechanics solver shown in
panel (a). Solid lines indicate a stiff formation (ER =40 GPa) and the dashed lines
indicate a soft formation (ER =5 GPa).
127
release rate along SC. The stress intensity factor for the case of a stiff formation is
generally greater because larger stresses develop under these more rigidly confined
conditions.
To more thoroughly investigate the effect of the formation stiffness on the energy
release rate due to MA, we plot g at complete hydration for a range of bulk moduli
in Fig. 5-3.
KR
The energy release rate for MA along both interfaces increases
substantially as ratio between rock and cement stiffnesses increases. Interestingly,
though the cement solid experiences bulk shrinkage, both interfaces are at risk of
debonding, even after the pressure drop has recovered 2. Here, it must be remembered
that (i) the initial pressure drop is administered differently along SC and RC in solving
the boundary value problem, as the formation undergoes a similar pressure change;
and (ii) the effective stress is calculated incrementally and in function of the solid
mass. Hence, while du* places SC in radial compression and RC in radial tension, dp
(a pressure increase) places both interfaces into tension. Only for high fluid exchange
coefficients, where the drop in fluid pressure in the cement is quickly recovered, is a
difference in the sign of the effective stress along SC and RC to be experienced3 . This
remarkable result implies that MA is a potential risk along both interfaces, even in
the event that expanding agents are used to reverse the sign of the eigenstress in the
solid skeleton.
5.3
Radial Fracture
A convenient method for evaluating elastic problems lacking axisymmetry is by means
of the Airy stress function U and complex variables. This approach was developed by
Muskhelishvili [581 and has been described in Section 4.1. It has found widespread
application in solving planar problems of fracture mechanics with specific application to the semi-analytical calculation of stress intensity factors for a radial crack in
an annular geometry [6] [14] [85]. Of apparent interest, the solution by Wang and
2
For a three-phase cylinder with a uniform bulk eigenstress in the intermediate matrix phase, if
one interface is under tension, the other is under compression.
3
This was previously noted in Section 4.3.1.
128
160
.
,
,
,
,2.5
140
2.0-
120
C 100
1.5
80
1.0
60
4040.
0.5-
20
-
-
0.5
micro-annulus along SC
micro-annulus along RC
05
2.5
0.0
1.5
1.0
2.0
--
-
KR/Kc
0.5
1.0
micro-annulus along SC
micro-annulus along RC
1.5
2.0
2.5
K1/ Kc
(b) stress intensity factor
(a) energy release rate
Figure 5-3: The (a) the energy release rate and (b) the stress intensity factor for microannulus formation along the steel-cement (SC) and rock-cement (RC) interfaces are
plotted for different ratios of the rock and cement bulk moduli KR/KC. The bulk
modulus of the of the cement, the pore-pressure, and the radial stress along the
interfaces have been calculated by the chemo-poromechanics solver and are evaluated
at complete hydration.
Shen [901 provides the stress intensity factor for an embedded radial fracture in the
intermediate matrix of a three-phase composite cylinder. However, in their approach
the inner inclusion is simply connected and does not contain a hollow interior as is
the case for our steel casing. Thus, to our knowledge no analytic solution exists that
has been adapted to the elastic boundary conditions and geometry prevailing in the
casing-sheath-formation system. Below, we follow the approach employed in many of
the works of Erdogan (see for instance [25]) and seek a solution to the energy release
rate of the system by first finding the Green's function for an edge dislocation in the
cement sheath. A boundary condition that relates the shear stress along the steelcement interface to the slip displacement jump is adopted from the solution by Wang
and Shen to simulate varying degrees interface damage Eo = x(u (R1 ) - u-(R 1 )).
Here, the damage parameter X defines the rigidity of the shear connection. Once at
hand, the Green's function may be integrated along the line of fracture to calculate
the crack opening displacement.
In calculating the energy release rate due a radial crack, the behavior of cement as a multi-level, chemo-poromechanics material is blended with linear fracture
129
mechanics. Hence, we construct the following solution procedure:
1. Connect the cement behavior to the chemo-poromechanics solver: Under reaction the cement paste stiffens, solid eigenstresses develop, and the pore-pressure
evolves. Utilizing the solution in Section 4.3, the tangential stress for a sheath
.
in good quality may be written in the form Eoo + p = I, + I2 /r 2
2. Construct regions of analytic continuity: A well-known simplification allows the
regions of steel, cement, and rock to be continued analytically across the SC
and RC interfaces. This allows the stress and displacement states to be written
in terms of a single potential (Dd.
3. Solve for the Green's function of an edge dislocation: By constructing the potentials bd as Laurent series (see Appendix B), the Green's function for an
edge dislocation is solved in a two-step process: First the stress due to an edge
dislocation in an infinite medium is calculated along SC and RC. Second, this
stress is superposed onto the boundary value problem of the cement sheath.
The boundary relations allow the coefficients of the Laurent series to be solved
as a decoupled system of equations.
4. Integrate the Green's function along the line of the crack: The Green's function, which represents a delta discontinuity in the displacement of the sheath,
may be integrated along the line of the radial crack to resolve the crack opening displacement. Hence, the unknown strength of the discontinuity along the
crack p(t) (herein termed the dislocation density) is solved in a singular integral
equation by relating the crack surface stress to the tangential stress calculated
in step 1.
5.3.1
Connection to the Chemo-Poro-Mechanics Solver
The expression in Eq. 5.2 of the energy release rate allows one to employ a linear
elastic approach derived through the method of superposition. The cement sheath,
undergoing stress and pressure developments, is split into two subproblems: (i) A
130
Table 5.1: Restrictions on the parameters 1, and I2 that ensure an effective hoop
stress in the tensile regime.
Sign of parameters
Restrictions
Max. extension of
Shape of function
enabling radial
cracking
crack
entire sheath
I1 > 0
12 > 0
convex
none
I1 > 0
I2 < 0
concave
crack initiation along
< 0
12 > 0
convex
I1 < 0
12 < 0
concave
I1
RC (uncommon)
I1
+
12
>
0
entire sheath is in
compression in the
0-direction
min
R,
,R
2}
no extension
continuous cement annulus, absent of defects, with hoop stresses E00 evolving due to
the uniform development of eigenstresses and pressure changes, and (ii) a sectionally
holomorphic annulus with a crack of length a, where an effective stress Eoo + p acts
on the crack surfaces. The effective stress must be opposite in sign and equal in
magnitude (less the added pressure term) to the stress evaluated in the defect-free
system. This establishes a hydrostatic pressure along the crack lips after superposing
the two subproblems, consistent with the assumed drained nature of the fracture
process. The presence of a single radial crack in sub-problem (ii) introduces a loss
of axisymmetry, posing the challenge and novelty of the approach. Additionally, it
should be noted that the propagation of a crack solely alters the stress state of (ii).
The poromechanics model detailed in Chapters 2 and 3 accounts for the interactive effects of hydration and fluid kinetics by evaluating the state equations incrementally. Thus, the macroscopic effective hoop stress in the uncracked cement annulus is
sought in classical form
EOO + P = I1 ( ) + 12( ),
r)
R, <r <R2
(5.20)
and is related to the effective eigenstress development (dE* + dp = (1 - b)(do* + dp))
131
by
I1
/
{1+2 [K(
J O
=
1 + 2 [K( ) +
+
2
(Qt)= {2G( )c
+
GQ)1 c
)+-G(
L3<
(_)}
dZ7
(5.21)
G() c()}
<
( )} dZ d*
(5.22)
{2G(d)c *(<)}
where it has been remembered that the radial displacement solution implemented in
our boundary value problem can be sought in the form u,( ) = C1()r + C2 ( )/r.
Within the context of our problem, I1 + I2 (Ri/r)2 must be positive at r = R,
in order for crack initiation to find opportunity.
Emperical evidence and chemo-
poromechanical simulations evince predominant radial fracture initiation along SC.
This is due to the disparity in the magnitude of the elastic moduli of sheath, casing,
and formation 1111. The contraction of the sheath around a stiff casing localizes the
greatest hoop stresses along SC (see Figure 4-2 for the stress at complete hydration).
Moreover, if a tension-to-compression transition exists, the crack may not propagate
beyond the inner tensile region. The conditions on I1 and I2 to entail a tensile regime
are summarized in Table 5.1.
In the sequel, we connect the effective hoop stress calculated in Eq.(5.20) and
resolve the crack opening opening displacement in sub-problem (ii). This enables us
to calculate the work required to advance the radial crack, and thus obtain the energy
release rate.
5.3.2
Method of Continuation
The primary drawback in constructing solutions for elasticity problems using the
Kolosov-Muskhelishvili formulae is that lengthy and complicated expressions typically
emerge. For this reason, one often seeks devices that simplify the solution approach.
England [281, in his book, presents the particularly favorable method of analytic
132
continuation, which allows the two potentials required to resolve a mechanical state
to be reduced to a single potential.
Consider an infinite plate with a circular hole of radius r. If we denote the region
of the of the plate by V- and the region of the hole by V+, then for every point z in
V- we can calculate an image point r2 /2 in the region V+. It is observed along the
boundary contour z E (V- n V+) that z
=
r 2 /-.
Now, the stress and displacement
of the plate are defined by Eq.(4.25) and Eq.(4.26), where the potentials <b(z) and
'(z)
are valid in z E V-. Consequently D(z) and '(z)
may be defined arbitrarily
for z E V+, and we can, for instance, express I(z) in V- in terms of 1D(z) in V+.
In the analysis of our wellbore liner, this scenario describes the case of the material
region of the rock. Here, the rock, with elastic parameters GR = ER/2(1 +
R
1/R)
and
3 - 4iR, borders the outer wall of the sheath and extends to infinity (z E R).
The regions defining the rock and its image (z E R+) are portrayed in Fig. 5-4c. If
we continue the function 4)R(z) across RC as
4)R(Z)
= -zA)(R/z)
-
'(R'/z),
z E R+
(5.23)
the resultant force along RC is given by (see Eq. 4.17):
f, + fy = 4)-(Z) - (D+(z).
(5.24)
The displacement is obtained as:
2GR(u, + iny) = KR4 -(Z) -
The superscripts
+ and
- -
'+(z).
(5.25)
indicate the direction from which the boundary is
.
being approached 4
Similar arguments follow for the annular domains of the steel casing (z c S) and
the cement sheath (z E C). Because these domains are multiply connected, two image
4+ (-) will be used to denote the left (right) side of the boundary contour with respect to a
counter-clockwise traversal.
133
regions a piece must be constructed. Take for example an annulus that is bounded by
two concentric circles of radii r 1 and r 2 that produce a region V. By continuing the
region of analyticity across z
to z = r2/r2 (z
=
=
i (z = r 2 ), the thickness of the annulus is extended
r2/ri) and the annulus is defined for V- U V U V+. Thus, we can
define
- '(r2/z)
IJ(r2/z),
-
if (r
/r2)
z|
r1
(Dd(Z) =
;
dz'~
for the interior V+ G
2/Z) -
if r2 < Iz|I
1XIF~r2/)
(r2/r2 <
d = S, C
(5.26)
r2 /ri)
continu-
(r2/1
IzI < ri) and exterior V- E (r2
z
ations of the steel V = S and the cement V = C. The two extended material domains
and their boundaries are depicted in Figures 5-4 a,b. Though of little mathematical
consequence, one should observe that several domains overlap (e.g., S- overlaps C).
By inverting the relations in Eq.(5.26) it is recognized that
TId(Z),
resolved in
the interior region, has two expressions:
-diI(z)
5d(Z)
(r/z)
d- (r'/z) - Yd (z)
(5.27)
in V
in V
Consequently, in order to ensure that the mechanical state is well defined,
TId(z)
must
satisfy the compatibility condition:
(2)-
5.3.3
(2/(r/z) +T
) (z) =0.
-
(5.28)
z
Green's Function for an Edge Dislocation
The Green's function, in application for the cement sheath, solves for a divergence
free stress state with a delta inhomogeneity positioned at z = t. Here, the delta
inhomogeneitiy describes the physical analogy of an edge dislocation. From a mathematical point of view this means that the traversal around the inhomogeneity in C
produces a jump in displacement that is proportional to the strength of the dislo134
R"
IS
Ro
R,
(a) steel casing
-----
R2
R,
C
W2
(b) cement sheath
RR
R2
R+
(c) rock formation
Figure 5-4: Regions of continuation.
135
cation. However, a single dislocation enacts a mono-valued jump, regardless of the
radius by which we circumscribe the dislocation. For our application in solving for
the fracture energy release rate we wish the discontinuity to be indicative of the crack
opening displacement. Hence, a gradient of dislocation strengths ferred to as dislocation pile-up -
sometimes re-
is positioned in the sheath to recreate the shape of
the crack. The great utility of the Green's function is that, once at hand, it can be
used to construct any arbitrarily shaped crack in the region V = C.
Again, a solution will be sought by superposing two sub-problems. In the first
sub-problem, the stress state due to an edge dislocation embedded in an infinite,
homogeneous medium (cement) will be denoted by E(l). By calculating the stress
E(1) along the location of the two interfaces of the sheath, SC and RC, the response
of the adjoining media can be determined by applying an equal and opposite loading
to a holomorphic 5 system. The stress resulting in this second sub-problem will be
denoted E(2), such that the superposed result,
Edis
-
E(1)
E(2),
(5.29)
produces the stress due to an edge dislocation for the particular boundary conditions
of the sheath.
For the purposes of our fracture analysis, the Green's function shall solve the
mechanical state of the cement sheath (i.e., produce expressions for <D(z) and 'I(z))
for an edge dislocation with a Burgers vector of unit value Jb2 + ibl = 1 placed
at z = t. Because the desired result is a radial crack along a single ray (where 0
is constant) and the displacement jump shall occur perpendicular to the crack, the
result is simplified by assuming the position of the dislocation along the real axis
a(t) = 0 with a Burgers vector pointing in the y-direction b, = 0.
Under these
simplifications, the potentials for the first sub-problem (an infinite cement medium)
5Holomorphicity implies single-valuedness in S, C, and R.
136
are well known and given as 1721,
(Z)
-
J(z)
=
(Z
-G(bx+ iby)
-- r7(K
G
log(z - t)
i7r(K + 1)
z
+ 1) Z -- t
Tr(K + 1)
G(bx- iby)
i7r(K + 1)
log(z - t)
(5.30a)
G
lo~
)=7(K +
1)
lo(zf)
zNj
(gz
t)-z - t)f
(5.30b)
where
3 - 4v has been chosen for the conditions of plane strain. It should be
K=
noted that 1(z) and T (z) are singular at z = t where z E C, and that additional
singularities exist in the image regions at z = R1 /t where z C C+ and z = R/t
where
z E C-. Substituting the potentials into
fX
+ ify = 1-(z) - D+(z)
(5.31)
and noting that 1(z) is defined by Eq. (5.26) for the continued regions, the resultant
force f E (' - n dz along the interfaces is given as:
{
- (z) (l
G
(+z)
log(z - t) + log (z-
-
log(t) +
+
log(z - t) + log (z-R2/t
-
log(t) +
+
)
)-AI/t
Z-R/t
z along SC
z along RC
(5.32)
Next, it is remembered that any constant term in the above does not alter the stress
state. Additionally, with the foresight of developing 4(z) into a Laurent series, only
the principal value of the expressions is of importance. This allows the superposed
result to be calculated by:
I
disdz
J
(1) + E(2)
(2 )(z) + Q (log(z - t) + log (z-/t
( 2 )(z) +
Q (log(z - t) + log
(Z-
137
z along SC
+
+
_
z along RC
(5.33)
where
=(R!- Rt 2 ) t3, and 72 = (Ri
-=
2-
Rit2/3).
Laurent Series
As was done in the analysis of the system with an eccentrically placed casing, a solution to the boundary value problem will be sought by developing
<P(2
(z) into a
Laurent series6 for the three material regions d = {S,C,R}. It must be remembered
that the continuation of the casing, sheath, and formation produces additional domains. Consequently, a total of 8 series are required to define the mechanical state
of the oil well system7 . In general, these can be written as
(p2 (Zc)
n=
a+Zn
if z E V+
an z
if Z E
n=0a-
V
if Z E V-
n"
and we define the coefficients for the three material regions d
a+ = A+
= An;
aa+ = BBan =B;
a = B; for the cement
a+
= CZ;
a-
=
(5.34)
A-
a. = Cn
=
S, C, R as
for the steel
(5.35)
for the rock.
The dash across the summation symbols in Eq.(5.34) indicates that the zeroth-order
term in the series is omitted from the sum. These terms describe rigid body displacements that are inconsequential to the stress/deformation states of the regions. As the
remainder of the analysis is largely concerned with the response of the boundaries to
the dislocation singularity, the indication
()
to denote the second sub-problem will be
omitted unless required for clarity, and all quantities shall refer to the cement sheath
unless s or R indicate reference to the steel or rock.
The poles in the physical and image regions due to the dislocation are described
'Appendix B provides a brief introduction to Laurent series.
73 series to represent <b(z) in C+, C, and C-, 3 series to represent <bs(z) in S+, S, and S, and 2
series to represent 4bR(Z) in R+, and R.
138
in Eq.(5.33); they are the terms that describe the stress state that must be superposed
onto the boundaries SC and RC. As we have opted to resolve the mechanical state by
the orthogonal terms of the Laurent series, the effect of the poles must similarly be
decomposed into powers of zn in order to measure the contribution of the frequency
e"O to the solution. One may readily access the following important power series
expansions,
log(1-
z)=
-
for jzj < 1
(5.36a)
for jzj < 1
(5.36b)
n=1
1
1
00
Zzn
Z
n=O
Using these relations and recognizing that zI < t and IzI > R'/t along SC and Iz| > t
and jzj < R
/t
along RC, we can recast the resultant force along the boundaries as
Edisdz
=
<(b(2)(Z)
+
(
+ E(dz
0(_1 [
+
(z_
(,)n +-1 E
(R )n
[
+
( )f
z along SC
-
2 (t)n+1
zn]
z along RC
(5.37)
With the necessary series expressions at hand, it remains only to define the boundary
relations that allow the unknown coefficients A+, An, A-, Bj, Bn, B-,C ,andC to
be solved as a system of equations.
Boundary Conditions
As usual, the boundary conditions ensure the traction and displacement continuity
along the SC and RC interfaces, and ensure a stress-free inner surface of the casing
and a vanishing far-field stress in the formation. However, because the radial crack
emanates from SC, the bond in proximity of the crack origin is necessarily damaged.
In this location, we wish to model the interfaces as sliding with respect to one another,
while nonetheless maintaining non-zero shear stress. A capable solution is adapted
139
from the work of Wang and Shen [90], where the shear stress is imposed proportional
to the tangential displacement jump:
E
zi = Rie
(zi) = E (zi) =xTUo] = x(UO (Zi) - U0(Zi))
(5.38)
Thus, the parameter x measures the rigidity of the shear connection. The asymptotic
cases x = 0 and x
-
oc assume a completely sliding and a rigidly bonded interface,
respectively.
Having continued the regions of definition for
across SC and RC, we may
4Dd
write the boundary conditions in the following way:
e Traction-free conditions along the inner surface of the steel casing (zo = Roe' 0 ):
E- (zi)dz = JI (zo) - %s (zo) = 0,
r=RO;
0<0<27T
(5.39)
. Traction continuity along SC (z,1 = R ie):
SE(zi) + iE+(z)dz=
f
E-(zi) + 1E;- (zl)dz
(5.40)
(zi) - D-(zi) = D-(zi) - @+(zi)
* Radial displacement continuity along SC (z, = Rieo):
Ur,(zi) = U-,(zi)
{z
1 R
[Ks(D (zi) + D- (zi)]} = 22G
(5.41)
{Z-1
[K4D(zl)
* Jump in the tangential displacement along SC (z 1
EgO(zi)= E-(z1 ) = X(
S{4'I-(zi) - 4b'+(zi)
I
=
{4 '+(zi) -
=
+
2i
2Gs
+(z')]}I
Rie):
(zi) - uf+(zi))
'()
=
2C
Xe.-
{
{I
Zi
, [K-(Zi) + 4+(z)]
[Ks 4(zi) + P -(zi)]
(5.42)
140
e
Traction continuity along RC (z 2 = R 2eio):
I
E+(z2 ) + iE%(z 2 )dz =
+(z 2 ) - (
J
-,(z 2 )+ iE -(z 2 )dz
(5.43)
r
(z2 ) = 4Fj(z 2 ) - 4I(z 2 ),
= R2 ;
0 < 0 < 2wr
o Displacement continuity along RC (z 2 = R2C 4):
)
uf,.(z 2 ) + irO(z2 ) =u (z 2 ) + 11U7(z2
20
)
r
r'(2) +'U
2[R(Z2) ' I z2
[WP+ (Z2) + 4)-(Z2)] = 2GI[KR4D-z2)
+
RR
2GR
R+(Z2)]
r=R;
1
0<0<27
(5.44)
* Zero far-field effective stress condition in the formation. This boundary condition truncates the positive powers of the Laurent series representation of
1
R(z)
in z E R and the negative powers in z E R+ in order to ensure a finite value at
z = oo and z = 0, respectively.
The displacement vector along SC is divided into its real and imaginary parts in
order to ascribe conditions on the the radial and tangential deformation separately.
In order to avoid decomposing the exponential e' 0 into its cosine and sine terms, we
isolate the real part of Eq.(5.41) by adding the complex conjugate to both sides and
(z) + ( - /)-(z) + z
(z)
(r
-
by simplifying the expression using Eq.(5.40):
In the above, the two bimaterial constants are given by q =
(5.45)
Gs++
KSG Gs
GsGs
and 0 = KGG
For the link between the tangential jump in displacement and the shear stress along
the interface, we isolate the imaginary component of Eq.(5.41) by subtracting the
141
complex conjugate terms on both sides. This yields
S['s+(z) +Fs+(z) + (7 =
+'s+
D'+(z) -
)Ps (z) -
Z)
[+(z) +
(n - b)%s(z)
(5.46)
4[-(z) -- @-(z)
where X = 2GRs
2GGS
Matrix Coefficients
The task that remains is the assembly of a system of equations that solves for the
Laurent series coefficients in 4)d; d =
{S,C,R}.
Before moving to compose these
matrices, we reduce the number of coefficients for the steel casing by applying several
relations a priori. In particular, the compatibility condition written in Eq. (5.28) links
the coefficients for the steel domain and its image regions by,
2R 2/Z2
s(Rj/z) - @s(R/z) +4(z)
= 0
z
An- RA + (R - R )(-n + 2)A-n+2 = 0
(5.47)
Similarly, the traction boundary condition (5.39) along the inner surface of the steel
imposes the relation,
A+ = An.
(5.48)
This reduces the previous relation to an expression for the coefficients of the interior
image region,
An- =
R2n
An -
(R 21R"
- RR 2)) (-n
+ 2)A-n+2.
(5.49)
Thus, it suffices to solve An for the steel.
It should be recognized that, in general, each of the boundary and compatibility
equations only links coefficients of order n to coefficients of order -n + 2 (unlike the
previous solution for the stress state in a sheath with an eccentrically placed casing).
This fortunate result allows us to provide an explicit expression for the matrices
142
that define the coefficients A,, B+, B,, Bn, CZ, and Cn. In conjunction with the
compatibility condition for 1(z) for the cement,
2
<D(Rf /z)
2
R2-R
D'4)(z) = 0
- (Ds(RO/z) + -1
(5.50)
(
-
the previously defined boundary conditions are solved by the following relations:
A1
A2
B1
B+
+
(
o The coefficients for terms of order n = 1 and n = 2 read as
B1
= (A ) 1(by);
B2
B-
B-2
C1
C2
=(A2)
(b2g);
(5.51)
SAll other coefficients of order n > 2 and -n + 2 < 0 are calculated by
An
A-n+2
Btn
Bin+2
Bn
=
(An)- 1 (bn)
(5.52)
B-n+2
Bn
B-n+2
Cz
)
C-n+ 2
where A 1 , bi, A 2 , b 2 , An, and b, are provided on the following page. This completely
defines the Green's function for an edge dislocation in the annular region of the cement
sheath.
143
(A 1 ) =
+2W2
1
-11G
- 1]
1
0
0
s/G
R 2 (R -R2)
0
0
0
(
0
0
0
0
-1
1/G -11GR
-1
(
(A 2 ) =
1
--
-1
W4) -Xl
0
X~
-
0
0
0
Q
(b)T =0
0
-3t
-
]
0
0
0
0
(5.53)
-1
1
11G -1GR
4
0
R2
1
i/G
R 4-1
0
2
n -- 0
)T =
1-W4
XR s+w4]+2(
)
2(1-W2)
4- rs
0
0
0)
0
1
(5.54)
(An)=
(2 - n)R( 2 -2n)
(2n-
2
-~~~~~~ ~
_
~
Z2)
n ,I-
(2
l) [n(1 - Z2) _
I + (6 -
(1_
1
2) - -- --
0
I
Tr
-
- .2n
1
72
>
1
I
(4-2n)
-
- ~ - ~ - - - ~ -2
2-2n>
F
-1
0
0
~~
o
0
0
0
0
0
0
I
0
I
0
-
0
1
0
I
-
R (-- -(1-C2)(2--n)
0
I
o
2
.2n]
I-+
-1
-
1
-
-----------------------
+ R
R
2n[(n+X(O-3r)1}
2
(-2-X)
-
2
2
0
L
0'
0
---- - - - - 0- - -- - - - - - -- - - - - - - - - - - - - - -i----
0
0I
0
I
1
4-
-I
0
-1
0
1
0
I-
0
1/G_
0
-1/GR
0
KIG
0
IIG
0
-KRIGR
0
0
I---
-I
0
-
0
1
-1
o0
I
-I
-
i
-1--1
-
0
0
I
I
(2-n+X(3--n))
0
0
X8R(2-2n)
-X
0
0
n) (1-.2)(2-n)(n+X(O-7))
-
-
[n-2-X(3-7)]
2
-
nw
2
2 2
R
X+n[n-l-X(-77)]
o
-
i
0
0
0
0
-
oI
i
i
R2n\ -L
0
I
I
0
I
R
R(
0/G
0
0
0
0
L2n)
-
4
1-n
(n-2)(R2- R
-
n(R -R
2)
I
1
0
L _ __0
1
L
0
_- _ .__- - L
-- R 2 4-2n)
-
oi
0
0
0
0
(5.55)
2
(b)T
In the above
=rKl
u
17
-y
_a
2n-4
n n-2
R( nt
2
)
R 2n-4
(n-2)t-2j
0
0
Q
[
+
_
--
n-
the expressions for the parameters r; = 'Gs+G
we restate
convenience,
= R =. For
2GG
2SG3GS'
rsG+Gs+1
(Ri - Rlt 2/t, and
'Y2
2R
R~ 2 /t
0
0
0
-GSGs
0)
(5.56)
=
XRs
Singular Integral Equation for the Crack Surface Bound-
5.3.4
ary Condition
The Green's function in the previous section solves the stress state in the cement
sheath for a single edge dislocation with a unit Burgers vector pointing in the yDue to the linearity of the response, the influence of a dislocation of
direction.
variable strength p pydis
=
/E(1)
commonly termed the dislocation density -
is measured by
+ PE(2). The versatility of the function is immediately recognized:
by relating the dislocation density to the crack opening displacement, a crack of
arbitrary shape may be represented in the cement region. It is noted that the Green's
function has presently been limited to discontinuities along the real axis, such that
a continuous arrangement of dislocations along the sheath thickness allows a radial
crack of any length and origin to be constructed (emanating from the proximal/distal
interface, or embedded).
While devising the circumstances of a radial crack for a known distribution of
the dislocation density p(t) is trivial, the risk of radial fracture for the hydrating
cement sheath must be assessed with respect to the poromechanical properties and
the eigenstress generation. Consequently, we are tasked with finding the distribution
that relates the tangential stress generation in a holomorphic specimen to the crack
surface traction in the impaired specimen. From the above, the hoop stress caused
by a dislocation at location r = x = t is resolved from the Kolosov-Muskhelishvili
formulae as
Ed
+ PE(2 where
PE(
p(t)E2 )
(r, t)}
=
pt(t)R {2#(r, t) + r#(r, t) +
=
P(t)R {24'(r, t) + r"(r, t) + '(r, t)}
(5.57)
= P(t)H(r,t).
Remembering that TJ(z, t) is calculated by either of the cases in Eq.(5.27), we may
choose
/ (z, t) = -D (R2 /-f, t)
145
Z
I'(z , t)
(5.58)
and the potentials read in terms of their Laurent series as
00
<D
B, (t)zZ
(z, t) = E
(5.59a)
n= -oo
XF
00
)
~zt) =-1: B()
(+j~
n1) Bn (t) z7
-1:
.
00
(5.59b)
fl-oc(
fl-o0
Whence, it follows from Eq.(5.57):
00
H(r, t)
B(t) [(n2 + n)rn-1 - (n2 - 2n)R2r"-3 ] + B+(t) [nR 2nr-"-1] . (5.60)
=
71=-cO
It will be remembered that our solution strategy superposes two elastic solutions: the
solution for the holomophic problem, and the solution with a crack whose surface
traction is measured by
-
(I1( ) + I 2 ( )/r 2 ). With the relevant expressions at hand,
such a boundary traction is guaranteed by setting
-2G
er(r
+ 1)
p (t)E((r,t)dt + P(t)E 2 (r, t)dt =
-
I 12
fP2
fP)2
IP2
d
t -
+
r
/
I
H(r,t)p(t)dt =rs
effect of dislocations in an infinite medium
I2
(5.61)
r2
boundary response
We notice that p(t) can be solved as a singular integral equation of the first kind,
where pi and P2 are the left and right crack tips, respectively. The integrals must be
.
taken in the Cauchy Principal Value sense8
Numerical solutions to the singular integral equation.
Much work has been devoted to the analysis of singular integral equations [591
and noteworthy contributions by Erdogan et al.
[291
[481,
give numerical approaches to the
The Cauchy principal value of an integral f f(x)dx- also termed the finite part of an integral
with a singularity at the end point x = b may be calculated as
8
-
f(x) dx
P.v.
a
lim
+0+ 1
146
f (x) dx.
(5.62)
solution of crack-type problems. In particular, they have adopted Gaussian quadrature formulas that utilize the orthogonality of the Jacobi polynomials to provide
high-accuracy solutions with a small number of quadrature points. In Appendix D,
it is shown that the fundamental function of the dislocation density is that of the
weight function for the Jacobi polynomials (Chebychev polynomials in particular),
such that the form of p(t) is sought as
P(t) = w(t)g(t) = (t - P1)"(P2 - t)'g(t)
(5.63)
and w(t) is the pre-determined weight function. It depends on two parameters, a and
3, that describe the nature of the crack tip singularities. In order to ascribe meaning
to these parameters, we investigate the physical interpretation of P(t). Because a
Burgers vector is defined as the integral of &ui/Ds taken counter clockwise around
the dislocation, it is readily understood that
P(r) =
ar".
(5.64)
defines the gradient of the displacement discontinuity. In other words, the dislocation
density p(t) describes the slope of the crack opening displacement. If the crack is
embedded in a homogeneous medium, p(t) will be singular at the points t = pi and
t = P2. For this scenario, the symmetry of the top and bottom crack surfaces and the
elliptical shape requires the opening displacement to terminate at an infinite slope;
this demands the negativity of a and 3. Case (a) in Fig. 5-5 shows the idealized shape
of a crack embedded in a homogeneous medium. On the other hand, if P(t) represents
an edge crack, where one of the end points of the crack, say t = pi, terminates at
a zero slope or a finite slope (for instance, if shear forces along the interface subdue
an unimpeded opening displacement), then the respective parameter is necessarily
non-negative, /3 > 0. Cases (c) and (d) in Fig. 5-5 show the idealized crack shapes
for these scenarios.
The numerical recipes by Erdogan et al. [291 will now be applied to the various
cases of a radial crack in our cement sheath. However, in order to relate the crack
147
of integration must be normalized to (-1,1). Thus, the integral equation is recast as
'2~
7r
T
_T
(T)
-p dT +
-
1 f(r(p). t(T))f(T)dT =
[
p
I
2
(5.65)
r (p) 2
where
H(r(p), t(T))
= -
(K+ 1)(P2 - P1)
4G
H(r(p), t(T))
(K + 1)
2G
(2 2+ 1) 1
2G
fA(r)
=ptr)
and the coordinates r and t have been parameterized using the following relations:
P2
2
T
+P1
P2-P1
P2-P1
2
P2+P1.
P2-P1
_ P2 - P1
+
P2
+P1
2
P2P1
+
2
P2-P1
(5.66a)
2
+
)
boundary condition in Eq. (5.61) to the general formulae cited in literature, the interval
.
2
(5.66b)
In the following we provide the numerical solutions to Eq.(5.61) for a crack that
originates at SC and propagates radially outward toward RC. Varying assumptions
are made for the shear traction continuity along SC.
Closed Crack Geometry: Rigid Shear Connection Along SC
The first case we consider is that of a rigid connection between steel and cement, for
which x -+ oc and an ellipsoidal crack shape ensures displacement continuity at the
proximal crack tip r = R1 . It should be noted that a shear stress singularity along
SC is necessitated to maintain the closure of the crack. As a result, the Fredholm
kernel H(p,
T)
in Eq.(5.65) becomes singular for p
= T =
-1.
Typically, numerical
methods are only designed to accommodate the Cauchy singularity
1/(T -
p) in the
first integral. Thus, additional attention must devoted in the numerical procedure to
deal with ft(p,
T).
With this in mind, we will first proceed by giving the well known
148
solution for a bounded Fredholm kernel; as would be appropriate for an embedded
crack, not in contact with SC or RC. Thereafter, the form of the weight function tb(T)
will be modified to provide more accurate results in the case that either of the crack
tips terminates at a bi-material interface.
As a first approximation, the symmetry of the problem allows us to estimate the
fundamental function of the crack by
(lT(()
=
,\(I
1
- 7) (1 + 7)
(5.67)
.
where a and 3 have been set to -0.5 in Eq.(5.63). Substituting the above expression
for
ft(T)
into Eq.(5.65), a system of linear algebraic equations can be developed by
discretizing
g(T)
and using a Gauss-Jacobi quadrature scheme [29]:
S- N
1
g(Ti)
1-1
Ti -Pk
+ 7rH(pk, Ti)
_
=
2-
ZI +
(5.68)
2
r(P )x2
Herein, the relevant orthogonal polynomials are of the Jacobi type and reduce to
Cliebyshev polynomials of the first and second kind -
denoted TN(T) and UN 1(p),
respectively. Their abscissas define the collocation points, which are given by,
Ti - OS 72i -1)
Ti = cos 7r
1=1 11..IN
(i=1,2,3 ... ,N)
N
c p)s(k =1,(2, =3,..., N).
Within the development of the numerical quadrature, TN(T) provides the N abscissas
for ri, and UN-1(p) provides N - 1 abscissas for Pk in the interval -1
< T < 1.
Hence, an additional relation is of need to obtain a unique solution. This condition
is given by the physics of the problem, where we require the closure of the crack
opening displacement -
i.e. the condition of single-valuedness of the displacement
upon crack traversal:
1
N
g(7j) = 0
- (T)dT ~
f222N
149
(5.70)
Table 5.2: The first root of the characteristic equation describing the singular behavior
of the crack tip terminating at the RC bi-material interface; assuming v = vR = 0.27
m=GR/G
-a
0.1
0.753
0.2
0.679
0.5
0.575
1.0
0.500
2.0
0.431
5.0
0.358
10.0
0.320
While the approach outlined above provides a good first estimate of
g(T)
(and
is accurate for an embedded crack), we will now seek a modification to the weight
function
')(T)
that will allow the effects of the Fredholm singularities to be taken
into consideration. It should be pointed out that if the fracture penetrates the entire
width of the sheath, Ht(p,
p
T)
will have two singularities located at p =
T
-1 and
T = 1.
Fortunately, several authors have investigated the details of this singular behavior 1161 [781, and Cook and Erdogan provide a mathematical formalism for the change
in the singular behavior for a crack terminating at the interface of two semi-infinite
half planes [22] [42]. By calculating the eigenvalues of a characteristic transcendental
function, the values of a and 3 are derived. This function is substantially complicated in the case of our annulus due to the Laurent series expression. But, because the
characteristic radial dimension of the sheath is much greater than the characteristic
dimension of the opening displacement, adoption of their formalism well approximates
the singular behavior for the crack tip ending at our curved surface at RC. Thus, from
their analysis a is the first root of [221,
2d, cos(7r(a + 1)) - d 2 (a + 1)2 - d3
150
=
0;
0 < -a < 1
(5.71)
where
di =
(m +Vs) (I + MK)
d2 = -4(m + Ks)(I
d3
=
(1 - m)(m -
-
m)
rs) + (1
+ mK)(m + Ks) - m(1 + K)(1 + mnK)
and rn = GR/G. Table 5.2 shows the values of a for varying ratios of the shear
modulus.
As the stiffness of the rock increases in comparison to the cement, the
strength of the singularity decreases. The rock is able to absorb more of the stress
and the load near the crack tip is dampened. It should be remembered that the
stiffening behavior of the cement will vary the ratio of the moduli, m, such that a
changes in the course of the reaction.
The application of this approach to the crack tip at SC is less justifiable, because
the steel casing is not suitable to the idealization of an infinite half space. Nonetheless, we will calculate
#
from the first root of Eq.(5.71), and are confident that this
approximation lends only minor error since (i) the high ratio of the moduli between
steel and cement will lead to a large absorption of the stresses by the steel near the
crack tip, and (ii) we are principally concerned with the stress intensity factor at the
distal tip. Therefore, the inaccuracies of the approximation are expected to decrease
as the crack length increases.
The fundamental function g(T) must now be approximated by means that utilize
the general form of the Jacobi polynomials P()
(x), for which Polyanin and Manzhi-
rov give a numerical method that is readily adopted to estimate Eq.(5.65)( [64];
Chapter 15):
N
-i^
wig(Ti)
1
Ti
--
Pk
+ 7TH(r(pk),(r)
k(+Pk)2
=
I,+
-
(k = 1, 2, 3, . . , N)
(5.72)
151
G1,
t =1
=0
1t
t =
(a)
W(t) =
K1
(U1+
W(t) =
-0t
-t)(1
(1-+ t)1
(G1,
i,
G2, 2)
W (t) =
(b)
(c)
Figure 5-5: Shapes for the fundamental function for the dislocation density P(t) cX
of a uniform surface pressure for (a) an embedded crack, (b) a
at"U under the loading
closed crack with the left tip ending at a bi-material interface, and (c) an edge crack.
71 ~ cos(0);
Pk
Wi
Cos(1k);
2
2N + a +3+1
ei dk
2a - 1 + 4i
~
7r
2N + a +/ + 12
2a + 1 + 4i 7r
2N + a + + 12
1 - T(1
- T,)"(1 + TF)"
The Open Crack Geometry: Complete Loss of Shear Along SC
In the event that the shear bond between steel and cement is lost entirely, the radial crack will assume the greatest opening displacement and, consequently, also the
greatest stress intensity factor. Where the proximal end of the crack is otherwise
restrained from opening, the loss of the interfacial bond allows the crack to open
unimpeded. Case (c) in Fig. 5-5 depicts the resulting surface displacements for an
edge crack, where x = 0 causes the steel and cement to slide without resistance.
Gupta and Erdogan proved analytically that the weight function of an edge crack
takes the form [341:
= 0.
7b(T) =X
152
(5.73)
The equation above ensures the finite slope of the surface displacement upon nearing
the steel interface. The square root singularity signifies that the crack tip is embedded
in a homogeneous medium. Now. it is desirable to extend the definition of P(t) beyond
the physical extent of the domain in order to make use of the Jacobian integration
formula with orthogonal polynomials P 1/2 ,-1/ 2 (x) (i.e, the Chebyshev polynomials
of the first kind, TN(x)). As was done in
[341,
we proceed by defining the following
normalization of the radial coordinate,
r(t) =
(5.74)
Pi
p(r ) =
P1
P2 -
P2 - P1
such that a continuation across the origin can be made,
(TT)
(-1< T<1)
; g(T) =g(-T)
=
01
( )
--
(-1<
< 0).
)
T)
(1 +
7-)(5.75)
The integral equation in Eq.(5.65) may now be recast as
1
I
2r
g(r)
_1
gT+
12
I
+ H(r(p), t(T)) dr
Ir I - p
(r(p), t(I )) d
(5.76)
I2
r(p)2
where
H(r(p), t(T))
(rK + 1)(P2 - P1)
2G
The procedure for solving the integral equation with
H(r(p), t(r)).
w(T) =
(1-T 2 )-
1
/2
was discussed
above with reference to an embedded crack with a closed geometry and may again be
adopted'.
9
The symmetry of the extension guarantees the closure of the crack shape. Instead, the first N
zeros of T2N+1 and U2N provide the collocation points and produce an N x N system of equations.
For additional details see Ref. [34].
153
5.3.5
Calculating the Stress Intensity Factor and the Surface
Displacement
The formal definition of the stress intensity factor for a crack whose tip is embedded
in the cement is given by
Krad _ lim V2w(p
2
r-*P2
where (Eoo(r, 0
=
- r) (Zoo(r, 0 = 0) + p),
(5.77)
0) + p) is the effective stress acting ahead of the crack. It is well
established that the singular stress behavior in proximity of the crack tip may be
expressed in terms of the cleavage stress 00, which takes the form [731
Olue(r)]
4G
'00(r)1r>b =
=1+K
Or
2G
1+Kr
2G
_
[29],
1+
r<b
uo(r)
(5.78)
r<b
0=0
r)
Thus, the stress intensity factor can be rewritten as
/rad
= lim
r-4P2
=
2
1+
2w(P2 - r)Co0
(5.79)
lim
r r-+P2
V27(P2 - r) P(r)l
and on substituting p(r) = h(r)(j' ) and h(r) = a-
g((r)), where ao= P2 -Pi
is the crack length, the above equation becomes,
Krad =
1+ rK
2iaoI g(1)|.
(5.80)
The value of a has been left ambiguous and depends on whether the proximal tip
adheres to an open or closed geometry10 . In either case, we obtain the same solution.
A numerical procedure for solving g(T) was given in the previous section.
10a = -1/2 for the open geometry and a =
f(G, Gs,
154
,,
Ks) for the closed geometry
Next, we must consider the case in which, the radial crack has fully penetrated the
cement sheath, and the crack tip concludes at an interface. Here the stress singularity
and the stress intensity factor according to Cook and Erdogan may be
has order -,
sought as [221,
yCad
= lim G*V2w(p
2
-
r) "Coo
lim GY-2wr(p 2
-
r) '|p(T(r))|
r
=
P2
(5.81)
r-= P2
where
G* = Gm (3 + 20)(1 + m) - (1 + 2/3)(m +
(m + KR)(1 + m,'K) sin(-F (1 + 0))
We remember that m = GR/G. In the above, the relation y(r) = (-)
Pi) 0 (P2 -
(5.82)
R)
('+8 g(T r))(r+
r) was used". Stress intensity factors were calculated for embedded cracks,
and edge cracks where Gs and GR were set to zero to resemble a "hollow" ring. Their
solution compared perfectly with the results given in [251.
Finally, the surface displacement of the crack is obtained by
UO
(r, 0)
1f&~[uJJ
=-dr=
1
-
P2
p(r)dr p < r < P2
(5.83)
and is readily calculated at the collocation points for an edge crack by Eq.(5.70):
1
UO(t(Tk))
where
T2 = cos
k
T(r)di
-
g(Ti)
(5.84)
k
1-
2Na++
(1
-
-
(])d
+
UO(t(Tk)) =
-
1
(- 7r). Similarly, the formula for the closed crack yields:
1
where wi
2
1
~
-Wi g(ri)
(5.85)
TI
.
"It is remembered that the open crack was scaled by
scaled by (P2 - p1)/ 2
155
(P2 - Pi),
whereas the closed crack was
000
800
-
6
(b)
x= 0
700.
a/(R2 -R )=0.2
a/(R2 -R 1 )=0.4
-
5
,9
-
4
-600
500
L4
400
a/(R2 -R)=0.6
-
a/(R2 -R,)=0.8
-
a/(R2 -R 1 )=1.0
2
S300
200
100
0.2
0.4
0.6
0.8
0.2
1.0
3
300
(C)
-
250
x-+o
2.5
,200
2.0
150
1
100
1 .0
1.5
-
--
0.6
0.8
1.0
a/(R2 -R
1.0 F
50
0
0.8
a/(R2 -R , )=0.2
a/(R2 -R , )=0.4
a/(R2 -R , )=0.6
a/(R2 -R , )=0.8
-
:
0.6
F
3.0
C4
0.4
0.5
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
Figure 5-6: The evolution of the (a) (c) energy release rate ((b) (d) stress intensity
factor) for a radial crack emanating from the steel-cement interface. The top panel
(a) (b) corresponds to the open crack geometry, and the bottom panel (c) (d) corresponds to the closed crack geometry. Values are consistent with the stress state shown
in 4-2b for ER = 40 GPa.
5.3.6
Sample Model Output: Evolution of Energy Release
Rate due to Radial Fracture
Two variants to the linear elastic fracture mechanics solution for the radial crack are
investigated in Fig. 5-6 by changing the condition of the shear bond between the steel
casing and the sheath. In particular, we plot the energy release rates and the stress
intensity factors for the limit states of a rigid bond (X -+ oc; red) and a loss of bond
(x = 0; blue) along SC. The stress development has been modeled from the input
parameters provided in Table 2.1 for stiff formation. The width of the lines indicates
the depth of crack penetration. As expected, the loss of shear along SC increases the
156
(a)700
600
-
(b)
x=
a/(R 2 -R 1)=0.2
-
5001
5
-
a/(R 2 -R
4
-
a/(R 2 -R 1)=0.6
-
a/(R 2 -R 1)=0.8
a/(R 2 -R 1)=1.0
1 )0.-4
r__"
CA
400
3
-
300
2
200[
100
0.6
0.8
0
1. 0
20
(d)
(c)
I
x-*o
--
0.2
-
2.5
S2.0
-
P
1005
501
-
0.6
0.8
1
0.6
0.8
1.0
0
3 C,
150
9
0.4
.
0.4
.
0.2
1.5
-
)=0.2
a/(R 2 -R 1)=0.4
a/(R 2 -R)=0.6
a/(R 2 -R 1)=0.8
a/(R 2 -Ri)=1.0
a/(R 2 -R
1
1.0
0.5
0.2
0.4
0.6
0.8
1.0
(n
0.2
0.4
Figure 5-7: The evolution of the (a)(c) energy release rate ((b)(d) stress intensity
factor) for a radial crack emanating from the steel-cement interface. The top panel
(a)(b) corresponds to the open crack geometry, and the bottom panel (c)(d) corresponds to the closed crack geometry. Values are consistent with the stress state given
by the input parameters in Table 2.1 and lowering the permeability to A = 1 x 10s/m.
157
energy release rate of the radial crack. Here, the system's compliance with respect
to a pressure along the crack surface is increased by the absence of a counteracting
shear stress. Thus, the crack opens wider, releasing more energy upon advancement
2
)
(grad X 'UO
Nonetheless, the same trends for both bond conditions are observed: (i) The
stiffening and eigenstress development in course of the reaction lead to an accumulation of elastic potential energy that accelerates toward the latter stages of hydration,
and (ii) the fracture process is generally unstable, because the energy release rate
increases upon crack propagation. Indeed, the energy release rate grows at a greater
rate the further the crack penetrates the sheath thickness. There are a few exceptions. At intermediate ages
(
-
0.4), as the pressure in the sheath recovers to the
formation pressure and the effective hoop stress transitions from the compressive to
the tensile regime, the sheath is temporarily in a stable fracture state. Additionally,
simulations not reported here, have shown that soft, permeable formations allow for
stable fracture propagation, though the stress build-up and risk of fracture is lower.
500
50(a)
3.5
,3.0
400
CN1
(b)
S 2.5
q
2.0
1.5
200
1.0
100
0.5
0
0.2
0.4
0.6
0.8
1.0
0
0
0.2
0.4
0.6
0.8
1.0
Figure 5-8: Diagrams depicting (a) the fracture toughness and (b) the critical stress
intensity factor for white ordinary Portland cement with a water-to-cement ratio of
w/c = 0.4. These results were obtained by the study of Hoover and Ulm [37].
Results for a low permeability formation, for which the pressure in the sheath
recovers more slowly, are shown in Fig. 5-7 (A has been lowered form 8 x 10-7 s/M
158
to 1 x 10-
s/m). Here, one readily notices the delayed response of the pressure in
driving the fracture process. Nonetheless, the rapidly hardening cement and continued
shrinkage of the solid skeleton continue to drive the fracture process, such that grad
increases monotonically. Interestingly, the closed crack shape allows the potential for
fracture Crad > 0 to be delayed slightly.
Lastly, we make reference of the experimental fracture toughness results by
Hoover and Ulm that were obtained from microscratch tests on hydrating white ordinary Portland cement (see Fig. 5-8). Fortunately, the fracture toughness evolves
with a concavity, while the energy release rate evolves with a convexity. Initially, the
cement rapidly gains a resistance to fracture, while the increase in elastic potential
energy is comparatively slow. At later stages of hydration, the trend is reversed: The
increase in toughness is small compared to the increase in the energy release rate. For
the results of the two sample simulations provided in Figs. 5-6 and 5-7, only a crack
with an open geometry and having penetrated over 80% of the sheath thickness is
expected to further advance near the end of the hardening process.
Sample Model Output: Energy Release Rate due to the
5.3.7
Loss of Shear Traction
In this section, we elaborate on the meaning of the two shear bond conditions. As
the open crack shape enforces a zero shear stress along steel and cement and the
closed crack shape maintains a rigid connection, their potential energy release during
fracture can be related to the energy stored in the bond. In particular, the debonding
energy for a radial crack that extends from p, to P2 may be written as
=
P2
goPen -
gcloseddr = 1
jP2
(i
+
(2 n open
_
closed1)
dr.
(5.86)
Herein, we have related the energy release rate to the work done by the surface stress,
I, +
2/r 2 , to open the crack by an amount
[uo1.
By integrating the energy release
rate along the propagated line of fracture we arrive at the total energy dissipated
during fracture. The difference between the energy quantities for the open and closed
159
Figure 5-9: Diagram depicting the potential energy stored in the shear connection
between the steel casing and the cement sheath upon developing a radial crack of
closed shape.
crack geometries measures the energy stored in the rigid shear bond (see the diagram
in Fig. 5-9).
For the standard model parameters in Table 2.1, Fig. 5-10 displays the crack
shapes of the two traction conditions along SC for increasing penetration depths.
Fig. 5-13 shows a direct comparison of the crack shape for x -+ oc and x = 0
at complete penetration.
(~
The maximum opening displacement of the edge crack
1.2 mm) reaches approximately four times that of the closed crack (~
0.3 mm).
Bachu and Bennion experimented with brine and CO 2 to evaluate the permeability
of wellbore liners in the lab. In good quality the liner's permeability was found to be
10-2 m2 , and in the presence of an annular gap and radial cracking, with apertures
ranging between 0.01 mm and 0.3 mm, the permeability increased to 10-1 m 2 [7].
Though precise conditions of the mechanics driving the fracture processes in their
experimental samples is unclear, it is of comfort that the results of our simulation
are of a similar order of magnitude. Additionally, as observed radial fractures of
160
wellbore liners likely preserve a partial bond along SC, only loosing connection locally
near the proximal crack tip, we can expect the most accurate model of the opening
displacement to lie between the limit cases.
Finally, the green line in Fig. 5-12 plots the potential energy stored in the rigid
connection along SC for an advancing crack. Remarkably, most of the energy release
for the edge crack scenario could be averted by improving the bond between steel and
cement.
5.4
Chapter Summary
In Chapter 5, we derived the energy release rate and fracture criteria for the failure
mechanisms most at risk of impairing the sheath's sealing function at early ages.
More precisely, Clapeyron's formula was employed to relate the external work done
to open the apertures. By deducing effective elastic stiffnesses for the casing and the
formation, the displacement of the interface boundaries was calculated upon complete
rupture of SC and RC. In the case of the radial crack, Muskhelishvili's method of
complex variables was used to derive the Green's function of an edge dislocation in
the sheath. Integrating the Green's function along the real axis, an integral equation
was developed for the dislocation density. Here, direct connection could be made
to the stress intensity factor of a radial crack covering any portion of the sheath's
thickness.
Key to an accurate description of the crack propagation, was the incorporation of
the drained nature of the fracture process. In this circumstance, the saturated cement
medium expels fluid into newly created fissures, which exerts additional pressure onto
the crack surfaces, exacerbating the risk of failure. Thus, the effective stress E + pl
calculated by the chemo-poro-elastic solver in Chapter 4 was incorporated as the
driving mechanism of fracture.
As follows from the investigation of the bulk stress development, fracture energy
release rates are greatest when the cement sheath is bounded by a low-permeable, stiff
rock. A central finding is that the risk of both micro-annulus formation and radial
161
fracture can be substantially mitigated by improving the bond along the interfaces.
162
1
4
1e-3
1.2-
1.0
0.8
~0.60.4
0.2
0..0
0.2
0.4
0.6
0.8
1.0
r-p,
P2 -Pi
3.51e-4
3.0
2.5
2.0
S1.56..J
0.5
0.0
-0.5-
X-+00
'..0.2 0.4 0.6
0.8 1.0
r-P1
P2 -Pi
Figure 5-10: The surface displacement uO(r) = uoj] for pi < r < P2 plotted along
the line of the crack pi < T< P2 for (a) the open geometry (the edge crack; X = 0)
and (b) the closed crack (x -+ oc) for a crack that has propagated the width of the
sheath. The thickness of the lines are proportional to the penetration depth of the
crack (P2 - pi)/(R 2 - R1 ). Values correspond to complete hydration of the cement
S= 1 and are consistent with the stress state shown in 4-2b for ER = 40 GPa.
163
(b)
2. 0 4le-3
x-0
-
1.5
L
1.0
0.5
0.8-.0
0.4
0.2
0.6
0.8
1.0
r-p,
P2 -P1
Figure 5-11: A comparison of the surface displacement uo(r) = 1 uoj along the crack
P1 < r < P2 between the open crack geometry (blue) and the closed crack geometry
(red). Values correspond to complete hydration = 1 and are consistent with the
stress state shown in 4-2b for ER = 40 GPa.
5
fr
gope"
4[
Jr
dr
gclosed
dr
Pi
2[
1J
0
0.2
0.4
0.6
0.8
1.0
r-pi
P2 -P1
-
Figure 5-12: The energy stored in the elastic shear bond between the steel and the
cement AS plotted in relation to the normalized penetration depth of the crack (r
Pi) / (P2 - Pi). Values correspond to complete hydration ( = 1 and are consistent with
the stress state shown in 4-2b for ER = 40 GPa.
164
4.0 le-4
-""
X-oo
3.5
3.0
2.5
2.0
1.5
1.0
0.5-
0.8.0
0.2
0.6
0.4
0.8
1.0
r-p,
P2 -P1
Figure 5-13: The crack surface displacement bounded by steel Es = 200GPa and
rock ER = 40 GPa. The y-axis has been elongated to show the effect on the surface displacement as the crack tips terminate at bimaterial interfaces with varying
stiffnesses.
165
166
Chapter 6
Conclusions and Perspectives
The health of wellbore cement liners is vital to the efficient extraction and ultimate
recovery of fossil fuel resources. A failed cement liner can cause an uncontrolled release
of pressurized oil and gas from the reservoir into overlying strata, the groundwater
aquifer, or to the surface. For natural gas wells, Howarth et al. [39] estimate that
between 3.6% and 7.9% of the methane from shale-gas production is lost to the
atmosphere due to venting and leaks.
The department of energy predicts that by 2035 the domestic production of
natural gas will grow by 20% with unconventional drilling techniques accounting for
75% of the total [3]. This means that the rate of oil and gas well constructions will
significantly increase from the -35,000/yr produced on average between 2001 and
2010. Moreover, as easy-to-reach oil supplies become depleted, drilling contractors
are posed with the challenge of cementing in more extreme temperature and pressure
environments. We are thus challenged to continuously improve our design practices
by incorporating advancements in science, engineering, and technology. Hence, the
design tools introduced in this thesis have expanded the capabilities of current models
by (i) incorporating cement as a chemo-poro-mechanics material, (ii) evincing the
driving mechanisms of the cement eigenstress developments in the solid and pore
phases and connecting them to a pressure state equation, and (iii) calculating the
structural failure criteria by utilizing new solutions to the stress and fracture energy
release rates.
167
6.1
Review and Results
6.1.1
Problem Synopsis
The early-age stress developments of cement pose a risk to the sealing function of a
wellbore liner. After the steel production channel is inserted into the wellbore hole, it
is stabilized by pumping cement slurry into the annular gap between its outer surface
and the hole diameter. As it cures, cement eigenstresses develop. Simultaneously, the
stoichiometry of the reaction, and the difference in the chemical potentials of H20
in the porespace and on the gelpore surfaces leads to the mobilization of water into
and onto the CSH structure. Consequently, the cement experiences a drop in pore
pressure, causing a flux of water into the sheath from the adjacent lying formation.
The coupling between the eigenstresses in the cement's solid skeleton and the pressure
change in the porespace, place the sheath at risk of fracture once the cement becomes
resistive to loading. Under this setting, the bulk reduction in volume acts against the
restraints of the steel and the rock to cause potentially fatal tensile loads.
6.1.2
Modeling Contributions and Findings
In this thesis, we incorporate the recent elucidations of the CSH constitution and
the origin of its eigenstress into a chemo-poro-elastic model of hydrating cement.
Molecular and mesoscale simulations, as well as state-of-the-art experiments, have
shown that the eigenstress is the result of a net-attractive potential between CSHspheres that remains after the cement has reached its hardened state [86]. Correlated
with the packing density, we were able to track the densification and eigenstress in
function of the degree of hydration.
The core of our material model was described by a poro-mechanical pressure
equation, which treated the coupled processes of the internal loading of the CSH solid
du*, the pore pressure dp, the densification of the reactants
eff,
and the prescribed
displacement of the REV boundary du (therein linked by the mean strain dEm).
This enabled us to bridge the chemical and physical changes to the porosity and
168
empowered the stress evolution to be tracked in course of the growth of the solid
skeleton by solving the pressure equation incrementally. It was discovered that the
drop in pressure at the early stages of hydration and subsequent increase in pressure
-
following a significant increase in the solid volume - locked the system into a
state of effective tensile stress. Here, the dynamics of the pressure variation were well
explained by a time ratio
hyd/Trfl
that compares the rate of the reaction to the fluid
velocity in the sheath.
With a material model at hand, the bulk stress development along the interfaces
was tracked in function of the hydration degree. The key finding is that low permeable,
stiff formations place the sheath at greatest risk of fracture, because (i) low permeable
formations delay the pressure recovery, increasing the magnitude of the final tensile
stress, and (ii) a rigid rock barrier prevents the cement to shrink in response to its
eigenstress generation. Additionally, a Laurent series solution to the stress state in
a sheath with an eccentrically placed casing, allowed the added risk of failure due
to the casing off-set to be estimated. For large off-sets, the greatest magnification of
stresses was found for the hoop stress in soft formations (the stress can be increased by
over 100% of the reference, concentric value). Moreover, non-negligible shear stresses
develop along SC and RC not otherwise present.
Finally, the bulk stress state was linked to the fracture scenarios of micro-annulus
formation along SC and RC and a single radial fracture emanating from SC. Due to the
role of the pressure evolution in the sheath, it was found that micro-annulus formation
is a risk along both interfaces, even for soft formations. Subsequently, the use of
expanding agents to negate the shrinkage of the solid skeleton does not guarantee
compressive stresses along RC and will further increase the risk of debonding along
SC.
The loss of axisymmetry for the stress state in a sheath with a a radial crack,
prompted use of Kolosov-Muskhelishvili formulae. Here, a solution was found for the
derivative of the opening displacement vector p(t), which was linked to the stress
intensity factor KIrad. Interestingly, the stress intensity factor was increased substantially by considering an open crack (where the shear bond between steel and cement
169
is lost) rather than a closed crack geometry. More succinctly, much elastic potential
energy is stored in the bond along SC and improving the adhesion between steel and
cement is expected to significantly decreased the risk of radial fracture.
6.2
Next Steps
Finally, we leave the reader with our contemplation of improvements to the model
that would enhance its proximity to the physical conditions encountered downhole
and during reaction:
" As many drilling contractors incorporate expansive agents into the cement
mix design to counteract bulk shrinkage, the phase morphology and upscaling
schemes should be modified to allow such inclusions.
" Due to the heat production during reaction, additional eigenstresses caused by
the temperature variation in the phases will be induced. These must be included
in the pressure state equation and the formulation of the bulk eigenstress.
" While ordinary Portland cement typically requires 7-days to attain 60%-75% of
its 28-day compressive strength [471, the high cost of drilling equipment (e.g.,
drilling rigs) incentivizes contractors to initiate the testing and production processes as early as 3-5 days after placement. Hence, our model output should
investigate the effects of pressure cycles applied to the inside of the casing in
course of the cement hydration.
" As it was concluded that much elastic potential energy is stored in the bond
between steel and cement during radial fracture of the closed geometry, it is of
interest to determine the effect of partial debonding on the energy release rate.
Does guaranteeing the partial adhesion of cement to steel substantially reduce
the risk of radial fracture? Or, is a predominance of the elastic potential energy
stored locally near the tip at SC?
170
Appendix A
Effective Stiffnesses of Steel Casing
and Rock Formation
In calculating the stress state for the concentric annulus (see Chapter 4), our expressions may be simplified by identifying effective stiffness constants for the steel and
rock formation. These measure the uniform stress along the boundaries required to
advance the SC and RC interfaces outward by a unit length. We can derive the
effective stiffnesses of the two media by considering the displacement solution of an
infinite cylindrical tube subjected to an interior pressure po and an exterior pressure
Pi:
Ur(2
R21
- 1
2(A + G)
+
Ur (r) = PO2
-U
2Gr )
1
P21
2-+
2 J- 1
r
(2(A + G)
R2
0
2Gr
(A.1)
where r is the radial coordinate, RO is the radius defining the interior surface, R1
is the radius defining the exterior surface,
= R 1 /RO, and A = K -
G is Lam6's
constant.
In the case of the steel casing, we consider a thick-walled cylinder with a zero
inside pressure (i.e. po
=
0). Therefore, noting that the radial stress on the outside
is balanced by the exterior pressure, we can write:
171
)_R R)(A.4)
(Trr(r
R1) = -p1 = 2 G
2 2 - 1) u, (r = R )
(Gs + 3Ks)(u
1
= fs
=(R(r
RA
U
(A.2)
In the above, is represents the effective stiffness of the casing. For the formation,
we must adapt Eq. A. 1 to the case of an infintely thick pressure vessel
('
-
00) whose
inner boundary is subjected to a pressure of po:
lim
zu o0
Urrr- O)
Ur(T
O)
RO
RO) =r(rPo
2GR
-GRUr(r
0
)R= Ro) =
R_
172
(A.3)
_
RUr (r
rT
= Ro)(A4
Appendix B
Laurent Series Representation of
Analytic Functions in the Complex
Plane
Often we wish to represent functions that are analytic everywhere except at some
point(s) or region(s) of the complex plane (see, for instance, Section 5.3.3). Here, it
may not be possible to employ a Taylor series in the neighborhood of the singularities
(It should be remembered that Taylor series contain only the positive powers of z).
Instead, we look to Laurent series, which contain both positive and negative powers
of z, to resolve these functions. Such a series is analytic in the region and on the
boundaries of a circular annulus, where r1 < Izf
<
r 2 and the center has been placed
at the origin. The general expression for a Laurent series of a complex-valued function
g(z)
=
g1 (z) + ig2 (z) in the complex plane is then given as,
00
g(z)
=
E
where ri < IzI < r2.
c"z
(B.1)
n=-oo
Herein, Ck are complex-valued coefficients.
In the special cases of a cicular disk
(lim r1 -+ 0) and an infinite plate with a hole (lim r2
-÷
o) the negative and positive
powers of the series are dropped, respectively, in order to ensure convergence.
In order to deduce an expression for the nth coefficient, the Cauchy integral
173
formula can be used to show that
cn =
f)dz
272 c zn+1
(B.2)
where C is any simple closed contour in the annulus that circumscribes the inner
boundary IzI = r1 (see e.g. Ref. [1], pg. 128 for a thorough derivation). However, in
the case that we wish to evaluate the coefficients from a circular contour for which g(O)
becomes dependent only on the angular component, a classical Fourier series emerges.
Using Eq.(B.1) we can proceed by multiplying both sides by Zn and integrating the
series over its interval of orthogonality, noting that
2
zn+k d
-
2
rn+ke(n+kOdO =
JO
{0.if n+ k y 0
27r, if n + k = 0
(B.3)
Hence, we recover the coefficients as,
Cn=
21
27rrT
-ing(z)dO.
j21
(B.4)
0
'The Cauchy integral formula states: For a simply connected region D, any holomorphic (i.e.
analytic) function f(z) in D can be evaluated at a point a as,
f(a) =
27ri
jC zf(z)dz
- a
where C is a closed contour forming the boundary of D.
174
Appendix C
Upscaling Poroelastic Constants
C.1
Level I: CSH Gel with Gelpore Pressure
The Level I state equations define changes in the stresses and the gel-porosity at
constant hydration degree. The equations for a basic pore-solid composite are given
by Dormieux et al. [271 and read as:
dEm = 1
I1I
J
dorm(z)dz
=
KjdEv + (1 - bi)du* - bidpg
(C.1)
V
d#l|' = bdE v
b1
bidEv +
(du-* + dpg)
s
1
-(d-* + dpg)
-s
(C.2)
where the gelporosity change (d#' = d(# - 0)) is due solely to the elastic response
of the material system. In the above k, is the bulk modulus of the CSH solid, a-*
is the eigenstress in the CSH solid, and pg is the gelpore pressure. The poroelastic
properties at this scale are well known such that the Biot coefficient and Biot modulus
are defined classically as:
bi =1 -K,
(C.3)
1
b1 0N1 ~ k
(C.4)
and
_
175
Level II: CSH Gel with Macro-Pores
C.2
At the second level, we consider the CSH gel as a matrix for the macro-pores. Here another pressure prevails, denoted p, such that the state equations must be redeveloped,
starting from the following distribution of stresses:
(C.5)
dum(z) = k(z)dE(z) + do&(z).
In the above, dcP(z) is the local eigenstress, such that
in Vhp
K
0
=
do(z) =
{27
inV,
(1
bi)du* - bidpg
-
k(z)
in Vhp
(C.6)
in V,
-dp
Application of Levine's Theorem allows the Level II homogenized eigenstress to be
sought as,
d E*-
I1
J
)dE* - b( dp
A(z) : duP(z)dz = (1 - b(d
11
(C.7)
1
whence, the incremental equations of state are given by:
do"l'
I
=
=
J
dom(z)dz = K 1dE' + (1 - b
(1 - #1)
1 = b, dEv'
-
dE
d#"l
Herein, d#'l
- b )do* - b
b, (dE*+ dp) +
dvg - b
dp
(do* + dpg)
b)dEv"-+ I (dE* + dp)
WI
NI,
(C.8)
(C.9)
(C.10)
is the change in the gel-porosity at Level II, and d4r j is the change
in the macro-porosity at Level II. Moreover, b
176
and b
are the Biot coefficients for
the gel and macro porespace, respectively. Thus,
Knl = (1 - b
(C.11a)
1 - -K
(C.11b)
,
0)
(1
1
Nn
(b)
K,
K1
(2))b,=
-
K:')
(C.11c)
Kn
)
K,
b =
(C.11d)
"
b
)K1
K,
It is convenient to recast the state equations for the changes in porosity as,
11
1
dOg = b()dEv, + NI do* + NI dpg + NIdp
11
do*pg
do3"|j = b dE +
+
1
I dp
(C.12)
(C.13)
N"udpg21 2
hN
where the various Biot moduli can be calculated from
1
(I - #")
N1
1
N
1
(1 - bi)
N11
(bj
+
Nl
N11
I
I
N=1 N~1
b, (I -- bl)
N,
(C.14a)
(C. 14b)
N)
bi
(C. 14c)
(C.14d)
N,
1
(C. 14e)
Nil
Throughout this thesis, we make the assumption that the gelpore pressure is in equilibrium with the capillary pressure, such that dpg = dp.
In this case, the state
equations simplify to:
= KndEv'
+
d
(1 - bn,)do-* - b,,dp
dr,*i
177
(C.15)
1
1
I dE*"
(C.16
I dp
)
' +
*
=b+d
d|
101 1
and
b = b
(1 - b ())b + b
+ b ()=
N" +
1
1
(I - #")(b-
=
1
= 1- K
1
(C. 17a)
01) + (1 - bl)(b (2
1
(C. 17b)
(C. 17c)
Thus, the Biot Modulus with respect to the development of pressure in the porespace
or eigenstress in the CSH solid may be approximated by the same formula given in
Eq.( C.17b).
C.3
Level III: Reinforcement by Rigid, Slippery inclusions (Anhydrous Cement Grains and NonReactive Additives), CSH Gel+Macropores
The strain distribution in the system at Level III is written as above in the form
do(z)
=
(C.18)
k(z)dem(z) + duP(z)
with
k(z) =Kil
00
in Vhp
+V
(C.19)
in Vinc = Vnr + V.
Here, the rigid inclusions include anhydrous cement grains and non-reactive inclusions, such as silicate particles. Homogenization using Levine's theorem yields,
dEl
=
I
dam(z)dz = K,,,dEv"I' + (1 - bl,)du* - b 3dpg - b (2dp
d*O
178
(C.20)
1
1
1
d$OII|
= (I - finc )d$ |g= bl d Ev"n + N"do* + Nildpg + N
d$"lll
=(1 - finc)dl|jbw
= b
W
~
dp
(C.21)
1
1
1
dEll + N"w~do-* + N1
N"'l dpg + N dp
(C.22)
where the Biot moduli are related to their Level II analogues by
1
(1 - finc)
1
__
(1
(C.23a)
fm c)
-
1 1___(1( c- f)
(C.23b)
(C.23c)
N
1N "-- (1-finc)
(C.23d)
NjIINII N11I
1
-I
(I - Nc)
f
(C.23e)
NI
N'I
Finally, considering the mass content in the gelpores and the macropores in relation
to the fluid state equation (Eq. 3.23), we obtain for a fluid of compressibility kfl:
dmg~
P
P
-
d$,' + $" kfldpg
bi E + 1
1
=b ld El" + -"do-* + M -dpg
=M'
dollg + $ Ikfldp
E
+ 1
=bjd El" +
-do-*
V
wI N"' 21 +
-
(C.24)
1
(C.25)
+N
1
-dpg
1
+
I
where M refers to a Biot modulus of the fluid-solid composite (contrary to the solid
Biot ModulusN, M considers volume changes due to the compressibility of the fluid).
Therefore,
1
1
M 11
1
M
II
+ $g kfl
(C.26)
1
NI + $O kfl
179
(C.27)
Lastly, we inform the reader of the simplification in the state equations by considering
the gelpore and macropore pressure changes equal dpg
d Li = KjdE
d#"
d# " + do"'
dm-b
pfl
=
dp:
+ (1 - biI)dor* - bldp
1
I du-* +
bmdEt" +
(C.28)
1
dp.
dE"1 Idu*+ 1
Mdp
NII
(C.29)
(C.30)
Throughout this thesis the bulk properties often omit the superscript /subscript III.
The total porosity is given as
#
= 11 = (1 - finc)#and the Biot coefficient and the
Biot moduli are calculated from
1
1
1
-
1
Ii
NvI,
1
1 XJII
1
1P
+
=
b~ + b
1
I=
Nw l
1 2III
-
(C.31)
1 - K1,
11
I
(finc)NIa*
(C.32a)
NI'*MNI
1
N"'
+ 2N
1
finc)
1
-
b
1
N,,
(C.32b)
(C.32c)
The primary results for the upscaling of the poroelastic properties summarized in
main text in Table 2.2.
180
Appendix D
Analytic Functions with Branch Cuts
This appendix gives an insight into why the solution to p(t) in Eq. (5.61) can be sought
as k(t)(1
+ t)'(1 - t),3 , where a =
with general coefficients -1
0.5 and 3
< a < 1 and -1
=
0.5. A more thorough analysis
< # < 1 is omitted, but is addressed
in Ref. [591. As the topics are fairly complex, additional background information can
be sought in Ref.
[58], and much of the derivation to follow has been adopted from
Ref. [72].
Defining an analytic function f(z) whose value is the solution to the Cauchy
integral,
2F i
) dt = f (z),
t- z
(D.1)
where L is a smooth cut in z and g(t) must be H6lder continuous 1 , the Plemelj
formulae give the solution to f(to) on approaching a point on L from the left (+) or
the right (-):
(D.2a)
f+(to) - f -(to) = g(to)
f+(to) + f -(to) =
j
g(to dt.
(D.2b)
These formulas suppose that g(to) is not a point of discontinuity and is not an end
point at which g(to) / 0.
Isee Muskhelishvili [59].
181
Now, suppose f(z) is a function analytic everywhere in a chosen domain D,
except along an arc L such that:
(D.3)
fT+(t) - f -g(t) = s(t)
Then the more general solution is attained by
f(z)
=
1
I
27ri IL
g t)dt + fo(z),
(D.4)
t- z
where fo(z) is holomorphic in D. As proved by Muskhelishvili [581 we may select a
function h(z) defined along the arc L,
1
h(z) =
(D.5)
(z - p 1 )(z - P2)
Since it can be shown that h(z) changes sign upon crossing the arc L,
h+ (z) + h-(z) = 0,
(D.6)
(
f(Z)
)
h(z)
+
the second equality in Eq.(D.2) is written as
Sh(z)
7=i
(z)
IL
g(t) dt.
t -z
zc L
(D.7)
Making the substitutions f,(z) = f(z)/h(z) and q(z) = fL g(t)/(t - z)dt we arrive at
(Z)
-
(Z)
q(z)
zcL
(D.8)
We can then show for Eq. (D.4), by the same approach used to arrive at the Plemelj
2
Choose L as a branch cut in the complex plane and use
right side of the branch cut.
182
h(z) to traverse from the left to the
formulas, that:
*(z) =
q(t)
1
2
27 IL h+(t)(t - z)dt+PO(z)
(D.9)
)
hf(z) I
q(t)
h(z)po(z)
f~z) 27T2= -dt +h+
(t) (t z
where po(z) must be a constant to preserve the order of f(z) and ensure its boundedness for large z. Finally, substitution of the above into Eq. (D.3) yields
___
_+___
g(to)= - h(o
g (to)r2
by choosing po = -k/27ri.
+()
JL
___
h+ (t) (t - to)
dt -
k
17r
h+(to)
(D.10)
k may be chosen so that g(t) is finite at one end of the
arc 3
3
For k = 0, the analysis above is representative of a closed contour crack along L (i.e., an
embedded crack). A similar analysis can be shown to hold for an edge crack, where
P
h(z) =
and
P2
is the location of the crack tip.
183
(D.11)
184
Bibliography
[1] M. J. Ablowitz and A. S. Fokas. Introduction and applications of complex variables, 2003.
(21
P. Acker. Micromechanical analysis of creep and shrinkage mechanisms. In F.-J.
Ulm, Z. P. Bazant, and F. H. Wittmann, editors, Creep, Shrinkage and Durability
Mechanics of Concrete and other quasi-brittle Materials, pages 15-25. Elsevier,
Oxford, UK, Cambridge, MA, August 2001.
[31 U.S. Energy Information Administration. Annual energy outlook 2014 with projections to 2040. http://www.eia.gov/f orecasts/archive/aeo14/, 2014. Accessed: 2015-05-15.
141
A. Albawi. Influence of thermal cycling on cement sheath integrity. Master's
thesis, Norwegian University of Science and Technology, June 2013.
[51 A. J. Allen, J. J. Thomas, and H. M. Jennings. Composition and density of
nanoscale calcium-silicate-hydrate in cement. Nature materials, 6(4):311-316,
2007.
[61 C. P. Andrasic and A. P. Parker. Dimensionless stress intensity factors for cracked
thick cylinders under polynomial crack face loadings. EngineeringFracture Mechanics, 19(1):187-193, 1984.
[7j S. Bachu and D. B. Bennion. Experimental assessment of brine and/or CO 2
leakage through well cements at reservoir conditions. International Journal of
Greenhouse Gas Control, 3(4):494-501, 2009.
185
[81 M. Bauchy, M. J. Abdolhosseini Qomi, C. Bichara, F.-J. Ulm, and R. J.-M.
Pellenq. Nanoscale structure of cement: viewpoint of rigidity theory. The Journal
of Physical Chemistry C, 118(23):12485-12493, 2014.
[91
D. P. Bentz, 0. M. Jensen, K. K. Hansen, J. F. Olesen, H. Stang, and C.-J.
Haecker. Influence of cement particle-size distribution on early age autogenous
strains and stresses in cement-based materials. Journal of the American Ceramic
Society, 84(1):129-135, 2001.
[10] Y. Benveniste. A new approach to the application of Mori-Tanaka's theory in
composite materials. Mechanics of materials, 6(2):147-157, 1987.
[11] A.-P. Bois, A. Garnier, G. Galdiolo, and J.-B. Laudet. Use of a mechanistic model
to forecast cement-sheath integrity. SPE Drilling & Completion, 27(02):303-314,
2012.
[121 A.-P. Bois, A. Garnier, F. Rodot, J. Saint-Marc, and N. Aimard. How to prevent loss of zonal isolation through a comprehensive analysis of microannulus
formation. SPE Drilling & Completion, 26(1):13-31, 2011.
[13j A. Bonett and D. Pafitis. Getting to the root of gas migration. Qilfield Review,
8(1):36-49, 1996.
[14] 0. L. Bowie and C. E. Freese. Elastic analysis for a radial crack in a circular
ring. Engineering Fracture Mechanics, 4(2):315-321, 1972.
[151 H. J. H. Brouwers. The work of Powers and Brownyard revisited: Part 1. Cement
and Concrete Research, 34(9):1697-1716, 2004.
[16] H. F. Bueckner. On a class of singular integral equations. Journal of Mathematical Analysis and Applications, 14(3):392-426, 1966.
.
[17] S. Chatterji and J. W. Jeffery. Crystal growth during the hydration of CaSO 4
1/2H 2 0. Nature, 200:463-464, November 1963.
186
118] G. Constantinides. The elastic properties of calcium leached cement pastes and
mortars : a multi-scale investigation. Master's thesis, Massachusetts Institute of
Technology, 2002.
[19] G. Constantinides. InvariantMechanical Propertiesof Calcium-Silicate-Hydrates
(C-S-H) in Cement-Based Materials: Instrumented Nanoindentationand Microporomechanical Modeling. PhD thesis, Massachusetts Institute of Technology,
2006.
[201 G. Constantinides and F.-J. Ulm. The effect of two types of csh on the elasticity
of cement-based materials: Results from nanoindentation and micromechanical
modeling. Cement and Concrete Research, 34(1):67-80, 2004.
[21] G. Constantinides and F.-J. Ulm. The nanogranular nature of C-S-H. Journal
of the Mechanics and Physics of Solids, 55(1):64-90, 2007.
1221 T. S. Cook and F. Erdogan. Stresses in bonded materials with a crack perpendicular to the interface. InternationalJournal of Engineering Science, 10(8):677697, 1972.
[231 C. E. Cooke, M. P. Kluck, and R. Medrano. Field measurements of annular pressure and temperature during primary cementing. Journal of Petroleum Technology, 35(08):1-429, 1983.
1241 0. Coussy. Poromechanics. John Wiley & Sons, Chichester, UK, 2004.
125] F. Delale and F. Erdogan. Stress intensity factors in a hollow cylinder containing
a radial crack. InternationalJournal of Fracture, 20(4):251-265, 1982.
[261 S. Diamond. The microstructure of cement paste and concrete-a visual primer.
Cement and Concrete Composites, 26(8):919-933, 2004.
&
[271 L. Dormieux, D. Kondo, and F.-J. Ulm. Microporomechanics. John Wiley
Sons, Chichester, UK, 2006.
187
128] Arthur Henry England. Complex variable methods in elasticity. Dover Publications, Inc., 2012.
129] F. Erdogan, G. D. Gupta, and T. S. Cook. Numerical solution of singular integral
equations. In Methods of analysis and solutions of crack problems, pages 368-425.
Springer, Netherlands, 1973.
[30] J. D. Eshelby. The determination of the elastic field of an ellipsoidal inclusion,
and related problems.
Proceedings of the Royal Society of London. Series A.
Mathematical and Physical Sciences, 241(1226):376-396, 1957.
[31J R. F. Feldman and P. J. Sereda. A model for hydrated portland cement paste
as deduced from sorption-length change and mechanical properties. Materiaux
et construction, 1(6):509-520, 1968.
1321 K. J. Goodwin and R. J. Crook. Cement sheath stress failure. SPE Drilling
Engineering, 7(4):291-296, 1992.
[33] E. Grabowski and J. E. Gillott. Effect of replacement of silica flour with silica
fume on engineering properties of oilwell cements at normal and elevated temperatures and pressures. Cement and Concrete Research, 19(3):333-344, 1989.
[34j G. D. Gupta and F. Erdogan. The problem of edge cracks in an infinite strip.
Journal of applied Mechanics, 41(4):1001-1006, 1974.
[35] Ch. Hellmich, F.-J. Ulm, and H. A. Mang.
Consistent linearization in finite
element analysis of coupled chemo-thermal problems with exo-or endothermal
reactions. Computational mechanics, 24(4):238-244, 1999.
[36] A. V. Hershey.
The elasticity of an isotropic aggregate of anisotropic cubic
crystals. Journal of Applied Mechanics, 21(3):236-240, 1954.
137]
C. G. Hoover and F.-J. Ulm. Experimental chemo-mechanics of early-age fracture
properties of cement paste. Cement and Concrete Research, 75:42-52, 2015.
188
[381 R. W. Howarth, A. Ingraffea, and T. Engelder. Natural gas: Should fracking
stop? Nature, 477(7364):271-275, 2011.
[39] R. W. Howarth, R. Santoro, and A. Ingraffea. Methane and the greenhouse-gas
footprint of natural gas from shale formations. Climatic Change, 106(4):679-690,
2011.
1401 MDM Energy Inc. Oil & gas procedures. http://www.mdmenergy.com/oil_
gas-procedure
.html,
2007. Accessed: 2014-12-05.
[411 A. R. Ingraffea, M. T. Wells, R. L. Santoro, and S. B. C. Shonkoff.
Assess-
ment and risk analysis of casing and cement impairment in oil and gas wells
in pennsylvania, 2000-2012. Proceedings of the National Academy of Sciences,
111(30):10955-10960, 2014.
[42] N. I. loakimidis and P. S. Theocaris. The numerical evaluation of a class of generalized stress intensity factors by use of the lobatto-jacobi numerical integration
rule. InternationalJournal of Fracture, 14(5):469-484, 1978.
143] IPCC. Climate change 2014: Mitigation of climate change. Contributionof Working Group III to the Fifth Assessment Report of the Intergovernmental Panel on
Climate Change, 2014.
[441
G. R. Irwin. Analysis of stresses and strains near the end of a crack traversing
a plate. Journal of Applied Mechanics, 1957.
[45] R. B. Jackson, A. Vengosh, T. H. Darrah, N. R. Warner, A. Down, R. J. Poreda,
S. G. Osborn, K. Zhao, and J. D. Karr.
Increased stray gas abundance in a
subset of drinking water wells near marcellus shale gas extraction. Proceedings
of the National Academy of Sciences, 110(28):11250-11255, 2013.
[461 H. M. Jennings. A model for the microstructure of calcium silicate hydrate in
cement paste. Cement and Concrete Research, 30(1):101-116, 2000.
189
[471
A. Kabir, M. Hasan, and M. K. Miah. Predicting 28 days compressive strength of
concrete from 7 days test result. In InternationalConference on Advances in Design and Construction of Structures. The Association of Civil and Environmental
Engineers, 2012.
[48]
A. C. Kaya and F. Erdogan. On the solution of integral equations with strongly
singular kernels. Quarterly of Applied Mathematics, 45(1):105-122, 1987.
[49] G. V. Kolosov. On an application of complex function theory to a plane problem
of the mathematical theory of elasticity. Yuriev, Russia, 1909.
[50] K. J. Krakowiak, J. J. Thomas, S. Musso, S. James, A.-T. Akono, and F.-J.
Ulm. Nano-chemo-mechanical signature of conventional oil-well cement systems:
Effects of elevated temperature and curing time. Cement and Concrete Research,
67:103-121, 2015.
[51] E. Kr6ner. Kontinuumstheorie der versetzungen und eigenspannungen, volume 5.
Springer, 1958.
[52]
H. A. Luo and Y. Chen. An edge dislocation in a three-phase composite cylinder
model. Journal of Applied Mechanics, 58(1):75-86, 1991.
[53] E. Masoero, E. Del Gado, R. J.-M. Pellenq, F.-J. Ulm, and S. Yip. Nanostructure
and nanomechanics of cement: polydisperse colloidal packing. Physical review
letters, 109(15):155503, 2012.
[54] E. Masoero, E. Del Gado, R. J.-M. Pellenq, Sidney Yip, and Franz-Josef Ulm.
Nano-scale mechanics of colloidal c-s-h gels. Soft Matter, 10(3):491-499, 2014.
[55]
P. J. McDonald, V. Rodin, and A. Valori. Characterisation of intra-and inter-cs-h gel pore water in white cement based on an analysis of nmr signal amplitudes
as a function of water content. Cement and Concrete Research, 40(12):1656-1663,
2010.
190
[56J
T. Mori and K. Tanaka. Average stress in matrix and average elastic energy of
materials with misfitting inclusions. Acta metallurgica, 21(5):571-574, 1973.
&
[571 T. Mura. Micromechanics of defects in solids, volume 3. Springer Science
Business Media, 1987.
[58]
N. 1. Muskhelishvili. Some basic problems of the mathematical theory of elasticity: fundamental equations, plane theory of elasticity, torsion, and bending.
Translated from the Russian by J. R. M. Radok. P. Noordhoff Ltd, 1953.
[59] N. I. Muskhelishvili. Singular integral equations: boundary problems of function
theory and their application to mathematicalphysics. Courier Corporation, 2008.
1601 E. B. Nelson. Well cementing. Newnes, 1990.
1611 A. M Neville. Properties of concrete. Pitman Publishing Limited, London, UK,
3 edition, 1981.
162] S. G. Osborn, A. Vengosh, N. R. Warner, and R. B. Jackson. Methane contamination of drinking water accompanying gas-well drilling and hydraulic fracturing.
Proceedings of the National Academy of Sciences, 108(20):8172-8176, 2011.
[63] B. Pichler, C. Hellmich, and J. Eberhardsteiner. Spherical and acicular representation of hydrates in a micromechanical model for cement paste: prediction
of early-age elasticity and strength. Acta Mechanica, 203(3-4):137-162, 2009.
[64] A. D. Polyanin and A. V. Manzhirov. Handbook of Integral Equations. CRC
Press, 2012.
[65] T. Powers. Absorption of water by portland cement paste during the hardening
process. Industrial & Engineering Chemistry, 27(7):790-794, 1935.
[661 T. C. Powers. Structure and physical properties of hardened portland cement
paste. Journal of the American Ceramic Society, 41(1):1-6, 1958.
[67] T. C. Powers and T. L. Brownyard. Studies of the physical properties of hardened
portland cement paste. Bulletin, 22, 1947.
191
168] M. J. A. Qomi, M. Bauchy, F.-J. Ulm, and R. J.-M. Pellenq.
Anomalous
composition-dependent dynamics of nanoconfined water in the interlayer of disordered calcium-silicates. The Journal of chemical physics, 140(5):054515, 2014.
169] M. J. A. Qomi, M. Bauchy, F.-J. Ulm, and R. J.-M. Pellenq. Polymorphism and
its implications on structure-property correlation in calcium-silicate-hydrates. In
Nanotechnology in Construction, pages 99-108. Springer, 2015.
[70] M. J. A. Qomi, K. J. Krakowiak, M. Bauchy, K. L. Stewart, R. Shahsavari,
D. Jagannathan, D. B. Brommer, A. Baronnet, M. J. Buehler, and S. Yip. Combinatorial molecular optimization of cement hydrates. Nature Communications,
5, 2014.
171] K. Ravi, M. Bosma, and 0. Gastebled. Improve the economics of oil and gas
wells by reducing the risk of cement failure. In IADC/SPE Drilling Conference.
Society of Petroleum Engineers, 2002.
1721 J. R. Rice. Mathematical analysis in the mechanics of fracture. Fracture: An
Advanced Treatise, 2:191-311, 1968.
[73] J. R. Rice. Some remarks on elastic crack-tip stress fields. InternationalJournal
of Solids and Structures, 8(6):751-758, 1972.
1741 J. Sanahuja, L. Dormieux, and G. Chanvillard. Modelling elasticity of a hydrating cement paste. Cement and Concrete Research, 37(10):1427-1439, 2007.
175] A. Sieminski. International energy outlook 2013. U. S. Energy Information
Administration, 2013.
1761 L. Stefan, F. Benboudjema, J.-M. Torrenti, and B. Bissonnette. Prediction of
elastic properties of cement pastes at early ages. Computational Materials Science, 47(3):775-784, 2010.
177] E. Stokstad.
Will fracking put too much fizz in your water?
344(6191):1468-1471, 2014.
192
Science,
[78]
D. 0. Swenson and C. A. Rau.
The stress distribution around a crack per-
pendicular to an interface between materials. InternationalJournal of Fracture
Mechanics, 6(4):357-365, 1970.
[791 J. H. Taplin. A method for following the hydration reaction in portland cement
paste. Australian Journal of Applied Science, 10(3):329-345, 1959.
[801 H. F. W. Taylor. Cement chemistry. Thomas Telford, 1997.
1811 P. D. Tennis and H. M. Jennings. A model for two types of calcium silicate
hydrate in the microstructure of portland cement pastes. Cement and Concrete
Research, 30(6):855-863, 2000.
[82] J. J. Thomas, S. A. FitzGerald, D. A. Neumann, and R. A. Livingston. State of
water in hydrating tricalcium silicate and portland cement pastes as measured
by quasi-elastic neutron scattering. Journal of the American Ceramic Society,
84(8):1811-1816, 2001.
183] J. J. Thomas, H. M. Jennings, and A. J. Allen. The surface area of cement paste
as measured by neutron scattering: evidence for two csh morphologies. Cement
and Concrete Research, 28(6):897-905, 1998.
[841 Stephen P Timoshenko and JN Goodier.
Theory of elasticity.
International
Journal of Bulk Solids Storage in Silos, 1(4), 2014.
1851 P. G. Tracy. Elastic analysis of radial cracks emanating from the outer and inner
surfaces of a circular ring. EngineeringFractureMechanics, 11(2):291-300, 1979.
[861 F.-J. Ulm, M. Abuhaikal, T. A. Petersen, and R. J.-M. Pellenq. Poro-chemofracture-mechanics... bottom-up: Application to risk of fracture design of oil
and gas cement sheath at early ages.
Computational Modelling of Concrete
Structures, page 61, 2014.
[87] F.-J. Ulm, G. Constantinides, and F. H. Heukamp. Is concrete a poromechanics
materials?aATa multiscale investigation of poroelastic properties. Materials and
Structures, 37(1):43-58, 2004.
193
[88] F.-J. Ulm, M. Vandamme, C. Bobko, A. J. Ortega, K. Tai, and C. Ortiz. Statistical indentation techniques for hydrated nanocomposites: concrete, bone, and
shale. Journal of the American Ceramic Society, 90(9):2677-2692, 2007.
189] G. J. Verbeck and R. H. Helmuth. Structures and physical properties of cement
paste. In Proceedings of the 5th InternationalSymposium on the Chemistry of
Cement, volume 3(1), pages 1-32, 1968.
[90] X. Wang and Y.-P. Shen. An edge dislocation in a three-phase composite cylinder
model with a sliding interface.
Journal of Applied Mechanics, 69(4):527-538,
2002.
[911 A. Zaoui. Continuum micromechanics: survey. Journal of Engineering Mechanics, 128(8):808-816, 2002.
192] M. Zhang and S. Bachu. Review of integrity of existing wells in relation to CO 2
geological storage: What do we know? InternationalJournal of Greenhouse Gas
Control, 5(4):826-840, 2011.
194