Microporoelastic Modeling of Organic-Rich Shales ARCIW
by
MASSACHUSETTS INSTITUTE
Siavash Khosh Sokhan Monfared
MA 052015
B.S., University of Oklahoma (2012)
LIBRARIES
MAY 0 5 2015
Submitted to the Department of Civil and Environmental Engineering
in partial fulfillment of the requirements for the degree of
Master of Science in Civil and Environmental Engineering
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
February, 2015
@
Massachusetts Institute of Technology 2015. All rights reserved.
Signature redacted
A uthor ....................
Department of Civil and Environmental Engineering
January 21, 2015
Signature redacted
C ertified by ............
.........
Franz-Josef Ulm
Professor of Civil and Environmental Engineering
Thesis Supervisor
Signature redacted
A ccepted by ..........
...............
Heidi M. Nepf
Donald and Martha Harleman Professor of Civil and Environmental
Engineering
Chair, Graduate Program Committee
2
Microporoelastic Modeling of Organic-Rich Shales
by
Siavash Khosh Sokhan Monfared
Submitted to the Department of Civil and Environmental Engineering
on January 21, 2015, in partial fulfillment of the
requirements for the degree of
Master of Science in Civil and Environmental Engineering
Abstract
Due to their abundance, organic-rich shales are playing a critical role in re-defining
the world's energy landscape leading to shifts in global geopolitics. However, technical challenges and environmental concerns continue to contribute to the slow growth
of organic-rich shale exploration and exploitation worldwide. The engineering and
scientific challenges arise from the extremely heterogeneous and anisotropic nature of
these naturally occurring geo-composites at multiple length scales. Specifically, the
anisotropic poroelastic behavior of organic-rich shales becomes of critical importance
for petroleum engineers. Thus, the focus of this thesis is to capture mechanisms of
first-order contribution to the effective anisotropic poroelasticity of organic-rich shales
which can pave the way for more efficient and effective exploration and exploitation. We introduce an original approach for micromechanical modeling of organicrich shales which accounts for the effect of organic maturity on the overall anisotropic
poroelasticity through morphology considerations. This morphology contribution is
captured by means of an effective media theory that bridges the gap between immature and mature systems through the choice of the system's microtexture; namely
a matrix-inclusion morphology (Mori-Tanaka) for immature systems and a polycrystal/granular morphology for mature systems. Also, we show that interfaces play a role
on the effective elasticity of mature organic-rich shales. The models are calibrated by
means of ultrasonic pulse velocity measurements of elastic properties and validated by
means of lab measured nanoindentation data. Sensitivity analyses using Spearman's
Partial Rank Correlation Coefficient show the importance of porosity and Total Organic Carbon (TOC) as key input parameters for accurate model predictions. These
models' developments provide a mean to define a "unique" set of clay elasticity. They
also highlight the importance of the depositional environment, burial and diagenetic
processes on overall mechanical and poromechanical behavior of organic-rich shales.
Thesis Supervisor: Franz-Josef Ulm
Title: Professor of Civil and Environmental Engineering
3
4
Acknowledgments
First and foremost, I would like to express my gratitude to my advisor, Franz-Josef
Ulm. His continuous support, encouragements and patience have been instrumental
to this work. His insightful perspective on a variety of topics leaves no dead ends for
his students. I have cherished every one of our discussions and will look forward to
many more during the course of my PhD.
I would also like to acknowledge my undergraduate mentor, Younane Abousleiman.
He re-kindled my desire for knowledge and helped me embark on a satisfying journey,
given all the challenges that I have encountered and I am expecting to face in the
future. I remain his mentee to date.
I am also thankful to the financial support provided by X-Shale Project and CSHHub. I wish to acknowledge the help provided and the insightful comments of Ronny
Hofmann of Shell International Exploration and Production as well as Romain Prioul
of Schlumberger-Doll Research Center. I am indebted to Alberto Ortega for being
forthcoming during the course of this work. He helped me navigate through his PhD
work on which my thesis is partly built on. Also, special thanks to Sara Abedi with
whom I held many stimulating discussions that shaped my ideas for this work.
Lastly, I am truly grateful for my parents, Mehdi and Taji.
Their unconditional
love and support give me a strength and confidence beyond imagination.
5
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6
Contents
Industrial Context & Research Motivations . . . . . . . . . . . . . .
20
1.2
Problem Statement & Research Objectives . . . . . . . . . . . . . .
21
1.3
T hesis O utline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
1.4
N otations
24
.
.
.
1.1
.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Multi-Scale Nature of Organic-Rich Shales
26
27
2.1
Depositional Systems . . . . . . . .
. . . . . . . . . . . . . .
27
2.2
M ineralogy
. . . . . . . . . . . . . .
28
2.3
Porosity .............
. . . .... . . ... ..
29
2.4
Organics & Organic Maturity . . .
. . . . . . . . . . . . . .
30
2.5
Chapter Summary
. . . . . . . . . . . . . .
33
.
. . . . . . . . .
.
.
.....
.
. . . . . . . . . . . . .
.
Elements of Multi-Scale Petrophysics of Organic-Rich Shales
35
3.1
Multi-Scale Structural Thought-Model of Organic-Rich Shales
35
3.1.1
Level 0: Clay . . . . . . . . . . . . . . . . . . . . . .
36
3.1.2
Level I: Clay, Kerogen & Porosity . . . . . . . . . . .
36
3.1.3
Level II: Porous Solid & Inclusions
. . . . . . . . . .
36
. . . . . . . . . . . . . . . . . . . . . . .
37
3.2
Chapter Summary
.
.
.
Multi-Scale Representation of Organic-Rich Shales
.
3
19
.
2
Introduction
.
II
18
.
1
General Presentation
.
I
7
Multi-Scale Material Characterization & Properties
. . . . . . . . .
40
4.2
Macroscopic C 13 Estimation . . . . . . . . . . . . .
. . . . . . . . .
43
4.3
Instrumented Nanoindentation . . . . . . . . . . . .
. . . . . . . . .
45
4.4
Calibration Data Sets . . . . . . . . . . . . . . . . .
. . . . . . . . .
47
4.5
Validation Data Sets . . . . . . . . . . . . . . . . .
. . . . . . . . .
52
4.6
Phase Properties
. . . . . . . . . . . . . . . . . . .
. . . . . . . . .
57
4.7
Chapter Summary
. . . . . . . . . . . . . . . . . .
. . . . . . . . .
58
.
.
.
.
.
Theoretical Background & Model Developments
59
Elements of Microporomechanics
61
Scale Separability Conditions
. . . . . . . . . . . . . . . . . .
. . .
61
5.2
Hom ogenization . . . . . . . . . . . . . . . . . . . . . . . . . .
. . .
62
5.3
Inclusion-Based Effective Estimates . . . . . . . . . . . . . . .
. . .
64
5.4
Hill Concentration Tensor . . . . . . . . . . . . . . . . . . . .
. . .
68
5.4.1
Spheroidal Inclusion in an Isotropic Medium . . . . . .
. . .
69
5.4.2
Spheroidal Inclusion in a Transversely Isotropic Medium
. . .
69
.
.
.
.
.
5.1
Approximation Schemes: Self-consistent and Mori-Tanaka
. .
. . .
71
5.6
Im perfect Interfaces . . . . . . . . . . . . . . . . . . . . . . . .
. . .
72
5.7
Chapter Summ ary
. . .
76
.
.
5.5
77
6.1
Hypothesis Testing: Maturity Induced Morphological Change
77
6.2
Basis of Design: A Bread Analogy . . . . . . . . . . . . . . . .
79
6.3
Imperfect Interfaces: Organic Maturity Evolution . . . . . . .
82
6.3.1
The Quenching Problem . . . . . . . . . . . . . . . . .
82
Immature Organic-Rich Shale . . . . . . . . . . . . . . . . . .
86
6.4.1
Volum e Fractions . . . . . . . . . . . . . . . . . . . . .
86
6.4.2
L evel I . . . . . . . . . . . . . . . . . . . . . . . . . . .
87
6.4.3
L evel II
. . . . . . . . . . . . . . . . . . . . . . . . . .
93
.
.
.
.
.
6.4
.
Microporoelastic Model for Organic-Rich Shales
.
.
. . . . . . . . . . . . . . . . . . . . . . . .
.
6
.
Elastic Waves in a Transversely Isotropic Medium .
III
5
39
4.1
.
4
8
6.7
IV
Volume Fractions . . . . . .
. . . . . . . . . . . . . . .
98
6.5.2
Level I . . . . . . . . . . . .
. . . . . . . . . . . . . . .
100
6.5.3
Level II . . . . . . . . . . .
. . . . . . . . . . . . . . . 100
Undrained Behavior . . . . . . . . .
. . . . . . . . . . . . . . . 107
6.6.1
Immature Organic-Rich Shale
. . . . . . . . . . . . . . . 10 8
6.6.2
Mature Organic-Rich Shale
. . . . . . . . . . . . . . . 108
.
.
.
.
6.5.1
Chapter Summary
. . . . . . . . .
. . . . . . . . . . . . . . .
Results
110
111
113
7.1
. . . . . . . . . . . . . . . . . . . . . .
113
7.1.1
Procedure . . . . . . . . . . . . . . . . . . .
113
7.1.2
Calibration Input . . . . . . . . . . . . . . .
115
7.1.3
Calibration Results . . . . . . . . . . . . . .
118
Validation . . . . . . . . . . . . . . . . . . . . . . .
122
7.2.1
Procedure . . . . . . . . . . . . . . . . . . .
122
7.2.2
Validation: Grain Scale Clay Properties (Level 0)
122
7.2.3
Validation: Indentation Data (Level I)
7.2.4
Validation: Dynamic Properties (Level II)
7.2.5
.
.
.
.
.
. . .
125
.
128
Discussions
. . . . . . . . . . . . . . . . . .
129
Chapter Summary
. . . . . . . . . . . . . . . . . .
130
.
.
.
7.3
C alibration
.
Model Calibration & Validation
7.2
Sensitivity Analysis
131
Quality Given Uncertainty in C!""............
. . . .
8.1
Inversion
8.2
Dependence of Output Variance to Different Input Parameters
8.3
132
139
Immature Organic-Rich Shale Model . . . . . . . . .
. . . .
141
8.2.2
Mature Organic-Rich Shale Model . . . . . . . . . . .
. . . .
145
8.2.3
Poroelastic Coefficients' Sensitivity Analyses . . . . .
Chapter Summary
150
.
.
.
8.2.1
. . . . . . . . . . . . . . . . . . . . . . .
.
8
98
.
7
. . . . . . . . . . . . . . .
.
6.6
Mature Organic-Rich Shale . . . . .
.
6.5
9
. . . .
150
V
Conclusions
160
9 Discussion of Results & Future Perspectives
161
9.1
Summary of Main Findings
. . . . . . . . . . . . . . . . . . . . . . .
162
9.2
Limitations & Future Perspectives . . . . . . . . . . . . . . . . . . . .
165
A Nomenclatures
167
10
List of Figures
1-1
Adopted Cartesian coordinate system in this thesis. . . . . . . . . . .
25
3-1
A schematic representation of multi-scale thought model discussed.
.
37
4-1
A typical nanoindentation load-displacement curve.
. . . . . . . . . .
45
4-2
Quality check of the elasticity data by comparing static and dynamic
stiffness coefficients. Sample B5 is not consistent with other samples
and thus it will not be considered for the subsequent analyses. Note
Cij values in (a),(b),(c) and (d) refer to the macroscopic elasticity. . .
6-1
51
Schematic of the multi-scale microporoelastic model for immature organicrich shales. Inclusion stiffness at level II is computed by homogenizing
the dominant non-clay minerals in a self-consistent manner. Following
the hypothesis of texture effect; Mori-Tanaka approximation scheme is
applied at each scale for homogenization. For immature systems, interfaces are considered to be perfect (perfect bonding) among different
constituents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
97
6-2
Schematic for the multi-scale microporoelastic model for mature organicrich shales. Inclusion stiffness at level II is computed by homogenizing
the dominant non-clay minerals in a self-consistent scheme. Following
the hypothesis testing approach regarding texture; a self-consistent approximation scheme is applied at each scale for homogenization. Furthermore, interfaces are considered to be slightly weakened at level
II following the the bread based analogy and the discussion accompanied with the quenching problem presented earlier. In addition, a
self-consistent morphology entails a self-consistent porosity distribution; hence porous inclusions.
7-1
. . . . . . . . . . . . . . . . . . . . . .
106
SEM images of a Haynesville shale samples. Based on these images, a
grain radius of 2 pm was chosen as the input for the imperfect interface model associated with the mature organic-rich shale model used
for downscaling macroscopic Haynesville elasticity data. Also, the existence of pores in the inclusion on the bottom right is a noteworthy feature, consistent with our self-consistent porosity distribution assumption.117
7-2
Measured vs predicted macroscopic elasticity of Woodford shale; representative of an immature organic-rich shale system. "dr" and "un"
refer to drained and undrained responses. . . . . . . . . . . . . . . . . 120
7-3
Measured vs predicted macroscopic elasticity of Haynesville shale; representative of a mature organic-rich shale system. "dr" and "un" refer
to drained and undrained responses . . . . . . . . . . . . . . . . . . .
7-4
121
Measured vs predicted indentation moduli for different shale formations in x
direction.
Marcellus*refers to computations considering'
negligible kerogen elasticity while Marcellus includes kerogen elasticity. See Section 7.2.3 for more details. . . . . . . . . . . . . . . . . . .
12
126
7-5
Measured vs predicted indentation moduli for different shale formations in
x3
direction. Marcellus* refers to computations considering
negligible kerogen elasticity while Marcellus includes kerogen elasticity. See Section 7.2.3 for more details. . . . . . . . . . . . . . . . . . .
8-1
127
Normal distributions prescribed to the macroscopic C11"" of each Woodford sample. These serve as inputs for assessing the influence of uncertainty in estimation of CIIu" on the "grain scale" values through the
model for immaure organic-rich shales. . . . . . . . . . . . . . . . . . 135
8-2
Histograms of output for each Woodford sample, i.e. stiffness coefficients of clay at level 0 ("grain scale"), obtained by introducing uncertainty in macroscopic CIt"" and the inversion of the macroscopic
elasticity through the model for immature organic-rich shales. ....
8-3
Fitted probability density function (PDF) for each Woodford smaple
and the "experimental" PDF obtained by Monte-Carlo simulations.
8-4
136
. 137
Fitted cumulative density functions (CDF) for each Woodford sample
and "experimental" CDF obtained from Monte-Carlo simulations. . . 138
8-5
PRCC result displaying the sensitivity of the outputs (defined on the
abscissa) to different input parameters (defined in the legend) for the
immature organic-rich shale model. . . . . . . . . . . . . . . . . . . . 143
8-6
PRCC result displaying the sensitivity of the indentation moduli (defined on the abscissa) to different input parameters (defined in the
legend) for the immature organic-rich shale model . . . . . . . . . . . 144
8-7
PRCC result displaying the sensitivity of the outputs (defined on the
abscissa) to different input parameters (defined in legend) for the mature organic-rich shale model assuming uniform distributions for model
parameters associated with imperfcet interface model. . . . . . . . . . 147
13
8-8
PRCC result displaying the sensitivity of the outputs (defined on the
abscissa) to different input parameters (defined in the legend) for the
mature organic-rich shale model assuming normal distributions for input parameters associated with imperfect interface model.
8-9
. . . . . . 148
PRCC result displaying the sensitivity of the indentation moduli, at
level I, to different input parameters (defined in the legend) for the
mature organic-rich model. Note interface parameters do not interfere
at level I. .......
.... .. . .............. . ... .. .149
8-10 PRCC result displaying the sensitivity of Biot modulus, N 1 , and Biot
pore pressure coefficients, a1 ,1 and a 3 , at level I of the immature
organic-rich shale model to the stochastically defined input parameters. 152
,
8-11 PRCC result displaying the sensitivity of overall Biot modulus, M 1
and Skempton pore pressure build-up coefficients, BI,
and B3, I, at
level I of the immature organic-rich shale model to the stochastically
defined input parameters.
. . . . . . . . . . . . . . . . . . . . . . . .
153
8-12 PRCC result displaying the sensitivity of Biot modulus, N11 , and Biot
pore pressure coefficients, ai,,
and a 3,11 , at level II of the mature
organic-rich shale model to the stochastically defined input parameters.
. .. ......
.
....
. .
.. . .............
.... .... . 154
,
8-13 PRCC result displaying the sensitivity of overall Biot modulus, M 11
and Skempton pore pressure build-up coefficients, B 1 ,11 and B 3 ,11 , at
level II of the immature organic-rich shale model to the stochastically
defined input parameters.
. . . . . . . . . . . . . . . . . . . . . . . .
155
8-14 PRCC result displaying the sensitivity of Biot modulus, N 1 , and Biot
pore pressure coefficients, aij and a 3 ,1 , at level I of the mature organicrich shale model to the stochastically defined input parameters.
. .
156
8-15 PRCC result displaying the sensitivity of overall Biot modulus, M 1 , and
Skempton pore pressure build-up coefficients, BI,
qnd B3, I at level
I of the mature organic-rich shale model to the stochastically defined
input param eters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
14
8-16 PRCC result displaying the sensitivity of Biot modulus, N11, and Biot
pore pressure coefficients, czi,u and a 3 ,11 , at level II of the mature
organic-rich shale model to the stochastically defined input parameters.
. . . .. . . ....
.. . ...
. . . . . . . . . . . . . . . . . . . . .
158
,
8-17 PRCC result displaying the sensitivity of overall Biot modulus, M 11
and Skempton pore pressure build-up coefficients, B 1,11 and B 3 ,11 , at
level II of the mature organic-rich shale model to the stochastically
defined input parameters.
. . . . . . . . . . . . . . . . . . . . . . . .
15
159
THIS PAGE INTENTIONALLY LEFT BLANK
16
List of Tables
2.1
Mineral densities used for volume fraction calculations [61][56]. ....
4.1
Mineralogy and kerogen content of Woodford shale samples in [mass
29
%] [95] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
4.2
Bulk density of Woodford shale samples [61]. . . . . . . . . . . . . . .
48
4.3
Porosity of Woodford shale samples in [%] [95].
. . . . . . . . . . . .
48
4.4
Reported elasticity of Woodford Shale samples [61]. . . . . . . . . . .
49
4.5
Mineralogy and kerogen content of Haynesville shale samples in [mass
%] [4 1] ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
4.6
Bulk density of Haynesville shale samples [411. . . . . . . . . . . . . .
50
4.7
Porosity of Haynesville shale samples in [%] 1411 . . . . . . . . . . . .
50
4.8
Calculated elasticity from measured UPV (except for C1g"" which was
estimated by method presented in Section 4.2 from data in Ref.
4.9
[411.
51
Indentation moduli of Woodford shale samples as reported in Ref. [611
except for a correction for sample A2. . . . . . . . . . . . . . . . . . .
53
4.10 Measured Haynesville indentation moduli in x, 1 2].
. . . . . . . . . .
53
4.11 Measured Haynesville indentation moduli in x 3 [21.
. . . . . . . . . .
54
4.12 Mineralogy and kerogen content of Barnett shale sample in [mass %]
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
4.13 Porosity of Barnett shale sample in [%] [41]. . . . . . . . . . . . . . .
54
. . . . . . . . . . . .
54
[4 1].
4.14 Indentation moduli of Barnett shale sample [2].
4.15 Mineralogy and kerogen content of Antrim shale sample in [mass %] [41]. 55
4.16 Porosity of Antrim shale sample in [%] [41].
17
. . . . . . . . . . . . . .
55
4.17 Indentation moduli of Antrim shale sample [2]. . . . . . . . . . . . . .
55
4.18 Mineralogy and kerogen content of Marcellus shale samples in [mass
%] [4 1] ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
4.19 Porosity of Marcellus shale sample in [% 1411. . . . . . . . . . . . . .
55
4.20 Measured indentation moduli in x, on Marcellus shale smaples
[2].
56
4.21 Measured indentation moduli in x 3 on Marcellus shale smaples
12]. .
56
4.22 (quasi-)isotropic elasticity of different minerals.
. . . . . . . . . . . .
6.1
Linear thermal expansion coefficients for various geomaterials. ....
7.1
Calculated inclusion volume fractions of Woodford shale samples (level
II). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2
. . . . . . . . . . . . . . . . . . . . . . . . . .
116
116
Calculated volume fractions of clay, kerogen and porosity of Haynesville
shale sam ples (level I).
7.5
1 16
Calculated volume fraction of porous inclusions of Haynesville shale
sam ples (level II). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4
85
Calculated clay and kerogen volume fractions and porosity of Woodford
shale sam ples (level I).
7.3
58
. . . . . . . . . . . . . . . . . . . . . . . . . .
116
"Grain scale" elasticity and interface parameters obtained by downscaling measured macroscopic elasticities of Woodford and Haynesville
shale samples. In the case of Haynesville, an inclusion grain radius of
2 /pm was used for the mature organic-rich shale model. . . . . . . . .
7.6
Computed "grain scale" indentation moduli (level 0) for clay values
obtained by inversion of measured elasticity as reported in Table 7.5.
7.7
119
Means and standard deviations of relative error between macroscopi. . . . . . . . . . .
119
Some reported anisotropic clay elasticity in the literature. . . . . . . .
124
cally measured and predicted elasticity (level II).
7.8
119
18
7.9
Riemannian distance between different elasticity tensors reported in
Table 7.8 and values obtained by downscaling Woodford and Haynesville macroscopic elasticity (see Table 7.5), as a metric to assess
the similarities between reported and obtained values in the Reimannian space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
124
7.10 Measured vs predicted Thomsen parameters for Woodford shale samples. 128
7.11 Measured vs predicted Thomsen parameters for Haynesville shale samp les.
8.1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
128
Defined mean and standard deviation for describing a normal distribution for each C 1"" of Woodford shale samples. These values are used
as inputs in the downscaling procedure to assess uncertainty in grain
scale clay elasticity (level 0). . . . . . . . . . . . . . . . . . . . . . . .
8.2
134
Means and standard deviations obtained for stiffness coefficients of
clay (level 0), after downscaling macrosopic elasticity through the microporoelastic model for the immature organic-rich shale system. Note
that the only input parameter defined stochastically was Cij"" for each
in our defined
to C.lY
Woodford shale sample. Also C is equivalent
ii
ij
multi-scale thought-model. . . . . . . . . . . . . . . . . . . . . . . . . 134
8.3
Stochastically defined input paramters for the immature microporoelastic model. The result used for PRCC analysis.
8.4
. . . . . . . . . . .
142
Stochastically defined input parameters for the mature microporoelastic model needed for PRCC analyses. We assumed a uniform distribution for interface parameters . . . . . . . . . . . . . . . . . . . . . . .
8.5
146
Stochastically defined input parameters for the mature microporoelastic model needed for PRCC analyses. We assumed a normal distribution for interface parameters . . . . . . . . . . . . . . . . . . . . . . .
8.6
146
Stochastically defined input parameters for the PRCC analysis of poroelastic coefficients for mature and immature organic-rich shale models.
19
151
Part I
General Presentation
20
Chapter 1
Introduction
A recently published report by the Energy Information Administration (EIA) [1] estimates 345 billion barrels [MMbbl of technically recoverable shale oil, and 7,299 trillion cubic feet [TCF] of technically recoverable gas, worldwide. However, there remain
many engineering obstacles for exploiting these reserves. The challenges at the core of
the exploration and exploitation of organic-rich shales are associated with the where
and how of an effective and efficient exploitation. The where question focuses on locating "sweet spots"; referring to the zone that yield the highest volume of recoverable
hydrocarbon. "Sweet spots" are characterized by a suit of reservoir quality parameters
such as total organic carbon (TOC), maturity (reservoir fluid to be recovered is highly
correlated to the organic maturity) and porosity. The how question deals with designing and optimizing hydraulically induced fractures in the identified "sweet spots" to
stimulate production from a shale formation with an intrinsically low permeability.
Addressing these questions requires advanced characterization and modeling techniques associated with the challenges and uncertainty inherent to characterization of
thousands of feet into the subsurface, accounting for spatial variations of mineralogy,
porosity, TOC and its maturity as well as elastic anisotropy. The focus of this work
is to develop a microporomechanics based model that accounts for organic maturity
and poroelastic anisotropy. While previous attempts have been made to model shales
and organic-rich shales (see e.g. [97],161],[27],[105],1107],[79],[451,[108],17],[42]),
linking
organic maturity to the overall anisotropic poroelasticity of organic-rich shales is not
21
a trivial exercise [1021,[71],[70].
In addition, establishing a framework for integrat-
ing data at multiple length scales and consolidating geologic knowledge to improve
constraints on a model's inputs while addressing uncertainty associated with such a
framework remain a challenging task for rock physicists [32]. Furthermore, the basic
elasticity of clay minerals is yet to be rigorously characterized. Reported values in
the literature cover a wide range (see e.g.[78],[61J, [28],[35],[1031,[47],[65],[112],[16]).
Establishing these values is of fundamental importance for geophysicists and geomechanicians.
The objective of a rock physics model is to predict lithology and reservoir fluid away
from well control. This requires the identification of a link between poroelastic properties, microtexture and seismic properties of rocks, and how they vary with geologic
age, depth, and location
[341.
In a multi-scale framework explored in this thesis,
microtexture and morphology are linked to acoustic properties; accounting for the
presence of organic /inorganic constituents, the organic maturity (i.e. mature or immature), imperfect interfaces and anisotropic poroelasticity. Ultimately, the results
can be integrated into a comprehensive rock physics model to infer reservoir rock physical parameters by post-processing inverted seismic data. Seismic inversion refers to
the process of estimating elastic properties in the subsurface from seismic data. During a post-processing step, the elastic properties in time or depth can be transformed
into reservoir properties
1.1
[33].
Industrial Context & Research Motivations
The relationship between inicrotexture and anisotropic poroelasticity, in the context
of modeling of organic-rich shales, accounting for organic maturity is yet to be rigorously defined (see e.g.
102],[71],[70],[32J,[46]).
Developing a model with accurate
nredictive capabilities could lead to estimation of reservoir quality parameters from
post processing inverted seismic data with a higher confidence. This can be a valuable
tool to assist with estimating probable, proved and technically recoverable reserves,
22
that are all critical for asset management decisions.
A model that links microtexutre to anisotroic poroelasticity and seismic properties of
organic-rich shales is valuable in many ways. It would enable an exploration geophysicist to infer physical rock parameters from seismic data inversion. By integrating these
information into a geomodel built based on the regional geology and constrained by
well-logs, mud-logs, as well as core measurements, a geomodeler can produce spatial
distribution of various physical properties in the 3D volume of a target formation. A
reservoir engineer can utilize such models, knowing the uncertainty associated with it,
to evaluate with a higher confidence the economical prospect and to forecast reservoir
performance. Drilling engineers can utilize spatial variations of elastic anisotropy to
re-fine in-situ stress estimations and to re-evaluate wellbore stability analyses in problematic zones. Geomechanicians can employ elastic property maps to estimate the
spatial variation of energy release rates as a robust way to assess "fracability" of the
formation of interest, accounting for organic-rich shale heterogeneity and anisotropic
poroelasticity. Lastly, completions engineer can integrate this information to re-fine a
hydraulic fracturing job's pump schedule, increasing the overall efficiency and efficacy
of the operations in terms of money spent and resources used (e.g. water).
1.2
Problem Statement & Research Objectives
Organic-rich shale is an extremely complex, naturally occurring geo-composite. The
intricacy of organic-rich shale, in the context of its mechanical and poromechanical
properties, originates from the presence of organic/ inorganic heterogeneities, their
interfaces, as well as the occurrence of porosity and elastic anisotropy at multiple
length scales. The heterogeneous nature of organic-rich shale and its anisotropic behavior pose some challenges for characterization, modeling and engineering design.
Engineers often resort to "field experiments" and statistical analysis to correlate parameters of the stimulation design (e.g.
number of stages, number of clusters per
stage, number of perforations per cluster, perforation depth, perforation angle, type
23
and amount of proppants used) to well performance to assess completions design efficacy without properly accounting for the roles of elastic anisotropy and organic-rich
shale heterogeneity. Formation evaluation techniques, using e.g. well-logs, do not provide any information beyond their limited depths of investigations. This may have
been sufficient for sandstone and limestone reservoir characterization, where lateral
variation of properties are much more pronounced than horizontal.
In the case of
shale, given its heterogeneous nature, petroleum engineers ideally need spatial variations of properties in the 3D volume of the zone of interest to minimize costs and
maximize productions. Although cores provide values with relatively low uncertainty
in measurement compared to field characterization techniques, they only represent
discrete points in space and they are very costly to obtain.
Seismic surveys, if in-
verted through a proper model, could be a useful tool that can provide a sense of
spatial variations of various properties in the target formation, though one needs to
be aware of their intrinsically low resolution due to their characteristically long wave-
lengths.
With this background in mind, the objective of this thesis is to design a microporomechanical model for organic-rich shales; accounting for the presence of organics/inorganics, their interfaces, organic maturity and elastic anisotropy at multiple
length scales. These are of fundamental importance for a good rock physics model.
Our philosophy to address the defined objective is that of a realist, acknowledging the
large number of factors (often statistically correlated) that influence mechanical and
poromechanical behavior of organic-rich shales and the limited number of analytical
tools available; we seek to capture mechanisms that have a first-order contribution
on the effective behavior.
first established in Ref.
The developments to be presented follow the framework
[971 and further explored in Ref. [611. The originality of
our approach in this multi-scale modeling framework is due to the attribution of the
influence of maturity on poroelastic anisotropy and acoustic properties to a mechanistically "effective" texture effect.
24
1.3
Thesis Outline
Following this introduction chapter, Chapter 2 is dedicated to establishing some basic understanding on depositional systems, burial and diagenetic processes. Herein,
we recognize that the understanding of the mechanisms that give rise to the intricate mechanical and poromechancial behavior of organic-rich shales are critically
important for modeling purposes. Specifically, mineralogy, porosity, organic contents
and processes responsible for organic maturation are discussed in this chapter. In
Chapter 3, a structural thought-model is established which serves as the basis for
formulating the multi-scale microporoelastic model for organic-rich shales. Chapter 4
introduces the data sets utilized for calibration and validation of the microporoelastic
model for mature and immature organic-rich shales. For calibration, we employed
macroscopic elasticity measured on samples of Woodford and Haynesville formations,
representing immature and mature organic-rich shale systems respectively, by means
of ultra-sonic pulse velocity (UPV). For validation, measured microscopic elasticity
by means of instrumented nanoindentation were utilized. The validation data set
includes measurements on Woodford, Haynesville, Marcellus, Barnett, and Antrim
formations. Also in this chapter, the techniques used for characterizing elastic properties of shales at different length scales are briefly discussed.
Lastly, the method
used for approximating the C 13 stiffness coefficient (at macroscopic length scale) is
presented. Chapter 5 develops the theoretical microporomechanics based tools that
will be employed for the subsequent model developments. This includes the required
elements of the theory of homogenization and the relevant approximation schemes
as well as imperfect interfaces in the framework of inclusion based effective media
theories. Chapter 6 is dedicated to explicit derivation of microporoelastic representation of organic-rich shale model for both mature and immature systems. Chapter
7 presents the calibration and validation results, including calibration procedure and
inputs as well as the means used to assess the quality of calibration and validation
steps. A comprehensive sensitivity analyses, using the Spearman's Partial Rank Correlation Coefficient (PRCC) is presented in Chapter 8. The objective of the sensitivity
25
analyses is to study the sensitivity of the model output to variations in input parameters. In addition, a case study is presented to assess how estimation of C 13 (at the
macroscopic level) would alter calibration results for clay stiffness, at the grain scale.
Chapter 9 presents comprehensive discussions regarding modeling results; followed by
concluding remarks, limitations and future perspectives.
1.4
Notations
In this report, we assume organic-rich shale elasticity to exhibit transversely isotropic
elasticity at all considered length scales. In the theoretical framework to be developed herein, a Cartesian coordinate system shall be adopted where for a transversely
isotropic medium, the x 3 axis is perpendicular to the bedding plane (plane of isotropy)
and xi and x 2 directions are parallel to the bedding planes and perpendicular to the
axis of rotational symmetry (see Figure 1-1). Throughout this thesis,
sors are denoted by either Blackboard Bold font (e.g. A,B,...)
coordinate indices (e.g. AijikBijk1,...).
A,B,...)
(e.g.
2 nd
4 th
order ten-
or their 4 Cartesian
order tensors are denoted by Bold face
or their 2 Cartesian coordinate components (e.g. Aij,Bij, . . ).
1
st or-
der tensors (i.e. vectors) are shown with an underline (e.g. A,B,...) or with their 1
Cartesian coordinate component (e.g. Ai,Bj...).
shown in regular font (e.g. A,B,...).
oth
order tensors (i.e. scalars) are
In addition, we define the stiffness tensor for
a transversely isotropic medium, in Voigt's notations1 , with a normalized tonsorial
basis (see [23][22]), as follows:
-
C1111,C 12
C11 22,C 13
-
C 11 3 3 , C 33 = C 3333 , C 44
26
-
C1313
-
C 232 3 , C66 =
- C 12
)
1
[Ci.]
ClI
C 12
C 13
0
0
0
C12
CII
C 13
0
0
0
Cl3
C1 3
C 33
0
0
0
0
0
0
0
0
0
0
0
0
2C 44
0
0
0
0
0
0
2C 44
=
2C 6 6 = C-C
12
x.
x2
Figure 1-1 - Adopted Cartesian coordinate system in this thesis.
27
(1.1)
Part II
Multi-Scale Nature of Organic-Rich
Shales
28
Chapter 2
Elements of Multi-Scale Petrophysics
of Organic-Rich Shales
Microporomechanical modeling starts by establishing the length scales associated with
pores and heterogeneities present in the material system to be modeled. Then, based
on experimental observations and mechanical testing at different length scales, a
multi-scale modeling approach can be developed with the goal of predicting microporomechanical behavior of organic-rich shales. The challenge in modeling lies in the
proper combination of the theoretical tools available and the dominating mechanisms
that contribute in first-order to the effective elasticity. With this focus in mind, this
chapter aims to establish a basic understanding of both available theoretical tools and
physical properties and processes of the system being modeled to best match the two
"worlds". Thus, this chapter is dedicated to establishing some basic understanding
of organic-rich shale petrophysics and to initiating a structural thought-model, discussed in the following chapter, which forms the backbone of subsequent theoretical
developments.
2.1
Depositional Systems
The successful exploitation of an organic-rich shale play is strongly dependent on
the depositional environment.
For example, success of Barnett shale in the Fort
29
Worth basin has been linked to its deep and rather rapid initial burial, leading to
early thermal maturation of the organic contents and subsequent uplifting over geological time, reducing the the in-situ stresses [1111 and lowering costs associated
with drilling and completions. Burial history and diagenetic/catagenetic processes,
governed by original mineralogy, fabric, texture, organic content, saturating fluids,
hydrology, geothermal gradient, rate and depth of burial control the formation of
multi-scale porosity, elastic anisotropy and organic maturity [51]. Many current shale
plays were deposited in the foreland basin setting. "Depositional processes associated with shale formation are not stratigraphically or spatially homogeneous nor are
all of them deposited by hemipelagic rain in quiet, deep marine environments where
sediment transportation occurs with processes such as hyperpycnal flows, turbidity
currents, storm and wave re-working and bottom hugging slope oceanic currents" [87].
The environmental factors associated with these processes such as temperature, pH
level, and the presence of electrolytes, affect clay mineral setting and organic matter evolution during initial deposition and burial stages. Advancements in imaging
techniques and computational methods have made it possible to capture geological
processes such as compaction, sorting and diagenesis on the overall effective behavior
of rocks (see e.g.
[77]),
though much more work is needed to establish and to refine
such approaches for organic-rich shales.
2.2
Mineralogy
Shale is a term that has been associated with a variety of fine grained rocks, composed
of particles with less than 4 ptm in diameter of characteristics length scale, but may
contain up to 62.5 ptm of silt-size particles, mixed with organic matter ranging from oil
prone algae and hervaceous to gas prone woody/coaly materials [681. The dominant
inorganic components of a shale formation at the time of deposition are clay minerals
such as illite, smectite, montmorillonite and kaolinite as well as quartz, feldspar,
calcite, and apatite.
Diagenesis provokes important changes in the mineralogical composition of shale.
30
Table 2.1 - Mineral densities used for volume fraction calculations [61][561.
Mineral
Quartz
Feldspar
Calcite
Illite
Kaolinite
Dolomite
Ankerite
Pyrite
Anatase
Barite
Muscovite
Albite
Microline
Gypsum
Sanidine
Siderite
Chlorite
pg[g/cc]
2.65
2.65
2.71
2.65
2.64
2.90
3.00
5.00
3.80
4.48
2.82
2.65
2.55
2.32
2.52
3.80
2.95
For example, with rise in temperature, smectite is transformed into illite. Quantities
of illite and chlorite tend to increase with deeper burial depths and a longer exposure time [61]. Porosity occurs at multiple-length scales in organic-rich shales. In
some cases, there are pores in microcrystalline pyrite grains
[52j.
The values for min-
eral densities reported in Table 2.1 are used to convert mineralogical composition of
organic-rich shales, often reported in mass percents and obtained by x-ray diffraction
(XRD) method, into volume fractions which carry the mechanistic contribution of
each phase onto the overall effective behavior of the composite.
2.3
Porosity
Porosity is defined as the ratio of volume of pore space over total volume. Traditionally, it is measured by density differences and fluid intrusion methods. However these
methods are not suitable to capture the wide pore size distribution in organic-rich
shale systems. Clarkson et al. 117] have shown that laboratory characterization of
pore space in organic-rich shale samples is not trivial due to its multi-scale nature.
31
One needs a combination of techniques to fully capture pore geometry and pore size
distribution at various length scales such as Small Angle Neutron Scattering (SANS),
Ultra-Small Angle Neutron Scattering (USANS), low pressure adsorption using N 2
and CO 2 gases, and high pressure mercury intrusion measurements to be able to gain
a comprehensive insight into organic-rich shale pore system.
2.4
Organics & Organic Maturity
Organic matter appearance can be traced back to the Precambrian era as various
plant and animal life forms started to emerge. Towards the end of Silurian period,
plants started to move inland. During the Carboniferous period, the first coal formations started to appear. Most source rocks date back to late Jurassic/early Cretaceous
period.
These geological time scales become relevant in the context of the deposi-
tional environments. The ideal environment for organic matter preservation is linked
to high organic productivity, optimum sedimentation rate and anoxic conditions
[83].
Kerogen is associated with organic matter in sediments insoluble in petroleum solvents, a characteristic that distinguishes it from bitumen
193]. Tissot [92] classified
the processes responsible for the evolution of organic matter maturity during burial
stages as follows:
Diagenesis: This process is associated with biogenic decay, catalyzed by bacteria
and abiogenic reactions which occur in shallow depths with normal temperatures and
pressures. During this process, methane, carbon dioxide, and water are given off by
the originally deposited organic matter, leaving behind what is called kerogen.
In
this process, oxygen content is reduced, leaving the Hydrogen:Carbon ratio (H:C)
unchanged.
Catagenesis: This phase is linked to petroleum release from kerogen as burial
continues and subsequent pressure and temperature increases, first oil and later gas is
generated. During this stage, the Hydrogen:Carbon (H:C) ratio decreases while the
Oxygen:Carbon (O:C) ratio remains mainly intact.
32
Metagenesis: This phase occurs at high pressure and high temperature environments (HP/HT). During this process, the last hydrocarbons (HC), generally methane,
are expelled. The H:C ratio keeps decreasing until the Carbon left is in the form of
graphite.
The rate of maturation is temperature, time, and possibly pressure dependent. A
crude estimation suggests significant oil generation between 60'C-1200 C and noticeable gas generation between 120'C-2250 C
[93].
It has been observed that the oc-
currence of porosity is more prevalent in mature kerogen compared to immature
kerogen 124]. This is consistent with physical intuition since immature kerogen can
be viewed as a pliable organic matter, with a polymeric amorphous structure (see
[68],[100],[101],[120],[119],
and [521) that "self-heals" as soon as a pore is formed.
This organic porosity formation has been contributed to thermal maturation and the
conversion of the organic matter [52]. The type of organic matter does not only depend on its initial composition, but also on the environment of deposition. Type I
and Type II kerogen are associated with algal and hervaceous materials; they exhibit
high H:C ratios and they will typically generate oil during the thermal maturation
due to increases in burial depth, temperature, pressure and exposure time. Type III
kerogen is largely composed of woody/coaly material and leads to (mostly) gas generation during thermogenic maturation. Tissot and Welte
[93] proposed
the following
classification for kerogen:
Type I: Rich in aliphatic chains and some aromatic nuclei, Type I kerogen exhibits high H:C ratio, with high potential for hydrocarbon generation.
Type II: Mainly composed of aromatic and naphthenic rings, with lower H:C
ratio as well as oil and gas potential relative to Type I kerogen. Type II kerogen is
associated with marine organic matter.
Type III: Rich in condensed polyaromatics and oxygenated functional groups
with minor aliphatic chains, Type III kerogen exhibits lower H:C ratio and higher
O:C ratio, relative to the other two types and a high tendency to produce gas at
33
greater depths.
Vernik and Landis [107], classified maturity of organics based on their vitrinite reflectance, %RO, in the following way:
Stage I: Compaction /early methane with %RO < 0.3
Stage II: HC generation/H 2 0 expulsion with %RO of 0.3-0.5
Stage III: Advanced HC generation, with %RO of 0.5-0.75
Stage IVa,b: Main stage HC generation/ primary migration with %RO 0.75-1.3
Stage V: Condensate and wet gas with %RO 1.3-2
Stage VI: Dry gas with %RO > 2
Bousige et al. [12] show that sp 2 /sp 3 hybridization ratio can be utilized as a geochemical indicator for kerogen's maturation.
The quantified relationship between
H:C, O:C and %RO can be represented in what is known as Van Krevelen diagaram
(see e.g. 182]). This diagram shows the evolution of immature kerogen with different
compositions and increasing levels of thermal maturity (%Ro) which is represented
as lines of isochors in the Van Krevelen diagram. With regards to organic density, although a variety of values have been used in the literature they lie in a narrow range.
Zhang and Leboeuf computed a density of 0.93 g/cc for Green River shale kerogen
and reported a measured value of 1.11 g/cc, after correction for pyrite and marcasite.
Robl et al. [75] reported the following densities for dimineralized kerogen: 1.05-1.15
g/cc for alginite, 1.2-1.3 g/cc for bituminite, 1.3-1.35 g/cc for vitrinite and over 1.35
g/cc for inertinite. A range of 1.1-1.4 g/cc is reported in Ref. 156], while Vernik and
Landis 11071 assumed a kerogen density of 1.25 g/cc for their calculations. Considering these values, we shall assume a kerogen density of 1.2 g/cc in what follows for
the subsequent analyses.
34
2.5
Chapter Summary
This chapter introduced some petrophysical notions that need to be considered when
modeling organic-rich shales. Understanding depositional system provides an insight
into the evolution of microstructural parameters with geological time.
For exam-
ple, environmental conditions such as pH level and subsurface temperature gradient
can influence the processes associated with organic maturation [87][821. In addition,
kerogen types and geochemical maturity indicators have been introduced. The discussed petrophycal notions lead to microstructural features that directly control the
mechanical and poromechanical behavior of organic-rich shales.
THIS PAGE INTENTIONALLY LEFT BLANK
36
Chapter 3
Multi-Scale Representation of
Organic-Rich Shales
Prior to any model developments for mature and immature organic-rich shale systems,
one needs to establish a framework that would form the backbone of the subsequent
model developments. The framework used in this report addresses both multi-scale
structure as well as morphological features of immature and mature organic-rich shale
systems; framed within a hypothesis testing approach.
This is accompanied by a
simple analogy between the process of baking bread and kerogen maturation which
together with structural though-model and the morphology-based hypothesis allow
us to develop maturity dependent microporoelastic models for organic-rich shale formations.
3.1
Multi-Scale Structural Thought-Model of OrganicRich Shales
Having established a basic understanding of the petrophysics (see Chapter 2) of
organic-rich shales; we will now define a structural thought model, inspired by the
original work of Ulm et al.
[971
and Ortega [61]. This structural thought model forms
the backbone of subsequent model developments.
37
3.1.1
Level 0: Clay
The lowest level considered is the solid building block of the model, representing clay
at the length scale where no experimental investigation can be made due to limitations
with resolution and isolation of a single clay mineral, clay's high affinity for water
and their platy geometry 128]. Clay is assumed to be transversely isotropic at this
level, consistent with its layered structure. This allows us to attribute anisotropy and
its evolution at different length scales to the intrinsic anisotropy of the clay. In what
follows, the stiffness associated with this level is denoted by Cd
3.1.2
(
Cclay).
Level I: Clay, Kerogen & Porosity
Level I is the scale of relevance to nanoindentation (10-7-10-6 m) where, based on
chemomechanical testings and micrograph observations (e.g. see [25]), a porous solid
composite is considered, with the solid components consisting of kerogen and clay.
Herein, stiffness associated with this level of the multi-scale structural thought model
is denoted by Chm.
3.1.3
Level 1I: Porous Solid & Inclusions
Level II defines the macroscopic scale (10-5-10-4 m) relevant to the wavelength ex-
plored by ultrasonic pulse velocity (UPV) measurements. At this level, the mechanistic contribution of silt inclusion grains, composed of dominant non-clay inorganic
minerals, to the effective elasticity is considered. The stiffness associated with this
level is denoted by
Chom
for drained behavior and
Chom
for undrained response.
Having established the tools for modeling mature and immature organic-rich shales
and with n strctnral
thgiicrht mcodl,
in plaie
Part III will friric
fn dvelpnning the
mathematical representation of the model for both mature and immature organic-rich
shales.
38
&
Level 1I:
Porous fabric
inclusion
Level 1:
Kerogen
& clay porous
fabric
Level 0:
Clay aggregates
-9
Log[m]
Figure 3-1 - A schematic representation of multi-scale thought model discussed.
3.2
Chapter Summary
This chapter presented a multi-scale structural thought-model which permits modeling the intricate and heterogeneous nature of organic-rich shales by separation of the
length scale of relevance to our micromechanics-based modeling efforts. Also, such
a multi-scale framework empowers us to utilize measured elasticity data at different
length scales for calibration and validation of our model.
39
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40
Chapter 4
Multi-Scale Material
Characterization & Properties
In order to calibrate and to validate microporoelastic models (to be developed in
subsequent chapters), different data sets representing samples from both mature and
immature shale plays are needed. In this chapter, two comprehensive data sets belonging to Woodford and Haynesville shales are presented. The macroscopic elasticity measured by means of ultrasonic pulse velocity (UPV) are used for calibration
of our model for both mature and immature organic-rich shales. For validation, instrumented nano-indentation data on Woodford, Haynesville, Marcellus, Antrim and
Barnett are utilized. In addition, the material constants used in the model and the
technique employed for C 13 estimation (at macroscopic length scale) are discussed.
In Chapter 8, a thorough sensitivity analysis regarding the effect of uncertainty associated with macroscopic C13 estimations and its effect on our models' predictive
capabilities, is presented.
41
4.1
Elastic Waves in a Transversely Isotropic Medium
Motion in a homogeneous anisotropic solid can be described by a set of linearized
partial differential equations as follows [691:
(4.1)
-
V - -+P
where Pb is the bulk density of the solid, u denotes displacement vector, and f is the
body force vector, per unit mass. The moment equilibrium condition requires:
(4.2)
O-ij = 17j i
And the strain field, displacement relation reads:
=
1 aui 8u
J
1+
2 Oxj
axi
-(
)
ij
(4.3)
Lastly, the constitutive equation for an anisotropic solid can be expressed as:
(4.4)
Manipulating (4.1),(4.2),(4.3) and (4.4), one can write (ignoring body forces):
x1uk
t
__
Ciikla XIOXj
-
b
t2
(4.5)
Taking advantage of proportionality for the solution of the wave equation (4.5) to
ei(at-kn.r) for a plane wave propagating in n; Christoffel's equation can be expressed
as:
2
v
k 2 (n.C.n).v = k 2 'v = pdf
(4.6)
where:
a
=
at
-
V
42
(4.7)
with k being the wave number, C denoting wave frequency, v symbolizing particle
field velocity and
F
representing Christoffel's matrix. Christoffel's equation (4.6) can
be written in the following form for determining acoustic wave velocity propagation:
det(A
k2 1) = 0
-
(4.8)
with A being defined as the undrained acoustic tensor:
(4.9)
Aij = -- njCijkjnj
POb
In the adopted Cartesian coordinate system (see Fig. 1-1), for a transversely isotropic
medium, n = (sin 0, 0, cos 0), with 0 = 0 corresponding to the material symmetry axis,
denoted by x 3 , parallel to the axis of rotational symmetry. One can express the normal
vector in matrix form, as [611:
[n]=
0
0
-n
0
n2
0
-n
2
0
0
ns3
2
1
0
2
-
1
ni
3V2
0
2
2
-n3
2d
(4.10)
0!
1-li~II
- nI v
2.
using this representation of the normal vector, (4.9) reads:
(C 1 3 + C 44 ) sin 0 cos 0
Pb
0
C 44 + sin 2 0(C 66 - C 44
0
(C13 + C44) sin 0 cos 0
0
C 3 3 + sin 2 0(C4 4 - C 33
)
[Aj]
)
0
)
C 44 + sin 2 O(C 1 1 - C 44
Therefore, for velocity of waves propagating in the plane of isotropy (xI, x 2 ), one can
write:
C6 6
v,-
(4.11a)
VsPOb
C 44
V3-
P0,b
43
(4.11b)
1
I
Vi
(4.1lc)
C1 1
Pb
Where subscripts p and s refer to compression and shear waves, respectively; while
the numbers in the subscripts denote the direction of wave propagation. Furthermore:
Vsi: pure shear mode polarized normal to the axis of symmetry
V, 3 : pure shear model polarized parallel to the axis of symmetry
Vpi: pure longitudinal mode in the bedding direction
and for the waves propagating in the direction parallel to the symmetry axis (i.e. x 3 ):
(4.12a)
Vs3 =
Vp
=
Pb
C33
(4.12b)
C33
Pb
Where:
Vs 3 :
pure shear mode polarized parallel to the axis of symmetry
Vp3 : pure longitudinal mode in the normal-to-bedding direction
Thus, in order to characterize the elasticity of any transversely isotropic medium,
i.e. organic-rich shales in our case, by means of ultrasonic pulse velocity (UPV), one
needs to make the following lab measurements:
cynamic = PbV
(4.13a)
Cdynamic = PbV 3
(4.13b)
Cdynamic =PbV
1
(4.13c)
V 3
(4.13d)
CPbnamic
Cdynamic
12
/(,,,2,QC __ j
C13
13
_V
C-C
44
2 uN33
Vdynamic
Cdynamic
=1C
-
_,;, 2 ,Q0
1 0112 uk44
2
(4.13e)
Cdynamic
66
,-XT 2 t;,
Pbv po)k 0
0
J"
N-11
-T-
u,-)44
-
Pb
x T2
V
p)
sin 0 cos 0
(4.13f)
44
Where:
Vpo: off-axis compressional wave velocity measurement (0 w.r.t the plane of isotropy,
(xI, x 2 )).
Cdynamc refers to stiffness coefficient characterized by means of UPV, rather than
quasi-static measurements, denoted by C uasistatic. Furthermore, Thomsen
[91]
in-
troduced parameters that quantify p-wave and s-wave anisotropy, known as C and
-y, respectively; in addition to 6*, which is associated with the shape of p-wave and
s-wave surfaces. These parameters are defined as [91]:
C
*=
2
2C23
33
4.2
C3 3
2C 33
(4.14a)
= C 66 - C 44
2C 44
(4.14b)
C11 -
[2(C 1 3 + C 4 4 ) 2 - (C 3 3 -
C 4 4 )(C 1 1 + C 3 3 -
2C 4 4 )]
(4.14c)
Macroscopic C 13 Estimation
Full elastic characterization of organic-rich shales in the laboratory has proven to
be a challenging task. Other than issues related to core preservation, which may
alter the mechanical integrity of the samples, and difficulties in boring samples due
to delicate nature of shales; determination of C 13 from (4.13f) requires an off axis
wave travel time measurement. Specifically, the current laboratory testing configuration for a sample under confining pressure, inside a pressure vessel and subjected
to uniaxial compression loading, inside a loading frame has made it nearly impossible to directly measure C 13 . There are some methods to estimate C 13 (see e.g.
1111[81][95][891). In this thesis, we will employ the modified ANNIE method [891 to
estimate macroscopic C 13 for Haynesville data1 . This method is based on combining both quasi-static measurements obtained through strain gages and extensometers
1Woodford elastic data have been published in the literature. The specific method used for C
13
estimation is presented in [951.
45
on samples cored in different directions, in addition to dynamic measurements using
ultrasonic pulse velocity (UPV) technique. From quasi-static measurement, one can
obtain 4 independent stiffness coefficients 2 in addition to
3-
Cquasi-static
C1
E_(1_-__
3_
(1 + V 12 )(E 3 (1 -
Cquasi-static
_
)
2Eiv
E1 E 3v 13
_
E 3 (1 -
v12)
Cquasi-static
-
_2)
12) -
11
-
2E,13
(4.15a)
(4.15c)
3
)
asi-static
2Eiv 3
E 3 (1 - V12) -
as follows:
(4.15b)
12
_
33
Cquasi-static
66
3
(1 + V1 2 )(2Eiv23 - E3 (1 - V12)
Cquasi-s t a t c
Cqt
12
Cquasi-static
(4.15d)
Cquasi-static
-
12
2
(4.15e)
where V12 and E1 correspond to Poisson's ratio and Young's modulus when the sample is loaded in the isotropic plane and V13 and E 3 are Poisson's ratio and Young's
modulus of the sample when loaded in the transverse (parallel to the axis of rotational symmetry and perpendicular to the plane of isotropy, i.e. the bedding-planes)
directions.
On the other hand, one can obtain 4 (out of 5) independent stiffness
coefficients from UPV measurements, as follows:
ck namic
c
namic
V2
=
PbV2a
cdynamic = Cdnamic -
2
(4.16a)
2
Cdynamic
(4.16b)
(4.16c)
Cdnaic
PV2
(4.16d)
Cdnamic
PbV2
(4.16e)
Note C 66 is not an independent stiffness coefficient.
46
Employing the modified ANNIE method [891, one can calibrate a newly introduced
parameter, ( by assuming that this parameters is invariant to the characterization
method. ( is defined as:
C 12
(4.17)
C13
thus, by calibrating ( using Cquasi-static and
Cuasi-static
one can estimate
Cdnanlic
from
cdynamic
12*
4.3
Instrumented Nanoindentation
Indentation allows one to to infer mechanical properties of the indented material employing continuum mechanics based contact solutions; specifically the contact problem
of an elastic half-space and a rigid axisynmetric indenter. Instrumented nanoindentation is performed by pushing an indenter tip, of known geometry, onto the surface
of the material of interest. From an analysis of the load-displacement curve, one can
obtain the indentation modulus, defined as:
900
-
750
-
600
-
450
300
-
el)
0
0
0
.WM&D0
- - - -
-
150-
100
Displacement,
200
300
h [nm]
Figure 4-1 - A typical nanoindentation load-displacement curve.
47
M=
where S
(4.18)
2 VA
dP.
dh' is the initial slop of the unloading phase of the load (P)-displacement
(h) curve and Ac is the projected contact area between indenter tip and sample
surface. Ac can be determined from contact depth to indentation depths relation,
through Galin-Sneddon solution [611. For the particular case of transversely isotropic
elasticity, the indentation moduli, in terms of stiffness coefficients, read [98]:
M3 =
2
C F C33 - C23
C
Cii
13
(
1
C44
M11
C 33
2
+
-I
(4.19a)
N/ClIC33 + C13)
-
Cii
C12M 3
(4.19b)
where M 3 and M 1 denote indentation moduli in x 3 and x, directions, respectively.
For heterogeneous materials with heterogeneities occurring at different length scales,
Ulm and co-workers (see e.g. [19][18]) developed a grid indentation technique. This
method is based on conducting a large number of tests over the surface of a heterogeneous material system. From a statistical point of view, the proper choice of
indentation depth and grid size, ensures that the volume probed is not altered by
previous indentations and other heterogeneities. In turn, this enables one to treat
every indentation as an independent "experiment", which paves the way for statistical analysis of the grid indentation data. Previous finite element method (FEM)
analysis shows that the volume probed by an indenter is 3 to 5 times larger than the
indentation depth [491. One needs also to ensure that the grid size, L, is much larger
than the footprint left by indentation in addition to a large number of experiments,
n. This can be summarized as:
h < d < L v/i
(4.20)
where h denotes indentation depths and d is the size of the material phase (i.e.
characteristic length of heterogeneities).
48
4.4
Calibration Data Sets
For calibration of our model for immature organic-rich shales, we employed a comprehensive published data set on Woodford shale (see Ref. [95][61]). Woodford formation
dates back to Upper (late) Devonian-Lower (early) Mississippian and it is mainly located in South-Central and South-Eastern Oklahoma. Chert, siltstone, sandstone,
dolostone and light colored shale are the dominant lithofacies in Woodford 1861. The
Woodford formation is identified, based on its lithology, by three members: Upper,
Middle, and Lower. Upper Woodford consists mostly of black shale with parallel laminations, phosphate nodules and it exhibits intermediate radioactivity. Middle Woodford is mainly black shale with significant pyrite and total organic carbon (TOC)
concentration. Lower Woodford contains a higher concentration of carbonates, silt
and sand relative to the other two members, and it overlies the Hunton group carbonates [951.
The published data corresponds to five samples obtained from cores taken from a
well drilled in Wyche county, Oklahoma as a part of a research project. Although no
geochemical analysis was performed on the samples to assess the maturity of their organic contents, Woodford shale in Wyche county, Oklahoma is indeed immature [76].
These samples were characterized in terms of mineralogy, porosity, and TOC. Their
macroscopic elasticity was characterized at the Integrated PoroMechanics Instiute at
the University of Oklahoma using the UPV technique, and nanoindentation experiments were performed at MIT. Mineralogy and TOC of each sample are reported in
Table 4.1. The data pertaining to bulk density and porosity are reported in Table
4.2 and Table 4.3, respectively.
The macroscopic elasticity of Woodford shale is
reported in Table 4.4.
In order to calibrate and to validate the model for mature organic-rich shales, a set
of data from Haynesville formation was used [41],[2]. Haynesville and Bossier formations date back to the late Jurassic and they were mainly deposited in East Texas and
North Louisiana salt basins during the opening of the Gulf of Mexico [86]. Lithology
49
Table 4.1 - Mineralogy and kerogen content of Woodford shale samples in [mass
[95].
Mineralogy
Quartz
Al
37
K-feldspar
Plagioclase
A2
31
A3
33
A4
34
2
-
2
2
2
3
3
3
3
3
Dolomite
0.5
3
1
-
6
Ankerite
Pyrite
2
9
4
13
2
10
7
6
8
3
A5
27
Kaolinite
1
-
-
-
3
Illite/IS
Other clay
Kerogen
26
3
17
25
3
17.5
27
4
18
31
5
12
29
4
14
Table 4.2 - Bulk density of Woodford shale samples 161].
Sample
pb[g/cC]
Al
A2
A3
A4
A5
2.21
2.18
2.18
2.26
2.11
Table 4.3 - Porosity of Woodford shale samples in [%1 [95].
Sample
Al
A2
A3
A4
A5
50
q5
16
21
16
19
21
%1
Table 4.4 - Reported elasticity of Woodford Shale samples [611.
Sample
Al
A2
A3
A4
A5
Ci""[GPa]
23.1
25.6
23.8
28
21.2
C ""[GPa]
6.9
6.1
6.2
7.5
6.3
C1""[GPa]
8.8
7.7
7.8
8.3
7.9
Cl un[GPa]
15.7
16
14.9
17.3
13.8
Cun[GPa]
5.2
6.6
5.3
5.6
4.9
of Haynesville is dominated by calcite and clay with some quartz and pyrite. Haynesville is considered to be a mature organic-rich shale [60].
The provided Haynesville data set (see [41]), accompanied by instrumented nanoindentation data published in Ref. 12] were employed for calibration and validation of
the microporoelastic model for mature organic-rich shales. The provided mineralogy
and TOC of Haynesville shale samples are reported in Table 4.5. Bulk density of each
sample is provided in Table 4.6 and porosity can be found in Table 4.7. The provided
macroscopic elasticity is reported in Table 4.8.
To assess the quality of the provided macroscopic data, dynamically measured elasticity of all seven samples were cross plotted against quasi-statically measured ones.
The results, shown in Figure 4-2, suggest that sample B5 is not consistent with other
samples. This could be due to a variety of reasons such as sample damage during
boring; thus altering its mechanical integrity, and/or presence of sea shell(s) inside
the sample; making it not representative of the reported petrophysical properties.
Therefore, sample B5 has been excluded from all analyses presented hereafter.
51
Table 4.5
-
Mineralogy and kerogen content of Haynesville shale samples in [mass
[41].
Mineralogy
Quartz
Feldspar
Carbonates
Other
Illite/IS
Illite
Kaolinite
Chlorite
Kerogen
B1
30
7
30
4
8.4
16.8
2.1
2.7
2.48
B2
27
9
22
4
15.6
19
1.5
1.9
3.34
B3
16
5
65
3
4.5
5.4
0.55
0.55
1.57
B4
20
6
51
3
7.2
9
0.8
3
2.65
B5
31
9
11
5
18.45
25.6
0.45
0.45
2.57
B6
32
11
9
4
17.6
21.9
1.3
2.1
3.30
B7
28
10
12
12
19.4
10.3
3.8
4.56
3.16
Table 4.6 - Bulk density of Haynesville shale samples f411.
Sample
BI
B2
B3
B4
B5
B6
B7
pb[g/cC]
2.54
2.48
2.59
2.53
2.48
2.47
2.47
Table 4.7 - Porosity of Haynesville shale samples in [%]
Sample
BI
B2
B3
B4
B5
B6
B7
52
0
6.64
7.36
4.61
5.77
6.03
7.16
7.59
1411
%]
Table 4.8 - Calculated elasticity from measured UPV (except for Cl,"" which was
estimated by method presented in Section 4.2 from data in Ref. [41].
Cl""[GPa]
20.5
19.9
13.4
20.3
11.06
18.3
18.5
C ""[GPa]
58.7
54.1
49.9
64.6
32.78
51.4
58.5
Sample
BI
B2
B3
B4
B5
B6
B7
C1
[GPa]
15.4
11.3
10.4
21.4
5.10
12.6
11.6
C '""[GPa]
33.8
33.1
41.9
58.7
19.15
30.3
35.1
C"[GPa]
14.7
15.7
15.3
20.7
7.68
13.6
14.6
60
81
IQ2
l&9B31
50
82
50-
-3
85
86
B7
86
40
40
B7
0~
C
30
30-
E
E
20-
20-
10-
10
0
10
20
30
40
50
6
Static C 11 [GPa]
0
0
30
20
10
0
50
60
(b)
(a)
60
40
Static C 1 2 [GPa]
ftB
00
B1
B2
B2
83
50
*
50
30
Wi
aCD
CL
8
40
40
1
0
30--
30CO
E
M
20--
20
10--
100)
4
102L0
0
10
20
30
40
50
0
60
Static C 3 3 [GPa]
10
20
30
40
50
60
Static C 6 6 [GPa]
(d)
(c)
Figure 4-2 - Quality check of the elasticity data by comparing static and dynamic
stiffness coefficients. Sample B5 is not consistent with other samples and thus it will
not be considered for the subsequent analyses. Note Ci values in (a),(b),(c) and (d)
refer to the macroscopic elasticity.
53
4.5
Validation Data Sets
For validation, in addition to data sets belonging to Woodford and Haynesville, additional data from Antrim, Marcellus and Barnett formations were utilized.
For
validation, we used measured nanoindenation data on the samples belonging to these
formations. The instrumented nanoindentation data of Woodford are published in
Ref. [611 and the indentation data for Haynesville, Antrim, Marcellus and Barnett
are reported in Ref.
[41].
[2]
and their mineralogy, porosity and TOC were obtained from
Furthermore, Abedi et al. [21 classified Antrim and Barnett as immature and
Marcellus as mature.
The measured indentation moduli are reported for Woodford samples in Table 4.9,
for Barnett in Table 4.14, for Antrim in Table 4.17, for Haynesville in Tables 4.10
and 4.11, and for Marcellus shale in Tables 4.20 and 4.21. The mineralogy, kerogen
content and porosity of each set of data is needed for predicting indentation moduli
using our model. The mineralogy, kerogen content and porosity of both Woodford
and Haynesville samples were reported in the previous section. The mineralogy and
kerogen content of Barnett is reported in Table 4.12, for Antrim they are summarized
in Table 4.15, and for Marcellus shale they are presented in Table 4.18. Lastly, the
measured porosity for Antrim, Barnett and Marcellus are reported in Tables 4.13,
4.16 and 4.19, respectively.
54
Table 4.9 - Indentation moduli of Woodford shale samples as reported in Ref. [611
except for a correction for sample A2.
Sample
Al
A2
A3
A4
A5
M1 [GPa]
10.79
11.95
10.24
11.98
7.33
3.39
3.1
3.24
4.24
2.57
M3 [GPa]
8.47 2.35
9.09 2.69
6.82 2.12
9.82 3.06
7.14 2.63
Table 4.10 - Measured Haynesville indentation moduli in x, [2].
Sample
M 1 [GPaj
BI
B2
B3
B4
B5
B6
B7
36.83 6.24
36.68 5.68
28.39 6.68
34.8 6.32
29.68 7.18
30.98 6.22
55
Table 4. 11
Measured Haynesville indentation moduli in x 3 12].
Sample
M 3 [GPa]
BI
23.04
B2
6.07
22.51 6.64
24.26 4.19
24 7.4
-
B3
B4
B5
B6
B7
Table 4.12
141].
22.84+7.99
24.22+9.33
22.36+7.52
19.85 6.88
21.09 5.94
21.41 6.75
Mineralogy and kerogen content of Barnett shale sample in [mass %1
C1
Mineralogy
Table 4.13
Quartz
29.73
Illite/IS
Chlorite
Albite
Calcite
Microline
Pyrite
Gypsum
Kerogen
39.67
2.11
2.2
2.64
3.25
0.53
7.83
12.2
Porosity of Barnett shale sample in
Sample
C1
Table 4.14
[%] 1411.
<7
7.3
Indentation moduli of Barnett shale sample [2].
Sample
C1
M 1 [GPa
17.37 4.02
56
M 3 [GPa]
11.78 2.45
Table 4.15
Mineralogy and kerogen content of Antrim shale sample in [mass %] [41].
Mineralogy
Quartz
Illite/IS
Chlorite
Albite
Dolomite
Pyrite
Sanidine
Kerogen
DI
40.91
25.57
5.84
3.47
4.38
3.11
7.95
9.61
Sample
DI
#
Table 4.16 - Porosity of Antrim shale sample in [%] [411.
8.8
Table 4.17 - Indentation moduli of Antrim shale sample [2].
Sample
Dl
M 3 [GPa]
12.31 2.91
Mi[GPa]
21.11 4.68
Mineralogy
Quartz
Illite/IS
Chlorite
Calcite
Dolomite
Pyrite
Plagioclase
Siderite
Anatase
Barite
Muscovite
Kerogen
Table 4.19
F1
19.70
23
6.2
F2
29.6
36.3
2.1
F3
36.2
31.8
0.4
-
3.1
3
30.6
4.4
1.5
3.2
0.20
1.4
8.7
6
0.7
0.40
1.5
11.7
5.6
0.3
0.50
-
1.5
-
Table 4.18 - Mineralogy and kerogen content of Marcellus shale samples in [mass %]
[41].
10.7
0.49
10.2
7.68
9
8.18
Porosity of Marcellus shale sample in
Sample
0
8.4
7.2
6.5
F1
F2
F3
57
[%] [41].
Table 4.20 - Measured indentation moduli in x, on Marcellus shale smaples [2].
Sample
F1
F2
F3
M 1 IGPa
45.74 9.81
41.70 6.32
53.37 7.32
52.6117.83
57.70 7.12
28.8115.04
35.30 6.39
33.0215.76
30.52 5.59
34.0617.23
28.17 5.39
29.4115.50
Table 4.21 - Measured indentation moduli in x 3 on Marcellus shale smaples 121.
Sample
M 3 [GPa]
34.5918.31
F1
F2
F3
40.9519.61
37.7416.41
40.5018.11
19.6613.44
25.1714.48
23.5114.24
23.8516.22
23.9215.28
23.1915.51
58
4.6
Phase Properties
The non-clay inorganics, present in both Woodford and Haynesville samples, exhibit
different mechanical properties and distributions in terms of mass percents. In order to account for the variations in compositions of silt grains, effective elasticity of
silt minerals are calculated based on a given composition rather than assuming that
inclusion grain is entirely composed of one type of mineral. Based on mineralogical distributions, measured by x-ray diffraction (XRD) and elasticity contrast (w.r.t.
other minerals); quartz, pyrite, calcite and feldspar were identified as dominant silt
minerals present in Woodford and Haynesville samples. Although these minerals have
different elastic symmetries, for geomechanics and geophysics based applications, it is
safe to assume that one is dealing with a conglomerate of mineral crystals rather than
a single one. With this reasoning in mind, the isotropic elasticity of calcite, quartz,
feldspar, and pyrite characterized here by their bulk modulus, K, and shear modulus,
G, is reported in Table 4.22.
In the forthcoming analyses, we will assume pore fluid to be water, with a bulk
modulus, Kf of 2.3 GPa [611. The choice of water for the pore fluid may seem invalid
given that pores in organic-rich shales are usually dominated by gas or light condensates. This will be addressed later on by modeling poroelastic constants for mature
and immature organic-rich shale systems and by performing a sensitivity analysis to
study the effect of pore fluid compressibility on poroelastic coefficients. In the case of
Haynesville, since calcite and feldspar elastic properties are similar and their contributions to the effective stiffness of the inclusion grains weighted by their normalized
volume fraction is low relative to quartz, feldspar is considered to be mechanistically
represented by calcite, in terms of elasticity.
3
We used indentation modulus reported in [13] and assumed a Poisson's ratio of v = 0.25 to back
calculate K and G.
59
Table 4.22 - (quasi-)isotropic elasticity of different minerals.
Phase
Calcite3
[13]
Quartz[56]
Feldspar[30]
Pyrite[1141
4.7
K [GPa]
G [GPa]
58.18
37.9
62
138.9
28.33
44.3
29.3
112.3
Chapter Summary
In this chapter, comprehensive data sets, to be used for calibration and validation
of our microporoelastic model, are presented for both mature and immature organicrich shales. These data sets include mineralogy, porosity, organic content. In addition, macroscopic elasticity, measured by means of ultrasonic pulse velocity (UPV)
and microscopic elasticity measured by instrumented nanoindentation are presented.
Furthermore, both UPV and instrumented nanoindentation techniques are discussed.
While the UPV measurements are used for calibration of the model for immature and
mature organic-rich shale systems, the instrumented nanoindentation data serve as an
independent tool for validation of our model in this multi-scale micromechanics-based
investigation of organic-rich shale systems.
60
Part III
Theoretical Background & Model
Developments
61
THIS PAGE INTENTIONALLY LEFT BLANK
62
Chapter 5
Elements of Microporomechanics
All the theoretical developments herein are achieved within the framework of linear microporoelasticity. The general approach for obtaining macroscopic constitutive
models is to solve a well defined boundary value problem on a Representative Elementary Volume (REV), with its domain denoted by Q consisting of r subdomains
(Q =Q U Q 2 U
3U . . U
Qr )
of micro-homogeneous phases. The REV is a statistical
representation of the heterogeneous material system being modeled. The constitutive
models can be obtained by relating (i.e. localizing) a macro scale stress field, E, or
strain field, E, to micro scale stress field, o- or strain field, e, by solving a well-defined
boundary value problem. While there exist a variety of analytical and semi-analytical
tools available to compute the homogenized response of these heterogeneous material systems, inclusion-based effective medium theories are employed herein to obtain
estimates of the effective composite response by linking the micro structure to the
elastic and poroelastic material behavior at different length scales; utilizing Eshelby's
landmark results
5.1
[29]
and the microporomechanics framework (see Ref. [26]).
Scale Separability Conditions
Before implementing the differential and integral tools of continuum mechanics to
solve a boundary value problem on a REV, one needs to ensure that the scale separability conditions are met [1181. The scale separability conditions require that the
63
characteristic length of the REV, t, to be much smaller than the characteristic length
of the structural system, L, and much larger than the characteristic length scale
of local heterogeneities, d ,where d must be much larger than the length scale below
which the tools of continuum mechanics; based on continuity of stress and strain fields
seize to be defined, denoted here by do. The exact quantification of "much larger" or
"much smaller" depends on the material system and application under consideration;
but as a rule of thumb, at least an order of magnitude ratio shall be maintained. It
is also required for f to be much smaller than the load fluctuation length, A. These
conditions are summarized as follows:
d < d < f < ,C(5.1a)
f< A
(5.1b)
In the forthcoming developments, infinitesimal deformation of the REV is assumed
at each considered length scale. Also in order to avoid geometrical non-linearities, it
is assumed that displacements induced by loading parameters are small.
5.2
Homogenization
We begin by considering homogeneous boundary conditions prescribed on the boundaries of a REV, OQ, by either imposing a uniform stress or a uniform strain field. In
the case of uniform stress boundary conditions, a traction, Td is prescribed on OQ:
Td E(x) . n
Vx E OQ
(5.2)
where E is the known macroscopic stress field and n is the unit outward normal to
the boundary. It can be shown:
jU)a (x)dQ
E =
64
(5.3)
where o(x) is a divergence free stress field', 2 and (...) stands for volume averaging.
In the case of homogeneous strain boundary conditions, a uniform displacement, (d
is prescribed on OQ:
d=
(5.4)
E(x) -n
where E is the known macroscopic strain field. Similarly it can be shown:
E
=
=
=e)
(5.5)
je(x)dQ
An important consequence of considering homogeneous boundary conditions is the
link between externally supplied work to a heterogeneous material system and the
sum of the internal energy of the phases present. Known as Hill Lemma, it is stated
as [26]:
(5.6)
(a:E) = (u) : (6) = E : E
This expression establishes the equivalency of macroscopic and microscopic strain energies. Also, an important result of Hill Lemma is that the stress and strain fields
need not to be associated, which will turn out of importance for upscaling the microporomechanical behavior of a multi-scale heterogeneous material system. In linear
elasticity, one can relate macroscopic stress (or strain) field to their microscopic counterpart through a 4 th order concentration tensor [118:
where A is the
4 th
Vx E Q
(5.7)
e(x) = A :E
Vx C Q
(5.8)
order strain concentration tensor and B is the
concentration tensor.
V -=(
a(x) = B :E
4 th
order stress
By invoking Hook's law and using (5.8), one can write the
a+ a+ a
=0
Ox1
IOx2
OX3
fBody forces,provided that E > pfl x 1, can be neglected where pf is the volume force and f
is the characteristics length scale of the REV. This is indeed satisfied for homogenization purposes
2
[26].
65
stress field in the rth phase of the REV as:
oU(x) = C'(x) : e' = C'(x) : Ar(x) : E
Vx (E
r
(5.9)
Then, application of (5.3) to (5.9) results in:
(5.10)
E(X) = (Cr(x) : Ar(X)), : E
thus, it is recognized that:
Chom
(Cr(x)
: A'(x))Q
(5.11)
Following a similar argument, one can develop the expression for the homogenized
compliance tensor as:
Shorn
(
(5.12)
r(x)):
where Q includes solid phases (Qs), as well as pore phase(s) (QP).
5.3
Inclusion-Based Effective Estimates
Next, Eshelby result 1291 is utilized in pursuit of an expression for localization tensors.
In his landmark paper, Eshelby
[29] solved for the strain field in an ellipsoidal inho-
mogeneity embedded (with perfect interfaces) in an infinite, linear elastic medium
subjected to uniform strain boundary conditions at infinity. His solution implied that
the strain field in the ellipsoidal inhomogeneity is constant when subjected to uniform
displacement boundary conditions at infinity. A summary of the problem statement
and solution is presented below:
U(x) = CS -
V - a = 0,
Vx E Q
I(x),
Vx E Q
+
= E - x,
66
x-
oc
(5.13)
where CS is the background stiffness, and al represents the fictitious stress field, characterizing the stress field perturbation due to the presence of the inclusion:
or,(_X) =
0,
Vx E QS
6C,
Vx E Q,
(5.14)
with Q = Qs + Q1, where Q' denotes the domain of inclusion and Qs represents the
volume not occupied by the inclusion.
C = C - Cs is the stiffness contrast between
the inclusion and the background matrix. The interesting result, due to Eshelby, is
that the strain is constant in any ellipsoidal inhomogeneity and it can be expressed
as:
eI(x) =
SEsh
(5.15)
:s : '(x) + E
where Ss = (Cs)- 1 is the compliance of the background matrix, while SEsh is the
Eshelby tensor.
The fictitious stress, al(x) also remains constant in the inclusion.
Combining (5.14) and (5.15), one can express a1 as:
a(x)
where the
4 th
1
[I+ 6Cs : SEsh : (Cs)- 1
: SC : E
order identity tensor is defined as I =ijkl ---
(5.16)
6
( ikj
+
6iioi)
and
6
ik
denotes the Kronecker delta. By expanding (5.16) into (5.15), one obtains:
EI(X)
+ SEsh : ((Cs<' : C'
]I)V'
-
: E
(5.17)
By comparison with (5.8), one can readily recognize the expression for the strain
localization tensor, A, which links a macroscopic strain field to its microscopic counterpart(s). Thus, A based on (5.17), for the inclusion, is defined as follows:
AI = [I + sEsh : ((Cs)-
: C'
-
li1f'
(5.18)
Now, utilizing results from Eshelby's solution, one can write the strain field in the rth
inhomogeneity in response to a macroscopically imposed strain field at infinity, E',
67
as:
Er(x) = [If + SEsh
s((C)l
: Cr
-
1)]
:E
(5.19)
where C' is the stiffness of the rth inclusion and Eo is the homogeneous macroscopic
strain field imposed on the boundaries of the REV at infinity. Exploiting the relationship between Eshelby tensor, SEsh and Hill concentration tensor, P 159][26], that
is:
Cssp:
SEsh
(5.20)
one can re-write (5.19) as:
er(x) = [II + P : (Cr
Cs)-1 : E
-
(5.21)
Now inserting (5.21) into (5.5), a link between the homogenized macroscopic strain,
E, and the prescribed homogeneous macroscopic strain, E' is established:
E
= ([I + p : (Cr - Cs)]) I : E
(5.22)
By substitution of (5.22) into (5.21), the generalized expression for the localization
tensor of the rth phase is obtained:
Ar
[ + p : (Cr - Cs)]-l : ([f + p : (Cr - Cs)]-1) 1
(5.23)
In the isotropic case (a special case of the generalized case presented here), one can
decompose (5.23) into volumetric and deviatoric components:
Ar = ArJ + ArK
(5.24)
where Ar and Ar represent the volumetric and deviatoric components of Ar, respectively. In addition:
S=Jijkl
-
K =68
Uijuk)
J
(5.26)
and;
Kr
Ar = (1 + 6(K
fr(1 + (
1))-[
Kr
1))1]
(5.27)
1))1]-1
(5.28)
r
Ar =
(1 + 3( G
fr(1 + i(r
1))- [
r
where d and / characterize the isotropic Hill concentration tensor, to be further discussed in Section 5.4.1. K and G represent the background bulk and shear moduli,
respectively; whereas Kr and G' denote bulk and shear moduli of the r'h phase.
Expansion of (5.11) with (5.23) results in:
Chom
cr : [E + p: (Cr _ Cs)]-l : k + P : (Cr_ Cs) - 1 ])'
(5.29)
which is the generalized expression for the effective (homogenized) stiffness of a heterogeneous composite. The morphological features such as aspect ratio, orientation
and geometry of each phase is condensed and represented in homogenization by Hill
concentration tensor. For the isotropic case, elastic stiffness, C can also be decomposed as follows:
C = 3KjI + 2GK
(5.30)
where K and G denote bulk and shear moduli, respectively. The effective bulk, Khom
and shear Ghom of a composite can be obtained as follows:
Khom =
rfKrAr
(5.31)
frGrAr
(5.32)
r
Ghom
__
r
where fr is the volume fraction associated with the rth phase and Ar and Ar are
defined in (5.27) and (5.28), respectively. One can re-write (5.23) as:
((Cr
-
CS) : [II + P : (C'
69
-
Cs)]- 1 ), = 0
(5.33)
For an isotropic morphology (e.g.
spheroidal grains/pores), which translates into
identical expression for P, (5.33) can be written as:
([II + P : (C'
-
Cs)-1)
= E
(5.34)
Thus, when all geometries and orientations associated with concentration tensors of
different phases in a heterogeneous material system are the same (i.e. isotropic morphology), in a self-consistent homogenization scheme, one can write (5.23) as:
A
5.4
[I + P : (Cr - Chonl
(5.35)
Hill Concentration Tensor
The generalized Hill concentration tensor can be defined as [1181
Pij-=-
Gik
(x-x')
(5.36)
OxjOx,12
) (ij)(kl)
where (ij) (kl) indicates symmetrization and Gij(x -x') is the
2 nd
order Green's tensor
for generalized linear, elastic, anisotropic media that expresses displacement at point
x due to a Dirac delta type point force at x'. For a transversely isotropic medium, the
solution for Green's function can be found in [661. Note that P is positive definite and
exhibits both major and minor symmetries. In a different form, Laws [50] expressed
the generalized Hill concentration as follows:
PijkI
with;
Mkil
/
=
16
(Akijkl
(5.37)
+ Mkjii + A4ik + A4jik)
(a2c 22Lj,2 ,2ba,2
3
a2 Cj, 22/)3/2
Fkjk
i jd S
(5.38)
parameters a1 , a2 , a3 are geometric degrees of freedom that constraint the topology of
,
an ellipsoid, dS7 is the surface element of an unit ellipsoid with components w 1 ,w 2 ,w 3
70
and Fik(Q) =Cijklsjji is known as the Christoffel matrix. One may be able to derive
the explicit expressions for (5.36) and (5.37), depending on the elastic symmetry of
the background medium, as well as the orientation and aspect ratio associated with
the inclusion. In what follows, we will present two well-known special cases; first the
solution for a spheroidal inclusion embedded in an isotropic medium and then the
solution for a spherical inclusion embedded in a transversely isotropic matrix.
5.4.1
Spheroidal Inclusion in an Isotropic Medium
The simplest expression for Hill concentration tensor is that of a spheroidal inclusion
embedded in an isotropic medium. It reads as follows (see e.g.[26]):
P =
+
3K
2G
K
(5.39)
where:
d =
3K
K
(5.40)
3K + 4G
6(K + 2G)
5(3K + 4G)
with K and G denoting the bulk and shear moduli of the background isotropic matrix,
respectively.
The proper choice of K and G in (5.31) and (5.32) would result in a
self-consistent or Mori-Tanaka homogenization scheme. As a reminder, J = Jijk1
-(Oij k1) and K=E- J.
3
5.4.2
Spheroidal Inclusion in a Transversely Isotropic Medium
Evaluation of Pik(Q) =Cijkiajai in (5.38) can be performed by the following matrix
operation:
[F] = [][C][C]
T
where [...]T stands for transpose. Writing Q in matrix form 161]:
71
(5.42)
[w]
=0
0
1
0
0
-w 2 V2
W2
0
2
2
1j
0
0 W3
12
0
-2v2
v/2
-
Wi
2
2/
2
0
(5.43)
V/2
-Cl
2.
with the unit vector C in spherical coordinates [37]; 0 E [0, 7] and
#
E [0, 27], being
defined as:
ci = sin 0 cos q5
(5.44)
sin 0 sin#
(5.45)
CO 2 =
(5.46)
Cj3 = cos 0
the none-zero terms in (5.37) and (5.38) lead to line integrals:
= cosO and d =
- sin OdO which can be evaluated numerically. Finally, in Voigt's notation the nonzero components of P reads [37][61]:
1 11
4 C C ...
12 4 4
2
- 44C
+4C33C+5
C11(4C
C-C
44
C 13 C 44 + 3
2
I~CC 33
-4
C1 2 C 33 - 5
2
((
-
13 C 44 -6
2
C1 C 33 - 6
2
44+6 6 44
6
C1 2 C 44 + 2
2
C23
1)2( + 1) 2 (C1 2 C 44
2C12 C44
+ 2C12 C 33 + ClIC 4 4 ...
D,
_-1
-
2
C1 1 C 44 + (2CC 3 3 - 22C23
1
P13
=
WC13
+ C44 )
1
2(-Cl
2
_a
1
P33 =
I
(5.48)
42C
C 4 4 - 2 2 C424
13 1 3 C3C44ds
-
I 02V1
+ C1(
D2
72
(5.47)
C C 44 + 3CUC 4 - 5CjjC44
4
d
-
I
4
+
-5-
. . 10
16
2 4 C23 +3
-
_iDi
1
. 4
1) (-84C33C44- 3(4C 12 C 33
.
16
( 2 -
J-
2
d
(5.49)
D 2
- C442)d
(5.50)
P44
... 4
6C
-C
(3 2C2i - 2
16
1
6
6
_ C21
1 - 4 C, 1 C 44 - 86 C1 3 C 44
C21
-
1= 1
If
3
_1
Di
6
33 C 44 + 3 C1 1 C 3 3
-
3 4 Cl11 + ClIC 2 2 +
-
C1C2
-
-
Di
...
6
C 12 C 3 3 + 4
4
C 12 C 13 - 2
2
C 12 C 13 + 2
6C
11
C 13 +.
Di
.. 2
4C
3
+ (4C
1 2 C3 3
+ 8
4C C
11 4 4
- 3
4C
- 4
2
C 1 C 44 +.
C 1 C1 3 + 3
4
Cl
11 C 33
(5.51)
... 84C13C44 -
4
4
C 12
-
D,
Di
.. 3
2
C
C 12 - 2 6 C 12 C 1 + 2
3
2
D,
C1 C13)
dc
where:
D-=( 2Cu - C11 - 2 2C44 - (2C 12 +C12)(D 2)
D2 =
-C33C44
4
(5.52)
+ 2(2C13C44 - (2C C33 - 2(4 C13 C44 + (4C C33...
(5.53)
. . +2(2C 1 C 44 +
5.5
1(C
3 -(CHC44
-
C[ 3
- CC44
Approximation Schemes: Self-consistent and MoriTanaka
There are different ways to approximate (5.29) since exact statistical distribution of
texture parameters are almost never available for a REV. The two approximation
schemes, with some physically meaningful interpretation, employed in this thesis are
the self-consistent and the Mori-Tanaka schemes. The self-consistent approximation
scheme was introduced and developed by Hershey [391, Kroner
[48],
Budiansky [14]
and Hill 1401. In the self-consistent scheme, one needs to set C' in (5.29) equal to
Chom
C
resulting in an implicit expression that is solved by iteration methods. Letting
Chom
physically implies that no particular phase plays a dominant role in
contributing to the effective stiffness of the composite. Thus one can see why this
is the method of choice for micromechanical modeling of polycrystalline materials
73
and materials of a granular nature [621 [96].
An interesting characteristic of the
self-consistent homogenization scheme is the prediction of a percolation threshold of
0.5, meaning that for a solid packing density below this limit, the polycrystalline
(or granular) system is unable to form a continuous force path (Hertzian contact
between grains) that would establish stiffness and strength of the system. The MoriTanaka approximation scheme was initially proposed by Mori and Tanaka 158] and
further developed by Beneviste [10]. The Mori-Tanaka approximation scheme can be
achieved by setting Cs in (5.29) equal to CM, where CM is the stiffness of the load
bearing phase which acts as the mechanistically dominant phase responding to some
loading parameters imposed at the boundaries of a heterogeneous material system.
This scheme is often associated with a "swiss-cheese" matrix-inclusion morphology.
5.6
Imperfect Interfaces
This section summarizes the developments published by Qu (see 172],[73]). In most
continuum mechanics treatments, interfaces are assumed to be perfect. However, in
reality, interfaces may play a significant role on the effective elasticity of a composite.
Thus, typical continuum mechanics approaches need to be refined by accounting for
the presence of imperfect interfaces. The interface model employed here introduces a
spring layer of vanishing thickness, with a characteristic compliance, w, between the
inclusion and the matrix. The objective is to seek an expression, based on Eshelby's
solution, to capture the effect of imperfect interfaces between an inclusion, belonging
to domain Q1, and the matrix on the effective elasticity. The continuity of traction and
displacement discontinuity across the interface can be mathematically summarized as:
AO-ijnj = [{-ij(x)|r+ - 0-ij(x)|-]nj = 0
(5.54)
Aui = ui(x) 1+ - ui(x) I - = wij jknk
(5.55)
where F denotes the discontinuity surface, nj is its unit outward normal vector component, and ui(x)|r+ and ui(x)|r- are the values of ui(x), the displacement discontinuity
74
vector, as x approaches the interface from outside and inside of the inclusion, respectively (same notion applies to oij(x)lr+ and o-ij(x)jr-). The compliance of the linear
spring layer is denoted by wij which is assumed to be symmetric and positive definite.
One can recover a perfect interface (i.e. full bonding between the inclusion and the
matrix) by setting wij-0, while wij
oc represents a complete de-bonding of the
-
inclusion from the matrix. Relative sliding can be further considered by decomposing
wij into tangential and normal components:
Wij = a6ij + (0 - a)ninj
(5.56)
oz and 0 represent the compliance in the tangential and the normal directions of the
interface, respectively.
It is important to note that material interpenetration may
occur for some non-zero values of 3, which would violate the strain compatibility
It was shown by Eshelby [29] that the total
requirements used in the derivation.
strain in an ellipsoidal inclusion is uniform if the eigenstrain distribution is uniform:
S
ijk (5.57)
k1
E
is the uniform eigenstrain and Eij is the total strain in the inclusion.
One can
refine (5.57) to account for imperfect bonding by introducing a surface integral over
the interface to collect contributions due to interface "imperfection":
-k Clmu
Eij = Sijkl
(5.58)
Au k(-)ijmn(X - x')nidF(x)
The first term is Eshelby's original solution (5.57), where
fijmn is related to Green's
function in an infinite domain, G, as follows:
4
jijmn (X) =
1iGm
""
-[
4
OxnOxj
(x) +
-
G
mj
DxXj
(x) +
-
75
G.
"'
xmdXj
(x) +
"n'
oxmxi
(x)]
(5.59)
Utilizing (5.55) in (5.58), one can write:
ij
-
(5.60)
Wkp7pq (x)'ijmn(x - x')nqnidF(x)
CkIrn
Furthermore, since:
(7ij =
(5.61)
ijkl(Uk,l - EkI)
one can write (5.60) in the form:
Eij =
iSE
+
I
. . - CklmnCpqst
For small values of
Wkp,
')nqnidF(x)...
ijmn(2 -
kp
*kmnCpgstEst
(5.62)
j
WkpEst 4'ijmn(X -
X')nnid7 (x)
which physically translates into slightly weakened interfaces,
the above expression can be approximated by perturbation methods. By iteration,
one can write (5.62) as:
E
")=+ij
I
ijkl kI + C.J
C ~mnEJpqstE*t
n
tf
CklmnCpqst
for n = 1, 2, 3...
.
j
4
kp
1
WkpE-
')nqnidf(x)...
ijmn(X -
jijmn(x_ -
(5.63)
x')nqnidf(x)
Thus, the leading order term of the solution for the total strain
field inside an ellipsoidal inclusion with slightly weakened interfaces reads:
I + CklmnCpqst*
-
SEs
j
I[CklmnCpqst
Hence, the modified Eshelby tensor,
W
f
STIIEs
-
(64ijmn()
'nqnidF(x)...)
kp 4 fijmn(N
-
Nxxnnldf(x)]
for an ellipsoidal inclusion with slightly
weakened interface, can be expressed as:
S{g/fkXsh
= sIsh
[CklmnCpqst
j
Wkpq'ijnn(N -
76
N')nqnldF(x)](IstkI
-
Cstkl)
(5.65)
It should be noted that strains are no longer uniform in this general case.
Let us
consider an ellipsoid defined by:
xi
X2 2
2
(5.66)
(X3 )2<
a2/
\al,
\a3/
where a,, a 2 , and a3 are three length parameters needed to define an ellipsoid in space.
In pursuit of obtaining the average strain of the inclusion, one can employ:
S
=
ijkl
Cmn
CklmnJ 2
ijmn(
fij
-
(5.67)
x)d(x)
Thus (5.65) can be re-written as:
Slksh
-
1
j
S
(x)dQ (x) = SiM + (Iijpq
- Sijpq)HpqrsCrsmn(Imnk
SEmshk) (5.68)
where:
Hijkl - aTijkl
(5.69)
+ (- - Ce)Qijkl
and;
Tijkl
163
16-F.
j
(6ikfijfil + 6jkfi1
Qij ki =
+ 6i1~kfj + 6jillkfli)n
nin
Z47F
nknin-
dO] sin Odo
where
(...)T
sin0 cosO
a2
\ a1
(5.71)
a3
T
(5.73)
/
a
=, , a
(5.70)
(5.72)
n = V'n
^~i
(sin bcos0 sin
dO] sin Od#
denotes transpose. Note, that H possesses both major and minor sym-
metries; i.e. Hijkl = Hjikl = H i=
Hjilk.
For the special case of spheroidal inclusions,
(5.70) and (5.71) reduce to, respectively:
1
a
77
(5.74)
Qijak
1
=
5a
(2Iijkl +
6ij 6 k1)
(5.75)
"a" is the inclusion grain radius, which introduces a length scale into the model.
Finally, invoking (5.20), one obtains the modified Hill concentration tensor:
pM =
SMEsh : Vs
(5.76)
This modified Hill concentration tensor allows one to to account for imperfect (to be
precise: slightly weakened) interfaces in a microporomechanics based framework.
5.7
Chapter Summary
We have presented in this chapter the theoretical tools that we will employ for model
developments for organic-rich shale systems. The homogenization theory is presented.
It is through the approximation schemes associated with the homogenization theory
that we capture morphological variations in organic-rich shale formations of different
maturity. Finally, imperfect interfaces are introduced as an additional tool that will
be utilized in our model development.
78
Chapter 6
Microporoelastic Model for
Organic-Rich Shales
The objective of this chapter is to integrate the materials presented in the previous
chapters into two multi-scale microporoelastic models representing mature and immature organic-rich shale systems. For each model, first, the volume fractions of the
considered geo-mechanistic phases are introduced, based on our hypothesis regarding
texture of these shale systems and the structural thought-model developed earlier.
Next, the formulations for computing effective anisotropic poroelasticity of organicrich shales, at each considered length scale, are derived.
Finally, all developments
are integrated to obtain the undrained behavior of these porous, naturally occurring
geo-composites.
6.1
Hypothesis Testing: Maturity Induced Morphological Change
The structure of kerogen evolves with burial depth and consequently pressure /temperature
change and hydrocarbon generation, going through complex physical and chemical
transformations. This evolution moves towards increasing aromatization and development of an ordered carbon structure, leading to graphite at the upper end of maturity.
79
Groups of aromatic sheets, making up the building block of kerogen, evolve from a
random distribution for immature kerogen to parallel stacks (an ordered structure)
as kerogen thermally matures, giving rise to a carbon order which becomes stronger
with increasing temperature
[93].
Immature organics, containing a large proportion
of aliphatically associated hydrogen, tend to have a lower density and to deform plastically [71]. "Hydrogen rich macerals (liptinite group), which yield high amounts of
oil upon heating show intense florescence, high reflectance, and a higher density as a
result of the aromatization processes (i.e. dyhydrogenation and cyclization) resulting
in more planer and aligned structure of the carbon-rich rings"[102. Morphology of
organic-rich shale goes through major transformation as kerogen matures. In immature organic-rich shale, kerogen seems to form a connected network (large kerogen
pockets). This network does not necessarily mean that kerogen constitutes a solid
framework, embedding inorganic components [801. As maturity progresses, the concentration of coarse grains seems to increase while kerogen pockets are reduced in size
and become dispersed in the matrix [71]. Ahmadov [5] reports that as maturation
progresses and hydrocarbons are generated, low aspect ratio organics are disconnected
and distributed as patches in an inorganic framework. Bousige et al. 112] report a
transition ductile-brittle cross over as kerogen matures. These observations are indeed consistent with physical intuition since with maturation and generation of oil,
and subsequently condensates and gas, initially long Carbon chains are broken into
smaller ones, their chemical structure changes and their structure is transformed into
a more ordered one. Considering these observations, we introduce change in morphology in our quest for maturity dependent modeling of organic-rich shale systems, in a
hypothesis testing framework, as follows:
Hypothesis 1: The first-order effect of kerogen maturity on overall effective elasticity
of organic-rich shale systems with low TOC can be captured as an effective texture
effect. Namely, a polycrystalline morphology is used for a mature organic-rich shale
system and a matrix-inclusion morphology is introduced for the immature organicrich system. This implies that for such systems, the elasticity of kerogen does not
80
play a role of first-order on the overall elasticity. However, with high TOC in mature
organic-rich shale systems, the effect of kerogen elasticity become of first-order. This
is not the case in immature systems as mature kerogen is arguably stiffer than an
immature one.
Hypothesis 2: Clays exposed to similar depositional and burial (digenetic) processes
exhibit (e.g. water salinity, pH, temperature variations, etc) similar elastic behavior.
6.2
Basis of Design: A Bread Analogy
From a physical analogy perspective, one may consider organic-rich shale systems as a
bread dough that has been put into a stone oven (inspired by a popular bread in Iran
called "Sangak"). Initially, the dough makes "smooth" contacts with stones in the
oven which in the modeling realn translates into perfect contacts. As the dough is
exposed to temperature over time (burial/ diagenetic processes over geological time),
air bubbles are formed (hydrocarbon generation) and popped; temporarily creating
pores that are "self-healed" moments later. This is a common occurrence at the initial
stage of baking (maturation), consistent with observation reported in Ref. [241 that
porosity in organics are dominantly observed in mature kerogen rather than immature
samples. As baking progresses, the composition and structure of dough transforms
into bread (mature kerogen), small pores start appearing and the "smooth" contact
between bread and stones are no longer in existence. When the bread is ready, one
needs to pull-out the stones that have been engulfed by the bread. If one meticulously
performs "experiments" with pulling-out stones, a positive correlation between the
ease of separation of stone, from bread, and the degree of "well-doneness" of the bread
can be established. This is indeed due to a contrast in the linear thermal expansion
coefficients (see e.g. 1115]) of a stone (its linear thermal expansion coefficient remains
more or less, constant during baking) and the bread; which evolves as the structure
of dough and its composition changes over time. As a result of this contrast, residual stresses are built up at the interface of bread/stone, and they lead to (partial)
81
de-bonding at the interface.
With this analogy in mind, a literature survey on mechanical characterization of
organic-rich shales suggests the prevalent presence of discontinuities in mature organicrich shale relative to immature systems.
The pioneering work of Vernik and his
colleagues in the 1990s on laboratory characterization of organic-rich shales (see
[107],[110],[1091 and [1041) suggests that the existence of micro-cracks in "mature,
kerogen-rich shales in-situ could be the rule rather than the exception". Pahanhi et
al. 167] reported, after a series of experiments on Green River shale samples using high
resolution synchrotron x-ray tomography, that cracks tend to nucleate and propagate
in the locally most heterogeneous areas and that they are usually not penny-shaped.
Padin et al.
[64]
report, while studying some samples from Eagle Ford shale, that
kerogen and calcite interface provides a plane of weakness for microfracture growth.
In addition, it has been observed that velocity anisotropy in organic-rich shale is
"mainly textural dependent rather than due to microcracks" 171]. Thus, for modeling purposes, based on the logic that if discontinuities are described and modeled
as microcracks, then they shall not remain open under high in-situ stresses in the
subsurface or under high confining pressures in lab (simulating in-situ conditions)
and following the bread analogy; imperfect interfaces will introduced as a modeling
ingredient for mature organic-rich shales.
With regards to grain and pore orientation distribution and aspect ratio; in what
follows we consider grains and pores to be spheroidal. This is based on the notion
that aspect ratios and grain orientations do not have a first-order contribution on the
overall poroelastic behavior of organic-rich shales, specifically in a porous solid with
a high packing density [63]. Vernik and Kachanov 11061 argue that in seismic and
sonic frequency range for low porosity rocks, (pore or crack) aspect ratios do not play
a role, at least not with a first-order contribution. In addition, introduction of aspect
ratio and orientation of grains and/or pores in the forward application of a model
leads to input parameters with high uncertainty attached to their characterized lab
82
values. Such modeling input parameters would introduce high uncertainty into the
model that may diminish its predictive capabilities. Also, these parameters could be
abused by treating them as "fitting" parameters to match predictions with observations, without much physical meaning attached to them.
The other modeling ingredient that we need to establish is morphology of mature
vs. immature systems. For an immature system, with kerogen best described as a
pliable, amorphous organic polymer which manifests itself in more or less connected
kerogen pockets
[71],
we employ a Mori-Tanaka scheme, suggesting that the inorganic
solids constitute the mechanistically dominant, load bearing phase in immature systems. This is consistent with our physical intuition that a phase composed of organic
polymers does not contribute, nearly as much as inorganics, to the effective elasticity
of immature organic-rich shale systems. Also, it is interesting to note that Vernik and
Kachnov [105j report that Mori-Tanaka scheme is a powerful tool for capturing poorly
consolidated sands. Indeed, one can think of kerogen maturation as a "consolidation"
process induced not only by physical but also chemical processes since the process
of kerogen maturation entails a volume change in the organics phase, a phenomenon
that has been extensively studied (see e.g.
several authors (see e.g.
[71][5])
[116],154]).
Indeed, it has been reported by
that with maturation and hydrocarbon generation,
large pockets of kerogen tend to break down, leading to a patchy distribution of small
pockets of kerogen, described as coaly/woody (see e.g. [68]) with a more orderly structure (see e.g. [931). As maturity progresses, the morphological evolution described
above encourages Hertzian contact between inorganic grains. This would constitute
a granular morphology best captured by a self-consistent scheme (see e.g.[961).
In summary, for immature organic-rich shales, a Mori-Tanaka scheme is employed,
where the interfaces are assumed to be perfect.
For mature organic-rich shales,
a self-consistent morphology is considered with imperfect interfaces.
In terms of
microtexture, an isotropic morphology is assumed; meaning that grains and pores
posses the same spheroidal morphology. Lastly, the anisotropic poroelastic behavior
83
of the organic-rich shales is contributed solely to the intrinsic anisotropy of clay particles which propagates, depending on the homogenization scheme, at each considered
length scale of the organic-rich shale models.
6.3
Imperfect Interfaces: Organic Maturity Evolution
The bread analogy discussed in the previous section highlights the role of imperfect
interfaces in mature organic-rich shale systems. In order to show more rigorously
why imperfect interfaces need to be considered in modeling mature organic-rich shale
systems; we will derive an expression for radial stresses at the interface of a matrix
and inclusion using the classical quenching problem. Then, using available data in
the literature, we show robustly that imperfect interfaces contribute to the effective
elasticity of mature organic-rich shale systems. Lastly, we will discuss a recent study of
mature and immature kerogen which agrees with our intuition regarding the physical,
chemical and structural evolution of kerogen with maturity.
6.3.1
The Quenching Problem
The "quenching problem" refers to the studying of the production of industrial ceramics composed of a matrix and inclusions. The manufacturing process comprises
of exposure to high temperature followed by a cooling period which leads to residual
stresses at the matrix-inclusion interface. Then, the stored energy at the interface
could be released by partial debonding.
The development herein follows closely Ref. [99]. We consider a spherical inclusion
of radius a embedded in an infinitely extended matrix, with zero initial stress field.
Both the matrix and the inclusion are assumed to h homogeneous linear isotropic
thermoelastic materials. The composite, i.e. matrix and inclusion, are subjected to a
uniform decrease in temperature (60). The goal is to derive the expression for strain
84
field and the radial stress, a.., generated by the cooling process.
In a spherical coordinate system, the radial displacement vector,
, only depends
on radial coordinate r:
(6.1)
= u(r)er
Thus, the strain components associated with it read:
du
(6.2a)
dr
-
8 rr
u
E00
= E
(6.2b)
r
The stress components for a linear thermoelastic material read:
=
(K
2
-
=
=
du
G
)
o
udu
+r2G
du
+2)
dr
r
2 G
3
)
rr
3
dr
-3ThKAO
dr
+2-) +2G-- 3aThKAO
r
r
(6.3a)
(6.3b)
Thus, the equilibrium equation (V - - = 0) reduces to:
dUrr
dr
2rr
-
0700 -
o "
= 0
r
(6.4)
Substituition of 6.3a, 6.3b into 6.4 leads to the following partial differential equation:
d (du
dr
dr
+2rU
(6.5)
0
The solution of 6.5 and subsequently the expressions for 6.2a, 6.2b and 6.3a, considering the boundary conditions read:
For r < a:
U = AincrAO
(6.6a)
Err = E0 = E, = AincAo
(6.6b)
Urr
= 3Kinc (Ainc
85
-
ai)
AO
(6.6c)
and for r > a:
u
Avr + B)AO
Am B
Err=
AO
(6.7b)
= EAM
+AO
E(A
1rr=
(6.7a)
(3KM(AM
-
(6.7c)
- 4 GM
T
B )AO
(6.7d)
Utilizing the continuity of both the radial stresses, i.e. 9rr (r = a) and the displacement u (r = a), the expressions for Ainc and BM are derived, as follows:
Ailc
BM
Thus, the expression for
Urr
=
(A
acK inc
+ 4GMaM
+G
(6.8a)
3Kinc + 4GM
M
3Kinc
K
(ah
-
Th-am
Th
(6.8b)
3Kinc + 4GM
(r = a) can be written as:
3Kinc4GM
rr (r =a) =3Kinc + 4GM (Th
9
--
where a positive value of radial stress, grr, at the inclusion /matrix boundary, i.e.
r-a, implies a tensile stress field.
Here, superscripts M and inc denote quantities
associated with matrix and inclusions, respectively.
orh and a i
represent coeffi-
cients of linear thermal expansion for matrix and inclusions respectively, and AO is
the change in temperature. Assuming that change of temperature is positive during
burial/ diagentic processes over geological times (i.e.
AO > 0), one can readily see
that a positive radial stress occurs when thermal expansion of the matrix is greater
than that of the inclusion, producing a tensile radial stress at the interface. Characteristic values of thermal expansion coefficients are given in Table 6.1. For inclusions,
taking chnrcoal as qn end memhr of mature kerogen and Green River as an immature example as well as kaolinite as the mineral making up the inorganic matrix; one
readily sees that at some point in geological times, the physical and chemical changes
86
in kergoen structure and composition lead to aM - ai" > 0 which in turn leads to
partial de-bonding at the interfaces between organic and inorganic constituents.
This illustrates the significance of burial/diagenetic processes on overall anisotropic
poroelasticity of organic-rich shales. In reality, organic-rich shale deposits, over geological time, not only go through burial; but also uplifting and erosion, entailing
a AO < 0. In addition, the geothermal gradient at the geological time and burial
location become a critical factor in terms of kerogen maturation and its effect on the
overall poroelastic behavior of organic-rich shales. Kerogen maturation is a complex
function of exposure time, pressure, temperature and composition that is subjected
to rapid physical and chemical changes (in geological time scale) as it generates hydrocarbons.
A recent study by Bousige et al.
[121 of the molecular structure of mature and
immature kerogens suggests that kerogen maturation is followed by a crossover from
plastic to brittle rupture mechanisms; consistent with our intuition based bread analogy. However, the thermal expansion coefficients of constructed molecular models for
mature and immature kerogens were not studied in this work.
Table 6.1 - Linear thermal expansion coefficients for various geomaterials.
Material
aCTh[mm/mCo]
Degassed Charcoal[8]
Charcoal (glassy)[120]
Charcoal (rubbery)[120]
Green River Kerogen-MD Simulation[119]
Green River Kerogen-Experiment[119]
Kaolinite (x 3 ) [57]
4.50
4t0.1
6 1
292+25
104 8
18.6 1.3
Kaolinite (xi)[571
5.2 1.7
9
Chlorite (x 3 )[57]
2.3
Chlorite (x 3 )[571
11.1 1.4
a Quartz[43
Feldspar[43]
24.3
14.1-15.6
87
6.4
Immature Organic-Rich Shale
This section presents the multi-scale microporoelastic model for immature organicrich shale system. Based on our hypothesis testing approach and the basis of design
discussed before and schematically shown in Figure 6-1, a Mori-Tanaka homogenization scheme, at each considered length scale, is employed to solve for effective elastic
behavior. Furthermore, for the immature systems, interfaces are assumed to be perfectly bonded.
6.4.1
Volume Fractions
Based on the structural thought model developed before, the volume fractions associated with mechanistic phases at each considered length scale introduced. For the
immature model, the following phases are considered at level II; required to satisfy
the imposed constraint:
fclay
/
+
fker
+
finc
+
= 1
(6.10)
represents the (measured) porosity at level II. The volume fraction of the rth phase
at level II is defined as:
fr = (1
mr/Pgr
-
(6.11)
K1-1 mi/pg,i
where mr is the mass percent of the rth phase,
Pg,r
is the grain density of this phase
and P stands for all minerals present plus kerogen. The inclusion volume fraction,
finc,
(1Et
1mj/pg'
(6.12)
consisting of all non-clay inorganic minerals is calculated as follows:
E1 mj/pg~i
finc
where N stands for all non-clay inorganic constituents.
At level I of the model, the solid volume fractions are defined as:
r =
1
88
fr inc
(6.13)
similarly, porosity at level I reads:
S= 1 - 0finc
_ic(6.14)
Based on the discussed structural thought-model, two solid phases and a pore phase
are defined at level I, satisfying the following constraint:
rqker +
+ P = 1
ciay
(6.15)
In order to obtain the homogenized elasticity of the inclusion, and following the discussion regarding the dominant (by mass percent and/or stiffness) non-clay minerals
based on Woodford mineralogy (see Table 4.1), the following normalized volume fractions are defined:
f pyrite
norm
1 nm
fquartz
and
fPyrite
f quartz
(6.17)
can be obtained using (6.11). It is required that:
qurtz
6.4.2
+
fquartz
f pyrite _ f quartz
fquartz
where
(.6
fpyrite
fpyrite
+
t
f Pyri e =
1
(6.18)
Level I
At level I of the microporoelastic model we consider a 3 phase composite consisting
of two solid phases (kerogen and clay) and a porous phase. Thus, the behavior of this
porous composite can be described using the classical poroelastic state equations:
E
(p -
=
Chom
: E - alp
p
: E + p(6.20)
o =
NT
89
(6.19)
where o - po1 is the Lagrangian porosity change.
a, is the
2 nd
order tensor of
Biot pore pressure coefficient at level I and N 1 is the Biot solid modulus at level
I. Equations (6.19) and (6.20) describe the poroelastic behavior of an REV, with its
domain denoted by Q, composed of pore space, QP = ooQ, and a solid domain defined
as
Qs
-
Qker
+
Qclay
- (1
-
O)Q.
Utilizing a continuous description of stress field in
a heterogeneous material system associated with the defined REV, one can write:
a(x)
C(x) : e(x) + UT (x)
Vx C Q
(6.21)
with the following distribution of elastic properties:
C(x)
Cker,
Vx c Qker
Cclay
Vx E Qclay
(6.22)
The eigenstress distribution reads:
-p 1,
T(x)
0,
Vx E
P
Vx
ker
(6.23)
Vx EQclay
thus, one can express the mechanics problem associated with finding the stress and
strain fields when the defined REV is subjected to macroscopic strain (E) and eigenstress
(OT)
fields, as:
V - a = 0,
Vx E Q
a = C(x) :e + UT(x),
Vx C Q
E - x,
Vx E-
(6.24)
Q
The variation of porosity at this level (and level II) is based on a Lagrangian porosity description
(as opposed to a Eularian definition). In the Lagrangian definition, current partial saturation relates
to fluid prior to any deformation, i.e. the undeformed initial porous volume. This is due to Coussy
[21].
90
since the defined mechanics problem is linear with respect to the loading parameters
(E and aT), the problem can be decomposed into two sub-problems. The solution of
each sub-problem will be superposed to obtain the final solution to (6.24):
A. The mechanics problem of the response of the REV to E
B. The mechanics problem of the response of the REV to
UT
The mechanics problem associated with the response of the REV to imposed E can
be summarized as:
V-oA
,
Vx E Q
a A = C(X): EA,
Vx E Q
A
=
=
(6.25)
Vx EQ
E x,
where superscript A refers to the first sub-problem. Utilizing (5.8), one can link the
loading parameter E to the local strain field as:
eA(x)
(6.26)
= A(x) : E
the corresponding macroscopic stresses can be obtained by using (6.26) into (6.21)
and considering (6.22) and (6.23). This leads to:
EKA _
=
(CCA(x))Q
:A Q : E
(6.27)
With this in hand, one can write the drained homogenized stiffness tensor for the
so-defined REV as:
Chom =
(C : A)
=
,clay Aclay
91
:
Cclay +
,kerAker
:
Cker
(6.28)
where:
Aclay = [I + P :
-
:
...
:
(Ccay
Cclay))-l
-
+
...
F
ker
-(6.29)
p
- Cclay)]l : LclayQ([I +
(Cclay
ker (I + P : (Cker - Cclay-l +l p +p : (C, - Cclay))-I1]l
Aker = [I + p : (Cker -
[1clay(I[
c:ay)
+ p : (Cclay
-
Cclay))-
+.
ker
Cker
:
-1
(I + P : (C er - CClay))+ p(I-+1+ : (C+p - Cclay))-(6.30)
..
Next, the second sub-problem associated with the mechanics response of the REV to
the application of the loading parameter aT is defined as follows:
V - UB
o.B
= C(X) : eB
+
= o,
VX
EQ
T(X),
VX
Q
B =
VX
0,
E
(6.31)
Q
with the subscript B denoting the sub-problem B. The macroscopic stress, EB, for the
second sub-problem is the average of the local stress field, uB, over the volume of the
REV (Q). Application of the Hill Lemma (5.6) to the local strain field of sub-problem
A (i.e. EA) and the local stress field associated with sub-problem B (i.e.
(UB :
A
)
B
UB)
leads to:
(6.32)
:E
By expanding (6.32) using (6.31), one then obtains:
(oB : eA)
EB
:C
A)
A:
+ (UT(x)
: EA)
(633)
A second application of the Hill Lemma to the local strain field associated with subproblem B, where
(eB)Q
= 0, and the local stress field of sub-problem A (i.e. aA
leads to:
KeB : C
EA)
A:
92
0
(6.34)
by combining (6.26),(6.33) and (6.34) one thus obtains:
EB _
.T (x) : A(x))Q
(6.35)
Expression (6.35) is known in micromechanics as Levin's theorem [26][118]. By definition of sub-problem B, EB represents pressure variations in the pore domain, EP,
under zero macroscopic strain conditions (E = 0).
Employing the distribution of
eigenstresses in (6.23), (6.35) can be expressed as:
EB _
-pfo
T
1: A"
(6.36)
where A* is the strain localization tensor associated with the pore inclusions, defined
as:
A
[I + P : (ClO
-
Cclay)] -I
ker (ff + P : (Cker -
one readily recognizes the
2 nd
:
IClay (
+ P
: (Ccay
(i[ + P : (C
Cclay))-l +
-
Cclay))-I+
(6.37)
Cclay))-I]-
order tensor of Biot pore pressure coefficient by com-
paring the microscopic eigenstresses (aT) with its macroscopic counterpart (ET):
a, =
where subscript
(...)1
po 1: A" =1: (i - (A)Qs)
refers to level I of the microporoelastic model.
(6.38)
Finally, the
superposition of the stress field solutions for the two defined sub-problems leads to
the first poroelastic state equation (6.19). Expanding (5.3) using (6.21) and utilizing
(A)Q = i, lead to:
(.B)
_
B)Qs
-
Pop 1= -pOZI
Now expanding (6.39) while using (6.38) entails:
n
93
(6.39)
with r denoting solid phases (i.e. clay, kerogen). One can readily see from (6.40) that:
(6.41)
(A' - I)
p
(ar)B
where (ar)B is the average stress associated with the rth solid phase, under conditions
pertaining to the sub-problem B. Also, from the compatibility requirement of strain
fields and utilizing E = 0, we have:
(6.42)
_ _B)Q
0o(Ep)B
Combining (6.42) with the second poroelastic state equation (6.20), one obtains:
((P - 9PO)B
-
1:
B6%)Q
p
(6.43)
N,
where:
(Cr)
(er)B
1
:
(6.44)
(Ur)B
Thus, (6.43) can be re-written as:
n
-
p
1: (,r)B
1:
(6.45)
-
(P - po) =
N,
r=1
Next, by substituting (6.39) into (6.45), the generalized expression for the skeleton
Biot modulus for a porous composite consisting of multiple solid phases at level I is
obtained:
NTIN
= 1:
r= 1
,r(Cr)-I : (1: (I -- Ar))
(6.46)
employing (6.46), the explicit expression for N 1 for our defined REV reads:
1N
NT
1:
[claysclay :
(1: (I
-
Aclay)) _
94
,kerSker
:
(1: ( f
-
Aker))]
(6.47)
6.4.3
Level II
Following the developed structural thought-model (see Section 3.1), shale at the
macroscopic level (i.e. level II) is considered to be a composite consisting of inclusion
grains and a porous solid fabric, upscaled from level I, phases. Thus, the REV at this
level consists of two domains: the porous solid denoted by
the inclusion domain,
Qinc
fincQ.
QhO"
(1
-
finc)Q
and
A continuous description of the stress field in the
heterogeneous material system under consideration reads as:
a(x)
C(x) :e(x) + oT (x)
Vx C Q
(6.48)
with the following spatial distribution of stiffness properties:
C(x) =
Ch"m
Vx E o1
Chom
in
Vx E Qinc
(6.49)
_
Chom denotes the effective elasticity of the inclusion grains. The eigenstress distribution is defined as:
(6.50)
UT (x)
E Qho"
Vx vXE
Qiflc
{0,P
where the expressions for a, (6.38) and Chom (6.28) were derived for the problem considered at level I. Similar to the mechanics problem introduced and solved for level I
of the multi-scale micromechanics model, the mechanics problem at level II is broken
down into two sub-problems as follows:
A'.The mechanics problem of the response of the REV to E
B'.The mechanics problem of the response of the REV to oT
For the first load case, the mechanics problem associated with the response of the level
II REV to E, one can relate the local strain field to the macroscopic field (prescribed),
95
I Isb
l A'
I
as follows:
Vx C Q
e A(x) = (A(x))o: E
(6.51)
Application of (5.3) to (6.48) and expanding it using (6.51), results in:
-A=(A' o : E
(6.52)
Vx C Q
where C or", the drained effective (homogenized) elasticity, reads:
Cho" = Chor
+ finc(Chomn
horn
: A'n
(6.53)
with:
Chom)]-l : [finc(
+ P:(Con"-C-
"n))-1
+
horn = [I + P : (Cho" -
(1 - finc)(E + P : (Cho" - Cho"))-l]
[f + p : (Chor
inc
_-
Ihon )]-1 : [finc(ll+
-
finc)(II
:(hon inc
.
-+ P
-
: (Chor"
Chor
(6.54)
-1+
)
horn =
inc
I
-
Cho"))-1]
I
(6.55)
The second sub-problem, denoted by B', corresponds to the case of a prescribed
eigenstress field, with zero macroscopic strain at the boundaries of the REV. Based
on Levin's theorem (6.35), one can write:
EB'
with al being the
2 nd
: A(x))Q - -pall
T =T(x)
(6.56)
order tensor of Biot pore pressure coefficients at level II of the
multi-scale model:
all
I - tie ciiaiige
g of porosity
wilere
a1 : (fI-
fincAhorn)
(6.57)
in the SUD-prODIemlA canl be expressed as:
(0 - Oo)A' - all : E
96
(6.58)
For sub-problem B', utilizing porosity variations at level I and employing the scaling
relations between volume fractions, i.e.
po =
00
one can express the macro(1 - finc)
scopic change in porosity under loading conditions dictated by sub-problem B', as
follows:
f inc) (
_(
-)'
- :1(Ehom
0o)B'
-
'
f')(a,
Utilization of the zero strain condition, i.e.
+
)
(o - #
IN,
(6.59)
0, into macroscopic stress (6.56)
(EB/)Q
results in:
EB
inc(inc)B'
_f
+ (1
-
finc) (orom B/
(6.60)
(1
-
:c(a
finc)(c"om - C
1
Bom
finC)&ip
Expanding (6.60) one step further using (6.57), leads to:
(1
-
(Con
finc)hom)B'
-
chorn
[-al
1:
(1
finc)ai]p
(6.61)
Finally, combining (6.61) with (6.59) and (6.58), the second poroelastic state equation
for level II is derived:
p
S- 00 = a
(6.62)
: E + Nil
where the Biot solid modulus at level II reads:
1
iN 1
:
finc
-
+f"o
=
chor
m
C)
-1
:a : ([ -A horn)
(6.63)
INI
and the homogenized inclusion stiffness, C O", in a self-consistent manner, is obtained
from:
Cc"'
3K
o"J
+ 2Gho"K
(6.64)
where:
fqOuarzKquartzAquartz + fnpyriteKpyrite Aprite
onm
norm
d
97
norm
d
(6.65)
(6.66)
and;
AV"la
=
Kquartz
(1 + L( Kp m
Kquartz
+
-
1))
[fnorm(1 +
(
1))'...
inc
-
APyr"
( KPyrite
(1 +
(+
V
Kquartz
K!m
inC
-
1))l
[fna(1
+
nKr
&( Khorn
-m 1))
inc
( .8
KPyri t e
~quartz
A quar t z
AP
Al"ri t
(1 +
G
"G G
h"
/3( G horn
1))l[ff"T1
inc
(.8
quartz
(1+ (Iorm
G3
=
(6.67)
mnc
+
. pyrite
Ko
"z(1 +
quartz
f3(
G
-
1))1...
(6.69)
1))-'..
(6.70)
inc
Thus, we have derived explicit expressions for computing the effective elasticity of the
inclusion grains, assumed to be composed of quartz and pyrite, based upon reported
mineralogical compositions as well as the (quasi-)isotropic elasticities associated with
these mineral crystals.
98
Inclusion
Inclusion
(effective)
Kerogen
Porosity
Clay
O
Perfect
Interface
Level
I
Level 0
Immature System
"Swiss Cheese"
Morphology
(Mori-Tanaka)
Figure 6-1 - Schematic of the multi-scale microporoelastic model for immature
organic-rich shales. Inclusion stiffness at level II is computed by homogenizing the
dominant non-clay minerals in a self-consistent manner. Following the hypothesis of
texture effect; Mori-Tanaka approximation scheme is applied at each scale for homogenization. For immature systems, interfaces are considered to be perfect (perfect
bonding) among different constituents.
99
6.5
Mature Organic-Rich Shale
As discussed previously, for the mature organic-rich shale systems, a self-consistent
morphology is assumed; entailing a self-consistent porosity distribution. In addition,
imperfect interfaces are considered between inclusion grains and the porous solid
fabric, at level II of the model, schematically shown in Figure 6-2.
In terms of
microporoelastic formulation, level I of the mature system is identical to the level I of
the immature model, with a slightly different set of definitions for volume fractions.
However, level II of the mature organic-rich shale model is different than that of
an immature one due to the porous inclusions; a consequence of considering a selfconsistent morphology. Thus, we will introduce new tools which enable us to derive
the effective poroelastic coefficients of two porous systems, assuming same pressure
field prevail in both.
6.5.1
Volume Fractions
With the assumption of a self-consistent morphology, and consequently self-consistent
porosity distribution for a mature organic-rich shale system, the volume fractions are
defined slightly different than that of the immature model. For the mature organicrich shale model, at level II, the following mechanistic phases are considered:
fclay + fker + fpor-inc
+
$s=
(6.71)
1
where:
fpor-inc
(6.72)
finc _ oinc
finc
finc
+
(6.73)
f ker _ f clayo
and;
(6.74)
-o#inc
q Ps +
# is the measured porosity,
$il" is the porosity associated with inclusion grains and oPs
is the porosity associated with the porous solid fabric. felay and
100
fker
can be computed
using (6.11), finc is defined in (6.12) and
r-inc
fp
denotes the volume fraction associated
with the porous inclusion, that carries the mechanistic contribution of the porous
inclusion onto the overall effective behavior of the composite at level II of the model.
The porosity at level I of this model, slightly different than the immature case, is
obtained from:
O7 PS
1
-
(6.75)
fpor-inc
Solid volume fractions at level I of the model are defined as:
r
=
_fr_(6.76)
1
-
f por-inc
(6.76)
+
(6.77)
where the following constraint is enforced:
'TIclay
+
,ker
The volume fractions relevant for obtaining the homogenized elasticity of the
porous inclusion, based on Haynesville mineralogy, are defined as follows:
inc
(6.78)
f por-inc
norm
mcalcite
fcalcite
norm
fpor-inc
fquartz
norm
f qurtz
fquartz
where
fquartz
and
fcalcite
(6.80)
fpor-inc
are obtained from (6.11); with the following constraint satis-
fied 2 :
cm
4Onorm
+
fcalcite
+norm
+
fquartz
norm
2
1
(6.81)
As a reminder, we considered feldspar phase to be mechanistically represented by calcite due
their relatively small elastic contrast
101
6.5.2
Level I
Although microporoelastic representation of level I of the mature organic-rich shale
is analogous to the immature case, the results are summarized in this section for
completion. Note that the definition of porosity at level I is slightly different than
that of the immature case. At level I, the homogenized response of clay, kerogen and
porosity in a self-consistent scheme reads:
Chom = (C : A)
=
lclayAclay
: Cclay + ,kerAker : Cker
(6.82)
utilizing (6.30):
Aclay= [E + P:
Aker =
+ P
(Cclay -
: (Cker
-
C')]-
1
(6.83)
(6.84)
Chom)-
the tensor of Biot pore pressure coefficients, from (6.40), reads:
a, = oo 1: A" =1: (If - (A)Qs)
(6.85)
where, the strain localization associated with pore domain denoted by A", is expressed
as:
Ap = [f[ + P : (Cc
-
(6.86)
Chof)]1
The Biot solid modulus, from (6.86), is expressed as:
1
=1: [rclaysclay : (1: (If
6.5.3
--
Aclay))
+ ykersker : (1: (-
Aker))]
(6.87)
Level II
The framework for deriving the microporoelastic formulation of the effective response
at level II of the mature organic-rich shale model, based on the structural thought
102
model presented previously, is the same as the one presented for immature systems.
However, extra steps need to be taken in order to obtain the poroelastic coefficients
of the two porous composites, i.e. a porous solid fabric upscaled from level I, and the
porous inclusion grains. The description of a continuous stress field in the REV at
level II of the mature organic-rich shale model reads:
U(x) = C(x) : e(x) + orT (x)
Vx E Q
(6.88)
which reads the same as (6.21), with different spatial distributions for stiffness:
vXE
chorn
C "
_E
Chom
por-incI
Chm -cis
(6.89)
Vx E Qpor-inc
-
C(x) =
horn
the elasticity associated with the porous inclusion grains. The eigenstress
distribution in this case reads:
{
aT (x)
Vxlp(6.90)
where the expressions for a, (6.38) and
considered at level I. apr-in
(E Qpor-inc
__por-incp
(6.28) were derived for the problem
Chm
denotes the tensor of Biot pore pressure coefficient of
the porous inclusion. Before proceeding further, the tools needed for homogenizing
poroelastic coefficients at level 1I of the mature organic-rich shale model are introduced. For the homogenized tensor of Biot pore pressure coefficients, the sub-problem
A defined in 6.4.2 is considered, again. This is the sub-problem associated with a mechanics response of a REV to a prescribed macroscopic strain field at the boundaries
(for
OT
= 0). Employing strain compatibility conditions and the second poroelastic
state equation, one can write:
n
5
n
r~7 a(,r) A r~r : A r
_ CO) A =KaeCA)
r
r
103
: E=ahom : E (6.91)
It is then immediately recognized from (6.91) that:
r
r
ihom
(6.92)
where r denotes solid phases. Similarly, in order to obtain Biot solid modulus, consider
the sub-problem B as defined in section 6.4.2. This sub-problem is associated with
the mechanics response of a REV to imposed eigenstresses in the pore space, while
the the macroscopic strain field at the boundaries remains zero. From (6.45):
(6.93)
((- Po)_OB Nhom
-
Thus;
1
1
Nhom
Nr
(6.94)
)s
Furthermore, since the inclusion grains are assumed to be isotropic, the tensor of Biot
pore pressure coefficients (6.38) reduces to:
a
1
=AO
(6.95)
thus, for the porous inclusion grain, it reads:
c
z in
where, utilizing (5.27), AVnr
is expressed as:
AO\'Pnc m
=
(
-
V
d)
V
K calcite
d
nm l
K!m)
inc
o
rm (I -
) 1 + ...
+dKquartz
I + f quarz (I
(
calcite ( 1
..
.. norrn
(6.96)
norm AVm 1n
apor-inc
(6.97)
K1c
il om
Kinc
In addition, the isotropic form of Biot solid modulus expressed in (6.46) reads:
-
"
fr(1
Z=
r
104
- Ar
)
1
Kr
(6.98)
I_
falcite
form
(I - Acalcite)
v)
Kcalcite
NPor-inc
fquartzl(ci
+
-
Thus, adapting (6.98) for the porous inclusion, as defined in our REV, leads to:
(6.99)
unorm
(6.99)
Kqu"rt
where in a self-consistent scheme, the volumetric component of strain localization
tensors associated with calcite and quartz phases read as follows:
AKcalcite
-
( Kcalcite
m
(1 +
1))
-
n'orm
d)(1
mnc
t
Khom
z(1
mfquar
=(1
Kquartz
(K
m
+
1)) 1
nq5rm(1
Kaceinc
(I +
( Kcalci)-1 + fquartz
"or"
Ke
in
inC
... fcalcite(
I+
orm
urt
11-1
Khom
Inc
Inc
Auar
(6.100)
Kquartz
-
-)
Kquartz )
6Z(
K "rr
.
d(
K ca cite )alit
+
~(1 ~+
norm
fcalcite
...
norm 1
(6.101)
1]1
Inc
with d being defined in (5.40). Now, accounting for imperfect interfaces (i.e. (5.76)),
at level II, one can write the drained effective stiffness tensor as:
Chom
II
.
.
__ fpor-incghom
por-inc
+
Aom
por-inc
(I -
: A hor
I
fpor-inc)chom
I
(6.102)
and;
Af
"' =
[R + PM : (Chom -
Ahom
por -inc
-=por-inc
(hom
M
(6.103)
Chom )]-1
_
hom
II
1
(6.104)
with poroelastic coefficients, associated with the upscaled porous fabric, expressed as:
inl
1
1
Nil
1
_ f
Ni
1
= C
-
fr-r-in
+
fPorinc
: [I
(6.105)
fPor-incAhom -c]
-
: [Chon
o-n
105
Chor"]-l : cI : (
1pri,
- A
o"
)
(6.106)
Utilizing (6.92), (6.96) and (6.105), the effective tensor of Biot pore pressure coefficients, at level II, reads:
(1
"hom
-
:A
fpor-inc)a
frncapr-inc
"+
:
(6-107)
Ahorn
Similarly, by combining (6.94),(6.99) and (6.106), the effective Biot solid modulus can
be expressed as:
1
Nh"m
where C
_"-n
1
fpor-inc
por-inc
_
(6.108)
NI,
+
.o
=
Npor-inc
reads:
3Khom
por-inc J+ 2Ghom
por-inc K
Chom
.
por-mc
(6.109)
with:
Khom
por-inc
Ghom
por-mec
f quartzquartzquartz
norm
v
+
fcalciteKcalcite Acalcite
norm
v
uartz
d
+
fcalciteGcalcite
quartzGquartz
norm
(6.110)
calcite
(6.111)
d
norm
and;
Aquartz
Kquartz
+
1))l[fjura(1
1) fnorm
-
VKhom
Khom
por-inc
por-inc
-
1))
1.
(6.112)
Kcalcite
c alcite
1)-
-
1))-l[furatz(1
norm
K hom
por -inc
))-
-
-
K quartz
K calcite1
+ fcalcite(l+ d(Khom
+ Onor m (1
-
Khom
por-inc
Kcalcite
Khom
por-inc
=(1
...
re(
+
...+
+
(6.113)
nI
+ Onormq i c . 0
por-mc
d
(;quartz
quartz
+
-
Ghom.
1por-inc
1)1)
[furatz(1
fnorm
+
Aquartz
(
(1 +Kquartz
(Ghom
por-inc
-Gcalcite
...
-
1))
1
(6.114)
11II
+(horn
Gpor-inc
106
norm
Acaci t e
=(1
+
Gquartz
G)(.
((homr
por inc
..
+
Gcalcite
coG
+ fnce(
+
calci tepor-inc
- 1)-
-ia
(Ghorn
(6.115)
+
1(
nor1
por-inc
The explicit expressions for computing the effective stiffness of porous inclusion grains
associated with the self-consistent morphology conclude the derivation of microporoelastic formulations for shale models. Next, we will consider the undrained behavior
of these porous composites.
107
Inclusion
Inclusion
(effective)
Kerogen
Level
II
Porosity
.s
Clay
O
Interface
Q
Perfect
Weakened
Interface
Level I
Level 0
Mature System
Particulate
Morphology
(Self-Consistent)
Figure 6-2 - Schematic for the multi-scale microporoelastic model for mature organicrich shales. Inclusion stiffness at level II is computed by homogenizing the dominant
non-clay minerals in a self-consistent scheme. Following the hypothesis testing approach regarding texture; a self-consistent approximation scheme is applied at each
scale for homogenization. Furthermore, interfaces are considered to be slightly weakened at level II following the the bread based analogy and the discussion accompanied
with the quenching problem presented earlier. In addition, a self-consistent morphology entails a self-consistent porosity distribution; hence porous inclusions.
108
6.6
Undrained Behavior
In this section, the tools for obtaining the undrained response of fully saturated porous
composites are introduced. For a fully saturated pore system, one can introduce the
lagrangian fluid mass content as 2.3:
m = (ppf(p)
(6.116)
The changes in fluid density as a function of pressure can be characterized by the
following linear state equation
[201:
Pf,o
=_1 + p
Kf
(6.117)
where Kf is the fluid bulk modulus and pf,o is the reference fluid density. Expansion
of (6.19) and (6.20) using (6.117) results in:
=Chom:
E - B(m -
o)
(6.118)
where:
( mo)= a : E + M
Pf,o
Chom
(6.119)
M
is the undrained effective stiffness, B is the
2 nd
order tensor of Skempton pore-
pressure build up coefficients and M is the overall Biot modulus. They can be expressed as:
Chom +
Chom
B
M
o
(Ma 0 a)
a
+
M
N
Kf
(6.120)
(6.121)
(6.122)
The tensor of Biot pore pressure coefficients, a, is essentially a correction factor for
the stress induced in the solid frame of a porous system due to variations in pore
pressure (see e.g.
1151,[26],[74]). On the other hand, Biot solid modulus (i.e. pore
compressibility), N, quantifies pore volume changes due to pore pressure variations,
109
under zero macroscopic strain boundary conditions. The tensor of Skempton pore
pressure build-up coefficients characterizes pore pressure variations due to stress application
[851,
essential for analyzing poroelastic effects such as pore pressure build
up and its dissipation due to some loading parameters, known as the Mandel-Cryer
effect (see e.g. Ref. [3]).
6.6.1
Immature Organic-Rich Shale
Thus, following (6.120), (6.121) and (6.122), one can obtain the undrained poroelastic
behavior at macroscopic scale, for the immature model, by computing:
Chom = C hor+
(Mal 9 all)
Bil= M,,Sho'" : all
1
MMll
where
Chorn
is defined in (6.53), Sh"m
(6.123)
(6.124)
1
N 1-+
Nil
(7!
Kf
(Ch"
)--1, al is defined in (6.57) and Ni, is
(6.125)
obtained from (6.63). One obtains:
Chom"= o" +(Mial 9a,,)
(6.126)
(6.127)
B, =
: a1
Ml
(6.128)
1 +1
N1 Kf
where C'1" is defined in (6.28), S "-
(Chor)
1,
a1 is defined in (6.38) and N1 can
be obtained from (6.47).
6.6.2
Mature Organic-Rich Shale
For the mature model, assuming the existence of a uniform pressure field in the two
porous systems introduced, the effective poroelastic behavior at the macroscopic scale
110
reads:
Cho"n
®& hor")
C"om + (Mho"'"
Mh"sh" : ahor"
Bil
1
1
M+
(6.130)
1
where Cho"n is defined in (6.102), Shom = (Ch" )-1
(6.129)
(6.131)
, Chom
is defined in (6.107) and
11 can be obtained from (6.108). Similarly, at level I, one can obtain the undrained
response of a porous solid, as follows:
Cfhom + (Mia1 0
Cho "
1
M1
where Ch,, is defined in (6.82), S
1
(6.132)
(6.133)
1,u
B,
&1)
MSho
: a,
1+
N1
Kf
(6.133)
(6.134)
= (Ci)-1, a, is defined in (6.85) and N1 can
be obtained from (6.87).
111
6.7
Chapter Summary
This chapter is dedicated to explicit derivation of multi-scale microporoelastic model
for immature and mature organic-rich shale systems. The main differences between
the model for immature and mature organic-rich shales are as follows:
1. Mori-Tanaka approximation scheme is used for immature organic-rich shale representing a "swiss-cheese" morphology while a self-consistent scheme is employed for
mature-organic rich shale representing a poly-crystalline morphology.
2. Weakened interfaces are introduced as an additional modeling tool for mature
organic-rich shale system following the reported observation of prevalent presence of
discontinuities in mature systems relative to immature ones and the presented bread
analogy.
3. The self-consistent morphology entails a self-consistent porosity distribution. Thus,
porosity at level I and level II of the multi-scale model for mature organic-rich shale
systems are equivalent.
112
Part IV
Results
113
THIS PAGE INTENTIONALLY LEFT BLANK
114
Chapter 7
Model Calibration & Validation
As it has been previously noted, direct measurement of clay minerals' elasticity remains a challenge that has led to a wide range of reported values in the literature
(see e.g.[78],[61], [28],135],[103],[471,[65],[112],[16]). Thus, one of the objectives of this
work is to pave the way for establishing a "unique" set of clay elasticity, which can
be utilized for geomechanics and geophysics-based applications. In this chapter, the
methodology used for downscaling macroscopic elasticity as well as the steps taken in
order to validate the values obtained grain scale values for clay elasticity are outlined.
7.1
7.1.1
Calibration
Procedure
Based on the hypothesis that the first-order contribution of kerogen maturity on the
effective anisotropic poroelasticity can be captured by considering a change in morphology, the objective is to calibrate Cla, using macroscopic elastic data belonging
to two different organic-rich shale formations (see Section 4.4), representing both
immature and mature kerogen systems, through the framework provided by our microporoelastic models (see Chapter 6).
This process was initiated with Woodford
shale data and the immature organic-rich shale model. The downscaling is achieved
by minimizing the frobenius norm between the experimentally measured values, i.e.
115
C""" (see Table 4.4) and the predicted undrained macroscopic elasticity,
Cdynayriic
i.e. ieC11horn
'Un (6.123), by changing Cclay(- C jkl)
.
In other words, the objective function
for minimization includes 5 degrees of freedom; summarized as:
min
Cclay
Chom
Woodford
(7.1)
Cdynamic
un
Woodford
F
where the frobenius norm is defined as:
||AlFl
=
Tr(A -A T )
(7.2)
-T
where A
is conjugate transpose of A and Tr stands for the trace. In addition, to
ensure positive definiteness of the clay's stiffness tensor, the objective function (7.1)
is subjected to the following constraints [61]:
CO + C 2 + CO 3 +
> 0
(7.3)
C11 + C12 + C033-
>0
(7.4)
CO1 - C1 2 > 0
(7.5)
(7.6)
C4 > 0
where;
=
-
(C I)2
(C 2 ) 2
8(C73 ) 2 + (CS 3 ) 2 + 2CO 1 C2 - 2CO1C93 - 2C22C0 3
(7.7)
This procedure is implemented using MATLAB's fmincon interior-point optimization
algorithm with GlobalSearch option [551 to ensure a global minimum when inverting
Woodford data.
Next, based on the hypothesis approach, the downscaled values
for clay from the immature model are employed as an initial guess for downscaling
Cl
s,-(see Table 4.8) through the model for mature systems.
In addition to
1(...)0 denotes "grain scale" elasticity associated with level 0 of the model, understood to be that
of clay mineral. Note that we do not make a distinction regarding clay mineral type at this point.
116
5 degrees of freedom, due to stiffness coefficients of the transversely isotropic clay,
interface normal (#) and tangential (a) compliances are introduced as additional
degrees of freedom in the minimization algorithm. In summary, the objective function
for downscaling Haynesville elastic data reads:
min
'cay
"
(HCh
Haynesville
C
iyn1e|F)
(7.8)
where Chorn is defined by (6.129).
7.1.2
Calibration Input
Implementation of (6.123) and (6.129) into the defined objective functions requires
as input the volume fractions of the phases present, their mechanical properties, in
addition to measured elasticity at macroscopic scale. Specifically, for the immature
organic-rich shale model, one needs
fi"c
(Table 7.1) and its (effective) elasticity, C Oc
(6.64), belonging to level II of the model. In addition, clay (jcay) and kerogen
( 1qker)
volume fractions are needed (Table 7.2), at level I. Similarly for the mature organicrich shale model, one needs
fPor-i"
(Table 7.3) and the (effective) elasticity associated
with it, Cpoinc, (6.109), at level II. For level I of the model, clay (Tclay) and kerogen
(,ker)
volume fractions, reported in Table 7.4, are the other input parameters. The
additional input parameter for the mature organic-rich shale model, in the comparison
to its immature counterpart, is the inclusion grain radius, a length scale introduced in
the model due to consideration of imperfect interfaces. In order to estimate this input
parameter, we used Scanning Electron Microscope (SEM) images of Haynesville shale 2
(see Figure 7-1) to estimate an average inclusion grain radius, a, of 2pm. In Chapter
8, the sensitivity of the multi-scale microporoelastic model for the mature organicrich shale with regards to the interface parameters, namely tangential (a) and normal
(P) compliances
as well as inclusion grain size are studied; addressing any concerns
regarding the sensitivity of our results to choice of the input parameters.
2
Courtesy of Amer Deirieh, PhD Candidate at the Department of Civil and Environmental
Engineering of Massachusetts Institute of Technology.
117
Table 7.1 - Calculated inclusion volume fractions of Woodford shale samples (level
II).
Sample
finc
Al
A2
A3
A4
0.35
0.33
0.33
0.36
A5
0.32
Table 7.2 - Calculated clay and kerogen volume fractions and porosity of Woodford
shale samples (level I).
Sample
rIc'ay
1ker
P
Al
A2
A3
A4
0.33
0.29
0.33
0.40
0.42
0.40
0.43
0.30
0.25
0.31
0.24
0.29
A5
0.37
0.32
0.31
Table 7.3 - Calculated volume fraction of porous inclusions of Haynesville shale sam-
ples (level II).
Sample
BI
B2
B3
B4
B6
B7
fPer-inc
0.66
0.57
0.86
0.75
0.52
0.57
Table 7.4 - Calculated volume fractions of clay, kerogen and porosity of Haynesville
shale samples (level I).
Sample
rIc'ay
,ker
P
1
B2
B3
0.79
0.78
0.72
0.14
0.15
0.23
0.07
0.074
0.05
B4
0.71
0.91
.06
B6
0.80
0.13
0.07
B7
0.78
0.14
0.08
118
(a)
(b)
Figure 7-1 - SEM images of a Haynesville shale samples. Based on these images,
a grain radius of 2 pm was chosen as the input for the imperfect interface model
associated with the mature organic-rich shale model used for downscaling macroscopic
Haynesville elasticity data. Also, the existence of pores in the inclusion on the bottom
right is a noteworthy feature, consistent with our self-consistent porosity distribution
assumption.
119
7.1.3
Calibration Results
The obtained
CcIay
value by downscaling Woodford data through the immature organic-
rich shale model; following the procedure outlined in Section 7.1 and using the input
parameters highlighted in Section 7.1.2, is reported in Table 7.5. These values were
used as initial guess for the downscaling of Haynesville macrosopic elasticity data
through the mature organic-rich shale model. The result for Cclay, from downscaling Haynesville macroscopic elasticity, along with interface compliances,
# and a, are
summarized in Table 7.5. Furthermore, clay elasticity was condensed into indentation
moduli using (4.19b) and (4.19a), denoted by m 3 and mi, "grain scale indentaion"
moduli in
x3
and x1 directions, respectively. For the clay elasticity obtained from
downscaling Woodford and Haynesville data, M 3 and mi are reported in Table 7.6.
The obtained values are in great agreement with the values reported in Ref. [2] obtained by back-analysis of nanoindentation data on a variety of organic-rich shale
samples. To evaluate the quality of inversion, the models' predictions, i.e. Cho, were
compared to the measured values, i.e. Canrd
the calibrated
Cclay,
daane. To be more specific,
obtained from downscaling through each model was used to com-
pute Chm, as defined for the immature organic-rich shale model using a Mori-Tanaka
homogenization and perfect interfaces, as well as the mature organic-rich shale model
with a self-consistent homogenization and imperfect interfaces. The results are displayed in Figure 7-2 and Figure 7-3. To further quantify the inversion quality, the
mean, 6, and the standard deviation, e, of relative error, ei; were computed, as
follows [38]:
ei =
(Xi - Yi)
j
n =
e. =
1
n -- I1
(7.9a)
ei
(7.9b)
(ei
(7.9c)
whereXX and V. represent model predictions and measurements, respectively, and
where n represents the number of samples in the data set. For both immature and
mature organic-rich shale models, 8 and e, are reported in Table 7.7.
120
The quality of inversion seems to be acceptable.
As expected, inversion result is
better for immature systems since the clay values were first obtained by downscaling
through the immature organic-rich shale model utilizing a global minimization algorithm. For the mature case, C13 seems to have the highest relative error. The possible
explanations for such behavior will be discussed in Chapter 8 where a comprehensive
sensitivity analyses is presented.
Table 7.5 - "Grain scale" elasticity and interface parameters obtained by downscaling
measured macroscopic elasticities of Woodford and Haynesville shale samples. In the
case of Haynesville, an inclusion grain radius of 2 pum was used for the mature organicrich shale model.
Woodford
106.5
47.8
63.3
74.8
10.9
-
Calibrated Parameters
CO,[GPa]
C1 2 [GPa]
C%[GPa]
Ci3 [GPa]
C 4t[GPa]
o[GPa]-1
#[GPa]- 1
Haynesville
105.6
47.8
64.0
73.7
8.2
1.7x 105.57 x 10-8
Table 7.6 - Computed "grain scale" indentation moduli (level 0) for clay values obtained by inversion of measured elasticity as reported in Table 7.5.
Formation
Woodford
Haynesville
mi [GPa]
61.8
56.8
121
m 3 [GPa]
37.6
32.2
Table 7.7 - Means and standard deviations of relative error between macroscopically
measured and predicted elasticity (level II).
Stiffness
Immature Model
e,
C-1
Mature Model
e,
15
-10
9
C12
-2
8
-1
15
C13
C11
C1
-2
14
51
40
-1
-1
15
20
12
-5
21
20
o0
30
C
dr
C1 2
OCdr
C33
0 Cdr
44
Cun
2
wo
12
0
un
13
25-
o
20C
V
110
(U
~15
sdr
ad
-
10-
C 33
C dr4
CU
6'105-
0
0
5
10
C.
15
20
2530
[GPa] Predicted
Figure 7-2 - Measured vs predicted macroscopic elasticity of Woodford shale; representative of an immature organic-rich shale system. "dr" and "un" refer to drained
and undrained responses.
122
o
0
S dr
12
o
70
dr
13
Cdr
33
60-
0
~
0
11
-~
L_
Cdr
44
Cun
50un
503
-
CI)
U
40 -3
12
Gun
13
dr
CU
044
-
30
20
10-
0
0
10
20
30
40
50
60
70
80
C.. [GPa] Predicted
Figure 7-3 - Measured vs predicted macroscopic elasticity of Haynesville shale; representative of a mature organic-rich shale system. "dr" and "un" refer to drained and
undrained responses.
123
7.2
Validation
7.2.1
Procedure
Several steps were taken to validate the obtained results from downscaling macroscopic elasticity.
First, the obtained Cclay values were compared to some available
values in the literature. Furthermore, this comparison was quantified by means of the
Riemannian distance between values reported in the literature and the values obtained
by downscaling. Then, the computed "grain-scale indentation moduli" reported in
Table 7.6 were compared to the results obtained from back-analysis of instrumented
nanoindentation data on samples from five different shale plays. Next, the obtained
values were upscaled to level I by computing
Chom
tensor. Then, they were compared,
in the condensed form of indentation moduli, M, and M 3 (see (4.19b) and (4.19a)),
to the available instrumented nanoindentaion data. Level I of our multi-scale microporomechanics based model does indeed represent the length scale relevant to
instrumented nanoindentation. This allows us to evaluate the validity of the results
using independently measured values. Finally, the predicted Thomsen parameters
[91] were compared with the measured values as a metric for assessing our models'
performance in predicting poroelastic aniosotropy.
7.2.2
Validation: Grain Scale Clay Properties (Level 0)
Following the outlined strategy for assessing the validity of the results, the values
obtained by inversion were compared to some values reported in the literature. The
literature data reported here (see Table 7.8) includes transversely isotropic clay elasticity values obtained by various techniques, at different length scales. Specifically, we
[79],
denoted as
and C 44 measured on mica muscovite by Tosaya
PRzhtv
[Y . Fro[ m experimet byritVe\/rni
194]
Tdind
CA,
from reported C11, C1 2 , C 3 3
,
used values gathered by Sayers in
and C 13 values of Alexandrov and
[1 f71
P-lakknlq sei
CmpTlsnc
Sayers [801 reports another set of stiffness values for clay-bearing inorganics, denoted
herein with CB. Chesnokov et al. 116] reported chlorite, kaolinite and illite-rich clay
124
stiffness values gathered from the work of Ref.
Cc, CD, and CE, respectively. Hantal et al.
[9]
[35]
[44];
and Ref.
denoted herein by
reported two different set of values
for illite by performing Molecular simulations employing ClayFF and ReaxFF force
fields; denoted herein by CF and CG, respectively. All these values are summarized
in Table 7.8. Furthermore, by means of Riemannian distance, a reliable metric which
is independent of coordinate system, invariant under inversion, and which preserves
the symmetry of the material being investigated [28], the comparison between these
results and some transversely isotropic clay elasticity values reported in the literature
and summarized it in Table 7.9 were quantified. The Riemannian distance between
matrix A 1 and A 2 is defined as:
m
dR(A1, A 2 )
where Ai denotes the
ith
=
|ln(A. 5 A-- 1A. 5HR
n
2
0.5
eigenvalue of A 1-'A 2 and m is number of eigenvalues. The
smaller the Riemannian distance, the closer the two elastic tensors.
Furthermore, the m, and the m 3 values, reported in Table 7.6, were compared to
the results obtained by Abedi et al. [21 from back analysis of instrumented nanoindentation data on few different organic-rich shale formations. The results obtained
from testing on samples from Haynesville, Marcellus, Fayetteville, Barnett and Antrim
shales suggest a "unique" pair of m 3 and m1 . Current results indicate an average m 3
value of 46 7 GPa and an average m, value of 63.5 7.3 GPa. The results, obtained
from an entirely independent method/set of data, reported in Table 7.6, are in great
agreement. More interestingly, the computed M3 for ClayFF and ReaxFF (both reported in Table 7.8) are 39.03
0.6 GPa and 39.85 1.1 GPa, respectively while the
computed m 3 associated with Woodford and Haynesville are 36.7 GPa and 32.2 GPa,
respectively.
125
Table 7.8 - Some reported anisotropic clay elasticity in the literature.
Literature
CA
Cc'ay [GPaj
ClaY [GPa]
Ccly [GPa]
Cday [GPa]
Cdy [GPaj
[781
178
42.4
14.5
54.9
12.2
CB [16]
CC [44]
CD 44]
CE [9
CF[35]
CG [35]
85.6
26.2
21.1
65.5
24.6
181.76
56.76
171.52
127.39
38.88
48.07
20.34
28.37
53.69
216 5
76 9
29 4
106.77
27.11
52.63
11.41
14.76
14.41
292.5 0.5
128.3+0.4
16.67 0.08
48.9+0.1
8.99 0.02
93 1
4.7 0.6
Table 7.9 - Riemannian distance between different elasticity tensors reported in Table 7.8 and values obtained by downscaling Woodford and Haynesville macroscopic
elasticity (see Table 7.5), as a metric to assess the similarities between reported and
obtained values in the Reimannian space.
CA
CB
CC
CD
CE
CF
CG
Woodford
Haynesville
-
0.7
1.8
0.4
1.5
0.9
1.0
0.8
2.5
1.5
2.8
2.1
1.7
2.3
2.1
1.7
CC
0.7
1.8
-
0.9
1.2
1.1
1.3
2.3
2.4
CD
CE
0.4
0.9
0.8
1.5
2.1
2.3
1.5
1.0
2.5
2.8
1.7
2.1
0.9
1.2
1.1
1.3
2.3
2.4
0.8
1.1
1.8
1.9
2.2
0.8
1.1
1.6
1.8
2.0
1.2
2.6
2.6
1.9
1.4
2.7
2.6
0.4
2.2
1.7
2.8
2.6
0.4
CF
CG
Woodford
Haynesville
-
1.6
2.0
1.4
1.7
126
-
1.2
2.7
2.8
-
CB
1.7
-
CA
7.2.3
Validation: Indentation Data (Level I)
For further analysis, Cclay, as reported in Table 7.5, and obtained by inversion of
macroscopic elastic data through the multi-scale microporoelastic models, is utilized
to compute Chm which in turn allows one to calculate indentation moduli, M 1 (4.19b)
and M 3 (4.19a), and enables one to compare the models' predictions for indentation
moduli (at level I) and laboratory measured instrumented nanoindentation data. To
be exact, the values reported in Table 7.5, obtained by downscaling Woodford macroscopic elastic data, are used as input for Cclay for a forward application of the both
mature and immature organic-rich shale model to compute C' m and subsequently M 1
and M 3 using (4.19b) and (4.19a), respectively. This would allow one to compare our
model predictions, with calibrated Cclay, against laboratory measured instrumented
nanoindentation. For this purpose, we utilized the validation data set presented before which included indentation data belonging to Woodford, Haynesville, Marcellus,
Antrim and Barnett. The results are presented in Figure 7-4 for M 1 and Figure 7-5
for M 3 comparison of measurements and predictions. Based on our hypothesis that
the first-order influence of maturation on poroelastic behavior of organic-rich shales
can be captured by considering a change in texture, the results presented in Figures
7-4 and 7-5 were computed assuming
|CkerH
jCceay
11 clayl
< 1. In fact, the process of organic
maturation presents a competition between an organic phase that is becoming more
stiff or as put by Bousige et al. [121, going through a ductile to brittle transition
while the volume fraction of the organic phase is decreasing as it gets decomposed
over geological time; producing oil and gas. For the case of Marcellus, while the
samples were identified as mature [2], the samples exhibit a high content of organics
relative to other samples used in this thesis. Thus, using the computations of [12], for
a kergoen density of 1.2 g/cc; an estimated bulk modulus, Kker, of 8 GPa and a shear
modulus, Gker, of 4 GPa; we computed M1 and M 3 for what is labeled as Marcellus
in Figures 7-4 and 7-5, while Marcellus* refers to results obtained with negligibale
kerogen elasticity.
127
-6
.)o
-I
0
-
C
0 0
')
0M
0
(9
IiOk-
[eBdD] V painsea~q
Figure 7-4 - Measured vs predicted indentation moduli for different shale formations
in x, direction. Marcellus* refers to computations considering negligible kerogen elasticity while Marcellus includes kerogen elasticity. See Section 7.2.3 for more details.
128
> L_
:3
LC)
-6a) E ~~
00C
L.
o.
i~ <
0
2
LO
0O
qz@0
C
a
0
LO)
CV)
CUO
C',
_0
00
LO
LC)
0
0
C0
CO
C0
00
C%4
[edO] !CVY painseen~
Figure 7-5 - Measured vs predicted indentation moduli for different shale formations
in X3 direction. Marcellus* refers to computations considering negligible kerogen elasticity while Marcellus includes kerogen elasticity. See Section 7.2.3 for more details.
129
7.2.4
Validation: Dynamic Properties (Level II)
Finally, Thomsen anisotropy parameters were used, as defined in (4.14a), (4.14b)
and (4.14c), to compare predictions, using Ch1
defined in (6.123) for the immature
organic-rich shale model, and (6.121) for its mature counterpart, against calculated
Thomsen parameters from reported elastic data. The result is reported in Table 7.10
for immature and Table 7.11 for mature organic-rich shale systems.
Table 7.10 - Measured vs predicted Thomsen parameters for Woodford shale samples.
Depth[ft]
Al
A2
A3
A4
A5
Table 7.11
ples.
-
Measured
Predicted
Measured
Predicted
Measured
Predicted
0.24
0.30
0.29
0.31
0.27
0.28
0.29
0.29
0.27
0.28
0.28
0.24
0.33
0.42
0.26
0.30
0.30
0.31
0.31
0.31
0.19
0.28
0.17
-0.02
0.27
0.17
0.18
0.17
0.16
0.16
Measured vs predicted Thomsen parameters for Haynesville shale sam-
Sample
BI
B2
B3
B4
B6
B7
6*
E
6*
C
Measured
Predicted
Measured
Predicted
Measured
Predicted
0.37
0.32
0.10
0.050
0.35
0.33
0.08
0.15
0.03
0.07
0.16
0.15
0.15
0.04
0.10
0.03
0.11
0.18
0.03
0.16
-0.13
-0.02
0.20
0.16
0.27
0.22
-0.09
0.07
0.25
0.02
0.19
0.16
0.30
0.23
0.15
0.16
130
7.2.5
Discussions
As one would expect, the Riemannian distance between grain scale clay elasticity
values obtaiend from downscaling Haynesville and Woodford measured macroscopic
elasticity is the shortest. It is interesting to note that values reported by Sayers [791
and those reported by Hantal et al.
[35], obtained by Molecular Dynamics, have
relatively short distance in the Riemannian space, highlighting their similarities in
elastic tensor structure. More interestingly, the next shortest distance between our
values and values reported in the literature is the illite-rich clay, denoted by
CE.
This
is significant result since both Woodford and Haynesville shales are illite-rich.
In terms of indentation moduli, the reported standard deviations, a measurement
directly linked to the realizability of indentation moduli in the framework of the grid
indentation technique and not the experimental error, bring models predictions into
an acceptable range.
Regarding anisotropy, the models' performance does an ac-
ceptable job in most cases, given that the anisotropy is solely attributed to intrinsic
clay anisotropy. However, improvements can be made by considering mechanisms of
higher order contributions to re-fine the presented analysis and to improve predictive
capabilities of the developed models.
The hypothesis of change in morphology of organic-rich shale as a geo-composite,
as maturity of kerogen changes proves to be consistent with the experimental observations by Prasad et al. [71] who suggested that the "distribution of kerogen and
grains undergo a major change as the maturity progresses". In the case of an immature system, our approach for choosing a Mori-Tanaka scheme is not only consistent
with the physical intuition, that clay assumes the role of the load bearing phase when
kerogen is immature and perceived as a pliable, amorphous, organic polymer. But
it is further justified when one looks into polymer/clay nanocomposites literature.
In the polymer/clay nanocomposites literature, Mori-Tanaka or some variation of it,
is almost universally used for micromechanical modeling due to excellent modeling
131
agreement with experimental results (see [84],[117],[361,and
[53]).
This ductile to
brittle transition has also been reported by Ref. [12]. Indeed, what our hypothesis
seems to capture in addition to a change in texture, is competition between kerogen
stiffening due to maturation and decomposition, i.e. lower volume fraction and hence
a lower contribution to the effective poroelasticity, of kerogen as it produces oil and
gas. This can be clearly seen in the case of Marcellus.
7.3
Chapter Summary
In this chapter, the model for mature and immature organic-rich shale was implemented. The model was calibrated by means of UPV data of Haynesville shale for
mature and Woodford shale for immature systems. Then, the calibrated values were
validated at three different length scales. First, the calibrated grain scale values were
compared to reported values in the literature. Next, employing the obtained grain
scale values predicted nanoindentation moduli were compared to measured indentation moduli on five different formations. Finally, a general trend is shown to be
captured by our model with regards to variation of Thomsen parameters with organic
maturity.
132
Chapter 8
Sensitivity Analysis
To assess the sensitivity of the developed model to variations in different input parameters, the result of a series of sensitivity analyses is presented in this chapter, by
considering various case scenarios. In the first case, the quality of inversion is studied,
given uncertainty in C" "' estimation at macroscale, applied to the matrix-inclusion
model representative of immature shale systems. For the second case, the sensitivity
of both mature and immature models, including Thomsen anisotropy parameters,
minimum horizontal in-situ stress and Vp3 /Vs 3 ratio are studied, in a forward application, to uncertainty in different input parameters by means of Spearman's Partial
Rank Correlation Coefficient (PRCC). In addition, the contribution of variance of
each input parameter, including pore fluid compressibility, on the overall normalized
variance associated with poroelastic coefficients, at each length scale, are investigated.
The combination of microporomechanical modeling and sensitivity analyses presented
here can be of great value in terms of practical application and identification of critical
subsurface parameters that need to be characterized.
Bandyopadhyay
[7]
has com-
piled and plotted the probability density function (PDF) of the reported macroscopic
elasticity in the literature on organic-rich shales.
His results suggest a multi-mode
distribution of elastic coefficients, possibly due to the inability to group the compiled
data based on kerogen maturity and TOC (due to incomplete published data sets).
In what follows, a normal distribution for the stiffness coefficients, porosity, TOC,
quartz elasticity and fluid bulk modulus is assumed, while both normal and uniform
133
distribution types are considered for the imperfect interface model parameters. After
studying the convergence, for each case studied, 1,000 Monte-Carlo simulations were
performed.
8.1
Inversion Quality Given Uncertainty in C1'un
Laboratory characterization of organic-rich shale (or any transversely isotropic media)
by ultra-sonic pulse velocity (UPV) experiments under confining pressure, for simulating in-situ conditions, imposes experimental constraints (not theoretical ones).
This is due to difficulties for off-axis travel time measurement that is needed, along
with the distance the waves travel, to characterize
CII.
An approximation can be
made by making a "bench off-axis measurement" on a block, outside of the loading
frame/cell. Though, in this case, the signal-to-noise ratio of traveling waves is low
due to the lack of confinement and "imperfect" contact of piezoelectric transducers
to the sample; thus making the process of picking wave arrival times challenging.
In addition, in such a set-up, there could be difficulties measuring the distance between two off-axis mounted piezoelectric transducers directly which itself becomes a
source of error in velocity calculations. Not to mention that such test would represent
surface conditions rather than subsurface conditions. Of course, one can avoid the
trouble of making off-axis measurements and to resort to empirical relationships for
estimating C1ju" based on a combination of static and dynamic data (e.g. Modified
ANNIE method 189]). In order to understand the effect of uncertainty in C11" on
downscaled particle properties at level 0, we introduce C1jj"" stochastically and introduce other parameters deterministically. The uncertainty is introduced by defining a
normal distribution for C11,un of each Woodford sample at macroscopic scale, with a
Ponstawnt Pooeiient of vaitoV:
V
o-
-
p134
0.15
(8.1)
where
- is the standard deviation and p denotes the mean, needed to characterize a
normal distribution. Inputs are thus of the form (see Table 8.1):
C11""--,
N(p, o-)
(8.2)
where N denotes a normal distribution and i represent different Woodford samples.
Then, by means of 1,000 Monte-Carlo simulations, applied to immature organicrich shale model, the histograms of the inverted results were obtained; as shown in
Figure 8-1. Next, normal distributions (based on histogram shape) were fitted to the
results and thus the mean and variance of the fitted distribution was obtained. The
results of this analysis are summarized in Table 8.2, and show, as expected, that the
highest impact of the uncertainty in C11,"" assessment is in the values of Co 2 and Co3
whereas C' 1 and Co are the least affected. On the other hand, the uncertainty in
the calibration w.r.t.
C13"" is stable in the sense that the maximum coefficient of
variation of the output is almost identical to the input. Thus, the result that we have
obtained for grain scale clay elasticity by downscaling macroscopic elasticity is not
very sensitive to the uncertainty associated with Ci"" estimation.
135
p
-
Table 8.1 - Defined mean and standard deviation for describing a normal distribution
for each Ci"" of Woodford shale samples. These values are used as inputs in the
downscaling procedure to assess uncertainty in grain scale clay elasticity (level 0).
Sample
Al
A2
8.8
7.7
1.32
1.15
A3
7.8
1.17
A4
A5
8.3
7.9
1.24
1.18
Table 8.2 - Means and standard deviations obtained for stiffness coefficients of clay
(level 0), after downscaling macrosopic elasticity through the microporoelastic model
for the immature organic-rich shale system. Note that the only input parameter
defined stochastically was CII"" for each Woodford shale sample. Also C is equivalent
to C"a in our defined multi-scale thought-model.
Elastic Coefficients
C 1 [GPa]
C2[GPa]
C1 3 [GPa]
Ci3 [GPa]
C 4 [GPa]
136
P
U
V
106.9
48.6
63.6
75.2
10.8
8.4
7.2
10.5
8.8
0.2
0.08
0.15
0.16
0.12
0.02
250,
250
200
200
150-
150
100-
L 100
50
50
14
0L
0L
4
14
6
5
8
7
II,un
Al
CIIun -
13
13
(a)
(b)
10
9
11
12
A2
-
300
250
250
200-
200C
a)
3 150-
= 1500r
a)
a)
LL
UL
100-
100
50
50
01
4
5
6
7
8
9
Ci ,un - A3
13
10
2
12
11
6
4
10
8
II,un
-
12
14
A4
13
(d)
(c)
300r
250[-
200
C
(D
: 150
U
1
100-
50
0L.'
3
4
5
6
8
7
CIIun - A5
13
9
10
11
12
(e)
Figure 8-1 - Normal distributions prescribed to the macroscopic C'I"" of each Woodford sample. These serve as inputs for assessing the influence of uncertainty in estimation of Ci"" on the "grain scale" values through the model for immaure organic-rich
shales.
137
200
200
o150
0150
CY
100
U
2 100
50
50
C
C
0
L
90
100
110
120
130
0
30
140
35
40
45
50
60
65
70
75
C12
(a)
C
a)
55
0
01i
(b)
250
250
2002
200
150
1501
CD
100
1001
50
M50
40
50
60
70
80
90
0
100
60
70
80
C0
C0
90
100
110
C3 3
C13
(d)
(C)
300
250-
200Cr
LiL
100-
50U(e
S510
1005
11
11.5
C4 4
(e)
Figure 8-2 - Histograms of output for each Woodford sample, i.e. stiffness coefficients
of clay at level 0 ("grain scale"), obtained by introducing uncertainty in macroscopic
CII"" and the inversion of the macroscopic elasticity through the model for immature
organic-rich shales.
138
--
Noral Fit -- -Simulation Result
Normal Fit - - - Simulation Result
-
0.
0.0 6
0.0 5
0.04-
0.0 4
0.03
(-3
oo- 0.0 3
0. 02
0.02
0 01-
0. 01
0
0
140
130
120
110
100
90
45
40
35
60
65
70
75
C0
12
(b)
(a)
-
55
50
C 011
Normal Fit - - - Simulation Result
Normal Fit - - - Simulation Result
-
0.045
0.04
0.04
0.035
0.035
0.03
0.03
0.025
c 0.025
0.02
0.02
0.015
0.015
0.01
0.005
'30
0.005
40
50
60
80
70
90
100
00
70
60
80
90
100
110
C 033
C0
13
(d)
(c)
-
Normal Fit - - - Simulation Result
2
1
0
0 .5
8
10
10.2
10.4
10.6
10.8
11
11.2
11.4
11.6
0
C44
(e)
Figure 8-3 - Fitted probability density function (PDF) for each Woodford smaple
and the "experimental" PDF obtained by Monte-Carlo simulations.
139
Normal Fit - - - Simulation Result
Normal Fit - - - Simulation Result
. .
-
-
1
0.8
0 .6
0.6
.4
0 .4
0.2-
0 .2
-
0 .8
0
U0
0
90
100
110
120
130
140
35
40
50
45
C0
(a)
-
55
60
65
70
75
110
120
CC012
(b)
Normal Fit - - - Simulation Result
--
--
1
Normal Fit - - - Simulation Result
1
0. 8-
0.8
0. 6-
0.6
0.
IL
0
U_
4-
0. 2-
0.4
0.2
_.,.,ovoooooooooo,,O,0000
30
40
50
60
70
80
90
100
40
50
60
70
80
C0
C0
90
100
C33
(d)
(c)
-
Normal Fit - - - Simulation Result
0.8
0.6
0
LL
0.4
0.2
0
.5
10
11
1M5
11.5
12
C4 4
(e)
Figure 8-4 - Fitted cumulative density functions (CDF) for each Woodford sample
and "experimental" CDF obtained from Monte-Carlo simulations.
140
8.2
Dependence of Output Variance to Different Input Parameters
In reality, most of the model input parameters (e.g. mineralogy mass percents, porosity, TOC, elasticity) are subjected to some degree of uncertainty, whether they are
measured in the field or characterized in the lab. In this section we intend to do a
simple study to assess the sensitivity of the models' output variance, namely indentation moduli at level I and macroscopic elasticity at level II, Thomsen anisotropy
parameters, Vp 3 /Vs
3
, minimum in-situ horizontal stress,
0h,
as well as poroelastic
coefficients to the variance associated with input parameters.
The solution for minimum horizontal in-situ stress in a transversely isotropic formation reads
[901:
h = C 13
C33
(03
-
CC2
a3p) + G19p + (CH
C2
~
I)E2 C
C3 3
(C12
- )E 1
(8.3)
C 33
where cr3 is overburden stress, PP is pore pressure, cs and a are Biot pore pressure coefficient in the
x3
and x, directions, respectively. E2 and El represent horizontal strains
due to tectonic activities in the
x2
and x, directions, respectively. In-situ stresses are
inputs into many geomechanics-based models, critical for wellbore (in)stability analyses, reservoir compaction/ subsidence problems as well as completions design. Ignoring
the elastic aniostropy when estimating in-situ stresses may lead to non-negligible errors. For example, in the case of Barnett shale, Waters et al. [1131 show that ignoring
anisotropy of organic-rich shale can lead to non-negligible error in estimating in-situ
stresses while Sone [88] attributes fluctuations in in-situ stresses to viscoelastic effects, without accounting for anisotropy. In the forthcoming analyses, the following
assumptions are made: tectonically non-active zones, i.e. E3 =
El
= 0, a hydrostatic
pore pressure gradient, i.e. 0.433 psi/ft, a lithostatic overburden gradient of 1 psi/ft
and a true vertical depth (TVD) of 10,000 ft.
141
Vp3 /Vs 3 along with Acoustic Impedance (Al) are important qualitative geophysical
metrics employed to identify compliant zones in subsurface. According to Ref. [1081,
all organic-rich shales with a TOC more than 3% are characterized by low Vp3 /Vs 3
ratio, in the range of 1.6-1.7. This ratio can be defined as:
Vp3 _
C33Pb
Vs3
C4/pb
(8.4)
In a forward application of both mature (self-consistent plus weakened interfaces) and
immature (Mori-Tanaka, perfect interfaces) models, grain (i.e. clay) elasticity, porosity at level II, TOC and quartz elastic properties are introduced stochastically, while
the rest of modeling parameters remained deterministic. For the specific case of mature organic-rich shale, the imperfect interface parameters were introduced stochastically. In the case of Spearman's Partial Rank Correlation Coefficient (PRCC) analysis
applied to poroelastic coefficients, a fluid bulk modulus was considered as an additional stochastically defined input. After establishing the inputs, by means of PRCC,
the normalized contribution of uncertain variables to the output variance were quantified. PRCC captures the degree of association between rankings, including both
linear and non-linear correlations, rather than actual variant values. The degree of
association between rankings is still considered a measure of association between samples as well as an estimate for the association of X and Y in a continuous bivariate
population. To understand what we mean by ranking, let us consider the following
continuous set of bivariate random variables 1311:
(XI1, Y2), (X2, Y2), ... ,- (X11,
Y11)
(8.5)
the correlation coefficient, i, for n pairs reads:
fr(Xi
- X)(Yi
E - 1 (Y,
- X)2 S(Xi
)
Y)2]0
(8.6)
where X and Y represent the arithmetic average of Xi and Yj, respectively. By sorting
X and Y observations from smallest to largest using integers 1,2,...,n, one can rank
142
each observation, based on its magnitude, relative to other samples present in a data
set. Assuming that marginal distribution of X and Y are continuous, then unique sets
of ranking must theoretically exist 131]. The result is called Spearman's coefficient of
Rank Correlation.
8.2.1
Immature Organic-Rich Shale Model
The PRCC analyses technique was applied to the immature model (see Section 6.4),
with the distribution for the input parameters are summarized in Table 8.3, and
were employed to generate 1,000 Monte Carlo simulations. The contribution of the
variations of each input parameters onto the variance of the output parameter; is
displayed in Figure 8-6 namely on Thomsen anisotropy parameters , macroscopic
elasticity, Vp 3 /Vs 3 , and
0h
is displayed in Figure 8-5. For the case of indentation
.
moduli, the result is displayed in Figure
The sensitivity analysis relevant for the set of input variables considered shows that
while TOC and porosity make up nearly 50% of the variance of the elastic stiffness
values, they barely affect the Thomsen parameters, Vp 3 /Vs
3
and Ch which are dom-
inated by the variance of the different solid (i.e. level 0) input parameters. On the
other hand, uncertainty in Co 3 and Co 4 seem to contribute the least to the macroscopic
elasticity. However, Co3 combined with Co3 make up nearly 50% of the normalized
variance of Vp 3 /Vs 3 and c7h. The contribution of C1 2 to the normalized variance of the
parameters studied, including Thomsen anisotropy parameters, Vp 3 /Vs 3 ratio, as well
as
Th,
is negligible. C
2
and C
4
have relatively low contribution to the normalized
variance of indentation moduli, while contribution due to C01 , Co 3 , Co 3 ,
#
and TOC
is nearly equally distributed. Variances associated with the elasticity of quartz has
almost no contribution to the normalized variance of the studied parameters.
143
Table 8.3 - Stochastically defined input paramters for the immature microporoelastic
model. The result used for PRCC analysis.
Input parameters
CO,[GPa]
C12[GPa]
C1 3 [GPa]
C3 3 [GPa]
CS3 [GPa]
0
TOC
KQuar tz[GPa]
GQuart z[GPa]
Distribution types
P
a-
V
Normal
Normal
Normal
Normal
Normal
Normal
Normal
Normal
Normal
106.5
47.8
63.3
74.8
10.9
0.13
15.7
37.9
44.3
10.65
4.78
6.33
7.48
1.09
0.03
3.14
3.79
4.43
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
144
04
0
Y
6C)
6oo
rrsl ipaig hestvt fteoupt dfndo h b
Fiur 8-7
scisa t difeen iputpaamtes (efne i th lged)forth initur ogaic
richshae mdel
145
I
I
0'-
00 CO
I
I
I
I
C0
0
I
04
0-
0-
- CO U) Mt CY) CNi C-
Figure 8-6 - PRCC result displaying the sensitivity of the indentation moduli (defined
on the abscissa) to different input parameters (defined in the legend) for the immature
organic-rich shale model.
146
8.2.2
Mature Organic-Rich Shale Model
Following the same procedure, a sensitivity analysis was performed for the mature
organic-rich shale model. In addition to the type of input parameters introduced in
Table 8.4; interface parameters, namely normal (/) and tangential (aZ) compliances of
the interface, and inclusion grain radius, were introduced stochastically. To further
understand the effect of distribution type of 3 and oz and inclusion grain radius, a,
on the overall variance, two studies were undertaken: one with the assumption of
uniform distributions (see Table 8.4) of these parameters and another where it is
assumed that these parameters are normally distributed (see Table 8.5). A uniform
distribution can be defined as:
X ~ U(A, B)
(8.7)
where A is the lower bound (LB) and B, the upper bound (UB). The results of
1,000 Monte Carlo simulations, using inputs reported in Table 8.4, with uniform
distribution of interface parameters, are displayed in Figure 8-7. Similarly, the results
of 1,000 Monte Carlo simulations for input parameters displayed in Table 8.5, with
normal distribution of interface parameters, are summarized in Figure 8-8. The result
for indentation moduli, independent of interface parameters, are displayed in Figure
8-9.
A close look at the results suggests that relative to porosity and TOC, the
contribution of interface modeling parameters to the overall variance is minimal,
for both uniform and normal distribution types. Also, the results indicate that the
distribution type of imperfect interface model parameters does not matter, in the
framework that has been defined for performing sensitivity analyses. In addition,
relative to an immature system, porosity,
#, and
TOC have significant contributions
to the normalized variance of macroscopic elasticity. Meanwhile, Co has minimal
impact on the Thomsen parameters, indentation moduli and Vp 3 /Vs 3 ratio.
For
indentation moduli, the contributions, other than that of Co4 , are almost equally
distributed. C4 4 contribution to the normalized variance of indentation moduli is the
least among other input parameters. Variances associated with the elasticity of quartz
has almost no contribution to the normalized variance of the studied parameters.
147
Table 8.4 - Stochastically defined input parameters for the mature microporoelastic
model needed for PRCC analyses. We assumed a uniform distribution for interface
parameters.
Input parameters
C, [GPa]
C 2 [GPa]
C 3 [GPa]
C 3 [GPa]
4~4 [GPa]
TOC
KQuartz [GPa]
GQuartz [GPa]
a[GPa]- 1
O[GPa]-'
a [m]
Distribution types
Normal
Normal
Normal
Normal
Normal
Normal
Normal
Normal
Normal
Uniform
Uniform
Uniform
a-
V
106.5
10.65
47.8
63.3
4.78
6.33
74.8
7.48
10.9
0.13
0.13
37.9
44.3
1.09
0.03
0.03
3.79
4.43
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
LB
UB
0
0
10-7
10-7
2x10-5
2x10-7
Table 8.5 - Stochastically defined input parameters for the mature microporoelastic
model needed for PRCC analyses. We assumed a normal distribution for interface
parameters.
Input parameters
Cii [GPa]
C%2[GPa]
C1 [GPa]
C 3 [GPa]
C 4 [GPa]
TOC
K3"3
[GPa]
G4"
[GPa]
a[GPa]-1
O[GPa]- 1
a1 [m
Distribution types
Normal
Normal
Normal
Normal
Normal
Normal
Normal
Normal
Normal
Normal
Normal
NmaL
I
106.5
10.65
47.8
4.78
63.3
10.9
0.13
0.13
6.33
7.48
1.09
0.03
0.03
37.9
3.79
74.8
44.3
4.43
1.7x 10-7
5.57 x 10-8
3.42 x 10-8
1.114x 10-8
2 x 1 n-6
I
II/\-U%
148
a-
4x 1
-
7
V
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.2
0.2
U.2
M
a
b|
-
oc
0
~
Figure 8-7 - PRCC result displaying the sensitivity of the outputs (defined on the
abscissa) to different input parameters (defined in legend) for the mature organic-rich
shale model assuming uniform distributions for model parameters associated with
imperfcet interface model.
149
N
CO
M
~0
i
0-
0
CD
C
I-
CD
LO
It
0
Figure 8-8 - PRCC result displaying the sensitivity of the outputs (defined on the
abscissa) to different input parameters (defined in the legend) for the mature organicrich shale model assuming normal distributions for input parameters associated with
imperfect interface model.
150
I
I
I
I
I
M
C14
04,
0
0
0--
000
r
CD
0
-
I
U')
-
I
--
~
I
I
C)
Figure 8-9 - PRCC result displaying the sensitivity of the indentation moduli, at level
I, to different input parameters (defined in the legend) for the mature organic-rich
model. Note interface parameters do not interfere at level I.
151
8.2.3
Poroelastic Coefficients' Sensitivity Analyses
Poroelastic coefficients are crucial in petroleum geomechanics for calculating effective
stresses, assessing wellbore (in)stability (e.g. see
[41), reservoir compaction /subsidence
analysis, as well as pore pressure build up and diffusion due to a point force (see [3]).
To better understand how clay and quartz elasticity, pore fluid compressibility, porosity and TOC variances contribute to the normalized variance of different poroelastic
coefficients, PRCC analysis for Biot tensor of pore pressure coefficient, a, Biot solid
modulus, N, Biot overall modulus, M and the tensor of the Skempton pore pressure
build-up, B, for both mature and immature models are performed. The inputs used
for the study are reported in Table 8.6. As expected, the results, for both the immature model (Figures 8-10, 8-11, 8-12 and 8-13) and the mature model (Figures 8-14,
8-15, 8-16 and 8-17) suggest that porosity and TOC variances have significant contributions to all poroelastic coefficients at both levels of the models. Also, the effect
of C2 and CO on the overall variance of the studied parameters, relative to other
inputs, seem to be minimal for both models. CS 3 and CO, become of significance for
the tensor of Skempton coefficients in x 3 and x, directions. Meanwhile, the prescribed
variance of fluid bulk modulus has minimal effect on the Skempton coefficients and
Biot overall modulus, at both levels, in x 3 and x1 directions. This is consistent with
the modeling results presented in Section 7.1.3, where negligible poroelastic effects
are predicted. This is partly due to low porosity in organic-rich shales as well as the
presence of a highly compliant organic phases, a consequence of the assumption of
negligible organic to inorganic stiffness ratio.
8.3
Chapter Summary
This chapter includes a thorough sensitivity analyses employing Monte-Carlo simulation technique Spearman's Partial Rank Correlation Coefficient (PRCC). First, the
sensitivity of calibration of the immature model to a given uncertainty in macroscopic
CIIun
is assessed. Next, sensitivity of various outputs such as as macroscopic elas-
ticity, Thomsen parameters, Vp/Vs, and minimum in-situ horizontal stress as well
152
Table 8.6 - Stochastically defined input parameters for the PRCC analysis of poroelastic coefficients for mature and immature organic-rich shale models.
Input parameters
CO1 [GPa]
C12 [GPa]
C 3 [GPa]
C3 [GPa]
CO44[GPa]
TOC
KQuartz [GPa]
GQuartz [GPa)
Kf [GPa]
Distribution types
PU
0-
Normal
Normal
Normal
Normal
Normal
Normal
Normal
Normal
Normal
Normal
106.5
10.65
47.8
74.8
4.78
6.33
7.48
10.9
0.13
0.13
37.9
44.3
1.09
0.03
0.03
3.79
4.43
V
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
2
0.8
0.25
63.3
as poroelastic coefficient needed to characterize the poroelastic behavior as captured
by mature and immature multi-scale organic-rich shale model, to variations in model
input parameters are assessed.
153
C0 EC2 MC3 MC3 MCI
$O TOC
0.90.80.70.6
0.50.4
0.30.2
0.101
0I
ce
3 ,j
a3,I
c o<EETOCEK
Ec
c
2
0
E
1
0.90.80.70.60.50.40.3
0.20.10
B1ij
BV,
*CD C 122ECo3ECo3
13
11
1
C444
4
0.90.80.7
0.6
0.5
0.4
0.3
0.2
0.1
SN
11
&1,IIa,1
$l
TOC
KQuartz
Guartz
fC0
HCC
3
0
CA
$
TOC
IKQuartz IGQuartz
1
0.90.80.70.6C."
0.50.40.30.2
0.1
0
B1 11
B3JI
K
c
C
C
C0
0.90.80.70.60.50.40.30.20.10
N
ali
C2
<I TOC
*CO
C
C
C
4
$
TOC
KI
1
0.90.80.70.60.50.40.30.20.10
Mi
B1i
B3,I
1
1
flCD
13
CEC33
C 44
044
$ TOC
KQuartz IGQuarz
1
0.90.80.70.60.50.4
0.30.20.10
N11
ei,II
aZ3,II
1
CD
11
C0CMC
12
1
C
TOC KQuartz GQuartzK
3
44E
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Mll
Biji
B11l
Part V
Conclusions
162
Chapter 9
Discussion of Results & Future
Perspectives
An original approach for micromechanical modeling of organic/ inorganic mixtures has
been presented, accounting for the maturity of organics and their effect on overall elasticity by attributing its contribution to morphology. Self-consistent and Mori-Tanaka
homogenization schemes have shown to be able to theoretically capture mechanisms
consistent with physical intuition, experimental observations and mechanical testings. Also, it has been shown that interfaces play a role on the effective elasticity of
mature organic-rich shales. This highlights the importance of integrating geological
knowledge into mechanics and sub-continuum mechanics based modeling efforts to
capture physical processes. Also, it has been elaborated that linking microtexture to
elasticity, combined with statistical analyses, can provide powerful engineering tools
for identification of important parameters which may be utilized as a means for generating property maps and the uncertainty associated with the quantities of interest,
for more efficient and effective exploration and exploitation of highly heterogeneous
organic shale systems.
163
9.1
Summary of Main Findings
The consistency between the result obtained in this work for grain scale clay elasticity
by downscaling macroscopic elasticity through our model for mature and immature
organic-rich shales, developed based on our hypothesis testing approach, and the reported clay anisotropic elasticity as discussed in Section 7.2.3, prediction of measured
nanoindentation moduli as discussed in Section 7.2.3 and overall agreement with observed variations of Thomsen parameters with maturity as presented in Section 7.2.4
hint at the existence of an invariant set of clay elasticity. In other words, the consistency between values obtained from two completely different experimental techniques
(i.e. UPV and instrumented nanoindentation), performed at different scales, with no
reason to agree a priori; and the reported values in the literature for clay elasticity
affirm our hypothesis of existence of a unique set of clay properties in organic-rich
shales with similar depositional environment. Furthermore, the contribution of burial
and diagenetic processes on the evolution of organic matter and on overall elasticity
of organic-rich shales is captured by introducing slightly weakened interfaces between
organics and inorganics.
The quality of inversion, given the values reported in Table 8.2, seems to be acceptable considering the uncertainty associated with C1", as defined in Table 8.1,
in the framework used for sensitivity analysis. Also, this variation partly explains
why predicted indentation moduli at level I do not perfectly agree with the measured
indentation values, as shown in Figures 7-4 and 7-5. In dealing with shales, one needs
to be aware of the extreme heterogeneity of organic-rich shales (at mm scale) and
their sensitivity to the environment around it when comparing modeling results to
measured values and to consider the possibility of local perturbations in grain orientations. That is, it is imperative in such studies for pictures to be taken from samples
in
every step of the preparation before, during and after mechanical testing.
By
now, it is well known that organic-rich shale cores need be preserved in inert environments, under controlled conditions, to preserve their mechanical integrity. Otherwise
164
comparison between theoretical predictions and laboratory measurements will not be
meaningful.
By means of the Riemannian distance, summarized in Table 7.9, the clay elasticity obtained by downscaling macroscopic elasticity of Woodford and Haynesville shales were
evaluated against the values reported in the literature (see Table 7.8). As one would
expect, the Riemannian distance between Haynesville and Woodford is the smallest, while the largest distance, d, is obtained between Clay FF
[351
and Haynesville.
Overall, Chesknokov et al. values 1161 are "closest" to Woodford and Haynesville
formations. It is interesting to note the similarity of values reported by Sayers [78]
and those obtained by means of molecular dynamics simulations by Hantal et al.[35].
The PRCC analyses for the mature model suggest that porosity, TOC, C' 1 , Co 3 , and
CO are the most influential factors on the output. The analyses for the mature
model show an insignificant contribution of interface parameters on the variance of
the elasticity and Thomsen parameters. This is of critical importance as it attests
that weakened interfaces capture a physical mechanism rather than serving as a tool
for fitting predictions to measured data. For indentation moduli, the uncertainty in
porosity and kerogen make up almost 50% of the contribution to the output variance.
In addition, advances in experimental techniques (both laboratory and simulations)
and the understanding of kerogen structure and organic /inorganic interfaces, may
enable one to obtain interface compliance independently. In regards to poroelastic
coefficients, porosity and TOC make up the majority of normalized variance, while
the contribution of Co2 and Co are negligible.
The combination of PRCC and the microporoelastic models, which link microstructure of organic-rich shales to their poroelastic behavior, can be a a powerful engineering tool. For example, it was shown that TOC, C1 3 and C03 make up to 75%
of the normalized variance associated with minimum horizontal stress,
accurate estimation of
Uh
9h.
Thus, for
in practice, one needs to best characterize these parame165
ters for a confident estimation. Given the limitations and economic constraints for
subsurface characterization, these types of tools can be used to increase a "return on
investment" by helping geoscientist focus on characterizing critical parameters. This
sort of approach can be extended to wellbore (in)stability analyses by linking radial
and hoop stresses to the microtexture, or hydraulic fracturing by linking microtexture
to a global energy release rate. Subsequently, property maps can be created for identification of most optimum zones for drilling and completions; giving rise to a new
set of advanced engineering tools that are needed for exploitation of highly intricate
organic-rich shales.
The presented models do an acceptable job in predicting overall anisotropy, given
that intrinsic clay anisotropy is the only source of anisotropy that propagates through
various length scales. It is interesting to note that the microporoelastic models for
organic-rich shales, representing asymptotic degrees of maturity, capture a decreasing
trend in anisotropy as one goes from a highly immature system to a highly mature
one. This is consistent with Ref. [1021, who reports that anisotropy increases from
immature to early mature organic-rich shales; but it starts to decrease as maturity
increases beyond a vitrinite reflectance, %Ro, of 0.65. This observation is captured
by the models since Haynesville, with a %RO ranging from 1-2
[601,
is predicted to be
less anisotropic than Woodford (see Table 7.10 and 7.11).
Although the model for mature organic-rich shales may be over predicting CI"",
one must note that in this model, all discontinuities present in a mature system is
captured by modeling them as imperfect interfaces. This is due to observations that
discontinuities exist along the interfaces and the rational that microcracks should have
no effect under high in-situ stresses in subsurface or high confining pressure in the
lab. However, microcracks may have a minimal effect of second-order nature. Indeed,
this would highly impact the prediction of C1"" in a model. The question is not a
theoretical one; but it is one of experimental nature: how would one properly allocate
the compliances induced by discontinuities in mature organic-rich shale systems to
166
imperfect interfaces and to microcracks?
9.2
Limitations & Future Perspectives
The modeling tools for extremely heterogeneous and anisotropic porous composites
are limited in the real of continuum mechanics. Also, the knowledge of the statistical
distribution of various phases in such complex composite has only recently become
accessible due to FIB-SEM and micro-CT imaging techniques.
This would open
new and very exciting doors for a modeler to gain a better understanding of the
natural truth and transform that perception into models that can better capture and
represent the underlying physics of the problem. Also, our understanding of kerogen
and its structural, physical and chemical evolution due to a variety of processes that
are amalgamated into the term "maturation" is very limited. Indeed, our improved
understanding the effect of maturation on kerogen physical, chemical and structural
properties can lead to more accurate models with improved predictive capabilities.
167
THIS PAGE INTENTIONALLY LEFT BLANK
168
Appendix A
Nomenclatures
Symbol
x =
Description
(x1, x 2 , x 3 )
Position vector in a Cartesian coordinate system
c = (C1,w 2 , w3 )
Unit vector in a Spherical coordinate system
n
Unit normal vector
a1 , a 2 , a 3
Semi-principal axes of an ellipsoid
f
Vector of body force per unit mass
Au
Displacement discontinuity vector
Q
Domain of a REV
Qr
rth subdomain within a REV
QI
= Qinc
Inclusion subdomain within a REV
QS
Solid subdomain within a REV
QP
Pore subdomain within a REV
F
Discontinuity surface
V
Divergence operator
...]- Inversion
operator
T
[]T
Transpose operator
||AllF
Frobenius norm of A
dR(A1A 2 )
Riemannian distance between A 1 and A 2
do
Length scale below which tools of continuum mechanics is not
applicable
169
d
Characteristic length scale of heterogeneities
Characteristic length scale of REV
12
Characteristic length scale of structural systems
A
Length scale associated with load fluctuations
L
Grid size associated with grid indentation technique
Td
Prescribed traction
Macroscopic stress field
a-
Microscopic stress field
Stress field associated with an inclusion
Strain field associated with an inclusion
Prescribed displacement vector
d
E
Macroscopic strain field
E
Microscopic strain field
K
Bulk modulus
Kr
Bulk modulus of the rth phase
bulk modulus of porous inclusion
KHomogenized
K
i
Homogenized bulk modulus of inclusion (at level II)
G
Shear modulus
Gr
Shear modulus of the rth phase
G
inc
(at level II)
Homogenized shear modulus of porous inclusion (at level II)
G ac"
Homogenized shear modulus of inclusion (at level II)
E1
Young's modulus in the isotropic plane
E3
Young's modulus in the transverse plane
V12
Poisson's ratio in the isotropic plane
V13
Poisson's ratio in the transverse plane
Chom
4 th
order effective (homogenized) stiffness tensor
Shom
4 th
order effective (homogenized) compliance tensor
cay
4 th
order clay stiffness tensor
Cker
4 th
order kerogen stiffness tensor
c"om
4 th
order homogenized stiffness tensor at level I
170
chom
11
4 th
order homogenized stiffness tensor at level II
Cinc
4 th
order stiffness tensor of inclusion
Cun
4 th
order Undrained stiffness tensor
CS = CM
4 th
order background (matrix) stiffness tensor
Cquasi-static
4 th
order quasi-statically measured stiffness tensor
Cdynamic
4 th
order dynamically measured stiffness tensor
W
2 nd
order interface compliance tensor
Interface tangential compliance
OZ
Interface normal compliance
a
Inclusion grain radius
M,
Thomsen parameters
Modified ANNIE calibration parameter
Indentation moduli of transversely isotropic elastic medium in x,
M3
"grain scale" indentation moduli of transversely isotropic elastic
M1
medium in x 3
M3
"grain scale" indentation moduli of transversely isotropic elastic
medium in x,
m3
Indentation moduli of transversely isotropic elastic medium in
h
Indentation depth
P
Indentation load
Ac
Contact area between indenting tip and the indented material
U7rr
Radial stress
O~h
Minimum horizontal stress
A
4 th
order strain localization tensor
B
4 th
order stress localization tensor
Ar
Volumetric component of strain concentration tensor of rth phase
Ar
Deviatoric component of strain concentration tensor of rth phase
I6ij =1
4 th
order identity tensor
2 nd
order identity tensor
Volumetric component of I
171
x3
K
fr
Deviatoric component of fI
fiflc
Inclusion volume fraction
fpor-i"c
Porous inclusion volume fraction
qr
Solid volume fraction of the rth phase (at level I)
1clay
Clay volume fraction (at level I)
qker
Kerogen volume fraction (at level I)
Solid volume fraction of the rth phase (at level II)
Porosity (at level II)
0finc
5ps
Inclusion grain porosity at level II
Porosity associated with porous solid at level II
Porosity (at level I)
f'
Chrisotffel's matrix
G
2 nd
order tensor of Green's function
sEsh
4 th
order Eshelby tensor
4 th
order Modified Eshelby tensor
MEsh
P
4th order Hill concentration tensor
PM
4 th
order modified Hill concentration tensor
Wave frequency
k
Wave number
Pg
Grain density
Pb
Bulk density
pf
Fluid density
Kf
Fluid bulk modulus
ni
Mineral mass percent
N1
Biot solid modulus at level I
N1,
Biot solid modulus at level II
Npor-inc
Biot solid modulus associated with porous inclusion
N1 "om
Homogenized Biot solid modulus
M1
Overall Biot (solid+fluid) modulus at level I
Ml
Overall Biot (solid+fluid) modulus at level II
172
Mho "Homogenized
overall Biot (solid+fluid) modulus at level II
B
2nd order tensor of Skempton Coefficients
B1 ,1
Skempton coefficient in plane of isotropy at level I
B 1,11
Skempton coefficient in plane of isotropy at level II
B 3,1
Skempton coefficient in transverse plane at level I
B 3,11
Skempton coefficient in transverse plane at level II
al
2 nd
order tensor of Biot coefficients at level I
all
2 nd
order tensor of Biot coefficients at level II
apor-inc
2 nd
order tensor of Biot coefficients associated with porous inclu-
sion
ahom
2 nd
O1,1
Biot coefficient in plane of isotropy at level I
ai,
Biot coefficient in plane of isotropy at level II
LU,1
Biot coefficient in transverse plane at level I
a3 ,1
Biot coefficient in transverse plane at level II
ej
Relative error
a
Mean of relative errors
11
Mean
0-
Standard deviation
aOTh
Coefficient of linear thermal expansion
e,
Standard deviation of relative errors
V
Coefficient of variation
order tensor of homogenized Biot coefficients at level II
173
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174
Bibliography
[1]
Technically recoverable shale oil and shale gas resources: An assessment of 137
shale formations in 41 countries outside the united states. Technical report,
U.S. Energy Information Administration.
Nanochemomechanical signature of organic-rich shales: A coupled indentation-edx
analysis-in preparation.
12] S. Abedi, M. Slim, R. Hofmann, T. Bryndzia, and F.-J. Ulm.
[31 Y. Abousleiman, A.H.-D. Cheng, L. Cui, E. Detournay, and J.-C. Rogiers.
Mandel's problem revisited. Geotechnique, 46(2):187-195, 1997.
[4] Y. Abousleiman and L. Cui. Poroelastic solutions in transversely isotropic media
for wellbore and cylnider. Int.J.Solids Structures, 35:4905-4929, 1998.
[5] R. Ahmadov. Micro-textural, elastic and transport properties of source rocks.
PhD thesis, Stanford University.
161 K.S. Alexandrov and T.V. Ryzhova. Elastic properties of rock forming minerals
ii. Geophysics, 9:1165-1168, 1961.
[7] K. Bandyopadhyay, R. Sain, E. Liu, C. Harris, A. Martinez, M. Payne, and
Y. Zhu. Rock property inversion in organic-rich shale: Uncertatinties, ambiguities, and pitfalls. In Annual Meeting SEG Extended Abstract.
181
D.H. Bangham and R.E. Franklin. Thermal expansion of coals and carbonised
coals. Proceedings of Royal Society A, 27:147-160, 1946.
[9] 1.0. Bayuk, M. Ammerman, and E.M. Chesnokov. Upscaling of elastic properties of anisotropic sedimentary rocks. J.Geophys.Int., 172:842-860, 2008.
[101 Y. Beneviste. A new approach to teh application of mori-tanaka's theory in
composite materials. Mechanics of Materials, 6:147-157, 1987.
111] J.G. Berryman and S. Nakagawa. Inverse problem in anisotropic poroelasticity:
Drained constants from undrained ultrasound measurements.
J.Acoust.Soc.Am., 2, February 2010.
175
[12] C. Bousige, C. Ghimbeu, V. Cathie, A.E. Pomerantz, A. Suleimnova,
G. Vaughan, G. Garbarino, M. Feygenson, C. Wildgruber, F.-J. Ulm, J.-M.
Pellenq, and B. Coasane. Realistic molecular model of mature and immature
kerogens in organic-rich shales-in preparation.
[13] M.E. Broz, R.F. Cook, and D.L. Whitney. Microhardness,toughness,and modulus of mohs scale minerals. American Mineralogist, 91:135-142, 2006.
114] B. Budiansky. On the elastic moduli fo some heterogeneous materials. Journal
of the Mechanics and Physics of Solids, 13(4):223-227, 1965.
Material coefficients of anisotropic poroelasticity.
J.Rock.Mech.Min.Sci., 34(2):199-205, 1997.
[151 A.H.-D. Cheng.
Int.
[161 E.M. Chesnokov, D.K. Tiwary, 1.0. Bayuk, M.A. Sparkman, and R.L. Brown.
Mathematical modelling of anisotropy of illite-rich shale. Geophysical Journal
International, 178:1625-1648, 2009.
[171 C.R. Clarkson, N. Solano, R.M. Bustin, A.M.M. Bustin, G.R.L. Chalmers,
L. He, and Y.B. Melnichenko. Pore structure characteriztion of north american
shale gas reservoirs using usans/sans,gas adsorption, and mercury intrusion.
Fuel, 103:606-616, 2013.
[181 G. Constantinides, K.S. Ravi Chandra, F.-J. Ulm, and K.J. Van Vliet. Grid
indentation analysis of composite microstructure and mechanics: Principles and
validation. Materials Science and Engineering, pages 189-202, 2006.
[191 G. Constantinides, F.-J. Ulm, and K. Van Vliet. On the use of nanoindentation
for cementitious materials. Materials and Structures, 36:191-196, 2003.
[20] 0. Coussy. Poromechanics. Wiley,Chichester.
[211 0. Coussy. Revisiting the constitutive equations of unsaturated porous solids
using a lagrangian saturation concept. International Journal for Numerical and
Analytical Methods in Geomechanics, 31(15):1675-1694, 2007.
[221 S.C. Cowin. A recasting of anisotropic poroelasticty in matrices of tensor components. Transport in Porous Media, 50:35-56, 2003.
123] S.C. Cowin and M.M. Mehrabadi. The structure of the linear anisotropic elastic
symmetries. Journal of Mechanics and Physics of Solids, 40(7):1459 1471, 1992.
124] M.E. Curtis, C.H. Sondergeld, and C.S. Rai. Relationship between organic shale
microstructure and hydrocarbon generation.
Conference, 2013. SPE 164540.
SPE Unconventional Resources
[25] A. Deirieh. Statistical nano-chemo-mechanical assessment of shale by wave
dispersive spectroscopy and nanoindentation. Master's thesis, Massachusetts
Institute of Technology.
176
1261 L. Dormieux, D. Kondo, and F.-J. Ulm. Microporomechanics. Wiley,Chichester.
[27] A. Draege, M. Jakobsen, and T.A. Johansen. Rock physics modelling of shale
diagenesis. Petroleum Geoscience, 12(4):49-57, 2006.
[28] D. Ebrahimi, R.J.-M. Pellenq, and A.J. Whittle. Nanoscale elastic properties
of montmorillonite upon water adsorption. Langmuir, 41:16855-16863, 2012.
[291 J.D. Eshelby. The determination of the elastic field of an ellipsoidal inclusion,
and related problems. Proceedings of Royal Society A, 241:376-396, 1957.
130] A.G. Every and A.K. McCurdy. Second and Higher Order Elastic Constants
of Crystals, in Numerical Data and Functional Relationships in Science and
Technology. Group III of Landolt Bornstein.
131] J.D. Gibbons and S. Chakraborti. Nanparametric Statistical Inference. Marcel
Dekker, Inc.
[32]
E. Gonzales, C. Torres-Verdin, and R. Hofmann. Introduction to this special
section: Rock physics. The Leading Edge, 33(3):221-360, 2014.
[331 E.F. Gonzalez, T. Mukerji, and G. Mavko. Seismic inversion combining rock
physics and multiple-point geostatistics. Geophysics, 73(1):R11-R21, 2007.
[34]
B. Gurevich. 12.s593,advances in seismic rock physics. e. Class Notes, Spring
2014, MIT.
135] G. Hantal, L. Brochard, H. Laubie, D. Ebrahimi, J.-M. Pellenq, F.-J. Ulm,
and B. Cosane. Atomic-scale modeling of elastic and failure properties of clays.
Molecular Physics:An International Journal at the Interface Between Chemistry
and Physics, 112:1294-1305, 2014.
[361 K. Hbaieb, Q.X. Wang, Y.H.J. Chia, and B. Cotterell. Modeling stiffness of
polymer/clay nanocomposites. Polymer, 48:901-909, 2007.
[37]
C. Hellmich, J.-F. Barthelemy, and L. Dormieux. Mineral-collagen interactions in elasticity of bone ultrastructure-a continuum micromechanics approach.
European Journal of Mechanics-A /Solids, 23(5):783-810, 2005.
[38]
C. Hellmich and F.-J. Ulm. Drained and undrained poroelastic properties of healthy and pathological bone: a poro-micromechanical investigation.
Transport in Porous Media, 58(3):243-268, February 2005.
[391
A.V. Hershey. The elasticity of an isotropic aggregate of anisotropic cubic
crystals. Journal of Applied Mechanics, 21:226-240, 1954.
[401 R. Hill. A self-consistent mechanics of composite materials.
Mechanics and Physics of Solids, 13(4):213-222, 1965.
[411 R. Hofmann. Private communication.
177
Journal of
1421 B.E. Hornby, L.M. Schwartz, and J.A. Hudson. Anisotropic effective-medium
modeling of the elastic properties of shales. Geophysics, 59:1570-1583, 1994.
[431
T. Houtari and I. Kukkonen. Thermal expansion of rocks:litearture survey and
estimation of thermal expansion coefficient for olkiluoto mica gneiss. Technical
report, Geological Survey of Finland.
[441
K.W. Katahara. Clay minerals elastic properties. Proceedings of the 66th SEG
Annual Meeting, Expanded Technical Program Abstracts, pages 1691-1694,
1996.
[45] Y. Khadeeva and L. Vernik. Rock-physics model for unconventional shales. The
Leading Edge, 33(3):221-360, 2014.
[461
J. Khazanehdari, N. Dutta, and J. Herwanger. The next generation of rock
physcs models. E&P Magazine, pages 409-416, September 2010.
[47]
M. Kopycinska-Muller, M. Prasad, U. Rabe, and W. Arnold. Elastic properties
of clay minerals determined by atomic force acoustic microscopy technique.
Acoustical Imaging, 28:409-416, 2007.
[48] E. Kroner.
Berechnung derelastischen konstanten des vielkristalls aus den
konstanten des einkristalls. Zeitschrift fur physik A Hadrons and Nuclei,
151(4):504-518, 1958.
[49] P.L. Larsson, A.E. Giannakopoulos, E. SoDerlund, D.J. Rowcliffe, and R. Vetergaard. Analysis of berkovich indentation. Internatinoal Journal of Solids and
Structures, pages 221-248, 1996.
[50] N. Laws and G.J. Dvorak. The effect of fiber breaks and aligned penny-shaped
cracks on the stiffness and energy release rates in unidirectional composites.
International Journal of Solids Structures, 23(9):1269-1283, 1987.
[51] R.G. Loucks, R.M. Reed, S.C. Ruppel, and U. Hammes. Spectrum of pore types
and networks in mudrocks and a descriptive classification for matrix-related
mudrock pores. AAPG Bulletin, 96(6):1071-1098, June 2012.
1521
MorpholR.G. Loucks, R.M. Reed, S.C. Ruppel, and D.M. Jarvie.
ogy,genesis,and distribution of nanometer-scale pores in siliceous mudstones of
the missippian barnett shale. Journal of Sedimentary Research, 79:848-861,
2009.
1531
J.J. Luo and I.M. Daniel. Characterizing and modeling of mechanical behavior
of polymer/clay nanocomposites. Composites Science and Technology, 63:16071616, 2003.
[541 A. Malthe-Srenssen, B. Jamtveit, and P. Meakin. Fracture patterns generated
by diffiusion controlled volume changing reactions. Physical Review Letters, 96,
June 2006.
178
[551 MATLAB. version R2013b. The MathWorks Inc., Natick, Massachusetts, 2013.
[56] G. Mavko, T. Mukerji, and J. Dvorkin. Rock Physics Handbook: Tools for
Seismic Analysis in Porous media. Cambridge University Press.
[57]
Thermal expansion of clay minerals.
H.A. McKinstry.
Mineralogist, 50, 1965.
The American
[58] 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.
[59]
T. Mura. Micromechanics and Defects in Solids. Martinus Nijhoff.
[60] J.A. Nunn. Burial and thermal history of the haynesville shale: Implications
for overpressure, gas generation, and natural hydrofracture. GCAGS Journal,
1:81-96, February 2012.
[611 J.A. Ortega. Microporomechanical Modeling of Shale.
sachusetts Institute of Technology.
PhD thesis, Mas-
162] J.A. Ortega, F.-J. Ulm, and Y. Abousleiman. The effect of nanogranular nature
of shale on their poroelastic behavior. Acta Geotechnica, 2:155-182, 2007.
[631 J.A. Ortega, F.-J. Ulm, and Y. Abousleiman. The effect of particle shape
and grain-scale properties of shale: A micromechanics approach. International
Journal for Numerical an Analytical Methods in Geomechanics, 34:1124-1156,
October 2009.
[641 A. Padin, A.N. Tutuncu, and S. Sonnenberg.
On the mechanisms of shale
microfracture propagation. SPE Hydraulic Fracturing Technology Conference,
February 2014. SPE 168624.
[651 A. Pal-Bathija, M. Prasad, H. Liang, M. Upmanyu, N. Lu, and M. Batzle.
Elastic properties of clay mienrals. Acoustical Imaging, 28:409-416, 2007.
166] Y.-C. Pan and T.-W. Chou. Point force solution for an infinite transversely
isotropic solid. Transactions of ASME, pages 608-612, December 1976.
[67] H. Panahi, P. Meakin, F. Renard, M. Kobchenko, J. Scheibert, A. Mazzini,
B. Jamtveit, A. Malthe-Sorenssen, and D.K. Dysthe. A 4d synchrotron xray-tomography study of the formation of hydrocarbon-migration pathways in
heated organic-rich shale. SPE Journal, 18:366-377, 2012.
[68] Q.R. Passey, K.M. Bohacs, W.L. Esch, R. Klimentidis, and S. Sinha. From oilprone source rock to gas producing shale reservoir-geologic and petrophysical
characterization of unconventional shale gas reservoirs. In International Oil and
Gas Conference and Exhibition in China.
[69] R.C. Payton. Elastic wave propagation in transversely isotropic media. Martinus Nijhoff Publishers.
179
[701
M. Prasad, K.C. Mba, and T.E. McEvoy. Maturity and impedance analysis of
organic-rich shales. SPE Reservoir Evaluation and Engineering, pages 533-543,
2011. SPE 123531.
[71] M. Prasad, T. Mukerji, M. Reinstaedtler, R.B. GmbH, and W. Arnold. Acoustic signatures, impedance microstructure, textural scales, and anisotropy of
kerogen-rich shale. In Annual Technical Conference and Exhibition SPE.
[721 J. Qu. The effect of slightly weakened interfaces on the overall elastic properties
of composite materials. Mechanics of Materials, 14(4):269-281, 1993.
[73] J. Qu. Eshelby tensor for an elastic inclusion with slightly weakened interface.
Journal of Applied Mechanics, 60(4):1048-1050, 1993.
[741
J.R. Rice and M.P. Cleary. Some basic stress diffusion solutions for fluidsaturated elastic porous media with compressible constituents. Reviews of
Geophysics and Space Physics, 14(2), 1976.
[751 T.L. Robl, D.N. Taulbee, L.S. Barron, and W.C. Jones. Petrologic chemistry
of a devonian type ii kerogen. Energy and Fuels 1, pages 507-513, 1987.
[76] A.M. Romero and R.P. Philp.
Organic geochemistry of the woodford
shale,southwestern oklahoma:how variable can shale be?
AAPG Bulletin,
96(3):493-517, 2012.
[77]
R. Sain, T. Mukerji, and G. Mavko. How computational rock-physcs tools can
be used to simulate geologic processes, understand pore-scale heterogeneity, and
refine theoretical models. The Leading Edge, 33(3):221-360, 2014.
[78] C.M. Sayers. The effect of anisotropy on the young's moduli and poisson's ratios
of shale. Geophysical Prospecting, 61:416-426, 2013.
[791 C.M. Sayers. The effect of kerogen on the the elastic anisotropy of organic-rich
shales. Geophysics, 78(2):D65-D74, March-April 2013.
[80] C.M. Sayers. The effect of kerogen on the avo response of organic-rich shales.
The Leading Edge, December 2013, VOLUME =.
[811 M. Schoenberg, F. Mui, and C.M. Sayers. Introducing annie: A simple threeparameter anisotropic velocity model for shales. Journal of Seismic Exploration,
5:35-49, 1996.
[82] J.S. Seewald. Organic-inorganic interactions in petroluem-producing sedimentary basins. Nature, 426, 2003.
[83] R.C. Selley. Elements of Petroleum Geology. Academic Press.
[841 N. Sheng, M.C. Boyce, D.M. Parks, G.C. Rutledge, J.I. Abes, and R.E. Cohen.
Multiscale micromechanical modeling of polymer/clay nanocomposites and the
effective clay particle. Polymer, 45:487-506, 2004.
180
[851 A.W. Skempton. The pore pressure coefficients a and b. Geotechnique, 4:143152, 1954.
[86] R.M. Slatt and Y. Abousleiman. Reservoir characterization i,ii. 2012 Academic
Year, Univeristy of Oklahoma.
[87] R.M. Slatt and Y. Abousleiman. Merging sequence stratigraphy and geomechanics for unconventional gas shales. The Leading Edge, (3), 2011.
[88] H. Sone. Mechanical Properties of Shale Gas Reservoir Rocks and Its Relation
to the In-situ Stress Variation Observed in Shale Gas Reservoirs. PhD thesis,
Stanford University.
Estimating horizontal stress from
[891 R. Suarez-Rivera and T.R. Bratton.
three-dimensional anisotropy.
Geophysical Prospecting, 2009.
US
Patent:2009/0210160A1.
[901 M.J. Thiercelin and R.A. Plumb. Core-based prediction of lithologic stress
contrasts in east texas formations. SPE Formation Evaluation, December 1994.
[91] L. Thomsen. Weak elastic anisotropy. Geophysics, 51(10):195-196, 1986.
[921 B.P. Tissot. The application of the results of organic chemical studies in oil and
gas exploration. Developments in Petroleum Geology, 1:53-62, 1977.
[93]
B.P. Tissot and D.H. Welte. Petroleum formation and Occurrence. SpringerVerlag.
[94]
C.A. Tosaya. Acoustical properties of clay-bearing rocks. PhD thesis, Stanford
University.
[951
M.H. Tran. Geomechanics field and laboratory characterization of woodford
shale. Master's thesis, The University of Oklahoma.
[96] F.-J. Ulm and Y. Abousleiman.
The nanogranular nature of shale.
Acta
Geotechnica, 1(265):77-88, 2006.
[97] F.-J. Ulm, G. Constantinides, A. Delafargue, Y. Abousleiman, R. Ewy, L. Duranti, and D.K. McCarty. Material invariant poromechanics properties of shales.
Proceedings of the 3rd Biot Conference on Poromechanics, 37(265):43-58, 2004.
[98] F.-J. Ulm, G. Constantinides, and F.H. Heukamp. Is concrete a poromechanics
material? -a multiscale investigation of poroelastic properties. Materials and
Structures, 37(265):43-58, 2004.
[99]
F.-J. Ulm and 0. Coussy. 1.570,mechanics and durability of solids ii. Materials
and Structures, 2013. Class Notes,Micromechanics and Durability of SolidsMIT.
181
11001
M. Vandenbroucke. Kerogen:from types to models of chemical structure. Oil
and Gas Science and Technology, 58(2):243-269, 2003.
[101] M. Vandenbroucke and C. Largeau. Kerogen origin,evolution and structure.
Organic Chemistry, 38:719-833, 2007.
1102] T. Vanorio, T. Mukerji, and G. Mavko. Elastic anisotropy, maturity, and maceral microstructure in organic-rich shales. In Annual Meeting SEG Extended
Abstract.
[103] T. Vanorio, M. Prasad, and A. Nur. Elastic properties of dry clay mineral
aggregates, suspensions and sandstones. Geophysics J. Int., 155:319-326, June
2003.
[1041 L. Vernik. Microcrack-induced versus intrinsic elastic anisotropy in mature hesoruce shales. Geophysics, 58(11):1703-1706, 1993.
[105] L. Vernik and M. Kachanov. Modeling elastic properties of siliciclastic rocks.
Geophysics, 75(6):E171-E182, 2010.
[1061 L. Vernik and M. Kachanov. On some controversial issues in rock physics. The
Leading Edge, pages 636-642, June 2012.
[107] L. Vernik and C. Landis. Elastic anisotropy of source rocks: Implications for
hydrocarbon generation and primary migration. AAPG Bulletin, 80(4):531544, 1996.
1108] L. Vernik and J. Milovac. Rock physics of organic shales. The Leading Edge,
pages 318-323, 2011.
[109] L. Vernik and A. Nur. Ultrasonic velocity and anisotropy of hydrocarbon source
rocks. Geophysics, 57(5):727-735, 1997.
[110] L. Vernik and L. Xingzhou. Velocity anisotropy in shales: A petrophysical
study. Geophysics, 62(2):521-532, 1997.
[1111] P.F. Wang and J.F.W. Gale. Screening criteria for shale-gas systems. Gulf
Coast Association of Geological Soceities Transactions, 59:779-793, 2009.
1112] Z. Wang, H. Wang, and M.E. Cates. Effective elastic properties of solid clays.
Geophysics, 66(2), March 2001.
[1131 G.A. Waters, R.E. Lewis, and D.C. Bentley. The effect of mechanical properties anisotropy in the generation of hydraulic fractures in organic shales. SPE
146776, 2011.
[114] M.L. Whiaker, W.L. Liping, and B. Li. Acoustic velocities and elastic properties
of pyrite (fes 2 ) to 9.6 gpa. Journal of Earth Science, 21(5):792-800, October
2010.
182
[115] P.J. Withers and H.K.D.H. Bhadeshia. Residual stress part 1-measurement
techniques. Material Science and Technology, 17, 2001.
[1161 B.I. Yakobson. Morphology and rate of fracture in chemical decomposition of
solids. Physical Review Letters, 67(12), Septermber 1991.
11171 L. Yaning, A.M. Waas, and E.M. Arruda. The effect of the interphase and strain
gradient on the elasticity of layer by layer (lbl) polymer/clay nanocomposites.
International Journal of Solids and Structures, 48:1044-1053, 2011.
[118] A. Zaoui. Continuum micromechanics:
Mechanics, 128:808-816, 2007.
A survey.
Journal of Engineering
[1191 L. Zhang and E.J. Leboeuf. A molecular dynamics study of natural organic
matter:1.lignin,kerogen and soot. Organic Chemistry, 40:1132-1142, 2007.
11201 L. Zhang, E.J. Leboeuf, and B. Xing. Thermal analytical investigation of
biopolymers and hiumic-and carbonaceous-based soil and sediment organic matter. Environmental Science and Technology, 41(14):4888-4894, 2007.
183