Master's Thesis-Petroleum Engineering

UNIVERSITY OF OKLAHOMA
GRADUATE COLLEGE
STUDY ON CHEMICAL ACTIVITY OF PIERRE SHALES AND ITS EFFECT ON
NEAR WELLBORE PORE PRESSURE DISTRIBUTION
A THESIS
SUBMITTED TO THE GRADUATE FACULTY
in partial fulfillment of the requirements for the
Degree of
MASTER OF SCIENCE
By
SABARISHA SUBRAMANIYAN
Norman, Oklahoma
2014

STUDY ON CHEMICAL ACTIVITY OF PIERRE SHALES AND ITS EFFECT ON
NEAR WELLBORE PORE PRESSURE DISTRIBUTION
A THESIS APPROVED FOR THE
MEWBOURNE SCHOOL OF PETROLEUM AND GEOLOGICAL ENGINEERING
BY
______________________________
Dr. Ahmad Jamili, Chair
______________________________
Dr. Deepak Devegowda
______________________________
Dr. Benjamin (Bor-Jier) Shiau

Dedicated to my parents and my brother

iv
Acknowledgements
I would like to thank god for making all my dreams come true. I would like to thank my
brother Suzanth for his valuable guidance in choosing my majors, his friend Sathish
Kulathu for inspiring me to become a Petroleum Engineer and also my parents for their
constant support and trust they had in me to send me to a different country for pursuing
my master’s degree.
I would like to express my sincere gratitude to my advisor Dr. Ahmad Jamili for giving
me this opportunity to do my research and for the trust he had on me though I came
from a different technical background. He has always encouraged me whenever I am
distressed about grades and has been a great source of inspiration. I would like to
appreciate him for the guidance and valuable suggestions he has given me throughout
my research. I would like to thank Dr. Deepak Devegowda for making me understand
the rudiments of Reservoir Engineering and also for his constant support throughout my
studies at OU. I would like to thank Dr. Ben Shiau wholeheartedly for being so kind and
supportive to join the committee and also for taking pains to go through my research at
the eleventh hour.
I would like to express my sincere appreciation to Aman Srivastava and K.V Srinivasan
for assisting me whenever I had the troubles of understanding the subject.
And now last but not the least I would like to say a big thank you to all my friends
Sumeer, Mounraj Aditya Srinivasan, Purnima, Dhivya, Reshma, Krithi and Pavithra for
tolerating me whenever I cribbed and for the ensuing valuable advice. A special thanks
to my best friends Arun and Kavitha for the awesome home food and for making me to
forget that I have a family in India.

v
Table of Contents
Acknowledgements ......................................................................................................... iv
List of Tables................................................................................................................... ix
List of Figures.................................................................................................................. xi
Abstract........................................................................................................................... xv
Chapter 1: Introduction..................................................................................................... 1
1.1: Significance of Shale in Oil and Gas Industry .................................................... 1
1.2: Geological significance and chemical composition of Shale.............................. 3
1.3: Wellbore Instability in shale................................................................................ 4
1.3.1: Principle of borehole stability .................................................................... 5
1.3.2: Causes for wellbore instability................................................................... 6
1.3.3: Classification of chemical instability mechanisms .................................... 6
1.4: Pierre Shale formation......................................................................................... 7
1.4.1: Stratigraphic positioning of Pierre Shale.................................................... 7
1.4.2: Mineralogical composition of Pierre Shale................................................ 8
1.5: Essentiality of the research.................................................................................. 8
1.6: Précis of the thesis............................................................................................. 10
Chapter 2: Literature Review ......................................................................................... 12
2.1: Clay colloidal chemistry.................................................................................... 12
2.2: Cation Exchange Capacity (CEC) of Shale....................................................... 14
2.3: Membrane Efficiency of Shales ........................................................................ 15
2.4: Water activity of shale....................................................................................... 18
2.5: Transport Mechanisms in Shale ........................................................................ 21

vi
2.5.1: Osmotic flow ............................................................................................ 22
2.5.2: Diffusional Flow....................................................................................... 24
2.5.3: Hydraulic flow.......................................................................................... 25
2.6: Shale consolidation theories.............................................................................. 25
2.6.1: Terzaghi and Biot’s poroelastic consolidation ......................................... 25
2.6.2: Porochemoelectroelastic consolidation .................................................... 26
2.7: Recent research on consolidation theory to study wellbore stability ................ 28
Chapter 3: Coupled flow formulations for Shale ........................................................... 32
3.1: Extension of classical thermodynamics for irreversible processes ................... 32
3.1.1: Local equilibrium ..................................................................................... 33
3.1.2: Linear phenomenological equations......................................................... 34
3.1.3: Validity of Onsager reciprocal relations .................................................. 34
3.2: Formulations of the irreversible processes........................................................ 34
3.2.1: Dissipation function for shale media........................................................ 35
3.2.2: Transport equations for shale media ........................................................ 37
3.3: Phenomenological coefficients in terms of field parameters ............................ 37
3.3.1: Conductivity coefficients ......................................................................... 37
3.3.2: Coupling coefficients ............................................................................... 38
3.4: Concentration of the ions, net charges and pH of the pore fluid in shale ......... 40
3.5: Alternative formulation of Electro-Chemico-Osmotic processes in Shale ....... 41
Chapter 4: Governing equations for modeling Shale consolidation............................... 44
4.1: Transport Equations .......................................................................................... 44
4.2: Driving force equation....................................................................................... 45

vii
4.3: Equations related to physical structure of shale ................................................ 45
4.4: Governing equations of the porous medium ..................................................... 49
4.4.1: Strain displacement equation: .................................................................. 49
4.4.2: Semi-static stress equilibrium equation:................................................... 49
4.4.3: Mass conservation equations:................................................................... 49
4.4.4: Conservation of charges ........................................................................... 50
4.5: Equations applied for field purposes................................................................. 50
Chapter 5: Mathematical model ..................................................................................... 53
5.1: Initial wellbore conditions and stress transformations...................................... 53
5.2: Initial conditions of the formation before drilling............................................. 55
5.3: Boundary conditions and Problem definition.................................................... 58
5.3.1: Interfacial and far field stresses of the porous medium and drilling fluid 58
5.3.2: Loading decomposition of the porous medium........................................ 61
Chapter 6: Superposed Solutions of the chemically active porous medium .................. 63
6.1: Poroelastic plane strain problem ....................................................................... 63
6.1.1: Far field conditions................................................................................... 63
6.1.2: Near wellbore conditions.......................................................................... 64
6.1.3: Mode 1-Elastic radial loading / far field isotropic stress.......................... 65
6.1.4: Mode 2- Virgin pore pressure/ diffusional loading.................................. 66
6.1.5: Mode 3- Far field stress deviator/ poroelastic deviatoric stress loading .. 68
6.2: Elastic Uniaxial stress problem ......................................................................... 70
6.3: Elastic anti- plane shear problem ...................................................................... 70
6.4: Superposed solutions of the shale medium ....................................................... 71

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Chapter 7: Numerical Modeling Results ........................................................................ 72
7.1: MATLAB coding structure ............................................................................... 72
7.2: Code Testing...................................................................................................... 74
7.2.1: Validation of results using analytical solutions of Nguyen et al., (2008) 74
7.2.2: Validation of results using Jaeger’s analytical solutions for 1-D
consolidation of poroelastic medium...................................................... 76
7.3: Case study-Pierre Shale formations .................................................................. 84
7.4: Simulation Results............................................................................................. 85
7.4.1: When mud activity is greater than pore fluid water activity .................... 85
7.4.2: When water activity of pore fluid is greater than the mud activity.......... 89
7.4.3: Time propagation of osmotic pore pressure generated ............................ 93
7.4.4: Effect of Mechanical Properties on Pore Pressures of Chemically Active
Shale ....................................................................................................... 95
7.4.5: Effect of Petrophysical and Surface Charge Properties on Pore Pressure
distribution near wellbore region............................................................ 98
Chapter 8: Conclusions................................................................................................. 106
8.1: Conclusions from the above study on Pierre Shale formations....................... 106
8.2: Limitations of the model ................................................................................. 108
8.3: Recommendation............................................................................................. 109
References .................................................................................................................... 110
Appendix A: Laplace Inversion-Stehfest’s Algorithm................................................. 120
Appendix B: Properties of the Berea Sand and Shale gas formations used for validation
of results ........................................................................................................... 121

ix
List of Tables
Table 1: Mineralogical composition of an average shale (Adapted from U.S Geoscience
News and Information, Geology.com) ............................................................................. 4
Table 2: Mineralogical composition of Pierre Shale formation (adapted from Aminul
Islam et al., 2013) ............................................................................................................. 8
Table 3: Cation Exchange capacities of different clays and sand present in Shale
(adapted from Stephens et al., 2009).............................................................................. 15
Table 4: Hierarchy of observations in developing model describing the flow in soil.... 29
Table 5: Modeling parameters for an inclined wellbore ................................................ 84
Table 6: Pore pressures of different models at various distances from wellbore for high
mud activity.................................................................................................................... 86
Table 7: Effective Radial Stresses of different models at various distances from the
wellbore for high mud activity ....................................................................................... 87
Table 8: Effective Tangential stresses of different models for high mud activity ......... 88
Table 9: Pore Pressure of different models for lower mud activity ............................... 90
Table 10: Effective Radial stresses of different models for lower mud activity............ 91
Table 11: Effective Tangential stresses of different models for lower mud activity ..... 92
Table 12: Time propagation of the osmotic pressures generated................................... 94
Table 13: Pore pressure distributions for Chemically Active Shales for various Shear
Modulus.......................................................................................................................... 95
Table 14: Pore Pressure distributions for Chemically Active Shales for various Bulk
Modulus.......................................................................................................................... 96
Table 15: Pore Pressures for different Poisson's ratio.................................................... 97

x
Table 16: Relation between membrane efficiency and CEC for porosity range 0 - 40%
........................................................................................................................................ 99
Table 17: Relation between membrane efficiency and CEC for porosity range 40 - 60%
...................................................................................................................................... 100
Table 18: Pore Pressure distributions for Pierre Shale formations of different porosities
and constant CEC = 36 meq/ 100 g.............................................................................. 101
Table 19: Pore pressure distributions for Pierre Shale formations with different surface
charges and constant porosity of 0.176 ........................................................................ 103
Table 20: Pore pressure distributions for Pierre Shale formations with various
permeabilities (in m2
) ................................................................................................... 104
Table 21: Properties of Berea sand used for results validation (Chareonwongsak, S., et
al. 2010)........................................................................................................................ 121
Table 22: Properties of Shale gas formations used for results validation (data from table
5)................................................................................................................................... 122

xi
List of Figures
Figure 1: Forecast of natural gas production from different rock sources until 2040
(EIA, Annual Energy Outlook, 2013) .............................................................................. 1
Figure 2: Comparison of prices of Brent crude oil and Henry Hub natural gas from 1990
to 2040 (EIA, Annual Energy Outlook, 2014) ................................................................. 2
Figure 3: Microstructures of shales based on maturity influencing the brittleness and
fracturability (U.S Geoscience News and Information, Geology.com) ........................... 3
Figure 4: Types of wellbore instability problems (Petrowiki.spe.org)............................. 5
Figure 5: Wellbore stresses after drilling (McLean, 1990) .............................................. 6
Figure 6: Cross section of Pierre Shale in northern Great Plains region (adapted from
Schultz et al., 1980).......................................................................................................... 7
Figure 7: Surface charge characteristics-Diffuse double layer diagram (Colloid
chemistry department, University of Szeged) ................................................................ 12
Figure 8: Membrane efficiency Vs Permeability when shale is contacting NaCl solution
(Collins et al., 2008). ...................................................................................................... 16
Figure 9: Membrane efficiency Vs Water activity when shale is contacting NaCl
solution (Collins et al., 2008) ......................................................................................... 16
Figure 10: Membrane efficiency Vs porosity when NaCl mud is used (Collins et al.,
2008)............................................................................................................................... 17
Figure 11: Variation of membrane efficiency with porosity for different clay types
(adapted from Keijzer et al., 2001)................................................................................. 18
Figure 12: Adsorption isotherm of shale (adapted from Chenevert 1970)..................... 19

xii
Figure 13: Effect of temperature on water activity of shale (adapted from Chenevert and
Strassner 1975) ............................................................................................................... 20
Figure 14: Effect of confining pressure on water activity of shale (adapted from Fonseca
and Chenevert 1996)....................................................................................................... 21
Figure 15: Representation of reversible and irreversible processes in classical
thermodynamics (Department of Physics, University of Manchester) .......................... 33
Figure 16: Stress Vs strain curve for ductile material (Ratner et al., 2004) ................... 46
Figure 17: Inclined wellbore with axes inclined to in-situ stresses (Adapted from Cui et
al., 1997)......................................................................................................................... 53
Figure 18: Physical model of inclined wellbore in local coordinate system (adapted
from Cui et al., 1997) ..................................................................................................... 54
Figure 19: Initial and after drilling conditions of shale (adapted from Nguyen et al 2010)
........................................................................................................................................ 55
Figure 20: Heaviside step function (www.intmath.comlaplace)................................... 60
Figure 21: Loading decomposition of an inclined wellbore (adapted from Cui et al.,
1997)............................................................................................................................... 62
Figure 22: Flowchart of the MATLAB coding .............................................................. 73
Figure 23: Pore pressure distributions near wellbore for high mud activity
(Abousleiman et al & Matlab results) ............................................................................ 74
Figure 24: Effective radial stresses near wellbore for high mud activity (Abousleiman et
al & Matlab results)........................................................................................................ 75
Figure 25: Effective tangential stresses near wellbore for high mud activity
(Abousleiman et al & Matlab results) ............................................................................ 75

xiii
Figure 26: Problem Description ..................................................................................... 76
Figure 27: Column displacement for a 1-D consolidation problem; (a) The initial
condition (undrained condition); (b) the column is subjected to a constant load, pressure
is increased (undrained condition); (c) Fluid is drained from the column and pressure is
decreased (drained condition) adapted from Charoenwongsa et al. (2010). .................. 77
Figure 28: Displacement of the top of the column with dimensionless time for Berea
Sand ................................................................................................................................ 78
Figure 29: Displacement of the top of the column with dimensionless time for Shale
formations....................................................................................................................... 79
Figure 30: Displacement of the top of the column with dimensionless time for various
permeabilities of Shale formation .................................................................................. 79
Figure 31: Displacement of the top of the column with dimensionless time for various
porosities of Shale formation.......................................................................................... 80
formations of various Bulk modulus.............................................................................. 82
formations of various Shear modulus............................................................................. 83
Figure 34: Pore Pressure distribution at time = 0.01 day (15 mins) for higher mud
activity............................................................................................................................ 86
Figure 35: Effective radial stresses around the wellbore after time = 0.01 day (15 mins)
for higher mud activity................................................................................................... 87
Figure 36: Effective tangential stresses around the wellbore when time = 0.01 day (15
mins) for higher mud activity......................................................................................... 88

xiv
Figure 37: Pore Pressure distribution near wellbore for time = 0.01 day (15 mins) for
lower mud activity.......................................................................................................... 90
Figure 38: Effective radial stresses around the wellbore for time = 0.01 day (15 mins)
for lower mud activity .................................................................................................... 91
Figure 39: Effective tangential stresses around the wellbore when t = 0.01 day (15 mins)
for lower mud activity .................................................................................................... 92
Figure 40: Time propagation of the osmotic pore pressure generated for lower mud
activity............................................................................................................................ 94
Figure 41: Pore pressure distribution for various Shear Modulus.................................. 95
Figure 42: Pore Pressure distribution for various Bulk Modulus................................... 96
Figure 43: Effect of Poisson's ratio on Pore Pressure near the wellbore for high mud
activity............................................................................................................................ 97
Figure 44: Variation of membrane efficiency with Cation Exchange Capacity for
porosity range of 0-30% ................................................................................................. 99
Figure 45: Variation of membrane efficiency with Cation Exchange Capacity for
porosity range of 40 - 60%........................................................................................... 100
Figure 46: Pore Pressure distribution near wellbore region for various porosities and
CEC = 36 meq/ 100 g (higher mud activity) ................................................................ 102
Figure 47: Pore Pressure distributions near wellbore region for various CECs and
porosity = 0.176 (higher mud activity)......................................................................... 103
Figure 48: Pore pressure distributions for Pierre Shale formations with various
permeabilities ............................................................................................................... 104

xv
Abstract
The pore pressure distributions and the mechanical response of the poroelastic medium
are generally calculated to study the wellbore instability problems that could be
encountered during drilling. The mechanical response of the porous medium is
generally studied using the Biot’s poroelastic model which considers only the fluid flow
due to hydraulic pressure gradient. However in case of shales which are chemically
active, having a negative colloid diffused on its surface, forms an electrical double layer
with the cations of the pore fluid. The double layer influences the movement of the fluid
due to the chemical potential and concentration gradients that are generated. Hence
shales exhibit the phenomenon of swelling and shrinking when it is separating fluids of
different salinities. The fluid movement due to the chemical potential and concentration
gradients which are often neglected have to be considered when the near wellbore pore
pressure distributions are studied to avoid wellbore instability. This thesis resumes the
analytical models developed for a porochemoelectroelastic medium and also the
solutions depicting the response of a chemically active porous media when an inclined
wellbore is drilled. The model is based on the assumptions that the formation is
isotropic, the electrical field developed is static, chemical reactions are uncoupled from
ion diffusion and also the advection of ions with pore fluid is negligible. This work
presents the pore pressure distributions and also the effective radial and tangential
stresses at various distances from the wellbore obtained using the porochemoelastic
model by assuming that the electrical potentials are in equilibrium in the
porochemoelectroelastic model developed in literature. The results are compared with
the results obtained using Biot’s model for drilling mud of various salinities. The

xvi
significance of the mechanical properties such as porosity, shear and bulk modulus,
Poisson’s ratio, the hydrological properties such as permeability and surface properties
such as Cation Exchange Capacity have been studied and the impact of the properties
on pore pressure distributions for a porochemoelastic medium has been quantified. The
quantification of the influence of the properties helps in controlling the parameters to
overcome the wellbore instability. The coding algorithm has also been presented along
with the methods used for validation of the results obtained. An unexpected surge or
drop in pore pressures near wellbore occurs when chemical activity of the formations
are considered. Using a water based mud with an optimum mud activity lower than the
pore fluid activity can prevent hydration of shale or shale fractures and wellbore
collapse. Formations which are highly porous, less permeable and with low CEC can
yield results close to poroelastic medium. However all shale formations have higher
percentage of smectite clay which has higher CEC and deviated significantly from
properties of poroelastic medium.

1
Chapter 1: Introduction
1.1: Significance of Shale in Oil and Gas Industry
Shales were left unexplored for a long time because of its low permeability which
prevents gas migration. However, the depreciation of the conventional sources of
energy, the sufficiency of shale and recent evolution in technology has made shale gas
reservoirs an ultimate source of energy for upcoming decades. The recent advancements
in horizontal drilling and the stimulation methods have facilitated to explore the shale
gas reservoirs and to enhance gas recovery rates. Unlike other unconventional
reservoirs, gas production in shale gas reservoirs does not decline once the production
has been stabilized - a major cause for the petroleum industries to prefer shale reservoirs
over other unconventional resources. This is because shales are more porous and less
permeable and needs to be stimulated to allow the flow of natural gas to the wells. Once
Figure 1: Forecast of natural gas production from different rock sources
until 2040 (EIA, Annual Energy Outlook, 2013)

2
stimulated, shale gas reservoirs are also produced like any other conventional gas wells
(CAPP). The U.S Energy Information Administration forecasts that there will be an
increase of 44% in natural gas production from 23.0 trillion cubic feet in 2011 to 33.1
trillion cubic feet in 2040 where 88% of the production is from the shale gas industry.
Due to the significant increase in the production of natural gas and the remarkable
price difference between the crude oil and the natural gas, the U.S economy has been
reshaped by replacing other fuels with natural gas in transportation, electricity and
industrial sectors.
The EIA, Annual Energy Outlook, 2014 has presented the prognosis of oil and gas price
ratio of the upcoming years based on which the decision of using Liquefied Natural Gas
as a railroad fuel has been made.
Figure 2: Comparison of prices of Brent crude oil and Henry Hub natural
gas from 1990 to 2040 (EIA, Annual Energy Outlook, 2014)

3
1.2: Geological significance and chemical composition of Shale
Shale is a generic term to characterize rocks which are capable of being split or divided
into thin sheets in the direction of the grain. Shale is a clastic sedimentary rock
comprising clay, quartz and calcite of varying compositions. Shale gas reservoirs are
sometimes called as self-sourcing rocks because of the presence of the significant
fraction of organic matter called Kerogen. Organic rich shales have 40% of its bulk
reservoir occupied by kerogen (Passey et al., 2010). Shale gas reservoirs are one of the
major unconventional sources for natural gas. The gas produced is partially from the
free gas present in pores of the rock matrix and also from the gas adsorbed to the
organic matter. In case of shale gas reservoirs, the gas formed in the source rocks does
not migrate because of the low permeability. Hence one of the key parameters
controlling the gas flow in shale reservoirs is its effective permeability induced by
natural and man-made fractures. Shales are classified based on the thermal maturity
attained by heat and pressure producing hydrocarbons. Dry gas shales are highly,
thermally mature while shales with wetter gas are less mature and the shales with oil
being the least matured.
Figure 3: Microstructures of shales based on maturity influencing the brittleness
and fracturability (U.S Geoscience News and Information, Geology.com)

4
Dark colored shales have higher percentage of organic debris thereby making them
eligible for oil and gas production. Haynesville Shale, Barnett Shale, Fayetteville Shale
are most common gas producing reservoirs while Eagle ford Shale and Bakken Shale
produce oil (U.S Geoscience News and Information, Geology.com). Nearly 50-60% of
shale is made up of clay, a negative charged colloid, which influences the chemical and
physical properties of the rock because of its high ability for water exchange. The
degree of influence depends on the types of clays like kaolinite, illite that have minor
impact and montmorillonite-smectite which has a higher tendency to change in volume
during water exchange poses serious threats during drilling operations.
1.3: Wellbore Instability in shale
Wellbore instability is an unfavorable condition during which the diameter of the
borehole is remarkably different from the estimated shape and size and loses its
structural wholeness. The wellbore instability leads to the following problems:
a. Enlargement of the borehole
b. Hole shrinkage
Table 1: Mineralogical composition of an average shale (Adapted from U.S
Geoscience News and Information, Geology.com)

5
c. Hole fracture
d. Hole collapse
1.3.1: Principle of borehole stability
Borehole stability is based on the principle of equilibrium between the rock strength and
the in-situ stresses existing before the drilling operations. However when a wellbore is
drilled, the equilibrium is disturbed. The disturbance is amplified due to chemical
interactions between the injected fluids and the formation fluid. The doubtfulness in
some of the input data has made many borehole stability simulation models
unsuccessful. One of the important parameters being the pore pressure is affected due to
rock dilation and contraction which is commonly found in shales.
Figure 4: Types of wellbore instability problems
(Petrowiki.spe.org)

6
1.3.2: Causes for wellbore instability
Reasons attributing to the wellbore instability are:
1. Mechanical instability due to disturbance of stress equilibrium
2. Chemical instability due to rock-chemical interaction
1.3.3: Classification of chemical instability mechanisms
The chemical instability mechanisms of the wellbore drilled can be classified as follows
(petrowiki.spe.org/borehole-instability):
1. Osmotic pressure
2. Capillary pressure
3. Near wellbore pressure diffusion
4. Mud invasion into the formation due to overbalanced drilling
Figure 5: Wellbore stresses after drilling (McLean, 1990)

7
1.4: Pierre Shale formation
1.4.1: Stratigraphic positioning of Pierre Shale
Pierre shale rocks were deposited in a north south oriented trough approximately 1000
miles extending from Canadian arctic to south of New Mexico and sometimes plunges
into Gulf of Mexico. The Pierre shale rocks form a sedimentary wedge thinning from
thousands of feet in west to 1000 ft. in the east. All the sediments derived were from the
west and most of the sediments in the Far East are finely grained shale (Schultz et al.,
1980). Normal and reverse faults exist in Pierre shale formation in central South Dakota
with dips ranging from 40ᵒ
to 90ᵒ
and vertical displacements range from less than 1 m to
37 m (Nichols et al., 1994).
Figure 6: Cross section of Pierre Shale in northern Great Plains
region (adapted from Schultz et al., 1980)

8
1.4.2: Mineralogical composition of Pierre Shale
Major proportion of the Pierre Shale formation consists of clay stone that has little
fissile and have few proportions of fissile, fine grained, clayey rock (Schultz et al.,
1980). The presence of Kaolinite and Chlorite is generally minor and constitute to 10%
of clay minerals, illite makes up to 15-20 % of clay minerals and more than half of the
total clay (about 30-60%) is made up of mixed-layer clays. Quartz makes up more than
a few percent of most of the bentonite, and plagioclase and biotite may be abundant
especially at the bottom of the bed (Schultz 1978). The Pierre shale contains more than
59% of smectite, which is calcium saturated montmorillonite. Pierre shale is expected to
have high cation-exchange capacity and swelling potential compared to other shales
(Richard et al., 1991).
1.5: Essentiality of the research
Wellbore instability is a frequent problem that occurs while drilling shale formations.
Consequences of the wellbore instability can lead to many hole problems which could
result in expensive drilling operations. In a field, the instability is contributed by both
the mechanical and chemical effects. However only the mechanical effect has been
quantified in the earlier research ignoring the time dependent effects due to pore fluid
migration (McLean et al., 1990). When the wells are in exploratory stage many cores
and logs are obtained in order to understand the mechanical and hydrological properties.
However these mechanical and hydrological properties are completely altered when a
Table 2: Mineralogical composition of Pierre Shale formation (adapted from
Aminul Islam et al., 2013)

9
shale formation is drilled because shale has a very rich content of clay which holds
complicated water molecules and also has complex interactions between the ions of the
pore fluid and drilling fluid. Moreover the swelling pressures are modified based on the
mud chemistry (Abass et al., 2006). Osmotic effects are common in shale which can act
as a semi permeable membrane separating fluids of different salinities and electro
chemical potentials affecting the pore pressure distribution and the effective stress of
the rock. In order to avoid cost overruns and non- productive time, a model predicting
the pressure and stress distributions, and also helpful in determining optimum chemical
properties for drilling fluid by including both chemical and mechanical instability
mechanisms is essential.
The clay on the surface of shale has negative charges which forms a diffuse layer and
results in electro-kinetic phenomena when influenced by external electrochemical
potential gradients of the ions of drilling and pore fluids. The macroscopic transport
equations for the electro-kinetic phenomena are formulated based on the non-
equilibrium thermodynamics for irreversible process (Yeung et al., 1993).
Hence a mathematical model predicting the pore pressure distributions, effective radial
and tangential stresses including the electro-kinetic phenomena to the existing Biot’s
theory of poromechanics was first developed by Nguyen Vinh and Abousleiman (2008).
Nguyen et al., (2008) have developed the model to study the mechanical response of a
chemically active and charged formation through a comprehensive research on transport
mechanisms of the ions developed based on non-equilibrium thermodynamics by Yeung
et al.,(1993).

10
Main objectives of the research:
1. To give a comprehensive idea on how the presence of clay has been an attribute
to many of the unknown characteristics of shale
2. To elucidate the coupled transport of ions, fluid and chemical species that
happens inside shale due to the existence of different driving forces.
3. To summarize the transport equations those are used for such coupled flows
through a semi permeable membrane.
4. To enumerate the flaws in original theory of poromechanics by Biot which helps
to comprehend the current consolidation theory including the effects of
electrokinetics as developed by Nguyen Vinh and Abousleiman (2008).
5. To understand the relationship between various properties of shale such as
Cation Exchange Capacity, water activity, porosity, permeability and
conductivity.
6. To identify the key parameters that influence the pore pressure distributions
significantly and can be controlled to prevent the wellbore instability.
1.6: Précis of the thesis
Chapter 1 gives an insight of the work carried out in this literature. Chapter 2 gives a
brief description of the petro-physical properties of shale affecting the wellbore stability
that needs to be considered and also the observations from previous research papers.
Chapter 3 lists out transport formulations for irreversible processes happening in shale
and also expressions for the coupling and conductivity coefficients in terms of field
parameters. Chapter 4 presents the mathematical model developed by Nguyen et
al.,(2008) based on the formulations of chapter 3 to simulate the pressure and stress

11
distributions. Chapter 5 describes the mechanical and hydrological properties of Pierre
shale (formation of consideration) and also enumerates the boundary conditions used by
Nguyen et al., (2008) to obtain the analytical solutions for the mathematical model
framed. Chapter 6 lists the superposed solutions of the model which is decomposed into
3 sub models of different loading conditions. Chapter 7 presents the plots obtained
using the mathematical model developed in MATLAB based on assumption of
equilibrium of electrical potentials to simplify the porochemoelectroelastic model to
porochemoelastic model followed by the results and discussion. Algorithm of the code
used for simulation has been presented along with the methods used for validating the
results. Chapter 8 resumes the conclusions based on the observations from the plots, the
limitations of the model and also the recommendations for improving the wellbore
stability based on the conclusions.

12
Chapter 2: Literature Review
2.1: Clay colloidal chemistry
Clays are charged due to isomorphic substitution of metal atoms inside the crystalline
structure and chemical reactions between the reactive silanol and aluminol groups
present on surfaces of shale and pore water (Revil et al., 2001). Fine pores and the
negative charge of clay on shale surface causes shale to behave like a semi permeable
membrane (Fritz et al., 1994).
The basic mechanism in electro-kinetic phenomena which occurs in soil due to presence
of clay is explained by the electrical double layer theory for charged surfaces immersed
in aqueous solutions. This colloidal chemistry of clay particles has to be considered as it
has negative surface charge which influences the motion of the particles. The aqueous
Figure 7: Surface charge characteristics-Diffuse double layer diagram
(Colloid chemistry department, University of Szeged)

13
solution in formation is the pore fluid which behaves like an electrolyte and dissociates
into ions in presence of the negatively charged shale surface. The ions of pore fluid
redistribute to neutralize the charges on the shale surface. The quantity of the
exchangeable cations needed to neutralize the charge deficiency of clay is described as
Cation Exchange Capacity (CEC) and is often expressed in milli-equivalents for 100 g
of dry clay (Akram et al., 2000). The cations of the pore fluid move towards the
negative surface charge. The cations are adsorbed and held tightly to form a thick layer
called the Stern layer. The stern layer is surrounded by diffuse layer where the ions are
in thermal motion. The stern layer and the diffuse layers are separated by a slipping
plane and the voltage possessed by the slipping plane is called the zeta potential (Van
Olphen, 1963).
The electrical potential decreases as the distance from the surface increases. The zeta
potential influences the velocity of particle motion in the electrical field. The zeta
potential depends on the Cation Exchange Capacity of the shale medium. Higher the
CEC more negative zeta potential is developed and the velocity is greater within the
electrical field (Van Olphen 1963). The semi permeability exhibited by shale due to
presence of clay when it is separating two fluids of different salinities establishes
Donnan equilibrium. This is because shale is impermeable to some solutes of bigger
size as a result of which ions redistribute between shale to be in chemical equilibrium.
However the ions on either side of shale are not in equilibrium electrically (Overbeek
1956). The important aspects of Donnan equilibrium are:
1. unequal distribution of ions
2. osmotic pressure

14
3. potential differences between the phases
2.2: Cation Exchange Capacity (CEC) of Shale
CEC is considered as an index of the dispersibility of the shale formation which affects
the properties of the drilling fluids and the wellbore stability (Gray and Darley, 1980).
CEC is expressed as milliequivalents per 100 g of clay. The cation exchange capacity is
a measurement of the exchangeable cations present in the clays of a shale sample. The
exchangeable cations are positive ions which neutralize the negative charges on the
shale surface. Most common cations are sodium, calcium, magnesium, iron, potassium
and most common exchangeable ions in shale are from bentonite, montomorilloite clays
(Stephens et al., 2009).
Several analytical methods to determine the CEC were proposed. Some of them are
calorimetric technique based on the cobalt hexamine trichloride depletion (Bardon et
al., 1983), ammonium acetate saturation method (Sumner et al., 1996) and the copper
complexes method (Meier et al., 1999), barium exchange methods (Hendershot and
Duquette 1986). The methylene blue test (MBT) recommended by the API is the most
commonly used method to determine CEC which requires 1 gram of fine, dry shale
powder dispersed in water with dispersant, sulfuric acid, hydrogen peroxide and boiled
for few minutes followed by cooling to room temperature and titration with methylene
blue solution. The end point is observed when the sample suspension droplet on the
filter paper becomes a blue halo surrounding the dyed solids (API recommended
practices 13 I, 2004).
The higher the value of CEC, higher the reactivity of shale. Sandstone and limestone are
non-reactive and have CEC less than 1 meq/ 100 g. Average reactive shale has CEC of

15
10 – 20 meq/ 100 g, and highly reactive shale has CEC more than 20 meq/ 100 g. The
shale gets its CEC value based on the percentage of the composition of the different
clays and sand with the following individual CEC (Stephens et al., 2009):
Table 3: Cation Exchange capacities of different clays and sand present in Shale
(adapted from Stephens et al., 2009)
Since the clays exhibit surface conductivity effect, the clays with larger surface area has
larger CEC values
2.3: Membrane Efficiency of Shales
The osmotic pressure generated in the shale formations is proportional to the shale
membrane efficiency which is estimated using the wireline measurements. The flow of
water and ions through the shale is controlled by the membrane efficiency. The
membrane efficiency of shale is influenced by the porosity and the permeability. When
variation of membrane efficiency with porosity and permeability was studied by Collins
et al., (2008), it was found that membrane efficiency is negatively related to the shale
porosity until porosity of 7.5% (figure 8) and is related to permeability of order of 0.1
nD (figure 7) while there are no significant changes beyond the threshold values of
porosity and permeability.
Types of clay / sand CEC in meq/ 100 g
Smectite 80-120
Illite 10-40
Kaolinite 3-15
Chlorite 10-40
Sand < 0.5

16
Figure 8: Membrane efficiency Vs Permeability when
shale is contacting NaCl solution (Collins et al., 2008).
Figure 9: Membrane efficiency Vs Water activity when
shale is contacting NaCl solution (Collins et al., 2008)

17
Figure 10: Membrane efficiency Vs porosity when NaCl mud
is used (Collins et al., 2008)
The above plots show that the rate of the solute transport is influenced by the porosity
and permeability until it reaches its threshold values. The membrane efficiency values
of shale to brine solutions range from 0-10% (Van Oort et al., 1996 and Collins et al.,
2008).
The membrane efficiency of shale increases with the increase in Cation Exchange
Capacity and is also proportional to ratio of (CEC/k) where ‘k’ is permeability. The
membrane efficiency is directly proportional to the hydrated diameters of the ions in the
drilling fluid (Al Bazali Talal. M et al., 2006).
Keijzer et al., (2001) represented membrane efficiency of shale as a function of porosity
for various clay types (as shown in figure 11) based on the empirical relation developed
by Fritz et al., (1986).

18
Figure 11: Variation of membrane efficiency with
porosity for different clay types (adapted from
Keijzer et al., 2001)
Figure 11 shows that the membrane efficiency increases with decrease in the porosity.
This is true because when the porosity decreases the clay membrane compacts and the
diffuse double layer overlap and the net negative potential developed excludes the
anions from the surface making the ion exclusion behavior of the membrane more ideal.
Similarly when the net negative charges on the clay surface increases the ion exclusion
behavior is more predominant in shale.
2.4: Water activity of shale
Water activity of the shale formation is an indicator of the state of hydration of the
shale, and its potentiality to absorb or lose water. The water activity is affected by
factors such as pressure, temperature, mineralogy, platelets spacing and pore fluid
compositions. The water activity is usually measured in a laboratory by using a

19
Figure 12: Adsorption isotherm of shale (adapted from Chenevert 1970)
hygrometer which measures the relative humidity of an enclosed space containing the
cuttings (Winston et al., 1960). The percentage of relative humidity divided by 100
gives an approximate value of water activity. A recent technology of using chilled
mirror to find water activity was developed (Jarrett et al., 2004). The most common
method to find the water activity of shale is by using adsorption isotherm method
(Chenevert 1970) which uses the desiccators with saturated salt solutions that maintain
the vapor pressure of water. The degree of adsorption depends on the amount of
moisture adsorbed onto the shale surface. This relationship is conveniently represented
in an adsorption isotherm which plots moisture adsorbed onto shale surface as a
function of relative vapor pressure at constant temperature. Though the adsorption
isotherm is distinct for each shale, there is a common observation of increase in the
number of adsorbed molecules with increasing pressure.

20
The figures 13 and 14 show that the water activity of shale increases with increase in
confining pressure and temperature.
The water activity of the drilling fluid is also calculated based on the correlation
𝑎 𝑤,𝑚𝑢𝑑 = 𝐶1 𝑒𝑥𝑝𝐶2 𝑇 … … … … … … … … … … … … … … … … … … … … … … … … … … … . (1)
Where C1 and C2 are constants depending on concentration of the salt in the solution
(Chenevert and Strassner 1975).
Figure 13: Effect of temperature on water activity of shale
(adapted from Chenevert and Strassner 1975)

21
The osmotic pressure is calculated from the activity of the shale and the drilling fluid as
shown in the equation 2. The osmotic potential of an ideal membrane is greater than the
osmotic potential of a non- ideal membrane. Equation 2 is adapted from Fritz et al 1981.
𝑃𝑤 = 𝜎 𝑚
𝑅𝑇
𝑉𝑤
ln (
𝑎 𝑤,𝑚𝑢𝑑
𝑎 𝑤,𝑠ℎ𝑎𝑙𝑒
) … … … … … … … … … … … … … … … … … … … … … … … … … (2)
Where
𝑃𝑤 = 𝑜𝑠𝑚𝑜𝑡𝑖𝑐 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒
𝜎 𝑚 = 𝑚𝑒𝑚𝑏𝑟𝑎𝑛𝑒 𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑐𝑦
𝑉𝑤 = 𝑚𝑜𝑙𝑎𝑟 𝑣𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑤𝑎𝑡𝑒𝑟 (0.018 𝑙/ 𝑚𝑜𝑙)
2.5: Transport Mechanisms in Shale
When the water based muds come in contact with the shale formations the ions and the
water flow into the shale due to different driving forces. The driving forces in shale are:
Figure 14: Effect of confining pressure on water activity of
shale (adapted from Fonseca and Chenevert 1996)

22
Hydraulic pressure difference, electrical potential difference, chemical potential
difference of water between mud and pore fluid and the concentration gradients
between the solutes of the mud and the pore fluid (Simpson et al., 2000).
2.5.1: Osmotic flow
The osmotic pressure developed in shale influences the transfer of solutes and the
associated water between the drilling mud and the pore fluid, through clay which acts as
a semi permeable membrane.
The osmotic pressure developed due to difference in the water activity of the drilling
mud and the pore fluid causes the flow of water from region of high water activity (low
salt concentration) to region of low water activity (high salt concentration) and the
process is termed as capillary osmosis. It is a reverse process for diffusion osmosis
discussed in section 2.5.2. Similarly osmotic pressure built due to electrical potential
gradient of the components in the two fluids causes ions to move from fluid of high
electrical potential to fluid of low electrical potential. The process is called
electrophoresis and is accompanied with associated water movement termed as electro-
osmosis. Hydration of shales makes it weaker thus aggravating the problems of
wellbore instability (Abass et al., 2006).
In 1947 Sitter figured that the changes in the salinity of subsurface water with depth is
due to the semi-permeable nature of shale. In 1965 Allen Young and Philip Low
demonstrated the osmotic flow of water through shale samples which causes the
subsurface pressure anomalies. Though Young et al., (1965) were able to justify the
osmotic phenomena in shales, the quantification of osmotic pressure was less compared
to the theoretical pressures associated with water flux. The reasons attributed for such

23
discrepancy are micro-cracks in the core samples, inability of the large weakly charged
pores to restrict the solute movement.
In 2000 Neuzil conducted tests on Pierre shale to study the influence of osmotic
pressure developed on fluid movement and also the longevity of the effect. The fluid
flow in shale can be expressed as (Bresler 1973)
𝑞 = −
𝑘
µ
𝜕𝑝
𝜕𝑟
+ 𝜎
𝑘
µ
𝜕𝜋
𝜕𝑟
… … … … … … … … … … … … … … … … … … … … … … … … … … … . (3)
The 1st
term in the equation is Darcy’s law generated due to pore pressure gradient and
the 2nd
term is for osmotic pressure gradient. The fluid flux stops when the osmotic
equilibrium is attained (q=0). The osmotic pressure built can be found at equilibrium by
integrating equation 3 (Neuzil 2000)
∆𝑝 = ∫ 𝜎(𝑐)
𝑑𝜋
𝑑𝑐
𝑐𝑚𝑖𝑛
𝑐𝑚𝑎𝑥
𝑑𝑐 … … … … … … … … … … … … … … … … … … … … … … … … … … . (4)
Π (measure of decrease in chemical potential of water due to solute presence) and σ
(osmotic efficiency) are functions of c (solute concentration).
Π is related to ‘c’ through water activity. However the relation between ‘c’ and σ is
complex and the relationship is established between σ and b√ 𝑐 by Bresler (1973), where
‘b’ is half distance between the platelets proving the dependency of osmotic efficiency
of shale on shale compaction and the solute concentration. The osmotic pressure
decreases only when the TDS differences that created it dissipate by diffusion and the
longevity is influenced by the membrane effective ionic diffusion coefficient 𝐷 𝑑
̂ .
𝐷 𝑑
̂ [
𝜕2
𝑐
𝜕𝑟2
+
1
𝑟
𝜕𝑐
𝜕𝑟
] =
𝜕𝑐
𝜕𝑡
… … … … … … … … … … … … … … … … … … … … … … … … … … . . (5)

24
The time taken for TDS difference to reduce to half its value for a shale of thickness ‘L’
and diffusion coefficient 𝐷 𝑑
̂ is given by Crank. J (1957)
𝑡 = 0.1 ∗
𝐿2
𝐷 𝑑
̂
… … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … (6)
The non -osmotic pressure decays 1000 times faster than osmotic pressure (Neuzil,
2000).
2.5.2: Diffusional Flow
Diffusional flow is opposite to capillary osmotic flow. By principle of Fick’s law of
diffusion, the solutes flow from the solution with high salt concentration (with low
chemical potential of water) to solution with low salt concentration (high chemical
potential of water) while water passes from solution with high chemical potential of
water to solution which has lower water chemical potential. The diffusional flow is
effective when the shale membrane efficiency is less and has high permeability to allow
the solutes to pass through them. The diffusional flow is significant in fractured or high
permeable formations affecting the shale stability compared to formations which are
less permeable. For example when the shale is in contact with the brine, the clay
structure is altered and destabilized due to the cation exchange reactions (Abass et al.,
2006). If the diffusional force is greater than the chemical osmotic force, invasion of
ions and water increases the pore pressure and water content of shale near the wellbore
region. Water and solutes can be transferred from mud to shale by diffusion osmosis
even when the hydraulic pressure difference and chemical osmotic pressure difference
does not exist. Addition of salt to a water based mud to reduce the water activity causes
an increase in the diffusional osmotic force affecting the desired increase in the
chemical osmotic force (Simpson et al., 2000). The concentration gradient in soil causes

25
a drag of bulk pore fluid by diffusional flow of species from concentrated to dilute
solutions. Though diffusional osmosis and chemical osmosis happen simultaneously,
chemical osmosis is significant in densely compacted formation with high cation
exchange capacity while diffusion osmosis is significant in less compacted formation
with less cation exchange capacity (Olsen et al., 1990). By Curie Prigogine principle,
there is no coupling between the scalar and vector quantities in isotropic medium.
Hence diffusion and chemical reaction are not coupled.
2.5.3: Hydraulic flow
The fluid flow in the subsurface is always considered to be an effect of pressure
differences between the drilling mud and the shale pore fluid which is governed by
Darcy’s law. The flow is usually directed from the mud to the shale formation during an
overbalanced drilling. Increase in mud weight of the drilling fluid increases the
confining pressure which stabilizes the shale formation. However it also increases the
shale hydration. The water based muds are at a disadvantage compared to oil based
muds which has threshold capillary pressure and prevents shale hydration. Water based
muds dissipate pressure because of the miscibility with shale pore water while oil based
muds don’t and have high radial stress (Abass et al., 2006). Hydraulic conductivity of
shale increases with the fractures created and is found to be thrice the magnitude of the
unfractured shale (McKay et al., 1993).
2.6: Shale consolidation theories
2.6.1: Terzaghi and Biot’s poroelastic consolidation
The phenomena of soil consolidation when load is applied was first proposed by
Terzaghi (1925). He assumed the soil to be a porous medium possessing elastic

26
properties and the pores are filled with water analogous to a rubber sponge. He
propounded that the rate at which the soil consolidates or settles depends on the rate at
which the water is squeezed out of the pores when a load is applied. He also formulated
equations representing the soil consolidation phenomena mathematically. However
Terzaghi’s formulations were limited to one dimensional consolidation for a fixed load.
Biot (1941) overcame these limitations by extending the equations to three dimensions
and also validated them for any load varying with time. He assumed the soil to have the
following properties:
a. Isotropy of soil
b. Reversibility of stress-strain relations under equilibrium conditions
c. Linearity of stress strain relationship
d. Smaller strains
e. Incompressible pore water
f. Water may contain air bubbles
g. Water flows through pores in accordance to Darcy’s law
2.6.2: Porochemoelectroelastic consolidation
Though the first theoretical investigation on electro-osmosis was started by Reuss
(1908) based on his observations on thin clay diaphragm it was not practically taken
into consideration for modelling transport equations for shale as it was considered to be
a poroelastic medium and the knowledge of colloidal chemistry of shale was
insufficient. Casagrande (1949) investigated on the electro osmotic transport of water in
soil and found the process to be constant irrespective of the soil types. Later when the
behavior of clay rich porous formations were studied it was found that shale behaves

27
like a semi permeable membrane restricting the transport of solute of some species in
pore fluid (Young and low 1965, Olsen 1969, Neuzil 2000). Esrig (1968) studied the
electro-kinetic consolidation of shale and predicted that the positive and negative pore
pressures develop because of the application of electrical field. The magnitude of the
pore pressures developed depends on the electric field applied and also on the geometry
of the electrodes. He also studied that the rate of consolidation due to electric potential
applied is very slow as the soil consolidation depends mainly on soil compressibility
and permeability. He developed a one dimensional solution for consolidation problem
but ignored ion transportation. Since shale has a higher percentage of clay which acts
like a semi permeable membrane and allows fluid flow due to non-hydraulic driving
forces, shale also exhibited swelling and shrinking phenomena termed osmosis when it
was brought in contact with aqueous solutions. The concept of osmosis has invoked to
explain the anomalous pressure behavior in subsurface aquifers (Neuzil 2000). The low
permeability and the negative charges on the surface of shale membrane are the reasons
for shale to have high membrane efficiency. The fluid components separated by shale
with different chemical potentials cause movement of chemical species along with
water which is termed as chemical-osmosis. Similarly when shale with pore fluid which
acts like an electrolyte with its ions is subjected to an electrical potential gradient, it
causes movement of ions along with water from fluid with high to low electrical
potentials. This is similar to the coupled flows that take place in soil with simultaneous
flows of water, ions and chemical species. Since then shale is considered as a
porochemoelectroelastic medium and the electrochemical interactions between the
fluids separated by shale and also with shale media which affects the pore pressure

28
distribution and effective stress has to be quantified for predicting the overall
mechanical response of shale. The transport formulations for the coupled irreversible
processes in soil derived based on non- equilibrium thermodynamics (Katchalsky and
Curran 1967, Yeung and Mitchell 1993) was adopted for explaining the transport
mechanisms in shale. The original poromechanics theory by Biot (1941) was
reformulated for porous medium to include the electrochemical interactions based on
non-equilibrium thermodynamics (Sachs et al., 1989, Corapcioglu 1991). The
analytical solutions explaining the mechanical response of the porochemoelectroelastic
medium were presented by Lui et al., 1999 and Van Meerveld et al., 2003. However the
solutions were restricted to one dimensional consolidation problem.
2.7: Recent research on consolidation theory to study wellbore stability
For many years significant efforts have been made to understand the rock mechanics
causing the wellbore instabilities. Before the advent of electrokinetic phenomena
happening in shale, it was considered that the swelling and shrinking of clays happened
only due to adsorption of water molecules onto shale surface. Later the analytical
solutions explaining the mechanical response of many semi permeable membranes were
incorporated to shale to understand its wellbore stability. However many analytical
solution developed were one dimensional and could not completely predict the
mechanical response. Ekbote and Abousleiman (2006) developed generalized a coupled
formulation for chemically active formation in which an inclined wellbore has been
drilled and is subjected to in-situ stresses. However this formulation ignored the
electrical coupling. Nguyen Vinh et al., (2008) developed the analytical solution for the
coupled porochemoelectroelastic processes when an inclined wellbore is drilled into an

29
isotropic formation. Nguyen Vinh et al., (2009) developed a finite element model to
study the responses for a naturally fractured porous medium using dual porosity dual
permeability method. Ghassemi et al., (2009) analyzed the pore pressure and stress
distribution around the wellbore based on a coupled thermo elastic model of chemically
active rock showing the interactions between temperature, stress and chemistry. Roshan
and Rahman (2010) have developed a finite element based chemo-thermo-poroplastic
model to study the stress and pore pressure distribution in naturally fractured shale
formations. Tran et al., (2013) have formulated the analytical solution for anisotropic
porochemoelectroelastic medium with a cylindrical wellbore drilled into the formation
and subjected to various loading conditions.
Table 4: Hierarchy of observations in developing model describing the flow in soil
YEAR AUTHOR FORMULATION LIMITATIONS
1908 Von Reuss
Initiated the investigation on electro-
osmosis in soil
Did not model
transport
equations due to
lack of knowledge
about colloidal
chemistry of soil
1925 Terzaghi
Soil is a porous medium with
elastic properties filled with
water. The rate of consolidation
depends on rate of liquid drop out
when load is applied
Limited to 1-D
consolidation for
fixed load
1941 Biot
Extended the existing Terzaghi's
equations to 3 dimensions and
validated for any load varying
with time
Later
reformulated to
include the
electrochemical
interactions
1949 Casagrande
Investigated on electro osmotic
transport of water in soil and
found that the porocess is
constant in all soils
1965,1969,
2000
Young and
low, Olsen,
Neuzil
Confirmed shale behaves like a
semi permeable membrane
restricting transport of solute of
some species

30
1967, 1993
Katchalsky
& Curran,
Yeung &
Mitchell
Developed formulations for
coupled irreversible transport
processes based on non
equilibrium thermodynamics
1968 Esrig
Studied electro-kinetic
consolidation and suggested that
negative pore pressure are due to
electrical field application.
Magnitude of pore pressure
depends on electrical field and
also on geometry of the
electrodes.
Developed a 1 D
solution for
consolidation
but ignored ion
transport
1989, 1991
Sachs et al.,
Corapcioglu
Reformulated Biot's theory
including electrochemical
interactions based on non
equilibrium thermodynamics
1999, 2003
Lui et al.,
Van
Meerveld et
al.,
Developed analytical solutions
explaining mechanical response
of porochemoelectroelastic
medium
But restricted to
1 dimensional
consolidation
2000 Neuzil
Found anamalous pressure
behavior is due to osmosis and
also framed equations to calculate
the presure developed and also
time to calculate the longevity of
the developed pressure
2006
Ekbote and
Abousleiman
Developed a generalized coupled
formulations for chemically
active formations with inclined
wellbore and subjected to in-situ
stress
Ignored
electrical
coupling and
simplified
isotropic
medium
2008
Nguyen and
Abousleiman
for coupled
porochemoelectroelastic medium
with inclined wellbore
Ignored thermal
effects and
simplified
isotropic
medium
2009
Nguyen and
Abousleiman
Studied the responses of the
naturally fractured coupled flow
porous medium using dual
porosity dual permeability
method
Ignored thermal
effects and
simplified
isotropic
medium

31
2009 Ghassemi
Analyzed pore presure and stress
distribution near wellbore for a
coupled thermo elastic chemically
active porous medium explaining
the relationship between the
tempertaure, stress and chemistry.
Ignored
electrical
coupling and
simplified
isotropic
medium
2010
Roshan and
Rahman
Developed
chemothermoporoplastic model to
study stress and pore pressure
distributions in naturaly fractured
shale formations
Ignored
electrical
coupling and
simplified
isotropic
medium
2013 Tran et al
for anisotropic
porochemoelectroelastic medium
with cylindrical wellbore
Ignored thermal
effects

32
Chapter 3: Coupled flow formulations for Shale
3.1: Extension of classical thermodynamics for irreversible processes
The coupled fluid, electrical and chemical flows in soil have been analyzed by Yeung
and Mitchell (1993) based on the concept of non-equilibrium thermodynamics to
comprehend the transport processes involved.
The coupled flow is understood as the linear relationship existing between the flow and
the corresponding driving forces or gradients based on the previous formulations such
as Darcy law relating the fluid flow and the pressure gradient, Fick’s law relating the
flow of individual species in solution and its concentration gradient, Fourier’s law
relating the heat flow and the temperature gradient.
The linear relationship existing between the coupled flows is given by the following
equation where the total flow of a particular type ‘i’ is related to the summation of the
contribution from all existing flows due to the corresponding gradients (Yeung and
Mitchell 1993).
𝐽𝑖
⃗⃗ = ∑ 𝐿𝑖𝑗 𝑋𝑗
𝑛
𝑗=1
… … … … … … … … … … … … … … … … … … … … … … … … … … … … … … . . (7)
Where 𝐿𝑖𝑗 is the coupling coefficient relating the flow of type ‘i’ with gradients ‘j’. The
magnitude of the coupling coefficient 𝐿𝑖𝑗 greatly influences the coupled flows and has
to be quantified for practical use of equation 7. The coupled flows in the shale system is
out of equilibrium and the formulations are framed based on non-equilibrium
thermodynamics. The transport coefficients can be expressed in terms of field
measurable parameters based on the concept of irreversible or non-equilibrium
thermodynamics (Gray 1966).

33
Non equilibrium thermodynamics differs from the classical thermodynamics by
defining state of the system at all times during the path of an irreversible process.
The classical thermodynamics can be extended to define the irreversible processes by
including few postulates such as (Yeung and Mitchell 1993)
1. Local equilibrium
2. Linear phenomenological equations
3. Validity of the Onsager reciprocal relations
3.1.1: Local equilibrium
The system under consideration can be divided into smaller segments where in each of
the segments the state functions exist and the segments are assumed to be in local
equilibrium as the state functions are continuous with space and time (infinitesimal
changes). Since process in each segment is in equilibrium state and can be considered
Figure 15: Representation of reversible and irreversible
processes in classical thermodynamics (Department of
Physics, University of Manchester)

34
reversible, classical thermodynamics is applied in each segment (Yeung and Mitchell
1993).
3.1.2: Linear phenomenological equations
The linear equation relating the flow of one type with the effective driving forces in the
system as shown in equation 7. The transport coefficients 𝐿𝑖𝑗 are also called as
phenomenological coefficients and are independent of the driving forces. If i = j then 𝐿𝑖𝑖
is the conductivity coefficient of the flows and if i ≠ j then 𝐿𝑖𝑗 are the coupling
coefficients relating flows of different types producing cross effects (Mitchell 1976).
3.1.3: Validity of Onsager reciprocal relations
Onsager (1931) put forward his theory of reciprocal relations based on the assumption
of local equilibrium and reversibility of the processes in small segments which states
that the phenomenological coefficients are symmetrical if they are not influenced by the
external magnetic field or Coriolis forces (Fitts 1962). This postulate helps to reduce the
number of coefficients from 𝑛2
to ((n+1)*n)/2.
𝐿𝑖𝑗 = 𝐿𝑗𝑖 … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … . . (8)
3.2: Formulations of the irreversible processes
The fluxes and the driving forces exist only when the state parameters deviate from the
equilibrium values. The driving forces and the fluxes are selected only if it is in
accordance with 2nd
law of thermodynamics (Yeung 1990).
𝑋𝑖
⃗⃗⃗ =
𝜕∆𝑠
𝜕𝛼𝑖
… … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … . (9)
Where
𝑋𝑖
⃗⃗⃗ = 𝑑𝑟𝑖𝑣𝑖𝑛𝑔 𝑓𝑜𝑟𝑐𝑒𝑠
𝜕∆𝑠 = 𝑐ℎ𝑎𝑛𝑔𝑒𝑠 𝑖𝑛 𝑒𝑛𝑡𝑟𝑜𝑝𝑦 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝑣𝑜𝑙𝑢𝑚𝑒 𝑓𝑟𝑜𝑚 𝑒𝑞𝑢𝑖𝑙𝑖𝑏𝑟𝑖𝑢𝑚 𝑣𝑎𝑙𝑢𝑒𝑠

35
𝜕𝛼𝑖 = 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 𝑖𝑛 𝑠𝑡𝑎𝑡𝑒 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟𝑠 𝑓𝑟𝑜𝑚 𝑒𝑞𝑢𝑖𝑙𝑖𝑏𝑟𝑖𝑢𝑚 𝑣𝑎𝑙𝑢𝑒𝑠
𝐽𝑖
⃗⃗ =
𝜕𝛼𝑖
𝜕𝑡
… … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … . (10)
From equation 9 and 10 the entropy produced per unit volume per unit time during the
irreversible process can be calculated as
𝑑𝑠
𝑑𝑡
= 𝜎 = ∑ 𝐽𝑖
⃗⃗ . 𝑋𝑖
⃗⃗⃗
𝑛
𝑖=1
… … … … … … … … … … … … … … … … … … … … … … … … … … … (11)
Based on equation 11 and the Gibbs equation relating the intensive properties of a
system in equilibrium undergoing reversible changes Katchalsky and Curran 1965
derived an expression similar to equation 11 to determine the local dissipation energy
per unit volume per unit time (Φ = σ *T).
𝛷 = ∑ 𝐽𝑖
⃗⃗ . 𝑋𝑖
⃗⃗⃗
𝑛
𝑖=1
… … … … … … … … … … … … … … … … … … … … … … … … … … … … … … (12)
Equation 12 is different from equation 11 by a factor of ‘T’. Hence under isothermal
conditions it is convenient to use Φ than σ (as Φ is related to Gibbs free energy while σ
is related to entropy as seen in s = q/ T).
The formulations are framed by a) estimating the dissipation function b) forming linear
phenomenological equations relating the predefined fluxes and driving forces c)
applying the Onsager’s reciprocal theory and d) expressions for phenomenological
coefficients in field parameters.
3.2.1: Dissipation function for shale media
Based on the coupled flows and the driving forces present in the soil, dissipation
function for the above irreversible processes was framed by Yeung and Mitchell 1993
as following:

36
𝛷 = ∑ 𝐽𝑖
⃗⃗ . 𝛻(−µ𝑖̃)
𝑛
𝑖=1
… … … … … … … … … … … … … … … … … … … … … … … … … … … . . (13)
Where
µ𝑖̃ 𝑖𝑠 𝑡ℎ𝑒 𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑐ℎ𝑒𝑚𝑖𝑐𝑎𝑙 𝑝𝑜𝑡𝑒𝑛𝑡𝑖𝑎𝑙 𝑜𝑓 𝑠𝑝𝑒𝑐𝑖𝑒𝑠 𝑖
µ𝑖̃ = µ𝑖 + 𝑧𝑖 𝐹𝐸 … … … … … … … … … … … … … … … … … … … … … … … … … … … … … . (14)
Where
µ𝑖 = 𝑐ℎ𝑒𝑚𝑖𝑐𝑎𝑙 𝑝𝑜𝑡𝑒𝑛𝑡𝑖𝑎𝑙 𝑜𝑓 𝑠𝑝𝑒𝑐𝑖𝑒𝑠 𝑖
𝑧𝑖 = 𝑐ℎ𝑎𝑟𝑔𝑒𝑠 𝑜𝑓 𝑠𝑝𝑒𝑐𝑖𝑒𝑠 𝑖
𝐹 = 𝐹𝑎𝑟𝑎𝑑𝑎𝑦′
𝑠 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
𝐸 = 𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐𝑎𝑙 𝑝𝑜𝑡𝑒𝑛𝑡𝑖𝑎𝑙 𝑜𝑓 𝑠𝑝𝑒𝑐𝑖𝑒𝑠 𝑖
The chemical potential of species ‘i’ has concentration dependent, temperature
dependent and pore pressure dependent components.
µ𝑖 = µ𝑖
𝑜
(𝑇) + 𝑉𝑖
̃ 𝑃 + µ𝑖
𝑐
… … … … … … … … … … … … … … … … … … … … … … … … … . (15)
Substitute equations 14 and 15 in 13 for isothermal conditions to get
ɸ = ( ∑ 𝐽𝑖 𝑉⃗ 𝑖
𝑛
𝑖=1 ). ∇ ( −𝑃) + ∑ 𝐽𝑖
𝑛
𝑖=1 . ∇ (−𝜇𝑖
𝑐
− 𝑧𝑖 𝐹𝐸) … … … … … … … … … … … … . . (16)
In equation 16 ( ∑ 𝐽𝑖 𝑉⃗ 𝑖
𝑛
𝑖=1 ) = 𝐽𝑣 (rate of volume of fluid flowing per unit area of an
element) and for isothermal conditions ∑ 𝑐𝑖
𝑛
𝑖=1 ∇ (𝜇𝑖
𝑐
+ 𝑧𝑖 𝐹𝐸) = 0 by Gibbs-Duhem
relation. By incorporating the above conditions and replacing the absolute diffusional
flow (𝐽𝑖) by relative diffusional flow (𝐽 𝑑) in equation 16 gives
ɸ = 𝐽𝑣 . ∇ ( −𝑃) + 𝐼. ∇ (−𝐸) + 𝐽 𝐶
𝑑
∇ (−𝜇 𝑐
𝑐) + 𝐽 𝑎
𝑑
∇ (−𝜇 𝑎
𝑐 ) … … … … … … … … … … . (17)
Where 𝐼 = ∑ (𝐽𝑖
𝑑𝑛−1
𝑖=1 𝑧𝑖 𝐹) is the electric current density passing through clay rich shale
(Fitts 1962).

37
3.2.2: Transport equations for shale media
Based on equation 17 and theory of linear phenomenological equations the flows and
the driving forces in shale are related as
𝐽𝑣 = 𝐿11∇(−𝑃) + 𝐿12∇(−𝐸) + 𝐿13∇(−𝜇 𝑐
𝑐) + 𝐿14∇(−𝜇 𝑎
𝑐 ) … … … … … … … … … … … (18)
𝐼 = 𝐿21∇(−𝑃) + 𝐿22∇(−𝐸) + 𝐿23∇(−𝜇 𝑐
𝑐) + 𝐿24∇(−𝜇 𝑎
𝑐 ) … … … … … … … … … … … (19)
𝐽𝑐
𝑑
= 𝐿31∇(−𝑃) + 𝐿32∇(−𝐸) + 𝐿33∇(−𝜇 𝑐
𝑐) + 𝐿34∇(−𝜇 𝑎
𝑐 ) … … … … … … … … … … . (20)
𝐽 𝑎
𝑑
= 𝐿41∇(−𝑃) + 𝐿42∇(−𝐸) + 𝐿43∇(−𝜇 𝑐
𝑐) + 𝐿44∇(−𝜇 𝑎
𝑐 ) … … … … … … … … … … . (21)
3.3: Phenomenological coefficients in terms of field parameters
Based on Onsager’s reciprocal theory as shown in equation 8, the phenomenological
coefficients are reduced from 16 to 10. Expressions of the coefficients in field
parameters are found from experiments by maintaining conditions such that 1 or 2
gradients disappear. The conductivity parameter of any flow type ‘i’ is defined as the
ratio of the flow of type ‘i’ per unit area to its corresponding gradient applied.
3.3.1: Conductivity coefficients
Hydraulic conductivity coefficient
𝐿11 =
𝑘ℎ
𝛾𝑡 𝑛
+
𝐿12 𝐿21
𝐿22
… … … … … … … … … … … … … … … … … … … … … … … … … … … . (22)
Where
𝑘ℎ = ℎ𝑦𝑑𝑟𝑎𝑢𝑙𝑖𝑐 𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑣𝑖𝑡𝑦 𝑜𝑓 𝑝𝑜𝑟𝑜𝑢𝑠 𝑚𝑒𝑑𝑖𝑎
𝛾𝑡 = 𝑢𝑛𝑖𝑡 𝑤𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑝𝑜𝑟𝑒 𝑓𝑙𝑢𝑖𝑑
𝑛 = 𝑝𝑜𝑟𝑜𝑠𝑖𝑡𝑦 𝑜𝑓 𝑠ℎ𝑎𝑙𝑒
kh varies from 1x10-13
to 1 m/s (Freeze et al., 1979).

38
Electrical conductivity coefficient is given by
𝐿22 =
𝜅
𝑛
… … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … . (23)
Where
𝜅 = 𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐𝑎𝑙 𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑣𝑖𝑡𝑦 𝑜𝑓 𝑠ℎ𝑎𝑙𝑒 𝑚𝑒𝑑𝑖𝑎
κ generally varies from 0.01 to 0.1 S/m (Yeung and Mitchell 1993).
Chemical conductivity coefficient is given by
𝐿33 =
𝐷𝑐
∗
𝐶𝑐
𝑅𝑇
… … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … (24)
𝐿44 =
𝐷 𝑎
∗
𝐶 𝑎
𝑅𝑇
. . … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … (25)
Where
𝐶𝑖 = 𝑐𝑜𝑛𝑐𝑒𝑛𝑡𝑟𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑐ℎ𝑒𝑚𝑖𝑐𝑎𝑙 𝑠𝑝𝑒𝑐𝑖𝑒𝑠 𝑖
𝑅 𝑎𝑛𝑑 𝑇 = 𝑢𝑛𝑖𝑣𝑒𝑟𝑠𝑎𝑙 𝑔𝑎𝑠 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 , 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 𝑟𝑒𝑠𝑝𝑒𝑐𝑡𝑖𝑣𝑒𝑙𝑦
𝐷𝑖
∗
= 𝑒𝑓𝑓𝑒𝑐𝑡𝑖𝑣𝑒 𝑑𝑖𝑓𝑓𝑢𝑠𝑖𝑜𝑛 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 𝑐ℎ𝑒𝑚𝑖𝑐𝑎𝑙 𝑠𝑝𝑒𝑐𝑖𝑒𝑠 𝑖
The effective diffusion coefficient of chemical species in shale is generally greater than
its value in aqueous solutions due to tortuosity (τ longer flow path in soil) which
generally varies from 1.2 to 2.8 (Gillham et al., 1982). The relation between effective
diffusion coefficient in soil and its diffusivity in solution is given by 𝐷𝑖
∗
= 𝐷𝑖 𝑛 𝜏
(Gillham et al., 1984). The diffusivity of ions in extremely dilute solutions at room
temperature ranges from 5 x 10-10
to 20 x 10-10
(Dean 1973).
3.3.2: Coupling coefficients
Coefficients of fluid flow due to applied electric field and vice versa
𝐿12 = 𝐿21 =
𝑘 𝑒
𝑛
… … … … … … … … … … … … … … … … … … … … … … … … … … … … … (26)

39
Where
𝑘 𝑒 = 𝑒𝑙𝑒𝑐𝑡𝑟𝑜 − 𝑜𝑠𝑚𝑜𝑡𝑖𝑐 𝑝𝑒𝑟𝑚𝑒𝑎𝑏𝑖𝑙𝑖𝑡𝑦
ke is the ratio of volumetric flow rate of fluid per unit area to applied electrical gradient
and ranges from 1 x 10-9
to 10 x 10-9
m2
/ V.s (Casagrande 1983).
Interionic-diffusional coefficients
The pore fluid in shale is highly saturated and can be considered as a dilute solution.
The interactions between ions in dilute solution is very weak such that the flow of ions
of one type is not influenced by the concentration gradient of other type of ions.
𝐿34 = 𝐿43 = 0 … … … … … … … … … … … … … … … … … … … … … … … … … … … … … . (27)
Coefficients of ionic migration due to applied electrical field and vice versa
The effective ionic mobility in soil (the velocity of ionic migration in soil when electric
field is applied) is less compared to ionic mobility in free aqueous solutions due to
tortuosity (Mitchell and Yeung 1990). The similarity between effective diffusion
coefficient and effective ionic mobility was observed by Koryta (1982) and he
propounded the validity of Nernst equation for the chemical species in pore fluids of
shale to relate the effective diffusion coefficient and ionic mobility as 𝑢𝑖
∗
= 𝐷𝑖
∗
|𝑧𝑖|𝐹/
𝑅𝑇. Based on these conceptions
𝐿23 = 𝐿32 =
𝐷𝑐
∗|𝑧 𝑐|𝐹𝐶𝑐
𝑅𝑇
… … … … … … … … … … … … … … … … … … … … … … … … … . (28)
𝐿24 = 𝐿42 =
𝐷 𝑎
∗|𝑧 𝑎|𝐹𝐶 𝑎
𝑅𝑇
… … … … … … … … … … … … … … … … … … … … … … … … … . (29)
Coefficients of ionic migration due to applied hydraulic gradient and vice versa
The coefficients depend on ω (membrane efficiency or reflection coefficient or osmotic
selectivity coefficient) which has been discussed in section 2.3.

40
𝐿13 = −
𝜔𝐶𝑐(𝐿11 𝐿22 − 𝐿12 𝐿21) − 𝐿12 𝐿23
𝐿22
… … … … … … … … … … … … … … … … … . . (30)
𝐿14 = −
𝜔𝐶 𝑎(𝐿11 𝐿22 − 𝐿12 𝐿21) − 𝐿12 𝐿24
𝐿22
… … … … … … … … … … … … … … … … … . . (31)
3.4: Concentration of the ions, net charges and pH of the pore fluid in shale
Based on the principle of conservation of mass for steady state flow, an equation to
determine the concentration of chemical species ‘i’ in the pore fluid as function of space
and time has been formulated by Yeung and Datla (1994).
𝜕𝑐𝑖
𝜕𝑡
= −∇. 𝐽𝑖 −
𝜌 𝑑𝑟𝑦
𝑛
𝜕𝑆𝑖
𝜕𝑡
− 𝐺𝑖 … … … … … … … … … … … … … … … … … … … … … … … (32)
Where
𝜌 𝑑𝑟𝑦 = 𝑑𝑟𝑦 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 𝑜𝑓 𝑚𝑎𝑡𝑟𝑖𝑥
𝑆𝑖 = 𝑚𝑎𝑠𝑠 𝑜𝑓 𝑐ℎ𝑒𝑚𝑖𝑐𝑎𝑙 𝑠𝑝𝑒𝑐𝑖𝑒𝑠 𝑎𝑑𝑠𝑜𝑟𝑏𝑒𝑑 𝑜𝑛 𝑠ℎ𝑎𝑙𝑒 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝑠ℎ𝑎𝑙𝑒 𝑚𝑎𝑠𝑠
𝐺𝑖 = 𝑠𝑜𝑢𝑟𝑐𝑒/𝑠𝑖𝑛𝑘 𝑖𝑛𝑑𝑖𝑐𝑎𝑡𝑖𝑛𝑔 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑟𝑒𝑚𝑜𝑣𝑎𝑙 𝑠𝑝𝑒𝑐𝑖𝑒𝑠 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝑣𝑜𝑙𝑢𝑚𝑒
In addition to electro-kinetic flows that happen when electric current is applied in
presence of the diffuse double layer some electro chemical reactions also take place
generating pH gradient across the shale. Changes in the pH gradient affects the
adsorption and desorption phenomena that particles undergo.
The pore fluid is acidic in the region of high cationic concentration and more alkaline in
the region of high anionic concentration. The entire system of shale along with pore
fluid is electrically neutral (Snoeyink and Jenkins 1980). The electrical neutrality of the
shale media which is disturbed by applying an electric field can be restored by
formation of hydrogen and hydroxyl ions (Yeung and Datla 1994).
The equation relating the net positive and negative charges is
𝐶 𝑇 + [𝐻+] = 𝐴 𝑇 + [𝑂𝐻−] … … … … … … … … … … … … … … … … … … … … … … … … . (33)

41
Where
𝐶 𝑇, 𝐴 𝑇 = 𝑡𝑜𝑡𝑎𝑙 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑎𝑛𝑑 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 𝑐ℎ𝑎𝑟𝑔𝑒𝑠 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝑣𝑜𝑙𝑢𝑚𝑒 𝑟𝑒𝑠𝑝𝑒𝑐𝑡𝑖𝑣𝑒𝑙𝑦
[𝐻+], [𝑂𝐻−] = 𝑐𝑜𝑛𝑐𝑒𝑛𝑡𝑟𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 ℎ𝑦𝑑𝑟𝑜𝑔𝑒𝑛 𝑎𝑛𝑑 ℎ𝑦𝑑𝑟𝑜𝑥𝑦𝑙 𝑖𝑜𝑛𝑠 𝑟𝑒𝑠𝑝𝑒𝑐𝑡𝑖𝑣𝑒𝑙𝑦
The total positive and negative charges in shale are given by the following equations
based on the assumption that the activity coefficients of chemical species are 1.
𝐶 𝑇 = ∑ 𝑧 𝑐,𝑖 𝑐 𝑐,𝑖
𝑛
𝑖=1
… … … … … … … … … … … … … … … … … … … … … … … … … … … … … (34)
𝐴 𝑇 = ∑ 𝑧 𝑎,𝑗 𝑐 𝑎,𝑗
𝑛
𝑗=1
… … … … … … … … … … … … … … … … … … … … … … … … … … . … … (35)
Equation 32 is evaluated for every time step and is used to calculate the net charges
from equation 34 and 35. Equations 33 and 36 are solved simultaneously to determine
the concentration of hydrogen ions as a function of space and time.
[𝐻+] + [𝑂𝐻−] = 𝐾 𝑤 … … … … … … … … … … … … … … … … … … … … … … … … … … . (36)
The values of pH generated can be found as a function of time and space using the
following equation
𝑝𝐻 = − log10[𝐻+] … … … … … … … … … … … … … … … … … … … … … … … … … … … . . (37)
3.5: Alternative formulation of Electro-Chemico-Osmotic processes in Shale
Corapcioglu (1991) developed a set of one dimensional governing equations for the
explaining the simultaneous flows of water, chemical species and electric current in a
porous medium based on the macroscopic conservation of mass equations and principle
of continuum, an alternative to the previous approach based on the irreversible coupled
flows. Some of the important equations from his literature are

42
Mass balance equation for the entire water phase
𝑘
µ 𝑓
𝜕2
𝑝
𝜕𝑥2
+ 𝑘ℎ𝑐
𝜕2
𝐶𝑠
𝜕𝑥2
+ 𝑘 𝑒
𝜕2
𝜑
𝜕𝑥2
+
𝐷
𝜌 𝑓
𝜕2
𝜌 𝑓
𝜕𝑥2
= (𝛼 + 𝑛𝛽)
𝜕𝑝
𝜕𝑡
… … … … … … … … … … … . (38)
Mass balance equation for the chemical species in water phase with an assumption
of zero net mass transfer rate due to reactions
𝜌 𝑓 𝐶𝑠 𝑘
µ 𝑓
𝜕2
𝑝
𝜕𝑥2
+ 𝜌 𝑓 𝐶𝑠 𝑘ℎ𝑐
𝜕2
𝐶𝑠
𝜕𝑥2
+ 𝜌 𝑓 𝐶𝑠 𝑘 𝑒
𝜕2
𝜑
𝜕𝑥2
+ 𝐷
𝜕2
𝜌 𝑓 𝐶𝑠
𝜕𝑥2
= 𝜌 𝑓 𝐶𝑠 𝛼
𝜕𝑝
𝜕𝑡
+ 𝑛
𝜕𝜌 𝑓 𝐶𝑠
𝜕𝑡
… . (39)
One dimensional model for current flow
𝜎ℎ
𝑔𝜌 𝑓
𝜕2
𝑝
𝜕𝑥2
+ 𝜎𝑒
𝜕2
𝜑
𝜕𝑥2
𝑘 𝑒𝑐
𝜕2
𝐶𝑠
𝜕𝑥2
= 0 … … … … … … … … … … … … … … … … … … … … … … (40)
Equation of state
𝜌 𝑓 = 𝜌 𝑓0 𝑒𝑥𝑝 [𝛽 𝑝(𝑝 − 𝑝0) + ∑ 𝑣𝑖(𝑚𝑖 − 𝑚𝑖0)
𝑖=𝑤,𝑠
] … … … … … … … … … … … … … … . (41)
Pore fluid viscosity
µ 𝑓 = 1.002 ∗ 10−3
(1 + 0.4819𝐶𝑠 + 0.2774𝐶𝑠
2
+ 0.7814𝐶𝑠
3
) … … … … … … … … . (42)
Where
𝐶𝑠 = 𝑚𝑎𝑠𝑠 𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑐ℎ𝑒𝑚𝑖𝑐𝑎𝑙 𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡 𝑖𝑛 𝑤𝑎𝑡𝑒𝑟 𝑝ℎ𝑎𝑠𝑒
𝜑 = 𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐𝑎𝑙 𝑝𝑜𝑡𝑒𝑛𝑡𝑖𝑎𝑙
µ 𝑓 , 𝜌 𝑓 = 𝑣𝑖𝑠𝑐𝑜𝑠𝑖𝑡𝑦 𝑎𝑛𝑑 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 𝑜𝑓 𝑤𝑎𝑡𝑒𝑟 𝑝ℎ𝑎𝑠𝑒 𝑟𝑒𝑠𝑝𝑒𝑐𝑡𝑖𝑣𝑒𝑙𝑦
𝐷 = 𝑚𝑒𝑐ℎ𝑎𝑛𝑖𝑐𝑎𝑙 𝑑𝑖𝑠𝑝𝑒𝑟𝑠𝑖𝑜𝑛 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡
𝑘 = 𝑝𝑒𝑟𝑚𝑒𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝑝𝑜𝑟𝑜𝑢𝑠 𝑚𝑒𝑑𝑖𝑢𝑚
𝑘 𝑒 = 𝑐𝑜𝑒𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 𝑒𝑙𝑒𝑐𝑡𝑟𝑜 𝑜𝑠𝑚𝑜𝑡𝑖𝑐 𝑝𝑒𝑟𝑚𝑒𝑎𝑏𝑖𝑙𝑖𝑡𝑦
𝑘 𝑒𝑐 = 𝑚𝑖𝑔𝑟𝑎𝑡𝑖𝑜𝑛 𝑝𝑜𝑡𝑒𝑛𝑡𝑖𝑎𝑙 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡
𝑘ℎ𝑐 = 𝑐ℎ𝑒𝑚𝑖𝑐𝑜 𝑜𝑠𝑚𝑜𝑡𝑖𝑐 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡

43
𝜎ℎ, 𝜎𝑒 = 𝑠𝑡𝑟𝑒𝑎𝑚𝑖𝑛𝑔 𝑐𝑢𝑟𝑟𝑒𝑛𝑡 𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑣𝑖𝑡𝑦, 𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐𝑎𝑙 𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑣𝑖𝑡𝑦 𝑟𝑒𝑠𝑝𝑒𝑐𝑡𝑖𝑣𝑒𝑙𝑦
𝑝, 𝑔 = 𝑝𝑜𝑟𝑒 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑎𝑛𝑑 𝑔𝑟𝑎𝑣𝑖𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑟𝑒𝑠𝑝𝑒𝑐𝑡𝑖𝑣𝑒𝑙𝑦
𝛼 = 𝑚𝑎𝑡𝑟𝑖𝑥 𝑐𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑖𝑏𝑖𝑙𝑖𝑡𝑦 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡
𝑛, 𝛽 = 𝑝𝑜𝑟𝑜𝑠𝑖𝑡𝑦 𝑎𝑛𝑑 𝑤𝑎𝑡𝑒𝑟 𝑝ℎ𝑎𝑠𝑒 𝑐𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑖𝑏𝑖𝑙𝑖𝑡𝑦
However the formulations framed for clay rich soil by Corapcioglu cannot be used
efficiently as the transport coefficient cannot be determined readily (Yeung and Datla
1995).

44
Chapter 4: Governing equations for modeling Shale consolidation
4.1: Transport Equations
Since the time Fourier formulated a linear relationship between the heat flow and the
temperature gradient, many researchers have tried to relate the flows and the driving
forces. However some researchers like Rouss, tried to relate the flows not only with the
driving forces but also with other gradients present in the system after observing the
flow of volume along with charges when electric field was applied and also same
happened with application of hydrostatic pressure. Similar phenomena was observed
even in Seebeck effect and Peltier effect, thus proving that in a slow process any flow is
related directly and linearly to both conjugated and non-conjugated forces. After Lord
Rayleigh formulated an equation relating the mechanical flows with all the mechanical
forces available in the system, Onsager extended this equation to accommodate the
thermodynamic forces and flows and named them as phenomenological equations as
discussed in chapter 3. The equations 18, 19, 20 and 21 formulated by Yeung and
Mitchell (1993) are the transport equations and are repeated below.
𝑞𝑖 = 𝐿11
𝜕(−𝑝)
𝜕𝑥𝑖
+ 𝐿12
𝜕(−𝜓)
𝜕𝑥𝑖
+ 𝐿13
𝑅𝑇
𝑚 𝑜
𝑎
𝜕(−𝑚 𝑎
)
𝜕𝑥𝑖
+ 𝐿14
𝑅𝑇
𝑚 𝑜
𝑐
𝜕(−𝑚 𝑐
)
𝜕𝑥𝑖
… … … … … … (43)
𝐼𝑖 = 𝐿21
𝜕(−𝑝)
𝜕𝑥𝑖
+ 𝐿22
𝜕(−𝜓)
𝜕𝑥𝑖
+ 𝐿23
𝑅𝑇
𝑚 𝑜
𝑎
𝜕(−𝑚 𝑎
)
𝜕𝑥𝑖
+ 𝐿24
𝑅𝑇
𝑚 𝑜
𝑐
𝜕(−𝑚 𝑐
)
𝜕𝑥𝑖
… … … … … … (44)
𝐽𝑖
𝑎,𝑑
= 𝐿31
𝜕(−𝑝)
𝜕𝑥𝑖
+ 𝐿32
𝜕(−𝜓)
𝜕𝑥𝑖
+ 𝐿33
𝑅𝑇
𝑚 𝑜
𝑎
𝜕(−𝑚 𝑎
)
𝜕𝑥𝑖
+ 𝐿34
𝑅𝑇
𝑚 𝑜
𝑐
𝜕(−𝑚 𝑐
)
𝜕𝑥𝑖
… … … … … (45)
𝐽𝑖
𝑐,𝑑
= 𝐿41
𝜕(−𝑝)
𝜕𝑥𝑖
+ 𝐿42
𝜕(−𝜓)
𝜕𝑥𝑖
+ 𝐿43
𝑅𝑇
𝑚 𝑜
𝑎
𝜕(−𝑚 𝑎
)
𝜕𝑥𝑖
+ 𝐿44
𝑅𝑇
𝑚 𝑜
𝑐
𝜕(−𝑚 𝑐
)
𝜕𝑥𝑖
… … … … … (46)

45
4.2: Driving force equation
Model is developed based on the following assumptions:
1. The pore fluid (NaCl solution) behaves like an electrolyte with a solvent and one
type of cations and anions.
2. The porous medium might be charged or neutral but the entire system is
electrically neutral.
The electrochemical potential of each ionic species in shale which is controlled by the
electrostatic potential field in addition to the existing chemical potential is given by
Katchalsky and Curran (1965).
µ 𝑟̃ = 𝑉 𝑟
𝑝 + 𝑅𝑇𝑙𝑛[𝑎 𝑟] + 𝑧 𝑟
𝐹𝜓 = 𝑉 𝑟
𝑝 + 𝑅𝑇𝑙𝑛[𝜍 𝑟
𝑚 𝑟] + 𝑧 𝑟
𝐹𝜓 … … … … … … … … . (47)
µ 𝑟̃ = 𝑡ℎ𝑒 𝑒𝑙𝑒𝑐𝑡𝑟𝑜 𝑐ℎ𝑒𝑚𝑖𝑐𝑎𝑙 𝑝𝑜𝑡𝑒𝑛𝑡𝑖𝑎𝑙 𝑜𝑓 𝑡ℎ𝑒 𝑓𝑙𝑢𝑖𝑑 𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡𝑠 ′𝑟′
𝑉 𝑟
= 𝑝𝑎𝑟𝑡𝑖𝑎𝑙 𝑚𝑜𝑙𝑎𝑟 𝑣𝑜𝑙𝑢𝑚𝑒
𝑝 = ℎ𝑦𝑑𝑟𝑜𝑠𝑡𝑎𝑡𝑖𝑐 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒
𝑅 𝑎𝑛𝑑 𝑇 = 𝑢𝑛𝑖𝑣𝑒𝑟𝑠𝑎𝑙 𝑔𝑎𝑠 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑎𝑛𝑑 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 𝑟𝑒𝑠𝑝𝑒𝑐𝑡𝑖𝑣𝑒𝑙𝑦
𝑎 𝑟
= 𝑐ℎ𝑒𝑚𝑖𝑐𝑎𝑙 𝑎𝑐𝑡𝑖𝑣𝑖𝑡𝑦, 𝑎 𝑝𝑟𝑜𝑑𝑢𝑐𝑡 𝑜𝑓 𝑎𝑐𝑡𝑖𝑣𝑖𝑡𝑦 𝑐𝑜𝑒𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝜍 𝑟
& 𝑚𝑜𝑙𝑒 𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛 𝑚 𝑟
𝐹 𝑎𝑛𝑑 𝜓 = 𝐹𝑎𝑟𝑎𝑑𝑎𝑦 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑎𝑛𝑑 𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐 𝑝𝑜𝑡𝑒𝑛𝑡𝑖𝑎𝑙 𝑟𝑒𝑠𝑝𝑒𝑐𝑡𝑖𝑣𝑒𝑙𝑦
𝑧 𝑟
= 𝑣𝑎𝑙𝑒𝑛𝑐𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑖𝑜𝑛𝑖𝑐 𝑠𝑝𝑒𝑐𝑖𝑒𝑠
4.3: Equations related to physical structure of shale
As mentioned in section 2.6.1, earlier equations developed to define the mechanics in
physical structure of soil based on Biot’s theory ignores the electrical and chemical
potentials developed in soil. However these equations can facilitate only poroelastic
medium unlike soil where electrochemical potentials play a significant role in fluid and

46
ion fluxes. Hence Biot’s poroelastic equations have to be extended to be compatible to
soil.
The area under the curve gives the amount of work or energy required per volume to
deform the specimen until it fails. The area under the curve is a product of stress and
strain which is in units of work. The movement of ions and charges of the electrolyte
solution inside the porous medium absorbs some potential energy and weakens the
porous medium. Coussy (2004) derived an expression for the change in free energy
density of soil which is porous and saturated with salt solution under isothermal
conditions and negligible deformation.
𝑑𝑊 = 𝜎𝑖𝑗 𝑑𝜀𝑖𝑗 − ∑ 𝑀 𝑟
𝑑µ 𝑟̃
𝑟=𝑎,𝑐,𝑓
… … … … … … … … … … … … … … … … … … … … … … . (48)
Figure 16: Stress Vs strain curve for ductile material (Ratner et
al., 2004)

47
Where
𝜎𝑖𝑗& 𝜀𝑖𝑗 = 𝑠𝑡𝑟𝑒𝑠𝑠 𝑡𝑒𝑛𝑠𝑜𝑟 & 𝑠𝑡𝑟𝑎𝑖𝑛 𝑡𝑒𝑛𝑠𝑜𝑟 𝑟𝑒𝑠𝑝𝑒𝑐𝑡𝑖𝑣𝑒𝑙𝑦
𝑀 𝑟
= 𝑚𝑎𝑠𝑠 𝑜𝑓 𝑓𝑙𝑢𝑖𝑑 𝑠𝑝𝑒𝑐𝑖𝑒𝑠 𝑖𝑛 𝑚𝑜𝑙𝑒𝑠 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝑣𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑟𝑜𝑐𝑘
The gradients of all the electrochemical potentials of the components in the pore fluid
are related to each other through Gibbs-Duhem equation (Katchalsky and Curran 1965).
−𝜙𝑑𝑝 + ∑ 𝑀 𝑟
𝑑µ 𝑟̃
𝑟=𝑎,𝑐,𝑓
= 0 … … … … … … … … … … … … … … … … … … … … … … … . . (49)
Assuming the pore space is completely saturated with the fluid and hence the porosity is
given by the following equation (Nguyen Vinh et al., 2010).
𝜙 = 𝑉 𝑠𝑜𝑙
= ∑ 𝑉 𝑟
𝑀 𝑟
𝑟=𝑎,𝑐,𝑓
… … … … … … … … … … … … … … … … … … … … … … … … . (50)
Substituting equation 48 in 49 gives
𝑑𝑊 = 𝜎𝑖𝑗 𝑑𝜀𝑖𝑗 − 𝜙𝑑𝑝 … … … … … … … … … … … … … … … … … … … … … … … … … … . (51)
Equation 51 is compared with the constitutive equations developed by Coussy (2004)
developed for an isotropic medium
𝑑𝜎𝑖𝑗 = 2𝐺𝑑𝜀𝑖𝑗 +
2𝐺𝜈
1 − 2𝜈
𝑑𝜀 𝑘𝑘 𝛿𝑖𝑗 + 𝛼𝑑𝑝𝛿𝑖𝑗 … … … … … … … … … … … … … … … … … (52)
𝑑𝜙 = −𝛼𝑑𝜀 𝑘𝑘 +
1
𝐾 𝜙
𝑑𝑝 … … … … … … … … … … … … … … … … … … … … … … … … … . (53)
Where
𝛼 = 𝑏𝑖𝑜𝑡′
𝑠 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑟 𝑝𝑜𝑟𝑒 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡
𝜀 𝑘𝑘 = 𝑣𝑜𝑙𝑢𝑚𝑒𝑡𝑟𝑖𝑐 𝑠𝑡𝑟𝑎𝑖𝑛 = 𝑠𝑢𝑚 𝑜𝑓 𝑙𝑖𝑛𝑒𝑎𝑟 𝑠𝑡𝑟𝑎𝑖𝑛
𝐺&𝜈 = 𝑠ℎ𝑒𝑎𝑟 𝑚𝑜𝑑𝑢𝑙𝑢𝑠 𝑎𝑛𝑑 𝑝𝑜𝑖𝑠𝑠𝑜𝑛 𝑟𝑎𝑡𝑖𝑜 𝑟𝑒𝑠𝑝𝑒𝑐𝑡𝑖𝑣𝑒𝑙𝑦
𝐾 𝜙 = 𝑝𝑜𝑟𝑒 𝑐𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑖𝑏𝑖𝑙𝑖𝑡𝑦
𝛿𝑖𝑗 = 𝑘𝑟𝑜𝑛𝑒𝑐𝑘𝑒𝑟 𝑑𝑒𝑙𝑡𝑎, 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒𝑠, 1 𝑖𝑓 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒𝑠 𝑎𝑟𝑒 𝑒𝑞𝑢𝑎𝑙 𝑒𝑙𝑠𝑒 0

48
Based on the statement in equation 50, the changes in porosity in equation 53 is
replaced by the total fluid content in the rock matrix which is given by
𝑑𝜁 =
𝑑𝑀 𝑠𝑜𝑙
𝜌 𝑜
𝑠𝑜𝑙 =
𝑑(𝜙𝜌 𝑠𝑜𝑙
)
𝜌 𝑜
𝑠𝑜𝑙 = 𝑑𝜙 + 𝜙 𝑜
𝑑𝜌 𝑠𝑜𝑙
𝜌 𝑜
𝑠𝑜𝑙 … … … … … … … … … … … … … … … … … (54)
Where
𝑀 𝑠𝑜𝑙
= 𝑡𝑜𝑡𝑎𝑙 𝑚𝑎𝑠𝑠 𝑜𝑓 𝑓𝑙𝑢𝑖𝑑 𝑖𝑛 𝑝𝑜𝑟𝑒𝑠 𝑖𝑛 𝑚𝑜𝑙𝑒𝑠 = ∑ 𝑀 𝑟
𝑟=𝑎,𝑐,𝑓
𝜌 𝑜
𝑠𝑜𝑙
= 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 𝑜𝑓 𝑡ℎ𝑒 𝑓𝑙𝑢𝑖𝑑 𝑖𝑛 𝑝𝑜𝑟𝑒𝑠 𝑖𝑛 𝑚𝑜𝑙𝑒𝑠/𝑚3
𝑑𝜌 𝑠𝑜𝑙
𝜌 𝑜
𝑠𝑜𝑙 =
1
𝐾𝑓
𝑑𝑝 … … … … … … … … … … … … … … … … … … … … … … … … … … … … … . (55)
By substituting equation 55 and 53 in 54 we get
𝑑𝜁 = −𝛼𝑑𝜀 𝑘𝑘 +
1
𝐾 𝜙
𝑑𝑝 + 𝜙 𝑜
1
𝐾𝑓
𝑑𝑝 … … … … … … … … … … … … … … … … … … … … . (56)
The coefficients of dp are grouped together under ground water storage coefficient M as
𝑀 =
1
𝐾 𝜙
+ 𝜙 𝑜
1
𝐾𝑓
… … … … … … … … … … … … … … … … … … … … … … … … … … … … . (57)
Like equation 54 the fluid content changes in each species of the fluid is given by
𝑑𝜁 𝑟
=
𝑑𝑀 𝑟
𝜌 𝑜
𝑠𝑜𝑙 =
𝑑(𝑚 𝑟
𝑀 𝑠𝑜𝑙
)
𝜌 𝑜
𝑠𝑜𝑙 = 𝑚 𝑜
𝑟
𝑑𝜁 + 𝜙 𝑜 𝑑𝑚 𝑟
… … … … … … … … … … … … … … … . . (58)
Where
𝑚 𝑟
= 𝑚𝑜𝑙𝑒 𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑓𝑙𝑢𝑖𝑑 𝑠𝑝𝑒𝑐𝑖𝑒𝑠
Substituting equations 56 and 57 in 58 gives
𝑑𝜁 𝑟
= 𝑚 𝑜
𝑟
(−𝛼𝑑𝜀 𝑘𝑘 +
1
𝑀
𝑑𝑝) + 𝜙 𝑜 𝑑𝑚 𝑟
… … … … … … … … … … … … … … … … … … . (59)
Equation 59 can be rewritten for anionic and cationic species as

49
𝑑𝜁 𝑎
= 𝑚 𝑜
𝑎
1
𝑀
𝑑𝑝) + 𝜙 𝑜 𝑑𝑚 𝑎
… … … … … … … … … … … … … … … … … … . (60)
𝑑𝜁 𝑐
= 𝑚 𝑜
𝑐
1
𝑀
𝑑𝑝) + 𝜙 𝑜 𝑑𝑚 𝑐
… … … … … … … … … … … … … … … … … … . (61)
Equations 52, 56, 60 and 61 are the important equations pertaining to the physical
structure of the porous medium. It is evident from equations 52 and 56 that the changes
in stress and fluid content of the porous medium are due to pore pressure changes and
are uninfluenced by the electrochemical potentials.
4.4: Governing equations of the porous medium
4.4.1: Strain displacement equation:
𝜀𝑖𝑗 = 0.5 ∗ (
𝜕𝑢𝑖
𝜕𝑥𝑗
+
𝜕𝑢𝑗
𝜕𝑥𝑖
) … … … … … … … … … … … … … … … … … … … … … … … … … . (62)
Where
𝑢𝑖 & 𝑢𝑗 = 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑛𝑡 𝑣𝑒𝑐𝑡𝑜𝑟𝑠 𝑖𝑛 𝑢𝑛𝑖𝑡 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛𝑠 𝑖, 𝑗 𝑟𝑒𝑠𝑝𝑒𝑐𝑡𝑖𝑣𝑒𝑙𝑦
4.4.2: Semi-static stress equilibrium equation:
𝜕𝜎𝑖𝑗
𝜕𝑥𝑖
= 0 … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … (63)
4.4.3: Mass conservation equations:
𝜕𝜁
𝜕𝑡
= −
𝜕𝑞𝑖
𝜕𝑥𝑖
… … … … … … … … … … … … … … … … … … … … … … … … … … … … … … . . (64)
For ionic species the volumetric flux is a product of molar volume and absolute flux
𝑞𝑖
𝑟
= 𝑉𝑜
𝑓
𝐽𝑖
𝑟
… … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … . (65)
Equation 64 is modified for ionic species by incorporating equation 65 and including
the advectional transport of ions (movement of ions along with horizontal movement of
fluid) when relative diffusional fluxes are considered is as following

50
𝑑𝜁 𝑎
= −𝑉𝑜
𝑓 𝜕𝐽𝑖
𝑎
𝜕𝑥𝑖
= −
𝜕
𝜕𝑥𝑖
(𝑉𝑜
𝑓
𝐽𝑖
𝑎,𝑑
+ 𝑚 𝑎
𝑞𝑖) … … … … … … … … … … … … … … … … … . (66)
𝑑𝜁 𝑐
= −𝑉𝑜
𝑓 𝜕𝐽𝑖
𝑐
𝜕𝑥𝑖
= −
𝜕
𝜕𝑥𝑖
(𝑉𝑜
𝑓
𝐽𝑖
𝑐,𝑑
+ 𝑚 𝑐
𝑞𝑖) … … … … … … … … … … … … … … … … … . (67)
4.4.4: Conservation of charges
Since the entire matrix with the pore fluid is electrically neutral without any charge
development, the field generated is assumed to be electro static such that the total
charge density is constant and does not vary with time (Corapcioglu 1991). Hence the
charge conservation equation is given by
𝜕𝜌𝑒
𝜕𝑡
= −
𝜕𝐼𝑖
𝜕𝑥𝑖
… … … … … … … … … … … … … … … … … … … … … … … … … … … … … … . (68)
The electrostatic condition and equation 68 makes 1st
derivative of equation 44 equal to
0.
𝐿21 𝛻2
𝑝 + 𝐿22 𝛻2
𝜓 + 𝐿23
𝑅𝑇
𝑚 𝑜
𝑎 𝛻2
𝑚 𝑎
+ 𝐿24
𝑅𝑇
𝑚 𝑜
𝑐 𝛻2
𝑚 𝑐
= 0 … … … … … … … … … … … (69)
The above equations are the governing equation of the porous medium saturated with
electrolytic solution including its electro chemical potentials.
The phenomenological coefficients/ transport coefficients depend on 5 field parameters
in case of poroelastic medium unlike porochemoelectroelastic medium which depends
on 10 field parameters.
4.5: Equations applied for field purposes
Combining equations 52 and 63, and Kronecker delta =1 as i = j we get
2𝐺
𝜕𝜀𝑖𝑗
𝜕𝑥𝑖
+
2𝐺𝜈
1 − 2𝜈
𝜕𝜀 𝑘𝑘
𝜕𝑥𝑗
+ 𝛼
𝜕𝑝
𝜕𝑥𝑗
= 0 … … … … … … … … … … … … … … … … … … … … . (70)
Differentiating equation 70 with respect to 𝑥𝑗

51
2𝐺
𝜕2
𝜀𝑖𝑗
𝜕𝑥𝑗 𝜕𝑥𝑖
+
2𝐺𝜈
1 − 2𝜈
𝜕2
𝜀 𝑘𝑘
𝜕𝑥𝑗 𝜕𝑥𝑗
+ 𝛼
𝜕2
𝑝
= 0 … … … … … … … … … … … … … … … … (71)
Differentiating equation 62 with respect to 𝑥𝑖 & 𝑥𝑗
𝜕2
𝜀𝑖𝑗
=
1
2
𝜕2
(
𝜕𝑢𝑖
𝜕𝑥𝑗
+
𝜕𝑢𝑗
𝜕𝑥𝑖
) =
𝜕2
(
𝜕𝑢𝑖
𝜕𝑥𝑖
) =
𝜕2
𝜀 𝑘𝑘
= 𝛻2
𝜀 𝑘𝑘 … … … … … (72)
The above strain equation can be further simplified by simultaneously solving equations
71 and 72 as following
𝛻2
(𝜀 𝑘𝑘 +
𝜂
𝐺
𝑝) = 0 … … … … … … … … … … … … … … … … … … … … … … … … … … … . (73)
Nguyen Vinh et al., 2010 derived the diffusion equations that can be used for practical
purposes by using equations 56, 60 61 & 43-46 in 64, 66, and 67.
−𝛼
𝜕𝜀 𝑘𝑘
𝜕𝑡
+
1
𝑀
𝜕𝑝
𝜕𝑡
= 𝐿11 𝛻2
𝑝 + 𝐿12 𝛻2
𝜓 + 𝐿13
𝑅𝑇
𝑚 𝑜
𝑎 𝛻2
𝑚 𝑎
+ 𝐿14
𝑅𝑇
𝑚 𝑜
𝑐 𝛻2
𝑚 𝑐
… … … … (74)
𝑚 𝑜
𝑎
(−𝛼
𝜕𝜀 𝑘𝑘
𝜕𝑡
+
1
𝑀
𝜕𝑝
𝜕𝑡
) + 𝜙 𝑜
𝜕𝑚 𝑎
𝜕𝑡
= 𝑉𝑜
𝑓
(𝐿31 𝛻2
𝑝 + 𝐿32 𝛻2
𝜓 + 𝐿33
𝑅𝑇
𝑚 𝑜
𝑎 𝛻2
𝑚 𝑎
+ 𝐿34
𝑅𝑇
𝑚 𝑜
𝑐 𝛻2
𝑚 𝑐
) + 𝑚 𝑜
𝑎
𝜕𝑞𝑖
𝜕𝑥𝑖
+ 𝑞𝑖
𝜕𝑚 𝑎
𝜕𝑥𝑖
… … … … … … … … … … … … … … … … … … … … … … … … … . (75)
𝑚 𝑜
𝑐
(−𝛼
𝜕𝜀 𝑘𝑘
𝜕𝑡
+
1
𝑀
𝜕𝑝
𝜕𝑡
) + 𝜙 𝑜
𝜕𝑚 𝑐
𝜕𝑡
= 𝑉𝑜
𝑓
(𝐿41 𝛻2
𝑝 + 𝐿42 𝛻2
𝜓 + 𝐿43
𝑅𝑇
𝑚 𝑜
𝑎 𝛻2
𝑚 𝑎
+ 𝐿44
𝑅𝑇
𝑚 𝑜
𝑐 𝛻2
𝑚 𝑐
) + 𝑚 𝑜
𝑐
𝜕𝑞𝑖
𝜕𝑥𝑖
+ 𝑞𝑖
𝜕𝑚 𝑐
𝜕𝑥𝑖
… … … … … … … … … … … … … … … … … … … … … … … … … . (76)
However based on the conclusions of Yeung and Datla (1994) that the hydraulic
diffusion is small compared to the ionic diffusion, the advective transport terms (last 2

52
terms in equations 75 and 76) are ignored and the electrostatic potential field is also
disregarded based on equation 69 to obtain equations with non -symmetric coefficients
to form a matrix.
−𝛼
𝜕𝜀 𝑘𝑘
𝜕𝑡
+
1
𝑀
𝜕𝑝
𝜕𝑡
= 𝐷11 𝛻2
𝑝 + 𝐷12 𝛻2
𝑝 𝑎
+ 𝐷13 𝛻2
𝑝 𝑐
… … … … … … … … … … … … … (77)
𝑚 𝑜
𝑎
(−𝛼
𝜕𝜀 𝑘𝑘
𝜕𝑡
+
1
𝑀
𝜕𝑝
𝜕𝑡
) +
𝜙 𝑜 𝑉𝑜
𝑓
𝑅𝑇
𝜕𝑝 𝑎
𝜕𝑡
= 𝐷21 𝛻2
𝑝 + 𝐷22 𝛻2
𝑝 𝑎
+ 𝐷23 𝛻2
𝑝 𝑐
… … … … . (78)
𝑚 𝑜
𝑐
(−𝛼
𝜕𝜀 𝑘𝑘
𝜕𝑡
+
1
𝑀
𝜕𝑝
𝜕𝑡
) +
𝜙 𝑜 𝑉𝑜
𝑓
𝑅𝑇
𝜕𝑝 𝑐
𝜕𝑡
= 𝐷31 𝛻2
𝑝 + 𝐷32 𝛻2
𝑝 𝑎
+ 𝐷33 𝛻2
𝑝 𝑐
… … … … . (79)
Where
𝑝 =
𝑅𝑇
𝑉
𝑚
From the above equations it is evident that the electrical transport coefficients are
considered only if there is a streaming flux due to the formation of diffuse double layer.
[
𝐷11 𝐷12 𝐷13
𝐷21 𝐷22 𝐷23
𝐷31 𝐷32 𝐷33]
=
[
𝜅 −𝜒𝜅 −𝜒𝜅
𝑚 𝑜
𝑎
(1 − 𝜒)𝜅 𝐷𝑒𝑓𝑓
𝑎
(
𝑉𝑜
𝑓
𝑅𝑇
) − 𝑚 𝑜
𝑎
(1 − 𝜒)𝜒𝜅 −𝑚 𝑜
𝑎
(1 − 𝜒)𝜒𝜅
𝑚 𝑜
𝑐
(1 − 𝜒)𝜅 −𝑚 𝑜
𝑐
(1 − 𝜒)𝜒𝜅 𝐷𝑒𝑓𝑓
𝑐
(
𝑉𝑜
𝑓
𝑅𝑇
) − 𝑚 𝑜
𝑐
(1 − 𝜒)𝜒𝜅
]
… . (80)
The equations developed so far in this chapter have been used to develop a
mathematical model for drilling an inclined wellbore in chemically active porous
medium.

53
Chapter 5: Mathematical model
5.1: Initial wellbore conditions and stress transformations
An inclined wellbore is drilled such that the axis of the wellbore is inclined to the in-situ
stresses. The Cartesian coordinate axes x’, y’, z’ overlap the principle axes of the far
field stresses Sx’
, Sy’, Sz’ as shown in figure 17.
The deviation of the wellbore from the vertical axis is called as inclination angle 𝜑 𝑦 and
the angle the wellbore makes with the axis along maximum horizontal in-situ stress is
called as azimuth angle 𝜑 𝑧.
The local coordinates are considered such that the axis of the wellbore is parallel to the
vertical axis z of the coordinate system xyz as shown in figure 18.
Figure 17: Inclined wellbore with axes inclined to in-
situ stresses (Adapted from Cui et al., 1997)

54
The in-situ stresses can be transformed from the Cartesian coordinates to local
coordinates. The transformation is described mathematically by direction cosines as
following (Fjaer et al 2008).
{
𝑆 𝑥
𝑆 𝑦
𝑆𝑧
𝑆 𝑥𝑦
𝑆 𝑦𝑧
𝑆 𝑥𝑧}
=
[
𝑙 𝑥𝑥′
2
𝑙 𝑦𝑥′
2
𝑙 𝑧𝑥′
2
𝑙 𝑥𝑥′ 𝑙 𝑦𝑥′
𝑙 𝑦𝑥′ 𝑙 𝑧𝑥′
𝑙 𝑧𝑥′ 𝑙 𝑥𝑥′
𝑙 𝑥𝑦′
2
𝑙 𝑦𝑦′
2
𝑙 𝑧𝑦′
2
𝑙 𝑥𝑦′ 𝑙 𝑦𝑦′
𝑙 𝑦𝑦′ 𝑙 𝑧𝑦′
𝑙 𝑧𝑦′ 𝑙 𝑥𝑦′
𝑙 𝑥𝑧′
2
𝑙 𝑦𝑧′
2
𝑙 𝑧𝑧′
2
𝑙 𝑥𝑧′ 𝑙 𝑧𝑧′
𝑙 𝑦𝑧′ 𝑙 𝑧𝑧′
𝑙 𝑧𝑧′ 𝑙 𝑥𝑧′]
[
𝑆 𝑥′
𝑆 𝑦′
𝑆𝑧′]
… … … … … … … … … … … … … … … … . … (81)
Where 𝑙𝑖𝑗′ is the cosine of the angle between the axes ‘i’ and ‘j’ and is given as
[
𝑙 𝑥𝑥′ 𝑙 𝑥𝑦′ 𝑙 𝑥𝑧′
𝑙 𝑦𝑥′ 𝑙 𝑦𝑦′ 𝑙 𝑦𝑧′
𝑙 𝑧𝑥′ 𝑙 𝑧𝑦′ 𝑙 𝑧𝑧′
] = [
𝑐𝑜𝑠𝜑 𝑧 𝑐𝑜𝑠𝜑 𝑦 𝑠𝑖𝑛𝜑 𝑧 𝑐𝑜𝑠 𝜑 𝑦 −𝑠𝑖𝑛𝜑 𝑦
−𝑠𝑖𝑛𝜑 𝑧 𝑐𝑜𝑠𝜑 𝑧 0
𝑐𝑜𝑠𝜑𝑧 𝑠𝑖𝑛𝜑 𝑦 𝑠𝑖𝑛𝜑 𝑧 𝑠𝑖𝑛𝜑 𝑦 𝑐𝑜𝑠𝜑 𝑦
] … … … … … … … … . . (82)
Figure 18: Physical model of inclined wellbore
in local coordinate system (adapted from Cui et
al., 1997)

55
5.2: Initial conditions of the formation before drilling
As mentioned in earlier chapters, the shale formation is negatively charged and when
the matrix is considered along with the pore fluid, the entire medium is electrically
neutral such that it satisfies the following condition
𝑧 𝑐
𝑚 𝑜
𝑐
+ 𝑧 𝑎
𝑚 𝑜
𝑎
+ 𝑧 𝑓𝑐
𝑚 𝑜
𝑓𝑐
= 0 … … … … … … … … … … … … … … … … … … … … … … … (83)
Where 𝑧 𝑐
is valence of cations and is positive and 𝑧 𝑎
, 𝑧 𝑓𝑐
are valences of the anions and
fixed charges of formation respectively which are negative values.
The mole fractions of cations, anions and the fixed charges on the formation cannot be
determined easily. It demands a complete knowledge of the cation exchange capacity
and pore fluid water activity which is equal to the water activity of the fluid in
Figure 19: Initial and after drilling conditions of shale
(adapted from Nguyen et al 2010)

56
equilibrium with it on other side of the porous medium (Overbeek 1956). The
electrochemical potentials of the initial ionic species in pore fluid are given as
µ̃ 𝑒𝑞
𝑓
= µ̃ 𝑜
𝑓
µ̃ 𝑒𝑞
𝑐
= µ̃ 𝑜
𝑐
µ̃ 𝑒𝑞
𝑓
= µ̃ 𝑜
𝑓
… … … … … … … … … … … … … … … … (84)
Equation 84 can be extended for a mud shale interface which avoids infinite ionic fluxes
by maintaining a continuous electrochemical potential for all fluid species.
µ̃ 𝑚𝑢𝑑
𝑓
= µ̃ 𝑠ℎ𝑎𝑙𝑒
𝑓
µ̃ 𝑚𝑢𝑑
𝑐
𝑐
µ̃ 𝑚𝑢𝑑
𝑓
𝑓
… … … … … … … … … … … (85)
Simultaneously solving equations 47 and 84, using the electrical neutrality of the outer
fluid which is in equilibrium and also by ignoring the electrical potential terms we get
expressions to determine the initial mole fractions of the cations and anions for a
monovalent salt which is further simplified by ignoring the negligible pressure
exponential term (Nguyen et al., 2010).
𝑚 𝑜
𝑐
𝑚 𝑜
𝑎
= (𝑚 𝑒𝑞
𝑠
)2
… … … … … … … … … … … … … … … … … … … … … … … … … … … … . (86)
Where
𝑚 𝑒𝑞
𝑠
= 𝑚𝑜𝑙𝑒 𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑠𝑜𝑙𝑢𝑡𝑒 𝑖𝑛 𝑓𝑙𝑢𝑖𝑑 𝑖𝑛 𝑒𝑞𝑢𝑖𝑙𝑖𝑏𝑟𝑖𝑢𝑚 𝑤𝑖𝑡ℎ 𝑝𝑜𝑟𝑒 𝑓𝑙𝑢𝑖𝑑
The mole fractions of the cations and the anions in the shale formation are given by the
following equations
𝑚 𝑠ℎ𝑎𝑙𝑒
𝑎
= 0.5 (−𝑚 𝑓𝑐
+ √(𝑚 𝑓𝑐)2 + 4(𝑚 𝑚𝑢𝑑
𝑠
)2) … . . … … … … … … … … … … … … … (87)
𝑐
= 0.5 (𝑚 𝑓𝑐
+ √(𝑚 𝑓𝑐)2 + 4(𝑚 𝑚𝑢𝑑
𝑠
)2) … … . … … … … … … … … … … … … … (88)
Where
𝑚 𝑚𝑢𝑑
𝑠
= 𝑚𝑜𝑙𝑒 𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑠𝑎𝑙𝑡/𝑠𝑜𝑙𝑢𝑡𝑒 𝑖𝑛 𝑑𝑟𝑖𝑙𝑙𝑖𝑛𝑔 𝑚𝑢𝑑
The above equations 87 and 88 are framed such that it satisfies the condition in equation
85.

57
For a monovalent salt in mud the mole fraction of cations and anions are same such that
𝑚 𝑚𝑢𝑑
𝑠
= 𝑚 𝑚𝑢𝑑
𝑎
= 𝑚 𝑚𝑢𝑑
𝑐
… … … … … … … … … … … … … … … … … … … … … … … … … (89)
The initial mole fraction of the ionic species can be calculated only if the mole fraction
of the fixed charges on the porous media is known.
𝑚 𝑓𝑐
= 10−2
∗
𝐶𝐸𝐶(1 − 𝜙 𝑜)𝜌𝑠 𝑉𝑜
𝑓
𝜙 𝑜
… … … … … … … … … … … … … … … … … … … … … (90)
Where
𝜌𝑠 = 𝑔𝑟𝑎𝑖𝑛 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 𝑖𝑛 𝑔/𝑐𝑐
𝑉𝑜
𝑓
= 𝑚𝑜𝑙𝑎𝑟 𝑣𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑤𝑎𝑡𝑒𝑟 𝑖𝑛 𝑠ℎ𝑎𝑙𝑒 𝑖𝑛 𝑙𝑖𝑡𝑟𝑒/𝑚𝑜𝑙
CEC has to be calculated before calculating mole fraction of the fixed charges. The
significance of CEC and different methods adopted to measure have been discussed in
detail in chapter 2.
Initially when the solution dissociates into its corresponding cations and anions, the
water activity is measured (water content of the fluid inside shale). Using this water
activity, the mole fraction of the salt in the fluid present outside the porous medium
which is in equilibrium with the pore fluid can be calculated as following
𝑎 𝑜
𝑓
= 1 − (𝑥 + 𝑦)𝑚 𝑒𝑞
𝑠
… … … … … … … … … … … … … … … … … … … … … … … … … … . (91)
Replacing 𝑚 𝑓𝑐
and 𝑚 𝑚𝑢𝑑
𝑠
in equation 87 with equations 90 and 91 we get
𝑎
= 0.5 (−10−2
∗
𝑓
𝜙 𝑜
+ √(10−2 ∗
𝑓
𝜙 𝑜
)2 + (1 − 𝑎 𝑜
𝑓
)2) … … … … … … … … (92)

58
Some of the inferences observed from equation 92 are
1. Geological membranes have pore sizes big enough to allow the passage of
largest hydrated radii of ions found in ground water. However the salt exclusion
behavior of clay is mainly due to the electrical restrictions operating within the
clay membrane structure. Hence the efficiency of the salt exclusion behavior for
clay is high when the clay has higher Cation Exchange Capacity. Ex: Smectite
membranes are inherently more ideal than kaolinite membranes (Marine & Fritz
1981).
2. If the porosity of the formation is very small approaching zero, the mole fraction
of anions decreases. This is because as the porous medium undergoes
consolidation/compaction the membrane efficiency increases excluding the ions
and allowing only the passage of fluids separated.
3. When the water activity 𝑎 𝑜
𝑓
approaches unity, the mole fraction of anions in
fluid is approaching zero. This is because if 𝑎 𝑜
𝑓
= 1 then the pore fluid is just
water.
5.3: Boundary conditions and Problem definition
5.3.1: Interfacial and far field stresses of the porous medium and drilling fluid
The total mole fraction of the ions in the formation is greater than the mole fraction of
the ions in the mud at the interface between the shale and drilling fluid. Hence a
pressure differential is generated which causes flow of water from one fluid to another.
The reason for this phenomenon is explained under section 2.1 in chapter 2 as a
consequence of Donnan equilibrium effect.
The pressure difference generated at the mud shale interface is

59
𝑝𝑠ℎ𝑎𝑙𝑒 − 𝑝 𝑚𝑢𝑑 =
𝑅𝑇
𝑉𝑜
𝑓
∗ (𝑚 𝑠ℎ𝑎𝑙𝑒
𝑎
+ 𝑚 𝑠ℎ𝑎𝑙𝑒
𝑐
− 2𝑚 𝑚𝑢𝑑
𝑠
) … … … … … … … … … … … … . . . (93)
∆𝑝 𝑚𝑢𝑑−𝑠ℎ𝑎𝑙𝑒 =
𝑅𝑇
𝑉𝑜
𝑓
∗ (√(𝑚 𝑓𝑐)2 + 4(𝑚 𝑚𝑢𝑑
𝑠
)2 − 2𝑚 𝑚𝑢𝑑
𝑠
) … … … … … … … … … … . . (94)
Pore pressure, normal and shear stresses in local coordinates at the far field conditions
when 𝑟 → ∞ are
𝜎𝑥𝑥 = 𝑆 𝑥 … … … … … … … … … … … … … … … … … … … … … … … … … … … … … . … … (95.1)
𝜎 𝑦𝑦 = 𝑆 𝑦 … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … (95.2)
𝜎𝑧𝑧 = 𝑆𝑧 … . … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … (95.3)
𝜏 𝑥𝑦 = 𝑆 𝑥𝑦. … … … … … … … … … … … … … . … … … … … … … … … … … … … … … … … (95.4)
𝜏 𝑦𝑧 = 𝑆 𝑦𝑧 … . … … … … … … … … … … … … … … … … … … … … … … … … … … … … … . (95.5)
𝜏 𝑥𝑧 = 𝑆 𝑥𝑧. . … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … . (95.6)
𝑝 = 𝑝 𝑜 … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … (95.7)
𝑝 𝑐
=
𝑅𝑇
𝑉𝑜
𝑓
𝑚 𝑜
𝑐
… … … … … … … … … … … … … … … … … … … … … … … … … … … … … . . (95.8)
𝑝 𝑎
=
𝑅𝑇
𝑉𝑜
𝑓
𝑚 𝑜
𝑎
… … … … … … … … … … … … … … … … … … … … … … … … … … … … … . (95.9)
Pore pressure, normal and shear stresses in cylindrical coordinates at the interface when
𝑟 = 𝑟𝑤 (wellbore radius) are
𝜎𝑟𝑟 = (𝜎 𝑚 + 𝜎 𝑑 cos(2(𝜃 − 𝜃𝑟)) 𝐻(−𝑡) + 𝑝 𝑚𝑢𝑑 𝐻(𝑡) … … … … … … … … … … … … (96.1)
𝜏 𝑟𝜃 = −𝜎 𝑑 sin(2(𝜃 − 𝜃𝑟)) 𝐻(−𝑡) … … … … … … … … … … … … … … … … … … … … . (96.2)
𝜏 𝑟𝑧 = (𝑆 𝑥𝑧 cos(θ) + 𝑆 𝑦𝑥sin(𝜃))𝐻(−𝑡) … … … … … … … … … … … … … … … … … … (96.3)
𝑝 = 𝑝 𝑜 𝐻(−𝑡) + (𝑝 𝑚𝑢𝑑 + ∆𝑝 𝑚𝑢𝑑−𝑠ℎ𝑎𝑙𝑒)𝐻(𝑡) … … … … … … … … … … … … … … … . (96.4)

60
𝑝 𝑎
=
𝑅𝑇
𝑉𝑜
𝑓
(𝑚 𝑜
𝑎
𝐻(−𝑡) + (𝑚 𝑚𝑢𝑑
𝑠
+ ∆𝑚 𝑚𝑢𝑑−𝑠ℎ𝑎𝑙𝑒
𝑎
)𝐻(𝑡) … … … … … … … … … … … … (96.5)
𝑝 𝑐
=
𝑅𝑇
𝑉𝑜
𝑓
(𝑚 𝑜
𝑐
𝐻(−𝑡) + (𝑚 𝑚𝑢𝑑
𝑠
𝑐
)𝐻(𝑡) … … … … … … … … … … … … (96.6)
Where H is Heaviside step function or unit step function which is discontinuous and is
used to represent a piecewise constant function such as
H (- t) = 1 if t < 0
H (- t) = 0 if t > 0
H is a function of time and the graph of unit step function is shown in the following
figure
In equations 96,
𝜎 𝑚 =
𝑆 𝑥 + 𝑆 𝑦
2
… … … … … … … … … … … … … … … … … … … … … … … … … … … … … … (97)
𝜎 𝑑 = 0.5√(𝑆 𝑥 − 𝑆 𝑦)2 + 4𝑆 𝑥𝑦
2 … … … … … … … … … … … … … … … … … … … … … … … (98)
Figure 20: Heaviside step function (www.intmath.comlaplace)

61
𝜃𝑟 = 0.5𝑡𝑎𝑛−1
(
2𝑆 𝑥𝑦
𝑆 𝑥 − 𝑆 𝑦
) … … … … … … … … … … … … … … … … … … … … … … … … . (99)
Where
𝜎 𝑚, 𝜎 𝑑, = 𝑚𝑒𝑎𝑛 𝑐𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑖𝑣𝑒 & 𝑑𝑒𝑣𝑖𝑎𝑡𝑜𝑟𝑖𝑐 𝑠𝑡𝑟𝑒𝑠𝑠𝑒𝑠 𝑟𝑒𝑠𝑝𝑒𝑐𝑡𝑖𝑣𝑒𝑙𝑦
𝜃𝑟 = 𝑝𝑟𝑖𝑛𝑐𝑖𝑝𝑙𝑒 𝑎𝑛𝑔𝑙𝑒 𝑖𝑛 𝑝𝑜𝑙𝑎𝑟 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠 (𝐶𝑢𝑖 𝑒𝑡 𝑎𝑙. , 1997).
The concentration differential generated at the shale mud interface is given by the
following equations
∆𝑚 𝑚𝑢𝑑−𝑠ℎ𝑎𝑙𝑒
𝑎
= 𝑚 𝑠ℎ𝑎𝑙𝑒
𝑎
− 𝑚 𝑚𝑢𝑑
𝑠
… … … … … … … … … … … … … … … … … … … … … (100)
∆𝑚 𝑚𝑢𝑑−𝑠ℎ𝑎𝑙𝑒
𝑐
= 𝑚 𝑠ℎ𝑎𝑙𝑒
𝑐
− 𝑚 𝑚𝑢𝑑
𝑠
… … … … … … … … … … … … … … … … … … … … … (101)
5.3.2: Loading decomposition of the porous medium
Due to the linearity of the governing equations of the chemically active porous medium
the boundary conditions can be split into 3 different problems, such that the solutions of
which can be superposed to obtain the final pore pressure and the effective radial and
tangential stresses (Cui et al., 1997). The mechanical loading of the porous medium is
decomposed into the following 3 problems:
1. Poroelastic plane strain problem
It is a state of strain when the normal strain to x-y plane and the shear strains are
zero (Plane strain module UCSB College of Engineering).
2. Elastic Uniaxial stress problem
It is the condition when the porous medium is subjected to tension by opposite
forces of magnitude acting along its axes.

62
3. Elastic anti-plane shear problem
It is a state of strain in the medium when all the stresses except the normal stress
and the shear stress to x-y plane are zero (Plane strain module UCSB College of
Engineering).
The decomposition of the loading problem can be shown diagrammatically as
Figure 21: Loading decomposition of an inclined wellbore
(adapted from Cui et al., 1997)

63
Chapter 6: Superposed Solutions of the chemically active porous
medium
6.1: Poroelastic plane strain problem
As mentioned in section 5.3.2 of chapter 5, the plane strain problem remains in a state
of strain during which the shear strains to x-y plane are zero such that the corresponding
shear stresses are also zero. In a plane strain problem, the far field conditions having
𝜎𝑧𝑧 , 𝜏 𝑦𝑧 , 𝜏 𝑥𝑧 and interfacial stress 𝜏 𝑟𝜃 become superfluous (Cui et al., 1997). The
analytical solution to the plane strain problem has been adapted from Detourney et al.,
(1988). The analytical solution was derived in Laplace space transform, based on the
assumption of plane strain conditions in the plane normal to the axis of the wellbore.
The solution can be used only if the coordinates are rotated about z axis such that x axis
is along the direction of minimum horizontal principal stress and y along the maximum
stress.
6.1.1: Far field conditions
𝜎𝑥𝑥 = 𝑆 𝑥 … … … … … … … … … … … … … … … … … … … … … … … … … … … … … . … . (102.1)
𝜎 𝑦𝑦 = 𝑆 𝑦 … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … . (102.2)
𝜎𝑧𝑧 = 2𝜈𝜎 𝑚 + 𝛼(1 − 2𝜈)𝑝 𝑜 … . … … … … … … … … … … … … … … … … … … … … . . (102.3)
𝜏 𝑥𝑦 = 𝑆 𝑥𝑦. … … … … … … … … … … … … … . … … … … … … … … … … … … … … … … . (102.4)
𝜏 𝑦𝑧 = 0 … . … … … … … … … … … … … … … … … … … … … … … … … … … … … … … . . (102.5)
𝜏 𝑥𝑧 = 0. . … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … . (102.6)
𝑝 = 𝑝 𝑜 … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … (102.7)
𝑝 𝑐
=
𝑅𝑇
𝑉𝑜
𝑓
𝑚 𝑜
𝑐
… … … … … … … … … … … … … … … … … … … … … … … … … … … … … (102.8)

64
𝑝 𝑎
=
𝑅𝑇
𝑉𝑜
𝑓
𝑚 𝑜
𝑎
… … … … … … … … … … … … … … … … … … … … … … … … … … … … … (102.9)
6.1.2: Near wellbore conditions
At the wellbore wall, the problem is analyzed based on the assumption that the drilling
operation is instantaneous only if the time required for drilling a distance which is 5
times the radius of the wellbore is smaller than the characteristic time given by
𝑡 𝑐 =
𝑤𝑒𝑙𝑙𝑏𝑜𝑟𝑒 𝑟𝑎𝑑𝑖𝑢𝑠
𝑔𝑒𝑛𝑒𝑟𝑎𝑙𝑖𝑧𝑒𝑑 𝑐𝑜𝑛𝑠𝑜𝑙𝑖𝑑𝑎𝑡𝑖𝑜𝑛 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡
… … … … … … … … … … … … … … . (103)
The physical version of the problem is facilitated by decomposing the problem further
into 3 modal conditions (Detournay et al., 1998) such as
1. Far field isotropic stress (axisymmetric)
2. Virgin pore pressure (axisymmetric)
3. Far field stress deviator (asymmetric)
For a poroelastic plane strain problem, the solution for the near wellbore conditions
includes the superposed solution of the 3 individual modal solutions.
𝜎𝑟𝑟 = 𝜎 𝑚 + 𝜎 𝑑 cos(2(𝜃 − 𝜃𝑟)) + 𝜎𝑟𝑟
1
+ 𝜎𝑟𝑟
2
+ 𝜎𝑟𝑟
3
… … … … … … … … … … … … . (104.1)
𝜏 𝑟𝜃 = −𝜎 𝑑 sin(2(𝜃 − 𝜃𝑟)) + 𝜏 𝑟𝜃
3
… … … … … … … … … … … … … … … … … … … … (104.2)
𝑝 = 𝑝 𝑜 + 𝑝2
+ 𝑝3
… … … … … … … … … … … … … … … … … … … … … … … … … … . . (104.3)
𝑝 𝑎
=
𝑅𝑇
𝑉𝑜
𝑓
𝑚 𝑜
𝑎
+ 𝑝 𝑎(2)
+ 𝑝 𝑎(3)
… … … … … … … … … … … … … … … … … … … … … … (104.4)
𝑝 𝑐
=
𝑅𝑇
𝑉𝑜
𝑓
𝑚 𝑜
𝑐
+ 𝑝 𝑐(2)
+ 𝑝 𝑐(3)
… … … … … … … … … … … … … … … … … … … … … … (104.5)
The Laplace solution of the volumetric strain which is in polar coordinates depending
only on time and radial distance is given by Carter et al., (1982) as following

65
𝐸̃ 𝑘𝑘 = − (
𝜂
𝐺
) 𝑃̃ + 𝐶 𝑜 𝑟−𝑛
… … … … … … … … … … … … … … … … … … … … … … … … . . (105)
Where
𝜂 = 𝑙𝑢𝑚𝑝𝑒𝑑 𝑝𝑜𝑟𝑜𝑒𝑙𝑎𝑠𝑡𝑖𝑐 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 =
𝛼(1 − 2𝜈)
2(1 − 𝜈)
… … … … … … … … … … … … (106)
𝐶 𝑜 = 𝐶 𝑜[𝑠] = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑒𝑑 𝑓𝑟𝑜𝑚 𝑏𝑜𝑢𝑛𝑑𝑎𝑟𝑦 𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛𝑠
Equation 105 is substituted in diffusion equations 77-79 to replace the volumetric strain
yielding a set of differential equations giving a coefficient matrix of
[𝑍] = [𝑌]−1[𝐷] … … … … … … … … … … … … … … … … … … … … … … … … … … … … . (107)
Where
[𝐷] = 𝑛𝑜𝑛𝑠𝑦𝑚𝑚𝑒𝑡𝑟𝑖𝑐 𝑙𝑢𝑚𝑝𝑒𝑑 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑚𝑎𝑡𝑟𝑖𝑥 𝑔𝑖𝑣𝑒𝑛 𝑏𝑦 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 80
[𝑌] =
[
1
𝑀
+
𝛼𝜂
𝐺
𝑚 𝑜
𝑎
(
1
𝑀
+
𝛼𝜂
𝐺
)
𝑚 𝑜
𝑐
(
1
𝑀
+
𝛼𝜂
𝐺
)
0
𝜙 𝑂 𝑉𝑜
𝑓
𝑅𝑇
0
0
0
𝜙 𝑂 𝑉𝑜
𝑓
𝑅𝑇 ]
… … … … … … … … … … … … … … … … … … … … (108)
The [𝑍] in equation 107 yields real values only if the matrix is positive definite.
However if the determinant and the elements along the leading diagonal are positive
then the matrix is positive definite (Johnson 1970).
6.1.3: Mode 1-Elastic radial loading / far field isotropic stress
For elastic radial loading the near wellbore conditions are given as
𝜎𝑟𝑟 = −𝜎 𝑚 + 𝑝 𝑚𝑢𝑑 … … … … … … … … … … … … … … … … … … … … … … … … … … (109.1)
𝜎𝑟𝜃 = 0 … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … (109.2)
𝑝 = 𝑝 𝑎
= 𝑝 𝑐
= 0 … … … … … … … … … … … … … … … … … … … … … … … … … … … (109.3)

66
The diffusional equation in pressure terms (equation 109.4) yields a classical Lamé
solution which is purely elastic (Detournay et al., 1988).
𝜕2
𝑝
𝜕𝑟2
+
1
𝑟
𝜕𝑝
𝜕𝑟
=
1
𝑐
𝜕𝑝
𝜕𝑡
… … … … … … … … … … … … … … … … … … … … … … … … … … (109.4)
𝜎𝑟𝑟
1
= −(𝜎 𝑚 − 𝑝 𝑚𝑢𝑑) (
𝑅 𝑤
2
𝑟2
) … … … … … … … … … … … … … … … … … … … … … … . (109.5)
𝜎 𝜃𝜃
1
= (𝜎 𝑚 − 𝑝 𝑚𝑢𝑑) (
𝑅 𝑤
2
𝑟2
) … … … … … … … … … … … … … … … … … … … … … … … (109.6)
In this mode of loading, rock deformation happens only due to deviatoric strain and the
pore pressure is not generated.
6.1.4: Mode 2- Virgin pore pressure/ diffusional loading
Mode 2 loading reduces to an uncoupled diffusional problem for pore pressure. In this
mode of loading the radial stresses are zero at the wellbore (Detournay et al., 1988).
𝜎𝑟𝑟 = 𝜎𝑟𝜃 = 0 … … … … … … … … … … … … … … … … … … … … … … … … … … … … (110.1)
Other boundary conditions at the wellbore are
𝑝 = ∆𝑝 … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … . (110.2)
𝑝 𝑎
= ∆𝑝 𝑎
… … … … … … … … … … … … … … … … … … … … … … … … … … … … … . . . (110.3)
𝑝 𝑐
= ∆𝑝 𝑐
… … … … … … … … … … … … … … … … … … … … … … … … … … … … … … . (110.4)
The unsteady state solutions in Laplace domain are listed below
𝑠𝑝̃(2)
= 𝑚11∆1 𝛷[𝜉1] + 𝑚12∆2 𝛷[𝜉2] + 𝑚13∆3 𝛷[𝜉3] … … … … … … … … … … … . . (110.5)
𝑠𝑝̃ 𝑎(2)
= 𝑚21∆1 𝛷[𝜉1] + 𝑚22∆2 𝛷[𝜉2] + 𝑚23∆3 𝛷[𝜉3] … … … … … … … … … … … (110.6)
𝑠𝑝̃ 𝑐(2)
= 𝑚31∆1 𝛷[𝜉1] + 𝑚32∆2 𝛷[𝜉2] + 𝑚33∆3 𝛷[𝜉3] … … … … … … … … … … … (110.7)
𝑠𝜎𝑟𝑟̃ (2)
= −2𝜂{𝑚11∆1 𝛯[𝜉1] + 𝑚12∆2 𝛯[𝜉2] + 𝑚13∆3 𝛯[𝜉3] … … … … … … … … … (110.8)

67
𝑠𝜎 𝜃𝜃̃ (2)
= 2𝜂{𝑚11∆1(𝛯[𝜉1] + 𝛷[𝜉1]) + 𝑚12∆2(𝛯[𝜉2] + 𝛷[𝜉1])
+ 𝑚13∆3(𝛯[𝜉3] + 𝛷[𝜉1])} … … … … … … … … … … … … … … … … . (110.9)
The above solution in Laplace domain can be inverted to time domain using the Stehfast
Algorithm described in Appendix A.
In the above solutions
𝜉𝑖 = √𝑠/𝜆𝑖 𝑤ℎ𝑒𝑟𝑒 𝜆𝑖 𝑖𝑠 𝑒𝑖𝑔𝑒𝑛 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 [𝑍]
{𝑚1𝑖, 𝑚2𝑖, 𝑚3𝑖} = 𝑐𝑜𝑟𝑟𝑒𝑠𝑝𝑜𝑛𝑑𝑖𝑛𝑔 𝑒𝑖𝑔𝑒𝑛 𝑣𝑒𝑐𝑡𝑜𝑟 𝑜𝑓 [𝑍]𝑤ℎ𝑒𝑟𝑒 𝑖 = 1,2,3
The functions ∆𝑖, 𝛷, 𝛯 are defined as following
∆1= [(𝑚22 𝑚33 − 𝑚23 𝑚32)∆𝑝 + (𝑚13 𝑚32 − 𝑚12 𝑚33)∆𝑝 𝑎
+ (𝑚12 𝑚23 − 𝑚13 𝑚22)∆𝑝 𝑐]/𝑚 … . … … … … … … … … … … … . . (110.10)
∆2= [(𝑚23 𝑚31 − 𝑚21 𝑚33)∆𝑝 + (𝑚11 𝑚33 − 𝑚13 𝑚31)∆𝑝 𝑎
+ (𝑚13 𝑚21 − 𝑚11 𝑚23)∆𝑝 𝑐]/𝑚 … . … … … … … … … … … … … . . (110.11)
∆3= [(𝑚21 𝑚32 − 𝑚22 𝑚31)∆𝑝 + (𝑚12 𝑚31 − 𝑚11 𝑚32)∆𝑝 𝑎
+ (𝑚11 𝑚22 − 𝑚12 𝑚21)∆𝑝 𝑐]/𝑚 … . … … … … … … … … … … … . . (110.12)
𝛷[𝑥] =
𝐾𝑜[𝑥𝑟]
𝐾𝑜[𝑥𝑅 𝑤]
… … … … … … … … … … … … … … … … … … … … … … … … … … . (110.13)
𝛯[𝑥] =
𝐾1[𝑥𝑟]
(𝑥𝑟𝐾𝑜[𝑥𝑅 𝑤])
−
𝑅 𝑤 𝐾1[𝑥𝑅 𝑤]
(𝑥𝑟2 𝐾𝑜[𝑥𝑅 𝑤])
… … … … … … … … … … … … … … … . … . (110.14)
Where
𝑚 = 𝑚11(𝑚22 𝑚33 − 𝑚23 𝑚32) − 𝑚12(𝑚21 𝑚33 − 𝑚23 𝑚31)
+ 𝑚13(𝑚21 𝑚32 − 𝑚22 𝑚31) … … … … … … … … … … … … … … … (110.15)
∆𝑝 = 𝑝 𝑚𝑢𝑑 + ∆𝑝 𝑚𝑢𝑑−𝑠ℎ𝑎𝑙𝑒 − 𝑝 𝑜 … … … … … … … … … … … … … … … … … … … . . (110.16)
∆𝑝 𝑎
=
𝑅𝑇
𝑉𝑜
𝑓
(𝑚 𝑚𝑢𝑑
𝑠
𝑎
− 𝑚 𝑜
𝑎
) … … … … … … … … … … … … … … … . (110.17)

68
∆𝑝 𝑐
=
𝑅𝑇
𝑉𝑜
𝑓
(𝑚 𝑚𝑢𝑑
𝑠
𝑐
− 𝑚 𝑜
𝑐
) … … … … … … … … … … … … … … … . (110.18)
𝐾 𝑛 = 𝑏𝑒𝑠𝑠𝑒𝑙 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑠𝑒𝑐𝑜𝑛𝑑 𝑘𝑖𝑛𝑑 𝑎𝑛𝑑 𝑜𝑟𝑑𝑒𝑟 ′𝑛′
6.1.5: Mode 3- Far field stress deviator/ poroelastic deviatoric stress loading
Mode 3 is a deviatoric loading and since the loading is asymmetric, the solution
includes a complete poroelastic coupling (Detournay et al., 1988). The boundary
conditions near the wellbore are (Nguyen et al., 2008)
𝜎𝑟𝑟 = −𝜎 𝑑 cos[2(𝜃 − 𝜃𝑟)] … … … … … … … … … … … … … … … … … … … … … … … (111.1)
𝜎𝑟𝜃 = 𝜎 𝑑 sin[2(𝜃 − 𝜃𝑟)] … … … … … … … … … … … … … … … … … … … … … … … . . (111.2)
𝑝 = 𝑝 𝑎
= 𝑝 𝑐
= 0 … … … … … … … … … … … … … … … … … … … … … … … … … … … (111.3)
The solutions in Laplace transform space domain are
𝑠𝑝̃(3)
= 𝜎 𝑑 {𝑚11 𝐷1 𝐾2[𝜉1 𝑟] + 𝑚12 𝐷2 𝐾2[𝜉2 𝑟] + 𝑚13 𝐷3 𝐾2[𝜉3 𝑟]
+ 𝐷4 𝑓1 (
𝑅 𝑤
2
𝑟2
)} 𝑐𝑜𝑠[2(𝜃 − 𝜃𝑟)] … … … … … … … … … … … … … … . (111.4)
𝑠𝑝̃ 𝑎(3)
+ 𝐷4 𝑓2 (
𝑅 𝑤
2
𝑟2
)} 𝑐𝑜𝑠[2(𝜃 − 𝜃𝑟)] … … … … … … … … … … … … … … . (111.5)
𝑠𝑝̃ 𝑐(3)
+ 𝐷4 𝑓3 (
𝑅 𝑤
2
𝑟2
)} 𝑐𝑜𝑠[2(𝜃 − 𝜃𝑟)] … … … … … … … … … … … … … … . (111.6)

69
𝑠𝜎𝑟𝑟̃
(3)
= −𝜎 𝑑 {2𝜂(𝑚11 𝐷1 𝛩[𝜉1] + 𝑚12 𝐷2 𝛩[𝜉2] + 𝑚13 𝐷3 𝛩[𝜉3])
− 2𝐺 (ℎ +
𝛼
𝜂
) 𝐷4 (
𝑅 𝑤
2
𝑟2
) − 𝐷5 (
𝑅 𝑤
4
𝑟4
)} 𝑐𝑜𝑠[2(𝜃 − 𝜃𝑟)] … … … … … (111.7)
𝑠𝜎 𝜃𝜃̃
(3)
= 𝜎 𝑑 {2𝜂(𝑚11 𝐷1 𝛱[𝜉1] + 𝑚12 𝐷2 𝛱[𝜉2] + 𝑚13 𝐷3 𝛱[𝜉3])
− 𝐷5 (
𝑅 𝑤
4
𝑟4
)} 𝑐𝑜𝑠[2(𝜃 − 𝜃𝑟)] … … … … … … … … … … … … … … … . (111.8)
𝑠𝜏 𝑟𝜃̃
(3)
= −𝜎 𝑑 {4𝜂(𝑚11 𝐷1 𝛺[𝜉1] + 𝑚12 𝐷2 𝛺[𝜉2] + 𝑚13 𝐷3 𝛺[𝜉3]) − 𝐺 (ℎ +
𝛼
𝜂
) 𝐷4 (
𝑅 𝑤
2
𝑟2
)
− 𝐷5 (
𝑅 𝑤
4
𝑟4
)} 𝑠𝑖𝑛[2(𝜃 − 𝜃𝑟)] … … … … … … … … … … … … … … … . (111.9)
Where
𝑓𝑖 = 𝛼[𝑌]−1
{1 𝑚 𝑜
𝑎
𝑚 𝑜
𝑐
} 𝑇
𝑓𝑜𝑟 𝑖 = 1, 2, 3 … … … … … … … … … … … … … … … … . (111.10)
ℎ =
𝜂𝑓1
𝐺
− 1 … … … . … … … … … … … … … … … … … … … … … … … … … … … … … (111.11)
{
𝐷1
𝐷2
𝐷3
} =
2
𝐺(ℎ +
𝛼
𝜂
)
[
𝑑11
𝑑21
𝑑31
𝑑12
𝑑22
𝑑32
𝑑13
𝑑23
𝑑33
]
−1
. {
𝑓1
𝑓2
𝑓3
} … … … … … … … … … … … … … … . (111.12)
𝑑𝑖𝑗 = 𝑚𝑖𝑗 𝐾2[𝜉𝑗 𝑅 𝑤] −
2𝜂
𝐺 (ℎ +
𝛼
𝜂
)
𝑓𝑖
𝐾1[𝜉𝑗 𝑅 𝑤]
𝜉𝑗 𝑅 𝑤
𝑓𝑜𝑟 𝑖, 𝑗 = 1, 2, 3 … … … … … … . (111.13)
𝐷4 = −
2
𝐺 (ℎ +
𝛼
𝜂
)
{1 + 𝜂 ∑[𝑚1𝑖 𝐷𝑖
3
𝑖=1
𝐾1[𝜉𝑖 𝑅 𝑤]
𝜉𝑖 𝑅 𝑤
]} … … … … … … … … … … … … … (111.14)
𝐷5 = 3 {1 + 2𝜂 ∑[𝑚1𝑖 𝐷𝑖
3
𝑖=1
(
𝐾1[𝜉𝑖 𝑅 𝑤]
𝜉𝑖 𝑅 𝑤
+
2𝐾2[𝜉𝑖 𝑅 𝑤]
(𝜉𝑖 𝑅 𝑤)2
]} … … … … … … … … … … . (111.15)
The functions Ω, Π, Θ are defined as

70
𝛺[𝑥] =
𝐾1[𝑥𝑟]
(𝑥𝑟)
+
3𝐾2[𝑥𝑟]
(𝑥𝑟)2
… … … … … … … … … … … … … … … … … … … … … … . (111.16)
𝛩[𝑥] =
𝐾1[𝑥𝑟]
(𝑥𝑟)
+
6𝐾2[𝑥𝑟]
(𝑥𝑟)2
… … … … … … … … … … … … … … … … … … … … … … . (111.17)
𝛱[𝑥] = 𝛩[𝑥] + 𝐾2[𝑥𝑟] … … … … … … … … … … … … … … … … … … … … … … … … (111.18)
6.2: Elastic Uniaxial stress problem
This loading yields a solution which is uniaxial and has constant vertical stress acting at
every point of the field. It is visualized that the field is under a constant uniaxial stress
without pore pressure. The solutions obtained are elastic and are time dependent.
The boundary conditions at the wellbore are
𝜎𝑟𝑟 = 0 … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … (112.1)
𝜏 𝑟𝜃 = 𝜏 𝑟𝑧 = 𝑝 = 𝑝 𝑎
= 𝑝 𝑐
= 0 … … … … … … … … … … … … … … … … … … … … . . . (112.2)
The far field boundary conditions
𝜎𝑥𝑥 = 𝜎 𝑦𝑦 = 0 … … … … … … … … … … … … … … … … … … … … … … … … … … … . . . (112.3)
𝜎𝑧𝑧 = 𝑆𝑧 − 2𝜈𝜎 𝑚 − 𝛼(1 − 2𝜈)𝑝 𝑜 … … … … … … … … … … … … … … … … … … … … (112.4)
𝜏 𝑥𝑦 = 𝜏 𝑥𝑧 = 𝑝 = 𝑝 𝑎
= 𝑝 𝑐
= 0 … … … … … … … … … … … … … … … … … … … … … (112.5)
The solutions for the elastic uniaxial stress problem are
𝜎𝑧𝑧 = 𝑆𝑧 − 2𝜈𝜎 𝑚 − 𝛼(1 − 2𝜈)𝑝 𝑜 … … … … … … … … … … … … … … … … … … … … (112.6)
6.3: Elastic anti- plane shear problem
Unlike the previous 2 problems, disturbance is introduced in elastic anti-plane shear
problem due to a sudden change in shear stress from 𝑆 𝑥𝑧 to 0 near the wellbore.
The boundary conditions at the wellbore are
𝜎𝑟𝑟 = 𝜏 𝑟𝜃 = 𝑝 = 𝑝 𝑎
= 𝑝 𝑐
= 0 … … … … … … … … … … … … … … … … … … … … … (113.1)
𝜏 𝑟𝑧 = (𝑆 𝑥𝑧 𝑐𝑜𝑠𝜃 + 𝑆 𝑦𝑧 𝑠𝑖𝑛𝜃)𝐻(−𝑡) … … … … … … … … … … … … … … … … … … . … (113.2)

71
The far field conditions are
𝜎𝑥𝑥 = 𝜎 𝑦𝑦 = 𝜎𝑧𝑧 = 𝜏 𝑥𝑦 = 𝑝 = 𝑝 𝑎
= 𝑝 𝑐
= 0 … … … … … … … … … … … … … … … (113.3)
𝜏 𝑦𝑧 = 𝑆 𝑦𝑧 … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … . (113.4)
𝜏 𝑥𝑧 = 𝑆 𝑥𝑧 … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … . (113.5)
The solutions are elastic and listed below
𝜏 𝑟𝑧 = (𝑆 𝑥𝑧 𝑐𝑜𝑠𝜃 + 𝑆 𝑦𝑧 𝑠𝑖𝑛𝜃) (1 − (
𝑅 𝑤
2
𝑟2
)) … … … … … … … … … … … … … … … … . (113.6)
𝜏 𝜃𝑧 = −(𝑆 𝑥𝑧 𝑠𝑖𝑛𝜃 − 𝑆 𝑦𝑧 𝑐𝑜𝑠𝜃) (1 + (
𝑅 𝑤
2
𝑟2
)) … … … … … … … … … … … … … … … . (113.7)
𝜎𝑟𝑟 = 𝜎 𝜃𝜃 = 𝜎𝑧𝑧 = 𝜏 𝑟𝜃 = 𝑝 = 𝑝 𝑎
= 𝑝 𝑐
= 0 … … … … … … … … … … … … … … … . (113.8)
6.4: Superposed solutions of the shale medium
The superposed solutions of 𝑝, 𝑝 𝑎
, 𝑝 𝑐
, 𝜎𝑟𝑟, 𝜏 𝑟𝜃 are obtained from equations 104 and
𝜏 𝑟𝑧, 𝜏 𝜃𝑧 are given by equations 113 while
𝜎 𝜃𝜃 = 𝜎 𝑚 − 𝜎 𝑑 cos(2(𝜃 − 𝜃𝑟)) + 𝜎 𝜃𝜃
1
+ 𝜎 𝜃𝜃
2
+ 𝜎 𝜃𝜃
3
… … … … … … … … … … … … (114.1)
𝜎𝑧𝑧 = 𝑆𝑧 − 2𝜈𝜎 𝑚 + 𝜈(𝜎𝑟𝑟 + 𝜎 𝜃𝜃) + 𝛼(1 − 2𝜈)(𝑝 − 𝑝 𝑜) … … … … … … … … … … (114.2)
The inverse of the Laplace transform in the above solutions can be obtained using
Stehfest’s Algorithm presented in Appendix A.

72
Chapter 7: Numerical Modeling Results
7.1: MATLAB coding structure
The following flow chart shows the coding structure used to perform the analysis. The
input parameters are those parameters which are obtained from the lab data after
examining the cores. The data used in the simulations have been taken from various
research papers on Pierre Shale and the references are tabulated above. The primary
calculated parameters are those calculated by using the input parameters and from
which the secondary parameters are also obtained.
If the secondary parameters calculated are zero (i.e., the wellbore is vertical) then the
diffusional loading and the deviatoric stress loading are ignored and if the parameters
are not zero, then the individual solutions of the diffusional loading and the deviatoric
loading are calculated using equations 110 and 111. These solutions are combined with
other individual solutions to get the superposed solutions of the model using equations
104, 113, 114.1 and 114.2.

73
PRIMARY CALCULATED
PARAMETERS
(mfc, ma, mc, ms
mud, lumped
coefficient, D, Y, Z matrices)
SECONDARY CALCULATED
PARAMETERS
(𝜎 𝑚, 𝜎 𝑑, 𝜃𝑟)
IF
𝜎 𝑑, 𝜃𝑟 = 0
Diffusional loading and
deviatoric stress loading are
included –
𝑝̃(2)
, 𝜎𝑟𝑟̃ (2)
, 𝜎 𝜃𝜃̃ (2)
, 𝑝̃(3)
, 𝜎𝑟𝑟̃ (3)
,
𝜎 𝜃𝜃̃ (3)
, 𝜏 𝑟𝜃̃ (3)
are calculated
Diffusional loading and
deviatoric stress loading are
ignored
Inverse of the individual solutions are
obtained and from which superposed
solutions are calculated using
equations 104, 113, 114.1 & 114.2
NO YES
INPUT PARAMETERS
(G, ν, α, Minv, k, σ, µ, χ, D, CEC,
a, Vf , po, ht, 𝜑 𝑦, 𝜑𝑧, T, ϕ)
Figure 22: Flowchart of the MATLAB coding

74
7.2: Code Testing
7.2.1: Validation of results using analytical solutions of Nguyen et al., (2008)
The code generated to calculate the pore pressure distributions, radial and tangential
stresses for Pierre Shale formations for the conditions of poroelastic medium and
chemopotential medium is validated by testing the data of offshore Western Africa
Shale used by Nguyen and Abousleiman (2008) and the results generated matches with
the analytical solutions of Nguyen et al., (2008). Fractional differences in the results
could be attributed to computer round off errors and physical modelling errors. The
denominator in the formula for finding the coefficient used in the Stehfest’s Laplace
inverse algorithm is erroneous in case of Nguyen et al., (2008).
28
29
30
31
32
33
34
35
36
1 1.1 1.2 1.3 1.4 1.5
Porepressure(Mpa)
r/rw
abousleiman PE model
Matlab PE model
abousleiman PC model
Matlab PC model
Figure 23: Pore pressure distributions near wellbore for high mud activity
(Abousleiman et al & Matlab results)

75
47
48
49
50
51
52
53
54
55
56
57
1 1.05 1.1 1.15 1.2 1.25
Effectivetangentialstress(Mpa)
r/rw
Matlab PC model
Matlab PE model
Figure 25: Effective tangential stresses near wellbore for high mud activity
-6
-4
-2
0
2
4
6
8
10
12
14
1 1.05 1.1 1.15 1.2 1.25
Effectiveradialstress(MPa)
r/rw
Matlab PC model
Matlab PE model
Figure 24: Effective radial stresses near wellbore for high mud activity

76
Figures 23, 24 and 25 show the pore pressure, effective radial and effective tangential
stress distributions near the wellbore for the Offshore Western Africa shales (data
obtained from Nguyen et al., 2008) and compares the results generated using the
MATLAB code with the results obtained by Nguyen et al., (2008) for the poroelastic
and porochemoelastic models.
7.2.2: Validation of results using Jaeger’s analytical solutions for 1-D consolidation of
poroelastic medium
Jaeger et al., (2007) has developed an analytical solution for 1 dimensional
consolidation of the poroelastic medium. 1 dimensional consolidation happens when a
porous permeable column undergoes uniaxial strain in the vertical direction only. A
constant load is applied at the top of the column, the fluid boundary pressure is set to
zero gauge after the load is applied and displacement takes place only vertically.
Figure 26: Problem Description

77
Jaeger’s analytical solution for 1 dimensional consolidation gives a relationship
between the displacement at the top of the column before and after the drained
conditions with respect to dimensionless time.
Equation describing the displacement of the column during different drainage
conditions is given as:
𝑤(𝑧 = 0, 𝑡) =
𝜎𝑒𝑥ℎ
(𝜆 + 2𝐺)
[1 −
𝛼 𝑃
2
𝑀
(𝜆 + 2𝐺 + 𝛼 𝑃
2
)
∑
8
𝑛2 𝜋2
𝑒𝑥𝑝 (
−𝑛2
𝜋2
𝑘𝑡
4𝜇𝑆ℎ2
)
∞
𝑛=1,3
] … … . . (115)
Where
𝑀 = 𝐵𝑖𝑜𝑡′
𝑠𝑀𝑜𝑑𝑢𝑙𝑢𝑠 =
1
𝜙𝑐𝑡
(𝑖𝑛 𝑃𝑎)
𝜎𝑒𝑥 = 𝑒𝑥𝑡𝑒𝑟𝑛𝑎𝑙 𝑙𝑜𝑎𝑑 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝑎𝑟𝑒𝑎 𝑎𝑡 𝑡ℎ𝑒 𝑡𝑜𝑝 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑜𝑙𝑢𝑚𝑛 (𝑃𝑎)
𝛼 𝑃 = 𝐵𝑖𝑜𝑡′
𝑠 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 (𝑑𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛𝑙𝑒𝑠𝑠)
𝜆 = 𝑙𝑎𝑚𝑒′
𝑠 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 (𝑑𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛𝑙𝑒𝑠𝑠)
Figure 27: Column displacement for a 1-D consolidation problem; (a) The initial
condition (undrained condition); (b) the column is subjected to a constant load,
pressure is increased (undrained condition); (c) Fluid is drained from the
column and pressure is decreased (drained condition) adapted from
Charoenwongsa et al. (2010).

78
𝑆 =
1
𝑀
+
𝛼 𝑃
2
(𝜆 + 2𝐺)
𝑖𝑠 𝑠𝑡𝑜𝑟𝑎𝑔𝑒 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 (𝑃𝑎−1)
A VBA code is used to generate the displacement Vs dimensionless time profiles for
Berea Sand and Pierre Shale formations to validate the observations from the
porochemopotential medium. The properties of Berea sand, the shale formation and
pore fluid are listed in appendix B.
Figures 28 and 29 show that the matching between analytical solution and numerical
solution is excellent. From figures 28 & 29 it is seen that the time taken for the column
displacement when the fluid drainage begins is around 1000 (dimensionless time) for
Berea sand (figure 28) and for Shale (figure 29) it is in the order of 1011
(dimensionless
time). The reason for such significant difference in time is due to the permeability of the
shale which is insignificant compared to the permeability of the Berea sand.
0.36
0.37
0.38
0.39
0.4
0.41
0.42
0.43
0.44
0.45
0.000001 0.0001 0.01 1 100 10000 1000000 100000000 1E+10
Verticaldisplcement(m)
Dimensionless time (kt/μSh2)
Berea Sand formations
Figure 28: Displacement of the top of the column with dimensionless time
for Berea Sand

79
After testing the validity of numerical solution, the effect of permeabilities, porosities,
bulk and shear modulus on the vertical displacement of the column with time for shale
formations are studied.
0.633918
0.63392
0.633922
0.633924
0.633926
0.633928
0.63393
0.000001 0.001 1 1000 1000000 1E+09 1E+12 1E+15
Verticaldisplacement(m)
k=E-15 m2
k=E-16 m2
k=E-17 m2
k=E-18 m2
k=E-19 m2
k=E-20 m2
for various permeabilities of Shale formation
0.633905
0.63391
0.633915
0.63392
0.633925
0.63393
0.000001 0.001 1 1000 1000000 1E+09 1E+12 1E+15
Shale formations
for Shale formations

80
Figure 30 shows vertical displacement as a function of dimensionless time for different
permeabilities from 10-15
to 10-20
m2
. As permeability decreases the time taking for
drainage of the pore fluid decreases and hence the displacement of the top column is
slow.
It is expected that the pore pressure decline for various permeabilities should give a plot
where the decline is very gradual at relatively higher permeabilities unlike the
significant drop for lower permeabilities. This happens because when rock is subjected
to vertical loads, the pore throats reduce in size reducing the pore pressure which in turn
reduces the absolute permeability of the rock more. This reduction in permeability is
more for rocks with low permeability than with relatively higher permeability (Corelab,
1983).
0.63385
0.63386
0.63387
0.63388
0.63389
0.6339
0.63391
0.63392
0.63393
0.63394
0.000001 0.001 1 1000 1000000 1E+09 1E+12 1E+15
por=0.05
por=0.1
por=0.2
por=0.3
por=0.4
for various porosities of Shale formation

81
Figure 31 shows the effect of porosity on vertical displacement. It is seen that as the
porosity of the shale is increased, time taken to attain the final displacement for fully
drained condition is decreasing. Hence a plot in which the pore pressure decline is
significant for lower porosities while it is gradual for higher porosities is expected.
Moreover, for the porosities above 30%, the curves overlap or the difference is
insignificant as seen in figures 31.
This is because for higher porosities, the membrane efficiency is very less and most of
the pore fluid (water) is lost along with the dominating process diffusion and has only
little amount to be drained and the pore throats are also relatively bigger which eases
the flow.
Similarly when shale formations with different bulk modulus and same hydrological
properties were tested (figure 32), the difference in the bulk modulus did not affect the
displacement of the column and also the drainage of the pore fluid with dimensionless
time is same for all shales with different bulk modulus. Hence the changes in bulk
modulus should not affect the pore pressure distributions significantly even for a
porochemopotential model. This is because bulk modulus is related to the size/volume
of the porous medium and is adjusted based on shear modulus and poisson’s ratio.
Though all the shales have different bulk modulus, the Poisson’s ratio is kept constant
(0.4) for all media and the strain produced by the effective stress is same for all media.

82
In the following figure 33, the displacement at the top of the column with dimensionless
time for shale formation with different shear modulus is plotted. Since the initial level
of the top of the column is slightly different for shales with different shear modulus, and
also it takes longer time to displace completely plotting on the same graph yields
straight lines and the paths of the displacement are not distinctive. The plots show that
though the varying shear modulus affects the initial level of the top of the shale columns
slightly, the time it takes to drain the pore fluid is same for all shales of different shear
modulus. Hence the changes in the shear modulus should not affect the pore pressure
distributions even for a porochemopotential model. This is due to the same reason that
the Poisson’s ratio of the shales remain constant (0.4) though they have varying shear
modulus.
Figure 32: Displacement of the top of the column with dimensionless
time for Shale formations of various Bulk modulus
0.771705
0.77171
0.771715
0.77172
0.771725
0.77173
0.771735
0.000001 0.001 1 1000 1000000 1E+09 1E+12 1E+15
2528 Mpa
3550 MPa
4800 MPa

83
1.267815
1.26782
1.267825
1.26783
1.267835
1.26784
1.267845
1.26785
1.267855
1.26786
0.000001 0.01 100 1000000 1E+10 1E+14
600 MPa
1.23904
1.239045
1.23905
1.239055
1.23906
1.239065
1.23907
1.239075
1.23908
1.239085
0.000001 0.01 100 1000000 1E+10 1E+14
Vertcialdisplacement(m)
700 MPa
1.21174
1.211745
1.21175
1.211755
1.21176
1.211765
1.21177
1.211775
1.21178
1.211785
0.000001 0.01 100 1000000 1E+10 1E+14
800 MPa
Figure 33: Displacement of the top of the column with
dimensionless time for Shale formations of various Shear
modulus

84
7.3: Case study-Pierre Shale formations
The impact of hydraulic fluid and ion transfers on pore pressure and field stresses can
be studied by considering an inclined wellbore such that it’s angle of inclination is zero
(simplified vertical wellbore)drilled in a Pierre Shale formation with hydrological and
mechanical properties and also the corresponding drilling mud with fluid properties as
listed in table 5. The wellbore has a radius of about 0.1 m and drilled to a depth of 2200
m. in the formation. The wellbore is filled with drilling mud which has NaCl and water
as its major constituents. The drilling mud has 1.07 SG and exerts a pressure of 23.09
MPa. The initial pore pressure of the formation is 21.4 MPa. The response of the
chemically active formation is studied with the chemo-potential model and is also
compared with the response observed with the poroelastic model. The pore pressure
values and effective stresses are calculated using equations 104.1, 104.3 and 114.1.
Table 5: Modeling parameters for an inclined wellbore
Parameters Values Units References Range
Shear Modulus 600 MPa
Brian Richards
1996
400-1000
MPa
(Aminul et
al., 2013)
Bulk modulus 4800 MPa
Brian Richards
1996
270 -12920
Mpa
(Aminul et
al 2013)
Poisson ratio 0.44 Dimensionless Calculated
0.095-0.5
(Aminul et
al 2013)
Pore pressure
coefficient
0.9 Dimensionless Ojala et al 2010
Storage coefficient
1/M
1/10000 1/MPa

85
Porosity 0.176 Dimensionless
Brian Richards
1996
5-30%
(Leonard et
al., 1980)
Grain density 2.37 g/cc Fjaer 2008 2.2-2.5 g/cc
Permeability 6.00E-21 Sq m Van Oort 1996
E-15 - E-20
sq m
(C.E.Neuzil
1994)
Reflection coefficient 0.8 Dimensionless
Brian Richards
1996
CEC 36 Meq/100grams Salisbury et al 1991
Water activity of pore
fluid
0.96 Dimensionless
Brian Richards
1996
Effective diffusion
coefficient Na+
1.33E-10 Sq m /s
Yeung and Datla
1995
Effective diffusion
coefficient Cl-
2.032 E-
10
Sq m /s
Yeung and Datla
1995
Temperature 82 Deg celsius Salisbury et al 1991
Pore pressure 21.4 MPa Salisbury et al 1991
Depth 2200 m Salisbury et al 1991
Mud pressure 23.09 MPa Calculated
Sv 54 MPa Salisbury et al 1991
SH , Sh 44 MPa Salisbury et al 1991
mfc
0.0719 Dimensionless
Calculated using
equation 90
ma
Calculated using
equation 87
mc
Calculated using
equation 88
7.4: Simulation Results
7.4.1: When mud activity is greater than pore fluid water activity
Case is studied initially by considering the water activity of the mud to be greater than
the activity of the pore fluid (𝑎 𝑚𝑢𝑑
𝑓
= 0.95 & 𝑎 𝑜
𝑓
= 0.90).

86
Table 6: Pore pressures of different models at various distances from wellbore for
high mud activity
r/rw Chemo-potential model Poroelastic model
1 29.15527596 23.0804834
1.05 26.77902361 22.56434067
1.1 24.7850834 22.1315479
1.15 23.31868117 21.81372694
1.2 22.37656028 21.60998242
1.25 21.84457593 21.49525859
1.3 21.57888174 21.43815837
1.35 21.46170565 21.4130831
1.4 21.41689644 21.40354763
1.45 21.40285249 21.40058473
1.5 21.39989314 21.39997299
1.55 21.39998866 21.40000067
Figure 34: Pore Pressure distribution at time = 0.01 day
(15 mins) for higher mud activity

87
Figure 35: Effective radial stresses around the wellbore
after time = 0.01 day (15 mins) for higher mud activity
Table 7: Effective Radial Stresses of different models at various distances from the
wellbore for high mud activity
r/rw Chemo-potential model Poroelastic model
1 -6.062535963 0.012256601
1.05 -1.683937225 2.48484765
1.1 2.026703295 4.609318723
1.15 4.976571229 6.399913187
1.2 7.211205213 7.894160507
1.25 8.87825482 9.14646442
1.3 10.14776927 10.21182142
1.35 11.15802705 11.13493808
1.4 12.00143565 11.94791538
1.45 12.73272203 12.67260975
1.5 13.38235492 13.32397986
1.55 13.96736031 13.91275415

88
Table 8: Effective Tangential stresses of different models for high mud activity
r/rw Chemo-potential model Poro-elastic model
1 37.2476444 42.15086983
1.05 37.16327313 40.61102243
1.1 37.05596742 39.26866972
1.15 36.7560978 38.05242313
1.2 36.22401086 36.92637127
1.25 35.51833297 35.88138971
1.3 34.72896588 34.91922096
1.35 33.93046203 34.0414189
1.4 33.16803007 33.24567355
1.45 32.46212312 32.52633357
1.5 31.8178382 31.87606895
1.55 31.23266019 31.28724465
Figure 36: Effective tangential stresses around the wellbore
when time = 0.01 day (15 mins) for higher mud activity

89
When the water activity in mud is higher than the water activity in the pore fluid, the
salt concentration in mud is lower than the salt content in pore fluid. Hence the mud
become less saline. The difference in the chemical potential of water on either side of
the clay membrane develops a driving force causing water to flow from wellbore
towards the formation.
However, in addition to chemical potential difference of water there exists chemical
potential difference of the chemical ions. Hence the ions try to flow from higher saline
fluid to fluid with less salinity. Based on the membrane efficiency and the surface
charge the ion movement is restricted and develops osmotic pressure in addition to the
pore pressure which is shown in figure 34. The chemo potential model considers the
osmotic pressure developed unlike the poroelastic model. The pressure developed is
required to attain chemical equilibrium on either sides of the membrane.
In figure 35, the corresponding effective radial stresses for chemo-potential model
plunges to negative values near the wellbore region. The negative values correspond to
tensile stresses. This is because of the additional pore pressure developed which is
explained earlier. In this case the radial tensile stresses weaken the radial compressive
stresses near the wellbore and causes fragmentation of the formation. Similarly in figure
36, the tensile tangential stresses weaken the compressive tangential stresses and causes
the formation to spall.
7.4.2: When water activity of pore fluid is greater than the mud activity
Case is studied by considering the water activity of the mud to be lower than the activity
of the pore fluid (𝑎 𝑚𝑢𝑑
𝑓
= 0.87 & 𝑎 𝑜
𝑓
= 0.90).

90
Table 9: Pore Pressure of different models for lower mud activity
1 15.3440468 23.06123779
1.05 17.20024172 22.55359367
1.1 18.75764076 22.12728497
1.15 19.90276243 21.81325024
1.2 20.63824489 21.61099865
1.25 21.0533837 21.4964338
1.3 21.26062054 21.43899468
1.35 21.35196149 21.41354242
1.4 21.38686381 21.40374846
1.45 21.39778965 21.40064946
1.5 21.40008548 21.39998187
1.55 21.40000729 21.39999416
Figure 37: Pore Pressure distribution near wellbore for
time = 0.01 day (15 mins) for lower mud activity

91
Table 10: Effective Radial stresses of different models for lower mud activity
1 7.748693203 0.031502206
1.05 7.790510164 2.495461521
1.1 7.892961104 4.613398101
1.15 8.207038147 6.400204454
1.2 8.759530983 7.892977342
1.25 9.485193242 9.145144187
1.3 10.29186883 10.21085846
1.35 11.10488291 11.1343658
1.4 11.87958247 11.94761167
1.45 12.59609656 12.67244991
1.5 13.24975272 13.32388232
1.55 13.84333827 13.91267761
Figure 38: Effective radial stresses around the wellbore for
time = 0.01 day (15 mins) for lower mud activity

92
Table 11: Effective Tangential stresses of different models for lower mud activity
1 48.39527937 42.16640378
1.05 44.99905302 40.61982992
1.1 42.08215953 39.27229406
1.15 39.69868404 38.05299333
1.2 37.81706932 36.92571796
1.25 36.34119194 35.88058616
1.3 35.16000976 34.91867259
1.35 34.18192954 34.04116111
1.4 33.34415649 33.24561434
1.45 32.60789785 32.52637642
1.5 31.9500928 31.87615045
1.55 31.35664856 31.28733294
Figure 39: Effective tangential stresses around the wellbore
when t = 0.01 day (15 mins) for lower mud activity

93
When the water activity in mud is lower than the activity of pore fluid then the salt
content in the mud is greater than the salt concentration in the pore fluid. Hence in this
case the drilling mud is more saline than the pore fluid. As a result, the water flows
from the formation back into the wellbore and the chemical ions try to move towards
the formation, the movement of which is restricted at the shale/ mud interface due to
diffuse double layer and surface charges. In the process of osmosis, part of pore
pressure at the wellbore is exerted to drilling mud as induced osmotic pressure to attain
chemical equilibrium at the interface. Hence there is drop in the pore pressure for
chemo potential model unlike poroelastic model which ignores it as shown in figure 37.
Figures 38 and 39 show the corresponding effective radial effective tangential stresses
near the wellbore when mud activity is lower. The chemical osmosis of water has
increased the effective stresses though the osmotic movement of the ions has decreased
it in counteraction. The opposing tensile and compressive stresses near the wellbore can
cause formation fragmentation.
7.4.3: Time propagation of osmotic pore pressure generated
Figure 40 shows the evolution of the pore pressure near the wellbore region with time
for various time intervals such as 0.001 day, 0.01 day, 0.1 day and 1 day and for the
case when mud activity is lower than the pore fluid activity. As the time elapses the
pore pressure near the wellbore region increases due to subsequent diffusion of ions into
the formation and also as the chemical potential difference of water has decreased with
time.

94
Table 12: Time propagation of the osmotic pressures generated
r/rw t=0.001 day t=0.01 day t=0.1 day t=1 day
1 15.34 15.34 15.34 15.34
1.05 19.98 17.20 16.03 15.64
1.1 21.29 18.76 16.69 15.93
1.15 21.40 19.90 17.29 16.20
1.2 21.40 20.64 17.85 16.46
1.25 21.40 21.05 18.37 16.71
1.3 21.40 21.26 18.83 16.95
1.35 21.40 21.35 19.25 17.18
1.4 21.40 21.39 19.61 17.39
1.45 21.40 21.40 19.93 17.60
1.5 21.40 21.40 20.20 17.80
1.55 21.40 21.40 20.44 17.99
Figure 40: Time propagation of the osmotic pore
pressure generated for lower mud activity

95
Figure 41: Pore pressure distribution for various Shear
Modulus
7.4.4: Effect of Mechanical Properties on Pore Pressures of Chemically Active Shale
Table 13: Pore pressure distributions for Chemically Active Shales for various
Shear Modulus
r/rw G=600 Mpa G=700 Mpa G=800 Mpa
1 29.155 29.157 29.158
1.05 26.779 26.795 26.809
1.1 24.785 24.809 24.831
1.15 23.319 23.343 23.366
1.2 22.377 22.396 22.415
1.25 21.845 21.858 21.870
1.3 21.579 21.586 21.593
1.35 21.462 21.465 21.469
1.4 21.417 21.418 21.420
1.45 21.403 21.403 21.404
1.5 21.400 21.400 21.400
1.55 21.400 21.400 21.400

96
Figure 42: Pore Pressure distribution for various Bulk
Modulus
Table 14: Pore Pressure distributions for Chemically Active Shales for various
Bulk Modulus
r/rw K=2528 Mpa K=3550 Mpa K=4800 Mpa
1 29.116 29.139 29.155
1.05 26.377 26.600 26.779
1.1 24.196 24.516 24.785
1.15 22.754 23.051 23.319
1.2 21.963 22.171 22.377
1.25 21.598 21.715 21.845
1.3 21.456 21.510 21.579
1.35 21.411 21.431 21.462
1.4 21.401 21.406 21.417
1.45 21.400 21.400 21.403
1.5 21.400 21.400 21.400
1.55 21.401 21.400 21.400

97
Table 15: Pore Pressures for different Poisson's ratio
r/rw ν = 0.2 ν = 0.3 ν = 0.4 ν = 0.5
1 29.01 29.06 29.12 29.21
1.05 25.58 25.91 26.45 27.59
1.1 23.19 23.58 24.29 26.13
1.15 21.99 22.25 22.84 24.86
1.2 21.54 21.67 22.02 23.83
1.25 21.42 21.46 21.63 23.03
1.3 21.40 21.41 21.47 22.45
1.35 21.40 21.40 21.42 22.05
1.4 21.40 21.40 21.40 21.78
1.45 21.40 21.40 21.40 21.61
1.5 21.40 21.40 21.40 21.51
1.55 21.40 21.40 21.40 21.45
Figure 43: Effect of Poisson's ratio on Pore Pressure
near the wellbore for high mud activity

98
Figures 41 and 42 show the pore pressure distributions near the wellbore region for the
case of high mud activity and for varying shear modulus and bulk modulus respectively.
For an increasing shear modulus the Poisson’s ratio decreases and it increases with the
increasing bulk modulus. For changes in the shear modulus, the reflecting variations in
the Poisson’s ratio is almost thrice the variations observed for the same changes in the
bulk modulus.
Though the alterations in shear modulus and bulk modulus affected the near wellbore
pore pressure distributions, the Poisson’s ratio is found to have more direct and
significant effect. In figure 43 it is observed that when the Poisson’s ratio of Pierre
Shale formation is 0.5 it becomes incompressible radially that more pressure is exerted
at the interface to allow osmotic flow of water and also decline of pore pressure is more
gradual and slow. However when the Poisson’s ratio decreases, the pore pressure
decreases faster for same distance from the wellbore when compared with
incompressible shale formation. The differences in the pore pressure distribution at a
particular distance from the wellbore is insignificant for lower Poisson’s ratios when
compared with the ratio of 0.5.
7.4.5: Effect of Petrophysical and Surface Charge Properties on Pore Pressure
distribution near wellbore region
The plots are applicable only for NaCl pore fluid of concentration 0-3 M /liter. Figures
44 and 45 show that the reflection coefficient or the membrane efficiency of Pierre
shale increases with increasing surface charges or Cation Exchange Capacity and also
with decreasing porosity. This is because when the surface charges increases, the net

99
negative potential increases causing the clay membrane to exclude the anions more
effectively. Similarly when the porosity decreases, the shale compacts and the diffuse
double layer overlap and deflect the anions.
Table 16: Relation between membrane efficiency and CEC for porosity range 0 -
40%
CEC φ = 0.0001 φ = 0.1 φ = 0.2 φ = 0.3
20 1 0.9591 0.90894 0.86256
40 1 0.97015 0.91578 0.86442
60 1 0.97775 0.92597 0.87032
80 1 0.98254 0.93588 0.87856
100 1 0.98572 0.94431 0.88764
0.84
0.86
0.88
0.9
0.92
0.94
0.96
0.98
1
1.02
0 20 40 60 80 100 120
Reflectioncoefficient
CEC (meq/100 g)
por=0.0001
por=0.1
por=0.2
por=0.3
Figure 44: Variation of membrane efficiency with Cation Exchange
Capacity for porosity range of 0-30%

100
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 20 40 60 80 100 120
Reflectioncoefficient
CEC (meq/100 g)
por=0.4
por=0.5
por=0.6
Table 17: Relation between membrane efficiency and CEC for porosity range 40 -
60%
CEC φ = 0.4 φ = 0.5 φ = 0.6
20 0.59079 0.49145 0.39271
40 0.58802 0.48643 0.38725
60 0.5915 0.48503 0.38372
80 0.60032 0.48712 0.38212
100 0.61313 0.49239 0.38242
From figures 44 and 45, it is observed that for Pierre Shale formations with same Cation
Exchange Capacity and having porosities in the range 0-30%, a difference in porosity of
0.1 gives a difference in membrane efficiency which is half of the difference observed if
the porosities are in the range of 40-60%.
Figure 45: Variation of membrane efficiency with Cation Exchange
Capacity for porosity range of 40 - 60%

101
Similarly the changes in membrane efficiency for a Pierre Shale formation with porosity
in the range 0-30% are significantly observed for varying Cation Exchange Capacity
unlike the clay membrane with the porosity in the range 40-60% for which the change
in the reflection coefficient is insignificant. The reasons for the observed facts are
attributed to the concept of double layer overlap and increase in net negative potential
for low porosity and high CEC respectively.
When the porosity of the Pierre Shale formations are very less in the orders of < 0.1 the
membrane efficiency or the reflection coefficient of the membrane is very high. So the
ion exclusion behavior is more effective and deflects the diffusion of ions through it.
Hence the pressure build up within the pores of formation is relatively higher compared
to pore pressures developed for increasing porosities at the same distance ratio from the
wellbore. This result is a consequence of the concept explained using figures 44 and 45.
Table 18: Pore Pressure distributions for Pierre Shale formations of different
porosities and constant CEC = 36 meq/ 100 g
r/rw φ = 0.1 φ = 0.2 φ = 0.3 φ = 0.4 φ = 0.5 φ = 0.6
1 38.36 27.82 24.96 23.91 23.47 23.26
1.05 33.22 25.85 23.87 23.14 22.83 22.69
1.1 28.89 24.20 22.95 22.49 22.30 22.21
1.15 25.69 22.98 22.28 22.02 21.91 21.86
1.2 23.61 22.21 21.84 21.71 21.66 21.63
1.25 22.42 21.77 21.60 21.54 21.52 21.50
1.3 21.82 21.55 21.48 21.46 21.45 21.44
1.35 21.55 21.45 21.43 21.42 21.42 21.41
1.4 21.44 21.41 21.41 21.41 21.40 21.40
1.45 21.41 21.40 21.40 21.40 21.40 21.40
1.5 21.40 21.40 21.40 21.40 21.40 21.40
1.55 21.40 21.40 21.40 21.40 21.40 21.40

102
Figure 46: Pore Pressure distribution near wellbore region for
various porosities and CEC = 36 meq/ 100 g (higher mud
activity)
When the Pierre Shale formations have a surface charge or Cation Exchange
Capacity in the orders of higher magnitude ≥ 40 meq/ 100 g, the membrane efficiency
or the reflection coefficient of the Pierre Shale formation is very high restricting the ion
movement through it, a result observed in figures 44 and 45. Hence more pressure is
developed and this pressure build up is relatively higher than the pore pressure
generated for Pierre Shale formation with lower CEC or surface charges at the same
distance ratio.

103
Table 19: Pore pressure distributions for Pierre Shale formations with different
surface charges and constant porosity of 0.176
r/rw CEC= 10 CEC= 20 CEC= 30 CEC=40 CEC=50
1 23.7 25.4 27.6 30.2 33.0
1.05 23.0 24.1 25.7 27.5 29.5
1.1 22.4 23.1 24.1 25.3 26.5
1.15 22.0 22.4 22.9 23.6 24.3
1.2 21.7 21.9 22.2 22.5 22.9
1.25 21.5 21.6 21.8 21.9 22.1
1.3 21.5 21.5 21.5 21.6 21.7
1.35 21.4 21.4 21.4 21.5 21.5
1.4 21.4 21.4 21.4 21.4 21.4
1.45 21.4 21.4 21.4 21.4 21.4
1.5 21.4 21.4 21.4 21.4 21.4
1.55 21.4 21.4 21.4 21.4 21.4
Figure 47: Pore Pressure distributions near wellbore region for
various CECs and porosity = 0.176 (higher mud activity)

104
Table 20: Pore pressure distributions for Pierre Shale formations with various
permeabilities (in m2)
r/rw k=0.6E-15 k=0.6E-16 k=0.6E-17 k=0.6E-18 k=0.6E-19 k=0.6E-20
1 29.16 29.16 29.16 29.16 29.16 29.16
1.05 29.06 29.02 28.95 28.77 28.27 26.78
1.1 28.97 28.90 28.75 28.41 27.44 24.79
1.15 28.88 28.77 28.56 28.06 26.66 23.32
1.2 28.80 28.66 28.38 27.72 25.94 22.38
1.25 28.72 28.55 28.21 27.41 25.28 21.84
1.3 28.64 28.44 28.05 27.10 24.69 21.58
1.35 28.56 28.34 27.89 26.81 24.16 21.46
1.4 28.49 28.24 27.73 26.53 23.69 21.42
1.45 28.42 28.14 27.58 26.27 23.28 21.40
1.5 28.36 28.05 27.44 26.01 22.93 21.40
1.55 28.29 27.96 27.30 25.77 22.63 21.40
Figure 48: Pore pressure distributions for Pierre Shale
formations with various permeabilities

105
As the pore pressure drops in case of chemically active Pierre Shale formations in
which the mud activity is greater than the pore fluid activity, the effective stresses
increase. As a consequence, the permeability of the formation drops. From figure 48, it
is observed that the drop in pore pressure is accompanied by a drop in permeability and
vice versa.
The reduction in permeability due to drop in pore pressure is significant for formations
of low permeability compared to formations of relatively higher permeability (Katz and
Lee., 1990).

106
Chapter 8: Conclusions
8.1: Conclusions from the above study on Pierre Shale formations
1. The poroelastic model used in modeling the pore pressure distributions in many
commercial software gives underestimated/ overestimated values as it considers
only the fluid movement due to hydraulic pressure difference unlike the Chemo-
potential model which includes the fluid and ion movement due to coupled
driving forces.
2. The unexpected surge or drop in pore pressures near the wellbore region causes
wellbore instability.
3. The important compositional properties of the pore fluid and drilling mud which
can affect the pore pressure distributions are water activity and the salt
concentration.
4. Based on the differences in the water activity of the pore fluid and drilling mud,
the water moves from fluid with higher water content to a fluid with lower water
content. This could lead to increased water saturation in the shales thereby
weakening the formation.
5. When water moves from mud to formation, the osmotic pressure is exerted on
the formation and so the pore pressure is higher than the simulated pressure
using poroelastic model near the wellbore. Similarly when the water moves from
formation to mud, the formation loses pressure to wellbore and so the value is
lower than the simulated value.

107
6. When the water moves in one direction, the flow is opposed by the flow of ions
due to differences in the concentration of the solutes in the solutions. Hence the
ion fluxes generate stresses opposing the stresses generated by the water flow
thereby either decreasing or increasing the net value depending on the direction
of the flow.
7. The effective radial and tangential stresses become less compressive due to the
opposing stresses from the ion movement which can cause rock failure and lead
to wellbore collapse.
8. The important mechanical parameter affecting the pore pressure distributions is
Poisson’s ratio. Changes in the shear modulus has greater impact on Poisson’s
ratio than the bulk modulus of same changes. Values of Poisson’s ratio closer to
0.5 makes the formation incompressible and the decline in the pore pressure is
very slow compared to formations with lower Poisson’s ratio (< 0.3).
9. Formations with lower porosities and higher Cation Exchange Capacity have
higher membrane efficiency and increases the pore pressure near wellbore or
decreases it for the case of higher mud activity or for the case of lower mud
activity respectively.
10. When the porosity of the formation is in range of 0-30% the changes in
membrane efficiency for changes in CEC are significant compared to a
formation with porosity in the range 40-60%.
11. The changes in the membrane efficiency for change in porosity for a formation
with porosity in range 0-30% and with particular CEC is half the changes

108
observed for the same change in porosity for a formation with porosity in range
40-60% and with the same CEC.
12. The pore pressure decline is very sharp and linear for formations of higher
permeability compared to formations of lower permeability. This is because
formations with less permeability takes more time to allow fluid flow which is
validated using Jaeger’s analytical solutions and so the pore pressure drop is
cumulative and declines gradually.
13. The pore pressure drop near the wellbore region is significant for formations
with lower permeability compared to relatively high permeable formations. This
is because the lower permeability formations undergo significant reduction in
permeability for changes in pore pressure unlike the formations with higher
permeability for the same pore pressure changes which is in accordance with
conclusions of Katz and Lee (1990).
8.2: Limitations of the model
1. The model has uncoupled the chemical reactions that take place inside the
formation from the ion diffusions based on the Curie Prigogine Principle which
is valid only for isotropic medium.
2. The model does not include the electro kinetic effect of the ions. It is based on
assumption of equilibrium of the electrical potential at the shale/ mud interface.
3. This model does not include the impact of thermal effects on pore pressure
distributions.

109
8.3: Recommendation
Based on the above study, it is recommended that the condition of maintaining low
water activity in drilling mud is optimal with minimal increase in the diffusion osmosis
and drilling fluid viscosity, as the strength of shale formation can be increased by back
flow of water from formation to wellbore which prevents wellbore instability and also
helps to control the rock failure due to effective tensile stresses compared to the
condition of higher water activity in the mud.

110
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120
Appendix A: Laplace Inversion-Stehfest’s Algorithm
This section presents the Stehfest’s method for numerical inversion of the Laplace
transform of form 𝑓̃[𝑠] of the function 𝑓𝑛 𝑢𝑚[𝑡] at any time ‘t’. The formula for the
inversion is as following
𝑓𝑛 𝑢𝑚(𝑡) =
𝑙𝑛2
𝑡
∑ 𝐺𝑖 𝑓̃(𝑖 ∗
𝑙𝑛2
𝑡
𝑁
𝑖=1
) … … … … … … … … … … … … … … … … … … … … … … (𝐴1)
Where
𝑁 = 𝑆𝑡𝑒ℎ𝑓𝑒𝑠𝑡 𝑛𝑢𝑚𝑏𝑒𝑟 𝑎𝑛𝑑 ℎ𝑎𝑠 𝑡𝑜 𝑏𝑒 𝑒𝑣𝑒𝑛 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 2 𝑎𝑛𝑑 20
The coefficient 𝐺𝑖 is given by
𝐺𝑖 = (−1)𝑖+
𝑁
2 ∑
𝑘
𝑁
2 (2𝑘)!
(
𝑁
2
− 𝑘) ! 𝑘! (𝑘 − 1)! (𝑖 − 𝑘)! (2𝑘 − 𝑖)!
min(𝑖,
𝑁
2
)
𝑘=(𝑖+1)/2
… … … … … … … … (𝐴2)

121
Appendix B: Properties of the Berea Sand and Shale gas formations
used for validation of results
Table 21: Properties of Berea sand used for results validation (Chareonwongsak,
S., et al. 2010)
Rock Properties Values Units
Porosity (ɸ) 0.19
Permeability (k) 0.00000000000010 m2
Rock Compressibility (Cɸ) 4.40E-10 Pa-1
Fluid Properties
Density (ρ) 1000 Kg/m3
Viscosity (μ) 0.0008900000000 Pa.s
Fluid compressibility (Cf) 0.0000000006000 Pa-1
1.04E-09
Mechanical Properties
Biot's coefficient (αP ) 0.8
Bulk modulus (K) 8000000000 Pa
Shear modulus (G) 6000000000 Pa
Lame's constant (λ) 4000000000 Pa
Poisson's ratio (ν) 0.20 Pa
Biot modulus (M) 5060728745 Pa
External stress (σex) 14200000 Pa
Reservoir dimensions
Height (H) 500 m
Incremental pressure Po 2988215.49 Pa
Storage coefficient S 0.00000000024 Pa-1

122
Table 22: Properties of Shale gas formations used for results validation (data from
table 5)
Rock Properties Values Units
Porosity (ɸ) 0.173
Permeability (k) 0.00000000000000000001 m2
Rock Compressibility (Cɸ) 3.00E-06 Pa-1
Fluid Properties
Density (ρ) 1000 Kg/m3
Viscosity (μ) 0.0008900000000 Pa.s
Fluid compressibility (Cf) 0.0000000006000 Pa-1
3.00E-06 Pa-1
Mechanical Properties
Biot's coefficient (αP ) 0.9
Bulk modulus (K) 4800000000 Pa
Shear modulus (G) 600000000 Pa
Lame's constant (λ) 4400000000 Pa
Poisson's ratio (ν)
0.44 Pa
Biot modulus (M) 1926397 Pa
External stress (σex) 14200000 Pa
Reservoir dimensions
Height (H) 500 m
Incremental pressure Po 4395.09 Pa
Storage coefficient S 0.00000051925 Pa-1

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Master's Thesis-Petroleum Engineering