Available via license: CC BYNCND
Content may be subject to copyright.
Journal Preproof
A costeffective chemothermoporoelastic wellbore stability model for mud weight
design during drilling through shale formations
Saeed Rafieepour, Siavash Zamiran, Mehdi Ostadhassan
PII: S16747755(20)30038X
DOI: https://doi.org/10.1016/j.jrmge.2019.12.008
Reference: JRMGE 646
To appear in: Journal of Rock Mechanics and Geotechnical Engineering
Received Date: 9 July 2019
Revised Date: 19 September 2019
Accepted Date: 9 December 2019
Please cite this article as: Rafieepour S, Zamiran S, Ostadhassan M, A costeffective chemothermo
poroelastic wellbore stability model for mud weight design during drilling through shale formations,
Journal of Rock Mechanics and Geotechnical Engineering, https://doi.org/10.1016/j.jrmge.2019.12.008.
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition
of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of
record. This version will undergo additional copyediting, typesetting and review before it is published
in its final form, but we are providing this version to give early visibility of the article. Please note that,
during the production process, errors may be discovered which could affect the content, and all legal
disclaimers that apply to the journal pertain.
© 2020 Institute of Rock and Soil Mechanics, Chinese Academy of Sciences. Production and hosting by
Elsevier B.V. All rights reserved.
A costeffective chemothermoporoelastic wellbore stability model for mud weight design during drilling through shale formations
Saeed Rafieepour
1,
*, Siavash Zamiran
2
, Mehdi Ostadhassan
3
1
University of Tulsa, Tulsa, OK, USA
2
Southern Illinois University Carbondale, Carbondale, IL, USA
3
University of North Dakota, Grand Forks, ND, USA
Abstract: Drilling through chemicallyactive shale formations is of special importance due to timedependent drilling fluidshale interactions. The
physical models presented so far include sophisticated input parameters, requiring advanced experimental facilities, which are costly and in most cases
unavailable. In this paper, sufficientlyaccurate, yet highly practical, models are presented containing parameters easilyderived from wellknown data
sources. For ion diffusivity coefficient, the chemical potential was formulated based on the functionality of water activity to solute concentration for
common solute species in field. The reflection coefficient and solute diffusion coefficient within shale membrane were predicted and compared with
experimental measurements. For thermallyinduced fluid flow, a model was utilized to predict thermoosmosis coefficient based on the energy of
hydrogenbond that attained a reasonablyaccurate estimation from petrophysical data, e.g. porosity, specific surface area (SSA), and cation exchange
capacity (CEC). The coupled chemothermoporoelastic governing equations were developed and solved using an implicit finite difference scheme.
MogiCoulomb failure criterion was adopted for mud weight required to avoid compressive shear failure and a tensile cutoff failure index for mud
weight required to prevent tensile fracturing. Results showed a close agreement between the suggested model and experimental data from pressure
transmission tests. Results from a numerical example for a vertical wellbore indicated that failure in shale formations was timedependent and a failure
at wellbore wall after 85 min of mudshale interactions was predicted. It was concluded that instability might not firstly occur at wellbore wall as most
of the conventional elastic models predict; perhaps it occurs at other points inside the formation. The effect of the temperature gradient between
wellbore and formation on limits of mud window confirmed that the upper limit was more sensitive to the temperature gradient than the lower limit.
Keywords: Chemothermoporoelastic wellbore stability; shalefluid interaction; chemoosmosis; thermoosmosis
1. Introduction
Maintaining stability of a wellbore and strengthening of the wellbore
wall have been widely studied in the literature (Gholilou et al., 2017;
Zhong et al., 2017, 2018; Gao and Gary, 2019; Singh et al., 2019a, b).
Among these, wellbore instability in shales is of prominent importance as
the shales are clayrich formations forming the most abundant
sedimentary rocks (around 70%). In addition to this, approximately 90%
of drilling problems are related to wellbore integrity issues in shale
formations (Rafieepour et al., 2015). These wellbore stability problems
are attributed to the shalefluid interactions that include various processes
such as chemoosmosis, thermoosmosis, electroosmosis, and swelling
due to cation exchanges. Involving such timedelayed interactions in
wellbore failure analysis refers to previous decades. However, oil industry
has identified such interactions formerly and has taken advantage of oil
based drilling fluids to overcome related problems including packing off,
hole enlargement, tight hole, etc. However, due to environmental issues
regarding oilbased muds especially in offshore environments, the use of
waterbased muds is inevitable regardless of cumbersome drilling
problems. The theory of irreversible or nonequilibrium thermodynamics
is a convenient means of analyzing simultaneous flow of water, heat, and
electricity in porous media. The literature is rich of the experimental and
theoretical studies on the coupled processes in porous media. For
example, Taylor and Cary (1960) reported that the experiments within
thermal, electrical, and chemical gradients were impressed across
saturated samples of a silt loam. Each of these gradients caused moisture
movement through the soil that gave rise to other coupling effects.
Katchalsky and Curran (1967) not only dealt with the fundamentals of the
theory, but also suggested some practical applications to scientists and
engineers. Gary (1966) experimentally investigate electrokinetics and
thermal coupling in fullysaturated sodium claywater systems. He also
took advantage of irreversible thermodynamics as a theoretical framework
for the investigation.
*Corresponding Author. Email address: saeedrafieepour@utulsa.edu
Coupling mutual effects of chemical potential and hydraulic pressure
gradient into solute and solvent transport in shale membrane lead to
decrease in shale strength due to nearwellbore pore pressure
increase/effective stress decrease. Earlier works lumped the effects of
solute transport and chemical potential gradient in the chemoosmosis
analysis. Chemical potential and membrane efficiencies were later
incorporated into the coupled transport relationships (Sherwood, 1993;
Yu, 2002). A complete chemical interaction based on nonequilibrium
thermodynamics and mixture theory was presented by Sherwood (1993).
A chemoporoelastic model was developed by Ghassemi and Diek (2002)
in which the chemical potential of drilling fluid and formation fluid were
described as a linear function of the mass fraction of solute. Ekbote and
Abousleiman (2005) also proposed an analytical solution to a linearized
anisotropic porochemoelastic model for inclined wellbores. Roshan and
Rahman (2011a) proposed a fullycoupled chemoporoelastic model for
estimation of stress profile around the wellbore.
On the other hand, thermalosmosis has been found to be an influence
factor on fluid flow through soil systems. Derjaguin and Sidorenkov
(1941) developed the physical principles of thermoosmosis at the
molecular and pore scales. Regarding wellbore stability analysis, thermo
osmosis has not been taken into account as much as chemoosmosis has
been given (Nguyen and Abousleiman, 2009; Roshan and Rahman, 2011b;
Chen et al., 2015; Ostadhassan et al., 2015; Rafieepour et al., 2015a, b;
Dokhani et al., 2016a). Moreover, in the studies that include thermal
effects in the transport of volume fluids, there have been less regarding
how thermoosmosis can be experimentally and theoretically estimated
(or quantified).
In this study, a comprehensive model of transport phenomena was
proposed involving shalefluid interactions based on nonequilibrium
thermodynamics. The governing equations were presented in cylindrical
coordinate systems and solutions were assumed as nonideal. Moreover, a
model was used for estimation of thermoosmosis coefficient based on
work done by Goncalves et al., (2012). Several other models were used
from the literature to estimate the reflection coefficient and the solute
permeability in shale membranes. Then, the proposed model was
compared with experimental results conducted by Ewy and Stankovich
(2000). The effects of coupled processes on pore pressure and wellbore
failure were also investigated. Based on the results, different wellbore
failure types were observed including failure at borehole wall, failure
inside the formation, and transient wellbore failure. Finally, the effect of
the temperature gradient between wellbore and formation on the upper
and lower limits of mud window was investigated. It should be noted that
in this study, it was assumed that the medium was isotropic. In the
literature, some studies have developed models for wellbore stability in
shale formations considering the anisotropy effects (e.g. Ostadhassan et
al., 2012; Zamiran et al., 2014, 2018; Dokhani et al., 2016b).
2. Theory and model formulation
The phenomenological equation describes the linear relationship
between the driving forces X
j
and fluxes J
j
(Bader and Kooi, 2005):
(1)
where
are the coupling coefficients that relate flows of type i to
gradients of type j. The driving forces in fluid flow through a porous
medium are usually pressure, salinity, electrical potential, and
temperature gradients. In near equilibrium thermodynamics where the
forces are small, the phenomenological equations are presented as
(Jarzyńska and Pietruszka, 2008):
(2)
where
,
, and
are the molar total solution flux per unit pore cross
sectional area, molar diffusional flux of solute per unit pore cross
sectional area relative to the solvent flow, electrical current, and heat flux,
respectively. Also, , , and are the hydraulic potential
gradient, osmotic pressure gradient, electrical potential gradient, and
temperature gradient, respectively.
is the phenomenological
coefficient which couples the fluxes with driving forces, with values
being obtained from experiments.
A description of the direct and indirect (coupled) flows in shale
formations and their corresponding driving forces is shown in Fig. 1. For
near equilibrium systems, i.e. infinitesimal macroscopic gradients, the
phenomenological coefficients can be considered to be constant and
therefore, the transport equations are linearized. It is noteworthy that near
equilibrium state, according to Onsager’s symmetry or reciprocal
relations (ORR), in the range of equilibrium state, the cross coefficients
must be equal to each other, i.e.
. In this study, it was assumed
that the net electrical current was equal to zero. According to Rafieepour
et al. (2015a, b), the Soret’s effect, Dufour effect, Seebeck effect, Peltier
effect, electroosmosis, streaming current electrophoresis, and diffusion
currents are negligible due to their small values.
Fig. 1. Schematic diagram of coupling mechanisms for a fully coupled system in shales (Rafieepour et al., 2015a, b).
In addition to this, several experimental observations confirm a
violation from the Onsager’s reciprocity (Ghassemi and Diek, 2002). It
was assumed that the flow of solute and electrical current were not
influenced by temperature gradients and the effects of hydraulic,
chemical, and electrical potential gradients on heat flux were negligible.
The heat flux was approximated by Fourier law of heat conductivity. It
should be noted that due to the low permeability of shale membranes, a
convective form of the heat transfer is negligible.
Under these assumptions, the coupled transport equations are as
follows:
!
""
"#
"$
#"
##
%
%
%
$$
!
!
(3)
where
""
,
"#
,
"$
,
#"
,
##
and
$$
are the coefficients of the
membrane for filtration, chemoosmotic, thermoosmotic, ultrafiltration,
ion diffusional, and heat transport, respectively. The overall solute flow
through the membrane can be written as the following form:
&
'
&
(
(4)
where '
&
represents the bulk solute concentration inside the membrane
system.
2.1. Chemicalosmosis
Osmotic pressure is the pressure required to prevent water from
flowing through a semipermeable membrane from a solution with low
salt concentration to a solution with high salt concentration. Fig. 2 shows
a semipermeable clay membrane with a concentration gradient across it.
The osmotic pressure can be presented as
)
*
+
,

.
/0
1
2
.
2
.
3
(5)
where R is the universal gas constant, is the average shale temperature,
2
.
is the average activity of water, and +
,.
is the average molar volume
of water. The superscripts 1 and 2 stand for water activities of the two
different solutions across the shale membrane. A shale membrane is ideal
only in the presence of waterbased muds containing low concentration of
chemical agents to prevent the solute from diffusing into the formation
and uniform small pore size distribution. This assumption should be
justified for oilbased muds or waterbased muds, with high concentration
of chemical agents such as sodium silicates. Moreover, this is the case
when a shale formation has a broad range of pore size distribution
including large pore throats, which provide significant permeability to the
solute constituent. Thus, chemical agents may not trap all the solute
molecules and usually a few of these may diffuse into formation and
influence fluid flow and consequently, change the pore pressure adjacent
to the wellbore. The interactions of solute with the pore walls increase as
the pore size is reduced that reduces the permeability of the membrane to
solute. This phenomenon can be exerted into Eq. (5) by multiplying the
right term of the equation by a coefficient called reflection coefficient
(
,
). This coefficient ranges from zero to one. For unconsolidated
sandstones that freely allow flow of water and solute, the reflection
coefficient is equal to zero and therefore, no osmotic flow can be found.
For real shales, it varies between zero and one, and for the ideal
membrane, it is equal to unity. Hence, the efficiency of clay membranes
can be defined by
,
4
)
)
5
6
7
8
"#
""
(6)
Fig. 2. Illustration of osmotic pressure.
Water activity of an aqueous solution is a function of solute
concentration, in mathematical form:
2
.
9
9
.
:
'
&
(7)
where 9
and 9
.
are the vapor pressures of the solution and pure water,
respectively. The water activity in electrolytic solutions can be
determined from experimental work. According to Dokhani et al. (2015),
moisture content has a significant effect on water activity of aqueous
solutions. The osmotic pressure gradient in rdirection can be written as
*
+
,

.
2
.
;
2
.
<
2
.
<=
*
+
,

.
:
>
'
&
:
'
&
<
'
&
<=
(8)
where 2
.
; is the derivative of 2
.
with respect to the solute concentration.
From Eqs. (3) and (4) and after manipulation, the following can be
derived:
&
'
&
?
,
@
,
'
&
"$
(9)
where @ is the solute permeability coefficient and can be defined as
@
'
&
1
""
##
"#
""
3
(10)
This coefficient represents the rate of solute diffusion across the
membrane. When @ is zero, there is no diffusional flow (e.g. ideal
membranes), and for nonselective membranes, we have @
##
'
&
.
Therefore, solute flow in semipermeable membranes for nonideal
solutions is given by
&
'
&
?
,
""AB
AC
D@(
"#
'
&
?
,
E
FG
H
IJ
K>L
M
KL
M
AL
M
AC
'
&
"$
(11)
Because shale formations are composed of small plates of clay
minerals and transport phenomena mostly take place close to the wellbore
wall, here for simplicity, it is assumed that transport process is only in r
direction and thus, the continuity equation is simplified to the following
relation:
<
'
&
N
<O
(
?
=
<
<=
=
&
%
(12)
where N is the porosity of formation.
After substitution of Eq. (11) (by ignoring the last term) into Eq. (12),
it can be reached to
?
,
'
&
""
1?=<
<=(<
<=
3(*
+
,.
?=<
<=PD@(
"#
'
&
?
,
E
Q
:
>
'
&
:
'
&
=
<
'
&
<=
R
N
<
'
&
<O
(13)
Eq. (13) is the diffusion equation of solute through the porous
medium. If solute transport via potential hydraulic gradient is neglected,
then solute transport through the shales follows the Fick’s law of mass
transfer by diffusion and becomes as
?
=
<
<=
4
=
S
;
8
<
'
&
<=
5
<
'
&
<O
(14)
where S;
8
T
UVV
WN is the effective diffusion coefficient and T
UVV
is the
function of the solute concentration given by the following equation:
T
UVV
*
+
,

.
:
>
'
&
:
'
&
D
@
(
"#
'
&
?
,
E
(15)
According to the above analysis, three phenomenological parameters
are used to describe the behavior of nonideal shale membrane systems
(
""
,
,
, @). The hydraulic permeability coefficient or the mechanical
filtration coefficient of a given membrane can be expressed as follows:
""
4
5
X
8
Y
Z[
\
]
(16)
where K is the hydraulic conductivity (e.g. in cm/s); and \ and ] are the
absolute permeability and viscosity, respectively. The reflection
coefficient is representative of maximum expected osmoticallyinduced
pressure. Marine and Fritz (1981) developed an equation for estimation of
the reflection coefficient given as
,
?
Y
&
*
.
(
?
P
D
*
.
'
^
_
W
'
^
`
(
?
E
(
*
.,
D
*
,
'
^
_
W
'
^
&
(
?
E
a
N
(17)
where Y
&
is the distribution coefficient of solute within the membrane
pores, which is the ratio of anion concentration within the membrane
pores ( '^
_
) to the mean solute concentration ( '^
&
); and '^
`
is the
concentration of cations within the membrane pores as '^
`
'^
_
(
''Z?N. The anion concentration can be obtained from Teorell
MeyerSiever model:
'
^
_
?
W
b
''
Z
?
N
(
?
W
b
c
''
Z
?
N
(
d
'
^
&
N
e
W
(18)
where CEC is the cation exchange capacity of the clay (eq/g), Z is the dry
density (g/cm
3
), N is the porosity of membrane, and '^
&
is the mean
concentration of solute on either side of the membrane (eq/cm
3
). For ideal
'
&
P2
'
&
P1
Shale
,*
+,. /012.
2.
3
Additional pressure
due to chemical
potential gradient
membranes, '^
_
is zero and thus the reflection coefficient is unity, while
for nonperm selective porous media, it is equal to '^
&
. If the value of '^
&
is
too large, then the anion concentration approaches '^
&
N. *
.
is the ratio of
frictional coefficients between cation and anion with water in the
membrane and is assumed to be 1.63. *
.,
is the ratio of frictional
coefficients between the cation and anion with the membrane structure
and is assumed to be 0.1. *
,
is the ratio of frictional coefficients between
anion and solid membrane matrix to anion and water in the membrane
structure and is considered to be 1.8. Detailed discussion is given in Fritz
(1986). The last phenomenological parameter is the solute permeability
coefficient. This parameter is a measure of the rate of solute diffusion
from the side with high concentration to the side with low concentration.
For ideal membranes, the solute can neither be transferred by advection
(
,
?) nor by diffusion ( @%). The value of @ depends on Y
&
,
frictional resistance within the membrane of anion with water (:
_.
) and of
anion with membrane structure (:
_,
). The relation is expressed as
@
Y
&
:
_.
(
:
_,
Y
&
:
_.
?
(
*
.,
(19)
If the membrane is nonpermselective (
,
%) and Y
&
?, which
occurs when porosity is very high and thus *
.,
f%, Eq. (19) reduces to
@
?
:
.
g
(20)
where :
.g
is the frictional coefficient between anion and water in free
solution defined as :
.g
*WT, where D is the diffusivity of solute. The
unit for @ is mol/(Pa m s).
2.2. Thermoosmosis
Gray (1966) observed some pressure buildup by applying a
temperature gradient across a clay sample which was indicated by the rise
and drop of water level in standpipes on either side. Taylor and Cary
(1960) outlined a theoretical analysis on the thermodynamics of
irreversible processes to evaluate coupled flows of heat and water in
continuous soil systems. The physical principles of thermoosmosis in
microscopic scales were proposed in 1941 (Derjaguin and Sidorenkov,
1941). Although there is some advancement in the mentioned literature,
there are still shortages for theoretical prediction of thermoosmosis.
Also, most of the theoretical models are in molecular and pore scale with
too many parameters that are not easily measurable. Thus, it is required to
upscale the microscopic thermoosmosis coefficient to obtain
macroscopic one for practical applications using ordinary and available
data. Goncalves and Tremosa (2010) obtained the following equation for
thermoosmosis process based on the volume averaging method:
"$
"$
\
)
h
]
(21)
where T is the absolute temperature (K), and )his the fluid–solid
interactionsinduced macroscopic specific enthalpy change (J/m
3
).
According to this study, flow of fluid occurs from the side with higher
temperature to the side with lower temperature in the clay membrane
(where )hi%). Goncalves and Tremosa (2010) developed a theoretical
expression by directly formulizing the enthalpy changes due to hydrogen
bonding modifications at the macroscopic scale. In the current study, the
same model is used. Based on thermodynamics interpretation by
Derjaguin and Sidorenkov (1941), the specific enthalpy is written as
)
h
j
'
kl
m
'
kl
n
)
h
kl
(22)
where )h
kl
is the energy required to break one mole of hydrogen bonds;
and '
kl
and '
kl
m
are the concentrations of hudrogen bonds in the bulk
system and pore fluid, respectively. The relations for these parameters can
be given as follows:
'
kl
o
.
p
'
p
o
kl
p
Wb(o
.
q
'
q
o
kl
q
Wb(
'
.
o
.
p
'
p
o
.
q
'
q
r
sqs
M
s
o
kl
m
(
s
M
s
o
kl
&
t b
u (23)
'
kl
m
o
.
p
S
&
o
kl
p
Wb(o
.
q
S
&
o
kl
q
Wb('
.
o
.
p
S
&
o
.
q
S
&
o
kl
m
b
u (24)
where o
.
p
and o
.
q
are the populations of H
2
O molecules in the first
hydration shells; '
.
, '
p
and v
q
are the concentrations of water, cation,
and anion constituents in the pores, respectively; S
&
is the concentration of
the cation and the anion in the bulk solution (concentration of solute);
o
kl
p
and o
kl
q
are the average numbers of hydrogen bonds per water
molecule in the first hydration shells; the coefficients of b and b
s
are the
average halfpore size and halfthickness of highlyordered water,
respectively; o
kl
&
and o
kl
m
are the average numbers of hydrogen bonds
per water molecule of bulk and highlyordered water, respectively; and
the coefficient of '
.
is the water concentration in pore space and can be
obtained using the following conservation relationship:
w
.
'
.
(
w
p
'
p
(
w
q
'
q
?
(25)
where w
x
, w
p
and w
q
are the molar volumes of the water, cation, and
anion, respectively. Using the Donnan equilibrium, the values of '
p
and
'
q
can be estimated as
'
p
y
z
{

}
~

•

(
S
&
(
z
{
}
~
•
,
'
q
y
z
{

}
~

•

(
S
&
z
€
}
•
•
(26)
where e is the elementary charge, o
_
is the Avogadro’s constant, and ‚
ƒ
is defined as the excess charge in a unit volume of porous medium per
associated water volume based on following equation:
‚
ƒ
„…
†
‡
?
N
Z
&
''
N
(27)
where Z
&
is the density of the solid. Mean halfpore size (b) can be
derived from the following equation having specific surface area (SSA),
considering a planeparallel conceptual geometry for the porous medium:
ˆ
N
?
N
Z
&
‰
&
(28)
By introducing Eq. (28) into Eq. (27), it is obtained that ‚
ƒ
„…†‡'' ‰
&
ˆ
uŠ ˆ
u and Š„…†‡''W‰
&
. In order to use this model
for estimation of thermoosmosis coefficient, it is only required to have
some petrophysical properties along with some molecular parameters. For
molecular properties, parameters in the literature can be used including
o
.
p
o
.
q
…, o
kl
p
b†b‹, o
kl
q
b†…Œ, o
kl
m
‡†Œ, +
p
b‡†•Ž?%
q•
m
3
/mol, +
q
?‹†dŽ?%
q•
m
3
/mol, +
.
?•Ž?%
q•
m
3
/mol, )h
kl
?d†Œ kJ/mol, o
kl
&
‡†‹Œ, and a mean value of 1 nm for b
s
.
2.3. Volume flow through membrane
According to Eq. (2), the relationship for overall volume flow (solvent
flow) is as follows:
""
(
"#
(
"$
""AB
AC
"#AX
AC
"$AG
AC
(29)
From Eq. (12),
"#
,
""
and also from chemical potential
gradient relation ( ]*W+
,.
D:>'
&
W:'
&
E•'
&
W•= ), the
following relation for volume flow can be derived:
""
‘
<9
<=
,
*
+
,

.
:
>
'
&
:
'
&
<
'
&
<=
’
"$
<
<=
(30)
The continuity equation for flow of solvent through the porous media
can be presented as
A“”
A•
(
CA
AC
=Z
% (31)
By introducing the Eq. (30) into Eq. (31) and by considering slightly
compressible fluid, the diffusivity equation for solvent flow is as follows:
""
r
CA–
AC
(
A

–
AC

t(
—
˜™
FG
H
IJ
CA
AC
c=
K>L
M
KL
M
AL
M
AC
e(
"$
r
CAG
AC
(
A

G
AC

tS
š
N
A–
A•
(32)
where S
š
is the total compressibility of the shale formation.
2.4. Heat flow in semipermeable membranes
As mentioned earlier, due to low permeability of shales, heat transport
via convection is negligible and it is assumed that the conduction heat
transfer is dominant which follows Fourier’s law:
\
$
(33)
where \
$
is the thermal conductivity coefficient. In this study, it is also
assumed that the heat transfer is only in rdirection and therefore, we have
<
<O
'
1
?
=
<
<=
(
<
<
=
3
(34)
This is the heat diffusivity equation and '
is the thermal diffusivity
coefficient given by'
\
$
WZS
›
.
2.5. Boundary and initial conditions
The geometry of borehole stability problem is presented in Fig. 3. To
determine the solute concentration, pressure, and temperature profiles,
nine initial and boundary conditions must be specified. These conditions
for the model can be summarized as follows:
=O
8
'
&
=O'
8
=O
8
œ•O%
=O
.
(
,
'
&
=O'
žV
=O
.
œ•=%Ÿ ¡Oi%
=O
8
'
&
=O'
8
=O
8
œ•=f(¢Ÿ ¡Oi% £ (35)
(a) (b)
Fig. 3. (a) Schematic diagram for a wellbore subjected to anisotropic in situ stress
field; and (b) Problem domains.
3. Stress distribution around wellbore
The general solution to the thermoporoelastic wellbore stability
model is determined via the superposition principle by combining the
mechanical, hydraulic, chemical, and thermal induced effects. The stress
strain relations for a chemoporothermoelastic medium are written as
¤
b
¥
r
¦
(
§
?
b
§
¦
¨¨
©
t
(
ª
©
(
b
¥
?
(
§
?
b
§
ª
,
‡
©
(36)
where ¤
, ¦
and ¦
¨¨
are the total stress, total strain, and volumetric
strain, respectively; and ª, G, ν, ª
,
and ©
are the BiotWillis effective
stress coefficient, shear modulus, Poisson’s ratio, volumetric thermal
expansion coefficient, and the Kronecker delta, respectively. In Eq. (36),
compressive stress is assumed to be positive. The complete form of the
stress components around the wellbore including thermoporoelastic
effects is expressed as follows (Li, 1998; Yu, 2002):
¤
CC
r
«
¬¬
p«

tr?
C
J

C

t(r
«
¬¬
q«

tr?(‡
C
J
®
C
®
d
C
J

C

t¯ °b±(
¤
²³
r?(‡
C
J
®
C
®
d
C
J

C

t°´0b±(
µq¶
q¶
C

·9
V
=O=•=(
¸µ
I
q¶
C
C
J
C

·
V
=O=•=(
.C
J

C

C
C
J
(37a)
¤
¹¹
r
«
¬¬
p«

tr?(
C
J

C

tr
«
¬¬
q«

tr?(‡
C
J
®
C
®
t¯ °b±
¤
²³
r?(‡
C
J
®
C
®
t°´0b±
µq¶
q¶
c
C

·9
V
=O=•=9
V
=O
C
C
J
e
¸µ
I
q¶
c
C

·
V
=O=•=
C
C
J
V
=Oe
.C
J

C

(37b)
¤
ºº
¤
ººq
»
§cbj¤
²²
»
¤
³³
»
n
C
J

C

¯ °b±(d¤
²³
gC
J

C

°´0b±e
µq¶
q¶
9
V
=O
¸µ
I
q¶
V
=O (37c)
¤
C¹
¤
¹C
r
«
¬¬
¼
q«

¼
tr?‡
C
J
®
C
®
(b
C
J

C

t°´0b±(¤
²³
g
r?‡
C
J
®
C
®
(
b
C
J

C

t¯ °b± (37d)
¤
º¹
¤
¹º
j¤
²º
»
°´0±(¤
³º
g
¯ °±nr?(
C
J

C

t (37e)
¤
ºC
¤
Cº
j¤
²º
»
¯ °±(¤
³º
g
°´0±nr?
C
J

C

t (37f)
where =
.
is the wellbore radius and ± is the angle around the wellbore.
4. Failure criterion
The MohrCoulomb (MC) failure criterion is widely used in the area
of borehole instability analysis. However, this criterion ignores the effect
of intermediate stress component. AlAjmi (2006) introduced a polyaxial
failure criterion that considers the effects of intermediate principal stress
in the shear failure analysis. This failure envelope has been used in this
study as defined by
½
g`š
2(ˆj¤
,
9n2
¾
S¯ °¿ ˆ
¾
°´0¿ (38)
where ¿ is the internal friction angle; ¤
,
is the effective normal stress
and ½
g`š
is the octahedral shear stress defined by
¤
,
«
À
p«
Á
½
g`š
Â¤
¤
(¤
¤
(¤
¤
(39)
where ¤
, ¤
and ¤
are the major, intermediate and minor principal
stresses, respectively.
A failure index (FI) is usually defined to evaluate the wellbore
conditions in terms of collapse and tensile failures. For any type of failure
to occur, FI must become negative. For the case of MogiCoulomb failure
criterion, the collapse FI is given as
Ã
2
(
ˆ
j
¤
,

9
n
½
g`š
(40)
This failure function was successfully performed in a field study in
offshore Iran for well path optimization (Rafieepour and Jalalifar, 2014).
Moreover, a tensile fracturing is probable when the minimum effective
principal stress (
eff
min
σ
) in the rock formation surpasses the tensile strength
(McLean and Addis, 1990). The breakdown FI is given by
eff
bd min 0
0
σ σ
= + ≤
T (41)
where
0
T
is the tensile strength of the rock.
5. Numerical method and computer implementation
Solute concentration, pressure, temperature, and stress distributions
are the four primary unknowns in the proposed model, which shall be
determined to evaluate the wellbore stability. The system of equations
presented is nonlinear due to the dependency of the coefficients to the
mentioned unknowns. Consequently, a numerical approach shall be
considered to solve for these unknowns under specific initial and
boundary conditions. There are advanced numerical schemes, e.g. finite
element method, which are suitable for solution of complex
geomechanical problems (Zhai et al., 2009). However, the finite element
method is a timeconsuming and computationally expensive approach.
For these reasons, the system of equations is solved using an implicit
finite difference scheme. The constant time step and mesh size finite
difference approach is adopted to solve the proposed initialboundary
value problem. The flowchart for wellbore stability analysis is
summarized in Fig. 4.
Fig. 4. Flowchart of wellbore stability model.
6. Results and discussion
As mentioned previously, estimation of thermoosmosis coefficient
for wellbore stability analysis in shale formations has not been taken into
account. The previous studies simply have presented assumed amounts
for this coefficient in the analyses without any discussion on how this
parameter can be estimated. In this investigation, the proposed model is
validated for prediction of thermoosmosis coefficient through
comparison with laboratory works conducted by Gray (1966) on saturated
claywater electrolyte systems. Gray (1966) performed various
experiments on a pure kaolinite. Based on Gray (1966), the sample was
firstly washed in a concentrated solution of sodium chloride, and after
drying at 230
°
F (110
°
C), it was mixed with 0.001 mol/L NaCl solution.
Subsequently, the clay sample was placed in a flow cell with pressure,
thermal, and electrical gradients applied on both ends. The temperature
gradient across the clay sample was established by internal heating and
cooling coils. The temperatures were monitored using a digital
thermometer. The resulting changes in pressure were indicated by the rise
and fall of water level in standpipes on both sides of the flow cell. The
temperatureinduced pressure was observed to be directly proportional to
the difference in water level in the two pipes. By applying a thermal
gradient of 1.08
°
C/m and a mean temperature of 26.8
°
C at the steady
state condition, the difference in water level in the two standpipes was
measured as 0.51 cm/
°
C. SSA value was estimated as 50 m
2
/g.
Consequently, the mean half pore size would be 4.094 nm. Using the data
shown in Table 1 and applying the thermoosmosis coefficient model, the
estimated value for
)h
is equal to
?Œ†?ÄÅWÆ
. Substitution of specific
enthalpy, mean temperature, permeability, and viscosity into Eq. (21)
results in
)ÇW)%†Œb¯ÆWÈ
which is 0.06
¯ÆWÈ
higher than
measured value for temperatureinduced pressure (about 12% error).
Then, the thermoosmosis coefficient is equal to
"$
b†%?Ž?%
q8
.
The negative value for the thermososmosis coefficient reveals that the
flow occurs from the cold side to the hot side. The above analysis shows
that the theoretical model based on the alteration in hydrogen bond
network of water molecules can admittedly predict the results of
experimental work conducted by Gray (1966).
Table 1. Parameters for kaolinite shale reported by Gray (1966).
Parameter Unit Value
μ
Pa s 1×10
3
k m
2
3.9×10
15
Water content % 34.4
NaCl concentration
mol/L 0.001
Mean temperature K 300
CEC meq/g 0.038
Specific gravity  2.63
The above analysis focused on estimation of the thermososmosis
coupling coefficient. The relations presented in section for chemo
osmosis and several pressure transmission tests conducted by Ewy and
Stankovich (2000) are utilized to estimate the reflection coefficient and
permeability coefficient of solute species. Ewy and Stankovich (2000)
performed a series of tests on shale samples under simulated in situ
conditions. They developed a technique for measuring changes in shale
pore pressure caused by simultaneous application of hydraulic and
osmotic gradients. They used three shale samples in their experiments
(A1, A2 and N1). Only shale samples of A2 and N1 showed significant
membrane behavior. The pore pressure at the outlet end was measured
and recorded continuously. The data used for shale sample N1 contacting
CaCl
2
solution are tabulated in Table 2. Based on these data, values of
0.068 and 5.3×10
13
mol/(Pa m s) were obtained for the reflection and
solute permeability coefficients, respectively. Ewy and Stankovich (2000)
estimated a reflection coefficient of 0.02 by contacting shale sample N1
contacting a 267 g/L CaCl
2
solution and plotting the osmotic pressure
versus the fluid activity difference. Results show good agreement between
modeling and experimental data.
Table 2. Parameters for shale N1, reported by Ewy and Stankovich (2000) and
estimated based on Simpson (1997).
Parameter Unit Value
c=k/(µC
t
ϕ) m
2
/s (1.764.41)×10
8
k m
2
(9.86939.44)×10
21
T
#_
#É

m
2
/s 1.321×10
9
'
žV
(CaCl
2
concentration) g/L (=2.41 mol/L) 267
Pore fluid concentration (CaCl
2
) mol/L 0.01
Molecular weight of CaCl
2
g/mol 110.98
Arithmetic mean concentration mol/L 1.21
Porosity  0.25
CEC meq/g 0.16
SSA m
2
/g 230
To understand the effect of chemoosmosis (under isothermal
conditions) on the pore pressure with time for
8
.

8
?ŒÊ°´œ0•
.
„•ŒÊ°´
(1 psi = 6895 Pa), the correlation between
water activity and solute concentration was used, as shown in Fig. 5.
Wellbore geometrical data used here are from Table 3. Fig. 6 indicates the
transient pore pressure profile for various exposure times. The pressure
inside the formation increases with time and sweeps farther distances into
the formation. However, after 15 h of exposure, the profile of pore
pressure is stable and has not been changed with time. This implies a
balanced condition of chemical activities of formation water and drilling
fluid in all points inside the formation. In the next step, a wellbore
stability analysis was performed based on the general coupled model
presented in previous sections.
Fig. 5. Water activity (2
.
) versus solute concentration for CaCl2.
Table 3. Rock and fluid properties, well and in situ stress data (Ewy and Stankovich,
2000; Yu, 2002; Rafieepour et al., 2015a, b).
Type of properties Parameter Unit Value
Rock mechanics and
petrophysical
Elastic modulus Pa 6.8929×10
9
Poisson’s ratio  0.25
Mean halfpore size nm 3.4
CEC meq/g 0.12
Š
C/m
2
0.27
SSA m
2
/g 43
Formation compressibility Pa
1
2.61×10
10
Rock density kg/m
3
19,771.36
Internal friction angle ° 30
Cohesion Pa 6.893×10
6
Porosity % 22
Biot’s coefficient  0.8
Fluid Fluid viscosity Pa s 3×10
4
Fluid compressibility Pa
1
4×10
10
Average molar volume of water
m
3
/mol 18×10
6
Universal gas constant J/(mol
K)
8.314
Others Depth m 1524
Well radius
m 0.127
Wellbore azimuth/inclination °/° 0/0
In situ stress components,
¤
ƒ

¤
k
œ0•
¤
MPa/m
0.285, 0.276,
0.249
Fig. 6. Pore pressure distribution of shale in contact with CaCl2 solution for
8
.

8
?ŒÊ°´œ0•
.
„•ŒÊ°´.
6.1. Wellbore failure modes
As mentioned before, wellbore failure occurs when FI<0. The FI is a
function of time and space and strongly depends on the state of effective
stress in the rock. Three different scenarios are probable for compressive
failure around a wellbore: transient (timedependent) borehole failure,
failure occurring at the wellbore wall, and failure occurring at some
distance from the wellbore wall (i.e. inside the formation).
6.1.1. Transient compressive failure
Effects of thermo and chemoosmosis for the case of heating
(negative temperature gradient) and
'
žV
Ë'
8
are shown in Fig. 7a and b.
The data used in the simulations can be found in Tables 3 and 4 and the
correlation of water activity and solute concentration in Fig. 5. As it can
be seen, fluid flow due to hydraulic pressure gradient is lower than any
other flow mechanisms. According to Fig. 7a, the combination of thermo
and chemoosmosis including hydraulic potential increases pore pressure
inside the formation and triggers instability of the formation around the
wellbore (peak pore pressure of 23.5 MPa). This can also be observed as
large negative values of collapse failure index (collapse inside the
formation) in Fig. 7b. In this example, the effect of chemoosmosis is
more visible than that of thermoosmosis (large chemical potential and
small temperature gradients).
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5
Water activity, aw
CaCl2Concentration, C(mol/L)
(a)
(b)
Fig. 7. Effect of thermo and chemoosmosis on the pressure profile for the case of
heating and '
žV
Ë'
8
: (a) Plot of pore pressure; and (b) Failure index.
Fig. 8ac shows the contour diagrams of collapse failure index values
around a vertical wellbore over elapsed time. This case was designed such
that a compressive failure is probable at the wellbore wall due to using a
drilling fluid with low density. As it can be seen, the wellbore is stable
initially but shortly after shaledrilling fluid exposure (transport of fluid
into the formation due to hydraulic transport, chemo and thermoosmosis
changes pore pressure), an unstable region is formed, and wellbore wall
fails after 85 min. It is also clear from these figures that failure around the
wellbore occurs along the direction of the minimum horizontal stress, due
to the horizontal stress anisotropy in the ground.
Table 4. Hydraulic, chemical and thermal properties (Ewy and Stankovich, 2000;
Yu, 2002; Rafieepour et al., 2015a, b).
Type of
properties
Parameter Unit Value
Hydraulic Formation pressure MPa/m 0.0105
Mud pressure MPa/m 0.0141
Permeability m
2
7.66×10
20
Chemical Estimated membrane efficiency  0.0763
Estimated solute permeability,
@
mol/(Pa m s) 3.9722×10
13
Formation/mud solute concentration mol/L 4, 0.1
Water molar volume L/mol 0.01802
Thermal Thermal diffusivity m
2
/s 9.5×10
6
Estimated value for L
PT
m
2
/(s K) 5.34×10
15
Surface temperature
o
C 24
Formation temperature
o
C 150, 100.55
Mud temperature
o
C 150, 100.55
Thermal expansion coefficient
o
C
1
18×10
6
Fluid specific heat capacity J/(kg K) 1674.72
Fluid specific heat capacity J/(kg K) 837.36
(a)
(b)
(c)
Fig. 8. Contour diagrams of failure index around the wellbore: (a) After 20 min; (b)
After 30 min; and (c) After 85 min and the wellbore failure begins.
6.1.2. Compressive failure at wellbore wall
An example of wellbore stability analysis for a vertical well with
anisotropic horizontal stresses is presented in Fig. 9. In this case, due to
low mud weight, wellbore failure begins at the wall. Moreover, due to
horizontal stress anisotropy, the failure at wellbore wall is not uniform
(symmetric) and the breakout extension is along the minimum horizontal
stress component. Also, from Fig. 9, the FI at outer boundary is equal to 7
MPa while its value in the inner boundary (wellbore wall) is –0.5 MPa. In
this case, the hydrostatic pressure of mud column is responsible for failure
at wellbore wall. The most suitable and effective remedial act in this
scenario is raising mud weight.
Fig. 9. Failure at wellbore wall.
6.1.3. Compressive failure inside the formation
Another case that can be considered is that a vertical wellbore is
drilled through a chemical active shale formation with a large chemical
potential/temperature gradient between wellbore and formation, i.e.
negative temperature (formation heating) and positive
concentration/negative water activity (
df sh
w w
>
a a
) gradients. In such
circumstances, fluid transport into the formation occurs under various
mechanisms including the direct flow of volume fluid due to hydraulic
potential gradient, and indirect flow due to chemo and thermoosmosis
gradients. The combination of these transport processes causes a
significant rise in pore pressure around the wellbore. Fig. 10a shows pore
pressure profile for the case of
w 0
∆ = − =
T T T
49.45
°
C and
df sh
s s s
∆ = − =
C C C 3.9 mol/L. Change in pore pressure profile re
distributes stress concentration around the wellbore. Stress concentrations
exceeding rock strength form a failure region inside the formation are
shown in Fig. 10b, with negative failure index values of about 3 MPa.
(a)
(b)
Fig. 10. (a) Pore pressure profile around the wellbore; and (b) Failure at some points
inside formation.
6.2. Effect of temperature gradient on critical mud weight
Fig. 11 shows the effect of the temperature difference between
wellbore and formation on the critical mud weights required for
prevention of borehole fracturing and collapse. As it can be seen,
increasing temperature increases both upper and lower limits of mud
window. However, the effect of temperature gradient on critical
breakdown pressure is higher than that of breakout mud weight. For
example, formation heating by 10
°
C causes increasing in fracturing mud
pressure by 1.6 MPa (a fracturing mud weight increase of 0.35 ppg, 1 ppg
= 0.001176 MPa/m) while it increases collapse pressure by 0.35 MPa (a
collapse mud weight increase of 0.0725 ppg).
Fig. 11. Critical mud weight with various temperature differences between wellbore
and formation.
7. Conclusions
In this paper, several models were presented for prediction of chemo
and thermoosmosis parameters from wellknown data sources. For ion
diffusivity coefficient, the chemical potential was formulated based on the
functionality of water activity to solute concentration for common solute
species in field. For thermallyinduced fluid flow, a model was utilized to
predict thermoosmosis coefficient based on the energy of hydrogenbond
that attained a reasonablyaccurate estimation from petrophysical data,
e.g. porosity, SSA, and CEC. A coupled chemothermoporoelastic model
was presented and the governing equations were solved using an implicit
finite difference scheme. Results confirmed that chemical and thermal
effects significantly influenced formation fluid content and displacement
field near the wellbore wall. It was found that various types of wellbore
failure may occur around the wellbore including failure at wellbore wall,
failure inside the formation, and timedependent failure. Wellbore failure
at wall occurred when the hydrostatic effects were dominant with no
transport of fluid into/out of formation. This is what most of the
conventional elastic models predict.
In this study, it was demonstrated that in chemically active shale
formation and when there was a temperature gradient, wellbore failure
may occur firstly at some points inside the formation rather than at
wellbore wall, i.e. for the case of formation heating and negative water
activity (
df sh
w w
>
a a
) gradient. Moreover, the effect of temperature gradient
on critical collapse and fracturing mud weights was investigated. Findings
showed that heating the formation increased both mud limits; however,
the breakdown mud weight was influenced more by temperature rise than
breakout mud weight. For wellbore instability problems in shales, there
have been several experimental studies which show the effectiveness of
nanoparticles on wellbore stability enhancement in shale formations,
based on the results from triaxial strength measurements (Gao et al.,
2016). These experimental investigations can be considered for further
study in this area.
Declaration of Competing Interest
The authors wish to confirm that there are no known conflicts of
interests associated with this publication and there has been no significant
financial support for this work that could have influenced its outcome.
Acknowledgments
The authors would like to acknowledge the financial and technical
supports from the Petroleum Engineering Department at the University of
North Dakota.
References
AlAjmi A. Wellbore stability analysis based on a new truetriaxial failure criterion. PhD
Thesis. Imperial College University; 2006.
Bader S, Kooi H. Modelling of solute and water transport in semipermeable clay
membranes: Comparison with experiments. Advances in Water Resources 2005; 28:
203–14.
Chen Y, Yu M, Takach N, Shi Z, Gao C. Hidden impact of existing mud loss on state of
stresses disclosed by thermalporoelastic modeling. In: Proceedings of the 49th US
Rock Mechanics/Geomechanics Symposium, San Francisco, USA. 2015. ARM A 15
152.
Derjaguin BV, Sidorenkov GP. On thermoosmosis of liquid in porous glass. Comptes
Rendus del’Academie des Sciences de’lURSS 1941; 32: 622–6
Dokhani V, Yu M, Takach NE, Bloys B. The role of moisture adsorption in wellbore
stability of shale formations: Mechanism and modeling. Journal of Natural Gas Science
and Engineering 2015; 27: 168–77.
Dokhani V, Ma Y, Yu M. Determination of equivalent circulating density of drilling fluids
in deepwater drilling. Journal of Natural Gas Science and Engineering 2016a; 34:
1096–105.
Dokhani V, Yu M, Bloys B. A wellbore stability model for shale formations: Accounting
for strength anisotropy and fluid induced instability. Journal of Natural Gas Science
and Engineering 2016b; 32: 174–84.
Ekbote S, Abousleiman Y. Porochemothermoelastic solution for an inclined borehole in a
transversely isotropic formation. Journal of Engineering Mechanics 2005; 131: 522–33.
Ewy RT, Stankovich RJ. Pore pressure change due to shalefluid interactions:
Measurements under simulated wellbore conditions. In: Proceedings of the 4th North
American Rock Mechanics Symposium. A.A. Balkema; 2000. p 147–54.
Fritz SJ. Ideality of clay membranes in osmotic processes: A review. Clays and Clay
Minerals 1986; 34(2): 21423.
Gao C, Miska SZ, Yu M, Ozbayoglu EM, Takach NE. Effective enhancement of wellbore
stability in shales with new families of nanoparticles. In: Proceedings of the SPE
Deepwater Drilling and Completions Conference. Society of Petroleum Engineers
(SPE); 2016.
Gao C, Gray KE. A workflow for infill well design: Wellbore stability analysis through a
coupled geomechanics and reservoir simulator. Journal of Petroleum Science and
Engineering 2019; 176: 27990.
Ghassemi A, Diek A. Porothermoelasticity for swelling shales. Journal of Petroleum
Science and Engineering 2002; 34: 123–35.
Gholilou A, Behnoud Far P, Vialle S, Madadi M. Determination of safe mud window
considering timedependent variations of temperature and pore pressure: Analytical and
numerical approaches. Journal of Rock Mechanics and Geotechnical Engineering 2017;
9(5): 90011.
Goncalves J, de Marsily G, Tremosa J. Importance of thermoosmosis for fluid flow and
transport in clay formations hosting a nuclear waste repository. Earth and Planetary
Science Letters 2012; 339–340: 1–10.
Goncalves J, Tremosa J. Estimating thermoosmotic coefficients in clayrocks: I.
Theoretical insights. Journal of Colloid and Interface Science 2010; 342(1): 166–74.
Gray DH. Coupled flow phenomena in claywater systems. PhD Thesis. The University of
Michigan; 1966.
Jarzyńska M, Pietruszka M. Derivation of practical KedemKatchalsky equations for
membrane substance transport. Old and New Concepts of Physics 2008; 5(3): 45974.
Katchalsky A, Curran PF. Nonequilibrium thermodynamics in biophysics. Harvard
University Press; 1967.
Li X. Thermoporomechanical modelling of inclined boreholes. PhD Thesis. The University
of Oklahoma; 1998.
Marine IW, Fritz SJ. Osmotic model to explain anomalous hydraulic heads. Water
Resources Research 1981; 17: 2382.
McLean MR, Addis MA. Wellbore stability analysis: A review of current methods of
analysis and their field application. In: Proceedings of the SPE/IADC Drilling
Conference. Society of Petroleum Engineers (SPE); 1990.
Nguyen VX, Abousleiman YN. The porochemothermoelastic coupled solutions of stress
and pressure with applications to wellbore stability in chemically active shale. In:
Proceedings of the SPE Annual Technical Conference and Exhibition. Society of
Petroleum Engineers (SPE); 2009.
Ostadhassan M, Zeng Z, Zamiran S. Geomechanical modeling of an anisotropic formation
 Bakken case study. In: Proceedings of the 46th US Rock Mechanics/Geomechanics
Symposium, Chicago. American Rock Mechanics Association (ARMA); 2012.
Ostadhassan M, Jabbari H, Zamiran S, Osouli A, Bubach B, Oster B. Probabilistic time
dependent thermochemoporoelastic borehole stability analysis in shale formations. In:
Proceedings of the 49th US Rock Mechanics/Geomechanics Symposium. San
Francisco; 2015.
Rafieepour S, Jalalifar H. Drilling optimization based on a geomechanical analysis using
probabilistic risk assessment, a case study from offshore Iran. In: Structures in and on
Rock Masses. CRC Press; 2014. p. 1415–22.
Rafieepour S, Ghotbi C, Pishvaie MR. The effects of various parameters on wellbore
stability during drilling through shale formations. Petroleum Science and Technology
2015a; 33: 1275–85.
Rafieepour S, Jalayeri H, Ghotbi C, Pishvaie MR. Simulation of wellbore stability with
thermohydrochemomechanical coupling in troublesome formations: An example
from Ahwaz oil field, SW Iran. Arabian Journal of Geosciences 2015b; 8: 379–96.
Roshan H, Rahman SS. A fully coupled chemoporoelastic analysis of pore pressure and
stress distribution around a wellbore in water active rocks. Rock Mechanics and Rock
Engineering 2011a; 44: 199–210.
Roshan H, Rahman SS. Analysis of pore pressure and stress distribution around a wellbore
drilled in chemically active elastoplastic formations. Rock Mechanics and Rock
Engineering 2011b; 44: 541–52.
Sherwood JD. Biot poroelasticity of a chemically active shale. Proceedings of the Royal
Society A: Mathematical, Physical and Engineering Sciences 1993; 440: 365–77.
Simpson JP. Studies of the effects of drilling fluid/shale interactions on borehole instability.
Gas Tips 1997; 3(2): 30–6.
Singh A, Rao KS, Ayothiraman R. An analytical solution for a circular wellbore using
MogiCoulomb failure criterion. Journal of Rock Mechanics and Geotechnical
Engineering 2019a; 11(6): 121130.
Singh A, Rao KS, Ayothiraman R. A closedform analytical solution for a circular opening
using DruckerPrager criterion under plane strain condition. Indian Geotechnical
Journal 2019b; 49: 437–54.
Taylor SA, Cary JW. Analysis of the simultaneous flow of water and heat with the
thermodynamics of irreversible processes. In: Proceedings of the 7th International
Conference on Soil Science, Madison. 1960. p. 80–90.
Yu M. Chemical and thermal effects on wellbore stability of shale formations. The
University of Texas at Austin; 2002.
Zamiran S, Osouli A, Ostadhassan M. Geomechanical modeling of inclined wellbore in
anisotropic shale layers of Bakken formation. In: Proceedings of the 48th U.S. Rock
Mechanics/Geomechanics Symposium. American Rock Mechanics Association
(ARMA); 2014.
Zamiran S, Rafieepour S, Ostadhassan M. A geomechanical study of Bakken Formation
considering the anisotropic behavior of shale layers. Journal of Petroleum Science and
Engineering 2018; 165: 56774.
Zhai Z, Zaki KS, Marinello SA, AbouSayed AS. Coupled thermoporomechanical effects
on borehole stability. In: Proceedings of the SPE Annual Technical Conference and
Exhibition. Society of Petroleum Engineers (SPE); 2009.
Zhong R, Miska SZ, Yu M. Modeling of nearwellbore fracturing for wellbore
strengthening. Journal of Petroleum Science and Engineering 2017; 38: 47584.
Zhong R, Miska SZ, Yu M, Ozbayoglu E, Takach NE. An integrated fluid flow and
fracture mechanics model for wellbore strengthening. Journal of Petroleum Science and
Engineering 2018; 167: 70215.
Saeed Rafieepour obtained his MSc degree in Petroleum Engineering from Sharif University of Technology, Iran, in 2008,
and his PhD in
Petroleum Engineering from the University of Tulsa, Tulsa, Oklahoma, USA, in 2017. He is currently affiliated as
an Assistant Professor of
Petroleum Engineering with the University of Tehran, Tehran, Iran. His research interests include (1)
drilling and well completion in partially
depleted reservoirs, (2) experimental and modeling of the thermohydromechanical (THM) processes in poroelastoplastic media, (3)
experimental
poromechanics, (4) wellbore stability in unconventional formations and high pressurehigh temperature (HPHT) environments, and (5) well testing
.
He has been participated in a large number of national and international projects and he has authored (coauthored) over 20 peerreviewed journal
and conference publications.