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A cost-effective chemo-thermo-poroelastic wellbore stability model for mud weight design during drilling through shale formations

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Drilling through chemically-active shale formations is of special importance due to time-dependent drilling fluid-shale 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, sufficiently-accurate, yet highly practical, models are presented containing parameters easily-derived from well-known 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 thermally-induced fluid flow, a model was utilized to predict thermo-osmosis coefficient based on the energy of hydrogen-bond that attained a reasonably-accurate estimation from petrophysical data, e.g. porosity, specific surface area (SSA), and cation exchange capacity (CEC). The coupled chemo-thermo-poroelastic governing equations were developed and solved using an implicit finite difference scheme. Mogi-Coulomb failure criterion was adopted for mud weight required to avoid compressive shear failure and a tensile cut-off 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 time-dependent and a failure at wellbore wall after 85 min of mud-shale 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.
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A cost-effective chemo-thermo-poroelastic wellbore stability model for mud weight
design during drilling through shale formations
Saeed Rafieepour, Siavash Zamiran, Mehdi Ostadhassan
PII: S1674-7755(20)30038-X
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 cost-effective chemo-thermo-
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.
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A cost-effective chemo-thermo-poroelastic 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 chemically-active shale formations is of special importance due to time-dependent drilling fluid-shale 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, sufficiently-accurate, yet highly practical, models are presented containing parameters easily-derived from well-known 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 thermally-induced fluid flow, a model was utilized to predict thermo-osmosis coefficient based on the energy of
hydrogen-bond that attained a reasonably-accurate estimation from petrophysical data, e.g. porosity, specific surface area (SSA), and cation exchange
capacity (CEC). The coupled chemo-thermo-poroelastic governing equations were developed and solved using an implicit finite difference scheme.
Mogi-Coulomb failure criterion was adopted for mud weight required to avoid compressive shear failure and a tensile cut-off 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 time-dependent and a failure
at wellbore wall after 85 min of mud-shale 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: Chemo-thermo-poroelastic wellbore stability; shale-fluid interaction; chemo-osmosis; thermo-osmosis
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 clay-rich 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 shale-fluid interactions that include various processes
such as chemo-osmosis, thermo-osmosis, electro-osmosis, and swelling
due to cation exchanges. Involving such time-delayed 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 oil-based muds especially in offshore environments, the use of
water-based muds is inevitable regardless of cumbersome drilling
problems. The theory of irreversible or non-equilibrium 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 fully-saturated sodium clay-water systems. He also
took advantage of irreversible thermodynamics as a theoretical framework
for the investigation.
*Corresponding Author. E-mail address: saeed-rafieepour@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 near-wellbore pore pressure
increase/effective stress decrease. Earlier works lumped the effects of
solute transport and chemical potential gradient in the chemo-osmosis
analysis. Chemical potential and membrane efficiencies were later
incorporated into the coupled transport relationships (Sherwood, 1993;
Yu, 2002). A complete chemical interaction based on non-equilibrium
thermodynamics and mixture theory was presented by Sherwood (1993).
A chemo-poroelastic 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 fully-coupled chemo-poroelastic model for
estimation of stress profile around the wellbore.
On the other hand, thermal-osmosis has been found to be an influence
factor on fluid flow through soil systems. Derjaguin and Sidorenkov
(1941) developed the physical principles of thermo-osmosis at the
molecular and pore scales. Regarding wellbore stability analysis, thermo-
osmosis has not been taken into account as much as chemo-osmosis 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 thermo-osmosis can be experimentally and theoretically estimated
(or quantified).
In this study, a comprehensive model of transport phenomena was
proposed involving shale-fluid interactions based on non-equilibrium
thermodynamics. The governing equations were presented in cylindrical
coordinate systems and solutions were assumed as non-ideal. Moreover, a
model was used for estimation of thermo-osmosis 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, electro-osmosis, streaming current electro-phoresis, 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, chemo-osmotic, thermo-osmotic, 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. Chemical-osmosis
Osmotic pressure is the pressure required to prevent water from
flowing through a semi-permeable membrane from a solution with low
salt concentration to a solution with high salt concentration. Fig. 2 shows
a semi-permeable 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 water-based 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 oil-based muds or water-based 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
"#
""
(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 r-direction 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 non-selective membranes, we have @
##
'
&
.
Therefore, solute flow in semi-permeable membranes for non-ideal
solutions is given by
&
'
&
?
,
""AB
AC
D@(
"#
'
&
?
,
E
FG
H
I-J
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
;
<
'
&
<=
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 non-ideal shale membrane systems
(
""
,
,
, @). The hydraulic permeability coefficient or the mechanical
filtration coefficient of a given membrane can be expressed as follows:
""
4
5
X
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 osmotically-induced
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-
Meyer-Siever 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 non-perm 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 non-permselective (
,
%) 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. Thermo-osmosis
Gray (1966) observed some pressure build-up 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 thermo-osmosis 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 thermo-osmosis.
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 thermo-osmosis coefficient to obtain
macroscopic one for practical applications using ordinary and available
data. Goncalves and Tremosa (2010) obtained the following equation for
thermo-osmosis process based on the volume averaging method:
"$
"$
\
)
h
]
(21)
where T is the absolute temperature (K), and )his the fluid–solid
interactions-induced 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 half-pore size and half-thickness of highly-ordered water,
respectively; o
kl
&
and o
kl
m
are the average numbers of hydrogen bonds
per water molecule of bulk and highly-ordered 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 half-pore size (b) can be
derived from the following equation having specific surface area (SSA),
considering a plane-parallel 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 thermo-osmosis 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
I-J
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 r-direction and therefore, we have
<
<O
'
1
?
=
<
<=
(
<
<
=
3
(34)
This is the heat diffusivity equation and '
is the thermal diffusivity
coefficient given by'
\
$
WZS
.
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 thermo-poro-elastic 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 chemo-poro-thermoelastic 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 Biot-Willis 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 thermo-poroelastic
effects is expressed as follows (Li, 1998; Yu, 2002):
¤
CC
r
«
¬¬
--
tr?
C
J
|
C
|
t(r
«
¬¬
--
tr?(
C
J
®
C
®
d
C
J
|
C
|
t¯ °(
¤
²³
r?(
C
J
®
C
®
d
C
J
|
C
|
t°´0(
µq¶
q¶
C
|
·9
V
=-O=•=(
¸µ
I
q¶
C
C
J
C
|
·
V
=-O=•=(
.C
J
|
C
|
C
C
J
(37a)
¤
¹¹
r
«
¬¬
--
tr?(
C
J
|
C
|
tr
«
¬¬
--
tr?(
C
J
®
C
®
t¯ °b±
¤
²³
r?(
C
J
®
C
®
t°´0b±
µq¶
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±(
²³
gC
J
|
C
|
°´0b±e
µq¶
q¶
9
V
=-O
¸µ
I
q¶
V
=-O (37c)
¤
¤
¹C
r
«
¬¬
¼
--
¼
tr?
C
J
®
C
®
(b
C
J
|
C
|
t°´0b±(¤
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
¤
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 Mohr-Coulomb (M-C) failure criterion is widely used in the area
of borehole instability analysis. However, this criterion ignores the effect
of intermediate stress component. Al-Ajmi (2006) introduced a poly-axial
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¤
,-
9n-2
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
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 Mogi-Coulomb 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 time-consuming 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 initial-boundary
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 thermo-osmosis 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 thermo-osmosis coefficient through
comparison with laboratory works conducted by Gray (1966) on saturated
clay-water 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
temperature-induced 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 thermo-osmosis 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 temperature-induced pressure (about 12% error).
Then, the thermo-osmosis coefficient is equal to
"$
b†%?Ž?%
q8
.
The negative value for the thermos-osmosis 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
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 thermos-osmosis
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 (1.76-4.41)×10
-8
k m
2
(9.869-39.44)×10
-21
T
#_
|
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 chemo-osmosis (under isothermal
conditions) on the pore pressure with time for
8
.
-
8
?ŒÊ°´-œ0•
.
„•ŒÊ°´
(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 half-pore 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
?ŒÊ°´-œ0•
.
„•ŒÊ°´.
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 (time-dependent) 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 chemo-osmosis 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 chemo-osmosis 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 chemo-osmosis is
more visible than that of thermo-osmosis (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 chemo-osmosis on the pressure profile for the case of
heating and '
žV
Ë'
8
: (a) Plot of pore pressure; and (b) Failure index.
Fig. 8a-c 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 shale-drilling fluid exposure (transport of fluid
into the formation due to hydraulic transport, chemo- and thermo-osmosis
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 thermo-osmosis
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 thermo-osmosis parameters from well-known 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 thermally-induced fluid flow, a model was utilized to
predict thermo-osmosis coefficient based on the energy of hydrogen-bond
that attained a reasonably-accurate estimation from petrophysical data,
e.g. porosity, SSA, and CEC. A coupled chemo-thermo-poroelastic 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 time-dependent 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
nano-particles 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.
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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 thermo-hydro-mechanical (THM) processes in poroelastoplastic media, (3)
experimental
poromechanics, (4) wellbore stability in unconventional formations and high pressure-high temperature (HPHT) environments, and (5) well testing
.
He has been participated in a large number of national and international projects and he has authored (co-authored) over 20 peer-reviewed journal
and conference publications.
... The transport of water through this membrane is governed by the osmotic pressure difference (Pankaj et al., 2016). Thus, osmotic pressure may be described as the pressure necessary to prevent water from flowing through a semi-permeable membrane between two solutions that differ in the amount of solute present in one of the solutions (Rafieepour et al., 2020). ...
Article
Wellbore stability in shale is a recurring crisis during oil and gas well drilling. The adsorption of water and ions from drilling fluid by shale, which causes clay swelling, is the primary cause of wellbore instability. Nanomaterials have been a subject of interest in recent years to be an effective shale inhibitor in drilling fluid, intending to minimize clay swelling. This article presents a comprehensive review of the current progress of nanoparticle role in water-based drilling fluid with regards to wellbore stability, reviewing the experimental methods, the effect of nanoparticles in drilling fluid, the mechanism of shale stability and the outlook for future research. This paper employed a systematic review methodology to highlight the progress of nanoparticle water-based drilling fluids in recent years. Previous studies indicated the current trend for drilling fluid additives was nanoparticles modified with surfactants and polymers, which minimize colloidal stability issues and enhance shale stability. A review of experimental methods showed that the pressure transmission test benefits shale stability assessment under reservoir conditions. Parametric analysis of nanoparticles showed that parameters such as concentration and size directly affected the shale stability even in high salinity solution. However, there is a lack of studies on nanoparticle types, with silica nanoparticles being the most popular among researchers. Nanoparticles enhance shale stability via physical plugging, chemical inhibition, and electrostatic interactions between surface charges. To better comprehend the influence of nanoparticles on shale stabilization, it is necessary to evaluate a wider range of nanoparticle types using the proper experimental techniques.
... Empirical models are based on experience, observation and laboratory tests, while deterministic models rely on analytical solutions and numerical methods. The simplest deterministic method commonly used in wellbore stability is to assess when stress at one or more locations on the wellbore wall exceeds the rock strength (Gholami et al., 2014;Rafieepour et al., 2020). This method assumes a linear elastic model and completely disregards about stress states beyond the rock strength, near the hole away from the wellbore wall and the feasibility of rock detachment. ...
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... Extreme Friction is one from the various problems encountered during oil/gas drilling operations that generate between two contacting bodies such as drill string and wellbore/casing during drill string rotation and/or between bit bearing during their relative motion. The friction generation, which results from the contact between wellbore/casing and drill string during its rotation during drilling called Torque, increases in case of complex wells drilling such as ultra-deep wells and extended reached wells led by deviated or horizontal wells whose need increases year by year all over the world to drill more than one wells from a single site, causes problems like drill string and casing wear, borehole instability, formation damage and pipe sticking, etc. that causes hindrance in safety, drilling efficiency and at extent, drilling failure [1] which further leads to increment in the overall cost of the drilling operation. The Friction problem can be overcome through the use of smooth drilling fluid during drilling that has high lubricity results in lower friction values. ...
... Although numerous researches have been done on the pore structure, interphase interaction, and flow mechanisms in macroscale [22][23][24][25][26], mathematical models capable of accurately characterizing the pore deformation, dynamic capillary force phenomenon, and oil-water two-phase transport at the nanopore scale for shale oil are still missing [27][28][29][30][31]. To solve this problem, Gassmann's method is modified to describe the change of pore radius and roughness in the dynamic pressure field for elastic deformation. ...
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Full-text available
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... This is due to numerous risks of losing the borehole integrity as shale swells when water invasion takes place into rock voids. Since the salinity in shales has a significantly higher ionic strength compared to the drilling mud, a substantial osmotic pressure builds, forcing the mud to be spontaneously sucked into the shale pores (HUANG et al., 2021;Rafieepour et al., 2020). Colloidal particles are introduced into drilling mud to establish a mud cake which prevents water movement towards the region surrounding the wellbore. ...
Chapter
The petroleum industry has been an ever-growing industry. New technologies are always being introduced to encompass the challenges that are encountered. Nanomaterials are being included in these technologies to improve the operation of different processes. Their distinctive physical and chemical characteristics encourage their use in different sectors such as the upstream, midstream, and downstream of the oil and gas industry. In this chapter, the nanomaterials that are utilized in the oil and gas industries are highlighted. Their implementation in various applications is also provided. These applications include hydrocarbon exploration, well drilling and completion, production operations, enhanced oil recovery mechanisms, transportation, and refining operations. There is also a discussion about existing problems and possibilities for future uses.
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Article
Chemical agents capable of sealing pores and micro-fractures can be used to solve wellbore instability in shales. Nanomaterials, as sealing agents, have been applied to test fluids, and their wellbore strengthening effects were evaluated by triaxial tests. A recommended workflow is given to first conduct pore pressure transmission tests, and then perform triaxial tests to evaluate the influence of drilling fluids (sealing agents) on the strength of shales. Due to the low permeability of shales, the small sample (0.25 in. height and one in. diameter) is used in pore pressure transmission tests. This research gives special loading to handle small samples in triaxial testing and innovative calculation procedure to produce stress-strain curves for the shales. After Pore Pressure Transmission (PPT) tests with nanoparticles of various sizes (10, 40 nm), concentrations (3%, 10%), and types (aluminum oxide, magnesium oxide), triaxial tests were performed on Mancos Shale and Eagle Ford Shale. The best combinations for increasing UCS values (Uniaxial Compressive Strengths) are 10% 40-nm Al2O3 for Eagle Ford Shale and 3% 40-nm MgO for Mancos Shale, according to our test matrix. As shown by the experiments, the application of the appropriate nanoparticles on shales will improve their strength and reduce the risk of wellbore collapse. The methodology developed in this research can be extended to triaxial testing on cuttings-size samples, which provides useful rock characterization information for samples that do not meet standard testing conditions.
Article
One way to reduce wellbore instability in shales is to use agents capable of sealing pores and micro-fractures. The main objectives are 1) to perform pore pressure transmission experiments with test fluids containing nanomaterials, and 2) to develop a model to characterize pore pressure transmission with pore plugging. Pore pressure transmission experiments were performed on two types of shales, Mancos Shale and Eagle Ford Shale, using test fluids containing nanoparticles. An orthogonal test matrix has been developed, which includes different nanoparticle sizes (10 nm, 20 nm, 40 nm), particle types (aluminum oxide, magnesium oxide) and particle concentrations (3 %, 10 %). A mathematical model that incorporates chemical potential, pore plugging, and time-dependent permeability is developed to characterize pore pressure response. The best combinations to decrease pore pressure at the equilibrium state, according to the test matrix, are 10 % 10 nm Al2O3 for Eagle Ford Shale and 10 % 30 nm Al2O3 for Mancos Shale. This nanoparticle-based plugging technique will minimize pore pressure transmission and delay the time to equilibrium, reducing hydration and swelling problems in shale formation. As a result, this research will help to better define drilling fluid properties to enhance wellbore stability in actual wells containing shales.
Conference Paper
Frequently, shales are treated as isotropic formations. However, organic rich shales are anisotropic due to their laminated structure and chemical properties. In this study, shale mechanical properties with respect to different bedding plane orientations was studied. The goal of this study is to evaluate anisotropic mechanical properties of shale by triaxial tests and to predict these properties by well logging data along with a 3-D transversely-isotropic wellbore stability analysis (WBS) in Bakken formation. Shale samples were prepared with bedding plane angles equal to 0, 45, and 90 degrees. Young's modulus, shear modulus, and Poisson's ratio in different directions were measured. Parameters of stiffness tensor were calculated by mechanical properties. The compressive strength of shale samples was measured under different confining pressures: 0, 500, 1000, and 1500 psi. Simple Plane of Weakness was applied to describe shear failure mechanism. Well logging data was used to connect experimental and field data. Next, a three-dimensional numerical wellbore stability model based on finite difference method was used to evaluate stress and deformation alterations due to drilling of vertical to extended reach wellbores in Bakken formation, Williston Basin, North Dakota. Both anisotropic and isotropic conditions were considered and the Mohr- constitutive relations were implemented in the numerical model. Vertical Young's modulus (perpendicular to the bedding plane) found to be smaller than horizontal Young's modulus (parallel to the bedding plane). Shale is a transversely isotropic rock; which isotropy plane is generally the bedding plane. The result of tests for stiffness tensor showed that the shale can be characterized by only five independent stiffness constants. The Simple Plane of Weakness model is suitable to estimate shale anisotropic compressive strength. The P-wave velocity calculated from the stiffness tensor is a used to connect the experimental data and field data. P-wave velocity is increasing as the bedding inclination angle increases. The predicted compressional wave velocity for a 45-degree inclination angle showed a perfect fit with the field logging data. Steps of inverse sonic log data to stiffness parameters were shown by a flow chart. From the WBS model, the effects of anisptropy on stress distribution and wellbore convergence was investigated for two different constitutive models: isotropic model (IM) and transversely isotropic model (TIM). It was found that neglecting shale anisotropy will lead to erroneous prediction of wellbore deformation.
Article
Full-text available
Deep wellbores/boreholes are generally drilled into rocks for oil and gas exploration, monitoring of tectonic stresses purposes. Wellbore and tunnel in depth are generally in true triaxial stress state, even if the ground is under axisymmetric loading condition. Stability of such wellbores is very critical and collapse of wellbore must be avoided. Mogi-Coulomb failure criterion is a better representation of rock strength under true triaxial condition. In this paper, an analytical solution is proposed using Mogi-Coulomb failure criterion. The solution is obtained for rock mass exhibiting elastic-perfectly plastic or elastic-brittle-plastic behaviour considering in-plane isotropic stresses. The proposed solution is then compared with exact analytical solution for incompressible material and experimental results of thick-wall cylinder. It is shown that the results obtained by the proposed analytical solution are in good agreement with the experimental results and exact analytical solution. A reduction of about 13%–20% in plastic zone from the proposed closed-form solution is observed, as compared to the results from the finite element method (FEM) based Mohr-Coulomb criterion. Next, the influences of various parameters such as Poisson’s ratio, internal pressure (mud weight), dilation angle, and out-of-plane stress are studied in terms of stress and deformation responses of wellbore. The results of the parametric study reveal that variation in the out-of-plane stress has an inverse relation with the radius of plastic zone. Poisson’s ratio does not have an appreciable influence on the tangential stress, radial stress and radial deformation. Dilation angle has a direct relation with the deformation. Internal pressure is found to have an inverse relation with the radial deformation and the radius of plastic zone.
Article
Full-text available
Wellbore and tunnel problems are of true triaxial stress state, even if the ground is under axisymmetric loading condition. A closed-form analytical solution is proposed using Drucker-Prager failure criterion. The solutions are obtained for rock-mass exhibiting elastic perfectly plastic or elastic-brittle-plastic behaviour. The proposed solution is then compared with the Finite Element analysis (FE-analysis) results. Parametric studies are also carried out. The results of the proposed analytical solution are found to be in good agreement with the FE-analysis results. The proposed analytical solution can thus be used for predicting the stresses and deformation of underground circular openings considering true triaxial stress state.
Article
Full-text available
Analysis of wellbore stability is such a key factor to have a successful drilling operation. Induced stresses are one of the main factors affecting wellbore instability and associated problems in drilling operations. These stresses are significantly impacted by pore pressure variation and thermal stresses in the fields. In order to eliminate wellbore instability problems, it is important to investigate the mechanisms of rock-fluid interaction with respect to thermal and mechanical aspects. In order to estimate the induced stresses, different mathematical models have been carried out. In this study, the field equations governing the problem have been derived based on the thermo-poroelastic theory and solved analytically in Laplace domain. The results are transferred to time domain using Fourier inverse method. Finite difference method is also utilized to validate the results. Pore pressure and temperature distribution around the wellbore have been focused and simulated here. Afterward, induced radial and tangential stresses for different cases of cooling and heating of formation are compared. In addition, the differences between thermo-poro-elastic and poro-elastic models in situation of permeable and impermeable wellbore are described. It is observed that cooling and pore pressure distribution reinforce the induced radial stress. Whereas, cooling can be a tool to control and reduce tangential stress induced due to invasion of drilling fluid. In the next step, safe mud window is obtained using Mohr-Coulomb, Mogi-Coulomb, and modified Lade failure criterion for different inclinations. Temperature and pore pressure distribution does not change the minimum allowable wellbore pressure significantly. However, upper limit of mud window is sensitive to induced stresses and it seems vital to consider changes in temperature and pressure to avoid any failures. The widest and narrowest mud windows are proposed by modified Lade and Mohr-Coulomb failure criterion, respectively.
Article
The traditional way of analyzing wellbore stability in depleted reservoirs is by assuming homogenous depletion and uniaxial strain. Then the analytical solution of stress path is obtained. Pore pressure depletion, however, is location and time dependent. The objective of this study is to develop a workflow for infill well design. More specifically, a wellbore stability analysis is developed for infill wells using a coupled geomechanics and reservoir simulator (CGRS) which considers lateral displacement and inhomogeneous depletion. Different shear failure criteria are utilized in a new CGRS wellbore stability model. The upper bounds of shear failure are given by Drucker-Prager Inscribes and Griffith Theory, while the lower bound is given by Drucker-Prager Circumscribe. CGRS wellbore stability model is compared with conventional wellbore stability model (based on the analytical solution of stress path). CGRS wellbore stability can quantify the influence of azimuth on minimum and maximum mud weight during the depletion when initial maximum horizontal stress equals minimum horizontal stress. In addition, CGRS can give the output of a location-dependent mud weight map for the entire reservoir. Neither of the above two functions can be realized by conventional wellbore stability model. Therefore, CGRS can provide dynamic information concerning where to drill, when to drill, and how to drill for infill wells.
Article
Fracture-based wellbore strengthening techniques are preventive methods that can reduce the cost of lost circulation and non-productive time. The mud weight window can be extended by plugging fractures with wellbore strengthening materials (WSM) in the near-wellbore region. To maximize the strengthening effect, accurate fracture geometry prediction is of critical importance to the design of WSM. This paper presents a novel, coupled fluid flow and fracture mechanics model for wellbore strengthening applications that accounts for near-wellbore-induced fracture behavior. For fluid flow, mass conservation is considered and momentum conservation is examined; the latter shows that pressure loss with near-wellbore fracturing is low. Thus, we can neglect the pressure drop in the fractures and assume the fluid pressure inside the fractures is equal to the wellbore pressure. The pressure-width relationship (rock elastic deformation) and stress intensity factor are obtained by a dislocation-based approach. For the fracture propagation criterion, the calculated stress intensity factor is compared with fracture toughness at each time step. The stress intensity factor and fracture reopening pressure (FROP) are verified with Tada's model and Feng's model, respectively. Then, simulation results are compared with the large leak-off solutions of the Perkins-Kern-Nordgren (PKN) fracture model. The simulation results reveal that the PKN model overestimates the fracture mouth width, fracture length, and wellbore pressure. Furthermore, the simulation results of wellbore pressure show a different trend. Therefore, we cannot directly use the PKN model to design wellbore strengthening applications. The main reason is the presence of wellbore can generate near-wellbore effects that cannot be disregarded. Finally, we conduct a comprehensive parametric study (i.e., fracture toughness, Young's modulus, Poisson's ratio, horizontal stress ratio, and permeability) on wellbore strengthening fracturing. The proposed model is useful for wellbore strengthening applications using the intentionally induced fractures (i.e., near-wellbore fracturing). Particle size distribution (PSD) of WSM can be designed based on the simulated fracture geometry. No complex model mesh generation or assignment of boundary conditions are needed, which are commonly used in finite element simulation or other numerical methods. The proposed model can also be used to optimize wellbore strengthening operations by performing sensitivity analysis.
Article
While drilling through depleted or partially depleted reservoirs, one may encounter a series of problems (e.g., lost circulation, non-productive time, etc.) due to narrow mud weight window (MWW). Fracture-based strengthening techniques used in the industry effectively increase fracture reopening pressure (FROP) and ultimately reduce the cost of associated problems. However, traditional analytical and numerical studies using these techniques do not consider the time effect and usually ignore the fluid dynamics. Thus, these deficiencies may result in inaccurate wellbore strengthening operations if no proper fracture diagnostic techniques are available to acquire the real-time fracture geometry. In this paper, a quasi-static, dislocation-based fracture model is extended using fluid mass conservation with leak-off. A fixed dimensionless fracture coordinate system is employed and a numerical simulation procedure is developed. The model is capable of predicting real-time fracture geometry (both fracture width and length) from given wellbore conditions. Hence, it could provide optimal particle size distribution (PSD) selection for wellbore strengthening applications. Two case studies are performed and results reveal different fluid controlling mechanisms during the fracture propagation, which are fluid storage in fractures and fluid leak-off to surrounding formation. Drilling through low permeability reservoirs (e.g., tight gas reservoirs or shale reservoirs) is different from drilling in conventional reservoirs because fast propagation of induced fractures may interact with natural fractures and result in severe lost circulation. This model is of critical importance in designing wellbore strengthening operations during drilling.