-
Research ArticleSimulation of Wellbore Stability
duringUnderbalanced Drilling Operation
Reda Abdel Azim
Chemical and Petroleum Engineering Department, American
University of Ras Al Khaimah, Ras Al Khaimah, UAE
Correspondence should be addressed to Reda Abdel Azim;
[email protected]
Received 13 June 2017; Accepted 2 July 2017; Published 15 August
2017
Academic Editor: Myung-Gyu Lee
Copyright © 2017 Reda Abdel Azim. This is an open access article
distributed under the Creative Commons Attribution License,which
permits unrestricted use, distribution, and reproduction in any
medium, provided the original work is properly cited.
The wellbore stability analysis during underbalance drilling
operation leads to avoiding risky problems. These problems
include(1) rock failure due to stresses changes (concentration) as
a result of losing the original support of removed rocks and (2)
wellborecollapse due to lack of support of hydrostatic fluid
column. Therefore, this paper presents an approach to simulate the
wellborestability by incorporating finite element modelling and
thermoporoelastic environment to predict the instability
conditions.Analytical solutions for stress distribution for
isotropic and anisotropic rocks are presented to validate the
presented model.Moreover, distribution of time dependent shear
stresses around the wellbore is presented to be compared with rock
shear strengthto select appropriate weight of mud for safe
underbalance drilling.
1. Introduction
Very recent studies highlighted that the wellbore
instabilityproblems cost the oil and gas industry above
500$–1000$million each year [1]. The instability conditions are
related torocks response to stress concentration around the
wellboreduring the drilling operation. That means the rock
maysustain the induced stresses and the wellbore may remainstable
without collapse or failure if rock strength is enormous[2].
Factors that lead to formation instability are coming fromthe
temperature effect (thermal) which is thermal diffusivityand the
differences in temperature between the drilling mudand formation
temperature.This can be described by the factthat if the drilling
mud is too cold, this leads to decreasingthe hoop stress.These
variations in hoop stress have the sameeffect of tripping while
drilling which generates swab andsurge and may lead to both tensile
and shear failure at thebottom of the well.
The interaction between the drilling fluidswith formationfluid
will cause pressure variation around the wellbore, whichresults in
time dependent stresses changes locally [3]. There-fore, in this
paper the interaction between geomechanics andformation fluid [4]
is taken into consideration to analyze timedependent rocks
deformation around the wellbore.
Another study shows that the two main effects causingcollapse
failure are as follows: (1) poroelastic influence ofequalized pore
pressure at the wellbore wall and (2) thethermal diffusion between
wellbore fluids and formationfluids [3–5].
Numerous scientists presented powerful models to simu-late the
effect of poroelastic, thermal, and chemical effects byvarying
values of formation pore pressure, rock failure situa-tion, and
critical mud weight [3, 6].These models mentionedthat controlling
the component of the water present in thedrilling fluid results in
controlling thewellbore stability.Moreor less, there are many
parameters that could be controlledduring the drilling operation as
unfavorable in situ condition[7, 8]. In addition, mud weight
(MW)/equivalent circulationdensity (ECD), mud cake (mud filtrate),
hole inclination anddirection, and drilling/tripping practice are
considered themain parameters that affect wellbore mechanical
instability[9, 10].
The factors that affect the mechanical stability aremembrane
efficiency, water activity interaction betweenthe drilling fluid
and shale formation, the thermal expan-sion, thermal diffusivity,
and the differences in temperaturebetween the drilling mud and
formation temperature [11, 12].
HindawiJournal of Applied MathematicsVolume 2017, Article ID
2412397, 12 pageshttps://doi.org/10.1155/2017/2412397
https://doi.org/10.1155/2017/2412397
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2 Journal of Applied Mathematics
This paper presents a realistic model to evaluate
wellborestability and predict the optimum ECD window to
preventwellbore instability problems.
2. Derivation of Governing Equation forThermoporoelastic
Model
The equations used to simulate thermoporoelastic couplingprocess
are momentum, mass, and energy conservation.These equations are
presented in detail in this section.
2.1. Momentum Conservation. The linear momentum bal-ance
equation in terms of total stresses can be written asfollows:
∇ ⋅ 𝜎 + 𝜌𝑔 = 0, (1)where 𝜎 is the total stress, 𝑔 is the gravity
constant, and 𝜌 isthe bulk intensity of the porous media. The
intensity shouldbe written for two phases, liquid and solid, as
follows:
𝜌 = 𝜑𝜌𝑙 + (1 − 𝜑) 𝜌𝑠. (2)Equation (1) can be written in terms of
effective stress asfollows:
∇ ⋅ (𝜎 − 𝑝𝐼) + 𝜌𝑔 = 0, (3)where 𝜎 is the effective stress, 𝑝 is
the pore pressure, and𝐼 is the identity matrix. This equation for
the stress-strainrelationship does not contain thermal effects and,
to includethe thermoelasticity, the equation can be written as
follows:
𝜎 = 𝐶 (𝜀 − 𝛼𝑇Δ𝑇 × 𝐼) , (4)where 𝐶 is the fourth-order stiffness
tensor of materialproperties, 𝜀 is the total strain, 𝛼𝑇 is the
thermal expan-sion coefficient, and Δ𝑇 is the temperature
difference. Theisotropic elasticity tensor 𝐶 is defined as
𝐶 = 𝜆𝛿𝑖𝑗𝛿𝑘𝑙 + 2𝐺𝛿𝑖𝑘𝛿𝑗𝑙, (5)where 𝛿 is the Kronecker delta and 𝜆
is the Lame constant. 𝐺is the shear modulus of elasticity. The
constitutive equationfor the total strain-displacement relationship
is defined asfollows:
𝜀 = 12 (∇→𝑢 + (∇→𝑢)𝑇) , (6)
where →𝑢 is the displacement vector and ∇ is the
gradientoperator.
2.2. Mass Conservation. The fluid flow in deformable
andsaturated porous media can be described by the
followingequation:
𝑆𝑠 𝜕𝑝𝜕𝑡 + 𝛽∇ ⋅ (𝜕→𝑢𝜕𝑡 ) + ∇ ⋅ 𝑞 − 𝛼𝑇 𝜕𝑇𝜕𝑡 = 𝑄, (7)
where 𝛽 is the Biots coefficient and assumed to be = 1.0 in
thisstudy, 𝑝 is the pore fluid pressure, 𝑇 is the temperature, 𝛼𝑇
is
the thermal expansion coefficient, 𝑞 is the fluid flux, and 𝑄
isthe sink/source, and 𝑆𝑠 is the specific storage which is
definedby
𝑆𝑠 = (1 − 𝜑𝐾𝑠 ) + (𝜑𝐾𝑙) , (8)
where 𝐾𝑠 is the compressibility of solid and 𝐾𝑙 is
thecompressibility of liquid. The fluid flux term (𝑞) in the
massbalance in (7) can be described by usingDarcy’s flow
equationbecause the intensity has been assumed constant in this
study:
𝑞 = −𝑘𝜇 (∇𝑝 − 𝜌→𝑔) , (9)where 𝑘 is the permeability of the
domain. The Cubic law isused in determining fracture
permeability.
2.3. Energy Conservation. The energy balance equation forheat
transport through porous media can be described asfollows:
(𝜌𝑐𝑝)eff 𝜕𝑇𝜕𝑡 + ∇ ⋅ 𝑞𝑇 = 𝑄𝑇, (10)where 𝑞𝑇 is the heat flux,𝑄𝑇 is
the heat sink/source term, and𝜌𝑐𝑝 is the heat storage and
equals
(𝜌𝑐𝑝)eff = 𝜑 (𝑐𝑝𝜌)liquid + (1 − 𝜑) (𝑐𝑝𝜌)solid . (11)In this
study, conduction and convection heat transfers areconsidered
during numerical simulation. The heat flux termin (10) can be
written as
𝑞𝑇 = −𝜆eff∇𝑇 + (𝑐𝑝𝜌)liquid V ⋅ 𝑇, (12)where V is the velocity of
the fluid. The first term on theright hand side of (12) is the
conduction term and the secondterm is the convective heat transfer
term and 𝜆eff is theeffective heat conductivity of the porous
medium, which canbe defined as
𝜆eff = 𝜑𝜆liquid + (1 − 𝜑) 𝜆solid. (13)2.4. Discretization of the
Equations. First one discretizesthe thermoporoelastic governing
equations by using Greens’theorem [13] to derive equations weak
formulations. Theweak form of mass, energy, and momentum balance in
(1),(7), and (10) can be written as follows, respectively:
∫Ω𝑤𝑆𝑠 𝜕𝑝𝜕𝑡 𝑑Ω + ∫Ω 𝑤𝑇𝛼∇ ⋅
𝜕→𝑢𝜕𝑡 𝑑Ω + ∫Ω 𝑤𝛽𝜕𝑇𝜕𝑡
− ∫Ω∇𝑤𝑇 ⋅ 𝑞𝐻𝑑Ω + ∫
Γ𝑞𝐻
𝑤 (𝑞𝐻 ⋅ 𝑛) 𝑑Γ− ∫Ω𝑤𝑄𝐻𝑑Ω = 0,
(14)
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Journal of Applied Mathematics 3
∫Γ𝑑
𝑤𝑏𝑚𝑆𝑠 𝜕𝑝𝜕𝑡 𝑑Γ + ∫Γ𝑑 𝑤𝛼𝜕𝑏𝑚𝜕𝑡 𝑑Γ + ∫Γ𝑑 𝑤𝛽
𝜕𝑇𝜕𝑡 𝑑Γ− ∫Γ𝑑
∇𝑤𝑇 ⋅ (𝑏ℎ𝑞𝐻) 𝑑Ω + ∫Γ𝑞𝐻
𝑤𝑏ℎ (𝑞𝐻 ⋅ 𝑛) 𝑑Γ+ ∫Γ𝑑
𝑤𝑞+𝐻𝑑Γ + ∫Γ𝑑
𝑤𝑞−𝐻𝑑Γ = 0,(15)
∫Ω𝑤𝑐𝑝𝜌𝜕𝑇𝜕𝑡 𝑑Ω + ∫Ω 𝑤𝑐𝑝𝜌𝑞𝐻 ⋅ ∇𝑇𝑑Ω− ∫Ω∇𝑤𝑇 ⋅ (−𝜆∇𝑇) 𝑑Ω + ∫
Γ𝑞
𝑇
𝑤 (−𝜆∇𝑇 ⋅ 𝑛) 𝑑Γ− ∫Ω𝑤𝑇𝑄𝑇𝑑Ω = 0,
(16)
∫Γ𝑑
𝑤𝑏𝑚𝑐𝑙𝑝𝜌𝑙 𝜕𝑇𝜕𝑡 𝑑Γ + ∫Γ𝑑 𝑤𝑐𝑙𝑝𝜌𝑙𝑏ℎ𝑞𝐻 ⋅ ∇𝑇𝑑Γ
− ∫Γ𝑑
∇𝑤𝑇 ⋅ (−𝑏𝑚𝜆𝑙∇𝑇) 𝑑Γ+ ∫Γ𝑞
𝑇
𝑤(−𝑏𝑚𝜆𝑙∇𝑇 ⋅ 𝑛) 𝑑Γ + ∫Γ𝑑
𝑤𝑞+𝑇𝑑Γ+ ∫Γ𝑑
𝑞−𝑇𝑑Γ = 0,
(17)
∫Ω∇𝑠𝑤𝑇 ⋅ (𝜎 − 𝛼𝑝𝐼) 𝑑Ω − ∫
Ω𝑤𝑇 ⋅ 𝜌𝑔𝑑Ω
− ∫Γ𝑡
𝑤𝑇 ⋅ →𝑡 𝑑Γ − ∫Γ𝑑
𝑤+𝑇 ⋅ →𝑡 +𝑑𝑑Γ− ∫𝑤−𝑇 ⋅ →𝑡 −𝑑𝑑Γ = 0,
(18)
where 𝑤 is the test function, Ω is the model domain, Γ isthe
domain boundary, 𝑡 is the traction vector, superscripts+/− refer to
the value of the corresponding parameters onopposite sides of the
fracture surfaces, respectively, 𝑆𝑠 is thespecific storage, 𝑛 is
the porosity, 𝑞𝐻 is the volumetric Darcyflux, 𝛽 is the thermal
expansion coefficient, 𝑄𝐻 is the fluidsink/source term between the
fractures, 𝑞𝑇 is the heat flux,𝑐𝑝 is the specific heat capacity, 𝑏𝑚
and 𝑏ℎ are mechanicaland hydraulic fracture apertures, 𝑄𝑇 is the
heat sink/sourceterm, 𝛼 is the thermal expansion coefficient, 𝜆 is
the thermalconductivity, and 𝑑 refers to the fracture plane.
Then the Galerkin method is used to spatially discretizethe weak
forms of (14) to (18). The primary variables of thefield problem
are pressure 𝑝, temperature 𝑇, and displace-ment vector 𝑢. All of
these variables are approximated byusing the interpolation function
in finite element space asfollows:
𝑢 = 𝑁𝑢𝑢,𝑝 = 𝑁𝑝𝑝,𝑇 = 𝑁𝑇𝑇,
(19)
Table 1: Reservoir inputs used for validation of poroelastic
numer-ical model using circular homogenous reservoir.
Parameter ValuePoisson ratio 0.2Young’s modulus 40GPaMaximum
horizontal stress 40MPa (5800 psi)Minimum horizontal stress 37.9MPa
(5500 psi)Wellbore pressure (𝑃𝑤) 6.89MPa (1000 psi)Initial
reservoir pressure (𝑃𝑖) 37.9MPa (5500 psi)Fluid bulk module (𝐾𝑓)
2.5 GPaFluid compressibility 1.0 × 10−5 Pa−1Biot’s coefficient
1.0Fluid viscosity 3 × 10−4 Pa⋅sMatrix permeability 9.869 × 10−18m2
(0.01md)Wellbore radius 0.1mReservoir outer radius 1000m
1000
800
600
400
200
010008006004002000
Y
ℎ
H
X
Figure 1: Two-dimensional circular reservoir shape used for
valida-tion of poroelastic numerical model with 𝜎𝐻 = 39.9MPa and 𝜎ℎ
=37.9MPa, 𝑃𝑟 = 37.9MPa, and Δ𝑝 = 31MPa.
where𝑁 is the corresponding shape function and 𝑢, 𝑝, and 𝑇are
the nodal unknowns values.
3. Validation of Poroelastic Numerical Model
The verification of poroelastic numerical model against
ana-lytical solutions (see Appendix) is presented in this section.
Atwo-dimensional model of circular shaped reservoir with anintact
wellbore of 1000m drainage radius and 0.1m wellboreradius is used
(see Figure 1).The reservoir input data used arepresented in Table
1. The numerical model is initiated withdrained condition obtained
by using Kirsch’s problem [14].These conditions with the analytical
solution equations for
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4 Journal of Applied Mathematics
Input data
Mesh construction and element numbering
Solve for temperature at element corners only based on the
previous calculated pore pressure
Solve for displacement and pore pressure
Apply patch convergent method to
calculate stress distribution at nodes
Required time step
reached?
Calculate stress
distribution at Gaussian
points
No
Iteration
Iterationconverge
Yes
No
Loop over each iteration required to
converge
Yes Print the results
< 0.0001
Figure 2: Flow chart describes how the nodal unknowns are solved
using iterations process.
drained condition for the given pore pressure, displacement,and
stresses [15, 16] are presented in the Appendix. Flowchart
describes the solution process for pressure and dis-placement for
poroelastic model and also for temperature forthermoporoelastic
frameworks is presented in Figure 2. Thenumerical results obtained
are plotted against the analyticalsolutions in Figures 3–6.
For the verification purpose, a number of assumptions
aremade.
Initial State. In this study, zero time (initial state) is
assumedto represent drained situation in which pore pressure
isstabilized.
Boundary Conditions. They are boundary conditions for
theporoelastic model in this model.
Rock and Fluid Properties. In the numerical model,
Young’smodulus, Poisson’s ratio, porosity, permeability, and total
sys-tem compressibility as well as viscosity of fluid are
assumed
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Journal of Applied Mathematics 5
0
1000
2000
3000
4000
5000
6000
Pore
pre
ssur
e (ps
i)
1 Day_Analytical1 Month_Analytical1 Year_Analytical
1 Day_Numerical1 Month_Numerical1 Year_Numerical
1 10 100 10000.1Radial position along x- axis (m)
1 hr_Analytical1 hr_Numerical
Figure 3: Pore pressure as a function of radius and time
inporoelastic medium with 𝜎𝐻 = 5800 psi and 𝜎ℎ = 5500 psi, 𝑃𝑟 =5500
psi, 𝑃𝑤 = 1000 psi, 𝑘𝑥 = 0.01md, and 𝑘𝑦 = 0.01md.
0
0.002
0.004
0.006
0.008
0.01
x, d
ispla
cem
ent (
m)
1 10 100 10000.1Radial position along x-axis (m)
1 Day_Analytical1 Month_Analytical1 Year_Analytical
1 Day_Numerical1 Month_Numerical1 Year_Numerical
1 hr_Analytical1 hr_Numerical
Figure 4: 𝑋-displacement along 𝑥-axis as a function of time
inporoelastic medium with 𝜎𝐻 = 5800 psi and 𝜎ℎ = 5500 psi, 𝑃𝑟 =5500
psi, 𝑃𝑤 = 1000 psi, 𝑘𝑥 = 0.01md, and 𝑘𝑦 = 0.01md.
to be independent of time and space in order to be
consistentwith the analytical solutions.
As can be seen from Figure 3, the numerical resultsmatch well
with the analytical solutions. Due to discontinuityof initial state
and the first time step in the numericalprocedure a small mismatch
is observed between numerical
0
1000
2000
3000
4000
5000
6000
7000
Radi
al st
ress
(Psi)
1 10 100 10000.1Radial position along x axis (m)
1 Day_Analytical1 Month_Analytical1 Year_Analytical
1 Day_Numerical1 Month_Numerical1 Year_Numerical
1 hr_Analytical 1 hr_Numerical
Figure 5: 𝑋-component of radial stresses as a function of time
inporoelastic medium with 𝜎𝐻 = 5800 psi and 𝜎ℎ = 5500 psi, 𝑃𝑟 =5500
psi, 𝑃𝑤 = 1000 psi, 𝑘𝑥 = 0.01md, and 𝑘𝑦 = 0.01md.
4000
4500
5000
5500
6000
6500
7000
Hoo
p str
ess (
Psi)
1 10 100 10000.1Radial position along x-axis (m)
1 Day_Analytical1 Month_Analytical1 Year_Analytical
1 Day_Numerical1 Month_Numerical1 Year_Numerical
1 hr_Analytical 1 hr_Numerical
Figure 6: 𝑋-component of tangential stresses as a function of
timein poroelastic medium with 𝜎𝐻 = 5800 psi and 𝜎ℎ = 5500 psi, 𝑃𝑟
=5500 psi, 𝑃𝑤 = 1000 psi, 𝑘𝑥 = 0.01md, and 𝑘𝑦 = 0.01md.
and analytical solutions for 𝑡 = 1 hr. It is evident that, foran
initial drained condition and horizontal permeabilityanisotropy, no
directional dependence of the change in porepressure is observed
despite the anisotropic horizontal stressstate.
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6 Journal of Applied Mathematics
3
2
1
00 0.5 1 1.5 2 2.5 3 3.5 4
Y
X
Figure 7: Pore pressure contour map after 1 hr of fluid
production(for near-wellbore region) with 𝜎𝐻 = 5800 psi and 𝜎ℎ =
5500 psi, 𝑃𝑟= 5500 psi, 𝑃𝑤 = 1000 psi, 𝑘𝑥 = 0.01md, and 𝑘𝑦 =
0.01md.
In Figure 4 the numerical results for displacement have asmall
mismatch with the analytical solutions. This is due tothe method
(Patch Recovery Method) that has been used todistribute initial
reservoir displacement and calculating thechange in in situ
stresses with time.
In Figure 5, the numerical results show a good agreementwith the
exact solutions for different time and orientations.For all cases,
as expected, 𝜎𝑥 approaches the maximumhorizontal in situ stress
(5800 psi) at far field (away fromwellbore). The discontinuity of
𝜎𝑥 at wellbore wall is dueto the imposed pressure boundary
condition. It is assumedthat wellbore pressure is equal to the
reservoir pressure atzero time in order to simulate drained initial
state. It is alsoobserved that as time progresses, the size of the
area, which isaffected by the change in 𝜎𝑥, increases. This is due
to changein pore pressure.
In Figure 6 numerical results of 𝜎𝑦 match well with thatof the
analytical solutions for different time. As expected, 𝜎𝑦approaches
the minimum horizontal in situ stress (5500 psi)at far field (away
from wellbore).
The results of pore pressure and effective stress after onehour
of production are presented in Figures 7–9 in reservoirentire
region. These figures (Figures 7–9) clarify how thepressure and
stresses are changing from the wall of thewellbore to the reservoir
boundary.
4. Failure Criteria
Shear failure will occur if
𝜎3 < −𝑇0, (20)where 𝜎3 the lowest principle is effective
stress and 𝑇0 is therock tensile strength.
Y
X
0.8
0.6
0.4
0.2
010.80.60.40.20
Figure 8: 𝑋-component of radial stresses contour map after 1 hr
offluid production (for near-wellbore region) with 𝜎𝐻 = 5800 psi
and𝜎ℎ = 5500 psi, 𝑃𝑟 = 5500 psi, 𝑃𝑤 = 1000 psi, 𝑘𝑥 = 0.01md, and 𝑘𝑦
=0.01md.
Y
X
0.8
0.6
0.4
0.2
010.80.60.40.20
Figure 9:𝑋-component of tangential stresses contourmap after 1
hrof fluid production (for near-wellbore region) with 𝜎𝐻 = 5800
psiand 𝜎ℎ = 5500 psi, 𝑃𝑟 = 5500 psi, 𝑃𝑤 = 1000 psi, 𝑘𝑥 = 0.01md,
and 𝑘𝑦= 0.01md.
Using Mohr-Coulomb criteria, shear failure criteria aremet
when
𝜏net = 12 cos𝜑 (𝜎1 (1 − sin𝜑) − 𝜎3 (1 + sin𝜑)) > 𝑆0,
(21)where 𝜎1 is the highest principle effective stress, 𝜏net is
thenet shear stress, 𝜑 is the angle of internal friction, and 𝑆0
isthe rock shear strength. Once the maximum principle
stresssurpasses the rock shear strength, rock failure takes place
atthewellbore.Therefore, evaluating the highest principle
stress
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Journal of Applied Mathematics 7
Overbalanced,support pressure
Underbalanced, no support pressure
11
33 PwPw
Shear yielding
Figure 10: Shear yielding occurs for underbalanced conditions
due to the absence of a support pressure on the borehole wall
[16].
is important criterion to predict rock failure for analysis
ofwellbore stability [16].
Drilling with underbalanced technique where thebottom-hole
pressure is lower than the formation porepressure regularly
promotes borehole instability. Thus, itis important to design and
determine the ideal range ofthe bottom-hole pressure during
underbalanced drillingoperation, to avoid generating hydraulic
fractures, differentialsticking, or undesirable level of formation
damage [17] (seeFigure 10).
5. Case Study
This test case has been taken from a field located in
southernpart of Iran. The operator is considered a well drilled at
anapproximate depth of 4000 ft. the recorded pore pressuregradient
from the DST test analysis is 7.7 lb/gal. The rockmechanical data
used for wellbore stability analysis aredetermined from triaxial
test on core samples and presentedin Table 2. The wellbore
stability analysis has been executedusing underbalance technique.
Therefore, the reduction inpore pressure during the drilling
process will directly affectthe horizontal and shear stresses. To
avoid either loss ofcirculation problems or borehole failure, the
mud pressureshould be less than the formation fracture pressure
andgreater than its collapse pressure. Therefore, it is mandatoryto
predict the changes of stresses values as reservoir pressuredrops.
In this case study, the mud weight recommended to beused is 5
lb/gal.
To do this analysis, a finite element mesh has beengenerated as
the one presented in Figure 1 and it was refinedaround the
wellbore. The boundary conditions have beenassigned to the model
and Mohr-Coulomb criteria [18] areused for the simulation to
predict the stresses and porepressure distribution with time around
the wellbore.
6. Results and Discussion
Breakout shear failure occurred during underbalance
drillingoperation; therefore, it is very important to predict the
failureat the wellbore wall using failure criteria. Because of
pore
Table 2: Case study input data.
Mechanical parametersPoisson ratio (]) 0.2Bulk Young’s modulus
(𝐸) 4.4GPaMaximum horizontal stress (𝜎𝐻) 16.8MPa (2436 psi)Minimum
horizontal stress (𝜎ℎ) 14MPa (2030 psi)Wellbore pressure (𝑃𝑤)
6.89MPa (1000 psi)Hydraulic parametersInitial reservoir pressure
(𝑃𝑖) 11.1MPa (1610 psi)Fluid bulk module (𝐾𝑓) 0.45GPaFluid
compressibility 1.0 × 10−5 Pa−1Biot’s coefficient 1.0Physical
parametersFluid viscosity 3 × 10−4 Pa⋅sFluid density 1111 kg/m3
Matrix permeability 9.869 × 10−18m2(0.01md)Porosity (𝜙)
0.1Wellbore radius 0.15mReservoir outer radius 1000mFormation
temperature (𝑇𝑓) 375 oKDrilling mud temperature (𝑇𝑚) 330 oKThermal
osmosis coefficient (𝐾𝑇) 1 × 10−11m2/s KThermal expansion
coefficient of fluid (𝛼𝑓) 3 × 10−4 1/KThermal diffusivity (𝐶𝑇) 1.1
× 10−6m2/sThermal expansion coefficient of solid (𝛼𝑠) 1.8 × 10−5
1/K
pressure dissipation, the failure becomes time dependent asthe
net shear stress increases with time at the borehole wall.
It can be seen from Figure 11 that the net shear stress isthe
lowest for the time before starting of the underbalancedrilling
operation.Then, at thewellborewall, it can be noticedthat the net
shear stress drops suddenly after 4 s of the drillingoperation. In
addition, at this time, the net shear stress valueis higher than
net shear stress for long time of the drilling
-
8 Journal of Applied Mathematics
Initial
10 days
−1
0
1
2
3
4
5
6
7
Net
shea
r stre
ss (M
Pa)
0.2 0.25 0.3 0.35 0.4 0.45 0.50.15Position on the y-axis (m)
30 hr8min
30 s7.5 s4 s
Figure 11: 𝑌-component of net shear stresses as a function of
timein poroelastic medium with 𝜎𝐻 = 16.8MPa and 𝜎ℎ = 14MPa, 𝑃𝑟
=11MPa, 𝑃𝑤 = 9.7MPa, 𝑘𝑥 = 0.01md, 𝑘𝑦 = 0.01md, and 𝑇𝑚 = 330K.
0
2
4
6
8
10
12
14
16
Max
imum
hor
izon
tal s
tress
(MPa
)
0.2 0.25 0.3 0.35 0.4 0.45 0.50.15Position on the y-axis (m)
Initial
10 days30 hr8min
30 s7.5 s4 s
Figure 12: 𝑌-component of maximum horizontal stresses as
afunction of time in poroelastic medium with 𝜎𝐻 = 16.8MPa and 𝜎ℎ=
14MPa, 𝑃𝑟 = 11MPa, 𝑃𝑤 = 9.7MPa, 𝑘𝑥 = 0.01md, 𝑘𝑦 = 0.01md, and𝑇𝑚=
330K.
operation. This effect of short time of drilling operation onthe
net shear stress value is uncertain, as this time may betoo short
to allow failure to be devolved. But, in this casestudy, by
comparing the net shear stress value with the rockshear strength,
we found its value lower than the rock shearstrength (14MPa). This
proves that the failure will not occurusing mud weight of 5 lb/gal.
Figures 12, 13, and 14 show 𝑦-stresses at the wellbore wall.
Distribution of the net shear stress along 𝑦-axis for thecooling
(𝑇𝑚 < 𝑇𝑓) effect of mud during the underbalancedrilling
operation is presented in Figure 15. From this figure,
−3
−2
−1
0
1
2
3
Min
imum
hor
izon
tal s
tress
(MPa
)
0.2 0.25 0.3 0.35 0.4 0.45 0.50.15Position on the y-axis (m)
Initial
10 days30 hr8min
30 s7.5 s4 s
Figure 13: 𝑌-component of minimum horizontal stresses as
afunction of time in poroelastic medium with 𝜎𝐻 = 16.8MPa and 𝜎ℎ=
14MPa, 𝑃𝑟 = 11MPa, 𝑃𝑤 = 9.7MPa, 𝑘𝑥 = 0.01md, 𝑘𝑦 = 0.01md, and𝑇𝑚 =
330K.
Y
X
0.5
0.4
0.3
0.2
0.1
00.50.40.30.20.10
Figure 14: 𝑌-component of net shear stresses at 𝑡 = 30 s
inporoelastic medium with 𝜎𝐻 = 16.8MPa and 𝜎ℎ = 14MPa, 𝑃𝑟 =11MPa,
𝑃𝑤 = 9.7MPa, 𝑘𝑥 = 0.01md, 𝑘𝑦 = 0.01md, and 𝑇𝑚 = 330K.
it can be seen that the net shear stress accumulated at
thewellbore wall and increases the probability of failure of
thewell. If the breakout occurs, it will initiate near the
wellborenot at the wall bore wall (see Figure 15). If there is a
breakout,the shear forces will cause rock to fall into the
wellboreand in this case the well status becomes unstable
(wellboreinstability). But, in this case study, the net shear
stress istoo low to cause failure and this well will not suffer
frominstability problems even for long drilling period. Figures
16,17, and 18 show 𝑦-stresses at the wellbore wall with the
effectof mud cooling.
The cooling process near the wellbore can alter thestresses
significantly and leads to increasing the total stressesand the
pore pressure drop inside the formation; those
-
Journal of Applied Mathematics 9
0.2 0.25 0.3 0.35 0.4 0.45 0.50.15Position on the y-axis (m)
−2
−1
0
1
2
3
4
5
6
Net
shea
r stre
ss (M
Pa)
Initial
10 days30 hr8min
30 s7.5 s4 s
Figure 15: 𝑌-component of net shear stresses as a function of
timein poroelastic medium with 𝜎𝐻 = 16.8MPa and 𝜎ℎ = 14MPa, 𝑃𝑟
=11MPa, 𝑃𝑤 = 9.7MPa, 𝑘𝑥 = 0.01md, 𝑘𝑦 = 0.01md, and 𝑇𝑚 = 300K.
0.2 0.25 0.3 0.35 0.4 0.45 0.50.15Position on y-axis (m)
−3
−2
−1
0
1
2
3
4
Min
imum
hor
izon
tal s
tress
(Mpa
)
Initial
10 days30 hr8min
30 s7.5 s4 s
Figure 16: 𝑌-component of minimum horizontal stresses as
afunction of time in poroelastic medium with 𝜎𝐻 = 16.8MPa and 𝜎ℎ=
14MPa, 𝑃𝑟 = 11MPa, 𝑃𝑤 = 9.7MPa, 𝑘𝑥 = 0.01md, 𝑘𝑦 = 0.01md, and𝑇𝑚 =
300K.
increasing in total stresses and pore pressure cause
increasingin the effective stresses near the wellbore (see Figures
15,17, and 18). As time increases, the mud temperature
willequilibrate with its surroundings so that the formationshigher
in the section being drilled are subjected to theincreased
temperature of the mud. Heating process leads toreducing the pore
pressure and net shear stresses near thewellbore (see Figures 11
and 13).
The formation cooling increases the pore pressure (seeFigure 19)
near the wellbore wall at the beginning of thedrilling operation
(for 4 s, 7.5 s, and 30 s). This is due tothermal osmosis process
that results in fluid movement out
0
2
4
6
8
10
12
14
16
Max
imum
hor
izon
tal s
tress
(Mpa
)
0.2 0.25 0.3 0.35 0.4 0.45 0.50.15Position on y-axis (m)
Initial
10 days30 hr8min
30 s7.5 s4 s
Figure 17: 𝑌-component of maximum horizontal stresses as
afunction of time in poroelastic medium with 𝜎𝐻 = 16.8MPa and 𝜎ℎ=
14MPa, 𝑃𝑟 = 11MPa, 𝑃𝑤 = 9.7MPa, 𝑘𝑥 = 0.01md, 𝑘𝑦 = 0.01md, and𝑇𝑚 =
300K.
0.2 0.25 0.3 0.35 0.4 0.45 0.50.15Position along y axis (m)
9.69.810
10.210.410.610.8
1111.211.411.6
Pore
pre
ssur
e (M
pa)
Initial
10 days30 hr8min
30 s7.5 s4 s
Figure 18: Pore pressure as a function of radius and time
inporoelastic medium with 𝜎𝐻 = 16.8MPa and 𝜎ℎ = 14MPa, 𝑃𝑟 =11MPa,
𝑃𝑤 = 9.7MPa, 𝑘𝑥 = 0.01md, 𝑘𝑦 = 0.01md, and 𝑇𝑚 = 300K.
of the formation. Then, the transient response causes
porepressure on the 𝑦-axis to decrease.
Figure 20 shows a relationship between the mud weightand
accumulated shear stress around the wellbore. It can beseen from
the figure that, with using mud weight of 7.5 ppg,the net shear
stress (16MPa) becomes greater than the rockstrength (14MPa).
Therefore, to avoid wellbore breakouts,Mohr-Coulomb failure
criterion indicates that the safe mudweight used in this case study
is between 5.5 and 7.5 ppg.
-
10 Journal of Applied Mathematics
Table 3: General description of the problem.
Inner boundary Outer boundary Wellbore pressure Pore pressure
Maximum horizontalstressMinimum horizontal
stress𝑟𝑤 𝑟𝑒 = ∞ 𝑃𝑤 𝑃 = 𝑃𝑖𝑖𝑛𝑡 𝜎𝐻 𝜎ℎ
Y
X
0.5
0.4
0.3
0.2
0.1
00.50.40.30.20.10
Figure 19: 𝑌-component of net shear stresses at 𝑡 = 30 s
inporoelastic medium with 𝜎𝐻 = 2436 psi and 𝜎ℎ = 2030 psi, 𝑃𝑟 =1610
psi, 𝑃𝑤 = 1421 psi, 𝑘𝑥 = 0.01md, 𝑘𝑦 = 0.01md, and 𝑇𝑚 = 300K.
Net
shea
r stre
ss (M
Pa)
02468
1012141618
4.5 5 5.5 6 6.5 7 7.5 84Mud weight (ppg)
Figure 20: Relationship between mud weight and net shear
stress(iteration process).
7. Conclusion
An integrated thermoporoelastic numerical model has
beenpresented in this paper to predict the stresses distribution
andthe instability problem around the wall of the wellbore.
Themodel has been validated against the analytical model.
Behaviour of the stresses around the wellbore in under-balance
drilling operation is very sensitive to the mudweight and
mechanical properties of the rock as well. Thepore pressure and
stresses around the wellbore are signifi-cantly affected by the
thermal effects. Thus, when the mudtemperature is lower than the
formation temperature, thepore pressure changes, and the net shear
stresses values areincreased around the wellbore which increase the
probability
ℎ
ℎ
HH
Pw
P = PCHCN
Figure 21: Schematic of the problem.
of occurrence of the instability problem, if its values
becomegreater than the rock shear strength.
Appendix
Elastic Deformation of a Pressurized Wellborein a Drained Rock
Subjected to Anisotropic InSitu Horizontal Stress (Kirsch’s
Problem)
General description of the problem is tabulated in Table 3and
schematic of the problem is illustrated in Figure 21. Thisproblem
accounts for the concept of effective stress.
(i) Analytical pressure is
𝑝 (𝑟, 𝑡) = 𝑝𝑖 + (𝑝𝑤 − 𝑝𝑖) 𝑔 (𝑟, 𝑡) . (A.1)(ii) Analytical radial
stress is
𝜎𝑟𝑟 (𝑟, 𝜃) = 𝜎𝐻 + 𝜎ℎ2 (1 −𝑟2𝑤𝑟2 )
+ 𝜎𝐻 − 𝜎ℎ2 (1 + 3𝑟4𝑤𝑟4 − 4
𝑟2𝑤𝑟2 ) cos (2𝜃)
+ 𝑝𝑤 𝑟2𝑤𝑟2 + 2𝜂 (𝑝𝑤 − 𝑝𝑖) 𝑟𝑤𝑟 ℎ (𝑟, 𝑡) .
(A.2)
-
Journal of Applied Mathematics 11
(iii) Analytical tangential stress is
𝜎𝜃𝜃 (𝑟, 𝜃) = 𝜎𝐻 + 𝜎ℎ2 (1 +𝑟2𝑤𝑟2 )
− 𝜎𝐻 − 𝜎ℎ2 (1 + 3𝑟4𝑤𝑟4 ) cos (2𝜃)
− 𝑝𝑤 𝑟2𝑤𝑟2
− 2𝜂 (𝑝𝑤 − 𝑝𝑖) (𝑟𝑤𝑟 ℎ (𝑟, 𝑡) + 𝑔 (𝑟, 𝑡)) ,𝑔 (𝑟, 𝑠) = 𝐾0 (𝜉)𝑠𝐾0
(𝛽) ,ℎ̃ (𝑟, 𝑠) = 1𝑠 [ 𝐾1 (𝜉)𝛽𝐾0 (𝛽) −
𝑟𝑤𝑟𝐾1 (𝛽)𝛽𝐾0 (𝛽)] .
(A.3)
(iv) Radial displacement is
𝑢𝑟 (𝑟, 𝜃) = 𝑟4𝐺 (𝜎𝐻 + 𝜎ℎ)(1 − 2V +𝑟2𝑤𝑟2 )
+ 𝑟4𝐺 (𝜎𝐻 − 𝜎ℎ)× (𝑟2𝑤𝑟2 (4 − 4V −
𝑟2𝑤𝑟2 ) + 1) cos (2𝜃)
− 𝑝𝑤2𝐺𝑟2𝑤𝑟 − 𝜂𝐺𝑟𝑤 (𝑝𝑤 − 𝑝𝑖) ℎ (𝑟, 𝑡) .
(A.4)
(v) Tangential displacement is
𝑢𝜃 (𝑟, 𝜃) = − 𝑟4𝐺 (𝜎𝐻 − 𝜎ℎ)⋅ (𝑟2𝑤𝑟2 (2 − 4V −
𝑟2𝑤𝑟2 ) + 1) sin (2𝜃) .(A.5)
(vi) Analytical temperature is
𝑇 (𝑟, 𝑡) = 𝑇𝑜 + (𝑇𝑤 − 𝑇𝑜) 𝐿−1 {1𝑠𝐾0 (𝑟√𝑠/𝑐0)𝐾0 (𝑟𝑤√𝑠/𝑐0)} ,
(A.6)
where 𝑔 is the Laplace transformation of 𝑔 and𝜉 = 𝑟√ 𝑠𝑐 ,𝛽 =
𝑟𝑤√𝑠𝑐 ,
(A.7)
and𝐾0 and𝐾1 are the first-order modified Bessel function ofthe
first and second kind. Laplace inversion is solved using themethod
presented byDetournay andCheng [15].The solutionin time is achieved
by the following formula.
The Laplace transformation can be inverted using
𝑓 (𝑟, 𝑡) ≈ ln 2𝑡𝑁∑𝑛=1
𝐶𝑛 ≈𝑓 (𝑟, 𝑛 ln 2𝑡 ) , (A.8)where (ln) represents the natural
logarithm and
𝐶𝑛 = (−1)𝑛+𝑁/2⋅ min(𝑛,𝑁/2)∑𝑘=⌊(𝑛+1)/2⌋
𝑘𝑁/2 (2𝑘)!(𝑁/2 − 𝑘)!𝑘! (𝑘 − 1)! (𝑛 − 𝑘)! (2𝑘 − 𝑛)! .(A.9)
Conflicts of Interest
The author declares that he has no conflicts of interest.
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