simulation of wellbore stability during underbalanced ......drilling with underbalanced technique...

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

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)

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


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


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 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 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 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.


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


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


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

)


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

)


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



−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.


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)


(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.

References

[1] S. Rafieepour, C. Ghotbi, and M. R. Pishvaie, “The effects ofvarious parameters onwellbore stability during drilling throughshale formations,” Petroleum Science and Technology, vol. 33, no.12, pp. 1275–1285, 2015.

[2] G.Chen,M. E. Chenevert,M.M. Sharma, andM.Yu, “A study ofwellbore stability in shales including poroelastic, chemical, andthermal effects,” Journal of Petroleum Science and Engineering,vol. 38, no. 3-4, pp. 167–176, 2003.

[3] H. Roshan and S. S. Rahman, “Analysis of pore pressure andstress distribution around awellbore drilled in chemically activeelastoplastic formations,”RockMechanics and Rock Engineering,vol. 44, no. 5, pp. 541–552, 2011.

[4] M. Hodge, K. L. Valencia, Z. Chen, and S. S. Rahman, “Analysisof time-dependent wellbore stability of underbalanced wellsusing a fully coupled poroelastic model,” in Proceedings of theSPE Annual Technical Conference and Exhibition, ATCE 2006:Focus on the Future, pp. 3274–3278, September 2006.

[5] M. Gomar, I. Goodarznia, and S. R. Shadizadeh, “A transientfully coupled thermo-poroelastic finite element analysis ofwellbore stability,” Arabian Journal of Geosciences, vol. 8, no. 6,pp. 3855–3865, 2014.

[6] M. Aslannezhad, A. Khaksar manshad, and H. Jalalifar, “Deter-mination of a safemudwindowand analysis ofwellbore stabilityto minimize drilling challenges and non-productive time,”Journal of Petroleum Exploration and Production Technology,vol. 6, no. 3, pp. 493–503, 2016.

[7] S. Salehi, G. Hareland, and R. Nygaard, “Numerical simulationsof wellbore stability in under-balanced-drillingwells,” Journal ofPetroleum Science and Engineering, vol. 72, no. 3-4, pp. 229–235,2010.

[8] W. K. Heiduc and S.-W. Wong, “Hydration swelling of water-absorbing rocks: a constitutive model,” International Journal forNumerical and Analytical Methods in Geomechanics, vol. 20, no.6, pp. 403–430, 1996.

[9] P. McLellan and C. Hawkes, “Borehole stability analysis forunderbalanced drilling,” Journal of Canadian Petroleum Tech-nology, vol. 40, no. 5, pp. 31–38, 2001.

[10] M. R. McLean and M. A. Addis, “Wellbore stability: theeffect of strength criteria on mud weight recommendations,”in Proceedings of the SPE Annual Technical Conference andExhibition, pp. 9–17, September 1990.


[11] M. F. Kanfar, Z. Chen, and S. S. Rahman, “Risk-controlledwellbore stability analysis in anisotropic formations,” Journal ofPetroleum Science and Engineering, vol. 134, pp. 214–222, 2015.

[12] T. Ma and P. Chen, “A wellbore stability analysis model withchemical-mechanical coupling for shale gas reservoirs,” Journalof Natural Gas Science and Engineering, vol. 26, pp. 72–98, 2015.

[13] S. De and K. J. Bathe, “The method of finite spheres,” Computa-tional Mechanics, vol. 25, no. 4, pp. 329–345, 2000.

[14] G. Kirsch, “Theory of elasticity and application in strength ofmaterials,” Zeitschrift des Vereins Deutscher Ingenieure, vol. 42,no. 29, pp. 797–807, 1898.

[15] E. Detournay and A.-D. Cheng, “Poroelastic response of a bore-hole in a non-hydrostatic stress field,” International Journal ofRockMechanics andMining Sciences & Geomechanics Abstracts,1988.

[16] H. Stehfest, “Algorithm 368: numerical inversion of Laplacetransforms,” Communications of the ACM, vol. 13, no. 1, pp. 47–49, 1970.

[17] B. S. Aadnoy and M. E. Chenevert, “Stability of highly inclinedboreholes (includes associated papers 18596 and 18736),” SPEDrilling Engineering, vol. 2, no. 4, pp. 364–374, 1987.

[18] R. T. Ewy, “Wellbore-stability predictions by use of a modifiedlade criterion,” SPE Drilling and Completion, vol. 14, no. 2, pp.85–91, 1999.

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