simulation of wellbore stability during underbalanced ......drilling with underbalanced technique...
TRANSCRIPT
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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 ยฉ 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$โ1000$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 [3โ5].
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:
โ โ ๐ + ๐๐ = 0, (1)where ๐ is the total stress, ๐ is the gravity constant, and ๐ isthe bulk intensity of the porous media. The intensity shouldbe written for two phases, liquid and solid, as follows:
๐ = ๐๐๐ + (1 โ ๐) ๐๐ . (2)Equation (1) can be written in terms of effective stress asfollows:
โ โ (๐ โ ๐๐ผ) + ๐๐ = 0, (3)where ๐ is the effective stress, ๐ is the pore pressure, and๐ผ 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:
๐ = ๐ถ (๐ โ ๐ผ๐ฮ๐ ร ๐ผ) , (4)where ๐ถ is the fourth-order stiffness tensor of materialproperties, ๐ is the total strain, ๐ผ๐ is the thermal expan-sion coefficient, and ฮ๐ is the temperature difference. Theisotropic elasticity tensor ๐ถ is defined as
๐ถ = ๐๐ฟ๐๐๐ฟ๐๐ + 2๐บ๐ฟ๐๐๐ฟ๐๐, (5)where ๐ฟ is the Kronecker delta and ๐ is the Lame constant. ๐บis the shear modulus of elasticity. The constitutive equationfor the total strain-displacement relationship is defined asfollows:
๐ = 12 (โโ๐ข + (โโ๐ข)๐) , (6)
where โ๐ข is the displacement vector and โ is the gradientoperator.
2.2. Mass Conservation. The fluid flow in deformable andsaturated porous media can be described by the followingequation:
๐๐ ๐๐๐๐ก + ๐ฝโ โ (๐โ๐ข๐๐ก ) + โ โ ๐ โ ๐ผ๐ ๐๐๐๐ก = ๐, (7)
where ๐ฝ is the Biots coefficient and assumed to be = 1.0 in thisstudy, ๐ is the pore fluid pressure, ๐ is the temperature, ๐ผ๐ is
the thermal expansion coefficient, ๐ is the fluid flux, and ๐ isthe sink/source, and ๐๐ is the specific storage which is definedby
๐๐ = (1 โ ๐๐พ๐ ) + (๐๐พ๐) , (8)
where ๐พ๐ is the compressibility of solid and ๐พ๐ is thecompressibility of liquid. The fluid flux term (๐) in the massbalance in (7) can be described by usingDarcyโs flow equationbecause the intensity has been assumed constant in this study:
๐ = โ๐๐ (โ๐ โ ๐โ๐) , (9)where ๐ 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:
(๐๐๐)eff ๐๐๐๐ก + โ โ ๐๐ = ๐๐, (10)where ๐๐ is the heat flux,๐๐ is the heat sink/source term, and๐๐๐ is the heat storage and equals
(๐๐๐)eff = ๐ (๐๐๐)liquid + (1 โ ๐) (๐๐๐)solid . (11)In this study, conduction and convection heat transfers areconsidered during numerical simulation. The heat flux termin (10) can be written as
๐๐ = โ๐effโ๐ + (๐๐๐)liquid V โ ๐, (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 ๐eff is theeffective heat conductivity of the porous medium, which canbe defined as
๐eff = ๐๐liquid + (1 โ ๐) ๐solid. (13)2.4. Discretization of the Equations. First one discretizesthe thermoporoelastic governing equations by using Greensโtheorem [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:
โซฮฉ๐ค๐๐ ๐๐๐๐ก ๐ฮฉ + โซฮฉ ๐ค๐๐ผโ โ
๐โ๐ข๐๐ก ๐ฮฉ + โซฮฉ ๐ค๐ฝ๐๐๐๐ก
โ โซฮฉโ๐ค๐ โ ๐๐ป๐ฮฉ + โซ
ฮ๐๐ป
๐ค (๐๐ป โ ๐) ๐ฮโ โซฮฉ๐ค๐๐ป๐ฮฉ = 0,
(14)
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Journal of Applied Mathematics 3
โซฮ๐
๐ค๐๐๐๐ ๐๐๐๐ก ๐ฮ + โซฮ๐ ๐ค๐ผ๐๐๐๐๐ก ๐ฮ + โซฮ๐ ๐ค๐ฝ
๐๐๐๐ก ๐ฮโ โซฮ๐
โ๐ค๐ โ (๐โ๐๐ป) ๐ฮฉ + โซฮ๐๐ป
๐ค๐โ (๐๐ป โ ๐) ๐ฮ+ โซฮ๐
๐ค๐+๐ป๐ฮ + โซฮ๐
๐ค๐โ๐ป๐ฮ = 0,(15)
โซฮฉ๐ค๐๐๐๐๐๐๐ก ๐ฮฉ + โซฮฉ ๐ค๐๐๐๐๐ป โ โ๐๐ฮฉโ โซฮฉโ๐ค๐ โ (โ๐โ๐) ๐ฮฉ + โซ
ฮ๐
๐
๐ค (โ๐โ๐ โ ๐) ๐ฮโ โซฮฉ๐ค๐๐๐๐ฮฉ = 0,
(16)
โซฮ๐
๐ค๐๐๐๐๐๐๐ ๐๐๐๐ก ๐ฮ + โซฮ๐ ๐ค๐๐๐๐๐๐โ๐๐ป โ โ๐๐ฮ
โ โซฮ๐
โ๐ค๐ โ (โ๐๐๐๐โ๐) ๐ฮ+ โซฮ๐
๐
๐ค(โ๐๐๐๐โ๐ โ ๐) ๐ฮ + โซฮ๐
๐ค๐+๐๐ฮ+ โซฮ๐
๐โ๐๐ฮ = 0,
(17)
โซฮฉโ๐ ๐ค๐ โ (๐ โ ๐ผ๐๐ผ) ๐ฮฉ โ โซ
ฮฉ๐ค๐ โ ๐๐๐ฮฉ
โ โซฮ๐ก
๐ค๐ โ โ๐ก ๐ฮ โ โซฮ๐
๐ค+๐ โ โ๐ก +๐๐ฮโ โซ๐คโ๐ โ โ๐ก โ๐๐ฮ = 0,
(18)
where ๐ค is the test function, ฮฉ is the model domain, ฮ isthe domain boundary, ๐ก is the traction vector, superscripts+/โ refer to the value of the corresponding parameters onopposite sides of the fracture surfaces, respectively, ๐๐ is thespecific storage, ๐ is the porosity, ๐๐ป is the volumetric Darcyflux, ๐ฝ is the thermal expansion coefficient, ๐๐ป is the fluidsink/source term between the fractures, ๐๐ is the heat flux,๐๐ is the specific heat capacity, ๐๐ and ๐โ are mechanicaland hydraulic fracture apertures, ๐๐ is the heat sink/sourceterm, ๐ผ is the thermal expansion coefficient, ๐ is the thermalconductivity, and ๐ 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 ๐, temperature ๐, and displace-ment vector ๐ข. All of these variables are approximated byusing the interpolation function in finite element space asfollows:
๐ข = ๐๐ข๐ข,๐ = ๐๐๐,๐ = ๐๐๐,
(19)
Table 1: Reservoir inputs used for validation of poroelastic numer-ical model using circular homogenous reservoir.
Parameter ValuePoisson ratio 0.2Youngโs modulus 40GPaMaximum horizontal stress 40MPa (5800 psi)Minimum horizontal stress 37.9MPa (5500 psi)Wellbore pressure (๐๐ค) 6.89MPa (1000 psi)Initial reservoir pressure (๐๐) 37.9MPa (5500 psi)Fluid bulk module (๐พ๐) 2.5 GPaFluid compressibility 1.0 ร 10โ5 Paโ1Biotโs coefficient 1.0Fluid viscosity 3 ร 10โ4 Paโ sMatrix permeability 9.869 ร 10โ18m2 (0.01md)Wellbore radius 0.1mReservoir outer radius 1000m
1000
800
600
400
200
010008006004002000
Y
โ
H
X
Figure 1: Two-dimensional circular reservoir shape used for valida-tion of poroelastic numerical model with ๐๐ป = 39.9MPa and ๐โ =37.9MPa, ๐๐ = 37.9MPa, and ฮ๐ = 31MPa.
where๐ is the corresponding shape function and ๐ข, ๐, and ๐are 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 Kirschโs 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 3โ6.
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, Youngโsmodulus, Poissonโs 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 ๐๐ป = 5800 psi and ๐โ = 5500 psi, ๐๐ =5500 psi, ๐๐ค = 1000 psi, ๐๐ฅ = 0.01md, and ๐๐ฆ = 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: ๐-displacement along ๐ฅ-axis as a function of time inporoelastic medium with ๐๐ป = 5800 psi and ๐โ = 5500 psi, ๐๐ =5500 psi, ๐๐ค = 1000 psi, ๐๐ฅ = 0.01md, and ๐๐ฆ = 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: ๐-component of radial stresses as a function of time inporoelastic medium with ๐๐ป = 5800 psi and ๐โ = 5500 psi, ๐๐ =5500 psi, ๐๐ค = 1000 psi, ๐๐ฅ = 0.01md, and ๐๐ฆ = 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: ๐-component of tangential stresses as a function of timein poroelastic medium with ๐๐ป = 5800 psi and ๐โ = 5500 psi, ๐๐ =5500 psi, ๐๐ค = 1000 psi, ๐๐ฅ = 0.01md, and ๐๐ฆ = 0.01md.
and analytical solutions for ๐ก = 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 ๐๐ป = 5800 psi and ๐โ = 5500 psi, ๐๐= 5500 psi, ๐๐ค = 1000 psi, ๐๐ฅ = 0.01md, and ๐๐ฆ = 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, ๐๐ฅ approaches the maximumhorizontal in situ stress (5800 psi) at far field (away fromwellbore). The discontinuity of ๐๐ฅ 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 ๐๐ฅ, increases. This is due to changein pore pressure.
In Figure 6 numerical results of ๐๐ฆ match well with thatof the analytical solutions for different time. As expected, ๐๐ฆapproaches 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 7โ9 in reservoirentire region. These figures (Figures 7โ9) clarify how thepressure and stresses are changing from the wall of thewellbore to the reservoir boundary.
4. Failure Criteria
Shear failure will occur if
๐3 < โ๐0, (20)where ๐3 the lowest principle is effective stress and ๐0 is therock tensile strength.
Y
X
0.8
0.6
0.4
0.2
010.80.60.40.20
Figure 8: ๐-component of radial stresses contour map after 1 hr offluid production (for near-wellbore region) with ๐๐ป = 5800 psi and๐โ = 5500 psi, ๐๐ = 5500 psi, ๐๐ค = 1000 psi, ๐๐ฅ = 0.01md, and ๐๐ฆ =0.01md.
Y
X
0.8
0.6
0.4
0.2
010.80.60.40.20
Figure 9:๐-component of tangential stresses contourmap after 1 hrof fluid production (for near-wellbore region) with ๐๐ป = 5800 psiand ๐โ = 5500 psi, ๐๐ = 5500 psi, ๐๐ค = 1000 psi, ๐๐ฅ = 0.01md, and ๐๐ฆ= 0.01md.
Using Mohr-Coulomb criteria, shear failure criteria aremet when
๐net = 12 cos๐ (๐1 (1 โ sin๐) โ ๐3 (1 + sin๐)) > ๐0, (21)where ๐1 is the highest principle effective stress, ๐net is thenet shear stress, ๐ is the angle of internal friction, and ๐0 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 Youngโs modulus (๐ธ) 4.4GPaMaximum horizontal stress (๐๐ป) 16.8MPa (2436 psi)Minimum horizontal stress (๐โ) 14MPa (2030 psi)Wellbore pressure (๐๐ค) 6.89MPa (1000 psi)Hydraulic parametersInitial reservoir pressure (๐๐) 11.1MPa (1610 psi)Fluid bulk module (๐พ๐) 0.45GPaFluid compressibility 1.0 ร 10โ5 Paโ1Biotโs coefficient 1.0Physical parametersFluid viscosity 3 ร 10โ4 Paโ sFluid density 1111 kg/m3
Matrix permeability 9.869 ร 10โ18m2(0.01md)Porosity (๐) 0.1Wellbore radius 0.15mReservoir outer radius 1000mFormation temperature (๐๐) 375 oKDrilling mud temperature (๐๐) 330 oKThermal osmosis coefficient (๐พ๐) 1 ร 10โ11m2/s KThermal expansion coefficient of fluid (๐ผ๐) 3 ร 10โ4 1/KThermal diffusivity (๐ถ๐) 1.1 ร 10โ6m2/sThermal expansion coefficient of solid (๐ผ๐ ) 1.8 ร 10โ5 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
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8 Journal of Applied Mathematics
Initial
10 days
โ1
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: ๐-component of net shear stresses as a function of timein poroelastic medium with ๐๐ป = 16.8MPa and ๐โ = 14MPa, ๐๐ =11MPa, ๐๐ค = 9.7MPa, ๐๐ฅ = 0.01md, ๐๐ฆ = 0.01md, and ๐๐ = 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: ๐-component of maximum horizontal stresses as afunction of time in poroelastic medium with ๐๐ป = 16.8MPa and ๐โ= 14MPa, ๐๐ = 11MPa, ๐๐ค = 9.7MPa, ๐๐ฅ = 0.01md, ๐๐ฆ = 0.01md, and๐๐= 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 ๐ฆ-stresses at the wellbore wall.
Distribution of the net shear stress along ๐ฆ-axis for thecooling (๐๐ < ๐๐) effect of mud during the underbalancedrilling operation is presented in Figure 15. From this figure,
โ3
โ2
โ1
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: ๐-component of minimum horizontal stresses as afunction of time in poroelastic medium with ๐๐ป = 16.8MPa and ๐โ= 14MPa, ๐๐ = 11MPa, ๐๐ค = 9.7MPa, ๐๐ฅ = 0.01md, ๐๐ฆ = 0.01md, and๐๐ = 330K.
Y
X
0.5
0.4
0.3
0.2
0.1
00.50.40.30.20.10
Figure 14: ๐-component of net shear stresses at ๐ก = 30 s inporoelastic medium with ๐๐ป = 16.8MPa and ๐โ = 14MPa, ๐๐ =11MPa, ๐๐ค = 9.7MPa, ๐๐ฅ = 0.01md, ๐๐ฆ = 0.01md, and ๐๐ = 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 ๐ฆ-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)
โ2
โ1
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: ๐-component of net shear stresses as a function of timein poroelastic medium with ๐๐ป = 16.8MPa and ๐โ = 14MPa, ๐๐ =11MPa, ๐๐ค = 9.7MPa, ๐๐ฅ = 0.01md, ๐๐ฆ = 0.01md, and ๐๐ = 300K.
0.2 0.25 0.3 0.35 0.4 0.45 0.50.15Position on y-axis (m)
โ3
โ2
โ1
0
1
2
3
4
Min
imum
hor
izon
tal s
tress
(Mpa
)
Initial
10 days30 hr8min
30 s7.5 s4 s
Figure 16: ๐-component of minimum horizontal stresses as afunction of time in poroelastic medium with ๐๐ป = 16.8MPa and ๐โ= 14MPa, ๐๐ = 11MPa, ๐๐ค = 9.7MPa, ๐๐ฅ = 0.01md, ๐๐ฆ = 0.01md, and๐๐ = 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: ๐-component of maximum horizontal stresses as afunction of time in poroelastic medium with ๐๐ป = 16.8MPa and ๐โ= 14MPa, ๐๐ = 11MPa, ๐๐ค = 9.7MPa, ๐๐ฅ = 0.01md, ๐๐ฆ = 0.01md, and๐๐ = 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 ๐๐ป = 16.8MPa and ๐โ = 14MPa, ๐๐ =11MPa, ๐๐ค = 9.7MPa, ๐๐ฅ = 0.01md, ๐๐ฆ = 0.01md, and ๐๐ = 300K.
of the formation. Then, the transient response causes porepressure on the ๐ฆ-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
stress๐๐ค ๐๐ = โ ๐๐ค ๐ = ๐๐๐๐๐ก ๐๐ป ๐โ
Y
X
0.5
0.4
0.3
0.2
0.1
00.50.40.30.20.10
Figure 19: ๐-component of net shear stresses at ๐ก = 30 s inporoelastic medium with ๐๐ป = 2436 psi and ๐โ = 2030 psi, ๐๐ =1610 psi, ๐๐ค = 1421 psi, ๐๐ฅ = 0.01md, ๐๐ฆ = 0.01md, and ๐๐ = 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
โ
โ
HH
Pw
P = P๏ผฃ๏ผจ๏ผฃ๏ผฎ
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 (Kirschโs 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
๐ (๐, ๐ก) = ๐๐ + (๐๐ค โ ๐๐) ๐ (๐, ๐ก) . (A.1)(ii) Analytical radial stress is
๐๐๐ (๐, ๐) = ๐๐ป + ๐โ2 (1 โ๐2๐ค๐2 )
+ ๐๐ป โ ๐โ2 (1 + 3๐4๐ค๐4 โ 4
๐2๐ค๐2 ) cos (2๐)
+ ๐๐ค ๐2๐ค๐2 + 2๐ (๐๐ค โ ๐๐) ๐๐ค๐ โ (๐, ๐ก) .
(A.2)
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Journal of Applied Mathematics 11
(iii) Analytical tangential stress is
๐๐๐ (๐, ๐) = ๐๐ป + ๐โ2 (1 +๐2๐ค๐2 )
โ ๐๐ป โ ๐โ2 (1 + 3๐4๐ค๐4 ) cos (2๐)
โ ๐๐ค ๐2๐ค๐2
โ 2๐ (๐๐ค โ ๐๐) (๐๐ค๐ โ (๐, ๐ก) + ๐ (๐, ๐ก)) ,๐ (๐, ๐ ) = ๐พ0 (๐)๐ ๐พ0 (๐ฝ) ,โฬ (๐, ๐ ) = 1๐ [ ๐พ1 (๐)๐ฝ๐พ0 (๐ฝ) โ
๐๐ค๐๐พ1 (๐ฝ)๐ฝ๐พ0 (๐ฝ)] .
(A.3)
(iv) Radial displacement is
๐ข๐ (๐, ๐) = ๐4๐บ (๐๐ป + ๐โ)(1 โ 2V +๐2๐ค๐2 )
+ ๐4๐บ (๐๐ป โ ๐โ)ร (๐2๐ค๐2 (4 โ 4V โ
๐2๐ค๐2 ) + 1) cos (2๐)
โ ๐๐ค2๐บ๐2๐ค๐ โ ๐๐บ๐๐ค (๐๐ค โ ๐๐) โ (๐, ๐ก) .
(A.4)
(v) Tangential displacement is
๐ข๐ (๐, ๐) = โ ๐4๐บ (๐๐ป โ ๐โ)โ (๐2๐ค๐2 (2 โ 4V โ
๐2๐ค๐2 ) + 1) sin (2๐) .(A.5)
(vi) Analytical temperature is
๐ (๐, ๐ก) = ๐๐ + (๐๐ค โ ๐๐) ๐ฟโ1 {1๐ ๐พ0 (๐โ๐ /๐0)๐พ0 (๐๐คโ๐ /๐0)} , (A.6)
where ๐ is the Laplace transformation of ๐ and๐ = ๐โ ๐ ๐ ,๐ฝ = ๐๐คโ๐ ๐ ,
(A.7)
and๐พ0 and๐พ1 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
๐ (๐, ๐ก) โ ln 2๐ก๐โ๐=1
๐ถ๐ โ๐ (๐, ๐ ln 2๐ก ) , (A.8)where (ln) represents the natural logarithm and
๐ถ๐ = (โ1)๐+๐/2โ min(๐,๐/2)โ๐=โ(๐+1)/2โ
๐๐/2 (2๐)!(๐/2 โ ๐)!๐! (๐ โ 1)! (๐ โ ๐)! (2๐ โ ๐)! .(A.9)
Conflicts of Interest
The author declares that he has no conflicts of interest.
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