Abstract
The loadings which act on the wellbore are more frequently dynamic than static, such as the surge/swab pressure caused by tripping operations. The changing rate of loading could induce a change in wellbore stress and result in wellbore instability. The conventional Kirsch solution to calculate wellbore stress is only applicable to the steady state without considering the coupled deformation–diffusion effect. To account for these deficiencies, this paper introduces a coupled poroelastodynamics model to obtain wellbore stress distribution under dynamic loading. The model is solved by the implicit finite-difference method with the surge/swab pressure caused by tripping operations taken as the specific dynamic loading source. Then, failure criteria are applied to analyze dynamic wellbore stability and the hemisphere plots of minimum mud density (MMD) to avoid wellbore collapse are generated. The effect of in-situ stress regimes, failure criteria, and permeable properties on the instability of the borehole has been investigated, and the applicability and accuracy of the conventional failure criteria for dynamic loading conditions have been studied by the extensive triaxial compression tests performed on Bedford limestone under different strain rates. Furthermore, two specific cases have been analyzed and results show that compared with the poroelastodynamics model, the conventional method either overestimates or underestimates the MMD, the difference of which can be as significant as 0.11 g/cm3. Not limited to the tripping operations, the developed poroelastodynamics model can be applied to any specific dynamic loading conditions with the fluid inertia neglected.
Similar content being viewed by others
Abbreviations
- MMD:
-
Minimum mud density
- MMD3DHoek–Bown :
-
Minimum mud density by 3D Hoek–Bown
- MMDMogi−Coulomb :
-
Minimum mud density by Mogi–Coulomb
- MMDModified Lade :
-
Minimum mud density by Modified Lade
- NF:
-
Normal fault
- NF-SS:
-
The transition between normal fault and strike-slip
- SS:
-
Strike-slip
- SS-TF:
-
The transition of the strike-slip and thrust fault
- TF:
-
Thrust fault
- N30°E:
-
The degree of 30° from North to East
- B:
-
Junction point
- Ba, Bb :
-
Upper and lower surface of the junction point B
- Aa :
-
Upper surface of the point A
- Cb :
-
Lower surface of the point C
- Q :
-
Flow rate
- Q Ba, Q Bb :
-
Flow rate at Ba, Bb
- Q t,Ba, Q t,Bb :
-
Flow rate at position Bb and Bb at time t
- Q t−1,Aa, Q t−1,Cb :
-
Flow rate at position Aa and Cb at time t − 1
- t :
-
Time
- p :
-
Pressure
- p t,Bb, p t,Ba :
-
Pressure of position Bb and Ba at time t
- p t−1,Aa, p t−1,Cb :
-
Pressure of position Aa and Cb at time t − 1
- p s :
-
Surface pressure
- S AB, S BC :
-
Characteristic impedances of sections AB and BC
- ΔAB :
-
The change of pipe cross-sectional area at junction B
- V t :
-
Pipe velocity at time t
- Δp :
-
The loss of pressure in the artificial orifice
- σ ij,j :
-
Stress gradient tensor
- σ ij :
-
Stress tensor
- σ r, σ θ, σ z :
-
Radial, tangential and vertical normal stress
- σ rθ, σ θz, σ rz :
-
Shear stress terms
- u i :
-
Displacement vector
- u r, u θ, u z :
-
Radial, tangential and vertical normal displacement
- ε ij :
-
Strain tensor
- ε r, ε θ, ε z :
-
Radial, tangential and vertical normal strain
- ε kk :
-
Volumetric strain
- r, θ, z :
-
Radial, tangential and vertical coordinate
- δ ij :
-
Kronecker delta
- ρ :
-
Density
- G, λ :
-
Lame parameters
- v :
-
Poisson’s ratio
- α :
-
Biot coefficient
- S z :
-
Specific storage at constant strain
- Q fs :
-
Explicit fluid sources
- k :
-
Permeability
- µ :
-
Fluid viscosity
- K u :
-
Undrained bulk modulus
- K :
-
Drained bulk modulus
- c v :
-
Hydraulic diffusivity
- σ r 1, σ θ 1, σ z 1 :
-
Stress solution of Eq. (12)
- σ r 2, σ θ 2, σ z 2 :
-
Far-field stress terms
- σ v :
-
Vertical stress
- σ H :
-
Maximum horizontal stress
- σ h :
-
Minimum horizontal stress
- θ :
-
Wellbore circumferential angle
- r w :
-
Wellbore radius
- \({\sigma }_{x}^{0}, {\sigma }_{y}^{0}, {\sigma }_{z}^{0}\),\({\sigma }_{xy}\),\({\sigma }_{xz}\),\({\sigma }_{yz}\) :
-
In-situ stress terms in the Cartesian coordinate
- \( {\text{Diff}}_{{{\text{ave}}}} \) :
-
Non-dimensional average prediction difference
- \({\text{Diff}}_{{{\text{abs}}}}\) :
-
Non-dimensional absolute prediction difference
- [(σ′1)pre]i :
-
Predicted major principal stress at failure
- [(σ′1)exp]i :
-
Experimental major principal stress at failure
- UCS:
-
Uni-axial rock strength
- n :
-
The total number of data points
- ξ :
-
Non-dimensional distance
- τ :
-
Non-dimensional time
- \(\phi\) :
-
Non-dimensional displacement
- \({S}_{\text{r}}\) :
-
Non-dimensional radial stress
- \(\chi\) :
-
Non-dimensional pore pressure
- c :
-
Compression wave velocity
- \({p}_{0}\) :
-
Static wellbore pressure
- ∅:
-
Porosity
References
Aadnoy BS, Chenevert ME (1987) Stability of highly inclined boreholes (includes associated papers 18596 and 18736). SPE Drill Eng 2:364–374. https://doi.org/10.2118/16052-PA
Al-Ajmi A (2006) Wellbore stability analysis based on a new true-triaxial failure criterion. Dissertation, KTH Royal Institute of Technology
Al-Ajmi AM, Zimmerman RW (2005) Relation between the Mogi and the Coulomb failure criteria. Int J Rock Mech Min Sci 42:431–439. https://doi.org/10.1016/j.ijrmms.2004.11.004
Alejano LR, Bobet A (2012) Drucker–Prager criterion. Rock Mech Rock Eng 45:995–999. https://doi.org/10.1007/s00603-012-0278-2
Bahrami B, Mohsenpour S, Miri MA, Mirhaseli R (2017) Quantitative comparison of fifteen rock failure criteria constrained by polyaxial test data. J Petrol Sci Eng 159:564–580. https://doi.org/10.1016/j.petrol.2017.09.065
Bradley WB (1979) Failure of inclined boreholes. J Energy Resour Technol 101:232–239. https://doi.org/10.1115/1.3446925
Burkhardt JA (1961) Wellbore pressure surges produced by pipe movement. J Petrol Technol 13:595–605. https://doi.org/10.2118/1546-G-PA
Cai M, Kaiser PK, Suorineni F, Su K (2007) A study on the dynamic behavior of the Meuse/Hhaute-Marne argillite. Phys Chem Earth 32:907–916. https://doi.org/10.1016/j.pce.2006.03.007
Cannon GE (1934) Changes in hydrostatic pressure due to withdrawing drill pipe from the hole. In: Drill and production practice. American Petroleum Institute
Chen M, Jin Y (2008) Rock mechanics in petroleum engineering. Science Press, China (Chinese edition)
Crespo FE, Ramadan MA, Arild S, Majed E, Mahmood A (2012) Surge-and-swab pressure predictions for yield-power-law drilling fluids. SPE Drill Complet 27:574–585
Detournay E, Cheng AD (1988) Poroelastic response of a borehole in a non-hydrostatic stress field. Int J Rock Mech Min Sci Geomech Abstr 25:171–182. https://doi.org/10.1016/0148-9062(88)92299-1
Ewy RT (1999) Wellbore-stability predictions by use of a modified Lade criterion. SPE Drill Complet 14:85–91. https://doi.org/10.2118/56862-PA
Fairhurst C (1967) Methods of determining in-situ rock stresses at great depths. Missouri River Division, US Army, Jefferson City
Feng Y, Li X, Gray KE (2018) An easy-to-implement numerical method for quantifying time-dependent mudcake effects on near-wellbore stresses. J Pet Sci Eng 164:501–514
Fjær E, Ruistuen H (2002) Impact of the intermediate principal stress on the strength of heterogeneous rock. J Geophys Res Solid Earth 107:ECV–E3. https://doi.org/10.1029/2001JB000277
Fjar E, Holt RM, Raaen AM, Risnes R, Horsrud P (2008) Petroleum related rock mechanics. Elsevier, Amsterdam
Fontenot JE, Clark RK (1974) An improved method for calculating swab and surge pressures and circulating pressures in a drilling well. Soc Pet Eng J 14:451–462. https://doi.org/10.2118/4521-PA
Gholami R, Moradzadeh A, Rasouli V, Hanachi J (2014) Practical application of failure criteria in determining safe mud weight windows in drilling operations. J Rock Mech Geotech Eng 6:13–25. https://doi.org/10.1016/j.jrmge.2013.11.002
Handin J, Heard HA, Magouirk JN (1967) Effects of the intermediate principal stress on the failure of limestone, dolomite, and glass at different temperatures and strain rates. J Geophys Res 72(2):611–640. https://doi.org/10.1029/JZ072i002p00611
Islam MA, Skalle P, Al-Ajmi AM, Soreide OK (2010) Stability analysis in shale through deviated boreholes using the Mohr and Mogi-Coulomb failure criteria. In: 44th US rock mechanics symposium and 5th US–Canada rock mechanics symposium. American Rock Mechanics Association
Jaeger JC, Cook NG, Zimmerman R (2009) Fundamentals of Rock Mechanics. Wiley, Oxford
Karlsen AG (2014) Surge and swab pressure calculation: calculation of surge and swab pressure changes in laminar and turbulent flow while circulating mud and pumping. Master thesis, Institutt for petroleumsteknologi og anvendt geofysikk
Lajtai EZ, Duncan ES, Carter BJ (1991) The effect of strain rate on rock strength. Rock Mech Rock Eng 24:99–109. https://doi.org/10.1007/BF01032501
Li HB, Zhao J, Li TJ (1999) Triaxial compression tests on a granite at different strain rates and confining pressures. Int J Rock Mech Min Sci 36:1057–1063. https://doi.org/10.1016/S1365-1609(99)00120-3
Liang C, Wu S, Li X, Xin P (2015) Effects of strain rate on fracture characteristics and mesoscopic failure mechanisms of granite. Int J Rock Mech Min Sci 76: 146–154. https://doi.org/10.1016/j.ijrmms.2015.03.010
Lubinski A (1988) Developments in petroleum engineering. Gulf Publishing Company, Houston
Lubinski A, Hsu FH, Nolte KG (1977) Transient pressure surges due to pipe movement in an oil well. Oil Gas Sci Technol 32:307–348. https://doi.org/10.2516/ogst:1977019
Ma T, Chen P, Yang C, Zhao J (2015) Wellbore stability analysis and well path optimization based on the breakout width model and Mogi–Coulomb criterion. J Pet Sci Eng 135:678–701. https://doi.org/10.1016/j.petrol.2015.10.029
Mahmutoğlu Y (2006) The effects of strain rate and saturation on a macro-cracked marble. Eng Geol 82:137–144. https://doi.org/10.1016/j.enggeo.2005.09.001
Maleki S, Gholami R, Rasouli V, Moradzadeh A, Riabi RG, Sadaghzadeh F (2014) Comparison of different failure criteria in prediction of safe mud weigh window in drilling practice. Earth Sci Rev 136:36–58. https://doi.org/10.1016/j.earscirev.2014.05.010
Meng M, Qiu ZS (2018) Experiment study of mechanical properties and microstructures of bituminous coals influenced by supercritical carbon dioxide. Fuel 219:223–238
Meng M, Baldino S, Miska SZ, Takach N (2019a) Wellbore stability in naturally fractured formations featuringdual-porosity/single-permeability and finite radial fluid discharge. J Petrol Sci Eng 174:790–803
Meng M, Zamanipour Z, Miska S, Yu M, Ozbayoglu EM (2019b) Dynamic stress distribution around the wellbore influenced by surge/swab pressure. J Pet Sci Eng 172:1077–1091. https://doi.org/10.1016/j.petrol.2018.09.016.
Mitchell RF (1988) Dynamic surge/swab pressure predictions. SPE Drill Eng 3:325–333. https://doi.org/10.2118/16156-PA
Mme U, Pal S (2012) Effects of mud properties, hole size, drill string tripping speed and configurations on swab and surge pressure magnitude during drilling operations. Inter J Pet Sci Technol 5:143–153
Rahimi R, Nygaard R (2015) Comparison of rock failure criteria in predicting borehole shear failure. Int J Rock Mech Min Sci 79:29–40. https://doi.org/10.1016/j.ijrmms.2015.08.006
Sadd MH (2009) Elasticity: theory, applications, and numerics. Academic Press, Burlington
Schanz M (2009) Poroelastodynamics: linear models, analytical solutions, and numerical methods. Appl Mech Rev 62:030803. https://doi.org/10.1115/1.3090831
Schuh FJ (1964) Computer makes surge-pressure calculations useful. Oil Gas J 31:96
Tran MH, Abousleiman YN, Nguyen VX (2010) The effects of low-permeability mudcake on time-dependent wellbore failure analyses. In: IADC/SPE Asia Pacific drilling technology conference and exhibition. Society of Petroleum Engineers. https://doi.org/10.2118/135893-MS
Ulusay R (2014) The ISRM suggested methods for rock characterization, testing and monitoring: 2007–2014. Springer, Berlin
Wang HF (2000) Theory of linear poroelasticity with applications to geomechanics and hydrogeology. Princeton University Press, Princeton
Wang Z, Miska S, Yu M, Takach N (2013) Effect of tripping velocity profiles on wellbore pressures and dynamic loading of drillstring. AGH Drill Oil Gas 30(1):269–286. https://doi.org/10.7494/drill.2013.30.1.269
Wrana B, Pietrzak N (2013) Influence of inertia forces on soil settlement under harmonic loading. Studia Geotechnica et Mechanica 35:245–258. https://doi.org/10.2478/sgem-2013-0019
Zamanipour Z (2015) Automation of tripping operations in directional wellbores. Master thesis, The University of Tulsa
Zamanipour Z, Miska SZ, Hariharan PR (2016) Effect of transient surge pressure on stress distribution around directional wellbores. In: IADC/SPE drilling conference and exhibition. Society of Petroleum Engineers. https://doi.org/10.2118/178831-MS
Zhang L (2008) A generalized three-dimensional Hoek–Brown strength criterion. Rock Mech Rock Eng 41:893–915. https://doi.org/10.1007/s00603-008-0169-8
Zhang L, Zhu H (2007) Three-dimensional Hoek–Brown strength criterion for rocks. J Geotech Geoenviron 133:1128–1135. https://doi.org/10.1061/(ASCE)1090-0241(2007)133:9(1128)
Zhang L, Cao P, Radha KC (2010) Evaluation of rock strength criteria for wellbore stability analysis. Int J Rock Mech Min Sci 47:1304–1316. https://doi.org/10.1016/j.ijrmms.2010.09.001
Zhang F, Kang Y, Wang Z, Miska S, Yu M, Zamanipour Z (2016) Real-time wellbore stability evaluation for deepwater drilling during tripping. In: SPE deepwater drilling and completions conference. Society of Petroleum Engineers. https://doi.org/10.2118/180307-MS
Zhang F, Kang Y, Wang Z, Miska S, Yu M, Zamanipour Z (2018) Transient coupling of swab/surge pressure and in-situ stress for wellbore-stability evaluation during tripping. SPE J 23(4):1019–1038. https://doi.org/10.2118/180307-PA
Zhao J (2000) Applicability of Mohr–Coulomb and Hoek–Brown strength criteria to the dynamic strength of brittle rock. Int J Rock Mech Min Sci 37:1115–1121. https://doi.org/10.1016/S1365-1609(00)00049-6
Zhou Y, Zhao J (2011) Advances in rock dynamics and applications. CRC Press, Boca Raton
Zienkiewicz OC, Chang CT, Bettess P (1980) Drained, undrained, consolidating and dynamic behavior assumptions in soils. Geotechnique 30:385–395. https://doi.org/10.1680/geot.1980.30.4.385
Zoback MD (2010) Reservoir geomechanics. Cambridge University Press, Cambridge
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendix
Appendix
1.1 A1 Numerical Solution Strategy of the Poroelastodynamics Model
For the governing equations of the poroelastodynamics model, the numerical solving method is preferred to be used. The reason is, on one hand, the pore pressure term in the displacement equation makes it difficult to find the analytical solution. On the other hand, the analytical solution is not suitable, since the dynamic surge/swab pressure changes arbitrarily which cannot be simulated simply by any specific function. To avoid the redundant unit conversion, non-dimensional quantities should be introduced to modify Eq. (12).
-
Non-dimensional distance: \(\xi =\frac{r}{{r}_{w}}\)
-
Non-dimensional displacement: \(\phi =\frac{2G}{{p}_{0}}\cdot \frac{{u}_{r}}{{r}_{w}}\)
-
Non-dimensional pore pressure: \(\chi =\frac{p}{{p}_{0}}\)
-
Non-dimensional time: \(\tau =\frac{ct}{{r}_{w}}\)
-
Non-dimensional radial stress: \({S}_{r}=\frac{{\sigma }_{r}}{{p}_{0}}\)
Introducing these non-dimensional quantities into Eq. (12):
The implicit difference method is used to differentiate the diffusion equation:
Since the governing equation is in cylindrical coordinates, it is better to transform the radial direction into Cartesian coordinates. The way is as follows:
Then, the diffusion equation in Eq. (18) becomes
Therefore, the implicit finite differential formula of Eq. (20) is
Through some derivations, we have
Implicit finite-difference method is also used to differentiate the displacement equation:
Using Eq. (19), the displacement equation in Eq. (23) could be converted into
Therefore, the implicit finite differential formula of Eq. (24) is
The tridiagonal matrix algorithm (TDMA), also known as the Thomas algorithm, is a simplified form of Gaussian elimination that can be used to solve Eqs. (22) and (26)
After rearrangement, we have
1.2 A2 Conventional Method of Calculating Wellbore Stress
The in-situ stress of the virgin formation for a deviated well is as follows (Fig. 24):
Under static loading conditions, the conventional solution for impermeable wellbore is provided by Fjar et al. (2008). It is very well-known and not include here. The conventional solution for permeable wellbore is (Chen 2008; Ma 2015)
where \({p}_{0}\) is the static wellbore pressure, and \(\varnothing\) is the porosity.
Rights and permissions
About this article
Cite this article
Meng, M., Zamanipour, Z., Miska, S. et al. Dynamic Wellbore Stability Analysis Under Tripping Operations. Rock Mech Rock Eng 52, 3063–3083 (2019). https://doi.org/10.1007/s00603-019-01745-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00603-019-01745-4