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Using different rock failure criteria in wellbore stability analysis
Geomechanics for Energy and the Environment 2 (2015) 15–21
Contents lists available at ScienceDirect
Geomechanics for Energy and the Environment journal homepage: www.elsevier.com/locate/gete
Using different rock failure criteria in wellbore stability analysis Ayub Elyasi, Kamran Goshtasbi ∗ Department of Mining Engineering, Tarbiat Modares University, Tehran 14115143, Iran
highlights • The minimum allowable mud pressure of Mogi–Coulomb is much less than the other two criteria. • The minimum and maximum allowable mud pressure resulted from Mohr–Coulomb and Hoek–Brown criteria are not significantly different.
• The Hoek–Brown criterion is more conservative than the other two criteria. • The mud window resulted from Mogi–Coulomb is the broadest and Hoek–Brown is the most limited.
article
info
Article history: Received 4 November 2014 Received in revised form 6 April 2015 Accepted 10 April 2015 Available online 23 April 2015 Keywords: Allowable drilling mud pressure Mohr–Coulomb Mogi–Coulomb Hoek–Brown Wellbore stability
abstract The minimum and the maximum allowable drilling mud pressures for two wellbores are calculated analytically by Mohr–Coulomb, Hoek–Brown and Mogi–Coulomb failure criteria and numerically using a Fish program which is implemented in FLAC. The allowable drilling mud pressures that resulted from these methods are compared with each other and then with the field data: two wellbores in two Iranian oilfields are investigated as case studies to assess the results. Finally the criterion that matches closely field data is identified. It is shown that the minimum allowable drilling mud pressure obtained from the Mogi–Coulomb criterion is close to actual data collected during drilling operation. Hence, this criterion is considered more appropriate for wellbore stability analysis than the other two criteria. © 2015 Elsevier Ltd. All rights reserved.
1. Introduction Underground formations are subjected to a vertical compressive stress caused by the weight of the overlying strata, and horizontal stresses due to the confining lateral restraints. Under the action of these in situ stresses, prior to drilling a borehole, the rock mass is in a state of equilibrium that will be destroyed by the excavation. When a borehole is drilled, the load carried by the removed rock is taken over by the adjacent rock to re-establish equilibrium. As a result, a stress concentration is produced around the well, and the in situ stresses are modified. If there is no support pressure introduced into the borehole, failure in the
∗
Corresponding author. Tel.: +98 21 82883377. E-mail address: [email protected] (K. Goshtasbi).
http://dx.doi.org/10.1016/j.gete.2015.04.001 2352-3808/© 2015 Elsevier Ltd. All rights reserved.
formation may take place. Therefore, maintaining equilibrium in the field to prevent rock failure requires the use of a support pressure which during drilling is provided by a pressurized fluid called ‘‘drilling mud’’ or by maintaining pressure properly during production. Wellbore instability during drilling is one of the most important issues for engineers in oil industry. These instability issues involve stuck pipe or tight hole, drilling mud loss or lost circulation, wellbore breakout or collapse and wall tensile fracture. Exact determination of the allowable drilling mud pressure needs application of a suitable rock failure criterion which considers the wellbore condition and the rock properties. The most common failure criterion that is used is Mohr–Coulomb. It was used by Fjaer et al.1 to determine the drilling mud pressure window; i.e. the minimum well pressure permitted to prevent hole collapse (or fluid
16
A. Elyasi, K. Goshtasbi / Geomechanics for Energy and the Environment 2 (2015) 15–21
influx) and the maximum well pressure permitted to prevent loss of fluid to the formation by flow into existing or induced fractures. When these limits are known, the well may be designed. Wiprut and Zoback,2 Zoback et al.3 employed Mohr–Coulomb failure criterion to determine the principal stresses around a well and analyze its stability. Vernik and Zoback4 found that using the Mohr–Coulomb criterion to relate the wellbore breakout to the in situ stresses in crystalline rocks does not provide realistic results. They recommended the use of a failure criterion which considers the influence of intermediate principal stress (σ2 ) on the rock strength. Song and Haimson5 conducted laboratory tests to investigate the wellbore breakouts in Westerly granite and Berea sandstone, and compared the observations with different failure criteria approximations. Al-Ajmi and Zimmerman6,7 developed Mogi–Coulomb failure criterion based on true-triaxial test results, in which the effect of σ2 on the stability of wellbores is considered. Ewy8 used modified Lade criterion for analyzing the wellbore stability. Wiebols and Cook9 proposed criterion based on the shear strain energy which consists of octahedral and normal shear stresses. In rock mechanics, there are some empirical failure criteria; among them Hoek–Brown failure criterion is the most popular. Pan and Hudson10 have reviewed several criteria which include the influence of the intermediate principal stress and have proposed a three dimensional variation of the Hoek–Brown criterion. Kim and Lade11 used the first and the third stress invariants and proposed a 3D criterion. In this paper, maximum and minimum allowable drilling mud pressures are determined analytically using three rock failure criteria, namely Mohr–Coulomb, Mogi– Coulomb and Hoek–Brown. The analytical results are compared to computation performed with the Finite Difference Method and the FLAC12 software employing a Fish program and actual field data. Moreover, implemented models are validated by comparing its result against analytical results. These three failure criteria are applied to obtain the allowable drilling mud pressure window related to two vertical wells from two Iranian oilfields. Finally, the results of this analysis are compared with the field data (actual drilling mud pressure). 2. Stresses at wellbore wall The stress distribution at the borehole wall in an infinite plate in one-dimensional tension was first published by Kirsch.15 The Kirsch formulas generalize easily to a vertical well with unequal far field stresses. As the well is drilled, the wellbore wall must bear stresses previously carried by the removed rock. This causes the stresses to concentrate about the wellbore. These stress concentrations depend on the wellbore orientation and the far field in situ stresses.13,14 Assuming that the vertical stress is a principal stress, the three principal stresses acting at the vertical wellbore wall are: 1. The effective radial stress (σrr ) which acts normal to the wellbore. 2. The effective axial stress (σzz ) which acts parallel to the wellbore axis and;
Fig. 1. Schematic view of wellbore breakout.17
3. The effective circumferential stress (σθ θ ), which acts orthogonal to σrr and σzz . According to the Kirsch solution, the three effective stresses at wellbore wall are given by15
σrr = Pw − P0 , σθ θ = σH + σh − 2(σH − σh ) cos 2θ − Pw − P0 , σzz = σv − 2ν(σH − σh ) cos 2θ − P0 ,
(1)
where σH is the maximum horizontal stress, σh is the minimum horizontal stress, Pw is the drilling mud pressure acting on the wellbore wall, P0 is the pore pressure inside the formation, ν is Poisson’s ratio of the rock and θ is measured from azimuth of maximum horizontal stress. Since oil wells are typically drilled overbalanced and drilling mud cake is formed at the wellbore wall, the assumption of impermeability is also generally considered valid for wellbore walls.16 During drilling operations two instability issues, wellbore collapse and Drilling Induced Tensile Fracture, are possible. From hydraulic fracturing test, it is known that the direction of the fracture is perpendicular to the minimum horizontal principal stress and parallel to the maximum horizontal principal stress (Figs. 1 and 2). By employing the laboratory test results, Hubbert and Willis18 confirmed that the work required to open a fracture by a particular length is proportional to the stress value acting perpendicular to the fracture plane. Since there are three principal stresses, six permutations are possible. According to Eq. (1) σrr and σθ θ are the functions of drilling mud pressure, Pw . When Pw increases, σrr also increases and σθ θ decreases toward the tensile strength. Therefore, the upper limit of the drilling mud pressure is associated with fracturing which is happening when σθ θ is lower than σrr and is equal to tensile strength of the rock. On the other hand, when Pw decreases, σθ θ increases toward the compressive strength. Thus, the lower limit of the drilling mud pressure is associated with wellbore collapse, in which σθ θ is greater than σrr .
A. Elyasi, K. Goshtasbi / Geomechanics for Energy and the Environment 2 (2015) 15–21
17
5. Hoek–Brown failure criterion In the generalized Hoek–Brown criterion, Hoek used Geological Strength Index (GSI), instead of Rock Mass Rating (RMR) proposed by Bieniawski.19–22 The generalized Hoek–Brown criterion has been defined as
a′′ σ′ σ1′ = σ3′ + σci mb 3 + s , σci
(5)
where mb is a reduced value of the material constant mi and is given by23
mb = mi exp
GSI − 100 28 − 14D
,
(6)
D is the disturbance factor of rock, s and a′′ are constants for the rock mass given by:
s = exp
GSI − 100
Fig. 2. Schematic view of tensile wall fracture.17
Three out of six permutations are associated with the wellbore breakout and the other three are associated with the wellbore wall fracture. Substituting three principal stresses into an appropriate rock failure criterion one can obtain the upper limit of the allowable drilling mud pressure, Pwmax , so that applying Pw > Pwmax results in tensile fracture. Lower limit of the allowable drilling mud pressure is Pwmin so that applying Pw < Pwmin results in wellbore breakout. To determine the maximum allowable drilling mud pressure three situations should be considered: (1) σrr ≥ σθθ ≥ σzz , (2) σrr ≥ σzz ≥ σθθ and (3) σzz ≥ σrr ≥ σθ θ . In addition, three other situations for determining the minimum allowable pressure are as follows: (1) σzz ≥ σθ θ ≥ σrr , (2) σθθ ≥ σzz ≥ σrr and (3) σθθ ≥ σrr ≥ σzz . 3. Mohr–Coulomb failure criterion A frequently used criterion is the Mohr–Coulomb criterion, which is based on the assumption that f (σn ) is a linear function of σn at failure
τ = c + σn tan ϕ,
(2)
where τ is the shear stress, C is the cohesion, σn is the normal strength and ϕ is the internal friction angle of the rock.
AI-Ajmi and Zimmerman6 simplified nonlinear Mogi criterion and defined linear Mogi criterion which is known as ‘‘Mogi–Coulomb’’. In (τoct − σm,2 ) space this criterion is given by
2 2
2 2
(3)
sin ϕ, (4) 3 where τoct is the octahedral shear stress, σm,2 is the mean normal stress, a and b are constants. a=
3
c cos ϕ,
σ2 + σ3 σm,2 = 2 √ b=
1 6
, (7)
(e−GSI /15 − e−20/3 ).
6. Geological background of studied area The Zagros fold-thrust belt stretches around 2000 km from south-eastern Turkey through northern Syria and Iraq to western and southern Iran.24 Structurally, the Zagros basin is placed in the north of the Arabian plate. Studied formations of the wells include Pabdeh, Gurpi, Ilam and Sarvak. The Zagros basin is part of the Tethys Ocean and is one of the most important petroleum reservoirs in the world.24 The geological history of this basin includes a long time subsidence and deposition which is interrupted by a short-term uplift. Its folding process began during Miocene and Pliocene anticlines and continued until now, hence forming a long anticline.25 These anticlines constitute most of oil traps in this basin. Fields 1 and 2 are part of Mansouri field. Mansouri is a great oilfield that is located in the northern Dezful subsidence at 60 km to the south of Ahvaz. It is surrounded by the Ahvaz field from northwest, Abteimoor field from west and Shadegan field from northeast. At oil–water contact surface, the Mansouri field has 39 km length and 3.5 km width. 6.1. Lithology of studied formations
4. Mogi–Coulomb failure criterion
τoct = a + bσm,2 , √
a′′ = 0.5 +
9 − 3D
Alavi24 studied the lithostratigraphic units of the Zagros thrust belt and published the description of its formations. The formations studied are briefly characterized as follows: Pabdeh (Upper Paleocene to lowermost Oligocene, 2902–3160 m):thin-bedded Globigerina-bearing deepmarine hemipelagic–pelagic calcareous shale, marl, and limemudstone with subordinate argillaceous limestone containing fish fossils; foredeepfacies grading toward the northeast into sandy Shahbazan dolomitic limestone and toward the southwest intosabhka-type Jahrum carbonates (Fig. 3).
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Table 1 Rock mass properties and in-situ stresses in wells 32 and 54.
Well 32 Well 54
σc (MPa)
υ
E (GPa)
C (MPa)
ϕ (degree)
σv (MPa)
σH (MPa)
σh (MPa)
18 16
0.33 0.25
10.5 8.5
3.7 4
41.5 35
72 84
48 82
43 78
Note: σC : uniaxial compressive strength, v : Poison’s ratio, E: Young’s modulus, C : Cohesion strength, ϕ : the internal friction angle, σv : vertical stress, σH : maximum horizontal stress and σh : minimum horizontal stress.
Gurpi (Upper Cretaceous; Santonian to Maastrichtian, 3160–3414 m): dark bluish gray, thin-bedded, deepmarine, Globigerina-bearing pelagic marl (and marly limestone) and hemipelagic claystone; foredeepfacies (resting uncomfortably on Sarvak and onlapping Ilam) resulting from prograde southwestward migration of proforelandzones (Fig. 3). Ilam (Oligocene to lowermost Miocene, 3414–3630 m): medium-bedded to thick-bedded, locally shelly or oolitic, nummulites-bearing limestones (grainstone, packstone, wackestone) shoaling upward above a thin basal conglomerate from fine-grained (low-energy) deep-marine marly limestone to high-energy shallow-marine skeletal grainstone; composed of a number of sequences; unconformity-bounded, highly prolific reservoir; interpreted as transgressive–regressive foredeepfacies of the proforeland basin (Fig. 3). Sarvak (Cretaceous, 3630–3768 m): the formation is a thick carbonated unit that was deposited in ‘‘Neotethys southern margin of Zagros’’ area. It is one of the most important hydrocarbon resources production horizons in Iran. Laboratory and field observation lead to recognition of four facies environments: open marine, shale, and lagoon in Coastal area of Fars, Khuzestan and Lurestan. The lower lithostratigraphic limit of Sarvak Formation, which is conformable and gradational, overlies the Kazhdumi Formation in section. Upper lithostratigraphic limit of that is secant with Gurpi Formation (Fig. 3). 7. Allowable drilling mud pressure determination in the case studies
the stability of wells. This program simulates the behavior of structures built of soil, rock or other materials that may undergo plastic flow when their yield limits are reached. Materials are represented by elements, or zones, which form a grid that is adjusted by the user to fit the shape of the object to be modeled. Each element behaves according to a prescribed linear or nonlinear stress/strain law in response to the applied forces or boundary restraints. The material can yield; flow and the grid can deform (in large-strain mode) and move with the material that is represented. The explicit, Lagrangian calculation scheme and the mixeddiscretization zoning technique used in FLAC ensure that plastic collapse and flow are modeled very accurately.12 The modeled wells are 42 cm in diameter and the initial and boundary condition for the model are as follows. The pore pressure and stresses shown in Table 1 are used to establish initial equilibrium conditions within the model. Also the lateral boundaries are fixed in the XY direction and submodel boundary conditions have been used to constrain the pore pressure during the steps created for the wellbore stability analysis and there is no flow across the boundary of the model. The numerical solution for well 32 is reached more quickly than the numerical solution for well 54. Note that the numerical calculation for well 32 requires about 11 000 steps, while the calculation for well 54 requires approximately 15 500 steps. This is because of the low strengths parameters of well 54 medium compared with well 32 medium. Also the in-situ stresses of well 54 medium are greater than the in-situ stresses of well 32 medium. 7.2. Results
As case studies, the wellbore stability analysis was carried out in two vertical wellbores (well 54 from field 1 and well 32 from field 2). The dominant rock in both wellbores consists of limestone but it had different properties. The concerned depths where the collapse is probable in wellbores 32 and 54 are 2648 m and 3485 m, respectively. Moreover, in the above mentioned depths the pore pressures are 11 and 30 MPa, respectively for wellbores 32 and 54. We calculated the geomechanical parameters of these rocks by using the laboratory tests. Vertical stresses, σv were determined from overburden density and depth, horizontal stresses were obtained from stress polygon constraining.3 Rock mass properties and in-situ stresses in two fields 1 and 2 are presented in Table 1. 7.1. Numerical modeling In this study, the FLAC numerical code which is based on the finite difference method is utilized in order to analyze
Maximum and minimum allowable drilling mud pressures are determined analytically using three rock failure criteria, namely Mohr–Coulomb, Mogi–Coulomb and Hoek–Brown. The analytical results are compared to computation performed with the Finite Difference Method using FLAC software and actual field data. These three failure criteria are applied to obtain the allowable drilling mud pressure window related to two vertical wells from two Iranian oilfields. Fig. 4(a) and (b) illustrates minimum (Pwmin ) and maximum (Pwmax ) allowable drilling mud pressure, within well 54 from field 1 in terms of θ which is measured from the azimuth of σH , respectively. Where P0 and σh parameters are pore pressure and the minimum horizontal stress, respectively. Constant parameters m and s in Hoek–Brown failure criterion are calculated using RocData software and the resulted values are 4.99 and 0.88 respectively. Analytical and numerical results using three criteria and FLAC software show monotonic increase of drilling mud pressure
A. Elyasi, K. Goshtasbi / Geomechanics for Energy and the Environment 2 (2015) 15–21
19
Fig. 3. Mesozoic–Cenozoic stratigraphy correlation chart of the Zagros basin.25
around wellbore from 0° to 90°. It can be found that numerical and analytical results using FLAC, Mohr–Coulomb (MC) and Hoek–Brown (HB) overestimate the Pwmin obtained from field data (actual drilling mud pressure). The Mogi–Coulomb (MG) criterion shows realistic result in this case. Analytical and numerical results using three criteria and FLAC software show increase of drilling mud pressure around wellbore from 0° to 90°. It can be concluded that numerical and analytical results using FLAC, MC, HB and MG overestimate the Pwmax . The actual drilling mud pressures during drilling in well 32 and 54 were 6.8 MPa and 25 MPa, respectively. This fact confirms that the Mogi–Coulomb has estimated a more
realistic drilling mud window rather than the other two criteria. According to Fig. 4(a), 25 MPa is within the stable range of the Mogi–Coulomb criterion and in the unstable zone of Hoek–Brown and Mohr–Coulomb criteria. It should be noticed that for Pwmax in Fig. 4(b) the results are larger than minimum horizontal stress (σh ). However, in practice Pwmax is considered to be equal to σh to prevent fracturing in wellbore wall. Fig. 5(a) and (b) demonstrates Pwmin and Pwmax around well 32 from field 2 in terms of θ , respectively. As previous case analytical and numerical results using three criteria and FLAC software show monotonic increase of drilling mud pressure around wellbore. Also it can be seen that numerical and analytical results using FLAC, MC, HB and MG overestimates the Pwmax .
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Fig. 4. Minimum (a) and maximum (b) allowable drilling mud pressure as a function of θ resulted from the Mohr–Coulomb (MC), the Mogi–Coulomb (MG) and the Hoek–Brown (HB) criteria and FLAC, well 54 from field 1.
Fig. 5. Minimum (a) and maximum (b) allowable drilling mud pressure as a function of θ resulted from the Mohr–Coulomb (MC), the Mogi–Coulomb (MG) and the Hoek–Brown (HB) criteria and FLAC, well 32 from field 2.
Moreover, as shown in Figs. 4 and 5, the results form analytical and numerical analysis for each failure criteria are very close and there is no significant discrepancy between the analytical results and the FLAC results. 8. Conclusion According to the obtained results, it can be concluded that:
• The minimum allowable drilling mud pressure of Mogi–Coulomb is much less than the other two criteria. In well 32 and 54, this minimum drilling mud pressure is even less than the pore pressure. • The minimum allowable drilling mud pressure resulted from Mohr–Coulomb and Hoek–Brown criteria are not significantly different. This similarity is also true about the maximum allowable drilling mud pressure. These values are very different from the Mogi–Coulomb criterion results. This is because these two criteria just consider the maximum and minimum principal stresses and not the mean principle stress, like Mogi–Coulomb, as well. • It can also be concluded that the Hoek–Brown criterion is more conservative than the other two and the drilling mud window that resulted from Mogi–Coulomb is the broadest and Hoek–Brown is the most limited.
• In well 32 and 54 all three criteria estimate the maximum allowable drilling mud pressure as greater than the minimum horizontal stress. References [1] Fjaer E, Holt RM, Horsrud P, Raaen AM, Risnes R. Petroleum Related Rock Mechanics. 2nd ed. Amsterdam, Netherlands: Elsevier; 2008. [2] Wiprut D, Zoback MD. Constraining the stress tensor in the Visund field, Norwegian North Sea: Application to wellbore stability and sand production. Int. J. Rock Mech. Min. Sci. 2000;37:317–336. [3] Zoback MD. Determination of stress orientation and magnitude in deep wells. Int. J. Rock Mech. Min. Sci. 2003;40:1049–1076. [4] Vernik L, Zoback MD. Estimation of maximum horizontal principal stress magnitude from stress-induced well bore breakouts in the Cajon Pass scientific research borehole. J. Geophys. Res. 1992;97: 5109–5119. [5] Song I, Haimson BC. Polyaxial strength criteria and their use in estimating in situ stress magnitudes from borehole breakout dimensions. Int. J. Rock Mech. Min. Sci. 1997;34(3–4):116.e1–116.e16. [6] Al-Ajmi AM, Zimmerman RW. Relation between the Mogi and the Coulomb failure criteria. Int. J. Rock Mech. Min. Sci. 2005;42:431–439. [7] Al-Ajmi AM, Zimmerman RW. Stability analysis of vertical boreholes using the Mogi–Coulomb failure criterion. Int. J. Rock Mech. Min. Sci. 2006;43:1200–1211. [8] Ewy RT. Wellbore-stability predictions by use of a modified Lade criterion. SPE Drill. Complet. 1999;14(2):85–91. [9] Wiebols GA, Cook NGW. An energy criterion for the strength of rock in polyaxial compression. Int. J. Rock Mech. Min. Sci. 1968;5(6): 529–549. [10] Pan XD, Hudson JA. A simplified three dimensional Hoek–Brown yield criterion. Rock Mechanics and Power Plants. In: Romana M, ed. Rotterdam: Balkema; 1988:95–103.
A. Elyasi, K. Goshtasbi / Geomechanics for Energy and the Environment 2 (2015) 15–21 [11] Sheorey PR. Empirical Rock Failure Criteria. Rotterdam, Netherlands: Balkema; 1997. [12] ITASCA, FLAC3D, Student License, Itasca Consulting Group, Inc. USA, 2006. [13] Amadei B, Stephansson O. Rock Stress and its Measurement. London: Chapman & Hall; 1997. [14] Jaeger JC, Cook NGW. Fundamentals of Rock Mechanics. 3rd ed. London, United Kingdom: Chapman & Hall; 1979. [15] Kirsch G. Die Theorie der Elastizitat und die Bedurinisse der Festigkeitslehre. Z. Ver. Dtsch. Ing. 1898;42:797–807. [16] Nelson EJ, Meyer JJ, Hillis RR, Mildren SD. Transverse drillinginduced tensile fractures in the West Tuna area, Gippsland Basin, Australia: implications for the in situ stress regime. Int. J. Rock Mech. Min. Sci. 2005;42(3):361–371. [17] M. Ask, In-situ stress at the Cote D’ivoire-Ghana marginal ridge from FMS logging in hole 959D, In: Proceedings of the Ocean Drilling Program, Scientific Results, Vol. 159: 1998.
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[18] Hubbert MK, Willis DG. Mechanics of hydraulic fracturing. Pettrans AIME 1957;210:153–163. [19] E. Hoek, D. Wood, S. Shah, A modified Hoek-Brown criterion for jointed rock masses, In: International ISRM Symposium on Rock Characterization, Chester, UK, September: 1992. [20] Hoek E. Strength of rock and rock masses. ISRM News J. 1994;2(2): 4–16. [21] Hoek E, Kaiser PK, Bawden WF. Support of Underground Excavations in Hard Rock. Rotterdam: Balkema; 1995. [22] Bieniawski ZT. Rock mass classification in rock engineering. Exploration for Rock Engineering, Proc. of the Symp, Vol. 1. In: Bieniawski ZT, ed. Balkema; 1976:97–106. [23] Benz T, Schwab R. A quantitative comparison of six rock failure criteria. Int. J. Rock. Mech Min. Sci. 2008;45:1176–1186. [24] Alavi M. Regional stratigraphy of the Zagorsk fold-thrust belt of Iran and its proforeland evolution. Am. J. Sci. 2004;304:1–20. [25] Mottiei H. Geology of Iran; petroleum geology of Zagros. Geol. Soc. Iran Publ. 1995;37(2):1009.
Contents lists available at ScienceDirect
Geomechanics for Energy and the Environment journal homepage: www.elsevier.com/locate/gete
Using different rock failure criteria in wellbore stability analysis Ayub Elyasi, Kamran Goshtasbi ∗ Department of Mining Engineering, Tarbiat Modares University, Tehran 14115143, Iran
highlights • The minimum allowable mud pressure of Mogi–Coulomb is much less than the other two criteria. • The minimum and maximum allowable mud pressure resulted from Mohr–Coulomb and Hoek–Brown criteria are not significantly different.
• The Hoek–Brown criterion is more conservative than the other two criteria. • The mud window resulted from Mogi–Coulomb is the broadest and Hoek–Brown is the most limited.
article
info
Article history: Received 4 November 2014 Received in revised form 6 April 2015 Accepted 10 April 2015 Available online 23 April 2015 Keywords: Allowable drilling mud pressure Mohr–Coulomb Mogi–Coulomb Hoek–Brown Wellbore stability
abstract The minimum and the maximum allowable drilling mud pressures for two wellbores are calculated analytically by Mohr–Coulomb, Hoek–Brown and Mogi–Coulomb failure criteria and numerically using a Fish program which is implemented in FLAC. The allowable drilling mud pressures that resulted from these methods are compared with each other and then with the field data: two wellbores in two Iranian oilfields are investigated as case studies to assess the results. Finally the criterion that matches closely field data is identified. It is shown that the minimum allowable drilling mud pressure obtained from the Mogi–Coulomb criterion is close to actual data collected during drilling operation. Hence, this criterion is considered more appropriate for wellbore stability analysis than the other two criteria. © 2015 Elsevier Ltd. All rights reserved.
1. Introduction Underground formations are subjected to a vertical compressive stress caused by the weight of the overlying strata, and horizontal stresses due to the confining lateral restraints. Under the action of these in situ stresses, prior to drilling a borehole, the rock mass is in a state of equilibrium that will be destroyed by the excavation. When a borehole is drilled, the load carried by the removed rock is taken over by the adjacent rock to re-establish equilibrium. As a result, a stress concentration is produced around the well, and the in situ stresses are modified. If there is no support pressure introduced into the borehole, failure in the
∗
Corresponding author. Tel.: +98 21 82883377. E-mail address: [email protected] (K. Goshtasbi).
http://dx.doi.org/10.1016/j.gete.2015.04.001 2352-3808/© 2015 Elsevier Ltd. All rights reserved.
formation may take place. Therefore, maintaining equilibrium in the field to prevent rock failure requires the use of a support pressure which during drilling is provided by a pressurized fluid called ‘‘drilling mud’’ or by maintaining pressure properly during production. Wellbore instability during drilling is one of the most important issues for engineers in oil industry. These instability issues involve stuck pipe or tight hole, drilling mud loss or lost circulation, wellbore breakout or collapse and wall tensile fracture. Exact determination of the allowable drilling mud pressure needs application of a suitable rock failure criterion which considers the wellbore condition and the rock properties. The most common failure criterion that is used is Mohr–Coulomb. It was used by Fjaer et al.1 to determine the drilling mud pressure window; i.e. the minimum well pressure permitted to prevent hole collapse (or fluid
16
A. Elyasi, K. Goshtasbi / Geomechanics for Energy and the Environment 2 (2015) 15–21
influx) and the maximum well pressure permitted to prevent loss of fluid to the formation by flow into existing or induced fractures. When these limits are known, the well may be designed. Wiprut and Zoback,2 Zoback et al.3 employed Mohr–Coulomb failure criterion to determine the principal stresses around a well and analyze its stability. Vernik and Zoback4 found that using the Mohr–Coulomb criterion to relate the wellbore breakout to the in situ stresses in crystalline rocks does not provide realistic results. They recommended the use of a failure criterion which considers the influence of intermediate principal stress (σ2 ) on the rock strength. Song and Haimson5 conducted laboratory tests to investigate the wellbore breakouts in Westerly granite and Berea sandstone, and compared the observations with different failure criteria approximations. Al-Ajmi and Zimmerman6,7 developed Mogi–Coulomb failure criterion based on true-triaxial test results, in which the effect of σ2 on the stability of wellbores is considered. Ewy8 used modified Lade criterion for analyzing the wellbore stability. Wiebols and Cook9 proposed criterion based on the shear strain energy which consists of octahedral and normal shear stresses. In rock mechanics, there are some empirical failure criteria; among them Hoek–Brown failure criterion is the most popular. Pan and Hudson10 have reviewed several criteria which include the influence of the intermediate principal stress and have proposed a three dimensional variation of the Hoek–Brown criterion. Kim and Lade11 used the first and the third stress invariants and proposed a 3D criterion. In this paper, maximum and minimum allowable drilling mud pressures are determined analytically using three rock failure criteria, namely Mohr–Coulomb, Mogi– Coulomb and Hoek–Brown. The analytical results are compared to computation performed with the Finite Difference Method and the FLAC12 software employing a Fish program and actual field data. Moreover, implemented models are validated by comparing its result against analytical results. These three failure criteria are applied to obtain the allowable drilling mud pressure window related to two vertical wells from two Iranian oilfields. Finally, the results of this analysis are compared with the field data (actual drilling mud pressure). 2. Stresses at wellbore wall The stress distribution at the borehole wall in an infinite plate in one-dimensional tension was first published by Kirsch.15 The Kirsch formulas generalize easily to a vertical well with unequal far field stresses. As the well is drilled, the wellbore wall must bear stresses previously carried by the removed rock. This causes the stresses to concentrate about the wellbore. These stress concentrations depend on the wellbore orientation and the far field in situ stresses.13,14 Assuming that the vertical stress is a principal stress, the three principal stresses acting at the vertical wellbore wall are: 1. The effective radial stress (σrr ) which acts normal to the wellbore. 2. The effective axial stress (σzz ) which acts parallel to the wellbore axis and;
Fig. 1. Schematic view of wellbore breakout.17
3. The effective circumferential stress (σθ θ ), which acts orthogonal to σrr and σzz . According to the Kirsch solution, the three effective stresses at wellbore wall are given by15
σrr = Pw − P0 , σθ θ = σH + σh − 2(σH − σh ) cos 2θ − Pw − P0 , σzz = σv − 2ν(σH − σh ) cos 2θ − P0 ,
(1)
where σH is the maximum horizontal stress, σh is the minimum horizontal stress, Pw is the drilling mud pressure acting on the wellbore wall, P0 is the pore pressure inside the formation, ν is Poisson’s ratio of the rock and θ is measured from azimuth of maximum horizontal stress. Since oil wells are typically drilled overbalanced and drilling mud cake is formed at the wellbore wall, the assumption of impermeability is also generally considered valid for wellbore walls.16 During drilling operations two instability issues, wellbore collapse and Drilling Induced Tensile Fracture, are possible. From hydraulic fracturing test, it is known that the direction of the fracture is perpendicular to the minimum horizontal principal stress and parallel to the maximum horizontal principal stress (Figs. 1 and 2). By employing the laboratory test results, Hubbert and Willis18 confirmed that the work required to open a fracture by a particular length is proportional to the stress value acting perpendicular to the fracture plane. Since there are three principal stresses, six permutations are possible. According to Eq. (1) σrr and σθ θ are the functions of drilling mud pressure, Pw . When Pw increases, σrr also increases and σθ θ decreases toward the tensile strength. Therefore, the upper limit of the drilling mud pressure is associated with fracturing which is happening when σθ θ is lower than σrr and is equal to tensile strength of the rock. On the other hand, when Pw decreases, σθ θ increases toward the compressive strength. Thus, the lower limit of the drilling mud pressure is associated with wellbore collapse, in which σθ θ is greater than σrr .
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5. Hoek–Brown failure criterion In the generalized Hoek–Brown criterion, Hoek used Geological Strength Index (GSI), instead of Rock Mass Rating (RMR) proposed by Bieniawski.19–22 The generalized Hoek–Brown criterion has been defined as
a′′ σ′ σ1′ = σ3′ + σci mb 3 + s , σci
(5)
where mb is a reduced value of the material constant mi and is given by23
mb = mi exp
GSI − 100 28 − 14D
,
(6)
D is the disturbance factor of rock, s and a′′ are constants for the rock mass given by:
s = exp
GSI − 100
Fig. 2. Schematic view of tensile wall fracture.17
Three out of six permutations are associated with the wellbore breakout and the other three are associated with the wellbore wall fracture. Substituting three principal stresses into an appropriate rock failure criterion one can obtain the upper limit of the allowable drilling mud pressure, Pwmax , so that applying Pw > Pwmax results in tensile fracture. Lower limit of the allowable drilling mud pressure is Pwmin so that applying Pw < Pwmin results in wellbore breakout. To determine the maximum allowable drilling mud pressure three situations should be considered: (1) σrr ≥ σθθ ≥ σzz , (2) σrr ≥ σzz ≥ σθθ and (3) σzz ≥ σrr ≥ σθ θ . In addition, three other situations for determining the minimum allowable pressure are as follows: (1) σzz ≥ σθ θ ≥ σrr , (2) σθθ ≥ σzz ≥ σrr and (3) σθθ ≥ σrr ≥ σzz . 3. Mohr–Coulomb failure criterion A frequently used criterion is the Mohr–Coulomb criterion, which is based on the assumption that f (σn ) is a linear function of σn at failure
τ = c + σn tan ϕ,
(2)
where τ is the shear stress, C is the cohesion, σn is the normal strength and ϕ is the internal friction angle of the rock.
AI-Ajmi and Zimmerman6 simplified nonlinear Mogi criterion and defined linear Mogi criterion which is known as ‘‘Mogi–Coulomb’’. In (τoct − σm,2 ) space this criterion is given by
2 2
2 2
(3)
sin ϕ, (4) 3 where τoct is the octahedral shear stress, σm,2 is the mean normal stress, a and b are constants. a=
3
c cos ϕ,
σ2 + σ3 σm,2 = 2 √ b=
1 6
, (7)
(e−GSI /15 − e−20/3 ).
6. Geological background of studied area The Zagros fold-thrust belt stretches around 2000 km from south-eastern Turkey through northern Syria and Iraq to western and southern Iran.24 Structurally, the Zagros basin is placed in the north of the Arabian plate. Studied formations of the wells include Pabdeh, Gurpi, Ilam and Sarvak. The Zagros basin is part of the Tethys Ocean and is one of the most important petroleum reservoirs in the world.24 The geological history of this basin includes a long time subsidence and deposition which is interrupted by a short-term uplift. Its folding process began during Miocene and Pliocene anticlines and continued until now, hence forming a long anticline.25 These anticlines constitute most of oil traps in this basin. Fields 1 and 2 are part of Mansouri field. Mansouri is a great oilfield that is located in the northern Dezful subsidence at 60 km to the south of Ahvaz. It is surrounded by the Ahvaz field from northwest, Abteimoor field from west and Shadegan field from northeast. At oil–water contact surface, the Mansouri field has 39 km length and 3.5 km width. 6.1. Lithology of studied formations
4. Mogi–Coulomb failure criterion
τoct = a + bσm,2 , √
a′′ = 0.5 +
9 − 3D
Alavi24 studied the lithostratigraphic units of the Zagros thrust belt and published the description of its formations. The formations studied are briefly characterized as follows: Pabdeh (Upper Paleocene to lowermost Oligocene, 2902–3160 m):thin-bedded Globigerina-bearing deepmarine hemipelagic–pelagic calcareous shale, marl, and limemudstone with subordinate argillaceous limestone containing fish fossils; foredeepfacies grading toward the northeast into sandy Shahbazan dolomitic limestone and toward the southwest intosabhka-type Jahrum carbonates (Fig. 3).
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Table 1 Rock mass properties and in-situ stresses in wells 32 and 54.
Well 32 Well 54
σc (MPa)
υ
E (GPa)
C (MPa)
ϕ (degree)
σv (MPa)
σH (MPa)
σh (MPa)
18 16
0.33 0.25
10.5 8.5
3.7 4
41.5 35
72 84
48 82
43 78
Note: σC : uniaxial compressive strength, v : Poison’s ratio, E: Young’s modulus, C : Cohesion strength, ϕ : the internal friction angle, σv : vertical stress, σH : maximum horizontal stress and σh : minimum horizontal stress.
Gurpi (Upper Cretaceous; Santonian to Maastrichtian, 3160–3414 m): dark bluish gray, thin-bedded, deepmarine, Globigerina-bearing pelagic marl (and marly limestone) and hemipelagic claystone; foredeepfacies (resting uncomfortably on Sarvak and onlapping Ilam) resulting from prograde southwestward migration of proforelandzones (Fig. 3). Ilam (Oligocene to lowermost Miocene, 3414–3630 m): medium-bedded to thick-bedded, locally shelly or oolitic, nummulites-bearing limestones (grainstone, packstone, wackestone) shoaling upward above a thin basal conglomerate from fine-grained (low-energy) deep-marine marly limestone to high-energy shallow-marine skeletal grainstone; composed of a number of sequences; unconformity-bounded, highly prolific reservoir; interpreted as transgressive–regressive foredeepfacies of the proforeland basin (Fig. 3). Sarvak (Cretaceous, 3630–3768 m): the formation is a thick carbonated unit that was deposited in ‘‘Neotethys southern margin of Zagros’’ area. It is one of the most important hydrocarbon resources production horizons in Iran. Laboratory and field observation lead to recognition of four facies environments: open marine, shale, and lagoon in Coastal area of Fars, Khuzestan and Lurestan. The lower lithostratigraphic limit of Sarvak Formation, which is conformable and gradational, overlies the Kazhdumi Formation in section. Upper lithostratigraphic limit of that is secant with Gurpi Formation (Fig. 3). 7. Allowable drilling mud pressure determination in the case studies
the stability of wells. This program simulates the behavior of structures built of soil, rock or other materials that may undergo plastic flow when their yield limits are reached. Materials are represented by elements, or zones, which form a grid that is adjusted by the user to fit the shape of the object to be modeled. Each element behaves according to a prescribed linear or nonlinear stress/strain law in response to the applied forces or boundary restraints. The material can yield; flow and the grid can deform (in large-strain mode) and move with the material that is represented. The explicit, Lagrangian calculation scheme and the mixeddiscretization zoning technique used in FLAC ensure that plastic collapse and flow are modeled very accurately.12 The modeled wells are 42 cm in diameter and the initial and boundary condition for the model are as follows. The pore pressure and stresses shown in Table 1 are used to establish initial equilibrium conditions within the model. Also the lateral boundaries are fixed in the XY direction and submodel boundary conditions have been used to constrain the pore pressure during the steps created for the wellbore stability analysis and there is no flow across the boundary of the model. The numerical solution for well 32 is reached more quickly than the numerical solution for well 54. Note that the numerical calculation for well 32 requires about 11 000 steps, while the calculation for well 54 requires approximately 15 500 steps. This is because of the low strengths parameters of well 54 medium compared with well 32 medium. Also the in-situ stresses of well 54 medium are greater than the in-situ stresses of well 32 medium. 7.2. Results
As case studies, the wellbore stability analysis was carried out in two vertical wellbores (well 54 from field 1 and well 32 from field 2). The dominant rock in both wellbores consists of limestone but it had different properties. The concerned depths where the collapse is probable in wellbores 32 and 54 are 2648 m and 3485 m, respectively. Moreover, in the above mentioned depths the pore pressures are 11 and 30 MPa, respectively for wellbores 32 and 54. We calculated the geomechanical parameters of these rocks by using the laboratory tests. Vertical stresses, σv were determined from overburden density and depth, horizontal stresses were obtained from stress polygon constraining.3 Rock mass properties and in-situ stresses in two fields 1 and 2 are presented in Table 1. 7.1. Numerical modeling In this study, the FLAC numerical code which is based on the finite difference method is utilized in order to analyze
Maximum and minimum allowable drilling mud pressures are determined analytically using three rock failure criteria, namely Mohr–Coulomb, Mogi–Coulomb and Hoek–Brown. The analytical results are compared to computation performed with the Finite Difference Method using FLAC software and actual field data. These three failure criteria are applied to obtain the allowable drilling mud pressure window related to two vertical wells from two Iranian oilfields. Fig. 4(a) and (b) illustrates minimum (Pwmin ) and maximum (Pwmax ) allowable drilling mud pressure, within well 54 from field 1 in terms of θ which is measured from the azimuth of σH , respectively. Where P0 and σh parameters are pore pressure and the minimum horizontal stress, respectively. Constant parameters m and s in Hoek–Brown failure criterion are calculated using RocData software and the resulted values are 4.99 and 0.88 respectively. Analytical and numerical results using three criteria and FLAC software show monotonic increase of drilling mud pressure
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Fig. 3. Mesozoic–Cenozoic stratigraphy correlation chart of the Zagros basin.25
around wellbore from 0° to 90°. It can be found that numerical and analytical results using FLAC, Mohr–Coulomb (MC) and Hoek–Brown (HB) overestimate the Pwmin obtained from field data (actual drilling mud pressure). The Mogi–Coulomb (MG) criterion shows realistic result in this case. Analytical and numerical results using three criteria and FLAC software show increase of drilling mud pressure around wellbore from 0° to 90°. It can be concluded that numerical and analytical results using FLAC, MC, HB and MG overestimate the Pwmax . The actual drilling mud pressures during drilling in well 32 and 54 were 6.8 MPa and 25 MPa, respectively. This fact confirms that the Mogi–Coulomb has estimated a more
realistic drilling mud window rather than the other two criteria. According to Fig. 4(a), 25 MPa is within the stable range of the Mogi–Coulomb criterion and in the unstable zone of Hoek–Brown and Mohr–Coulomb criteria. It should be noticed that for Pwmax in Fig. 4(b) the results are larger than minimum horizontal stress (σh ). However, in practice Pwmax is considered to be equal to σh to prevent fracturing in wellbore wall. Fig. 5(a) and (b) demonstrates Pwmin and Pwmax around well 32 from field 2 in terms of θ , respectively. As previous case analytical and numerical results using three criteria and FLAC software show monotonic increase of drilling mud pressure around wellbore. Also it can be seen that numerical and analytical results using FLAC, MC, HB and MG overestimates the Pwmax .
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Fig. 4. Minimum (a) and maximum (b) allowable drilling mud pressure as a function of θ resulted from the Mohr–Coulomb (MC), the Mogi–Coulomb (MG) and the Hoek–Brown (HB) criteria and FLAC, well 54 from field 1.
Fig. 5. Minimum (a) and maximum (b) allowable drilling mud pressure as a function of θ resulted from the Mohr–Coulomb (MC), the Mogi–Coulomb (MG) and the Hoek–Brown (HB) criteria and FLAC, well 32 from field 2.
Moreover, as shown in Figs. 4 and 5, the results form analytical and numerical analysis for each failure criteria are very close and there is no significant discrepancy between the analytical results and the FLAC results. 8. Conclusion According to the obtained results, it can be concluded that:
• The minimum allowable drilling mud pressure of Mogi–Coulomb is much less than the other two criteria. In well 32 and 54, this minimum drilling mud pressure is even less than the pore pressure. • The minimum allowable drilling mud pressure resulted from Mohr–Coulomb and Hoek–Brown criteria are not significantly different. This similarity is also true about the maximum allowable drilling mud pressure. These values are very different from the Mogi–Coulomb criterion results. This is because these two criteria just consider the maximum and minimum principal stresses and not the mean principle stress, like Mogi–Coulomb, as well. • It can also be concluded that the Hoek–Brown criterion is more conservative than the other two and the drilling mud window that resulted from Mogi–Coulomb is the broadest and Hoek–Brown is the most limited.
• In well 32 and 54 all three criteria estimate the maximum allowable drilling mud pressure as greater than the minimum horizontal stress. References [1] Fjaer E, Holt RM, Horsrud P, Raaen AM, Risnes R. Petroleum Related Rock Mechanics. 2nd ed. Amsterdam, Netherlands: Elsevier; 2008. [2] Wiprut D, Zoback MD. Constraining the stress tensor in the Visund field, Norwegian North Sea: Application to wellbore stability and sand production. Int. J. Rock Mech. Min. Sci. 2000;37:317–336. [3] Zoback MD. Determination of stress orientation and magnitude in deep wells. Int. J. Rock Mech. Min. Sci. 2003;40:1049–1076. [4] Vernik L, Zoback MD. Estimation of maximum horizontal principal stress magnitude from stress-induced well bore breakouts in the Cajon Pass scientific research borehole. J. Geophys. Res. 1992;97: 5109–5119. [5] Song I, Haimson BC. Polyaxial strength criteria and their use in estimating in situ stress magnitudes from borehole breakout dimensions. Int. J. Rock Mech. Min. Sci. 1997;34(3–4):116.e1–116.e16. [6] Al-Ajmi AM, Zimmerman RW. Relation between the Mogi and the Coulomb failure criteria. Int. J. Rock Mech. Min. Sci. 2005;42:431–439. [7] Al-Ajmi AM, Zimmerman RW. Stability analysis of vertical boreholes using the Mogi–Coulomb failure criterion. Int. J. Rock Mech. Min. Sci. 2006;43:1200–1211. [8] Ewy RT. Wellbore-stability predictions by use of a modified Lade criterion. SPE Drill. Complet. 1999;14(2):85–91. [9] Wiebols GA, Cook NGW. An energy criterion for the strength of rock in polyaxial compression. Int. J. Rock Mech. Min. Sci. 1968;5(6): 529–549. [10] Pan XD, Hudson JA. A simplified three dimensional Hoek–Brown yield criterion. Rock Mechanics and Power Plants. In: Romana M, ed. Rotterdam: Balkema; 1988:95–103.
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