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202 Part III Underbalanced Drilling Systems
The stresses at the same point during drilling of the hole are expressed as
8
σ r = p bh
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>
<
σ θ = σ x + σ y − 2ðσ x − σ y Þ cosð2θÞ − 4σ xy sinð2θÞ − p bh
(9.50)
σ a = σ V − ν½2ðσ x − σ y Þ cosð2θÞ + 4σ xy sinð2θÞ
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>
:
σ θa = 2½σ yz cosðθÞ − σ xz sinðθÞ
where σ r , σ θ ,and σ a are stresses in the radial, tangential (loop), and axial
directions, respectively. It can be shown that the loop stress σ θ reaches its
maximum at θ =90° and θ =270°.The principalstressesatthe pointof
concern are expressed as
r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
σ 1 = σ θ + σ a + σ θ − σ a + σ 2 aθ (9.51)
2 2
r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
σ 2 = σ θ + σ a − σ θ − σ a + σ 2 (9.52)
2 2 aθ
and
(9.53)
σ 3 = σ r
The maximum and minimum stresses among the three principal stresses
are expressed as
(9.54)
σ max = maxðσ 1 ,σ 2 ,σ 3 Þ
and
(9.55)
σ min = minðσ 1 ,σ 2 ,σ 3 Þ
The Mohr-Coulomb failure criterion can be written as
′ π + 2φ ′ 2 π + 2φ
σ max ≤ 2S o tan + σ min tan (9.56)
4 4
in which S o is the cohesive strength and ϕ is the friction angle. The
effective stress is expressed as
′
σ = σ − α B p pore (9.57)
where α B is the poroelastic constant taking a value between rock porosity
and 1, averaging at 0.72.
