ISSN : 2277-1328 (Online) - ISRM

Study of Wellbore Stresses and Stability based on a Hollow Cylinder Model
11
...(6)
Note that S o
can be linked to C o
and ϕ through S o
= C o
/2q 1/
2
where q = [(μ 2 +1) 1/2 + μ] 2 = tan 2 (π/4 + ϕ/2].
Fig. 1 : Comparison of different rock failure criteria
in the π-plane.
Mohr proposed that when shear failure takes place across
a plane, the normal stress σ n
and the shear stress τ across
that plane are related by
|τ| = S o
+ μs n
...(1)
where S o
= cohesion and μ = the coefficient of internal
friction of the material which is related to the angle of
internal friction ϕ of that material by μ = tanϕ. Since the
sign of τ only affects the sliding direction, only the
magnitude of τ matters. Presented in terms of principal
stresses, the Mohr-Coulomb criterion is:
σ 1
= C o
+ σ 3
tan 2 β ...(2)
where σ 1
= the major principal effective stress at failure,
σ 3
= the least principal effective stress at failure, C o
= the
uniaxial compressive strength, and β gives the orientation
of the failure plane and is related to internal friction angle
ϕ as β = π/4 + ϕ/2. In some formulations tan 2 β is replaced
by q, where q = [(μ 2 +1) 1/2 + μ] 2 .
The Lade criterion [9] is a three-dimensional failure criterion.
Originally proposed for cohessionless sands, the criterion
was then adopted for analyzing rocks with finite values of
cohesion and tensile strength [10] and such a formulation
was later linked [11] with standard rock mechanics
parameters such as ϕ and S o
to obtain:
where I 1
’ and I 3
’ are stress invariants
...(3)
I 1
’ = [(σ 1
+ S – P p
)+(σ 2
+ S – P p
) +(σ 3
+ S – P p
) ...(4)
I 3
’ = [(σ 1
+ S – P p
)(σ 2
+ S – P p
) (σ 3
+ S – P p
) ...(5)
where S and η are material constants, and P p
is the pore
pressure. The parameter S is related to the cohesion of
the rock, while η represents the internal friction. These
parameters can be derived directly from the Mohr-Coulomb
cohesion S o
and internal friction angle ϕ by
The typical behaviour of the Modified Lade criterion is
shown in Figs 1 and 2. In principal stress space criterion [3]
has the form of a convex, triangularly shaped cone. The
parameter η determines the shape of the cross-section in
the π-plane: increasing values of h correspond to the crosssectional
shape changes from circular to triangular with
smoothly rounded edges, Fig. 1. When plotted in σ 1
- σ 2
space, the criterion first predicts a strengthening effect
with increasing intermediate principal stress s 2
followed
by a slight reduction in strength once s 2
becomes ‘‘too
high’’, Fig. 2. Thus, the Modified Lade criterion seems to
provide a good alternative to the Mohr-Coulomb criterion.
Fig. 2 : Behaviour of different failure criteria in σ 1
-σ 2
space
for different values of confining stress σ 3
.
The extended von Mises yield criterion, or the Drucker-
Prager criterion, was originally developed for soil
mechanics. The yield surface of that criterion in principal
stress space is a right circular cone equally inclined to
the principal-stress axes, Fig. 1. The intersection of the
p-plane with this surface is a circle and the Drucker-Prager
yield function has the form:
1/2
J 2
= k + α 1
J 1
...(7)
where
J 1
= (σ 1
+σ 2
+σ 3
)/3 ...(8)
J 2
= [(σ 1
-σ 2
)+(σ 2
-σ 3
) +(σ 1
-σ 3
) 2 ]/6 ...(9)
The material parameters α 1
and k can be determined from
the slope and the intercept of the failure envelope plotted
1/2
in the J 1
- J 2
space: α 1
is related to the internal friction of
the material and k is related to its cohesion. In this way,
the Drucker-Prager criterion can be compared to the Mohr-
Coulomb criterion.
The Drucker-Prager criterion can be further divided into an
outer bound criterion (or Circumscribed Drucker-Prager) and
an inner bound criterion (or Inscribed Drucker-Prager). These
Volume 1 No. 2 July 2012