Question

In a reverse fault stressed area, a horizontal wellbore has been drilled parallel to maximum horizontal stress (VH). Consider two 180 deg angle apart completion strategies: perforation to the sides of the wellbore and perforation to the top and bottom of the wellbore. Using a schematic Mohr Circle plot explain which completion method results in a longer stable perforation tunnel.

Answer 1

If a piece of rock is subject to sufficiently large stresses, a failure will occur. This implies that the rock changes its shape permanently, and possibly also falls apart. The condition is accompanied with a reduced ability to carry loads. Rock failure is an important phenomenon also for petroleum related rock mechanics, as it is the origin of severe problems such as borehole instability and solids production. It is therefore useful to be able to predict under which conditions a rock is likely to fail A more general and frequently used criterion is the Mohr–Coulomb criterion, which is based on the assumption that f (σ´) is a linear function of σ´: τ| = S0 + μ σ´ Here μ is the coefficient of internal friction. The latter term is clearly chosen by analogy with sliding of a body on a surface, which to the first approximation is described by Amontons’ law: τ = μ σ´

As we have the Mohr–Coulomb criterion, and a Mohr’s circle that touches the failure line. The angle ϕ defined in the Figure is called the angle of internal friction (or friction angle) and is related to the coefficient of internal friction by tan theta = μ

Note that the Tresca criterion can be considered as a special case of the Mohr–Coulomb criterion, with ϕ = 0. The intersection point between the Mohr–Coulomb failure line and the normal stress axis is of no practical interest in itself, as the point is inaccessible due to tensile failure. However, for some purposes it is convenient to make use of the parameter A defined as the distance from the intersection point to the origin The parameter is called the attraction. The attraction is related to the other Mohr–Coulomb parameters by A = S0 cot theta

Also as we know the angle 2β, which gives the position of the point where the Mohr’s circle touches the failure line. It can be seen from the figure that the shear stress at this point is: |τ| = 0.5 (σ´1 - σ´3) sin 2β

While the normal stress is

σ´= 0.5 (σ´1 + σ´3) +0.5 (σ´1 - σ´3) cos2β

Also, we see that β and theta are related by

theta + π/2= 2β

Since β is the angle for which the failure criterion is fulfilled, β gives the orientation of the failure plane we have that β = π/4 + theta/2

The allowable range for ϕ is from 0° to 90° (in practice the range will be smaller, and centered on approximately 30°), hence it is clear that β may vary between 45° and 90°. It is concluded that the failure plane is always inclined at an angle smaller than 45° to the direction of σ´ , this clearly shows schematically how the failure planes may be oriented in a rock described by the Mohr–Coulomb criterion. One important point to note is that β is given solely by theta, which is a constant in the Mohr–Coulomb criterion. Thus the orientation of the failure plane is independent of the confining stress. This is a special feature for the Mohr–Coulomb criterion. Experiments often show that the failure angle decreases with increasing confining pressure, in particular at low confining pressures.