In this paper, we address the theoretical possibility of drilling with mud weights in excess of the least principal stress for cases of particularly high pore pressure or severe wellbore instability. Because lost circulation caused by hydraulic fracturing is to be avoided, we consider three critical wellbore pressures, pfrac, plink, and pgrow. Tensile fractures initiate at the wellbore wall at pfrac, link up to form large axial fractures that are subparallel to the wellbore axis at plink, and propagate away from the wellbore at pgrow. It is obvious that lost circulation cannot occur if the wellbore pressure during drilling is below pfrac. However, even if pfrac is exceeded and tensile fractures are initiated at the wellbore wall, fracture propagation (and, hence, lost circulation) will be limited as long as the wellbore pressure is below plink. Finally, if the wellbore pressure is greater than plink, the fractures will not grow away from the wellbore (and significant lost circulation will not occur) if the wellbore pressure is below pgrow, which must exceed (if only slightly) the least principal stress. In general, our modeling shows that pfrac and plink can be maximized by drilling the wellbore in an optimally stable orientation, and pgrow can be maximized with noninvading drilling muds that prevent fluid pressure from reaching the fracture tip. We apply the model that uses in-situ stress data collected in real fields, such as the South Eugene Island field in the Gulf of Mexico and the Visund field in the northern North Sea.


In this paper, we theoretically investigate the circumstances under which it may be possible to drill with mud weights in excess of the least principal stress in extreme drilling environments. To accomplish this, we must avoid lost circulation caused by the propagation of hydraulic fractures. We consider the case of an arbitrarily oriented well with a perfect mudcake such that in the absence of hydraulic fracturing, no drilling fluids leave the wellbore. We consider a three-fold strategy to maximize, to the greatest degree possible, mud weights during drilling. To accomplish this, we use the following facts.

  • Wellbore pressure at fracture initiation varies with the wellbore deviation and azimuth.1–4

  • Because deviated wells are generally not parallel to one of the principal stresses, multiple tensile fractures form in an en-echelon pattern on opposite sides of the wellbore wall.5 Wellbore pressure and orientation determine whether these multiple fractures link up.6

  • When drilling with high solids water-based muds, pressures in the wellbore may not reach the fracture tip because of the narrow width of the fracture and the bridging of solids within it.7,8

Taking these facts into account, we develop a theoretical model to estimate the critical pressures that dominate hydraulic fracture initiation and propagation. The pressure necessary to initiate fractures at the wellbore wall is first. Second is what is required to link the inclined tensile fractures near the wellbore wall, and third is what is required to extend the fracture instability away from the wellbore. To the degree to which we can predict these pressures, we can potentially raise mud weights to deal with problems of extreme wellbore instability, especially in cases of extremely high pore pressure, in which the difference between the pore pressure and the fracture gradient is quite small. We apply the model shown here to the stress state encountered in real fields, such as the Visund and the South Eugene Island fields, and demonstrate how to maximize wellbore pressures during drilling by controlling wellbore orientation and mud parameters.

Fracture Model
Tensile Fracture Initiation.

Let us consider the stress state in a plane tangent to an arbitrarily oriented wellbore (Fig. 1). In the plane, the tangential stress st is given by

  • Equation 1

where st deviates by the angle ? from the wellbore axis, and s??, szz, and s?z=the stress components at the wellbore wall in a wellbore cylindrical coordinate system (r, ?, z) (see Refs. 9 and 10). Compressive stresses are positive in this paper. Note that s??, szz, and s?z change with wellbore orientation and the state of remote stresses, but s?? also changes with wellbore pressure. According to the Terzaghi effective stress law, the effective stress component of st is given by

  • Equation 2

where st=the effective stress, and pp=the formation pore pressure. The minimum value stmin is given by

  • Equation 3

Note that stmin is a function of ?. If stmin goes into tension at certain angles of ?, tensile wall fractures will initiate at these points, provided stmin overcomes the tensile strength of rock, T0.1–5 In other words, the tensile fractures will initiate when the following occurs.

  • Equation 4

T0 is frequently expected to be very close to zero because some pre-existing flaws or irregularities are usually present on the wellbore wall.11 In this paper, T0 is assumed to be zero. The wellbore pressure at fracture initiation (i.e., wellbore pressure that satisfies Eq. 4) is hereafter referred to as the initiation pressure and is denoted as pfrac.

Every two points on the wellbore wall, identified by ? and ?+180°, have the same s??, szz, and s?z, but with opposite signs (see Refs. 9 and 10). Eq. 3 shows that the sign of s?z does not play a role in the value of stmin because s?z appears in a quadratic form. This means that if stmin satisfies Eq. 4, the fracture initiates normal to stmin at ?=?f and at ?=?f+180°. The fractures deviate from the wellbore axis by the angle ?f (see Fig. 2), where

  • Equation 5

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