The fracturing of a deviated well and the deviation of a hydraulically induced fracture plane were analyzed by applying three-dimensional (3D) elasticity theory. It was demonstrated that, under the influence of the off-plane shear-stress components, the fracture lines on the well surface were inclined at an angle with respect to the direction of the well axis and that the plane initiated from these fracture lines deviates from the plane formed by these initial fracture lines.

The subject of fracturing a wellhole in a rock medium has been studied extensively by many authors. In the classic analysis, the rock was regarded as an elastic medium, and by assuming that the well was under a plane-strain condition, the breakdown pressure for the wellhole was determined by equating the circumferential stress at the well surface to the tensile failure stress of the rock medium. Later. Haimson and Fairhurst considered the effect of porosity and pore pressure by applying poroelasticity to the problem in these studies, however, the well axis was aligned with the direction of the overburden-stress component; the wellhole was thus under the action of two horizontal in-situ stresses.

The well axis is not aligned with the direction of the in-situ-stress components in the rock medium in a deviated well. When the well axis is deviated from the direction of the in-situ-stress component, the medium surrounding the wellhole is under the combined action of normal and shear stresses. The wellhole is thus in a 3D stress state. In the analysis of failure of inclined boreholes, the importance of shear stresses was recognized by Bradley. Formulas for stress components around the hole given by Bradley were later used by Aadnoy in an analysis of stability of deviated boreholes.

In this study, the fracturing of a deviated wellhole was analyzed by applying 3D elasticity theory. A set of formulas for stress components around the hole was derived. It will be shown in a later section that the off-plane normal and shear stresses are important stress components that influence both the magnitude of the breakdown pressure as well as the position and orientation of the fracture on the well surface. The deviation of a hydraulically induced fracture plane was also studied by applying the minimum-strain-energy density criterion. The results from this study demonstrate that when two initial fracture lines on the well surface were not aligned with the well axis, the fracture plane initiated from these fracture lines deviated from the plane formed by these two initial fracture lines.

Coordinate System. Referring to Fig. 1, the coordinate (1,2,3) is aligned with the direction of the corresponding principal in-situ stresses, 1, 2, and 3, respectively. The axis of the deviated well is defined by a vector n (), as shown. For convenience, rectangular (xyz) and cylindrical () coordinate systems shown in Fig. 1 are used to describe the stress components around the well-hole. The x-y plane is perpendicular to the axis of the well. The x-y coordinates are so orientated that the × and y axes coincide with the 1 and 2 axes when the well is at the vertical position. The axis of the cylindrical coordinate is aligned with the z axis, and the angular variation, is measured counterclockwise from the × axis, as shown in Fig. 1. To establish a transformation relationship between the x, y, z coordinate and the 1,2,3 coordinate, the original 1,2,3 coordinate is rotated through the following sequence.

First, rotate the 1,2,3 coordinate about the 3 axis at angle alpha to the x1 y1 z1 coordinate.

Second, rotate the x1 y1 z1 coordinate about the y1 axis atangle to the x2 y2 z2 coordinate.

Finally, rotate the x2 y2 z2 coordinate about the z2 axis at angle -alpha to the xyz coordinate.

The directional cosine matrix between the xyz coordinate and the 1,2,3 coordinate can thus be written as follows:xyz

(1)

Stress Distribution Around a Wellhole. The in-situ stresses, expressed in the xyz, coordinate system can now be calculated by applying the directional cosine matrix (Eq. 1) to the following transformation formula(2)

The wellhole is therefore under the combined action of stresses, as shown in Fig. 2. Referring to the cylindrical coordinate (, ), the stress distribution around the wellhole was derived. The procedures are outlined in the following sections.

1. The expressions of the stress distribution around a hole caused by the internal pressure, p, and the in-plane stresses, and, shown in Fig. 2 are readily available. The stress distribution is written as follows.(3a)

and(3b)

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