## Info

Figure 2.9 Ratio of horizontal to vertical stresses obtained from in situ stress measurements (after Brown and Hoek, 1978).

Firstly, if the vertical stress (SJ is the least principal stress (S3), a horizontal fracture will form in the wall of the borehole, at A in Figure 2.10, when the bore-hole fluid pressure p is

Figure 2.9 Ratio of horizontal to vertical stresses obtained from in situ stress measurements (after Brown and Hoek, 1978).

Firstly, if the vertical stress (SJ is the least principal stress (S3), a horizontal fracture will form in the wall of the borehole, at A in Figure 2.10, when the bore-hole fluid pressure p is where T0 is the tensile strength of the rock. These conditions may frequently be satisfied in shallow boreholes.

However, here we are concerned with determining which of the horizontal stresses represents the least principal stress, S3. These conditions are most frequently encountered in relatively deep bore-holes (i.e. greater than 2 km).

The theory relating to the second situation, in which the vertical stress is greater than at least one of the horizontal principal stresses, was first given by Hubbert and Willis (1957) and is also set out in Jaeger and Cook (1969). It can be shown, that the tangential stress St in the surface of the borehole corresponding to the fluid pressure (p) and .V, and Ss is given by:

This expression varies from a maximum of 3S1-S3-p, when 0=p/2, to a minimum of 3S3-S1-p, when 0=0. If tensile failure will take place along a radial plane normal to the axis of least principal stress (B in Figure 2.10). The value of T0, of course, is obtained from tests conducted in the laboratory on specimens taken from the test horizon.

If the axis of the vertical principal stress is not exactly parallel to the borehole, the geometry of the borehole will determine the orientation of the fracture at the borehole, but the induced hydraulic fracture will curve into the plane, normal to the axis of the least principal stress, away from the borehole. The

Figure 2.10 Simplified diagram of a borehole (b.h.) with a sealed section between upper and lower packers. Fluid pressure is generated between the packers until the borehole fails. Near the surface, the rock tends to develop a horizontal hydraulic fracture (HHF)—A. At depth, where the vertical stress is higher than the lateral rock pressure, a vertical hydraulic pressure fracture (VHF) usually forms—B.

deviation between the borehole fracture and the true orientation of the hydraulic fracture away from the borehole will not usually be known.

### 2.4.3 Break-out

A further source of information regarding the direction of action of the greatest horizontal principal stress is obtained from the shape, in cross-section, of deep, unlined bore-holes. These, of course, are usually drilled by the oil and gas industry. From elasticity theory (Obert and Duvall, 1967; Jaeger and Cook, 1969), it can be shown that the stress concentrations around a hole subjected to biaxial compression, normal to the axis of the hole, is such that the maximum reduction in stress occurs in the hole where the axis of maximum principal stress meets the hole boundary. A simplistic representation of how and where tensile and compressive stresses, which develop about a small hole in a disk subjected to uniaxial compression, is shown in Figure 2.11a.

Deep drill holes in the crust contain drilling muds at relatively high pressure. It can be inferred from Figure 2.11b that, because of the stress concentration that occurs, the walls of the drill hole will fail in compression and spall into the drill hole, as indicated in the shaded area of Figure 2.11b. This spalling, or break-out, causes the hole to develop an elliptical cross-section, with the long axis coincident with the axis of minimum horizontal stress. From surveys of borehole break-out, reasonable estimates can be obtained of the regional trends of the axes of greatest and least horizontal stress (Cowgill, 1994).