## Hydraulic Fracturing

Figure 8-15a. Failure modes at the wall. (From Maury and Sauzay.27b Copyright 1987 by SPE-AIME.)

Strike of Tensile Fault

<j Greatest Horizontal Stress

Least Hoop Stress

Least Horizontal Strr'--

Greatest Hoop SHo*1;

Figure 8-16. Stress concentrations in a region of tensile faulting. In a region of compressive faulting, the location of the greatest and least stresses with respect to the strike would be reversed.

indicates unstable conditions. Figure 8-19 shows the effect of mud density and hole angle on stability.

Bradley also showed that the stress cloud method could be used to demonstrate the stress field in the vicinity of a salt dome.29

The Influence of Hydraulic Pressure Gradient on Hole Stability

So far we have neglected the effect of fluid flowing into or out of the formation on the stress field around the borehole. Nevertheless, it has been theoretically19 and experimentally6 shown that the hydraulic pressure gradient thus created can materially influence hole stability.

The magnitude and direction of flow in the pores of the formation is a function of the pressure differential, Ap, between the pressure exerted by the mud column, pw, and the formation pore pressure, i.e., Ap = pw — pf.

At the instant of penetration Ap is applied at the face of the borehole, but in the course of time a pressure gradient is established in the pores of the formation. When equilibrium conditions have been reached, the pore pressure, pT, at any radius, r, from the center of the hole is given by the well known radial flow equation:

where [i is the viscosity of the fluid, q is the rate of flow per unit vertical thick ness, k is the permeability of the formation, and rw the diameter of the borehole

Figure 8-17. Stress states about a bore bole for regional-stress ratios aB/aA of 1.4, 2.0, and 3.0. (From Hubbert and Willis.2 Copyright 1959 by SPE-AIME.)

At the wall of the hole Ap = —pf in the case of a hole being drilled with air. In high permeability formations the resulting inflow of fluid will be excessive, and air-drilling must be discontinued, but in low permeability formations the rate of inflow may be tolerable. The resulting distribution of pr at two time intervals is shown in Figure 8-20a. The effect of this hydraulic gradient is to increase the

Mohr Failure Curve /

### Effective Confining Stress

Figure 8-18. Stress clouds for three borehole stresses, pw3>pw2>pwi. other conditions remaining the same. (After Bradley,:26 Courtesy of the Oil and Gas Journal.)

hoop stress gradient, and hence to destabilize the hole. In air-drilled holes such hydraulic gradients are of no great consequence, because inflow rates are low, but similar gradients are established in producing wells, and, where inflow rates are high, may be a major factor affecting hole stability.15

In wells drilled with liquids, the density of the mud column is normally maintained great enough to ensure that pw exceeds Pf by a safe margin, so that the flow of fluid is from the hole to the formation, thereby reducing the hoop stress. If no mud cake is established, as when drilling with brine, the hydraulic pressure gradient decreases with r, as shown in Figure 8-20b, but the pressure drop across an element on the wall of the hole is small. If, however, there is mud in the hole, a filter cake is established and virtually the whole pressure drop is then on the face of the formation (see Figure 8-20c) thus preventing the caving of sand grains and fractured shale fragments.

### Occurrence of Plastic Yielding in the Field

Pure plastic yielding as discussed earlier in this chapter occurs only with two types of formation: rock salt and soft unconsolidated shales. Except for anomalies in the earth's stress field in the vicinity of salt domes, there is little problem in drilling salt beds encountered at shallow depths. Even if the yield stress is exceeded, the rate of deformation is slow, and the hole may be kept to gauge by reaming. Reaming does not significantly affect the strength of the borehole, because the amount reamed off is small relative to the width of the plastic zone, and readily compensated for by a slight increase in the outer radius. Because of

Effective Confining Stress

Figure 8-18. Stress clouds for three borehole stresses, pw3>pw2>pwi. other conditions remaining the same. (After Bradley,:26 Courtesy of the Oil and Gas Journal.)

Depth—1,000's of Feet 30 28 26 24 22 20 18 16 14 12 10

CD 10

0 0