One of the main functions of rock mechanics research has been to find ways of determining in-situ stresses. Many methods have been suggested, the most significant ones calling for measurements inside boreholes. These methods usually employ some instrumentation for the purpose of measuring hole deformation. Several years ago, a new method was suggested by Seheidegger,1 based on previous work by Hubbert and Willis2 It was the method of "hydraulic fracturing," which had been introduced by the oil industry in 1948 for the purpose of oil-well production stimulation. Basically, hydraulic fracturing consists of sealing off a section of a borehole, pumping in a fluid, and pressurizing it until fracture occurs. If pumping is continued vigorously, the fracture is opened up and extended. In an oil field, such an artificial fracture increases the overall permeability of the formation and usually enhances production. During the entire operation, the variation of pressure with time is ordinarily recorded. Hubbert,2 - Scheidegger,1 Kehle,3 and others have shown that the recorded pressures can be theoretically related to the magnitudes of the principal in-situ stresses; the orientation of the fracture can often be used to determine the direction of the principal stresses. The advantage of hydraulic fracturing over the present in-situ stress determination methods is simplicity: no sophisticated instrumentation is required inside the borehole; hence, the stresses can be measured at any depth. Moreover, if the formation is impermeable to the fracturing fluid, no elastic constants of the rock are required in calculating the stresses, a factor that not only simplifies the problem, but renders the results more reliable. It was felt, however, that as most rocks are porous-permeable the influence of stresses due to fluid flow into the formation should also be considered in determining the in-situ stress distribution. Furthermore, although there exists quite an extensive literature on the theoretical basis of hydraulic fracturing, very little had been done to verify the results experimentally. The present chapter, based mainly on the Ph.D. thesis undertaken by one of the authors,4 attempts to extend the criterion for hydraulic fracturing and to report on some laboratory tests on simulated boreholes. Some interesting field results are also mentioned.
In order to establish the stress distribution in a formation and relate it to hydraulic fracturing pressures some assumptions are made regarding the materials involved. The rock is brittle elastic, homogeneous, isotropic, linear, and porous. The injected fluid flows through the pores according to Darcy's law. Following Biot,5 a complete analogy exists between the elasticity of a porous material, like the one just described, and thermoelasticity. Hence, solutions to problems in thermoelasticity may be used to obtain solutions to problems of porous materials. Prior to drilling of the borehole, the state of stress at a point, situated at a depth D from the surface, is generally nonhydrostatic. To simplify the problem, it is assumed that one of the principal tectonic stresses (S33) acts in the vertical direction. This is justified in most formations (see Anderson6). Taking tensile stresses as positive, the larger horizontal tectonic stress is S22 and the smaller is S11.
