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Hubbert and Willis (1957) developed the first realistic model relating the recorded hydraulic fracturing test variables to the in situ state of stress in rock.

At the borehole wall the tangential stress at the two points aligned perpendicular to the minimum horizontal stress, Sh, will be the first to meet this criterion as the test-interval pressure is raised. A hydraulic fracture will thus initiate and extend in the direction of the maximum horizontal stress, SH.

With these assumptions, Hubbert and Willis (1957) were able to obtain an elastic solution relating the hydraulic fracturing initiation pressure Pc (also called critical or breakdown pressure) and the two principal horizontal stresses, Sh and SH.

where T is the tensile strength of the rock.

In their paper, Hubbert and Willis assumed that T is negligible at great depth because of preexisting fissures traversing the rock and did not incorporate it in the above fracturing criterion .

Hubbert and Willis suggested that the least horizontal stress is equal to the wellbore pressure required to extend the hydraulic fracture while holding it open. Kehle (1964) was more precise about the magnitude of Sh and suggested that it was equal to the shut-in pressure, Ps, or the minimum pressure needed to keep the fracture open against the fracture-normal stress (equal to Sh) after pumping has been stopped:

Thus, knowledge of four test variables (Pc, Ps, Po and T) is required to calculate the two horizontal principal stresses provided the above assumptions hold.

To avoid the need for determining T, one of the most ambiguous rock mechanical properties (Hudson and Fairhurst, 1969), Bredehoeft et al. (1976) suggested replacing the breakdown pressure Pc with the fracture reopening (or refrac) pressure, Pr, obtained in subsequent pressurization cycles. Under the assumption that the hydraulic fracture closes completely between pressurization cycles and that the state of stress around the borehole thus returns to its pre-test condition, equation Hubbert and Willis equation reduces to:

Typical Hydrofrac Pressure-Time Record

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One of the simplifying assumptions made in the elast model is that no injected fluid penetrates into the surrounding rock. This condition may be correct in oil wells lined with impermeable mud cake, but certainly inaccurate in clean open holes of the type commonly used for stress measurements. The radial outward flow of the injected fluid into the rock pores creates an additional stress field around the borehole. Haimson and Fairhurst (1967) invoked the theory of poroelasticity (Biot, 1941) to incorporate the effect of the injection fluid permeation on the stress distribution around the borehole, and obtained the following hydraulic fracturing criterion:

a is the Biot poroelastic parameter (Biot and Willis, 1957), and is defined as a = 1 - Cr/Cb, where Cr is rock matrix compressibility and Cb is rock bulk compressibility; n is the Poisson's ratio for the rock.

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Conventional elastic and poroelastic criteria assume that the rock subject to hydraulic fracturing behaves as a continuous medium. However, the presence of natural cracks in the test interval may violate this simplifying assumption. Abou-Sayed et. al. (1978) introduced a fracture mechanics approach to the hydraulic fracturing criterion assuming the existence of arbitrarily oriented cracks in rock. Further assuming that there is always a symmetrical double crack of length a (typical of the rock type) which extends from the hole wall in the direction of SH. Rummel and Winter (1983) and Rummel (1987) derived a fracture mechanics solution that can be used to calculate SH. They suggest that hydraulic fracturing will occur when the mode I (opening) stress intensity factor KI at the tip of the crack reaches a critical value (fracture toughness KIC). The peak pressure Pc recorded on the pressure-time plot is then interpreted as equal to:

where f*, g*, h* are dimensionless stress intensity functions calculated in terms of the normalized crack length a/r, where r is the radius of the borehole. For comparison with the classical elastic solution (Hubbert and Willis, 1957) the first term of the equation f*KIC r is equivalent to the apparent hydraulic fracturing tensile strength of the rock.

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A novel approach to calculating in situ stresses from hydraulic fracturing data has been proposed by Cornet and Valette (1984). It is the Fracture Pressurization Method, originally called 'hydraulic tests on preexisting fractures' (HTPF) by Cornet (1986). By using the general theory of stress in 3-dimensional space, the fracture-normal stress can be formulated as a nonlinear function of depth and fracture orientation, expressed in the form of 6 unknown parameters which uniquely define the stress tensor. Unlike the conventional models, the fracture pressurization method neither invokes the stress-strain relations nor assumes the idealistic (homogeneous, isotropic, and linear elastic) rock properties. The only assumptions made where the fracture pressurization criterion is used are:

  • The shut-in pressure is equal to the magnitude of the stress normal to the pressurized fracture Sn even if it is not oriented along Sh.

  • Each non-zero component of the stress tensor varies linearly with depth.
  • Sv is taken equal to the weight of the overburden and assumed to be a principal component of the in situ stress tensor.

Under these assumptions, the state of stress at depth D can be represented by the following tensor (Cornet and Valette, 1984):

where g is the mean weight density of the rock; D is the depth; (S) is symmetric with four independent components; (A) is the stress tensor at the surface (D=0); and (B) is a tensor representing the linear variation of the stress components with depth. McGarr (1980) had shown that the stress linear relationship with depth is a direct result of the equations of static equilibrium when the horizontal stresses are uniform over large area.

Using the definitions in the above equations A1 and A2 are the principal horizontal stresses at the surface (D=0). Denoting the direction of A1 with respect to north as l A, eigenvalues of (B) as B1 and B2, and the direction of the eigenvector corresponding to B1 with respect to A1 as l B. Then each normal (i=j) and shear (i¹ j) stress component Sij (i,j=x,y) can be related to the six unknown principal components A1,A2,B1,B2,l A and l B by (Jaeger and Cook, p. 24-26, 1976):

The measured attitude (dip f i, dip direction y i) of the pressurized fracture defines the unit normal vector (n) of each fracture-normal stress Sn,i across ith fracture, measured by the respective shut-in pressure Ps,i can be related to each stress component of (S):

By conducting a minimum of six pressurization tests in separate intervals within the same stress regime, the six unknown parameters (A1, A2, B1, B2, l A and l B) in the fracture-normal stress equation can be determined, using one of several available least squares techniques. The six parameters completely define three independent components (Sxx, Syy, Sxy) of horizontal stresses as a function of depth. Thus, the horizontal principal stresses and their directions are calculated as follows:

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