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# Historical, Theoretical and Practical Perspectives on Hydraulic Fracturing - History and Applications of Hydraulic Fracturing

Mechanics of Hydraulic Fracturing

There are many robust theoretical outlines and empirical datasets in the literature that investigate the mechanics of HF. In almost all of these studies, the Mohr-Coulomb failure criterion is used as the basis for investigation. This geometric approach yields equations that describe the shear and normal stresses acting upon a plane of arbitrary orientation in a differential stress field. As a theoretical criterion, assumptions of isotropy and linear elasticity are required. The Mohr diagram (figure 1) is a graphical representation of the results of this approach which are used to define a linear failure envelope (the linear nature of the envelope does not continue in the tensile or plastic regimes of deformation). This envelope is empirically derived for rocks. The reader is encouraged to review Goodman (1989) or Jaeger and Cook (1969) for in-depth derivations of these equations.

When working in these natural systems, the stresses utilized are most commonly the effective stresses. This convention recognizes the fact that rocks in the subsurface are affected by both the pressure of the solid constituent, but also a fluid pressure (commonly hydrostatic) that pervades the system. Effective stress is the axial compressive stress (from the solid constituent) minus the fluid pressure. The inability of fluids to transmit shear stress is the true crux of the HF method. If the fluid pressure experienced by a rock is either naturally or artificially increased, the effective normal stress is decreased. The diameter of the Mohr circle on the Mohr diagram is unchanged in this situation and is, in effect, shifted to the left by an interval equal in magnitude to the fluid pressure. In this way, the circle can be shifted out of the region of stability and brought into contact with the failure envelope to induce fracturing.

Figure 1 Typical Mohr envelope for rock. From Hubbert and Willis, 1957.

Figure 2 portrays this shifting Mohr circle and its ability to induce fractures. This figure was published by Cosgrove (1995), accompanying a discussion about the predicted orientation and spatial organization of the induced hydraulic fractures. His overview of the phenomenon emphasized the fact that the expression of hydraulic fracturing was dependent upon the magnitude of the differential stress, the orientation of the principal stress axes, and the material parameters of the rock (especially its cohesive strength). The nature of the fracture pattern will be different if the fractures occur in shear (Figure 2, circle 1) than if they occur in tension (Figure 2, circles 2-4). Shear fracture orientations are material dependent and their orientation is described by the angle of a radius line connecting the tangent point where the circle intersects the failure envelope and the circle's center. High differential stresses in the tensile regime will produce regular, parallel arrays of fractures that are normal to the least principle stress. When the state of stress approaches hydrostatics (differential stress approaches zero), many randomly oriented fractures will be generated.

Figure 2 The effect of fluid pressure on the Mohr circle. From Cosgrove, 1995.

This problem of the orientation of fractures produced is one commonly discussed in the literature, and it is generally accepted that fractures should occur along planes normal to the least principal stress. Minimum injection pressures, then, should be equal to the least principal stress. The determining factor is the regional principal stress directions. Figure 3 is a depiction of the Andersonian theory of faulting that predicts three types of faults by varying which stress is orthogonal to the free surface of the earth. In strike-slip regimes as shown on the left, the intermediate stress is vertical. In compressive regimes, the minimum stress is vertical, as shown in the center. Finally, in extension regimes as on the right, the maximum stress is vertical. Natural and induced fractures will try to orient themselves such that they open in the direction of minimum stress.

Figure 3 Depiction of Andersonian theory of faulting.

In regions characterized by normal faulting, vertical fractures will form with injection pressures of magnitudes smaller than the overburden. In regions characterized by thrust faulting, horizontal fractures will form with injection pressures equal to the overburden. In practice, with the addition of pore pressure, the effective stresses needed to accomplish either vertical or horizontal fracture are often lower than this theoretical minimum. Regardless, the main point is that the orientation of induced fractures is controlled by the preexisting stress field of the rocks into which the fluid is injected, not other factors such as the penetrability of the fracturing fluid that have been suggested. Hossain et al. (2000), run numerical simulations wherein they vary the orientation of a wellbore within each of the above stress states and show that for a given set of ambient stress conditions, there are optimal wellbore orientations that will decrease both the breakout and injection pressures required to successfully fracture a rock mass (figure 4).

Figure 4 Normalized fracture initiation pressure vs. wellbore deviation under severe normal faulting stress conditions (Hossain, 2000).

Another way of looking at the problem is calculating the radial stresses caused by the presence of a borehole in a homogeneous, elastic medium, and then looking at the effects of pressure from within the borehole and how the two pressures interact. This is done in a seminal paper by Hubber and Willis (1957). Using the model of an infinite plate containing a circular, smooth-walled hole, whose axis is perpendicular to the plate, they conclude that stress concentrations induced by the presence of such a hole are local. Within a few borehole diameters, the stresses rapidly approach the undisturbed regional stresses. When a fluid pressure is applied from within the borehole, the compressive stresses around the borehole are all reduced until the minimum stress is reached and the compressive stress across some plane in the walls of the hole is reduced to zero. This is called the breakout pressure. Figure 5 depicts the behavior of two types of borehole during treatment. 4a depicts a high breakout pressure that is required to initially induce and propagate a fracture. 4b depicts a constant fluid pressure required to open and propagate pre-existing discontinuities.

Figure 5 Pressure vs. time graph of HF treatments with (a) induced fracture and (b) exploitation of pre-existing fracture. (Hubbert and Willis, 1957)

Once a fracture has been opened, the injection pressure required to keep the fracture open is then equal to the component of the undistorted stress field normal to the plane of the fracture. Any stress marginally larger than this will propagate the fracture indefinitely as long as the stress can be effectively transferred to the crack tip. Figure 6 depicts this phenomenon.

Figure 6 Depiction of stresses in a borehole in the vicinity of a crack when pressure on crack walls is greater than the ambient stress. (Hubbert and Willis, 1957)

A final theoretical observation considers the penetrability and viscosity of the fracturing fluid. When pressure is considered Hubbert and Willis (1957) show that the pressure required of both a penetrating and a non-penetrating fluid to hold a fracture open once it has been initiated is exactly equal. To achieve breakdown, however, a penetrating fluid effectively reduces the ambient stress concentrations at the face of a borehole. This is the only true theoretical difference: the pressure required to initiate fracture is less with a penetrating fluid. Ishida et al. (2004) show that variations in the viscosity of the fracturing fluid have an effect on the fracture network and the modes of failure of induced fractures. An experimental approach that uses both high and low viscosity fluids finds that viscous oils tend to generate thick and planar cracks with few branches that fail in tension, whereas low viscosity fluids generate thin and wavelike cracks with many secondary branches that fail in shear.

To summarize this section, the parameters that will influence hydraulic fracture initiation, propagation, and orientation from a borehole in the subsurface include the following: wellbore fluid pressure, mechanical properties of the formation, prevailing in-situ stress regime in the formation, orientation of the wellbore, and fluid penetrability and viscosity.