M. C. Thiercelin, Schlumberger Dowell J.-C. Roegiers, University of Oklahoma
3A. Mechanics of hydraulic fracturing
Hubbert and Willis (1957) introduced several key concepts that explain the state of stress underground and its influence on the orientation of hydraulic fractures. Reviewed here are the fundamental experiments that Hubbert and Willis per-formed to validate these concepts.
State of stress underground
The general state of stress underground is that in which the three principal stresses are unequal. For tectonically relaxed areas characterized by normal faulting, the minimum stress should be horizontal; the hydraulic fractures produced should be vertical with the injection pressure less than that of the overburden. In areas of active tectonic compression and thrust faulting, the minimum stress should be vertical and equal to the pressure of the overburden (Fig. 3-20). The hydraulic fractures should be horizontal with injection pres-sures equal to or greater than the pressure of the overburden.
To demonstrate these faulting conditions, Hubbert and Willis performed a sandbox experiment that reproduces both the normal fault regime and the thrust fault regime. Figures 3A-1 and 3A-2 show the box with its glass front and contain-ing ordinary sand. The partition in the middle can be moved from left to right by turning a hand screw. The white lines are plaster of paris markers that have no mechanical significance.
As the partition is moved to the right, a normal fault with a dip of about 60° develops in the left-hand compartment, as shown in Fig. 3A-1. With further movement, a series of thrust faults with dips of about 30° develops in the right-hand com-partment, as shown in Fig. 3A-2.
The general nature of the stresses that accompany the failure of the sand is shown in Fig. 3A-3. The usual
conven-tion is adopted of designating the maximum, intermediate and minimum principal effective stresses by σ1´, σ2´ and σ3´, respectively (here taken as compressive). In the left-hand compartment, σ3´ is the horizontal effective stress, which is reduced as the partition is moved to the right, and σ1´ is the vertical effective stress, which is equal to the pressure of the overlying material minus the pore pressure. In the right-hand compartment, however, σ1´ is horizontal, increasing as the partition is moved, and σ3´ is vertical and equal to the pres-sure of the overlying material minus the pore prespres-sure. The third type of failure, strike-slip faulting, is not demonstrated in the sandbox experiment.
Next, the combination of shear and normal stresses that induce failure must be determined. These critical effective stress values can be plotted on a Mohr diagram, as shown in Fig. 3A-4. The two diagonal lines form the Mohr envelopes of the material, and the area between them represents stable combinations of shear stress and normal effective stress, whereas the area exterior to the envelopes represents unsta-ble conditions. Figure 3A-4 thus indicates the stability region within which the permissible values of σn´ and τare clearly defined. The stress circles can then be plotted in conjunction with the Mohr envelopes to determine the conditions of fault-ing. This is illustrated in Fig. 3A-4 for both normal and thrust faulting. In both cases, one of the principal effective stresses
Figure 3A-1. Sandbox experiment showing a normal fault.
Figure 3A-2. Sandbox experiment showing a thrust fault.
Figure 3A-3. Approximate stress conditions in the sand-box experiment.
σ1 σ3
σ3 σ1
Sand
Figure 3A-4. Mohr diagram of the possible range of hori-zontal stress for a given vertical stress σv´. The horizontal stress can have any value ranging from approximately one-third of the normal stress, corresponding to normal faulting, to approximately 3 times the vertical stress, cor-responding to reverse faulting.
σ1´ σ3´
σv´
Mohr envelope
0 τ
φφ
2a 2a
σ
Points of fracture
φ φ
3A. Mechanics of hydraulic fracturing (continued) is equal to the overburden effective stress σv´. In the case of normal faulting, the horizontal principal stress is progressively reduced, there-by increasing the radius of the stress circle until the circle touches the Mohr envelopes. At this point, unstable conditions of shear and nor-mal effective stress are reached and faulting occurs on a plane mak-ing an angle of 45° + φ/2 with the minimum stress. For sand with an angle of internal friction of 30°, the normal fault would have a dip of 60°, which agrees with the previous experiments. The minimum princi-pal effective stress would reach a value at about one-third of the value of the overburden effective stress (Eq. 3-58). For the case of thrust faulting, the minimum principal stress would be vertical and remain equal to the overburden pressure while the horizontal stress is pro-gressively increased until unstable conditions occur and faulting takes place on a plane making an angle of 45° + φ/2 with the minimum prin-cipal stress or 45° – φ/2 with the horizontal. For sand, this would be a dip of about 30°, which again agrees with the experiment. Failure occurs when the maximum horizontal effective principal stress reaches a value that is about 3 times the value of the overburden effective stress (Eq. 3-59). The intermediate stress, which is the mini-mum horizontal stress, is not defined by this process.
From these limiting cases and for a fixed effective vertical stress σv′, the effective horizontal stress may have any value between the extreme limits of 1⁄3and 3 times σv′.
Orientation of hydraulic fractures
The second important contribution of Hubbert and Willis’ work con-cerns the orientation of hydraulic fractures. When their paper was pre-sented, technical debate was occurring on the orientation of hydraulic fractures. A theoretical examination of the mechanisms of hydraulic fracturing of rocks led them to the conclusion that, regardless of whether the fracturing fluid was penetrating, the fractures produced should be approximately perpendicular to the axis of minimum princi-pal stress.
To verify the inferences obtained theoretically, a series of simple laboratory experiments was performed. The general procedure was to produce fractures on a small scale by injecting a “fracturing fluid” into a weak elastic solid that had previously been stressed. Ordinary gel-atin (12% solution) was used for the solid, as it is sufficiently weak to fracture easily, molds readily in a simulated wellbore and is almost perfectly elastic under a short-time application of stresses. A plaster of paris slurry was used as the fracturing fluid because it could be made thin enough to flow easily and once set provided a permanent record of the fractures produced. The experimental arrangement consisted of a 2-gal polyethylene bottle, with its top cut off, used to contain a glass tubing assembly consisting of an inner mold and concentric outer cas-ings. The container was sufficiently flexible to transmit externally applied stresses to the gelatin. The procedure was to place the glass tubing assembly in the liquid gelatin and after solidification to withdraw the inner mold leaving a “wellbore” cased above and below an open-hole section.
Stresses were then applied to the gelatin in two ways. The first way (Fig. 3A-5) was to squeeze the polyethylene container laterally, thereby forcing it into an elliptical cross section and producing a com-pression in one horizontal direction and an extension at right angles in the other. The minimum principal stress was therefore horizontal, and vertical fractures should be expected, as observed in Fig. 3A-6. In other experiments, the container was wrapped with rubber tubing stretched in tension, thus producing radial compression and vertical extension. In this case, the minimum principal stress was vertical, and a horizontal fracture was obtained.
From these analyses and experiments, Hubbert and Willis con-cluded that
• the state of stress, and hence the fracture orientations, is gov-erned by “incipient failure” (i.e., faulting) of the rock mass
• in areas subject to active normal faulting, fractures should be approximately vertical
• in areas subject to active thrust faulting, fractures should be approximately horizontal.
Figure 3A-5. Experimental arrangement for pro-ducing the least stress in a horizontal direction.
Favored fracture direction
Least principal stress
Figure 3A-6. Vertical fracture produced under stress conditions illustrated in Fig. 3A-5.