The application of mechanics to fracturing was cat-alyzed by the horizontal orientation of fractures implied in the Stanolind patent and the desire of sev-eral operators to avoid paying the nominal patent royalty of $25–$125, based on volume (C. R. Fast, pers. comm., 1997). Significant research activity was conducted to show that fractures can be vertical, as is now known to be the general case for typical fractur-ing conditions. The fracture orientation debate even-tually led to a lawsuit that was settled before the trial ended. The settlement accepted the patent and nomi-nal royalty payments and stipulated that other service companies receive a license to practice fracturing.
However, the royalty benefits were more than nomi-nal to Stanolind because about 500,000 treatments were performed during the 17-year period of the patent (C. R. Fast, pers. comm., 1997). Key to the favorable settlement for Stanolind was its well-documented demonstration of a horizontal fracture in the Pine Island field (see fig. 7-1 in Howard and Fast, 1970).
The central issue in the orientation debate was the direction of the minimum stress. The pressure required to extend a fracture must exceed the stress acting to close the fracture. Therefore, the fracture preferentially aligns itself perpendicular to the direc-tion of minimum stress because this orientadirec-tion pro-vides the lowest level of power to propagate the frac-ture. The minimum stress direction is generally hori-zontal; hence, the fracture plane orientation is gener-ally vertical (i.e., a vertical fracture). The preference for a horizontal fracture requires a vertical minimum stress direction.
In the following review, the orientation considera-tion is expanded to also cover the state of stress in more general terms. The stress at any point in the var-ious rock layers intersected by the fracture is defined by its magnitude in three principal and perpendicular directions. The stress state defines not only the frac-ture orientation, but also the fluid pressure required to propagate a fracture that has operational importance, vertical fracture growth into surrounding formation
layers and stress acting to crush proppant or to close etched channels from acid fracturing. The crushing stress is the minimum stress minus the bottomhole flowing pressure in the fracture. The orientation debate resulted in three papers that will remain signifi-cant well into the future.
The first paper to be considered is by Harrison et al. (1954). Some of the important points in the paper are that the overburden stress (vertical stress σv) is about 1 psi per foot of depth, fracturing pressures are generally lower than this value and therefore frac-tures are not horizontal, and an inference from elas-ticity that the minimum horizontal stress is
(1) where Ko= ν/(1 – ν) = 1⁄3for ν= 1⁄4(see Eq. 3-51).
Using Poisson’s ratio νof 1⁄4, Harrison et al. con-cluded that the horizontal stress is about one-third of the vertical stress and therefore fractures are vertical.
Appendix Eq. 1 provides the current basis for using mechanical properties logs to infer horizontal stress, with Poisson’s ratio obtained from a relation based on the shear and compressional sonic wave speeds (see Chapter 4). Another assumption for Appendix Eq. 1 is uniaxial compaction, based on the premise that the circumference of the earth does not change as sediments are buried to the depths of petroleum reservoirs and hence the horizontal components of strain are zero during this process. Therefore, Ap-pendix Eq. 1 provides the horizontal stress response to maintain the horizontal dimensions of a unit cube constant under the application of vertical stress.
However, there is one problem with this 1954 conclusion concerning horizontal stress. Appendix Eq. 1 is correct for the effective stress σ′but not for the total stress σthat governs fracture propagation:
σ′= σ– p, where p is the pore pressure, which also has a role in transferring the vertical stress into hori-zontal stress as explicitly shown by Appendix Eq. 2.
Harrison et al. (1954) correctly postulated that shales have higher horizontal stresses and limit the vertical fracture height. The general case of higher stress in shales than in reservoir rocks was a necessary condi-tion for the successful applicacondi-tion of fracturing because fractures follow the path of least stress. If the converse were the general case, fractures would prefer to propagate in shales and not in reservoir zones.
Harrison et al. also reported the Sneddon and Elliott (1946) width relation for an infinitely ing pressurized slit contained in an infinitely extend-ing elastic material. This framework has become the basis for predicting fracture width and fracturing pressure response (see Chapters 5, 6 and 9). They used the fracture length for the characteristic, or smaller and finite, dimension in this relation. Sel-ecting the length for the characteristic dimension resulted in what is now commonly termed the KGD model. Selecting the height, as is the case for a very long fracture, is termed the PKN model. These mod-els are discussed in the next section and Chapter 6.
Harrison et al. considered a width relation because of its role in fracture design to determine the fluid volume required for a desired fracture extent.
The role of volume balance (or equivalently, the material balance in reservoir terminology) is an essential part of fracture design and fracture simula-tion code. As shown schematically on the left side of Appendix Fig. 2, each unit of fluid injected Viis either stored in the fracture to create fracture volume or lost to the formation as fluid loss. (However, Harrison et al.’s 1954 paper does not discuss fluid loss.) The stored volume is the product of twice the fracture half-length L, height hfand width w. If the latter two dimensions are not constant along the frac-ture length, they can be appropriately averaged over the length. The half-length is then obtained by sim-ply dividing the remaining volume, after removing the fluid-loss volume, by twice the product of the σh=Koσv,
Appendix Figure 2. Volume balance for fracture place-ment (equation from Harrington et al., 1973) (adapted courtesy of K. G. Nolte and M. B. Smith, 1984–1985 SPE Distinguished Lecture).
Geometry
hf 2L w
Fluid loss CL √ t
Volume
Proppant
Pad
Proppant (% area = η) η =2hfwL
Vi
L = Vi
2hf(w + CL√8t )
average height and the average width. The fluid-loss volume depends on the fluid-loss surface area, or a height-length product. Furthermore, as shown on the right side of Appendix Fig. 2, the ratio of stored to total volume is termed the fluid efficiency ηand dir-ectly affects the proppant additional schedule (Har-rington et al., 1973; Nolte, 1986b) (see Sidebar 6L).
The second paper to be discussed from the orienta-tion era is by Hubbert and Willis (1957). The lessons from this paper extend beyond fracturing and into the area of structural geology. This work provides simple and insightful experiments to define the state of in-situ stress and demonstrate a fracture’s prefer-ence to propagate in the plane with minimum stress resistance. For the latter experiments, the “forma-tion” was gelatin within a plastic bottle preferentially stressed to create various planes of minimal stress.
They also used simple sandbox experiments to demonstrate normal and thrust faulting and to define the state of stress for these conditions (see Sidebar 3A). They showed that Ko, or equivalently the hori-zontal stress, within Appendix Eq. 1 is defined by the internal friction angle (ϕ= 30° for sand) and is
1⁄3for the minimum stress during normal faulting and 3 for the maximum stress during thrust faulting. For the normal faulting case and correctly including pore pressure in Appendix Eq. 1, the total minimum hori-zontal stress becomes
(2) where Ko= 1⁄3with ϕ= 30°. For this case the horizon-tal stress is much less than the vertical stress except in the extreme geopressure case of pore pressure approaching overburden, which causes all stresses and pore pressure to converge to the overburden stress. For the thrust faulting case, the larger horizon-tal stress (i.e., for the two horizonhorizon-tal directions) is greater than the overburden and the smaller horizon-tal stress is equal to or greater than the overburden.
Both the extreme geopressure case and an active thrust faulting regime can lead to either vertical or horizontal fractures. The author has found Appendix Eq. 2 to accurately predict the horizontal stress in tec-tonically relaxed sandstone formations ranging from microdarcy to darcy permeability. The accuracy at the high range is not surprising, as the formations approach the unconsolidated sand in the sandbox experiments. The accuracy obtained for microdarcy-permeability sands is subsequently explained.
Hubbert and Willis also provided an important set of postulates: the rock stresses within the earth are defined by rock failure from tectonic action and the earth is in a continuous state of incipient faulting.
From this perspective, the stress is not governed by the behavior of the intact rock matrix, but by an active state of failure along discrete boundaries (e.g., by sand grains within fault boundaries, which explains the application of Appendix Eq. 2 to micro-darcy-permeability sandstones). This insightful con-clusion about the role of failure is at the other extreme of the behavior spectrum from the elastic assumptions that Poisson’s ratio (Appendix Eq. 1) governs the horizontal stress and that failure has no effect on the stress. This extreme difference in the assumptions for Appendix Eqs. 1 and 2 is often overlooked because of the similar value of Ko= ~1⁄3 obtained in the case of a tectonically relaxed region and Poisson’s ratio near 1⁄4. However, the role of elas-ticity becomes important in thrusting areas (see Section 3-5.2) because of the difference in horizontal stress resulting for layers with different values of Young’s modulus (stiffness). More of the tectonic action and higher levels of stress are supported by the stiffer layers.
Additional considerations for horizontal stress out-lined by Prats (1981) include the role of long-term creep. Creep deformation allows relaxation of the stress difference between the overburden and hori-zontal stresses, thereby enabling the horihori-zontal stress to increase toward the larger vertical stress governed by the weight of the overburden. This effect is well known for salt layers that readily creep and can col-lapse casing by transferring most of the larger over-burden stress into horizontal stress. The role of stress relaxation is an important mechanism for providing favorable stress differences between relatively clean sands governed by friction (i.e., Appendix Eq. 2) with minimal creep and sediments with higher clay con-tent. In the latter case, the clay supports some of the intergranular stresses. The clay structure is prone to creep that relaxes the in-situ stress differences and increases the horizontal stress for a clay-rich formation.
Hence, both clay content and Poisson’s ratio pro-duce the same effect on horizontal stress. Because clay content also increases Poisson’s ratio, there is a positive correlation of clay content (creep-induced stress) to larger Poisson’s ratios (and elastic stress, from Appendix Eq. 1) inferred from sonic velocities.
The implication of the correlation is that clay-rich σh=(σv+2p) 3,
formations can also have horizontal stresses greater than those predicted by either Appendix Eq. 1 or 2, which is consistent with the general requirement to calibrate elastic-based stress profiles to higher levels of stress (e.g., Nolte and Smith, 1981). The correla-tion of clay and Poisson’s ratio links the conclusions of Hubbert and Willis and Prats that horizontal stress is governed primarily by nonelastic effects and the general correlation between the actual stress and elastic/sonic-based stress profiles.
The third significant paper from this period is by Lubinski (1954). He was a Stanolind researcher who introduced the role that poroelasticity can have in generating larger stresses during fracturing. (Poro-elasticity could increase horizontal stress and lead to horizontal fractures, as in the Stanolind patent.) Lubinski presented poroelasticity within the context of its analogy to thermoelasticity. His use of the ther-mal stress analogy facilitates understanding the poro-elastic concept because thermal stresses are generally more readily understood than pore stresses by engi-neers. The analogy provides that when pore pressure is increased in an unrestrained volume of rock, the rock will expand in the same manner as if the tem-perature is increased. Conversely, when the pore pressure is lowered, the rock will contract as if the temperature is lowered. When the rock is con-strained, as in a reservoir, a localized region of pore pressure change will induce stress changes: increas-ing stress within the region of increasincreas-ing pore pres-sure (e.g., from fracturing fluid filtrate or water
injection) and decreasing stress within the region of decreasing pore pressure (e.g., production). The long-term impact of Lubinski’s paper is that the importance of poroelasticity increases as routine fracturing applications continue their evolution to higher permeability formations. This is apparent from the thermal analogy—as the area of expansion increases the induced stresses also increase. For poroelasticity, the area of significant transient change in pore pressure increases as the permeability
increases (see Section 3-5.8).
Appendix Fig. 3 shows an example of significant poroelasticity for a frac and pack treatment in a 1.5-darcy oil formation. The time line for the figure begins with two injection sequences for a linear-gel fluid and shows the pressure increasing to about 7500 psi and reaching the pressure limit for the oper-ation. During the early part of the third injection period, crosslinked fluid reaches the formation and the pressure drops quickly to about 5600 psi (the native fracturing pressure) and remains essentially constant during the remainder of the injection.
The first two injections, without a crosslinked-fluid filtrate (or filter cake) to effectively insulate the for-mation (as in the thermal analogy) from the increas-ing injection pressure, resulted in pore pressure increases of significant magnitude and extent within the formation. The pore pressure increase provides up to a 1900-psi horizontal and poroelasticity stress increase that extends the fracturing pressure beyond the operational limit, leading to the shut-in for the
Appendix Figure 3. High-permeability frac and pack treatment (Gulrajani et al., 1997b).
10,000
8000
6000
4000
2000
0
Bottomhole pressure, BHP (psi)
50
40
30
20
10
0
Injection rate (bbl/min)
0 0.5 1.0 2.0 13.0 13.5 14.0
Time (hr)
Linear gel
Linear gel Crosslinked gel
Step rate
Injection
BHP Injection rate
Step rate Minifracture Propped fracture
Injection
second injection. This increase is about one-third of the native stress. However, during the two subsequent injections the insulating effect of the crosslinked fluid’s internal cake and filtrate allows fracture exten-sion within essentially the native stress state. The pressure drop supported by the cake and filtrate is about 1300 psi, as reflected by the rapid pressure decrease after the third injection. This decrease occurs because of the rapid closure and cessation of fluid loss (that activated the pressure drop), which is the same reason that surface pressure decreases at the cessation of injection and loss of pipe friction. The last injection for the proppant treatment is also of interest because of the absence of a poroelastic effect during the initial linear-gel injection. This observation indicates that the insulating effect remained effective from the prior injection of crosslinked fluid.
For a normally pressured and tectonically relaxed area, the maximum increase in horizontal stress before substantial fracture extension is about one-third of the native horizontal stress (Nolte, 1997), as was found for the case shown in Appendix Fig.
3. Also, for any pore pressure condition in a relaxed area, the stress increase will not cause the horizontal stress to exceed the overburden (i.e., cause horizontal fracturing). However, as the example shows, without fluid-loss control, poroelasticity can significantly increase the fracturing pressure and extend it beyond operational limits for high-permeability reservoirs.