The propagation and geometry of hydraulic fractures are strongly controlled by the downhole state of stress.
In particular, it is generally accepted that the degree of fracture containment is determined primarily by the in-situ stress differences existing between layers. In the absence of a meaningful stress contrast, other mecha-nisms such as slip on bedding planes (Warpinski et al., 1993) and fracture toughness contrast (Thiercelin et al., 1989) can have a role. Moreover, hydraulic fractures propagate, in most cases, normal to the minimum stress direction. Consequently, knowledge of the minimum stress direction allows prediction of the expected direc-tion of the hydraulic fracture away from the wellbore.
Stresses in the earth are functions of various para-meters that include depth, lithology, pore pressure, structure and tectonic setting. A typical example from the Piceance basin in Colorado (Warpinski and Teufel, 1989) is shown in Fig. 3-18. The stress regime in a given environment depends, therefore, on regional considerations (such as tectonics) and local considera-tions (such as lithology). Understanding the
interac-tion between regional and local considerainterac-tions is important as it controls the stress variation between layers. In some stress regimes the adjacent layers are under higher stress than the pay zone, enhancing frac-ture height containment; in others, the adjacent layers are under lower stress than the pay zone, and fracture propagation out of the zone is likely, limiting lateral fracture penetration. Key regional stress regimes and
Figure 3-18. Stress profile for Well MWX-3 (Warpinski and Teufel, 1989).
5000 7600
6000 7000 8000
7400 7200 7000 6800 6600 6400 6200
Stress (psi)
Depth (ft)
Mudstone Sandstone
the consequences of these regimes on the local state of stress in a reservoir are reviewed in the following.
These regimes lead to the introduction of simple stress models that allow making rough estimates of the stress profile as a function of depth and rock properties.
These models can also be used to obtain a calibrated stress profile from log and stress measurement infor-mation, as shown in Chapter 4. The influence of the variation of temperature and pore pressure on the state of stress is also analyzed. Finally, the influence of industrial intervention on the state of stress is pre-sented. Intervention includes drilling a hole and depleting or cooling a formation.
3-5.1. Rock at rest
One stress regime is when the rock is under uniaxial strain conditions (i.e., there is no horizontal strain any-where). To estimate the state of stress that is generated under this regime, it is assumed that the rock is a semi-infinite isotropic medium subjected to gravita-tional loading and no horizontal strain.
Under these conditions, the vertical stress is gener-ated by the weight of the overburden and is the maxi-mum principal stress. Its magnitude, at a specific depth H, is estimated by
(3-50) where ρis the density of the overlying rock masses and g is the acceleration of gravity. The value of this stress component is obtained from the integration of a density log. The overburden gradient varies from about 0.8 psi/ft in young, shallow formations (e.g., Gulf Coast) to about 1.25 psi/ft in high-density formations. Assuming that quartz has a density of 165 lbm/ft3, the overburden gradient ranges between the well-known values of 1.0 and 1.1 psi/ft for brine-saturated sandstone with porosity ranging between 20% and 7%, respectively.
With uniaxial strain assumed, the other two princi-pal stresses are equal and lie in the horizontal plane.
If they are written in terms of effective stress, they are a function of only the overburden:
(3-51) where Kois the coefficient of earth pressure at rest and σh´ is the minimum effective horizontal stress.
Assumptions about rock behavior can be used to
esti-mate values of Ko. However, stress predictions using these assumptions must be used with great caution and may not be applicable in lenticular formations (Warpinski and Teufel, 1989). Nevertheless, they are useful for understanding the state of stress in the earth and can be used as a reference state (Engelder, 1993).
With the assumption of elasticity and for the bound-ary conditions outlined previously, Kois
(3-52) and the relationship between the total minimum hori-zontal stress σhand the overburden σvis, after re-arranging and using the Biot effective stress for σ´,
(3-53) The dependence of horizontal stress on rock lithol-ogy results from the dependence of Poisson’s ratio ν on rock lithology. In most cases, the model predicts that sandstones are under lower stress than shales as Koin sandstones and shales is about equal to 1⁄3and 1⁄2, respectively. The use of Eq. 3-53 to obtain stress pro-files in relaxed basins is presented in Section 4-5.2.
More complex elastic models that are associated with this stress regime have been developed to consider rock anisotropy (Amadéi et al., 1988) and topography (Savage et al., 1985).
For purely frictional materials, Kocan be approxi-mated by (1 – sinφ) (Wroth, 1975), which gives the following relationship for the total stresses:
(3-54) whereφis the angle of internal friction of the rock (Eq. 3-41), of the order of 20° for shales and 30° for sandstones. In this expression, the Terzaghi effective stress concept prevails because this case involves fric-tional behavior.
This equation implies that rocks with a high value of friction angle are under lower stress than rocks with low value of friction angle; i.e., in general, sandstones are under lower stress than shales. The observation that models based on elasticity and models based on frictional behavior give the same trend of stress contrast always occurs, although the fundamental assumptions for these models have nothing in common.
For purely viscous materials (salt), Kois simply equal to 1 and the state of stress is lithostatic (Talobre, 1957, 1958):
(3-55) (a lithostatic state of stress as such does not require the uniaxial strain condition, and therefore, it defines a stress regime by itself).
Over geologic time, rock experiences, in various combinations and degrees, diverse mechanical behav-iors and various events. Behavbehav-iors include elastic, frictional and viscous behaviors, and events include the occurrence of tectonic strain, variation of pore pressure and temperature, erosion and uplift. As reviewed by Prats (1981), these mechanisms lead to deviations from these simple reference states, some of which are briefly reviewed here.
3-5.2. Tectonic strains
Tectonic stresses and strains arise from tectonic plate movement. In this section, the notion of tectonic strain is introduced, which is a quantity added to or sub-tracted from the horizontal strain components. If incremental tectonic strains are applied to rock forma-tions, these strains add a stress component in an elas-tic rock as follows:
(3-56)
(3-57) where dεHand dεhare the (tectonic) strains with dεH>
dεh. The resulting stress increments are not equal, with dσH> dσh, where dσHis the stress increment gener-ated in the dεHdirection and dσhis the stress increment generated in the dεhdirection. These relations are obtained by assuming no variation of the overburden weight and provide a dependence of stress on Young’s modulus E. This means that the greater the Young’s modulus, the lower the horizontal stress if the strains are extensive and the higher the horizontal stress if the strains are compressive. To understand this mecha-nism, the different layers can be compared to a series of parallel springs, the stiffness of which is propor-tional to Young’s modulus as depicted in Fig. 3-19.
This model is actually a good qualitative description of the state of stress measured in areas in which com-pressive tectonic stresses occur. The model can account for situations where sandstones are under higher horizontal stress than adjacent shales (Plumb et al., 1991; see also Chapter 4). The overburden stress
is a principal stress but not necessarily the maximum.
The state of stress described in this section cannot be considered to define a particular stress regime (although one could speak of compressional stress regime) as it does not define a reference state. Only if the strains are high enough for the rock to fail are ref-erence states obtained, as discussed in the next section.
3-5.3. Rock at failure
If the strains are high enough, the rock fails either in shear or in tension. Three stress regimes can be defined if the rock fails in shear. These stress regimes are associated with the three classic fault regimes (Anderson, 1951): normal, thrust and strike-slip fault regimes (Fig. 3-20). Stresses can be estimated by the adapted shear failure model. The simplest shear fail-ure model that applies to rocks is the Mohr-Coulomb failure criterion. A stress model based on this criterion assumes that the maximum in-situ shear stress is gov-erned by the shear strength of the formation (Fenner, 1938). Hubbert and Willis (1957) used this criterion and sandbox experiments in their classic paper on rock stresses and fracture orientation (see Sidebar 3A). As presented in Eq. 3-42, the Mohr-Coulomb failure criterion can be written to give σ1at failure in terms of σ3. In sandstones and shales, Nϕis about equal to 3 and 2, respectively.
If failure is controlled by slip along preexisting sur-faces, the compressive strength σccan be assumed negligible. However, a residual strength may still exist. The angle of internal friction φis usually mea-sured by using ultimate strength data as a function of the confining pressure obtained during triaxial testing.
This angle can also be measured by using residual σh≈σv
d E
d E
h h d H
σ ν ε ν
ν ε
≈ − +
−
1 2 1 2
d E
d E
H H d h
σ ν ε ν
ν ε
≈ − +
−
1 2 1 2 ,
Figure 3-19. By analogy, the stiffer the spring, the more load it will carry.
Formation A
Formation B
Formation C
Stiff plate Stiff
fixed plate
Constant displacement
strength data as a function of the confining pressure obtained during triaxial testing once the sample has failed. Using the residual angle of friction rather than the angle of internal friction in a failure stress model should be more consistent with the assumption that the minimum stress is controlled by friction along pre-existing planes. Generally, the residual angle of fric-tion is smaller than or equal to the internal angle of friction.
If the formation is in extension (i.e., normal fault regime, Fig. 3-20), the vertical stress is the maximum principal stress. The minimum principal stress is in the horizontal plane and is therefore σh. Equation 3-42 becomes
(3-58)
Figure 3-20. The three fault regimes (Anderson, 1951).
Normal fault regime
Thrust fault regime
Strike-slip fault regime
σv = σ1
σv = σ3
σv = σ2
σH = σ2 σh = σ3
σh = σ2
σh = σ3
σH = σ1
σH = σ1
σ σ
ϕ
h p v
N p
− ≈ 1
(
−)
,in which the effect of strength is neglected. An equa-tion similar to Eq. 3-51 can be retrieved. However, if the rock is at failure, the coefficient of proportionality cannot be considered as a coefficient of earth stress at rest. The most surprising and confusing result is that, in practice, Eqs. 3-53 and 3-58 give similar predic-tions, especially if, in the elastic model, αis assumed equal to 1. The coefficient of proportionality in sand-stones and shales is, whether elasticity or failure is assumed, about equal to 1⁄3and 1⁄2, respectively. This similarity has been demonstrated in more detail for one area of East Texas by Thiercelin and Plumb (1994b).
If the formation fails under compressive tectonic strain, the maximum principal stress is in the horizon-tal plane and is therefore σH. In the thrust fault regime, the minimum principal stress is the vertical stress (Fig. 3-20):
(3-59) In this case, σhis the intermediate principal stress and is equal to or greater than the vertical stress.
Horizontal hydraulic fractures could be achieved.
Thus, the principal stresses can be estimated and ordered by looking at the fault regime. In practice, these considerations must be checked with downhole measurements, as the state of stress may deviate from the expected ordering of stresses because of stress his-tory. These models assume that the fault plane was created under the current tectonic setting; i.e., the nor-mal to the fault plane makes an angle (π/4 + φ/2) with the direction of the maximum principal stress. Pre-existing faults can be reactivated under a state of stress that differs from the one that created them. A Mohr-Coulomb stability criterion can still be applied, but Eq. 3-42 must be modified to take into consideration that the fault plane orientation was not induced by the current state of stress.
Another stress regime is associated with tensile fail-ure. Tensile failure is sometimes observed downhole, although it appears to contradict the general compres-sional regime of the earth. This mode of failure sim-ply states that σ3– p = 0 (by neglecting the tensile strength of the rock) and may be suspected if it is observed from downhole images that the normal to the plane of the preexisting fractures is the direction of minimum stress. This condition can occur in exten-sional regions with overpressured zones (where the pore pressure tends to be the value of the minimum
stress component) or when the in-situ stress ratio is too large. As the rock is close to a uniaxial state of stress, this regime can occur only in rocks with a com-pressive strength high enough to avoid normal fault-ing (as a rule of thumb, the uniaxial compressive stress must be equal to or greater than the effective overburden stress). This condition is achieved for tight gas sandstones in some areas of the Western United States and East Texas.
Failure models also have an important role in provid-ing bounds for the in-situ stress. They represent a limit state above which the rock is unstable in the long term.
In the extension regime in particular, it is unlikely that a minimum stress value below the value predicted by the failure model can be obtained.
3-5.4. Influence of pore pressure
It is of interest to understand what happens when depleting or injecting into a reservoir. Elastic models with uniaxial strain conditions can be applied with some confidence, as the variation of stresses occurs over a short period of geologic time, although it is always necessary to double check the assumptions because failure models could well be the real physical mechanism, as shown in the following.
If the material behaves elastically, and assuming uniaxial strain conditions, Eq. 3-53 gives
(3-60) The range of 2ηis approximately between 0.5 and 0.7. Geertsma (1985) demonstrated the applicability of this model to stress decrease during depletion.
A failure model can also be applied. For example, Eq. 3-58 gives
(3-61) If the coefficient of friction is 30°, the coefficient of proportionality is 0.67. As previously, a strong sim-ilarity exists between the predictions from the elastic and failure models. To use a failure model, however, requires checking that the effective state of stress sat-isfies the failure criterion prior to and during the varia-tion of pore pressure. The effective stresses increase during depletion, although the total minimum stress σhdecreases.
Field data generally support the predictions of these models and show that variation in the minimum stress σH − ≈p Nϕ
(
σv−p)
.dσh=2ηdp.
d N
N dp
σh ϕ ϕ
= −1 .
ranges from 46% to 80% of the change in pore pres-sure (Salz, 1977; Breckels and van Eekelen, 1982;
Teufel and Rhett, 1991).
3-5.5. Influence of temperature
Temperature variation also changes the state of stress (Prats, 1981). Cooling happens during uplift or the injection of a cool fluid. This induces an additional stress component in the horizontal plane, which using the uniaxial strain assumption again is
(3-62) where dT is the temperature variation and αTis the linear thermal expansion coefficient. In this case, an influence of Young’s modulus on the state of stress is also obtained. Cooling the formation reduces the nor-mal stress; hence, cool-water injection could lead to tensile fracturing of the formation in the long term.
3-5.6. Principal stress direction
Figure 3-20 indicates the expected direction of the minimum stress as a function of the fault regime (Anderson, 1951). In practice, it is observed that at shallow depths the minimum principal stress is the vertical stress; i.e., a hydraulic fracture is most likely to occur in a horizontal plane. The transition between a vertical minimum principal stress and a horizontal minimum principal stress depends on the regional sit-uation. In an extension regime, however, the mini-mum stress direction can be expected to be always in the horizontal plane, even at shallow depths. This is usually not observed, probably because of the exis-tence of residual stresses and because vertical stress is usually the minimum principal stress at shallow depths. In normally pressured sedimentary basins, the minimum stress is most probably in the horizontal plane at depths greater than 3300 ft (Plumb, 1994b).
Stress rotation may also occur because of topology.
However, at great depths, rotation is induced mainly by fault movement. In some situations, overpressur-ization has been observed to generate a change in the ordering of stress, with the value of the minimum hor-izontal stress higher than that of the vertical stress.
Finally, changes in structural or stratigraphic position can locally affect the stress direction dictated by the
far-field stress and the stress value. An example is the stress field at the top of the Ekofisk formation, where the maximum principal horizontal stress is oriented perpendicular to the structure contour around Ekofisk dome (Teufel and Farrell, 1990).
3-5.7. Stress around the wellbore
So far, only the far-field stress components resulting from geologic contributions or reservoir production have been considered. In addition, the magnitude and orientation of the in-situ stress field can be altered locally, as a result of excavation. These induced stresses usually result in large stress concentrations, differing significantly from the original values.
Drilling a borehole, for example, distorts the preexist-ing stress field. The followpreexist-ing expressions can be obtained for the stresses around the wellbore, where σxand σyare principal stresses in the x-y plane, pwis the wellbore pressure, rwis the wellbore radius, and r is the distance from the center of the well (Fig. 3-21):
(3-63)
Figure 3-21. Stress concentration around a circular hole in the absence of wellbore pressure.
σy
To derive these expressions, it is assumed that the rock remains linear elastic, the borehole is drilled par-allel to one of the principal stress directions, and the wellbore fluid pressure pwdoes not penetrate the rock (e.g., because of the presence of mudcake). At the borehole wall (i.e., r = rw), the following expressions are obtained:
(3-64) Considering only the directions parallel and per-pendicular to the minimum stress direction (i.e., θ= 0 and θ= π/2, respectively), these expressions further simplify:
(3-65) (3-66) As an example, consider the case of 3000-psi well-bore pressure in equilibrium with the pore pressure of the reservoir and values of 3500 psi for σxand 5000 psi for σy. The equations lead to maximum values for the effective tangential stress (σθ– p) of 5500 psi in com-pression (θ= 0°) and 500 psi in tension (θ= 90°). The latter result indicates the possibility for the occurrence of tensile failure in a direction perpendicular to the min-imum stress, solely as a result of drilling the borehole.
A hydraulic fracture is induced by increasing the wellbore pressure pwup to the point where the effec-tive tangential stress (σθ– p) becomes equal to –To. If σx= σh, this happens at θ= 90° (where the stress concentration induced by the far-field state of stress is minimum), which means that fracture initiates in a direction perpendicular to the minimum horizontal stress direction. Fracture initiation at the breakdown pressure pifis, therefore, obtained when (Hubbert and Willis, 1957)
(3-67) These induced stresses diminish rapidly to zero away from the wellbore. Consequently, they affect the pressure to induce a fracture, but not the propagation of the fracture away from the wellbore wall.
If the wellbore fluid penetrates the formation, poro-elastic effects must be taken into account to calculate the stress concentration around the wellbore. In
partic-ular, σθat the wellbore becomes a function of time if
partic-ular, σθat the wellbore becomes a function of time if