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# Subsurface stress and pore pressure

Geological effects can impact the design and successful completion of oil, gas, and geothermal wells. Understanding the stresses and pore pressures within the subsurface are important to development of a geomechanical model that can guide well design as part of an integrated process to minimize cost and maximize safety.

## Overview

Forces in the Earth are quantified by means of a stress tensor, in which the individual components are tractions (with dimensions of force per unit area) acting perpendicular or parallel to three planes that are in turn orthogonal to each other. The normals to the three orthogonal planes define a Cartesian coordinate system (x1, x2, and x3). The stress tensor has nine components, each of which has an orientation and a magnitude (see Fig. 1.a). Three of these components are normal stresses, in which the force is applied perpendicular to the plane (e.g., S11 is the stress component acting normal to a plane perpendicular to the x1-axis); the other six are shear stresses, in which the force is applied along the plane in a particular direction (e.g., S12 is the force acting in the x2-direction along a plane perpendicular to the x1-axis). In all cases, Sij = Sji, which reduces the number of independent stress components to six.

At each point there is a particular stress axes orientation for which all shear stress components are zero, the directions of which are referred to as the “principal stress directions.” The stresses acting along the principal stress axes are called principal stresses. The magnitudes of the principal stresses are S1, S2, and S3, corresponding to the greatest principal stress, the intermediate principal stress, and the least principal stress, respectively. Coordinate transformations between the principal stress tensor and any other arbitrarily oriented stress tensor are accomplished through tensor rotations.

It has been found in most parts of the world, at depths within reach of the drill bit, that the stress acting vertically on a horizontal plane (defined as the vertical stress, Sv) is a principal stress. This requires that the other two principal stresses act in a horizontal direction. Because these horizontal stresses almost always have different magnitudes, they are referred to as the greatest horizontal stress, SHmax, and the least horizontal stress, SHmin (Fig. 1.b).

The processes that contribute to the in-situ stress field primarily include plate tectonic driving forces and gravitational loading (see Table 1). Plate driving forces cause the motions of the lithospheric plates that form the crust of the Earth. Gravitational loading forces include topographic loads and loads owing to lateral density contrasts and lithospheric buoyancy. These are modified by the locally-acting effects of processes such as volcanism, earthquakes (fault slip), and salt diapirism. Human activities such as mining and fluid extraction or injection can also cause local stress changes. Because the largest components of the stress field (gravitational loading and plate driving stresses) act over large areas, stress orientations and magnitudes in the crust are remarkably uniform (Fig. 2). However, local perturbations, both natural and manmade, are important to consider for application of geomechanical analyses to drilling and reservoir engineering (Fig. 3).

## Relative magnitudes of the principal stresses in the earth

The vertical stress can be the greatest, the intermediate, or the least principal stress. In 1924, Anderson developed a classification scheme to describe these three possibilities, based on the type of faulting that would occur in each case (Table 2 and Fig. 4). A normal faulting regime is one in which the vertical stress is the greatest stress. When the vertical stress is the intermediate stress, a strike-slip regime is indicated. If the vertical stress is the least stress the regime is defined to be reverse. The horizontal stresses at a given depth will be smallest in a normal faulting regime, larger in a strike-slip regime, and greatest in a reverse faulting regime. In general, vertical wells will be progressively less stable as the regime changes from normal to strike-slip to reverse, and consequently will require higher mud weights to drill.

## Pore pressure

Pore pressure is the pressure at which the fluid contained within the pore space of a rock is maintained at depth. In the absence of any other processes, the pore pressure is simply equal to the weight of the overlying fluid, in the same way that the total vertical stress is equal to the weight of the overlying fluid and rock (Fig. 5). This pressure is often referred to as the “hydrostatic pressure.” A number of processes can cause the pore pressure to be different from hydrostatic pressure. Processes that increase pore pressure include:

• Undercompaction caused by rapid burial of low-permeability sediments
• Lateral compression
• Release of water from clay minerals caused by heating and compression
• Expansion of fluids because of heating
• Fluid density contrasts (centroid and buoyancy effects)
• Fluid injection (e.g., waterflooding)

Processes that decrease pore pressure include:

• Fluid shrinkage
• Rock dilation
• Reservoir depletion

Because pore pressure and horizontal stresses are interrelated, changes in pore pressure also cause similar changes in stress. While the exact relationship depends on the properties of the reservoir, it is reasonable to assume that the change in horizontal stress is approximately two-thirds of the change in pore pressure (see Eq. 1 and Fig. 6). This leads to a considerable reduction in leakoff pressure in a depleted reservoir and an increase in horizontal stress where pore pressure increases.

if

and ....................(1)

where v is Poisson's ratio, and α (= 1 - Kdry/Kgrain) is the Biot poroelastic coefficient, which varies between zero for a rock that is as stiff as the minerals of which it is composed and one for most sediments, which are much softer than their mineral components. It is important to note that Eq. 1 cannot be used to calculate the relationship between pore pressure and stress in the Earth that develops over geological time because in that case the assumptions used to derive the equation are not valid.

## Effective stress

The mathematical relationship between stress and pore pressure is defined in terms of effective stress. Implicitly, the effective stress is that portion of the external load of total stress that is carried by the rock itself. The concept was first applied to the behavior of soils subjected to both externally applied stresses and pore pressure acting within the pore volume in a 1924 paper by Terzaghi as ....................(2)

where σij is the effective stress, Pp is the pore pressure, δij is the Kronecker delta (δij = 1, if i = j, δij = 0 otherwise), and Sij represents the total stresses, which are defined without reference to pore pressure. While it is sometimes necessary to use a more exact effective stress law in rock (σij = Sijδij α Pp , where α is Biot’s coefficient and varies between 0 and 1), in most reservoirs it is generally sufficient simply to assume that α = 1. This reduces the effective stress law to its original form (Eq. 2). When expanded, the Terzaghi effective stress law becomes

and ....................(3)

The concept of effective stress is important because it is well known from extensive laboratory experiments (and from theory) that properties such as velocity, porosity, density, resistivity, and strength are all functions of effective stress. Because these properties vary with effective stress, it is therefore possible to determine the effective stress from measurements of physical properties such as velocity or resistivity. This is the basis for most pore-pressure-prediction algorithms. At the same time, effective stress governs the frictional strength of faults and the permeability of fractures.

## Constraints on stress magnitudes

If rock were infinitely strong and contained no flaws, stresses in the crust could, in theory, achieve any value. However, faults and fractures exist at all scales, and these will slip if the stress difference gets too large. Even intact rock is limited in its ability to sustain stress differences. It is possible to take advantage of these limits when defining a geomechanical model for a field when other data are not available.

### Stress constraints owing to frictional strength

One concept that is very useful in considering stress magnitudes at depth is frictional strength of the crust and the correlative observation that, in many areas of the world, the state of stress in the crust is in equilibrium with its frictional strength. Because the Earth's crust contains widely distributed faults, fractures, and planar discontinuities at many different scales and orientations, stress magnitudes at depth (specifically, the differences in magnitude between the maximum and minimum principal effective stresses) are limited by the frictional strength of these planar discontinuities. This concept is schematically illustrated in Figs. 7.a and 7.b. In the upper part of the figure, a series of randomly oriented fractures and faults is shown. Because this is a two-dimensional (2D) illustration (for simplicity), it is easiest to consider this sketch as a map view of vertical strike-slip faults. In this case, it is the difference between σHmax (SHmaxPp) and σHmin (SHminPp) that is limited by the frictional strength of these pre-existing faults. In other words, as σHmax increases with respect to σHmin, a subset of these pre-existing faults (shown in light gray) begins to slip as soon as its frictional strength is exceeded. Once that happens, further stress increases are not possible, and this subset of faults becomes critically stressed (i.e., just on the verge of slipping). The lower part of the figure illustrates using a three-dimensional (3D) Mohr diagram, the equivalent 3D case.

The frictional strength of faults can be described in terms of the Coulomb criterion, which states that faults will slip if the ratio of shear to effective normal stress exceeds the coefficient of sliding friction (i.e., τ/σn = μ); see Fig. 8. Because for essentially all rocks (except some shales) 0.6 < μ < 1.0, it is straightforward to compute limiting values of effective stresses using the frictional strength criterion.

This is graphically illustrated using a 3D Mohr diagram as shown in the lower part of Fig. 7. 2D Mohr diagrams plot normal stress along the x-axis and shear stress along the y-axis. Any stress state is represented by a half circle that intersects the x-axis at σ = σ3 and σ = σ1 and has a radius equal to (σ1σ3)/2. A 3D Mohr diagram plots three half circles the endpoints of which lie at values equal to the principal stresses and the radii of which are equal to the principal stress differences divided by 2. Planes of any orientation plot within and along the edges of the region between the circles at a position corresponding to the values of the shear and normal stresses resolved on the planes. Planes that contain the σ2 plot along the largest circle are first to reach a critical equilibrium.

The critically stressed (light gray) faults in the upper part of the figure correspond to the points (also shown in light gray) in the Mohr diagram, which have ratios of shear to effective normal stress between 0.6 and 1.0. It is clear in the Mohr diagram that for a given value of σHmin, there is a maximum value of σHmax established by the frictional strength of pre-existing faults (the Mohr circle cannot extend past the line defined by the maximum frictional strength).

The values of S1 and S3 corresponding to the situation illustrated in Fig. 7 are defined by ....................(4)

That is, it is the effective normal stress on the fault (the total stress minus the pore pressure) that limits the magnitude of the shear stress. Numerous in-situ stress measurements have demonstrated that the crust is in frictional equilibrium in many locations around the world (Fig. 9). This being the case, if one wished to predict stress differences in-situ with Eq. 4, one would use Anderson's faulting theory to determine which principal stress (i.e., SHmax, SHmin, or Sv) corresponds to S1 or S3, depending of course on whether it is a normal, strike-slip, or reverse-faulting environment, and then utilize appropriate values for Sv and Pp (the situation is more complex in strike-slip areas because Sv corresponds to neither S1 nor S3). Regardless of whether the state of stress in a given sedimentary basin reflects the frictional strength of pre-existing faults, the importance of the concept illustrated in Fig. 7 is that at any given depth and pore pressure, once we have determined the magnitude of the least principal effective stress using minifracs or leakoff tests (σHmin in a normal or strike-slip faulting case), there is only a finite range of values that are physically possible for σHmax.

Eq. 4 defines the upper limit of the ratio of effective maximum to effective minimum in-situ stress that is possible before triggering slip on a pre-existing, well-oriented fault. The in-situ effective stress ratio can never be larger than this limiting ratio. Therefore, all possible stress states must obey the relationship that the effective stress ratios must lie between 1 and the limit defined by fault slip as shown in Eq. 5. ....................(5)

These equations can be used along with the Andersonian definitions of the different faulting regimes (Table 1) to derive a stress polygon, as shown in Fig. 10. These figures are constructed as plots at a single depth of SHmax vs. SHmin. The shaded region is the range of allowable values of these stresses. By the definitions of SHmax and SHmin, the allowable stresses lie above the line for which SHmax = SHmin. Along with the pore pressure, Sv, shown as the black dot on the SHmax = SHmin line, defines the upper limit of SHmax [the horizontal line at the top of the polygon, for which σHmax/σv = f(μ)], and the lower limit of SHmin [the vertical line on the lower left of the polygon, for which σv/σHmin = f(μ)]. The third region is constrained by the difference in the horizontal stress magnitudes [i.e., σHmax/σHmin < f(μ)]. The larger the magnitude of Sv, the larger the range of possible stress values; however, as the pore pressure increases, the polygon shrinks, until at the limit when Pp = Sv, all three stresses are equal.

It is important to emphasize that the stress limit defined by frictional faulting theory is just that—a limit—and provides a constraint only. The stress state can be anywhere within and along the boundary of the stress polygon. As discussed at length later, the techniques used for quantifying in-situ stress magnitudes are not model based, but instead depend on measurements, calculations, and direct observations of wellbore failure in already-drilled wells in the region of interest. These techniques have proved to be sufficiently robust that they can be used to make accurate predictions of wellbore failure (and determination of the steps needed to prevent failure) with a reasonable degree of confidence.

### Stress constraints owing to shear-enhanced compaction

In weak, young sediments, compaction begins to occur before the stress difference is large enough to reach frictional equilibrium. Therefore, rather than being at the limit constrained by the frictional strength of faults, the stresses will be in equilibrium with the compaction state of the material. Specifically, the porosity and stress state will be in equilibrium and lie along a compactional end cap. The physics of this process is discussed in Rock properties

Constraints, based on compaction, define another stress polygon similar to the one shown in Fig. 10. It is likely that in regions such as the Gulf of Mexico, and in younger sediments worldwide where compaction is the predominant mode of deformation, this is the current in-situ condition. Unfortunately, while end-cap compaction has been studied in the laboratory for biaxial stress states (σ1 > σ2σ3), there has been little laboratory work using polyaxial stresses (σ1σ2σ3), and there have been relatively few published attempts to make stress predictions using end-cap models. Also, it is important to apply end-cap analyses only where materials lie along a compaction curve, and not to apply these models to overcompacted or diagenetically modified rocks. If the material lies anywhere inside the region bounded by its porosity-controlled end cap, this constraint can be used only to provide a limit on stress differences.

## Nomenclature

 S1 = greatest principal stress, MPa, psi S2 = intermediate principal stress, MPa, psi S3 = least principal stress, MPa, psi SHmin = least horizontal stress, MPa, psi, lbm/gal SHmax = greatest horizontal stress, MPa, psi, lbm/gal Sij = component of the stress tensor acting in the xj direction on a plane perpendicular to xi, Pa, psi Sji = component of the stress tensor acting in the xi direction on a plane perpendicular to xj, MPa, psi x1, x2, x3 = Cartesian coordinate system x, y, δij = Kronecker delta (δij = 1, if i = j; δij = 0 otherwise) Δ = operator indicating a change in a parameter (Δ Pp is change in Pp) μ = coefficient of sliding friction on a pre-existing weak plane, where μ = tanΦ σ1, σ2, σ3 = maximum, intermediate, and least effective stresses, MPa, psi σij = effective stress acting in the i direction on a plane perpendicular to the j direction, MPa, psi

### Subscripts

 i = index j = index