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, S_ij = S_ji, which reduces the number of independent stress components to six.

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  Fig. 1.a—Definitions of the stress tensor in Cartesian coordinates, tensor transformation through direction cosines, the principal stress axes (courtesy GeoMechanics Intl. Inc.).

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, S_v) 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, S_Hmax, and the least horizontal stress, S_Hmin (Fig. 1.b).

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  Fig. 1.b—The vertical (S_v) and horizontal maximum and minimum stresses (S_Hmax and S_Hmin), which are usually, but need not be, principal stresses (courtesy GeoMechanics Intl. Inc.).

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).

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  Fig. 2—World stress map showing orientations of the greatest horizontal stress, S_Hmax, where it has been measured using wellbore breakouts or inferred from earthquakes. Also shown are the boundaries of the major tectonic plates. Colors of the symbols indicate the relative stress magnitudes (light gray, normal; gray, strike-slip; black, reverse or unknown). This figure was produced using software and data available from the World Stress Map Project website.

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  Fig. 3—Orientation of maximum horizontal stress within the San Joaquin basin of south-central California derived from breakouts. Although within individual fields stress orientation is quite uniform, it varies systematically across the region (modified after Castillo and Zoback).

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  Table 1-Sources Of Stress In The Earth

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.

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  Table 2-Definitions Of S₁ And S₃ Foe Andersonian Faulting Classifications

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  Fig. 4—Diagram illustrating the three faulting states based on Andersonian faulting theory (courtesy GeoMechanics Intl. Inc.).

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
• Unloading
• Rock dilation
• Reservoir depletion

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  Fig. 5—Illustration of pore pressure in permeable rock under hydrostatic pressure (courtesy Geo-Mechanics Intl. Inc.).

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)

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  Fig. 6—In a laterally infinite reservoir where L>>h, the relationship between a change in pore pressure and the resulting change in stress is defined in Eq. 1 (courtesy GeoMechanics Intl. Inc.).

where v is Poisson's ratio, and α (= 1 - K_dry/K_grain) 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 σ_ij is the effective stress, P_p is the pore pressure, δ_ij is the Kronecker delta (δ_ij = 1, if i = j, δ_ij = 0 otherwise), and S_ij 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 (σ_ij = S_ij – δ_ij α P_p , 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 σ_Hmax (S_Hmax – P_p) and σ_Hmin (S_Hmin – P_p) that is limited by the frictional strength of these pre-existing faults. In other words, as σ_Hmax increases with respect to σ_Hmin, 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.

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  Fig. 7.a—Map view of theoretical faults and fractures. The fractures and faults shown in gray are optimally oriented to slip in the current stress field (courtesy GeoMechanics Intl. Inc.).

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  Fig. 7.b—Mohr diagram showing poles to the critically stressed faults and fractures for 0.6 < μ < 1.0 (courtesy GeoMechanics Intl. Inc.).

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/σ_n = 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.

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  Fig. 8—Sliding on faults is limited by the ratio of the shear stress (τ) to the effective normal stress (σ_n) on the fault plane, as defined by the Coulomb criterion: τ_n = μ, where μ, is the coefficient of sliding friction (courtesy GeoMechanics Intl. Inc.).

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 (σ₁ – σ₃)/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 σ_Hmin, there is a maximum value of σ_Hmax 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., S_Hmax, S_Hmin, or S_v) 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 S_v and P_p (the situation is more complex in strike-slip areas because S_v 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 (σ_Hmin in a normal or strike-slip faulting case), there is only a finite range of values that are physically possible for σ_Hmax.

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  Fig. 9—Stress measurements made in brittle rock (dots) reveal that in most of the world, the crust is in a state of frictional equilibrium for fault slip for coefficients of sliding friction between 0.6 and 1.0 as measured in the laboratory (modified after Townend and Zoback).

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 S_Hmax vs. S_Hmin. The shaded region is the range of allowable values of these stresses. By the definitions of S_Hmax and S_Hmin, the allowable stresses lie above the line for which S_Hmax = S_Hmin. Along with the pore pressure, S_v, shown as the black dot on the S_Hmax = S_Hmin line, defines the upper limit of S_Hmax [the horizontal line at the top of the polygon, for which σ_Hmax/σ_v = f(μ)], and the lower limit of S_Hmin [the vertical line on the lower left of the polygon, for which σ_v/σ_Hmin = f(μ)]. The third region is constrained by the difference in the horizontal stress magnitudes [i.e., σ_Hmax/σ_Hmin < f(μ)]. The larger the magnitude of S_v, the larger the range of possible stress values; however, as the pore pressure increases, the polygon shrinks, until at the limit when P_p = S_v, all three stresses are equal.

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  Fig. 10—This figure shows construction of the polygon that limits the range of allowable stress magnitudes in the Earth’s crust at a fixed depth and corresponding magnitude of S_v). It is a plot of S_Hmax vs. S_Hmin as constrained by the strength of well-oriented, pre-existing faults. The limits are constrained by Eq. 4, with S₁ and S₃ defined by Andersonian faulting theory, as shown in Table 2 (courtesy GeoMechanics Intl. Inc.).

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 > σ₂ ≅ σ₃), there has been little laboratory work using polyaxial stresses (σ₁ ≠ σ₂ ≠ σ₃), 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

Noteworthy papers in OnePetro

E. Skomedal, H.P. Jostad and M.H. Hettema 1999. Feasibility Study of the Stability of Openhole Multilaterals, Cook Inlet, Alaska, SPE Mid-Continent Operations Symposium, 28-31 March. 52186-MS. http://dx.doi.org/10.2118/52186-MS.

G.N. Boitnott, T.W. Miller, and J.L. Shafer 2009. Title Pore-Pressure Depletion and Effective Stress Issues in the Gulf of Mexico's Lower Tertiary Play, SPE Annual Technical Conference and Exhibition, 4-7 October. 124790-MS. http://dx.doi.org/10.2118/124790-MS.

External Links

Heidbach, O., Tingay, M., Barth, A., Reinecker, J., Kurfeß, D., and Müller, B. 2008. The World Stress Map database release http://dx.doi.org/10.1594/GFZ.WSM.Rel2008

See also

Building geomechanical models

Stress impact on rock properties

Stress strain relationships in rocks

Pore pressure prediction using seismic

Pore pressure prediction using acoustic logging

Pore fluid effects on rock mechanics

PEH:Geomechanics_Applied_to_Drilling_Engineering

PEH:Rock_Properties

Page champions

Fersheed Mody, Ph.D., P.E.