Geomechanics: elastic to stress and fracture

From velocities to elastic moduli

For an isotropic elastic rock, $V_P$, $V_S$ and density give every elastic modulus:

Poisson’s ratio depends on $V_P/V_S$ alone ($V_P/V_S = 2$ is $\nu = 1/3$ exactly), so an inversion that delivers $Z_P$, $Z_S$ and $\rho$ delivers $E$ and $\nu$ at every voxel. For the Part 5 brine-saturated sandstone ($V_P$ = 3300 m/s, $V_S$ = 1810 m/s, $\rho$ = 2.30 g/cm³) they are $\mu$ = 7.5 GPa, $K$ = 15.0 GPa, $E$ = 19.4 GPa and $\nu$ = 0.285.

These are dynamic moduli. A seismic or sonic wave strains the rock by about a microstrain, briefly. Loading in a press, or in the earth, strains it a thousand times more and slowly; microcracks and grain contacts close and slide, and the rock responds softer. A stress model needs static moduli, and they are lower than the dynamic ones, most of all in weak, porous or cracked rock. Correlations bridge the gap. Figure 8.1 uses the linear fit of Eissa and Kazi (1988), $E_s = 0.74\,E_d - 0.82$ with both in GPa, which takes the sandstone’s 19.4 GPa to 13.5 GPa. It is one fit among many, with real scatter; a field calibrates its own against core.

Poisson’s ratio holds a further trap. A seismic wave passes too fast for the pore fluid to move, so the dynamic $\nu$ of a brine-saturated rock is raised by the stiffness of the brine, while gas, nearly as compressible as empty pore space, leaves a gas sand’s dynamic $\nu$ close to its drained frame’s. The $\nu$ a stress model needs is the drained, static one.

Reading Figure 8.1

The figure takes a rock from the book’s rock library, with the same velocities and densities as Part 5. Plate (a) places every library rock by its dynamic $\nu$ and $E$ (halite is left out: it creeps and obeys no elastic stress model), marks the one chosen with a large dot, and drops an open ring to its static $E$; the dashed lines are Rickman’s brittleness index, discussed below. Plate (b) runs the stress model of the next box down a column of the chosen rock under an overburden of 2.3 g/cm³, the canon of the site’s geomechanics book: $S_v$ is 67.7 MPa at 3000 m, where the hydrostatic pressure is 30.3 MPa. Plate (c) draws the Mohr circles of the three effective principal stresses at 3000 m against a Coulomb failure line. The headline value is $S_{hmin}$ at 3000 m, and the last readout is the further change of pore pressure that would first bring the rock to failure; the action on plate (c) applies that change when it lies within the slider’s range.

By default, brine-saturated sandstone with a pore pressure of 35.3 MPa, $\alpha$ = 0.80 and tectonic strains of 100 and 400 microstrain gives $S_{hmin}$ = 47.1 MPa and $S_{Hmax}$ = 50.3 MPa, a normal-faulting state 22.7 MPa of pore pressure from tensile opening. Swap the brine for gas and the dynamic $\nu$ falls from 0.285 to 0.065: the model then puts $S_{hmin}$ at 32.4 MPa, below the pore pressure, a state no rock at rest holds. The model has left its ground, and the figure says so. The exercises under the figure walk through this and four other cases.

From moduli to stress: the uniaxial-strain model

Three numbers set the stress in a basin: the vertical stress $S_v$ and the two horizontal stresses, $S_{Hmax}$ and $S_{hmin}$. The vertical one is the weight of the rock above, $S_v = \int_0^z \rho\, g\, dz$, integrated from a density log. The horizontal ones need a model. The simplest is uniaxial strain, or the bilateral constraint: a layer buried under a wide basin cannot expand sideways, so as the overburden loads it, its horizontal effective stress grows by $\nu/(1-\nu)$ of the vertical. Eaton (1969) wrote the result for fracture-gradient prediction; with Biot’s coefficient and the poroelastic tectonic-strain terms (Thiercelin and Plumb 1994; Blanton and Olson 1999) it reads

The first two terms are the bilateral constraint. They make $S_{hmin}$ climb with the pore pressure, which is why overpressured basins have higher fracture gradients, but by less than the pore pressure itself, so the least effective stress falls. The tectonic terms scale with the static $E$, so a strain every layer shares raises the stress most in the stiffest layers: in Figure 8.1, tectonic strains of 100 and 800 microstrain add 16.2 MPa to $S_{hmin}$ in limestone and 1.4 MPa in soft shale. Contrasts in $\nu$ matter too: medium shale, with $\nu = 1/3$, carries 3.3 MPa more $S_{hmin}$ than the brine sandstone, and that contrast is what contains a hydraulic fracture’s height.

No term of this model is measured by seismic. The strains are found by fitting the model to a measured $S_{hmin}$ from leak-off tests or minifracs, and $S_{Hmax}$ is checked against breakouts.

The stress regimes

The order of the three stresses sets the faulting regime (Anderson 1951):

• Normal faulting, $S_v \gt S_{Hmax} \gt S_{hmin}$: extensional basins and passive margins (the Gulf of Mexico, much of the North Sea); the hanging wall moves down dip.
• Strike-slip faulting, $S_{Hmax} \gt S_v \gt S_{hmin}$: transform margins (San Andreas, Anatolia); faults slip horizontally.
• Reverse faulting, $S_{Hmax} \gt S_{hmin} \gt S_v$: compressional belts (the Rocky Mountain foothills, the Andes); the hanging wall moves up dip, as on thrusts.

Hydraulic fractures open against the least principal stress. In the normal and strike-slip regimes that is $S_{hmin}$, so fractures are vertical and strike along $S_{Hmax}$; in the reverse regime it is $S_v$, and shallow fractures can be horizontal. In Figure 8.1 the regime is computed, not chosen: stiff rocks under strong tectonic strain move from normal into strike-slip faulting, and dolomite with both strains at 800 microstrain reaches reverse faulting.

Effective stress and the pore-pressure lever

The rock frame carries the effective stress, the total stress less the share the pore fluid holds up:

Biot’s coefficient $\alpha = 1 - K_{dry}/K_{min}$ (Biot and Willis 1957) is near 1 in soft, porous rock and lower in stiff, tight rock; for the Part 5 clean-sand frame it is 0.82. An error of 0.2 in $\alpha$ moves the Biot effective stress by $0.2\,P_p$, 7 MPa at the 35.3 MPa pore pressure of Figure 8.1. Deformation follows Biot’s form, but rock failure follows Terzaghi’s, $\sigma' = S - P_p$ (Jaeger, Cook and Zimmerman 2007; Zoback 2007), which is why Figure 8.1 uses $\alpha$ in its stress model and the full pore pressure in its Mohr circles.

Production lowers the pore pressure and raises every effective stress; injection lowers them. In a reservoir wide enough to stay in uniaxial strain the horizontal stresses follow the pore pressure, by the stress-path coefficient $\gamma = \alpha(1-2\nu)/(1-\nu)$, while $S_v$ stays put (Segall and Fitzgerald 1998). Depletion therefore raises $\sigma_v'$ by the full drawdown and $\sigma_h'$ by only $1 - \gamma$ of it, and the Mohr circle grows as it moves. Where $\gamma$ is large it grows fast enough to reach the failure line, and a depleting reservoir in a normal-faulting basin faults.

The Mohr-Coulomb failure criterion

A rock fails in shear when the shear stress on some plane reaches a strength that grows with the effective normal stress on that plane:

Here $S_0$ is the cohesion, several MPa or more for intact rock and close to zero on an existing fault, and $\mu = \tan\phi$ the coefficient of friction, with $\phi$ the friction angle. Byerlee (1978) found $\mu$ between 0.6 and 0.85 for most rocks; clay-rich fault gouge is far weaker. The Mohr circle of $\sigma_1'$ and $\sigma_3'$ holds every pair of $\sigma_n'$ and $\tau$ acting on the planes through a point. If it reaches the line, some plane is at failure, the one whose normal lies at $45^\circ + \phi/2$ from $\sigma_1$. For a circle of centre $c$ and radius $r$, the stress it still lacks is $S_0\cos\phi + c\sin\phi - r$.

• A larger differential stress, $\sigma_1' - \sigma_3'$, makes a larger circle that reaches higher toward the line.
• A higher pore pressure moves the whole circle left, toward lower $\sigma_n'$, where the line is lower: injection brings rock toward failure.
• When the least effective stress reaches zero (taking the tensile strength as zero, as for rock with natural fractures; for intact rock, with a tensile strength of several MPa, that is conservative), the rock opens in tension: a hydraulic fracture.
• Faults have little cohesion, so they reach the line with smaller circles than intact rock. Much of the crust holds critically stressed faults that a few MPa of injected pressure can reactivate.

Brittleness indices and their caveats

Completion engineers rank intervals by a brittleness index, and the most used elastic one is that of Rickman et al. (2008): rescale $E$ between 1 and 8 Mpsi (6.9 to 55 GPa) and $\nu$ between 0.40 and 0.15, and average the two, in percent:

High $E$ and low $\nu$ score as brittle. The index is a useful screen and a poor measurement, for well-known reasons:

• Its bounds were chosen for the Barnett shale. A rock outside them is extrapolated or clipped, and the same rock scores differently against another basin’s bounds.
• Applied to log or seismic moduli, as in Figure 8.1, it reads dynamic values, while a fracture responds to static ones. Rickman’s calibration is often quoted with static moduli, and fed the static $E$ the brine sandstone scores 30 rather than 36.
• The dynamic $\nu$ reads the pore fluid: in Figure 8.1 the same sandstone frame scores 36 with brine and 76 with gas.
• It ignores the stress. Whether a fracture grows tall or stays contained is set by the contrast in $S_{hmin}$ between layers, not by an index.
• Brittleness is not a material constant: the same rock deforms more ductilely under a higher confining stress.

What a seismic interpreter can and cannot infer

From prestack inversion an interpreter can map the dynamic $E$ and $\nu$, and with them relative stiffness, the contrasts in $\nu$ that hint at stress barriers, and a brittleness screen, all at seismic resolution and with density the least certain of the three inputs. Velocity-based methods add a pore-pressure estimate, calibrated at wells.

Seismic alone cannot give the static moduli, which need core; the strength, which needs core or log correlations; Biot’s $\alpha$; the tectonic strains; or any absolute stress. $S_v$ comes from integrated density logs, $S_{hmin}$ from leak-off tests and minifracs, $S_{Hmax}$ from breakouts and drilling-induced fractures, and the orientation of the stresses from image logs and, where it can be measured, azimuthal anisotropy (Section 8.3). A geomechanical cube built from seismic is a seismic-guided interpolation of a model calibrated at wells, and its maps should say so.

Workflow, from QI cube to geomechanical deliverable

• Input: the elastic cubes ($Z_P$, $Z_S$, $\rho$) of Section 7.3, and a pore-pressure cube from velocity analysis, basin modelling or mud weights, calibrated at wells.
• Moduli: the dynamic $E$ and $\nu$ at every voxel, converted to static ones with a correlation calibrated on core.
• Stresses: $S_v$ from integrated density; $S_{hmin}$ and $S_{Hmax}$ from the uniaxial-strain model, with $\alpha$ and the tectonic strains fitted to leak-off tests, minifracs and breakouts.
• Failure: at every voxel, the Mohr circles of the effective stresses against a strength model, giving the margins to shear and tensile failure, breakout risk and the mud-weight window.
• Scenarios: depletion and injection along the reservoir’s stress path, to find where and when failure risk emerges.
• Deliverables: stress and failure-margin cubes, mud-weight windows, well trajectories, and fracture-azimuth and containment maps.

Drilling, completions and reservoir engineering consume these. In mature fields the model is coupled to the reservoir simulator and run through time, the bridge to Section 8.2.