Vertical Stress

Summary

The Vertical stress at a depth is the weight of everything above it, the integral of the bulk density log with depth, plus the water column offshore. It is the best constrained of the three principal stresses, and every other stress in the model is built on it. The work is in the density above the top of the log, where an assumed trend is used.

Inputs and outputs

Equations

The vertical stress is the integral of bulk density times gravity over depth. In field units, with density in g/cm³, depth in feet and stress in psi, the conversion factor is 0.4335 psi/ft per g/cm³ (62.43 lb/ft³ divided by 144):

In practice the integral is a trapezoid sum over the log samples, starting from the stress at the top of the log:

Above the top of the log the density is extrapolated. The calculator uses an exponential compaction trend from a near-surface density \(\gmRhoS\) to a deep density \(\gmRhoD\) over a compaction length \(\gmLc\), which has a closed-form integral. Offshore, the water column of depth \(\gmHw\) and density 1.03 g/cm³ adds its own weight:

The equivalent mud weight and the mean overburden density use the total depth \(z + \gmHw\) below sea level, and the factor 0.052 psi/ft per ppg:

Single-value calculator

Behavior

Vertical stress increases with depth, and its slope is the bulk density. At 5000 ft with the default trend the stress is 4606 psi (17.72 ppg, mean density 2.13 g/cm³); at 10000 ft it is 9794 psi (18.83 ppg); at 20000 ft it is 20786 psi (19.99 ppg). The curve is steeper at depth than at the top, because the compacted rock is denser: the gradient averaged over the whole column rises from 0.92 psi/ft at 5000 ft to 1.04 psi/ft at 20000 ft. A water column of 5000 ft at the same 10000 ft below the mudline lowers the equivalent mud weight from 18.83 to 15.42 ppg, because water is lighter than rock, though it adds 2233 psi of stress. A mean density of about 2.31 g/cm³ gives the 1 psi/ft rule of thumb.

Parameter guidance

Density log. Use the repaired, environmentally corrected bulk density. Fill washout intervals by a local trend and do not integrate spikes. A bias in density that persists over a long interval, for example from a poor calibration, accumulates in the integral, while random noise averages out.

Density above the log. The shallow section, from the surface or sea floor to the top of the density log, has to be assumed. Options, in rising order of effort: a constant mean density or gradient; a fitted compaction trend such as the exponential here, with the near-surface and deep values set to match the logged density at the top of the log; or a density derived from seismic or check-shot velocity, for which Gardner's relation, roughly \(\rho \approx 0.31\,V_p^{0.25}\) with \(V_p\) in m/s and \(\rho\) in g/cm³, is the usual form. The shallow part matters most offshore, where the sediments above the log can be a large share of the column. A trend must be continuous with the log at the join. Pore pressure models also need this overburden, see Pore Pressure.

Water and air gap. Offshore, add the water column at 1.03 g/cm³ (about 1.0 to 1.05 depending on salinity), and refer all depths to the same datum. The equivalent mud weight is conventionally referred to the depth below the rig floor, so state the datum.

Deviated wells. Use true vertical depth, not measured depth, as the integration variable.

Worked example

Integrating a short density log by the trapezoid rule, starting from a stress at the top of the log obtained from an assumed constant density of 2.0 g/cm³ above it, and checking against the closed-form trend of the calculator:

import math
k = 0.4335                                   # psi/ft per g/cm3
# (depth ft, bulk density g/cm3): top of log at 4000 ft
log = [(4000, 2.15), (5000, 2.25), (6000, 2.33), (7000, 2.40), (8000, 2.45)]
sv = k * 2.0 * 4000                          # assumed 2.0 g/cm3 above the log
print(f"Sv at top of log (4000 ft) = {sv:.0f} psi = {sv / (0.052 * 4000):.2f} ppg")
for (z0, r0), (z1, r1) in zip(log, log[1:]):
    sv += k * 0.5 * (r0 + r1) * (z1 - z0)
    print(f"Sv at {z1} ft = {sv:.0f} psi = {sv / (0.052 * z1):.2f} ppg")
# closed-form compaction trend at 10000 ft, land well
z, r0, rd, L = 10000.0, 1.9, 2.6, 6000.0
sv_trend = k * (rd * z - (rd - r0) * L * (1 - math.exp(-z / L)))
print(f"trend: Sv(10000 ft) = {sv_trend:.0f} psi, {sv_trend / (0.052 * z):.2f} ppg, {sv_trend / z:.3f} psi/ft")
print(f"psi/ft for a constant 2.31 g/cm3: {k * 2.31:.3f}")

Output

Sv at top of log (4000 ft) = 3468 psi = 16.67 ppg
Sv at 5000 ft = 4422 psi = 17.01 ppg
Sv at 6000 ft = 5414 psi = 17.35 ppg
Sv at 7000 ft = 6440 psi = 17.69 ppg
Sv at 8000 ft = 7491 psi = 18.01 ppg
trend: Sv(10000 ft) = 9794 psi, 18.83 ppg, 0.979 psi/ft
psi/ft for a constant 2.31 g/cm3: 1.001

Assumptions and limitations

• The stress is the weight of the overburden at the point, and the Earth is laterally uniform. Near salt, steep topography or large density contrasts, it is a 3D stress problem.
• The density log is the bulk density of the rock with its fluid, free of washout and invasion bias, and covers the interval. Missing sections are filled by a trend.
• The extrapolated trend above the log is representative. Its error enters every deeper depth as a fixed offset.
• Constants: 0.4335 psi/ft per g/cm³ is exact for water at 1 g/cm³; the ppg conversion with 0.052 differs from it by about 0.1 percent (8.345 ppg per g/cm³, times 0.052, is 0.434).

QC checks

• The vertical stress gradient is between about 0.9 and 1.1 psi/ft (17 to 21 ppg) onshore at depths below a few thousand feet. Gradients well outside that range need an explanation.
• The stress curve is monotonic and smooth. A step in stress means a density spike or a gap in the log.
• The stress is continuous at the join between the trend and the log. A jump indicates mismatched density.
• Compare with a seismic or regional overburden gradient, and with the overburden used in the pore pressure model. They must be the same curve in both places.
• Offshore, the equivalent mud weight at a given depth is lower than onshore for the same rock, by an amount that grows with water depth.

Going Deeper

Vertical stress is usually treated as a principal stress, which is exact for a flat, laterally uniform overburden and an approximation otherwise. Its uncertainty is typically a few percent, the smallest of the three stresses, but in the deepwater and subsalt settings where the shallow section is thin or the extrapolation is long, it can be several percent and it feeds directly into pore pressure and the horizontal stresses. Because the lower and upper bounds of the mud weight window are differences between stresses of similar size, small relative errors in Sv matter more than the percentages suggest.

References

1. Zoback, M.D., 2007. Reservoir Geomechanics. Cambridge University Press.
2. Fjær, E., Holt, R.M., Horsrud, P., Raaen, A.M. and Risnes, R., 2008. Petroleum Related Rock Mechanics, 2nd edition. Elsevier.

Python reference implementation

Related methods

• Horizontal Stress Methods
• Near-Wellbore Stress State
• Breakout and Breakdown Pressures
• Mud Weight Window
• Pore Pressure
• Dynamic Elastic Moduli