Equivalent Depth

Summary

The equivalent depth method finds the shallower depth at which normally compacted shale has the same log value as the overpressured shale, assumes the Vertical effective stress is the same at both, and sets the pore pressure from it. It needs only a trendline and the stress profile, and no empirical exponent. Use it where the overpressure mechanism is disequilibrium compaction and a trend can be defined.

Inputs and outputs

Equations

The method assumes that two shales with the same slowness have the same effective stress. For the observed slowness \(\dtc\) the equivalent depth \(\ppZe\) is where the normal trendline equals the observation. With the exponential sonic trendline from the normal compaction page it can be found in closed form:

The normal slowness \(\ppDtn\) at the observed depth, for comparison, is the same trendline evaluated at \(\zdepth\). At the equivalent depth the pore pressure is hydrostatic, so the effective stress there is the overburden at \(\ppZe\) less the hydrostatic pressure \(\ppPh\) at \(\ppZe\). With mean gradients of the overburden \(\ppGob\) and of water \(\ppGh\), it is:

The pore pressure at the observed depth \(\zdepth\) is the overburden there less this effective stress:

The equivalent depth is limited to zero or above. With the overburden stress known as a profile, replace the gradient products with the overburden at each depth: \(\ppPp = \ppSv(\zdepth) - \left[\ppSv(\ppZe) - \ppPh(\ppZe)\right]\).

Single-value calculator

Behavior

Pore pressure is hydrostatic where the slowness equals the normal value at the depth (81.2 µs/ft at 10,000 ft with the default trendline) and increases as the slowness grows. At 10,000 ft with an overburden gradient of 0.95 psi/ft, a slowness of 100 µs/ft gives 6,171 psi, 110 µs/ft gives 6,760 psi (13.0 ppg, with an equivalent depth of 5,649 ft) and 120 µs/ft gives 7,259 psi. The overburden gradient shifts the whole curve: each 0.05 psi/ft changes the result by about 500 psi at 10,000 ft, because the overburden term is carried in full, and a smaller part comes from the equivalent-depth stress term.

Parameter guidance

The method has the trendline parameters of the normal compaction page and the two gradients. The gradient of the overburden at the equivalent depth may differ from the gradient at the depth of interest, since stress gradient increases with depth; use the overburden profile from the overburden page at both depths when it is available. The method assumes the trend is correct everywhere, so check that the trend in the normal section is fitted in clean shale only. The equivalent depth can also be read graphically, from the log display, by tracing a horizontal line from the observed value to the trend and reading its depth; the closed form does the same.

Worked example

A slowness of 110 µs/ft at 10,000 ft, with a mudline slowness of 190, a deep value of 50 µs/ft, a rate of 0.15 per kft, an overburden gradient of 0.95 psi/ft and a water gradient of 0.465 psi/ft:

import math
z, dt = 10000.0, 110.0
dt_ml, dt_inf, c = 190.0, 50.0, 0.15
g_ob, g_h = 0.95, 0.465
dtn = dt_inf + (dt_ml - dt_inf) * math.exp(-c * z / 1000)
ze = -math.log((dt - dt_inf) / (dt_ml - dt_inf)) / c * 1000
sigma_e = ze * (g_ob - g_h)
pp = g_ob * z - sigma_e
ph = g_h * z
print(f'normal slowness at {z:.0f} ft = {dtn:.1f} us/ft; observed {dt:.0f}, so the log is slow by {dt - dtn:.1f}')
print(f'equivalent depth = {ze:.0f} ft')
print(f'effective stress at {ze:.0f} ft = {ze:.0f} x ({g_ob} - {g_h}) = {sigma_e:.0f} psi')
print(f'overburden at {z:.0f} ft = {g_ob * z:.0f} psi, pore pressure = {pp:.0f} psi = {pp / (0.052 * z):.2f} ppg')
print(f'overpressure = {pp - ph:.0f} psi above hydrostatic ({ph:.0f} psi)')

Output

normal slowness at 10000 ft = 81.2 us/ft; observed 110, so the log is slow by 28.8
equivalent depth = 5649 ft
effective stress at 5649 ft = 5649 x (0.95 - 0.465) = 2740 psi
overburden at 10000 ft = 9500 psi, pore pressure = 6760 psi = 13.00 ppg
overpressure = 2110 psi above hydrostatic (4650 psi)

Assumptions and limitations

• Overpressure comes from disequilibrium compaction: effective stress on the shale is the same at the two depths when the log value is the same. Unloading mechanisms break this assumption.
• The normal trend is valid at the equivalent depth. If the trend is extrapolated from a short normal interval its error is multiplied in the equivalent depth.
• Gradients of overburden and water are constant between the surface and the depth, or the full profiles are used.
• The same lithology and temperature history at the two depths; otherwise equal slowness does not mean equal stress.
• The overburden is correct at the depth of interest.

QC checks

• The equivalent depth is shallower than the observed depth wherever the pressure is above hydrostatic, and equals it at the trend.
• The equivalent depth increases smoothly in shale and does not jump between sand and shale.
• The result is consistent with the Eaton result from the same trend, and with measured pressures in sands.
• The calculated pressure never exceeds the overburden.
• An equivalent depth of zero (slowness at or above the mudline value) is flagged for review, not accepted as valid.

Going Deeper

The method is credited to Foster and Whalen's 1966 work with resistivity trends, and is also called the depth-of-equal-effective-stress or Terzaghi method. Its attraction is that it needs no empirical exponent, only the trendline and the overburden. It shares Eaton's main limitation, that it assumes loading, and also that a log value at or above the trend has no equivalent depth deeper than the point. Modified forms use a vertical-effective-stress gradient that changes with depth, and apply it to resistivity and density as well as sonic. In practice the method is the quickest first check on any Eaton result since both are built on the same trendline.

References

1. Foster, J.B. and Whalen, H.E., 1966. Estimation of formation pressures from electrical surveys, offshore Louisiana. Journal of Petroleum Technology, 18(2), 165–171.
2. Terzaghi, K., 1943. Theoretical Soil Mechanics. John Wiley & Sons, New York.
3. Zhang, J., 2011. Pore pressure prediction from well logs: methods, modifications, and new approaches. Earth-Science Reviews, 108(1–2), 50–63.

Python reference implementation

Related methods

• Eaton (Sonic and Resistivity)
• Bowers
• Overburden and Hydrostatic Pressure
• Normal Compaction Trendlines
• Loading vs Unloading