Eaton (Sonic and Resistivity)

Summary

Eaton's method estimates Pore pressure from the departure of a shale log property from its normal compaction trend. The pressure is the overburden less an effective stress, and the effective stress is the normal value scaled by the ratio of observed to normal property raised to an empirical exponent. Use it for sonic or resistivity in shale when the overpressure comes from disequilibrium compaction.

Inputs and outputs

Equations

Eaton's equations scale the normal effective stress, the difference between the overburden and the hydrostatic pressure, by a power of the ratio between the observed log value and its normal trend value. For the sonic log the ratio is normal over observed slowness:

For the resistivity log the ratio is observed over normal resistivity:

The exponents \(\ppXs\) and \(\ppXr\) have standard starting values of 3 and 1.2. Both are calibrated for each basin from measured pressures. The result in psi is converted to an equivalent mud weight at the depth:

and limited to values of zero or above. Where the log equals its normal value the equations return the hydrostatic pressure; where it departs far enough the pressure approaches the overburden.

Single-value calculator

Behavior

Pore pressure is hydrostatic where the observed slowness equals the normal slowness (75 µs/ft here) and rises as the log becomes slower than normal. The exponent sets how fast: at 100 µs/ft, an exponent of 2, 3 and 4 gives 8,126, 8,945 and 9,559 psi, so each unit of exponent is worth roughly 600 to 800 psi. This is why the exponent has to be calibrated: a wrong value changes the result by hundreds of psi for the same log. With the default inputs at 12,000 ft (overburden 11,400 psi, hydrostatic 5,580 psi, slowness 90 against a normal 75 µs/ft), the sonic gives 8,032 psi or 12.87 ppg, and a resistivity of 1.4 ohm·m against a normal 2.0 gives 7,606 psi or 12.19 ppg. Changing the resistivity exponent from 1.0 to 1.5 moves the result from 7,326 to 7,991 psi.

Parameter guidance

Overburden and hydrostatic come from the overburden and hydrostatic page. Normal value is read from the normal compaction trendline at the depth, not a single constant: the calculator takes the trend value for the depth of interest. Exponents. 3 for sonic and 1.2 for resistivity are the values Eaton found for the US Gulf Coast. Other basins need different values: calibrate them by adjusting the exponent until the calculated pressure matches measured pressures in sands (formation tester points) and mud weights or kick data. Values of 0.5 to 1.5 for resistivity and 1.5 to 4 for sonic are found in practice. Which shale points. Use only clean shale. In sand, silt and carbonate the log does not follow the shale trend, so the computed pressure is meaningless; either null those intervals or fill them with hydrostatic pressure. Resistivity. Use the deep resistivity corrected to a common temperature, and avoid hydrocarbon-bearing or conductive-mineral shale.

Worked example

The sonic and resistivity forms at 12,000 ft, with the unit conversion to equivalent mud weight, and the effect of a change in each exponent.

sv, ph, z = 11400.0, 5580.0, 12000.0   # psi, psi, ft
dt, dtn = 90.0, 75.0                   # us/ft
rt, rn = 1.4, 2.0                      # ohm.m
def eaton_dt(x): return sv - (sv - ph) * (dtn / dt) ** x
def eaton_rt(x): return sv - (sv - ph) * (rt / rn) ** x
emw = lambda p: p / (0.052 * z)
print(f'hydrostatic = {ph:.0f} psi = {emw(ph):.2f} ppg;  overburden = {sv:.0f} psi = {emw(sv):.2f} ppg')
print(f'normal effective stress = {sv - ph:.0f} psi')
print(f'sonic ratio (DTn/DT) = {dtn / dt:.4f}; cubed = {(dtn / dt) ** 3:.4f}')
print(f'sonic Pp = {eaton_dt(3):.0f} psi = {emw(eaton_dt(3)):.2f} ppg')
print(f'resistivity ratio (R/Rn) = {rt / rn:.3f}; to the 1.2 = {(rt / rn) ** 1.2:.4f}')
print(f'resistivity Pp = {eaton_rt(1.2):.0f} psi = {emw(eaton_rt(1.2)):.2f} ppg')
for x in (2.0, 3.0, 4.0):
    print(f'sonic exponent {x:.0f}: {eaton_dt(x):.0f} psi ({emw(eaton_dt(x)):.2f} ppg)')
# limiting case: a log equal to its normal value gives hydrostatic pressure
print(f'DT = DTn gives {sv - (sv - ph) * 1.0:.0f} psi')

Output

hydrostatic = 5580 psi = 8.94 ppg;  overburden = 11400 psi = 18.27 ppg
normal effective stress = 5820 psi
sonic ratio (DTn/DT) = 0.8333; cubed = 0.5787
sonic Pp = 8032 psi = 12.87 ppg
resistivity ratio (R/Rn) = 0.700; to the 1.2 = 0.6518
resistivity Pp = 7606 psi = 12.19 ppg
sonic exponent 2: 7358 psi (11.79 ppg)
sonic exponent 3: 8032 psi (12.87 ppg)
sonic exponent 4: 8593 psi (13.77 ppg)
DT = DTn gives 5580 psi

Assumptions and limitations

• Overpressure comes from disequilibrium compaction (undercompaction): the rock is less compacted than its depth implies and effective stress has been on the loading curve. Fluid expansion and unloading mechanisms are under-predicted. See the loading vs unloading page.
• The normal trend is correct, and the log values come from clean shale of the same type as the trend.
• The exponents are valid for the basin and the formation. The standard values come from one basin.
• The overburden and hydrostatic pressure are correct; an error in the overburden is carried into the result in full.
• Resistivity is not affected by changes in pore water salinity, temperature, hydrocarbons or conductive minerals, other than those corrected for.

QC checks

• Calculated pressure equals the hydrostatic pressure in the interval used to fit the trend, and is nowhere above the overburden.
• The result matches measured pressures in sands (formation tester points) and mud weights at the same depths. This is the final test, and the exponent is calibrated against it.
• Sonic and resistivity results agree in trend; a large difference points to a trend or temperature correction problem in one of them.
• The result is not computed or not used in sand, carbonate or coal intervals. Spikes in it follow lithology changes.
• Pressure does not fall below hydrostatic without a reason such as depletion; if it does, the trend is too fast or the exponent too large.

Going Deeper

Eaton's 1972 and 1975 papers presented the relations as ratios of gradients, with the exponents found by matching Gulf Coast pressure data. The equation is empirical but works in many basins because effective stress controls both compaction and the log response. Modified forms replace the constant normal value with a trendline that changes with depth, and several authors adjust the exponent with depth or lithology. A clear limitation is that it is a single-valued function of the log: it cannot represent unloading, where the log value stays high while the stress falls, and it is why Bowers' method, which has an explicit unloading curve, is used where fluid expansion is suspected.

References

1. Eaton, B.A., 1975. The equation for geopressure prediction from well logs. SPE 5544, SPE-AIME Fall Meeting, Dallas, TX.
2. 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

• Bowers
• Equivalent Depth
• Overburden and Hydrostatic Pressure
• Normal Compaction Trendlines
• Loading vs Unloading