Bowers

Summary

Bowers' method models velocity as a function of Vertical effective stress on a virgin loading curve, inverts it to get the effective stress from the observed velocity, and subtracts that from the overburden. A second curve describes unloading, so it can also be used where fluid expansion has caused overpressure. Use it with a sonic or seismic velocity when unloading is suspected or when a calibration of the effective stress curve is available.

Inputs and outputs

Equations

Velocity in ft/s comes from slowness in µs/ft by \(\ppVel = 10^6/\dtc\). The virgin (loading) curve relates velocity to effective stress, in psi:

Inverting it gives the effective stress for an observed velocity:

If the rock has been unloaded from a maximum effective stress \(\ppSmax\), which corresponds to the velocity \(\ppVmax\) on the virgin curve at the onset of unloading, the unloading curve is:

and, solved for effective stress where \(\ppVel < \ppVmax\):

The pore pressure follows from the Terzaghi relation and is expressed as an equivalent mud weight:

Single-value calculator

Behavior

The default curve has an effective stress of 3,969 psi at 10,000 ft/s (100 µs/ft) and 5,061 psi at the onset of unloading, 11,000 ft/s. With the overburden at 11,400 psi and 12,000 ft the virgin curve gives 7,431 psi (11.91 ppg) at 100 µs/ft and 10,118 psi (16.21 ppg) at 140. Switching on the unloading curve changes nothing at velocities above 11,000 ft/s (slowness below 90.9 µs/ft), because the rock there is on the virgin curve. Below that velocity it raises the pressure sharply, to 8,960 psi (14.36 ppg) at 100 µs/ft and 11,318 psi (18.14 ppg) at 140, close to the overburden of 11,400 psi. This is the difference between loading and unloading seen from the same log. The calculator returns an empty result where the velocity is at or below the mudline velocity.

Parameter guidance

Virgin curve (V0, A, B). Calibrate from a normally pressured interval, or from a well where effective stress is known from measured pressures and a density-based overburden, by regression of velocity on effective stress. The mudline velocity V0 is near 5,000 ft/s for water-saturated, unconsolidated sediment. Values of A near 10 and B near 0.7 to 0.8 (A of 5 to 20, B of 0.6 to 0.9) are of the order found in Gulf of Mexico shale, and are starting points only. Different basins need different values; do not carry them over without calibration. Unloading (U, Vmax). Vmax is the velocity on the virgin curve where the observed velocity first leaves the loading trend; it is read from the velocity-depth or velocity-density cross-plot where the velocity reverses or stops increasing. U describes how strongly velocity is retained during unloading: a value of 1 is the loading curve, and published values for different basins are of order 3 to 8. Calibrate from measured pressures in the unloaded interval. Use the unloading curve only below the onset depth. Units. The equations use ft/s and psi. Convert slowness in µs/ft with \(10^6/\Delta t\). In SI units use consistent values of V0, A and B for m/s and MPa.

Worked example

A slowness of 100 µs/ft at 12,000 ft with an overburden of 11,400 psi. The slowness is converted to velocity, then the loading and unloading results are compared.

dt = 100.0                       # us/ft
v = 1e6 / dt                     # ft/s
v0, a, b, u, vmax = 5000.0, 10.0, 0.75, 3.0, 11000.0
sv, z = 11400.0, 12000.0
s_max = ((vmax - v0) / a) ** (1 / b)
s_load = ((v - v0) / a) ** (1 / b)
s_unl = s_max * ((v - v0) / (vmax - v0)) ** (u / b)
print(f'V = 10^6 / {dt:g} = {v:.0f} ft/s')
print(f'maximum effective stress at Vmax = {s_max:.0f} psi')
for name, s in (('loading', s_load), ('unloading', s_unl)):
    pp = sv - s
    print(f'{name:9s}: effective stress {s:6.0f} psi, Pp = {pp:6.0f} psi = {pp / (0.052 * z):5.2f} ppg')
# check: the unloading curve passes through (Vmax, s_max)
print(f'unloading stress at Vmax = {s_max * 1.0 ** (u / b):.0f} psi')
print(f'velocity at 5000 psi on the virgin curve = {v0 + a * 5000.0 ** b:.0f} ft/s')

Output

V = 10^6 / 100 = 10000 ft/s
maximum effective stress at Vmax = 5061 psi
loading  : effective stress   3969 psi, Pp =   7431 psi = 11.91 ppg
unloading: effective stress   2440 psi, Pp =   8960 psi = 14.36 ppg
unloading stress at Vmax = 5061 psi
velocity at 5000 psi on the virgin curve = 10946 ft/s

Assumptions and limitations

• Velocity depends on vertical effective stress alone (and on the history of that stress), through the curves given. Lithology, temperature, cementation and diagenesis are assumed to have no further effect, which fails in deep, cemented or chemically altered rock.
• The virgin and unloading curves are calibrated for the basin and the lithology. The same A, B and U do not transfer between basins.
• Velocity comes from clean shale. Sands and carbonates do not follow the curve.
• The unloading curve begins at the maximum effective stress, found from the onset of velocity reversal, and the same U applies throughout the unloaded interval.
• The overburden is correct. Pore pressure is an overburden minus an effective stress, so any error in the overburden is carried in full.

QC checks

• Effective stress is positive, increases with depth in loading intervals and is less than the overburden.
• Pressure equals hydrostatic in the interval used to calibrate the virgin curve.
• Calculated pressures match measured pressures in sands and mud weights at the same depth, in the unloaded and in the loaded intervals.
• Where unloading is used, Vmax lines up with a visible reversal or flattening of velocity, and the resulting pressure does not exceed the overburden.
• Compare with Eaton's result: the two should agree in the loading interval, and Bowers' should be higher where unloading is applied.

Going Deeper

Bowers developed the method from Gulf of Mexico data to handle overpressure mechanisms other than undercompaction. Its central idea is that effective stress, and not depth, controls velocity, and that velocity does not recover on unloading as it was lost on loading, so a log in an unloaded zone cannot be interpreted with a loading relationship. The method needs the overburden and an effective stress calibration, which makes it more demanding than Eaton's, but it works naturally in terms of stress and can be used with seismic interval velocity. The velocity-density cross-plot, with loading and unloading paths in different directions, is the usual way of recognizing unloading before the curve is chosen. The relations with exponent U make no allowance for a gradual onset; real data often show a transition.

References

1. Bowers, G.L., 1995. Pore pressure estimation from velocity data: accounting for overpressure mechanisms besides undercompaction. SPE Drilling & Completion, 10(2), 89–95.
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

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