Bowers and the Unloading Limb
Learning objectives
- Write Bowers' virgin curve: velocity is the mudline value plus a power law in effective stress
- Anchor the mudline velocity at 5000 ft/s, where effective stress is zero
- Follow the unloading limb, a flatter path the rock takes when it is unloaded rather than loaded
- Explain why unloading makes a given low velocity mean a much lower effective stress, so Eaton under-predicts and Bowers does not
Velocity Remembers Its History
Eaton assumes velocity is a one-to-one function of effective stress: know one, know the other. Bowers' 1995 method breaks that assumption in the one case where it fails, and the failure is instructive. His virgin curve, the path a rock follows while it is being loaded for the first time, is a power law: , with the velocity climbing from a mudline value of about 5000 ft/s at zero effective stress. That is the one firm anchor here, the speed of unconsolidated sediment at the seafloor; the coefficients and are basin-calibrated fits, and Gulf of Mexico values are representative but not universal, so a real study fits them to local data rather than borrowing them. On the virgin curve, Eaton and Bowers agree.
Now unload the rock. If overpressure is generated not by trapping the load but by adding fluid, gas cracking, aquathermal expansion, the effective stress falls from a value it had already reached. The rock does not retrace the virgin curve on the way down; it follows a flatter unloading limb, because the grain contacts that stiffened under the past maximum stress do not soften back symmetrically. Velocity comes down only a little as effective stress drops a lot. The consequence is the whole point: at a given observed velocity, the virgin curve reads one effective stress, but the unloading limb reads a much lower one, and a much lower effective stress means a much higher pore pressure. Eaton, which only knows the virgin curve, reads the observed velocity off it and reports the higher, safer, wrong effective stress. Bowers, recognizing the unloading limb, reports the true one.
Reading the Trend to Choose the Method
How do you know a rock has been unloaded? The velocity reversal of the last section: a zone where velocity actually decreases with depth. Loading can only slow velocity's rise, never reverse it, so a reversal is the signature of an unloading mechanism, and it is a signal to switch from Eaton to Bowers before the pressure is dangerously under-called. The two methods together form the field standard: fit a virgin curve to the normally compacted section, watch for a reversal, and where you see one, invoke the unloading limb with its extra parameter, the unloading exponent, that measures how flat the return path is. This is the deepest reason pore-pressure prediction is a skilled craft and not a formula: the same velocity number means two different pressures depending on the rock's history, and only the shape of the trend, read by a geoscientist, tells you which. The next section leaves the log behind for the real-time signals at the bit.
References
- Bowers, G. L. (1995). Pore pressure estimation from velocity data: Accounting for overpressure mechanisms besides undercompaction. SPE Drilling & Completion, 10(2), 89-95.
- Bowers, G. L. (2002). Detecting and predicting abnormal pore pressure. The Leading Edge, 21(2), 174-177.
- Zoback, M. D. (2007). Reservoir Geomechanics. Cambridge University Press.