In the mathematical modeling of bit penetration rate for tri-cone roller bits in permeable formations, virtually all of the current techniques assume that the differential pressure between the bottom-hole wellbore pressure and the formation is a “static” value. This work shows that the appropriate differential pressure is a dynamic quantity, because for overbalanced drilling, fluid filtrate from the wellbore requires a finite time to flow into the formation, producing a changing pressure gradient ahead of the bit. Moreover, this dynamic gradient is directly dependent upon the rate of drill bit penetration, which is in turn dependent upon the dynamic gradient itself. Accordingly, coupled penetration rate and dynamic gradient equations must be solved, which frequently result in the prediction of higher drilling penetration rates than when the static gradient is used. The appropriate dynamic differential pressure equations are developed and applied to an example drilling situation. It is shown that with water-based drilling fluids, for rock with permeability greater than a few microdarcies at virtually all penetration rates, and for penetration rates less than 3 m/h (9.84 ft/h) at permeabilities greater than 1 μd (microdarcy), the dynamic differential pressure is significantly less than the static differential pressure. Accordingly, using the conventional static differential pressure results in the prediction of penetration rates that are much too low. Moreover, using measured penetration rates from the field, the conventional approach yields predicted in-situ rock strength that is much too high.

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