Accurate prediction of shut-in and flowing bottom hole pressures in inclined holes presents a challenge in aerated mud drilling. It is highly desirable to develop a simple and accurate hydraulics equation for this purpose. This paper fills the gap.

A closed form hydraulics equation was developed in this study on the basis of recent experiments on multiphase flow in an inclined well model. The newly developed hydraulics equation is a 4-phase model considering injected liquid, injected gas, formation fluid influx, and cuttings entrained at bottom hole during drilling.

The equation was first calibrated using data measured from a full-scale research borehole. It was then tested with two field cases covering a deep (9831 ft) vertical well and a shallow (697 ft vertical depth) horizontal well drilled with aerated muds. The results show 8.66% and 0.81% error of the equation in bottomhole pressure prediction for the deep and shallow well, respectively. The equation was also compared with two commercial software packages (S1 and S2) using measured bottombhole pressure from another well. It indicates that the flowing bottomhole pressure calculated by the equation is comparable to that by S1 and much more accurate than that by S2. Sensitivity analyses with the equation show that gas injection rate affects "static" pressure more than flowing pressure in the annular space in the tested data range. The new model indicates very high Equivalent Circulating Density (ECD) and low Equivalent Mud Weight (EMW) in shallow depth near surface. ECD and EMW versus depth in a 6.37"×3.5" annulus was generated for field applications.

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