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.


The drilling operations where the drilling fluid pressures in the borehole are intentionally maintained to be less than the pore pressure in the formation rock in the open-hole section is called Underbalanced drilling (UBD). The low borehole pressures are achieved by using lightened drilling fluids such as air, gas, foam, aerated oil, and aerated mud. The aerated mud is often the candidate for drilling permeable zones to handle significant formation fluid influx to the wellbore during UBD.1,2

Good designs are the key to the successful aerated mud drilling operations. Severe wellbore damages and failures can result from poor designs. Pressures in the annular space during drilling and after circulation break play vitally important rules in controlling borehole instability during aerated mud drilling, especially in inclined holes. As aerated mud is a compressible fluid, special care needs to be taken in hydraulics calculations. This is mainly because the frictional and hydrostatic pressure components influence each other through the pressure-dependent fluid density. Sophisticated numerical simulators are required to perform accurate computations. Both steady state flow and transient flow simulators are available in drilling industry for aerated mud drilling hydraulics calculations.3–7 Unfortunately, the results from these simulators are frequently conflicting8 due to assumptions that were made in mathematical formulations. It is, therefore, highly desirable to develop a simple and reliable hydraulics equation.

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