A closed form hydraulics equation was derived in this study to predict bottom hole pressures during drilling ahead and circulation break when aerated liquids are used for drilling inclined boreholes. The analytical model was developed on the basis of bubbly flow. It considers effects of injected liquid, injected gas, formation fluid influx, and cuttings entrained at bottom hole on the bottom hole pressure. Different rheological models are used for aerated liquids with and without foaming agents.

The new equation together with selected rheological models was tested with data from two field-drilling cases. In the first case, a shallow (697 ft vertical depth) horizontal well was drilled with aerated mud. The difference between the measured and equation-calculated bottom hole pressures is less than 1%. In the second case, a vertical well section was drilled with stable foam from 860 m to 884 m. The difference between the measured and equation-calculated bottom hole pressures is less than 2%.


Aerated mud and foam have been utilized as drilling fluids in recent years for drilling low-pressure zones to reduce lost circulation and formation damage.1,2 Good hydraulics designs are the key to the successful aerated mud and foam drilling operations. Severe wellbore damages and failures can result from inaccurate borehole pressure predictions. Pressures in the annular space during drilling and after circulation break play vitally important rules in controlling borehole instability during aerated mud and foam drilling, especially in inclined holes.

As aerated mud and foam are compressible fluids, frictional and hydrostatic pressure components influence each other through the pressure-dependent fluid density. Sophisticated numerical simulators are required to perform accurate computations of the borehole pressures. 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 conflicting8,9 due to assumptions that were made in mathematical formulations. It is, therefore, highly desirable to develop a simple and reliable hydraulics equation.

Assuming bubbly flow of liquid-gas-solid mixture, a closed form hydraulics equation, coupling the frictional and hydrostatic pressure components, was developed in this study. The equation has been tested with data from field cases covering a horizontal well drilled with aerated mud and a vertical well drilled with stable foam. Comparison of the calculated and measured bottom hole pressures indicates excellent accuracy of the hydraulics equation. To demonstrate the applications of the newly developed hydraulics equation, Equivalent Circulating Density (ECD) and Equivalent Mud Weight (EMW) versus depth in a 6.37"x3.5" annulus have been generated for aerated mud and foam drilling operations.


The following assumptions were made in the model development:

  • bubbly flow prevails in the annular space; and

  • no slipping effect between phases.

Lage and Time's work5 indicates that bubbly flow exists when the gas/liquid ratio is less than unity. It also shows that dispersed bubble flow occurs for superficial liquid velocities greater than 6 ft/sec and the superficial gas velocities as high as 12 ft/sec.

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