An Analytical Solution for Aerated Mud and Foam Drilling Hydraulics in Deviated Holes
Authors: Sun K. , Guo B. , Ghalambor A. 
Published: March 2010

Abstract
A closed form hydraulics equation was derived in this study to predict bottomhole pressure during drilling in inclined boreholes. The analytical model was developed on the basis of bubbly flow. It considers the effects of injected liquid, injected gas, formation fluid influx, and drill cuttings on bottomhole pressure. Different friction factor correlations are used for aerated liquids with and without foaming agents.
The equation was tested with data from two field-drilling cases involving a shallow (212 m vertical depth) horizontal well drilled with aerated mud and a vertical well drilled with stable foam from 860 m to 884 m. The differences between the measured and equation-calculated bottomhole pressures are less than 2% in both cases. This work provides drilling engineers with an easy-to-use tool for designing aerated fluid drilling hydraulics.

Introduction
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). Good hydraulics designs are key to successful aerated mud and foam drilling operations. Severe wellbore damage and failure can result from inaccurate borehole pressure predictions. Pressure in the annulus during drilling and after a circulation break plays a vitally important role 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 pressure-dependent fluid density. Sophisticated numerical simulators are essential to perform accurate predictions of the borehole pressure. Both steady state flow and transient flow simulators are available in the drilling industry for aerated mud drilling hydraulics calculations(2–5). Unfortunately, the results from these simulators are frequently conflicting due to assumptions that were made in mathematical formulations. It is therefore highly desirable to develop a simple and reliable hydraulics equation for the purpose.
Assuming bubbly flow of a liquid-gas-solid mixture, a closedform hydraulics equation, coupling the frictional and hydrostatic pressure components, was developed in this study. The equation has been tested with data from field cases which include a horizontal well drilled with aerated mud and a vertical well drilled with stable foam. Comparison of the calculated and measured bottomhole pressures expresses the accuracy of the hydraulics equation. To demonstrate the application of the newly developed hydraulics equation, Equivalent Circulating Density (ECD) and Equivalent Mud Specific Gravity (EMW) vs. depth in a 16.18 cm × 8.89 cm (6.37 in x?3.5 in) annulus have been generated for aerated mud and foam drilling operations.

Hydraulics Model
Assumptions
The following assumptions were made in the model development:Bubbly flow prevails in the annular space; and,There is no slipping effect between phases.
Lage and Time's work(3) 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 1.85 m/sec and the superficial gas velocities as high as 3.7 m/sec. The research work by Sunthankar et al.(6) on multiphase flow in an inclined well model confirmed the results of Lage and Time in that bubbly flow exists in the annular space when the in situ gas-liquid ratio is less than unity. It can be shown that the in situ gas-liquid ratio is greater than 1 only in a small portion of borehole sections (near surface) in aerated mud and foa