A Generalized and Consistent Pressure Drop and Flow Regime Transition Model for Drilling Hydraulics
Authors: Bailey W.J. , Peden J.M. 
Published: December 2007

Summary
Generalized laminar and turbulent pressure loss formulations are presented that are suitable for both annular and circular conduits. In so doing, a new rheological parameter is introduced that provides a consistent link between turbulent and laminar flow calculations and flow regime transition criteria.
The approach presented is independent of the form of the rheological model of choice, it makes no assumptions about conduit geometry, can include the presence of a laminar plug flow region, accommodates drillpipe rotation effects and accounts for the effects of annular eccentricity. Simple, explicit and general formulations for approximating pressure losses over measurement while drilling tools and positive displacement motors are also presented to augment the suite of expressions required for comprehensive hydraulic calculation over the circulating system. Furthermore, procedures for establishing realistic bounds of confidence for laminar pressure loss functions are presented, enabling computation of the degree of certainty attributable to such a calculation. This procedure for establishing confidence intervals is applicable to any function requiring fitted parameters, hence it has a broader application potential than just drilling hydraulics. The model demonstrates good agreement with measured data from studies involving drilling fluids flowing in both circular and annular conduits.

Introduction
Accurate quantification of system losses enhances the validity of calculated equivalent circulating density (ECD) and other drilling parameters and indicators.1 Advantages of more precise ECD estimates include proper maintenance of the integrity of underbalanced conditions;2 prevention of lost circulation;3 an indication of the likelihood of a potential kick4 and improvement of overall drilling efficiency through improved hole cleaning.5
Some non-Newtonian turbulent flow friction factor correlations may be unsuitable for certain applications due to the particular fluid types and to the conduit geometry upon which they were constructed. Consequently, confidence based on quantities calculated using conventional pressure loss functions may be low, and have potentially detrimental implications for primary and incidental calculated quantities. This has an impact in slimhole (high aspect ratio, di/do) drilling operations where predictions made using conventional laminar and turbulent flow functions may be unrealistic6 (annular losses in such systems may be as much as 80% of total system parasitic loss7).
In this article we present a comprehensive suite of general formulations suitable for hydraulic calculations for conventional and nonconventional (slimhole, underbalanced and horizontal) wells. The role of the rheological model is discussed first to establish its proper application within the context of the proposed model. In the Generalized Rheological Parameter section the in-situ flow behavior index is introduced with appropriate solution procedures. In the Regime Transitions section we present transition criteria employing the generalized rheological parameter and a modified generalized Reynolds number (GRN). Generalized laminar flow functions are presented next in separate sections for circular and concentric annular conduits, respectively. The formulations given make no a priori assumptions of the form of the rheological model used, annular aspect ratio and the presence (or absence) of a region of plug flow. Turbulent flow is covered in the Turbulent Flow Friction Factor section where we outline an explicit, fitted, non-Newtonian friction factor correlation using quantities defined in this article. Consideration of the effects of drillstring rotation and annular eccentricity are covered in their respective sections while explicit and general correlations for approximating measurment while drilling (MWD) and positive displacement motor (PDM) tool losses are presented in the Downhole Equipment section. Finally, a mathematical treatment of fitted laminar flow functions is defined, thereby allowing calculation of confidence intervals over pressure drop estimates that are performed. This has beneficial implications for ECD estimates particularly when working within a narrow operating window (or in an underbalanced one) where accurate knowledge of ECD is necessary.

Rheological Models
Pivotal to the drilling hydraulics calculation is the rheological model used to characterize the (typically pseudoplastic) non-Newtonian behavior of the drilling fluid or cement slurry. However, no single rheological model is able to accurately represent the flow behavior of most pseudoplastic and yield-pseudoplastic fluids over the full spectrum of shear rates.8 Only very few rheological models have gained any widespread acceptance within the petroleum industry, the two most common being the Bingham plastic9 and the power law (sometimes referred to as the Ostwald-de Waele model, 10,11 although one similarly constructed was presented independently by Farrow and Lowe12 at the same time). To a lesser extent, the models of Casson,13 Herschel-Bulkley14 and Robertson-Stiff 15,16 are also employed. The popularity of these models can be explained by the tractable form of their respective laminar flow functions17 and the ease with which model parameters may be estimated.3,18 To compensate for these limitations, alternative three- or four-parameter models have been proposed to the industry (along with associated papers evaluating their accuracy and validating their use): for example, the Collins-Graves, 19,20 a "linear annular shear" model21 (essentially a modified Bingham plastic) and an n-order polynomial model.22 In more extensive literature many other models have been presented.23,24 Notable among these is the model of Sisko,25 given by Eq. 1, which has been conclusively shown to furnish the most consistently accurate fluid characterization from a large number of candidate functions26 and should be considered the "default" model in any hydraulic calculation. In this article we use the Sisko model which is given by (1)τ=aγ˙+bγ˙c, where a, b, and c are model parameters requiring fitting. The form of this, and other, models makes the derivation of tractable expressions for pressure drop as a function of flow rate (the flow function) nontrivial or impossible. This possibly explains their limited acceptance and application in the field. The inclusion of this model, however, is realistic since the ready availability of rig site computers now makes the requirement for simple parameter estimation and tractable flow functions redundant and provides a platform conducive to more rigorous analysis.