# A New Model for Laminar, Transitional, and Turbulent Flow of Drilling Muds

- Authors
- T.D. Reed (Conoco Inc.) | A.A. Pilehvari (U. of Tulsa)
- DOI
- https://doi.org/10.2118/25456-MS
- Document ID
- SPE-25456-MS
- Publisher
- Society of Petroleum Engineers
- Source
- SPE Production Operations Symposium, 21-23 March, Oklahoma City, Oklahoma
- Publication Date
- 1993

- Document Type
- Conference Paper
- Language
- English
- ISBN
- 978-1-55563-490-2
- Copyright
- 1993. Society of Petroleum Engineers
- Disciplines
- 1.4.4 Drill string dynamics, 1.11.5 Drilling Hydraulics, 1.10 Drilling Equipment, 1.11 Drilling Fluids and Materials, 4.2 Pipelines, Flowlines and Risers, 1.11.2 Drilling Fluid Selection and Formulation (Chemistry, Properties), 1.6 Drilling Operations
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Abstract

The concept of an "Effective" diameter is introduced for the flow of drilling muds through annuli. This new diameter accounts for both annular geometry and the effects of a non-Newtonian fluid. It provides the link between Newtonian pipe flow and non-Newtonian flow through concentric annuli. The method is valid in any flow regime and can be used to determine whether a non-Newtonian flow is laminar, transitional, or turbulent. An analytical procedure is developed for computing frictional pressure gradients in all three flow regimes. The analysis also quantifies how flow transition is delayed by increasing the yield stress of a fluid. In addition, it is shown that transition in an annulus is delayed to higher pump rates as the ratio of inner to outer diameter increases. Furthermore, the method accounts for wall roughness and its affects on transitional and turbulent pressure gradients for non-Newtonian flow through pipes and concentric annuli. Finally, the method runs on a 386 PC in only a few seconds.

INTRODUCTION

The standard API methods for drilling hydraulics assume either a Power Law or a Bingham Plastic rheology model. In reality, most drilling muds correspond much more closely to the Modified Power Law or Herschel-Bulkley rheological model. This distinction is particularly important for the annular geometries typical of normal drilling conditions where shear rates are usually low and the Power Law underestimates and Bingham Plastic model overestimates frictional pressure drops. This paper also shows that these two classical models, respectively, underestimate and overestimate pump rates required for transition from laminar to turbulent flow. Although there have been a number of papers published on the laminar flow of yield-pseudoplastics through annuli,^{1-9} none claim to be uniformly valid for laminar, transitional, and turbulent flow. This can be accomplished by introducing the concept of an "Effective" diameter which accounts for both annular geometry and the effects of a non-Newtonian fluid. This diameter also enables the inclusion of the effects of wall roughness on frictional pressure gradients and the process of flow transition. Furthermore, the resulting model can be used to determine whether flow of a non-Newtonian fluid through a pipe or concentric annulus is laminar, transitional, or fully turbulent.

A derivation of the model is presented in the Appendix. Some additional background and results from the model are given in the following discussions.

NEWTONIAN FLOW

**Relationship Between Pipe and Annular Flows.**

A considerable number of "equivalent" diameters have been proposed over the years for flow through conduits other than circular pipes.^{10-12} The purpose of defining such a diameter is to introduce definitions of friction factor and Reynolds number that will enable application of the well-known relations for pipe flow to other geometries. In particular, the objective is to be able to calculate what the frictional pressure drop will be for a given fluid at a particular flow rate.

As shown in the Appendix, the friction factor for a concentric annulus should be based on the Hydraulic Diameter, D_{hy}, which is simply the difference between the inner and outer diameters. The Reynolds number should be based on an Equivalent Diameter equal to the square of Lamb's Diameter, D_{L}^{2}, divided by D_{hy}, see Appendix Eqs. A-12, 13 and 14. When the friction factor and Reynolds number are defined in this manner, the classical relations for Newtonian pipe low, e.g., the Moody Diagram, can be applied directly to concentric annuli. In fact, Jones and Leung^{13} have proven that these definitions also apply to fully-turbulent flow through an annulus. This was demonstrated by showing that a variety of experimental data for concentric annuli agree with the Colebrook Equation for turbulent pipe flow, which is equivalent to the fully turbulent and hydraulically rough portions of the Moody Diagram.

Relationship Between Pipe and Annular Flows.

A considerable number of "equivalent" diameters have been proposed over the years for flow through conduits other than circular pipes.^{10-12} The purpose of defining such a diameter is to introduce definitions of friction factor and Reynolds number that will enable application of the well-known relations for pipe flow to other geometries. In particular, the objective is to be able to calculate what the frictional pressure drop will be for a given fluid at a particular flow rate.

As shown in the Appendix, the friction factor for a concentric annulus should be based on the Hydraulic Diameter, D_{hy}, which is simply the difference between the inner and outer diameters. The Reynolds number should be based on an Equivalent Diameter equal to the square of Lamb's Diameter, D_{L}^{2}, divided by D_{hy}, see Appendix Eqs. A-12, 13 and 14. When the friction factor and Reynolds number are defined in this manner, the classical relations for Newtonian pipe low, e.g., the Moody Diagram, can be applied directly to concentric annuli. In fact, Jones and Leung^{13} have proven that these definitions also apply to fully-turbulent flow through an annulus. This was demonstrated by showing that a variety of experimental data for concentric annuli agree with the Colebrook Equation for turbulent pipe flow, which is equivalent to the fully turbulent and hydraulically rough portions of the Moody Diagram.

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