A Generic Model for Calculation of Frictional Losses in Pipe and Annular Flows
Authors: Johansen S.T. , Skalle P. , Sveen J. 
Published: March 2010

Abstract
A theoretical foundation of a new simulator for hydraulic calculations of mud transport is presented. It is demonstrated that it is possible to design a simulator based on theoretical concepts taken from fluid dynamics. The theory is based on a simple turbulence model, which is calibrated against direct simulations of turbulent flows, and a simple expression for suppression of turbulence at low wall strain-rates. The model predicts turbulent non-Newtonian flows, provided rheology data is available. This is an attractive feature of the model as no general non- Newtonian flow correlations are available in the literature.
The output from the model is wall shear stresses and pressure drops, while the most important input is the fluid data, and in particular the fluid rheology and its dependency on pressure and temperature. Due to instabilities caused by the drill-string and variable surface roughness at the contacting walls, it is assumed that the flow is unstable or quasi-turbulent, even for low Reynolds numbers well below 1,000. Also, effects of rotation and heat transport are treated in the same fundamental manner. The effects of rotation are included by solving for the tangential velocity in addition to the axial velocity. It is shown that rotation may have a profound impact on the pressure drop.
The suggested methodology is expected to increase the precision of predictions and reduce the costs of lab testing and traditional curve fitting to analytical models. The model is verified against experimental data. For most situations, we obtained good agreement with data, as well as for laminar/turbulent transitional flows.
Finally, the paper indicates that traditional Fann readings, using 4 points, are inadequate for an accurate rheology determination. In order to improve the hydraulic model, more focus on rheology data is needed.

Introduction
An important aspect of hydraulic optimization is an accurate estimation of the equivalent circulating density (ECD), especially in long, narrow wells. For this purpose, several simulators for hydraulic calculations of drilling wells have been developed(1).
One frequent problem in such simulations is the quality of rheological data that affects the selection of the rheological model. A common approach is to let the user select a rheological model from a set of predefined rheologies(2, 3). Another approach is to curve fit the data to a best fit with a selected rheology model(4). Denis and Guillot(5) found that the Fann viscometer systematically seems to overpredict pressure losses. They tested one pipe viscometer and two rotational viscometers and recommended applying a high quality viscometer instead of the inaccurate Fann viscometer. In another work, Efaghi et al.(6) had access to the training well at Louisiana State University and recorded pressure losses of mud in a 27/8 in., 3,000 ft. deep tubing, comparing Bingham and Power law rheological models. Laminar pressure losses were not predicted accurately since the viscometer shear rate was not representative of the actual laminar flow in the tubing.
Valuable data of turbulent flow of non-Newtonian fluids in rough pipes was recorded by Szilas, Bobok, and Navratil(7).