Abstract

In order to determine laminar friction losses of non-Newtonian drilling fluids it is standard practice to utilise a simple rheological model with parameter estimation via explicit solution of the required number of simultaneous equations. This limits the choice of available models and, together with the fitting procedure, frequently results in poor fits. Numerous alternative models have been proposed, but their worth has never been clearly demonstrated. Here we consider the rheological models to form the deterministic part of a stochastic model hence acknowledging that observed data arise from randomness due to the system. Parameter estimates are obtained by nonlinear least squares using data readings from eight standard Fann viscometer settings. Results are presented for 20 rheological models on 414 Fann viscometer data sets that encompass a variety of different drilling fluids. The goodness of fit of the models is assessed by comparing boxplots of the resulting residual mean square values. Several of the alternative models are shown to be consistently capable of excellent shear stress prediction over all of the shear rate range expected during drilling operations. The use of statistical methodology allows us to go beyond mere data relationship estimation: it permits the calculation of confidence intervals and tests of significance of functions of the model parameters, for example equivalent circulating density. The validity of such calculations relies on the extent to which the behaviour of the nonlinear least squares estimates approximates the asymptotic properties of being unbiased, normally distributed and minimum variance estimators. The estimation behaviour of the ‘best’ fitting models is analysed and a reparameterised form of the Sisko model is identified as having reasonable behaviour. Thus, despite the small data size, when using this model form the subsequent estimates, confidence intervals and tests of significance of functions of the model parameters can be confidently reported.

Introduction

Drilling hydraulic calculations require the use of a rheological model to characterise the non-Newtonian behaviour of drilling fluids. It is standard practice in the field to use Bingham Plastic1,2 and/or Power Law (Ostwald-de Waele)3,4 models in drilling hydraulics calculations. This is due to the simplicity of their resultant flow equations and the ease by which model parameters may be estimated by explicit solution of the required number of simultaneous equations. However, the resulting fits are poor and rarely capable of accurate fluid characterisation over the full range of shear rates encountered during drilling operations. Typically, the Power Law model is considered to provide better characterisation at lower shear rates, whilst Bingham Plastic provides improved precision at higher shear rates. This belief has resulted in the common practice of applying the Power Law over the annulus and Bingham Plastic in the drillpipe and bottom hole assembly (BHA). This custom results in a discontinuity in the application of rheological models over a continuous hydraulic system. Also, it encourages the use of the Power Law in the annular region around the BHA when it may not be appropriate due to high local fluid shearing. Hence, this somewhat arbitrary application can have consequences for parasitic pressure loss prediction (bit optimisation) and equivalent circulating density (ECD) estimation.5

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