Novel Equation for the Prediction of Rheological Parameters of Drilling Fluids in an Annulus M. Nasiri and S. N. Ashrafizadeh* Research Lab for AdVanced Separation Processes, Department of Chemical Engineering, Iran UniVersity of Science and Technology, Narmak, Tehran 16846, Iran Several well-known correlations such as the Bingham-plastic, power-law, and Herschel-Bulkley models have been used so far to determine the rheological parameters of drilling fluids. For some particular fluids, however, even a three-parameter model such as Herschel-Bulkley does not exhibit appropriate behavior. On the other hand, determination of the rheological parameters by numerical methods such as nonlinear regression may provide meaningless values, i.e. negative yield stresses. This is particularly notable in determination of yield stress, which identifies the capacity of the drilling fluid to carry the cuttings. In this work, a new equation has been developed which is capable of determining the rheological parameters and, more particularly, the yield stress of drilling fluids. It is demonstrated that the developed correlation improves the prediction of the rheological parameters of the fluids by including a logarithmic term. The velocity profiles and pressure drop values obtained for several drilling fluids in an annulus geometry exhibit the suitability of this novel equation in comparison with the previously mentioned equations. 1. Introduction Drilling fluids typically used in drilling gas/oil wells are emulsion-suspension systems to which various viscosifiers and surface active reagents may be added to enhance the fluid’s performance. The flow characteristics of such a suspension, which is essentially regarded as a non-Newtonian fluid, are largely governed by the chemical properties of the colloidal bentonite clay particles that form a network with certain strength. 1 Three major categories of non-Newtonian fluids are basically recognized, namely, time-independent, time-dependent, and viscoelastic. 2 The time-independent category has received a substantial degree of attention in comparison with the other two categories. A large majority of drilling fluids falls into this category. 3 In time-dependent flow behavior, the apparent viscosity at a fixed shear rate does not remain constant but varies to some maximum or minimum with the duration of shear. If the apparent viscosity decreased with flow time, the fluid is known to be thixotropic. 1 It is generally accepted that drilling fluids can be typified by the Bingham-plastic model. 4,5 The Bingham-plastic model is defined by the relationship of eq 1: The Bingham-plastic model of flow differs most notably from a Newtonian fluid by the presence of a yield stress. A Bingham- plastic fluid will not flow until the applied shear exceeds a minimum value that is known as the yield stress. Once the yield stress exceeds the mentioned minimum, changes in shear stress are proportional to changes in shear rate and the constant of proportionality is called the plastic viscosity. As it will be discussed, the Bingham-plastic model usually does not ac- curately represent drilling fluids at low shear rates. The power-law model is defined by the relationship of eq 2: 5-7 The power-law model is frequently more convenient than the Bingham-plastic model. The power-law model demonstrates the behavior of a drilling fluid at low shear rates more accurately. However, this model does not include a yield stress and therefore can give poor results at extremely low shear rates. Therefore, either of the two models, i.e. the Bingham-plastic and power- law models, is inefficient at low shear rates. A typical drilling fluid exhibits behavior intermediate between the Bingham-plastic and the power-law models. The Herschel-Bulkley model eq 3, which is a hybrid of the Bingham-plastic and power-law models, includes three param- eters. 8 This model is in fact a power-law model with a yield stress. The model yields mathematical expressions relating flow rate to pressure drop that are not readily solved analytically but can be solved using nonlinear regression methods. The three-parameter Herschel-Bulkley model provides an appropriate relation between the shear stress and shear rate. This model is also in much stronger agreement with the rheological data of the fluid; especially at low shear rates. The latter is particularly important to horizontal directional drilling (HDD) where the flow regime is laminar and thus has a low shear rate. More complex four-parameter or even five-parameter models have also been proposed by Shulman, 9 Mnatsakanov et al., 10 and Maglione et al. 11 Detailed descriptions of the various rheological models have been proposed, and derivations of the appropriate flow equations have been given by Bird et al. 12 and Maglione et al. 13 The latter models are more accurate in predicting the behavior of drilling fluids than the two-parameter models; which are widely accepted at present. However, there is not wide acceptance and wide application of the more complex models because of the difficulty in finding analytical solutions for the differential equations of motion and because of the complexity of the calculations for the derivation of the appropriate hydraulic parameters such as Reynolds number, flow velocity profiles, circular and annular pressure drops, and criteria for transition from laminar to turbulent flow. 14 * To whom correspondence should be addressed. E-mail: ashrafi@ iust.ac.ir. Tel.: +98 (21) 77240402. Fax: +98 (21) 77240309. τ ) τ 0 + ηγ ˙ (1) τ ) C'γ ˙ n (2) τ ) τ 0 + kγ ˙ n (3) Ind. Eng. Chem. Res. 2010, 49, 3374–3385 3374 10.1021/ie9009233  2010 American Chemical Society Published on Web 02/22/2010