(83a) A Multivariate Herschel-Bulkley Rheological Model for Nanoparticle-Based Drilling Fluids

Injection fluids used during oil and gas drilling require specific rheological properties, and a bentonite-based drilling fluid enhanced with iron nanoparticles has been analysed to investigate its suitability for high-performance drilling applications. Accordingly, it is necessary to model the shear stress and viscosity in terms of the fluid shear rate, temperature and concentration of iron nanoparticles (Dandekar, 2013). By developing models relating these variables it is possible to examine how the viscoplastic fluid behaves under different drilling conditions and if it can reliably ensure process performance. Under steady shear, flow behaviour will vary with concentration and temperature (Andertova & Rieger, 2009), and careful tuning of rheological characteristics is critical to the design of the process. The currently prevalent tri-parametric model relating shear stress and shear rate is the Herschel-Bulkley model (Kelessidis et al., 2006; Kelessidis & Maglione, 2008; Pouyafar & Sadough, 2013) which combines the effects of power-law and Bingham plastic behaviour in a fluid. The parameters estimated from experimental campaigns are temperature- and concentration- dependent, and literature models are usually condition-dependent. Therefore, it is important to develop a unified model including all three independent variables (shear rate, temperature and iron nanoparticle concentration) in order to describe in detail the rheological behaviour of the injection fluid, which is of critical importance during drilling operations.

The literature has been extensively surveyed to identify the most widespread and suitable models of drilling fluids (Nguyen & Boger, 1992; Balmforth et al., 2014). A Grace M3600 Fann type rotational viscometer has been used to measure shear stress and viscosity at three temperatures and several iron nanoparticle concentrations (Yan & James, 1995). Multivariate nonlinear least squares regression (Berge, 1993) has been employed in order to determine the most accurate model (Puxty et al., 2005) for shear stress and viscosity as functions of shear rate, temperature and additive concentration; the multi-parametric estimation problem has been subsequently solved using the novel experimental data, in order to compute the optimal set of parameters which minimise the sum of squared errors (SSE) and maximise the respective coefficient of determination, R²(Graybill & Iyer, 1994). Ensuring that a global optimum is achieved is of particular importance, because the strongly nonlinear multivariate expression may induce trapping in various local minima; the multivariate nonlinear regression has thus been performed using a systematic strategy and several starting parameter sets to confirm optimality. The multivariate and multi-parametric nonlinear estimation models have been plotted against the experimental data, the standard error for shear stress and viscosity at each combination of independent variables has been computed, and error bars have been obtained to illustrate uncertainty.

Nonlinear model plots have been analysed to show that injection fluids are shear thinning fluids with yield stress which follow the Herschel-Bulkley fluid model for shear stress and the power law model for viscosity (Abu-Jdayil & Ghannam, 2014). The multiplicative rule has been found to provide reliable estimates, combining the Herschel-Bulkley form with an Arrhenius model for temperature and a nonlinear model for concentration dependence. Higher temperatures cause increased standard error for shear stress and viscosity, indicating that temperature affects the structure and rheology of the fluid significantly, increasing the expected prediction uncertainty. Further investigation of concentration effects for various additives has also been conducted, indicating that the model already provides reliable fluid behaviour prediction via quantitative estimation of shear stress and viscosity, in a wide spectrum of experimental conditions relevant to drilling.

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