Water- and oil-based drilling fluids are polydisperse non-Newtonian systems, the stress state of which is adequately described by the Herschel – Bulkley rheological model. This study hypothesizes that cuttings transport efficiency can be improved by selecting the most effective combination of the three parameters of the rheological model – yield stress, consistency index, and flow behavior index – when designing drilling fluid properties. The effective parameter combination of the Herschel – Bulkley model for achieving a uniform velocity profile was determined using correlation and regression analysis methods as well as machine learning techniques. The computational part of the work was performed in the Wolfram Mathematica symbolic calculation package. Deterministic regions of the dependence of the velocity profile uniformity index on the rheological coefficients were identified. For practical engineering calculations, a linear mathematical model was constructed to represent the relationship between the modified excess coefficient and the parameters of the three-parameter Herschel – Bulkley rheological equation. The proposed methodology can be recommended for designing new drilling fluid systems and testing existing ones under given wellbore cleaning conditions.

Modern drilling fluids are non-linear viscous media with an initial shear stress. In classical scientific works on hydromechanical modeling of drilling fluids motion in pipes and annular channels the Shvedov – Bingham approximation and Ostwald – de Waale power-law model were used, which did not fully account for behavior of technological fluids in a wide range of shear rates. This article presents a numerical solution for a mathematical model of drilling fluid motion of the three-parameter Herschel – Bulkley rheological model in the annular space of the well. The Herschel – Bulkley model in the rheological equation takes into account the presence of initial shear stress and a tendency for viscosity to change with shear rate, which distinguishes it from the Ostwald – de Waale and Shvedov – Bingham models. The target function in solving the equation of motion is the velocity distribution in the radial direction of the upward flow of the flushing fluid. The analysis of obtained solution is based on the theory of velocity profile influence on quality of cuttings removal during wellbore cleaning. Due to peculiarities of mathematical statement of the task, which supposes necessity of differential equation of motion solution, Wolfram Mathematica computational software has been used as a calculation tool. The analysis of numerical solution allowed to draw conclusions about the possibility of its application in evaluation of velocity profile when drilling fluid moves in annular space of the well. The possibility for application of modified excess coefficient as a relative quantitative parameter for evaluation of velocity profile uniformity was substantiated.