Model for estimating the medium cross-sectional area and the pressure surface of a deformed pipe section with wellbore walls

Abstract

. This article addresses a relevant scientific and applied problem in oil and gas engineering: the development of mathematical tools for the precise determination of the geometric parameters of contact between the drill string and the borehole walls. During the construction of deep deviated and horizontal wells, the drill string, under the action of significant axial and bending loads, undergoes deformations that lead to its local embedding in the filter cake and rock formation. Traditional approaches, based on point contact modelling or approximate empirical relationships, do not allow for an adequate assessment of the adhesive forces and viscous resistance, which significantly influence energy consumption and the efficiency of axial force transmission to the bit. This paper proposes an innovative geometric model in which the local contact zone of the deflected section of the drill bit is identified as part of the surface of a torus cut out by a circular cylinder, which models the borehole. This approach allowed the complex curvature of both surfaces to be taken into account. Using methods of parameterisation and analysis of singular points, the authors established analytical relationships for determining the coordinates of the intersection boundary and formulated integral equations for calculating the area of the contact surface and the area of the mid-section. Since the obtained integrals are not expressed in elementary functions, the Simpson numerical method was applied to solve them, implemented in Python and Maple code. Numerical modelling carried out for various pipe sections of coiled tubing (CT), drill pipe (DP) and weighted drill pipe (WDP) sections revealed important patterns: it was established that the contact area exceeds the area of the mid-section cross-section by 2–3 orders of magnitude. It has been demonstrated that as the insertion depth increases fivefold, the cross-sectional area increases by approximately ninefold, indicating a critical increase in hydrodynamic resistance. For cases of small deformations, the authors have proposed simplified parabolic formulas, which facilitate practical engineering calculations. The results of the study form the basis for improving methods for calculating the total drag forces on the drill string and for designing wellbore profiles with large zenith angles, thereby minimising the risk of tool snagging.