About Discretization of the Horizontal Drill-String Model with Uncertainty on the Frictional Force by the Lumped Parameters Method

Abstract

The dynamics of the horizontal drill string considering frictional forces between the column and the borehole is investigated. A model of longitudinal vibrations of columns proposed in T.G. Ritto et al. is considered. The investigated model has a nonlinear character, and is modelled by the lumped parameters method. The drill-string is discretized with 100 nodes of lumped masses. Displacement of the bit, bit speed and also the ratio between the output power (obtained from the bit-rock interaction) and the input power is defined by the authors. The frictional force between the column and the borehole is relevant and uncertain. The frictional coefficient is modelled as a random field with exponential autocorrelation function. The obtained results are qualitatively and quantitatively consistent with the results of Ritto, where the dynamic model calculations are carried out by the finite element method.

Modelling of Horizontal Drill String Motion by the Lumped-Parameter Method

“…The verification of the obtained results with the results of works [1] and [20] was carried out. In [1], the numerical modelling of the drill string motion was realized by the finite element method.…”

“…The dashed black line shows the results of T.G. Ritto [1], solid red line is the results of L. Khajiyeva, A. Sergaliyev [20], dotted black one is the results of this work.…”

“…Ritto et al in the work [1] by the finite element method. Here, the authors of the work, as in [20], use LPM, which is an effective method for the numerical analysis of such dynamical systems. Due to the inhomogeneity of the drill string loading, the mathematical model is written in accordance with the drilling equipment loading scheme (Figure 1) as follows:…”

“…where p in (t) is the input power, p out (t) is the output power. The number of the drill string splitting nodes was taken, as in [20], equal to N = 101. The research results, which are longitudinal displacement of the drill string at an interval of 10 s, are shown in Figure 2.…”

“…In [1], the numerical modelling of the drill string motion was realized by the finite element method. The authors of [20] used LPM, and the numerical solution of the mathematical model was found in the symbolic mathematics package Wolfram Mathematica (WM). Here, numerical modelling was conducted in C++.…”