# Mathematical Model and Experimental Investigation of Bit-Bounce in Horizontal Oil Well Drillstring

Research Article - Mechanical Engineering

## Abstract

The bit-bounce is detrimental to the service life of drillstring and bit. This study highlights the influence of axial excitation and torsional excitation on the bit-bounce in horizontal well. Because of the complicated surroundings of downhole, Hamilton principle, a kind of energy method, is employed in this paper to derive the motion equation of the drillstring, which is discreted through finite element method. The impact–friction between the drillstring and borehole wall, and bit–rock and fluid–structure interaction are also modeled in this paper. The research results show that both axial excitation and torsional excitation greatly affect the dynamic behavior of the drillstring. Both the amplitude and the intensity of bit-bounce reduce with the development of the axial excitation and torsional excitation. And the test results are in agreement with the numerical simulation. The results of both have been mutually verified.

## Keywords

Coupled vibrations Hamilton principle Drillstring Qualitative analysis Horizontal well

## List of Symbols

θx

Angular displacement along x direction

θy

Angular displacement along y direction

θz

Angular displacement along z direction

u

Displacement along x direction

v

Displacement along y direction

w

Displacement along z direction

U

Strain energy

T

Kinetic energy

W

Work done by the external forces

Fg

Work done by gravity

Fif

Work done by impact–friction

Ff

Work done by fluid–structure interaction

WBR

Work done by bit–rock interaction

n

Number of elements

le

Length of element

ξ

Local coordinate

qe

Displacement vector of element

$${\dot{\mathbf{q}}}_{{\mathbf{e}}}$$

The first derivative with time

$${\mathbf{N}}_{{\mathbf{u}}}^{\prime }$$

The first derivation of shape function with ξ

δue

Variation of the displacement ue

r

rgap

Gap between the drillstring and borehole wall

Re

Rh

kim

Impact factor

$$\varPsi_{\text{im}} \left( r \right)$$

Determination coefficient of the drillstring

μ

Coefficient of kinetic friction between drillstring and borehole wall

Fn

Contact force in normal direction

xn

Coordinate at node n

Tc

Cutting torque

Wc

Cutting weight

a

ɛ

Intrinsic specific energy

ζ

A number characterizing the orientation of the cutting force

σ

d

Instantaneous cutting depth

Bn

Nu

Shape function of u

$${\mathbf{N}}_{{\theta_{x} }}$$

Shape function of θx

Fin

Normal component of hydrodynamic forces

Fit

Tangential component of hydrodynamic forces

FA

Lateral non-viscous hydrodynamic force

FN

Frictional viscous hydrodynamic force

Mf

Mass per unit length

pi

Pressure of internal flow

pe

Pressure of external flow

Ui

Flow speed of internal flow

Uex

Flow speed of external flow

ρf

Density of drilling mud

FL

Viscous force

fz

Viscous force

Cf

Viscosity damping coefficient

Df

Hydraulic diameter

ppump

Output pressure of pump

r

Angular velocity vector

I

Cross-sectional moment of inertia matrix

I1

Polar moment of inertia

I2

Cross-sectional moment of inertia

V

Volume of drillstring

$${\varvec{\upvarepsilon}}$$

Strain tensor

$${\varvec{\upsigma}}$$

Stress tensor

$${\mathbf{F}}_{{{\mathbf{LIN}}}}$$

Vector of linear force

$${\mathbf{F}}_{{{\mathbf{NL}}}}$$

Vector of nonlinear force

$$\varTheta_{1}$$

$$\varTheta_{2}$$

$$\varOmega_{x}$$

Driving angular speed

V0

Initial axial speed

Δt

Time step

α

Integral coefficient

β

Integral coefficient

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