On Nonlinear Spatial Vibrations of Rotating Drill Strings under the
Effect of a Fluid Flow
ASKAR K. KUDAIBERGENOV, ASKAT K. KUDAIBERGENOV, L. A. KHAJIYEVA
Department of Mathematical and Computer Modeling,
Al-Farabi Kazakh National University,
71 al-Farabi Ave., Almaty, 050040,
KAZAKHSTAN
Abstract: - In this article, the development and subsequent numerical analysis of a nonlinear mathematical
model of the drill-string dynamics taking into account the effect of a drilling fluid flow and the gravitational
energy of the system is carried out. Spatial lateral vibrations of the drill string modeled as a rotating elastic rod
are studied. The developed nonlinear model generalizes and refines the well-known linear models of rod
vibrations with the considered effect. The obtained numerical results demonstrate the influence of geometric
nonlinearity, the gravitational energy of the drilling system, additional Coriolis and centrifugal forces as well as
the parameters of the fluid flow on spatial vibrations of the drill strings. It allows for giving some
recommendations on the choice of the drilling system parameters for ensuring safe drilling operations.
Key-Words: - mathematical model, drill string, rod, nonlinear, vibration, fluid flow.
Received: January 25, 2023. Revised: April 17, 2023. Accepted: May 13, 2023. Published: June 27, 2023.
1 Introduction
Modeling the motion of drill strings is a highly
nonlinear and quite complex process from
mathematical and physical points of view since their
dynamics include all three main types of vibrations:
lateral, longitudinal, and torsional ones, [1].
Amongst them, the lateral mode is said to be the
most dangerous that causes breakdowns of drill
strings and failure of drilling rigs, [2].
A number of works have been devoted to the
problems of drill-string vibrations with a fluid flow,
which is one of the main factors ensuring the
efficiency of the entire process of drilling oil and
gas wells. The impact of drilling fluid properties
along with formation boundaries on the transmission
of acoustic waves in a drill string was studied in,
[3]. A slight linear decrease in the acoustic wave
amplitude was observed when increasing the drilling
fluid damping. In, [4], the authors examined the
influence of the non-Newtonian drilling mud
rheology on the vertical drill bit vibrations using the
Herschel-Bulkley and Casson models and
determined the parameters for the significant
reduction of chatter. A stability map of the drill-
string stick-slip vibrations based on its nonlinear
distributed axial-torsional model with a rate-
independent bit-rock interaction was developed in,
[5]. The analysis allowed a better understanding of
coupled self-excited vibrations in the drill string
considered.
In, [6], the authors presented a comprehensive
review of papers related to drill pipe failures
indicating all metallurgical and mechanical aspects
of these undesirable phenomena. Separate attention
was paid to the corrosive and erosive behavior of
drilling mud that might damage the internal coating
of drill pipes. This problem was also investigated in,
[7], [8]. In [7], the authors, in particular, considered
the comparative design approach that normalized
several factors inducing drill-string fatigue. An
innovative approach for improving the directional
well cleaning process by controlling drill-string
buckling was proposed in [9]. In [10], the authors
studied the stability of the lateral vibration of drill
strings and found that the interaction between the
drill string and the drilling fluid had a great impact
on its buckling and dynamics. To solve the problem,
the conventional Galerkin method was utilized.
In, [11], the authors suggested a method for
estimation of the 3D static stress-strain state of a
drill string and marine riser under their contact
interaction. The stress-strain state of a rotating drill
string affected by drilling mud and external forces
for nonlinear and linear models was analyzed in
[12], using the maximum stress intensity criterion.
However, despite the existence of works in this
direction, the problem of spatial vibrations of drill
strings considering nonlinear effects and the
influence of the drilling fluid flow is still
insufficiently studied. In contrast to many articles
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
DOI: 10.37394/232011.2023.18.8
Askar K. Kudaibergenov,
Askat K. Kudaibergenov, L. A. Khajiyeva
E-ISSN: 2224-3429
75
Volume 18, 2023
(for instance, [13], [14]), where the linear models
are utilized, the current research involves not only
the effect of geometric nonlinearity along with the
action of external forces but also the gravitational
energy of the system.
Therefore, this work aims to study drill-string
nonlinear spatial lateral vibrations taking into
account the action of the drilling fluid flow along
with the effect of the gravitational energy of the
system. To demonstrate the significance of
considering nonlinearity when modeling the drill-
string dynamics, the comparative analysis of the
developed nonlinear model with its linearized
version is carried out. Moreover, the investigation of
the impact of the fluid flow on the drill-string spatial
vibrations allows for giving recommendations on
the optimal choice of parameters of the drilling fluid
for performing safe drilling operations.
2 Nonlinear Mathematical Model
The drill string is modeled as a homogeneous
isotropic elastic rod of the constant cross-section
with a length of l and rotating with an angular speed
Ω. The upper end of the rod is subjected to a
longitudinal compressive load
3,N x t
equal to the
reaction of the lower end to the bottom of the well,
and a torque
3,M x t
that causes the rod torsional
deformation. An incompressible fluid flow moves
along the internal tube of the drill string in the
positive direction of the longitudinal axis
3
x
.
The design scheme of the drill string taking into
account the mentioned effects is shown in Fig.1.
The hypothesis of plane sections and the main
provisions of the V.V. Novozhilov nonlinear theory
of elasticity, [15], are utilized. We use the second
system of simplifications of the nonlinear theory,
according to which the relative elongations, shears,
and angles of rotation are assumed to be
infinitesimal.
The rod displacement vector is decomposed into
two components
13
,u x t
and
23
,u x t
. The
position vector of the rod and its velocity vector are
determined from the following relations:
13
3 1 3 2 3 2 1
3
23 23
3
,
, , ,
,,
u x t
x t u x t u x t x
x
u x t x
x
1
r i i
i
(1)
12
2 1 2
22
12
1 2 3
33
uu
uu
t t t
uu
xx
x t x t
1
r
v i i
i
(2)
where
2 2 3
( ), ( ),tt
11
i i i i i
are unit vectors.
Fig. 1: The drill string scheme.
The strain components of the rod are given in the
form:
22
1 1 2 2
11 12 22
3 3 3 3
13 23
22
22
1 2 1 2
33 1 2
22 33
33
11
, , ,
22
0,
1,
2
u u u u
x x x x
u u u u
xx xx
xx
(3)
using which the elastic potential Ф for the case of
the rod vibrations in the considered planes,
13
Ox x
and
23
Ox x
, is written as
2
222
1 2 1 1
2
33 3
2
2 2 2
2 1 2
2 1 2
2 2 2
3 3 3
2
2
u u u
Gx
xx x
u u u
x x x
x x x
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2 2 2 2
1 2 1 2
3 3 3 3
2
22
12
12
22
33
3
2
,
u u u u
G
x x x x
uu
xx
xx
(4)
where
,
2 1 1 1 2
EE
G
are Lame
parameters.
The velocity vector of the drilling fluid in
accordance with the work of [16], is defined as
follows:
,
f
V
f
vv τ
(5)
where
f
V
is the fluid flow velocity,
t
r
τ
is the
direction vector tangential to the rod axis.
To derive a nonlinear mathematical model, the
Ostrogradsky-Hamilton variational principle is
applied. According to this principle, the following
condition must be satisfied:
2
1
00,
t
t
J T U dt
(6)
where T and
0
U
are kinetic and potential energies,
is the potential of external forces, taking into
account the effect of the longitudinal load, torque, as
well as the gravitational energy of the drill string
and the pressure of the fluid flow. The expressions
for the energies and the potential can be found in
[17].
Thus, the nonlinear mathematical model of the
drill-string spatial lateral vibrations, allowing for the
influence of the fluid flow and the effect of external
loads is obtained in the form [12]:
22
44
2
1 1 2
3
4 2 2 2 3
3 3 3
3
11
3
3 3 3 3
2
12
3 3 3
22
12
1
2
,
,1
56
21
2
xx
ff
u u u
EI I M x t x
x x t x
uu
EA
N x t
x x x x
EA uu
x x x
uu
A A u
t
t
2
4 4 4
1 1 1
4 3 2 2
3 3 3
22
21 1 2
233
3
2
11
32
33
2
22
0,
fx
f f f f f
ff
u u u
Ix x t x t
u u u
A V V V
x t x
x
uu
A A g l x
xx
(7)
11
1
44
2
2 2 1
3
4 2 2 2 3
3 3 3
3
22
3
3 3 3 3
2
21
333
22
21
2
2
44
22
43
33
,
,1
56
21
2
2
xx
ff
fx
u u u
EI I M x t x
x x t x
uu
EA
N x t
x x x x
EA uu
xxx
uu
A A u
t
t
uu
Ix x t
42
22
3
22
22 2 1
233
3
2
22
32
33
22
0,
f f f f f
ff
u
xt
u u u
A V V V
x t x
x
uu
A A g l x
xx
where E is Young’s modulus,
12
,
xx
II
axial inertia
moments,
the drill string density,
Poisson’s
ratio, A the cross-sectional area of the drill string,
f
the fluid density,
f
A
and the internal cross-
sectional area of the drill string.
The boundary conditions corresponding to the
simply supported rod are given as
21
1 3 2 3 3 3
22
1 3 2 3
22
33
33
, , 0 0, ,
,,
0
0, .
xx
u x t u x t x x l
u x t u x t
EI EI
xx
x x l
(8)
The initial conditions are written as follows:
1 3 2 3
1 3 2 3
12
, , 0 0 ,
,,
, 0 ,
u x t u x t t
u x t u x t
C C t
tt
(9)
where
1,2
i
Ci
are constants.
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E-ISSN: 2224-3429
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Neglecting the terms responsible for the
contribution of geometric nonlinearity, the effect of
the longitudinal load and the torque leads to a model
obtained by [13], without considering the axial
deployment of the rod.
3 Method of Solution
Direct integration of the obtained governing
equations, complicated by geometric nonlinearity,
causes great mathematical difficulties. Therefore, to
solve the nonlinear model (7)-(9), the Bubnov-
Galerkin method, which allows reducing partial
differential equations (PDEs) to ordinary differential
equations (ODEs), is utilized. According to this
approach, the desired solutions are represented as
expansions into finite series in terms of the basis
functions:
3
13 1
3
23 1
, sin ,
, sin .
n
i
i
n
i
i
ix
u x t f t
l
ix
u x t g t
l
(10)
It is assumed that the longitudinal load changes
with time according to a sinusoidal law and is
written in the following form:
30
, sin ,
t
N x t N N t
(11)
where
0
N
is the constant component of the external
load,
t
N
is the variable one, and
is the frequency
of the load, which depends on the angular speed
.
The action of the torque is assumed to be constant
and distributed along the length of the rod,
3,M x t M
.
We also take into account that the axial moments of
inertia are equal to each other; hence, the stiffness of
the rod relative to the axes
1
x
,
2
x
is constant, and
due to the homogeneity of the material properties,
does not change along the length of the drill string,
12
xx
EI EI EI
.
Substituting the displacement projections
13
,u x t
and
23
,u x t
, given in the form (10), into Eqs. (7)
and considering the first three terms of the series
3n
, a system of six second-order nonlinear
ODEs with respect to the functions
i
ft
, 1,3
i
g t i
is obtained. Accounting for a larger
number of terms in (10) gives only a slight
refinement of the solution, requiring, at the same
time, much more time for conducting the numerical
experiment, which is inappropriate from the
“implementation time–computational accuracy”
viewpoint and practically unreasonable. For reasons
of space, the nonlinear ODEs are not presented in
this article but can be easily determined.
Then, the obtained ODE system is solved
numerically using the stiffness switching method
including the eighth-order explicit Runge-Kutta
method and the linearly implicit Euler method. The
stiffness switching method is applied since the
studied equations belong to the class of stiff
equations, the numerical solution of which is
accompanied by the non-uniform change at different
intervals. Thus, the solution to the problem includes
so-called boundary (inner) layers characterized by
very fast change of the solution on a small interval,
followed by intervals of slow evolution (quasi-
stationary mode). Other variations of the stiffness-
switching method can be found in the work of [18].
Another reason for using the stiffness switching
method is its high efficiency compared to other
numerical methods when solving the considered
type of stiff equations, [19].
4 Numerical Results and Discussions
The software package used to analyze the numerical
results is Wolfram Mathematica.
To demonstrate the need of considering nonlinear
terms in model (7), the comparative analysis of the
model with its linear analog is carried out (Fig.2).
The values of the main parameters of the drill string,
external loads, and drilling fluid are given in Table
1. Spatial lateral vibrations of the steel rod in the
cross-section
0.49zl
with
12
0.01m sCC
are
considered.
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Table 1. Parameters of the drilling system
System parameter
Value
Drill string length, l
100m
The angular speed of rotation,
1.5rad s
Young’s modulus, E
11
2.1 10 Pa
Drill string density,
3
7800kg m
Poisson’s ratio,
0.28
The outer diameter of the drill string,
D
3
63.5 10 m
Wall thickness, h
3
4.5 10 m
Drill string cross-sectional area, A
32
0.834 10 m
Longitudinal load, constant part,
0
N
3
1.2 10 N
Longitudinal load, variable part,
t
N
3
10 N
Torque, M
4
10 N m
Fluid density,
f
3
1800kg m
Internal cross-sectional area,
f
A
32
2.33 10 m
Fluid flow speed,
f
V
3.5m s
As illustrated in Fig.2, the use of the linear
model for modeling the drill string vibrations under
the effect of the fluid flow results in the sharp rise of
the vibration amplitude with time, whereas the
application of the nonlinear model shows that the
oscillatory process remains stable. It indicates the
importance of accounting for nonlinear terms in
mathematical models to obtain more accurate
numerical results. It also confirms the results of,
[12], where the stress-strain state of rotating drill
strings with drilling mud was studied.
a)
b)
Fig. 2: Results of the comparative analysis for the
linear (dotted) and nonlinear (solid) mathematical
models in the a)
13
Ox x
and b)
23
Ox x
planes.
As stated before, the nonlinear model (7)
includes the effect of the gravitational energy of the
drill string and the fluid flow. Fig. 3 demonstrates
the influence of the gravitational energy on the drill
string lateral vibrations at the same values of the
system parameters. As can be seen from the
obtained graphs, taking into account the
gravitational energy of the system leads to a
noticeable increase in the amplitude of the drill
string spatial lateral vibrations in both planes.
a)
b)
Fig. 3: Drill string vibrations in the a)
13
Ox x
and b)
23
Ox x
planes with (dotted) and without (solid)
gravitational energy of the system.
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Also, the effect of the additional Coriolis and
centrifugal forces on the drill string dynamics was
studied (Fig.4). When these additional forces and
the gravitational energy of the drill string and the
fluid flow are not considered, the vibration
amplitude over time becomes higher than that
obtained by solving the nonlinear model (7). This
finding provides further evidence for using the full
nonlinear model for studying the drill string spatial
vibrations with the fluid flow.
The form of bending of the drill string axis
under the effect of the drilling fluid and the
gravitational energy is presented in Fig.5. As the
graph shows, the bending of the drill string axis is
observed in its lower part, which is related to the
effect of the gravitational energy of the system.
a)
b)
Fig. 4: Comparative analysis of the nonlinear model
(dotted) and the one without the effect of the
gravitational energy and additional forces (solid) in
the a)
13
Ox x
and b)
23
Ox x
planes.
Fig. 5: Drill string spatial bending at the time
moment t=68s.
Further, the change in the drilling fluid parameters
is investigated. The influence of the fluid flow speed
on the drill string nonlinear lateral vibrations are
illustrated in Fig.6. As can be seen from the graphs
below, a three-fold increase in the fluid speed up to
10.5m s
f
V
has a minor impact on the vibration
amplitude, whereas the further speed increase to
21m s
f
V
results in the sharp rise of the
amplitude. It shows that the high speed of the
drilling fluid may be one of the reasons leading to
the collapse of the borehole walls and the curvature
of the drill string axis and confirms the need to
regulate the speed regimes of the borehole cleaning.
a)
b)
Fig. 6: Influence of the fluid flow speed on the drill-
string vibrations in the a)
13
Ox x
and b)
23
Ox x
planes:
3.5m s
f
V
(solid),
10.5m s
f
V
(dashed),
21m s
f
V
(dotted).
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Figure 7 and Figure 8 demonstrate the effect of
the drilling fluid density on the oscillatory process.
At a relatively low speed of the fluid flow (
3.5m s
f
V
), the comparative analysis of the drill
string spatial vibrations at varying values of the
fluid density from
3
1000kg m
f
to
3
1600kg m
f
and then to
3
2200kg m
f
shows that the use of industrial water with density
3
1000kg m
f
yields the maximum values of the
vibration amplitude, slightly surpassing the
amplitude of the drill string vibrations under the
effect of the weighted fluid with density
3
2200kg m
f
(Figure 7).
a)
b)
Fig. 7: Influence of the drilling fluid density on the
drill-string vibrations in the a)
13
Ox x
and b)
23
Ox x
planes:
3
1000kg m
f
(solid),
3
1600kg m
f
(dashed),
3
2200kg m
f
(dotted).
On the other hand, when the flow speed
increases up to
20m s
f
V
, the utilization of the
industrial water allows the performing of drilling
operations with the smallest vibration amplitude
compared to other drilling fluids (Figure 8). In turn,
the largest amplitude of the lateral vibrations is
observed when the weighted drilling fluid is applied.
Another advantage of industrial water is its
availability and low cost. Having a low viscosity, it
successfully flushes the mud from the bottom of the
well and cools down the bit.
a)
b)
Fig. 8: Influence of the drilling fluid density on the
drill-string vibrations in the a)
13
Ox x
and b)
23
Ox x
planes when the flow speed increases,
20m s
f
V
3
1000kg m
f
(solid),
3
1600kg m
f
(dashed),
3
2200kg m
f
(dotted).
5 Conclusion
In this work, spatial lateral vibrations of rotating
drill strings under the effect of a drilling fluid flow
were considered. Using the main provisions of the
V.V. Novozhilov nonlinear theory of elasticity and
the Ostrogradsky-Hamilton variational principle, a
nonlinear mathematical model of the drill-string
vibrations accounting for the effect of the fluid flow
was developed.
The comparative analysis of the developed
nonlinear model and its linear analog showed the
importance of adding nonlinear terms to the
mathematical models for obtaining more accurate
numerical results. Also, the effect of the
gravitational energy of the drill string and the fluid
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DOI: 10.37394/232011.2023.18.8
Askar K. Kudaibergenov,
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Volume 18, 2023
flow and the influence of additional Coriolis and
centrifugal forces included in the model on the drill-
string spatial lateral vibrations was analyzed. It was
obtained that taking into account the gravitational
energy of the system led to a noticeable increase in
the drill-string vibration amplitude in both planes.
When the additional forces and the gravitational
energy were not considered, the research results
demonstrated the increase in the amplitude of
vibrations compared to that obtained by solving the
full nonlinear model over time.
The study of the influence of the drilling fluid
parameters on the drill string vibrations revealed
that the high speed of the drilling fluid flow could
be the reason for the vibration amplitude sharp rise
and confirmed the need to regulate the speed
regimes of the borehole cleaning. It was also shown
that the industrial water, having a low density
compared to other drilling fluids, allowed for
conducting the drilling process with relatively small
vibrations of the drill string at the high speed of the
fluid flow.
Our work has some limitations. The most
important one lies in the fact that the interaction of
the drill string with the borehole wall was not
considered. It would allow restricting the drill-string
vibrations and make the problem closer to the real
drilling process. Therefore, to further our research,
we are going to study the drill-string nonlinear
dynamics and its stability, taking into account the
intermittent contact with the borehole, and also
consider the effect of a two-directional fluid flow.
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WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
DOI: 10.37394/232011.2023.18.8
Askar K. Kudaibergenov,
Askat K. Kudaibergenov, L. A. Khajiyeva
E-ISSN: 2224-3429
82
Volume 18, 2023
[15] Novozhilov V.V., Foundations of the
nonlinear theory of elasticity, Dover
Publications, New York, 1999.
[16] Païdoussis M.P., Fluid-structure interactions:
slender structures and axial flow, 2nd ed.,
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[17] Kudaibergenov Askar K., Khajiyeva L.A.,
Modelling of Drill String Nonlinear Dynamics
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[19] Kudaibergenov A.K., Kudaibergenov Ask.K.,
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Russian).
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
-Askar K. Kudaibergenov developed the nonlinear
mathematical model of the drill string vibrations.
-Askat K. Kudaibergenov conducted the numerical
analysis of the drill string vibrations.
-L.A. Khajiyeva was responsible for the
conceptualization and methodology of the studied
problem.
Sources of funding for research presented in a
scientific article or scientific article itself
This research is funded by the Science Committee
of the Ministry of Education and Science of the
Republic of Kazakhstan, grant number
AP09261135.
Conflict of Interest
The authors have no conflicts of interest to declare
that are relevant to the content of this article.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
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WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
DOI: 10.37394/232011.2023.18.8
Askar K. Kudaibergenov,
Askat K. Kudaibergenov, L. A. Khajiyeva
E-ISSN: 2224-3429
83
Volume 18, 2023
