Research Journal of Applied Sciences, Engineering and Technology 4(19): 3822-3829, 2012
ISSN: 2040-7467
© Maxwell Scientific Organization, 2012
Submitted: April 25, 2012
Accepted: May 13, 2012
Published: October 01, 2012
Dynamic Response Analysis of Leaching Tube Used in Salt Cavern Gas Storage
Tongtao Wang, Xiangzhen Yan, Xiujuan Yang and Hongduan Huang
College of pipeline and civil engineering, China University of Petroleum, Qingdao 266580,
Shandong, China
Abstract: Fluid-structure coupling finite element model is proposed to analysis the dynamic response of
leaching tube in the construction of salt cavern gas storage. The stress and deformation of leaching tube
induced by the carrying fluid flow are studied. Meanwhile, an experiment on the dynamic response of
plastic tube caused by carrying fluid flow is carried out to verify the calculation accuracy of the proposed
model. Experimental results prove that the finite element model proposed in the study has a high accuracy,
which can well characterize the stress and deformation of leaching tube under the action of fluid flow. The
errors between experimental and numerical results are about 5%, which can satisfy the engineering
accuracy demands. Excited pressure produced by fluid flow has significant effects on the stress and
deformation of leaching tube string, which is the main cause of tube damage. Therefore, low injected
freshwater velocity is proposed at the initial stage and then it is increased slowly. Moreover, rapidly
opening or closing the valve should be avoided.
Keywords: Dynamic response, fluid-structure coupling, leaching tube, numerical simulation, underground
salt cavern gas storage
INTRODUCTION
Underground salt cavern is one of the most popular
methods to storage natural gas, which has the
advantages of safety, high efficiency and economy
(Wang et al., 2010a, b; 2011a, b). Water-soluble
method is the main way to construct underground salt
cavern at present. In the method, fresh water is injected
through the leaching tube and then the brine is
discharged through the annulus (Fig. 1). Due to the
viscous force produced by water, the leaching tube
takes place nonlinear vibration, which causes the
vibration of tube becoming complicated. Meanwhile,
the vibration will lead to the alternative bending stress
in the leaching tube which may result in large
deformation and fatigue damage and may cause the
failure of leaching tube.
The vibration of pipe caused by carrying fluid flow
is a common engineering problem which exists widely
in the water pipeline, oil pipeline, steam pipeline,
water-pump pipeline and the injection piping system of
rocket, etc. Many scholars studied the problems in the
relative fields and obtained some useful conclusions,
for example, (Herrmann and Nemat, 1967) put forward
nonlinear dynamic model of cantilevered pipeline,
which ignored the effects of gravity and considered the
relationship between curvature and moment was linear.
Fig. 1: Sketch map of leaching tube used in the construction
of salt cavern gas storage by water-soluble method
Corresponding Author: Xiangzhen Yan, College of pipeline and civil engineering, China University of Petroleum,
Qingdao 266580, Shandong, China
3822
Res. J. Appl. Sci. Eng. Technol., 4(19): 3822-3829, 2012
model and got some useful conclusions. Subsequently,
some scholars believed that the model proposed by
Walker and Phillips (1977) studied the interaction
between fluid and pipeline by plane axial symmetry
them was more suitable for the vibration analysis of
pipe within high speed fluid flow. Wylie and Streeter
(1978) introduced the water hammer theory into the
pipeline vibration caused by carrying fluid flow and
analyzed the pipeline vibration laws. However, the
effects of fluid-structure coupling were not taken into
account in the model, ignoring the influences of
pipeline vibration on the fluid pressure. Wiggert et al.
(1985) analyzed the vibration of pipeline with elbow
causing by carrying fluid flow and believed that the
Poisson effect on pipeline vibration mode was relative
significant. Lesmez et al. (1990) carried out the fluidsolid coupling analysis of pipeline by introducing
transient vibration impact items into the dynamic
vibration equation of pipeline. Nevertheless, the solving
processes were complicated and with a bad
convergence. Païdoussis (1998) analyzed the dynamics
problems of cantilevered pipeline under nonlinear
constraint and steady carrying flow fluid condition. Lee
and Kim (1999) deduced the fully coupled finite
element equations of pipeline dynamic vibration, which
considered the effects of pipeline circumferential strain
caused by fluid pressure and then developed the
corresponding calculation software. Gibbs and Qi
(2005) analyzed the pipeline vibration under the action
of carrying fluid flow and studied the effect of bend on
pipeline vibration. Ni et al. (2011) put forward a semianalytical method to describe the vibration of pipeline
causing by carrying fluid flow under different
constraint conditions. Moreover, he gave the free
vibration frequency of pipeline and the critical flow
velocity. Vibration of leaching tube string caused by
carrying fluid flow was a critical issue to the
construction of underground salt cavern gas storage,
which had caused many serious consequences (Li et al.,
2012), such as broken, large deformation, fatigue
damage and dropout, etc. However, little attentions
have been attracted.
The motives of the study are to establish the fluidstructure coupling finite element model and to analysis
the dynamic response laws of leaching tube.
Furthermore, the leaching tube of salt cavern gas
storage located at Jiangsu province of China is
simulated as examples. The encouraging results are
obtained, which can provide fundamental data and
references to the field predictions and monitors of
leaching tube dynamic response.
Fig. 2: Vibration model of leaching tube caused by carrying
fluid flow
METHODOLOGY
Finite element governing equations of fluidstructure coupling: The nonlinear dynamic response
mechanical model of the leaching tube induced by
carrying fluid flow is shown in Fig. 2 and the equation
of carrying fluid flow is written as:
&& + ρ Br&& + q = 0
HP + AP& + EP
0
(1)
1
T
T
where, H = ∫∫∫ ∇N ∇N d Ω ; A = ∫∫ NN d SR ;
C
Ω
SR
E=
1
C2
B = ( ∫∫
SI
∫∫∫ NN
T
Ω
1
NN T d S F ;
g ∫∫
SF
⎧ N1 ( x, y , z ) ⎫
⎪
⎪
NN d S I ) Λ ; N = ⎪ N 2 ( x , y , z ) ⎪ ,
⎨
⎬
K
⎪
⎪
⎪⎩ N m ( x , y , z ) ⎭⎪
T
shape function; P
3823 dΩ+
⎧ P1 ( t )
⎪ P (t)
⎪
2
= ⎨
K
⎪
⎩⎪ P m ( t )
⎫
⎪
⎪
⎬ ,
⎪
⎭⎪
which is the
Res. J. Appl. Sci. Eng. Technol., 4(19): 3822-3829, 2012
∂r ∂ r ∂ r
∇= i + j + k ; Ω
∂x ∂y ∂z
is the volume of fluid; S I , S F , S R are the surface area
which is the pressure vector;
of the interface of fluid and structure, free surface and
infinite boundary respectively; C is the sound velocity
in the fluid, C = K / ρ ; ρ is the fluid density; K is
the bulk modulus of fluid; Λ is coordinate
transformation matrix, which is a rectangular array and
composed by the direction cosines of
and zero;
S I outer normal
q0 is the input incentive vector; &r& is the
displacement of the structure element on the surface of
S Ie .
Thus, the virtual work of the fluid dynamic
pressure subjected to the interface is expressed as:
δ We = − ∫∫ δ U ne P*( e ) d S Ie = −δ U neT ( ∫∫ N Se N SeT d S Ie ) Pe
S Ie
T
f P( e ) = −( ∫∫ N Se N Se
d S Ie ) Pe
(2)
M s is the structural mass matrix; Cs is the
K s is the structural
structure joint at the interface of fluid and structure;
T
f P( e ) = −ΛT f Pn( e ) = −ΛT ( ∫∫ N Se N Se
d S Ie ) Pe
Within the entire range of fluid-solid interface
f p is a key parameter in the fluid-structure
coupling equation, which controls the calculating
the
derivation
of
fp
is
emphatically discussed in the study. Fluid dynamic
pressure can be obtained by the virtual work principle
theory, expressed as:
P
*( e )
= Ne Pe(t )
T
(3)
where, Pe is the node pressure vector of fluid
element; Ne is the vector of the shape function of
fluid element.
Assuming the node virtual displacement vector
value is
δU
(e)
n
(7)
S Ie
f0
is the dynamic load vector subjected to the structure
joints.
Therefore,
(6)
The nodal generalized force vector is obtained by
converting this nodal normal force in the direction of
the general coordinate system, which can be expressed
as:
stiffness matrix; r is the displacement vector; fp is the
dynamic pressure vector of the fluid loaded on the
accuracy.
(5)
S Ie
M s &&
r + Cs r& + K s r + f p + f 0 = 0
structural damping matrix;
S Ie
Then, the normal vector of element node at the
equivalent position of the fluid dynamic pressure is:
acceleration vector of structure joint.
The vibration equation of leaching tube string is:
where,
N Se is the shape function vector of the nodal
where,
SI ,
the nodal dynamic pressure vector of fluid loaded on
(e)
the structure is obtained by gathering f P on S I into
node group, expressed as:
f p = ∑ f P( e ) = ∑ − BeT { Pe} = − BeT P
e
(8)
e
where,
Be =
∫∫ N
S Ie
Se
N STe d S Ie Λ , BeT = Λ T
∫∫ N
Se
T
N Se
d S Ie
S Ie
The motion equation of structure contacting with
fluid is obtained by substituting Eq. (8) into (2), written
as:
M s &&
r + C s r& + K s r − B T P + f 0 = 0
(9)
at the fluid-structure interface along
normal direction, the normal virtual displacement at the
interface can be written as below:
δ U n( e ) = N STe δ U ne = δ U ne N S e
(4) In Eq. (9), P indicates the motion of structure with
the characteristic of fluid-structure coupling. Equation
(1) and (9) are the discrete form of fluid-structure
coupling motion equation. Ultimately, fluid-structure
coupling motion equation can be written as below:
3824 Res. J. Appl. Sci. Eng. Technol., 4(19): 3822-3829, 2012
&& + ρ Br&& + q = 0
⎪⎧ HP + AP& + EP
0
⎨
T
r + Cs r& + K s r − B P + f 0 = 0
⎪⎩ M s &&
string, m; EI is the tube stiffness, N.m2; A is the inner
sectional area of tube, m2; f is the string natural
(10)
The finite element governing equation of fluidstructure interaction is:
T &&
&&
⎪⎧M f P + K f P = Ff − ρ0 R U
(11)
⎨
⎪⎩MsU&& + KsU = Fs + RP
&& is the normal acceleration vector of node on
where, U
S I ; R is the coupling matrix.
Table 1: Material properties and dimension of plastic tube in the
experiment
Material
PPR plastic
Internal diameter of plastic tube/mm
25.000
External diameter of plastic tube/mm
32.000
Tube length/m
8.000
Water weight in per meter tube string/kg
0.491
Elastic modulus/MPa
850.000
910.000
Tube material density/ (kg/m3)
3
1000.000
Water density/ (kg/m )
Dynamic viscosity coefficient/ (kg/m.s)
8.899×10-4
The loads are considered loading at the inaction
surface of fluid and structure and the functions of node
freedom. Then, Eq. (11) can be written in the form of
matrix, expressed as:
⎡M f
⎢
⎣ 0
ρ0 RT ⎤ ⎪⎧P&& ⎪⎫ ⎡K f
0 ⎤ ⎧P ⎫ ⎪⎧Ff ⎪⎫
⎥ ⎨ &&⎬ + ⎢
⎥⎨ ⎬ = ⎨ ⎬
Ms ⎦ ⎪⎩U ⎪⎭ ⎣−R Ks ⎦ ⎩U ⎭ ⎪⎩Fs ⎪⎭
(12)
The fluid-solid coupling problem is the synthetic
solution of structure motion and wave equations. The
fluid-structure interface is serviced as the solution
boundary. The response of structure depends on the
load produced by the fluid flow, while the fluid
pressure is also affected by the vibration of structure.
Experimental verification: In order to verify the
accuracy of the proposed numerical model, experiments
are carried out. A plastic tube is selected for the
experiment and the material parameters and dimensions
are shown in Table 1. One of the tube ends is fixed
while the other end is free. The deformation and
vibration of the tube under different fluid speed
conditions are tested by dual-channel data transducers
at positions A and B (Fig. 3). Meanwhile, the finite
element numerical simulation model of the tube is
established by ANSYS and CFX based on the fluidstructure coupling principles of Eq. (1) to (12). The
displacement and vibration of A and B are monitored in
the numerical simulations. The experimental and
numerical results are compared with each others as
shown in Fig. 4.
The critical velocity of carrying fluid flow and
vibration frequency of tube string under the action of
fluid can be obtained by Eq. (13) and (14):
where,
vc is the critical velocity of fluid causing tube
string sympathetic vibration, m/s; L is the length of tube
Fig. 3: Tube vibration test experimental model and transducer
position
12
π ⎛ EI ⎞
vc = ⎜⎜ ⎟⎟
L ⎝ ρA⎠
(13)
12
2
f ⎡ ⎛v⎞ ⎤
= ⎢1−⎜ ⎟ ⎥
f1 ⎢ ⎜⎝vc ⎟⎠ ⎥
⎦
⎣
(14)
frequency, f = 2π m k , Hz; m is the tube string mass,
kg; k is the linear stiffness coefficient of tube string,
N.m; f1 is the tube string vibration frequency under the
action of fluid, Hz.
3825 Res. J. Appl. Sci. Eng. Technol., 4(19): 3822-3829, 2012
Numerical results
Measured results
Numerical results
Measured results
60
30
40
Displacement/mm
Displacement/mm
20
10
0
-10
0
-20
-40
-20
-30
0.0
20
0.1
0.2
0.3
0.4 0.5
Time/s
0.6
0.7
0.8
-60
0.0
0.1
0.2
0.3
0.4 0.5
Time/s
0.6
0.7
0.8
Fig. 4: Numerical and experimental results of tube string deformation when the flow rate is 1 m/s
According to Eq. (13)-(14) and the data in Table 1,
the critical velocity of plastic tube string in the
experiment is 3.68 m/s and the natural vibration
frequency is 5.12 Hz. The displacements and vibration
cycle of the monitoring points A and B are contrasted
by experiments and numerical simulations when the
fluid flow velocity is 1 m/s (Fig. 4).
As shown in Fig. 4, the string deformation values
of numerical simulation and monitoring have preferably
coherence, indicating that the numerical model
proposed in the study has high calculation accuracy.
Meanwhile, the theoretical value of tube string
vibration period obtained by Eq. (14) is about 0.19 s,
while that of the numerical simulation and experiment
is about 0.20 s. It indicates the numerical simulation
and experiment are reliable. The numerical model
proposed in the study is precise and can satisfy the
accuracy requirement of field engineering.
Applications:
Numerical model: In the section, the leaching tube of a
salt cavern gas storage located at Jiangsu province of
China is simulated as examples. The effects of the
speed of carrying fluid flow and time on the
deformation and stress of leaching tube string are
studied. The structure of the leaching tube string is
shown in Fig. 1 and its dimensions and material
properties are shown in Table 2. According to the fluidsolid coupling principle of string and fluid in Eq. (1) to
(12), the fluid-structure coupling finite element model
of the leaching tube is established by ANSYS and CFX
(Fig. 5). The model adopts Solid 186 element and uses
the sweeping way to divide grid with 16400 elements
and 90528 nodes. The fluid model is imported into to
the ICEM to divide grid and then the grid is imported
Table 2: Material properties and size of leaching tube string of salt
cavern gas storage
P100-grade
Material
steel
Inner diameter of leaching tube/mm
124.26
External diameter of leaching tube/mm
139.70
Length of leaching tube string/m
100.00
Water weight of per meter leaching tube string/kg
12.13
Elastic modulus/GPa
210.00
7850.00
Tube material density/ (kg/m3)
1000.00
Water density/ (kg/m3)
8.899×10-4
Dynamic viscosity coefficient/ (kg/m3)
into CFX for pre-processing. In order to improve the
solving accuracy, the fluid adopts the hexahedral grid
with 83443 elements and 133744 nodes.
RESULTS AND DISCUSSION
Figure 6 shows the effect of carrying fluid flow
speed on the stress distribution of leaching tube along
the depth direction at different times. It can be seen
from the Fig. 6. the stress increases with the increase of
fluid velocity, which increases violently at the initial
stage of injecting fluid. Moreover, the stress
distributions of the whole string appear extremely
uneven. Meanwhile, the max stress occurs near the
fixed end while the min stress appears at the free end.
With the increase of time, the stress of the fixed end
decreases gradually and the stress of the whole tube
string becomes evener. The location of the max. stress
of is not at the fixed end (location of casing shoe) but a
certain distance away from it when the vibration of
leaching tube become stable. It is mainly because the
small distance between the leaching tube and casing
which has a certain constraint on the vibration of
leaching tube. The top of the leaching tube cannot move
freely, resulting in that the bend stress achieves the
3826 Res. J. Appl. Sci. Eng. Technol., 4(19): 3822-3829, 2012
(a) Grids of leaching tube string with 16400 elements and
90528 nodes
(b) Grids of fluid with 83443 elements and 133744 nodes
Fig. 5: Fluid-structure coupling finite element model of leaching tube string used in the construction of salt cavern gas storage by
water-soluble method
2.0 m/s
1.0 m/s
100
Depth/m
80
60
40
20
0
0
40
160
120
200
80
Von miss stress/MPa
240
2.0%
1.0%
5.0%
3.5%
2.5%
120
100
80
Depth/m
5.0 m/s
3.5 m/s
2.5 m/s
120
60
40
20
0
280
0
240
160
Von mises stress/MPa
80
(a) t = 0.03 s
2.0%
1.0%
100
Depth/m
80
60
40
20
0
0
60
240
120
180
Von mises stress/MPa
400
(b) t = 0.06 s
300
5.0 m/s
3.5 m/s
2.5 m/s
120
2.0 m/s
1.0 m/s
100
80
Depth/m
5.0%
3.5%
2.5%
120
320
60
40
20
0
360
(c) t = 0.18 s
0
90
270
180
Von mises stress/MPa
(d) t = 0.8 s
Fig. 6: Stress distribution of leaching tube under different fluid velocities
3827 360
450
Res. J. Appl. Sci. Eng. Technol., 4(19): 3822-3829, 2012
maximum at the location with a certain distance away
from the casing shoe. From the above analyses, we can
see the most dangerous position of the leaching tube
appears at a distance away from the leaching tube top
and the most dangerous time is at the beginning of
injection. Therefore, the safety of leaching tube near the
casing shoe should be mainly checked in design.
Meanwhile, the freshwater injecting velocity should be
reduced at the initial stage and then increased gradually
with the increasing time. In order to reduce the
influences of exited pressure, rapidly opening or closing
the valve are avoided. Figure 7 presents the effect of
carrying fluid flow speed on the horizontal deformation
120
2.0%
1.0%
5.0%
3.5%
2.5%
100
120
100
80
60
40
20
0
2.0%
1.0%
5.0%
3.5%
2.5%
Depth/m
Depth/m
80
distribution of leaching tube along the depth direction at
different times. The deformation of leaching tube
increases with the increasing fluid velocity. The
deformation presents fluctuation from the beginning to
the steady vibration state. Fluid velocity within the
leaching tube has a great impact on the leaching tube;
especially on the middle part. Generally, the cycle of
salt cavern construction lasts for 1200 to 1500 days,
which may cause the leaching tube occur fatigue failure
at the middle location. The fatigue check is proposed to
the middle part of leaching tube string in the actual
design. Meanwhile, it also can be seen from Fig. 7 that
the position of the max. deformation gradually moves
60
40
20
0
-40
-30
10
-10
0
Displacement/mm
-20
20
30
-100
-75
-50
(a) t = 0.03 s
2.0%
1.0%
5.0%
3.5%
2.5%
100
Depth/m
80
60
40
20
0
-60
-30
0
30
60
Displacement/mm
50
75
100
(b) t = 0.06 s
90
5.0%
3.5%
2.5%
120
2.0%
1.0%
100
80
Depth/m
120
0
25
-25
Displacement/mm
120
60
40
20
0
-160
-120
(c) t = 0.18 s
-80
-40
0
Displacement/mm
(d) t = 0.8 s
Fig. 7: Horizontal deformation distribution of leaching tube under different fluid velocities
3828 40
80
Res. J. Appl. Sci. Eng. Technol., 4(19): 3822-3829, 2012
from top to bottom of the leaching tube string as the
time increasing.
CONCLUSION
•
•
•
Fluid-structure coupling finite element model is
proposed to analysis the dynamic response of
leaching tube in the construction of salt cavern gas
storage. Meanwhile, laboratory experiment of tube
vibration caused by carrying fluid flow is carried
out to verify the accuracy of the numerical model
proposed in the study.
The results of laboratory experiment show that the
proposed numerical model can describe the
vibration and deformation laws of tube caused by
carrying fluid flow. The errors between the
experimental and numerical results are about 5%,
which can meet the accuracy requirement of actual
engineering problem.
The simulation results indicate that the deformation
of the leaching tube middle and bottom changes
violently at the beginning time, which is mainly
caused by the exited pressure. As the increase of
time, the stress and deformation become stable.
The max. stress appears at the location with a
distance away casing shoe and the max.
deformation locates the tube bottom.
ACKNOWLEDGMENT
The authors wish to acknowledge the financial
support of China Postdoctoral Science Foundation
funded project (No. 2012M511557) and Post-doctor
Innovation Research Program of Shandong Province
(No.201102033).
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