*Corr. Author’s Address: Northeast Petroleum University, Xuefu Road 99#, Daqing, China, 1538937270@qq.com 483
Strojniški vestnik - Journal of Mechanical Engineering 69(2023)11-12, 483-496 Received for review: 2023-05-02
© 2023 The Authors. CC BY 4.0 Int. Licensee: SV-JME Received revised form: 2023-07-24
DOI:10.5545/sv-jme.2023.627 Original Scientific Paper Accepted for publication: 2023-09-13
Optimization Method of Multi-parameter Coupling  
for a Hydraulic Rolling Reshaper Based on Factorial Design
Li, H.F. – Luo, M. – Xu, T.T. – Li, Q.Z. – Hou, Y .M.
Hongfei Li – Min Luo – Tingting Xu – Qiaozhen Li – Yanming Hou
Northeast Petroleum University, School of Mechanical Science and Engineering, China
A hydraulic rolling reshaper is an advanced shaping technology with superior protection for casings, and the structural parameters of the 
reshaper affect its shaping effect on deformed casing directly. To improve the shaping capacity of the reshaper, a multi-parameter coupling 
optimization method of hydraulic rolling reshaper is proposed to optimize the design of the factors with significant influence under the premise 
of screening multi-structural parameters. In this paper, according to the working principle of the reshaper, considering the contact nonlinearity 
between the hydraulic rolling reshaper and deformed casing, as well as the material nonlinearity of the casing, a parametric finite element 
model of the hydraulic rolling reshaper repairing the shrinkage deformation of casings was developed. The remarkable factors were screened 
by factorial design, the sample points were generated by optimal Latin hyper-cube design (OLHD), and the response surface models were 
established by stepwise regression. Therefore, with the maximum plastic deformation of casings as the objective function, the maximum 
equivalent stress, residual stress, and the plastic deformation of casings as the constrained conditions, an optimized mathematical model 
for a reshaper was constructed, and the genetic algorithm (GA) is performed to obtain the optimal combination of parameters. The results 
showed that the optimal reshaper made the shaping process safe and effective, the plastic deformation of casings after single shaping was 
increased by 11.38 %, and the shaping effect was better (96.48 %), which can effectively improve the safety performance and shaping ability 
of the reshaper.
Keywords: hydraulic rolling reshaper, structural optimization, factorial design, orthogonal test
Highlights
• 	 Optimization of a hydraulic rolling reshaper used to repair deformed casings in oilfields has been performed by numerical 
simulation. 
• 	 Considering both the material and contact nonlinearities, a parametric finite element model of a hydraulic rolling reshaper 
repairing shrinkage deformation of casings was developed.
• 	 An orthogonal test was performed to analyse the multi-geometric parameters and find five significant parameters of the 
reshaper on the shaping effect.
• 	 Nonlinear interactive regression models of response variables were established using stepwise regression based on the 
sampling and calculation of OLHD.
0  INTRODUCTION
In recent years, with the development of downhole 
casing shaping technology, higher requirements have 
been put forward for the safety and efficiency of the 
shaping tools. As one of the advanced shaping tools to 
repair deformed casings, the hydraulic rolling reshaper 
has been widely used because of its advantages in 
protecting the damaged casing, fewer multiple trips 
to the well, and low construction cost. Fig. 1 shows 
the working principle of the hydraulic rolling reshaper 
when repairing horizontal wells. According to its 
working principle, the work section directly contacts 
the inner wall of the deformed casing to transmit the 
repairing load provided by the ground devices, and 
its structural dimensions directly impact the shaping 
effect of the casing.
For the structural optimization of shaping tools, 
the current research focuses on traditional shaping 
tools, such as the pear-shaped tube expander and 
the eccentric tube expander. Jiang [1] discussed the 
influence of the angle of the shaping tool on the 
shaping effect by establishing the mechanical and 
finite element model of the deformed casing of the 
pear-shaped tube expander, and the reasonable taper 
angle of the expander is 30° to 40°. Chen et al. [2] 
used an ANSYS / LS-DYNA explicit structural 
dynamic analysis module to calculate the stress of 
different cone angles under different diameters and 
Fig. 1.  Working principle of hydraulic rolling reshaper

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)11-12, 483-496
484 Li, H.F. – Luo, M. – Xu, T.T. – Li, Q.Z. – Hou, Y.M.
determined the reasonable cone angle of the eccentric 
expander under different diameters. Bai [3] used 
the numerical simulation method to conduct finite 
element analysis on the shaping process of four types 
(spherical, curved, stepped and conical), and obtained 
the conclusion that the shaping effect of a conical 
shaping cone is the best and the best cone angle is 15° 
to 20°. In addition, most of the studies on hydraulic 
rolling reshapers are focused on analysing its 
shaping process and shaping law by using numerical 
simulation combined with laboratory tests. Lin et al. 
[4] obtained the relationship between load and torque 
required for shaping casing based on the established 
mechanical model of repairing elliptic deformation 
casing combined with laboratory tests. Deng et al. [5] 
established the calculation method of shaping force 
required by rolling reshaper to repair the deformed 
casing, and verified the reliability of the calculation 
method through tests. Luo et al. [6] took a single ball 
on the hydraulic rolling reshaper to perform a radial 
extrusion test on the casing, obtained the relationship 
between the shaping force and the radial displacement 
of the inner wall of the casing, and established a 
corresponding finite element model for verification. 
Xu et al. [7] gave a research method for the single 
shaping limit of casing and established the finite 
element model of roll-shaping shrinkage deformation 
casing to obtain the single shaping limit of casing with 
different axial deformation lengths.
In summary, regarding the optimization of shaping 
tools, most researchers used the method of fixing 
other variables and changing a geometric parameter to 
explore the influence of this parameter on the shaping 
effect. The research process is equivalent to the 
“selection” rather than “optimization” of parameters. 
Moreover, there are few reports on the optimization of 
hydraulic rolling reshaper, and the present research [8] 
only optimizes the structural parameters of a part of 
the reshaper, and the influence of other parameters on 
the shaping effect has not been analysed or screened.
Therefore, to improve the shaping ability of 
the hydraulic rolling reshaper in the premise of 
ensuring the safety bearing capacity of the casing, it 
is necessary to perform structural optimization with 
full consideration of the key structural parameters of 
the reshaper. However, the structural complexity of 
the reshaper and the high nonlinearity of its contact 
with the inner wall of the casing make it difficult 
to calculate and evaluate the shaping effect using 
the traditional analytical method, which needs to 
be simulated by numerical simulation. Based on 
this, taking the shrinkage deformation casing as 
an example, the process of repairing the deformed 
casing by hydraulic rolling reshaper is simulated 
and analysed, and the deformed state and stress 
distribution of the casing during the shaping process 
are explored. Subsequently, a multi-parameter 
coupling optimization method of a hydraulic rolling 
reshaper based on the factorial design is proposed. 
The response surface model is constructed by stepwise 
regression, and the optimal structural parameters of 
the reshaper are obtained by genetic algorithm (GA).
1  RELATED OPTIMIZATION THEORY
1.1  Optimized Mathematical Model
Generally, an optimization problem with M 
optimization objectives and N design variables can be 
simplified into an optimized mathematical model:
min, ,, ,, ,,
,,
fx fxfx fxmM
gx i
m
i
     




 
12
12
01 2

s.t. ,, ,
,, ,,
,,,,,
,
minm ax



s
hx jk
xx xxx
xx xn
j
rN
nn n
 
 
 
01 2
12
1 12 ,, ,  N
min, ,, ,, ,,
,,
fx fxfx fxmM
gx i
m
i
     




 
12
12
01 2

s.t. ,, ,
,, ,,
,,,,,
,
minm ax



s
hx jk
xx xxx
xx xn
j
rN
nn n
 
 
 
01 2
12
1 12 ,, ,  N
, (1)
where M is the total number of optimization 
objectives, f
m
(x) is the m
th
 objective function, 
g
i
(x) = 0 is the i
th
 equality constraint, h
j
(x )  ≥  0  is  the j
th
 
inequality constraint, x is the N-dimensional design 
variable, x
nmax
 and x
nmin
 are the upper and lower limits 
of the feasible region of the n
th
 design variable.
1.2 Design of Experiment
The Design of Experiment involved in this paper 
includes a factorial design and an Optimal Latin 
hyper-cube design (OLHD).
1.  Factorial design
 A factorial design is a screening experimental 
design method that aims to find the design 
variables involved in optimization by screening 
out remarkable factors from multi-factors [9] to 
[11]. To judge the influence of different factors 
on the response and reduce the number and cost 
of experiments, an orthogonal test is chosen to 
analyse and screen the related influencing factors. 
The different values of influencing factors are 
called levels. According to the orthogonality, a 
representative combination of levels is selected 
from a full-scale test to form an orthogonal table 
for experimental design, and multi-factors are 
screened to simplify the subsequent optimization.

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)11-12, 483-496
485 Optimization Method of Multi-parameter Coupling for a Hydraulic Rolling Reshaper Based on Factorial Design 
2.  Optimal Latin hyper-cube design
 An optimal Latin hyper-cube design (OLHD) 
is a sampling experimental design method that 
improves the uniformity of the traditional Latin 
hyper-cube design. It can reflect the relationship 
between factors and responses more realistically 
and accurately, with better space-filling and 
equilibrium. Fig. 2 is quoted from [12], Fig. 2a 
shows the test points randomly generated by the 
Latin hyper-cube design, and Fig. 2b shows the 
more uniformly distributed test points generated 
by OLHD.
a)       b) 
Fig. 2.  Latin hyper-cube design; a) traditional; and b) optimal
1.3 Polynomial Fitting Based on Response Surface 
Methodology
Response surface methodology is a statistical 
method to obtain the relationship between multiple 
influencing factors and responses in complex systems 
so as to establish polynomial regression models [13]. 
The basic idea is to select suitable sample points in 
the design space and calculate the responses, use 
polynomial fitting to obtain explicit formulas between 
variables and responses in complex systems, and 
finally, the structural parameters are optimized based 
on the optimization algorithms.
Due to the complex structure of the hydraulic 
rolling reshaper and the large number of related 
variables, the stepwise regression is chosen for fitting. 
Stepwise regression is a multiple linear regression 
method [14]. However, considering only the linear 
relationship between variables and responses will 
reduce the fitting effect of the model, so in order to 
consider the nonlinear relationship between variables, 
interactions, and responses, it is necessary to 
generalize linearity to nonlinearity by making a power 
series of single-factor variables and considering 
interactions to form a new set of variables before 
stepwise regression. In the process of regression, 
statistical tests should be carried out on each variable 
in each step, remarkable variables are automatically 
screened and added; the variables are introduced into 
the regression model in turn according to the influence 
degree of each variable on the response. There are two 
reference values for judging the significance, which 
are the values of inclusion (α
c
) and deletion (α
d
). If the 
p-value of the variable is less than or equal to α
c
, the 
variable is automatically identified and added to the 
regression model; if the p-value is greater than α
d
, the 
variable is eliminated. The final regression model is 
obtained when the p-values of all variables not in the 
model are greater than α
c
 and less than or equal to α
d
.
There are two coefficients to determine the 
fitting effect of the model: the coefficient of 
determination (R
2
) and the degree-of-freedom 
adjusted coefficient of determination (R
adj
2
), and the 
calculation methods are shown in Eqs. (2) and (3). R
2
 
is used to represent the goodness of fit for the 
regression models, and the higher the value, the higher 
the goodness of fit of the models. R
adj
2
 includes the 
number of design variables in the model, which can 
better reflect the pros and cons of the regression model 
[15]. Moreover, the closer R
2
 and R
adj
2
 are to 1, the 
better the fitting effect of the model is.
 R
SS
SS
E
T
2
1  , (2)
 R
SS tw
SS t
adj
E
T
2
1
1

 
 
, (3)
where t is the total number of samples, and w is the 
total number of terms in the regression model.
2  OVERALL RESEARCH IDEA
The optimization flow of the hydraulic rolling 
reshaper is shown in Fig. 3. Firstly, a preliminary 
analysis of the reshaper is required to determine the 
geometric parameters of the structure and the response 
variables of the optimization problem. Then, enter 
the experimental design stage, the orthogonal test is 
used to sample and obtain the different combinations 
of multi-geometric parameters, and the corresponding 
responses are solved by invoking the parametric 
finite element model repeatedly. Therefore, the 
factorial design of multi-geometric parameters is 
performed to screen out the remarkable factors 
and determine the design variables involved in the 
subsequent optimization. On this basis, OLHD is 
used for scientific sampling in the feasible region, and 
the responses of the sample points are calculated by 
invoking the finite element model, and the response 
surface model is established based on responses. Thus, 
with the maximum plastic deformation of the casing 
as the objective function, the maximum equivalent 
stress, residual stress, and the plastic deformation of 
the casing as the constrained conditions, an optimized 

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)11-12, 483-496
486 Li, H.F. – Luo, M. – Xu, T.T. – Li, Q.Z. – Hou, Y.M.
mathematical model for reshaper is constructed. 
Thereafter, GA is used to optimize and evaluate 
the shaping effect of the reshaper before and after 
optimization, and finally the optimal structure of the 
reshaper is obtained. 
3  MECHANICAL ANALYSIS OF SHAPING PROCESS
3.1  Parametric Finite Element Model
Fig. 4 is the basic structure of the hydraulic rolling 
reshaper. It is mainly composed of a work section, a 
charging section, and a connective section. The rolling 
balls in the work section are the key component in 
direct contact with the inner wall of the casing, and 
they are also the main working body to transmit 
the shaping load and achieve the repair goal, so the 
reshaper is simplified into the multi-stage tapered 
structure shown in Fig. 4. There are 7-stage balls in 
the reshaper, and the first 6 stages are called “shaping 
stage”, which are used to roll the inner wall of the 
casing to achieve the repairing goal and diameters of 
each stage increase in steps with the conical degree of 
1.145°. Moreover, the 7
th
 stage is called “reinforced 
stage”, which is used to strengthen the shaping effect. 
Furthermore, the total length of the reshaper is 300 
mm, and the axial distance of each stage is 50 mm. 
The rolling balls (the outer diameter is 20.64 mm) are 
Fig. 3.  Optimization flow chart

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)11-12, 483-496
487 Optimization Method of Multi-parameter Coupling for a Hydraulic Rolling Reshaper Based on Factorial Design 
embedded in the grooves located at the work section, 
and the exposed size of the ball relative to the surface 
of the reshaper is 5 mm. The main body diameter of 
the 1
st
 stage is 100 mm (the maximum equivalent 
diameter is 110 mm), and the diameters of the 6
th
 
and 7
th
 stages are 110 mm (the maximum equivalent 
diameter is 120 mm).
The shrinkage deformation casing is the object 
of this research, and the ideal casing with material 
P110 and specification of 5
1/2
 in ( Φ 139.7 mm × 9.17 
mm) is selected. The axial deformation length of the 
casing is 300 mm, and the minimum diameter (110 
mm) is located in the middle of the casing. To avoid 
the influence of the boundary effect, both ends of the 
deformed section are connected with intact casing 
(length is 100 mm), as shown in Fig. 5. Before the 
shaping process begins, the section of the 1
st
 stage on 
reshaper coincides with the A-A section shown in Fig. 
5.
Fig. 4.  Structural parameters of hydraulic rolling reshaper
Fig. 5.  Structural parameters of shrinkage deformation casing
According to the API 5CT standard for casing 
material properties, the P110 casing with an elastic 
modulus of 210 GPa, Poisson's ratio of 0.3, yield limit 
σ
s
 of 851 MPa, and strength limit σ
b
 of 933 MPa is 
taken as an example for calculation. In addition, the 
multilinear isotropic hardening model is used to 
characterize the stress-strain relationship of casing 
[16], as shown in Fig. 6.
To simplify the calculation, the assumptions have 
been given for the simulation as follows. (1) Simplify 
the reshaper into a rigid body. (2) Ignore the influence 
of temperature on the mechanical properties of the 
material during the shaping process. (3) The central 
axis of the reshaper always coincides with the axis 
of the casing during the shaping process. (4) Ignore 
the wall thickness variation of the deformed casing. 
(5) Ignore the influence of the initial residual stress of 
deformed casing on the shaping effect.
Fig. 6.  Stress-strain curve of casing
According to the structural characteristics of the 
reshaper, parametric modelling is carried out on the 
basis of clarifying the geometric relationship between 
the structural parameters, and the casing is discretized 
into solid elements by the finite element method. 
Moreover, target elements and contact elements are 
created on the outer surface of the reshaper and the 
inner surface of the casing, respectively. Considering 
the computational cost and accuracy comprehensively, 
the global grid density is determined, and the key 
geometric features (such as the contact interface 
between the reshaper and the inner wall of the casing) 
are refined with the local grid refinement. According 
to the loading features of the integral structure, the end 
faces on both sides of the casing are set as fully fixed 
constraints. The axial translational velocity of the 
reshaper is v = 1 mm/s, the rotating angular velocity 
is ω = 0.48 rad/s, and the friction coefficient between 
the outer surface of the reshaper and the inner wall of 

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)11-12, 483-496
488 Li, H.F. – Luo, M. – Xu, T.T. – Li, Q.Z. – Hou, Y.M.
the casing is 0.1. Therefore, the mechanical model and 
finite element model of the hydraulic rolling reshaper 
shaping deformed casing are shown in Fig. 7.
a) 
b) 
Fig. 7.  Models of the shaping process;  
a) mechanical model; and b) finite element model
3.2  Solving Algorithm
To reflect the material nonlinearity of the casing 
and the contact nonlinearity of the contact interface 
involved in the shaping process, the overall 
mechanical equilibrium equation of the shaping 
process is determined as
 K+ K= F+F
CC
    



, (4)
where δ is the node displacement array, K( δ) is the 
elastic-plastic stiffness matrix of casing material, 
K
C
( δ) is the contact stiffness matrix, F is nodal force 
array, F
C
 is the contact force array.
Since the elastic-plastic analysis is involved, 
and the nonlinearity of the contact area and contact 
pressure on the contact interface, the solution uses 
intermediate sub-steps, and the convergence of each 
sub-step is achieved by stepwise iterative solution, so 
as to track the load path correctly and complete the 
overall calculation. Therefore, the modified Newton-
Raphson method is chosen to solve the nonlinear 
problems involved in this paper. The reliability of the 
algorithm in solving the double nonlinear problems 
involved in the shaping process has been verified 
by Xu et al. [7] in our research group through the 
combination of numerical simulation and experiment.
3.3  Results and Discussion 
The shaping process is calculated for 700 s, and the 
maximum equivalent stress and radial displacement of 
the deformed casing under different times are shown 
in Fig. 8. 
The figures show that when the reshaper moves 
to 136.56 s, the 1
st
 stage ball of the reshaper starts 
to contact with the inner wall of the casing, and the 
casing begins to deform elastically. At 166.55 s, the 
equivalent stress of the casing exceeds the yield limit, 
and the casing produces plastic deformation. As the 
reshaper keeps moving forward, the stress and strain 
of the casing increase, and the plastic deformation 
area gradually expands. The maximum equivalent 
stress of the casing is 933 MPa at 315.08 s, and the 
maximum radial displacement of the casing is 5.133 
mm at 427.37 s. When the reshaper moves to 587.56 
s, the rolling balls are completely separated from the 
inner wall of the casing, the repairing load is unloaded, 
and the shaping process is completed. At this time, 
the residual stress of the casing is 809.38 MPa, and 
the radial displacement is rebounded by 0.213 mm 
due to elastic strain, so the plastic deformation of the 
casing after single shaping is 4.920 mm. Hence, the 
equivalent stress and radial displacement nephogram 
of the casing at important times are shown in Fig. 
9. However, it should be noted that the maximum 
equivalent stress of the casing reaches the strength 
a)      b)
Fig. 8.  Maximum equivalent stress and radial displacement of the casing under different times;  
a) maximum equivalent stress; and b) maximum radial displacement

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)11-12, 483-496
489 Optimization Method of Multi-parameter Coupling for a Hydraulic Rolling Reshaper Based on Factorial Design 
limit during the shaping process at 315.08 s, and the 
casing will be destroyed in the actual construction, so 
this reshaper cannot repair this deformed casing. 
4  FACTORIAL DESIGN
By analysing the mutually restrictive relationship 
between the geometric parameters of the reshaper, the 
following eight geometric parameters are selected as 
influencings to participate in the factorial design, as 
shown in Fig. 10: main body diameter of the 1
st
 stage 
(D
1
), reshaper’s taper (α), length of the shaping stage 
(L), helix angle (β, replaced by the pitch H while 
modelling, β=arctan(H/πD
1
)), diameter of the ball 
(D
b
), number of balls each stage (s), total number of 
stages of reshaper (k), exposed size of the ball (h).
a) 
b) 
Fig. 10.  Schematic diagram of related geometric parameters;  
a) front view of the reshaper; and b) B-B section
Considering the trafficability of the reshaper 
and the rationality of the structure, combined with 
the requirements of GB/T 308.1-2013, the range of 
eight influencing factors is obtained by preliminary 
analysis. According to the central composite design 
method [17] and [18], three levels of eight influencing 
factors are selected for the 8-factor 3-level orthogonal 
test, and the values are shown in Table 1. 
Table 1. Factor values for the orthogonal test
No. Factors Unit
Level
1 2 3
1 L mm 220 230 240
2 D
1
mm 97.5 98.5 99.5
3 α ° 1.15 1.2 1.25
4 h mm 3.5 4 4.5
5 H mm 400 600 800
6 D
b
mm 20.64 21.43 22.22
7 k stage 6 7 8
8 s piece 4 5 6
To reduce the downhole frequency of the reshaper 
and shorten the construction period of workover, it is 
necessary to ensure that the plastic deformation of 
the casing after single shaping is the largest under 
the safety of the casing when evaluating the shaping 
effect of the reshaper. At the same time, the residual 
stress of the shaped casing will also weaken its own 
load-bearing capacity, so it is necessary to minimize 
the residual stresses of the shaped casing. Therefore, 
the maximum equivalent stress (p
max
), residual stress 
(p
r
), and plastic deformation of the casing after 
Fig. 9.  Von-Mises stress and radial displacement nephogram at important times in repairing process;  
a) Von-Mises stress nephogram at 315.08 s; b) Von-Mises stress nephogram at 587.56 s;  
c) radial displacement nephogram at 427.37 s; and d) radial displacement nephogram at 587.56 s

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)11-12, 483-496
490 Li, H.F. – Luo, M. – Xu, T.T. – Li, Q.Z. – Hou, Y.M.
the absolute value of the t-value obtained by the t-test, 
and the ordinates are arranged in descending order 
of the value of the standardized effect. Based on the 
significance level α of 0.05, the statistical significance 
threshold of 2.101 is obtained, which corresponds to 
the critical value of the t-value of the red reference 
line in Fig. 11. The variable is statistically significant 
when p ≤  0.05  or  the  bar  in  the  Pareto  diagram  exceeds 
the reference line
 
[19]. Hence, it can be seen from the 
chart that the geometric parameters that have a great 
single shaping (d) are selected as response variables. 
The table of the 8-factor 3-level orthogonal test is 
generated by SPSS, and 27 combined schemes are 
obtained. By invoking the parametric finite element 
model, the responses corresponding to each combined 
scheme are obtained, as shown in Table 2.
The p-value and t-value of each factor for 
the three response variables are shown in Table 3, 
and the Pareto diagrams are shown in Fig. 11. The 
abscissa “standardized effect” in Fig. 11 represents 
Table 2.  Schemes and responses of orthogonal test
Sample 
No.
Geometric parameters Responses
L [mm] D
1 
[mm] α [°] h [mm] H [mm] D
b
 [mm] k [stage] s [piece] p
max
 [MPa] p
r
 [MPa] d [mm]
1 220 99.5 1.2 4 800 21.43 7 5 880.05 692.54 0.84
2 230 97.5 1.15 4.5 800 20.64 8 6 797.49 328.73 0.38
3 240 97.5 1.15 4 600 20.64 7 5 585.84 215.39 0.17
4 220 97.5 1.25 3.5 600 21.43 8 5 18.177 2.9887 0.01
5 240 99.5 1.15 4.5 600 22.22 7 4 930.36 752.92 1.68
6 220 99.5 1.25 4 600 20.64 8 4 905.80 670.05 0.95
7 230 97.5 1.25 4.5 400 21.43 7 4 889.39 677.74 0.58
8 220 99.5 1.15 4 400 22.22 6 6 856.19 656.66 0.58
9 230 99.5 1.2 3.5 600 21.43 6 4 847.77 650.73 0.41
10 240 98.5 1.25 3.5 800 22.22 6 4 836.07 520.20 0.34
11 220 98.5 1.2 4.5 800 20.64 7 4 895.91 692.24 0.63
12 230 99.5 1.25 3.5 400 20.64 7 6 870.60 705.28 0.72
13 240 99.5 1.25 4.5 800 20.64 6 5 931.53 806.39 1.91
14 230 98.5 1.25 4 400 22.22 7 5 875.67 667.63 0.75
15 230 97.5 1.2 4.5 600 22.22 6 5 849.1 606.62 0.54
16 240 97.5 1.2 4 400 22.22 8 4 760.19 322.70 0.21
17 230 98.5 1.15 4 800 21.43 8 4 793.61 418.97 0.30
18 230 98.5 1.2 4 600 20.64 6 6 847.55 577.83 0.47
19 220 98.5 1.15 4.5 400 21.43 6 5 870.77 708.42 0.55
20 240 98.5 1.2 3.5 400 20.64 8 5 738.45 300.02 0.25
21 220 97.5 1.2 3.5 800 22.22 7 6 0 0 0
22 240 97.5 1.25 4 800 21.43 6 6 847.05 536.95 0.47
23 240 98.5 1.15 3.5 600 21.43 7 6 636.87 230.57 0.64
24 230 99.5 1.15 3.5 800 22.22 8 5 782.71 403.77 0.37
25 220 98.5 1.25 4.5 600 22.22 8 6 892.80 679.19 1.13
26 220 97.5 1.15 3.5 400 20.64 6 4 0 0 0
27 240 99.5 1.2 4.5 400 21.43 8 6 927.77 809.90 1.93
Fig. 11.  Pareto charts of response variables

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)11-12, 483-496
491 Optimization Method of Multi-parameter Coupling for a Hydraulic Rolling Reshaper Based on Factorial Design 
influence on the response variables are D
1
, h, L, α, and 
k, so these five parameters are selected as the design 
variables for optimization.
Table 3.  The p-value and t-value of the response variables
Vari-
ables
p
max
p
r
d
p-value t-value p-value t-value p-value t-value
L 0.028 2.39 0.3 1.07 0.01 2.88
D
1
0.001 4.07 0 9.39 0 6.97
α 0.313 1.04 0.001 4.21 0.044 2.17
h 0.001 4.16 0 8.82 0 6.54
H 0.975 -0.03 0.239 -1.22 0.749 -0.32
D
b
0.792 0.27 0.405 0.85 0.908 0.12
k 0.735 -0.34 0.007 -3.06 0.802 0.25
s 0.818 -0.23 0.63 -0.49 0.238 1.22
5  OPTIMIZATION DESIGN
5.1  Elements of the Mathematical Model
The optimized mathematical model includes three 
elements: objective function, design variable and 
constraint.
1.  Objective function
The key to the hydraulic rolling reshaper is to 
ensure that the casing has a large plastic deformation 
after single shaping (d) under safe conditions to 
ensure a rapid and effective workover. Therefore, d is 
selected as the objective function.
2.  Design variable
Based on the requirements for performance and 
dimensions of the reshaper and its trafficability of 
deformed casing, the value ranges of the 5 design 
variables will be determined separately.
(1)  Main body diameter of the 1
st
 stage (D
1
)
At present, the maximum equivalent diameter of 
the reshaper is 120 mm, and it determines the diameter 
of the casing through which the reshaper can pass and 
the shaping effect on the casing. After analysis, the 
maximum equivalent diameter is affected by three 
geometric parameters: main body diameter of the 1
st
 
stage (D
1
), exposed size of the ball (h), and reshaper's 
taper (α), so the main body diameter of the 1
st
 stage is 
determined first. According to the working principle 
of the reshaper, the balls will retract into the grooves 
after the centre shaft is lifted up, and Fig. 12 shows 
the minimum limit state for the main body diameter of 
the 1
st
 stage when the balls are not exposed.
As can be seen from the figure, the contact point 
at the bottom corner of the groove is point B, point O 
is the centre of the centre shaft, point A is the contact 
point between the ball and the bottom surface of the 
groove, and the diameter of the ball is D
b
. According 
to the geometric relationship in the figure, 
DD
1
1
2 







tan
b
, and the diameter of the ball is 
20.64 mm, α = 30°. After calculation, the minimum 
D
1
 is 77.03 mm, and the safety factor is taken as 1.1, 
the minimum value of D
1
 is 84.73 mm which is 
rounded up to 85 mm. 
Based on past experience, the smaller the 
diameter of the work section, the easier to wedge 
into the minimum diameter of the casing for repair. 
Therefore, the main body diameter of the 1
st
 stage is 
100 mm as the upper limit of this design variable, so 
the range of D
1
 is determined to be 85 mm ≤ D
1 
≤ 100 
mm.
Fig. 12.  Minimum limit state for the main body diameter  
of the 1
st
 stage
(2)  Exposed size of the ball (h)
The balls will be extruded from the grooves when 
the charging section pushes the centre shaft forward, 
so the exposed size of the ball has a certain impact on 
the shaping effect of the casing, if it is too small, the 
shaping ability of the reshaper will be reduced, and the 
desired effect will not be achieved. Simultaneously, 
in order to ensure that the balls cannot fall out of 
the grooves, the exposed size of the ball should not 
be greater than 1/4 of its diameter, the range of h is 
determined to be 3 mm ≤ h ≤ 5.16 mm.
(3)  Length of the shaping stage (L)
The balls on the work section are spirally 
distributed and build the helical line by controlling 
the pitch and cycle number in parametric modelling. 
Therefore, the number of turns of the spiral is set to 
one, and the helix angle is replaced by the pitch for 
representation. To facilitate the subsequent analysis, 
the shaping stage of the reshaper is expanded along 
the ring direction of the 1
st
 stage, and the side unfolded 
drawing of spirally arranged balls on the shaping stage 
is obtained, as shown in Fig. 13.

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)11-12, 483-496
492 Li, H.F. – Luo, M. – Xu, T.T. – Li, Q.Z. – Hou, Y.M.
Fig. 13.  Side unfolded drawing of spirally arranged balls  
on the shaping stage
The reference circle is established by the 
circumference of the 1st stage, and the circumference 
of the reference circle is πD
1
 after expansion, so 
the conversion relationship between pitch (H) and 
helix angle (β) is β = arctan(H/πD
1
). For structural 
parameters of the reshaper, D
1
 = 100 mm, H = 600 
mm, so the helix angle is equal to 62.36°. Take 
the geometric relationship between the 1
st
 and 2
nd
 
balls as an example (Fig. 14). Combined with the 
practical processing technology, at least 1/2 diameter 
processing allowance should be left between each 
stage, MN = 30.96 mm, so the minimum interval 
between each stage is MN sinβ. After calculation, the 
minimum interval between each stage is 27.43 mm.
Fig. 14.  Schematic diagram of the geometric relationship between 
the 1st and 2nd balls
The hydraulic rolling reshaper is one of the 
main shaping tools for repairing deformed casing 
in horizontal wells. To improve the applicability of 
the reshaper so that it can pass through a horizontal 
well with a smaller radius, the length of the reshaper 
should be reduced on the current basis. The total 
number of shaping stages should be less than 9 
(250 mm / 27.43 mm = 9.11) with 8 intervals, so the 
length of the shaping stage should be greater than 
27.43 × 8  = 219.44 mm and rounded upward to 220 
mm. Hence, the range of L is determined to be 220 
mm ≤  L ≤  250 mm.
(4)  Reshaper’s taper (α)
The inner diameter of the intact casing is 121.36 
mm, and if the exposed size of the ball reaches the 
maximum (5.16 mm), the maximum diameter of the 
main body is 111.04 mm. According to the expression 
of the reshaper’s taper:  
 





arctan
DD
L
k 1
. (D
k
 
indicates the main body diameter of the k
th
 stage), the 
main body diameter of the 1
st
 stage ranges from 85 
mm to 100 mm through the analysis of point (1). After 
calculation, when D
1
 = 85 mm, α ≤  3.387°, when 
D
1
 = 100 mm, α ≤  1.437°. The current reshaper's taper 
is 1.145°, in order to obtain the remarkable result of 
shaping, the range of α is determined to be 
1.145° ≤  α ≤  1.437°.
(5)  Total number of stages of reshaper (k)
The balls in the reinforced stage play a role in 
consolidating the shaping effect, so it is necessary to 
set up 1 stage to strengthen. For the shaping stage, 
in addition to the first and final stages of the shaping 
stage, at least two stages of balls should be set between 
these two stages to ensure that the reshaper has a more 
uniform shaping effect during the spinning process, 
so the total number of stages of reshaper should be at 
least 5 stages, as shown in Fig. 15.
Fig. 15.  Structure diagram of 5-stage reshaper
According to the analysis of point (3), the 
minimum interval between each stage is 27.43 mm, 
and the number of shaping stages is not higher than 
9. In addition, it is necessary to add another stage for 
reinforcement; then, the total number of stages of 
reshaper should not be higher than 10, so the range of 
k is determined to be k = 5, …, 10.
3.  Constraint condition
According to the simulation results in Section 2.2, 
the non-optimized shaper cannot repair the deformed 
casing with the axial deformation length of 300 mm 
and the minimum diameter of 110 mm. To ensure the 
integrity of the casing, the maximum equivalent stress 
of the casing is required to be lower than the strength 
limit, and the residual stress of the casing after 

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)11-12, 483-496
493 Optimization Method of Multi-parameter Coupling for a Hydraulic Rolling Reshaper Based on Factorial Design 
shaping is lower than the yield limit. Considering a 
certain safety margin (1 %), the maximum equivalent 
stress of the casing is lower than 923.67 MPa, and the 
residual stress is lower than 842.49 MPa. Therefore, 
the maximum plastic deformation of casing that can 
be shaped is 5.68 mm ((121.36 – 110) / 2 = 5.68 mm), 
and d should be less than 5.68 mm.
5.2  Response Surface Model
OLHD is used to conduct scientific sampling in the 
feasible region of the design variables, and 50 
sample points are obtained, which are brought into 
the finite element calculation module to calculate 
the corresponding responses, and the nonlinear 
interactive regression model of three response 
variables is established by stepwise regression, as 
shown in Eqs. (5) to (7). R
2
 and R
adj
2
 of the three 
regression models are shown in Table 4, and it is 
considered that the regression models have high 
accuracy.
pL D
hh
max
  

1639917 7983 16495 1436683
29129 3837 574 2
1
22

 .7 79 5
6467
288 22
21 6 7975 14220
7975 63
1
1
1
.
.
.
LD
LL hhD
Dh h

  

 
 .. ., 51 37 4
1
LD Lh   (5)
pD h
kD D
r
  



2668966 72108 197139
14 37 640
32808
314 4
1
1
2
1

 .. 4 4054
507 1 861 20 91
1
11
32
Dh
hD Dh

  ., . (6)
 
dL D
hh
  

58 12 0 0221 0 50495 4 476
05 70 1845
1
2
.. ..
...

 (7)
Table 4.  R
2
 and R
adj
2
 of the regression model
Responses R
2
 [%]
R
adj
2
 [%]
p
max
99.92 99.83
p
r
97.39 95.76
D 99.55 99.44
To summarize, combined with Eqs. (5) to (7), 
the multi-parameter coupling optimization of the 
hydraulic rolling reshaper is transformed into an 
optimized mathematical model with constraints as
 
max
s.t .67 MPa
42.49 MPa
mm
m
r
dLDh k
p
p
d
(, ,,)
.
.
max
,
1
923
8
56 8
220




m mm m
mm mm
mm mm





L
D
h
k
250
85 100
1 145 1 437
35 16
5
1
..
.
,...,


1 10
. (8)
5.3  Comparison of Optimization Results
For the optimization problem of hydraulic rolling 
reshaper, the geometric parameters have both 
continuous and discrete variables, which will lead to 
a higher degree of nonlinearity in the optimization 
process. For such highly nonlinear optimization 
problems, a genetic algorithm is more advantageous 
[20]. The number of the initial population is 5000, 
the samples’ number for one iteration is 1000, the 
maximum allowable Pareto percentage is 70 %, 
the stable convergence percentage is 2 %, and the 
maximum number of iterations is 20 [21] and [22]. 
The iteration converges when the number of iterative 
evaluations is 13556, and the optimal solution of this 
optimization problem is obtained. The comparison 
with the structural parameters before the reshaper 
optimization is shown in Table 5.
Table 5.  Comparison of structural parameters of reshaper before 
and after optimization
Geometrical parameters
Non-
optimization
Optimization
Design 
variables
L [mm] 250.00 238.68
D
1
 [mm] 100 100
α [°] 1.145 1.345
h [mm] 5.000 5.131
k [stage] 7 7
Other 
variables
Axial distance of each 
stage [mm]
50.00 47.74
Length of reshaper [mm] 300.00 286.42
Equivalent diameter of 
stage 1 [mm]
110.00 110.26
Maximum equivalent 
diameter [mm]
120.00 121.41
The parametric finite element model is called 
again to simulate the shaping process of the deformed 
casing with an axial deformation length of 300 mm 
and a minimum diameter of 110 mm. The minimum 

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)11-12, 483-496
494 Li, H.F. – Luo, M. – Xu, T.T. – Li, Q.Z. – Hou, Y.M.
diameter in the middle of the casing is a dangerous 
position in the shaping process; its shaping effect 
is also worthy of attention. Fig. 16 shows the 
distribution of plastic deformation at the minimum 
diameter of the casing before and after optimization. 
It can be seen that the shaping effect of the casing is 
more uniform while the plastic deformation of the 
casing is expanded. The maximum equivalent stress 
and radial displacement of the casing before and after 
optimization under different times are shown in Fig. 
17, and the comparison of the evaluation parameters is 
shown in Table 6.
Table 6.  Comparison of the results of evaluation parameters before 
and after optimization
Results  
comparison
Is the casing 
reshaped 
safely?
Effect of 
reshaping
[%]
p
max
[MPa]
p
r
[MPa]
d
[mm]
Optimization Yes 96.48 922.97 799.25 5.480
Non- 
optimization
No / 933.00 809.38 4.920
Degree of 
optimization [%]
/ / -1.08 -1.25 11.38
Depending on the above chart, it can be seen that 
the optimal reshaper can safely repair the shrinkage 
deformation casing with a minimum deformation 
diameter of 110 mm and an axial deformation length 
of 300 mm. The shaping effect of the casing is the ratio 
of d to the maximum plastic deformation of the casing 
that can be shaped, so the shaping effect reaches 96.48 
%. In addition, compared with the non-optimized 
reshaper, p
max
 is reduced by 1.08 %, p
r
 is reduced by 
1.25 %, and the area of high-stress distribution on the 
Table 7. Comparison of evaluation parameters for repairing deformed casings before and after optimization
Minimum diameter 
[mm]
Evalua tion para meters Non-optimi zation Optimi zation
Degree of optimi zation 
[%]
Effect of reshap ing 
[%]
112
p
max
 [MPa] 933.00 920.58 -1.33
96.30 p
r
 [MPa] 793.38 804.97 1.46
d [mm] 3.725 4.507 20.99
114
p
max
 [MPa] 932.19 920.1 -1.30
93.75 p
r
 [MPa] 805.72 805.63 -0.01
d [mm] 2.758 3.450 25.09
116
p
max
 [MPa] 930.45 917.76 -1.36
91.49 p
r
 [MPa] 805.27 779.03 -3.26
d [mm] 1.676 2.452 46.30
118
p
max
 [MPa] 852.77 908.2 6.50
93.21 p
r
 [MPa] 564.26 739.6 31.07
d [mm] 0.542 1.566 188.93
deformed casing is reduced. Also, d is increased by 
11.38 %, which can effectively improve the efficiency 
of workover.
Fig. 16.  Distribution of plastic deformation at the minimum 
diameter of the casing before and after optimization
Furthermore, the reshaping effect of the casings 
with different deformation diameters before and 
after optimization is calculated, and the results and 
comparisons of the evaluation parameters are shown 
in Table 7. It can be seen from the table that the 
optimized reshaper can safely repair the deformed 
casing, and the reshaping effect of the casing and the 
single shaping ability of the reshaper are improved.
6 CONCLUSIONS
The present work presents a multi-parameter coupling 
optimization method for the complex structure of a 

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)11-12, 483-496
495 Optimization Method of Multi-parameter Coupling for a Hydraulic Rolling Reshaper Based on Factorial Design 
hydraulic rolling reshaper. The deformed state and 
stress distribution of the casing during the shaping 
process are discussed in detail, and an optimization 
method for screening multi-factors is given. Some 
interesting conclusions can be drawn from the results:
1. Based on the structural characteristics and 
working principle of the hydraulic rolling 
reshaper, considering the dual nonlinear 
characteristics of the contact and material, a 
parametric finite element model of hydraulic 
rolling reshaper repairing shrinkage deformation 
casing is developed.
2. The orthogonal test is performed to analyse 
and screen the eight geometric parameters of 
the reshaper, and the following five structural 
parameters are determined which have a 
significant effect on the shaping effect: the main 
body diameter of the 1
st
 stage, exposed size 
of the ball, the reshaper’s taper, length of the 
shaping stage, and the total number of stages of 
the reshaper.
3. By analysing the coupling relationship between 
structural parameters and combining it with the 
processing requirements, the value range of the 
five significant factors is reasonably determined, 
the regression model of the responses is obtained 
by stepwise regression method, and the multi-
parameter coupling optimization mathematical 
model of a hydraulic rolling reshaper is obtained.
4. The optimal structure of the reshaper is obtained 
after optimization: the main body diameter of 
the 1st stage is 100 mm, the exposed size of the 
ball is 5.131 mm, the reshaper’s taper is 1.345°, 
the length of the shaping stage is 238.68 mm, 
and the total stages of reshaper is 7. Moreover, 
the optimized reshaper can protect the casing 
better and improve the shaping effect of the 
casing with different deformation diameters, 
which can effectively improve the efficiency of 
the workover.
In addition, to facilitate the calculation, the 
numerical simulation in this paper has been simplified 
to a certain extent. To better reflect the stress change 
and distribution law in the shaping process, more 
factors that may affect the shaping effect can be 
considered, such as: considering the rolling contact of 
the ball with the casing and the body of the reshaper, 
considering the change of high-temperature material 
properties in the shaping process and the influence of 
the initial residual stress of the casing on the shaping 
effect, etc., so as to further improve the level of 
simulation and improve the optimization results.
7  ACKNOWLEDGEMENTS
The authors are thankful for the financial support of 
the National Natural Science Foundation of China 
[51674088] and the Natural Science Foundation of 
Heilongjiang [LH2021E011].
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