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Multi-objective optimization and experiment of nylon cord rubber in expandable packer

Multi-objective optimization and experiment of nylon cord rubber in expandable packer Nylon cord rubber has the advantages of small residual deformation and is easy to lift and lower the tubing string in low- permeability oil and gas reservoirs. However, it is associated with low-pressure resistance and poor sealing performance. To enhance the performance of nylon cord rubber, a three-dimensional numerical model of the nylon cord rubber was established and its accuracy experimentally determined. The Plackett–Burman test, the Steepest climbing test and the Response surface method were used to acquire the polynomial response surface model connecting structural parameters with bearing and sealing pressure. Using genetic algorithms, optimal structural parameters of nylon cord rubber were determined depending on field operation. The reliability of the optimized results was verified by laboratory tests. It was shown that after optimiza- tion, the bearing capacity of the expandable packer increased by 25% while the sealing performance increased by 66%. In addition, the bearing pressure was 70 MPa while the sealing pressure was 50 MPa. These measurements effectively met the on-site requirements of high-pressure and fine fracturing in low-permeability oil and gas reservoirs. Keywords Low-permeability reservoirs · Nylon cord rubber · Numerical simulation · Response surface method · Multi- objective genetic algorithm · Laboratory test List of symbols 1 Introduction X Angle, degree X Layer Low-permeability oil and gas reservoirs have become X Spacing, mm important exploratory and development fields in the X R ubber thickness, mm world (Zou et al. 2017; Mu and Ji 2019). Horizontal well X Shoulder angle, mm small-hole fracturing technology is an important produc- P Bearing pressure, MPa tion stimulation measure for low-permeability oil and gas max C Sealing pressure, MPa reservoirs (Lei et al. 2018; Qu et al. 2019; Agarwal et al. max DF Degree of freedom 2019). In reservoirs, the expansion packer determines SS Mean bias, MPa the outcomes of the fracturing technology. The expand- MS Mean square, MPa able packer isolates the tubing and the borehole wall and F Statistical magnitude then forms an annular space in Fig. 1. The annular space separates the oil and gas to achieve a layered fracture. As the core component of the expandable packer, the rubber affects fracturing during construction (Guo and Gao 2013). Currently, overlapped steel belts and steel cord rubber cylinders are used in rubber, but the residual deformation is large. When the pipe string is lifted up, Edited by Xiu-Qiu Peng the steel belt and steel cord are more likely to become stuck in the casing leading to underground accidents * Han-Xiang Wang (Patel et al. 2019a, b). Nylon cord rubber has the advan- wanghxupc@163.com tages of small residual deformation and is easy to lift and College of Mechanical and Electronic Engineering, China lower the tubing string in low-permeability oil and gas University of Petroleum (East China), Qingdao 266580, reservoirs (Zhong et al. 2015; Akhtar et al. 2018; Pradie Shandong, China Vol.:(0123456789) 1 3 270 Petroleum Science (2021) 18:269–284 et al. 2008). However, it is associated with low-pressure- both the axial and hoop compressive stress concentrations bearing and poor sealing performance. It is, therefore, were generated in the thread teeth edges near the contact important to analyze and optimize the structural param- surfaces of threads. This was associated with the mutual eters of the nylon cord rubber in expandable packer to squeezing of box and pin thread teeth during experimental meet the on-site requirements of high-pressure and fine verification. Qamar et al. (2009), Al Ramadan et al. (2019), fracturing in low-permeability oil and gas reservoirs (Li Daou et al. (2014) mechanically tested and characterized et al. 2017; Tian et al. 2019). an inert water-swelling elastomer that had been developed Studies on the sealing properties and optimization of by a local petroleum development firm. The elastomer the structural parameters of packer rubber through theo- was tested for hardness, compression at different tempera- retical calculation, numerical simulation and experimental tures and for different periods of time, tensile strength at verification are being done. For theoretical calculation, Al- different strain rate, tensile properties regarding fracture Hiddabi et al. (2015), Al-Abri et al. (2015), Renaud et al. strength and percent elongation, and swelling ratio. Ahmed (2009) investigated the deformation of an elastomer seal et al. (2019a, b), Al Ramadan et al. (2019) performed the confined between a metal tube and a rigid casing. They verification process and tests by critically reviewing the showed the effect of the geometry of the elastomer geom- literature, current regulations, and applicable industrial etry and its material properties on sealing performance standards in order to develop testing protocols for the in terms of maximum sealing pressure. Alkharusi et al. investigation of the performance of common elastomeric (2011), Gajewski et al. (2015), Akhtar et al. (2018) investi- seals that are used in a liner hanger seal assembly. Dong gated the effects of the material and geometrical properties et al. (2020), Fothergill (2003) determined seal failures of the elastomer on its sealing performance under different of the rubber tube at high temperatures and studied the loading conditions. In addition, they investigated the effect constitutive model parameters of the rubber tube through of radial strain and annular fluid pressure on the sealing the rubber thermal aging experiments. The effects of key performance. Agata et al. (2013) described the material and parameters of the rubber tube-casing gap, the dip angle form factors that regulated the ability of the pipe to expand. of the adjacent rubber tube contact surface, and the ini- These factors included the influence of axial restraint dur - tial setting load on the sealing performance of the packer ing expansion and the post-expansion collapse resistance under high temperature conditions were analyzed. Chen of solid expandable tubulars. Banks et al. (2002) studied et al. (2019), Grelle et al. (2019) determined the effects of the compression of rubber bonded to rigid metal plates different stress conditions and the speed of lifting or lower - of different geometry (rectangular and V-shaped blocks). ing the pipe on the weakness of the rubber matrix. Zhang and Wang (2016), Zhang et al. (2018) while relying The above described studies determined contact stress on the laws of momentum and energy conservation and the distribution of the rubber cylinder under different structural transient heat transfer property between the wellbore fluid parameters, and optimized the rubber structure with the and the annulus fluid developed a calculation model of the maximum contact stress as the goal. However, the tearing temperature and pressure fields on single-layer and multi- failure caused by excessive internal stress of the rubber was layer annuli. Cavalaro and Aguado (2012) characterized not considered. These studies did not consider the effect of the behavior of the packer under simple stress (normal) cord arrangement parameters on the performance of the rub- and under coupled stresses (normal and tangential) as well ber. This resulted in large variations between the simulation as proposed mathematical constitutive models to describe results and the actual condition. In addition, the optimization both behaviors. Their results indicated that the packer pre- method utilized the maximum contact stress as the single sented a nonlinear almost elastic mechanical behavior from optimization goal and ignored the internal failure stress. This the second load cycle onwards. For numerical simulation, could not improve the overall performance of the expand- Hu et al. (2017, 2018), Patel et al. (2019a, b) studied the able packer. In this paper, the numerical simulation model of influence of three rubber materials on sealing performance nylon cord packer was established by considering the inner of packing element in the compression packer. Wang et al. cord action of the drums. With the aim of sealing perfor- (2017), Lan et al. (2019) studied the structure of the packer mance and pressure-bearing performance, a combination of rubber with different materials and optimized the struc- response surface method and a genetic algorithm were used ture of its sealing unit. Based on the simulation results, to optimize the design of nylon cord rubber. Specifically, the the best inner groove shape and the structure of shoulder Plackett–Burman test was used to determine the climbing protection were confirmed. Furthermore, the validity of direction of influencing factors. The steepest climbing test the simulation results was experimentally confirmed. Xu was used to obtain the center of the response surface. The et al. (2017) investigated the structural response of tubulars response surface method was used to acquire the polyno- to stress evolution, deformation of the thread joint and to mial response surface model connecting the cord angle, the the effect of expansion cone geometry. It was found that number of cord layers and the cord spacing with bearing and 1 3 Petroleum Science (2021) 18:269–284 271 relieved, the packer barrel automatically shrank to complete the unsealing by depressurizing the ground. The tearing failure under excessive setting pressure and sealing failure caused by low contact pressure of the expand- able packer were common reasons for failure. To enhance Oil tube the total working performance, we improved the bearing pressure and sealing pressure of the expandable packer. Safety joint The bearing pressure represented the ultimate setting pres- sure before packer failure which can expand the packer and encapsulate the annular space between the casing and the Hydraulic 2# oil layer center pipe. At the same time, the sealing performance of anchor the expandable packer depended on the contact pressure. Increasing the contact pressure enhanced the sealing perfor- mance. When analyzing the expandable packer, the maxi- mum contact pressure along the sealing surface was consid- Expanded packer ered to be the sealing pressure under the setting pressure. The working condition of the expandable packer is shown in Fig. 3. In order to improve the bearing pressure and sealing pres- Sand blast 1# oil layer sure, the nylon cord rubber adopted the nylon cord as skel- packer eton, which was composed of an inner and an outer rubber. The structural parameters of the nylon cord rubber used in the oilfield are shown in Table  1. Laboratory tests indicated Plug that the maximum set pressure of the current expandable packer was 60 MPa and the maximum sealing pressure was 30 MPa. These pressures could not meet the site require- ments of low-permeability oil and gas reservoirs with a set pressure of 70 MPa and a sealing pressure of 50 MPa. There- fore, the parameters of cord arrangement and barrel structure Fig. 1 The schematic diagram of fracturing string should be optimized. 2.2 Establishment of the finite element model sealing performance. Optimal structural parameters of the nylon cord rubber were determined according to the require- The finite element model of the nylon cord rubber is shown in Fig.  4. Due to the symmetry of expandable packer, a ments of field operations. To meet the high-pressure and fine fracturing on-site use requirements, the reliability of the quarter of the three-dimension finite element model was developed with the Y-axis as the symmetrical axis. The optimized results was verified by laboratory tests. SOLID185 element was used for the plastic tube, center tube, sleeve, and upper and lower joints while the REINF265 2 Structure and analysis method element was used as the polyamide cord reinforcement mate- rial. The 70 MPa setting load was applied inside the expand- 2.1 Structure of nylon cord rubber tube able packer. According to the indoor material test (ASTM D573-04 2015), the material parameters of nylon cord rub- The nylon cord packer is shown in Fig. 2. It is composed of ber cylinder are shown in Table 2 (Wang et al. 2020). In terms of material failure, the third strength and maxi- the upper joint, upper steel bowl, vulcanized core, central tube, rubber cylinder, lower cylinder liner and lower joint. mum stress criteria were utilized for stress analysis of expandable packer in formula (1). In particular, shear fail- At work, the expansion fluid was pumped into the nylon cord packer rubber from central tube through the ground ure was common inside the expandable packer. The third strength criterion states that the plastic flow occurs when booster pump. When the pressure difference between the internal and external oil pipes reached the packer starting the maximum shear stress reaches its shear strength. This phenomenon was attributed to the failure mechanism of pressure, the rubber tube expanded. After the pressure was 1 3 272 Petroleum Science (2021) 18:269–284 12 34 56 78 91011121314 (a) (b) (c) Cord layer Cord angle Cord spacing Cord angle Rubber thickness (d) Fig. 2 Structure diagram of the downhole expandable packer. 1-upper joint; 2-upper steel bowl; 3-screw; 4-plug nail; 5-vulcanized core; 6-cen- tral tube; 7-cylinder; 8-vulcanized core; 9-cylinder liner; 10- Piston; 11-lower cylinder liner; 12-”O” seal; 13-lower steel bowl; 14-lower joint the rubber. The maximum stress criterion which states that material damage occurs when the maximum stress stretches Annular space Center pipe Setting Contact pressure pressure Rubber Casing Fig. 3 The working condition of expandable packer Table 1 Structure parameter of nylon cord packer to the tensile strength was associated with damage to the fiber. According to the above material criteria, the different Parameters Value parameters of nylon cord rubber were optimized to meet Cord angle, degree 35 the site use requirements of low-permeability oil and gas Cord layer 10 reservoirs (70 MPa setting pressure and 50 MPa sealing Cord spacing, mm 2 pressure). Shoulder angle, degree 45 Rubber thickness, mm 18 1 3 Petroleum Science (2021) 18:269–284 273 (a) Physical model level was marked (+ 1). The low-level mark was (− 1). The optimal level of each factor was determined and the key fac- 50 mm 350 mm tors selected. The Plackett–Burman test parameter design is shown in Table 3. 500 mm 2.3.2 The Steepest Climbing Test Design (b) Simulation model 2 5 7 1 3 4 6 The steepest climbing test design utilizes the direction of the gradient of the test value as the climbing direction. It deter- mines the step size of the change according to the effective value of each factor, which can quickly and economically approach the optimal response area. During the test, and based on Plackett–Burman test results, the design direction of the influencing factors was changed, and the steepest (c) Cord model climbing test of the pressure resistance performance P max and sealing performance C of the nylon cord rubber was max determined. 2.3.3 Response Surface Test Design Fig. 4 Nylon cord rubber finite element model The response surface test was designed by experiment- 2.3 Optimal design test methods ing on a set of sample points in a specified design space. The global approximation function of the system can be Because the response surface model has a high fitting accu- approximated to replace the actual response surface. In an racy in the center neighborhood while the fitting accuracy engineering optimization design, the response relationship outside the center neighborhood is low, the fitting equation between the response target and the design variables can be was almost meaningless. Using the ANSYS software, the achieved through the response surface test, and the design Plackett–Burman test was used to screen for key factors, the variables under the optimal objective function can then be steepest climbing test was applied to approximate the best area, and a polynomial model connecting the influencing Table 3 Plackett–Burman test design factors with pressure resistance and contact performance of Impact factor Level the nylon cord rubber was established by response surface method. Parameters Low level (− 1) High level 2.3.1 Plackett–Burman Test Design (+ 1) Cord angle 20 30 During field investigations, the cord angle, the number of Cord layer 2 4 cord layers, cord spacing, the end inclination angle, and Cord spacing 2 3 the thickness of the barrel were the test factors. Each test Shoulder angle 45 60 factor was taken at two levels for the test design. The high Rubber thickness 17 19 Table 2 Material parameters of nylon cord rubber Component Material Density, g/cm Elasticity modu- Constitutive model Poisson’s ratio Tensile lus, MPa strength, MPa Nylon cord Nylon 1.15 2.80 × 10 – 0.34 190 Rubber HNBR 1.56 – Yeoh Model – 17 C = 0.15 C = − 1.29 C = 0.61 Central tube-casing pipe 45MnMo7 7.8 2.06 × 10 – 0.24 – Upper and lower joints 4145H 7.8 2.06 × 10 – 0.24 – 1 3 66 mm 29 mm 37.5 mm 274 Petroleum Science (2021) 18:269–284 determined.The second-order polynomial model is often the respectively. The independent variables are coded according construction of the response surface approximation model. to formula (2). The distribution of test points is shown in The approximation is the relationship between the system Fig. 5 while the design of response surface test parameters input and the response target. The basis function is shown is illustrated in Table 4. in formula (1): X =(x − x )∕Δx (2) i 1 0 k k k whereby X is the encoding value of independent variable, x y = 𝛽 + 𝛽 x + 𝛽 x + 𝛽 x x i 1 0 i i ii ii i j i (1) i=1 i=1 i=1 is the true value of the independent variable at the test center i<j point, and Δx is the step size of the independent variable. whereby β is the interaction coefficient between offset, linear offset and second-order offset; k is the total number of design variables; Y is the predicted response value. Table 4 Response surface test design Box-Behnken Design (BBD) is a common test design Test Parameters Central point Low level High level method for response surface testing. The number of tests is compact, and the economy is good. It is suitable for opti- 1 X 17.5 17 18 mization experiments with 2 to 5 factors. Each factor takes 2 X 7 6 8 3 levels, with 0 as the center point, (+  1) and (−  1) are 3 X 1.75 1.7 1.8 the high and low values corresponding to the cubic points, Determine the type of optimization problem constraint Gamultiobj solve Creat initial population Get the optimal Whether or not to exit solution Pareto Population evolution next generation Plot optimization results Judge termination condition Fig. 5 Flow chart of multi-objective genetic algorithm 1 3 Petroleum Science (2021) 18:269–284 275 2.3.4 Genetic Algorithm Optimization lower end of the packer was then connected with blind plug- ging to block its outlet. The entire expandable packer was Genetic algorithm is a type of adaptive artificial intelligence soaked in water at 18–28 °C for 24 h. It was then installed in technology that simulates the evolutionary processes of bio- the casing. The pressure pump (100 MPa) was connected to logical organisms and the solution of extreme values. Its the center pipe through the center pipe sealing cover while basic idea is an algorithm that searches for the optimal solu- the pressure pump (60 MPa) was connected with an annular tion formed by simulating the genetic mechanism of nature between the casing and tubing through the casing sealing and biological evolutionary theory. It is suitable for solv- ing complex nonlinear and multidimensional optimization (a) Install problems. Multi-objective genetic algorithm, as a fast and effective global optimization algorithm, has a brisk running speed. The solution set has the advantage of excellent con- vergence. The specific optimization process was as shown in Fig. 5. (b) Soak 2.4 Laboratory test To verify the accuracy of the simulation results, indoor tests of the nylon cord rubber before and after the optimization were performed. The dimensions of the experimental setup and the simulation model were consistent, and the specific (c) Pressure test constitution of the experimental setup was as follows: (1) 1/2 Packer experimental rig including 5 casing (with an outer 7/8 diameter of 139.7 mm), 2 tubing (with an outer diameter of 78.6 mm), flange and plug hear; (2) K344-114 expansion packer on which a rubber be can installed; (3) High-pressure (d) Check pumps (100 MPaand 60 MPa); (4) Oil bath device and pipe line (Wang et al. 2020). The schematic diagram of the labo- ratory test setup was as shown in Fig. 6, and the process of experimental measurement was as shown in Fig. 7. Briefly, the rubber was assembled on the expandable packer with an amount of butter being applied on the sealing groove and Fig. 7 Polyamide cord rubber test diagram engine oil being applied on the surface of the thread. The Pressure gage End cap Top 5 1/2 connection casing Drainage K344-114 Rubber Throttle 27/8 tubing Tube Pump Pump (100 MPa) (60 MPa) Casing pipe Thermometer Casing pressure Casing pressure Water tank Oil bath tank Tubing pressure Fig. 6 Laboratory equipment layout diagram 1 3 276 Petroleum Science (2021) 18:269–284 cover. The pressure gauges of the center and casing pipelines Table 5 Plackett–Burman test results were set and used to observe the working conditions of the Test Angle, Layer Spacing, mm Rubber Shoulder expandable packer. In addition, the casing valve was closed degree thickness, angle, mm and the center pipe valve opened. The high-pressure liquid mm was driven through the high-pressure pump (100 MPa) into 1 20 4 3 60 17 the center pipe. After pressing the inside packer within a 2 30 4 2 60 17 pressure range of 0–80 MPa and stabilizing the pressure for 3 20 2 2 45 17 5 min every 10 MPa, the working conditions of the packer 4 20 4 2 45 17 could be observed. If the pressure gauge in tube pipeline 5 20 2 3 60 19 rose steadily and remains steady for 5 min, the pressure test 6 30 2 3 45 17 pump continued to exert pressure based on the existing pres- 7 20 4 3 45 19 sure. When the pressure gauge experienced a rapid decline 8 30 4 3 45 19 with sharp fluctuations, the pressure test pump stopped and 9 30 2 2 45 19 the packer was considered to have been damaged. The maxi- 10 30 2 3 60 17 mum setting pressure before packer failure during the period 11 30 4 2 60 19 of pressure stabilization was known as the bearing pressure. 12 20 2 2 60 19 After pressing the packer from the center pipe at 10 MPa to the bearing pressure, the center pipe valve was closed and the casing valve opened. The sealing pressure under die ff rent cord spacing and shoulder angle exhibited positive effects setting pressures was measured. To be specific, the high- pressure liquid was driven through the high-pressure pump while the cord angle, cord layers and shoulder angle exhib- ited negative effects. The influential order of each factor on (60 MPa) into the annulus between center pipe and casing from 5 to 60 MPa for 5 min every 5 MPa. If the pressure sealing performance was: cord layer > cord angle > cord spacing > rubber thickness > shoulder angle. gauge in the casing pipeline rises steadily and remains stead for 5 min, the pressure test pump continued to exert pressure Above all, it can be seen that nylon cord parameters (cord angle, cord layer and cord spacing) exerted a more obvious based on the existing pressure. When the pressure gauge experienced a rapid decline with sharp fluctuations, the pres- effect on bearing pressure and sealing pressure when com- pared to rubber parameters (rubber thickness and shoulder sure test pump was stopped and the packer was considered to have been damaged. The maximum annual working pressure angle). This could be attributed to the fact that nylon cord was the frame of expandable packer on which the external before packer failure during the period of pressure stabiliza- tion was shown in the pressure gauges in casing pipeline. pressure was exerted. In addition, changes in cord angle, cord layer and cord spacing played a significant role in bear - It was described as the sealing pressure under the setting pressure. After the test, pressure was released and the packer ing pressure and sealing pressure. Therefore, we determined the inu fl ence of nylon cord arrangement on the bearing pres - was restored to its original state. The failure morphology of expandable packer was observed. sure and sealing pressure of the expandable packer and other factors were in agreement with the site. It can be found that with an increase in cord angle, the radial stiffness of the rubber grew. This led to cord breakage and low sealing pres- 3 Results and discussion sure that resulted to poor bearing pressure. Meanwhile, as the cord layer increased, the vertical stiffness of the packer 3.1 Plackett–Burman test results grew synchronously. This improved the bearing pressure. Considering the high vertical stiffness, the sealing pressure The Plackett–Burman test results are shown in Table 5. The diagrammatic presentation of the Pareto standardization roughly reduced under the same setting pressure. Further- more, the bearing pressure experienced a reduction when effect of pressure-bearing performance is shown in Fig.  8. The number of cord layers exhibited a positive effect while the cord spacing grew due to lower vertical stiffness of the nylon cord in the expandable packer. In contrast, it improved the cord angle, cord spacing, end face inclination, and packer thickness exhibited negative effects. The influential order of the sealing pressure. In field practice, the cord angle and the cord space should be reduced. To ensure pressure perfor- each factor on the pressure-bearing performance was: cord angle > cord layer > cord spacing > rubber thickness > shoul- mance, few cord layers should be selected. der angle. From the Pareto chart of the standardization effect of the sealing performance, it can be seen that the 1 3 Petroleum Science (2021) 18:269–284 277 2.447 2.45 Angle Layer Layer Angle Spacing Spacing Rubber Rubber thickness thickness Positive effect Positive effect Shoulder Shoulder angle angle Negative effect Negative effect 0123456 70 2 468 10 12 14 Standardization effect Standardization effect (a) Pressure performance (b) Sealing performance Fig. 8 Pareto diagram of the normalized effect of impact factors as shown in Table 7. The experimental data were fitted by 3.2 T he steepest climbing test results polynomial regression analysis to determine the effects of the independent variables (X , X , X ) on pressure perfor- These results are shown in Table  6. It is shown that the 1 2 3 pressure-bearing performance P and sealing performance mance P . The polynomial response surface model is max max given as follows: C of Test 4 met the site requirements. However, before max Test 4, the pressure-bearing performance P was lower max P =−175 + 27.7X ++29.01X − 101X − 0.736X max 1 2 3 compared to the field setting pressure. After Test 4, the seal 2 3 − 2.1354X + 27X + 0.038X X − 1.821X X + 3.07X X 1 2 1 3 2 3 performance C of the cartridge was lower compared 2 3 max (3) to the field sealing pressure. Therefore, optimal structural parameters were obtained between Tests 3 and Test 4, and the response areas of the cord angle, the number of cord Table 7 Plackett–Burman test result table layers, and the cord spacing were [17, 18], [6, 8], and [1.7, Test X , X X , mm P , MPa C , MPa 1 2 3 max max 1.8], respectively. degree 1 0 1 − 1 78.2 51.91 3.3 Response surface test results 2 0 0 0 74.6 54.35 3 − 1 1 0 77.8 51.99 On the basis of the corresponding surface method, the 4 − 1 0 1 74 54.14 Minitab software was used to generate a test plan table and 5 0 1 1 77 51.92 record the test results for each group of factor combinations 6 1 − 1 0 66.5 56.95 7 0 0 0 74.5 54.3 8 0 0 0 74.5 54.3 Table 6 The steepest climb test parameters 9 0 − 1 1 66.4 56.96 Test X , X X , mm P , MPa C , MPa 1 2 3 max max 10 − 1 0 − 1 75.9 54.38 degree 11 0 0 0 74.5 54.3 1 20 2 2 20.7 66.2 12 − 1 − 1 0 68.5 57.12 2 19 4 1.9 37.1 61.9 13 1 0 1 72.9 54.22 3 18 6 1.8 65.9 56.9 14 0 0 0 74.5 54.3 4 17 8 1.7 78.8 51.9 15 0 − 1 − 1 75.3 54.33 5 16 10 1.6 86.4 47.7 16 1 1 0 76.9 51.92 6 15 12 1.5 94.4 45.7 17 1 0 − 1 74.9 53.83 1 3 Impact factor Impact factor 278 Petroleum Science (2021) 18:269–284 Table 8 Results of variance analysis of pressure-bearing performance Table 9 Variance of sealing performance 2 2 Type DFSS, MPa MS, MPa F P > F Type DFSS, MPa MS, MPa F P > F r r Module 9 202.22 22.47 1158.98 0.000 Module 9 44.45 4.94 1147.95 0.000 Linear 3 1.71 0.57 29.32 0.000 Linear 3 0.06 0.02 4.80 0.04 X 1 0.12 0.11 5.91 0.045 X 1 0.00 0.00 0.01 0.911 1 1 X 1 1.33 1.32 68.33 0.000 X 1 0.06 0.06 13.49 0.008 2 2 X 1 0.02 0.02 0.77 0.408 X 1 0.00 0.00 0.26 0.625 3 3 Interaction 3 0.09 0.03 1.52 0.291 Interaction 3 0.10 0.04 8.23 0.011 X X 1 0.0 0.00 0.06 0.812 X X 1 0.00 0.00 0.13 0.730 1 2 1 2 X X 1 0.0 0.00 0.44 0.529 X X 1 0.09 0.10 22.16 0.002 1 3 1 3 X X 1 0.08 0.01 3.87 0.09 X X 1 0.00 0.00 0.0 0.980 2 3 2 3 Square 3 18.53 6.18 318.67 0.000 Square 3 0.22 0.07 17.39 0.001 2 2 X 1 0.12 0.12 6.25 0.041 X 1 0.02 0.02 4.56 0.070 1 1 2 2 X 1 17.24 17.25 889.57 0.000 X 1 0.19 0.19 43.68 0.000 2 2 2 2 X 1 0.02 0.02 0.84 0.389 X 1 0.03 0.03 7.64 0.028 3 3 Residual 7 0.14 0.02 Residual 7 0.03 0.00 – – Vector quasi 3 0.13 0.04 21.28 0.006 Vector quasi 3 0.03 0.01 18.75 0.008 Pure error 4 0.01 0.00 Pure error 4 0.00 0.00 – – The results of the analysis of variance are shown in As shown in Table 9, the interaction term X X exhibited 1 3 Table 8, F = 1158.98 > F (9,7) = 6.72, P > F < 0.001. The a significant effect on the pressure-bearing performance. To 0.01 r regression model in which the linear and square effects were determine the influence of the changing trend of influencing obvious is shown to be highly significant. The coefficient of factors and their interactions on contact stress, the number determination is R = 99.85% indicating that 99.85% of the of cord layers was taken as the center level. The contour data can be interpreted by this model, while 0.15% of the map of the other two factors was drawn according to the variance values cannot be interpreted by this model. This regression Eq. (4). shows that the actual measured value was highly correlated In general, high contact pressure represented the better with the predicted value, and the model had a high accuracy. sealing performance of expandable packer. During param- The test data were fitted by polynomial regression anal- eter optimization, the structure with high sealing pressure ysis to determine the effect of the independent variables should be selected. The influence of the cord angle is shown (X , X , X ) on the seal performance C . The polynomial in Fig. 9. As the cord angle increased, contact pressure ini- 1 2 3 max response surface model is given as follows: tially increased and then reduced. At 18°, contact pressure was high, representing a better sealing performance. Reduc- C = 67 − 0.62X − 6.07X + 27.7X − 0.296X max 1 2 3 1 tion in cord spacing led to an increase in contact pressure. 2 3 + 0.2229X − 38.2X + 0.0262X X + 6.08X X The high contact pressure reached a plateau when the cord 1 2 1 3 2 3 angle was nearly 18° while the cord spacing was about 15°. − 0.019X X 2 3 There was an “isolate land” when the cord angle was nearly (4) 20° and the cord spacing was about 16.l°. The results of the analysis of variance are shown in Table, F = 1147.95 > F (9,7) = 6.72, P > F < 0.001. The model is 0.01 r 3.4 Genetic algorithm optimization results highly significant. The linear and square effects were obvi- ous. The coefficient of determination was R = 99.85%. This With the nylon cord rubber pressure-bearing performance illustrated that 99.85% of the data could be interpreted by and sealing performance as optimization targets, and cord the model. angle, cord layer and cord spacing as variables the model 1 3 Petroleum Science (2021) 18:269–284 279 21 56.97 56.87 20 56.77 56.68 19 56.58 56.48 18 56.38 56.29 17 56.19 1.51.6 1.71.8 1.9 Cord spacing, mm Fig. 9 Cord angle and cord spacing contour map was transformed into the minimum problem of solving func- cord layers, and a cord spacing of 1.6 mm, the best solution tions (−P ) and (−C ). The optimization proposition can from the best area of Pareto and the optimal parameters of max max be expressed as: the nylon cord rubber could be acquired. These results are shown in Fig. 10. min −P max min −C max 3.5 Design Analysis of results s.t. P ≥ 70 max before and after optimization C ≥ 50 (5) max 0 ≤ X ≤ 45 Figure  11 shows changes in nylon cord and rubber cord 0 ≤ X ≤ 8, X = 2n, n ∈ Z + ; 2 2 arrangement before and after optimization. Figure 12 shows 0 ≤ X ≤ 3 changes in stress and contact stress of the nylon cord rubber before and after optimization. Figure  9 shows that as the When the multi-objective genetic algorithm was opti- sealing pressure increased, rubber stress and contact stress mized, the population number was 50, the evolution number increased simultaneously. After optimization, the rubber was 80, the crossover probability was 0.8, and the mutation stress significantly reduced while the contact stress sig- probability was 0.1. The results show that in the evolution nificantly increased. This was conducive for improving the of 55 generations, the Pareto optimal solution was obtained. rubber service life and working reliability. Compared to the According to the actual working parameters at the site, when rubber material, the inner cord of the rubber exhibited a sig- the set pressure was 70 MPa and the sealing pressure was nificant stress concentration that was prone to strength fail- 50 MPa, the pressure and sealing performance of the rub- ure. This outcome was used to ascertain the pressure-bearing ber met the requirements for use. At a cord angle of 16°, 6 1 3 Cord angle, degree Contact pressure, MPa 280 Petroleum Science (2021) 18:269–284 Best area 01224364860728496 108 Pressure performance, MPa Fig. 10 Multi-objective genetic algorithm optimization results performance of the nylon cord rubber. This indicates that 3.6 Results verification the pressure-bearing capacity of the optimized rubber was gradually enhanced, resulting in the increase in cord stress 3.6.1 Response Surface Model Verification and the decrease in rubber stress. By comparing and analyzing the working performance The ANSYS software was used to establish a three-dimen- of the nylon cord rubber before and after optimization, the sional mechanical finite element model based on the struc- pressure-bearing performance of the drum after optimiza- tural parameters of the nylon cord rubber. The pressure- tion was 75 MPa. This was a 25% increase compared to resisting performance of the rubber after optimization and the pressure-bearing performance before optimization. The the sealing performance of the drum under a 70 MPa setting sealing performance of the drum after optimization was pressure was analyzed. The obtained results and the response 57.5 MPa. This was a 66% in the sealing performance that surface model (3) and (4) results were compared to verify was observed before optimization (34.5 MPa). the accuracy of the response surface model (Table 10). The results indicated that the response surface model was similar 1 3 Sealing performance, MPa Petroleum Science (2021) 18:269–284 281 Original rubber parameters 45 45 Optimized rubber parameters 18 18 1.6 Cord Cord Cord Rubber Shoulder angle Layer spacing thickness angle Fig. 11 Comparison of structural parameters before and after optimization to the ANSYS simulation results, and the error was within 3.6.2 Laboratory Test Verification 10%. Figure 13 shows that the increase in sealing pressure was directly proportional to the increase in setting load. For the pre-optimized rubber, the bearing pressure of the packer was 60 MPa while the sealing pressure of the packer was 40 280 Rubber stress, MPa Contact stress, MPa Optimized rubber stress, MPa Optimized contact stress, MPa 35 60 Cord stress, MPa Optimized cord stress, MPa (75,57.5) Cord allowable stress 160 20 (60,34.5) 15 ` Rubber allowable stress 10 20 30 40 50 60 70 80 10 20 30 40 50 60 70 80 Setting pressure, MPa Setting pressure, MPa (a) stress (b) contact stress Fig. 12 Changes in stress and contact stress of nylon cord rubber before and after optimization 1 3 Rubber stress, MPa Value Cord stress, MPa Contact stress, MPa 282 Petroleum Science (2021) 18:269–284 1. Using the field packer as an example, a REFINE265 unit Table 10 Response surface model validation was applied in a three-dimensional numerical simulation Type Response surface Simulation Error, % model of the nylon cord rubber. The maximum pres- model, MPa model, MPa sure performance of the cartridge was 60 MPa while Setting pressure 71.02 75 5 the maximum sealing performance was 30 MPa. The Contact pressure 57.42 52.3 9.78 simulation results were consistent with the test results. These outcomes verified the accuracy of the simulation model. 30 MPa. For the optimized rubber, the bearing pressure of 2. Using Plackett–Burman test, steepest climbing test and the post-optimized cartridge was 70 MPa while the sealing response surface test, the multi-objective optimization pressure was 50 MPa. The error was within 40% as shown in of the nylon cord rubber was performed with the pres- Fig. 14. The test results show that the optimized nylon cord sure-bearing and sealing performance as the objective rubber effectively complied with the requirements of low- functions. The accuracy of the response surface model permeability oil and gas reservoirs with a 70 MPa setting obtained was superior simulated with ANSYS, which pressure and 50 MPa sealing pressure. remains within 90%. 3. Based on the actual working parameters in the field, the optimal combination of nylon cord rubber tube structure 4 Conclusions parameters (the cord angle (16°), cord layers (6), and cord spacing (1.6 mm)) was determined. It was shown A novel numerical simulation and optimization method was that the optimized rubber bearing pressure performance used to evaluate the performance of the expandable packer. increased to 75 MPa. This was a 25% increase compared To meet the requirements of high-pressure and fine fractur - to the pre-optimized rubber. The sealing performance ing, the reliability of the optimized results was verified by is increased to 57.5 MPa, which was a 66% increase laboratory tests. compared to than that before. Simulation of optimized packer Experiment of optimized packer Simulation of packer Experiment of packer 10 20 30 40 50 60 70 80 Setting pressure, MPa Fig. 13 Rubber simulation versus experiment comparisons before and after optimization 1 3 Working pressure, MPa Petroleum Science (2021) 18:269–284 283 0.50 The error precentage after opimization 0.45 The error percentage before optimization 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 10 20 30 40 50 60 70 Setting pressure, MPa Fig. 14 The error percentage of sealing performance before and after optimization 4. The laboratory tests showed that the maximum set pres- References sure of the barrel after optimization was 70 MPa while Agarwal K, Kegel J, Ballard B. Evolving completion designs to opti- the maximum working pressure was 50 MPa. 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Multi-objective optimization and experiment of nylon cord rubber in expandable packer

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Springer Journals
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Copyright © The Author(s) 2021
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1672-5107
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1995-8226
DOI
10.1007/s12182-020-00539-6
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Abstract

Nylon cord rubber has the advantages of small residual deformation and is easy to lift and lower the tubing string in low- permeability oil and gas reservoirs. However, it is associated with low-pressure resistance and poor sealing performance. To enhance the performance of nylon cord rubber, a three-dimensional numerical model of the nylon cord rubber was established and its accuracy experimentally determined. The Plackett–Burman test, the Steepest climbing test and the Response surface method were used to acquire the polynomial response surface model connecting structural parameters with bearing and sealing pressure. Using genetic algorithms, optimal structural parameters of nylon cord rubber were determined depending on field operation. The reliability of the optimized results was verified by laboratory tests. It was shown that after optimiza- tion, the bearing capacity of the expandable packer increased by 25% while the sealing performance increased by 66%. In addition, the bearing pressure was 70 MPa while the sealing pressure was 50 MPa. These measurements effectively met the on-site requirements of high-pressure and fine fracturing in low-permeability oil and gas reservoirs. Keywords Low-permeability reservoirs · Nylon cord rubber · Numerical simulation · Response surface method · Multi- objective genetic algorithm · Laboratory test List of symbols 1 Introduction X Angle, degree X Layer Low-permeability oil and gas reservoirs have become X Spacing, mm important exploratory and development fields in the X R ubber thickness, mm world (Zou et al. 2017; Mu and Ji 2019). Horizontal well X Shoulder angle, mm small-hole fracturing technology is an important produc- P Bearing pressure, MPa tion stimulation measure for low-permeability oil and gas max C Sealing pressure, MPa reservoirs (Lei et al. 2018; Qu et al. 2019; Agarwal et al. max DF Degree of freedom 2019). In reservoirs, the expansion packer determines SS Mean bias, MPa the outcomes of the fracturing technology. The expand- MS Mean square, MPa able packer isolates the tubing and the borehole wall and F Statistical magnitude then forms an annular space in Fig. 1. The annular space separates the oil and gas to achieve a layered fracture. As the core component of the expandable packer, the rubber affects fracturing during construction (Guo and Gao 2013). Currently, overlapped steel belts and steel cord rubber cylinders are used in rubber, but the residual deformation is large. When the pipe string is lifted up, Edited by Xiu-Qiu Peng the steel belt and steel cord are more likely to become stuck in the casing leading to underground accidents * Han-Xiang Wang (Patel et al. 2019a, b). Nylon cord rubber has the advan- wanghxupc@163.com tages of small residual deformation and is easy to lift and College of Mechanical and Electronic Engineering, China lower the tubing string in low-permeability oil and gas University of Petroleum (East China), Qingdao 266580, reservoirs (Zhong et al. 2015; Akhtar et al. 2018; Pradie Shandong, China Vol.:(0123456789) 1 3 270 Petroleum Science (2021) 18:269–284 et al. 2008). However, it is associated with low-pressure- both the axial and hoop compressive stress concentrations bearing and poor sealing performance. It is, therefore, were generated in the thread teeth edges near the contact important to analyze and optimize the structural param- surfaces of threads. This was associated with the mutual eters of the nylon cord rubber in expandable packer to squeezing of box and pin thread teeth during experimental meet the on-site requirements of high-pressure and fine verification. Qamar et al. (2009), Al Ramadan et al. (2019), fracturing in low-permeability oil and gas reservoirs (Li Daou et al. (2014) mechanically tested and characterized et al. 2017; Tian et al. 2019). an inert water-swelling elastomer that had been developed Studies on the sealing properties and optimization of by a local petroleum development firm. The elastomer the structural parameters of packer rubber through theo- was tested for hardness, compression at different tempera- retical calculation, numerical simulation and experimental tures and for different periods of time, tensile strength at verification are being done. For theoretical calculation, Al- different strain rate, tensile properties regarding fracture Hiddabi et al. (2015), Al-Abri et al. (2015), Renaud et al. strength and percent elongation, and swelling ratio. Ahmed (2009) investigated the deformation of an elastomer seal et al. (2019a, b), Al Ramadan et al. (2019) performed the confined between a metal tube and a rigid casing. They verification process and tests by critically reviewing the showed the effect of the geometry of the elastomer geom- literature, current regulations, and applicable industrial etry and its material properties on sealing performance standards in order to develop testing protocols for the in terms of maximum sealing pressure. Alkharusi et al. investigation of the performance of common elastomeric (2011), Gajewski et al. (2015), Akhtar et al. (2018) investi- seals that are used in a liner hanger seal assembly. Dong gated the effects of the material and geometrical properties et al. (2020), Fothergill (2003) determined seal failures of the elastomer on its sealing performance under different of the rubber tube at high temperatures and studied the loading conditions. In addition, they investigated the effect constitutive model parameters of the rubber tube through of radial strain and annular fluid pressure on the sealing the rubber thermal aging experiments. The effects of key performance. Agata et al. (2013) described the material and parameters of the rubber tube-casing gap, the dip angle form factors that regulated the ability of the pipe to expand. of the adjacent rubber tube contact surface, and the ini- These factors included the influence of axial restraint dur - tial setting load on the sealing performance of the packer ing expansion and the post-expansion collapse resistance under high temperature conditions were analyzed. Chen of solid expandable tubulars. Banks et al. (2002) studied et al. (2019), Grelle et al. (2019) determined the effects of the compression of rubber bonded to rigid metal plates different stress conditions and the speed of lifting or lower - of different geometry (rectangular and V-shaped blocks). ing the pipe on the weakness of the rubber matrix. Zhang and Wang (2016), Zhang et al. (2018) while relying The above described studies determined contact stress on the laws of momentum and energy conservation and the distribution of the rubber cylinder under different structural transient heat transfer property between the wellbore fluid parameters, and optimized the rubber structure with the and the annulus fluid developed a calculation model of the maximum contact stress as the goal. However, the tearing temperature and pressure fields on single-layer and multi- failure caused by excessive internal stress of the rubber was layer annuli. Cavalaro and Aguado (2012) characterized not considered. These studies did not consider the effect of the behavior of the packer under simple stress (normal) cord arrangement parameters on the performance of the rub- and under coupled stresses (normal and tangential) as well ber. This resulted in large variations between the simulation as proposed mathematical constitutive models to describe results and the actual condition. In addition, the optimization both behaviors. Their results indicated that the packer pre- method utilized the maximum contact stress as the single sented a nonlinear almost elastic mechanical behavior from optimization goal and ignored the internal failure stress. This the second load cycle onwards. For numerical simulation, could not improve the overall performance of the expand- Hu et al. (2017, 2018), Patel et al. (2019a, b) studied the able packer. In this paper, the numerical simulation model of influence of three rubber materials on sealing performance nylon cord packer was established by considering the inner of packing element in the compression packer. Wang et al. cord action of the drums. With the aim of sealing perfor- (2017), Lan et al. (2019) studied the structure of the packer mance and pressure-bearing performance, a combination of rubber with different materials and optimized the struc- response surface method and a genetic algorithm were used ture of its sealing unit. Based on the simulation results, to optimize the design of nylon cord rubber. Specifically, the the best inner groove shape and the structure of shoulder Plackett–Burman test was used to determine the climbing protection were confirmed. Furthermore, the validity of direction of influencing factors. The steepest climbing test the simulation results was experimentally confirmed. Xu was used to obtain the center of the response surface. The et al. (2017) investigated the structural response of tubulars response surface method was used to acquire the polyno- to stress evolution, deformation of the thread joint and to mial response surface model connecting the cord angle, the the effect of expansion cone geometry. It was found that number of cord layers and the cord spacing with bearing and 1 3 Petroleum Science (2021) 18:269–284 271 relieved, the packer barrel automatically shrank to complete the unsealing by depressurizing the ground. The tearing failure under excessive setting pressure and sealing failure caused by low contact pressure of the expand- able packer were common reasons for failure. To enhance Oil tube the total working performance, we improved the bearing pressure and sealing pressure of the expandable packer. Safety joint The bearing pressure represented the ultimate setting pres- sure before packer failure which can expand the packer and encapsulate the annular space between the casing and the Hydraulic 2# oil layer center pipe. At the same time, the sealing performance of anchor the expandable packer depended on the contact pressure. Increasing the contact pressure enhanced the sealing perfor- mance. When analyzing the expandable packer, the maxi- mum contact pressure along the sealing surface was consid- Expanded packer ered to be the sealing pressure under the setting pressure. The working condition of the expandable packer is shown in Fig. 3. In order to improve the bearing pressure and sealing pres- Sand blast 1# oil layer sure, the nylon cord rubber adopted the nylon cord as skel- packer eton, which was composed of an inner and an outer rubber. The structural parameters of the nylon cord rubber used in the oilfield are shown in Table  1. Laboratory tests indicated Plug that the maximum set pressure of the current expandable packer was 60 MPa and the maximum sealing pressure was 30 MPa. These pressures could not meet the site require- ments of low-permeability oil and gas reservoirs with a set pressure of 70 MPa and a sealing pressure of 50 MPa. There- fore, the parameters of cord arrangement and barrel structure Fig. 1 The schematic diagram of fracturing string should be optimized. 2.2 Establishment of the finite element model sealing performance. Optimal structural parameters of the nylon cord rubber were determined according to the require- The finite element model of the nylon cord rubber is shown in Fig.  4. Due to the symmetry of expandable packer, a ments of field operations. To meet the high-pressure and fine fracturing on-site use requirements, the reliability of the quarter of the three-dimension finite element model was developed with the Y-axis as the symmetrical axis. The optimized results was verified by laboratory tests. SOLID185 element was used for the plastic tube, center tube, sleeve, and upper and lower joints while the REINF265 2 Structure and analysis method element was used as the polyamide cord reinforcement mate- rial. The 70 MPa setting load was applied inside the expand- 2.1 Structure of nylon cord rubber tube able packer. According to the indoor material test (ASTM D573-04 2015), the material parameters of nylon cord rub- The nylon cord packer is shown in Fig. 2. It is composed of ber cylinder are shown in Table 2 (Wang et al. 2020). In terms of material failure, the third strength and maxi- the upper joint, upper steel bowl, vulcanized core, central tube, rubber cylinder, lower cylinder liner and lower joint. mum stress criteria were utilized for stress analysis of expandable packer in formula (1). In particular, shear fail- At work, the expansion fluid was pumped into the nylon cord packer rubber from central tube through the ground ure was common inside the expandable packer. The third strength criterion states that the plastic flow occurs when booster pump. When the pressure difference between the internal and external oil pipes reached the packer starting the maximum shear stress reaches its shear strength. This phenomenon was attributed to the failure mechanism of pressure, the rubber tube expanded. After the pressure was 1 3 272 Petroleum Science (2021) 18:269–284 12 34 56 78 91011121314 (a) (b) (c) Cord layer Cord angle Cord spacing Cord angle Rubber thickness (d) Fig. 2 Structure diagram of the downhole expandable packer. 1-upper joint; 2-upper steel bowl; 3-screw; 4-plug nail; 5-vulcanized core; 6-cen- tral tube; 7-cylinder; 8-vulcanized core; 9-cylinder liner; 10- Piston; 11-lower cylinder liner; 12-”O” seal; 13-lower steel bowl; 14-lower joint the rubber. The maximum stress criterion which states that material damage occurs when the maximum stress stretches Annular space Center pipe Setting Contact pressure pressure Rubber Casing Fig. 3 The working condition of expandable packer Table 1 Structure parameter of nylon cord packer to the tensile strength was associated with damage to the fiber. According to the above material criteria, the different Parameters Value parameters of nylon cord rubber were optimized to meet Cord angle, degree 35 the site use requirements of low-permeability oil and gas Cord layer 10 reservoirs (70 MPa setting pressure and 50 MPa sealing Cord spacing, mm 2 pressure). Shoulder angle, degree 45 Rubber thickness, mm 18 1 3 Petroleum Science (2021) 18:269–284 273 (a) Physical model level was marked (+ 1). The low-level mark was (− 1). The optimal level of each factor was determined and the key fac- 50 mm 350 mm tors selected. The Plackett–Burman test parameter design is shown in Table 3. 500 mm 2.3.2 The Steepest Climbing Test Design (b) Simulation model 2 5 7 1 3 4 6 The steepest climbing test design utilizes the direction of the gradient of the test value as the climbing direction. It deter- mines the step size of the change according to the effective value of each factor, which can quickly and economically approach the optimal response area. During the test, and based on Plackett–Burman test results, the design direction of the influencing factors was changed, and the steepest (c) Cord model climbing test of the pressure resistance performance P max and sealing performance C of the nylon cord rubber was max determined. 2.3.3 Response Surface Test Design Fig. 4 Nylon cord rubber finite element model The response surface test was designed by experiment- 2.3 Optimal design test methods ing on a set of sample points in a specified design space. The global approximation function of the system can be Because the response surface model has a high fitting accu- approximated to replace the actual response surface. In an racy in the center neighborhood while the fitting accuracy engineering optimization design, the response relationship outside the center neighborhood is low, the fitting equation between the response target and the design variables can be was almost meaningless. Using the ANSYS software, the achieved through the response surface test, and the design Plackett–Burman test was used to screen for key factors, the variables under the optimal objective function can then be steepest climbing test was applied to approximate the best area, and a polynomial model connecting the influencing Table 3 Plackett–Burman test design factors with pressure resistance and contact performance of Impact factor Level the nylon cord rubber was established by response surface method. Parameters Low level (− 1) High level 2.3.1 Plackett–Burman Test Design (+ 1) Cord angle 20 30 During field investigations, the cord angle, the number of Cord layer 2 4 cord layers, cord spacing, the end inclination angle, and Cord spacing 2 3 the thickness of the barrel were the test factors. Each test Shoulder angle 45 60 factor was taken at two levels for the test design. The high Rubber thickness 17 19 Table 2 Material parameters of nylon cord rubber Component Material Density, g/cm Elasticity modu- Constitutive model Poisson’s ratio Tensile lus, MPa strength, MPa Nylon cord Nylon 1.15 2.80 × 10 – 0.34 190 Rubber HNBR 1.56 – Yeoh Model – 17 C = 0.15 C = − 1.29 C = 0.61 Central tube-casing pipe 45MnMo7 7.8 2.06 × 10 – 0.24 – Upper and lower joints 4145H 7.8 2.06 × 10 – 0.24 – 1 3 66 mm 29 mm 37.5 mm 274 Petroleum Science (2021) 18:269–284 determined.The second-order polynomial model is often the respectively. The independent variables are coded according construction of the response surface approximation model. to formula (2). The distribution of test points is shown in The approximation is the relationship between the system Fig. 5 while the design of response surface test parameters input and the response target. The basis function is shown is illustrated in Table 4. in formula (1): X =(x − x )∕Δx (2) i 1 0 k k k whereby X is the encoding value of independent variable, x y = 𝛽 + 𝛽 x + 𝛽 x + 𝛽 x x i 1 0 i i ii ii i j i (1) i=1 i=1 i=1 is the true value of the independent variable at the test center i<j point, and Δx is the step size of the independent variable. whereby β is the interaction coefficient between offset, linear offset and second-order offset; k is the total number of design variables; Y is the predicted response value. Table 4 Response surface test design Box-Behnken Design (BBD) is a common test design Test Parameters Central point Low level High level method for response surface testing. The number of tests is compact, and the economy is good. It is suitable for opti- 1 X 17.5 17 18 mization experiments with 2 to 5 factors. Each factor takes 2 X 7 6 8 3 levels, with 0 as the center point, (+  1) and (−  1) are 3 X 1.75 1.7 1.8 the high and low values corresponding to the cubic points, Determine the type of optimization problem constraint Gamultiobj solve Creat initial population Get the optimal Whether or not to exit solution Pareto Population evolution next generation Plot optimization results Judge termination condition Fig. 5 Flow chart of multi-objective genetic algorithm 1 3 Petroleum Science (2021) 18:269–284 275 2.3.4 Genetic Algorithm Optimization lower end of the packer was then connected with blind plug- ging to block its outlet. The entire expandable packer was Genetic algorithm is a type of adaptive artificial intelligence soaked in water at 18–28 °C for 24 h. It was then installed in technology that simulates the evolutionary processes of bio- the casing. The pressure pump (100 MPa) was connected to logical organisms and the solution of extreme values. Its the center pipe through the center pipe sealing cover while basic idea is an algorithm that searches for the optimal solu- the pressure pump (60 MPa) was connected with an annular tion formed by simulating the genetic mechanism of nature between the casing and tubing through the casing sealing and biological evolutionary theory. It is suitable for solv- ing complex nonlinear and multidimensional optimization (a) Install problems. Multi-objective genetic algorithm, as a fast and effective global optimization algorithm, has a brisk running speed. The solution set has the advantage of excellent con- vergence. The specific optimization process was as shown in Fig. 5. (b) Soak 2.4 Laboratory test To verify the accuracy of the simulation results, indoor tests of the nylon cord rubber before and after the optimization were performed. The dimensions of the experimental setup and the simulation model were consistent, and the specific (c) Pressure test constitution of the experimental setup was as follows: (1) 1/2 Packer experimental rig including 5 casing (with an outer 7/8 diameter of 139.7 mm), 2 tubing (with an outer diameter of 78.6 mm), flange and plug hear; (2) K344-114 expansion packer on which a rubber be can installed; (3) High-pressure (d) Check pumps (100 MPaand 60 MPa); (4) Oil bath device and pipe line (Wang et al. 2020). The schematic diagram of the labo- ratory test setup was as shown in Fig. 6, and the process of experimental measurement was as shown in Fig. 7. Briefly, the rubber was assembled on the expandable packer with an amount of butter being applied on the sealing groove and Fig. 7 Polyamide cord rubber test diagram engine oil being applied on the surface of the thread. The Pressure gage End cap Top 5 1/2 connection casing Drainage K344-114 Rubber Throttle 27/8 tubing Tube Pump Pump (100 MPa) (60 MPa) Casing pipe Thermometer Casing pressure Casing pressure Water tank Oil bath tank Tubing pressure Fig. 6 Laboratory equipment layout diagram 1 3 276 Petroleum Science (2021) 18:269–284 cover. The pressure gauges of the center and casing pipelines Table 5 Plackett–Burman test results were set and used to observe the working conditions of the Test Angle, Layer Spacing, mm Rubber Shoulder expandable packer. In addition, the casing valve was closed degree thickness, angle, mm and the center pipe valve opened. The high-pressure liquid mm was driven through the high-pressure pump (100 MPa) into 1 20 4 3 60 17 the center pipe. After pressing the inside packer within a 2 30 4 2 60 17 pressure range of 0–80 MPa and stabilizing the pressure for 3 20 2 2 45 17 5 min every 10 MPa, the working conditions of the packer 4 20 4 2 45 17 could be observed. If the pressure gauge in tube pipeline 5 20 2 3 60 19 rose steadily and remains steady for 5 min, the pressure test 6 30 2 3 45 17 pump continued to exert pressure based on the existing pres- 7 20 4 3 45 19 sure. When the pressure gauge experienced a rapid decline 8 30 4 3 45 19 with sharp fluctuations, the pressure test pump stopped and 9 30 2 2 45 19 the packer was considered to have been damaged. The maxi- 10 30 2 3 60 17 mum setting pressure before packer failure during the period 11 30 4 2 60 19 of pressure stabilization was known as the bearing pressure. 12 20 2 2 60 19 After pressing the packer from the center pipe at 10 MPa to the bearing pressure, the center pipe valve was closed and the casing valve opened. The sealing pressure under die ff rent cord spacing and shoulder angle exhibited positive effects setting pressures was measured. To be specific, the high- pressure liquid was driven through the high-pressure pump while the cord angle, cord layers and shoulder angle exhib- ited negative effects. The influential order of each factor on (60 MPa) into the annulus between center pipe and casing from 5 to 60 MPa for 5 min every 5 MPa. If the pressure sealing performance was: cord layer > cord angle > cord spacing > rubber thickness > shoulder angle. gauge in the casing pipeline rises steadily and remains stead for 5 min, the pressure test pump continued to exert pressure Above all, it can be seen that nylon cord parameters (cord angle, cord layer and cord spacing) exerted a more obvious based on the existing pressure. When the pressure gauge experienced a rapid decline with sharp fluctuations, the pres- effect on bearing pressure and sealing pressure when com- pared to rubber parameters (rubber thickness and shoulder sure test pump was stopped and the packer was considered to have been damaged. The maximum annual working pressure angle). This could be attributed to the fact that nylon cord was the frame of expandable packer on which the external before packer failure during the period of pressure stabiliza- tion was shown in the pressure gauges in casing pipeline. pressure was exerted. In addition, changes in cord angle, cord layer and cord spacing played a significant role in bear - It was described as the sealing pressure under the setting pressure. After the test, pressure was released and the packer ing pressure and sealing pressure. Therefore, we determined the inu fl ence of nylon cord arrangement on the bearing pres - was restored to its original state. The failure morphology of expandable packer was observed. sure and sealing pressure of the expandable packer and other factors were in agreement with the site. It can be found that with an increase in cord angle, the radial stiffness of the rubber grew. This led to cord breakage and low sealing pres- 3 Results and discussion sure that resulted to poor bearing pressure. Meanwhile, as the cord layer increased, the vertical stiffness of the packer 3.1 Plackett–Burman test results grew synchronously. This improved the bearing pressure. Considering the high vertical stiffness, the sealing pressure The Plackett–Burman test results are shown in Table 5. The diagrammatic presentation of the Pareto standardization roughly reduced under the same setting pressure. Further- more, the bearing pressure experienced a reduction when effect of pressure-bearing performance is shown in Fig.  8. The number of cord layers exhibited a positive effect while the cord spacing grew due to lower vertical stiffness of the nylon cord in the expandable packer. In contrast, it improved the cord angle, cord spacing, end face inclination, and packer thickness exhibited negative effects. The influential order of the sealing pressure. In field practice, the cord angle and the cord space should be reduced. To ensure pressure perfor- each factor on the pressure-bearing performance was: cord angle > cord layer > cord spacing > rubber thickness > shoul- mance, few cord layers should be selected. der angle. From the Pareto chart of the standardization effect of the sealing performance, it can be seen that the 1 3 Petroleum Science (2021) 18:269–284 277 2.447 2.45 Angle Layer Layer Angle Spacing Spacing Rubber Rubber thickness thickness Positive effect Positive effect Shoulder Shoulder angle angle Negative effect Negative effect 0123456 70 2 468 10 12 14 Standardization effect Standardization effect (a) Pressure performance (b) Sealing performance Fig. 8 Pareto diagram of the normalized effect of impact factors as shown in Table 7. The experimental data were fitted by 3.2 T he steepest climbing test results polynomial regression analysis to determine the effects of the independent variables (X , X , X ) on pressure perfor- These results are shown in Table  6. It is shown that the 1 2 3 pressure-bearing performance P and sealing performance mance P . The polynomial response surface model is max max given as follows: C of Test 4 met the site requirements. However, before max Test 4, the pressure-bearing performance P was lower max P =−175 + 27.7X ++29.01X − 101X − 0.736X max 1 2 3 compared to the field setting pressure. After Test 4, the seal 2 3 − 2.1354X + 27X + 0.038X X − 1.821X X + 3.07X X 1 2 1 3 2 3 performance C of the cartridge was lower compared 2 3 max (3) to the field sealing pressure. Therefore, optimal structural parameters were obtained between Tests 3 and Test 4, and the response areas of the cord angle, the number of cord Table 7 Plackett–Burman test result table layers, and the cord spacing were [17, 18], [6, 8], and [1.7, Test X , X X , mm P , MPa C , MPa 1 2 3 max max 1.8], respectively. degree 1 0 1 − 1 78.2 51.91 3.3 Response surface test results 2 0 0 0 74.6 54.35 3 − 1 1 0 77.8 51.99 On the basis of the corresponding surface method, the 4 − 1 0 1 74 54.14 Minitab software was used to generate a test plan table and 5 0 1 1 77 51.92 record the test results for each group of factor combinations 6 1 − 1 0 66.5 56.95 7 0 0 0 74.5 54.3 8 0 0 0 74.5 54.3 Table 6 The steepest climb test parameters 9 0 − 1 1 66.4 56.96 Test X , X X , mm P , MPa C , MPa 1 2 3 max max 10 − 1 0 − 1 75.9 54.38 degree 11 0 0 0 74.5 54.3 1 20 2 2 20.7 66.2 12 − 1 − 1 0 68.5 57.12 2 19 4 1.9 37.1 61.9 13 1 0 1 72.9 54.22 3 18 6 1.8 65.9 56.9 14 0 0 0 74.5 54.3 4 17 8 1.7 78.8 51.9 15 0 − 1 − 1 75.3 54.33 5 16 10 1.6 86.4 47.7 16 1 1 0 76.9 51.92 6 15 12 1.5 94.4 45.7 17 1 0 − 1 74.9 53.83 1 3 Impact factor Impact factor 278 Petroleum Science (2021) 18:269–284 Table 8 Results of variance analysis of pressure-bearing performance Table 9 Variance of sealing performance 2 2 Type DFSS, MPa MS, MPa F P > F Type DFSS, MPa MS, MPa F P > F r r Module 9 202.22 22.47 1158.98 0.000 Module 9 44.45 4.94 1147.95 0.000 Linear 3 1.71 0.57 29.32 0.000 Linear 3 0.06 0.02 4.80 0.04 X 1 0.12 0.11 5.91 0.045 X 1 0.00 0.00 0.01 0.911 1 1 X 1 1.33 1.32 68.33 0.000 X 1 0.06 0.06 13.49 0.008 2 2 X 1 0.02 0.02 0.77 0.408 X 1 0.00 0.00 0.26 0.625 3 3 Interaction 3 0.09 0.03 1.52 0.291 Interaction 3 0.10 0.04 8.23 0.011 X X 1 0.0 0.00 0.06 0.812 X X 1 0.00 0.00 0.13 0.730 1 2 1 2 X X 1 0.0 0.00 0.44 0.529 X X 1 0.09 0.10 22.16 0.002 1 3 1 3 X X 1 0.08 0.01 3.87 0.09 X X 1 0.00 0.00 0.0 0.980 2 3 2 3 Square 3 18.53 6.18 318.67 0.000 Square 3 0.22 0.07 17.39 0.001 2 2 X 1 0.12 0.12 6.25 0.041 X 1 0.02 0.02 4.56 0.070 1 1 2 2 X 1 17.24 17.25 889.57 0.000 X 1 0.19 0.19 43.68 0.000 2 2 2 2 X 1 0.02 0.02 0.84 0.389 X 1 0.03 0.03 7.64 0.028 3 3 Residual 7 0.14 0.02 Residual 7 0.03 0.00 – – Vector quasi 3 0.13 0.04 21.28 0.006 Vector quasi 3 0.03 0.01 18.75 0.008 Pure error 4 0.01 0.00 Pure error 4 0.00 0.00 – – The results of the analysis of variance are shown in As shown in Table 9, the interaction term X X exhibited 1 3 Table 8, F = 1158.98 > F (9,7) = 6.72, P > F < 0.001. The a significant effect on the pressure-bearing performance. To 0.01 r regression model in which the linear and square effects were determine the influence of the changing trend of influencing obvious is shown to be highly significant. The coefficient of factors and their interactions on contact stress, the number determination is R = 99.85% indicating that 99.85% of the of cord layers was taken as the center level. The contour data can be interpreted by this model, while 0.15% of the map of the other two factors was drawn according to the variance values cannot be interpreted by this model. This regression Eq. (4). shows that the actual measured value was highly correlated In general, high contact pressure represented the better with the predicted value, and the model had a high accuracy. sealing performance of expandable packer. During param- The test data were fitted by polynomial regression anal- eter optimization, the structure with high sealing pressure ysis to determine the effect of the independent variables should be selected. The influence of the cord angle is shown (X , X , X ) on the seal performance C . The polynomial in Fig. 9. As the cord angle increased, contact pressure ini- 1 2 3 max response surface model is given as follows: tially increased and then reduced. At 18°, contact pressure was high, representing a better sealing performance. Reduc- C = 67 − 0.62X − 6.07X + 27.7X − 0.296X max 1 2 3 1 tion in cord spacing led to an increase in contact pressure. 2 3 + 0.2229X − 38.2X + 0.0262X X + 6.08X X The high contact pressure reached a plateau when the cord 1 2 1 3 2 3 angle was nearly 18° while the cord spacing was about 15°. − 0.019X X 2 3 There was an “isolate land” when the cord angle was nearly (4) 20° and the cord spacing was about 16.l°. The results of the analysis of variance are shown in Table, F = 1147.95 > F (9,7) = 6.72, P > F < 0.001. The model is 0.01 r 3.4 Genetic algorithm optimization results highly significant. The linear and square effects were obvi- ous. The coefficient of determination was R = 99.85%. This With the nylon cord rubber pressure-bearing performance illustrated that 99.85% of the data could be interpreted by and sealing performance as optimization targets, and cord the model. angle, cord layer and cord spacing as variables the model 1 3 Petroleum Science (2021) 18:269–284 279 21 56.97 56.87 20 56.77 56.68 19 56.58 56.48 18 56.38 56.29 17 56.19 1.51.6 1.71.8 1.9 Cord spacing, mm Fig. 9 Cord angle and cord spacing contour map was transformed into the minimum problem of solving func- cord layers, and a cord spacing of 1.6 mm, the best solution tions (−P ) and (−C ). The optimization proposition can from the best area of Pareto and the optimal parameters of max max be expressed as: the nylon cord rubber could be acquired. These results are shown in Fig. 10. min −P max min −C max 3.5 Design Analysis of results s.t. P ≥ 70 max before and after optimization C ≥ 50 (5) max 0 ≤ X ≤ 45 Figure  11 shows changes in nylon cord and rubber cord 0 ≤ X ≤ 8, X = 2n, n ∈ Z + ; 2 2 arrangement before and after optimization. Figure 12 shows 0 ≤ X ≤ 3 changes in stress and contact stress of the nylon cord rubber before and after optimization. Figure  9 shows that as the When the multi-objective genetic algorithm was opti- sealing pressure increased, rubber stress and contact stress mized, the population number was 50, the evolution number increased simultaneously. After optimization, the rubber was 80, the crossover probability was 0.8, and the mutation stress significantly reduced while the contact stress sig- probability was 0.1. The results show that in the evolution nificantly increased. This was conducive for improving the of 55 generations, the Pareto optimal solution was obtained. rubber service life and working reliability. Compared to the According to the actual working parameters at the site, when rubber material, the inner cord of the rubber exhibited a sig- the set pressure was 70 MPa and the sealing pressure was nificant stress concentration that was prone to strength fail- 50 MPa, the pressure and sealing performance of the rub- ure. This outcome was used to ascertain the pressure-bearing ber met the requirements for use. At a cord angle of 16°, 6 1 3 Cord angle, degree Contact pressure, MPa 280 Petroleum Science (2021) 18:269–284 Best area 01224364860728496 108 Pressure performance, MPa Fig. 10 Multi-objective genetic algorithm optimization results performance of the nylon cord rubber. This indicates that 3.6 Results verification the pressure-bearing capacity of the optimized rubber was gradually enhanced, resulting in the increase in cord stress 3.6.1 Response Surface Model Verification and the decrease in rubber stress. By comparing and analyzing the working performance The ANSYS software was used to establish a three-dimen- of the nylon cord rubber before and after optimization, the sional mechanical finite element model based on the struc- pressure-bearing performance of the drum after optimiza- tural parameters of the nylon cord rubber. The pressure- tion was 75 MPa. This was a 25% increase compared to resisting performance of the rubber after optimization and the pressure-bearing performance before optimization. The the sealing performance of the drum under a 70 MPa setting sealing performance of the drum after optimization was pressure was analyzed. The obtained results and the response 57.5 MPa. This was a 66% in the sealing performance that surface model (3) and (4) results were compared to verify was observed before optimization (34.5 MPa). the accuracy of the response surface model (Table 10). The results indicated that the response surface model was similar 1 3 Sealing performance, MPa Petroleum Science (2021) 18:269–284 281 Original rubber parameters 45 45 Optimized rubber parameters 18 18 1.6 Cord Cord Cord Rubber Shoulder angle Layer spacing thickness angle Fig. 11 Comparison of structural parameters before and after optimization to the ANSYS simulation results, and the error was within 3.6.2 Laboratory Test Verification 10%. Figure 13 shows that the increase in sealing pressure was directly proportional to the increase in setting load. For the pre-optimized rubber, the bearing pressure of the packer was 60 MPa while the sealing pressure of the packer was 40 280 Rubber stress, MPa Contact stress, MPa Optimized rubber stress, MPa Optimized contact stress, MPa 35 60 Cord stress, MPa Optimized cord stress, MPa (75,57.5) Cord allowable stress 160 20 (60,34.5) 15 ` Rubber allowable stress 10 20 30 40 50 60 70 80 10 20 30 40 50 60 70 80 Setting pressure, MPa Setting pressure, MPa (a) stress (b) contact stress Fig. 12 Changes in stress and contact stress of nylon cord rubber before and after optimization 1 3 Rubber stress, MPa Value Cord stress, MPa Contact stress, MPa 282 Petroleum Science (2021) 18:269–284 1. Using the field packer as an example, a REFINE265 unit Table 10 Response surface model validation was applied in a three-dimensional numerical simulation Type Response surface Simulation Error, % model of the nylon cord rubber. The maximum pres- model, MPa model, MPa sure performance of the cartridge was 60 MPa while Setting pressure 71.02 75 5 the maximum sealing performance was 30 MPa. The Contact pressure 57.42 52.3 9.78 simulation results were consistent with the test results. These outcomes verified the accuracy of the simulation model. 30 MPa. For the optimized rubber, the bearing pressure of 2. Using Plackett–Burman test, steepest climbing test and the post-optimized cartridge was 70 MPa while the sealing response surface test, the multi-objective optimization pressure was 50 MPa. The error was within 40% as shown in of the nylon cord rubber was performed with the pres- Fig. 14. The test results show that the optimized nylon cord sure-bearing and sealing performance as the objective rubber effectively complied with the requirements of low- functions. The accuracy of the response surface model permeability oil and gas reservoirs with a 70 MPa setting obtained was superior simulated with ANSYS, which pressure and 50 MPa sealing pressure. remains within 90%. 3. Based on the actual working parameters in the field, the optimal combination of nylon cord rubber tube structure 4 Conclusions parameters (the cord angle (16°), cord layers (6), and cord spacing (1.6 mm)) was determined. It was shown A novel numerical simulation and optimization method was that the optimized rubber bearing pressure performance used to evaluate the performance of the expandable packer. increased to 75 MPa. This was a 25% increase compared To meet the requirements of high-pressure and fine fractur - to the pre-optimized rubber. The sealing performance ing, the reliability of the optimized results was verified by is increased to 57.5 MPa, which was a 66% increase laboratory tests. compared to than that before. Simulation of optimized packer Experiment of optimized packer Simulation of packer Experiment of packer 10 20 30 40 50 60 70 80 Setting pressure, MPa Fig. 13 Rubber simulation versus experiment comparisons before and after optimization 1 3 Working pressure, MPa Petroleum Science (2021) 18:269–284 283 0.50 The error precentage after opimization 0.45 The error percentage before optimization 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 10 20 30 40 50 60 70 Setting pressure, MPa Fig. 14 The error percentage of sealing performance before and after optimization 4. The laboratory tests showed that the maximum set pres- References sure of the barrel after optimization was 70 MPa while Agarwal K, Kegel J, Ballard B. Evolving completion designs to opti- the maximum working pressure was 50 MPa. 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