
==== Front
Heliyon
Heliyon
Heliyon
2405-8440
Elsevier

S2405-8440(24)12372-9
10.1016/j.heliyon.2024.e36341
e36341
Research Article
Comprehensive analysis of the effect of structural parameters on erosion wear, structural stress, and deformation of high-pressure double-elbow in shale-gas fracturing
Yang Siqi yangsiqi@cnpc.com.cn
a⁎
Fan Jianchun fjc19090@126.com
b
Zhao Nan zhn2009@petrochina.com.cn
c
Yang Jiakun woshiyizuzu@163.com
d
Xu Changfeng xucf@petrochina.com.cn
c
Lu Junan lujunan-tlm@petrochina.com.cn
e
Zou Guanggui 13579049370@163.com
e
Wang Jianjun wangjianjun005@cnpc.com.cn
a
Dai Siwei 903160272@qq.com
b
Zhou Binchao zhou_bingchao@163.com
f
a State Key Laboratory of Oil and Gas Equipment, CNPC Tubular Goods Research Institute, Xi'an, 710076, China
b Key Laboratory of Oil and Gas Safety and Emergency Technology, China University of Petroleum, Beijing, 102249, China
c Gas Storage Co., Ltd., PetroChina Xinjiang Oilfield Company, Hutubi, 831200, China
d PipeChina ZhongYuan Gas Storage Limited Liability Company, Puyang, 457001, China
e PetroChina Tarim Oilfield Company, Korla, 841000, China
f Mechanical Engineering College, Xi'an Shiyou University, Xi'an, 710065, China
⁎ Corresponding author. yangsiqi@cnpc.com.cn
14 8 2024
30 8 2024
14 8 2024
10 16 e363414 7 2024
13 8 2024
13 8 2024
© 2024 The Authors
2024
https://creativecommons.org/licenses/by-nc/4.0/ This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).
In field hydraulic fracturing operation of shale gas development, the high pressure and large displacement liquid-particle two-phase fracturing fluid can be forced to change direction many times through high-pressure double-elbow, and be transported from the outlet pipeline of the fracturing pump to the main pipeline. The high-pressure double-elbow is prone to be affected by erosion wear and Fluid-Structure Interaction (FSI), resulting in perforation and fracture, posing a potential safety threat to field operation. In this study, we conducted the erosion wear experiments on 35CrMo steel used for high-pressure double-elbow in shale-gas fracturing. The erosion rates under different impact angles and flow velocities were obtained, and proposed a novel model of erosion prediction for high-pressure double-elbow. Then the numerical investigation was employed to conduct a comprehensive analysis of erosion wear, structural stress and deformation by the coupling of Computational Fluid Dynamics (CFD) and Finite Element Analysis (FEA). The effects of structural parameters such as connection straight pipe length, pipe inner diameter and fluid turning direction were discussed. The results indicate that with the increase of connection straight pipe length, the flow erosion decreases first then varies little, and the deformation gradually increases. Slight erosion wear but large structural stress and deformation in major inner diameter pipe. And the minimum degree of erosion and flow-induced deformation present with the fluid turning direction of double-elbow as 0°. The study can provide references for the design, installation and detection of high-pressure double-elbow and ensure safety in the process of shale gas fracturing.

Keywords

Erosion prediction
Double-elbow
Fluid-structure interaction
Structural parameters
Numerical simulation
==== Body
pmc1 Introduction

Shale gas as an important unconventional energy, has attracted worldwide attention in recent years [1,2]. China's shale gas development has developed rapidly, its ‘factory mode’ fracturing operation can produce efficiently but also bring flinty challenges of long-term continuous operation and complicated loading for fracturing equipment [3]. As a key fracturing equipment, the high-pressure manifold system is used to convey pulsating large displacement and high-pressure fracturing fluid pumped by fracturing truck to the down-hole to produce artificial micro-cracks [4]. The manifold not only withstanding FSI under high pressure, but also suffering solid particle erosion from fracturing fluid, this results in erosion wear, flow-induced deformation and stress concentration [[5], [6], [7]]. Beneath these coupling factors, the manifold system, especially the parts which alter velocity or flow direction, is confronted with an extremely high accident risk [8]. High-pressure double-elbow plays a vital role in the manifold system, from Fig. 1, we can see that a double-elbow is used to connect the outlet pipeline of fracturing pump and the main pipeline, flow direction of fracturing fluid that flow inside the double-elbow has changed many times. According to on-site accident statistics, this structure is the most prone to severe failure such as perforation and fracture. In case of these problems, leakage of fracturing fluid under high pressure would constitute a major threat to site fracturing operation and result in operator injury and environmental pollution. Therefore, it's urgently necessary to comprehensively analyze the flow erosion, structure stress and deformation of double-elbow under fracturing process.Fig. 1 Photograph of high-pressure double-elbow.

Fig. 1

Flow erosion wear is an important reason for material failure in many engineering fields, and therefore this subject has increasingly drawn interests from more and more researchers. Numerous studies have been conducted by experiment methods and numerical simulations on flow erosion of elbows [[9], [10], [11], [12]]. For the experimental study, Yoganandhet al. [13] conducted orthogonal erosion tests on four influencing factors, including flow velocity, impact angle, slurry concentration, and particle size. The results showed that the contribution of flow velocity and impact angle to the erosion wear of the materials was 60 % and 21 %, respectively. In regard to impact velocity, many experimental studies have shown that the higher the impact velocity of a particle, the higher its kinetic energy when impacting the target material [[14], [15], [16], [17]]. In terms of impact angle, most studies have found that for ductile materials, the erosion wear rate increases first and then decreases with the increase of impact angle [18,19]. It has been observed that the impact angles causing the most severe erosion wear generally occur between 20° and 40° [20,21]. For the numerical simulation, Zahedi et al. [22] created a database to study the flow erosion of elbows with different fluid conditions and sand particles properties by using CFD simulations. Parsi et al. [23,24] carried out a CFD simulation of solid particle erosion in gas-dominant multiphase flow, reported that the maximum erosion occurred at the top of elbow extrados, and investigated the effects of particle velocity, particle distribution and particle size on the erosion mechanism. However, in the research related to erosion wear, most studies focused on altering flow parameters, while the impact of structural parameters on erosion has received relatively less attention. As for the high-pressure double-elbow in shale-gas fracturing, considering the fluid properties and particle properties are basically determined by formation condition and reservoir micro-cracks, which are difficult to change as our wish. Thus, it is imperative to analyze the impact of structural parameters on erosion rates in order to mitigate the erosion wear of pipe fittings.

Furthermore, strong FSI occurs at double-elbow pipe wall with the internal large displacement and high-pressure fracturing fluid, resulting in flow-induced stress and deformation. Therefore, it is necessary to consider the FSI severity of double-elbow at the same time. With the development of CFD and Computational Structural Dynamics (CSD), CFD-CSD method has gradually become a popular solution to study FSI in pipelines [[25], [26], [27]]. However, coupling studies on flow erosion and FSI effects are limited. Zhu et al. have made great contributions in this area through analysis of gas-solid flow of the single-elbow used in the gas well development [[28], [29], [30], [31]]. They researched the effects of gas flow velocity, fluctuation period and particle concentration on flow erosion and structure displacement. However, unlike a single-elbow, the FSI effects faced by a double-elbow are more intricate. At present, the relevant research is always focus on the single elbow, but there is little analysis on the double-elbow. Therefore, for high-pressure double-elbows that are prone to erosion and deformation damage during fracturing operations, it is necessary to employ the CFD-CSD method to comprehensively analyze the behavior of erosion, structural deformation, and stress.

Consequently, in this study, the erosion experiments on 35CrMo steel with different impact angles and flow velocities were carried out. Additionally, a novel model of erosion prediction for high-pressure double-elbow was established. Then the numerical investigation was employed, and several vital results such as erosion rate, structural equivalent stress and flow-induced deformation of double-elbow were simulated. The effects of connection straight pipe length, pipe inner diameter and fluid turning direction on flow erosion, structure stress and deformation are discussed respectively by a series of numerical experiments. This research can provide reference for the optimized structural design, installation method, and detection strategy of high-pressure double elbows in shale gas fracturing process.

2 Fundamental theory

In this paper the fundamental theory of the numerical simulation includes four different models: continuous phase flow, particle motion, erosion prediction and structure motion.

2.1 Continuous phase flow

Generally, turbulence incessantly perturbs particles within the flow field, resulting in intricate and unpredictable particle trajectories. This diffusion effect contributes to a more uniform distribution of particles throughout the flow field. In turbulent flows, the velocity distribution of particles frequently exhibits non-uniformity, particularly in the near-wall region, where the effect of turbulent shear stress significantly alters the particle velocities. The turbulence motion of unsteady continuous flow in double-elbow can be described by Unsteady Reynolds-Averaged-Navier-Stokes (URANS) equations. Continuity equation and Navier-Stokes (N–S) equation are expressed by Eqs. (1), (2), respectively [32,33]:(1) ∂ui‾∂xi=0

(2) ∂ui‾∂t+uj‾∂ui‾∂xj+∂ui′uj′‾∂xj=−1ρf∂p‾∂xi+νfδ2ui‾δxjδxj+fi

where ui‾ and uj‾ denote the average flow velocities in i and j directions respectively, xi and xj denote the space coordinate in i and j directions respectively, t means the flow time, p‾ means the average flow pressure, ρf and νf represent density and Kinematic viscosity coefficient of flow fluid respectively, ui′uj′‾ is Reynolds stress term. fi is volumetric force in i direction.

The URANS equations can be closed by using the Re-Normalized Group (RNG) k-ε turbulence model with rotation correction. The RNG k-ε model provides improved accuracy for swirling flows and has an additional term in the turbulent dissipation rate transport equation that also improves its performance for rapidly strained flows [34]. Transport equations of turbulence kinetic energy (k) and turbulence dissipation rate (ε) are expressed by Eqs. (3), (4), respectively [35]:(3) ∂(ρfk)∂t+∂(ρfkui‾)∂xi=∂∂xj[(μ+μtσk)∂k∂xj]+Gk−ρfε

(4) ∂(ρfε)∂t+∂(ρfεui‾)∂xi=∂∂xj[(μ+μtσε)∂ε∂xj]+C1εεkGk−C2ε*ρfε2k

In which,(5) C2ε*=C2ε+Cμη3(1−η/η0)1+βη3

(6) η=(2Sij⋅Sij)1/2kϵ

(7) Sij=12(∂ui‾∂xj+∂uj‾∂xi)

where μ and μt are the dynamic viscosity and turbulence viscosity of flow fluid respectively, Gk means generation item of turbulence kinetic energy, C2ε* is corrected items of C2ε and C1ε, C2ε, Cμ, σk, σε , η0 and β are empirical constants given as 1.48, 1.68, 0.0845, 0.7194, 0.7194, 4.38 and 0.012.

2.2 Particle motion

Particles in fracturing fluid are regard as discrete phases and described by DPM, the trajectory of particles can be tracked by motion equations of particles in the Lagrangian framework [36]:(8) du⇀pdt=f⇀D+f⇀P+f⇀VM+f⇀G

in which:(9) f⇀D=3μρpdp2CdRep4(u⇀−u⇀p)

(10) f⇀P=(ρρp)∇Ps

(11) f⇀VM=12ρd(u⇀−u⇀p)ρpdt

(12) f⇀G=(ρp−ρ)ρpg⇀

where u⇀ and u⇀p are the velocity vectors of liquid and particle respectively, f⇀D is the drag force per unit mass, f⇀P is the pressure gradient force per unit mass, f⇀VM is the virtual mass force per unit mass, f⇀G is the buoyancy force per unit mass, μ is the liquid viscosity,Ps is the static pressure, g⇀ is the gravity acceleration vector, Rep is the particle Reynolds number:(13) Rep=ρdp|u⇀p−u⇀|μ

Cd is the drag coefficient:(14) Cd=m1+m2Rep+m3Rep2

where m1, m2 and m3 are constants, which are given by Morsi and Alexander [37].

The fluid carrier influences the dispersed phase via drag and turbulence, and the particles in turn influence the carrier fluid via the reduction in mean momentum and turbulence. The two-way coupling is used to solve the interaction between the particles and the liquid.

The particle reflection cases were calculated using the particle-wall rebound model proposed by Forder [38] with the following restitution coefficient equations:(15) en=0.988−0.78θ+0.19θ2−0.024θ3+0.027θ4

(16) et=1−0.78θ+0.84θ2−0.21θ3+0.028θ4−0.022θ5

2.3 Erosion prediction

When considering the slurry erosion of carbon steel, the E/CRC erosion model, advocated by the University of Tulsa, stands as the most suitable prediction. Derived from numerous direct impingement experiments conducted on plate specimens, this model has garnered widespread acceptance for forecasting erosion in elbows, straight pipes and branch pipes [[39], [40], [41]]. Mathematically, the E/CRC model can be formulated as [42,43]:(17) ER=C(BH)−0.59FsupnF(θ)

F(θ)=∑i=15Eiθi

where ER represents the erosion rate (mg/mg), calculated as the ratio of the target material's mass loss to the impacting particles' mass. C is a constant (C = 2.17E-07 for carbon steel). The particle sharpness factor FS varies depending on the particle shape, ranging from 0.2 for rounded sand particles, 0.53 for semi-rounded particles, to 1 for sharp particles. BH is the Brinell hardness of the target material, up is the particle impact velocity, n is the velocity exponent (n = 2.41). θ is the impact angle (rad), F(θ) is the impact angle function, Ei are the function coefficients and the values are listed in Table 1.Table 1 Values of parameters in E/CRC model.

Table 1E1	E2	E3	E4	E5	
5.4	−10.11	10.93	−6.33	1.42	

In this investigation, we conducted erosion tests on 35CrMo steel samples, specifically used for high-pressure double-elbow applications. The mechanical properties and nominal chemical composition of 35CrMo are summarized in Table 2. These experiments served as the foundation for the development of a novel erosion model, building upon the established E/CRC model framework.Table 2 Mechanical properties and nominal chemical composition of 35CrMo.

Table 2Density, kg/m3	Yield Strength, MPa	Tensile Strength, MPa	Elastic Modulus, GPa	Brinell hardness	Reduction of area	Extensibility	
(a) Mechanical properties of 35CrMo steel	
7850	835	985	207	229	45 %	≥12 %	
	C	Si	Mn	Cr	Mo	P	S	Ni	Cu	
(b) Nominal chemical composition of 35CrMo steel	
wt%	0.33–0.38	0.15–0.35	0.70–0.90	0.80–1.10	0.15–0.25	≤0.035	≤0.040	≤0.030	≤0.030	

To conduct the erosion experiments, we utilized a liquid-solid two-phase flow circulating jet erosion testing apparatus, whose schematic diagram is depicted in Fig. 2(a). The whole erosion testing system included the solid-liquid mixing tank, slurry pump, hydraulic servo system, recirculation pipeline, erosion chamber and control&data acquisition cabinet. The geometric layout of the specimen employed in these erosion tests is illustrated in Fig. 2(b). For all experiments, spherical ceramsite sand was chosen as the erodent due to its prevalent usage in actual hydraulic fracturing scenarios (See Fig. 2(c)). The mechanical properties and nominal chemical composition of the ceramsite sand are summarized in Table 3. Prior to commencing the tests, the ceramsite sand particles were carefully sieved to ensure a uniform size range of 300–500 μm.Fig. 2 Schematic diagram of the experimental setup. (a) erosion testing system, (b) geometric layout of the specimen, (c) spherical ceramsite sand.

Fig. 2

Table 3 Mechanical properties and nominal chemical composition of the ceramsite sand.

Table 3Density kg/m3	Average size μm	Sphericity	Roundness	Mohs hardness	Breakage rate, %	
(a) Mechanical properties of ceramsite sand	
1800	400	0.9	0.9	8	86 MPa ≤ 8	
	Al2O3	SiO2	Fe2O3	TiO2	MnO	K2O	CaO	Others	
(b) Nominal chemical composition of ceramsite sand	
wt%	65.82	14.94	9.43	3.88	2.94	0.85	0.77	1.37	

During the experimental process, the ceramsite sand particles and water were thoroughly mixed in the solid-liquid mixing tank. The resulting sand-carrying liquid then entered the erosion chamber through the slurry pump and recirculation pipeline, impinging on the specimen in the form of a jet to cause erosion. After impacting the specimen, the sand-carrying liquid returned to the solid-liquid mixing tank, forming a continuous cycle. The impact velocity was adjusted to range from 7.5 m/s to 20 m/s, and the impact angle varied from 0° to 90°, the average particle size was 400 μm, and the particle mass concentration was 10 %. Each erosion test lasted for 60 min. The specimen weight was measured before and after the erosion test, and the erosion rate obtained from the experiment was calculated as the ratio of the specimen weight loss (mg) to the mass of the impacting particles (mg). For each operating condition, three repeated tests were conducted to ensure the reliability of the test results.

Fig. 3 depicts the variation of erosion rate with different impact angles, where the flow velocity was set at 15 m/s during the experiment and the impact angle ranged from 0° to 90°. As can be observed from the figure, the erosion rate initially increases and then decreases with the increase in impact angle. The maximum erosion rate occurs at a 30° impact, while the minimum erosion rate is observed at a 90° impact. According to the research by Wang et al. [44], the impact angle of fluid on elbow sections ranges from 20° to 30°, resulting in a significant increase in erosion rate at the elbow compared to straight pipe. In addition, Fig. 3 shows the SEM micrographs of the eroded region of the target material at impact angles of 30° and 90°. At low impact angles of and 30°, the entire region was covered with narrow furrows. Notably, the lips discernible on both lateral edges and the anterior terminus of these furrows were primarily attributable to the accumulation of metal particles, a phenomenon stemming from the extrusion of material particles. Conversely, At the high angle impact of 90°, the entire region was covered with indentation craters generated by plastic deformation of the material.Fig. 3 Variation of erosion rate under different impact angles.

Fig. 3

During the hydraulic fracturing, the fracturing fluid displacement varies significantly in different stages. The variation of erosion rate with different flow velocities is shown in Fig. 4, where the impact angle was set at 30° and the flow velocity varied from 5 m/s to 25 m/s during the experiment. It is clearly demonstrated that the effect of flow velocity on erosion wear is significant, and the erosion wear rate rapidly increases with the increase in flow velocity. The relationship between the above two parameters can be fitted with a first-order power function, with a power exponent value of 2.08, which is consistent with the range corresponding to ductile metals [45,46]. In addition, Fig. 4 shows the SEM micrographs of the eroded region of the target material at flow velocities of 10 m/s and 20 m/s. At a low flow velocity of 10 m/s, the size of the furrows was relatively small. As the flow velocity increased, the micro-scale cutting effects inflicted on the target material by the impact particles became progressively pronounced, leading to a substantial enlargement in the dimensions of the formed furrows. By analyzing the variety of the microstructure of eroded region at different flow velocities, it is discernible that an increase in flow velocity lead to a more intense erosion wear, aligning well with the recorded erosion rate results.Fig. 4 Variation of erosion rate under different flow velocities.

Fig. 4

Based on the erosion data and E/CRC model, the new erosion model can be written as follows:(18) ER0=C′BH−0.59FSF′(θ)Vn′

(19) F′(θ)=a0+a1cos(wθ)+b1sin(wθ)+a2cos(2wθ)+b2sin(2wθ)

where F′(θ) is the revised impact angle function obtained by the polynomial fitting of erosion data at different impact angles, and it can be can be expressed as a two-term Fourier series, the values of ai, bi and w are listed in Table 4, n’ is the revised velocity exponent (n’ = 2.08) obtained by the exponential fitting of erosion data at different flow velocities, C′ is the revised constant whose value can be deduced from the erosion data (in our model, C' = 2.25 × 10−6).Table 4 Values of parameters in the new erosion model.

Table 4a0	a1	b1	a2	b2	w	
−0.507	−0.143	1.831	0.649	0.306	1.781	

The experimental results are compared to the corresponding values derived from the novel erosion model, as illustrated in Fig. 5. A remarkable congruence is discernible between the experimental values and model predictions. Consequently, the new erosion model can be stands poised to be integrated into CFD simulation.Fig. 5 Comparison of experimental data and model prediction results.

Fig. 5

2.4 Structure motion

According to the basic theory of theoretical mechanics, the structure motion of double-elbow is established as follow:(20) mpy″+cpy′+kpy=f(t)

where mp, cp and kp represent the mass, damping and stiffness of structure respectively, y″, y′ and y denote the acceleration, velocity and displacement of structure respectively, and f means the force of structure.

Moreover, according to the theory of elasticity, the pipe wall structure stress under internal pressure load can be expressed by the Lamé equation:(21) σr=1−Ro2r2Ro2Ri2−1p

(22) σθ=Ro2r2+1Ro2Ri2−1p

where σr and σθ are the radial stress and circumferential stress of pipe wall respectively, Ro and Ri are the external and internal radius of elbows respectively, r is the distance to the central axis of pipe, p means the flow pressure.

3 Numerical simulation

3.1 Geometry modeling construction

A three-dimensional geometric model has been established, which is consistent with the double-elbow structure used in the field. The geometric model contains five parts: inlet straight pipe, the first elbow, connection straight pipe, the second elbow and outlet straight pipe. Fig. 6 shows the double-elbow schematic diagram of standard case, and Table 5 shows the designed structural parameters in standard case.Fig. 6 3D schematic diagram of double-elbow structure.

Fig. 6

Table 5 Important designed structural parameters of double-elbow in standard case.

Table 5Structural parameters	Design data	
Inlet straight pipe length(Li)/mm	1500	
Outlet straight pipe length(Lo)/mm	1500	
Internal diameter(Di)/mm	70	
External diameter(Do)/mm	106	
Curvature radius(R)/mm	R = 2Di = 140	
Connection straight pipe length(Lc)/mm	Lc = 1.5Do = 159	

To study the influence of structural parameters, connection straight pipe length, internal diameter and fluid turning direction of double-elbow are set as variables, and some simulation cases are established. Table 6 shows the involved structural parameters of all simulation cases. Case 1 is the standard case, case 2 to 7 are used to evaluate the effect of connection straight pipe length, case 8 to 11 are used to evaluate the effect of internal diameter, case 12 to 13 are used to evaluate the effect of fluid turning direction.Table 6 Simulation cases.

Table 6Case	Dimensionless connection straight pipe length (Lc/Do)	Internal diameter(Di)/mm	Fluid turning direction (θ)/°	
1	1.5	70	90	
2	1	70	90	
3	2	70	90	
4	3	70	90	
5	4	70	90	
6	5	70	90	
7	6	70	90	
8	1.5	50	90	
9	1.5	60	90	
10	1.5	80	90	
11	1.5	90	90	
12	1.5	70	0	
13	1.5	70	180	

3.2 Computational mesh

ANSYS Meshing is employed to perform mesh generation for fluid and solid regions. Fig. 7 shows the computation mesh of the two regions of standard case. A mesh-independence test is conducted to select the appropriate mesh size for simulation, and the comparison of the maximum erosion rate and structural deformation of the elbows with four representative grids is summarized in Table 7. For the erosion rate, it can be noted that a maximum difference of 10.4 % exists between M1 and M2, and a minimum difference of 1.55 % is present between M3 and M4. For the structural deformation, it can be seen that the four representative grids have a relatively small impact on the simulation results. In this study, a comprehensive consideration of two-way FSI across both fluid and solid regions necessitated careful attention to the long iterative calculation cycle. To strike a balance between accuracy and CPU time efficiency, M3 is chosen as the computational mesh for all simulations. Specifically, the fluid and solid regions are discretized into 46222 and 28187 elements, respectively.Fig. 7 Computational mesh used for the simulation.

Fig. 7

Table 7 Mesh-independent tests: the maximum erosion rate and structural deformation of the elbows (standard case).

Table 7Mesh	Elements in fluid region	Elements in solid region	Maximum erosion rate (kg·m−2·s−1)	Percentage changes	Maximum deformation(mm)	Percentage changes	
M1	12712	8012	8.10E-05	/	4.34E-01	/	
M2	21350	12862	9.04E-05	10.4 %	4.42E-01	1.81 %	
M3	46222	28187	9.50E-05	4.84 %	4.37E-01	1.14 %	
M4	104023	60720	9.65E-05	1.55 %	4.33E-01	0.92 %	

3.3 Boundary conditions

Based on the practical conditions, the boundary conditions of the flow field are set up. The inlet and outlet boundary are assumed to be velocity-inlet and pressure-outlet conditions, respectively. The inlet pulsating fluid is consistent with the output flow of the quintuple cylinders fracturing pump used in the field, which is 1.876+0.04726sin(5πt) m3/min. Turbulence intensity is set to 5 %. And spherical particles are uniformly injected with the same velocity as the liquid, the reflection condition is adopted on the particle collision. The properties of liquid and particles in the flow field are shown in Table 8. The pressure at the outlet is set to 80 MPa to simulate high-pressure environment.Table 8 The properties of liquid and particles in the flow field.

Table 8	Fluid (water)	Particle(sand)	
Density ρp (kg/m3)	998.2	1650	
Dynamic viscosity μ (Pa·s)	1.003 × 10−3		
Particle size dp (μm)		1000	
Mass concentration Mp (kg/m3)		18	

For pipe structure, the inner wall is set as FSI interface and considered to be no-slip. The pipe inlet and outlet boundary can be assumed to be fixed support. The material of pipe wall is selected as 35CrMo steel, the density is 7850 kg/m3, the Young's modulus is 212 GPa, and the Poisson's ratio is 0.286.

3.4 Numerical procedure

Fig. 8 shows the flowchart of numerical procedure, which can be divided into pre-precessing, CFD/Finite Element Method (FEM) calculation and post-precessing. All programs are implemented on ANSYS multi-physical platform. Fluid and solid regions are processed in FLUENT and ANSYS mechanical analysis module respectively. The two-way FSI calculation between these two regions is completed in System coupling module, and the results of the flow field and structural field are obtained.Fig. 8 The flowchart of numerical procedure.

Fig. 8

RNG k-ε turbulence and DPM model are applied in FLUENT. In the spatial discretization settings, pressure terms and convection terms can be processed based on second-order discretization scheme and second-order upwind scheme. In addition, dynamic mesh is set on the inner wall to accommodate the mesh deformation caused by FSI. On the fluid-solid interfaces, the force and displacement of on the two boundaries are taken as variables to calculate iteratively and the time step is set to 0.005 s. The calculation is completed after 1.0 s, and the completed simulation results are used for study.

3.5 Model validation

Considering standard case as the study model, in flow field simulations, the flow field velocity can be obtained by solving N–S governing equations. For validating the numerical model, the inlet section of the fluid region is selected as the reference-plane, and the check-plane is located at 400 mm from the reference-plane.

Based on the unsteady Bernoulli equation [47], velocity in check-plane should meet:(23) du(t)dt∫xrxcdx+(uc2(t)2−ur2(t)2)+Pc(t)−Pr(t)ρf=0

Where u(t) represents the fluid time-varying velocity equation, xc and xr are the positions of check-plane and reference-plane respectively, Pc(t) and Pr(t) are the section-averaged pressure of the check-plane and reference-plane for time t respectively, uc(t) represents the theoretical result of the velocity in check-plane for time t, and ur(t) is the velocity in inlet section which has been set.

The calculation time is set as 0.25 s, 0.5 s, 0.75 s and 1.0 s respectively, as shown in Table 9, there is a good agreement with the simulation and theoretical results of the velocity in check-plane at these calculation times, and the maximum relative error is 1.46 %. Therefore, the simulation model used in this paper is available.Table 9 Simulation and theoretical results in check-plane.

Table 9Calculation time (s)	Simulation result of velocity (m/s)	Theoretical result of velocity (m/s)	Relative error (%)	
0.25	8.132	8.194	0.76	
0.5	8.583	8.658	0.87	
0.75	8.129	8.180	0.62	
1.0s	8.281	8.404	1.46	

4 Analysis of results

4.1 Standard case results

Fig. 9 presents the pressure distribution of the whole double-elbow and the velocity distribution at the first and second elbows. It can be seen that the pressure distribution in the straight pipe is uniform, and there is a significant pressure gradient in each elbow. The maximum and minimum pressure turn up on the extrados and intrados of the first elbow respectively, which directly leads to the formation of eddy currents at the elbow. Fig. 10 shows the secondary flow vector in a cross-section normal to the pipe axis at the entrance and the exit of the two elbows. Due to the centrifugal effect and radial pressure gradient, the vortices are formed when the high-speed fracturing fluid passes through the two elbows, and the secondary flow caused by vortices continues to develop with the mainstream. Under the action of centrifugal force and secondary flow, the particles mostly affect the extrados of the two elbows. According to the study by Wang et al. [48], the impact angle of particles on the elbow is less than 30°, resulting in particles primarily eroding the pipe wall in a plowing manner with a small angle at the elbow, thereby causing severe erosion in this area.Fig. 9 Pressure and velocity distribution of double-elbow.

Fig. 9

Fig. 10 Secondary flow vector in a cross-section normal to the pipe axis at (a) the first elbow entrance, (b) the first elbow exit, (c) the second elbow entrance, (d) the second elbow exit.

Fig. 10

In addition, the erosion rate contour and the particle trajectory are shown in Fig. 11, and it can be found that the erosion severity of the second elbow is higher than that of the first elbow. The behavior of dispersed particles in fluid flow can be explained by analyzing the particle Stokes number (St), which is defined as the ratio of the particle response time to the fluid travel time and can be expressed as:(24) St=ρpdp2u18μDi

Fig. 11 Erosion rate contour and particle trajectory.

Fig. 11

For St ≪1, the particle response time is much less than the fluid travelling time. Thus there is sufficient time for particles to respond to changes in the flow field, and they can follow the fluid flow closely. However, for St≫1, the case is opposite. Thus the particles will move independently of the fluid flow.

Based on the fluid operating conditions, the St can be calculated to be 0.01. It can be seen from the distribution of particle trajectory that the particles at the first elbow are evenly distributed in the circumferential direction in the pipe before impacting the pipe wall. Therefore, the first elbow is impacted relative uniformly by particles in a large area. Due to the St≪1, under the influence of centrifugal effects and secondary flows, after passing through the first elbow, the particles flow along the outer side of the connecting straight pipe with the fluid and impact on the specific relatively small area of the extrados of the second elbow, which is verified by the second flow vector plot as shown in Fig. 10(d). Therefore, the maximum corrosion rate at the second elbow is higher than that at the first elbow. Consequently, in field operations, the risk of perforation due to erosive wear at the second elbow is higher than that at the first elbow.

The flow-induced deformation of high-pressure double-elbow, as shown in Fig. 12, it can be seen that the two elbows and connection straight pipe have large deformation with the flow force acting on the pipe wall. Furthermore, due to the centrifugal effect and pressure gradient, the extrados of the elbows is affected by relatively large flow force, which can also be confirmed in Fig. 12, the deformation of the outside of pipe wall is larger than that of the inside. Fig. 13 shows the equivalent stress distribution of high-pressure double-elbow. Generally, the inner wall of pipe is subjected to greater stress than the outer wall. Thus, the fatigue crack may be generated from the inner wall first. The stress of the elbow intrados is greater than that of the elbow extrados. And the maximum equivalent stress occurs at the inner wall of the fixed end of the inlet and outlet, with a value of 334 MPa. Therefore, the anchorage end of the inlet and outlet straight pipe, should be given more attention in the process of design and operation.Fig. 12 Flow-induced deformation of high-pressure double-elbow.

Fig. 12

Fig. 13 Equivalent stress distribution of high-pressure double-elbow.

Fig. 13

4.2 Effect of the connection straight pipe length

The contours of erosion rate with different dimensionless connection straight pipe length (Lc/Do), as shown in Fig. 14. It can be clearly detected that the erosion rate gradually decreases with the increase of connection straight pipe length when Lc/Do ≤ 4. This is because the particles development more fully after flowing through the first elbow and are more evenly distributed in the circumferential direction of the connection straight pipe before impacting the second elbow with increasing the connection straight pipe length. Thus the erosion rate at the second elbow is reduced. When Lc/Do > 4, the erosion rate changes little with the length of the connection straight pipe increases, which is due to the fully flow development when the connection straight pipe is long enough. In addition, as the erosion rate of the second elbow decreases, the maximum erosion rate of the two elbows gradually approaches. Fig. 15 shows the relationship between the maximum erosion rate (emax) of the double-elbow and the connection straight pipe length (Lc/Do). It can be seen that the maximum erosion rate decreases linearly as the increase of connection straight pipe length when Lc/Do ≤ 4. And the prediction formula is established as follow:(25) emax=−4.16×10−3(LcDo)+4.93×10−2

Fig. 14 Effect of connection straight pipe length on flow erosion.

Fig. 14

Fig. 15 The maximum erosion rate (emax) versus with connection straight pipe length (Lc/Do).

Fig. 15

The flow-induced deformation of double-elbow with different dimensionless connection straight pipe length (Lc/Do) as shown in Fig. 16. Due to the change of structural stiffness, the deformation degree of double-elbow increases as the connection straight pipe length increases. The most severe flow-induced deformation happens in the longest connection straight pipe (Lc/Do = 6) with the maximum deformation arriving at 0.505 mm. Fig. 17 shows the maximum deformation (demax) versus with the connection straight pipe length (Lc/Do), and the prediction formula is obtained by using polynomial fitting, which can be expressed as:(26) demax=0.392+0.032(LcDo)−0.00231(LcDo)2

Fig. 16 Effect of connection straight pipe length on flow-induced deformation.

Fig. 16

Fig. 17 The maximum deformation (demax) versus with connection straight pipe length (Lc/Do).

Fig. 17

The results of erosion rate and flow-induced deformation are considered comprehensively. It is better to design the dimensionless connection straight pipe length as Lc/Do = 4. In this way, the erosion wear of high-pressure double-elbow can be alleviated without severe flow-induced deformation.

4.3 Effect of the pipe inner diameter

As shown in Fig. 18, the contours of erosion rate with different inner diameter pipes indicate that the erosion degree of high-pressure double-elbow changes greatly with the variation of pipe inner diameter. The maximum erosion rate, 5.90e-04 kg m−2•s−1, occurs in the minimum inner diameter pipe (Di = 50 mm), which is nearly 39 times than that in the maximum inner diameter pipe (Di = 90 mm). This can be explained by the reason that, on the one hand, an increase in the pipe inner diameter leads to an enlargement of the impingement area at the elbow, which facilitates a more dispersed erosion effect. This expansion of the impact zone contributes to a mitigation of the erosion rate at the elbow by distributing the erosive forces over a larger area. On the other hand, when the pipe diameter is reduced while maintaining a constant inlet mass flow rate, the flow velocity experiences a marked increase. According to the kinetic energy equation, this acceleration of the fluid endows particles with greater kinetic energy, augmenting their impact force and subsequently exacerbating the erosion wear at the elbow. The two distinct effects arising from variations in pipe diameters underpin the differing mechanisms that govern the influence of pipe inner diameters on erosion rates. The relationship between the maximum erosion rate (emax) of the double-elbow and the pipe inner diameter (Di), as shown in Fig. 19, it can be obviously detected that the erosion severity decreases exponentially with the increase of pipe inner diameter, the fitting formula is expressed as:(27) emax=149.5exp(−Di7.88)+0.01

Fig. 18 Effect of pipe inner diameter on flow erosion.

Fig. 18

Fig. 19 The maximum erosion rate (emax) versus with pipe inner diameter (Di).

Fig. 19

Although the fluid flowing in the small inner diameter pipe has a large kinetic energy, the internal pressure load changes little with the pipe inner diameter under the relatively high internal pressure. According to the theory of elasticity, the stress in pipe wall increases with the increasing of the pipe inner diameter with the same wall thickness. The effect of the inner diameter of double-elbow on the equivalent stress of the pipe wall, as shown in Fig. 20, the variation trend is consistent with the theoretical analysis, which indicates that the larger the inner diameter of the pipe, the greater the equivalent stress. The flow-induced deformation of double-elbow presents the same tendency as the equivalent stress, as shown in Fig. 21, the maximum value of which increases from 0.296 mm to 0.632 mm as the pipe inner diameter increases from 50 mm to 90 mm. Fig. 22 shows the relationship between the maximum deformation (demax) of the double-elbow and the pipe inner diameter (Di), and the prediction formula is obtained by using polynomial fitting, which can be expressed as:(28) demax=0.232−0.002Di+7.59×10−5Di2

Fig. 20 Effect of pipe inner diameter on structural stress.

Fig. 20

Fig. 21 Effect of pipe inner diameter on flow-induced deformation.

Fig. 21

Fig. 22 The maximum deformation (demax) versus with pipe inner diameter (Di).

Fig. 22

At the same mass flow rate, increasing the pipe inner diameter can dramatically reduce the erosion wear of the double-elbow, but it will also result in increased the structural stress and flow-induced deformation. Therefore, the pipe inner diameter should not be too large or too small.

4.4 Effect of the fluid turning direction

By changing the dihedral angle of the two elbows, the different fluid turning directions are obtained. The contours of erosion rate and particle trajectory of double-elbows with different fluid turning directions, as shown in Fig. 23. It is shown that erosion becomes the most severe as the fluid turning direction is 180°. This is due to the reason that, after flowing through the first elbow, the particles are widely distributed in the outside of connection straight pipe and directly impact the extrados of the second elbow along the connection straight pipe, which is confirmed in the particle trajectory. The particle trajectory analysis also explains that why the erosion becomes the slightest as the fluid turning direction is 0°, due to the extrados of two elbows is not on the same side of the connection straight pipe, only a few particles can affect the extrados of the second elbow. In this case, the erosion rate of the second elbow is lower than that of the first elbow. In addition, the structural stiffness changes with the dihedral angle of the two elbows, which effect the structural flow-induced deformation. As shown in Fig. 24, the maximum deformation of double-elbow appears at 90° dihedral angle. It is reasonable to design the fluid turning direction of double-elbow as 0° based on the comprehensive consideration of the erosion rate and flow-induced deformation.Fig. 23 Effect of fluid turning direction on flow erosion.

Fig. 23

Fig. 24 Effect of fluid turning direction on flow-induced deformation.

Fig. 24

5 Conclusions

During the operation of shale gas fracturing, the high-pressure double-elbow is constantly subjected to the impact of the high internal pressure and large displacement solid-liquid two-phase flow, which can easily lead to severe erosion damage and flow-induced deformation of the pipe wall. Therefore, the aim of this study was to investigate the effect of structural parameters on erosion wear, structural stress, and deformation of high-pressure double-elbow in complex coupled environments. A new erosion prediction model suitable for 35CrMo steel used in high-pressure double-elbow was proposed by conducting erosion experiments at different impact angles and flow velocities. Then the numerical investigation was employed based on the coupling of DPM and FSI methods. A series of simulations are conducted to predict the influence of connection straight pipe length, pipe inner diameter and fluid turning direction. From the present investigation, the following conclusions are drawn.Ⅰ According to the erosion wear experimental results, the erosion wear rate variations of 35CrMo steel under different impact angles and flow velocities were obtained. As the impact angle increased, the erosion rate initially rose and then decreased, with the maximum and minimum erosion rates occurring at impact angles of 30° and 90° respectively. Furthermore, with the augmentation of flow velocity, the erosion rate increased in a power function fashion. Based on the experimental results, a novel erosion prediction model for was proposed for 35CrMo steel used in high-pressure double-elbow, formulated within the framework of the E/CRC model. Upon validation with experimental values, the prediction model exhibited a high degree of accuracy, with the coefficient of determination (R2) exceeding 0.95.

Ⅱ From the numerical simulation results of the double-elbow structure, it can be seen that: the extrados of elbow is more susceptible to erosive damage due to the particles impact by centrifugal force, and the maximum erosion rate of the second elbow is greater than that of the first elbow. The maximum structural stress is located at the inner wall surface of the anchor end of the inlet and outlet straight pipe, while the two elbows and connection straight pipe have large flow-induced deformation.

Ⅲ The erosion wear gradually decreases with the increasing of connection straight pipe length as Lc/Do ≤ 4, but as Lc/Do ＞4, it tends to be stable and the maximum erosion rate of the two elbows gradually approaches. The flow-induced deformation of the pipe wall increases as the connection straight pipe length increases. Considering together the erosion wear and deformation, it is better to design the Lc/Do = 4.

Ⅳ Both the flow erosion and FSI of double-elbow are sensitive to the pipe inner diameter. Although the erosion severity can be decreased exponentially with the increasing of pipe inner diameter, it can also result in the increase of the structure stress and flow-induced deformation. Therefore, the pipe inner diameter should not be too large or too small. Furthermore, the fluid turning direction of double-elbow also effects the erosion wear and flow-induced deformation, flow erosion and deformation become the slightest as the dihedral angle between two elbows is 0°.

In future research, we will consider the impact of the significant local stress on the pipe body of high-pressure double-elbows under complex fracturing conditions on erosion wear, structural stress, and deformation. Additionally, the consideration of particle shape, type, and rotational motion in the fluid on the aforementioned results will also be studied, which can more accurately assist field operators in safely maintaining high-pressure double-elbows.

CRediT authorship contribution statement

Siqi Yang: Writing – original draft, Conceptualization. Jianchun Fan: Formal analysis. Nan Zhao: Validation. Jiakun Yang: Methodology. Changfeng Xu: Project administration. Junan Lu: Methodology. Guanggui Zou: Project administration. Jianjun Wang: Writing – review & editing. Siwei Dai: Validation. Binchao Zhou: Software.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This research is supported by 10.13039/501100012166 National Key R&D Program of China  (No.2023YFC3009200 ), Key Science and Technology Projects for Basic and Prospective Research of CNPC (No.2023ZZ11 ), MIIT Fundamental public service platform for industrial technology (No. 2023-273-1-1 ), 10.13039/100014717 National Natural Science Foundation Project of China (No. 52175208 ).
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