Introduction

In the process of oilfield development, waterflooding technology is widely used to supplement formation energy and enhance oil recovery (EOR) efficiency in oilfields (Reis 2002; Liu et al. 2013; Moradi et al. 2020). Accurate control of the water injection rate of the separate-layer water injection is the key issue of the water injection technology (Alhuthali et al. 2010; Zhang et al. 2020). The injection rate of each layer is controlled by the downhole injection mandrel (Liu et al. 2017; Ming et al. 2023). It can effectively alleviate inter-layer interference caused by reservoir non-homogeneity. In order to improve the operating efficiency of the water injection system and reduce the production energy consumption of the water injection system, research on efficient water injection mandrel of the water injection system is one of the important aspects for oilfield development (Ruan et al. 2021).

The injection mandrel has been developed from the traditional hollow injection mandrel to the current eccentric intelligent injection mandrel (Liang et al. 2011; Gomes Filho and Sampaio 2023). Liang et al. introduced the role of the eccentric injection mandrel. Through the injection mandrel, the injection pressure and flow rate of the high-permeability interval can be controlled, and the flow rate of the low-permeability interval can be increased (Liang et al. 2011). Fakhruzan et al. explained the technical principle of bridge eccentric injection mandrel and applied it in Kanali Asam field. The injection target of each layer was achieved by optimizing injection pressure and nozzle size (Fakhruzan et al. 2017). Cui et al. calculated the flow field of different roundness and size of the throttle tube for the front throttle injection mandrel, performed a comparative analysis of pressure loss and velocity, and determined the optimal tube diameter and number of roundness for different pressure losses based on the test results (Cui et al. 2020). The previous studies on the flow field and total pressure loss in the eccentric injection mandrel were less. Traditional experimental methods are widely used in the research, while relatively few detailed studies are conducted from numerical simulation methods, resulting in poor repeatability and high cost. The gap between this study and previous research is the analysis of flow characteristics and structure of the injection mandrel through computational fluid dynamic (CFD) simulation.

The eccentric injection mandrel is equipped with automatically adjustable nozzles, which allow the nozzle opening to be adjusted without human intervention, thus controlling the injection rate of each zone (Jia et al. 2012; Yu et al. 2017; Lyu et al. 2018). The smaller the nozzle throttling differential pressure of the injection mandrel, the lower the required injection pressure at the wellhead, and the smaller the energy consumption of the surface water injection system (Liu et al. 2017). Han et al. introduced the structure, principle and analysis of a new type of adjustable nozzle, and established a design calculation model through relevant performance tests (Han and Feng 2013). Liu et al. designed a downhole water injection mandrel with an adjustable choke. Following the V-shaped design, the choke was highly adaptable to testing and adjusting of minor water flow. With small losses and small torque for adjusting, the adjustable downhole water injection mandrel can be operated more easily under identical differential pressure (Liu et al. 2015). However, the structures of the adjustable nozzles studied in the above studies are all sleeve-piston type regulating valves. The gap between the sleeve and the plunger would cause a certain flow loss. Nozzle leakage makes it difficult to accurately adjust the injection rate of the injection zone, resulting in inaccurate test data of the well. Using a flat adjustable water nozzle can effectively solve the above problems. But there is less research on flat adjustable water nozzles at present. The shape of the injection mandrel nozzle is critical to control the flow rate and has been less studied yet. Therefore, it is necessary to optimize the structure of the water injection mandrel with a flat nozzle so that it has a good adjustable performance to ensure accurate control of stratified water injection. This can improve water injection efficiency and reduce water waste.

In this paper, an eccentric injection mandrel injection process is directly simulated by using FLUENT software, and orthogonal tests are conducted to optimize the structure in combination, which greatly saves the test time and cost. Firstly, through simulation and experiment, the flow rate and pressure loss of nozzles under different openings are obtained, and the consistency between simulation and experiment is verified. Secondly, five CFD models for different shapes of the adjustable nozzle were established. The results show that the semicircle nozzle has the smallest differential pressure loss, and the flow rate is linearly increasing with the opening degree; therefore, the eccentric injection mandrel with the semicircle nozzle has the best regulation performance. Finally, three prominent influencing factors (i.e., number of injection holes, injection hole diameter, and nozzle shape) were selected to optimize the injection mandrel. This study can provide clear insights into the optimized methods of injection mandrel for intelligent measurement and regulation, and improve accurate control of downhole wireless intelligent water injection. Therefore, this research can improve oil recovery efficiency in oilfields. The flowchart of this study is shown in Fig. 1.

Fig. 1
figure 1

The overall flowchart of structure optimization for eccentric injection mandrel

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Theoretical basis of water injection mandrel

In the process of water injection deployment application, due to the influence of the injected water quality, reservoir pressure, nozzle flow resistance, the nozzle clogging, and the difficulty in achieving reasonable flow allocation problems often occur. The appropriate nozzle structure is selected based on the actual situation of the water injection well flow allocation. Generally, the nozzle structure with the characteristics of a large flow adjustment range, easy adjustment of the flow area, and large hydraulic radius is selected. At the same time, the impact and wear of the fluid on the nozzles should be considered. Through comprehensive analysis, a new eccentric injection mandrel with fan-shaped adjustable water nozzle proposed by Petroleum Exploration and Production Research Institute was chosen in this paper. The cross-sectional dimensions of the water nozzle and flow channel are shown in Fig. 2.

Fig. 2
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Sectional view of water nozzle and flow channel

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The CFD simulations were conducted using FLUENT software, which is a widely recognized and established tool in the field of fluid dynamics (Kargarpour 2019; Hussein et al. 2019). CFD simulations are used to numerically simulate the dynamic characteristics of the water before and after passing through the nozzles. The relationship between nozzle opening and injection rate is determined, which provides the theoretical basis for automatic control of water injection rate.

The cross-sectional shape of the adjustable water nozzle will change when the opening is different. The flow area characteristic of the nozzle changes greatly, and its discharge coefficient \(C_{d} \left( x \right)\) is not a specific value (Hollingshead et al. 2011). It changes continuously with the change of the nozzle opening. According to fluid dynamics, the following formula is obtained:

$$ Q = C_{{\text{d}}} \left( x \right)A\left( x \right)\sqrt {\frac{2\Delta P}{\rho }} $$
(1)

where Q is the flow rate through the nozzle, m3/d; A(x) is the area of the flow channel of the nozzle, m2; \(\rho\) is the fluid density, kg/m3; and \(\Delta P\) is the differential pressure formed by the water nozzle, MPa.

If the discharge coefficient \(C_{{\text{d}}} \left( x \right)\) and flow rate Q are known, the differential pressure (flow resistance) formed by the water nozzle can be calculated as follows:

$$ \Delta P = \frac{\rho }{2}\left[ {\frac{Q}{{C_{{\text{d}}} \left( x \right)A\left( x \right)}}} \right]^{2} . $$
(2)

From Eq. (1), it is necessary to establish the mathematical relationship between the flow area A(x) and the opening x of the nozzle. The total flow area under the fully open state of the nozzle is A0. The opening x of the adjustable water nozzle is the ratio of the cross-flow area of the throttle valve to the total throttle port area of the throttle valve, expressed in percentage form as follows:

$$ x = \frac{A\left( x \right)}{{A_{0} }} \times 100\% . $$
(3)

As shown in Fig. 3, the corresponding rotation angle of the water nozzle opening x is α, R = 6 mm, \(\theta_{1} = 78^{ \circ }\), \(\theta_{2} = 97^{ \circ }\). The calculation of the flow area A(x) is shown in Eq. (4). The relationship between the flow area A(x) and the opening x of the nozzle and the rotation angle α is shown in Fig. 4.

$$ A\left( x \right) = \left( {\frac{78 - \alpha }{{360}}} \right)\pi R^{2} - R^{2} \sin \left( {\frac{78 - \alpha }{2}} \right)\cos \left( {\frac{78 - \alpha }{2}} \right) + \frac{{\left[ {R\sin \left( {\frac{78 - \alpha }{2}} \right)} \right]^{2} }}{{\tan \left( {\frac{97 - \alpha }{2}} \right)}} $$
(4)
Fig. 3
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Calculation diagram of flow area of nozzle

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Fig. 4
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The relationship between the flow area A(x) and the opening x of the nozzle and the rotation angle

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For a fixed nozzle opening, its discharge coefficient is a constant value, which can be measured through experiments. As for the variable water nozzle, its cross-sectional shape, and flow area change with the opening x, so finite element method is used to simulate and calculate the discharge coefficient \(C_{{\text{d}}} \left( x \right)\).

Simulation modeling of eccentric injection mandrel

Model building and meshing for adjustable water nozzle

The finite element model for adjustable water nozzles was established, as shown in Fig. 5. The intelligent adjustable nozzle adopts a flat fan-shaped nozzle, which is mainly composed of the regulating valve (driven by a motor to rotate axially) and the lower plugger. The fluid flows in from the side inlet of the regulating valve and flows out through the fan-shaped holes of the plugger. The injection water flows in through 36 small holes on the wall and out through the lower fan overflow channel of the nozzle. The adjustable water nozzle has eleven openings ranging from 5 to 100%. This design can not only meet the requirements of large displacement downhole injection, but also solve the injection problem caused by large differences between layers. The three-dimensional geometric model of the adjustable nozzle is imported into the meshing software to create the fluid boundary, and a CFD model is generated as shown in Fig. 5. The computational fluid domain is used to represent the internal flow channels of the adjustable nozzle.

Fig. 5
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Geometric model of the water nozzle and flow channel

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For the complexity of the water flow process, the following assumptions are made for the modeling and simulation: (1) Water is an incompressible fluid with a density of 998 kg/m3 (Hussein et al. 2019); (2) the flow field is a single-phase flow with only one direction; and (3) there is no heat conduction in the flow field.

The boundary conditions of the simulated flow field are set as follows: (1) The inlet boundary condition is velocity inlet, and the inlet pressure is determined by the specific situation; (2) the outlet boundary is pressure outlet; (3) the wall is no slip; (4) the standard k-ε turbulence model is used (Lyu et al. 2018); and (5) the SIMPLE algorithm is used for pressure coupling. Unstructured tetrahedral mesh is used for meshing, as shown in Fig. 6.

Fig. 6
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Computational model mesh of the water nozzle and flow channel

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By obtaining the differential pressure of the section 4 mm in front of the water nozzle and 18 mm behind the water nozzle under different mesh numbers, the influence of different mesh numbers on the simulation results is compared. The cross-section is shown in Fig. 7. When the nozzle opening is 100% and the appropriate number of grids is selected, the differential pressure under different mesh numbers shows a trend of increasing first and then stabilizing, as shown in Fig. 8. Considering the accuracy and efficiency of the simulation calculation, the number of grids is set to 548,261.

Fig. 7
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Schematic diagram of nozzle pressure monitoring surface

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Fig. 8
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The pressure difference between monitoring surfaces 1 and 2 under different mesh numbers

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Simulation of water nozzle flow characteristics

Figures 9 and 10 show the velocity and pressure contours under the condition of the flow rate 30 m3/d and the flow area 38.77 mm2. The pressure loss is mainly concentrated at the minimum flow cross-section. The pressure loss is mainly used to overcome the local friction loss and increases the fluid velocity. In addition, the static pressure at the outlet of the water nozzle increases slightly due to the increasing flow channel. The local friction pressure loss of the fluid at this time is less than the pressure increase caused by the reduction in velocity. This phenomenon conforms to Bernoulli’s principle.

Fig. 9
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Velocity contour of water nozzle and its front and back flow channels (m/s)

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Fig. 10
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Pressure contour of water nozzle and its front and back flow channels (Pa)

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By analyzing the velocity contour, it is found that the velocity of the water flow changes significantly when water starts to passes the nozzle. Due to the vortex and gravity, the local maximum in the vertical section appears in the portion passing the nozzle, and then, the velocity begins to slow down. By analysis the pressure contour, the pressure loss when the water flows through the water distributor is obtained, and the pressure in the water distributor flow channel gradually decreases along the water flow direction.

The nozzle model simulation can obtain the flow rate at a certain opening and a certain differential pressure, which can be brought into Eq. (1) to obtain the discharge coefficients for different differential pressures and nozzle openings. In this paper, the discharge coefficients at different openings of the water nozzles are shown in Fig. 11.

Fig. 11
figure 11

Simulate the discharge coefficient under the opening of the water nozzle

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According to the discharge coefficient data calculated by the computational fluid dynamics simulation, the relationship curve between the discharge coefficient and the opening of the water nozzle can be fitted. Spline difference is used for polynomial fitting of simulation data. The relationship between the opening of the water nozzle x and the discharge coefficient \(C_{{\text{d}}} \left( x \right)\) is obtained as follows:

$$ C_{{\text{d}}} \left( x \right) = - 5.4312x^{4} + 11.7278x^{3} - 8.3833x^{2} + 2.4289x + 0.4389. $$
(5)

Simulation model verification

Combining the conditions of water injection wells and the applicable characteristics of the nozzle, this paper carried out fluid simulations and tests with nozzle openings of 12.5%, 25%, 50%, 75%, and 100%. The schematic diagram of the experiments is shown in Fig. 12. The pressure loss of the water nozzle at different openings and different flow rates was obtained. As shown in Fig. 13, the following conclusions can be drawn:

  1. (1)

    By comparing the simulation and experiment errors between the nozzle openings at 12.5–100%, the error is the smallest when the nozzle openings 12.5%, and the corresponding flow rate control accuracy is also the highest. In the simulation, water is assumed to be incompressible, and the fluid–structure interaction (FSI) between the nozzle and the water is not considered. There will be pressure and flow rate measurement errors in the experiment. All of these will lead to errors between the experiments and simulations.

  2. (2)

    When the differential pressure between the front and rear of the nozzle is small, the difference between the experimental flow and the simulated flow is small. When the differential pressure between the nozzle increases, the error ratio of the flow rate increases. The density of compressible water in the experiment decreases gradually with the increasing pressure. As the pressure difference increases, the FSI effect becomes more obvious, resulting in greater pressure loss. These lead to large flow rate differences between the results in the simulation and experiment. The maximum error occurs when the nozzle is fully open, which is 8.29%. Therefore, the consistency between simulation and experiment is verified.

  3. (3)

    According to the water injection situation, it is necessary to select the appropriate water nozzle opening and water injection differential pressure, so that it can achieve relatively accurate control at both small and large flow rates.

Fig. 12
figure 12

Schematic diagram of experiment

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Fig. 13
figure 13

Comparison of experiment and simulation nozzle under different openings of nozzle

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Structural optimization of eccentric injection mandrel

The eccentric injection mandrel regulates the flow mainly by changing the nozzle opening, and the flow regulating regulation performance is inseparable from the shape of the nozzle: (1) Under certain flow conditions, different shapes of nozzles will produce different inlet and outlet differential pressures. The smaller pressure loss, which is conducive to the injection effect of the eccentric injection mandrel; and (2) under the constant nozzle opening, the better the linearity of the flow characteristic curve of the nozzle, the more conducive to multi-zone flow regulating for the eccentric injection mandrel.

Water nozzle shape optimization

Common nozzle shapes for water distribution have fan, fan ring, triangle, ellipse, and semicircle. And the discharge characteristics of these five nozzle shapes were analyzed to obtain the differential pressure and flow rate of each nozzle in different openings in the paper. It assumes that under the condition of the opening 100%, the water nozzles of these five nozzle shapes have the same total area, stroke, and distance between the overflow areas of the left and right.

Triangle-shaped adjustable water nozzle

The side length of the equilateral triangle is 6.69 mm. CFD simulations are performed for the triangular nozzle at the openings 25%, 50%, 75%, and 100%, respectively, to obtain the relationship between its differential pressure and flow rate with different openings, as shown in Fig. 14.

Fig. 14
figure 14

Triangle-shaped water nozzle

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When the adjustment valve clockwise rotation α angle, the blue part is the adjusted nozzle overflow area A, as shown in Eq. (6).

$$ A = \left[ {\frac{{7.29\sin \left( {24.69^{ \circ } - \alpha } \right)}}{{\sin \left( {\alpha + 60^{ \circ } } \right)}} + 3.35} \right]\sqrt {\left[ {\frac{{\frac{{7.29\sin \left( {24.69^{ \circ } - \alpha } \right)}}{{\sin \left( {\alpha + 60^\circ } \right)}} + 3.35}}{{2\sin \left( {30^{ \circ } - \alpha } \right)}}} \right]^{2} - \left[ {\frac{{7.29\sin \left( {24.69^{ \circ } - \alpha } \right)}}{{2\sin \left( {\alpha + 60^{ \circ } } \right)}} + 1.67} \right]^{2} } $$
(6)

With the nozzle opening increases, the throttle differential pressure of triangular nozzle decreases significantly. For the triangular nozzle, its characteristic is easy to control the flow in the small opening range.

Fan-shaped adjustable water nozzle

As shown in Fig. 15, the fan-shaped nozzle is the initially selected structure with a circular angle of \(\theta_{2}\) = 97°. The relationship between the differential pressures and flow rates was obtained at different openings.

Fig. 15
figure 15

Fan-shaped nozzle

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When the adjustment valve is rotated clockwise α angle, the blue part is the adjusted nozzle overflow area A, as shown in Eq. (4). When the opening of the fan nozzle \(x \le 50\%\), the differential pressure becomes larger. The overall differential pressure control range of the fan nozzle is better than that of the triangle nozzle. When the opening \(x \ge 75\%\), the opening has poor control of flow rate.

Fan ring-shaped adjustable water nozzle

Fan ring nozzle outer radius is 5 mm, and inner diameter is 1.5 mm. Its circular angle of \(\theta_{2}\) = 97.64° is chosen. And the relationship between the differential pressure and flow rate at different openings is obtained, as shown in Fig. 16.

Fig. 16
figure 16

Fan ring-shaped nozzle

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When the adjustment valve clockwise rotation α angle, the blue part is the adjusted nozzle overflow area A, as shown in Eq. (7).

$$ A = 38.77 - 0.13\alpha \pi $$
(7)

The curve of the fan ring nozzle is similarly to that of the triangular nozzle. However, the overall differential pressure control of the fan ring water nozzle is better than that of the triangle water nozzle.

Ellipse-shaped adjustable water nozzle

An ellipse-shaped adjustable water nozzle with a long axis of 11.2 mm and a short axis of 4.4 mm is chosen. And the relationship between differential pressure and flow rate at different openings is obtained, as shown in Fig. 17.

Fig. 17
figure 17

Ellipse-shaped nozzle

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When the adjustment valve is rotated clockwise α angle, the blue part is the adjusted nozzle overflow area A, as shown in Eq. (8).

$$ \theta = \arcsin \frac{{2.66\sin \left( {55.8 - \alpha } \right)}}{{\sqrt {9.33 - 7.98\cos \left( {55.8 - \alpha } \right)} }} - 90^{ \circ } $$
(8)

From Eq. (8), the angle \(\theta\) can be found, and then, the cross-flow area A can be found by integrating.

$$ A = - 24.64\int\limits_{{ - \frac{\pi }{2}}}^{{\frac{\pi }{2} - \theta }} {\sin^{2} \alpha \;{\text{d}}\alpha } $$
(9)

Under the larger opening range, the differential pressure is very small. Such as 75% opening and 100% opening, the differential pressure is basically the same, the two curves are close to overlap. The above differences are mainly due to the structure of the elliptical nozzle, when the regulating valve through the short axis of the ellipse, the nozzle overflow area changes very little. This shape of nozzle characteristics is easy to regulate the flow rate with a small opening. Especially when the opening is greater than 75%, it is difficult to achieve precise control of the flow rate by adjusting the spout nozzle opening.

Semicircle-shaped adjustable water nozzle

A semicircle-shaped adjustable water nozzle with a radius of 3.51 mm is chosen. And the relationship between differential pressure and flow rate at different openings is obtained, as shown in Fig. 18.

Fig. 18
figure 18

Semicircle-shaped nozzle

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When the adjustment valve clockwise rotation α angle, the blue part is the adjusted nozzle overflow area A, as shown in Eq. (10).

$$ A = 38.77 - 0.068\pi \left[ {\arcsin \frac{{3.82\sin \left( {66.76 - \alpha } \right)}}{3.51} - 90^{ \circ } } \right] $$
(10)

The semicircle nozzle has the best overall control of differential pressure, but also has the same problem of poor control of flow rate under the larger openings.

The selection principle of the injection mandrel nozzle is to require the smallest possible throttling differential pressure. In order to visually compare the relationship between the flow and differential pressure of different nozzle shapes, Fig. 19 shows the calculation results of the flow and differential pressure at the different openings. As shown from Fig. 19: (1) The throttling differential pressure of triangular nozzle is the largest, and significantly higher than that of the other shapes nozzle, the throttling differential pressure of semicircle nozzle is the smallest; and (2) when \(x \ge 75\%\), the differential pressure between the ellipse nozzle and the semicircle nozzle gradually becomes smaller. In summary, the overall performance of the semicircle nozzle is relatively better.

Fig. 19
figure 19

Relationship curve between flow rate and differential pressure of different shaped nozzles

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Analysis of orthogonal test results

In the actual water injection process, in addition to the shape of the nozzle, other structural parameters of the eccentric injection mandrel can also affect the oil recovery rate.

The orthogonal test selects the number of injection holes, the diameter of the injection holes, and the shape of the nozzle as influencing factors, each factor takes five level values, see Table 1, designed the orthogonal test table of \(L_{25} \left( {5^{3} } \right)\). The average shear stress on the tube flow and the differential pressure between the inlet and outlet of the eccentric injection mandrel are used as optimization targets to study the degree of influence of these three factors on the average shear stress and the differential pressure between the inlet and outlet of nozzle, and to determine the best combination of structural parameters (van Ginkel and Kroonenberg 2021; Su et al. 2022).

Table 1 Factor levels for structural parameters of eccentric injection mandrel
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For the simulation analysis, steady-state simulation analysis is used to improve efficiency. Each test is set up with the pressure at the inlet 15 MPa and the nozzle opening at 100%. The test results are obtained by simulation calculations, as shown in Table 2.

Table 2 Results of orthogonal test
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In the analysis of variance (Ahmed et al. 2020), the empty column is used as the error column for the calculation of the random effects. Analysis of variance is obtained in Table 3. Checking the F distribution table shows that when the number of factor degrees of freedom is 4, and the error degrees of freedom are 8:

$$ F_{0.1} \left( {4,8} \right) = 2.81,\;F_{0.05} \left( {4,8} \right) = 3.84,\;F_{0.025} \left( {4,8} \right) = 5.05. $$
Table 3 Analysis of variance table
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A schematic diagram of the relationship between factors and indicators was drawn to illustrate the trend of the test results with each factor considered at different levels, as shown in Fig. 20.

Fig. 20
figure 20

Effect curve between factors and average shear stress (a) and differential pressure (b)

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Figure 20a shows that the main order of the influence of each parameter on the average shear stress is as follows: C–A–B-empty column, indicating that the shape of the nozzle has the greatest influence on the average shear stress to which the wall is subjected, and the injection hole’s diameter has the least influence. However, Fig. 20b shows that the order of influence of each parameter on the differential pressure between the inlet and outlet is: C-A-empty column-B, indicating that the shape of the nozzle has the greatest influence on the differential pressure of inlet and outlet, and the diameter of the injection hole has the least influence. And the extreme difference of the empty column is similarly to the number of injection holes and the extreme difference of the injection hole’s diameter.

From Table 3, the conclusions can be drawn:

  1. (1)

    Factor A: number of water injection holes

The number of injection holes has a small effect on the differential pressure between the inlet and outlet, and can even be treated as an error column. Comprehensive analysis shows that the increase in the number of injection holes can reduce the shear stress on the fluid, so the number of injection holes can be increased.

  1. (2)

    Factor B: water injection hole diameter

Combined with the effect curve analysis, increasing the injection hole’s diameter makes the inlet velocity decrease significantly. However, when the hole’s diameter is larger than 3 mm, the decrease is small. And with the increase in the hole’s diameter, it will make the pipe wall thinner and cannot bear the larger pressure. Therefore, the size of the water injection hole can be adjusted appropriately according to the actual needs.

  1. (3)

    Factor C: nozzle shape

Combined with the effect curve analysis, changing the shape of the nozzle can greatly improve the average shear stress and the size of the inlet and outlet differential pressure. The triangular nozzle will make the injection mandrel produce a large pressure loss, and the internal fluid is damaged by large shear stress.

In summary, during the optimization of the eccentric injection mandrel, the average shear stress and the differential pressure between the inlet and outlet are used as the evaluation indexes of its performance, and the comparative analysis shows that the 14th test solution has the best performance. No. 14 means that the eccentric injection mandrel with the number of injection holes n = 36, the injection hole diameter d = 3.2 mm, and the shape of the nozzle is semicircle. Compared with the original model with n = 36, d = 3 mm, and the nozzle shape of fan, the nozzle shape changes, and the diameter of injection hole increases slightly. The number of injection holes remains the same.

The comparative analysis with the original model is shown in Table 4. The 14th test shows a substantial decrease in the inlet and outlet pressure differential of 48.12%. A smaller decrease is in the inlet velocity and average shear stress of 12.16% and 1.5%, respectively. The conclusion shows that the optimized model greatly reduces the pressure loss of the eccentric injection mandrel, and the shear stress damage of the water flow in the nozzle was alleviated.

Table 4 Performance comparison
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Performance comparison analysis of eccentric injection mandrel before and after optimization

The above optimized parameter scheme is given by orthogonal analysis with the average shear stress and the differential pressure between the inlet and outlet as the optimization objectives. Through analysis shows that the optimized parameter model reduces the pressure loss and shear stress on the eccentric water distributor when the wall thickness \(\delta\), pressure p at the inlet, and nozzle opening are certain.

The transient simulations of the model before and after optimization are carried out under the condition of the pressure at the inlet p = 15 MPa and the nozzle opening 100%, and the differential pressure curve between the inlet and outlet is shown in Fig. 21a. The differential pressure of the optimized model is greatly reduced, where the differential pressure reaches a steady value at 0.12 s. The differential pressure is 0.21 MPa before optimization. After optimization, the differential pressure reaches 0.09 MPa, and the differential pressure decrease reaches 57.14%. The fluid average shear stress is shown in Fig. 21b, the shear stress of the model after optimization has been reduced slightly. It reached a stable value of about 0.32 s, the maximum shear stress before optimization is 236.37 Pa. After optimization, the maximum shear stress reached 227.21 Pa. The shear stress decrease reaches 3.88%.

Fig. 21
figure 21

Comparison curve of differential pressure (a) and average shear stress (b) between inlet and outlet before and after optimization

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Now setting the flow rate Q = 10–90 m3/d. The performance of the optimized model and the original model at different flow rates is analyzed using steady state. Figure 22a shows that the differential pressure of the optimized model is smaller than that of the original model at each flow rate, and the reduction increases with the increase in the flow rate. When Q = 90 m3/d, the differential pressure before and after optimization is 1.63 MPa and 0.67 MPa, and the decrease amplitude is 58.89%. Figure 22b shows that by increasing the flow rate, the shear stresses of the optimized model are reduced, and the reduction of shear stresses is more obvious at high flow rates. At Q = 60 m3/d, the shear stresses before and after optimization are 422.45 Pa and 350.71 Pa, respectively, with a decrease of 16.98%.

Fig. 22
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Variation curve of model differential pressure (a) and shear stress (b) with flow rate before and after optimization

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At a small flow rate, the average shear stress of the model before and after optimization changes little. With the continuous increase in the flow rate, the variation range of average shear stress increases. This may be due to the increase in flow rate, resulting in an increase in the internal flow velocity of the eccentric injection mandrel. As the optimized model increases the water injection hole diameter and changes the shape of the nozzle, the effect of the change in the two parameters on the reduction of shear stress is more significant at high flow rates. The above analysis shows that the optimized model is significantly better than the original model.

Conclusions

In this paper, a computational fluid dynamics model of a new eccentric injection mandrel with adjustable water nozzles was established using FLUENT software. The flow characteristics of injection mandrel were simulated and analyzed, and the simulation results were verified through experiment. The results can be summarized as follows:

  1. (1)

    The pressure loss curves of the water nozzle under different openings and flow rates obtained by the simulation are consistent with that of the experiment. At the same opening, there is a good linear relationship between the square root of the pressure loss and the flow rate, which conforms to the local hydraulic loss formula of fluid dynamics. The simulation and experimental design and methods of the water nozzle established in this article are also applicable to developing other water nozzles.

  2. (2)

    The CFD simulations were employed to analyze and optimize the nozzle shapes (fan, fan ring, triangle, ellipse, and semicircle). The semicircle nozzle demonstrated the smallest differential pressure and a strong linear relationship between flow rate and opening degree, establishing it as the optimal design for efficient water injection regulation.

  3. (3)

    The structure of the eccentric injection mandrel was optimized through orthogonal tests. The test results show that the number of injection holes is 36, the diameter of injection holes is 3.2 mm, and the shape of nozzle is semicircle, the eccentric injection mandrel has relatively better performance. After optimization, the average shear stress on the fluid is reduced by 3.88%. The eccentric injection mandrel exhibited a 57.14% reduction in the differential pressure between the inlet and outlet, contributing to lower required injection pressures and reduced energy consumption in the surface water injection system.

  4. (4)

    This research introduces an innovative approach to water injection technology, utilizing CFD simulations and orthogonal tests to optimize the eccentric injection mandrel's design. It is beneficial to achieve more precise water injection rate control, thus reducing water wastage. Considering the compressibility of water, the density of water will change under different pressure conditions, which will affect the differential pressure between the inlet and outlet of the water nozzle. In addition, it will also cause fluid–structure interaction problem need to be studied. In the future, the optimized eccentric injection mandrel will be processed, and on-site water injection test research will be carried out to verify its performance under actual conditions.