Pressurization Characteristics and Proppants Transport of Pulse Jet Fracturing with CFD-DEM Coupling Method

**CFD/DEM Coupling Method**

In the process of pulsed jet fracturing, the interaction between fracturing fluid and proppants in wellbore and perforation is a complex fluid-solid coupling problem. As a typical two-phase flow of pulsed jet fracturing, both solid domain and fluid domain solutions should be established. The main governing equations are as follows:

The Governing Equations of the Fluid Domain: The continuity equation in differential form is established with the micro-element control body. The mass of the net outflow of the micro-element body must be equal to the mass reduction in the micro-element body, according to the law of mass conservation. Equation (1) can be obtained as follows:

Continuity equation:

$$\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{u}) = 0$$

Equation (1) is the partial differential equation form of the continuity equation based on the infinitesimal micro cluster model with fixed spatial position. The law of conservation of momentum is a law that any flowing system must satisfy. Fluid micro cluster are subjected to volumetric and surface forces in the x, y, and z directions. The total force is obtained by adding the volume and surface forces in the three directions. According to the conservation of mass, we can get equation (2) as follows:

Momentum equation:

$$-\frac{\partial \rho}{\partial x} + \frac{\partial \tau_{xx}}{\partial x} + \frac{\partial \tau_{yx}}{\partial y} + \frac{\partial \tau_{zx}}{\partial z} + \rho f_x = \frac{\partial (\rho u)}{\partial t} + \nabla \cdot (\rho u V)$$

where $t$ is the time, $s$; $\rho$ is the particle density, kg/m$^3$, $u$ is the fluid velocity vector, m/s, $\tau$ is the viscous stress tensor.

The law of conservation of energy is the fundamental law that must be satisfied by the heat exchange system. It can be expressed as: The power of volume force, surface force, and net heat flow into the micro-element body are added to determine the rate of energy change in the micro-element body. Equation (3) can be obtained as follows:

$$\rho \frac{De}{Dt} = \rho q + \frac{\partial}{\partial x} \left( \lambda \frac{\partial T}{\partial x} \right) + \frac{\partial}{\partial y} \left( \lambda \frac{\partial T}{\partial y} \right) + \frac{\partial}{\partial z} \left( \lambda \frac{\partial T}{\partial z} \right) - \rho \left( \frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} + \frac{\partial u}{\partial z} \right) + \tau_{xx} \frac{\partial u}{\partial x} + \tau_{yy} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial z} + \tau_{xy} \frac{\partial u}{\partial x} + \tau_{zy} \frac{\partial u}{\partial y} + \tau_{xz} \frac{\partial u}{\partial z} + \tau_{yz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau_{zz} \frac{\partial u}{\partial y} + \tau

m

Schiller-Nauman correlation is applicable to spherical solid particles, liquid droplets and small diameter bubbles. The settings are:

( ) 24 0.687 3 1 0.15Re Re 10 p p Rep d 3 0.44 Re 10 p   

$$ C _ {\mathrm {d}} = \left\{ \begin{array}{l l} \frac {2 4}{\mathrm {R e} _ {\mathrm {p}}} \left(1 + 0. 1 5 \mathrm {R e} _ {\mathrm {p}} ^ {0. 6 8 7}\right) & \mathrm {R e} _ {\mathrm {p}} \leq 1 \\ \end{array} \right. $$ (15) >

The pressure gradient force _F_p in equation (7) is defined as:

p p static F V P = − ∇ (17) where _V_p is the volume of particles, m3. And Ñ_P_static is the gradient of static pressure in the continuous phase.

The virtual mass force _F_vm in equation (7) is defined as:

$$ F _ {\mathrm {v m}} = C _ {\mathrm {v m}} \rho V _ {\mathrm {p}} \left(\frac {D v}{D t} - \frac {d v _ {\mathrm {p}}}{d t}\right) $$ (18) where _C_vm is the virtual mass coefficient. The default value of 0.5 for this coefficient is for a sphere with uniform, inviscid, incompressible flow.

The gravity _F_g in equation (8) is defined as:

$$ F _ {\mathrm {g}} = m _ {\mathrm {p}} g \tag {19} $$

where g is the gravitational acceleration vector, m/s2.

The force _F_MRF in the moving reference coordinate system in Formula (8) is defined as:

( ) ( ) 2 p p MRF F m r v ω ω ω = × × + ×     (20)

where ω is the angular velocity vector of the rotation reference coordinate system, rad/s. And r is the distance vector to the rotation axis, m.

The coulomb force _F_Co in equation (8) is defined as:

$$ F _ {\mathrm {C o}} = q E \tag {21} $$

The user-defined volume force Fu in equation (8) is defined

as:

$$ F _ {\mathrm {u}} = V _ {\mathrm {p}} f _ {\mathrm {u}} \tag {22} $$

where fu is the volume force per unit volume specified, N/m3.

CFD/DEM Coupling Model

Geometry of Numerical Model: The numerical model of pulse jet fracturing, which is composed of the pulse jet fracturing tool, wellbore annular and perforation, is first established by using CFD-DEM coupling method, as shown in Figure 1(a). To investigate the flow characteristic of the fracturing fluid and proppants, a single section of the numerical model is selected as the research goal Figure 1(b), which is named as a single section of the pulse jet fracturing model (SPJFM). The internal chamber of the single section of the pulse jet fracturing tool (SPJFT), the inlet channel section of the nozzle, the convergence section of the nozzle, the outlet channel section of the nozzle, the wellbore annulus section, and the perforation section are all included in the meshing results of the SPJFM, which are displayed in Figure 1(c). The internal chamber of the SPJFT is a cylindrical shape, with a diameter of 40mm and a length of 40mm (Figure 1(b) and Figure 1(c)). The cylindrical inlet channel section of nozzle has a 10mm diameter and a 16mm length, connecting with SPJFT. The conical convergence section of nozzle has a length of 7mm, with a convergence angle of 13.5°. The outlet section of nozzle has a 4mm diameter and a 2mm length. The high- pressure fracturing fluid would come out from the nozzle outlet into the wellbore annular, which has an inner diameter of 90 mm and an outer diameter of 114.3 mm. And then the jet will pressurize in the perforation section with a diameter of 15mm and length of 88.85mm. The distance between the perforation outlet and the referenced axis of the SPJFT is 146mm. The simulation boundary conditions at the inlet and outlet of SPJFM is both set as pressure boundary, and the numerical wall is all smooth wall. The prismatic layer grid and boundary layer grid are used to mesh the fluid model, and the ultimate number of grids is 146376 (Figure 1(c)).

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Boundary Modeling of CFD-DEM: The Reynold Average Navier-Stokes (RANS) model and the k-turbulence model are used for modeling in this study, and the implicit unsteady solver is used for solution. The simulation model establishes the CFD-DEM coupling model to investigate the inter- phase interactions in the SPJFM channel. In the inter-phase interactions, consider the four typical interactions: particle and particle, particle and boundary, particle and fluid. In the parameters interaction between particle and particle, including the linear spring settings and rolling resistance, are decided based on many times practice. The ratio factor of rolling resistance is 0.001, and the linear spring’s tangential and normal stiffness are 50,000 N/m. The equivalent diameter of the proppants is 0.18 mm. The drag coefficient is calculated by using the Schiller-Nauman model, the virtual mass is set to 0.5 kg and the volume forces is set into 0 N/ mm3. The model parameters of the SPJFM is illustrated as shown in Table 1.

proppants size Young’s modulus (MPa)

$$ P _ {\mathrm {j}} = P _ {\mathrm {m}} \sin \left(2 \pi f t\right) + P _ {0} \tag {23} $$ Among them, the pressure amplitude _P_m is 15MPa, the angular velocity ω is 1rad/s, the average pressure _P_0 is 45MPa, and the outlet is set to the pressure outlet

Mesh Independence and Validation

numbers determines the optimal number of meshes for all subsequent simulations and studies. The Table 3 shows the results comparison of velocity and total pressure relative error under different mesh numbers.

Mesh Independence: We should verify the mesh independence first. Comparing the results of different mesh

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Validation: The pressure drop at the nozzle is changed by changing the pressure amplitude _P_m at the inlet and the average pressure _P_0 accordingly. Qu, et al. [32] studied the pressure distribution in the perforation under different nozzle pressure drop by setting the pressure of the nozzle in the annulus section to 15MPa. The study showed that the nozzle diameter is constant, and the pressure in the perforation increased with the nozzle pressure drop. In our study, the nozzle inlet pressure is 40MPa, the annular pressure is 15MPa, and the pressure drop of the nozzle is 25MPa. The overall pressure trend is similar to that obtained by Qu, et al. [32], Cheng, et al. [33] and Shi, et al. [34]. It is manifested as the pressure decreases first as the axial distance increases, and then gradually rises to the same as the outlet pressure after being reduced to the minimum pressure, confirming the model’s accuracy (Figure 3).

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Figure 3: Comparison results on pressure of our study with Qu, et al. [33] (The annular pressure is 15MPa and the pressure drop of nozzle is 25MPa), Cheng, et al. [34] (The nozzle inlet pressure is 55MPa, the annular pressure is 20MPa and the pressure drop of nozzle is 35MPa) and Shi, et al. [35] (The pressure drop of nozzle is 25MPa) under different conditions.

Effect of Pressure Amplitude (Pm)

The Variation of Jet Velocity (V): Figure 4 shows the velocity variation at wellbore section and perforation section velocity under different pressure amplitude. The results indicated that the maximum velocity along the jet axis sharply rises as the pressure amplitude increases. When the pulse pressure amplitude is 65MPa, the velocity of jet core reaches a maximum value of 269.9m/s. It is pointed out that the jet velocity is 135m/s obtained at the position of 21.44mm along the jet axis, which is equivalent value of 0.5_V_max at the position of 21.23% jet length. For all cases, the maximum velocity all rapidly reduced to the same value at the outlet of the perforation. The maximum velocity of the jet along the perforation inlet to the perforation outlet along the jet axis is gradually decreasing under the same pressure amplitude. And the maximum jet velocity increases with the pressure amplitude increases. Compared the pulse jet and non-pulse jet, it is found that when the pressure amplitude _P_m is 15MPa, the jet velocity of different perforated positions in SPJFT is greater than non-pulsed under this parameter.

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In order to explore the velocity variation deeply, Figure 5 shows the variation of jet velocity along the X axis to various distance under different pressure amplitudes. It is found that when the pulse pressure amplitude increases from 50MPa to 65MPa, the maximum velocity amplitude varied from 234.7m/s to 269.9m/s, reaching increment of 15% in the maximum velocity amplitude. When the pressure amplitude _P_m has the maximum value of 15MPa, the maximum velocity of pulse case and non-pulse case is 258.72m/s and 221.70m/s respectively. For all cases, the maximum velocity rapidly reduced to 17m/s at the outlet section of the perforation. The maximum velocity reduction of the pulsed case is greater than the non-pulsed case under the same pressure amplitude.

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The Variation of Total Pressure (P): The variation of total pressure at wellbore section and perforation section, respectively, under different pressure amplitude is shown in Figure 6. It can be seen that the maximum total pressure along jet axis steeply rises as the pressure amplitude increasing.

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When the pulse pressure amplitude reaches to the maximum jet pressure amplitude of 65MPa, the total pressure of jet core reaches the maximum value of 61.7MPa. It is pointed out that the jet total pressure is 30.85MPa at the position of 15.23mm along the jet axis, which is correspondingly the

0.5_P_max at the 15.08% of the entire jet length. Particularly, at the outlet of the perforation, the maximum total pressure of all case rapidly reduced to the same value because of the same outlet pressure boundary setting. The maximum total pressure of the jet along the perforation axis has gradual reduction under  the  same  pressure amplitude according to the color map in Figure 6(a). And when the pressure amplitude varied from 50MPa to 65MPa, it is found that there is a significant increasing of total pressure. Compared the pulse jet fracturing and non-pulse jet fracturing, the results reveal that the total pressure of pulse jet fracturing in SPJFM is greater than that of non-pulsed jet fracturing.

It is necessary to investigate the total pressure variation along the X axis under different pressure amplitudes (Figure 7). According to Figure 7, it is found that when the pulse pressure amplitude rises from 50MPa to 65MPa, the maximum total pressure has a 25.7% increment, rising from 49.1MPa to 61.7MPa. When the pressure amplitude _P_m has the value of 15MPa, we could obtain the maximum value of total pressure of 44.9MPa in non-pulse case. And the maximum value of total pressure of 57.5MPa in pulse case, which is 4.1% larger than the non-pulse case at the same jet parameters. Thereby, we can conclude that the total pressure reduction of pulsed case is always more significant than the non-pulse case under the same pressure amplitude.

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Proppants Transport in the SPJFM: Figure 8 shows the proppants transport in the SPJFM under different pressure amplitudes. Compared the results under different amplitude pressure, it can be seen that there are 67, 64, 57 and 65 proppants accessing in the perforation, respectively. And the proportion of proppants entering the perforation is 22.3%, 21.3%, 19% and 21.7%, respectively. It is concluded that the number of transporting proppants would decrease firstly and then increase finally, with increasing amplitude pressure. As a compared group, it is interesting that the 52 proppants, which is 16.67% proportion of total proppants, transport into the perforation under the non-pulsed pressure of 45MPa. Also, it can be seen that there is a large number of proppants blocking the perforation inlet of the SPJFM. The number of blocking proppants at the perforation inlet of non-pulse jet fracturing is more obvious than that of pulsed case jet fracturing, which means the pulse jet could relieve the proppants blocking in the perforation entrance section.

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Effect of Angular Velocity (Ω)

The Variation of Jet Velocity (V): Figure 9 shows the influence of angular velocity on jet velocity at wellbore and perforation section. The results suggest that the jet core velocity reaches a maximum value of 258.7 m/s when the angular velocity ω is 1 rad/s, and the jet velocity is 129.35m/s at 19.85mm along the jet axis, which is equivalent to 0.5_V_max at the position of 19.65% jet length. There is a smaller value of jet velocity than all others when the angular velocity ω is 10 rad/s. The jet velocity is 97.05 m/s at the position of 12.47mm along the jet axis, which is correspondingly the 0.5_V_max at the 12.35% of the entire jet length. As can be seen from Figure 9(b), when the angular velocity ω is 1rad/s, the velocity of jet core reached the maximum value at the nozzle outlet, and the velocity of the jet has the most clearly downward trend through the entire perforation. When the angular velocity ω is 10rad/s, there is a minimum value at the nozzle outlet. It is concluded that when the angular velocity varies from 0.01rad/s to 10rad/s, the maximum velocity increases firstly and then decrease. The comparison between pulse jet fracturing and non-pulse jet fracturing indicates that at a pulsed pressure amplitude of 60MPa, the jet velocity of different perforated positions in SPJFT is greater than that of the non-pulsed jet under the same angular velocity.

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Figure 9: The influence of angular velocity on jet velocity at wellbore and perforation section. (Pm=15MPa; P0=45MPa; d=4mm; Dp=15mm) Figure 10 shows the variation of jet velocity along the X axis to various distances under different angular velocities so as to explore the angular velocity variation deeply. It is pointed out that the maximum velocity that can be achieved by the jet varies at different angular velocities. When the angular velocity ω is 1rad/s, 10rad/s, the maximum velocity of the jet is 258.7 m/s and 194.1 m/s, respectively. And the difference is 33.3%. Thus, we can conclude that there is an optimal angular velocity to get the maximum value of jet velocity.

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The Variation of Total Pressure (P): Figure 11(a) can be seen as the influence of angular velocity on total pressure at wellbore and perforation section. The total pressure of jet core reaches the maximum value of 57.51MPa when the angular velocity ω is 1rad/s. It is noted that the jet total pressure is 28.75MPa at 13.82mm along the jet axis, which corresponds to the 0.5_P_max at 13.68% of the entire jet length. When the angular velocity ω is 10 rad/s, the total pressure of jet core reaches the minimum value of 18.39MPa. And the total pressure of jet core obtains the position of 5.74mm along the jet axis, which is an equivalent value of 0.5_P_max at the position of 5.7% jet length. From Figure 11(b), it can be seen that when the angular velocity ω is 1rad/s, the total pressure of jet core reached the maximum value at the nozzle outlet, and it has the steepest decrease trend of the entire perforation compared with other tree groups. The nozzle outlet has a minimum total pressure when the angular velocity ω is 10rad/s. Compared with the pulse jet and non-pulsed jets, it is found that when the pulsed pressure amplitude is 60MPa, the total pressure of different perforated positions in SPJFT is greater than the non-pulsed jet under the same parameter.

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Figure 11: The influence of angular velocity on total pressure at wellbore and perforation section. (Pm=15MPa; P0=45MPa; d=4mm; Dp=15mm) It is necessary to explore the variation of total pressure along the X axis under different angular velocity (Figure 12). According to Figure 12, when the angular velocity is 1 rad/s and 10rad/s, the maximum and lowest total pressures are

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determined to be 57.51MPa and 36.77MPa, respectively. In addition, the overall pressure was lowered by 56.4%. As a result, we may conclude that the greatest total pressure can be obtained by varying the angular velocity.

Proppants Transport in The SPJFM: Figure 13 depicts the transit of proppants in the SPJFM at various angular velocity. When the findings for various angular velocity are compared, it is discovered that there are 57, 62, 59, and

62 proppants entering the perforation, correspondingly. In addition, 19%, 20.67%, 19.67%, and 20.67% of proppants penetrate the perforation, correspondingly. The findings show that there is an ideal angular velocity for getting the most proppants into the SPJFT. At the same time, it is obvious that there is a considerable number of proppants obstructing around the perforation inlet of the SPJFM in the non-pulsed jet fracturing. As a result, the pulse jet fracturing helps to remove the proppants that are clogging the perforation inlet.

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Effect of Average Pressure (P0)

The Variation of Jet Velocity (V): The variation of velocity at wellbore section and perforation section under different average pressure can be seen in Figure 14. The results show that with the average pressure increasing from 35MPa to 65MPa, the maximum velocity along jet axis sharply increases. When the maximum total pulse pressure (_P_j) varies from 50MPa to 80MPa, the velocity of jet core reaches a maximum value of rises from 229.9m/s to 308.4m/s.

In Figure 14(a), the jet velocity rises from 114.95m/s to 154.20m/s from 15.47mm to 24.85mm along the X axis, representing the corresponding value of 0.5Vmax at 15.3% and 24.6% of the total jet length, respectively. We can see the maximum jet velocity increases with the average pressure increasing. Compared the pulse jet and non-pulse jet, it is shown that when the average pressure _P_0 is 45MPa, the jet velocity of different perforated positions in SPJFT is greater than non-pulsed under the same average pressure.

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As can be seen from Figure 15, it is shown the variation of jet velocity along the X-axis to various distance under different average pressure. As the average pressure increases, the pulse jet’s maximum velocity can be achieved and increases gradually. It is found that when the pulse pressure increases from 50 MPa to 80 MPa, the maximum

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velocity amplitude increases from 229.9 m/s to 308.4 m/s, an increase of 34.1% in the maximum velocity amplitude. Under the same average pressure, the pulsed case has a faster maximum velocity decrease than the non-pulsed case, which is beneficial to reduce the proppants blocking at the inlet of perforation.

The Variation of Total Pressure (P): It can be seen through Figure 16 that the variation of total pressure at wellbore section and perforation section under different average pressure. As the average pressure increases, the maximum total pressure along jet axis rapidly increases. When the pulse pressure increases from 50MPa to 80MPa, the total pressure of jet core reaches a maximum value varied from 47.5MPa to 77.5MPa. The total pressure is 23.75MPa at 10.94mm and 38.75MPa at 18.71mm along the jet axis, respectively, equivalent to the 0.5_P_max at 10.83% and 18.52% of the entire jet length. Simultaneously, according to the color map in Figure 16(a), the maximum total pressure of the jet from the perforation inlet to the perforation outlet along the jet axis decline gradually under the same average pressure. Also, it can be seen that the maximum total pressure of the jet increases as the intermediate pressure increases. At a moderate pressure of 45MPa, the total pressure at different perforation locations is greater than the non-pulsed case at the same parameter setting.

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Figure 16: Variation of total pressure at wellbore section and perforation section under different average pressure. (Pm=15MPa; ω=1rad/s; d=4mm; Dp=15mm) Figure 17 depicts the change of total pressure along the X-axis under various average pressures to examine the impact of average pressure on total pressure. When the pulse pressure climbs from 50MPa to 80MPa, the maximum total pressure rises from 47.5MPa to 77.5MPa, representing a 63.2% increase in maximum total pressure. When the average pressure is 45 MPa, the maximum total pressure in the pulsed scenario is 57.5 MPa. The non-pulsed case, on the other hand, is 44.9MPa, which is 28.7% less than the pulse jet fracturing. As a result, even if the end total pressure is 10MPa, the pulse jet fracturing might produce a higher maximum total pressure in the hole. In other words, as the average pressure rises, so does the maximum total pressure of the jet, resulting in a positive contribution to fracture pressurization in the hole.

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Proppants Transport in the SPJFM

As can be seen from Figure 18, it shows the proppants transport in the SPJFM under different average pressure. When the number of proppants carrying under different average pressure are compared, it is clear that there are 66, 57, 64 and 72 at pulse pressures of 50MPa, 60MPa, 70MPa and 80MPa, respectively. And the proportion of proppants entering the perforation is 22%, 19%, 21.3% and 24%, respectively. We can find that the transporting proppants would first decrease and then increase with the average pressure rising. As a compared group, it is worth noting that 52 proppants, 16.67% of total proppants, transport into the perforation under non-pulsed pressure jet. It is found that the number of blocking proppants at the perforation inlet of non-pulse jet fracturing is greater than that of pulsed case jet fracturing, showing that the pulse jet could alleviate proppants blocking in the perforation entrance section.

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Effect Of Nozzle Diameter (D)

The Variation Of Jet Velocity (V): It can be seen through Figure 19 that the variation of velocity at wellbore section and perforation section under different nozzle diameter. It is shown that the maximum velocity of the jet at the nozzle outlet increase with the increasing nozzle diameter. When the nozzle diameter is 4mm, the jet core reaches a maximum of 258.7m/s and when the nozzle diameter is 0.5mm, the jet core reaches a maximum of 118.72m/s. It is pointed out that the jet velocity is 129.35m/s at 19.65% jet length when the nozzle diameter is 4mm, equivalent to 0.5_V_max. Also, when the nozzle diameter is 0.5mm, the jet velocity is 59.36m/s, which is equivalent to 0.5_V_max at the position of 0% jet length. It is concluded that when the nozzle diameter is 4mm, the maximum velocity of the jet reduces more rapidly from Figure 19(a). It can be seen that the maximum velocity of the jet increases as the nozzle diameter increases when it is the different parameter. When the nozzle diameter is 4mm, the maximum jet velocity of different perforated parts is greater than the non-pulsed case under the same parameter and the velocity at the perforation outlet is same.

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Figure 19: Variation of velocity at wellbore section and perforation section under different nozzle diameter. (Pm=15MPa; ω=1rad/s; P0=45MPa; Dp=15mm) Figure 20 shows the variation of jet velocity along the X-axis to various distances under different nozzle diameters. With the nozzle diameter increasing, the maximum velocity of the pulse jet gradually increases. When the nozzle diameter increases from 0.5mm to 4mm, the maximum velocity increases from 59.36m/s to 129.35m/s, an increase of 117.9% in the maximum velocity. The maximum velocity

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decay trend in the pulsed case is greater than in the non- pulsed case under the same nozzle diameter. Consequently, we can conclude that the larger the nozzle diameter, the greater the initial jet velocity from the SPJFT axis and the greater the jet velocity to the nozzle outlet and perforation outlet.

The Variation of Total Pressure (P): The fluctuation of total pressure at the wellbore and perforation sections with varying nozzle diameters, as shown in Figure 21. It can be shown that when the nozzle diameter is less than 4mm, the maximum total pressure of the jet at the nozzle outlet rises as the nozzle diameter grows. When the nozzle diameter rises from 0.5mm to 2mm, maximum total pressure of the jet core varies from 57.62MPa to 57.76MPa. It is noted that the total pressure is 28.81MPa at -2.53mm of the jet axis and 28.88MPa at 2.96mm of the jet axis, respectively, which corresponds to the 0.5_P_max at -2.5% and 2.93% of the total jet length, respectively. Compared with other nozzle diameters, when the nozzle diameter is 4mm, the total pressure is 28.75MPa at 13.82mm of the jet axis, which is equivalent to 0.5_P_max at the position of 13.68% of the entire jet section. When the nozzle diameters are 0.5mm, 1mm and 2mm, respectively, the total pressure changes slightly. When the nozzle diameter is 4mm, the total pressure of the jet in different perforation section is greater than the non-pulsed case under the same nozzle diameter. It can be inferred that the maximum total pressure rises with the increasing nozzle diameter at the nozzle outlet.

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Figure 21: Variation of total pressure at wellbore section and perforation section under different nozzle diameter. (Pm=15MPa; ω=1rad/s; P0=45MPa; Dp=15mm) As can be seen in Figure 22, it shows the variation of total pressure along the X-axis to various distances under different nozzle diameters. When the nozzle diameter increases from 0.5mm to 2mm, the maximum total pressure increases from 57.62MPa to 57.76MPa, the maximum total pressure increases by 0.24%. When the nozzle diameter is 4mm, the maximum total pressure that can be achieved in the pulse case is 57.5MPa, and the maximum total pressure of non- pulse case is 44.9MPa, which has a large gap of 28.06% of the maximum total pressure. It can be observed that the smaller the nozzle diameter, the earlier and faster the total pressure reduce. The pulsed case decreases faster than the non-pulsed case when the nozzle diameter remains constant.

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Proppants Transport in the SPJFM: It shows the comparison of the proppants transport in the SPJFM under different nozzle diameters in Figure 23. In the pulsed jet case, when the nozzle diameters are 0.5mm, 1mm, 2mm and 4mm, respectively, the number of proppants entering the perforation are 0, 89, 202 and 57, and the ratio of proppants entering the perforation is 0, 6.15%, 21.7% and 19%. As the diameter of the nozzle increases, the proportion of proppants entering the perforation first increases and then decreases.

In the non-pulsed case, with a nozzle diameter of 4 mm, the number of proppants entering the perforation is 52 and the proportion of proppants entering the perforation is 16.67%. It can be seen that the non-pulsed jet fracturing delivers more blocking proppants at the perforation inlet than pulsed case jet fracturing does. When the proppants are of a certain size and the nozzle diameter is very small, it is difficult for the proppants to enter the perforation.

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Effect of Perforation Diameter (Dp)

The Variation of Jet Velocity (V): Figure 24 shows the variation of velocity at wellbore section and perforation section under different perforation diameters. When the perforation diameter is 10mm, the jet core reaches a maximum velocity of 277.08m/s; When the perforation diameter is 25mm, the jet core reaches a maximum velocity of 249.16m/s. The jet velocity is 138.54m/s when the perforation diameter is 10mm, which is equivalent to 0.5_V_max at the position of 14.7% jet length, measured at 14.85mm along the jet axis. And when the perforation diameter is 25mm, the jet velocity is 124.58m/s, which is equivalent to 0.5_V_max at the position of 18.4% jet length, measured at 18.55mm along the jet axis. With a perforation diameter of 10 mm, the maximum velocity of the jet decreases more rapidly.

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Figure 24: Variation of velocity at wellbore section and perforation section under different perforation diameter. (Pm=15MPa; ω=1rad/s; P0=45MPa; d=4mm) When the perforation diameter is 15mm, the maximum jet velocity of the various perforation sections is greater than that of the non-pulsed case with the same perforation diameter. Finally, it is concluded that the maximum velocity of the jet gradually decreases as the perforation diameter increases.

Figure 25 displays the variation of jet velocity along the X-axis to various distances under different perforation diameters. The maximum velocity of pulse jet is different from another perforation diameter. When the perforation diameter varied from 10mm to 25mm, the maximum velocity decreased from 277.08m/s to 249.16m/s, and the maximum velocity reduced by 11.21%. It can be found that when the diameter of the perforation is larger than 10mm, there is a smaller value of the maximum velocity than in other cases. When the perforation diameter remains constant, the velocity of the pulsed case reduces faster than that of the non-pulsed case.

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The Variation of Total Pressure (P): In Figure 26, there is the variation of total pressure at wellbore section and perforation section under different perforation diameters. When the perforation diameter is 15mm, the jet core reaches a maximum total pressure of 57.5MPa. It is pointed out that the total pressure is 28.75MPa at 13.82mm along the jet axis, which corresponds to the 0.5_P_max at 13.68% of the total jet length. And when the perforation diameter is 10mm, the total pressure is 28.57MPa at 9.28 mm along the jet axis, which is an equivalent value of 0.5_P_max at the position of 9.2% jet length. Under non-pulsed pressure, when the perforation diameter is 15mm, the total pressure of the jet core is 22.45MPa, and the length is 6.83mm, which is equivalent to the 0.5_P_max at 6.76% of the entire jet length.

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Figure 26: Variation of total pressure at wellbore section and perforation section under different perforation diameter. (Pm=15MPa; ω=1rad/s; P0=45MPa; d=4mm) When the perforation diameter is 15mm, the total pressure of the jet in different perforation sections under the pulsed condition is greater than the non-pulsed case with the same parameter setting. In the case of pulse jet, the change of the perforation diameter has little effect on the maximum total pressure of the jet at the nozzle outlet, and the degree of change in the total pressure is small.

Figure 27 shows the variation of total pressure along

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the X-axis to various distances under different perforation diameters. Among them, when the perforation diameter is varied from 10mm to 15mm, the maximum total pressure increases from 57.14MPa to 57.5 MPa, and the maximum total pressure increases by 0.63%. It can be obtained that the smaller the perforation diameter, the earlier and faster the reduction of the total pressure. When the perforation diameter is fixed, the total pressure in the pulsed case decreases faster than in the non-pulsed case.

Proppants Transport in the SPJFM: As shown in Figure 28, there is the comparison of the proppants transport in the SPJFM under different perforation diameters. The number of proppants entering the perforation under pulse conditions are 21, 57, 103, and 126, respectively, and the proportion of proppant entering the perforation is 7.29%, 19%, 31.12%, and 33.24% for perforations with diameters of 10mm, 15mm, 20mm, and 25mm, respectively. When the perforation diameter is 15mm in the non-pulsed case, the number of proppants entering the perforation is 52, and the proportion of proppants entering the perforation is 16.67%. It can be found that as the diameter of the perforation increases, the number of proppants entering the perforation also increases. Simultaneously, when all other parameters are maintained, the smaller the diameter of the perforation, the less proppant enters the perforation, and some of the proppant appears in the annular section.

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