# Propagation Mechanism of Pressure Waves during Pulse Hydraulic Fracturing in Horizontal Wells
## 2 Fluid Transient Dynamics Model
### Momentum Equation  
In Figure 4, we show a situation of taking a fluid element within the working pipe at an arbitrary location, wherein the inclination angle of the wellbore is \(\theta\). The forces acting on the fluid element include the pressure at both ends of cross-section One and cross-section Two, the self-weight of the fluid element, and the friction between the fluid element and the inner wall of the working pipe. When the force's direction is the same as the fluid movement direction within the element, the force magnitude is positive; otherwise, the force magnitude is negative. Based on [PERSON]'s second law, the motion equation for the compressible fluid within the working pipe could be established. At cross-section One, the forces acting on the microelement are as follows:  
\[F_{1}=p_{t}A_{t}, \tag{13}\]  
At cross-section Two, the forces acting on the microelement are as follows:  
\[F_{2}=(p_{t}+\frac{\partial p_{t}}{\partial x}dx)A_{t}, \tag{14}\]  
The component of gravity force in the fluid flow direction for the microelement is  
\[F_{m}=\rho_{t}gA_{t}dx\sin\theta_{x}, \tag{15}\]The friction force acting on the microelement due to the inner wall of the working pipe is  
\[F_{f}=\tau_{t}\pi D_{t}dx, \tag{16}\]  
According to [PERSON]'s second law, the motion equation for the compressible fluid within the working pipe can be obtained as  
\[\rho_{t}A_{t}dx(\frac{\partial v_{t}}{\partial t}+v_{t}\frac{\partial v_{t}}{ \partial x})=F_{1}-F_{2}+F_{m}-F_{f}, \tag{17}\]  
After simplification, the motion equation for the compressible fluid within the working pipe becomes  
\[\rho_{t}A_{t}dx(\frac{\partial v_{t}}{\partial t}+v_{t}\frac{\partial v_{t}}{ \partial x})=p_{t}A_{t}-[p_{t}A_{t}+\frac{\partial(p_{t}A_{t})}{\partial x}dx] -\tau_{t}\pi D_{t}dx+\rho_{t}gA_{t}dx\sin\theta_{x}, \tag{18}\]  
where \(p_{t}\) is the injection pressure within the microelement, \(\tau_{t}\) is the shear stress between the fluid inside the microelement and the inner wall of the working pipe, \(D_{t}\) is the inner diameter of the working pipe, \(g\) is the gravitational acceleration, and \(\theta_{x}\) is the angle between the axis of the pipe and the horizontal plane. In the study of constant-flow hydraulic fracturing, the change rate of the fracturing fluid velocity inside the working pipe is not significant. Therefore, the shear stress between the fluid and the inner wall of the working pipe can be investigated under the assumption of steady-state flow. At this point, the shear stress between the fluid and the inner wall of the working pipe can be calculated using the Darcy formula. The Darcy formula is as follows:  
\[\tau_{t}=\frac{\rho_{t}v_{t}|v_{t}|f_{d}}{8}, \tag{19}\]  
where \(f_{d}\) is the constant friction coefficient. In constant-flow hydraulic fracturing, the rate of change in the actual fracturing fluid velocity is relatively small, and the error caused by simplifying the shear stress using the Darcy formula is acceptable. However, in variable-flow hydraulic fracturing, the injection rate of the fracturing fluid varies frequently, leading to rapid changes in the velocity. Continuing to use the Darcy formula to calculate the shear stress in this case would result in significant errors. Therefore, the influence of non-steady friction should be considered in variable-flow hydraulic fracturing operations. At present, the main non-steady friction models include the [PERSON] weighted function model, the [PERSON] model, and modified models based on these two models. In [PERSON]'s  
Figure 4: Schematic diagram of fluid element force analysis. modified model, the friction coefficient consists of two parts: the constant friction coefficient and the non-steady friction coefficient. This formulation facilitates subsequent programming. The non-steady friction coefficient considers the influence of instantaneous local acceleration and instantaneous convection acceleration on friction loss.