# Propagation Mechanism of Pressure Waves during Pulse Hydraulic Fracturing in Horizontal Wells
## 2 Fluid Transient Dynamics Model
### Continuity Equation  
The fluid element shown in Figure 3 is taken at any position within the operating column; according to the principle of mass conservation, the difference in fluid mass flowing into the element from cross-section One and flowing out of the element from cross-section Two within a unit time dt is equal to the change in fluid mass within the element during the same time interval. Based on this, a continuity equation can be established.  
The fluid mass flowing into the microelement from cross-section One can be calculated using the following formula:  
\[m_{1}=\rho_{t}v_{t}A_{t}dt, \tag{1}\]  
The fluid mass flowing into the microelement from cross-section Two can be calculated using the following formula:  
\[m_{2}=(\rho_{t}+\frac{\partial\rho_{t}}{\partial x}dx)(v_{t}+\frac{\partial v _{t}}{\partial x}dx)A_{t}dt, \tag{2}\]  
where \(\rho_{t}\) is fluid density in element bodies, \(v_{t}\) is velocity of axial fluid motion in element bodies, \(A_{t}\) is cross-sectional area of working column, \(t\) is time, \(x\) is length of pipe.  
The difference between the mass of fluid flowing into the element body from cross-section One and out of the element body from cross-section Two per unit time \(dt\) is  
\[dm=m_{1}-m_{2}=\rho_{t}v_{t}A_{t}dt=-\frac{\partial(\rho_{t}v_{t}A_{t}dt)}{ \partial x}dx, \tag{3}\]  
The change in fluid mass within the element body per unit time \(dt\) is  
\[dm=\frac{\partial(\rho_{t}v_{t}A_{t}dx)}{\partial t}dt, \tag{4}\]  
This is easy to obtain according to the principle of conservation of mass:  
\[\frac{\partial v_{t}}{\partial x}+\frac{v_{t}}{\rho_{t}}\frac{\partial\rho_{t }}{\partial x}+\frac{1}{\rho_{t}}\frac{\partial\rho_{t}}{\partial t}=0, \tag{5}\]  
Figure 3: Schematic diagram of fluid element continuity analysis.  
which is obtained according to the differential operator method:  
\[\frac{d\rho_{t}}{dt}=\frac{\partial\rho_{t}}{\partial t}+\frac{\partial\rho_{t}}{ \partial x}\frac{dx}{dt}=\frac{\partial\rho_{t}}{\partial t}+v_{t}\frac{ \partial\rho_{t}}{dx}, \tag{6}\]  
\[\frac{dp_{t}}{dt}=\frac{\partial p_{t}}{\partial t}+\frac{\partial p_{t}}{ \partial x}\frac{dx}{dt}=\frac{\partial p_{t}}{\partial t}+v_{t}\frac{ \partial p_{t}}{dx}, \tag{7}\]  
Substituting Equation (6) into Equation (5) gives  
\[\frac{d\rho_{t}}{dt}+\rho\frac{\partial v_{t}}{\partial x}=0, \tag{8}\]  
The fluid in the operational pipe column satisfies the linear elasticity eigen structure relationship, which is obtained as  
\[\frac{d\rho_{t}}{\rho_{t}}=\frac{dp_{t}}{K_{t}}, \tag{9}\]  
where \(K_{t}\) is the volumetric modulus of elasticity of the fluid in the operating column.  
The functions that represent \(\rho_{t}\) as \(p_{t}\) are  
\[\rho_{t}=\rho_{0}e^{\frac{1}{K_{t}}(p_{t}-p_{0})}, \tag{10}\]  
where \(\rho_{0}\) is fluid density at standard pressure \(p_{0}\).  
Taking the derivative of Equation (9) with respect to time \(t\), and combining Equations (7) and (8), the continuity equation for the compressible fluid within the working pipe can be obtained:  
\[\rho_{t}c^{2}\frac{\partial v_{t}}{\partial x}+\frac{\partial p_{t}}{\partial t }+v_{t}\frac{\partial p_{t}}{\partial x}=0, \tag{11}\]  
where \(c\) is propagation velocity of pressure wave.  
\[c=\sqrt{\frac{K_{t}}{\rho_{t}}}, \tag{12}\]