# Transient Friction Analysis of Pressure Waves Propagating in Power-Law Non-Newtonian Fluids
## 3 Results and Discussion
### Pressure Waves Transmitted inside the Drill String  
In the continuous wave information transmission during the measurement while drilling (MWD), the pressure waves inside the drill string can be regarded as a harmonic periodic flow, \(\rho P_{p}\text{exp}(i\omega t)\), superimposed on a steady pipe flow, which is the steady state of the startup flow shown in Equation (23). The velocity fluctuation of the periodic flow is smaller than that of the steady pipe flow. Therefore, in the flow of pressure wave transmission, the viscosity of power-law non-Newtonian fluids only changes along the radial direction and does not change with time. For the pressure waves superimposed on the steady pipe flow, the pressure gradient can be expressed as  
\[\frac{\partial p}{\partial x}=\rho\big{(}P_{c}+P_{p}\text{exp}(i\omega t)\big{)} \to F(s)=\rho\bigg{(}\frac{P_{c}}{s}+\frac{P_{p}}{s-i\omega}\bigg{)}, \tag{25}\]  
where \(P_{p}\) is the amplitude of the pressure waves, \(P_{c}\) is the average of the pressure pulsation, and \(\omega\) is the angular frequency of the pressure waves. The average of the pressure pulsation \(P_{c}\) has the same meaning as the constant pressure gradient in the startup flow, which is given in Equation (23). The first term on the right side of Equation (25) represents the pressure gradient for the startup flow, and its inverse Laplace transform is Equation (24). The second term on the  
Figure 3: Comparison of analytical and numerical results for startup flow under different power-law indexes \(n\). (**a**) \(n=0.25\), (**b**) \(n=0.5\), (**c**) \(n=1.5\), (**d**) \(n=1.75\). right side of Equation (25) is the pressure gradient for periodic flow, and its inverse Laplace transform is  
\[ \begin{split} u(r,t)&=\frac{P_{p}}{i\omega}\Bigg{(} \frac{2r^{h-1}}{R^{h-1}(\sigma-2)}+1\Bigg{)}+\frac{P_{p}}{i\omega}\Bigg{(}1- \frac{J_{-\sigma}\Bigg{(}\sqrt{-\frac{i\omega}{\upsilon}}\frac{1}{h}r^{h} \Bigg{)}r^{h-1}}{J_{-\sigma}\Bigg{(}\sqrt{-\frac{i\omega}{\upsilon}}\frac{1}{h }R^{h}\Bigg{)}R^{h-1}}\Bigg{)}\exp(i\omega t)\\ &\quad-P_{p}\sum\limits_{k=1}^{\infty}\frac{J_{-\sigma}\Bigg{(} \lambda_{k}\Big{(}\frac{r}{R}\Big{)}^{h}\Bigg{)}r^{h-1}}{J_{1-\sigma}(\lambda _{k})\lambda_{k}R^{h-1}\Bigg{(}\lambda_{k}^{2}\frac{vh^{2}}{2R^{2h}}+\frac{i \omega}{2}\Bigg{)}}\exp\Bigg{(}-\frac{\lambda_{k}^{2}}{R^{2h}}vh^{2}t\Bigg{)}.\end{split}  \tag{26}\]  
This study mainly discusses the influence of the magnitude of transient friction on the attenuation of the pressure waves, and the leading or lagging of the phase has no effect on the attenuation of the pressure waves. Therefore, for the periodic flow, only the real part of Equation (26) has been considered. For a long-time steady oscillation, the transient term, which is the third term on the right side of Equation (26), can be neglected. For Newtonian fluids, Equation (26) is an exact solution for pulsating laminar flow in a circular pipe without assumptions, which can be used to verify the accuracy of the numerical results. Figure 4 shows a comparison between the numerical and theoretical solutions of Newtonian fluids. It can be seen from the figure that when the frequency is low, the numerical and theoretical solutions are in good agreement. However, as the frequency increases, the numerical solution of the flow velocity at the center of the pipe is higher than the theoretical solution. Compared with the finite difference method, the time step size of the implicit weighted residual method has little effect on the results. However, reducing the time step can still reduce the difference between numerical and theoretical results, as shown in Table 4.  
According to Equation (26), the shape of the velocity profile is independent of the amplitude of pressure fluctuation, \(P_{p}\). However, the numerical results indicate that the shape of the velocity profile is related to the amplitude of the pressure fluctuation when \(n\
eq 1\).