# Transient Friction Analysis of Pressure Waves Propagating in Power-Law Non-Newtonian Fluids
## 2 Mathematical Model and Solution Method
### Governing Equations  
As shown in Figure 1, the pressure waves transmitted in the drill string can be described using a one-dimensional model with time, \(t\), and distance along the axis of the drill string, \(x\), as independent variables, where the positive direction of \(x\) is downward. In this model, pressure, \(p\), density, \(\rho\), mean sectional velocity, \(U\), and sectional area, \(A\) are the dependent variables. For the fluid microelement in Figure 1a, by applying the principlesof continuity and momentum to a segment \(dx\) of the drilling fluids, the basic governing equations can be derived. They are the equation of motion  
\[\frac{\partial\rho AU}{\partial t}+\frac{\partial\rho AU^{2}}{\partial x}=-2\pi R \tau_{w}-A\frac{\partial p}{\partial x}+\rho Ag, \tag{1}\]  
and the equation of continuity  
\[\frac{\partial\rho A}{\partial t}+\frac{\partial\rho AU}{\partial x}=0, \tag{2}\]  
where \(R\) is the inner radius of the drill string, \(g\) is the acceleration due to gravity, and \(\tau_{w}\) is the shear stress on the inner wall of the drill string, which is determined by the radial gradient of axial velocity at the wall. The expression for the mean sectional velocity \(U\) is  
\[U=U(x,t)\ 0=\frac{\int_{0}^{R}2\pi ru(x,r,t)\mathrm{d}r}{A}, \tag{3}\]  
where \(u\) is the velocity that varies with time, the radial coordinate, and the axial coordinate. When analyzing the transient friction of pipe flow in this study, \(u\) is independent of \(x\). The reason for this is that the wall shear stress \(\tau_{w}\) is related to the radial velocity gradient \(\partial u/\partial r\) at the wall on the cross-section at position \(x\). The purpose of this paper is to obtain the transient  
Figure 1: Pressure waves transmitted in drill string. (**a**) Fluid microelement segment used for pressure wave transmission analysis and flow parameters that do not vary radially. (**b**) Flow parameters of cylindrical microelement in transient friction analysis. The blue straight arrows represent the direction of liquid flow. The red curved arrows represent the propagation of pressure waves. The arrows in (**a,b**) represent the direction of forces. friction on a certain cross-section, which is determined by the radial distribution of the velocity \(u\) on the cross-section. Therefore, we only need to take a certain section to represent all of the pipe sections, without considering its specific location. According to the method in this paper, the friction on a specific cross-section can be determined in Equations (1) and (2). According to Equation (1), the attenuation of pressure waves is determined by the dissipation term \(2\pi R\tau_{w}\) at the right hand of Equation (1). In other words, the friction loss during pressure wave transmission is generated by the shear stress at the wall, which depends on the velocity gradient at the wall. In the derivation of this velocity gradient for unsteady laminar flow, the motion equation for parallel axisymmetric flow of incompressible fluids will be used, as shown in Figure 1b. The pressure and density are assumed to be constant throughout the cross-section, but the axial velocity, \(u\), is a function of time, \(t\), and radial coordinate, \(r\), then  
\[\rho\frac{\partial u}{\partial t}=-\frac{\partial p}{\partial x}-\frac{ \partial\tau}{\partial r}-\frac{\tau}{r}, \tag{4}\]  
where \(\tau\) is the shear stress. In oil and gas engineering, the rheological properties of most drilling fluids conform to the power-law model, which is widely used in drilling engineering [23; 24]. For power-law non-Newtonian fluids, the shear stress can be represented by a generalized Newtonian fluid as a function of the apparent viscosity and the shear rate  
\[\tau=-\mu_{a}\frac{\partial u}{\partial r}=-K\bigg{|}\frac{\partial u}{ \partial r}\bigg{|}^{n-1}\frac{\partial u}{\partial r}, \tag{5}\]  
where \(\mu_{a}\) is the apparent viscosity, \(K\) is the consistency coefficient, \(n\) is the power-law index, and \(\partial u/\partial r\) is the radial gradient of axial velocity.