# Transient Friction Analysis of Pressure Waves Propagating in Power-Law Non-Newtonian Fluids
## 1 Introduction  
Mud pulse telemetry (MPT) is a reliable real-time communication technology that transmits data obtained by downhole sensors to the surface during oil and gas drilling [1, 2, 3]. This technology generates a series of encoded pulse signals inside the drill string by controlling the pressure changes of the drilling fluids [4, 5]. These pressure changes are then transmitted to the surface along the drilling fluid column inside the drill string and are sensed by a pressure sensor mounted on the standpipe [6, 7]. Due to fluid friction, pressure waves experience severe energy loss during long distance transmission, resulting in very weak pressure signals being received by the surface pressure sensor. In the analysis of pressure waves' propagation characteristics, modeling unsteady friction remains a great challenge [8]. In order to achieve pressure waves with a higher signal-to-noise ratio in the surface pressure sensor under various drilling conditions, both theoretical and experimental works have been conducted in pressure wave transmission modeling. [PERSON] and [PERSON] [9] developed a numerical solution for the governing equations of unsteady pipe flow, in which a constant friction coefficient was used. [PERSON] et al. [10] utilized an attenuation equation derived by [PERSON] for Newtonian fluids, in which the viscosity of the mud remains constant regardless of the shear rate. [PERSON] et al. [11] developed the transfer functions for different components in the pressure wave transmission channel based on a frequency-dependent friction model for Newtonian fluids. [PERSON] and [PERSON] [12] treated the pressure wave channel as a cylindrical multi-layered waveguide consisting of inner mud, adrill string, annular mud, and the formation, and investigated the coupling effect between the layers. [PERSON] et al. [13] discussed the propagation and attenuation of pressure waves transmitted in wellbore gas-liquid two-phase flow. [PERSON] et al. [14] developed a transmission model for MPT by considering the effect of the boundary layer thickness on viscous friction, where a constant dynamic viscosity was used in the model. The studies mentioned above ignored the non-Newtonian fluid characteristics of drilling fluids and did not consider the change in fluid viscosity with a velocity gradient. Therefore, the propagation characteristics of pressure waves in drilling fluids could not be accurately evaluated. There are few studies on the propagation characteristics of pressure waves that consider drilling fluids as non-Newtonian fluids, but the research on pressure waves' propagation in non-Newtonian fluids in other fields can provide a reference for the study of MPT. [PERSON] et al. [15] developed a frequency-dependent friction model for non-Newtonian drilling fluids, in which the shear rate of the Newtonian fluids was substituted into the constitutive equation of non-Newtonian fluids to calculate the frequency-dependent friction. [PERSON] et al. [16] numerically studied the pressure transmission in Bingham fluids compressed within a closed pipe and showed that the attenuation of pressure waves is significantly increased by increasing the fluid shear stress. Subsequently, [PERSON] et al. [17] proposed a mathematical model to simulate the pressure propagation in non-Newtonian drilling fluids and solved the model by using the method of characteristics. [PERSON] et al. [18] analyzed the propagation of the periodic mud pressure waves in Bingham non-Newtonian drilling fluids by correcting the friction factor of Newtonian fluids. [PERSON] et al. [19] investigated the influence of secondary viscosity on pressure pulses, describing the energy loss due to the compressibility of liquid. [PERSON] et al. [20] established a 2D dimensionless model for the power-law fluids hammer problem and showed more consistent results with their experiments than the 1D model by [PERSON] et al. [21].