## 2 Modeling  
A simplified analytical model was developed to calculate the pressure loss gradient for the power-law and yield-power-law fluid rheological models. This model applies to fluid flow within concentric and eccentric annular geometries under steady-state conditions. The analysis assumes laminar flow, and fluid is characterized by incompressibility. Additionally, \begin{table}
\begin{tabular}{p{113.8 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt}} \hline \hline
**Authors** & **Research Focus and Parameters/Models** & **Fluid Rheology** & **Key Findings and Remarks** \\ \hline \multirow{6}{*}{[PERSON] et al. (2010) [11]} & Pressure Variations Resulting from & & \\  & String Pipe Reciprocation, Flow Slot & & \\  & Analysis, Conservation Equations, & Yield Power Law & \
\begin{tabular}{l} Concentration on the effects of reciprocating pipe in \\ steady-state flow. \\ \end{tabular} \\  & Input Fluid Properties, Borehole & & \\  & Geometry, and Pipe Velocity & & \\  & Investigation of Eccentric and & & \\  & Concentric Annuli, Closed-End Pipes, & & Herschel–Bulkley analysis of \\  & and the Impact of Pipe Diameter & & \\  & Ratios and Fluid Rheological & & \\  & Properties & & \\  & Concentric and Eccentric Annuli, & & \\  & Reciprocating Pipe, Temperature & & \\  & Effect, Fluid Properties, and Borehole & & \\  & Geometry & & \\  & Eccentric Annulus, Diameter Ratios, & & \\  & Fluid Behavior & & \\  & Drill String Rotations, Eccentricity, & & \\  & Buckling Configurations & & \\  & Concentric Annulus, Diameter Ratio, & & \\  & Fluid Behavior Index, and Varying & Yield Power Law &
\begin{tabular}{l} Comparative analysis of \\ drilling parameters. \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of literature reviewed swab and surge models. it considers a scenario without fluid circulation, in which the pipe moves vertically within the wellbore. A no-slip boundary condition is assumed at the interface between the pipe's outer diameter wall and the wellbore's inner wall. Figure 1 provides a visual representation of the drill string's motion within the wellbore, which induces the movement of the annular fluid. As shown in the figure, \(\mathrm{V_{p}}\) is the speed of the string, \(\mathrm{r_{p}}\) is the outer radius of the pipe, and \(\mathrm{R_{h}}\) is the inner radius of the wellbore. Due to pressure difference, fluid flows from high pressure to low pressure. The fluid flow due to the external force is resisted by the shear force in the fluids. By employing the force balance analysis, the differential pressure over length can be related to the shear stress as:  
\[\frac{\Delta P}{2l}r=\tau \tag{1}\]  
This paper assumes that the fluid's rheological behavior resembles the yielded power-law model. Consequently, we utilize the Herschel-Bulkley model ([PERSON] et al., 2004) ([PERSON] et al., 2004) ([PERSON] et al., 2004)  
\[\tau=\tau_{y}+k\dot{\gamma}^{n}\] (2a) Equating Equations (1) and ( 2a ), we obtain: \[\frac{\Delta P}{2l}r=\tau_{y}+k\dot{\gamma}^{n}\] (2b) where \[\tau_{y}\] is the yield stress, \[k\] is the consistency index, and \[n\] is the flow index. The shear rate is the negative of the velocity gradient, \[\dot{\gamma}=-\frac{du}{dr}\] \[\int du=-\int\left(\frac{\Delta P}{2 kl}r-\frac{\tau_{y}}{k}\right)^{1/n}dr \tag{3}\]  
Figure 1: Tripping out while rig pump shut off. Integrating, we obtain:  
\[\frac{\Delta P}{2l}r=\tau \tag{1}\]0  
The boundary condition stipulates that the fluid velocity at \(r=r_{p}\) matches the pipe speed (\(V_{p}\)), while at the wellbore wall (i.e., \(r=R_{h}\)), the velocity equals zero.