# Modeling of Swab and Surge Pressures: A Survey
## 2 Literature Review of Five Prominent Works on Swab and Surge Models
### Experimental Study of Swab and Surge Pressures in Horizontal and Inclined Wells
For static inner pipe, the pressure \begin{table}
\begin{tabular}{l l l} \hline
**Notation** & **Variable** & **Equations** \\ \hline \(d_{p}\) & Pipe diameter. & (53), (54) \\ \hline \(d_{h}\) & Hole/casing diameter. & (53), (54) \\ \hline \(n\) & Fluid behavior index. & (53), (54), (56) \\ \hline \(R\) & Reduction factor. & (53), (54) \\ \hline \(h(\theta)\) & Element slot thickness. & (55)–(58) \\ \hline \(r_{o}\) & Inner radius of outer pipe. & (55) \\ \hline \(r_{i}\) & Outer radius of inner pipe. & (55) \\ \hline \(E\) & Fractional eccentricity. & (55) \\ \hline \(C\) & Radius clearance \((r_{o}-r_{i})\). & (55) \\ \hline \(V_{p}\) & Pipe velocity. & (56) \\ \hline \(k\) & Consistency Index. & (56) \\ \hline \(\pi_{2}\) & Dimensionless plug thickness. & (57)–(59), (62) \\ \hline \(y_{1}\) & Lower limit of region II. & (57) \\ \hline \(y_{2}\) & Upper limit of region II. & (57) \\ \hline \(\bar{y}_{1}\) & Dimensionless lower boundary limit of region II. & (57), (59), (62) \\ \hline \(\bar{y}_{2}\) & Dimensionless upper boundary limit of region II. & (57) \\ \hline \(\tau_{o}\) & Yield stress. & (58) \\ \hline \(\Delta L\) & Slot length/hole depth. & (58) \\ \hline \(\Delta P\) & Pressure drop. & (58) \\ \hline \(\pi_{1}\) & Dimensionless pressure. & (59), (62) \\ \hline \(b\) & Constant. & (59), (62) \\ \hline \(\bar{q}_{t}\) & dimensionless total flow rate. & (60)–(62) \\ \hline \(q\) & Actual flow rate. & (61), (63) \\ \hline \(W\) & Slot width. & (61), (64) \\ \hline \(V_{p}\) & Pipe velocity. & (61), (63) \\ \hline \(H\) & Slot thickness. & (61), (64) \\ \hline \(d_{p}\) & Pipe diameter. & (63), (64) \\ \hline \(d_{h}\) & Wellbore diameter. & (64) \\ \hline \(K\) & Diameter ratio \(\left(K=d_{p}/d_{h}\right)\) & (67) \\ \hline \(\varepsilon\) & Fractional eccentricity. & (67) \\ \hline \end{tabular}
\end{table} 
Table 4: The variables used in the model by [PERSON] et al. [10]. loss in the eccentric annulus is calculated for a pressure drop of the concentric annulus by Equation (53) and the reduction factor by Equation (54). Concept of Narrow-Slot Model: For simplification of mathematical analysis of annular flow, numerous approximations have been developed which consider different mud rheology, for example, Newtonian, Bingham Plastic, Power land, and yield power law. A commonly used model in the industry is a narrow-slot model, in which a rectangular slot represents the concentric annulus can. Model Formulation: In this study, the narrow-slot modeling technique developed [PERSON] [12] has been adopted to predict swab and surge pressures. The eccentric annulus is divided into numerous concentric annuli with a variable annular clearance. Each concentric annulus is treated separately and is represented by its annular clearance, which is a function of pipe eccentricity and angular position. Flow is determined in every discrete element as a narrow slot with a constant slot height \(h\) depending on angular location \(\theta\) and eccentricity. The slot height expression [12] is given by Equation (55). The dimensionless surge pressure \((\pi_{1})\) and dimensionless exponent (\(b=\frac{n+1}{n}\)) in model development are given in Equation (56). The dimensionless plug thickness \((\pi_{2})\) is determined from the dimensionless plug-boundary limits (\(\bar{y}_{1}\) and \(\bar{y}_{2}\)) is given by Equation (57). For the momentum balance, a relationship between dimensionless plug thickness and surge pressure gradient is given by Equations (58) and (59). Flowrate Analysis is given by the Equation (60), in which \(\bar{q}_{t}\) is a dimensionless total flowrate and is given by Equation (61). By putting the quantities of dimensionless velocities integrating the equation, [PERSON] and [PERSON] [13] developed a dimensionless expression, Equation (62), to compute the flow rate. In the case of the closed-end pipe string, the flow rate in the annular is equal to the displaced fluid during the tripping operation.