# Modeling of Swab and Surge Pressures: A Survey
## 2 Literature Review of Five Prominent Works on Swab and Surge Models
### Automatic Prediction of Downhole Pressure Surges in Tripping Operations
The resultant pressure force is given by Equation (32). As the authors treated the volumetric flow rates as state variables, they therefore presented the momentum as Equation (33). Conservation of Momentum in the string: The whole surrounding wall of the drill string moves uniformly. Therefore, the theory for calculating the viscous friction force is simpler than the annulus one. As the diameters \(D_{IM},D_{II},D_{IC},D_{IB}\) across the drill string are not uniform, to deal the diameters non-uniformity, the authors derived an effective radius. This work first calculated the approximated values of the respective radii and obtained \(R_{IMFL},R_{IFFL},R_{ICFL},R_{IBFL}\) for laminar flow and \(R_{IMFT},R_{IFFT},R_{ICFT},R_{IBFT}\) for turbulent flow. Similarly, they computed the effective radii representing the secondary part of the entire volume inside the drill string. This work expressed the friction forces as given in Equation (34). This work also computed the pressure forces by Equation (35), and the momentum equation inside the drill string is given by Equation (36). Changes in Hydrostatic Pressure: As the BHA moves in the annulus, the pressure sensors also follow the BHA. The authors computed the change in hydrostatic pressure caused by the BHA movement. They left out the initial component of hydrostatic pressure \begin{table}
\begin{tabular}{l l l} \hline \hline
**Notation** & **Variable** & **Equations** \\ \hline \(Q_{n}\) & Flow through the bit nozzle. & (22)–(24) \\ \hline \(V_{I}\) & String inside volume. & (22)–(24) \\ \hline \(A_{IM}\) & String internal cross-section. & (22)–(24) \\ \hline \(\rho\) & Mud density. & (26) \\ \hline \(A_{n}\) & Total cross-section of all nozzles. & (26) \\ \hline \(C_{n}\) & Discharge coefficient to account for unmodeled effects. & (26) \\ \hline \(A_{u1(j)}\), \(A_{u2(j)}\) & The boundary surface areas. & (31), (34) \\ \hline \(\tau_{u1(j)}\), \(\tau_{u2(j)}\) & Wall share stress that were calculated from respective & (31), (34) \\  & radius, \(R_{1(j)}\) or \(R_{2(j)}\). & (32) \\ \hline \(L_{2(j)}\) & Length of secondary part. & (32) \\ \hline \(h_{2(j)}\) & Annular gap of secondary part. & (32) \\ \hline \(m_{1(j)}\) & Mass of main part. & (33), (36) \\ \hline \(m_{2(j)}\) & Mass of secondary part. & (33), (36) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The variables used in the model by [PERSON] et al. [8]. The Automatic Pressure Estimation: Since there are a lot of uncertainties in their model, to achieve a good agreement with the data, it is important to have some calibration of the model. One of the problems the authors pointed out is that during tripping operations, no real-time measurements are often available due to limitations imposed by mud pulse telemetry, which cannot measure the pressures when the pumps are off. However, the model can be calibrated for consecutive runs if pumps are turned on before tripping operations. The authors employed the Ensemble Kalman Filter for the downhole pressure prediction. In addition, they computed the change in hydrostatic pressure caused by the BHA movement. They left out the initial component of hydrostatic pressure in our computations. The change in hydrostatic pressure is given by Equation (37), in which the string position is give by \(\frac{d}{dt}(\Delta z)=v_{s}\). Analysis of the Work by [PERSON] et al. By discretization, the authors supposed that the non-linear, time-invariant system [8] can be described as:  
\[x(t+1)=f(x(t),u(t))+w(t)\]  
\[y(t)=h(x(t))+v(t)\]  
where \(x\in R^{n_{x}}\) is the state of the system, \(u\in R^{n_{u}}\) is the input, and \(y\in R^{n_{y}}\) is the output. The objective of their EnKF is to obtain the estimate \(\hat{x}(t)\) of the true state \(x(t)\) using the measurements \(y(t)\) so that \(\text{tr}\left(E\big{[}(x(t)-\hat{x}(t))(x(t)-\hat{x}(t))^{T}\big{]}\right)\) is minimized. This work assumed that at initial time \(t\), there is an ensemble of \(N\) forecasted state estimates with random sample error.