# Modeling of Swab and Surge Pressures: A Survey
## 2 Literature Review of Five Prominent Works on Swab and Surge Models
### Surge and Swab Modeling for Dynamic Pressures and Safe Trip Velocities
#### 2.1.1 [PERSON]'s Model  
Unsteady transient flow in the borehole due to pipe movement generates pressure surges that propagate at the sound velocity, \(C\). For a given set of initial boundary conditions, the transient nature of a pipe movement in the borehole with a certain cross-section area, \(A\), depends on the fluid density, fluid compressibility, conduit expansion ability, and friction (resistance to flow).  
The model of this work [4] is shown in Equations (1) to (6). This model shows the fundamental equations for the transient pressure and average flow rate in a vertical well with a constant area of cross-section were obtained from the momentum balance, flow continuity, and equations of state.  
\[\frac{\partial p}{\partial z}+\frac{p}{A}\frac{\partial g}{ \partial t}+h_{f}\big{(}q,v_{p}\big{)}=0 \tag{1}\] \[\frac{\partial p}{\partial t}+s\cdot c\frac{\partial q}{\partial z }=0\] (2) \[S=Pc/A\] (3) \[c=\sqrt{g/\rho(\alpha+\beta)}\] (4) \[\Delta P=\frac{1}{\alpha}\frac{\partial V}{V}\] (5) \[\Delta P=\frac{1}{\alpha}\frac{A_{p}\cdot\Delta L}{A_{h}\cdot L} \tag{6}\]  
This work solved the first two equations for the known initial and the boundary conditions at the endpoints, considering every section in a borehole filled with drilling fluid and in communication with the other sections. Also, the endpoints boundary conditions were determined from pressure variations and the flow continuity.  
The impedance of surge is given by Equation (3). The sonic speed of propagation, \(C\), is given by Equation (4). For calculating acoustic speed, \(C\), a compressibility factor \(2.7\times 10^{-6}\) (psi) of water at 122 deg F and 7255 psi was used. The expansibility factor for various conduits is derived from the theory of elasticity.  
This work computed the friction pressure term by utilizing the Power-Law drilling fluid model. In addition, the frictional pressure loss term in the Power-Law model for the static pipe and fluid in the annulus was modified for incorporating the moving pipe effect on the average cross-section fluid speed and flow rate.  
For the turbulent fluid, the Equation (1) becomes non-linear due to the nature of the friction term. This work also employed the characteristic popular numerical technique used for computing problems of unsteady flow. This method essentially converts Equations (1) and (2) to ordinary differential equations while using the finite difference scheme for converting these equations to algebraic form.  
The effect of initial pressure required to break the mud gel were computed from the fluid compressibility. According to the definition of fluid compressibility if a fluid of volume is compressed by a small volume, this can result in an increase in the pressure and is given by Equation (3).  
Suppose a pipe with a cross-section area, \(A_{p}\), is moved a small distance \(\Delta L\), in an open hole with cross-section area, \(A_{h}\), and length \(L\), from the bottom of the pipe to the wellbore. The resulting increase in pressure \(\Delta P\) due to compressing the fluid below the pipe is given by Equation (4).  
From the computed result, the model assumes that the effect of gel breaking the mud on the initial pressure is insignificant. Therefore, the initial conditions are taken as zero.  
The calculated pressures and flow rates along the fluid line (pipe moving the hole) at \(t=\Delta t\) are used as the initial inputs to calculate the pressures and flow rates at time \(t=2\Delta t\). This computation is continued to calculate the pressures and the flow rates at a different time span along the pipe and the borehole.