## 2 Swab Surge Modeling
### Physics-Based Modeling
#### 2.1.3 Fluid Models  
The swab and surge simulations were based on the hydraulic model. The model required defining PVT for density predictions and rheological models. Including tripping speeds, the density and viscosity parameters adjusted for the given temperature and pressure were used to calculate the pressure loss during swab and surge operations.  
Fluid PVT Models  
We used the empirical Glassse oil density model formulated based on North Sea oil for density calculation. For the water density model, we used the Dodson and Standing empirical model.  
Rheology Models  
Three rheological models were implemented in DrillbenchTM software [19]. These were Bingham plastic (BP), power law (PL), and Robertson-Stiff (RS) models for the evaluation of swab and surge simulations.  
Bingham plastic is a two-parameter rheology model. The model assumes a constant viscosity as the shear rate increases. The shear stress and shear rate vary linearly with excess yield stress. The model equation is as follows [21; 22]:  
\[\tau=\text{PV}\dot{\gamma}+\text{YS} \tag{1}\]  
where PV is the plastic viscosity, \(\dot{\gamma}\) is the shear rate, and \(\text{YS}\) is the yield stress.  
The plastic viscosity and yield stress values were calculated from the 600 and 300 rpm dial readings of the Fann viscometer as:  
\[\text{PV}[\text{cP}]=\theta_{600}-\theta_{300} \tag{2}\]  
\[\text{YS}\Big{(}\text{lbf}/100\text{ft}^{2}\Big{)}=2\theta_{300}-\theta_{600} \tag{3}\]  
Figure 2: Experimental well structure and well trajectory.  
Unlike the Bingham plastic model, the shear stress-shear rate of the drilling fluid described by the power law excludes yield stress, and the viscosity decreases as the shear rate increases. The power law model is as follows [21; 22]:  
\[\tau=k\gamma^{n} \tag{4}\]  
where n and k are the flow and the consistency index parameters, respectively. The values were quantified based on curve fitting between the model and from the 600 and 300 RPM Fann dial readings as:  
\[n=3.321\log\left(\frac{\theta_{600}}{\theta_{300}}\right) \tag{5}\]  
\[k\left(\mathrm{lbf}/100\mathrm{ft}^{2}\right)=\frac{\theta_{300}}{511^{n}}= \frac{\theta_{600}}{1022^{n}} \tag{6}\]  
The Robertson-Stiff model is a three-parameter model. The model describes drilling fluids and cement slurries. The viscosity decreases as the shear rate increases [23].  
\[\tau=A\left(\dot{\gamma}+C\right)^{B} \tag{7}\]  
where A, B, and C are model parameters. \(A\) and \(B\) are similar to the \(k\) and \(n\) parameters of the power law model. However, the RS model exhibits yield stress, \(\tau_{\mathrm{o}}=AC^{B}\). C is the shear strain correction factor and can be determined using the interpolation method.  
\[C=\frac{\left(\gamma_{\mathrm{min}}\gamma_{\mathrm{max}}-\gamma^{*^{2}}\right) }{2\gamma^{*}-\gamma_{\mathrm{min}}\gamma_{\mathrm{max}}} \tag{8}\]  
The shear rate, \(\gamma^{*}\), can be calculated using the interpolation method, which is the corresponding value of \(\tau^{*}\). The shear stress, \(\tau^{*}\), is the geometric mean between the maximum and the minimum, given as:  
\[\tau^{*}=\sqrt{\tau_{\mathrm{min}}\tau_{\mathrm{max}}} \tag{9}\]