# Examining Pipe-Borehole Wall Contact and Pullback Loads for Horizontal Directional Drilling
## Abbreviations  
\(A_{i}\) (\(i=1,2\ldots 8\)) dimensionless coefficient required to be determined/fitted  
\(A_{p}\) projected area of the pipe-borehole wall contact area  
\(A_{r}\) real contact area  
\(B\) function determined by \(n\) and gamma function  
\(B_{0}\) function determined by \(b\)  
\(b\) auxiliary variables, \(b=\tan(\epsilon/2)\)  
\(C\) correction factor related to the eccentricity \(e\) and the overcut ratio \(R_{2}/R_{1}\)  
\(c\) cohesion  
\(E_{1}\) elastic modulus of the inner cylinder  
\(E_{2}\) elastic modulus of the outer cylinder  
\(E^{*}\) equivalent elastic modulus  
\(E^{*}_{1}\) modified Young's modulus of the inner cylinder  
\(e\) eccentricity  
\(i\) number of linear calculation units before calculation point  
\(k\) dimensionless coefficient  
\(l\) semi-chord length corresponding to the angle \(\epsilon\)  
\(N\) number of linear calculation units  
\(n\) pressure distribution exponent  
\(p\) external load per unit length of the pipeline  
\(P_{k}\) external load per unit length of the calculation unit \(k\)  
\(p_{0}\) maximum contact pressure accounting for the drilling fluid pressure  
\(p_{c}\) contact pressure  
\(q_{0}\) maximum contact pressure without drilling fluid pressure  
\begin{tabular}{l l} \(R\) & relative curvature of the contact \\ \(R_{1}\) & radius of the pipe \\ \(R_{2}\) & radius of the borehole \\ \(S\) & length of the pipeline entering the borehole \\ \(S_{0}\) & length of the pipeline at the ground \\ \(S_{k}\) & length of the calculation unit \(k\) \\ \(T_{i}\) & pullback loads for the pipeline to be pulled back to the end of calculation unit \(i\) \\ \(T_{ipf}\) & drilling fluid drag force \\ \(T_{ig}\) & friction force between the pipeline and the ground or roller \\ \(T_{is}\) & friction force between the pipeline and the borehole wall \\ \(T_{iw}\) & pipeline's weight \\ \(T_{k-1}\) & pullback loads for the pipeline to be pulled back to the end of calculation unit \(k-1\) \\ \(T_{s}\) & Pipe-borehole wall friction force per unit length of the pipe \\ \(u_{1}\) & radial displacement of the inner cylinder at the contact point \\ \(u_{2}\) & radial displacement of the outer cylinder at the contact point \\ \(v\) & drilling fluid flow velocity \\ \(v_{1}\) & Poisson's ratio of the inner cylinder \\ \(v_{2}\) & Poisson's ratio of the outer cylinder \\ \(w\) & net weight of the unit length of the pipe \\ \(x\) & distance between the point on the contact surface and the symmetry axis \\ \(\alpha\) & real contact area ratio \\ \(\alpha_{0}\) & Dundurs's material parameters \\ \(\beta\) & angle at which points on the interface depart from the central line \\ \(\beta_{0}\) & Dundurs's material parameters \\ \(\Gamma\) & gamma function \\ \(\Delta p\) & maximum contact pressure increment \\ \(\Delta\Gamma_{k}\) & pressure difference in the concentric annulus \\ \(\Delta\Gamma_{f}\) & pressure difference in HDD borehole \\ \(\Delta\Gamma_{k}\) & external load increment caused by the capstan effect \\ \(\Delta P_{sti}\) & external load increment caused by pipeline stiffness \\ \(\Delta R\) & radial clearance of the pipe and borehole, \(\Delta R=R_{2}-R_{1}\) \\ \(\delta\) & compressive displacement \\ \(\delta_{k}\) & compressive displacement of the calculation unit \(k\) \\ \(\epsilon\) & semi-angle of contact corresponding to the whole contact arc \\ \(\theta_{0}\) & angle between the ground and the horizontal \\ \(\theta_{k}\) & inclination angle of the calculation unit \(k\) \\ \(\theta_{k-1}\) & inclination angle of the calculation unit \(k-1\) \\ \(\lambda\) & distance between centers of pipe and borehole \\ \(\mu\) & dynamic viscosity of drilling fluid \\ \(\mu_{a}\) & coefficient of friction applicable at the surface before the pipe enters borehole \\ \(\sigma_{0}\) & drilling fluid pressure \\ \(\sigma_{a}\) & normal stress \\ \(\tau_{0}\) & yield point \\ \(\tau_{f}\) & shear strength of the borehole wall \\ \(\tau_{s}\) & shear (or friction) stress to break the junction (or adhesion) \\ \(\varphi\) & internal friction angle \\ \end{tabular}