## 2 Bidirectional FSI Numerical Model
### Dynamic Modeling
#### 2.1.2 Drill String-Wellbore Wall Contact Force  
The contact relationship between the drill string and wellbore wall, as illustrated in Figure 1a, represents an internal contact with strong nonlinearity. Based on the contact radius relationship within the contact area, the contact between the drill string and borehole wall can be equivalently simplified as the contact between a cylindrical object with an equivalent radius \(R_{0}\) and a flat surface, as shown in Figure 1b. The correlation used to determine this equivalent radius \(R_{0}\) is expressed as Equation (9) [51]. \[\frac{1}{R_{0}}=\frac{1}{R_{2}}-\frac{1}{R_{1}} \tag{9}\]  
where \(R_{1}\) denotes the curvature radius of the wellbore (concave surface) and \(R_{2}\) represents the curvature radius of the drill pipe (convex surface). In this study, the drill string outer diameter is 71 mm (thus, \(R_{2}=35.5\) mm), and the wellbore inner diameter is 76 mm (thus, \(R_{1}=38\) mm). Substituting these values into Equation (9), the equivalent contact radius R0 is calculated to be 539.6 mm. According to the Hertzian contact theory, it is assumed that mutual interpenetration occurs between two contacting bodies under applied loads, and the contact forces between contacting bodies can be derived using the generalized Hooke's law and the stress-strain relationships of materials. To facilitate the derivation of the governing equations, a representative example--the contact problem of two elastic spheres with different diameters--is employed in the subsequent analysis. After establishing the generalized contact force formula, the equivalent contact radius between the drill string and the wellbore wall will be substituted into the final expression. As shown in Figure 2, the geometric relationships within the contact area are defined under the assumption that mutual indentation occurs due to the applied load. Sphere \(A\) and sphere \(B\) have respective initial (undeformed) radii of curvature \(R_{1}\) and \(R_{2}\). The \(x\)-\(y\) plane, serving as the reference plane, is defined as the plane passing through the center \(O\) of the circular contact area. The radius of the contact area is \(a\). The total indentation depth \(\delta\) represents the total relative displacement by which the two spheres approach each other along the \(z\)-axis from their initial state of tangential contact. \(\delta\) is the sum of the indentation \(\delta_{1}\) of sphere \(A\) and \(\delta_{2}\) of sphere \(B\) (i.e., \(\delta=\delta_{1}+\delta_{2}\)). At \(r=a\), \(z_{1}\) is the normal distance from a point on the undeformed surface of sphere \(A\) to its original vertex tangent plane. Similarly, \(z_{2}\) is the corresponding sagitta (profile height) for sphere \(B\) at \(r=a\). \(w_{1}\) and \(w_{2}\) denote the normal distances from points on the deformed surfaces of sphere \(A\) and sphere \(B\), respectively, at the radial distance \(r=a\), to the common \(x\)-\(y\) plane. Their sum represents the normal gap between the two deformed surfaces at the edge of the contact radius a. Based on these geometric definitions, the relationship among the parameters \(w_{1}\), \(w_{2}\), \(z_{1}\), \(z_{2}\), and \(\delta\) at \(r=a\) is given by Equation (10). Thus, a quantitative relationship is established for the gap (\(w_{1}+w_{2}\)) between the two deformed bodies at the edge of the contact area (\(r=a\)), the total indentation depth \(\delta\), and the initial geometric parameters of the contacting bodies. \[w_{1}+w_{2}=\delta-(z_{1}-z_{2}) \tag{10}\]  
The equivalent contact radius and equivalent elastic modulus of the two contacting bodies are formulated in Equation (11) [52,53]. According to the Hertzian theory, the pressure distribution across the contact surface is semi-ellipso  
Figure 1: Contact relationship between drill pipe and wellbore wall: (**a**) internal contact relationship; (**b**) equivalent representation of the contact between the drill string and wellbore wall. Figure 2: Geometric relationship of the contact area. denoted as \(Q\).