This paper will discuss a purpose written fully 3D Finite Element Analysis (FEA) model which analyzes the bending of pipe inside a wellbore. The yielding and residual curvature of CT can be included in this model. The model is also able to calculate the onset of buckling. This paper presents the equations and methodology used in the development of the model. Results from the model are compared with analytical solutions currently in use. Examples studies performed with the model are presented.


Several types of numerical methods are currently used by various models to calculate the behavior of pipe inside a wellbore or pipeline when subjected to axial and torsional loads. The most common type of model is a "soft-string" model which calculates the forces on the pipe sequentially, beginning at one end with a known force and calculating in increments to the other end. These models are accurate and useful for many types of problems. However, they do not include the bending stiffness of the pipe itself in their analysis. There are some types of problems in which the bending stiffness is important.

So called "stiff-string" models are much less common than the soft-string models. Stiff-string models do include the bending stiffness of the pipe. Special stiff-string models have been written for solving special problems, such as drilling bottom hole assembly (BHA) analysis. Some of these models use finite difference and finite element analysis techniques. Reference 1 discusses the various model types.

Of the numerical modeling techniques, finite element analysis (FEA) is usually considered to be the best. However, there are several challenges in implementing FEA has kept it from being the most common type of tubing forces model. Most numerical techniques are either a "bottom up" or a "top down" calculation. An FEA solution solves for all the unknown displacements of the pipe in the wellbore simultaneously.

One of the biggest challenges for any stiff-string model is knowing at a specific point if the pipe is touching the wall of the wellbore or not. If not, the pipe is free to move. If it is touching, some wall contact force (WCF) is being applied to the pipe. It is challenging for any numerical technique when any sudden change or step function exists. This is possibly the biggest challenge for implementing an FEA tubing forces model.

This challenge, and several other challenges were overcome in the development of this model. The following sections will outline the FEA equations used, and discuss how the various challenges were overcome. Then example studies are presented showing how the model is used.

Finite Element Model
Local Element Stiffness Matrix

Figure 1 shows a single beam element, i, in its local coordinate system. The x axis of the local coordinate system runs through the center of the beam. The y and z axis are defined as shown. Note that small characters x,y and z are used to denote the local coordinate system.

At each end of the element there is a node. Nodes are shared between two adjacent elements for connectivity. This element has node number n on one end and node n+1 on the other end. Each node may move in 6 ways, known as degrees of freedom (DOF). 3 of these DOF are displacements, u, along the local x,y and z axis. The other 3 DOF are rotations, ua, about the three local axes. Note that the double arrows shown in Figure 1 represent rotation about that axis.

Figure 2 shows two elements, i and I+1, with their respective local coordinate systems, connected at node n, and located in a global coordinate system X,Y and Z. The global X coordinate is defined as pointing downward, toward the center of the earth. The nodes are placed on the centerline of the wellbore, as shown. The inclination, a, of each element is the average of the wellbore inclinations at the two adjacent nodes. Likewise, the azimuth angle, ?, (not shown) is the average of the wellbore azimuth angle at each node.

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