Most rod pump simulation tools interpolate the wellbore path between survey points using straight-line segments. For vertical wells and gently deviated wells with wide survey spacing, this linear approximation introduces negligible error. The wellbore path between stations is close enough to a straight line that the simplification does not materially affect the simulation output.
For deviated wells with tight doglegs - which now represent the majority of active wells in the Permian, Eagle Ford, Bakken, and most other producing basins - the limitations of linear interpolation become significant. This article explains the technical basis for cubic spline interpolation, where it produces meaningfully different results, and what that means for rod string design decisions.
How linear interpolation introduces error
Consider a well with survey stations every 100 feet through a build section where inclination changes from 5 degrees to 45 degrees over 800 feet. Linear interpolation connects each pair of stations with a straight segment, creating a polygon that approximates the actual wellbore curve.
The actual wellbore does not change direction in discrete steps at station boundaries. It follows a continuous curve. Linear interpolation introduces an artificial discontinuity at every station - a sharp angle change that does not exist in the real well. These discontinuities create corresponding artifacts in the stress calculation: predicted side loads and rod-tubing contact forces that are concentrated at station boundaries rather than distributed along the actual curve.
The magnitude of this error depends on two factors: survey spacing and dogleg severity. At 50-ft step lengths with moderate doglegs (2 to 3 degrees per 100 feet), the effect is present but may fall within engineering tolerance. At 100-ft step lengths with severe doglegs (above 5 degrees per 100 feet), the geometry becomes a crude approximation that can shift predicted stress peaks by 50 feet or more from their actual locations.
What cubic spline interpolation changes
Cubic spline interpolation fits a smooth, continuous curve through the survey points. Between each pair of stations, the interpolation uses a third-degree polynomial that matches position, direction, and rate of curvature change. The resulting trajectory has continuous first and second derivatives everywhere - no artificial angle discontinuities at station boundaries.
For the wave equation solver, this produces three specific improvements. First, side load calculations are based on the actual curvature distribution rather than an approximated polygonal path. In a build section with 6-degree-per-100-ft dogleg severity, the difference in predicted contact force can reach 10 to 20 percent compared to linear interpolation. Second, the stress distribution along the rod string is continuous rather than showing artificial concentrations at station boundaries, providing a more accurate picture of fatigue damage accumulation. Third, rod guide and centralizer placement recommendations reflect the real geometry rather than interpolation artifacts.
When the difference is significant
Linear and cubic spline interpolation converge as survey spacing decreases. With MWD data at 10-ft intervals through a build section, both methods produce nearly identical results because the segments are short enough that straight lines closely approximate the actual curve.
The difference becomes significant in three scenarios. First, when survey spacing is wide relative to the curvature - a 100-ft survey interval through a 5-degree-per-100-ft dogleg means the wellbore turns 5 degrees between stations. Linear interpolation treats this as an abrupt bend at each station. Cubic splines distribute the curvature continuously across the interval. Second, when designing through tight doglegs where rod-tubing wear and failure rates are highest. These are the wells where interpolation accuracy has the greatest impact on design decisions. Third, when optimizing rod guide placement - if the predicted stress peak is displaced from the actual peak by the interpolation method, the guide is placed where it provides less benefit.
For vertical wells or wells with long tangent sections and mild build rates, linear interpolation remains adequate. The approximation error is within normal engineering tolerance for these geometries.
The relationship between interpolation and step length
Cubic spline interpolation delivers the most value when combined with shorter step lengths. RodSim allows step length configuration down to 10 feet, compared to the 50-ft default in most legacy tools. The combination of cubic spline geometry and 10-ft steps provides up to 100 times the resolution on the wellbore trajectory compared to linear interpolation at 50-ft steps.
A practical approach is 10-ft steps through the build section and 25 to 50-ft steps through the tangent, which balances resolution with computation time. With cloud-based processing, the simulation still completes in seconds regardless of step count.
Implications for rod string design
If you are designing rod strings for deviated wells using a tool with linear interpolation at 50-ft steps, the predicted stress distribution may not accurately represent the actual loading conditions in the wellbore. This does not necessarily mean the designs are wrong - conservative safety factors may compensate. But it means the simulation is working with an approximation that can be improved.
The improvement is most relevant for wells with unexplained rod failures. When a well repeatedly parts rods in a location that the simulation does not flag as high-stress, interpolation error is one of the first factors to investigate. Running the same well with cubic spline interpolation and shorter step lengths frequently reveals stress concentrations that the coarser model averaged out.
RodSim Professional includes cubic spline interpolation and configurable step length as standard capabilities. Engineers can run the same well at different resolutions and compare the results to quantify how much the interpolation method affects their specific design conditions.
