The directional survey defines the geometry of the wellbore. Every calculation that follows in a rod pump simulation - rod weight distribution, tension profiles, side load magnitudes, buckling tendency, dynamometer card shape - depends on how accurately that geometry is represented between survey stations. The interpolation method used to construct a continuous wellbore path from discrete survey measurements is therefore not a secondary concern. It is foundational to the accuracy of everything the simulation produces.
Three interpolation methods are commonly used in rod pump simulation software: minimum curvature, linear (tangential), and cubic spline. Each makes different assumptions about the wellbore path between survey stations, and each produces different geometry - which in turn produces different load and stress predictions. The differences are negligible in vertical wells and significant in deviated wells with tight doglegs. This article quantifies those differences.
Minimum curvature: the industry standard for survey calculation
Minimum curvature is the standard method for calculating wellbore position from survey measurements. It was formalized by Saag and Pennebaker (1979) and adopted by the SPE as the recommended calculation method. The approach assumes the wellbore follows a circular arc between each pair of survey stations, constrained to pass through both measured inclination and azimuth values.
The ratio factor F in the minimum curvature method accounts for the dogleg angle between stations. For small dogleg angles (under 0.25 degrees), F approaches 1.0 and the method converges with the balanced tangential method. For larger dogleg angles, F adjusts the position calculation to follow the circular arc rather than a straight line, producing more accurate TVD, northing, and easting values than simpler methods.
Minimum curvature is used almost universally for survey position calculations - it is what your well planner, your drilling engineer, and your directional drilling service company all use to determine where the wellbore is in three-dimensional space. It is well validated, widely implemented, and produces position accuracy within the measurement uncertainty of the survey instruments themselves.
However, position accuracy and curvature accuracy are not the same thing. Minimum curvature provides an accurate estimate of where the wellbore is at each survey station and at the midpoint between stations. It does not necessarily provide an accurate representation of how the curvature varies continuously between stations - and it is the curvature distribution, not the position, that drives the side load and contact force calculations in rod pump simulation.
Linear interpolation: simplicity at the cost of curvature accuracy
Linear interpolation (sometimes called the tangential or straight-line method) connects each pair of survey stations with a straight segment. Inclination and azimuth change abruptly at each station boundary. Between stations, the wellbore path has zero curvature - it is treated as a straight pipe.
This creates a polygonal approximation of the actual wellbore. At each station, there is a discrete angle change - a mathematical discontinuity in the first derivative of the path. The magnitude of this discontinuity equals the dogleg angle between the two adjacent segments. For a well building from 5 degrees to 45 degrees over 800 feet with stations every 100 feet, the interpolated path has eight sharp bends where the real wellbore has a continuous curve.
For position calculation, linear interpolation is the least accurate of the three methods. For rod pump simulation, the inaccuracy manifests differently: the predicted curvature is concentrated at station boundaries rather than distributed along the actual curve. This produces artificial contact force peaks at stations and underestimates contact forces between stations. The net effect on rod string stress predictions depends on the survey spacing, the dogleg severity, and the location of taper transitions relative to the dogleg zones.
Most legacy rod pump simulation tools use linear interpolation because it is computationally simple and was adequate when the majority of rod-pumped wells were vertical or near-vertical. As the industry has shifted toward deviated and horizontal wells, the limitations of this approximation have become more consequential.
Cubic spline interpolation: continuous curvature between stations
Cubic spline interpolation fits a third-degree polynomial between each pair of survey stations, with continuity constraints on the function value, first derivative, and second derivative at each station. The result is a wellbore path that is smooth everywhere - no discontinuities in position, direction, or curvature at station boundaries.
The physical basis for this approach is straightforward: wellbores do not change direction in discrete jumps. The bottomhole assembly follows a continuous trajectory governed by the formation properties, the BHA configuration, and the drilling parameters. Between survey stations, the curvature varies smoothly. Cubic spline interpolation approximates this smooth variation more closely than either minimum curvature or linear methods.
For rod pump simulation specifically, the advantage is in the contact force calculation. Side load on the rod string is proportional to the product of rod tension and wellbore curvature. With linear interpolation, curvature is zero between stations and infinite (in theory) at stations - the contact force prediction is a series of impulses at station locations. With cubic spline, curvature varies continuously, and the contact force prediction is a smooth distribution that more closely reflects the actual loading conditions.
Quantitative comparison: a field example
Consider a well in the Delaware Basin with the following geometry: vertical to 2,000 ft, building from 2,000 to 4,500 ft at an average rate of 5.4 degrees per 100 ft to a maximum inclination of 68 degrees, then tangent from 4,500 to 8,200 ft. Survey stations are at 100-ft intervals through the build section and 200-ft intervals through the tangent. Total measured depth is 8,200 ft with a TVD of approximately 6,800 ft.
The rod string is a three-taper design: 1-inch Grade D from surface to 3,200 ft, 7/8-inch Grade D from 3,200 to 5,800 ft, and 3/4-inch Grade D from 5,800 to pump depth at 8,100 ft. The well produces 120 barrels of fluid per day with a pump speed of 6 SPM and a 168-inch stroke.
Running the wave equation solver with linear interpolation at 50-ft steps, the predicted peak side load occurs at 2,800 ft MD - coinciding with a survey station at the midpoint of the build section. The predicted peak contact force is 2.3 lbs/ft. The stress at the first taper transition (3,200 ft) is 78% of the modified Goodman limit.
The same well with cubic spline interpolation at 10-ft steps shows a different picture. The peak side load shifts to 2,650 ft MD - between two survey stations, at a location where the curvature rate is highest based on the spline fit. The peak contact force is 1.9 lbs/ft (17% lower than the linear prediction), but the force distribution is broader, covering approximately 400 ft of the build section rather than being concentrated at a single station. The stress at the first taper transition changes to 83% of the modified Goodman limit - higher than the linear prediction because the cubic spline captures bending stress contributions that the linear model misses.
The implication for rod guide placement is direct: linear interpolation suggests placing guides at 2,800 ft (the station with the highest predicted load), while cubic spline suggests guides distributed across the 2,500 to 2,900 ft interval where the actual contact forces are concentrated. The second approach better addresses the real wear mechanism.
When interpolation method matters and when it does not
For vertical wells (inclination under 5 degrees) and gently deviated wells (maximum dogleg under 2 degrees per 100 ft), all three interpolation methods produce effectively identical simulation results. The curvature is low enough that the differences in how it is represented between stations fall within normal engineering tolerance.
The differences become significant when dogleg severity exceeds 3 degrees per 100 ft, when survey spacing is wide relative to the curvature (100 ft or more through a build section), when rod guide or centralizer placement is being optimized, or when unexplained rod failures are occurring in build or turn sections. These are exactly the wells where design accuracy has the highest economic impact.
For wells with MWD data at fine intervals (10 to 30 ft through the build section), all interpolation methods converge because the segment lengths are short enough that straight lines closely approximate the actual curve. The interpolation method matters most when the data is sparse relative to the curvature - which is the common case for older wells, wells without MWD, and wells where only a gyroscopic survey at wide intervals is available.
Step length and interpolation: a coupled effect
The simulation step length - the distance between calculation points in the wave equation solver - interacts with the interpolation method. With linear interpolation, reducing the step length below the survey spacing does not improve the curvature representation because the geometry between stations is a straight line regardless of how many calculation points are placed along it. Shorter steps improve the wave equation solution accuracy but not the geometry accuracy.
With cubic spline interpolation, shorter step lengths do improve the simulation because the spline provides a continuously varying curvature that can be sampled at each calculation point. A 10-ft step length on a cubic spline captures curvature variations that a 50-ft step length on the same spline would average over. The combination of cubic spline interpolation and short step lengths provides the highest-fidelity geometry representation currently available in commercial rod pump simulation.
Practical recommendations
For new well designs in deviated wells: use cubic spline interpolation with 10-ft step lengths through the build and turn sections, and 25 to 50-ft steps through the tangent section. This provides the most accurate curvature representation with reasonable computation time.
For validation of existing designs: run the same well with both linear and cubic spline interpolation and compare the stress predictions at taper transitions and in dogleg zones. If the results differ by more than 5% at any critical location, the cubic spline result is more likely to reflect actual wellbore conditions.
For failure investigation: when rod failures occur in build or turn sections at locations the simulation does not flag as high-stress, interpolation error should be one of the first factors investigated. Running the well at higher resolution with cubic spline frequently reveals stress concentrations that coarser models miss.
RodSim uses cubic spline interpolation with configurable step length as standard. Engineers can run the same well at multiple resolutions and compare the results to quantify the sensitivity of their specific designs to the interpolation method.
