Wellbore Trajectory Modeling with Cubic Splines
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Solution Overview
Problem
Conventional wellbore trajectory models, such as the minimum curvature model, fail to accurately represent the smoothness of drillstring trajectories, leading to inaccuracies in modeling drillstring bending and shear forces, and neglect contact forces due to discontinuous curvature and bending moments.
Innovation Solution
A computer-implemented method using tangent vector interpolation functions to model wellbore trajectories, transforming the torque-drag drillstring model into a full stiff-string formulation, ensuring continuity of bending moment proportional to curvature, and determining wellbore trajectories with cubic splines that satisfy the Frenet equation.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Ease of manufacture
If the minimum curvature model is used to model wellbore trajectories, then the model is simple and commonly used, but the curvature is discontinuous at survey points leading to inaccurate representation of drillstring trajectory smoothness
Solution Approach 1:
The patent changes the mathematical parameters of the trajectory model from piecewise circular arcs (minimum curvature model) to cubic spline functions. This parameter change enables the model to achieve C2 continuity (continuous curvature) while maintaining computational feasibility, thereby resolving the contradiction between model simplicity and trajectory smoothness representation.
Solution Approach 2:
The patent uses composite mathematical formulations combining cubic spline functions with drillstring mechanics equations. This composite approach integrates geometric smoothness requirements with mechanical behavior constraints, achieving both trajectory accuracy and physical realism without excessive computational complexity.
2Adaptability or versatility
If the standard torque-drag model is used, then the model is general and widely applied, but it neglects bending moment and contact forces due to discontinuous curvature
Solution Approach 1:
The patent ensures continuity of bending moment and curvature throughout the wellbore trajectory by using cubic spline functions. This continuity allows the torque-drag model to accurately compute contact forces at all points, including where the wellbore contacts the drillstring, thereby improving measurement precision while maintaining general applicability.
Solution Approach 2:
The patent introduces cubic spline trajectory models as an intermediary between the drillstring mechanics and the wellbore geometry. This intermediary provides smooth, continuous curvature that enables accurate calculation of bending moments and contact forces, bridging the gap between general model applicability and precise force measurement.
3Device complexity
If the drillstring trajectory is assumed to be the same as the wellbore trajectory, then the modeling is simplified, but the bending moment and shear forces cannot be determined correctly
Solution Approach 1:
The patent changes the trajectory model parameters to cubic splines that satisfy both geometric constraints and mechanical equilibrium equations. This allows the drillstring and wellbore trajectories to be treated as consistent with each other while enabling accurate determination of bending moments and shear forces through the continuous curvature provided by the spline formulation.
Data Source
AI summary
Systems and methods for modeling wellbore trajectories, which can be used to model corresponding drillstring trajectories and transform the torque-drag drill string model into a full stiff-string formulation.