Calculating Borehole Trajectories Using the Minimum Curvature Method in TypeScript

In the oil and gas industry, accurately determining the path of a borehole is crucial for efficient drilling operations. One of the most reliable methods to calculate this path is the Minimum Curvature method. This method provides a smooth, accurate representation of the borehole trajectory using periodic survey points. In this blog post, we’ll explore how to implement the Minimum Curvature method in TypeScript, including the key formulas and a step-by-step code example.

Understanding the Minimum Curvature Method

The Minimum Curvature method is designed to create a smooth curve that represents the borehole trajectory. It uses survey points, which are measurements taken at various depths along the borehole, to calculate this curve. The primary parameters involved are:

• Dogleg Severity (DL): This measures the change in the borehole's direction between two survey points.

• Ratio Factor (RF): This factor helps to convert inclination and azimuth changes into a smooth curve.

Key Formulas

Before we dive into the code, let’s look at the essential formulas used in the Minimum Curvature method:

Implementing the Minimum Curvature Method in TypeScript

Now, let’s see how we can implement these formulas in TypeScript. We will create a function that calculates the new coordinates of a borehole segment based on two survey points.

interface SurveyPoint {
  azimuth: number; // in degrees
  inclination: number; // in degrees
  measuredDepth: number;
}

interface Coordinates {
  x: number;
  y: number;
  z: number;
}

function toRadians(degrees: number): number {
  return degrees * (Math.PI / 180);
}

function calculateDL(azimuth1: number, azimuth2: number, inclination1: number, inclination2: number): number {
  const deltaAzimuth = toRadians(azimuth2 - azimuth1);
  const deltaInclination = toRadians(inclination2 - inclination1);
  return Math.sqrt(deltaAzimuth * deltaAzimuth + deltaInclination * deltaInclination);
}

function calculateRF(dl: number): number {
  return 2 * (Math.sin(dl / 2) / dl);
}

function calculateCoordinates(
  point1: SurveyPoint,
  point2: SurveyPoint,
  initialCoordinates: Coordinates
): Coordinates {
  const deltaMD = point2.measuredDepth - point1.measuredDepth;
  const dl = calculateDL(point1.azimuth, point2.azimuth, point1.inclination, point2.inclination);
  const rf = calculateRF(dl);

  const incl1 = toRadians(point1.inclination);
  const incl2 = toRadians(point2.inclination);
  const azm1 = toRadians(point1.azimuth);
  const azm2 = toRadians(point2.azimuth);

  const x = initialCoordinates.x + (deltaMD / 2) * (Math.sin(incl1) * Math.cos(azm1) + Math.sin(incl2) * Math.cos(azm2)) * rf;
  const y = initialCoordinates.y + (deltaMD / 2) * (Math.sin(incl1) * Math.sin(azm1) + Math.sin(incl2) * Math.sin(azm2)) * rf;
  const z = initialCoordinates.z + (deltaMD / 2) * (Math.cos(incl1) + Math.cos(incl2)) * rf;

  return { x, y, z };
}

// Example usage
const point1: SurveyPoint = { azimuth: 0, inclination: 0, measuredDepth: 0 };
const point2: SurveyPoint = { azimuth: 30, inclination: 10, measuredDepth: 100 };
const initialCoordinates: Coordinates = { x: 0, y: 0, z: 0 };

const newCoordinates = calculateCoordinates(point1, point2, initialCoordinates);
console.log(newCoordinates);

Explanation of the Code

1. SurveyPoint Interface: Defines the structure of a survey point, including its azimuth, inclination, and measured depth.

2. Coordinates Interface: Defines the structure for storing the x, y, and z coordinates.

3. toRadians Function: Converts degrees to radians, which is necessary for trigonometric calculations.

4. calculateDL Function: Computes the Dogleg Severity between two survey points.

5. calculateRF Function: Computes the Ratio Factor based on the Dogleg Severity.

6. calculateCoordinates Function: Calculates the new coordinates (x, y, z) using the Minimum Curvature method based on the initial and final survey points and the initial coordinates.

Usage Example

In the example usage, we define two survey points and calculate the new coordinates using the calculateCoordinates function. This demonstrates how the Minimum Curvature method can be implemented in a real-world scenario.

Conclusion

The Minimum Curvature method is a powerful tool for accurately determining borehole trajectories in the oil and gas industry. By implementing this method in TypeScript, engineers can leverage modern programming practices to ensure precise and efficient drilling operations. The provided code offers a robust starting point for further development and integration into more comprehensive drilling software solutions.