## Minimum Curvature

The minimum curvature method is similar to the radius of curvature method in that it assumes that the wellbore is a curved path between the two survey points. The minimum curvature method uses the same equations as the balanced tangential multiplied by a ratio factor, which is defined by the curvature of the wellbore. Therefore, the minimum curvature provides a more accurate method of determining the position of the wellbore. Like the radius of curvature, the equations are more complicated and not easily calculated in the field without the aid of a programmable calculator or computer.

The balanced tangential calculations assume the wellbore course is along the line 1iA + AI2. The calculation of the ratio factor changes the wellbore course to I1B + BI2 which is the arc of the angle B. This is mathematically equivalent to the radius of curvature for a change in inclination only.

So long as there are no changes in the wellbore azimuth, the radius of curvature and minimum curvature equations will yield the same results. If there is a change in the azimuth, there can be a difference in the calculations. The minimum curvature calculations assume a curvature that is the shortest path for the wellbore to incorporate both surveys. At low inclinations with large changes in azimuth, the shortest path may also involve dropping inclination as well as turning. The minimum curvature equations do not treat the change in inclination and azimuth separately.

The tangential and average angle methods treat the inclination and azimuth separately. Therefore, larger horizontal displacements will be calculated. The radius of curvature method assumes the well must stay within the survey inclinations and will also yield a larger horizontal displacement though not as large as the tangential and average angle.

The minimum curvature equations are more complex than the radius of curvature equations but are more tolerant. Minimum curvature has no problem with the change in azimuth or inclination being equal to zero. When the wellbore changes from the northeast quadrant to the northwest quadrant, no adjustments have to be made. The radius of curvature method requires adjustments. If the previous survey azimuth is 10o and the next survey is 355°, the well walked left 15o. The radius of curvature equations assume the well walked right 345o which is not true. One of the two survey azimuths must be changed. The lower survey can be changed from 355o to -5o, then the radius of curvature will calculate the correct coordinates. c

Minimum Curvature Equations

ANorth = AMD X [(Sin/2 X CosA2) + (Sin/; X CosA;)] X FC 2

AEast = AMD X [(Sin/2 X SinA2) + (Sin/; X SinA;)] X FC 2

D1 = Cos(/2 - /;) - {Sin/2 X Sin/; X [1-CosA - A;)]} D2 = Tan-1 X SQRT [(1/D12) - 1] FC = 2/D2 X Tan (D2/2)

Note: Inclination and azimuth values must be in radians only.

Table 1 shows survey data used to illustrate the difference in the calculation methods. Table 2 and 3 is a summary of the results.

 MD (ft) Inclination Azimuth MD (ft) Inclination Azimuth (degrees) (degrees) (degrees) (degrees) 0 0 0 2900 30.60 22.00 1000 0 0 3000 30.50 22.50 1100 3.00 21.70 3100 30.40 23.90 1200 6.00 26.50 3200 30.00 24.50 1300 9.00 23.30 3300 30.20 24.90 1400 12.00 20.30 3400 31.00 25.70 1500 15.00 23.30 3500 31.10 25.50 1600 18.00 23.90 3600 32.00 24.40 1700 21.00 24.40 3700 30.80 24.00 1800 24.00 23.40 3800 30.60 22.30 1900 27.00 23.70 3900 31.20 21.70 2000 30.00 23.30 4000 30.80 20.80 2100 30.20 22.80 4100 30.00 20.80 2200 30.40 22.50 4200 29.70 19.80 2300 30.30 22.10 4300 29.80 20.80 2400 30.60 22.40 4400 29.50 21.10 2500 31.00 22.50 4500 29.20 20.80 2600 31.20 21.60 4600 29.00 20.60 2700 30.70 20.80 4700 28.70 21.40 2800 31.40 20.90 4800 28.50 21.20
 Method TVD (ft) North (ft) East (ft) Tangential 4364.40 1565.23 648.40 Balanced Tangential 4370.46 1542.98 639.77 Average Angle 4370.80 1543.28 639.32 Radius of Curvature 4370.69 1543.22 639.30 Minimum Curvature 4370.70 1543.05 639.80
 Method A TVD A North A East Tangential -6.30 +22.18 +8.60 Balanced Tangential -0.24 -0.07 -0.03 Average Angle +0.10 +0.23 -0.48 Radius of Curvature -0.01 +0.17 -0.50 Minimum Curvature +0.00 +0.00 +0.00

TABLE 8.4—COMPARISON OF ACCURACY OF VARIOUS CALCULATION METHODS (after Craig and Randall1)

TABLE 8.4—COMPARISON OF ACCURACY OF VARIOUS CALCULATION METHODS (after Craig and Randall1)

Direction: Due north

Survey interval: 100 It

Rate of build: 3"7100 tt

Total inclination 60° at 2.000 It

Total Vertical Depth. North Displacement. Difference From Difference From

Direction: Due north

Survey interval: 100 It

Rate of build: 3"7100 tt

Total inclination 60° at 2.000 It

Total Vertical Depth. North Displacement. Difference From Difference From

 Calculation Method Actual (ft) Actual (ft) Tangential 1,628-61 -25 38 998.02 + 43 09 Balanced tangential 1.653 61 - 0.38 954.72 - 0.21 Angle-averaging 1.654.18 + 0 19 955.04 + 0.11 Radius of curvature 1.653.99 0.0 954.93 0 0 Minimum curvature 1.653.99 0.0 954.93 0 0 Mercury* 1.153.62 - 0.37 954.89 0.04 "Fifteen tool survey :ool

+2 -2

### Responses

• Violet
What is minimum curvature method calculation?
9 years ago
• yusef
How do you find a factor ratio in minimum curvature method?
8 years ago
• Amanda
What is minimum radius of curvature?
3 years ago
• fre-qalsi
When was minimum curvature invented?
7 months ago