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.
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 |
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