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Chapter 2

S URVEY C ALCULATIONS

INTRODUCTION Directional surveys are taken at specified intervals in order to determine the position of the bottom of the hole relative to the surface location. The surveys are converted to a North-South (N-S), East-West (E-W) and true vertical depth coordinates using one of several calculation methods. The coordinates are then plotted in both the horizontal and vertical planes. By plotting the survey data, the rig personnel can watch the progress of the well and make changes when necessary to hit a specified target. There are several methods that can be used to calculate survey data; however, some are more accurate than others. Some of the most common methods that have been used in the industry are: 1. Tangential, 2. Balanced Tangential, 3. Average Angle, 4. Radius of Curvature and 5. Minimum Curvature Of these methods, the tangential method is the least accurate, and the radius of curvature and the minimum curvature are the most accurate. The industry uses primarily minimum curvature. The first three calculation methods are based on the trigonometry of a right triangle; therefore, a review of these trigonometric functions would be in order. By definition, a right triangle has one angle which is equal to 90°. The sum of the other two angles is 90°. Therefore, the sum of all three angles is 180°. Referring to the triangle in Figure 2-1, the angles are A, B, and C with C being the right angle (90°). C = 90° A + B = 90° A + B + C = 180° In Figure 2-1, the length of the triangle sides are designated a, b, and c. Therefore we can say that for a right triangle: a 2 + b 2 = c 2 when c is the hypotenuse of the triangle. The hypotenuse is always the side opposite the right angle (90º). The length of the hypotenuse can be determined by rearranging the equation to read:

c = a2 + b2

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2-1

Horizontal and Directional Drilling Chapter 2

Figure 2-1. Right Triangle

The following equations also apply to a right triangle. Sine of angle A Sin A =

opposite side a = hypotenuse c

Cosine of angle A Cos A =

adjacent side b = hypotenuse c

Tangent of angle A Tan A =

2-2

opposite side a = adjacent side b

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Horizontal and Directional Drilling Survey Calculations

Sine of angle B opposite side b = hypotenuse c

Sin B =

Cosine of angle B Cos B =

adjacent side a = hypotenuse c

Tangent of angle B Tan B =

opposite side b = adjacent side a

The values of sine, cosine and tangent of angles from 0 to 90o are given in the Appendix.

Example 2-1 Given:

Well XYZ in Figure 2-2, assume the triangle represents the plan view of a well. In this well, B is the surface location and A is the position of the bottom of the hole. The length "b" would then be the East coordinate and is equal to 450 feet. The length "a" would be the North coordinate and is equal to 650 feet.

Determine:

1.

The closure distance (length “c”), and

2.

The closure direction (angle B).

Solution:

To aid in solving the problem, a plan view similar to Figure 2-2 should be constructed and labeled. Then, use the trigonometric functions of a right triangle to solve the problem. 1.

Calculate the closure distance:

c 2 = a2 + b2

c = a2 + b2 c=

(450)2 + (650)2

c = 790.57ft 2.

Calculate the closure direction. The direction of a borehole is always given in azimuth from 0° to 360° or from the north or south such as:

N 48°13'W

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2-3

Horizontal and Directional Drilling Chapter 2

N 10° 43'E S 42°0'E S 24°32'W

In this example, angle "B" would be the closure direction. Solving for angle "B": Sin B =

opposite side 450' = hypotenuse 790.57'

Sin B = 0.5692 . B = 34.70 ° (See Appendix for Sine table and interpolate)

Figure 2-2. Horizontal Plan View of Well XYZ North: 650’, East: 450’, o Closure Distance: 790.57’, Closure Direction: N34 42’E (Azimuth 34.70º)

2-4

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Horizontal and Directional Drilling Survey Calculations

The closure direction can be expressed in azimuth as 34.70° or it can be expressed in the coordinate system. Converting the decimal to minutes: Minutes = (decimal)(60) Minutes = (0.70) (60) Minutes = 42' Therefore, the closure distance and direction are: 790.57' and N34°42' E. Presented next is a brief explanation of the most commonly used survey calculation methods and the appropriate calculations.

TANGENTIAL At one time the tangential method was the most widely used because it was the easiest (Table 2-1). The equations are relatively simple, and the calculations can be performed easily in the field. Unfortunately, the tangential method is the least accurate method and results in errors greater than all the other methods. The tangential method should not be used to calculate directional surveys. It is only presented here to prove a point. The tangential method assumes the wellbore course is tangential to the lower survey station, and the wellbore course is a straight line. Because of the straight line assumption, the tangential method yields a larger value of horizontal departure and a smaller value of vertical displacement when the inclination is increasing. This is graphically represented in Figure 2-3.

Figure 2-3. Illustration of Tangential Calculation Method

In Figure 2-3, Line AI 2 is the assumed wellbore course. The dashed line AB is the change in true vertical depth and the dashed line BI 2 is the departure in the horizontal direction. The opposite is true when the inclination is decreasing. In Type I, III and IV holes, the error will be significant. In a Type II hole, the error calculated while increasing angle will be offset by the error calculated while decreasing angle but only when the build and drop rates are comparable.

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2-5

Horizontal and Directional Drilling Chapter 2

With the tangential method, the greater the build or drop rate, the greater the error. Also, the distance between surveys has an effect on the quantity of the error. If survey intervals were 10 feet or less, the error would be acceptable. The added expense of surveying every 10 feet prohibits using the tangential method for calculating the wellbore course especially when more accurate methods are available. The North-South, East-West coordinates are determined by assuming the horizontal departure of the course length is in the same direction as the azimuth recorded at the lower survey station, but this assumption is wrong. The actual wellbore course will be a function of the upper and lower survey stations. Therefore, the tangential method results in an additional error because an error already exists due to the method used to calculate the horizontal departure. The error is compounded when the North-South, East-West coordinates are calculated. Table 2-1. Directional Survey Calculation Formula

Tangential ∆TVD = ∆MD × Cos I 2

Equation 2-1

∆North = ∆MD × Sin I 2 × Cos A2

Equation 2-2

∆East = ∆MD × Sin I 2 × Sin A2

Equation 2-3

Balanced Tangential

∆TVD =

∆MD (Cos I1 + Cos I 2 ) 2

∆North =

∆East =

Equation 2-4

∆MD [(Sin I1 × Cos A1 ) + (Sin I 2 × Cos A2 )] 2 ∆MD [(Sin I1 × Sin A1 ) + (Sin I 2 × Sin A2 )] 2

Equation 2-5

Equation 2-6

Average Angle

⎛I + I ⎞ ∆TVD = ∆MD × Cos ⎜ 1 2 ⎟ ⎝ 2 ⎠

Equation 2-7

⎛ A + A2 ⎞ ⎛I + I ⎞ ∆North = ∆MD × Sin⎜ 1 2 ⎟ × Cos ⎜ 1 ⎟ 2 ⎠ ⎝ ⎝ 2 ⎠

Equation 2-8

⎛ A + A2 ⎞ ⎛I + I ⎞ ∆East = ∆MD × Sin⎜ 1 2 ⎟ × Sin⎜ 1 ⎟ 2 ⎠ ⎝ ⎝ 2 ⎠

Equation 2-9

2-6

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Horizontal and Directional Drilling Survey Calculations

Radius of Curvature

∆TVD =

(180)(∆MD)(SinI 2 − SinI1 ) π (I 2 − I1 )

∆North =

∆East = ∆DEP =

r =

(180 )2 (∆MD )(Cos I1 − Cos I 2 )(Sin A2 − Sin A1 ) π 2 (I 2 − I1 )(A2 − A1 )

180 2 (∆MD )(Cos I1 − Cos I 2 )(Cos A1 − Cos A2 )

π 2 (I 2 − I1 )(A2 − A1 )

180 (∆MD )(CosI 1 − CosI 2 ) π (I 2 − I1 )

180 (π )(DLS )

∆MD =

Equation 2-10

Equation 2-11

Equation 2-12

Equation 2-13 Equation 2-14

I 2 − I1 Br

Equation 2-15 Minimum Curvature

⎛ ∆MD ⎞ ∆TVD = ⎜ ⎟(Cos I1 + Cos I 2 )(FC ) ⎝ 2 ⎠

Equation 2-16

⎛ ∆MD ⎞ ∆North = ⎜ ⎟[(Sin I 2 × Cos A2 ) + (Sin I1 × Cos A1 )](FC ) ⎝ 2 ⎠

Equation 2-17

⎛ ∆MD ⎞ ∆East = ⎜ ⎟[(Sin I 2 × Sin A2 ) + (Sin I1 × Sin A1 )](FC ) ⎝ 2 ⎠

Equation 2-18

D1 = Cos (I 2 − I1 ) − {Sin I 2 × Sin I1 × [1 − Cos (A2 − A1 )]}

Equation 2-19

⎛ 1 ⎞ D 2 = Tan −1 ⎜ 2 ⎟ − 1 ⎝ D1 ⎠

Equation 2-20

FC =

2 ⎛ D2 ⎞ × Tan⎜ ⎟ D2 ⎝ 2 ⎠

Equation 2-21

Note: Use inclinations and azimuths in radians only Calculations for Closure

⎛ East ⎞ Closure Direction = Tan −1⎜ ⎟ ⎝ North ⎠

Closure Distance =

(North )2 + (East )2

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Equation 2-22 Equation 2-23

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Horizontal and Directional Drilling Chapter 2

Vertical Section VS = Cos (Az vs − Az cl ) × (Closure Distance )

Equation 2-24

Dogleg Severity ⎛ 100 ⎞ −1 DLS = ⎜ ⎟Cos {(Sin I1 × Sin I 2 )[(Sin A1 × Sin A2 ) + (Cos A1 × Cos A2 )] + (Cos I1 × Cos I 2 )} Equation 2-25 ⎝ ∆MD ⎠

or 2

⎡ ⎛ A − A1 ⎞⎤ ⎡ ⎛ I − I ⎞⎤ 200 DLS = Sin −1 (Sin I1 )(Sin I 2 )⎢Sin⎜ 2 ⎟⎥ + ⎢Sin⎜ 2 1 ⎟⎥ ∆MD 2 ⎠⎦ ⎣ ⎝ ⎣ ⎝ 2 ⎠⎦

2

Equation 2-26

BALANCED TANGENTIAL The balanced tangential method is similar to the tangential method in that the wellbore course is determined by the tangent to the angle. The difference between the two methods is the balanced tangential uses both the upper and lower survey stations. The top half of the wellbore course is approximated by the upper inclination line I1A in Figure 2-4 and the lower half of the wellbore course is approximated by the lower inclination line AI 2 . The azimuth is approximated in the same manner. Both the upper and lower portions of the assumed wellbore course are in error, but the errors are opposite and will nearly cancel each other. Therefore, the balanced tangential method is accurate enough for field applications. The balanced tangential equations are more difficult to perform (Table 2-1) and are more likely to result in an error because of mechanical mistakes while making the calculations.

Figure 2-4. Illustration of Balanced Tangential Calculation Method

2-8

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Horizontal and Directional Drilling Survey Calculations

AVERAGE ANGLE When using the average angle method, the inclination and azimuth at the lower and upper survey stations are mathematically averaged, and then the wellbore course is assumed to be tangential to the average inclination and azimuth. The calculations are very similar to the tangential method (Table 2-1), and the results are as accurate as the balanced tangential method. Since the average angle method is both fairly accurate and easy to calculate, it is the method that can be used in the field if a programmable calculator or computer is not available. The error will be small and well within the accuracy needed in the field provided the distance between surveys is not too great. The average angle method is graphically illustrated in Figure 2-5. The average angle method does have problems at low inclinations with large changes in azimuth so it should not be used for vertical wells.

I1 + I 2 2

Figure 2-5. Illustration of Average Angle Calculation Method

RADIUS OF CURVATURE The radius of curvature method is currently considered to be one of the most accurate methods available. The method assumes the wellbore course is a smooth curve between the upper and lower survey stations. The curvature of the arc is determined by the survey inclinations and azimuths at the upper and lower survey stations as shown in Figure 2-6. The length of the arc between I1 and I 2 is the measured depth between surveys. In the previous methods, the wellbore course was assumed to be one or two straight lines between the upper and lower survey points. The curvature of the wellbore course assumed by the radius of curvature method will more closely approximate the actual well; therefore, it is more accurate. Unfortunately, the equations are complicated (Table 2-1) and are not easily calculated in the field without a programmable calculator or computer. A closer inspection of the radius of curvature equations show that if the inclination or azimuth are equal for both survey points, a division by zero will result in an error. In Figure 2-6 the radius, r, will become infinitely long. In that case, the minimum curvature or average angle

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2-9

Horizontal and Directional Drilling Chapter 2

methods can be used to make the calculations. It is also possible to add a small number (such as 1 x 10-4) to either survey point. The resulting error will be insignificant. Generally, the radius of curvature calculations are used when planning a well. Using one of the three previous methods to plan a well will result in substantial errors when calculating over long intervals. This will be further explained in the section on planning a well.

Figure 2-6. Illustration of Radius of Curvature Calculation Method

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 equations can be found in Table 2-1. Figure 2-7 is a graphic representation of the minimum curvature calculations. The balanced tangential calculations assume the wellbore course is along the line I1A + AI 2 . The calculation of the ratio factor changes the wellbore course to I1B + BI 2 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 as do the radius of curvature calculations.

2-10

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Horizontal and Directional Drilling Survey Calculations

Figure 2-7. Illustration of Minimum Curvature Calculation Method Table 2-2. Surveys for a Near Vertical Well

MEASURED DEPTH

DRIFT ANGLE

DRIFT AZIMUTH

(feet)

(degrees)

(degrees)

0.00

0.00

0.00

100.00

1.00

94.80

200.00

1.50

140.00

300.00

1.75

186.00

400.00

1.50

120.00

500.00

2.00

240.00

600.00

2.00

350.00

700.00

1.50

260.00

800.00

1.25

200.00

900.00

1.75

180.00

1,000.00

1.50

340.00

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 10° and the next survey is 355°, the well walked left 15º. The radius of curvature equations assume the well walked right 345° which is not true. One of the two

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2-11

Horizontal and Directional Drilling Chapter 2

survey azimuths must be changed. The lower survey can be changed from 355° to -5°, and then the radius of curvature will calculate the correct coordinates. Table 2-2 shows survey data for a near vertical well to 1,000 feet. The survey data exhibits large changes in azimuth which is common in near vertical wells. Figure 2-8 shows a plot of that survey data. Note that the minimum curvature calculations always yield the least amount of departure. There will also be a slight difference in TVD. The minimum curvature calculations are recommended for near vertical wells and for the vertical portions of a directional well. 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 10° and the next survey is 355°, the well walked left 15º. The radius of curvature equations assume the well walked right 345° which is not true. One of the two survey azimuths must be changed. The lower survey can be changed from 355° to -5°, and then the radius of curvature will calculate the correct coordinates.

Figure 2-8. Plan View

2-12

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Horizontal and Directional Drilling Survey Calculations

Example 2-2 Given:

The survey data for Directional Well No. 1 are shown in Table 2-3.

Determine:

The wellbore position at each survey point using the tangential, balanced tangential, average angle, radius of curvature, and minimum curvature method. Table 2-3. Survey for Example 2-2

MEASURED DEPTH

DRIFT ANGLE

DRIFT AZIMUTH

(feet) 0.00 1,000.00 1,100.00 1,200.00 1,300.00 1,400.00 1,500.00 1,600.00 1,700.00 1,800.00 1,900.00 2,000.00 2,100.00 2,200.00 2,300.00 2,400.00 2,500.00 2,600.00 2,700.00 2,800.00 2,900.00 3,000.00 3,100.00 3,200.00 3,300.00 3,400.00 3,500.00 3,600.00 3,700.00 3,800.00

(degrees) 0.00 0.00 3.00 6.00 9.00 12.00 15.00 18.00 21.00 24.00 27.00 30.00 30.20 30.40 30.30 30.60 31.00 31.20 30.70 31.40 30.60 30.50 30.40 30.00 30.20 31.00 31.10 32.00 30.80 30.60

(degrees) N 0.00E N 0.00E N21.70E N26.50E N23.30E N20.30E N23.30E N23.90E N24.40E N23.40E N23.70E N23.30E N22.80E N22.50E N22.10E N22.40E N22.50E N21.60E N20.80E N20.90E N22.00E N22.50E N23.90E N24.50E N24.90E N25.70E N25.50E N24.40E N24.00E N22.30E

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2-13

Horizontal and Directional Drilling Chapter 2

Solution:

N21.70E N20.80E N20.80E N19.80E N20.80E N21.10E N20.80E N20.60E N21.40E N21.20E

31.20 30.80 30.00 29.70 29.80 29.50 29.20 29.00 28.70 28.50

3,900.00 4,000.00 4,100.00 4,200.00 4,300.00 4,400.00 4,500.00 4,600.00 4,700.00 4,800.00

Tangential Method

At 0 and 1,000 feet the inclination is 0°, therefore, the wellbore position is 0’ North and 0’ East. A survey at 1,100 feet shows the inclination to be 3o in the N21.7E direction (Azimuth = 21.7). Calculate the position of the wellbore at 1,100 feet. (The nomenclature is defined at the end of the chapter.) ∆MD = MD2 − MD1

∆MD = 1,100 − 1,000 ∆MD = 100'

The direction of the wellbore is given as N21.7E; however, in the equations, “A” must equal a value between 0° and 360° where: N = 360° or 0° E = 90° S = 180° W = 270º When referring to the hole direction as being N21.7E, it is 21.7° East of North. Therefore, the azimuth is equal to 0+21.7 or 21.7º. Using the tangential method, calculate ∆TVD ∆TVD = (∆MD )(Cos I 2 ) ∆TVD = (100 )(Cos 3°)

∆TVD = 99.86ft

2-14

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Horizontal and Directional Drilling Survey Calculations

Calculate the true vertical depth. TVD 2 = ∆TVD + TVD1 TVD 2 = 99.86 + 1,000 TVD 2 = 1,099 .86ft

Calculate ∆North ∆North = (∆MD )(Sin I 2 )(Cos A2 ) ∆North = (100 )(Sin 3° )(Cos 21 .7°) ∆North = 4.86'

Calculate the North coordinate. North 2 = ∆North + North1 North 2 = 4.86'+0' North 2 = 4.86ft

Calculate ∆East ∆East = (∆MD )(Sin I 2 )(Sin A2 ) ∆East = (100 )(Sin 3°)(Sin 21 .7 ) ∆East = 1.94'

Calculate the East coordinate. East 2 = ∆East + East 1 East 2 = 1.94'+0' East 2 = 1.94'

Calculate the position of the wellbore at the next survey point of 1,200’. ∆MD = MD2 − MD1

∆MD = 1,200'−1,100' ∆MD = 100'

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2-15

Horizontal and Directional Drilling Chapter 2

The direction is N26.5E at 1,200 feet; therefore, the azimuth is 26.5º. ∆TVD = (∆MD )(Cos I 2 ) ∆TVD = (100 )(Cos 6° ) ∆TVD = 99 .45' TVD 2 = ∆TVD + TVD1 TVD 2 = 99 .45'+1,099 .86' TVD2 = 1,199 .31' ∆North = (∆MD )(Sin I 2 )(Cos A2 ) ∆North = (100 )(Sin 6°)(Cos 26 .5° ) ∆North = 9.35'

North 2 = ∆North + North1 North 2 = 9.35'+4.86' North2 = 14.21' ∆East = (∆MD )(Sin I 2 )(Sin A2 ) ∆East = (100 )(Sin 6° )(Sin 26.5° )

∆East = 4.66ft East 2 = ∆East + East 1 East 2 = 4.66'+1.94' East 2 = 6.60ft

The same calculations are made at each survey depth, and the results are shown in Table 2-4.

Balanced Tangential Method

Calculate the position of the wellbore at 1,300 feet using the balanced tangential method given the values at 1,200 feet from Table 2-5.

2-16

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Horizontal and Directional Drilling Survey Calculations

∆MD = MD2 − MD1

∆MD = 1,300'−1,200' ∆MD = 100ft The azimuth at 1,300 feet is 23.30º. ⎛ ∆MD ⎞ ∆TVD = ⎜ ⎟(Cos I 2 + Cos I1 ) ⎝ 2 ⎠

⎛ 100 ⎞ ∆TVD = ⎜ ⎟(Cos 9° + Cos 6°) ⎝ 2 ⎠ ∆TVD = 99 .11'

TVD 2 = ∆TVD + TVD1 TVD 2 = 99 .11'+1,199 .59' TVD 2 = 1,298 .70ft ⎛ ∆MD ⎞ ∆North = ⎜ ⎟(Sin I1 × Cos A1 + Sin I 2 × Cos A2 ) ⎝ 2 ⎠

⎛ 100 ⎞ ∆North = ⎜ ⎟(Sin 6° × Cos 26 .5° + Sin 9° × Cos 23 .30 ° ) ⎝ 2 ⎠ ∆North = 11 .86'

North 2 = ∆North + North1 North 2 = 11.86'+9.54' North 2 = 21 .40' ⎛ ∆MD ⎞ ∆East = ⎜ ⎟(Sin I1 × Sin A1 + Sin I 2 × Sin A2 ) ⎝ 2 ⎠

⎛ 100 ⎞ ∆East = ⎜ ⎟(Sin 6° × Sin 26.5° + Sin 9° × Sin 23.30°) ⎝ 2 ⎠ ∆East = 5.43' East 2 = ∆East + East 1

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2-17

Horizontal and Directional Drilling Chapter 2 Table 2-4. Survey Calculations for Directional Well No. 1 using the Tangential Method

MEASURE DDEPTH

INCLINATION

(feet)

(degrees)

AZIMUTH

COURSE LENGTH

TRUE VERTICAL DEPTH

(degrees)

(feet)

(feet)

RECTANGULAR COORDINATES NORTH

EAST

VERTICAL SECTION 10 DEG

DOGLEG SEVERITY

(feet)

(deg/100’)

0.00 0.00 3.00 6.00 9.00

0.00 0.00 21.70 26.50 23.30

0.00 1000.00 100.00 100.00 100.00

0.00 1,000.00 1,099.86 1,199.32 1,298.08

0.00 0.00 4.86 14.22 28.58

0.00 0.00 1.94 6.60 12.79

0.00 0.00 5.12 15.15 30.37

0.00 0.00 3.05 3.02 3.03

1,400.00 1,500.00 1,600.00 1,700.00 1,800.00

12.00 15.00 18.00 21.00 24.00

20.30 23.30 23.90 24.40 23.40

100.00 100.00 100.00 100.00 100.00

1,395.90 1,492.49 1,587.60 1,680.96 1,772.31

48.08 71.86 100.11 132.74 170.07

20.00 30.24 42.76 57.56 73.71

50.83 76.01 106.01 140.72 180.29

3.05 3.08 3.00 3.00 3.02

1,900.00 2,000.00 2,100.00 2,200.00 2,300.00

27.00 30.00 30.20 30.40 30.30

23.70 23.30 22.80 22.50 22.10

100.00 100.00 100.00 100.00 100.00

1,861.41 1,948.01 2,034.44 2,120.69 2,207.03

211.64 257.56 303.94 350.69 397.43

91.96 111.74 131.23 150.60 169.58

224.40 273.06 322.11 371.51 420.84

3.00 3.01 0.32 0.25 0.23

2,400.00 2,500.00 2,600.00 2,700.00 2,800.00

30.60 31.00 31.20 30.70 31.40

22.40 22.50 21.60 20.80 20.90

100.00 100.00 100.00 100.00 100.00

2,293.11 2,378.82 2,464.36 2,550.34 2,635.70

444.50 492.08 540.25 587.97 636.65

188.98 208.69 227.76 245.89 264.47

470.56 520.84 571.59 621.74 672.90

0.34 0.40 0.51 0.65 0.70

2,900.00 3,000.00 3,100.00 3,200.00 3,300.00

30.60 30.50 30.40 30.00 30.20

22.00 22.50 23.90 24.50 24.90

100.00 100.00 100.00 100.00 100.00

2,721.77 2,807.94 2,894.19 2,980.79 3,067.22

683.84 730.73 777.00 822.50 868.12

283.54 302.96 323.47 344.20 365.38

722.69 772.24 821.36 869.77 918.38

0.98 0.27 0.72 0.50 0.28

3,400.00 3,500.00 3,600.00 3,700.00 3,800.00

31.00 31.10 32.00 30.80 30.60

25.70 25.50 24.40 24.00 22.30

100.00 100.00 100.00 100.00 100.00

3,152.93 3,238.56 3,323.37 3,409.26 3,495.34

914.53 961.15 1,009.41 1,056.19 1,103.29

387.72 409.95 431.84 452.67 471.99

967.96 1,017.74 1,069.06 1,118.75 1,168.48

0.90 0.14 1.07 1.22 0.89

3,900.00 4,000.00 4,100.00 4,200.00 4,300.00

31.20 30.80 30.00 29.70 29.80

21.70 20.80 20.80 19.80 20.80

100.00 100.00 100.00 100.00 100.00

3,580.87 3,666.77 3,753.37 3,840.23 3,927.01

1,151.42 1,199.28 1,246.03 1,292.64 1,339.10

491.14 509.32 527.08 543.86 561.51

1,219.21 1,269.51 1,318.62 1,367.44 1,416.26

0.67 0.61 0.80 0.58 0.51

4,400.00 4,500.00 4,600.00 4,700.00 4,800.00

29.50 29.20 29.00 28.70 28.50

21.10 20.80 20.60 21.40 21.20

100.00 100.00 100.00 100.00 100.00

4,014.05 4,101.05 4,188.80 4,276.51 4,364.40

1,385.04 1,430.65 1,476.03 1,520.74 1,565.23

579.24 596.56 613.62 631.14 648.40

1,464.58 1,512.50 1,560.16 1,607.23 1,654.04

0.33 0.33 0.22 0.49 0.22

0.00 1,000.00 1,100.00 1,200.00 1,300.00

2-18

Copyright © 2004 OGCI/PetroSkills. All rights reserved

Horizontal and Directional Drilling Survey Calculations Table 2-5. Survey Calculations for Directional Well No. 1 using the Balanced Tangential Method

MEASURED DEPTH

INCLINATION

(feet)

(degrees)

AZIMUTH

COURSE LENGTH

TRUE VERTICAL DEPTH

(degrees)

(feet)

(feet)

RECTANGULAR COORDINATES

NORTH

EAST

VERTICAL SECTION 10 DEG

DOGLEG SEVERITY

(feet)

(deg/100’)

0.00 1,000.00 1,100.00 1,200.00 1,300.00

0.00 0.00 3.00 6.00 9.00

0.00 0.00 21.70 26.50 23.30

0.00 1000.00 100.00 100.00 100.00

0.00 1,000.00 1,099.93 1,199.59 1,298.70

0.00 0.00 2.43 9.54 21.40

0.00 0.00 0.97 4.27 9.69

0.00 0.00 2.56 10.14 22.76

0.00 0.00 3.05 3.02 3.03

1,400.00 1,500.00 1,600.00 1,700.00 1,800.00

12.00 15.00 18.00 21.00 24.00

20.30 23.30 23.90 24.40 23.40

100.00 100.00 100.00 100.00 100.00

1,396.99 1,494.20 1,590.04 1,684.28 1,776.63

38.33 59.97 85.98 116.43 151.41

16.39 25.12 36.50 50.16 65.64

40.60 63.42 91.01 123.37 160.51

3.05 3.08 3.00 3.00 3.02

1,900.00 2,000.00 2,100.00 2,200.00 2,300.00

27.00 30.00 30.20 30.40 30.30

23.70 23.30 22.80 22.50 22.10

100.00 100.00 100.00 100.00 100.00

1,866.86 1,954.71 2,041.23 2,127.57 2,213.86

190.86 234.60 280.75 327.31 374.06

82.84 101.85 121.49 140.92 160.09

202.34 248.73 297.58 346.81 396.18

3.00 3.01 0.32 0.25 0.23

2,400.00 2,500.00 2,600.00 2,700.00 2,800.00

30.60 31.00 31.20 30.70 31.40

22.40 22.50 21.60 20.80 20.90

100.00 100.00 100.00 100.00 100.00

2,300.07 2,385.96 2,471.59 2,557.35 2,643.02

420.97 468.29 516.16 564.11 612.31

179.28 198.83 218.22 236.82 255.18

445.70 495.70 546.22 596.66 647.32

0.34 0.40 0.51 0.65 0.70

2,900.00 3,000.00 3,100.00 3,200.00 3,300.00

30.60 30.50 30.40 30.00 30.20

22.00 22.50 23.90 24.50 24.90

100.00 100.00 100.00 100.00 100.00

2,728.74 2,814.85 2,901.06 2,987.49 3,074.00

660.24 707.29 753.87 799.75 845.31

274.01 293.25 313.22 333.83 354.79

697.79 747.47 796.80 845.57 894.08

0.98 0.27 0.72 0.50 0.28

3,400.00 3,500.00 3,600.00 3,700.00 3,800.00

31.00 31.10 32.00 30.80 30.60

25.70 25.50 24.40 24.00 22.30

100.00 100.00 100.00 100.00 100.00

3,160.08 3,245.75 3,330.96 3,416.31 3,502.30

891.33 937.84 985.28 1,032.80 1,079.74

376.55 398.83 420.90 442.26 462.33

943.17 992.85 1,043.40 1,093.91 1,143.62

0.90 0.14 1.07 1.22 0.89

3,900.00 4,000.00 4,100.00 4,200.00 4,300.00

31.20 30.80 30.00 29.70 29.80

21.70 20.80 20.80 19.80 20.80

100.00 100.00 100.00 100.00 100.00

3,588.10 3,673.82 3,760.07 3,846.80 3,933.62

1,127.35 1,175.35 1,222.65 1,269.33 1,315.87

481.56 500.23 518.20 535.47 552.69

1,193.85 1,244.36 1,294.06 1,343.03 1,391.85

0.67 0.61 0.80 0.58 0.51

4,400.00 4,500.00 4,600.00 4,700.00 4,800.00

29.50 29.20 29.00 28.70 28.50

21.10 20.80 20.60 21.40 21.20

100.00 100.00 100.00 100.00 100.00

4,020.53 4,107.69 4,195.07 4,282.66 4,370.46

1,362.07 1,407.84 1,453.34 1,498.38 1,542.98

570.37 587.90 605.09 622.38 639.77

1,440.42 1,488.54 1,536.33 1,583.70 1,630.64

0.33 0.33 0.22 0.49 0.22

Copyright © 2004 OGCI/PetroSkills. All rights reserved.

2-19

Horizontal and Directional Drilling Chapter 2

East 2 = 5.43'+4.27' East 2 = 9.70ft

The same calculations are made at each survey depth, and the results are shown in Table 2-5. Average Angle Method

Calculate the position of the wellbore at 1,400 feet using the averaging angle method and the survey data at 1,300 feet in Table 2-6. ∆MD = MD2 − MD1

∆MD = 1,400'−1,300' ∆MD = 100ft The azimuth at 1,400 feet is 20.30º. ⎛I + Ι ⎞ ∆TVD = ∆MD × Cos ⎜ 1 2 ⎟ ⎝ 2 ⎠

⎛ 9° + 12° ⎞ ∆TVD = 100 × Cos⎜ ⎟ 2 ⎝ ⎠

∆TVD = 98.33ft TVD 2 = ∆TVD + TVD1 TVD 2 = 98 .33'+1,298 .80' TVD2 = 1,397.13ft

⎛ A + A2 ⎞ ⎛I + I ⎞ ∆North = ∆MD × Sin⎜ 1 2 ⎟ × Cos ⎜ 1 ⎟ 2 ⎠ ⎝ ⎝ 2 ⎠

⎛ 9° + 12° ⎞ ⎛ 23.3° + 20.3° ⎞ ∆North = 100 × Sin⎜ ⎟ × Cos⎜ ⎟ 2 2 ⎝ ⎠ ⎝ ⎠

∆North = 16.92ft North 2 = ∆North + North 1 North 2 = 16 .92 '+ 21 .57 ' North 2 = 38.49ft

2-20

Copyright © 2004 OGCI/PetroSkills. All rights reserved

Horizontal and Directional Drilling Survey Calculations

⎛I + I ⎞ ⎛ A + A2 ⎞ ∆East = ∆MD × Sin⎜ 1 2 ⎟ × Sin⎜ 1 ⎟ 2 ⎝ 2 ⎠ ⎝ ⎠

⎛ 9° + 12° ⎞ ⎛ 23.3° + 20.3° ⎞ ∆East = 100 × Sin⎜ ⎟ × Sin⎜ ⎟ 2 2 ⎝ ⎠ ⎝ ⎠

∆East = 6.77ft East 2 = ∆East + East 1 East 2 = 6.77'+9.19' East 2 = 15.96ft

The same calculations are made at each survey depth, and the results are shown in Table 2-6.

Radius of Curvature Method

Calculate the position of the wellbore at 1,500 feet using the radius of curvature method and the survey data at 1,400 feet in Table 2-7. ∆MD = MD2 − MD1

∆MD = 1,500'−1,400' ∆MD = 100ft The azimuth at 1,500 feet is 23.30°.

∆TVD =

(180)(∆MD)(SinI 2 − SinI1 ) π (I 2 − I1 )

∆TVD =

(180 )(100 )(Sin15° − Sin12°) π (15 − 12)

∆TVD = 97.23ft TVD 2 = ∆TVD + TVD1 TVD 2 = 97 .23'+1,397 .08' TVD 2 = 1,494 .31ft

Copyright © 2004 OGCI/PetroSkills. All rights reserved.

2-21

Horizontal and Directional Drilling Chapter 2

∆North =

(180 )2 (∆MD )(Cos I1 − Cos I 2 )(Sin A2 − Sin A1 ) π 2 (I 2 − I1 )(A2 − A1 )

∆North =

(180 )2 (100 )(Cos12° − Cos15°)(Sin 23.3° − Sin 20.3°) π 2 (15 − 12)(23.3 − 20.3 )

∆North = 21.67ft North 2 = ∆North + North1 North 2 = 21 .67'+38 .47' North 2 = 60 .14ft

∆East =

(180 )2 (∆MD )(Cos I1 − Cos I 2 )(Cos A1 − Cos A2 ) π 2 (I 2 − I1 )(A2 − A1 )

∆East =

(180 )2 (100 )(Cos12° − Cos15°)(Cos 20.3° − Cos 23.3°) π 2 (15 − 12 )(23.3 − 20.3 )

∆East = 8.67ft East 2 = ∆East + East 1 East 2 = 8.67'+15 .95' East 2 = 24.62ft

The same calculations are made at each survey depth, and the results are shown in Table 2-7.

Minimum Curvature Method

Calculate the position of the wellbore at 1,600 feet using the minimum curvature method and the survey data at 1,500 feet in Table 2-8. ∆MD = MD2 − MD1

∆MD = 1,600'−1,500' ∆MD = 100ft The azimuth at 1,600 feet is 23.90°.

2-22

Copyright © 2004 OGCI/PetroSkills. All rights reserved

Horizontal and Directional Drilling Survey Calculations

For the minimum curvature method, all the data must be changed to radians. I1 = 15 °

or

(15)(π ) = 0.2618radians

I 2 = 18 °

or

(18)(π ) = 0.3142radians

A1 = 23 .30 °

or

(23.30)(π ) = 0.4067 radians

A2 = 23 .90 °

or

(23.90)(π ) = 0.4171radians

180

180

180

180

Calculate the ratio factor. D1 = Cos (I 2 − I1 ) − {Sin I 2 × Sin I1 × [1 − Cos (A2 − A1 )]} D1 = Cos (0.3142 − 0.2618 ) − {Sin 0.1342 × Sin 0.2618 × [1 − Cos (0.4171 − 0.4067 )]} D1 = 0.9986

⎛ 1 ⎞ D 2 = Tan −1 ⎜ 2 ⎟ − 1 ⎝ D1 ⎠ 1 ⎞ ⎛ D 2 = Tan −1 ⎜ ⎟ −1 2 ⎝ 0.9986 ⎠ D 2 = 0.0538

FC =

2 ⎛ D2 ⎞ × Tan⎜ ⎟ D2 ⎝ 2 ⎠

FC =

2 ⎛ 0.0538 ⎞ × Tan⎜ ⎟ 0.0538 ⎝ 2 ⎠

FC = 1.0002408 ⎛ ∆MD ⎞ ∆TVD = ⎜ ⎟(CosI1 + CosI 2 )(FC ) ⎝ 2 ⎠ ⎛ 100 ⎞ ∆TVD = ⎜ ⎟(Cos 0.2618 + Cos 0.3142)(1.0002408 ) ⎝ 2 ⎠

Copyright © 2004 OGCI/PetroSkills. All rights reserved.

2-23

Horizontal and Directional Drilling Chapter 2

∆TVD = 95.87ft TVD 2 = ∆TVD + TVD1 TVD 2 = 95 .87'+1,494 .31' TVD 2 = 1,590 .18ft

⎛ ∆MD ⎞ ∆North = ⎜ ⎟[(Sin I 2 × Cos A2 ) + (Sin I1 × Cos A1 )](FC ) ⎝ 2 ⎠

⎛ 100 ⎞ ∆North = ⎜ ⎟[(Sin0.3142 × Cos 0.4171) + (Sin0.2618 × Cos 0.4067)](1.0002408) ⎝ 2 ⎠

∆North = 26.02ft North 2 = ∆North + North1 North2 = 26.02'+59.98' North 2 = 86 .00ft ⎛ ∆MD ⎞ ∆East = ⎜ ⎟[(Sin I 2 × Sin A2 ) + (Sin I1 × Sin A1 )](FC ) ⎝ 2 ⎠ ⎛ 100 ⎞ ∆East = ⎜ ⎟[(Sin 0.3142 × Sin 0.4171) + (Sin 0.2618 × Sin 0.4067 )](1.0002408 ) ⎝ 2 ⎠ ∆East = 11.38ft

East 2 = ∆ East + East 1 East 2 = 11 .38'+25 .12' East 2 = 36.50ft

The same calculations are made at each survey depths, and the results are shown in Table 2-8.

2-24

Copyright © 2004 OGCI/PetroSkills. All rights reserved

Horizontal and Directional Drilling Survey Calculations Table 2-6. Survey Calculations for Directional Well No. 1 using the Average Angle Method

AZIMUTH

COURSE LENGTH

TRUE VERTICAL DEPTH

(degrees)

(degrees)

(feet)

(feet)

0.00 0.00 3.00 6.00 9.00

0.00 0.00 21.70 26.50 23.30

0.00 1000.00 100.00 100.00 100.00

0.00 1,000.00 1,099.97 1,199.66 1,298.80

0.00 0.00 2.57 9.73 21.57

0.00 0.00 0.49 3.70 9.19

0.00 0.00 2.62 10.23 22.84

0.00 0.00 3.05 3.02 3.03

1,400.00 1,500.00 1,600.00 1,700.00 1,800.00

12.00 15.00 18.00 21.00 24.00

20.30 23.30 23.90 24.40 23.40

100.00 100.00 100.00 100.00 100.00

1,397.13 1,494.36 1,590.25 1,684.51 1,776.90

38.49 60.17 86.19 116.65 151.64

15.96 24.63 36.00 49.66 65.16

40.68 63.53 91.14 123.50 160.65

3.05 3.08 3.00 3.00 3.02

1,900.00 2,000.00 2,100.00 2,200.00 2,300.00

27.00 30.00 30.20 30.40 30.30

23.70 23.30 22.80 22.50 22.10

100.00 100.00 100.00 100.00 100.00

1,867.16 1,955.04 2,041.55 2,127.89 2,214.19

191.11 234.86 281.01 327.57 374.32

82.36 101.39 121.02 140.45 159.63

202.50 248.90 297.76 346.99 396.35

3.00 3.01 0.32 0.25 0.23

2,400.00 2,500.00 2,600.00 2,700.00 2,800.00

30.60 31.00 31.20 30.70 31.40

22.40 22.50 21.60 20.80 20.90

100.00 100.00 100.00 100.00 100.00

2,300.40 2,386.29 2,471.92 2,557.68 2,643.35

421.23 468.55 516.43 564.37 612.57

178.82 198.37 217.76 236.36 254.72

445.88 495.88 546.39 596.84 647.50

0.34 0.40 0.51 0.65 0.70

2,900.00 3,000.00 3,100.00 3,200.00 3,300.00

30.60 30.50 30.40 30.00 30.20

22.00 22.50 23.90 24.50 24.90

100.00 100.00 100.00 100.00 100.00

2,729.07 2,815.19 2,901.39 2,987.82 3,074.34

660.51 707.56 754.14 800.02 845.58

273.55 292.80 312.76 333.38 354.34

697.98 747.65 796.99 845.75 894.26

0.98 0.27 0.72 0.50 0.28

3,400.00 3,500.00 3,600.00 3,700.00 3,800.00

31.00 31.10 32.00 30.80 30.60

25.70 25.50 24.40 24.00 22.30

100.00 100.00 100.00 100.00 100.00

3,160.41 3,246.08 3,331.30 3,416.66 3,502.64

891.60 938.12 985.56 1,033.08 1,080.02

376.09 398.38 420.45 441.81 461.88

943.36 993.04 1,043.60 1,094.11 1,143.82

0.90 0.14 1.07 1.22 0.89

3,900.00 4,000.00 4,100.00 4,200.00 4,300.00

31.20 30.80 30.00 29.70 29.80

21.70 20.80 20.80 19.80 20.80

100.00 100.00 100.00 100.00 100.00

3,588.45 3,674.17 3,760.42 3,847.15 3,933.97

1,127.64 1,175.64 1,222.95 1,269.63 1,316.17

481.12 499.78 517.75 535.02 552.24

1,194.05 1,244.57 1,294.27 1,343.24 1,392.07

0.67 0.61 0.80 .0.58 0.51

4,400.00 4,500.00 4,600.00 4,700.00 4,800.00

29.50 29.20 29.00 28.70 28.50

21.10 20.80 20.60 21.40 21.20

100.00 100.00 100.00 100.00 100.00

4,020.88 4,108.04 4,195.42 4,283.01 4,370.80

1,362.37 1,408.14 1,453.64 1,498.68 1,543.28

569.93 587.45 604.64 621.93 639.32

1,440.64 1,488.76 1,536.55 1,583.91 1,630.85

0.33 0.33 0.22 0.49 0.22

MEASURED DEPTH

INCLINATION

(feet)

0.00 1,000.00 1,100.00 1,200.00 1,300.00

Copyright © 2004 OGCI/PetroSkills. All rights reserved.

RECTANGULAR COORDINATES NORTH

EAST

VERTICAL SECTION 10 DEGREES

DOGLEG SEVERITY

(feet)

(deg/100’)

2-25

Horizontal and Directional Drilling Chapter 2 Table 2-7. Survey Calculations for Directional Well No. 1 using the Radius of Curvature Method

MEASURED DEPTH

DRIFT ANGLE

DRIFT AZIMUTH

COURSE LENGTH

TRUE VERTICAL DEPTH

(feet)

(degrees)

(degrees)

(feet)

(feet)

RECTANGULAR COORDINATES NORTH

EAST

VERTICAL SECTION 10 DEG

DOGLEG SEVERITY

(feet)

(deg/100’)

0.00 1,000.00 1,100.00 1,200.00 1,300.00

0.00 0.00 3.00 6.00 9.00

0.00 0.00 21.70 26.50 23.30

0.00 1000.00 100.00 100.00 100.00

0.00 1,000.00 1,099.63 1,199.63 1,298.77

0.00 0.00 2.56 9.71 21.55

0.00 0.00 0.49 3.69 9.19

0.00 0.00 2.60 10.21 22.82

0.00 0.00 3.00 3.02 3.03

1,400.00 1,500.00 1,600.00 1,700.00 1,800.00

12.00 15.00 18.00 21.00 24.00

20.30 23.30 23.90 24.40 23.40

100.00 100.00 100.00 100.00 100.00

1,397.08 1,494.31 1,590.18 1,684.43 1,776.81

38.47 60.14 86.16 116.62 151.60

15.95 24.62 35.99 49.64 65.15

40.65 63.50 91.10 123.47 160.61

3.05 3.08 3.00 3.00 3.02

1,900.00 2,000.00 2,100.00 2,200.00 2,300.00

27.00 30.00 30.20 30.40 30.30

23.70 23.30 22.80 22.50 22.10

100.00 100.00 100.00 100.00 100.00

1,867.06 1,954.93 2,041.44 2,127.78 2,214.08

191.06 234.81 280.96 327.52 374.27

82.35 101.37 121.01 140.44 159.61

202.46 248.85 297.70 346.93 396.30

3.00 3.01 0.32 0.25 0.23

2,400.00 2,500.00 2,600.00 2,700.00 2,800.00

30.60 31.00 31.20 30.70 31.40

22.40 22.50 21.60 20.80 20.90

100.00 100.00 100.00 100.00 100.00

2,300.29 2,386.18 2,471.81 2,557.57 2,643.24

421.18 468.50 516.37 564.32 612.52

178.80 198.35 217.74 236.34 254.70

445.82 495.83 546.34 596.79 647.45

0.34 0.40 0.51 0.65 0.70

2,900.00 3,000.00 3,100.00 3,200.00 3,300.00

30.60 30.50 30.40 30.00 30.20

22.00 22.50 23.90 24.50 24.90

100.00 100.00 100.00 100.00 100.00

2,728.96 2,815.08 2,901.28 2,987.71 3,074.23

660.46 707.50 754.08 799.96 845.53

273.53 292.78 312.74 333.36 354.32

697.92 747.59 796.93 845.70 894.21

0.98 0.27 0.72 0.50 0.28

3,400.00 3,500.00 3,600.00 3,700.00 3,800.00

31.00 31.10 32.00 30.80 30.60

25.70 25.50 24.40 24.00 22.30

100.00 100.00 100.00 100.00 100.00

3,160.30 3,245.97 3,331.19 3,416.54 3,502.53

891.55 938.06 985.50 1,033.02 1,079.96

376.07 398.36 420.43 441.79 461.86

943.31 992.98 1,043.54 1,094.04 1,143.76

0.90 0.14 1.07 1.22 0.89

3,900.00 4,000.00 4,100.00 4,200.00 4,300.00

31.20 30.80 30.00 29.70 29.80

21.70 20.80 20.80 19.80 20.80

100.00 100.00 100.00 100.00 100.00

3,588.33 3,674.05 3,760.30 3,847.03 3,933.85

1,127.58 1,175.58 1,222.89 1,269.57 1,316.11

481.10 499.76 517.73 535.00 552.22

1,193.99 1,244.50 1,294.21 1,343.18 1,392.00

0.67 0.61 0.80 0.58 0.51

4,400.00 4,500.00 4,600.00 4,700.00 4,800.00

29.50 29.20 29.00 28.70 28.50

21.10 20.80 20.60 21.40 21.20

100.00 100.00 100.00 100.00 100.00

4,020.76 4,107.92 4,195.30 4,282.89 4,370.69

1,362.31 1,408.08 1,453.57 1,498.62 1,543.22

569.90 587.43 604.62 621.91 639.30

1,440.57 1,488.69 1,536.48 1,583.85 1,630.79

0.33 0.33 0.22 0.49 0.22

2-26

Copyright © 2004 OGCI/PetroSkills. All rights reserved

Horizontal and Directional Drilling Survey Calculations Table 2-8. Survey Calculations for Directional Well No. 1 using the Minimum Curvature Method

AZIMUTH

COURSE LENGTH

TRUE VERTICAL DEPTH

(degrees)

(degrees)

(feet)

(feet)

0.00 1,000.00 1,100.00 1,200.00 1,300.00

0.00 0.00 3.00 6.00 9.00

0.00 0.00 21.70 26.50 23.30

0.00 1000.00 100.00 100.00 100.00

0.00 1,000.00 1,099.95 1,199.63 1,298.77

0.00 0.00 2.43 9.54 21.40

0.00 0.00 0.97 4.27 9.70

0.00 0.00 2.56 10.14 22.76

0.00 0.00 3.05 3.02 3.03

1,400.00 1,500.00 1,600.00 1,700.00 1,800.00

12.00 15.00 18.00 21.00 24.00

2030 23.30 23.90 24.40 23.40

100.00 100.00 100.00 100.00 100.00

1,397.08 1,494.31 1,590.18 1,684.44 1,776.81

38.34 59.98 86.00 116.45 151.44

16.40 25.12 36.51 50.17 65.65

40.61 63.44 91.03 123.40 160.54

3.05 3.08 3.00 3.00 3.02

1,900.00 2,000.00 2,100.00 2,200.00 2,300.00

27.00 30.00 30.20 30.40 30.30

23.70 23.30 22.80 22.50 22.10

100.00 100.00 100.00 100.00 100.00

1,867.06 1,954.93 2,041.45 2,127.79 2,214.08

190.90 234.66 280.81 327.37 374.12

82.86 101.88 121.51 140.94 160.11

202.39 248.78 297.64 346.87 396.23

3.00 3.01 0.32 0.25 0.23

2,400.00 2,500.00 2,600.00 2,700.00 2,800.00

30.60 31.00 31.20 30.70 31.40

22.40 22.50 21.60 20.80 20.90

100.00 100.00 100.00 100.00 100.00

2,300.29 2,386.19 2,471.81 2,557.58 2,643.25

421.02 468.34 516.22 564.16 612.36

179.30 198.86 218.25 236.85 255.20

445.76 495.76 546.27 596.72 647.38

0.34 0.40 0.51 0.65 0.70

2,900.00 3,000.00 3,100.00 3,200.00 3,300.00

30.60 30.50 30.40 30.00 30.20

22.00 22.50 23.90 24.50 24.90

100.00 100.00 100.00 100.00 100.00

2,728.96 2,815.08 2,901.29 2,987.72 3,074.23

660.30 707.35 753.92 799.80 845.37

274.03 293.28 313.24 333.86 354.82

697.86 747.53 796.86 845.63 894.14

0.98 0.27 0.72 0.50 0.28

3,400.00 3,500.00 3,600.00 3,700.00 3,800.00

31.00 31.10 32.00 30.80 30.60

25.70 25.50 24.40 24.00 22.30

100.00 100.00 100.00 100.00 100.00

3,160.31 3,245.98 3,331.20 3,416.55 3,502.54

891.39 937.90 985.34 1,032.86 1,079.80

376.57 398.86 420.92 442.28 462.36

943.23 992.91 1,043.47 1,093.97 1,143.68

0.90 0.14 1.07 1.22 0.89

3,900.00 4,000.00 4,100.00 4,200.00 4,300.00

31.20 30.80 30.00 29.70 29.80

21.70 20.80 20.80 19.80 20.80

100.00 100.00 100.00 100.00 100.00

3,588.34 3,674.06 3,760.31 3,847.04 3,933.87

1,127.42 1,175.42 1,222.72 1,269.40 1,315.94

481.59 500.26 518.23 535.50 552.71

1,193.91 1,244.43 1,294.13 1,343.10 1,391.92

0.67 0.61 0.80 0.58 0.51

4,400.00 4,500.00 4,600.00 4,700.00 4,800.00

29.50 29.20 29.00 28.70 28.50

21.10 20.80 20.60 21.40 21.20

100.00 100.00 100.00 100.00 100.00

4,020.77 4,107.94 4,195.31 4,282.90 4,370.70

1,362.14 1,407.91 1,453.40 1,498.45 1,543.05

570.40 587.93 605.12 322.41 639.80

1,440.49 1,488.61 1,536.40 1,583.77 1,630.71

0.33 0.33 0.22 0.49 0.22

MEASURED DEPTH

INCLINATION

(feet)

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RECTANGULAR COORDINATES NORTH

EAST

VERTICAL SECTION 10 DEG

DOGLEG SEVERITY

(feet)

(deg/100’)

2-27

Horizontal and Directional Drilling Chapter 2

The results of the survey calculations for Directional Well No. 1 in Example 2-2 are compared in Table 2-9 and Table 2-10. The comparison shows a significant difference when using the tangential method. The difference is much less pronounced with the other four methods. Table 2-10 shows the difference in the calculated TVD, North and East assuming the minimum curvature method is the most accurate. The average angle, balanced tangential and radius of curvature methods are all within one foot of each other at total depth. It must be remembered that as the distance between surveys increases, the average angle and balanced tangential errors will increase significantly. Table 2-9. Comparison of the Survey Calculation Methods for Example 2-2 Results

METHOD

TVD Feet 4,364.40 4,370.46 4,370.80 4,370.69 4,370.70

Tangential Balanced Tangential Average Angle Radius of Curvature Minimum Curvature

NORTH Feet 1,565.23 1,542.98 1,543.28 1,543.22 1,543.05

EAST Feet 648.40 639.77 639.32 639.30 639.80

Table 2-10. Relative Difference between the Survey Calculation Methods for Example 2-2 Results

METHOD

Tangential Balanced Tangential Average Angle Radius of Curvature Minimum Curvature

DIFFERENCE IN TVD Feet -6.30 -0.24 +0.10 -0.01 +0.00

DIFFERENCE IN NORTH Feet 22.18 -0.07 +0.23 +0.17 +0.00

DIFFERENCE IN EAST Feet +8.60 -0.03 -0.48 -0.50 +0.00

CLOSURE AND DIRECTION The line of closure is defined as "a straight line, in a horizontal plane containing the last station of the survey, drawn from the projected surface location to the last station of the survey." The line of closure is identified in Figure 2-9. Simply stated, the closure is the shortest distance between the surface location and the horizontal projection of the last survey point. The closure is always a straight line because a straight line is the shortest distance between two points. The closure is the polar coordinates at a given survey point as opposed to north and east being rectangular coordinates. When defining closure, the direction must also be given. Without indicating direction, the bottomhole location projected in a horizontal plane could be anywhere along the circumference of a circle with the radius of the circle being equal to the closure distance. The direction and closure exactly specifies where the bottom of the hole is located in relation to the surface location.

2-28

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Horizontal and Directional Drilling Survey Calculations

Figure 2-9. Graphic Representation of the Difference between Closure Distance and Vertical Section in the Horizontal Plane

The closure distance and direction are calculated using the following equations: Closure direction = Tan −1

Closure distance =

East North

Equation 2-27

(North)2 + (East )2

Equation 2-28

These are the same equations used for calculating an angle and hypotenuse of a right triangle.

Example 2-3 Given:

To illustrate the use of these equations, the closure and direction of the Directional Well No. 1 in Example 2-2 for the results of the minimum curvature method are calculated below From Table 2-7, the coordinates of the last survey point in the example well are: North = 1,543.05 ft East = 639.80 ft

Closure distance= (North) + (East ) 2

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2

2-29

Horizontal and Directional Drilling Chapter 2

Closure distance= (1,543.05 )2 + (639.80 )2 Closure distance =1,670 .43 ft

⎛ East ⎞ Closure direction =Tan −1⎜ ⎟ ⎝ North ⎠ ⎛ 639.80 ⎞ Closure direction =Tan −1 ⎜ ⎟ ⎝ 1,543.05 ⎠ Closure direction = 22.52 ° or 22 ° 31' Since the bottomhole location is in the northeast quadrant, the closure distance and direction are: 1,670.43 ft

N22.52E

Then, the horizontal projection of the bottom of the hole is 1,670.43 feet away from the surface location in the N22.52E direction.

VERTICAL SECTION The vertical section is the horizontal length of a projection of the borehole into a specific vertical plane and scaled with vertical depth. When the path of a wellbore is plotted, the vertical section is plotted versus TVD. The closure distance cannot be plotted accurately because the plane of closure (closure direction) can change between surveys. The vertical plot of a wellbore is in one specific plane. Figure 2-9 graphically shows the difference between the closure distance and vertical section. The closure distance and vertical section are equal only when the closure direction is the same as the plane of the vertical section. The vertical section azimuth is usually chosen as the azimuth from the surface location to the center of the target. If multiple targets are present and changes in azimuth are required to hit each target, the vertical section azimuth is usually chosen as the azimuth from the surface location to the end of the wellbore. The vertical section is calculated from the closure distance and direction. The equations for calculating the vertical section can be seen in Table 2-1 and are as follows: VS = Cos (Azvs − Azcl ) × (Closure distance )

Equation 2-29

Example 2-4 Given:

The data of Directional Well No. 1 from the previous examples. The plane of the vertical section is 10°.

Calculate:

2-30

The vertical section at the last survey point.

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Horizontal and Directional Drilling Survey Calculations

From the previous example: Closure distance = 1,670.43 feet Closure direction = 22.52° Calculate the vertical section: VS = Cos (Azvs − Azcl ) × (Closure distance ) VS = Cos (10 − 22 .52 ) × (1,670 .43 ) VS = 1,630.71ft

Therefore, the distance of 1,630.71 feet would be plotted on the vertical section. Figure 2-10 and Figure 2-11 are respectively the plan view and vertical section for Example 2-2.

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2-31

Horizontal and Directional Drilling Chapter 2

Plan View 1800

1600

1400

North, feet

1200

1000

800

600

400

200

0 0

200

400

600

800

1000

1200

1400

1600

East, feet Figure 2-10. Plan View for Directional Well No. 1 of Example 2-2

2-32

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Horizontal and Directional Drilling Survey Calculations

Vertical Section 0

500

1000

True Vertical Depth, feet

1500

2000

2500

3000

3500

4000

4500

5000 0

500

1000

1500

2000

2500

3000

3500

Vertical Section, feet Figure 2-11. Vertical Section for Directional Well no. 1 in Example 2-2

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2-33

Horizontal and Directional Drilling Chapter 2

PROBLEMS 1. Given the following survey data, calculate the ∆TVD, ∆North and ∆East using the average angle and radius of curvature methods. MD1 = 1000 feet I1 = 0º A1 = S42W

MD2 = 2000 feet I2 = 40º A2 = S42W

2. Given the following rectangular coordinates, calculate the vertical section of the survey point if the vertical section azimuth is 215º. North = -1643.82 feet and East = -822.16 feet 3 Given the following survey data, calculate the ∆TVD, ∆North and ∆East using the average angle, radius of curvature and minimum curvature methods. MD2 = 200 feet I2 = 1º A2 = 180º

MD1 = 100 feet I1 = 1º A1 = 0º

NOMENCLATURE

2-34

A1

=

Azimuth angle at upper survey point.

A2

=

Azimuth angle at lower survey point.

Azcl

=

Azimuth of closure or closure direction (0 to 360), degrees

Azvs

=

Azimuth of Vertical Section (0 to 360), degrees

Br

=

Build rate.

D1

=

Intermediate calculation in minimum curvature method.

D2

=

Intermediate calculation in minimum curvature method.

DLS

=

Dogleg severity in degrees per 100 feet.

FC

=

Ratio factor for minimum curvature

I1

=

Inclination angle at upper survey point, degrees

I2

=

Inclination angle at lower survey point, degrees

MD

=

Measured depth.

r

=

Radius of curvature.

VS

=

Vertical Section length

∆

=

Change in parameter

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Horizontal and Directional Drilling Survey Calculations

∆ DEP

=

The change in the horizontal departure.

=

The East coordinate at a survey point

∆East

=

The change in East coordinates between survey points.

∆MD

=

The measured distance along the wellbore course between survey points.

North

=

The North coordinate at a survey point

∆North

=

The change in North coordinates between survey points.

∆TVD

=

The change in true vertical depth between survey points.

1

=

Subscript denotes upper or previous survey point

2

=

Subscript denotes lower or last survey point

East

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2-35