Directional Surveying Calculations (Minimum Curvature Method)

Surveying is an inseparable part of directional drilling.

Surveys are recorded at regular intervals while drilling.

Reasons for Taking Surveys

1. To allow accurate determination of well coordinates at a series of measured depths and determine the current location.
2. To plot the well path over the measured depth.
3. To measure the inclination and azimuth at the bottom of the hole and hence determine where the well is heading.
4. To determine the orientation of tool face of deflection tools or steerable systems.
5. To locate dog legs and allow calculation of dogleg severity values.

Accurate Knowledge of the Course of a Borehole is Necessary:

1. To hit geological target.
2. To avoid collision with other near by wells.
3. To define the target of a relief well in the event of a blowout.
4. To provide a better definition of geological and reservoir data to allow for optimization of production.
5. To fulfill the requirements of local legislation if any.

For a directional driller, to successfully drill a well to the specified targets, all that's required is inclination and azimuth.

Now a days, there  are many types of advanced tools used and those along with the directional survey, provides the required geophysical characteristics of the well.

There are

1. MWD (Measurement While Drilling) tools and 
2. LWD (Logging While Drilling) tools. 

• As the name suggests, MWD tools mainly measures the values of inclination and azimuth while drilling whereas, LWD tools in addition to it measures geophysical characteristics of the formations encountered while drilling.
• LWD tools are more advance and sophisticated. Use of these tools eliminate the need of separate wireline logging; thereby saving rig time.
• Although hiring of LWD tools and engineers are costly as compared to MWD's.

Survey Calculations

Directional survey in terms of 'Inclination' & 'Azimuth' of a wellbore at certain 'Measured Depth' is taken. This information is then used to calculate the actual position of the wellbore relative to the surface location. When assuming an idealized well path between the two survey stations, many mathematical models can be used such as;

1. Tangential method
2. Balanced Tangential method
3. Average Angle method
4. Radius of Curvature method
5. Minimum Curvature method

Among above mathematical models, 
Tangential method is the least accurate, it assumes a straight line well path taking into consideration the inclination and azimuth at upper survey station and lower station is not accounted.

Balanced Tangential Method takes into account the upper and lower survey station and approximates well path by two  equal straight line segments. The upper line segment is defined by inclination and azimuth at upper survey station and the respective values at lower survey station.

Average Angle Method assumes one straight line defined by averaging inclination and azimuth at both survey stations, intersects both upper and lower survey stations.

Radius of Curvature Method assumes that well path is not a straight line but a circular arc tangential to inclination and azimuth at each survey station.

Minimum Curvature Method is the most accurate, it further adds a Ratio Factor to smoothen the spherical arc formed by using radius of curvature method. This is practically used and accepted calculation method among all and deserve to be discussed in detail.

Survey station is the measured depth at which survey is taken.

Course length (CL) is the difference between two survey stations.

Minimum Curvature Method

The inclination and azimuth at each survey station define two vectors namely inclination vector (lying in the vertical plane) and azimuthal vector (lying in the horizontal plane); and both are tangential to the wellbore trajectory. The only other piece of information available from a survey is the course length (the difference in survey measured depths) between the two stations. Minimum Curvature Method most accurately creates idealized well path between the upper and lower stations. The accuracy of the final coordinates generated by it approximates the actual trajectory of the borehole.


Let's explain the formula's used with an example. 

Consider inclination and azimuth at these two survey stations.

Azimuth of target is 316°.
Determine next set of values?

Sol'n:

Upper Survey Station (MD₁) is at 1914.75m
I₁ = 13.6°; A₁ = 315.2°; TVD₁ = 1827.53m; N/S₁ = 311.70m;
E/W₁ = -299.27m; VS₁ = 432.11m; CD₁ = 432.11m; CA₁ = 316.17°

Lower Survey Station (MD₂) is at 1940.30m
I₁ = 10.7°; A₁ = 314°


Course Length (CL) = Δ MD = MD₂ – MD₁

                                = 1940.30 – 1914.75 = 25.55m

Dog Leg = cos^–1 [{sinI₁ × sinI₂ × cos(A₂ – A₁)} + {cosI₁ × cosI₂}]

= cos^–1[{sin(13.6) × sin(10.7) × cos(314 – 315.2)} + {cos(13.6) × cos(10.7)}]

= 2.91
DLS = (DL × 30)/CL, when calculated per 30m.
DLS = (DL × 100)/CL, when calculated per 100ft.

DLS = (2.91 × 30)/25.55 = 3.42

Ratio Factor (R.F) is simply a smoothing factor used in the following calculations. It has no other significance.

RF = Tan(DL/2) × (180/π) × (2/DL)

      = Tan(2.91/2) × 180/π × (2/2.91)

      = 1


“0” Dogleg Exception
When the inclination and the direction do not change between two survey stations, the dogleg and dogleg severity are equal to 0. When the dogleg is equal to 0, the formula for ratio factor (R.F.) is undefined. In this case, simply assign the ratio factor the value of 1.0.

Change in N/S coordinate

Δ N/S = [(sinI₁ × cosA₁) + (sinI₂ × cosA₂)] [R.F. × (ΔMD/2)]

= [(sin(13.6) × cos(315.2)) + (sin(10.7) × cos(314))] [1 × (25.55/2)]
= 3.78
Total N/S (or) (N/S)₂ = (N/S)₁ + ΔN/S  

                                   = 311.7 + 3.78
                                   = 315.48


Change in E/W coordinate

Δ E/W = [(sinI₁ × sinA₁) + (sinI₂ × sinA₂)] [R.F. × (ΔMD/2)]
= [(sin(13.6) × sin(315.2)) + (sin(10.7) × sin(315))] [1 × (25.55/2)]
= –3.79

Total E/W (or) (E/W)₂ = (E/W)₁ + ΔE/W 

                                     = –299.27 + –8.04

                                     = –303.06


Change in TVD

Δ TVD = [cosI₁ + cosI₂] [R.F. × (Δ MD/2)]
             = [cos(13.6) + cos(10.7)] [1 × (25.55/2)]
             = 24.97

Total TVD (or) TVD₂ = TVD₁ + ΔTVD
                                    = 1827.53 + 24.97
                                    = 1852.50

Closure Distance

CD = [(N/S)2_Total + (E/W)2_Total]^1/2

      =  [(315.48)2 + (–303.06)2]^1/2
      =  437.46


Closure Azimuth
CA = Tan^–1[(E/W) _Total / (N/S) _Total]
       = Tan^–1[–303.06 / 315.48]
       = –43.84°
       = 360°– 43.84°
       = 316.16°

Note:
If the given target azimuth lies in b/w 
0° to 90°, then CA = Tan^–1[(E/W) _Total / (N/S) _Total]
90° to 180°, then CA = 180° –  Tan^–1[(E/W) _Total / (N/S) _Total]
180° to 270°, then CA = 180° + Tan^–1[(E/W) _Total / (N/S) _Total]
270° to 360°, then CA = 360° – Tan^–1[(E/W) _Total / (N/S) _Total]


Directional Difference (DD) is the angle between target azimuth and closure azimuth.
DD = Azimuthtarget – CA
       = 316° – 316.6°

           = –0.16°

VS = CD × cos(DD)
      = 437.46 × cos(–0.16°)
      = 437.46 

Hence,

Note:

The calculation for dogleg and dogleg severity, closure and vertical section do not change when different survey methods are used.

To summarize:

*Course Length (CL) = Δ MD = MD₂ – MD₁

*Dog Leg = cos^–1 [{sinI₁ × sinI₂ × cos(A₂ – A₁)} + {cosI₁ × cosI₂}]

  DLS = (DL × 30)/CL, when calculated per 30m.

  DLS = (DL × 100)/CL, when calculated per 100ft.

*RF = Tan(DL/2) × (180/π) × (2/DL)

*Δ N/S = [(sinI₁ × cosA₁) + (sinI₂ × cosA₂)] [R.F. × (ΔMD/2)]

  Total N/S (or) (N/S)₂ = (N/S)₁ + ΔN/S  

*Δ E/W = [(sinI₁ × sinA₁) + (sinI₂ × sinA₂)] [R.F. × (ΔMD/2)]

  Total E/W (or) (E/W)₂ = (E/W)₁ + ΔE/W 

*Δ TVD = [cosI₁ + cosI₂] [R.F. × (Δ MD/2)]

  Total TVD (or) TVD₂ = TVD₁ + ΔTVD

*CD = [(N/S)2_Total + (E/W)2_Total]^1/2

*CA = Tan^–1[(E/W) _Total / (N/S) _Total]
  If the given target azimuth lies in b/w 
   0° to 90°, then CA = Tan^–1[(E/W) _Total / (N/S) _Total]
   90° to 180°, then CA = 180° – Tan^–1[(E/W) _Total / (N/S) _Total]
   180° to 270°, then CA = 180° + Tan^–1[(E/W) _Total / (N/S) _Total]
   270° to 360°, then CA = 360° – Tan^–1[(E/W) _Total / (N/S) _Total]

*DD = Azimuthtarget – CA

  VS = CD × cos(DD)

These days Directional Drillers need not perform manual calculations instead, many high-end software are used for well planing such as COMPASS, WELL PATH, WIN SERVE, etc.
But, remembering these formulas is useful in the due course.

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Well co-ordinates

Physical representation of a well on the earth's surface can be done by-

1. Geographical co-ordinates
2. Rectangular co-ordinates
3. Polar co-ordinates

1. Geographical Co-ordinates

Let's discuss representation of well location in Geographical co-ordinates.

Latitude (Parallels) & Longitude (Meridian) represents the geographical co-ordinates of the earth. We can determine any location on the earth with the help of these.

Latitude (φ) & Longitude (λ) is the angle between N/S point & E/W point on the earth's surface respectively and center of the earth.

Equator is the reference point in latitudes and designated as 0°; it divides earth into Northern Hemisphere (0° to 90°N) & Southern Hemisphere (0° to 90°S).

There are two main points in longitude -

1. Prime Meridian (0°)
2. International Date line (180°)

Prime Meridian divides earth into Eastern Hemisphere (0° to 180°E) and Western Hemisphere (0° to 180° W).

Degrees of latitude and longitude can be further sub divided into minutes and seconds.

1 degree (°) = 60 minutes (')

and 1 minute (') = 60 seconds (")

There are three basic ways to display geographical co-ordinates:

1. Degrees, Minutes, Seconds (D M S)

2. Degrees, Minutes as decimal (D M.m)

3. Numerical (in degrees)

Example:

Amsterdam co-ordinates might be written as 52°22'25'' N, 4°53'27" E .

Degrees and decimal minutes: 52°22.416' N, 4°53.45' E 

(lat = 52°, 22' + 25/60'; lat = 4°, 53' + 45/60')

Degrees can also be expressed as decimals: 52.3736° N, 4.890° E

(lat = 52°+ 22/60°+ 25/3600°; lat = 4°+ 53/60° + 45/3600°)

Note: Latitude is written first, followed by longitude.

One degree of latitude is approximately 69 miles;

a minute of latitude is approximately 1.15 miles;

a second of latitude is approximately 0.02 miles.

However, one degree of longitude varies in size. At the equator, it's approximately 69 miles and the size decreases to zero as the meridians meet at the pole.

2. Rectangular Co-ordinates

In 1637 Rene Descartes, a French mathematician and philosopher, developed a method of associating the points on a plane with pairs of numbers. By drawing two number lines or axes, perpendicular at the 0 point or origin, a Rectangular Co-ordinate system is formed.

 

A point on this coordinate plane is associated with a pair of numbers called an Ordered Pair. The first number in the pair corresponds to the projection of the point on the horizontal or x-Axis. The second number corresponds to the projection of the point on the vertical or y-Axis. 

Points P and Q are associated with the ordered pairs (1,2) and (2,-3) respectively. Such ordered pairs are called Rectangular Co-ordinates.

This Rectangular Co-ordinate system has been adopted in directional drilling for several purposes. The easiest being, determining bottom hole location of the well w.r.t well head or rig location

In this case a Rectangular Co-ordinate system is set up by, 

y-axis being replaced by N/S co-ordinates, 

x-axis being replaced by E/W co-ordinates and 

0 or origin being replaced by well head.

Note: N/S co-ordinates are represented first followed by E/W co-ordinates.

North is +ve number that indicates the distance NORTH from the well head/rig location, while a -ve number indicates a distance SOUTH.

East is +ve number that indicates the distance EAST from the well head/rig location, while a -ve number indicates a distance WEST.

Both these form an ordered pair and represents the bottom hole well location with reference to well head/rig location.

N/S & E/W are represented on a Plan View or Horizontal Plot.

N/S & E/W values will be respectively,

• +ve & +ve for well direction from 0° to 90°.
• -ve & +ve for well direction from 90° to 180°
• -ve & -ve for well direction from 180° to 270°
• +ve & -ve for well direction from 270° to 360/0°

Example:

Horizontal Plots of four wells are shown. Locate N/S & E/W coordinates.

(N/S,E/W) Co-ordinates at;

A = (60,60) 

B = (75,-15)

C = (-60,-45)

D = (75,-75)

3. Polar Co-ordinates

In directional drilling, polar co-ordinates are used for representing location of a point as a distance away from the origin and a direction away from the origin.

Origin is taken as well head or rig location;
Closure distance gives the distance away from the origin; and
Closure azimuth gives the direction of the well.

Although points are not plotted on polar graphs, polar co-ordinates are used to locate bottom hole closure. 

Example:

Find bottom hole location of four wells from same well head/rig location from the horizontal plot.

Point A is at 20 m @ 330°
Point B is at 25 m @ 125°
Point C is at 25 m @ 250°
Point D is at 15 m @ 245°
Point E is at 30 m @ 90°

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