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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Basic Terminologies

Below are listed few of the basic terminologies commonly used in directional drilling-

Inclination-

Inclination is the measurement of angle (in degrees) between true vertical (local gravity vector) and the well bore axis at any given point.

Inclination values varies from 0° (for vertical well) to 90° (for horizontal well).

Plot of  inclination values can be seen from Section View or Vertical Plot and type of well-profile can be depicted from it.

Azimuth (or Direction)-

Azimuth (or Direction) is the angle in degrees between the horizontal component of the well bore axis and the reference north.

Azimuth value varies from 0° to 360° in clock-wise direction.

North is represented as 0° or 360°.

East as 90°

South as 180°

West as 270°.

VERTICAL PLOT/ SECTION VIEW

Measured Depth (MD)-

It's the depth measured along the actual well bore from the surface reference point to the survey point. 

MEASURED DEPTH, INCLINATION & AZIMUTH ARE THE SET OF MEASURED DATA FROM WHICH OTHER VALUES CAN BE CALCULATED.

True Vertical Depth (TVD)-

TVD is the vertical depth measured from the surface reference point to the survey point. 

This depth is always calculated from the deviation survey.

Kick-off Point (KOP)-

KOP is the point in the well bore where change in inclination and orientation of well in a particular direction is initiated.

Building Section-

Building section ranges from KOP to the point, along measured depth of the well where maximum planned angle of the well is achieved.

End of Building Section (EOB) is the point along the measured depth of the well, where maximum planned angle is achieved and it marks the completion of building section or start of Holding section.

Vertical Section (VS)/Departure-

VS is the perpendicular distance (or shortest distance) between the survey point and the vertical axis from the rig's surface reference in a Vertical Plot/Section View.

Build-up Rate (BUR) or Drop-off Rate-

It's the rate at which change in inclination occurs and is calculated from one survey point to another survey point.

Build-up rates are shown in +ve.

Drop-off rates are shown in -ve.

They are measured in deg/100 ft or deg/30 m.

Turn Rate-

It's the rate at which there's change in the direction of the well bore and is calculated from one survey point to another survey point.

Turn rates are shown in +ve if the azimuth turns to right from the previous reading and -ve if left.

They are measured in deg/100 ft or deg/30 m.

Holding (or) Tangent Section-

Holding section specifies to the portion of the well bore along the measured depth where, there is no change in inclination and/or azimuth.

Generally it ranges from End of Build (EOB) to End of Hold (EOH) or Start of Drop (SOD).

Tangent (or Drift) Angle is the inclination of the holding section of the well.

It's also referred as tangent section because it forms tangent to the arc formed by building or dropping section of the well.

Dropping Section

Dropping section refers to the portion of the well from where there's start in decrease of inclination to the final decreased inclination.

Generally. it ranges from End of Hold (EOH) or Start of Drop (SOD) to End of Drop (EOD).

Dog leg & Dog leg Severity (DLS)-

Dog legs are referred as the crooked place in the course of well bore where profile of the well changes rapidly. High dog legs are generally associated with unwanted problems in the hole.

In directional drilling, since the changes in the well bore profile are made intentionally there has to be ways of measuring these changes.

Dog leg severity measures the changes in inclination and/or azimuth from one survey point to another survey point.

It's measured in deg/100 ft or deg/30 m.

North/South (N/S) and East/West (E/W)-

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

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

N/S and E/W are the RECTANGULAR CO-ORDINATES and are represented on a Plan View or Horizontal Plot.

Eg: 

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

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

Closure Distance (CD) & Closure Direction/Azimuth (CA)-

Closure Distance & Closure Direction is the distance and direction of a straight line drawn from the surface reference of rig location to a rectangular co-ordinates on a horizontal plane.

In general, 

VS/Departure/Drift will be approximately equal or near to the CD values.

Latitude & Longitude-

They are the imaginary lines on the surface of the earth running from North-South (longitude) and East-West (Latitude).
These are used to represent the GEOGRAPHICAL CO-ORDINATES of the well.

COMPARISON-

	LATITUDE	LONGITUDE
Direction	East-west, parallel to the equator	North-south; converging at the poles and widest at the equator
Parallel lines	Yes	No
Range	0 to 90° North and South	0 to 180° East and West
Denoted by	Greek letter phi (Φ)	Greek letter lambda (λ)
Hemisphere	All locations along a common latitude fall in the same hemisphere of the earth (northern or southern)	Locations along a common longitude may be in different hemispheres.
Denotes distance from	Equator (north or south)	Prime Meridian (east or west)
Time zone	Locations that share the same latitude do not necessarily fall into the same time zone	All locations on the same longitude fall in the same time zone
Number of lines	180	360
Notable lines	Equator,  Tropic of Cancer, Tropic of Capricorn	Greenwich Meridian
Applications	Classifying temperature zones	Classifying time zones