### DirDrilling2

```Petroleum Engineering - 406
LESSON 19
Survey Calculation Methods
LESSON 11
Survey Calculation Methods
Balanced Tangential
Minimum Curvature
– Kicking Off from Vertical
– Controlling Hole Angle (Inclination)
Homework
Chapter 8 “Applied Drilling Engineering”,
( first 20 pages)
Assumption: The wellbore follows a
smooth, spherical arc between survey
points and passes through the measured
angles at both ends.
(tangent to I and A at both points
1 and 2).
Known: Location of point 1, MD12 and
angles I1, A1, I2 and A2
Length of arc of circle, L = Rrad
A1
1
 MD = R1 (I2-I1) (rad)
I2 -I1
North
R1
I1
A1
I2
East
2
North
East
Radius of Curvature - Vertical Section
In the vertical section, MD = R1(I2-I1)rad
π
MD = R1 (
) (I2-I1)deg
180
R1=
180
( π
)(
I1
MD
)
I2  I2
ΔVert  R1 sin I2  R1 sin I1
 180 


 π 
R1
 Vert
 ΔMD 

 sin I2  sin I1 
 I2  I1 
I2-I1
MD
I2
Vertical Section
I1 I2
Δ Horiz  R1 cos I1  R1 cos I2
Δ Horiz  R1 cos I1  cos I2 
180

π
 ΔMD 

 (cos I1  cos I2 )
 I2  I1 
R1
R1
MD
I2
 Horiz
Horizontal Section
N
A2
L2 = R2 (A2 - A1)RAD
2
A1
1
L2
North
East
R2
A2
180
so, R2 
π
A2-A1
A1
O
 L2 


 A 2  A1  DEG
East = R2 cos A1
- R2 cos A2
= R2 (cos A1 - cos A2)
East = R2 (cos A1 - cos A2)
180
R2 
π
East =
L2
 L2 


 A 2  A1 
180

π
 ΔMD 

 (cos I1  cos I2 )
 I2  I1 
MD cos I1  cos I2  cos A 1  cos A 2  180 


I2  I1  A 2  A1 
 π 
2
North = R2 (sin A2 - sin A1)
180
R2 
π
North =
 L2 


 A 2  A1 
180
L2 
π
 ΔMD 

 (cos I1  cos I2 )
 I2  I1 
MD cos I1  cos I2  sin A 2  sin A 1 
I2  I1  A 2  A1 
 180 


 π 
2
MD  cos(I1 )  cos(I2 )  sin( A 2 )  sin( A1 )
North 
(I2  I1 )  ( A 2  A1 )
MD  cos(I1 )  cos(I2 )  cos( A1 )  cos( A 2 )
East 
(I2  I1 )  ( A 2  A1 )
MD  sin(I2 )  sin(I1 )
Vert 
(I2  I1 )
If I1 = I2, then:
 sin A 2  sin A1 

North = MD sin I1 
A 2  A1


East = MD sin
 cos A1  cos A 2
I1 
A 2  A1

Vert = MD cos I1



If A1 = A2, then:
North = MD cos A1
 cos I1  cos I2 


I2  I1


East = MD sin A1
 cos I1  cos I2 


I2  I1


sin
I

sin
I


2
1
Vert = MD 



I

I
2
1


Radius of Curvature - Special Case
If I1 = I2 and A1 = A2
North = MD sin I1 cos A1,
East = MD sin I1 sin A1
Vert = MD cos I1
Balanced Tangential Method
MD
MD
Vert 
cos I1 
cos I2
2
2
1
I1
MD
2
I2
0
MD
MD
Horiz 
sinI1 
sinI2
2
2
MD
2
I2
Vertical Projection
I2
Balanced Tangential Method
Horiz. 2
N
A1
E
North  Horiz.1cos A1  Horiz.2 cos A 2
MD
sinI1 cos A1, sinI2 cos A 2 

2
East  Horiz.1sin A1  Horiz.2 sin A 2
MD
sinI1 sin A1, sinI2 sin A 2 

A2
2
Horiz.1
Horizontal Projection
Balanced Tangential Method - Equations

MD
North 
sin I1 cos A 1  sin I2 cos A 2
2

MD
East 
sin I1 sin A 1  sin I2 sin A 2
2

MD
Vert 
cos I2  cos I1
2



Minimum Curvature Method
This method assumes that the wellbore
follows the smoothest possible circular arc
from Point 1 to Point 2.
This is essentially the Balanced Tangential
Method, with each result multiplied by a
ratio factor (RF) as follows:
Minimum Curvature Method - Equations


MD
North 
sin I1 cos A 1  sin I2 cos A 2 RF
2


MD
East 
sin I1 sin A 1  sin I2 sin A 2 RF
2


MD
Vert 
cos I2  cos I1 RF
2
Minimum Curvature Method
PS  SR
RF 
Arc PQR
 DL 
 DL 
r tan 
  r tan 

2 
2 



r DL
DL = b
O
r
P
r
Q
2
DL
RF 
tan
DL
2
DL
2
S
DL
R
The Dogleg angle, DL, is calculated as follows :
cos (DL)  cos (I2  I1 )  sin I1 sin I2 1  cos ( A 2  A1 )


Fig 8.22
b
A curve
representing a
wellbore between
Survey Stations A1
and A2.
b  b(A, I)
Tangential Method
North  MD  sin(I2 )  cos( A 2 )
East  MD  sin(I2 )  sin( A 2 )
Vert  MD  cos(I2 )
Balanced Tangential Method
MD
sin(I1 )  cos( A1 )  sin(I2 )  cos( A2 )
North 
2
MD
sin(I1 )  sin( A1 )  sin(I2 )  sin( A 2 )
East 
2
MD
cos(I2 )  cos(I1 )
Vert 
2
Average Angle Method
 I1  I2
North  MD  sin
 2
 I1  I2
East  MD  sin
 2

 A1  A 2 
  cos

2




 A1  A 2 
  sin

2



 I1  I2 
Vert  MD  cos

 2 
MD  cos(I1 )  cos(I2 )  sin( A 2 )  sin( A1 )
North 
(I2  I1 )  ( A 2  A1 )
MD  cos(I1 )  cos(I2 )  cos( A1 )  cos( A 2 )
East 
(I2  I1 )  ( A 2  A1 )
MD  sin(I2 )  sin(I1 )
Vert 
(I2  I1 )
Minimum Curvature Method
MD
sin(I1 )  cos( A1 )  sin(I2 )  cos( A 2 )  RF
North 
2
MD
sin(I1 )  sin( A1 )  sin(I2 )  sin( A 2 )  RF
East 
2
Vert 
MD
cos(I1 )  cos(I2 )  RF
2
Mercury Method
North 
 MD  STL 

sin(I1 )  cos( A1 )  sin(I2 )  cos( A 2 )  STL  sin(I2 )  cos( A 2 )
2


East 
 MD  STL 

sin(I1 )  sin( A1 )  sin(I2 )  sin( A 2 )  STL  sin(I2 )  sin( A 2 )
2


Vert 
 MD  STL 

cos(I2 )  cos(I1 )  STL  cos(I2 )
2


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