Appendix 2

343

APPENDIX 2 MATHEMATICS O F MANUAL DIP COMPUTATION (reproduced from “Fundamentals of dipmeter interpretation” by courtesy of Schlumberger)

1, = a tan

o [1

-

cos(

$

-

$11

The displacements measured between dip curves are, therefore:

Figure A2-1 shows a section of the borehole traversed by bedding plane B. Hole axis OA is supposed vertical in this section, leading to the determination of apparent dip 8 and apparent azimuth cp. A reference plane D O F is drawn perpendicular to OA. As electrodes 1, 2 and 3 travel upward, they encounter plane B at elevations I , , I, and I, above DOF. Axis OD is in the reference plane containing Axis OA and electrode 1. The sonde is assumed not to rotate as it crosses plane B. A plane drawn perpendicular to B through axis OA cuts B along its apparent line of greatest slope, and DOF through line OM, where M is in the downdip direction. Angle (OMC) is the apparent dip angle B and angle (DOM) is the apparent azimuth cp, counted positive clockwise from D to M. Let a be the borehole radius ( a = d J2):

Knowing a , this system can be solved for 8 and cp. The method of solution used is graphical and uses * the concept of combined displacement. We can write:

I, = a t a n 8 [ 1 - c o s + ]

K

=$a

Z,

= cos

I

[ (‘3” a tan 0 cos cp - cos [ (“3”

11 $11

h , - , = I , - 1, = a tan B cos cp - cos -- cp h , . 3= I , - I ,

=

--

Noting that s i n ( 2 ~ / 3 )= - s i n ( 4 ~ / 3 )= d v 2 and cos(2a/3) = c o s ( 4 ~ / 3 )= - 1/2 this transforms to:

(3)

tan 8 = combined displacement. 1 . q~- - sin cp

0

1 . Z, = cos cp + sin cp

0

and A

h,., = K X I , h ,-3

=K

x I,

Equations 2a and 3a are the equations of a set of ellipses as shown in Fig. A3-la. Each ellipse corresponds to a value of the combined displacement. Carry displacements h 1 - ,and h , - 3 on their respective axes. If electrode 2 is “up” with respect to electrode 1, h , - , is positive, and similarly for electrode 3 and h,.3 *. The representative point ( A l . * , h ,-,) falls between two ellipses. Combined displacement K and apparent azimuth C#I are read. Note that whenever one displacement is equal to zero, (or both displacements are equal), the other displacement is (or both displacements are) equal to the combined displacement. The combined displacement is now carried into

* In another approach, the sign convention is the opposite, “up” is negative, “down” is positive. The reasoning remains the same.

344

Fig. A3-lb which solves the equation:

K

= +a tan

8

for 8, the apparent dip

Example: data of Appendix 3, Method 1. Displacement between curves: Curves I and 2 2 up = 3.0 mm Curves I and 3 3 up = 1.5 mm Figure A3-la gives K = 2.65 mm on 1/20 scale So on scale 1: K = 2.65 X 20 = 53 mm and a = 4.125 X 25.4

104.775 mm. So we have: 2 K = 106 tan 8 = - 0.33723 3a 314.325 which gives: =

~

e = 18038’ as apparent dip. Compare with graphical methods explained in Appendix 3.