1 14 Fractures ond hemisphericul projectionpfd = Bn0 x (n/180); 1 = sin(an)cos(Bn); m = cos(an)Co~(B~); and
n = sin@,). The mod function is a 'remainder' function, and ensures
that the result lies between 0" and 360". Notice that all angles are con-
verted to radians, as these are the units for trigonometrical functions in
most computer packages. The calculations are shown in the table below.
Borehole:
Degrees Radians Components
as Bs us b, I, m, n,
136 55 2.374 0.960 0.398 -0.413 0.819Fracture data:
Degrees Radians Components 1 Weighted comp.
B (Y" pn an pn 1 m n I COS@ w I' m' n'
201 39 21 51 0.367 0.890 0.226 0.588 0.777 I 0.484 2.066 0.466 1.214 1.606
213 50 33 40 0.576 0.698 0.417 0.642 0.643 I 0.428 2.338 0.975 1.502 1.503
215 63 35 27 0.611 0.471 0.511 0.730 0.454 I 0.274 3.645 1.863 2.660 1.655
230 52 50 38 0.873 0.663 0.604 0.507 0.616 I 0.536 1.866 1.127 0.945 1.149
247 42 67 48 1.169 0.838 0.616 0.261 0.743 I 0.746 1.340 0.825 0.350 0.996
253 28 73 62 1.274 1.082 0.449 0.137 0.883 I 0.846 1.183 0.531 0.162 1.044
Sum 2.822 2.865 4.116 I
Mean 0.470 0.478 0.686 I
Normalized mean 0.490 0.498 0.715!5.787 6.834 7.952
0.964 1.139 1.325
0.483 0.571 0.664The mean of the components as calculated is not an orientation vector,
as its magnitude is not equal to unity: dO.47O2 + 0.47S2 + 0.6862 = 0.959.
The final row normalizes the mean by dividing the components by 0.959
so that the magnitudeis equal tounity: JO.49O2 + 0.49@ + 0.719 = 1.000.
Because sampling bias correction is not required in the first part of
the calculation, having found the components of the mean we calcu-
late the orientation of the mean using the formulae cr, = atan2(rn,I)
and Bn = asin(n). The result is an = atan2(0.498,0.490) = 44.5" and
pn = asin(0.715) = 45.6'. Converting these to a mean dip direction
and dip angle gives an orientation of24.5/44.4, or more appropriately
25/44.
To compute the corrected mean, we weight each fracture orientation
according to the cosine of the angle it makes to the borehole axis.
This cosine is given as the scalar product of the normal vector and the
borehole vector, i.e. for fracture i it is cos 0, = Zi .I, +mi em, +ni en,, and the
reciprocal of this is the weighting factor, i.e. w = 1/ cos 6. The weighted
components are given by the product of this factor and the unweighted
components. Again, the mean of these components has to be normalized
before it can be used to calculate the orientation, but once this has been
done we find that the corrected mean normal has an Orientation of
an = atan2(0.571,0.483) = 40.2' and Bn = asin(0.664) = 41.6". In terms of
dip direction and dip angle this is an orientation of 20.3/48.4, or more
appropriately 220/48.