True Vertical Depth, True Vertical Thickness And True Stratigraphic Thickness Logs


Holt, O.R.; Schoonover, L.G.; Wichmann, P.A.

Transactions of the SPWLA Eighteenth Annual Logging Symposium: 1-19

1977


The increased numbers of wells both onshore and offshore which must be directionally drilled to reach their intended objective has increased significantly in the past ten years. These directionally drilled wells show bed thickness to be too great, the amount depending upon the direction of dip of the measured formation and the drift angle and direction of the borehole. To rectify these differences in thickness for mapping and reserves estimation, differentapproaches have been used. The most frequently used method is the True Vertical Depth log which computes each point to the depth it would appear if the hole were drilled vertically. For mapping purposes, this approach is better than the information derived from the original open hole log, but is not accurate enough in highly deviated wells. The True Vertical Thickness log is derived by computing each bed as though the borehole passed through it in a vertical direction. This method results in improved data for reserves determination and also better precision for subsurface mapping points. The True Stratigraphic Thickness log is computed to provide the actual thickness of a given formation perpendicular to the bedding planes. The True Vertical Thickness, True Stratigraphic Thickness, and True Vertical Depth logs can be presented on the same plot. However, it is usually best to plot each as an individual log to reduce the amount of information which must be evaluated. The dip angle and direction of target formations must be known before the True Stratigraphic Thickness log can be run. This information can be most reliably derived from Diplog data. However, dip- data from other sources can be used in the program if a Diplog is not available. A review of the mathematics involved in the computation and some computed examples are included in the paper.

SPWLA
EIGHTEENTH
ANNUAL
LOGGING
SYMPOSIUM,
JUNE
5-8,
1977
True
Vertical
Depth,
True
Vertical
Thickness
and
True
Stratigraphic
Thickness
Logs
by
0.
R.
Holt,
L.
G.
Schoonover
&
P.
A.
Wichmann
Dresser
Atlas
-
Houston,
Texas
ABSTRACT
The
increased
numbers
of
wells
both
onshore
and
offshore
which
must
be
directionally
drilled
to
reach
their
intended
objective
has
increased
significantly
in
the
past
ten
years.
These
directionally
drilled
wells
show
bed
thickness
to
be
too
great,
the
amount
depending
upon
the
direction
of
dip
of
the
measured
formation
and
the
drift
angle
and
direction
of
the
borehole.
To
rectify
these
differences
in
thickness
for
mapping
and
reserves
estimation,
different
approaches
have
been
used.
The
most
frequently
used
method
is
the
True
Vertical
Depth
log
which
computes
each
point
to
the
depth
it
would
appear
if
the
hole
were
drilled
vertically.
For
mapping
purposes,
this
approach
is
better
than
the
information
derived
from
the
original
open
hole
log,
but
is
not
accurate
enough
in
highly
deviated
wells.
The
True
Vertical
Thickness
log
is
derived
by
computing
each
bed
as
though
the
borehole
passed
through
it
in
a
vertical
direction.
This
method
results
in
improved
data
for
reserves
determination
and
also
better
precision
for
subsurface
mapping
points.
The
True
Stratigraphic
Thickness
log
is
computed
to
provide
the
actual
thickness
of
a
given
formation
perpendicular
to
the
bedding
planes.
The
True
Vertical
Thickness,
True
Stratigraphic
Thickness,
and
True
Vertical
Depth
logs
can
be
presented
on
the
same
plot.
However,
it
is
usually
best
to
plot
each
as
an
individual
log
to
reduce
the
amount
of
information
which
must
be
evaluated.
The
dip
angle
and
direction
of
target
formations
must
be
known
before
the
True
Stratigraphic
Thickness
log
can
be
run.
This
information
can
be
most
reliably
derived
from
Diploedata.
However,
dip
data
from
other
sources
can
be
used
in
the
program
if
a
Diplog
is
not
available.
A
review
of
the
mathematics
involved
in
the
computation
and
some
computed
examples
are
included
in
the
paper.
INTRODUCTION
Programs
are
now
available
for
computing
vertical
thickness
and
true
stratigraphic
thickness
from
sets
of
well
logs.
These
particular
computations
are
of
considerable
usefulness
to
those
concerned
with
determining
reserves
in
an
area
which
is
characterized
by
wells
with
high
borehole
angles
and/or
steep
dips.
The
program
Dresser
Atlas
now
uses
takes
the
well
information
from
magnetic
tapes
produced
at
the
well
site
or
from
logs
digitized
at
the
office.
The
information
output
from
the
computer
can
be
any
one
or
all
of
the
following:
(1)
the
true
vertical
depth
plot,
(2)
the
true
vertical
thickness
plot,
which
adds
all
the
computed
vertical
thicknesses
of
the
encountered
beds
to
produce
a
log,
or
(3)
a
stratigraphic
thickness
log,
which
adds
the
normal
(or
horizontal)
bed
thicknesses
of
the
formations
encountered.
-
1
-
SPWLA
EIGHTEENTH
ANNUAL
LOGGING
SYMPOSIUM,
JUNE
5-8,
1977
THE
SOFTWARE
I.
Methods
of
Computation
1.
The
most
accurate
method
of
computing
this
type
of
log
is
the
radius
of
curvature
method.
The
borehole
angle
and
direction
are
assumed
to
take
all
the
values
between
two
assigned
points.
The
formation
dip
angle
and
direction
are
assumed
to
change
at
specific
depths.
This
type
of
computation
is
illustrated
in
figure
1.
FIGURE
1
RADIUS
OF
CURVATURE
Angle
assumes
each
value
between
A
and
B.
For
each
depth
there
is
a
discrete
value
of
angle.
D
I
DEPTH
2.
Various
Types
of
Tangential
Methods
Tangential
methods
of
computation
are
not
as
accurate
as
the
radius
of
curvature
in
most
cases.
This
is
due
to
the
method
of
picking
points.
The
tangential
method
uses
abrupt
changes
in
borehole
angle
and
direction
between
two
points.
Obviously
if
the
points
computed
are
close
together
the
tangential
method
can
be
nearly
as
accurate
as
the
radius
of
curvature.
For
longer
intervals,
the
difference
in
accuracy
between
the
two
methods
becomes
greater.
Three
methods
of
picking
points
for
tangential
computations
are
shown
in
figure
2.
The
value
of
the
TVD
presentation
is
in
areas
where
correlation
is
difficult
because
of
elongated
sections.
Using
these
elongated
sections--
without
correction
--leads
to
drastic
overestimation
of
pay
thickness
and,
therefore,
of
reserves.
II.
Options
in
Program
1.
The
curves
are
relatively
well
fixed.
We
would
normally
be
plotting
the
SP,
16
inch
normal,
resistivity
and
conductivity
curves.
However,
any
log
which
has
been
properly
digitized
can
be
processed
through
the
program.
2.
Tick
marks
can
be
made
to
show
the
number
of
feet
of
drilled
depth,
the
vertical
depth,
and
stratigraphic
(bed
thickness)
depth.
The
tick
marks
can
be
made
at
any
reasonable
interval,
and
selected
intervals
can
be
labelled
with
depths.
3.
The
log
can
be
plotted
on
any
of
the
following
parameters:
(a)
Vertical
Thickness
(b)
Stratigraphic
Thickness
(c)
True
Vertical
Depth
The
parameters
which
are
not
used
as
the
base
plot
scale
can
be
indicated
by
tick
marks.
4.
The
depth
numbers
are
written
with
the
plotter.
The
size
of
the
numbers
can
be
changed
if
required.
I
0
-
2
-
D
9
1
C
i
B
1
1
1
T
1
LOW
TANGENTIAL
All
values
between
A
and
B
are
computed
as
being
equal
to
A.
D
t
I
I
C
B
HIGH
TANGENTIAL
All
values
between
A
and
B
are
computed
as
being
equal
to
B.
SPWLA
EIGHTEENTH
ANNUAL
LOGGING
SYMPOSIUM,
JUNE
5-8,
1977
DEPTH
DEPTH
D
f
B
4
I
AVERAGE
TANGENTIAL
METHOD
All
values
between
A
and
B
are
consid-
ered
to
be
midway
between
A
and
B.
DEPTH
FIGURE
2
-
3
-
SPWLA
EIGHTEENTH
ANNUAL
LOGGING
SYMPOSIUM,
JUNE
5-8,
1977
III.
Parameters
Required
for
Computation
1.
Borehole
angle
and
direction
-
This
can
be
derived
from
a
Diploeor
from
multi-shot
information
on
a
well.
2.
Formation
dip
angle
and
direction
-
This
information
can
be
supplied
by
the
customer
from
mapped
information,
or
it
can
be
derived
from
a
Diplog
recorded
on
the
specific
well.
3.
Set
of
digitized
logs
for
the
well
in
question:
(a)
digitized
in
field
or
(b)
digitized
in
office
IV.
Relationship
of
Computed
Parameters
The
relationship
of
the
various
computed
parameters
is
shown
in
figure
3.
This
figure
shows
a
case
in
which
the
borehole
angle
and
the
formation
dip
are
in
the
same
direction.
A
=
Drilled
thickness
of
the
formation
B
=
Vertical
thickness
of
the
formation
(computed)
C
=
Stratigraphic
(true
bed)
thickness
of
the
formation
D
=
True
vertical
depth
increment
added
by
the
formation
to
a
true
vertical
depth
log
A
=
Drilled
Thickness
(Logged
depth)
B
=
True
Vertical
Thickness
C
=
True
Bed
Thickness
D
=
True
Vertical
Depth
Increment
for
this
bed
TARGET
FORMATION
B
J
FIGURE
3
-
4
-
SPWLA
EIGHTEENTH
ANNUAL
LOGGING
SYMPOSIUM,
JUNE
5-8,
1977
In
figure
4
the
same
type
of
drawing
is
shown
in
which
the
dip
of
the
formation
is
in
a
direction
nearly
opposite
to
that
of
the
borehole
inclination
direction.
A
=
Drilled
Thickness
(Logged
depth)
B
=
True
Vertical
Thickness
C
=
Stratigraphic
Thickness
D
=
True
Vertical
Depth
Increment
for
this
bed
TARGET
FORMATION
t
D
C
FIGURE
4
SPWLA
EIGHTEENTH
ANNUAL
LOGGING
SYMPOSIUM,
JUNE
5-8,
1977
LOG
EXAMPLES
OF
THE
TECHNIQUE
To
illustrate
the
magnitude
of
the
corrections
necessary
to
show
data
from
deviated
holes
more
in
relation
to
a
vertical
altitude,
consider
the
schematic
that
is
Figure
5.
The
5
lines
represent
borehole
traces
in
wells
varying
from
vertical
all
the
way
to
a
60°
deviation
from
vertical.
Each
successively
numbered
line
represents
a
15°
increasing
increment
of
borehole
deviation.
POINT
A
1
2
3
4
5
0
°
15
°
30
0
45
°
60
°
FIGURE
5
-
Schematic
of
Five
Borehole
Traces
In
Figure
6,
we
have
shown
a
TVD
of
the
S.P.
for
each
of
the
deviated
boreholes.
The
same
log
trace,
shortened
only
for
each
successive
TVD
correction,
is
used
for
illustrative
purposes.
The
3800
foot
depth
is
assumed
to
be
the
point
common
to
each
borehole
(Point
A
from
Figure
5).
These
five
examples
cover
the
normal
range
in
which
we
would
be
requested
to
run
True
Vertical
Depth
logs.
Examination
of
the
five
logs
indicates
that
the
amount
of
productive
formation
indicated
on
a
log
run
in
a
highly
deviated
well
can
be
greatly
exaggerated.
It
is
necessary
for
a
reservoir
analyst
to
have
correct
information
for
reserves
studies,
therefore,
the
True
Vertical
Depth
Log
is
a
necessity
for
adequate
evaluation
of
reserves
in
formations
cut
by
high
angle
boreholes.
The
relative
depth
of
the
various
formations
are
also
changed
by
the
True
Vertical
Depth
computation.
This
depth
change
can
affect
the
structural
maps
of
the
field
under
investigation.
Therefore,
True
Vertical
Depth
logs
are
also
necessary
for
good
structural
control
in
areas
which
must
be
developed
using
highly
deviated
wells.
-
6
-
SPWLA
EIGHTEENTH
ANNUAL
LOGGING
SYMPOSIUM,
JUNE
5-8,
1977
VERTICAL
15
°
DEVIATION
30
°
DEVIATION
-
45
°
DEVIATION
2
3
60
°
DEVIATION
1
t
o
1
1
1
1
U.
FIGURE
6
-
S.P.'s
Shortened
by
the
TVD
Calculation
by
Successive
15
°
Borehole
Deviation
Increment
Increases.
-
7
-
SPWLA
EIGHTEENTH
ANNUAL
LOGGING
SYMPOSIUM,
JUNE
5-8,
1977
Figure
7
is
a
section
of
log
on
which
all
three
computations,
True
Vertical
Depth,
True
Vertical
Thickness
and
True
Stratigraphic
Thickness,
have
been
made.
The
depth
column
and
the
standard
5"
presentation
feature
the
True
Vertical
Thickness
computation,
which
most
engineers
consider
the
preferred
solution.
True
Stratigraphic
(or
normal)
Thickness
is
indicated
by
the
10'
incremental
ticks
and
hundred
foot
depth
numbers
on
the
right
hand
edge
of
the
log.
True
Vertical
Depth
is
indicated
by
similar
ticks
and
depth
numbers
at
the
left
hand
edge
of
the
depth
track,
and
the
original
measured
depths
are
similarly
shown
at
the
right
hand
edge
of
the
depth
track.
This
represents
a
large
amount
of
data
and
the
various
solutions
can
be
optionally
shown
as
individual
logs.
Table
I
summarizes
the
borehole
deviation
and
formation
dip
data
used
for
this
solution
along
with
thickness
calculations
for
the
gross
intervals
between
points
of
borehole
deviation
and/or
formation
dip
change.
Cumulative
T.V.T.
and
T.V.D.
numbers
are
also
shown.
TABLE
I
TRUE
VERTICAL
THICKNESS
PROGRAM
MEASURED
DEPTH
TRUE
VERT.
DEPTH
TRUE
VERT.
THICKNESS
TRUE
BED
THICKNESS
BOREHOLE
ANGLE
BOREHOLE
DIRECTION
DIP
ANGLE
DIP
DIRECTION
2064.0
2041.42
1981.01
1942.21
16.0
234.0
22.0
250
2249.0
2219.34
2110.72
2033.93
16.0
231.0
45.0
250
2405.0
2369.68
2220.93
2111.97
15.5
234.0
45.0
250
2613.0
2571.47
2375.55
2221.26
14.0
230.0
45.0
250
2737.0
2692.82
2473.67
2290.65
11.8
227.0
45.0
250
2832.0
2786.13
2549.78
2344.43
11.0
231.0
45.0
250
2972.0
2924.32
2667.34
2427.62
9.3
226.0
45.0
250
3134.0
3083.91
2772.55
2472.00
9.8
228.0
65.0
250
3260.0
3207.60
2847.71
2503.81
11.0
230.0
65.0
250
3352.0
3297.84
2900.10
2526.06
11.3
239.0
65.0
250
3478.0
3420.70
2965.40
2553.64
12.8
245.0
65.0
260
3590.0
3530.23
3023.92
2578.38
12.3
244.0
65.0
244
3892.0
3824.72
3233.28
2707.17
12.8
250.0
52.0
250
4015.0
3943.99
3315.09
2758.63
14.3
253.0
51.0
253
4200.0
4121.87
3432.17
2833.82
16.0
253.0
50.0
253
4383.0
4295.83
3538.76
2902.41
18.0
252.0
50.0
252
4494.0
4400.52
3599.20
2941.31
19.5
252.0
50.0
252
4586.0
4485.66
3643.46
2969.80
22.3
252.0
50.0
242
4804.0
4676.37
3710.17
3012.68
29.0
252.0
50.0
242
4984.0
4832.96
3773.14
3055.67
29.5
255.0
47.0
245
5141.0
4964.63
3814.52
3083.86
33.0
256.0
47.0
246
5265.0
5066.22
3841.03
3101.89
35.0
258.0
47.0
248
5416.0
5189.11
3871.31
3122.53
35.5
257.0
47.0
247
5592.0
5330.64
3911.02
3149.67
36.5
257.0
47.0
232
5780.0
5473.51
3935.34
3166.26
40.5
258.0
47.0
233
5933.0
5589.93
3955.16
3179.72
40.5
260.0
47.0
235
6121.0
5735.07
3983.93
3199.40
39.5
259.0
47.0
234
6308.0
5880.36
4014.97
3220.51
39.0
260.0
47.0
235
6490.0
6021.77
4045.08
3141.02
39.0
260.0
47.0
235
Please
note
that
on
this
well
both
the
borehole
and
the
formation
dips
are
generally
trending
in
the
same
direction,
and
that
both
reach
relatively
large
values.
In
this
extreme
case,
the
apparent
elongation
of
a
section
can
be
quite
drastic.
This
is
typified
by
Figure
8
from
the
same
borehole.
We
have
shown
a
pay
section
near
the
bottom
of
the
hole
where
these
effects
are
most
pronounced.
The
original
logged
IEL
data
is
shown
to
the
left
of
the
figure
and
the
accompanying
T.V.T.
presentation
on
the
right.
The
shortening
of
the
pay
section
on
the
T.V.T.
most
dramatically
illustrates
what
has
been
previously
discussed.
-
8
-
SP
WLA
EIGHT
EEN
TH
ANNU
AL
L
OGGIN
G
SYM
POS
IU
M,
JUN
E
5-
8,
1977
TRUE
STRATIG
APHIC1
1
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,._
E
.
%
0
VI
Tr
I
8
8
MEASURED
Tr
DEPTH
,
0
I
Tr
_
I.
_
NI
Tr
Tr
1.---.......
I I I
TRUE
VERTICAL
THICKNESS
;
3400
I I I
3500
I
I I
I I
J
J
I
I
I
L
I
I
I,
I
I
I
J
L
I
„I
I
I
J
L.
I
J
I
I,
TRUE
VERTICAL
DEPTH
0
0
SPWLA
EIGHTEENTH
ANNUAL
LOGGING
SYMPOSIUM,
JUNE
5-8,
1977
0
0
0
LU
3800
3900
In
00
Cl
0
8
N
00
LU
8
LU
00
0
LC;
v
••
5700 5800
5900
41'
FIGURE
8
-
10
-
SPWLA
EIGHTEENTH
ANNUAL
LOGGING
SYMPOSIUM,
JUNE
5-8,
1977
CONCLUSIONS
Steeply
dipping
formations
and/or
those
drilled
in
highly
deviated
boreholes
present
a
highly
distorted
picture
of
a
reservoir,
and
can
cause
significant
mapping
anomalies
if
not
understood
and
recognized.
The
suite
of
depth
correction
and
log
replotting
programs
presented
in
this
paper
provide
the
necessary
tools
for
easily
identifying
and
solving
all
the
problems
attendant
to
these
situations.
O.R.
Holt
P.A.
Wichmann
BIOGRAPHIES
OLIN
R.
HOLT
is
presently
Manager
of
the
Dresser
Atlas
Log
Analysis
Center
in
Houston.
He
has
held
this
position
since
January
1,
1977.
Before
the
present
assignment
Mr.
Holt
was
Chief
Log
Analyst
in
charge
of
Diplog
Processing
and
Interpretation
in
the
Log
Analysis
section.
Prior
to
the
present
position
Mr.
Holt
has
been
a
geologist
for
the
Indiana
Geological
Survey,
a
logging
engineer
and
Log
Analyst
for
the
Birdwell
Division
of
Seismograph
Service
Corporation,
and
a
geologist
and
well
log
analyst
for
Sinclair.
He
joined
PGAC
as
a
sales
engineer
in
1965
and
was
named
Regional
Log
Analyst
for
the
California
region
in
1967.
He
is
a
member
of
The
SPWLA,
AAPG,
and
AIME.
LARRY
G.
SCHOONOVER
joined
Dresser
Atlas
in
1970
and
is
currently
Manager
of
Informa-
tion
Services.
His
duties
include
supervision
of
computer
programming,
computer
maintenance,
and
some
types
of
processing
for
the
data
centers.
Mr.
Schoonover
recieved
a
MS
degree
in
Mathematics
from
the
University
of
Houston
in
1972.
He
is
a
member
of
the
SPWLA
and
has
written
and
presented
several
papers
to
the
SPWLA,
AIME,
and
other
technical
societies.
P.
A.
WICHMANN
is
the
Director
of
Account
Management
for
Dresser
Atlas
in
Houston.
A
1958
honors
graduate
in
Petroleum
Engineering
from
the
Colorado
School
of
Mines,
he
worked
for
the
Shell
Oil
Company
from
that
date
until
1965
in
a
variety
of
positions,
including
Petrophysical
Engineer.
At
that
time
he
went
to
work
for
Dresser
as
a
Research
Log
Analyst
for
Lane-Wells.
In
1968
he
became
Chief
Log
Analyst
for
Dresser
Atlas,
was
promoted
to
Manager
of
Log
Analysis
in
1975,
and
assumed
his
present
duties
in
January,
1977.
Mr.
Wichmann
has
been
active
in
the
SPWLA
for
a
number
of
years
and
has
been
on
the
National
Board
of
Directors.
Paul
is
the
current
Executive
Secretary
of
the
National
and
President
of
the
local
Houston
Chapter.
He
has
authored
or
co-authored
about
35
technical
papers,
including
16
that
have
appeared
in
SPWLA
publications.
He
has
also
spoken
at
numerous
international
and
local
SPWLA
and
other
technical
society
meetings.
Resides
the
SPWLA,
he
is
a
member
of
the
CW
LS,
the
SPE
of
AIME,
Tau
Beta
Pi,
and
is
a
Registered
Engineer
in
the
State
of
Texas.
SPWLA
EIGHTEENTH
ANNUAL
LOGGING
SYMPOSIUM,
JUNE
5-8,
1977
True
Bed
Thickness
VERTICAL
NORMAL
ATBT
DIP
ANGLE
ATVT
ATBT
=
cos
(0
)
.
ATVT
where,
0
is
the
formation
dip
angle
ATVT
is
the
change
in
true
vertical
thickness
ATBT
is
the
change
in
true
bed
thickness.
True
Vertical
Thickness
For
the
derivation
of
the
TVT
equations
we
will
need
the
following
formulae.
Consider
a
plane
which
passes
through
the
point
(0,
0,
0).
It
will
have
the
general
form
A.X+B•Y=Z.
And
suppose
further
that
the
formation
dip
angle
of
interest
is
0
and
the
dip
direction
is
W.
Let
A=
sin
hi
tan
0
B
=
cos
4/
tan
0
Then
the
dip
direction
of
the
plane
AX
+
BY
=
Z
will
be
tan
-1
(A/B)
=
tan-1
(
sin
'Y
tan
0
)
cos
NI/
tan
0
=
hi
SPWLA
EIGHTEENTH
ANNUAL
LOGGING
SYMPOSIUM,
JUNE
5-8,
1977
The
dip
angle
of
the
plane
will
be
arc
tangent
of
the
z
coordinate
at
the
point
on
the
unit
circle
in
the
dip
direction.
This
point
is
(sing',
cos'').
Thus,
the
dip
angle
is
tan
-1
(A
.
sin*
+
B
.
cos+)
=
tan
-1
(sin
2
lif
.
tan()
+
cos
2
'I'
.
tan0)
=
tan
-1
((sin
2
4f
+
cos
2
4')
tan())
=
tan
-1
(tan
0)
=
0
With
this
formulation,
we
now
have
a
representation
of
our
geological
formation
as
a
plane.
We
will
now
consider
the
dip
of
a
line
in
this
plane
that
is
not
in
the
direction
of
dip.
Let
this
line
be
in
direction
a.
The
dip
angle
5
of
the
line
R
would
be
=
tan
-1
(A
.
sina
+
B
cosa)
=
tan
-1
(sin*
sina
tan0
+
cos*
cosa
tan0)
=
tan
-1
((sin*
sina
+
cos*
cosa)
tan0)
=
tan
-1
(cos
(gI'—a)
tan())
.
(Note:
an
approximation
of
this
which
is
useful
on
small
calculators
is
=
cos
('lf—a)
.
0.
This
is
reasonably
accurate
for
0
<
20°
).
-
14
-
SPWLA
EIGHTEENTH
ANNUAL
LOGGING
SYMPOSIUM,
JUNE
5-8,
1977
Now
we
will
consider
the
case
of
a
borehole
intersecting
a
dipping
bed.
Let
klf
0
a
,
a
X
AD
be
the
formation
dip
direction,
be
the
formation
dip
angle,
be
the
borehole
direction,
be
=
a'
+
180°,
be
the
borehole
drift
angle,
is
the
change
in
depth,
and
A
TVT
is
the
change
in
true
vertical
thickness.
VERTICAL
BOREHOLE
VERTICAL
AD
X
0
STRIKE
90-a
DIP
a
is
given
by
the
formula
we
have
just
derived
a
=
tan
-1
(cos
(T—a)
tan0).
3,
the
angle
that
we
are
interested
in
would
be
given
by
13
=
90
+
a
X
,
and
'Y=
180
0
—a+X—X
=
180
90
—a+
X
X
=
90
a
N
A
TVT
SPWLA
EIGHTEENTH
ANNUAL
LOGGING
SYMPOSIUM,
JUNE
5-8,
1977
By
the
"law
of
sines"
ATVT
AD
sing
sirry
A
D
sing
A
TVT
=
sing
sin
(g)
=
sin
(90
+
a
X)
=
cos
(
a
X)
=
cos
(tan
-1
(cos
(xli
a)
tang)
X)
sin
(5)
=
sin
(90
a
)
=
COS
(
a
)
=
cos
(tan
-1
(cos
(1If
a)
tang)
A
TVT
cos
(tan
-1
(cos
(kV
a)
tans)
Examples
of
solutions:
Case
1
%If
=
270°
0
=
10°
a'
=
90°
a
=
270°
X
=
10°
A
D
=
100
°
BOREHOLE
E
A
TVD
=
100
.
cos
(10)
=
98.5
AD
0
ATVT
or
SO
A
D
cos
(tan
-1
(cos
OP
a)
tans)
X)
-
16
-
SPWLA
EIGHTEENTH
ANNUAL
LOGGING
SYMPOSIUM,
JUNE
5-8,
1977
A
TVT
=
100
.
cos
(tan
-1
(cos
(270
270)
tan
(10)
10)
cos
(tan
-1
(cos
(270
270)
tan
(10))
100
.
cos
(0)
100
=
101.5
.
cos
(10)
.985
A
TBT
=
cos
(10)
TVT
=
.095
.
101.5
=
100
.
Case
2
BOREHOLE
=
0
0
ATVT
AD
a'
=
90
a
=
270
=
10
A
D
=
100
A
TVD
=
100
.
cos
(10)
=
98.5
A
TVT
cos
(tan
-1
(cos
(0
270)
tan
(0)))
100
.
cos
(—
10)
100
.
98.5
=
cos
(0)
1
98.5
100
cos
(tan
-1
(cos
(0
270)
tan
(0))
10)
A
TBT
=
98.5
.
cos
(0)
=
98.5
SPWLA
EIGHTEENTH
ANNUAL
LOGGING
SYMPOSIUM,
JUNE
5-8,
1977
Case
3
=
90
0
=
10
a'
=
90
a
=
270
X
=
10
AD
=
100
ATVT
ATVD
=
100
cos
(10)
=
98.5
100
cos
(tan
-1
(cos
(90
270)
tan
(10))
10)
ATVT
cos
(tan
-1
(cos
(90
270)
tan
(10)))
100
cos
(—
10
10)
cos
(—
10)
100
cos
(20)
.939
=
100
(
.985
)
=
95.4
A
TBT
=
cos
(10)
.
95.4
=
93.9
Case
4
w
B
0
R
E
H
E
‘If
=
90
=
20
a'
=
90
X
=
70
AD
=
100
a
=
270
BOREHOLE
w
ATBT
AD
cos
(10)
-
18
-
SPWLA
EIGHTEENTH
ANNUAL
LOGGING
SYMPOSIUM,
JUNE
5-8,
1977
A
TVD
=
100
.
cos
(70)
=
34.2
ATVT
100
.
cos
(tan
-1
(cos
(90
270)
tan
(20))
70)
cos
(tan
-1
(cos
(90
270)
tan
(20)))
100
cos
(-20
70)
cos
(90)
=
100
cos
(-20)
cos
(20)
ATVT
is
actually
undetermined.
In
this
case,
the
triangle
used
in
the
derivation
collapses
to
a
single
line.
A
TBT
=
cos
(20)
0
=
0
A
TBT
is
also
undetermined.