The log analyst and the programmable pocket calculator; Part IV, Finding true stratigraphic thickness and true vertical thickness of dipping beds cut by directional wells


Bateman, R.M.; Konen, C.E.

The Log Analyst 20(2): 3-6

1979


THE
LOG
ANALYST
AND
THE
PROGRAMMABLE
POCKET
CALCULATOR
By
R.
M.
BATEMAN
and
C.
E.
KONEN
Amoco
International
Oil
Company
Part
VI
Finding
True
Stratigraphic
Thickness
and
True
Vertical
Thickness
of
Dipping
Beds
Cut
by
Directional
Wells
ABSTRACT
Offshore
exploration
and
development
requires
multiple
deviated
wells
to
be
drilled
from
fixed
platforms.
As
a
result
the
thickness
of
a
formation
as
measured
from
a
log
made
in
a
deviated
hole
reflects
neither
its
true
strati-
graphic
thickness
nor
its
true
vertical
thickness.
The
geologist
who
tries
to
trace
a
given
bed
across
the
field
finds
it
difficult
to
deduce
whether
variations
in
bed
thickness
measured
from
logs
are
due
to
legitimate
bed
thickness
variations
or
whether
they
are
geometrical
distortions
due
to
hole
devia-
tion
and
bed
dip
effects.
Likewise,
the
engineer
trying
to
compute
reserves
is
faced
with
the
same
type
of
problem.
Measured
bed
thickness
can
be
less
than
true
vertical
bed
thickness
in
some
cases
and
greater
in
other
cases.
A
need
for
a
quick
and
simple
solution
to
the
problem
is
evident,
and
a
method
of
using
a
programmable
pocket
calculator
'
(PPC)
will
be
presented
to
satisfy
the
demands
of
both
the
explorationist
and
the
engineer.
QUANTIFYING
THE
PROBLEM
TST
=
MT
Cos
a
p
(1)
Vertical
Well
Bore
Figure
1
illustrates
the
case
of
well
deviated
from
the
vertical
crossing
a
horizontal
bed
of
measured
thickness
MT.
If
the
well
enters
the
bed
at
A
and
leaves
the
bed
at
B,
the
measured
thickness
is
AB
and
the
True
Stratigraphic
Thick-
ness
(TST)
is
related
to
the
measured
thickness
by:
Dipping
Bed
Horizontal
Bed
TST
B
FIGURE
1
THE
LOG
ANALYST
3
FIGURE
2
Figure
2
illustrates
the
case
of
a
straight
hole
crossing
a
bed
of
measured
thickness
MT
dipping
from
the
hori-
zontal.
If
the
hole
enters
the
bed
at
A
and
leaves
the
bed
at
B,
then
the
measured
thickness
is
AB,
and
the
True
Strati-
graphic
Thickness
is
given
by:
a
Deviated
Well
Bore
Note
that
Equations
3
and
4
only
hold
for
the
case
where
bed
dip
azimuth
and
well
deviation
azimuth
are
exactly
the
same.
In
the
general
case
the
well
azimuth
is
different
from
the
bed
azimuth,
and
the
simple
relationships
deduced
for
the
unique
case
illustrated
in
Figure
3
do
not
hold.
a
Deviated
Well
Bore
SOLUTION
TO
THE
PROBLEM
Dipping
Bed
aT
FIGURE
3
TST
=
MT
Cos
)
3
(2)
Note
that
where
the
well
is
vertical
the
azimuth
of
the
dipping
bed
is
immaterial
and
that
where
the
bed
is
hori-
zontal,
the
well
azimuth
is
likewise
not
required
for
calculating
TST.
Figure
3
illustrates
the
case
of
a
well
deviated
from
vertical
crossing
a
bed
dipping
)3°
from
the
horizontal.
Note
that
both
the
bed
azimuth
and
the
well
azimuth
are
the
same
for
this
particular
illustration.
Again,
the
measured
thickness
is
AB.
It
is
of
interest
to
note
that
the
true
vertical
depth
difference
between
A
and
B
is
AC
where
AC
=
AB
Cos
a.
However,
the
engineer
is
interested
in
the
True
Vertical
Thickness
(TVT)
given
by
AE,
and
the
geologist
is
seeking
the
True
Stratigraphic
Thickness
given
by
BD.
For
the
case
where
both
well
and
bed
dip
are
oriented
in
the
same
azimuth
it
is
a
trivial
exercise
to
deduce
TVT
(distance
AE
on
Figure
3)
and
TST
(distance
BD
on
Figure
3).
TST
=
MT
Cos
(a
+
13)
(
3
)
TVT
=
TST
=
MT
Cos
(a
+
/3)
(4)
Cos
)3
Cos
)3
The
practicing
log
analyst
who
has
had
to
deal
with
dipmeter
related
problems
will
appreciate
the
similarity
between
structural
dip
subtraction
and
the
solution
devised
by
the
present
authors.
In
a
previous
publication
)
a
method
of
using
a
PPC
for
structural
dip
subtraction
was
described.
The
basic
idea
is
to
consider
the
well
as
one
dip
vector
and
the
bed
as
another
and
perform
a
"dip
rotation."
In
this
method
the
bed
is
brought
back
to
the
horizontal
and
the
well
deviation
and
azimuth
assume
new
values,
in
the
geographical
frame
of
reference,
while
maintaining
the
same
orientation
relative
to
the
bed
itself.
The
reader
may
better
understand
the
principle
by
imagining
a
pencil
(representing
the
well)
stuck
through
a
slab
of
cheese
(representing
the
dipping
bed).
If
the
cheese
is
initially
dipping
to
north
and
the
pencil
transverses
the
cheese
deviated
to
the
west,
then
the
result
of
placing
the
cheese
on
a
horizontal
surface
will
be
to
move
the
azimuth
of
the
pencil
from
west
to
somewhere
north
of
west
and,
at
the
same
time,
to
increase
the
deviation
of
the
pencil
from
vertical.
An
alternative
approach
would
be
to
allow
the
pencil
to
rotate
until
it
was
vertical
in
which
case
the
cheese
slab
would
dip
at
an
increased
angle
of
dip
to
some
point
west
of
north.
Either
of
these
rotations
can
be
made
by
treating
it
as
a
structural
dip
subtraction.
The
manipulation
can
be
made
graphically
by
using
a
stereo
net
with
a
rotating
overlay,
or
it
can
be
made
mathematically
using
the
Cosine
Law
of
Spherical
Triangles.
Once
the
bed
is
rotated
back
to
horizontal
(or
the
hole
back
to
vertical)
then
the
simple
relationships
illustrated
in
Figures
1
and
2
can
be
used.
Since
the
new
azimuth
of
the
bed
or
the
hole
are
not
required,
the
only
parameter
needed
is
the
new
bed
dip
or
the
new
hole
deviation,
depending
on
which
approach
is
used.
For
the
purposes
of
illustration,
we
will
rotate
the
bed
back
to
horizontal.
If,
a °
=
well
deviation
from
vertical,
=
bed
dip
from
horizontal,
HAZ°
=
hole
azimuth,
DAZ°
=
dip
azimuth,
and
MT
=
measured
bed
thickness,
then
the
well
deviation
from
vertical
after
the
bed
has
been
rotated
to
horizontal
is
given
by
a'
where:
Cos
a'
=
Cos
a
Cos
/3
Sin
a
Sin
/3
4
MARCH-APRIL,
1979
MT
FLAGS
TRIG
DMA
'
Bottom
E
Trim.
C
Well
Dev.
Welt
Az.
°
R:g
RIP
A
mp
TST
2
Top
Bottom
Well
Dev.
Well
Az
pip
Az.
St
,
TVD
Diff.
'EST
Sn
so
S3
5
Dip
S7
SO
55
52
3
FIX
60
SCI
0
ENG
0
DEG
M
GRAD
0
RAD
0
ON
OFF
o
11
0
0 0
2
0 0
3
0 0
Cos
(HAZ
DAZ)
(
5
)
INPUT
Top,
Bottom
Well
Deviation
(a),
Hole
Azimuth
(HAZ)
Bed
Dip
(01,
Dip
Azimuth
(DAZ)
If
the
bed
is
now
horizontal
then
the
true
stratigraphic
bed
thickness
is
given
by:
TST
=
MT
Cos
a'
(6)
or
TST
=
MT
[Cosa
Cos
fl
Sin
a
Sin
/3
Cos
(HAZ
DAZ)]
(
7
)
and
the
true
vertical
thickness
of
the
bed
is
given
by:
TVT
=
TST
(8)
Cos
/3
or
TVT
=
MT
[Cos
a
Sin
a
Tan
Cos
(HAZ
DAZ)]
(
9
)
A
simple
PPC
program
to
handle
this
computation
is
illustrated
in
the
accompanying
flowchart
and
program
step
listing
suitable
for
an
HP-97
calculator.
MT
=
Bottom—Top
TVD
Diff
=
MT
Cos
Ot
TST
=
MT
(Cosa.
Cos
—Sin
a•
Sin
p
cos
(HAZ-DAZ))
TVT
=
TST/Cos
(
Display
Results
FLOW
DIAGRAM
The
program
is
arranged
so
that
the
operator
enters
6
items.
1.
The
measured
top
of
the
formation
2.
The
measured
bottom
of
the
formation
3.
Well
deviation
from
vertical
4.
Well
azimuth
5.
Bed
dip
6.
Bed
azimuth
Program
Haling
STEP
KEY
ENTRY
KEY
CODE
STEP
KEY
ENTRY
KEY
CODE
pel
*LOLA
21
11
802
0701
35
01
003
RT.
24
084
.1818
21
12
805
ST02
35
82
806
RT.
24
807
*UGC
21
13
888
141T
-41
809
3T03
35
83
011
01
-31
811
ST04
35
84
812
12114
24
813
44810
21
14
814
x:t
-41
015
3T05
35
85
0'6
01
-31
81
-
5106
35
06
818
RTH
24
019
4161E
21
15
228
RCL2
36
02
02'
RCL1
36
81
122
-
-45
821
STOO
35
08
024
RCL3
36
03
25
COS
42
26
4
-35
27
0T07
35
87
28
1CL3
36
83
29
COS
42
38
RCL5
36
05
31
COS
42
32
K
-35
33
RCL4
36
84
34
RCL6
36
86
35
-
-45
36
COS
42
37
8013
36
83
38
SIR
41
39
x
-35
48
RCL5
36
85
41
SIN
41
42
x
-35
43
-
-45
44
RCLO
36
88
45
x
-35
46
STO8
35
08
47
XCL5
36
85
46
COS
42
45
4
-24
OSO
54
ST09
35
89
51
SIC
16-11
52
RCL1
36
81
53
POI
-14
54
033.2
36
02
55
POTS
-14
56
SELO
36
84
COMMENTS
Top
Bottom
Hole
Deviation
Mole
Azimuth
Bed
Dip
Dip
Azimuth
STEP
KEY
ENTRY
KEY
CODE
COMMENTS
857
FCC?
36
07
056
RCL1
36
86
055
RCL9
36
89
060
FRST
16
-
14
861
17:11
24
062
64
51
COMMENTS
STEP
KEY
ENTRY
KEY
CODE
170
100
190
200
210
220
080
100
110
Display
Results
120
1
00
Measured
Thickness
TVD
Difference
TST
TVT
COMMENTS
REGISTERS
LABELS
FLAGS
SET
STATUS
THE
THE
LOG
ANALYST
S
True
Stratigraphic
Thickness
and
True
Vertical
Thickness
R.
M.
Bateman
Date
9/5/78
Amoco
International
Oil
Company
Chicago
sww
Illinois
]gyp
Code
60601
Program
Title
Nemo
Address
Coy
PmgramDescdpeogEgueflone,Vonegles,eM.
The
program
solves
the
problem
of
finding
the
True
StraLlgraphic
Thickness
and
the
True
Vertical
Thickness
of
a
formation
from
a
knowledge
of
the
well
deviation
and
azimuth,
the
bed
dip
and
azimuth
and
the
measured
bed
thickness.
If
well
is
deviated
a
°
from
vertical
in
direction
HAZ
and
bed
dips
0
°
from
horizontal
In
direction
DAZ
then
if
well
enters
bed
at
A
and
exits
at
a
the
following
relations
hold
Measured
Thickness
(MT)
-
AB
True
Vertical
Depth
Difference
between
A
and
B
-
MT
Cosa
True
Stratigraphic
Thickness
MT(Cos
a
Cos
$
-
Sin
a
Sint)
Cos
ULU
-
au>)
True
Vertical
Thickness
.
TST/Cos
0
Operating
Limns
and
Mulling*
Assumptions
made
are
that
the
bed
is
of
uniform
thickness
with
upper
and
lower
boundaries
parallel.
If
(04
)
is
greater
than
90
°
the
well
enters
the
bed
from
the
underside.
In
that
case
answers
for
TST
and
TVT
are
given
as
negative
numbers.
Program
111eseriplitm
_
DO
NOT
USE
THIS
SPACE
The
computed
results
will
display:
a.
The
measured
thickness
b.
The
True
Vertical
Depth
Difference
between
the
top
and
the
bottom
of
the
bed
c.
The
True
Stratigraphic
Thickness
d.
The
True
Vertical
Thickness
A
typical
case
is
illustrated
for
the
user
to
check
his
transposition
of
the
algorithms
to
his
particular
model
of
PPC.
User
Insiruelions
I
Wneliatiin
]Bed
Well
Bed
Azimuth Azimuth
+
STEP
aystaucnous
INPUT
DATA,UNITS
KEYS
OUTPUT
OATITUNITS
I
2
3
A
.
5
6
7
Measured
depth
of
formation
top
Measured
depth
of
formation
bottom
Well
deviation
from
vertical.
Well
azimuth
in
degrees
from
North
Feet
Feet
°
A
I
B
I
entail
enteil
D
E
A
B
entee
C
E
C
I
entee.
D
I I
Bed
dip,
from
horizontal
°
Top
Bed
in
deszoee
from
Worth
_azimuth
°
Hit
E
for_printer
diepley,
(NN-97)
or
sequential
d415).ey
(HV-67)
Bottom
MT
TVD
Diff
TST
TVT
Worked
Example
1
2
3
4
5
6
7
Well
Deviation
30
.
to
128
0
Azimuth
Bed
Dip
25
Co
45
Azimuth
Top
of
bed
5642'
Bottom
of
bed
5878'
5642
5878
30
128
2
_
25
45
Hit
E
5642.00
5878.00
236.00
204.58
179.16
197.68
OPERATING
LIMITS
It
is
assumed
that
the
true
stratigraphic
thickness
is
the
same
at
the
point
the
well
enters
the
bed
as
it
is
when
the
well
leaves
the
bed.
It
is
also
assumed
that
the
well
deviation
and
azimuth
do
not
change
as
the
well
crosses
the
bed.
If
the
bed
is
highly
dipping,
it
is
possible
that
the
well
will
in
fact
enter
the
bed
from
the
"bottom"
side
and
exit
from
the
"top"
side.
If
this
occurs,
the
answers
for
TST
and
TVT
will
appear
as
negative
numbers.
/Top
I
Bottom
Bottom
MT
Diff
TOT
TVT
CONCLUSION
WORKED
EXAMPLE
Well
is
deviated
30°
to
128°
azimuth.
Bed
is
dipping
25°
to
45°
azimuth.
Top
of
the
bed
is
measured
on
the
log
at
5,642
feet.
Bottom
is
found
at
5,878.
The
PPC
once
again
provides
a
simple
and
practical
solution
to
a
complex
problem
which
to
date
has
only
been
soluble
via
some
rather
questionable
nomograms.
While
the
program
discussed
is
illustrated
by
use
of
an
HP
calculator,
the
authors
find
no
reason
why
other
similar
makes
and
models
of
PPC
should
not
be
used
provided
the
trigonometric
functions
behave
in
the
same
way
that
the
HP
functions
do
for
limiting
cases
such
as
negative
angles.
The
computed
results
show:
REFERENCES
Measured
thickness
=
236.00'
1.
Bateman,
R.
M.,
and
Konen,
C.
E.:
"The
Log
Analyst
and
the
TVD
difference
=
204.38'
Programmable
Pocket
Calculator
Part
III
Dipmeter
TST
=
179.16'
Computation",
The
Log
Analyst,
Volume
XIX,
No.
1,
January-
TVT
=
197.68'
February,
1978.
6
MARCH-APRIL,
1979