Use of the curvature method to determine true vertical reservoir thickness


Rivero, R.T.

Journal of Petroleum Technology 23(4): 491-496

1971


A generalized curvature method of directional survey interpretation can be shown to converge into the Tangential method for curvatures close to zero. True vertical thickness corrections may be made based on directional surveys, taking into account the slant of the hole and the dip of the formation.

Use
of
the
Curvature
Method
To
Determine
True
Vertical
Reservoir
Thickness
R.
T.
Rivero,
SPE-AIME,
Atlantic
Richfield
Co.
Introduction
Net
pay
is
one
of
the
most
important
factors
in
deter-
mining
equity
participation
formulas.
In
the
past,
it
was
picked
directly
from
the
logs
of
reasonably
ver-
tical
holes
with
relatively
negligible
error.
In
direc-
tionally
drilled
wells,
net
pay
must
be
selected
by
cor-
recting
the
observed
log
thickness
of
a
zone
to
the
value
that
would
have
been
logged
if
the
well
had
penetrated
the
zone
vertically
through
the
point
it
pierced the
top
of
the
bed.
These
corrections
must
take
into
account
the
borehole
inclination
as
well
as
the
dip
and
strike
of
the
formation.
They
may
be
made
using
the
true
coordinates
of
the
points
at
which
the
top
and
bottom
of
the
pay
zone
are
penetrated
by
the
well.
These
coordinates
can
be
obtained
by
the
Tangential
(straight
line)
method
of
directional
survey
interpretation,
which
in
certain
cases
has
doubtful
ac-
curacy.
In
those
instances
when
the
segments
between
survey
stations
are
curved,
the
Tangential
method
will
cause
errors
that,
because
of
their
cumulative
effects,
may
become
substantial.
To
correct
for
the
effects
of
these
errors,
Wilsonl
presented
an
improved
method
of
directional
survey
computation,
which
he
called
the
"Radius
of
Curva-
ture".
He
derived
the
necessary
equations
for
those
cases
in
which
the
curvature
of
the
wellbore
was
either
on
a
vertical
or
on
a
horizontal
plane,
pointing
out
that
the
Tangential
method
was
materially
a
different
interpretive
technique.
Wilson's
solutions
may
be
generalized.
A
general
set
of
equations
of
the
Curvature
method
of
direc-
tional
survey
interpretation
can
be
shown
to
converge
into
the
equations
of
the
Tangential
method
when
the
borehole
curvature
either
is
zero
or
approaches
zero.
The
purpose
of
this
paper
is
to
derive
an
analytical
expression
for
true
vertical
formation
thickness
when
a
bed
of
known
dip
and
strike
is
pierced
by
a
slanted
well.
Since
the
accuracy
of
these
calculations
is
de-
pendent
upon
the
true
location
of
points
on
the
well-
bore,
the
Curvature
method
affords
at
this
time
a
computational
technique
that
is
superior
to
those
ex-
isting
heretofore.
The
wide
availability
of
computers
makes
the
application
of
the
equations
an
easy
task.
Directional
Survey
Interpretation
For
every
survey
station,
three
items
of
information
are
recorded
by
companies
engaged
in
running
direc-
tional
surveys:
(1)
the
depth
at
which
the
instrument
was
stopped;
(2)
the
drift
(inclination)
of
its
axis
off
the
vertical;
and
(3)
the
drift
direction
on
a
horizontal
plane.
Two
sources
of
possible
error
are
immediately
ap-
parent.
First,
the
tool
may
not
have
been
centralized;
that
is,
its
axis
may
not
have
coincided
with
that
of
the
wellbore.
Second,
the
measured
depth
may
have
been
in
error
because
of
cable
stretch
or
because
the
cable
was
not
centered
in
the
wellbore.
Other
sources
of
error
inherent
in
particular
recording
instruments
may
also
become
apparent.
However,
even
if
these
sources
of
error
are
ignored,
it
is
possible
to
introduce
a
computational
error
during
the
interpretation
of
the
survey,
if
it
is
assumed
that
survey
points
are
con-
nected
by
straight
lines.
The
accuracy
of
this
method
A
generalized
curvature
method
of
directional
survey
interpretation
can
be
shown
to
converge
into
the
Tangential
method
for
curvatures
close
to
zero.
True
vertical
thickness
corrections
may
be
made
based
on
directional
surveys,
taking
into
account
the
slant
of
the
hole
and
the
dip
of
the
formation.
APRIL,
1971
491
increases
as
the
spacing
between
survey
points
de-
creases,
but
very
close
spacing
may
not
be
econom-
ically
feasible.
To
improve
the
accuracy
of
computation,
Wilson
5
introduced
the
Radius
of
Curvature
method
of
inter-
pretation,
basing
it
on
the
assumption
that
the
well-
bore
has
a
constant
curvature
either
on
a
horizontal
or
on
a
vertical
plane.
A
measure
of
the
curvature
is
obtained
from
the
change
in
drift
angle
and
direction
from
one
survey
station
to
the
next.
Wilson's
analysis
may
be
expanded
into
sets
of
solutions
that
cover
those
instances
when
the
curvature
of
the
wellbore
is
not
on
a
vertical
plane.
From
the
observed
drift
angles
(0)
and
their
directions
(0)
at
two
consecutive
survey
points,
a
and
b,
four
possible
cases
may
be
envisioned:
Case
1:
(4.b
Oa)
0
0;
(eb
0
Case
2:
(sbb
Oa)
0
0;
(Ob
0
a
)
=
0
Case
3:
(cpb
Oa)
=
0;
(Bb
ea)
0
Case
4:
(fkb
Oa)
=
0;
(0
b
Oa)
=
0
The
derivation
of
the
solutions
for
each
of
these
cases
is
presented
in
Appendix
A.
The
solutions
for
each
of
the
four
cases
in
terms
of
incremental
dis-
tances
in
the
x,
y
and
z
directions
are
as
follows:
Cases
I
and
2.
(kb
0
Oa
Lb
La
(sin
cpb
sin
Oa)
(1)
Zb
Za
ekb
Case
1
8b0Oa[
a
<(Bb
Oa)
<
irl
(Lb
La)(cos
O
a
cos
(4)(cos
0
a
cos
Ob)
((kb
cfra)(eb
0a)
(2)
(L
b
L
a
)(cos
¢
a
cos
4n)(sin
B
b
sin
Oa)
[44
sba)(Ob
a)
Case
2
eb
=
O
a
Xb
X
a
(L
b
La)(cos
¢
a
cos
4.b)(sin
Bb)
((In
Oa)
(L
b
L
a
)(cos
cos
4b)(cos
Bb)
Yb
Ya
(54
Oa)
Cases
3
and
4.
4,b
=
Zb
Za
=
(Lb
La)(cos
4n)
Case
3
Ob
0
Oa
[—
<
(gb
00
<
7ri
Xb
Xa
(fib
Oa)
(Lb
La)(sin
Ob)(cos
Oa
(Lb
La)(sin
Ob)(sin
Ob
Yb
Ya
(Ob
ea)
Case
4
Ob
Oa
X
b
X
a
=
(Lb
La)(sin
4n)(sin
B
b
)
.
.
(9)
492
Yb
Ya
=
(Lb
La)(sin
4b)(cos
0b)
.
.
(10)
It
should
be
emphasized
that
the
expected
change
in
drift
direction
azimuth
may
introduce
a
sizable
error
if
it
is
not
treated
adequately.
This
change
is
limited
to
±
180°
(or
±
ir
radians),
since
any
change
greater
than
this
would
imply
a
change
in
the
opposite
direction.
For
example,
if
O
a
=
350°
and
B
b
=
2°,
the
resulting
Oh
O
a
)
is
12°
and
not
—348°.
This
is
a
reasonable
assumption
as
long
as
the
drilled
section
is
short
and
as
long
as
it
is
physically
impos-
sible
for
the
drill
pipe
to
bend
—348°.
The
expressions
in
Eqs.
6,
9,
and
10
are
those
used
by
the
Tangential
method
of
directional
survey
in-
terpretation.
These
are
applicable
only
when
the
well-
bore
curvature
is
zero.
In
practice,
when
the
curvature
approaches
zero,
the
error
introduced
in
each
segment
by
straight-line
representation
is
rather
small.
Fig.
1
shows
that
an
error
of
only
1
ft
per
100
ft
of
depth
is
introduced
by
using
the
Tangential
method
in
a
borehole
segment
with
a
curvature
of
10°/100
ft.
Insignificant
as
this
error
may
seem,
its
cumulative
effect
over
the
entire
length
of
the
borehole
may
result
in
drastic
discrepancies
as
to
the
location
of
the
bot-
tom
of
the
hole.
It
may
be
contended
that
a
1-percent
error
correction
is
well
beyond
the
accuracy
of
the
recording
instrument.
Fig.
1
shows
the
variance
that
can
be
expected
for
instrument
inclination
errors
of
±0.5°.
This
range
of
error
is
within
the
maximum
range
in
most
instruments,
yet
it
does
not
change
the
significance
of
the
error
appreciably.
As
a
further
analysis,
it
is
possible
to
extend
the
work
done
by
Walstrom
et
al.
2
to
determine
the
degree
of
certainty
that
can
be
associated
with
the
computed
bottom-hole
locations
for
a
given
tool.
This
task,
however,
is
be-
yond
the
scope
of
this
paper.
It
should
be
recognized
that
other
approaches
to
directional
survey
interpretation
are
still
open,
For
instance,
instead
of
a
constant
wellbore
curvature
between
two
survey
points,
it
may
be
useful
to
con-
sider
a
changing
curvature
and
analyze
three
or
four
points
at
a
time.
The
possible
improvements
in
accu-
racy
to
be
gained
with
these
or
other,
more
sophisti-
cated,
methods
must
be
considered
in
the
light
of
possible
sources
of
error.
Any
interpretive
technique
must
take
into
account
its
inherent
model
errors
as
well
as
the
limitations
of
the
recording
instruments.
Correction
of
Vertical
Thickness
The
problem
of
correcting
for
true
vertical
thickness
arises
when
it
is
desirable
to
compute
net
pay
from
apparent
log
intervals
recorded
on
directionally
drilled
holes.
The
correct
net
pay
is
used
in
many
reservoir
engineering
calculations
and
in
determining
equity
participation
formulas.
It
must
be
computed
taking
into
account
the
effect
of
the
inclination
of
the
wellbore
as
well
as
the
strike
and
dip
of
the
reser-
voir
beds.
Eq.
11
eves
an
expression
for
true
vertical
thick-
ness
derived
in
detail
in
Appendix
B.
ht
=
(zd
zo)
tan
ad
[(Yd
yo)
cos
Y
(x
d
xo)
sin
yl
(11)
JOURNAL
OF
PETROLEUM
TECHNOLOGY
Xb
X
a
Ya
cos
Ob)
sin
0a)
The
subscripts
o
and
d
ind)cate
the
top
and
bottom
of
the
bed,
respectively.
Fig.
2
shows
the
computed
true
vertical
thickness
of
a
pay
zone
where
the
well
has
a
constant
45°
inclination
and
a
N
45°
E
bearing,
as
a
function
of
the
formation
dip
and
its
direction.
The
observed
log
thickness
of
this
hypothetical
well
is
a
constant
100
ft.
The
dip
angle
was
changed
from
to
10°,
and
the
azimuth
of
its
direction
from
to
360°.
It
should
be
noted
that
only
when
the
direc-
tion
of
the
well
parallels
the
strike
of
the
bed,
true
vertical
thickness
is
independent
of
bed
dip.
In
Fig.
2,
this
occurred
at
135°
and
again
at
315°.
Obviously,
when
the
dip
is
zero
the
observed
log
thickness
must
be
corrected
only
by
the
inclination
of
the
wellbore.
Another
observation
can
be
made
on
Fig.
2.
If
the
abscissa
is
expressed
as
degrees
clockwise
from
the
direction
of
the
well,
the
same
curves
apply
to
any
direction
the
well
may
take.
For
example,
consider
a
well
that
has
an
observed
100
ft
of
log
interval,
a
45°
inclination
and
a
S
30°
E
direction.
Assume
that
the
formation
being
penetrated
is
dipping
10°
with
an
80°
azimuth.
The
dip
azimuth
is
therefore
290°
ahead
of
the
direction
of
the
well.
Starting
at
45°
on
the
abscissa
of
Fig.
2,
the
290°
would
correspond
to
335°.
At
this
point
the
corrected
vertical
thickness
would
be
approximately
66.3
ft.
To
apply
Eq.
11
one
must
know
the
coordinates
y
o
,
zo,
Xd,
Yd,
and
zd.
These
points
can
be
obtained
from
an
analysis
of
directional
survey
data
by
expand-
ing
the
curvature
method
to
points
located
within
two
consecutive
survey
stations.
This
procedure,
shown
in
Appendix
B,
involves
obtaining
the
inclination
and
the
directional
azimuth
angles
ell
and
0
for
points
o
and
d.
Once
these
angles
have
been
determined,
it
is
1
.6
1.4
I.2
1
.0
06
-
0.5•
0.6
0.4
0.2
0
0
2
4
6
10
12
WELLBORE
CURVATURE
DEG
/100
FT.
Fig.
1—Error
introduced
by
Tangential
method
of
interpretation.
possible
to
compute
the
true
coordinates
at
those
points
and
apply
Eq.
11.
When
the
directional
survey
points
are
not
too
far
apart,
there
is
an
alternate
method
that
may
be
fol-
lowed
to
deter-
4
the
location
of
points
n
and
d.
This
method
may
be
followed
in
most
cases
without
introducing
appreciable
error,
if
the
curvature
is
not
excessive
and
if
the
true
coordinates
of
the
survey
points
a
and
h
have
been
fixed
by
the
Curvature
method
of
analysis.
In
this
case
it
may
be
sufficient
to
obtain
the
incremental
coordinates
by
proportions.
Thus:
Ld
L
o
Zd
Zo
(Zb
Zo)
L
b
+
L
d
Lo
Lb
La
Ld
L
o
(Yb
Yo)
Lb
La
Eqs.
12
through
14
may
be
useful
for
hand
calcu-
lations.
It
may
also
be
desirable
to
compute
h
t
using
the
average
inclination
and
direction
of
the
borehole
through
the
pay
interval.
In
this
case
Eq.
11
may
be
rewritten:
h
t
=
(L
a
L
O
)[coscp
tan
ot
a
sin
43
cos
(E—
y)l,
(15)
where
ckd
cflo
tiSb
2
2
7
_
Od
+
O
o
Ob+
O
a
2 2
Eq.
17
should
be
used
with
care,
realizing
the
in-
herent
errors
that
may
be
committed
by
averaging
angles.
Obviously
the
average
of
359°
and
is
not
180°
(due
south)
but
(due
north).
100
FT.
LOG
INTERVAL
45•
WELL
INCLINATION
AT
A
N.45*
E.
DIRECTION
VI
I2
,--
1
DIP
1:
/
/
.
0'
N
\
/
7
130
78
/
...._..
5
.
\
ii
76
74
72
....
...'''
7
0
D.
412'..s
iA
it\
'.....--
68
66
-
N.,...
,
.N
,//
1
\\
\
',...
............
\\
64
\
/
\
.........../
/
N
62
1 /4
60
\
/ I
4
........
0
45
90
135
180
225
270
315
380
DIP
DIRECTION
AZIMUTH-DEGREES
Fig.
2—Vertical
thickness
correction.
xd
x
o
Yd
Yo
(x
o
x
o
)
.
(16)
. .
(17)
APRIL,
1971
493
Eq.
A-4
gives
the
incremental
true
vertical
depth,
applicable
when
4'b
is
different
from
Oa.
To
derive
expressions
for
departure,
assume
that
the
projection
of
ab
on
a
horizontal
plane
has
a
con-
stant
radius
of
curvature;
then,
by
definition,
d0
=
Bb
Oa
constant
.
. . .
(A-5)
Nomenclature
a,
o,
b,
d
=
points
of
the
wellbore
A'
B'
C'
1
A,
B,
C,
D
_
coefficients
of
equations
of
bed
planes
true
vertical
thickness,
ft
h
t
=
drilled
(measured)
depth,
ft
L=
horizontal
departure,
ft
s=
distance
along
the
east-west
axis,
IL
x
(east
is
+x)
y
=
distance
along
the
north-south
axis,
ft
(north
is
+y)
z
=
vertical
depth,
ft
ad
=
formation
dip
angle,
degrees
y
=
dip
azimuth,
degrees
=
drift
direction
azimuth,
radians
=
wellbore
drift
(inclination)
angle,
radians
References
1.
Wilson,
G.
J.:
"An Improved
Method
for
Computing
Di-
rectional
Surveys",
J.
Pet.
Tech.
(Aug.,
1968)
871-876.
2.
Walstrom,
J.
E.,
Brown,
A.
A.
and
Harvey,
R.
P.:
"An
Analysis
of
Uncertainty
in
Directional
Surveying",
I.
Pet.
Tech.
(April,
1969)
515-523.
3.
Woods,
R.:
Analytic
Geometry,
Revised
ed.,
The
Mac-
millan
Co.,
New
York
(1948)
262.
4.
Middlemiss,
Russ
R.:
Differential
and
Integral
Calculus,
2nd
ed.,
McGraw-Hill
Book
Co.,
Inc.,
New
York
(1946)
150.
5.
Wilson,
G.
J.:
"Radius
of
Curvature
Method
for
Comput-
ing
Directional
Surveys",
paper
presented
at
SPWLA
Ninth
Annual
Logging
Symposium,
New
Orleans,
June
23-26,
1968.
APPENDIX
A
Derivation
of
the
Generalized
Curvature
Equations
Assume
that
the
drilled
length
(Lb
La)
is
divided
into
an
infinite
number
of
se
c
ti
o
ns
dT.
,
nne
of
which
is
drawn
in
Fig.
3.
From
calculus,
it
can
be
written
dL
dL
dz
dO
_
dz
dO
(A-1)
Assume
also
that
the
curvature
of
the
entire
section
ab
is
constant.
Then,
by
definition,
dL
Lb
La
dO
_
ctib
constant
.
(A-2)
By
inspection
of
Fig.
3,
dz
(A-3)
C°S
=
dL
Substitution
of
Eqs.
A-2
and
A-3
into
Eq.
A-1
yields
(kb
cos
d
cos
o
Lb
L
a
dZ
Separating
variables
and
integrating,
ds
Sb
Sa
This
assumption
contains
the
condition
that
7r
<
(B
b
O
a
)
<
a
,
(A-6)
since
the
change
in
9
is
not
expected
to
exceed
180°.
From
inspection
of
Fig.
3
we
can
write
sin
0
dL
(A-7)
ds
dx
sin
=
ds
(A-8)
dv
cos
9
=
ds
(A-9)
Also,
from
calculus
we
have
ds
_
dL
ds
(A-10)
do
dL
Substitution
of
Eqs.
A-2
and
A-7
into
Eq.
A-10
yields
ds
L
b
La
\
dcs
95b
/
Separating
variables
and
integrating,
b
r
06
a
ds
(
1
;
4
_
ika
1
s
i
n
dO
"
b
La
)
Lb
La
(cos
¢
a
cos
4
,
b)
.
(A-11)
Sb
S
a
Also,
by
calculus,
d0
dx
d0
(A-12)
ds
dx
Substitution
of
Eqs.
A-5
and
A-8
into
Eq.
A-12
yields
eb
ea
sin
g
de
S
a
ud
Separating
variables
and
integrating,
y
(North)
ii‘t)
J
.
dz
Lb
La
ekb
a
Or,
I
costt.
d
4
414
a,
b
=
Survey
points
.
(A-4)
Lb
--
La
Zb
Za
(sin
Sin
Oa)
S
6
a
Fig.
3—Graphical
representation
of
a
wellbore
segment.
494
JOURNAL
OF
PETROLEUM
TECHNOLOGY
b
03
J
dx
sb
3
4
:
1
j
sin
0
dO
B
b
e
a
a
Xb
Xa
Gi
Sb
S
n
a
a
)
(cos
ea
—cos
Ob)
.
(A-13)
Finally,
substituting
the
value
of
(sb
sti)
from
Eq.
A-11
into
Eq.
A-13,
(T
b
T
a
)(cns
0,
a
cos
f
i
r
.
b
yrn
s
cos
0
b
)
(44
4
1
/
4
)(0b
Oa)
(A-14)
Eq.
A-14
gives
the
true
surface
displacement
along
the
east-west
direction
(east
is
positive).
To
obtain
that
along
the
north-south
direction
we
again
write
from
calculus
dO
_
dy
dO
ds
ds
dy
Substitution
of
Eqs.
A-5
and
A-9
into
Eq.
A-15
yields
Of)
Oa
cos
B
Sb
Sa
Sena
rafing
variables
and
integrating,
dy
sb
Sa
a
61
f
cos
0
dO
Ob
O
--
4
1-:-
h
A
Y6
Ya
e
kbUI
Ub
--
Daj
b
ua
Substitution
of
the
expression
for
(sb
s
a
)
found
in
Eq.
A-11
into
Eq.
A-16
gives
(Lb
La)(cos
4a
cos
cpb)(sin
sin
Oa)
(4n,
4
,
a)(96
ea)
(A-17)
which
is
the
north-south
true
displacement
(north
is
positive).
Before
we
complete
these
equations,
we
must
con-
sider
those
instances
in
which
the
change
in
either
drift
angle
or
its
azimuth
is
zero.
That
is,
,-kb
Oa,
or
Bp
=
O
a
.
If
both
these
conditions
exists,
we
know
intuitively
that
the
Curvature
method
must
yield
the
equations
of
the
Tangential
method.
Before
proceeding,
therefore,
let
us
prove
two
equalities.
Jim
(
sin
e
l)
sin
O a
)
cos
Bp
.
(A-18)
hm
COS
0
a
COS
b)
.
-
SM
Bb
Ob
Oa
k
Ob
Oa
To
prove
these
indeterminate
forms,
let
O.
be
con-
stant
and
let
B
b
approach
it.
Then,
by
L'Hospital
theorem,'
him
sin
Ory
Sin
O.
eb
—b
O
a
k
Ob
Oa
liM
r
d
sin
B
b
sin
O.
B
a
—>
O
a
IdO
b
(
eb
ea
)1
COS
B
b
which
proves
Eq.
A-18.
Similarly
in
Eq.
A-19,
him
(cos
O.
cos
B
b
)
e
b
—b
O
a
k
B
b
Oa
him
[
d
(cos
9,
2
cos
Bb
)
=
Bb
Oa
`
deb
Bb
a
Sin
Ub
which
proves
Eq.
A-19.
We
may
use
Eqs.
A-18
and
A-19
to
simplify
Eqs.
A-4,
A-14
and
A-17.
Thus,
using
these
generalized
expressions,
we
may
write
solutions
to
all
four
cases
shown
in
the
text.
APPENDIX
B
Vertical
Thickness
Correction
Equation
Let
us
represent
the
top
of
the
bed
by
a
plane
in
space
with
a
general
equation:
Ax
+
By
+
C7
D
=
0
(B
-1
)
If
the
origin
(0,
0,
0)
is
located
on
this
plane,
it
can
be
shown
that
D
=
0
and
Eq.
B-i
reduces
to
Ax
+
By
+
Cz
=
0
(B-2)
The
bottom
of
the
bed
may
also
be
represented
by
a
plane
parallel
to
the
top
one.
It
must
satisfy
the
condition
that'
A
_
B
_
C
A'
B'
C'
(B-3)
We
may
write
the
equation
of
a
particular
plane
by
making
Eq.
B-3
equal
to
unity.
Thus
A
=
A'
B
=
B'
(B-
4
)
C
=
C'
If
this
condition
exists,
and
the
second
plane
does
not
coincide
with
the
first,
its
generalized
equation
meet
he
that
chnurn
nn
Pn
R
1
with
n
n
The
vertical
distance
separating
these
two
planes
may
be
computed
by
solving
for
the
z
intercept
of
the
second
plane.
Thus,
for
x
=
0
and
y
=
0,
Cz
+
D
=
0
and
D
z
(B-5)
To
apply
these
concepts
to
true
vertical
thickness
of
a
bed
pierced
by
a
directional
well,
we
may
trans-
late
the
origin
to
that
point
where
the
well
penetrates
the
top
of
the
bed.
With
a
knowledge
of
the
dip
and
strike
of
the
bed
and
the
drilled
thickness,
we
can
determine
the
equation
of
the
top
plane
and
solve
for
that
of
the
bottom
plane.
With
a
previous
knowledge
of
the
spatial
location
of
points
above
and
below
the
bed,
obtained
from
the
curvature
analysis,
we
may
obtain
the
true
coordinates
of
the
points
o
and
d
at
which
the
well
penetrates
the
top
and
the
bottom
of
the
bed,
respectively.
Let
us
consider
Fig.
4,
where
directional
survey
points
a
and
b
are
above
and
below
the
bed
bound-
aries,
respectively.
Since
by
definition
the
segment
from
a
to
b
has
a
constant
curvature,
we
may
write
L
b
La
Lo
La
j6b
Oa
Xb
Xa
(A-15)
dy
a
!A
14\
to
-
1,JJ
Yb
rya
=
eb—b
O
a
Ob
Oa
.
(A-19)
(B-6)
or
APRIL,
1971
495
y
(North)
PLAN
VIEW
POINT
2
x
(East)
POINT
I
S
I
d
POINT
3
At
)
SID
E
Fig.
4—Representation
of
two
parallel
planes
in
space.
(L
o
La)(41b
Oa)
+
ai
a
.
(Lb
La)
.
(B-7)
Similarly,
add
La)(4
0
0
Oa)
+
. .
(B
-
8)
(Lb
La)
The
expressions
for
the
location
of
points
o
and
d
are
similar
to
those
given
in
the
text
as
Eqs.
1
through
10,
with
the
subscripts
a
and
b
replaced
by
the
sub-
scripts
o
and
d.
The
equation
of
the
top
plane
is
determined
by
manipulating
the
dip
angle
w
and
its
azimuth
y
to
determine
three
points.
From
Fig.
5,
Point
1:
x1=
0,
y1=
0,
zi
=
0
Point
2:
x
2
=
sin
y
Y2
=
COS
y
z2
=
tan
ad
Point
3:
x
2
=
sin
(y
+
90
°
)
y3
=
cos
(y
90
°
)
z3
=
0
By
substituting
these
values
intn
Fo
r
R-2
and
solv-
ing,
we
may
find
expressions
for
A,
B,
and
C.
=
Y2
Z3
Y3
Z2
=
cos
(-y
+
90
°
)
tan
ad
z2
x2
=
Z2
X3
Z3
X2
Z3
X3
=
tan
cci
sin
(y
+
90
°
)
ti
Fig.
5—Dip
and
strike
representation.
Since
we
know
the
location
of
one
point
on
the
bottom
plane,
we
may
determine
D
in
Eq.
B-1
thus:
x'
=
Xd
X0
(B-12)
Y
t
=
Yd
Yo
(B-13)
z'
=
zd
zo
(B-14)
Solving
for
D
in
Eq.
B-1
and
substituting
the
values
found
for
y',
and
z',
we
obtain
D=
(Ax'
+
By'
+
Cz')
D
=
(—
(xd
xa)
cos
(y
+
90
°
)
tan
w
+
(Yd
yo)
tan
ad
sin
(y
+
90
°
)
+
(zd
zo)
[sin
-y
cos
(y
+
90
°
)
sin
(y
+
90
°
)
cos
yn
(B-15)
And
substituting
Eqs.
B-15
and
B-11
into
Eq.
B-5
yields
the
expression
for
the
corrected
vertical
thick-
ness:
(zd
zo)
+
((tan
ad
[(Yd
Yo)
sin
(y
+
90
°
)
(xd
x
o
)
cos
(y+90°)])÷
(sin
y
cos
(y+90°)
sin
(y
+
90
°
)
cos
y})
(B-16)
Using
the
formulas
for
sums
of
angles,
it
is
possible
to
reduce
Eq.
B-16
to
that
given
in
the
text
as
Eq.
11.
For
the
sake
of
simplicity,
it
was
assumed
that
the
top
and
bottom
of
the
formation
were
parallel.
This
condition
is
not
necessary
for
the
application
of
Eq.
11,
and
only
the
dip
and
strike
of
the
bottom
of
the
bed
must
be
known.
As
long
as
the
translated
origin
is
located
on
the
top
plane,
its
dip
and
strike
are
not
significant,
since
the
analysis
could
have
been
based
on
an
imaginary
plane
parallel
to
the
bottom
of
the
bed_
J
P
T
Original
manuscript
received
in
Society
of
Petroleum
Engineers
office
March
17,
1970.
Revised
manuscript
received
Nov.
2,
1970.
Paper
(SPE
3076)
was
presented
at
SPE
45th
Annual
Fall
Meeting,
held
in
Houston,
Oct.
4-7,
1970.
v
Copyright
1971
Am
Institute
of
Mining,
Metallurgical,
and
Petroleum
Engineers,
Inc
This
paper
will
be
printed
in
Transactions
volume
251,
which
will
cover
1971.
A=
Y2
Z2
Y3
Z3
B=
(B-9)
(B-10)
I
x
3
X2
y21
C
y3
=
x2
Y3
X3
Y2
=
sin
y
cos
(y
+
90')
sin
(y
+
90
°
)
cos
y
(B-11)
496
JOURNAL
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TECHNOLOGY