Note: Descriptions are shown in the official language in which they were submitted.
8 ~ ~
,
BOREHOLE SURVEY APPARATUS AND METHOD
SPECIFICATIO~
_
This invention relates to an apparatus and
method for surveying a borehole or the like with
measurements of gravity components to provide a
representation of the borehole trajectory with respect
to a known ground reference point, such as ground O
where the borehole starts.
Surveying of a borehole or the like is often ac-
complished by an inqtrument or probe which move~
through the borehole and measures inclination and
azimuth angles at successive points. Inclination, the ;
angle by which the borehole tangent deviates from the
vertical, may be measured with a pendulum or acceler-
ometer. Azimuth, the angle of the borehole w~th
respect to a reference direction, such as north, is
typically measured with a magnetic or gyroscopic
compass. These angles, together with the distance
along the borehole, are used to determine coordinates
of points along the borehole with respect to the
reference, ground 0.
- ...... . .
8 ~ 3
-- 2 --
A pendulum for measuring inclination may take
the form of a linear servoed accelerometer which
responds to the acceleration of gravity. Servoed ac-
celerometers are available which are small, rugged and
accurate. The measurement of azimuth is not so simple.
Magnetic ~ornpasses or other devices for measuring the
earth's magnetic field are subject to errors causea by
magnetic anomalies in the ground. Gyroscopic compasses
have several drawbacks including large size, bearing
wear, sensitivity to shock, drift and precession
errors and the requirement for a long settling period
for stabili~ation when a measurement is made.
In accordance with the present invention, an
apparatus and method are provided with which the
borehole trajectory may be determined from gravity
component measurements, as made with the linear
servoed accelerometers mentioned above, and from dis-
tance along the borehole. The survey measurements are
made as the probe is moved through the borehole pro-
viding a data output from which the borehole trajec-
tory is determined. The speed and accuracy of a sur-
vey based on servoed accelerometer measurements ar
surpasses that which is achieved with other
instruments. In addition, the probe utili~ing
accelerometers and not requiring a compass for azimuth
measurement may be contained in a smaller diameter
housing and is more rugged.
One feature of the invention is a borehole sur-
vey apparatus or ~ensing probe having first and second
pairs of accelerometers with the input axes of each
pair defining a measurement plane. The accelerometers
are mounted to pass through the borehole with the two
rneasurement planes spaced apart and perpendicular to
their respective local borehole tangent and to the
borehole axis. The two pairs of accelerometers are
joined by a flexible~ torsionally stiff connector~ to
-- 3
follow the borehole trajectory while rnaintaining a
fixed angular relationship between the accelerometer
pairs, about the borehole axis. The apparatus further
includes means for deriving from the accelerometer
signals a representation of the borehole trajectory.
Another feature of the invention is that the
probe has first and second sections ~ith one set o~
accelerometers mounted in each section. The sections
are joined by a connector which is torsionally stiff
about its longitudinal axis and flexible about axes at
right angles thereto as a cable or pipe. This con-
nector insures that the distance along the borehole
between probe sections remains fixed; that the joint
between the two sections is stiff în torsion; and that
each set of accelerometers follows the local borehole
axis and is free to rotate about the borehole a~is.
More particularly, the two probe sections are joined
by a cable or pipe which is resilient to bend along
the borehole axis but is rigid with respect to tor-
sional stresses about its axis. ~lternatively, thetwo sets of accelerometers may have a common housing
which is flexible to follow bends in the borehole, but
torsionally stiff to resist twisting.
A further feature of the invention is the method
of surveying the angles oE inclination and azimuth
along a borehole, including measuring the acceleration
of gravity (along orthogonal axes) in planes trans-
verse to the borehole axis at successive points along
the borehole, generating signals representing the
acceleration, and deriving from the acceleration
signals a representation of the borehole trajectory.
The inclination of the borehole at each measure-
ment plane can be found from the vector sum of the two
components of earth' 8 gravity measured by the cor-
responding accelerometer pair. This has been known in
1 ;1668~3
-- 4 --
1 the prior art. A principal feature of the .invention is that
the incremental change in azimuth of the borehole between the
two measurement planes may be determined from the four output
signals of the two accelerometer pairs and -the distance along
the borehole between measurement planes. The borehole tra-
jectory.may be determined in terms of inclination, azimuth
and borehole length.
~ore.particularl~, as the probe is moved along the
-borehole, gravity measurements from the two accelerometer
pairs and a correlated measurement.of distance along the
~rajectory of the hole are made, from which the course of
the ~orehole in three~dimensions:is.determined.
Further features and advantages of the invention
will readily be apparent from the following specification
and from the drawings, in which:.
. ~igure 1 is a broken diagram of an apparatus embody-
ing the`.invention, including sections through a borehole
showing the sensing probe,
Figure.2 is a-block diagram of the accelerometers
and a.circuit for transmitting acceleration signals to the
`surface;
. Figures 3-12 are..geometric diagrams which illus-
trate deriYation of the inclination and incremental azimuth
-angles and the borehole position coordinates from the ac-
celeration and b.orehole.distance measurements;
Flgure 13 is..a developmenk.tree showing in chart
form the derivation illustrated geometrically in Figures 3-12;
- Figure 14 is a block diagram illustrating a system
for deriving the borehole.trajector~ from the accelerometer
-signal-s with measurements taken at positions spaced apart
a distance equal to the spacing of the accelerometer pairs.
. Figure 15A is a matrix/vector pxogram diagram; and
...: Figure 15B is an illustration of a local
-coordinate transformation matrix.
.
,.
" ~ 18~3
-- 5 --
1 The invention is described herein in connection
with a borehole as for an oil or gas well. It can be
used for other mining or civil engineering applica-tions
such as surve~ing subterranean structures as a mine shaft,
for example. Reference to a borehole in the claims shall
be broadly construed unless the context requires a different
interpretation. Derivation of a representation of the
borehole trajectory may be accomplished, for example, in the
form of three dimensional coordinates or of a plot of an
existing borehole to determine its physical location.
~he trajectory representation may also be derived as the
borehole is drilled to monitor the drilling operation and
to enable an operator or drill controller to direct the
drill along a desired path. The invention is not limited
to a particular trajectory representation.
In Figure 1, a borehole 20 extends downwardly
from the point 20a at the ground surface and is lined
with a casing 21. The sensing probe has a Eirst section
22 and a second section 23 spaced therefrom, the two
sections being joined by a cable or pipe 24. The
- -sensing probe is lowered into the borehola on a hoisting
cable 25,which also includes conductors for supplying
electrical power to the probe and for directiny signals
- - from the probe to circuitry above ground at the well head.
A,pair of accelerometers (not shown in Figure 1
are located in the first probe section 22 and preferably
have their sensitive axes X, Y at right angles to each
other defining a measurement plane at right angles to the
longitudinal axis of the probe section.
. .
-- 6 --
The probe axis corresponds with the axis of the bore-
hole. Similarly, a pair of accelerometers (not shown
in Figure l) in the second section 23 have their sen-
sitive axes X, Y at right angles to each other defin-
ing a measurement plane at right angles to the longi-
tudinal axis of the probe section 23 and the borehole
axis.
As will be explained below, the borehole posi-
tion coordinates are determined from the inclination
angle with respect to the gravity vector and an an-
gular measure of the zenith of each measurement plane.
These angles are readily ana accurately determined
from measurement of the gravity vector with orthog-
onally positioned accelerometers in a measurement
plane at right angles to the borehole axis. However,
measurement of the gravity vector with any pair of
accelerometers whose axes are sensitive to independent
vectors in a plane having a known attitude in the
borehole (i.e., the sensitive axes are neither colin-
ear nor parallel) may be transformed geometrically to
the inclination and zenith angle measurements.
In a typical probe the diameter oE the section
housings is of the order of 2-3 inches and two servoed
accelerometers cannot be mounted ~ide-by-side.
Accordingly, the accelerometers in a pair are phys-
ically spaced apart axially of sections 22, 23, but
are sufficiently close together as compared with the
spacing between the accelerometer pairs to be con-
sidered coplanar.
A third accelerometer could be added to each set
with its sensitive axis at right angles to the axes of
the other accelerometers of the pair as indicated by
Z, Z'. The third accelerometers afford an improvement
in accuracy, and enable operation if an X or Y
accelerometer malfunctions.
8 ~ '~
~ 7 ~
Cable or pipe 24 iq fixed at each end to the
probe sections 22, 23 and serves to space the sections
apart a predetermined distance in borehole 20. The
cable 24 is flexible to follow bends in the borehole
but resists torsional stresses to pre~ent rotation of
one section with respect to the other. This maintains
a preestablished relationship between the acceler-
ometer axes X, X' and Y, Y'. Preferably, with the
probe sections 22, 23 axially aligned, axes X, X' are
parallel with each other and de~ine a plane through
the longitudinal axis of the probe. Similarly, axes Y,
Y' are parallel and define a second p~ane through the
probe axis, at right angles to the first. It is not
essential that the corresponding axes be parallel,
only that they have a fixed relationship. However,
processing of the signals developed by the acceler-
ometers is simplified if the sensitive axes are
nominally parallel.
Each of the accelerometers is preferably a
linear, servoed accelerometer with an associated
electronic circuit (not shown) which generates an an-
alog signal having an amplitude representing the com~ !
ponent of gravity acceleration along the sensitive
axis of the accelerometer. Jacobs U.S. patent
3,702,073 shows such an accelerometer. An electronic
circuit in the probe, to be described in more detail
below, multiplexes the analog signals, converts them
to digital form and couples them through conductors in
hoisting cable 25 with circuitry at the well head.
The acceleration signals are connected with the data
input of data storage unit 26. The output of data
storage unit 26 is connected with a processor 27
which, as will appear, derives a representation of the
borehole trajectory. A transducer 28 associated with
hoisting cable 25 provides a signal ~L to the
&; 8 ~ ~
8 -
processor 27 indicating the position of the sensing
probe in the borehole.
Keyboard and display 30 i9 connected with data
processor 27. A representation of the borehole
trajectory may be displayed as in terms of coordînate
dimensions in a three axis system. The keyboard
provides for operator input and control. The
representation of the borehole trajectory may be
printed or recorded for futur~ use. Means for
performing these functions are known and are not
illustrated in the drawings.
The sections 22, 23 of the sensing probe have
cylindrical pressure housings. Resilient central-
izers 31 on the outside of the housings engage the
inner wall of borehole liner 21, positionir~g the
housings so that their longitudinal axes coinciae sub
stantially with that of the borehole. Lower section
23 of the probe has a housing divided into two parts,
32, 33. Cable 24 is connected with the upper end of
housing part 32. Accelerometers X', Y' are located in
housing part 32. The second housiny part 33 of the
second probe section 23 has the centralizers 31 there-
on and is long enough to maintain proper alignment
with the borehole. The housing parts 32, 33 are joined
by a swivel connector (not shown) so that housing part
32 can rotate freely with respect to part 33 to main-
tain the desired alignment with the upper probe sec-
tion 22.
The borehole survey is carried out by causing
the probe to move through the borehole from one end to
the other in either direction while data is collected
and processed. The survey may be conducted as the
probe is lowered into the borehole or as it is raised
from the bottom. For increased accuracy, data may be
8 ~ 3
g
collected as the probe moves in each direction ana the
survey results averaged.
The borehole azimuth is referenced to the out-
side world by establishing an initial a~imuth con-
dition of the probe at the surface. For example, the
probe may be physically aligned with a fixed benchmark
and the alignment verified with a surveying instrument
35.
Figure 2 illustrates diagrammatically the
accelerometers and signal handling circuitry in the
probe. Upper probe section 22 contains the X, Y and Z
accelerometers which have analog signal outputs ax,
ay, az. The lower probe section 23 has X', Y', Z'
accelerometers with analog outputs ax~, ay~, az~.
Power from a surface source 37 is connected
through the hoisting cable 25 with a power supply 38
in the probe. The analog accelerometer signals are
connected with sample and hold circuits 39, 39' and
are multiplexed through analog to digital convertPrs
40, 40' to a signal control 41 through which they are
transmitted to the surface. Signal control 41
provides timing for the sample and hold circui-ts 39,
39' and the A/D converters 40, 40'. Signals ~rom
cable length transducer 28 are correlated with the
accelerometer signals to identify the point along the
borehole at which each set of signals is taken.
A source of error in the survey may be minimized
by providing temperature sensors 42, 42' in each probe
section together with temperature controls 43, 43' to
maintain ~he temperature of temperature sensitive ele-
ments within desired limits. Analog temperature sig-
nals t, t' are sampled and transmitted to the sur-
face with the acceleration signals. The temperature
signal~ are utilized in a temperature compensation
circuit 26' to minimize further any temperature er-
ror.
166~3
-- 10 --
The probe sections 22, 23 must be long enough to
maintain alignment between the section axes and the
borehole axis. The maximum length is limited by the
minimum radius of bend in the borehole liner. Within
these limits a typical probe section is between 2 and
20 feet long. The distance between accelerometer
pairs should be at least 10-15 feet. The maximum
spacing is dictated by handling problems. A typical
probe is between 50 and 150 feet long~
Figures 3-12 illustrate the geometric relation-
ships which underlie the derivation of the borehole
trajectory from the gravity component signals provided
by the two pairs of accelerometers. Figure 13 shows
in chart form some of the relationships. Following is
a tabulation of notations and terminologies used in
the drawings and in the subsequent discussion.
0 ground reference
NEG unit direction vectors North,
East, Downward ~gravity)
NnEnGn coordinates of the center
n f circle C~ with re
spect to NEG coordinate
system
C borehole curve
C upward projection o~ borehole
Cn unit circle at the nth cross
section of the borehole
Cnl or unit circle at the n+l cro~
Cn+l section of the
borehole
_n center of Cn
n upward projection of n
Q distance from Cn ~ Cn'
along the borehole curve C
XnYn two orthogonal accelerometers
at n
,
~n'Yn~ two orthogonal accelerometers
at n~ such that when the
curve C is a straight line,
the sensitive axes of Xn and
Xnl point at the same direc-
tion. Similarly, the sensi-
tive axes of Yn and Yn~
point in the same direction
ax ay acceleration signals from
axn-ayn~ Xn Yn Xnl Yn
Zn zenith on Cn, the point on
Cn closest to the surface
in unit vector from n to
jn unit horizontal vector 90
clockwise from in looking
down the borehole
kn~ kn- local unit vector tangent to
the borehole axis at n~
n~ in the plane defined by
Qnn'
n the point on Cn marked by
Zn or in
90n the point on Cn pointed by
~n
Q center of the borehole curve
with radius rn between n
and n'
An In azimuth and inclination of
borehole axi~ at n relative
to ground O using NEG
coordinate system
In In' inclination of circles Cn Cnt
nn' the vector from n to
n'
n' ~the vector nn' in NEG
coordinate system
~n angle from zenith Zn to
Xn accelerometer axis
bearing angle of the direc-
tion of bending from Zn to
n' :
bending angle of the borehole
from Qn to n'
- 12 -
r radius of the borehole curve
from n to n'~ equal to
Y ~ 'n ' = ~ - ~
~ a quantity used in the
geometric analysis
g gravity constant
Mn transformation matrix between
(i, j, k)nl and(i, i, k)n
Mn+l transformation matrix between
(i, j, k)n~and (N,E,G); note
that (i, j, k)n~ = (i, j,
k)n+l
Figure 3 is a three dimensional diagram with a
rectangula~ coordinate system NEG having an origin at
ground reference 0. Borehole curve C extends down-
wardly under the northeast quadrant. Curve C is a
projection of the borehole curve on the ground. The
coordinates NE define a horizontal plane at the ground
surface. G extends downwardly at right angles thereto
and represents gravity direction. The circles Cn
and Cn~ represent unit circles with centers on the
borehole curve at n and n'~ The planes of the
circles are normal to the borehole curve and the cir-
cles are spaced apart along the borehole a distance
Q, equal to the spacing between accelerometer pairs
in the sensin~ probe. It is assumed that the borehole
curve between n and n' is a circular arc of
radius rn about a center Qn~ Figure 4.
The sensing probe is moved through the borehole
and readings are taken from the two pairs of acceler-
om~ters at successive sensor positionsspaced apart a
distance Q, equal to the spacing between the sensor
pairs. As will be explained below, the inclination of
the borehole at each accelerometer position and the
change in azimuth angle between accelerometer posi-
tions can b~ determined from the accelerometer read-
~ ~68~
- 13 ~
ings. If the measurements start at ground reference 0
and the azimuth is known at that point, the azimuth
may be determinea for any point along the borehole by
summing the incremental azimuth figures. Measurement
may start at ground reference 0 and proceed to the
bottom of the borehole or may start at the bottom of
the borehole and continue up to ground reference. In
the latter case, the determination of the actual bore-
hole azimuth at the various positions is not known un-
til the survey is completed and the cumulative incre-
mental azimuth measurement is summed with the azimuth
at ground reference.
The inclination and azimuth angles and the
distance along borehole curve C for points on the
curve may be used to derive an identification of the
location of each borehole point in the rectangular
coordinate system NEG.
The acceleration signals ax and ay from a
pair of orthogonal accelerometers determine the in-
clination I of the plane of the accelerometers and the
orientation angle ~ between the zenith or point on the
unit circle closest to the ground and the sensitive
axis Gf the X accelerometer. In Figure 6 unit circle
CH is hori~on~al and Cn is tilted with respect
thereto about a diameter in~ ~in. Figure 7 is a
further detail of Figure 6 looking perpendicularly at
the vector Xn. It will be seen that,
the X-accelerometer si~nal
aXn = g cos ~n sin In
the Y-accelerometer has reading
Yn g cos (~n~) sin In
= -g sin ~n sin In
and
-ay
tan ~n ~ a~
As the accelerometer signals aXn and ayn are
known, both Wn and In can be determined. These
- 14 -
determinations are made for the unit circles Cn and
Cn,. From this information and the assuMption that
the borehole follows the arc of a circle between posi-
tions n and n ', the change in azimuth from n to n' may
also be determined.
More particularly,
aXn2 ~ ayn2 = g2(cos2~n ~ sin2 ~n) sin2In
Thus
(ax 2 + ay 2)1/2 = g sin In
or
In = Arc sin
This gives inclination of the borehole at n. That
of n~ is calculated similarly. This is represented
at step 43, Figure 13.
In Fiyure 8, three concentric circles are shown:
the circle CH is horizontal, or parallel to the
ground, the circle Cn is perpendicular to the bore-
hole at n with zenith Zn and is tilted with re-
spect to CH about an axis defined by in and ~~n
The circle Cnl is perpendicular to khe borehole at
n' with zenith Zn'~ and obtained by turning ~he
circle Cn about Vn and ~Vn at an angle 2~.
Circle Cnl intersects the circle CH at in' and
~inl- The turning point Tn on Cn is 90 apart
from Vn and ~Vn. Corresponding to the 2~ angle
turning, the point Tn on Cn is moved to U~ on
Cnl. Thus, both Tn and Un are 90 from Vn. In
figure 8, ~ is the angle between Zn and Tn and~ is
the angle between Zn' and Unl.
Figure 10 shows the circles Cn and Cnl
superimposed, looking along the axis of the borehole.
' .: . -
-
,
6 8 /.~ 3
. ,
- 15 -
In Figures 8 and 10, it will be seen that
= / ZnOTn = / inOVn
and
~ = /ZnOun = /inlovn
Thus, the zenith shift
Y = ~n~~n' = /znox-/
=/ ZnZn '
=/inojnl
(after turning the
right angle /znoin
clockwise by angle Y)
The spherical triangle of Figure 9 lies at the
right hand side of the circles of Figure 8. In this
triangle:
Let A = ~~In~ a - a
B = In b =
C = 2~
By spherical sine law:
sin a sin b
sin A sin B
or
sin a= sin
sin (~~In') sin In
Si~ce Y = ~
sin a = _in (_a - y)
sin Inl sin In
sin a sin In = sin Inl (sin~ C05 Y - COS ~ sin~ )
sina (sin In ~ sin In cosY ) = -cosa sin Y sin I
Thus:
sin Y sin I
tan ~ =
cos~ sin Inl - sin In
As Y = ~n~~n' all of the quantities on the right
hand side of the equation are known rom the four
accelerometer signals and tan a and ~ may be
determined.
,
~ 16~8~13
-16-
Al~o with reference to the spherical triangl~ o
Figure 9, the bending angle Z may be determined as
foll.ows using a spherical triangle law~
cot C
sin 1/2 (a~b)
tan 172 (A-B) sin 1/2 (a-~h~
5Since C = 2~, we have
sln 1 ~ (~ ~) tan
sin ( ~ ~ 2)
cot 1 (In + In')
sin Y 2
(sin ~ cos Y _ cos ~ sin Y)
= _ 2 _ 2 cot 12 (In ~ In')
sin y
= cos ~ (tan ~ . cot Y _ 1) cot 1 (In -~ In~)
Since all of the quantities on the right hand
side of the equation are known, the angle ~ can be
calculated.
The three quantities ~, and ~, step 44, Figure
13, are known. The geometric significance of ~, the
bearing angle of the lower probe center n~ looking
straight down along the borehole tangent at upper
probe center n~ is illustrated in Figure 5. The
benaing angle 2~ is illustrated in Figures 3 and 4
showing how much the borehole cross section C~ has
turned relative to the cross section Cn.
The position of vectors i, j and k, Figures 3
and 8, may be related for successive circles ~y
coordinate transformation matrices as follows:
- 17 -
- n n'
Mn Mn
NEG )(i,j,k)n ~(i,j,k)n'
Mn+l
so that
Mn+l = Mn ~n
The ma-trix Mn has already been obtained in a
previous measurement and calculation. It is necessary
only to derive the matrix Mn~. Based on Figure 8,
the three circle picture, the expression of vectors
(Unl Vn, kn~) in terms of (in~in~lkn) i5
Un = in(cos ~n cos 2 e n)
+ in (sin ~n cos 2 ~n)
+ kn (-sin 2 ~n)
Vn = in (-sin ~n)
+ jn (cos ~n)
+ kn ()
kn = in ~cos ~n sin 2 e n)
~ jn (sin ~n sin 2 ~n)
+ kn (cos 2 e n)
The coordinate transformation Mn which relates
the two vectors (in~ in~ kn) and (in-, inl~
knl~ in Figure 8 is obtained via the (Unl Vn,
kn) symbols.
nl = cos ~nUn ~ sin ~nVn
= in (cos ~n cos un cos 2~n + sin ~n sin ~n)
+ jn (cos ~n sin ~n cos 2~ n ~ sin ~n cos ~n)
~ kn (-cos ~n sin 2 en)
in' = sin ~nUn + cos ~nVn
= in (sin ~n cos ~n cos 2~n ~ cos ~n sin ~n)
+ in (sin ~n sin ~n cos 2en
+ cos ~n co~ ~n~
+ kn (-sin ~n sin 2 en)
s ~
- 18 -
knl = in (cos ~n 6in 2 ~n)
+ in (sin ~n sin 2 ~n)
+ kn (cos 2 ~n~
This means the coordinate transformation matrix Mn
(step 45, Figure 13) can be constructed:
~inl~ ~all al2 al3~ ~in~ ~in
in ' ¦ = a21 a22 a23 ¦ jn = Mn in
kn' J a31 a32 a33 ~kn kn
where all = cos ~ cos ~ cos 23 ~ sin ~ sin
10al2 = cos ~ sin ~ cos 23 - sin ~ cos
.
In practice, the processor will store the
coordinate transformation matrix from previous local
coordinates (i, j, k) in the ground zero global
coordinate system tWEG). That means the computer
already knows the matrix Mn where
fin ~ ~ bll bl2 bl3~ N~ ~ N
in = b21 b22 b23 E ~ = Mn E
kn b31 b32 b33 GJ G J
In order to update the transformation matrix Mn to
20Mn~l, which transforms the local coordinates (inl,
inl~ knl) into the global coordinate ~iystem (NEG)
determine the matrix product
Mn+l = MnMn
See Figure 13, step 46.
In Figure 11, looking along the borehole in the
direction of the vector ~krl~ Figures 3, 4 and 5,
assume that the borehole from n to n' has a
bearing clockwise ~n degrees from zenith Zn; and
that borehole bends along a circular path through an
arc 2 ~n .
If Q is the borehole length from n to n'~
the local position vector nn' from n to n'
(step 47, Figure 13) may be expressed,
: , .
8 fl~ 3
(nn~) = in ( Q~ cos ~n sin2 ~n)
+ jn ( ~ sin n sin2 ~n)
+ kn (~ sin ~n sin2 ~n)
To write the column vector nn' in the ~EG
coordinate system:
nnl = Mn(nn~)(Step 48, Figure 13)
As seen in Figure 12,
n+l = n' = n ~ nn'
where n is stored from previous calculations. The
~ location of n~ relative to the ground reference O
is thus determined.
With the vector n' pointing to the position
n'~ the azimuth An~ (see Figure 3) may be
expressed:
tan An, = n
Nn l
-- (Nn~2 + En-2)1/2
tan In~ ~ Gnl
where (Nnl, Enl, Gnl) are the coordinates of the
vector n' in NEG system with ground O as reerence
(step 49, Figure 13).
The derivation of the borehole trajectory from
gravity vector signals is preferably performed by a
programed digital processor. Figures 14 and 15 are
diagrammatic charts illustrating derivation of a
representation of the trajectory in NEG coordinates.
The illustration and description assume the use of
accelerometer signals from positions spaced apart at
distance Qin the borehole.
~he scalar inputs to Figure 14 are the digital
gravity vertor signals ax, ay and ax~, ay~.
Each of the blocXs of the diagram indicates algebraic-
ally or in words the function performed thereby. The
program will be described in general terms and related
to certain of the geometric explanations given above.
¦ ~ 66~ ~3
-- ~o --
At step 50, ax and ay are combined with
gravity g and an arc sin function is utilized at step
51 to obtain the inclination angle I for one position
in the borehole. Similarly, at steps 52, 53, ax-
and ay are utilized to derive I', the inclination
at the second point of the borehole. At step 54, the
ratio of ax to ay is taken: and at step 5S, the
arc tangent further gives a measure of the angle ~,
see Figures 3, 6 and 8. Similarly, ax and ay
are combined at steps 56, 57 to provide a signal re-
presenting the angle ~'. At step 58, the difference
~- ~' provides the angle Y , the shift in zenith
between successive positions along the borehole, see
Figure 10. The inclination angles I, I' and zenith
shift angle r are combined at steps 60, 61 to
determine the angle ~ representing the bearing of the
borehole between successive positions. At steps 62,
63 ~ is combined with the inclination angles I, I' and
shift angle Y to derive the bending angle ~.
The scalar quantities ~, ~, Y and I provide
inputs for the matrix/vector program illustrated dia-
grammatically in Figure 15. In the notation used in
Figure 15, M represents a borehole local coordinate
transformation matrix from (i, j, k)n~ to (i, j,
k)n and Mnl is the global coordinate trans-
formation matrix from (i, j, k~n~ to ~N, E, G).
The initial azimuth Ao for the probe is de-
termined as by the surveying instrument 35 and this
information is provided as an input to the system
through keyboard 30'. At step 70 a global matrix
Mo(Ao, Io)~ defines the starting position ~or the
probe. The form o the matrix Mo is indicated in
the footnote * to Figure 15. For the first measurement
position or n = 0, the matrix Mo fr~m 70 is con-
nected through gate 71 with matrix multiplier 72
~ ~8~3
.
- 21 -
The angles ~ and y are subtracted at step 73
to provide the angle ~ which is further combined with
and ~at step 74 to provide the matrix Mn which has
the form indicated in the footnote ** to Figure 15.
The matrix Mn is multiplied by the matrix Mn at 75
to provide a transformed global matrix Mn~l for the
next position along the borehole. This matrix is
delayed at step 76 and is coupled through gate 77 to
matrix multiplier 72 when n is lor greater, becoming
Mn for the succeeding measurement.
~ and ~are combined at step 78 to provide the
vector nn' which is multiplied by matrix Mn
at step 72, see Figures 11 and 12. The output of this
multiplication, nn' is connected with a vector
adder 80 where it is summed with the NEG coordinates
for the point n. At the first measurement location
(the borehole at the surface), these coordinates are
000. The result of the vector addition is the set of
NEG coordinates representing a point on the borehole.
This result is also connected through a unit delayor
step 81 as an input to the vector adder 80 for the
next measurement position. The successive sets of NEG
coordinates developed from successive accelerometer
measurements provide a representation of the borehole
trajectory.
The survey instrument described herein utilizing
servoed accelerometers provides reliable results so
long as the borehole is not within about one degree of
true vertical or true horizontal. If these conditions
are encountered, the accelerometer measurements should
be supplPmented with some other measurement of the
borehole trajectory.
, '
