CA1166843A - Borehole survey apparatus and method - Google Patents
Borehole survey apparatus and methodInfo
- Publication number
- CA1166843A CA1166843A CA000387999A CA387999A CA1166843A CA 1166843 A CA1166843 A CA 1166843A CA 000387999 A CA000387999 A CA 000387999A CA 387999 A CA387999 A CA 387999A CA 1166843 A CA1166843 A CA 1166843A
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- CA
- Canada
- Prior art keywords
- borehole
- axis
- accelerometer
- along
- accelerometers
- Prior art date
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- 238000000034 method Methods 0.000 title claims abstract description 10
- 239000000523 sample Substances 0.000 claims abstract description 53
- 230000005484 gravity Effects 0.000 claims abstract description 22
- 230000008859 change Effects 0.000 claims abstract description 5
- 238000005259 measurement Methods 0.000 claims description 34
- 230000001133 acceleration Effects 0.000 claims description 13
- 238000005070 sampling Methods 0.000 claims 1
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- 238000009795 derivation Methods 0.000 description 5
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- 101100384865 Neurospora crassa (strain ATCC 24698 / 74-OR23-1A / CBS 708.71 / DSM 1257 / FGSC 987) cot-1 gene Proteins 0.000 description 2
- 238000004364 calculation method Methods 0.000 description 2
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- 230000002596 correlated effect Effects 0.000 description 2
- 238000013500 data storage Methods 0.000 description 2
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- 230000001186 cumulative effect Effects 0.000 description 1
- 230000003111 delayed effect Effects 0.000 description 1
- 238000005553 drilling Methods 0.000 description 1
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- 238000000844 transformation Methods 0.000 description 1
- 229910052725 zinc Inorganic materials 0.000 description 1
Classifications
-
- E—FIXED CONSTRUCTIONS
- E21—EARTH OR ROCK DRILLING; MINING
- E21B—EARTH OR ROCK DRILLING; OBTAINING OIL, GAS, WATER, SOLUBLE OR MELTABLE MATERIALS OR A SLURRY OF MINERALS FROM WELLS
- E21B47/00—Survey of boreholes or wells
- E21B47/02—Determining slope or direction
- E21B47/022—Determining slope or direction of the borehole, e.g. using geomagnetism
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- Geology (AREA)
- Life Sciences & Earth Sciences (AREA)
- Engineering & Computer Science (AREA)
- Physics & Mathematics (AREA)
- Mining & Mineral Resources (AREA)
- Geophysics (AREA)
- Environmental & Geological Engineering (AREA)
- Fluid Mechanics (AREA)
- General Life Sciences & Earth Sciences (AREA)
- Geochemistry & Mineralogy (AREA)
- Geophysics And Detection Of Objects (AREA)
- Length Measuring Devices With Unspecified Measuring Means (AREA)
- Analysing Materials By The Use Of Radiation (AREA)
Abstract
ABSTRACT OF THE DISCLOSURE
An apparatus and method for surveying a borehole, mine shaft or the like to determine its trajectory. A sensing probe with two spaced sets of accelerometers measures components of the gravity vector along orthogonal axes at successive positions along the borehole. The two sets of accelerometers are joined by a connector such as a pipe or a cable which is flexible to bend but is torsionally stiff.
The angular orientation of the two accelerometer sets relative to each other about the borehole axis is fixed by the connector so that any difference in ori-entation of the two sets is a function of the local trajectory of the borehole. Two sets of accelerometer output signals representing gravity vector components at positions spaced apart along the axis of the borehole are utilized to derive the borehole inclina-tion at each position as well as the change in borehole azimuth angle between the positions of the accelerometer sets. The accelerometer signals, together with a signal representing the position of the probe along the length of the borehole, are com-bined to provide a three dimensional representation of the borehole trajectory, relative to a reference point, which may be the termination of the borehole at the surface.
An apparatus and method for surveying a borehole, mine shaft or the like to determine its trajectory. A sensing probe with two spaced sets of accelerometers measures components of the gravity vector along orthogonal axes at successive positions along the borehole. The two sets of accelerometers are joined by a connector such as a pipe or a cable which is flexible to bend but is torsionally stiff.
The angular orientation of the two accelerometer sets relative to each other about the borehole axis is fixed by the connector so that any difference in ori-entation of the two sets is a function of the local trajectory of the borehole. Two sets of accelerometer output signals representing gravity vector components at positions spaced apart along the axis of the borehole are utilized to derive the borehole inclina-tion at each position as well as the change in borehole azimuth angle between the positions of the accelerometer sets. The accelerometer signals, together with a signal representing the position of the probe along the length of the borehole, are com-bined to provide a three dimensional representation of the borehole trajectory, relative to a reference point, which may be the termination of the borehole at the surface.
Description
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 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 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 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 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.
. .
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 ~ ~
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 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' 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 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 . , 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 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:
- 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 ~
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 .
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.
, '
, 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 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 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 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 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.
. .
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 ~ ~
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 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' 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 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 . , 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 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:
- 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 ~
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 .
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.
, '
Claims (14)
1. A borehole survey apparatus, comprising:
a first pair of accelerometers having sensi-tive axes defining a first plane;
a second pair of accelerometers having sensi-tive axes defining a second plane;
means mounting both pairs of accelerometers to pass through said borehole with the planes of the ac-celerometers at an angle to the longitudinal axis of the borehole and spaced apart along the axis of the borehole and with the angular alignment of the accelerometer pairs about the borehole axis fixed with respect to each other; and means for deriving from each accelerometer a signal representing the component of gravity along the sensitive axis of the accelerometer.
a first pair of accelerometers having sensi-tive axes defining a first plane;
a second pair of accelerometers having sensi-tive axes defining a second plane;
means mounting both pairs of accelerometers to pass through said borehole with the planes of the ac-celerometers at an angle to the longitudinal axis of the borehole and spaced apart along the axis of the borehole and with the angular alignment of the accelerometer pairs about the borehole axis fixed with respect to each other; and means for deriving from each accelerometer a signal representing the component of gravity along the sensitive axis of the accelerometer.
2. The borehole survey apparatus of claim 1 including:
means for deriving from said accelerometer signals at positions spaced along the borehole a signal representing the inclination angle of the borehole at each position; and means for deriving from said accelerometer signals a signal representing the incremental azimuth angle of the borehole between positions.
means for deriving from said accelerometer signals at positions spaced along the borehole a signal representing the inclination angle of the borehole at each position; and means for deriving from said accelerometer signals a signal representing the incremental azimuth angle of the borehole between positions.
3. The borehole survey apparatus of claim 1 including:
means for deriving a signal representing the distance from a reference of each position along the borehole; and means for deriving from the accelerometer and distance signals the coordinates of the borehole positions with respect to said reference.
means for deriving a signal representing the distance from a reference of each position along the borehole; and means for deriving from the accelerometer and distance signals the coordinates of the borehole positions with respect to said reference.
4. A borehole survey apparatus, comprising:
a sensing probe to be moved through the borehole, having a first section with an axis extending along the borehole axis, a second section spaced from said first section and having an axis extending along the borehole axis, and means joining said two sections maintaining a fixed spacing between them, said means being flexible to bend along the axis of the borehole as said first and second sections change position relative to each other with changes of inclination and azimuth of the borehole, said joining means resisting rotation of one section with respect to the other about the borehole axis to maintain the angular alignment about the borehole axis of the sections with respect to each other;
a first pair of accelerometers in said first section, having their sensitive axes at right angles to define a sensitive plane at right angles to the axis of the borehole;
a second pair of accelerometers in said second section, having their sensitive axes at right angles to define a sensitive plane at right angles to the axis of the borehole; and means for deriving from each accelerometer a signal representing the component of gravity along the sensitive axis thereof.
a sensing probe to be moved through the borehole, having a first section with an axis extending along the borehole axis, a second section spaced from said first section and having an axis extending along the borehole axis, and means joining said two sections maintaining a fixed spacing between them, said means being flexible to bend along the axis of the borehole as said first and second sections change position relative to each other with changes of inclination and azimuth of the borehole, said joining means resisting rotation of one section with respect to the other about the borehole axis to maintain the angular alignment about the borehole axis of the sections with respect to each other;
a first pair of accelerometers in said first section, having their sensitive axes at right angles to define a sensitive plane at right angles to the axis of the borehole;
a second pair of accelerometers in said second section, having their sensitive axes at right angles to define a sensitive plane at right angles to the axis of the borehole; and means for deriving from each accelerometer a signal representing the component of gravity along the sensitive axis thereof.
5. The borehole survey apparatus of claim 4 in which, with the sensing probe sections aligned, the sensitive axis of each of the first pair of accelerometers is coplanar with the sensitive axis of the corresponding accelerometer of the second pair.
6. The borehole survey apparatus of claim 4 having a third accelerometer in each probe section with a sensitive axis along the axis of the section.
7. The borehole survey apparatus of claim 4 having:
a housing for each section; and means for centering the housings in the borehole.
a housing for each section; and means for centering the housings in the borehole.
8. The borehole survey apparatus of claim 7 in which said housings are free to rotate in the borehole.
9. The borehole survey apparatus of claim 4 having:
a housing for each sensor probe section;
and in which the means for joining the section is a connector fixed at each end to one of said housings, said connector having an axis which follows the axis of the borehole, the connector being rigid with respect to twisting about its axis and resilient to bend along the borehole axis as said housings shift with respect to each other at different positions along the borehole.
a housing for each sensor probe section;
and in which the means for joining the section is a connector fixed at each end to one of said housings, said connector having an axis which follows the axis of the borehole, the connector being rigid with respect to twisting about its axis and resilient to bend along the borehole axis as said housings shift with respect to each other at different positions along the borehole.
10. The borehole survey apparatus of claim 4 including means for deriving from the accelerometer signals a representation of the borehole trajectory.
11. The method of surveying a borehole which comprises:
measuring the acceleration of gravity along two different axes at successive pairs of points along said borehole the points of each pair of points being spaced apart along the borehole, the axes at each point of a pair of points having a known relation to each other and to the borehole;
generating signals representing said accelera-tions;
providing a measure of distance of said points along the borehole; and deriving from said acceleration signals and the measure of distance a representation of the borehole trajectory.
measuring the acceleration of gravity along two different axes at successive pairs of points along said borehole the points of each pair of points being spaced apart along the borehole, the axes at each point of a pair of points having a known relation to each other and to the borehole;
generating signals representing said accelera-tions;
providing a measure of distance of said points along the borehole; and deriving from said acceleration signals and the measure of distance a representation of the borehole trajectory.
12. The method of surveying a borehole of claim 11 in which the two different axes at each point are orthogonal axes which define a plane at right angles to the axis of the borehole.
13. The method of surveying a borehole of claim 11 in which the representation of the borehole trajectory is in terms of coordinates related to a reference point.
14. The method of surveying a borehole of claim 11 in which:
two sets of spaced apart accelerometers are moved through the borehole; and successive measurements are made by sampling signals from the accelerometers.
two sets of spaced apart accelerometers are moved through the borehole; and successive measurements are made by sampling signals from the accelerometers.
Applications Claiming Priority (2)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| US20009680A | 1980-10-23 | 1980-10-23 | |
| US200,096 | 1980-10-23 |
Publications (1)
| Publication Number | Publication Date |
|---|---|
| CA1166843A true CA1166843A (en) | 1984-05-08 |
Family
ID=22740315
Family Applications (1)
| Application Number | Title | Priority Date | Filing Date |
|---|---|---|---|
| CA000387999A Expired CA1166843A (en) | 1980-10-23 | 1981-10-15 | Borehole survey apparatus and method |
Country Status (12)
| Country | Link |
|---|---|
| JP (1) | JPS6015883B2 (en) |
| AU (1) | AU533909B2 (en) |
| CA (1) | CA1166843A (en) |
| DE (1) | DE3135743C2 (en) |
| FR (1) | FR2492882A1 (en) |
| GB (1) | GB2086055B (en) |
| IT (1) | IT1142908B (en) |
| MX (1) | MX150339A (en) |
| NL (1) | NL8104801A (en) |
| NO (1) | NO813568L (en) |
| SE (1) | SE8105959L (en) |
| ZA (1) | ZA817068B (en) |
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| US5493909A (en) * | 1991-01-30 | 1996-02-27 | Mitsubishi Denki Kabushiki Kaisha | Method of and an apparatus for detecting control information |
| CN110906850A (en) * | 2019-12-04 | 2020-03-24 | 昆明理工大学 | Accurate positioning device and method for inclined deep hole sensor |
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1981
- 1981-08-18 AU AU74273/81A patent/AU533909B2/en not_active Ceased
- 1981-09-09 DE DE3135743A patent/DE3135743C2/en not_active Expired
- 1981-10-01 GB GB8129597A patent/GB2086055B/en not_active Expired
- 1981-10-08 SE SE8105959A patent/SE8105959L/en not_active Application Discontinuation
- 1981-10-13 ZA ZA817068A patent/ZA817068B/en unknown
- 1981-10-15 CA CA000387999A patent/CA1166843A/en not_active Expired
- 1981-10-15 JP JP56163573A patent/JPS6015883B2/en not_active Expired
- 1981-10-21 IT IT49530/81A patent/IT1142908B/en active
- 1981-10-22 FR FR8119842A patent/FR2492882A1/en active Granted
- 1981-10-22 NO NO813568A patent/NO813568L/en unknown
- 1981-10-23 MX MX189789A patent/MX150339A/en unknown
- 1981-10-23 NL NL8104801A patent/NL8104801A/en not_active Application Discontinuation
Cited By (2)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| US5493909A (en) * | 1991-01-30 | 1996-02-27 | Mitsubishi Denki Kabushiki Kaisha | Method of and an apparatus for detecting control information |
| CN110906850A (en) * | 2019-12-04 | 2020-03-24 | 昆明理工大学 | Accurate positioning device and method for inclined deep hole sensor |
Also Published As
| Publication number | Publication date |
|---|---|
| DE3135743C2 (en) | 1984-12-06 |
| FR2492882A1 (en) | 1982-04-30 |
| ZA817068B (en) | 1982-09-29 |
| MX150339A (en) | 1984-04-17 |
| JPS6015883B2 (en) | 1985-04-22 |
| NL8104801A (en) | 1982-05-17 |
| FR2492882B1 (en) | 1984-12-28 |
| GB2086055B (en) | 1985-04-17 |
| AU7427381A (en) | 1982-07-01 |
| GB2086055A (en) | 1982-05-06 |
| IT8149530A0 (en) | 1981-10-21 |
| DE3135743A1 (en) | 1982-05-19 |
| SE8105959L (en) | 1982-04-24 |
| IT1142908B (en) | 1986-10-15 |
| JPS57100308A (en) | 1982-06-22 |
| NO813568L (en) | 1982-04-26 |
| AU533909B2 (en) | 1983-12-15 |
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