CA2112777A1 - Calibration correction method for magnetic survey tools - Google Patents
Calibration correction method for magnetic survey toolsInfo
- Publication number
- CA2112777A1 CA2112777A1 CA002112777A CA2112777A CA2112777A1 CA 2112777 A1 CA2112777 A1 CA 2112777A1 CA 002112777 A CA002112777 A CA 002112777A CA 2112777 A CA2112777 A CA 2112777A CA 2112777 A1 CA2112777 A1 CA 2112777A1
- Authority
- CA
- Canada
- Prior art keywords
- borehole
- vector
- earth
- measurement
- magnetic field
- Prior art date
- Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
- Abandoned
Links
Classifications
-
- E—FIXED CONSTRUCTIONS
- E21—EARTH DRILLING; MINING
- E21B—EARTH DRILLING, e.g. DEEP 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
Abstract
ABSTRACT OF THE DISCLOSURE
A method for determining the orientation of the axis of a borehole with respect to an Earth-fixed reference coordinate system at a selected series of locations in the borehole, the borehole having a trajectory, and adapted to receive a drill string, comprising defining a model for the influence of magnetic interference from elements of the drill string on the measurement of components of the Earth's magnetic field in the borehole in terms of an unknown vector, a measurement vector, and a measurement matrix relating the unknown vector and the measurement vector;
measuring at two or more selected locations along the borehole trajectory:
1) at least one of:
i) two cross-borehole components of the Earth's gravity field at the selected locations in the borehole, and ii) two cross-borehole components and an along-borehole component of the Earth's gravity field at the selected locations in the borehole;
2) two cross-bore hole components of the Earth's magnetic field and an along-borehole component of the Earth's magnetic field at the selected locations;
computing the elements of the measurement vector and the measurement matrix from the measured Earth's gravity and Earth's magnetic field components for the selected locations; solving for the unknown vector using the measurement vector And the measurement matrix; computing for each of the selected locations corrected values for the Earth's magnetic field components using the unknown vector, the model, and the measured Earth's magnetic field components; and determining a value for the azimuthal orientation of the borehole axis at each the selected locations using the corrected values for the Earth magnetic field components and the measured gravity components.
A method for determining the orientation of the axis of a borehole with respect to an Earth-fixed reference coordinate system at a selected series of locations in the borehole, the borehole having a trajectory, and adapted to receive a drill string, comprising defining a model for the influence of magnetic interference from elements of the drill string on the measurement of components of the Earth's magnetic field in the borehole in terms of an unknown vector, a measurement vector, and a measurement matrix relating the unknown vector and the measurement vector;
measuring at two or more selected locations along the borehole trajectory:
1) at least one of:
i) two cross-borehole components of the Earth's gravity field at the selected locations in the borehole, and ii) two cross-borehole components and an along-borehole component of the Earth's gravity field at the selected locations in the borehole;
2) two cross-bore hole components of the Earth's magnetic field and an along-borehole component of the Earth's magnetic field at the selected locations;
computing the elements of the measurement vector and the measurement matrix from the measured Earth's gravity and Earth's magnetic field components for the selected locations; solving for the unknown vector using the measurement vector And the measurement matrix; computing for each of the selected locations corrected values for the Earth's magnetic field components using the unknown vector, the model, and the measured Earth's magnetic field components; and determining a value for the azimuthal orientation of the borehole axis at each the selected locations using the corrected values for the Earth magnetic field components and the measured gravity components.
Description
- 21~2777 ~ACRGROUND OF TH~ INVENTION
It is generally well known that magneti~
survey tools are disturbed in varying waya by anomalous magnetic fields assoc~ated with fixed ox induced magnetic fields in elements of the drill string. It i5 further well known that the predominant error component lies along the axis of the drill qtring. This latter fact i8 the basis for several patented procedures, to eliminate the along-axis field errors in 3-magnetometer survey tools. Among these are U.S. patents:
4,163,324 to Russell et al.
4,433,491 to Ott et al.
4,510,696 to Roesler 4,709,486 to Walters 4,682,421 to Van Dongen et al.
4,761,889 ~o Cobern et al.
4,819,336 to Russel 5,155,916 to Engebretson U.K. patents 2,138,141A to Russell et al. and 2,185,580 to Russell.
U.S. Patent 5,155,916 to Engebret60n provides a method for error reduction in compensation for magnetlc lnterference.
All o2 these methods, in e~fect, ignore the output of the along-axis magnetometer, except perhap~
for selectlng a slgn for a ~quare root computation.
They providQ an azlmuth re3ult by computatlon of a synthetic solutlon, either~
1) by using only the two cross-axls 21~ 2777 magnetometer~ and known characteristic~ o~ the Earth field, or 2) by using the crosB-axl8 components and an along-axis component computed ~rom the cross-axis components and known characterlstics of the Earth'o field. ~ ~-Most o~ the~e require, as the known characteri~tic3 of the Earth ~ield, one or ~ore o~ the following:
1) field magnitude 2) dlp angle 3) horizontal component 4) vertical component.
~he Naltsrs method reguires, as known characteristics of the Earth field, only that~
1) the fleld ~agnitude i~ constant in the survey area~
2) the dip angle i8 constant in the survsy area.
ThQ fact that these quantlties are constant is all that is reguired. The value o~ the con~tant is not needed but is dQrived within the correction algorithm.
Since all of these compensation methods use, in effect, a computed along-axis component, all of the~
break down ~or cases o~ borehole hlgh inclination angles in a generally Ea~t/We~t dlrection. Thl- ~s because the cross-axls msa~urement plane ~or such conditlon tsnd- to be aligned 80 a~ to contain both the gravlty and Earth ~leld vector~, and thus measure~ents in this plane provide a poor nea~ur- o~ the cross 2i1 2r~77 product o~ the Earth field and gravity vectors. The cross product vector of the ~wo reference vector~ i3 the vector that actually contains the directional reference information.
The actual degradation of accuracy at high inclinations in the East/Weet direction for the previously cited methods depends both on the inherent accuracy of the ~ensors in the ~urvey tool and on the accuracy of the required knowledge o~ the Earth field characteri~tics.
To provide a mechanization for a magnetometer survey tool that does not seriously degrade in accuracy at borehole higher inclination angles near the East/WTst direction, it is found to be nece~sary to provide a method and means to calibrate the error~ in the along-axi~ magnetometer 80 that accurate measurements can be made with it. Thi~ ia in direct contra~t with existing ~ethod~ that substitut~ co~puted values for along-axls measurements.
There i8, therefore, need to provide a calibration method for an along-axis ~agnetometer in a magnetlc survey tool to correct anomalous magnetic effect~ ln a drlll strlng and thereby to pernit accurate measurements of the along-axis component of the Earth magnetio field. Such accurat~ ~ea~urement of the along-axis component then per-it~ accurate co~putatlon o~ azimuthal direction lndependent of inclinatlon and directlon.
-`- 2112777 SUMMARY OF TH~ INVENTION
The drill string anomalou~ magnetizatlon i8 :
composed of both a fixed component re~ulting from permanently magnetized elements in the bottom hole ~ -assembly and the drill string, and an induced component resulting from ~he interaction of ~oft magnetic materials with the Earth field The along-axi~
component of the induced field can be expected to be proportional to the along-axi~ component of the Earth field This mo~el of a fixed error and an along-axis induced field proportional to the true along-axis Earth field can be interpreted as simply alterlng the basic along-axis magnetometer's bias or o~set error, and its scale factor for measuring the Earth field component In its simplest form, the present invention ;~
provide~ a method, includ~ng the steps Or determining a set of along-axis magnetometer errors at different points along the borehole path by any of the well known methods, and then fltting these errors to a model, as referred to, for bias and scale factor 80 that ac~urate along-axis mea~urements can be computed using the determined bias and scalQ ~actor value~
In a more generalized embodlment, the inventlon provldes a method to callbrate the ef~ect~ of ~Agnetic lnter~erence from the drlll strlng that includes modellng the lnterference e~fects as an unknown ve¢tor that lnclude~ a~ elements the ano~alous scale factor and blas e~ect~; maklng a serles measurement~ at a number of different survey locations - S - ~ :.
along the borehole; forming from the measurement data a measurement vector and a measure~ent matrix relating the mea~urement vector to the unknown vector; and solving the unknown vector. The element~ of the s unknown vector may then be used to compute accurate along-axis measurements and for quality control purposes.
These and other ob~ects and advantages of the invention, a~ well as the details of an illu~tratlve embodi~ent, will be more fully understood from ths following specification and drawing~, in which:
~RAWqNG D~SCRIPTION
Fig. 1 shows a typical borehole and drill string, $ncluding a magnetic survey tool;
Fig. 1~ shows a survey tool in a drill collar, as used in Fig. l;
Fig. 2a shows the influence o~ a piece Or high permeability magnetic material when placed parallel to an orlginally undisturbed magnetic field;
Fig. 2_ shows the influence of a piece Or high permeab~llty magnetic material when placed perpendicular to an orlginally undisturbed magnetic field; and Flg~. 3a, 3~, 3, and 3_ show a coordinate set in relation to ~ borehole and an Earth-fix~a ;~
coordinato ~et.
211277~
Dl~rAII~D DESCRIPTION
Fig. 1 shows a typical drllllng rig 10 and borehole 13 in ~ection. A magnetic survey tool 11 is ~hown contain~d in a non-magnetic drill collar 12 S (made, for example, of ~onel or other non-magnetlc material) extending in line along the borehole 13 and the drill string 14. ~he ~agnetic ~urvey tool i~
generally of the type de~cribed ln U.S. Patent 3,862,499 to Isham et al., lncorporated herein by reference. It contains three nominally orthogonal magnetometers and three no~inally ortho~onal ~ -accelerometer6 for ~ensing components o~ the Earth'~
magnetic and gravity ~ields. The drill string 14 above the non-~agnetic collar 12 i8 of ferromagnetic material ~for example ~teel) having a high permeability compared to the Earth ~urrounding the borehole and the non- ;
magnetic coll~r. There may, or may not, be other ferromagnetlc materlals contalned in the drill assembly 15 below the non-magnetlc collar. It 1~ generally well known that the ~erromagnetic material~ above, and -po~slbly below, the non-magnetic collar 12 cau~e anomalles in the Earth's magneti¢ ~ield ln the reglon of the survey tool that in turn cause errors in the measurement o~ the azi~uthal direction of the ~urvey tool.
It 1~ further well know that such ano~alies ~ay lnclude both rixed and lnduced error riold~, the --fixed error rields resulting from residual ~agn~tic erfects in th~ ~erro~agnetic materials and th~ lnduced 2~12777 error f~elds resulting fro~ di~tortlon o~ the Earth'~
true field by the high-permeability ~erromagnetic mater~als. It i8 also well known from both theoretical considerations and experiment that the predominant error field lies along the direction o~ the drill string. It i8 this latter knowledge that the predo~sinant error lies along the drill string direction that has led to all of the previou~ly cited methods to eliminate ~uch an error component. As previously tated, all such ~ethods discard the measurement along the drill string axi~ and find either a two-component solution or a three-component solutlon in which the third component i~ computed mathematically. All of these previous methods, therefore, result in ~ignificant error when the borehole path approaches a near-horizontal, near East/West direction.
Figs. 2a and 2b show the effects o~ a long ; piece of metal 16 of hlgh permeability, lmmersed ~n an initlally uni~orm nagnetic field. The ~ield lines 17 are dlstorted by the pr~sence of the high permeability materlal. In Fig. 2a, th~ plece 16, generally tubular, i~ shown placed parallel to the original field, and in Fig. 2k perpendicular to the original ~ield. ~ ~-A~ can be ~een fro~ the figures, there i~
2S con~iderable increa~e in the den~ity of field l~nes near the end~ o~ piec~ 16, in region- 18, in Fig. 2 when the piece 16 i~ parallel to the original ~leld.
In Flg. 2k, when the piece 16 1~ perpendicular to the original ~leld, there i8 only a ~mall increase in the den~ity o~ the field lines in the reglon~ 18. Further, ~S, . '~: ' . ' ' ' ., . ! . , " , , 2~12777 : ::
it can be noted that the field lines along thq line of the axis 19, of piece 16, in regions 18, have the same direction as the original ~ield lines. It may be verified either analytically or experimentally that for any arbitrary orientation of the piece 16 to the original field, the end result will be the superposition of the effect~ of the two components that may be resolved as along-axis and cross-axis to the piece.
A similar pattern to Fig. 2a (except that the field patterns close to loop from one end to the other) results if the piece 16 contains residual, permanent magnetic materials having poles lying along the axis 19. These patterns generally presented here are the basis for the previously cited correction algorithm~
used to avoid errors from magnetic e~fQcts in the drill string and bottom hole assembly. As previou~ly cited, ;~
the assumption used is that the along-borehole error is the predominant error, and that by not using the measurement along the borehole axi~, the error i8 avoided.
It may be shown either analytically or experimentally that the magnitude of the field anomalies shown in Figs. 2a and 2_ are linearly proportional to the original, undi~turbed ~ield aa long as the permeabillty o~ the piece 16 is constant with ~ield strength. Further, for the general case, tho ~ield along the axis 19 will be directly proportional to the cosine o~ the angle between the axls 19 and the total ~ield vector o~ the original, undisturbed ~ield.
~` 2112777 Figs. 3a, 32b, 3c, and 3d show an x, y, z coordinate set and the direction of a borehole axis 20, that i8 assumed to be colinear with the drill string 14 of Fig. l. Defining the Earth's magnetic field as the vector 2H having components Hx, ~, Hz, along the three axes of the survey tool ll, the measurements of the three magnetometers in the survey tool will be:
x-Magnetometer Hx (l) y-Magnetometer ~ ~2) z-Magnetometer Hz (3) in the absence of any di~turbances from magnetic materials in the drill 6tring.
Similarly, defining the Earth's gravity as the vector G, the measurements of the three accelerometers in the survey tool will be~
x-Accelerometer Gx (4) ~ ~
y-Accelerometer Gy (5) ~ -z-Accelerometer Gz (6) In Fig. 3, starting wlth the three-axls, Earth-fixed coordinate set, N, 2~, 2Q--(representlng North, East, and Down) where the underllne represent~
a unit vector ln the dirèction given, the orientation of the set of tool axes 2~2, y, ~ 18 deflned by a series of rotatlon angles, AZ, TI, HS (representing AZimuth, TIlt, and HighSlde). In thls nomenclature, x i8 rotated by HS from the vertlcal plane, y 18 normal to x, and z 1~ down along the borehole ax~s. The formulatlon of the calculation of azimuth, adapted from U.S. Patent 3,862,499, is:
2il2777 AZ = ArCtan (HX*Sin(HS)+P~,*C08(HS) ) .
COB(~I) *HX*CO~(HS)_HY*Sin(HS) )+HZ*Sin(TI) In this equation, Hx, ~, and Hz are the three magneto~eter-measured component The angle~ ~I and HS
are solved for from tha thr~e accelerometer-mea~ured components by well known methods in previous steps.
If there are induced field and permanent field effects from materials in the drill string, defined as HI and HF, re~pectively, and the symmetries are as discussed in Fig. 2 above, then the x- and y-magnetometer measurements will remain as above, but the ;~
z-magnetometer measure will become~
z-Magnetometer Hz+HI+HF (8) However, the induced field, HI, wa~ previously stated to be proportional to the original field along th~ axis of the magnetic material and, therefore, it must be - proportional to Hz. If one describes the proportionality by a constant, RI, then:
HI ~ KI*Hz (g and the z-magnetometer measurement then becomes:
z-~agnetometer (l+KI)*Hz+HF (10) This shows that the output of the z-magnetometer mea~urement may be lnterpreted ~ust like the other two measurement~, but that the scale factor of the measurement i~ now tl+KI); and there 1~ an off~et or blas-type o~ error, HF, added to the maasurement. I~
the value~ o~ KI and HF could be determined, then the z-magnetometer output could be used in azi~ut~
computation without error and the magnetic in~luence of the drill ~tring could be avoided without encountarlng the problem o~ lncreasing error as the hlgh tilt, East/West condition i8 approached.
There 1B no way that the two unknown~, RI and HF, can be determined ~rom a ~ingle set o~ mQasuxements at ona survey sta~ion. However, ~rom a series of two or more measurements at different locations along the borehole where the z-axis components of the Earth'~
field, Hz, are different, a eolution for the two unknowns may be found. A series of mea~urements may be ;~
expressed as:
Hz~(~ z(l)*~l+XI)+H
Hzm ( 2 ) ~ HZ ~ ;! ) * t l+KI ) +HF
Hzm(3) - HZ(3)~1+KI)+HF (11) Hzm(4) ~ HZ(4)*(1+KI)+HF
Hzm(5) 3 Hz(5)~(l+RI)+HF
.
Hzm(n) - Hz(n)*(l+KI)+HF
where Hzm(n) represents the n-th measurement at the n-th location along the borehole of the z-axis magnetlc field component and Hz(n) represent~ the corresponding n-th true s-axis co~ponent o~ the Earth'o true ~ield, not includlng th- anomalle~ re~ultlng ~rom magnetlc materlals in the drlll string or other bottom hola asseably components.
The prevlou~ly cited aethods ~or correctlon o~ aagnetic erroro do not use the z-axi~ mea~ur~ent.
They do, however, elther compute a z-axi~ component or compute ~n azlmuth wlthout such a component (from whlch a z-axlo component may be co~puted). Since, except ~or '' region~ near high inclination East/West, the azimuth results have been shown to produce reasonably accurate results, it follows that such computed z-axis components are much more accurate than measursd z-axis s components. Thus, in the above series o~ measurements Hzm(n), i~ the corresponding Hz(n) values are computed by any of the cited methods, the set of measurement equations may be solved for the two unknowns, KI and F- -Since there are two unknowns, a minimum o~
two measurements (to provide two eguations) is required. For example:
Hz~(l) = Hz(l)*(l+KI)+HF (12) Hzm(2) - HZ(2)*(1+KI)+HF (13) may be solved to obtain:
X = (Hzm(l) - Hzm(2)) 1 (14) (Hz(l) - Hz(2)) H Hz(l)*Hzm(2) - HZ(2)*Hzn(l) (15) Hz(l) - Hz(2) In general, if there are more measurements than there are unknowns, the system of measurement equations i8 said to be overdetermined. However, considering various errors that may be involved in the measurement or computation proces~ed, it i~ ~tlll desirable to uBe a~ much mea~urement data a~ po~ible to minimize error~ in the unknown~' value~ ~ought.
This i~ the classical problem o~ parameter estimatlon that ha~ been addressed in many ~leld~. One well-known method leading to what i~ known a~ a Hleast-~quared-211277~
error result" i8 shown below. ~:~
The set of measurements Hzm(~ Hz~(n) can berepresented as the n-element vector -zm~ called "the measurement vector", where the vector notation 18 indicated by the underscore.
The unknown quantities (l+KI) and ~F may be represented as a 2-element vector x. These vectors may be related by writing:
Hz~ Hx + y (16) where H, a matrix called the measurement matrix, is an nx2 matrix:
Hz(l) 1 Hz(2) Hz(3) Hz(4) 1 ~:
H = . (17) :
Hz(n) 1 ~:
and where ~ is the unknown vector:
XI : :~
' (18) HF
and y is a vector of measurement ~nolse~. The solution deslred i8 that for the "best" estimate of x, minlmlzing the effects o~ the measure~ent ~noisen.
When the ~best~ criteria 1~ defined as that solution, :~: .
that minimizQs the sum of the squarQs o~ the elements f ~zm ~ ~, where the symbol ^ over ~ lndicates the best estimate Or ~, then lt may be ~hown that - 14 - - :
:
2112777 ~ ~
X ~ (HTH)-l HT Hzm (19) where HT i8 the transpose of th~ nx2 matrix H and (HTH)-l is the matrix inverse of the matrix HTH. ~-The method shown above, as represented by equations (11) through (19), will result in some error in the determination of the desired unknown KI and HF
that depends on errors in the reference values of the Earth's magnetic field, since errors in these quantities will produce some error in the computed "true" values of Hz(n). Such reference-induced errors are the accuracy limiting factors in the correction algorithms of the previously cited patents.
A method that does not depend on Earth' 8 field reference may be found by generalizing the problem. In general, a series of measurements of some quantity, for example z, can be represented as the value, for example x, plus some unknown measurement error, for example v. The ~eries of measurements may ;~
be written in vector/matrix notatlon as:
z - H ~ ~ v (20) zl ' `' ' ` ~ :~
z2 :
where: ~ - z3 an n-element measurement vector (21) ;~ :
~.
~:
zn 21~2777 ~
~, x - ~n m-element unknown vector (22) XDI
H i8 an n- by m-element me~surement matrix (23) vl v2 v - v3 an n-ele~ent measurement error vector ~24) ' . , vn Given these definltions, consider an unknown vector x defined as:
HTotal HVertical x - HNorth (25) : -~
XI
~: HF ~ -~
20 where, in addition to the previously defined KI and HF:
H~otal i8 the tot~l Earth field ~ :
magnitude at the borehole :: locatlon.
~Vertical i8 the vertical component of the total Earth field.
th i8 the horizontal North component o~ the rield~
Here the Earth-field guan~ities are added to th~
unknown vector and will be estimated, ~long wlth tho z-axis ~agnetometer wale factor and bia-, XI and HF. ~-2~ 27~7 A new measurement vector z and a new measurement matrix H are required for this expanded problem. Also, since there are mors unknowns, more equations and more survey location~ will be required.
Define the two vectors of unit length, ~ and g as:
H (26) Hx^ + Hy~ + HzA
G
_ (27) Gx^ + Gy~ + Gz^
That is, the vector ~ i5 a unit vector in the sa~e direction as the Earth's magnetic field vector H, as measured by the magnetometers; and ~ is a unit vector in the same direction as the Earth's gravity field G, as measured by the accelerometers. The vector g i5 thus along the direction D (Down) in Fig. 3a.
Now de~ine a unit vector r as: -~
r ~ _ (28) Ig X E21 That i5, the vector ~ is a vector that i8 the vector cross product of ths vectors ~ and ~ divided by the absolute magnltude of the ~ame vector cross product.
Thu~, ~ i8 a unit vector; and, ~ince, by deflnltion, the vectors ~ and Ç, and thereby the vector~ p and g li~ in the North-South plan~ o~ Fig. 3~, th~ vector E
is in the E (East) direction o~ Fig. 3a.
La~tly, define a vector 8 as:
X ~ (29) ~1~2777 That 18, the vector B i~ the vector cro~s product o~
the vector~ r and ~, and is thus a unit vector in the N
(North) direction in Flg. 3a.
Each o~ the three vectors, ~, g, and ~, ha~
three components--one component along the x-axis, one component along the y-axis, and one component along the z-axis. The three component~ for the vector ~ are defined a~ Px, py~ and pz. The three co~ponents ~or the vector ~ are defined a~ qx, qy, and qz. The three components for the vector g are defined as ~x~ ~y~ and 8z~
With these three unit vector definitions for 2, ~, and B, these three vectors may be computed at e~ch survey location from the mea~ured H and G vector~
lS Then, three elements of the total measurement vector z ~--may be computed for each survey locatlon. For exa~ple, consider the three eleuents of the vector z for the first location, which are to be computed a~
Z t z(2) ~ ~m- g t30) Z(3) ' ~m- 8 That is, each of these three elements is ~ust the vector dot product of the measured magnetlc field vector ~ and the ~, ~, and B vector~ defined and computed, as shown $n eguation~ (26) through (29).
ztl) i~ thu8 a mea8ure o~ the total magnetlc field;
z(2) i~ a measure of the vertical component o~ that field: and z~3) i~ a Dea~ure o~ the horizontal North component Or that total ~ield. -~
A~o, for each survey ~tation, three rows Or -~
2~12777 the measurement matrix H that relates the ~easurement vector z to the unXnown vector x may be computed in terms of the measured magnetic field component~ and the element~ of the vectors ~, ~, and 8. These three rows, for the unknown vector _, as defined by equation (25) are:
1 0 0 pz*Hz pz 0 1 0 qz*Hz gz ~31) o 0 1 sz*Hz 8z Since there are five unknown quantities in ths unknown vector x, the quantities ~hown in equations (30) and (31) are not sufficient to solve for the unknown vector x by using equation (19). A minimum of three survey locations i9 recommended. More locations will increase the accuracy of the determination of the unknown vector _.
As previously stated, three elements of the measurement vector z are computed a~ in equation (30);
and three rows of the measurement matrix H are computed a~ in equation (31) for each survey location. I~ the recommended minimum of three survey locations is used, the meaeurement vector ~ becomes a nine element vectdr, and the measurement matrix H becomes a nine row by ~ive column matrlx. If six survey location~ were used, the mea~urement vector would have èighteen element~, and the ~easurement matrix would have eighteen row~ and still five column~.
When sufficient survey locations have been accumulated, then the unknown vector ~ olved for using equatlon (19~ with the measurement vector ~
21~27~7 substituted for Hzm~ This then provides results that indicate HT0tal~ the total maqnetic field; HVerti~a the vertical component; HNorth~ the horizontal component; KI, the desired anomalous scale ractOr caused by the induced magnetization from drill ~tring elements: and HF, the desired anomalous bias or offset ~-resulting fro~ the fixed magnetization in the drill string elements.
The values--HTOtal, Hvertical, and HNorth may be used for quality control purposes, or as input reference data to any of the previously cited patent methods of interference compensation that require input data on the local Earth magnetic field. For quality control purposes, these values may be compared to ;~
reference values obtained from maps or Earth-field computer models. If the values depart significantly from the reference values, it indicates either a possibl~ failure in the sensor~ ln the ~urvey tool or a ~-significant geophysical local varlation in the Earth's field. Either one of these possibilities alerts the survey operator to possible serious survey error.
The values of KI and HF may be used to compute corrected values of the z-axis magnetometer measure of the Earth fleld for each survey location as~
HZc(n) ~ ( Hzm(n) HF ) (32) 1 + KI
where: Hzc(n) 1~ the corrected z-axl~ value ~or locatlon n Hzm(n) 18 the measured z-axls value ~or locatlon n ` 2~2777 Thls corrected valus for location n may then b~
combined with the measured x-axis and y-axls measured magnetic components to solve for the borehole azlmuth at location n, a~ shown in equation (7).
Some care is needed in select~on of the locations that are to be used in the solution for the model unknowns. As is well known, if the borehole path is a straight line, all of the measurement equations are highly correlated, and there will be no viable ~olution for the unknowns. In effect, all of the equations are equivalent and, therefore, the requirement that the number of equations must equal or exceed the number of unknowns is not met. In general, the methods of selecting the equations for solution of simultaneous equations are well known; and methods are known for the analysis of probable errors in the solution for unknowns from multiple equations.
Further, in the implementation of the methods described here, it may be found that some measurements or est$mates of the along-borehole magnetic field are more accurate than others in the 6eries of locations ~o be used for solving for the unknowns. In this event, the well-known method~ of "welghtQd" solut~on for unknowns may be used wherein the mor~ accurate data ~8 HweightedH more heavily ln the solution to ~inim~ze errors.
The vector v of measurement error~, defined at equation (24), may ~e further characterized in ~-;
general by a matrix computed from its elements that $8 usually designated a~ the covariance matrix of the :~
'2112777 error vector an~ i~ often designated by the letter R.
This matrix is computed as the expected value of the matrix product of the vector v and its transpo~e.
Thus~
R s E( y * vT ) whe-e: E designates the expected value of the product Superscript T denotes the transpose.
With this definition and the terms deflned above, it may be shown that the optimum e~timate o~ the unknown elements in the vector x that minimize~ the 6um of the ~ :
squared errors in the estimate i~ given by~
x = ( HT~R-l*H )-1 * ( HT*R-l ) * z (34) -;
where: * denotes matrix product Superscxipt T is transpose Superscript -1 denotes matrix inverse ~he actual values of the elements of the ~easurement noi6e vector y are now known. I~ they were ~ ~ -known, the values could be subtracted from the elements of the measurement vector and the proble~ could then be solved with errorless measurement data. ~owever, the expected statistical value of the element~ of y can be computed by error analysis of the elements of the measurement vector. Such errors will, in general, depend on the error~ ln all of the sensors and on the orientation of the borehole with respect to the Earth coordinate set. When these expected values havs been determined, the covariance ~atrix R can be determined using equation (33); and then the unknown vector ~ ~ay be determined using equation (34).
~1~2777 Both equations (19) and (34) are well known means to obtain so-called "least ~quared error"
estimates of unknowns from an overdetermined set of equations. They require computer storage and manipulation of all of the data included in the solution. Recursive "least squared error" formulation~
that work incrementally on the data, for example as each new survey location data becomes available, rather than waiting for the complete data set from all locations to be used, are well known. One of the better Xnown 6uch recursive methods i~ the method called "Kalman filtering", after its developer Dr.
R. E. Kalman. This method i8 described ln Chapter 4 of the book Applied Optimal Estimation, Arthur Gelb et al., The M.I.T. Press, Cambridge, Massachusetts. In this method, the input sensor data is processed as it is received and a continuing estimate of the unknown vector and lt~ covarlance matrix i~ computed. The recursivs formulation el$minates the need to provide ever increasing ~torage and to process matrix computation~ o~ ever growing matrix dimensions as larger nu~bers o~ data sets (survey locations) are addQd to the computation. This mechanization 1~ a real time formulation that at each cycle (each new survey location) provides an optimal estimate o~ the unknown~.
Additionally, it may be desirable to use more complex models ~or the magnetic inter~erence. For example, the equivalent ~cale ~actor anomaly may require non-llnear term~ or temperature dependent terms to achieve higher accuracle~ under some conditions.
2~2777 .`~
Also, the unknown vector may be expanded to include anomalous scale factor and bias terms for the cross-borehole magnetometers. Thi~ would account for cases in which the drill string magnetic interference was not principally along the axial direction as originally assumed. As the unknown vector i8 expanded, the measurement matrix must be expanded 80 that the number of columns in it is equal to the number of elements in the unXnown vector. The general method above may ~till be used, but it must be recognized that more unknowns lead to a need for ~ore independent measurement equations from which elements o~ the measurement vector are to be computed.
Another solution to the problem of obtaining the best result~ from a series of~multiple measurements is that Xnown as ~Optimal Linear SmoothingH, as described in Chapter 5 of the book Applied Optimal ~;
EstimatiQn, Arthur Gelb et al., The M.I.T. Press, Cambridga, Massachusetts. In this formulation, all of the measurement data i8 comblned to provide an optimal estimate of all elements of the state vector describing the physical process at each point within the serles of measurement~. Smoothing, a~ derined in the clted reference, is a non-real-time data processing scheme that uses all measurements. The general methodology 18 generally related to the well-known Kalman filtering methods. In optimal smoothing, in errect, Xalman f~ltering is applied to the data set both rrOm the first data point rorward in time to the location of interest; and from the last data point backwards in ~ 2112777 time to the same location of interest.
In its simplest forn then, the essential elements of the invention described herein are:
1) Select a model for the magnetic interference effecte of the drill string and other bottom hole assembly components on the z-axi~
magnetometer measurements.
2) Make a seriee of 3-axie measurements of the total magnetic field at different locations along the borehole path.
3) Compute, by any of a variety to known methods, an estimate of the true Earth 18 magnetic field along the borehole axis at each of the different locations.
154) Using the model selected and the ~eries of borehole axis measurements and estimates, solve for -~
the coefficients of the model.
It is generally well known that magneti~
survey tools are disturbed in varying waya by anomalous magnetic fields assoc~ated with fixed ox induced magnetic fields in elements of the drill string. It i5 further well known that the predominant error component lies along the axis of the drill qtring. This latter fact i8 the basis for several patented procedures, to eliminate the along-axis field errors in 3-magnetometer survey tools. Among these are U.S. patents:
4,163,324 to Russell et al.
4,433,491 to Ott et al.
4,510,696 to Roesler 4,709,486 to Walters 4,682,421 to Van Dongen et al.
4,761,889 ~o Cobern et al.
4,819,336 to Russel 5,155,916 to Engebretson U.K. patents 2,138,141A to Russell et al. and 2,185,580 to Russell.
U.S. Patent 5,155,916 to Engebret60n provides a method for error reduction in compensation for magnetlc lnterference.
All o2 these methods, in e~fect, ignore the output of the along-axis magnetometer, except perhap~
for selectlng a slgn for a ~quare root computation.
They providQ an azlmuth re3ult by computatlon of a synthetic solutlon, either~
1) by using only the two cross-axls 21~ 2777 magnetometer~ and known characteristic~ o~ the Earth field, or 2) by using the crosB-axl8 components and an along-axis component computed ~rom the cross-axis components and known characterlstics of the Earth'o field. ~ ~-Most o~ the~e require, as the known characteri~tic3 of the Earth ~ield, one or ~ore o~ the following:
1) field magnitude 2) dlp angle 3) horizontal component 4) vertical component.
~he Naltsrs method reguires, as known characteristics of the Earth field, only that~
1) the fleld ~agnitude i~ constant in the survey area~
2) the dip angle i8 constant in the survsy area.
ThQ fact that these quantlties are constant is all that is reguired. The value o~ the con~tant is not needed but is dQrived within the correction algorithm.
Since all of these compensation methods use, in effect, a computed along-axis component, all of the~
break down ~or cases o~ borehole hlgh inclination angles in a generally Ea~t/We~t dlrection. Thl- ~s because the cross-axls msa~urement plane ~or such conditlon tsnd- to be aligned 80 a~ to contain both the gravlty and Earth ~leld vector~, and thus measure~ents in this plane provide a poor nea~ur- o~ the cross 2i1 2r~77 product o~ the Earth field and gravity vectors. The cross product vector of the ~wo reference vector~ i3 the vector that actually contains the directional reference information.
The actual degradation of accuracy at high inclinations in the East/Weet direction for the previously cited methods depends both on the inherent accuracy of the ~ensors in the ~urvey tool and on the accuracy of the required knowledge o~ the Earth field characteri~tics.
To provide a mechanization for a magnetometer survey tool that does not seriously degrade in accuracy at borehole higher inclination angles near the East/WTst direction, it is found to be nece~sary to provide a method and means to calibrate the error~ in the along-axi~ magnetometer 80 that accurate measurements can be made with it. Thi~ ia in direct contra~t with existing ~ethod~ that substitut~ co~puted values for along-axls measurements.
There i8, therefore, need to provide a calibration method for an along-axis ~agnetometer in a magnetlc survey tool to correct anomalous magnetic effect~ ln a drlll strlng and thereby to pernit accurate measurements of the along-axis component of the Earth magnetio field. Such accurat~ ~ea~urement of the along-axis component then per-it~ accurate co~putatlon o~ azimuthal direction lndependent of inclinatlon and directlon.
-`- 2112777 SUMMARY OF TH~ INVENTION
The drill string anomalou~ magnetizatlon i8 :
composed of both a fixed component re~ulting from permanently magnetized elements in the bottom hole ~ -assembly and the drill string, and an induced component resulting from ~he interaction of ~oft magnetic materials with the Earth field The along-axi~
component of the induced field can be expected to be proportional to the along-axi~ component of the Earth field This mo~el of a fixed error and an along-axis induced field proportional to the true along-axis Earth field can be interpreted as simply alterlng the basic along-axis magnetometer's bias or o~set error, and its scale factor for measuring the Earth field component In its simplest form, the present invention ;~
provide~ a method, includ~ng the steps Or determining a set of along-axis magnetometer errors at different points along the borehole path by any of the well known methods, and then fltting these errors to a model, as referred to, for bias and scale factor 80 that ac~urate along-axis mea~urements can be computed using the determined bias and scalQ ~actor value~
In a more generalized embodlment, the inventlon provldes a method to callbrate the ef~ect~ of ~Agnetic lnter~erence from the drlll strlng that includes modellng the lnterference e~fects as an unknown ve¢tor that lnclude~ a~ elements the ano~alous scale factor and blas e~ect~; maklng a serles measurement~ at a number of different survey locations - S - ~ :.
along the borehole; forming from the measurement data a measurement vector and a measure~ent matrix relating the mea~urement vector to the unknown vector; and solving the unknown vector. The element~ of the s unknown vector may then be used to compute accurate along-axis measurements and for quality control purposes.
These and other ob~ects and advantages of the invention, a~ well as the details of an illu~tratlve embodi~ent, will be more fully understood from ths following specification and drawing~, in which:
~RAWqNG D~SCRIPTION
Fig. 1 shows a typical borehole and drill string, $ncluding a magnetic survey tool;
Fig. 1~ shows a survey tool in a drill collar, as used in Fig. l;
Fig. 2a shows the influence o~ a piece Or high permeability magnetic material when placed parallel to an orlginally undisturbed magnetic field;
Fig. 2_ shows the influence of a piece Or high permeab~llty magnetic material when placed perpendicular to an orlginally undisturbed magnetic field; and Flg~. 3a, 3~, 3, and 3_ show a coordinate set in relation to ~ borehole and an Earth-fix~a ;~
coordinato ~et.
211277~
Dl~rAII~D DESCRIPTION
Fig. 1 shows a typical drllllng rig 10 and borehole 13 in ~ection. A magnetic survey tool 11 is ~hown contain~d in a non-magnetic drill collar 12 S (made, for example, of ~onel or other non-magnetlc material) extending in line along the borehole 13 and the drill string 14. ~he ~agnetic ~urvey tool i~
generally of the type de~cribed ln U.S. Patent 3,862,499 to Isham et al., lncorporated herein by reference. It contains three nominally orthogonal magnetometers and three no~inally ortho~onal ~ -accelerometer6 for ~ensing components o~ the Earth'~
magnetic and gravity ~ields. The drill string 14 above the non-~agnetic collar 12 i8 of ferromagnetic material ~for example ~teel) having a high permeability compared to the Earth ~urrounding the borehole and the non- ;
magnetic coll~r. There may, or may not, be other ferromagnetlc materlals contalned in the drill assembly 15 below the non-magnetlc collar. It 1~ generally well known that the ~erromagnetic material~ above, and -po~slbly below, the non-magnetic collar 12 cau~e anomalles in the Earth's magneti¢ ~ield ln the reglon of the survey tool that in turn cause errors in the measurement o~ the azi~uthal direction of the ~urvey tool.
It 1~ further well know that such ano~alies ~ay lnclude both rixed and lnduced error riold~, the --fixed error rields resulting from residual ~agn~tic erfects in th~ ~erro~agnetic materials and th~ lnduced 2~12777 error f~elds resulting fro~ di~tortlon o~ the Earth'~
true field by the high-permeability ~erromagnetic mater~als. It i8 also well known from both theoretical considerations and experiment that the predominant error field lies along the direction o~ the drill string. It i8 this latter knowledge that the predo~sinant error lies along the drill string direction that has led to all of the previou~ly cited methods to eliminate ~uch an error component. As previously tated, all such ~ethods discard the measurement along the drill string axi~ and find either a two-component solution or a three-component solutlon in which the third component i~ computed mathematically. All of these previous methods, therefore, result in ~ignificant error when the borehole path approaches a near-horizontal, near East/West direction.
Figs. 2a and 2b show the effects o~ a long ; piece of metal 16 of hlgh permeability, lmmersed ~n an initlally uni~orm nagnetic field. The ~ield lines 17 are dlstorted by the pr~sence of the high permeability materlal. In Fig. 2a, th~ plece 16, generally tubular, i~ shown placed parallel to the original field, and in Fig. 2k perpendicular to the original ~ield. ~ ~-A~ can be ~een fro~ the figures, there i~
2S con~iderable increa~e in the den~ity of field l~nes near the end~ o~ piec~ 16, in region- 18, in Fig. 2 when the piece 16 i~ parallel to the original ~leld.
In Flg. 2k, when the piece 16 1~ perpendicular to the original ~leld, there i8 only a ~mall increase in the den~ity o~ the field lines in the reglon~ 18. Further, ~S, . '~: ' . ' ' ' ., . ! . , " , , 2~12777 : ::
it can be noted that the field lines along thq line of the axis 19, of piece 16, in regions 18, have the same direction as the original ~ield lines. It may be verified either analytically or experimentally that for any arbitrary orientation of the piece 16 to the original field, the end result will be the superposition of the effect~ of the two components that may be resolved as along-axis and cross-axis to the piece.
A similar pattern to Fig. 2a (except that the field patterns close to loop from one end to the other) results if the piece 16 contains residual, permanent magnetic materials having poles lying along the axis 19. These patterns generally presented here are the basis for the previously cited correction algorithm~
used to avoid errors from magnetic e~fQcts in the drill string and bottom hole assembly. As previou~ly cited, ;~
the assumption used is that the along-borehole error is the predominant error, and that by not using the measurement along the borehole axi~, the error i8 avoided.
It may be shown either analytically or experimentally that the magnitude of the field anomalies shown in Figs. 2a and 2_ are linearly proportional to the original, undi~turbed ~ield aa long as the permeabillty o~ the piece 16 is constant with ~ield strength. Further, for the general case, tho ~ield along the axis 19 will be directly proportional to the cosine o~ the angle between the axls 19 and the total ~ield vector o~ the original, undisturbed ~ield.
~` 2112777 Figs. 3a, 32b, 3c, and 3d show an x, y, z coordinate set and the direction of a borehole axis 20, that i8 assumed to be colinear with the drill string 14 of Fig. l. Defining the Earth's magnetic field as the vector 2H having components Hx, ~, Hz, along the three axes of the survey tool ll, the measurements of the three magnetometers in the survey tool will be:
x-Magnetometer Hx (l) y-Magnetometer ~ ~2) z-Magnetometer Hz (3) in the absence of any di~turbances from magnetic materials in the drill 6tring.
Similarly, defining the Earth's gravity as the vector G, the measurements of the three accelerometers in the survey tool will be~
x-Accelerometer Gx (4) ~ ~
y-Accelerometer Gy (5) ~ -z-Accelerometer Gz (6) In Fig. 3, starting wlth the three-axls, Earth-fixed coordinate set, N, 2~, 2Q--(representlng North, East, and Down) where the underllne represent~
a unit vector ln the dirèction given, the orientation of the set of tool axes 2~2, y, ~ 18 deflned by a series of rotatlon angles, AZ, TI, HS (representing AZimuth, TIlt, and HighSlde). In thls nomenclature, x i8 rotated by HS from the vertlcal plane, y 18 normal to x, and z 1~ down along the borehole ax~s. The formulatlon of the calculation of azimuth, adapted from U.S. Patent 3,862,499, is:
2il2777 AZ = ArCtan (HX*Sin(HS)+P~,*C08(HS) ) .
COB(~I) *HX*CO~(HS)_HY*Sin(HS) )+HZ*Sin(TI) In this equation, Hx, ~, and Hz are the three magneto~eter-measured component The angle~ ~I and HS
are solved for from tha thr~e accelerometer-mea~ured components by well known methods in previous steps.
If there are induced field and permanent field effects from materials in the drill string, defined as HI and HF, re~pectively, and the symmetries are as discussed in Fig. 2 above, then the x- and y-magnetometer measurements will remain as above, but the ;~
z-magnetometer measure will become~
z-Magnetometer Hz+HI+HF (8) However, the induced field, HI, wa~ previously stated to be proportional to the original field along th~ axis of the magnetic material and, therefore, it must be - proportional to Hz. If one describes the proportionality by a constant, RI, then:
HI ~ KI*Hz (g and the z-magnetometer measurement then becomes:
z-~agnetometer (l+KI)*Hz+HF (10) This shows that the output of the z-magnetometer mea~urement may be lnterpreted ~ust like the other two measurement~, but that the scale factor of the measurement i~ now tl+KI); and there 1~ an off~et or blas-type o~ error, HF, added to the maasurement. I~
the value~ o~ KI and HF could be determined, then the z-magnetometer output could be used in azi~ut~
computation without error and the magnetic in~luence of the drill ~tring could be avoided without encountarlng the problem o~ lncreasing error as the hlgh tilt, East/West condition i8 approached.
There 1B no way that the two unknown~, RI and HF, can be determined ~rom a ~ingle set o~ mQasuxements at ona survey sta~ion. However, ~rom a series of two or more measurements at different locations along the borehole where the z-axis components of the Earth'~
field, Hz, are different, a eolution for the two unknowns may be found. A series of mea~urements may be ;~
expressed as:
Hz~(~ z(l)*~l+XI)+H
Hzm ( 2 ) ~ HZ ~ ;! ) * t l+KI ) +HF
Hzm(3) - HZ(3)~1+KI)+HF (11) Hzm(4) ~ HZ(4)*(1+KI)+HF
Hzm(5) 3 Hz(5)~(l+RI)+HF
.
Hzm(n) - Hz(n)*(l+KI)+HF
where Hzm(n) represents the n-th measurement at the n-th location along the borehole of the z-axis magnetlc field component and Hz(n) represent~ the corresponding n-th true s-axis co~ponent o~ the Earth'o true ~ield, not includlng th- anomalle~ re~ultlng ~rom magnetlc materlals in the drlll string or other bottom hola asseably components.
The prevlou~ly cited aethods ~or correctlon o~ aagnetic erroro do not use the z-axi~ mea~ur~ent.
They do, however, elther compute a z-axi~ component or compute ~n azlmuth wlthout such a component (from whlch a z-axlo component may be co~puted). Since, except ~or '' region~ near high inclination East/West, the azimuth results have been shown to produce reasonably accurate results, it follows that such computed z-axis components are much more accurate than measursd z-axis s components. Thus, in the above series o~ measurements Hzm(n), i~ the corresponding Hz(n) values are computed by any of the cited methods, the set of measurement equations may be solved for the two unknowns, KI and F- -Since there are two unknowns, a minimum o~
two measurements (to provide two eguations) is required. For example:
Hz~(l) = Hz(l)*(l+KI)+HF (12) Hzm(2) - HZ(2)*(1+KI)+HF (13) may be solved to obtain:
X = (Hzm(l) - Hzm(2)) 1 (14) (Hz(l) - Hz(2)) H Hz(l)*Hzm(2) - HZ(2)*Hzn(l) (15) Hz(l) - Hz(2) In general, if there are more measurements than there are unknowns, the system of measurement equations i8 said to be overdetermined. However, considering various errors that may be involved in the measurement or computation proces~ed, it i~ ~tlll desirable to uBe a~ much mea~urement data a~ po~ible to minimize error~ in the unknown~' value~ ~ought.
This i~ the classical problem o~ parameter estimatlon that ha~ been addressed in many ~leld~. One well-known method leading to what i~ known a~ a Hleast-~quared-211277~
error result" i8 shown below. ~:~
The set of measurements Hzm(~ Hz~(n) can berepresented as the n-element vector -zm~ called "the measurement vector", where the vector notation 18 indicated by the underscore.
The unknown quantities (l+KI) and ~F may be represented as a 2-element vector x. These vectors may be related by writing:
Hz~ Hx + y (16) where H, a matrix called the measurement matrix, is an nx2 matrix:
Hz(l) 1 Hz(2) Hz(3) Hz(4) 1 ~:
H = . (17) :
Hz(n) 1 ~:
and where ~ is the unknown vector:
XI : :~
' (18) HF
and y is a vector of measurement ~nolse~. The solution deslred i8 that for the "best" estimate of x, minlmlzing the effects o~ the measure~ent ~noisen.
When the ~best~ criteria 1~ defined as that solution, :~: .
that minimizQs the sum of the squarQs o~ the elements f ~zm ~ ~, where the symbol ^ over ~ lndicates the best estimate Or ~, then lt may be ~hown that - 14 - - :
:
2112777 ~ ~
X ~ (HTH)-l HT Hzm (19) where HT i8 the transpose of th~ nx2 matrix H and (HTH)-l is the matrix inverse of the matrix HTH. ~-The method shown above, as represented by equations (11) through (19), will result in some error in the determination of the desired unknown KI and HF
that depends on errors in the reference values of the Earth's magnetic field, since errors in these quantities will produce some error in the computed "true" values of Hz(n). Such reference-induced errors are the accuracy limiting factors in the correction algorithms of the previously cited patents.
A method that does not depend on Earth' 8 field reference may be found by generalizing the problem. In general, a series of measurements of some quantity, for example z, can be represented as the value, for example x, plus some unknown measurement error, for example v. The ~eries of measurements may ;~
be written in vector/matrix notatlon as:
z - H ~ ~ v (20) zl ' `' ' ` ~ :~
z2 :
where: ~ - z3 an n-element measurement vector (21) ;~ :
~.
~:
zn 21~2777 ~
~, x - ~n m-element unknown vector (22) XDI
H i8 an n- by m-element me~surement matrix (23) vl v2 v - v3 an n-ele~ent measurement error vector ~24) ' . , vn Given these definltions, consider an unknown vector x defined as:
HTotal HVertical x - HNorth (25) : -~
XI
~: HF ~ -~
20 where, in addition to the previously defined KI and HF:
H~otal i8 the tot~l Earth field ~ :
magnitude at the borehole :: locatlon.
~Vertical i8 the vertical component of the total Earth field.
th i8 the horizontal North component o~ the rield~
Here the Earth-field guan~ities are added to th~
unknown vector and will be estimated, ~long wlth tho z-axis ~agnetometer wale factor and bia-, XI and HF. ~-2~ 27~7 A new measurement vector z and a new measurement matrix H are required for this expanded problem. Also, since there are mors unknowns, more equations and more survey location~ will be required.
Define the two vectors of unit length, ~ and g as:
H (26) Hx^ + Hy~ + HzA
G
_ (27) Gx^ + Gy~ + Gz^
That is, the vector ~ i5 a unit vector in the sa~e direction as the Earth's magnetic field vector H, as measured by the magnetometers; and ~ is a unit vector in the same direction as the Earth's gravity field G, as measured by the accelerometers. The vector g i5 thus along the direction D (Down) in Fig. 3a.
Now de~ine a unit vector r as: -~
r ~ _ (28) Ig X E21 That i5, the vector ~ is a vector that i8 the vector cross product of ths vectors ~ and ~ divided by the absolute magnltude of the ~ame vector cross product.
Thu~, ~ i8 a unit vector; and, ~ince, by deflnltion, the vectors ~ and Ç, and thereby the vector~ p and g li~ in the North-South plan~ o~ Fig. 3~, th~ vector E
is in the E (East) direction o~ Fig. 3a.
La~tly, define a vector 8 as:
X ~ (29) ~1~2777 That 18, the vector B i~ the vector cro~s product o~
the vector~ r and ~, and is thus a unit vector in the N
(North) direction in Flg. 3a.
Each o~ the three vectors, ~, g, and ~, ha~
three components--one component along the x-axis, one component along the y-axis, and one component along the z-axis. The three component~ for the vector ~ are defined a~ Px, py~ and pz. The three co~ponents ~or the vector ~ are defined a~ qx, qy, and qz. The three components for the vector g are defined as ~x~ ~y~ and 8z~
With these three unit vector definitions for 2, ~, and B, these three vectors may be computed at e~ch survey location from the mea~ured H and G vector~
lS Then, three elements of the total measurement vector z ~--may be computed for each survey locatlon. For exa~ple, consider the three eleuents of the vector z for the first location, which are to be computed a~
Z t z(2) ~ ~m- g t30) Z(3) ' ~m- 8 That is, each of these three elements is ~ust the vector dot product of the measured magnetlc field vector ~ and the ~, ~, and B vector~ defined and computed, as shown $n eguation~ (26) through (29).
ztl) i~ thu8 a mea8ure o~ the total magnetlc field;
z(2) i~ a measure of the vertical component o~ that field: and z~3) i~ a Dea~ure o~ the horizontal North component Or that total ~ield. -~
A~o, for each survey ~tation, three rows Or -~
2~12777 the measurement matrix H that relates the ~easurement vector z to the unXnown vector x may be computed in terms of the measured magnetic field component~ and the element~ of the vectors ~, ~, and 8. These three rows, for the unknown vector _, as defined by equation (25) are:
1 0 0 pz*Hz pz 0 1 0 qz*Hz gz ~31) o 0 1 sz*Hz 8z Since there are five unknown quantities in ths unknown vector x, the quantities ~hown in equations (30) and (31) are not sufficient to solve for the unknown vector x by using equation (19). A minimum of three survey locations i9 recommended. More locations will increase the accuracy of the determination of the unknown vector _.
As previously stated, three elements of the measurement vector z are computed a~ in equation (30);
and three rows of the measurement matrix H are computed a~ in equation (31) for each survey location. I~ the recommended minimum of three survey locations is used, the meaeurement vector ~ becomes a nine element vectdr, and the measurement matrix H becomes a nine row by ~ive column matrlx. If six survey location~ were used, the mea~urement vector would have èighteen element~, and the ~easurement matrix would have eighteen row~ and still five column~.
When sufficient survey locations have been accumulated, then the unknown vector ~ olved for using equatlon (19~ with the measurement vector ~
21~27~7 substituted for Hzm~ This then provides results that indicate HT0tal~ the total maqnetic field; HVerti~a the vertical component; HNorth~ the horizontal component; KI, the desired anomalous scale ractOr caused by the induced magnetization from drill ~tring elements: and HF, the desired anomalous bias or offset ~-resulting fro~ the fixed magnetization in the drill string elements.
The values--HTOtal, Hvertical, and HNorth may be used for quality control purposes, or as input reference data to any of the previously cited patent methods of interference compensation that require input data on the local Earth magnetic field. For quality control purposes, these values may be compared to ;~
reference values obtained from maps or Earth-field computer models. If the values depart significantly from the reference values, it indicates either a possibl~ failure in the sensor~ ln the ~urvey tool or a ~-significant geophysical local varlation in the Earth's field. Either one of these possibilities alerts the survey operator to possible serious survey error.
The values of KI and HF may be used to compute corrected values of the z-axis magnetometer measure of the Earth fleld for each survey location as~
HZc(n) ~ ( Hzm(n) HF ) (32) 1 + KI
where: Hzc(n) 1~ the corrected z-axl~ value ~or locatlon n Hzm(n) 18 the measured z-axls value ~or locatlon n ` 2~2777 Thls corrected valus for location n may then b~
combined with the measured x-axis and y-axls measured magnetic components to solve for the borehole azlmuth at location n, a~ shown in equation (7).
Some care is needed in select~on of the locations that are to be used in the solution for the model unknowns. As is well known, if the borehole path is a straight line, all of the measurement equations are highly correlated, and there will be no viable ~olution for the unknowns. In effect, all of the equations are equivalent and, therefore, the requirement that the number of equations must equal or exceed the number of unknowns is not met. In general, the methods of selecting the equations for solution of simultaneous equations are well known; and methods are known for the analysis of probable errors in the solution for unknowns from multiple equations.
Further, in the implementation of the methods described here, it may be found that some measurements or est$mates of the along-borehole magnetic field are more accurate than others in the 6eries of locations ~o be used for solving for the unknowns. In this event, the well-known method~ of "welghtQd" solut~on for unknowns may be used wherein the mor~ accurate data ~8 HweightedH more heavily ln the solution to ~inim~ze errors.
The vector v of measurement error~, defined at equation (24), may ~e further characterized in ~-;
general by a matrix computed from its elements that $8 usually designated a~ the covariance matrix of the :~
'2112777 error vector an~ i~ often designated by the letter R.
This matrix is computed as the expected value of the matrix product of the vector v and its transpo~e.
Thus~
R s E( y * vT ) whe-e: E designates the expected value of the product Superscript T denotes the transpose.
With this definition and the terms deflned above, it may be shown that the optimum e~timate o~ the unknown elements in the vector x that minimize~ the 6um of the ~ :
squared errors in the estimate i~ given by~
x = ( HT~R-l*H )-1 * ( HT*R-l ) * z (34) -;
where: * denotes matrix product Superscxipt T is transpose Superscript -1 denotes matrix inverse ~he actual values of the elements of the ~easurement noi6e vector y are now known. I~ they were ~ ~ -known, the values could be subtracted from the elements of the measurement vector and the proble~ could then be solved with errorless measurement data. ~owever, the expected statistical value of the element~ of y can be computed by error analysis of the elements of the measurement vector. Such errors will, in general, depend on the error~ ln all of the sensors and on the orientation of the borehole with respect to the Earth coordinate set. When these expected values havs been determined, the covariance ~atrix R can be determined using equation (33); and then the unknown vector ~ ~ay be determined using equation (34).
~1~2777 Both equations (19) and (34) are well known means to obtain so-called "least ~quared error"
estimates of unknowns from an overdetermined set of equations. They require computer storage and manipulation of all of the data included in the solution. Recursive "least squared error" formulation~
that work incrementally on the data, for example as each new survey location data becomes available, rather than waiting for the complete data set from all locations to be used, are well known. One of the better Xnown 6uch recursive methods i~ the method called "Kalman filtering", after its developer Dr.
R. E. Kalman. This method i8 described ln Chapter 4 of the book Applied Optimal Estimation, Arthur Gelb et al., The M.I.T. Press, Cambridge, Massachusetts. In this method, the input sensor data is processed as it is received and a continuing estimate of the unknown vector and lt~ covarlance matrix i~ computed. The recursivs formulation el$minates the need to provide ever increasing ~torage and to process matrix computation~ o~ ever growing matrix dimensions as larger nu~bers o~ data sets (survey locations) are addQd to the computation. This mechanization 1~ a real time formulation that at each cycle (each new survey location) provides an optimal estimate o~ the unknown~.
Additionally, it may be desirable to use more complex models ~or the magnetic inter~erence. For example, the equivalent ~cale ~actor anomaly may require non-llnear term~ or temperature dependent terms to achieve higher accuracle~ under some conditions.
2~2777 .`~
Also, the unknown vector may be expanded to include anomalous scale factor and bias terms for the cross-borehole magnetometers. Thi~ would account for cases in which the drill string magnetic interference was not principally along the axial direction as originally assumed. As the unknown vector i8 expanded, the measurement matrix must be expanded 80 that the number of columns in it is equal to the number of elements in the unXnown vector. The general method above may ~till be used, but it must be recognized that more unknowns lead to a need for ~ore independent measurement equations from which elements o~ the measurement vector are to be computed.
Another solution to the problem of obtaining the best result~ from a series of~multiple measurements is that Xnown as ~Optimal Linear SmoothingH, as described in Chapter 5 of the book Applied Optimal ~;
EstimatiQn, Arthur Gelb et al., The M.I.T. Press, Cambridga, Massachusetts. In this formulation, all of the measurement data i8 comblned to provide an optimal estimate of all elements of the state vector describing the physical process at each point within the serles of measurement~. Smoothing, a~ derined in the clted reference, is a non-real-time data processing scheme that uses all measurements. The general methodology 18 generally related to the well-known Kalman filtering methods. In optimal smoothing, in errect, Xalman f~ltering is applied to the data set both rrOm the first data point rorward in time to the location of interest; and from the last data point backwards in ~ 2112777 time to the same location of interest.
In its simplest forn then, the essential elements of the invention described herein are:
1) Select a model for the magnetic interference effecte of the drill string and other bottom hole assembly components on the z-axi~
magnetometer measurements.
2) Make a seriee of 3-axie measurements of the total magnetic field at different locations along the borehole path.
3) Compute, by any of a variety to known methods, an estimate of the true Earth 18 magnetic field along the borehole axis at each of the different locations.
154) Using the model selected and the ~eries of borehole axis measurements and estimates, solve for -~
the coefficients of the model.
5) Correct the z-axis magnetometer data using the coefficients determined in step 4) above for each survey location.
6) Solve for the azimuthal orientation of the borehole at each location ueing the corrected magnetometer data ~or each eurvey location.
In more generalized method, the essential elements of the invention are:
1) Select a model ~or the magnetic lnterference e~fects of the drill strinq and other bottom hole as~embly components on the magnetometer measurements and define an unknown vector that contains the elements of the selected model.
,-. -~ 25 ~
r ~
.. . .. . . . . . .
211~777 2) Make a series of 3-axis measurements of the total magnetic field at different locations along the borehole path.
3) Compute for the serie~ o~ dlfferent locations a measurement vector and a measurement matrix relating the measurement vector to the unknown vector.
4) Using the measurement vector and the measurement matrix, solve for the unknown vector to determine the element~ of the 6elected model.
5) Correct the magnetometer data using the coefficients determined in step 4) above ~or each survey location.
6) Solve for the azimuthal orientation of -the borehole at each location using the corrected magnetometer data for each survey location.
In more generalized method, the essential elements of the invention are:
1) Select a model ~or the magnetic lnterference e~fects of the drill strinq and other bottom hole as~embly components on the magnetometer measurements and define an unknown vector that contains the elements of the selected model.
,-. -~ 25 ~
r ~
.. . .. . . . . . .
211~777 2) Make a series of 3-axis measurements of the total magnetic field at different locations along the borehole path.
3) Compute for the serie~ o~ dlfferent locations a measurement vector and a measurement matrix relating the measurement vector to the unknown vector.
4) Using the measurement vector and the measurement matrix, solve for the unknown vector to determine the element~ of the 6elected model.
5) Correct the magnetometer data using the coefficients determined in step 4) above ~or each survey location.
6) Solve for the azimuthal orientation of -the borehole at each location using the corrected magnetometer data for each survey location.
Claims (22)
1. A method for determining the orientation of the axis of a borehole with respect to an Earth-fixed reference coordinate system at a selected series of locations in the borehole, the borehole having a trajectory, and adapted to receive a drill string, comprising the steps of:
a) defining a model for the influence of magnetic interference from elements of the drill string on the measurement of components of the Earth's magnetic field in the borehole in terms of an unknown vector, a measurement vector, and a measurement matrix relating the unknown vector and the measurement vector, b) measuring at two or more selected locations along the borehole trajectory:
1) at least one of:
i) two cross-borehole components of the Earth's gravity field at said selected locations in the borehole, and ii) two cross-borehole components and an along-borehole component of the Earth's gravity field at said selected locations in the borehole;
a) defining a model for the influence of magnetic interference from elements of the drill string on the measurement of components of the Earth's magnetic field in the borehole in terms of an unknown vector, a measurement vector, and a measurement matrix relating the unknown vector and the measurement vector, b) measuring at two or more selected locations along the borehole trajectory:
1) at least one of:
i) two cross-borehole components of the Earth's gravity field at said selected locations in the borehole, and ii) two cross-borehole components and an along-borehole component of the Earth's gravity field at said selected locations in the borehole;
2) two cross-borehole components of the Earth's magnetic field and an along-borehole component of the Earth's magnetic field at said selected locations;
c) computing the elements of said measurement vector and said measurement matrix from said measured Earth's gravity and Earth's magnetic field components for said selected locations;
d) solving for said unknown vector using said measurement vector and said measurement matrix, e) computing for each of said selected locations corrected values for the Earth's magnetic field components using said unknown vector, said model, and said measured Earth's magnetic field components, f) and determining a value for the azimuthal orientation of said borehole axis at each said selected locations using said corrected values for the Earth's magnetic field components and said measured gravity components.
2. The method of claim 1 wherein said model and said unknown vector include:
a) an anomalous scale factor value for said along-borehole measurement of the Earth's magnetic field component in the along-borehole direction, b) an anomalous bias or offset value for said along-borehole measurement of the Earth's magnetic field component in the along-borehole direction.
c) computing the elements of said measurement vector and said measurement matrix from said measured Earth's gravity and Earth's magnetic field components for said selected locations;
d) solving for said unknown vector using said measurement vector and said measurement matrix, e) computing for each of said selected locations corrected values for the Earth's magnetic field components using said unknown vector, said model, and said measured Earth's magnetic field components, f) and determining a value for the azimuthal orientation of said borehole axis at each said selected locations using said corrected values for the Earth's magnetic field components and said measured gravity components.
2. The method of claim 1 wherein said model and said unknown vector include:
a) an anomalous scale factor value for said along-borehole measurement of the Earth's magnetic field component in the along-borehole direction, b) an anomalous bias or offset value for said along-borehole measurement of the Earth's magnetic field component in the along-borehole direction.
3. The method of claim 1 wherein said model and said unknown vector include the magnitude of the Earth's total magnetic field, excluding said effects of said drill string magnetic interference.
4. The method of claim 1 wherein said model and said unknown vector include the magnitude of the vertical component of the Earth's total magnetic field, excluding said effects of said drill string magnetic interference.
5. The method of claim 1 wherein said model and said unknown vector include the magnitude of the horizontal component of the Earth's total magnetic field, excluding said effects of said drill string magnetic interference.
6. The method of claim 1 wherein said measurement vector comprises the differences between said measured along-borehole magnetic field components and estimates of the true value of the along-borehole magnetic field components at said selected locations.
7. The method of claim 1 wherein said measurement vector includes the computed total values of the measured Earth's magnetic field at said selected locations.
8. The method of claim 1 wherein said measurement vector includes the computed vertical component values of the measured Earth's magnetic field at said selected locations.
9. The method of claim 1 wherein said measurement vector includes the computed horizontal component values of the measured Earth's magnetic field at said selected locations.
10. The method of claim 1 wherein the unknown vector is solved for by employing multiple simultaneous equations relating the measurement vector to the unknown vector.
11. The method of claim 1 wherein the unknown vector is solved for by employing a "least squared error" computation using said measurement vector and said measurement matrix relating said measurement vector to said unknown vector.
12. The method of claim 1 wherein the unknown vector is solved for by employing a "weighted least squared error" computation using said measurement vector, said measurement matrix relating said measurement vector to said unknown vector and the co-variance matrix of said measurement vector.
13. The method of claim 1 wherein the unknown vector is solved for by employing a Kalman filtering computation using said model, said unknown vector, said measurement vector, and said measurement matrix.
14. The method of claim 1 wherein the unknown vector is solved for by employing an "optimal linear smoothing" computation using said model, said unknown vector, said measurement vector, and said measurement matrix.
15. The method of claim 1 wherein said model includes a fixed field component resulting from permanently magnetized elements in at least one of the following:
i) the drill string, which is metallic, ii) a bottom hole metallic assembly in the borehole.
i) the drill string, which is metallic, ii) a bottom hole metallic assembly in the borehole.
16. The method of claim 1 wherein said model includes an induced field component resulting from the interaction of soft magnetic materials in the borehole with the Earth's magnetic field.
17. The method of claim 1 wherein the borehole trajectory has a portion that extends in a near horizontal, East-West direction.
18. The method of claim 1 including a survey tool for performing said method, and including locating said tool in said portion of said borehole.
19. The method of determining the orientation of the axis of a borehole in the Earth, that includes a) determining a set of along-axis magnetometer errors at different points along the borehole, b) determining a model composed of a fixed magnetic component and an induced magnetic component, associated with magnetization in the borehole, c) and fitting said errors to said model for determining bias and scale factors, d) whereby accurate, along-axis measurements can be computed using the determined bias and scale factor values.
20. The method of claim 1 including employing magnetometers to effect said b) 2) step measuring, wherein said along-borehole component is represented by:
(1+KI)*HZ+HF
wherein the quantity (1+KI) is the scale factor of the measurement and the quantity HF is a bias type of correction, and said method step, including said measurements at said two or more locations, derive values for (1+KI) and HF.
(1+KI)*HZ+HF
wherein the quantity (1+KI) is the scale factor of the measurement and the quantity HF is a bias type of correction, and said method step, including said measurements at said two or more locations, derive values for (1+KI) and HF.
21. The method of claim 20 wherein a corrected value Hzc(n) for a Z-axis magnetometer is obtained for each survey location, according to:
where Hzm(n) is the measured z-axis value for location n.
where Hzm(n) is the measured z-axis value for location n.
22. The method of claim 21 wherein borehole azimuth AZ then obtained according to:
AZ = Arctan
AZ = Arctan
Applications Claiming Priority (2)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| US08/023,387 | 1993-02-26 | ||
| US08/023,387 US5321893A (en) | 1993-02-26 | 1993-02-26 | Calibration correction method for magnetic survey tools |
Publications (1)
| Publication Number | Publication Date |
|---|---|
| CA2112777A1 true CA2112777A1 (en) | 1994-08-27 |
Family
ID=21814794
Family Applications (1)
| Application Number | Title | Priority Date | Filing Date |
|---|---|---|---|
| CA002112777A Abandoned CA2112777A1 (en) | 1993-02-26 | 1994-01-04 | Calibration correction method for magnetic survey tools |
Country Status (3)
| Country | Link |
|---|---|
| US (1) | US5321893A (en) |
| CA (1) | CA2112777A1 (en) |
| GB (1) | GB2275536B (en) |
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| CN104912542A (en) * | 2015-05-19 | 2015-09-16 | 中国地质科学院探矿工艺研究所 | Inclinometer calibration device for simulating nonmagnetic ultrahigh temperature condition |
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-
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- 1994-02-25 GB GB9403672A patent/GB2275536B/en not_active Expired - Fee Related
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| CN104912542A (en) * | 2015-05-19 | 2015-09-16 | 中国地质科学院探矿工艺研究所 | Inclinometer calibration device for simulating nonmagnetic ultrahigh temperature condition |
Also Published As
| Publication number | Publication date |
|---|---|
| GB2275536B (en) | 1997-01-22 |
| GB2275536A (en) | 1994-08-31 |
| GB9403672D0 (en) | 1994-04-13 |
| US5321893A (en) | 1994-06-21 |
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