spe-67616-pllall

Accuracy Prediction for DirectionalMeasurement While Drilling

H.S. Williamson, SPE, BP

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SummaryIn this paper a new method for predicting wellbore position ucertainty which responds to the current needs of the industrdescribed. An error model applicable to a basic directional msurement while drilling~MWD! service is presented and used fillustration. As far as possible within the limitations of space, tpaper is a self-contained reference work, including all the nesary information to develop and test a software implementatiothe method. The paper is the product of a collaboration betwthe many companies and individuals cited in the text.

IntroductionAs the industry continues to drill in mature oil provinces, the duchallenges of small geological targets and severe well congesincrease the importance of quantifying typical wellbore positioerrors. The pioneering work of the 1970’s culminated in the paby Wolff and de Wardt.1 Their approach, albeit extensively modfied and added to, has remained the de facto industry standathis day. At the same time, various shortcomings of the methave been identified,2-4 but are not discussed further here.

In recent years, a number of factors have created the oppnity for the industry to develop an alternative method:

d risk-based approaches to collision avoidance and targetting require position uncertainties with associated confidenceels, something which Wolff and de Wardt specifically avoided

d changing relationships brought about by integrated servcontracts have forced directional drilling and survey companieshare information on tool performance;

d the development of several new directional software produand their integration with subsurface applications has providednecessity and the opportunity to develop new means of comnicating and visualizing positional uncertainty.

This paper provides a three-part response to this need.

1. Error Model for Basic MWD. This is based on the currenstate of knowledge of a group of industry experts. There are seral reasons why directional MWD is the most suitable survservice to illustrate a new method of error modeling. The erbudget is dominated by environmental effects, so that accudifferences attributable to tools alone are minimal. It is the surtool of choice for most directional wells, where position uncetainty is of greatest concern. The physical principles of its opetion, including the navigation equations, are in the public doma

2. Mathematical Basis.This is a rigorous description of thpropagation of errors in stationary tools. Fit-for-purpose ermodels using the same basis are in development for inertialcontinuous gyroscopic tools, although some simplification acompromise are inevitable. A rigorous treatment of continuosurvey tools would probably have too restricted a cognoscenbe practical.

3. Standard Examples and Results.Despite the apparent simplicity of the Wolff and de Wardt method, different softwarimplementations generally give subtly different results. Whileeffort has been made in this paper to provide a comprehendescription of the new method, there will surely remain so

Copyright © 2000 Society of Petroleum Engineers

This paper (SPE 67616) was revised for publication from SPE 56702 first presented at the1999 SPE Annual Technical Conference and Exhibition held in Houston, Texas, 3–6 Oc-tober. Original manuscript received for review 31 January 2000. Revised manuscript re-ceived 26 July 2000. Paper peer approved 14 August 2000.

SPE Drill. & Completion15 ~4!, December 2000

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areas of ambiguity or confusion. In such cases, reproduction ofnumerical results at the end of the paper will act as a powecriterion for ‘‘validation.’’

Genesis of the Work.The content of this paper is the fruit of twcollaborative groups.

ISCWSA.The Industry Steering Committee on Wellbore Suvey Accuracy is an informally constituted group of companies aindividuals established following the SPWLA Topical Conferenon MWD held in Kerrville, Texas in late 1995. The group’s broaobjective is ‘‘to produce and maintain standards for the indusrelating to wellbore survey accuracy.’’ Much of the content of thpaper, and specifically the details of the basic MWD error modhad its genesis in the group’s meetings, which were distinguisby their open and cooperative discussions.

Four Company Working Group.The ISCWSA being too largea forum to undertake the detailed mathematical development oerror propagation model, this was completed by a small workgroup from Sysdrill Ltd., Statoil, Baker Hughes INTEQ, and BExploration. The mathematical model created by the groupdescribed below has been made freely available for use byindustry.

Assumptions and DefinitionsThe following assumptions are implicit in the error models amathematics presented in this paper.

d Errors in calculated well position are caused exclusivelythe presence of measurement errors at wellbore survey statio

d Wellbore survey stations are, or can be modeled as, thelement measurement vectors, the elements being along-depth,D, inclination, I, and azimuth,A. The propagation math-ematics also requires a toolface angle,a, at each station.

d Errors from different error sources are statistically indepedent.

d There is a linear relationship between the size of each msurement error and the corresponding change in calculatedposition.

d The combined effect on calculated well position of any nuber of measurement errors at any number of survey stationequal to the vector sum of their individual effects.

No restrictive assumptions are made about the statistical dibution of measurement errors.

Error Sources, Terms and Models.An error source is a physicaphenomenon which contributes to the error in a survey tool msurement. An error term describes the effect of an error sourca particular survey tool measurement. It is uniquely specifiedthe following data:

d a name;d a weighting function, which describes the effect of the erroe

on the survey tool measurement vectorp. Each function is re-ferred to by a mnemonic of up to four letters.

d A mean value,m.d A magnitude,s, always quoted as a 1 standard deviati

value.d A correlation coefficientr1 between error values at surve

stations in the same survey leg.~In a survey listing made up oseveral concatenated surveys, a survey leg is a set of contigsurvey stations acquired with a single tool or, if appropriatesingle tool type.!

d A correlation coefficientr2 between error values at survestations in different survey legs in the same well.

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d A correlation coefficientr3 between error values at survestations in different wells in the same field.

To ensure that the correlation coefficients are well defined, ofour combinations are allowed.

Propagation Mode r1 r2 r3Random~R! 0 0 0Systematic~S! 1 0 0Well by well ~W! 1 1 0Global ~G! 1 1 1

r1, r2, andr3 are to be considered properties of the error souand should be the same for all survey legs.

An error model is a set of error terms chosen with the aimproperly accounting for all the significant error sources whaffect a survey tool or service.

An Error Model for ‘‘Basic’’ MWDFor the survey specialist in search of a ‘‘best estimate’’ of potion uncertainty it is tempting to differentiate minutely amontools types and models, running configurations, bottomholesembly ~BHA! design, geographical location and several othvariables. While justifiable on technical grounds, such anproach is impractical for the daily work of the well planner. Thtime needed to find out these data for historical wells, andmany planned wells, is simply not available.

The error model presented in this section is intended torepresentative of MWD surveys run according to fairly standquality procedures. Such procedures would include rigorousregular tool calibration, survey interval no greater than 100nonmagnetic spacing according to standard charts~where no axialinterference correction is applied!, not surveying in close proxim-ity to existing casing strings or other steel bodies, and passtandard field checks on the G total, B total, and dip.

The requirement to differentiate between different services mbe met by defining a small suite of alternative error models.amples covered in this paper are application or not of an ainterference correction and application or not of a BHA sag crection.

Alternative models would also be justified for in-field refeenced surveys, in-hole~gyro! referenced surveys, and deptcorrected surveys.

The model presented here is based on the current statknowledge and experience of a number of experts. It is a starpoint for further research and debate, not an endpoint.

Sensor Errors. MWD sensors will typically show small shifts inperformance between calibrations. We may make the assump

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222 H. S. Williamson: Accuracy Prediction for MWD

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that the shifts between successive calibrations are representof the shifts between calibration and field performance. On tbasis, two major MWD suppliers compared the results of succsive scheduled calibrations of their tools. Paul Rodney exami288 pairs of calibrations, and noted the change in bias~i.e., offseterror!, scale factor, and misalignment for each sensor. WaPhillips did the same for 10 pairs of calibrations, except that ssor misalignments were not recorded.

Andy Brooks has demonstrated that if a sensor is subjectscale error and two orthogonal misalignments, all independentof similar magnitude, the combination of the three error termsequivalent to a single bias term. This term need not appear exitly in the error model, but may be added to the existing bias teto create a ‘‘lumped’’ error. This eliminates the need for 20 exweighting functions corresponding to sensor misalignments.

The data from the MWD suppliers suggest that in-service ssor misalignments are typically smaller than scale errors. Aresult, only a part of the observed scale error was ‘‘lumped’’ wthe misalignments into the bias term, leaving a residual scale ewhich is modeled separately. In this way, four physical errorseach sensor were transformed into two modeled terms. The rewere as follows.

Error SourceWeightingFunction Magnitude

PropagationMode

Accelerometer biases ABX,Y,Z 0.004 ms22 SAccelerometer scale

factors ASX,Y,Z 0.005 SMagnetometer biases MBX,Y,Z 70 nT SMagnetometer scale

factors MSX,Y,Z 0.0016 S

These figures include errors which are correlated betweensors, and which therefore have no effect on calculated inclinaand azimuth~the exception being the effect of correlated magntometer errors on interference corrected azimuths!. It could beargued that the magnetometer scale factor errors in partic~which may be influenced by crustal anomalies at the calibrasites! should be reduced to account for this.

BHA Magnetic Interference. Magnetic interference due to steein the BHA may be split into components acting parallel~axial!and perpendicular~cross axial! to the borehole axis.

Axial Interference.Several independent sets of surface mesurements of magnetic pole strengths have now been made.served root-mean-square~RMS! values are the following.

Pin Box

Item RMS Pole Strength „Sample Size… SourceDrill collar 505 mWb ~8! Grindrod and Wolff~Ref. 5!

605 mWb ~11! 435 mWB ~11! Lotsberg~Ref. 5!511 mWb ~4! McElhinney*

Stabilizer 177mWb ~6! Grindrod and Wolff396 mWb ~10! 189 mWb ~10! Lotsberg369 mWb ~5! 408 mWb ~10! McElhinney

Motor 340mWb ~12! 419 mWb ~10! Lotsberg

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Oddvar Lotsberg also computed pole strengths for 41 BHfrom the results of an azimuth correction algorithm. The RMpole strength was 369mWb ~micro-Webers!.

These results suggest that 400mWb is a reasonable estimate fothe 1 s.d. pole strength of a steel drillstring component whfurther information is lacking. This is useful information for BHAdesign, but cannot be used for uncertainty prediction withou

*Minutes of the 7th Meeting of the ISCWSA, Houston, 9 October 1997.

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value for the nonmagnetic spacing distance. Unfortunately, this no ‘‘typical’’ spacing used in the industry, and we must finanother way to estimate the magnitude of this error source.

A well-established industry practice is to require nonmagnespacing sufficient to keep the azimuth error below a fixed toance~typically 0.5° at 1 s.d.! for assumed pole strengths andgiven hole direction. This tolerance may need to be compromiin the least favorable hole directions. For a fixed axial interferefield, and neglecting induced magnetism, the azimuth errostrongly dependent on hole direction, being proportional

SPE Drill. & Completion, Vol. 15, No. 4, December 2000

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sin I sinAm . Thus to model the azimuth error in uncorrected sveys, we require a combination of error terms which predicts zerror if the well is vertical or magnetic north/south, predicts errsomewhat greater than the usual tolerance if the well is near hzontal and magnetic east/west or predicts errors near the utolerance for other hole directions.

These requirements could be met by constructing some acial weighting function, but this would violate our restrictionphysically meaningful error terms. A constant error of 0.25° andirection-dependent error of 0.6° sinI sinAm is perhaps the beswe can achieve by way of a compromise. It is legitimate to csider these values representative of 1 standard deviation, sincpole strength values which underlie the nonmagnetic spacingculations are themselves quoted at 1 s.d.

Both error terms may be propagated as systematic, althothere is theoretical and observational evidence4 that this error isasymmetric, acting in the majority of cases to swing magnesurveys to the north in the northern hemisphere. Givingdirection-dependent term a mean value of 0.33° and a magniof 0.5° reproduces this asymmetry~with about 75% of surveysbeing deflected to the north!, while leaving the root-mean-squarerror unchanged.

Axial interference errors are not modeled for surveys whhave been corrected for magnetic interference.

Cross-Axial Interference.Cross-axial interference from thBHA is indistinguishable from magnetometer bias, and propagin the same way. Anne Holmes** analyzed the magnetometebiases for 78 MWD surveys determined as a by-product of a mtistation correction algorithm. Once a few outliers, probably dto magnetic ‘‘hot spots’’ and hence classified as gross errors,been eliminated, the remaining observations gave a RMS valu57 nT. This figure is somewhat smaller than the 70 nT attributato magnetometer bias alone. The conclusion must be that craxial interference does not, on average, make a significant cobution to the overall MWD error budget, and may be safely lout of the model.

Tool Misalignment. Misalignment is the error caused by thalong-hole axis of the directional sensor assembly being ouparallel with the center line of the borehole. The error maymodeled as a combination of two independent phenomena:

BHA Sag.This is due to the distortion of the MWD drill collaunder gravity. It is modeled as confined to the vertical planeproportional to the component of gravity acting perpendicularthe wellbore~i.e., sinI!. The magnitude of the error depends oBHA type and geometry, sensor spacing, hole size and sevother factors. Two-dimensional BHA models typically calculainclination corrections of 0.2° or 0.3° for poorly stabilized BHAin horizontal hole.6 For well stabilized assemblies the valueusually less than 0.15°. In the absence of better information,~at 1 s.d.! may be considered a realistic input into the basic ermodel.

Sag corrections, if they are applied, are calculated on the ounjustified assumptions of both the hole and stabilizers bein

** Minutes of the 8th Meeting of the ISCWSA, Trondheim, 19 February 1998.

H. S. Williamson: Accuracy Prediction for MWD

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gauge. Data comparisons by the author suggest a typicalciency of 60% for these corrections, leaving a post-correctionsidual sag error of 0.08°.

Assuming similar BHA’s throughout a hole section, all BHsag errors may be classified as systematic.

Radially Symmetric Misalignment.This is modeled as equallylikely to be oriented at any toolface angle. John Turvill madeestimate of its magnitude based on the tolerances on severalcentric cylinders.

d Sensor package in the housing. Tolerances on three comnents are clearance, 0.023°, concentricity, 0.003°, and straightof sensor package, 0.031°.

d Sensor housing in the drill collar. For a probe mounted incentralized, retrievable case, 0.063°.

d Collar bore in the collar body. Typical MWD vendors’ tolerance is 0.05°.

d Collar body in the borehole. The API tolerance on collstraightness equates to 0.03°. MWD vendors’ specificationstypically somewhat more stringent.

d The root sum square of these figures is 0.094°. Being baon maximum tolerances, it is probably an overestimate for stalized rotary assemblies.

d An analysis by the author of the variation in measured incnation over 46 rotation shots produced a root-mean-squarealignment of 0.046°. Simulations show that within this figurabout 0.007° is attributable to the effect of sensor errors.

d An additional source of misalignment, collar distortion ouside the vertical plane due to bending forces, may be estimusing three-dimensional BHA models and 0.04° seems to btypical value. This error differs from those above by not rotatiwith the tool. It should therefore strictly have its own weightinfunction. Being so small, it seems justifiable on practical~if nottheoretical! grounds, to include it with the other sources of radally symmetric misalignment. This leaves us with an estimatethe error magnitude of 0.06°. This figure may be a significunderestimate where there is an aggressive bend in the BHAprobe-type MWD tool is in use. This error term may be consered systematic.

Magnetic Field Uncertainty. For basic MWD surveys, only thevalue assumed for magnetic declination affects the computedmuth. However, conventional corrections for axial interferenrequire estimates of the magnetic dip and field strength. Any ein these estimates will cause an error in the computed azimu

A study by the British Geological Survey and commissionedBaker Hughes INTEQ6 investigated the likely error in using aglobal geomagnetic model to estimate the instantaneous ammagnetic field downhole. Five sources of error were identifimodeled main field vs. actual main field at the base epoch, meled secular variation vs. actual secular variation, regular~diurnal!variation due to electrical currents in the ionosphere, irregutemporal variation due to electrical currents in the magnetosphand crustal anomalies.

By making a number of gross assumptions, and by considetypical drilling rates, this author has distilled the results of tstudy into a single table.

Error Source

Error Magnitude

PropagationMode

Declination„deg…

Dip„deg…

Total Field„nT…

Main field model 0.012* 0.005 3 GSecular variation 0.017* 0.013 10 GDaily variation 0.045** 0.011** 11** R/S†

Irregular variation 0.110** 0.043** 45** R/S†

Crustal anomaly 0.476 0.195 120 G

*Below 60° latitude N or S.

** At 60° latitude N or S.†Daily and irregular variation are partially randomized between surveys. Correlations between consecutive stations are approximately 0.95 and 0.5 for the two error sources.

SPE Drill. & Completion, Vol. 15, No. 4, December 2000 223

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The dominant error source is crustal anomalies caused by ving magnetization of rocks in the Earth’s crust. The figures shoare representative of those of the North Sea. Some areas, palarly those at higher latitudes and where volcanic rocks are cloto the surface, will show greater variation. Other areas, whsedimentary rocks dominate, will show less.

In the absence of any other information, the uncertainty inestimate of the magnetic field at a given time and place proviby a global geomagnetic model may be obtained by summingabove terms statistically. There is one complication: some accmust be taken of the increasing difficulty of determining declintion as the horizontal component of the magnetic field decreaThis can be achieved by splitting this error into two componenone constant and one inversely proportional to the horizontaljection of the field,BH . For the purposes of the model, the sphas been defined somewhat arbitrarily, while ensuring thattotal declination uncertainty at Lerwick, Shetland (BH515 000 nT) is as predicted by the BGS study~0.49°!. Beingdominated by the crustal anomaly component, all magnetic fi

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errors may be considered globally systematic and summarizethe following.

Error SourceWeightingFunction Magnitude

PropagationMode

Declination~constant! AZ 0.36° GDeclination

~BH dependent! DBH 5000° nT GDip angle MFD 0.20° GTotal field MFI 130 nT G

Along-Hole Depth Errors. Ekseth7 identified 14 physical sourceof drill-pipe depth measurement error, wrote down expressionpredict their magnitude and, by substituting typical parameter vues into the expressions, predicted the total error for a numbedifferent well shapes. He then proposed a simplified model offour terms, and chose the magnitudes of each to match thedictions of the full model as closely as possible. The results was follows.

Error Source

ErrorProportional

to

Error Magnitude „1 s.d…Propagation

ModeLand Rig Floating RigRandom reference 1 0.35 m 2.2 m RSystematic reference 1 0 m 1 m SScale D 2.431024 2.131024 SStretch type D.Dv 2.231027 m21 1.531027 m21 G

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For the purposes of the basic model, the values for the land~or, equivalently a jackup or platform rig! may be chosen. Thestretch-type error, which dominates the other terms in deep wmodels two physical effects, stretch and thermal expansion ofdrill pipe. Both of these effects generally cause the drill stringelongate, so it may be appropriate to apply this term as a bias~seebelow!. If this is done, a mean value of 4.431027 m21 should beused, since Ekseth effectively treated his estimates of these eas 2 s.d. values.

Errors Omitted From the Basic MWD Model. Some errorsknown to affect MWD surveys have nonetheless not beencluded in the basic error model.

Tool Electronics and Resolution.The overall effect on accuracy caused by the limitations of the tool electronics and the relution of the tool-to-surface telemetry system is not considesignificant. Such errors will tend to be randomized over long svey intervals.

External Magnetic Interference.Ekseth7 discussed the influ-ence of remanent magnetism in casing strings on magneticveys, and gave expressions for azimuth error when drilling oua casing shoe and parallel to an existing string. Although certanot negligible, both error sources are difficult to quantify, aequally difficult to incorporate within error modeling software.seems preferable to manage these errors by applying qualitycedures designed to limit their effect.

Effect of Survey Interval and Calculation Method.Themethod presented in this paper relies on the assumption thaerror-free measurement vectorp will lead to an error-free well-bore position vectorr . If minimum curvature formulas are usefor survey calculation, this assumption will only be true when twell path between stations is an exact circular arc. The resulerror may be significant for sparse data, but may probablyneglected so long as the station interval does not exceed 100

Gravity Field Uncertainty.Differences between nominal anactual gravity field strengths will typically have no effect oMWD accuracy since only the ratio of accelerometer measuments is used in the calculation of inclination and azimuth.

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Gross Errors.Any attempt at a comprehensive discussionMWD error sources must at least acknowledge the possibilitygross errors, sometimes called human errors. These errors lacpredictability and uniformity of the physical terms discussabove. They are therefore excluded from the error model, withassumption that they are adequately managed through procesprocedure.

Propagation MathematicsThe mathematical algorithm by which wellbore positional unctainty is generated from survey error model inputs is based onapproach outlined by Brooks and Wilson.3 The development ofthis work described here was carried out by the working groreferred to in the Introduction.

A physical error occurring at a survey station will result in aerror, in the form of a vector, in the calculated well position. FroRef. 3:

ei5s i

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whereei is a vector-valued random variable~a vector error!, s i isthe magnitude of theith error source,]p/]« i is its ‘‘weightingfunction’’ and dr /dp describes how changes in the measuremvector affect the calculated well position. It is sufficient to assuthat the calculated displacement between consecutive surveytions depends only on the survey measurement vectors at ttwo stations. WritingDr k for the displacement between survestationsk21 andk, we may thus express the~1 s.d.! error due tothe presence of thei th error source at thekth survey station in thel th survey leg as the sum of the effects on the precedingfollowing calculated displacements:

ei ,l ,k5s i ,l S dDr k

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wheres i ,l is the magnitude of thei th error source over thel thsurvey leg, andpk is the instrument measurement vector at thekthsurvey station.


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The total position error at a particular survey stationK in sur-vey legL will be the sum of the vector errorsei ,l ,k taken over allerror sourcesi and all survey stations up to and includingK. Theuncertainty in this position error is expressed in the form ocovariance matrix:

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wherer(e i ,l 1 ,k1,e i ,l 2 ,k2

) is the correlation coefficient between thvalue of thei th error source at thek1th station~in the l 1th leg! andthek2th station~in the l 2th leg!. In practice, it is more conveniento sum separately the contributions of errors with different progation characteristics. Details are in Appendix A.

Weighting Functions. The weighting function for a particular error source is a 331 vector, the elements of which describe teffect of a unit error on the measured along-hole depth, inclinaand azimuth. For example, the weighting functions for constandBH-dependent magnetic declination errors are

]p

]eAZ5F 0

01G , ~4!

]p

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01/~B cosQ!

G . ~5!

For BHA sag and direction-dependent axial magnetic interence they are

]p

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]p

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]p

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]p

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]p

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G . ~10!

Weighting Functions For Sensor Errors.Tool axes and tool-face angle,a, are defined inFig. 1. There are 12 basic sensor errsources~a bias and scale factor for each of three acceleromeand three magnetometers! and each requires its own weightinfunction. These are obtained by differentiating the standard ngation equations for inclination and azimuth:

I 5cos21S Gz

AGx21Gy

21Gz2D , ~11!

Am5tan21S ~GxBy2GyBx!AGx21Gy

21Gz2

Bz~Gx21Gy

2!2Gz~GxBx1GyBy!D , ~12!

and making use of the inverse relations

Gx52G sin I sina, ~13!

Gy52G sin I cosa, ~14!

Gz5G cosI , ~15!


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Bx5B cosQ cosI cosAm sina2B sinQ sin I sina

1B cosQ sinAm cosa, ~16!

By5B cosQ cosI cosAm cosa2B sinQ sin I cosa

2B cosQ sinAm sina, ~17!

Bz5B cosQ sin I cosAm1B sinQ cosI . ~18!

Taking theX-accelerometer bias (ABX) as an example,

]I

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G3 D52cosI sina

G, ~19!

and, similarly,

]Am

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G.

~20!

The appropriate weighting function is therefore

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G F 02cosI sina

~cosI sinAm sina2cosAm cosa!tanQ1cot I cosaG .

~21!Effect of Axial Interference Correction.When a simple axial

magnetic interference correction is applied, Eq. 12 is no lonused, and different weighting functions are required for senerrors. The following analysis is by Andy Brooks.

Details of the interference corrections differ from methodmethod, but since all such methods suffer from similar limitatioit is reasonable to characterize them all with a single examMethods which ignore theBZ measurement and find the solutiowhich minimizes the vector distance between the computedexpected values of the magnetic field vector will satisfy Eqs.and 17 and

~B cosQ2B̂ cosQ̂!21~B sinQ2B̂ sinQ̂!25minimum, ~22!

whereB̂ andQ̂ are the estimated values of total field strength adip angle, respectively. Solving these three equations for azimleads to

P sinAm1Q cosAm1R sinAm cosAm50, ~23!

where

P5~Bx sina1By cosa!cosI 1B̂ sinQ̂ sin I cosI , ~24!

Q52~Bx cosa2By sina!, ~25!

R5B̂ cosQ̂ sin2 I . ~26!

Fig. 1–Definition of tool sensor axes and toolface angle.


TABLE 1– ERROR SOURCE WEIGHTING FUNCTIONS NOT GIVEN IN THE TEXT

Sensor Errors (without axial interference correction)

ABX1

G F 02cos I sin a

~cos I sin Am sin a2cos Am cos a!tan Q1cot I cos aG ASX F 0

sin I cos I sin2 a2~tan Q sin I~cos I sin Am sin a2cos Am cos a!1cos I cos a!sin a

GABY

1

G F 02cos I cos a

~cos I sin Am cos a1cos Am sin a!tan Q2cot I sin aG ASY F 0

sin I cos I cos2 a2~tan Q sin I~cos I sin Am cos a1cos Am sin a!2cos I sin a!cos a

GMBX F 0

0~cos Am cos a2cos I sin Am sin a!/~B cos Q!

GMSX F 0

0~cos I cos Am sin a2tan Q sin I sin a1sin Am cos a!~cos Am cos a2cos I sin Am sin a!

GMBY F 0

02~cos Am sin a1cos I sin Am cos a!/~B cos Q!

GMSY F 0

02~cos I cos Am cos a2tan Q sin I cos a2sin Am sin a!~cos Am sin a1cos I sin Am cos a!

GABZ

1

G F 02sin I

tan Q sin Isin Am

G ASZ F 02sin I cos I

tan Q sin I cos I sin Am

G MBZ F 00

2sin I sin Am /~B cos Q!G

MSZ F 00

2~sin I cos Am1tan Q cos I!sin I sin Am

GSensor errors (with axial interference correction)

ABIX1

G F 02cos I sin a

@cos2 I sin Am sin a~tan Q cos I1sin I cos Am!2cos a~tan Q cos Am2cot I!#/~12sin2 I sin2 Am!G

ABIY1

G F 02cos I cos a

@cos2 I sin Am cos a~tan Q cos I1sin I cos Am!1sin a~tan Q cos Am2cot I!#/~12sin2 I sin2 Am!G

ASIX F 0sin I cos I sin2 a

2sin a@sin I cos2 I sin Am sin a~tan Q cos I1sin I cos Am!2cos a~tan Q sin I cos Am2cos I!#/~12sin2 I sin2 Am!G

ASIY F 0sin I cos I cos2 a

2cos a@sin I cos2 I sin Am cos a~tan Q cos I1sin I cos Am!1sin a~tan Q sin I cos Am2cos I!#/~12sin2 I sin2 Am!G

MSIX F 00

2~cos I cos Am sin a2tan Q sin I sin a1sin Am cos a!~cos I sin Am sin a2cos Am cos a!/~12sin2 I sin2 Am!G

MSIY F 00

2~cos I cos Am cos a2tan Q sin I cos a2sin Am sin a!~cos I sin Am cos a1cos Am sin a!/~12sin2 I sin2 Am!G

MBIX F 00

2~cos I sin Am sin a2cos Am cos a!/@B cos Q~12sin2 I sin2 Am!#G

ABIZ1

G F 02sin I

@sin I cos I sin Am~tan Q cos I1sin I cos Am!#/~12sin2 I sin2 Am!G

MBIY F 00

2~cos I sin Am cos a1cos Am sin a!/@B cos Q~12sin2 I sin2 Am!#G

ASIZ F 02sin I cos I

@sin I cos2 I sin Am~tan Q cos I1sin I cos Am!#/~12sin2 I sin2 Am!G

Magnetic field errors (with axial interference correction)

MFI F 00

2sin I sin Am~tan Q cos I1sin I cos Am!/@B~I2sin2 I sin2 Am!#G MDI F 0

02sin I sin Am~cos I2tan Q sin I cos Am!/~12sin2 I sin2 Am!

G
226 H. S. Williamson: Accuracy Prediction for MWD SPE Drill. & Completion, Vol. 15, No. 4, December 2000

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The sensitivities of the computed azimuth to errors in the senmeasurements are found by differentiating Eq. 23.

Magnetic Field Uncertainty.The weighting function for mag-netic declination error is given above. Those for magnetic fistrength and dip angle, which are required when an axial magninterference correction is in use, are derived by differentiating23 with respect toB̂ andQ̂.

Misalignment Errors.Brooks and Wilson3 model tool axialmisalignment as two uncorrelated errors corresponding to thxandy axes of the tool. Their expressions for the associated innation and azimuth errors lead directly to the following weightifunctions:

]p

]eMX5F 0

sina2cosa/sin I

G , ~27!

]p

]eMY5F 0

cosasina/sin I

G . ~28!

Table 1 contains expressions for all the weighting functions ncited in this section which are required to implement the ermodels described in this paper.

Calculation OptionsThe method of position uncertainty calculation described heremits a number of variations. It can still claim to be a standardthat selection of the same set of conventions should always ythe same results.

Along-Hole Depth Uncertainty. The propagation model described above is appropriate for determining the position untainty of the points in space at which the survey tool came~or willcome! to rest. These may be called uncertainties ‘‘at survey stions.’’

Thorogood2 argued that it is more meaningful to compute tposition uncertainties of the points in the wellbore at the alohole depths assigned to the survey stations. These may be cthe uncertainties ‘‘at assigned depths.’’ This approach allocomputation of the position uncertainty of points~such as picksfrom a wireline log! whose depths have been determined indepdently of the survey. Thorogood made this calculation by defina weighting function incorporating the local build and turn ratesthe well. The approach described in Appendix A achievessame result without the need for a new weighting function.

The results of the two approaches differ only in the along-hcomponent of uncertainty. The along-hole uncertainty at a surstation includes the uncertainty in the station’s measured dewhile the uncertainty at an assigned depth does not.

The correct choice of approach depends on the engineeproblem being tackled, in many cases it is immaterial. The usewell-designed directional software need not be aware of the is

Fig. 2–Plan view of standard well profiles.


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Survey Bias.Not to be confused with sensor biases~which mightbetter be termed offset errors!, survey bias is the tendency for thmost likely position of a well to differ from its surveyed positionThe only bias term defined by Wolff and de Wardt was for manetic interference in ‘‘poor magnetic’’ surveys. The claims fstretch and thermal expansion of the drill pipe to be treated aserrors are at least as strong.

Some vendors of directional software have neglected to mosurvey bias on the grounds that such errors should be correfor, and that engineers do not like/understand them. The firstjection can be countered by the observation, ‘‘yes, but theynot,’’ the second by careful software design.

The sign convention for position bias is from survey to molikely position ~i.e., opposite to the direction of the error!. Sincethe drill pipe generally elongates downhole, most likely depthsgreater than survey depths and bias values are positive. Fordrillstring interference, most likely azimuths are greater than svey azimuths when the weighting function, sinI sinAm , is posi-tive, so bias values are again positive~at least in the northernhemisphere!. The additional mathematics required to model svey bias is included in Appendix A.

Calculation Conventions.The calculation of position uncertaintrequires a wellbore survey consisting of discrete stations, eacwhich has an associated along-hole depth, inclination, azimand toolface angle. Clearly, these data will not be availablemany cases, and certain conventions are required wherebysumed values may be calculated. The following are suggeste

Along-Hole Depth.For drilled wells, actual survey stationshould be used. For planned wells, the intended survey inteshould be determined, and stations should be interpolated awhole multiples of this depth within the survey interval. Typcally, an interval of 100 ft or 30 m should be used. For well plathe way points should be included as additional stations.

Inclination and Azimuth. For drilled wells, measured valueshould be used. For planned wells, the profile should be intelated at the planned survey station depths using minimum cuture.

Toolface.If actual toolface angles are available, they shouldused. If not, several means of generating them are possible.

d Random number generation. Possibly close to reality, butsults are not repeatable and will tend to be optimistic.

d Worst case. Several variations on this idea are possible,each will require some additional calculation. The principlequestionable, and the computational overhead is probably nottified.

d Borehole toolface~i.e., the up-down left-right change in borehole direction!. This angle bears little relation to survey tool orentation, but is at least well defined, and may be computedrectly from inclination and azimuth data. This approach will teto limit the randomization of toolface dependent errors, givingconservative uncertainty prediction. This is the convention use

Fig. 3–Vertical section plot of standard well profiles. Note thedifferent section azimuths.


TABLE 2– STANDARD WELL PROFILES

ISCWSA No. 1–North Sea extended reach wellLat.560°N, Long.52°E, G59.80665 ms22, B550,000 nT, Q572°,d54°W, Station interval530 m, Vertical Section Azimuth575°

MD(m)

Inc(deg)

Azi(deg)

North(m)

East(m)

TVD(m)

VS(m)

DLS°/30 m

0.00 0.000 0.000 0.00 0.00 0.00 0.00 0.001200.00 0.000 0.000 0.00 0.00 1200.00 0.00 0.002100.00 60.000 75.000 111.22 415.08 1944.29 429.79 2.005100.00 60.000 75.000 783.65 2924.62 3444.29 3027.79 0.005400.00 90.000 75.000 857.80 3201.34 3521.06 3314.27 3.008000.00 90.000 75.000 1530.73 5712.75 3521.06 5914.27 0.00

ISCWSA No. 2–Gulf of Mexico fish-hook wellLat.528°N, Long.590°W, G59.80665 ms22, B548,000 nT,Q558°, d52°E, Station interval5100 ft, Vert. Sect. Azim.521°MD(ft)

Inc(deg)

Azi(deg)

North(ft)

East(ft)

TVD(ft)

VS(ft)

DLS°/100ft

0.00 0.000 0.000 0.00 0.00 0.00 0.00 0.002,000.00 0.000 0.000 0.00 0.00 2,000.00 0.00 0.003,600.00 32.000 2.000 435.04 15.19 3,518.11 411.59 2.005,000.00 32.000 2.000 1,176.48 41.08 4,705.37 1,113.06 0.005,525.54 32.000 32.000 1,435.37 120.23 5,153.89 1,383.12 3.006,051.08 32.000 62.000 1,619.99 318.22 5,602.41 1,626.43 3.006,576.62 32.000 92.000 1,680.89 582.00 6,050.92 1,777.82 3.007,102.16 32.000 122.000 1,601.74 840.88 6,499.44 1,796.70 3.009,398.50 60.000 220.000 364.88 700.36 8,265.27 591.63 3.00

12,500.00 60.000 220.000 21,692.70 21,026.15 9,816.02 21,948.01 0.00

ISCWSA No. 3–Bass Strait designer wellLat.540°S, Long.5147°E, G59.80665 ms22, B561,000 nT, Q5270°,d513°E, Station interval530 m, Vertical Section Azimuth5310°

MD(m)

Inc(deg)

Azi(deg)

North(m)

East(m)

TVD(m)

VS(m)

DLS°/30 m

0.00 0.000 0.000 0.00 0.00 0.00 0.00 0.00500.00 0.000 0.000 0.00 0.00 500.00 0.00 0.00

1100.00 50.000 0.000 245.60 0.00 1026.69 198.70 2.501700.00 50.000 0.000 705.23 0.00 1412.37 570.54 0.002450.00 0.000 0.000 1012.23 0.00 2070.73 818.91 2.002850.00 0.000 0.000 1012.23 0.00 2470.73 818.91 0.003030.00 90.000 283.000 1038.01 2111.65 2585.32 905.39 15.003430.00 90.000 283.000 1127.99 2501.40 2585.32 1207.28 0.003730.00 110.000 193.000 996.08 2727.87 2520.00 1197.85 9.004030.00 110.000 193.000 721.40 2791.28 2417.40 1069.86 0.00

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the examples which follow. Formulas for borehole toolfacegiven in Appendix B.

Standard ProfilesAt the Eighth Meeting of the ISCWSA participants were set ttask of designing a number of well profiles suitable for testsoftware implementations of the error models and propagamathematics, studying and highlighting the behavior of differerror models ~magnetic and gyroscopic! and individual errorsources, demonstrating to a nonspecialist audience the unceties to be expected from typical survey programs.

The ideas generated at the meeting were used to devise athree profiles:ISCWSA No. 1: an extended reach well in the North Sea,ISCWSA No. 2: a ‘‘fish-hook’’ well in the Gulf of Mexico, witha long turn at low inclination, andISCWSA No. 3: a ‘‘designer’’ well in the Bass Strait, incorporaing a number of difficult hole directions and geometries.

Figs. 2 and 3 illustrate the test profiles in plan and sectioTheir full definition, given inTable 2, includes location, gravityand magnetic fields, survey stations, toolface angles and dunits.


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Example Results

The error models for basic and interference-corrected MWD hbeen applied to the standard well profiles to generate posiuncertainties in each well. The results of several combinationstabulated inTable 3.

Examples 1 and 2 compare the basic and interference-corremodels in well ISCWSA No. 1. Being a high inclination werunning approximately ENE, the interference correction actuadegrades the accuracy. The results are plotted inFig. 4. Examples3 to 6 all represent the basic MWD error model applied to wISCWSA No. 2. They differ in that each uses a different permtation of the survey station/assigned depth and symmetric esurvey bias calculation options. The variation of lateral unctainty and ellipsoid semimajor axis, characteristic of a fish-howell, is shown inFig. 5. Finally, example 7 breaks well ISCWSANo. 3 into three depth intervals, with the basic and interferencorrected models being applied alternately. This example iscluded as a test of error term propagation.

The results in Table 3 were computed by the author, and hbeen independently verified by Anne Holmes, Steve Grindrand Andy Brooks. Exact duplication of these results is a powe


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and demanding test for implementations of the method and mels described in this paper.

Conclusions and RecommendationsThis paper, and the collaborative work which it describes, eslishes a common starting point for wellbore position uncertaimodeling. The standardized elements are a nomenclature~see be-low!, a definition of what constitutes an error model, mathemaof position uncertainty calculation, an error model for a badirectional MWD service,Table 4, a set of well profiles for in-

Fig. 4–Comparison of basic and interference corrected MWDerror models in well ISCWSA No. 1.

Fig. 5–Variation of lateral uncertainty and ellipsoid semimajoraxis in a fish-hook well, ISCWSA No. 2.


230 H. S. Willi

TABLE 4– SUMMARY OF BASIC MWD ERROR MODELS

WeightingFunction Basic Model

With AxialCorrection

PropagationMode

Sensors

ABX 0.004 ms22 SABY 0.004 ms22 SABZ 0.004 ms22 SASX 0.0005 SASY 0.0005 SASZ 0.0005 SMBX 70 nT SMBY 70 nT SMBZ 70 nT SMSZ 0.0016 SMSY 0.0016 SMSZ 0.0016 SABIX 0.004 ms22 SABIY 0.004 ms22 SABIZ 0.004 ms22 SASIX 0.0005 SASIY 0.0005 SASIZ 0.0005 SMBIX 70 nT SMBIY 70 nT SMSIX 0.0016 SMSIY 0.0016 SMisalignmentSAG 0.2° 0.2° SMX 0.06° 0.06° SMY 0.06° 0.06° SAxial magnetic interferenceAZ 0.25° SAMID 0.6° S or B*DeclinationAZ 0.36° 0.36° GDBH 5000°nT 5000°nT GTotal magnetic field and dip angleMDI 0.20° GMFI 130 nT GAlong-hole depthDREF 0.35 m 0.35 m RDSF 2.431024 2.431024 SDST 2.231027 m21 2.231027 m21 G or B**

*When modeled as bias: m50.33°, s50.5°.

** When modeled as bias: m54.431027 m21, s50.

i

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ical

d

vestigating error models, and a set of results for testing softwimplementations.The future work which these standards were designed to facilincludes

d establishment of agreed error models for other surveyvices, including in-field referencing and gyroscopic tools and

d interchangeability of calculated position uncertainties amosurvey vendor, directional drilling company, and operator.

Useful though this work is, it is only a piece in a large jigsapuzzle. Taking a wider view, the collaborative efforts of the etended survey community should now be directed towards sdardization of quality assurance measures, strengthening thebetween quality assurance specifications and error model paeters, and better integration of wellbore position uncertainty wthe other aspects of oil field navigation.

NomenclatureISCWSA Nomenclature*

D 5 along-hole depth, m, ft

amson: Accuracy Prediction for MWD

are

tate

er-

ng

wx-an-linkam-ith

I 5 wellbore inclination, degA 5 wellbore azimuth, deg

Am 5 wellbore magnetic azimuth, dega 5 toolface angle, degd 5 magnetic declination, degQ 5 magnetic dip angle, degB 5 magnetic field strength, nTG 5 gravity field strength, ms22**

X,x,Y,y,Z,z5 tool reference directions; see Fig. 1

*Adopted by ISCWSA participants as a standard for all techncorrespondence.** The international standard value forG of 9.806 65 ms22 wasused in the calculation of the results in Table 3.

Special Nomenclature

b 5 component of wellbore position bias vectorB̂ 5 estimated magnetic field strength

BH 5 horizontal component of magnetic fielstrength


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†C‡ 5 wellbore position uncertainty covariance matre 5 1 s.d. vector error at an intermediate station

e* 5 1 s.d. vector error at the station of interestE 5 sum of vector errors from slot to station of in

terest« 5 particular value of a survey error

H,L 5 used in calculation of the toolfacem 5 bias vector error at an intermediate station

m* 5 bias vector error at the station of interestM 5 wellbore position bias vectorm 5 mean of error values 5 standard deviation of error value, component

wellbore position uncertaintyp 5 survey measurement vector (D,I ,A)

P,Q,R 5 intermediate calculated quantitiesr 5 wellbore position vector

Dr k 5 increment in wellbore position between statiok21 andk

r 5 correlation coefficientQ̂ 5 estimated magnetic dip anglev 5 along-hole unit vectorw 5 factor relating error magnitude to uncertainty

measurement

Subscripts and Counters

hla,HLA 5 borehole referenced framei 5 a survey error termk 5 a survey stationK 5 survey station of interest

Kl 5 number of stations in thel th survey legl 5 a survey leg

L 5 survey leg containing the station of interestnev 5 earth-referenced frame

Superscripts

dep 5 at the along-hole depth assigned to the survstation

rand 5 random propagation modesvy 5 at the point where the survey measureme

were takensyst 5 systematic propagation modewell 5 well by well or global propagation mode

AcknowledgmentsThe author thanks all participants in the ISCWSA for their enthsiasm and support over several years and in the review ofpaper.

Particular contributions to the MWD error model were madeJohn Turvill and Graham McElhinney, both now with PathFindEnergy Services, formerly Halliburton Drilling Systems; WayPhillips, Schlumberger, Paul Rodney and Anne Holmes, SpeSun Drilling Services; and Oddvar Lotsberg, formerly of BakHughes INTEQ.

Participants in the working group on error propagation wDavid Roper, Sysdrill Ltd.; Andy Brooks and Harry WilsonBaker Hughes INTEQ; and Roger Ekseth, formerly of Statoil.

The author also wishes to thank BP for their permissionpublish this paper.

References1. Wolff, C.J.M. and de Wardt, J.P.: ‘‘Borehole Position Uncertaint

Analysis of Measuring Methods and Derivation of Systematic ErModel,’’ JPT ~December 1981! 2339.

2. Thorogood, J.L.: ‘‘Instrument Performance Models and Their Appcation to Directional Survey Operations,’’SPEDE~December 1990!294.

3. Brooks, A.G. and Wilson, H.: ‘‘An Improved Method for ComputinWellbore Position Uncertainty and Its Application to Collision anTarget Intersection Probability Analysis,’’ paper 36863 presentedthe SPE 1996 SPE European Petroleum Conference, Milan, I22–24 October.


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-

of

s

n

ey

ts

u-this

byere

rry-er

re,

to

–or

li-

dat

aly,

4. Dubrule, O. and Nelson, P.H.: ‘‘Evaluation of Directional SurveErrors at Prudhoe Bay,’’SPEDE~September 1987! 257.

5. Grindrod, S.J. and Wolff, J.M.: ‘‘Calculation of NMDC Length Required for Various Latitudes Developed From Field MeasurementsDrill String Magnetisation,’’ paper 11382 presented at the SPE 19IADC/SPE Drilling Conference, New Orleans, 20–23 February.

6. Macmillan, S.et al.: ‘‘Error estimates for geomagnetic field valuecomputed from the BGGM,’’ British Geological Survey TechnicReport No. WM/93/28C~1993!.

7. Ekseth, R.: ‘‘Uncertainties in Connection with the DeterminationWellbore Positions,’’ PhD dissertation, Norwegian U. of Science aTechnology, Trondheim, Norway~1998!.

Appendix A: Mathematical Description of PropagationModelThe total position uncertainty at a survey station of interest,K ~insurvey legL! is the sum of the contribution from all the activerror sources. It is convenient computationally to group the esources by their propagation type and to sum them separatel

Vector Errors at the Station of Interest. Recall that the vectorerror due to the presence of error sourcei at stationk is the sum ofthe effect of the error on the preceding and following survey dplacements:

ei ,l ,k5s i ,l S dDr k

dpk1

dDr k11

dpkD ]pk

]« i. ~A-1!

Evaluating this expression using the minimum curvature wtrajectory model is cumbersome. There is no significant lossaccuracy in using the simpler balanced tangential model:

Dr j5D j2D j 21

2 F sin I j 21 cosAj 211sin I j cosAj

sin I j 21 sinAj 211sin I j sinAj

cosI j 211cosI j

G . ~A-2!

The two differentials in the parentheses in Eq. A-1 may thenexpressed as

dDr j

dpk5FdDr j

dDk

dDr j

dI k

dDr j

dAkG ~ j 5k,k11!, ~A-3!

dDr k

dDk5

1

2 F sin I k21 cosAk211sin I k cosAk

sin I k21 sinAk211sin I k sinAk

cosI k211cosI k

G , ~A-4a!

dDr k11

dDk5

1

2 F2sin I k cosAk2sin I k11 cosAk11

2sin I k sinAk2sin I k11 sinAk11

2cosI k2cosI k11

G , ~A-4b!

dDr j

dI k5

1

2 F ~D j2D j 21!cosI k cosAk

~D j2D j 21!cosI k sinAk

2~D j2D j 21!sin I k

G ~ j 5k,k11!, ~A-5!

dDr j

dAk5

1

2 F2~D j2D j 21!sin I k sinAk

~D j2D j 21!sin I k cosAk

0G ~ j 5k,k11!. ~A-6!

For the purposes of computation, the error summation terminat the survey station of interest. Vector errors at this stationtherefore given by

ei ,L,K* 5s i ,L

dDrK

dpK

]pK

]« i. ~A-7!

The notationei ,L,K* indicates that a measurement error at this stion affects only the preceding survey displacement. In whatlows we reserve the notationei ,l ,k for vector errors at intermediatestations, which affect both the preceding and following displaments.

Undefined Weighting Functions.For some combinations oweighting function and hole direction, one component of the m


232 H. S. Willi

TABLE A-1– ERROR VECTORS IN VERTICAL HOLE WHERE WEIGHTING FUNCTIONIS SINGULAR

Sensor errors (with or without axial interference correction)ABXorABIX

ei , l ,k5s i , l~Dk112Dk21!

2G F2sin~A1a!cos~A1a!

0G ABY

orABIY

ei , l ,k5s i , l~Dk112Dk21!

2G F2cos~A1a!2sin~A1a!

0G

Misalignment errors

MX ei , l ,k5s i , l~Dk112Dk21!

2 F sin~A1a!2cos~A1a!

0G MY ei , l ,k5

s i , l~Dk112Dk21!

2 F cos~A1a!sin~A1a!

0G

en

n

r

n

cu

t

r

storrom

r-of

a

es-

, it

surement vector~usually the azimuth! is highly sensitive tochanges in hole direction and the vector]p /]« i is apparentlyundefined. There are two cases.

Vertical Hole. In this case, dr /dp is zero but the vectorsei ,l ,k

andei ,L,K* are still finite and well defined. They may be computby forming the products of Eqs. A-1 and A-7 algebraically aevaluating them as a whole. Take as an example the weighfunction for anX-axis radially symmetric misalignment. Substituing the expression for]p/]«MX , Eq. 27, and the well trajectorymodel equations, Eqs. A-3 to A-6, into Eqs. A-1 and A-7, asettingI equal to zero gives

ei ,l ,k5s i ,l~Dk112Dk21!

2 F sin~A1a!

2cos~A1a!

0G ~A-8!

and

ei ,L,K* 5s i ,L~DK2DK21!

2 F sin~A1a!

2cos~A1a!

0G . ~A-9!

There are similar expressions forY-axis axial misalignment andX- andY-axis accelerometer biases. These are given inTable A-1.Equivalent expressions may be used for evaluating bias vectothe vertical hole, withmi ,l ,k , mi ,L,K* , m i ,l andm i ,L substituted forei ,l ,k , ei ,L,K* ands i ,l ands i ,L , respectively.

Other Hole Directions.Some error sources really are ubounded in certain hole directions. The examples in this papersensor errors after axial interference correction in a horizontalmagnetic east/west wellbore, a so-called ‘‘90/90’’ well. In sucases, the assumptions of linearity break down, and compposition uncertainties are meaningless. Software implementatshould include an error-catching mechanism for this case.

Summation of Errors. Vector errors are summed into positiouncertainty matrices as follows.

Random Errors.The contribution to survey station uncertainfrom a randomly propagating error sourcei over survey legl ~notcontaining the station of interest! is

@C# i ,lrand5(

k51

Kl

~ei ,l ,k!"~ei ,l ,k!T, ~A-10!

and the total contribution over all survey legs is

@C# i ,Krand5(

l 51

L21

@C# i ,lrand1(

k51

K21

~ei ,L,k!"~ei ,L,k!T

1~ei ,L,K* !"~ei ,L,K* !T. ~A-11!

Systematic Errors.The contribution to survey station uncetainty from a systematically propagating error sourcei over sur-vey leg l ~not containing the station of interest! is

amson: Accuracy Prediction for MWD

dd

tingt-

d

s in

-are

andhted

ions

n

y

-

@C# i ,lsyst5S (

k51

Kl

ei ,l ,kD "S (k51

Kl

ei ,l ,kD T

, ~A-12!

and the total contribution over all survey legs is

@C# i ,Ksyst5(

l 51

L21

@C# i ,lsyst

1S (k51

K21

ei ,L,k1ei ,L,K* D "S (k51

K21

ei ,L,k1ei ,L,K* D T

. ~A-13!

Well by Well and Global Errors.Each of these error types isystematic among all stations in a well. The individual vecerrors can therefore be summed to give a total vector error fslot to station:

Ei ,K5(l 51

L21 S (k51

Kl

ei ,l ,kD 1(k51

K21

ei ,L,k1ei ,L,K* . ~A-14!

The total contribution to the uncertainty at survey stationK is

@C# i ,Kwell5Ei ,K"Ei ,K

T . ~A-15!

Total Position Covariance.The total position covariance at suvey stationK is the sum of the contributions from all the typeserror source:

@C#Ksvy5(

i PR@C# i ,K

rand1(i PS

@C# i ,Ksyst1 (

i P$W,G%@C# i ,K

well , ~A-16!

where the superscriptsvy indicates the uncertainty is defined atsurvey station.

Survey Bias.Error vectors due to bias errors are given by exprsions entirely analogous with Eqs. A-1 and A-7:

mi ,l ,k5m i ,l S dDr k

dpk1

dDr k11

dpkD ]pk

]« i, ~A-17!

mi ,L,K* 5m i ,L

dDrK

dpK

]pK

]« i. ~A-18!

The total survey position bias at survey stationK, MKsvy , is the

sum of individual bias vectors taken over all error sourcesi, legsl and stationsk:

MKsvy5(

iS (

l 51

L21 S (k5 l

Kl

mi ,l ,kD 1(k51

K21

mi ,L,k1mi ,L,K* D . ~A-19!

Position Uncertainty and Bias at an Assigned Depth.Definingthe superscriptdepto indicate uncertainty at an assigned depthmay be shown that

ei ,L,K* dep5ei ,L,K* svy2s i ,Lwi ,L,KvK , ~A-20!

ei ,l ,kdep5ei ,l ,k

svy , ~A-21!


u

nsr-

is

esthe

i-ced

pal

ol-

wherewi ,L,K is the factor relating error magnitude to depth mesurement uncertainty andvK is the along-hole unit vector at station K. Figs. A-1 and A-2 illustrate these results. Substitutinthese expressions into Eqs. A-12 to A-16 yields the positioncertainty at the along-hole depth assigned to each survey sta

Survey bias at an assigned depth is calculated by substituthe following error vectors into Eq. A-19:

mi ,L,K* dep5mi ,L,K* svy2m i ,Lwi ,L,KvK , ~A-22!

mi ,l ,kdep5mi ,l ,k

svy . ~A-23!

Fig. A-1–Vector errors at the last station „point of interest … dueto an along-hole depth error at the last station.

Fig. A-2–Vector errors at the last station „point of interest … dueto an along-hole depth error at an earlier station.


a--gn-

tion.ting

Relative Uncertainty Between Wells.When calculating the un-certainty in the relative position between two survey statio(KA ,KB) in wells ~A,B!, we must take proper account of the corelation between globally systematic errors. The uncertaintygiven by

@C#svy@rKA2rKB

#5@C#KA

svy1@C#KB

svy2(i PG

$~Ei ,KA!"~Ei ,KB

!T

1~Ei ,KB!"~Ei ,KA

!T%. ~A-24!

The relative survey bias is simply

M svy@rKA2r kB

#5MKA

svy2MKB

svy . ~A-25!

Substitution of Eqs. A-20 to A-23 into these expressions givthe equivalent results at the along-hole depths assigned tostations.

Transformation Into Borehole Reference Frame.The resultsderived above are in an Earth-referenced frame~north, east, ver-tical, subscriptnev!. The transformation of the covariance matrces and bias vectors into the more intuitive borehole referenframe ~highside, lateral, along hole, subscripthla! is straightfor-ward:

@C#hla5@T#T@C#nev@T#, ~A-26!

F bH

bL

bA

G5Mhla5@T#TMnev , ~A-27!

where

@T#5F cosI K cosAK 2sinAK sin I K cosAK

cosI K sinAK cosAK sin I K sinAK

2sin I K 0 cosI K

G ~A-28!

is a rotation matrix. Uncertainties and correlations in the princiborehole directions are obtained from

sH5A@C#hla@1,1# etc., ~A-29!

rHA5@C#hla@1,2#

sHsLetc. ~A-30!

Appendix B: Calculation of Toolface AngleThe following formulas may be used to calculate borehole toface angle from successive surveys:

HK5sin I K cosI K212sin I K21 cosI K cos~AK2AK21!, ~B-1!

LK5sin I K21 sin~AK2AK21!, ~B-2!

if HK.0, aK5tan21~LK /HK!, ~B-3!

if HK,0, aK5tan21~LK /HK!1180°, ~B-4!

if HK50, aK5270°, 0° or 90° asLK

,0, LK50 or LK.0. ~B-5!

SI Metric Conversion Factorsft 3 3.048* E201 5 m

*Conversion factor is exact. SPEDC

Hugh. S. Williamson is a well positioning specialist with BP’s Up-stream Technology Group in Sunbury-on-Thames, U.K. e-mail:[email protected]. He holds a degree in mathematics fromCambridge U. and a degree in engineering surveying and ge-odesy from Nottingham U.


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