Accuracy Prediction for Directional MWD

8/9/2019 Accuracy Prediction for Directional MWD

1/16


2/16

2 HUGH S. WILLIAMSON SPE 56702

individuals established following the SPWLA Topical Conference on

MWD held in Kerrville, Texas in late 1995. The Group’s broad

objective is “to produce and maintain standards for the Industry relating

to wellbore survey accuracy”. Much of the content of this paper, and

specifically the details of the basic MWD error model, had its genesis in

the Group’s meetings, which were distinguished by their open and co-

operative discussions.

Four Company Worki ng Group. The ISCWSA being too large a

forum to undertake the detailed mathematical development of an error

propagation model, this was completed by a small working group from

Sysdrill Ltd., Statoil, Baker Hughes INTEQ and BP Exploration. The

mathematical model created by the group and described below has been

made freely available for use by the Industry.

Assumptions and Definitions

The following assumptions are implicit in the error models and

mathematics presented in this paper:

1. Errors in calculated well position are caused exclusively by the

presence of measurement errors at wellbore survey stations.

2. Wellbore survey stations are, or can be modelled as, three-elementmeasurement vectors, the elements being along-hole depth, D,

inclination, I , and azimuth A. The propagation mathematics also requires

a toolface angle, τ, at each station.3. Errors from different error sources are statistically independent.

4. There is a linear relationship between the size of each measurement

error and the corresponding change in calculated well position.

5. The combined effect on calculated well position of any number of

measurement errors at any number of survey stations is equal to the

vector sum of their individual effects.

No restrictive assumptions are made about the statistical distribution

of measurement errors.

Error sources, terms and models. An error source is a physical phenomenon which contributes to the error in a survey tool

measurement. An error term describes the effect of an error source on a

particular survey tool measurement. It is uniquely specified by the

following data:

• a name• a weighting function, which describes the effect of the error ε onthe survey tool measurement vector p. Each function is referred to by a

mnemonic of up to four letters.

• a mean value, µ.• a magnitude, σ, always quoted as a 1 standard deviation value.• a correlation coefficient ρ1 between error values at survey stationsin the same survey leg. (In a survey listing made up of several

concatenated surveys, a survey leg is a set of contiguous surveystations acquired with a single tool or, if appropriate, a single tool type).

• a correlation coefficient ρ2 between error values at survey stationsin different survey legs in the same well.

• a correlation coefficient ρ3 between error values at survey stations

in different wells in the same field.

To ensure that the correlation coefficients are well defined, only f

combinations are allowed.

Propagation Mode ρρ 1 ρρ 2 ρρ 3

Random (R) 0 0 0

Systematic (S) 1 0 0

Per-well (W) 1 1 0

Global (G) 1 1 1

ρ1, ρ2 and ρ3 are to be considered properties of the error source, a

should be the same for all survey legs.

An error model is a set of error terms chosen with the aim

properly accounting for all the significant error sources which affe

survey tool or service.

An Error Model for “Basic” MWD

For the survey specialist in search of a “best estimate” of posit

uncertainty it is tempting to differentiate minutely between tools typ

and models, running configurations, BHA design, geographical locat

and several other variables. While justifiable on technical grounds, su

an approach is impractical for the daily work of the well planner. T

time needed to find out this data for historical wells, and for ma

planned wells, is simply not available.

The error model presented in this section is intended to

representative of MWD surveys run according to fairly standard qual

procedures. Such procedures would include

• rigorous and regular tool calibration• survey interval no greater than 100 ft• non-magnetic spacing according to standard charts (where no ax

interference correction is applied)

• not surveying in close proximity to existing casing strings or ot

steel bodies• passing standard field checks on G-total, B-total and dip.The requirement to differentiate between different services may be m

by defining a small suite of alternative error models. Examples cove

in this paper are:

• application or not of an axial interference correction• application or not of a BHA sag correctionAlternative models would also be justified for:

• In-field referenced surveys• In-hole (gyro) referenced surveys• Depth-corrected surveys

The model presented here is based on the current state of knowledge a

experience of a number of experts. It is a starting point for furtresearch and debate, not an end-point.

Sensor Errors. MWD sensors will typically show small shifts

performance between calibrations. We may make the assumption t


3/16

SPE 56702 ACCURACY PREDICTION FOR DIRECTIONAL MWD 3

the shifts between successive calibrations are representative of the shifts

between calibration and field performance. On this basis, two major

MWD suppliers compared the results of successive scheduled

calibrations of their tools. Paul Rodney examined 288 pairs of

calibrations, and noted the change in bias (ie. offset error), scale factor

and misalignment for each sensor. Wayne Phillips did the same for 10

pairs of calibrations, except that sensor misalignments were not

recorded.

Andy Brooks has demonstrated that if a sensor is subject to a scale

error and two orthogonal misalignments, all independent and of similar

magnitude, the combination of the three error terms is equivalent to a

single bias term. This term need not appear explicitly in the error model,

but may be added to the existing bias term to create a “lumped” error.

This eliminates the need for 20 extra weighting functions corresponding

to sensor misalignments.

The data from the MWD suppliers suggest that in-service sensor

misalignments are typically smaller than scale errors. As a result, only a

part of the observed scale error was “lumped” with the misalignments

into the bias term, leaving a residual scale error which is modeled

separately. In this way, four physical errors for each sensor were

transformed into two modeled terms. The results were as follows:

Error Source weighting

function

magnitude prop.

mode

Accelerometer biases ABX,Y,Z 0.0004 g S

Accelerometer scale factors ASX,Y,Z 0.0005 S

Magnetometer biases MBX,Y,Z 70 nT S

Magnetometer scale factors MSX,Y,Z 0.0016 S

These figures include errors which are correlated between sensors, and

which therefore have no effect on calculated inclination and azimuth (the

exception being the effect of correlated magnetometer errors on

interference corrected azimuths). It could be argued that themagnetometer scale factor errors in particular (which may be influenced

by crustal anomalies at the calibration sites) should be reduced to

account for this.

BHA magnetic interference. Magnetic interference due to steel in

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

measurements of magnetic pole strengths have now been made.

Observed root-mean-square values are:

Item Pin Box Source

RMS pole strength(sample size)

Drill collar 505 µWb (8) Grindrod, Wol605 µWb (11) 435 µWb (11) Lotsberg5

511 µWb (4) McElhinney7

Stabiliser 177 µWb (6) Grindrod, Wol396 µWb (10) 189 µWb (10) Lotsberg369 µWb (5) 408 µWb (10) McElhinney

Motor 340 µWb (12) 419 µWb (10) LotsbergOddvar Lotsberg also computed pole strengths for 41 BHAs from

results of an azimuth correction algorithm. The RMS pole strength w

369 µWb (micro-Webers).These results suggest that 400 µWb is a reasonable estimate for th

s.d. pole strength of a steel drill string component where furt

information is lacking. This is useful information for BHA design, b

cannot be used for uncertainty prediction without a value for n

magnetic spacing distance. Unfortunately, there is no “typical” spac

used in the Industry, and we must find another way to estimate t

magnitude of this error source.

A well-established Industry practice is to require non-magne

spacing sufficient to keep the azimuth error below a fixed toleran

(typically 0.5° at 1 s.d.) for assumed pole strengths and a given hdirection. This tolerance may need to be compromised in the le

favourable hole directions. For a fixed axial interference field, a

neglecting induced magnetism, azimuth error is strongly dependent

hole direction, being proportional to sin I sin Am. Thus to model

azimuth error in uncorrected surveys, we require a combination of er

terms which

• predicts zero error if the well is vertical or magnetic north/south• predicts errors somewhat greater than the usual tolerance if the wis near horizontal and magnetic east/west

• predicts errors near the usual tolerance for other hole directions.These requirements could be met by constructing some artific

weighting function, but this would violate our restriction to physicameaningful error terms. A constant error of 0.25° and a directidependent error of 0.6°sin I sin Am is perhaps the best we can achieve way of a compromise. It is legitimate to consider these valu

representative of 1 standard deviation, since the pole strength valu

which underlie the non-magnetic spacing calculations are themselv

quoted at 1 s.d.

Both error terms may be propagated as systematic, although there

theoretical and observational evidence4 that this error is asymmetr

acting in the majority of cases to swing magnetic surveys to the north

the northern hemisphere. Giving the direction-dependent term a me

value of 0.33° and a magnitude of 0.5° reproduces this asymmetry (w

about 75% of surveys being deflected to the north), while leaving

root-mean-square error unchanged.Axial interference errors are not modelled for surveys which have be

corrected for magnetic interference.

Cross-Axi al I nterf erence Cross-axial interference from the BHA

indistinguishable from magnetometer bias, and propagates in the sa


4/16


way. Anne Holmes8 analysed the magnetometer biases for 78 MWD

surveys determined as a by-product of a multi-station correction

algorithm. Once a few outliers - probably due to magnetic “hot-spots”

and hence classified as gross errors - had been eliminated, the remaining

observations gave an RMS value of 57nT. This figure is somewhat

smaller than the 70nT attributable to magnetometer bias alone. The

conclusion must be that cross-axial interference does not, in the average,

make a significant contribution to the overall MWD error budget, and

may be safely left out of the model.

Tool Misalignment. Misalignment is the error caused by the along-

hole axis of the directional sensor assembly being out-of-parallel with

the centre line of the borehole. The error may be modeled as a

combination of two independent phenomena:

BHA sag is due to the distortion of the MWD drill collar under

gravity. It is modelled as confined to the vertical plane, and proportional

to the component of gravity acting perpendicular to the wellbore (ie.

sinI ). The magnitude of the error depends on BHA type and geometry,

sensor spacing, hole size and several other factors. Two-dimensional

BHA models typically calculate inclination corrections of 0.2° or 0.3°for poorly stabilised BHAs in horizontal hole5. For well stabilisedassemblies the value is usually less than 0.15°. In the absence of better information, 0.2° (at 1 s.d.) may be considered a realistic input into the basic error model.

Sag corrections, if they are applied, are calculated on the often

unjustified assumptions of both the hole and stabilisers being in gauge.

Data comparisons by the author suggest a typical efficiency of 60% for

these corrections, leaving a post-correction residual sag error of 0.08°.Assuming similar BHAs throughout a hole section, all BHA sag errors

may be classified as systematic.

Radially symmetri c misalignment is modelled as equally likely to

be oriented at any toolface angle. John Turvill made an estimate of its

magnitude based on the tolerances on several concentric cylinders:• Sensor package in housing. Tolerances on three components are(i) clearance, 0.023°, (ii) concentricity, 0.003°, (iii) straightness of sensor package, 0.031°.• Sensor housing in drill collar. For a probe mounted in acentralised, retrievable case, 0.063°.• Collar bore in collar body. Typical MWD vendors’ tolerance is0.05°.• Collar body in borehole. The API tolerance on collar straightness equates to 0.03°. MWD vendors’ specifications aretypically somewhat more stringent.

The root-sum-square of these figures is 0.094°. Being based onmaximum tolerances, it is probably an over-estimate for stabilised rotary

assemblies.An analysis by the author of the variation in measured inclination over

46 rotation shots produced a root-mean-square misalignment of 0.046°.Simulations show that within this figure, about 0.007° is attributable to

the effect of sensor errors.

An additional source of misalignment - collar distortion outside

vertical plane due to bending forces - may be estimated using

dimensional BHA models. 0.04° seems to be a typical value. This erdiffers from those above by not rotating with the tool. It sho

therefore strictly have its own weighting function. Being so small

seems justifiable on practical (if not theoretical) grounds, to include

with the other sources of radially symmetric misalignment. This leav

us with an estimate for the error magnitude of 0.06°. This figure may

a significant underestimate where there is an aggressive bend in the BH

or a probe-type MWD tool is in use. This error term may be consider

systematic.

Magnetic field uncertainty. For basic MWD surveys, only the va

assumed for magnetic declination affects the computed azimu

However, conventional corrections for axial interference requ

estimates of the magnetic dip and field strength. Any error in th

estimates will cause an error in the computed azimuth.

A study commissioned from the British Geological Survey by Ba

Hughes INTEQ9 investigated the likely error in using a glo

geomagnetic model to estimate the instantaneous ambient magnetic fi

downhole. Five sources of error were identified:

• Modelled main field vs. actual main field at base epoch• Modelled secular variation vs. actual secular variation• Regular (diurnal) variation due to electrical currents in ionosphere

• Irregular temporal variation due to electrical currents in magnetosphere

• Crustal anomaliesBy making a number of gross assumptions, and by considering typi

drilling rates, the current author has distilled the results of the study i

a single table:

Error Source error magnitude prop.

declination dip totalfield

mode

Main field model 0.012°* 0.005° 3 nT GSecular variation 0.017°* 0.013° 10 nT GDaily variation 0.045°† 0.011°† 11 nT† R/S‡Irregular variation 0.110°† 0.043°† 45 nT† R/S‡Crustal anomaly 0.476° 0.195° 120 nT G

* below 60° latitude N or S† at 60° latitude N or S‡ daily and irregular variation are partially randomised between surveCorrelations between consecutive stations are approximately 0.95 a0.5 for the two error sources.

The dominant error source is crustal anomalies, caused by vary

magnetisation of rocks in the Earth’s crust. The figures shown

representative of the North Sea. Some areas, particularly those at hig

latitudes and where volcanic rocks are closer to the surface, will sho

greater variation. Other areas, where sedimentary rocks dominate, w

show less.


5/16


In the absence of any other information, the uncertainty in an estimate

of the magnetic field at a given time and place provided by a global

geomagnetic model may be obtained by summing the above terms

statistically. There is one complication - some account must be taken of

the increasing difficulty of determining declination as the horizontal

component of the magnetic field decreases. This can be achieved by

splitting this error into two components: one constant and one inversely

proportional to the horizontal projection of the field, BH. For the

purposes of the model, the split has been defined somewhat arbitrarily,

while ensuring that the total declination uncertainty at Lerwick, Shetland

(BH = 15000nT) is as predicted by the BGS study (0.49°). Beingdominated by the crustal anomaly component, all magnetic field errors

may be considered globally systematic and summarised thus:

Error Source weighting

function

magnitude prop.

mode

Declination (constant) AZ 0.36° GDeclination (BH-dependent) DBH 5000°nT G

Dip angle MFD 0.20° GTotal field MFI 130nT G

Along-Hole Depth Errors. Roger Ekseth10

identified 14 physical

sources of drill-pipe depth measurement error, wrote down expressions

to predict their magnitude, and by substituting typical parameter values

into the expressions predicted the total error for a number of different

well shapes. He then proposed a simplified model of just four terms,

and chose the magnitudes of each to match the predictions of the full

model as closely as possible. The results were as follows.

Error Source error error magnitude (1 s.d.) prop.

proportional land rig floating rig mode

to

Random ref. 1 0.35 m 2.2 m R

Systematic ref. 1 0 m 1 m S

Scale D 2.4×10-4 2.1×10-4 SStretch-type D.V 2.2×10-7 m-1 1.5×10-7 m-1 G

For the purposes of the basic model, the values for the land rig (or,

equivalently a jack-up or platform rig) may be chosen. The stretch-type

error, which dominates the other terms in deep wells, models two

physical effects - stretch and thermal expansion of the drill pipe. Both

these effects generally cause the drill string to elongate, so it may be

appropriate to apply this term as a bias (see below). If this is done, a

mean value of 4.4×10-7 m-1 should be used, since Ekseth effectivelytreated his estimates of these errors as 2 s.d. values.

Errors omitted from the Basic MWD Model. Some errors known to

affect MWD surveys have nonetheless not been included in the basic

error model.

Tool electronics and resoluti on. The overall effect on accuracy

caused by the limitations of the tool electronics and the resolution of

tool-to-surface telemetry system is not considered significant. Su

errors will tend to be randomised over long survey intervals.

External magnetic in terf erence. Ekseth10

discusses the influen

of remanent magnetism in casing strings on magnetic surveys, and giv

expressions for azimuth error when drilling (a) out of a casing shoe a

(b) parallel to an existing string. Although certainly not negligible, b

error sources are difficult to quantify, and equally difficult

incorporate within error modelling software. It seems preferable

manage these errors by applying quality procedures designed to lim

their effect.

Ef fect of sur vey interval and calcul ation method . The meth

presented in this paper relies on the assumption that error-f

measurement vectors p will lead to an error-free wellbore position vec

r. If minimum curvature formulae are used for survey calculation, t

assumption will only be true when the well-path between stations is

exact circular arc. The resulting error may be significant for sparse da

but may probably be neglected so long as the station interval does

exceed 100 ft.

Gravity field uncertain ty . Differences between nominal and actgravity field strengths will typically have no effect on MWD accura

since only the ratio of accelerometer measurements are used in

calculation of inclination and azimuth.

Gross err ors . Any attempt at a comprehensive discussion of MW

error sources must at least acknowledge the possibility of gross erro

sometimes called human errors. These errors lack the predictability a

uniformity of the physical terms discussed above. They are therefo

excluded from the error model, with the assumption that they

adequately managed through process and procedure.

Propagation Mathematics

The mathematical algorithm by which wellbore positional uncertainty

generated from survey error model inputs is based on the approaoutlined by Brooks and Wilson

3. The development of this w

described here was carried out by the Working Group referred to in

Introduction.

A physical error occurring at a survey station will result in an err

in the form of a vector, in the calculated well position. From ref. 3:

er

p

pi i

i

d

d = σ

∂

∂ε…..(1)

where ei is a vector-valued random variable, (a vector error ), σi is

magnitude of the ith error source, ∂p/∂εi is its “weighting function” adr/dp describes how changes in the measurement vector affect

calculated well position. It is sufficient to assume that the calcula

displacement between consecutive survey stations depends only on

survey measurement vectors at these two stations. Writing ∆rk for displacement between survey stations k -1 and k , we may thus expr

the (1 s.d.) error due to the presence of the ith error source at the k


6/16


7/16


( ) P B B I B I I x y= + +sin cos cos $ sin $ sin cosτ τ Θ …(24)

( )Q B B x y= − −cos sinτ τ …(25)

R B I = $ cos $ sinΘ 2 …(26)

The sensitivities of the computed azimuth to errors in the sensor

measurements are found by differentiating (23).

Magnetic F ield Uncertainty . The weighting function for magneticdeclination error is given above. Those for magnetic field strength and

dip angle, which are required when an axial magnetic interference

correction is in use, are derived by differentiating (23) with respect to$ B and $Θ .Misalignment Errors . Brooks and Wilson

3 model tool axial

misalignment as two uncorrelated errors corresponding to the x and y

axes of the tool. Their expressions for the associated inclination and

azimuth errors lead directly to the following weighting functions

∂

∂ε τ

τ

p

MX I

=−

0

sin

cos / sin

∂

∂ε τ

τ

p

MY I

=

0

cos

sin / sin

…(27,28)

Table 2 contains expressions for all the weighting functions not cited in

this section which are required to implement the error models described

in this paper.

Calculation Options

The method of position uncertainty calculation described here admits a

number of variations. It can still claim to be a standard, in that selection

of the same set of conventions should always yield the same results.

Along-Hole Depth Uncertainty. The propagation model described

above is appropriate for determining the position uncertainty of the

points in space at which the survey tool came (or will come) to rest.

These may be called uncertainties “at survey stations”.Thorogood

2 argues that it is more meaningful to compute the position

uncertainties of the points in the wellbore at the along-hole depths

assigned to the survey stations. These may be called the uncertainties

“at assigned depths”. This approach allows computation of the position

uncertainty of points (such as picks from a wireline log) whose depths

have been determined independently of the survey. Thorogood made

this calculation by defining a weighting function incorporating the local

build and turn rates of the well. The approach described in Appendix A

achieves the same result without the need for a new weighting function.

The results of the two approaches differ only in the along-hole

component of uncertainty. The along-hole uncertainty at a survey

station includes the uncertainty in the station’s measured depth, while

the uncertainty at an assigned depth does not.

The correct choice of approach depends on the engineering problem

being tackled - in many cases it is immaterial. The user of well-designed

directional software need not be aware of the issue.

Survey Bias. Not to be confused with sensor biases (which mig

better be termed offset errors), survey bias is the tendency for the m

likely position of a well to differ from its surveyed position. The on

bias term defined by Wolff and deWardt was for magnetic interference

“poor magnetic” surveys. The claims for stretch and thermal expans

of drill-pipe to be treated as bias errors are at least as strong.

Some vendors of directional software have neglected to model surv

bias on the grounds that (a) such errors should be corrected for and

engineers don’t like/understand them. The first objection can

countered by the observation “yes, but they aren’t!”, the second

careful software design.

The sign convention for position bias is from survey to most lik

position (ie. opposite to the direction of the error). Since drill p

generally elongates downhole, most likely depths are greater than surv

depths and bias values are positive. For axial drillstring interferen

most likely azimuths are greater than survey azimuths when

weighting function, sin I sin Am is positive, so bias values are ag

positive (at least in the northern hemisphere). The additio

mathematics required to model survey bias is included in Appendix A

Calculation conventions. The calculation of position uncertai

requires a wellbore survey consisting of discrete stations, each of wh

has an associated along-hole depth, inclination, azimuth and toolfa

angle. Clearly, these data will not be available in many cases, and cert

conventions are required whereby assumed values may be calculat

The following are suggested.

Along-hole depth. For drilled wells, actual survey stations sho

be used. For planned wells, the intended survey interval should

determined, and stations should be interpolated at all whole multiples

this depth within the survey interval. Typically, an interval of 100 fe

or 30 metres should be used.

I nclination and azimuth. For drilled wells, measured values sho be used. For planned wells, the profile should be interpolated at

planned survey station depths using minimum curvature.

Toolface. If actual toolface angles are available, they should be us

If not, several means of generating them are possible:

• Random number generation. Possibly close to reality, but results not repeatable and will tend to be optimistic.

• Worst-case. Several variations on this idea are possible, but each wrequire some additional calculation. The principle is questionable, a

the computational overhead is probably not justified.

• Borehole toolface (ie. the up-down-left-right change in borehdirection). This angle bears little relation to survey tool orientation, b

is at least well-defined, and may be computed directly from inclinat

and azimuth data. This approach will tend to limit the randomisation

toolface dependent errors, giving a conservative uncertainty predicti

This is the convention used in the examples at the end of the pap

Formulae for borehole toolface are given in Appendix B.


8/16


Standard Profiles

At the 8th meeting of the ISCWSA participants were set the task of

designing a number of well profiles suitable for:

• testing software implementations of the error models and propagation mathematics

• studying and highlighting the behaviour of different error models(magnetic and gyroscopic) and individual error sources

• demonstrating to a non-specialist audience the uncertainties to beexpected from typical survey programs.

The ideas generated at the meeting were used to devise a set of three

profiles:

ISCWSA#1: an extended reach well in the North Sea

ISCWSA#2: a “fish-hook” well in the Gulf of Mexico, with a long

turn at low inclination

ISCWSA#3: a “designer” well in the Bass Strait, incorporating a

number of difficult hole directions and geometries.

Figs. 2, 3 illustrate the test profiles in plan and section. Their full

definition, given in Table 4, includes location, magnetic field, survey

stations, toolface angles and depth units.

Example Results

The error models for basic and interference-corrected MWD have been

applied to the standard well profiles to generate position uncertainties in

each well. The results of several combinations are tabulated in Table 5.

Examples 1 and 2 compare the basic and interference-corrected models

in well ISCWSA#1. Being a high inclination well running approximately

east-north-east, the interference correction actually degrades the

accuracy. The results are plotted in Fig 4. Examples 3 to 6 all represent

the basic MWD error model applied to well ISCWSA#2. They differ in

that each uses a different permutation of the survey station/assigned

depth and symmetric error/survey bias calculation options. The

variation of lateral uncertainty and ellipsoid semi-major axis,

characteristic of a “fish-hook” well, is shown in Fig 5. Finally, example7 breaks well ISCWSA#3 into 3 depth intervals, with the basic and

interference-corrected models being applied alternately. This example is

included as a test of error term propagation.

Taken together, the examples form a demanding test set for

implementations of the method and models described in this paper.

Conclusions and Recommendations

This paper, and the collaborative work which it describes, establishes a

common starting point for wellbore position uncertainty modelling. The

standardised elements are:

• a nomenclature (see below)• a definition of what constitutes an error model• mathematics of position uncertainty calculation• an error model for a basic directional MWD service• a set of well profiles for investigating error models• a set of results for testing software implementations

The future work which these standards were designed to facilitate

includes:

• establishment of agreed error models for other survey servicincluding in-field referencing and gyroscopic tools.

• interchangeability of calculated position uncertainties betwsurvey vendor, directional drilling company and operator.

Useful though this work is, it is only a piece in a larger jigsaw. Tak

a wider view, the collaborative efforts of the extended surv

community should now be directed towards:

• standardisation of quality assurance measures• strengthening the link between quality assurance specificatio

and error model parameters

• better integration of wellbore position uncertainty with the otaspects of oilfield navigation..

Acknowledgments

The author thanks all participants in the ISCWSA for their enthusia

and support over several years and in the review of this paper.

Particular contributions to the MWD error model were made by Jo

Turvill and Graham McElhinney, both now with PathFinder Ener

Services, formerly Halliburton Drilling Systems; Wayne Philli

Schlumberger Anadrill; Paul Rodney and Anne Holmes, Sperry-S

Drilling Services; and Oddvar Lottsberg, formerly of Baker Hugh

INTEQ.

Participants in the Working Group on error propagation were Da

Roper, Sysdrill Ltd; Andy Brooks and Harry Wilson, Baker Hugh

INTEQ; and Roger Ekseth, formerly of Statoil.The results in Table 5 were checked by Jerry Codling, Landmark.

The author also wishes to thank BP Amoco for their permission

publish this paper.

Nomenclature

ISCWSA Nomenclature*

D along-hole depth I wellbore inclination

A wellbore azimuth

Am wellbore magnetic azimuth

τ toolface angle

N north co-ordinate

E east co-ordinate

V true vertical depth

δ magnetic declination

Θ magnetic dip angle

B magnetic field strength

G gravity field strength

X, x, Y, y, Z, z

tool reference directions - see fig. 1.

* adopted by ISCWSA participants as a standard for all technical

correspondence.


9/16


Special Nomenclature

b component of wellbore position bias vector

$ B estimated magnetic field strength

C wellbore position uncertainty covariance matrix

e 1 s.d. vector error at an intermediate statione 1 s.d. vector error at the station of interest

E sum of vector errors from slot to station of interestε particular value of a survey error

H,L used in calculation of toolface

m bias vector error at an intermediate stationm bias vector error at the station of interest

M wellbore position bias vector µ mean of error value

σ standard deviation of error value, component of wellbore

position uncertainty

p survey measurement vector ( D, I , A)

P,Q,R intermediate calculated quantitiesr wellbore position vector

∆rk increment in wellbore position between stations k -1 and k ρ correlation coefficient$Θ estimated magnetic dip angle

v along-hole unit vector

w factor relating error magnitude to uncertainty in measurement

subscripts and counters

hla borehole referenced frame

i a survey error term

k a survey station

K survey station of interest

K l number of stations in l th survey leg

l a survey leg L survey leg containing the station of interest

nev earth-referenced frame

superscripts

dep at the along-hole depth assigned to the survey station

rand random propagation mode

svy at the point where the survey measurements were taken

syst systematic propagation mode

well per-well or global propagation mode

References

1. Wolff, C.J.M. and de Wardt, J.P., Borehole Position Uncertainty -

Analysis of Measuring Methods and Derivation of Systematic Error Model, JPT pp.2339-2350, Dec. 1981

2. Thorogood, J.L., Instrument Performance Models and their

Application to Directional Survey Operations, SPEDE pp.294-298,

Dec. 1990.

3. Brooks, A.G. and Wilson, H., An Improved Method for Comput

Wellbore Position Uncertainty and its Application to Collision a

Target Intersection Probability Analysis, SPE 36863, EUROPE

Milan, 22-24 Oct 1996.

4. Dubrule, O., and Nelson, P.H., Evaluation of Directional Surv

Errors at Prudhoe Bay, SPE 15462, 1986 ATCE, New Orleans, O

5-8.

5. Minutes of the 7 th Meeting of the ISCWSA, Houston, 9 O

1997.

6. Grindrod, S.J. and Wolff, J.M., Calculation of NMDC Len

Required for Various Latitudes Developed From Fi

Measurements of Drill String Magnetisation, IADC/SPE 113

1983 Drilling Conference, Houston.

7. Minutes of the 6 th Meeting of the ISCWSA, Vienna, 24-25 J

1997.

8. Minutes of the 8th Meeting of the ISCWSA, Trondheim, 19 F

1998.

9. Macmillan, S., Firth, M.D., Clarke, E., Clark, T.D.G. a

Barraclough, D.R., “Error estimates for geomagnetic field valu

computed from the BGGM”, British Geological Survey Techni

report WM/93/28C, 1993.

10. Ekseth, R, Uncertainties in Connection with t

Determination of Wellbore Positions, ISBN 82-471-0218

ISSN 0802-3271, PhD Thesis no. 1998:24, IPT report 1998:2, T

Norwegian University of Science and Technology, Trondhe

Norway.

Appendix A Mathematical Description of Propagati

Model

The total position uncertainty at a survey station of interest, K

survey leg L) is the sum of the contribution from all the active er

sources. It is convenient computationally to group the error sources

their propagation type and to sum them separately.

Vector errors at the station of interest. Recall that the vector er

due to the presence of error source i at station k is the sum of the eff

of the error on the preceding and following survey displacements:

er

p

r

p

pi l k i l

k

k

k

k

k

i

d

d

d

d , , ,= +

+σ

∂

∂ε

∆ ∆ 1 …(A-1)

Evaluating this expression using the minimum curvature w

trajectory model is cumbersome. There is no significant loss of accura

in using the simpler balanced tangential model:

∆r j j j

j j j j

j j j j

j j

D D I A I A

I A I A I I

= −

+

++

−

− −

− −

−

11 1

1 1

1

2

sin cos sin cos

sin sin sin sincos cos

…(A-2)

The two differentials in the parentheses in (A-1) may then

expressed as:


10/16


d

d

d

dD

d

dI

d

dI

j

k

j

k

j

k

j

k

∆ ∆ ∆ ∆r

p

r r r=

where j = k , k+1 …(A-3)

and d

dD

I A I A

I A I A

I I

j

k

j j j j

j j j j

j j

∆r=

− −

− −− −

− −

− −

−

1

2

1 1

1 1

1

sin cos sin cos

sin sin sin sin

cos cos

…(A-4)

( )( )

( )

d

dI

D D I A

D D I A

D D I

j

k

j j k k

j j k k

j j k

∆r=

−−

− −

−

−

−

1

2

1

1

1

cos cos

cos sin

sin

…(A-5)

( )( )

d

dA

D D I A

D D I A j

k

j j k k

j j k k

∆r=

− −

−

−

−1

20

1

1

sin sin

sin cos

…(A-6)

For the purposes of computation, the error summation terminates at

the survey station of interest. Vector errors at this station are therefore

given by:

e rp

pi L K i L

K

K

K

i

d d

, , ,= σ ∂∂ε

∆ …(A-7)

The notation ei L K , , indicates that a measurement error at this station

affects only the preceding survey displacement. In what follows we

reserve the notation ei,l,k for vector errors at intermediate stations, which

affect both the preceding and following displacements.

Undefined weighting functions. For some combinations of weighting

function and hole direction, a component of the measurement vector

(usually azimuth) is highly sensitive to changes in hole direction and the

vector ∂p/∂εi is apparently undefined. There are two cases:Vertical hole . In this case, dr/dp is zero and the vectors ei,l,k and

ei L K , , are still finite and well-defined. They may be computed by

forming the product (A-1) algebraically and evaluating it as a whole.

Take as an example the weighting function for an x-axis radially

symmetric misalignment. Substituting the expression for ∂p/∂ε MX (27)and the well trajectory model equations (A-3 to A-6) into (A-1) and (A-

7), and setting I equal to zero gives

( ) ( )

( )e i l k i l k k D D

A

A, ,,

sin

cos= −

+− +

+ −σ τ

τ1 1

20

…(A-8)

and

( ) ( )

( )ei L K i L K K D D

A

A, ,,

sin

cos= −

+

− +

−σ τ

τ1

20

…(A-9)

There are similar expressions for Y-axis axial misalignment and X- and

Y-axis accelerometer biases. These are given in Table 3. Equivalent

expressions may be used for evaluating bias vectors in vertical hole, w

m i l k , , , m i L K , , , and µi,l substituted for ei l k , , , ei L K , , , and

respectively.

Other hol e dir ecti ons. Some error sources really are unbounded

certain hole directions. The examples in this paper are sensor errors af

axial interference correction in a horizontal and magnetic east/w

wellbore - a so-called “90/90” well. In such cases, the assumptionslinearity break down, and computed position uncertainties

meaningless. Software implementations should include an error-catch

mechanism for this case.

Summation of errors. Vector errors are summed into posit

uncertainty matrices as follows.

Random err ors. The contribution to survey station uncertainty fr

a randomly propagating error source i over survey leg l (not contain

the station of interest) is:

( ) ( )C e ei l rand

i l k i l k T

k

K l

, , , , ,.=

=

∑1

…(A-10)

and the total contribution over all survey legs is:

( ) ( ) ( ) ( )C C e e e ei K rand

i l rand

l

L

i L k i L k T

k

K

i L K i L K T

, , , , , , , , , ,. .= + +=

−

=

−

∑ ∑1

1

1

1

…(A-11)

Systematic errors. The contribution to survey station uncertai

from a systematically propagating error source i over survey leg l (

containing the point-of-interest) is:

C e ei l syst

i l k k

K

i l k k

K T l l

, , , , ,.=

= =∑ ∑

1 1

…(A-12)

and the total contribution over all survey legs is:

C C e e e ei K syst

i l syst

l

L

i L k i L K

k

K

i L k i L K

k

K T

, , , , , , , , , ,.= + +

+

=

−

=

−

=

−

∑ ∑ ∑1

1

1

1

1

1

…(A-13)

Per-Well and Global errors. Each of these error types is systema

between all stations in a well. The individual vector errors can theref

be summed to give a total vector error from slot to station

E e e ei K i l k k

K

l

L

i L k

k

K

i L K

l

, , , , , , ,=

+ +

==

−

=

−

∑∑ ∑11

1

1

1…(A-14)

The total contribution to the uncertainty at survey station K is

C E Ei K well

i K i K T

, , ,.=…(A-15)

Total position covariance. The total position covariance at surv


11/16


station K is the sum of the contributions from all the types of error

source:

{ }

C C C C K svy

i K rand

i Ri K

syst

i S

i K well

i W G

= + +∈ ∈ ∈∑ ∑ ∑, , ,

,

…(A-16)

where the superscript svy indicates the uncertainty is defined at a

survey station.

Survey bias. Error vectors due to bias errors are given by expressions

entirely analogous with (A-1) and (A-7):

mr

p

r

p

pi l k i l

k

k

k

k

k

i

d

d

d

d , , ,= +

+µ

∂

∂ε

∆ ∆ 1 …(A-17)

mr

p

pi L K i L

K

K

K

i

d

d , , ,= µ

∂

∂ε

∆…(A-18)

The total survey position bias at survey station K , M K svy

, is the sum

of individual bias vectors taken over all error sources i, legs l and

stations k :

M m m m K svy

i l k k

K

l

L

i L k k

K

i L K i

l

=

+ +

==

−

=

−

∑∑ ∑∑ , , , , , ,11

1

1

1

…(A-19)

Position uncertainty and bias at an assigned depth

Defining the superscript dep to indicate uncertainty at an assigned

depth, it may be shown that:

e e vi L K dep

i L K svy

i L i L K K w, , , , , , ,= −σ …(A-20)

e ei l k dep

i l k svy

, , , ,= …(A-21)

where wi,L,K is the factor relating error magnitude to measurement

uncertainty and v K is the along-hole unit vector at station K . Figs. 6, 7

illustrate these results. Substituting these expressions into (A-12 to A-16) yields the position uncertainty at the along-hole depth assigned to

each survey station.

Survey bias at an assigned depth is calculated by substituting the

following error vectors into (A-19):

m m vi L K dep

i L K svy

i L i L K K w, , , , , , ,= − µ …(A-22)

m mi l k dep

i l k svy

, , , ,= …(A-23)

Relative uncertainty between wells. When calculating the

uncertainty in the relative position between two survey stations

( K A, K B) in wells ( A,B), we must take proper account of the correlation

between globally systematic errors. The uncertainty is given by:

[ ]

( ) ( ) ( ) ( )

C r r

C C E E E E

svy K K

K

svy

K

svy

i K i K

T

i K i K

T

i G

A B

A B A B B A

−

= + − +

∈

∑ , , , ,. .…(A-24)

The relative survey bias is simply:

[ ]M r r M M svy K K K svy K svy A B A B− = − …(A-25)

Substitution of equations (A-20) to (A-23) into these expressio

gives the equivalent results at the along-hole depths assigned to

stations.

Transformation into Borehole Reference Frame. The resu

derived above are in an Earth-referenced frame (North, East, Vertic

subscript nev). The transformation of the covariance matrices and b

vectors into the more intuitive borehole referenced frame (Highsi

Lateral, Along-hole - subscript hla ) is straightforward:

C T C ThlaT

nev= …(A-26)

b

b

b

H

L

A

hlaT

nev

= =M T M …(A-27)

where

T =−

−

cos cos sin sin cos

cos sin cos sin sin

sin cos

I A A I A

I A A I A

I I

K K K K K

K K K K K

K K 0

…(A-28)

is a transformation matrix. Uncertainties and correlations in the princi

borehole directions are obtained from:

[ ]σ H hla= C 1 1, etc. …(A-29)

[ ]ρ

σ σ HAhla

H L

=C 1 2,

etc. …(A-30)

Appendix B Calculation of Toolface Angle

The following formulae may be used to calculate a synthetic toolfa

angle from successive surveys:

( ) H I I A A I I K K K K K K K = − −− − −sin cos cos sin cos1 1 1 …(B-1)

( ) L I A A K K K K = − −sin sin 1 …(B-2)

If H K > 0, τ K = tan-1( L K / H K ) …(B-3)

If H K < 0, τ K = tan-1( L K / H K ) + 180° …(B-4)

If H K = 0, τ K = 270°, 0° or 90° as L K < 0, L K = 0 or L K > 0...(B-5)


12/16


Table 1—Summary of Basic MWD Error Models

Weighting

Function

Basic

model

with axial

correction

Prop.

Mode

Weight.

Func.

Basic model with axial

correction

Prop.

Mode

Sensors Misalignment

ABX 0.0004 g S SAG 0.2° 0.2° S ABY 0.0004 g S MX 0.06° 0.06° S ABZ 0.0004 g S MY 0.06° 0.06° S ASX 0.0005 S

ASY 0.0005 S Axial magn etic interference

ASZ 0.0005 S AZ 0.25° SMBX 70 nT S AMID 0.6° S or B*MBY 70 nT S

MBZ 70 nT S Declinat ion

MSX 0.0016 S AZ 0.36° 0.36° GMSY 0.0016 S DBH 5000°nT 5000°nT GMSZ 0.0016 S

ABIX 0.0004 g S Total magnet ic f ield and d ip angle

ABIY 0.0004 g S MDI 0.20° G ABIZ 0.0004 g S MFI 130 nT G

ASIX 0.0005 S

ASIY 0.0005 S Along-hole depth

ASIZ 0.0005 S DREF 0.35 m 0.35 m R

MBIX 70 nT S DSF 2.4 × 10-4 2.4 × 10-4 SMBIY 70 nT S DST 2.2 × 10-7 m-1 2.2 × 10-7 m-1 G or B†MSIX 0.0016 S * when modelled as bias: µ = 0.33°, σ = 0.5°MSIY 0.0016 S † when modelled as bias: µ = 4.4 × 10-7 m-1, σ = 0

Table 2—Error Source Weighting Functions not Given in the Text

Sensor Errors (without axial interference correction)

ABX

( )

10

G I

I A A I m m

−− +

cos sin

cos sin sin cos cos tan cot cos

τ

τ τ τΘ

ASX

( )( )

02

sin cos sin

tan sin cos sin sin cos cos cos cos sin

I I

I I A A I m m

τ

τ τ τ τ− − +

Θ

ABY

( )

10

G I

I Am Am I

−

+ −

cos cos

cos sin cos cos sin tan cot sin

τ

τ τ τΘ

ASY

( )( )

02

sin cos cos

tan sin cos sin cos cos sin cos sin cos

I I

I I A A I m m

τ

τ τ τ τ− + −

Θ

MBX

( ) ( )

0

0

cos cos cos sin sin / cos A I A Bm mτ τ−

Θ

MSX

( )( )

0

0

cos cos sin tan sin sin sin cos cos cos cos sin sin I A I A A I Am m m mτ τ τ τ τ− + −

Θ

MBY

( ) ( )

0

0

− +

cos sin cos sin cos / cos A I A Bm mτ τ Θ

MSY

( )( )

0

0

− − − +

cos cos cos tan sin cos sin sin cos sin cos sin cos I A I A A I Am m m mτ τ τ τ τΘ

AB

Z

10

G I

I Am

−

sin

tan sin sinΘ

ASZ0

−

sin cos

tan sin cos sin

I I

I I AmΘ

MBZ

( )

0

0

−

sin sin / cos I A Bm Θ

MSZ

( )

0

0

− +

sin cos tan cos sin sin I A I I Am mΘ


13/16


Table 2—Cont.

Sensor Errors (with axial interference correction)

ABIX

( ) ( )( ) ( )

10

1

2 2 2G

I

I A I I A A I I Am m m m

−

+ − − −

cos sin

cos sin sin tan cos sin cos cos tan cos cot / sin sin

τ

τ τΘ Θ

ABIY

( ) ( )( ) ( )

10

12 2 2

G I

I A I I A A I I Am m m m

−

+ + − −

cos cos

cos sin cos tan cos sin cos sin tan cos cot / sin sin

τ

τ τΘ Θ

ASIX

( ) ( )( ) ( )

0

1

2

2 2 2

sin cos sin

sin sin cos sin sin tan cos sin cos cos tan sin cos cos / sin sin

I I

I I A I I A I A I I Am m m m

τ

τ τ τ− + − − −

Θ Θ

ASIY

( ) ( )( ) ( )

0

1

2

2 2 2

sin cos cos

cos sin cos sin cos tan cos sin cos sin tan sin cos cos / sin sin

I I

I I A I I A I A I I Am m m m

τ

τ τ τ− + + − −

Θ Θ

MSIX

( )( ) ( )

0

0

12 2− − + − −

cos cos sin tan sin sin sin cos cos sin sin cos cos / sin sin I A I A I A A I Am m m m mτ τ τ τ τΘ

MSIY

( )( ) ( )

0

0

12 2− − − + −

cos cos cos tan sin cos sin sin cos sin cos cos sin / sin sin I A I A I A A I Am m m m mτ τ τ τ τΘ

MBIX

( ) ( )( )

0

0

1 2 2− − −

cos sin sin cos cos / cos sin sin I A A B I A

m m mτ τ Θ

ABIZ

( )( ) ( )

10

12 2

G I

I I A I I A I Am m m

−

+ −

sin

sin cos sin tan cos sin cos / sin sinΘ

MBIY

( ) ( )( )

0

0

1 2 2− + −

cos sin cos cos sin / cos sin sin I A A B I A

m m mτ τ Θ

ASIZ

( )( ) ( )

0

12 2 2

−

+ −

sin cos

sin cos sin tan cos sin cos / sin sin

I I

I I A I I A I Am m mΘ

Magnetic Field Errors (with axial interference correction)

MFI

( ) ( )( )

0

0

12 2− + −

sin sin tan cos sin cos / sin sin I A I I A B I A

m m mΘ

MDI

( ) ( )

0

0

1 2 2− − −

sin sin cos tan sin cos / sin sin I A I I A I Am m mΘ

Table 3—Error Vectors in Vertical Hole where Weighting Function is Singular

Sensor Errors (with or without axial interference correction)

ABX

or ABIX

( ) ( )( )e

i l k i l Dk Dk

G

A

A, ,

,sin

cos= + − −

− ++

σ τ

τ1 1

20

ABY

or ABIY

( ) ( )( )e

i l k i l Dk Dk

G

A

A, ,

,cos

sin= + − −

− +− +

σ τ

τ1 1

2

0

Misalignment Errors

MX ( ) ( )

( )e i l k i l k k D D

A

A, ,,

sin

cos= −

+− +

+ −σ τ

τ1 1

20

MY ( ) ( )

( )ei l k i l k k D D

A

A, ,,

cos

sin= −

++

+ −σ τ

τ1 1

20


14/16


Table 4—Standard Well Profiles

ISCWSA #1 - North Sea Extended Reach Well

Lat. 60° N, Long. 2°E, Total Field, 50,000 nT, Dip 72°, Declination4°W, Station interval 30m, VS Azimuth 75°MD Inc Azi North East TVD VS DLS

m deg deg m m m m °/30m 0.00 0.000 0.000 0.00 0.00 0.00 0.00 0.00

1200.00 0 .000 0.000 0.00 0.00 1200.00 0.00 0 .00

2100.00 60.000 75.000 111.22 415.08 1944.29 429.72 2.00

5100.00 60.000 75.000 783.65 2924.62 3444.29 3027.79 0.00

5400.00 90.000 75.000 857.80 3201.34 3521.06 3314.27 3.00

8000.00 90.000 75.000 1530.73 5712.75 3521.06 5914.27 0.00

ISCWSA #2 - Gulf of Mexico Fish Hook Well

Lat. 28° N, Long. 90°W, Total Field 48,000nT, Dip 58°, Declination2°E, Station interval 100ft, VS Azimuth 21°MD Inc Azi North East TVD VS DLS

ft deg deg ft ft ft ft °/100ft 0.00 0 .000 0 .000 0 .00 0.00 0 .00 0.00 0.00

2000.00 0 .000 0.000 0.00 0 .00 2000.00 0.00 0 .00

3600.00 32.000 2.000 435.04 15.19 3518.11 411.59 2 .00

5000.0032.000 2.000 1176.48 41.08 4705.37 1113.06 0 .00

5525.5432.000 32.000 1435.37 120.23 5253.89 1383.12 3.00

6051.0832.000 62.000 1619.99 318.22 5602.41 1626.43 3.00

6576.6232.000 92.000 1680.89 582.00 6050.92 1777.82 3.00

7102.1632.000 122.000 1601.74 840.88 6499.44 1796.70 3.00

9398.5060.000 220.000 364.88 700.36 8265.27 591.63 3 .00

12500.00 60.000 220.000-1692.70 -1026.15 9816.02 -1948.01 0.00

ISCWSA #3 - Bass Strait Designer Well

Lat. 40°S, Long. 147°E, Total Field 61,000nT, Dip -70°, Declination 13°E, Station interval 30m, VS Azimuth 310°MD Inc Azi North East TVD VS DLS

m deg deg m m m m °/30m 0.00 0 .000 0.000 0.00 0 .00 0 .00 0 .00 0 .00

500.00 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 2.50

1700.00 50.000 0.000 705.23 0.00 1412.37 570.54 0.00

2450.00 0.000 0.000 1012.23 0.00 2070.73 818.91 2.00

MD Inc Azi North East TVD VS DLS

m deg deg m m m m °/30m 2850.00 0.000 0.000 1012.23 0.00 2470.73 818.91 0.00

3030.00 90.000 283.0001038.01 -111.65 2585.32 905.39 15.00

3430.00 90.000283.000 1127.99 -501.40 2585.32 1207.28 0.00

3730.00110.000193.000 996.08 -727.87 2520.00 1197.85 9.00

4030.00110.000193.000 721.40 -791.28 2417.40 1069.86 0.00

Table 5—Calculated Position Uncertainties (at 1 standard deviation)

• uncertainty at tie-line (MD=0) is zero • stations interpolated at whole multiples of station interval using minimum curvature

• instrument toolface = borehole toolface

uncertainties along

borehole axes

correlations between

borehole axes

survey bias along

borehole axesNo. Well Depth interval(s) Model Option σ H σ L σ A ρ HL ρ HA ρ LA b H b L b A1 #1 0 m - 8000 m basic S, sym 20.11 m 84.33 m 8.62 m -0.015 +0.676 -0.003

2 #1 0 m - 8000 m ax-int S, sym 20.11 m 196.41 m 8.62 m -0.006 +0.676 +0.004

3 #2 0 ft - 12500 ft basic S, sym 16.17 ft 29.66 ft 10.12 ft +0.032 -0.609 +0.060

4 #2 0 ft - 12500 ft basic D, sym 16.17 ft 29.66 ft 9.16 ft +0.032 -0.426 +0.084

5 #2 0 ft - 12500 ft basic S, bias 15.69 ft 27.41 ft 8.61 ft +0.052 -0.602 +0.157 -6.79 ft -12.41ft +11.70 ft

6 #2 0 ft - 12500 ft basic D, bias 15.69 ft 27.41 ft 8.50 ft +0.052 -0.569 +0.160 -6.79 ft -12.41ft -4.76 ft

7 #3

(1) 0 m - 1380 m

(2) 1410 m - 3000 m

(3) 3030 m - 4030 m

basic

ax-int

basic

S, sym

S, sym

S, sym 5.64 m 5.76 m 9.59 m -0.186 -0.588 +0.297

Key to error models: basic Basic MWD

ax-int Basic MWD with axial interference correction

Key to calculation options: S, sym Uncertainty at survey station, all errors symmetric (ie. no bias)

S, bias Uncertainty at survey station, selected errors modelled as biases (see table 1)D, sym Uncertainty at assigned depth, all errors symmetric (ie. no bias)

D, bias Uncertainty at assigned depth, selected errors modelled as biases (see table 1)


15/16


Y-axis

X-axis

Z-axis(down hole)

HighSide

τ

τ = toolface angle

Fig.1 Definition of tool sensor axes and toolface angle

1000m

-1000m

-1000m

1500m

1000m

6000m5500m

1500m1000m

East

N o r t h

ISCWSA#1

ISCWSA#2

ISCWSA#3

Fig.2 Plan view of standard well profiles

Vertical Section

T r u e V e r t i c a l D e p t h

2000m

4000m

60004000m-1000m 2000m

ISCWSA#1VS Azi = 75 o

ISCWSA#3VS Azi = 310o

ISCWSA#2VS Azi = 21 o

Fig.3 Vertical section plot of standard well profiles. Note

different section azimuths.

1 s . d . L

a t e r a l U n c e r t a i n t y ( m )

0

40

200

160

120

80

2000 80060004000

Measured Depth (m)

6

4

2

1200 210018001500

Example 1 : Basic MWD

Example 2 : MWD with axial interference

correction

inset

“corrected” model is

marginally more

accurate at low

inclination

“corrected” model

deteriorates rapidly

near “90/90”

Fig.4 Comparison of basic and interference corrected MWD

error models in well ISCWSA#1


16/16


0 1200080004000

Measured Depth (ft)

1 s . d .

U n c e r t a i n t y ( f t )

10

40

30

20

0

lateral uncertainty andellipsoid semi-major

axis are equal while

azimuth is constant

at mid-turn, lateral

direction co-incides

with ellipsoid minor axis

ellipsoid semi-major axis

reduces as well returns below surface location

Example 4 : Ellipsoid semi-major axis

Example 4 : Lateral uncertainty

Fig.5 Variation of lateral uncertainty and ellipsoid semi-

major axis in a fish-hook well - ISCWSA#2

e ek dep

k svy=depth error at

earlier station

= σ i i k w ,

(2) vector errors for last survey

station and last assigned depth

due to depth error at earlier station

must therefore be the same:

(1) with no depth error at last station,

true positions at survey station and

at its assigned depth coincide

Recorded (and calculated)

survey station position

True position where

tool came to rest

True well position at depth

assigned tosurvey station

True well path

Calculated well path

depth error atlast station

=

e K svye e v K

dep K svy

i i K K w= − σ ,

σi i K w ,

vector error at last

assigned depth due todepth error at last station

=

vector error at last

station due to deptherror at last station

=

Recorded (and calculated)survey station position

True position wheretool came to rest

True well position at depth

assigned tosurvey station

True well path

Calculated well path

Fig.6 Vector errors at the last station (point of interest) d

to an along-hole depth error at the last station.

Fig.7 Vector errors at the last station (point of interest) due to an along-hole depth error at a

previous station.

Documents

Accuracy Prediction for Directional MWD