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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
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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
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4 HUGH S. WILLIAMSON SPE 56702
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.
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SPE 56702 ACCURACY PREDICTION FOR DIRECTIONAL MWD 5
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
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SPE 56702 ACCURACY PREDICTION FOR DIRECTIONAL MWD 7
( ) 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.
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8 HUGH S. WILLIAMSON SPE 56702
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.
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SPE 56702 ACCURACY PREDICTION FOR DIRECTIONAL MWD 9
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:
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10 HUGH S. WILLIAMSON SPE 56702
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
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SPE 56702 ACCURACY PREDICTION FOR DIRECTIONAL MWD 11
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)
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12 HUGH S. WILLIAMSON SPE 56702
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
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SPE 56702 ACCURACY PREDICTION FOR DIRECTIONAL MWD 13
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
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14 HUGH S. WILLIAMSON SPE 56702
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)
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SPE 56702 ACCURACY PREDICTION FOR DIRECTIONAL MWD 15
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
8/9/2019 Accuracy Prediction for Directional MWD
16/16
16 HUGH S. WILLIAMSON SPE 56702
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.