# Accuracy Prediction for Directional MWD

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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.

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 on the 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 stations in the same survey leg. (In a survey listing made up of several

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

•  a correlation coefficient ρ2 between error values at survey stations in 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 correction Alternative 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 furt research 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 the magnetometer 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, Wol 605 µWb (11) 435 µWb (11) Lotsberg5

511 µWb (4) McElhinney7

Stabiliser 177 µWb (6) Grindrod, Wol 396 µWb (10) 189 µWb (10) Lotsberg 369 µWb (5) 408 µWb (10) McElhinney

Motor 340 µWb (12) 419 µWb (10) Lotsberg Oddvar 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 a

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