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Similar survey equations exist for gyros tools and these MWD equations change if axial interference corrections are made.
The weighting functions can be derived from these equations by taking the partial derivatives of the survey equations with respect to the error source. Examples of the derivations can be found in the SPE papers, whilst all of the error sources and weighting functions for both models are given in the appendices. We now consider another example to illustrate this point:
Example 2: Sensor error - Z-Accelerometer Bias If we consider the effect of a z-accelerometer bias error; instead of reading the correct value of 𝐺 𝑧
𝑡𝑟𝑢𝑒 the tool will
actual give:
𝒔𝒄𝒂𝒍𝒆𝒇𝒂𝒄𝒕𝒐𝒓
𝑮 𝒛 𝒎𝒆𝒂𝒔𝒖𝒓𝒆𝒅 =(𝟏+𝑮 𝒛
𝒕𝒓𝒖𝒆 +𝑮
𝒛 𝒃𝒊𝒂𝒔
)𝑮 𝒛
𝑠𝑐𝑎𝑙𝑒𝑓𝑎𝑐𝑡𝑜𝑟
𝑏𝑖𝑎𝑠 and 𝐺 𝑧
where 𝐺 𝑧
represent the residual errors of the survey tool after calibration.
This equation represents a fairly standard, first-order method for modelling the output of a sensor (almost any type of sensor), when we know that it will not give perfect output.
The MWD model has an error source, coded ABZ for the z-accelerometer bias which corresponds to this 𝐺 𝑧 𝑏𝑖𝑎𝑠 term. From the MWD survey equations above, we can see that the G z term appears in both the inclination and azimuth equations. dngsdon’ h ny ff c onm su dd ph. o h M w gh ngfunc on s0, but the inclination and azimuth weighting functions are determined by taking the partial derivates of these survey equations with respect to G z i.e. for ABZ the weighting functions are: How , h cc l om
𝐬𝐢𝐧 𝑰 𝑮
, 𝒕𝒂𝒏𝚯 𝒔𝒊𝒏𝑰 𝒔𝒊𝒏𝑨 𝒎 𝑮 ]
[𝟎,
Since we are assuming that this z-accelerometer bias is due to residual tools errors, this error source will have the same value for each survey taken with that tool. However, if we change tools, the error will change. Hence the propagation mode is systematic (the same at each survey station in a run, but different for different legs and wells).
The model gives ABZ a magnitude of 0.004 ms -2 , meaning that for a number of properly calibrated tools if we ran a numb of ss,w ’d xp c og G uss nds bu onof sulsw h s nd dd onof0.004ms -2 .
Gyro tools can be designed a little differently – some systems also have a cluster of three accelerometers and the inclination weighting function will be the same as the MWD case (this is the gyro AXYZ-ZB term). Other gyro tools only have x and y-accelerometers and use the assumed total gravity value, and therefore these tools would be modelled without a z-accelerometer bias term. We can see that the error sources which are included in any particular survey tool model depend on the design of that tool.
Ap1.2.5 Error Propagation The core equation for the propagation of the errors from the error source through to the survey position error is:
𝒅𝒓 𝒅𝒑
𝝏𝒑 𝝏𝜺 𝒊
𝒆 𝒊 = 𝝈 𝒊
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