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Appendix A:
Details of the Mathematical Derivations
A1 - Details of the Propagation Mathematics Recall from section Ap1.2.5 that the core equation for the propagation of the errors from the error source through to the survey position error is: 𝒆 𝒊 = 𝝈 𝒊 (A.1) Where: e i is the size of the error in NEV axis due to error source i at the current survey station (a 3x1 vector) 𝜎 𝑖 is the magnitude of the ith error source (a scalar) 𝜕𝑝 𝜕𝜀 𝑖 are the weightings functions, the effect of the ith error source on the survey measurements; md, inc and azimuth. (a 3x1 vector) 𝑑𝑟 𝑑𝑝 is the effect of the survey errors in md, inc and az on the wellbore position in the NEV axis, (i.e. a 3x3 𝒅𝒓 𝒅𝒑 𝝏𝒑 𝝏𝜺 𝒊
𝒅𝑵 𝒅𝑴𝒅 𝒅𝑬 𝒅𝑴𝒅 𝒅𝑽 𝒅𝑴𝒅
𝒅𝑵 𝒅𝑰𝒏𝒄 𝒅𝑬 𝒅𝑰𝒏𝒄 𝒅𝑽 𝒅𝑰𝒏𝒄
𝒅𝑵 𝒅𝑨𝒛 𝒅𝑬 𝒅𝑨𝒛
matrix
)
𝒅𝑽 𝒅𝑨𝒛 ]
[
So we need to be able to calculate the 𝑑𝑟 𝑑𝑝 matrix. Given that the survey measurements define the depth, inclination and azimuth at either end of an interval the position over an interval will depend on the survey measurements at two stations. If we write ∆𝑟 𝑘 for the displacement between survey station k-1 and k and hence ∆𝑟 𝑘+1 for the displacement between stations k and k+1, then we can split 𝑑𝑟 𝑑𝑝 in to the variation over the preceding and following survey intervals
and write:
𝒅∆𝒓 𝒌 𝒅𝒑 𝒌
𝒅∆𝒓 𝒌+𝟏 𝒅𝒑 𝒌
𝝏𝒑 𝒌 𝝏𝜺 𝒊
𝒆 𝒊,𝒍,𝒌 = 𝝈 𝒊,𝒍 (
)
+
Where: e i,l,k is the error due to the i th error source at the k th survey station in the l th survey leg 𝑑∆𝑟 𝑘 𝑑𝑝 𝑘 is the effect of the errors in the survey measurements at station k, on the position vector from survey station k-1 to survey station k and similarly 𝑑∆𝑟 𝑘+1 𝑑𝑝 𝑘 is the effect of the errors in the survey measurements at station k, on the position vector from survey station k to survey station k+1 Although minimum curvature is the preferred method for calculating the wellbore positions, it is simpler to use the balanced tangential method to determine 𝑑∆𝑟 𝑘 𝑑𝑝 𝑘 and there is no significant loss of accuracy. The balanced tangential model gives us the following equation for the displacement between any two survey stations j-1 and j in nev -axes:
[ 𝒔𝒊𝒏𝑰 𝒋−𝟏 𝒄𝒐𝒔𝑨 𝒋−𝟏 +𝒔𝒊𝒏𝑰 𝒋 𝒄𝒐𝒔𝑨 𝒋 𝒔𝒊𝒏𝑰 𝒋−𝟏 𝒔𝒊𝒏𝑨 𝒋−𝟏 +𝒔𝒊𝒏𝑰 𝒋 𝒔𝒊𝒏𝑨 𝒋 𝒄𝒐𝒔𝑰 𝒋−𝟏 +𝒄𝒐𝒔𝑰 𝒋 ]
∆𝑵 ∆𝑬 ∆𝑽
𝑫 𝒋 −𝑫 𝒋−𝟏 𝟐
∆𝒓 𝒋 =[
]=
(A.3)
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