Electromagnetic method for obtaining dip azimuth angle
An electromagnetic method for obtaining a dip azimuth angle from downhole electromagnetic measurements includes acquiring electromagnetic measurement data in a subterranean borehole from at least one measurement array. The electromagnetic measurement data is processed by a least squares method to obtain the dip azimuth angle. Related systems and apparatuses are also disclosed herein.
1. A method for computing a dip azimuth angle from downhole electromagnetic measurements, the method comprising:
(a) placing an apparatus to conduct electromagnetic measurements in a subterranean borehole such that the apparatus is positioned next to a formation of interest;
(b) applying one of a time varying electric current and a time varying alternating current to at least one antenna in the apparatus;
(c) acquiring a plurality of voltage measurements at a corresponding plurality of measurement arrays deployed in the subterranean borehole, wherein the arrays are placed within the apparatus;
(d) causing a processor to process the voltage measurements acquired in (a) to obtain least squares complex fitting coefficients defined by a preset voltage measurement equation at a receiving antenna for the apparatus;
(e) causing the processor to process the least squares complex fitting coefficients to obtain the dip azimuth angle by minimizing a weighted average of squares of the plurality of voltage measurements;
(f) transmitting the obtained dip azimuth angle to a surface location; and
(g) plotting the dip azimuth angle versus a borehole depth at which the dip azimuth angle was calculated.
2. The method of claim 1 , wherein at least one of the measurement arrays used to acquire the voltage measurements in (c) has a tilted receiver antenna.
3. The method of claim 1 , wherein at least one of the measurement arrays used to acquire the voltage measurements in (c) has an axial or a transverse transmitter antenna.
4. The method of claim 1 , wherein a least square weighted sum of squared residuals for at least one voltage measurement is computed according to the following equation:
L
=
∑
n
w
n
V
n
2
wherein L represents the weighted sum of squared residuals, V n represent the plurality of voltage measurements n, and w n represent statistical weights assigned to each of the plurality of voltage measurements n.
5. The method of claim 4 , wherein the processor processes the least square weighted sum of squared residuals and a plurality of the complex fitting coefficients of a voltage signal acquired in (d) to obtain the dip azimuth angle in (e).
6. The method of claim 5 , wherein the dip azimuth angle is computed in (e) according to the following equation:
L=P+S cos 2(φ−α)
wherein L represents the weighted sum of squared residuals, α represents the dip azimuth angle, φ represents a tool face angle, and P and S represent weighted averages of the complex fitting coefficients.
7. The method of claim 6 , wherein the weighted averages of the complex fitting coefficients P and S are computed according to the following equations:
P
=
1
2
∑
n
[
b
n
2
+
c
n
2
]
S
=
Q
2
+
R
2
wherein:
Q
=
1
2
∑
n
w
n
[
b
n
2
-
c
n
2
]
R
=
∑
n
w
n
real
(
b
n
*
c
n
)
and wherein w n represent statistical weights assigned to each of a plurality of voltage measurements n acquired in (c), and b n and c n represent the complex fitting coefficients for each of the plurality of voltage measurements.
8. The method of claim 1 , wherein the dip azimuth angle is computed according to the following equation:
α
=
1
2
arctan
2
(
R
,
Q
)
∈
[
-
π
2
,
π
2
]
wherein α represents the dip azimuth angle, and Q and R represent the least squares coefficients of the plurality of voltage measurements obtained in (c).
9. The method of claim 1 , further comprising:
(h) causing the processor to compute a confidence interval for the dip azimuth angle obtained in (e).
10. The method of claim 9 , wherein the confidence interval is computed according to the following equation:
2
Δ
α
=
1
R
2
+
Q
2
(
Q
·
Δ
R
-
R
·
Δ
Q
)
wherein 2Δα represents the confidence interval, R and Q represent least squares coefficients of the plurality of voltage measurements obtained in (c), ΔR and ΔQ represent standard deviations of R and Q.
11. The method of claim 10 , wherein upper and lower bounds of the confidence interval are computed according to the following equations:
errhi=α+ abs (Δα)
errlo=α− abs (Δα)
wherein errhi and errlo represent the upper and lower bounds of the confidence interval and α represents the dip azimuth angle obtained in (e).
12. The method of claim 1 , wherein said processing in (d) and (e) is performed using a downhole processor.
13. The method of claim 1 , further comprising:
(h) further processing the dip azimuth angle at the surface location to obtain a subsequent drilling direction for the subterranean borehole.