# Susceptibility and Remanent Magnetization Estimates from Orientation Tools in Borehole Imaging Logs
## 2 Materials and Methods  
The total field \(F\) measured inside a borehole can be related to the Earth's magnetic field \(H\) and to the total magnetization \(M_{T}\) of a rock as:  
\[F=(H+M_{T})=H+(M_{H}+M_{R})=H+(\chi H+M_{R}), \tag{1}\]  
where \(M_{H}\) is the component of magnetization produced in response to the Earth's magnetic field, \(M_{R}\) is the natural remanent magnetization and \(\chi\) is the rock's magnetic susceptibility. The natural remanent magnetization is a permanent magnetization that remains in the rock even in the absence of an inducing field, recording its magnetization history. The magnetic susceptibility is a dimensionless physical property of the rock that in cases of sufficiently weak fields, like the Earth's magnetic field, represents a constant of proportionality relating the induced magnetization and the applied field. Since the induced magnetization is essentially parallel to the applied field, the magnetic susceptibility is a scalar quantity for most of rocks. Equation (1) is of fundamental importance because it represents the basis for all estimates carried in this study. As Equation (1) is a vector sum, it can only be correctly done with the knowledge of the three parameters that describe a magnetic field vector: magnitude \(F\), inclination \(I\) and declination \(D\). Alternatively, the sum can be simplified by decomposing each vector in its components in the x-, y- and z-directions as defined in Figure 2a. However, the triaxial magnetometers used in image logs to determine tool orientation and/or well direction measure the three components of the magnetic field \(F\) with respect to the tool coordinate system instead of true geographic coordinate system. In order to correctly perform the vector sum in Equation (1) it is necessary to project the field components from the tool's system into the geographic system described in Figure 2a. In the tool orientation system, the x-axis points to the direction of a reference marker, called P1 or Pad #1, located in the plane perpendicular to the z-axis, which for vertical wells correspond to the borehole axis (Figure 2b). As the tool rotates during logging operations, the tool's x-axis points not to north as defined in Figure 2a, but to different directions. Consequently, the magnitude of acquired horizontal components do not represent the true amplitude in north and east direction. Figure 3a shows the magnetic field acquired in a typical well as an example of the variation in amplitude of the horizontal components caused by the tool's rotation. The strong fluctuations occurring in the horizontal components of the acquired magnetic field need to be corrected before calculations. According to [20] the orientation of the tool in the geographical coordinate system can be described by four angles: Sonde Deviation (SDEV), Hole Azimuth (HAZI), Relative Bearing (RB) and Rotation of P1 from North (P1 NO), as illustrated in Figure 2b. Because P1 NO angle represents the rotation of P1 from north, its measurements provide the declination of the tool's x-axis along the logged interval. It is possible to compute the angle between tool's x-axis and the direction of measured field using the relations between magnetic field and its components shown in Figure 2a. The sum of these two angles provide the true magnetic declination of the total inducing field. The calculation of the true declination allows us to compute the correct \(F_{x}\) and \(F_{y}\) components by applying again the relations in Figure 2a using the correct angles this time. Figure 3b shows the field components after amplitude correction. As expected, the strong fluctuations present in the horizontal components of Figure 3a were significantly reduced. Once the true amplitude of the field components has been recovered, estimates for the magnetic susceptibility \(\chi\) can be made using two different approaches. The first and more simplistic case is that of sedimentary rocks.