Figure 3. Rand X-components of apparent conductivities and with respect to the relative permittivity.

Table 1. List of effective dielectric constants inducing 10% discrepancy on apparent conductivity and with and without dielectric enhancement.

given by Equation (8). The abnormally large dielectric constant is frequency dependent. Thus smaller dielectric constants are in need to reach the same level dielectric effect.

On the other hand, the effective dielectric constants on X-signal of apparent conductivity and are much smaller than on R-signal. Thus we know that imaginary components are more sensitive to dielectric effect than the real parts.

3.1.2 Case II:

Secondly, we investigate the relationship between coil spacing and the dielectric effect with the same isotropic formation. Table 2 lists the corresponding coil spacing between transmitter and main receiver and bucking coils.

Figure 4 compares Rand X-components of apparent conductivities and with respect to the coil spacing for small permittivity () and large dielectric constant (), respectively. A significant nonlinear discrepancy on X-signals with and without large permittivity is shown and changed into negative

Table 2. List of coil spacing from the transmitter to main receiver and bucking coil.

Figure 4. Rand X-components of apparent conductivities and with respect to coil spacing. The conductivity is 0.1 S/m. The frequency is 26 KHz.

signs.

3.2. Example 2

In this example, we consider a TI formation and figure out dielectric enhancement on anisotropic conductivity in layered laminated formation. The frequency is 26 KHz.

3.2.1. Case I:

Vertical conductivity is assumed as 0.1 S/m and horizontal conductivity is changing from 0.1 S/m to 10 S/m. The simulation results of coplanar component with respect to horizontal conductivity are presented in Figure 5.

In Figure 5, the x axis is the anisotropic ratio. As shown in Figure 5, weak discrepancy on the apparent conductivities and with and without the dielectric effect is observed. In order to illustrate explicitly, we present relative error in Figure 6.

Figure 6. shows that with the increased horizontal conductivity, dielectric effect on both X-and R- and is decreased. According to the conductive property of laminated anisotropic medium, the horizontal conductivity is dominated by salt water zone for laminated formation and thus we can infer that dielectric effect on, may be attenuated in the water-bearing zone.

3.2.2. Case II:

Next, we restore the horizontal conductivity to be

Figure 5. Rand X-components of apparent conductivities and with respect to anisotropic ratio. The vertical conductivity is a constant and set as 0.1 S/m. The horizontal conductivity is various from 0.1 S/m to 1 S/m. The frequency is 26 KHz.

Figure 6. The relative error of apparent conductivities and with and without dielectric enhancement.

0.1 S/m and alter vertical conductivity from 0.001 S/m to 0.1 S/m. Figure 7 compares apparent conductivities and with and without dielectric effect in a similar way as shown in Figure 6. In this example, the distance from the transmitter to the main receiver and bucking coil are 21 inch, 15 inch, respectively.

Since the horizontal conductivity is constant, Rand X- are independent to the vertical conductivity. Thus we know the discrepancy of the coaxial component is only related to dielectric effect. We have observed that X- is negative when is 50,000, which is caused by large vertical permittivity.

Figure 8 presents the absolute relative differences of apparent components with and without the dielectric effect, respectively. With larger anisotropic ratio, the relative differences on Xand R- are increasing. In the laminated anisotropic medium, the vertical conductivity is dominated by a hydrocarbon zone, and we can conclude that dielectric effect on would be boosted by the hydrocarbon-bearing zone.

3.3. Example 3

Cross components are helpful for detecting the formation boundary; therefore, efficiently help us solve multilayer inversion problem. Hence in this section, we will discuss how the dielectric effect takes effect in the deviated well.

We now assume one homogenous anisotropic medium, whose anisotropic ratio is 10 (σ_{h} = 1 S/m, σ_{v} = 0.1 S/m).

Figure 7. Rand X-components of both and with respect to anisotropic ratio. The vertical conductivity is various. The horizontal conductivity is 0.1 S/m. The frequency is 26 KHz.

Figure 8. The absolute relative difference of Rand X-components of both and when or 50,000, respectively, with respect to anisotropic ratio.

The frequency is 26 KHz. Figure 9 shows off-diagonal apparent responses, , with respect to dipping angle α.

The R-signal of and from normal and large permittivities are perfectly coincide with each other; and therefore, R-signal of both and are independent with dielectric effect in any deviated well. We observe one interesting phenomenon in the X-signals of both and. If the well is slightly deviated (α ≤ 10˚) or highly deviated (α ≥ 85˚), the discrepancy on

Figure 9. Rand Xcomponents of and with respect to the dipping angle, α. The horizontal conductivity and the vertical conductivity are 1 S/m, and 0.1 S/m, respectively. The frequency is 26 KHz. The distance from the transmitter to the main receiver and bucking coil are 21inches, 15inches, respectively.

X-signal of and is negligible. However, in the medium range (10˚ < α < 85˚), X-signal of or with is differentiated from small permittivity (). Thus we can infer that in a vertical or highly deviated borehole, the dielectric effect can be ignored on the cross components, during inversion, even though large dielectric constants do exist.

4. Conclusions

Previous work has proven that the dielectric effect causes the negative X-signal of array induction logging. We extend this discussion to triaxial induction logs and discuss the dielectric effect on the coplanar, coaxial, and cross components.

A 1-D synthetic forward model for the triaxial induction tool in the homogenous medium is explained and implemented. We employ the triaxial tool includes bucking coils to eliminate direct coupling between the transmitters and main receivers.

Sets of asymptotic analysis are illustrated. We find that the dielectric effect may cause negative signs on the imaginary components of both coplanar and coaxial responses of triaxial induction logs and enhance the real component. The dielectric effect is enhanced by high operation frequency as well as long coil spacing.

In the laminated anisotropic medium, the hydrocarbon-bearing zone manifests the dielectric effect, whereas the water-bearing zone weakens the dielectric effect. Additionally, the dielectric effect plays a negligible impact on the cross components, for vertical or highly deviated well.

5. Acknowledgements

The authors are indebted to the Well Logging Laboratory for their kind permission to provide the forward model code.

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