Investigation of Effects of Large Dielectric Constants on Triaxial Induction Logs ()
1. Introduction
Plentiful experiments have shown that at a low frequency, dielectric constants are significantly increased. For instance, Sengwa-Soni showed that the relative dielectric permittivity, εr, of water-saturated carbonate could be higher than 1000 when its frequency is at 100 Hz [1]; Ahualli discovered that in the highly concentrated colloidal fluid, εr could be more than 1000 at the frequency of 300 Hz [2]. Specifically, the dielectric constant can reach up to 50,000 in shale formations with metallic particles at an induction frequency range of (25 - 100 KHz) [3].
The reason is very complex and still under study since dielectric enhancement violates the equal distribution principle. The most common explanation is a double electric layer model [4]. A new rock model that simulates conductive grain enclosed by a super thin, non-conductive coating was found to be effective in explaining dielectric enhancement in rocks [5]. It is the low-conductivity membrane that causes dielectric enhancement at low frequencies in rock formations.
Induction logging tools operating at a low frequency (10 KHz - 400 KHz) is one important approach to detect the resistivity of the earth formations. In the induction operation range, the dielectric effect is negligible within the range of typical formation conductivities (0.5 - 5 S/m), because the displacement current is relatively smaller than the conduction current. However, in the recent years within the current decade, strange induction logs with large negative imaginary signals (also called as X-signals) in array induction logs have been encountered and successfully explained by large dielectric constants (εr > 10,000) [6,7].
Responses from traditional array induction logging are essentially coaxial components which transmitters and receivers are placed along the same axis. Because of the structure limitation, the traditional induction tool is not able to detect anisotropy. Advanced induction technology has been developed and is known as a triaxial induction tool. It is comprised by three mutually perpendicular pairs of transmitters and receivers and capable to collect multi-directional electrical information. Responses on a triaxial tool can be categorized into three aspects: coaxial, coplanar, and cross components. In fact, the coplanar responses play an important role in determining anisotropic resistivity. Some cross components give contributions to detect formation boundary. The inversion approach has been developed to detect dipping angle based on both coplanar and cross components [8-11]. However, the dielectric effect on coplanar and cross components is still unknown. Hence it is worthy to investigate the dielectric enhancement effect on those components, based on the structure of triaxial induction logging tool. The purpose of this paper is to discuss the effect of large dialectic constants on a triaxial induction logging tool. A 1-D synthetic model of a triaxial tool in homogenous transverse isotropic (TI) formation at an arbitrary dipping angle is employed to simulate the dielectric effect on triaxial responses. An asymptotic analysis approach is implemented to investigate dielectric effect.
2. 1-D Modeling
A basic structure of the triaxial induction tool consists of three orthogonal transmitters and three orthogonal receivers oriented at x, y, and z direction, as shown in Figure 1(a). Since the transmitter and receiver coils are infinitely small, we can treat them as magnetic dipoles. The equivalent dipole model is shown in Figure 1(b). For industry standard wire line triaxial tools, bucking coils placed between the transmitters and the receivers are always implemented to eliminate the direct coupling radiated from the transmitters.
A 3 × 3 tensor apparent conductivities 
is measured at each pair of transmitter-receiver spacing,
(1)
(2)
where 
is the measured apparent conductivity at the j-th receiver from the i-th transmitter.
Consider a triaxial tool in a 1-D TI medium. The orientation of transmitter and receiver is arbitrary with respect to formation coordinate. Figure 2 shows two types of coordinates in the whole system: the formation coordinates (unprimed) and the sonde coordinate (primed). The symbol α is a dipping angle between the Z axis and the 
axis. The symbol β is an azimuthal angle between the x axis and the projection of transmitter coils on the X-Y plane. The symbol γ represents a rotation angle that transmitter Tx is deviated from the 
axis.
The transformation of magnitudes between the bedding-plane coordinates X, Y, Z and the tool-system coordinates
, 
, and 
is affected by the same rotation matrix as We assume that 
denotes the magnetic moment in the sonde coordinate, given by
(3)
Then in the formation coordinate, the equivalent magnetic source M is obtained by
(4)
In the formation coordinate, Maxwell equations due to
(a) (b)
Figure 1. Basic structure of a triaxial induction tool [12]. (a) The original model; (b) The equivalent model.
a magnetic source M are shown as,
(5)
and
(6)
where
, M, 
are defined as
(7)
(8)
(9)
Note that the dielectric permittivity 
and electric conductivity tensor 
are combined into a single, complex-valued conductivity
. In terms of Equations (5) and (6), we can analytically solve magnetic fields in a homogenous formation. The details of the derivation are omitted here and can be referred to [13].
Magnetic responses in the sonde system is easily derived by multiplying the inverse of rotation matrix to magnetic components in formation coordinate, as
(10)
where 
is the magnetic filed is defined as
(11)
Figure 2. The relationship between tool coordinate and formation coordinate.
The magnetic fields in Equation (11) are consisted by direct coupling and the induced secondary fields. The latter one is dominated by the conductivity of the formation and always overwhelmed by the direct coupling. As we mentioned before, bucking coils are implemented to distract direct coupling from the total fields and leave the secondary fields. We can adjust the distance, turns or windings of the bucking coils to balance off direct fields. In this paper, we take use of spacing to eliminate direct coupling [10], shown as
(12)
where 
is the final magnetic filed and l1, l2 are distance to bucking coil and main receiver.
Finally we can find the apparent conductivity tensor 
with respect to Equation (12), as
(13)
K is the conversion matrix given by tool specific configuration [12], shown as,
(14)
Specifically, if the well is vertical, the apparent conductivity tensor 
would be a diagonal matrix, as
(15)
In Equation (15), 
are coplanar components. In TI medium, 
and 
are the same as each other because of the symmetric resistivity in horizontal plane. Thus in the following part, we only need to discuss
. 
is basically the coaxial response, which is the same as from an array induction tool.
In the deviated well, the apparent conductivity tensor 
is given by
(16)
with two nonzero cross components 
due to nonzero dipping angle. It is found that the horn effect on 
and 
is an important indicator of formation boundary.
3. Results and Discussion
3.1. Example 1
In the first example, we assume a homogenous isotropic formation, whose conductivity is 0.1 S/m. We set the tool movement trajectory perpendicular to formation, namely, zero dipping angle. The distance between transmitter and main receiver and bucking coil are 21 inch, 15 inch, respectively. Without specification, the same distance between transmitter and main receiver and bucking coil are the same as in the first example.
3.1.1 Case I: f = 26 KHz, 52 KHz, 104 KHz
In Figure 3, we present coplanar component 
and coaxial component 
versus permittivity 
(1 - 50,000) at 26 KHz, 52 KHz and 104 KHz, respectively.
Simulation results in Figure 3 reveal that dielectric enhancement does take effect on apparent conductivity 
and
. X-signals of both apparent conductivities 
and 
are decreased and become negative at higher permittivity
. Meanwhile, R-signal increases significantly with the increased permittivity
. Therefore, we predict dielectric enhancement can also cause negative sign on the imaginary components of apparent conductivity 
and 
whereas the real components are positively enhanced.
Then in Table 1, we summarize the minimum values of permittivity 
that induce 10% discrepancy on apparent conductivity 
and 
with and without dielectric enhancement. Specifically, we name those dielectric constants as effective dielectric constants.
According to Table 1, effective dielectric constants on the real parts of apparent conductivity 
and 
are reverse proportional to frequency. Therefore, we infer that higher frequency manifests dielectric effect, which obeys the definition of the complex conductivity 
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