Study on Ultra-deep Azimuthal Electromagnetic Resistivity LWD Tool by Influence Quantification on Azimuthal Depth of Investigation and Real Signal

Abstract

This paper proposes a new tool design of ultra-deep azimuthal electromagnetic (EM) resistivity logging while drilling (LWD) for deeper geosteering and formation evaluation, which can benefit hydrocarbon exploration and development. First, a forward numerical simulation of azimuthal EM resistivity LWD is created based on the fast Hankel transform (FHT) method, and its accuracy is confirmed under classic formation conditions. Then, a reasonable range of tool parameters is designed by analyzing the logging response. However, modern technological limitations pose challenges to selecting appropriate tool parameters for ultra-deep azimuthal detection under detectable signal conditions. Therefore, this paper uses grey relational analysis (GRA) to quantify the influence of tool parameters on voltage and azimuthal investigation depth. After analyzing thousands of simulation data under different environmental conditions, the random forest is used to fit data and identify an optimal combination of tool parameters due to its high efficiency and accuracy. Finally, the structure of the ultra-deep azimuthal EM resistivity LWD tool is designed with a theoretical azimuthal investigation depth of 27.42–29.89 m in classic different isotropic and anisotropic formations. This design serves as a reliable theoretical foundation for efficient geosteering and formation evaluation in high-angle and horizontal (HA/HZ) wells in the future.

Appendix

We set ρ = e^x, k_ρ= e^−y in Eq. (13) and multiple by e^x on both sides (Anderson 1979):

$$e^{x} {\text{g}}\left( {e^{x} } \right) = \int_{ - \infty }^{\infty } {e^{ - y} } f\left( {e^{ - y} } \right)e^{x - y} J_{\nu } \left( {e^{x - y} } \right){\text{d}}y.$$

(20)

When G(x)= e^xg(e^x), F(y)= e^−yf(e^−y), and H_v(y)= e^yJ_v(e^y), Eq. (20) can be translated into convolution form as follows:

$$G\left( x \right) = \int_{ - \infty }^{\infty } {F\left( y \right)H_{v} \left( {x - y} \right){\text{d}}y} ,$$

(21)

where G can be seen as the system respond of input function F and linear filtering function H_v. Then, the sinusoidal interpolation function is as follows (Ghosh 1971):

$$T({\text{t}}){ = }\frac{{\sin \left[ {\frac{\pi }{\Delta } \cdot (t - n\Delta )} \right]}}{{\frac{\pi }{\Delta } \cdot (t - n\Delta )}},$$

(22)

where Δ is the sampling interval and defined as $P(x) = \frac{\sin (\pi x)}{\pi x}$.

Then, Eq. (22) can be written as: $T({\text{t) = }}P\left( {\frac{t}{\Delta } - n} \right)$. The approximate value of F(y) is F^*(y) which is calculated by the sample of Eq. (21) based on Eq. (22) (Johansen and Sørensen 1979):

$$\begin{aligned} F^{*} (y) = & \sum\limits_{n = - \infty }^{\infty } {F(n\Delta )T\left( y \right)} , \\ { = } & \sum\limits_{n = - \infty }^{\infty } {F(n\Delta )P\left( {\frac{y}{\Delta } - n} \right)} . \\ \end{aligned}$$

(23)

According to Eqs. (21) and (23), we can obtain G^*(x) which is the approximate value of G(x) (Johansen and Sørensen 1979):

$$\begin{aligned} G^{*} (x) = & \int_{ - \infty }^{\infty } {F^{*} (y)H_{v} (x - y)} {\text{d}}y, \\ { = } & \int_{ - \infty }^{\infty } {\left[ {\sum\limits_{n = - \infty }^{\infty } {F(n\Delta )} P\left( {\frac{y}{\Delta } - n} \right)} \right]H_{v} (x - y){\text{d}}y} , \\ = & \sum\limits_{n = - \infty }^{\infty } {F(n\Delta )} H_{v}^{*} (x - n\Delta ), \\ \end{aligned}$$

(24)

where $H^{*} (x) = \int_{ - \infty }^{\infty } {P\left( {\frac{y}{\Delta }} \right)H_{v} (x - y){\text{d}}y}$ is the filtering coefficient.

Actually, the filtering coefficient is in exponential decay when nΔ → ∞. Therefore, Eq. (24) can be solved under the condition of limited filter length. We set x = mΔ (m is an integer), n′ = m–n and substitute Eq. (24) into Eq. (25) to greatly reduce the amount of calculation (Hu and Nie 1998):

$$G^{ *} \left( {m\Delta } \right){ = }\sum\limits_{{n^{\prime} = N_{1} }}^{{N_{2} }} {F[(m - n^{\prime} )\Delta ]} H_{\text{v}}^{ *} (n^{\prime}\Delta ),$$

(25)

where the sum limit N₁, N₂ is defined by the characteristic of function $f\left( {k_{\rho } } \right)$ and filter functions H ^*_v (y) when $k_{\rho } \to \infty$ and $y \to \pm \infty$, respectively.