Introduction

Resistivity LWD tool has become an important facility for real-time well-site data acquisition and interpretation, scene decision making, and geological-oriented drilling guidance. Therefore, it plays an increasingly important role in geosteer drilling, real-time formation comparative evaluation and complex oil reservoir development (Shi 2002). As a result, it is important to increase the measuring accuracy and improve the response features of the resistivity LWD tool. Novel numerical simulation technology of resistivity LWD tool response can provide important theoretical guidance for designing high-accuracy electrical logging instruments and building accurate measurement data interpretation method. So it is of important significance to research high effective numerical simulation algorithms of resistivity LWD tool (Sun et al. 2008; Wei et al. 2010).

Many researches have been conducted to develop numerical simulation algorithms for resistivity LWD tool response in recent years (Gao et al. 2010). The most frequently used algorithms in the forward numerical simulation include finite difference method (Teixeiraa and Chew 2004), integral equation method (Hue et al. 2005), numerical mode matching method (Tan et al. 2007), successive approximation method (Gao et al. 2010), transmission line matrix (Li and Liu 2005), FFT transform method (Zhang and Liu 2003), and finite element method (FEM) (Wang et al. 2006). Although these conventional algorithms have their own strong points in the field of forward numerical simulation of the resistivity LWD tool response, approximation accuracy and convergence speed are still not enough for these algorithms. In addition, the weakness of low computational efficiency, poor iterative performance and dynamic numerical simulation difficulties also restrict these conventional algorithms to be used further in wider fields. The higher-order vector FEM was introduced in 1980s by Babuška and Suri (Demkowicz et al. 2002; Pardo et al. 2007), and used in simulation of resistivity logging instrument response by Demkowicz and Vardapetyan (Šolín et al. 2008; Tomáš et al. 2007). Its advantage over other numerical methods is an unconditional exponential convergence, even for problems with singular solutions. Now, higher-order vector FEM theoretical foundations are well established and the algorithm is well used for oil field (Chen et al. 2011; Ma et al. 2012; Li et al. 2012). As the formations surrounding the borehole having influencing factors, the change of well geological conditions and complex stratigraphic geometrical structures become more complex, it makes the numerical modeling of the apparatus responses a demanding business in terms of computation speed and accuracy. Higher-order vector FEM in the convergence speed and the calculation accuracy have certain advantages. Therefore, higher-order vector FEM is suitable for simulating the responses of the resistivity LWD tool. Thus, based on the higher-order vector FEM, the influence of T–R 1 spacing, R 1–R 2 spacing, structure of antenna and transmitted signal frequency on the tool response were discussed, and the electromagnetic response of resistivity LWD tool in tilted anisotropic formation was also calculated.

Theory and mathematical modeling

Figure 1 shows the basic structure model of a resistivity LWD tool. In Fig. 1, T is the transmitter coil, R 1 and R 2 denote two receiver coils, L 1 (T–R 1 spacing) is the distance from T to R 1, L 2 (R 1–R 2 spacing) is the distance from R 1 to R 2, r is the radius of the transmitter coil and n 1 denotes the number of transmitter coil turns, r R is the radius of the receiver coil and n 2 is the number of receiver coil turns, I T is excitation current intensity in the transmitter coil and I T = I e iωt.

Fig. 1
figure 1

Model of the resistivity LWD tool. a The basic structure of the resistivity LWD tool (Oxz axis). b The electromagnetic field analysis model with drill rod, transmitter coil, borehole wall and formations (Oxy axis)

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For resistivity LWD tool, the transmitter coil can be considered as an excitation source, and the alternating current in the transmitter coil may produce a high-frequency electromagnetic field around the wellbore. The electric field and magnetic field may reciprocally transform each other in space. Assuming that the resistivity LWD tool operates at a high-frequency condition, because the induced electromotive force of the receiving coils is generated by the alternating current, the solving electromagnetic field is a time-harmonic field. As the alternating current exists in the interior of the transmitter coil, a time-varying magnetic field must exist in the formation around the wellbore; thus, the induced magnetic field near the receiver coils should be a time-harmonic electromagnetic field. Based on the Maxwell equation, the wave equation of electric field E in anisotropic formation can be obtained and shown in Eq. (1):

$$ \varDelta \times \left( {\frac{1}{{\mu_{\text{r}} }}\varDelta \times \varvec{E}} \right) - k_{0}^{2} \varepsilon_{r} \varvec{E = } - j\omega \mu_{0} \varvec{J}, $$
(1)

where E denotes the electric field, ε r denotes the composite dielectric constant, μ r and μ 0 denote the relative magnetic permeability and permeability of free space, respectively, and k 0 denotes the wave number. Using Eq. (1), we can obtain the functional expression,

$$ F(\varvec{E}) = \frac{1}{2}\iiint_{V} {\left[ {\frac{1}{{\mu_{r} }}(\varDelta \times \varvec{E}) \cdot (\varDelta \times \varvec{E}) - k_{0}^{2} \varvec{E} \cdot \varepsilon \cdot \varvec{E}} \right]\text{d} V + j\omega \mu_{0} \iiint_{V} \varvec{E} \cdot \varvec{J}\text{d} V}, $$
(2)

where V denotes the solution domain of the finite element.

According to Eq. (2), we can get

$$ F(\varvec{E}) = \sum\limits_{e = 1}^{M} {F_{e} (\varvec{E})} = \frac{1}{2}\sum\limits_{e = 1}^{M} {\iiint_{V} {\left[ {\frac{1}{{\mu_{r} }}(\varDelta \times \varvec{E}) \cdot (\varDelta \times \varvec{E}) - k_{0}^{2} \varvec{E} \cdot \varepsilon_{r} \cdot E} \right]}} dV - j\omega \mu_{0} \sum\limits_{e = 1}^{M} {\iiint_{V} {\varvec{E} \cdot J{\text{d}}V}}, $$
(3)

where e denotes the each split unit, M the total number of units, and J the Jacobian factor.

In each unit, the curl of scattered field can be expressed as

$$ \varDelta \times \varvec{E} = \sum\limits_{i = 0}^{{N_{u} - 1}} {\sum\limits_{j = 0}^{{N_{v} }} {\sum\limits_{k = 0}^{{N_{w} }} {a_{uijk} \varDelta } } } \times \varvec{f}_{uijk} + \sum\limits_{i = 0}^{{N_{u} }} {\sum\limits_{j = 0}^{{N_{v} - 1}} {\sum\limits_{k = 0}^{{N_{w} }} {a_{vijk} \varDelta } } } \times \varvec{f}_{vijk} + \sum\limits_{i = 0}^{{N_{u} }} {\sum\limits_{j = 0}^{{N_{v} }} {\sum\limits_{k = 0}^{{N_{w} - 1}} {a_{wijk} \varDelta } } } \times \varvec{f}_{wijk} , $$
(4)

where the curl of the basis function can be expressed as

$$ \left\{ {\begin{array}{*{20}l} {\varDelta \times \varvec{f}_{uijk} = \frac{1}{J}\left[ {u^{i} P_{j} (v)\frac{{{\text{d}}P_{k} (w)}}{{{\text{d}}w}} \cdot \frac{{{\text{d}}\varvec{r}}}{{{\text{d}}v}} - u^{i} \frac{{{\text{d}}P_{j} (v)}}{{{\text{d}}v}}P_{k} (w) \cdot \frac{{{\text{d}}\varvec{r}}}{{{\text{d}}w}}} \right]} \hfill \\ {\varDelta \times \varvec{f}_{vijk} = \frac{1}{J}\left[ {v^{j} P_{k} (w)\frac{{{\text{d}}P_{i} (u)}}{{{\text{d}}u}} \cdot \frac{{{\text{d}}\varvec{r}}}{{{\text{d}}w}} - v^{j} \frac{{{\text{d}}P_{k} (w)}}{{{\text{d}}w}}P_{i} (u) \cdot \frac{{{\text{d}}\varvec{r}}}{{{\text{d}}u}}} \right]} \hfill \\ {\varDelta \times \varvec{f}_{wijk} = \frac{1}{J}\left[ {w^{k} P_{i} (u)\frac{{{\text{d}}P_{j} (v)}}{{{\text{d}}v}} \cdot \frac{{{\text{d}}\varvec{r}}}{{{\text{d}}u}} - w^{k} \frac{{{\text{d}}P_{i} (u)}}{{{\text{d}}u}}P_{j} (v) \cdot \frac{{{\text{d}}\varvec{r}}}{{{\text{d}}v}}} \right]} \hfill \\ \end{array} } \right.. $$
(5)

Substitution of Eqs. (5) and (4) into Eq. (3) gives

$$ F(\varvec{E}) = \frac{1}{2}\sum\limits_{e = 1}^{M} {(a^{e} )^{\text{T}} } K^{e} a^{e} - \sum\limits_{e = 1}^{M} {(a^{e} )^{\text{T}} } b^{e} , $$
(6)

where K e denotes the unit matrix and b e the vector.

$$ K^{e} = \iiint_{{V^{e} }} {\left[ {\frac{1}{{\mu_{\text{r}}^{e} }}(\varDelta \times \varvec{f}^{e} ) \cdot (\varDelta \times \varvec{f}^{e} )^{\text{T}} - k_{0}^{2} \varvec{f}^{e} \cdot \varepsilon_{r}^{e} \cdot (\varvec{f}^{e} )^{\text{T}} } \right]}{\text{d}}V , $$
(7)
$$ b^{e} = - j\omega \mu_{0} \iiint_{{V^{e} }} {[\varvec{f}^{e} \cdot J]}{\text{d}}V . $$
(8)

According to Eq. (6), we can get the linear equations

$$ Ax = B , $$
(9)

where \( A = \sum\nolimits_{e = 1}^{M} {K^{e} } \), \( x = \sum\nolimits_{e = 1}^{M} {a^{e} } \) and \( b = \sum\nolimits_{e = 1}^{M} {b^{e} } \). Solving Eq. (9), the electric field E at each measurement point can be obtained.

Instrument parameter calibration

When an electromagnetic wave propagates in the porous media near the wellbore formation, the signal amplitude attenuation is different because there are different resistivities among different propagation media. Based on that, by detecting the amplitude ratio and phase difference of induction electromotive force received by two receiver coils mounted on the resistivity LWD tool, the resistivities of different media can be obtained according to the measurement model of the tool. The antenna system is the most important part in the resistivity LWD tool, which includes an electromagnetic wave transmitter and receiver sensors. The detection depth and resolution are influenced by the coil spacing of the antenna system. The basic structures of the resistivity LWD tool and the formation model are shown in Fig. 1. The drill collar is highly conductive, and the transmitter and receiver coils are equipped with a magnetic buffer at 10−4 S/m. The relative permeability is 104, the conductivity of the drill collar is 106 S/m, the conductivity of mud in the borehole is 10 S/m, the vertical depth of the formation model z is equal to 9.0 m, and the relative dielectric constant of formation ε r is equal to 1. The conductivity of each layer is shown in Table 1.

Table 1 Parameters of the formation model
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T–R 1 spacing of the antenna system

The initial T–R 1 spacing is L 1, which is equal to 0.4 m, and the initial R 1–R 2 spacing is L 2, which is equal to 0.2 m. The transmitting frequency f is equal to 2 MHz, and the excitation current intensity I T is equal to 1.0 A. By using higher-order vector FEM to build the numerical model, we fix the distance of two receiver coils, change the T–R 1 spacing L 1, and then investigate the measurement results of the resistivity LWD tool. When L 1 is equal to 0.4, 0.8, 1.2 and 1.4 m, respectively, the corresponding amplitude ratios and phase differences of the electromagnetic wave signal received by two receiver coils are shown in Fig. 2.

Fig. 2
figure 2

Amplitude ratios and phase differences by changing the T–R 1 spacing

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In Fig. 2, the amplitude ratios and phase differences of electromagnetic wave signals received by two receiver coils are increased when the T–R 1 spacing of antenna system is increased, but the phase differences are not influenced by T–R 1 spacing strongly. When the T–R 1 spacing increases between 0.4 and 1.4 m, the polarization angles emerge in the formation interface of the amplitude ratios and phase difference curves noticeably. When T–R 1 spacing is larger than 1.2 m, in the high-resistivity thin layer, the polarization angles are not clear enough in amplitude ratios and the phase difference curves because of the influence of the skin effect on receiving signals.

R 1–R 2 spacing of the antenna system

The initial R 1–R 2 spacing is L 2, which is equal to 0.2 m; the transmitting frequency f is equal to 2 MHz, and the excitation current intensity I T is equal to 1.0 A. We fix the T–R 1 spacing, which is equal to 0.8 m, change the R 1–R 2 spacing L 2, and then investigate the measurement results of the resistivity LWD tool. When L 2 is equal to 0.2, 0.6, 1.0 and 1.2 m, respectively, the corresponding amplitude ratios and phase differences of the electromagnetic wave signal received by two receiver coils are shown in Fig. 3.

Fig. 3
figure 3

Amplitude ratios and phase differences by changing the R 1–R 2 spacing

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In Fig. 3, in the high-resistivity layer, the amplitude ratios of electromagnetic wave signals received by two receiver coils increased when the R 1–R 2 spacing of the antenna system is increased. But in the low-resistivity layer, the amplitude ratios decreased when the R 1–R 2 spacing of the antenna system increased. The phase differences of the electromagnetic wave signals received by two receiver coils increased when the R 1–R 2 spacing of the antenna system increased. When the R 1–R 2 spacing increased between 0.2 and 1.2 m, the polarization angles which emerged in the formation interface of the amplitude ratios and phase difference curves were not clear enough. That is to say, by using amplitude ratios and phase differences curve, we cannot indicate the formation interface accurately and timely. With the increase of R 1–R 2 spacing, the resolution of the resistivity LWD tool is decreased, especially to the high-resistivity thin layer. That is to say, such a situation is unfavorable to perform geosteering drilling and thin-layer resolution. It can be seen from Figs. 2 and 3 that the change in the T–R 1 spacing of the antenna system has obvious influence on the investigation depth and formation resolving power of the resistivity LWD tool. The change in the R 1–R 2 spacing can influence the resolution and geosteering capabilities of the resistivity LWD tool. Therefore, the detection accuracy and detection depth of the resistivity LWD tool would be strongly affected by the antenna system. Based on numerical experiments, we obtain that the optimum T–R 1 spacing is equal to 0.8 m, and the optimum R 1–R 2 spacing is equal to 0.2 m.

Transmitting frequency of the antenna system

According to the measurement results of “ T–R 1 spacing of antenna system” and “ R 1–R 2 spacing of antenna system”, we obtain that the optimum T–R 1 spacing is equal to 0.8 m and the optimum R 1–R 2 spacing is equal to 0.2 m. So, we fix the T–R 1 spacing and R 1–R 2 spacing. When L 1 is equal to 0.8 m and L 2 is equal to 0.2 m, we change the transmitting frequency to investigate the response of the resistivity LWD tool. When the transmitting frequency f is equal to 200, 600 kHz, 1 and 2 MHz, respectively, the measurement results of the amplitude ratios and the phase differences of electromagnetic wave signal received by two receiver coils are shown in Fig. 4.

Fig. 4
figure 4

Amplitude ratios and phase differences by changing the transmitting frequency

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Figure 4 shows that the amplitude ratios and phase differences of the signals in two receiver coils increase gradually as the transmitting frequency increases from 200 kHz to 2 MHz, and the variations are more obvious in the case of low-resistivity formation. When the transmitting frequency is larger than 600 kHz, the polarization angle emerge in amplitude ratios and phase difference curves noticeably. According to Fig. 4, we find that the optimum transmitting frequency is equal to 2 MHz, because when f is equal to 2 MHz the amplitude ratios and phase difference curves have the best resolution. When the transmitting frequency is fixed at 2 MHz and the dielectric constant of an earth formation is changed during the procedure of numerical simulation, there are also changes in the amplitude ratios and phase differences of the receiving signal and these variations in the high-resistivity formation are more sensitive than in low-resistivity formation. However, increase the transmitting frequency can enhance the detection resolution of low-resistivity formation, but the transmitting frequency is not the larger the better. Because with the the increase of the transmitting frequency, the calculated amount of higher-order vector FEM must increase, the computation errors may also increase. In practice, improvement in the transmitting frequency may lead to skin effect influence on the transmitter coil and receiving signals; therefore, the precision of the measurement results will be impacted.

Skin effect of the antenna system

In practice, the high-resistivity layer is generally the oil layer, so we can use the resistivity LWD tool to find the high-resistivity layer. But the resolution and accuracy of the resistivity LWD tool decide the accuracy of the measurement results, especially the high-resistivity thin layer. Therefore, the high precision resistivity LWD tool should be designed, and the antenna system is a key part in the design of the tool. Because the actual instrument structure is fixed and it is hard to be changed, such as T–R 1 spacing and R 1–R 2 spacing, the transmitting frequency can be easily adjusted. Based on “Transmitting frequency of antenna system”, we obtain that if the transmitting frequency is low, the current is uniformly distributed in the internal parts of the transmitter coil, and the transmitter coil is not strongly affected by the skin effect. In conditions of high frequency, the current gradually concentrates on the surface of the transmitter coil and the internal current of the transmitter coil is obviously decreased. That is to say, the maximum current density is distributed at the surface of the transmitter coil. Therefore, in the condition of high frequency, the transmitter coil is strongly affected by the skin effect. So, in the high-frequency condition, the influence of the skin effect should be considered.

In “ T–R 1 spacing of antenna system”, “ R 1–R 2 spacing of antenna system and “Transmitting frequency of antenna system”, we obtain that in the antenna system the optimum T–R 1 spacing is equal to 0.8 m, the optimum R 1–R 2 spacing is equal to 0.2 m, and the optimum transmitting frequency is equal to 2 MHz. Therefore in this section, we use these optimum parameters to build up the numerical model. As the resistivity LWD tool moves along the vertical direction (z-axis) in a borehole, when z = 5.5 m (high-resistivity thin layer) and the transmitting frequency f is equal to 200, 600 kHz, 1 and 2 MHz, respectively, the distribution of the electric field to each transmitting frequency is shown in Fig. 5.

Fig. 5
figure 5

Potential distribution and mesh subdivision. a–d The left plot represents the distribution of the electric field (unit: V/m) around the transmitting coil in the high-resistivity thin layer; the right plot represents the final mesh by using the higher-order vector finite element method (Oxy axis and z = 5.5 m). Different colors indicate different polynomial orders of approximation, ranging from 1 to 5; the transmitting frequency f is equal to 200, 600 kHz, 1 and 2 MHz, respectively

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It is shown in Fig. 5 that as the transmitting frequency increases, the detection depth of the instrument also increases. As the transmitting frequency increases from 200 kHz to 2 MHz, the induced electric field around the transmitter coil is significantly enhanced. When the transmitting frequency is low, such as 200 kHz, the transmitting current is evenly distributed within the transmitting coils, and the higher-order vector FEM needs less degree of freedoms, calculation time and iterative numbers to compute the electric field strength in the computational domain. When the transmitting frequency is high, such as 2 MHz, and the transmitting current is gradually concentrated on the transmitter coil surface, then the transmitter coils may be influenced by the skin effect noticeably. The higher-order vector FEM needs more degrees of freedom, calculation time and iterative numbers to compute the electric field strength in the computational domain. However, in the high-frequency condition, the higher-order vector FEM needs more degrees of freedom, calculation time and so on. That is to say, this may increase the calculation error. In practice, the detection depth of the resistivity LWD tool is an important performance indicator, and the mud filtrate in the borehole has a certain degree of absorption of electromagnetic signal, which may reduce the induced electric field strength around the transmitter coil and reduce the detection accuracy and detection depth of the tool. Therefore, the transmitting frequency should be increased. But when the transmitting frequency is too high, a part of the current will be dissipated in the form of heat in the transmitter coil, which may lead to significantly reduced electromagnetic signal intensity, thus the detection accuracy and detection depth of the instrument may be strongly affected. When the transmitting frequency f is equal to 400, 800 kHz, 1, 2 and 5 MHz, respectively, the measurement results of the real part and imaginary part of the electromagnetic wave signals received by one receiver coil are shown in Fig. 6.

Fig. 6
figure 6

Real part and imaginary part curves of the receiver coil R 1 at f = 400 kHz, δ = 1.045 × 10−4 m; f = 800 kHz, δ = 0.738 × 10−4 m; f = 1 MHz, δ = 0.661 × 10−4 m; f = 2 MHz, δ = 0.467 × 10−4 m; f = 5 MHz, δ = 0.295 × 10−4 m. Here, f denotes the transmitting frequency and δ the skin depth

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It is shown in Fig. 6 that with the increases of transmitting frequency, the real part of the electromagnetic wave signals received by the receiver coil R 1 increases, but the imaginary part of the electromagnetic wave signals received by the receiver coil R 1 decreases. That is to say, with the increases of transmitting frequency, the resolution of instrument to the high-resistivity thin layer will gradually increase. When the transmitting frequency f is equal to 5 MHz, the imaginary part of electromagnetic wave signals is not approximately a straight line, that is to say, the transmitter coil is strongly affected by the skin effect. As the transmitting frequency increases, the potential differences received by two receiver coils gradually increase, and then at the interface of formations the distribution of electric field is uneven. Therefore, the polarization angle emerging in amplitude ratios and phase differences are easily influenced by different transmitting frequencies. If the transmitting frequency is very large, the transmitter coil may be noticeably affected by the skin effect; so it may lead to the distortion caused by the polarization angle, which may affect the accuracy of the measurement results.

Geosteering capabilities of the resistivity LWD tool

When an electromagnetic wave propagates in the porous media near the borehole formation, the signal amplitude attenuation is different because there are different resistivities among different propagation mediums. Based on that, by detecting the amplitude ratios and phase differences of induction electromotive force received by two receiver coils mounted on the resistivity LWD tool, the resistivities of different media can be reflected according to the measurement model of the tool.

Influence of tilted formation

The structure of the resistivity LWD tool is shown in Fig. 7. In Fig. 7, the drill collar is highly conductive, and the transmitter and receiver coils are equipped with a magnetic buffer at 10−4 S/m. The relative permeability is 104, the conductivity of the drill collar is 106 S/m and the conductivity of the mud in the borehole is 10 S/m. According to “ T–R 1 spacing of antenna system”, “ R 1–R 2 spacing of antenna system”, “Transmitting frequency of antenna system” and “Skin effect of antenna system”, we obtain that in the antenna system the optimum T–R 1 spacing is equal to 0.8 m, the optimum R 1–R 2 spacing is equal to 0.2 m, the optimum transmitting frequency is equal to 2 MHz and the optimum excitation current intensity I T is equal to 1.0 A. Therefore, in this section we use these optimum parameters to build up the numerical model. The conductivity of each formation is shown in Table 1.

Fig. 7
figure 7

Structure of the resistivity LWD tool in tilted formation (Oxz axis)

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Assume that the resistivity LWD tool moves upward along a vertical direction (z-axis) in a borehole. As receiver coils are placed without any tilt around a cylindrical steel mandrel, the dip angle α of the formation is equal to 15°, 30°, 45° and 60°, respectively. The amplitude ratios and the phase differences of the electromagnetic wave signal received by two receiver coils are shown in Fig. 8.

Fig. 8
figure 8

Amplitude ratios and phase differences in tilted formation

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Generally, we can provide timely geosteering drilling information according to the polarization angles whether appearing on the amplitude ratio curves or the phase difference curves corresponding to the interface between two formations. It can be seen from Fig. 8 that although there are narrow polarization angles occurring in the amplitude ratio and the phase difference curves, respectively, because the polarization angles appear on the curves corresponding to both the interface from low-resistivity formation to high-resistivity formation and the interface from high-resistivity formation to low-resistivity formation, it is very difficult to determine the drilling direction in time. Therefore, as the receiver coils are placed without any tilt around a cylindrical steel mandrel, the geosteering drilling information cannot be provided accurately.

Directional resistivity LWD in the tilted formation

Assume that the transmitter coil is placed without any tilt around a cylindrical steel mandrel and the two receiver coils are tilted by about 45° with respect to the cross section of the cylindrical steel mandrel. In that case, considering that the dip angle α of the tilted formation is equal to 15°, 30°, 45° and 60°, respectively, the amplitude ratios and the phase differences of the electromagnetic wave signal received by two receiver coils are shown in Fig. 9.

Fig. 9
figure 9

Amplitude ratios and phase differences in tilted formation with tilted receiver coils

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It is shown in Fig. 9 that tilts of the two receiver coils can affect the judgment of low-resistivity formation and high-resistivity formation. In the condition of tilted receiver coils, when the dip angle of the formation increases, the resolution of instrument measurement results in the low-resistivity formation and high-resistivity formation gradually increasing. According to the polarization angles in amplitude ratios and phase difference curves, we can indicate the interface between the two formations clearly. However, as shown in Fig. 9, with the increase of dip angle of the formation, the variations of amplitude ratios and phase differences received by two receiver coils all present an upward trend. But in amplitude ratios and phase difference curves, we can find that with the increase of dip angle of the formation, the polarization angles corresponding to the interface from low-resistivity formation to high-resistivity formation gradually disappear. Conversely, the polarization angles corresponding to the interface from high-resistivity formation to low-resistivity formation become more and more obvious. Based on that, the drilling direction can be determined in time according to the polarization angles whether appear on the amplitude ratios and the phase difference curves corresponding to the interface between two formations, then the geosteering drilling information can be accurately provided by resistivity LWD tool.

Based on the higher-order vector FEM, the numerical simulation model of the resistivity LWD tool can be built, and when the dip angle α of the tilted formation is equal to 15°, 30°, 45° and 60°, respectively, the average value of measurement results of the resistivity LWD tool are shown in Table 2, where D denotes the degrees of freedom, T the calculation time, N the iterative numbers and P the global errors.

Table 2 Instrument response of the resistivity LWD tool
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It can be seen from Table 2, by using higher-order vector FEM to build the simulation model of resistivity LWD tool and calculate the electric field strength around the receiver coils, when the dip angle of formation α = 15°, the model needs an average number of DOFs used in calculation of 27,138; the average calculation time is 9.59 s; the average number of iterations is 10, and the average global error is 0.65 %. When the dip angle of formation α = 30°, the model needs an average number of DOFs used in the calculation of 33,167; the average calculation time is 11.73 s; the average number of iterations is 13, and the average global error is 0.77 %. When the dip angle of formation α = 45°, the model needs an average number of DOFs used in calculation of 49,272; the average calculation time is 25.66 s; the average number of iterations is 17, and the average global error is 0.82 %. When the dip angle of formation α = 60°, the model needs an average number of DOFs used in the calculation of 57,163; the average calculation time is 30.15 s; the average number of iterations is 22, and the average global error is 0.93 %. In Table 2, when the dip angle of formation α = 60°, the model needs more DOF, calculation time, iterations and global errors. Because in order to ensure a higher solution accuracy, the grids refinement at the regions with a strong electric field changes and at the interface of formations would be strengthen, therefore a large amount of DOFs and calculation time would be used.

Conclusions

In this paper, by using higher-order vector FEM, we have successfully investigated the impact of the measurement results by changing the T–R 1 spacing, R 1–R 2 spacing, transmitting frequency and the structure of resistivity LWD tool. The antenna system is the important part of the resistivity LWD tool, and the measurement results of the instrument would be strongly affected by different T–R 1 spacing, R 1–R 2 spacing and transmitting frequency. Numerical simulation results show that the change of T–R 1 spacing has obvious influence on the investigation depth and detecting precision of the resistivity LWD tool, and the change of R 1–R 2 spacing may affect the resolution of the thin-layer distinguish. The change of signal source transmitting frequency can influence the judgment of the formation interface, and high transmitting frequency can improve the instrument resolution of high-resistivity thin layer. The structure of the antenna system can also affect the measurement results of the resistivity LWD tool, and we can use the asymmetry of polarization angles that appear on the amplitude ratios and the phase difference curves to provide accurate geosteering drilling information.

In practice, because the geological condition around the wellbore is complex, it may affect the electromagnetic response of the resistivity LWD tool, so that the instrument parameters need to be adjusted. Thus, it can be seen that the study of the numerical simulation for resistivity LWD tool response has important significance to guide the development of high-precision resistivity LWD tool and build accurate resistivity LWD measurement data interpretation method. But by using conventional resistivity LWD tool logging curve, field engineers are hard to judge the bit position, also the reservoir drilling rate must be declined. But directional resistivity LWD tool logging curve can provide the bit position information, well deviation information, azimuth information and tool surface information; thus the directional resistivity LWD tool is important for the directional drilling construction operation, and the next research focusing on the resistivity LWD tool will optimize the structure of the tool, reducing the length of the instrument and improving the performance index of the tool.