1. Introduction
The meaning of dynamic measurement while drilling (MWD) is to obtain real-time measurement data about the drill bit’s posture during the drilling process. Then, the obtained measurement data are transmitted to the ground through wireless transmission in real time, which provide reference opinions for the next steps of the staff operations. The acquisition of these measurement data is achieved by converting the analog measurement signals of various sensors in the measuring instrument into digital signals, and then calculating them based on physical models. Then, with the help of the drilling fluid (mud) inside the drill string, it is transmitted to the ground. Finally, the ground receives the mud pressure waveform through a pressure sensor, and decodes the waveform to obtain parameters.
During the signal transmission process, vibrations such as cutting the rock stratum, and the collision between the drill string and the borehole wall can cause strong mechanical vibrations in the drilling tool assembly. Therefore, the pressure wave of the mud pulse signal detected by the pressure sensor contains a large amount of noise signals, resulting in an extremely low signal-to-noise ratio (SNR) of the downhole attitude measurement signal [
1,
2]. The related research shows that the amplitude of the near-bit vibration signal is generally about 10 g, and the maximum can reach 30 g, while the amplitude of useful acceleration sensor signal is generally less than 1 g. Therefore, the SNR of the characteristic signal is usually as low as −20 dB, or even lower [
3,
4]. The weak acceleration signal is annihilated in the strong vibration background noise, which interferes with the accuracy of tool attitude measurement seriously, and even makes the dynamic MWD invalid [
5,
6,
7]. Therefore, how to detect weak useful signals from strong noise backgrounds with unfixed frequencies has become a major issue in mud pulse-transmission technology.
The digital filter based on time–frequency analysis was first applied to the noise suppression of MWD signals. Tu [
8] develops a set of ground decoding systems and uses finite impulse response (FIR) digital filtering to make noise reduction on the mud pulse signal. The field experiment results show that the developed device can correctly extract and recognize the mud pulse signal with a simple and practical decoding process. Zhao [
9] uses a combination of linear filtering and a nonlinear “flat-top elimination” method to process the signals collected at the drilling field. The comprehensive comparison with the processing results of the linear filtering method shows that this method can more effectively remove the noise and other interference of drilling pump.
Digital filtering is a signal processing method based on noise suppression. Its realization is based on the premise that the spectrum of the characteristic signal does not overlap with the spectrum of noise interference. Under such conditions, it can retain useful signals and filter out irrelevant noise components during the filtering process [
10]. However, in the actual collected MWD signals, the frequency distribution of noise signals is very complex due to the variety of interference sources, and there is bound to be a part close to the frequency of the characteristic signal [
11,
12]. Therefore, while suppressing noise, the characteristic signal will inevitably be suppressed or damaged, and even lead to invalid MWD.
In order to improve the accuracy of attitude measurement for the drilling tool, some scholars refer to the attitude measurement principle in the field of integrated navigation. The advanced filtering algorithm is applied to the downhole sensor signal to eliminate the noise interference maximum and realize its optimal estimation. Ref. [
13] proposed a linear random model by using a three-axis accelerometer and a three-axis magnetometer to form an MWD system, and then the state of the model is estimated by the Kalman Filter (KF) algorithm. However, the drilling tool is affected by many kinds of vibration during the drilling process, and shows strong nonlinear characteristics. Therefore, there is an obvious system model error between the linear stochastic system and the real drilling dynamic system, while the KF algorithm is only applicable to linear systems, and it cannot be used for iterative calculation of multi-sensor nonlinear systems.
Aiming at the above problems, refs. [
14,
15,
16] established a nonlinear model of downhole attitude measurement based on quaternion, and used the unscented Kalman Filter (UKF) and its improved algorithm to filter the vibration interference signal. The simulation results prove the effectiveness of this method. However, the application of UKF needs to meet the assumptions that the system state model and measurement model are accurately known. For the actual measured system in the drilling process, it is difficult to accurately establish an appropriate system model due to the influence of the complex underground environment. The predetermined system model has uncertainties such as system parameter mutation, instantaneous interference, unknown statistics of system noise, unknown drift, etc., which leads to the decline of the estimation accuracy of the filter. Even filtering divergence occurs, which makes the filter lose its estimation function [
17]. Therefore, the existing dynamic measurement methods based on multi-sensor combined measurement and optimal estimation algorithms have defects and limitations.
With the rapid development of science and technology, some emerging signal processing methods have been applied to weak signal detection in dynamic MWD [
18,
19,
20]. Ref. [
21] proposes a method of downhole mud pulse signal recognition based on deep learning. The detection model is composed of a wavelet neural network and automatic encoder, which can achieve a good recognition effect. However, this method has the disadvantages of a difficult selection of wavelet parameters and dependence on subjectivity. Moreover, the SNR of the signal that can be processed is high, and the engineering applicability needs to be further verified. Ref. [
22] adopts the set empirical mode decomposition (EEMD) method to establish the noise reduction and shaping algorithm of pulse signals. Then, the judgment criteria of the excellent noise reduction algorithm were presented for pulse signals while drilling was based on the algorithm approximation index and correlation index. The simulation results show that the proposed method is reasonable and effective. However, when the method is applied, it needs to reconstruct the signal one by one to calculate the mean square error value, making the algorithm less efficient. Moreover, the algorithm is unstable when building different low-pass filtering algorithms based on the natural mode.
In recent years, the profound study for nonlinear science has provided a new way for the researcher to understand, analyze and solve the problem of weak signal detection. The discovery of nonlinear dynamic phenomena such as chaos, fractal and stochastic resonance poses a powerful challenge to traditional signal processing methods [
23,
24,
25,
26]. Applying chaos theory to signal detection is a new research field rising from the 1990s. Compared with the traditional weak signal detection method, the weak signal detection method based on the chaotic phase transition can detect a lower SNR of the signal, and is not limited by the statistical characteristics of noise [
27,
28].
As a typical nonlinear model that can produce chaotic effects, the second-order Duffing system can realize the transformation of the output state under the conditions of appropriate parameters [
29,
30]. When the Duffing system is in the critical state from chaotic to large-scale periodic, it has the outstanding features of being sensitive to weak periodic signals and immune to noise signals. In addition, the greater the noise intensity, the more stable the chaotic state, and the higher the accuracy of weak signal detection through system phase transition [
31,
32,
33]. This abnormal effect and advantage make the Duffing system important in the practical application of weak signal detection. Therefore, the application of chaos theory to the detection of weak signals while drilling has great research space and application value.
In conclusion, this paper presents a weak signal detection method for dynamic MWD based on the chaotic effect of the array Duffing system. Using the transition of the output state of the array Duffing system, the frequency detection and parameter estimation of the downhole attitude measurement signal are carried out, and then the complete sensor signal is obtained. Finally, real-time and accurate measurement of attitude parameters for drill tool is realized.
3. Parameter Estimation
It can be seen from the above analysis that the frequency value of the MWD signal in the strong noise background can be identified using the array Duffing equations combined with the scale transformation. However, in order to calculate the real-time attitude information of the drilling tool, it is also necessary to determine the amplitude and phase value of the characteristic signal, so as to recover the complete signal waveform. To solve this problem, an amplitude and phase synchronization estimation method based on the array Duffing system will be presented in this section.
According to the analysis results in the previous section, when Equation (11) holds, the output state of the Duffing system will change from chaotic to large-scale periodic. Assuming that λ
1 represents the driving signal amplitude when the output state of the Duffing system just changes from chaotic to large-scale periodic, and then Equation (11) can be rewritten as follows.
where the
λ1 could be determined by observing the output status of the Duffing chaos system.
After
λ1 is determined, Equation (14) is a binary equation with an amplitude
γ and phase
θ of characteristic signals as variables. Therefore, another binary equation about amplitude and phase could be established in a similar way, and then the amplitude and phase values of the characteristic signal can be obtained by solving the binary function equations. It should be noted that since the variation range of
θ is from −π to π, the absolute value of
θ is solved through the binary function equation instead of
θ. To further determine the value of
θ, one more equation needs to be added. Therefore, aiming at the problem of parameter estimation to be solved in this section, a chaotic detection model based on array Duffing system is established as follows.
Firstly, the initial phase of the drive signal is set to 0, 0.5π and π, as shown in Equation (15). Then, the characteristic signal is input into the array Duffing equations in Equation (15), respectively, and the amplitude of the driving signal is adjusted step by step. Finally, the output state of the Duffing system is observed and the amplitude of the system is recorded when the phase transition occurs.
When the output state of the array Duffing system changes from the chaos to large-scale period, its driving signal amplitude is expressed as
λ1,
λ2 and
λ3. Therefore, the array Duffing equations for the amplitude
γ and phase
θ of the characteristic signal can be expressed as follows.
The following results can be obtained by solving the above equation.
The equations above is the estimation formula for the amplitude and phase of the characteristic signal. It can be seen that the array Duffing system used for parameter estimation in this section and the array Duffing system used for frequency detection in the previous section are derived based on Equation (10).
But the difference is that in
Section 2.2.2, in order to analyze the influence of the initial phase of the characteristic signal on the frequency detection, the range of the initial phase value is determined by giving the amplitude of the driving signal and the characteristic signal, and then the all-phase array Duffing system for frequency detection is obtained. However, the model of the array Duffing system is first given in this section and the amplitude of its driving signals are determined by observing the output state of the array system. Finally, the amplitude and phase values of the characteristic signal are determined by solving Equation (16).
Based on the analysis above, the specific process of the chaotic effect-based improved array Duffing systems for dynamic MWD signal detection is as follows.
Step 1 Determine the parameters of Duffing system. For example, in Equation (3), the damping ratio k is set to 0.5, the angular frequency of the drive signal is set to 1 rad/s, the initial value of the system (x(0), x’(0)) is set to (0,0), and the amplitude of the drive signal λ is set to 0.72.
Step 2 Input the noise characteristic signal into Equation (7), and set the initial phase angle of the driving signal to be 0, ±2π/3 (the amplitude of the characteristic signal is required to be not less than 0.02), from which the array Duffing system for frequency detection is obtained.
Step 3 By introducing the transformation coefficient R, the characteristic signal with sampling frequency of fs and angular frequency of ω is updated to the signal with sampling frequency of fs/R and angular frequency of ω/R. Then, the array Duffing equation for frequency detection is solved with the calculation step T1 = R/fs, and the phase trajectory of the output state of the array Duffing system is obtained.
Step 4 Adjust the transformation coefficient R and observe the output state of the above array system. As long as one of the phase trajectories jumps from the chaotic state to large-scale periodic state, it means that the value of the transformation coefficient at this time is the frequency value of the characteristic signal.
Step 5 After determining the frequency value of the characteristic signal, the array Duffing system for parameter estimation is designed according to the initial phase angle of the driving signal. At this time, the initial phase angles of the system drive signal are 0, π/2 and π, respectively. The frequency of the characteristic signal is reconstructed according to the transformation coefficient, that is, the frequency value is scaled on the time axis, so it does not affect the initial phase angle.
Step 6 Input the characteristic signal with noise after the previous step into the array Duffing system for parameter estimation, as shown in Equation (15), and observe the phase trajectory of output state of the Duffing system. The amplitude of the driving signal of the array Duffing system is trimmed to determine the corresponding driving signal amplitude while each Duffing equation jumps from the chaotic state to the large-scale periodic state, and is marked as λ1, λ2 and λ3 in turn.
Step 7 Finally, λ1, λ2 and λ3 is substituted into Equation (17) to obtain the amplitude and phase of the MWD signal.