In order to grasp the downhole situation immediately, logging while drilling (LWD) technology is adopted. An oil drilling technology is used to obtain the physical parameters of the rock formation or drilling parameters measured while drilling and propagating data to the ground.^[1] So far, the main wireless transmission technologies are mud pulse, electromagnetic (EM) telemetry and acoustic telemetry. Unfortunately, the throughput of the mud pulse system is too low to satisfy the volume of data and its magnitude is only about a few bits per second. An EM system will fail if the formation resistivity is low or the resistivity profile is too complex. The attenuation of wave is severe and the transmission distance is short.^[2] Both methods can rarely be applied successfully to modern drilling; the third method, called acoustic telemetry, can potentially overcome the shortcomings of them.

In the acoustic telemetry system, the downhole information sensed by the sensor is converted into electrical signals, which are carried by acoustic stress waves. The waves propagate along the drill strings until the signals are picked up by a surface receiver. This transmission does not depend on the drilling fluid, and is not affected by the properties of the formation. Its advantages, such as simple device framework, low cost, and high data rate, means it has broad application prospects in underbalanced drilling conditions. Therefore, the acoustic telemetry technology has become a hot research point in oil field drilling telemetry. If the technology is successfully used in LWD, there are two problems that need to be solved. One is how to choose the carrier frequency. Previous studies^[3–6] investigated the transmission of acoustic waves along the drill pipes and drew the conclusion that the channel formed by the drill string has a comb filter structure with the passband and stopband showing up in turn. The other is how to select an efficient signal modulation/demodulation method and a signal detection technology.^[7–10] How the signal is successfully transmitted to the ground is critical for acoustic telemetry technology.

Binary phase shift keying (BPSK) is a digital two-phase modulation scheme which modulates carrier waves based on a binary baseband signal in acoustic telemetry. It possesses high anti-noise performance and bandwidth utilization efficiency. In Ref. [7] the acoustic telemetry tool schematic was studied by the XACT Institute. The voltage across the piexoelectric tranducer (PZT) stack was modulated to generate an acoustic carrier wave with data encoded via BPSK. Through field testing it was proved that real-time annular pressure data were consistently transmitted to the surface utilizing acoustic telemetry. A BPSK signal was successfully transmitted at a depth of 3083 ft (1 ft = 3.048 × 10⁻¹ m), with carrier frequencies being 830 Hz, 1030 Hz, and 1230 Hz.^[8] A bit rate of 30 bps could be achievable.

In the present paper, a new BPSK demodulation scheme based on Duffing chaos is investigated. Chaos is a form of nonlinear movement^[11,12] and the Duffing system describes the motion of the system of nonlinear elasticity, in which the driving force is sinusoidal. Therefore, it can be widely used in signal detection.^[13–20] In Ref. [13], a quantitative detection method for weak sinusoidal signals based on the sensitivity of chaotic parameters in a particular state was proposed. The simulation experiments showed that the immunity to noise was enhanced with the increase of amplitude detected with guaranteed precision. Authors in Ref. [15] studied the weak sinusoidal signal detection under strong ocean background noise based on the intermittent chaos Duffing oscillator using Hilbert transform. The results proved that SNR was increased by 4.4 dB compared with those obtained by using the basic methods. Authors in Ref. [16] studied the stochastic resonance characteristic of a two-dimensional (2D) Duffing oscillator under the adiabatic assumption. In large parameter conditions, it revealed the mechanism of signal detection by Duffing oscillator stochastic resonance. In Ref. [17] a new method of detecting weak pulse signals based on an extended-Duffing oscillator was proposed. The approach could effectively expand the frequency detection range for weak signal detection. Authors in Ref. [19] developed a stochastic averaging procedure for the Duffing–Rayleigh system by using the generalized harmonic functions. The results showed that the fractional order, the coefficient of the fractional derivative, the noise intensity, and the natural frequency played important roles in the response of the system.

Only the Holmes Duffing equation was extensively studied in the above references. In Ref. [21], the simulation showed when the order of the Duffing equation was increased, the sensitivity of the signal detection was enhanced and the working stability was better. Therefore, the equation is advanced, a high-order system is given in order to enhance the signal detection capability and it is realized through building a virtual circuit by using EWB in this paper. A new BPSK demodulation scheme is proposed based on the intermittent chaos phenomena of the high-order Duffing system. The approach helps to detect the BPSK signal transmitted from the bottom successfully.

The rest of the paper is organized as follows. In Section 2, the high-order Duffing system is described and the main parameters are given according to the simulation results. In Section 3, the detection principle of an acoustic telemetry signal based on the high-order Duffing system is studied. The results are analyzed in Section 4 and the conclusions are drawn in the last section.

The Holmes Duffing equation is described as follows:

2.1. Dynamic behavior analysis of the high-order Duffing system

Equation (2) is rewritten in the form of a state space equation:

Since r is a constant, the state of the system changes with parameter A. When A ≠ 0, the system presents complex dynamic states and goes through a periodic oscillation state, homoclinic-orbit state, bifurcation state, and lastly into the chaotic state. When A is equal to A_d (the bifurcation threshold), the state of the system changes from a critical chaotic state to a large periodic state. Further if the value of A is increased to a value that is slightly larger than the bifurcation threshold, the system reaches a large periodic state. In this paper, the damping ratio r is given as 0.5. The system angular frequency is 1 rad/s and the initial values are [x(0),x′(0)] = [0,0]. The simulation step is set to be 0.01. The simulation is carried out. The system bifurcation diagram is shown in Fig. 1, and the simulation results of the critical chaotic state and the large periodic state are shown in Fig. 2. According to these results, the bifurcation threshold is determined to be A_d = 0.7369.

Fig. 1.

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	Fig. 1. Bifurcation diagram of the system under consideration.

Fig. 2.

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	Fig. 2. Large periodic state when Ad = 0.7369: (a) the critical chaotic state, and (b) the large periodic state.

The simulation results show that extremely small changes of A can cause fundamental changes of the system state. When the system is driven to the critical chaotic state, if the value of A fluctuates around the threshold, the Duffing system would be driven to a chaotic state or large periodic state. It is intermittent chaos, which is the principle of signal detection based on the Duffing system.

Specifically, firstly, adjust the amplitude of the driving force to make the Duffing system go into the critical chaotic state. Secondly, add the BPSK signal as the perturbation of the driving force to the left of the Duffing equation. Finally, observe the change of the system state and thereby demodulate the BPSK signal based on the sensitivity of the Duffing oscillator to the perturbation.

2.2. Virtual circuit implementation of the high-order Duffing system

To enhance the sensitivity of signal detection, the Duffing equation is improved. The intermittent chaos phenomenon is simulated to obtain critical parameters. The Duffing system is realized through building a virtual circuit by using EWB. The schematic circuit diagram is shown in Fig. 3, in which ideal operational amplifiers, multipliers, resistances and capacitors are used. The value of the amplitude of the periodic driving force is set to be 18 V, and the frequency is set to be 1 Hz. By adjusting the circuit parameters, the possibility to implement such a physical system is investigated. The simulation results are shown in Fig. 4, and the system is at the chaotic state according to the X–Y diagram.

Fig. 3.

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	Fig. 3. Schematic circuit diagram of the high-order Duffing system.

Fig. 4.

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	Fig. 4. Simulation results of X–Y diagram.

Assume that the BPSK signal ‘0/1’ is sent from the bottom of the well, and set the system parameters as the above to make the system meet the intermittent chaos conditions.

Fig. 9.

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	Fig. 9. Timing simulation diagram while transmitting ‘0’ when t ∈ (0,600).

When the signal sent is ‘0’, the intermittent chaos phenomenon is shown in Fig. 9 with t ∈ (0,600). In order to demonstrate the system is chaotic in a certain time series, the dynamic behavior of the Lyapunov exponent is simulated. The result is shown in Fig. 10. When A > A_d, the system is converted from the chaotic state into the large periodic state; when A < A_d, the system is transformed from the large periodic state into the chaotic state. The dynamic behavior is consistent with the principles discussed above.

When t = 200 nearly, the system is converted from the chaotic state into the large periodic state. When t = 430 nearly, the system is transformed from the large periodic state into the chaotic state. The variables crossing the zero-point in the interval t ∈ (180,450) are recorded. The results are shown in Tables 1 and 2. It is found that the critical time point is t₁₁ = 189.08 when the system is transformed from the large periodic state into the chaotic state and the time point is t₁₂ = 443.72 when the system is converted from the chaotic state into the large periodic state. Using this method the amount of computation to find the critical time point is reduced. Substituting t₁₁ and t₁₂ into Eq. (11), with k = 1 and Δφ = − 0.05 ≈ 0, the signal sent is determined to be ‘0’.

Fig. 10.

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	Fig. 10. Dynamic behavior of the Lyapunov exponent while transmitting ‘0’.

Table 1.

Table 1.

Table 1. Variables crossing zero-point statistics when the system is transformed from the chaotic state into the large periodic state while transmitting ‘0’. . t 177.72 179.68 189.54 198.08 201.36 204.40 207.30 210.16 213.11 216.20 X(t) –0.001 –0.003 0.003 0.003 –0.003 –0.006 –0.001 0.001 0.001 0.003

Table 1. Variables crossing zero-point statistics when the system is transformed from the chaotic state into the large periodic state while transmitting ‘0’. .

Table 2.

Table 2.

Table 2. Variables crossing zero-point statistics when the system is converted from the large periodic state into the chaotic state while transmitting ‘0’. . t 420.50 423.71 426.92 430.16 433.42 436.73 440.12 443.72 491.89 493.83 X(t) –0.004 –0.004 –0.002 –0.004 0.002 –0.004 0.002 0.003 0.000 0.000

Table 2. Variables crossing zero-point statistics when the system is converted from the large periodic state into the chaotic state while transmitting ‘0’. .

Fig. 11.

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	Fig. 11. Timing simulation diagram while transmitting ‘1’.

Similarly, when the signal sent is ‘1’, the intermittent chaos phenomenon is shown in Fig. 11 with the interval t ∈ (0,600).

The variables crossing the zero-point in the interval t ∈ (350,650) are recorded. The results are shown in the following Tables 3 and 4. The critical time points are t₂₁ = 355.16 and t₂₂ = 600.80. By substituting t₂₁ and t₂₂ into Eq. (11), with k = 2 and Δφ = 3.00 ≈ π, the signal sent is determined to be ‘1’.

Table 3.

Table 3.

Table 3. Variables crossing zero-point statistics when the system transformed from the chaotic state into the large periodic state while transmitting ‘1’. . t 336.75 346.61 355.16 358.44 361.47 364.38 367.24 370.19 373.28 376.41 X(t) 0.001 –0.004 0.003 –0.003 0.005 0.000 0.000 0.002 0.003 –0.003

Table 3. Variables crossing zero-point statistics when the system transformed from the chaotic state into the large periodic state while transmitting ‘1’. .

Table 4.

Table 4.

Table 4. Variables crossing zero-point statistics when the system is converted from the large periodic state into the chaotic state while transmitting ‘1’. . t 584.00 587.24 590.50 593.80 597.20 600.80 648.96 650.91 660.77 669.32 X(t) –0.001 –0.004 0.002 0.005 0.002 0.003 0.000 0.000 –0.003 0.002

Table 4. Variables crossing zero-point statistics when the system is converted from the large periodic state into the chaotic state while transmitting ‘1’. .

The accuracy of the simulation is affected by the simulation step. The step cannot be exactly chosen so that the time for the variable to cross the zero point is picked up each time. In this paper, the time when the variable crosses the zero-point is determined by the absolute value of the variable. If the variable changes from a negative value to a positive value, the crossing zero-point is where its absolute value is closest to zero. Therefore, when variables of the crossing zero-point equidistance method are used to judge the two states of the system, there are errors and Δφ calculated is an approximate value. But the phases of the BPSK signal are just 0 and π, so the error can be limited in a permissible range. To reduce the error, a smaller simulation step should be used. As a result, the calculating increases accordingly.

In the present work, the dynamic behavior of a high-order Duffing system is studied. The virtual circuit is built, and the intermittent chaos phenomenon is investigated. Based on the above, a new demodulation method of an acoustic telemetry BPSK signal is proposed. The system variables crossing zero-point equidistance method is adopted to find t₁ and t₂ and the phase difference can be obtained from Eq. (11). Finally, the digital signal transmitted from the bottom of the well is determined to be ‘0’ or ‘1’. The simulation results show that the demodulation method is feasible. There is a lot of noise produced in the underground operation, but the Duffing system is immune to noise. Therefore, further studies are needed to find how to detect the BPSK signal from the noise using this method and to achieve the real applications of the downhole data detection.