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Marine riser is a key equipment in offshore drilling operation, and failure of the riser can lead to drilling moratorium; in severe cases, it may cause oil and gas leaks. In this paper, the time-dependent boundary conditions of the riser and the randomness of wave load are considered to improve the calculation efficiency and accuracy of the dynamic response of the jack-up riser. Based on the Euler–Bernoulli beam theory, an analytical method to determine the response of the jack-up riser subjected to the random wave load was established by the Mindlin–Goodman method in the frequency domain, and an experiment was carried out to verify it. The research shows that transverse dynamic response is the main component of the transverse response of the riser, and the method proposed is feasible to calculate the transverse response of the riser.

In this study, the jack-up riser is taken as the research object. As the main equipment of offshore oil exploration and development, the jack-up platform plays a leading role in the continental shelf waters. Jack-up platform which can decrease the lateral rigidity of the platform is designed to adapt to the growth of oil and gas demand in deeper water. Under the random environment loads, random dynamic response is produced by the platform. Although the marine riser is a key equipment in offshore drilling, it is a weak component. As the working depth of the platform increases, the dynamic response of the riser becomes more and more complicated, which may lead to operational accidents, cause huge economic losses, and even threaten the lives of operators and the marine environment. Therefore, it is necessary to study the dynamic response characteristics of the jack-up riser. In the sea area with the depth of 100-200 m, subsea blowout preventers (BOP) are used to drill well (Figure

The working conditions of the jack-up platform with subsea BOP.

For the transverse vibration of the marine riser, some scholars have studied the natural vibration characteristics and forced vibration response of the riser. Ignoring the variable tension force, Clauss et al. [

The above literature review presents that most scholars used the numerical methods to investigate the lateral vibration response of marine risers. In their research, the average axial force and harmonic functions are used to establish the mathematical model of risers, and the quasistatic method is mainly used to solve the lateral vibration response of risers. However, neglecting the randomness of load and boundary conditions will reduce the calculation accuracy of the vibration response. Based on the current research achievements, the average axial force is also applied in this paper to establish the mathematical model of a riser; but we consider the time-dependent boundary conditions of the riser and the randomness of the wave load in this paper. We propose an analytical method to determine the random lateral vibration response by the Mindlin–Goodman method. The research results can further improve the random vibration theory of beams; in terms of engineering application, this method can provide technical support to evaluate whether a special sea area and its sea conditions are available for the jack-up, can adjust the installation sequence of the riser, and improve the service life of the marine riser system. The rest of this paper is organized as follows: an analytical procedure and an experiment are elaborated in Section

The motion of a jack-up is treated as the time-dependent boundary condition of the riser in this research, thus it should be studied first. Figure _{e} is the equivalent mass of the platform, _{e} is the equivalent damping of the jack-up leg, _{e} is the bending stiffness of the jack-up leg, and

The sketch map of the jack-up drilling system.

The frequency response function of the platform is_{p} is the natural frequency of the platform. In this study, the Pierson–Moskowitz spectrum _{s} is the significant wave height of the wave, ^{2}/s^{4}. Based on the Morison equation and Borgman’s linearization method, the total wave force spectrum of one leg of a jack-up can be obtained by the integral along the direction of the water depth, as shown in the following equation:_{D} is the drag force coefficient, _{M} is the inertia force coefficient, _{o} is the outer diameter of the leg, ^{2} is the variance of horizontal speed of the water particle, which is the function of the height

Substituting equation (

According to the definition of power spectral density function and the relationship between autocorrelation function and power spectral density function, the total wave force spectrum of the platform _{p}(

By using equations (_{u}(

When a jack-up platform uses subsea BOP to drill an exploration well, the top end of the riser is connected to the platform by a ball joint and a tension system, and the bottom end of the riser is connected to the subsea drilling system through a ball joint. In our research, several assumptions are applied as follows: (1) joints of the top end and the bottom end are modelled as a hinge; (2) the geometry characteristics and material properties of the riser are considered constant; (3) linearly varying axial force is replaced by the average axial force. Based on the above assumptions, the Euler–Bernoulli beam theory is adopted to establish the transverse motion equation of the riser as follows (without considering the internal damping of material):_{m} is the average axial tensile force,

The time-dependent boundary conditions of the riser at

By using the Mindlin–Goodman method, the lateral deformation of the riser is decomposed into quasistatic displacement _{s}(_{d}(

The coefficient _{n} (_{n} (

The corresponding boundary conditions for the riser are expressed in the following equations:

Substituting equation (

Based on equation (

The substitution of equations (

The transverse dynamic displacement of the riser satisfies the homogeneous boundary condition, which makes the mode shape function of the riser conform to the following equation:

Substituting equation (

By applying the orthogonality conditions, equation (

Then, the power spectral density functions of the transverse dynamic displacement can be formulated as

The cross spectral density function between

By using the definition of autocorrelation function and the Wiener–Khinchin principle, the power spectral density functions of the quasistatic displacement is derived as follows:

Based on the above analysis, the power spectral density function of the beam’s transverse displacement can be obtained by the following equation:

According to the relationship between displacement and stress, the frequency response function of bending stress response

On the basis of equations (_{d}(

According to equations (_{s}(

Combining equations (

According to the summation formula of spectrum, the power spectral density function of the bending stress of the riser can be obtained by combining equations (

Based on the Mindlin–Goodman method, we propose an analytical method to obtain the random dynamic response of the Euler beam with axial load and time-dependent boundary conditions. Subsequently, an experiment is carried out to determine the effectiveness of our method. Overall scheme of the experiment is (a) solve the dynamic stress response of the Euler beam under the sinusoidal constant frequency excitation based on our proposed method. (b) The stress response of the beam resulted from the above sinusoidal excitation is tested by the experimental system. (c) Make a contrastive analysis of the stress response data obtained by the proposed method and the experimental test. Figure

Diagram of the experiment system. 1, computer; 2, data analyzer; 3, power amplifier; 4, vibration exciter; 5, beam; 6, strain gage.

In order to better observe the experimental phenomena and collect more accurate experimental data, we constantly adjust the frequency and amplitude of the sine constant frequency excitation and then choose the following expression

The beam in the experiment is fixed at

The static influence function of the beam can be obtained by the initial parameter method, and the function of the beam’s pseudostatic displacement can be derived according to equation (

Then, the modal coordinate equation of the beam can be derived as

The material of the beam is 45 carbon steel, and the diameter of the beam is 6 mm. Since the series in equations have a fast convergence rate, the first five orders of the series are enough for the analysis. Therefore, the first five order natural frequencies of the beam and weight coefficients are calculated first, which are listed in Tables

First five order natural frequencies of the beam in the experiment.

1 | 2 | 3 | 4 | 5 | |
---|---|---|---|---|---|

_{n} (rad/s) | 82.43 | 329.9 | 742.34 | 1319.8 | 2062.2 |

First five order load coefficients of the beam in the experiment.

1 | 2 | 3 | 4 | 5 | |
---|---|---|---|---|---|

Ψ_{n} | −0.0033 | −0.0066 | −0.0101 | 0.0444 | 0.0547 |

Based on the above analysis, the total transverse displacement of the beam in the experiment is

From the relationship between displacement and stress, the stress response of the beam can be obtained as

As the beam is cylindrical, its curved surface causes initial deformation of the strain gauge; therefore, this part of stress caused by the deformation should be subtracted from the original data. To ensure a fixation at the bottom, we installed a pin at the connection between the beam and the base, which generates bending stress; this part of stress caused by installation should be removed from the initial stress too. In Figure

Stress response curves of the beam.

The stress response curve obtained by experiment basically coincides with the theoretical calculation data in the main vibration response time period of the beam, which demonstrates that the proposed method is effective in solving the random response of a jack-up riser. Engineers can use our method to conduct a preliminary study of the marine riser, and if they want to accurately obtain the dynamic response of the riser at a certain position, they need to establish the motion equation at that point.

We take a typical riser with the length of 22.86 m and an external diameter of 1.372 m as the example, and the dynamic response of the riser with subsea BOP under the random wave load is investigated in our research. The parameters of the platform are _{e} = 6.48 × 10^{6} kg, _{e} = 4.71 × 10^{6} N/m, _{e} = 8.77 × 10^{5} N m/s, _{o} = 3.62 m, ^{8} N m^{2}, _{m} = 3 × 10^{3} N, _{s} = 10 m, _{D} = 2.0, _{M} = 2.0, ^{3}, and ^{2}.

The first five order generalized load coefficients of the riser are calculated and listed in Table

Generalized load coefficients of the riser with subsea BOP.

1 | 2 | 3 | 4 | 5 | |
---|---|---|---|---|---|

_{n} | 0.6366 | −0.3183 | 0.2122 | −0.1592 | 0.1273 |

Stress response spectrum caused by the transverse dynamic responses (_{s} = 10 m).

Real part of the stress cross spectrum (_{s} = 10 m).

The total stress spectrum (_{s} = 10 m).

Based on the above research results, the mean square deviation of the random displacement response and the stress response of the riser are calculated and depicted in Figures

Standard deviation of the displacement response with the subsea BOP (_{s} = 4.52 m).

Standard deviation of the stress response with the subsea BOP (_{s} = 4.52 m).

In this paper, a frequency domain analysis method is proposed to solve the random dynamic response of the jack-up riser with subsea BOP and time-dependent boundary conditions based on the Mindlin–Goodman method. This method can also be used to calculate the lateral response of the Euler beam with other boundary conditions. In addition, an experimental system was established and a response test experiment of the Euler beam was finished. The experimental results show that the method is effective for solving the transverse vibration response of the Euler beam. Through the case study, it is found that the lateral dynamic response of the jack-up riser is dominated by the transverse dynamic responses, the minimum displacement response amplitude appears in the middle of the riser system, and the maximum stress response amplitude occurs at both ends of the riser system.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

The authors acknowledge the Ministry of Industry and Information Technology of the PR China for supporting this study through the project “Research on key technologies of integrated dismantling equipment for ultra-large offshore oilfield facilities” with the grant number 0300/05M1903005A.