Improved Fatigue Reliability Analysis of Deepwater Risers Based on RSM and DBN

Abstract

The fatigue reliability assessment of deepwater risers plays an important role in the safety of oil and gas development. Physical-based models are widely used in riser fatigue reliability analyses. However, these models present some disadvantages in riser fatigue reliability analyses, such as low computational efficiency and the inability to introduce inspection data. An improved fatigue reliability analysis method was proposed to conduct the fatigue reliability assessment of deepwater risers. The data-driven models were established based on response surface methods to replace the original physical-based models. They are more efficient than the physics-based model, because a large number of complex numerical and iterative solutions are avoided in fatigue reliability analysis. The annual crack growth model of the riser based on fracture mechanics was established by considering the crack inspection data as a factor, and the crack growth dynamic Bayesian network was established to evaluate and update the fatigue reliability of the riser. The performance of the proposed method was demonstrated by applying the method to a case. Results showed that the data-driven models could be used to analyze riser fatigue accurately, and the crack growth model could be performed to analyze riser fatigue reliability efficiently. The crack inspection results update the random parameters distribution and the fatigue reliability of deepwater risers by Bayesian inference. The accuracy and efficiency of fatigue analysis of deepwater risers can be improved using the proposed method.

1. Introduction

Deepwater risers are important equipment connecting the floating platform with subsea wellhead during offshore oil and gas development. Their main function is to provide fluid passage between the surface platform and subsea wellhead in drilling and production processes [1]. Deepwater risers are sensitive to dynamic loads, and fatigue damage due to ocean waves and movements of the floating platform is one of their main failure modes. The long term accumulation of fatigue damage leads to the fatigue fracture of risers and even leads to major accidents [2]. Several accidents, including oil leaks and subsea wellhead breakage, have occurred due to riser fatigue [3,4]. Consequently, studying the fatigue reliability of risers during drilling and production processes is essential.

Fatigue analyses are the basis of fatigue reliability analysis for deepwater risers. Physical-based models are widely used in riser fatigue analyses at present. The fatigue analysis schemes used by these models can be divided into frequency and time domain methods. The frequency domain method is rarely used because of possible linearization errors in the calculation. By contrast, the time domain method is often employed in riser fatigue analyses due to its high computational accuracy. In the time domain method, the dynamic response of the riser system is first analyzed using the finite element method, the finite difference method, and other approaches [5,6]. Second, the dynamic response is processed through the rain flow counting method. Lastly, fatigue damage is calculated based on the S–N curve or fracture mechanics [7,8]. The time domain fatigue of deepwater risers, including drilling, steel catenary, and flexible risers, has been widely investigated in previous studies.

Research on riser fatigue reliability has mainly focused on catenary and top-tensioned risers. The process of fatigue reliability analysis is relatively complex. Generally, the assessment of the fatigue capacity and fatigue failure of risers is conducted by applying two different methodologies, namely, crack growth rate curves using the fracture mechanics model and S–N curves using the Palmgren–Miner damage model [9,10]. During the fatigue reliability analysis of risers, the randomness of general dynamic model parameters, such as wall thickness, the additional mass coefficient, and the drag force coefficient, is often considered [11,12,13,14]. The uncertainty of the fatigue performance parameters of the S–N curve method and that of the Paris parameters of the fracture mechanics method are also considered [15,16,17,18]. In particular, the uncertainty of geotechnical model parameters and the nonlinear behavior of the pipe–seabed interaction are also considered when evaluating the fatigue reliability of catenary risers in the contact area [19,20,21]. The uncertainty of these parameters results in the uncertainty of riser fatigue. Fatigue assessment using certain parameters only cannot meet the needs of safety design and riser assessment. The fatigue reliability of risers must be assessed reasonably based on statistical theory [22]. Several practical reliability analysis approaches, including numerical simulation and analytical algorithms, have been developed for riser fatigue reliability assessment. The Monte Carlo method belongs to the former type of reliability analysis, whereas the first order and second order reliability methods belong to the latter type. The three methods are widely used, and each has its own advantages and disadvantages. The Monte Carlo method has the highest solving accuracy but the lowest solving efficiency, and the first order reliability method has a solving performance that is opposite to that of the Monte Carlo method; the second order reliability method is relatively mediocre when compared with the two other methods [23,24].

In previous research, physical-based models were widely used for fatigue and reliability analyses of risers. However, the calculation for single fatigue analysis using physical-based models is time consuming [25]. The calculation cost of fatigue reliability analysis using physical-based models increases significantly with the increase in the number of random parameters and the degree of random distribution of the parameters. These reasons account for the low efficiency of the fatigue reliability analysis of risers. Therefore, some scholars have attempted to improve the efficiency of fatigue reliability analysis by reducing the time of single fatigue analysis or the scale of fatigue analysis [26,27]. Data-driven models have been introduced as substitutes for time consuming physical-based models [25,28,29,30]. Notably, risers need to be inspected in a fixed time to guarantee their safety, and using the inspection results to update the fatigue reliability is essential to the safety of risers in the process of drilling and production. A problem at present is the difficulty of introducing inspection results into riser fatigue reliability analysis. The dynamic Bayesian network (DBN) is an optional method for solving this difficult problem. The DBN has increasingly been applied to probability analysis because of its functions of forward and backward analyses [31]. Fatigue reliability can be calculated based on the prior probabilities of root nodes in forward analysis, and the posterior reliability can be updated based on a few observations. Thus, the DBN was selected for riser fatigue reliability analysis and updating based on the inspection results in this study.

In this study, data-driven models based on the RSM were established to substitute for physical-based models in the fatigue analysis of risers. The direct mapping between riser dynamic analysis parameters and fatigue damage was established by the data-driven models, which can greatly improve the efficiency of fatigue analysis. The annual fatigue crack growth model was then established based on fracture mechanics theory and data-driven models that consider the crack inspection data as a factor. The crack growth DBN model was then established to obtain the fatigue reliability of the risers. The crack inspection data of the risers were introduced into the DBN model to update the fatigue reliability of the risers.

The rest of the paper is organized in the following manner. The riser fatigue reliability analysis methodology is introduced in Section 2. The physical-based models are introduced in Section 3. The data-driven models established by RSM are introduced in Section 4, and the fatigue reliability analysis method based on DBN is introduced in Section 5. A case is introduced for fatigue reliability analysis and updating in Section 6. The conclusions are summarized in Section 7.

2. Methodology

The methodology for the fatigue reliability analysis of risers is illustrated in Figure 1. Physical-based models were used to establish fatigue datasets of risers based on the S–N curve method. Data-driven models were established based on the fatigue datasets as a replacement for physical-based models to improve fatigue analysis efficiency. However, the fatigue response surface model based on S–N curve method was unsuitable for fatigue reliability analysis and updating of a cracked riser. The riser annual fatigue crack growth model based on fracture mechanics theory was established. Then, fatigue reliability analysis was performed by DBN method based on the fatigue crack growth model and inspection results. Bayesian forward inference was adopted for riser fatigue reliability analysis, and Bayesian backward inference was applied for fatigue reliability updating based on the inspection results. The physical-based models, data-driven models, and crack growth DBN model are introduced in detail in Section 3, Section 4 and Section 5, respectively.

3. Physical-Based Models

Figure 2 shows a deepwater drilling platform/riser coupling system that comprises a platform, a drilling riser system, and tensioners. The top end of the riser system is hung on the drilling platform via the tensioners and upper flex joint, and the bottom end of the riser system is connected to the Blowout preventer (BOP), the subsea wellhead, and the conductor via the lower marine riser package (LMRP). The riser system vibrates transversely and axially under environment loads and platform motions.

The transverse vibration equation of the riser system can be written as [32,33]

$$m(x)\frac{{\partial }^{2}y}{\partial t^2}+c\frac{\partial y}{\partial t}+\frac{{\partial }^2}{\partial x^2}\left(EI(x)\frac{{\partial }^{2}y}{\partial x^2}\right)-\frac{\partial }{\partial x}\left(T(x)\frac{\partial y}{\partial x}\right)=F_{\mathrm{sea}}(x,t)$$

(1)

where m is the mass per unit length of the riser, c is the damping coefficient, E is the elastic modulus of the riser, I(x) is the second moment of area, T(x) is the effective axial tension, and F_sea(x,t) is the marine load applied per unit length of the riser.

The axial vibration equation of the riser system can be written as [32,33,34]

$$\frac{\partial }{\partial x}\left(EA(x)\frac{\partial u(x,t)}{\partial x}\right)dx-c\frac{\partial u(x,t)}{\partial t}-m\frac{{\partial }^{2}u(x,t)}{\partial t^2}=F+mg$$

(2)

where u(x,t) is the axial vibration, F is the axial hydrodynamic load on the riser system, and A(x) is the area of the riser cross-section.

The effective axial tension, T(x), can be expressed as

$$T(x)=T_{\mathrm{top}}-{\displaystyle \underset{0}{\overset{x}{\int }}(F+mg)dx}$$

(3)

where T_top is the tension at the top of the riser system, and L is the length of the riser system.

The marine load applied per unit length of the riser, F_sea(x,t), can be written as

$$F_{\mathrm{sea}}(x,t)=\frac{\pi }{4}\rho C_{\mathrm{M}}{D}_{\mathrm{h}}{}^2{\dot{u}}_{\mathrm{w}}-\frac{\pi }{4}\rho C_{\mathrm{a}}{D}_{\mathrm{h}}{}^2\ddot{y}+\frac{1}{2}\rho D_{\mathrm{h}}{C}_{\mathrm{d}}(u_{\mathrm{w}}+u_{\mathrm{c}}-\dot{y})|u_{\mathrm{w}}+u_{\mathrm{c}}-\dot{y}|$$

(4)

where ρ is the density of seawater, C_M is the coefficient of inertia force, C_a is the coefficient of additional mass (C_a = C_M-1), D_h is the hydrodynamic outer diameter, C_d is the coefficient of drag force, u_w is the water point velocity, ${\dot{u}}_{\mathrm{w}}$ is the water point acceleration, u_c is the current velocity, $\dot{y}$ is the velocity of the riser, and $\ddot{y}$ is the acceleration of the riser.

The top end of the risers is connected to the platform by the upper flex joint, and the bottom end is connected to the LMRP via the lower flex joint, as is shown in Figure 2. Therefore, the top and bottom boundary conditions of the riser system are expressed as [35]

$$\{\begin{array}{l}\begin{array}{l}u(x,t){|}_{x=0}=S_{\mathrm{px}}(t)\\ y(x,t){|}_{x=0}=S_{\mathrm{py}}(t)\end{array}\hfill \\ EI(x)\frac{{\partial }^{2}y(x,t)}{\partial x^2}|{}_{x=0}=K_{\mathrm{u}}\frac{\partial y(x,t)}{\partial x}|{}_{x=0}\hfill \end{array}$$

(5)

$$\{\begin{array}{l}\begin{array}{l}u(x,t){|}_{x=L}=0\\ y(x,t){|}_{x=L}=0\end{array}\hfill \\ EI(x)\frac{{\partial }^{2}y(x,t)}{\partial x^2}|{}_{x=L}=K_{\mathrm{b}}\frac{\partial y(x,t)}{\partial x}|{}_{x=L}\hfill \end{array}$$

(6)

where K_b is the rotation stiffness of the lower flexible joint, K_u is the rotation stiffness of the upper flexible joint, L is the length of the risers, S_px(t) is the floating platform’s heave displacement, and S_py(t) is the floating platform’s drift distance. S_px(t) and S_py(t) can be calculated by Equation (7) [36].

$$S_{\mathrm{p}}(t)={\displaystyle \sum _{n=0}^{N\mathrm{w}}{S}_{\mathrm{n}}\cos (k_{\mathrm{n}}{S}_{\mathrm{p}}(t)-{\omega }_{\mathrm{n}}t+{\varphi }_{\mathrm{n}}+{\alpha }_{\mathrm{n}})}$$

(7)

where S_n is a single wave’s amplitude, N_w is the number of random waves, k_n is the number of constituent waves, ω_n is the wave frequency, φ_n is the initial phase angle of the wave, and α_n is the phase angle between the platform movement and wave.

On the basis of the motion equation of the riser system, the finite element method was used to establish the finite element analysis model of the riser system [37,38]. The whole dynamical equation of the riser system is written as

$$[M]\left\{\ddot{u}\right\}+[C]\left\{\dot{u}\right\}+[K]\left\{u\right\}=\left\{F\right\}$$

(8)

where {u} is the displacement vector of the riser system; [M], [C], and [K] are the mass, damping, and stiffness matrices of the riser system, respectively; and {F} is the load vector of the riser system.

The dynamic response of the risers was obtained by solving the whole motion finite element model of the riser system. The rain flow counting method was then used to count the fatigue stress time history and obtain the stress cycle and amplitude [7,8].

$$D_{\mathrm{L}}=\frac{1}{n_{\mathrm{f}}}=\frac{{\displaystyle \sum {\left(\Delta {\sigma }_{\mathrm{i}}\right)}^{m_{\mathrm{f}}}}}{C}$$

(9)

where D_L is the long term fatigue damage, n_f is the fatigue life, and C and m_f are the S–N curve method parameters.

4. Data-Driven Models

4.1. RSM

The numerical simulation of the nonlinear dynamic equation of the risers in Equation (8) is complex, which is the main reason a fatigue analysis based on physical-based models is time consuming. The application of data-driven models based on the RSM in the fatigue damage calculation of risers can solve the low efficiency problem of physical-based models. In practical application, the response surface is often constructed by polynomials of different orders, which is the most common application of the RSM. Inaccurate calculations will occur if lower order polynomial functions are used to deal with complex nonlinear problems. However, the calculation efficiency will be reduced due to the need to determine multiple parameters and conduct multiple tests if a higher order response surface equation is established. Considering the above two factors, the response function of the quadratic polynomial was used to establish the riser fatigue data-driven models in this paper, because it can solve nonlinear problems and reduce the number of experiments [28]. The second order response surface model is written as

$$D_{\mathrm{L}}={\beta }_0+{\displaystyle \sum _{\mathrm{i}=1}^k{\beta }_{\mathrm{i}}{x}_{\mathrm{i}}}+{\displaystyle \sum _{\mathrm{i}=1}^k{\beta }_{\mathrm{ii}}{x}_{\mathrm{ii}}{}^2}$$

(10)

where β is the coefficient of the response surface, x refers to random parameters, and k is the number of parameters for response surface fitting.

4.2. Fatigue Crack Growth Modeling

The fatigue damage of risers is often calculated based on the S–N curve. However, riser inspection results are generally represented as fatigue crack size. Using the inspection results to update riser fatigue reliability is difficult when the fatigue damage is calculated based on the S–N curve. Riser fatigue damage based on the S–N curve needs to be transformed into fatigue damage based on fracture mechanics. Fracture mechanics theory is a mathematical method to study the functional relationship among crack shape, load magnitude, and number of cycles. The growth of fatigue cracks for cyclic loading is commonly described by the Paris law. The crack growth rate is defined as follows [39]

$$\frac{da}{dN}=C_0{(\Delta K)}^m$$

(11)

where a is the crack size, N is the cycle number, C₀ and m are fracture mechanics method parameters, and ΔK is the amplitude of the stress intensity factor.

The amplitude of the stress intensity factor describes the stress condition around crack tips, and it can be written as

$$\Delta K=Y(\zeta )\Delta \sigma \sqrt{\mathsf{\pi }a}$$

(12)

where Y(ζ) is a geometric function related to crack size, and Δσ is the applied stress range.

Risers are generally subjected to a combination of axial tensile and bending loads. A riser crack is considered a type I crack, because the load direction is perpendicular to the crack surface, and the upper and lower sides of the crack open relative to each other under the combined load. The geometric function Y(ζ) is shown as [40,41]

$$Y\left(\zeta \right)=1.122-1.4\left(\zeta \right)+7.33{\left(\zeta \right)}^2-13.08{\left(\zeta \right)}^3+14{\left(\zeta \right)}^4$$

(13)

where ζ is the ratio of crack depth a of the riser to wall thickness t.

The crack growth due to a single cycle load is described as

$$\frac{da}{dN}=C_0{(Y(\zeta )\Delta \sigma \sqrt{\pi a})}^m$$

(14)

The amount of crack growth in a year is written as

$$a_{\mathrm{yn}}={\displaystyle \sum _{\mathrm{i}=1}^k{C}_0{\left(Y\left(\xi \right)\Delta {\sigma }_{\mathrm{i}}\sqrt{\pi a_{\mathrm{y}-1}}\right)}^m}$$

(15)

where k is the cycle load number in a year.

The geometric function is assumed to be constant in one year. The crack growth in a year is then simplified as

$$a_{\mathrm{yn}}=C_0{\left(Y\left(\xi \right)\sqrt{\pi a_{\mathrm{y}-1}}\right)}^m{\displaystyle \sum _{\mathrm{i}=1}^k{\left(\Delta {\sigma }_{\mathrm{i}}\right)}^m}$$

(16)

On the basis of the S–N curve and the data-driven model in Equation (10), the relationship between ∆σ and the established fatigue data-driven models can be obtained by substituting Equation (10) into Equation (9) as follows:

$$D_{\mathrm{L}}=\frac{1}{n_{\mathrm{f}}}=\frac{1}{C}{\displaystyle \sum _{\mathrm{i}=1}^k{\left(\Delta {\sigma }_{\mathrm{i}}\right)}^{m_{\mathrm{f}}}}={\beta }_0+{\displaystyle \sum _{\mathrm{i}=1}^k{\beta }_{\mathrm{i}}{x}_{\mathrm{i}}}+{\displaystyle \sum _{\mathrm{i}=1}^k{\beta }_{\mathrm{ii}}{x}_{\mathrm{ii}}{}^2}$$

(17)

S–N curve parameter m_f in Equation (9) serves as an intermediate calculation variable and has the same value as fracture mechanics parameter m. Therefore, the sum of the stress amplitude is shown as

$${\displaystyle \sum _{\mathrm{i}=1}^k{\left(\Delta {\sigma }_{\mathrm{i}}\right)}^m}=C\left({\beta }_0+{\displaystyle \sum _{\mathrm{i}=1}^k{\beta }_{\mathrm{i}}{x}_{\mathrm{i}}}+{\displaystyle \sum _{\mathrm{i}=1}^k{\beta }_{\mathrm{ii}}{x}_{\mathrm{ii}}{}^2}\right)$$

(18)

The annual crack growth can be obtained by substituting Equation (18) into Equation (16) as follows:

$$a_{\mathrm{yn}}=C_0{\left(Y\left(\xi \right)\sqrt{\pi a_{y-1}}\right)}^{m}C\left({\beta }_0+{\displaystyle \sum _{\mathrm{i}=1}^k{\beta }_{\mathrm{i}}{x}_{\mathrm{i}}}+{\displaystyle \sum _{\mathrm{i}=1}^k{\beta }_{\mathrm{ii}}{x}_{\mathrm{ii}}{}^2}\right)$$

(19)

Crack depth is the sum of all crack extension depths, and it is expressed as

$$a_{\mathrm{y}}=a_{\mathrm{y}-1}+a_{\mathrm{yn}}$$

(20)

where a_y is the crack depth in the current year, and a_y-1 is the crack depth in the previous year.

The crack size increases with time in the process of crack growth. When crack size reaches its failure limit, riser failure occurs. The failure criterion is described as fatigue crack depth being greater than the calculated critical crack size of the risers. The corresponding fatigue crack limit state equation is shown as

$$Z=a_{\mathrm{c}}-a_{\mathrm{y}}$$

(21)

where Z is the fatigue failure function, and a_c is the critical size of riser fatigue crack failure.

Generally, the Level 2A critical assessment criterion in BS 7910 [42] is used to obtain the critical size a_c when material information is complete, and it is expressed as

$$\{\begin{array}{ll}{K}_{\mathrm{r}}=(1-0.14L_{\mathrm{r}}^2)\{0.3+0.7{\mathrm{e}}^{(-0.65L_{\mathrm{r}}^6)}\}\hfill & L_{\mathrm{r}}\le L_{\mathrm{rmax}}\hfill \\ K_{\mathrm{r}}=0\hfill & L_{\mathrm{r}}>L_{\mathrm{rmax}}\hfill \end{array}$$

(22)

where K_r is the fracture ratio, L_r is the ratio of applied load to yield load, and L_rmax is the permitted limit of L_r.

These parameters are defined in detail as

$$L_{\mathrm{rmax}}=\frac{{\sigma }_{\mathrm{Y}}+{\sigma }_{\mathrm{u}}}{2{\sigma }_{\mathrm{Y}}}$$

(23)

where σ_Y is the yield stress, and σ_u is the ultimate stress of the material.

$$L_{\mathrm{r}}=\frac{{\sigma }_{\mathrm{ref}}}{{\sigma }_{\mathrm{Y}}}$$

(24)

where σ_ref is the reference stress.

$$K_{\mathrm{r}}=\frac{K_{\mathrm{I}}}{K_{\mathrm{mat}}}$$

(25)

where K_I is the stress intensity factor, and K_mat is the material fracture toughness (if a valid K_Ic is available, K_mat should be taken as K_Ic).

The reference stress, σ_ref, is defined as

$${\sigma }_{\mathrm{ref}}=\frac{P_{\mathrm{m}}\{\pi (1-\frac{a}{t})+2(\frac{a}{t})\sin (\frac{c_{\mathrm{L}}}{r_{\mathrm{m}}})\}}{(1-\frac{a}{t})\{\pi -(\frac{c_{\mathrm{L}}}{r_{\mathrm{m}}})(\frac{a}{t})\}}+\frac{2P_{\mathrm{b}}}{3{(1-a^{″})}^2}$$

(26)

where P_m is the primary membrane stress, P_b is the primary banding stress, r_m is the riser mean radius, a is the crack depth, c_L is the half of crack length, and $a^{″}$ is written as

$$\{\begin{array}{cc}{a}^{″}=\frac{\frac{a}{t}}{\{1+(\frac{t}{c_{\mathrm{L}}})\}}& \pi r_{\mathrm{m}}\ge c_{\mathrm{L}}+t\\ a^{″}=(\frac{a}{t})(\frac{c_{\mathrm{L}}}{\pi r_{\mathrm{m}}})& \pi r_{\mathrm{m}}<c_{\mathrm{L}}+t\end{array}$$

(27)

The stress intensity factor in the Level 2 critical assessment criterion is written as

$$K_{\mathrm{I}}=\{{(Y\sigma )}_{\mathrm{p}}+{(Y\sigma )}_{\mathrm{s}}\}\sqrt{\pi a}$$

(28)

where (Yσ)_p and (Yσ)_s represent contributions from primary and secondary stresses, respectively.

$$\begin{array}{l}{(Y\sigma )}_{\mathrm{p}}=M_1{f}_{\mathrm{w}}[k_{\mathrm{tm}}{M}_{\mathrm{km}}{M}_{\mathrm{m}}{P}_{\mathrm{m}}+k_{\mathrm{tb}}{M}_{\mathrm{kb}}{M}_{\mathrm{b}}\{P_{\mathrm{b}}+(k_{\mathrm{m}}-1)P_{\mathrm{m}}\left\}\right]\\ {(Y\sigma )}_{\mathrm{s}}=M_{\mathrm{m}}{Q}_{\mathrm{m}}+M_{\mathrm{b}}{Q}_{\mathrm{b}}\end{array}$$

(29)

where Q_m is the secondary membrane stress, and Q_b is the secondary bending stress. For the value of parameters M₁, f_w, M_m, M_b, M_km, M_kb, k_tm, k_tb, and k_m, readers may refer to BS7910.

5. Dynamic Bayesian Network

The Bayesian network (BN) is utilized to analyze the fatigue reliability of risers due to its forward and backward inference capability. The BN is a combination of probability and graph theories and is widely used in data analysis [43]. The structure of a BN model is clearly represented by graph theory, and the problems to be solved are analyzed by probability theory [44,45,46]. A basic BN is shown in Figure 3.

Each node is attached to a probability distribution. Root nodes A and B are attached to their marginal probability distributions P(A) and P(B), respectively. The non-root node X (C, D, E) is attached to the conditional probability distribution P(X|pa(X)), and pa(X) is the father node of node X [47]. A logical analysis is performed by analyzing the relationship between nodes in the Bayesian model.

Prior and posterior probabilities are relative to a certain set of evidence in Bayesian theory. In the presence of two different random variables A and D, A = a is an event, and D = d is evidence. The probability estimation P(A = a) of event A = a before considering evidence is called prior probability. Conversely, the probability estimation P(A = a|D = d) of event D = d after considering evidence is called posterior probability. The relation between prior and posterior probabilities is described as

$$P(A=a|D=d)=\frac{P(A=a)P(D=d|A=a)}{D=d}$$

(30)

where P (D = d A = a) is the likelihood of event A = a and can be called L (A = a| D = d). In reality, the likelihood of the event is easier to obtain than the posterior probability.

The DBN is an extension of static Bayesian network modeling in the time dimension. The state of the DBN is updated over time to analyze dynamic stochastic processes. Some assumptions need to be made before establishing a DBN model:

• The conditional probability is uniformly stable for all variable t over a finite period of time.
• The DBN is a Markov process. Future probabilities are determined by present probabilities and not by past probabilities, namely, P(Y[t+1]|Y[1],Y[2],…,Y[t]) = P(Y[t+1]|Y[t]).
• The conditional probability of adjacent time slices is stable. P(Y[t+1]|Y[t]) is independent of time.

The DBN is composed of a prior network and a transfer network. The prior network is a joint probability distribution defined at the initial state. The transfer network is the conditional transition probability P(Y[t+1]|Y[t]) defined by the variables Y[t] and Y[t+1]. The DBN is shown in Figure 4.

A DBN model of riser fatigue crack was established in this study to analyze the fatigue reliability of risers, as is shown in Figure 5. The crack depth in Equation (21) is the parent node, and the other variables are the child nodes. The distribution of fatigue crack size can be calculated through forward Bayesian inference. The inspection results can also be introduced to the DBN model, and the influence of the inspection results on riser failure probability is examined using backward Bayesian inference.

6. Case Analysis

6.1. Response Surface Analysis

A riser system in the South China Sea was employed as an example for case analysis. The dynamic response analysis parameters are listed in

Table 1. Dynamic response analysis parameters.

Parameters	Value
Length of each deepwater riser	22.86 m
Length of deepwater riser system	1531.62 m
Riser outer diameter	0.5334 m
Wall thickness	22.225 mm
Top tension	6 MN
Element number of single riser	20
Time step size	0.02

. The wave conditions for the fatigue analysis are listed in

Table 2. Wave conditions.

Sea condition number	1	2	3	4	5	6
Wave height (m)	1.23	2.58	3.91	4.95	5.95	6.53
Wave period (s)	3.5	4.5	5.5	6.5	7	7.5
Probability of occurrence (%)	3.99	13.48	38.32	32.56	10.64	1.01

. The response amplitude operator (RAO) was used for platform motion simulation and is shown in Figure 6.

The physical-based equations shown in Equation (8) are often solved with the generalized α method, the Newmark β method, or other numerical integral methods. Some software programs, such as Orcaflex and Deepriser, have also been developed especially for riser dynamic analysis. Orcaflex was used for the current analysis due to its ability to analyze the dynamic response and calculate the fatigue damage of risers. The fatigue damage of the deepwater riser system is shown in Figure 7.

It can be seen from Figure 7 that a large portion of the fatigue damage was concentrated at the top of the riser system. The main reason is that the top of the riser system is directly subjected to dynamic wave loads. The following fatigue sensitivity and reliability analysis was conducted at the largest fatigue point. Riser wall thickness t, additional mass coefficient C_a, S–N curve coefficient C, and drag coefficient C_d were adopted as random parameters. The distributions, mean values, and coefficients of variation (COVs) of these random parameters are listed in

Table 3. Random parameters distributions.

Parameters	Distribution	Mean (μ)	COV
Wall thickness t	Normal	22.225 mm	0.05
Additional mass coefficient Ca	Lognormal	1	0.05
S–N curve coefficient C	Lognormal	1.023 × 1012	0.05
Drag force coefficient Cd	Normal	1	0.05

[15,48].

μ_Xi-1.5σ_Xi and μ_Xi+1.5σ_Xi were adopted as critical values of each parameter X_i. One of the parameters was set to μ_Xi±1.5σ_Xi, and the others were set to be the mean value in one calculation round. The fatigue damage of the risers under two critical values was calculated. The other parameters were calculated in the same manner. The standard deviation of riser fatigue under each parameter is shown as follows

$${\sigma }_{X_{\mathrm{i}}}=\sqrt{{{\displaystyle \sum \left(\frac{\partial D_{\mathrm{L}}}{\partial x_{\mathrm{i}}}\right)}}^2{\sigma }_{x_{\mathrm{i}}}{}^2}$$

(31)

The sensitivity of riser fatigue to each parameter is expressed as

$$I_{\mathrm{i}}=\frac{{\left(\frac{\partial D_{\mathrm{L}}}{\partial x_{\mathrm{i}}}\right)}^2{\sigma }_{x_{\mathrm{i}}}^2}{D_{\mathrm{L}}^2}$$

(32)

The sensitivity of riser fatigue considering t, C_a, C, and C_d as random parameters and the others as the mean values, respectively, is shown in Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12.

The parameter sensitivity analysis results showed that wall thickness t had the greatest influence on fatigue damage, followed by additional mass coefficient C_a, S–N curve parameter C, and drag coefficient C_d. Drag coefficient C_d and S–N curve parameter C had an almost negligible influence on the fatigue damage of risers relative to the two other parameters. Therefore, wall thickness t and additional mass coefficient C_a were employed as the parameters for establishing the response surface model of fatigue damage. The random variable μ±hσ was also chosen, with h being 1.5. The parameters are shown in

Table 4. Data for the response surface model of fatigue.

Variable number	1	2	3	4	5
Wall thickness t (mm)	22.225	20.558	20.558	23.892	23.892
Additional mass coefficient Ca	1	0.925	1.075	0.925	1.075
Fatigue damage	0.002476	0.003056	0.003839	0.001683	0.002266

[49].

A fatigue damage data-driven model was established using the polynomial method shown in Equation (33) to determine the relationship between riser fatigue damage and random variables. Then, the riser fatigue crack growth model shown in Equation (34) was obtained with this data-driven model.

$$D_{\mathrm{L}}=0.0495-2.0641t+36.4980t^2-0.0429C_{\mathrm{a}}+0.0237C_{\mathrm{a}}{}^2$$

(33)

$$a_y=a_{y-1}+C_0{Y}^{m_0}(a){(\sqrt{\pi a_{y-1}})}^{m_0}C{D}_L$$

(34)

Five groups of fatigue damage based on data-driven and physical-based models are listed in

Table 5. Verification of the data-driven model of fatigue.

Variable number	1	2	3	4	5
Wall thickness t (mm)	22.225	21.114	21.114	23.336	23.336
Additional mass coefficient Ca	1	0.95	1.05	0.95	1.05
Data-driven models	0.002454	0.002824	0.003274	0.001842	0.002292
Physical-based models	0.002476	0.002735	0.003298	0.001909	0.002363
Relative error	0.1222%	4.391%	1.588%	2.714%	1.003%

and were compared with one another. The results of the two methods are essentially the same. The fatigue response surface method had high computational speed, because the response surface model had no integration or differentiation process. The established response surface model could be used for the subsequent fatigue reliability analysis due to its high calculation accuracy and efficiency.

6.2. Fatigue Reliability Analysis

In the process of critical crack assessment, initial crack depth a₀ is assumed to be 1 mm, and initial crack length c₀ is assumed to be 2 mm. The yield stress σ_Y of the riser is 555 MPa, the ultimate stress of the material σ_u is 625 MPa, and the outer diameter is 263.5 mm. According to Equation (22) to Equation (29), the riser reaches the critical crack depth of 12.5792 mm in the 40th year. The key parameters of the data-driven models were adopted to calculate the fatigue failure probability of deepwater risers on the basis of the established response surface model. The key parameters are listed in

Table 6. Basic parameters for fatigue reliability analysis.

Variable	Distribution	Mean	COV
S–N curve parameter C	Constant	1.023 × 1012	-
Fracture mechanics parameter m	Constant	3	-
Fracture mechanics parameter C0	Constant	2.3 × 10−12	-
Crack depth a0	Normal	1 mm	0.1
Additional mass coefficient Ca	Log-normal	1	0.2
Wall thickness t	Normal	22.225 mm	0.05

.

Each variable was discretized, because continuously distributed variables cannot be calculated with the DBN. Variables C_a and t were normally distributed. [μ−1.5σ, μ+1.5σ] was used as the truncated boundary, and the outer part was ignored in discrete processing. The variables were divided into 10 segments. Variable a was in half of the normal distribution, and a was divided into 20 segments. The sum of all segments is equal to critical crack size a_c. The segments that exceeded the critical crack size indicated riser fatigue failure. The riser fatigue crack limit state equation in Equation (21) was then transformed into the BN for fatigue reliability analysis, as indicated in Figure 13. The label of each root node represented the variables in the response surface model. The distribution of fatigue cracks could be calculated with the BN.

The BN was transformed into the DBN, because fatigue crack increases with time, as is shown in Figure 14. The total analysis time of the established DBN was set to 40 years, and the analysis time intervals were set to 1 year. The conditional probability of a was determined with Equation (34). A probability table was also established based on the corresponding distribution of discrete variables. The probability distribution of crack depth at each detection time could be calculated via Bayesian forward inference.

The fatigue failure probability was calculated efficiently by using the established DBN, as is shown in Figure 15. The riser fatigue failure probability was low and changed slightly in the initial time for the small crack size. Then, the crack size and corresponding failure probability increased gradually with service time, and, finally, the crack reached the critical crack size, and the fatigue failure of the riser occurred. The riser system should be inspected before the riser failure probability reaches the probability limit and maintained afterward to avoid major accidents.

6.3. Fatigue Reliability Updating

Risers are inspected periodically in actual riser engineering to guarantee their safety. The probability of detection (POD) is often used to describe the probability of detecting the corresponding crack size with a certain detection method under a given condition. It can be regarded as a function of defect crack size and shown as

$$POD\left(a\right)=1-1.2\exp (-0.3a)$$

(35)

Two groups of inspection results (Test1: detection and Test2: no detection) of the riser fatigue crack were studied. Test1 means a fatigue crack is detected, and Test2 means no fatigue crack is detected. The inspection results were introduced into the fatigue crack DBN model, as is shown in Figure 16.

Parameters t and C_a and riser failure probability can be updated based on the inspection results in the backward inference of the DBN model. The inspection results were introduced for parameter updating in the 20th year, as is shown in Figure 17.

Figure 17a,b show that the introduction of inspection results exerted a great influence on parameter t. The updated parameter t could not maintain a normal distribution after the inspection results were introduced. The inspection results had little influence on parameter C_a, and updated parameter C_a almost maintained a lognormal distribution after the inspection results were introduced. Figure 17c shows that the inspection results had a great influence on the fatigue failure probability of the deepwater risers. For Test1 (detection), the updated fatigue reliability of the riser at this time was lower than the value before the inspection, and the failure probability would be higher than before the inspection, because the fatigue crack was detected. On the contrary, the updated fatigue failure probability of the risers in Test2 (no detection) was lower than the value before the update, because the fatigue crack was not detected in the defect inspection. Riser fatigue reliability can be evaluated realistically by integrating simulation models and inspection results.

7. Conclusions

This study attempts to solve a challenging problem: the fatigue reliability analysis of deepwater risers. An improved fatigue reliability analysis method based on RSM and the DBN for deepwater risers was proposed. A more efficient data-driven model was established based on RSM compared with physical-based models. A crack growth model of the riser was established by considering the crack inspection data as a factor. The crack DBN model was employed to conduct a fatigue reliability analysis and updating. The combination of data-driven models and the crack growth model could solve the problems of traditional methods in the fatigue reliability analysis of risers, such as low computational efficiency and the inability to introduce inspection data.

The whole fatigue reliability analysis process includes fatigue damage analysis, fatigue damage modeling, and fatigue reliability analysis. First, the fatigue damage of the riser system was analyzed based on physical-based models. Second, a data-driven model for riser fatigue damage was established based on the fatigue damage analysis results, and, further, a riser fatigue crack growth model was established. Lastly, a fatigue crack DBN model was established for fatigue reliability analysis based on the fatigue crack growth model, and the fatigue reliability of the riser was updated by taking the crack inspection data as evidence.

The proposed method was applied to the fatigue reliability analysis and updating of deepwater risers. The results showed that fatigue reliability could be calculated efficiently through dynamic Bayesian forward inference. Moreover, the distribution of random variables and fatigue reliability could be updated based on the inspection results via the backward inference of the DBN model. The proposed method could analyze and update the fatigue reliability of deepwater risers accurately and efficiently.