Dimensionless groups governing response of steel catenary risers

Ocean Engineering ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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Dimensionless groups governing response of steel catenary risers Lucile M. Quéau n, Mehrdad Kimiaei, Mark F. Randolph Centre for Offshore Foundation Systems- M053, University of Western Australia, 35 Stirling Highway, Crawley, Perth, WA 6009, Australia

art ic l e i nf o

a b s t r a c t

Article history: Received 24 October 2012 Accepted 30 March 2013

Steel catenary risers (SCRs) are one of the most cost effective types of risers in deep water, but their design, particularly in the touchdown zone (TDZ), is challenging. Design uncertainties arise due to limited understanding of the influence of various parameters on the SCR response, pertaining to the SCR geometry and structural properties, environmental loading and the seabed characteristics. An improved framework that provides quantitative guidance on how each parameter affects the overall structural response, and in particular fatigue damage in the touchdown zone, would facilitate design and increase confidence in the use of SCRs. This paper details the application of dimensional analysis to SCR behaviour. Suitable dimensionless groups are proposed and validated through a series of numerical tests, comparing the response of similar SCR systems defined by appropriate scaling of parameters. Due to the fundamental effect of the seabed response on fatigue damage, two types of soil models are considered in the study: a linear and a nonlinear model. The study provides a framework for design of SCRs, facilitating sensitivity analyses, enabling better control on parameters and reducing the number of simulations required. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Steel catenary risers Dimensional analysis Fatigue Structural response Dynamic amplification factor

1. Introduction Offshore exploration and production of oil and gas continue to increase and move into ever deeper water. As a result, conventional flexible risers have largely been replaced by steel catenary risers (SCRs), which are one of the most cost effective types of risers (Bai and Bai, 2005; Campbell, 1999). Fatigue design of SCRs is a complex and crucial challenge as SCR failure would have significant environmental and economic consequences. Vessel motions and hydrodynamic loading result in two critical fatigue areas: at the vessel hang-off point, and in the touchdown zone (TDZ—the area of dynamic riser–soil interaction (Bridge, 2005)). Indeed, in the TDZ, cyclic motions of the SCR lead to complex interactions between the seabed and the riser. In most design, numerical analysis of riser–soil interaction is based on a linear soil model, although difficulties in assessing suitable values of soil stiffness, which have been shown to vary with the amplitude of cyclic displacement (Clukey et al., 2008, 2007, 2005), have encouraged development of more sophisticated nonlinear soil models (Aubeny and Biscontin, 2009; Bridge and Howells, 2007; Randolph and Quiggin, 2009). The riser–soil interaction is complicated by the likelihood of extreme soil degradation in the touchdown zone during significant storms, with intervening periods where reconsolidation of the soil will occur. This type of behaviour is not captured by existing soil models, although recent

n

Corresponding author. Fax: +61 8 6488 1044. E-mail address: [email protected] (L.M. Quéau).

experimental data suggests gradual hardening of the soil following initial cyclic degradation, with diminishing effects of each episode of cyclic motions (Hodder et al., 2013). Despite previous parametric studies, there is still limited understanding of the effect of individual parameters pertaining to soil response, riser characteristics (e.g. material properties, geometry etc.) and environmental loading, on SCR fatigue design (Kimiaei et al., 2010; Xia et al., 2008). Sensitivity studies to investigate the influence of parameters, particularly where their value is uncertain, are recommended in riser design guidelines (DNV, 2010), in order to improve understanding of the overall SCR behaviour, provide more accurate modelling and increase confidence in design. To conduct pertinent sensitivity analyses, it is necessary to identify the dimensionless groups of parameters that influence key response variables (Palmer, 2008), which are typically the pipe stress or curvature for overall integrity, and the cyclic stress range for fatigue assessment. Establishment of these groups can be achieved by performing dimensional analysis. As noted by Pesce et al. (1998), non-dimensional parameters are intrinsically “generally valid”. A limited set of dimensionless groups have been used in analytical studies of SCRs (Pesce et al., 2006, 1998), for the post-processing of results from numerical analyses (Aubeny and Biscontin, 2009; Pesce et al., 1998; Randolph and White, 2008) and recently in the scaling of a riser model to ensure similarity in modal responses, and hence overall riser dynamics (Rateiro et al., 2012). However, for more robust dimensional analysis, it is necessary to identify the entire set of dimensionless groups pertinent to a problem (Taylor, 1974). Sensitivity analyses

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consist in investigating, and ideally quantifying, the effects that changes in one or more dimensionless group of input parameters have on a specific dimensionless group of output parameters. For SCRs, dependencies between variables are not systematically obvious. Indeed some care is needed in choosing an appropriate set of dimensionless groups. Modification of a particular input parameter may otherwise lead to unwanted changes in several dependent dimensionless groups, which impedes interpretation of results due to a combination of contributions. An appropriate complete set of dimensionless groups allows clarification of relationships between parameters, thus facilitating pre-processing of input data and postprocessing of results. It also provides a framework for meaningful comparisons of SCR behaviour. The principal purpose of this paper is to establish an appropriate and complete list of dimensionless groups governing SCR static and dynamic response to motions applied at the hang-off point. This study is part of ongoing research aiming to simplify fatigue design of SCR systems by means of dynamic amplification factors (DAFs) (Quéau et al., 2011). DAFs allow the determination of maximum dynamic response amplitudes directly from the analytically derived static response, reducing the need to perform time consuming dynamic analyses in the early stages of design. The usefulness of this approach in SCR studies has been explored previously by the authors through pilot sensitivity analyses for a few key input parameters. Defining the entire set of dimensionless groups that impact SCR behaviour will assist future sensitivity analyses aimed at establishing quantitative relationships between input and output dimensionless groups. Dimensional analysis, using the traditional π-theorem, is applied to the structural response of SCRs, considering both linear and nonlinear soil models. The study is mainly focused on fatigue design in the touchdown zone and, therefore, the cyclic stress range and DAF are of primary interest. The logic behind the selection of the dimensionless groups is presented first, and then numerical analyses are performed to verify that all the dimensionless groups have been identified.

2. Dimensional analysis theory Dimensional analysis aims to clarify dependencies between magnitudes of quantities pertaining to a physical problem by considering their dimensions. The method is based on the principle of homogeneity, which states that equations involving magnitudes of quantities must be dimensionally homogeneous (Massey, 1971). Dimensional analysis is particularly suitable for the study of complex phenomena, involving numerous independent variables for instance, for which complete theoretical treatment is challenging (Massey, 1971; Palmer, 2008; Taylor, 1974). This method is used when researching a relation of the form expressed as Q 1 ¼ f ðQ 2 ; Q 3 ; …; Q n Þ

ð1Þ

where Q1 represents the magnitude of the output and Qi are the magnitudes of the inputs pertinent to the problem; f stands for “some function of” (Massey, 1971). The first step of the method is to select the appropriate initial quantities (output and inputs). This task is challenging as it is necessary that one and only one relationship exists between the quantities selected (Palmer, 2008; Taylor, 1974). The second step consists of grouping the original quantities into dimensionless groups, πi, in order to form a new relationship, such as π 1 ¼ f ðπ 2 ; π 3; …; π n−m Þ

ð2Þ

which contains all the information of Eq. (1) (Taylor, 1974). The number of independent dimensionless groups that can be formed is determined using the Vaschy–Buckingham (or π) theorem. It states

that “if m is the number of distinct fundamental magnitudes required to express the dimensional formulae of all the n magnitudes, then these n magnitudes may be grouped into n–m independent dimensionless terms” (Massey, 1971). Several mathematical methods exist to find preliminary dimensionless groups. As any combination of dimensionless groups is also dimensionless, these preliminary dimensionless groups may be combined together to obtain more meaningful groups for the purpose of a particular study. Although dimensional analysis alone cannot usually lead to the complete solution of a problem, it does increase understanding and can guide the design of experiments and numerical simulations. Moreover, the influence of certain variables, as well as some direct relationships, may emerge from the analysis (Massey, 1971; Palmer, 2008; Taylor, 1974).

3. Dimensional analysis of SCR response 3.1. Main assumptions The study has adopted some minor idealisations and simplifications in order to render the problem manageable, while still allowing broad application: 1. The study is restricted to 2D conditions, with SCRs response under in-plane motions only. 2. Loading of SCRs is represented by harmonic motions of the floating vessels through the procedure suggested by Kimiaei et al. (2010). Linear waves identified as the wave packs (WPs) constitute the input motions. They are represented by application of a time varying sinusoidal motion of the vessel, making an angle θm with the vertical and characterised by heave amplitude (H) and period (T) of the motion (i.e. maximum input velocity of 2πH/T). 3. The choice of the seabed model is of prime importance when studying the fatigue life of SCRs in the TDZ, as it can affect analysis results (Bridge and Howells, 2007; Clukey et al., 2007). The study focuses on elastic seabed response, where the soil is characterised by a linear stiffness (linear seabed model hereafter). However, equivalent dimensionless groups are also proposed for the nonlinear seabed model of Randolph and Quiggin (2009). Linear seabed models are easy to use, and indeed represent current industry practice. However, they are limited in their ability to accommodate realistic soil response, where the secant stiffness decreases with increasing magnitude of displacement. Nonlinear soil models are relatively new and although they capture some important features of the riser–soil interaction (e.g. variation of soil stiffness, soil suction during uplift etc.) their utilisation is more problematic (e.g. Bridge and Howells, 2007). 4. The current profile in the sea column is omitted (simplification also used by Rateiro et al. (2012)), the drag coefficient is set to a constant value and the rotational stiffness at the hang-off point is neglected (simplifications also used by Xia et al. (2008)). Furthermore, the flow rate of the content, coating and structural damping are not taken into account either. 5. All riser analyses have been carried out using the OrcaFlex software (Orcina, 2009) where the Randolph–Quiggin nonlinear seabed model has been implemented, in addition to a standard linear seabed model.

3.2. Choice of pertinent variables This research aims to identify the non-dimensional groups of parameters that play a role in SCRs behaviour for design improvement

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purposes (fatigue design in the TDZ in particular). SCRs response is characterised by measurement of the total axial stress (s), which combines axial force and bending moment effects, for a given angular position on the riser circumference. Hence, the dimensional analysis method is applied to the SCR total axial stress and consists in listing the complete set of independent variables that impact its amplitude and then categorising them into dimensionless groups. An inventory of all the parameters likely to influence s was made and some were eliminated to remove any redundancy. The final selection of parameters influencing the SCR stress when static and dynamic motions are applied to the vessel, together with notations, are shown in Table 1, while Fig. 1 gives an illustration of some of the key SCR parameters. Certain parameters commonly used in SCR studies are treated here as dependent input parameters because they can be determined using the independent parameters in Table 1. For instance, the inside diameter (Di), the second moment of inertia (I) and the content density (ρcont) can be calculated (trivially) by means of Di ¼ Do –2wt

ð3Þ

π ðD4 −D4i Þ I¼ 64 o

ð4Þ

ρcont ¼

4p πgDi

− 2

Do 2 Di 2

" ρsteel 1−

Di 2 Do 2

!

#

3

Catenary-based solutions that ignore the riser bending stiffness, as well as the riser–soil interaction, only give approximate solutions for the SCR behaviour. In spite of their limitations, however, catenary solutions provide good approximations of the overall geometry because SCRs have a high aspect ratio of suspended length over the outside diameter (Bridge, 2005). The hang-off angle (θHO), arc length (St) to the touchdown point at equilibrium (i.e. before any motion is applied to the vessel), see Fig. 1, and the horizontal offset to the TDP (Xt) at equilibrium may be estimated from analytical catenary solutions as T o ≈pΔz

sin θHO 1−sin θHO

ð7Þ

St ≈Δz

cos θHO 1−sin θHO

ð8Þ

X t ≈Δz

arcsinhðcot θHO Þsin θHO 1−sin θHO

ð9Þ

In this paper, unless otherwise specified, these parameters have been determined accurately using the static solution in OrcaFlex. 3.3. Determining the dimensionless groups

−ρwater

ð5Þ

An important variable is the characteristic length (λ), defined as sffiffiffiffiffi EI ð6Þ λ¼ To which approximates the distance between the actual and the ideal cable touchdown point (TDP—where the SCR first contacts the seabed) (Pesce et al., 1998), and hence provides a scale for the touchdown zone where the bending stiffness of the SCR affects the shape.

For a linear seabed model the Vaschy–Buckingham theorem predicts that 18 dimensionless groups can be formed from the 21 parameters. The output group, s/E, may therefore be expressed as a function, f, of 17 groups, chosen to facilitate sensitivity analyses, as rffiffiffiffiffiffiffiffiffiffi  s H H ρsteel Do Do p To ks ¼f ; Δθm ; ; ν; ; ; ; ; μ; ; C D ; C A ; E Δz T E Δz wt EΔz E EΔz2  ρsteel gΔzρsteel s t ; β; ð10Þ ; ; Δz T ρwater E

Table 1 Selected parameters for the dimensional analysis of SCR stress. Category

Variable

Symbol

Unit

Output Input motion

Axial stress Heave amplitude Cyclic period Angle of the motion (with the vertical) Outer diameter Wall thickness Young's modulus Density Unit submerged weight Poisson's ratio Vertical difference between positions of SCR ends Horizontal tension component Friction coefficient Soil stiffness Friction coefficient Soil undrained shear strength at mudline Soil undrained shear strength gradient Saturated soil density Penetration resistance parameter Penetration resistance parameter Soil buoyancy factor Normalised maximum stiffness Suction resistance ratio Normalised suction decay distance Normalised repenetration offset after uplift Drag coefficient Added mass coefficient Water density Gravity acceleration Arc length (from touchdown point) Angular position on the SCR circumference for stress checking Time

s H T θm Do wt E ρsteel p ν Δz To m ks m sum ρ ρsoil a b fb Kmax fsuc λsuc λrep CD CA ρwater g s β t

N/m2 m s – m m N/m2 kg/m3 N/m – m N – N/m2 – N/m2 N/m3 kg/m3 – – – – – – – – – kg/m3 m/s2 m – s

Riser characteristics

SCR geometry Seabed: Linear seabed model Seabed: Nonlinear seabed model

Hydrodynamic

Planet Earth Output location and time

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Fig. 1. SCR geometry and key parameters: (a) Side view of SCR configuration; (b) Riser cross-section; and (c) One cycle of sinusoidal input motion.

Table 2 Dimensionless groups for SCR stress. Category

Dimensionless group

Symbol

Comment

Output Input motion

s/E H/Δz Δθm

π1 π2 π3 π4

Strain in the riser wall Motion amplitude; vertical and horizontal perturbations of the catenary are defined by π2 and π3 Motion direction relative to the hang-off angle (i.e. Δθm ¼ θm−θHO)—non-dimensional Velocity of input motion relative to wave propagation speed in SCR

π5 π6 π7 π8 π9 π10 π11 π12 π13 π14 π15 π16 π17 π18

Riser outside diameter relative to water depth Riser outside diameter relative to wall thickness Riser unit submerged weight Poisson's ratio—non-dimensional Pipeline tension Soil friction coefficient—non-dimensional Soil stiffness Drag force coefficient—non-dimensional Added mass coefficient—non-dimensional Relative steel and water densities Gravitational force Location along the SCR Angular location on the SCR circumference—non-dimensional Time during one cycle of applied motion

Riser characteristics

SCR geometry Linear seabed model Hydrodynamic

Planet earth Output location and time

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðH=TÞ ρsteel =E Do/Δz Do/wt p/(EΔz) ν To/(EΔz2) μ ks/E CD CA ρsteel/ρwater gΔzρsteel/E s/Δz β t/T

For the nonlinear seabed model, the linear stiffness, ks, is replaced by 10 parameters that quantify the non-linear response (Randolph and Quiggin, 2009), leading to the following proposed set of groups: s E

¼f



H H Δz ; Δθ m ; T

qffiffiffiffiffiffiffi

ρsteel Do Do p To E ; Δz ; wt ; EΔz ; ν; EΔz2

a; b; f b ; K max ; f suc ; λsuc ; λrep ; C D ; C A ;

; μ; sum E ;

ρΔz ρsoil E ; ρwater

;

ρsteel gΔzρsteel s t ; β; ; ; Δz T ρwater E

 ð11Þ

The key parameters of the nonlinear model are the (linearised) shear strength profile of su ¼sum+ρz, where sum is the mudline intercept and ρ the strength gradient with depth, z, and the soil density, ρsoil (from which the effective unit weight may be determined). The other parameters are internal dimensionless model parameters that control the interaction.

Table 2 introduces the dimensionless groups for the linear seabed model and comments on their physical significance. Some of the dimensionless groups used in previous work are not represented in Eq. (10), but their effects are accounted for as they can be obtained by combining selected groups (e.g. nondimensional soil stiffness, λ2ks/To (Pesce et al., 1998; Rateiro et al., 2012; Randolph and White, 2008) or normalised pipeline tension, To/λp (Langner, 2003; Randolph and White, 2008)). Similarly, the fundamental axial and bending stiffnesses (Rateiro et al., 2012) are not explicitly used within the set of dimensionless groups but their effects are normalised. The dimensionless groups established here aim to isolate the key individual parameters so that the effect of each may be explored independently in future sensitivity studies. That is why the π-theorem is used rather than the alternative method involving physical considerations (as suggested by Rateiro et al. (2012)).

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For fatigue design purposes, the stress range (Δs) is the key input parameter. The stress range is the difference between maximum and minimum stress (i.e. Δs ¼smax−smin) occurring at a given location during one cycle of motion. Hence Δs/E replaces s/E as the key output group for fatigue analyses and is, by definition, independent from the normalised time π18 ¼t/T. The proposed approach to simplify fatigue design of SCRs relies on the use of a dynamic amplification factor (DAF); this is defined as the ratio of the maximum dynamic stress range, denoted as MaxΔsdyn, to the maximum static stress range, denoted as MaxΔssta, occurring in the TDZ under application of a given wave pack (Quéau et al., 2011). The DAF can therefore be defined as a function of all the dimensionless groups that influence Δs, with the exception of the normalised time π18 and the normalised location π16 (although DAF could be insensitive to some of these groups as will be investigated in future work). 3.4. Numerical models for verification of the relevance of dimensionless groups A series of numerical analyses were performed to verify results of the dimensional analysis with respect to the total axial stress and stress range for SCRs, and at the same time establish the relative accuracy of the dynamic analysis software OrcaFlex. Validation of the dimensionless groups was performed by comparing the strain (s/E) and strain range (Δs/E) for different SCR models that have identical values of dimensionless groups in Eqs. (10) and (11), but obtained through different values of input variables. 3.4.1. Linear seabed model Three distinct base cases (BCs) are defined (BC1, BC2 and BC3) and, for each of them, several model tests (MTs) with distinct sets of input parameters but the same dimensionless group values for

5

the particular BC, were established: MT1-1, MT2-1, MT3-1 for BC1; MT1-2, MT2-2 for BC2; and MT1-3, MT2-3 for BC3. Two MTs are defined for both BC2 and BC3 whereas an additional MT (MT3-1) is used for BC1 for specific reasons detailed later on. Therefore, three groups of equivalent models are defined (equivalent groups 1–3). Table 3 summarises the values of the input parameters used to define the 10 models as well as values of some key dependent parameters. In all these analyses, the vessel motion was applied parallel to the riser tangential line at the hang-off point (i.e. Δθm set to zero). This is based on the findings of Kimiaei et al. (2010), that the tangential component of motion alone could adequately represent the fatigue damage resulting from wave loading in the TDZ, but the restriction does not affect the conclusions from the paper. Fig. 2 shows the SCR shapes for the different models. Dimensionless groups in models within each equivalent group are presented in Table 4. The dimensionless groups that normalise motion period, location and time along the riser (π4 and π16 to π18), and non-dimensional input parameters that were not varied (π3, π8, π10, π12 and π13), are omitted in Table 4 as they are consistent across all models. BC1 is an example SCR for which parameters were defined from an in-service SCR connected to a semi-submersible in the Gulf of Mexico. BC2 and BC3 models were established from BC1 by changing some of the dimensionless groups through variations among the underlying input parameters (π11 ¼ ks/E, π9 ¼ To/EΔz2 and π6 ¼Do/wt) while having identical values of π2, π5, π7, π14 and π15 (see Table 4). This is to test the validity of Eq. (10) for different soil stiffness, smaller hang-off angle and lower bending stiffness than in BC1. MTs are established from their respective BCs by using scale factors (α). There are different ways of scaling some of the parameters while ensuring constant dimensionless groups. It was chosen mostly to artificially vary the value of gravity acceleration, g (i.e. scaling it by 1/α) in order to obtain identical dimensionless groups within each equivalent set. In this approach, the

Table 3 Characteristics of the models for linear soil behaviour. Model name

Input parameters

Dependent parameters

Equivalent group 1

Equivalent group 2

Equivalent group 3

BC1

MT1-1

MT2-1

MT3-1

BC2

MT1-2

MT2-2

BC3

MT1-3

MT2-3

H (m) T (s)

1 T

2 2T

10 10 T

1 T

2 2T

10 10 T

1 T

2 2T

10 10 T

Δθm (1) t (s)

0 t

0 2t

0 10 t

0 t

0 2t

0 10 t

0 t

0 2t

0 10 t

Δz (m) T0 (kN) s (m) Do (m) wt (m) E (kPa) ρsteel (kg/m3) p (kN/m) ν ks (kPa) m CD CA ρwater (kg/m3) g (m/s2) β (1)

981.886 164.341 s 0.228 0.025 2.12E+8 7850 0.817 0.293 22.8 0.5 1.2 1 1025 9.807 0

1963.772 657.362 2s 0.456 0.05 2.12E+8 7850 1.634 0.293 22.8 0.5 1.2 1 1025 4.903 0

9818.86 16434.056 10 s 2.28 0.25 2.12E+8 7850 8.17 0.293 22.8 0.5 1.2 1 1025 0.981 0

0.5 pffiffiffiffiffiffiffi 0:5 T 0 pffiffiffiffiffiffiffi 0:5 t 490.943 20.543 0.5 s 0.114 0.0125 1.06E+8 7850 0.204 0.293 11.4 0.5 1.2 1 1025 9.807 0

981.886 164.341 s 0.228 0.025 2.12E+8 7850 0.817 0.293 228 0.5 1.2 1 1025 9.807 0

1963.772 657.362 2s 0.456 0.05 2.12E+8 7850 1.634 0.293 228 0.5 1.2 1 1025 4.903 0

9818.86 16434.059 10 s 2.28 0.25 2.12E+8 7850 8.17 0.293 228 0.5 1.2 1 1025 0.981 0

981.886 132.974 s 0.228 0.019 2.12E+8 7850 0.817 0.293 22.8 0.5 1.2 1 1025 9.807 0

1963.772 531.896 2s 0.456 0.037 2.12E+8 7850 1.634 0.293 22.8 0.5 1.2 1 1025 4.903 0

9818.86 13297.409 10 s 2.28 0.186 2.12E+8 7850 8.17 0.293 22.8 0.5 1.2 1 1025 0.981 0

I (m4) Di (m) θHO (1) λ (m) ρcont (kg/m3) St (m) Xt (m)

8.34E-5 0.178 9.797 10.371 0 1168 496

1.33E-3 0.356 9.797 20.741 0 2335 992

8.34E-1 1.78 9.797 103.707 0 11675 4958

5.21E-06 0.089 9.797 5.185 0 584 248

8.34E-5 0.178 9.797 10.371 0 1171 500

1.33E-3 0.356 9.797 20.741 0 2342 999

8.34E-1 1.78 9.797 103.707 0 11712 4995

6.75E-5 0.191 8.18 10.371 1025 1135 432

1.08E-3 0.382 8.18 20.741 1025 2270 863

6.75E-1 1.909 8.18 103.707 1025 11352 4316

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Fig. 2. SCR shapes in (a) Equivalent group 1, (b) Equivalent group 2 and (c) Equivalent group 3.

Table 4 Dimensionless groups values in models within equivalent groups.

π2 (n10−3) π5 (n10−4) π6 π7 (n10−12) π9 (n10−13) π11 (n10−7) π14 π15 (n10−4)

Table 6 Characteristics of the model for study with nonlinear (NL) seabed.

Equivalent group 1

Equivalent group 2

Equivalent group 3

1.018 2.322 9.12 3.925 8.041 1.075 7.659 3.565

1.018 2.322 9.12 3.925 8.041 10.755 7.659 3.565

1.018 2.322 12.29 3.925 6.506 1.075 7.659 3.565

Table 5 Wave pack periods (T) for each load case (LC). π4 (n10−5)

T (s)

LC1 LC2 LC3 LC4 LC5 LC6 LC7 LC8 LC9 LC10 a

BCia

MT1-i

MT2-i

MT3-1

4 6 8 10 15 20 30 50 100 1000

8 12 16 20 30 40 60 100 200 2000

40 60 80 100 150 200 300 500 1000 10000

2.83 4.24 5.66 7.07 10.61 14.14 21.21 35.36 70.71 707.11

4.811 3.207 2.405 1.924 1.283 0.962 0.641 0.385 0.192 0.019

i: Index of equivalent group (i∈{1, 2, 3}).

submerged weight of the riser and wave pack period are scaled by α, tension by α2 whereas E, ks and s are unchanged between MTs and BCs. MT1s and MT2s were established following this technique and using scale factors of 2 and (a deliberately extreme) 10 respectively to get a priori very dissimilar models. Therefore, values of some input parameters in the MTs are not necessarily realistic (e.g. Do ¼ 2.28 m in MT2-1). However, it is also possible to scale the models by keeping g unchanged. This alternative method is illustrated and tested by means of MT3-1, using a scale factor of 0.5. Consequently, the submerged weight is in this case scaled by α2, period by α0.5, tension by α3 and E, ks and s are scaled by α; this is consistent with Rateiro et al. (2012). Every other quantity has the same scale factor in the two approaches, established in accordance with the dimensionless groups. To confirm the dimensionless groups for dynamic and static loadings applied to SCRs, 10 load cases (LCs) were used. Each LC

T (s) for WPs of LC1 T (s) for WPs of LC10 sum (kPa) ρ (kPa/m) ρsoil (kg/m3) St (m) Xt (m) a b fb Kmax fsuc λsuc λrep sum/E ρΔz/E ρsoil/ρwater

BC1 NL

MT1-1 NL

MT2-1 NL

4 1000 0 1.5 1500 1161 489

8 2000 0 0.75 1500 2322 978 6 0.25 1.5 200 0.2 1 0.3 0 6.95E-06 1.463

40 10,000 0 0.15 1500 11609 4892

refers to application of distinct WPs with “equivalent characteristics” in models within an equivalent group (i.e. harmonic motion of the vessel with similar amplitude of H, similar period of T and making a similar angle θm with the vertical). These values are not necessarily the same in all the models. θm is kept constant between models in the same equivalent group (so as to get appropriate To value and same π9) but H and T values in the MTs are deduced from those in the corresponding BC by the scale factor, so that values of π2 and π4 respectively for a given LC are identical between equivalent models. For all BC models, the WPs used in all LCs have the same H value (1 m in this study, as presented in Table 3) and different T values ranging from 4 s (representing a dynamically loaded system) to 1000 s (representing a statically loaded system by virtually suppressing inertia and damping effects). Values of π2 and π9 are presented in Table 4 whereas T and π4 values for each LC are reported in Table 5.

3.4.2. Nonlinear seabed model For the nonlinear soil model, results were checked by means of three models forming one equivalent group. The models BC1, MT11 and MT2-1, used in Equivalent group 1 for the linear seabed cases, were modified to create their homologues with the nonlinear soil (namely BC1 NL, MT1-1 NL and MT2-1 NL respectively). The nonlinear soil parameters and the values of any parameters that are different from those for the linear seabed (already listed in Table 3) are presented in Table 6, together with the associated

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dimensionless groups sum⧸Ε, ρΔz/E and ρsoil/ρwater. The two extreme LCs (shortest and longest periods, T) were simulated for each model, in each case applying cycles on an initially undisturbed seabed in order to reach a steady-state response. The nondimensional parameters of the seabed model were set to their recommended values for fatigue analysis (Randolph and Quiggin, 2009).

4. Validation of dimensionless groups 4.1. Results for the linear seabed model Different levels of analyses were performed to validate the suitability of the suggested dimensionless groups. Preliminary verifications were carried out to confirm that the SCR configurations were indeed similar. Stress and stress ranges were then compared between different models within each equivalent group. 4.1.1. SCRs geometry For each equivalent group, the normalised shapes of the modelled SCRs at equilibrium are represented in Fig. 3 plotting the normalised vertical coordinate (Zn) against the normalised horizontal coordinate (Xn) of the nodes located along each SCR (note the markers in Fig. 3 are only used to differentiate the curves and do not represent corresponding nodes in the different models). Zn is defined as Zn ¼ z/Δz and Xn as Xn ¼x/Xt where z and x are the vertical and horizontal coordinates of the nodes measured from the TDP, as illustrated in Fig. 1. It is seen that for models within each equivalent group, the SCR shapes correlate well once normalised, whereas they are evidently different in Fig. 2, prior to normalisation. Although only an approximation, the catenary solution for the SCR shape also predicts these similarities. The solution may be expressed as     To px z¼ cosh −1 ð12Þ To p This expression may be rewritten, using dimensionless groups, as

" pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! ! # 1 þ 2π 9 =π 7 π9 cosh arcsinh X n −1 Zn ¼ π 9 =π 7 π7

ð13Þ

The two dimensionless groups, π7 and π9 were identical for all models within each equivalent group, therefore giving identical normalised shapes in Fig. 3. The smaller hang-off angle for Equivalent group 3 has the effect of increasing the curvature of the SCR shape (and decreasing the horizontal tension at TDP).

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The declination angle (angle between the riser's longitudinal axis and the vertical axis) at any location along the riser length has also been compared for the different models for dynamic and static loading cases, as illustrated in Fig. 4a–b for Equivalent group 1. The normalised times π18 were chosen arbitrarily as 0.25 and 0.75, corresponding to the vessel being at its highest and lowest positions respectively (see Fig. 1c). To investigate the declination distribution along the complete riser, it is convenient to normalise the location as Sn ¼ s/St. It is found that the declination angle at any normalised location on the riser is similar. Also, declination angle for a given location during an entire cycle of dynamic and static motion (i.e. π18 ¼t/T ranging from 0 to 1) was compared for Equivalent group 1, as shown in Fig. 4c. The normalised location for comparison of results was chosen arbitrarily at the equilibrium TDP (Sn ¼0), which is typically close to the position of maximum fatigue damage. Variations of declination angle are similar during the entire cycle of sinusoidal motion. Therefore, it seems that normalised shapes are similar for the undisturbed SCRs and stay similar during a given sinusoidal perturbation applied at the SCR hang-off point. From this, it is inferred that SCR normalised displacements are the same between equivalent models. Shifting of the TDP is a fundamental aspect of the SCR response, as it drives curvature variations that lead to fatigue damage (Pesce et al., 2006). Similar changes in horizontal tension, arc length to TDP, curvature and, hence, similar structural responses should be predicted when the parameters are scaled through the defined non-dimensional groups. Testing of Eq. (10) is performed through detailed investigation of the SCR absolute riser strain (s/E) and, then, of SCR strain range (Δs/E).

4.1.2. Axial stress From a fatigue perspective, the variation of total axial stress during cyclic motions of the SCR for every location (along the riser length and around the section circumference) should be considered. Initially, the normalised stress (π1 ¼s/E) responses are compared for different models, for a specific normalised location and also in a specific circumferential position, during one cycle of motion. Provided the normalised stress responses for equivalent models match during an entire cycle of motion, then the maxima and minima of s/E, and thus the normalised stress range and fatigue damage, will also match. Stress results from the different models are compared at the bottom of the riser cross section (β¼01) and at the equilibrium TDP (Sn ¼0). The normalised stress, s/E, for dynamic (short period motion, LC1) and static (long period motion, LC10) loading are plotted in Fig. 5 during one full cycle. In each subplot, the normalised stress responses match very well, validating the adequacy of the dimensionless groups in Eq. (10).

Fig. 3. Normalised SCR shapes in (a) Equivalent group 1, (b) Equivalent group 2 and (c) Equivalent group 3.

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Fig. 5 also indicates the effect of the input excitation period (T) on the time (t) when the maximum and minimum axial stresses occur during each motion cycle. For the statically loaded case (LC10) in each equivalent group, the normalised stress response is directly in phase with the input sinusoidal motion, with the maximum stress after one quarter of a cycle when the vessel is at its highest position (see Fig. 1c). Input motion and stress response are, however, not in phase at the selected location for the dynamically loaded case (LC1). The dynamic characteristics of the system have caused a time lag between the maximum stress in the SCR and the maximum input motion, but have also led to the minimum stress advancing relative to the lowest position of the vessel. 4.1.3. Stress range The normalised stress range (Δs/E) is a function of the same set of dimensionless groups as shown for s/E in Eq. (10) (with the exception of π18 ¼t/T by definition of Δs). As all these dimensionless groups remain unchanged in models from the same equivalent group, the same Δs/E is expected. At first, the normalised stress range distribution along the entire riser length is compared between equivalent models. Then, the maximum normalised stress range occurring in the TDZ, denoted as MaxΔsTDZ/E hereafter, is investigated in greater detail as it is critical in determining the SCR fatigue life. The results presented here are limited to only one location around the SCR circumference (β ¼01), although in a proper fatigue analysis involving out of plane motions as well as those in a vertical plane, the maximum stress range anywhere around the SCR circumference would need to be assessed. Fig. 6 shows Δs/E (for β¼ 01) for a dynamic (LC1) and static (LC10) motion for each equivalent group. The normalised SCR responses in the different models match well for both LCs, with MaxΔsTDZ/E obtained at the same normalised location for equivalent SCR models, as anticipated in Eq. (10). Table 7 summarises the

maximum stress range values for all LCs in each equivalent group. Close agreement is found for the MTs compared with the corresponding BC results, with a maximum relative difference of  0.4%. The maximum differences arise from the cases with shortest excitation period, and result mainly from numerical sources. To keep reasonable calculation times, identical log sample interval, time step size and numerical damping for all LCs were used in the analyses. That is why, overall, better agreement is found for longer periods. Table 8 illustrates the effect of numerical settings for the example of LC1 for BC3 and MT1-3. By scaling the log sample interval, the inner time step and the critical damping in MT1-3 (creating MT1-3n), the difference with BC3 for this dynamic case has been reduced from 0.4% to −0.001%. However, calculation time has doubled. The observation of similar overall trends of normalised stress range along the riser length within each equivalent group for dynamic and static motions, and the match of maximum values in the TDZ, tend to validate Eq. (10). Consequently, similar DAF values for every WP within each equivalent group are expected, hence contributing to the confirmation of the dimensionless groups influencing DAF. Moreover, close agreement of results within Equivalent group 1 demonstrates the accuracy of the two proposed scaling approaches (i.e. varying or constant gravity acceleration). 4.2. Results for the nonlinear seabed model Results of MaxΔsTDZ/E (for β¼01) for two different load cases in equivalent nonlinear seabed models (BC1 NL, MT1-1 NL and MT21 NL) are presented and compared in Table 9. The MT results match the BC results well (with a relative difference less than 0.5%). More tests were also performed to verify the relevance of the proposed dimensionless groups by comparing key dimensionless parameters inherent in the nonlinear soil model. Fig. 7

Fig. 4. Declination results for Equivalent group 1 : (a) normalised time, π18 = 0.25; (b) normalised time, π18 = 0.75; and (c) normalised arc length, Sn=0.

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represents a typical hysteretic plot of the seabed response at the TDP for equivalent models during a cycle of dynamic motion (LC1). Good agreement is shown between the different models.

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Another important feature that nonlinear seabed models aim to capture is the formation and development of trenches. Normalising trench depth and dimensions by the riser outside diameter

Fig. 5. Dimensional analysis results for (a) Equivalent group 1, (b) Equivalent group 2 and (c) Equivalent group 3: comparison of normalised stress during one cycle of motion; β¼ 01 and π16 ¼ 0.

Fig. 6. Normalised stress range distributions for (a) Equivalent group 1, (b) Equivalent group 2 and (c) Equivalent group 3; β¼ 01.

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Table 7 Comparison of maximum Δs in the TDZ for models with linear seabed; β ¼ 01. Equivalent group 1 BC1

MT1-1

Max ΔrTDZ/E (n10−4)

Max ΔrTDZ/E (n10−4)

MT2-1 Relative difference with BC1 results (%)

Max ΔrTDZ/E (n10−4)

MT3-1 Relative difference with:

Max ΔrTDZ/E (n10−4)

BC1 MT1-1 results (%) results (%) LC1 LC2 LC3 LC4 LC5 LC6 LC7 LC8 LC9 LC10

5.26 3.80 3.34 3.04 2.45 2.03 1.73 0.97 1.08 1.16

5.26 3.81 3.34 3.03 2.45 2.03 1.73 0.97 1.08 1.16

0.00 0.17 0.00 −0.24 −0.11 −0.01 0.00 0.00 0.00 0.00

5.27 3.81 3.34 3.03 2.45 2.03 1.73 0.97 1.08 1.16

0.16 0.18 0.01 −0.23 −0.11 −0.02 −0.01 −0.01 −0.01 −0.01

0.16 0.02 0.01 0.00 0.00 −0.01 −0.01 −0.01 −0.01 −0.01

5.25 3.81 3.34 3.03 2.45 2.03 1.73 0.97 1.08 1.16

Relative difference with:

BC1 MT1-1 results (%) results (%)

MT2-1 results (%)

−0.18 0.06 0.01 −0.21 −0.11 0.01 0.01 0.00 0.00 0.18

−0.34 −0.12 0.00 0.02 0.00 0.03 0.03 0.01 0.01 0.19

−0.18 −0.10 0.01 0.02 0.00 0.02 0.01 0.00 0.00 0.17

Equivalent group 2 BC2

MT1-2

Max ΔrTDZ/E (n10−4)

Max ΔrTDZ/E (n10−4)

MT2-2 Relative difference with BC2 results (%)

Max ΔrTDZ/E (n10−4)

Relative difference with: BC2 MT1-2 results (%) results (%)

LC1 LC2 LC3 LC4 LC5 LC6 LC7 LC8 LC9 LC10

6.08 4.33 3.82 3.51 2.91 2.45 2.10 1.21 1.35 1.44

6.10 4.33 3.82 3.51 2.90 2.45 2.10 1.21 1.35 1.44

0.22 0.05 0.05 −0.18 −0.09 0.02 0.00 0.00 0.00 0.00

6.10 4.34 3.82 3.51 2.90 2.45 2.10 1.21 1.35 1.44

0.27 0.11 0.07 −0.16 −0.09 0.01 −0.01 0.00 −0.01 0.00

0.04 0.07 0.02 0.02 0.00 −0.01 −0.01 0.00 −0.01 0.01

Equivalent group 3 BC3

MT1-3

Max ΔrTDZ/E (n10−4)

Max ΔrTDZ/E (n10−4)

MT2-3 Relative difference with BC3 results (%)

Max ΔrTDZ/E (n10−4)

Relative difference with: BC3 MT1-3 results (%) results (%)

LC1 LC2 LC3 LC4 LC5 LC6 LC7 LC8 LC9 LC10

6.01 4.45 3.91 3.55 2.83 2.40 2.06 1.16 1.27 1.37

6.04 4.45 3.91 3.55 2.83 2.40 2.06 1.16 1.27 1.37

0.40 0.03 0.01 0.00 −0.12 −0.01 −0.01 −0.01 −0.02 0.00

6.04 4.45 3.91 3.55 2.83 2.40 2.06 1.16 1.27 1.37

0.41 0.04 0.02 0.01 −0.12 −0.03 −0.02 −0.02 −0.03 −0.01

0.01 0.00 0.01 0.01 0.00 −0.02 −0.01 −0.01 −0.01 −0.01

Table 8 Influence of numerical settings on stress range results.

Log sample interval (s) Line target damping (% critical damping level; inversely proportional to excitation period) Inner time step (fraction of shortest natural period) Max ΔsTDZ/E (n10−4) Relative difference with BC3 results

BC3

MT1-3

MT1-3n

0.1 10 10 6.0112 N/A

0.1 10 10 6.0351 0.398%

0.2 5 20 6.0111 −0.001%

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Table 9 Comparison of maximum Δs in the TDZ for models with nonlinear (NL) seabed.

LC1 LC10

BC1 NL

MT1-1 NL

Max ΔsTDZ/E (n10−4)

Max ΔsTDZ/E (10−4)

4.97 1.15

4.99 1.15

MT2-1 NL Relative difference with BC1 NL results (%)

0.32 −0.05

Max ΔsTDZ/E (10−4)

4.99 1.15

Relative difference with: BC1 NL results (%)

MT1-1 NL results (%)

0.48 0.01

0.16 0.05

5. Potential applications of framework

Fig. 7. Seabed responses at TDP for the nonlinear soil model.

Fig. 8. Normalised trench profiles for the nonlinear soil model.

(Do) is common in SCR studies (e.g. Bridge and Howells, 2007; Clukey et al., 2007). Here, the normalised trench profile is defined as the plot of maximum penetration during one cycle of motion divided by Do, versus the normalised arc length (Sn). Normalised trench profiles are represented in Fig. 8 and match well for all the models, therefore validating Eq. (11).

Establishment of suitable non-dimensional groups will assist future numerical and experimental studies investigating SCR dynamics. Indeed, defining appropriate scaling allows for optimisation of the number of simulations, as similar stress range and, therefore, similar fatigue life can be predicted with models having the same nondimensional groups. Moreover, although the non-dimensional groups impacting the stress range have been identified, their individual contributions to the stress range amplitude have yet to be determined. This represents the next stage, where sensitivity analyses will be undertaken, with the results synthesised to allow the dynamic stress range to be expressed directly in terms of the non-dimensional groups listed in Eq. (10). The non-dimensional groups presented in this paper will also assist future studies aiming to quantify DAF sensitivity to SCR system parameters and environmental loading, and hence will contribute in simplifying structural analyses of SCRs. Fig. 9 illustrates how dimensionless groups can be used to investigate the sensitivity of the fatigue life of SCRs to a chosen non-dimensional input (or group of non-dimensional inputs), considering for example the effect of the normalised linear soil stiffness (π11). The variation of maximum normalised stress ranges against normalised excitation periods (π4), which reflect the maximum velocity of the riser motion, as well as the DAF for equivalent groups 1 and 2 (i.e. BC1 and BC2), are shown in Fig. 9. The soil stiffness is the only difference between BC1 and BC2 (π11 ¼ ks/E for BC2 is 10 times higher than for BC1). Fig. 9a confirms that increasing the soil stiffness has a significant effect on amplification of the induced maximum stress range and therefore reduction of the fatigue life of the system, as noted previously in the literature (e.g. Bridge et al., 2004). Also, it is shown in Fig. 9b that, overall, DAF sensitivity to the non-dimensional soil stiffness is limited. Differences between DAF values at a given π4 are smaller than 10% with the highest differences observed for motions with short periods (high π4 values) and almost no difference for motions with longer periods. The softer soil modelled in BC1 leads to slightly higher DAF values for dynamic cases, which is consistent with previous results presented by Quéau et al. (2011). Further work would aim to test the robustness of results on the soil stiffness effect by using new numerical models with different values of dimensionless groups, and more especially to perform sensitivity studies for the other non-dimensional groups of parameters that have been shown to influence the SCR fatigue life. The non-dimensional groups also facilitate the design of experiments. Two different scaling approaches were tested to represent typical experimental practice. The main approach consists in varying the gravity acceleration so as to perform centrifuge tests. However, for practical reasons, experiments may also be carried out in 1g by means of the proposed alternative scaling approach. This study has focused on fatigue issues in the TDZ, but the results presented here also establish dynamic similarity along the entire riser. Hence the proposed dimensionless groups could be used for other problems such as analysis of vortex induced vibration (VIV). However, for VIV studies it may be more appropriate to: (i) use a varying drag

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Fig. 9. Normalised linear soil stiffness (π11) effect on dynamic amplification; β ¼01.

coefficient, and therefore scale the Reynolds number through scaling of the water kinematic viscosity (as length and velocity are already similar with the defined non-dimensional groups), (ii) account for the effect of structural damping and scale it accordingly (Rateiro et al., (2012)).

6. Conclusions It is commonly acknowledged that uncertainties are present in SCR design, and it remains difficult to quantify the level of accuracy in fatigue estimates. In consequence, design guidelines recommend that sensitivity analyses are performed for any given design, varying the key parameters that govern SCR behaviour. The study developed in this paper details the application of dimensional analysis to identify the entire set of non-dimensional groups that control the normalised stress and stress ranges within an SCR. The aim has been to facilitate sensitivity analyses, and also to lay the ground work for subsequent studies to evaluate functional relationships between the input nondimensional groups and the output normalised stress ranges and resulting dynamic amplification factors. This has the potential to simplify the estimation of SCR fatigue life within the touchdown zone. Non-dimensional groups influencing the dynamic and static response of SCRs to top excitations, and subsequently the DAF values, were identified for linear and nonlinear seabed models. The study was limited to in-plane motions of the riser resulting from sinusoidal motion of the vessel. Appropriate scaling factors were determined in order to obtain equivalent models with similar shapes and structural responses for a given input motion. The proposed non-dimensional groups will allow improvement of riser sensitivity analyses through better control on input parameters, reduction in the number of simulations, together with easier comparisons and interpretation of the results. The groups will also assist in the planning of 1g and centrifuge experiments through appropriate scaling of parameters that play a critical role in riser behaviour. Detailed investigation of the influence of each dimensionless group on the dynamic stress ranges and fatigue life of SCR systems is currently underway, using the validated non-dimensional groups from this study.

Acknowledgements This research is being undertaken with funding from the Lloyd's Register Foundation. The Lloyd's Register Foundation funds education,

training and research programmes in transportation, science, engineering, technology and the safety of life, worldwide for the benefit of all. This research forms part of the activities of the Centre for Offshore Foundation Systems (COFS), established under the Australian Research Council's Research Centres Programme and currently supported as a node of the Australian Research Council Centre of Excellence for Geotechnical Science and Engineering. The first author acknowledges her research studentship support from The University of Western Australia.

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