Practical Airgap Prediction for Offshore Structures

Two new methods are proposed to predict airgap demand. Airgap demand is is the maximum expected increase in the water surface elevation caused incident waves interacting with an offshore structure. The first new method enables inclusion of some second-order effects, though it is based on only first-order diffraction results. The method is simple enough to be practical for use as a hand-calculation in the early stages of design. Two existing methods of predicting airgap demand based on first-order diffraction are also briefly presented and results from the three methods are compared with model test results. All three methods yield results superior to those based on conventional post-processing of first-order diffraction results, and comparable to optimal post-processing of second-order diffraction results. A second new method is also presented; it combines extreme value theory with statistical regression to predict extreme airgap events using model test data. Estimates of extreme airgap events based on this method are found to be more reliable than estimates based on extreme observations from a single model test. This second new method is suitable for use in the final stages of design. Page 1 of 29 Bert Sweetman Introduction and Background Present airgap design methodologies for floating structures are not standard and rely heavily on empirical knowledge and model tests. Large production semi-submersibles and novel large-volume offshore structures provide design challenges that are well outside the present design experience base. Unlike drilling semisubmersibles, these vessels are generally required to remain on station throughout the most severe weather. Urgency is added by the fact that airgap design problems (wave impacts) have been encountered on large North Sea semi-submersibles, including the Veslefrikk B platform in the Norwegian sector of the North Sea, the vessel which is the subject of the analysis and model tests presented in this paper. High volume structures, whether fixed or floating, complicate the airgap calculation by significantly diffracting the incident waves. For these structures, ignoring diffraction effects is non-conservative in that diffraction effects generally worsen the airgap demand. Large-volume floating structures, including semisubmersibles and floating production, storage and offloading vessels (FPSO’s), offer the most significant challenge. Two distinct hydrodynamic effects are observed: (1) global forces and resulting motions are significantly affected by diffraction; and (2) the local wave elevation, η(t), is also significantly influenced by diffraction. For semisubmersibles, these wave amplification effects are most extreme at locations above a pontoon and/or near a major column. Here, four methods for prediction of airgap demand without use of second-order diffraction are presented and compared with model test results. First, a theoretical overview and motivation section outlines which physical terms are included in conventional firstand second-order diffraction analysis and the implications of the assumptions implicit to various post-processing methods. Some background is then presented which is relevant to all of the methods presented here. This background includes airgap notation, model testing issues and methodologies and some statistical background. Two existing and one new method of predicting airgap demand using results from only first-order diffraction are then presented and critically compared with model test data. It is concluded that second-order effects should be included in airgap prediction throughout the early stages of design, but that these methods are inadequate for final design. Another new methodology is then presented which is useful for final design. In the new methodology extreme value theory and regression techniques are combined to predict extremes from model test results. Theoretical Overview and Motivation The disturbed water surface in the presence of the vessel, η(t), is assumed to be a sum of incident and diffracted waves, ηi and ηd, each of which is a sum of firstand second-order components: η(t) = ηi(t) + ηd(t) (1) ηi(t) = η1,i(t) + η2,i(t) (2) Page 2 of 29 Bert Sweetman ηd(t) = η1,d(t) + η2,d(t) (3) This assumption is consistent with most state-of-the-art nonlinear hydrodynamic analyses, which employ second-order perturbation solutions. Combining Eqs. 1–3: η(t) = η1,i(t) + η1,d(t) + η2,i(t) + η2,d(t) (4) In a conventional second-order analysis, all four terms in the sum are considered. In a first-order analysis, only the first two terms are considered, i.e., second-order effects in both the incident wave and diffraction processes are neglected. In the “hybrid” methods investigated here, second-order effects of the incident waves are considered, while second-order diffraction effects are neglected. Thus, the methods proposed here lie between firstand second-order analysis. Three methods of applying first-order diffraction results to predict airgap demand including some secondorder effects are compared in this paper. The difference between the two existing methods is in the application of the Stokes second-order wave component. In the method denoted “Stokes second-order,” a square matrix of Stokes second-order transfer function components is developed and applied as in Eqs. 18–21 (to be discussed). In the method based on narrow-band theory, the extreme of the total wave process, η(t), is assumed to coincide in time with that of η1(t); it is therefore only necessary to model second-order effects during that single, largest wave cycle, so the Stokes second-order contribution is calculated for only that largest wave. Special consideration of only the single largest wave cycle is conceptually similar to the design wave approach. The new method proposed here, “First-Order with Statistical Correction,” implicitly assumes that the airgap demand resulting from the incident wave is statistically similar to the incident wave and applies a statistical correction to approximate the non-linear effects. The advantage of any of these “hybrid” approaches is computational simplicity: η1,d requires only (relatively straightforward) linear diffraction analysis, and η2,i, is available analytically from second-order Stokes theory. The latter two methods are of sufficient computational simplicity to allow inclusion of second-order effects as a hand-calculation, while the first of the three methods requires solution of an eigenvalue problem. Second-order diffraction analyses are available (e.g. [3]) but these second-order methods are difficult to apply and presently lack widespread use and verification in modeling the nonlinear diffracted wave surface. Standard airgap response prediction uses linear theory, which generally does not effectively reproduce measurements from model tests [2, 4–6]. First-order diffraction is often selected over the more powerful second-order because of its relative simplicity, ease of use and robustness of the solution. While second-order diffraction effects are expected to better reflect observed data, these radiation/diffraction panel calculations have been found sometimes to over-predict airgap demand [4, 7]. Neglecting second-order diffraction substantially simplifies computations but is clearly expected to underestimate the spatial variability in diffraction effects; such an underestimation has previously been shown for this platform [2]. The results presented later in Figure 5, however, show that optimal use of second-order Page 3 of 29 Bert Sweetman diffraction results also underestimates this spatial variability. A goal here is to understand the numerical impact of this simplification through use of three different stochastic models. Airgap Notation and Modeling Issues Figure 1 shows a schematic view of a semi-submersible platform, both before and after waves are applied. In the absence of waves, the still-water airgap distance is denoted a0. In the presence of waves, η(t) denotes the wave surface elevation relative to a vertically fixed observer at a particular horizontal location along the moving structure. The corresponding vertical motion of the platform is denoted δ(t). If η = δ, the airgap would remain equal to its still-water value, a0. More generally, the airgap response a(t) will be reduced from a0 by the difference, η(t)− δ(t): a(t) = a0 − [η(t)− δ(t)] (5) Deck impact occurs if the airgap a(t) < 0. Among the various terms in Eq. 5, the vertical offset δ(t) is perhaps the most straightforward to model. Linear diffraction results may often suffice to accurately model this offset. In contrast, the free surface elevation, η(t), generally shows nonlinear behavior—and hence represents a non-Gaussian process. Modeling attention is therefore focused here on η(t). This separation of vessel motions from diffraction effects on the water surface is fairly standard in hydrodynamic postprocessing, and is consistent with typical hydrodynamic diffraction analysis. This separation also makes these methods applicable to fixed offshore structures. Description of Model Test Data Test data used here for verification of the theoretical models come from a 1:45 length-scale model of Veslefrikk, which was tested in the wave tank at Marintek using various types of irregular waves [8]. Figure 2 shows a plan view of the platform, together with the 9 locations for which the airgap responses have been measured as a function of time. Note that airgap probes with lower numbers are generally further up-stream, i.e., closer to the wave generator. All tests studied here apply long-crested waves traveling along the diagonal of the structure. The platform rigid-body motions in heave, roll, and pitch—denoted ξ3, ξ4, and ξ5—have also been recorded. Sign conventions must be consistent with the model test data; here, ξ3 is positive in the upward direction, ξ4 is positive rolling to port, and ξ5 is positive pitching bow-down. The measured motions permit estimation of the net vertical displacement, δ(t), at any location (x, y) of interest: δ(t) = ξ3(t) + y · sin(ξ4(t))− x · sin(ξ5(t)) (6) The time-history of the elevation of the water-surface relative to a fixed observer is inferred using Eq. 5 and Page 4 of 29 Bert Sweetman the measured airgap, a(t), and the estimate of δ(t) as: η(t) = a0 − a(t) + δ(t) (7) Table 1 summarizes the geometric properties of the platform as configured for the tests used here. Prior to the model test, waves are first generated in the model test basin in absence of the model. The incident wave, ηi(t), is measured at location 7 (Figure 2), where the platform is to be centered. Following common practice, wave histories have been generated from a stationary random process model, applied over a fixed “seastate” duration of Tss=3 hours. Its spectral density function, Sη(f), is described by the significant wave height Hs=4ση, the peak spectral period, TP , and the spectral peakedness factor γ associated with the JONSWAP spectrum (e.g. [9] from [10]). Table 2 describes the Hs, TP , and γ values for each of the three test conditions. The peakedness factor, γ, is the ratio of the maximum spectral density to that of the corresponding Pierson-Moskowitz spectrum (e.g. [9]). The γ reported for the bimodal spectrum is an approximation of the peakedness parameter for an equivalent JONSWAP spectrum. An actual JONSWAP spectrum with the parameters indicated for this bimodal spectrum would not be realizable due to wave breaking. Statistical Background: Prediction of Extremes Three of the four methods presented in this paper predict extreme values by first predicting the desired fractile of a Gaussian process (e.g., fractiles of the maximum response over a fixed duration) and then relying on the Hermite model to transform that fractile of the standard normal distribution to the equivalent fractile of a non-Gaussian response. Gaussian Extremes Extreme values of a random variable which is assumed to follow a standard-normal, or Gaussian, distribution can be predicted by assuming upcrossings of high levels of u(t) follow a Poisson process (e.g. [11] from [12, 13]) so that: P [Umax ≤ u] = e−ν0Te −u2/2 (8) where ν0 is the average up-crossing rate and T is the duration. The number of cycles, N , is given by N = ν0 × T . Setting the probability P in Eq. 8 to p, the resulting fractile is given by: umax,T,p = √√√√2 ln   N ln ( 1 p )   (9) Consistent with Type I extreme value theory (e.g. [14] from [15]), the mean of umax,T,p is assumed to be well-approximated by its p = 0.57 fractile, so the expected value of the maximum of the Gaussian process Page 5 of 29 Bert Sweetman can be estimated by: E[Umax] = √ 2 ln ( N 0.562 ) (10) It has been shown [16] that for problems such as those presented here, the result of Eq. 9, which is strictly based on the Poisson and Gumbel models, differs only slightly (e.g. 0.2%) from the classic Eq. 11 (e.g., [17]): E[Umax] ≈ √ 2 lnN + 0.577 √ 2 ln N (11) The expected value of the first-order maximum wave elevation is then: E[η1,max] = ση1E[Umax] + mη1 (12) The result of Eq. 10 or 11 is an estimate of the expected value of the maximum of a normal distribution which can be used directly in the Hermite model to predict non-Gaussian extreme values. Non-Gaussian Extremes The Hermite model assumes the non-Gaussian process η(t) to be a cubic transformation of a standard Gaussian process u(t). Hermite transformation, g(u), can be conveniently written as a polynomial [18]: η = g(u) = mη + κση1 [u + c3(u 2 − 1) + c4(u − 3u)] (13) For statistically “softening” processes, i.e. those with coefficient of kurtosis greater than 3 (α4 > 3), the variance of η is preserved by setting κ = [ 1 + 2c3 + 6c 2 4 ]−1/2 (14) in which the coefficients c3 and c4 control the shape of the distribution, i.e. preserve the desired skewness (α3) and kurtosis (α4). For processes that have relatively mild deviations from Gaussian behavior: c3 = α3 4 + 2 √ 1 + 1.5(α4 − 3) ≈ α3 6 (15) c4 ≈ α4 − 3 24 (16) Assuming the same transformation g in Eq. 13 applies at every point in time, including those points which are the maxima: E[ηmax] = g(E[Umax]) (17) in which E[Umax] can be conveniently estimated using Eq. 10. Prediction of Extremes from First-order Diffraction Analysis Two methods proposed by Sweetman and Winterstein [1] include non-linear, non-Gaussian effects as part of post-processing linear diffraction results. These methods are briefly discussed here for comparison with a Page 6 of 29 Bert Sweetman newly proposed method. In [1], the incident waves were considered a principal source of non-Gaussian effects; these effects were estimated by applying complete Stokes second-order transfer functions to the results of first-order diffraction analysis. The method is relatively rigorous from a physical standpoint: those terms being neglected from a complete second-order analysis are clearly identified. Unfortunately, the method is somewhat computationally intensive: it requires generation of a square matrix of second-order transfer functions which are combined with the transfer functions from first-order diffraction and the target seastate. An eigenvalue problem is then solved to determine the non-Gaussian statistical properties of the response. A less computationally intensive approach based on narrow-band theory was also proposed in the same work [1]. This method assumes the maximum event as predicted from first-order theory coincides with that in second-order theory. Accordingly, a Stokes second-order correction is calculated at a single frequency and is applied to only the single maximum event predicted by first-order theory. The resulting method is computationally simple, but is an overly simplistic representation of the physical phenomenon. Also, use of the narrow-band assumption, which implies a worst-case phase locking, often over-predicts the observed maxima of the airgap demand. A new method is proposed here in which a purely statistical correction based on the non-Gaussianity inherent to the incident waves is applied to the maximum airgap response process. The method is similar to that based on the Stokes second-order correction in that the incident waves are assumed to be a principle source of non-Gaussianity, but here the non-Gaussian effects are captured directly as statistical quantities, rather than physically through Stokes second-order transfer functions. The resulting method is computationally simple, yet does not rely on the narrow-band assumption. Stokes Second-Order Correction In this method post-processing first-order diffraction results, the only non-linear effects considered are those associated with the incident waves. This method is relatively rigorous from a physical standpoint in that the terms being neglected from a complete second-order analysis are clearly identified: in Eq. 4, all terms are included except η2,d, which is assumed equal to zero. This method was previously proposed by Sweetman and Winterstein [1]. In modelling non-linear systems, it is common to employ Volterra series that enable estimation of a response quantity as a sum of firstand second-order transfer functions. For floating structures, these transfer functions are generally obtained from second-order hydrodynamic diffraction software packages such as WAMIT [3]. Here, Volterra series are used to calculate the free-surface elevation of the sea, η. η(t) = η1(t) + η2(t) = η1(t) + η2+(t) + η2−(t) (18) where η2+(t) and η2−(t) are the second-order sum and difference frequency contributions to the response. Page 7 of 29 Bert Sweetman Each of these quantities can be written in terms of firstand second-order transfer functions: