تجزیه و تحلیل حساسیت جهانی از تاثیر ناخالصی بر نارسایی خط لوله CO2

This paper describes the testing, comparison and application of global sensitivity techniques for the study of the impact of the stream impurities on CO2 pipeline failure. Global sensitivity analysis through non-intrusive generalised polynomial chaos expansion with sparse grids is compared to more common techniques and is found to achieve superior convergence rate to crude Monte Carlo, quasi-Monte Carlo and EFAST for functions with up to a moderate level of “roughness”. This methodology is then applied to the hypothetical full bore rupture of a 1 km CO2 pipeline at 150 bara and 283.15 K. The sensitivity of the ensuing outflow to the composition of a quaternary mixture of CO2 with N2, CH4 and O2 as representative stream impurities. The results indicate that the outflow rate is highly sensitive to the composition during the early stages of depressurisation, where the effect of the impurities on phase equilibria has a significant impact on the outflow.

As part of the carbon capture and sequestration (CCS) chain, pressurised pipelines are considered to be the most practical and efficient means for transportation of the large amounts of CO2 captured from fossil fuel power plants for subsequent sequestration [22]. It is inevitable that such pipelines will cover distances of several hundreds of kilometres, possibly at line pressures above 100 bar. Given that CO2 gas is an asphyxiant at concentrations higher than 7% [25], the safety of CO2 pipelines is of paramount importance and indeed pivotal to the public acceptability of CCS as a viable means for tackling the impact of global warming [22]. The outflow and its variation with time following pipeline failure dictates the resulting atmospheric dispersion of the escaping inventory, an example of which can be observed in Fig. 1. These data govern all the consequences associated with the pipeline failure, including minimum safe distances to populated areas and emergency response planning. Full-size image (18 K) Fig. 1. CO2 outflow and dispersion during pipeline decompression courtesy of Dalian University. Figure options Naturally a great deal of uncertainty is present due to the many possible circumstances in which a failure occurs including failure type, i.e. puncture or full bore rupture, initial failure pressure and temperature and variations in the captured stream composition due to fluid stream sources based on differing capture methods (i.e. pre-combustion, post-combustion or oxyfuel) [12] and post-capture processing. The use of predictive models to examine the sensitivity of the consequences of pipeline failure to these variations is standard within a quantitative risk assessment [23]. Such an analysis is often conducted using a one-factor-at-a-time (OAT) methodology [23] and [47], but as discussed by Saltelli and Annoni [35] this technique assumes an underlying linear behaviour, which is unlikely to be the case in such complex systems. To avoid such an assumption about the underlying model, a global sensitivity analysis (GSA) is required. GSA is concerned with quantifying how the variation in the model's output depends on different sources of variation over the entire parameter space, here treated as random input data, by providing quantitative importance measures that relate the variance of the output with each input variable. This form of analysis of model sensitivity has been applied to parts of the CCS chain, to the geological storage of CO2 by Kovscek and Wang [24] where the effect of porosity and permeability on reservoir performance was assessed, and widely applied in environmental engineering. For example, Cea et al. [10] studied the effects of aleatoric and epistemic uncertainty on a water quality model for evaluating biological pollutant concentration. Given the complexity of the fluid and thermodynamic behaviour of the flow following a pipeline failure, substantial resources are required for its computation [28], and as a result, the application of GSA has been considered impractical. However, the success of recent work [6] and [28] to decrease the computational expense of each simulation enables one to calculate the total sensitivities. In this work GSA is applied using a sensitivity measure proposed by Sobol' [41] to gain a better understanding of the effect of impurities on the outflow following pipeline failure. The Sobol' method is related to analysis of variance (ANOVA) and decomposes the model variation into a number of effects that represent the influence of each input, represented by a probability distribution, and their interactions. Many methods have been proposed to compute the integrals required to calculate these effects, of these the most widely applied are Monte Carlo sampling and the Extended Fourier Amplitude Sensitivity Test (EFAST) [30]. These approaches usually require large sample sizes to provide accurate estimations of the sensitivities, making them impractical when the underlying model is computationally expensive. Sudret [44] proposed a procedure for the computation of the Sobol' sensitivity measures through the approximation of the model's output by a polynomial expansion, known as generalised polynomial chaos (gPC) [19]. The gPC expansion is a linear combination of suitable global polynomial approximations in probability space, for which the statistical moments, expected value and variance, are known exactly from the coefficients of the expansion (see also [13] and [16]). The family of orthonormal mono-dimensional polynomials is selected in accordance with the general Askey scheme [50] with respect to the probability measure of each random input variable. The gPC expansion may be constructed intrusively by a Galerkin projection reformulation of the underlying problem or through non-intrusive approaches such as projection and regression (see [5] and [45]). In recent years stochastic collocation [4] and [49] has been applied to build sparse gPC expansions on tensor grids for high dimensional random input data (see e.g. [9]), to mitigate the so-called “Curse of Dimensionality”. This method constructs an approximative function that is a sum of Lagrangian interpolants on a set of points, which is known as a sparse grid (see [8]) (originally introduced by Smolyak [39] for multi-dimensional integration). The approximative function can be converted into the form of a gPC expansion. Formaggia et al. [18] applied GSA with gPC expansion derived from the stochastic collocation method to a basin-scale geochemical compaction model and advocated its applicability to models subject to high dimensional random input data. This sparse gPC expansion potentially requires far fewer function evaluations than the other methods identified above, meaning that the use of GSA for complex numerical models, such as that required for modelling the discharge following pipeline failure, may be tractable. The paper is organised as follows. Section 2 presents a review of a particular decomposition of a multi-variate function (Section 2.2). It is then shown how this expansion is used to define the Sobol' sensitivity indices (Section 2.3) and a number of common methods (i.e. Monte Carlo, EFAST and gPC) for calculating the Sobol' indices are presented. These methodologies are then tested against two benchmark test functions, and a family of test functions constructed to investigate the robustness of gPC (Section 2.4). The test functions constructed exhibit near discontinuous behaviour, and further difficulty is induced with additive artificial white noise. In Section 3 the most efficient of these techniques, in terms of convergence per number of function evaluations, is applied to a pipeline failure scenario. An extensively validated pipeline decompression model is presented in Section 3.1, while the uncertainty in the likely composition of a CO2 stream is discussed in Section 3.2. Firstly, Monte Carlo simulation is used to estimate the outcome probability distribution and perform a crude sensitivity analysis with scatter plotting. The final analysis serves as a framework for future work on consequence analysis for pipeline failure under uncertainty. Finally conclusions resulting from this work are drawn in Section 4.

In the CCS chain, pressurised pipelines employed for the transportation of the captured CO2 for subsequent sequestration will inevitably contain a range of stream impurities. The above presents a significant challenge given the established marked impact of the type and composition of the stream impurities on the safe and economical pipeline transportation of CO2. Predictive models utilised for design and risk assessment of such a system have been exploited for studying the sensitivity to such inherent variations. However, in the context of CO2 pipeline transportation, given the very large number of potential variables, and the complexity of the models required, mean that the computational cost for a full global sensitivity analysis will be prohibitive. In this paper commonly applied methods of GSA, i.e. MC, QMC and EFAST were first reviewed. Additionally, a gPC technique based on sparse grids was described. This formulation allows the simple evaluation of the Sobol' indices, while the sparse grid sampling greatly reduces the number of sample evaluations required. These methodologies were then applied to two benchmark test problems found in the literature and a further problem constructed to replicate discontinuous behaviour. In the former the results indicated that the sparse grid based gPC method was able to achieve machine precision accuracy. For the latter problem it was observed that as the function became “rougher” the convergence rate decreased, and in the extreme case presented convergence was not observed. The addition of increasing levels of noise to this problem, used to replicate numerical error in computational simulations, showed that the gPC performed moderately well with low levels of noise, but again fails to converge as this was increased. In comparison the other methods tested require substantially larger sample sizes to achieve equivalent accuracy. On the other hand, as expected, the performance of MC/QMC was almost unaffected by the applied numerical noise. In summary, the gPC outperformed the other methods tested in all cases in terms of convergence per number of function evaluations, except in a single case where a very high level of noise was present. The gPC technique was then applied to an analysis of an hypothetical pipeline failure under uncertainty in CO2 mixture composition. An initial uncertainty analysis showed a variation in the outflow rate (after 1 s) of >10%>10%. Clearly, given that the outflow rate largely dictates the resulting dispersion, this level of variation has significant implications for the emergency response planning. Furthermore, scatter plotting showed that of the three impurities considered (N2, CH4 and O2) only N2 had a linear impact on the outflow rate. The results of the gPC for the full decompression showed three distinct regimes of behaviour in which it was found that generally N2 had the greatest impact on the outflow rate. In particular the second regime, in which the Sobol' indices show a relatively stable behaviour, appears to be the most for assessing the overall importance of each component. In conclusion, it should be noted that the CO2 impurities sensitivity analysis performed in this study primarily focused on pipeline transportation issues. Although in a wider context, the proposed sensitivity analysis could serve as part of a techno-economic analysis of the impact of impurities for the entire CCS chain.