Compliant vertical access riser assessment: DOE analysis and dynamic response optimization

Applied Ocean Research 41 (2013) 28–40

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Compliant vertical access riser assessment: DOE analysis and dynamic response optimization Michele A.L. Martins* , Eduardo N. Lages, Eduardo S.S. Silveira Scientific Computing and Visualization Laboratory, Federal University of Alagoas, Macei´ o, AL, Brazil

a r t i c l e

i n f o

Article history: Received 28 October 2012 Accepted 9 February 2013 Keywords: Compliant vertical access risers Design of experiments Dynamic response optimization

a b s t r a c t As a contribution to the deepwater oil and gas industry, this paper addresses the use of optimization techniques together with a design of experiments (DOE) assessment, as a way of automating the design of compliant vertical access risers (CVARs) while also leading to an optimal riser configuration based on some desired efficiency parameters. The CVAR is a new riser concept that can improve the structural performance of the production system and also provide several operational benefits. The DOE is a statistical technique that provides an objective measure of how design parameters are correlated and the effective contribution of each one at the riser performance. Based on such a study some general conclusive remarks on the global behavior of CVAR will be presented. Such results also play an important role for the optimization process, as it can highlight significant design parameters, enabling design simplifications and efficiency improvement. For optimization assessment, geometric parameters are taken as the design variables and the design constraints consider both structural integrity and operational criteria. A multi-objective approach is considered taking into account the structural performance and geometric criteria. Optimal solution is obtained by NSGA-II method. Extreme and operational environmental conditions of a Brazilian offshore field are used as the base case.  c 2013 Published by Elsevier Ltd.

1. Introduction New discoveries of oil and gas continue to be made at an increasing rate in several deepwater areas around the world. It is known that the biggest oil reserves are currently on the continental shelf, in deep (300–1500 m) and ultra-deepwaters (above 1500 m), as Brazilian pre-salt fields. However, there are several technological challenges for oil exploitation in such fields, e.g. instability of the salt layer, high temperatures and pressures, high waterdepths, among others. Concerning increased waterdepths the implementation of structural systems for offshore oil production becomes even more complex. In such a scenario, commonly associated with severe environmental load conditions, one of the greatest challenges is the definition of a riser system capable of withstanding the high stresses imposed on the structure. Production risers can be installed at different geometric configurations, depending upon several factors and key field parameters, such as water depth, environmental conditions, installation aspects, platform concept, production rates, well pressure and temperature, among others. Each riser configuration varies in complexity and has its own set of advantages and disadvantages. Many different proposed and existing riser systems have been extensively discussed for * Corresponding author. Tel.: +55 82 32141303. E-mail addresses: [email protected] (M.A. Martins) [email protected] (E.N. Lages) [email protected] (E.S.S. Silveira).

c 2013 Published by Elsevier Ltd. 0141-1187/$ - see front matter  http://dx.doi.org/10.1016/j.apor.2013.02.002

ultra-deepwater applications [1–3]. To absorb the motions of the floater without exceeding the risers structural capacity so that it can withstand the imposed loads, special motion compensation systems or compliant riser configurations could be used. Conventionally, steel catenary risers (SCR) are used. They are free hanging steel pipes suspended from the platform to the wellhead in a catenary shape and run along the seafloor for a distance in order to absorb heave motion imposed by the platform. Fatigue induced by VIV (vortex induced vibrations) and heave motion are serious concerns in the design of this kind of riser system. Furthermore, at deepwaters they can also be subjected to high concentrated tension, especially at the top end connection. To improve riser performance, the addition of external buoyancy to the riser structure, modifying its static shape and dynamic response is common. These kinds of risers are usually called wave systems and have gained popularity as a viable solution to improve fatigue and strength performance at the touchdown zone compared to simple catenary risers. The characteristic compliance of such risers configuration is a key feature to properly absorb floater motions by change of geometry, without use of heave compensation systems. Thus, such systems present a more favorable structural behavior compared to the usual free-hanging catenary, under environmental loadings of waves, marine current, and the motions imposed by the platform. Another alternative of riser configuration is the top tensioned riser (TTR) that consists of a steel vertical pipe tensioned by the platform that connects the subsea well bore to the floating production vessel,

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enabling dry tree production. This configuration has the advantage of low costs but is very sensitive to heave motion, requiring the use of tensioners and heave compensators in order to maintain the safety of the riser and avoid buckling. Other known riser concept is the hybrid riser. In general, it consists of a vertical steel bundle of pipes supported by external buoyancy. The required compliancy is provided using short flexible pipe jumpers located near the surface to accommodate relative motions between the riser and the floating production unit. Studies had already shown that the hybrid arrangements are technically feasible and cost effective [4]. The impact of ultra-deepwaters on the risers design leads to the need of the development of new alternatives for riser configuration, enabling to overcome the barriers imposed by deepwater fields for oil exploitation in an efficient, safe and economic way. An example of a non-classical riser configuration is the compliant vertical access riser (CVAR): a lower-tension riser configuration fixed to a floating platform while still providing dry tree vertical access to the well bore. Its characteristic geometrical shape is achieved by the addition of special elements, such as syntactic buoyancy and additional weight, properly positioned along the riser. The main feature of CVARs is a nearly neutrally buoyant central region near an inflection point that tends to isolate the motions of the upper region from those of the lower region of the riser. The definition of the CVAR geometric configuration is dependent upon several design variables, further detailed, which increases the complexity of its design, compared to the design of a simple catenary riser, for example. Therefore, as the number of design variables increases, the number of possible configurations increases as well. This implies a growth of the search domain of the problem as a well as its complexity. This complexity is directly related to the time and computational cost demand. Due to this aspect, a methodology focusing on the global analysis and optimal design of the CVAR is presented. This methodology comprises a design of experiments assessment followed by an optimization design process. It is intended to give directions and knowledge of the relative importance of each design variable on the preliminary CVAR design and also introduce an automate methodology for the design of this kind of structure based on optimization procedures. Such methodology leads to a feasible riser configuration that is also the best one according to a given objective measure of efficiency in a short time and automatically. Hence, the use of optimization techniques simplifies and accelerates the design of such a structure. Details on how to perform each step of the entire structure design itself can be found mainly in the standards and recommended industry practices, such as [5,6]. As illustration on a typical deepwater CVAR a generic case has been defined and discussed here.

2. Compliant vertical access risers CVAR is an underdevelopment alternative to oil exploitation at deep and ultra-deepwaters, as the pre-salt fields. This riser configuration has not been installed yet, but studies prove its technical and economic feasibility. The concept was proposed in a patent application in 1988 [7], when a flexible riser arrangement associated with specific equipment to enable vertical access to the production well bore was idealized. A study considering CVAR system coupled to a floating production storage and offloading (FPSO) unit was made [8], and demonstrated its feasibility and also the possibility of using it with dry completion, turning this new configuration into a very economically attractive alternative for oil exploitation in deepwater fields. The same assessment was made considering different floating units, as semi-submersible platforms, deep draft caisson vessel (DDCV), floating production unit (FPU) and FPSO at depths of 1500–3000 m, and the CVAR system viability was demonstrated [9–11]. Work reference [12] carried out an extensive study on the global behavior of a generic

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Fig. 1. Compliant vertical access riser concept.

compliant composite riser for deep and ultra-deepwaters considering harsh environmental conditions such as found at, e.g. the Ormen Lange field offshore Norway. Analysis and large scale tests of CVAR concept were also undertaken in order to improve and verify the riser analysis methods, comparing the simulation results given by engineering specific software with the results of physical experiments [13,14]. The obtained results showed the great influence and importance of considering the effect of VIV to the global response of the riser. An important contribution on the design and structural behavior of CVARs was also made [15] regarding a comprehensive study of this new configuration, addressing the conceptual development of CVARs, details of the analysis and simulation and the risk assessment associated with its installation and operating conditions. 2.1. General description The CVAR system consists on a rigid riser with a peculiar geometry (Fig. 1) obtained by the provision of syntactic buoyancy modules fitted on the lower section of the riser and an additional weight on the upper section of the riser. These elements ensure that the risers ends remain mainly vertical, therefore allowing direct connection at its top end to the floating unit and direct access to the well bore. Due to CVAR characteristic compliancy, any top tensioning system, production jumpers or heave compensator is not required, as those used in TTR designs, since the shape of the riser itself is capable of absorbing the relative heave motion. The buoyancy modules must be defined to provide a thrust effort high enough to ensure that the effective tension in the lower slick pipe section always remains sufficiently high in order to avoid excessive bending stress at the riser base. As well explained by [12] the magnitude of the lateral deflection of a riser depends upon the tension level in the riser that can be provided by the non-linear geometrical stiffness effect applying some pretension to the structure. This applied pretension will increase the geometrical stiffness and as a consequence it will reduce the magnitude of the lateral deflection for equivalent lateral loading. For the case of CVAR concept as well, a pretension effect is obtained by the addition of the extra heavy weight and the buoyancy segments. The geometry of the system also allows direct intervention procedures in the well bore, since curvature constraints are satisfied,

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enabling workover operations to be performed directly from the production platform. Such condition strongly depends on riser geometric parameters. Thus, this feature eliminates the need of hiring specific units for that matter, making this new riser concept economically attractive. Therefore, the CVAR system has the potential to reduce the completion and well intervention costs. The characteristic compliancy of the CVAR system, required to enable direct connection of the CVAR to the platform unit as well as to the well bore, is achieved by providing an excess length of pipe, known as riser overlength, associated with a horizontal offset between the top and base connections. It is common practice to represent the riser overlength by the parameter known as overlength fraction, defined as the ratio of the overlength to the straight-line distance between the hang-off position and the subsea connection point. This parameter defines the degree of compliance of the CVAR system. The larger the over-length fraction the easier it is to keep the extreme stresses within limits, for higher motion floating units, once it allows the riser to better absorb the platform movements without causing the occurrence of high stress values. Thus, the system compliance is one of the most important design parameters of the CVAR system in a way that sufficient compliance must exist in order to satisfy the extreme response criteria when the FPU offsets to the near and far relative positions. Increased compliance, however, must be carefully analyzed to limit the potential for riser-to-riser and mooring lines contact. The horizontal offset of CVAR systems enables the wellheads to be offset by a considerable distance from the platform. This horizontal distance between the wellhead and the floating unit can be in the range of 10–100% of the water depth [11], thus enabling direct vertical access to subsea wells located at relatively large offsets. Due to the benefits provided by its differentiated geometry and composition, CVAR can be considered a potentially enabling technology for deep and ultra-deepwater field development.

3. Analysis and design methodology The aim of the global analysis methodology is to establish the important effects caused by the CVAR global behavior. These effects are used as the basis for the riser design playing an important role in the optimization design problem definition. The methodology applied follows a systematic procedure as outlined below:

1. 2. 3. 4.

5. 6. 7. 8. 9.

Identify all relevant input design variables; Identify all relevant operation conditions and limit states; Identify all relevant loads; Formulate the design process as an optimization problem (by the proper definition of design variables, constraints and objective functions); Establish load cases that will yield responses which reflect extreme combined load effects and represent all relevant limit states; Perform the design of experiments assessment; Evaluate DOE results of the riser global behavior and if necessary adjust formulation defined at step 4; Perform the optimization procedure conducting global riser analyses for the established load cases; Obtain the optimal riser configuration and check if all constraints are satisfied.

To conduct the above described methodology we made use of two computer programs: Anflex [16], developed by Petrobras, for the ® static and dynamic simulations of the riser, and modeFRONTIER [17], which consists of a multidisciplinary optimization program, for both DOE and optimization analysis.

4. Design of experiments Riser dynamics can be highly non-linear due to large displacements, potential soil–riser, riser–riser, and vessel–riser contacts and hydrodynamic effects. Therefore it is not easy to predict riser behavior, especially for complex configurations such as the CVAR system. Parametric studies are often undertaken to better understand the influence of a particular system characteristic or a design parameter on the riser dynamics. A parametric study is a collection of possible configurations generated by varying one or more simulation variables. Such studies often involve innumerous individual cases. Each of these cases must be properly simulated, considering static and dynamic analyses, which can be an extremely time consuming activity with great computational effort associated. An efficient alternative for that matter is the design of experiments (DOE) methodology for sensitivity analysis [18]. In statistics, DOE is usually used in controlled experiments. The original use of DOE refers to methods used to obtain the most relevant qualitative information from a database of experiments by making the smallest possible number of experiments. According to such method, experiments are planned in a way that redundant observations can be eliminated without any loss of representativeness, reducing the number of tests in order to provide information on the major interactions between the variables. Statistical DOE is usually part of the optimization process and it should always be applied before the actual optimization phase as it can be useful to establish the relationship between the measured responses of interest and the process factors (design variables) being studied. The factors may have individual effects on the response (referred to as main effects) or may have effects that are interdependent (referred to as interaction effects). Since the designed experiments are generated on the basis of statistical theory, confidence in the results obtained must be properly defined. By this study we aim at stressing the fact that, before employing a search strategy by an optimization procedure, it may be useful or even essential to carry out a preliminary exploration of the design space. This can be well done by the use of statistical DOE analysis. Such preliminary study is able to let the user build some understanding of the behavior of the objective functions and constraints and how those response parameters are related to input design variables, to provide an initial population of candidate designs and to also shrink the range of variations or even reduce the number of variables. To properly conduct the DOE methodology one has to follow several steps. It starts by identifying the input variables and the response parameters of interest on which the sensitivity will be assessed. For each input variable, a number of levels are defined that represent the range for which the effect of that variable is desired to be known. Experiments are then defined based on a specific experimental design method, which defines each input parameter for each run of the experimental tests. Responses obtained in each run of the test are assessed, observing the possible differences for different groups of the input changes. These differences are then attributed to individual input variables acting alone (main effect) or in combination with another input variable (interaction effect). Finally, the significance of the observed effect is assessed by means of statistical hypothesis test. There are several existing DOE methods and they can be categorized as follows: screening, known to be useful for obtaining information about the problem and about the design space; fractional and full factorial, indispensable in order to perform a good statistical analysis of the problem and also for studying the effect of the variables on the responses, as well as the effects of interactions between variables; response surface methodology (RSM), or metamodels, are mathematical, statistical, numerical models for time consuming, computationally exhaustive functions. An RSM offers a guess of the value of the unknown function at not yet evaluated points of the parameters space, on the basis of assumptions on the response function. Since

M.A. Martins et al. / Applied Ocean Research 41 (2013) 28–40

this study intends to perform a statistical analysis of the sensitivity of the CVAR design in terms of its design variables, a factorial method is adopted. 4.1. Fractional factorial method In order to determine whether any change in design variables affects the results of the CVAR system, the most intuitive approach would be to vary the factors of interest in a full factorial design, that is, to try all possible combinations of settings. This would work fine and provide the desired observations; however, the number of necessary experimental runs increases geometrically with the number of variables. Since each run is very time-consuming and associated to a great computational cost, it is often not feasible to execute such many test runs. Instead, a fractional factorial design method is used here. The fractional factorial method defines experimental designs consisting of a carefully chosen fraction of the experiments defined by a full factorial method. Such fraction is chosen so as to exploit the sparsity-of-effects principle capable to provide representative information about the most important effects related to the problem studied, while using a reduced fraction of the effort of a full factorial design in terms of experimental runs and computational resources. Fractional designs are usually expressed using the notation k(n − p) , where k is the number of levels of each factor investigated, n is the number of factors investigated, and p describes the size of the fraction of the full factorial used. A two-level design is usually enough for evaluating factors effect in many scientific problems [19]. Experimenters evaluating process changes are often interested in the factor effect directions that lead to process improvement. Regarding fractional factorial design, a half-fraction of the 2k design is usually adopted as it involves running only half of the treatments of the full factorial design. For this study, comprising eleven design variables, a full factorial design would define 2048 experiments (211 ). A half-fraction of this design is the design in which only half of such experiments would be required, equivalent to 1024 test runs. For experiments with many factors, two-level full factorial designs can lead to large amounts of data. However, in many practical cases, higher order interactions have no distinguishable effects on a response. As a result, a well-designed experiment can use fewer runs for estimating model parameters without any loss of representativeness. 4.2. Statistical hypothesis test A statistical hypothesis test is a method of making decisions using data obtained from a controlled experimental run. In this study the data are obtained from static and dynamic simulations of a CVAR subjected to functional and environmental loads. Each experiment defined by fractional factorial method represents a specific CVAR configuration to be run. The purpose is to verify the significance of the design variables and their interaction on response parameters. In statistics, a result is called statistically significant if it is unlikely to have occurred only by chance, according to a pre-determined threshold probability, the significance level, also known as Type I error. Such critical tests are also called tests of significance, and when such tests are available one may discover whether a second sample is or is not significantly different from the first. In other words, this test will evaluate if the mean response with respect to the upper level factors are significantly different from those with respect to the lower level factors. Statistical hypotheses testing and confidence interval estimation of parameters are the fundamental methods used for comparative experiment assessment, as is the interest in this study. To perform such analysis, one has to formulate the hypothesis to be tested: the null and alternative hypotheses, denoted H0 and H1 , respectively.

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Null hypothesis corresponds to a general or default position, and it is the statement that one wants to deny. Alternative hypothesis predicts the results from the experiment when the null hypothesis is denied. For example, the null hypothesis might be that there is no relationship between two response parameters,

H0 : μ+ = μ− ,

(1)

H1 : μ+ = μ− ,

(2)

where μ+ is the mean of specific response in the upper level of domain of the input variable, and μ− is the mean of specific response in the lower level of domain of the input variable. It is important to highlight that the null hypothesis can never be proven. A set of data can only reject a null hypothesis or fail to reject it. In other words, if the evaluated factor reveals no statistically significant effect on the riser response, it does not mean that the factor is not important and that the null hypothesis is true, it only means that there is not enough evidence to reject the null hypothesis. The decision of rejecting or not the null hypothesis can be taken based on a confidence interval. Here we consider a confidence interval of 95%, corresponding to a significance equal to 0.05. In order to assess the significance of the design variables effect on the response parameters a Students t-test is here considered. This test is used to evaluate if two sets of data differ significantly. Based on a two-sample pooled t-test with equal variances the test statistic is defined as   X + − X −  , (3) t= S P (1/n+ ) + (1/n− ) where n+ and n− are the numbers of values in the upper and lower parts of domain of the input variable; X + and X − are the means of the values for the response parameters in the upper and lower parts of domain of the input variable; SP is the general variance given by S 2P =

2 S+ =

2 S− =

2 + 2 (n− − 1) S− (n+ − 1) S+ , n− + n+ − 2

n+  i=1

x+ − X +

2

n+ − 1 n−  i=1

x− − X −

n− − 1

(4)

,

(5)

,

(6)

2

where x− and x+ are the upper and lower output variables, respec2 and S 2 are the variances of the population for the tively, and S+ − response output variable in the upper and lower parts of domain of the input variable. 5. Optimization methodology The ever increasing industry demand to lower production costs without compromising quality and efficiency has stimulated engineers to look for the adoption of decision making methods, such as optimization, to design and produce products and systems both economically and efficiently. Different methods have been developed and the optimization methodology has reached a degree of maturity over the last years. Successful applications examples are shown in a wide spectrum of industries, including aerospace, automotive and petroleum industries [20–25]. Examples of optimization technique applied to riser design can be found in [26]. Generally, optimization refers to the act of obtaining the best result under given circumstances done by the selection of the best element from some set of available alternatives. The main goal of such methodology can be either to minimize the effort required or to maximize any desired benefit. Such efficiency measures should be expressed as a real function of certain decision variables and so optimization consists of the process of finding variables values, from within an allowed

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set, that give the maximum or minimum value of the expressed real function. A constrained optimization problem, as general engineering problems, can be stated as find arg min f (x) ,

(7)

subject to the constraints gi (x) ≤ 0, h j (x) = 0,

i = 1, . . . , ng , j = 1, . . . , nh ,

lbk ≤ xk ≤ ubk ,

k = 1, . . . , nx ,

(8) (9) (10)

where x is the design vector comprising the decision variables, f (x) is termed the objective function, gi (x) and h j (x) are known as inequality and equality constraints, respectively and lbk and ubk are the lower and upper boundaries of the design variables, respectively. nx is the number of design variables, ng is the number of inequality constraints and nh is the number of equality constraints. Design variables consist of parameters whose values can be freely varied by the designer to define the designed structure, taken here as the riser configuration. Design constraints represent the restrictions that must be satisfied to produce an acceptable design. In general, there will be more than one acceptable design, and the purpose of optimization is to choose the best one among those based on the defined objective functions. The definition of the objective function is governed by the nature of each problem. In many engineering structural design problems, the objective is usually taken as the minimization of cost or minimization of structural stress or effort. However, there may be cases where the optimization with respect to a particular criterion may lead to results that may not be satisfactory with respect to another criterion. For the optimization of production riser configurations, design variables comprise geometrical parameters that define the riser configuration, such as total length, outer diameter and buoyancy diameter. Operational and structural criteria, e.g. admissible stresses, maximum curvature and minimum tension are usually taken as design constraints. For such problems, literature presents different objective functions approaches: cost minimization [27,28], minimization of the maximum dynamic stress amplitude along the riser and minimization of the maximum static stress along the riser [29]. 5.1. Multi-objective optimization In some situations, the need to consider more than one criterion to be satisfied simultaneously may occur. Such problems, involving multiple objective functions, are usually called multi-objective problems [30,31]. Also known as multi-criteria problems, they can be found in various fields wherever optimal decisions need to be taken in the presence of trade-offs between two or more conflicting objectives. A multi-objective problem comprising n objectives can be written as find arg min f (x) = [ f1 (x) , f2 (x) , · · · , fn (x)] ,

(11)

where f (x) is the objective vector and fi (x) is the ith objective function. For nontrivial multi-objective problems, one cannot identify a single solution that simultaneously optimizes each objective. While searching for solutions, multi-objective algorithms find solutions such that, it is impossible to improve one criterion without worsening another one. Such solutions are called non-dominated or Pareto designs. Finding such non-dominated solutions is the goal when setting up and solving a multi-objective optimization problem. In the absence of any further information, one of the Pareto optimal solutions cannot be said to be better than the other. This demands the algorithm to find as many Pareto solutions as possible and then assess such solutions to choose only one. To solve multi-objective

optimization problems, several methods have been proposed. They are mainly categorized as classical and evolutionary methods. Classical optimization methods suggest scalarizing the set of objectives into a single objective by multiplying each objective with a user supplied weight [32–34]. These weight values depend on the relative importance of each objective function. A list of a few commonly used classical multi-objective optimization methods can be found in [30]. On the other hand, evolutionary algorithms are based on natural evolutionary principles, such as reproduction, mutation and selection, to define the optimization process. Candidate solutions to the optimization problem play the role of individuals in a population, and the fitness function determines the environment within which the solutions live or not. Evolution of the population then takes place after the repeated application of the cited biological operators. Among multi-objective optimization methods are VEGA [35], MOGA [36], NPGA [37], NSGA [38], NSGA-II [39], SPEA [40]. Ref. [41] presents a comparison among different evolutionary algorithms for multi-objective problem solution. Evolutionary multi-objective optimization methods have been used in solving riser design problems [42]. Based on well succeeded applications with the use of the method NSGA-II for riser optimization, this work makes use of this specific method for the CVAR optimization. 5.2. NSGA-II Elitist non-dominated sorting genetic algorithm (NSGA-II) is a well-known multi-objective optimization algorithm whose main feature is the use of Pareto dominance relation and a diversity maintenance mechanism for fitness evaluation together with some sort of elitism. Initially some solutions are randomly generated to form an initial population. After the population is initialized it is then sorted into fronts based on non-domination characteristic of each individual. The first front comprises all non-dominant individuals in the current population; the second front is defined by the individuals dominated by those in the first front only and the definition of the following fronts goes so on. On the sequence, each individual in each front is assigned rank values (fitness). The assigned rank of each individual is used as the primary criterion in the parent selection. In addition to fitness value a new parameter called crowding distance is calculated for each individual, being a measure of how close an individual is to its neighbors. Therefore, individuals with the same rank value are evaluated by the crowding distance criterion. By this criterion, individuals in less crowded regions in the objective space are viewed as being better than other individuals in more crowded regions if they have the same rank value. For the definition of the next generation population, considering an initial population comprising N individuals, by applying selection, crossover and mutation operations, a N “children” population is generated. The generated population is added to the current parent population, generating a 2N preliminary population. Each individual in this enlarged population is evaluated by the rank assignment procedure and the crowding distance in the same manner as in the parent selection phase. The final population set for the next generation is constructed by choosing the best N individuals from the enlarged population. 6. Case study Given the complexity of the CVAR system and motivated by the lack of knowledge about this relatively new riser configuration, this paper presents in the first place a sensitivity analysis of the CVAR system based on statistical DOE methodology. Such analysis can provide information about the contribution of each variable on the problem

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studied. In addition, using DOE one can better understand the CVAR behavior under functional and extreme environmental loads. Additionally, with the increasing complexity of the risers configuration and the constant challenge of minimizing costs and improving the performance of structures, the use of an optimization methodology is of great importance. Based on this motivation, after DOE analyses, the design of the CVAR studied here is also evaluated by means of an optimization methodology. To perform sensitivity analysis a fractional factorial design of 211−1 is considered that represents a half fraction of a full factorial design with two levels (high and low) comprising eleven design variables. The results obtained from the main effects and first order interactions were tested for significance using the Student t-test. An important feature of the problem treated here is that it consists of an optimization problem via simulation, associated with a considerable computational effort, since for each configuration analyzed during the process it is necessary to simulate the static and dynamic behavior of the riser. In order to improve the performance of the solution, the whole analysis was carried out in the computer cluster of the Scientific Computing and Visualization Laboratory (LCCV) of the Federal University of Alagoas (UFAL). The cluster installed on LCCV/UFAL has 172 nodes with two processors per node and 4 cores per processor, comprising a total of 1376 cores 2.8 GHz Intel Nehalem. The cluster also has a numerical processor of 20 teraflops, a graphics processor of 32 teraflops, an interconnection network of 40 gigabits per second and a 50 terabytes parallel file system. The use of the cluster promoted a significant gain on the efficiency of the analysis, since it allowed the simulation to be run in parallel. To cope with demanded computational costs one can also make use of advanced techniques, such as response surface methodology or artificial neural networks to efficiently predict riser response. The main idea of such methodology is to represent a dynamic response parameter of the structure by means of an approximate simple mathematical model instead of performing the complete long term simulation. It performs a nonlinear mapping of the current and past system excitations to produce subsequent system response for the random dynamic analysis of mooring lines and risers, for example [43]. The study performed here made use of two computer programs: Anflex, developed by Petrobras, for the static and dynamic simula® tions of the riser, and modeFRONTIER , which consists of a multidisciplinary optimization program, for both DOE and optimization analysis. 6.1. Model description A rigid CVAR is analyzed. It is considered to be installed in a waterdepth of 2200 m connected to the central point of a monocolumn platform. The azimuth of the line is set equal to 135◦ (SW). Riser pipe is modeled with beam finite elements and buoyant and additional weight segments are modeled as distributed forces at the elements. Simulations consider irregular waves defined by the Jonswap Spectrum, and dynamic movements are defined by the equivalent harmonic. Geometrical and physical properties of the riser are summarized in Table 1. For the riser simulation, the riser is considered operating at three different conditions, in order to consider different riser content density in operational conditions: riser conveying internal production fluid, riser conveying water and empty riser. It is important to impose significant key parameters variation to the analyzed model as a way of verifying if the riser configuration is robust and can absorb the variation effects without significantly changing the riser global response. The buoyant segment pipe is composed of five different diameter modules, the diameter of the first section being the largest and the diameter of the subsequent floaters modules decreasing up to the last one. This gradual diameter variation provides curvature reduction

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Table 1 Riser properties. Inner diameter (in.) Steel Young module (kN/m2 ) Steel specific weight (kN/m3 ) Steel yield strength (kN/m2 )

10.75 2.07e+008 77.00 551580.56

at the intermediate riser region and, therefore, stresses reduction. The floaters modules are equally spaced 0.5 m from each other. The length and diameter of the floaters modules are equal. Additionally, the entire riser is assumed to have a thermal insulation coating with 2 in thickness and 7 kN/m3 specific weight. At the bottom end and the riser top there are stress-joints. The former is composed by titanium, with Young modulus equal to 1.14E8 kN/m2 , since this region is subjected to higher efforts; and the latter is made of steel, with Young modulus equal to 2.07E8 kN/m2 . For structural modeling, the riser bottom connection is considered clamped and the top end connection has its movement prescribed by vessel motion. 6.2. Environmental load conditions Design sea states for riser simulation are obtained from the document I-ET3A26.00PPC1000941001 [44]. The environmental actions of wave and current are considered acting in a collinear direction to the riser on the near and far relative positions. According to previous analyses, those were the critical directions on risers assessment. The design basis includes metocean conditions for both extreme storm and operational conditions. The top connection of the riser is subject to the monocolumn prescribed motions, based on specific RAO platform data. The extreme environmental scenario comprises the combined action of centenary wave and decennial current. Sea states for operation scenario simulation comprise the combined action of wave and current both with annual return period. Static offset for extreme analysis is taken as 8.5% of waterdepth, i.e., 187 m. Annual static offset is equivalent to 4.5% of waterdepth. A previous study was conducted that allowed the identification of critical sea states for the CVAR design. This helped to reduce the computational cost demanded by the analysis of the numerous sea states. Based on these preliminary results, the analysis considered only the four most critical sea states regarding centenary near condition, centenary far condition, annual near condition and the annual far condition. 6.3. Optimization problem formulation Based on the CVAR geometrical and operational characteristics and on the efficiency measures of interests, the optimization problem was then formulated leading to the definition of the design variables, design constraints and objective functions. 6.3.1. Design variables It is considered that the riser comprises five segments with constant properties, which are: lower slick pipe, buoyant segment, intermediate slick pipe, heavy coated pipe and upper slick pipe (Fig. 2). To define each of these specific riser segments it is necessary to determine several geometrical parameters, which are further detailed. It is important to highlight that the proper definition of the input design variables of the problem is essential for the correct evaluation of the design optimization. The more variables, more complex becomes the problem once the design space increases. On the other hand, not considering any significant variable can affect the quality of the problem solution, once the lack of variables restricts the search space and may exclude viable solutions that eventually can determine

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6.4. Design constraints

Fig. 2. CVAR composition.

the optimal configuration. Thus, it is necessary to carefully assess the design variables, selecting only the most representative ones. In general, the CVAR design comprises the definition of the following eleven parameters, taken here as design variables: bottom stress-joint length (bottom TSJ), top stress-joint length (top TSJ), lower slick pipe length (lower pipe), buoyant segment length (buoyant pipe), floater module diameter (floater diam), heavy coated pipe length (heavy pipe), heavy coating wall thickness (heavy th), overlength fraction (OF), well horizontal offset (pOffset), expressed as a proportion of the water depth, upper slick pipe length (upper pipe), and outer diameter of the riser (OD). The lower slick pipe length, the buoyant segment length, the heavy coated pipe length and the upper slick pipe length are all expressed as a proportion of the riser total length. Risers inner diameter is usually defined according to specific conditions of each exploration well, taken into account the required flow, admissible pressure drop and the composition of the mixture found in the field. It is a complex problem once it involves multiphase flow (oil, gas, sand) at high temperature gradients. Thus, the design of the internal diameter is a detailed and complex task that is outside the scope of this work. For this reason, the inner diameter is here considered constant and predetermined. Some preliminary design tests were performed concerning the CVAR configuration, which provided some early observations about its behavior. It was noticed that by varying the floaters modules diameters along the buoyant segment length, one can obtain a significant reduction of stresses at the intermediate transition riser section and also an increase of the curvature radius at this region, creating a smooth transition between the weighted and the buoyant part of the riser. This should be done to avoid detrimental local response along the riser. In the study presented here it is considered that the buoyant segment is composed by five different floater modules diameters that decrease linearly from bottom to top. This would mean an increase in the number of design variable; however, some simplifications were considered in order to efficiently perform this initial study: the length of each of the five sections of the buoyant segment is expressed as proportional to the total buoyant segment length and this proportion is a constant predefined parameter; the diameters of the three intermediate floater sections are expressed as a function of the first and last floaters diameters in a way that diameters decrease linearly from bottom to top; the diameter of the smallest float is fixed, being equal to the lower limit of the defined range for the floater diameter variable; the diameter of the first floater (the greater one) is taken as a design variable varying according to a predefined range. Stress-joint thickness, that defines the diameter of the conic base, is considered constant and only stress-joint length is considered as a design variable, once previous study showed that this was the most significant one.

Constraints considered in this study are based on structural criteria of the riser system with respect to riser feasibility, comprising the evaluation of the von Mises maximum stress and minimum effective tension along the riser, as structural design parameters, and minimum radius of curvature along the riser, as operational criteria. The curvature constraint is imposed as a way of guaranteeing the passage of downhole equipment such as sand screens and sub-surface valves for workover operations. Stress-joints at the top and bottom tend to have the highest curvatures for large offset cases. However, a larger inner diameter can be provided if necessary at the stress-joints to allow the passage of tools or equipments [11]. More detailed information regarding CVAR related workover operation can be found at [11,8]. For each configuration evaluated in the optimization process, static and dynamic simulations are performed in Anflex. The responses are then compared with the constraint limits, thus evaluating the feasibility of the riser configuration. According to stress criterion, maximum stress throughout the riser shall not exceed 80% of yield strength steel for extreme analysis. For operating conditions analysis, the admissible stress is 67% of the yield strength of steel. The minimum effective tension assessment is considered in order to avoid buckling occurrence. Minimum accepted radius of curvature is 90 m [8]. 6.5. Objective functions The optimality of each configuration is evaluated by means of an efficiency measure of interest, which characterizes the objective function. The proper definition of the objective function is decisive on the optimization results. Most of the studies on rigid risers optimization use the cost of the material as merit function to be minimized [26– 28]. However, the total cost of a riser cannot be easily calculated and this parameter goes beyond the amount of material used, including also several aspects, e.g. transport, installation, operation and maintenance. Once the service life of a riser is normally around 20 years, the cost of maintenance and operation, which take place throughout its life, tend to be larger than the price due to the material and installation only. In practice, a greater stress relief at the top, obtained by the use of larger portions of floaters (that would result in a higher material cost), could extend the service life of the structure and also reduce the need for maintenance, thereby reducing the overall cost of system. Therefore, it would be more appropriate to consider objective functions based also on structural criteria. Based on that, this paper discusses the use of a multi-objective optimization approach for the CVAR design. Two objective functions are simultaneously considered: the total riser volume (comprising the steel tube, the buoyancy modules and the additional weight volumes) and the minimum stress utilization factor along the riser, given by the ratio between the in service stress and the admissible stress (Eq. (12)). For scale reasons, the total riser volume is normalized by the total floater volume considering that the whole riser has floaters attached (Eq. (13)). When considering the optimization procedure it is intended to minimize riser volume, for costs reduction, and maximize the stress utilization factor, as a way of maximizing the material strength used capacity.   σserv , (12) fob jI = min σadm fob jII =

Vfloater + Vaddweight + Vsteel

, L riser floater diam2 − O D 2

(13)

where σ serv is the in service stress (kN/m2 ), σ adm is the admissible stress (kN/m2 ), Vfloater is the floater volume (m3 ), Vaddweight is the additional weight volume (m3 ), Vsteel is the steel pipe volume (m3 ) and Lriser is the total riser length (m).

M.A. Martins et al. / Applied Ocean Research 41 (2013) 28–40

Fig. 3. Variables effect size with respect to the stress utilization factor.

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Fig. 5. Variables effect size with respect to the radius of curvature. Table 2 Student significance for hypothesis test regarding the minimum stress utilization factor. Design variable OD floater diam lower pipe pOffset heavy th OF top TSJ buoyant pipe bottom TSJ upper pipe heavy pipe

Effect size −1.002 −0.377 0.338 0.305 0.269 0.220 −0.128 −0.106 0.054 0.035 −0.034

Significance 0.00 0.001 0.005 0.369 0.021 0.092 0.168 0.206 0.342 0.395 0.399

t-Student 8.939 2.980 2.585 0.332 2.040 1.337 0.963 0.821 0.405 0.265 0.255

Table 3 Student significance for hypothesis test regarding the volume factor. Design variable Fig. 4. Variables effect size with respect to the volume factor.

7. Results and discussion Considering the problem formulated herein, that consists of the design of CVARs, a sensitivity analysis was carried out through the use of a statistical design of experiments methodology using fractional factorial design of 211−1 experiments. At first, only the effect size of factors (design variables) regarding of responses of interest were assessed (Figs. 3–5). This chart shows the importance of each input variable related to the overall project. Although effect size information is of great importance in this study, it is necessary to evaluate the significance of these results. For that matter this study considers the Student’s t-test for testing hypothesis and evaluating the significance of the results. The null hypothesis states that the variable does not produce significant effect on the response, while the alternative hypothesis says otherwise, i.e., the variable has a significant effect. A confidence interval of 95% is considered here, which represents a significance threshold of 0.05. The significance, also called p-value, represents the Type I error, the probability of making a mistake by rejecting the null hypothesis when it is true. Thus, if the calculated p-value is greater than 0.05, the null hypothesis is not rejected, which implies that the variable being tested is not significant. In cases where p-value is less than 0.05, the null hypothesis is rejected and one can therefore say that the variable being tested is significant. Table 2 presents the result data obtained for the hypothesis tests on the effect size of the minimum stress utilization factor, Table 3

floater diam pOffset buoyant pipe OF heavy th heavy pipe OD bottom TSJ upper pipe lower pipe top TSJ

Effect size −0.253 −0.204 0.196 −0.047 0.038 0.022 −0.020 −0.011 −0.011 0.007 −0.006

Significance 0.00 0.110 0.00 0.033 0.057 0.179 0.205 0.316 0.317 0.378 0.389

t-Student 16.004 1.229 10.033 1.847 1.587 0.918 0.825 0.479 0.476 0.310 0.281

presents the result data obtained for the hypothesis tests on the effect size of the volume factor, Table 4 presents the result data obtained for the hypothesis tests on the effect size of the minimum radius of curvature, Table 5 presents the result data obtained for the hypothesis tests on the effect size of the maximum combined von Mises stress, and Table 6 presents the result data obtained for the hypothesis tests on the effect size of the minimum effective tension. These tables show information about the effect size, significance and t-Student test statistic. The t-Student critical test statistic is 1.65, considering a confidence interval of 95%. In Tables 2–6 the variables for which the null hypothesis was not rejected are highlighted on bold format, that is, those non-significant responses to the response parameter in question. Effect size indicates the kind of relationship between the factor and the response variable: negative values indicate that the relationship is inverse. Significance indicates the p-value, that is, the probability of obtaining a test statistic at least as extreme as the one observed

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Table 4 Student significance for hypothesis test regarding the minimum radius of curvature. Design variable pOffset OF buoyant pipe floater diam heavy th bottom TSJ lower pipe heavy pipe OD top TSJ upper pipe

Effect size −39.682 24.051 16.929 16.657 −3.647 −3.620 −3.365 2.418 −2.181 1.010 −0.543

Significance

t-Student

0.050 0.00 0.00 0.00 0.149 0.151 0.168 0.247 0.282 0.387 0.438

1.651 5.456 5.315 5.204 1.040 1.033 0.960 0.683 0.577 0.287 0.154

Table 5 Student significance for hypothesis test regarding the maximum combined von Mises stress. Design variable OF pOffset buoyant pipe floater diam OD top TSJ heavy pipe bottom TSJ lower pipe upper pipe heavy th

Effect size 813359.374 -806386.679 597580.417 557430.985 −399817.526 195882.152 173740.470 −158800.470 −79001.8443 −73866.3267 −50753.7307

Significance 0.050 0.181 0.00 0.00 0.001 0.064 0.090 0.108 0.269 0.283 0.347

t-Student 6.476 0.910 4.926 4.557 3.064 1.527 1.343 1.237 0.614 0.573 0.393

Table 6 Student significance for hypothesis test regarding the minimum effective tension. Design variable floater diam OD buoyant pipe heavy pipe OF pOffset top TSJ bottom TSJ upper pipe heavy th lower pipe

Effect size −245.698 200.739 −125.100 −110.812 −53.263 −41.393 −33.765 31.251 −29.263 −24.914 24.395

Significance 0.00 0.00 0.00 0.002 0.059 0.440 0.199 0.217 0.232 0.266 0.271

t-Student 6.868 4.928 3.210 2.804 1.575 0.150 0.844 0.782 0.732 0.623 0.610

assuming that the null hypothesis is true. Small p-values provide evidence against the null hypothesis indicating that the observed data are unlikely when the null hypothesis is true. In such cases design variables are considered significant, meaning that the observed difference is not likely due to chance. According to the information provided by Tables 2–6, it is verified that the design variables top stress-joint length (top TSJ), bottom stress-joint length (bottom TSJ) and upper slick pipe length (upper pipe) do not provide evidence enough to be considered significant for all response parameters evaluated. Thus, it is possible to consider them as constant parameters for subsequent analysis, like an optimization process, without any loss of representativeness of the problem formulated. Furthermore, it is necessary to evaluate the interaction effect between design variables before any conclusion is taken. For that purpose, a half normal plot of effects is evaluated. This is a plot of the absolute value of the effect estimates against cumulative normal probabilities. A normal plot is useful to distinguish between the most significant (upper right) and less significant (lower left) effects, considering both main and first order interaction effects. The existence of a strong interaction means that the effect of a variable depends on the response of another variable.

Fig. 6. Half normal plot chart with respect to the minimum stress utilization factor along the riser.

Fig. 6 illustrates that the following effects produced significant effects on the stress utilization factor response and should be included in the statistical model: interaction effect between outer riser diameter and well horizontal offset, the individual main effect of the outer diameter, the combined interaction effect of the outer diameter and floater module diameter. Note that direct effect of the variables on the response is represented by black squares, that is, when the variable value increases the response parameter increases as well. Inverse relation between design variables and responses is represented on half plot charts with white squares. The results are consistent regarding the expected structural behavior of the riser. The increase in the risers outer diameter results in increased stiffness, reducing the service stress which in turn reduces the stress utilization factor. Thus, the inverse relationship for outer diameter and stress utilization factor is justified. Regarding the floaters module diameters, the increase of this variable results in increased volume of the float and thereby increasing the thrust. Therefore, there is a relief of the weight supported by the riser, reflecting the reduction of operating stress, justifying the inverse relationship between the variable floater diameter and the stress utilization factor. As seen in Fig. 7 the variables floater module diameter, horizontal well offset and buoyant segment length are those that have greater influence on the behavior of CVARs with respect to the volume factor. This is justified by the fact that these three variables cause the greatest impact on the geometry, reflecting consequently on the total volume of the structure. According to the results the variables floater module diameter, horizontal well offset and the interaction between the floater diameter and the well offset have an inverse relationship with the volume factor response parameter. On the other hand, the buoyant segment length and the interaction between this same variable and the horizontal well offset have a direct relationship with the volume factor response. This relationship is justified by the way this response parameter was defined (Eq. (13)), in which the denominator comprises the diameter of the floater and the length of the riser. Thus, increasing the horizontal offset leads to an increase on the length of the riser and as a consequence the volume factor decreases. At the same time, as the buoyant segment length increases the floaters volume also increases leading to the direct effect of the growth of the volume factor. Fig. 8 presents the half normal plot of the effect size of design variables effect in relation to the radius of curvature. The following design variables are observed as the most significant effects: horizontal well offset, overlength fraction and the interaction effect between these two variables. It appears that increasing the horizontal offset causes a decreasing of the radius of curvature, an undesirable situation as it

M.A. Martins et al. / Applied Ocean Research 41 (2013) 28–40

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Fig. 7. Half normal plot chart with respect to the riser volume factor.

Fig. 9. Half normal plot chart with respect to the combined von Mises stress.

Fig. 8. Half normal plot chart with respect to the minimum radius of curvature.

Fig. 10. Half normal plot chart with respect to the minimum effective tension.

causes increased stress and also overwhelm workover operations. Fig. 9 presents the half normal plot of the effect size of design variables effect in relation to the maximum combined von Mises stress. The results show a similar behavior compared to the half plot regarding the minimum radius of curvature. The following design variables are observed as the most significant effects: horizontal well offset, overlength fraction and the interaction effect between these two variables. Hence, risers maximum stress is related to minimum radius of curvature, that is as the latter reduces the former increases since the bending effort increases. The half normal plot of the effect size of design variables effect in relation to the minimum effective tension is presented in Fig. 10. The results show that the most significant effects are those related to the following design variables: floater module diameter, outer diameter, the interaction effect between the overlength fraction and the floater diameter and the interaction effect between the outer diameter and the well horizontal offset. According to results obtained from this sensitivity analysis, the variables upper stress-joint length, bottom stress-joint length and upper slick pipe length have neither significant main effect nor first order relevant interaction effect with respect to the response parameters considered. In other words, there is no evidence that a change on any of these three variables will cause a relevant change on the response parameters. For that reason, they can be considered constant, reducing the subsequent optimization search domain and as a consequence leading to a better performance of the optimization process.

In the sequence, optimization analysis was performed using the NSGA-II method which consists of an elitist genetic algorithm that allows an efficient multi-objective optimization process. The application of such method requires the definition of specific parameters. For the optimization analysis performed here a population of 50 individuals and 100 generations, a cross-over probability of 90% and a mutation probability of 10% were considered. A multi-objective problem seeking to minimize the volume factor and maximize the minimum stress utilization factor is considered. By this approach we seek to achieve the maximum material used capacity simultaneously with the minimum material cost, equivalent to the minimum amount of material needed to obtain a feasible CVAR configuration, i.e., that properly satisfy previously established structural and operational constraints. This approach considers the following optimization problem:  (14) find arg min fobj = fob jI , − fob jII such that VME100 ≤ 0.8σyield ,

(15)

VME1 ≤ 0.67σyield ,

(16)

FX ≥ 0,

(17)

RC ≥ 90 m,

(18)

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Fig. 11. Summarized simulations status.

Fig. 13. Multi-objective optimization results.

Fig. 12. Constraints violations.

where VME100 is the admissible stress for extreme analysis, VME1 is the admissible stress for operational conditions, FX is the admissible limit for the effective tension response and RC is the minimum admissible radius of curvature. Fig. 11 presents a summarized status of the performed simulations indicating proportionally the occurrence of feasible and unfeasible configurations as well as those in which it was impossible to converge to a solution resulting, thus in error. For the riser configurations that turned out to be unfeasible, Fig. 12 quantifies proportionally the violated constraints. It is possible to observe that the most critical constraints regarding the CVAR design are the maximum von Mises stress and the minimum bend radius. It is also important to highlight that significant constraints violations occur at the static analysis already, regarding as well the minimum radius of curvature and maximum stresses. Obtained optimization results are shown in Fig. 13 where the set of optimal designs is highlighted with a square contour and it is also possible to identify the feasible and unfeasible analyzed configurations. The CVAR configurations highlighted with a square contour are those belonging to the Pareto front, i.e., the nondominated optimal solutions. Unlike expected, the Pareto frontier is found not well delineated and does not meet the desired criteria of diversity in multiobjective approaches in general. This result is explained by the fact that the two objectives considered are not in conflict, i.e., when maximizing the stress utilization factor along the riser one is simultaneously minimizing the bend radius. This can be seen by regression line also plotted in Fig. 13. Thus, the optimization process converged to very close solutions. Additionally, the optimization process history shows that for the stress utilization factor, a gain of approximately 35% was achieved

Fig. 14. Optimization process history regarding the stress utilization factor.

Fig. 15. Optimization process history regarding the volume factor.

(Fig. 14) while for volume factor response a reduction of approximately 46% was achieved (Fig. 15). In Figs. 14 and 15 it is observed that the search process starts in a region of the search space that results in lower values of the stress

M.A. Martins et al. / Applied Ocean Research 41 (2013) 28–40

Fig. 16. (a) Neutral mean position; (b) annual near relative position; (c) centenary near relative position; (d) annual far relative position; and (e) centenary far relative position.

Table 7 Optimized riser parameters. Parameter

Value

botton TSJ (m) top TSJ (m) slick pipe (%) OF (%) OD (m) buoy diam 1 (m) buoy diam 2 (m) buoy diam 3 (m) buoy diam 4 (m) buoy diam 5 (m) heavy th (m) buoyant pipe (%) lower pipe (%) upper pipe (%) pOffset (%) heavy pipe (%)

20 20 40 7 0.32 1.5 1.22 1.08 0.94 0.8 0.1 30 15 5 40 30

utilization factor and as the optimization process progresses configurations are found which result in higher values of this measure, as desired, since the intention is to maximize this factor to ensure the best use of the material. The same occurs with the volume factor. Initially the exploration of the search space results in riser configurations with high values of this factor and throughout the process the method leads to lower volume values as desired. As mentioned previously, in multi-objective problems several compromise solutions are obtained. The choice of a particular solution is defined by considering some additional criteria. As an example, among the Pareto solutions obtained in this case study, the one that results in a larger radius of curvature is chosen. This configuration is shown in Fig 16. The optimized riser parameters are shown in Table 7. 8. Conclusions The CVAR concept is assessed in this paper. The study of this relatively new configuration is motivated by the advantages provided that can overcome the challenges of ultra-deepwater oil production. Due to the lack of knowledge about this riser configuration and its design complexity, a sensitivity analysis based on statistical design of experiments methodology is proposed in order to provide a better understanding about the riser behavior under functional and environmental loads. In the sequence an optimization approach is considered for the design process, providing an efficient methodology that can indicate optimal solutions. Riser simulations are performed. The primary goal is to evaluate the statistical significance of the effect that a particular factor (design

39

variable) exerts on the response parameters. As a consequence important and unimportant factors are distinguished. Through the performed analysis it was found that for different response parameters, different values of significance are obtained for the design variables, i.e., each variable has a particular effect on each response of interest. In general, the variables that proved to be most significant were the outer diameter of the riser, the diameter of the floaters module, the horizontal well offset, the buoyant segment length and the overlength fraction. The outer diameter of the riser has a relevant effect on the response since it directly affects the submerged weight of the riser as well as the structure stiffness. As the outer diameter increases, the riser stiffness increases as well, causing a stress and curvature reduction along the riser. On the other hand, the increased outer diameter leads to increased weight of the submerged structure, thereby increasing stresses, especially at the top end connection. The variation of the external diameter has an effect even on the drag effort, which is the resisting force to the relative movement between the riser and water. With increasing diameter of the riser, the contact area with the external fluid increases, elevating the friction that contributes to structure damping. Finally, increasing the outer diameter of the riser increases the volume of water displaced, which directly influences the inertia of the system. The diameter and length of the floater also have an impact on the structural behavior of the riser. With the increase in the volume of the float caused by the increase of these two variables, there is a reduction in submerged structure weight and also a reduction of the curvature along the riser, leading to a decrease on stresses and an increase on the radius of curvature. However, the increase in the volume of the float results in increased effective tension in the lower slick pipe which can result in high stress at this region. Regarding overlength fraction, which determines the total length of the riser, it appears that, when at near relative position, the shorter the line the lower the stresses, since the radius of curvature will be greater. In contrast, a short line causes high stress at the top of the riser when this is at the far relative position. Thus, it is necessary to measure the length of the riser so that it is long enough to avoid high tension effort and meet the criteria of tension at the top connection when at far position while still guaranteing small curvatures when at near position. Hence, sufficient compliance must exist in the CVAR system in order to satisfy the extreme response criteria when the FPU offsets to the near and far positions. Increased compliance, that leads to greater riser length, however, must be balanced by the need to limit the potential for riser-to-riser and mooring lines contact. It was also observed that for a certain length, there is a proper (optimal) horizontal offset. If offset is too large, bending moment at the top and bottom end becomes too large as well. On the other hand, if offset is too small, bending moment around the midpoint becomes too large. If the bending stress is almost the same, shorter length risers are preferred since they lead to reduced costs. Likewise, through this study, it was also possible to identify the unimportant variables to the CVAR design, i.e., those that have little impact in all response parameters evaluated, namely: the bottom stress-joint length, the upper stress-joint length and the upper slick pipe length. Since they have little effect on the response, these variables may be considered constant on the CVAR design formulation, allowing a reduction of the optimization search domain and hence providing a significant gain on the process efficiency. Finally it must be highlighted that statistically designed experiments are a powerful tool for improving the efficiency of experimentation. NSGA-II method was applied for the optimization process and a set of optimal solutions was obtained. By the optimization approach it was possible to achieve a gain of approximately 35% for the stress utilization factor response and a reduction of 46% for the volume factor response. In general, real engineering problems are complex and difficult to

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model. Nonlinear problems are usually represented by multimodal and non-continuous functions. In the context of offshore structures design, as risers, the importance of the study and application of optimization techniques are highlighted. In such area, in which the design possibilities are numerous and where it is necessary to perform numerous structure simulations, the application of optimization techniques allows to automate the design process improving its efficiency. Thus, it becomes possible to obtain a response considered to be optimal in terms of a certain merit function, while still capable of ensuring technical and economic feasibility of the system, and in the shortest time possible. Acknowledgments We acknowledge the permission granted by Petrobras regarding the use of the software Anflex, as well as the company sponsoring of research projects at LCCV/UFAL. We also acknowledge the companies ESSS and ESTECO for the support and for providing the ® modeFRONTIER license used in this project. The authors would also like to acknowledge the Brazilian national agency of petroleum, natural gas and biofuels, ANP, the sponsor of studies and projects, FINEP, and the Ministry of Science and Technology, MCT, for training and research funding through the Human Resources Program of the ANP for the oil and gas sector PRH/ANP/MCT.

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