Offshore Hazards: Assessing the Impact of Icebergs on Offshore Production Platforms, Part 2

Dmitry A. Onishchenko (Gazprom VNIIGAZ LLC) Part 1

4 Iceberg Movement Model

An important aspect when modelling iceberg movement is the identification of two scales: global and local. Let’s select, with the centre at the platform location point, a “local” domain, the area of which we express as A_reg (Fig. 1).
At the global scale level, we estimate the probability (frequency) of the icebergs arriving at this area, and at the local scale level, we estimate the conditional probability of an iceberg/platform collision (provided that the iceberg has arrived within the domain boundaries). The dimensions of the region must be such that the archive data would provide reliable evidence of the facts of seeing the icebergs within the selected domain. Usually, such domains have meridians and parallels as boundaries. The archive data on icebergs in the Barents sea are provided for the domains occupying 2 degrees of latitude and 5 degrees of longitude (Fig. 2, [16]). Let λ denote the rate of occurrence for icebergs arriving at the selected region. Note that the said parameter may be assessed using archive data, but, in principle, it can be estimated through modelling of the entire iceberg’s “life” using the Monte-Carlo method, from the moment of its formation near one of the outlet glaciers, tracking its drift trajectory under the influence of flow fields and wind within the framework of general regional models of “atmosphere-ice-ocean” type. The estimation using the second option above requires, naturally, colossal computer time consumption and credible information on the flow fields and winds.

Let’s introduce the following symbols: D – the size of the iceberg entering the region A_reg (e.g. typical or maximum plane size at the water level); v – the average speed of the iceberg movement within the domain; t_res – the duration of the icebergs stay within the domain; ℓ – the total length of the iceberg travel distance within the domain. In the general case, all the four above-mentioned parameters are random. It’s often assumed that the values D and v are independent in the probability sense (which, of course, requires to be substantiated in the specific conditions) and are described with the corresponding density functions w_D(x) и w_v(x). At the same time, the values v, t_res and ℓ are invariably dependent, as they are connected by an evident relation

ℓ = v.t_res.

Further construction of the model depends on the hypotheses of probability distributions of these variables, including on which of them should be adopted as the basic ones. For example, in the “rain drops” model [13], when there are grounds to assume that all the icebergs in the region of study carry out predominantly rectilinear movements along the selected direction (Fig. 3), the speed v should be adopted as the basic variable. With that in mind, ℓ ≡ L₂ = const. If the icebergs movement is not rectilinear (Fig. 1) or is sufficiently chaotic, the typical approach to modelling in this case would be an assumption that the time t_res is distributed under the exponent rule
F_t (x)=1-e^-μx, where

μ=1⁄t ̅_res

is a reciprocal of the average value of the time spent by the iceberg within the domain. Usually, the flow of arriving icebergs is considered a Poisson input, and then it is possible to show that the average (statistical) value n ̅ of the number of icebergs in domain A_reg at an arbitrary time point is calculated from the formula

n ̅=λ/μ .(10)

For quality comparison of different regions in terms of the level of iceberg hazard, we introduce the notion of spatial density of icebergs (the average number of icebergs within the given domain at a certain time point, referred to the area):

ρ=n ̅/A_reg .(11)

Let’s move on to the calculation of the probability of an iceberg/platform collision. For the duration of the iceberg stay in the region it travels over the distance ℓ = v t_res and “sweeps” the area A_sw = ℓ D (Fig. 1).

A collision occurs when (and only when) the distance between the “centres” of the platform and the iceberg turns out to be less than (W +D)/2, where W is a characteristic size of the platform at the waterline level. If this is interpreted as the platform centre (point С in Fig. 1) being within a band of width W +D and length ℓ, which in some random manner is located within the domain, then the conditional probability p_imp (D,v) of collision between the iceberg arriving at the region and the platform (provided that its diameter is D, and the speed is v) may be estimated from the expression

p_imp (D,v)=(D+W)/L₁ =(D+W)L₂/L₁ L₂ =A_sw/A_reg.(12)

For the rain drops model, in accordance with the above,
ℓ = L₂, and for the chaotic movement model ℓ = v/μ.

Now we can determine a conditional probability P_Imp of the collision of the platform with an iceberg of random (unknown) size and speed, provided that it arrived at the region under consideration:

P_Imp=∫₀^Dmax∫₀^vmax p_imp (x₁,x₂ ) w_D (x₁) w_v (x₂)dx₁ dx₂ ,(13)

where D_max characterises the maximum iceberg that can reach the boundaries of the region under consideration (taking account of the geographic and climatic features of the region), and v_max is the maximum possible, by physical limits, speed of the iceberg drift. Note that the used approach allows to quite simply take into account reducing the collision probability owing to the monitoring system and active action on the icebergs for the purpose of their diversion. If we assume that the efficiency of such a system in relation to an iceberg of size D and speed v is characterised by a failure probability γ(D,v)), the formula for conditional probability of a possible collision would become

P_Imp=∫₀^Dmax∫₀^vmax p_imp (x₁,x₂ )γ(x₁,x₂ )w_D (x₁)w_v (x₂)dx₁ dx₂ .

If the available statistical data is insufficient to determine the relationship between the iceberg speed and its size, and if, for simplicity, we assume that the success rate of the operations aimed at impacting the iceberg does not depend on its size and speed and equals γ=const, we can obtain the following expression for estimation of the conditional probability of the collision of the arriving iceberg with the platform:

P_Imp=γ(D  ̅  +W)v   ̅ t ̅_res/A_reg ,(14)

where the line above means the mean value of the corresponding random variable.

As noted earlier, the collision probability alone is not enough for setting the design load. For the purposes of design, it is necessary to design the probability of more complex random events that characterise not only the fact of collision, but the parameters of the colliding iceberg too. Let us give you some typical examples.

Within the framework of the described approach we can, for instance, estimate the conditional probability P_Imp* (D) of a collision between the platform and an iceberg whose size is above the established value D, provided that in the region an iceberg with unknown parameters has turned up (in the absence of a system of active impact on icebergs):

P_Imp* (D)=(1/A_reg)  v ̅  t ̅ _res ∫_D^Dmax(x₁+W)w_D (x₁)dx₁ (15)

This value monotonically decreasing from P_Imp to 0 as D rises from 0 to D_max.

To find the unconditional probability of the platform/iceberg collision, the frequency of icebergs arrival at the region of study should be taken account of. If we use a Poisson flow model with intensity λ, then, as is known, the probabilities of occurrence of precisely k icebergs (k = 0, 1, 2, …) during the time interval T are calculated from the formula

P_arr (k,T)=(λT)^k e^-λT /k!

Now we can show that the unconditional probability of the platform collision within the time period T with an iceberg, whose size is above the established value D, is calculated by the relationship

P(D,T)=1-exp[-λTP_Imp* (D)](16)

(if D=0, we obtain an unconditional probability of collision of the platform during the time period T with a random iceberg).

In fact, the found relationship (16) is true for both rare events and frequent ones (see [17]), of course, within the framework of the assumptions made. However, for rare events, which include a collision with an iceberg, the value P_Imp* (D) is very small, and that allows to write down the following approximation formula:

P(D,T)≈λTP_Imp* (D),

whose accuracy is not worse than  (λTP_Imp*)².

Note that within the framework of the described approach we can estimate various other indicators of the iceberg hazard such as the probabilistic function of distribution of kinetic energy for an iceberg colliding with the platform:

F_K* (z)≡Pr{K<z / Imp}= 

=∫₀^Dmax∫₀^vmax∫₀^kmax(x₁+W)/(D ̅ +W) F_h (z/ ( x₃ x¹² x²² )) w_D (x₁)
w_v (x₂ ) w_k (x₃)dx₁ dx₂ dx₃ .

It is assumed that the kinetic energy K is calculated from the formula

K=¹/₂ mv²=¹/₂  (πD²/4) k’hρ_ice v²=kρ_ice D²hv²

where h is the total height of the iceberg, D is its diameter at the waterline level, v is the speed, k is the form coefficient, and ρ_ice is the density of the iceberg’s ice. All the values except the last one are assumed as random with known probability distributions: distribution function F_h (x₄) and density functions w_D (x₁),w_v (x₂),w_k (x₃) respectively.

To obtain numerical estimates, the relevant source data is required. The issue of obtaining such data, analysis of its credibility and completeness is the subject of a separate review.

5 Assessment of the Frequency of Iceberg to Platform Collisions

Let’s run estimates of probability of the platform/iceberg collision for the Shtokman field area located in the Barents Sea. The analysis of archive data for the years 1888-1991 [8,16] has demonstrated that the total number of icebergs observed during the entire 100-year period in the region being a geographic square 72-74⁰ north, 40-45⁰ east, of area A_reg = 36,000 m² (Fig. 2), may be estimated at 30. This corresponds to the frequency of the icebergs arrival of 0.3 per year. In 2003, a regular ice research expedition visiting the Shtokman gas condensate field [16] recorded an abnormal outbreak of icebergs: the total number of the recorded icebergs and their fragments was 41. By combining the information in one array, we assume the average rate of iceberg occurrence λ = 0,7/T₁, where T₁ is the duration of one year in the selected units of time.

Let’s assume the average size of the icebergs at the waterline level as D = 100 m, and the diameter of the floating platform as W = 50 m. The average drift velocity is estimated at v_drift = 0.2 m/s. Then, provided that the iceberg movement is rectilinear, the maximum duration of its stay in the region under consideration is 12 days. Observations of icebergs in the Barents Sea demonstrate that their trajectories are significantly different from rectilinear. An estimated value was found for the tortuousness of icebergs’ trajectories for the central part of the Barents Sea: k_curve  = 5 [16]. For further calculations, let’s assume three characteristic values of the average duration of iceberg stay in region A_reg:

t_res = 20, 40, 60 days.

The corresponding values of spatial density for icebergs calculated from the formula (11) are

ρ = 1,1.10^-12; 2,1.10^-12; 3,2.10^-12 (17)

Using the ratio (14), we find that the probabilities of collision with an iceberg arriving at the region for these three cases are 0.0014, 0.0029 and 0.0043. Taking account of the frequency of the icebergs arrival at the square under consideration, we obtain that the anticipated numbers of collisions with an iceberg during a year for a platform located in the Shtokman field are 0.001, 0.002 and 0.003 respectively. We will show below that the found values are in good agreement with the results of estimations made when evaluating the iceberg hazard for Canadian offshore fields (honouring the rate of occurrence of the arriving icebergs).

We should note that in the works of AARI specialists based on the same source data, the estimated frequency of iceberg collisions is once every 35 years [16] (in the work [8], a slightly different estimate is given, once every 104 years), which is equivalent to the probability of collision during a year approximately equal to 0.03 (and 0.01, respectively). Both values obtained using the methodology different from the described in this work, are one order of magnitude greater than those above. Also, the said works contain an estimate according to which in the Shtokman field area, on the average once every five years, with 95% probability, one should expect the arrival of 9 to 19 icebergs during a year, and once every 10 years – from 12 to 26 icebergs. In view of the above, it appears that in the said works the degree of the iceberg hazard expressed in probabilistic indicators is considerably overestimated. It seems that this is a consequence of the chosen approach that does not reflect all the specific aspects of the problem.

It would be interesting to compare the iceberg occurrence rate data with those for the Grand Banks of Newfoundland (east coast of Canada). There, icebergs occur much more often. For example, in the square 46-47.5⁰ north,  47.5-49.5⁰ west, of 26,000 km² area, the average number of observed icebergs is 72.6 in April and 88.9 in May [7]. Note that there is a substantial irregularity of the icebergs arrival at the region under consideration. For example, in 1984, the 48⁰ north parallel was crossed by over 2200 icebergs, and in 1966 no such cases at all were recorded, which, in comparison with the massive arrival of icebergs at the Shtokman field area recorded in 2003 [8], as it seems, allows to draw a conclusion that such situations, despite having low probabilities of occurrence, are still typical for the problem under study.

For the square 46-47⁰ north,  48-49⁰ west with the area 8500 km², where the White Rose field is located, when modelling the iceberg hazard, the average annual number of icebergs arriving at the said square is estimated at 70. The spatial density of icebergs for this region is ρ = 2,5.10^-10, which is two orders of magnitude greater than the corresponding indicator (17) for the Shtokman gas condensate field area.

According to the calculations made at the pre-design phase, for an individual iceberg, the probability of collision with a floating platform or FPSO vessel installed in the White Rose field is in the order of 0.002…0.003 (as is known, the FPSO concept was implemented in the field). Taking account of the rate of occurrence for the arriving icebergs, the annual frequency of iceberg collisions is 0.14…0.21. This is almost two orders of magnitude greater than the obtained above estimates for the Shtokman field, and it is in full agreement with the fact that the rate of occurrence of icebergs in the Grand Banks of Newfoundland is 100+ times greater than the same indicator for the Shtokman field.

Furthermore, the development plan for the White Rose field provides for a possible collision with an iceberg weighing up to 100,000 tons. Taking account of the application of the iceberg monitoring system and, where necessary, measures of making active impact on the icebergs (the efficiency of the entire system being estimated at 85-90%), and also potential disconnection of risers and anchor lines in emergency situations (the reliability of this technical operation is assumed as 99%), the resulting probability of the platform collision during a year with an iceberg whose weight exceeds 100,000 tonnes is estimated at (1,2…1,6).10^-4, which is regarded as meeting the safety criterion.

The Hibernia oil field is located 315 km south-east off the coast of the island of Newfoundland, just a bit south of the White Rose field. The frequency of iceberg occurrence here is much lower. Hibernia Gravity Base Structure (GBS) is the first platform (and the only one in the world so far) designed to survive a collision with an iceberg. The design iceberg, by the strength criterion (collision with which the platform must withstand without any damage), weighs 1 million tons, and the repeatability of such an event is estimated as once every 500 years. The design iceberg, by the safety criterion (when colliding with which the platform can sustain local damage that can be remedied by a later repair), weighs 6 million tons, and the repeatability of such an event is estimated as once every 10,000 years. The frequency of iceberg collisions estimated for the Hibernia platform is in the order of 1 event per 10 years, which is equivalent to the collision probability of 0.1 during a year. Moreover, the design provides for an ongoing ice monitoring programme based on the availability of systems of early warning, tracking and active action on icebergs aimed at reducing the probability of iceberg/platform collisions. For over 10 years of operation of the Hibernia platform no collisions have been reported, though in the work [15] within the framework of the approach honouring a quite large probability of omitting smaller icebergs in the course of the monitoring implementation, thus considerably increasing the forecast value of the number of arriving icebergs, an estimated frequency of iceberg collisions was obtained at 0.3 events per year, which, again, is considerably higher than that for the conditions of the Shtokman field.

Conclusion

From the results of a comparative analysis of the “deterministic” (traditional) and “probabilistic” design approaches, a conclusion has been drawn that the main hindrance for the so-called probabilistic design approach may be the lack of source data on probabilistic distributions of the determining parameters. At the same time, the example of the task to estimate the iceberg hazard has demonstrated the efficiency of application of probabilistic approaches to the tasks of a certain class.  An example of a model construction and analysis has been given that allows to calculate various indicators of the iceberg hazard for offshore platforms, including the probability of collision with an iceberg whose certain parameters (e.g. diameter, weight or kinetic energy) have values above the established ones. The estimated values for probability of iceberg/platform collision are given for the Shtokman field conditions, which are an order of magnitude smaller than the similar values obtained earlier by other authors. In order to apply the described methodology at the design phase of the Shtokman field development, it is necessary to carry out a more detailed statistical analysis of the available database of icebergs observations in the Barents Sea, so that we could determine with an adequate degree of reliability of probabilistic distributions of iceberg parameters used within the framework of the proposed approach.

Part 1