CHAPTER 5. CASE STUDY - DRAGGED ANCHOR INTERFERENCE ASSESSMENT
5.5 ANCHOR HIT CRITERIA CHECK
When moving ship suddenly loses its anchor, the anchor-chain arrangement will stabilize after some time, but it will never be perfectly vertical due to an interaction between anchor-chain arrangement and surrounding water. The questions that have to be answered here are what a shape of the chain will take place and what a distance between the anchor and vessel itself will be (Figure 45). In case, if this distance (determined by Y-axis) is less than the water depth at the location of pipeline, it is obvious that the anchor does not have a potential to approach and hit the line.
Assuming that all the ships passing the Pipeline 1 route are moving with the constant velocity, one shall see that the steady flow will force the anchoring system in the opposite direction. Since the flow is steady one, the acceleration term becomes equal to zero. That is why an added mass and inertia terms are not considered in this case. Moreover, the drag force of the anchor is assumed to be negligible in comparison with its weight.
Consequently, the equation of the motion can be written in accordance with the 2nd Newton’s Law as follows (De Silva, 2007):
II
I
71
∑ 𝐹⃗ = 𝑚𝑎⃗(𝑙, 𝑡) = 0⃗⃗ (44) 𝑚𝑎⃗(𝑙, 𝑡) = 𝜕
𝜕𝑙(𝑇𝑡⃗) + 𝑓𝑛𝑛⃗⃗ + 𝑓𝑡𝑡⃗ + 𝑚𝑔𝑘⃗⃗ (45) 0⃗⃗ = 𝜕
𝜕𝑙(𝑇𝑡⃗) + 𝑓𝑛𝑛⃗⃗ + 𝑓𝑡𝑡⃗ + 𝑚𝑔𝑘⃗⃗ (46) 𝑚 – mass of the chain per unit length;
𝑔 – acceleration of gravity;
𝑎⃗(𝑙, 𝑡) – acceleration of the chain;
𝑙 – coordinate along the anchor chain;
𝑇 – tension force;
𝑓𝑛 – normal drag force per unit length;
𝑓𝑡 – tangential drag force per unit length;
𝑛⃗⃗ – normal unit vector;
𝑡⃗ – tangential unit vector;
𝑘⃗⃗ – unit vector in the direction of gravity.
Figure 45: Anchor-chain system configuration illustration
As seen from the formula, there are two components of drag force: normal (dominant) and tangential. The formulas on both of them are presented below:
𝑓𝑛 = 𝐶𝐷𝑛 ∙ 𝜌𝑤 ∙𝐷
2 ∙ 𝑣2∙ 𝐶𝑜𝑠2𝛼 (47) 𝑓𝑡= 𝐶𝐷𝑡∙ 𝜌𝑤 ∙𝐷
2 ∙ 𝑣2∙ 𝑆𝑖𝑛2𝛼 (48) CDn – normal drag coefficient;
CDt – tangential drag coefficient;
ρw – seawater density;
D – anchor chain diameter;
72 v – water flow velocity;
𝛼 – angle between the vertical axes and tangential vector.
Normal and tangential drag coefficients of the stud-link and stud less chains are given in the Offshore Standard DNV-OS-E301 (2010) (Table 20).
Table 20: Anchor chain drag coefficients
Chain types Normal drag coefficient Tangential drag coefficient
Stud-link 2.6 1.4
Stud less 2.4 1.15
To continue with the calculations needed for the criteria check, the equation of motion should be rewritten in the scalar form:
{
𝑑𝑇
𝑑𝑙 − 𝐶𝐷𝑡∙ 𝜌𝑤 ∙𝐷
2 ∙ 𝑣2∙ 𝑆𝑖𝑛2𝛼 − 𝑚 ∙ 𝑔 ∙ 𝐶𝑜𝑠𝛼 = 0
−𝑇 ∙𝑑𝛼
𝑑𝑙 + 𝐶𝐷𝑛 ∙ 𝜌𝑤 ∙𝐷
2 ∙ 𝑣2∙ 𝐶𝑜𝑠2𝛼 − 𝑚 ∙ 𝑔 ∙ 𝑆𝑖𝑛𝛼 = 0 (49) It is a system of two ordinary differential equations that can be solved numerically to find 𝑇(𝑙) and 𝛼(𝑙). For that purpose, two initial conditions must be specified:
𝑇(0) = 𝑊𝑎𝑛𝑐ℎ𝑜𝑟 (50) 𝛼(0) = 0 (51) 𝑊𝑎𝑛𝑐ℎ𝑜𝑟 – weight of the anchor in the water:
𝑊𝑎𝑛𝑐ℎ𝑜𝑟 = 𝑚𝑎𝑛𝑐ℎ𝑜𝑟∙ 𝑔 ∙ (1 − 𝜌𝑤
𝜌𝑠𝑡𝑒𝑒𝑙) (52) In addition to the previous system, the relation between the Cartesian coordinates and angle 𝛼 is to be included either:
{
𝑑𝑥
𝑑𝑙 = 𝑆𝑖𝑛𝛼
𝑑𝑦
𝑑𝑙 = 𝐶𝑜𝑠𝛼 (53) Where the initial conditions are:
𝑥(0) = 0 (54) 𝑦(0) = 0 (55) Thus, solving the system of four ordinary differential equations with specified set of initial conditions, it is possible to understand how an equilibrium configuration of anchor-chain arrangement looks like. Moreover, the distance between the towed anchor and the ship is determined as well. For the present case study, the anchor towing depth calculation can be done by applying MATLAB software. With this objective in view, an effort has been made to create appropriate code. An ode-45 function, based on an explicit Runge-Kutta formula, is included.
According to the Equipment Specification Letter grouping (Figure 43) all 49 various ELs are
73 taken for the calculation: for every EL the combination of corresponding anchor mass; chain length, chain diameter; and average speed (in m/s) are put into the prepared code. All the results have been exported to the EXCEL sheet in order to use them for the development of essential graphs and future analyses.
Obtained data helps to identify, whether the anchoring systems of given passing ships are capable of hitting the Pipeline 1 or not. The anchor towing depth variations in terms of different Equipment Letter groups are illustrated in Figures 46-48.
Figure 46: Tow depth of anchors classified by Small Letters and water depth lines (dashed) of certain KP sections
Figure 46 shows that not all the anchors classified by Small Letters will reach the sea bottom in case if passing ships lose their anchors. The majority of intersections are observed only where the water depth value is about 140 m and less.
Figure 47 demonstrates the same sort of estimation for Capital Letter class of anchors. One can see that this kind of anchors have the potential to reach even deeper sections of 250 m depth. It is quite obvious, because larger anchors have longer and heavier chains, which easily approach
Anchor Tow Depth (Small Anchors)
efg
74 Figure 47: Tow depth of anchors classified by Capital Letters and water depth lines (dashed) of
certain KP sections
Finally, an assessment of the 3rd anchor EL group is presented in Figure 48. In this case the anchor tow depth is increased up to 280 m. So, almost all the pipeline sections can be reached and hit by huge anchoring equipment of crossing ships.
Figure 48: Tow depth of anchors classified by Letters marked with star and water depth lines (dashed) of certain KP sections
-400
Anchors Tow Depth (Large Anchors)
AB
Anchor Tow Depth (Huge Anchors)
A*
75 Summarizing the part of the hit criteria check, it should be said that all the anchors with the EL starting from “l” are dangerous for the Pipeline 1. Thus, not all the anchors pose threat to the pipeline resting on the seabed. The key elements here are anchor mass, chain length, and ship speed. The larger the anchor, the longer the chain will be. The anchor hanging on the longer chain has more chances to approach the seabed. Regarding vessel speed, it also affects the anchor towing depth value. In case of high velocity the anchor-chain arrangement will stabilize at less depth than in case of low velocity. A combination of anchor size, chain length, water depth and vessel speed is of great importance for the assessment of anchor towing depth while the ship is underway.
In order to verify the results of analytical solution explained above, it has been decided to take a model scale test. Detailed description of taken experiment is given hereinafter.