It was my intention to write only two blogs on cable lay, but here’s the third one. There’s a lot of interesting stuff to investigate. We’ve only touched upon regular lay, maybe we’ll dive into pull-ins later as well.

In the last blog post we’ve looked at cable response due to vertical chute motions. It was found that the velocity was a good predictor for typical cable responses, such as curvature (bend radius), compression and maximum tension. In order to find the underlying relation, we performed numerical forced heave oscillation tests in OrcaFlex—a vessel with a cable running over its chute was given a sinusoidal heave motion with various amplitudes and periods. This is, for obvious reasons, a strong simplification to investigate correlations between motions and cable response. In the same blogpost results were also shown for 100 seeds from irregular waves. In this post we’ll dive deeper into a realistic response, investigating effects such as wave spreading, vessel heading relative to waves, currents, water depth and more. We also propose a method that can be used to obtain a design heave velocity, that can be used as a limit instead of allowable significant wave height.

Before we start, I want to share some findings from the convergence study I did. I varied the line segment length, expecting that below a certain length the response remains stable (converges). I’ve plotted the convergence of curvature and compression to the line segment size below.

The curvature converges well, but the compression results seem odd. The case with 0.1m line segments actually caused compression vibrations that did not show for longer elements, which can be seen in the time traces below, where the dip to approximately -8kN can be seen. The only solution to solve these spikes was to use a smaller time step. What makes this difficult, is that we have to decide if it is a numerical or physical vibration. We’ll settle for engineering judgement here. I’ve chosen to continue with 0.2m line elements in the sag bend, and 1m elements in most of the line part on the sea bed and in the upper part of the catenary; the time step is 0.05s.

So, we finally can run a set of sea states, or rather a batch. I’ve chosen sea states between 1.5 and 2.5m Hs, with varying periods and relative headings (head to stern seas in steps of 45 degrees). I run simulations of 3 hours, the irregular waves are modeled with a JONSWAP spectrum. It’s not the most detailed batch in terms of environmental load cases, but for the sake of this blog it’s fine.

The results are post-processed from the simulations using the following steps:

- Define zero crossing cycles in the chute velocity time trace and find the negative velocity amplitude
- Find the maximum curvature in this period for all nodes in the sag-bend
- Obtain the maximum value from this set of nodes

This approach means that you can get a lot of information from one simulation. Instead of only looking at one maximum event, we suddenly look at a maximum event about every 5 seconds. So we suddenly have approximately 2000 times more information than before. As in the last post, we’ll focus on curvature only. I’ve plotted the maximum curvature to the negative velocity amplitudes below.

We see a few things:

- In general a trend is clearly visible: up until approximately -1.3m/s it is almost linear, beneath that velocity the trend deviates up to a plateau
- The relative wave direction has an influence on the response, g. the beam sea case (90deg) deviates from the other data, giving significantly higher negative chute velocities for a certain allowable bend radius

With respect to the trend, in the lower velocities the cable dynamically follows a catenary. In the higher velocities the cable cannot keep up the pace of the rapidly downwards moving chute. The reason that the cable cannot keep up is because of drag. The terminal velocity (maximum downward velocity this cable can reach) in this example is 1.6m/s. However, it takes some time to reach the terminal velocity, making the cable at a certain negative velocity slower to react than the chute. This causes a buildup of compression in the cable, as this article points out. These compression waves lead to high curvature and are illustrated in the following video. The curvature caused by such a compression wave is limited: the cable has reached terminal velocity; it doesn’t matter anymore if the chute moves downward faster. However, the numerical results are questionable and the solution is avoid these high responses.

In order to understand the effect of the relative wave direction on the cable curvature, I’ve disabled wave kinematics by cutting off the wave velocities lower than 5m under the free surface. If wave kinematics in the sag bend cause the cable to curve more or less in certain wave directions, the effect should vanish if we cut off the wave velocities. This is precisely what we see in the figure below. The scatter has reduced and the results now lie much closer to each other. The remaining scatter is caused by numerous other effects, such as the other ship motions (e.g. surge).

Cutting off wave velocities is obviously not a practical solution. However, in reality a wave system is not unidirectional. Especially the wind seas we model with a JONSWAP commonly have a lot of energy spread out over directions. I’ve run the same batch with wave spreading (cos^{N}, with a realistic spreading coefficient of N = 4). The spread of wave energy has a similar effect of reducing scatter in the cable curvature.

I’ve also had a look at the effect of water depth, layback and current. For the water depth variations I’ve kept the back tension (horizontal force component in the catenary) equal. This results in the same catenary for all water depths. This seems to prove the hypothesis: results from all water depths overlap quite well, as the catenary is similar the right hand side is similar. Further, for large amplitudes the same limiting bending radius is reached. So we’ve taken water depth out of the equation as well.

The effect of varying layback is shown in the figure below. We see that the behavior in the “catenary zone” is different; the larger the departure angle (with respect to the water line; larger angle means more slack), the steeper the curve.

I’ve also checked the effect of current on the cable response by imposing a 1 knot current is with a 1/7^{th} power law. Results are plotted below. Head current significantly decreases workability in this case; the steepness in the catenary is the same as when no current is present, but the curve has shifted to the right. At smaller velocity amplitudes the limiting bend radius will be achieved. The opposite holds for stern current. I expect that by adding some pretension to the head current case, and lowering the pretension for stern current, it should be possible to overlap all cases. I haven’t researched this yet, though.

By doing all this, we’ve proven that it is possible to reduce the number of parameters required for an analysis significantly. Instead of running various significant wave heights, directions, periods and water depths, it is only necessary to run a set of catenaries. Further, it takes the vessel and loading condition out of the equation. To fully benefit from this method, an allowable chute velocity should be determined which can be used for offshore decision making.

I’ve divided the chute velocity amplitudes in small bins. For each bin I check how many cases exceed the 3m MBR criteria. This gives us the curve shown below. I think this is a very powerful curve, as it can be used to determine a design limit, as well as it provides a strong method for monitoring. By measuring chute velocities, we can determine probability of having exceeded limits. The design limit obtained from this curve lies very close to the value we find with the imposed chute motions.

Finally, we can draw the following conclusions:

- Chute motions give a highly accurate representation of the cable response and can be used to better maintain cable integrity during lay operations, by using motion forecasting and monitoring.
- A good estimate for cable failure by bend radius can be derived from the chute vertical velocity. The value can be obtained by superimposing harmonic motions.
- The limit for chute motions, where a cable criterion is exceeded, is independent of vessel and loading condition.
- In a more detailed approach a motion limit can be determined, although this is very similar to the value obtained from superimposing motions to the CLV. Engineering can be simplified strongly, by using the imposed motion approach.
- It should be possible to make an analytical model of the cable response, for small velocities it follows the catenary equation, and for larger velocities it converges to the same value, independent of water depth or lay back, and possibly current.