## Line theory: Seabed tangential resistance |

This feature is primarily designed to assist with more advanced pipelay modelling, such as lateral breakout or axial slip from a subsea trench. It is envisaged that this feature will be used in conjunction with restarts to model phenomena such as thermal expansion/contraction, pipe walking, shutdown/startup cycles, berm formation etc.

A pipe or line attracts resistive forces as it moves across the surface of the seabed. We call these forces *tangential* because they act in the tangent plane to the seabed at the point of contact; this distinguishes them from the *normal* reaction, out of the seabed, which is the force that resists penetration of the line deeper into the substrate (sand/soil). Tangential resistance is often modelled entirely as Coulomb friction. However, this neglects more complex interactions with the seabed, such as the accumulation of soil being pushed ahead of the line. Seabed tangential resistance profiles provide a general and flexible way of capturing these additional seabed forces, at least to a first approximation.

A key phenomena in subsea pipeline modelling is breakout/slip of the line from its as-laid position on the seabed. Breakout takes place perpendicular to the axis of the line and occurs when the lateral forces on the line are enough to overcome its initial embedment. Slip is the analog of breakout along the axis of the line (i.e. axially). After the line has escaped its initial embedding, it continues to push soil ahead of its motion, which is often modelled as a small, but constant, residual resistance. Breakout (or slip) can be approximated by a characteristic trilinear resistance profile, with an initial peak representing the force needed to breakout, followed by a reduction in resistance down to the residual constant value.

Figure: | A trilinear resistance profile |

The y-axis on this graph shows non-dimensional *normalised resistance*, $\mu(\Delta)$ – also known as *effective friction coefficient* – which is the ratio of the magnitude of the lateral (axial) resistance to the magnitude of the normal resistance. The x-axis is lateral (axial) displacement, $\Delta$.

More advanced models exist, such as the Verley-Sotberg model, discussed in DNVGL-RP-F114, which considers the work done displacing the soil to update the resistance profile dynamically. Such models are not currently implemented by OrcaFlex.

Once initial breakout/slip has been achieved, future shutdown/startup cycles may lead to oscillatory motion of the pipe over the seabed. At the extremes of motion, *berms* of soil will form gradually as the line traverses backwards and forwards over the same patch of seabed. Such effects may be captured by restarting OrcaFlex with successive resistance profiles that encapsulate the growth of these berms.

Note: | Seabed resistance models are designed to be used in conjunction with Coulomb friction. Coulomb friction can be disabled, if desired, by simply setting the seabed friction coefficients to zero on the line type data form. |

In OrcaFlex, each node of a line can be assigned a number of lateral and axial tangential resistance profiles. Each profile specifies normalised resistance, $\mu(\Delta)$, as a function of displacement, $\Delta$. The total contact force, $\vec{F}$, can be written as the sum of the normal reaction, $\vec{V}$, plus the tangential reaction (in the tangent plane to the seabed), $\vec{H}$. \begin{equation} \vec{F} = \vec{H} + \vec{V} \end{equation} $\vec{H}$ can be further split into its axial and lateral components: \begin{equation} \vec{H} = \vec{H}_l + \vec{H}_a = H_l (\Delta_l) \, \hat{\vec{l}} + H_a (\Delta_a) \, \hat{\vec{a}} \end{equation} where $\hat{\vec{l}}$ and $\hat{\vec{a}}$ are the current lateral and axial directions of the line, respectively. These are defined relative to some reference configuration of the line, which is either its original as-laid (or parent model) position (for a static analysis), or its position at the start of the time-step (for a dynamic analysis). If the seabed is curved, then these directions are further projected into the plane of the seabed at the instantaneous contact point.

$\Delta_l$ and $\Delta_a$ are the lateral and axial displacements, respectively. These are accumulated by integrating the motion of the node relative to the reference directions. For example, in a dynamic analysis, if a node moves 2m in the lateral direction, $\hat{\vec{l}}$, in a time-step, then $\Delta_l$ will increase by 2m.

$H_l (\Delta_l)$ and $H_a (\Delta_a)$ are the scalar lateral and axial resistance forces, respectively. There is no particular difference between the lateral and axial models, so let us henceforth just consider a generic profile, $H(\Delta)$, which we write as:
\begin{equation}
H(\Delta) = \tilde{\mu} (\Delta) \, V
\end{equation}
where $V$ is the magnitude of the normal reaction, $\vec{V}$, and $\tilde{\mu} (\Delta)$ is a function that is closely related to $\mu(\Delta)$, the user-defined normalised resistance profile. For simplicity, let us consider the case where $\Delta$ has been increasing. $\tilde{\mu} (\Delta)$ is given by:
\begin{eqnarray}
\tilde{\mu}(\Delta) = \left\{
\begin{array}{ll}
\mu( \Delta - \Delta_o ) & \Delta \geq \Delta_\max \\ & \\
\lambda \, ( \Delta - \Delta_\times ) & \Delta_\times \leq \Delta \leq \Delta_\max \\ & \\
-\mu( \Delta_\times - \Delta ) & \Delta \leq \Delta_\times
\end{array}
\right.
\end{eqnarray}
where $\lambda$ is the user-defined *unloading stiffness*, $\Delta_\max$ is the *maximum excursion* reached so far, $\Delta_o$ is the *profile origin* and $\Delta_\times$ is given by:
\begin{equation}
\Delta_\times = \Delta_\max - \frac{ H( \Delta_\max ) }{\lambda}
\end{equation}
These quantities need some further explanation and are best visualised by the following figure:

Figure: | OrcaFlex's tangential resistance model |

The dotted blue line here is the original, user-defined, normalised resistance profile, $\mu(\Delta)$, with its origin moved to the point $\Delta_o$. When the node first makes contact with the seabed, both $\Delta_o$ and $\Delta_\max$ are set to zero and, as the node translates to the right, it feels resistance exactly as defined by the profile of $\mu(\Delta)$. Additional complications arise if the node's motion changes direction between one time-step and the next. In this case we broadly adopt the reversal policy of DNV, supplemented by a user-specified unloading stiffness, $\lambda$, as suggested in Ballard *et al*. Let us assume that $\Delta$ has hitherto been increasing monotonically, to reach a value $\Delta_\max$ at the end of the previous time-step. In the next time-step, the node changes direction and the displacement decreases: $\Delta \lt \Delta_\max$. Instead of simply reversing along the user-defined profile, we instead linearly reduce $H$ from its maximum value along a line with gradient $\lambda$. This represents the assumption that the motion of the line has smoothed the soil in its wake.

Note: | By default, the unloading stiffness is set equal to the initial slope of $\mu(\Delta)$, but it can be independently set to be any positive value. |

If the resistance unloads completely, so that the x-axis is crossed, then we reinitialise the profile in the other direction, redefining the origin via $\Delta_o \rightarrow \Delta_\times$. There is some debate in the literature as to which policy to adopt in this scenario; in OrcaFlex, we most closely follow the DNV approach.

The red line on the graph represents the actual normalised resistance profile experienced by the line at some instant in time, with current state variables $\Delta_\max$ and $\Delta_o$. Moving to the right, $\Delta > \Delta_\max$, simply continues along the user-defined profile; the central section, $\Delta_\times \leq \Delta \leq \Delta_\max$, is linear with the specified unloading stiffness; and the left-hand section, $\Delta \lt \Delta_\times$, is simply the original profile reversed.

At the end of a time-step, the values of $\Delta_\max$ and $\Delta_o$ are updated according to which section of the red curve the node has converged to: \begin{eqnarray} \Delta_o &=& \left\{ \begin{array}{ll} \Delta_o & \Delta > \Delta_\times \\ \Delta_\times & \Delta \leq \Delta_\times \end{array} \right. \\ \Delta_{\max} &=& \left\{ \begin{array}{ll} \Delta_\max & \Delta_\times \lt \Delta \leq \Delta_\max \\ \Delta & \textrm{otherwise} \end{array} \right. \end{eqnarray} If the node switches onto the reverse curve – below the x-axis – then the above formulae are modified in the obvious way to reflect that the motion is now trending left rather than right (e.g. $\Delta_\max$ is effectively reinterpreted as a $\Delta_\min$).

Note: | In many analyses, this reversal mechanism will not be relevant because the displacement of the line will be monotonic by construction. Cyclic response of the line will instead be managed by instigating a succession of restarts; however, OrcaFlex must cater for all possibilities and handle reversal sensibly. |