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Line theory: Buried lines |
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Line theory: Buried lines |
Upheaval buckling (UHB) of a buried line occurs when its cover (the material under which the line is buried) provides insufficient resistance to prevent the line from bowing upwards. Such buckling tends to occur when the pipe undergoes cycles of thermal expansion and contraction during production: small upwards deformations of the pipe develop at first, and these can become reinforced over subsequent cycles, eventually leading to a catastrophic buckle. By modelling cover resistance forces, engineers may be able to add extra mitigation (such as overlaying additional concrete mattresses) at the points where upheaval buckling is most likely to occur.
Upheaval buckling and cover resistance forces can be modelled in OrcaFlex using line covered sections. A covered section is an interval on the line that is considered to have been buried under a covering material whose properties are encoded in a cover type object. OrcaFlex's cover resistance model comprises two distinct, but related, aspects:
We define the total cover force (per unit length) on the line, $F$, by combining these two contributions: \begin{equation} F = R_\downarrow - R_\uparrow \end{equation} such that $F$ is positive if the net force is downwards, and negative if the net force is upwards.
| Note: | The model is defined on a per unit length basis, which means that the cover force varies with arc length along the line. In practice, this means that the model is node-based, with a different resistance acting at each node. For the purposes of this theory topic, we often refer to 'the line' for simplicity, but it should be remembered that we generally only mean one particular cross-section of the line. |
| Since cover properties are a function of arc length, the cover above each line node moves with that node when it translates. The cover resistance always acts vertically and is a one dimensional function of the vertical separation between the line and the seabed. Motions of the node that are tangent to the seabed do not change the vertical separation and therefore do not change the value of the cover resistance. |
Line covered sections are primarily aimed at modelling lines embedded in soil (although the feature can also be used to model uplift resistance forces from other sources). There are several mechanisms by which covering soil can prevent line uplift:
For the present, we will work in general terms and not tie ourselves into any particular physical mechanism. The following diagram illustrates the quantities involved in OrcaFlex's uplift resistance model:
| Figure: | Uplift resistance nomenclature |
| Warning: | If a node lies between two segments that have different outer contact diameters, then the diameter used to define the underside of the line will be different from the covered line diameter, $D$, used in the cover force calculations. The former is the maximum of the two segments' outer contact diameters; the latter is their weighted mean. This policy has been chosen to match the seabed contact model when moving downwards through the cover, and also to avoid significantly overestimating the uplift resistance at such a node. The effects of this policy can be reduced by using a finer discretisation of the line. |
The uplift resistance, $R_\downarrow$, is computed using the following formula: \begin{equation} R_\downarrow = R_c \, r \end{equation} where $R_c$ is a characteristic resistance scale (per unit length) and $r$ is a non-dimensional normalised resistance. What follows is a high-level discussion of these quantities; specific formulae for different types of cover, along with associated demonstration models, can be found by referring to the buried line examples topic.
The resistance scale, $R_c$, is often set equal to the peak resistance, $R_p$, which is the maximum resistance to uplift (per unit length) that the soil can apply to the line. A safety factor, $f_\textrm{safety}$, can also be included by setting $R_c = f_\textrm{safety} R_p$.
$R_c$ is commonly a function of $D$ and $H$, although more complicated models exist that also have $z$-dependence. In OrcaFlex, $R_c$ can be:
For example, a common DNV assumption for drained soil (see DNVGL-RP-F114) is: \begin{equation} R_p(D, H) = \gamma' \left[ D H + \left(\frac12 - \frac{\pi}{8} \right) D^2 + f \left(H + \frac{D}{2} \right)^2 \right] \end{equation} where $\gamma'$ is the submerged weight of the soil (per unit volume) and $f$ is known as the uplift resistance factor.
Usually, the line must mobilise by a small amount before the full peak resistance begins to act. The normalised resistance, $r = R_\downarrow/R_c$, is a non-dimensionalised quantity corresponding to the proportion of the resistance scale that has been mobilised. It is defined in OrcaFlex by a tabular variable data function, $r_0(x)$, and a user-specified unloading stiffness, $\lambda$. $r_0(x)$ determines how quickly the uplift resistance builds up to its maximum value following the onset of uplift; it will usually be a bilinear or trilinear curve, defined as a function of normalised mobilisation, $x = z/d_m$, where $d_m$ is a parameter known as the mobilisation scale. $\lambda$ determines what happens when the pipe descends back into the soil from its point of maximum uplift, $z_\textrm{max}$. There is no clear guidance in the literature regarding unloading. The unloading stiffness has therefore been included to provide maximum flexibility, and for consistency with the equivalent data for seabed tangential resistance models. It is envisaged that $\lambda$ will usually be set to 0 or N/A in the present upheaval buckling context.
Given $r_0(x)$ and $\lambda$, the normalised resistance, $r(x)$, is given by: \begin{equation} r(x) = \begin{cases} r_0(x) & x \geq x_\textrm{max} \; \textrm{or} \; \lambda = \textrm{N/A} \\ r_0(x_\textrm{max}) - \lambda (x_\textrm{max} - x) & x_\times \leq x < x_\textrm{max} \; \textrm{and} \; \lambda \neq \textrm{N/A} \\ 0 & x < x_\times \; \textrm{and} \; \lambda \neq \textrm{N/A} \end{cases} \end{equation} where $x_\textrm{max} = z_\textrm{max}/d_m$ and $x_\times = x_\textrm{max} - r_0(x_\textrm{max}) / \lambda$. The relationship between $r(x)$, $r_0(x)$ and $\lambda$ are illustrated in the following figure:
| Figure: | Typical trilinear uplift resistance |
The $r_0(x)$ normalised resistance curve ramps the uplift resistance from zero to $R_c$ over a distance $d_m$. The red curve, $r(x)$, includes an unloading stiffness, $\lambda$. $r(x)$ will remain equal to $r_0(x)$ as long as the mobilisation, $x$, is monotonically increasing, but may diverge from $r_0(x)$ if $x$ ever decreases.
| Notes: | The normalised resistance can never be negative but it can be greater than one. |
| The normalised resistance can also be set equal to a constant value. There is no unloading in this case. |
The mobilisation scale, $d_m$, non-dimensionalises the $x$-axis of the normalised resistance curve, $r(x)$. It is typically set to the peak mobilisation, $d_p$ (the distance that the pipe must uplift by in order to mobilise the peak soil resistance), although this is not mandatory. It is commonly a function of $D$ or $H$, and varies by soil type. In OrcaFlex, $d_m$ can be:
For example, Thusyanthan et al., 2010 proposed the following equation for loose sands: \begin{equation} d_m(D, H) = \frac{D}{50} \, \exp \left( \frac{H}{2D} \right) \end{equation}
There is no special handling when the top of the line leaves the defined cover. OrcaFlex will simply continue to treat the line as if it were still covered. However, OrcaFlex can optionally report a warning if this should occur, which lets you know that the line has achieved break out. This may indicate that extra protective measures are needed to guard against upheaval buckling. The practical reason for continuing to apply cover resistance after break out is that zeroising it would result in a discontinuous force model, leading to insurmountable convergence problems. An alternative would be to ramp the cover resistance down to zero in some way, but this would be rather arbitrary. Instead, we have opted to give the user responsibility for handling this by incorporating any drop off in resistance into their cover load model. The default behaviour in OrcaFlex can be visualised as the line lifting a column of soil as it rises out of the cover, with this column forever remaining on top of the line, even after break out. You can implement alternative schemes by using a cover load model with $z$-dependence, an example of which can be found here.
| Note: | No account is taken of the presence of the seabed in the uplift resistance calculation. Any residual uplift resistance will unload in the usual way when the line descends below its initial, as-buried position. |
If the line is buried with a layer of cover between the bottom of the pipe and the seabed, then this will act to prevent the line from descending deeper into the cover. This can happen for any part of the line that was not in contact with the seabed when it was buried. More likely, it will be the result of the surrounding soil infilling any voids created beneath the line as it uplifts. This can introduce an asymmetry, whereby the upwards motion of a pipe during its operational phase is not matched by its downward motion during shutdown. This phenomenon is known as pipe ratcheting and is a major contributor to upheaval buckling: the points on a pipe where ratcheting has previously occurred are much more likely to bow upwards in subsequent operational cycles. Furthermore, the curvature at each ratcheting point tends to increase in each successive cycle, until the line eventually buckles catastrophically. The extent to which ratcheting can occur depends on the type of soil within which the line is embedded. In particular, it depends upon the critical friction angle of the covering material: fine sands are more likely to infill than sticky clays.
Two data items, downward stiffness and ratcheting length, control the downward resistance on a buried line.
The soil beneath the line is treated as an elastic solid for purposes of computing downward resistance. It can be considered as a simple spring whose stiffness is either:
In both cases, the resulting downward resistance (per unit contact area) can be expressed as a function $u(q)$.
The ratcheting length, $d_r$, controls how quickly the cover infills beneath the line during uplift. A value of infinity means that the covering material never infills, so ratcheting will not occur; a value of zero means that any void beneath the line is instantly infilled, so there will be immediate resistance to downward motion once uplift reverses. More generally, the ratcheting length can be:
The downward resistance model is most easily described by working in terms of $b$, the height of the bottom of the pipe above the seabed. $b$ is related to the mobilisation, $z$, via: \begin{equation} b = z - p_\textrm{ab} \end{equation} Given $u(q)$ and $d_r$, the downward resistance (per unit length), $R_\uparrow(b)$, is computed as follows: \begin{equation} R_\uparrow(b) = \begin{cases} D \, u(q) & b < b_\textrm{infill} \; \textrm{and} \; b_\textrm{infill} > 0 \\ 0 & \textrm{otherwise} \end{cases} \end{equation} where
These quantities are illustrated by the following diagram:
| Figure: | Downward resistance nomenclature |
$b_\textrm{infill}$ defines the infill level, below which the cover will exert an upwards force. Prior to mobilisation, $b_\textrm{infill}$ will lie directly beneath the pipe (i.e. $b_\textrm{infill} = b_\textrm{ab}$); however, once mobilisation begins, $b_\textrm{infill}$ will rise as the soil to the sides of the pipe falls into the space created beneath it. $b_\textrm{infill}$ can be visualised as being dragged upwards by the pipe, a distance $d_r$ beneath its bottom edge. The various $\min$ and $\max$ conditions in the formulae benefit from further explanation.
The formula $b_\textrm{infill} = \max(\min(b_\textrm{max}, b_\textrm{top}) - d_r, b_\textrm{ab})$ constrains $b_\textrm{infill}$ to lie in the range $b_\textrm{ab} \leq b_\textrm{infill} \leq b_\textrm{top} - d_r$. This means:
The expression $R_\uparrow(b)$ when $b < b_\textrm{infill}$ and $b_\textrm{infill} > 0$ is interpreted as follows:
The formula $q = b_\textrm{infill} - \max(b, 0)$ means that the downward resistance provided by the cover is truncated once the bottom of the pipe makes contact with the seabed. This ensures that there is no double-counting between the cover and seabed downward resistance forces, resulting in a continuous force profile as the line transitions between the cover and the seabed.
It is possible to input bespoke formulae for $R_c$, $d_m$ and $d_r$ using a cover load model; and a bespoke formula for $H$ using a cover height model. Cover models are best illustrated by reading the buried line examples topic. One scenario where you are forced to use cover models is when you have a multilayer cover model, such as the one outlined in Thusyanthan et al., 2008, which describes a model comprising separate backfill and rockdump layers. When you have multiple layers of cover, $H$ becomes a vector quantity, having one component per layer of the cover. This can only be achieved by defining a cover height model that returns $H$ as a tuple or list. Similarly, any cover load models must take into account that $H$ is now a Python iterable containing multiple entries. How this works in practice is best illustrated by studying the backfill plus rockdump example model.
As well as having multilayer cover types, it is also possible to define multiple cover types acting in parallel at a given node. In this case, the resultant resistance forces are summed and it is assumed that there are no interactions between the different cover types. This can be useful as a way of applying additional upheaval buckling mitigation – such as laying concrete mattresses over the points on the line where buckling is most likely to occur – to an existing buried line.
In a restart analysis, the internal history state variables of the cover resistance calculation, $b_\textrm{ab}$ and $z_\textrm{max}$, are persisted from the parent model, even if the properties of the covering material have been changed. However, switching to use an entirely different cover type, or changing the as-buried seabed penetration, means that any existing state is discarded and the line is considered to have been freshly reburied for the purposes of the restart. The same effect can be achieved by marking the cover type reference, or the as-buried seabed penetration data, as changed on the line data form.
