New Target position
5.10 ENVIRONMENT THEORY
5.10.1 Buoyancy Variation with Depth
The buoyancy of an object is normally assumed to be constant and not vary significantly with position. The buoyancy is equal to ρVg, where ρ is the water density, V is the volume and g is the acceleration due to gravity. In reality the buoyancy does vary due to the following effects:
If the object is compressible then its volume V will reduce with depth due to the increasing pressure.
The water density ρ can vary with position, either because of the compressibility of the water, or else because of temperature or salinity variations. Normally the density increases with depth, since otherwise the water column would be unstable (the lower density water below would rise up through the higher density water above).
For buoys and lines these effects can be modelled in OrcaFlex.
Note: The bulk modulus and density variation facilities in OrcaFlex only affect the buoyancy of objects.
OrcaFlex does not allow for compressibility or density variation when calculating hydrodynamic effects such as drag, added mass, etc. The calculation of hydrodynamic effects use the uncompressed volume and a nominal sea density value, which is taken to be the density value at the sea density origin.
Compressibility of Buoys and Lines
All things are compressible to some extent. The effect is usually not significant, but in some cases it can have a significant effect on the object's buoyancy. To allow these effects to be modelled, you can specify the compressibility of a 3D Buoy, 6D Buoy or Line Type by giving the following data on the object's data form.
Bulk Modulus
The bulk modulus, B, specifies how the object's volume changes with pressure. If we denote by V the compressed volume of the object then V is given by:
V = V0(1-P/B)
where V0 is the uncompressed volume at atmospheric pressure, and P is the pressure excess over atmospheric pressure.
The bulk modulus has the same units as pressure F/L2 and the above formula can be thought of as saying that the volume reduces linearly with pressure, and at a rate that would see the object shrink to zero volume if the pressure ever reached B. For an incompressible object the bulk modulus is infinity, and this is the default value in OrcaFlex.
The above formula breaks down when P>B. In this case OrcaFlex uses a compressed volume V of zero. However, the relationship between pressure and volume would become inaccurate well before the pressure exceeded the bulk
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Theory, Environment Theory modulus. In practise B is normally very large, so the object normally only experiences pressures that are small compared to B.5.10.2 Current Theory
Extrapolation
In the presence of waves, the current must be extrapolated above the still water level; in OrcaFlex we adopt the convention that the surface current applies to all levels above the still water level.
If a sloping seabed is specified, the boundary is inconsistent with a horizontal current. This effect is not usually important and is uncorrected in OrcaFlex. The current at the greatest depth specified is applied to all greater depths.
Interpolated Method
Horizontal current is specified as a full 3D profile, variable in magnitude and direction with depth. The profile should be specified from the still water surface to the seabed. Linear interpolation is used for intermediate depths. If the specified profile does not cover the full depth then it is extrapolated (see Extrapolation above).
Power Law Method
Current direction is specified and does not vary with depth. Speed (S) varies with position (X,Y,Z) according to the formula:
S = Sb + (Sf - Sb) x ((Z-Zb) / (Zf-Zb))1/Exponent where
Sf and Sb are the current speeds at the surface and seabed, respectively, Exponent is the power law exponent,
Zf is the water surface Z level,
Zb is the Z level of the seabed directly below (X,Y).
Note: If Z is below the seabed (e.g. has penetrated the seabed) then the current speed is set to Sb and if Z is above the surface (e.g. in a wave crest) then current speed is set to Sf.
5.10.3 Seabed Theory
The seabed reaction force is the sum of a penetration resistance force in the seabed normal direction and a friction force in the direction tangential to the seabed plane and towards the friction target position. If explicit integration is used in the dynamic analysis then, in addition, seabed damping forces are applied in the normal and tangential directions.
The penetration resistance force depends on the choice of seabed model used – for details see either Linear Seabed Model Theory or Non-linear Soil Model Theory. For details of the friction force see Friction Theory.
Objects Affected
3D Buoys and 6D Buoys, lines and drag chains interact with the seabed. Other objects are not affected by it.
A line interacts when one of its nodes penetrates the seabed. The seabed reaction forces are calculated using the penetration of the lower outer surface of the line (based on the line type contact diameter) and the seabed forces are applied at that point. The seabed lateral friction force is calculated using the line type seabed friction coefficient.
A 3D Buoy interacts when the buoy origin penetrates the seabed. The seabed reaction forces are calculated using the penetration of the buoy origin, and are applied at the buoy origin. The seabed friction force is calculated using the buoy seabed friction coefficient.
A 6D Buoy interacts when any of its vertices penetrates the seabed. Each penetrating vertex experiences its own seabed normal reaction and lateral friction force, based on the penetration of that vertex and displacement of that vertex from its friction target position, and the forces are applied at that vertex. This gives a model where each vertex behaves rather like a pad (such as the landing pad on a lunar module).
Drag chain interaction with the seabed is calculated differently – see drag chain seabed interaction.
Linear Seabed Model Theory
In the Linear seabed model the seabed behaves as a linear spring in the normal direction, with spring strength equal to the Normal seabed stiffness specified in the seabed data.
Normal Seabed Stiffness Force
The normal stiffness reaction force has magnitude = KnAd and is applied in the outwards normal direction, where:
Kn = seabed normal stiffness A = penetrator contact area
Theory, Environment Theory
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For details on how the penetrator contact area is calculated see 3D Buoy Theory, 6D Buoy Theory and Line Interaction with Seabed and Solids.
Normal Seabed Damping Force
If implicit integration is used in the dynamic analysis then no seabed damping forces are applied. If explicit integration is used in the dynamic analysis then, seabed damping forces are applied in the normal and tangential directions, as follows.
The normal seabed damping force is only applied when the penetrating object is travelling into the seabed, not when it is coming out of the seabed. It is applied in the seabed outward normal direction and has magnitude Dn given by:
Dn = λ.2(MKnA)½Vn if Vn > 0 Dn = 0 if Vn ≤ 0
where
λ = seabed percent critical damping / 100
M = mass of the object (e.g. the mass of a node of a line) Kn = seabed normal stiffness
A = penetrator contact area
Vn = component of velocity normal to the seabed, positive when travelling into the seabed and negative when coming out.
The tangential seabed damping force, Dt, is applied in the direction opposing the tangential component of the velocity of the penetrator. It is given by
Dt = -λ2(MKtA)½Vt
where
Kt = seabed shear stiffness
Vt = vector component of penetrator velocity tangential to the seabed.
For details on how the penetrator contact area is calculated see 3D Buoy Theory, 6D Buoy Theory and Line Interaction with Seabed and Solids.
5.10.4 Seabed Non-Linear Soil Model Theory
The non-linear soil model has been developed in collaboration with Prof. Mark Randolph FRS (Centre for Offshore Foundation Systems, University of Western Australia). It is a development from earlier models that proposed and used a hyperbolic secant stiffness formulation, such as those proposed by Bridge et al and Aubeny et al.
For details of the data used by the non-linear soil model and its suitability for different seabed types see Non-linear Soil Model.
Note: The non-linear soil model is currently experimental and we are working on comparing the model against experimental results for pipe-seabed contact. Please contact Orcina if you have any feedback and comments on the model or ideas for improvement.
Full details of the non-linear soil model are given in Randolph and Quiggin (2009). The main aspects of the model are:
It models the seabed normal resistance using four penetration modes, as shown in the diagram below.
In each penetration mode the seabed reaction force per unit length, P(z), is modelled using an analytic function of the non-dimensionalised penetration z/D, where z = penetration and D = penetrator contact diameter. See Penetration Resistance Formulae below.
In Not In Contact mode the resistance P(z) is zero. In the other 3 modes the formula for P(z) uses a term of hyperbolic form, which provides a high stiffness response for small reversals of motion, but ensures that as the penetration z increases or decreases from its value when this episode of penetration or uplift started, then the resistance P(z) asymptotically approaches the soil ultimate penetration resistance (for penetration) or ultimate suction resistance (for uplift) at that penetration depth.
These features are now described in more detail in the sections below.
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Theory, Environment Theory Penetration ModesFigure: Soil Model Penetration Modes
The penetration mode of a given penetrator (e.g. a node on a line, a vertex of a 6D Buoy or the origin of a 3D Buoy) is determined by its penetration, z, and in the dynamic analysis by whether the penetration has increased or decreased since the previous time step. The details are as follows.
In the static analysis the Uplift and Repenetration modes are not used and the penetration mode is set to Initial Penetration if the penetration is +ve or to Not In Contact otherwise. The effect of this is that the static position found is based on the assumption that any static penetration occurred as a single progressive penetration, and it does not allow for the effect any of uplift and repenetration that might have occurred during first installation. OrcaFlex cannot allow for such effects since it only has limited information about how the line was originally laid.
The dynamic simulation starts from the results of the static analysis, so the penetrator starts the simulation in either Not In Contact mode or Initial Penetration mode. If it starts in Not In Contact mode then it changes to Initial Penetration mode the first time the penetration becomes +ve.
Once initial penetration has occurred in the dynamic simulation, the penetrator then stays in Initial Penetration mode until it starts to lift up, and it then changes to Uplift mode.
The penetrator then stays in Uplift mode until either the penetration falls to zero, in which case it breaks contact and changes to Not In Contact mode, or else until the penetration starts to increase again, in which case it changes to Repenetration mode. Similarly, Repenetration mode persists until the penetrator starts to lift up again, when it changes to Uplift mode. So if the penetrator stays in contact with the seabed but oscillates up and down then it switches back and forth between Uplift and Repenetration modes.
If the penetrator breaks contact and then later makes contact again then it enters Repenetration mode, not Initial Penetration mode. This is because the model assumes that second and subsequent periods of contact are making contact with the same area of seabed as was previously disturbed by the initial penetration.
Ultimate Resistance Limits
The resistance formulae are arranged so that as penetration z increases (for penetration) or decreases (for uplift) then the resistance asymptotically approaches the ultimate penetration resistance Pu(z) (for penetration) or the ultimate suction resistance Pu-suc(z) (for uplift). These ultimate penetration and suction asymptotic limits are given by
Pu(z) = Nc(z/D)su(z)D P (z) = -f P(z)