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Effective tension
In case of an immersed riser with an internal fluid inside, the system "riser + internal fluid" in water may be modelled by an equivalent beam taking account of the total apparent weight.
The resulting tension of the model is the effective tension which is linked to the real tension (or true tension) acting on the riser alone with the following relation :
with
\(P_i\) : internal fluid pressure
\(S_i\) : internal section of the studied system
\(P_e\) : external fluid pressure
\(S_e\) : external section of the studied system
For rigid risers, the combination of the keywords RIGIPRO, FLUID-IN et *RISER enables to play with various parameters so as to get the total apparent weight of the system :
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internal riser section \(S_i\),
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additional mass,
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varying pressure.
Note that the word system may stand for a riser, its internal fluid, buoyancy modules, kill and choke lines, drilling string
Once the required apparent weight is obtained, the resulting tension is the effective tension for the whole system. The problem therefore is to come back to the stresses on each component.
The tangential \(\sigma_t\) and the radial \(\sigma_{r}\) stresses due to pressure are introduced using the well-known expressions of Lam in elasticity.
The true axial stress \(\sigma_{real}\) is deduced from the effective stress \(\sigma_{eff}\) :
On another hand, due to the Poisons coefficient, the fluid pressures have an effect on the axial strain. The well-known expressions of Lam actually write :
With this formula, the axial strain may be expressed as a function of the the true axial stress \(\sigma_{real}\), the tangential stress and the radial stress \(\sigma_{r}\) :
DeepLines is a finite element program and is classically based on the nodal positions which stand for the unknowns of the problem. An iterative procedure is used to find the quilibrium positions for which the associated internal forces and moments balance the external efforts applied on the line. At each time step, the calculations give the total axial strain \(\varepsilon_{axial} = \varepsilon_{mechanical}- \frac{2\nu} E \sigma_p\) and \(\varepsilon_{mechanical}\) is converted into axial true tension.
The effect of pressure is taken into account for rigid pipes. For flexible ones, this is optional and when selected, the geometrical characteristics of the pipe are required.
It is recommended to be careful with the stress post-processing from the results of the equivalent beam. So far, in DeepLines, the stress post- processing is performed with the following assumptions :
For drilling risers:
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The external diameter in the term \(P_eS_e\) is the bare risers outer diameter even for buoyant joints since most of time, some water may be found in the gap.
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The contribution of the kill & choke lines is ignored.
For buoyancy modules:
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A complete calculations of the stresses in both the riser and the buoyancy module should be conducted so as to know which is the pressure transfered to the riser. This requires some data on the buoyancy foam.
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Considering no radial displacement at the risers outer section leads to the following expression of the radial stress : \(\sigma = 2 \frac{(\nu-1)P_er_e^2}{-2\nu r_0^2 + r_e^2 + r_0^2}\) where \(r_o\) and \(r_e\) is the buoyancy modules inner and outer radii.
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Finally, with a Poissons coefficient of 0.5 for the foam the whole external pressure is transferred to the risers outer surface. The stress post-processing is then performed with the bare risers outer diameter.
For flexible risers:
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For real flexible structures, a specific post-processing is required to get the stresses in the various layers.
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Nevertheless, stresses may be post-processed with the diameter and thickness specified for the post-processing with DeepLinesGUI. Otherwise, we assume that the global stiffness define a consistent set of data which allows to come back to the outer and inner diameters of the riser. For example, the steel area is computed as \(A_{steel} = \frac{EA}{E}\).
