Effects from Internal Fluid Flow (Slug Flow)

The riser system model includes specification of internal fluid flow by:

Parameter Symbol Dimension

Cross section area of internal volume

\(\mathrm {A_i}\)

\(\mathrm {L^2}\)

Density

\(\mathrm {\rho _i}\)

\(\mathrm {M/L^2}\)

Velocity

\(\mathrm {v_i}\)

\(\mathrm {L/T}\)

Pressure gradient due to flow resistance

\(\mathrm {\rho d_i}\)

\(\mathrm {F/L^3}\)

Pressure at end 1, including the effect of flow density

\(\mathrm {\rho _{1i}}\)

\(\mathrm {F/L^2}\)

The following simplifications are made:

  1. The fluid is incompressible.

  2. The velocity and pressure distribution are assumed to be un-affected by the riser motion.

  3. The pressure is included only in order to account for possible future pressure-dependent stiffness of nonbounded flexible pipes and for axial wall force evaluation.

The internal flow as modelled will affect the riser configuration and behaviour in the following ways:

  1. The effect of weight and mass

Vertical force per unit length due to internal fluid:

\[q_i=A_i\rho _ig\]

This weight will affect the effective tension and thus the configuration of the riser. The weight will also affect the internal pressure distribution.

  1. The effect of internal pressure

The bending stiffness of the nonbounded flexible pipes is assumed to be dependent upon the pressure differential,

\[\begin{array}{l} p_d=p_e-p_i\\\\ p_e=\begin{cases}p_wg(-z),&z < 0 \\ 0,&z\geq0\end{cases}\\\\p_i=p_{1i}-\rho _igz-p_{di}s\end{array}\]

where

  • \(\mathrm {\rho _w,\rho _i}\) = density of water and internal fluid, respectively

  • \(\mathrm {p_{di}}\) = pressure drop per unit length due to flow resistance

  • \(\mathrm {s}\) = coordinate along the riser

  • \(\mathrm {p_{1i}}\) = internal pressure at lower end (end 1)

This information is not utilized in the present version.

  1. The static effect of the impulse and centrifugal force

The centrifugal force effect may be included in the riser loads. This force depends upon local curvature, velocity and mass of fluid flow.

RIFLEX TheoryManual 42 v0 869
Figure 1. Centrifugal force effect

The centrifugal force per unit length can be expressed by:

\[\begin{array}{l}q_c=m_iv_i^2/R\\m_i=\rho _iA_i\end{array}\]

This force will act radially on any curved part of the riser. However, the force will also contribute to increase the effective riser tension:

\[f_e=f_{eo}+mv_i^2\]

Comparing the two cases a) and b) in Figure 1 for a case with zero bending stiffness, the radii of curvature can be expressed by:

\[\begin{array}{l}\textrm{a)}\quad R_o=(f_{eo}/q_o)\\\\\textrm{b)}\quad R=(f_{eo}+mv_i^2)/(q_o+mv_i^2/R)=f_{eo}/q_o=R_o\end{array}\]

Thus, the centrifugal force will not contribute to change the static configuration. The only effect is the increase of wall tension due to the impulse force, according to Equation (4) written:

\[\boldsymbol{q_I}=-m_i\boldsymbol{\dot v}\]

where \(\mathrm {\boldsymbol{\dot v}}\) is the acceleration vector of the fluid. Assuming stationary, incompressible flow, the acceleration can be written:

\[\boldsymbol{\dot v}=v_i^2\frac{\partial \boldsymbol{e}}{\partial s}+\boldsymbol{\ddot r}-2v_i(\boldsymbol{\omega }\times \boldsymbol{e})\]

where

  • \(\mathrm {\frac{\partial \boldsymbol{e}}{\partial s}}\) = curvature vector

  • \(\mathrm {\boldsymbol{r}}\) = acceleration of pipe

  • \(\mathrm {\boldsymbol{\omega }}\) = rotation of pipe

  • \(\mathrm {\boldsymbol{e}}\) = pipe tangent unit vector

  • \(\mathrm {s}\) = curve coordinate along the pipe

Thus, the following force terms are derived:

\[\boldsymbol{q_I}=-m_i\boldsymbol{\ddot r}-m_iv_i^2\frac{\partial \boldsymbol{e}}{\partial s}-2m_iv_i(\boldsymbol{\omega }\times \boldsymbol{e})\]

The force terms accounts for:

  1. Force due to pipe acceleration

  2. Force due to centripetal acceleration

  3. Force due to pipe rotation (Coriolis force)

The present program version includes only the first term. It is included directly in the mass matrix of the elements.

A special slug modelling option is included to simulate the effect of a relatively short fluid slug moving through a riser. This is done by updating the mass matrix according to a specified slug speed in the pipe.

The second term is identical with the static centrifugal force discussed previously, but the curvature will be time-dependent in the dynamic case.

The third term may have more complicated effects.

Both the second and the third terms may be included directly as forces in the time domain analysis in a possible future extension of the program.