LedaFlow is a new dynamic multiphase flow simulator for wells and flowlines. Based on models that are closer to the actual physics of multiphase flow, LedaFlow provides a step change in fidelity, quality, accuracy and flexibility over current generation multiphase flow simulation technology. This increase in model definition provides the engineer with much greater understanding of the flow in wells and pipelines.
Ledaflow development started in 2001 in collaboration between Total, ConocoPhillips and SINTEF. In 2009, Kongsberg joined as the industrialization and commercialization partner. First commercial release was in June 2011 and in March 2013 the 4th version, release 1.3 was issued.
http://www.kongsberg.com/en/kogt/offerings/software/ledaflow/
The flow regimes are model elements that describe the geometrical distribution of the various phases. In LedaFlow there are two sets of flow regimes: the gas-liquid flow regime and the oil-water flow regime.
The gas-liquid flow regime describes the geometrical configuration of the gas and liquid. Here the liquid is considered as a mixture of oil and water with its own internal structure represented by the oil-water flow regime. There are three basic gas-liquid flow regimes:
In separated flow, the gas and liquid flow in different zones. The liquid may be a mixture of oil and water, which may be fully mixed, completely or partially separated. The gas zone may have droplets of oil and water and the liquid zone may have gas bubbles. Gas entrainment in the liquid film is only modelled in slug capturing.
The separated flow regime is sub-divided into two categories:
In bubbly flow the gas is assumed to exist only in the form of bubbles entrained inside the liquid, so there is no gas zone. In three-phase flow with separated oil and water, we assume that the geometric distribution of oil and water is the same as in separated flow and that the gas bubbles are uniformly distributed across the pipe cross-section.
Hydrodynamic slug flow is an intermittent transient flow regime, where flow consists of liquid slugs with regions of separated flow in between. In the liquid slugs the local flow regime is assumed to be bubbly. Between the slugs the local flow regime is assumed to be separated. The models used in these two regions are consistent with the modelling used in the bubbly flow and separated flow.
In the standard LedaFlow model, hydrodynamic slug flow is modelled on a sub-grid level, i.e. the slugs are not resolved on the grid. Instead, the steady-state solution to the slug flow equations (called the Unit Cell Model, or UCM) is calculated, and the results are subsequently used to calculate the appropriate friction terms, etc. in the transient code.
The UCM consists of 34 equations and 34 unknowns in the case of three-phase flow. The unknown quantities are:
The UCM equations for three-phase flow are:
Slug flow is selected if a physical solution to the UCM is found, that is a solution with a slug fraction between 0 and 1. If no such solution can be found the flow regime is deemed either bubbly flow or separated flow. The flow regime selection logic within LedaFlow is formulated so that the UCM equations are only solved if the flow regime is slug flow.
The logic can be summarised by:
In LedaFlow we have two basic oil-water flow regimes:
The oil-water regime is modelled as mixed if the absolute value of the pipe angle exceeds 9 degrees in separated flow or 15 degrees in slug/bubbly flow. In addition, if the no-slip phase fraction of either oil or water is less than 10-5 , the oil-water regime is also assumed to be mixed.
For the gas phase
\begin{equation}
\frac{\partial (V_{g}\rho _{g})}{\partial t}=-\frac{1}{A}\frac{\partial (AV_{g}\rho _{g}v_{g})}{\partial z}+\psi _{g}+G_{g}
\end{equation}
For the liquid droplets
\begin{equation}
\frac{\partial (V_{D}\rho _{L})}{\partial t}=-\frac{1}{A}\frac{\partial (AV_{D}\rho _{L}v_{D})}{\partial z}-\psi _{g}\frac{V_{D}}{V_{L}+V_{D}}+\psi _{e}-\psi _{d}+G _{D}
\end{equation}
For the liquid film
\begin{equation}
\frac{\partial (V_{L}\rho _{L})}{\partial t}=-\frac{1}{A}\frac{\partial (AV_{L}\rho _{L}v_{L})}{\partial z}-\psi _{g}\frac{V_{D}}{V_{L}+V_{D}}-\psi _{e}+\psi _{d}+G _{L}
\end{equation}
$g= Gas$
$D=Droplets$
$L=liquid\;film$
$V=Volume\;fraction\;(-)$
$v=velocity\;m/s$
$\rho=density\;(kg/m^3)$
$A=cross-sectional\;area\;of\;the\;pipe\;(m^2)$
$\psi_g=mass\;transfer\;rate\;between\;phases\;(kg/m^3/s)$
$\psi_e=entrainment\;rate\;of\;liquid\;droplets\;(kg/m^3/s)$
$\psi_d=deposition\;rate\;of\;liquid\;droplets\;(kg/m^3/s)$
$G=possible\;mass\;source\;of\;a\;particular\;phase\;(kg/m^3/s)$
For the gas phase
\begin{equation}
\frac{\partial (V_{g}\rho _{g}v_{g})}{\partial t}=-V_{g}\left (\frac{\partial p}{\partial z} \right )-\frac{1}{A}\frac{\partial (AV_{g}\rho _{g}v_{g}^{2})}{\partial z}\\
-\lambda _{g}\frac{1}{2}\rho_{g} \left |v_{g} \right |v_{g}\frac{S_{g}}{4A}-\lambda _{i}\frac{1}{2}\rho_{r} \left |v_{r} \right |v_{r}\frac{S_{i}}{4A}+V_{g}\rho_{g}g\,cos\alpha+\psi_{g}v_{a}-F_D
\end{equation}
For the liquid droplets
\begin{equation}
\frac{\partial (V_{D}\rho _{L}v_{D})}{\partial t}=-V_{D}\left (\frac{\partial p}{\partial z} \right )-\frac{1}{A}\frac{\partial (AV_{D}\rho _{L}v_{D}^{2})}{\partial z}\\+V_{D}\rho_{L}g\,cos\alpha+\psi_g \frac{V_D}{V_L+V_D}v_a+\psi_{e}v_{i}-\psi_{d}v_{D}+F_D
\end{equation}
Eqs. (4) and (5) are combined to yield the following equation.
\begin{equation}
\frac{\partial }{\partial t}\left ( V_g \rho_g v_g+V_D\rho_L v_D \right )=-\left ( V_g+V_D \right )\left ( \frac{\partial p}{\partial z} \right )-\frac{1}{A}\frac{\partial }{\partial z}\left ( AV_g\rho_gv_g^{2} +AV_D\rho_L v_D^2\right )-\lambda_g\frac{1}{2}\rho_g\left | v_g \right |v_g\frac{S_g}{4A}\\-\lambda_i\frac{1}{2}\rho_g\left | v_r \right |v_r\frac{S_i}{4A}+\left ( V_g\rho_g+V_D\rho_L \right )\,g\,cos\alpha+\psi_g\frac{V_D}{V_L+V_D}\,v_a+\psi_e\,v_i-\psi_d\,v_D
\end{equation}
For the liquid film
\begin{equation}
\frac{\partial }{\partial t}\left ( V_L\rho_Lv_L \right )=-V_L\left ( \frac{\partial p}{\partial z} \right )-\frac{1}{A}\frac{\partial }{\partial z}\left ( AV_L\rho_Lv_L^2 \right )-\lambda_L\frac{1}{2}\rho_L\left | v_L \right |v_L\frac{S_L}{4A}+\lambda_i\frac{1}{2}\rho_g\left | v_r \right |v_r\frac{S_i}{4A}\\+V_L\rho_L\,g\,cos\alpha-\psi_g\frac{V_L}{V_L+V_D}v_a-\psi_ev_i+\psi_dv_D-V_Ld\left(\rho_L-\rho_g \right )\,g\,\frac{\partial V_L}{\partial z}\,sin\alpha
\end{equation}
where,
$v_a=v_L \;\;for\;\; \psi_g>0\;\; \left(and\; evaporation\; from\; the \;liquid \;film\right)$
$v_a=v_D \;\; for \;\; \psi_g>0\;\; \left(and\; evaporation \; from \; the \;liquid\;droplets\right)$
$v_a=v_g\;\; for \;\; \psi_g<0\;\;\left(condensation\right)$
$alpha\;=\;pipe\; inclination\;from\; the\;vertical\;\left(rad\right)$
$p\; = \; pressure\; \left(Nm^{-2}\right)$
$v_r\;= \;relative\; velocity\; \left(ms^{-1}\right)$
$G\;=\;internal\; mass\; source\; - assumed\; to\; enter\; perpendicular\; to\; the\; pipe\; wall\;, carrying\; no\; moment$
$ S_g\;=\;wetted\; perimeter\; of\; the\; gas\; \left(m\right)$
$S_L\;=\;wetted\; perimeter\; of\; the\; liquid \; \left(m\right)$
$S_i\;=\;wetted\; perimeter\; of\; the\; interface \; \left(m\right)$
The relative velocity is given by the following equation
\begin{equation}
v_g=R_D\left( v_L+v_r\right)
\end{equation}
For the conservation of energy a single mixture energy balance equation is applied
\begin{equation}
\frac{\partial }{\partial t} \left [m_g\left( E_g+\frac{1}{2}v_g^{2}+gh \right)+m_L\,\left(E_L+\frac{1}{2}v_L^{2}+gh \right ) +m_D \left ( E_D+\frac{1}{2}v_D^{2}+gh \right )\right ]\\ =-\frac{\partial }{\partial z}\left [ m_gv_g\left ( H_g+\frac{1}{2}v_g^{2} +gh\right ) + m_Lv_L\left ( H_L+\frac{1}{2}v_L^{2} +gh\right )+ m_Dv_D\left ( H_D+\frac{1}{2}v_D^{2} +gh\right ) \right ]\\+H_S+U
\end{equation}
where,
$m_f=V_f\rho_f\; for \; the \; phase\; f$
$E=internal\; energy\; per \; unit\; mass\; \left( Jkg^{-1} \right)$
$U=heat\; transfer\; per\; unit\; volume\; from\; the \; pipe\; walls\; \left(Jm^{-3} \right)$
Work between the gas and liquid phases are usually negligible when compared to the heat transfer from the pipe walls.
All fluid properties, e.g. densities, compressibilities, viscosities, surface tension, enthalpies, heat capacities, and thermal conductivities, can be provided to LedaFlow by a file that contains a table in which these fluid properties of the phases are given for a number of thermodynamic conditions, i.e. pressure and temperature.
The fluid properties during a simulation are found by interpolating in this table.
The two-fluid model, as formulated above, provides a set of coupled first-order, nonlinear, one-dimensional partial differential equations. LedaFlow uses a finite difference scheme on a staggered grid for the spatial discretization. A semi-implicit time integration method used. The time step is limited by the average phase velocities based on the mass-transfer criterion
\begin{equation}
\Delta t \leq min_{j}\left( \frac{\Delta z_j}{\left | v_{fj} \right |}\right)
\end{equation}
where,
$\Delta z= \:length\: of\: a\: grid\: cell\: (m) $
$v_f = velocity\: of\: phases\:\: f$
$_j=index \: to \: identify\: a\: grid\: cell$