Many industrial processes in petroleum engineering involve reactions occurring between cocurrent liquid and gas downflow in fixed bed reactors. Reaction rates in these processes are strongly affected by the phases distribution and more generally by the hydrodynamic multiple features occuring in a bed scale and summarized in the review of Maiti et al. . Due to the flow complexity, most previous studies focused on describing two-phase flow at the bed-scale without taking into account phenomena occuring at the grain-scale. The effect of fluid properties and injection conditions on pressure drop, holdup and flow regimes at the bed scale were described and correlated but local flow heterogeneities cannot be studied in this way.
Considering that the packed bed can be seen as a network of sites (void space between particules) and bonds, the two-phase flow can be understood in terms of well-defined processes at the level of elementary channels. The advantage of a network representation of a packed bed is its ability to model the simultaneous presence of different flow regimes ("trickling", "pulsing", "bubbling"...) within the same environment. This network approach was first proposed by Melli and Scriven  describing experimentally constitutive laws in bonds using elongated constricted tubes.
Constitutive laws of the network model are developped in this study using direct numerical simulations at the grain scale. Forming the basis of the model, the interface-diffuse method "Volume Of Fluid" implemented in Ansys Fluent is first validated quantitately for different geometries, meshes and numerical schemes.To validate the overall method, we first study the spread of a liquid jet in a two-dimensional array of cylinders (Fig.3). This configuration allows to reproduce the hydrodynamics multiplicity observed in packed beds while allowing a three-way study : experimental visualizations, direct numerical simulations and network simulations.
In order to extend the two-dimensional validated network model to a three-dimensional case, a study by CT imaging of the spread of a liquid jet on a real stack of grains is presented.
2. Pore-Scale Study
As a first step, the "Volume Of Fluid" method is quantitatively validated performing a comparison between direct numerical simulations and analytic solution of dynamic film occurring during the drainage of a wetting phase in a capillary tube . The "Volume Of Fluid" method is able to simulate a film flow accurately under some conditions such as a minimum number of cells in the film or the choice of a numerical scheme being a compromise between solution stability and interface conservation. Once validated, the method is used to simulate two-phase flow in constrictions between two particles and the different flow regimes observed in a packed bed are reproduced (Fig.1). Regime flow maps depending on liquid and gas flow rates are plotted for different constriction radii. The simulations allow determining local regime transition criteria such as a critical saturation or liquid Reynolds number which are then incorporated in the network model. In anticipation of the development of constitutive laws for the three-dimensional network model, we discuss the differences between a purely 2D and an axisymmetric constriction.
3. The two-dimensional validation case
3.1. Network model
To compute the velocity and pressure fields for each phase in the packed bed represented as a network of sites and bonds, we consider that a site is a circular void space (spherical in 3D) characterized by an average pressure and saturation while the passage between the pores are tubes characterized by a tube saturation. Considering that both phases are present in the tubes, the flow rate of each phase is calculated by assuming that the basic flow regime in the passage is trickling. Assuming a two-phase Poiseuille velocity profile within the tube, it is possible to define a flow rate for each phase depending on the pressure gradient, saturation and gravity. The error induced by this assumption can be measured using direct numerical simulations in constrictions. The variables at each site, saturation and pressure, are obtained by solving the liquid mass balance and the overall mass balance and the saturation in each bond can be obtained solving the liquid mass balance in the bond.
The model complexity is due to the fact that regime-change laws introduced by the pore-scale study can change abruptly the model coefficients. One example is the creation of a liquid bridge preventing the flow of the gas phase when the saturation in a bond reaches a critical value. An example of simulation of the spread of a liquid jet is presented in Fig.2.
3.2. Experimental process
The experimental process consists of a random array of cylinders with a diameter of 5mm placed between two plates (7cm by 7cm), all made of polydimethylsiloxane (PDMS). The spacing between the 2 plates is 2mm as the average spacing between cylinders and the experiment boundaries are open to ambient air. The wetting phase is injected at the top of the array and we can observe the spread of the liquid jet through the array of cylinders. As shown in Fig.3, the simultaneous presence of the different flow regimes identified in the literature is observed with this experiment as well as local regime flow transitions.
3.3. Model validation
The experiment field remains relatively small and it is possible, although expensive, to simulate directly with the "Volume Of Fluid" method the spread of the liquid jet (Fig.3). The two-dimensional representation of the array of cylinders is validated performing a comparison between experimental visualizations and direct numerical simulations. Direct numerical simulations can then be used to study the stabilizing effect of the gas flow with direct simulations (which would be much more time consuming and expensive to set up experimentally). Finally, experimental visualizations and direct numerical simulations are used to validate the two-dimensional version of the network model.
4. Extension to real 3D packed beds
To validate the three-dimensional network model, the spread of a liquid jet on a fixed bed pilot was studied by CT and tomography imaging. The microtomography imaging provides the topology of the stack of grains in order to construct the representative network of the void space. CT imaging has a lower resolution but a shorter acquisition time and can capture the spread of the liquid jet (Fig.4). Effects related to the pre-wetting of the fixed bed are highlighted in this experiment and requires a more comprehensive study to be integrated in the network model.
5. Conclusions and Perspectives
The choice of an array of cylinders as a simplified representation of a fixed bed was useful to test a three-way approach which would be particularly complicated to set-up for a real stack of grain. From an experimental point of view, it was possible to observe hydrodynamic multiple features in a sort of fixed bed with a camera. Although passages between cylinders are in three dimensions, it is possible to reproduce the flow by performing direct simulations in two dimensions and by studying the gas flow effect. Finally, the two-dimensional case validates a network approach to simulate two-phase ow in fixed bed.
For a future validation of the three-dimensional network model, the topology of a fixed bed is extracted by microtomography imaging and the spread of a liquid jet is captured by CT imaging.
R. Maiti and R. Khanna and K.D.P Nigam, Hysteresis in Trickle-Bed Reactors : A review, Ind. Eng. Chem. Res., 45, 5185-5198, 2006.
Tomas R. Melli and L. E. Scriven, Theory of Two-Phase Cocurrent Downflow in Network of Passages, Ind. Eng. Chem. Res., 30, 951-969, 1991.
F.P. Bretherton, The Motion of Long Bubbles in Tubes, 1960.