M Benioug

F Golfier

P Fischer

C Oltean

M A Buès

X Yang
email: xfyang@bnu.edu.cn

Interaction between Biofilm Growth and NAPL Remediation: A Pore-scale Study

Keywords: NAPL biodegradation, lattice Botzmann method, immersed boundary method, biofilm growth, porous media, cellular automata

In this paper, we introduce a pore-scale model to study the interaction between biofilm growth and nonaqueous-phase-liquid (NAPL) dissolution. Liquid flow and dissolved NAPL transport are coupled with a biofilm growth model to correctly describe the complex dynamics of the processes including fluid flow, NAPL dissolution/biodegradation and biofilm growth. Fluid flow is simulated using an immersed boundary-lattice Boltzmann model; while solute transport is described by a cut-cell finite volume method. A uniform dissolution approach is also adopted to capture the temporal evolution of trapped blobs. Spatio-temporal distribution of the biomass are investigated using a cellular automaton algorithm combined with the immersed boundary method. Simulations focused on the dissolution of NAPL in both abiotic and biotic conditions are conducted to assess the capability of the model. In abiotic conditions, we analyze the effects of the hydrodynamic regimes and the spatial distribution of NAPL blobs on the dissolution rate under different assumptions (blob size and Péclet number). In biotic conditions, a series of scenarios are also investigated (spatial distribution, reaction kinetics and NAPL-induced toxicity). Finally, this model is used to evaluate the pore scale relevance of a local equilibrium assumption between fluid phase and biofilm phase in the vicinity of the NAPL source.

Introduction

Non-aqueous phase liquids (NAPLs) are important but challenging problems for traditional groundwater remediation [START_REF] Kent | Remediation of NAPL-contaminated aquifers: is the cure worth the cost?[END_REF]Bianchi , 2001, Lee et al., 2007). NAPLs are the hydrocarbons that include the halogenated chemical solvents and petroleum products (such as PCB, PAH, BTEX, etc.) and often contaminate the subsurface after an accidental spill or oil tank leakage [START_REF] Mercer | A review of immiscible fluids in the subsurface: Properties, models, characterization and remediation[END_REF]Cohen, 1990, Miller et al., 1998).

These contaminants remain trapped in the form of immiscible droplets within the aquifers, becoming a persistent source of contamination that is difficult to remediate. Predicting the fate of the contaminants requires characterizing all the mechanisms involved in the coupled processes, especially biodegradation, which can occur in the vicinity of the contaminated source or further, within the dissolved plume (Fig. 1). One of the major difficulties to model such processes and advance the understanding of NAPLs transport in porous media with biofilm (the bacterial phase) is the complex interplay between biological and physicochemical phenomena [START_REF] Vega | Nickel and cadmium ions inhibit quorum sensing and biofilm formation without affecting viability in Burkholderia multivorans[END_REF][START_REF] Brackman | Use of quorum sensing inhibitors to interfere with biofilm formation and development in Burkholderia multivorans and Burkholderia cenocepacia[END_REF][START_REF] Chmielewski | Biofilm formation and control in food processing facilities[END_REF][START_REF] Eberl | A new deterministic spatio-temporal continuum model for biofilm development[END_REF][START_REF] Krefta | Individual-based modelling of biofilms[END_REF], Picioreanu et al., 2000a[START_REF] Eberl | A three-dimensional numerical study on the correlation of spatial structure, hydrodynamic conditions, and mass transfer and conversion in biofilms[END_REF][START_REF] Mah | Mechanisms of biofilm resistance to antimicrobial agents[END_REF][START_REF] Decho | Microbial biofilms in intertidal systems: an overview[END_REF][START_REF] Wingender | What are bacterial extracellular polymeric substances?[END_REF].These processes also have multiscale nature in both time and space [START_REF] Van Loosdrecht | Mathematical modelling of biofilm structures[END_REF], Picioreanu et al., 2000b[START_REF] Kreft | BacSim, a simulator for individual-based modelling of bacterial colony growth[END_REF]. The study of NAPL biodegradation in porous media is of growing interest due to many environmental applications where biofilm plays an essential role, such as for example, the bioremediation of contaminated sites [START_REF] Al-Bader | Biofilm comprising phototrophic, diazotrophic, and hydrocarbonutilizing bacteria: a promising consortium in the bioremediation of aquatic hydrocarbon pollutants[END_REF][START_REF] Singh | Biofilms: implications in bioremediation[END_REF] or the development of bio-barriers for aquifer protection [START_REF] Huang | Efficiency of active barriers attaching biofilm as sediment capping to eliminate the internal nitrogen in eutrophic lake and canal[END_REF][START_REF] Seo | The application of a mulch biofilm barrier for surfactant enhanced polycyclic aromatic hydrocarbon bioremediation[END_REF].

In recent decades, extensive efforts have been devoted to study contaminant transport in porous media with biofilms. Nevertheless, whether is at the pore scale or macroscopic scale, these studies [START_REF] Ebigbo | A coupled pore-scale model for methanogenic microbial activity in underground hydrogen storage[END_REF][START_REF] Golfier | Biofilms in porous media: Development of macroscopic transport equations via volume averaging with closure for local mass equilibrium conditions[END_REF][START_REF] Kreft | BacSim, a simulator for individual-based modelling of bacterial colony growth[END_REF] were focused, in the most part, on the areas away from the source of contamination. Additionally most of the previous work were interested in dissolution of residual NAPL saturation and limited to abiotic conditions [START_REF] Mateas | In situ stabilization of NAPL contaminant source-zones as a remediation technique to reduce mass discharge and flux to groundwater[END_REF][START_REF] Imhoff | An experimental study of complete dissolution of a nonaqueous phase liquid in saturated porous media[END_REF][START_REF] Powers | An experimental investigation of nonaqueous phase liquid dissolution in saturated subsurface systems: Transient mass transfer rates[END_REF][START_REF] Powers | An experimental investigation of nonaqueous phase liquid dissolution in saturated subsurface systems: Steady state mass transfer rates[END_REF][START_REF] Miller | Dissolution of trapped nonaqueous phase liquids: Mass transfer characteristics[END_REF]. Despite the importance of these contributions to understand the mechanisms involved during biotic or abiotic dissolution of NAPLs, it is still challenging to predict the fate of the contaminant without considering the biological activities close to the source of contamination. At present, to our best 2 knowledge, no numerical study has modeled the bacterial growth mechanisms at the pore scale, specifically those involving NAPL sources, which are indeed believed to be hot spots of the biological activities.

Many studies [START_REF] Paulsen | Visualisation of bacterial degradation and mobilisation of oil in a porous medium[END_REF][START_REF] Yang | Biologically enhanced dissolution of tetrachloroethene DNAPL[END_REF][START_REF] Armstrong | Investigating the pore-scale mechanisms of microbial enhanced oil recovery[END_REF] have shown that bacterial metabolism of the oil-degrading bacteria leads to the production of biosurfactant products in the presence of NAPLs. These products are active surfactants that reduce the interfacial tension at the surface of the pure NAPL droplet so as to destabilize and break it into small droplets evacuated by the flow. Experiments performed in 2D transparent micromodels by [START_REF] Armstrong | Investigating the pore-scale mechanisms of microbial enhanced oil recovery[END_REF] compared the effectiveness of the abiotic and biotic degradation of NAPLs and the role of each parameter on the macroscopic process. The authors concluded that biodegradation is optimal in the presence of biosurfactant-producing biomass, benefiting simultaneously from the enhanced mobilization and consumption effects of NAPLs. The results of [START_REF] Paulsen | Visualisation of bacterial degradation and mobilisation of oil in a porous medium[END_REF] also deserve to be reported as they highlighted the impact of biosurfactant production on the dissolution process of a hydrocarbon phase. [START_REF] Paulsen | Visualisation of bacterial degradation and mobilisation of oil in a porous medium[END_REF] experimentally investigated the influence of biosurfactants on a NAPL droplet at the pore scale. The laboratory images showed that the growth of the biofilm with associated biofilaments developed preferentially around and even on the droplet. An attempt to upscale this bio-enhanced NAPL dissolution was proposed by [START_REF] Bahar | An upscaled model for bio-enhanced NAPL dissolution in porous media[END_REF]. Nevertheless, NAPLs also constitute a xenophobic agent for bacteria that may potentially inhibit their growth. In their experiment, [START_REF] Singh | Kinetics of trichloroethylene and toluene toxicity to Pseudomonas putida F 1[END_REF] followed the evolution of the concentration of living and dead bacteria around a droplet of TCE placed in the presence of Pseudomonas sp. Initially, biofilm was developed uniformly at the periphery of the droplet. After 5 minutes, living bacteria migrated gradually away from TCE. At the same time, an increase in the number of dead cells in the vicinity of the droplet was observed. This area of dead cell accumulation kept increasing with time. This experiment highlighted the influence of the toxic effect of NAPLs, which, at high concentrations, caused the death of nearby bacteria. The influence of toxicity caused by the dissolution enhanced through the biodegradation of NAPLs was also studied at macroscopic scale via numerical simulations (Chu et al., 2004, Gallo andS.M. Hassanizadeh, 2002).

The aim of this paper is to present a numerical model with the capability to predict the biodegradation of the NAPLs in porous medium at the pore scale. First, we conduct numerical simulations to study the dissolution of NAPLs in abiotic conditions under different hypotheses (blob size and Péclet number).

Second, we focus on the phenomenon of NAPL dissolution in the presence of bacteria. Different conditions are considered (spatial distribution, reaction kinetics, toxicity effect), and their impact on the dissolution process are assessed. Finally, the current pore-scale model is used to evaluate the relevance of a local equilibrium assumption between the fluid and the biofilm phases close to the NAPL source..

Numerical Model

The numerical model proposed here extends the previous biofilm growth model [START_REF] Benioug | An immersed boundary-lattice Boltzmann model for biofilm growth in porous media[END_REF] by introducing an additional NAPL phase. Since in the present study, we investigate the mutual interaction between NAPL phase and bacterial activity, four phases are included in the current model: solid (s), fluid (f ), biofilm (b) and NAPL (h). In addition to the assumptions introduced in [START_REF] Benioug | An immersed boundary-lattice Boltzmann model for biofilm growth in porous media[END_REF] (monospecies biofilm, nutrient transport only by diffusion in the biofilm phase), we further assume that:

• NAPL is assumed trapped as an immobile phase within the water-wet porous matrix.

• Residual NAPLs are trapped in the form of perfectly circular blobs (in 2D). In addition, flows passing through the NAPL interface (in the case of a triple interface with the fluid and the biofilm for the same blob) are supposed not to be sufficient to overcome the forces of surface tension (no breakage of the blobs). In other words, the radius of each NAPL blob decreases over time as a function of the sum of the fluxes exchanged with the surrounding medium.

• NAPL is a pure phase composed of a single component which is biodegradable, and acting as the electron donor in the redox process. Other substances (electron acceptor, mineral salts, amino acids) that are necessary for the survival of bacteria are assumed to be in excess.

• Diffusion of water within the NAPL phase is negligible.

• Biofilm may develop at the fluid and NAPL interfaces. Bacteria produce biosurfactants whose concentration is assumed to be constant and are present only within the biofilm phase (no transport of the biosurfactants produced into the fluid phase and no variability of the interfacial conditions as a function of the surfactant concentration).

• Only the increase in solubility of the dissolved hydrocarbon (via the partition coefficient K bh ) in the presence of biosurfactants is considered. The change of the surface tension is not taken into account.

Removal of NAPL droplets by water flow is also assumed to be negligible in the dissolution process.

• Biodegradation of the dissolved hydrocarbon within the biofilm phase is governed by the Haldane kinetics to account for the inhibition effects [START_REF] Saravanan | Growth kinetics of an indigenous mixed microbial consortium during phenol degradation in a batch reactor[END_REF].

Fluid Flow

In the fluid phase (f ), we consider steady-state, incompressible and Newtonian flow at low Reynolds number. The mass balance and momentum conservation equations are described by the Stokes equations as follows:

∇ • u = 0, (1)
ρ f u • ∇u = -∇p + µ f ∇ 2 u, (2)
and no-slip boundary conditions for fluid-solid interfaces A f s , fluid-biofilm A f b and fluid-NAPL A f h are employed as:

B.C.1 u = 0, at A f b (3) B.C.2 u = 0, at A f s (4) B.C.3 u = 0, at A f h (5)
where u, p, µ f , ρ f are respectively the velocity, pressure, dynamic viscosity and fluid density. A ij represents the interface between i and j phases (i, j = f (fluid), b (biofilm), s (solid), h (NAPL)). We must keep in mind that, with the hypothesis of the quasi-steady-state processes of NAPL dissolution and bacterial growth, interface A f i between both regions is assumed to evolve with time but sufficiently slowly so that a no-slip condition can be adopted.

A non-boundary conforming numerical method is used to treat these dynamically changing pore-scale geometries due to NAPL dissolution and biofilm growth. Hereafter, the momentum equations are solved by using the immersed boundary-lattice Boltzmann (IB-LB) model presented in [START_REF] Benioug | Numerical efficiency assessment of IB-LB method for 3D pore-scale modeling of flow and transport[END_REF] and already applied in [START_REF] Benioug | An immersed boundary-lattice Boltzmann model for biofilm growth in porous media[END_REF]. A direct-forcing immersed boundary (IB) model originally derived by ( Mohd-Yusof , 1997) is coupled with the lattice Boltzmann method (LBM) to keep the fluid-solid interface sharp and maintain a direct control on numerical accuracy. Basically, it means that the fluid-biofilm-NAPL boundaries, which may evolve in time, are defined on a fixed grid mesh. A boundary force density term that acts as a forcing term is introduced into the momentum equation, Eq. (2), so that the no-slip condition is still satisfied on the immersed boundaries. For more details on the LB and IB methods, one can refer to [START_REF] Succi | The lattice Boltzmann equation for fluid dynamics and beyond[END_REF][START_REF] Khosronejad | Curvilinear immersed boundary method for simulating coupled flow and bed morphodynamic interactions due to sediment transport phenomena[END_REF][START_REF] Rosis | Harmonic oscillations of laminae in non-Newtonian fluids: A lattice Boltzmann-Immersed Boundary approach[END_REF]. To avoid excessive computational complexity, however, a piecewise linear interface construction (PLIC) approach is adopted so that the immersed boundaries still remain parallel to the facets of the cells. More details about these numerical methods and their validation can be found in [START_REF] Benioug | Numerical efficiency assessment of IB-LB method for 3D pore-scale modeling of flow and transport[END_REF].

Solute transport

Transport of dissolved hydrocarbon within this three-phase system is governed by the following conservation equations:

∂c f ∂t + ∇ • (uc f ) = ∇ • (D f ∇c f ), in the fluid phase (6) ∂c b ∂t = ∇ • (D b ∇c b ) -R b ,
in the biofilm phase ( 7)

∂c h ∂t = 0, in the NAPL phase (8) c f = K bf c b , at A f b (9) D f ∇c f • n f b = D b ∇c b • n f b , at A f b (10) c f = K f h c h , at A f h (11) D f ∇c f • n f h = D h ∇c h • n f h , at A f h (12) D f ∇c f • n f h = -c h • w f h • n f h , at A f h (13) c b = K bh c h , at A bh (14) D b ∇c b • n bh = D h ∇c h • n bh , at A bh (15) D b ∇c b • n bh = -c h • w bh • n bh , at A bh (16) 0 = D f ∇c f • n f s , at A f s (17) 0 = D b ∇c b • n bs , at A bs (18) 0 = D h ∇c h • n hs , at A hs ( 19
)
where c i represents the hydrocarbon concentration within the phase i, u is the velocity field as predicted from Eqs. (1)-( 5) and n ij are the externals normal to the interfaces between phases i and j. D i is the diffusion coefficient of hydrocarbon species in the phase i. K ij is the partition coefficient between the two phases i and j. R b is the reaction term. w hj is the local displacement velocity of the A hj interface. Note that c h is constant and corresponds to the density of the pure-phase NAPL compound.

Unlike the previous model [START_REF] Benioug | An immersed boundary-lattice Boltzmann model for biofilm growth in porous media[END_REF], it is important to emphasize that the computation of the transport equations is performed here in transient conditions. Indeed, since the transport of the dissolved hydrocarbon is directly related to the temporal evolution of the NAPL droplets, steady-state solution of the transport equations is never reached. However, we retain the hypothesis of quasi-steady-state transport processes with respect to the displacement kinetics of mobile interfaces (as well as the hypothesis of quasisteady-state of the flow) by sequentially solving the transport (Eqs. ( 6)-( 15)), the NAPL dissolution (Eqs.

(13)-( 16)) and the bacterial growth equations.

The values of the partition coefficients K f h and K bh are different in order to take into account the presence of biosurfactants at the biofilm-NAPL interface. We usually have the correlation as:

K f h K bh (20)
It should also be noted that those three partition coefficients are not strictly independent. In order to satisfy the thermodynamic equilibrium condition of the system at rest, the equilibrium partition coefficient between the fluid and biofilm phases follows the relationship as:

K bf = K f h K bh (21)
which can be derived from Eqs. (9 -14) as:

K bf = c f c b (22) = K f h c h K bh c h (23)
Boundary conditions governing the evolution of the interface of the NAPL phase (Eqs. ( 13)-( 16)) tend to link the interfacial diffusive flux to the displacement velocity w hj of the interface A hj , and are valid for the tracer transport (low concentration of dissolved hydrocarbon). The values of w f h and w bh are not necessarily identical between the fluid phase and the biofilm phase due to the variations in solubility. In the case of a NAPL droplet having an interface shared both by the biofilm and the fluid phases, this may theoretically lead to a deformation of the shape of the trapped droplet. However, Eqs. ( 13)-( 16) do not take into account the surface tension that are supposed to counterbalance the deformation in theory. We will return in the next section on the approach to solve this problem and the numerical methods describing the dissolution of NAPL blobs.

The major difficulty in the spatial discretization of the mass conservation equations, Eqs. ( 6)-( 15), is the presence of physical boundaries immersed in cells with 3 phases. In the case of the fluid-biofilm system studied in [START_REF] Benioug | An immersed boundary-lattice Boltzmann model for biofilm growth in porous media[END_REF], this problem was solved by a volume-of-fluid (VOF) type model based on a local equilibrium assumption between the two phases (also called "single-field model", see [START_REF] Marschall | Numerical simulation ofspecies transfer across fluid interfaces in free-surface flows using OpenFOAM[END_REF]. In the presence of an additional NAPL phase, however, this assumption is getting more complicated. Considering the high concentration gradients existing in the field (the only sources of solute are from trapped droplets), the hydrodynamic conditions for which the NAPLs would be close to local equilibrium conditions with fluid and/or biofilm (even only in the vicinity of immersed boundaries) appear to be extremely limited. Based on this consideration, we opt for a reconstruction method of interfacial fluxes at the immersed boundaries (denoted by FV-reconstruction) (also referred to as the Cartesian cut-cell method with cell merging approach, see [START_REF] Ingram | Developments in Cartesian cut cell methods[END_REF]. The finite volume method (FVM) is used to discretize the equations. In practice, this consists of modifying the volume of the cell in the vicinity of the immersed boundary. The control volume of the immersed cell is then reduced to the volume occupied by the majority phase (i.e., the volume fraction of which is the largest) while the neighboring cell is enlarged by the remaining volume fraction (Fig. 2). This modification impacts not only the volume and the interfacial surfaces of the cells in contact with the immersed boundary but also the calculation of the interfacial fluxes.

This reconstruction step, which can be particularly delicate in 3D, is quite straightforward in our case due to the assumption of piecewise-plane interfaces, as illustrated in Fig. 2. We refer the readers to [START_REF] Benioug | Numerical efficiency assessment of IB-LB method for 3D pore-scale modeling of flow and transport[END_REF] for more details on the methodology used in the case of a two-phase system. When the three phases (and therefore two mobile interfaces) are present within the same cell, a local equilibrium assumption is first used between the fluid and biofilm phases. The interface between the NAPL and the fluid-biofilm mixed phase, whose properties depend on the volume fractions of the two phases, is then modeled by the Cartesian cut-cell method described above.

We illustrate here in brief the finite volume discretization of the diffusive flux, over the one-dimensional immersed cell depicted in Fig. 2. First, a single-domain description is used to replace Eqs. ( 6)-( 8):

∂c ∂t + ∇ • (uc) = ∇ • (D∇c) -R b (24)
where U, D and R are piecewise functions. Discretization of the diffusive flux leads to the following piecewise integration for the immersed cell i (Fig. 2):

i+ 1 2 i-1 2 ∂ ∂x D ∂c ∂x dx = i-1 2 +φ i-1 2 ∂ ∂x D h ∂c h ∂x dx + i+ 1 2 i-1 2 +φ ∂ ∂x D f,b ∂c f,b ∂x dx = D f,b ∂c f,b ∂x i+ 1 2 -D h ∂c h ∂x i-1 2 (25)
c f,b and D f,b , are respectively, the concentration and diffusion coefficient of the fluid or biofilm phase, depending on the type of cell i + 1. By using the Taylor series expansions and the appropriate boundary conditions at the immersed boundary (Eqs. ( 9)-( 15)). Equation ( 25) can be rewritten as:

i+ 1 2 i-1 2 ∂ ∂x D ∂c ∂x dx = 2D i D i+1 ∆x K i,i+1 c| i+1 -c| i D i+1 (1 -2φ(i)) + K i,i+1 D i (1 + 2φ(i)) - 2D i D i-1 ∆x c| i -K i-1,i c| i-1 D i-1 (1 -2(1 -φ(i))) + K i-1,i D i (1 + 2((1 -φ(i))) (26)
where K i,j is the partition coefficient between the phases of grid i and j.

In the temporal discretization, the transport time step t CF L is determined from the velocities at the nodes based on the Courant conditions (or CF L conditions for Courant-Friedrichs-Lewy) to avoid oscillation problems:

t CF L = CF L max(U max x /∆ x , U max y /∆ y ) (27)

NAPL Dissolution

As mentioned in the previous sections, differences in the solubility of the hydrocarbon in the presence of the fluid and biofilm phases will lead to variations in the interfacial fluxes that may be significant. In the case of a NAPL droplet in contact with these two phases, this may generate spatial changes in the dissolution rate over the periphery of the interface. Theoretically, this phenomenon should be compensated by the surface tension forces that have a stabilizing effect on the shape of the interface. If these forces are prevalent, the fluctuations in the dissolution rate will be distributed over the entire circumference so as to preserve a circular droplet. Otherwise, the differences between the dissolution rates will lead to a rupture of the interface that influences the formation of two new droplets. However, since we have deliberately chosen to ignore these surface tension effects, it is necessary to consider an alternative way in order to preserve a realistic modeling approach of the dissolution of the trapped droplet. A so-called uniform dissolution model has been developed.

This method distributes the dissolution uniformly around the droplet interface. In this way, the blob retains a circular shape throughout the dissolution process. Once the concentration field is calculated, the mass of NAPLs dissolved during the time t CF L , M diss blob is determined for each NAPL blob from Eqs. ( 13)-( 16). This dissolved quantity M diss blob is obtained directly from the calculation of the diffusive fluxes at the interfaces:

M diss blob = t CF L • A blob -c h w h • n h dA (28) = t CF L • A blob D h ∇c h • n h dA (29)
where A blob is the exchange surface of the NAPL blob. After summing up the fluxes through each grid-cell boundary, we can determine the volume of the blob that is dissolved during the t CF L time step as well as the remaining volume V t+t CF L blob as:

V diss blob = M diss blob c h (30) V t+t CF L blob = V t blob -V diss blob (31)
where V t blob is the volume of the NAPL blob at the time t (before the dissolution step) and V diss blob is the volume dissolved during the time step t CF L . From the blob's original circular shape, it is then sufficient to update its radius R t+t CF L from the following relation:

R t+t CF L = V t+t CF L blob π in 2D (32)
Once the blob is reduced, the new blob of radius R t+t CF L is re-projected on the mesh in order to determine the remaining NAPL cells as well as the volume fractions φ h updated from the cells through the newlyformed interface. The immersed cells previously belonged to the NAPL phase but no longer within in the radius (R t+t CF L ) are then transformed into fluid or biofilm as a function of the neighboring cells.

Bacterial growth model and inhibitory effect

The time evolution of the biomass concentration ρ bio is governed by the Haldane model [START_REF] Saravanan | Growth kinetics of an indigenous mixed microbial consortium during phenol degradation in a batch reactor[END_REF], which is developed to study the impact of NAPL toxicity on biomass. The Haldane model is widely used to represent the influence of inhibition effect on the kinetics of the bacteria growth, including the conditions with the presence of a biodegradable substrate but toxic for high concentrations [START_REF] Kumar | Biodegradation kinetics of phenol and catechol using Pseudomonas putida MTCC 1194[END_REF]. The reaction term R b in Eq. ( 7) is written in the form of:

R b = ρ bio µ max C C + K + C 2 Ki , (33)
and the bacterial growth can be calculated from the following equation:

∂ρ bio ∂t = F λ ρ bio µ C C + K + C 2 Ki -k d ρ bio ( 34
)
where ρ bio , µ, K are, respectively, the biomass concentration, the maximum reaction rate and the halfsaturation constant. F λ is the stoichiometric coefficient of the biological reaction, k d is the extinction coefficient of the bacteria and K i is the inhibition coefficient. For illustrative purposes, Fig. 3 compares the evolution of the reaction rate R b predicted by the Monod and Haldane models as a function of substrate concentration. As a reminder, Monod kinetics is classically described by:

R b = ρ bio µ max C C + K , ( 35
)
We can see the impact of the inhibition effect by the decrease of the reaction rate when the concentration is increased (Fig. 3). The biomass concentration ρ bio calculated at each time step from Eq. ( 34) is then used to determine the value of the biofilm volume fraction φ per cell (the biomass density is supposed to be constant in the whole domain). The net change in biomass concentration is calculated over a reference time step of growth (t growth ) and corresponding to an increase in φ of 0.01 based on the maximum reaction rate µ max . As soon as ρ bio exceeds locally, within a biofilm cell, an arbitrary value denoted by ρ biomax (corresponding to a value of φ = 1) defined as the biomass density is distributed spatially by using the cellular automata model . The algorithm introduced in (Benioug et al., 2017) will be briefly explained here. Basically, in each grid cell (x, y), as soon as φ(x, y) > 1 (i.e., ρ bio (x, y) > ρ biomax ), the excess of biomass ∆ρ bio = ρ bio (x, y) -ρ biomax is moved towards an adjacent fluid cell, randomly chosen with equal probability, and this cell becomes a partially-filled cell with a volume fraction of φ = ∆ρ bio ρ biomax . This algorithm is repeated for each cell in turn until all the excess biomass is redistributed.

Finally, the general algorithm of our model including fluid flow, chemical species transport, biofilm growth and extinction, and NAPL dissolution is illustrated in Fig. 4. Note that the biofilm growth model has been previously validated from qualitative comparison with benchmark cases of the literature in [START_REF] Benioug | An immersed boundary-lattice Boltzmann model for biofilm growth in porous media[END_REF].

Results and discussion

In order to demonstrate the capability of the current model in simulating the complex and dynamic processes in porous media (e.g. NAPL blobs reduction, biofilm growth), as well as its predictability to study the impact of the main bio-physico-chemical mechanisms on the dissolution of the residual hydrocarbon phase, we test our dissolution model under various hydrodynamic conditions. The influence of different parameters (such as Péclet number P e, Damköhler number Da, heterogeneity, etc.) will thus be tested separately.

To facilitate the analysis and the interpretation of results, numerical data will be reported in a dimensionless form as described below:

x = x l , y = y l , t = t D f l 2 , c i = c i c h , (i = f, b, h) (36) p = p l µ f u moy , u = u u moy , ρ bio = ρ bio ρ biomax , D i = D i D f , (37)
K = K c h , K i = K i c h , µ = µ l 2 D f , k d = k d l 2 D f , w ij = w ij l D f ( 38
)
where ρ biomax , u moy and l respectively represents the maximum biomass concentration, the averaged fluid velocity and the characteristic length of the medium. We also introduce the following dimensionless numbers, Péclet number:

P e = u moy • l D f (39)
Damköhler number:

Da = µ max • l 2 • ρ biomax D f K = µ max K Γ with Γ = ρ biomax c h (40)
The influence of fluid flow and substrate consumption are then investigated through different values attributed to the dimensionless Péclet and Damköhler numbers.

NAPL dissolution within a single pore

As the first step, we need to test our model by simulating the dissolution process with and without biodegradation in a simple pore structure -a single NAPL droplet in a narrow channel. The computational

domain Ω = [0, lx] × [0, ly] (lx × ly = 30 × 10mm 2 ) is discretized with uniform mesh ((N x -1) × (N y -1))
with a resolution of ∆x = lx N x-1 = ∆y = ly N y-1 , where N x and N y are the number of nodes along the (x) and (y) directions. A velocity profile is imposed at the inlet (x = 1); while a constant pressure at the outlet (x = 3). No slip-boundary conditions are introduced at the wall boundaries (y = 0 and y = 1). The NAPL droplet is in circular shape with a radius of 0.21, with its center located at the coordinates (1, 0.5).

The bacteria corresponding to 0.08% of total volume are randomly distributed on the NAPL droplet surface as shown in Fig. 5a (biomass is presented here in white). The input data are summarized in Table 1.

Figures 5b-5c show the NAPL droplet dissolution with and without biodegradation. A reduction over time in the size of the NAPL droplet is captured while its circular shape being maintained, which is in agreement with our uniform dissolution approach. If the NAPL blob gradually decreases in both cases, for biotic and abiotic conditions, its decrease is much faster in the presence of bacteria. At the time t = 500, the volume fraction of the NAPL is equal to 0.16 in abiotic condition, while it is equal to 0.1 in biotic conditions. This acceleration of NAPL degradation in the presence of bacteria is due to the biosurfactant-induced change of the partitioning coefficient on the surface of the contaminant droplet.

From the above attempt, our model is able to correctly describe the physics of the NAPL dissolution.

However, as the dissolution process depends on the velocity field, we need to assess the mesh-dependency of our numerical results. Three other meshes are considered for this purpose (450 × 150, 225 × 75, and 25 × 75).

The decrease of the NAPL blob volume over time is illustrated in Figure 5d, which is almost identical for the meshes (450 × 150, 300 × 100, 225 × 75) proving our numerical simulations are mesh independent. When continuing to degrade the mesh size, the surface of the NAPL blob which drives the dissolution rate is too overestimated in spite of using immersed boundaries. However, even for the mesh (75 × 25), with less than 10 cells along the initial diameter, the numerical error on the total dissolution time is lower than 10% (t = 2400 against t = 2650 for the refined solution). This comparison guided our spatial discretization process and in the following simulations, we do not go below this lower limit for mesh discretization (i.e., 10 cells along the initial NAPL blob diameter).

D f 10 -9 m 2 /s D f = 1 D b 0.25 × 10 -9 m 2 /s D b = 0.25 K f h - 6 × 10 -4
Péclet number, (P e) -0.3

K 0.24 mg/l K = 3 × 10 -4 k d 2 × 10 -14 s k d = 2 × 10 -7 K i 0.64 mg/l K i = 8 × 10 -4 K bh - 6 × 10 -3 F λ - 0.3
Damköhler number, (Da) -0.1, 10

NAPL dissolution in porous media

We consider now a representative volume of a 2D porous medium (Fig. 6). The domain has a size of lx × ly = 12 × 12mm 2 with an initial porosity 0 of 0.67 and an initial NAPL saturation s 0 = 4.2%. The pore throat characteristic length, l f , is estimated to be 0.3 mm. An initial velocity profile is imposed at the inlet of the domain (i.e., at x = 0) while the pressure is kept constant at the outlet (i.e., at x = 1).

Periodic conditions are introduced at the boundaries at y = 0 and y = 1.

NAPL dissolution without biodegradation

In this section, we examine the influence of the spatial distribution of NAPL droplets and the flow velocity on the dissolution within the porous medium. We initially consider three NAPL droplets trapped in the pores of the medium as shown in Figure 6. At this point, there is no bacterial population that can biodegrade these organic compounds. The values of input data used for this simulation are presented in Table 2.

D f 10 -9 m 2 /s D f = 1 D b 0.25 × 10 -9 m 2 /s D b = 0.25 K f h - 6 × 10 -3
NAPL blobs location (x center , y center ) -A(0.27, 0.17 Figure 7e shows a nearly linear correlation between the droplet radius and time, which is in agreement with the experimental results obtained by [START_REF] Kennedy | A pore-scale investigation of mass transport from dissolving DNAPL droplets[END_REF] and [START_REF] Poesio | Dissolution of a Liquid Microdroplet in a Nonideal Liquid-Liquid Mixture Far from Thermodynamic Equilibrium[END_REF]. This relationship reveals that the dissolution kinetics are independent of the size of the NAPL droplets, at least with the numerical conditions imposed. Figure 7f confirms such result by reducing the volume of the dissolved NAPL in a quadratic way. This nonlinear decrease is obviously related to the reduction of the exchange surface between the NAPL blob and the surrounding fluid that leads to a decrease in the interfacial fluxes exchanged between two phases, which also agrees with the observations of [START_REF] Kennedy | A pore-scale investigation of mass transport from dissolving DNAPL droplets[END_REF].

On the other hand, if each NAPL blob evolves constantly over time, variations in the rate of dissolution between the droplets are analyzed. Figures 7e-7f show that droplet B dissolves a bit more rapidly while droplet A dissolves before droplet C. This disparity is justified by their spatial distribution and the establishment of preferential flow paths linked to the heterogeneity of the medium at the pore scale. Indeed, since the blob C is located downstream of the dissolution plume of blob A, dissolved contaminant is advected by the bulk fluid flow in the vicinity of blob C. This plume channeling generates low concentration gradients around this blob and hence a substantial decrease of the dissolution rate (Figs. 7c-7e). In contrast, droplet B is trapped in a macropore where the flow velocities are greater, which leads to an increase in the concentration gradient and therefore in the flow exchanged. These results are also in agreement with the experimental observations in the literature [START_REF] Kennedy | A pore-scale investigation of mass transport from dissolving DNAPL droplets[END_REF]Lennox , 1997, Imhoff et al., 1994).

Influence of the flow velocity

Flow velocity plays a non-negligible role in the dissolution of the NAPL droplets as it directly impacts the mass transfer to the bulk fluid. To investigate the impact of flow velocity, we compare our simulated results under two flow rates. The same simulation setup is kept here as in the previous section except for two parameters, namely the size of the droplets and the Péclet number. The radius of the NAPL droplets are now: R A = 0.05, R B = 0.055, R C = 0.06. Figure 8 shows the dissolution of these blobs for two Péclet numbers, respectively P e = 0.5 and P e = 10 and at two different moments, t = 5 and t = 51.

A closer observation of the concentration fields indicates that the spreading of the NAPL plume is strongly influenced by the Péclet number (P e = 10: Figs. 8b,8f; P e = 0.5: Figs. 8d,8h). When the number of Péclet increases, the lateral extension of the plume is reduced. From a macroscopic point of view, this classical behavior translates into in an increase in the dispersion coefficient with the Péclet number [START_REF] Powers | An experimental investigation of nonaqueous phase liquid dissolution in saturated subsurface systems: Steady state mass transfer rates[END_REF][START_REF] Ahmadi | Calculation of the effective properties describing active dispersion in porous media: from simple to complex unit cells[END_REF]. In other words, the increase in the Péclet number involves strong concentration gradients around the NAPL droplets and thus an increase of the dissolved NAPL flux. This increase obviously leads to a more rapid dissolution of the residual saturation zones (Figs. 8a,8e). As we can see, at t = 51, the dissolution of the NAPL droplets is practically completed for P e = 10 (Fig. 8e) while all three droplets are still visible for P e = 0.5 (Fig. 8g). For low Péclet numbers, indeed, the diffusive effects are still strongly predominant. The exchanges are limited by the external mass transfer and the dissolution of the trapped droplets is relatively slower (Figs. 8c,8g).

Upscaling from the pore-scale to Darcy-scale (over the entire porous sample), Fig. 9 shows the reduction of the total volume fraction of the trapped NAPLs over time for both flow velocities. An approximately 6% increase in the apparent rate of dissolution was observed for P e = 10 compared to P e = 0.5. This increase may seem small given the increase in velocity by a factor of 20 but it may be explained by the very low saturation of NAPLs initially present in the medium (s 0 = 4.2%). Note that [START_REF] Ahmadi | Calculation of the effective properties describing active dispersion in porous media: from simple to complex unit cells[END_REF] obtained comparable results in terms of order of magnitude for mass transfer coefficient variations from 2D unit cell calculations (increase of the exchange coefficient by a factor of about 10% for the same range of variation of P e when the NAPL saturation is less than 5%).

Bio-enhanced NAPL dissolution and biofilm growth

In this section, the same porous media domain is initially injected with bacteria capable of biodegrading the hydrocarbon. The distribution of the oil phase is similar to that of the previous section. Bacteria, present in sessile form, initially occupy 13.7% of the pore volume, distributed randomly on the solid surfaces.

Simulation parameters are shown in Table 3. The color scale used to represent the different grids is detailed in Fig. 10h and will be kept for the rest of the study. It should be noted here that we have represented the fluid/biofilm immersed cells as biofilm cells in order to limit the number of colors used and to facilitate the readability of the figures.

First, we considered cases with relatively low Damköhler and Péclet numbers, i.e., Da = 20 et P e = 0.5.

Moreover, in order to study the influence of the solubility increase of dissolved contaminants in contact with bacteria, we have taken a high coefficient of inhibition, i.e., K i = 2 × 10 -3 . This high value of K i allows us to eliminate the toxicity effects of NAPLs so that the bacteria can grow even in the presence of an excess of substrate. In other words, this hypothesis favors bacterial growth in the vicinity (or even in contact) of the NAPL blobs.

c h 800 mg/l c h = 1 D f 10 -9 m 2 /s D f = 1 D b 0.25 × 10 -9 m 2 /s D b = 0.25 K f h - 6 × 10 -3
Péclet number, (P e) -0.5

Damköhler number, (Da) -20

K 0.8 mg/l K = 10 -3 k d 2 × 10 -14 s k d = 2 × 10 -7 K i 1.6 mg/l K i = 2 × 10 -3 K bh - 6 × 10 -2 F λ - 0.3 source).
Taking into account both the initial spatial distribution of NAPL and biofilm, and preferential flows (highlighted in the previous paragraph), droplet C is the first blob reached by the bacterial development,

i.e., towards t = 15, whereas droplet B is the last (at t = 20). Once the biofilm is in contact with NAPL, the dissolution process accelerates under the effect of biosurfactants that increase the contaminant solubility.

It is thus observed that droplet C, although it is the widest (R C = 0.06), is dissolved first.

In order to confirm this hypothesis, we compare in Figs. 10e-10g the evolution of the NAPL blob radius with and without the presence of the bacteria. We first note that, before the contact between NAPL and biofilm, dissolution kinetics are almost the same in biotic and abiotic conditions. However, once the biofilm adhere to the surface of the droplet of NAPL, the dissolution rate increases significantly. On one hand, the contact with the bacteria favors the solubility of the hydrocarbon due to the surfactants. On the other hand, the bacterial assimilation of the organic contaminant accentuates the concentration gradients and therefore the interfacial fluxes. This so-called 'bio-enhanced dissolution' phenomenon has been already identified by previous studies (Seagren et al., 1994, Yang and[START_REF] Yang | Biologically enhanced dissolution of tetrachloroethene DNAPL[END_REF].

Reactions induced by bacteria can impact the dissolution rate of NAPLs via the increase of the biomass concentration which plays an essential role in the spatial and temporal evolution of the pore structure.

Knowing the most influential parameters on the change of this rate of dissolution constitutes an important step in modeling the NAPLs. Thus, in the following sections, our interest will be focused on the influence of the Damköhler number and the coefficient of inhibition on the dissolution process.

Impact of Damköhler number

In this paragraph we keep the same setup as used previously (Tab. 3), with the exception of the Damköhler number set at Da = 50. This high value translates into an increase in the reaction kinetics and therefore in the amount of biodegradated NAPL compared to the results obtained at Da = 20. It should also favor an increased biomass production in the immediate vicinity of NAPL blobs.

Our simulated results, illustrated in Figures 11a-11d, confirm such hypothesis. A more rapid growth of biomass is shown compared to the results observed at Da = 20 in Fig. 10. In fact, the blobs of NAPL are in contact with the bacterial filaments much more quickly, i.e., at t = 7 for Da = 50 against t = 15

for Da = 20. Similarly, at t = 24 (Fig. 11c), practically all the blobs are dissolved (against t = 38 for Da = 20). This behavior is also verified in Figs. 11d-11f where we compared the evolution of the different blobs for two Damköhler numbers. The dissolution is faster when Da increases for all droplets.

As long as the biofilm is not in contact with the NAPL droplets, the increase of the dissolution rate with Damköhler number is low. The gap is strongly accentuated, however, with the biofilm adhesion to the NAPL blobs. Biodegradation, which is conditioned by the Da value, then plays a preponderant role in the dissolution process by increasing the concentration gradient and consequently the dissolved NAPL fluxes.

However, the influence of structural heterogeneity seems to dissipate with the increase of Da. Indeed, the dissolution of the droplet C, which was favored by the preferential flow effects at low Da (Fig. 10), seems to be somewhat delayed here. As it can be seen in Fig. 11a, the bio-clogging of the pores in the vicinity of the trapped droplets plays as barriers to the flow and leads to a reduction of the velocity (and thus the local Péclet number) near the water/NAPL interface. Too high values of Damköhler number could thus have a paradoxical effect and slow down the dissolution process instead of accelerating it by bio-clogging the pores at the entrance of the porous medium and thus limiting the availability of trapped NAPL droplets.

Impact of toxicity

The impact of NAPL toxicity on bacterial growth is taken into account in our model through the use of Haldane kinetics [START_REF] Saravanan | Growth kinetics of an indigenous mixed microbial consortium during phenol degradation in a batch reactor[END_REF]. The reaction rate is thus no longer only controlled by the Damköhler number and the half-saturation constant as in the Monod kinetics model but also by the inhibition coefficient K i which plays an essential role in the bacterial growth/decay when the tolerance threshold for the substrate concentration is exceeded. In this section, we use the same data as those summarized in Table 3, with the exception of the values of the inhibition K i and extinction k d coefficients. We first keep the value of the extinction coefficient as constant (k d = 10 -7 ) and study the influence of the inhibition coefficient. The results indicate that bacterial growth is strongly influenced by the inhibition coefficient. The more the K i decreases, the slower the bacterial growth moves towards the NAPL blobs. Thus, for the numerical conditions used, at K i = 2 × 10 -3 , the NAPL blobs are practically covered with bacteria and completely dissolved at t = 38. On the other hand, when K i = 6 × 10 -4 , the bacteria grow preferentially far from the NAPL blobs and do not adhere to the oil phase until about t = 38 (Fig. 12b).

This effect is more obvious when we increase the value of the extinction coefficient at k d = 2 × 10 -4 . The comparison performed for the same K i but for two different values of k d is presented in Figs. 12b and12c.

For a high value of k d , bacterial growth in the vicinity of the NAPL is no longer slowed down but clearly halted, the bacterial mortality rate become higher than the growth rate above a threshold concentration of dissolved hydrocarbon. This effect can clearly be seen in Fig. 12d that indicates a decrease in the biomass concentration when bacteria are too far (dissolved substrate concentration too low) or too close (inhibitory effect) to the pollutant source. This creates an exclusive zone around the trapped droplets, which would become larger when K i is smaller and k d is larger.

Upscaling of pore-scale simulations and local mass equilibrium assessment

Numerical experiments allow us to better understand different mechanisms involved and their influence on the process of natural attenuation (favored or not by the biological activity) of a hydrocarbon source trapped within a porous medium. Another advantage of using numerical models at the pore scale is that they can validate or invalidate the relevant simplifying hypotheses in the development of macroscopic models.

For example, a classical approach used in the modeling of contaminant transport is the hypothesis of local equilibrium between the fluid phase and the NAPL phase [START_REF] Kaluarachchi | Modeling multi-component organic chemical transport in three-fluid-phase porous media[END_REF]Parker , 1990, Ahmadi et al., 2001). This approach considers that the fluid phase surrounding the entrapped hydrocarbon droplets is close to the thermodynamic equilibrium with the NAPL phase. In other words, the dissolved hydrocarbon concentration is close to the maximum solubility at any point in the domain. Thus, solute transport in the water/NAPL system can be described by a single concentration governed by a single transport equation. This hypothesis is not always verified, since many experimental studies [START_REF] Miller | Dissolution of trapped nonaqueous phase liquids: Mass transfer characteristics[END_REF][START_REF] Powers | Theoretical study of the significance of nonequilibrium dissolution of nonaqueous phase liquids in subsurface systems[END_REF] have shown a non local-equilibrium behavior for which the dissolution process is remarkably longer. In this case under abiotic conditions, the hypothesis of local equilibrium between fluid and NAPL phases is irrelevant given the low saturation of NAPL. However, the suitability of the local equilibrium assumption between the fluid and biofilm phases should be assessed. It would simplify the development of the macroscopic model by switching from the three-phase system to a two-phase NAPL/water-biofilm problem. The hypothesis of local equilibrium between biofilm and fluid has been successfully validated under certain conditions in the case of dissolved contaminant transport [START_REF] Golfier | Biofilms in porous media: Development of macroscopic transport equations via volume averaging with closure for local mass equilibrium conditions[END_REF]. The results of this study showed that low Péclet and Damköhler numbers are generally required to limit the occurrence of high gradients and to apply this hypothesis. However, the above work was based on an arbitrary spatial distribution of biomass which can strongly impact the occurrence of concentration gradients. We will try, in this last section, to evaluate the relevance of this hypothesis through simulations at the pore scale with our coupled model.

The geometry considered here is shown in Fig. 13a, and is constructed by duplicating four times the domain used in the previous sections. This makes the length of the domain much larger than the porescale characteristic lengths, which is enough to separate the scales and to upscale reliable correlations. The porosity of the domain and the NAPL saturation are also modified to allow a more even distribution of NAPL blobs and avoid artificially biased growth of the bacterial population. The domain now has an initial porosity ( 0 ) of 0.65 and a NAPL saturation (s 0 ) of 21%, distributed in (17 × 4) blobs as shown in Fig. 13a.

The porous medium that we have defined above is inoculated with bacteria capable of biodegrading the hydrocarbon. The bacteria, present in sessile form, initially occupy 3.8% of the pore volume, distributed randomly on the solid and NAPL blob surfaces. Figure 13a shows the initial geometry of the medium inoculated by the bacteria. Biomass is illustrated here in white to facilitate readability. The simulation parameters are given in Table 4. From the beginning, for both Da values, the ratio R eq increases around 1.1. We then observe a rapid to the characteristic dimensionless numbers of physical processes as classically identified but also to the pore topology. A more homogeneous distribution of biofilm related to a greater saturation of the oil phase would be necessary to approximate the local equilibrium conditions. We can also notice that, at the beginning of the simulation, the decrease of the R eq is almost identical for both Da numbers, due to the low initial value of the volume fraction of biofilm. When the biofilm begins to develop, we observe a significant divergence between the two values of R eq . R eq decreases to about 0.75 for Da = 0.1, and to 0.41 for Da = 60, which can be classically explained by an increase of the non-equilibrium phenomenon for higher value of Damköhler number.

Conclusions

In this paper, we develop a 2D mathematical model integrating the mechanisms associated with the dissolution and biodegradation of NAPL phase in porous medium. Pore-scale equations of fluid flow and solute transport are combined with a biofilm growth model. Fluid flow is simulated with an immersed boundary-lattice Boltzmann model while transport of dissolved hydrocarbon is described using a Cartesian cut-cell method. A cellular automaton algorithm based on the immersed boundary method previously developed in [START_REF] Benioug | An immersed boundary-lattice Boltzmann model for biofilm growth in porous media[END_REF] is used to describe the spreading and distribution of biomass. A uniform dissolution approach, which we consider as the most suitable for the envisaged simulations, is adopted to describe the evolution of the residual NAPL droplets over time.

The first set simulations focus on NAPL dissolution in abiotic conditions with analysis under different hypotheses (droplets size and Péclet number) and the effect of the spatial distribution of NAPL droplets on the dissolution rate. The results obtained are consistent with those from the literature. In a second step, we are interested in the bioenhancement of NAPL dissolution. Different conditions are also considered (spatial distribution, reaction kinetics, toxicity), and their impacts on the process of dissolution in the presence of bacteria are investigated. Finally, this pore-scale model is used to assess the suitability of a local equilibrium assumption between the fluid-biofilm phases in the vicinity of the NAPL sources. Despite the

Figure 1 :

 1 Figure 1: Sketch of an aquifer contaminated by NAPLs.

Figure 2 :

 2 Figure 2: Extent of the cell i + 1 (fluid/biofilm phase) and reduction of the cell i (NAPL phase).

Figure 3 :

 3 Figure 3: Reaction rate as a function of substrate concentration predicted by the Monod and Haldane model (µ = 0.3h -1 , K = 44.9mg/l, K i = 525mg/l)

Figure 4 :

 4 Figure 4: Algorithm of the dissolution and biodegradation model of a hydrocarbon phase at the pore scale

Figure 5 :

 5 Figure 5: a) The initial geometry of the computational domain: red -solid phase, blue -fluid phase, yellow -NAPL, blue cyan -NAPL/fluid grids and white -biofilm; b) Geometry of the medium at t = 500 with the presence of the biofilm (in green); c) Geometry of the medium at t = 500 without biofilm; d) Temporal variation of the NAPL blob volume.

Figure 6 :

 6 Figure 6: The initial geometry of the computational domain: red -solid phase, blue -fluid phase, yellow -NAPL and blue cyan -NAPL/fluid grids.

  We consider the same radius for three NAPL blobs R A = R B = R C = 0.055 (dimensionless, equal to their physical values divided by the length of the domain). Results illustrated in Figs.7b-7dshow reasonable behavior of the flow. The velocity field is in agreement with the assumptions imposed: the upstream fluid flow bypass the trapped droplets of NAPL behaved as obstacles and form perturbations behind. A consistent pattern is also observed for the concentration fields (Figs.7a-7c): (i) the hydrocarbon plume resulting from the dissolution is well entrained by the flow downstream of the NAPLs and follows the preferential flow paths between the solid grains and (ii) the plume of dissolved contaminant reabsorbs as the dissolution of the droplets progresses. Finally, NAPL blobs, which gradually decrease over time, retain in circular shapes, as assumed by our model. In other words, the mechanisms behind the dissolution of the NAPL are correctly taken into account by our model demonstrated by reasonable and consistent flow/concentration field results.

Figure 7 :

 7 Figure 7: (a, c) Hydrocarbon concentration, NAPL blobs are white; (b, d) Geometry of the porous medium and the velocity field. a), b) at t = 5; c), d) at t = 45; e) Temporal variation of NAPL blobs radius; f) Temporal variation of NAPL blobs volume.

Figure 8 :

 8 Figure 8: Evolution of the transport processes (a, c, e, g) and associated concentration fields (b, d, f, h) at two different times for two Péclet numbers: a), b) t = 5 -P e = 10; c), d) t = 5 -P e = 0.5; e), f) t = 51 -P e = 10; g), h) t = 51 -P e = 0.5.

Figure 9 :

 9 Figure 9: Temporal variation of the NAPL volume fraction for two Péclet numbers: P e = 10, P e = 0.5

Figures

  Figures 10a-10d indicate that the biofilm grows preferentially near the NAPL blobs. Conversely, few biofilm cells develop far away from the residual NAPL droplets. Over time, the volume of biomass increases and the biofilm approaches the NAPL blobs until reaching the interface. This preferential growth clearly reflects bacterial growth where the substrate concentration is maximum (NAPL blobs are the only substrate

Figure 10 :

 10 Figure 10: Evolution of the transport processes for Da = 20 (a, b, c, d) : a) t = 15, b) t = 20, c)t = 30, d)t = 38; temporal variation of blobs radius with/without biofilm (e, f, g): e) blob A, f) blob B, g) blob C; h) color scale of grid types.

Figure 11 :

 11 Figure 11: Evolution of the transport processes for Da = 50 (a, b, c) : a) t = 7, b) t = 14 c) t = 24; temporal variation of blobs radius with two Da values (d, e, f): d) blob A, e) blob B, f) blob C.

Figure 12 :

 12 Figure 12: Evolution of the transport processes (a, b, c) : a) t = 38,K i = 2 × 10 -3 , k d = 1 × 10 -7 b) t = 38, K i = 6 × 10 -4 , k d = 1 × 10 -7 c) t = 38, K i = 6 × 10 -4 , k d = 9 × 10 -3 ; d) biomass concentration: t = 38, K i = 6 × 10 -4 , k d = 9 × 10 -3 .

  Figures 13b-13c represent the results for two Damköhler numbers Da = 0.1, Da = 60 at t = 138.The biofilm is rapidly developing preferentially in the vicinity of NAPL blobs leading to the formation of a biomass ring around the droplets of NAPL. At this stage, the process of dissolution is highly accelerated by increasing the solubility and bio-assimilation of the dissolved hydrocarbon. In order to determine whether the observed behavior approximates local equilibrium conditions, it must be verified that the averaged concentrations in the fluid and biofilm phases, respectively C moy f

Figure 13 :

 13 Figure 13: a) Initial geometry; b) Geometry of the porous medium for P e = 0.6 and Da = 0.1 at t = 138; c) Geometry of the porous medium for P e = 0.6 and Da = 60 at t = 138

Figure 14 :

 14 Figure 14: The evolution of Req according to the NAPL saturation.

Table 1 :

 1 Physical and biological parameters used in the single pore simulations.

	Parameter	physical/biological	dimensionless value
		value
	lx	30 mm	lx = 3
	ly	10 mm	ly = 1
	N x	301
	N y	101
	c h	800 mg/l	c h = 1
	C eq	0.48 mg/l	C eq = 6 × 10 -4

Table 2 :

 2 Physical parameters used in the simulations.

	Parameter	Physical value	Dimensionless
			value
	Length of the domain, lx	12 mm	lx = 1
	Width of the domain, ly	12 mm	ly = 1
	Number of mesh cells in x	116
	direction, N x
	Number of mesh cells in y	116
	direction, N y
	c h	800 mg/l	c h = 1

Table 3 :

 3 Physical and biological parameter values used in the simulations

	Parameter	physical/biological	dimensionless value
		value
	lx	12 mm	lx = 1
	ly	12 mm	ly = 1
	N x	116
	N y	116

Table 4 :

 4 Physical and biological parameters used in the simulations.

Acknowledgements M. Benioug and X. Yang are funded by the Recruitment Program of Global Experts in China. This work was partially supported by the French National Research Agency (ANR) through the MOBIOPOR project, with the reference ANR-10-BLAN-0908 and was performed as part of the French Scientific Interest Group-Industrial Wasteland (GISFI) program. This project has also received partial funding from the European Union's Horizon 2020 research and innovation program through the PROTINUS project under Grant Agreement No. 645717.


ARCADE, Federico Tajariol

ARCADE, Jean-Michel Adam

Michel Dubois

Seeing the learner during a mediated tutoring session: influence of non-verbal cues on tutoring dialogue

Keywords: video-person, video-actions, tutoring dialogue, kinesic and ostensive-inferential cues

This paper deals with the effect of non-verbal cues on a video-mediated tutoring situation. We will first state communication theoretical underpinnings and then we will focus on the role of kinesic cues and ostensive-inferential cues in communication and mediated activities. After that we will describe the coding scheme we used to analyze the tutoring dialogue and we will present two experimental studies that evaluate the effects of these nonverbal cues on the following measures: the tutor's proactive behaviour, the tutor-student's mutual understanding, the efficiency of the dialogue and the intrinsic tutoring speech acts. We will finally conclude with some considerations for the design of interfaces to support video mediated tutoring dialogue.

Introduction

Communication technologies are more and more used in mediated human activities, like in project meeting [START_REF] Tang | Why do users like video video? Studies of multi-media supported collaboration[END_REF], in repairing equipment [START_REF] Kraut | Visual information as a conversational resource in collaborative physical tasks[END_REF] and also in teaching activities [START_REF] Collis | Tele-learning in a digital world: The future of distance learning[END_REF]. Nowadays, audio-video technologies allow distant participants to see one another during the interaction [START_REF] Heath | Reconfiguring media space : supporting collaborative work[END_REF] and even to show one another the actions they are doing in their working environment [START_REF] Nardi | Video-as-Data : Technical and social aspects of a collaborative multimedia application[END_REF]. Nevertheless, several researches illustrate that the richness media framework [START_REF] Daft | Organisational information requirements, media richness and structural design[END_REF] is not a completely adequate solution for the special needs in mediated activities [START_REF] Whittaker | Theories and methods in mediated communication[END_REF]. From a pragmatic point of view, we do consider that affordances of communication tools are different according to the type of conveyed information [START_REF] Clark | Grounding in communication[END_REF] and the specific human activity. In the next section we will define a background about communication from face-to-face interaction studies. Next, we will describe our experimental method and the two studies we conducted about the affordances of two non-verbal cues (kinesic and ostensive-inferential) on mediated tutoring dialogue. The outcomes show that the efficiency of tutoring dialogue is different according to the conveyed nonverbal cue and we suggest some principles to improve the design of interfaces supporting the mediated tutoring dialogue.

Background Communication as a joint action

We communicate with another person to perform different activities, as planning, bargaining, creating, teaching, etc. Therefore we use language to act on our environment [START_REF] Austin | How to do things with words[END_REF] and to make our addressee believe by saying words [START_REF] Searle | Speech Acts, An Essay in the Philosophy of Language[END_REF]. When we communicate with our partner, we cannot directly know his (her) thoughts, feelings and intentions. We are just able to infer them by interpreting our partner's utterances [START_REF] Sperber | Relevance: Communication and cognition[END_REF] and (her) his non-verbal behaviour [START_REF] Argyle | The Central Europe Experiment : looking at persons and looking at things[END_REF]. Nevertheless, communication is not just a oneway human activity, where the speaker sends content and the addressee receives it. Communication is rather a joint action as playing a piano duet [START_REF] Clark | Using languages[END_REF]. For instance, the tutoring dialogue is a cooperative action [START_REF] Fox | The Human Tutorial Dialogue Project: Issues in the Design of Instructional Systems[END_REF], where a tutor chooses the content of his (her) contribution according to the knowledge (s)he supposes student owns. Both tutor and student coordinate their turns to ground on a mutual understanding (dis-) confirming that they understand what has been said [START_REF] Clark | Referring as a collaborative process[END_REF].

Communication as a multimodal process

When the speaker and the addressee are in a face-to-face setting, they enjoy the multimodal properties [START_REF] Mcgurk | Hearing lips and seeing voices[END_REF]. In fact, participants produce and interpret verbal and non-verbal cues. In the non-verbal cues class, we distinguish kinesic cues (e.g., facial expressions, postures, etc.) and ostensive-inferential cues (actions and deictic gestures). The former ensures the speaker-listener turns transitions and informs the participants about the other person's feelings and intentions [START_REF] Amado | Dynamique des communications dans les groupes[END_REF]; the latter facilitates the verbal referring process, helps participants to coordinate their actions and to anticipate the other's needs [START_REF] Nardi | Video-as-Data : Technical and social aspects of a collaborative multimedia application[END_REF]. For instance, in the tutoring dialogue, the tutor infers when and how to help students without disturbing by observing facial expressions [START_REF] Fox | The Human Tutorial Dialogue Project: Issues in the Design of Instructional Systems[END_REF]. So, we can suppose that in video-mediated communication setting it is sufficient for distant participants to have all non-verbal cues to act properly.

Non-verbal cues in mediated human communication

Nevertheless, in the literature related to the video-mediated communication, the effects of non-verbal cue are still discussed. On the one hand, several studies show that non-verbal cues do not always support participants to better perform their mediated activities [START_REF] Anderson | Impact of video-mediated communication on simulated service encounters[END_REF][START_REF] Daly-Jones | Some advantages of video-conferencing over high-quality audio conferencing : fluency and awareness of attentional focus[END_REF][START_REF] Doherty-Sneddon | Face-to-face interaction and video mediated communication : a comparison of dialogue structure and co-operative task performance[END_REF][START_REF] O'malley | Comparison of face-to-face and video-mediated communication[END_REF]. On the other hand, other studies show that kinesic and ostensive-inferential cues help distant participants to establish a mutual understanding [START_REF] Doherty-Sneddon | Face-to-face interaction and video mediated communication : a comparison of dialogue structure and co-operative task performance[END_REF][START_REF] Isaacs | What video can and cannot do for collaboration[END_REF][START_REF] Whittaker | Shared workspaces: how they work and when are they useful[END_REF] and that ostensive-inferential cues support participants to perform a mediated activity more quickly [START_REF] Fussell | Where do helpers look? Gaze targets during collaborative physical tasks[END_REF][START_REF] Gergle | Language efficiency and visual technology: minimizing collaborative effort with visual information[END_REF]. We outline that these studies focus much more on the nature of nonverbal cues than on the socio-cognitive properties of a specific human activity. Following this point of view, we consider the synchronous tutoring activity [START_REF] Chi | Eliciting self-explanations improves understanding[END_REF][START_REF] Fox | The Human Tutorial Dialogue Project: Issues in the Design of Instructional Systems[END_REF] and their socio-cognitive categories to analyze the effect of non-verbal cues on the efficiency and the contents of tutoring dialogue.

Research Problem

Our main research question is the following: which type of non-verbal cue improves the efficiency of tutoring dialogue? Would it be better if the tutor and the student see each other at a distance (kinesic cues)? Would it be better if the tutor observes the student's actions to improve (her) his learning (ostensive-inferential cues)? To answer these questions, we set two experimental studies: the first one was dedicated to understand the affordance of kinesic cues; the second one was oriented to understand the affordance of ostensiveinferential cues and to compare their effects on tutoring dialogue. We will describe in the next section the experimental tasks and apparatus, as well as the measures we collected.

Method

In this section, we will describe the experimental task, the apparatus, the procedure and the dependant measures, which are common to the two studies.

Task

Tutors and students were involved in a sort of practical pedagogical work. One tutor had to help two students to learn basic commands of HTML language and to create an easy web page. In fact, they had to edit some sentences of a text in bold and italic, as well as building an internal link. The tutor could help and communicate with only one student at once, as in a dyadic tutoring situation: in fact, when the tutor was talking to a student, the other one could not hear their dialogue.

Apparatus

The tutor and the two students were in three separate rooms, each one was equipped with a personal computer (central unit and monitor) and another monitor was used to show the person. Anytime the tutor wanted to communicate with a student, (s)he chose the student by pressing a button. Both students could ask the tutor to help them anytime, sending him/her a standard message by means of synchronous chat software. Each computer was equipped with a web browser and a simple text editor.

Procedure

Each experimental session lasted nearly an hour. In the preparing phase (15 minutes), the researchers introduced the tutor to the two students and the researchers explained the aims of the experimentation. Then, the tutor and each student settled down in three separate rooms. Each subject filled in a consent form and was briefed on the main functions of the communicating technical apparatus. Next, each student answered a pre-test to evaluate his/her HTML knowledge. Then the experimental session started and lasted 35 minutes. At the beginning, each student received a four-page HTML manual, which contained some HTML basic commands. Of course, the students needed their tutor's help to design the web page. The tutor had to help each student spontaneously or when a student asked for help. In the last phase, each subject was asked to complete a questionnaire to evaluate the quality of the interaction with others, before he/she was debriefed and then dismissed.

Measures

We videotaped the 24 experimental sessions that we transcribed verbatim. Researches concerning the common grounding processes in communication [START_REF] Clark | Using languages[END_REF], the affordances of video information in mediated communication [START_REF] Kraut | Visual information as a conversational resource in collaborative physical tasks[END_REF][START_REF] Karsenty | Cooperative work and shared visual context : an empirical study of comprehension problems in side-by-side and remote help dialogues[END_REF] and the tutoring dialogues [START_REF] Chi | Quantifying qualitative analyses of verbal data : a practical guide[END_REF] inspired us the following measures: a) the proactive behaviour of the tutor : we distinguished the tutors' spontaneous interventions towards the students as proactive interventions from the tutors' reactive interventions when they had replied to a student's call; b) the learning score : we measured the score of the web page that the students had realized1 ; c) the mutual understanding : we developed a coding scheme to categorize verbal markers the students and tutors had used in common ground process ( 2 -Intrinsic tutor's and student's tutoring speech acts Three trained coders analysed the transcriptions of the dialogue. We calculated the reproducibility test [START_REF] Carletta | The reliability of a dialogue structure coding scheme[END_REF] to evaluate the reliability of coding: we obtained an average value for the students' speech acts (K=.58) and a high value for the tutors' speech acts (K=.82).

Experimental Studies

In the next two sections we will describe two studies: first, to understand the effects of the video-person (kinesic cues) and second, the study concerning the effects of video-actions (ostensive-inferential cues) on tutoring dialogue.

Experiment 1: what are the effects of seeing our partner on tutoring dialogue? Hypotheses

The tutor's behaviour was expected to be more proactive when a video-person link was available, because the tutor could observe the student's face and infer his/her difficulties and motivation during the practical work. We expected the efficiency of the dialogue to be improved when a video-person link was available, because the subjects could better coordinate their contributions to the dialogue by seeing each other's face and producing fewer speech acts. Furthermore, we also predicted the students' performance to improve because the tutors could better help and encourage the students while they performed the task.

Subjects

As tutors, we recruited twelve subjects, half men and half women (age M=30.9, S.D.=7.7). All of them were computer scientists owning good skills at HTML programming. Two of them were associate professors, whereas the others were Ph.D. students with almost one-year teaching experience in computer science. They were paid 25 euros each for their participation. As students, we recruited forty-eight undergraduate students in psychology (thirty-six women and twelve men; age M=24.1, S.D.=6.2), all of them unskilled at HTML programming. For their participation, they received a credit for their social psychology course.

Experimental conditions

We set the following conditions: c1) audio only: the tutor chose the student s/he wanted to talk to. c2) audio & video-person: the tutor and each of the two students could see each other's face and upper torso on a personal monitor, and they communicated by means of the audio channel. Each tutor performed in both conditions (within-participants experimental design), whereas half students were assigned to one of the two conditions (between-participants design). We controlled that the age of the students involved in the two conditions were equivalent (t(46)=1.318;n.s.) and so was the HTML knowledge (t(46)=0.09;n.s.). To guard against order effects, we counterbalanced the tutors' performing order setting. From these outcomes, it seems that communicative affordances of kinesic cues are the following: the tutor more often takes the floor spontaneously, (s)he produces fewer verbal markers to check students' understanding and (s)he more willingly encourages them. Furthermore, the tutors and students spend more time talking about the ongoing task.

Experiment 2: the effects of observing students' actions on tutoring dialogue.

The task, the procedure and the measures of this second experiment were the same than for the first one. In addition, we revised the coding scheme to include deictic utterances the tutors and students had used in the mutual understanding process. Deictic utterances were used to refer to the objects they had seen. We coded deictic expressions (i.e. T: "you move the icon down", S: "here?", T: "yes") and pronouns (i.e., T: "take this and move it on your desktop").

Hypotheses

We expected the tutor to be more proactive when (s)he could observe the students' actions, because (s)he was always aware of their difficulties and (s)he could take the floor before they asked for help. Mutual understanding was expected to be easier when the tutor could observe the student's actions, because they shared a visual environment. Moreover, we expected the tutor and students to describe less explicitly their actions and all related objects, so they would produce more deictic utterances. We also predicted the student's performance to improve when the tutor could monitor them during the practical work session, because the tutor could choose the most appropriate help according to their needs.

Subjects

We recruited the same tutors (N = 12) as for the first study, whom we paid 25 euros each for their participation. We also recruited seventy-two undergraduate students in psychology (fifty women and twenty-two men; age M=23.8, S.D.=5.1), all of them unskilled at HTML programming. They received a credit for their social psychology course.

Experimental conditions

We set the three following conditions: c1) audio & video-person: the tutor and the students could see each other's face and upper torso on one screen and they communicated by means of the audio channel; c2) audio & video-actions: the tutor and the students could communicate by the audio channel, and the tutor could observe both students' computer screens (by means of VNC software). So, the tutor's computer screen was split in two windows showing the students' desktops and (s)he could observe their activity and make the decision to help them at will; c3) audio & video-person & video-actions: the tutor could observe both students' face and upper torso and their computer screens, while the verbal communication between the tutor and each student was supported by the audio channel. As for the first study, each tutor performed task in all conditions (within-participants design) and we counterbalanced the order of the tutors' performance to guard against order effects. About students, they were equally shared between the three experimental conditions. We controlled the age (F(2, 69) = 1.73; n.s.) and the HTML knowledge (F(2, 69) = 0.16; n.s.) of each group of students.

Main Outcomes

We conducted an analysis of variance (ANOVA) and a priori comparisons between conditions. a) About the tutor's proactive behaviour, the number of tutor' The aim of these experiments was to understand communicative affordances of kinesic cues (videoperson) and ostensive-inferential cues (video-actions) on a same activity, the mediated tutoring dialogue. The outcomes of study 1 show that when the tutor and the student can see each other's face, the tutor more often takes the floor spontaneously, (s)he produces fewer verbal markers to check the student's understanding and (s)he more willingly encourages him/her. Outcomes from study 2 partially corroborate study 1, while confirming some previous researches [START_REF] Doherty-Sneddon | Face-to-face interaction and video mediated communication : a comparison of dialogue structure and co-operative task performance[END_REF][START_REF] Fussell | Where do helpers look? Gaze targets during collaborative physical tasks[END_REF]. For example, the tutor's communicative behaviour is more proactive when kinesic cues are available. On the contrary, it appears that ostensive-inferential cues let the tutor and the student focus on intrinsic learning contents and improve their mutual understanding. Moreover, the student's performance is better and the efficiency of the dialogue is higher. For the design of the interface supporting mediated tutoring dialogue, these main outcomes imply that: If we want the tutor to be proactive, it is better to let the tutor and the student see each other; If we prefer the student to improve learning and the dialogue to be more efficient, it is better to let the tutor observe the student's ongoing actions. However, these suggestions have to be taken according to the limits of our studies. The first concerns the content of tutoring interaction: we are planning to conduct another experimentation in which the content will be rather declarative (about the comprehension of a dialogue and text in a foreign language). The second limit is related to the number of students who were involved in each experimental session. We think that if a tutor had to assist more students at the same time, it would be necessary to investigate the form of ostensive-inferential cues. For instance, we are going to study a visualisation form by which the tutor can just observe the steps twelve students perform during a practical work.

s

Figure 1 -

 1 Figure 1 -Experimental conditions of study 1: audio only (left) vs. audio & video-person (right). Main outcomes We conducted independent t-test and we calculated the value of the effect size d. a) There were significantly more tutors' proactive interventions in the audio & video-person than in the audio only condition (M = 4.6 (2) vs. 1.8 (0.9), t (22) = 4.27; p<.001, d = 1.8). b) Concerning the learning score, no significant differences were found between the two conditions (audio only M=4.2(1.4) vs. audio & video-person M = 4.6 (1.5), t(22)=0.83 ; n.s., d =1.1). c) Concerning the mutual understanding, statistical analysis showed that tutor had significantly produced fewer verbal markers to check students' understanding in audio & video-person than in audio only condition (M=3.9 (3.6) vs. 10 (6.4), t(22)=2.85; p<.01, d=1.17). About other students' and tutors' verbal markers, we did not find any significant differences between the two conditions. d) Concerning the speech acts referred to tutoring dialogue, we first presented outcome about tutors' speech acts and then about students' speech acts. i) Encouraging acts -Tutors produced significantly more speech acts to encourage students in audio & video-person than in audio only condition (M=3.2(1.3) vs. 1.2(1.2), t(22)=2.9; p <.01, d=1.14). * The number of tutors' speech acts oriented to know students' ongoing task was significantly higher in audio & video-person than in audio only condition (M=60.7 (33.2) vs. 35.3 (8.1), t(22)=4.27; p<.02, d=1.05). ii) Consequently, to inform their tutors about their ongoing task, the students significantly resorted to more speech acts in audio & video-person than in audio only condition (M=63.6 (28.4) vs. 43.8 (14.4), t(22)=2.15; p <.05, d=0.88). No significant differences between the two conditions were found about other speech acts.

Figure 2 -

 2 Figure 2 -Experimental conditions study 2: audio & video-person (up) vs. audio & video-person (left) vs. audio & video-person & video-actions (right).

Table 1 )

 1 ;

	Role	Category	Utterance/Example
		To accept	T: "yes" "ok, right"
		student's
	Tutor	utterance To check	T: "is it clear now?",
		student's	"is it ok?"
		understanding
		To accept	S: "yes", "ok"
		tutor's
	utterance Student To check	S: "could you repeat,
		tutor's	please?", "what?"
		utterance

Table 1 -Mutual understanding coding scheme d

 1 ) the intrinsic speech acts concerning tutor-student dialogue : we classified the tutors' and students' speech acts which were related to the essential nature of the tutoring dialogue (Table2).

	Role	Category	Utterance/Example
		To find out	T: "Did you try
	Tutor	student's ongoing	moving it on the red
		task	icon?"
		To help student	T: "Close the

  spontaneous interventions is significantly different between conditions (F(2, 33) = 4.671; p <.02, η 2 =0.22). In fact, the tutor is significantly more proactive in audio & video-person & video-actions than in audio & video-The Tutor's encouraging speech acts -A one-way analysis of variance shows significant differences between conditions (F(2, 33) = 4.813; p <.01, η 2 = 0.22). The Tutors more often encourage the students in audio & video-person than in audio & video-actions condition (M=3.2 About the student's speech acts to give their tutor information about the ongoing task: the oneway analysis of variance shows significant differences between conditions (F(2, 33) = 4.529; p <.02, η 2 =0.21). the Students less often give their tutors information about their ongoing task in audio & video-person & video-actions than in audio & video-person condition (M=26.5 (14.1) vs. 43.8 (14.4), t(22)=2.5; p<.01, d=1.21) and less often in audio & video-actions than in audio & video-person condition (M=27.7 (18.4) vs. 43.8 (14.4), t(22)=2.6; p<.005, d=0.97). All other students' speech acts are substantially equivalent between conditions.

	(2.3) vs. 0.8 (1.1), t(33)=2.69; p<.01, d=1.33) and more often in audio & video-person & video-
	actions than in audio & video-actions (M=3.2 (2.8) vs. 0.8 (1.1), t(33)=2.69; p <.01, d=1.12).
	ii) Measures	Categories	(c1) audio & video-	(c2) audio & video-	(c3) audio & video-person & video-	Comparisons
			person	actions	actions
							C3 > C1
	Tutor's	Tutor's spontaneous	M = 4.2	M = 4.6	M = 6.6		(p<.007)
	proactivity	interventions	(2.2)	(2)	(1.9)		C3 > C2
							(p<.02)
							C2 > C1
	Mutual	Deictic terms	M = 90.9	M = 125.2	M = 134.3		002, (p<.005)
	d=0.88) and better in audio & video-actions than in audio & video-person condition (M=5.5 (1.2) understanding (29.3) (21.6) (46.2) C3 > C1
	vs. 4.5 (1.5), t(69)=2.4; p <.02, d=0.73).					(p<.01)
	C3 > C1 c) About mutual understanding, the number of deictic utterances is significantly different between Learning Student's task performance M = 4.5 M = 5.5 M = 5.9 (p<.002) conditions (F(2, 33) = 5.453; p <.01, η 2 = 0.99). A-priori comparisons show that the tutors and the students produce significantly more deictic utterances in audio & video-actions than in audio & score (1.5) (1.2) (1.6) C2 > C1
	video-person (M=125.2 (21.6) vs. 90.9 (29.3), t(20.2)=3.27; p <.005, d=1.33) and more in audio & (p<.02)
	video-person & video-actions than in audio & video-person (M=134.3 (46.2) vs. 90.9 (29.3), C1 > C2 Tutor
	t(18.6)=2.75; p <.01, d=1.12). About the tutors' and students' verbal markers targeted to mutual To find out student's M = 35.3 M = 27.4 M = 26.6 (p<.01)
	understanding, outcomes do not show any significant differences between conditions. ongoing task (8.1) (7.6) (6.8)	C1 > C3
	d) About tutoring speech acts, we will first present outcomes for tutors (i) and then for students (ii). (p<.005)
	i) One-way ANOVA shows that the number of tutor's speech acts to find student's ongoing task is C2 > C1
	significantly different between conditions (F(2,33)=4.529; p <.02, η 2 = 0.21). The Tutor produces fewer speech acts oriented to the student's task in audio & video-actions than in audio & video-M = 14.8 M = 40.2 M = 37.9 (p<.000) To help student (10.4) (12.9) (13.8) C3 > C1 Intrinsic
	person condition (M = 27.4 (7.6) vs. 35.3 (8.1), t(22)=2.8; p<.01, d=1) and fewer in audio & (p<.000) tutoring
	audio-person & audio-actions than in audio & video-person condition (M = 26.6 (6.8) vs. 35.3 C1 > C2 speech acts
	(8.1), t(22)=2.6; p<.005, d=1.16). * the Tutor's speech acts to help the student. A one-way analysis of variance shows significant M = 3.2 M = 0.8 M = 3.2 (p<.01) To encourage student (2.3) (1.1) (2.8) C3 > C2
	(p<.01) differences between conditions (F(2, 33) = 15.248; p<.000, η 2 =0.48). There are fewer help speech acts from the tutor in audio & video-person than in audio & video-actions (M=14.8 (10.4) vs. 40.2 (12.9), t(33)=4.99; p <.000, d=2.18) and fewer in audio & video-person than in audio & video-person & video-actions (M=14.8 (10.4) vs. 37.9 (13.8), t(33)=4.55; p <.000, d=1.91). Students C1 > C2 To give tutor M = 43.8 M = 27.7 M = 26.5 (p<.005) information about (14.4) (18.4) (14.1) C1 > C3 ongoing task (p<.01)

actions condition (M = 6.6 (1.9) vs.

[START_REF] Heath | Reconfiguring media space : supporting collaborative work[END_REF]

.6 (2), t(33)=2.366; p <.02, d=1.02) and than in audio & video-person condition (M = 6.6 (1.9) vs. 4.2 (2.2), t(33)=2.859; p <.007, d=1.16). b) The students' web page score is significantly different between conditions (F(2, 69) = 5.776; p <.005, η 2 = 0.14). The students perform significantly better in audio & video-person & videoactions than in audio & video-person condition (M = 5.9 (1.6) vs. 4.5 (1.5), t(69)=3.27; p<.*

Table 3 -Main outcomes study 2 Conclusions

 3

We mark 1 point for a right tag and 0.4 points for a right tag which was badly inserted in the html code.