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In this paper, we are interested by the dissolution of NAPL (Non-Aqueous Phase Liquid) contaminants in heterogeneous soils or aquifers. The volume averaging technique is applied to 2D systems with Darcy-scale heterogeneities. A large-scale model is derived from a Darcy-scale dissolution model in the case of small and large Damkholer numbers, i.e., for smooth or sharp dissolution fronts. The resulting models in both cases have the mathematical structure of a non-equilibrium dissolution model. It is shown how to calculate the resulting mass exchange and relative permeability terms from the Darcy-scale heterogeneities and other fluid properties. One of the important finding is that the obtained values have a very different behavior compared to the Darcy-scale usual correlations. The large scale correlations are also very different between the two limit cases. The resulting large-scale models are compared favorably to Darcy-scale direct simulations.

The mechanisms of migration in aquifers of partially miscible pollutants to water (oil, hydrocarbons…), later denoted NAPL for Non-Aqueous Phase Liquid, are well-known. Two- and three-phase flow mechanisms lead to a more or less dispersed distribution of the organic phase in the aquifer. The prediction of the contaminant plume requires in general solving the transport equations completed with proper initial and boundary conditions. In practice, two classes of model are available. The first class makes use of the assumption of local equilibrium, which translates the fact that the average, Darcy- scale concentrations are close to the pore-scale thermodynamic equilibrium concentrations at the interface between the two phases. Indeed, concerning the modeling of dissolution (water/NAPL transfer), many laboratory works ( [

Recently, experimental works of NAPL dissolution in heterogeneous porous media [

In recent laboratory experiments, Yra [

The one-dimensional problem was the subject of a first study [

In this paper, we consider a stratified model of a heterogeneous porous medium typical of geological systems. The medium is composed of stacks of the elementary pattern described in

are filled initially with water and NAPL. The typical system is made of two different strata (ω, η) of the same length l. Each stratum contains initially NAPL at residual saturation _{ω}, ε_{η} and constant permeability k_{ω}, k_{η}.

We do not present here the development of the Darcy-scale dissolution model since very detailed and complete developments are available in the literature ( [

Dissolution equations:

Large scale Darcy law:

Boundary conditions are:

‒ at the north and south boundaries of the domain,

‒ at the east boundary,

‒ at the west boundary one imposes, the Dirichlet condition as well for the pressure as for the concentration:

In the above equations,

with f_{i}, the volume fraction and K_{i} the permeability of the i-stratum respectively, I the tensor identity and j the unit vector along the y-axis. The resolution of Equations (1) to (6’’) requires the knowledge of the evolution of the large scale average mass exchange coefficient,

The mass exchange coefficient in the transport equation for the pollutant is a simplified representation of complex pore-scale mechanisms: diffusion of the pollutant at the interface towards the porous media, coupling with convection, and geometrical evolution of the interface due to dissolution. Several works were undertaken to estimate this coefficient ( [

In previous works ( [

very small Damkhöler numbers. In this work, we extend these investigations to the cases of flow parallel to the strata. One presents below the calculation for the Darcy-scale local equilibrium cases. It is supposed here that the porous medium is initially at Darcy-scale local equilibrium in the two strata illustrated in _{f} is higher in one of the strata. Here one will suppose that the dissolution front is faster in the ω-stratum. If a certain volume of stratum is taken, one has the two following situations described below.

・ Here we assume that the dissolution front is in the first stratum of the unit cell and there is not yet dissolution in the 2nd stratum. This situation is illustrated by

If we suppose,

The variation of this mass of NAPL in the cell is:

where

tion of mass is equal to the mass flux of NAPL going out of the unit cell through section A (Cf.

where _{f}:

Following the equation of dissolution, one writes:

where v_{w} is the volume of the water phase. Combining Equations (17), (18) and (19), one still writes:

However, the volume of water in the cell is evaluated as follows

Thus,

In addition, according to (11) and (13) one has

Moreover, according to (8) and (13), one can express

Combination of the relations (13), (23) and (24) in (22) gives finally

This correlation is very different from those available in the literature ( [

・ A situation when the dissolution front has left the first stratum and dissolution starts in the second stratum. This second situation is illustrated in

The mass of NAPL contained in the unit cell, assuming small

This mass varies in the cell as follows:

With

tion is equal to the mass flux of NAPL going out of the unit cell through section A (Cf.

By the same argument as in the preceding case, one obtains the correlation of mass exchange

In this case, the volume of the water phase can be calculated as follows

Moreover,

And

Taking account Equations (35) and (36), Equation (33) becomes

This correlation is valid for

Ultimately, one finds the same relations as in 1D [

The problem considered is that to estimate the relative permeabilities during the dissolution of NAPL in a stratified porous medium. The flow, parallel with the strata (Cf.

The coefficient of mass exchange in this case is available in the literature [

and the calculation of

1)

2) we impose an increment of average dissolution, i.e.,

3) we calculate

4) we calculate

5) we calculate then

6) we reiterate by going to the step 1.

However a scaling of the Darcy law is essential to close Equation (3). The oil phase being trapped, the flow in each stratum is comparable to a monophasic flow, but for the fact that the permeability will change with the evolving oil saturation. The problem is thus identical to that schematized on _{wω} and K_{wη}. The scaling of the problem makes it possible to homogenize the system with the large scale averaging volume V_{∞} and to estimate the average large scale

Neglecting gravity effects, we write as follows the local scale total flow problem:

The scaling of Darcy law for heterogeneous systems is largely developed by Quintard et al., ( [

where

In which

Initially the stratified system satisfies the conditions of

We present in

In _{ω} and α_{ω}_{η} the corresponding values of the mass exchange coefficients in the stratum ω and in the stratum η respectively, _{rwω} and k_{rwη} the relative permeabilities of water in each stratum, _{ω} and f_{η} are the domain volume fractions such as f_{ω} + f_{η} = 1 with f_{ω} = V_{ω}/V_{∞} and f_{η} = V_{η}/V_{∞}.

For a flow which is parallel to the strata, all these characteristics are calculated for the second iteration by the formulas below.

Iter. 1 | ||||||||||

Iter. 2 | Cf. system of Equations (50) | |||||||||

… |

with

Thus, in a more general way, if one gives oneself a certain evolution at local Darcy scale of saturation at large scale, of saturation in each stratum, of the coefficient of exchange in each stratum, of the average exchange coefficient at large scale and of the relative permeability of Corey type, all the relations (50) can be evaluated at the time step n + 1. The relative permeability and the large scale mass exchange coefficient are respectively estimated by the following expressions:

Moreover, the relative permeabilities at large scale depend on saturation according to Equation (51). In conclusion in this paragraph, it is interesting to stress that the correlations on the large scale mass exchange coefficient obtained in this study differ from the traditional correlations of mass transfer available in literature ( [

where n < 1 and 0.6 ≤ m ≤ 0.9.

Indeed, the correlations (25) (37) and Equation (40) are very different from the correlation (53).

With these estimates of the effective properties, the large-scale equations of dissolution are now under a closed form, i.e., purely expressed in terms of large-scale variables. They are solved numerically in this section for the particular situations considered in this work. A comparison is made between Darcy-scale and large-scale results in order to validate the proposed models. The calculations were carried out using a proprietary program DISSOL 2D conceived for this study and based on a classical finite volume formulation [

The numerical experiments were carried out using the mass exchange coefficient of the relations (25) and (37). This coefficient corresponds to the dissolution of a NAPL in a stratified system characterized by a unit cell including two strata and having each one, a rather large mass exchange coefficient so that the dissolution front is sharp and characteristic of a local equilibrium dissolution. Simulations at the two scales were carried out with the data of

・ Concentration fields at large scale

Water injection at x = 0, under an entry pressure of Pe = 10^{5} Pa in the direction of the flow (the pressure at the exit of the domain is taken to be zero), starts the dissolution phenomenon.

‒ the zone of dissolution increases with time. In the interior of this zone which separates pure water from polluted water, the concentrations are lower than the equilibrium concentration;

Darcy scale parameters | Large scale parameters | Physical properties of the phases | |||||
---|---|---|---|---|---|---|---|

Porosity/stratum | Residual saturation/stratum strate | Exchange coefficient/stratum (s^{−1}) | Coefficient of permeability/stratum (m^{2}) | Average porosity | Initial average saturation | Average permeability (m^{2}) | |

‒ the concentration at the downstream of this zone remains equal to the equilibrium concentration. Moreover, water with weak concentration in pollutant advances in front of the dissolution front in the strata removed from pollutant in order to contribute to decrease the concentration of pollutant in water polluted in the downstream;

‒ the process is very long.

・ Large scale saturation fields

In this paragraph, we presents the results of the 2D calculations carried out by supposing that the Damkhöler numbers are very small; i.e., the mass exchange coefficient used is that developed in paragraph 3.2. The modeled situation corresponds to the dissolution of NAPLs in a stratified porous medium (Cf.

The experiments carried out made it possible to highlight the evolution following the

average saturation of mass exchange coefficients in each stratum and that at large scale and relative permeabilities.

・ Evolution of mass exchange coefficients

Darcy scale parameters | Large scale parameters | Physical properties of the phases | |||||
---|---|---|---|---|---|---|---|

Porosity/stratum | Residual saturation/stratum | Exchange coef./stratum (s^{−}^{1}) | Coefficient of permeability/stratum (m^{2}) | Average porosity | Initial average saturation | Average permeability (m^{2}) | |

is higher in the stratum η least permeable. This situation explains certainly the difficulties of starting of dissolution due to the importance of heterogeneities of the medium.

・ Evolution of large scale relative permeabilities

We illustrate in _{rwω} and k_{rw} and the large scale relative permeability

・ Effects of heterogeneities on the large-scale effective properties

A fine analysis of the effects of heterogeneities of the medium on the large-scale effective properties is essential. Indeed, in the case being studied, NAPL mass flux are controlled at the same time by the differential distribution of the residual pollutant in the medium and by the heterogeneity.

The essential effective properties that we established in this study to model NAPL dissolution in heterogeneous systems are the large-scale mass transfer coefficient and the large-scale effective permeability (Cf. Equations and system of Equations (25), (37), (40), (51) and (52)). We examine in this paragraph, how the heterogeneities of the medium influence these parameters.

‒ Effects of heterogeneities on the large-scale relative permeabilities

the medium containing more NAPL which is more permeable with water.

‒ Effects of heterogeneities on the large-scale mass transfer coefficients

The transfer is more significant when the volume fraction of the stratum ω increases, this means increase of mass of pollutant according to the data of

As in the former studies [

‒ quantity of residual pollutant contained in poral space;

‒ installation of this pollutant in the porous medium, certainly marked by the influence of heterogeneities or hydrodynamic instabilities.

f_{ω} = 0.3 | f_{ω} = 0.5 | f_{ω} = 0.7 | |
---|---|---|---|

ε_{ω} | 0.32 | 0.32 | 0.32 |

ε_{η} | 0.4 | 0.4 | 0.4 |

S_{or}_{ω} =S_{or1} | 0.24 | 0.24 | 0.24 |

S_{or}_{η} =S_{or2} | 0.2 | 0.2 | 0.2 |

Total mass of NAPL in the unit cell: M_{NAPL}/unit volume | 0.0470 × ρ_{o}C_{oeq} | 0.0784 × ρ_{o}C_{oeq} | 0.1098 × ρ_{o}C_{oeq} |

In this paper we have proposed a large-scale model describing NAPL dissolution for stratified heterogeneous systems.

Two limit dissolution cases were studied: a local equilibrium case and a local non- equilibrium case. Both situations lead to a large-scale model of local non-equilibrium type. The construction of the large-scale effective parameters in the local equilibrium case is specific of flows parallel to a stratified system. While the results for the local non-equilibrium case is not specific to 2D or 3D systems, nor a given type of hetero- geneity, our application was also presented for flows parallel to a stratified system.

The proposed models have been validated by a comparison of the results of direct Darcy-scale simulations to homogenized simulations. Our results confirm the interest of using local non-equilibrium models for large-scale simulations of contaminated aquifers. In addition, the correlations for the obtained large-scale effective parameters, at least for the 2D stratified systems investigated here, are rather different from most of the correlations suggested by laboratory experiments at, therefore, a small scale. This should be taken into account when designing large-scale models for practical situations.

In perspective, it would be interesting to carry out 2D/3D calculations for more complex heterogeneous media such as nodular media. We believe the results for small Da numbers can be extended in a straightforward manner. However, it must be understood that our results for the case of large Da numbers, i.e., sharp fronts propagating through the system, cannot be in principle extended to more general heterogeneities.

Mabiala, B., Nsongo, T., Tomodiatounga, D.N., Tathy, C. and Nganga, D. (2017) Two-Dimensional Modeling of the NAPL Dissolution in Porous Media: Heterogeneities Effects on the Large Scale Permeabilities and Mass Exchange Co- efficient. Computational Water, Energy, and Environmental Engineering, 6, 56-78. http://dx.doi.org/10.4236/cweee.2017.61005

*refers to large-scale properties

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