Relation between fractional flow models and fractal or long-range 2-D permeability fields

International audienceFractional flow models introduced by Barker (1988) have been increasingly popular as means of interpreting nonclassical drawdown curves obtained from well tests. Fractional flow models are intrinsically isotropic scaling models depending to first order on two exponents n and dw expressing the dimension of the structure available to flow and the flow slowdown, respectively. We study the fractional flow induced either by geometrically scaling structures such as Sierpinski- and percolation-like fractal media or by hydraulically scaling media such as long-range continuous correlated media. First, percolation and Sierpinski structures have two well-separated dw values in the range [2.6, 3] and [1.9, 2.5], respectively. The bottlenecks, characteristic of percolation, induce a more anomalous transport (larger dw values) than the impervious zones present at all scales of Sierpinskis. Second, the realization-based values of n and dw depend both on global and on local characteristics like the fractal dimension and the permeability around the well, respectively. Finally, solving the inverse problem on anomalous transient well test responses consists in comparing the (n, dw) realization-based values with field data. Indeed, well tests performed from a unique pumping well must be taken as realization-based results. For the site of Ploemeur (Brittany, France), from which n and dw have been previously determined (Le Borgne et al., 2004), the only consistent model is given by the continuous multifractals. However, the values obtained from continuous multifractals cover most of the (n, dw) plane, and realization-based results are not selective in terms of model. So this should be replaced by the comparison of (n, dw) values averaged over different pumping well locations, which however requires a significantly larger quantity of field tests

On the validity of effective formulations for transport through heterogeneous porous media

International audienceGeological heterogeneity enhances spreading of solutes, and causes transport to be anomalous (i.e., non-Fickian), with much less mixing than suggested by dispersion. This implies that modeling transport requires adopting either stochastic approaches that model heterogeneity explicitly or effective transport formulations that acknowledge the effects of heterogeneity. A number of such formulations have been developed and tested as upscaled representations of enhanced spreading. However, their ability to represent mixing has not been formally tested, which is required for proper reproduction of chemical reactions and which motivates our work. We propose that, for an effective transport formulation to be considered a valid representation of transport through Heterogeneous Porous Media (HPM), it should honor mean advection, mixing and spreading. It should also be flexible enough to be applicable to real problems. We test the capacity of the Multi-Rate Mass Transfer (MRMT) to reproduce mixing observed in HPM, as represented by the classical multi-Gaussian log-permeability field with a Gaussian correlation pattern. Non-dispersive mixing comes from heterogeneity structures in the concentration fields that are not captured by macrodispersion. These fine structures limit mixing initially, but eventually enhance it. Numerical results show that, relative to HPM, MRMT models display a much stronger memory of initial conditions on mixing than on dispersion because of the sensitivity of the mixing state to the actual values of concentration. Because MRMT does not restitute the local concentration structures, it induces smaller non-dispersive mixing than HPM. However long-lived trapping in the immobile zones may sustain the deviation from dispersive mixing over much longer times. While spreading can be well captured by MRMT models, non-dispersive mixing cannot

Connectivity-consistent mapping method for 2-D discrete fracture networks

International audienceWe present a new flow computation method in 2-D discrete fracture networks (DFN) intermediary between the classical DFN flow simulation method and the projection onto continuous grids. The method divides the simulation complexity by solving for flows successively at a local mesh scale and at the global domain scale. At the local mesh scale, flows are determined by classical DFN flow simulations and approximated by an equivalent hydraulic matrix (EHM) relating heads and flow rates discretized on the mesh borders. Assembling the equivalent hydraulic matrices provides for a domain-scale discretization of the flow equation. The equivalent hydraulic matrices transfer the connectivity and flow structure complexities from the local mesh scale to the domain scale. Compared to existing geometrical mapping or equivalent tensor methods, the EHM method broadens the simulation range of flow to all types of 2-D fracture networks both below and above the representative elementary volume (REV). Additional computation linked to the derivation of the local mesh-scale equivalent hydraulic matrices increases the accuracy and reliability of the method. Compared to DFN methods, the EHM method first provides a simpler domain-scale alternative permeability model. Second, it enhances the simulation capacities to larger fracture networks where flow discretization on the DFN structure yields system sizes too large to be solved using the most advanced multigrid and multifrontal methods. We show that the EHM method continuously moves from the DFN method to the tensor representation as a function of the local mesh-scale discretization. The balance between accuracy and model simplification can be optimally controlled by adjusting the domain-scale and local mesh-scale discretizations

Reply to comment by A. Fiori et al. On ''Asymptotic dispersion in 2D heterogeneous porous media determined by parallel numerical simulations''

International audienceThe comment by A. Fiori, G. Dagan, and I. Jankoviæ[Fiori et al., 2008] compares numerical results [de Dreuzy etal., 2007] on the longitudinal asymptotic dispersion coefficientto a self-consistent solution [Fiori et al., 2003

Percolation parameter and percolation-threshold estimates for 3D random ellipses with widely-scattered distributions of eccentricity and size

In fractured materials of very low matrix permeability, fracture connectivity is the first-order determinant of the occurrence of flow. For systems having a narrow distribution of object sizes (short-range percolation), a first-order percolation criterion is given by the total excluded volume that is almost constant at threshold. In the case of fractured media, recent observations have demonstrated that the fracture-length distribution is extremely large. Because of this widely-scattered fracture-length distribution, the classical expression of the total excluded volume is no longer scale invariant at the percolation threshold and has no finite limit for infinitely large systems. Thus, the classical estimation method of the percolation threshold established in short-range percolation becomes useless for the connectivity determination of fractured media. In this study, we derive a new expression of the total excluded volume that remains scale invariant at the percolation threshold and that can thus be used as the proper control parameter, called parameter of percolation in percolation theory. We show that the scale-invariant expression of the total excluded volume is the geometrical union normalized by the system volume rather than the summation of the mutual excluded volumes normalized by the system volume

Statistical characteristics of flow as indicators of channeling in heterogeneous porous and fractured media

International audienceWe introduce two new channeling indicators Dic and Dcc based on the Lagrangian distribution of flow rates. On the basis of the participation ratio, these indicators characterize the extremes of both the flow-tube width distribution and the flow rate variation along flow lines. The participation ratio is an indicator biased toward the larger values of a distribution and is equal to the normalized ratio of the square of the first-order moment to the second-order moment. Compared with other existing indicators, they advantageously provide additional information on the flow channel geometry, are consistently applicable to both porous and fractured media, and are generally less variable for media generated using the same parameters than other indicators. Based on their computation for a broad range of porous and fracture permeability fields, we show that they consistently characterize two different geometric properties of channels. Dic gives a characteristic scale of low-flow zones in porous media and a characteristic distance between effectively flowing structures in fractured cases. Dcc gives a characteristic scale of the extension of high-flow zones in porous media and a characteristic channel length in fractured media. Dic is mostly determined by channel density and permeability variability. Dcc is, however, more affected by the nature of the correlation structure like the presence of permeability channels or fractures in porous media and the length distribution in fracture networks

Is the Dupuit assumption suitable for predicting the groundwater seepage area in hillslopes?

International audienceMany physically based hydrological/hydrogeological models used for predicting groundwater seepage areas, including topography-based index models such as TOPMODEL, rely on the Dupuit assumption. To ensure the sound use of these simplified models, knowledge of the conditions under which they provide a reasonable approximation is critical. In this study, a Dupuit solution for the seepage length in hillslope cross sections is tested against a full-depth solution of saturated groundwater flow. In homogeneous hillslopes with horizontal impervious base and constant-slope topography, the comparison reveals that the validity of the Dupuit solution depends not only on the ratio of depth to hillslope length d/L (as might be expected), but also on the ratio of hydraulic conductivity to recharge K/R and on the topographic slope s. The validity of the Dupuit solution is shown to be in fact a unique function of another ratio, the ratio of depth to seepage length d/LS. For d/LS0.2, it increases dramatically. In practice, this criterion can be used to test the validity of Dupuit solutions. When d/LS increases beyond that cutoff, the ratio of seepage length to hillslope length LS/L given by the full-depth solution tends toward a nonzero asymptotic value. This asymptotic value is shown to be controlled by (and in many cases equal to) the parameter R/(sK). Generalization of the findings to cases featuring heterogeneity, nonhorizontal impervious base and variable-slope topography is discussed

Des pollutions suivies à la trace par Jocelyne Erhel et Jean-Raynald de Dreuzy. Des déchets à vie longue, entretien avec Jérôme Jaffré, propos recueillis par Dominique Chouchan.

National audienceL'accès à l'eau potable est un privilège si l'on songe qu'un milliard d'humains en est encore dépourvu. Les recherches visent à produire les connaissances susceptibles d'aider à réduire la dégradation des nappes phréatiques

Impact of fractures on diffusion dominated reactive transport in porous media: application to the study of a radioactive waste storage

International audienceImpact of fractures on diffusion dominated reactive transport in porous media: application to the study of a radioactive waste storag

Influence of fracture scale heterogeneity on the flow properties of three-dimensional discrete fracture networks (DFN)

International audienceWhile permeability scaling of fractured media has been so far studied independently at the fracture- and network- scales, we propose a numerical analysis of the combined effect of fracture-scale heterogeneities and the network-scale topology. The analysis is based on 2 106 discrete fracture network (DFNs) simulations performed with highly robust numerical methods. Fracture local apertures are distributed according to a truncated Gaussian law, and exhibit self-affine spatial correlations up to a cutoff scale Lc. Network structures range widely over sparse and dense systems of short, long or widely distributed fracture sizes and display a large variety of fracture interconnections, flow bottlenecks and dead-ends. At the fracture scale, accounting for aperture heterogeneities leads to a reduction of the equivalent fracture transmissivity of up to a factor of 6 as compared to the parallel plate of identical mean aperture. At the network scale, a significant coupling is observed in most cases between flow heterogeneities at the fracture and at the network scale. The upscaling from the fracture to the network scale modifies the impact of fracture roughness on the measured permeability. This change can be quantified by the measure a2, which is analogous to the more classical power-averaging exponent used with heterogeneous porous media, and whose magnitude results from the competition of two effects: (i) the permeability is enhanced by the highly transmissive zones within the fractures that can bridge fracture intersections within a fracture plane; (ii) it is reduced by the closed and low transmissive areas that break up connectivity and flow paths. Citation: de Dreuzy, J.-R., Y. Méheust, and G. Pichot (2012), Influence of fracture scale heterogeneit