R Le Goc

J.-R De Dreuzy

P Davy

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

Keywords: flow channels, heterogeneous media, connectivity, fracture network, channeling indicators

We introduce two new channeling indicators D ic and D cc 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. D ic gives a characteristic scale of low-flow zones in porous media and a characteristic distance between effectively flowing structures in fractured cases. D cc gives a characteristic scale of the extension of high-flow zones in porous media and a characteristic channel length in fractured media. D ic is mostly determined by channel density and permeability variability. D cc 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.

Introduction

Spatial heterogeneity in hydraulic conductivity affects fluid flow and solute transport in complex natural media like fractured media [START_REF] Tsang | Flow Channeling in heterogeneous fractured rocks[END_REF], alluvial systems [START_REF] Fogg | Connected-network paradigm for the alluvial aquifer system[END_REF] and strongly heterogeneous porous media [START_REF] Moreno | Flow channeling in strongly heterogeneous porous media[END_REF] and has been a subject of research for decades ( [START_REF] De Marsily | Dealing with spatial heterogeneity[END_REF] and references therein). It is a function of contrasts between high permeability and low permeability values. As flow tends to avoid low-k zones for high-k zones, heterogeneity induces the development of preferential flow paths [START_REF] Knudby | On the relationship between indicators of geostatistical, flow and transport connectivity[END_REF][START_REF] Koltermann | Heterogeneity in sedimentary deposits: a review of structure imitating, process-imitating and descriptives approaches[END_REF] also called "paths of least resistance" [START_REF] Tsang | Flow channeling in a single fracture as a two dimensional strongly heterogeneous permeable medium[END_REF], along which flow is focused. Their effects on upscaled/effective hydrologic properties have been observed in laboratory and numerical studies. Fogg [START_REF] Fogg | Groundwater flow and sand body interconnectedness in a thick, multiple aquifer system[END_REF] performed a numerical study on the hydraulic conductivity distribution in the Wilcox aquifer and suggests that flow is mainly controlled by the continuity and connectivity of sand deposits rather than by local hydraulic conductivity values. Hanor [START_REF] Hanor | Effective hydraulic conductivity of fractured clay beds at a hazardous waste landfill; Louisiana Gulf Coast[END_REF] drew similar conclusions for the Livingston site. Silliman [START_REF] Silliman | An interpretation of the difference between aperture estimates derived from hydraulic and tracer tests in a single fracture[END_REF] illustrated the formation of preferential flow paths with laboratory experiments. [START_REF] Knudby | Binary upscaling -the role of connectivity and a new formula[END_REF][START_REF] Ronayne | Effective permeability of porous media containing branching channel networks[END_REF] showed how the estimate of aquifer properties, like the effective permeability of a system, should take channeling into account. Ronayne et al. [START_REF] Ronayne | Identifying discrete geologic structures that produce anomalous hydraulic response: an inverse modeling approach[END_REF] used statistical channeling properties to estimate aquifer parameters in a system affected by channeling. Similarly, Kerrou et al. [START_REF] Kerrou | Issues in characterizing heterogeneity and connectivity in non-multiGaussian media[END_REF] showed that not accounting explicitly for channeling in a sequential self-calibration approach resulted in flow underestimation and strong deviations in capture zone estimates. Trinchero et al. [START_REF] Trinchero | Point-to-point connectivity, an abstract concept or a key issue for risk assessment studies?[END_REF] showed that for moderate heterogeneities, both the connectivity of high-k values and apparent porosity are key in predicting transport times efficiently. Although channeling is important for flow and transport properties, its quantification remains a matter of debate. Two types of indicators have been proposed: indicators derived from the comparison of upscaled hydraulic properties with their small-scale counterparts, and statistical indicators calculated from the permeability and flow fields. The first category of estimators is based on hydraulic properties that are sensitive to channeling. The simplest estimator is the effective permeability, K eff , known to be sensitive to flow organization [START_REF] Guswa | On using the equivalent conductivity to characterize solute spreading in environments with low-permeability lenses[END_REF]. In 2D multi-log-Gaussian isotropic weakly-correlated fields, the equivalent permeability is equal to the geometric mean K eff =K g [START_REF] Matheron | Elements pour une théorie des milieux poreux[END_REF]. If the connectivity of the higher-K zones is greater than that of the lower-K zones, K eff is larger than K g [START_REF] Sánchez-Vila | Scale effects in transmissivity[END_REF] within the limit that ≤ (where K a is the arithmetic mean) [START_REF] Warren | Flow in heterogeneous porous media[END_REF]. The type of average measured by the power averaging exponent CF 1 [START_REF] Desbarats | Spatial Averaging of Hydraulic Conductivity in Three-Dimensional Heterogeneous Porous Media[END_REF][START_REF] Journel | Power Averaging for Block Effective Permeability, SPE California Regional Meeting[END_REF] has thus been considered as a measure of channeling [START_REF] Knudby | On the relationship between indicators of geostatistical, flow and transport connectivity[END_REF]:

∶ = 1 (1)
CF 1 varies between -1 and 1 for the harmonic and arithmetic means, respectively, and is equal to zero for the geometric mean corresponding to isotropic weakly-correlated multi-Gaussian fields. As transport is also strongly affected by channeling, breakthrough curve properties have been proposed as estimators of the channeling degree [START_REF] Wen | Numerical modeling of macrodispersion in heterogeneous media -a comparison of multi-Gaussian and non-multi-Gaussian models[END_REF]. Knudby and Carrera [START_REF] Knudby | On the relationship between indicators of geostatistical, flow and transport connectivity[END_REF] used the ratio CT 1 of the average arrival time to the time at which 5% of the solute have broken through the domain boundary t 5 :

= . 2

When preferential flow paths exist, t 5 becomes much smaller than , CT 1 increases and the field should be considered as increasingly connected. The apparent hydraulic diffusivity has been proposed as an intermediary characteristics between flow and transport connectivities [START_REF] Knudby | On the use of apparent hydraulic diffusivity as an indicator of connectivity[END_REF]. Park et al. [START_REF] Park | A study of preferential flow in heterogeneous media using random-walk particle tracking[END_REF] suggested that the normalized travel time and distance be used to investigate preferential flow.

The second category of estimators uses statistical characteristics of the permeability field or of the flow field. N-point spatial connectivity statistics are dedicated to the measurement of connectivity and were applied to permeability fields to estimate the presence of high-k connected patterns [START_REF] Journel | Focusing on spatial connectivity of extreme-valued attributes : stochastic indicator models of reservoir heterogeneities[END_REF][START_REF] Krishnan | Spatial connectivity : from variograms to multiple-point measures[END_REF]. Western et al. [START_REF] Western | Toward capturing hydrologically significant connectivity in spatial patterns[END_REF] used a directional multi-point geostatistical indicator and showed that it could capture the difference between random and channeled fields with similar k-distributions, unlike non-directional indicators. Frippiat et al. [START_REF] Frippiat | Anisotropic effective medium solutions of head and velocity variance to quantify flow connectivity[END_REF] suggested that the presence of preferential flow paths or flow barriers could be identified using head and flow variances, since head variance is negatively correlated to connectivity while flow variance is positively correlated to the effective permeability increase. Bruderer-Weng et al. [START_REF] Bruderer-Weng | Relating flow channelling to tracer dispersion in heterogeneous networks[END_REF] used the multifractal spectrum of the flow field to quantify channeling in heterogeneous pipe networks. The distribution of flow has also been used for quantifying channeling in fractured networks [START_REF] De Dreuzy | Hydraulic properties of two-dimensional random fracture networks following a power law length distribution 1. Effective connectivity[END_REF].

The multiplicity of the proposed indicators shows that channeling cannot be restricted to a single simple characteristic. The concept of channeling also strongly depends on the application targeted. The relevant use of channeling indicators probably differ between flow and transport applications [START_REF] Scheibe | Scaling of flow and transport behavior in heterogeneous groundwater systems[END_REF]. In this study, we focused first on the geometrical characterization of channels, i.e. on the channels themselves rather than on their consequences in terms of flow or transport. In this respect, the first category of indicators based on equivalent medium properties are limited by the fact that they measure the consequences of channeling rather than channeling itself. The limitation of the indicators based on permeability statistics arises from the measurement of a single cause of channeling cause (the connectivity of high-K zones) where channeling is also induced by the variability of permeability [START_REF] Le Borgne | Characterization of the velocity field organization in heterogeneous media by conditional correlation[END_REF]. The advantage of those indicators based on the statistical properties of the flow field is the measurement of channeling itself. As opposed to the multifractal dimensions and the variance of head or flow, we look for indicators based on the geometrical properties of the channels that additionally identify channeling consistently in both porous and fractured media.

Even though channeling occurs under many different circumstances, it has two recurrent characteristics. First, flow is localized within a few structures. Second, channeling locally maintains high flow rates over long distances. On the basis of these two characteristics, we aimed at defining quantitative channeling indicators that met the three following constraints.

First, they must be globally consistent with the visually intuitive classification of channeling.

Second, they must provide a quantification of channeling. Third, they must be applicable simultaneously to porous and fractured media.

We define two new indicators in section 2. We compute their value for the broad range of synthetic fields introduced in section 3. In section 4, we analyze first their consistency with the expected ranking of channeling and then their dependency on the permeability correlation structures. Finally, we compare them to other existing indicators in section 5.

Flow-based indicators

A channeled medium is defined as a medium where flow is localized within a few structures and where preferential flow locally maintains high flow rates over long distances. To this end, we defined two channeling indicators, one quantifying the localization of flow within the system and the other quantifying the continuity of flow paths. Since the proposed indicators were not straightforward, we introduce them using preliminary attempts based on simpler quantities. The objective was to show the relevance of the more complex indicators finally adopted. The first indicator should characterize the relative volume occupied by the high-flow zones. The simplest indicator could be the relative volume occupied by flows larger than a given threshold value. Although simple, this indicator depends on the arbitrary choice of the threshold value. Rather than a deterministic indicator, we looked for a statistical characteristic biased toward the higher values of the flow distribution. Since the moments of the flow distribution M k (Φ) are increasingly sensitive to the highest values with increasing orders k, the idea was to compare moments of increasing orders like in the participation ratio S 2 [START_REF] Davy | Localization and fault growth in layered brittle-ductible systems: Implications for deformations of the continental lithosphere[END_REF][START_REF] Sornette | Fault Growth in Brittle-Ductile Experiments and the Mechanics of Continental Collisions[END_REF] equal to :

S 2 (Φ) = M 1 (Φ) 2 /(M 0 (Φ)⋅M 2 (Φ)) (3)
where Φ stands for the spatial distribution of flow rates. For Φ discretized on a domain of n cells of volumes V i , M k (Φ) writes:

" # $ = % & ' # × ' ) '* (4)
where ϕ i is the mean value of Φ over the grid cell i. Table 1 shows S 2 values for usual distributions. When the distribution variability vanishes, S 2 tends to 1. By contrast, S 2 systematically decreases with higher variability whatever the distribution type. For the lognormal distribution, S 2 is solely function of the lognormal variance.

Whereas S 2 (Φ) gives indications about the surface occupied by the largest flow rates, it does not account for the distribution of this surface within the domain. Consequently, we did not use S 2 (Φ) but S 2 (W n ), where W n is the distribution of flow-tube widths carrying all the same fraction 1/n of the total flow. Since S 2 (W n ) is biased toward the larger W n values , it characterizes the extension of the low-flow zones and hence the distance between main flow channels. We defined and computed W n in the specific context of permeameter-like boundary conditions defined for a square domain by fixed heads on two opposite sides and no flow on the other sides. The definition of W n may also be adapted for different boundary conditions. In convergent flow conditions, W n would be defined by the distance between flow lines normalized by the distance to the well. With permeameter-like boundary conditions, we first determined n equivalent flow tubes defined as the tubes carrying all the same fraction 1/n of the total flow (Figure 1, middle column). Then, we computed the participation ratio S 2 (W n ) on the flow-tube width distribution.

We introduce the meaning of S 2 (W n ) with the case of p regularly spaced flow tubes of width L/p, where L is the system size, and n-p flow tubes of negligible width within the channels.

+ ,- . = 1 / • 1 2 3 4 (5)
D ic ranges from L/n in homogeneous fields to L in a unique channel conveying all the flow, for which S 2 (W n )=1/n. L/n can be a priori interpreted as a channel resolution. The selection of an appropriate value for n will be investigated at the beginning of section 4. D ic is a characteristic distance between channels. However, it does not provide any information on the channel persistence throughout the system. For example, in Figure 1 the field at the top and the field in the middle have two different flow fields with about the same D ic /L ratio equal to 0.09, but high flow rates are visually maintained over a longer distance in the middle field than in the top field.

We looked for a second indicator designed to differentiate these two fields by characterizing the distance over which flow rates are continuously high. We first tried the Lagrangian correlation length of flow rates. It was however not consistent with our intuition of channel persistency. For example, the correlation length of the rearranged field in Figure 1 (middle) is smaller than the correlation length of the non-rearranged field (Figure 1, top). The correlation length is not only sensitive to the large flow rates but also to all other values. It thus fails to characterize high-flow zone connectivity. Like for the previous indicator, computing the Lagrangian correlation length from the sole velocities larger than a given threshold faces the same problem of the arbitrary choice of the threshold. Moreover, the channels may display some discontinuities that hinder the relevance of a threshold (Figure 1, right bottom). We found that characterizing flow channel discontinuities is easier than flow channel persistence since discontinuities are more localized. Large values of the spatial derivatives of flow rates are more localized at the entrance and exit of channels than in the remaining of the field. On the contrary, the variations of flow rates are smaller and distributed evenly in non-channeled media. To characterize the distribution of the flow transitions taken as the Lagrangian derivatives of flow rates Φ ', we used again the participation ratio S 2 on Φ ' (right column in Figure 1). Numerically, S 2 (Φ ') was calculated according to (3) from the moments of the distribution of Φ ' discretized along the flow lines:

" 5 ′ = % % 7 , 8 9 & , 8 7 , 8 9 5 :
,* ; 8* [START_REF] De Dreuzy | Hydraulic properties of two-dimensional random fracture networks following a power law length distribution 1. Effective connectivity[END_REF] with j the flow line index, p the number of flow lines, 7 , 8 the i th position along the flow line j, m the number of positions along the flow line, & , 8 and ∆s i j the flow rate variation and distance between two consecutive points. Flow lines were computed by using a particletracking algorithm. We chose p equal to 10 4 and m so that ∆s i j was of the order of the grid cell size after ensuring that larger p and m values did not modify the results. S 2 (Φ ') defines a characteristic distance l eff over which flow rates are actually variable divided by the average flow line length (L') [START_REF] Sornette | Fault Growth in Brittle-Ductile Experiments and the Mechanics of Continental Collisions[END_REF]. By contrast, a characteristic scale D cc over which flow rates are only slowly varying is function of L'-l eff :

D cc /L'=(L'-l eff )/L'= (1-S 2 (Φ ')) (7)
D cc will be taken as a characteristic channel length. In media with discontinuous flow paths, Φ ' has a narrow and spatially uniform distribution, leading to S 2 (Φ ')=1 and D cc /L' = 0. In highly-channeled media, the distribution Φ ' contains values close to zero except at the channel extremities, leading to small S 2 (Φ ') values and D cc /L' close to 1. For example, the D cc /L' value for the middle field in Figure 1 is equal to 0.72. It is larger than the value of 0.43

for the field at the top of Figure 1 following the intuition that persistence is larger in the middle field.

D ic /L and D cc /L' are statistically-derived indicators designed to characterize flow localization and flow continuity. They are dimensionless quantities ranging between 0 and 1 that can be used to compare channeling in different systems. In section 3, we define a broad range of synthetic porous fields and fracture networks in which D ic and D cc will be computed in section 4.

Tested media and computational methods

Indicators D ic and D cc will be compared in the different synthetic fields displaying various connectivity degrees presented in this section.

Field generation and flow computational method

Simulations were performed in four steps, consisting in the generation of the tested fields, simulation of flow, derivation of the flow lines and computation of the different indicators.

The generation of the multi-Gaussian porous fields was performed via a Fourier transform [START_REF] Gutjahr | Fast Fourier transforms for random field generation[END_REF] using the software FFTW [START_REF] De Dreuzy | Asymptotic dispersion in 2D heterogeneous porous media determined by parallel numerical simulations[END_REF][START_REF] Frigo | The design and implementation of FFTW3[END_REF]. Some fields are then rearranged according to the rearrangement methods described in [START_REF] Zinn | When good statistical models of aquifer heterogeneity go bad: a comparison of flow, dispertion and mass transfer in connected and multivariate Gaussian hydraulic conductivity fields[END_REF] and [START_REF] Knudby | On the relationship between indicators of geostatistical, flow and transport connectivity[END_REF] if required. In order to avoid side effects, particularly with large correlation lengths, all fields were generated within a 1280x1280 grid of which the central part (128x128) was kept for the analysis, so L=128 was the characteristic system size. The original field was taken with a log-k mean equal to zero, a variance (σ y 2 )

equal to 1 and 3 and a correlation length (λ) equal to 8 and 64. Fracture networks were generated within a system size equal to 10 * l min , where l min is the size of the smallest fracture.

In porous media, the flow equation was discretized on the structure of the medium according to a finite volume framework with harmonic inter-cell permeabilities [START_REF] Kruel-Romeu | Calculation of internodal transmissivities in finite difference models of flow in heterogeneous porous media[END_REF]. As previously mentioned, permeameter-like boundary conditions were imposed on the sides of the domain, i.e. fixed heads on two opposite borders and no flow on the others. The discretized flow equations ended up to a linear system A.x=b solved by the multifrontal method implemented in the software UMFPACK [START_REF] Davis | Algorithm 832: UMFPACK, an unsymmetric-pattern multifrontal method[END_REF]. Flow lines were constructed using a particle-tracking algorithm. Particles were injected through a vertical segment positioned in a central part at a distance of one correlation length from the inlet and proportionally to flow in order to avoid boundary effects [START_REF] Beaudoin | An Efficient Parallel Particle Tracker for Advection-Diffusion Simulations in Heterogeneous Porous Media in[END_REF][START_REF] Tompson | Numerical-simulation of solute transport in 3dimensional, randomly heterogeneous porous-media[END_REF]. Indicators were computed from 500 Monte-Carlo realizations for each tested case.

Description of the test cases

We used a broad range of 2D synthetic porous and fractured media characterized by the histogram of their permeability distribution and their connectivity structures. The synthetic porous media have all a lognormal permeability distribution of variance σ y 2 , where y stands for log(k) and k is the permeability, but differ by their correlation structure (see Figure 2). For the same correlation length λ, we used six correlation patterns. The tested structures are identified by P as in porous and one or two other letters specifying the correlation pattern. The first two fields have Gaussian and exponential correlation structures (PG and PE in Figure 2).

The two next ones result from the rearrangement methods by Zinn and Harvey [START_REF] Zinn | When good statistical models of aquifer heterogeneity go bad: a comparison of flow, dispertion and mass transfer in connected and multivariate Gaussian hydraulic conductivity fields[END_REF], yielding fields of highly-connected high or low permeabilities (PC+ and PC-on Figure 2). The two last ones are Gaussian-correlated fields to which are added highly permeable fracture-like structures oriented parallel to the average head gradient and spanning either half of the system (PF on Figure 2) or the whole system (PF2 on Figure 2) [START_REF] Knudby | On the relationship between indicators of geostatistical, flow and transport connectivity[END_REF]. These rearrangement methods provide different flow distributions (Figure 3). The C+ method increases the mean flow rate compared to the Gaussian correlated field, while the F method adds a second peak of larger flow rates to the histogram. Note, however, that the rearrangement methods do not modify the permeability histogram.

Flow channeling was also observed in fractured media because of both the fractures and the network-scale heterogeneities [START_REF] Tsang | Flow Channeling in heterogeneous fractured rocks[END_REF]. In this paper, we concentrated on the network-scale complexity stemming at first from the power-law fracture length distribution:

n(l)~l -a ( 8
)
where a is a characteristic exponent between 1.5 and 4 [START_REF] Bonnet | Scaling of fracture systems in geological media[END_REF]. We chose five types of fracture networks differing by their fracture length and transmissivity distributions and by their density. They are identified by the letter F followed by three additional letters. The first one, FTL0, corresponds to fracture networks at percolation threshold (structures just connected) with a power-law length distribution yielding to large fractures corresponding to a equal to 2.0 (Figure 4). The four other networks show a smaller probability of occurrence of large fractures (a=3.5) and are respectively at threshold (FTS0, Figure 4) and dense with a density three times as large as that of threshold. The dense fracture networks differ by their fracture transmissivity distributions of lognormal standard deviation σ y equal to 0 in FDS0, 1 in FDS1 and 2 in FDS2. Flow fields displayed in Figure 4 (second column) show different flow structures from highly-channeled (FTL0, FTS0 and FDS2) to well-distributed (FDS0).

Results

After a visual inspection of the different test cases (Figure 2 and Figure 4), we ranked them by their apparent channeling degree (Table 2). The order was derived separately in the porous and fracture cases according to the flow-tube widths and regularity. In porous media, this order is consistent with CF 1 values (Table 3). All results discussed in the following paragraph are given in Table 3.

Relation between D ic and the number of considered flow tubes

The interchannel distance D ic (5) depends on the proportion of flow used to define a channel.

If n is the number of flow tubes, each flow tube carries 1/n part of the total flow. Figure 5 displays the relation between D ic and 1/n. When 1/n tends to 1, D ic tends to L, meaning that no channel contains all the flow by itself. For the smallest values of 1/n, the fracture cases reach a plateau characteristic of the smallest distance between flowing fractures. 1/n can be interpreted as characteristic of the smallest flow channel that can be identified. In Figure 5, we chose a value of n=20, for which all test cases have a characteristic interchannel distance larger than the interfracture distance. The value of D ic remains dependant of n but the relative order for the different test cases remains the same whenever n≤20. We will thus compare D ic values between test cases rather than their absolute values in single test cases. The chosen value of n is the flow-tube resolution. If D ic <L/n, the medium will be considered as homogeneous in the sense that inter-channel distances are smaller than L/n. The value of n should be increased for distinguishing closer channels. or PF2 with σ y 2 =3. The latter case corresponds to extreme channeling for which flow is both highly localized and highly continuous in a very small number of channels.

Channeling characteristics D ic and D cc

Relation between channeling characteristics and k-field parameters

In this section, we look for a finer understanding of indicators D ic /L and D cc /L' by analyzing their dependence on the structures of the porous (Figure 89) and fractured test cases (Figure 101112). We then comment on the variation trends and amplitudes. D ic /L systematically increases with more heterogeneity. In fact, D ic /L increases with σ 2 in porous media (Figure 8)

and with σ y 2 in fracture networks (Figure 12). Larger σ 2 values imply that flows focus in sparser transmissivity zones. D ic /L also systematically decreases in denser fracture networks, i.e. when increasing the number of connected parallel fracture paths (Figure 11). Increasing the probability of occurrence of long fractures with smaller a values yields similar causes and effects (Figure 10). Similarly, increasing the correlation length λ in porous media from small to median values induces more channeling. The sole non-obvious D ic /L variation is its decrease from intermediary to large correlation lengths, approximately from λ~L/8 to λ=L/2 (Figure 9). This may be due to two reasons. First, the standard deviation of D ic /L steeply increases with λ to the point where its variations become smaller than its variability. Second, when the large correlation length λ is comparable to the system size L, for example for λ=L/2, the system becomes more homogeneous and channels are more regularly distributed, explaining the smaller value of D ic /L. Apart from these side effects, D ic /L is first determined by the density of potential channels given by the correlation length in porous media and by fracture density and length distributions. Among the potential channels, only those made up of the higher permeabilities lead to effective channels. This is confirmed by the variation ranges of D ic /L presented in Table 4. The largest variation ranges are due to the fracture density first and to the difference of D ic /L values between porous and fracture cases (Figure 6). They cover at least two thirds of the full variation of D ic /L. The variation ranges according to σ y 2 are smaller but not negligible and account on average for less than half of the full variation range of porous and fracture cases.

D cc /L' is less variable than D ic /L. It varies significantly only in PF as a function of σ 2 (Figure 8), in FDS as a function of σ y 2 (Figure 12), in PG as a function of λ (Figure 9) and in FD0 as a function of a (Figure 10). We argue in the following that the sole genuine variation is the last D cc /L' decreases with the correlation length. This counter-intuitive result is only apparent and mostly due to the simultaneous variations of L and λ in our simulation settings. In fact, we have found that D cc /L' depends more on L/λ than on λ. The range of variations of D cc /L' is reduced from 0.25 to 0.09 when decreasing the range of variations of L/λ from 64 to 8.

Disregarding these dummy variations, the sole genuine variation that is not a transition is the increase in D cc /L' with smaller a values for dense networks (FD0). What fundamentally changes in the latter case is the nature of the correlations. D cc /L' seems to be more affected by the nature of the correlation than by the variability of permeability. Aside from the large variation ranges due to a transition from porous-like to fracture-like media marked in grey in Table 4, it is the sole case where the variation range of D cc /L' is significant. More precisely, it is of the order of three quarters of the full range of variations in all fracture networks, where all other cases are restricted to one quarter. The channel continuity measured by D cc /L' is thus much more influenced by the nature of the correlation structure than by the other parameters including the permeability variability, the fracture density and the correlation length. D cc /L' can be considered as an indicator of the nature of correlation. Finally, the absence of correlation between variability and D cc /L' expresses that there is a fundamental limit in channeling related to the local permeability structure rather than to the permeability variability.

Discussion

As concluded in the previous section, the D cc /L' ratio depends much more on the nature of the correlation than on the other parameters. The continuity of channels is at first a function of the occurrence of underlying-connected permeability structures. D ic /L is more intuitively a function of the density of paths and of the selection of the highest permeability paths.

Although D ic /L and D cc /L' have been defined as statistical characteristics, they are still rough estimates of the geometrical characteristics of the fields as shown in Figure 2. We recall that D ic has been computed with a separation of flow tubes into 20 parts and thus measures statistical properties of structures carrying at least 1/20 of the total flow. D ic is only slightly variable for porous media and gives a characteristic scale of the low-flow zones (blue patches). In simple multiGaussian cases (PG), D ic is close to the correlation length of the velocity field. D ic is much more variable in the fracture networks where it is not too far from the distance between effectively flowing structures. On the other hand, D cc is more variable in the porous media than in the fracture networks. In the porous media, it gives a characteristic scale of the extension of the high-flow zones (red flow tubes in Figure 2) and in fractured media, it gives a characteristic length of flow channels.

As displayed in Figure 7, D cc /L' and D ic /L are not strongly correlated. Their dependences on the different model parameters (Figure 8 to Figure 12) explain the lack of strong correlation.

Consequently, D ic and D cc actually do measure two different channeling characteristics that are only weakly interdependent. In other words, they complementarily characterize channeling.

Figure 13 displays the relation between the new indicators D ic /L and D cc /L' and the existing indicators CF 1 and CT 1 [START_REF] Knudby | On the relationship between indicators of geostatistical, flow and transport connectivity[END_REF]. CT 1 does not systematically identify the sparse fracture cases (FTS0 and FTL0). Furthermore, CT 1 values within a single case are highly variable in channeled media (Figure 14). that the correlation structures are better distinguished. For example, PC+ and PF have very close CF 1 values (Figure 6) but differ by their D cc /L' values. This difference stems from the more contrasted channels in the PF configuration relative to the PC+ configuration, despite the similar channeling intensity. However, it must be noted that CF 1 better distinguish PF from PF2 than D ic /L and D cc /L. The second advantage of D cc /L' over CF 1 is that it provides information on the flow structure in both porous and fracture cases D ic /L and D cc /L' could also be readily computed in anisotropic and 3D fields. We expect both anisotropy and 3D to increase D cc /L' without significantly modifying D ic /L. D ic /L will remain linked to the volume of the low-flow zones, the characteristic size of which will not be strongly modified. However, we expect anisotropy to increase D cc /L' just by the effect of the higher velocity correlations in the flow direction. 3D could also potentially provide longer and more tortuous channels around the low-flow zones, and hence increase D cc /L'.

The derivation of D ic /L and D cc /L' in natural cases is more difficult because of the lack of data that would lead to their direct estimate. They could however be inferred from the geometrical and hydraulic characteristics of the permeability field either with the results of this study or with closer simulations. It would be interesting, in field cases, to condition the estimation of these indexes on permeability and flow values and thus to lower the non-negligible variability displayed in Figure 14.

Summary and Conclusion

Channeling has been observed both in field and in synthetic contexts. However, its defined in [START_REF] Knudby | On the relationship between indicators of geostatistical, flow and transport connectivity[END_REF]. Note that D ic is available only in porous cases.

Figure 6a and

 6a Figure 6a and Figure 6b display the two new indicators D ic /L and D cc /L' in highlyheterogeneous porous cases (σ y 2 =3) and fracture cases. PG and PE configurations have very close values of D ic and D cc despite the visual ranking, meaning that the short-range

  one. The two first variations are due to the transition from porous to fractured cases. Beyond the transition (σ y 2 >1), D cc /L' is almost constant. In porous-fracture fields, small σ 2 values correspond to almost pure porous cases without fractures whereas high σ 2 values lead to fracture-like cases. In dense fracture networks with σ y 2 =0, the fracture network looks like a porous medium. Increasing σ y 2 triggers channeling while keeping D cc /L' almost constant.

  CT 1 is thus not a good channeling indicator. The correlation of CF 1 with D cc /L' is apparently better than with D ic /L. D cc /L' < 0.5 corresponds to weakly negative CF 1 values, indicating flow fields slightly more influenced by low-permeability zones. D cc /L' > 0.5 corresponds to positive CF 1 values indicating flow fields more influenced by the high-permeability zones making up the channels. The advantage of D cc /L' over CF 1 is

Figure 1 :Figure 2 :

 12 Figure 1: Permeability fields

Figure 3 :FDS2Figure 4 :Figure 5 :Figure 6 :Figure 7 :

 34567 Figure 3: Probability density function of the logarithm of flow rates in the classical multi-Gaussian fields with a Gaussian correlation (PG) and in the fields with the two rearrangement methods PC+ and PF. The C+ rearrangement method globally shifts the flow rate distribution to higher flow values while the F rearrangement introduces a second peak of higher flow values.

Figure 8 :

 8 Figure 8: Variation of D cc /L' versus D ic /L in porous test cases PG and PF for varying permeability standard deviations σ y . σ y values are given next to the corresponding symbols.

Figure 9 :

 9 Figure 9: Variation of D cc /L permeability correlation length L' versus D ic /L in porous test cases PG an gths λ. λ values are given next to the correspon

Figure 10 :

 10 Figure 10: D cc /L' versus D ic /L in fractured test cases with varying power-law length exponents a. Values of a are given next to the corresponding symbols.

Figure 11 :Figure 12 :Figure 13 :

 111213 Figure 11: D cc /L' versus D ic /L in fractured test cases with varying fracture densities d. Values of d are given next to the corresponding symbols. Density is measured as the percentage of fractures above percolation threshold. It is 0 at percolation threshold and 100 in networks having twice as much fractures as in networks at percolation threshold.

Table 3 :

 3 Mean and variance on 500 realizations for the different indicators and test cases. N/A stands for indicators that cannot be computed in 644

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Table 4 :

 4 Variation range (maximal minus minimal values) of D ic /L and D cc /L' according to porous and fracture parameters. High values in grey are due to transitions from porous-like to fracture-like structures rather than to variations with σ y 2 .

Acknowledgements

The French National Research Agency ANR is acknowledged for its financial founding through the MOHINI project (ANR-07-VULN-008) and for its contribution to the development of numerical methods through the MICAS project (ANR-07-CIS7-004). The authors thank two anonymous reviewers for their highly detailed and constructive reviews.


Space in Language and Cognition : Explorations in

), 'is a major ethnocentric error. ' Instead, SILC (22) proposes to show that 'there is diversity in cognition just as there is diversity in language, clothing, hairstyle, kinship practices, modes of subsistence, ritual and religion'. The second chapter presents different frames of reference used by humans while reviewing the concepts used in variously combined oppositions to describe these frames of reference in different fields (e.g. 'egocentric' vs. 'allocentric' or 'viewer-centred' vs. 'object-centred' vs. 'environment-centred') as well as the linguistic expression of these frames of reference in a cross-linguistic perspective. In that respect, Levinson (34) remarks that 'the analysis of spatial terms in familiar European languages remains deeply confused, and those in other languages almost entirely unexplored', and he goes on to propose a ternary typology, 'intrinsic',

'relative' and 'absolute', each of these categories comprising several related yet semantically distinct subcategories. This typology is based on the work of the Language and Cognition Group at the Max Planck Institute for Psycholinguistics in Nijmegen (headed by S. Levinson) on (xix) 'over forty, mostly unwritten languages spoken in small-scale, traditional societies.' The three linguistic frames of reference are then distinguished according to a number of parameters, and aligned according to the concepts previously reviewed, which leads Levinson to take up Molyneux's question to Newton (if a blind man, who knew by touch the difference between a cube and a sphere, had his sight restored, would he recognize the selfsame objects under his new perceptual modality or not?). The conclusion offered in SILC is that if frames of reference are not entirely translatable from one to the other, then 'representational systems of different kinds, specialized to different sensory modalities (like visual memory) or output systems (like gesture and language), may be capable of adopting different frames of reference' (60); i.e. the answer is 'yes'. Chapter three deals with linguistic diversity in a survey that is not exhaustive (such an exhaustive survey is presented, along with its methodology, in a companion book, Grammars of Space, Levinson & Wilkins 2006) but offers enough evidence of radical differences in this respect to show, in the author's opinion (111), 'that the assumption of uniformity [in spatial semantics] is entirely mistaken'. This is illustrated, in particular, by the fact that some frames of reference are prevalent in certain languages and not in others; but frames of reference are here studied in a wider approach of spatial semantics, along with location systems in which no frame of reference is used, placenames (where the figure is located at a named place/ground), deixis (where the figure is located relative to the ground) and topology (where the figure is located as contiguous to the ground), and motion. Consideration is given to the respective advantages and disadvantages of the different systems. The most simple system, naming the location, is very well for underpopulated areas, but would be cognitively too costly elsewhere. In most cases, deixis crucially lacks angular specifications, so that (70) 'deictics are often (and often obligatorily) accompanied by gesture.' Topology also involves a loss of information regarding angular and distance specifications, but may come first in language acquisition. It has also been shown that in many languages, some prepositions and adverbs used in frames of reference originate in topological notions, as in the English 'back', first designating a body part, then a part of an object, then the space contiguous to that part, and eventually the space behind it (106). As for spatial systems involving frames of reference, the greater complexity of (ternary) relative systems compared to (binary) intrinsic ones raises the question of the reason why languages would develop the former. The answer offered in SILC is threefold: First, not all objects offer distinguishable facets suitable for intrinsic orientation; secondly, relative systems allow logical inferences such as 'if A is to the left of B and B to the left of C, then A is to the left of C'; thirdly, these systems correspond directly to visual experience. And finally, an absolute frame of reference is prevalent in some languages and even, in the case of the Australian language Guugu Yimithirr, the only frame of reference available. This system also allows complex logical inferences and (91) 'is by far the most elegant solution to the problem of angular descriptions on the horizontal'. There are, in Levinson's words, 'only two catches' in the absolute system; the first is that it does not capture egocentric constancies (on which the directionality of our writing systems rely, among other things), and the second is that speakers have to keep constant track of their bearings. Chapter four gives us some insight into the 'cognitive style' of two so-called 'absolute communities', speakers of Guugu Yimithirr (an Australian language, henceforth GY) and of Tzeltal (a Mayan language) and investigates the hypothesis that non-verbal cognition might evidence characteristics similar to those of the languages spoken in these communities. In order to study non-verbal cognition (mostly spatial memory), special tests have been devised, which are presented in some detail in this chapter. GY is a language almost completely absolute in spatial description, with the exception of (120) 'a few body-part terms which have slightly extended spatial uses, for example walu (temple), baaru (front), buthiil (nose), all used to indicate the front of a man or animal, usually in conjunction with an orientational specification by cardinal direction.' GY makes essential use of cardinal directions, which actually refer to quadrants (or edges). This is somewhat misleading, because what in SILC is glossed as 'north' seems to really mean 'somewhere approximately between north-west and north-east'. The orientational terms appear in various derivations to constitute a remarkably detailed absolute system. Also remarkable is the absence of relative terms. There is, for example, no spatial term for 'right' or 'left'. Not surprisingly, GY speakers seem to keep constant track of their bearings and are impressively able to point immediately to known locations in various circumstances (127). Levinson has tested the coding of visual perception in GY speakers using, in particular, what he calls the 'rotation paradigm', a test he designed himself in 1992: if, facing north, a subject is asked to remember the picture of an arrow pointing to the right, hence to the east, and the subject is then made to turn round 180° to face a similar table, a 'relative coder' will show an arrow pointing right, whereas an 'absolute coder' will show an arrow pointing left (and still pointing east). The results of several experiments seem to show a majority of 'absolute coders' in that community. Tzeltal possesses both intrinsic terms and absolute terms, but the only absolute orientation is on the north-south axis. East and west are both glossed as 'across'. Members of the Tenejapan community, who speak Tzeltal, also tend to show a surprising incapacity at distinguishing between images which are symmetrical on a vertical axis, and this, it seems, independently of cardinal orientation. And a majority of the Tzeltal speakers tested seem to be 'absolute coders'. From these results, SILC concludes that there is the possibility of a 'Whorfian effect' in the relation between language and cognition. Chapter five of SILC is concerned with the hypothesis of spatial language influencing spatial non-linguistic cognition, but presents the results of a study investigating another (171) 'more easily directly tested, and crucially a weaker correlational hypothesis', namely that the frames of reference used in a language to describe specific scenes are likely to correlate with the nonlinguistic coding of the same scenes. This chapter starts with a description of the methods used to test the correlation hypothesis in field conditions on small-scale, traditional communities. An important consideration is given to the way of assessing test results that do not fit into the expected relative/absolute dichotomy. In SILC, the data are analysed along a 'relative-to-absolute' gradient (Levinson suggests the hypothesis of a competition between possible frames of reference for each informant and in each trial, p.178). These data seem to confirm the correlation hypothesis. But the study also serves to eliminate different factors on spatial coding, such as ecological conditions (communities living in similar conditions may use different frames of reference), gender, age and literacy (although literacy does seem to play some role in the results). Competing factors having been more or less brushed aside, Levinson is able to conclude that (214) 'linguistic determinism seems the most likely explanation for the correlation.' Chapter six investigates the cognition of navigation. The literature on animal navigation is very large, but we know very little about the human cognitive aspects involved in wayfinding. One major aspect of the problem, though, is that (218) 'human groups vary enormously in their navigational systems and abilities: navigation is quite largely a cultural matter.' One cannot, therefore, talk of human navigation in quite the same way as one would talk of the salmon's or the stork's navigations. The point, here again, is to show a correlation between the characteristics of a given language and the navigational abilities of its speakers, and the working hypothesis is that (227) 'speakers of absolute-coding languages should be better dead reckoners', a fact that the reader will accept readily, as this ability has already been presented as characteristic of GY and Tzeltal speakers (see chapter four). But this study on three absolute communities, GY, Tenejapan (Tzeltal speakers) and Hai///om bushmen compared to two relative communities (Dutch and English) seems to be the first systematic study on the subject. And as expected, the results of this study shows that people in absolute communities are markedly better at pointing toward a specific location than people belonging to relative communities (although there are some differences in the degree of accuracy in different absolute communities, Tzeltal speakers are not as good dead reckoners as the other two communities). Also under investigation in this chapter are gestures made by speakers of absolute and relative communities. The outcome is that there seems to be a typical gesture system in the former, which include for instance large gestures with extended arms, the frequent use of both hands, and pointing behind oneself and though oneself, the speaker's trunk remaining mostly stable through sequences of gestures. This leads the author to infer the use of absolute mental maps by these speakers, as opposed to so-called 'strip maps' by relative speakers. In the final chapter, the author attempts to elaborate a model of the human mind's representational system drawn from the observations presented in the preceding chapters. The main point is that there cannot be a single spatial representation in the mind, there must be many different representations, e.g. visual imagery, mental maps and propositional representation. Absolute and relative orientation for instance, which can coexist, imply two different kinds of representations that are not directly translatable into one another. The implications of all this are described as 'Whorfian perspectives' but the theory as such (301) 'is not Whorfian in any strict sense.' The main argument is that (ibid.) 'language is an output system. The output must meet the local semantic requirements. Consequently, the input to language production must code for the right distinctions.' And consequently (302), 'semantic parameters are not universal.' To account for the possibility of universals (the three spatial frames of reference seem to be such universals) and what is described in SILC as a variety of cognitive styles, the author presents in the last section of this chapter his 'co-evolution hypothesis'. According to this theory, already envisaged by biologists R. Dawkins and J. Thompson, and anthropologist D. Sperber (319), 'both genome and culture are conceived of as vertically transmitted lines of self-replicating and modifying information, which interact with one another in systematic ways.' Assessment: the overall impression given by SILC is that of a rich, complex and thoughtprovoking book. As an interdisciplinary work, it reaches significantly beyond the scope of cognitive linguistics, convincingly calling up theoretical resources from anthropology, biology, philosophy and psychology. Every point is discussed in detail and taken up from various angles (except, understandably, for the concluding hypothesis); this richness and complexity also has its downside, as it entails a certain amount of repetitiveness, and sometimes backward logic. Something presented as established, e.g. that (218) 'one cannot talk of human navigation in the same breath as one might talk of the navigation of the arctic tern' will often be taken up later as mere conjecture, as in the mention of observations (241) 'rendering talk of a single uniform kind of basic human navigation suspect.' Apart from the obvious complexity of the question of space, this 'patchwork effect' is probably partly due to the fact that SILC integrates research conducted by different teams, in different locations, with different methods and over many years. The unifying thread is the aim to show how language may influence the way we think. A highly creditable point in SILC is that although the author's conviction on the subject is palpable, he never tries to downplay the difficulties raised by methods used, a limited amount of available data or the interpretation of particular results. The size of the groups tested is particularly crucial, as human navigation abilities are highly variable, but also eminently trainable. Also, one might suspect in SILC an overestimation of the 'otherness' of exotic communities, as is often the case in Whorfianrelated approaches. 'Otherness' is not always exactly what or where it is taken to be. For instance, when Levinson, to illustrate absolute thinking, notes (1) that in Balinese, to lose the north means to be crazy, the French reader cannot help exclaim that the French language, although undoubtedly a relative language in Levinson's classification, has precisely the same expression, perdre le nord (to lose the north), and also perdre la boussole (to lose one's compass). Even though SILC will probably not convince those who are not ready to be convinced (as Levinson himself states, p.303, 'all arguments here are controversial, including the ones in this book'), it is sure to give every reader a lot to think about; and it will be followed by more studies to confirm -or try to disprove -the results presented here. And so, there is no doubt that SILC is a major contribution to the debate about the mind-language relation.