Samer Majdalani
email: samer.majdalani@msem.univ-montp2.fr

Solute transport in periodical heterogeneous porous media: Importance of observation scale and experimental sampling

Keywords:

This paper focuses on the effects of the observation scale and sampling on the dispersion of tracers in periodical heterogeneous porous media. A Model Heterogeneous Porous Medium (MHPM) with a high degree of heterogeneity was built. It consists of a preferential flow path surrounded by glass beads. 44 tracer experiments were carried out on several series of periodic MHPM to investigate the effect of the observation scale on solute dispersion. Each series was replicated several times, allowing for a statistical description of the unit transfer function of the MHPM. No significant trend was found for the dispersion coefficient as a function of the size of the MHPM. However, given the variability of the breakthrough curves from one experiment replicate to another, under-sampling might easily lead to conclude that the dispersion coefficient is variable with distance. Depending on the samples used, it would be as easy to (wrongly) detect an increasing trend as to detect a decreasing one. A confidence interval analysis of the experimental breakthrough curves in the Laplace space shows that (i) there exists a model with scale independent parameters that can describe the experimental breakthrough curves within the limits of experimental uncertainty, (ii) this model is not the Advection-Dispersion (AD) model, (iii) the modelling error of the AD model decreases with the number of periods, (iv) the size of the Reference Elementary Volume for the dispersion coefficient is between 10 and 20 periods. The effects of sampling prove to override those of scaling. This, with the invalidity of the AD model, leads to question attempts to calibrate and/or identify trends in the dispersion coefficient at intermediate scales from a limited number of experiment replicates.

Introduction

Understanding solute transport in porous media is important to predict the fate of contaminants in natural soils. Natural soils are highly heterogeneous and implicate various transport mechanisms. Tracer experiments on laboratory soil columns made a big contribution to identify these mechanisms. The most widespread transport model for inert solute is the Advection-Dispersion (AD) model. Fitting this model against real-world tracer tests has been reported to induce an increase in the dispersion coefficient with the travelled distance (see e.g. [START_REF] Dagan | Flow and Transport in Porous Formations[END_REF][START_REF] Gelhar | A critical review of data on field-scale dispersion in aquifers[END_REF][START_REF] Zhou | Scale-dependent dispersion in soils: an overview[END_REF]. The variability of the dispersion coefficient with distance has motivated a strong interest for alternative models. Recently, laboratory tracer experiments have been used as a benchmarking basis to test the respective accuracy of a variety of normal and anomalous transport models (see e.g. the review in [START_REF] Gao | Comparison of alternative models for 17 simulating anomalous solute transport in a large heterogeneous soil column[END_REF]. It has also been suggested by some authors that the dispersion coefficient in mobile-immobile models should be made exponentially dependent on distance in order to accurately reproduce experimental breakthrough curves in laboratory column experiments [START_REF] Gao | A new mobile-immobile model for reactive solute transport with scale-dependent dispersion[END_REF]. A number of laboratory tracer experiments in porous media are available from the literature (Silliman et al., 1998). Laboratory experiments on heterogeneous porous media are very few in comparison to those done on homogeneous porous media. One cause might be the practical difficulty to conceive and construct heterogeneous porous media. [START_REF] Silliman | Laboratory evidence of the scale effect in dispersion of solutes in porous media[END_REF] used fine sand inclusions (small cubes) with coarse sand surrounding. They found a change in the shape of the breakthrough curve at each of five measurement sections, and inferred a continuous increase in dispersivity with distance. [START_REF] Saiers | Colloidal silica transport through structured, heterogeneous porous media[END_REF] used a coarse sand inclusion (tubule) with finer sand surrounding. The inclusion constituted a preferential flow path and contributed to more than 60% of the mass balance. The breakthrough curves exhibited multiple inflection points that cannot be accounted for by the classical Advection Dispersion (AD) model. In comparison to other studies, the work of [START_REF] Saiers | Colloidal silica transport through structured, heterogeneous porous media[END_REF] is distinguished by the high degree of heterogeneity of the porous media. This fact is reflected in the shape of the breakthrough curves. Much more than a mere tailing effect, the breakthrough curves exhibited a clear two-step shape. [START_REF] Li | Mass transfer in soils with local stratification of hydraulic conductivity[END_REF] used coarse inclusions (polyethylene porous cylinders) with finer sand surrounding. The resulting breakthrough curves exhibited asymmetry. [START_REF] Huang | Experimental investigation of solute transport in large, homogeneous and heterogeneous, saturated soil columns[END_REF] used a 12.5 m long column with a wide range of soil materials. Their breakthrough curves exhibited irregular and nonsigmoidal distributions and were in most cases poorly described by the AD model. [START_REF] Sternberg | Laboratory observation of nonlocal dispersion[END_REF] and [START_REF] Irwin | Experimental investigation of characteristic length scale in periodic heterogeneous porous media[END_REF] were the first to use a periodical heterogeneous porous media, where the heterogeneity consisted of a succession of several glass bead sets tested in several serial orders. The breakthrough curves were used to infer the variation of the dispersion coefficient with distance. Unfortunately, the breakthrough curves are not shown in the publications and no estimate is provided for the goodness-of-fit of the AD model. It is thus impossible to determine whether fitting the coefficients of the AD model was meaningful. However, both studies present the results of dispersion estimations. [START_REF] Sternberg | Laboratory observation of nonlocal dispersion[END_REF] found that dispersion does not necessarily increase with scale, but that it can increase and decrease depending on the particular arrangement of properties in the medium. They deduced that the system length was unimportant in observing the scale effect. Conversely, in the study of [START_REF] Irwin | Experimental investigation of characteristic length scale in periodic heterogeneous porous media[END_REF], dispersion appeared to be scale dependent up to a certain travel distance. But, quoting the authors' own words, "because of the scatter in the data, it could be argued that the apparent increase in longitudinal dispersion is simply a result of the variation in the data". This raises the issue of sampling effects. [START_REF] Greiner | Magnetic resonance imaging of paramagnetic tracers in porous media: Quantification of flow and transport parameters[END_REF] used aggregates of lower permeability in a surrounding of higher permeability. [START_REF] Niehren | Artificial colloid tracer tests: development of a compact on-line microsphere counter and application to soil column experiments[END_REF] used cylindrical cellpore filters (less permeable) in a quartzsand surrounding (more permeable). Tran [START_REF] Ngoc | Two-scale modeling of solute dispersion in unsaturated double-porosity media: Homogenization and experimental validation[END_REF] used spherical clay inclusions (micro porosity) surrounded by sand (macro porosity). [START_REF] Danquigny | Laboratory tracer tests on three-dimensional 14 reconstructed heterogeneous porous media[END_REF] performed tracer tests on two kinds of heterogeneous porous material: a channel structured medium (5.6m×1m×1m), with channels crossing the whole tank, and a statistically correlated random structure. They found that the breakthrough curves of the statistically correlated random structure could be fitted with the AD model (at least for long distances), but that the fit is quite poor for channel structured model due to the non-Gaussian distribution of the concentration. [START_REF] Zin | Experimental visualization of solute transport and mass transfer processes in two-dimensional conductivity fields with connected regions of high conductivity[END_REF] designed an artificial porous medium with several conductivity contrasts, using low conductivity inclusions (small beads) surrounded by a high conductivity matrix (large beads). They found that the breakthrough curves of the low-contrast medium showed no tailing, while those of highcontrast medium showed tailing. [START_REF] Golfier | Comparison of theory and experiment for solute transport in weakly heterogeneous bimodal porous media[END_REF] used fine sand inclusions (lenses) with coarse sand surrounding. Their breakthrough curves showed asymmetry. The low degree of heterogeneity in the studies of [START_REF] Sternberg | Laboratory observation of nonlocal dispersion[END_REF] and [START_REF] Irwin | Experimental investigation of characteristic length scale in periodic heterogeneous porous media[END_REF] might have led to some contradictory conclusions concerning the dispersion scale dependence. As to the study of [START_REF] Saiers | Colloidal silica transport through structured, heterogeneous porous media[END_REF], the authors were not mainly interested in the problematic of dispersion scale dependence, but rather in colloidal mobilization and transport in a structured porous media. In the light of the abovementioned studies, it appears that studying the effect of heterogeneity on solute transport requires two conditions: (i) a high degree of heterogeneity of the porous media (as the experimental model of [START_REF] Saiers | Colloidal silica transport through structured, heterogeneous porous media[END_REF]), and (ii) a periodical heterogeneous porous media (as the experimental protocol of [START_REF] Sternberg | Laboratory observation of nonlocal dispersion[END_REF] and [START_REF] Irwin | Experimental investigation of characteristic length scale in periodic heterogeneous porous media[END_REF]). The first condition guarantees that the effects of material heterogeneity are clearly visible, while the second allows the influence of scale effects to be assessed. This also allows for a better quality of the experimental data, with easily replicable experiments yielding reliable statistics. In this study, we conceived a Model Heterogeneous Porous Medium (MHPM) with a high degree of heterogeneity. The purpose is to (i) enrich the experimental database of laboratory tracer tests with high quality data, (ii) answer the following six questions. (Q1) Can a scaling trend be observed for the dispersion coefficient in this highly heterogeneous MHPM? (Q2) If this is not the case, can the variations of the dispersion with the observation scale be attributed to experimental artefacts such as sampling effects?

(Q3) Does there exist a model with uniform coefficients (thus scale-independent) to describe adequately the present MHPM response? (Q4) Can the AD model be considered valid at the scale of this MHPM? (Q5) If it cannot, does a scale exist above which the AD model can be deemed a satisfactory approximation of the experimental response of the MHPM? (Q6) Can more advanced models such as the Mobile-Immobile (MI) or Continuous Time Random Walk (CTRW) models be considered as better candidates than the AD model to simulate smalldistance and short-time MHPM behaviour? The detailed characteristics of the experimental setup is described in Section 2 of the paper. The results of experimental breakthrough curves are shown in Section 3. Such experiments are usually termed Intermediate Scale Experiments (ISE) in that they involve a scale larger than the typical size of the heterogeneity but they do not achieve the size of the Reference Elementary Volume (REV) above which the continuous medium formalism should be expected to be valid [START_REF] Bear | Dynamics of fluids in porous media[END_REF]. Section 3 is also devoted to an analysis of the transfer function of the MHPM. Section 4 is devoted to discussion. A point-by-point answer to the six questions above is given throughout Section 3 and 4.

Materials and methods

Model heterogeneous porous media

The MHPM consists of a set of PVC columns (10 cm in diameter, 15 cm in length each) containing 1 mm glass spheres (40 % porosity) surrounding a cylindrical cavity (2.5 cm in diameter, 10 cm in length) placed in the centre of the column (Figure 1). The cylindrical cavity has 100 % porosity and thus acts as a wide pore that induces preferential water and solute transfer. The pore volume V 0 of one MHPM period is about 461 ml. The water and solute flow enters and leaves the column by 4 inlets and outlets (4 mm in diameter). A 500 µm sieve prevents the glass spheres from clogging the inlets and outlets, possibly blocking the flow. Figure 2 shows real captures of the MHPM. The incoming discharge (imposed by a peristaltic pump) is equally distributed between the 4 inlets. It is expected that the flow of the central inlet will mainly transfer through the central cavity, while the flow of the three remaining inlets will mainly transfer through the surrounding glass beads. After carrying out the series of experiments, four randomly chosen columns were disassembled. In each of them, the position of the sieve was compared to what it was before the experiment. No detectable displacement was noted.

Experimental setup

Step tracing experiments are done by injecting salty water (deionised water supplemented with NaCl at C 0 = 0.1 Mol/l) in a study column initially containing deionised water. A flow of 7.5 l h -1 salty water is induced into the study column until the outlet concentration C stabilizes to C 0 . In these experiments, the stable C 0 level was generally reached after injecting three times the pore volume V 0 of the study column. The outlet concentration is measured using a conductimeter (WTW TetraCon 325) and saved on a data logger (Campbell CR1000) every 5 s. The cumulated outlet volume V is measured by weighing the effluent every 5 s and saved on the data logger (Figure 3). The study column consists of a series of N connected MHPM. In this study, four series were investigated with N = 3, 5, 10 and 15. The series with N = 3 corresponds to the smallest scale of periodical heterogeneity because the first MHPM has a divergent flow inlet and a parallel flow outlet while the last MHPM has a parallel flow inlet and a convergent flow outlet. Only the in-between MHPM have the same parallel flow inlets and outlets (Figure 3). Therefore, N = 3, 5, 10, and 15 correspond to 1, 3, 8, and 13 periodical heterogeneities. In this study, 15 MHPM columns were built, named A, B, C,…, N, O. For each series, four replicates of the step tracing experiment were conducted for four 90° rotations of the study column, from which a mean breakthrough curve was deduced. The purpose of the 90° rotation is to eliminate biases resulting from a possible interaction between density-driven effects and the 120° spacing between the column inlets and outlets. The various column combinations used for the various experiments are summarized in Table 1. 1. Experimental configurations. L is the total length of the porous media.

N V 0 (ml) L (m)

Experimental results

Experimental breakthrough curve

For each value of N, four experimental signals are computed: the average breakthrough concentration ) (t C , its standard deviation C  , the minimum and maximum breakthrough concentrations, defined as

  ) ( max ) ( ), ( min ) ( , ) ( ) ( 1 1 , ) ( 1 ) ( max min 1 2 2 1 t C t C t C t C t C t C R t C R t C i i i i R i i C R i i            (1)
where C i (t) is the concentration at time t for the ith replicate and R is the number of replicates. The standard deviation and the difference between the minimum and maximum signals provide a measure of experiment variability. Figure 4 (top) shows the experimental signals obtained for the four values N = 3, 5, 10 and 15. As expected from previously published studies [START_REF] Danquigny | Laboratory tracer tests on three-dimensional 14 reconstructed heterogeneous porous media[END_REF][START_REF] Huang | Experimental investigation of solute transport in large, homogeneous and heterogeneous, saturated soil columns[END_REF][START_REF] Saiers | Colloidal silica transport through structured, heterogeneous porous media[END_REF], the breakthrough curve for small N strongly differs from the analytical S-shaped solution of the AD model (see the graphs for N = 3, N = 5). As N increases, however, the breakthrough curves tend to behave similarly to a classical advection-dispersion model output (N = 10 and N = 15 in Figure 4). It is also worth stressing the strong variability of the breakthrough curve from one replicate to another for a given N. The amplitude of the standard deviation

) (t C 
shown in Figure 4 is consistently about 5×10 -2 regardless of N. The amplitude of the min-max interval seems to decrease for larger N, but this might be due only to the smaller number of experiments (the min and max indicators being biased contrarily to the standard deviation). This experimental result has two main consequences. First, it clearly shows that the observed behaviour (atypical or typical breakthrough curve) depends on the observation scale. Second, it leads to question the use of the AD approach to model solute transfer in porous media for small spatial scales. Indeed, the AD is adequate to reproduce breakthrough curves with a typical 'S' shape but not those with atypical behaviour. Several studies on heterogeneous porous media have already shown that the AD model is inadequate to describe the resulting breakthrough curves [START_REF] Saiers | Colloidal silica transport through structured, heterogeneous porous media[END_REF][START_REF] Huang | Experimental investigation of solute transport in large, homogeneous and heterogeneous, saturated soil columns[END_REF][START_REF] Danquigny | Laboratory tracer tests on three-dimensional 14 reconstructed heterogeneous porous media[END_REF][START_REF] Li | Mass transfer in soils with local stratification of hydraulic conductivity[END_REF]. Additional information is brought by the Laplace transform of dC*/dt (t)

    0 d ) exp( ) ( * ) ( t st t C s s P (2)
where s is the Laplace variable. Since the input concentration is a unit step, P(s) is the Laplace transform of the response of the N columns to a concentration unit pulse. The graphs P(s) are shown in Figure 4 (bottom). The graphs are drawn for Hz 1 . 0  s because the sampling time step is 5 seconds, thus making the numerical computation of (2) inaccurate for larger s. The response P(s) behaves as a decreasing exponential for large s, as indicated by the linear asymptote in the graphs (P is plotted using logarithmic coordinates). This is the typical signature of a delay. That the asymptotic linear behaviour is reached for smaller s as N increases indicates that advection phenomena become predominant as N increases, a well-known feature of the advection-dispersion model for increasing Peclet number values. This, however, is not sufficient to validate the AD model, an issue that is addressed in Section 4.1.

Scaling trend for the dispersion coefficient

Questions (Q1) and (Q2) are answered assuming that the AD model is valid, with governing equations

0 2 2          x C D x C u t C (3a)   L x x C , 0 0 ) 0 , (    (3b)
where D is the dispersion coefficient, and u the flow velocity. Analytical solutions are available for an upstream boundary condition in the form of a step flux function [START_REF] Ogata | A solution of the differential equation of longitudinal dispersion in porous media[END_REF])

    2 2 erfc exp 2 erfc ) , ( 0 0 ) , 0 ( 0 2 / 1 2 / 1 C Dt ut x D ux Dt ut x t x C t t C                                       (4)
The unit response to a Dirac flux pulse is obtained as the derivative of (4) with respect to time

                                      Dt ut x ut x Dt ut x ut x Dt C t x C t C t C 4 ) ( exp 4 ) ( exp ) ( 4 ) , ( ) ( ) , 0 ( 2 2 2 / 3 2 / 1 0 0   (5)
The Laplace transform of the solution (5) can also be obtained by solving equation (3) in the space-Laplace domain:

2 / 1 2 4 1 , 2 1 exp 1 2 ) , ( ˆ                 s u D F F D ux F s x C (6)
The experimental dispersion coefficient D is computed using a formula derived from the analytical solution of the advection-dispersion equation [START_REF] Bear | Dynamics of fluids in porous media[END_REF]):

2 2 t L t t t Lu t t t D                     (7)
used by e.g. [START_REF] Irwin | Experimental investigation of characteristic length scale in periodic heterogeneous porous media[END_REF] and [START_REF] Sternberg | Laboratory observation of nonlocal dispersion[END_REF]. Equation ( 5) was applied to the experimental configurations shown in Table 1. The detailed results are given in the Appendix (Tables A1 to A4), only statistics are presented here. For each value of N, the average, standard deviation, minimum and maximum of D obtained from the various possible configurations are provided in Table 2. Several remarks can be made at this stage. Firstly, the variations in the average D with L are smaller than the standard deviation of D for any given L. Since the individual variations between the replicates for a given N override by far any visible trend, it can be concluded that no significant trend of distance-dependent dispersion D(L) is detected given the experimental uncertainty. Secondly, given the range of variations between the minimum and maximum values, it is possible to find a sequence among all the experimental values in Tables A1-4 such that an increasing trend D(L) is detected. An example of such a sequence is (ABC1, ABCDE2, A-J4, A-O4). But it would be just as easy to detect a decreasing trend D(L) by choosing the values in Tables A1-4 appropriately, for instance the sequence (KLM4, FGHIJ3, K-O2, A-O3). Such trends, however, would only be sampling artefacts. It can be concluded as an answer to question (Q1) that, in the limit of experimental uncertainty, no significant scaling trend can be identified for the dispersion coefficient in the present MHPM. As an answer to question (Q2), sampling artefacts may easily lead to the conclusion that the dispersion coefficient is subjected to a scaling trend in this MHPM. It is stressed that only the systematic replication of the tracer experiments and a high-resolution sampling allows this question to be answered with a satisfactory degree of confidence.

L (m)

N D (m 2 s -1 ) average D (m 2 s -1 ) standard deviation D (m 2 s -1 ) minimum D (m 2 s -1 ) maximum 0.45 3 1.08×10 -4 3.65×10 -5 5.60×10 -5 1.69×10 -4 0.75 5 1.36×10 -4 6.37×10 -5 7.40×10 -5 3.29×10 -4

1.50 10 1.18×10 -4 3.38×10 -5 6.58×10 -5 1.81×10 -4 2.25 15 1.18×10 -4 3.00×10 -5 8.43×10 -5 1.67×10 -4

Table 2. AD model. Dispersion coefficients obtained using equation ( 7). See the detailed results in the Appendix.

Unit transfer function

The purpose of the present subsection is to answer question (Q3 the Laplace transform of the ith replicate for a number N of columns. A minimum-maximum confidence interval is defined for P with the following lower and upper bounds:

i N R i N i N R i N P P P P , , , 1 max , , , , 1 min , max , min 
     (8)
where N denotes the number of columns used in the experiment and i denotes the replicate. This allows a confidence interval to be defined for the unit Laplace transform of a single MPMH period. Let r be the Laplace transform of a single MHPM period for a Dirac concentration input. The lower and upper bounds defined in (8) yield a confidence interval  

  N M N M N M r r N M r     , , ) , ( ) , ( max min (9a)     N M P P N M r P P N M r M N M N M N M N                      , , , ,
    N M r r N M r r N M N M , min , , max max , 2 min , 1   (10)
The confidence intervals overlap provided that

s s r s r   ) ( ) ( 2 1
. Figures 5a-f show the min-max confidence intervals derived from equation (9b) for all the possible (M, N) combinations. It is noted that the confidence interval becomes narrower as the number (N -M) of periods used to determine the unit transfer function increases, thus indicating the convergence of the estimates (9b). The intersection of all the confidence intervals is shown in Figure 5g. The non-zero width of the interval   2 1 , r r indicates that a transfer function with scale-independent parameters can indeed be found within the range N = 3 to N = 15. In other words, it is possible to find a scalar transport model with spatially uniform parameters that yields simulation results lying within the experimental uncertainty. This model, however, is not necessarily the classical AD model. This issue is explored in the next section.

Discussion

Validity of the AD model

The purpose of the present section is to answer questions (Q4-5). If the AD model is valid at the scale of N MHPM periods, then there exists a combination (D, u) such that [START_REF] Dagan | Flow and Transport in Porous Formations[END_REF] where l = 15 cm is the length of a single MHPM period. Clearly, this condition is not satisfied for N = 1. Indeed, for large s, the analytical solution (6) behaves as

s s r s Nl C s r N N    ) ( ) , ( ) ( 2 1
          2 / 1 2 / 1 2 / 1 2 / 1 2 / 1 2 / 1 exp ) , ( 2 s D Nl s D u s Nl C s D u F (13)
a function with a horizontal asymptotic direction in semi-logarithmic coordinates, while r 1 (s) and r 2 (s) take the form of straight lines for large s (see Figure 5g). Consequently, the function

/ 1 0 2 2 d ) ( 1         S s s S L  (13a)             ) ( ) , ( îf ) ( ) , ( ˆ) ( ) , ( ) ( if 0 ) ( ) , ( îf ) ( ) , ( ) ( 2 2 2 1 1 1 s r s Nl C s r s Nl C s r s Nl C s r s r s Nl C s r s Nl C s N N N N N N  ( 13b
)
where S = 0.1 Hz is the upper bound of s. Eq. ( 13b) gives a zero error if the solution of the AD model lies between r 1 and r 2 in the Laplace space, that is, if the AD solution falls within the range of the experimental uncertainty.

Table 3 shows the combinations (D, u) that minimize the L 2 -norm (13a) for various values of N. Figure 6a-g shows the corresponding plots in the Laplace space. The L 2 -norm for N = 1 is large, indicating that the AD model provides a poor description of the transfer process at the scale of a single column. However, the L 2 -norm decreases quickly with N, which is an indication that the AD model gains worth with the number of MHPM periods. The driving mechanism for this is interesting to notice. As shown by Figures 6a-g, the AD transfer function lies outside the confidence interval for s > 3×10 -2 Hz in all graphs. In other words, in terms of relative error, the AD model is equally inaccurate for all values of N. However, the amplitude of the transfer function decreases quickly as N increases. The absolute value of the error thus becomes smaller as N increases.

N D u L 2 Figure 1 1.57×10 -4 1.98×10 -3 2.13×10 -3 6a 3 2.85×10 -4 1.18×10 -3 1.27×10 -4 6b 5
2.36×10 -4 9.98×10 -4 1.70×10 -6 6c 10 2.24×10 -4 7.97×10 -4 1.75×10 -10 6d 15 2.07×10 -4 7.71×10 -4 6.51×10 -15 6e 20 2.07×10 -4 7.71×10 -4 2.045×10 -18 6f 30 2.07×10 -4 7.71×10 -4 2.66×10 -25 6g Table 3. AD model. Optimal parameter set for the objective function (13a-b).

Note that the value for D is found to be consistently about 2×10 -4 m 2 s -1 (except for N = 3), while formula (7) gives an average value of about 1.2×10 -4 m 2 s -1 (see Table 2). This striking difference between the two sets of values can be explained by the calibration approach. The values given in Table 1 are computed separately for each individual experiment, under the assumption that the AD model provides an accurate description of the transport processes for each experiment. However, a (D, u) combination calibrated for a given replicate will fail totally to predict the behaviour of another replicate. In contrast, the dispersion coefficient values in Table 3 account for the variability of experimental uncertainty. It is not requested that the AD model should satisfy each individual experiment as accurately as possible, but that its response should lie within the intersection of all the confidence intervals associated with different MHPM lengths. As a consequence, smoother unit responses (thus larger dispersion coefficient values) are allowed compared to the former approach. Moreoever, both calibration approaches use different variables (pointwise C(t) values in the first approach, ) , ( ˆs Nl C over an interval in the second approach).

Validity of alternate transport models

This section is motivated by Question (Q6). Since the classical AD model fails to reproduce the unit pulse response of the column, the question arises whether alternate transport models may be more successful. To start with, examining Figure ( 5g) indicates an asymptotically linear behaviour in semilogarithmic coordinates. In other words, the unit transfer function is asymptotically equivalent to a decreasing exponential in the Laplace space. The experimental unit transfer function thus verifies the following property:

          t f Bx t H t x C Bxs A s x C t s         0 , exp , ˆ (14)
where A and B are constants and H is Heaviside's step function. In other words, the signal propagates at a finite velocity B -1 , with a breakthrough concentration rising from 0 at t = Bx to 1 for

  t
. The inadequacy of the AD model for small times and distances precisely stems from the infinite front propagation speed associated with the Fickian flux model. The Mobile-Immobile (MI) model [START_REF] Van Genuchten | Mass transfer studies in sorbing porous media: II. Experimental evaluation with tritium (3H2O)[END_REF] is one of the earliest and most probably best-known non-Fickian models for dispersion. Later on, generalizations such as the multirate model (Haggeerty and Gorelick, 1995) have been proposed. The governing equations for the MI model are:

  m i m m C C k x C D x C u t C            2 2 (15a)   i m i C C k t C       1 (15b)
where C i and C m are respectively the concentrations in the immobile and mobile regions, k is an exchange rate constant and  is the relative volume fraction of the mobile region. Applying the Laplace transform in time, eliminating

C i yields           s k s k a C D s a x C D u x C m m m         1 1 , 0 dd d d 2 2 (16a)       D u D as D u x s C s x C m m 2 2 , exp , 0 , ˆ2 / 1 2                      (16b)
For large s, the following equivalence is obtained:

                  x D s A s x C a s m s 2 / 1 2 / 1 exp , 1 (17)
leading to a horizontal asymptotic direction in the semi-logarithmic plots of Figures 4 to 6. Consequently, the ) , ( ˆs x C m curve (16) cannot remain between the min-max bounds of the experimental confidence interval for large s, thereby failing to account for the experimental MHPM behaviour for small times. The modelling community's attention has been drawn over the past years to anomalous transport models (see e.g. [START_REF] Gao | Comparison of alternative models for 17 simulating anomalous solute transport in a large heterogeneous soil column[END_REF] for a review). Anomalous transport models use a non-Fickian model for dispersion. The underlying assumption is that the characteristic time scale and/or the characteristic length of the solute movement are infinite. Under such conditions, the assumption of Brownian movement is no longer valid and long-term memory effects appear. The general form for anomalous transport models is

                            1 , 1 1 0 1 0 , 0 1 1           x C t D x C t u t C (18)
where

k k t   /
stands for the kth-order fractional time derivative (see e.g. [START_REF] Oldham | The fractional Calculus[END_REF]Podlubny, 1999 for definitions). For

1 , 1    
, Equation ( 18) is obtained assuming that the probability density function w of the displacement time and d of displacement length obey the following asymptotic behaviours for large time and distances [START_REF] Metzler | The random walk's guide to anomalous diffusion: a fractional dynamics approach[END_REF]:

              1 1 , x B x d t A t w x t (19) The combinations   1 , 1       and   1 , 1    
correspond respectively to the so-called Galileanvariant and Galilean-invariant fractional diffusion models presented in Metzler et al. (2002) under the Continuous Time Random Walk (CTRW) formalism. For 1 



, so-called Levy walks are obtained. As shown in [START_REF] Klafter | Stochastic pathway to anomalous difusion[END_REF], that solute particles may cover arbitrary long distances within a displacement (thus leading to infinite velocities, a physical impossibility) may be balanced by a coupling between the pdfs w(t) and d(x). Depending on the respective natures of w and d, any anomalous dispersive behaviour may be obtained, from subdiffusive to superdiffusive. Note that the case  

1 , 1      
has been investigated by [START_REF] Pachepsky | Simulating scale-dependent solute transport in soils with the fractional advective-dispersive equation[END_REF] and applied to column experiment simulation by [START_REF] Gao | Comparison of alternative models for 17 simulating anomalous solute transport in a large heterogeneous soil column[END_REF]. The behaviour of the analytical expression given by Pachepsky et al. (2009) is not easy to analyze, even in the Laplace space. However, a qualitative reasoning suffices to discard this option to model the present experiments. The Levy flight model yields infinite signal propagation velocities, just as does the standard, Fickian model. Consequently, for large s, this model also lies outside the experimental unit response of the MHPM in the Laplace space. The case  will therefore not be analyzed hereafter. For 1 



, applying the Laplace transform to (18) gives

0 d d d d 1 12 2      C D s x C s D u x C      (20a)                                     1 2 / 1 2 2 2 2 2 2 , exp , 0 , ˆs D u D s s D u x s C s x C (20b)
which gives the following equivalences

                       x D s A s x C D
Although attractive in theory, anomalous transport models may fail in a number of cases, not because they are not valid in general, but because they are not applied at the right space/time scales. Fractional derivatives are introduced in anomalous transport models under the assumption of heavy-tailed waiting time and travel distances probability density functions. The CTRW analysis [START_REF] Klafter | Stochastic pathway to anomalous difusion[END_REF][START_REF] Metzler | The random walk's guide to anomalous diffusion: a fractional dynamics approach[END_REF] shows that fractional derivatives stem naturally from this assumption as the Laplace variable s tends to zero. For the CTRW analysis to be valid, the observation time and/or travelled distance must be sufficient for a significantly large number of solute molecule jumps to occur. Moreover, the solute must cover a sufficient number of medium heterogeneities so that the ergodicity of the transport process be ensured. This underlying assumption is in contradiction with the present experimental setup, that obviously deals with space and time scales smaller than those required for ergodicity. As mentioned by [START_REF] Gao | Comparison of alternative models for 17 simulating anomalous solute transport in a large heterogeneous soil column[END_REF], the reason why anomalous transport models may succeed better to reproduce experimental column behaviour may simply be that they incorporate more degrees of freedom than classical transport models. As stressed by these authors, more degrees of freedom do not necessary induce a better model.

Conclusions

This paper presents the results of intermediate scale tracer experiments carried out using a model heterogeneous porous medium with a variable length. The periodic character of the medium allows the stationarity of the medium properties to be preserved. The large number of replicates (44 experiments for 4 different column lengths) allows sampling artefacts to be minimized and reliable statistics to be derived. A confidence interval for the unit transfer function of a single period of the model medium is derived from a Laplace transform analysis. In the light of this work, the answers to questions (Q1-6) raised in the Introduction are the following. (A1) No clear scaling trend can be identified for the dispersion coefficient within the range of experimental uncertainty for the present MHPM. (A2) Given the strong individual variability observed from one replicate to another, sampling artefacts may lead to wrongly conclude that the dispersion coefficient is scale-dependent. (A3) There exists a model with scale-independent parameters that can provide a correct description of the solute transfer in the MHPM from N = 3 to N = 15. (A4) This model is not the AD model, because it does not allow the experimental breakthrough curves to be reproduced accurately within the range of the experimental uncertainty. (A5) The AD model can be deemed to provide an accurate description of the present MHPM after the solute has been transported over at least 10 (and most probably 20) periods. The size of the REV for this MHPM is thus between 10 and 20 MHPM periods. (A6) The MI and CTRW models will fail to reproduce the small-time and short-distance experimental behaviour of the MHPM because of the second-order derivative used in the modelling of dispersion. We should like to stress the strong variability of the experimental results, even in the case of the present artificial MHPM, that was intentionally made strictly periodic. This should make modellers extremely cautious when dealing with natural media, the geometry of which is often poorly known and does not exhibit the degree of determinism of the present MHPM. This makes the systematic replication of breakthrough experiments indispensable in seeking reliable conclusions on scaling effects, as well as in model calibration attempts. It is a useful reality check of the assumptions of stationarity and ergodicity [START_REF] Kulasiri | Stochastic dynamics -modelling solute transport in porous media[END_REF] of the AD model. From a more general perspective, one might also question the relevance of investigating the scaling behaviour of the AD model for spatial scales where it is invalid. This questioning also holds for alternate models such as anomalous transport models. Lastly, the present work shows the usefulness of a high resolution sampling in analyzing the MHPM transfer functions (each of the breakthrough curves used in this study is made of 500 to 1500 records). Further research focuses on (i) an experimental design from which the boundary effects arising from the inlet distribution is absent, which would allow the unit response r to be derived directly for a single

  Indeed, the convolution property of the Laplace transform leads to

  transfer function with scale-independent parameters exists, all the confidence intervals possible combinations (M, N) should overlap. Denoting respectively by r 1 and r 2 the lower and upper bounds of the intersection of all the confidence intervals, one has by definition

  all s and the AD model is invalid to some extent. This is illustrated by Figure6a, where the confidence interval   plotted for N = 1. The degree of accuracy of the AD model is assessed by computing the L2-norm of the difference between the experimental transfer function and the theoretical one in the Laplace space:

  2

  Figure1. (a) A model heterogeneous porous media (MHPM) with 4 inlets and outlets that occupy the centre of gravity and the 3 vertices of an equilateral triangle. The centre of gravity is on the central axis of the column and the 3 vertices are 1 cm away from the column periphery. (b) The internal cavity is a tube that is punched on top and bottom and covered with a 500 µm sieve. The sieve impedes the glass spheres from entering inside the cavity and guaranties that only water (and solute) can transfer through the cavity.

  Figure5. Unit response of a column inferred from the Laplace transforms of the experimental signals.

  Figure6. Best AD model fits in the Laplace space for various of N.

  ). Since the various replicates C i (t) all give different time series for a given number N of columns, the unit transfer function is different for each replicate. The proposed approach consists in determining the envelope of all possible C(t) signals for a Dirac input. It is obvious from Definition (1) that neither the lower bound C min nor the upper bound C max of this envelope do correspond to a single replicate signal C i (t), the purpose being only to determine the range of possible variations of C(t) for each time t. For this, it is necessary to estimate the unit transfer function of a single MHPM period. Denote by P N,i

Acknowledgements

7

The authors would like to thank Dr Philippe Ackerer, Research Director at LHYGEES Strasbourg, 8

France, and Jonas Darey for assisting in the experimental work.

Appendix. Dispersion coefficient -detailed results


Mouncef Sedrati
email: mouncef.sedrati@univ-ubs.fr

Edward J Anthony
email: anthony@cerege.fr

Confronting coastal morphodynamics with counter-erosion engineering: The emblematic case of Wissant Bay, Dover Strait

Keywords: Coastal erosion, coastal engineering, coastal morphodynamics, macrotidal coast, beach nourishment, beach rotation, Wissant Bay

Wissant Bay is a picturesque and highly frequented French coastal resort comprising beaches, dunes, marshes, and bold capes facing the Dover Strait. Situated at the southern approaches to the North Sea, the 8 km-long bay has, arguably, the most rapidly eroding shoreline in metropolitan France. Retreat has largely affected much of the bay shoreline west of Wissant town, with parts of this sector having lost up to 250 m in the last fifty years, whereas a much smaller sector east of the town is a zone of accretion. Various dune, beach and nearshore morphodynamic studies conducted over the last decade have identified chronic sand bleeding from the western sector and longshore transport to the east, within a framework of what appears to be an ongoing shoreline rotation process within a dominant longshore sediment transport cell between the headland of Cape Gris Nez to the west and the bold chalk cliffs of Cape Blanc Nez to the east. Retreat of the narrowing beach-dune barrier poses a threat in the coming years, as there is a likelihood of it being breached by storms. The seawall protecting Wissant town has also been repeatedly damaged since 2000 due to the chronic sand deficit. These changes involve interactions between a nearshore sand bank, a complex macrotidal beach comprising multiple subtidal to intertidal bars and troughs subject to strong longshore sand transport especially during storms, and aeolian dunes. The nearshore bank acts as a dissipater of incident storm wave energy and as a sand source for the multi-barred beaches and dunes, and has been strongly impacted by past massive aggregate extraction. The bank is, in turn, part of a larger system of mobile banks reworked by storms and tidal currents within the framework of a sand circulation system between the eastern English Channel and the southern North Sea. The aim of this work is to confront knowledge acquired on the morphodynamics of the bay with an engineering plan proposed to counter erosion and reestablish shoreline stability. The plan is based essentially on the creation of an 'equilibrium' beach profile, capable of withstanding storms, comprising an enlarged upper beach berm, and constructed through beach nourishment from a nearshore source located 20 km east of Wissant Bay. The plan has not been implemented because of cost. Even if it were to be implemented, its efficiency seems very doubtful because the beach profile simulations on which it is based neglect the complex multi-barred morphology and the overwhelming dominant longshore transport over bars during storms. The plan is also geared towards resolving a local problem of erosion that is embedded in the larger and rather complex spatiotemporal morphodynamics and sediment transport mechanisms evoked above. Wissant Bay is emblematic of the problems of erosion facing many communes in France, and elsewhere. The fight against shoreline erosion generally starts with the commonly insurmountable hurdle of fund-raising for costly engineering proposals that are not always based on a clear grasp of the embedded scales of change affecting the coast.

Introduction

One reason for the failure of many operations aimed at countering shoreline erosion is that their conception and implementation are commonly not based on a sufficient knowledge of processes shaping the coast at various morphodynamic scales. In many situations, the links between short-to long-term geomorphic change in small segments of coast and the way these are embedded in, and controlled by, larger-scale aspects of coastal change, are not always well apprehended [START_REF] Gelfenbaum | Large-scale coastal change in the Columbia River littoral cell: An overview[END_REF]. From a coastal management point of view, a first step towards a better understanding of this scale relationship has been the coastal cell concept, commonly used in a sediment budgetary framework in which process gradients may or may not be ignored, the emphasis being on definition of each coastal cell and on the net gains and losses of sediment within each cell (e.g., [START_REF] Bray | Littoral cell definition and budgets for central southern England[END_REF][START_REF] Cooper | Appraisal and evolution of the littoral 'sediment cell' concept in applied coastal management: Experiences from England and Wales[END_REF]Patsch and Griggs, 2008;[START_REF] Anfuso | An integrated approach to coastal erosion problems in northern Tuscany (Italy): Littoral morphological evolution and cell distribution[END_REF][START_REF] Van Rijn | Coastal erosion and control[END_REF]. This is a useful approach but even where coastal cell definition may appear simple, the task of simply delineating the shoreline and constraining the processes operating both across shore and alongshore in such cells may turn out to be difficult [START_REF] Gelfenbaum | Large-scale coastal change in the Columbia River littoral cell: An overview[END_REF]. In particular, the sediment dynamics of alluvial coasts subject to large tidal ranges can be quite complex because of potentially strong tidal currents, modulation of wave action by tides, and large variations in the 'shoreline' controlled by tidal range. This situation can be even more so where large stocks of loose mobile sediments are available and constantly reworked by processes generated by waves and currents over large shorefaces, as in the English Channel, the southern North Sea and some of the marginal seas in Asia. The ensuing morphodynamic adjustments between such sediment stocks and the forcing agents can lead to particularly complex situations where sediment cells are hard to define and where apprehension of coastal change requires crossing short-to long-term series of observations at various spatial and temporal scales. In such situations, sound solutions to problems of local coastal erosion require taking into account the complex, larger-scale morphodynamic background. These issues are examined here using the example of Wissant Bay, a highly frequented resort in the Dover Strait (Fig. 1a), set in a particularly complex environment in terms of coastal processes. The bay shoreline has varied significantly over the last half century, with certain sectors, hitherto prograding, now in a particularly critical condition in terms of erosion. A detailed plan has been set up for the implementation of engineering solutions. Meanwhile, numerous studies conducted over the last decade have progressively highlighted the complex patterns of shoreface and shoreline change at both the small-scale level of the Wissant town front, where erosion poses a direct hazard, and at the larger bay scale. This larger scale is embedded in the hydrodynamic and sediment circulation system of the Dover Strait. In this paper, we will briefly summarise the recent findings on the morphodynamics of the bay, confront these with the proposed engineering solutions, and then discuss why Wissant Bay is emblematic of many communes, in France, and elsewhere, where the fight against erosion is commonly stalled by fund-raising for costly engineering proposals that are not always based on a clear understanding of the embedded scales of change affecting coasts.

Wissant Bay

Beach, dune and shoreface morphology Wissant Bay is, in terms of setting and morphological diversity, commonly considered as one the most picturesque sites on the southern North Sea coast, occupying an 8 km-long mild embayment between bedrock cliffs. Wissant town has functioned as a resort since the 19th century. It is extremely popular, and its beaches, dunes, marshes, and bold capes facing the Dover Strait offer a variety of tourism-based recreation and leisure activities. The bay is also part of a protected site. The dunes in Wissant Bay form a linear barrier 100 to 300 m wide and with a maximum inland height of about 20 m. They impound marshes of ecological value. The beach is characterised by multiple intertidal to subtidal bars and troughs (ridges and runnels). In this transition zone between the eastern English Channel and the southern part of the North Sea, the gently sloping shallow shoreface extending seaward of the beach bars and troughs is characterized by prominent tidal sand banks and ridges that are particularly well developed as the narrow Dover Strait opens up on the epicontinental southern North Sea (Fig. 1a). These banks are up to several kilometres long and have heights of up to 10 m. They practically impinge on the beach in places. These elongated sand bodies are commonly oriented WSW-ENE, roughly parallel to sub-parallel to the coastline. One of these tidal banks, the Line Bank, lies just off Wissant Bay (Fig. 1b). The importance of these nearshore banks for coastal stability in the southern North Sea has been emphasised in a number of recent studies [START_REF] Anthony | Embryo dune development on a large actively accreting macrotidal beach: Calais, North Sea coast of France[END_REF][START_REF] Anthony | Shoreface sand supply and mid-to late Holocene aeolian dune formation on the storm-dominated macrotidal coast of the southern North Sea[END_REF][START_REF] Héquette | The influence of nearshore sand bank dynamics on shoreline evolution in a macrotidal coastal environment, Calais, Northern France[END_REF][START_REF] Anthony | Storms, shoreface morphodynamics, sand supply, and the accretion and erosion of coastal dune barriers in the southern North Sea[END_REF]. Sand banks modulate both wave dissipation patterns and onshore sand supply for coastal dune accumulation. They are consequently involved in longshore variations in shoreline accretion and erosion, the latter occurring where bank sand supply and protection from waves are lacking. The banks can also weld onto the coast, leading to locally important coastal accretion at multi-decadal timescales. [START_REF] Aernouts | L'évolution du rivage et des petits fonds en Baie de Wissant pendant le XXè siècle, Pas-de-Calais, France[END_REF] showed, from differential bathymetric analysis, that the Line Bank underwent erosion during the 20 th century, losing over 1 million m 3 of sand over a 90-year period. This loss has been due essentially to massive aggregate extraction, especially in the western sector of the bank closest to the coast. This practice was prohibited in the 1980s but the damage had already been done. The lowering of the Line Bank by seabed mining not only curtailed sand supply to the western part of Wissant Bay, but may also have induced a process of erosion that became self-maintained following the prohibition of extraction. The dynamics of the Line Bank are likely to be embedded in a process of longer-term larger-scale storm-and tide-controlled sand migration from the eastern English Channel towards the Dover Strait and the southern North Sea, a process that could explain a west-east gradient of decreasing bank lowering.

Hydrodynamic context

Wissant Bay lies in a typical mixed storm-wave-and tide-dominated environment subject to a complex pattern of time-varying influences of tides and storms, in addition to wind-forced flows [START_REF] Héquette | Sediment transport under wave and current combined flows on a tide-dominated shoreface, northern coast of France[END_REF]. Winds are dominantly from southwest and northeast, but the strongest winds mostly originate from west to southwest. The hydrodynamic context is that of a short-fetch, storm-wave environment, characterised by marked short-term (order of days to weeks) fluctuations in wave height (Fig. 2a). The dominant waves are from southwest to west, originating from the English Channel, followed by waves from the northeast to north, generated in the North Sea. Breaking waves are essentially from a north-northeast to northwest window, although the dominant deepwater directions are from both north and west. The tidal regime in the region is semi-diurnal and macrotidal, the tidal range in Wissant Bay being about 6.5 m during spring tides. In calm weather, current directions are closely conditioned by the tide, with dominantly longshore eastward-directed flood directions and westward ebb directions (Fig. 2b). Strong winds enhance ebb or flood current speeds when blowing in the same direction, or limit, and even prevent tidal reversal when blowing in the opposite direction in the bay (Sedrati, 2006), but flow is more commonly flood-dominated. During conditions of significant wind stress (sustained wind speeds > 10 m.s -1 ), the peak current speeds can be two to three times higher than 'normal' (tide-generated) peak spring tide speeds (~ 0.45 m.s -1 ). Longshore currents, especially setting east, can become particularly strong during storms as a result of direct wind stress [START_REF] Sedrati | Storm-generated morphological change and longshore sand transport in the intertidal zone of a multi-barred macrotidal beach[END_REF] and reinforcement by longshore gradients in radiation stress that divert, alongshore, offshore mean currents generated as waves pass over or break over the sand banks [START_REF] Anthony | Storms, shoreface morphodynamics, sand supply, and the accretion and erosion of coastal dune barriers in the southern North Sea[END_REF]. Storms may add up to 1 m of surge above high-tide swash excursion levels.

Morphodynamic sectors

Wissant Bay comprises three sectors (Fig. 1b): a 6 km-long strongly eroding western sector, comprising Dune du Châtelet and Dune d'Aval, where the foredune has retreated by up to 250 m between 1949 and 2000 (Fig. 3), following an early period of stability and even progradation [START_REF] Aernouts | L'évolution du rivage et des petits fonds en Baie de Wissant pendant le XXè siècle, Pas-de-Calais, France[END_REF][START_REF] Chaverot | Changes in storminess and shoreline evolution along the northern coast of France during the second half of the 20th century[END_REF], a short central sector, Wissant town, fronted by a seawall that has held the shoreline, and an equally short accreting eastern sector, Dune d'Amont. [START_REF] Sedrati | Sediment dynamics and morphological change on the upper beach of a multi-barred macrotidal foreshore, and implications for mesoscale shoreline retreat: Wissant Bay, northern France[END_REF] showed that a retreat rate of the dune front of up to 4 m in the eroding western sector can occur in just 24 hours during severe storms associated with high surge levels (up to 1 m) and high spring tides. This value is equivalent to the annual mean shoreline retreat rate calculated for the Dune d'Aval sector by [START_REF] Aernouts | L'évolution du rivage et des petits fonds en Baie de Wissant pendant le XXè siècle, Pas-de-Calais, France[END_REF]. This suggests a highly rhythmic foredune retreat that depends on the right combination of storm waves, spring tidal range and storm surge conditions. This erosion has led to the cropping out of peat on the beach representing former backbarrier vegetation. Erosional in the past, the eastern sector of the bay is now a zone of deposition, characterised by significant foredune growth and active formation of embryo dunes [START_REF] Anthony | Short-term beach-dune sand budgets on the North Sea coast of France: sand supply from shoreface to dunes and the role of wind and fetch[END_REF]. In terms of the overall shoreline dynamics, the western and central parts of the bay constitute a long updrift erosional sector linked to a short downdrift depositional sector in the east. Figure 4 shows unpublished beach profiles that summarise these three sectors. The first and the last of these three profiles start from the dune front. The Wissant profile starts from the seawall protecting the town. The Dune d'Aval profiles show a typical system of bars and troughs the changes of which reflect both cross-shore beach mobility and longshore bar mobility. Two significant aspects associated with the profile changes at this site are the low elevation of the beach close to the dune front and the marked retreat of the latter (Fig. 4a). At a distance of 100 m from the survey origin point, the beach profile fluctuations occur within an envelope of nearly 4.5 below the high water neap tide level. The dune front shows nearly monotonous retreat over the survey period, exceeding 20 m. The Wissant townfront profile shows much larger fluctuations than the previous profile (Fig. 4b). The bar and trough morphology is, consequently, much more pronounced. At a distance of 100 m from the survey point on the seawall, the beach fluctuations are up to 4.5 m below the level of high neap tides. The envelope of fluctuations at the foot of the seawall attained up to 3 m during the survey period. The Dune d'Amont profile shows a system of more subdued bars with a milder envelope of fluctuation (Fig. 4c). The envelope below the mean high water neap tide level within the 100 m distance is about 2 m. The net sand budget changes of the three profiles over the survey period are depicted in Fig. 5. Dune d'Aval and Wissant town front show a fluctuating but strongly negative sand budget over much of the survey period, whereas Dune d'Amont beach shows a fluctuating but net positive budget with a single negative value in 1998. The three profiles in Wissant Bay show envelope patterns and net budget changes that reflect the west to east gradient in shoreline mobility highlighted by [START_REF] Chaverot | Changes in storminess and shoreline evolution along the northern coast of France during the second half of the 20th century[END_REF]. The profile of the beach in the eroding western sector of the bay is significantly lower than in the accreting sector in the east. This difference in elevation also goes with much larger bar and trough fluctuations in the eroding sector, which clearly incorporates Wissant town front. Photographs of the town front shoreline in 1952 and 1986 show a much more accreted beach with sand covering the seawall, which is clearly visible in the earlier 1909 photograph (Fig. 6a). This suggests that sand was still accumulating in this central sector nearly 30 years ago, a situation that contrasts with the beach lowering and repeated damage to the seawall since 2000 (Fig. 6d-f).

Proposed management solutions to erosion

The investigations summarised above have thrown light on the large-scale processes involved in sediment movements in which are embedded shorter-term changes such as those depicted by the beach profiles (Fig. 4). To summarise these findings, we note active sand transfer over the shoreface of the bay from west to east within at least the last three decades, resulting in sustained erosion of the western bay foredunes and lowering of the beach. The retreat constitutes a threat in the coming years, because of the likelihood of storm breaching of the narrowing dune barrier. In essence, this entails eventual failure of the retreating dune front in the west, leading to marine invasion of a backbarrier marsh (Fig. 1b) of high ecological value. Wissant town is also becoming increasingly vulnerable to submersion. At the scale of the entire bay shoreline, the only sector where efforts at containing erosion have been deployed is the town front with its seawall. Figure 6a shows an inclined seawall in 1906 that served as a coastal defence structure in replacement of the mobile dunes and as a seafront promenade as the town prospered essentially from winter tourism. The wall assumed a military defence function during World War 2 when the Atlantic Wall was built along large parts of the French coast, including Wissant Bay, by the German army. Much of this Atlantic wall has been eroded. Remnants still subsist on the beach in the western sector, whereas remnants in the eastern accreting sector have been largely buried by aeolian dune accretion. Since 2000, the seawall fronting the town has been damaged on several occasions by storms, and undermined by beach lowering at its base (Fig. 6d-f). The difference in profile morphology between Dune d'Aval and Wissant town (Fig. 4) appears to reside essentially in the presence of this seawall, which has acted as a rampart against shoreline retreat, notwithstanding the progressive beach lowering. Hence the exclusion of this sector (Fig. 3) from the multi-decadal shoreline mobility survey conducted by [START_REF] Chaverot | Changes in storminess and shoreline evolution along the northern coast of France during the second half of the 20th century[END_REF]. The fight against erosion in Wissant Bay has been mainly focussed on maintaining the seawall. Damages have been repaired in piecemeal fashion following severe storm attacks (Fig. 6), as the wave of erosion, largely ignored as long as it concerned only the uninhabited Dune du Châtelet and Dune d'Aval sector, started affecting the town front more actively. Wissant Bay is a protected site of natural value, and this drastically reduces the possibility of implementing heavy defence structures such as breakwaters and concrete or rock groynes. The installation of a low-cost but more ecologically 'friendly' defence system of wooden dykes was envisaged but rapidly abandoned by the communal authorities because it was viewed as nefarious to tourism and recreation. The most comprehensive management options have been proposed by CETMEF ( 2004), a state-run agency in fluvial and coastal engineering expertise, and by SOGREAH ( 2006), a private hydraulic engineering firm. In a global study that recommended beach nourishment as the primary solution, CETMEF (2004) stipulated that a policy of 'no intervention' constituted a threat to the commune. This report also recongnised the limited utility of reinforcing the seawall by emplacing rock armouring because of the rampant erosion and the high cost of this defence option, as well as the relatively low potential efficiency of wooden groynes, and the problems such structures will pose for various beach activities such as kiting, kite-surfing, sand yachting and speed sailing. CETEMEF (2004) proposed sand nourishment over a distance of 2.5 km covering the central sector (Wissant town front) and extending 200 m west of the seawall. This solution was subsequently also retained by [START_REF] Sogreah | Requalification du site de la Baie de Wissant -Réensablement de la partie centrale[END_REF]. Much of the expertise proposed was based on defining an 'equilibrium' beach profile to be attained through nourishment using a variant of the SBEACH (Storminduced BEAch Change) Model of the US Army Corps of Engineers [START_REF] Larson | Sbeach: Numerical model for simulating storm-induced beach change[END_REF][START_REF] Rosati | Numerical model for simulating storm-induced beach change[END_REF] that simulates cross-shore beach, berm, and dune erosion produced by storm waves and high water levels. [START_REF] Sogreah | Requalification du site de la Baie de Wissant -Réensablement de la partie centrale[END_REF] proposed an initial renourishment to the tune of 300,000 m 3 in order to rebuild the upper beach lowered by erosion and to attain this equilibrium profile, and examined the possibility of the adjunction of groynes to hold the recharged sand against drift to the east, but this latter proposal was discarded on the grounds evoked above. An alternative solution of complementary annual nourishment to the tune of 15,000 m 3 was thus proposed. Another complementary option consisted in installing a beach drainage system using the Ecoplage ® procedure to help maintain recharged sand. The nourishment project also recommended the dismantling of the last blockhouses subsisting as part of the Atlantic seawall, and dune rehabilitation in the eroding Dune d'Aval sector. The estimated cost of these operations ranged from 6.2 to 9.05 M€, depending on optional adjunctions, maintenance and annual sand recharge over 20 years. The plan also highlighted the numerous risks of failure and uncertainties related to the operations proposed.

Discussion: confronting engineering options with processes and scales of shoreline change

Wissant Bay is an emblematic example of the problems facing many coastal communes in France wherein several layers of difficulties have resulted in stalted shoreline protection and in failure in the fight against erosion. A major problem, which is the focus of this paper, is that of confronting such solutions with morphodynamics of the coast at various scales. Confronting management options with the processes and scales involved in shoreline change is often a major challenge. This is particularly well illustrated in Wissant Bay, arguably the most strongly eroding coast in metropolitan France. The engineering solutions proposed to stave off erosion in Wissant Bay do not sufficiently take into account the complexity of the sediment dynamics of the bay. This large tidal-range setting is subject to important tide-, wind-and storm-wave-controlled fluxes that are embedded in a larger-scale sediment circulation system between the eastern English Channel and the North Sea. The shoreline rehabilitation plan proposed by SOGREAH ( 2006) is strongly centred around simulations of an equilibrium beach profile (corresponding essentially to the Wissant town front sector (Fig. 4b)) that includes the construction, via beach nourishment, of an enlarged berm, in front of the failing seawall, capable of withstanding storm attack, and the stability of which will be further enhanced by the Ecoplage operation. However, the maintenance of a complex multi-barred profile of the beach in Wissant Bay is not considered in these simulations which are indeed unlikely to adequately replicate such complex beach morphology. Storm wave dissipation is assured by: (1) the Line Bank offshore, (2) inshore by the multiple subtidal to intertidal bars, and (3) the aeolian foredune front, and not just by an enlarged beach berm and upper beach drainage. Furthermore the main beach morphodynamic process during such storms is strongly hinged on longshore sand transport over the bars, rather than on offshore sand losses. Figure 7 shows unpublished calculated sand transport rates over the multi-barred beach using the formulation by van Rijn (1990), based on wave and current data acquired in 2005 (Sedrati, 2006). The results highlight the overwhelming importance of longshore transport relative to cross-shore transport. [START_REF] Sedrati | Storm-generated morphological change and longshore sand transport in the intertidal zone of a multi-barred macrotidal beach[END_REF] showed from high-resolution beach topographic changes that these longshore transport conditions are considerably reinforced during storms. [START_REF] Anthony | Storms, shoreface morphodynamics, sand supply, and the accretion and erosion of coastal dune barriers in the southern North Sea[END_REF] has recently suggested that this longshore transport may be reinforced by longshore gradients in radiation stress generated by 3D changes in the morphology of the Line Bank from west to east. These gradients divert offshore storm flows alongshore, hence preventing sand loss offshore but strengthening sand transfers from west to east. The large-scale interactions encompassing the Line Bank offshore clearly illustrate here the difficulty, predicted by [START_REF] Sipka | Les cellules sédimentaires du littoral du Nord Pas-de-Calais, Nord de la France. Définition et applications[END_REF], of using the cell concept as a management tool on the macrotidal coasts of the Dover Strait and the southern North Sea where cell boundaries are hard to delimit both alongshore and seaward. It seems very likely that the progressive erosion affecting Wissant Bay is part of the long-term (multidecadal) shoreline response to the lowering of the Line Bank, diminishing both its capacity to dissipate storm waves and to supply sand to the beaches and foredunes in the west. As lowering of the Line Bank has occurred, storm wave energy dissipation has been largely transferred to the bar-trough beach and the foredune front, notably in the deeper western sector. [START_REF] Aernouts | L'évolution du rivage et des petits fonds en Baie de Wissant pendant le XXè siècle, Pas-de-Calais, France[END_REF] showed from a SWAN wave propagation model simulation that incident wave energy over the bank had increased in 2002 relative to 1977 due to the lower bank surface. The beach and dunes adjacent to the deeper western end of the bank have, therefore, been rapidly retreating. Although the thrust of the anti-erosion plan proposed by SOGREAH ( 2006) is on cross-shore dissipation of storm wave energy, the plan does recognise the need to contain longshore sand transport but only via a short-term (with costs exponentially increasing from five to 20 years) renourishment plan. There is no vision of what may happen beyond this period. The past changes in shoreline retreat and advance that have affected Wissant Bay have been viewed in terms of a multidecadal to secular shoreline rotation process [START_REF] Sedrati | Sediment dynamics and morphological change on the upper beach of a multi-barred macrotidal foreshore, and implications for mesoscale shoreline retreat: Wissant Bay, northern France[END_REF]). There appears to be little scope for reversal of this sand transfer process in the future, given the effectively strong large-scale residual drift to the east. Rotation is generally reversible, but also operational on shorter timescales of seasons (e.g., [START_REF] Norcross | Annual and interannual changes on a reef-fringed pocket beach: Kailua Bay, Hawaii[END_REF][START_REF] Jeanson | Wave characteristics and the morphology of pocket beaches fronted by a coral reef-lagoon system, Mayotte Island, Indian Ocean[END_REF] to years [START_REF] Thomas | Medium time-scale behaviour of adjacent embayed beaches: Influence of low energy external forcing[END_REF]. The strong drift to the east that reinforces the destabilisation of the western sector of the bay is also due to combined wave, tide and wind-induced currents, thus differentiating the Wissant Bay system from many of the beach rotation examples described in the literature, driven by changes in combined incident wave energy and direction. Since one of the two trailing edges of the Line Bank is close inshore in the Dune d'Amont sector, it probably provides sand for the coastal dunes and shelters the shore from the larger storm waves. The loss of sand by this bank and the scale of longshore sand transfer to the east may suggest that even an expensive long-term renourishment solution based on constant truckload transfers to the western beach sector of sand accumulating feeding foredune accretion in the east may not be tenable in terms of cost, nor given the ecological status of the bay dunes. Without a consideration of the rehabilitation or renourishment of the Line Bank sand source and wave energy dissipater, such a renourishment solution may also not be enough to counter the progressive retreat of the dunes in the west of the bay. The anti-erosion and rehabilitation plan proposed by [START_REF] Sogreah | Requalification du site de la Baie de Wissant -Réensablement de la partie centrale[END_REF] does not concern aeolian dune rehabilitation in the eroding western sector. Two other problems facing the fight against erosion in Wissant Bay are a clear definition of responsibilities in coastal management and in the implementation of engineering solutions against erosion, and finding funds to finance such engineering solutions. Lobbying against erosion has been spearheaded by the landowners' association in Wissant Bay. A French law of 1807 stipulated the setting up of landowner associations that had to bear, proportionately to the interests of each landowner, all costs of defence or maintenance works carried out on the shore or along river banks. Exceptions concerned sectors where government interests were concerned, thereby providing a source of state subsidy for such works. Enforcement of this law was never really assured and under the continuing pressure of coastal urbanisation, generally spurred on by large-scale lucrative estate acquisition and development, the state decided, in the Law of 1973, to allow individual communes or unions of communes to undertake coastal defence works where this was deemed necessary to preserve the common interest. This is presently the situation in France, where the municipality or commune bears the costs of local defence operations, with the possibility of additional funding by the Regional Council. Authorisations concerning the implementation of defence works on urbanised coasts, and decisions as to whether state funding is appropriate, are taken by a regional engineer delegate of the state directorate responsible jointly for matters of environment, territorial management, and housing. State funding is exceptional. The problem is compounded in Wissant Bay by the high costs involved in even implementing a plan such as that proposed by [START_REF] Sogreah | Requalification du site de la Baie de Wissant -Réensablement de la partie centrale[END_REF].

van Rijn (2011) indicated as an acceptable cost a range of 100-150 € per metre of coast per year over a 20-year period for 100 km of the Holland coast, a country where the fight against coastal erosion is as much a tradition as a national priority. The cost of the proposed Wissant Bay plan is 125-362 € per metre of coast over a 20 year period depending on options and effective maintenance operations, a cost well beyond the possibilities of the commune. Effective implementation of the plan has been delayed by lack of funds and it is doubtful whether the plan will be ever implemented. Hence the continuing erosion and the piecemeal repairs to the seawall.

Conclusion

The case of Wissant Bay illustrates a common situation in France, and in many other countries, where problems of shoreline defence and rehabilitation are considered in a short-term, piecemeal perspective that is often hinged on local problems that are not viewed within the larger context of spatiotemporal shoreline change and the processes involved. In France, there has generally been lacking an overall view of management practice in terms, for instance, of coastal sediment cells, although this situation has been changing in the last few years. As a result, the spread of beach erosion has commonly been aggravated by individual communal efforts lacking a common view of what exactly is happening, and the effects on downdrift sectors, of engineering structures implanted in updrift sectors. The problem has been exacerbated by the high costs involved in implementing a plan against erosion. The implementation of the anti-erosion operations proposed by SOGREAH ( 2006) has been delayed by lack of funds and it is doubtful whether these operations will ever see the light of day. Should one conclude from this that the cost of defending Wissant is too high relative to the value of the resort? This raises the delicate issue of considering other management options such as set-back lines (e.g., [START_REF] Ferreira | An integrated method for the determination of set-back lines for coastal erosion hazards on sandy shores[END_REF], options that may not be readily accepted by the communal authorities of Wissant. The synopsis shows a shift from a situation of relative sand abundance in front of the wall, with sand even masking it (1952,1986), to one of beach erosion and damage to the seawall between 2002 and 2010. Note the rock armouring emplaced to protect the front of the wall in the 2010 photograph. Figure 7. Sand transport rates calculated using the van Rijn (1993) formula, from data obtained from current meters deployed in January 2005 over two bars submerged at high tide in the eroding Dune d'Aval sector, with peaks expressing the dominant longshore transport (a); the strong relationship between these rates and longshore current velocities (b). From Sedrati (2006).

Figure captions

 captions Figure captionsFigure 1. Wissant Bay in the southern North Sea facing the Dover Strait (a); sketch showing the shoreline status of Wissant Bay and the Line Bank offshore (b).Figure 2. Offshore wave heights over a one-year period from the Sandettie lightship (UK Met Office) (a), and a two-week record of wind conditions, inshore water levels, wave conditions, and mean longshore and cross-shore currents in the eroding Dune d'Aval sector, January 2005. From Sedrati, 2006. Figure 3. Rates of shoreline change in Wissant Bay calculated for various time slices between 1949 and 2000, showing the strong fluctuations both alongshore and over time. From Chaverot et al. (2008). The townfront seawall has acted as a rampart against shoreline retreat. Note that longshore transect gaps in top figure are not reproduced in bottom figure.Figure 4. An 8-year dataset of beach profiles representing the three sectors of the bay: (a) Dune d'Aval, (b) Wissant town front, (c) Dune d'Amont. The data reported here concern monitoring between 1996 and 2005 at intervals of six months. Profile data are missing for the years from 2001 to 2003. The profiles were surveyed using a high-resolution TC 407 Leica total station with errors within  3 mm for distance and elevation and 0.0015° for direction. An uncertainty margin of 5 cm, covering both field measurement and interpolation errors and uncertainties, was applied in the treatment of the raw profile data. All surveys were referenced to IGN 69 benchmarks of the French national datum. Figure 5. Profile volume changes over the 8-year survey for the three sectors of the bay: (a) Dune d'Aval, (b) Wissant town front, (c) Dune d'Amont. Figure 6. Photographs of the seawall fronting Wissant town: (a) 1906; (b) 1952; (c) 1986; (d) 2002; (e) 2007; (f) 2010.The synopsis shows a shift from a situation of relative sand abundance in front of the wall, with sand even masking it(1952, 1986), to one of beach erosion and damage to the seawall between 2002 and 2010. Note the rock armouring emplaced to protect the front of the wall in the 2010 photograph. Figure7. Sand transport rates calculated using the van Rijn (1993) formula, from data obtained from current meters deployed in January 2005 over two bars submerged at high tide in the eroding Dune d'Aval sector, with peaks expressing the dominant longshore transport (a); the strong relationship between these rates and longshore current velocities (b). FromSedrati (2006).

Figure 2 .

 2 Figure captionsFigure 1. Wissant Bay in the southern North Sea facing the Dover Strait (a); sketch showing the shoreline status of Wissant Bay and the Line Bank offshore (b).Figure 2. Offshore wave heights over a one-year period from the Sandettie lightship (UK Met Office) (a), and a two-week record of wind conditions, inshore water levels, wave conditions, and mean longshore and cross-shore currents in the eroding Dune d'Aval sector, January 2005. From Sedrati, 2006. Figure 3. Rates of shoreline change in Wissant Bay calculated for various time slices between 1949 and 2000, showing the strong fluctuations both alongshore and over time. From Chaverot et al. (2008). The townfront seawall has acted as a rampart against shoreline retreat. Note that longshore transect gaps in top figure are not reproduced in bottom figure.Figure 4. An 8-year dataset of beach profiles representing the three sectors of the bay: (a) Dune d'Aval, (b) Wissant town front, (c) Dune d'Amont. The data reported here concern monitoring between 1996 and 2005 at intervals of six months. Profile data are missing for the years from 2001 to 2003. The profiles were surveyed using a high-resolution TC 407 Leica total station with errors within  3 mm for distance and elevation and 0.0015° for direction. An uncertainty margin of 5 cm, covering both field measurement and interpolation errors and uncertainties, was applied in the treatment of the raw profile data. All surveys were referenced to IGN 69 benchmarks of the French national datum. Figure 5. Profile volume changes over the 8-year survey for the three sectors of the bay: (a) Dune d'Aval, (b) Wissant town front, (c) Dune d'Amont. Figure 6. Photographs of the seawall fronting Wissant town: (a) 1906; (b) 1952; (c) 1986; (d) 2002; (e) 2007; (f) 2010.The synopsis shows a shift from a situation of relative sand abundance in front of the wall, with sand even masking it(1952, 1986), to one of beach erosion and damage to the seawall between 2002 and 2010. Note the rock armouring emplaced to protect the front of the wall in the 2010 photograph. Figure7. Sand transport rates calculated using the van Rijn (1993) formula, from data obtained from current meters deployed in January 2005 over two bars submerged at high tide in the eroding Dune d'Aval sector, with peaks expressing the dominant longshore transport (a); the strong relationship between these rates and longshore current velocities (b). FromSedrati (2006).

Figure 4 .

 4 Figure captionsFigure 1. Wissant Bay in the southern North Sea facing the Dover Strait (a); sketch showing the shoreline status of Wissant Bay and the Line Bank offshore (b).Figure 2. Offshore wave heights over a one-year period from the Sandettie lightship (UK Met Office) (a), and a two-week record of wind conditions, inshore water levels, wave conditions, and mean longshore and cross-shore currents in the eroding Dune d'Aval sector, January 2005. From Sedrati, 2006. Figure 3. Rates of shoreline change in Wissant Bay calculated for various time slices between 1949 and 2000, showing the strong fluctuations both alongshore and over time. From Chaverot et al. (2008). The townfront seawall has acted as a rampart against shoreline retreat. Note that longshore transect gaps in top figure are not reproduced in bottom figure.Figure 4. An 8-year dataset of beach profiles representing the three sectors of the bay: (a) Dune d'Aval, (b) Wissant town front, (c) Dune d'Amont. The data reported here concern monitoring between 1996 and 2005 at intervals of six months. Profile data are missing for the years from 2001 to 2003. The profiles were surveyed using a high-resolution TC 407 Leica total station with errors within  3 mm for distance and elevation and 0.0015° for direction. An uncertainty margin of 5 cm, covering both field measurement and interpolation errors and uncertainties, was applied in the treatment of the raw profile data. All surveys were referenced to IGN 69 benchmarks of the French national datum. Figure 5. Profile volume changes over the 8-year survey for the three sectors of the bay: (a) Dune d'Aval, (b) Wissant town front, (c) Dune d'Amont. Figure 6. Photographs of the seawall fronting Wissant town: (a) 1906; (b) 1952; (c) 1986; (d) 2002; (e) 2007; (f) 2010.The synopsis shows a shift from a situation of relative sand abundance in front of the wall, with sand even masking it(1952, 1986), to one of beach erosion and damage to the seawall between 2002 and 2010. Note the rock armouring emplaced to protect the front of the wall in the 2010 photograph. Figure7. Sand transport rates calculated using the van Rijn (1993) formula, from data obtained from current meters deployed in January 2005 over two bars submerged at high tide in the eroding Dune d'Aval sector, with peaks expressing the dominant longshore transport (a); the strong relationship between these rates and longshore current velocities (b). FromSedrati (2006).

Acknowledgements

We thank three anonymous reviewers and Norbert Psuty for their salient suggestions for improvement.