Marc Fleurbaey
email: mfleurba@princeton.edu.

Stéphane Zuber
email: stephane.zuber@univ-paris1.fr.

A new puzzle in the social evaluation of risk

Keywords: social welfare, risk, social rationality, Pareto, fairness

We highlight a new paradox for the social evaluation of risk that bears on the evaluation of individual well-being rather than social welfare, but has serious implications for social evaluation. The paradox consists in a tension between rationality, respect for individual preferences, and a principle of informational parsimony that excludes individual risk attitudes from the assessment of riskless situations. No social evaluation criteria can satisfy these three principles. This impossibility result has implications for the evaluation of social welfare under risk, especially when the preferences of some individuals are not known. It generalizes existing impossibility results, while relying on very weak principles of social rationality and respect for individual preferences. We explore the possibilities opened by weakening one of our three principles and discuss the advantages and drawbacks of these different routes.

1 Introduction [START_REF] Harsanyi | Cardinal welfare, individualistic ethics, and interpersonal comparisons of utility[END_REF] theorem states that, in the context of risk, if social evaluation and individual preferences all take the form of an expected utility, and social evaluation respects individual preferences by satisfying a version of the Pareto principle, then social evaluation must take the form of a weighted sum of the individuals' von Neumann-Morgenstern (VNM) utilities. While Harsanyi and many scholars celebrated this theorem as vindicating utilitarianism, others have argued that one could relax the expected utility assumption for social welfare in order to accommodate considerations of ex ante fairness [START_REF] Diamond | Cardinal welfare, individualistic ethics, and interpersonal comparisons of utility. Comment[END_REF], or that one could weaken the Pareto principle in order to accommodate considerations of ex post fairness [START_REF] Fleurbaey | Assessing risky social situations[END_REF]). Ultimately, one can read this literature as revealing a tension between social rationality, respect for individual preferences, and fairness. In this literature, fairness is defined as aversion to inequality of some sort (either over individual prospects or over final consumption levels). This aversion to inequality may differ from what is implied by utilitarianism: under utilitarianism, there is no aversion to inequalities in individual expected utilities; on the other hand, inequality aversion over final consumption levels is exactly determined by individuals' aversion to risk (in the sense of Arrow and Pratt).

In this paper, we present an even more fundamental tension, which bears on the assessment of individual well-being rather than social welfare. It is a tension between rationality, respect for individual preferences, and a principle of informational parsimony that excludes individual risk attitudes from the assessment of riskless situations. The principles of rationality and respect for preference principles that we adopt in this paper are Documents de travail du Centre d'Economie de la Sorbonne -2020. 28 fairly intuitive and compelling: in particular, they are milder than other requirements in the literature and avoid objections of spurious unanimity [START_REF] Mongin | Spurious unanimity and the Pareto principle[END_REF]. The principle of the informational parsimony is less familiar and we need to justify why it is also worth considering.

The principle of informational parsimony states that, in situations not involving risk, we do not need to know people's attitudes to risk in order to compare individual well-being levels.

Harsanyi's utilitarianism relies on individuals ' Von Neumann Morgenstern (henceforth VNM) utilities, and thus on attitudes towards risk, for interpersonal comparisons. Many applied allocation problems, however, are studied without incorporating risk in the analysis.

Sometimes analysts (e.g., Stern 2017) invoke risk attitudes to calibrate the utility functions that are used, even if there is no explicit risk in the contemplated problem. But most often, risk-free problems are studied on the sole basis of preferences over the dimensions at hand, if only because preferences over extra dimensions are not observable in the context that is studied. Many evaluation criteria in cost-benefit analysis and in fair allocation theory rely solely on individual ordinal preferences over commodities and do not bother to examine individual risk attitudes. Our paper shows that either these evaluation criteria are deeply flawed, or they cannot be extended to risky contexts without clashing with compelling principles of rationality and respect for preferences.

Although our main argument is directed towards the evaluation (and comparison) of individual well-being, it has serious implications for the evaluation of collective risk. Indeed, such evaluation generally relies on the evaluation of the situation of the individuals involved in the problem, at least if some unanimity principle like the Pareto principle holds. Our paper Documents de travail du Centre d'Economie de la Sorbonne -2020.28 thus also contributes to the literature on the aggregation of preferences under risk and uncertainty and presents a new impossibility result in that context.

The paper is structured as follows. In Section 2 we introduce and prove the impossibility result. In Section 3 we discuss the various options that are available when one of the conditions of the result is relaxed. Section 4 examines how this problem bears on the evaluation of social, as opposed to individual, situations. Section 5 contains a discussion and a tentative conclusion.

The impossibility

Consider a single individual who is facing either sure options 𝑥 in a set 𝑋 or acts 𝑎 that map any states of the world 𝑠 from a finite set 𝑆 into the set 𝑋. Acts are the standard way of representing risk in the [START_REF] Savage | The Foundations of Statistics[END_REF] approach, and we assume that there are only finitely many states for simplicity. For each state 𝑠 and for an act 𝑎, 𝑎(𝑠) is the option obtained in state 𝑠 when choosing 𝑎: it is therefore an element of 𝑋. For simplicity, we assume that each state 𝑠 has a well-defined positive probability. 1 We assume that the individual's preferences over 𝑋 and over 𝐴 can be represented with the help of a VNM utility function 𝑢. The individual thinks that 𝑥 is at least as good as 𝑥′ if and only if 𝑢(𝑥) ≥ 𝑢(𝑥 , ), and that 𝑎 is at least as good as 𝑎′ if and only if 𝐸𝑢(𝑎) ≥ 𝐸𝑢(𝑎 , ), where 𝐸𝑢 denotes the expected utility.

We do not actually need to assume that individuals are themselves expected utility maximizers. We only posit that the evaluation of risky individual prospects relies on expected utility, and this can be a normative stance for the evaluator rather than an empirical assumption about how individuals behave. When we talk about respecting individuals' risk attitudes in this paper, this is more fundamentally about respecting the best evaluation of their interests, not necessarily about respecting their possibly flawed attitudes.

For instance, they may be ignorant of the probabilities and exhibit ambiguity aversion, or display loss aversion and status quo bias. The best evaluation of their interests is supposed to eliminate such behavioral phenomena and take the most rational approach. 2 The problem we consider is to assess the well-being of this individual in situations described by an option (a sure one or a risky prospect) and a VNM utility function. We want to allow for the possibility that the individual may have different utility functions. Such a possibility would be essential to make interpersonal comparisons,3 but it is also important for a single individual, for instance if we think of assessing the situation of a future individual whose preferences are yet unknown. The concrete question that motivated our research on this problem is the question of sustainability when we do not know whether future generations will care about the environment more than the present generation or will adapt to the degraded environment that we prepare for them.

Formally, we seek an ordering ≽ over situations 𝑥, 𝑢 and 𝑎, 𝑢 . Here are the three principles we would like the ordering ≽ to satisfy. Hereafter, the notation 𝑎, 𝑢 ≽ 𝑎′, 𝑢′ 2 Our main argument would however carry over to situations where individuals' best interest itself may even not be represented by an expected utility. The expected utility model is convenient because it identifies a specific utility function u that represents people risk attitudes. The description of comparative risk attitudes beyond the expected utility model are more complicated (see for instance [START_REF] Chateauneuf | More pessimism than greediness: A characterization of monotone risk aversion in the rank-dependent expected utility model[END_REF] for the rank-dependent expected utility model).

means that an individual with VNM utility function 𝑢 facing an act 𝑎 is better-off than an individual with VNM utility function 𝑢 , facing an act 𝑎 , .

The first principle is a very basic rationality principle in the presence of risk. It stipulates that if an option is better than another in all possible states, then it is better. This is not a substantive principle, it just imposes consistency between how the ordering assesses prospects ex ante and how it assesses final consequences ex post.

Statewise dominance: For all 𝑎, 𝑎 , , 𝑢, 𝑢 , :

if 𝑎(𝑠), 𝑢 ≽ 𝑎′(𝑠), 𝑢′ for all 𝑠, then 𝑎, 𝑢 ≽ 𝑎′, 𝑢′ .

The second principle is that for a fixed utility function, one should respect the individual's preferences over acts. Since there is only one individual under consideration, the objections against the Pareto principle that invoke inequalities do not apply here. We will discuss how social considerations, in particular inequalities, can be taken into consideration in section 4 of this paper.

Pareto: For all 𝑎, 𝑎 , , 𝑢: 𝐸𝑢(𝑎) ≥ 𝐸𝑢(𝑎 , ) if and only if 𝑎, 𝑢 ≽ 𝑎 , , 𝑢 . This Pareto principle is related to the "acceptance principle" put forward by Harsanyi in his impartial observer's argument for utilitarianism [START_REF] Harsanyi | Rational Behavior and Bargaining Equilibrium in Games and Social Situations[END_REF]. The acceptance principle applies to situations where the impartial observer knows which identity she will assume for sure. It requires that, in this case, the impartial observer's preferences must coincide with that individual's preferences that are represented by the individual's expected utility.

Observe that the Pareto principle does not imply that ex ante evaluations always rely on an Documents de travail du Centre d'Economie de la Sorbonne -2020.28 expected value of ex post consequences (or, said differently, that the social criterion satisfies the VNM independence principle). The Pareto principle implies that social comparisons take the expected utility form only for comparisons of acts involving the same individual but not when we compare two individuals. In section 3, we provide an example of social comparisons satisfying the Pareto principle but not Statewise dominance (and therefore they are not of the expected utility type).

The last principle says that only preferences over 𝑋 matter for the assessment of sure options. In other words, risk attitudes are irrelevant in this context. Formally, this is expressed by considering utility functions that have the same preference ordering over 𝑋.

We thus introduce the following terminology: we say that two utility functions 𝑢 and 𝑢 , are ordinally equivalent on 𝑋 if for all 𝑥, 𝑥′ in 𝑋, 𝑢(𝑥) ≥ 𝑢(𝑥 , ) if and only 𝑢′(𝑥) ≥ 𝑢′(𝑥 , ). When 𝑢 and 𝑢 , are ordinally equivalent on 𝑋, they yield the same preference ordering over 𝑋.

Then, the next principle states that they yield equivalent situations for sure options (the symbol ∼ denotes the "equally good" relation generated by ≽).

Irrelevance of risk attitudes in absence of risk:

For all 𝑢, 𝑢 , :

If 𝑢 and 𝑢 , are ordinally equivalent on 𝑋, then for all 𝑥 in 𝑋, 𝑥, 𝑢 ∼ 𝑥, 𝑢′ .

The above principle is related to a widely-used principle in the theory on fair allocation of goods, namely the principle of equal treatment of equals (see Thomson, 2011). Equal treatment of equals stipulates that two individuals with the same preferences should be treated in the same way (and thus have the same well-being level at a given allocation).

Irrelevance of risk attitudes in absence of risk adds the condition that the preferences Documents de travail du Centre d'Economie de la Sorbonne -2020.28 considered in that case are only those on sure options. This is consistent with our observation, made in the introduction, that risk-free problems are usually studied on the sole basis of preferences over the risk-free outcomes, without any considerations of the risk preferences. A similar line of argument has appeared in [START_REF] Chambers | When does aggregation reduce risk aversion?[END_REF].

Irrelevance of risk attitudes in absence of risk can also be related to the idea, put forward by [START_REF] Moreno-Ternero | The veil of ignorance violates priority[END_REF], that priority should be given to individuals on the basis of some objective notion of wellbeing when there is no risk. Irrelevance of risk attitudes in absence of risk also implies that we have some notion of wellbeing for non-risky outcomes that is independent of the risk attitudes as represented by the VNM utility function. In our case, contrary to [START_REF] Moreno-Ternero | The veil of ignorance violates priority[END_REF], this wellbeing measure may depend on preferences over non-risky outcomes and sure options.

We can now state our impossibility result.

Theorem: There is no ordering that satisfies Statewise dominance, Pareto and Irrelevance of risk attitudes in absence of risk.

Proof:

The proof is very simple. Consider a pair of acts 𝑎, 𝑎′, and two VNM functions 𝑢, 𝑢′ that represent the same preferences over 𝑋, but not the same risk attitudes, so that 𝐸𝑢 𝑎 = 𝐸𝑢(𝑎 , ) but 𝐸𝑢′(𝑎) ≠ 𝐸𝑢′(𝑎 , ). By Pareto, one has 𝑎, 𝑢 ∼ 𝑎′, 𝑢 and not 𝑎, 𝑢′ ∼ 𝑎′, 𝑢′ .

Pick a state 𝑠. By irrelevance of risk attitudes in absence of risk, 𝑎(𝑠), 𝑢 ∼ 𝑎(𝑠), 𝑢′ . Since this is true for every 𝑠, by statewise dominance one has 𝑎, 𝑢 ∼ 𝑎, 𝑢′ . By the same reasoning, 𝑎′, 𝑢 ∼ 𝑎′, 𝑢′ . This shows that risk attitudes are irrelevant even in the context of risk. Documents de travail du Centre d'Economie de la Sorbonne -2020.28

Recall that 𝑎, 𝑢 ∼ 𝑎′, 𝑢 . By transitivity, one therefore has 𝑎, 𝑢′ ∼ 𝑎′, 𝑢′ , which yields a contradiction. QED.

This proof shows that one does not need the full force of the Pareto condition to obtain the impossibility. It would be sufficient to posit that risk attitudes do make a difference in at least one case, since the bulk of the proof consists in showing that risk attitudes never matter.

The logic of the argument is that after uncertainty has been resolved, one should use the evaluation that one would use in absence of risk. But if such evaluation does not rely on risk attitudes, then statewise dominance implies that risk attitudes never matter. What is not intuitive about the result is that it seems that it should be possible to invoke risk attitudes at least when trading-off the possible outcomes in different states. But the evaluation bears on pairs (act,utility function). Statewise dominance implies that if changing the individual's risk attitude in such a pair never makes a difference in any state of the world, it cannot make a difference at all.

The next section explores the options that become available when one of the three conditions is dropped.

Possibilities

Let us first examine what is possible when statewise dominance is dropped. In fact, there is a very precise possibility, because the other two conditions leave only one class of criteria possible. Rather than stating a characterization theorem, which would involve more notations and formalism, let us explain the reasoning. First, irrelevance of risk attitudes in Documents de travail du Centre d'Economie de la Sorbonne -2020. 28 absence of risk implies that the evaluation of situations 𝑥, 𝑢 does not depend on risk attitudes. This is given by an ordering ≽ such that 𝑥, 𝑢 ∽ 𝑥, 𝑢 , whenever 𝑢 and 𝑢 , are ordinally equivalent and 𝑥, 𝑢 ≽ 𝑥 , , 𝑢 whenever 𝑢(𝑥) ≥ 𝑢(𝑥 , ).

On the other hand, under some basic continuity assumptions, that will not be spelled out here in detail, for every risky situation 𝑎, 𝑢 , there is at least one sure option 𝑥(𝑎, 𝑢) that is equally good for the individual, i.e., such that 𝑢 𝑥(𝑎, 𝑢) = 𝐸𝑢(𝑎). Let us call this 𝑥(𝑎, 𝑢) a "certainty-equivalent" of the situation 𝑎, 𝑢 . The Pareto condition implies that 𝑥(𝑎, 𝑢), 𝑢 ∼ 𝑎, 𝑢 . Therefore, only one type of evaluation is possible: for every risky situation, find a certaintyequivalent, and evaluate the certainty-equivalent situations with a criterion that is independent of risk attitudes. 4 This defines a class of orderings, and in this class the various orderings differ by the criterion that is used for riskless situations. But once the criterion for riskless situations is chosen, the evaluation of risky situations is fully determined.

Let us note that, although it violates statewise dominance in general, this approach is consistent with statewise dominance when we consider situations with exactly the same preferences. Indeed, in that case a situation will be better if the certainty equivalent gives higher utility (in terms of the VNM utility function), which is equivalent to the expected utility being higher -but this is only true because the two situations have the same VNM utility function. 4 Formally, the criteria would be as follows: for all 𝑎, 𝑎 , , 𝑢 and 𝑢 , , (𝑎, 𝑢) ≽ (𝑎 , , 𝑢 , ) if and only if 𝑥 𝑎, 𝑢 , 𝑢 ≽ 𝑥 𝑎 , , 𝑢 , , 𝑢 , . When 𝑢 = 𝑢 , , the condition boils down to 𝑢 𝑥 𝑎, 𝑢 ≥ 𝑢 𝑥 𝑎 , , 𝑢 , which by definition of the certainty-equivalent can be written 𝐸𝑢(𝑎) ≥ 𝐸𝑢(𝑎 , ). The Pareto principle is satisfied. When 𝑎 = 𝑎 , = 𝑥, we have 𝑥 𝑎, 𝑢 = 𝑥 𝑎 , , 𝑢 , = 𝑥 and thus the condition is equivalent to (𝑥, 𝑢) ≽ (𝑥, 𝑢 , ). Documents de travail du Centre d'Economie de la Sorbonne -2020. 28 What can go wrong with this approach? After all, certainty-equivalence is not uncommon in welfare economics, and there is for instance a popular criterion, the [START_REF] Epstein | Substitution, risk Aversion, and the temporal behavior of consumption and asset returns: A theoretical framework[END_REF] utility, that makes heavy use of it for the assessment of acts unfolding over time in a sequential way. What is bizarre about this approach, in our setting, is the following. Consider 𝑢 that is more risk-averse than 𝑢′, 5 and an act 𝑎 that gives a better outcome in every state of the world than 𝑎′. Since 𝑢 is more risk-averse than 𝑢′, it is possible that the certaintyequivalent of 𝑎, 𝑢 is worse than the certainty-equivalent of 𝑎′, 𝑢′ . Is it really true, then, that 𝑎, 𝑢 is worse than 𝑎′, 𝑢′ in spite of its guaranteeing a better outcome? This may not seem outlandish in view of the greater risk aversion with 𝑢, but this violation of statewise dominance is bound to raise thorny issues when evaluating well-being along a lifetime. If the certainty-equivalent is computed successively at different periods, time inconsistent evaluations are likely to occur. If, in order to avoid such problems, one wants to only refer to the certainty-equivalent computed at a single period, this opens a wedge between the evaluation and the available information at the time of evaluation.

Let us illustrate this problem with a simple case where options are income levels and there are two equally likely states of the world. One can then assume that VNM functions are increasing functions of income, so that all VNM functions are ordinally equivalent (they all have the same ordering of income levels: more income is always better). Consider the 5 Risk-aversion here is defined here in the sense of [START_REF] Arrow | Essays in the Theory of Risk Bearing[END_REF] and Pratt (1964), and generalized to multidimensional outcomes by [START_REF] Kihlstrom | Risk aversion with many commodities[END_REF]. These definitions hold for the expected utility model. The literature in philosophy often considers risk-aversion in a non-expected utility model, where probabilities are being transformed in some way (see Buchak, 2013). Some people argue that we should only talk about risk aversion in the second case [START_REF] Hansson | Risk aversion as a problem of conjoint measurement[END_REF]. In this paper, we adopt a more general view that risk-aversion can be defined more generally with respect to some relevant metric (in our example, this metric is money). Act 𝑎′ 9000 37000

Clearly, with the information that we may have after the state of the world is known, the options produced by act 𝑎 are better: in state 1 and state 2 it produced more income, which is the only relevant dimension for social evaluation (we assume that individuals are the same in all respects: they have the same needs and none of them deserves more). It may thus appear appealing to prefer act 𝑎. But this is not necessarily the case if we adopt the criterion based on the certainty-equivalent. Assume for instance that 𝑢 is such that 𝑢 𝑥 = 𝑥 6 7 for all income levels 𝑥, while 𝑢 , 𝑥 = 𝑥. It can be showed that the certainty equivalent of 𝑎, 𝑢 corresponds to an income of 22500, while the certainty-equivalent of 𝑎′, 𝑢′ is 23000.

Given that the only criterion we have that is independent of risk attitudes is to compare income levels, we should thus prefer 𝑎′, 𝑢′ based on the information we have before the state of the world is known. This inconsistency between the social evaluation before or after the state of the world is known, which is due to a violation of statewise dominance, seriously undermines the credibility of any evaluation.

To be more concrete, let us consider the case of an education program that can reduce the risk aversion of future people. This program has a cost that will be borne by the future people themselves. This cost makes them ultimately worse off, and we know this for sure ex satisfies statewise dominance because it relies on expected utility, and it satisfies irrelevance of risk attitudes in absence of risk by construction.

The example above is just one example, but to get the entire class of solutions that are available with the weakening of Pareto to riskless options, one only has to include all decision criteria under risk that satisfy statewise dominance but do not take the form of an expected utility.

Nevertheless, the fact that this approach does not respect risk attitudes in general may make it evaluate risky prospects in a starkly different way than the individual. Unless there is a reason not to trust risk attitudes, this seems quite disrespectful. Imagine for instance that the normalized utilities are risk neutral in some relevant sense whereas the individual is very risk averse. Such an approach would then vastly overestimate the well-being of the individual when facing very high risks with positive expected payoffs but substantial probability of a high loss.

Finally, the third possibility consists in dropping Irrelevance of risk attitudes in absence of risk. The simplest criterion is then to rely simply on 𝐸𝑢(𝑎) for the evaluation of (𝑎, 𝑢), but this may be simplistic, because the VNM functions are defined by risk attitudes only up to an affine transform. Therefore, it is probably more interesting to think that, again, a normalization is in order, though one that respects not only preferences on 𝑋 but also risk attitudes. Concretely, this means that, for each utility function 𝑢, there would be two numbers 𝛼 > > 0 and 𝛽 > , such that: 𝑎, 𝑢 ≽ 𝑎 , , 𝑢′ if and only if 𝐸 𝛼 > 𝑢 𝑎 + 𝛽 > ≥ 𝐸 𝛼 >, 𝑢′ 𝑎 , + 𝛽 >, (𝑢′) . An analysis of the normalization of VNM utilities in applications of utilitarianism is made in [START_REF] Fleurbaey | Fair utilitarianism[END_REF].

Documents de travail du Centre d'Economie de la Sorbonne -2020.28

The approach defined in the previous paragraph would be the only acceptable one if statewise dominance was strengthened into requiring that the ordering ≽ be itself representable by an expected utility. Since it seems reasonable to us that expected utility is the most rational criterion under risk, we consider that this approach is therefore the one that deserves most attention when irrelevance of risk attitudes in absence of risk is dropped.

One may wonder if, like Pareto, irrelevance of risk attitudes in absence of risk could be weakened rather than completely dropped. But for riskless options, the formula 𝛼 > 𝑢 𝑥 + 𝛽 > depends on the curvature of 𝑢 in a crucial way and it appears impossible to define an interesting subset of cases in which risk attitudes would not influence the evaluation.

Implications for social evaluation

The impossibility highlighted in this paper is really about the evaluation of individual situations, but it undermines the construction of social preferences, because social preferences rely on interpersonal comparisons as an essential building block.

One can also reformulate the impossibility theorem in terms of the evaluation of social risk based on some social preference ordering. Let us thus seek an ordering, still denoted ≽, over social situations, which are either sure situations 𝑥, 𝑢 B B∈D or acts 𝑎, 𝑢 B B∈D , where 𝑖 is the index denoting an individual and 𝑃 is the population.

The axiom of Statewise dominance can be kept unchanged, except for notation details. The axiom of Irrelevance of risk attitudes in absence of risk can also be kept, specifying that each But how should inequalities be measured? Let us assume that there is a given measure of well-being for interpersonal comparisons, a specific utility function 𝑣 B for every 𝑖. In all generality, this function may or may not be identical to 𝑖's VNM utility. Since it is about comparing individuals in final states, one should expect that, in the presence of the axiom of Irrelevance of risk attitudes in absence of risk, the function 𝑣 B does not depend on risk attitudes. But we do not make this assumption at the outset. However, we assume that for every 𝑢 function there is only one 𝑣 function.

We obtain the following Pareto axiom:

Pareto for equal risk: For all 𝑎, 𝑎 , , 𝑢 B B∈D such that for all 𝑠, all 𝑖, 𝑗 ∈ 𝑃, 𝑣 B 𝑎(𝑠) = 𝑣 I 𝑎(𝑠) and 𝑣 B 𝑎′(𝑠) = 𝑣 I 𝑎′(𝑠) :

-𝑎, 𝑢 B B∈D ≽ 𝑎′, 𝑢 B B∈D if for all 𝑖, 𝐸𝑢 B 𝑎 ≥ 𝐸𝑢 B 𝑎 , ;

-𝑎, 𝑢 B B∈D ≻ 𝑎′, 𝑢 B B∈D if, in addition, for some 𝑖, 𝐸𝑢 B 𝑎 > 𝐸𝑢 B 𝑎 , .

We now have the following impossibility.

Theorem: There is no ordering that satisfies the three conditions.

Proof: The proof, whose details are omitted here, mimics the previous proof. [START_REF] Fleurbaey | Fair management of social risk[END_REF]. If Statewise dominance is strengthened to an Expected utility requirement for the social ordering, Fleurbaey and Zuber (2017, Theorem 1) show that we obtain a specific class of criteria that work in two steps. First, in each state of the world we compute an equally distributed equivalent option, that is, an egalitarian option (in the sense of the 𝑣 B functions) that is equivalent from the social point of view.

Then we take an affine aggregation of individual expected utility at this egalitarian prospect, using the actual VNM functions of individuals. Documents de travail du Centre d'Economie de la Sorbonne -2020.28

Discussion and conclusion

The conundrum raised by our impossibility theorem relates to two strands of the literature.

As already mentioned in the introduction, Harsanyi's theorem [START_REF] Harsanyi | Cardinal welfare, individualistic ethics, and interpersonal comparisons of utility[END_REF]) reveals a tension between social rationality, Pareto and fairness (both ex ante and ex post). In this tension, fairness is meant to refer to inequality aversion either with respect to individual expected utilities (ex ante fairness) or to final well-being (ex post fairness). Our axiom of irrelevance of risk attitudes in absence of risk is not about inequality aversion at all, but it does relate to fairness indirectly. As discussed before, in the theory of fair allocation, interpersonal comparisons are typically made in terms of bundles of commodities and ordinal preferences, and therefore such comparisons ignore risk attitudes when there is no risk in the contemplated allocation problem. [START_REF] Fleurbaey | Fair management of social risk[END_REF] variant of Harsanyi's result mentioned as the end of the previous section applies the weighted utilitarian sum of utilities not to individual bundles, but to hypothetical final allocations in which inequalities have been eliminated, in every state of the world, while preserving social welfare as defined for riskless allocations, i.e., ignoring risk attitudes. This offers a nice combination of the standard theory of fair allocation for riskless settings and Harsanyi's theorem suitably watered down. But it was proposed for a situation in which the composition of the population and individual preferences are fixed. This reconciliation is seriously undermined by our result here, in the perspective of dealing with preference changes and changes in the composition of the population.

The other strand of the literature to which our result relates has been looking at the tension between dominance principles and context-dependence principles. The main context in Documents de travail du Centre d'Economie de la Sorbonne -2020. 28 which this tension has been studied (Fleurbaey 2007, Pattanaik and[START_REF] Pattanaik | Minimal relativism, dominance, and standard of living comparisons based on functionings[END_REF] involves interpersonal comparisons of bundles of commodities where the context-dependence idea is represented by the Pareto principle: when preferences change, interpersonal comparisons are altered. In this context, the relevant dominance principle is bundle dominance: if one individual is better off in every dimension, this individual is better off overall.

The bundle dominance principle is endorsed in the literature in various contexts. In particular, Sen (1992) defends it as a cornerstone of the evaluation of capabilities. The tension between this principle and Pareto comes from the fact that for certain preferences, bundles 𝑥 and 𝑦 may be equally good, whereas for a second type of preferences, bundles 𝑥 N and 𝑦 O may be equally good, and this clashes with the dominance principle if bundle 𝑥 N dominates 𝑥 while 𝑦 dominates 𝑦 O . This is illustrated in Figure 1, where bundles are bundles of two commodities and the curves are indifference curves for individuals with utility 𝑢 and 𝑢 , over bundles. In Figure 1, the solid line describes all bundles equally good as 𝑥 with utility function 𝑢, while the dotted line describes all bundles equally good as 𝑥 N with utility function 𝑢 , . By bundle dominance (𝑥 N , 𝑢 , ) is better than (𝑥, 𝑢) and (𝑦, 𝑢) is better than (𝑦 O , 𝑢 , ), simply because there are more commodities in 𝑥 N than in 𝑥, and there are more commodities in 𝑦 than in 𝑦 O . But by Pareto indifference (𝑥, 𝑢) is as good as (𝑦, 𝑢) (𝑥 and 𝑦 are on the same indifference curve for 𝑢), and (𝑥 N , 𝑢 , ) is as good as (𝑦 O , 𝑢 , ). We thus obtain an inconsistency in the social evaluation.

In our paper, we also have the Pareto principle, but bundle dominance is replaced in our analysis by Statewise dominance applied not to physical payoffs but to the evaluation itself, which is a very substantial weakening. As discussed in the previous section, there is no direct clash between Pareto and Statewise dominance, and it is only the third axiom of irrelevance of risk attitudes in absence of risk that triggers a conflict. This third axiom is in fact embedded in the usual bundle dominance principles, in which the statement "more in one dimension" is tacitly assumed to depend only on physical characteristics of the bundle and not on the preferences of the individual over the whole bundle. In our setting, the axiom requires the evaluation of the final situation in every state of the world not to depend on risk attitudes. Fleurbaey and Blanchet (2013) criticize the bundle-dominance principle for failing to take account of the problem of fit between preferences and bundles. Having more in every dimension may not mean much if associated with different preferences for which the composition of the contemplated bundle is bad. A better paid banker may be worse off than a less well paid colleague if the banker would rather be an artist. We think, however, that this criticism does not apply to our setting. Statewise dominance is a pure rationality Documents de travail du Centre d'Economie de la Sorbonne -2020.28 principle and has nothing to do with the fit between preferences and prospects. Irrelevance of risk attitudes in absence of risk also cannot be related to the fitting issue either, because it simply says that overall preferences (over prospects) do not matter when risk is absent, i.e., in a particular context.

It is in fact impossible to adapt our formalism to the bundle evaluation problem, because in our setting we can equate a final consequence with a sure situation once uncertainty is resolved, whereas in the bundle context, it would make no sense to imagine living in a world in which only one dimension exists (say, income) and the other dimensions have vanished.

There is, however, something similar to the artist-banker example in our setting. This is the case of a risk averse individual who faces a better prospect but is actually worse-off because he is more risk-averse, so that the risk in the prospect is a "bad fit" for his risk attitude.

Supposing that there is something true about this assessment, there are two ways of accommodating it here. One consists in dropping Statewise dominance and evaluating certainty-equivalent situations. A more risk-averse individual can have a lower certaintyequivalent due to risk aversion. But we have seen that violating Statewise dominance leads to practical difficulties because of the induced irrationality in the evaluation.

The other possibility is to drop irrelevance of risk attitudes in absence of risk and posit that a risk-averse individual who ends up being lucky may still have to be considered relatively badly off, compared to an equally lucky but less risk-averse individual, because of risk aversion embodied in the utility function. This means that risk aversion should be incorporated into the evaluation of sure options, and be viewed as a burden even in absence of risk. It then becomes a burden in the presence of risk by the expected value of the burden it has in every possible final consequence. As shown in [START_REF] Fleurbaey | Fair utilitarianism[END_REF], it is Documents de travail du Centre d'Economie de la Sorbonne -2020.28 possible to scale VNM utilities so that more risk averse preferences yield a lower utility for every sure option. In that case, we obtain that VNM utilities, once properly normalized are (interpersonally comparable) measures of individual well-being. 6 But one way also want to argue against the risk averse individual example above and reply that there is no specific burden to be attached to risks borne by such risk averse persons.

The fact that some people are more risk averse than others may just be that they rely on different reasons when making their choices in risky situations (for instance thinking in terms of threats rather than thinking in terms of opportunities). If we follow [START_REF] Mongin | Spurious unanimity and the Pareto principle[END_REF] in defining spurious unanimity as situations where unanimous preferences are based on mutually incompatible reasons, and if we admit that the Pareto principle has less appeal in situations of spurious unanimity, we may prefer not to rely on risk attitudes for well-being assessment. In that case, we may prefer to retain Statewise dominance and Irrelevance of risk attitudes in absence of risk.

In conclusion, it seems to us that there are two reasonable escapes from the impossibility presented in this paper. The first one consists in accepting to use VNM utilities in interpersonal comparisons, even in the absence of risk, and to scale them in a way that makes risk aversion appear as a burden, while retaining expected utility as the valuation measure in order to abide by the most basic principles of rationality and respect for preferences and risk attitudes. The second one consists in basing the social evaluation on the expected value of some utility function that may not correspond to the individual VNM 6 Adler (2012, 2019) also defends a measure of individual well-being based on VNM utilities but he has a different line of argument: VNM utilities are useful even in absence of risk because they convey a notion of well-being that includes, via risk aversion, the diminishing marginal utility phenomenon. Adler's scaling makes more risk-averse individual appear better off between two reference situations. Here we suggest instead that risk aversion is a burden for every situation. Documents de travail du Centre d'Economie de la Sorbonne -2020.28 functions but that is consistent with individual ordinal preferences over options. As suggested by the reference to spurious unanimity above, the choice of the appropriate utility function in that case will be based on reasons that are deemed socially acceptable. To the best of our knowledge, there exists no theory of reasonable social VNM utility functions.

One possibility would have reference risk attitudes be an aggregation of individual ones (for instance taking a weighted sum of individuals VNM utility functions). Another possibility, defended by Adler in a different context, would be for the social criterion to be risk neutral with respect to an appropriate wellbeing measure for sure alternatives [START_REF] Adler | Measuring Social Welfare: An Introduction[END_REF]. 7

  individual's VNM can change without affecting the ordering on sure situations, provided individual preferences over sure situations remain unchanged. The Pareto axiom is more problematic. When should we respect individual risk attitudes when there are several individuals and the risky prospect may generate different levels of inequality? Fleurbaey (2018) reviews several axioms proposed in the literature and Documents de travail du Centre d'Economie de la Sorbonne -2020.28concludes that the least controversial consists in restricting the application of the Pareto principle to situations which are as close as possible to single-individual situations. Such situations are characterized by the absence of inequalities in every state of the world. If there is risk, everyone is then "in the same boat."

  Consider two egalitarian acts (acts such that the options obtained in each state of the world are egalitarian, in the sense of the 𝑣 B functions). Consider a utility profile 𝑢 B B∈D where all individuals have the same VNM function, and another utility profile 𝑢 B , B∈D where all individuals have the same VNM function, which is ordinally equivalent to that in the first profile but involves different risk attitudes. Assume that the two acts are indifferent with the first utility profile, but that one act is preferred to the other with the second profile. By Pareto, the social preference is just the same as the individual preference. Documents de travail du Centre d'Economie de la Sorbonne -2020.28 By Irrelevance of risk attitudes in absence of risk and Statewise dominance, once again, risk attitudes do not matter, which contradicts the application of the Pareto principle in the previous paragraph. QED. As we did for the evaluation of individual risk, we can try to look for possibilities by weakening one of the axioms. Salient solutions, similar to those discussed in Section 3, are as follows: a) If we drop Statewise dominance, we can focus on the distribution of certaintyequivalent situations over the population. For instance, if options are just income levels, one could adopt an additive separable social welfare function that would consist in the sum of some concave transformation of certainty-equivalent incomes. One then obtains a version of ex ante prioritarianism as discussed by Adler and Treich (2017). b) If we weaken Pareto by restricting it to riskless situations, we can say that individual well-being is measured by the expected value of some normalized utility function that normalizes risk attitudes among people having ordinally equivalent utilities over sure options. Then taking a sum of these individual wellbeing measures provides a social welfare function that satisfies Statewise dominance (actually the stronger expected utility requirement), Irrelevance of risk attitudes in absence of risk and the weakened Pareto requirement. c) If we drop Irrelevance of risk attitudes in absence of risk, we are left with Pareto for equal risk and Statewise dominance. Social orderings satisfying these two requirements are extensively studied in

Figure 1 :

 1 Figure 1: The conflict between bundle dominance and Pareto indifference

  Documents de travail du Centre d'Economie de la Sorbonne -2020.28 following table describing the incomes produced by two acts act 𝑎 and 𝑎′ in the two states of

	the world:
		State 1	State 2
	Act 𝑎	10000	40000

  The program may raise their certainty-equivalent, but it seems odd to want to implement the program.

	Violations of Statewise dominance can trigger violations of time consistency, because the ex
	ante evaluation (not knowing the state of the world) may induce planning certain future
	actions which, ex post (knowing the state of the world), appear dominated. Suppose the
	education program taken as an example in the previous paragraph is planned for the next
	generation because it enhances its certainty-equivalent. Once more information about the

Documents de travail du Centre d'Economie de la Sorbonne -2020.

28

ante. situation of the next generation is made available, however, it appears harmful to pursue the program as planned and the program would better be cancelled. The key rationality problem with a violation of time consistency is that it is known in advance that the chosen plan will be cancelled later in every state of the world. Why, then, plan a policy that will be repealed for sure? It thus seems to us that dropping Statewise dominance is probably the least appealing route: its seems preferable to consider dropping either Pareto or Irrelevance of risk attitudes in absence of risk. Let us first consider the Pareto principle. Instead of completely abandoning the Pareto principle, we can try to weaken it and retain Pareto only for riskless options 𝑥, 𝑢 . The VNM utility function 𝑢 is a particular cardinal representation of preferences over riskless options, and we can choose another representation 𝑢 that satisfies two conditions: (i) 𝑢 and 𝑢 are ordinally equivalent on 𝑋; (ii) whenever 𝑢, 𝑢′ are ordinally equivalent on 𝑋, one has 𝑢 = 𝑢 , .

In other words, the transformation 𝑢 respects the preferences over 𝑋 but eliminates differences in risk attitudes by changing the cardinal scale of utility measurement. Consider the ordering defined by: 𝑎, 𝑢 ≽ 𝑎 , , 𝑢′ if and only if 𝐸𝑢(𝑎) ≥ 𝐸𝑢′(𝑎′). This criterion Documents de travail du Centre d'Economie de la Sorbonne -2020.

28

If we consider instead a situation of uncertainty where probabilities are not well-defined, individuals may have different beliefs, and impossibility may already arise on the sole basis of the Pareto principle: see[START_REF] Mongin | Consistent Bayesian aggregation[END_REF]. To focus on our main argument and contribution here, we do not consider this issue in this paper. Documents de travail du Centre d'Economie de la Sorbonne -2020.28

This paper thus follows the idea by[START_REF] Fleurbaey | Universal social orderings: An integrated theory of policy evaluation, inter-society comparisons, and interpersonal comparisons[END_REF] of developing "Universal social orderings" for inter-personal and inter-society comparisons. Documents de travail du Centre d'Economie de la Sorbonne -2020.28

Acknowledgements

We would like to thank Jean Baccelli, John Broome, Paul Egré, Simon Grant, Philippe Mongin, Klaus Nehring, Wlodek Rabinowicz and Katie Steele and audience at the London School of Economics and the Australian National University for their comments on this paper.


Anis Younes

Marwan Fahs
email: fahs@unistra.fr

Behzad Ataie-Ashtiani

Craig T Simmons

Effect of distance-dependent dispersivity on density-driven flow in porous media

Keywords: Density driven flow, saltwater intrusion, leachate transport, variable dispersion, asymptotic model, global sensitivity analysis

In this study, the effect of distance-dependent dispersion coefficients on density-driven flow is investigated. The linear asymptotic model, which assumes that dispersivities increase linearly with distance from the source of contamination and reach asymptotic values at a large asymptotic distance, is employed. An in-house numerical model is adapted to handle distancedependent dispersion. The effect of asymptotic-dispersion on aquifer contamination is analyzed for two tests: (i) a seawater intrusion problem in a coastal aquifer and (ii) a leachate transport problem from a surface deposit site. Global Sensitivity Analysis (GSA) combined with the Polynomial Chaos Expansion (PCE) surrogate modelling is conducted to assess the influence of the dispersion coefficients on the contamination plume for both configurations.

For the seawater intrusion problem, the results show that the length of the toe is mainly controlled by the asymptotic transverse dispersivity whereas the spread of the concentration is sensitive to the asymptotic longitudinal dispersivity and the asymptotic dispersivity distance.

The latter is the most important parameter controlling the amount of salt which intrudes into the aquifer. For the leachate transport problem, the results show that the asymptotic longitudinal dispersivity coefficient does not affect the concentration distribution. The asymptotic dispersivity distance has a strong effect on the total amount of contaminant that enters the aquifer. This effect can be three times more important than the effect of the asymptotic transverse dispersivity. These findings are likely to be helpful for the investigation and management of density-driven flow problems.

Introduction

Density-driven flow (DDF) is a particular configuration of transport in porous media in which the fluid concentration causes a change in groundwater density which can significantly affect the flow dynamics. DDF can be encountered in several applications related to contaminant transport in aquifers. Among these applications, a well-known problem is the contamination of coastal aquifers by saltwater intrusion [START_REF] Werner | Seawater intrusion processes, investigation and management: Recent advances and future challenges[END_REF] which is a major concern around the world. Another important example is groundwater contamination by leachates from surface industrial waste and landfills [START_REF] Frind | Simulation of long-term transient density-dependent transport in groundwater[END_REF]. Managing and predicting the evolution of pollutants in such situations require accurate numerical simulations. In most DDF models, dispersion is ruled using a velocity-dependent dispersion tensor involving constant coefficients characterizing mixing in the longitudinal (parallel to the flow) and transverse (orthogonal to flow) directions. In the last decades, many studies have shown that this conventional approach cannot satisfactorily represent field transport especially for aquifers with spatial heterogeneity (Pickens and Grisak, 1981a). Alternative approaches have developed such as stochastic models (e.g. [START_REF] Gelhar | A critical review of data on field-scale dispersion in aquifers[END_REF]Zhang, 2002, Kerrou and[START_REF] Kerrou | A numerical analysis of dimensionality and heterogeneity effects on advective dispersive seawater intrusion processes[END_REF][START_REF] Pool | Effects of tidal fluctuations and spatial heterogeneity on mixing and spreading in spatially heterogeneous coastal aquifers[END_REF] or continuous time random walk methods (e.g. [START_REF] Berkowitz | Anomalous transport in laboratory-scale, heterogeneous porous media[END_REF][START_REF] Dentz | Time behavior of solute transport in heterogeneous media: transition from anomalous to normal transport[END_REF]). However, these methods usually require sufficient field measurements to formulate statistical structure and are known to be computationally expensive [START_REF] Wang | Specifying Scale-dependent Dispersivity in Numerical Solutions of the Convection-Dispersion Equation[END_REF]. Such difficulties have motivated using the conventional dispersion approach, but by considering that the dispersivity values are temporal or scale dependent (Pickens and Grisak, 1981a). In other words, the longitudinal and transverse dispersion coefficients are not constant but can vary with the distance from the source of contamination. Indeed, in a tracer test, [START_REF] Molz | An Examination of Scale-Dependent Dispersion Coefficients[END_REF] found that dispersivity is not constant but increases with the travel distance because of the scale dependence of dispersivities. This phenomenon has been observed both in field-scale transport (e.g. Pickens and Grisak 1981a;[START_REF] Gelhar | A critical review of data on field-scale dispersion in aquifers[END_REF]; Schulze-Makuch, 2005) and laboratoryscale transport (e.g., [START_REF] Silliman | Laboratory evidence of the scale effect in dispersion of solutes in porous media[END_REF][START_REF] Khan | A laboratory study of the dispersion scale effect in column outflow experiments[END_REF][START_REF] Huang | Experimental investigation of solute transport in large, homogeneous and heterogeneous, saturated soil columns[END_REF][START_REF] Vanderborght | Review of Dispersivities for Transport Modeling in Soils[END_REF]. According to [START_REF] Gao | A mobile-immobile model with an asymptotic scale-dependent dispersion function[END_REF], the scale dependence of dispersivity can be related to different processes such as the heterogeneity of the porous media at different scales [START_REF] Gelhar | A critical review of data on field-scale dispersion in aquifers[END_REF][START_REF] Huang | Evidence of one-dimensional scale-dependent fractional advection-dispersion[END_REF], the fractal nature of the pore space in the aquifer [START_REF] Wheatcraft | An explanation of scale-dependent dispersivity in heterogeneous aquifers using concepts of fractal geometry[END_REF] or the anomalous transport [START_REF] Cortis | Anomalous Transport in "Classical" Soil and Sand Columns[END_REF]. [START_REF] Mishra | Analysis of solute transport with a hyperbolic scaledependent dispersion model[END_REF] showed that a hyperbolic dispersivity-distance function allows a good fitting of the data estimated from a natural gradient tracer experiment. Kangle et al. [START_REF] Hunt | Scale-Dependent Dispersion from a Pit[END_REF][START_REF] Chen | Analytical power series solutions to the twodimensional advection-dispersion equation with distance-dependent dispersivities[END_REF] or radially convergent divergent flow fields (e.g. [START_REF] Chen | A Laplace transform power series solution for solute transport in a convergent flow field with scale-dependent dispersion[END_REF][START_REF] Chen | Evaluation of longitudinal and transverse dispersivities/distance ratios for tracer test in a radially convergent flow field with scaledependent dispersion[END_REF][START_REF] Chen | Analysis of solute transport in a divergent flow tracer test with scale-dependent dispersion[END_REF]. To the best our knowledge, investigation of distance-dependent dispersion coefficients in cases involving complex velocity fields, such as in DDF problems, have not been undertaken.

The aim of this work is to incorporate distance-dependent dispersion in a DDF model and to investigate the effect of dispersion parameters on contaminant transport. As conceptual models, we consider (i) the Henry problem describing seawater intrusion (SWI) in a coastal aquifer [START_REF] Henry | Effects of dispersion on salt encroachment in coastal aquifers[END_REF] and (ii) the leachate transport problem proposed by [START_REF] Frind | Simulation of long-term transient density-dependent transport in groundwater[END_REF] to investigate the leachate plume from a surface deposit site. The effect of distance-dependent dispersivities on the aquifer contamination is investigated using Global Sensitivity Analysis (GSA) combined with Polynomial Chaos Expansion (PCE) surrogate modelling [START_REF] Sudret | Global sensitivity analysis using polynomial chaos expansions[END_REF][START_REF] Fajraoui | Reactive Transport Parameter Estimation and Global Sensitivity Analysis Using Sparse Polynomial Chaos Expansion[END_REF][START_REF] Fajraoui | Analyzing natural convection in porous enclosure with polynomial chaos expansions: Effect of thermal dispersion, anisotropic permeability and heterogeneity[END_REF][START_REF] Mara | Addressing factors fixing setting from given data: A comparison of different methods[END_REF].

Methods

The mathematical model and numerical code

The mathematical model for water movement through porous media is based on the mass conservation equation and Darcy's law [START_REF] Guevara Morel | Systematic investigation of non-Boussinesq effects in variable-density groundwater flow simulations[END_REF]:

0 hC S t C t              q (1) 0 0 0 g hz              qk (2)
where  is the fluid density [ML -3 ],

S the specific mass storativity related to head changes [L - Flow and transport equations are coupled via the linear mixture density equation:

  0 1 0 C        (5)
where 1  is density of contaminant.

In this work, we assume that the longitudinal and transverse dispersion coefficients are a function of the distance from the source of contamination. Distance-dependent dispersivities are generally ruled using one of the four types of functions suggested by [START_REF] Pickens | Scale-dependent dispersion in a stratified granular aquifer[END_REF] including linear, parabolic, asymptotic and exponential functions. The linear distancedependent dispersivity function has been largely used in the literature (e.g. [START_REF] Pang | Solutions and verification of a scale-dependent dispersion model[END_REF][START_REF] Gao | A new mobile-immobile model for reactive solute transport with scale-dependent dispersion[END_REF][START_REF] Guerrero | Analytical solution for one-dimensional advection-dispersion transport equation with distance-dependent coefficients[END_REF][START_REF] Chen | Analytical power series solutions to the twodimensional advection-dispersion equation with distance-dependent dispersivities[END_REF]. However, this function seems to be unphysical, because field observations show that a constant dispersivity could be asymptotically reached [START_REF] Gelhar | A critical review of data on field-scale dispersion in aquifers[END_REF]Pickens and Grisak, 1981a). Huang et al.

(1995) used a linear-asymptotic distance-dependent function where the dispersivity value increases linearly with the transport distance and reaches an asymptotic value at a certain large distance. The linear-asymptotic model was adopted by [START_REF] You | New solutions for solute transport in a finite column with distance-dependent dispersivities and time-dependent solute sources[END_REF] and is employed in this work:

  0 0 0 0 0 0 L,T L,T L,T , ,            x ( 6
)
where corresponds to the distance from the source of contamination, 0 elementwise and their values are calculated using (Eq. 4) where corresponds to the distance from the center of each element to the source of contamination.

The Henry saltwater Intrusion Problem

Real applications of SWI at a field scale are increasingly reported in the literature. However, in several theoretical and applied studies, SWI is often investigated based on the hypothetical Henry problem (Henry, 1964) (Figure 1a). This problem represents a common benchmark that is widely used for multiple purposes as understanding physical processes, numerical model The Henry problem represents SWI in a vertical cross-section of a confined coastal aquifer where an inland freshwater flow is in equilibrium with seawater that intrudes into the aquifer from the seaside due to its higher density (Figure 1a). The first studies on the  and the asymptotic distance 0 . Note that the longitudinal and transverse dispersivities are assumed to be independent. The corresponding uncertainty ranges (Table 2) are sufficiently large to explore the role of each parameter.

  3 L  m 1 H  m -molecular diffusion coefficient
L  [m] [0.1, 1.0] 0 T  [m] [0.04, 1.0] 0 [m] [0, 2.0]

The leachate transport problem in unconfined aquifer

This problem was proposed by [START_REF] Frind | Simulation of long-term transient density-dependent transport in groundwater[END_REF] to investigate groundwater contamination by leachates from sanitary landfills or industrial waste disposal sites. A typical problem is considered (Figure 1b) where a disposal site unprotected from precipitation is situated above the water table in a rectangular unconfined aquifer of 3000 m length and 24m thickness.

The parameters and boundary conditions are described in Table 3. -the first-order sensitivity index,

  i i i V E y V S V y V        (4)
-the total sensitivity index,  

T i i i E V y V ST V y V        ( 5
)
where y is the model output,  is the set of parameters

  00 0 ,, LT   ,   E is the mathematical expectation (the average),   V is the mathematical variance, E     and V    
are their respective conditional forms. 2 and Table 4 respectively.

Results for the Henry SWI problem

A mesh converged solution is obtained a uniform triangular mesh formed by 4800 elements.

The effect of the dispersion parameters on saltwater intrusion is investigated based on the following metrics (see Figure 2).

-The positions 0.1 0.5 , XX and 0.9

X of respectively the 10%, 50% and 90% isochlors at the aquifer bottom. Note that the 0.5 X is related to the well-known length of the toe toe L which is the distance between the seaside boundary and 0.5 X .

-The spread of the concentration 0.9 0.1 S L X X  which corresponds to the distance between the 10% and the 90% isochlors at the bottom of the domain.

-The total mass in the domain.

-The values of steady state concentration at the following selected points:   The distribution of the variance of the concentration shows that high variances regions are located at the center of the domain near the bottom of the aquifer (Figure 3b). This makes sense as the length of the toe is mainly controlled by the dispersion processes [START_REF] Abarca | Anisotropic dispersive Henry problem[END_REF][START_REF] Fahs | The Henry problem: New semianalytical solution for velocity-dependent dispersion[END_REF]. Indeed, it is well known that low dispersion increases the buoyancy forces compared to dispersion effects and yields much more intrusion near the bottom of the aquifer [START_REF] Younes | A semi-analytical solution for saltwater intrusion with a very narrow transition zone[END_REF]. This explains the high variance region near A1 in Figure 3b.

Significant variability of the salinity can be observed also at top of the aquifer near the seaside (Figure 3b). In this zone, the groundwater flow is discharging to the sea. Thus, the salinity of this zone is mainly due to dispersion. The dashed contour in Figure 3b shows the region where the effect of dispersion parameters is significant which corresponds to 5% of the standard deviation of the concentrations.

The spatial maps of first order Sobol' indices representing the sensitivity of the salinity distribution to 0 L  , 0 T  and 0 are plotted in the Figure 4. For the region of significant variance (the region delimited by the dashed lines), Figure 4a shows that 0 L  has a negligible effect on salinity distribution, except around A2 where a moderate effect can be observed. The parameter 0 T  is the most influential parameter as its zone of high sensitivity is situated in the region of high variability (Figure 4b). Significant sensitivity to 0 T  is observed around A3 but the highest sensitivity area is situated in the mixing zone near A1. This is in agreement with the results of

Sensitivity of the SWI metrics

The Sobol' indices for the concentration at the observation points (A1, A2, A3) as well as for the SWI metrics are depicted in Table 5. This table also gives the mean value and the standard deviation for all quantities of interest. . In this region, moderate interactions occur between parameters The most influential parameter in this region is 0   3 0.6 S  followed by the parameter 0

T    2 0.33 S 
. The parameter 0

L  is irrelevant   1 0.02 S 
. Interactions between the three dispersion parameters are not significant -The 50% isochlor intersects the substratum at an average distance of 2.0 m. The dispersion parameters have a strong effect on that position since the standard deviation is significant (std=0. [START_REF] Fajraoui | Analyzing natural convection in porous enclosure with polynomial chaos expansions: Effect of thermal dispersion, anisotropic permeability and heterogeneity[END_REF]). As 0.1 X , 0.5 X is mainly controlled by the parameter 0

T    2 0.73 S 
. The parameters 0

L    1 0.07 S  and 0   3 0.13 S 
have a limited effect.

Because the 50% isochlor is closer to the sea than the 10% isochlor, 0.5 X is more sensitive to 0 than 0.1 X . Small interactions are observed between the dispersion . Marginal effects of the three sensitive dispersion parameters on 0.9 X are plotted in the Figure 5. The sensitivity of 0.9 X to 0 L  (Figure 5b) has a positive slope as an increase of 0 L  induces an increase of the spreading of the concentration front resulting in an increase of 0.9 X . The sensitivity of 0.9 X to 0 T  (Figure 5c) is similar to that observed for 0.5 X . This demonstrates that saltwater intrusion is mainly controlled by the asymptotic transverse dispersivity. A decrease of 0 T  induces more intrusion which results in a decrease of 0.5 X and 0.9 X . The sensitivity of the intrusion to 0 T  is more pronounced for small values of this parameter. The 0.9 X varies almost linearly with a negative slope with respect to the parameter 0 (Figure 5d).

Indeed, the 90% isochlor is located near the sea boundary (the average 0.9 X is 2.66m)

where the effect of 0 is significant (see Figure 4c). In that region, the increase of 0 yields less dispersion effects which results in more saltwater intrusion and hence a decrease in 0.9 X .

-For the spread of the concentration (Ls), the most influential parameter is 0

L    1 0.5 S  , followed by 0   3 0.41 S 
. The sensitivity of S L to 0 L  is much more important than that of 0.1 X and 0.9 X . Furthermore, although 0 T  is influential on 0.9 X and on 0.1 X , it has no effect on

S L   2 0.05 S 
. The interactions between parameters are almost absent 5e). This confirms that the spreading is directly proportional to the value of 0 L  . The sensitivity of S L to 0 has a negative slope (Figure 5f). The increase of 0 yields less dispersion which results in less spreading of the concentration front and hence a reduction in S L .

-The standard deviation of the total mass in the aquifer is around 10% of its mean value.

Significant interactions occur between the dispersion parameters . This shows that 0 plays the most important role on the amount of mass which has intruded into the domain, followed by

Results of the leachate transport problem

Starting with no leachate in the aquifer, the leachate transport problem is simulated for 24 years.

A mesh converged solution is obtained using a uniform triangular mesh formed by 14400 triangular elements. A hundred simulations were performed using independent random parameter values generated inside the intervals given in Table 4. The mean leachate plume is shown in the Figure 6a. The leachate enters the aquifer due to dispersion and vertical infiltration.

Within the aquifer, the leachate plume moves to the right side due to the hydraulic gradient between left and right sides. A stable flow is obtained for all explored dispersivity values because of (i) the large dispersion and (ii) the weak density difference between the contaminant and freshwater. . In this region, strong interactions occur between these two parameters This study showed that distance-dependent dispersion coefficients can significantly affect contaminant distribution in aquifers in the case of density-driven flow. It demonstrates the advantage of using GSA with PCE surrogate modeling for such investigation since it allows to determine, for each parameter, the regions of high influence and the regions where the effect of the parameter is insignificant. It also allows to determine regions of high interactions between parameters and to explore the marginal effect of sensitive parameters on the model output.

  The simulation of DDF problems is based on coupling Darcy's groundwater flow equation to the solute transport equation via a state relation expressing the density as a function of solute concentration. Transport of solute in the aquifer is ruled by advection, representing the solute displacement by the mean fluid flow, and by dispersion, which accounts for solute spreading caused by velocity variations due to the heterogeneity of the porous medium at different scales (Liu et Kitandis, 2013; Kitanidis, 2017; Dai et al., 2020). Dispersion processes have been found to play a major role in DDF problems as they cause mixing between different fluids. The effect of dispersion on DDF has been widely investigated in the literature. For instance, Abarca et al. (2007) studied the effect of dispersion on DDF in the context of seawater intrusion and showed that when dispersion is taken into account, concentration isolines resemble those observed in real coastal aquifers. Emami-Meybodi (2017) studied instabilities driven by dispersion for an unstable DDF problem with mixed convective flow. Wen et al. (2018) defined a dispersive Rayleigh number and investigated the effect of dispersion on the Rayleigh-Darcy convection problem. Fahs et al. (2020) investigated the effect of dispersion on thermal DDF problem.

(

  1996) provided a one-dimensional analytical solution with linear asymptotic dispersion. Chen et al. (2003, 2007) investigated distance-dependent dispersion for convergent and divergent flow fields with linear scale-dependent dispersion. Chen et al (2008a) studied one-dimensional transport with hyperbolic asymptotic dispersivity function. Pérez Guerrero and Skaggs (2010) derived a general analytical solution for one-dimensional transport with distance-dependent coefficients. Gao et al. (2010, 2012) investigated mobile-immobile transport model with asymptotic scale-dependent dispersivity. You and Zhan (2013) developed semi-analytical solutions for solute transport in a finite column with linear asymptotic and exponential distancedependent dispersivities and time-dependent sources. Thus, in the literature, the effect of asymptotic dispersivity has been essentially investigated for simplified situations of 1D transport (e.g. Basha and El-Habel, 1993; Yates, 1992; David-Logan, 1996; Pang and Hunt, 2001; Chen et al., 2003; Pérez Guerrero and Skaggs, 2010; Sharma and Abgaze, 2015, Wang et al., 2019), 2D problems with a uniform flow field (e.g.



  1 ], h the equivalent freshwater head [L], t the time [T],  the porosity [-], C the relative concentration [-], q the Darcy's velocity [LT -1 ], 0  the density of the displaced fluid [ML -3 ], g the gravity acceleration [LT -2 ],  the fluid dynamic viscosity [ML -1 T -1 ], k the permeability tensor [L 2 ] and z the depth [L] taken positive upwards. The contaminant transport in porous media is based on the solute mass conservation equation: the longitudinal and transverse dispersion coefficients [L], m D the pore water diffusion coefficient [L 2 T -1 ] and I the unit tensor. The associated boundary conditions of the flow-transport system (1)-(3) are of Dirichlet, Neuman or mixed type.

L

  longitudinal and transverse dispersion coefficients and 0 is the asymptotic distance after which both longitudinal and transverse dispersivities reach their -transport system is solved with an advanced in-house numerical model using triangular meshes[START_REF] Ackerer | Efficient approximations for the simulation of density driven flow in porous media[END_REF]. The flow equations (Eqs. 1-2) are solved by the mixed finite element method(Younes et al., 2010). The transport equation (Eq. 3) is solved by combining two numerical methods: Discontinuous Galerkin (DG) method for solving advection and Multipoint Flux Approximation (MPFA) method for solving dispersion. Coupling between flow and transport equations is performed using the non-iterative scheme proposed in Younes and Ackerer (2010) with proper time management. This scheme was shown to be highly efficient and more accurate than the standard iterative procedure. The in-house code has been validated by comparison against semi-analytical solutions in[START_REF] Fahs | The Henry problem: New semianalytical solution for velocity-dependent dispersion[END_REF]. Performance and robustness of the code has been highlighted in[START_REF] Shao | A 3-D Semianalytical Solution for Density-Driven Flow in Porous Media[END_REF] by comparison against COMSOL Multiphysics. In this work, the in-house code is modified to handle distance-dependent dispersion coefficients. Both longitudinal and transverse dispersivities are defined

  verification, and parameter sensitivity analyses. A detailed review of the different use of the Henry problem as a surrogate representation of SWI can be found in Werner et al. (2013) and Fahs et al. (2018).

Figure 1 .

 1 Figure 1. (a) Henry problem domain and boundary conditions; (b) The leachate transport problem[START_REF] Frind | Simulation of long-term transient density-dependent transport in groundwater[END_REF]. Following[START_REF] Simpson | Improving the worthiness of the Henry problem as a benchmark for density-dependent groundwater flow models[END_REF], we decrease the freshwater recharge by half to

1 

 1 flow -hydrostatic pressure at the right hand side -constant flux at the inland boundary: along the top and bottom Boundary conditions for transport -0  = 1000 kg/m 3 on the left boundary. -= 1025 kg/m 3 on the right boundary -zero concentration gradient along the top and bottom The numerical model is employed to analyze the saltwater intrusion by considering that uncertainty of model outputs is associated with the following dispersion parameters: the asymptotic longitudinal dispersivity 0 L  , the asymptotic transverse dispersivity 0 T

Table 3 . 1 -L  , 0 T  and 0 )Table 4 . 3 .

 310043 Parameters and boundary conditions for the leachate transport problem. -permeability kx= 0.3262 x 10 -10 m 2 kz= 0.3262 x 10 -11 m 2 -porosity -length of the aquifer -height of the aquifer 0boundary conditions for flow -fixed head at the left (h0=0) and right (hL=-17.5m) hand sides -constant flux at the top: 30 z q  cm/year -no flow along the bottom -boundary conditions for transport -fixed relative concentration (C=C0) at the top with C0=1 for 12 x x x  C0=0 elsewhere with 1 120 x  m and 2 920 x  m -fixed relative concentration (C=0) at the left and right hand sides -density of the contaminant 1 1007.1   kg/m 3 -zero concentration gradient along the bottom Large uncertainty ranges (Table 4) are associated to the dispersion parameters ( 0 in order to investigate the role of each of them. Note that because the leachate transport problem has larger dimensions than the synthetic Henry problem, the dispersion parameters are allowed to have larger values than previously. Uncertainty ranges of the dispersion parameters for the leachate transport problem. Global sensitivity analysis Effect of the dispersion parameters on DDF is investigated using global sensitivity analysis (GSA). To this aim, the variance-based sensitivity indices of Sobol' (Sobol', 2001) are computed using Polynomial Chaos Expansion (PCE). The Sobol' indices measure the contribution of an input (alone or by interactions with other inputs) to the output variance. They are well adapted for GSA since they do not require any assumption of monotony or linearity of the model (Saltelli et al., 2006). Two Sobol' indices are noteworthy:

iL  , 0 T.

 0 represents one of the parameters, and i   stands for the set of parameters  without the parameter i  . The first-order sensitivity index (or main effect index) by which the variance of y is reduced when the true value of i  is known. The total sensitivity index the remaining uncertainty in y after all parameters are known except i  . It evaluates the contribution of i  to the response variance, including its interactions with the other parameters (i.e., ~i  ). If interactions between parameters are negligible, we have 1 marginal effect   i y  of the parameter i  on the model output y enables to investigate the range of variation of y with respect to i In this work, we use the Polynomial Chaos Expansion (PCE) surrogate modeling to infer sensitivity indices (Fajraoui et al., 2012, 2017; Younes et al., 2016; Shao et al., 2017). PCE allows an efficient evaluation of the Sobol' indices, since they can be easily calculated using the PCE coefficients (Sudret, 2008). Since we deal with only three uncertain parameters ( 0 and 0 ), we use a full surrogate PCE of order 4. The number of polynomial coefficients in the expansion is therefore 7The coefficients of the surrogate PCE are calculated by a least-square technique minimizing the sum of squared errors between model responses and the PCE. To this aim, a hundred of evaluations of the Henry and leachate transport problems are performed using parameter values randomly generated in the intervals of variation given in Table

Figure 2 . 4 . 1 Figure 3 .

 2413 Figure 2. Seawater intrusion metrics used for the assessment of effects of the dispersion parameters.

FahsT.Figure 4 .

 4 Figure 4. Spatial maps of the first-order sensitivity indices: (a) sensitivity of salinity distribution to 0 L  , (b) sensitivity of salinity distribution to 0 T  and (c) sensitivity of salinity distribution to 0 . Dashed lines limit the zone of high variability (5% of standard deviation).

-

  with the results discussed previously based on the spatial maps of Sobol' indices (Figure4).-Around A2, located near the sea boundary, high concentrations can be observed (mean=0.79). The concentration has slight variability (std=0.1) which indicates that SWI reaches this point whatever the values of the dispersion parameters. The most influential parameter near A2 is 0   The point A3 is located near the top of the domain and close to the sea boundary. The standard deviation (std=0.11) of concentrations is relatively significant (mean=0.27).

T

  isochlor intersects the substratum at an average distance of 1.39 m from the sea boundary. 0.1 X has high variability (std=0.17m). The parameter 0 influential since their total sensitivity indices are significant  1

X when varying 0 T 5 X increases with 0 TFigure 5 .T on 0. 5 X

 05055 Figure 5. Marginal effect of: (a) 0 T  on 0.5 X , (b) 0 L  on 0.9 X , (c) 0 T  on 0.9 X , (d) 0 on

Figure 6 .

 6 Figure 6. The leachate transport problem at 24 years: (a) Spatial map of the mean concentration values and (b) Spatial map of variance of concentration. The dashed contour delimits the region of high variances (5% of standard deviation).

Figure 7 LL

 7 Figure7shows the spatial distributions of the first-order Sobol' indices. For the region of

Figure 7 .T

 7 Figure 7. Spatial map of the first order sensitivity indices for the leachate transport problem: a) sensitivity to 0 L  , b) sentitivity to 0 T  and c) sensitivity to 0 .

Figure 8 .L

 8 Figure 8. Spatial map of the total Sobol' index of 0 L  for the leachate transport problem.

Figure 9 .

 9 Figure 9. Spatial map of the interaction 1

S and 3 SL  , 0 TT 2

 302 represent the first order Sobol' indices for the sensitivity to 0 and 0 . 'mean' and 'std' represent the mean value and the standard variation for the quantities of interest.-The concentration around A1 is first influenced by 0 and 0 have close first-order sensitivity indices 

- 0 L has no effect   1 0.TT

 01 The total mass in the system has a mean of 10,667 and a significant variance of 4,058.The parameter S  . The parameter 0 has a strong effect on the total mass value. This effect   The marginal effects of the parameters 0 and 0 on the total mass are plotted in Figure10. The sensitivity of the total mass to 0 depicts a curve with a positive slope since the total mass increases as 0 T  increases. The marginal effect of 0 is represented by a negative slope curve which is consistent with physics. The leachate plume, and consequently the total mass, increase as dispersion increases (i.e. 0 decreases).

Figure 10 . 0 L , asymptotic transverse dipersivity 0 TT but by 0 L

 10000 Figure 10. Marginal effects of 0 T  , and 0 on the total mass for the leachate transport problem.

Table 1 .

 1

	3m

Table 1 .

 1 Parameters and boundary conditions for the Henry problem.

	permeability	k= 1.0204 x 10 -9 m 2
	porosity	0.35
	length of the aquifer
	height of the aquifer

Table 2 .

 2 Uncertainty ranges of the dispersion coefficients for the Henry problem.

	Parameter	Uncertainty Range
	0

Table 5 .

 5 Sensitivity

	of the concentration at the observation points (A1, A2 and A3) and of
	the SWI metrics. 1 S , 2 S and 3 S represent the first order Sobol' indices for the sensitivity to

Table 6 .

 6 The results of this table show that:

									3
		mean	std	1 S	S	2	S	3	1   i	S	i
	A1	0.19	0.2	0.0	0.29	0.58	0.87
	A2	0.52	0.3	0.	0.51	0.46	0.97
	A3	0.29	0.13	0.0	0.37	0.39	0.76
	Total mass	10667	4058	0.0	0.24	0.74	0.98

Table 6 .

 6 Sobol's indices for the concentration at the observation points (A1,..,3) and for the total mass in the domain for the leachate transport problem. 1 S , 2

Acknowledgments

This work was partially supported by the Tunisian-French joint international laboratory NAILA (http://www.lmi-naila.com/). Marwan Fahs would acknowledge the support from the national school of water and environmental engineering of Strasbourg through the research project PORO6100. Behzad Ataie-Ashtiani and Craig T. Simmons acknowledge support from the National Centre for Groundwater Research and Training, Australia. Behzad Ataie-Ashtiani also appreciates the support of the Research Office of the Sharif University of Technology, Iran. The data used in this work are available on the GitHub repository: https://github.com/fahs-LHYGES