Elias Bouacida
email: elias.bouacida@univ-paris8.fr

Renaud Foucart
email: r.foucart@lancaster.ac.uk

Maya Jalloul
email: m.jalloul@lancaster.ac.uk

Decreasing Differences in Expert Advice: Evidence from Chess Players *

Keywords: decreasing differences, expert, advice, chess, control JEL-Codes: C78, C91, C93, D91, J24, O33

We study the impact of external advice on the relative performance of chess players. We asked players in chess tournaments to evaluate positions in past games and allowed them to revise their evaluation following advice from a high or a low ability player. While our data confirms the theoretical prediction that high-quality advice has the potential to act as a "great equalizer," reducing the difference between high and low ability players, this is not what happens in practice. This is in part because our subjects ignore too much of the advice they receive, but also because low ability players pay -either due to overconfidence or intrinsic preference -a higher premium than high ability ones by following their initial idea instead of high-quality advice.

Introduction

Does offering high quality advice help reduce the gap between good and bad experts? In theory, the answer is yes: the benefit from being able to rely on outside advice is higher if your own expertise is lower. In a pre-registered 1 lab-in-the-field experiment with chess players participating in tournaments in Lebanon, we find that it is only true to a limited extent. High quality advice has the potential to benefit low ability players significantly more than high ability ones, but in practice it fails to make any significant difference. Our subjects reveal such a high preference for following their first idea and ignoring additional information that they forego a large share of the potential gains from the advice. Lower ability players end up paying the highest premium from ignoring good advice.

A major promise of the increasing digitalization of sectors such as legal studies, computer science, cancer diagnostic, or surgery, is to allow average practitioners to benefit from increased access to the knowledge of the very best in their fields. This is particularly true with the emergence of Artificial Intelligence (AI), which has shown its potential to improve the performance of lawyers [START_REF] Choi | Ai assistance in legal analysis: An empirical study[END_REF], programmers [START_REF] Peng | The impact of ai on developer productivity: Evidence from github copilot[END_REF], writers [START_REF] Noy | Experimental evidence on the productivity effects of generative artificial intelligence[END_REF], customer support [START_REF] Brynjolfsson | Generative AI at work[END_REF], and consultants [START_REF] Dell'acqua | Navigating the jagged technological frontier: Field experimental evidence of the effects of ai on knowledge worker productivity and quality[END_REF] in routine tasks.

Perhaps the most striking result from this emerging literature is the potential for such advice to act as a "great equalizer:" if everyone has access to the same high quality external input, this should indeed benefit us all, but mostly those with lower expertise in the first place. This property is known as decreasing differences and theoretically holds for most ways of matching expertise [START_REF] Chade | Matching information[END_REF]: the marginal impact of the quality of advice is decreasing in the ability of the expert who receives it. Other examples include production in garment factories [START_REF] Hamilton | Team incentives and worker heterogeneity: An empirical analysis of the impact of teams on productivity and participation[END_REF][START_REF] Adhvaryu | No line left behind: Assortative matching inside the firm[END_REF] and student coursework in universities [START_REF] Fischer | When, and why, teams benefit from self-selection?[END_REF]. A more general study of the US labour market shows that lower ability workers typically benefit more from the being part of a team with a high ability partner [START_REF] Herkenhoff | Production and learning in teams[END_REF].

The decision to ignore one's own signal and follow the advice of others is typically studied in economics in the context of information cascades [START_REF] Anderson | Information cascades in the The American economic review[END_REF][START_REF] Kübler | Limited depth of reasoning and failure of cascade formation in the laboratory[END_REF], and there is evidence that subjects often like to bet on themselves even when it is optimal not to do so [START_REF] Weizsäcker | Do we follow others when we should? a simple test of rational expectations[END_REF]. In psychology, a large literature studies how subjects tend to give a sub optimal weight on advice in their decision-making [START_REF] Bailey | A meta-analysis of the weight of advice in decision-making[END_REF][START_REF] Bonaccio | Advice taking and decision-making: An integrative literature review, and implications for the organizational sciences[END_REF]. This result is also linked to the idea of preference for decision rights or control premium [START_REF] Bartling | The intrinsic value of decision rights[END_REF][START_REF] Owens | The control premium: A preference for payoff autonomy[END_REF] ; and the "illusion of control" (Langer, 1975; Sloof and von Siemens, 1 The pre-registration is available at: https://aspredicted.org/124_MSY 1 2017) where subjects are overconfident when they make the decision themselves. A recent literature also studies whether people react differently to human advice and to the one provided by AI [START_REF] Candrian | Rise of the machines: Delegating decisions to autonomous ai[END_REF][START_REF] Hertz | Good advice is beyond all price, but what if it comes from a machine[END_REF].

We partnered with a local academy to run our experiment alongside chess tournaments in several locations in Lebanon in the Summer of 2023. We paid the participation fee of our subjects to the tournament, as well as variable monetary incentives depending on their performance. The main task our subjects had to perform was to evaluate the pawn advantage of 20 chess positions -a measure of which player is better positioned to win the game, and by how much -taken from a large database of past games. For each position, we first asked our subjects to make their own evaluation, by choosing one of four possible answers. We then provided them with the evaluation of an external adviser for the same position, and asked our subjects to evaluate it again. Subject choices were not visible to the experimenters and remained anonymous. Our paper is thus closer to the literature on advice than control, as our subjects could not delegate their decision to the adviser, but we offered them to reconsider their own decision based on external advice. Hence, they had to write their evaluation in the same way regardless of whether they took the advice into account. Both pre-and post-advice evaluations were selected for payment with equal probability.

One of our advisers is an International Master, with an Elo rating placing him among the top 6,000 players in the world, and better rated than all of our subjects. The second adviser is an everyday chess player with no formal rating, placing him at the bottom of our subjects. In one treatment, we disclosed the rating of both advisers, but only told our subjects the advice came from one of them with equal probability. In the other treatment, we also informed the subjects of which expert the advice came from.

As expected, the advice of our International Master (75% of correct answers) is much better than the advice of our everyday player (15% of correct answers). As from our pre-registration, we defined as "high ability" our subjects with an official FIDE rating in the top half of our sample, and as "low ability" those in the bottom half. The results are similar if we define as "high ability" those with a rating and low ability the unrated, as 55 out of our 102 subjects had an official FIDE rating. Before receiving the advice, high ability subjects had a rate of correct answers of 41.8%, and low ability ones of 31.2%.

In our main pre-registered test, we find no evidence of decreasing differences, with a slightly and non-significantly higher share of good answers when the matching of subjects with advisers is disassortative -low ability subjects with our best adviser and high ability ones with our worst one -than in the opposite case (40.5% versus 38.4%). We then ran a similar exercise by comparing the rate of good answers when observing our best adviser as compared to no advice at all, and do not find any statistically significant difference either (42.5% versus 40.1%).

The potential for decreasing differences in our setting is however real, and strikingly different from the observed behaviour of our subjects: by blindly following all advice, no subject would answer correctly less than 45% of the time. By following exclusively the advice of the International Master and answering at random when the adviser is the everyday player, one could expect 50% of correct answers. Yet, our subjects display a strong preference for following their own idea: a large majority of the evaluations remain unchanged after observing advice different from their initial evaluation. Even among our lower ability subjects, 52.9% of the evaluations remain unchanged after observing a different advice from an International chess Master.

Our paper contributes to the scientific literature on decreasing differences, advice and control discussed above. The main novelty of our research is to explicitly measure the potential for decreasing differences in a context where our subjects have a certain level of expertise on their topic and have to confront that expertise with a possibly better external advice. It serves as a cautionary tale for views that greater access to advice will compress the productivity distribution and reduce inequality between workers: while good advice may benefit the less able more, there is no reason to believe that the ability to identify and follow good advice is homogeneously distributed in the population. If the most able are also the most able to use advice, it is far from obvious that inequality is reduced.

We also contribute to the literature on control and advice by providing results from a non-WEIRD (White, Educated, Industrialized, Rich, and Democratic) sample [START_REF] Henrich | Most people are not weird[END_REF], as our subjects live in a Middle-Eastern country in the midst of a banking and political crisis. Finally, this paper is part of a literature using chess players to study human decisions, such as strategic behaviour in sequential games [START_REF] Levitt | Checkmate: Exploring backward induction among chess players[END_REF], gender differences in risk-taking [START_REF] Gerdes | Strategic behavior across gender: A comparison of female and male expert chess players[END_REF], social norms and the gender gap [START_REF] Dilmaghani | The gender gap in competitive chess across countries: Commanding queens in command economies[END_REF], or the role of superstars [START_REF] Bilen | The queen's gambit: Explaining the superstar effect using evidence from chess[END_REF].

The rest of the paper is organised as follows. We start in Section 2 by providing some background about the nature of the tasks, our measure of ability, and why we should expect to observe decreasing differences in the impact of the quality of advice. In Section 3, we describe our experimental protocol, procedures and pre-registered outcomes. We present the results in Section 4 and conclude in Section 5.

Tasks, Ability, and Decreasing Differences

In this section, we describe our main task of evaluating a position in terms of pawn advantage. We then explain the measure we use to rank subjects by ability group, the Elo rating. Finally, we outline the theoretical argument for why we should expect decreasing differences in our context.

Pawn Advantage

We asked subjects to evaluate chess positions, which is a description at a given point of a game of the positions of the pieces on the board. We show in Figure 1 one of the positions used in our experiment, exactly as we showed it to our subjects. Positions are evaluated using the notion of pawn advantage, a measure of which player (White or Black) is better placed to win the game. We chose 20 positions from past games of chess using the Chessbase Mega database 2023, and picked half of them with a pawn advantage of 0.7 (a slight advantage) and the other half with 2.4 (a large advantage), either for Black (-2.4 and -0.7) or for White (0.7 and 2.4). 2 The task for our subjects was to identify the correct evaluation out of the four possible ones (in Figure 1, the correct answer is -0.7).

While evaluations are not an exact science, contemporary chess engines converge towards almost identical pawn advantages. There is thus no ambiguity as to which of the four evaluations is the correct answer.

2 Chess players are in general reluctant to translate pawn advantages into winning probabilities, one reason being that there are not two but three possible outcomes in chess: a win, a loss, or a draw.

According to one measure however (suggested by Sune Fischer and Radu Pannan based on 405,460 past games), a pawn advantage of 0.7 correspond to a 60% probability of win and of 2.4 to a 80% probability of win -counting a draw as half a win. In our selection of positions, we followed this statistical regularity: of the games with a pawn advantage of ±2.4, 7 ended with a win for the advantaged player, 2 with a draw, and 1 with a loss ; of the games with a pawn advantage of ±0.7, 3 ended with a win for the advantaged player, 6 with a draw, and 1 with a loss.

To measure the influence of advice on evaluations, subjects were asked to evaluate positions first without advice, and then got a chance to revise their answer after seeing the advice of either an unrated player or of a highly rated player, based on a measure called the Elo rating.

Elo Rating

In order to rank our subjects and advisers by their estimated ability, we use their Elo Rating, a system created by Arpad Elo to compute the relative skill level of a player. When two players play against each other in a tournament registered with the international chess federation FIDE, the winner gains Elo points, and the loser loses points. The number of points gained and lost depends on the difference in ratings and on the expected outcome.

Any player with a rating strictly lower than 1000 is considered as unrated by the FIDE (and in our sample). As a rule of thumb, a difference of 100 points in the Elo rating means that the best rated player is expected to win 5 out of 8 games. While Elo is an imperfect measure of ability, it is taken seriously by players. In our case, the average rating of our rated players is 1490, while our best subject is in the range 2100-2200. We plot the Elo distribution of our subjects in Figure 5 in Appendix A. It should thus be clear to all our subject that our high ability adviser with a rating higher than 2300 is more likely than them to correctly evaluate a position. It should also be clear that our low ability adviser, described as an "an unrated player, who plays regularly for fun" is not of strictly higher ability than any of our subjects who all take part in a registered tournament.

Decreasing Differences in the Return from Expert Advice

Consider two subjects l and h, with perfect information about their own probability of successfully solving a task p i , i ∈ {l, h}, as well as the probability of the low L and high H ability advisers to do so q L < q H . In the case in which subjects do not know the identity of the adviser -but know both are equiprobable -we denote by q = q L +q H 2 this probability.

Unless all subjects follow (or ignore) all types of advice, we should observe strictly decreasing differences if subjects correctly infer the probabilities and maximize their expected probability of finding the correct answer.

Define by f (i, j) the probability that subject i solves a task correctly after observing advice j and assume that q L < p l < p h < q H . If subjects want to maximize their probability of success and know the identity of the adviser,

f (l, L) = p l , f (h, L) = p h , and f (l, H) = f (h, H) = q h .
It is easy to see that in that case, the function displays decreasing differences:

f (l, H) -f (l, L) > f (h, H) -f (h, L),
as the expression simplifies to p l < p h . This statement is equivalent to saying that Negative Assortative Matching (NAM) of subjects to advisers yields a higher expected share of correct answers than Positive Assortative Matching (PAM),

f (l, H) + f (h, L) 2 > f (l, L) + f (h, H) 2 .
The same result holds when considering the case of unknown advisers if p l < q < p h , so that type l subjects follow all advice and type h do not follow any. In that case,

f (l, L) = q L , f (h, L) = f (h, H) = p h , and f (l, H) = q H .
The condition for decreasing differences is then q H > q L , and the difference between N AM and P AM is higher than with known advisers. The reason is that a good adviser then not only helps more the low ability subjects, but it also protects them from following bad advice. Finally, if q ≥ p h or q ≤ p l , the differences are constant and the probability of a correct answer in NAM is the same as in PAM. This result is trivial, as it simply states that if all subjects follow all advice, they also solve all problems with the same probability, and if they ignore all advice, the quality of advice has no influence on their success.

By the same logic, we can compare advice from H and no advice at all, where f (i, 0) = p i is the probability of the answer of subject i being correct before advice. With known adviser, the result is identical to the one above, as f (i, L) = p i for both types of subjects.

With unknown adviser, there are always decreasing differences unless all advice is ignored.

If q < p h , the condition becomes q H > p l . If q ≥ p h , it is p h > p l .

There are however two main biases and preferences that could influence our theoretical result of decreasing differences in the experiment. The first is that our subjects do not have full information on their probability of success and the one of their advisers. If lower performing subjects are also more overconfident than high ability ones, they may benefit relatively less from advice. The second is preference for following their initial idea: if lower ability subjects value more strongly keeping their first answer than high ability ones, they are less likely to follow advice for a given expected gain, decreasing the potential for advice to act as a great equalizer.

The Experiment

We ran the experiment during the Summer of 2023 in several cities in Lebanon, alongside tournaments organised by a local academy. 3 Our subjects were regular participants in tournaments, and had therefore a certain level of expertise in the game. We describe their self-reported demographic characteristics in Table 6 in Appendix A. All subjects received the experimental material written both in English and Arabic.

Protocol

We recruited subjects before the tournament through the organizing chess academy and paid for their registration (around $5) as a participation fee. The experiment took part in a separate room at times where our subjects were not playing. Each subject was randomly allocated either to a treatment with or without information on the adviser. Subjects received tasks booklets and answer sheets (see Figure 2) upon being seated. There were two rounds of tasks, each corresponding to solving ten positions. In each round, subjects were given 8 minutes to complete their evaluation of each position among the possible choices (-2.4, -0.7, +0.7, or +2.4) on the left part of their answer sheets. Then, they were provided with the answers of one of our two advisers for the same questions (Figure 3). They were given 4 minutes to look back at their answers, compare with the advice, and complete the right part of the answer sheet with their possibly updated evaluations. The tasks booklets were labelled 1 or 2 corresponding to the known or unknown adviser condition, and a or b corresponding to the order in which the advice was received, where a means the H adviser for the first round of ten evaluations and the L adviser for the second a total of 103 subjects. Our total sample is however n = 102 as, in line with our pre-registration, we removed observations for which no choice were made and one of our subjects did not write anything in the second part of the answer sheet. The project has received IRB approval from Lancaster University.

one. In the known adviser condition, we told subjects that the answers we gave them were coming from "a player with a rating of 2335" for the round in which their adviser was H, and from "an unrated player, who plays regularly for fun" when their adviser was L. In the unknown adviser condition, we told them that "With equal probability, the player has a rating of 2335, or it is an unrated player, who plays regularly for fun".

Figure 3: A sheet containing advice for one round.

After solving the two rounds of evaluations, subjects completed a short demographic questionnaire as well as questions about their stated preference for control (5 questions borrowed from [START_REF] Burger | The desirability of control[END_REF]. We provide all the experimental material in Appendix C. All the sessions were administered by one of the co-authors of this study (Maya Jalloul), who read the experimental material and ensured no one could cheat.

On top of the participation fee, we picked one of the 40 evaluations of each subject at random (20 evaluations before advice, and 20 after) and paid a variable amount of $10 if the answer was correct. 4 We only knew the subject number, and not their identity. We communicated a list of payments and subject numbers to the organizing chess academy, who then processed the payments based on a list they made allocating participant numbers to individuals. At the end of the experiment, they destroyed these lists.

Pre-Registered Hypotheses

Our main pre-registered outcome was the existence of decreasing differences, tested by comparing the probability that an answer is correct under Negative Assortative Matching of subjects to advisers with the probability that it is correct under Positive Assortative Matching.

In line with our pre-registration, we divided our sample of n = 102 into two groups of equal size, based on their Elo rating, and removed those questions for which subjects did not answer. As 54 subjects had a formal Elo rating, the results are almost identical when considering a dichotomy rated/unrated instead. The subjects in the lower ability group are denoted as l and those in the higher ability group as h. We verify empirically that our division reflects an average difference in ability to solve our tasks by comparing their respective share of correct answers pre-advice (see Table 1). We also see that, on average, our L adviser is less likely to answer correctly than our l subjects, who are themselves less likely to do so than h subjects. Finally, our H adviser performs much better than the average h subject, so that on average q L < p l < p h < q H . We averaged the share of correct answers of l subjects after observing H advice and of h subjects after observing L advice to have our measure of Negative Assortative Matching.

We then did the same for PAM with l having observed L advice and h having observed H advice. Our main test was to see if, as in Section 2.3, NAM indeed generates a significantly higher share of correct answers than PAM. Our alternative main pre-registered test was to do the same exercise using the pre-advice evaluation instead of the revised evaluation following L advice. We had also pre-registered to look at the same question for the two experimental conditions separately.

Results

Individual Performance

We present the result of our main pre-registered test in the first part of Table 2. Looking at all subjects accross treatments, we see that matching high ability subjects with low ability advice and low ability subjects with high ability advice (NAM) yields on average 40.5% of correct answers, only slightly more than the 38.4% of correct answers when matching subjects and advisers assortitatively (PAM). When looking at the two experimental conditions separately, we see that the difference is never significant, and that, when the adviser is known, the proportions are almost equal. In the second part of Table 2, we look at our alternative pre-registered test in which we replace low ability advice by the pre-advice evaluations, we see that the difference between NAM and PAM is not significant either.

While we cannot rule out that some decreasing differences exist, our sample size should have been sufficient to identify any large effect. To give an idea of our statistical Following the logic of Section 2.3, we can get an intuition of the lower and upper bound for the difference between NAM and PAM if payoff-maximizing subject followed a simple decision rule. As an upper bound, in the unknown adviser treatment, if all lsubjects follow all advice, and all h-subject ignore all advice, the difference between NAM and PAM would be of 30 percentage points. As a lower bound, in the known adviser treatment, if all subjects ignore L-advice and follow H-advice, the difference would be of 5 percentage points. This contrasts with the non-statistically significant difference of 2.5pp and 2.7pp found in our experiment.

Pooling across treatments, both our high ability (from 41.2% to 50.8%) and low ability (from 32.9% to 42.5%) subjects see their share of correct answers increase by 9.6 percentage points after seeing good advice. After seeing advice from our low ability adviser, the share of good answers of our low ability subjects drops by 3.3pp, to 26.1%, while for our high ability subjects the drop is by 3.9pp to 38.4%. The negative impact of L-advice is largely a mechanical result following from the fact that our low ability adviser performed worse than the expected result of someone answering at random.

To see why despite a large initial difference in the ability of our subjects high quality advice fails to play the role of a great equalizer, we look at the share of evaluations remaining unchanged after observing advice. Perhaps the most striking fact is that advice remains largely ignored. A first descriptive statistics is that, overall, around three-quarters of the choices are unchanged after observing advice (we report in Table 7 in Appendix B.1 the figure for each treatment and type).

Unchanged advice can however be for several reasons, one of them being that if a subject's answer is identical to the adviser's there is no reason to modify it. The first three columns of Table 3 indeed show, unsurprisingly, that when their initial answer is the same as the adviser, our subjects tend to keep it. 5 The last four columns show what happens when they differ. Participants mostly, and correctly, ignore advice from our L-adviser, although almost 10% of our high ability subjects update their evaluation following advice from an adviser they should expect to be worse than them, in line with [START_REF] Schultze | On the inability to ignore useless advice[END_REF] who shows that some subjects feel the need to incorporate even useless advice. Low ability subjects ignore advice from unknown advisers around twothirds of the time, as compared to around 80% for the high ability subjects. Our low ability subjects ignore (46.8%) or even move further away (1.8%) from our H-adviser roughly half of the time, only slightly less than our high ability subjects. Given the large difference in the pre-advice results of both types of subjects, this choice either implies that low ability subjects are much more overconfident than high ability ones, or that they have a much larger intrinsic preference for not following advice.

To see how much payment subjects left on the table by ignoring advice, we compare in the next section their choices in our experiment with two "heuristics" of always following or ignoring some type of advice.

Heuristics and Premium for Ignoring Advice

Our first "Elo" heuristic is arbitrary and purely deterministic. We assume that, when they know the adviser type, all subjects follow H advice and ignore L advice. When the adviser is unknown, our high ability subjects ignore all advice, and our low ability subjects follow all advice. While this approach has the advantage of being simple and corresponds to information known ex-ante by the subjects, it does not rely on actual probabilities of success.

Our second "Probability" heuristic is based on actual probabilities, an information not available to our subjects. We approximate a subject i's probability of evaluating a position correctly p i by their share of correct answers pre-advice, and, similarly, the probability for experts to do so q L and q H , with q = q L +q H 2 the probability for unknown advice. This approach differs from the one of section 2.3, as we do not look at different probabilities at the question level, but at the individual level for a given round of ten positions, so that this "first-best" way of incorporating advice follows a simple decision rule: if p i > q H , ignore all advice ; if p i ∈ (q, q H ), only follow the known advice of H ; if p i ∈ (q L , q), follow all advice, known or unknown, except for the advice of L ; and if p i < q L , follow all advice.

For each subject, we pick these heuristics and see how many correct answers they could have achieved by following them. While imperfect (and not part of our pre-registration), this method gives us an illustrative idea of the potential of advice and its role as a great equalizer. We cannot rule out that subjects could have done even better if they were able to identify the questions for which they are particularly confident to have a correct answer for instance.

We measure in Table 4 the premium subjects are paying in order to ignore high-quality advice, defined as the difference (in percentage points) in the share of correct answers post H-advice if they followed our heuristics and in the experiment. Accross treatments, our low ability subjects would have a 30 percentage points higher share of correct answers after H-advice following the Probabilistic heuristics than they did in the experiment, as compared to 22.0 for our high ability subjects. Using the Elo heuristic, these number are 32.2pp and 9.0pp respectively. Both differences are statistically significant. We can thus conclude that, be it because of overconfidence or intrinsic preference for keeping their original answer, our low-ability subjects ended up paying a higher premium than high-ability ones for ignoring good advice.

We then aggregate both types of advice to see how this result is affected by the possibility of receiving low quality advice. We plot on Figure 4 the share of correct answers before advice, post-advice in the experiment, and post-advice following our two heuristics. The left-hand side corresponds to the treatment with unknown adviser. There, our two heuristics are radically different. The reason is that our H-adviser performed so much better than all our subjects that many of our best subjects would have been better off following all advice. This implies that, following our Elo heuristics, advice is more than a great equalizer, it leads to our low-ability subjects performing better than the high-ability ones. Under our probabilistic heuristics however, our best subjects continue to perform better than the low ability ones, even after following advice. This is because, in order to benefit from good advice, subjects have to also follow some very bad oneas our L-adviser performed worst than randomness -and decrease their performance.

Thus, the top performers among our high ability subjects ignore advice and continue to overperform. In the case with known adviser, the two heuristics are roughly similar: it is in the interest of almost everyone to ignore L-advice and to follow H-advice. The initial difference between the share of correct answers of high and low ability subjects is of 10 percentage points, and increases to 10.9pp post-advice in the experiment, as low ability subjects do not follow enough H-advice while nonetheless following some L-advice.

In contrast, by following our heuristics, this difference could have decreased to between 5.8pp (Probabilistic heuristic) and 6.6pp (Elo heuristic). We report those differences for both treatments in Table 9 in Appendix B.2.

Correct Answers and Preference for Control

We start by constructing an index of the stated preference for control by aggregating the answers to our questions borrowed from [START_REF] Burger | The desirability of control[END_REF]. We find that stated preference for control is correlated with the probability of a subject keeping their answers, after controlling for subject and answer characteristics (see Table 8 in Appendix B.1).

We then present our main regression Table 5. Following our pre-registration, we use both our binary definition of low and high-ability subjects, and a continuous measure based on the Elo rating, and run regressions for the known and unknown adviser treatment.

As expected, a high ability adviser typically benefits subjects, and the better rated subjects are more likely to evaluate positions correctly. The interaction term between the adviser type and the rating of our subjects gives an alternative measure of the existence of decreasing differences. As in the main tests with two categories of subjects, it is not significant. We control for individual characteristics in Table 10 in Appendix B.2.

Conclusions

Digitalization and the development of Artifical Intelligence promise to give broad access to high quality specialist advice. In theory, one of the main consequences of this evolution is a compression in the distribution of productivity, reducing the difference between the best and worst performers. the literature on advice and preference for control tells us that subjects may simply not take up this advice.

In this paper, we used a sample of subjects with specialist knowledge in their topic in a natural setting -chess players evaluating chess positions during a chess tournament -to learn more about the "great equalizer" potential of advice. While we find evidence that improving the quality of advice could benefit low ability players more, most of the potential benefit of advice is wasted by subjects choosing to keep their initial evaluation.

This preference for following their own expertise hurts low ability subjects the most, as they had the most to gain. The fact that low-ability subjects are also those payingeither due to overconfidence or intrinsic preference -the highest premium to follow their initial evaluation is consistent with the idea that the most able subjects are also the most able to follow advice. It could also be the case that ability in chess is not exogenous, and that the best rated players are precisely those who were able to listen to advice during their training.

Our experiment suggests that existing evidence of decreasing differences based on the routine use of AI advice to improve writing or the gathering of information may not translate easily to sectors such as medicine or engineering where subjects have expertise and may want to trust it more at the moment of the decision than any advice they receive. Good advice however still plays the important role of protecting subjects from bad advice. In a world in which advice is, on average, correct, lower ability subjects are more at risk of using bad advice by accident. In terms of public policy, this means that digital literacy and the importance of telling good from bad advice will become even more important when people start routinely relying on it.

Among the limitations of our paper is the fact that we do not distinguish between advice from humans and from computers. We did so because, since the landmark victory of chess engine Deep Blue versus the then world champion Garry Kasparov in 1997, chess players see algorithmic analysis of the games as the gold standard. This is precisely the reason why we could use the chess engines evaluation of the pawn advantage in our chosen positions as the unambiguously correct answer. Further studies of subject specialists such as our chess players would benefit from comparing computer-based advice and human one and see whether decreasing differences are more pronounced with the latter.

B Additional Results

B.1 Keeping your Answer

Table 7 shows that subjects correctly change their answers more after seeing H-advice.

They fail to change them as often as they should however. It also shows that lower Elo players change their answers more often that higher Elo ones, which is expected. Table 8 shows that higher Elo subjects tend to stick with their answers more often lower rated ones. When knowing the adviser, the baseline is that it is the L adviser, and subjects

correctly keep their answer more often. On the other, as shown by the interaction terms between the H adviser and the Known Adviser, then subject change more often their answer when they know it is of good quality. (3) and (4) add demographic controls but restrict the sample. 1 Absolute distance from the correct answer in pawn advantage.

B.2 Correct Answers

In Table 9, we see that high ability subjects have higher rate of correct answers than low ability ones. The only exception is the Elo heuristic, because in the Unknown Adviser treatment, h type subjects are penalized by never following the H advice at all. The probability heuristic tells us that they should sometimes follow the mixed advice. The regression in Table 8 shows that higher Elo subjects tend to keep their answer more.. Importantly, to be followed by subjects, the H advice has to be revealed as one, as shown by the interaction term beteen H adviser and Known Adviser, compared to the insignificant H adviser term alone.

Table 9: Difference (in percentage points) between the average share of correct answer of h and l type subjects, before advice, post advice, and following our two heuristics.

Treatment

Before After Probabilistic Elo All 10.3 9.4 6.5 2.2 Unknown Adviser 10.7 7.6 7.0 -3.5 Known Adviser 10.0 10.9 5.8 6.6

The regression in Table 10 adds to the regression in Table in 5 demographic controls.

The results are consistent between the two and the additional control variables are not significant. In parenthesis on the same line are the p-value, below the standard error.

C Experimental Instructions (Online Appendix)

Figure 1 :

 1 Figure 1: A position in a game of chess, as shown to our subjects.

Figure 2 :

 2 Figure 2: The answer sheet for the first round. The left-hand side was completed before seeing the advice, and the right-hand side after.

Figure 4 :

 4 Figure 4: Share of correct answers before advice, post advice, and following our heuristics, in our two experimental conditions

Table 1 :

 1 Share of correct answers before advice.

	Player type
	Subjects l	31.2%
	Subjects h	41.8%
	Adviser L	15.0%
	Adviser H	75.0%

Table 2 :

 2 Main pre-registered test: comparing the share of correct answers post-advice under Negative Assortative Matching (NAM) and Positive Assortative Matching (PAM)

	of subjects to advisers.
	Treatment	NAM PAM P-value 1
	Main test: H vs L-advice
	All	40.5% 38.4% 0.365
	Unknown Adviser 38.1% 35.2% 0.384
	Known Adviser	42.6% 41.3% 0.711
	Alternative test: H vs No-advice
	All	42.5% 40.1% 0.301
	Unknown Adviser 41.5% 36.9% 0.165
	Known Adviser	43.3% 43.0% 0.951

1 P-value of the two-sided two sample test of equal proportion between NAM and PAM.

Table 3 :

 3 How do subjects react to the advice received, depending on agreeing or not with

					it?
			Agree Agree Before 2 Disagree		Disagree Before 2
	Type Treatment Before 1 Keep React Before 1 Keep Follow Closer Further
	l	Know H	33.5	93.0	7.0	66.5	46.8	39.2	12.3	1.8
	l	Know L	27.8	98.6	1.4	72.2	78.8	14.0	7.3	0.0
	l	Unknown	29.4	94.0	6.0	70.6	64.3	27.3	6.3	2.2
	h	Know H	41.8	96.5	3.5	58.2	52.5	42.5	2.5	2.5
	h	Know L	26.8	95.9	4.1	73.2	83.9	9.5	4.0	2.5
	h	Unknown	33.0	97.3	2.7	67.0	79.7	15.3	3.3	1.7

1 Bold font: percentage of identical and different pre-advice answer than the adviser.

2 Normal font: percentage of kept or changed answer, conditional on pre-advice answer being identical to adviser's; percentage of kept or changed answer (following, getting closer, or further away from advice) conditional on pre-advice answer being different from adviser's.

Table 4 :

 4 Difference (in percentage points) between the average share of correct answers of h and l type subjects having received H-advice, following our heuristics and in the experiment.

			Types
	Treatment Heuristics	l	h	P-value 1
	Unknown Probabilistic 32.4 23.0	0.056
	Known	Probabilistic 27.7 21.1	0.134
	All	Probabilistic 30.0 22.2	0.014
	Unknown Elo	36.8 -5.7 <0.001
	Known	Elo	27.7 21.1	0.139
	All	Elo	32.2 9.0	<0.001
	Premia paid for ignoring H-advice are given in
	percentage points.
	1 P-value of the two-sided two sample t-test of
	equal control premium between the h and l Elo
	types.

Table 5 :

 5 Regression for the share of correct answers, with fixed effects at the position level. Robust standard errors clustered at the position level. In parenthesis on the same line are the p-value, below the standard error.

		Grouped by ELO	Continous ELO
		Known	Unknown	Known	Unknown
	H Adviser	0.147 (0.021) 0.066 (0.192)	0.23 (0.002)	0.17 (0.010)
		(0.058)	(0.049)	(0.065)	(0.059)
	Low Elo	-0.140 (0.001) -0.118 (0.053)
		(0.036)	(0.057)
	Elo			1.1 × 10 -4 (<0.001) 8.7 × 10 -5 (0.027)
				(2.2 × 10 -5 )	(3.6 × 10 -5 )
	H Adviser×Low Elo 0.056 (0.267) 0.096 (0.093)
		(0.049)	(0.055)
	H Adviser×Elo			-6.8 × 10 -5 (0.071) -7.2 × 10 -5 (0.056)
				(3.5 × 10 -5 )	(3.6 × 10 -5 )
	Std.Errors	by: position	by: position	by: position	by: position
	Num.Obs.	1064	912	1064	912
	R2	0.145	0.082	0.147	0.085
	R2 Adj.	0.127	0.059	0.129	0.063
	Notes:

Table 7 :

 7 Share of identical answers for l and h subjects after observing different types of

			advice.
	Type L-Advice H-advice Unknown Advice
	l	80.4%	61.5%	69.4%
	h	85.0%	69.6%	83.5%

Table 8 :

 8 Regression for keeping the answer after receiving the advice, with fixed effects at the position level. Robust standard errors clustered at the position level. In parenthesis on the same line are the p-value, below the standard error.

		(1)	(2)	(3)	(4)
	Distance Correct 1	-0.043 (0.001)	-0.0475 (0.001)	-4.0e-02 (0.002)	-4.5e-02 (0.002)
		(0.012)	(0.0123)	(1.1e-02)	(0.01212)
	H Adviser	-0.019 (0.544)	-0.0085 (0.795)	-1.9e-02 (0.544)	-8.2e-03 (0.808)
		(0.031)	(0.0322)	(3.1e-02)	(0.03339)
	Known Adviser	0.066 (0.009)	0.0414 (0.089)	7.9e-02 (0.004)	4.9e-02 (0.052)
		(0.022)	(0.0230)	(2.4e-02)	(0.02387)
	High Elo	0.083 (<0.001)	0.0702 (0.003)
		(0.019)	(0.0205)
	H Adviser×Known Adviser -0.168 (<0.001) -0.1685 (<0.001) -1.7e-01 (<0.001) -1.7e-01 (<0.001)
		(0.038)	(0.0380)	(3.8e-02)	(0.03812)
	Elo			1.7e-04 (0.010)	2.2e-04 (0.002)
				(6.1e-05)	(0.00006)
	Elo×rated			4.7e-06 (0.881)	-1.2e-05 (0.685)
				(3.1e-05)	(0.00003)
	Control Index		0.0331 (0.090)		3.4e-02 (0.087)
			(0.0185)		(0.01901)
	Male		0.0562 (0.092)		4.3e-02 (0.163)
			(0.0316)		(0.02957)
	Age		0.0012 (0.267)		9.2e-04 (0.396)
			(0.0011)		(0.00107)
	Std.Errors	by: position	by: position	by: position	by: position
	Num.Obs.	1999	1749	1999	1749
	R2	0.064	0.069	0.073	0.080
	R2 Adj.	0.052	0.055	0.061	0.065
	Notes:

Table 10 :

 10 Regression for the share of correct answers, with fixed effects at the position level and controlling for individual characteristics.

		Grouped by Elo	Continous Elo
		Known	Unknown	Known	Unknown
	H Adviser	0.1533 (0.020)	0.0701 (0.225)	0.31017 (0.050)	0.34626 (0.048)
		(0.0606)	(0.0559)	(1.5e-01)	(0.16387)
	Low Elo	-0.1656 (<0.001) -0.1366 (0.051)
		(0.0346)	(0.0657)
	Elo			0.00030 (<0.001) 0.00026 (0.029)
				(5.9e-05)	(0.00011)
	H Adviser×Low Elo	0.0460 (0.368)	0.0924 (0.146)
		(0.0500)	(0.0610)
	H Adviser×Elo			-0.00011 (0.311) -0.00018 (0.167)
				(1.0e-04)	(0.00013)
	Male	-0.0473 (0.186) -0.0869 (0.107) -0.07296 (0.047) -0.09827 (0.068)
		(0.0345)	(0.0514)	(3.4e-02)	(0.05081)
	Age	-0.0036 (0.053) -0.0035 (0.072) -0.00259 (0.191) -0.00360 (0.077)
		(0.0018)	(0.0019)	(1.9e-03)	(0.00193)
	Control Index	0.0169 (0.601)	0.0576 (0.165)	0.00247 (0.939)	0.05682 (0.167)
		(0.0317)	(0.0399)	(3.2e-02)	(0.03953)
	Std.Errors	by: position	by: position	by: position	by: position
	Num.Obs.	960	800	960	800
	R2	0.168	0.086	0.168	0.091
	R2 Adj.	0.145	0.056	0.146	0.061
	Robust standard errors clustered at the position level.

The exact dates are August 15, August 20, September 2, and September 17, 2023. In line with the pre-registration, we stopped recruiting participants when we reached 100 subjects, so that we recruited

Given the difficult banking situation in Lebanon and the fact that some of our subjects were minor, we did not pay subjects directly in cash but with monetary vouchers for subsequent tournaments or other spending on the day.

We remove a small number of null answers from this table, because we have no distance from the answer for them. We therefore slightly overestimate the agreement percentage before receiving the advice.

Appendix

A Sample Description

Table 6 shows that most of our subjects are young men. Figure 5 that most of our subjects are rated, but the mode is not being rated. The proportion of unrated players means that the low Elo group is almost all made of unrated players. [1000, 1200) [1200, 1400) [1400, 1600) [1600, 1800) [1800, 2000) [2000, 2200) Rapid Elo Share of Subjects بنفسي قراراتي باتخاذ أستمتع أنا . I would rather someone else took over the leadership role when I'm involved in a group project.

أشارك عندما القيادي الدور آخر شخص يتولى أن أفضل جماعي مشروع في . There are many situations in which I would prefer only one choice rather than having to make a decision.

فقط ًا واحد ا خيارً فيها أفضل التي المواقف من العديد هناك قرار اتخاذ إلى االضطرار من ً بدال .

A1i

Here are ten positions that occurred in real chess games which have been chosen from a dataset of previous games from the Mega Database 2023.

We will ask you to evaluate 20 games over two rounds: 1 and 2. We will pick one of your evaluations at random and you will receive a voucher of $10 if your answer was correct.

Please complete Round 1, Part 1 of the Response Sheet by indicating for each game your best estimate of the pawn advantage, which can be +0.7, -0.7, +2.4, or -2.4. Please check the box corresponding to your choice (only one possible answer). Note that the positions have a pawn advantage of ±0.7 and one of ±2.4 with equal probability.

A1i

We will now provide you with some additional information about the ten positions.

We have asked a player with a rating of 2335 to evaluate the ten games in the same conditions as you. You can find their prediction in the table below.

Looking

A1ii

We will now provide you with some additional information about the ten positions.

We have asked an unrated player, who plays regularly for fun, to evaluate the ten positions in the same conditions as you. You can find their predictions in the table below.


Philippe Parnaudeau

Eric Lamballais

Dominique Heitz

Jorge H Silvestrini

Combination of the immersed boundary method with compact schemes for DNS of flows in complex geometry

come   L'archive ouverte pluridisciplinaire

CONTEXT OF THE STUDY

Despite the continual progress of computers, direct and large eddy simulation of turbulent flows in complex geometries remains a difficult task. For each flow configuration, a compromise must be specifically determined in order to correctly describe the physics of the flow for a reasonable computational cost (speed, memory requirement, code complexity). The choice of the computational grid is well known to be crucial for the determination of this compromise. In order to take accurately small details of the geometry into account, the most popular method is to generate a sophisticated grid following the body geometry, despite the fact that such grids are frequently strongly distorted, resulting in a degradation of the numerical accuracy associated with a significant increase of the global computational cost.

An alternative method to avoid the drawbacks of the body fitted approach consists in extending capabilities of codes based on simplified grids via the use of the 'immersed boundary method'. The basic idea of this technique is to mimic the effect of a solid surface on the fluid through a forcing applied in the body region. This operation is performed by additional terms introduced in Navier-Stokes equations. Various formulations are proposed in the literature with various names like 'virtual boundary method', 'fictitious domain method' or 'penalization method'.

Short review of immersed boundary methods. Here, we use the generic term 'immersed boundary method' introduced by [START_REF] Peskin | Flow patterns around heart valves: a numerical method[END_REF] where this idea was employed to consider the full interaction between elastic solids and the fluid. For more simplified situations where the motion of solid surfaces is a known of the problem, three types of forcing can be distinguished: (i) feedback forcing [START_REF] Goldstein | Modeling a no-slip boundary condition with an external force field[END_REF], (ii) algebraic forcing [START_REF] Arquis | Sur les conditions hydrodynamiques au voisinage d'une interface milieu fluide-milieu poreux : application à la convection naturelle[END_REF], (iii) direct forcing [START_REF] Tremblay | DNS of flow around a circular cylinder at a subcritical reynolds number with cartesian grid[END_REF][START_REF] Fadlun | Combined immersed-boundary finite-difference methods for three-dimensional complex flow simulations[END_REF]. The feedback forcing is based on an artificial term that can 'freeze' efficiently the fluid in the body region through a damping oscillation process. The algebraic forcing is a simplified form of the feedback one where the time integral term is suppressed. With this simplification, it is possible to establish a physical analogy where the forcing can model realistically a porous medium, the limit case of a zero porosity corresponding to the modelling of a solid obstacle1 . Unfortunately, feedback and algebraic forcings have a common drawback related to their numerical stability properties. Schematically, both methods lead to a severe additional restriction on the time step to maintain very low residual velocities in locations where no-slip conditions are expected. In order to avoid this limitation (often very expensive in terms of computational cost), the use of the direct forcing technique is very attractive. In this third method, which introduces no additional numerical stability restriction, the boundary condition is ensured in a quite straightforward way by prescribing directly the velocity in forcing region at each step of the time integration. Finally, note that it can be easily shown [START_REF] Lamballais | Direct numerical simulation of interactions between a mixing layer and a wake around a cylinder[END_REF] that feedback and algebraic methods can behave asymptotically (for a 'vanishing porosity') like a forcing method when their common modelling term is time integrated with a forward (implicit) Euler scheme.

Goals of the present study.

In this paper, we are interested in the strategy where the direct forcing is combined with centred finite difference schemes of high accuracy. Such a combination is a priori problematic due to the dis-continuities of the velocity derivatives created by the forcing. More precisely, the sudden application of the forcing inside the virtual boundary guarantees only the continuity of the solution whatever the spatial resolution is. The numerical code used for this study solves the incompressible Navier-Stokes equations discretized on a Cartesian collocated grid with the aid of th order compact schemes. Despite their very favourable accuracy properties, these finite difference schemes are a priori not well suited for the numerical treatment of a discontinuity, even if the jump condition concerns only the first derivative. This problem is related to the quasi-spectral behaviour of compact schemes that leads to spurious oscillations in a similar way as spectral methods in presence of discontinuities (Gibbs phenomenon). In preliminary calculations based on the same numerical code as here, [START_REF] Lamballais | Direct numerical simulation of interactions between a mixing layer and a wake around a cylinder[END_REF] reported that the creation of spurious oscillations in the neighbourhood of the obstacle was increased when a direct forcing was used instead of a feedback one. Since this problem was not mentioned in previous studies based on second order accurate codes [START_REF] Tremblay | DNS of flow around a circular cylinder at a subcritical reynolds number with cartesian grid[END_REF][START_REF] Fadlun | Combined immersed-boundary finite-difference methods for three-dimensional complex flow simulations[END_REF][START_REF] Piomelli | Numerical simulations using the immersed boundary technique[END_REF], it was concluded by [START_REF] Lamballais | Direct numerical simulation of interactions between a mixing layer and a wake around a cylinder[END_REF] that the spurious oscillations were a consequence of the spectral-like nature of the spatial discretization.

In this work, we propose a direct forcing method better suited for compact schemes. Basically, the idea is to create inside the body a flow preserving the no-slip condition at the surface but reducing the step-like change of the first derivative of velocities across the immersed boundary. This modification is shown to improve results both qualitatively and quantitatively.

FORMALISM

Schematically, the direct method consists in the application of the velocity condition in the forcing region. The target velocity field is a priori a known of the problem, at least at the locations where boundary conditions are expected. The discrete integration of the incompressible Navier-Stokes equations with a Adams-Bashforth scheme gives

where is the modified pressure ( and are the pressure and the constant density respectively) while is the vorticity field. The direct forcing term takes simply the expression [START_REF] Fadlun | Combined immersed-boundary finite-difference methods for three-dimensional complex flow simulations[END_REF] with in the body region and everywhere else. Note that for simplicity, we consider only a motionless body surface (no time dependence of the mask function ). In this case, the simplest method to ensure no-slip conditions is to use a zero target velocity field, namely . As already discussed in the preceding section, this simplified forcing generates discontinuities on the first derivative of velocities that are problematic when spectral or spectral-like schemes are used. In order to avoid this difficulty, the approach proposed in this study consists in estimating a target velocity field that allows the no-slip condition at the boundary while reducing the presence of discontinuities.

In order to do this, a first possibility is to estimate as the reverse to the flow immediately outside the body. For instance, for a circular cylinder of diameter , a quite natural choice for the target velocity is where are cylindrical coordinates associated to the circular body geometry. This type of forcing has already been tested by previous authors [START_REF] Tremblay | DNS of flow around a circular cylinder at a subcritical reynolds number with cartesian grid[END_REF][START_REF] Fadlun | Combined immersed-boundary finite-difference methods for three-dimensional complex flow simulations[END_REF] but only for the prescription of the inner velocities at the closest grid points to the body surface. In the present context of compact schemes, such a selective action would not be efficient enough due to the nonlocal character of the derivative estimation. Then, we define a target velocity in the full body domain as [START_REF] Goldstein | Modeling a no-slip boundary condition with an external force field[END_REF] where the modulation function [START_REF] -R. Kevlahan | Computation of turbulent flow past an array of cylinders using a spectral method with brinkman penalization[END_REF] is adjusted to ensure the regularity of inner velocities and to avoid the singularity at . Naturally, other choices of are possible provided that the three following conditions are verified (i)

: accurate reverse condition near the body surface, (ii)

: singularity cancellation and (iii) with moderate first and second derivatives for . For instance, is another suitable modulation function. Here, we choose the expression (5) because it cancels slightly more rapidly the target velocity when , but it should be recognized that this choice is a partly arbitrary.

Another point is that it is possible to apply the reverse condition only to the tangential component of the velocity, the normal component being simply cancelled. Such a forcing leads to concentric streamlines inside the cylinder. Note this cancellation of the normal velocity inside the body does not create additional discontinuities due to the incompressibility condition at the boundary that guaranties a zero normal derivative for this velocity component. In preliminary calculations (not presented here), we observed that both treatments of the internal normal velocity (reverse condition or cancellation) lead to equivalent results. In the data presented in the following, the normal inner velocity was maintained to zero except for the case of the tapered cylinder where the reverse condition was directly applied at the vectorial level. In complex geometry, the reverse condition is easier to implement because no projection of the velocity vectors is necessary in order to distinguish normal and tangential components. Naturally, the estimation of the target velocity using (4) on a Cartesian grid needs to perform interpolations. Here, we only use a multilinear interpolation to prescribe the internal flow from the knowledge of the external one. Despite the second order accuracy of such a procedure (which is significantly lower than to the formal accuracy of the numerical code itself), it will be shown in the following that important benefits can be obtained from the use of th order compact schemes.

An important point is that the target velocity field is not a priori divergence free. For this reason, the verification of the incompressible condition must be discarded inside the body by allowing a mass source/sink in region where . Following the approaches proposed by [START_REF] Tremblay | DNS of flow around a circular cylinder at a subcritical reynolds number with cartesian grid[END_REF][START_REF] Kim | An immersed-boundary finite-volume method for simulations of flow in complex geometries[END_REF], a first possibility is to impose . Here, we use a slightly modified condition that was found to reduce more efficiently oscillations in the vicinity of the obstacle.

In the framework of the fractional step method, several adjustments are necessary in order to eliminate the various couplings introduced by the forcing method. In this context, a three step advancement yields (6) (7) [START_REF] Mittal | On the inclusion of three-dimensional effects in simulations of two-dimensional bluff-body wake flows[END_REF] with the associated forcing term [START_REF] Peskin | Flow patterns around heart valves: a numerical method[END_REF] where the target velocity is estimated by [START_REF] Piomelli | Numerical simulations using the immersed boundary technique[END_REF] In first analysis, it can be expected that the splitting error introduced by the use of in [START_REF] Kim | An immersed-boundary finite-volume method for simulations of flow in complex geometries[END_REF] and in (10) instead of and respectively is only without any consequence on the final order of the time advancement.

The last step is to derive a Poisson equation compatible with the condition

. Here, we propose to use the approximation that leads to the pressure equation [START_REF] Schneider | Adaptative wavelet simulation of a flow around an impulsively started cylinder using penalisation[END_REF] where the conventional Poisson equation is recovered for whereas inside the body, the condition yields the Laplace equation. Note that other variations on the method are possible, especially concerning the correction step (8) that can be conditioned by as in [START_REF] Tremblay | DNS of flow around a circular cylinder at a subcritical reynolds number with cartesian grid[END_REF] where the pressure cells are explicitly masked (in this case, a two step splitting is only necessary). In this work, the correction by pressure gradients is performed everywhere in the computational domain via a fractional method in three steps.

RESULTS

Steady 2D wake at . The benefit of the use of a non-zero target velocity is shown in figure 1 where a constant flow around a 2D cylinder is considered. Two cases are compared depending on the target velocity treatment that can be zero (no internal flow) or given by ( 4) for the tangential component. The comparison between both forcing methods shows clearly the improvement offered by the use of a reverse flow inside the body. First, the examination of the longitudinal velocity profiles (1c view) obtained in each case shows clearly the more realistic near-body behaviour of the velocity when the first derivative discontinuity is avoided. The improvement of the near-body data is confirmed by the examination of vorticity isocontours (1b view). Note in particular the reduction of spurious vorticity when a reverse flow is imposed Table 1. Comparison of statistical results obtained from various combinations between the forcing method and the numerical code accuracy. All DNS are performed using a computational domain discretized on grid points. Data of DNS II' are from [START_REF] Lamballais | Direct numerical simulation of interactions between a mixing layer and a wake around a cylinder[END_REF]. The streamwise location of the cylinder is . inside the cylinder. Quantitatively, by comparison to previous [START_REF] Ye | An accurate cartesian grid method for viscous incompressible flow with complex immersed boundaries[END_REF] or highly resolved results, we observe that the characteristic length scales of the flow are predicted more accurately when using the new forcing with the present spatial resolution. For instance, we verified that this new method allows a satisfactory prediction of the length of the wake bubble that is ( ), in good agreement with [START_REF] Ye | An accurate cartesian grid method for viscous incompressible flow with complex immersed boundaries[END_REF] who found . In contrast, the use of a zero target velocity field leads to an overestimation of 10%, i.e., (

DNS

). Finally, note that we have verified that both forcing methods allow the convergence towards the correct length of the wake bubble while reducing considerably the spurious vorticity in the neighbourhood of the cylinder.

Unsteady 3D wake at . In this section, we compare four DNS combining three different forcing methods with two numerical codes based on or (compact) centred finite difference schemes (see table 1 and its caption for more details about the simulation parameters). Concerning the length scale selection, similar trends as for are recovered for this unsteady case at higher Reynolds number. The formation length (deduced from the streamwise location where the Reynolds stress reaches its maximum) is better predicted using the forcing (4) whereas the use of leads to a typical 10% overestimation, in agreement with previous observations of [START_REF] Lamballais | Direct numerical simulation of interactions between a mixing layer and a wake around a cylinder[END_REF] based on DNS using the feedback forcing method. Compared to reference values, an improvement of the Strouhal number prediction is also obtained. However, the comparison between DNS III and IV shows that these improvements are not obtained when schemes are used, this insensitivity to the forcing method (with or without internal flow) being consistent with the previous observations of [START_REF] Fadlun | Combined immersed-boundary finite-difference methods for three-dimensional complex flow simulations[END_REF]. Present conclusions can be confirmed by the examination of and profiles presented in figure 2 for each case. The improvement offered by the use of compact schemes is clearly shown, especially when the forcing with internal flow is applied (DNS I). Conversely, the overestimation of the longitudinal velocity fluctuations obtained for both DNS III and IV emphasizes the interest of the use of highly accurate schemes, even if the formal accuracy is significantly lower due to the forcing method itself. Note that this present overestimation is in agreement with the results of [START_REF] Piomelli | Numerical simulations using the immersed boundary technique[END_REF] who used also a numerical code based on schemes. Physically, at marginal resolution, the combination of schemes with a direct forcing method seems to inhibit the 3D motions near the cylinder. This phenomenon can be shown not only by statistical results but also through instantaneous visualizations. For instance, figure 3 from its value for a conventional wake. Note that such a local selection leads to high frequencies in the low region for case A and the opposite situation for case B. Naturally, this selection cannot be purely local due to the preservation of the coherence of the flow motion in -direction. The vortical organization obtained for each case is presented in figure 4. Despite the similarities between these two flow configurations, the local mechanisms of vortex shedding lead to the formation of well marked cells in case A whereas in case B, the main effects are linked to the selection of oblique structures (note however that a cellular pattern of vortex shedding can also be identified for case B by means of a frequency analysis [START_REF] Silvestrini | Direct numerical simulation of oblique vortex shedding from a cylinder in shear flow[END_REF]). The occurrence of dislocations (phase breaking, tearing of vortices) can be observed in both cases, but these phenomena are found to be more frequent for case A. A quantitative comparison between these two flows (frequency analysis, mean and fluctuating motions) is currently in progress.

Figure 1 .

 1 Figure 1. Comparison between two direct forcing methods with and without internal flow at. The computational domain is discretized on grid points and the numerical code is based on compact schemes.

Figure 2 .

 2 Figure 2. Profiles of longitudinal velocity statistics at different streamwise locations . -: DNS I; : DNS II; : DNS III; . . : DNS IV; : reference data obtained by spectral DNS based on a cylindrical grid [8].

Figure 3 .

 3 Figure 3. Isosurfaces of vorticity modulus . Left: DNS I, right: DNS III.

Figure 4 .

 4 Figure 4. Comparison of instantaneous vorticity visualizations. The computational domain is discretized on grid points with and . Data of Case B are from [12] who used a feedback forcing method.

This physical analogy can be exploited advantageously in order to interpret the meaning of the residual flow inside the modelled body (establishment of a d'Arcy law where velocity and pressure gradient are proportional) and to make easier the computation of the associated drag and lift[START_REF] Schneider | Adaptative wavelet simulation of a flow around an impulsively started cylinder using penalisation[END_REF]. Another advantage of this second forcing is related to its algebraic nature that offers the possibility to study theoretically its asymptotic convergence towards the case of a purely solid wall[START_REF] Angot | A penalization method to take into account obstacles in incompressible viscous flows[END_REF][START_REF] -R. Kevlahan | Computation of turbulent flow past an array of cylinders using a spectral method with brinkman penalization[END_REF].

Acknowledgments

Calculations were carried out at the IDRIS. This study was partially supported by the Région Poitou-Charentes and the CNRS.


Stéphane Bonelli
email: stephane.bonelli@aix.cemagref.fr

Didier Marot
email: didier.marot@iutsn.univ-nantes.fr

Fabien Ternat
email: fternat@gmail.com

Nadia Benahmed
email: nadia.benahmed@cemagref.fr

D Maro

F Terna!

S Bonelli Cemagref

Hydraulic criteria for internal erosion in cohesive soils

  L'archive ouverte pluridisciplinaire

Introduction

Erosion can only develop if two vital conditions are met: particles must be torn off and must be transported. Four types of erosion process, internal to the work, have been identified [START_REF] Fell | Internal erosion of dams and their foundations[END_REF]]: 1) suffusion, which affects the soil structure, 2) evolution of defects in the soil matrix (holes, cracks), 3) backward erosion, 4) contact erosion between two soils.

After exposing a new approach concerning the critical erosion shear stress for cohesive soil, the present paper presents some recent results concerning types 1 ), 2) and 3): the suffusion and backward erosion processes, internal to the soil, and the piping erosion process, external to the soil and internal to the work.

Critical Erosion Shear Stress for Cohesive Particles

Basically, the critical erosion shear stress of cohesive sediments depends on the granulometry (i.e. particle size and shape) and cohesive particle content whose diameter is finer than 2µm [START_REF] Graf | Hydraulics of Sediment Transport[END_REF]. A strong dependence on the consolidation degree of the water/sediment mixture has also been highlighted [START_REF] Mitchener | Erosion of mud/sand mixtures[END_REF]. This consolidation degree can be related to various experimental measurements such as porosity or water content measurements [Sunborg, 1956;[START_REF] Migniot | A study of the physical properties of different very fine sediment and their behaviour under hydrodynamic action[END_REF]. The Atterberg limits are also useful, as featured on Table .1, but specific interpretation of this parameter is required to relate it to mechanical properties of the sediment. Particle whose diameter is smaller than 2µm are suggested to interact by the mean of the Van der Waals force. Its expression between two spherical particles of diameters d1 and d 2, separated with the distance d;, [lsraelachvili, 1985] can be written in the following way:

(1)

Recent work has tried to evaluate this particular cohesion force [START_REF] Terna | Erosion of self-weight consolidated cohesive sediments[END_REF]. The main difficulty remains in the determination of the interparticle distance d;, which has been linked to the porosity n, considering a crystalline modelling of the particle network. This modelling provides, on the one hand, the expected link between interparticle distance and the porosity, but on the other hand, it also provides a multiplicative factor C1, called coordination, that is used to account for multiple interactions. For instance, here is the expression of the interparticle distance between two same size particles:

(

) 2
where nmax and nmin are respectively the maximum and the minimum values of the porosity, corresponding respectively to the most compacted and the loosest states of the sediment, n is the local porosity. The coordination factor is considered to vary like the cube of the ratio between the two particle sizes:

(3)

Finally, the resulting cohesion force becomes:

(4)

Once determined, the cohesion force is included into the usual force balance considered to assess the critical shear stress [START_REF] Dade | Predicting erosion resistance of muds[END_REF][START_REF] Graf | Hydraulics of Sediment Transport[END_REF]Wiberg, 1987]. The erosion criterion is expressed in the following form:

(5)

Where <Dis the internal angle, F0 is the drag force of the flow, FL is the lift force of the flow, Fw the buoyant weight of the particle and Fe the cohesion force. The drag force must account for the granulometry of the eroded surface, which is achieved in the drag coefficient [START_REF] Graf | Hydraulics of Sediment Transport[END_REF].

The lift force is also described and can be accurately determined [START_REF] Yalin | Mechanics of Sediment Transport[END_REF][START_REF] Saffman | The lift force on a small sphere in a shear flow[END_REF].

Note that hydrodynamic forces are valid in the viscous domain (Re ' <1), determining thus the validity domain of the present model. The granulometry of the eroded surface is accounted by means of the a coefficient, characterizing the relative position of the particle related to the sediment/water interface. There is little controversy concerning the buoyant weight of the particle, which can be obtained by the product of the particle volume with its specific weight. All these contributions are adjusted with shape factors, related to each force: k0 for the drag force, kL for the lift force, kw for the buoyant weight. Gathering all those information leads to an expression of the critical shear stress for the erosion of a spherical cohesive particle, in terms of dimensionless Shields parameter,

• in function of the particle Reynolds number Re ' :

T* = H, ell•:]) .- 1 + _!l_ I.; .tan<!> ( F.
) a+ f;tan @.Re"" + :3.��a

.Re*2 Fiv ( 6
)
where a is a constant of the drag coefficient. This expression features the rule of the cohesion function (1+Fe/Fw) that increases the value of the erosion threshold since Fe becomes comparable to Fw. Let's note that the determination of the erosion threshold does not present any difficulty to be solved when Fe is null, but a numerical resolution must be done when not.

To illustrate these approaches, here are examples of possible results. Firstly, it is interesting to have a look over the cohesion function that features a critical diameter smaller than a few microns, where cohesion between two particles becomes efficient (Figure 1). When multiple interactions are accounted for, cohesion affects all the particle range present in the granulometry, revealing a kind of cementation of coarse particles by fine ones (Figure 2).

It is then possible to obtain the critical erosion velocity u ' (m/s), defined by 1 ' =pw.u • 2 in function of the particle diameter d (Figure 3). -+-n =0.6 ...... n =0.7

•-•-n =0.8 ---n=0.9

-n=1

Cohesion range 1QjL_ -�����'-'--' '-=-""-'"'-""-'""""" ..... -e--n = 0.5 -1-n = 0.6

""" l'l = 0.7 -<>-n =0.5 -+-n = 0.6 ...... n =0.7 All the parameters values required for the application of the presented modelling are given on Table 2. This section reveals a first simple approach for assessing the critical erosion threshold of cohesive sediments. It is deduced from a force balance that includes the cohesion force, attributed to the presence of clay particles. This force, supposed to be the surface Van der Waals interaction, is directly dependent on the compaction degree and the particle size. The present model integrates all these declarations and introduces the porosity as a main parameter. Results reproduce well the main declarations of the literature: cohesion force appears to come from the presence of clay particles, and thanks to the coordination number, coarse particles are also affected. The usual critical shear stress magnitude is assessed in good agreement with the literature. This model is interesting because of its simple parameterization by means of granulometry and porosity. The geometric factor including the unknown coordination appears also as a parameter, but whose value includes measurement and modelling errors. This original work results from the gathering of many models, which may be ameliorated. At the moment and to our knowledge, no modelling pretends to assess the coordination of such a wide range granulometric mixture. Moreover, cohesion is here attributed to the presence of clay particles, and is reduced to one of its components, the Van der Waals force. Other forces should also be incorporated to refine the results, such as double layer or chemical interactions. [-]

•-•-n = 0.8 ---n=0.9 -n=1 ------------------------- 10 " -----------------......i.--------' 10- 7 'lO- f . 10- 5 10- 4 d(m)
•-•-n = 0.8 --• n = 0.9 -n =l --•--� -• ------"" ' 'I 0 -4������-����-���������� 1� 1� 1� 1� 10 - d{m)
A 37,49

[-] k o 0,4 [-] k, 30 [ -] k w 1 [ -] nmax 1 [-] nmin 1-16 [-]
3

Internal Erosion in Cohesive Soils without Crack

The two main phenomena responsible for erosion of particles in uncracked soils are backward erosion and suffusion. In backward erosion particles are detached from the downstream surface by the outward seepage. The suffusion process is similar but the coarse particles form a matrix and erosion is only of the finer particles in the pore space between the larger particles.

3.1

Criteria Soil structure can be analyzed into two groups: a primary structure and a secondary structure [START_REF] Kenney | Internal stability of granular filters[END_REF]. The primary structure consists of grains, which are in contact with each other and provide primary resistance to erosion, compressibility and shear strength. If these grains are eroded there are changes to the soil resistance and this may cause collapses.

The secondary structure, on the other hand, is composed of grains, which are in the spaces between the primary grains and, which may be displaced under the action of mechanical (vibration) or hydraulic (flow) stresses.

On the assumption that bigger grains can hinder the erosion of smaller grains, [START_REF] Kenney | Internal stability of granular filters[END_REF] have developed a method for assessing whether soils are internally unstable based on the shape of the coarse or the fine low-content grading curve. However, the method does not apply to clay soils.

In order to characterize the initiation of internal erosion for cohesive soils, Reddi et al. [2000] have developed an expression of the hydraulic shear stress 1: [START_REF] Fell | Internal erosion of dams and their foundations[END_REF] where L'.p/L is the average pressure gradient, 'A is the intrinsic permeability, n is the porosity, k is the engineering hydraulic permeability, Tlw and Yw are the water viscosity, and specific weight, respectively.

3.2

The Triaxial Device

The developed experimental device can be used to study the initiation of suffusion and backward erosion for sandy-clay samples (Figure 4).

Results

The material used is a washed Loire sand (grain density: 26 kN/m 3 ) with a grain size distribution within the range 80µm-1mm (dso = 440µm, uniformity coefficient: 3.125). The clay consists of kaolinit e, with liquidity and plasticity limits are 55% and 22%, respectively.

In order to improve understanding of the phenomena, a distinction is made between the tests during which only clay particle migration is initiated, and the tests during which the transport of both clay particles and sand grains is observed. Maximum erosion rates per sample sections are preferred to some cumulative eroded mass information because the erosion rate here reaches its maximum value very quickly, which therefore means that it does not depend on the test duration.

Clay Erosion

From the beginning of the test, the mass flow increases until reaching a maximum value, qsmax• It then decreases asymptotically toward zero. The permeability remains constant when no erosion occurs, but it decreases where erosion has been initiated. The erosion occurring within the clay fraction does not affect the size analysis nor the volume of the samples significantly.

Consequently, according to the previously defined terminology, this phenomenon, characterized by some diffuse mass losses can be called suffusion.

The impact of three different parameters on the initiation of suffusion has been examined:

-kaolinite content: 5, 10, 20 and 30%;

-hydraulic gradient ranging between 5 and 160 m/m; -confining pressure a3: 100, 150, 200 and 250 kPa.

The range of hydraulic gradient was chosen relatively large to include the possible reduction of flow path in an earth structure by backward erosion phenomena. In this case, the local gradient can be much higher than the global one. The rate of suffusion increases according to the hydraulic gradient as follows:

qsmax = 16. 6 ( 10 °•02 (i-S) -1 ) (8)
where i=�p/(ywl) is the hydraulic gradient.

Obtained results show that, depending on the hydraulic gradient, the erosion of the soils studied decreases as a function of the clay content according to:

for i = 20m/m : qsmax = -0.06 %clay + 1.28 for i = 60m/m : qsmax = -0.13 %clay + 2.85 for i = 1 OOm/m : qsmax = -0. 17 %clay + 5. 15

(9) (10) (11)
In a general way, the erosion rate doubles when the clay content changes from 20 to 10%.

Figure 5 represents the evolution of the maximum erosion rate according to the hydraulic shear stress Eq. ( 7) for a3=100 kPa.

The initial porosity depending on the consolidation, studying the effects of the confining pressure is essential. For sand specimens subjected to oedometer confinement conditions, Papamichos et al. [2001] observe that the maximum erosion rate growth according to the axial pressure applied is destabilizing. The present tests conducted under isotropic confinement with a 20 m/m hydraulic gradient and a 10 % clay content, reveal some opposite results (Figure 6). !:; 6 ., "'

. To obtain the same maximum erosion rate, the hydraulic shear stress has to increase as a function of the confining pressure.

3 5 4 .; 3 c 2 - 0 ,-• 0 0,5 • 10% kaolinite � 20% kaolinite " 30% kaolinite 1,5 2 '(Pa) 2,5 • • 3 3,5
This approach makes it possible to identify the minimum stress levels, which depend on both clay contents and confining pressures and below which suffusion will not occur for the soils tested. For example, we obtained for a 3 =100 kPa Figure 7 shows the effects of the confining pressure on the maximum erosion rate (determined by weighing regarding sand erosion). This confirms that when the confining pressure rises, sand erosion within the samples tends to increase.

Piping Erosion in Cohesive Soils

Piping erosion is internal to the work, but external to the soil: this is a surface erosion. A large literature on surface erosion exists in the field of hydraulics and river engineering [START_REF] Graf | Hydraulics of Sediment Transport[END_REF], [Valin, 1977] The results of the test are given in terms of the flow rate versus time curve with a constant pressure drop. Therefore, the flow rate is used as an indirect measurement of the erosion rate.

For further details about this test, see [START_REF] Wan | Investigation of internal erosion and piping of soils in embankment dams by the slot erosion test and the hole erosion test[END_REF].

Results

The scaling law is now compared with previously published data [START_REF] Wan | Investigation of internal erosion and piping of soils in embankment dams by the slot erosion test and the hole erosion test[END_REF]. Analysis were performed in 18 tests, using 9 different soils (clay, sandy clay, clayey sand or silty sand).

The initial radius and the length of the pipe were R 0 =3 mm and L=117 mm. Table 1 contains particle size distribution, and critical stress and Fell erosion index.

Figure 9 gives the effect of erosion process as the flow rate in relation to time, and shows that the use of ter leads to efficient dimensionless scaling. Without this scaling, multiple graphs would be necessary to provide clarity of presentation. Scaled radius are plotted as a function of the scaling time in Figure 10. Nearly all the data can be seen to fall on a single curve. This graph confirms the validity of the scaling law [START_REF] Mitchener | Erosion of mud/sand mixtures[END_REF].

Therefore, it is recommended to use hole erosion tests and the scaling law in order to evaluate the piping erosion parameters on any sample of cohesive soil from a site.

16

• HDHET001

• HDHET005

14

• HDHET006

• HDHETOO?

I0 4 t

 4 ,, :�����rro-�---�,.,.,.. -�===:o::!: o::;i

Figure 2 :

 2 Figure 2: Evolution of the cohesion function 1 +FC/FW vs. particle diameter d for different porosity values.

Figure 3 :

 3 Figure 3: Evolution of the critical erosion velocity u* vs. particle diameter d (m) for different porosity values.

Tab. 2 :

 2 Numerical values of the parameters for the erosion threshold resolution

  Optical t:em:01 �+---�t-Tnnk re"�::==+--++ !vlass balance

Figure 4 :

 4 Figure 4: Schematic representation of the experimental triaxial cell equipped with the two controllers, effluent weight measurement and optical sensor mechanism (GeM). The device placed in a temperature-controlled chamber (20 ± 0.5 °C) consists of three modified triaxial cells. These cells have been modified to let the flow come through up to the core of the sample. So as to avoid all unwanted disturbances on the samples, saturation, consolidation, hydraulic and mechanical test stages are carried out inside the same cell without deconfining the samples. The carrying out of long-lasting tests is possible thanks to the automation of both the monitoring and the data acquisition. The use of three cells simultaneously makes it possible to reduce the duration of the test program. The detection of erosion in the effluent is performed using optical aids and by weighing the amount of grains in the eroding fluid. The internal erosion critical gradient can be assessed from the effluent instantaneous optical analysis. In order to address internal erosion development, injection volume flow rates and obtained mass flow measurements are compared. A detailed description is given by Bendahmane et al. [2006].

Figure 5 :(

 5 Figure 5: Maximum erosion rate v.s. hydraulic shear stress according to clay content (cr 3 = 100 kPa).

3

 3

Figure 6 :

 6 Figure 6: Maximum erosion rate v.s. hydraulic shear stress according to confining pressure (10% clay content) Depending on the hydraulic gradient value, the linear decrease of the maximum erosion rate according to the confining pressure as follows: for i = 20 m/m : qsmax = -0.006cr 3+1.256 for i = 60 m/m : qsmax = -0.013cr3+2.877

Tcr= 0 . 13

 13 Pa with a 1 0 % clay content; Tcr = 0.23 Pa with a 20 % clay content; Tcr = 0.32 Pa with a 30 % clay content. For a3 = 200 kPa and a 10 % clay content, Tcr is approximately 0.42 Pa. These values are several orders of magnitude greater than those in the case of surface erosion experiments, referring notably to the results from tests done in rotating cylinders [Arulanandan and Perry, 1983] or HET [Reddi et al., 2000]. However these values are four times smaller than those from Reddi et al. [2000]'s measurements in the case of internal erosion experiments. The direct application of our results and Reddi et al. [2000]'s seems to be difficult for the moment.The result difference observed between the samples tested here, and the Ottawa sand + kaolinite mixtures tested by[START_REF] Reddi | Comparison of internal and surface erosion using flow pump test on a sand-kaolinite mixture[END_REF] can be accounted for by many reasons. First the characteristics of the filter can either be open or be a porous stone may be responsible for these differences. The sand itself could also play a role, as grain sizes and grain angularities were different in our and their experiments.The result difference observed between the samples tested here and Papamichos et al.[2001]'s specimens can be accounted for by many reasons: the characteristics of the pressure first, which can be isotropic or axial, of the samples, which can be made of a sandy-clay mixture or composed of sand only, then of the phenomena examined which are interpreted to be suffusion or backward erosion, and the sand grain angularity.

3. 3 . 2

 32 Clay and Sand ErosionConsidering two samples, both with a 10 % clay content and consolidated at 150 kPa. The first sample subjected to the action of a hydraulic gradient of 60 m/m suffers some clay erosion, whose extent is measured using the optical sensor, which gives a clay eroded mass of 60 mg.The second sample, on the other hand, subjected to a 140 m/m hydraulic gradient, gives a weighed eroded mass (clay and sand) of approximately 40 g.The quantity of effluents achieved here (17% of the sample initial volume) produces a substantial volume variation within the sample, which finally collapses revealing then a significant change in the erosion mechanisms. Clay and sand particles are discharged from the soil downstream and along the upstream line through a backward erosion mechanism occurring within the sample.

Figure 7 :

 7 Figure 7: Maximum erosion rate v.s. confining pressure (10% clay content)

  and in the field of poromechanics and petroleum engineering[START_REF] Papamichos | Volumetric sand production model and experiment[END_REF]]. In the field of geomechanics, several experimental methods have been developed for simulating the piping erosion process experimentally, with particular attention focussed on the hole erosion test. The experience acquired on more than 200 tests on several soils has confirmed what an excellent tool this test can be for quantifying the rate of piping erosion in a soil.

Figure 8 :

 8 Figure 8: Schematic representation of the hole erosion test apparatus (Cemagref)

Figure 9 :Figure 10 :

 910 Figure 9: Hole Erosion Tests (symbols) versus scaling law (continuous lines). Dimensionless flow rate is shown as a function of dimensionless time_

  ., t--'111' .... i-e ..... ... ..... -e--4 Ht'

	10-•	10-	0	d(m)	10-:	10-	4
	Figure 1: Evolution of the cohesion function 1 +Fc/Fw vs. particle diameter d for different
	porosity values.
		Jncreasing , co.mpactlon

Table 1 :

 1 Properties of soils samples, critical stress and Fell erosion index

		Soil	%	%	%	%	cc (Pa)	le
			Gravel Sand Fines <2µm
	Lyell	silty sand	1	70	29	13	8	2
	Fattorini	medium plasticity sandy clay	3	22	75	14	6	3
	Pukaki	silty sand	10	48	42	13	13	3
	Jindabyne	Clayey sand	0	66	34	15	6-72 3-4
	Bradys	high plasticity sandy clay	1	24	75	48	50 -76	4
	Shellharbour	high plasticity clay	1	11	88	77	99 -106 4
	Waranga	Low plasticity clay	0	21	79	54	106	4
	Matahina	Low plasticity clay	7	43	50	25	128	4
	Hume	Low plasticity sandy clay	0	19	81	51	66 -92 4-5

Acknowledgement

This research project is sponsored by the French National Research Agency (ERINOH).

Literature

4.1

Criteria

Piping occurs if Po>rc where Po is the initial tangential shear stress exerced by the piping flow on the soil, and 'c is the critical stress. The scaling law of radius time evolution is [START_REF] Bonelli | On modelling of piping erosion[END_REF] with

where ter is the characteristic time of piping erosion, Ro is the initial radius, L'. p is the pressure drop, L is the hole length, Pd is the dry soil density, and Ce is the Fell coefficient of soil erosion.

The later is similar to the [START_REF] Temple | Headcut development in vegetated earth spillways[END_REF] coefficient of erosion kd, as kd= Ce /pd. The Fell erosion index is le=-log(Ce) (Ce given in s/m).

4.2

Influence of Soil Properties

All test results - [START_REF] Temple | Headcut development in vegetated earth spillways[END_REF] jet tests, Briaud EFA [2006] tests or [START_REF] Wan | Investigation of internal erosion and piping of soils in embankment dams by the slot erosion test and the hole erosion test[END_REF] hole erosion tests -give a interesting relationship between the critical stress and the coefficient of erosion, or equivalently between the critical stress and the erosion rate index: le is proportional to log(,0). The greater the critical stress, the greater the erosion rate index (the slower the erosion).

It is well known fact that different soils erode at different rate. Attemps were made to correlate erosion parameters -critical stress and coefficient of erosion -to common geotechnical or chemical soil properties in hope that simple equations could be developed for everyday use.

As a matter of fact, the erosion strength has been found to increase with: 1) the dry density of the soil, 2) the percent clay. This is illustrated by the following correlation [START_REF] Temple | Headcut development in vegetated earth spillways[END_REF]]: [START_REF] Papamichos | Volumetric sand production model and experiment[END_REF] However, for others parameters like the plasticity index, the liquid limit, the percent fines, the shear strength or the salinity, the situation is not clear. Due to the interdependence of these properties, however, it is clear that erosional strength of a soil cannot be represented accurately by the algebraic sum of the contribution of each of its properties. All attemps failed to reach a reasonable correlation coefficient value [START_REF] Briaud | Chapter 9 -Erosion tests on New Orleans levee samples, in Investigation of the performance of the New Orleans protection systems in Hurricane Katrina[END_REF]. It is strongly recommended carrying out hole erosion tests rather than using correlations [START_REF] Wan | Investigation of internal erosion and piping of soils in embankment dams by the slot erosion test and the hole erosion test[END_REF]].

The Hole Erosion Test

The hole erosion test was designed to simulate piping flow erosion in a hole. This test is not new [START_REF] Lefebvre | Erosivity of natural intact structured clay : evaluation[END_REF]]. An eroding fluid is driven through the soil sample to initiate erosion of the soil along a pre-formed hole (Figure 8).