Rim Lahmandi-Ayed
email: rim.lahmandi.ayed@gmail.com

Mohamed Salah Matoussi

Short and Long term Effects of Water Markets

Keywords: JEL Classification: Q25, C72, O33 Water markets, Water scarcity, Production efficiency, Total production, Private investment, Social costs, Transaction costs

Introduction

In several countries having limited water resources, water is being sold cheaply, which encourages water wasting behaviour. The design of an adequate pricing to urge users to take water scarcity into account, implying necessarily a substantial rise of prices, would probably meet resistance from users and would moreover require accurate information on demands. Water markets have been precisely suggested to manage more efficiently water resources while ensuring traditional users' approval, without requiring information on demands. Each user is given a water use right on the basis of his/her historical use at the historical price. An exchange of these rights is allowed, which should lead to a new water price better revealing the water value as it results from a balance of water offer and demand. It should be expected that water markets would result in a more efficient use of water, an improvement of farmers' situation, an improvement of total production thus in an improvement of social welfare. In this paper, we aim to check whether these expectations are always theoretically founded, taking into account the transaction cost of water markets and the social cost due to the inactivity of some farmers that may result from water markets.

In several countries and regions around the world, although water provision requires high costs, water is being attributed through quotas to farmers at very low prices. Some farmers are using water inefficiently (with regard to its real value) while other more efficient farmers are not being satisfied with the quantities received. A water market would result in a balance between offer and demand, thus avoiding the rationing of farmers, and in water transfer from the least to the most efficient farmers (Lahmandi-Ayed and [START_REF] Matoussi | Selection through water markets[END_REF]. When the efficiency difference between farmers is sufficiently high, some farmers among the most inefficient ones would stop production activity. This very advantage in terms of water use may have highly negative socioeconomic impacts. First inefficient farmers becoming inactive, are very likely to move to cities with all the related families thus increase the number of rural exoders with all their urban, social, political and economic problems mainly in the developing countries. Second they are unemployed persons. The possibility of conversion to another activity is only theoretical for the major part of farmers who are poorly educated thus hardly convertible. Third from the viewpoint of junior farmers, the introduction of a water market results in the introduction of an intermediary in water sale, thus in a rise of water prices. Within a short period, junior farmers will become themselves "senior", and may dislike that a difference remain, which may result in a tenseness in farmers' relationship. Finally people may not think acceptable that a resource involving high public costs, be offered graciously to inefficient farmers that will live on its sale or rent. Hence there is a real cost to put some farmers outside the productive sector which is taken simply into account in this paper.

In the long run water markets are expected to urge farmers to invest in water saving technologies to sell the water saved, which would mitigate at least some of the short term drawbacks. The few empirical observations on the question are contradictory. According to [START_REF] Hearne | Water allocation and water markets: an analysis of gains from trade in Chile[END_REF], concerning the Chilean water market experience, farmers sell in general a part of their use rights, which allows them to invest in new irrigation techniques that conserve better the resource and results in improving the global production without requiring new mobilizations harmful to the environ-ment. However, according to Bauer (1997) concerning the same experience, "despite expectations on the contrary, there has been almost no private investment in irrigation technology for the purpose of selling rights to the water saved". In Most of Chile, the author observes that water use efficiency remains at its traditional level and that flood irrigation remains the dominant practice. Therefore the question of whether water markets foster private investments in water saving technologies, needs to be explored from a theoretical viewpoint.

We consider a simple model involving unequally efficient farmers. We suppose that water resources are being rationed. In the short run farmers have only the possibility of exchanging water. We prove that in the short run water markets improve farmers' profits thus production efficiency but do not necessarily improve total production. Taking into account the transaction cost of water markets and the social cost due to the inactivity of some farmers which may result from water markets, we prove that water markets do not necessarily improve social welfare. In the long run farmers have the possibility of investing in water saving technologies before exchanging water rights. We prove that water markets do not always foster private investments when compared to the status quo (situation without water market).

The related literature. Water markets look very similar to tradeable pollution permits. Thus it would be natural to compare our results concerning the investment decisions of farmers in the long run with the papers dealing with the innovation in pollution abating technologies within polluting industries, even if a main difference exists compelling to deal separately with each sort of problem: a pollution permit is not an input as it is the case for water rights. [START_REF] Malueg | Emission Trading and the Incentive to Adopt New Pollution Abatement Technology[END_REF] and [START_REF] Milliman | Firm Incentives to Promote Technological Change in Pollution Control[END_REF] prove that the introduction of tradeable pollution permits may actually decrease some firms' incentives to adopt more effective pollution control technologies. But both papers examine the question at the firm's level. Jung, Krutilla and Boyd (1996) extend Milliman and Prince's approach from the firm to the industry level. They prove that auctioned permits provide the most incentive effects to promote the development and adoption of advanced pollution abatement technology. But Jung et al. model the pollution permits price mechanism only roughly while in our model the water price is function of the investment decisions since the demand of each farmer is function of his/her level of investment. The more recent paper of [START_REF] Requate | Incentives to innovate under emission taxes and tradeable permits[END_REF] examines the incentives to innovate (adopt a cleaner technology) within a polluting industry under emission taxes and tradeable permits. Requate proves that "there is no unique ranking between those tools, i.e. neither does one of the two tools provide a higher incentive to adopt a new technology in all cases". His conclusion is close in spirit to our own one in that tradeable rights do not necessarily provide higher incentives to innovate. His model is richer than the previous ones in that he models more richly the pollution price mechanism and takes the feedback on the ouput market into account. The weakness of the previous paper lies nevertheless in the hypothesis that only one firm has the possibility of adopting the new cleaner technology, while the possibility of investing in water saving technologies is given in our paper to all inefficient farmers.

The paper is organized as follows. Section 2 describes the model. Section 3 deals with the short term effects of water markets. Section 4 deals with their long term effects mainly in terms of private investment. Section 5 concludes. An appendix contains the proofs.

The model 2.1 The general framework

Consider n farmers (i = 1, .., n) producing a homogenous farm good sold at an exogenous price p. This is a simplifying but reasonable hypothesis as a water market is local while the output trade may be made at a national or international level. The model accounts for the effect of a local switch to water market within a given group of farmers sharing a given water resource, which thus has a negligible effect on the output price.

The limiting factor is supposed to be water thus the produced quantity and costs may be expressed function of the quantity of water used. More precisely, when the quantity of water x i is applied by farmer i, f i (x i ) is the quantity of farm good produced, while C i (x i ) is the total cost stemming from inputs other than water (fertilizers, labor, chemicals...). Functions f i and C i are naturally supposed to be increasing. We suppose that they are twice continuously differentiable with

∂ 2 f i ∂x 2 i ≤ 0 hence f i is concave, while ∂ 2 C i ∂x 2 i ≥ 0 hence C i is convex,
and that at least one among the two inequalities is strict. Denote by Ω the total exogenous offer. The authority assigns a given quota ω i to each farmer i. We have Ω = n i=1 ω i . We refer to status quo the situation without water market. In this situation, each farmer may buy a quantity of water up to his/her quota ω i , at the historical price p h = 0. Hypothesis p h = 0 supposed for simplicity is close to reality in several developing countries as Tunisia or Egypt for instance. Farmer i's profit in the status quo is given by:

π i (x i ) = pf i (x i ) -C i (x i ).
The solution of the first order condition π i (x i ) = pf i (x i ) -C i (x i ) = 0 is denoted by xs i . The demand of farmer i if he/she were not constrained by his/her quota is: x s i = max(x s i , 0). Taking farmer i's quota into account, his/her actual demand d s i is given by:

d s i =    x s i if x s i < ω i ω i if x s i ≥ ω i We suppose that n i=1 x s i > Ω. ( 1
)
This inequality means that the total desired quantity is not satisfied, which would be natural in a scarcity context. It will also ensure as we will see the existence of a price equilibrium when a water market is introduced. This implies that there exists necessarily some rationed farmers, i.e. such that x s i > ω i . In the welfare analysis, we focus on the implied group of farmers and the authority who organizes the water market (who is inactive in the status quo). This partial analysis is a first step to have an idea on the effect of water markets on the agricultural sector taking into account their direct costs. The explicit consideration of interests outside the group of farmers and the organizing authority would require at least to model adequately the output market thus for instance to deal with water market within the other groups of farmers and to have information on the demand side. This would be a more complete treatment of the subject and may be a further step of the present research. However the demand side is implicitly dealt with through the examination of the effect of water market on total production.

The social welfare in the status quo is given by

W s = n i=1 π i (d i ).
Suppose now that a water market is introduced. We suppose that this involves a transaction cost T , often mentioned in the literature. The transaction cost T is a global cost corresponding to all the costs incurred by the organizing authority to set up the water market and make it work. In this paper, we suppose that farmers do not pay fees to the organizing authority when they participate in the water market. Cost T thus corresponds someway to a cost assumed by the remaining of the society in order to promote water markets. Indeed when a new practice is introduced, even if it is expected to do only good, the mere mentioning of a fee to be paid may give rise to suspicion thus may discourage the adhesion of farmers. Thus we think that the transaction cost should be incurred entirely by the organizing authority at least for some period of time if it wants to promote water markets. If farmers were to pay fees, the sharing of the transaction cost is not a simple issue and the conclusions of the paper may change substantially depending on the way those fees are calculated, another issue which can be dealt with in a next work.

Section 3 deals with the short term effects of water markets. Farmers have only the possibility after receiving their quotas to exchange water, which results in a new price p e . We suppose that they are price-takers.

Under some conditions water markets result in the inactivity of some farmers. We suppose that this inactivity involves a social fixed cost S per each inactive farmer. Cost S may be viewed as the cost necessary to solve all the problems implied by the inactivity of one farmer (rural exodus, conversion to another activity...)1 .

A specification of the model for the long run

Section 4 deals with the long term effects of a water market. We suppose that in the long run farmers have the possibility of making investments to "improve their productivities" before water exchange. To deal simply with the long run, we choose a particular specification of the relevant functions.

We suppose that all farmers produce with the same cost function C i (x i ) = cx2 i and that the n farmers are divided into two homogeneous groups. The first group involves q "inefficient farmers" (q ≥ 2), i = 1, • • • , q, i.e. that have the same function f 1 (x) = δ 1 x, thus the same low "productivity" δ 1 . The second group involves m "efficient farmers", i = q + 1, • • • , q + m, "up-to-date" i.e. that have the best possible productivity δ 2 , implying function f 2 (x) = δ 2 x. Parameters δ 1 and δ 2 correspond to the quantity of ouput obtained per unit of water used, respectively by inefficient and efficient farmers. The unit of the farm good is chosen such that δ 2 = 1.

In order to check whether the long term effects of water markets mitigate the short term ones, the difference between the two productivities is chosen such that only the efficient farmers would remain productive in the short run if a water exchange were to take place 2 . For simplicity, we consider a specific numerical case: cΩ mp = 1/4 and δ 1 = 1/3; and we take the case of farmers having the same quota:

ω i = ω. Hence, the relation cΩ mp = 1/4 becomes cω p = m 4(m + q) . (2)
Moreover we suppose that m ≤ 2q.

Equality ( 2) together with Inequality (3) imply that in the status quo a farmer maximizes his/her profit when he/she uses his/her whole quota (See the proof of Proposition 2). This means that all farmers are being rationed in the status quo.

In the long run the inefficient farmers have the possibility of improving their productivities. To move from productivity δ 1 to some productivity µ with δ 1 ≤ µ ≤ 1, an inefficient farmer must invest I(µ) = a(µ -δ 1 ). Parameter a > 0 may be called the intensity of investment and measures how intensive an investment must be to move from the initial productivity to some given new one. We suppose that efficient farmers cannot improve their productivites, they have the best available one.

Inefficient farmers are involved in a non-cooperative game in which they choose simultaneously their investment levels (or equivalently the level of their new productivities). In the second step or the "exchange" step, after observing the choices of the first step, with the new productivities, a water exchange occurs between farmers, supposing that farmers have a competitive behaviour.

The short run

In this section we compare the status quo with a water market in the short run from the following viewpoints: the profit of each farmer and production efficiency, total production and social welfare. We prove that a water market always improves each farmer's profit thus production efficiency but does not necessarily improve total production and social welfare.

When a water market is introduced, the profit of farmer i is given by:

π i = p e (ω i -x i ) + pf i (x i ) -C i (x i ).
p e (ω i -x i ) is the part of the profit coming from the exchange (to be further referred to as "exchange profit") which may be either positive if the farmer is globally a water seller or negative if the farmer is globally a water buyer; while pf i (x i ) -C i (x i ) is the profit coming from production (to be referred to as "production profit").

Denote by x i (p e ) the solution of the first order condition:

pf i (x i ) -C i (x i ) = p e .
The demand of farmer i is given by d i (p e ) = max(x i (p e ), 0). A price equilibrium p * e satisfies:

n i=1 d i (p * e ) = Ω.
Result 1 (Existence of price equilibrium) Under the general framework of the model, there exists a unique price equilibrium.

Result 1 is needed to ensure that the demonstrated properties have cases of application. It holds for general specifications of f i and C i under the general hypotheses supposed in the model.

Result 2 (Profits, Production efficiency) Under the general framework of the model, the water market improves each farmer's profit thus improves production efficiency Result 2 is a natural one for, with a water market each farmer has the possibility of applying exactly quantity d s i (his/her demand in the status quo), which gives him/her a production profit equal to his/her profit in the status quo, and selling the difference ω i -d s i at price p e , which gives him/her an additional exchange profit. Water markets offer indeed more possibilities to farmers by releasing the constraint on demands, which necessarily improves the situation of each one of them. Consequently, if they are given the choice, farmers have always interest to participate in the water market.

Denote by A the difference between the sum of profits with a water market at price equilibrium and the sum of profits in the status quo. Necessarily A > 0. Hence in terms of production, a water market improves production efficiency in the sense that it improves the profit of the whole sector as it improves each farmer's profit.

However the improvement of production efficiency is not synonymous of the improvement of total production. This is illustrated through the following simple example.

Result 3 (Total production) Suppose that there are two farmers i = 1, 2. Farmer i has function

f i (x i ) = δ i x i , the cost function C i (x i ) = c i x 2
i and the quota ω i = Ω/2. Let δ 1 = 1, δ 2 = 3, c 1 = 0.5, c 2 = 2.9 and Ω p = 1. With these values of the parameters, the switch to water market decreases total production relative to the status quo.

Note that we consider the case of two farmers for exposition simplicity. But the same type of result may be proved for two groups of homogenous farmers. The reason that total production decreases with the chosen example is that the most productive farmer has also the highest cost. Thus there is a shift to more production by the farmer with lowest productivity and lowest cost 3 . Obviously the result is not a general one 4 but is provided only say that it is a possible case. As cost C i corresponds to the costs implied by the inputs other than water (labor, fertilizers...), the quantities to be used with a given amount of water thus the implied cost C i may depend on several parameters as the land's slope or the soil's nature. Thus all combinations are a priori possible (high productivity with high costs, low productivity with low costs, high productivity with low costs...) including the given example.

Although a water market always increases the profit of each farmer thus always increases total profit (which is true in particular with the chosen example), this does in no way imply an increase of total production. In short terms, the example provided by Result 3 together with the general Result 2 show that the sector may end up earning more and producing less after setting up a water market! Finally, in terms of welfare, if only the welfare of incumbent farmers is considered, it is obvious that a water market improves the social surplus, since it improves the profit of each one of them. But if the transaction and the inactivity costs are considered, a water market has an ambiguous effect on social surplus.

If equilibrium involves h inactive farmers, the surplus difference between both situations W w -W s = A -hS -T.

The water market improves the social welfare only if:

A -hS -T > 0
The last inequality allows to see simply the advantages and disadvantages of water markets, as far as only farmers and the organizing authority are concerned. A water market improves the farmers' profit but involves costs and may result in the inactivity of some farmers. A net benefit would result only if the gain outweighs the losses.

But number h is endogenous. The consideration of a specific example allows to calculate that number and derive conclusions on whether a water market improves welfare, depending on exogenous parameters.

An example

Lahmandi-Ayed and [START_REF] Matoussi | Selection through water markets[END_REF] studied the case of linear functions f i (x i ) = µ i x i , µ i being the productivity of farmer i, and a quadratic cost function C(x i ) = cx 2 i . Farmers are supposed to have the same cost function and to differ only by their functions f i . Thus a farmer with a higher productivity is more efficient than a farmer with a lower productivity. Suppose that farmers are ordered as follows:

µ 1 ≤ µ 2 • • • ≤ µ n .
And denote by (u i ) 1≤i≤n defined by:

u i = j=n j=i+1 (µ j -µ i )
At equilibrium "Productive" farmers are those who produce a positive quantity of the farm good and "non-productive" farmers those who make profit only from water resale. Lahmandi-Ayed and [START_REF] Matoussi | Selection through water markets[END_REF] proved Proposition 1 stated below. [START_REF] Matoussi | Selection through water markets[END_REF] At the competitive equilibrium, 1) if 2cΩ/p > u 1 then all the farmers are productive (thus h = 0).

(Lahmandi-Ayed and

2) if 2cΩ/p ≤ u n-1 , then only Farmer n, the most efficient one, is productive (thus h = n -1).

3) otherwise, there exists some integer 2 ≤ i ≤ n -1 such that u i < 2cΩ/p ≤ u i-1 , in which case only farmers j = i, • • • , n are productive (thus h = i -1).

What is particular here is that the water market improves not only production efficiency as we defined it, but total production, as water is transferred from the least to the most efficient farmers in terms of production. Moreover in the first case where the difference in productivities is small enough, all farmers remain active.

Consider now the specification of the model adopted for the long run which is a particular case of the above one. The parameters of the example have been chosen such that only the efficient farmers are active in the short run if a water market is set up (thus h = q). After calculations, the improvement of the sum of profits due to the water market is equal to A = (5/12) qpΩ m+q . The water market improves the social welfare only if5 :

(5/12)

qpΩ m + q -T -qS > 0. ( 4
)
Denote by α = (5/12)pΩ. Inequation 4 is equivalent to

q < q = α -T -mS + [(α -T -mS) 2 + 4ST m] 1/2 2S ,
which may also be written as:

S < 1 q [(5/12) qpΩ m + q -T ].
Therefore there is an overall benefit in the short run from a water market only if inefficient farmers are not too numerous and/or the cost due to inactivity of a farmer is not too high. Otherwise, the social welfare worsens with a water market relative to the status quo. Indeed in the opposite case, the improvement of the farmers' profit implied by a water market is not enough to outweigh the too high social costs due to those inactive farmers.

The long run

We now suppose that farmers have the possibility of investing in water saving technologies as in the specification described in the model for the long run. As in the short run case, it is easy check that a water market improves each farmer's profit thus production efficiency. Hence each farmer has interest to participate in the water market if he/she is given the choice. However we prove that a water market does not always encourage private investments relative to the status quo.

Result 4 (Long run, farmers' profits and production efficiency) Under the specification for the long run, a water market improves each farmer's profit thus production efficiency.

Each farmer has always the possibility within a water market to keep his/her initial productivity, apply the same quantity of water as the status quo and sell the remaining quantity, which gives him/her a profit at least equal to his/her profit in the status quo. The improvement of each farmer's profit naturally implies the improvement of the sector's profit.

It remains now to check whether a water market encourages investments. We consider the investment decisions respectively in the status quo then with a water market. Then the investment decisions in both situations are compared.

Proposition 2 (Investment decisions in the status quo) Under the specification for the long run, in the status quo, -If pω > a all inefficient farmers choose µ = 1.

-If pω ≤ a all inefficient farmers choose µ = 1/3, i.e., no investment is made.

Suppose now that a water market is set up. Proposition 3 provides the investment decisions of farmers in such conditions. Before doing so, we need the following lemma.

Lemma 1 Under the specification for the long run, an inefficient farmer necessarily chooses productivity µ = 1/3 or productivity µ = 1. Thus after the investment choice, productive farmers have necessarily a productivity equal to 1.

Lemma 1 states that an inefficient farmer actually makes his/her choice between keeping his/her initial productivity and moving to the best one. He/she never makes intermediary decisions. This is because the investment is linear w.r.t. the difference between productivities. This result allows to simplify the comparison between the water market and the status quo in terms of private investment, as it amounts to the comparison of the number of investing farmers in both situations.

Proposition 3 (Investment decisions with a water market) Under the specification for the long run, let (u t ) m≤t≤m+q-1 be the sequence defined by:

u t = (3/2)mpω( 1 2t(t + 1) + (m + q) 4(t + 1) 2 ) (5)
Three cases are possible: u m ≤ a, there is a unique Nash equilibrium in which all inefficient farmers keep their productivity at its initial level.

-a < u m+q-1 , there is a unique Nash equilibrium in which all inefficient farmers move to productivity 1.

-Otherwise, there exists some integer h such that u m+h ≤ a < u m+h-1 , in which case the set of Nash equilibria involves all the q-uples in which there are exactly h investing farmers who move to productivity 1 and (q -h) farmers that keep their initial productivity.

The more a is high, the more investments are intensive. Hence, when a is too high (a ≥ u m ), it is interesting for no inefficient farmer to invest. When a is sufficiently small (a < u m+q-1 ), all inefficient farmers will move to the best productivity. For intermediary values of a, it is interesting to make the investment only if those who have made so are not so numerous. Indeed if the number of investing productive farmers is too large, the water demand is high, making more interesting to be just a water retailer without production activity thus to make no investment.

We are now ready to compare the investment decisions in both situations. The ranking of pω, u m and u m+q-1 plays a key role in the comparison of the investment decisions (Remember that the sequence (u m ) is defined in Proposition 3.)

We prove by a simple calculation that we always have:

pω > u m+q-1 .
We also prove that pω < u m is equivalent to6

5 3 + 10 3m + 2 3m 2 < q m .
Result 5 (Comparison) Under the specification for the long run, for the comparison of the incentives for private investment between the status quo and a water market, two cases are possible.

1. When q m < 5 3 + 10 3m + 2 3m 2 , the number of investing farmers in the status quo is always greater than or equal to that number with a water market. In this case, a water market provides lower incentives for private investments.

When 5

3 + 10 3m + 2 3m 2 < q m , two subcases have to be distinguished.

(a) When a < pω the number of investing farmers with a water market is less than or equal to that number in the status quo.

(b) When a ≥ pω, the number of investing farmers with a water market is greater than or equal to that number in the status quo.

From Result 5 the consideration of effects of water markets in terms of private investments turns out to be more complicated than expected by a rough intuitive approach. In some cases water markets may dissuade farmers from investing while the same farmers would invest if no water market is set up. These findings which could not be easily guessed in advance, may be explained as follows.

With a water market, the profit of an (initially) inefficient farmer comes from production and water resale. The decision of investing in new technologies will be taken only if water resale is not very profitable. Water markets result in two contradictory effects. First a water market better reveals the value of water through the water equilibrium price, thus urges farmers to save water through adequate investments. At the same time, the possibility of water resale may dissuade farmers from production and investment. Without a water market, inefficient farmers make profit only from production. The decision of investment is taken if it is profitable in terms of production only.

In the first case ( q m < 5 3 + 10 3m + 2 3m 2 ), the proportion of initially inefficient farmers is relatively low, then the second effect is always the winner. Indeed in this case water being scarce because provided by few farmers, water resale is profitable and inefficient farmers are more willing to rely on water resale than on production, to make profit. The existence of a water market always reduces the number of investing farmers when compared to the status quo.

In the second case ( q m > 5 3 + 10 3m + 2 3m 2 ), the number of inefficient farmers is relatively high, the choice between water resale and investment-production becomes more complicated. The two effects have comparable consequences. Hence a priori the number of investing farmers with a water market may be greater or smaller than that number in the status quo. For sufficiently intensive investments and/or sufficiently low output or quota (pω < a < u m ), it appears that the first effect is stronger than the second one. This can be explained in a double way. Considering the production and investment side, a high intensity of investment and/or a low output price have two contradictory effects. Investment and production are not profitable as production is costly w.r.t. entries, reducing the incentive for investment. This direct effect raises water offer, lowers the water price thus making the water resale less profitable and production more attractive! An equilibrium between the two effects occurs when some farmers among the inefficient ones invest and the remaining ones continue to sell their water rights. The explanation is simpler when we view the result in terms of quotas. When the quota is low, the available water is scarce, which makes its resale interesting and lowers the incentives for investment and production within a water market.

Concluding Remarks

To conclude, the claims made about the effects of water markets in the short and long run, are not always theoretically founded. Several concluding remarks and perspectives for future research may be driven from the analysis.

Water markets improving each farmer's profit thus the whole sector's profit, ensure traditional farmers' support and are likely to improve the agricultural sector's situation in countries where farmers are among the poorest people, without increasing the water thus without harming the environment. The same amount of water is used more efficiently through water markets. However only traditional farmers are considered in the model. The results may change if "potential farmers" are considered. A water market may even dissuade new farmers from entering in the sector as it results in a rise of water prices, while they would be younger, more highly educated, thus more aware of the modern techniques than traditional farmers. A further interesting step to this research would be to deal with the effects of water markets considering a group of farmers of two types: traditional farmers with historical use rights and potential ones with no use rights.

However the improvement of production efficiency does not necessarily imply an improvement of total production. We indeed prove that a water market may not improve total production, which may be problematic in a scarcity or poverty context or when the considered farm good has a strategic or food security role (for instance some kinds of cereals in Tunisia).

Taking into account the transaction cost of water markets and the social cost due to the inactivity of the least efficient farmers, we prove that a water market does not necessarily improve social surplus. The welfare analysis would be more complete if the price of the farm good is endogenized and the demand side of the farm good properly taken into account. This would be another possible further research step.

Finally, a water market sometimes discourages private investments. This last result, contrary to the roughly intuitive expectations but consistent with some observations on water market experiences, proves that the short term drawbacks of a water market are not always mitigated by its long term effects.

Appendix: Proofs

Proof of Result 1. First note that x s i = d i (0). Inequality (1) is then equivalent to n i=1 d i (0) > Ω. Equation pf i (x i )-C i (x i ) = p e defines implicitly xi (p e ) as a continuous decreasing function of p e . d i is then a continuous non-increasing function of p e , and so is n i=1 d i . On the other hand there exists pe such that for all p e > pe , for all i = 1, .., n, pf i (0) -C i (0) -p e < 0, then π (x i ) < 0 for all x i ≥ 0, which implies d i (p e ) = 0, so n i=1 d i (p e ) = 0. Consequently, there exists a unique p * e > 0 such that n i=1 d i (p * e ) = Ω.

Proof of Result 2. As d i (p e ) maximizes the profit of farmer i with a water market, it ensures to him/her a better profit than d s i his/her demand in the status quo, which is written as

pf i (d i (p e )) -C i (d i (p e )) + p e (ω i -d i (p e )) ≥ pf i (d s i ) -C i (d s i ) + p e (ω i -d s i ).
As p e (ω i - which says that the profit of farmer i is better under water market than in the status quo. Summing these inequalities, together with n i=1 d i (p e ) = n i=1 ω i , we prove that a water market improves the profit of the whole sector.

d s i ) ≥ 0 since d s i ≤ ω i , we have pf i (d s i ) -C i (d s i ) + p e (ω i -d s i ) ≥ pf i (d s i ) -C i (d s i ),
Proof of Result 3. With these values, we have first: x s i > ω i , so that in the status quo d s i = ω i = Ω/2 and total production equals P s = (δ 1 + δ 2 )Ω/2. With a water market, we have δ 2 -δ 1 < 2c 2 Ω p , which ensures the activity of both farmers at equilibrium. Demands of farmers are respectively given by the following:

d 1 = pδ 1 -pδ 2 + 2c 2 Ω 2(c 1 + c 2 )
and

d 2 = pδ 2 -pδ 1 + 2c 1 Ω 2(c 1 + c 2 ) .
Total production is then given by

P w = δ 1 d 1 + δ 2 d 2 = p(δ 2 -δ 1 ) 2 + 2Ω(c 2 δ 1 + c 1 δ 2 )
2(c 1 + c 2 ) .

P w < P s is equivalent to δ 2 -δ 1 < Ω(c 2 -c 1 ) p , which holds with the chosen values of the parameters.

Proof of Result 4. Denote by d i (µ i , µ -i ) the farmer i's demand of water when he/she chooses µ i and the other farmers choose µ -i . Denote by µ * the q-uple of productivities chosen by farmers at equilibrium. We have the following:

pf i (d i (µ * i , µ * -i )) -C i (d i (µ * i , µ * -i )) -I(µ * i ) + p e (ω i -d i (µ * i , µ * -i )) ≥ pf i (d i (δ 1 , µ * -i )) -C i (d i (δ 1 , µ * -i )) + p e (ω i -d i (δ 1 , µ * -i )) (6) ≥ pf i (d s i ) -C i (d s i ) + p e (ω -d s i ) (7) ≥ pf i (d s i ) -C i (d s i ). ( 8
)
Inequality 6 holds as µ * i is a best reply to µ * -i thus is better than the initial productivity δ 1 . Inequality 7 holds as quantity d i (δ 1 , µ * -i ) is chosen to maximise farmer i's profit within a water market thus is better than d s i the farmer's demand in the status quo.

This proves that a water market improves each farmer's profit. Now summing the inequalities

pf i (d i (µ * i , µ * -i )) -C i (d i (µ * i , µ * -i )) -I(µ * i ) + p e (ω i -d i (µ * i , µ * -i )) ≥ pf i (d s i ) -C i (d s i ),
there exists some integer h such that u h ≤ a < u h-1 . In this case the best reply of Farmer i is to move to productivity 1 if the number of other productive farmers is less or equal to h -1, and to keep his/her productivity if this number is greater or equal to h.

Proof of Result 5.

1) When q m < 5 3 + 10 3m + 2 3m 2 , we have: u m+q-1 < u m < pω.

• When a < pω all farmers invest in the status quo. However with a water market all of them invest only when a < u m+q-1 . number of investing farmers with a water market is at most equal to q -1 when u m+q-1 < a.

• When a > pω > u m > u m+q-1 , whether or not a water market is implemented, no farmer invests.

2) When q m > 5 3 + 10 3m + 2 3m 2 , we have: u m+q-1 < pω < u m . Denote by n wm the number of investing farmers with a water market and by n s the number of investing farmers in the status quo. Results are summarized in the table below:

a n wm Comparison n s a < pω ≤ q ≤ q pω < a < u m > 0 > 0 a > u m 0 = 0

  implying pf i (d i (p )) -C i (d i (p e )) + p e (ω i -d i (p e )) ≥ pf i (d s i ) -C i (d s i ),

See the introduction for a more detailed explanation.

This is a consequence of a result obtained by Lahmandi-Ayed and[START_REF] Matoussi | Selection through water markets[END_REF] recalled in Proposition 1 later in this paper.

An anonymous referee is thanked for this remark.

In the example provided in Proposition 1 a water market increases total production.

Note that Inequality 4 involves now only exogenous parameters.

Note that when m is very large this comes roughly to the comparison of q/m with 5/3.

Proof of Proposition 2: Denote by µ the productivity chosen by an inefficient farmer. His/her profit when he allocates quantity x to production, is given by:

His profit is maximum at x = ω. Indeed, first order condition yields: x * = pµ/2c ≥ p/6c, to be compared to ω. But relation [START_REF] Burness | Appropriative Water Rights and the Efficient Allocation of Resources[END_REF] equivalent to ω = pm/4c(m + q) together with inequation (3), implies x * ≥ ω.

After calculations, the profit is given by:

which implies the result.

Proof of Lemma 1: Consider an inefficient farmer. Denote by µ the productivity he/she chooses at the second step. Suppose that µ is such that he/she remains non-productive in the exchange step, his/her profit is then given by:

But p e is independent of µ, thus on this segment, the profit is a decreasing linear function w.r.t. µ.

Suppose now that µ is such that he/she becomes productive in the exchange step. Let t be the number of productive farmers other than the considered one. Note that t is greater than m for, efficient farmers are productive.

The equilibrium water price is given by:

α being the sum of productivities of all the other productive farmers.

The productive quantity is given by:

The total profit of the considered farmer is given by:

The coefficient of µ 2 has the same sign as 1 -t+2 2t+2 which is non negative for all t ≥ 0.

Hence, the profit of the farmer a continuous function that is linear decreasing then convex. It reaches its maximum at 1/3 or at 1.

Proof of Proposition 3:

Consider an inefficient farmer i ∈ {1, .., q}. Let t be the number of productive farmers (in the first and the second group other than the considered one). According to Lemma (1), they have necessarily a productivity equal to 1. Farmer i makes his/her choice between µ i = 1/3 and µ i = 1.

If µ i = 1/3, then the equilibrium water price is equal to

The corresponding profit is given by:

using the relation cΩ mp = 1/4, this profit becomes

If µ i = 1, the equilibrium water price is equal to

and the productive quantity is equal to

The corresponding profit is equal to

Using Equality (2), this profit is equal to

The difference is then given by:

Farmer i moves to productivity µ i = 1 if a < u t . Note that (u t ) m≤t≤m+q-1 is a decreasing sequence. Three cases are then possible: If u m ≤ a, then for all t, u t ≤ a. This implies that whatever the number of other investing farmers, Farmer i has no interest to invest.

If a < u m+q-1 then for all t, a < u t . This implies that whatever the number of other investing farmers, Farmer i invests.


A Perrot

D Rangeard

F Menasria

S Guihéneuf

Strategies for optimizing the mechanical strengths of raw earth-based mortars

Keywords: Earth-based materials, flax fibers, woven flax fabric, alginate

Strategies for optimizing the mechanical strengths of raw earth-based mortars

I Introduction

Earthen construction has recently regained much attention in the building industry due to its low environmental impact and recyclability [START_REF] Azeredo | Applicability of rheometers to characterizing earth mortar behavior. Part I: experimental device and validation[END_REF][START_REF] Bui | Compression behaviour of nonindustrial materials in civil engineering by three scale experiments: the case of rammed earth[END_REF][START_REF] Aubert | Towards a simple compressive strength test for earth bricks?[END_REF][START_REF] Moevus | Earthen construction: an increase of the mechanical strength by optimizing the dispersion of the binder phase[END_REF]. The development of earthen construction is still limited because of the time required for the material to harden and by the difficulty to achieve a mix-design that allows for both fast casting and sufficient strength in the dry state.

In order to address both problems and to improve mix-design of earth-based mortars, a recent trend has been to apply scientific knowledge and expertise, developed by the concrete industry, to earthen construction [START_REF] Moevus | Earthen construction: an increase of the mechanical strength by optimizing the dispersion of the binder phase[END_REF][START_REF] Gnanli | Development of a Self-Compacted Clay based Concrete, rheological, mechanical and environmental investigations[END_REF][START_REF] Perrot | Linking rheological and geotechnical properties of kaolinite materials for earthen construction[END_REF][START_REF] Landrou | Clay particles as binder for earth buildings materials: a fresh look into rheology of dense clay suspensions[END_REF][START_REF] Ouellet-Plamondon | Self-Compacted Clay based Concrete (SCCC): proof-of-concept[END_REF].

One of the options adopted from concrete technology is the optimization of the granular skeleton [START_REF] Moevus | Earthen construction: an increase of the mechanical strength by optimizing the dispersion of the binder phase[END_REF] which increases the dry density of earth mortars and thus its mechanical strength.

Another option is to use a coagulant, a biopolymer or a hydraulic binder in order to shorten the material's hardening stage [START_REF] Landrou | Lime as an Anti-Plasticizer for Self-Compacting Clay Concrete[END_REF][START_REF] Walker | Properties of some cement stabilised compressed earth blocks and mortars[END_REF][START_REF] Reddy | Cement stabilised rammed earth. Part A: compaction characteristics and physical properties of compacted cement stabilised soils[END_REF][START_REF] Reddy | Cement stabilised rammed earth. Part B: compressive strength and stress-strain characteristics[END_REF][START_REF] Khelifi | Design of clay/cement mixtures for extruded building products[END_REF][START_REF] Khelifi | Mechanical enhancement of cementstabilized soil by flax fibre reinforcement and extrusion processing[END_REF][15][START_REF] Miccoli | Static behaviour of rammed earth: experimental testing and finite element modelling[END_REF][START_REF] Van Damme | Earth concrete. Stabilization revisited[END_REF][START_REF] Pinel | Towards poured earth construction mimicking cement solidification: demonstration of feasibility via a biosourced polymer[END_REF]. Another solution, emanating from concrete mix design, is to improve the earth mortar's workability to enable the possibility of making an extrudable, flowable and even castable earth mortar [START_REF] Moevus | Earthen construction: an increase of the mechanical strength by optimizing the dispersion of the binder phase[END_REF][START_REF] Gnanli | Development of a Self-Compacted Clay based Concrete, rheological, mechanical and environmental investigations[END_REF][START_REF] Perrot | Linking rheological and geotechnical properties of kaolinite materials for earthen construction[END_REF]. The objective of using a dispersant is to deflocculate the micro-sized clay-based structures in order to reduce the interaction force between clay particles. Deflocculation can be obtained by using a dispersant which acts just like a superplasticizer on cement particles in concrete [START_REF] Perrot | Linking rheological and geotechnical properties of kaolinite materials for earthen construction[END_REF][START_REF] Roussel | Steady state flow of cement suspensions: A micromechanical state of the art[END_REF][START_REF] Kjeldsen | Relating the molecular structure of combtype superplasticizers to the compression rheology of MgO suspensions[END_REF][START_REF] Flatt | Dispersion forces in cement suspensions[END_REF][START_REF] Perrot | Yield stress and bleeding of fresh cement pastes[END_REF][START_REF] Zingg | The microstructure of dispersed and nondispersed fresh cement pastes -New insight by cryo-microscopy[END_REF]. This option is also expected to lead to a reduction of the material porosity and therefore improve material strength and durability [START_REF] Moevus | Earthen construction: an increase of the mechanical strength by optimizing the dispersion of the binder phase[END_REF].

In this study, the first option was used to obtain a material with a continuous particle size distribution, leading to a dense matrix, by adding fine and coarse sand. Two materials were used, a commercial kaolin powder mixed with sand and an earth-based material which is known to be suitable for cob construction. The earth-based mortar can be either compacted or cast by adding a dispersant hexametaphosphate (HMP) thus making the mortar flowable. In this study, HMP was used as a dispersant because it has been shown to be more efficient than cement superplasticizer for reducing the yield stress of kaolin mortar [START_REF] Perrot | Linking rheological and geotechnical properties of kaolinite materials for earthen construction[END_REF]. Finally, alginate was also added in order to obtain a rapid strength gain of the earth. The effects of alginate and dispersant on the compressive strength of the materials were also tested.

Various water to dry material mass ratios were tested to find an optimal value that gave the highest mortar densities for different forming process. An assessment of the material behavior was also made to see if it was influenced by the forming process.

Following the powder mix-design, reinforcement strategies were tested using dispersed flax fibers and woven fabrics made of flax fibers. These different reinforcement strategies were tested within the framework of this study. Following a description of the materials and methods, the following evaluations were carried out on the earth-based mortars; the effects of particle size distribution optimization; the influence of processing and bio-based reinforcements (alginate or flax individual fibers or woven fabrics) on ductility, compressive and flexural strength of earth-based mortars. Finally, the combined effects of the different reinforcement types were assessed.

II Materials and methods

II.1 Materials

The kaolin clay used was provided by Imerys and came from the "Kaolins de Bretagne" quarry at Ploemeur, France. The kaolin clay had a specific gravity of 2.65 and a specific surface area of 10 m²/g (data provided by the supplier). The largest clay grain size was approximately 10 m and the mean kaolin grain size approximately 4m. The particle size was measured using a laser particle size analyzer and is set out in figure 1. Before measurement, the sample was dispersed in a suspension of distilled water with a deflocculant agent (sodium hexametaphosphate Na-HMP) at 2,500 rpm stirrer speed. The suspension was placed in the measurement cell, and an ultrasonic stimulation was carried out for 30 seconds before measurement.

The mortar used in the tests consisted of a mixture of kaolin clay powder, sand and water. The clay was mixed with two different sands: fine sand, with grain size ranging from 63 to 200 m, and coarser sand with grain size ranging from 0 to 4 mm. The grain size distributions of the sands are also plotted in figure 1. The fine sand particle size distribution was obtained by laser diffraction and the coarse sand distribution was obtained by sieving.

Fig. 1: Particle size distribution of the powders and Saint-Sulpice-La-Forêt soil (cob earth)

In order to optimize the particle size distribution (PSD) of the kaolin-based mix, the method proposed by Dreux and Gorisse [START_REF] Dreux | Nouveau guide du béton et de ses constituants[END_REF] to determine the optimized ratio of sand to gravel was used to obtain the densest aggregates packing. As three different particle types were used, a two steps computation was performed. Firstly, the kaolin clay and the fine sand PSD were used to obtain the optimized PSD of the clay/fine sand mix (binary mix). The optimized PSD of the binary mix was then used together with the PSD of the coarse sand to find the optimized PSD of the ternary mix of clay/fine sand/coarse sand.

The final PSD of the ternary mix of dry particles was optimized using Dreux and Gorisse method [START_REF] Dreux | Nouveau guide du béton et de ses constituants[END_REF] which is plotted in figure 2. This composition allows for the best mechanical strength with this ternary mix (see next section and in [25]). The mix was composed by mass (and volume, as all particles have almost the same density) as follows 17% kaolin, 23% fine sand and 60% coarse sand. A plateau in the PSD can be seen between 10 and 60 m, as neither the kaolin nor the sand have those particle sizes.

The second material tested was a cob earth from Saint-Sulpice-La-Forêt (Ille et Vilaine, France). It was a fine soil with 70% of particles finer than 10 m (the PSD is plotted on figure 1 and was obtained in the same way as the kaolin PSD). The soil particles were a mix of quartz, and various types of clay: kaolinite, illite and smectite; this being determined by XRD analysis before and after thermolysis at 550°C. The Plasticity Index of this soil was 21 with a liquid limit of 48% and a plastic limit of 27%. The alginate used in the study was a white powder of alginic salt Cimalgin HS3® provided by Cimaprem (Redon, France). It is designed to make high strength gel for arts and molding applications. Dosage rates used ranged from 1% to 5% of particles smaller than 10 m. The HS3 product is mostly composed of alginate salt with an "on demand" calcium release agent that allows the monitoring of the duration of the alginate gel network creation. Alginate can form a cross-linked isotropic insoluble gel when a soluble form of alginate nucleates with divalent metal cations, like Ca 2+ , which can be found in earth-based materials. Chains of alginate make junctions by intercalating divalent cations creating a sort of egg-box connection [START_REF] Funami | Rheological properties of sodium alginate in an aqueous system during gelation in relation to supermolecular structures and Ca 2+ binding[END_REF]. In order to aid the dispersion of alginate within the earth material, the HS3 powder is firstly mixed with an equal mass of water. Hexametaphosphate (HMP) was used in this study as a dispersant to make a mortar flowable.

For the kaolin mortar, a dosage of 0.25 % of the kaolin clay content was tested, this being based on previously obtained results on the effect of dispersant on the rheological behavior of clay pastes [START_REF] Perrot | Linking rheological and geotechnical properties of kaolinite materials for earthen construction[END_REF]. For the cob earth from Saint-Sulpice-La-Forêt, the mass ratio of HMP was 0.3% of the dry material. The choice of this dosage was made using the same methodology as that used for the kaolin mortar in [START_REF] Perrot | Linking rheological and geotechnical properties of kaolinite materials for earthen construction[END_REF].

The dispersed flax fibers (Marylin variety) used in this study was supplied by the CTLN® Company (Le Neubourg, France). The fibers, shown on Figure 3, were scutched, carded and cut into 4 mm lengths. These flax fibers were the same as those used in the study by Bourmaud et al. [27]. The volume fraction of fibers ranging from 1 to 4% was used in the present study. The woven fabric of flax fibers was isotropic with a mesh size of 4 mm. The mass per unit area of the fabric textile was 500g/m². Dispersed flax fibers and woven fabric of flax fibers were tested as reinforcements (Figure 3).

II.2 Methods

The earth samples were prepared using high capacity Hobart mixer. Dry powder was firstly introduced into the bowl. When using fiber reinforced material, dry fibers were mixed with the dry powder by hand until a homogeneous material with well dispersed fibers was obtained. Water was then added, the quantity being that necessary to attain the targeted water content. The mixing procedure consisted of a 4 minutes low velocity mixing stage, followed by a high velocity mixing stage of 5 minutes. Between these two stages, the bowl was scraped in order to ensure that no unmixed material remained adhered to the bowl. After mixing, the material was placed in a closed container for 24 hours to allow homogenization.

The alginate solution was prepared just before casting and/or compaction and was mixed with the wet earth using a mortar hand mixer for 4 minutes.

Cylindrical samples of 50 mm diameter and height were compacted using static compression with a 200 kN loading frame at a maximum vertical stress of 45 MPa. To compare the effect of the compaction technique a Proctor-type dynamic compaction was used to make 150 mm diameter and height samples. Samples were compacted in CBR mold, in five layers with 56 blows of CBR ram for each layer. For cast samples, cement mortar 40x40x160 mm 3 moulds were used.

Because all the compressive tests were carried out with samples of an aspect ratio of one (sample height to diameter for cylindrical samples or sample side length for prismatic samples), it is considered that mechanical measurements can be compared even if their geometries are different. The comparison between results obtained with those different sizes and shapes of samples was acceptable as it has been shown that at a same aspect ratio, the strength variation due to shape variation is lower than 5% [START_REF] Yi | Effect of specimen sizes, specimen shapes, and placement directions on compressive strength of concrete[END_REF]].

Samples were then conserved in a 50°C temperature controlled store until weight stabilization. Such a temperature allows a relatively fast drying and ensures that only free water leaves the sample. After this curing step, once the sample weight was stabilized, it was considered that the samples were dry. The compressive strength of the sample was then measured using a 50 kN loading frame. The test was carried out at a constant velocity of 1 mm/min which is in agreement the French national standard recommendation for soils testing XP CEN ISO TS 17892-7 (the test duration ranging between 2 and 15 minutes, maximum strain rate of 2% per minute).

III Effect of particle size distribution and processing

III.1 Effect of particle size distribution

The particle size distribution of the mortar was designed to obtain the highest density once placed. Concepts from concrete mix-design were used in order to obtain the densest packing.

It is also important to note that the binder content (here the kaolin clay) must be kept higher than a critical value that will ensure sufficient cohesion within the sample.

It is interesting to note that using the Dreux and Gorisse method, the amount of binder is equal to 17% of the total mass of powder. This value can be considered as lower bound when dealing with clay content commonly found in earth construction.

In order to emphasize that this step of PSD optimization is paramount, clay samples, of binary mixes (fine sand + kaolin) and of ternary mixes (coarse sand + fine sand + kaolin), were prepared with different water contents ranging from 5% to 10% , these were then compacted in a CBR mould in 5 layers using 56 blows per layer. The samples were weighted after drying and mechanically tested to measure their compressive strength.

Figure 4 shows the compressive strength as a function of the dry density of the samples.

Results show that the widening of a continuous particle size distribution increases dry density of the samples and therefore their compressive strength, as commonly observed [START_REF] Morel | Compressive strength testing of compressed earth blocks[END_REF][START_REF] Bruno | Briques de terre crue: procédure de compactage haute pression et influence sur les propriétés mécaniques[END_REF].

Fig. 4: Compressive strength as a function of dry density for kaolin, binary mix (fine sand+kaolin) and ternary mix (coarse sand+fine sand+kaolin)

In the following sections of this paper, only the ternary mix was used.

It is interesting to note that we attempted to add a limestone filler with a particle size distribution ranging from 10 to 63 m (ie in the range of the size not present in the PSD shown in figure 3). Using this fourth material, optimized PSD provides kaolin volume fraction of around 10% that was not sufficient to obtain a cohesive product after casting or compaction. This result is in agreement with that which can be found in the literature suggesting that a minimal content of clay is needed to make earth-based materials for construction [START_REF] Maniatidis | A review of rammed earth construction[END_REF][START_REF] Delgado | The selection of soils for unstabilised earth building: A normative review[END_REF][START_REF] Mchenry | Adobe and rammed earth buildings: design and construction[END_REF].

Another crucial point is that for the cob earth the results were opposite: Using the same methodology, we obtained a mix of 40% cob earth, 29% fine sand and 31% coarse sand. Such mixes result in an increase in the dry density of the mixture, initial values ranging between 1780 and 1850 kg/m 3 which increased to values ranging from 1950 to 2050 kg/m 3 . However, this densification of the dry matrix leads to a decrease of the compressive strength from above 8 MPa to values of the order of 2.5 MPa. Therefore, for the cob earth which contains clay that can induce relatively high compressive strength (8 MPa for an earth-based materials), the strength of the material appears to be directly linked to the clay content (Figure 5). The effect of the densification of earth-based material strengthening directly depends on the type of clays contained in the earth. With pure kaolinite, which has a low compressive strength (around 500 kPa), a densification of the matrix with sand, leads to a significant increase in strength. As opposed to this, the high strength cob earth content directly governs the strength of the samples.

III.2 Effect of processing and dispersant

This part of the study is focused on the kaolin mortar and it is to be noted that the same kind of trend has been observed for the cob earth.

Figure 6 shows the evolution of the dry density of cast and compacted samples with respect to initial water content. The curves show that the optimal water content for obtaining the densest material depends on the forming process. Dynamic compaction required the smallest water content, in this case 5.5% of the dry particles mass, whereas casting required more water (14%) to obtain the densest material, and was therefore the material with the best mechanical properties. Between these two forming process, static compaction (maximum load of 25 MPa) showed a maximum dry density for an initial water content of 10%. These results indicate that the rheology (or the consistency) of the wet material must be optimized with regard to the forming process.

It is also interesting to note that the maximum density obtained for each type of tested forming process was in the same range (between 2050 and 2080 kg/m 3 ). Using higher energy of compaction would result in a higher density as proposed by [START_REF] Bruno | Mechanical behaviour of hypercompacted earth for building construction[END_REF]. Fig. 6: Influence of the initial water content (water added during mixing) on the dry density of the samples for the three forming process. . For the optimized water content for the static compaction at 25 MPa, it was observed that an increase of the vertical stress from 25 to 45 MPa led to an increase of the dry density from 2060 to 2100 kg/m3. Thus, for the static compaction process, the final dry density was also depended on the loading.

The curve of the compressive strength versus the dry density is shown in figure 7. All the results are grouped around two lines, the first relate to the mixes containing HMP and the second to the mixes without HMP. For a same density, mixes with HMP present a higher mechanical strength. This increase could be due to a better dispersion of the clay particles, as shown by [START_REF] Moevus | Earthen construction: an increase of the mechanical strength by optimizing the dispersion of the binder phase[END_REF][START_REF] Perrot | Linking rheological and geotechnical properties of kaolinite materials for earthen construction[END_REF] , that would improve the mechanical properties of the dry materials. It can be seen that the compressive strengths of the dry samples increases from 0.5 MPa to 2 MPa, for mixes without HMP, and from 0.8 MPa to 2.8 MPa, for mixes containing HMP, when the dry density increases from 1800 to 2080 kg/m 3 . This trend can be explained by a better distribution of the clay particles with HMP. HMP reduces interaction forces between clay particles in a fresh state and allows for an increase of the distance at the contact point between particles and a decrease of the average pore size [START_REF] Perrot | Linking rheological and geotechnical properties of kaolinite materials for earthen construction[END_REF]. In this case, the effect of HMP would be the same as the one of superplasticizer within cement paste [START_REF] Zingg | The microstructure of dispersed and nondispersed fresh cement pastes -New insight by cryo-microscopy[END_REF].

It is also worth noting that the forming process does not influence the mechanical strength.

Two samples made with different forming processes, both having the same dry density, will have the same compressive strength.

IV Fibers and fabrics reinforcement

IV.1 Woven fabric

The woven flax fabric was placed between layers of earth-based material, compacted or cast in the surface perpendicular to the direction of loading during compressive strength measurements. All test samples had an aspect ratio of 1. The number of woven fabric sheets in a sample, tested in this study, ranged from 1 to 4. The sample containing one woven fabric present two layers of material with a dimensionless height of 0.5 (ratio of the layer height to the total sample height). In a general form, the height of cast and compacted earth-based materials is equal to 1/(n+1) where n is the number of woven fabric sheets. It is to be noted that in order to obtain layers of equal height a target mass of earth-based material for a layer was computed assuming the density obtained with no woven fabric.

In Figure 8, the dimensionless compressive strengths of samples with woven fabrics (i.e. the ratio of the compressive strength to the compressive strength of the samples without fabric) are plotted versus the dimensionless height of earth-based material layer. It appears that the reinforcement effect of woven fabric did not depend on the forming process.

For the tested materials (kaolin mortar and cob earth), it appears that there is a relationship between the reinforcement effect of the woven fabric and the height of an elemental layer of earth-based material between woven fabric sheets. These results are in agreement with results previously obtained in the work of P'kla [START_REF] Pkla | Comparaison de méthodes d'essais de mesures des caractéristiques mécaniques des mortiers de terre[END_REF][START_REF] Kla | Caractérisation en compression simple des blocs de terre comprimée (BTC): application aux maçonneries[END_REF] and more generally with works on the reinforcement of soils by geotextiles and geomembranes [37,[START_REF] Lawrence | 8 -High performance textiles for geotechnical engineering: geotextiles and related materials[END_REF]. To conclude, it is interesting to note that the maximum compressive strengths of the sample was obtained with 4 woven fabric sheets (the maximum number of fabric sheets tested); the compressive strength were 4.5 MPa for cast samples, 4.35 MPa for compacted samples for the kaolin mortar and 17.6 MPa for the cast cob earth. Another benefit of woven fabric is that the large increase in compressive strength is accompanied by a decrease in dry density of the sample; dry densities are reduced from 2080 to 2050 kg/m 3 in the case of compacted samples and from 2070 to 1930 kg/m 3 in the case of cast samples).

Flax fibers

Flax fibers were mixed with the dry powder before the addition of water. Volume fractions of fiber ranging from 1 to 4% were tested in this study. Figure 10 shows the evolution of dimensionless compressive strength of the sample versus the fiber volume fraction added to the earth-based materials. It can be seen that in the case of kaolin mortar, the increase of the compressive strength does not depend on the forming process but is directly linked to the fiber content; the higher the fiber content, the higher the compressive strength. This observation is generally true for the tested materials provided that the fibers do not cause workability issues as is commonly observed in fiber-reinforced concrete rheology [START_REF] Perrot | Structural build-up of rigid fiber reinforced cement-based materials[END_REF][START_REF] Martinie | Rheology of fiber reinforced cementitious materials: classification and prediction[END_REF].

For the statically compacted samples with 4% volume content of flax fiber, the compressive strength was 5.6 MPa, which is more than twice the compressive strength obtained without fibers. Moreover, it is important to note that the dry density of this sample decreased from 2070 to 1920 kg/m 3 .

For kaolin mortar, the increase in compressive strength with fiber content is quite surprising because it is not observed with cement-stabilized clay or concrete [START_REF] Khelifi | Mechanical enhancement of cementstabilized soil by flax fibre reinforcement and extrusion processing[END_REF]. However, in this case, the tested mineral matrix has a lower mechanical strength and the mechanical contribution resulting from the friction/shear at the fiber/matrix interface becomes a non-negligible factor in compressive strength. This latter supposition is supported by the fact that in the case of cob earth, which has a higher compressive strength, the contribution of fibers to the compressive strength is less than that seen in the case of kaolin mortar, see Figure 10. This observation indicates that the reinforcement contribution of fiber to the overall compressive strength depends on the relative strength of the fiber interface with respect to the strength of the matrix; that is to say the lower the matrix strength the higher the fiber reinforcement's contribution to strength.

V. Effect of alginate biopolymer

Once the samples were dry compressive strengths of the samples are measured once the samples were dry to provide a reference state for further testing. Figure 11 shows the evolution of the compressive strength with respect to sample dry density for kaolin mortar samples made with different amounts of alginate; Figure 12 shows a similar evolution for the cob earth from Saint-Sulpice-La-Forêt.

For kaolin mortar, the addition of alginate at a 1% dosage did not change the mechanical strength of the sample. In this case, the amount of alginate may not be sufficient to build a gel network able to reinforce the material. However, for higher dosage, the compressive strength of the sample increased with the alginate dosage. For a 3% dosage, and a dry density of 2050 kg/m 3 , the addition of alginate more than doubled the compressive strength (from 2.25 MPa to 5.9 MPa). At a dosage of 5%, the compressive strength reached its maximum value of 7.1

MPa. It would be of interest to test higher dosages in further work to verify if increased dosage result in a higher compressive strength and what is the critical dosage for which the compressive strength reaches a plateau.

For the cob earth from Saint-Sulpice-La-Forêt, the addition of alginate did not change the compressive strength of the samples. Figure 12 shows that the evolution of sample compressive strength with respect to dry density was not influenced by the addition of alginate in the range of dosage tested (from 0 to 4%). It can be assumed that structure of the material without alginate is stronger (or at least of equal strength) than the alginate gel network. Therefore, it is noted that the addition of alginate did not modify the mechanical strength of the samples as it did not reinforce the soils particles network that has stronger interactions; this maybe due to the presence of swelling clays like smectite at a dry state (high suction effect).

By comparing the effect of alginate addition on the compressive strength of both materials, it can be concluded that the ability of alginate to strengthen an earth-based materials depends on its dosage and on the mineralogy of the clay particles in the soil. Figure 13 summarized the effect of alginate on both tested materials with respect to its dosage. On the other hand, in the case of the soil from Saint-Sulpice-La-Forêt that contains swelling clay (smectite), alginate does not increase the compressive strength of the material, even at high dosage. The gel strength appears to be weaker than the interaction within clay particle network.

VI Coupled reinforcement effects

This section reports on a study made of the effect of coupled reinforcement strategies used to strengthen the material. This work focused on kaolin mortar because only the fabric sheets have been shown able to increase the compressive strength of cob earth whereas for the kaolin paste, all types of reinforcements have strengthened the material.

VI.1 Alginate and flax fibers

Figure 14 shows the evolution of the compressive strength of the kaolin mortar sample with respect to dry density. Firstly, it can be seen that the addition of flax fibers induces a decrease in the sample density. This decrease can be attributed to two phenomena: a workability decrease due to the fibers (leading to higher void content) and to the low density of the fibers.

Secondly, the mechanical strength of the samples only increases with the addition of fibers for an alginate dosage of 1%. It appears that for higher dosage, fibers do not increase the compressive strength of the samples because the cohesion of the material is higher than the interfacial strength between fiber and mineral matrix as shown in Menasria et al. [25]. For dosages of 3 and 5% of alginate, the addition of fibers appears to have no effect as the mechanical strengths of samples with and without fibers lie along the same strength density curve for each fiber dosage (Figure 14). Figure 15 shows force displacement curves obtained during the compressive strength measurements with and without fibers for alginate dosages of 3 and 5%. If the curves for samples with and without fibers are compared, it can be see that the elastic parts of the curve before the peak are close, thus indicating that fiber addition does not significantly influence the elastic modulus. The load peak is higher for samples without fibers, as confirmed by Figure 8, however, after the load peak; the load decrease is much slower for samples with fibers. Such type of behavior indicates an increase in the ductility of the sample; this is a very interesting material property for earth construction in a seismic area, for example.

Fig. 15: Force versus displacement curves obtained during the compressive strength measurements on kaolin mortar with and without fiber for alginate dosages of 3 and 5%.

Figure 16 shows the evolution of flexural strength of the kaolin mortar samples with respect to dry density, for different dosages of alginate (1, 3 and 5%) with and without fibers. The results are close to those obtained for compressive strength. Once again, the beneficial effect of fibers on the strength is obtained only for the lowest alginate dosage for which the mineral matrix is the weakest. For higher dosages of alginate, the influence of fibers is not so obvious but the flexural strength remains constant even if the dry density decreases with the addition of fibers. This type of behavior can be attributed to the level of tensile strength of the flax fibers that can act in tension when embedded in a low-strength material. We can see that samples without fiber exhibit a brittle behavior with a sudden failure obtained after the load peak whereas the samples with fibers shows a post-peak behavior with a smooth decrease of the load due to the continuous failure of the interfaces between fibers and matrix [START_REF] Khelifi | Mechanical enhancement of cementstabilized soil by flax fibre reinforcement and extrusion processing[END_REF]. In order to complete this study of reinforcement of earth-based materials, a study was carried out on the effect of combining alginate, fibers and woven fabrics on the compressive strength of the kaolin mortar. A kaolin mortar with a water content of 12% containing HMP and an alginate dosage of 3% and with a volume fraction of flax fiber of 4% was cast in 40x40x160 mm 3 mould in five layers separated by the tested woven fabric. After the drying stage, the sample was tested in compression. The compressive stress vs. the deformation curve (Figure 18) shows that the compressive strength of the material was quite high 17.5

MPa. However, the elastic modulus remained only at a value of 110 MPa, which is quite low in comparison with the elastic modulus of concrete at the same level of compressive strength. This shows that the reinforcement strategies and their combination help to delay the failure of the samples by giving some bridging between clay particles provided by alginate or fibers. However, it clearly appears that these reinforcements do not affect the elastic modulus, which is always of the order of kaolin without reinforcement (150 MPa for the same material without fibers and alginate).

VII Conclusions

In this article, several strategies for optimizing the mechanical resistance of the earth material were tested. It appears that the effect of the reinforcement depends on the nature of the soil tested and of the mineralogical nature of the clay. Firstly, we have been able to assess the influence of dry density and dispersion of clay particles (by using HMP and granular optimization) for a cob earth and a kaolin mortar. In the case of kaolin mortar, which had the highest dry density, optimization of the compressive strength was possible whereas in the case of cob earth containing other types of clay (smectite), it was observed that the compressive strength is governed by the clay content and could not be optimized.

An interesting result shows that the compressive strengths of the dry samples do not depend on the forming process but only on the dry density of the material.

The benefit provided by alginate and textile reinforcements depended on the nature of the earth. While their actions are highly beneficial for kaolin-based soils, it is almost negligible for materials containing swelling clay that initially have a high initial resistance (>8 MPa) in the dry state.

Finally, by combining all types of reinforcement strategies, it was possible to make a material with high compressive strength (18 MPa) and high deformability.

Fig. 2 :

 2 Fig. 2: Particle size distribution of the kaolin mortars (binary and ternary mixes) and Saint-

Fig. 3 :

 3 Fig. 3: Dispersed flax fibers and woven flax fiber fabric.

Fig. 5 :

 5 Fig. 5: Compressive strength as a function of dry density for cob earth with and without

Fig. 7 :

 7 Fig. 7: Evolution of the compressive strength of the sample after drying versus the dry density

Fig. 8 :

 8 Fig. 8: Dimensionless compressive strength of fabric reinforced samples versus the

Figure 9

 9 Figure9shows a fabric reinforced sample before and after a compressive strength test.

Fig. 9 :

 9 Fig. 9: Observation of the dynamically compacted samples with 4 woven fabric sheets before

Fig. 10 :

 10 Fig. 10: Dimensionless compressive strength of fiber reinforced samples versus the volume

Fig. 11 :

 11 Fig. 11: Compressive strengths versus dry density curves of cast kaolin mortar samples with

Fig. 12 :

 12 Fig. 12: Compressive strengths versus dry density curves of cast cob earth (Saint-Sulpice-La-

Fig. 13 :

 13 Fig. 13: Evolution of compressive strength of tested materials with alginate dosage at a fixed dry density (extrapolated value at 2050 kg/m 3 for the kaolin mortar and 1850 kg/m 3 for the

Fig. 14 :

 14 Fig. 14: Evolution of sample compressive strength of kaolin mortar with dry density for

Fig. 16 :

 16 Fig. 16: Evolution of compressive strength of samples of kaolin mortar with dry density for

  Figure17shows the force displacement curves measured during the flexural strength tests.

Fig. 17 :

 17 Fig. 17: Force versus displacement curved obtained during the flexural strength

Fig. 18 :

 18 Fig. 18: Stress versus deformation curves obtained during the compressive strength

Acknowledgement: The authors would like to thank the Ecomaterre project led and managed by the IAUR (Institut d'Aménagement et d'Urbanisme de Rennes) and the ACROTERRE association for its action of structuring regional research on earth construction.