Chloé Duvivier Duvivier
email: chloe.duvivier@u-clermont1.fr

Mary-Françoise Renard

Shi Li

Mary -Françoise

Renard February

Cerdi

Duvivier Chloé

Professeur Li Shi

Renard Mary-Françoise

Patrick Plane

Catherine Araujo Bonjean

Are workers close to cities paid higher non-agricultural wages in rural China?

Keywords: Remoteness, wage differentials, regional labor market, China Codes JEL / JEL codes : J31, O53, R10, R23

come

Introduction

Reforms in China have allowed for a fall of nearly half a billion in the number of poor people over the past thirty years (World Bank, 2009). However, many rural people remain poor and poverty is strikingly concentrated in isolated rural areas (World Bank, 1992;[START_REF] Jalan | Geographic poverty traps? a micro model of consumption growth in rural China[END_REF]. Other studies, both on developed (Pardridge and Rickman, 2008) and developing countries [START_REF] Bird | Livelihoods and chronic poverty in semi-arid Zimbabwe[END_REF], also highlight that poverty increases with remoteness.

A large strand of the literature argues that non-agricultural employment enables rural households to get out of poverty. Indeed, non-agricultural work can enable households both to raise their income and to reduce its instability [START_REF] Ellis | Household strategies and rural livelihood diversification[END_REF]. This is particularly true for China, where farm size is extremely small1 and thus, where farmers have few opportunities to generate agricultural income. Thanks to the economic reforms implemented in China over the last thirty years, nowadays, most rural households are involved in some kind of non-agricultural activity [START_REF] Liu | Nonagricultural employment determinants and income inequality decomposition[END_REF]. However, non-agricultural employment has developed unevenly across rural areas, leading to a huge increase in intrarural inequality [START_REF] Scott | Rural industrialization and increasing inequality: Emerging patterns in China's reforming economy[END_REF][START_REF] Kung | So what if there is income inequality? the distributive consequence of nonfarm employment in rural China[END_REF]. More specifically, distance to urban areas plays a significant role in determining the level of non-agricultural employment. As peri-urban rural areas benefit from low transport costs and from a significant transmission of ideas, they benefit from more non-agricultural employment opportunities than areas further away from cities [START_REF] Henderson | Geography and development[END_REF]. This has been demonstrated specifically for China [START_REF] Knight | Chinese peasant choices: Migration, rural industry or farming[END_REF][START_REF] De Janvry | The role of non-farm incomes in reducing rural poverty and inequality in China[END_REF] and also for several other developing countries [START_REF] Corral | Rural nonfarm incomes in Nicaragua[END_REF][START_REF] Ferreira | Rural nonfarm activities and poverty in the brazilian northeast[END_REF][START_REF] Micevska | Rural nonfarm employment and incomes in the Himalayas[END_REF][START_REF] Winters | Assets, activities and rural income generation: Evidence from a multicountry analysis[END_REF][START_REF] Deichmann | Urban proximity, agricultural potential and rural non-farm employment: Evidence from Bangladesh[END_REF][START_REF] Jonasson | How important are locational characteristics for rural nonagricultural employment? Lessons from Brazil[END_REF].

The present work aims at studying more deeply how remoteness affects non-agricultural employment by investigating whether remote workers are trapped in low-paid non-agricultural jobs. Therefore, unlike previous studies, the discussion here is not about the level but about the kind of non-agricultural employment that remote workers manage to get. Looking at this issue in China is particularly relevant for three reasons. First, given the in-stitutional restrictions on labor mobility (hukou)2 , local conditions play a very significant role in determining rural workers' earnings and well-being [START_REF] Xia | Diversify and prosper: Peasant households participating in emerging markets in northeast rural China[END_REF].

Second, if on average non-agricultural activities are much more income-generating than agricultural activities3 , there is a huge variation in the remuneration of non-agricultural wage-employment. There are even low-paid non-agricultural jobs where earnings are lower than agricultural earnings [START_REF] Lanjouw | Rural nonagricultural employment and poverty in Ecuador[END_REF], so that one cannot assume a priori that nonagricultural employment enables workers to raise their income. Third, it has been recognized that there are persistent spatial differences in wages, especially between workers in urban areas and workers in remote areas [START_REF] Hanson | Firms, Workers, and the Geographic Concentration of Economic Activity[END_REF].

According to the 2002 and 2007 Chinese Household Income Project (CHIP) surveys, there are both high differences in wages across rural areas and high differences in wages between suburban villages and other villages in China. On the one hand, in 2002 the average daily wage4 was 2.5 times higher in the ninth decile than in the first decile. Even if the gap narrowed slightly5 from 2002 to 2007, the average daily wage was still two times higher in the ninth decile than in the first decile in 2007. This data shows that intra-rural wage inequality and intra-urban wage inequality are of comparable magnitude (see [START_REF] Combes | Urbanization and Agglomeration Economies in China[END_REF] for data on intra-urban wage inequality). On the other hand, suburban villages benefit from higher wages, as the average daily wage in these villages was about 1.25 higher than in other villages.

In spite of these spatial variations in wages in rural China, to our knowledge, no empirical evidence exists on the effect of urban proximity on rural non-agricultural wages in China 6 , although there is some empirical evidence for other developing countries. On the one hand, non-agricultural earnings tend to be higher in rural areas closer to urban centers and roads [START_REF] Corral | Rural nonfarm incomes in Nicaragua[END_REF][START_REF] De Janvry | Income strategies among rural households in Mexico: the role of off-farm activities[END_REF][START_REF] Micevska | Rural nonfarm employment and incomes in the Himalayas[END_REF]. However, these studies estimate the determinants of annual non-agricultural earnings which depend on both the intensity of participation in the non-agricultural sector and on the hourly wage. As urban proximity increases the intensity of participation in the non-agricultural sector [START_REF] Knight | Chinese peasant choices: Migration, rural industry or farming[END_REF], one cannot infer from these studies that workers close to urban areas are paid higher wages. On the other hand, others assess whether workers closer to cities have a higher probability of being involved in high-paid 7 jobs and find mixed evidence. [START_REF] Deichmann | Urban proximity, agricultural potential and rural non-farm employment: Evidence from Bangladesh[END_REF] estimates that high-paid jobs are concentrated in rural areas surrounding urban centers in Bangladesh. In contrast, [START_REF] Jonasson | How important are locational characteristics for rural nonagricultural employment? Lessons from Brazil[END_REF] find that there is no clear relation, as both high-paid and low-paid jobs are concentrated around urban agglomerations.

In China, non-agricultural wages are likely to be lower in remote rural areas. First, the type of industry varies across rural areas, and high-return sectors are more likely to concentrate in suburban areas. Indeed, in areas located far away from cities, non-agricultural employment often consists of small scale manufactures that specialize in modest articles.

On the contrary, in areas close to cities, the production is much more technologically sophisticated and is tied to urban production, through subcontracting and technical assistance to urban firms [START_REF] Naughton | The Chinese Economy: Transitions and Growth[END_REF]. Second, nowadays Chinese suburban areas are highly urbanized with a high level of industrial development, densely concentrated industries and competitive industrial clusters, so that they are very similar to cities [START_REF] Naughton | The Chinese Economy: Transitions and Growth[END_REF]. Suburban villages are therefore likely to benefit from some kinds of agglomeration economies, leading to higher labor productivity and so, to higher real wages 8 . However, the huge labor surplus in rural China could also impede wages from rising close to urban areas. Moreover, most of the production in rural industries is intensive in low-skilled labor so that human capital externalities may not be significant, limiting the scope of agglomeration economies. As a result, one cannot say a priori whether or not rural areas closer to cities benefit from higher wages -this issue requires an empirical assessment.

7 Non-agricultural jobs are high-paid if the hourly wage falls above the earnings of wage laborers in agriculture.

8 [START_REF] Duranton | Micro-foundations of urban agglomeration economies[END_REF] present the theoretical micro-foundations of agglomeration economies and [START_REF] Rosenthal | Evidence on the nature and sources of agglomeration economies[END_REF] review the empirical literature on agglomeration economies.

Using the 2002 rural survey of the Chinese Household Income Project, we estimate the determinants of hourly wages, controlling both for the potential selection bias and for a range of characteristics at the individual and village levels.

We find that remote workers suffer both from scarcer opportunities to work out of agriculture locally and from lower local wages. Specifically, workers in suburban villages are paid about 15% more for one hour worked. Moreover, the result holds even after controlling for differences in living costs between suburban and other areas, which suggests that urban proximity leads to higher wages in real terms. We also find that migration enables remote workers to partially compensate for lower local wages, suggesting that restrictions on migration hurt remote workers more than other workers.

The rest of the paper proceeds as follows. Section 2 presents the data and Section 3 the methodology used. We describe the results in Section 4 and finally, we conclude and propose some policy recommendations in Section 5.

Data

To carry out the empirical analysis, we use the 2002 rural survey of the Chinese Household Income Project 9 . This survey was conducted by the Chinese Academy of Social Sciences and investigates households' conditions in 2002. The database is composed both of an individual, a household and a village level survey. Thus, we benefit from detailed information on individual labor allocation and from household and village characteristics.

In addition, this is a nationally representative survey which investigates 37,969 individuals of 9,200 households from 961 villages belonging to 122 counties (xiàn) of 22 provinces 10 .

As a result, we benefit from a huge variability in terms of remoteness-proximity to urban areas, contrary to most micro-economic studies.

9 A detailed description of the data can be found in Gustafson et al. (2008). We do not use the 2007 CHIP survey as there is no detailed information on rural non-agricultural work to calculate hourly wages. 10 The "province" and the "county" correspond respectively to the first and third levels of administrative division in China. The sample includes the provinces of Beijing, Hebei, Shanxi, Liaoning, Jilin, Jiangsu, Zhejiang, Anhui, Jiangxi, Shandong, Henan, Hubei, Hunan, Guangdong, Guangxi, Chongqing, Sichuan, Guizhou, Yunnan, Shaanxi, Gansu and Xinjiang.

We restrict the CHIP sample to workers. Every individual above 15 years old who reports having earned some income or having spent some time working is considered as a worker. We further distinguish between agricultural workers (those who work on-farm or as a farm-employee) and non-agricultural workers (those who work out of agriculture).

Non-agricultural workers are composed of wage-earners and self-employed workers. are non-agricultural wage-earners working locally 12 . According to the data, a significant share of the labor-force (39%) is involved in some kind of non-agricultural work, which is very consistent with previous findings (de [START_REF] De Brauw | The evolution of China's rural labor markets during the reforms[END_REF][START_REF] Knight | Chinese peasant choices: Migration, rural industry or farming[END_REF][START_REF] Shi | Choices between different off-farm employment sub-categories: An empirical analysis for Jiangxi province, China[END_REF]. In addition, nearly one-third of workers is diversified, as they work both in and out of agriculture. Finally, given land rights reallocation and the scarcity of non-agricultural jobs in rural China, a large share of the labor force remains in agriculture (especially on-farm).

11 According to 2007 Chinese Household Income Project data, more than 90% of migrant rural workers leave their local countryside to work in towns or cities.

12 On the 37,969 individuals surveyed, 7,869 are children and 30,100 are adults. 26,065 adults are workers and 4,035 are inactive. However, we have missing information on place of work, labor time and/or wage for 949 workers so that our sample is composed of 25,116 workers. Finally, there are 2464 individuals for whom explanatory variables are missing. As a result, the estimates are undertaken on a sample of 22,652 workers.

[Table 1]

Dependent Variable: Non-agricultural Hourly Wages

To study whether spatial differences in wages exist between remote rural areas and other ones, we estimate the determinants of individual non-agricultural hourly wage (NAHW). This variable is calculated using information on wages and labor time for non-agricultural wage earners. To ensure robustness, we calculate the individual nonagricultural hourly wage in two ways. First, we consider both an individual's primary and secondary activities. In this case, the individual non-agricultural hourly wage is calculated as:

N AHW i = 2 k=1 W ki D ki * H ki * 1 k
where k refers to the non-agricultural wage employment of individual i (k = 0 if the individual is a full-time agricultural worker; k = 1 if the individual has one non-agricultural wage activity and k = 2 if the individual is engaged in non-agricultural wage-employment both as a primary and secondary activity). W ki are the annual earnings13 of individual i derived by working in activity k; D ki is the number of days worked during the year in activity k and H ki the number of hours worked per day in activity k.

Second, we follow [START_REF] Deichmann | Urban proximity, agricultural potential and rural non-farm employment: Evidence from Bangladesh[END_REF] and only consider an individual's primary occupation. In this case, the individual non-agricultural hourly wage is calculated as:

N AHW i = W i D i * H i

Variable of Interest: Remoteness from Urban Centers

When measuring distance to urban centers, two elements must be considered: (1) how to measure distance? (2) Whether or not different urban centers can have varying impacts.

As Bird and Sheperd (2003) highlight, remoteness can result from physical distance (number of kilometers) and/or from frictional distance (due to bad road quality or natural conditions such as mountainous areas or floods). To take into account both of these dimensions, distance is sometimes measured by travel time [START_REF] Jacoby | Access to markets and the benefits of rural roads[END_REF][START_REF] Fafchamps | The spatial division of labour in Nepal[END_REF]. However, travel time indicators may suffer from endogeneity14 so that other studies prefer using indicators of physical distance (Deichamnn et al., 2009).

In addition, the effect of urban centers on rural non-agricultural employment is expected to vary according to their size. In Nepal, for example, rural household' labor allocation is affected both by the distance to city and by the population of the city: bigger cities tend to have a higher impact [START_REF] Fafchamps | The spatial division of labour in Nepal[END_REF]. In the same way, other studies distinguish between major urban centers and other urban municipalities [START_REF] Deichmann | Urban proximity, agricultural potential and rural non-farm employment: Evidence from Bangladesh[END_REF] or between rural towns and cities [START_REF] De Janvry | Income strategies among rural households in Mexico: the role of off-farm activities[END_REF][START_REF] Lanjouw | Non-agricultural earnings in peri-urban areas of Tanzania: evidence from household survey data[END_REF]. Thus in China, proximity to the county seat and to the city significantly increases the number of days worked out of farm whereas proximity to smaller towns has no effect [START_REF] Knight | Chinese peasant choices: Migration, rural industry or farming[END_REF].

Here we use the two different indicators of remoteness that are available in the 2002 CHIP survey: the distance to rural town and the distance to city. On the one hand, the distance to rural town is measured by the distance, in kilometers, between a worker's village and the nearest county seat. On the other hand, the distance to city is measured by a dummy variable which is equal to 1 if the village is not a suburb of a large or middle city (jiaoqu) and 0 if it is a suburban village. Thus, these two indicators of interest enable us to check whether different types of urban areas have a heterogeneous impact. Indeed, agglomeration economies are likely to be much more bigger in the vicinities of cities, which are larger, and the economy of which is much less dependant on agriculture than that of rural towns. Moreover, these indicators are measures of physical distance and not of travel time, which protects us from endogeneity problems. However, this raises concerns about the accuracy of the distance measure as it does not take into account frictional distance. To solve this problem, we introduce two variables to control for frictional distance: a measure of road access and of the topographical conditions of the village (see Appendices A and B).

As a result, our indicators of interest enable us to estimate the effect of physical remoteness after controlling for frictional distance. Note that our indicators of interest account both for lower transaction costs and for demand-side effects (size of the local market) arising from urban proximity. In other words, we only test whether urban proximity leads to higher wages, without assessing which transmission channels lead to higher wages, this being well beyond the scope of this paper15 .

[Table 2] Table 2 gives descriptive statistics on the hourly wages in yuan according to the distance to urban areas. According to the table, hourly wages broadly decrease with distance to county seat. The relation between hourly wages and distance to city seems stronger given that workers in suburban villages are paid significantly higher hourly wages.

Methodology

Baseline specification

To test whether workers further away from urban centers are paid lower wages, we estimate an income function. Contrary to previous studies [START_REF] Corral | Rural nonfarm incomes in Nicaragua[END_REF][START_REF] De Janvry | Income strategies among rural households in Mexico: the role of off-farm activities[END_REF][START_REF] Jonasson | How important are locational characteristics for rural nonagricultural employment? Lessons from Brazil[END_REF], to test the effect of distance on rural non-agricultural earnings, we do not use as dependant variable total non-agricultural earnings. Indeed, total non-agricultural earnings are determined both by hourly earnings and by the intensity of participation in the activity. As remoteness significantly reduces the intensity of participation in non-agricultural activities [START_REF] Knight | Chinese peasant choices: Migration, rural industry or farming[END_REF], it would over-estimate the effect of remoteness on earnings. Therefore, we use as dependant variable the individual non-agricultural hourly wage (NAHW) presented in Section 2.2.

As not every rural worker derives income from non-agricultural wage-employment, we estimate the two-step Heckman selection model [START_REF] Heckman | Sample selection bias as a specification error[END_REF] in order to correct for the potential selection bias. The first step consists in estimating a probit model of participation in non-agricultural wage-employment (selection model). In the second step, we estimate an augmented Mincerian hourly earnings function [START_REF] Mincer | Schooling, experience and earnings[END_REF] by the Ordinary Least Squares:

ln N AHW i = α + β 1 Dist T v + β 2 Dist C v + δX i + ζX v + γλ i + ε i (1)
where i refers to the worker and v to the village. Dist T and Dist C are respectively distance to the nearest rural town and distance to city, our two indicators of interest. ε is the error term, β, δ and ζ are vectors of unknown coefficients, associated with the explanatory variables, which must be estimated. We control for variables both at the individual (X i )

and village (X v ) levels. As control variables, we follow the literature and introduce a set of characteristics which are expected to affect the level of non-agricultural hourly wage [START_REF] Li | Economic transition and returns to education in China[END_REF][START_REF] Hering | Market access and individual wages: Evidence from China[END_REF][START_REF] Démurger | Earnings differentials between the public and private sectors in China: Exploring changes for urban local residents in the 2000s[END_REF]. We control for worker's age and its square, education, experience and its square, gender and dummy variables to control whether the worker is from an ethnic minority and whether he is a member of the Communist Party. Controlling for individual characteristics, we address the issue that workers may sort spatially according to their characteristics16 [START_REF] Combes | Spatial wage disparities: Sorting matters![END_REF]. In this way, we are able to clearly separate the effect of location variables from the effect of workers characteristics.

We introduce two more variables, at the village level, to control for frictional distance as discussed in Section 2.3.: a dummy variable to control for the topography of the village (plain, hilly area or mountainous area) and a dummy variable indicating whether or not a road reaches the village. Province and regional (East, Center, West) dummies are introduced to control for differences in development, living costs, endowments and policies.

In addition, as wages are expected to be lower in poorer areas, we introduce a dummy variable indicating whether the village is in a province level poverty township. Finally, to correct for the potential sample selection bias, the inverse Mills ratio λ, generated from the 1 st step probit model, is introduced among the determinants of income. If λ is significant, it indicates that common factors influence both the participation in the non-agricultural sector and the hourly wage earned from this sector, so that the errors of the two equations are correlated. The Heckman selection model, by introducing the inverse Mills ratio among the determinants of income, enables us to take into account this correlation and thus, leads to consistent estimations. As identifying restriction 17 , we use the quantity of land per capita in the worker's household. Indeed, this should decrease rural workers participation in the non-agricultural sector without affecting the wage level.

Definition and descriptive statistics of the variables are given in Appendices A and B.

Controlling for higher living costs close to urban centers

In equation ( 1) we control for differences in living costs by introducing provincial dummies. However, living costs are also likely to vary within a given province, and especially between remote rural areas and other ones. As wages are expected to be an increasing function of living costs, and as living costs are expected to be higher close to urban areas, the coefficient associated with the variables of interest could be over-estimated [START_REF] Hering | Market access and individual wages: Evidence from China[END_REF].

To control for differences in living costs between remote and other areas, we calculate an index of living costs at the village level. The index is calculated using information on the market price, in yuan per kg, of six non-staple foods 18 (meat, eggs, edible oil, sugar, vegetables, fruit and melons). As a result, we further analyze the effect of urban proximity on hourly wages by adding the index of living costs among the determinants of income:

ln

N AHW i = α + β 1 Dist T v + β 2 Dist C v + ηLC v + δX i + ζX v + γλ i + ε i (2)
where LC refers to the index of living costs at the village level. As the higher the living costs are, the higher the wage should be, we expect η to be positive. 17 The identification of the Mills ratio implies that all the explanatory variables in the income function must be included in the selection model. In addition, at least one explanatory variable must be introduced in the selection model but not in the income model. 18 We do not use information on the market prices of fish and shellfish because of too many missing values. Moreover, market prices of non-staple foods are reported at the household level. As market prices are likely to vary across villages and to avoid measurement errors, we construct an index at the village level. First, for each of the six non-staple foods, we calculate the average of its market price at the village level. Second, we create the living cost index by averaging the market price of the six non-staple foods.

4.1. Are there spatial differences in wages in rural China?

Table 3 presents the estimation results of equation (1) (estimations (1) and ( 3)) and equation (2) (estimations ( 2) and ( 4)) for local non-agricultural wage-earner. As explained in Section 2.2, we calculate the worker's non-agricultural hourly wage in two different ways:

first, we consider both a worker's primary and secondary occupation (estimations (1) and

(2)) and second, we only consider the worker's primary occupation (estimations (3) and ( 4)).

The inverse Mills ratio is significant in all estimations, suggesting that the Heckman selection model is appropriate. A simple OLS estimation of the income function, without correction for the sample selection, would have led to biased estimates. Regarding the determinants of participation in non-agricultural wage-employment (selection model), the results are consistent with previous findings [START_REF] Zhao | Labor migration and earnings differences: The case of rural China[END_REF][START_REF] Xia | Diversify and prosper: Peasant households participating in emerging markets in northeast rural China[END_REF][START_REF] Guang | Migration as the second-best option: Local power and off-farm employment[END_REF][START_REF] Liu | Nonagricultural employment determinants and income inequality decomposition[END_REF][START_REF] Démurger | Rural households' decisions towards income diversification: Evidence from a township in northern China[END_REF]. Educated workers, men and party members have a higher probability of working out of agriculture as wage earners. Moreover, participation in non-agricultural activities decreases with land holdings and is lower in poor townships. Finally, our results confirm that workers closer to urban areas benefit from a higher probability of being involved in the non-agricultural sector, as estimated by [START_REF] Knight | Chinese peasant choices: Migration, rural industry or farming[END_REF] and [START_REF] De Janvry | The role of non-farm incomes in reducing rural poverty and inequality in China[END_REF].

Turning to the income equation, it appears that, as for urban areas [START_REF] Li | Economic transition and returns to education in China[END_REF][START_REF] Hering | Market access and individual wages: Evidence from China[END_REF][START_REF] Démurger | Earnings differentials between the public and private sectors in China: Exploring changes for urban local residents in the 2000s[END_REF], hourly wages in rural China are an increasing function of a worker's age, education and experience. Men and party members also benefit from higher wages. Regarding our two indicators of interest, remote workers are paid lower wages. The result is particularly robust for workers living in the vicinity of a middle or a large city given that the coefficient is significant at the 1% level in all fourth cases. Specifically, workers not living in a suburban rural areas are paid from 12.5% to 15.8% less for one hour worked. Regarding the distance to rural town, a 1-kilometer increase in the distance to the county seat decreases hourly wage by 0.2%. However, the result is not so strong for rural towns given that the coefficient is not significant in estimation (2). This is not surprising given that agglomeration economies are likely to be much stronger in the vicinity of cities where population, population density and industrial density are higher than in towns. Controlling for the higher living costs due to urban proximity does not change the results. Workers are paid more where living costs are higher and, as living costs are higher close to urban areas, this leads to a small decrease in the coefficient associated with the indicators of interest. However, the coefficients remain negative and statistically significant. This suggests that spatial differences in real wages exist across rural areas, according to their distance to urban areas. Finally, the results regarding frictional distance variables (road and topography) are broadly relevant. As for physical distance, wages decrease with frictional distance. Indeed, participation and wages are significantly higher in villages linked by a road. In addition, living in a mountainous area has a negative, but not robust, effect on wages.

[Table 3] 4.2. Robustness Checks: controlling for endowments Differences in regional characteristics are one major source of spatial differences in wages, as endowments, such as a favorable climate, can affect workers' productivity [START_REF] Hanson | Firms, Workers, and the Geographic Concentration of Economic Activity[END_REF]. Endowments not only refer to natural conditions (climate, natural resources) but also include institutions and technology [START_REF] Combes | Spatial wage disparities: Sorting matters![END_REF]. Moreover, endowments are one major source of spatial agglomeration, so that they may be correlated with our indicators of urban proximity, leading to estimation bias. According to [START_REF] Hering | Market access and individual wages: Evidence from China[END_REF], endowments are likely to vary across Chinese provinces so that provincial dummies should control for such differences. However, to ensure robustness, we successively carry out two more tests to control for endowments. First, we estimate equation ( 2) substituting provincial dummies with county level dummies. Second, we follow [START_REF] Fally | Economic geography and wages in Brazil: Evidence from micro-data[END_REF] and estimate equation ( 2) by excluding of our analysis sectors which depend on natural resources19 .

Estimation results are reported in Table 4. Introducing other controls for endowments does not change the findings given that the coefficients associated with the indicators of remoteness remains negative and significant (in fact, the magnitude of the coefficients even increases). These robustness checks confirm that results are not driven by differences in endowments and that the closer to urban areas, the higher the hourly wages.

[ [START_REF] Zhao | Labor migration and earnings differences: The case of rural China[END_REF][START_REF] Guang | Migration as the second-best option: Local power and off-farm employment[END_REF]. Moreover, Chinese remote workers, who have scarcer opportunities to work out of agriculture locally, would be more likely to migrate than workers close to urban areas [START_REF] Knight | Chinese peasant choices: Migration, rural industry or farming[END_REF]. Thus, migration could be a way for remote workers to get access to better paid non-agricultural jobs. In this case, even if wages are higher in rural areas close to cities, remote workers could compensate by migrating more to cities. Finally, what really matters in terms of well-being is to check whether remote workers are, on average, paid lower-wages 21 . To test whether, and to what extent, remote workers manage to compensate for lower local wages through migration, we estimate equation ( 2) on the whole sample of rural non-agricultural wage-earners. As it has been shown that remote workers migrate more and that migration is better paid, we expect the coefficients associated with the variables of interest to decrease and/or to lose their significance.

The results are reported in Table 5. Several interesting results can be found when considering both local and migrant wage-earners. First of all, regarding the determinants of participation, the coefficients associated with the variable male increases whereas 20 This is confirmed in our sample in which a large proportion of non-agricultural workers are migrants (considering individuals' primary occupation, around 30% of non-agricultural workers are migrants).

21 Note, however, that beyond the question of earnings, migration leads to many costs such as separation with family and inferior work and living conditions in cities [START_REF] Zhao | Labor migration and earnings differences: The case of rural China[END_REF][START_REF] Guang | Migration as the second-best option: Local power and off-farm employment[END_REF]. As a result, migration is very often a "second best" choice for Chinese rural workers who prefer to work locally, even for lower wages, rather than to migrate to urban areas.

the sign associated with age begins negative. This confirms that migration is mainly composed of young men. In addition, in the selection model, the coefficients of interest decrease and are insignificant in two cases. Thus, when considering both migratory and local non-agricultural rural workers, location variables are less significant determinants of participation than when one only considers local work. In other words, remote rural workers, who face scarcer opportunities to work out of agriculture locally, do engage more in migration than workers closer to cities. These results highlight that the determinants of participation vary according to the category of non-agricultural work considered as estimated by [START_REF] Shi | Choices between different off-farm employment sub-categories: An empirical analysis for Jiangxi province, China[END_REF].

Second, turning to the income equation, location variables are less important determinants of wages when considering both migrants and local workers. On the one hand, the coefficient associated with rural town is insignificant in all cases. By migrating more, remote workers manage to compensate for the difference in remuneration they suffer in comparison to that of workers closer to rural towns. On the other hand, both the magnitude and the level of significance of the coefficient associated with distance to city decreases. However, the coefficient remains negative and significant in every fourth cases which suggests that, even when migration is taken into account, rural workers living close to cities benefit from higher real wages. Thus, workers far away from cities only manage to compensate partially for lower local wages by migrating more than workers closer to urban areas.

These results underline that restrictions on migration hurt remote workers more than other workers because they suffer both from fewer opportunities to work out of agriculture locally and from lower wages.

[Table 5]

Conclusion

Since the beginning of the economic reforms, rural workers have diversified out of agriculture, which has enabled most of them to get out of poverty. However, not all workers benefit from these new employment opportunities. Besides, one major source of the rise of intra-rural inequality is the uneven development of non-agricultural activities across rural areas. More specifically, non-agricultural work is concentrated in rural areas surrounding urban centers. Previous studies focus on the nexus between rural workers' location and their participation in non-agricultural employment. This paper attempts to study more deeply how remoteness affects non-agricultural employment by investigating whether remote workers engaged in different types of non-agricultural work and particularly in poorly remunerated jobs. In this way, we try to provide additional explanations for why poverty is concentrated in remote rural areas. This issue is particularly relevant for China, where birthplace still plays a significant role in determining an individual's place of work, earnings and well-being.

Contrary to previous studies on rural non-agricultural earnings, we do not estimate the determinants of annual non-agricultural wages, which depend on both the intensity of participation in the non-agricultural sector and on the hourly wage. As urban proximity increases the intensity of participation in non-agricultural work, it would over-estimate the effect of urban proximity on earnings. As a result, we estimate the determinants of non-agricultural hourly wage, controlling for a broad set of individual and village level characteristics. We find that remote workers are paid lower real wages. The result is particularly strong for workers who reside far away from cities. By demonstrating that non-agricultural wages vary according to the distance from urban centers, we shed additional light on intra-rural inequality and on the geographic repartition of poverty in China.

In this context, rural development policies not only must pay attention to the individual determinants of job access and earnings but also to their spatial determinants, in order to reduce poverty and inequality in rural China.

In addition, we find that remote workers manage to compensate, at least partially, for lower local wages by migrating more than workers closer to urban areas. This last result highlights that restrictions on migration hurt remote workers more than other workers because they suffer from scarcer opportunities to work out of agriculture and are paid lower wages locally. Thus, one way to reduce intra-rural inequality and poverty in remote rural areas would be to facilitate labor mobility. However, such a policy must come together with policies aiming at developing non-agricultural job opportunities in remote areas, in order to avoid that remote areas be drained of their most efficient workers. For the selection model, the marginal effects at the mean values of the independent variables are given. The marginal effect gives the change in the probability of participation given a small change in an explanatory variable (or given a change from 0 to 1 for a dichotomous variable).

Table 4 ]

 4 4.3. Does migration enable remote workers to compensate for lower wages?Until now, we have tested for spatial differences in wages across rural areas so that we have only considered local workers' wages. However, non-agricultural employment is composed both of local and migratory work, the latter being a very significant component of rural non-agricultural work in China[START_REF] De Brauw | The evolution of China's rural labor markets during the reforms[END_REF][START_REF] Shi | Choices between different off-farm employment sub-categories: An empirical analysis for Jiangxi province, China[END_REF] 20 . It has been shown that migratory work has traditionally been better paid than local non-agricultural work

Table 1 :

 1 Employment in rural China

	Party member	0.08	0.27	0.17	0.37	0.06	0.23
	Male	0.53	0.50	0.75	0.43	0.46	0.50
	Minority	0.12	0.32	0.07	0.25	0.13	0.34
	Land per capita	1.92	2.10	1.55	1.71	2.03	2.18
	Distance to town	24.37	20.99	19.81	16.64	25.66	21.90
	Distance to city	0.93	0.25	0.89	0.32	0.94	0.23
	Road	0.96	0.19	0.97	0.18	0.96	0.19
	Topography	1.73	0.77	1.63	0.74	1.76	0.78
	Township	0.14	0.34	0.09	0.29	0.15	0.36
	Living costs	4.65	0.60	4.69	0.58	4.63	0.61
	Total		22652		4986		17666

Table 2 :

 2 Non-agricultural hourly wages and distance to towns and cities in rural China

			1 ary and 2 ndary activities		Only 1 ary activity
		Mean Std. Dev. Median Difference	Mean Std. Dev. Median	Difference
	Distance to county seat
	0-20 km	3.07	3.64	2.34		3.02	3.46	2.34
	21-40 km	3.42	5.34	2.37		3.36	5.27	2.38
	41-60 km	3.05	4.32	2.28		2.87	2.92	2.28
	61-80 km	2.15	1.43	1.96		2.15	1.49	1.96
	More than 80 km	2.58	1.77	2.21		2.33	1.47	2.14
	Suburban village
	Yes	3.51	4.70	2.50	0.44***	3.47	4.71	2.50	0.46***
	No	3.07	4.03	2.29	(-2.37)	3.01	3.77	2.29	(-2.58)
	Note: *, **, *** indicate significance at the 10%, 5% and 1% levels respectively. t-statistics in parenthesis.

Table 3 :

 3 Urban proximity and local non-agricultural wages in rural China

		1 ary and 2 ndary activities			Only 1 ary activity
		(1)		(2)		(3)		(4)
		Selection	Income	Selection	Income	Selection	Income	Selection	Income
	Individual characteristics
	Age	0.012***	0.054***	0.012***	0.054***	0.008***	0.055***	0.008***	0.055***
		(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)
	Age 2	-0.014***	-0.061***	-0.014***	-0.062***	-0.010***	-0.063***	-0.010***	-0.063***
		(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)
	Education	0.010***	0.037***	0.010***	0.037***	0.009***	0.041***	0.009***	0.041***
		(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)
	Experience	0.043***	0.052***	0.043***	0.050***	0.037***	0.065***	0.037***	0.063***
		(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)
	Experience 2	-0.105***	-0.111***	-0.105***	-0.107***	-0.089***	-0.144***	-0.088***	-0.137***
		(0.000)	(0.003)	(0.000)	(0.003)	(0.000)	(0.001)	(0.000)	(0.001)
	Party member	0.079***	0.180***	0.078***	0.175***	0.064***	0.202***	0.064***	0.196***
		(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)
	Male	0.137***	0.349***	0.138***	0.352***	0.113***	0.388***	0.114***	0.389***
		(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)
	Minority	-0.001	-0.043	-0.004	-0.050	-0.015	-0.047	-0.017	-0.055
		(0.911)	(0.375)	(0.758)	(0.310)	(0.202)	(0.386)	(0.131)	(0.312)
	Village characteristics
	Distance town	-0.001***	-0.002*	-0.001***	-0.001	-0.001***	-0.002**	-0.001***	-0.002**
		(0.000)	(0.079)	(0.000)	(0.101)	(0.000)	(0.032)	(0.000)	(0.041)
	Distance city	-0.079***	-0.136***	-0.076***	-0.125***	-0.072***	-0.158***	-0.069***	-0.147***
		(0.000)	(0.003)	(0.000)	(0.005)	(0.000)	(0.002)	(0.000)	(0.004)
	Road	0.037**	0.120**	0.034**	0.105*	0.047***	0.207***	0.044***	0.193***
		(0.014)	(0.037)	(0.022)	(0.068)	(0.001)	(0.002)	(0.002)	(0.003)
	Topography	0.001	-0.019	-0.0004	-0.021	-0.006	-0.038**	-0.007	-0.040**
		(0.886)	(0.275)	(0.933)	(0.231)	(0.185)	(0.045)	(0.113)	(0.035)
	Township	-0.048***	-0.168***	-0.044***	-0.153***	-0.030***	-0.188***	-0.025***	-0.172***
		(0.000)	(0.000)	(0.000)	(0.000)	(0.001)	(0.000)	(0.006)	(0.000)
	Living costs			0.140***	0.579***			0.153***	0.563***
				(0.000)	(0.000)			(0.000)	(0.000)
	Household characteristic
	Land per capita	-0.005***		-0.005***		-0.006***		-0.006***
		(0.002)		(0.002)		(0.001)		(0.000)
	Constant	-2.889***	-1.490***	-3.827***	-2.433***	-2.589***	-1.915***	-3.682***	-2.824***
		(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)
	Mills ratio		0.361***		0.354***		0.521***		0.509***
			(0.004)		(0.005)		(0.001)		(0.001)
	Regional dummies	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes
	Provincial dummies	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes
	Observations	22652	22652	22652	22652
	Uncensored observations	4986	4986	4535	4535
	P-value of Wald test	0.000	0.000	0.000	0.000
	Note: *, **, *** indicate significance at the 10%, 5% and 1% levels respectively. P-values in parenthesis.

Table 4 :

 4 Robustness checks: controlling for endowments

		Robustness check 1: county level dummies	Robustness check 2: sectors that do not depend on NR a
		1 ary and 2 ndary activities	Only 1 ary activity	1 ary and 2 ndary activities	Only 1 ary activity
		(1)		(2)		(3)		(4)
		Selection	Income	Selection	Income	Selection	Income	Selection	Income
	Individual characteristics
	Age	0.012***	0.054***	0.008***	0.054***	0.011***	0.055***	0.008***	0.056***
		(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)
	Age 2	-0.015***	-0.061***	-0.011***	-0.061***	-0.013***	-0.063***	-0.010***	-0.064***
		(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)
	Education	0.010***	0.039***	0.009***	0.042***	0.010***	0.039***	0.009***	0.044***
		(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)
	Experience	0.041***	0.049***	0.035***	0.057***	0.041***	0.051***	0.035***	0.063***
		(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)
	Experience 2	-0.102***	-0.106***	-0.086***	-0.129***	-0.098***	-0.108***	-0.084***	-0.137***
		(0.000)	(0.001)	(0.000)	(0.001)	(0.000)	(0.003)	(0.000)	(0.001)
	Party member	0.079***	0.179***	0.064***	0.187***	0.077***	0.184***	0.064***	0.205***
		(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)
	Male	0.147***	0.361***	0.122***	0.387***	0.128***	0.344***	0.106***	0.375***
		(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)
	Minority	0.006	0.060	0.002	0.055	0.004	-0.022	-0.011	-0.022
		(0.704)	(0.356)	(0.915)	(0.435)	(0.724)	(0.657)	(0.337)	(0.685)
	Village characteristics
	Distance town	-0.002***	-0.002*	-0.001***	-0.002**	-0.001***	-0.001*	-0.001***	-0.002**
		(0.000)	(0.051)	(0.000)	(0.024)	(0.000)	(0.084)	(0.000)	(0.031)
	Distance city	-0.067***	-0.170***	-0.051***	-0.172***	-0.075***	-0.123***	-0.068***	-0.143***
		(0.000)	(0.003)	(0.002)	(0.005)	(0.000)	(0.007)	(0.000)	(0.006)
	Road	0.021	0.059	0.038**	0.121*	0.027*	0.098*	0.038***	0.187***
		(0.184)	(0.316)	(0.014)	(0.064)	(0.065)	(0.089)	(0.007)	(0.004)
	Topography	0.014*	0.037	0.014**	0.038	-0.006	-0.036**	-0.012***	-0.059***
		(0.059)	(0.152)	(0.049)	(0.165)	(0.186)	(0.047)	(0.007)	(0.003)
	Township	-0.007	-0.073*	0.001	-0.106**	-0.044***	-0.165***	-0.024***	-0.179***
		(0.504)	(0.092)	(0.888)	(0.021)	(0.001)	(0.000)	(0.006)	(0.000)
	Living costs	0.102**	0.417**	0.107**	0.386**	0.146***	0.573***	0.156***	0.549***
		(0.029)	(0.016)	(0.015)	(0.037)	(0.000)	(0.000)	(0.000)	(0.000)
	Household characteristic
	Land per capita	-0.001		-0.002		-0.005***		-0.006***
		(0.726)		(0.217)		(0.000)		(0.000)
	Constant	-3.681***	-1.983**	-3.597***	-2.222**	-3.829***	-2.458***	-3.675***	-2.818***
		(0.000)	(0.015)	(0.000)	(0.011)	(0.000)	(0.000)	(0.000)	(0.000)
	Mills ratio		0.359***		0.472***		0.365***		0.513***
			(0.002)		(0.001)		(0.005)		(0.001)
	Regional dummies	Yes	Yes	Yes	Yes

Table 5 :

 5 Migrants and local workers

		1 ary and 2 ndary activities			Only 1 ary activity
		(1)		(2)		(3)		(4)
		Selection	Income	Selection	Income	Selection	Income	Selection	Income
	Individual characteristics
	Age	-0.011***	0.048***	-0.010***	0.045***	-0.014***	0.043***	-0.014***	0.042***
		(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)
	Age 2	0.003*	-0.054***	0.003	-0.050***	0.007***	-0.050***	0.007***	-0.048***
		(0.077)	(0.000)	(0.186)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)
	Education	0.014***	0.028***	0.015***	0.029***	0.012***	0.029***	0.013***	0.030***
		(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)
	Experience	0.091***	0.040***	0.091***	0.041***	0.077***	0.050***	0.076***	0.045***
		(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.001)
	Experience 2	-0.245***	-0.091***	-0.244***	-0.096***	-0.206***	-0.118***	-0.205***	-0.109***
		(0.000)	(0.005)	(0.000)	(0.002)	(0.000)	(0.003)	(0.000)	(0.003)
	Party member	0.077***	0.123***	0.082***	0.117***	0.054***	0.131***	0.058***	0.114***
		(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)
	Male	0.217***	0.261***	0.227***	0.260***	0.178***	0.278***	0.187***	0.265***
		(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)
	Minority	-0.037**	-0.087**	-0.017	-0.028	-0.054***	-0.110***	-0.016	-0.045
		(0.013)	(0.022)	(0.434)	(0.587)	(0.000)	(0.009)	(0.439)	(0.400)
	Village characteristics
	Distance town	-0.001***	0.0003	-0.001***	0.0001	-4.35e-04**	0.0003	-0.001**	0.0003
		(0.000)	(0.541)	(0.000)	(0.795)	(0.026)	(0.581)	(0.021)	(0.651)
	Distance city	-0.049***	-0.073**	-0.041	-0.137***	-0.047***	-0.071*	-0.018	-0.124**
		(0.003)	(0.043)	(0.102)	(0.005)	(0.002)	(0.062)	(0.427)	(0.014)
	Road	0.024	0.195***	0.038*	0.110**	0.037**	0.242***	0.054***	0.143***
		(0.204)	(0.000)	(0.063)	(0.016)	(0.035)	(0.000)	(0.006)	(0.004)
	Topography	0.015**	-0.021	0.019*	0.031	0.005	-0.028*	0.018*	0.032
		(0.012)	(0.147)	(0.052)	(0.152)	(0.393)	(0.070)	(0.059)	(0.148)
	Township	-0.014	-0.145***	0.001	-0.071**	0.002	-0.154***	0.006	-0.100***
		(0.235)	(0.000)	(0.921)	(0.020)	(0.834)	(0.000)	(0.618)	(0.002)
	Living costs	0.004	0.377***	-0.065	0.317**	0.053	0.361***	-0.006	0.314**
		(0.925)	(0.000)	(0.275)	(0.014)	(0.185)	(0.000)	(0.912)	(0.018)
	Household characteristic
	Land per capita	-0.009***		-0.004*		-0.009***		-0.006**
		(0.000)		(0.073)		(0.000)		(0.014)
	Constant	-1.357***	-1.675***	-1.142*	-1.302*	-1.291***	-1.763***	-1.285**	-1.352*
		(0.000)	(0.000)	(0.051)	(0.071)	(0.000)	(0.000)	(0.026)	(0.064)
	Mills ratio		0.213***		0.221***		0.297***		0.264***
			(0.006)		(0.003)		(0.003)		(0.006)
	Regional dummies	Yes	Yes	Yes	Yes	Yes	Yes

The arable land area in China is far below the world average as it was only 0.11 hectare per capita in

[START_REF] Tan | Urban land expansion and arable land loss in China-a case study of Beijing-Tianjin-Hebei region[END_REF].

For a review of the hukou system, see[START_REF] Chan | Is China Abolishing the Hukou System?[END_REF].

Wage-employment is estimated to be paid more than twice, and self-employment three to five times as much as agricultural work in China[START_REF] Kung | Off-farm labor markets and the emergence of land rental markets in rural China[END_REF].

The average daily wage refers to the daily wage for temporary workers who perform local labor in towns and villages.

Note that the 2002 and 2007 CHIP surveys were not carried out in the same villages so that the narrowing in wage differentials could be lead by differences in villages surveyed between 2002 and 2007.

Most studies focus on spatial differences in wages in urban China[START_REF] Hering | Market access and individual wages: Evidence from China[END_REF][START_REF] Combes | Urbanization and Agglomeration Economies in China[END_REF].

Following Hering and Poncet (2010) and[START_REF] Démurger | Earnings differentials between the public and private sectors in China: Exploring changes for urban local residents in the 2000s[END_REF], it includes the basic wage, bonuses, in-kind earnings and subsidies and pension income.

Indeed, rural roads are likely to be built where there is a developed non-agricultural sector. As a result, the placement of roads in these areas reduce the travel time to urban centers, which would lead to estimation bias.

Very few empirical studies on rural non-agricultural earnings separately assess the different effects arising from urban proximity. A notable exception is[START_REF] Jonasson | How important are locational characteristics for rural nonagricultural employment? Lessons from Brazil[END_REF] who use a collection of variables to separately estimate demand-side effects and transaction costs effects.

Note that we also partially address the issue of the spatial sorting of workers, carrying out the analysis on the sample of local workers.

Specifically, these workers are no longer considered as wage-earners (their hourly wage is set to zero) and are part of the censored observations.

Remerciements

We thank M. Fafchamps, S. Guillaumont-Jeanneney and the participants of the 8 th International Conference on the Chinese Economy (Clermont-Ferrand, October 2011) for their helpful comments.

For the selection model, the marginal effects at the mean values of the independent variables are given. The marginal effect gives the change in the probability of participation given a small change in an explanatory variable (or given a change from 0 to 1 for a dichotomous variable). For the selection model, the marginal effects at the mean values of the independent variables are given. The marginal effect gives the change in the probability of participation given a small change in an explanatory variable (or given a change from 0 to 1 for a dichotomous variable).


Laurent Talon

Harold Auradou

Alex Hansen

Relation Between First Arrival Time and Permeability in Self-Affine Fractures with Areas in Contact

Keywords: 83.80.Hj -Suspensions, dispersions, pastes, slurries, colloids 46.50.+a -Fracture mechanics, fatigue and cracks 62.20.mt -Cracks

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Due to their role in the flow properties of tight and low permeability reservoirs such as shale gas reservoirs and carbonate reservoirs, and on contaminant transport e.g. in connection with waste storage, the study of transport in fractures is still a very vigorous field [1][START_REF]Committee on Fracture Characterization and Fluid Flow Rock Fractures and Fluid Flow: Contemporary Understanding and Applications[END_REF][START_REF] Holditch | [END_REF][START_REF] Wang | SPE, Hydraulic Fracturing Technology Conference[END_REF]. Most present theoretical efforts attempts to relate the transport properties of fractures to the statistics of the aperture fields through analytical models based on statistical averages, weak disorder perturbation expansions [START_REF] Keller | [END_REF], mean-field approximations or simplified aperture models [6]. We also mention the work of Zhan and Yortsos [7] where a method to deduce the heterogenities of a permeability distribution from the concentration arrival time field was proposed.

Due to the surface roughness, i.e., the heterogenities of the aperture field, these relations provide satisfactory results only over a finite range of conditions and do not permit to predict the behavior of a fracture with large heterogenities in aperture field. One of the main difficulties is to correctly take into account the increasing influence of the contact area as the fracture aperture is decreased [8][9][10][11]. We analyze in this Letter the dispersion problem at finite Péclet number and identify the proper aperture measure for this problem, taking into account severe heterogeneities such as large contact zones. Our main focus is on the breakthrough time, i.e., the time at which the tracer appears at a given position. The surprising result that we find is that this aperture is the same as the one controlling the permeability [12,13].

There are now numerous experimental studies and field observations that demonstrate that natural fractures have a self-affine roughness [14][15][16][17][18] -for a review, see Bonamy and Bouchaud [19]. Self-affine fractures are characterized by a scaling invariance of the statistical properties of the surface roughness under a rescaling of the distances by a factor λ in the average fracture plane and a rescaling λ ζ of the heights. Here ζ is the Hurst or roughness exponent which takes value close to 0.8 for rocks such as granite [20] and values close to 0.5 for porous rocks like sandstone [21,22].

We consider in this work synthetic self-affine fracture surfaces that have been generated using a Fourier method [START_REF] Turcotte | Fractals and Chaos in Geology and Geophysics[END_REF][START_REF] Sahimi | [END_REF]. The fracture is modeled by matching the fracture surface with an opposite flat surface. Since we use the Reynolds approximation, this correctly models flow in fractures as it is only the aperure that enters the flow equations. We define the fracture aperture H( r) as the distance between the two surfaces at position r. In the present work, the rough surface progressively approaches the flat surface one and the aperture of overlapping regions is set to zero. Hence, H( r) > 0 where there is no overlap and H( r) = 0 where there is overlap.

The flow field is determined for a fixed pressure difference between the fracture inlet and outlet by solving the finite differenced Reynolds equations through LU decomposition. The total volume entering the fracture per width and time at the inlet is proportional to the pressure ∆P over the fracture, and is given by

Q = - K Lµ ∆P , (1)
where K is the permeability and µ the viscosity of the fluid. L is the length of the fracture. The breakthrough time is typically obtained by summing along each streamline the local time of convection This method does, however, take into account diffusion between and along the streamlines. We instead discretized velocity field on a square lattice with nearest-neighbor and next nearest-neighbor connections. If we assume that the dispersion time between neighboring nodes is the inverse of the velocity component along the vector between them, we have then introduced diffusion into the dispersion process. This is so since a given tracer particle will not follow the streamlines but move between the nodes via projected velocity vectors. We add to this description the analysis of Stern [28] of the first arrival time of a diffusive process with or against a convective velocity field, making it possible to tune the Péclet number. In order to find the first arrival time, we use the optimal path algorithm of Hansen and Kertesz [29]. We have verified our algorithm by comparing it to a two-dimensional Lattice Boltzmann method [30]. Fig. 1 shows the tracer concentration in gray levels at breakthrough with Péclet number P e = 10 based on the Lattice Boltzmann method. In the following, we do not discuss any dependence of our results on the Péclet number. The numerical experiments we report have been done at P e = 10 as a reasonable value. Other values add nothing significant.

Transport properties of fractures are often characterized by equivalent or apparent apertures -such as the hydraulic, mass balance or electrical apertures -which refers to the aperture of a fracture with flat and parallel walls having the same property as the original fracture. In practice, equivalent apertures are estimated from hydraulic and conservative tracer tests. The mass balance aperture b m is defined as the ratio between the fluid flux Q and the averaged fluid velocity u [31,32]. In practice, the average fluid velocity u equals the average of the velocities of all fluid particles and should be derived from the average residence time determined from the momentents of the time distribution of the measured tracer breakthrough curve [31]. Here we connect the first arrival time, τ min , to the average fluid velocity u by the expression u = L/τ min , as proposed by Guimerà and Carrera [32]. Since the pressure gradient is kept fixed, the mass balance aperture b m = Q/u is proportional to Kτ min .

Before considering two-dimensional fractures, i.e., fractures where the aperture is orthogonal to a two-dimensional fracture plane, we consider a one-dimensional version of the problem, i.e., a fracture where the aperture is orthogonal to a fracture line. We introduce a cartesian coordinate system with the x axis along the line which now consitutes the flat surface. Let us set a = min x H(x). We then define

h(x) = H(x) -a . (2)
When a ≤ 0, the fracture is closed and hence the permeability is zero. For positive a, it is this parameter that controls the permeability of the fracture in the lubrication approximation [13]. The permeability is in this limit given by the expression

L K = L 0 dξ k(cξ ζ + a) 3 , (3)
where k and c are two parameters. c, the topothesis which characterizes the roughness of the aperture field, is a length scale, and k has the units of permeability. For large a, this gives rise to the scaling relation

K ∼ La 3 , (4)
whereas for intermediate a, we find

K ∼ La 3-1/ζ . (5)
For small a, the permeability is completely controlled by the region where h(x) = 0, and the continuum approach behind eq. ( 3) breaks down. We then find that the permeability is given by

K ∼ L 0 a 3 . (6)
We now calculate min using lubrication approximation and in the infinite Péclet number limit where diffusion is The first tracer to the rough channel is the one which has traveled along the streamline located midway between the walls i.e. where the velocity is at its maximum. The time this has taken is τ min , and it is given by

τ min = L 0 dx u(x) , (7)
where u(x) is the maximal velocity at position x along the channel which is proportional to the flow rate over the local aperture in the lubrication aproximation. We have thus

τ min ∝ 1 Q L 0 (h(x) + a) dx . (8)
Combining eqs.( 1) and ( 8), we get

τ min K = Lµ ∆P L 0 (h(x) + a) dx . (9)
This integral may be performed by using order statistics [13]: We order the function h

(x) → h[ξ] = h(x[ξ]) such that h[ξ 1 ] ≤ h[ξ 2 ] when ξ 1 ≤ ξ 2 .
For a self-affine profile we have that h[ξ] ∼ ξ ζ , and eq. ( 8) becomes

τ min K = Lµ ∆P L 0 cξ ζ + a dξ = µ ∆P cL 2+ζ + aL 2 , ( 10
)
where c is a constant. Hence, we have the central result for a one-dimensional channel

τ min K = A + Ca , (11)
where A ∝ L 2+ζ and C ∝ L 2 . Hence, we see that it is the minimum aperture a which controls the first arrival time τ min . This is the same aperture that controls the permeability, see eqs. ( 4) -( 6). This is a somewhat surprising result, since at the minimal apperture location, because of mass conservation, the flow rate is maximal. Consequently, the time in the bottle neck effect has a small contribution to the integral eq. ( 7). However, the bottle neck controls the total flow rate, and this, in turn, controls the first arrival time.

As described in Talon et al. [12,13], the extrapolation of the bottle neck effect to a two-dimensional fracture is not straight forward since the minimal aperture point is easily bypassed by the flow. However, it is possible to generalize the concept to two dimensions by replacing the minimal aperture a by the minimal path aperture. In order to introduce this concept, we orient our fracture such that the one of the edges parallel to the average flow direction follows the x axis. The y axis follows the edge where the tracer is injected and the z axis is orthogonal to the average fracture plane. Hence, 0 ≤ x ≤ L and 0 ≤ y ≤ W . We define C(x) as a path starting at (x, y = 0) and ending at (x ′ , y = W ) without crossing itself. Hence, we define the quantity

B(x) = 1 W L min C(x) C(x) d ℓ • e y ( ℓ) 3 1/3 . ( 12
)
This is the minimal average fracture opening over all paths starting at (x, 0) and ending anywhere along the opposite edge at y = W . This quantity corresponds to H(x) in the one-dimensional case. The minimal path aperture is defined as The solid curves are for τminK -A vs. bc, whereas the broken curve is for τminK -A vs. a. We determine A by varying it until we obtain the best possible power law. In both figures, the straight portions of the curves have unit slope as indicated in eq. ( 19). The curves are based on one sample of size 512 × 512 for each roughness.

b c = min x B(x) , (13)
corresponding to the smallest aperture a in the one-dimensional case. We finally define

b(x) = B(x) -b c , (14)
in the same way we defined h(x) in eq. ( 2) in the one-dimensional case. The central idea in Talon et al. [13] was that the two definitions b c and b(x) could replace a and h(x) in the one-dimensional case in the permeability integral [START_REF] Holditch | [END_REF]

K ∼ W Lb 3-1/β c , (15)
whereas the large and small scale regimes become respectively

K ∼ W Lb 3 c , (16)
and

K ∼ W L 0 b 3 c . ( 17
)
Numerical experiments based on solving the Kirchhoff equations give

K ∼ W Lb 2.25±0.02 c for ζ = 0.8 , W Lb 2.16±0.02 c for ζ = 0.3 , (18)
for the intermediate regime. The results are very close to the prediction of eq. (15). By following exactly the same procedure for the first arrival time, i.e., replace h(x) by b(x) and a by b c in eq. ( 8), we find where A ∝ W L 2+ζ and C ∝ W L 2 . Fig. 3 shows A and C as a function of L verifying these two scaling laws. We show in fig. 2, τ min K as a function of b c and of a. We see that the linearity of τ min K is verified for the entire range of b c values, whereas it is only true for large values of a. This is where a and b c begin to coincide. Hence, we have verified that the miminal path aperture b c controls both the permeability K and the minimal time τ min . This, together with eq. ( 19) constitute two main results of this Letter.

τ min K = A + Cb c , (19)
Since the first arrival time and the permeability are controlled by the same aperture, it is possible to eliminate the aperture between them. Hence, we may express the first arrival time directly in terms of the permeability by combining eqs. ( 16) - (19). We show in Fig. 4 τ min vs. K for two roughnesses, ζ = 0.8 and ζ = 0.3. We expect that for small K, τ min ∼ A/K 1 , where A is defined in eq. (19). For large K, we expect τ min ∼ C/K 2/3 . For intermediate K, we expect τ min ∼ A/K + C/K 0.56 when ζ = 0.8 and τ min ∼ A/K + C/K 0.54 for ζ = 0.3. C is defined in eq. ( 19). Straight lines with the appropriate slopes have been added in Fig. 4. For the intermediate region, we have used only the term proportional to C.

We have in this Letter shown that the permeability and the first arrival time in dispersive processes are controlled by the same aperture length scale in self-affine fractures. We have also shown that the functional relation between the first arrival time and this aperture is linear, see eq. (19). The appropriate aperture is the minimal path aperture defined in eqs. ( 14) and (13). It is a generalization of the concept of the narrowest constriction that controls both the permeability and the first arrival time in one-dimensional fracture systems. Whereas the scaling properties we report are specific to self-affine aperture fields, the method of analyis based on optimal paths is not. of Geological Complex System" and the Réseaux de Thématiques de Recherches Avancées "Triangle de la Physique" for financial support.

Fig. 1 :

 1 Fig.1: Fluid flows in a self-affine aperture field seen from above. The curves are the streamlines of the flow field. The streamlines have been found using the Kirchhoff method. We have then used the Lattice Boltzmann method to simulate dispersion fixing the Péclet number at 10. The tracer concentration is shown on a grey scale where darker means higher concentration. Areas where the aperture is zero -i.e., the fracture surfaces are in contact -are shown as black.

Fig. 2 :

 2 Fig. 2: Loglog plot of τminK -A as function of a and bc for ζ = 0.3 (upper) and ζ = 0.8 (lower).The solid curves are for τminK -A vs. bc, whereas the broken curve is for τminK -A vs. a. We determine A by varying it until we obtain the best possible power law. In both figures, the straight portions of the curves have unit slope as indicated in eq. (19). The curves are based on one sample of size 512 × 512 for each roughness.

  After ordering b(x) → b[ξ], we find that b[ξ] ∼ ξ β , where β = 1.5 for ζ = 0.8 and β = 1.2 for ζ = 0.3. The intermediate scaling regime (5) then is replaced by

Fig. 3 :

 3 Fig. 3: Scaling of coefficients A (upper figure) and C (lower figure) defined in eq. (19). The straight lines are best fits and have slopes 3 for the ζ = 0.8 data (+) and 2.2 for the ζ = 0.3 data (×) in the upper figure. The theoretical values are 2.8 and 2.3 respectively. In the lower figure, the best fit has slope 2.0 for both the ζ = 0.8 and ζ = 0.3 data. The theoretical value is 2.

Fig. 4 :

 4 Fig. 4: First arrival time τmin vs. permeability K for ζ = 0.8 (upper figure) and ζ = 0.3 (lower figure). Each curve is based on one sample of size 512 × 512. The slopes of the straight lines are given in the main text.

We thank J. P. Hulin for many interesting discussions. A. H. thanks the Université de Paris-Sud 11 for financial support. H. A. and L. T. thanks the PICS "The Physics