Dohy Hong
email: dohy.hong@n2nsoft.com

Danny De Vleeschauwer
email: danny.de_vleeschauwer@alcatel-lucent.com

Francois Baccelli
email: francois.baccelli@ens.fr

A chunk-based caching algorithm for streaming video

Keywords: caching, popularity, streaming. I

It is customary nowadays that large web objects are cached somewhere close to the user. This saves traffic upstream of the cache and offers the users a better responsiveness. Caching algorithms typically rank the objects in some way and cache the top-ranked objects. In this paper we study a scenario in which a requested video is (instantaneously) streamed to the user and in which the video library is highly dynamic: new videos are frequently introduced, get popular, get consumed and fade away. Caching streaming videos differs from caching traditional web objects as the former are consumed as their information trickles in, while the latter have to be downloaded (almost) completely before they can be consumed. We develop a caching algorithm specifically for streaming video taking into account the dynamicity of the library. First we make sure that its ranking algorithm can follow the dynamicity of the library (better than traditional algorithms can). Second we segment each video in chunks and propose a new algorithm to rank these chunks. We compare the performance of caching based on this new ranking algorithm with traditional caching algorithms and show that chunking is most beneficial.

INTRODUCTION

Packet-based networks, in particular the Internet, are more and more used to distribute video. Channel-based networks (e.g., terrestrial, satellite, cable networks) over which traditionally video was delivered can rely on some form of broadcast, i.e., a number of channels are offered side by side over a shared medium and users can tune in to one of these channels. While this service is also offered over a packet-based network via multicast trees (a service, which is often referred to as "linear programming IPTV"), packet-based networks more easily support personalized on-demand video services: a video library is offered via a catalogue (hosted on a server) and a separate flow is set up for each requesting user.

In such an on-demand system the amount of video traffic on the distribution network is in principle proportional to the number of users, while in linear programming IPTV it was proportional to the number of channels. Since there are many more users than channels, this leads to a prohibitive increase of the capacity requirement. Fortunately, most videos in the library are of interest to more than one user and although the users do not consume a particular video at exactly the same time any more, that video is still consumed over a limited time span [START_REF] Avramova | Analysis and modeling of video popularity evolution in various online video content systems: power-law versus exponential decay[END_REF]. So, if it can be identified which are the most popular videos at any moment in time, these could be cached close to the user; transporting them only once to this cache (and as many times as needed to the users served by that cache), would save a considerable amount of transport capacity on the distribution network.

Based on a similar reasoning, caches are deployed over the Internet to offer users a faster response when downloading large web objects. Since the fair share (e.g., offered by the Transport Control Protocol (TCP)) on the path from the cache to the user is larger than the fair share end-to-end, the web object could be downloaded and consumed faster. Such a set of strategically deployed caches (referred to as a content distribution network) are well understood nowadays: all large web objects (for which a content owner pays) are cached and so these objects are served with less waiting time for the user. Note that the primary aim is the increased responsiveness, while the decrease in transport capacity is a secondary consequence.

Caching for streaming differs in a number of aspects from caching large web objects. First, as indicated above, the prime aim of caching for streaming is different: it envisions decreasing the required transport capacity on the distribution network as much as possible. Second, the dynamicity of the video library has a different character: videos are introduced, get popular and fade away in popularity. As a result a video object is consumed over a relatively short time span (while traditional web objects are requested more or less uniformly over prolonged periods). Moreover, video objects are usually much larger than traditional web objects. For these reasons it is very important to store the right content at the right time in the caches in video streaming. However, the biggest difference is that a video stream is consumed very soon after it has been requested. In contrast to traditional web objects (including downloaded video and to a lesser extent progressively downloaded video), which need to be downloaded completely before they are consumed, a video stream is consumed as the information trickles in. For that reason segmenting the video in chunks will turn out to be beneficial.

The rest of this paper is organized as follows. The next section describes some related work. In Section III the principles upon which our new caching algorithm is based are highlighted. Section IV explains the simulator we used and compares the performance of our new algorithm with traditional algorithms. Finally, in Section V we draw the main conclusions.

II. RELATED WORK

Caching algorithms have been used to decide what to store in web proxies [5][9], in CPU caches [START_REF] Megiddo | Outperforming LRU with an adaptive replacement cache algorithm[END_REF] and in surrogate servers in a video on-demand application [3][6]. In all of these cases the purpose of caching is to reduce the response time of a request for data. Deploying caches between the data server and the data client(s) alleviates the processor of the data server and reduces the load on a capacity-constrained link somewhere between the data server and the data client. In this paper we mainly concentrate on reducing the load on the network.

Caching algorithms (also often referred to as cache replacement algorithms) take decisions based on observations. Most algorithms use recency and/or frequency of requests for an object or a combination thereof to rank the objects. The ones that are ranked the highest (and which are implicitly assumed to be requested the most frequently over the next period) are cached. The ranking is updated regularly (e.g., at each request time or at constant intervals). Often this ranking (or estimated request rate) is used in a cost function that trades off the transport capacity saved with the storage cost incurred by caching the object [START_REF] Borst | Self-organizing algorithms for cache cooperation in content distribution networks[END_REF].

Least-recently used (LRU) [START_REF] Hasslinger | Efficiency of caches for content distribution in the Internet[END_REF], which is solely based on recency, suffers from attaching too much importance to unpopular objects. Even an unpopular object jumps to the first rank upon a request for it and subsequently takes a long time to be evicted from the cache, where it occupies a place that could have been used by another object more fruitfully. Leastfrequently used (LFU) [START_REF] Shi | An Applicative Study of Zipf's Law on Web Cache[END_REF], which is solely based on frequency, is sluggish as it attaches as much importance to ancient as to recent history. Often an aging factor is used to improve its performance. In [START_REF] Avramova | Performance Analysis of a Caching Algorithm for a Catch Up Television Service[END_REF][6] it is proved that by tuning the aging factor the algorithm can be made to behave like LRU or like LFU and that often there is an optimal aging factor in between these two extremes for which the hit ratio is maximal. More complex algorithms to determine the ranking (or equivalently the expected request rate) have been studied: based on a combination of LRU and LFU [START_REF] Megiddo | Outperforming LRU with an adaptive replacement cache algorithm[END_REF] or based on a neural network [START_REF] Cha | I tube, you tube, everybody tubes: analyzing the world's largest user generated content video system[END_REF].

The main difference between the algorithm developed in this paper and the algorithms discussed above is that our algorithm takes the time structure of video objects to be cached into account. Indeed, the fact that streaming video is consumed as it trickles in opens up new possibilities for improving the caching performance: if a set of users are consuming a streaming video, each being at a different point in time, it is known which parts of the video will still be visited and which parts are superfluous. A method based on a similar premise is discussed in [START_REF] Wu | Segment-based proxy caching of multimedia streams[END_REF], but concentrates on the initial portions of the video, while our method encompasses the complete video.

III. CACHING ALGORITHM

Figure 1 shows the set up that we consider in this paper. A video library is offered to a set of users from an origin server. A cache, which serves a subset of the users, is deployed on the edge of the distribution network. The aim of the cache is to reduce the traffic on the distribution network. The video library considered in this paper is highly dynamic. New videos are introduced frequently. Just after the introduction time of a new video it is most popular, and the more time elapses since its introduction time, the less frequently it is asked. This popularity evolution is typical for a catch-up television service as studied in this paper (see [START_REF] Avramova | Analysis and modeling of video popularity evolution in various online video content systems: power-law versus exponential decay[END_REF]). Each video has its own popularity decay curve that follows either an exponential or a power law. Similar consumption patterns have been observed for other video services, e.g., for YouTube in [START_REF] Cha | I tube, you tube, everybody tubes: analyzing the world's largest user generated content video system[END_REF]. In this paper all videos are assumed to be encoded at the same bit rate and to have the same length (of 120 minutes).

Origin server

Distribution network

The cache is not large enough to store all videos. Therefore a selection needs to be made of which videos to cache at any moment in time. For that purpose the caching algorithm ranks the videos and caches the ones that are ranked in the first L positions.

The caching algorithm that we develop in this paper is based on two principles: 1) scoring videos based on requests for them and 2) segmenting the videos in chunks, which we combine with the knowledge that chunk m+1 of a given video will be requested with a high probability in the near future if chunk m of that video is currently streamed to some user. To be able to separate the benefits both principles bring, we first consider a simplified version of our caching algorithm without chunking, which makes decisions by ranking videos in their entirety. For the full version of our caching algorithm all videos are segmented in chunks of equal duration. In this case the decision which chunk to cache is based on the ranking of the chunks. Both versions of the algorithms are described in more detail in the next paragraphs.

A. Scoring

The simplified version of our algorithm is based on keeping a score S k for each video k (k=1, 2, …, K). Each time video k is requested, it gets a bonus, i.e., its score is increased by an amount A, while the score of all other videos is decreased by 1.

When a new video is requested for the first time its score is initialized to a value B. Moreover, the score is kept within the interval [-C,C], by truncating it to these bounds if adding a value A or decreasing it by 1 would move it outside this interval. The simplified version of our algorithm re-ranks at each request time the videos based on these scores S k and the first L ranked videos are cached. Hence, at these request instants one of the following three events can occur 1. The requested video resided already in the cache (an event that is referred to as a "hit"). In this case the caching algorithm updates the ranking, but no videos are evicted from the cache. Moreover, the video is served from the cache. Note that no flow is needed on the distribution network.

2. The requested video did not reside in the cache and the caching algorithm decides to cache the video, because this request makes that this video gets upgraded to a rank in the first L positions.

The server copies the video into the cache, i.e., the information trickles in at the video rate, thus consuming one flow over the distribution network. This also makes that the video that resided at rank L (i.e., the last rank) just before the current request, is gradually being evicted from the cache (in fact it is being overwritten by the new video).

As the new video information trickles in (and is copied into the cache, overwriting the video at position L), it is forwarded towards the requesting user.

3. The requested video did not reside in the cache and the caching algorithm decides that the requested video does not need to be cached this time. The ranking is updated, and (since the algorithm did not cache the video) the requested video has a rank larger than L. The video is served from the origin server, thus consuming one flow over the distribution network.

In this paper the value S k is maintained for every video ever seen, but in practice videos that would have a too low value S k (indicating that they would not have been visited for a long time) would be deleted from the list. If there is a tie between two videos (i.e., if they have equal S k values) the video with the lowest k value takes precedence.

B. Number of guaranteed hits

The full version of our caching algorithm also maintains the values S k in just the same way as described above. Moreover each video is segmented in M chunks and each chunk inherits the score S k from the video it belongs to. For each chunk m of video k a value N k,m is maintained that accumulates the number of guaranteed hits this chunk will have, knowing which videos are currently watched by the users (and assuming that no user aborts watching a video). Figure 2 illustrates that the value N k,m indicates how many times that particular chunk m of video k, will be consumed in the near future (given the current user behavior). This counter N k.m is maintained as follows:

1. The values N k,m are increased by 1 for all values of the index m, each time video object k is requested by a user.

2. The value of N k,m is decreased by 1 after a user watching video object k has consumed chunk m.

3. Remark that if a user aborts viewing video object k before the end of the video (or uses other trickplay commands like "rewind" or "fast rewind"), the values N k.m need to be updated accordingly: e.g., in case a user aborts watching a video for each m larger than or equal to the current chunk the user was watching before he or she aborted, the N k,m value needs to be decreased by 1. However, note that "abort", "rewind" or "fast rewind" events do not occur in our simulations. For the rest this chunk-based caching algorithm (i.e., the full version of our caching algorithm) operates in a similar way as the simplified version of our algorithm: at each request time for a chunk, one of the three types of events described in the previous paragraph occurs (i.e., a cache hit, a cache miss combined with a cache update or a cache miss without a cache update). The only difference is that the ranking is now based on comparing the values N k,m . (the higher the value N k,m the higher the rank of the chunk (k,m)) while the values S k are merely used as tie-breakers. If after ranking chunks based on both N k,m and S k there is still a tie, chunks are ranked based on their chunk number.

Figure 3 illustrates that traditional caching algorithms operating on chunks do not take into account the time structure of the video (i.e., the fact that chunk (m+1) necessarily follows chunk m) and therefore often make wrong caching decisions. This figure shows a snapshot of video k currently being consumed by four users. The chunk (m 1 +1) has a low recency value, as "user a" consumed this chunk a long time ago and "user b" did not reach it yet, so that a traditional caching algorithm (e.g., LRU) operating on chunks would currently not cache it. It can be seen that this chunk is about to be visited by "user b" and two additional users in the near future, such that it merits to be cached, which a caching algorithm based on N k,m would do. This qualitative reasoning will be further quantified in Section IV.

As stated before the most efficient caching algorithm is the one that, at any moment in time, caches those chunks that are most likely to be consumed in the near future. It is clear that in order to determine this likelihood a suitable combination of consumption of videos in the past and currently ongoing videos should be made. Our algorithm is an example of such a combination, where the knowledge related to the currently consumed videos gets priority over the historical video consumption. Other combinations are possible too.

IV. SIMULATIONS

This section briefly describes the simulator and then uses this simulator to assess the performance of the caching algorithm described in Section III.

A. Simulator

We have developed a time-driven (slotted) simulator to assess the performance of this caching algorithm. The time slot is chosen small enough so that it has no impact.

We study a catch-up television service as an example in this paper. We took inspiration for the process with which new videos are introduced in the library and how requests for existing videos are generated in the observation of a real-life system analyzed in [START_REF] Avramova | Analysis and modeling of video popularity evolution in various online video content systems: power-law versus exponential decay[END_REF]. In the simulator videos are introduced into the video library at time instants generated by a Poisson process with arrival rate 10/day. So, on average ten new videos are introduced in the library per day.

The aging process of the videos is as follows. After the introduction time of video k a user can request video k to be streamed to him or her. These requests for video k are generated according to a non-stationary Poisson process with arrival rate (which is referred to as the "demand curve for video k" in this paper) that is specific to video k. This demand curve for video k is randomly drawn at the introduction time of video k in the simulator. Overall the demand curve exposes an exponential decay, but there may be some moments where the content popularity gets a boost. In particular we model a demand curve as ρ(t)⋅exp(-t/τ) (t>0 and 0 otherwise), with ρ(t) a piece-wise constant function. The decay time τ is uniformly drawn between 1 day and 3 days and the initial request rate ρ(0) is uniformly drawn in the interval [43/day,129/day]. Ninety percent of the videos have a constant ρ(t), and hence, have a pure exponentially decaying request rate. The remaining ten percent of the videos are (randomly selected to be) upgraded to become "popular" videos. For those the (randomly selected) decay time τ is divided by 2 and the initial request rate ρ(0) is multiplied by 10. Moreover, for these "popular" movies, the demand curve is boosted after a t=(j⋅7) days to a value 10⋅ρ(0)/(5⋅j) (with j=1, 2, …). In particular, for these popular videos the function ρ(t) is a piece-wise constant function specified as: ρ(t)=10⋅ρ(0) for 0<t<7 and ρ(t)=10⋅ρ(0)/(5⋅j)⋅exp(j⋅7/(τ/2)) for j⋅7<t<(j+1)⋅7 (with j=1, 2, …). This behavior we introduced because we noticed that in the catch-up television service studied in [START_REF] Avramova | Analysis and modeling of video popularity evolution in various online video content systems: power-law versus exponential decay[END_REF] the previous episode of the series gets a boost when the next one is aired (see also [START_REF] Vleeschauwer | Performance of caching algorithms for IPTV on-demand services[END_REF]). Figure 4 shows two typical demand curves, one for a "popular" video and one for an un"popular" video.

Once the demand curve associated with video k is determined, requests for video k are generated by a nonstationary Poisson process with an arrival rate equal to the demand curve. This is just an example of a (realistic) way to generate requests. The simulator can work with requests generated in any other way, in particular with a list of requests observed in an actually deployed video-on-demand system. This list of requests (generated internally or read from a list) drives the simulator. In each slot the simulator takes the following actions. First the requests for videos are generated as described above. If there are requests for video k in this slot the values of S k and N k,m are updated as described in Section III. All new requests are added to the list of ongoing requests. Then, for all ongoing requests it is checked whether chunk (k,m) ends, in which case its corresponding value N k,m is updated as described in Section III. Subsequently it is tested for each ongoing request whether or not the requested information (i.e., that part of the chunk that needs to be served in the current slot) resides in the cache or needs to be retrieved from the origin server, and in the latter case whether or not it needs to be cached (based on the decision made by the caching algorithm). If the requested information resides in the cache the counter H that maintains the number of hits is increased. Additionally a counter N is maintained that keeps the total number of times information was requested. The average hit ratio which is reported at the end of the simulation is defined as H/N and indicates the percentage with which the average traffic volume on the distribution network can be decreased by deploying a cache.

Since new videos are introduced at a rate of 10 videos per day and each video is consumed over the time span of a few days, we intuitively expect that a cache size of a few tens of videos will suffice to obtain a high hit ratio [START_REF] Vleeschauwer | Performance of caching algorithms for IPTV on-demand services[END_REF]. Therefore we study caches that range in size from 1 to 100 videos.

Finally, in all simulations, we set the parameters A, B and C as follows: A=2⋅L, C=60⋅L and B is set equal to A at the beginning of the simulation and later follows the average of S k in particular B=max{A,average{S k }} (where the average is only calculated based on videos that reside in the cache). This is just one, albeit a robust way, to choose these parameters. In all tests we have performed we have observed that the choices of the parameters A, B and C have a second order impact on the cache hit ratio. In particular, the major conclusion that we draw in this paper, i.e., that a caching algorithm can benefit from the knowledge that chunks in a video are consumed one after the other, is insensitive to the choice of the parameters A, B and C. We leave the fine tuning of those parameters for future work.

B. Performance assessment

In this section we assess the performance of the proposed caching algorithm.

First we compare the simplified version of our caching algorithm, which does not segment the video in chunks, with traditional algorithms, which also operate on entire videos. Figure 5 illustrates that our scoring algorithm (labeled "CC-120") outperforms LRU (labeled "LRU-120") considerably for small cache sizes. Figure 6 shows that for small cache sizes the performance of our algorithm increases by more than 100% with respect to LRU. The reason is that our algorithm unlike LRU does not promote a video to the first rank each time it is requested. For LFU (labeled "LFU-120") the window size was optimally tuned: 12h yielded the best performance and the optimum was broad. It can be seen in Figure 6 that our algorithm gains about 10% with respect to LFU. Finally, Figure 5 also shows the optimal hit ratio (labeled "HR-120"). To obtain this optimal hit ratio the videos are ranked in each slot based on the (theoretical) request rate with which requests were generated and the L top ranked videos are cached. In fact this is the same procedure as in our algorithm except that in the optimal algorithm the ranking is not based on S k but on the theoretical request rates. These theoretical request rates are in principle not known to the caching algorithm, but if they were, this would be the best performance that a caching algorithm working on entire videos can reach. It can be seen that the simplified version of our caching algorithm, which is solely based on scoring videos, almost reaches the optimal performance in case entire videos are cached.

Cache size [number of objects]

Gain in hit ratio

Gain CC-120/LRU-120

Gain CC-120/LFU-120 Figure 6. Performance gain of our simplified algorithm (CC) with respect to LRU and LFU operating on entire videos Second, we consider the full version of our caching algorithm. We segment the video in chunks of 1 minute each and compare our full version with LRU and LFU (labeled "LFU-1"), which also operate on chunks. Figure 7 and 8 illustrate that our algorithm (labeled "CC-1") outperforms LRU (labeled "LRU-1") and LFU (labeled "LFU-1") where the latter again used an optimal window of 12h, even more than in the case when all algorithms work on entire videos, especially for small cache sizes. Apparently taking into account the time structure of the video, as our algorithm does presents more valuable information on the chunks that will be retrieved next than the knowledge related to how long ago or how many times these chunks were consumed in the past. Part of this may be due to the fact that in our model a video is always completely consumed whenever it is started. However, our algorithm can be extended for cases in which there is a non-negligible probability that a user abandons watching a video. Also in this case the algorithm will rank the chunks to be cached based on combining past behavior (similar to S k ) and ongoing behavior (i.e., the fact that for currently consumed videos there is a high likelihood that the next parts will be consumed too, similar to N k,m ). If the probability of abandoning the video is small enough, the latter will have to be weighted more than the former. Also in this case of chunk-based algorithms we consider the performance of a caching algorithm where the S k values are not estimated but are taken equal to the true request rates (labeled "HR-1"). In particular, such an algorithm operates almost exactly the same as the full version of our caching algorithm: it ranks the chunks based on N k,m as well, but instead of using S k as tie-breaker, it uses the theoretical request rates for this purpose. Remember that this strategy was optimal in case entire videos were cached, but this is no longer the case for chunk-based caching. Nevertheless it can be seen that our algorithm almost reached the performance of this ideal algorithm which shows that we do not lose a lot of performance by estimating rather than knowing S k .

Cache size [number of objects]

Hit ratio Figure 9 illustrates the gain of chunking. It compares the hit ratio of Figure 5 with the one of Figure 7. For the three considered algorithms (LRU, LFU and CC) there is a performance gain. For small cache sizes the gain is largest for the full version of our algorithm. For large cache sizes chunking does not improve the performance of our algorithm a lot, but notice that the hit ratio of the simplified version of our algorithm already approaches 100% for such cache sizes as shown in Figure 5. To further explain why the full version of our caching algorithm has a better performance than traditional ones and to verify the arguments made when discussing Figure 3, we consider two additional statistics. The first one assesses the hit ratio per chunk number. The second one pertains to the nature of the chunks that are evicted from the cache.

HR-1 LRU-1 LFU-1 CC-1
Figure 10 shows the hit ratio associated with chunk number m (m=0, 1, …, 119). Remember that we segmented each video in M=120 chunks. The hit ratio associated with chunk number m, is the fraction of times a chunk with chunk number m could be served from the cache. Remark that, since users do not abandon watching a video, chunks with chunk number m 1 are requested an equal number of times as chunks with chunk number m 2 as all chunks of a particular video are requested an equal amount of times. Hence, the overall hit ratio (shown in Figure 7) is the average over all possible values m of the hit ratio associated with chunk number m. It can be seen that, for small cache sizes (L=5), the full version of our algorithm has the tendency to cache chunks with a higher chunk number, i.e., it has the tendency to cache chunks towards the end of the video. This is a direct consequence of the fact that our algorithm anticipates better the immediate future. For larger cache sizes (L=10), the chunk distribution flattens a bit as a function of the chunk number and the difference between the full version of our caching algorithm and LRU and LFU is less pronounced. In the case of a large cache (L=50), all chunks, except the very unpopular ones, are in the cache and since each of them is visited once per request for a video, this results in a more or less flat distribution. Remark that for all cache sizes, our algorithm ("CC") achieves a better hit ratio associated with chunk number m than LRU and LFU for all values of the chunk number m. This illustrates why the full version of our algorithm outperforms traditional caching algorithms. Figure 11 illustrates the nature of the chunks that are evicted from the cache. We make a distinction between the chunks based on their N k,m value. This figure shows that the full version of our algorithm (by construction) very rarely removes chunks for which the N k,m value is non-zero, while LRU and LFU do so. This corroborates the argument made when discussing Figure 3.

V. CONCLUSIONS

In this paper we proposed and evaluated a caching algorithm specifically targeted at streaming video. Each video is segmented in chunks and the algorithm makes caching decisions for these chunks. Traditional caching algorithms rank the chunks based on how often they were visited in the past and cache the highest ranked chunks (until the cache is full). Our algorithm also ranks the chunks and caches the top-ranked chunks, but it ranks them differently. The ranking is primarily based on the number of times the chunks will be visited in the near future. This value is determined in an efficient way by observing which chunks of which videos are currently being watched combined with the knowledge that when chunk m of video k is currently being consumed all chunks with a higher chunk number than m of that video k will be visited in the near future (under the assumption that the users do not abort watching a video). As tie-breaker for chunks that have the same value, our algorithm uses the historical consumption pattern (which is similarly, but better, measured than in traditional algorithm). We have proved via simulations that chunking combined with this new strategy of ranking the chunks, yields a considerable performance increase, especially for small cache sizes.

Figure 1 .

 1 Figure 1. Cache network configuration

1 2 3 Figure 2 .Figure 3 .

 1323 Figure 2. Maintaining Nk,m for video k

Figure 4 .

 4 Figure 4. Two typical demand curves

Figure 5 .

 5 Figure 5. Comparison of our simplified algorithm (CC) with LRU, LFU and the optimal algorithm operating on entire videos

Figure 7 . 1 Figure 8 .

 718 Figure 7. Comparison of our full algorithm (CC) with LRU, LFU and the ideal algorithm operating on chunks of 1 minute

Figure 9 .

 9 Figure 9. Performance gain of using chunks of 1 minute rather than caching the video as a whole.

Figure 10 .

 10 Hit ratio as a function of the chunk number for our full algorithm (CC), LRU and LFU.

  Figure 11. Percentage of removed chunks as a function of their number of guaranteed hits (NGH), i.e., Nk,m value, for our full algorithm (CC), LRU and LFU.

Part of the research leading to these results has received funding from the European Union's Seventh Framework Programme ([FP7/2007-2013]) under grant agreement n° 248775.


Christophe Tournassat
email: c.tournassat@brgm.fr

Hélène Gailhanou

Catherine Crouzet

Gilles Braibant

Anne Gautier

Arnault Lassin

Philippe Blanc

Eric C Gaucher

Two cation exchange

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

The objective of the present study was to provide an up-to-date cation exchange constant database for Na, K, Ca and Mg on illite surfaces. Cation exchange databases are easily found in chemical reaction databases provided with speciation calculation codes, such as PHREEQC2 [START_REF] Parkhurst | User's guide to phreeqc -a computer program for speciation, batch-reaction, one-dimensional transport, and inverse geochemical calculations[END_REF]. However, (i) these databases contain no or few references to published results and (ii) the tabulated selectivity coefficients should be considered for montmorillonite only.

Knowledge of reliable cation exchange selectivity coefficients for major cations on illite is of importance in the context of nuclear waste deep disposal studies in order to simulate the porewater solute composition. The exchanger composition provides information about the relative amount of Na + , K + , Ca 2+ , Mg 2+ in solution:

NaX + K + ⇔ KX + Na + K Na/K Reaction 1

2 NaX + Ca 2+ ⇔ CaX 2 + 2 Na + K Na/Ca Reaction 2

2 NaX + Mg 2+ ⇔ MgX 2 + 2 Na + K Na/Mg Reaction 3

where X -represents one mole of the exchanger. The exchanger composition, an easily measurable parameter in compacted clayey rocks, is then an "image" of the porewater major cation composition, which cannot be directly characterized by water extraction and concentration measurements due to the low water content of these rocks [START_REF] Bradbury | A Physicochemical characterisation and geochemical modelling approach for determining porewater chemistries in argillaceous rocks[END_REF][START_REF] Pearson | Mont Terri project -Geochemistry of water in the Opalinus clay formation at the Mont Terri Rock Laboratory[END_REF]GAUCHER et al., Accepted). The following calculation shows how to obtain the Na + concentration assuming that (i) the exchanger composition is known, (ii) the sum of the equivalent concentrations of Na + , K + , Ca 2+ , Mg 2+ is equal to the sum of the equivalent concentrations of Cl -and SO 4 2-(i.e. neglecting H + and other minor cation contributions) and (iii) the exchange selectivity coefficients are known.

In the following, the Gaines and Thomas convention is used to express the equilibrium equation with the exchanger phase (e.g. [START_REF] Sposito | The thermodynamics of soil solution[END_REF]. According to this convention, the activity of an exchanged species is equal to the equivalent fraction occupied by the cation on the exchanger phase, times an activity coefficient. This activity coefficient is here set equal to 1 for the analysis that follows (Eqs. 1 to 8).

Exchange reaction equilibria then lead to the following equations:

{ } { } 2 2 2 / 2 NaX MgX Mg Na ex E E Mg Na K × = + + E q . 1 { } { } 2 2 2 / 2 NaX CaX Ca Na ex E E Ca Na K × + + E q . 2 { } { } NaX KX K Na ex E E K Na K × = + + / E q . 3
where values in brackets are representative of species activities and E i is representative of the equivalent fraction on the exchanger of surface species i. The concentrations of each solution species as a function of the Na + concentration are then: where γ i is the activity coefficient of the species i in solution. These coefficients can be obtained using Debye-Hückel , Davies or Pitzer equations.

By combining Eqs. 4 to 6 with the electro-neutrality relationship:

] [ ] [ ] [ 2 ] [ 2 ] [ 2 ] [ 2 2 2 4 + + + + - - + + × + × = × + K Na Mg Ca SO Cl
Eq. 7

where values in square brackets are concentrations, one obtains: Bicarbonate concentrations are neglected in the electro-neutrality relationship, because they were found to represent a minor fraction (~ 2%) of the anionic charge in the considered clayey systems (e.g. GAUCHER et al., Accepted, for a Callovian-Oxfordian formation with total normality about 0.05-0.1 mol/L, or [START_REF] Pearson | Mont Terri project -Geochemistry of water in the Opalinus clay formation at the Mont Terri Rock Laboratory[END_REF] for an Opalinus Clay formation). Second order equations similar to Eq. 8 can be derived for K, Ca and Mg. One obtains five second order equations with ten unknown parameters (the five concentrations and the associated activity coefficients). Debye-Hückel, Davies or Pitzer (for concentrated solutions) equations give five other independent equations for the activity coefficients. Hence, the mathematical system is fully constrained (ten unknowns with ten independent equations) and the Na, K, Ca and Mg concentrations can be calculated, based on the exchanger composition and the Cl -and SO 4 2-concentrations (and possibly other anions that contribute significantly to the total anion equivalent concentration). However, a good precision on the

0 ] [ 2 ] [ ] [ 1 ] [ ] [ 2 2 4 / / /

Materials and experimental methods

Chemicals. All solutions and suspensions were prepared with Millipore Milli-Q 18 MΩ water. NaCl, KCl, MgCl 2 and CaCl 2 solutions were prepared from analytical grade salts.

Clay material preparation. IMt-2 clay sample material was obtained from the Source Clays Repository (http://www.clays.org/sourceclays/SourceClaysCCM.html). After saturating the suspension with NaCl (0.5 mol/L), it was successively treated with a H 2 O 2 /HNO 3 mixture (pH ~ 2) to remove carbonate and organic matter impurities, then with a mixture of sodium hydrosulphite/HCl (pH 3) to remove iron and manganese oxi(hydroxi)des. Each of the preceding treatments was followed by careful washing with deionised water. The clay was then washed with a mixture of NaCl (1 mol/L) and HCl (10 -4 mol/L). The fine fraction (< 2 µm) of the sample was then isolated by sedimentation, redispersed in NaCl (1 mol/L) and washed with deionised water until the supernatant exhibited a negative response to the silver nitrate test.

Cation exchange isotherms. Na-K, Na-Ca and Na-Mg exchange isotherms were performed at a total normality of ~ 0.005 mol/L in chloride anionic background. The chloride anionic background was preferred to a perchlorate background since it better represents the natural systems studied in our laboratories. Moreover, the effect of cation-anion pairing is expected to be limited at this concentration in the light of results obtained with montmorillonite (SPOSITO et al., 1983a;SPOSITO et al., 1983b;[START_REF] Fletcher | The chemical modeling of clay/electrolyte interactions for montmorillonite[END_REF]TOURNASSAT et al., 2004b;[START_REF] Charlet | Fe(II)-Na(I)-Ca(II) cation exchange on montmorillonite in chloride medium; evidence for preferential clay adsorption of chloride -metal ion pairs in seawater[END_REF][START_REF] Ferrage | Evidence for calciumchloride ion pairs in the interlayer of montmorillonite and implications on hydration state. A XRD profile modelling approach[END_REF]. The method described in TOURNASSAT et al., 2004b was used to perform exchange isotherms, with the exception of the use of cobalt hexamine instead of ammonium acetate as an extracting agent of exchanged cations [START_REF] Ciesielski | Determination of cation exchange capacity and exchangeable cations in soils by means of cobalt hexamine trichloride. Effects of experimental conditions[END_REF]. The pH of the experiment was 5.

Presentation, thermodynamic convention and interpretation of the results.

The cation exchange selectivity coefficient is representative of the relative affinity of the exchanger for two considered cations. In the case of a Na/K or Ca/K exchange, selectivity coefficient values above 1 would denote an affinity for K greater than for Na and Ca, meaning a "preference" of the surface for K. The selectivity coefficient is not a thermodynamic exchange constant, since it varies as a function of the exchanger composition (e.g. [START_REF] Jensen | Potassium -calcium exchange equilibria on a montmorillonite and a kaolinite clay[END_REF].

In the following, two main thermodynamic conventions for cation exchange selectivity coefficients will mainly be used: the Gaines and Thomas convention described above and the Vanselow convention (e.g. [START_REF] Sposito | The thermodynamics of soil solution[END_REF][START_REF] Mcbride | Environmental Chemistry of Soils[END_REF]. The Vanselow selectivity coefficient of a monovalent-monovalent exchange (e.g. Na-K) is:

{ } { } + + × × = K N Na N K Na K K Na v / E q . 9
and the Vanselow selectivity coefficient of a divalent-monovalent exchange (e.g. Ca-K) is:

{ } { } 2 2 2 / + + × × = K N N Ca K Ca K K Ca v E q . 1 0
where N i represents the molar fraction of the cation i on the exchanger and where values in brackets represent the activity of solute species. Knowing the cation exchanger composition makes it possible to switch easily from one convention to the other: in the case of homovalent exchange, Gaines and Thomas and Vanselow selectivity coefficient values are identical and for monovalent / divalent exchange, the following formula applies:

Ca Na Ca Na Ca Na v Ca Na GT N E E N K K × × × = 2 2 / / E q . 1 1
On the one hand, the Gaines and Thomas convention is very useful in the calculation approach shown by Eqs. 1 to 8. Moreover, this calculation convention is available in all numerical speciation codes (e.g. CRUNCH, STEEFEL, 2001, GEOCHEMIST WORKBENCH, BETHKE, 1996, PHREEQC2, PARKHURST and APPELO, 1999), as opposed to the Vanselow convention (especially in PHREEQC2). On the other hand, the Vanselow convention is the only one that enables the calculation of exchange species activity coefficients directly as a function of their variation with changes in the exchanger composition [START_REF] Sposito | The chemistry of soils[END_REF], an approach that is developed at the end of this paper. The data in the figures are always presented at least in the Vanselow convention in order to enable comparison between the figures. The exchanger composition is expressed in the following in terms of the equivalent exchanged fraction of the cations participating in the exchange process, i.e. E Na , E K , E Ca and E Mg [START_REF] Sposito | The thermodynamics of soil solution[END_REF]. The activity of solute species was computed using PHREEQC v2.12 and the Llnl.dat database provided with the software (PARKHURST and APPELO, 1999, http://wwwbrr.cr.usgs.gov/projects/GWC_coupled/phreeqc/).

Experimental results

The complete set of data obtained in this study and from the literature is given in the table of Electronic Appendix 1. It can be seen from this table that the K v values calculated for divalent/monovalent cation exchange on the basis of the data reported by [START_REF] Thellier | Quaternary cation exchange on Silver Hill illite[END_REF]THELLIER and SPOSITO, 1989b) are different from the values reported by these authors by a factor 2, probably due to a misuse of equivalent/kg instead of mole/kg for divalent cations in Thellier's K v calculation.

Results for Na/K exchange selectivity coefficients are shown in Fig. 1. K v Na/K is shown to vary as a function of the exchanger occupancy. This has already been shown for Na/K exchange on montmorillonite with the same trend of K v Na/K decreasing as a function of increasing E K [START_REF] Jensen | Potassium -calcium exchange equilibria on a montmorillonite and a kaolinite clay[END_REF]. One can also note a large scattering of results in the low K occupancy zone observed little effect of pH on the Na-Ca selectivity coefficient [START_REF] Thellier | Proton effects on quaternary cation exchange and flocculation of Silver Hill illite[END_REF]. The nature of the extracting reagent could also have an effect on the K desorption (cobalt hexamine vs. ammonium). Finally, we used the < 2 µm fraction whereas Thellier and Sposito used a < 115 µm fraction. This latter effect is certainly predominant with an increase in the effect of edge sorption sites as the particle size decreases.

(E K < 0.2),
Results for Na/Ca exchange selectivity coefficients are shown in Fig. 2. The trend observed on the data as a function of exchanger occupancy is in good agreement with the data taken from [START_REF] Thellier | Quaternary cation exchange on Silver Hill illite[END_REF]THELLIER and SPOSITO, 1989b) under similar ionic strength conditions. Once again, one can observe considerable variability in the selectivity coefficient as a function of the exchanger composition. Note that the Na plus Ca occupancy never reaches 100% because of the presence of K in all of the experiments: the presence of K in solution and on the exchanger cannot be avoided due to its presence in the structural formula of illite and the weathering of the mineral.

In the exchanger occupancy region, where Na or Ca account for almost all the cations present, the experimental points describe a bell-shaped curve. Similar observations are also made for Na/Mg and Ca/Mg exchange (not shown here, see Electronic Appendix 1). In the middle region (both E Na and E Ca values are above 0.2), the exchange can be considered to be ideal with a constant selectivity coefficient value.

These new cation exchange data, together with data taken from the literature, illustrate the need to derive reliable parameters for calculating the exchange selectivity coefficients for illite. Two models are proposed in the following. The first is a multi-site approach and the second is a one-site approach, each of them having their own applicability domains.

Multi-site modelling approach

A multi-site modelling approach for cation exchange processes on illite has been shown to be successful in predicting Cs + sorption under various conditions of pH and ionic strength (e.g. [START_REF] Brouwer | Cesium and Rubidium ion equilibria in illite clay[END_REF][START_REF] Poinssot | Experimental and modelling studies of caesium sorption on illite[END_REF][START_REF] Zachara | Sorption of Cs+ to micaceous subsurface sediments from the Hanford site, USA[END_REF][START_REF] Liu | A cation exchange model to describe Cs + sorption at high ionic strength in subsurface sediments at Hanford site, USA[END_REF] or to explain the variation in CEC measurements as a function of ionic strength with the 22 Na isotope dilution technique [START_REF] Baeyens | Cation exchange capacity measurements on illite using the sodium and cesium isotope dilution technique: effects of the index cation, electrolyte concentration and competition: modeling[END_REF].

A similar approach was therefore adopted here in order to fit both the results of our experiments and the data obtained from literature. Data obtained in this study were first fitted with a two-site model, referred to as ≡Illa -and ≡Illb -, in agreement with the description of illite reactive surfaces with the presence of frayed edge and basal surface sites (e.g. [START_REF] Poinssot | Experimental and modelling studies of caesium sorption on illite[END_REF][START_REF] Zachara | Sorption of Cs+ to micaceous subsurface sediments from the Hanford site, USA[END_REF].

Relative amounts of these two sites and Gaines and Thomas (e.g. SPOSITO, 1981) Na-K, Na-Ca and Na-Mg exchange selectivity coefficients were fitted concomitantly using the freely available parameter minimization PEST software [START_REF] Doherty | PEST Model-independent parameter estimation[END_REF] in combination with PHREEQC v2.12 (PARKHURST and APPELO, 1999) and the Llnl.dat chemistry database for solute species. Solute concentrations were used as input data and the difference between measured and calculated equivalent fractions of Na, Ca, K, and Mg on the exchanger were minimized by varying the fitting parameters given above. A second round of parameter minimization was then performed, where the difference between the measured and calculated natural logarithms of the Vanselow selectivity coefficients were minimized. This second round gave results identical to the first one. This second round of minimization was necessary to enable the comparison of the results of the two-site model with the results of the one-site model that was fitted only with the difference between measured and calculated natural logarithm of the Vanselow selectivity coefficients (see below, one-site model approach section).

Data from the literature were then simulated. The same selectivity coefficients were used, but the relative proportion of the sites was varied (while maintaining a constant total CEC) to capture the change in sorption behaviour as a function of the particle size distribution (or maximum particle size) considered. Only data at total normality (TN) > 0.005 mol/L and with a complete description of the exchanger were used, i.e. excluding data where K occupancy is not available. The results of this fitting approach are given in Table 1 and Fig. 3.

The data of this study are reproduced by the model with quite low standard deviations on the difference between the measured and simulated equivalent fraction of each cation

( N x mean i i - = σ
where x i is the observable parameter, mean is the mean of this parameter and N the number of scores): σ XNa = 0.052, σ XK = 0.037, σ X2Ca = 0.054 and σ X2Mg = 0.061.

Data from the literature are reproduced with the following standard deviations on the difference between the measured and simulated equivalent fraction of each cation: σ XNa = 0.066, σ XK = 0.020, σ X2Ca = 0.073 and σ X2Mg = 0.057 for data acquired at TN = 0.005 mol/L, σ XNa = 0.166, σ XK = 0.031, σ X2Ca = 0.178 and σ X2Mg = 0.077 for data acquired at TN < 0.005 mol/L down to 0.001 mol/L (not shown), and σ XNa = 0.114, σ XK = 0.009, σ X2Ca = 0.102 and σ X2Mg = 0.105 for data acquired at TN > 0.005 up to 0.1 mol/L (most of the data were given without K occupancy measurement, not shown). One notes that the data at TN < 0.005 mol/L are poorly reproduced as compared to the other data, especially for Na and Ca. Incidentally, the change in selectivity coefficients for Na-Ca exchange for example is very significant between TN = 0.002 mol/L and TN = 0.005 mol/L (see Fig. 4, ~ 1 log 10 unit difference).

Between 0.005 mol/L and 0.0092 mol/L, the change in the selectivity coefficient value is within the uncertainty of the measurement. This behaviour explains why the two-site model selectivity coefficients derived at 0.005 mol/L enable the results at TN > 0.005 mol/L to be predicted with good confidence, bearing in mind that most of the data obtained at these total normalities were obtained without K occupancy measurements.

Recently, [START_REF] Liu | A cation exchange model to describe Cs + sorption at high ionic strength in subsurface sediments at Hanford site, USA[END_REF] have proposed a two-site model explicitly taking into account the variation in the water activity in order to simulate the changes in Cs-Na and Cs-K exchange selectivity coefficients as a function of ionic strength. However, this method was applied to ionic strengths ranging from 0.01 to 5 mol/L, where the water activity value exhibits large variations. In the present study, water activity is nearly constant between 0.001 mol/L and 0.01 mol/L. Moreover, in their model, one of the "driving forces" of the selectivity coefficient change as a function of ionic strength is also the difference in the number of moles of water per equivalent exchange capacity. This difference is high when a Na + ion is replaced by a Cs + ion because of the peculiar hydration properties of Cs at phyllosilicate surfaces. On the contrary, the difference when Na + is replaced by Ca 2+ is expected to be small. Hence, no attempt was made to try to reproduce with the Liu et al. model the ionic strength effect that is observed in this study.

The modelling problem concerning data at low ionic strength could not have been fixed with the two-site model approach even by considering activity coefficients in the form of Davies or (Wateq) Debye-Hückel activity coefficients as proposed by the PHREEQC v2.12 capabilities.

The Wateq Debye-Hückel activity coefficient is of the following form

I b I Ba I Az i i i i + + - = 0 2 10 1 log γ E q . 1 2
where γ i is the activity of the exchanged species i, A and B are constants dependent only on temperature, a i 0 and b i are exchanged species specific parameters, Ι is the ionic strength and z i is the number of equivalents of exchanger in the exchange species i, e.g. 1 for NaX and 2 for CaX 2 (PARKHURST and APPELO, 1999, pp 89). In the case of a Na→Ca exchange, one can evaluate the effect of Wateq Debye-Hückel activity coefficients by considering the variations -Δlog 10 K GT (I) -they impose on the Gaines and Thomas selectivity coefficient as a function of the ionic strength:

( I b b I Ba I Ba I A I K XNa Ca X Ca X XNa NaCa GT × - + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + - + × = Δ 2 1 1 1 1 2 ) ( log 2 2 0 0 10 ) Eq. 13
A judicious choice of parameter b X2Ca and b XNa could help to fit the difference observed between the selectivity coefficient values obtained at ionic strength values 0.001 and 0.005 mol/L. However, these b X2Ca and b XNa parameters are related to a linear behaviour as a function of ionic strength. This linear behaviour is not in agreement with the observation that shows a very sharp change between 0.003 and 0.005 mol/L and almost no change below and above 0.005 mol/L. A sensibility test was performed on the influence of a 0 parameters on the Δlog 10 K GT (I) value as a function of ionic strength. Considering two ionic strengths of 0.001 and 0.005 mol/L together with a 0 parameters varying from 0 to 100 for XNa and X 2 Ca, one obtains a maximum difference of 0.03 log 10 unit in Δlog 10 K GT (I), which is clearly not sufficient to explain the experimental difference, the order of magnitude of which is about one log 10 unit.

In Table 1, one notes that only the Na-K exchange selectivity coefficient reaction has different values for the two sites ≡Illb -and ≡Illa -. For Na-Ca and Na-Mg exchange, the selectivity coefficient value remains constant and the variation in the selectivity coefficient as a function of the exchanger composition cannot be reproduced. Indeed, the bell-shaped curve observed for example for Na→Ca exchange, at a single ionic strength, cannot be reproduced by considering a multi-site model, whatever the number of sites considered in the calculation.

Considering a population of sites with decreasing affinity for Ca, one can easily show that these sites will be filled by order of affinity with Ca as a function of Ca occupancy of the exchanger, thereby creating a monotonic decrease in the global Na→Ca selectivity coefficient when the equivalent exchanged fraction of Ca increases. Note that it is well known that the use of a constant Gaines and Thomas selectivity coefficient leads to a Vanselow selectivity coefficient that varies slowly with the charge fraction of adsorbed divalent cations. However, this effect is not sufficient to model the sharp part of the curve at high bivalent cation exchanger loading (Fig. 2). Fig. 2 None of these representations enables a monotonic decrease in the selectivity coefficient as a function of the exchanger occupancy to be obtained, with the exception of the Rothmund and Kornfeld convention. However, this thermodynamic convention led to a large scattering between experimental points even for the points originating from the same experiment.

The increase in the Na→Ca (or Na→Mg) selectivity coefficient concomitant to the increase in the Ca content on the exchanger signals the presence of a cooperative effect for exchanged Ca. We propose here two possible explanations for this effect. The first one relies on probabilities of exchange site combinations, given that Ca needs to exchange with two Na sites (Fig. 5).

The probabilities of the occurrence of configurations 1, 2 and 3 are directly linked to the cation exchanger composition. Assuming a pure Na/Ca system, and a random distribution of Na and Ca on the exchanger, one obtains the following probabilities of occurrence for the 3 situations given in Fig. 5:

2 1 2 / ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + = Ca Na Na E E E P E q . 1 4 ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + × ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + × = 2 / 2 / 2 / 2 2 Ca Na Ca Ca Na Na E E E E E E P E q . 1 5 2 3 2 / 2 / ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + = Ca Na Ca E E E P E q . 1 6
Now, assuming that configuration 2 is favoured thermodynamically compared to configuration 1, it is expected that the value of the selectivity coefficient of the exchange process is correlated with the probability P 2 . In Fig. 6, it can be seen that the shape of the variation in the P 2 probability value as a function of the Ca equivalent fraction is similar to that observed for the selectivity coefficient in Na/Ca exchange experiments (Fig. 2).

A second explanation is linked to the flocculation behaviour of illite. Thellier and Sposito (THELLIER and SPOSITO, 1989a) have studied the effect of electrolyte concentration and exchangeable cations on the flocculation of Silver Hill Illite. As with montmorillonite (e.g. [START_REF] Schramm | Influence of exchangeable cation composition on the size and shape of montmorillonite particles in dilute suspension[END_REF], they have shown that bivalent cations favour the flocculation of illite particles as compared to monovalent cations and that turbidity is developed gradually as the equivalent fraction of bivalent cations decreases from the 0.8 value. This effect of the bivalent cation on the flocculation of illite suspension is perfectly in agreement with an increase in the selectivity coefficient for Na→Ca exchange as a function of the increase in the exchanged Ca, when considering the effect of stacking of the illite particles (Fig. 7). As already depicted by other authors for smectite monolayers (e.g. MCBRIDE, 1994, Fig. 7), the stacking of illite external basal surfaces might favour divalent and multivalent cations as compared to monovalent cations. This last explanation would also be in agreement with the effect of ionic strength on the Na/Ca and Na/Mg selectivity coefficients. Fig. 8 shows the effect of the ionic strength on the percentage of light transmission through a suspension of Na/Ca exchanged illite (data from THELLIER and SPOSITO, 1989a). The plotted experimental points correspond to conditions where equivalent bivalent cation fractions are ~ 0.5. Under these conditions, the solution composition is mainly dominated by Na + cations (> 97%). The light transmission is related to the flocculation (high percentages) or dispersion (low percentages) of the clay suspension (THELLIER and SPOSITO, 1989a). One can not fail to note the sharp transition at approximately 4 mmol/L, in agreement with the sharp variation observed for the Na/Ca and Na/Mg selectivity coefficient in the same ionic strength range value.

The stacking of illite particles then seems to have a marked effect on the selectivity coefficient for monovalent/bivalent cation exchange, but this cannot be taken into account in the above multi-site approach since the nature and the properties of the sites change as a function of the exchanger composition.

From this modelling result analysis, it appears that, concerning the prediction of the cation exchanger composition, the two-site model with constant Gaines and Thomas selectivity coefficients proposed in [START_REF] Comans | Sorption of cesium on illite: Nonequilibrium behaviour and reversibility[END_REF], 1.5% (Illite, [START_REF] Poinssot | Experimental and modelling studies of caesium sorption on illite[END_REF]. The proportion of K + high affinity sites is at least twice as high as the reported proportion of Cs + high affinity sites. This proportion can also be compared to the Type II sites and planar sites described by [START_REF] Baeyens | Cation exchange capacity measurements on illite using the sodium and cesium isotope dilution technique: effects of the index cation, electrolyte concentration and competition: modeling[END_REF] for Na → Ca+Mg exchange: 20% and 80%

respectively. The Illb -sites considered in the present model are then rather more similar to type II sites described in the literature than frayed edge sites whose influence is not visible in the Na-Ca-Mg-K exchange experiments, due to their low proportion. A third type of site should then certainly be added to our model. However, the data available in the literature do not enable its properties towards major cations (Na, Ca, Mg, K) to be calculated.

The exchange selectivity constants for Na → Ca exchange can also be compared with those given by Baeyens and Bradbury: log 10 K GT (Na → Ca) = 0.30 for planar sites (80% of the sites) and log 10 K GT (Na → Ca) = 1.90 for type II sites (20% of the sites). The values obtained in our study lie in the middle: log 10 K GT (Na → Ca) = 0.53 for both sites. This could be explained by the fact that the values derived in [START_REF] Baeyens | Cation exchange capacity measurements on illite using the sodium and cesium isotope dilution technique: effects of the index cation, electrolyte concentration and competition: modeling[END_REF] were not obtained from direct measurements of all the cations present on the exchanger and that K was never considered in their calculation. If K had been considered, their deduced amount of exchanged Ca would have been lowered and then the selectivity coefficient would have been lowered as well. The same type of problem may also have applied in the study carried out by [START_REF] Zachara | Sorption of Cs+ to micaceous subsurface sediments from the Hanford site, USA[END_REF]. They treated their experiments as binary (e.g., Cs-Ca), when in fact K and even Na were almost certainly present, or in the case of Cs-Na, as binary when K, Mg, and Ca were present. This is a probable explanation for the underprediction of Cs retardation in column experiments using batch-derived data where no data on other cations (e.g. other than Cs and Na) were available.

Note that for TN > 0.2 mol/L in pure anionic chloride medium, the formation of chloridecation pairs on the illite surface could explain the deviation of the model result towards the high TN data available in [START_REF] Shainberg | Sodium/calcium exchange in montmorillonite and Illite suspension[END_REF]: in these experiments the illite affinity for Ca increased when the TN increased. A Gaines and Thomas selectivity coefficient of 10 3.5 for Na → CaCl + exchange reaction has been derived to fit these nine data points (not shown). This high selectivity coefficient value is in agreement with high selectivity coefficient values derived for chloride-cation pair exchange on montmorillonite surfaces (SPOSITO et al., 1983b;SPOSITO et al., 1983c;TOURNASSAT et al., 2004a;[START_REF] Charlet | Fe(II)-Na(I)-Ca(II) cation exchange on montmorillonite in chloride medium; evidence for preferential clay adsorption of chloride -metal ion pairs in seawater[END_REF][START_REF] Ferrage | Evidence for calciumchloride ion pairs in the interlayer of montmorillonite and implications on hydration state. A XRD profile modelling approach[END_REF].

As explained in the introduction, our interest lies primarily in the prediction of the solution composition as a function of the exchanger composition rather than conversely. It is possible to apply the procedure outlined in the introduction with a two-site model instead of a one-site model, provided that the ratio between the two types of site is known.

The prediction capability of the two-site model (inverse modelling) was then tested using the parameters listed in Table 1 for the literature data, keeping the total normality of the solution at the experimental value. All calculations were performed with PHREEQC v2.12 and the Hence, based on the experimental cation exchange occupancy and the two-site model presented here, the potassium concentration can be calculated with an accuracy of ± ~ 1 log 10 unit. Other elements are calculated with an accuracy of ± ~ 0.3-0.6 log 10 unit. The concentration prediction capability of the present two-site model is then not very accurate, especially for potassium whose concentration can be underestimated or overestimated by a factor of ten. On the contrary, the direct modelling of cation exchange occupancy leads to rather good results as long as one remains within the range of the site occupancy given previously. One should note that the sites proportion is the key factor governing the success of the model.

One-site modelling approach

The uncertainty on the concentration prediction capability of the two-site model presented above leads us to model the data in a different way: the changes in selectivity coefficient are interpreted here as a continuous change in surface species activity coefficient. The use of a one-site model for cation-exchange on illite represents a departure from the approach used in all recent studies. Although questionable from a theoretical point of view, it will be shown in the following that this approach allows a better concentration prediction capability than a twosite model with constant selectivity coefficients.

Given the multi-site microscopic reality of the exchange process on illite surfaces, it is unlikely that the following proposed one-site method is thermodynamically correct. Instead, it should be regarded as an empirical method to approach the exchange selectivity coefficients as a function of exchanger composition and ionic strength.

Activity coefficients and exchange equilibrium constants calculation in binary

exchanges. The calculation method presented here concerns only monovalent and divalent species. Instead of using a generic term for chemical species, we have chosen to present the calculations with Na, K and Ca as example cases. More generic representations of these calculations can be found in for example [START_REF] Chu | The thermodynamics of ternary cation exchange systems and the subregular model[END_REF].

Let us consider the Ca-K divalent-monovalent exchange reaction:

CaX 2 + 2 K + ⇔ Ca 2+ + 2 K X R e a c t i o n 4
The constant associated with this exchange is:

{ } { } { } { } 2 2 2 2 / + + × × = K CaX KX Ca K K Ca E q . 1 7
The surface species activities are given by the following relationship

{ } CaK K K f N KX × = E q . 1 8
where N K is the molar fraction of the surface species KX (in this particular case

[ ] [ ] [ ] 2 CaX KX KX N K + =
), and is the activity coefficient of the surface species KX that depends on the exchanger composition. By introducing the Vanselow selectivity coefficient [START_REF] Vanselow | Equilibria of the base-exchange reaction of bentonites, permutites, Soil colloids and zeolites[END_REF], one derives

CaK K f CaK Ca CaK K K Ca v K Ca f f K K 2 / / × = . E q . 1 9
At constant T and P, and considering no variation in the surface water activity, the Gibbs-Duhem equation, applied to the exchanger phase only, gives (SPOSITO, 1989, see Electronic

∫ × + × - = × Ca E Ca K Ca v K Ca v Ca CaK K dE K K E f 0 ' / / ln ln ln 2 E q . 2 0 ∫ - × = K E K K Ca v K Ca v K CaK Ca dE K K E f 0 ' / / ln ln ln E q . 2 1 ∫ ∫ - = = 1 0 ' / 1 0 ' / / ln ln ln Ca K Ca v K K Ca v K Ca dE K dE K K E q . 2 2
Similar equations are obtained in the case of a homovalent exchange (e.g. Na/K or Ca/Mg, See Electronic Appendix 2)

Eqs. 20 to 22 can be evaluated by finding an expression that relates ln K v and the exchanger composition [START_REF] Sposito | The chemistry of soils[END_REF][START_REF] Agbenin | Potassium-calcium and potassium-magnesium exchange equilibria in an acid savanna soil from northern Nigeria[END_REF]. The curves obtained in this study for Na/Ca exchange (in the presence of a minor presence of K + on the exchanger) are very sharp near to the 0 and 1 values for E Ca or E Na respectively (Fig. 2). For practical reasons, it is then necessary to introduce into the equation a term that is easily integrated and that can account for the sharp shapes of the curves. It was found that exponential terms were far more efficient than high order polynomial terms because exponential terms do not introduce oscillations of the fitting curve in the central region of the curve, where E Ca or E Na stand between 0.3 and 0.6

K Ca v K / ln
By considering the following equation:

) 1 ( 3 3 2 2 1 0 / ln K CaK K CaK E c CaK E b CaK K CaK K CaK K CaK CaK K Ca v e C e B E a E a E a a K - - - × + × + + + + = E q . 2 3
and by applying it to other binary exchanges, it should be possible to fit reasonably well even very complicated shaped curves.

v K ln

One can then obtain:

( ) ( ) CaK CaK c CaK CaK b CaK CaK CaK CaK CaK CaK K Ca e c C e b B a a a a K - - - × + - × + + + + = 1 1 4 3 2 ln 3 2 1 0 / E q . 2 4 CaK c CaK CaK CaK CaK CaK CaK K E c CaK CaK CaK K E b K CaK K CaK K CaK CaK Ca c e C b B c C C E e b B B E e E a E a E a f CaK K CaK K CaK - - - - × + - ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - × × + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + × × + × + × + = = ) 1 ( 4 3 3 2 2 1 4 3 3 2 2 ln E q . 2 5 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + - - × × + + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - + - × × + - × × + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ × × + × + × × - × + × + × = - - - - CaK CaK Ca CaK E c CaK CaK CaK CaK Ca CaK E b b CaK CaK CaK Ca CaK CaK Ca CaK CaK CaK Ca CaK K c c E C e c C b b E B e e b B a E a a E a a a E f Ca CaK Ca CaK CaK 1 1 1 1 2 1 3 8 1 6 2 6 1 3 2 4 1 ln ) ( 1 3 4 3 2 3 3 2 1 2 E q . 2 6
The same type of equations are obtained for ln K Na/K , ln f Na and ln f K (see Electronic Appendix 2 for more details).

Given the above formula, the activity coefficient of surface species should be calculated on the basis of the K v data given as a function of the exchanger composition only. However, based on literature data, we have shown that the ionic strength has an influence on the selectivity constant value. This information partially invalidates the above approach. The effect of ionic strength is likely to originate in changes in the electrostatic field near to the surface, creating conditions favourable or not to the aggregation of particles (see discussion in the multi-site modelling approach section). Considering the role of ionic strength, Eq. 19 must be rewritten:

Ca K K Ca v K Ca g g K K 2 / / × = E q . 2 7
with g a function depending on the exchanger composition and the ionic strength. This means that all of the polynomial and exponential coefficients given in Eqs. 24 to 26 are a function of ionic strength. It is not now conceivable to define such a function for all of the necessary parameters. However, as a simplification, one can assume that:

h f g × =
where h is a function of ionic strength (I) only. Defining the reference state at I = 0, the h function should converge to 1 when I converges to 0. By analyzing data from Thellier and Sposito (THELLIER and SPOSITO, 1989b), one should also note that the logarithm of h function ratios converge to one other limit at I ~ 0.004-0.005, then describing a sigmoid curve, in correlation with the flocculation behaviour of illite suspension (THELLIER and SPOSITO, 1989a). They have also shown that this flocculation behaviour is a function of the bivalent cation exchanged fraction.

However, no attempt was made to calculate explicitly all h functions as a function of their relative cations. The logarithm of h function ratios were approached with a complementary error function (erfc) with the form of:

) ( ln ln NaK NaK NaK NaK Na NaK K I erfc h h γ β α - × = - , E q . 2 8
for monovalent / monovalent interactions or

) ( ln ln 2 CaK CaK CaK CaK Ca CaK K I erfc h h γ β α - × = - × . E q . 2 9
for monovalent / divalent interactions.

These functions must be considered as a very convenient way of trapping the ionic strength effect. The Na-K, Ca-Na, and Mg-Na systems were described with such functions leading to nine new fitting parameters and allowing other systems (Ca-Mg, Ca-K and Mg-K) to be described by combinations of the former ones.

Activity coefficient and exchange equilibrium constant calculations in multicomponent

exchanges. [START_REF] Chu | The thermodynamics of ternary cation exchange systems and the subregular model[END_REF] have demonstrated that ternary exchange thermodynamic parameters cannot be deduced from binary parameters only.

However, we decided voluntary to oversimplify our system given (i) that it already contains a lot of fitting parameters and (ii) that a one-site true thermodynamic description is certainly not possible due to the multi-site nature of the exchange process.

We assume here firstly that the activity coefficients for Na, K, Ca and Mg exchanged species in a multicomponent exchange can be approximated from the individual binary contributions without crossing terms, i.e.:

CaMg Ca CaNa Ca CaK Ca NaKMgCa Ca f f f f ln ln ln ln + + = E q . 3 0
However, in a Ca-Mg exchange experiment in the presence of Na and K, one can note that the ln K v Ca/K value depends only on E Na and E K , not on the ratio between E Mg and E Ca (Fig. 9).

This means that Ca and Mg have a joint similar effect on the Ca-K exchange selectivity coefficient. This effect is taken into account by considering the following combination:

CaMgK K NaK K CaMgNaK K f f f ln ln ln + = E q . 3 1
where is the weighted mean of and (see Electronic Appendix 2 for the full formula).

CaMgK K f ln CaNa K f ln CaMg K f ln
With this approach we then obtained four mathematical expressions for the activity coefficients of exchanged Na, K, Ca and Mg:

CaMgK K NaK K CaMgNaK K f f f ln ln ln + = E q . 3 2 CaMgNa Na NaK Na CaMgKNa Na f f f ln ln ln + = E q . 3 3 CaMg Ca CaNa Ca CaK Ca MgKNaCa Ca f f f f ln ln ln ln + + = E q . 3 4 CaMg Mg MgNa Mg MgK Mg CaKNaMg Mg f f f f ln ln ln ln + + = E q . 3 5
The efficiency of the model to capture the variations in the selectivity coefficients as a function of exchanger composition and ionic strength was then tested.

Modelling of experimental and literature results

Equilibrium constants and activity coefficients were obtained from the polynomial and exponential coefficients as described above. The fitting approach also takes into consideration one additional constraint concerning the exchange equilibrium constants consistency, which is the constant additivity, i.e. for a Ca/Na/K system:

ln K Ca/Na + 2 × ln K Na/K = ln K Ca/K E q . 3 6
The least square fitting approach of these parameters was broken down into several steps:

• Firstly, only data at TN = 0.005 mol/L were considered and the a, B, b, C and c parameters were fitted for Na, K and Ca, considering that the parameters for Mg were identical to Ca.

• Secondly, the Mg parameters were fitted together with the former parameters obtained for Na, K and Ca.

• Finally, the α, β and γ parameters, concerning ionic strength effects, were fitted for the system Ca-Na, Na-K and Mg-Na separately before a last fitting round concerning all of the parameters.

This approach was reiterated until the fit was found to be satisfactory. Note that the fitted systems are always multi-component (most of them are quaternary systems) and thereby highly non-linear. Reiteration of the fitting procedure was then necessary. The parameters obtained from this regression analysis are given in Tables 2 and3. Of course, we are aware that these coefficients are not definitive and certainly not the best solutions, given the high number of fitting parameters. Complementary experiments are needed to better constrain them, especially Ca-K and Mg-K binary exchanges at different ionic strengths and with data at high K surface coverage. The result of the fit is presented in Figs. 10 and 11 together with standard deviation of the simulation as compared to the experimental data.

The concentration prediction capability of the model was then tested using the same procedure as that used for the two-site model: the TN was kept at the experimental TN and the selectivity coefficients were calculated as a function of the normality and the exchanger composition. A Gaines and Thomas convention was used because the calculations were performed with PHREEQC v2.12. Gaines and Thomas selectivity coefficients were recalculated according to Eq. 11.

Results are shown in Fig. 12 [START_REF] Shainberg | Sodium/calcium exchange in montmorillonite and Illite suspension[END_REF][START_REF] Sposito | Sodium-calcium-magnesium exchange on Silver Hill Illite[END_REF]. The proposed one-site model thereby makes it possible to recover the concentrations of the major cations in a solution in contact with an illite whose exchanger composition is known.

Application to natural and engineered systems

Natural systems with a clay fraction constituted only by illite are not common. Here we consider the example of illite rich layers in the Callovian-Oxfordian formation for the calculation of exchanger selectivity coefficients towards Na, Ca, Mg and K. Data on exchanger composition and ionic strength were obtained from [START_REF] Gaucher | ANDRA underground research laboratory: Interpretation of the mineralogical and geochemical data acquired in the Callovian-Oxfordian Formation by investigative drilling[END_REF] and are given in Table 4. Selectivity coefficients in the Vanselow convention were calculated using the model described in this paper (Table 4). Associated concentrations in porewater can then be calculated, provided the total normality is known and the sulphate and chloride anion proportions are known. In agreement with Gaucher et al. (GAUCHER et al., Accepted), we considered a solute S(+6)/Cl(-1) equivalent ratio of 2 as an approximation. The results of the calculation (Table 4) shows that the porewater composition is likely to be constituted mainly of Na cations (~ 54% of the cation equivalent concentration) then Ca cations (~ 30%), Mg cations (~ 14%), K cations representing less than 2% of the cation equivalent charge with concentrations ranging from 1.4 to 2 mmol/L. Future porewater extraction results from the Bure underground research laboratory will help to verify the accuracy of the model.

Conclusions

We have shown that our one-site model considering the ionic strength effect and variations in the surface species activity coefficients is efficient for the determination of water major cation composition for a wide range of exchanger compositions. For natural systems, this model still needs to be refined in order to consider the presence of smectite and illite/smectite rather than pure illite. Moreover, the compaction effect deserves further study as a similar effect of compaction is expected as compared to that of ionic strength. It is obvious that this model will be the most efficient for illite exhibiting the same shape parameters, provided the proportions of the different types of sites are controlled by the morphology of the particles. As a matter of interest, the data fitted here concern only Silver Hill Illite (Imt-1 and 2, Montana) and Fithian Illite (Illinois).

For the prediction of the exchanger composition as a function of the solution cation concentrations, a multi-site model as presented in the first part of this paper is certainly more appropriate, even if some features of the exchange isotherm, e.g. the "bell-shaped" aspect of the curve, cannot be predicted. Ideally, we believe that the two approaches should be combined in order to take into account the peculiar behaviour of Cs or K cations on high affinity sites such as frayed edge sites.

Tables

Table 1. Two-site model parameters for the modelling of Na-K-Ca-Mg exchange experiments.

Data from the literature are taken from: [START_REF] Shainberg | Sodium/calcium exchange in montmorillonite and Illite suspension[END_REF][START_REF] Sposito | Sodium-calcium-magnesium exchange on Silver Hill Illite[END_REF][START_REF] Thellier | Quaternary cation exchange on Silver Hill illite[END_REF]THELLIER and SPOSITO, 1989b;[START_REF] Thellier | Proton effects on quaternary cation exchange and flocculation of Silver Hill illite[END_REF] [START_REF] Shainberg | Sodium/calcium exchange in montmorillonite and Illite suspension[END_REF][START_REF] Sposito | Sodium-calcium-magnesium exchange on Silver Hill Illite[END_REF][START_REF] Thellier | Quaternary cation exchange on Silver Hill illite[END_REF]THELLIER and SPOSITO, 1989b;[START_REF] Thellier | Proton effects on quaternary cation exchange and flocculation of Silver Hill illite[END_REF]. Symbols hide each other and therefore are not explained here in relation to the data source. Full lines represent the 1:1 relationship.

Figure captions

  selectivity coefficient value is needed to obtain reliable concentration values. The present work aims at obtaining these selectivity coefficients for Na, K, Ca and Mg on illite. Data collected from the literature is widely used in the following work together with new exchange data obtained in our laboratories. Data from the literature always originated from exchange experiments where both the solution and the exchanger composition were measured. New exchange data were obtained in the framework of the French Nuclear Waste Management Agency (ANDRA) investigation program on the clayey formation surrounding the French Underground Laboratory (URL) in Bure (Meuse Haute-Marne, France).

  also shows the variation in the selectivity coefficient as a function of the equivalent fraction of exchanged calcium in Na/Ca exchange experiments using various thermodynamic conventions: Vanselow, Gaines and Thomas, Gapon and Rothmund and Kornfeld (see SPOSITO, 1981 and Appelo C. A. J. and Parkhurst D. L. Calculating cation exchange with PHREEQC (Version 2), http://www.xs4all.nl/~appt/pub/).

  Llnl.dat associated database for solute species. When dealing with the literature data at TN ≥ 0.005 mol/L, the standard deviations on the difference between the experiment concentration values and the modelled concentration values in log 10 units are as follows:σ [Na+] = 0.32, σ [K+] = 1.31, σ [Ca2+] = 0.43 and σ [Mg2+] = 0.34. Surprisingly, this standard deviation is higher for K + concentration predictions than for the other cations despite the fact that the standard deviation on the difference between the measured and simulated equivalent fraction of the model was the lowest for K exchanger occupancy in the direct modelling. Most of this deviation originates from a "Ca-Mg" exchange experiment in THELLIER and SPOSITO, 1988, with a low K content. If this experiment is neglected, the standard deviation on the difference between the experimental concentration values and the modelled concentration values in log 10 units becomes σ [K+] = 0.63. When considering the total dataset from the literature, standard deviations are similar: σ [Na+] = 0.30, σ [K+] = 1.16 (0.61 without the Ca-Mg experiment cited above), σ [Ca2+] = 0.64 and σ [Mg2+] = 0.51.

  together with the standard deviation of the simulation as compared to the experimental data. Note that the K concentrations in the Na-Ca and Na-Mg exchange experiment of the present study have been discarded, as for the presentation of the two-site model result. On the contrary, results concerning Na, Ca and Mg concentrations of the present study are also presented. Significant deviations are observed mainly for Mg concentrations, but the precision of the concentration prediction is far better than the precision obtained with the two-site model. For the two-site model we obtained the following standard deviations on the difference between the experiment concentration values and the modelled concentration values in log 10 units σ [Na+] = 0.30, σ [K+] = 1.16, σ [Ca2+] = 0.64 and σ [Mg2+] = 0.51. With the present one-site model, one obtains: σ [Na+] = 0.13, σ [K+] = 0.10, σ [Ca2+] = 0.30 and σ [Mg2+] = 0.45, highlighting the improvement of the fit, especially for the potassium concentration prediction. Even points that lack data measurement for K + in solution and on the exchanger are well predicted (data from

Fig. 1 .Fig. 8 .

 18 Fig. 1. Vanselow selectivity coefficient of Na/K exchange as a function of the relative

Fig. 9 .

 9 Fig. 9. ln K v Ca/K values as a function of E Ca in a Ca-Mg exchange experiment. Circles and

Fig. 10 .

 10 Fig. 10. Comparison of measured and calculated ln K v values for Na/K, Na/Ca, Na/Mg, Ca/Mg, K/Ca and K/Mg exchange reactions at TN ≥ 0.005 mol/L. Circles and crosses:

  Fig.1

  Fig. 12

  the data points obtained in this study for the Na/Ca and Na/Mg experiments having the highest values and showing no clear trend. This behaviour could be due to several experimental reasons. The first is the possible effect of pH (the pH of the present experiment was 5 whereas the pH of the Thellier and Sposito experiment was 7), although Thellier et al.

  Table1can be used with good confidence for TN > 0.005 mol/L and for exchanger occupancy as follows: E Na ∈ [0.1; 0.7], E K ∈ [0.05; 1], E Ca ∈ [0.25; 0.9], E Mg ∈ [0.25; 0.9]. However, the ratio between the two types of site is needed. For natural systems, we think it likely that the values given in Table1for the literature data (0.93 and 0.07) should be used in the absence of other information. The values for the present study were obtained on the 2 µm fine fraction instead of the 115 µm fraction for most other studies, a fraction that is more representative of a natural illite. The proportion of K + high affinity sites in the present study (sites ≡Illb -= 7 -12%) can be compared with the proportion of Cs + high affinity sites in other literature studies: 0.045%(Hanford sediment, LIU et al., 2004), 0.4 + 3 ~ 3.4% (Illite,

Table 2 .

 2 . Ionic strength coefficients (Eqs. 28 and 29).

		≡Illa -	≡Illb -
	This study	Literature data	This study	Literature data
		Site proportions
	0.88	0.93	0.12	0.07
	Exchange selectivity coefficients
	Na-K	0.52	6.7
	Na-Ca	0.53	0.53
	Na-Mg	0.55	0.55

Table 3 .

 3 Polynomial and exponential coefficients obtained for Ca/K, Ca/Na, Na/K, Mg/Ca, Mg/Na and Mg/K exchange reactions, together with calculated thermodynamic exchange reaction equilibrium constants.

	Ca/K exchange
	Polynomial coefficients Exponential coefficients	ln	K	CaK
	0	a	CaK	1.43	CaK B	7.39
	CaK CaK a a 2 1	0.293 0.162	CaK b CaK C	1.055 -0.837	6.13
	3	a		CaK	0.222	c	CaK	7.078
	Ca/Na exchange
	Polynomial coefficients Exponential coefficients	ln	K	CaNa
	0	a	CaNa	1.34	CaNa B	254
	CaNa CaNa a a 2 1	-0.342 -0.816	CaNa b CaNa C	58.77 -0.026	6.08
	3	a	CaNa	3.542	c	CaNa	0.010
	Na/K exchange
	Polynomial coefficients Exponential coefficients	ln	K	NaK
	0	a	CaK	-0.83	CaK B	3.478
	CaK CaK a a 2 1	-1.964 10.08	CaK b CaK C	2.520 -211.4	0.025
	3	a		CaK	-3.57	c	CaK	111.6
	Mg/Ca exchange
	Polynomial coefficients Exponential coefficients	ln	K	MgCa
	0	a	NaK	0.40	B	NaK	0.575
	NaK NaK a a 2 1	0.000 0.000	NaK NaK C b	0.546 -23.98	0.593
	3	a		NaK	0.000	c	NaK	97.20
	Mg/Na exchange
	Polynomial coefficients Exponential coefficients	ln	K	MgNa
	0	a		MgNa	3.98	B	MgNa	257.14
	1 2	MgNa MgNa a a	-9.05 6.506	MgNa MgNa C b	55.11 -0.329	6.68
	3	a		MgNa	2.850	c	MgNa	0.009
	Mg/K exchange
	Polynomial coefficients Exponential coefficients	ln	K	MgK
	0	a		MgK	2.17	B	MgK	7.65
	1 2 a a MgK MgK	0.287 0.160	MgK MgK C b	1.234 -0.636	6.72
	3	a		MgK	0.220	c	MgK	7.044

Table 4 .

 4 Cation exchange composition and associated total normality (TN) of selected illitic samples[START_REF] Gaucher | ANDRA underground research laboratory: Interpretation of the mineralogical and geochemical data acquired in the Callovian-Oxfordian Formation by investigative drilling[END_REF].

	Litho-	Sample	Core levels (m)	Cation exchange occupancy (eq/kg)	TN (mmol/L)
	facies	name
				Ca	Mg	K	Na
	C2b1	EST05703	493.47 493.67	0.065	0.033	0.012	0.024	85
		EST05712	495.50 495.70	0.061	0.032	0.012	0.024	93
		EST05724	498.47 498.67	0.066	0.034	0.013	0.027	80
		EST05738	501.50 501.70	0.066	0.034	0.012	0.025	101
		EST05748	503.93 504.13	0.064	0.033	0.013	0.025	136

Table 5 .

 5 Decimal logarithm of cation exchange selectivity coefficients for selected samples (Table4) in Gaines and Thomas convention associated with calculated concentrations of Na, K, Ca and Mg in the porewater of the samples.

				Gaines and Thomas exchange
	Litho-	Sample	TN					Calculated concentrations
					selectivity coefficients
	facies	name	(mmol/L)						(mmol/L)
					(log 10 scale)
				Na/Ca	Na/Mg	Na/K	Na	K	Ca	Mg
	C2b1	EST05703	85	0.67	0.78	1.23	45.7	1.4	13.0	6.0
		EST05712	93	0.67	0.78	1.23	49.9	1.5	14.1	6.7
		EST05724	80	0.68	0.79	1.22	46.5	1.4	10.9	5.1
		EST05738	101	0.67	0.78	1.23	51.8	1.5	16.2	7.6
		EST05748	136	0.67	0.78	1.23	63.4	2.0	24	11.3

Acknowledgements

This research was funded by the French National Radioactive Waste Management Agency (ANDRA) and the French Geological Survey (BRGM) in the framework of the BRGM-ANDRA scientific partnership (THERMOAR project under the coordination of Dr. E. Jacquot and Dr. E. Gaucher). H.G. acknowledges ANDRA for her Ph.D. financial support, under the supervision of Dr. N. Michau. Dr. C.A.J. Appelo, two anonymous reviewers and Pr. Sposito, Associate Editor, are gratefully acknowledged for their constructive comments in reviewing this paper.

Fig. 5 ---

Dispersed suspension

Illite