Matthieu Constant
email: mconstan@univ-mlv.fr

Anthony Sigogne
email: sigogne@univ-mlv.fr

Patrick Watrin
email: patrick.watrin@uclouvain.be

Discriminative Strategies to Integrate Multiword Expression Recognition and Parsing

The integration of multiword expressions in a parsing procedure has been shown to improve accuracy in an artificial context where such expressions have been perfectly pre-identified. This paper evaluates two empirical strategies to integrate multiword units in a real constituency parsing context and shows that the results are not as promising as has sometimes been suggested. Firstly, we show that pregrouping multiword expressions before parsing with a state-of-the-art recognizer improves multiword recognition accuracy and unlabeled attachment score. However, it has no statistically significant impact in terms of F-score as incorrect multiword expression recognition has important side effects on parsing. Secondly, integrating multiword expressions in the parser grammar followed by a reranker specific to such expressions slightly improves all evaluation metrics.

Introduction

The integration of Multiword Expressions (MWE) in real-life applications is crucial because such expressions have the particularity of having a certain level of idiomaticity. They form complex lexical units which, if they are considered, should significantly help parsing.

From a theoretical point of view, the integration of multiword expressions in the parsing procedure has been studied for different formalisms: Head-Driven Phrase Structure Grammar [START_REF] Copestake | Multiword Expressions: Linguistic Precision and Reusability[END_REF], Tree Adjoining Grammars [START_REF] Schuler | Tree-rewriting models of multi-word expressions[END_REF], etc. From an empirical point of view, their incorporation has also been considered such as in [START_REF] Nivre | Multiword units in syntactic parsing[END_REF] for dependency parsing and in [START_REF] Arun | Lexicalization in crosslinguistic probabilistic parsing: The case of French[END_REF] in constituency parsing. Although experiments always relied on a corpus where the MWEs were perfectly pre-identified, they showed that pre-grouping such expressions could significantly improve parsing accuracy. Recently, [START_REF] Green | Multiword Expression Identification with Tree Substitution Grammars: A Parsing tour de force with French[END_REF] proposed integrating the multiword expressions directly in the grammar without pre-recognizing them. The grammar was trained with a reference treebank where MWEs were annotated with a specific non-terminal node.

Our proposal is to evaluate two discriminative strategies in a real constituency parsing context: (a) pre-grouping MWE before parsing; this would be done with a state-of-the-art recognizer based on Conditional Random Fields; (b) parsing with a grammar including MWE identification and then reranking the output parses thanks to a Maximum Entropy model integrating MWE-dedicated features. (a) is the direct realistic implementation of the standard approach that was shown to reach the best results [START_REF] Arun | Lexicalization in crosslinguistic probabilistic parsing: The case of French[END_REF]. We will evaluate if real MWE recognition (MWER) still positively impacts parsing, i.e., whether incorrect MWER does not negatively impact the overall parsing system. (b) is a more innovative approach to MWER (despite not being new in parsing): we select the final MWE segmentation after parsing in order to explore as many parses as possible (as opposed to method (a)). The experiments were carried out on the French Treebank [START_REF] Abeillé | Building a treebank for French[END_REF] where MWEs are annotated.

The paper is organized as follows: section 2 is an overview of the multiword expressions and their identification in texts; section 3 presents the two different strategies and their associated models; section 4 describes the resources used for our experiments (the corpus and the lexical resources); section 5 details the features that are incorporated in the models; section 6 reports on the results obtained.

Multiword expressions 2.1 Overview

Multiword expressions are lexical items made up of multiple lexemes that undergo idiosyncratic constraints and therefore offer a certain degree of idiomaticity. They cover a wide range of linguistic phenomena: fixed and semi-fixed expressions, light verb constructions, phrasal verbs, named entities, etc. They may be contiguous (e.g. traffic light) or discontinuous (e.g. John took your argument into account). They are often divided into two main classes: multiword expressions defined through linguistic idiomaticity criteria (lexicalized phrases in the terminology of [START_REF] Sag | Multiword Expressions: A Pain in the Neck for NLP[END_REF]) and those defined by statistical ones (i.e. simple collocations). Most linguistic criteria used to determine whether a combination of words is a MWE are based on syntactic and semantic tests such as the ones described in [START_REF] Gross | Lexicon Grammar. The Representation of Compound Words[END_REF]. For instance, the utterance at night is a MWE because it does display a strict lexical restriction (*at day, *at afternoon) and it does not accept any inserting material (*at cold night, *at present night). Such linguistically defined expressions may overlap with collocations which are the combinations of two or more words that cooccur more often than by chance. Collocations are usually identified through statistical association measures. A detailed description of MWEs can be found in [START_REF] Baldwin | Multiword Expressions. Handbook of Natural Language Processing[END_REF].

In this paper, we focus on contiguous MWEs that form a lexical unit which can be marked by a part-ofspeech tag (e.g. at night is an adverb, because of is a preposition). They can undergo limited morphological and lexical variations -e.g. traffic (light+lights), (apple+orange+...) juice -and usually do not allow syntactic variations1 such as inserts (e.g. *at cold night). Such expressions can be analyzed at the lexical level. In what follows, we use the term compounds to denote such expressions.

Identification

The idiomaticity property of MWEs makes them both crucial for Natural Language Processing applications and difficult to predict. Their actual identification in texts is therefore fundamental. There are different ways for achieving this objective. The simpler approach is lexicon-driven and consists in looking the MWEs up in an existing lexicon, such as in [START_REF] Silberztein | INTEX: an FST toolbox[END_REF]. The main drawback is that this procedure entirely relies on a lexicon and is unable to discover unknown MWEs. The use of collocation statistics is therefore useful. For instance, for each candidate in the text, [START_REF] Watrin | N-gram frequency database reference to handle MWE extraction in NLP applications[END_REF] compute on the fly its association score from an external ngram base learnt from a large raw corpus, and tag it as MWE if the association score is greater than a threshold. They reach excellent scores in the framework of a keyword extraction task. Within a validation framework (i.e. with the use of a reference corpus annotated in MWEs), [START_REF] Ramisch | mwetoolkit: a framework for multiword expression identification[END_REF] developped a Support Vector Machine classifier integrating features corresponding to different collocation association measures. The results were rather low on the Genia corpus and [START_REF] Green | Multiword Expression Identification with Tree Substitution Grammars: A Parsing tour de force with French[END_REF] confirmed these bad results on the French Treebank. This can be explained by the fact that such a method does not make any distinctions between the different types of MWEs and the reference corpora are usually limited to certain types of MWEs. Furthermore, the lexicon-driven and collocation-driven approaches do not take the context into account, and therefore cannot discard some of the incorrect candidates. A recent trend is to couple MWE recognition with a linguistic analyzer: a POS tagger [START_REF] Constant | MWU-aware Partof-Speech Tagging with a CRF model and lexical resources[END_REF] or a parser [START_REF] Green | Multiword Expression Identification with Tree Substitution Grammars: A Parsing tour de force with French[END_REF]. Constant and Sigogne (2011) trained a unified Conditional Random Fields model integrating different standard tagging features and features based on external lexical resources. They show a general tagging accuracy of 94% on the French Treebank. In terms of Multiword expression recognition, the accuracy was not term.

clearly evaluated, but seemed to reach around 70-80% F-score. [START_REF] Green | Multiword Expression Identification with Tree Substitution Grammars: A Parsing tour de force with French[END_REF] proposed to include the MWER in the grammar of the parser. To do so, the MWEs in the training treebank were annotated with specific non-terminal nodes. They used a Tree Substitution Grammar instead of a Probabilistic Context-free Grammar (PCFG) with latent annotations in order to capture lexicalized rules as well as general rules. They showed that this formalism was more relevant to MWER than PCFG (71% Fscore vs. 69.5%). Both methods have the advantage of being able to discover new MWEs on the basis of lexical and syntactic contexts. In this paper, we will take advantage of the methods described in this section by integrating them as features of a MWER model.

3 Two strategies, two discriminative models

Pre-grouping Multiword Expressions

MWER can be seen as a sequence labelling task (like chunking) by using an IOB-like annotation scheme [START_REF] Ramshaw | Text chunking using transformation-based learning[END_REF]. This implies a theoretical limitation: recognized MWEs must be contiguous. The proposed annotation scheme is therefore theoretically weaker than the one proposed by [START_REF] Green | Multiword Expression Identification with Tree Substitution Grammars: A Parsing tour de force with French[END_REF] that integrates the MWER in the grammar and allows for discontinuous MWEs. Nevertheless, in practice, the compounds we are dealing with are very rarely discontinuous and if so, they solely contain a single word insert that can be easily integrated in the MWE sequence. [START_REF] Constant | MWU-aware Partof-Speech Tagging with a CRF model and lexical resources[END_REF] proposed to combine MWE segmentation and part-of-speech tagging into a single sequence labelling task by assigning to each token a tag of the form TAG+X where TAG is the part-ofspeech (POS) of the lexical unit the token belongs to and X is either B (i.e. the token is at the beginning of the lexical unit) or I (i.e. for the remaining positions): John/N+B hates/V+B traffic/N+B jams/N+I. In this paper, as our task consists in jointly locating and tagging MWEs, we limited the POS tagging to MWEs only (TAG+B/TAG+I), simple words being tagged by O (outside): John/O hates/O traffic/N+B jams/N+I. For such a task, we used Linear chain Conditional Ramdom Fields (CRF) that are discriminative prob-abilistic models introduced by [START_REF] Lafferty | Conditional random Fields: Probabilistic models for segmenting and labeling sequence data[END_REF] for sequential labelling. Given an input sequence of tokens x = (x 1 , x 2 , ..., x N ) and an output sequence of labels y = (y 1 , y 2 , ..., y N ), the model is defined as follows:

P (y|x) = 1 Z(x) N t=1 exp K k=1 λ k f k (t, x, y t , y t-1 )
where Z(x) is a normalization factor depending on x. It is based on K features each of them being defined by a binary function f k depending on the current position t in x, the current label y t , the preceding one y t-1 and the whole input sequence x. The tokens x i of x integrate the lexical value of this token but can also integrate basic properties which are computable from this value (for example: whether it begins with an upper case, it contains a number, its tags in an external lexicon, etc.). The feature is activated if a given configuration between t, y t , y t-1 and x is satisfied (i.e. f k (t, y t , y t-1 , x) = 1). Each feature f k is associated with a weight λ k . The weights are the parameters of the model, to be estimated. The features used for MWER will be described in section 5.

Reranking

Discriminative reranking consists in reranking the nbest parses of a baseline parser with a discriminative model, hence integrating features associated with each node of the candidate parses. [START_REF] Charniak | Coarse-to-Fine n-Best Parsing and MaxEnt Discriminative Reranking[END_REF] introduced different features that showed significant improvement in general parsing accuracy (e.g. around +1 point in English). Formally, given a sentence s, the reranker selects the best candidate parse p among a set of candidates P (s) with respect to a scoring function V θ :

p * = argmax p∈P (s) V θ (p)
The set of candidates P (s) corresponds to the n-best parses generated by the baseline parser. The scoring function V θ is the scalar product of a parameter vector θ and a feature vector f :

V θ (p) = θ.f (p) = m j=1 θ j .f j (p)
where f j (p) corresponds to the number of occurrences of the feature f j in the parse p. According to [START_REF] Charniak | Coarse-to-Fine n-Best Parsing and MaxEnt Discriminative Reranking[END_REF], the first feature f 1 is the probability of p provided by the baseline parser. The vector θ is estimated during the training stage from a reference treebank and the baseline parser ouputs.

In this paper, we slightly deviate from the original reranker usage, by focusing on improving MWER in the context of parsing. Given the n-best parses, we want to select the one with the best MWE segmentation by keeping the overall parsing accuracy as high as possible. We therefore used MWE-dedicated features that we describe in section 5. The training stage was performed by using a Maximum entropy algorithm as in [START_REF] Charniak | Coarse-to-Fine n-Best Parsing and MaxEnt Discriminative Reranking[END_REF].

Resources 4.1 Corpus

The French Treebank2 [FTB] [START_REF] Abeillé | Building a treebank for French[END_REF] is a syntactically annotated corpus made up of journalistic articles from Le Monde newspaper. We used the latest edition of the corpus (June 2010) that we preprocessed with the Stanford Parser preprocessing tools [START_REF] Green | Multiword Expression Identification with Tree Substitution Grammars: A Parsing tour de force with French[END_REF]. It contains 473,904 tokens and 15,917 sentences. One benefit of this corpus is that its compounds are marked. Their annotation was driven by linguistic criteria such as the ones in [START_REF] Gross | Lexicon Grammar. The Representation of Compound Words[END_REF]. Compounds are identified with a specific non-terminal symbol "MWX" where X is the part-of-speech of the expression. They have a flat structure made of the part-of-speech of their components as shown in figure 1. The French Treebank is composed of 435,860 lexical units (34,178 types). Among them, 5.3% are compounds (20.8% for types). In addition, 12.9% of the tokens belong to a MWE, which, on average, has 2.7 tokens. The non-terminal tagset is composed of 14 part-of-speech labels and 24 phrasal ones (including 11 MWE labels). The train/dev/test split is the same as in [START_REF] Green | Multiword Expression Identification with Tree Substitution Grammars: A Parsing tour de force with French[END_REF]: 1,235 sentences for test, 1,235 for development and 13,347 for training. The development and test sections are the same as those generally used for experiments in French, e.g. [START_REF] Candito | Improving generative statistical parsing with semi-supervised word clustering[END_REF].

Lexical resources

French is a resource-rich language as attested by the existing morphological dictionaries which include compounds. In this paper, we use two largecoverage general-purpose dictionaries: Dela [START_REF] Courtois | Un système de dictionnaires électroniques pour les mots simples du franc ¸ais[END_REF][START_REF] Courtois | Dictionnaire électronique DELAC : les mots composés binaires[END_REF] and Lefff [START_REF] Sagot | The Lefff, a freely available, accurate and large-coverage lexicon for French[END_REF]. The Dela was manually developed in the 90's by a team of linguists. We used the distribution freely available in the platform Unitex3 [START_REF] Paumier | Unitex 3.9 documentation[END_REF]. It is composed of 840,813 lexical entries including 104,350 multiword ones (91,030 multiword nouns). The compounds present in the resources respect the linguistic criteria defined in [START_REF] Gross | Lexicon Grammar. The Representation of Compound Words[END_REF]. The lefff is a freely available dictionary4 that has been automatically compiled by drawing from different sources and that has been manually validated. We used a version with 553,138 lexical entries including 26,311 multiword ones (22,673 multiword nouns). Their different modes of acquisition makes those two resources complementary. In both, lexical entries are composed of a inflected form, a lemma, a part-of-speech and morphological features. The Dela has an additional feature for most of the multiword entries: their syntactic surface form. For instance, eau de vie (brandy) has the feature NDN because it has the internal flat structure noun -preposition de -noun.

In order to compare compounds in these lexical resources with the ones in the French Treebank, we applied on the development corpus the dictionaries and the lexicon extracted from the training corpus. By a simple look-up, we obtained a preliminary lexicon-based MWE segmentation. The results are provided in table 1. They show that the use of external resources may improve recall, but they lead to a decrease in precision as numerous MWEs in the dictionaries are not encoded as such in the reference corpus; in addition, the FTB suffers from some inconsistency in the MWE annotations. In terms of statistical collocations, Watrin and Franc ¸ois (2011) described a system that lists all the potential nominal collocations of a given sentence along with their association measure. The authors provided us with a list of 17,315 candidate nominal collocations occurring in the French treebank with their log-likelihood and their internal flat structure.

MWE-dedicated Features

The two discriminative models described in section 3 require MWE-dedicated features. In order to make these models comparable, we use two comparable sets of feature templates: one adapted to sequence labelling (CRF-based MWER) and the other one adapted to reranking (MaxEnt-based reranker). The MWER templates are instantiated at each position of the input sequence. The reranker templates are instantiated only for the nodes of the candidate parse tree, which are leaves dominated by a MWE node (i.e. the node has a MWE ancestor). We define a template T as follows:

• MWER: for each position n in the input sequence x, T = f (x, n)/y n

• RERANKER: for each leaf (in position n) dominated by a MWE node m in the current parse tree p,

T = f (p, n)/label(m)/pos(p, n)
where f is a function to be defined; y n is the output label at position n; label(m) is the label of node m and pos(p, n) indicates the position of the word corresponding to n in the MWE sequence: B (starting position), I (remaining positions).

Endogenous Features

Endogenous features are features directly extracted from properties of the words themselves or from a tool learnt from the training corpus (e.g. a tagger). Word n-grams. We use word unigrams and bigrams in order to capture multiwords present in the training section and to extract lexical cues to discover new MWEs. For instance, the bigram coup de is often the prefix of compounds such as coup de pied (kick), coup de foudre (love at first sight), coup de main (help). POS n-grams. We use part-of-speech unigrams and bigrams in order to capture MWEs with irregular syntactic structures that might indicate the idiomacity of a word sequence. For instance, the POS sequence preposition -adverb associated with the compound depuis peu (recently) is very unusual in French. We also integrated mixed bigrams made up of a word and a part-of-speech. Specific features. Due to their different use, each model integrates some specific features. In order to deal with unknown words and special tokens, we incorporate standard tagging features in the CRF: lowercase forms of the words, word prefixes of length 1 to 4, word suffice of length 1 to 4, whether the word is capitalized, whether the token has a digit, whether it is an hyphen. We also add label bigrams. The reranker models integrate features associated with each MWE node, the value of which is the compound itself.

Exogenous Features

Exogenous features are features that are not entirely derived from the (reference) corpus itself. They are computed from external data (in our case, our lexical resources). The lexical resources might be useful to discover new expressions: usually, expressions that have standard syntax like nominal compounds and are difficult to predict from the endogenous features. The resources are applied to the corpus through a lexical analysis that generates, for each sentence, a finite-state automaton TFSA which represents all the possible analyses. The features are computed from the automaton TFSA. Lexicon-based features. We associate each word with its part-of-speech tags found in our external morphological lexicon. All tags of a word constitute an ambiguity class ac. If the word belongs to a compound, the compound tag is also incorporated in the ambiguity class. For instance, the word night (either a simple noun or a simple adjective) in the context at night, is associated with the class adj noun adv+I as it is located inside a compound adverb. This feature is directly computed from TFSA. The lexical analysis can lead to a preliminary MWE segmentation by using a shortest path algorithm that gives priority to compound analyses. This segmentation is also a source of features: a word belonging to a compound segment is assigned different properties such as the segment part-of-speech mwt and its syntactic structure mws encoded in the lexical resource, its relative position mwpos in the segment ('B' or 'I'). Collocation-based features. In our collocation resource, each candidate collocation of the French treebank is associated with its internal syntactic structure and its association score (log-likelihood). We divided these candidates into two classes: those whose score is greater than a threshold and the other ones. Therefore, a given word in the corpus can be associated with different properties whether it belongs to a potential collocation: the class c and the internal structure cs of the collocation it belongs to, its position cpos in the collocation (B: beginning; I: remaining positions; O: outside). We manually set the threshold to 150 after some tuning on the development corpus.

All feature templates are given in table 2.

Endogenous Features w(n + i), i ∈ {-2, -1, 0, 1, 2} w(n + i)/w(n + i + 1), i ∈ {-2, -1, 0, 1} t(n + i), i ∈ {-2, -1, 0, 1, 2} t(n + i)/t(n + i + 1), i ∈ {-2, -1, 0, 1} w(n + i)/t(n + j), (i, j) ∈ {(1, 0), (0, 1), (-1, 0), (0, -1)} Exogenous Features ac(n) mwt(n)/mwpos(n) mws(n)/mwpos(n) c(n)/cs(n)/cpos(n)
Table 2: Feature templates (f ) used both in the MWER and the reranker models: n is the current position in the sentence, w(i) is the word at position i; t(i) is the partof-speech tag of w(i); if the word at absolute position i is part of a compound in the Shortest Path Segmentation, mwt(i) and mws(i) are respectively the part-of-speech tag and the internal structure of the compound, mwpos(i) indicates its relative position in the compound (B or I).

6 Evaluation

Experiment Setup

We carried out 3 different experiments. We first tested a standalone MWE recognizer based on CRF.

We then combined MWE pregrouping based on this recognizer and the Berkeley parser 5 [START_REF] Petrov | Learning accurate, compact and interpretable tree annotation[END_REF] trained on the FTB where the compounds were concatenated (BKYc). Finally, we combined the Berkeley parser trained on the FTB where the compounds are annotated with specific non-terminals (BKY), and the reranker. In all experiments, we varied the set of features: endo are all endogenous features; coll and lex include all endogenous features plus collocation-based features and lexicon-based ones, respectively; all is composed of both endogenous and exogenous features. The CRF recognizer relies on the software Wapiti 6 [START_REF] Lavergne | Practical Very Large Scale CRFs[END_REF] to train and apply the model, and on the software Unitex [START_REF] Paumier | Unitex 3.9 documentation[END_REF] to apply lexical resources. The part-of-speech tagger used to extract POS features was lgtagger 7 [START_REF] Constant | MWU-aware Partof-Speech Tagging with a CRF model and lexical resources[END_REF]. To train the reranker, we used a MaxEnt algorithm 8 as in [START_REF] Charniak | Coarse-to-Fine n-Best Parsing and MaxEnt Discriminative Reranking[END_REF]. Results are reported using several standard measures, the F 1 score, unlabeled attachment and Leaf Ancestor scores. The labeled F 1 score [F1] 9 , defined by the standard protocol called PARSEVAL [START_REF] Black | A procedure for quantitatively comparing the syntactic coverage of English grammars[END_REF], takes into account the bracketing and labeling of nodes. The unlabeled attachement score [UAS] evaluates the quality of unlabeled 5 We used the version adapted to French in the software Bonsai [START_REF] Candito | Improving generative statistical parsing with semi-supervised word clustering[END_REF] We trained the parser as follows: right binarization, no parent annotation, six split-merge cycles and default random seed initialisation (8). 6 Wapiti can be found at http://wapiti.limsi.fr/. It was configured as follows: rprop algorithm, default L1-penalty value (0.5), default L2-penalty value (0.00001), default stopping criterion value (0.02%).

7 Available at http://igm.univmlv.fr/˜mconstan/research/software/.

8 We used the following mathematical libraries PETSc et TAO, freely available at http://www.mcs.anl.gov/petsc/ and http://www.mcs.anl.gov/research/projects/tao/ 9 Evalb tool available at http://nlp.cs.nyu.edu/evalb/. We also used the evaluation by category implemented in the class EvalbByCat in the Stanford Parser.

dependencies between words of the sentence 10 . And finally, the Leaf-Ancestor score [LA] 11 [START_REF] Sampson | A test of the leafancestor metric for parsing accuracy[END_REF] computes the similarity between all paths (sequence of nodes) from each terminal node to the root node of the tree. The global score of a generated parse is equal to the average score of all terminal nodes. Punctuation tokens are ignored in all metrics. The quality of MWE identification was evaluated by computing the F 1 score on MWE nodes. We also evaluated the MWE segmentation by using the unlabeled F 1 score (U). In order to compare both approaches, parse trees generated by BKYc were automatically transformed in trees with the same MWE annotation scheme as the trees generated by BKY.

In order to establish the statistical significance of results between two parsing experiments in terms of F 1 and UAS, we used a unidirectional t-test for two independent samples 12 . The statistical significance between two MWE identification experiments was established by using the McNemar-s test [START_REF] Gillick | Some statistical issues in the comparison of speech recognition algorithms[END_REF]. The results of the two experiments are considered statistically significant with the computed value p < 0.01.

Standalone Multiword recognition

The results of the standalone MWE recognizer are given in table 3. They show that the lexicon-based system (lex) reaches the best score. Accuracy is improved by an absolute gain of +6.7 points as compared with BKY parser. The strictly endogenous system has a +4.9 point absolute gain, +5.4 points when collocations are added. That shows that most of the work is done by fully automatically acquired features (as opposed to features coming from a manually constructed lexicon). As expected, lexiconbased features lead to a 5.3 point recall improvement (with respect to non-lexicon based features) whereas precision is stable. [START_REF] Gillick | Some statistical issues in the comparison of speech recognition algorithms[END_REF], except lex/all and all/coll; lex/coll is "border-line". The results of the systems based on the Stanford Parser and the Tree Substitution Parser (DP-TSG) are reported from [START_REF] Green | Multiword Expression Identification with Tree Substitution Grammars: A Parsing tour de force with French[END_REF].

Combination of Multiword Expression Recognition and Parsing

We tested and compared the two proposed discriminative strategies by varying the sets of MWEdedicated features. The results are reported in table 4. Firstly, we note that the accuracy of the best realistic parsers is much lower than that of a parser with a golden MWE segmentation13 (-2.65 and -5.92 respectively in terms of F-score and UAS), which shows the importance of not neglecting MWE recognition in the framework of parsing. Furthermore, pre-grouping has no statistically significant impact on the F-score14 , whereas reranking leads to a statistically significant improvement (except for collocations). Both strategies also lead to a statistically significant UAS increase. Whereas both strategies improve the MWE recognition, pre-grouping is much more accurate (+2-4%); this might be due to the fact that an unlexicalized parser is limited in terms of compound identification, even within nbest analyses (cf. Oracle in table 6). The benefits of lexicon-based features are confirmed in this experiment, whereas the use of collocations in the reranking strategy seems to be rejected. Table 7 shows the results by category. It indicates that both discriminative strategies are of interest in locating multiword adjectives, determiners and prepositions; the pre-grouping method appears to be particularly relevant for multiword nouns and adverbs. However, it performs very poorly in multiword verb recognition. In terms of standard parsing accuracy, the pre-grouping approach has a very heterogeneous impact: Adverbial and Adjective Modifier phrases tend to be more accurate; verbal kernels and higher level constituents such as relative and subordinate clauses see their accuracy level drop, which shows that pre-recognition of MWE can have a negative impact on general parsing accuracy as MWE errors propagate to higher level constituents.

Conclusions and Future Work

In this paper, we evaluated two discriminative strategies to integrate Multiword Expression Recognition in probabilistic parsing: (a) pre-grouping MWEs with a state-of-the-art recognizer and (b) MWE identification with a reranker after parsing. We showed that MWE pre-grouping significantly improves compound recognition and unlabeled dependency annotation, which implies that this strategy could be useful for dependency parsing. The reranking procedure evenly improves all evaluation scores. Future work could consist in combining both strategies: pre-grouping could suggest a set of potential MWE segmentations in order to make it more flexible for a parser; final decisions would then be made by the reranker.

Figure 1 :

 1 Figure 1: Subtree of MWE part de marché (market share): The MWN node indicates that it is a multiword noun; it has a flat internal structure N P N (noun -preprosition -noun)

Table 1 :

 1 Simple context-free application of the lexical resources on the development corpus: T is the MWE lexicon of the training corpus, L is the lefff, D is the Dela. The given scores solely evaluate MWE segmentation and not tagging.

		T	L	D	T+L	T+D	T+L+D
	recall	75.9	31.7	59.0	77.3	83.4	84.0
	precision	61.2	52.0	55.6	58.7	51.2	49.9
	f-score	67.8	39.4	57.2	66.8	63.4	62.6

  : http://alpage.inria.fr/statgram/frdep/fr stat dep parsing.html.

	The	original	version	is	available	at:
	http://code.google.com/p/berkeleyparser/.

Table 3 :

 3 The more precise system is the base one because it almost solely detects compounds present in the training corpus; nevertheless, it is unable to capture new MWEs (it has the 10 This score is computed by using the tool available at http://ilk.uvt.nl/conll/software.html. The constituent trees are automatically converted into dependency trees with the tool Bonsai. /www.cis.upenn.edu/˜dbikel/software.html. lowest recall). BKY parser has the best recall among the non lexicon-based systems, i.e. it is the best one to discover new compounds as it is able to precisely detect irregular syntactic structures that are likely to be MWEs. Nevertheless, as it does not have a lexicalized strategy, it is not able to filter out incorrect candidates; the precision is therefore very low (the worst). MWE identification with CRF: base are the features corresponding to token properties and word ngrams. The differences between all systems are statistically significant with respect to McNemar's test

	11 Leaf-ancestor	assessment	tool	available	at
	http://www.grsampson.net/Resources.html
	12 Dan	Bikel's	tool		available	at
	http:/

Table 4 :

 4 Table 5 compares the parsing systems, byshowing the score differences between each of the tested system and the BKY parser. Parsing evaluation: pre indicates a MWE pregrouping strategy, whereas post is a reranking strategy with n = 50. The feature gold means that we have applied the parser on a gold MWE segmentation.

	Strat. Feat. Parser	F 1	LA	UAS	F 1 (MWE)
	-	-	BKY	80.61	92.91	82.99	71.1
	pre	-	BKYc 75.47	91.10	76.74	0.0
	pre	endo	BKYc 80.23	92.69	83.62	74.9
	pre	coll	BKYc 80.32	92.73	83.77	75.5
	pre	lex	BKYc 80.66	92.81	84.16	77.4
	pre	all	BKYc 80.51	92.77	84.05	77.2
	post	endo	BKY	80.87	92.94	83.49	72.9
	post	coll	BKY	80.71	92.85	83.16	71.2
	post	lex	BKY	81.08	92.98	83.98	74.5
	post	all	BKY	81.03	92.96	83.97	74.3
	pre	gold	BKYc 83.73	93.77	90.08	95.8

Table 5 :

 5 Comparison of the strategies with respect to BKY parser.

Table 6 :

 6 Reranker F 1 evaluation with respect to n and the types of features. The F 1 (MWE) is given in parenthesis.

Table 7 :

 7 Evaluation by category with respect to BKY parser. The BKY column indicates the F 1 of BKY parser.

	cat	#gold BKY	endo	lex	endo	lex
				(pre)	(pre)	(post)	(post)
	MWET	4	0.0	N/A	N/A	N/A	N/A
	MWA	22	37.2	+15.2	+21.3	+0.9	+4.7
	MWV	47	62.1	-9.7	-13.2	+1.7	+2.5
	MWD	24	62.1	+7.3	+10.2	0.0	+1.2
	MWN	860	68.2	+4.0	+7.0	+1.7	+4.2
	MWADV	357	72.1	+3.8	+6.4	+3.4	+4.1
	MWPRO	31	84.2	-3.5	-0.9	0.0	0.0
	MWP	294	79.1	+4.3	+5.8	+0.4	+1.1
	MWC	86	85.7	+0.9	+3.7	+0.2	+1.0
	Sint	209	47.2	-7.7	-8.7	+0.1	-0.2
	AdP	86	48.8	+1.2	+3.0	+3.4	+5.1
	Ssub	406	60.8	-1.1	-1.1	-0.3	-0.5
	VPpart	541	63.2	-2.8	-2.1	-0.5	-1.6
	Srel	408	74.8	-3.4	-3.5	-0.3	-0.6
	VPinf	781	75.2	0.0	-0.1	-0.3	-0.3
	COORD	904	75.2	+0.2	+0.4	-0.3	-0.4
	PP	4906	76.7	-0.8	-0.3	+0.5	+0.7
	AP	1482	74.5	+3.2	+3.9	+0.7	+1.6
	NP	9023	79.8	-1.1	-0.8	+0.1	+0.2
	VN	3089	94.0	-2.0	-1.0	0.0	0.0

Such MWEs may very rarely accept inserts, often limited to single word modifiers: e.g. in the short term, in the very short

http://www.llf.cnrs.fr/Gens/Abeille/French-Treebankfr.php

http://igm.univ-mlv.fr/˜unitex

http://atoll.inria.fr/˜sagot/lefff.html

The F1(MWE) is not 100% with a golden segmentation because of tagging errors by the parser.

14 Note that we observe an increase of +0.5 in F1 on the development corpus with lexicon-based features.

Acknowlegments

The authors are very grateful to Spence Green for his useful help on the treebank, and to Jennifer Thewissen for her careful proof-reading.


Matthieu Constant
email: mconstan@univ-mlv.fr

Isabelle Tellier
email: isabelle.tellier@univ-paris3.fr

Evaluating the Impact of External Lexical Resources into a CRF-based Multiword Segmenter and Part-of-Speech Tagger

Keywords: Multiword Expressions & Collocations, Part Of Speech tagging, Statistical and machine learning methods, Conditional Random Fields, Morphosyntactic lexicons, lexical segmentation

This paper evaluates the impact of external lexical resources into a CRF-based joint Multiword Segmenter and Part-of-Speech Tagger. We especially show different ways of integrating lexicon-based features in the tagging model. We display an absolute gain of 0.5% in terms of f-measure. Moreover, we show that the integration of lexicon-based features significantly compensates the use of a small training corpus.

Introduction

Coupling external lexicons with an annotated corpus to train discriminative models is a recent trend in Natural Language Processing, e.g. [START_REF] Mccallum | Early results for named entity recognition with conditional random fields, feature induction and web-enhanced lexicons[END_REF] for Named Entity Recognition. Some studies have shown that it can significantly improve the accuracy of Part-of-Speech (POS) Tagging -e.g. [START_REF] Denis | Coupling an annotated corpus and a morphosyntactic lexicon for state-of-the-art pos tagging with less human effort[END_REF] for Frenchas it helps to deal with unknown words 1 . Similarly, [START_REF] Constant | Intégrer des connaissances linguistiques dans un CRF : application à l'apprentissage d'un segmenteur-étiqueteur du français[END_REF] and [START_REF] Constant | Intégrer des connaissances linguistiques dans un CRF : application à l'apprentissage d'un segmenteur-étiqueteur du français[END_REF] described different methods to use external Multiword Unit (MWU) lexicons in order to improve the CRF 2 -based joint task of MWU segmentation and POS tagging. They show that it helps to recognize unknown segments. Neverthess, the different methods are sometimes hardly comparable because experiments have been conducted in different environments 3 . The objective of this paper is four-fold : -synthesize the different possible methods for coupling external lexicons with an annotated corpus for the joint task of MWU segmentation and POS tagging ; -evaluate them in a uniform environment on two different versions of the reference annotated corpus ; -compare with a sequential approach (segmentation followed by tagging) ; -release a fully parameterized LGLPL-licensed software resource 'lgtagger' (http ://igm.univmlv.fr/˜mconstan/research/software) 4 . The first section describes the task and the resources used. We then present the different methods for coupling these resources to train a single linear CRF model. In the last section, we detail the experiments undertaken and comment the results. All experiments were carried out on French.

1. Words that are absent of the training corpus. 2. Conditional Random Fields [START_REF] Lafferty | Conditional random fields : Probabilistic models for segmenting and labeling sequence data[END_REF] 3. Different CRF training softwares, different versions of the training corpus 4. Some CRF features implemented in this tool were directly inspired by the ones in the tool SEM : http ://www.univorleans.fr/lifo/Members/Isabelle.Tellier/SEM.html.

Task and resources

We describe here the joint task of MWU segmentation and POS tagging, and the resources we have used. We take advantage of the fact that French is a language for which various MWU lexicons are available, as well as a fully POSannotated corpus where MWUs are marked.

Joint MWU Segmentation and POS tagging

Our joint task consists in segmenting and labelling lexical units including multiword ones. By using an IOB 5 scheme [START_REF] Ramshaw | Text chunking using transformation-based learning[END_REF], it is equivalent to labelling simple tokens. Each token is labeled by a tag of the form X+B or X+I, where X is the POS label of the lexical unit and the suffix B indicates that the token is at the beginning of the lexical unit, while the suffix I indicates an internal position. Suffix O is useless as the end of a lexical unit corresponds to the beginning of another one (suffix B) or the end of a sentence. Such a procedure therefore determines lexical unit limits, as well as their POS.

French Treebank

The French Treebank (FTB) (Abeillé et al., 2003) is a syntactically annotated corpus made of journalistic texts from Le Monde newspaper. We used two different versions : (i) the 2005 version ; (ii) the recent version dedicated to tagging and parsing experiments (Candito and Crabbé., 2009). We have uniformized the POS tagsets for both versions 5. I : Inside (segment) ; O : Outside (segment) ; B : Beginning (of segment) with 29 tags. Version (i) is composed of 19,490 sentences and 569,080 units including 32,975 multiword ones (i.e. 5.8%). Version (ii) is smaller and has a lower proportion of MW units : it contains 12,351 sentences and 350,931 units including 10,785 multiword units (i.e. 3.1%). MW units are of different types : compound words -e.g. parce que (because), flambant neuf (brand new), dans l' immédiat (right now) -and named entities -e.g. Europe de l'Est (Eastern Europe) , Jean-Pierre (John Peter) -.

Lexical resources

The lexical resources used are of two types : morphological electronic dictionaries and strongly lexicalized local grammars. They are freely available in the software 'lgtagger' under LGPL-LR license. Morphological electronic dictionaries are lists of lexical entries of simple and compound forms. They contain : the general-language dictionary DELA of 976,000 entries [START_REF] Courtois | Un système de dictionnaires électroniques pour les mots simples du français[END_REF][START_REF] Courtois | Dictionnaire électronique DELAC : les mots composés binaires[END_REF], the general-language dictionary Lefff of 579,000 entries [START_REF] Sagot | The lefff, a freely available, accurate and large-coverage lexicon for french[END_REF] and the toponym dictionary Prolex of 123,000 entries [START_REF] Piton | The prolex data base : Toponyms and gentiles for nlp[END_REF]. Strongly lexicalized local grammars are factorized sets of multiword entries in the form of finite-state graphs [START_REF] Gross | The construction of local grammars[END_REF]. For the purpose of this paper, we used 42 graphs recognizing numerical determiners and some Named Entities like organization or location names.

In our experiments, we concatenated the information coming from all these resources, whitout preserving the memory of their origin.

Coupling external lexicons and annotated corpus to train a CRF model

Linear CRFs are the best current statistical models to learn to annotate sequences. Their interest also relies in the fact that they allow to take into account a great number of various features. The features can be computable intrinsic properties of the data as well as external information [START_REF] Mccallum | Early results for named entity recognition with conditional random fields, feature induction and web-enhanced lexicons[END_REF]. We detail different solutions to integrate them.

Linear CRF Models

Linear chain Conditional Ramdom Fields (CRF) are discriminative probabilistic models introduced by [START_REF] Lafferty | Conditional random fields : Probabilistic models for segmenting and labeling sequence data[END_REF] for sequential labelling. Given an input sequence of tokens x = (x 1 , x 2 , ..., x N ) and an output sequence of labels y = (y 1 , y 2 , ..., y N ), the model is defined as follows :

P(y|x) = 1 Z(x) N t=1 exp K k=1 λ k f k (t, x, y t , y t-1 )
where Z(x) is a normalization factor depending on x. It is based on K features, each of them being a binary function f k depending on the current position t in x, the current label y t , the preceding one y t-1 and the whole input sequence x. This means that any computable property x i of x can be taken into account in features : the lexical value of x at any position (not only at position t), whether this value begins with an upper case, contains a number, etc. The feature is activated concerning t, y t , y t-1 and x is satisfied. Each feature f k is associated with a weight λ k . The weights are the parameters of the model, to be estimated. The features belong to the model and should be provided by the users. But softwares implementing CRF models help users : they usually only require to provide feature templates which are automatically instanciated into as many features as positions in the training data they can apply. We detail this point in the following subsection, because it is crucial to understand how this mechanism allows to integrate external information.

(i.e. f k (t, y t , y t-1 , x) = 1) if a given configuration x 0 1 x 1 1 • • • x p 1 y 1 . . . x 0 t-1 x 1 t-1 • • • x p t-1 y t-1 x 0 t x 1 t • • • x p t y t . . .

Feature templates

In CRF++ 6 and Wapiti 7 , some of the best known (and, in the case of Wapiti, most efficient) implementations of linear CRFs, training data (x, y) must be provided in files of the form of Figure 1.

In this kind of "tabular-like" files, each line corresponds to a position t in a sentence. Sequences of tokens (i.e. labeled sentences) are just consecutive lines, two distinct sentences being separated by a blank line. At a position t, the first p+1 columns display the computable properties x i t , 0 ≤ i ≤ p of x t which can be taken into account in features, and the last one contains the value y t of the correct label. Figure 1 also illustrates the notion of template. A template can be seen as a configuration of holes which can be positioned on any line of the file. At any given position, it is able to select the not-empty values of the files that appear in its holes. Each of these positions produces a feature, as a conjonction of all observed values. Of course, to respect the constraints of linear CRFs, the hole configurations can take any form on the first p + 1 columns, but can only integrate a unique value y t ("unary features") or a sequence of two consecutive values y t-1 and y t ("binary features") on the last one. The template of Figure 1, applied on the example sentence of section 2.1. at the position t = 6 where x 0 t is the lexical value of x at position t, x 1 t the property for x t of belonging to a punctuation list and x 2 t = x p t the property of beginning with an upper case, provides the following feature :

f k (t, y t , y t-1 , x) = 1 if (y t =DET+B) ∧ (x 0 t ='son') ∧ (x 1 t = 0) ∧ (x 2 t = 0) ∧ (x 0 t-1 =',') ∧ (x 1 t-1 = 1) ; f k (t, y t , y t-1 , x) = 0 otherwise
The usual trick of CRFs consists in using the same set of 6. http ://crfpp.googlecode.com/svn/trunk/doc/index.html 7. http ://wapiti.limsi.fr/ features for each position t, so the value of the position itself (t = 6) is not a criterion in the definition of the feature.

Integrating external resources into a CRF

To take into account external resources in a learning process using CRF models, many different solutions are possible. To illustrate them, we take again the labeled example of section 2.1.. A lexical resource will provide information about the set of every possible POS category that the lexical units of this sentence can have : -quant à : prep -quant : prep -à : prep -la : cn, det, pro -technique : cn -... The set of categories of the resource is not necessary the same as the one of the labeled example, this is why we write them with lower cases. Some of the solutions proposed further require that the set of labels match, others don't. The first possible use of a resource is called Filtering. It requires that the set of labels match. The process of filtering consists in using the resource to limit the search space of possible labelings. Instead of searching for the y that maximizes P λ (y|x), we search for the best y among those compatible with the resource which maximizes this value. In other words, incompatible labelings are discarded. The filtering step is, in this case, independent of the learning step, and can be applied before of after it. A second more interesting approach consists in taking into account the resource during the learning step, by integrating the information it contains into the tabular-like file. Even then, there are various possible options. Three of them have been identified : -Learn-concat : a single column is added to the file, which contains a string concatenating in a fixed order each possible category associated with the token in the resource. This string thus becomes a new property of the token. This solution does not require at all that the sets of labels are the same. ... This strategy requires that the sets of labels match. It has the advantage of adding possible instances of associations between a token and a label, which may not appear otherwise in the examples. But it can perturb the probability distributions of these associations, if the set of examples is small. By adding a large number of new examples, it may also slow down the learning step. Furtermore, feature templates with "holes" outside of the current position (in particular, every binary feature templates) will not apply, as each of these new examples is surrounded by blank lines.

Evaluation

We trained different CRF models with the software Wapiti [START_REF] Lavergne | Practical very large scale CRFs[END_REF] using the algorithm rprop. The standard feature templates (i.e. without lexicon-based ones) are defined in the table below : w t stands for the token at the relative position t from the current token ; l t is the label at the relative position t.

w t = X, t ∈ {-2, -1, 0, 1, 2} &l 0 = L Lowercase form of w 0 = W &l 0 = L Prefix of w 0 = P with |P| < 5 &l 0 = L Suffix of w 0 = S with |S | < 5 &l 0 = L w 0 contains a hyphen &l 0 = L w 0 contains a digit &l 0 = L w 0 is capitalized &l 0 = L w 0 is all in capital &l 0 = L w 0 is capitalized and BOS 8 &l 0 = L w 0 is part of a multiword &l 0 = L w i w j = XY, ( j, k) ∈ {(-1, 0), (0, 1), (-1, 1)} &l 0 = L l -1 = L ′ &l 0 = L Table 1 -Feature templates without lexicon-based ones
We used all the resources described in subsection 2.3. and we tested two ways of integrating lexicon-based features : Learn-concat and Learn-bool (cf. subsection 3.3.). Each model was trained on 80% of the FTB with an Intel(R) Core(TM)2 Quad CPU Q9400 @ 2.66GHz including 3.6 Gb memory. Each model combined about 40 million (resp. 25 million) features and its training duration was 2h-2h20 (resp. 1h-1h10) for version (i) of the FTB (resp. version ii). Three different approaches were evaluated : (a) baseline approach ; (b) 1-model approach, (c) 2-model approach. The baseline approach (a) consists in first performing a naive lexical segmentation (including MW recognition) and then in tagging the segmented text with a standard POS-tagger.

The segmentation phase is based on a lexical analysis that generates a finite automaton of all possible analyses for each sentence. We evaluated the different approaches by cross-validation, by using a standard f-score on the lexical unit segments (uniformely combining recall and precision). We computed the global tagging score and the tagging score of the MWUs (in parenthesis). Results are provided in table 2. Note that the Filtering and Learn-ex methods described in subsection 3.3. were not tested here, because [START_REF] Constant | Intégrer des connaissances linguistiques dans un CRF : application à l'apprentissage d'un segmenteur-étiqueteur du français[END_REF] have shown that they are less efficient than the other two.

We can first observe that the segmentation has a cost of around 1.8 and 3.8 points (97.8% for standard tagging vs. 96.0% and 94.0% for joint segmentation and tagging). The integration of large-coverage lexicons in the learning step makes the results improve by 0.5 points in the best case. Part-of-Speech Tagger. We especially show different ways of integrating lexicon-based features in the model. We display an absolute gain of 0.5% in terms of f-measure. Moreover, we show that the integration of lexicon-based features significantly compensates the use of a small training corpus.

Summary and Future Work

Future work would consist in adapting such approach to chunking. We are also willing to integrate new types of features computed from a symbolic rule-based tagger.

A . Abeillé, L. Clément, and F. Toussenel. 2003

Figure 1 -

 1 Figure 1 -A File of Labeled Examples with an instanciated Feature Template

Figure 2 -Figure 3 -

 23 Figure 2 -Overall tagging score evolution according to the size of the training corpus. The size is indicated in terms of percentage of the version (ii) of the French Treebank. We applied 5-fold cross-validation with (resp. without) corresponds to the experiment with lexical resources (resp. without).

  This strategy does not require either that the sets of labels are the same. But the generation of every possible feature template in this case is potentially explosive, as every possible combination of columns should give rise to a distinct template.-Learn-ex : each possible couple made of a token and a category seen in the resource can be considered as a new example of the training set.

							This strategy implements the
						idea that a resource is a collection of observed possible
						instances. In this case, the training set receives new lines
						of the form :
						Quant	PREP+B
						à	PREP+I
						Quant	PREP+B
						à	PREP+B
						la	CN+B
						la	DET+B
						la	PRO+B
						technique CN+B
		The training file will then become :
	Quant	prep			PREP+B
	à	prep_prep+I PREP+I
	la	cn_det_pro		DET+B
	technique cn			CN+B
	...
	Note that when the resource includes multiword units,
	they are treated the same way as in the rest of the corpus,
	i.e. whith an additional I label for internal units (the B
	label is implicit everywhere else).
	-Learn-bool : each possible category mentioned in the re-
	source is considered as a new boolean property taking
	the value 1 if it is a possible category for this token, 0
	otherwise. The training file will then become :
		det cn prep prep+I ...
	Quant	0	0	1	0	PREP+B
	à	0	0	1	1	PREP+I
	la	1	1	0	0	DET+B
	technique 0	1	0	0	CN+B
	...

  . Building a treebank for french. In A. Abeillé, editor, Treebanks. Kluwer, Dordrecht. M. Candito and B. Crabbé. 2009. Improving generative statistical parsing with semi-supervised word clustering. In Proceedings of the 11th International Conference on Parsing Technology (IWPT'09), pages 138-141. M. Constant and A. Sigogne. 2011. MWU-aware part-ofspeech tagging with a CRF model and lexical resources.

				lexicon-based features
			FTB (i)	FTB (ii)
	Approach	Resource	Learn-concat	Learn-bool	Learn-concat	Learn-bool
	Baseline	no	92.7		93.8
			(71.0)		(64.9)
	1-model	no	93.5		95.5
			(73.2)		(76.0)
	1-model	yes	94.0	93.8	96.0	95.7
			(75.1)	(74.0)	(78.2)	(76.7)
	2-model	yes	94.0	93.9	96.0	95.9
			(75.1)	(74.2)	(78.1)	(76.8)


Ahmad Pouya
email: ahmad.pouya@enpc.fr

Minh-Ngoc Vu

Numerical modelling of steady-state flow in 2D cracked anisotropic porous media by singular integral equations method

Keywords: porous media, cracks, steady state flow, singular integral equations, effective permeability

The equations governing plane steady-state flow in heterogeneous porous media containing curved-line intersecting cracks (Pouya and Ghabezloo 2010) and the potential solution obtained for these equations are considered here. The theoretical results are first completed for the mass balance at crack intersections points. Then, a numerical procedure based on a singular integral equations method is described concretely to derive this solution for cracked materials. Closed-form expressions of elementary integrals for special choice of collocation points lead to a very quick and easy numerical method. It is shown that this method can be applied efficiently to the study of the steady state flow in cracked materials with anisotropic matrix permeability and a dense distribution of curved-line intersecting cracks. Some applications of this method to the permeability of cracked materials are given.

Introduction

The discontinuities such as fractures and cracks have generally a great influence on the effective permeability of fractured rocks and porous materials. Some physical and engineering problems such as the contaminant transport in micro-cracked rocks or reservoir study in petroleum industry require the modelling of the fluid flow in porous media containing cracks or fractures. The problem of flow in a porous matrix containing cracks with mass exchange between matrix and cracks has been intensively investigated in recent years by different approaches (see the reviews given by [START_REF] Sánchez-Vila | A synthesis of approaches to upscaling of hydraulic conductivities[END_REF], [START_REF] Goméz-Hernández | Upscaling hydraulic conductivities in heterogeneous media: an overview[END_REF] and [START_REF] Renard | Calculating equivalent permeability: a review[END_REF]). Two-dimensional flow in and around a facture embedded by an infinite porous medium, under a constant pressure gradient, has in several cases derived analytically by means of some geometrical simplifications. In particular, the fracture has been sometimes modelled as a flattened ellipsoidal inclusion [START_REF] Zimmerman | Effective conductivity of a two-dimensional medium containing elliptical inhomogeneities[END_REF] or as a two-dimensional lens (Mityushev and Adler, 2006) filled with a Darcy type material. In the present paper, we only consider the case of zero-thickness fractures with no pressure jump between the two faces of the fracture. This corresponds to an assumption of infinite transverse permeability of the fracture and excluding the cases where fractures act as an impermeable membrane like some cases of shear faults [START_REF] Manzocchi | Faults and fault properties in hydrocarbon flow models[END_REF]. The effective permeability of materials with this kind of cracks has been studied by many numericalempirical approaches. A more rigorous approach based on the homogenization scheme has been given by [START_REF] Shafiro | Anisotropic effective conductivity of material with nonrandomly oriented inclusions of diverse ellipsoidal shapes[END_REF], [START_REF] Dormieux | Approche micromécanique du couplage perméabilité-endommagement[END_REF] and [START_REF] Barthélémy | Effective permeability of media with a dense network of long and micro fractures[END_REF] in which the crack is assimilated to oblate ellipsoidal inclusions obeying to a Darcy's law. This powerful approach easily provides results for the effective permeability of 3D cracked materials, but it can not take into account crack intersections explicitly. Regarding the numerical modelling, the most difficult question is to generate a consistent mesh for a randomly fractured medium. A robust and automated algorithm was presented first by [START_REF] Koudina | Permeability of three-dimensional fracture networks[END_REF] to triangulate a polygonal fracture network; and then was extended for the fracture network and the embedding matrix as described by [START_REF] Bogdanov | Effective permeability of fractured porous media in steady state flow[END_REF]. The flow equations are then discretized by means of the finite volume method as to compute the effective permeability of fractured porous media [START_REF] Bogdanov | Effective permeability of fractured porous media in steady state flow[END_REF][START_REF] Bogdanov | Effective permeability of fractured porous media with power-law distribution of fracture sizes[END_REF][START_REF] Mourzenko | Permeability of isotropic and anisotropic fracture networks, from the percolation threshold to very large densities[END_REF].

Following a different way, [START_REF] Liolios | A solution of steady-state fluid flow in multiply fractured isotropic porous media[END_REF] presented a mathematical formulation for plane steady-state flow in a cracked porous matrix where the cracks are considered as zero thickness discontinuity lines. However, their method, using complex number potentials, was restricted to isotropic matrix and excluded crack intersections. More recently, [START_REF] Pouya | Flow around a crack in a porous matrix and related problems[END_REF] used a direct approach to model plane steady-state fluid flow in an anisotropic porous body containing curved line cracks with possible intersections. This approach provided a general potential solution for the fluid pressure in the matrix function of discharge along cracks. The flow in the cracks is modelled by a Poiseuille type law, a linear relation between the discharge and the pressure gradient along the crack. This model of cracks will be designated as Poiseuille crack to be distinguished from ellipsoidal cracks obeying to Darcy's law. Moreover, closed-form solutions were derived for the case of a single crack in an infinite matrix and applied to the problem of permeability upscaling.

In the present paper, we first complete the work of [START_REF] Pouya | Flow around a crack in a porous matrix and related problems[END_REF] by the mass balance equation at crack intersection points. Then a numerical method is developed to solve the general system of singular equations for the case of an infinite body containing a dense family of curved line and intersecting cracks. This method is based on the resolution of the equations for a finite number of collocation points. A special choice of collocation points is given to simplify the computation. Besides, all the elementary integral terms are calculated explicitly. This leads to a highly efficient and fast calculation method. After presenting the method, a successful validation is first presented by comparing the numerical results obtained for a single superconductive crack with the closed-form solution for this case. Then, the flow is modelled around several curvilinear and intersecting cracks and the masse balance is checked carefully at intersection points. After that, the effective permeability of a material containing a random crack distribution, inspired from geological observations on a rock formation, is studied for the illustration of the method. Finally, the effective permeability of a periodic crack network is calculated with this method and the result is compared to those obtained by theoretical methods.

Governing equations

In this section, we briefly present the basic results obtained by [START_REF] Pouya | Flow around a crack in a porous matrix and related problems[END_REF]. An infinite heterogeneous porous body Ω containing a crack network is considered. The cracks are numbered by m and denoted by Γ m (Fig. 1). The crack Γ m is geometrically represented by a smooth curve m z of the curvilinear abscise s. A crack can end in the matrix or intersect other cracks at its extremities. The intersection points of two or more cracks and the extremities of cracks ending in the matrix constitute a set of singular points denoted by S. The set of points on the cracks are denoted by Γ, therefore, S ⊂ Г. The matrix corresponds to Ω-Г.

q(x) p(x) Ω ∂ p Ω Γ p(x), v(x)
∂ v Ω

Fig. 1 Heterogeneous porous material containing micro-cracks

The fluid flow in the matrix is assumed to be governed by Darcy's law:

Γ - Ω ∈ ∀x ( ) ( ). ( ) k v x x p x = - ∇ (1)
where, v(x) designates the fluid velocity, k the local permeability and ∇p(x) the pressure gradient at x.

In the absence of point sources, mass conservation in the matrix reads:

Γ - Ω ∈ ∀x ∇.v(x) = 0
1994) in which the discharge q in the crack is proportional to the pressure gradient along the crack line:

Γ ∈ ∀s ( ) ( ). s q s c s p = - ∂ (3)
where c is the hydraulic conductivity of the crack depending on its hydraulic aperture or its fill materials. According to the Poiseuille's law c=e 3 /(12µ) where e is the hydraulic aperture of the crack and µ the fluid viscosity. Superconductive cracks correspond to the limite case of ( ∞ → c

). In this case the pressure is constant along the crack curve.

The equation governing the crack-matrix mass exchange at regular points on the crack (excluding intersection points) is established by considering mass balance in a fragment of the crack comprised between abscises s and s+ds (Fig. 2a). It reads:

Γ ∈ ∀s ( ) . ( ) ( ) 0 s v z n s q s + ∂ = (4)
In this equation, z is the point on the crack at the curvilinear abciss s, ) (s n is the unit normal vector to the crack curve oriented from the side Г -to Г + (Fig. 2a) and ( )

( ) ( ) v z v z v z + - = -
is the discontinuity or jump of fluid velocity across the crack.

At crack intersections, analysing the mass balance in a circle centred at the intersection point, [START_REF] Pouya | Flow around a crack in a porous matrix and related problems[END_REF] established the following relation for the fluid velocity at this point:

S z ∈ ∀ ( ) 0 . ( ) 0 j j v x q x z   ∇ + δ -=     ∑ (5)
where, 0 j q designates the outgoing discharge on the crack branch number j, x is a point in a small neighbourhood of z containing only z as singular point (Fig. 2b), and δ represents the Dirac distribution.

s s+ds The body is submitted at infinity to a pressure field p ∞ (x):

Γ + q(s+ds) q(s) Γ - n v - v + x x D Γ j+1 v 0 j q 1 0 j q + D j,j+1 Γ j n ∂D 1 j D q + j D q ∂D j,j+1 n n n v - v + (a) (b)
[ ] 0 ) ( ) ( = -∞ ∞ → x p x p Lim x (6)
The field p ∞ satisfies ∆p ∞ =0. In the most general case of effective permeability modelling, p ∞ corresponds to constant pressure gradient, i.e. p ∞ (x)=A.x where A is a constant vector.

For the case of uniform matrix permeability k, a general potential solution for these equations was rigorously established by [START_REF] Pouya | Flow around a crack in a porous matrix and related problems[END_REF] with the following expression:

p(x) = p ∞ (x)+ m m m m 2 m m 1 ( ) ( ) . ( ) 2 ( ) -1 -1 k . k . x z s q s t s ds x z s Γ - πκ   -   ∑ ∫ (7)
In this relation, x is the current point in the porous body, z m (s) is the point on the crack Г m at the curvilinear abscise s, q m (s) is the discharge along the crack m, t m (s) is unit vector tangent to this crack at the point z m (s) and κ is the square root of determinant of k: κ = k . If the discharge q(s) in ( 7) is replaced by its expression function of p(s), for instance by (3), and the collocation point x is taken on the cracks, then a system of equations is obtained that allows us to determine the pressure p on the crack network. It is worth noting that this system of equation involves only the pressure on the crack lines and thus reduces the dimension of the problem from 2 to 1. This simplifies considerably the numerical resolution method in the same way that Boundary Elements method simplifies the numerical resolution compared to Finite Elements or Finite Volume methods.

Mass balance at intersection points

A deeper mathematical analysis of the mass balance condition at intersection points allows going further than equation ( 5) and demonstrates that at these points we have:

0 0 j j q = ∑ (8)
This relationship has an important physical interpretation and consequences for numerical modelling that will be detailed further. To demonstrate it, let us consider a small domain D surrounding an intersection point z with its boundary designated by ∂D (Fig. 2b). If there are J intersecting cracks at z, then D is divided into J sub-domains D j,j+1 bounded by the cracks Γ j and Γ j+1 (with the convention Γ J+1 = Γ 1 to have cyclic notation). The mass exchange between D and the matrix contains two parts: one by flow through the matrix on ∂D with fluid velocity v, and the other by the cracks Γ j at their intersection with ∂D where discharge is designated by D j q . Therefore, the mass balance for this domain reads:

. 0

D j j D v n ds q ∂ + = ∑ ∫ (9)
Now, we consider the mass balance in the subdomain ∂D j , j+1 . It reads:

, 1 1 . . . 0 D D j j j j D v n ds v n ds v n ds + + + - ∂ Γ Γ - + = ∫ ∫ ∫ (10)
where, D j Γ represents the portion of Γ j included in D, n is the unit normal vector on the integration line. On ∂D j,j+1 , n represents the outward unit normal, whereas on Γ j , it is oriented from the surface Γ j to the domain D j , j+1 . On Γ j the fluid velocity is discontinuous: v -and v + represent the velocity at respectively the negative and positive sides of this surface orientated by n (Fig. 2b). This convention determines the sign of the two last integrals in (10). When summing this equation on j, the contribution of the two last integrals can be transformed into expressions involving the jump of v on Γ j . Hence, the following equation is then found:

. . 0

D j j D v n ds v n ds ∂ Γ - = ∑ ∫ ∫ (11)
Now, by using (4) the second integral in this equation can be changed to write:

0 . ( ) D j D j j j j v n d s q q Γ = - - ∑ ∑ ∫ (12)
Introducing ( 12) and ( 11) into (9) results in (8). The physical interpretation of ( 8) is that the masse balance condition at crack intersection points is independent from the crack-matrix mass exchange. This relation is the same that one would find for a crack network in impermeable material. Moreover, this result also allows extending the Cauchy integration to intersection points by elimination of the singular contribution of different crack branches at these points to the integral (7). This equation will be verified numerically in Section 5.

Numerical resolution

By replacing q(s) in ( 7) by its expression (3), a system of integro-differential equations is obtained and then solved in order to determine the pore pressure and the fluid flow fields. This system allows us to investigate in a first stage only the pressure along crack lines and determining it by resolving the system of singular integral equations. In the second stage, the integral equation ( 7) with the known expression of the discharge q(s) will be used to calculate the pressure field at an arbitrary point of the matrix. It is worth noting that the values of q(s) in the first stage can be directly used for permeability upscaling and the second stage is not necessary for this purpose. This is an interesting feature of the numerical method presented here as it will be seen in section 6.

The principle for resolving the singular integral equations by collocation method can be found in [START_REF] Bonnet | Equations Intégrales et Éléments de Frontière[END_REF]. We propose in this paper a special choice of interpolation functions and collocation points as well as analytical expressions for elementary integrals that lead to an efficient and fast resolution method. The numerical method presented in the following assumes a constant value for the cracks conductivity c but it can be easily extended to variable conductivities for cracks.

Discretization and Interpolation

As a consequence for the theoretical formulae (7), numerical calculation only requires a mesh generation on the cracks which must be consistent by taking into account crack intersection points. Thus, cracks are discretized by a number of intervals, called in the sequel « elements ». The set of intervals on all cracks define globally N elements numbered by n and denoted by E n (1≤n≤N). With these notations, Eq. ( 7) can be written as:

p(x) 1 . N n n A x I = = + ∑ (13)
where,

n n n n 2 n ( ) ( ) ( ) . . . ( ) - =   -   ∫ n E x z s I x q s t ds x z s -1 -1 k k (14)
In order to define the interpolation functions of variable on the elements, we distinguish two types of elements: current element on the crack and extremity elements where the cracks end in the matrix. On the current elements, a linear interpolation of pressure is used; hence, q is constant on the element (Fig. 3). For extremity elements, general theoretical results concerning the velocity field singularity close to a crack tip leads to suppose that v varies as r 1/2 around the crack extremity points [START_REF] Muskhelishvili | Singular Integral Equations[END_REF].

Besides, based on the analytical solutions of fluid flow in superconductive cracks [START_REF] Pouya | Flow around a crack in a porous matrix and related problems[END_REF], we choose an interpolation function for discharge q(s) with a variation as s 1/2 where s is the distance from crack extremity. This leads to a variation as s 3/2 for p(s) on the extremity elements. We designate by L the length of the element and by p 1 and p 2 its nodal pressures at curvilinear abscises s = 0 and s = L. Then the interpolation functions for the pressure and discharge on current elements, left extremity elements (tip at s = 0) and right extremity elements (tip at s = L) take, respectively, the following expressions:

p(s) = p 1 + 2 1 - p p L s; q(s) = -c 2 1 - p p L (15) p(s) = p 1 + 3 2 1 3 - p p s L ; q(s) = -a s (16) p(s) = p 2 - 3 2 1 3 - - p p L s L ; q(s) = -a - L s (17)
where: The collocation method consists in enforcing exactly the singular integral equation ( 7) at a finite number of points called collocation points. In its simplest form and in most studies in the literature, these collocation points are chosen to be the nodes used to discretize the geometry. Herein, we propose another way to choose the collocation points that easily allows the introdution of analytical expressions for the elementary integrals and thus the simplification of the calculation. At first, a collocation point is taken per element and if the collocation point x is on the element E n , I n (x) becomes a singular integral. For the current element, a collocation point x n is placed naturally in the middle of the element because this implies:

a = 2 1 3 3 2 - p p c L (18)
I n (x n ) = 0 (19)
This equation results from the linearity of pressure interpolation on the current elements and simplifies the calculations. It suggests also choosing the collocation point on the extremity element such that the same equation ( 19) will be obtained.

Consider a left-extremity element E n with endpoints z 1 and z 2 and take the collocation point:

x = z(s x ) = z 1 + (s x /L) (z 2 -z 1 ) (20)
with 0 < s x < L. Substituting Eqs.( 16), ( 18) and (20) into Eq.( 14), then I n (x) can be integrated explicitly. As a result, equation ( 19) is fulfilled for a value of s x /L = ξ where ξ is the solution of the equation:

1 2 1 Log + ξ = -ξ ξ (21)
Numerical computation leads to ξ ≈ 0.694817. In a similar way, the condition s x /L = 1-ξ is obtained with the same numerical value for ξ for right-extremity elements.

In conclusion, on the element E n , a collocation point

x n = z 1 + ξ n (z 2 -z 1
) is chosen with, ξ n = ξ, ξ n = 0.5 and ξ n = 1-ξ for respectively left-extremity, current and right-extremity elements.

It should be noted that depending on the crack network connectivity, the number of elements can be smaller than the number of nodes. Hence, it is not generally sufficient to take one collocation point per element to solve the system of equations that has the nodal pressures as unknown variable because the number of equations can be smaller than the number of unknowns. The difference between the number of nodes and elements depends on the connectivity properties of the crack network and is not easy to determine. Nevertheless, a summary count of nodes and elements could help to determine an optimum choice of collocation points. As a matter of fact, each element is always related to two nodes, whereas, except for crack extremity points, a node is related to two or more elements. Thus, the number of elements is more than the number of nodes that are not crack extremities. Therefore, if, to the set collocations points x n defined here-above (the same number than the elements), we add the crack extremity points, the number of collocation points will become always greater than the number of nodes. In this way, we can have more equations than unknowns, and then the least square fitting method will be required to solve the problem. This method will be presented later in the paper.

Explicit expression for elementary integrals

In the literature on boundary elements method, computation of regular elementary integrals is usually based, as well as in finite element method, on the use of Gaussian quadrature formulas. Contrariwise, the Cauchy Principal Value definition for the singular elementary integral is the standard technique in the boundary integral analysis (Guiggiani, 1991). However, there are still other techniques available to deal with the singular integral such as: regularization technique (Tanaka et al. 2002) that permits treating weak singularity; boundary limit approach [START_REF] Sutradhar | Symmetric Galerkin Boundary Element Method[END_REF]) that defines the boundary equation as a limit case of x approaches the boundary from either interior or exterior and which is efficient for symmetric Galerkin boundary elements method.

Otherwise, the numerical difficulty appears when x is too close to E n . In this case, the integrals are not strictly singular but nearly singular and can be computed by some transformation techniques that allow the concentration of Gauss's points in the vicinity of the orthogonal projection of x on the considered element. In this study, all elementary integrals are explicitly computed by an analytical procedure which aims to a high speed convergence of numerical computation and avoids also the numerical integration difficulties.

According to the relative position of the point x c and the element E n , we can distinguish five types of elementary integrals I n (x c ) to be calculated: (i) x c = x n is the collocation point defined here above on the interval E n , (ii) x c is the extremity point on the extremity element E n , (iii) x c is an intersection point (iv) x c is on the element E n but is not an extremity nor endpoint, (v) x c is not on E n .

For the first case, we have already chosen x n verified I n (x n )=0. For the second case, the integrant function is singular at the point x c , but remains integrable because its singularity is the order of s -1/2 . In the third case, x n is an intersection point between several cracks or common points between several elements E n . Each I n (x c ) is singular in this case, nevertheless, the sum of singularities vanishes because of the mass balance (8).

For the fourth case, the integrant function is singular at x c , but can be integrated in Cauchy principal value sense. Finally, the integrand function is not singular in the last case. For determination of the nodal pressure, or for the first stage of our numerical method (see the beginning of Section 4), the system of equations derived from ( 13) involves only the first and second cases listed above. In these cases, I n (x c ) can be expressed by the analytical expressions. Hence, equation ( 13) leads to a matrix equation in which all components of the matrix are computed analytically. This provides a very fast calculation method for determining the nodal pressures in the first stage. In the case (iii) listed above, x c is a nodal point and its pressure in deduced from the results of the first stage. The pressure for the case (iv) is deduced by interpolation functions (15) to (17) from the nodal pressure. For the last case (v), analytical expressions for elementary equations can also be derived that simplify the calculations. Analytical expressions of different cases of elementary integrals are given in Appendix.

Construction and resolution of linear system of equations

The set of M scalar equations (13) for N nodal pressure unknowns have the following matrix structure:

F=HP=C+GP (22)
where, F(M×1) is the column of pressure values at collocation points, P(N×1) is the column of unknown nodal pressure, C(M×1) is column of infinite pressure field p ∞ applied to collocation points, H(M×N) is established from the interpolation functions in Eqs.( 15)-( 17), G(M×N) is computed from the elementary integrals in right-hand side of equation ( 7) which is set up through the assembly operation that is carried out by two nested loops: the first one is over all collocation points and the second one over all elements.

As a consequence, the system (22) consists of M equations and N unknowns with M>N.

However, an approximate solution can be computed by using the least squares fitting method which is performed by minimizing |(H-G)P -C| 2 on the P values. This leads to the equation:

KP=Y (23)
where:

K = (H-G) T (H-G); Y = (H-G) T C (24)
In this way, finally, the system of equations ( 13) is reduced to a linear system of N equations on N unknown nodal pressures.

Numerical examples

In this section, we present some examples that illustrate the numerical method obtained in precedent sections. The algorithm described in precedent section was been implemented in a Fortran code that generates automatically a consistent mesh to discretize the crack network. Then, as described here above, the nodal pressures are determined from (24) in a first stage and inserted in (13) to determine the pressure at an arbitrary point.

Straight single crack in an infinite plane

In order to validate the numerical solution, the fluid flow around a single straight crack in an infinite homogeneous matrix was modeled and compared to the analytical and semi-analytical results obtained by [START_REF] Pouya | Flow around a crack in a porous matrix and related problems[END_REF]. Consider an infinite matrix, with uniform and isotropic permeability k, containing a straight crack of length 2L and conductivity c, occupying the interval [-L,L] on the x-axis of coordinate system. This infinite body is submitted to a farfield pressure with a uniform gradient A = (a,0) parallel to the crack:

p ∞ (x) = a.x (25)
The following dimensionless parameters are introduced:

( , 0) , , ( ) 2 c x p x X P X kL L aL λ λ = = = π (26)
Fig. 4 displays, for different values of λ , the pressure and the discharge obtained along the crack line. Since the problem is symmetric with respect to the y-axis, only the solution on the right half-crack (0≤X≤1) is presented in the figure. It is important to note that, for every value of 0 > λ , at the crack tip, the discharge q as well as the pressure gradient in the crack are null. The pressure field obtained for the limit case of superconductive crack (c→∞ or λ→∞, numerically λ = 10 5 ) is presented in Fig. 5. This figure shows the singularity of the pressure gradient in the neighbourhood of the crack tip. The numerical results obtained here are in perfect agreement with the semi-analytical or close-formed solutions given by [START_REF] Pouya | Flow around a crack in a porous matrix and related problems[END_REF] .

Curvilinear cracks with intersection

The following example of several straight or curvilinear intersecting cracks allows us to illustrate the benefit of the new numerical method compared to previous works, and also to numerically check the mass balance equation ( 8). The geometry of the cracks is presented in the Fig. 6, with details for elements surrounding the intersection point on the right. The pressure field has been calculated here for an anisotropic matrix (k xx = 2, k yy =0.5) under the limit condition (25). The pressure field obtained in the matrix is presented in the left of Fig. 6. This example illustrates two improvements of the method compared to that presented by [START_REF] Liolios | A solution of steady-state fluid flow in multiply fractured isotropic porous media[END_REF]: extension to intersecting cracks and to anisotropic matrix.

Note that the limit condition of pressure is given by p(x) = ax at infinity and not on the boundary of the domain presented in the figure. On this boundary the pressure is affected by the presence of the cracks. This shows another advantage of the method: in numerical calculations for determination of the effective permeability of cracked materials, one often seeks for the flow in a family of cracks in an infinite matrix subjected at infinity to a uniform pressure gradient. This is the concept of an inclusion in an infinite matrix. But to determine numerically the flow field, generally an REV (Representive Element Volume) is considered and a pressure p(x) =ax is imposed on the boundary of this REV (Long et al., 1982, Pouya and[START_REF] Pouya | Permeability of 3D discontinuity networks: new tensors from boundaryconditioned homogenization[END_REF]. This does not correspond to the theoretical condition of a farfield pressure gradient. The method presented here, based on equation ( 7), allows naturally imposing a uniform pressure gradient at infinite boundary.

Here is also an occasion to check if the numerical results satisfy well the mass balance (8) at the intersection points. Note that equation ( 8) has not been included in the system of equations determining the numerical results and then mass balance ( 8) is not an a priori condition imposed on these results.

Fig. 6 The Pressure field p(x,y) in the anisotropic matrix However, the numerical discharges obtained on the elements related to the intersection point in the right-side of Fig. 6 (the node j=122), show that this mass balance is very accurately satisfied. As a matter of fact, let the error estimation be defined by:

b j b b j q E q = ∑ (27)
where b j q is the discharge outgoing from the node j on the branch b (one of the four elements in the case considered here), and b j q is the average absolute value of theses discharges. In the case considered here above, we found at the intersection point j =122 E ≈ 0.004 and E ≈ 0.012 for an isotropic and anisotropic matrix, respectively Thus, the mass balance is fulfilled with a good accuracy. The same result is obtained for all the intersection points that were analyzed.

Application to the permeability of cracked porous media

The numerical method established here above specially becomes interesting and efficient compared to other methods in cases where a great number of cracks are present. We show in this section its application to compute the permeability of cracked materials which needs the modelling of the flow in the presence of a crack distribution.

A homogeneous domain Ω, containing a population of internal curvilinear cracks Γ m , is under the influence of the linear pressure condition x A x p . ) ( = applied on its boundary Ω ∂ . A general and classical result for the case of cracks with zero or negligible thickness is that the effective permeability k e is the sum of the matrix permeability k and a contribution of cracks k c ;

k e =k+k c (28)
The crack contribution k c can be deduced from the balance of discharge in the cracks under a farfield pressure gradient applied to the matrix from the following relation [START_REF] Bogdanov | Effective permeability of fractured porous media in steady state flow[END_REF][START_REF] Liolios | A solution of steady-state fluid flow in multiply fractured isotropic porous media[END_REF][START_REF] Pouya | Flow around a crack in a porous matrix and related problems[END_REF]:

1 . k m c m qtds A Γ = - Ω ∑ ∫ (29)
In this formula, A is the pressure gradient imposed on the boundary Ω ∂ that contains the cracks Γ m . It is sufficient to compute the left-side of (29) for two different directions of A to be able to determine k c . The result can be compared to that predicted by approximate closed-form solutions for the case of straight cracks. Let Γ m designate a family of cracks with length 2L m , orientation t m , conductivity c m and density (number of crack centres per unit area) ρ m . In the case of randomly oriented cracks, k c is isotropic:

k c = k c δ δ δ δ (30)
and then k e = k e δ δ δ δ where, according to [START_REF] Pouya | Flow around a crack in a porous matrix and related problems[END_REF], the Mori-Tanaka and self consistent estimates, k e are respectively given by (31a) and (31b):

2 m m e m m m m m L c k k L c L πκ = + ρ + πκ ∑ (31a) 2 e m m e m m m e m m L c k k L c L πκ = + ρ + πκ ∑ (31b)
with κ = k , κ e = k e . In this case κ = k and κ e = k e .

For a network of parallel cracks, with unit vector t, we have:

k c = k c t⊗t (32)
and then:

k e = k n⊗n + k e t⊗t (33)
where, the Mori-Tanaka and self consistent estimates of k e are deduced respectively from (34a) and (34b):

2 2 e cL k k L c L π κ = + ρ + π κ (34a) 2 2 e e e cL k k L c L π κ = + ρ + π κ (34b)
where, κ e = . e k k . Thus, the self-consistent scheme (34b) leads to the third-degree equation for the variable k e .

The dilute Mori-Tanaka estimations (31a) and (34a) do not take into consideration the interaction between cracks whereas this interaction is implicitly taken into account in some approximate forms of the self consistent scheme and fully in the numerical modelling. In the following, the theoretical results will be compared to the results obtained by the numerical method in the two cases that of porous media containing a distribution of randomly oriented cracks and that of a regular network of parallel cracks.

Randomly oriented cracks

A geological rock formation is considered containing several families of fractures. Each family of fractures is characterized by an average length, a mean orientation and a density that are determined from in situ geological observations [START_REF] Billaux | Hydrogéologie des milieux fracturés. Géométrie, connectivité et comportement hydraulique[END_REF][START_REF] Gervais | Modélisation géométrique d'un réseau de fractures dans un massif rocheux stratifié. Application aux carrières marbrières de Comblanchien (Côte d'Or[END_REF]. The rock formation is represented by an infinite porous matrix with uniform and isotropic permeability and containing several families of cracks randomly generated by stochastic laws. A domain of 10×10 (Fig. 7 -left) in the cracked matrix is used for the numerical modelling. In this domain, fractures are generated according to statistical laws reproducing the geological data. This domain is submitted to the boundary conditions (25). A smaller area is taken in which the pressure field is computed and used for the equivalent permeability calculation. (Fig. 7 -right). The numerical method is very easy to use since it does not require meshing the whole surface as it would be required for Finite Element or Finite Volume methods. The pressure field obtained by this method is represented in Fig. 7 -right. The conductivity of cracks relative to the matrix permeability is given by λ = 1 where dimensionless variable λ is defined by ( 26). For a crack family of density (number of crack centres per unit area) ρ=2.5, average length L = 0.35 and of uniformly distributed orientation (so globally isotropic), the Mori-Tanaka estimate (31a) provides k e /k = 1.38, the self consistent estimate, k e /k = 1.53 and the numerical calculation k e /k = 1.59. The difference between the numerical and Mori-Tanaka estimation reveals the effect of the interaction between cracks. The selfconsistent estimate takes approximately the crack interaction into account and is, for this reason, closer to the numerical result.

Periodic distribution cracks

An infinite porous matrix containing a periodic distribution of cracks parallel to the xaxis is considered (Fig. 8). The matrix has a uniform and isotropic permeability k, and is submitted to the farfield pressure gradient (25). The cracks have the length 2L and the distance 2b between their extremities in the x-direction and h between their centres in the y-direction. The crack density is then:

1 2 ( ) N h b L ρ = = Ω + (35)
where, N is number of cracks on Ω. The mean value of qds Γ ∫ is numerically computed on an array of (m×m) cracks at the centre of the domain with m=1,2,3… Greater is m, more accurately the interaction between cracks is taken into account. As predicted by the formula (33), the effective permeability is changed only in the direction parallel to cracks. Fig. 10 displays the effective permeability computed by the numerical method compared to Mori-Tanaka and self consistent estimates. The three curves begin with the same slope at ρ=0, and this shows that the theoretical estimates give the exact variation of the permeability at first order development for small ρ values. The dilute Mori-Tanaka estimate remains a linear function of ρ because it does not take into consideration the cracks interaction. It can be noticed that the self-consistent estimate, that is supposed to take into account in some extent the cracks interactions, is strangely less accurate that the Mori-Tanaka estimate. This result is in contrast with that obtained in precedent section for randomly oriented cracks. However, it is well known that the self-consistent scheme improves the estimation only in the case of a random orientation of the heterogeneity [START_REF] Dormieux | Approche micromécanique du couplage perméabilité-endommagement[END_REF]. It can be shown that the in the case of parallel cracks must lead to a smaller equivalent permeability than that one obtained by the Mori-Tanaka estimate, and to prove that the numerical results presented here give the good trend of equivalent permeability versus density. This can be done by using the solution obtained for the pressure field around a single crack as shown in Fig. 10. Fig. 10 displays the pressure field in the infinite matrix containing only one of the cracks among the network presented in Fig. 8 (crack number 1). The pressure field, before this crack is placed in the matrix, is represented by the straight line corresponding to the farfield gradient. The bold line displays the pressure field on line containing a first crack placed in the matrix and the two other lines, the pressure on the lines parallel to the first line and distant from it of h and 2h. When the second and third cracks are placed in the matrix at a distance h and 2h from the first crack (cracks number 2 and 3 in Fig. 8), they are placed in the pressure field locally represented by the curve y = h and y = 2h in the figure, hence, in a local pressure field with smaller gradients compared to the farfield gradient encountered by the first crack. The qds Γ ∫ determining the contribution of these cracks to the global permeability is therefore smaller than that obtained for the first crack. This reasoning is somehow approximate since it takes into account the effect of the presence of the first crack on the pressure field encountered by the following cracks but not the effects of the following cracks on the first one. However, it explains well why the crack interaction tends to decrease the slope of the equivalent permeability curve versus the crack density, as found by the numerical results shown in Fig. 7. It also confirms that the self consistent scheme is here deficient.

Conclusions

Recent theoretical advances in the modelling of flow in cracked porous materials [START_REF] Pouya | Flow around a crack in a porous matrix and related problems[END_REF] were in some points completed here and then used to establish a numerical method for the study of these materials. This method was validated in the simple cases of a finite size crack in an infinite matrix for which closed-form solutions are available. It was shown that this method can be easily applied to the determination of the steady state flow in the materials with an anisotropic permeability, containing a dense distribution of curved-line cracks that intersect together. Some examples showed the efficiency of this method to determine the equivalent permeability of micro-cracked porous materials.

The efficiency of the numerical method is partly due to special choice of collocation points and to using closed-form expressions of elementary integrals. The theoretical result obtained on the mass balance at intersection points allows establishing a probably more efficient numerical method in which nodes are chosen as collocation points and using yet analytical expressions for elementary integrals. Furthermore, the theoretical results and the numerical algorithm may be extended to the 3D steady-state flow problem. These extensions are pursued in our ongoing research and will be presented in future papers.

Although these methods include some simplificative assumptions, like the uniform permeability of the matrix and no pressure jump between the two faces of the fracture, they can be usefully applied to many current research and engineering problems. The determination of the equivalent permeability of fractured oil reservoirs, of geological formations used for CO 2 sequestration and of damaged zones around underground galleries for nuclear waste disposal are some possible examples of applications that we undertake in parallel researches.

Appendix: Analytical expression of elementary integrals

The element E n between the endpoints z 1 , z 2 is considered with the unit tangent vector designated by t. An arbitrary point in the plane of the crack is designated by x. It can be shown that the elementary integral in Eq.( 13) can be calculated analytically with the interpolation functions ( 15)-( 17) for current elements and extremity elements as following:

Current element where c stands for the hydraulic conductivity of crack; p 1 , p 2 the nodal pressures of the element E n , z 0 =z 1 and ε =1 for the left extremity element, z 0 =z 2 and ε =-1 for the right extremity elements.

I n (x) = 2

Fig. 2 a

 2 Fig. 2 a Crack-matrix mass exchange at a regular point on a crack. b Crack-matrix mass exchanges at an intersection point of several cracks

Fig. 3

 3 Fig. 3 Interpolation of the discharge (a) and pressure (b) a long a discretized crack

Fig. 4 Fig. 5

 45 Fig. 4 Pressure and discharge along the crack line plotted for different values of λ. Left: pressure P λ along the x-axis, right: discharge q in the crack (0<X<1)

Fig. 7

 7 Fig. 7 Cracks network in geological rock formation

Fig. 8

 8 Fig. 8 Periodic distribution of parallel cracks in infinite porous media

Fig. 9

 9 Fig. 9 Comparison of effective permeability in the direction x obtained by three different ways:numerical method and Mori-Tanaka and self-consistent estimations (λ=1)

Fig. 10

 10 Fig. 10 Pressures along the lines at different level of y these are parallel to the axe x on whichan isolated crack is placed (λ=∞)

(2)The flow model in the cracks is described commonly by a Poiseuille type law[START_REF] Louis | A study of groundwater flow in jointed rock and its influence on the stability of rock mass[END_REF][START_REF] Norton | Transport phenomena in hydrothermal systems: the nature of porosity[END_REF][START_REF] Witherspoon | Validity of cubic law for fluid flow in a deformable rock fracture[END_REF][START_REF] Guéguen | Introduction to Physics of Rocks[END_REF]