Martine Vanhove Llacan

Mohamed-Tahir Hamid

Olfactory, gustatory and tactile perception in Beja (North-Cushitic)

Keywords: Perception, semantics, syntax, metaphor, Beja 1

This chapter studies the perception words associated with the so-called "lower senses" (olfaction, gustatory and tactile perceptions) in Beja, a Cushitic language spoken mainly in Sudan, from three different perspectives. The first one concerns the organization of the related lexicon from the point of view of word categories, and their semantic aspects along a positive / negative and more intense / less intense continuum.

Numerically, it shows a prevalence of the verb category over nouns and adjectives, and a more varied lexicon for smell, than for taste and touch. The second one provides a study of the syntactic constructions of perception words. It shows a prevalence of source-oriented constructions except in the tactile domain. The third one deals with the metaphorical extensions of the perception words for each sensory modality. The last section discusses the Beja findings in a broader comparative and typological perspective.

Introduction

Beja is the sole member of the North-Cushitic branch of the Afroasiatic phylum. It is mainly spoken in Eastern Sudan. The 1993 census, the last to contain a question about language, gives a total number of approx.

1,100,000 speakers. There are probably more now. Dialectal differences are not numerous, mostly lexical and phonological, and they do not jeopardize inter-comprehension.

The data used for this chapter are taken from various sources: texts of the central and southern varieties recorded in Sudan between 2000 and 2011 by the first author, dictionaries (Reinisch 1895, northern variety;Roper 1928, central variety), and elicitation with the second author who is an anthropologist and a native speaker of the southern variety. It was performed during his stay in Paris in 2016, but there was time only for basic elicitation, and none to use the stimuli that deal with the traditional five senses, in particular the olfactory kit designed at the MPI [START_REF] Majid | The language of olfaction[END_REF]). Consequently, it was not possible to answer the research question about the terminological and constructional consistency within the Beja community for describing sensory experiences.

After a brief outline of the main characteristics of the Beja morphosyntax necessary for a better understanding of the examples (Section 2), this chapter investigates the three lower senses of olfaction (Section 3), taste (Section 4) and touch (Section 5), because there is some overlap between the three modalities. For all senses the organization of the lexicon is discussed, then their syntactic constructions, and finally their metaphorical extensions. The conclusion (Section 6) discusses the Beja findings in a broader comparative and typological perspective.

The Beja vocabulary associated with the three lower senses is not particularly elaborated although not very poor: All in all, 56 terms were recorded, but this figure goes down to eighteen when one considers the number of different roots, one root being often used for the four word categories involved: verb, noun, adjective and manner converb. We are far from the richness of certain Bantu languages such as Lumongo [START_REF] Hulstaert | Dictionnaire lomongo-français[END_REF] or Setswana [START_REF] Creissels | Dictionnaire françaissetswana, setswana-français. Mogoditshane[END_REF]. Moreover, Beja has neither phonemic specificity nor morphological dedicated category for the expression of the lower senses, such as ideophones (which do not constitute a word class in Beja), categorical hybrids or invariant nouns, as is often the case in the languages of Africa for which we have available studies, e.g. in Kapsiki (van Beek 1992), Tarok [START_REF] Blench | Issues in African Languages and Linguistics: Essays in Honour of Kay Williamson [Special Issue of the Nigerian Language Studies[END_REF], Luwo [START_REF] Storch | Haptische, visuelle und olfaktorische Sprachen: Westnilotische Wahrnehmungen[END_REF], Kambaata [START_REF] Treis | Perception verbs and taste adjectives in Kambaata and beyond[END_REF], or Siwu [START_REF] Dingemanse | The meaning and use of ideophones in Siwu[END_REF][START_REF] Dingemanse | The semantic structure of sensory vocabulary in an African language[END_REF]).

Some basic aspects of Beja morphosyntax

Beja has a rich and complex morphology, both verbal and nominal. Part of the lexicon is organized, as in Arabic, with a system of roots, stems and patterns based on a combination of ablaut, affixes and reduplications that form nouns, adjectives, plurals, verb inflection and derivation. The morphology is also concatenative, partly with portmanteau morphemes.

Beja is a marked nominative language, with three lexical cases, nominative, accusative and genitive. They are marked on clitic determiners for the first two (overtly expressed only if the syllabic structure of the noun permits), and on the dependent noun for the last one. Pronouns have in addition dative and ablative/locative 1 cases, but dative, ablative and locative are expressed by (clitic) postpositions for nouns.

1 For sake of brevity, this set of pronouns is only glossed ABL (ablative) in the glossing lines. Syntactically, Beja is largely head final. The canonical word order is (X)(S)(O)V and the clause order is dependentmain clause. But word and clause order may vary for pragmatic reasons.

Olfaction

Organization of the lexicon

The vocabulary of olfaction is the most elaborated among the lower senses:

27 items, representing almost half of the total vocabulary of the lower senses (48.20%), and exactly half of its number of roots (9/18).

Table 1 presents the olfactory lexicon organized on the vertical axis on a scale of intensity, and on the horizontal axis by hedonic polarity, positive vs negative (attractive/pleasant vs repellent/unpleasant smells), and word category. One root, ɖʔaːʃ 'to exhale a light smell', is an Arabic loan. 3 3 The Sudanese Arabic noun daʕaːʃ, a poetic term, means precisely 'smell of a mixture of rain, dust and fodder grass; smell of rainy clouds beginning to The verb category is the largest (18 items: 9 roots and 1 compound), followed by nouns (12 items: 4 roots and 1 compound), while the adjective and manner converb (used as attributive adjective) categories are the poorest (6 items: 4 roots).

There are more roots to express intense negative odors: four (ɖimi 'stink', kʷimha 'smell bad', ʔabis 'exhale a fermented smell', harrʔaːk 'smell of burning'), while there is only one (ʃikʷaːn 'emit a pleasant odor, smell nice, perfume o.s.') to express intense positive odors. This difference reflects a general tendency across languages (Boisson 1997: 9).

Within the neuter scale of intensity, words denoting pleasant smells predominate over unpleasant ones: One root (fʔa 'smell, sniff'), and two derived verb forms (ʃa~ʃiːk-am 'to smell, scent, inhale', and ʃiːk-am-s 'to make sniff') are exclusively dedicated to pleasant odors. Two verbs of the same root (ʃiːk 'to snuffle, sniff' and ʃiːk-am 'to sniff') can only marginally be associated with unpleasant smells,4 in the sense that these activity verbs are used when the speaker is trying to figure out if a smell is pleasant as expected, even though an unpleasant one could be detected. Two verbs, baraːm 'to smell of', and ɖʔaːʃ 'to exhale a light smell', can be used for both hedonic polarities, but the noun form baraːm 'smell, odor' is usually shape; nice smell; cold wind that blows after a rain fall' ('Awn Al-Sharif Qasim 2002: 333).

qualified by the repellent manner converb, ɖimj-a 'stinky', when referring to a repulsive odor.

The lower part of the intensity scale shows a complex construction involving the activity verb ʈʔa 'to hit' and the neuter noun baraːm 'smell, odor': baraːm ʈʔa 'to exhale a light smell'. The corresponding noun is a compound which nominalizes the verb ʈʔa 'to hit' by means of the dummy noun na(ːj) 'thing': ʈʔa-naːj 'light smell'. Although not strictly basic terms since they are bi-lexemic, the constructions are felt as belonging to the basic set of olfactory verbs by the second author. These complex forms are reminiscent of the 'hit' origin of certain (intense and unpleasant) smell terms in some Indo-European languages [START_REF] Buck | A Dictionary of Selected Synonyms in the Principal Indo-European Languages: A Contribution to the History of Ideas[END_REF][START_REF] Buck | A Dictionary of Selected Synonyms in the Principal Indo-European Languages: A Contribution to the History of Ideas[END_REF]: 1023-1024;

Boisson 1997: 7). The differences are that in Beja the constructions are evaluated as non-intense, and can be used for both attractive and repulsive smells.

Two experience verbs in the negative polarity are related to specific kinds of smell: ʔabis 'to exhale a fermented smell', perceived as intense, and harrʔaːk 'to smell of burning' 5 which is only perceived as slightly unpleasant although intense. 5 It is unsure if this verb is an Arabic loan from haraqa 'burn'. If a change from q/g in Arabic to k in Beja is attested, the addition of a laryngeal stop ʔ is unattested in borrowings from Arabic.

Morphologically, the adjective and the manner converb of the upper part of the intensity scale display the augmentative suffixes -loːj and -l. It was not possible to elicit these suffixes with the other olfactory adjectives and manner converbs. Below are two examples taken from texts.

( Within the verb derivational system, all verb forms are basic forms (i.e. underived), except three in the neuter lines. ʃiːk-am 'to sniff' is the middle derived form of ʃiːk, ʃiːk-am-s its middle and causative (see ex. 3 below), and ʃa~ʃiːk-am 'to smell, scent, inhale' is the pluractional of the middle form. The fact that the latter is perceived as a neuter verb form in terms of intensity implies that its semantic value has nothing to do with intensity, one of the possible meanings of pluractionals, but only with plurality of events, be they repeated or successive actions, or distributive actions with several localizations or goals.7

(3) i=ginuːf-i=da

DEF.M=nose-GEN=DIR daː-s-i=t be_there-CAUS-AOR.3SG.M=COORD ʃiːk-am-s-eːn oː=doːr sniff-MID-CAUS-IPFV.3PL DEF.SG.M.ACC=time i=ga~gam-i=eːb REL.M=PLAC~shout-AOR.3SG.M=REL.M
'They used to put it (the dry meat) under its nose and make it (the baby) smell it when it used to cry a lot.' (in order to calm the baby down) (BEJ_MV_NARR_33_Meat_11)

Semantically, there is one dialectal difference. The verb ʃiːk is only known with the meaning 'to snuff; to chew tobacco' in the southern variety. This dialect uses instead the middle form ʃiːk-am (known also in the other two varieties), which precisely means 'to sniff for a long time, taking a long breath, in order to determine the nature of the thing smelled' (e.g. a farmer in order to know if the sorghum is ripe or not, or if the milk, cream or ghee is good or not). The precise meaning of fʔa is also 'to smell, sniff in order to determine the nature of the thing smelled', but the odor is necessarily pleasant. The neuter verb ɖʔaːʃ 'to exhale a light smell', also means 'to exhale an unexpected smell'.

The Beja lexicon shows a semantic connection between olfaction and weather terminology. The generic and neuter noun baraːm 'smell, odor' also means 'wind', and another neuter noun dʔaːʃ 'light smell' also means 'breeze'. Another noun for 'breeze' is the compound baraːm-ʈʔa (lit.

smell/odor/wind-hit) at the bottom of Table 1, which also shows the semantic link between weather terms and olfaction.

An adjective and a manner converb are used to qualify odor terms.

The adjective comes from the temperature semantic field: 8 lʔa 'cold', and can apply to both an attractive and a repellent smell: lʔa baraːm-i=t ʃakʷiːn

[cold smell-GEN=F perfume] 'a perfume with a light (unpleasant or not) smell' (lit. of a cold smell). Temperature terms are also sporadically associated with olfaction in other languages of the world, with different shades of meaning: English and Serbian (Rasuli 2015: 288), Ukranian (Kryvenko 2015: 321), Finnish (Juvonen and Nikunlassi 2015: 525), and

Indonesian (Siahaan 2015: 695). Olfaction is associated to a manner converb, ʔakr-a 'strong' denoting physical strength (which also applies to natural elements such as 'wind', 'storm'), an association not uncommon cross-linguistically, such as in French: ʔakr-a baraːm-i=t ʃakʷiːn [be strong-CVB.MNR smell-GEN-F perfume] 'a perfume with a strong smell; an intoxicating perfume'.

8 For a preliminary experiment concerning the association of olfaction and temperature terms in diverse languages, see [START_REF] Wnuk | Hot and cold smells: Odor-temperature associations across cultures[END_REF].

On the other hand, the intense positive and negative adjectival and converb forms, ʃikʷaːn 'emitting a pleasant odor; perfumed', and ɖimj-a On the contrary, the three activity verbs, fʔa 'to smell, to sniff', ʃiːk and ʃiːk-am 'to sniff' are not source-oriented. They are transitive, the subject (represented in the verb inflexion in ex. 12 and 13) is the experiencer, and the source of the odor is the direct object.

'

Taste

Organization of the lexicon

In terms of the number of lexical items, taste is the second sensory modality.

There are five gustatory roots which produce sixteen items, representing 28.5% of the total lexicon of the lower senses. These lexical items are distributed across word categories as follows: seven verbs for four roots, six nouns for four roots, four adjectives for four roots and one manner converb for one root. Three roots (marked with # in The taste experience verb dʔam 'to taste of' is related to the Arabic root ṬʕM 'taste'. It can be made active via the causative derivation: dʔam-s 'to taste, make taste', which is a polite way to invite people to eat, when used in the imperative paradigm: dʔam-s-a 'please eat!' (lit. taste!), instead of the straightforward order tam-a 'eat!'. 10 Its masdar is dʔam-si-ti '(act of) tasting' and dʔam 'taste' is the generic noun for the sensory modality. There is dialectal variation for this Arabic borrowing. [START_REF] Roper | TuBeḍawiɛ". An Elementary Handbook for the Use of Sudan Government Officials[END_REF], who described the central variety, does not provide the underived verb form (which is used today in this dialect), and the root is even absent from 10 dʔam has no imperative. 11 jam 'water', a masculine plural noun, becomes feminine (a common way of forming diminutives) when designating a 'well' by contiguity extension.

Syntactic constructions

The neuter gustatory verb dʔam 'to taste of', although basically a sourceoriented experience verb, can be used in three different constructions:

intransitively (19, 20), as expected, with or without an ablative/locative 3 rd person pronoun, which refers to the source or stimulus of the taste, but also transitively (21, 22) with an inanimate source as the subject.

Furthermore, the semantics of the utterance varies with the syntactic construction. When the ablative/locative pronoun is used, the type of taste is considered as an intrinsic gustatory property of the entity it refers to (19,21). If this entity is lexically expressed, it is syntactically an object in the accusative case (21), as expected from a transitive construction. On the contrary, the entity is the subject if the gustatory property is not considered as intrinsic, and the source is the object of the verb ( 22) or if the noun is used in a copular expression instead of the verb ( 23). nifir 'be sweet', hamir 'be sour' and hami 'be bitter' are all intransitive stative verbs, and are all source-oriented.

Metaphors

Taste-based metaphors are much richer than olfactory-based ones, and involve all word categories. Twelve metaphors have been recorded so far.

Depending on the context, dʔam, as a verb and a masdar, gives rise to metaphors in four semantic domains.

The first one refers to guessing. It makes use of the verb form in a construction with a benefactive auxiliary, where the source is the subject and the experiencer is expressed by an ablative/locative pronoun. 'hot'. 12 The experiencer is the subject.

(31) nabʔa na dʔam-s-ja hot thing taste-CAUS-PFV.3SG.M 'He suffered' (lit. 'he tasted a hot thing')

The gustatory adjectives can be used to characterize voice types, a case of synaethesia to the hearing domain, and characters or behaviors, bad or good.

Even though hamir 'sour' is experienced as pleasant in the sensory modality domain, it gives rise to a negative metaphor: hamr-a his (lit. sour voice) is always felt negatively and refers to a too deep voice. It is said of a male voice, also when not perfectly in tune. hami his (lit. bitter voice) is also felt negatively and simply refers to the unpleasant voice of a singer.

nafir 'sweet' provides a pleasant judgement as in (32), where his 'voice' is in the genitive case (the same syntactic construction applies to the other two adjectives), in an attributive construction with a copula.

12 Note that in many other languages (see e.g. the articles in Koptjevskaja-Tamm 2017), hot temperature is associated with pain.

(32) nafir his-iː=b=i sweet voice-GEN=INDF.M=COP.3SG

'He has a nice voice' (lit. he is a of sweet voice)

Applied to bhali 'words', the adjective nafir 'sweet', as in nafir bhali 'sweet words', implies that the speaker's formulation and choice of words perfectly respects the Beja socio-discursive and stylistic rules (cp. ex. 30).

Only nafir and hami are used to qualify positively and negatively, respectively, someone's character or the behavior. The head noun is also in the genitive, in an attributive construction with a copula.

(33) hami dab-iː=b=i bitter character-GEN=INDF.M=COP.3SG

'He has a bad character' (lit. he is of bitter character) (34) nafir dab-iː=b=i sweet character-GEN=INDF.M=COP.3SG

'He is easy-going' (lit. he is a of sweet character)

Touch

Organization of the lexicon

The tactile sense is lexically the less represented. It is expressed by four roots and thirteen items (23.20% of the lexicon of the lower senses): two roots in the verb (seven items) and noun (two items) categories, and two other roots (two items) in the adjective category. They are presented below in Table 3, grouping together the tokens belonging to the same root, regardless of intensity and hedonic polarity which are irrelevant for this sensory modality at the root level. The basic transitive verb is tha 'to touch', masdar tha-ti. It has four derived forms: causative tha-s, middle tha-m, reciprocal-sociative tha-s-am, pluractional tha~tha 'to frisk, palpate'. 13 The transitive verb ʃiːʃ is specifically 'to run one's hand on s.th'. It is translated as 'fühlen' ('to feel') by Reinisch. Roper only provides the 13 Strik Lievers (2007: 178) is right in remarking that 'touch' verbs "do not describe a perception, but an action causing a perception".

pluractional form ʃa~ʃiːʃ 'to grope'. It has a masdar form, ʃa~ʃiːʃ-ti 'groping, passing the hand repetitively on something to caress, examine a content…'.

Two adjectives are typically related to the tactile domain: lamsa 'smooth'

(which is a borrowing with a metathesis from the Arabic adjective ʔamlas 'smooth', and has no corresponding verb or noun form), and nʔakʷ 'tender'.

The antonym of the former is formed by a negative verb phrase lamsaː=b ki-iki 'rough, rugged, coarse' (lit. it is not smooth). The second adjective can also be used as a stative verb: nʔakʷ 'to be tender'.

Syntactic constructions

Contrary to olfaction and taste, both tactile verbs are transitive and experiencer-oriented.

( The expression for a tough character, ruthless, pitiless, akra mana 'strong bowels', is not based on the gustatory lexicon (but the body part remains).

Conclusion

Compared with other non-Indo-European languages such us Amis (Formosan, Austronesian) [START_REF] Lee | Lexical categories and conceptualization of olfaction in Amis[END_REF], Aslian languages (Austronesian) [START_REF] Burenhult | Olfaction in Aslian ideology and language[END_REF], Totonac (Mixe-Zoque, Mexico) [START_REF] Aschmann | Totonac categories of smell[END_REF], Hawaiian (Polynesian, Austronesian) (Boisson 1995: 9), Jahai (Mon-Khmer) (Majid & Burenhult 2014), or the African languages mentioned in Section 1, the lexicon of the lower senses is not very elaborate in Beja. Still some interesting results emerged from our preliminary investigation.

The organization of the lexicon across word categories shows a prevalence of the verb category over nouns and adjectives (and manner converbs), and a more varied lexicon for the olfactory modality than for gustatory and the tactile ones.

In the domain of repellent smells, noteworthy is the existence of a specialized mono-lexemic root for a fermented smell, recorded only in one (unspecified) language in Boisson's (1997: 41) sample of 60 languages.

Still, Beja is one more illustration of a general tendency across languages [START_REF] Buck | A Dictionary of Selected Synonyms in the Principal Indo-European Languages: A Contribution to the History of Ideas[END_REF]Boisson 1997: 38) to have a terminological prevalence (in terms of different roots and meanings) of repellent smells over pleasant ones.

The connection between weather and temperature terminology and part of the olfactory lexicon, be it the polysemy with 'wind', 'weather', 'breeze', or the use of the temperature qualia 'cold', is also remarkable.

Whether the weather polysemy is due to language contact with Sudanese Arabic or not is difficult to assert. To the best of our knowledge, it is not attested in East-Cushitic or Central Cushitic, the closest relatives of Beja within the Cushitic branch, but this pattern also exists in unrelated language, e.g. in other African languages such as Dendi (Songhay, Nilo-Saharan), Bwamu (Gur, Niger-Congo), Sereer and Peul (Atlantic, Niger-Congo) (see Segerer & Flavier 2011[START_REF] Vanhove | The Beja corpus[END_REF]. Moreover, 'wind' is considered as the source of olfactory words in a few Indo-European languages [START_REF] Buck | A Dictionary of Selected Synonyms in the Principal Indo-European Languages: A Contribution to the History of Ideas[END_REF][START_REF] Buck | A Dictionary of Selected Synonyms in the Principal Indo-European Languages: A Contribution to the History of Ideas[END_REF]:

1024). For lack of comparative data with other Cushitic languages, it is impossible to assess which of the two meanings is the source of a possible semantic evolution in Beja at the origin of the actual polysemy. On the other hand, the link between 'wind' and 'smell' in Beja bears no trace of a possible ancient meaning 'type of odor blown by the wind' lexicalized in unrelated languages such as Hawaiian (Boisson 1995: 39).

For the gustatory domain, one can remark that most of the roots are borrowings from the contact language, Sudanese Arabic, which probably mirrors the fact that the traditional diet of the Beja people is not very diverse. Nevertheless, it is interesting to note that Beja is one of these languages where evaluative taste words conflate (excess of) bitterness and (excess of) salt (see [START_REF] Myers | The taste-names of primitive peoples[END_REF][START_REF] Majid | Language does provide support for basic tastes [Commentary on A study of the science of taste: On the origins and influence of the core ideas by Robert P. Erickson[END_REF], for the various semantic partitioning and groupings of taste words across languages). Krifka (2010: 239), basing his remark on English and German data only, postulates a general positive hedonic implicature for general words denoting the sense of taste. If this would need to be confirmed by empirical data from other languages, at least Beja shows the same partitioning as English and German.

As far as the lexicon is concerned, the tactile domain can be considered as poor in Beja.

Regarding the syntactic constructions, we can recall that verb constructions are predominantly source-oriented for the olfactory and gustatory domains, while the tactile one is active.

Since [START_REF] Sweetser | From Etymology to Pragmatics[END_REF] study of the semantic changes in the domain of English senses where she established the metaphorical mappings of the lower senses as going from TASTE to LIKES and DISLIKES, TOUCH to FEELINGS, and SMELL to 'DISLIKEABLE' FEELINGS, further studies (e.g. Ibarretxe-Antuñano 1999 for English, Spanish and Basque) have

shown other metaphorical extensions that, even in English, go beyond these semantic targets.

In Beja, olfaction develops metaphors for a very restricted number of both positive and negative feelings and emotions, as in the three languages mentioned by Ibarretxe-Antuñano.

In spite of a rather poor lexicon, and a not very varied diet, it is the gustatory sense that is most often recruited for metaphorical extensions, including for domains where olfaction is recruited in other languages, such as guessing or suspicion (see e.g. [START_REF] Viberg | The verbs of perception: A typological study[END_REF]: 158, based on Ullmann 1957: 266ff). Similarly, the temperature adjective for 'hot' is recruited for a taste-based metaphor where other languages apply it to olfaction words [START_REF] Wnuk | Hot and cold smells: Odor-temperature associations across cultures[END_REF]). It should be noted that the 'cold' adjective applies to olfaction, without a metaphorical extension of the smell word itself.

As for the tactile domain, which is lexically even poorer than taste, metaphorical extensions show quite a large variety, which includes the semantic fields of cognition, psychological states and feelings, physical pain, money or goods reception, irregular activities. Some of them are known in unrelated languages such as French for the physical pain and the reception of money or goods, or Lussusu (also known as Lussese), a Bantu language, for bewitchment (Thanassoula 2012: 318). Others are more unexpected from a western perspective such as 'irregularity' expressed by a transparent compound involving success or failure of the touch activity, 'intelligence' which combines tenderness with an Arabic borrowing, fam 'understanding', or 'pity' which associates an internal body part and softness. Viberg (1984: 159) recalls [START_REF] Ullmann | Principles of Semantics[END_REF] study of transfers from one perception domain to another in a corpus of 19 th century poetry in which, on the basis of frequency, he proposed the following hierarchy: touch > hear > taste > scent > sound > sight Even though we could not conduct a frequency analysis, it is interesting to note that as far as the number of synaethesias is concerned, Beja does not confirm this hierarchy, with three metaphorical transfers to another sensory modality for taste, and none for touch and smell.

  There are two genders, masculine and feminine, two numbers, singular and plural, and two types of articles. The definite ones are proclitic, the indefinite are enclitic. They are overtly marked for gender, number and case, again if the syllabic structure of the noun permits. Nouns may thus be used in the bare form without any overtly expressed determiners. 2 Verbs have finite and nonfinite forms. Verb paradigms are marked for aspect, mood, person, number and gender. Finite forms have three aspects: imperfective, perfective and aorist. Complex predicates are used to express other aspects, tenses and modalities, and for light verb constructions. They are formed with auxiliaries or the copula. Beja has two morphologically based verb classes: A small majority is conjugated with prefixes (and suffixes in the plural); it is historically the oldest verb class. The other verb class is conjugated with suffixes only. There are six to seven derived verb stems (depending on the verb class): intensive, pluractional, middle, passive (for the prefix class), reciprocal-sociative, causative, double causative. Combinations of two, sometimes three, derived forms are possible. Nonfinite forms amount to four converbs: general, sequential, simultaneity and manner. The manner converb can also be used as an adjective modifier. 2 For sake of clarity, the syntactic function of bare nouns in the examples is marked in the glossing line in between brackets: [SBJ] for the subject function and [OBJ] for the object function.

  DEF.SG.M.ACC=flower smell-PFV.1SG 'I smelled the flower' (14) oː=faːr ʃiːk-am-an DEF.SG.M.ACC=flower sniff-MID-PFV.1SG 'I sniffed the flower' Odor expressions can be constructed with the transitive verb ʈʔa 'to hit'. Contrary to the above transitive olfactory verbs, and like the intransitive verbs, the subject is the source of the odor. But unlike the intransitive verbs, the experiencer is expressed by an object pronoun, not an ablative/locative one. Such a construction is of course linked to the fact that ʈʔa is transitive. (15) ʔakr-a=t ʃakʷiːn be_strong-CVB.MNR=INDF.F fragrance ti-ʈʔa=heːb 3SG.F-hit\PFV=OBJ.1SG 'I smelled an intoxicating perfume' (lit. a strong perfume hit me) 3.3. Metaphors Metaphors based on olfaction are not numerous. It is actually the sensory modality which produces the lowest number of metaphors. The three attested metaphors are linked to the semantic domains of feelings and emotions. They are based on the intensity neuter verbs, the active fʔa 'to smell, to sniff', and the source-oriented baraːm 'to smell of'. fʔa can be used with the meaning 'to kiss' in between lovers, when the verb is preceded by the ablative/locative 3 rd person pronoun: hoːj fʔaː 'to kiss' (lit. to smell from it); the subject is the agent. (16) ahmad lajlaː=b hoːj Ahmad Layla=INDF.M.ACC ABL.3 fʔi-ja sniff-PFV.3SG.M 'Ahmad kissed Layla' (lit. A. sniffed from/inside L.) The colexification of 'smell' and 'kiss' is widespread in the languages of Southeast Asia (see Shapper, this volume); it seems to be only sporadically attested in the languages of Africa. A search in the RefLex database (Segerer & Flavier 2011-2018), which contains lexicons of 782 African languages, provided only five other languages of the Niger-Congo phylum with this colexification: Dii (Adamawa), Bambara and Soninke (Mande), Wolof and Fula (Atlantic). 9 baraːm 'to smell of' is used to express yearning or anticipation of pleasant events, the latter being close to 'sense something intuitively' as described by 9 There are more examples of African languages (15) which colexify 'kiss' with 'taste' or 'suck'. Ibarretxe-Antuñano (1999) for English, Spanish and Basque. Like in the concrete sense of the sensory modality, the syntactic construction makes the source of the emotion the subject of the verb and the ablative/locative pronoun refers to the experiencer. (17) w=haːʃ hoːj baraːm-ja DEF.M.NOM=country ABL.3 smell_of-PFV.3SG.M 'He was homesick' (lit. the country smelled from/inside him) (18) ramadaːn hoːsi baraːm-ja Ramadan ABL.1SG smell_of-PFV.3SG.M 'I felt Ramadan coming' (lit. Ramadan smelled from/inside me)

'

  DEF.PL.NOM=tear\PL salt[OBJ] taste_of-IPFV.3SG.M 'The tears taste of salt' The food has a taste of burnt' (lit. is a burnt taste) When a human experiencer is present in the utterance, two constructions are possible, either with the basic neuter verb dʔam, or with the masdar of the positive and negative verbs. With the neuter verb, the construction is intransitive, the experiencer is represented by a directional construction with the postposition dhaːj (var. dh before pronouns) and the subject is the source of the taste. sugar in the food' (lit. sugar tastes to me in the food) With the masdars, the experiencer is the subject of the transitive verb miri 'to find', and the gustatory masdar is the object head of a genitive noun phrase. (25) sukʷkʷar=t-i nafruːr sugar=INDF.F-GEN sweetness ti=mʔari=t=iːb a-marri DEF.F=food=INDF.F=LOC.SG 1SG-find\IPFV 'I taste sugar in the food' (lit. I find a sweetness of sugar in the food)

( 26 )

 26 daːji ʃaga hoːj dʔam-jaː-j good work ABL.3 taste_of-PFV.3SG.M-L ji-hiː 3SG.M-give\PFV 'He smelled a good bargain' (lit. a good work tasted from/inside him) Positive and negative feelings and emotions can also be expressed with the noun form dʔam as the object of the verb miri 'find' whose subject is the experiencer. The ablative/locative pronoun this time refers to the source. The precise type of feeling is determined by the context. The construction denotes suspicion in (27) (where dʔam is the head of a genitive phrase), and pleasure in (28-29). The source of the feelings is represented by the ablative/locative pronoun in (27) and (28) and by a full lexeme with taste ABL.3 3SG.M-find\PFV 'He smelled the trap' (lit. he found a taste of trap from/inside it) in spite of what I did before, now, slowly, I really liked it' (lit. I found taste from/inside it) (BEJ_MV_NARR_03_Camel_208-210) (29) i=neːw-a=eːb dʔam miri-tiːt DEF.M=insult-PL=LOC.PL taste find-CVB.SEQ 'After she was pleased with the insults…' (lit. she found taste in the insults) (BEJ_MV_NARR_25_Chief_074-075) The last semantic domain of a taste-based metaphor is speech. The metaphor is constructed with the noun form dʔam, which is the object of a possession verb. This metaphor is linked to the socio-discursive and stylistic rules of speech of the Beja society. (30) i=bhali=aːkna dʔam DEF.M=word\PL=POSS.2PL.NOM taste ki=iː-bar-iːn NEG.IPFV=3-have\AOR-PL 'Your words are improper' (lit. your words have no taste) (Roper 1928: 171) = your words do not respect the Beja sociodiscursive and stylistic rules With the causative derived verb form dʔam-s, another metaphor of feeling is built, namely a metaphor of suffering (physical or psychological) which includes a synaethesia from the touch modality: The object of the verb consists of the dummy noun na 'thing' and the temperature adjective nabʔa

Table 1 .

 1 The olfactory lexicon of Beja

		Positive			Negative
		verb	noun	adjective	verb	noun	CVB.MNR
				ʃikʷaːn-loːj			ɖimj-a-l
				'perfumed-			'stink-
				AUG'			CVB.MNR-
							AUG'
		ʃikʷaːn	ʃakʷiːn	ʃikʷaːn	ɖimi 'stink' ɖimj-aːj=t	ɖimj-a
		'emit a	'fragrance'	'perfumed'		'stink-	'stink-
		pleasant				N.AC=F'	CVB.MNR'
		odor'
					kʷimha	kʷimha	kʷimha-a
	+intense				'smell bad'	'bad smell'	'smell bad-
							CVB.MNR'
					ʔabis	ʔabis	ʔabis-a
					'exhale a	'fermented	'exhale a
					fermented	smell'	fermented
					smell'		smell-
							CVB.MNR'
					harrʔaːk
					'smell of
					burning'
		baraːm	baraːm		baraːm	ɖimja
		'smell of'	'odor'		'smell of'	baraːm
						'stinking
						odor'
		fʔa 'smell'	fʔa-ti
			'smell-
			N.AC'
		ɖʔaːʃ	ɖʔaːʃ 'light		ɖʔaːʃ
		'exhale a	smell'		'exhale a
	neuter	light smell' ʃiːk 'sniff'			light smell' (ʃiːk 'sniff')
		ʃiːk-am			(ʃiːk-am
		'sniff-MID'			'sniff-
					MID')
		ʃa~ʃiːk-am
		'PLAC-
		sniff-MID'
		ʃiːk-am-s
		'sniff-MID-
		CAUS'
		baraːm ʈʔa	ʈʔa-naːj		baraːm ʈʔa	ʈʔa-naːj
	-intense	'exhale a	'light		'exhale a	'light
		light smell'	smell'		light smell'	smell'

  stinky', are used to qualify the intensity neuter noun baraːm 'smell, odor'

		(9) eːga	baraːm-iːni
	3.2.	Syntactic constructions smoke smell_of-IPFV.3SG.M
			'It smells of smoke' (lit. smoke smells)
	as attractive or repellent: ʃikʷaːn baraːm 'a nice smell', ɖimj-a baraːm 'a ʃikʷaːn 'to emit a pleasant odor', baraːm 'to smell of', and ɖʔaːʃ 'to exhale a (10) amba baraːm-iːn
	stinky smell'. light smell' are intransitive experience verbs and are all source-oriented. shit\PL smell-AOR.3PL
	Nouns of olfaction (masdars, i.e. action nouns) are derived from the They are non-agentive and the subject is always the source of the odor. 'It was smelling of shit' (lit. shit smelled)
	corresponding verb forms. One example is in (4). Below are examples with the neuter verb baraːm. The experiencer is either (BEJ_MV_NARR_18_Hunchback_416)
	(4) t=hoːj=t=aːk an ablative/locative pronoun (6-8), or not mentioned at all (9-10). hallaːn
	DEF.F=belly=INDF.F=POSS.2PL.NOM now (6) uː=faːr hoːsi baraːm-ja (11) and (12) are examples with the pleasant and unpleasant verbs of
	olfaction.	ɖimij-aːj=t	ʔataːb=t=u
		(11) ʃakʷiːn=t		ɖʔaːʃ-tini
		(7) ʔakr-a=t fragrance=INDF.F exhale_a_light_smell-IPFV.3SG.F ʃakʷiːn hoːsi
			be_strong-CVB.MNR=INDF.F 'A perfume exhales a light smell'	fragrance	ABL.1SG
		baraːm-tini (12) uː=buːn	harrʔaːki-ja
			smell_of-IPFV.3SG.F DEF.SG.M.NOM=coffee smell_of_burning-PFV.3SG.M
			'I smell an intoxicating perfume' (lit. a strong perfume smells 'The coffee smelt of burning'
			from/inside me)
		(5) ʔabis (8) misuːsannaːj=t-i=t	di-jaː=t	ɖimj-aːj	hoːsi
			fermented_smell carrion=INDF.F-GEN=INDF.F stink-N.AC ABL.1SG say-CVB.MNR=REL.F
			lam=t=i baraːm-ta
			sauce=INDF.F=COP.3SG smell_of-PFV.3SG.F
			'It is a sauce whose smell is unpleasantly fermented' 'I smelled the stinking smell of the carrion' (lit. the stinking of
			the carrion smells from/inside me)

stink-N.AC=INDF.F full=INDF.F=COP.3SG 'Your belly now is full of stinky things' (BEJ_MV_NARR_32_King_52-55) ʔabis 'to exhale a fermented smell' is commonly used in light verb constructions with the verb di 'say' instead of the ordinary verb paradigms, to express a particularly intense and unpleasant smell as in (5). DEF.SG.M.NOM=flower ABL.1SG smell_of-PFV.3SG.M 'I smelled the flower.' (lit. the flower smelled from/inside me)

Table 2 .

 2 Table 2) are borrowings from Arabic. The lexicon is listed in Table 2 below by word category on the horizontal axis and hedonic polarity on the vertical axis. The gustatory lexicon of Beja

		Verb	Noun	Adjective	Manner converb
		nifir 'be sweet'	nafruːr	nafir 'sweet'
			'sweetness'
		si-nafir
	Positive	'sweeten' #hamir 'be			#hamr-a 'sour'
		sour'
		#s-hamir 'make
		sour'
		#dʔam 'taste of'	#dʔam 'taste'/	#taːmi 'sweet'
		#dʔam-s 'taste'	#taːmi
	Neuter		'sweetness'
			#dʔam-si-ti
			'taste'
	Negative	#hami 'be	#hami-jaːj	#hami 'bitter'
		bitter'	'bitterness'
			siraːj 'very bad	siraːj
			taste'	'disgusting'

  [START_REF] Reinisch | Wörterbuch der Beḍawiye-Sprache[END_REF] who instead provides the form t mi, a noun and an adjective, 'süβ; süβigkeit' ('sweet; sweetness'), placed under the main heading tam 'to eat', which Reinisch relates to Arabic ṬʕM (as dʔam), and

	salted river), t=hami jam 'the well of bitter water'. 11 When talking about
	camel or cow milk, which traditionally constitute, with sorghum gruel, the
	basic diet of the Beja people, hamir typically applies to fermented milk with
	cognate forms in various Semitic languages (including Ethio-Semitic Tigre a good taste, while hami applies to a bad milk that has turned bitter. The
	and Gə'əz). Obviously, there are several layers of borrowings that took third evaluative taste is siraːj, a noun and an adjective, which indicates an
	place in Beja, which gave different phonetic outputs and different shades of extreme negative degree of bitterness, a very bad taste, in particular for too
	meaning. salty food, again conflating bitterness and saltiness. The last gustatory root,
	Beja distinguishes four physiological evaluative tastes.
	hami 'to be bitter', an unpleasant one, which also conflates with the
	meaning 'to be too salty' applied to food and water alike. Whether this
	distinction is also made by other Beja speakers and varieties is open to
	further investigation. Both Roper's and Reinisch's dictionaries only mention
	the meaning 'salty, brackish' for water: w=hami bhar 'the sea' (lit. the

hamir 'to be sour', its causative s-hamir, and the manner converb hamr-a 'sour' come from the Arabic root XMR, which among other things means 'ferment; yeast'. 'Yeast' is the meaning of the Beja derived noun hamiːr, not 'sourness', hence its absence form Table

2

. hami 'bitter', hami 'to be bitter', hami-jaːj 'bitterness' (lit. to be bitter + masdar suffix -jaːj) are related by

[START_REF] Reinisch | Wörterbuch der Beḍawiye-Sprache[END_REF]

to the Arabic root ĦMḌ of ħaːmiḍ 'sour', implying that the final consonant was lost in Beja. While Reinisch translates both hamir and hami by 'sour' and 'bitter', the second author of this chapter makes a clear distinction between hamir 'to be sour', a pleasant taste, and like the preceding one, cannot be traced back to Arabic or Ethio-Semitic: nifir 'to be sweet', causative si-nafir 'to sweeten', adjective nafir 'sweet', masdar nafruːr 'sweetness'. They all apply to sweet food, or to any kind of pleasant food with a good taste.

Unlike in Kambaata, a Highland East-Cushitic language

(Treis 2010: 239)

, there is no specific root for 'tasteless'. Beja uses either a negative noun phrase borrowed from Arabic: b(i)liːdʔam < Arabic bila ṭaʕam (lit. without taste) 'tasteless' (used as an adjective), or a negative verb phrase with a possession verb: dʔam ki=i-bari 'tasteless' (lit. it does not have taste). The latter construction also means 'odorless', a case of cross-modal matching (or synaethesia), these matchings being common cross-linguistically for the sensory modalities (e.g.

[START_REF] Viberg | The verbs of perception: A typological study[END_REF] Boisson 1997: 5; Levinson & Majid 2014: 414)

.

Table 3 .

 3 The tactile lexicon of Beja

	Verb	Noun	Adjective
	tha 'touch'	tha-ti 'touching'
	tha-s
	tha-m
	tha-s-am
	tha~tha
	ʃiːʃ 'run one's hand on s.th'
	ʃa~ʃiːʃ	ʃa~ʃiːʃ-ti 'groping'
	nʔakʷ 'be tender'		nʔakʷ 'tender'

lamsa 'smooth'

This is why they are in parentheses in Table1.

See Vanhove (2017) and[START_REF] Mattiola | The conceptual space of pluractional constructions[END_REF] for pluractionality in Beja.

Acknowledgments

Our thanks are due to the Beja speakers who provided the corpus examples, Ahmed Abdallah Mohamed-Tahir, his mother and siblings, the CNRS research unit Llacan and the Labex EFL who funded this research and the Paris stay of the second author, our colleague Guillaume Segerer from Llacan who attracted our attention to the data of Setswana and Lomongo, and the two anonymous reviewers for their constructive comments and suggestions.


J Aleksans
email: janis.aleksans@ucd.ie

D Koehn

R Toussaint

G Daniel

Janis Orcid

Aleksans
email: j.aleksans@gmail.com

Daniel Orcid

Koehn

SIMULATING HYDRAULIC FRACTURING: FAILURE IN SOFT VERSUS HARD ROCKS

Keywords: hydrofracturing, numerical modelling, Young's modulus, microseismicity

In this contribution we discuss the dynamic development of hydraulic fractures, their evolution and the resulting seismicity during fluid injection in a coupled numerical model. The model describes coupling between a solid that can fracture dynamically and a compressible fluid that can push back at the rock and open fractures. With a series of numerical simulations we show how the fracture pattern and seismicity change depending on changes in depth, injection rate, Young's Modulus and breaking strength. Our simulations indicate that the Young's Modulus has the largest influence on the fracture dynamics and the related seismicity. Simulations of rocks with a Young's modulus smaller than 10 GPa show dominant mode I failure and a growth of fracture aperture with a decrease in Young's modulus. Simulations of rocks with a Young's modulus higher than 10 GPa show fractures with a constant aperture and fracture growth that is mainly governed by a growth in crack length and an increasing amount of mode II failure. These results are very important for the prediction of fracture dynamics and seismicity during fluid injection, especially since we see a transition from one failure regime to another at around 10 GPa, a Young's modulus that lies in the middle of possible values for natural shale rocks.

Introduction

Hydraulic fracturing is the process of creating cracks within a rock volume by increasing the 2 fluid pressure until the rock fractures (Hubbert & Willis, 1959;[START_REF] Nordgren | Propagation of a vertical hydraulic fracture[END_REF]Valko & 3 Ecomides, 1995). It plays an important role in the geophysical, geomechanical and structural 4 mechanics of the Earth's crust in a wide variety of geological settings [START_REF] Fyfe | Fluids In The Earth's Crust: Their Significance In Metamorphic[END_REF] and has a 5 growing interest in geotechnical and industrial applications [START_REF] Baria | HDR/HWR reservoirs: 31 concepts, understanding and creation[END_REF]; Urbancic et 6 al., 1999; [START_REF] Warpinski | Apparatus and method 10 for monitoring underground fracturing[END_REF]. A hydrofracture develops if the fluid overpressure exceeds Fractures typically grow once the critical stress intensity factor is reached for materials 13 [START_REF] Inglis | Stress in a plate due to the presence of sharp corners and cracks[END_REF][START_REF] Griffith | The phenomena of rupture and flow in solids, Philosophical transactions of the 17 royal society of london[END_REF][START_REF] Irwin | The effect of size upon fracturing[END_REF]). The stress intensity factor can be viewed as the 14 prefactor of the stress singularity around a crack tip in an elastic rock, which is dependent on 15 the crack length and its opening width or aperture [START_REF] Anderson | Fracture Mechanics: Fundamentals and Applications[END_REF]. At this point the rock [START_REF] Griffith | The phenomena of rupture and flow in solids, Philosophical transactions of the 17 royal society of london[END_REF] properties such as the Young's modulus and Poisson ratio become important because they 17 will influence the elastic response of the rock to the stress around the crack [START_REF] Ohta | Effect of Young's modulus on basic crack propagation 11 properties near the fatigue threshold[END_REF] 18 [START_REF] Tandaiya | Effect of Poisson's ratio 37 on crack tip fields and fracture behavior of metallic glasses[END_REF]. Especially the Young's modulus is very variable depending on the 19 rock type. It can vary from around 6 GPa for clay minerals [START_REF] Prasad | Measurement of Young's modulus of 18 clay minerals using atomic force acoustic microscopy[END_REF], to up to 8 20 GPa to 50 GPa for shale [START_REF] Sayers | The effect of anisotropy on the Young's moduli and Poisson's ratios of shales[END_REF] and up to 70 GPa for limestones [START_REF] Sachpazis | Correlating Schmidt hardness with compressive strength and Young's modulus 27 of carbonate rocks[END_REF]).

21

The Young's modulus is the main component, which defines the compressibility of the 22 material and thus has a major role in controlling the shape of the fracture. If the rock is very 23 soft the crack will become elliptical with a large aperture, whereas it remains thin when the 24 rock is hard.

25

Fracturing usually occurs in extensional or shear mode depending on the setting and balance 26 between the differential stress σ 1 -σ 3 as well as the least compressive stress in a rock if it 27 becomes tensile [START_REF] Cobbold | Seepage forces, important factors in the formation of 38 horizontal hydraulic fractures and bedding-parallel fibrous veins ('beef'and 'conein-cone[END_REF]. In most cases in the Earth's crust, a triaxial 28 stress distribution due to the gravitational loading prompts shear opening and for tensile 29 fractures to develop an additional stress is required. This stress is thought to come from fluid 30 overpressure that builds up in the pore space and pre-existing fractures in the rock (Sachau et 31 al., 2015), or is increased during hydraulic pumping.

32

Stresses that build up in a fluid saturated rock must be divided into stresses that act on the 33 solid and fluid and those that act only on the solid [START_REF] Von Terzaghi | Principles of soil mechanics[END_REF]. Stresses acting only 34 on the solid are termed effective stress (𝜎 𝑖𝑗 ′ ) and can be described by Terzhagi's law:

35

𝜎 𝑖𝑗 ′ = 𝜎 𝑖𝑗 -𝑃 𝑓 𝛿 𝑖𝑗

36

(1)

37
where 𝜎 𝑖𝑗 is the stress tensor, 𝑃 𝑓 is fluid pressure and 𝛿 𝑖𝑗 is Kronecker delta with positive sign Coloumb failure envelope in the tensile regime, the rock will fail (Fig. 1).

2 However, this representation is an oversimplification and Cobbold and Rodriguez (2007) and however the circle is shrinking at the same time.

9

Hydrofracturing is a complex interaction process and consists of at least a two way feedback representing the fluid (Fig. 2) and is part of the "Elle" package [START_REF] Bons | Microdynamics simulation[END_REF]. In the model the triangular network is constructed of disk-shaped particles of constant radius 23 interconnected by springs (discrete element model or DEM). Such model configuration in 2D 24 mimics the isotropic elastic behavior of solid materials and can be used to model deformation 25 problems in systems described by linear elastic theory [START_REF] Flekkøy | Modeling hydrofracture[END_REF] (4)

14
where φ is the porosity, β is the fluid compressibility, P is the fluid pressure deviation from 𝐾(𝜌) = 𝑑 2 (𝜙) 3 180(1 -𝜙) 2

24

(5) 25 where d is the particle diameter and 1/180 is an empirical constant for packing of spheres 26 [START_REF] Carman | Fluid flow through granular beds[END_REF][START_REF] Ghani | Dynamic Development of 14 Hydrofracture[END_REF][START_REF] Ghani | Dynamics of hydrofracturing and permeability 12 evolution in layered reservoirs[END_REF].

27

Deformation mechanics are driven by the momentum exchange between the two phases, solid 28 and fluid. The total force applied on the particles is compiled from three main constituents where xi and xj are the positions of the connected particles, 𝑘 𝑖𝑗 is the elastic (spring) constant

6
of their contact, 𝑛 ̂𝑖𝑗 is unit vector pointing from the centroid of particle i to particle j and the Lastly the gravity force that is acting on every particle is calculated according to

18 19 𝑓 𝑖 𝑔 = 𝜌 𝑠 𝜋𝑅 𝑖 2 𝑔𝐶 20 ( 9
)
21
where 𝜌 𝑠 denotes the solid mass density, 𝑅 𝑖 2 = 𝑟 𝑖 2 𝑆, where S is the dimension of the real 22 system (1000m), g is the gravitational acceleration vector and C =2/3 is a scale factor (after 23 [START_REF] Ghani | Dynamics of hydrofracturing and permeability 12 evolution in layered reservoirs[END_REF] used to acquire a compatible one dimensional lithostatic stress that can be 24 applied to an isotropic 2D linear elastic solid.

25

The buoyancy term itself can be written according to Ostrogradsky's theorem as

26

-∫ 𝜌𝑓 𝑔𝑧 𝑛 ̂𝑑𝐴 = -𝜌 𝑓 𝜋𝑅 𝑖 2 𝑔𝐶 (Mory, 2013), so that the total effect of gravity, the direct one 27 plus the buoyancy effect, is (𝜌 𝑠 -𝜌 𝑓 )𝜋𝑅 𝑖 2 𝑔𝐶 , which incorporates the effects that gravity has The solid porosity that is used for the fluid pressure evolution in the fluid lattice is described 12 Once a fracture occurs, it is possible to calculate the seismic moment of the event. To do this 13 we assume that the fracture plane lies perpendicular to the midpoint of the spring connecting

14
the two fracturing particles. The total amount of movement can be calculated and then 15 subdivided into ut and us which represent pure tensile and pure shear movement respectively.

16

For simplicity we assume that the fracture area is equal to the sum of radii of particles broken The figure shows the decomposition of total particle displacement (Utot) into shear (Us) and tensile 31 components (Ut), which are respectively parallel and perpendicular to the imagined failure surface.

32

The failure surface lies perpendicularly to the broken elastic spring. Lastly, as we have both pure 33 tensile and pure shear movements, we introduce the concept of movement mode ratio which is 34 defined as Us/Ut . In this case ratios smaller than 1 are dominated by extensional fracturing whereas 35 ratios larger than 1 become shear fracture dominated 36

The numerical models have been extensively used in the literature and have been shown to Therefore, we assume that at these small time steps the model behaves quasi-continuously. The model has the following parameters: it is assumed to simulate a system that is 1 km 2 in 13 area represented by 400x400 particles. The sides and the bottom of the model box are 1). increasing with the magnitude of the event (Fig 6a). However, this trend is not linear and it 30 can be seen that there are some mixed mode Imode II events that have a very low 31 magnitude. Figure 6 further illustrates the bulk of the events record on the negative side of The given model is limited in the possibility to observe aseismic events due to logging only 1 the position of the particles at the end of each time step. Therefore, the dynamics of the 2 particle movement during the relaxation phase of the lattice are not observed or recorded.

3 There is however significant potential for aseismic activity. To illustrate this we recorded the 4 average amount of movement of particles located along the fracture walls during the time 5 steps when no active fracturing takes place (Fig. 7). Figure 7 shows results from the default 6 model setup but identical trends are observable in all control parameter configurations. The 7 amount of movement is an order of magnitude lower than that during time steps when bond 8 breakage occurs, however it follows the same pattern where there is greater increase in values 9 in the initial stages of fracturing and subsequent decline in the later stages. This would be 10 indicative of the hypothesis that large amounts of energy are being released aseismically 11 alongside the energy released during active fracturing and may not be directly observed. (Fig. 9a), the distribution of fracture wall displacement or aperture (Fig. 9b), the distribution 36 of the amount of broken bonds for single fracture events (Fig. 9c) and the distribution of The moment magnitude (Fig. 9a) changes in a complicated, non-linear manner as a function 17 of changes in the Young's Modulus. From 4.5 to 10 GPa the moment magnitude decreases 18 followed by an increase towards higher Young's modulus values. The average displacement 19 of the fractures shows clearly that the displacement median/mode is becoming smaller with 20 an increase of the Young's Modulus until 12.5 GPa from where it stays constant (Fig. 9b).

21

The overall distribution becomes smaller with increasing Young's modulus leading to a 22 constant displacement at high values. The fracture area shows a minor initial decrease from 23 4.5 GPa to 10 GPa and then an increase in area (Fig. 9c). The increase in fracture area is most 24 pronounced from 20 to 30 and 40 GPa. Simulations with the highest elastic constants seem to 25 have events where a larger amount of bonds break accounting for longer fractures and thus a 26 larger fracture area. A change of the Young's modulus has a strong influence on the fracture 27 mode ratio during the simulations (Fig. 9d). At low Young's modulus (from 4.5 to 10 GPa) 28 the fracture mode is mainly extensional with a relatively narrow distribution whereas higher 29 Young's moduli (12.5 to 40 GPa) show a much higher amount of shear fractures in addition 30 to mixed extensional/shear failure. The transition between these two regimes is very rapid 31 between the values of 10 and 12.5 GPa followed by unchanging behaviour at greater values.

32

The mean value of the logistic distribution is almost constant, around 1 to 3 (0.3-0.5) ratio of 33 shear versus tensile movement, for the lower Young's moduli, and becomes constant again, 34 around 1 to 1 (1.0) ratio, once the fracture mode changes at higher Young's moduli. In To see how the change in overburden pressure would influence the fracturing process, the 2 depth of the simulation was changed from 3 km to 1 km and the Young's modulus was varied 3 between the simulations in a similar fashion to the default case. Reducing the overburden 4 produced a significant decrease in fracturing in general, so that sometimes the results yielded 5 as much as ten times less fractures than at 3 km. The lower overburden stress led to an 6 increased porosity, which also meant that the fluid pressure gradients were lower and less 7 fractures developed (Fig. 5b). This resulted in a smaller amount of data points and thus more 8 noise in the data. However, when analysing the fracture displacement or aperture data, a very 9 similar trend to the 3 km cases can be observed where there is a rapid decrease of wall 10 displacement with an increase in Young's modulus followed by a steady aperture with the The other parameters (moment magnitude, mode ratio, fracture area), when analysed, 13 produced a high amount of noise in the results and thus showed very high uncertainty and no 14 clear relation with changes in Young's modulus. In a third set of simulations we changed the Young's modulus in simulations with a lower 17 injection rate of 50000Pa/30s. Changing the injection rate from 80000Pa/30s to 50000Pa/30s did not change the behaviour 24 of the simulations much. Both displacement and area change trends showed the same pattern 25 that was observed at higher injection rate with a change in behaviour at 10 GPa. The moment 26 magnitude pattern was almost identical in both cases with differing injection rates. The only 27 significant change was the movement mode pattern where the behaviour changes at 9 GPa 28 rather than 10 GPa. The data line is uneven as rather than going through a single fracture, it crosses a fracture The σyy (vertical) stress behavior is different for soft and hard rocks. In the case of a soft rock 1 it tends to be slightly extensional at the boundary of the fracture and then changes to 2 compressive further away. When looking at the hard rock, the stress is mainly compressive 3 with a small tensile area at the very center and the switch from tensile to compressive is more

29

The resolution of the model (amount of particles in the simulation) has an influence on the 30 amount of fractures that can be produced in the model but the crack tip stress is not directly 31 dependent on the particle size. The stress at the crack tip is given by the extension or shearing 32 of springs which is only determined by the relaxation threshold that is used to solve the stress 33 field, which is in the order of 4 magnitudes smaller than the particle size. Figure 16 shows 34 that the fracture network itself does not change in geometry at different resolutions, except 35 for the fact that a higher resolution simulation produces a more detailed fracture network with 36 more branches. shows a high resolution simulation of the hard rock. In this case the ratio between the fluid input area 45 and the particle size is changed to twice the size in the horizontal and vertical direction, meaning that 46 fluid is injected in four cells and not only in one cell as is the case with a) and b). The fracture pattern 47 in the high resolution simulation is the same as in b) but twice the size illustrating that the scale and 1 pattern of the fracture network is not dependent on the resolution of the model. The number of fractures that develop is a non-linear function of the breaking strength.

7

  the sum of the least compressive stress and the tensile strength of the rock (Cobbold & 8 Rodrigues, 2007). During industrial hydrofracturing large quantities of fluid are pumped into 9 a finite pore volume generating an overpressure and an outward push, which will lead to 10 brittle failure and the development of fractures (Pearson, 1981; Zoback & Harjes, 1997; 11 Niebling et al., 2012).

12

 12

38 conversion.

 38 The most common way to illustrate the 2D stress state within a rock is by using a 39 Mohr diagram plotting shear against normal effective stress, with σ 1 and σ 3 at the right hand 40 side of the diagram in the compressive state. 𝑃 𝑓 reduces the value of the effective stress and 3 pushes the Mohr circle towards the left hand side. Once the circle transects the Mohr-1

3

  Ghani et al. (2015) have shown with experiments and simulations that an increase in fluid 4 overpressure in a simple sedimentary basin will lead to a local decrease of the differential 5 stress of the effective stress tensor due to seepage forces. In this case the fluid overpressure 6 has an anisotropic effect on the effective stresses. The Mohr circle for the effective stress 7 state as such is still moving towards the left hand side into the regime of tensile failure,8

10 between( 21 [Figure 1 ] 22 Fig. 1 a

 10211221 Fig. 1 a) Standard Mohr circle diagram of effective stress showing how an increase in fluid pressure 23 shifts the Mohr circle from its initial compressive stress state at the right hand side to the left into the 24 tensile regime leading to failure; Note that the differential stress remains constant. b) Mohr circle 25 diagram of effective stress in the model illustrating the anisotropy of stress changes when fluid 26 overpressure or pressure gradients are taken into account. Locally the system will fluidize, the 27 differential stress goes to zero or becomes very small leading to mainly mode I failure 28 29

10 In

 10 order to numerically simulate the development of hydrofractures we employ a discrete 11 element model (Latte), which is built upon a 2D triangular setup coupled with a square grid 12

Fig. 2

 2 Fig. 2 Schematic illustration of the DEM (discrete element model) grid overlying the fluid pressure 17 nodes. The DEM grid is triangular whereas the fluid grid is square and larger than the DEM. Both 18 grids are mapped onto each other using tent functions to minimize grid effects. The fluid grid is 19 stationary whereas the DEM grid can move 20 21 22

15 hydrostatic,

 15 and us is the solid velocity field. K and μ stand for permeability and fluid 16 viscosity respectively. The left hand side of the equation is the Lagrangian derivative of pore 17 pressure following the solid matrix, the first term on the right expresses the Darcy fluid 18 pressure diffusion relative to the particles, the last term is a source term. The source term 19 expresses pressure change as a function of a change in the solid if particles move apart in the 20 local reference of the Darcy flow (Ghani et al., 2013). The Kozeny-Carman relation is used to 21 express K as a function of local porosity φ 22 23

7 13 ( 8 ) 14 where

 13814 sum runs over all the connected neighbours j. Once the change in relative particle positions 8 occurs, the resulting force can be decomposed into tensile and shear components. Further 9 details on this methodology can be found in work by Sachau & Koehn (2014). 10 The fluid force fp that acts on the surface normal dA of the unit cell is a function of the fluid 11 flow due to the pressure gradient and is described as 12 𝑓 𝑝 = -∫ 𝑃 0 𝑛 ̂𝑑𝐴 P0 = P + ρf gz is the local total fluid pressure, z depth, ρf density of the fluid and g 15 gravity constant. It is the sum of a term due to viscous forces arising in the case of fluid flow 16 through the solid, -∫ 𝑃𝑛 ̂𝑑𝐴, and a buoyancy term, -∫ 𝜌𝑓 𝑔𝑧 𝑛 ̂𝑑𝐴.

17

 17

28 on

 28 both solid and fluid: this drives the effective stress field σ' =(ρs -ρf)gz in the system in a 29 hydrostatic situation(Niebling et al., 2010a).

30

 30

31 by 34 Finally 4 7 particle over the four nearest fluid grid nodes 8 𝑠

 3134478 the solid mass fraction of solid particles within a fluid cell and changes due to 32 compression, solid movement or fracturing. The solid movement itself between two 33 deformation steps affects the source term in equation 4. , the model is set up in a way that the fluid continuum grid overlies the DEM so that 1 their boundaries coincide. Since the fluid grid is set to be twice as large as the DEM matrix, 2 the model uses the "cloud in the cell" method to facilitate the two-way interaction between 3 the porous matrix and the hydrodynamic phase[START_REF] Ghani | Dynamic Development of 14 Hydrofracture[END_REF][START_REF] Johnsen | Pattern formation during air 37 injection into granular materials confined in a circular Hele-Shaw cell[END_REF][START_REF] Vinningland | Granular 5 rayleigh-taylor instability: Experiments and simulations[END_REF][START_REF] Fyfe | Fluids In The Earth's Crust: Their Significance In Metamorphic[END_REF]. The interaction between the porous solid and hydrodynamic 5 phases is accomplished through the projection operator from the discrete space to the fluid 6 grid space by the help of the smoothing function s(ri-r0), which distributes the weight of the x,z) and ro(xo,zo) are the positions of the particle and the continuum node 11 respectively, w1 = |x -xo| and w2 = |z -zo| are the relative distances.

17 across 19 𝑀 21 where 22 ( 29 Fig. 3

 17192122293 Fig.3Calculation of the Moment Tensor after failure takes place (spring between particles breaks).30The figure shows the decomposition of total particle displacement (Utot) into shear (Us) and tensile 31 components (Ut), which are respectively parallel and perpendicular to the imagined failure surface.32The failure surface lies perpendicularly to the broken elastic spring. Lastly, as we have both pure 33 tensile and pure shear movements, we introduce the concept of movement mode ratio which is 34 defined as Us/Ut . In this case ratios smaller than 1 are dominated by extensional fracturing whereas 35 ratios larger than 1 become shear fracture dominated 36

1

  reproduce experimental and analytical solutions for fracturing(Niebling, 2012), roughness 2 growth (Koehn et al., 2003, 2012) and solid fluid interactions in granular (Niebling et al., 3 2010a, 2010b) and solid media (Flekkøy et al., 2002, Ghani et al., 2013, 2015) as well as in 4 landslides (Parez et al., 2016) or soil liquefaction (Clément et al., 2018; Zeev et al., 2017). 5 Since time steps in the model are discrete they had to be calibrated in order to avoid 6 overstepping the breaking threshold by unreasonably large amounts, which would lead to the 7 model stepping controlling the behavior. The time step was decreased until changes in stress 8 and magnitudes as well as fracturing behavior became independent of the model step size.

9

 9

  14 represented by fixed walls that are frictionless. The upper boundary is open, and force is 15 applied controlled as a function of gravity of the overlying rock units. Fluid is injected in one 16 fluid cell at the centre of the lower part of the model (Fig. 4). Time steps in the model are 30 17 seconds each and the models typically run 10'000 time steps. The fluid has a density of 1000 18 kg/m 3 and a viscosity of 8.90 × 10 -4 Pa•s, which are the properties of water at temperature of 19 25 o C. Variables that were changed include depth (overload varying from 1 to 3 km with a 20 density of overlying sediments of 2.5 × 10 3 kg/m 3 ), initial porosity, fluid injection rates, 21 Young's modulus and brittleness (breaking strength and its distribution) of the rock. The aim 22 was to study how these changes affect the tensile/shear movement ratio distribution, the 23 magnitude and the fracture pattern. A setting with 3 km depth, 0.33 Poisson's ratio, 6 GPa 24 Young's modulus, ~1% porosity, 34 MPa tensile strength and 80000Pa/30s injection rate was 25 our default setting. This setting was chosen because the mechanical properties of the rock are 26 close to those of a typical shale, yet the increased depth guaranteed an increased principal 27 stress and thus a larger amount of fracturing, which provides more data. Variations of these 28 settings were performed in order to study the effects of the different variables on the 29 hydrofracturing (Table

8 9Fig. 4 29 [

 8429 Fig.4 The setup of the numerical model is 2D, bounded by fixed walls on the right and left hand sides 10 and the bottom while a stress corresponding to the weight of the overburden is applied from the top. 11Fluid is injected at the centre of the lower part of the model 12 13

30 31Fig. 5

 305 Fig. 5 a) The model shows four stages of fracture evolution: 1) build up of stress, 2) fracturing, 3) 32 residual fracturing, 4) seeping. Timing and intensity of the fracturing depends upon the parameters, 33 for the default setting time step intervals for these stages are 1) 0-1000, 2) 1000-4000, 3) 4000-7500, 34 4) >7500. Changing the parameters of the model moderates the timing and intensity of the fracturing, 35 however the overall pattern is preserved in all cases. b) Figure shows changes in the vertical fluid 36 pressure gradient averaged over the entire system. The model starting parameter is hydrostatic 37 pressure. 10 KPa value is calculated as a difference between top and bottom boundary of the 38 simulation box, which is 3 and 4 km. In the default setting the fluid pressure gradient is less 39 susceptible to changes due to a low initial porosity and relatively high fluid injection rate. Once the 40

13 14 [

 1314 Figure 6]

Fig. 6 a 26 Figure

 626 Fig. 6 a) Figure shows the moment magnitude of the fracturing events in a model versus the mode of 18 fracturing in a default setup. All results produce an observable trend of an initial moment magnitude 19 increase as fracturing becomes more shear-like followed by a decrease of magnitude and increase of 20 shear component. This shows that there is no simple linear relationship between the amount of shear 21 in the fracture and the moment magnitude of the event. b) Figure illustrates the evolution of moment 22 magnitude values as a function of time for a model run in a default setup. The greatest moment 23 magnitude values can be observed during the phase of the most active fracturing (time-step 1000-24 4000) and tend to decrease towards the later phases of the model evolution 25 26

32 the

 32 magnitude scale. The few events that do record on the positive side do not overstep a 33 magnitude of 0.5 and thus are very low. Figure 6b illustrates the evolution of moment 34 magnitude values as a function of time for a model run. Fracturing events with higher 35 magnitudes occur early at stages of most active fracturing and magnitudes decrease towards 36 the later stages of the fracture development.

14 15Fig. 7 22 23

 14722 Fig. 7 Average displacement of particles during the time steps when active fracturing (seismic) takes 16 place versus average displacement of all the particles across the fracture walls during time steps when 17 no active fracturing is occurring (aseismic) 18 19 20 21

37 movement 40 [

 3740 mode ratios in model runs (mode I versus mode II, Fig.9d). The plots on the right 38 hand side of the figure show the evolution distributions' modes/means when changing 39Young's modulus from soft rock setting through to hard rock setting.Figure 8]

Fig. 8 4 [Figure 9 ] 5 Fig. 9

 84959 Fig. 8 Displacement data for experiments with E=4.5 GPa (left) and E=30.0 GPa is binned according 1 to the percentage of the total data falling within each of the intervals. Logistic distribution probability 2 density functions are then calculated for data point distributions 3 4

35 summary

 35 an increase in Young's modulus has a strong influence on the mode of fracturing 36 changing from extensional mode dominated at Young's moduli below 10 GPa to mixed mode 37 with a higher amount of shear fractures at higher values. The behaviour of the moment 38 magnitude as a function of Young's modulus is complex. Fracture wall displacement or 39 aperture of the fractures decreases with an increase of Young's modulus and becomes 40 stationary at high values (>20 GPa). The amount of broken bonds or the fracture area is not 41 changing much for small Young's modulus values but increases from values of 20 GPa and 42 higher.

11 turning

 11 point at around 15 GPa (Fig 10).

12

 12

Fig. 10

 10 Fig. 10 Logistic distribution curves showing average fracture wall displacements for a series of 21 Young's moduli at depth of 1 km. The behaviour is very similar to the 3 km deep reference case 22 23

29 30 [Figure 11 ] 31 Fig. 11

 2930113111 Fig.11The figure shows the amount of bonds broken during an experiment for a series of Young's 32 moduli normalized by the tensile breaking strength of the rock. Increasing Young's modulus and 33 decreasing the breaking strength both lead to more fractures 34[START_REF] Clément | Sinking during earthquakes: 36 Critical acceleration criteria control drained soil liquefaction[END_REF]

8 22 𝜎′ 26 Fig. 12 Figure 34 Figure 12 5

 82226123412 Fig. 12 Figure shows the calculated crack tip stress from the crack aperture and length as a function of 27 Young's Modulus (E). Both crack aperture and length values are derived as means/modes using the 28 logistic distribution function for all the data. The plot shows two regimes, below 10 to 11 MPa 29 Young's Modulus the crack tip stress increases rapidly, whereas at Young's Moduli higher than 11 30 MPa the crack tip stress increase is only minor. These two regime represent two failure regimes, 31 below the critical Young's Modulus the fractures are soft and open whereas at higher values the 32 aperture is constant the crack propagates by an increase in length 33 34

14 [

 14 Figure 13]

15 16Fig. 13 19 [

 151319 Fig. 13 Figure shows schematic placement of the profiles used to calculate the difference in σxx 17 (horizontal) and σyy (vertical) stress between the first and last time steps in the experiments. The 18 profiles cover fracture networks rather than single fractures and thus show fluctuations in stress 19

23

  Figure 15.

26 [

 26 Figure 15]

27 28 Fig. 15 Figure

 2815 Fig. 15 Figure shows schematic illustration of stress in soft and hard rocks. Arrows pointing towards 29 the centre of fracture indicate compressive stress while arrows pointing outwardsextensional 30

34

 34

Fig. 14

 14 Fig.[START_REF] Pearson | The relationship between microseismicity and high pore pressures during 15 hydraulic stimulation experiments in low permeability granitic rocks[END_REF] Profiles of change in σxx (horizontal) and σyy (vertical) stress throughout the experiment along the horizontal (left) and vertical (right) lines of profile. The horizontal profile shows an increase in σxx stress on each side of the fractured area representing a compaction zone, which is much more pronounced in the soft rock. The vertical profile shows a significant difference of σxx stress for soft versus hard rocks with hard rocks showing only compression whereas soft rocks have a tensile region at the top of the fracture. Both vertical profiles are anisotropic reflecting gravity

4 abrupt.

 4 At the crack tip soft and hard rocks behave quite differently, especially at the top of 5 the fractured area. Both soft and hard rock scenarios show a bimodal behavior where both σxx 6 and σyy stresses change from extensional to compressive closer to the end of the fractured 7 area. However, in the case of soft rocks, the σxx stress becomes sharply extensional just 8 outside the fractured area, while in the case of the hard rock scenario, no sudden change in 9 stress is observed. This backs up the theory that fracture development in the case of soft rocks 10 is mostly governed through lateral extension and increase of the fracture aperture as the stress 11 regimes readily allow for lateral expansion at the crack tip. The relatively higher σxx stress 12 values at the sides of the fracture suggest extensive expansion in that direction. In the case of 13 the hard rock, the length is the key value behind the crack growth. In the case of the soft rock 14 the fracture tip is already in extension before the crack propagates and the fluid pushes 15 outwards to pull the solid apart for further fracturing. This behavior is very similar to 16 fracturing in granular media where grains are being pushed apart (Eriksen et al. 2017, 2018, 17 Niebling et al. 2012) and would represent mode I extensional failure. In contrast the hard 18 rock fracture tip is actually under compression. The stress concentration in the hard rock case 19 is very high at the crack tip, so that almost any tensile stress at the tip exceeds the breaking 20 strength and leads to fracture propagation. The effect of these different mechanisms results 21 in different slopes for hard rocks in Figure 12.

22

 22

Figure 15

 15 Figure 15 illustrates that in soft rocks the cracks are shorter and have rounder shape. In this

37 [

 37 Figure 16]

38 39Fig. 16 Figure

 3816 Fig. 16 Figure shows the fracture patterns (blue particles) in detail for a hard (50 GPa) versus a soft 40 (5GPa) rock. a) Simulation of a soft rock with four distinct shear fractures and a central mode I 41 region. b) Simulation of a hard rock with a much more pronounced central fractured area in 42 comparison to the soft rock. c) shows details of the central parts where distinct openings can be seen 43 in the soft rock in the central part representing an opening elliptical crack when the rock is soft. d) 44shows a high resolution simulation of the hard rock. In this case the ratio between the fluid input area 45 and the particle size is changed to twice the size in the horizontal and vertical direction, meaning that 46 fluid is injected in four cells and not only in one cell as is the case with a) and b). The fracture pattern 47

4

 4 In this contribution we show numerically how hydrofracturing during fluid injection develops 5 in 4 distinct stages: 1) build up of stress, 2) fracturing, 3) residual fracturing, 4) seeping. The 6 Young's modulus of rocks and their breaking strength change the fracturing behavior 7 resulting in two predominant fracturing mechanisms. In the cases when the fractured rock is 8 soft, fracturing occurs due to a critical tensile strength or cohesion being exceeded, change in 9 the stress at the crack tip as a function of the Young's Modulus is then governed by the 10 change in the fracture aperture and the fracture tip is constantly under extension. When the 11 rock is hard, the fracture aperture is small and constant and the fracture length becomes the 12 key driver of the stress change. The fracture tip is under compression and fracturing is driven 13 strongly by stress intensification at the tip, causing fractures to propagate in more shear like 14 failure. 15 We show that the relationship between Young's modulus and the failure mechanism works 16 independently of the tensile strength of the rock as changes in Young's modulus for rocks 17 with different tensile strengths produce the same relative increase in the amount of fracturing.

  18

19

  Changes in secondary parameters such as the amount of overburden or fluid injection rate 20 have an impact on the amount of fractures that develop but have little to no effect on the 21 described failure mechanisms. The transition between the failure due to a critical tensile 22 strength for soft rocks versus a strong dependence on stress intensification for hard rock lies 23 at a Young's modulus of about 10 GPa in the simulations. Since Young's moduli for natural 24 shales vary from 8 to 50 GPa one has to be careful to use the right failure criterion depending 25 on the shale type.

  Click here to access/download;Figure;Fig1.eps Fluid pressure node of the fluid phase grid Disk shaped particle lattice is overlaying fluid pressure grid Elastic springs conecting solid state particles Solid lattice and fluid grid interact via tent function of solid particle's distance to four closest grid nodes Click here to access/download;Figure;Fig2.eps access/download;Figure;Fig7.eps Click here to access/download;Figure;Fig8.eps

FigureFigure

  FigureClick here to access/download;Figure;Fig9.eps

. The intrinsic 26 stiffness coefficient k is governed by the macro-scale parameters E and v (Young's modulus 27 and Poisson ratio) through the consistency measures of strain energy between the 2D elastic

	1	as a sum of tensile (Ut) and shear (Uc) energies (Sachau & Koehn, 2014). There are separate
	2	critical values for both tensile (Ec,σ) and shear (Ec,τ) criterions and their relationship can be
	3	expressed as
	4	𝑈 𝑠 𝐸 𝑐	+	𝑈 𝑡 𝐸 𝑐	= (	𝜎 𝑛 𝜎 0	) 2	+ (	𝜏 𝜏 0	2 )	= 1
	5											(3)
	6	which describes an ellipse in σn-τ space (Sun & Jin, 2012). Therefore, the failure can occur as
	7	a combination of both shear and tensile factors, or if one of the two components is absent, it
	8	will be equivalent to pure tensile or pure shear failure.
	9	The 2D DEM lattice is overlain by a square grid of fluid pressure nodes. The fluid pressure
	10	diffusion is derived from mass conservation of the fluid and solid using Darcy's law to
	11	express the seepage velocity through the porous medium (Ghani et al., 2013, 2015)
	12
	13	𝜙𝛽(𝜕 𝑡 𝑃 + 𝑢 𝑠 ∇ • 𝑃) = ∇ • [(1 + 𝛽𝑃)	𝐾 𝜇	∇𝑃] -(1 + 𝛽𝑃)∇ • 𝑢 𝑠
	28
	29	lattice of the triangular network and solid continua (Flekkøy et al., 2002)
	30				𝑘 =	√3 2	𝐸𝑙
	31											(2)
	32	where l stands for the thickness of the two-dimensional model. The model runs as a step
	33	function and therefore plain strain deformation is produced for each time step Δt. At the end
	34	of every time step all springs are checked and if the predetermined stress threshold for a
	35	spring is reached, it breaks releasing elastic energy, the spring is removed from the lattice and
	36	a fracture forms. The elastic energy contained in the removed spring is further redistributed
	37	among the neighboring springs via the relaxation algorithm. The breakage occurs once either
	38	the critical tensile normal stress σ0 or the critical shear stress τ0 is reached. In order to include
	39	combinations of tensile and shear failure it is assumed that the critical strain value Ec is seen

  𝑓 𝑝 and external force 𝑓 𝑔 applied due to the gravity 34 and large scale tectonic strain on the external boundaries. In this instance 𝑓 𝑒 is expressed as a

	29 30 31 32 33 2 3 4 spring or repulsive), fluid pressure force 1 𝐹 𝑛 𝑖 = 𝑓 𝑒 + 𝑓 𝑝 + 𝑓 𝑔 where 𝑓 𝑒 is the interaction force between the particles (either due to a connection with the function of the spring elasticity constant k and an equilibrium distance between particles a, which equals the sum of the initial radii of the two connected particles 𝑓 𝑒 = ∑ 𝑘 𝑖𝑗 (|𝑎 ⃗ 𝑖𝑗 | -|𝑥 ⃗ 𝑖 -𝑥 ⃗ 𝑗 |) • 𝑗 𝑛 ̂𝑖𝑗 5	(6) (7)

Table 1

 1 The default setting for the hydrofracturing experiments was set to 3 km depth, 0.3 Poisson's 2 ratio, 6 GPa Young's modulus, ~1% porosity, 34 MPa tensile strength and 8000Pa/30s injection.3During the experiments the Young's modulus was varied in addition to the depth, injection rate and

	30
		Depth (m)	Young's Moduli (GPa)	Poisson's Ratio	Porosity	Injection rate	Tensile Strength (MPa)	No. of simulations counducted
	Default		6
	Young's					80000Pa/30s
	Modulus	3000
	Lower rate injection		9; 10; 12.5; 4.5; 6; 7; 8;	0.33	~1%	50000Pa/30s	34	10x10000 time steps (each
	Shallower depth	1000	15; 20; 30 ;40					setting)
	Variable tensile strength	3000				80000Pa/30s	27.2; 34; 44.2; 54.4

Acknowledgments

We would like to extend our gratitude to Natural Environment Research Council (NERC) and especially to NERC Center for Doctoral Training in Oil & Gas for provided funding of the project within which this research has been carried out.


BERMOND, François Karine Bruyere

François Bermond

Robert Bouquet

Yves Caire

Michelle Ramet

Eric J Voiglio

Jean-Pierre Verriest

Bruyere Karine

Bermond François

Bouquet Robert

Caire Yves

Ramet Michelle

Voïglio Eric

VERRIEST Jean-Pierre France

Human face response to 30°degree oriented impacts

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The development of the air bag resulted in decreased severity of injuries of the head and face [START_REF] Krafft | The risk of skull/brain injuries in modern cars, 16th International Technical Conference on Experimental Safety Vehicles[END_REF]]. Using the Abbreviated Injury Scale (AIS), fractures of facial bones are often coded as minor or moderate. Yet, the personal consequences o f faces injuries are considerable because of the long-term cosmetic repercussions.

Based on research on the prevention of severe injuries to the head, injury criteria have been established. The National Highway Traffic Safety Administration (NHTSA) proposed the Head Injury Criterion: it uses time duration of impact and linear acceleration of the center of gravity of the head. Past research has also proposed fracture tolerances for major facial bones using different impactor sizes. These reference data was summarised in a SAE report in 1980 [SAE, 1980]. Later, this data was completed [START_REF] Allsop | Facial Impact Response. A Comparison Of The Hybrid III Dummy And Human Cadaver[END_REF] and overall facial response analyzed. Since contact on the face with the steering wheel is still most frequent [START_REF] Thomas | The cause of head injuries in real world car crashes[END_REF], several studies analyzed facial bone fractures following dynamic contact with the steering wheel or substitute [START_REF] Nyquist | Facial impact tolerances and response, 30th Stapp Car Crash Conference[END_REF]] [Allsop et al., 1988] [Yoganandan et al., 1988] [Bermond et al., 1999].

Energy absorbed during the crush of facial bones has an influence on both facial injuries and overall head dynamics. Thus, the mechanical response of the face and its injury mechanisms must be considered to define injury criteria for head impacts. Moreover, the knowledge of mechanical response of face is essential to assess and improve the biofidelity of mechanical and numerical models of the head. Recent works performed as part of the European project called Advanced Crash Dummy Research for Injury Assessment in frontal test conditions (ADRIA) brought new data concerning dynamic response of malar bone in comparison with frontal bone under impact with a steel cylinder [ADRIA, 1998] [Bermond et al., 1999].

This paper compares impacts on the maxilla bone with the frontal bone. The aims of the analysis were firstly to propose corridors of the dynamic behaviour of head during impact on these bones, and secondly to define injury risk curves.

Material and Method

PMHS Preparation

Since 1995, 69 tests have been performed on 18 PMHS. For each PMHS, a set of anthropometric data was collected to characterise the whole corpse (height, weight) and the head (circumference, height and weight). The Frankfurt plane was materialised by four small lead balls, placed at infraorbital notches and auditory meati. The anatomical frame was defined as shown on Figure 1. The centre of gravity of the head (G) was defined according to the results of [START_REF] Beier | Center of gravity and moments of inertia of human heads[END_REF] with x = 8.3mm, y = 0 and z = -31.2 mm in the anatomical frame.

Impact conditions

A horizontal guided impactor was used. To simulate steering wheel rim, the extremity of the impactor was equipped with a horizontal steel cylinder with a diameter equal to 2.25 cm. This rigid portion directly contacted the face of the subject during tests. The weight of impacting mass was 17 kg as used since 1989 at INRETS [START_REF] Welbourne | A comparison of human facial fracture tolerance with the performance of a surrogate test device[END_REF]] [Césari et al., 1989] [Bermond et al., 1999].

During tests, subject was seated, linked to the armchair by a strap and a cervical collar was used in order to hold the head in good position (Frankfurt plane in the horizontal) and to reduce the neck's flexibility. The direction of impact on maxilla or frontal bones was chosen as 30°-angled from the mid-sagittal section plane in order to obtain impact conditions closer to real impact conditions, where the seat belt leads to the rotation of the upper torso and the head.

The impactor energy taken from the mass and the velocity of the impactor was chosen on the basis of results from previous tests [START_REF] Césari | Experimental evaluation of human facial tolerance to injuries[END_REF] and in a way to observe either facial bone fractures or not. Some subjects have been submitted to both right and left impacts on maxilla. In these cases, the low energy impacts were performed first and it has been verified that they were subinjurious impacts. Some subjects have been submitted to both impacts on frontal bone and maxilla. In this case, it was verified that the impact on frontal bone would not affect the integrity of the maxilla region.

Instrumentation

The head was equipped with a metallic rig bearing 4 blocks of triaxial accelerometers located on the top, rear, left and right of the rig (Figure 2). This rig was screwed onto the head and weighed 0.34 kg, which was generally lower than 10% of the head mass. The location of the centre of gravity of the head was adjusted by considering this additional weight. The impact force and the speed and acceleration of the impactor were also measured.

The data was prefiltered at 2.5kHz and digitised at a sampling rate of 10kHz. Then, data was filtered according to the ISO normalisation: accelerations of the head were filtered with a FIR1000 filter and data from the impactor was filtered at 180Hz.

Analysis

Linear acceleration of the centre of gravity of the head The 4 triaxial accelerometers were used to calculate the linear acceleration at the centre of gravity of the head during the impact. The method used was the N×1 method, developed by APR [START_REF] Oudenard | The use of a multiaccelerometric method in automotive safety tests[END_REF]. Computing is based on equations for the head motion, considering it as a rigid body. These equations contain 6 unknown parameters, which are the linear and angular accelerations of the centre of gravity of the head along X, Y and Z axis. Using information supplied by N accelerometers with N ≥ 6 leads to a set of N equations which are resolved by an explicit integrating method.

The N×1 method requires the orientation and position of each accelerometric sensors relative to the centre of gravity in the head anatomical frame. This information was provided by X-Ray photographs of the head and rig system, taken in frontal and sagittal planes before the test. The use of a metallic rig made the orientations and positions of the accelerometric sensors easier to obtain from X-Ray photographs.

The linear acceleration of the centre of gravity of the head was plotted as a function of time. Response corridors were proposed at several energy levels for frontal bone and maxilla bone. The maximum value of the linear acceleration was also considered.

HIC criterion

The HIC criterion was computed according to the NHTSA definition [NHTSA, 1997]:

( ) ( ) 5 . 2 t t 1 2 1 2 2 1 dt t a t t 1 t t HIC         - - = ∫
, where a(t) is the linear acceleration of the centre of gravity of the head and t 1 and t 2 are two points in time which are separated by not more than a 35 ms time interval.

Impact force

The impact force was computed from the measured force of impactor, considering the mass of the impacting cylinder. As for the linear acceleration of the head, response corridors were established for impact force time-histories. The maximum value of the impact force was also considered.

Medical Investigation

Autopsy of head included an inspection of the soft tissue injuries of the face, and of the skull and facial bones. Injuries were coded according to the AIS90 [START_REF] Ais | The Abbreviated Injury Scale 1990 Revision[END_REF].

Statistics

Statistical analysis was performed using the SAS® Software. The relationship between the injury risk and the measured parameters was evaluated by the test of maximum likelihood. Logistic regressions were also evaluated by test of maximum likelihood, considered as significant for a probability p ≤ 0.05.

Results

In this paper, we present the results from 34 tests on frontal bone including 5 tests with impactor energy close to 280J and from 11 tests on maxilla including 7 tests with impactor energy close to 280 J.

Dynamic response of the head Time histories of linear acceleration of the centre of gravity of the head are given for impact on the frontal bone and the on maxilla in case of impactor energy close to 280J (Figures 3 and4). Results in terms of maximum linear acceleration, HIC and maximum impact force are given in Table 1 for the same impactor energy level (close to 280 J). These results showed higher values in case of impact on frontal bone. Observed injuries Localised abrasions or wounds of the skin were always observed and coded as AIS 1. Impacts on maxilla bone conducted to a caving in of the maxilla body and more severe impact also conducted to complete fracture of the palatine bone and/or displacement of the face with rupture of the suture between frontal and zygomatic bones. In case of severe impact on frontal bone, fractures of this bone and of the orbital cavity were observed. For both impacts on frontal bone and maxilla, an impactor energy close to 280 J seems to correspond to a juxtainjurious level, with AIS=1 (no fracture) or more (bone fracture).

Injury risk curves

In order to assess the measured parameters as injury risk criteria, their values have been compared to the risk of injury, considering it equal to 100% for AIS ≥ 2. For impacts on maxilla, risk distributions obtained did not allow the interpolation of the injury risk function by logistic curves since the probability of dependence was not significant. For frontal bone impact, the probability of dependence was significant for the impactor energy and the impact force. Logistic regressions are illustrated by Figure 5. Considering the impactor energy as injury criterion, a 50% injury risk is obtained for a value of 265 J. For the impact force, it corresponds to a value of 7500 N.

Discussion and conclusion

Such study using PMHS present some bias. First, it implies variations in anthropometry and in the biological tissue properties. The bone strength, which depends on many parameters (such as age, sex, living conditions, bone geometry), directly influences the mechanical behaviour and strength of the studied body part. Thus, mean PMHS behaviour is considered by building corridors and statistics are performed to obtain trends of this behaviour. Secondly, differences in the mechanical behaviour of PMHS and living persons exist, namely due to the lack of muscle tone in PMHS. However, the PMHS is still the best substitute of the human body to study particularly the injury mechanisms.

In our study, corridors proposed for the head response to facial impacts can be used as recommendations for the mechanical and numerical models. These models must be particularly biofidelic in case of juxta-injuruious impact in order to predict the injury risk.

Moreover, the comparison of the head mechanical response when maxilla or frontal bone is impacted with the same impactor energy, showed the protection of the intra-cranial contents by the facial bones crushing. This comparison also shows that the injury risk curve with 50% of injury risk for HIC=1000 can not be applied in case of impact on face. Thus, new risk curve using HIC and/or new injury criteria must be defined for facial impacts.

For impact on maxilla, injury risk distribution with regard to the measured parameters did not show any threshold and did not allow us to establish logistic regressions. It seems that the impact severity in terms of impactor energy or impact force corresponds to a 50% injury risk. In order to define injury risk curve for impact on this facial bone, more injurious tests will be performed. For impact on frontal bone, it has been possible to define injury risk curve with impactor energy and impact force as injury criteria. As expected, the impact force level obtained for 50% injury risk on frontal bone (7500 N) is higher than the one obtained by Yoganandan et al. for 50% injury risk in case of steering wheel impact on zygoma (1500 N) [START_REF] Yoganandan | Human Facial Tolerance to Steering Wheel Impact : A Biomechanical Study[END_REF].

In conclusion, despite the variations in the mechanical responses of PMHS and the necessary caution in applying the results to living persons, trends in the mechanical response of the human face to impacts with a substitute of steering wheel have been shown. Moreover, the comparison of the head behaviour when maxilla or frontal bone is impacted showed the protection of the intracranial contents by the facial bones crushing. In further research concerning the face response to impact, impacts with flat surface and/or airbag will have to be considered.

Figure 1 :Figure 2 :

 12 Figure 1 : Anatomical frame of the head (O = origin of the anatomical frame, G = center of gravity of the head)

Figure 3 :Figure 4 :

 34 Figure 3 : Maxilla bone impact : linear acceleration of the centre of gravity of the head versus time plots and proposed response corridor

Figure 5 :

 5 Figure 5 : Injury risk curve for impact on frontal bone using a) impactor energy, b) impact force, as injury criteria

Table 1 :

 1 Impact on the frontal bone and the maxilla with impactor energy close to 280 J : dynamic response of the head and impact force

		Maxilla bone (n=7)	Frontal bone (n=5)
		Mean (SD)	Range	Mean (SD)	Range
	Impactor energy (J)	283 (16)	[248 -298]	282 (3)	[279 -287]
	Max. impact force (N)	2409 (622)	[1802 -3331]	6818 (1144)	[5149 -7866]
	Max. linear acceleration
	of the cog of the head	1291 (403)	[561 -1810]	2948 (719)	[1972 -3542]
	(m/s²)
	HIC	288 (142)	[87 -479]	1569 (840)	[548 -2747]

SD : standard deviation; cog : centre of gravity

Acknowledgement

We would like to thank Pierre LAPELLERIE, Michel MARAIS and Gérard GOUTELLE for their work in collecting data, photos and measurements during the tests and Sabine COMPIGNE for her work in statistical analysis.