Eloi Magnin

Julie Monnin

Sylvie Chokron

Servane Mouton

Catherine Thomas Antérion

Sandrine Basaglia-Pappas

Laure Pisella

Isabelle Bernard

Olivier Martinaud

Mathilde Sauvée

Aurélie Richard-Mornas

Ilham Ryff

Neurodevelopmental disorders modify neurodegenerative diseases: a medical history to screen in memory clinics

RÉSUMÉ

Les troubles neurodéveloppementaux sont fréquents en population générale et perdurent souvent tout au long de la vie même s'ils peuvent être masqués par des mécanismes de compensation. Ces troubles sont pourtant souvent méconnus ou sous-estimés chez le sujet adulte et ce, même lorsqu'ils ont été diagnostiqués pendant l'enfance. Ils peuvent aussi interagir de manière complexe avec les processus neurodégénératifs, modifiant ainsi les présentations habituelles de ces pathologies. Prendre en compte cette dimension neurodéveloppementale au cours des consultations mémoire permet de mieux appréhender les différentes trajectoires cognitives tout au long de la vie. Cette nouvelle dimension vient enrichir l'idée d'une démarche intégrative en adoptant une approche médicale cognitivocomportementale de précision qui se veut personnalisée.

Mots clés :

Troubles neurodéveloppementaux ; maladie neurodégénérative ; démence atypique, adulte.

ABSTRACT

Neurodevelopmental disorders are frequent in the general population and are often lifelong conditions despite sometimes being masked by compensation mechanisms. These conditions are often unknown or underestimated in adults, even when diagnosed in childhood.

Neurodevelopmental disorders share similarities with and frequently interact in a complex way with neurodegenerative disorders. Considering these aspects during memory clinic assessments can provide a new perspective on lifelong neurocognitive trajectories. Assessing both neurodevelopmental and neurodegenerative dimensions is challenging but should improve diagnostic accuracy. It is therefore necessary to understand the lifelong specific neurocognitive trajectory of each patient in order to develop personalized and focused cognitive medicine and care.

Keywords: Neurodevelopmental disorders; neurodegenerative disease; atypical dementia

POINTS CLES

 Les troubles neurodéveloppementaux chez l'adulte sont fréquents, souvent méconnus et leur impact au quotidien est souvent sous-estimé.

 Les troubles neurodéveloppementaux perdurent souvent tout au long de la vie mais peuvent être masqués par des mécanismes de compensations.

 Les troubles neurodéveloppementaux modifient l'expression des pathologies neurodégénératives  Leur prise en compte en consultation mémoire est complexe mais permet d'affiner les diagnostics et d'adapter les prises en charge.

Introduction

Les troubles neurodéveloppementaux constituent un groupe de syndromes définis dans le DSM-V qui se manifestent tôt au cours de l'enfance par un développement cérébral atypique amenant à des fonctionnements différents ou des dysfonctions. Cela processus en terme de causalité, de facteurs de risques, de facteur de protection de comorbidités, restant encore largement méconnus, les relations sont souvent descriptives et/ou hypothétiques (figure 1).

Les différentes formes cliniques atypiques des pathologies neurodégénératives seraient associées de manière préférentielle avec certains syndromes neurodéveloppementaux, en lien avec leurs points communs cognitifs et structurels.

Aphasie progressive primaire

Marcel Mesulam a évoqué, dès les années 90, une possible association entre les aphasies progressives primaires et la dyslexie développementale, la considérant comme un facteur de risque possible [START_REF] Mesulam | Spectrum of primary progressive aphasia[END_REF]. Les premiers éléments en faveur de cette interaction sont rapportées seulement Les troubles des apprentissages mathématiques, quant à eux, correspondent souvent à un dysfonctionnement des régions pariétales [START_REF] Molko | Functional and structural alterations of the intraparietal sulcus in a developmental dyscalculia of genetic origin[END_REF][START_REF] Lehman | Cortical visual impairment in children: identification, evaluation and diagnosis[END_REF]. Ces régions occipito-pariétales sont également les zones de neurodégénérescence impliquées dans les atrophies corticales postérieures [START_REF] Crutch | Consensus classification of posterior cortical atrophy[END_REF]. Une étude portant sur 95 patients présentant une atrophie corticale postérieure a montré une surreprésentation d'antécédents neurodéveloppementaux comparativement à des patients atteints de maladie d'Alzheimer de présentation typique amnésique (13.6% versus 1%; p <0.001) [START_REF] Miller | Prevalence of Mathematical and Visuospatial Learning Disabilities in Patients With Posterior Cortical Atrophy[END_REF]. Les patients présentant une atrophie corticale postérieure avec un trouble neurodéveloppemental avaient également une atrophie plus asymétrique et prédominante à droite en comparaison à des patients sans antécédent neurodéveloppemental [START_REF] Miller | Prevalence of Mathematical and Visuospatial Learning Disabilities in Patients With Posterior Cortical Atrophy[END_REF]. Comme pour les aphasies progressives primaires, les patients avec un antécédent neurodéveloppemental présentaient des performances cognitives meilleures que les patients sans antécédent neurodéveloppemental au MMSE, fluences sémantiques et empans de chiffres envers, suggérant une moindre diffusion des lésions ou des mécanismes de compensations déjà en place limitant les déficits.

Un antécédent de TDAH a été rapporté avec un cas d'atrophie corticale postérieure questionnant sur un lien possible entre ces deux syndromes [START_REF] Seifan | Childhood Learning Disabilities and Atypical Dementia: A Retrospective Chart Review[END_REF]. Les troubles attentionnels visuels sont des symptômes communs aux deux entités [START_REF] Redondo | Visual Perceptual Skills in Attention-deficit/Hyperactivity Disorder Children: The Mediating Role of Comorbidities[END_REF][START_REF] Crutch | Consensus classification of posterior cortical atrophy[END_REF], tout comme, d'un point de vue structurel, les anomalies du lobe pariétal et la dysrégulation des noyaux gris centraux incluant notamment le thalamus, un relais visuel important [START_REF] Redondo | Visual Perceptual Skills in Attention-deficit/Hyperactivity Disorder Children: The Mediating Role of Comorbidities[END_REF][START_REF] Miller | Prevalence of Mathematical and Visuospatial Learning Disabilities in Patients With Posterior Cortical Atrophy[END_REF][START_REF] Chokron | The inter-relationships between cerebral visual impairment, autism and intellectual disability[END_REF][START_REF] Crutch | Consensus classification of posterior cortical atrophy[END_REF]. On soulignera également que la maladie à corps de Lewy peut se présenter initialement comme une atrophie corticale postérieure [START_REF] Crutch | Consensus classification of posterior cortical atrophy[END_REF] ce qui supporte l'existence d'un lien entre ce syndrome, l'étiologie sousjacente parfois à son origine, comme la maladie à corps de Lewy, et le TDAH.

Une étude pangénomique exploratoire chez 293 patients avec atrophie corticale postérieure suggère l'implication de trois gènes dont deux (SEMA3C et CNTNAP5) qui pourraient contribuer également à des processus en lien avec le développement cérébral [START_REF] Schott | Genetic risk factors for the posterior cortical atrophy variant of Alzheimer's disease[END_REF]. SEMA3C est impliqué dans le développement cortical et hippocampique, ainsi que la maturation du système visuel et CNTNAP5 est associé avec le trouble du spectre de l'autisme, ayant des interactions complexes avec les troubles neurovisuels de l'enfant [START_REF] Chokron | The inter-relationships between cerebral visual impairment, autism and intellectual disability[END_REF], et la maladie bipolaire, pathologie psychiatrique ayant aussi une forte composante neurodéveloppementale et des lien complexe avec les pathologies neurodégénératives [START_REF] Masouy | Bipolar disorder and dementia: where is the link?[END_REF].

Diagnostic différentiel : des troubles neurodéveloppementaux méconnus peuvent mimer des syndromes neurodégénératifs au cours du vieillissement

Les troubles neurodéveloppementaux ne sont pas toujours diagnostiqués dans l'enfance [START_REF] Barbiero | The submerged dyslexia iceberg: how many school children are not diagnosed? Results from an Italian study[END_REF] et par la suite à l'âge adulte, d'autant plus chez les patients les plus âgés pour qui ces diagnostics n'étaient pas autant explorés à l'époque de leur enfance qu'actuellement [START_REF] Magnin | Attention-deficit/hyperactivity disorder during adulthood[END_REF] le TDAH ou le trouble du spectre de l'autisme ont été associés à ces présentations atypiques [START_REF] Valente | Phenocopy syndrome of behavioral variant frontotemporal dementia: a systematic review[END_REF].

Evoquer un trouble neurodéveloppemental de l'adulte est important pour faire le diagnostic différentiel mais également identifier ces éventuelles comorbidités associées à des pathologies neurodégénératives (figure 2) risquant d'avoir plus souvent des présentations atypiques et ce, dans le but d'affiner les diagnostics et les prises en charge personnalisées.

Ceci représente un véritable défi en raison notamment de la nécessité d'une anamnèse très exhaustive, d'un suivi clinique et neuropsychologique à long terme qui se révèle être des outils diagnostiques majeurs pour ces situations complexes [START_REF] Magnin | Attention-deficit/hyperactivity disorder during adulthood[END_REF].

Diagnostiquer un trouble neurodéveloppemental au cours du vieillissement normal et/ou son éventuelle association à une authentique pathologie neurodégénérative ?

Le diagnostic de certitude pour les pathologies neurodégénératives et pour les troubles neurodéveloppementaux est post-mortem, à l'examen neuropathologique, ou génétique, avec l'identification de mutation causale de la symptomatologie [START_REF] Mckhann | The diagnosis of dementia due to Alzheimer's disease: recommendations from the National Institute on Aging-Alzheimer's Association workgroups on diagnostic guidelines for Alzheimer's disease[END_REF]. The accumulation of symptoms can therefore have a significant effect on autonomy (dotted grey line), and together could be wrongly identified as a severe neurocognitive disorder.

Diagnosis is therefore complicated because of comorbidities, aggravating/confounding factors, and differential diagnoses. The red star represents the example of a patient with an aging neurodevelopmental disorder, which could be wrongly diagnosed as a neurodegenerative disorder if the neurodevelopmental disorder is not identified. The neurodevelopmental disorder could also be forgotten about or underestimated (diagnostic error represented by the red arrow).

  pas clairement élucidé. Du point de vue structural, la modification du réseau neuronal du langage avec une moindre spécificité hémisphérique gauche chez ces patients mal latéralisés peut participer à la fragilisation des fonctions langagières incluant la mémoire sémantique. La perte d'asymétrie du planum temporale et de la pars triangularis, rapportée chez les sujets mal latéralisés, est ainsi associée à de moindres performances verbales[START_REF] Nair | Sex, Age, and Handedness Modulate the Neural Correlates of Active Learning[END_REF].Maladie de Parkinson et Maladie à corps de LewyUn possible fonctionnement prémorbide évoquant un TDAH, évalué par l'auto-questionnaire de la WURS, qui explore la sévérité des symptômes du TDAH avec des échelles de Likert de 0 (« pas du tout ou légèrement présent ») à 4 (« beaucoup »), est rapporté presque 3 fois plus souvent chez des patients présentant une maladie à corps de Lewy (48% des 109 patients) que chez des patients avec une maladie d'Alzheimer ou chez des sujets contrôles (environ 15% pour les deux groupes)[START_REF] Golimstok | Previous adult attention-deficit and hyperactivity disorder symptoms and risk of dementia with Lewy bodies: a case-control study[END_REF]. De même, sans pour autant dépasser les seuils diagnostiques des échelles diagnostiques, des symptômes d'inattention et d'hyperactivité dans l'enfance sont aussi modérément plus important chez les patients présentant une maladie de Parkinson par rapport à des sujets contrôles, (score à la WURS 14.4+/-13 vs 9.8+/-10 ; p = 0.01, valeur seuil > 30)[START_REF] Walitza | Association of Parkinson's disease with symptoms of attention deficit hyperactivity disorder in childhood[END_REF]. Les points communs entre TDAH et ces deux types de synucléinopathie sont cliniques et physiopathologiques. D'une part, ils partagent des symptômes cognitifs tels que les troubles attentionnels et le syndrome dysexécutif[START_REF] Callahan | Adult ADHD: Risk Factor for Dementia or Phenotypic Mimic?[END_REF]. D'autre part, le vaste réseau attentionnel fronto-pariétal[START_REF] Lin | Altered resting-state frontoparietal control network in children with attention-deficit/hyperactivity disorder[END_REF] et le dysfonctionnement dopaminergique au sein des circuits fronto-striato-cerebelleux participant au contrôle cognitif[START_REF]The neurobiology and genetics of Attention-Deficit/Hyperactivity Disorder (ADHD): what every clinician should know[END_REF][START_REF] Redondo | Visual Perceptual Skills in Attention-deficit/Hyperactivity Disorder Children: The Mediating Role of Comorbidities[END_REF] sont impliqués à la fois dans le TDAH et dans ces pathologies neurodégénératives.Atrophie corticale postérieureLes atrophies corticales postérieures sont fréquemment associées à des troubles neurodéveloppementaux[START_REF] Miller | Prevalence of Mathematical and Visuospatial Learning Disabilities in Patients With Posterior Cortical Atrophy[END_REF]: d'une part les troubles des apprentissages mathématiques, et d'autre part les troubles neurovisuels de l'enfance. Les symptômes neurovisuels et la dyscalculie sont des dysfonctions cognitives communes à ces entités[START_REF] Chokron | The inter-relationships between cerebral visual impairment, autism and intellectual disability[END_REF][START_REF] Crutch | Consensus classification of posterior cortical atrophy[END_REF]. Les troubles neurovisuels de l'enfance, survenant au cours du développement et pouvant mimer une déficience intellectuelle et/ou des signes autistiques correspondent, anatomiquement, souvent à une dysfonction cérébrale postérieure en particulier des régions occipitales et/ou pariétales.

Figure 1 .

 1 Figure 1. Quelques hypothèses d'interactions entre les troubles neurodéveloppementaux et

Figure 1 .

 1 Figure 1. A selection of non-mutually exclusive hypotheses concerning the interactions

Figure 2 :

 2 Figure 2 : Courbe représentant les symptômes des patients avec (bleu) et sans (marron)

Figure 2 :

 2 Figure 2: Curve showing the lifetime trajectory of symptoms of patients with (blue) and

Impact neurodéveloppemental sur l'expression clinique des pathologies neurodégénératives

	La prévalence des formes atypiques non amnésiques au sein de la maladie d'Alzheimer est
	d'environ 15% [11], elle est similaire à celle des troubles neurodéveloppementaux en
	peut entrainer des C'est pourquoi ces patients, dont l'antécédent neurodeveloppemental est souvent méconnus population générale [2] ceci soulève l'hypothèse que les troubles neurodéveloppementaux
	troubles cognitifs et/ou comportementaux pouvant retentir sur différents champs de la vie chez l'adulte, peuvent présenter au cours des évaluations en consultation mémoire adulte des pourraient être un facteur de risque de développement d'une présentation atypique de
	quotidienne profils cognitifs atypiques constitutionnels qui n'impliquent pas pour autant une pathologie (familial, social, scolaire, professionnel) [1]. Les troubles pathologie neurodégénérative [7]. En effet, avoir un possible antécédent de trouble
	neurodéveloppementaux sont très fréquents en population générale avec une prévalence neurodégénérative. Ces trajectoires cognitives différentes peuvent aussi perturber les neurodéveloppemental est plus souvent observé (Odd Ratio = 13.1 ; 95% IC 1.3-128.4) chez
	trajectoires habituelles du vieillissement normal et, le cas échéant, d'authentiques processus les patients ayant des présentations cliniques atypiques et focales de maladie d'Alzheimer
	neurodégénératifs, modifiant ainsi leur présentation clinique [6-7]. De nombreuses comme une atrophie corticale postérieure ou une aphasie progressive primaire (7 patients sur
	hypothèses quant aux interactions possibles entre ces 2 situations aux deux extrêmes de la vie 17, i.e. 41%) que chez les patients présentant des formes typiques amnésiques (8 patients sur
	existent mais ne sont pas encore élucidées (figure 1). Le défi sera pour les cliniciens d'être 68, i.e. 11%, p = 0.02) [7]. Il est possible que le fait d'avoir constitutionnellement un profil
	capables d'avoir une représentation cognitive globale de la personne et d'intégrer à leur cognitif neurodéveloppemental atypique, avec une structuration différente des réseaux
	démarche clinique ce fonctionnement cognitif différent afin de pouvoir évoquer ce diagnostic neuronaux (anomalies de développement corticale, connectivité différente), puisse conduire à
	différentiel ou cette comorbidité au cours de la démarche diagnostique [8]. des processus neurodégénératifs également atypiques.
	Cette dimension neurodéveloppementale au sein des pathologies neurodégénératives Cependant, même en tenant compte de la fréquente méconnaissance des troubles
	commence à être étudiée et décrite dans la littérature (for a review see [9]). L'objectif de cette neurodéveloppementaux, tous les patients présentant des formes atypiques ne semblent pas
	revue est de décrire les associations entre troubles neurodéveloppementaux et maladies forcément présenter d'antécédent de ce type ce qui suggère des interactions complexes et
	neurodégénératives. Cette approche poursuit tout d'abord un but fondamental, celui de probablement multifactorielles dans ce cadre. Les mécanismes sous-tendant ces différents
	comprendre le lien fonctionnel et neuro-anatomique entre ces deux entités. D'autre part,
	cette démarche vise un but clinique à long terme, en consultations mémoire adulte, celui de
	sensibiliser les professionnels de santé quant à l'importance du repérage des troubles
	neurodéveloppementaux à tout âge [10].

évaluée à 15% dans certains pays

[START_REF] Dietrich | Principles and practices of neurodevelopmental assessment in children: lessons learned from the Centers for Children's Environmental Health and Disease Prevention Research[END_REF]

, ce qui en fait un enjeu de santé publique, restant cependant peu connu des médecins s'occupant d'adultes ou de personnes âgées. Plusieurs présentations cliniques sont décrites telles que les troubles spécifiques des apprentissages (dyslexie ou dyscalculie par exemple), le trouble déficitaire de l'attention avec ou sans hyperactivité (TDAH), le trouble développemental du langage (anciennement appelé dysphasie développementale), le trouble de développement de la coordination (anciennement appelé dyspraxie développementale) et le trouble du spectre de l'autisme. Plusieurs d'entre eux peuvent s'associer et le terme de '"multidys" est de plus en plus utilisé pour rendre compte de ces différentes co-morbidités.

Souvent considérés comme des troubles relevant de la pédiatrie et/ou de la pédopsychiatrie, avec simplement un retard dans le développement qui se comblera avec le temps, cette condition cognitive et ce fonctionnement neurologique perdurent pourtant souvent tout au long de la vie, avec simplement des mécanismes de compensation ou d'adaptation plus ou moins coûteux en terme de surcharge cognitive. Au quotidien par exemple, des sujets adultes dyslexiques, au cours de lectures, ou des sujets adultes présentant un TDAH, au cours de remplissages de formulaires administratifs, pourront présenter un ralentissement en raison d'une stratégie de rétrocontrôle par vérification pour limiter leurs erreurs.

[START_REF] Peterson | Longitudinal Stability of Phonological and Surface Subtypes of Developmental Dyslexia[END_REF][START_REF] Kessler | Structure and diagnosis of adult attention-deficit/hyperactivity disorder: analysis of expanded symptom criteria from the Adult ADHD Clinical Diagnostic Scale[END_REF][START_REF] Chokron | Approche neuropsychologique des troubles des apprentissages[END_REF]

. Ainsi face à un évènement de vie (comme une naissance ou un deuil) ou évènement de santé (trouble du sommeil, troubles métaboliques), retentissant sur les ressources cognitives, ces mécanismes peuvent être décompensé et les symptômes et gènes au quotidien peuvent se manifester de manière plus importante pouvant mimer une pathologie évolutive neurodégénérative.

  enfance par exemple (i.e. développant ainsi une autre composante de la réserve cognitive). Ce « fardeau » compensatoire permanent pourrait expliquer la précocité des troubles cliniques par un phénomène de consommation des ressources cognitives ou surcharge cognitive alors que les patients non dyslexiques sont encore asymptomatiques.

	Certains gènes de susceptibilité impliqués dans la dyslexie développementale (KIAA0319 et
	CNTNAP2) sont d'ailleurs associés à une atrophie hémisphérique gauche plus importante
	chez les patients ayant une dégénérescence lobaire fronto-temporale [16-17]. Cependant,
	dans ces études, la présentation initiale des patients ayant ces gènes n'incluait pas de manière
	plus fréquente une aphasie progressive primaire (c'est-à-dire une forme langagière de
	dégénérescence fronto-temporale) que des formes comportementales.
	Enfin, la dyslatéralité développementale (i.e. absence de préférence manuelle droite type
	gaucherie ou ambidextrie), un autre fonctionnement développemental atypique ne pouvant
	Outre les altérations cérébrales structurales communes, le contexte génétique pourrait
	participer à ces associations, puisqu'une surreprésentation de troubles du langage
	développementaux et de fragilité du réseau neuronal du langage hémisphérique gauche,

[START_REF] Walitza | Association of Parkinson's disease with symptoms of attention deficit hyperactivity disorder in childhood[END_REF]

ans plus tard et concernent le sous-type logopénique d'aphasie progressive primaire pour lequel un antécédent de dyslexie développementale est retrouvé deux fois plus souvent que dans la population générale

[START_REF] Miller | Handedness and language learning disability differentially distribute in progressive aphasia variants[END_REF]

. Le point commun identifié dans les deux entités est le déficit cognitif similaire qui touche la mémoire à court terme auditivo-verbale, et plus spécifiquement, la boucle phonologique. De plus, les dysfonctionnements structuraux associés à ces deux troubles sont souvent observés dans le même territoire anatomique : le carrefour pariétotemporal gauche. C'est le cas des anomalies corticales de développement pour la dyslexie, et de l'atrophie avec accumulation de lésions neuropathologiques de pathologie neurodégénérative (le plus souvent de maladie d'Alzheimer biologique) pour l'aphasie progressive primaire logopénique, Dans cette étude

[START_REF] Miller | Handedness and language learning disability differentially distribute in progressive aphasia variants[END_REF]

, l'atrophie hémisphérique gauche des portions postérieures des gyri temporaux moyens et supérieurs était moindre chez les patients présentant une aphasie progressive primaire ayant eu une dyslexie développementale en comparaison à ceux sans trouble neurodéveloppemental. Les patients dyslexiques avaient déclaré des symptômes plus précocement (en moyenne 6.2 ans) mais la sévérité de leurs troubles au MMSE et aux tâches de dénomination était moindre que pour les patients sans trouble neurodéveloppemental. Ces interactions complexes peuvent suggérer que les patients dyslexiques seraient symptomatiques avec une moindre charge lésionnelle neurodégénérative (i.e. moindre atrophie) et présenteraient donc une moindre réserve cognitive focalisée à un domaine cognitif spécifique (i.e. au niveau de la boucle phonologique) constituant ainsi un « point faible » ou une fragilité d'origine neurodéveloppementale avec une addition des déficits constitutionnels et acquis. Ils présentent cependant également plus de mécanismes de compensation, acquis tout au long de la vie par eux-même ou grâce à une prise en charge orthophonique ou neuropsychologique dans l'Cependant, ces mécanismes compensatoires deviennent un atout secondairement pour faire face à la pathologie, limitant l'impact de ce processus neurodégénératif sur leur fonctionnement cognitif aux évaluations neuropsychologiques (i.e. moindre déficit cognitif).

Dans une autre étude

[START_REF] Miller | Cortical developmental abnormalities in logopenic variant primary progressive aphasia with dyslexia[END_REF]

, l'exploration neuropathologique de trois patients aux antécédents de dyslexie ayant eu une aphasie progressive primaire logopénique montre l'association dans la région périsylvienne de dysplasies corticales focales (i.e. delaminations corticales, neurones géants, immatures, dysplasiques, hétérotopiques ou en surnombre dans les différentes couches corticales) avec une concentration plus importante de lésions de maladie d'Alzheimer. Ces résultats ajoutent à l'hypothèse du « point faible » ou de la fragilité d'origine neurodéveloppementale, la possibilité d'un emballement neurodégénératif localisé au niveau de la zone d'anomalie développementale. associés à un moindre volume et une moindre connectivité au repos, est rapportée dans les familles de patients ayant une aphasie progressive primaire

[START_REF] Weintraub | Familial language network vulnerability in primary progressive aphasia[END_REF]

. pourtant pas être considéré réellement comme un trouble car peu handicapant au quotidien, est souvent associé aux dyslexies développementales

[START_REF] Eglinton | Handedness and dyslexia: a meta-analysis[END_REF]

. Cet antécédent est aussi surreprésenté dans les formes sémantiques d'aphasie progressive primaire

[START_REF] Miller | Handedness and language learning disability differentially distribute in progressive aphasia variants[END_REF]

. Ce lien n'est

  . De plus, même si un trouble neurodéveloppemental a été évoqué dans l'enfance, il n'est souvent pas rapporté spontanément par les patients dans leurs antécédents car il est considéré comme un problème scolaire passé plutôt qu'une histoire médicale pouvant encore avoir des conséquences, notamment sur une évaluation cognitive en consultation mémoire[START_REF] Magnin | Attention-deficit/hyperactivity disorder during adulthood[END_REF][START_REF] Magnin | Neurodevelopmental and neurodegenerative similarities and interactions: a point of view about lifelong neurocognitive trajectories[END_REF]. , amnésique ou multidomaine ou encore à une aphasie progressive primaire par exemple[START_REF] Magnin | Neurodevelopmental and neurodegenerative similarities and interactions: a point of view about lifelong neurocognitive trajectories[END_REF][START_REF] Callahan | Adult ADHD: Risk Factor for Dementia or Phenotypic Mimic?[END_REF]. Bien entendu, la recherche systématique d'une cause neurodégénérative surajoutée reste nécessaire avant de considérer que ces profils sont expliqués par un trouble

	neurodéveloppemental isolé [35].
	En cas de présentations initialement considérées comme des troubles neurocognitifs légers,
	des aphasies progressives primaires, des atrophies corticales postérieures ou des
	dégénérescences lobaires fronto-temporales qui se révèlent non-évolutives au cours d'un
	suivi clinique et neuropsychologique à long terme, le diagnostic de pathologie
	neurodégénérative suspectée doit être remis en question et, un trouble neurodéveloppemental
	et amener les patients à consulter en centre mémoire. Si leur fonctionnement cognitif
	atypique de base est méconnu, le profil cognitif présentant des déficits constitutionnels par
	rapport à la norme peut être considéré, à tort, comme un déclin récent et ainsi les classer dans
	la population à risque de développer une maladie neurodégénérative selon le DSM-V [8-9,

Et pourtant, même s'il peut être compensé, le fonctionnement cognitif atypique constitutionnel des patients avec un trouble neurodéveloppemental perdure tout au long de la vie. Au cours du vieillissement, habituellement associé à des troubles attentionnels légers, un discret syndrome dysexécutif et un ralentissement dans le traitement de l'information, le fonctionnement cognitif de ces patients peut parfois se modifier, en engendrant une moindre efficacité des mécanismes de compensation par un phénomène de surcharge ou d'épuisement des ressources cognitives

[START_REF] Laasonen | Developmentally impaired processing speed decreases more than normally with age[END_REF]

. Des facteurs généraux influençant la cognition peuvent s'ajouter et ainsi consommer une partie des ressources cognitives, notamment en population gériatrique fragile, tels qu'un syndrome d'apnée du sommeil, de la iatrogénie, des troubles de l'humeur ou encore des troubles endocriniens

[START_REF] Magnin | Neurodevelopmental and neurodegenerative similarities and interactions: a point of view about lifelong neurocognitive trajectories[END_REF]

. Une plainte cognitive peut ainsi s'installer 21, 34]. En fonction des troubles cognitifs et/ou comportementaux observés d'origine neurodéveloppementale méconnue, il est possible de conclure à un trouble neurocognitif léger exécutifsous-jacent devrait être systématiquement recherché. Les phénocopies de forme comportementale de dégénérescence fronto-temporale illustrent bien cette situation. Il s'agit de présentations cliniques sans anomalie d'imagerie associée, n'évoluant pas ou très peu dans le temps au cours du suivi et pour lesquelles l'étiologie sous-jacente reste incertaine. Outre des pathologies neurodégénératives très lentement évolutives ou des pathologies psychiatriques d'apparition tardive, des troubles neurodéveloppementaux méconnus tels que

  Dans la majorité des situations in vivo, le diagnostic n'est que probabiliste à partir de classifications clinicoradiologiques permettant parfois d'évaluer le risque « possible » et « probable » d'avoir ce diagnostic, en fonction du nombre de critères présentés par les patients. Très peu de biomarqueurs biologiques ou d'imagerie fiables existent pour améliorer cette acuité Les troubles neurodéveloppementaux peuvent faire partie des facteurs confondants et/ou des diagnostics différentiels participant à certaines de ces erreurs diagnostiques[8, 43- 44]. Ainsi, la précision diagnostique des évaluations cognitives de l'adulte et de la personne âgée pourrait être améliorée par un repérage systématique des troubles neurodéveloppementaux et de leur impact sur les profils clinique et cognitif des patients. ou comorbides (comme par exemple des troubles de personnalité, un trouble du spectre de la schizophrénie ou encore un trouble de l'humeur). Le développement de biomarqueurs d'imagerie ou biologiques est nécessaire pour améliorer la précision et le degré de confiance de ces diagnostics[START_REF] Scassellati | Biomarkers and attentiondeficit/hyperactivity disorder: a systematic review and meta-analyses[END_REF][START_REF] Serrallach | Neural Biomarkers for Dyslexia, ADHD, and ADD in the Auditory Cortex of Children[END_REF]. En combinaison avec les biomarqueurs des maladies neurodégénératives, les doubles diagnostics pourraient aussi être confortés et posés plus précocement sans nécessité d'un long suivi dans l'attente d'un déclin.

	questionnaire de dépistage court tel que le 30-Item Self-Report Questionnaire évaluant les
	difficultés en langage, mathématique, attention et fonction exécutive [10], et d'adjoindre a
	ces explorations, une évaluation neuropsychologique à la recherche de troubles cognitifs
	spécifiques soutenant ces hypothèses. Actuellement les outils sont rares et peu traduits en
	La confusion diagnostique dans les pathologies neurodégénératives est fréquente, à la fois entre les différentes étiologies neurodégénératives et syndromes cliniques mais aussi avec des diagnostics différentiels tels que des pathologies psychiatriques ou des troubles neurodéveloppementaux méconnus (figure 2). C'est un problème majeur des consultations mémoire avec un taux d'erreur pour le diagnostic de maladie d'Alzheimer évalué entre 20 et 30%. Il se retrouve dans le cadre de l'activité clinique habituelle, et peut avoir des conséquences délétères par excès ou par défaut sur les prises en charge médicamenteuses et médicosociales de ces patients. Il est également présent dans le cadre de la recherche clinique avec de possibles inclusions à tort pouvant induire une perte de puissance statistique avec un « bruit de fond » pouvant mettre en péril certaines découvertes, notamment thérapeutiques alors d'avoir une médecine de précision personnalisée dans le domaine neurocognitif [8-9]. Le rôle probablement complexe des troubles neurodéveloppementaux dans le mode évolutif rapide ou lent, et/ou l'apparition précoce ou tardive des processus neurodégénératifs, périodes de suivi. Poser un diagnostic de trouble neurodéveloppemental chez un adulte, et encore plus chez une personne âgée, repose sur une évaluation anamnestique a posteriori de symptômes présents depuis l'enfance (par exemple des retards dans les acquisitions du langage ou de la marche, des redoublements en primaire) et des facteurs de risque au cours du développement précoce (par exemple des troubles périnataux ou des antécédents familiaux de troubles neurodéveloppementaux). L'absence d'informations extérieures sur cette période de vie (parents, fratrie, carnet de santé, bulletins scolaires) est fréquente, surtout en population gériatrique, et vient compliquer cette démarche diagnostique chez des patients consultant pour des plaintes cognitives, potentiellement moins informatifs ou anosognosiques. Cela nécessite d'explorer systématiquement la période développementale par un repérage et de compléter ensuite par une investigation exhaustive comprenant l'utilisation de questionnaires et de trames d'anamnèse semi-structurée traduits en français et validés pour cette population, comme l'échelle de la WURS [45] et l'anamnèse semi-structurée de DIVA [46] pour une évaluation d'un TDAH de l'adulte, le Quotient du spectre de l'autisme (QA) [47] ou un français, n'explorent pas de tous les troubles neurodeveloppementaux, et leur validation est parfois peu robuste, surtout chez l'adulte et d'autant plus chez la personne âgée ou la personne ayant une pathologie neurodégénérative surajoutée [8]. De plus, il n'existe pas de consensus quant aux tests neuropsychologiques à utiliser chez cette population adulte. En conclusion, il existe des interactions complexes entre les troubles neurodéveloppementaux et les pathologies neurodégénératives mais ces résultats reposent encore sur de petites séries de patients, nécessitant confirmation sur de plus grandes cohortes, avec des outils et procédures diagnostiques pour repérer les troubles neurodéveloppementaux nécessitant d'être validés et harmonisés [7-10]. Le diagnostic de trouble neurodéveloppemental est aujourd'hui uniquement basé sur la clinique et l'évaluation cognitive avec des outils et des stratégies à améliorer pour cette population. Une vision pluridisciplinaire et pluri-professionnelle est fréquemment nécessaire au cours de la démarche diagnostique avec notamment l'appui de [38-42]. Connaître les trajectoires neurocognitives des patients tout au long de leur vie permettrait nécessitera la mise en place de grandes cohortes prospectives longitudinales sur de longues Figures et Légendes :

diagnostique, en dehors de ceux développés pour la maladie d'Alzheimer (imagerie TEP amyloïde, imagerie TEP Tau, dosage des protéines Tau et amyloïdes dans le liquide cérébrospinal, génotype APOe) qui ne sont pas toujours applicables à la pratique clinique quotidienne.

Comme cela a déjà été évoqué, une anamnèse très détaillée et, dans la mesure du possible, une hétéroanamnèse fiable auprès de proches informatifs sur la période de l'enfance, le suivi à long terme clinique et neuropsychologique, constituent des points clefs pour identifier ces situations

[START_REF] Magnin | Attention-deficit/hyperactivity disorder during adulthood[END_REF]

. Une stabilité à long terme du profil cognitif avec une anamnèse évocatrice d'un trouble neurodéveloppemental suggèrent ainsi que celui-ci est relativement isolé, alors qu'un déclin progressif des performances cognitives semble plus évocateur d'une composante neurodégénérative surajoutée

[START_REF] Deb | Dementia screening questionnaire for individuals with intellectual disabilities[END_REF]

.

psychiatres et/ou psychologues de l'adulte formés aux diagnostics de TDAH et/ou de TSA et sensibilisés aux autres troubles neurodéveloppementaux mais aussi pour identifier d'éventuels diagnostics différentiels


Antonio Rodríguez De Castro

Extending Darcy's law to the flow of yield stress fluids in packed beds: method and experiments

A large number of complex fluids commonly used in industry exhibit yield stress, e.g., concentrated polymer solutions, waxy crude oils, emulsions, colloid suspensions and foams. Yield stress fluids are frequently injected through unconsolidated porous media in many fields such as soil remediation and reservoir engineering, so modelling their flow through this type of media is of great economic importance. However, obtaining macroscopic laws to model non-Newtonian flow poses a considerable challenge given the dependence of the viscosity of the fluid on pore velocity. For this reason, no macroscopic equation is currently available to predict the relationship between injection flow rate and the pressure drop generated during the flow of a yield stress fluid without using any adjustable parameter. In this work, a method to extend Darcy's equation to the flow of yield stress fluids through model unconsolidated porous media consisting of packs of spherical beads is presented. Then, the method is experimentally validated through comparison with a total of 572 experimental measurements obtained during the flow of a concentrated aqueous polymer solution through different packs of glass spheres with uniform size. An improved prediction of the pressure drop-flow rate relationship is achieved by taking into account the non-linear relationship between apparent shear rate and average pore velocity.

Introduction

The flow of complex fluids in unconsolidated porous media is involved in many economically important industrial applications, e.g., remediation of polluted soils [START_REF] Gastone | Green stabilization of microscale iron particles using guar gum: Bulk rheology, sedimentation rate and enzymatic degradation[END_REF], Enhanced Oil Recovery (EOR) [START_REF] Wang | Advances in improved/enhanced oil recovery technologies for tight and shale reservoirs[END_REF], rock fracturing [START_REF] Roustaei | Non-Darcy effects in fracture flows of a yield stress fluid[END_REF] and liquid food engineering [START_REF] Welti-Chanes | Transport phenomena in food engineering: basic concepts and advances[END_REF]. Numerous complex fluids are shear-thinning, showing a decrease in shear viscosity as the applied shear rate is increased. Shear-thinning fluids are extensively used in petroleum engineering and soil remediation to improve the microscopic sweep of the reservoir through stabilization of the injection front [START_REF] Lake | Enhanced Oil Recovery[END_REF][START_REF] Silva | The effect of system variables on In situ sweep-efficiency improvement via viscosity modification[END_REF][START_REF] Wever | Polymers for enhanced oil recovery: a paradigm for structure-property relationship in aqueous solution[END_REF]. In some cases, fluids with shear-rate dependent viscosity additionally present a yield stress, i.e., a threshold value in terms of shear stress below which they do not flow. In the specific field of petroleum engineering, the drilling fluids injected into rocks for the drilling of wells are often designed so as to have a yield stress in order to prevent cutting from settling when circulation stops [START_REF] Lavrov | Non-Newtonian fluid flow in rough-walled fractures: a brief review[END_REF][START_REF] Coussot | Yield stress fluid flows: A review of experimental data[END_REF]. Some examples of yield stress fluids used in oil industry include emulsions, drilling muds, polymeric gels such as Carbopol, hevy oils and foams [START_REF] Talon | Effective rheology of Bingham fluids in a rough channel[END_REF]. Furthermore, a number of fracturing fluids used in hydraulic fracturing as gelling agents exhibit a yield stress designed to enhance proppant transport [START_REF] Roustaei | Non-Darcy effects in fracture flows of a yield stress fluid[END_REF]. Among them, guar, hydroxypropyl guar, carboxymethyl hydroxypropyl guar, hydroxyethyl cellulose, and polyacrylamide are of particular relevance [START_REF] Belyadi | Hydraulic Fracturing in Unconventional Reservoirs. Theories, Operation,s and Economics Analysis[END_REF].

Predicting the pressure drop of a yield stress fluid flowing through unconsolidated porous media is especially important, given that a great E-mail address: antonio.rodriguezdecastro@ensam.eu number of petroleum reservoirs are located in unconsolidated formations [START_REF] Peng | Borehole casing failure analysis in unconsolidated formations: a case study[END_REF][START_REF] Pang | The study on permeability reduction during steam injection in unconsolidated porous media[END_REF]. The majority of laboratory experiments in this field have been performed using beds of spherical beads, which represent an idealization of unconsolidated porous media [START_REF] Rao | Viscous non-Newtonian flow in packed beads: effects of column walls and particle size distribution[END_REF][START_REF] Tiu | Flow of viscoelastic polymer solutions in mixed beds of particles[END_REF][START_REF] Basu | Wall effect in laminar flow of non-Newtonian fluid through a packed bed[END_REF]. The Newtonian case is generally well understood in this type of porous media and allows process design calculations with acceptable levels of accuracy. The extensive literature regarding the flow of fluids with complex rheology through unconsolidated packed beds was critically reviewed by [START_REF] Chhabra | Flow of non-Newtonian fluids in fixed and fluidised beds[END_REF] , and [START_REF] Sochi | Non-Newtonian flow in porous media[END_REF] analysed the available models for describing non-Newtonian single-phase flow in porous media. More recently, Rodríguez de Castro and Radilla (2017a) extended Forchheimer's law and Ergun's equation to the flow of fluids with shear-ratedependent viscosity through packs of glass spheres with uniform size. The latter authors determined the accuracy of the extended laws under creeping and inertial regimes from comparison with a full set of experiments. Nevertheless, the yield stress effect was not addressed in the preceding works and obtaining a macroscopic law for the flow of yield stress fluids has proved to be a stumbling-block. [START_REF] Talon | On the determination of a generalized Darcy equation for yield-stress fluid in porous media using a Lattice-Boltzmann TRT scheme[END_REF] performed Lattice-Boltzmann simulations to solve 2D flow of Bingham yield stress fluids in porous media, and distinguished three different flow regimes. These regimes corresponded to 1) the flow of a single pore, 2) progressive pores opening and 3) flow of all pores. Furthermore, [START_REF] Chevalier | Darcy's law for yield stress fluid flowing through a porous medium[END_REF] focused on obtaining a generic relationship between flow rate and pressure drop applicable to the Darcian flow of yield stress fluids through packed beds. These authors proposed a macroscopic equation in which only the parameters of the rheological law of the injected fluid, the diameter of the beads and two coefficients related to the internal structure of the porous medium were used as inputs. Then, [START_REF] Chevalier | Breaking of non-Newtonian character in flows through a porous medium[END_REF] conducted NMR experiments which contributed to elucidate the structural parameters appearing in this generic law. However, the determination of these coefficients is still unclear and the proposed formula presents the inconvenient of assuming linear relationship between shear rate and Darcy velocity in definition of the apparent shear rate in the porous medium. Moreover, the extension of Darcy's equation to the flow of yield stress fluids proposed by [START_REF] Chevalier | Darcy's law for yield stress fluid flowing through a porous medium[END_REF] still has to be confirmed by further laboratory experiments given the serious lack of reliable experimental data in the literature [START_REF] Lavrov | Non-Newtonian fluid flow in rough-walled fractures: a brief review[END_REF][START_REF] Coussot | Yield stress fluid flows: A review of experimental data[END_REF].

Inspired by the broad interest of extending Darcy's law to the flow of yield stress fluids through model unconsolidated porous media, the objective of this work is to provide a straightforward procedure to predict the relationship between pressure gradient and flow rate. The proposed method is also evaluated through comparison with experimental data. To do so, a series of flow experiments through four different packs of mono-size spherical glass beads were carried out using concentrated aqueous solutions of xanthan biopolymer presenting a yield stress. The effect of beads size on the accuracy of the predictions is then assessed and discussed.

In the area of non-Newtonian flow in porous media, still very open, it is crucial to base the interpretations and the modelling on solid observations. The experimental details concerning the injection of xanthan gum solutions in different types of porous media have been carefully evaluated and discussed in the past: Rodríguez de Castro (2014), Rodríguez de Castro et al. (2014Castro et al. ( , 2016[START_REF] Rodríguez De Castro | Using Xanthan Gum solutions to characterize porous media with the yield stress fluids porosimetry method: robustness of the method and effects of polymer concentration[END_REF], Rodríguez de Castro and Radilla (2017b) . In all these preceding works, the same aqueous polymer solution was used and the experimental aspects were thoroughly addressed, including a discussion on the rheological model, the existence of a plateau viscosity, the capability of the fluid to emulate yield stress behaviour and the interactions between fluid and porous medium (polymer retention, mechanical degradation and polymer adsorption). For this reason, it was decided in the present work to capitalize the knowledge acquired from the preceding research by using the same extensively investigated xanthan gum solution.

Predicting the flow of yield stress fluids in packed beds

Previous attempts to extend Darcy's law to the flow of yield stress fluids

Herschel-Bulkley empirical law [START_REF] Herschel | Konsistenzmessungen von Gummi-Benzollösungen[END_REF] is commonly used to describe the rheological behaviour under shear of a large group of time-independent yield stress fluids. This law can be written as follows:

{ = 0 + ̇γ for > 0 ̇ = 0 for ≤ 0 (1)
where is the shear stress experienced by the fluid at a given shear rate ̇, 0 is the yield stress, a is the consistency and n is the flow index of the fluid. In the case of shear-thinning yield stress fluids, n is inferior to unity. The three parameters are generally obtained by fitting the data obtained by measuring the shear rate ̇ as a function of the applied shear stress with a rheometer. Several attempts have been made to obtain a macroscopic law linking the injection flow rate to the resulting pressure drop during the flow of yield stress fluids in porous media [START_REF] Pascal | Nonsteady flow through porous media in the presence of a threshold gradient[END_REF]Pascal, 1983;[START_REF] Al-Fariss | Flow through porous media of a shear-thinning liquid with yield stress[END_REF][START_REF] Chase | Correlation for yield stress fluid flow through packed beds[END_REF][START_REF] Coussot | Yield stress fluid flows: A review of experimental data[END_REF]. Pascal modified Darcy's law by introducing a threshold pressure gradient ∇ P t to account for the yield-stress [START_REF] Pascal | Nonsteady flow through porous media in the presence of a threshold gradient[END_REF]

: ∇ = 0 √
. ∇ P t is directly proportional to 0 and inversely proportional to the square root of the absolute permeability K . However, Pascal's relationship presents the serios drawback of including a dimensionless constant that must be empirically determined for each fluid-medium pair. Also, it only applies to the case n = 1. Indeed, the existence of experimentally adjustable parameters with no clear physical meaning as inputs, which impedes direct computational predictions, is a major drawback of most available macroscopic flow expressions. In this regard, [START_REF] Shahsavari | Mobility and pore-scale fluid dynamics of rate-dependent yield-stress fluids flowing through fibrous porous media[END_REF] conducted numerical simulations providing analytical expressions for such parameters in the particular case of fibrous materials, without including any specific dependence of these coefficients on the injection velocity. Only a few experimental works exist for the flow of yield stress fluids in porous media [START_REF] Al-Fariss | Flow through porous media of a shear-thinning liquid with yield stress[END_REF][START_REF] Chase | Correlation for yield stress fluid flow through packed beds[END_REF][START_REF] Chevalier | Darcy's law for yield stress fluid flowing through a porous medium[END_REF][START_REF] Chevalier | Breaking of non-Newtonian character in flows through a porous medium[END_REF]Rodríguez de Castro, 2016 ), and the ranges of variation of the flow rate are usually narrow. These experimental works showed that the relationship between the absolute value of the pressure gradient ∇ P and the absolute value of Darcy velocity u is of the same form as the constitutive equation of the fluid, i.e. ∇ P = ∇ P t + Cu n with C being a parameter that depends on the porous medium and the boundary conditions.

More recently, [START_REF] Chevalier | Darcy's law for yield stress fluid flowing through a porous medium[END_REF] presented a simple approach to extend Darcy's law to the flow of yield stress fluids. This general law contains a yielding term which may be simply expressed as a function of the yield stress of the material and the bead size:

∇ = 0 + ( ) (2)
with ΔP being the absolute value of the pressure drop through the packed bed of length L , ∇ = Δ the magnitude of the pressure gradient, Q the volume flow rate, A the cross-sectional area, u = Q/A the absolute value of the Darcy velocity and d s the diameter of the spherical beads. The latter authors initially stated that and in Eq. ( 2) , should be universal factors for the flow through spherical beads. The first coefficient is related to the path of maximum width throughout the porous medium while the second coefficient reflects the pore size distribution. However, on the basis of the results obtained by NMR measurements, it was subsequently shown that and are two dimensionless coefficients depending only on the distribution of shear rate intensity and on the coefficient n, which are in turn fluid-dependent [START_REF] Chevalier | Breaking of non-Newtonian character in flows through a porous medium[END_REF]. Also, u/d s was considered to be the apparent shear rate for the flow through such a porous medium, which is a serious flaw of Eq. ( 2) . Indeed, the apparent shear rate was shown not be proportional to u in the case of yield stress fluids flowing at low and moderate flow rates ( Rodríguez de Castro and Radilla, 2017b ).

The first (yielding) term on the right hand side of Eq. ( 2) corresponds to the critical pressure gradient below which no flow occurs. The second term is velocity-dependent, and expresses the additional viscous pressure drop above the yielding pressure once the fluid is flowing. [START_REF] Chevalier | Darcy's law for yield stress fluid flowing through a porous medium[END_REF] experimentally determined the values of and , obtaining = 12 for a Carbopol aqueous solution and = 5.5 for a water-in-oil emulsion, which did not permit to validate the universality of this coefficient. In contrast, these authors found that = 85 for both types of fluids.

New approach to extend Darcy's law to the flow of yield stress fluids

As mentioned above, despite the method presented by [START_REF] Chevalier | Darcy's law for yield stress fluid flowing through a porous medium[END_REF] being a valid approach, the choice of d s as characteristic length in the definition of the apparent shear rate and the non-dependence on injection velocity remain debatable. Also, the values of and are not easily predictable. For these reasons, the objective of this subsection is to present a method to simply predict the u vs. ∇ P relation by properly defining the actual shear rate and the shear viscosity of the fluid in the porous medium.

Darcy's law [START_REF] Darcy | Les Fontaines Publiques De La Ville De Dijon[END_REF] describes the single-phase flow of incompressible Newtonian fluids through porous media at low values of Reynolds number:

∇ = = (3)
where is the shear viscosity of the injected fluid, and K is the intrinsic permeability. Moreover, Kozeny-Carman equation allows to predict K from the porosity of the bed and the diameter of the beads using hydraulic radius theory:

= 3 2 36 ( 1 -) 2 (4)
being the Kozeny-Carman constant, the value of which is generally set to = 5 in packs of spheres [START_REF] Kaviany | Principles of Heat Transfer in Porous Media[END_REF].

Previous works have shown that some concentrated polymer solutions are yield stress fluids [START_REF] Song | Rheology of concentrated xanthan gum solutions: steady shear flow behavior[END_REF][START_REF] Carnali | A dispersed anisotropic phase as the origin of the weak-gel properties of aqueous xanthan gum[END_REF][START_REF] Withcomb | Rheology of xanthan gum[END_REF][START_REF] Economides | Reservoir Stimulation, Third edition[END_REF]Khodja, 2008;[START_REF] Benmouffok-Benbelkacem | Non-linear viscoelasticity and temporal behavior of typical yield stress fluids. Carbopol, Xanthan and Ketchup[END_REF]. The steady-state shear flow of these solutions can be well described by the Herschel-Bulkley law ( Eq. ( 1) ). A practical approach to study the flow of complex fluids through a porous medium consists in defining an equivalent viscosity eq as being the quantity that must replace the viscosity in Darcy's law to result in the same pressure drop actually measured [START_REF] Tosco | Extension of the Darcy-Forchheimer law for shear-thinning fluids and validation via pore-scale flow simulations[END_REF]:

= ∇ (5)
In order to predict eq from the constitutive equation of the fluid, an apparent shear rate in the porous medium ̇ has to be determined first. Assuming a bundle-of-capillaries model ̇ is usually taken as four times the average pore velocity 4 u / divided by the average pore throat radius (characteristic length of the microscopic flow) [START_REF] Chauveteau | Basic Rheological behavior of Xanthan polysaccharide solutions in porous media: effects of pore size and polymer concentration[END_REF][START_REF] Chauveteau | Rodlike polymer solution flow through fine pores: influence of pore size on Rheological behavior[END_REF][START_REF] Sheng | Modern Chemical Enhanced Oil Recovery, Theory and Practice[END_REF]. r can be estimated from the permeability K and the porosity of the porous medium, as proposed by [START_REF] Kozeny | Ueber kapillare Leitung des Wassers im Boden[END_REF] using a bundle-of-capillaries model:

= √ 8 (6)
According to the preceding definition, ̇ can be expressed as:

̇ = 4 = α √ 2 √ (7)
where is an empirical shift factor known to be a function of both the bulk rheology of the fluid and the tortuosity of the packed bed [START_REF] Chauveteau | Rodlike polymer solution flow through fine pores: influence of pore size on Rheological behavior[END_REF][START_REF] Sorbie | The rheology of pseudoplastic fluids in porous media using network modeling[END_REF][START_REF] López | Predictive network modeling of single-phase non-Newtonian flow in porous media[END_REF][START_REF] López | Department of Earth Science[END_REF][START_REF] Comba | Rheological characterization of xanthan suspensions of nanoscale iron for injection in porous media[END_REF]. Therefore, ̇ corresponds to the wall shear rate in a pore section of radius . This definition of apparent shear rate is in contrast with the one used by [START_REF] Chevalier | Darcy's law for yield stress fluid flowing through a porous medium[END_REF] , in which d s is taken as characteristic length instead of

√ √ 2
. For the creeping flow of Herschel-Bulkley fluids, eq can be obtained from Eqs. ( 1) and ( 7) :

= 0 √ √ 2 + ( √ 2 √ ) -1 (8)
Keeping in mind the objective to propose a prediction method, analytical expressions for the calculation of must be provided. In order to obtain such expressions, let us focus now on the determination of the wall shear rate in circular channels. For the steady flow of an incompressible fluid through a circular channel of radius , the wall shear stress w is related to the pressure gradient ∇ P as follows:

= ∇ 2 (9)
Using Eqs. ( 3) , ( 8) , ( 9) can be written as:

= ( 0 ̇ + ̇ -1 ) 2 = 1 2 √ 2 √ 0 + 2 -3 2 +1 2 -1 2 1- (10)
For a constant viscosity incompressible fluid, the wall shear rate ̇, is given by ̇, =

4

, where N is the shift factor for the injection of a Newtonian fluid. N is related to the tortuosity of the fluid flow through the packed bed and its value was shown to be 0.69 for spherical beads [START_REF] Christopher | Power-law flow through a packed tube[END_REF][START_REF] Shenoy | Non-Newtonian Fluid Heat transfer in porous media[END_REF]. However, this value N = 0.69 has been contested by some authors [START_REF] James | The laminar flow of dilute polymer solutions through porous media[END_REF]Chaveteau, 1982 ). For this reason, in this work N will be considered first as unknown and will be determined through fitting to the experimental u vs. ∇ P data. Then, the obtained N will be compared to the values previously reported in the literature.

The wall shear rate for the flow of liquids with a shear-ratedependent viscosity can be calculated by using the Weissenberg-Rabinowitsch-Mooney equation [START_REF] Rabinowitsch | Ueber die viskositat und elastizitat von solen[END_REF][START_REF] Mooney | Explicit formulas for slip and fluidity[END_REF]:

̇ = ̇, 3 [ 2 + d ln ̇, d ln ] = ̇, 3 ⎡ ⎢ ⎢ ⎣ 2 + d ( ln ̇, ) d d ( ln ) d + ( ln ) d ⎤ ⎥ ⎥ ⎦ = ̇, 3 ⎡ ⎢ ⎢ ⎣ 2 + d ( ln ̇, ) d ( ln ) + ( ln ) d d ⎤ ⎥ ⎥ ⎦ (11)
where is a function of u . Weissenberg-Rabinowitsch-Mooney equation is commonly used to calculate the wall shear rate of complex fluids with non-parabolic velocity profiles, including yield stress fluids [START_REF] Macosko | Rheology: principles, Measurements and Applications[END_REF][START_REF] Steffe | Rheological Methods in Food Process Engineering, Second Edition Freeman Press[END_REF][START_REF] Pipe | High shear rate viscometry[END_REF][START_REF] Sochi | Analytical solutions for the flow of Carreau and cross fluids in circular pipes and thin slits[END_REF]. The following assumptions are used in the derivation of Eq. ( 11) : incompressible fluid, steady state, laminar flow regime, no wall-slip, no end-effects, unidirectional flow, temperature is constant and properties are not a function of time or pressure [START_REF] Steffe | Rheological Methods in Food Process Engineering, Second Edition Freeman Press[END_REF]. For a Herschel-Bulkley fluid, Eq. ( 11) becomes:

̇ = 4 3 ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ 2 + ( 0 + 2 2 ( √ ) ) 2 2 ( √ ) -2 ( 0 -2 2 ( -1 ) ( √ ) n ) ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ (12)
From Eq. ( 12) , it can be deduced that becomes the constant value

α * = 3 ( 2 + 1
) for very high values of u . By combining Eqs. ( 7) and ( 12) , the following differential equation is obtained, which allows the determination of as a function of u :

= 2 √ 2 √ 3 √ ⎛ ⎜ ⎜ ⎜ ⎝ 2 + ( 0 + 2 2 ( √ ) ) 2 2 ( √ ) -2 ( 0 -2 2 ( -1 ) ( √ ) ) ⎞ ⎟ ⎟ ⎟ ⎠ (13)
For the simpler case of a power-law fluid ( 0 = 0), Eq. ( 14) leads to

= 3 ( 2 + 1
) , which becomes = N for a Newtonian fluid. Therefore, is a constant parameter only if 0 = 0.

Eq. ( 13) can be numerically solved within a given range of u to obtain the relation between and u . Then, the obtained relation can be used in Eq. ( 8) to obtain eq . Once eq has been determined, it can be entered in Eq. ( 3) , leading to the extension of Darcy's law ( Eq. ( 14) ):

∇ = = 1 + 2 -1 (14) with 1 = 0 √ √ 2 √ and 2 = +1 2 ( 2 ) -1 2 .
It is reminded that the value of N is considered first as unknown and must be obtained by fitting Eq. ( 14) to the experimental ( u i , ∇ P i ) data. This is achieved by finding the value of N that minimizes the sum = ∑ =1 ( |∇ -∇ ( ) | ×∇ ) , with N being the number of experimental data.

By comparing the method presented in this subsection with the works of [START_REF] Chevalier | Darcy's law for yield stress fluid flowing through a porous medium[END_REF] , it can be deduced from Eqs. ( 2) and ( 14) that:

= 1 2 2 1 2 1 2 (15)
Moreover, Eq. ( 4) can be used together with Eq. ( 15) to express as a function of only and , obtaining:

= 2 1 2 1 2 3 ( 1 -) = 2 1 2 5 1 2 3 ( 1 -) (16)
It can be concluded from the preceding equation that is a constant at high injection flow rates, given that the value of is also constant ( = * ). Therefore, the first (yielding) term on the right-hand side of Eq. ( 2) can be considered a constant at high flow rates. However, this term depends on u at low and moderate values of u , which was not taken into account in the work of [START_REF] Chevalier | Darcy's law for yield stress fluid flowing through a porous medium[END_REF] .

Also, the following relationship can be obtained from comparison between Eqs. ( 2) and ( 14) :

= 2 -1 2 -1 2 +1 2 -1 +1 (17)
Analogously, Eq. ( 4) can be used together with Eq. ( 17) to express as a function of only and , obtaining:

= 2 3 +1 2 3 +1 +1 2 ( 1 -) +1 2 +1 -1 = 2 3 +1 2 3 +1 5 +1 2 ( 1 -) +1 2 +1
-1 (18)

Eq. ( 18) shows that is also contant at high flow rates, while being a function of u at moderate and low flow rates. Moreover, is not a function of d s , but depends on the fluid properties through . This is contrast to the claim of [START_REF] Chevalier | Darcy's law for yield stress fluid flowing through a porous medium[END_REF] according to which and are universal factors for a porous medium composed of an assembly of spheres.

It should be kept in mind that elongational flows during the injection of solutions of polymers presenting a certain degree of flexibility through porous media are known to induce extra pressure losses with respect to pure shear flow [START_REF] Rodríguez | Flow of polymer solutions through porous media[END_REF][START_REF] Müller | The rheology of polymer solutions in porous media[END_REF][START_REF] Nguyen | Flexible Polymer Chains in Elongational Flow: Theory and Experiement[END_REF][START_REF] Seright | New Insights into polymer rheology in porous media[END_REF][START_REF] Amundarain | Solutions of xanthan gum/guar gum mixtures: shear rheology, porous media flow, and solids transport in annular flow[END_REF]. This is a result of the formation of transient entanglements of polymer molecules due to the action of the extensional component of the flow. In the present work, we first hypothesize that the deviation of the experimentally measured pressure drop with respect to the viscous pressure drop are negligible. This hypothesis is then validated through analysis of the experimental results.

Experimental methods and materials

Experimental ∇ P vs. u measurements were performed by injecting a xanthan gum aqueous solution (yield stress fluid) through four packs of spherical glass beads. Flow experiments with filtered water (Newtonian fluid) were also performed by following the procedure presented by Rodríguez de Castro and Radilla (2017a) in order to determine the permeability of the packed beds. The glass beads were first placed into transparent acrylic glass cylinders and then compactly packed by means of vibration with a sieve shaker. The inner diameter of the acrylic glass cylinders was D = 5 cm and the diameter of the glass spheres used in each of the four columns was uniform, with ds = 1 mm, 3 mm, 4 mm and 5 mm in each case. The length of the column was L = 20 cm.

Two different configurations were used depending on the involved flow rates. For 0.12 L/h ≤ Q ≤ 6 L/h, the injection circuit was open and the fluid was injected through the packed beds at the selected flow rate using a dual piston pump (Prep Digital HPCL pump, A.I.T., France). For 9 L/h ≤ Q ≤ 250 L/h, the fluid was injected through a closed circuit using a volumetric pump as performed by Rodríguez de Castro and Radilla (2017a) . A photo showing the experimental setup is provided as supplementary material (Fig. S1). Details of the experimental setup and procedure, including the working ranges of the instruments and the measurement uncertainties were provided by Rodríguez de Castro and Radilla (2017a) . The ranges of u imposed during the experiments with each packed bed are listed in Table 1 .

Xanthan biopolymer is a microbial high molecular weight exopolysaccharide produced by fermentation of X. campestris bacteria

Table 1

Range of average velocities imposed during the flow of the yield stress fluid through the packed beds as a function of d s . N is the number of experimental ∇ P vs. u data (four repetitions for each data).

d s

Range of u (m/s) N 1 mm 1.7 × 10 -5 -3.5 × 10 -3 27 3 mm 1.7 × 10 -5 -3.5 × 10 -2 44 4 mm 1.7 × 10 -5 -2.8 × 10 -2 39 5 mm 1.7 × 10 -5 -2.8 × 10 -2 33 ( Garcia-Ochoa et al., 2000;Palaniraj and Javarman, 2011;[START_REF] Kumar | Application of xanthan gum as polysaccharide in tissue engineering: a review[END_REF]. In solution state, an isolated macromolecule of this polymer is more or less rigid and with a typical contour length of 1 μm [START_REF] Mongruel | Axisymmetric orifice flow for measuring the elongational viscosity of semi-rigid polymer solutions[END_REF] Sixty litres of aqueous solution were prepared with xanthan gum concentration C p = 7000 ppm and the rheogram was obtained following the procedure presented by Rodríguez de Castro and Radilla (2017b) . Eq. ( 1) was then used to fit the rheogram ( Rodríguez de Castro et al., 2014 ) giving 0 = 7.4 Pa, a = 0.37 Pa s n and n = 0.52. The rheogram of the solution and the Herschel-Bulkley fit are provided in Fig. 1 . The dynamic viscosity of water (solvent) was measured to be 0.0011 Pa s and the densities of both the water and the xanthan gum solution were taken as 1000 kg/m 3 . Moreover, the rheograms of several effluent fluid samples were characterized and compared to that of the inflowing fluid at the highest injection flow rates in order to assess polymer degradation and retention on the pore walls. No significant difference was observed between the rheograms, proving that polymer degradation and polymer retention can be neglected.

Despite the used glass beads being quite coarse as compared to most natural granular media, the explored sizes fall within the range of grain sizes reported for coarse sand and fine gravel, which are widely investigated in hydrologic applications [START_REF] Morris | Summary of hydrologic and physical properties of rock and soil materials, as analyzed by the hydrologic laboratory of US geological survey 1948-60[END_REF]. Moreover, these beads sizes are commonly used in previous research, e.g. [START_REF] Dukhan | Experimental flow in various porous media and reconciliation of Forchheimer and Ergun relations[END_REF] , so this choice facilitates comparison with literature data. Furthermore, the use of smaller beads may result in polymer retention, which was not observed in the present experiments.

Results

The flow experiments were conducted for both fluids (water and yield stress fluid) and were repeated four times. The number of repetitions for yield stress fluid injection through each packed bed corresponds to 4 × N (ranging from 108 to 176) as listed in Table 1 . The 4 × N measures for each packed bed were considered to be an experimental set. A total of 572 measurements were performed during the flow experiments with the yield stress fluid.

Experimental determination of and k

The weight of each packed bed was measured before and after saturation with water in order to determine from the difference in mass. Also, the procedure followed by Rodríguez de Castro and Radilla (2017a) to Table 2 and K for the four different beads sizes. 1 mm 0.36 ± 3% 5.9 × 10 -10 m 2 ± 2% 3 mm 0.34 ± 3% 5.1 × 10 -9 m 2 ± 6% 4 mm 0.35 ± 3% 9.5 × 10 -9 m 2 ± 7% 5 mm 0.34 ± 3% 1.3 × 10 -8 m 2 ± 6% 2 together with the associated uncertainties (95% confidence interval).

Shear viscosity of the yield stress fluid in the porous media

Eq. ( 13) was numerically solved within the involved range of u for both all the investigated packed beds using an implicit Runge-Kutta method. The resulting versus u functions are represented in Fig. 2 and the results obtained for N are listed in Table 3 as a function of d s . It is noted that the value of N was close to 0.68 (average value) in all the tested porous media for the polymer solution used in the present work. This is in very good agreement with the results of [START_REF] Christopher | Power-law flow through a packed tube[END_REF] , who obtained N = 0.69. . Consequently, the boundary condition ( u = 10 5 u * ) = * was used to numerically solve Eq. ( 13) . The obtained * values are also listed in Table 3 and are all close to 0.88 (average value). Regarding the sensitivity of to the microstructure of the packed bed, it can be deduced from Fig. 2 that higher values of d s (coarser microstructure) result in higher values of .

The value of ̇ corresponding to each Darcy velocity u was calculated with Eq. ( 7) following two different approaches. First, a constant value of , named was determined for each porous medium by calculating the shift factor in terms of shear rate which led to the best superposition between the "in situ " eq vs. ̇ data and the bulk rheological law ( Eq. ( 1) ). The obtained values for are shown in Table 3 . The second approach consisted in using the ( u ) function obtained from Eq. ( 13) . The results of both approaches are presented in Fig. 3 , together with the bulk rheological law ( Eq. ( 1) ). In this figure, it can be observed that eq is close to Eq. ( 1) at high values of u for both the constant and the variable-methods. However, this is not the case at low and moderate values of u for which eq approaches better Eq. ( 1) with the variable-method. Also, eq is expected to be greater than the bulk viscosity at high values of u in the presence of important inertial effects [START_REF] Tosco | Extension of the Darcy-Forchheimer law for shear-thinning fluids and validation via pore-scale flow simulations[END_REF][START_REF] Rodríguez De Castro | Non-Darcian flow experiments of shearthinning fluids through rough-walled rock fractures[END_REF]. The fact that no important deviation of eq with respect to pm is observed in the present experiments reflects that inertial effects are not significant. Moreover, Fig. 3 shows that the shear rates involved in the flow through all porous media are within the same range as those measured with the rheometer during characterization of the fluid's shear viscosity.

Previous attempts to extend Darcy's law to the flow of yield stress fluids

The values of and were determined by fitting the experimental results presented in this work to Eq. ( 2) through minimization of the sum of the absolute values of the differences between fit and experimental data. The obtained values are listed in Table 4 , showing that and are porous medium-dependent as experimentally determined. Also, it is remarked that the values of these coefficients may depend on the range of imposed u , as they are obtained through fitting to experimental data. This dependence on u is taken into account by the new method proposed in the present work, as explained in Section 2.2 . The results of fitting Eq. ( 2) to the experimental data are shown in Fig. 4 . Moreover, the

Table 4

Values of and used in Eq. ( 2) for the four packed beds. average errors of these fits are presented in Table 5 for different ranges of u . It is observed that the resulting fits are accurate within a large range of u . However, a major drawback of this method is that and need to be experimentally determined, which impedes prediction of the u vs. ∇ P relation. It is worth mentioning that the errors obtained by using = 5.5 and = 85 as proposed in the work of [START_REF] Chevalier | Darcy's law for yield stress fluid flowing through a porous medium[END_REF] are too big in the case of the present experiments and lead to very inaccurate predictions.

Table 5

Average errors obtained by fitting Eq. ( 2) to the experimental ∇ P vs. u data, with the variable-method and with the fixed-method for different ranges of u .

Range of u (m/s) Average error using Eq. ( 2

Experimental validation of the new prediction method

Eq. ( 14) was used to predict the relation between ∇ P and u for the injection of the 7000-ppm solution through the four packed beds. The obtained predictions are presented in Fig. 5 together with the experimental results of measurements performed in the present work. In this figure, the errors bars correspond to a 95% confidence interval as explained in Section 3 . From these results, the accuracy of the proposed methods for the prediction of ∇ P as a function of u during the flow of yield stress fluids through packed beds of spherical beads can be assessed. Fig. 5 shows that the variable-approach provides more accurate predictions within the low and moderate u regions, which is in agreement with the arguments presented above. However, a less important difference is obtained between both methods for the highest values of u . The average errors obtained with the variable-method and the fixed-method for different ranges of u are summarized in Table 5 . It is observed that the variable-method successfully predicts the ∇ P-u relationship for the flow of the yield stress fluid through the four packed beds, even though the obtained predictions are slightly less accurate in the case of d s = 1 mm. The overestimation of ∇ P reported in Figs. 4 and5 for the lowest flow rates may be related to the longer times needed to achieve stationary measurements of pressure drop within this region. This effect is similar as the one reported for rheological measurements at low shear rates, as shown in Fig. 1 . This is a consequence of the viscosity of the fluid continuously increasing over time as illustrated in Fig. S2. This effect will be discussed in Section 5 . "

As mentioned above, becomes the constant value * = 3 ( 2 + 1 ) when

≫ * = 0 1 n 1 2 1 2 2 1∕2 1 *
. This means that Eq. ( 14) presents a constant yielding term of value 1 and a constant "consistency "t e r m of value C 2 n -1 for u ≫ u * . In other words, Eq. ( 14) has the same form as Herschel-Bulkley empirical law ( Eq. ( 1) ) only if the preceding condition is met. Therefore, the threshold Reynolds number Re * above which the extended Darcy's law for Herschel-Bulkley ( Eq. ( 14) ) fluids has the same form as Herschel-Bulkley equation is given by: *

= * √ * ( 19
)
where * is the shear viscosity of the fluid in the porous media at u * . Re * is represented as a function of K for the four packed beds in Fig. 6 showing linear relationship.

It is worth mentioning that, in spite of the negligible influence of inertial effects on the pressure drop vs. flow rate relationships in the case of the highly-viscous xanthan gum solutions used in this work, the procedure presented in Section 2.2 . is also valid to extend Forchheimer equation [START_REF] Forchheimer | Wasserberwegng durch Boden[END_REF] to the case of yield stress fluids. This is explained by the fact that the inertial coefficient appearing in Forchheimer equation does not depend on the shear rheology of the injected fluid as numerically [START_REF] Firdaouss | Nonlinear correction to Darcy's law at low Reynolds numbers[END_REF]Yadzchi and Luding, 2012;[START_REF] Tosco | Extension of the Darcy-Forchheimer law for shear-thinning fluids and validation via pore-scale flow simulations[END_REF] and experimentally [START_REF] Rodríguez De Castro | Non-Darcian flow experiments of shearthinning fluids through rough-walled rock fractures[END_REF]2017a;2017b ) proved in previous works. However, it must be Fig. 6. Threshold Reynolds number Re * ( Eq. ( 19) ) for the injection of the yield stress fluid through the four porous media. Squares represent experimental data and the solid line represents their linear adjustment obtained through standard Least Squares regression: Re * = 3.2 × 10 6 K ( m 2 ). noted that even for Newtonian fluids, the macroscopic transport equations governing inertial regime are still under debate in the literature. In particular, it was demonstrated that whereas Forchheimer regime is always well identified for inertial flow in disordered porous media, its appearance in ordered media is strongly dependent on the microstructure and the orientation of the pressure gradient [START_REF] Lasseux | On the stationary macroscopic inertial effects for one phase flow in ordered and disordered porous media[END_REF][START_REF] Agnaou | Origin of the inertial deviation from Darcy's law: an investigation from a microscopic flow analysis on two-dimensional model structures[END_REF]. Therefore, a non-linear dependence of apparent viscosity on pore scale velocity is expected to increase the complexity of the problem.

Discussion

The values of N obtained for all the tested porous media were always very close to 0.69, which was the value theoretically predicted by [START_REF] Christopher | Power-law flow through a packed tube[END_REF]. It should be noted that although N = 0.69 is valid for the present experiments, this value must still be confirmed by further experiments in different yield stress fluid-packed beds combinations before declaring that it is a universal constant. Nevertheless, it can be firmly stated that the results reported in this work are a highly promising step in this direction.

It can be deduced from Eq. ( 14) that can be considered a constant value * = 3 ( 2 + 1 ) in the high flow rates region, i.e., when

≫ * = 0 1 1 2 1 2 2 1∕2
1 , and Eq. ( 14) can be written as:

∇ = ∇ 0 + (20) with ∇ 0 = 3 0 √ √ 2 √ ( 2+ 1 ) and = +1 2 ( 2 ) -1 2 [ 3 ( 2 + 1 ) ] -1
. This is in agreement with the results of [START_REF] Talon | Effective rheology of Bingham fluids in a rough channel[END_REF] , who stated that u scales linearly as ( ∇ P -∇ P 0 ) in the case of a Bingham fluid ( n = 1) flowing at high u through a one-dimensional channel. Also, [START_REF] Nash | The effect of microstructure on models for the flow of a Bingham fluid in porous media: one-dimensional flows[END_REF] showed that the manner in which flow begins once the threshold pressure gradient is exceeded strongly depends on the channel size distribution of the porous media. The same authors [START_REF] Talon | Effective rheology of Bingham fluids in a rough channel[END_REF][START_REF] Nash | The effect of microstructure on models for the flow of a Bingham fluid in porous media: one-dimensional flows[END_REF] proved that ∇ P 0 is higher than the actual threshold pressure, which is consistent with our results given that increases as u tends to zero ( Fig. 2 ).

A model to accurately predict the flow of yield stress and Carreau fluids through rough-walled fractures by using N = 1 was presented and experimentally validated in a previous work ( Rodríguez de Castro and Radilla, 2017b ). It is reminded here that N is a tortuosity-related factor, so media with different tortuosity may lead to different values of N . Indeed, the effective average pore throat radius , which takes into account the tortuosity of the medium, can be defined as = [START_REF] Christopher | Power-law flow through a packed tube[END_REF]Chaveteau, 1982 ). Given that the tortuosity of the flow paths in a packed bed is higher than in a fracture, a lower value of N is expected for packed beds.

In the case porous media with more complex pore size distributions, the flow is highly conditioned by the narrowest flow paths at low flow rates and the representative pore section should be smaller than . In this regard, the full set of equations presented in Section 2.2 should be reconsidered as the current method is not able to capture the influence of pore size distribution. Nevertheless, the use of the present method with more complex porous media should still be useful to predict the relationships between ∇ P and u with higher accuracy than the existing methods which use a constant viscosity value.

The existence of yield stress was challenged by [START_REF] Barnes | The yield stress myth?[END_REF] and has been discussed for more than 30 years. As explained by [START_REF] Møller | Origin of apparent viscosity in yield stress fluids below yielding[END_REF] , the supporters of the existence of yield stress commonly argue that the viscosity increases very sharply in some materials as the stress decreases towards the yield stress. However, other researchers claim that only a finite and constant viscosity (Newtonian plateau of viscosity) is observed below a certain stress. In particular, [START_REF] Barnes | The yield stress myth?[END_REF] used stress-controlled rheometers to show that at low enough shear rates, viscosity reaches a Newtonian plateau for Carbopol and other fluids which had traditionally been considered to have a yield stress. They argued that any material flows providing enough observation time and sufficiently sensitive measuring equipment. In stark contrast with [START_REF] Barnes | The yield stress myth?[END_REF] and [START_REF] Møller | Origin of apparent viscosity in yield stress fluids below yielding[END_REF] experimentally showed that such Newtonian plateau is the consequence of non-steady-state measurements. They demonstrated that for stresses below the yield stress, viscosity is "a priori " unbounded and increases continuously, though slowly, if enough time is allowed. They effectively observed an increase in viscosity even after 100 s. In other words, they found that viscosity is time dependent and tends to infinity below the yield stress. In the case of the present xanthan gum solutions, the evolution of viscosity over time was measured for 1000 s under a shear stress of 0.5 Pa (below the yield stress) using a rheometer equipped with cone/plate geometry. The results are provided as supplementary material (Fig. S2), showing that viscosity does not attain a constant value and continues to increase after that time.

One may wonder whether the proposed procedure is simpler than performing a numerical solution to the actual flow equations, without invoking a bundle-of-capillaries approximation. In this sense, it should be highlighted that performing a numerical solution to the actual flow equations would imply using the size distribution of the flow paths as an input for the model, which is rarely available in real applications. It is reminded that the objective of this work is to present a simple method to predict the pressure drop for the flow of yield stress fluids through packed beds. Therefore, using hardly accessible inputs as needed to perform a numerical solution to the actual flow equations is not a valid approach.

It is noted that in our experiments with yield stress fluids, the total pressure drop through the porous media was successfully predicted from the value of K obtained from water injection without any significant deviation. Therefore, similarly to the case of previous flow experiments with shear-thinning fluids without yield stress ( Rodríguez de Castro and Radilla, 2017a ), no appreciabe effect of elongational viscosity has been observed in the present work. Also, wall-effect issue during shear-thinning creeping flow in packed beds was previously addressed in the literature. On this subject, [START_REF] Rao | Viscous non-Newtonian flow in packed beads: effects of column walls and particle size distribution[END_REF] studied the effects of column walls and particle size distribution on the flow ratepressure drop relationship, proposing a wall correction method and confirming the applicability of the mean hydraulic radius of the particles to characterize a bed of mixed size spheres. The latter authors showed that wall-effect is less significant in the case of shear-thinning fluids than in the Newtonian case. In the present experiments, the porosity of all packed beads is 0.35 ± 0.01 and the experimentally measured permeability is very close to Kozeny-Carman prediction for the largest beads (3.6% difference). Therefore, there is no evidence of significant walleffect affecting pressure drop vs. flow rate relationship.

Summary and conclusions

A simple approach to extend Darcy's law to the flow of yield stress fluids through packed beds has been presented in this work. This method takes into account the non-proportional relationship between the apparent shear rate in the porous medium ̇ and average pore velocity u . Only the porosity and the permeability K of the porous medium (exclusively for high flow rates) are used as inputs of the method, together with the Herschel-Bulkley parameters of the fluid ( 0 , a, n ).

The following procedure to predict ∇ P as a function of u is proposed:

(1) Determine the shear-rheology parameters of the fluid using a rheometer: ( 0 , a, n ). ( 2) Measure the porosity of the packed beds, e.g., from difference in mass before and after saturation with water. Note that the usual values are close to ∼ 35%. (3) Measure K from Newtonian-flow experiments. Alternatively, K can be estimated from Kozeny-Carman equation ( Eq. ( 4) ) or determined by other techniques (e.g., x-ray However, the cited methods provide estimates with very different accuracy, which can be roughly estimated to ∼5% for experimental assessment, ∼10% for Kozeny-Carman and ∼20% for tomography. (4) Calculate the values of ( u ):

4.1) When low and moderate values of u are involved, solve the following differential equation ( Eq. ( 13) ) to obtain ( u ): ) . 4.2) Use Eq. ( 8) to compute eq as a function of u, , K , 0 , a and n :

= 2 √ 2 √ 3 √ ⎛ ⎜ ⎜ ⎝ 2 + ( 0 + 2 2 ( √ ) )
= 0 √ √ 2 + ( √ 2 √
) -1

(5) Use Eq. ( 14) to calculate ∇ P as a function of u :

∇ = = 1 + 2 -1 with 1 = 0 √ √ 2 √ and 2 = +1 2 ( 2 ) -1 2 .
Flow experiments of yield stress fluids covering a wide range of u were performed in order to assess the accuracy of the predictions obtained using the proposed method, showing good agreement between model and experiments and negligible inertial effects within the explored range of u . Consequently, Darcy's law provides accurate u-∇ P predictions in contrast to the case of less concentrated solutions with no yield stress in which inertial effects were significant ( Rodríguez de Castro and Radilla, 2017a ).

As an important industrial application, the extended Darcy's law can be included in computational studies of large-scale non-Newtonian flow in unconsolidated porous media. The conclusions of this work have now to be assessed using real granular media. Also, future numerical studies should be performed in order to provide deeper insight into the physical mechanisms governing the non-proportional relationship between ̇ and u .

Fig. 1 .

 1 Fig. 1. (a) Rheogram of the injected yield stress fluid and (b) viscosity vs. shear rate relationship. The void circles represent experimental data and black solid lines represent the fitted curves using the Eq. (1) .

Fig. 2 .

 2 Fig. 2. ( u ) functions as numerically obtained from Eq. (13) corresponding to the injection of the 7000-ppm solution through the four packed beds. Purple colour corresponds to d s = 1 mm, red colour corresponds to d s = 3 mm, black colour corresponds to d s = 4 mm and blue colour corresponds to d s = 5 mm. Solid lines represent the computed ( u ) functions and dashed lines represent .

Fig. 3 .

 3 Fig. 3. eq for the flow of the yield stress fluids through the four packed beds: (a) d s = 1 mm, (b) d s = 3 mm, (c) d s = 4 mm, (d) d s = 5 mm. Symbols represent predictions and solid lines represent bulk viscosity as obtained from the rheometer ( Eq. (1) ).

Fig. 4 .

 4 Fig. 4. ∇ P as a function of u and the corresponding fits obtained with Eq. (2) for (a) d s = 1 mm, (b) d s = 3 mm, (c) d s = 4 mm, (d) d s = 5 mm.

Fig. 5 .

 5 Fig. 5. ∇ P as a function of u and the corresponding predictions obtained with Eq. (14) for (a) d s = 1 mm, (b) d s = 3 mm, (c) d s = 4 mm and (d) d s = 5 mm. Symbols represent experimental measurements, red solid lines represent the predictions obtained with the variable-method and blue dashed lines represent the predictions obtained with the fixed-method.

√

  and T being the tortuosity factor( Christopher

  N = 0.68 is proposed, based on the results of the present experiments and previous theoretical works. When only high values of u are involved (

  and a transverse size of approximately 2 nm. The stiffness of xanthan macromolecules leads to high levels of shear viscosity and highly shear-thinning behaviour of semidilute solutions in water. For this reason, the shear rheology of xanthan gum solutions is well described by the Herschel-Bulkley model ( Eq. (1) ) under steady-state conditions[START_REF] García-Ochoa | Apparent yield stress in xanthan gum solutions at low concentrations[END_REF][START_REF] Song | Rheology of concentrated xanthan gum solutions: steady shear flow behavior[END_REF] Rodríguez de Castro et al. 2014, 2016, 2018; Rodríguez de Castro and Radilla, 2017b ). However, rigorously speaking, they should be referred to as pseudo-yield stress fluids. The capacity of xanthan gum solutions to emulate the shear rheology of a yield stress fluid and the effects of polymer concentration was experimentally assessed by[START_REF] Rodríguez De Castro | Using Xanthan Gum solutions to characterize porous media with the yield stress fluids porosimetry method: robustness of the method and effects of polymer concentration[END_REF] , concluding that concentrated solutions ( ∼7000 ppm) behave similarly to a yield stress fluid due to high viscosity values at low shear rates.

Table 3

 3 Values of N , * and obtained for the four packed beds.

	d s	N	*
	1 mm	0.62	0.81	1.07
	3 mm	0.66	0.86	1.01
	4 mm	0.63	0.82	1.00
	5 mm	0.80	1.04	1.25
	determine K from injection experiments with water was applied to the
	present measurements. The obtained values and for and K are listed
	in Table

Acknowledgments

The author would like to acknowledge Pr. Giovanni Radilla for providing the equipment used in the experimental part of this work. The author also wishes to thank Frédéric Bastien for his technical support throughout the experimental campaign. Powered by TCPDF (www.tcpdf.org) Powered by TCPDF (www.tcpdf.org) Powered by TCPDF (www.tcpdf.org) Powered by TCPDF (www.tcpdf.org) Powered by TCPDF (www.tcpdf.org) Powered by TCPDF (www.tcpdf.org) Powered by TCPDF (www.tcpdf.org) Powered by TCPDF (www.tcpdf.org) Powered by TCPDF (www.tcpdf.org) Powered by TCPDF (www.tcpdf.org) Powered by TCPDF (www.tcpdf.org) Powered by TCPDF (www.tcpdf.org)

Supplementary materials

Supplementary material associated with this article can be found, in the online version, at doi: 10.1016/j.advwatres.2019.01.012 .