Pollution des sols et interaction avec la biosphère
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1 Pollution des sols et interaction avec la biosphère Mazen SAAD Ecole Centrale de Nantes Laboratoire de Mathématiques Jean Leray FRANCE Mathématiques Pour La Planète La Rochelle 17 octobre 2013
2 Plan 1 Transport d éléments radioactifs 2 Interaction avec la biosphère 3 Stockage du CO2 Mazen Saad (ECN) Stockage et interaction La Rochelle 17 octobre / 31
3 Transport d éléments radioactifs Stockage des déchets radioactifs GdR MoMaS initié par l ANDRA Simuler la fuite éventuelle des radionucléides Mazen Saad (ECN) Stockage et interaction La Rochelle 17 octobre / 31
4 Transport d éléments radioactifs Stockage des déchets radioactifs Plan du site de stockage Mazen Saad (ECN) Stockage et interaction La Rochelle 17 octobre / 31
5 Transport d éléments radioactifs Modélisation de transport des éléments radioactifs Simuler une fuite éventuelle des radionucléides de leur site de stockage Sol : populations Sous sol : radionucleides Site de stockage Une seule phase EAU dans un milieu poreux hétérogène Des composés chimiques miscibles à l eau. Mazen Saad (ECN) Stockage et interaction La Rochelle 17 octobre / 31
6 Transport d éléments radioactifs Modèle mathématique Déplacement de m-radionuclides de concentration c = (c 1,..., c m) T ( ) ( tg k x, ck + div ck V ) ( ( div D k x, ck, V ) ) c k (t, x) ( ) ( ) + λ k G k x, ck λ l R k,l G l x, cl = fk, k = 1,..., m, (1) l<k div V = 0, et V = 1 ( ) µ(c) K p ρ(c) g avec G k (x, c) = φ(x)c + ( 1 φ(x) ) ρ s(x)f k (x, c) Adsorption : F k (x, c) Transport : c k V Diffusion-Dispersion : D k = d mi d + d s(x, V ) Décroissance radioactive : λ k Filiation : R k,l (structure triangulaire) f k : terme source (2) Mazen Saad (ECN) Stockage et interaction La Rochelle 17 octobre / 31
7 Transport d éléments radioactifs Stockage des déchets radioactifs Site de Bure Site de stockage : m 2 Plusieurs strates : argile, calcaire 2 composés radioactifs : Iode 129 I, Plutonium 242 Pu Adsorption dans l argile : 0, 001 pour 129 I pour 242 Pu. Décroissance : T 1 ( 129 I )= 1, années; T 2 ( 242 Pu)= 3, années. Ecoulement de droite à gauche Mazen Saad (ECN) Stockage et interaction La Rochelle 17 octobre / 31
8 Fuite de radioactifs Transport d éléments radioactifs e-05 8e-05 7e-05 6e-05 5e-05 4e-05 3e-05 2e-05 1e e+06 Termes sources pour 129 I (gauche) et 242 Pu (droite) Mazen Saad (ECN) Stockage et interaction La Rochelle 17 octobre / 31
9 Transport d éléments radioactifs Domaine Mazen Saad (ECN) Stockage et interaction La Rochelle 17 octobre / 31
10 Transport d éléments radioactifs Maillage Mazen Saad (ECN) Stockage et interaction La Rochelle 17 octobre / 31
11 Transport d éléments radioactifs Iode 129 au Temps = années Mazen Saad (ECN) Stockage et interaction La Rochelle 17 octobre / 31
12 Transport d éléments radioactifs Iode 129 au Temps = années Mazen Saad (ECN) Stockage et interaction La Rochelle 17 octobre / 31
13 Transport d éléments radioactifs Iode 129 au Temps = années Mazen Saad (ECN) Stockage et interaction La Rochelle 17 octobre / 31
14 Transport d éléments radioactifs Iode 129 au Temps = 10 6 années Mazen Saad (ECN) Stockage et interaction La Rochelle 17 octobre / 31
15 Transport d éléments radioactifs Plutonium 242 au Temps = 10 6 années Mazen Saad (ECN) Stockage et interaction La Rochelle 17 octobre / 31
16 Transport d éléments radioactifs Plutonium 242 au Temps = années Mazen Saad (ECN) Stockage et interaction La Rochelle 17 octobre / 31
17 Interaction avec la biosphère Interaction avec la biosphère Prévoir l impact sur une population d une exposition à des radionucléides Contamination d une chaîne alimentaire par des radionucléides Couplage entre milieu poreux et dynamique des populations Sol : populations Sous sol : radionucleides Site de stockage Trois espèces : Végétaux, Animaux, Super-prédateur(Humain?) Exposition des Végétaux aux radionucléides par le sol Contamination des animaux sains par prédation des végétaux exposés Contamination des supers prédateurs sains par contact avec les animaux exposés. Mazen Saad (ECN) Stockage et interaction La Rochelle 17 octobre / 31
18 Interaction avec la biosphère Modèles de Contamination et de prédation Trois espèces : V : Végétaux ; V = V S + V E A : Animaux ; A = A S + A E P : Super-prédateur ; P = P S + P E Mazen Saad (ECN) Stockage et interaction La Rochelle 17 octobre / 31
19 Interaction avec la biosphère Modèles de Contamination et de prédation Trois espèces : V : Végétaux ; V = V S + V E A : Animaux ; A = A S + A E P : Super-prédateur ; P = P S + P E P S Population Saine ; P E Population Exposée tp S d P S = r P ( 1 P K P )P S β P A E P S tp E d P E = r P ( 1 P K P )P E + β P A E P S β P : taux de contamination des super prédateurs par consommation des animaux infectés Mazen Saad (ECN) Stockage et interaction La Rochelle 17 octobre / 31
20 Interaction avec la biosphère Modèles de Contamination et de prédation A S Animaux Sains ; A E Animaux Exposés pv S pv E ta S div(d A S ) = m A A S + e A S + (1 β A )e A S δa S P S 1 + qv S 1 + qv E pv S pv E pv E ta E div(d A E ) = m A A E + e A E + e A E + β A e A S δa E P S 1 + qv S 1 + qv E 1 + qv E β A : taux de contamination des animaux par consommation des végétaux Mazen Saad (ECN) Stockage et interaction La Rochelle 17 octobre / 31
21 Interaction avec la biosphère Modèles de Contamination et de prédation V S Végétaux Sains ; A E Végétaux Exposés ( tv S = r V 1 V ) V S pv S A ρ(c)v S K V 1 + qv S ( tv E = r V 1 V ) V E pv E A+ρ(c)V S K V 1 + qv E ρ(c) : taux de transmission de la concentration des radionulcéides (présents en surface) aux végétaux. Mazen Saad (ECN) Stockage et interaction La Rochelle 17 octobre / 31
22 Interaction avec la biosphère Interaction radioactifs Végétaux/Animaux Contamination des végétaux par de l iode radioactive 129 I. La trace de la concentration à la surface est moyennée sur 15m de profondeur. Exercice Couplex 1 apres ans Sol : populations pendant 300 ans Marne y Calcaire 20 km A & V Argilite Calcaire 300 m z Zone d exposition 25 km y 695 m 40 km x Densités d individus exposés d animaux A E : Mazen Saad (ECN) Stockage et interaction La Rochelle 17 octobre / 31
23 Interaction avec la biosphère Quantités totales en fonction du temps pour l espèce A (gauche) et l espèce V (droite) La ligne solide représente les individus sains, la ligne pointillée la classe des exposés. Mazen Saad (ECN) Stockage et interaction La Rochelle 17 octobre / 31
24 Interaction avec la biosphère Radioactifs Végétaux/Animaux/Humains Intrusion d un super-prédateur dans une zone isolée et non infectée. 20 km Animaux + Vegetaux Zone du super predateur (K=1000) Zone d exposition 40 km Contamination des humains par déplacement des animaux infectés Densités d individus exposés P E du super-prédateur P, après 100,150, 200, 300 ans de contamination Mazen Saad (ECN) Stockage et interaction La Rochelle 17 octobre / 31
25 Stockage du CO2 Stockage du CO2 CO2 capture and storage. The process consists of capturing waste CO2 and transporting it to a storage site. Various forms have been conceived for storage CO2 into deep geological formations : CO2 is sometimes injected into declining oil fields to increase oil recovery. This option is attractive because the geology of hydrocarbon reservoirs is generally well understood and storage costs may be partly offset by the sale of additional oil that is recovered. Model 1 : Gas 90% CO2 = Oil Gas model. two compressible flow Model 2 : Dissolution of CO2 in water. Two compressible and partially miscible flow Mazen Saad (ECN) Stockage et interaction La Rochelle 17 octobre / 31
26 Stockage du CO2 Two-phase two-component flow in porous media TWO PHASES TWO COMPONENTS POROUS MEDIA MODEL Discretize a general model for two phase partially miscible flow in porous media : CO2 storage in a liquid gas reservoir. Mazen Saad (ECN) Stockage et interaction La Rochelle 17 octobre / 31
27 Stockage du CO2 CO2 in liquid and gas form After the CO2 injection, several different trapping mechanisms lead to an entrapment of the CO2. Shortly after the injection, structural trapping through caprocks is the most important factor. Later solubility trapping, where CO2 is dissolved into water, and residual trapping get more important. After several thousand years, there could also occur mineral trapping caused by geochemical reactions To simulate the process of dissolution of CO2, a multiphase flow equation with equilibrium phase exchange is used. The CO2 storage can be modeled with two components (CO2 and water) in two phases (liquid and gas). Mazen Saad (ECN) Stockage et interaction La Rochelle 17 octobre / 31
28 Mathematical model Stockage du CO2 We consider herein a porous medium saturated with a fluid composed of : two phases : liquid (α = l) and gas (α = g) two components in each phase : CO2 (β = c) and water (β = w) The component CO2 is present in the two phases : In liquid form : dissolved CO2 In gas form : CO2 in the gas phase The component water exists only in liquid form (no vapor water) Mazen Saad (ECN) Stockage et interaction La Rochelle 17 octobre / 31
29 Stockage du CO2 Two-phase two-component model: Notation s l, s g : liquid and gas phase saturations p l, p g : liquid and gas phase pressures Phases occupy the entire pore space: s l + s g = 1 ρ l, ρ g : liquid and gas phase densities ρ c l, ρc g : density of dissolved CO2 and density of the CO2 in the gas phase Xl c = ρ c l /ρ l : mass fraction of the CO2 in the liquid phase X w l = ρ w l /ρ l : mass fraction of the water in the liquid phase Xl c + Xl w = 1 Xg c = ρc l /ρ l : mass fraction of the CO2 in the gas phase Capillary pressure between liquid pressure and gas pressure : p c(s l ) = p g p l Mazen Saad (ECN) Stockage et interaction La Rochelle 17 octobre / 31
30 Stockage du CO2 Two-phase two-component model: Assumption Isothermal flow (T is a constant) Perfect gas law for CO2 gas phase ρ c g = Mc RT pg where M c is the molar mass of the CO2 and R is the perfect gas constant Equilibrium between dissolved CO2 and gas CO2 is prescribed by Henry s law, ρ c l = M c H c p g where H c is the Henry s constant. Mazen Saad (ECN) Stockage et interaction La Rochelle 17 octobre / 31
31 Mathematical model Stockage du CO2 The mass conservation of each component : CO2: Φ t(ρ c l s l + ρ c g s g ) + div(ρ c l V l + ρ c g Vg + Jc l + J c g ) = Qc Water: Φ t(ρ w l s l ) + div(ρ w l V l + J w l ) = Q w where diffusion of the components β = c, w in the phase α = l, g is modeled by J β α = ΦsαDα ρβ α, J c α + Jw α = 0 Darcy s law for the velocity of each phase V α = K krα(sα) µ α ( p α ρ αg) Mazen Saad (ECN) Stockage et interaction La Rochelle 17 octobre / 31
32 CO2 field Stockage du CO2 CO2 in the dissolved form. Mazen Saad (ECN) Stockage et interaction La Rochelle 17 octobre / 31
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