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Energy Approach to Poromechanics Energy Approach To Biot Medal Lecture 2003 Energy Approach to Poromechanics Olivier COUSSY NAVIER INSTITUTE Energy Approach to Poromechanics • drying and shrinkage • rocks, soils • wetting and swelling • concretes • temperature rise and build-up of pressure • woods • hydraulic diffusion and subsidence •foams • freezing and spalling • bones • capillarity and cracking • living tissues • electro-osmosis and consolidation •etc. •etc. Outlines : 1. Revisiting the Energy Approach of Saturated Poroelasticity 2. Energy Approach to Partially Saturated Poroelasticity 3. From Poromechanics to the Mechanics of Colloids 1 Revisiting Biot’s Energy Approach to Saturated Poroelasticity Open system: Element of sk fl m fl dΩ0 Solid Saturating Porous = + fluid mass material Skeleton fluid content Assuming no p dissipation σ dε + f + fl dm −dF = 0 ij ij fl fl ρfl Strain work Helmoltz Free energy infinitesimal change of Free energy Work to make the the open system supplied mass enter the by convection porous element g fldm fl Gibbs potential Separate equality for the skeleton Mesoscopic σij dεij + g fl dm fl −dF = 0 information dpfl m fl = φ ρfl dg fl = ρfl Fsk = F − ffl m fl Fluid state equations Skeleton free energy σij dεij + pfl dφ −dFsk = 0 Strain work Related to the skeleton only Lagrangean porosity 2 Linear Poroelasticity and Poroplasticity (saturated) σij dεij + pfl dφ −dFsk = 0 ∂Fsk ∂Fsk State equations: σij = pfl = ∂εij ∂φ LINEAR POROELASTICITY σ = Kε −bpfl φ − φ0 = pfl /N +b ε sij = 2G eij POROPLASTICITY p εij → εij − εij p ∂f p ∂f f (σij ,pfl ) ≤ 0 dε = dλ , dφ = dλ p ij φ → φ − φ ∂σij ∂pfl Plastic criterion State equations Flow rule Biot’s effective stress − ()σ + bp l = −Kε Experimental validation ? − σ 35 Loading 30 − (σ + bp ) 70 l [Bouteca, 1994] 60 25 50 Tavel limestone 40 20 30 15 20 b = 0.63 K =19500 MPa 10 p fl 10 0 K = 52700 MPa 020406080 5 − ε s 0 0 0.0005 0.001 0.0015 0.002 σ + pfl − pfl σ K p = 0 p 0 ≤ b =1− ≤ 1 fl + pfl = fl Ks 1 2 σ +p pfl σ 1 1 fl ε = − ε = 2 ε = + pfl − 1 + K = K K K K s s σ +pfl non-linear skeleton, linear solid matrix ε1 = KSEC ()σ +pfl 3 dσ = Kdε −bdpfl Dormieux et al., 2002 K = K()σ + pfl b =1− K()σ + pfl /Ks Terzaghi’s effective stress 10 103 8 103 (MPa) 3 K 6 10 pfl = 51 MPa Bemer et al., 2001 pfl = 1 MPa Sandstone specimen 4 103 2 103 − σ’ = − (σ + pfl) 0 0204060 Extension of Energy Approach to Unsaturated Poroelasticity Liquid water saturation dΩ t (1−n)dΩt n Sl dΩt n (1−Sl )dΩt sk l g Porous Solid Liquid Gaseous element = skeleton + water + mixture σij dεij + (Sl pl + Sg pg )dφ − φpcdSl −dFsk = 0 Total stress Averaged pressure * p pc = pg − pl Capillary pressure Fsk = Fsolid + φU gas γ Skeleton Solid Interface γ sg lg energy energy Energy liquid Capillary γ sl meniscus ∂Fsk * ∂Fsk ∂Fsk σij = p = − φpc = ∂εij ∂φ ∂Sl 4 Unsaturated Poroelasticity: Equivalent Pore Pressure π Solid Interface Coussy et ∂Fsk φpc = − matrix energy Dangla, 2002 ∂Sl F = F ε ,φ + φU S Energy separation pc = pc ()Sl sk solid ( ij ) ( l ) ∂Fsolid ∂Fsolid σij dεij + πdφ −dFsolid = 0 σij = π = ∂εij ∂φ p* Averaged pore pressure ( = Bishop’s equivalent pore pressure) π = Sl pl + Sg pg −U ≡ Equivalent pore pressure Sl U = − pc ()s ds ∫1 σ = Kε −bπ ! p (MPa) h (%) c r p = 0.1MPa = p (neglected) 300 100 atm g σ + p* (Bishop’s effective stress) 75 ij 200 * (here) σij + π = σij + p −U 50 100 MPa 25 120 −π = −p* + U 0 0 100 0 20 40 60 80 100 S (%) 80 l U 60 * Sl −p U = − pc ()s ds 40 ∫1 20 0 Cement paste 20 40 60 80 100 Sl (%) 5 w (kg/m-3) Sorption curve1/3 (10−6) (Swelling) 200 2000 Experiment 150 1500 Predicted 100 1000 50 500 0 0 0 20406080100 0 20 40 60 80 100 hr (%) hr (%) Cellular cement fiber composite h 0 b RT r w b b = 0.79 = × × ∫ dh = π nK M h h K Carmeliet, 1999 v r From poromechanics to mechanics of colloids At constant solution - - -- -- -- pressure and decreasing + ++ - + +++ salt concentration: - + - + + - + +- - +++- +- - - ++- - - +- +-- - - +- - - - - + ++- -- SWELLING - +- - - + - + ++- -- + -- + - - - + - +++ - + - + - - + - - +++ - - - - -+ + + + 1 ρ + - - - - - φ e + + + + σ dε + p dφ + p d −dF = 0 = eff A >1 + - - - - - ij ij w e sk A ρ + + - + - + - + - A e -+ + + + + - - - - - + + + + + - - - - - dσ = Kdε −dpw −bedpe + + + + + φ dpe eff d = be dε + pe = 2RT c A N εε 0RT ∂(φ/A) A = 2 eff dε = dφ ⇒ b = 2F c e ∂φ pe 6 From poromechanics to mechanics of colloids d q q -- ++-- s× = φ - ++ - + 2 +++- +- - - +- +-- - - - + ++- -- - -- + - z + - + -- p = 2RT ceff +++ e −d /2 +d / 2 Solution pressure d()σ + pw + pe = K dε −dpsw psw = RTc+ (0)()+ RTc− 0 − pe psw = −∫()1−be dpe ∂()φ / A ∂p sw b = or b = + 1 e ∂φ e ∂p pe e φ Poromechanics Porophysical Chemistry 1 b e 0.8 φ dσ = 0.1 0.5 1. 1.5 = 0.6 sA Kdε −dp −b dp w e e 0.4 0.2 0 10−2 10−1 100 101 q q = 2F Ac eff Ceff ranging from 10-4 to 10-4 mol/l εε RT . ranging from 30 nm to 1 nm A = 0 2F 2c eff q = 0.01to 0.2C ×m −2 7 As a conclusion build a bridge between … Upscaling methods thermodynamics POROPHYSICAL CHEMISTRY Micro/Nano Scales POROMECHANICS back analysis Macro Scale Many Thanks To Many People and specially to … P. Acker V. Baroghel-Bouny X. Chateau L. Dormieux P. Dangla T. Lassabatère F. Ulm 8.
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