Applications of Poromechanics to Energy Engineeering
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Applications of Poromechanics to Energy Engineeering Chloé Arson Assistant Professor Texas A&M University, Department of Civil Engineering C. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 1 / 66 Introduction Modeling Thoughts... « Toute loi physique, étant une loi approchée, est à la merci d’un progrès qui, en augmentant la précision des expériences, rendra insuffisant le degré d’approximation que comporte cette loi ; elle est essentiellement provisoire. L’appréciation de sa valeur varie d’un physicien à l’autre, au gré des moyens d’observation dont ils disposent et de l’exactitude que réclament leurs recherches ; elle est essentiellement relative. » “Every law of Physics, being approximated, is at the mercy of progress, which, by increasing the accuracy of experiments, will make the degree of approximation of this law insufficient ; [every law of Physics] is essentially temporary. The appreciation of its value varies from one physicist to the other, depending on the means of observation that they have, and on the exactness required by their research work ; [every law of Physics] is essentially relative.” Pierre Duhem, La Théorie Physique. Son Objet - Sa Structure. Chap. 5, x 3, Revue de Philosophie, 1906-1914. C. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 2 / 66 Introduction 1 Thermodynamic Framework of Poromechanics 2 Performance Assessment of Heat Exchanger Piles 3 Modeling Damage in Porous Media 4 Study of the EDZ around Nuclear Waste Disposals 5 Prediction of Permeability in Fractured Rock C. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 3 / 66 Poromechanics 1 Thermodynamic Framework of Poromechanics General Thermodynamic Framework Application to Constitutive Modeling of Porous Media 2 Performance Assessment of Heat Exchanger Piles A Review of Geothermal Systems Preliminary Results and Research Plans 3 Modeling Damage in Porous Media State of the Art Limitations and Challenges An Alternative for Saturated Rock : Double Effective Stress 4 Study of the EDZ around Nuclear Waste Disposals Outline of the THHMD Model Study of Nuclear Waste Disposals with the THHMD Model 5 Prediction of Permeability in Fractured Rock A New Model of Permeability for Cracked Porous Rock Results : Simulation of Triaxial Compression Tests C. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 4 / 66 Poromechanics General Thermodynamic Framework Thermodynamics of Open Systems [Coussy, 2004] First Law of Thermodynamics (porous solid filled with a non reactive fluid mixture) _ _ _ _ _ K + Eint = Etot = Pmec + Echem + Q _ _ X _ _ Pmec = Pdefo + K ) Eint = Pdefo + µj Nj + Q j _ _ _ X _ _ Ψ = Eint − TS ) Ψ + T S + ST = Pdefo + µj Nj + Q j Second Law of Thermodynamics _ S_ Q > T Inequality of Clausius-Duhem X _ _ _ Φ = Pdefo + µj Nj − ST − Ψ > 0 j Thermodynamic potentials depend on state variables and internal variables. C. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 5 / 66 Poromechanics General Thermodynamic Framework State Variables for Non-Isothermal Unsaturated Porous Media 2 miscible pure fluids (liquid, gas), small deformation, absence of body forces. Inequality of Clausius-Duhem [Coussy, 2004] Φ = Φs + Φl + Φg + ΦT > 0 s s s d d φl d φg dT d Ψs Φs = σ : dt + pl dt + pg dt − Ss dt − dt r Φl = [−rX pl ] · φl Vl r Φg = [−rX pg ] · φg Vg Q ΦT = − T · rX T “deformation” “stress” e σ e φl pl e φg pg TSs C. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 6 / 66 Poromechanics General Thermodynamic Framework Thermodynamic Conjugation Relationships The free energy of the solid skeleton is sought in the form : e e e Ψs = Ψs( ; φl ; φg; T ; χ) χ : vector containing all internal variables of interest, e.g. damage (Ω), hardening variables such as the equivalent plastic strain (γp) Mechanical dissipation inequality (in the absence of plastic porosity changes): p e ! e ! dφe d @Ψs d @Ψs dφl @Ψs g @Ψs dT @Ψs dχ σ : + σ − + pl − + pg − − Ss + − > 0 @e @φe @φe @ @χ dt dt l dt g dt T dt dt In the absence of irreversible microstructure change... e ! e ! e @Ψ d @Ψ dφ @Ψ dφg @Ψ dT σ − s + p − s l + p − s − S + s = 0 e l e g e s @ dt @φl dt @φg dt @T dt ... from which thermodynamic conjugation relationships are deduced... @Ψ @Ψ @Ψ @Ψ σ = s ; p = s ; p = s ; S = − s e l e g e s @ @φl @φg @T ... and the reduced dissipation inequality is obtained : dp dχ @Ψ σ : + ξ · 0 ; ξ = − s dt dt > @χ C. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 7 / 66 Poromechanics Application to Constitutive Modeling of Porous Media Tangent Properties [Coussy, 2004] Expression of the skeleton free energy ) REV’s tangent properties. Example : thermo-poro-elasticity (saturated porous medium) e with a free energy sought in the form Ψs = Ψs( ; p; T ) : 2 2 2 σ_ = @ Ψs : _e + @ Ψs p_ + @ Ψs T_ @e2 @e@p @e@T σ_ = D : _e − Bp_ − α (D : δ) T_ D : stiffness tensor ; B : Biot’s tensor ; α : thermal expansion coefficient p_ φ_ = b : _e + − 3α T_ N φ 1=N : inverse of Biot’s modulus ; αφ : coefficient for volumetric thermal dilation due to porosity changes only T_ S_ = (D : δ) α : _e − 3α p_ + C s φ T C : skeleton volumetric heat capacity C. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 8 / 66 Poromechanics Application to Constitutive Modeling of Porous Media Flow Properties [Coussy, 2004] Conduction laws ensure the positivity of the fluid and thermal dissipations in the absence of TH couplings : “Darcy’s” law : v = φ Vr = k l :[−r p ] l l l ηl X l ) Φ = [−r p ] · v = k l :[r p ]:[r p ] ≥ 0 l X l l ηl X l X l v = φ Vr = k g :[−r p ] g g g ηg X g ) Φ = [−r p ] · v = k g :[r p ]:[r p ] ≥ 0 g X g g ηg X g X g Fourier’s law : Q = −λrX T λ ) ΦT = T rX T · rX T ≥ 0 with positive permeabilities and thermal conductivity C. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 9 / 66 Poromechanics Application to Constitutive Modeling of Porous Media General case of an isothermal solid saturated with water and air Typical Constitutive Relationships (hyp. incompressible solid grains, σnet = σ − pg δ, s = pg − pl ) _ Stress state law : σ_ net = D : _ − α (D : δ) T − βs_ Heat flow equation : ∗ QT = − λT rT + hfg ρw V vap + ρvap V a ∗ + ρw CPw V w + ρw CPvap V vap + ρa CPa V a (T − T0) Moisture transfer equation : Q wtot = Q w + Q vap = ρw (−DT · rT + DP · rs − Kw · rz) Air flow equation : 1 @pa(x;T (x)) pa V a = − Ka · rT (x) − Ka · r − Ka · rz γa @T (x) γa Balance equations Solid skeleton balance equation : r · σ + f = 0 @ET Energy conservation : @t + r · QT = 0; ET = CT (T − T0) + n (1 − Sw ) ρvap hfg @ρm ∗ Moisture mass conservation : @t + r · ρw V w + V vap = 0 Air mass conservation : @ ∗ @t [n ρa (1 − Sw + HSw )] + r · [ρa V a] + r · [ρa H V w ] − r · ρw V vap = 0 C. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 10 / 66 Poromechanics Application to Constitutive Modeling of Porous Media Choice of the stress state variables for unsaturated porous media number of stress variables ? A long-standing debate [Fredlund and Morgenstein 1977, Houlsby 1997] Example for incompressible solid grains : 0 Bishop’s effective stress : σ = (σ − pg δ) + χ (pg − pl ) δ independent state variables : (σ − pg δ; pg − pl ) or (pg − pl ; σ − pl δ) or (σ − pl δ; σ − pg δ) nature of the state variables ? No unique formulation [Coussy, 2004 ; Dangla, 2010]. Example for elastic isothermal unsaturated porous media, based on the separation of energies : e e Ψs ; φ, Sl = s ; φ + φ U (φ, Sl ) e 8 @ s( ,φ) σ = > @e > > < @ e,φ @(φ U(φ,Sl )) s( ) π = Sl pl + Sg pg − @φ = @φ > > > e > @U( ,φl ,φg ) : pc = − @Sl C. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 11 / 66 Geothermal Piles 1 Thermodynamic Framework of Poromechanics General Thermodynamic Framework Application to Constitutive Modeling of Porous Media 2 Performance Assessment of Heat Exchanger Piles A Review of Geothermal Systems Preliminary Results and Research Plans 3 Modeling Damage in Porous Media State of the Art Limitations and Challenges An Alternative for Saturated Rock : Double Effective Stress 4 Study of the EDZ around Nuclear Waste Disposals Outline of the THHMD Model Study of Nuclear Waste Disposals with the THHMD Model 5 Prediction of Permeability in Fractured Rock A New Model of Permeability for Cracked Porous Rock Results : Simulation of Triaxial Compression Tests C. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 12 / 66 Geothermal Piles A Review of Geothermal Systems Forms of Geothermal Energy [Lund, 2007] comes “from the decay of the naturally occurring isotopes of uranium, thorium and potassium in the earth” natural thermal reservoirs above 10km depth : 1:3 × 1027J ' 3 × 1017 oil barrels geothermal resources > 150oC : electrical power plants geothermal resources < 150oC : direct use for heating and cooling Geothermal Resources Direct Use of Geothermal Energy (2005) C. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 13 / 66 Geothermal Piles A Review of Geothermal Systems GCHP with Heat Exchanger Piles ground = heat reservoir high thermal conductivity of ground and grout ) foundation piles used as heat exchangers ) thermo-mechanical behavior of the foundation ? heat stored in summer, pumped in winter ) annual energy balance ? ) design and performance ? C.