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, § 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 = [−∇X pl ] · φl Vl r Φg = [−∇X pg ] · φg Vg Q ΦT = − T · ∇X 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 :[−∇ p ] l l l ηl X l ⇒ Φ = [−∇ p ] · v = k l :[∇ p ]:[∇ p ] ≥ 0 l X l l ηl X l X l v = φ Vr = k g :[−∇ p ] g g g ηg X g ⇒ Φ = [−∇ p ] · v = k g :[∇ p ]:[∇ p ] ≥ 0 g X g g ηg X g X g Fourier’s law :

Q = −λ∇X T λ ⇒ ΦT = T ∇X T · ∇X 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 ∇T + 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 · ∇T + DP · ∇s − Kw · ∇z) Air ﬂow equation : 1 ∂pa(x,T (x)) pa V a = − Ka · ∇T (x) − Ka · ∇ − Ka · ∇z γa ∂T (x) γa

Balance equations Solid skeleton balance equation : ∇ · σ + f = 0

∂ET Energy conservation : ∂t + ∇ · QT = 0, ET = CT (T − T0) + n (1 − Sw ) ρvap hfg ∂ρm ∗ Moisture mass conservation : ∂t + ∇ · ρw V w + V vap = 0 Air mass conservation : ∂ ∗ ∂t [n ρa (1 − Sw + HSw )] + ∇ · [ρa V a] + ∇ · [ρa H V w ] − ∇ · 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 = ψs , φ + φ U (φ, Sl )

e  ∂ψs( ,φ) σ =  ∂e    ∂ψ 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

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. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 14 / 66 Geothermal Piles A Review of Geothermal Systems Why using GCHP in buildings ? [DoE Buildings Data Book]

chap_1_chart1.jpg 673!240 pixels 11/13/11 11:04 AM

73% of U.S. electrical energy 55% of U.S. natural gas

consumed by U.S. buildings in 2008, mainly for heating and lighting

C. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 15 / 66

http://buildingsdatabook.eren.doe.gov/images/chap_1_chart1.jpg Page 1 of 1 Geothermal Piles A Review of Geothermal Systems Thermodynamic Principle of a Heat Pump

C. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 16 / 66 Geothermal Piles A Review of Geothermal Systems Thermodynamic Balance of a GCHP

C. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 17 / 66 Geothermal Piles A Review of Geothermal Systems Typical Earth-Coupled HVAC System [Akrouch, 2011]

C. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 18 / 66 Geothermal Piles A Review of Geothermal Systems Examples of GCHP Systems [Hughes, 2008 (Oak Ridge Nat. Lab.)]

In average, a U.S. household consumes 8,900 kWh of electrical energy per year.

Cooling down by 1oC the volume of soil below a typical one-family house (100 m2) over a pile depth of 12m could bring up to 1,000 kWh.

C. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 19 / 66 Geothermal Piles Preliminary Results and Research Plans Preliminary Numerical Results [Berns and Arson, 2011]

Equivalent Simulation with Theta-Stock FEM code axis-symmetric [Gatmiri and Arson, 2008]. model inspired from work done at Oakridge National 1D mesh, i.e. ﬁxed depth Laboratory [Shonder 3 thermo-elastic materials : grout, ﬁlm, soil and Beck, 2000]. E E ν σ = (1+ν) + (1+ν) (1−2ν) Tr () δ

E + 3(1−2ν) (1 − n)αs∆T δ

q = −(1 − n)λs∇T film : fictitious material with mechanical properties of the grout and thermal properties of water free heating and cooling : temperature imposed in the film ' typical atmospheric temperature recorded in Texas

C. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 20 / 66 Geothermal Piles Preliminary Results and Research Plans Thermal Inﬂuence Zone in Cooling Mode

initial temperature of the ground : 22 oC pile diameter : 1.1m thermal zone of inﬂuence > 1 pile diameter

C. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 21 / 66 Geothermal Piles Preliminary Results and Research Plans 1-year cycle for typical Texan temperatures

initial temperature of the ground : 22 oC pile diameter : 1.1m thermal zone of inﬂuence > 1 pile diameter ground heating, no energy balance

C. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 22 / 66 Geothermal Piles Preliminary Results and Research Plans TAMU Liberal Arts Building Site [Briaud, Akrouch, Arson, Sanchez] foundation piles of the Liberal Arts Building : 20 feet deep below the pile cap reinforcement : steel cage of 6 #6 rebars + a single # 9 steel rebar in the middle 3 piles equipped with a High Density Polyethylene (HDPE) U-shaped tube average distance between the HDPE pipes along the pile : 20 cm 6 thermistors in each experimental pile + in 3 close boreholes (at same depths)

C. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 23 / 66 Geothermal Piles Preliminary Results and Research Plans Preliminary Experimental Results

C. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 24 / 66 Geothermal Piles Preliminary Results and Research Plans Research Prospectives techniques to measure ground thermal properties in the ﬁeld full scale heating and cooling tests on the foundation of the Liberal Arts Building inﬂuence of thermal cyclic loading on the thermo-hydro-mechanical behavior of unsaturated expansive clays and on soil/structure interactions energy pile group effects in cooling mode cost assessment

[Li et al., 2010] [Bourne-Webb et al., 2009]

C. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 25 / 66 Damage 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

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 26 / 66 Damage Poromechanics State of the Art Framework of Fracture Networks

Multimodal models [Pruess et al. 1990] several connected networks forming a unique equivalent network (e.g. pores of the intact matrix + cracks) 1 pressure head h for the whole network, 1 single balance equation ∂θw (h) ∂h ∂ ∂ = · K w (h) · (h + z) ∂h ∂t ∂x ∂x

homogenized, equivalent retention properties θw (h) and kR (h) θw (h) = wf θw,f (h) + (1 − wf ) θw,m(h) kR (h) = wf kR,f (h) + (1 − wf ) kR,m(h)

Multicontinuum systems several networks behaving as separate entities coupled balance equations [Gwo et al. 1995, Vogel et al. 2000] permeability : retention properties ? equivalent pressure head ? [Gerke and Van Genuchten 1993, Zimmermann et al. 1996]

C. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 27 / 66 Damage Poromechanics State of the Art Framework of Fracture Mechanics

study of the initiation, propagation and coalescence of cracks initiation and propagation criteria based on energy balance at crack tips requires the determination of stress intensity factors (e.g., Grifﬁth theory)

Limitations : scale dependence, no direct relation to elastic properties at the REV scale An alternative approach based on a combination of fracture and damage mechanics : characterization of a damaged zone around cracks with a damaged (effective) stress variable [Valko and Economides, 1994] [Valko and Economides, 1994]

C. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 28 / 66 Damage Poromechanics State of the Art Framework of Continuum Damage Mechanics

Micro-mechanical models effective material area decrease, redistribution of stresses [Kachanov, 1992] stress in the intact solid matrix 6= stress in the cracked material, effective stress state variables apply to a fictitious undamaged system counterpart [de Borst et al. 1999] the size of the Representative Elementary Volume needs to be defined to homogenize the fields of variables and the damaged properties [Nemat-Nasser and Hori, 1994 ; Dormieux et al., 2006]

Phenomenological models

1 “averaged” variables, directly deﬁned at the REV scale : damage = crack density tensor (often) ; effective stress (applying to the undamaged matrix) 2 reduced dissipation inequality obtained from the Inequality of Clausius-Duhem

e e σ : ˙ − Ss dT − Ψ˙ s , T , Ω ≥ 0 =⇒ Y : Ω˙ ≥ 0

3 stress/strain relations derived from the expression of the free energy

C. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 29 / 66 Damage Poromechanics State of the Art Formulation of Phenomenological Damage Models

1 “averaged” variables, directly defined at the REV scale Crack density tensor, sufficient to characterize the influence of damage on stiffness is cracks do not interact [Kachanov, 1992] : N 3 1 X k k k Ω = Dc = l n ⊗ n VREV k=1 Effective stress defined with a damage operator [Hansen and Schreyer 1994] σ˜ = M (Ω): σ 2 reduced dissipation inequality obtained from the Inequality of Clausius-Duhem e e σ : ˙ − Ss dT − Ψ˙ s , T , Ω ≥ 0 =⇒ Y : Ω˙ ≥ 0 3 stress/strain relations derived from the expression of the free energy, in particular : ∂Ψ (e, T , Ω) σ = s = D (Ω): e − α (D (Ω): δ) T ∂e 4 the Principle of Equivalent Elastic Energy is often used to determine D (Ω) : e ∗ e Ψs ( , T , Ω) + Ψs (σ, T , Ω) = σ : ∗ ∗ −1 T Ψs (σ, T , Ω) = Ψs (σ˜, T , Ω = 0) ⇒ D (Ω) = M (Ω) : D0 : M (Ω)

C. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 30 / 66 Damage Poromechanics Limitations and Challenges

A Lack of Experimental Data

lab protocols available for mechanical damage parameters [Halm and Dragon, 2002] and poro-elastic parameters [Gatmiri, 1997]

determination of retention properties from the Pore Size Distribution curve [Romero and Jommi, 2008] : bimodal porosity, but for undamaged material

permeability assessment based on scalar variables (no ﬂow orientation) [Dal Pont et al., 2004], or on postulates on the cracks shape and density [Shao et al., 2005 ; Maleki and Pouya, 2010]

C. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 31 / 66 Damage Poromechanics Limitations and Challenges Limitations of Models of Damaged Tangent Properties

Damage in “unsaturated” behaviour laws

Bishop’s effective stress concept [Shao et al. 2005] 0 dry σij = σij − b [Sw pw + (1 − Sw ) pa] δij

0 ∂Ψs(,Ω) σij = ∂ − b [Sw pw + (1 − Sw ) pa] δij no consensus on the dependence of b to damage only 1 damage model formulated in independent state variables [Lu et al. 2006] no assessment of the model thermodynamic consistency recent developments on chemo-thermal processes in cementitious materials based on a scalar damage variable [Gawin et al., 2003 ; Schreﬂer and Pesavento, 2004]

C. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 32 / 66 Damage Poromechanics Limitations and Challenges Limitations of Models of Damaged Transport Properties

Damage in fluid transfer laws Fracture network theories. Relative permeability computed by integration of retention variables. Intrinsic permeability assessed for a fixed porous network (rigid solid skeleton, no damage evolution) ⇒ uncoupled, purely "hydraulic" approach Coupled poro-elastic models based on state surfaces. [Gatmiri, 1997-2008] ⇒ no damage Existing continuum damage models. Biot’s effective stress [Yang et al. 2007]. Micro-flow homogenization [Shao et al. 2005].

C. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 33 / 66 Damage Poromechanics Limitations and Challenges Damage is non local

Damage is non local, i.e. Ω(x) inﬂuences ﬁelds of variables at x + dx [Bazant, 1991]

integral formulations : usually, non-local deformation or non-local energy release rate [Jirasek, 1998] Z α∞(kx − ξk) f (x) = α(x, ξ)f (ξ)dξ, α(x, ξ) = R α∞(kx − ξk)dξ Vtot Vtot differential formulations : usually, introduction of the gradient of deformation [Askes and Sluys, 2002] or the gradient of damage [Frémond and Nedjar, 1996]

d(x) d 2(x) 1 Z +l/2 ( + ) = ( ) + + 2 + ( 2), ( ) = ( + ) x s x s s 2 o s x x s ds dx dx l −l/2 l2 d 2(x) ⇒ (x) = (x) + + o(l2) 24 dx 2 microstructure-enriched models [Mindlin, 1964 ; Germain, 1973(a,b)] : usually second-gradient models (micro-translation), Cosserat models (micro-rotation)

C. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 34 / 66 Damage Poromechanics Limitations and Challenges Characteristic length in the expression of the free energy

residual crack openings scaled by crack surface energy fracture energy stored / square of the characteristic length

energetic term for residual strains after unloading [Halm and Dragon, 1998] 1 Ψ(, Ω) = : D (Ω): − gΩ : 2

surface energy due to crack opening [Hansen and Schreyer, 1994] 1 Ψ , Ω, Ωh = : D (Ω) + H Ωh +γ Ω : Ω 2 D

energy related to the zone of inﬂuence of damage [Frémond and Nedjar, 1998] 1 k Ψ(, ω, ∇ω) = : D : + W (1−ω) −M [log(|ω|) − ω + 1]+ (∇ω)2 2 0 2

C. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 35 / 66 Damage Poromechanics An Alternative for Saturated Rock : Double Effective Stress

Modeling Viscoplastic Damage in Saturated Rock [Dufour et al., 2011]

Objective : modeling creep processes in the Excavation Damaged Zone

Framework : isothermal saturated rock

e e vp Helmholtz free energy for the solid skeleton : Ψs ( , Φ , Ω, γ ) : e vp ˙ e ˙ vp ˙ σ :(˙ + ˙ ) + pw Φ + Φ − Ψs ≥ 0

vp ˙ vp vp σ : ˙ + pw Φ + Y : Ω˙ + k γ˙ ≥ 0 ∂Ψ ∂Ψ ∂Ψ ∂Ψ σ = s , p = s , Y = − s , k = − s ∂e w ∂Φe ∂Ω ∂γvp convex, positive-deﬁnite dissipation potential sought in the form φ (σ, pw , Y , k), with: ∂φ ∂φ ∂φ ∂φ ˙vp = , Φ˙ vp = , Ω˙ = − , γ˙ vp = − ∂σ ∂pw ∂Y ∂k

C. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 36 / 66 Damage Poromechanics An Alternative for Saturated Rock : Double Effective Stress

Problem : what is the driving force in the dissipation potential ? (1/3)

in Continuum Damage Mechanics : Principle of Equivalent Elastic Energy :

e 1 e e 1 T −T Ψs ( , Ω|σ) = 2 : D (Ω): = 2 σ : D (Ω): σ

1 T −T e = 2 σ˜ : D0 : σ˜ = Ψs ( , Ω = 0|σ˜) σ˜ = M (Ω): σ

in Damage Mechanics coupled to poro-elasticity [Coussy, 2004] :

e σ = D (Ω): − pw B (Ω)

0 e σ = σ + pw B (Ω) = D (Ω):

C. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 37 / 66 Damage Poromechanics An Alternative for Saturated Rock : Double Effective Stress

Problem : what is the driving force in the dissipation potential ? (2/3)

...in Damage Mechanics coupled to poro-elasticity...

... either a decomposition of the free energy is postulated...

∗ e ∗ e ∗ Ψs , pw , Ω = Ψs1 , Ω + Ψs2 (pw )

... and Biot’s tensor can be expressed as a function of the drained damaged elastic tensor...

1 B (Ω) = δ − D (Ω): δ 3 Ks ... or this decomposition is not postulated...... and Biot’s tensor is expressed as a function of the undrained damaged stiffness tensor [Shao, 1998]

C. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 38 / 66 Damage Poromechanics An Alternative for Saturated Rock : Double Effective Stress

Problem : what is the driving force in the dissipation potential ? (3/3)

in poro-(visco)-plasticity, it is often postulated that :

˙ vp vp Φ = B0 : ˙

As a result : 0 σ = σ + pw B0

in Damage Poro-(visco)-plasticity, should it be...

σeff = M (Ω): σ + pw B0 ?

σeff = M (Ω):(σ + pw B0)? something else ? ? ?

C. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 39 / 66 Damage Poromechanics An Alternative for Saturated Rock : Double Effective Stress Modeling Approach : Concept of Double Effective Stress

Dependence of the visco-plastic porosity change rate to damage :

Φ˙ vp = B (Ω): ˙vp

(Classical) decomposition of the free energy of the solid skeleton :

∗ e vp ∗ e ∗ vp Ψs ( , pw , Ω, γ ) = Ψs1 ( , pw , Ω) + Ψs2 (γ )

C. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 40 / 66 Damage Poromechanics An Alternative for Saturated Rock : Double Effective Stress Modeling Approach : Concept of Double Effective Stress

Dependence of the visco-plastic porosity change rate to damage : Φ˙ vp = B (Ω): ˙vp (Classical) decomposition of the free energy of the solid skeleton : ∗ e vp ∗ e ∗ vp Ψs ( , pw , Ω, γ ) = Ψs1 ( , pw , Ω) + Ψs2 (γ )

vp ˙ vp vp σ : ˙ + pw Φ + Y : Ω˙ + k γ˙ ≥ 0 vp vp (σ + pw B (Ω)) : ˙ + Y : Ω˙ + k γ˙ ≥ 0 σ0 : ˙vp + Y : Ω˙ + k γ˙ vp ≥ 0 “double effective stress” ∗ e vp ∗ e 0 ∗ vp Ψs ( , pw , Ω, γ ) = Ψs1 ( , Ω |σ ) + Ψs2 (γ ) ∗ e vp ∗ e 0 ∗ vp Ψs ( , pw , Ω, γ ) = Ψs1 ( , |M (Ω): σ ) + Ψs2 (γ )

σeff = M (Ω):(σ + pw B (Ω))

vp vp vp ∂φ σeff : ˙ + k γ˙ ≥ 0, ˙ = ∂σeff σeff is “conjugate” to the viscoplastic strain rate by the dissipation potential. C. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 40 / 66 Nuclear Waste Disposals

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

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 41 / 66 Nuclear Waste Disposals Outline of the THHMD Model

Independent State Variables

1 Assumption : incompressible solid phase. Clausius-Duhem Inequality :

(σij − paδij ) ∆ji +(pa − pw )∆ (−nSw ) −S∆T − ∆Ψs (ij , nSw , T , Ωij ) ≥ 0

2 3 independent strain variables : mechanical strain Mij , capillary strain Sv and thermal strain Tv ...... conjugate to 3 independent stress variables : net stress σ”ij = σij − paδij , suction s = pw − pa, and thermal stress pT :   σ”ij ↔ M ij s ↔ Sv  pT ↔ Tv

3 Thermodynamic decomposition of the total strain tensor : 1 1 d = de + dd + δ de + dd + δ de + dd ij Mij Mij 3 ij Sv Sv 3 ij Tv Tv e : elastic, d : non-elastic 4 Clausius-Duhem Inequality written in terms of stress/strain products : σ”ij ∆Mij +s ∆Sv +pT ∆Tv − ∆Ψs Mij , Sv , Tv , Ωij ≥ 0

C. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 42 / 66 Nuclear Waste Disposals Outline of the THHMD Model

Stress/Strain Relationships

3 components of Helmholtz free energy :

Ψs(Mkl , Sv , Tv , Ωkl ) =

1 1 1 2 Mji Deijkl (Ωpq ) Mlk + 2 Sv βs (Ωpq ) Sv + 2 Tv βT (Ωpq ) Tv

gS gT −gM Ωij Mji − 3 δij Ωji Sv − 3 δij Ωji Tv

Conjugation Relationships :

∂Ψ ( , , ,Ω ) ∂Ψ ( , , ,Ω ) s Mkl Sv Tv kl s Mkl Sv Tv kl σ”ij = ∂ , s = ∂ Mij Sv

∂Ψ ( , , ,Ω ) ∂Ψ ( , , ,Ω ) s Mkl Sv Tv kl s Mkl Sv Tv kl pT = , Yd = − ∂Tv ij ∂Ωij

Damaged Rigidities Computed by the Principle of Equivalent Elastic Energy [Codebois et Sidoroff 1982]

C. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 43 / 66 Nuclear Waste Disposals Outline of the THHMD Model

Damage Evolution Law

inﬂuence of tensile mechanical stress, thermal expansion and capillary pore shrinkage: g g Y + = g + + S − δ + T + δ d1ij M M ij 3 Sv ij 3 Tv ij

a unique damage criterion [Dragon et Halm 1996] : r 1 + + fd Yd pq , Ωpq = Tr Y Y − C0 − C1δij Ωji 2 d1 ij d1 ji after applying the consistency rules : ∂fd Yd , Ωpq dΩ = dλ pq ij d ∂Y + d1 ji

C. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 44 / 66 Nuclear Waste Disposals Outline of the THHMD Model

Transfer Equations

Liquid Water Flow :

ΨR (θw ) dσ(T ) 1 σ(T ) Vwi = − Kwij ∇ (T )j + Kwij ∇ (s)j − Kwij ∇ (z)j σ(Tref ) dT γw σ(Tref )

Inﬂuence of Damage on the intrinsic permeability :

Kwij = kr (Sw , T ) Kintij (n, Ωpq )

h intact rev dg frac i Kwij = kr (Sw , T ) Kij (n ) + Kij n , Ωpq

Assumption : laminar ﬂow in the 3 equivalent cracks of the REV [Shao et al. 2005] :

3 −2/3 5/3 dg frac π γw 4/3 2 X k k k Kij n , Ωrs = χ b d δij − ni nj 12 µw (Tref ) k=1 b : internal length parameter

C. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 45 / 66 Nuclear Waste Disposals Study of Nuclear Waste Disposals with the THHMD Model

Inﬂuence of the Damage Parameters [Pollock 1986]

Sw0 = 0.15

C. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 46 / 66 Nuclear Waste Disposals Study of Nuclear Waste Disposals with the THHMD Model

Inﬂuence of the Internal Length Parameter [Pollock 1986]

dg frac K n , Ωrs = 2ij

−2/3 π γw χ4/3 b2 12 µw (Tref )

P3 k 5/3 k k × k=1 d δij − ni nj

dg = max δ Ω = . δ Kij Kw, dg ij for ij 0 95 ij ⇒ computation of b

C. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 47 / 66 Damaged Permeability

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

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 48 / 66 Damaged Permeability A New Model of Permeability for Cracked Porous Rock

Main Assumptions [Arson and Pereira, 2011]

Cracks do not interact ⇒ damage = crack-density tensor [Kachanov, 1992]

3 X Ω = d k nk ⊗ nk k=1

Cracks do not intersect but are connected to the natural pores:

0 c K w = Kw + Kw Cracks and natural pores are connected, but do not overlap:

Vv = Vp + Vc

Flow in the natural pores and cracks is modeled as a laminar ﬂow in parallel cylinders. For an isotropic model [Garcia-Bengochea et al., 1979]: γ 1 Z ∞ k = Φ f (r)r 2dr w 8µ R ∞ 0 f (r)dr 0

C. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 49 / 66 Damaged Permeability A New Model of Permeability for Cracked Porous Rock Size Distributions of the Natural Pores and Cracks (1/3)

Volumetric frequency of the pores of radius r :

2 f (r) = L α(r) π r α(r) = αp(r)+ αc(r)

Micro/macro relationship for a unit REV (L = 1):

p c Z rmax Z rmax 2 2 αp(r)πr dr = Vp αc(r)πr dr = Vc p c rmin rmin

p p c c rmin, rmax , rmin, rmax : min. and max. radius values for natural pores and cracks αp : frequency of occurrence of natural pores of radius r in the REV αc : frequency of occurrence of cracks of radius r in the REV

p c Z rmax Z rmax 2 2 π Np pp(r)r dr = Vp π Nc pc(r)r dr = Vc p c rmin rmin

Np, Nc : number of natural pores and cracks in the REV

C. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 50 / 66 Damaged Permeability A New Model of Permeability for Cracked Porous Rock Size Distributions of the Natural Pores and Cracks (2/3)

Natural Pores : bell-shaped distribution [Van Genuchten, 1980 ; Alves et al., 1996]

( ( − )2 p p √1 exp − r m ifr ≤ r ≤ r ( ) = s 2π 2 s2 min max pp r p p 0 if r < rmin or if r > rmax

C. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 51 / 66 Damaged Permeability A New Model of Permeability for Cracked Porous Rock Size Distributions of the Natural Pores and Cracks (3/3)

Cracks : exponential distribution [Maleki, 2004]

( 1 r c c λ exp − λ ifr min ≤ r ≤ rmax pc(r) = c c c c 0 if r < rmin or if r > rmax

C. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 52 / 66 Damaged Permeability A New Model of Permeability for Cracked Porous Rock

Updating kw with the State Variables γ 1 Z ∞ k = Φ f (r)r 2dr w 8µ R ∞ 0 f (r)dr 0

Knowing the State Variables ij and Ωij at the current iteration : 1 updating porosity Φ (hyp. incompressible solid grains) :

VREV −ii = 0 − 1 = ∆Φ VREV

2 updating the volumetric frequency f (r)

2 f (r) = (αp(r)+ αc(r)) π r

⇒ How to update αp(r) and αc(r) with the current state variables ? Z ∞ Z ∞ 2 2 αp(r)πr dr = Vp αc(r)πr dr = Vc 0 0

⇒ How to update Vp and Vc with the current state variables ?

C. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 53 / 66 Damaged Permeability A New Model of Permeability for Cracked Porous Rock Types of Deformations

el d id ∆Vp = −Tr ∆Vc = −Tr D (Ω): = −gΩ

C. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 54 / 66 Damaged Permeability A New Model of Permeability for Cracked Porous Rock Algorithm (1/2)

1 compute the current increment of damage dΩ(k) 2 compute the increment of stress dσ(k). For a strain-controlled test : ! ∂D Ω(k−1) dσ(k) = D Ω(k−1) : d(k) + : (k−1) : dΩ(k) −gdΩ(k) ∂Ω

(k) 3 update of the total porous volume Vv . For a strain-controlled test : (k) (k−1) (k) (k) (k) = + d , Vv = −Tr + Φ0

(k) 4 update of the volume occupied by cracks Vc : (k) −1 del = D Ω(k−1) : dσ(k)

d (k) (k) el (k) (k) d (k) d = d − d , Vc = −Tr

(k) 5 update of the volume occupied by natural pores Vp : (k) (k) (k) Vp = Vv − Vc

C. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 55 / 66 Damaged Permeability A New Model of Permeability for Cracked Porous Rock Algorithm (2/2)

6 update the mean radius of natural pores m(k) : p Z rmax (k) 2 (k) 1 (r − m ) 2 Vp = π Np √ exp − r dr p 2 s2 rmin s 2π

Np and s are ﬁxed parameters, determined in the algorithm initialization (k) 7 update the number of cracks present in the REV Nc c Z rmax (k) (k) 1 r 2 Vc = π Nc exp − r dr c λc λc rmin

λc is a ﬁxed parameter, determined in the algorithm initialization (k) (k) (k) 8 update αp (r) and αc (r), update the “volumetric frequency” f (r) :

(k) (k) (k) 2 f (r) = αp (r) + αc (r) πr

(k) 9 update hydraulic conductivity kw : γ 1 Z ∞ k (k) = Φ + Tr −(k) f (k)(r)r 2dr w 8µ 0 R ∞ (k) 0 f (r)dr 0 C. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 56 / 66 Damaged Permeability A New Model of Permeability for Cracked Porous Rock Initialization Process

1 solve the following system of equations for the initial mean radius of the

natural pores( m0), for the standard deviation of the natural pore radius (s, ﬁxed), and for the number of natural pores present in the REV( Np, ﬁxed):  p 0 R rmax 0 2  V = Φ = π N p p (r)r dr  p 0 p r p  min   p m ≈ R rmax p0(r)rdr 0 r p p  min   p  R rmax 0  1 ≈ p pp(r)dr rmin

2 solve the following equation for the crack characteristic length( λc, ﬁxed):

c Z rmax λc ≈ pc(r)rdr c rmin

C. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 57 / 66 Damaged Permeability A New Model of Permeability for Cracked Porous Rock Summary [Arson and Pereira, 2011]

4 input parameters : p p c c rmin, rmax , rmin, rmax

3 model parameters computed in the algorithm initialization : mathematical conditions + knowledge of the initial porosity

Np, s, λc

2 variables updated with deformation and damage :

m, Nc

C. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 58 / 66 Damaged Permeability Results : Simulation of Triaxial Compression Tests

Main Material Parameters (Granite, σc = 0)

E (Pa) ν (-) g (Pa) C0 (Pa) C1 (Pa) e0 (-) 8.01 1010 0.28 −3.3 108 1.1 105 2.2 106 0.008

p p c c rmin (µm) rmax (µm) rmin (µm) rmax (µm) 0.01 1 0.1 10

Reference Results [Halm and Dragon, 2002 ; Arson and Gatmiri, 2010]

7 8 x 10 x 10 16 4

14 3.5

12 3

10 2.5 (Pa) (Pa)

rr 8 rr 2 σ σ − ε ε − zz rr zz σ 6 zz σ 1.5 ε ε rr zz 4 1

2 0.5

0 0 −1 −0.5 0 0.5 1 1.5 2 −4 −2 0 2 4 6 ε −3 ε −3 x 10 x 10

σc = 0 MPa σc = 20 MPa C. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 59 / 66 Damaged Permeability Results : Simulation of Triaxial Compression Tests Deviatoric Stress and Damage Evolutions

1400

1200 0.5

1000 0.4

800

(−) 0.3 3 Ω

q (MPa) 600

0.2 400

0.1 200

0 0 0 0.005 0.01 0.015 0.02 0 0.005 0.01 0.015 0.02 0.025 ε (−) ε (−) 1 1

C. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 60 / 66 Damaged Permeability Results : Simulation of Triaxial Compression Tests

Variations of the Model Variables m and Nc

−7 8 x 10 x 10 5.5 18

16 5 14

4.5 12

4 (−) c N m (m) 8

3.5 6

4 3 2

2.5 0 0 0.005 0.01 0.015 0.02 0 0.005 0.01 0.015 0.02 ε (−) ε (−) 1 1

C. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 61 / 66 Damaged Permeability Results : Simulation of Triaxial Compression Tests Porous Volume Changes

0.016 1.01 Vv Vp 0.014 1.008 Vc

0.012 1.006

0.01 1.004 ) 3 ) 3

0.008 (m 1.002 REV Vx (m V 0.006 1

0.004 0.998

0.002 0.996

0 0.994 0 0.005 0.01 0.015 0.02 0 0.005 0.01 0.015 0.02 ε ε (−) (−) 1 1

C. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 62 / 66 Damaged Permeability Results : Simulation of Triaxial Compression Tests Impact on Permeability

−6 −6 10 10

−7 −7 10 10 (m/s) (m/s) w w k k

−8 −8 10 10

−9 −9 10 10 0 0.005 0.01 0.015 0.02 0 0.1 0.2 0.3 0.4 0.5 0.6 ε (−) Ω (−) 1 3

C. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 63 / 66 Damaged Permeability Results : Simulation of Triaxial Compression Tests Possible Application : Hydraulic Fracturing

Prospective research : permeability of rock damaged by hydraulic fracturing, from the laboratory scale to the reservoir scale

crack initiation, propagation, bifurcation, coalescence damage due to tectonic stresses (mechanical) and injected ﬂuid pressure (hydraulic) crack opening and closure inﬂuenced by the presence of a propping agent (sand)

[Busetti et al., 2010]

C. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 64 / 66 1 multi-dimensional : damage anisotropy crack interaction, compression unilateral effects, rotation of damage directions

2 multi-physics : THMC couplings, healing non-mechanical origin of cracking, deﬁnition of healing and recovery, damaged tangent and transport properties

3 multi-scale : microstructure, upscaling method growth of pores and defects, connectivity, internal length(s) evolution

Conclusion Research Plans...

geothermal foundations nuclear waste disposals oil and gas exploitation by hydraulic fracturing high-pressure gas storage

CO2 sequestration tunneling and mining subsidence problems mineral exploitation, ore forming mechanisms fault processes, diapirs

C. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 65 / 66 1 multi-dimensional : damage anisotropy crack interaction, compression unilateral effects, rotation of damage directions

2 multi-physics : THMC couplings, healing non-mechanical origin of cracking, deﬁnition of healing and recovery, damaged tangent and transport properties

3 multi-scale : microstructure, upscaling method growth of pores and defects, connectivity, internal length(s) evolution

Conclusion Research Plans...

Intellectual Merit Broader Impacts

geothermal foundations nuclear waste disposals oil and gas exploitation by hydraulic fracturing high-pressure gas storage

CO2 sequestration tunneling and mining subsidence problems mineral exploitation, ore forming mechanisms fault processes, diapirs

C. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 65 / 66 2 multi-physics : THMC couplings, healing non-mechanical origin of cracking, deﬁnition of healing and recovery, damaged tangent and transport properties

3 multi-scale : microstructure, upscaling method growth of pores and defects, connectivity, internal length(s) evolution

Conclusion Research Plans...

Intellectual Merit Broader Impacts

CO2 sequestration tunneling and mining subsidence problems mineral exploitation, ore forming mechanisms fault processes, diapirs

C. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 65 / 66 3 multi-scale : microstructure, upscaling method growth of pores and defects, connectivity, internal length(s) evolution

Conclusion Research Plans...

Intellectual Merit Broader Impacts

1 multi-dimensional : damage anisotropy geothermal foundations crack interaction, compression unilateral nuclear waste disposals effects, rotation of damage directions oil and gas exploitation by hydraulic fracturing 2 multi-physics : THMC couplings, healing high-pressure gas storage non-mechanical origin of cracking, deﬁnition of healing and recovery, damaged tangent CO2 sequestration and transport properties tunneling and mining subsidence problems mineral exploitation, ore forming mechanisms fault processes, diapirs

C. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 65 / 66 Conclusion Research Plans...

Intellectual Merit Broader Impacts

3 multi-scale : microstructure, upscaling subsidence problems method mineral exploitation, ore growth of pores and defects, connectivity, forming mechanisms internal length(s) evolution fault processes, diapirs

C. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 65 / 66 Conclusion

Questions ?

C. Arson (Texas A&M University) Poromechanics & Energy Engineering January 6th, 2012 66 / 66