ALERT Geomaterials 25 th ALERT Workshop, Aussois, France 29 Sept.-1 Oct. 214 Multiphysicscouplings and stability in geomechanics Jean Sulem, Ioannis Stefanou Laboratoire Navier, Ecole des Ponts ParisTech, IFSTTAR, CNRS UMR 825, Université Paris-Est, Marne-la-Vallée, France jean.sulem@enpc.fr, ioannis.stefanou@enpc.fr Aknowledgments: N. Brantut, V. Famin, A. Schubnel, M. Veveakis
Seismic event: Unstable slip along a pre-existing mature fault (stick-slip instability) Faultzone embeddedin an elastic crustal block Conceptual model of a spring slider system τ Key factor for the nucleation of seismic slip : Softeningof the shearresistance of the fault zone stable unstable δ 2
Exampleof coupled processes during seismic slip
8 18 Temperature ( C) 6 4 7 km depth f=.1; k f =1-19 m 2 h=5mm T Pp 16 14 12 1 8 pore pressure (MPa) Thermal decomposition of carbonates in fault zones: Slip-weakening and temperature-limiting effects 2 6 12 1 2 3 4 time (s) 16 1 14 -The endothermic chemical reaction limits the co-seismic temperature increase Shear stress (MPa) 8 6 4 7 km depth f=.1; k f =1-19 m 2 h=5mm τ P p 12 1 8 pore pressure (MPa) -Pore pressure exhibits a maximum and then decreases due to the reduction of solid volume (pore pressure pulse) -Weakening/restrengthening of the shear stress 2 1 2 3 4 time (s) 6 Sulem& Famin, (29), J. Geoph. Res.
A key parameter: Size of the localizedzone
ADIABATIC SHEAR BANDING
Adiabatic shear-banding (1/3) τ γ& T Strain rate hardening τ c τ ( γ&,t ) and Temperature softening γ τ H = ( > ) & γ τ ξ = ( < ) T Simple model: : strain rate hardening modulus : temperature softening coefficient (& & ) ( ) τ = τ + H γ γ + ξ T T s T s : threshold for temperature softening
Adiabatic shear-banding (2/3) We consider a shear experiment at constant strain rate Assumption: all the frictional energy is converted into heat ( ( s )) ρct& = τγ& = τ + ξ T T & γ at t = : T = T and τ = τ s z u z (z,t) & γ = V h σ V/2 h ρc : specific heat Integrating the above differential equation we get: τ = + ξ ξγ& ρc T Ts 1 exp t ξγ& ξ τ c = τ exp t = τ exp γ ρc ρc x τ -V/2 τ c γ for : + and τ t T T s τ ξ
Linear stability analysis of adiabatic shear-banding (1/5) To study the stability of adiabatic shear banding we perform a linear perturbation analysis of the uniform solution T ( z, t) = T ( t) + T ( z, t) V ( z, t) = & γ z + V ( z, t) τ ( z, t) = τ ( t) + τ ( z, t) 1 1 1 z T1 ( z, t) = Aexp( st) exp 2π i λ z V1 ( z, t) = Bexp( st) exp 2π i λ z τ1( z, t) = Dexp( st) exp 2π i λ s is the growth coefficient in time of the instability λ=h/n is the wave length of the instability (N=1,2, wave number)
Governing equations(with heat diffusion term) Constitutive equation: c th : thermal diffusivity Stability analysis of adiabatic shear-banding (2/5) (& & ) ( ) τ = τ + H γ γ + ξ T T Energy balance: 2 T ρc T& cth τγ 2 z = & Momentum balance: τ V = ρ z t s 2π i ξ H 1 λ 2 A 2π 2π i ρc s + cth τ & γ B = λ λ D 2π i ρs λ For non trivial solution :(A,B,C) (,,) 2 2 4 2 2π 2π 2π ρcρ s + ρc ( ρcth + H ) ξργ& s + ξτ + ρccthh = λ λ λ Stability condition: All the roots of the characteristic equation have a negative real part
Stability analysis of adiabatic shear-banding (3/5) For strain rate and temperature softening (H < and ξ < ), the uniform solution is unstable for all wave lengths of the perturbation For strain rate hardening and temperature softening (H > and ξ < ) Stability condition: ξτ 2 2π + cthhρc > λ < λcr, with λcr = 2 λ π cthhρc ξτ Perturbations with wavelengths shorter than the critical value will decay exponentially; those greater than the critical value will grow exponentially. Only shear zones with a thickness h < λ cr /2 will support stable homogeneous shear.
Stability analysis of adiabatic shear-banding (4/5) Numerical example c th 2 1mm / s ρc = 2.7MPa/ C τ H = 7MPa =.5MPa.s ξ =.5MPa/ C λ λ λ cr cr cr = = 2π ( τ 7MPa) ( τ ) = cth H ρc ξτ = =.39mm Growth coefficient s (7) (3) (1) (5) τ λ cr λ λ cr For the growth coefficient s of the instability reaches a maximum for a particular wave length λ Shear will localize in narrow zones with a size controlled by the wave length with fastest growth coefficient of the instability
Stability analysis of adiabatic shear-banding (5/5) Mode (1) Evolution in time and spaceof the strain rate (3) (1) Mode (3)
THERMAL PRESSURIZATION OF THE PORE FLUID : A MECHANISM OF THERMAL SOFTENING
z x Undrained adiabatic shearing of a saturated rock layer u z (z,t) u x (z,t) Solution: σ n -V/2 ( σ n ) ( σ n p ) V/2 p, T h & γ = f Λ p = p + p 1 exp( γ & t) ρc f Λ = + 1 exp( & γ ) Λ ρc T T t V h p T mass balance: = Λ t t T 1 V energy balance: = ( σ n p) f t ρc h / 2 & γ f = f + H log & γ is the rate-dependent friction coefficient Λ = λ f n λ is the coefficient of thermal pressurization (typical values:.1 to 1 MPa/ C) n β + β f The pore-pressure increases towards the imposed normal stress σ n. In due course of the shear heating and fluid pressurization process, the shear strength τis reduced towards zero. 15
Linearstability analysis of undrained adiabatic shearing of a saturated rock layer Stability condition: Rice et al., (214), J. Geoph. Res. λ < λ, with λ = 2π cr cr ( c ) th + chy H ρc f Λ f + H & γ ( 2 ) Only shear zones with a thickness h < λ cr /2 will support stable homogeneous shear. Competing processes: Fluid and thermal diffusion and rate-dependent frictional strengthening tend to expand the localized zone, while thermal pressurization tends to narrow it. For representative material parameters at a seismogenic depth of 7km, V=1m/s, h=1mm λ / 2 = 3µ m for intact material, 23 µ m for damaged material cr The localizedzone thickness may becomparable withthe gouge grain size 16
LOCALIZED ZONE THICKNESS AND MICROSTRUCTURE Stability analysis of undrained adiabatic shearing of a rock layer with Cosserat microstructure Sulem, Stefanou, Veveakis, (211), Granular Matter Strainhardeningelasto-plasticity for 2D Cosseratcontinuum
Rock layer at great depth (7km) Cosserat continuum Cosserat terms are active only in the post localization regime Microinertia Wave length selection is obtained (wave length with fastest growth in time) Cauchy continuum The critical hardening modulus for instability is unchanged (h cr =.15). The growth coefficient tends to infinity for the infinitely small wave length limit (ill-posedness).
The selected wave length decreases with decreasing hardening modulus and reaches a minimum (for λ~2). λ is the wave length normalized by the internal length of the Cosserat model R (grain size). With R=1µm(grain size for highly finely granulated fault core) the obtained localized zone thickness is about 1mm which is compatible with field observations of localized shear zones in broader damaged fault zones.
Effectof chemicalreactions such as thermal decomposition of minerals on the localized zone thickness t t z 2 p T p = Λ + chy 2 Veveakis, Sulem, Stefanou, (212), J. Stuct. Geol. Mass balance: Pore pressure diffusion and generation + 1 ρ β f m t 1 β d * * n t d Thermal pressurization Fluid diffusion Fluid production Porosity increase (solid decomposition) Energy balance T T 1 v 1 H m t z ρc z M ρc t 2 = cth + τ 2 CaCO 3 d Heat diffusion Frictional heat Heat consumedin the chemical reaction
Example: Thermal decomposition of Carbonate For representativematerial parameters at a seismogenic depthof 7km, a selected wave length of 4-5 * Cosserat length scale is obtained The shear band thickness is initially controlled by the thermal pressurization and canbe reducedwhen the temperature of the slippingzone reachesthe critical temperature of mineral decomposition (about 7 C for carbonates)
CHEMICAL DEGRADATION AND COMPACTION INSTABILITIES IN GEOMATERIALS http://sydney.edu.au/engineering/civil/people/das.shtml Stefanou& Sulem, (214), J. Geoph. Res
Theoretical framework for deformation bands formation Criterion for compaction bands without dissolution: β + µ 3 Associate plasticity: β = µ = 3 2 Issen& Rudnicki(2) ζ is a chemical softening parameter (e.g. Grain Dissolution) e.g. Nova et al. (23), Int.J.Num.An.Meth.Geom. Xie et al. (211) Int.J.Rock Mech. Min. Sc, Hu & Hueckel (27), Int.J.Num.An.Meth.Geom. 23
Distinction of scales Macro-scale / Elementary volume (REV) - Constitutive behavior - Momentum balance - Mass balance Micro-scale / Single Grain - Reaction kinetics of dissolution - Grain crushing 24
Reaction kinetics (micro-scale) solid +solvent solution ( 3) ( 1) ( 2) e.g. dissolution of quartz SiO (solid)+2h O(liquid) 2 2 4 4 H SiO (aqueous solution) or carbonate ( ) CaCO (solid)+h CO (aqueous solution) Ca HCO (aqueous solution) 3 2 3 3 2 w2 S w2 = k 1 eq t e w2 w 2 k e is the mass fraction of the dissolution product in the fluid is a reaction rate coefficient is the void ratio S 1 D is the specific area of a single grain of diameter D 25
Evolution of the effective grain size Grain crushing (micro-scale) a = D D a + E T Lade et al. J. (1996), J.Geotech. Engrg. or S = S + E 1 T a Baud et al. (29),1mm a E T is a material constant which describes grain crushability is the total energy density given to the 26 system
Constitutive behavior (macro-scale) Modified Cam-Clay plasticity model f 2 2 q M p p p c + ( ) = c R R ( ) p p p p ζ κ Non local chemical softening ζ µ ρ w = eζ t µ ρ t M 3 f 2 2 s w M 2 1 = w + V T V T w 2 2 2 2dV w2 lc 2 z l c characteristic length 27
Mass balance (macro-scale) p f 1 ε w = c p c t n t t 2 2 hy X f p, ch β f chy n β f c p, ch p f ε is the hydraulic diffusivity is the porosity is the fluid compressibility is the chemical pressurization coefficient is pressure of the fluid is the volumetric strain 28
Linear stability analysis of oedometric compaction h (, ) = + % (, ) u z t u u z t z z z h (, ) = + % (, ) p z t p p z t f f f h (, ) = + % (, ) w z t w w z t 2 2 2 st z u% z ( z, t) = Ue cos λ st z p% f ( z, t) = Pe sin λ st z w% 2 ( z, t) = We sin λ s is the growth coefficient of the perturbation (Lyapunov exponent) 29
Numerical example : Compaction banding in a reservoir Carbonate grainstone Elastic constants K = 5GPa G = 5GPa Cam clay yield surface p R MPa p = 3MPa M =.9 Physical properties c hy = 1 m s n =.25 3 2 1 5 D =,2mm Chemical parameters k = 1 mols m 6 1 2 Grain crushing parameter: Initial stress state at 1.8km (oedometric) σ n p f 45MPa 18MPa a =.5 MPa 3
Linear Stability Analysis & zones of instability µ = β = 3 2 Instability region Tendency for grain crushing 1 a Instability region becomes larger 31
Oedometric stress path
Wave length selection 33
Wave length selection influence of the hydraulic diffusivity 34
Wave length selection influence of grain crushing and dissolution rate a 35
Conclusions - Importance of coupled processesin seismicslip: shearheating, pore fluid pressurization, thermal decomposition of minerals - Thermal decompositionof mineralscanexplainthe lackof pronounced heat outflow along major tectonic faults - A key parameter: thicknessof the localizedzone (competition of several length scales) - Thickness and periodicity of compaction bands: a new chemomechanicalmodel withstrongcouplingthat accounts for the increase of dissolution kinetics rate because of grain crushing 36