Biot Poromechanics in Earthquake and Faulting Phenomena · Department of Civil Engineering and Engineering Mechanics, 19 October 2005 Biot Poromechanics in Earthquake and Faulting

Biot Lecture, Columbia University, Department of Civil Engineering and Engineering Mechanics,

19 October 2005

Biot Poromechanics in Earthquake and Faulting Phenomena

James R. Rice

Division of Engineering and Applied Sciences, andDepartment of Earth and Planetary Sciences

Harvard University

Examples, pore fluid interactions with deformation and failure in earth materials

• Large scale pore fluid processes in the seismogenic lithosphere:

Elevated pore pressure in fault zones.Solitary waves of pore pressure change; pulses of fluid outflow.Non-volcanic seismic tremors downdip of seismogenic subduction interface.

• Poroelasticity in crustal materials:

Post-seismic poroelastic deformation.Poroelastic effects in earthquake interactions, stress transfers, aftershocks.Alteration of effective stress during rupture by dissimilarity of properties across fault plane

• Fluid interactions with shear rupture in fault gouge:

Extreme shear localization.Dilatant stabilization of slip; relation to lack of shallow EQ nucleation?Thermal pressurization of pore fluid by shear heating.Partial melting of fault zone, pseudotachylytes.

• Other:

Sediment liquefaction in cyclic shearing.Landslide to debris-flow transition.…

Linear Poroelasticity

[consistent with σ ijdεij + pd(m / ρ f ) = perfect differential]

m − mo = ρ fα(εkk +α

Ku − Kp)

σ ij = (K −2

3G)δijεkk + 2Gεij − αδij p

Poroelasticity

Linear quasi-static version -- Terzaghi (1923), Biot (1935, 1941, 1973)(notation here of Rice & Cleary, 1976)

[ Ku = another new poroelastic constant = undrained (m = const.) bulk modulus; Ku > K ]

[ K = drained (p = const.) bulk modulus, G = shear modulus, α = new Biot poroelastic constant ]

= density of purefluid at pressure p

(total; no need to introduce separate stresses on solid and fluid)

Elastic isotropic constitutive relations:

∇ ⋅σ = 0

∇ ⋅ q +∂m

∂t= 0 ε = sym(∇ u)

q = −ρ f k

η f∇ p

• infinitesimal geometry changes of solid

(equilibrium, neglecting body force or describing perturbation from gravity-loaded state)

(q = mass flux of fluid relative to solid)

•

• • (u = solid displacement)

(Darcy, neglecting body force or describing perturbation from gravity-loaded state)•

Governing field equations:

(αhy∇2 −∂∂t

)(∇ ⋅ u +α

Ku − Kp) = 0

Final set of pde’s (in u and p):

αhy =k

η f

(Ku − K )(K + 4G / 3)

α 2(Ku + 4G / 3)

(K +

1

3G)∇ (∇ ⋅ u) + G∇2u − α∇ p = 0 ,

Other poroelastic parameters:

[ νu = undrained (m = const.) Poisson ratio; ν = drained (p = const.) Poisson ratio; 1/2 > νu > ν ][ B = Skempton coefficient, (dp)undrained = – B d(σkk)/3; 0 ≤ B < 1 ]

νu =ν + (1− 2ν )(Ku − K ) / 3Ku

1− (1− 2ν )(Ku − K) / 3Ku , B =

Ku − K

αKu

α = 1−KKs

,

For homogeneous and interconnected fluid phase, and solid phase which responds isotropically to pure pressure with same bulk modulus Ks at all points:

Ku = K +α2KsK f

nKs + (α − n)K fn =

mρ f

=fluid volume

unit (reference) volume≈ porosity

hydraulic diffusivity

Example: Dislocation suddenly introduced, at t = 0, along semi-infinite fault (Rice & Cleary, 1976, generalizing Nur and Booker solution for incompressible constituents):

F(∞) = 1 (undrained limit, t = 0+ , or x → ∞) , F(0) =1−νu

1 −ν< 1 (drained limit, t → ∞ , or x → 0+ )

τ =Gδ

2π (1− νu )xF

x2

4αhyt

, F(λ) = 1−

νu − ν1− ν

1 − exp(−λ)

λδ = slip, uniform

τ

x

Compressed,causesuplift near surface,and P increase.

Expanded, causessubsidence near surface,and P decrease.

Elementary model of a finte fault (slipped and then locked):

But both effectsdiminish with time(due to undrained todrained transition).Subsidence Uplift Uplift Subsidence

δ

τ

t

t

∆τ o∆τ ∞

∆τ ∞ =1− νu

1− ν∆τ o < ∆τ o

Contributes to aftershocks alongrupture zone?

Jonsson,Segall,Pedersen.Bjornsson,Nature(2003)

Co-seismicwell levelincreases

(black circles)and decreases(white circles)

Jonsson,Segall,Pedersen.Bjornsson,Nature(2003)

Jonsson,Segall,Pedersen.Bjornsson,Nature(2003)

Mature crustal fault zones

Some observations on their structure,composition, and hydrologic properties

J.Geophys. Res. (1993)

10s-100s mm(But principalfailure surfaceis much thinner,typically < 1-5 mm!)

1-10 m(Sometimes describedas foliated gouge, orfor some faults, simply as gouge.)

30-100 m

(Damage ≈ highly cracked rock.Zone with macrofaults or fracturesextends ~ 10x further.)

F. Chester, J. Evans and R. Biegel, J. Geoph. Res., 98 (B1), 771-786 (1993)

Chester, F. M., and J. S. Chester, Ultracataclasite structure and friction processes of the Punchbowl fault, San Andreas system, California, Tectonophysics, 295 (1-2): 199-221,1998

Prominent slip surface (pss) is located in the center of the layer and identified by the black arrows.(Exhumed from 2-4 km depth. Total slip ≈ 44 km. "Several km" of slip in earthquakes on the pss.)

From J. S. Chester and D. L. Goldsby,

SCEC Ann. Rpt., 2003(also, Chester et al.,

EOS, Trans AGU, 2003)

Punchbowl Faultprominent slip

surface

mm-thick layer;crystal-latticepreferred orientation(evidence: uniformbirefringence,bright layer);contains distinct microscopicslip surfaces, 100-300 µm thick

MedianTectonic

Line Fault(MTL), Japan

h ≈ 3 mmWibberley (priv. commun., 2003)

Median TectonicLine Fault, Japan

Gas permeability data from thefirst pressure cycle; p = 20 MPa.

Permeability,k ( m2 )

(Wibberley and Shimamoto, JSG, 2003) permeability of clay gouge containing the central slip zone, Median Tectonic Line Fault, Japan

Physics of Fault Zone Processes during Seismic Slips

Questions:

• How does shear stress (τ) vary with slip (δ) during earthquakes? Focus is on weakening during rapid, large slip δ on mature faults, i.e.,

δ >> 0.01-0.1 mm (the slip range at which earthquakes are thought tonucleate, according to rate & state concepts and lab-based properties).

• What are the physical mechanism of weakening during slip? Suggested here: Primary mechanisms are

•Thermal pressurization of pore fluid, and only this one discussed today•Flash heating at highly stressed frictional contacts.

Both seem to be important. At sufficiently large slip, others mechanisms may become important too:

•Melting if large enough normal stress (deeper slip),•Gel formation in lithologies of high silica content.

• What fracture energy (G) is implied by the τ vs. δ relation? Important because we can thereby test any proposed τ vs. δagainst seismic constraints on G.

Background for theoretical modeling of stress vs. slip relation:

Field and lab observations, exposures of mature, highly slipped fault zones:

• Slip in individual events is localized to a thin shear zone (h < 1-5 mm)within a finely granulated (ultracataclastic, possibly clayey) fault core that is of order 10s to 100s mm thickness, with low permeability(estimatedk ~10–20 m2 at mid-seismogenic depths)

[that despite the existence of much wider (~1-100 m) damage zones with granulation, pervasive cracking and/or minor faulting]

Hypotheses:

• Earthquake failure occurs in a water-saturated fault zone (a porousgranular material in the shallow to middle crust).

• It has material properties (permeability, porosity, poroelastic moduli)like those inferred from lab studies of fault materials from the Median Tectonic Line (MTL), Nojima and Hanaore faults in Japan.

[locations for which relatively complete data exists]

Governing equations, 1-space-dimension shearing field, constant normal stress σn:

m =mass of pore fluid

unit (reference) vol. of porous medium= ρ f ntemperature

porepressure

velocity

heatflux

fluidmass flux

• Constitutive relation (for shear) :

∂u

∂y=

&τµ

+ &γ ( &γ = inelastic shear rate)

Friction law : τ = f ⋅ (σn − p) if &γ ≠ 0 ( &γ ≥ 0).

• Equations of motion (= equilibrium equations) :

∂σ yy

∂y= 0 ,

∂σ yx

∂y= 0

⇒ σn ≡ −σ yy = const. , τ ≡ σ yx = τ (t)

Thermalpressurizationof pore fluid

τσn

y

x

qh (y , t) q f ( y , t)

T ( y, t ) , p ( y , t )

u ( y, t)

m(y , t)

τ [= σ yx ]

σ n [= −σ yy ]

• Energy equation :

τ &γ = ρc∂T

∂t+

∂qh

∂y , qh = −K

∂T

∂y;

ρc ≈ 2.7 MPa/ºC ; αth =K

ρc≈ 0.5-0.7 mm2 /s.

• Fluid mass conservation :

∂m

∂t+

∂q f

∂y=0 , q f = −

ρ f k

η f

∂p

∂y ⇒

∂ p

∂t= Λ

∂T

∂t−

1

β∂n pl

∂t+ αhy

∂2 p

∂y2; αhy =

k

η f β,

Λ ≈ 0.3-1.0 (MPa/ºC), β ≡ n(β f + βn ) = 5.5-30 × 10−11/Pa;

β f ,βn = fluid compressibility, pore space expansivity.

n =volume of pore - fluid

unit (reference) vol. of porous mediumm = ρ f n

Ý m = Ý ρ f n + ρ f Ý n = Ý ρ f n + ρ f ( Ý n el + Ý n pl )

ρ f = ρ f β f &p − ρ f λ f&T , &nel = nβn &p + nλn

&T

m / ρ f = β( &p − Λ &T ) + &n pl ,

where β = n(β f + βn ) and Λ =λ f − λn

β f + βn .

Diffusivity αhy =k

η f β

Some calculations:

inelastic dilatancy rate

A perspective on shear localization in fluid - infiltrated granular media

Assume that all inelastic dilatancy ∆n pl is over at small shear.

Then the governing equations for p and T are:

1

ρcτ (t)&γ (y, t) =

∂T

∂t− αth

∂2T

∂y2

Λ∂T

∂t=

∂p

∂t− αhy

∂2 p

∂y2 .

τ (t) = f [σn − p(y, t)] if &γ (y, t) ≠ 0

Question : What type of solutions exist, if we assume that f = constant? Answer:

Either

(i) p(y,t) is spatially uniform, p(y,t) = p(t), ⇒ T (y, t) = T (t), ⇒ &γ (y, t) = &γ (t);

i.e., no fluid flow (undrained), no heat flow (adiabatic), homogeneous strain

(too idealized to be realistic, and has been proven

to be unstable to perturbations [Rice and Rudnicki, 2005]),

or

(ii) &γ (y,t) = 0 except at the isolated position(s) y where p(y,t) = global maximum;

&γ (y,t) = V (t)δDirac(y) for global max at y = 0 [V (t) = slip rate].

Slip on a plane at slip rate V ( Thickness h of shearing layer assumed

small compared to boundary layers where p and T increase ) :

• In y > 0 , ∂T

∂t= αth

∂2T

∂y2 and

∂p

∂t− Λ

∂T

∂t= αhy

∂2 p

∂y2 .

• On y = 0± , qh = − K∂T

∂y= ±

f (σn − p)V

2 ; q f = −

ρ f k

η f

∂p

∂y= 0 .

• Assumes all dilatancy ∆n pl (distributed) is over at small slip [pamb → po = pamb −∆n pl

β].

Simple solution : For V ≡ dδ / dt = constant, and f = constant,

we solve for the fields T (y,t) and p(y,t), and hence p(0,t), to evaluate

τ = τ (δ ) = f σn − p(0,t)( ) (where slip δ = Vt) :

τ (δ )=f (σn − po )expδ

L *

erfcδ

L *

,

where L* =4

f 2ρc

Λ

2 αhy + αth( )2V

.

[Mase & Smith, 1987; Rice, 2005]

Slip on a Plane, stress vs. slip (V = const.):

τ

f (σn − po )=exp

δL *

erfcδL *

; L* =

4

f 2ρc

Λ

2 αhy + αth( )2V

.

Note apparent multi-scale nature of the slip-weakening; no well-defined Dc:

0.4 mm3 mm

4 mm30 mm

40 mm0.3 m

0.4 m3 m

L* = 4 mm L* = 30 mm

On slip plane, T − Tamb = ∆Tmax 1−τ

f (σn − po )

, ∆Tmax =

f 2VL *

4αth

σn − po

ρc

ρc

Λ

2

αhy + αth( )2 ≈60 mm2 /s (low end) ⇒ L* ≈ 4 mm, if V = 1 m/s and f = 0.25.

450 mm2 /s (high end) ⇒ L* ≈ 30 mm, if V = 1 m/s and f = 0.25,

or L* ≈ 50 mm, if V = 1 m/s and f = 0.20.

How large is L*?

Evaluations for 7 km depth, typical centroidal depth of crustal rupture zone;σn ≈ overburden = 196 MPa, po = pamb = hydrostatic = 70 MPa, Tamb = 210 ºC:

Part of L* based on poro-thermo-elastic properties of fault gouge [Rice, 2005, to JGR]:

Assuming relatively intact gouge, MTL properties

Accounting approximately fordamage at the rupture front

and during subsequent shear,kdmg = 5-10 k, βd

dmg =1.5-2 βd

• V = 1 m/s is the average ratio of slip to slip duration at a point, for the 7 slipinversions discussed in [Heaton, 1990], range is 0.56 to 1.75 m/s, average is 1.06 m/s).

• f = 0.25 represents effects of flash heating, like in high-speed friction experiments [Tullis and Goldsby, 2003; Prakash, 2004].

L* =4

f 2ρc

Λ

2 αhy + αth( )2V

.

Estimate of fresh damage zones [Rice, Sammis & Parsons (BSSA, 2005) fracture model, based on Broberg (GJRAS, 1978; book, 1999), Freund (JGR, 1979), Heaton (EPSL, 1990), & Poliakov, Dmowska & Rice (JGR, 2002)]

To formulate as an elasticity problem for a steady-state field, dependent on x – vr t and y, we specify:

fpeak ≈ 0.6 e.g., 0.2from Heaton (1990)

vr , L , δ , τ peak

−σ yyo

, τ res

τ peak , Ψ and one of

R

L or

τ0 − τ res

τ peak − τ res .

denotes parameter estimated from seismic slip inversion, Heaton (1990) -- L, δ and vr

Smax

Ψ

Stresses off the main fault plane: Why we have to worry about effects of fresh damage on k and β

Procedure:

(1) Solve the 2D elasticity problem,calculate stresses σαβ .

(2) Check if the σαβ would cause Coulomb shear failure [τ > (− σn)x tan(31º)] on someOrientation, or cause tensile failure [σ2 > 0].

Poroelastic effects included; Skempton B = 0.6 [∆p = – B ∆(σ11 + σ22 + σ33 )/3 ]

[Rice et al. (BSSA, 2005), buildingon Poliakov et al. (JGR, 2002)]

YELLOW = shear failure RED = tensile failureFor R/L = 0.1, and τr / τp = 0.2 (nearly complete strength loss)

Scale length in plots:

(Ro*)avg = 20-30 m,

Ro* = 1-70 m,

fitting model to Heaton(1990) earthquake set,assuming fp = 0.6 andhydrostatic initial porepressure.

Smax

Ψ

(Rice, 2005,to JGR)

(Rice, 2005,to JGR)

General definition of fracture energy associated with slip weakening on a fault:

G ≡0δ large∫ [τ( ′ δ ) −τ res]d ′ δ τ(δlarge) ≈ τres( ) .

When a uniform residual level τres has not been reached at maximum slip δ,

a consistent definition is:

G = G(δ) ≡ 0δ∫ [τ ( ′ δ ) − τ(δ)]d ′ δ .

Relation of fracture energy to slip weakening properties:

τ o

τ peak

τ res

=

stress

τ

slip, δ

fracture

energy G

Stress, τ

Slip, δ

Fracture energy G

τ = τ(δ)initial stress

residual strength(may not be well defined)

peak strength

Does not include contributions to G from inelastic deformation near the fault plane. How large?

Slip on a Plane, energy release rate

G = G(δ )=0

δ∫ [τ ( ′δ ) − τ (δ )]d ′δ = f (σn − po )L * exp

δL *

erfcδL *

1 −

δL *

− 1 + 2δ

π L *

For δL *

>> 1, G → 2(σn − po )ρc

Λ

αtht

π+

αhyt

π

(indep. of f and V! -- t = slip duration)

View onto fault plane during rupture:

wL

l

previously slipped, now re-locked; slip = δ

* vr

not yet slipped

currently slipping

w

l

Fault along Earth’s surface

*nucleationregion l x w ruptured

Seismic estimates of fracture energy (G):

Method A Use of seismic slip inversion results for large sets of earthquakes:

A.1: Fit of seismic slip inversion results from Heaton (PEPI, 1990) to a steady state, self-healing, slip pulse model (Rice, Sammis and Parsons, BSSA, 2005),

A.2: Tinti, Spudich and Cocco (JGR, in press 2005), use of kinematic slip (δ) inversions, smoothed, to get stress (τ) histories too, then use of

G =0

δ∫ τ ( ′δ ) − τ (δ )( )d ′δ

Fracture energies G and slips δ, large earthquakes (arranged in order of slip magnitude)

(Rice, 2005,to JGR)

Method B (Abercrombie and Rice, GJI, 2005),Use of radiated energy, moment, and source area (hence stress drop and slip):

Notation :

Es = radiated seismic energy S∫ 0

∞∫ ρ(du / dt)2 dtdS

A = rupture area (from corner frequency or duration)

δ = final slip (from moment Mo=µδ A)

′δ = variable slip as event develops

τ0 = initial shear stress

τ1 = final static shear stress (stress drop τ0 − τ1 ∝ µδ / A )

τ (δ )[= τdyn ] = stress in last increment of dynamic slip

Energy balance :

τ0 + τ1

2δ

A =0

δ∫ τ ( ′δ )d ′δ

A + Es = G + τdynδ( )A + Es

since G =0

δ∫ [τ ( ′δ ) − τ (δ )]d ′δ =

0

δ∫ [τ ( ′δ ) − τdyn ]d ′δ

⇒ τ0 − τ1

2δ = G + (τdyn − τ1)δ +

Es

A , or

′G ≡ G + (τdyn − τ1)δ = (τ0 − τ1) −2µEs

Mo

δ2

; ′G ≈ G and ′G = G if τdyn = τ1

[We find, with the Madariaga (1979) estimate of τ0 − τ1 , that µEs / Mo ≈ 0.1(τ0 − τ1) .]

Comparison, predictions of G from the slip on a plane model with seismic estimates, for:

• The large earthquake data set for G from seismic slip inversions for 12 events (shownas ovals here, one symbol per event), and

• A data set for G' for small and large events based on radiated energy, moment, andseismic source dimension [Abercrombie & Rice, 2005]; G' ≈ G, and G' = G if stress during last increments of slip = final static stress after rupture (no overshoot/undershoot).

(Rice, 2005, to JGR)

(Rice & Rudnicki, in progress, 2005)

Configurational stability of spatially uniform, adiabatic, undrained, shear:(Motivation: Why do zones of localized slip have the thickness that they do?)

Governing equations for shearing velocity V(y, t), shear stress τ(y, t), pore pressure p(y, t), and temperature T(y, t):

∂τ∂y

= 0 (inertia irrelevant) , σn = const.

τ =f (∂V / ∂y)(σn − p) , ′f (...) > 0

τ∂V

∂y= ρc

∂T

∂t+

∂qh

∂y , qh = −ρcαth

∂T

∂y

∂m

∂t≡ ρ f β

∂p

∂t− Λ

∂T

∂t

= −∂q f

∂y , q f = −ρ f βαhy

∂p

∂y

The spatially uniform solution:

V (y, t) = V0(y) = γ oy (γ o = uniform shearing rate),

p(y, t) = p0(t) , τ (y, t) = τ0(t) = f (γ o )[σn − p0(t)],

σn − p0(t) = [σn − p0(0)]exp(−Hγ ot) [call this σ0(t)]

where H ≡f (γ o )Λ

ρc≈ 0.1-0.3 f (γ o ),

T (y,t) = T0(t) , ρcdT0(t) / dt = f (γ o )σ0(t)γ o

Simple rate-strengtheningfriction model; approximatelyvalid only in stable regions inwhich rupture cannot nucleate,but may propagate through (orin unstable regions that have shear-heated to a frictionally stable T range).

(Fuller rate-state description,with localization limiter, mustbe used in regions of unstable,rate-weakening, friction -- butlocalization is expected then.)

τ(y,t)V(y,t)= γoy

y

Linearized perturbation about time-dependent spatially uniform solution:

V (y, t) = γoy + V1(y, t) , p(y, t) = p0(t) + p1(y,t) ,

T (y, t) = T0(t) + T1(y, t) , f = f (γo) + ′ f (γo)∂V1(y,t) / ∂y

∂∂y

− f (γ o )p1 + ′f (γ o )∂V1

∂yσ0(t)

= 0

f (γ o )σ0(t)∂V1

∂y− f (γ o )p1γ o + ′f (γ o )

∂V1

∂yσ0(t)γ o = ρc

∂T1

∂t− αth

∂2T1

∂y2

∂p1

∂t− Λ

∂T1

∂t= αhy

∂2 p1

∂y2

Nature of solution with spatial dependence exp(2πiy/λ):

σ 0(t)∂V1(y, t)

∂y , p1(y,t) , T1(y, t) ∝ exp(st)exp(2πiy / λ)

σ 0(t) ∝ exp(−Hγot) ⇒ ∂V1(y, t)

∂y ∝ exp[(s+ Hγo)t]exp(2πiy / λ)

s = s(λ) satisfies:

Typically of order 20-60,in results oflow shear-rateexperiments.

Dynamic disksimulations[Chevoir et al., 05]z > 1000 foreffective stress> 10 MPa

zHγ os = s +4π 2αth

λ2

s +

4π 2αhy

λ2

where z =f (γ o )

γ o ′f (γ o )=

f

a − b≈

0.6

0.015= 40

Condition for linear instability of flow profile ∂V1

∂y→ ∞

:

Re(s) +f Λρc

γ o > 0 ⇒ λ > λcr ≡ 2π(αth + αhy )ρc

(z + 2) f Λγ o

For shear of layer of thickness h γ o =V

h

: λ > λcr ≡ 2π(αth + αhy )ρch

(z + 2) f ΛV

Comment: Near λ = λcr , Im(s) ≈ zf Λρc

αthαhy

αth + αhy

V

h (oscillatory; unloading?)

Implications:

• Even with velocity strengthening [with large z (= f / [γ df/dγ] ), e.g., of orderz > 10], we must expect large shear strain to be confined to a thin zone, less thandiffusion penetration distances of heat and fluid in moderate and larger events.

• Justifies use of model based on slip on a plane.

• Observed 1-5 mm deformed zone thickness in gouge may be a precursor thickness(i.e., λcr based on an initial, broad h) not the thickness of the large shear zone.

Possible self - consistent estimate of shear layer thickness h at large shear :

Set γ o =V

h, λcr ≈ h ⇒ h ≈

4π 2(αth + αhy )ρc

(z + 2) f ΛV

Results (using z = 40, V = 1 m/s, αth = 0.7 mm2 /s, ρc = 2.7 MPa/ºC):

Low end (Λ = 0.70 MPa/ºC, αhy = 1.5 mm2 /s): h = 4.4 µm / f = 11 µm (if f = 0.4).

High end (Λ = 0.34 MPa/ºC, αhy = 3.5 mm2 /s): h = 31 µm / f = 78 µm (if f = 0.4).

σ0yy = – σo

σ0xy = τo

2

1

+–

Damagefringes (alsodissimilar)

Dissimilarelastic materials"1" and "2" meetat slip surface

(Rudnicki & Rice, 2005)

For steadily propagating mode II ruptures, in form slip = δ = δ(x – v t),on an interface between elastically dissimilar solids(where v = rupture propagation speed, called vr earlier),Weertman (1980) showed that

Note that – dδ /dx = V / v where V = slip rate. Much recent study ofthis case (Adams, 1995, 1998; Andrews & Ben-Zion, 1997; Harris & Day, 1997;Cochard & Rice, 2000; Ben-Zion & Huang, 2001; Brietzke & Ben-Zion, 2005).

When slip surface is bordered by damage fringes, of dissimilarporoelastic properties, which undergo undrained response exceptin a narrow boundary layer where the p discontinuity is reconciled,p and effective tensile stress on the slip plane are (Rudnicki & Rice, 2005)

Both W(v) and µ∗(v) reverse sign when the materials on the two sides are interchanged; effective tension can either increase(destabilizing) or decrease (stabilizing) in different cases.

Number on W(v) / 2 µ2 curves is ratio k+ β+ / k– β–

Case shown: µ1 = 0.75 µ2 , cs1 = 0.90 cs2 , B+ = B– = 0.6

(tensile stress σ0yy is negative)

W(v) / 2 µ2= k+ β+ / k– β–W(v) / 2 µ2

and

µ∗(v) / µ2

Consequence of strong dependenceof permeability k on effective stress:

Lithostatic pore pressure gradients

Solitary waves of pressure increase

z z

σσ

Earth’s surface

Fault

σ

σ

σ

phydrostat

p and σ

pore pressure p

lithostatic stress σ

V

effective stressσ = σ − p

solitary waveof p increase

p

p

Rice, Fault Mech. Transp. Prop. Rocks, 1992:

Upflow fromdeep fluid source

Mass conservation and Darcy flow in one-dimensional vertical seepage[with p = p(z,t)] in a fault zone channel of constant (irrelevant) width:

∂∂z

−ρ f k

ηw

∂p

∂z+ ρ f g

+

∂m

∂t= 0

k = k(σ ), m = m(σ ),

σ = σ (z) = total normal stress, satisfies (approximately)dσdz

= −ρ totg, so that

∂p(z,t)

∂z+ ρ f g = −

∂σ (z, t)

∂z− γ ,where γ = (ρtot − ρ f )g.

∴∂∂z

ρ f k(σ )ηw

∂σ ∂z

+ γ

+

∂m(σ )∂t

= 0

Time-independent steady flow solution on z < 0, with σ = 0 at z = 0: σ satisfies

k(σ ) dσ dz

+ γ

= constant =

ηw

ρ fq f , where q f is steady mass upflow rate

.

⇒ As z → −∞, dp

dz→

dσdz

= −ρtotg (i.e., p gradient → lithostatic).

Rice, Fault Mech. Transp. Prop. Rocks, 1992:

As z → −∞, σ → σ o, a constant, where γk(σ o ) =ηw

ρ fq f ;

0

σ (z)∫ k( ′ σ )d ′ σ k ( ′ σ ) − k(σ o )

= −γz

σ = σ − p (and σ & σ positive in compression).

z

σ

Revil & Cathles, "Fluid transport by solitary waves along growing faults: A field example from the South Eugene Island Basin, Gulf of Mexico", EPSL, 2002

σ − phydrostatic

p − phydrostatic

σ − phydrostatic

Solitary wave solutions on

Both k(σ ) and m(σ ) decrease monotonically with increasing σ , but k(σ ) is the more strongly varying,

k(σ ) ∝ d 3 and m(σ ) ∝ d, where d is a crack aperture, decreasing monotonically with increasing σ :

Example:

k (σ )− k(σ 2 )k(σ 1) − k(σ 2 )

=d 3 − d2

3

d13 − d2

3=

d − d2

d1 − d2

(d 2 + dd2 + d22)

(d12 + d1d2 + d2

2 )<

d − d2

d1 − d2=

m(σ ) − m(σ 2 )m(σ 1)− m(σ 2 )

for all d1 > d > d2.

in the precise sense that we assume k (σ )− k(σ 2 )k(σ 1) − k(σ 2 )

<m(σ )− m(σ 2)m(σ 1) − m(σ 2 )

for all σ 1 < σ < σ 2.

−∞ < z < +∞, in form σ = σ (t − z /V ) , to

∂∂z

ρ f k(σ )ηw

∂σ ∂z

+γ

+

∂m(σ )∂t

= 0 :

and V =ρ f γηw

k (σ f ) − k(σ o )

m(σ f ) − m(σ o )> 0 (only upward propagation possible).

Result : Solutions exist if and only if σorig >σ final (i.e., corresponds to a p increase)

Rice, Fault Mech. Transp. Prop. Rocks, 1992:

Haney, Snieder, Sheiman & Losh, "A moving fluid pulse in a fault zone", Nature, 2005(Related study by Revil & Cathles, "Fluid transport by solitary waves along growing faults:

A field example from the South Eugene Island Basin, Gulf of Mexico", EPSL, 2002)

Pulsed fluid flow at the toe of theBarbados accretionary wedge

[Henry, J. Geophys. Res., 105 (B11), 2000]Packer testsin drillholes

Lab tests (nobig fissures)

Deepsubmersibletests

Drillholedata here

Melt as the pore fluid

Shear heating: Sometimes fault zones get hot enough that they (partially) melt

pictures only

Rapidly resolidified melt (now a glass, called pseudotachylyte) which was extruded -- onecould say hydraulically fractured -- into cracks in material bordering a melting fault zone:

Otsuki, Monzawa & Nagase, Fluidization and melting of fault gouge, J. Geophys. Res. (2003):

9 pseudotachylyte-generating events:

• Resolidified shear zones marked P1, …, P9; noor minimal overlap.

• All 9 shear zones fit within a 20 mm width! -- aswell as 2 more not shown.

• Individual zones have h < 2 mm, often < 1 mm.

Examples shown, pore fluid interactions with deformation and failure in earth materials

• Poroelasticity in crustal materials:

Post-seismic poroelastic deformation.Alteration of effective stress during rupture by dissimilarity of properties across fault plane

• Fluid interactions with shear rupture in fault gouge:

Extreme shear localization.Thermal pressurization of pore fluid by shear heating, likely a primary weakening mechanism.

• Large scale pore fluid processes in the seismogenic lithosphere:

Solitary waves of pore pressure change; pulsed fluid outflow.