Dilatancy/Compaction and Slip Instabilities of Fluid Infiltrated Faults

Vahe GabuchianGE169B/27701/25/2012Dr. LapustaDr. Avouac

• Experimental results of frictional behavior for porous materials• Constitutive equations stemming from experimental observations• 1-DOF spring slider system used as the model for the problem setup• Derivation of the lumped parameter set of equations describing the system• How does proposed model compare to the experimental results?• Linearized stability analysis and it’s significance• Implications for nucleation of earthquakes

List of references used1) Dilatancy, compaction, and slip instability of a fluid-infiltrated fault, Segall, P. and Rice, J.R.,Journal of Geophysical Research, Vol. 100, No.

B11, Pages 22,155-22,171, November 10, 1995.2) Dilatant strengthening as a mechanism for slow slip events, Segall, P., Rubin, A.M., Bradley, A.M., and Rice, J.R., Journal of Geophysical

Research, Vol. 115, B12305, 2010.3) Frictional behavior and constitutive modeling of simulated fault gouge, Marone, C., Raleigh, C.B., and Scholz, C.H., Journal of Geophysical

Research, Vol. 95, No. B5, Pages 7007-7025, May 10, 1990.4) Creep, compaction and the weak rheology of major faults, Sleep, N.H. and Blanpied, M.L., Nature, Vol. 359, 22 October, 1992, Pages 687-

692.5) An earthquake mechanism based on rapid sealing of faults, Blanpied, M.L., Lockner, D.A., and Byerlee, J.D., Nature, Vol. 358, 13 August,

1992, Pages, 574-576.

Experiments on dilatancy and compaction of Marone, et al, apply step changes to load velocity (up/down) and measure porosity (cylinder height) and friction coefficient.

• Frictional coefficient and porosity evolve to steady state values (μ μss and ϕ ϕss)• Step increase in velocity promotes dilatancy, step decrease promotes compaction• The evolution length scale is approximately the same suggests physics are related• Higher loads shift porosity and frictional coefficient up to higher values

Fault Gouge Experiments Link Frictional Resistance μ and Porosity ϕ

Need to model porosity changes with changes in system parameters (i.e. slip velocity, etc)

€

˙ φ = −v

dc

φ − φss( )

€

φ =φ0 +ε lndc

v0θ

⎛

⎝ ⎜

⎞

⎠ ⎟

Both approaches lead to models that are nearly identical but are exactly equal at steady state.Here we are only considering the plastic effects of porosity, elastic effects will be considered.

• ϕ is a function of slip velocity v (coming from the “critical state concept” in soil mechanics where a steady state value is postulated)• v ϕ and v ϕ • Introducing the dilatancy coefficient, ε

• Assume that steady state porosity varies as a function of state variable θ rather than velocity• Same length scale, dc, makes proposed form physically reasonable

€

φss = φ0 +ε lnv

v0

⎛

⎝ ⎜

⎞

⎠ ⎟

€

θ ss =dc

v

Constitutive Equations Based on Experimental Observation

Approach 1

€

φ =φ v( ) Approach 2

€

φ =φ θ( )

Dilatancy coefficient

Classic 1-DOF spring-slider model used to relate slip/slip rate u/v to system properties such as system stiffness, k, pore pressure, p, and frictional laws.

Rate and State friction coefficientFrictional resistance depends on sliding velocity, v, and history of slip given by a state variable, θ. Allows for strengthening during no-slip conditions.

Frictional resistance

Driving force

Stability analysis has been done for this system with p = const (drained case).

€

kcrit = −1

dc

∂τ ss

∂ lnv=

σ − p( ) b − a( )

dc

k < kcrit Unstable k > kcrit Stable

Large stiffness (high k) favors stable sliding. High pore pressure (large p) favors stable sliding.

Model for Development of 1-DOF Spring Slider System

The model is quasi- static: inertial effects are ignored (mass of block ignored).

€

μ =μ0 + a lnv

v0

⎛

⎝ ⎜

⎞

⎠ ⎟+ b ln

θ

θ 0

⎛

⎝ ⎜

⎞

⎠ ⎟

€

dθ

dt=1−

θv

dc

€

τ = σ −p( )μ

€

dτ

dt= k v∞ − v( )

(1)

(2)

(3)

(4)

(5)

(4)A lumped parameter model is derived by combining equations (4) (3) (2) (1).

€

˙ φ = φβφ ˙ p + ˙ φ plastic

Governing Equations for Fluid: Linking Dilatancy, ϕ, to Pore Pressure, p

€

∂qi

∂x i

+ ˙ m = 0

€

qi = −ρ 0

κ

v

∂p

∂x i

€

˙ m = ρ ˙ φ + φ ρβ f ˙ p ( )

€

βf =1

ρ

⎛

⎝ ⎜

⎞

⎠ ⎟∂ρ

∂p

⎛

⎝ ⎜

⎞

⎠ ⎟

Conservation of mass

Darcy’s law

Evolution of fluid mass

Compressibility of the fluid

(1)

(2)

(3)

Distinguishing between elastic and plastic pore compressibility (elastic is only due to volumetric strains while plastic refers to irreversible volume changes due to shear motion)

€

βφ =1

φ

⎛

⎝ ⎜

⎞

⎠ ⎟∂φ

∂p

⎛

⎝ ⎜

⎞

⎠ ⎟

€

˙ φ = ˙ φ elastic + ˙ φ plastic

Defining ELASTIC PORE COMPRESSIBILITY

Can write as

€

˙ φ elastic

€

˙ φ elastic =∂φ

∂p˙ p

€

c∇2 p −∂p

∂t=

˙ φ plastic

β

β = φ β f + βφ( )

c =κ

vβ

Assume the pressure to follow a simpler model and introduce a length scale, L

€

c * p∞ − p( ) −∂p

∂t=

˙ φ plastic

β

c* =c

L2 =κ

vβL2

€

c∇2 p = cp∞ − p

L2 = c * p∞ − p( )

Pore pressure satisfies diffusion equation in lumped parameter model with forcing term .

€

˙ φ plastic

Neglecting poroelastic coupling

System of 5 equations with 5 variables

€

dθ

dt= ˙ θ θ ,v( )

dτ

dt= ˙ τ v( )

dφ

dt= ˙ φ φ,v( )

dp

dt= ˙ p p,φ,v( )

dv

dt= ˙ v p,φ,v,θ( )

€

dθ

dt= −

v

dc

θ −θ ss( )

dτ

dt= k v∞ − v( )

dφ

dt= −

v

dc

φ − φss( )

dp

dt= −

˙ φ

β− c * p∞ − p( )

dv

dt= ˙ v p,φ,v,θ( )

Full Set of Governing Equations

Variable Physical Quantification

τ Shear resistance

θ State variable

v Slip rate

p Pore pressure

ϕ Porosity

(1)

(2)

(3)

(4)

(5)

€

vss = v∞

θ ss =dc

v∞

pss = p∞

φss = ε lnv∞

v0

⎛

⎝ ⎜

⎞

⎠ ⎟

μ ss = μ0 + a − b( )lnvss

v0

⎛

⎝ ⎜

⎞

⎠ ⎟

τ ss = σ − p∞( )μ ss

Steady state values of variables are:

• The two results (dashed and solid line) represent two models of ϕ (Approach 1 and Approach 2).• The steady state solutions are exactly identical.• Small differences exist in the evolutionary portions in porosity and nearly no deviations in the frictional coefficient• Model and experimental results give a good qualitative and quantitative match

Model Agrees Well with Experimental Results of Marone, et. al.

System of first order ODEs is solved numerically.• Step increase/decrease of v0 (1 μm/s 10 μm/s 1 μm/s)• Confining eff. pressure of 150 MPa• a = 0.010• b = 0.006• dc = 0.02 mm• ε = 1.7e-4

The drained case (p = const) has already been solved (Ruina’s spring slider model yields a kcrit and system stability behavior can be analyzed). Now allow pressure to be a system variable and perform a linear stability analysis for an undrained system.Linearize the system of equations about the steady state condition.At equilibrium Fspring= Ffric. resistance :

Assume solutions of the form plug into the linearized equations, generate a characteristic equation for s, and solve for the roots.

€

Δp = Pest , Δθ = Θest , Δv = Ve st , Δφ = Φest ,

The system has stable slip if all Re(s) < 0 and unstable slip if any Re(s) > 0.

Stability Analysis of an Undrained System

€

σ −p( )μ v,θ( ) = k v∞t − u( )

€

σ −p( )a

v∞ Δ˙ v = − σ − p( )bv∞

dc

Δ ˙ θ + μ ssΔ˙ p − kΔv

Δ ˙ θ = −v∞

dc

Δθ −1

v∞ Δv

Δ ˙ φ = −v∞

dc

Δφ +ε

dc

Δv

Δ˙ p = −Δ ˙ φ

β− c * Δp

Δ˙ u = Δv

€

σ −p( )μ v,θ( ) = k v∞t − u v( )[ ]

˙ θ θ ,v( ) = −v

dc

θ −θ ss( )

˙ φ φ,v( ) = −v

dc

φ − φss( )

˙ p p,φ,v( ) = c * p∞ − p( ) −˙ φ

β

˙ u v( ) = v

Similar to Ruina’s drained stability analysis, a kcrit is found and is given by

Linear Stability Analysis Results

€

kcrit =σ − p( ) b − a( )

dc

−εμ ss

βdc

F c *( )

F c *( ) =1+ λ + γ

2−

1+ λ + γ( )2

4− γ

⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥

λ =β σ − p( )a

μ ssε

ξ 2

ξ +1

γ =β σ − p( ) b − a( )

μ ssε

1

ξ +1

ξ =c * dc

v∞ =κdc

υ βL2 and c* =c

L2 =κ

υβL2

Value c* represents the ratio of permeability κ and the product of viscosity and the lumped parameter, υβ. The units of c* are ~ 1/t. The ratio v∞/dc is the inverse of θss is ~ 1/t. Thus ξ is the ratio of the characteristic time for state evolution to characteristic time for pore fluid diffusion. Note that if c* ∞, γ 0, and F(c*) 0 and recovers kcrit drained.For values of k slightly smaller than kcrit a velocity perturbation causes decaying oscillations in stress, sliding velocity, porosity, and pore pressure and the converse for values slightly larger than kcrit. Persistent oscillations exist for k = kcrit.

Undrained Drained€

Drained 0 ≤ F c *( ) ≤1 Undrained

Numerical Simulations of Governing Equations

• Results show that decaying oscillations exist for case A (k/kcrit = 1.05) Stable

• Slowly growing sustained oscillations exist for case B (k/kcrit = 0.95) Limit cycle

• Slowly growing sustained oscillations exist for case C (k/kcrit = 0.75) Limit cycle

• Seeing larger magnitude sustained oscillations for case D (k/kcrit = 0.40) Limit cycle

• Results show that oscillations rapidly increase for case E (k/kcrit = 0.30) Unstable

Nucleation Size hcrit in a Continuum: Relation to kcrit

Extending this 1-DOF spring-slider system to a continuum gives a feel of how the system spring stiffness, k, relates to the critical crack length, hcrit.

Dh

τ

τ

Δδ

The stiffness of a patch in a continuous media is given by:

€

k =Δτ

Δδ

The strain is proportional to:

€

ε ~Δδ

hThe change in shear stress goes as:

€

Δτ ~ Gε

€

k ~Gε

hε=

G

h

Drained case (p = const)

€

hcrit ~dcG

σ − p( ) b − a( )

Undrained case (fluid trapped)

€

hcrit ~dcG

σ − p( ) b − a( ) −εμ ss /β

€

kcrit ~G

hcrit

€

h > hcrit Unstable

h < hcrit Stable

• As p increases, (σ – p) decreases and hcrit increases• If hcrit is too large no instability can occur• Undrained case has even larger hcrit

In Sleep and Blanpied (1992) as p σ, (σ – p) 0 and hcrit ∞

Quantitative Analysis of hcrit

• Sleep and Blanpied (1992) undrained case

Question: if we would like hcrit small enough, how small can (σ – p) be?

Using the values of parameters from Segall and Rice (1995)

€

σ −p( )

€

=

€

dcG

hcrit b − a( )

€

εμss

β b − a( )

€

+Any hcrit Any dc What does (σ – p) have to be?

hcrit < 1 km dc= 10 μm (σ – p) > 87 MPa

hcrit < 1 km dc= 1 mm (σ – p) > 100 MPa

hcrit < 1 m dc= 10 μm (σ – p) > 210 MPa

ε (dilatancy coefficient) 1.7 x 10-4

μss (steady state friction coeff.) 0.64

β (compressibility lumped parameter) 5 x 10-4 MPa-1

(b – a) (rate and state law coefficients) 0.0025

G (shear modulus of porous material) 3 x 104 MPa