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Dilatancy /Compaction and Slip Instabilities of Fluid Infiltrated Faults. Vahe Gabuchian GE169B/277 01/25/2012 Dr. Lapusta Dr. Avouac. Experimental results of frictional behavior for porous materials Constitutive equations stemming from experimental observations
Dilatancy/Compaction and Slip Instabilities of Fluid Infiltrated Faults
List of references used
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.
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.
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.
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.
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.
Needto model porosity changes with changes in system parameters (i.e. slip velocity, etc)
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.
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.
The model is quasi- static: inertial effects are ignored (mass of block ignored).
Rate and State frictioncoefficient
Frictional resistance depends on sliding velocity, v, and history of slip given by a state variable, θ. Allows for strengthening during no-slip conditions.
Stability analysis has been done for this system with p = const (drained case).
Large stiffness (high k) favors stable sliding. High pore pressure (large p) favors stable sliding.
Governing Equations for Fluid: Linking Dilatancy, ϕ, to Pore Pressure, p
Conservation of mass
Evolution of fluid mass
Compressibility of the fluid
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)
Neglecting poroelastic coupling
Defining ELASTIC PORE COMPRESSIBILITY
Can write as
A lumped parametermodel is derived by combining equations (4) (3) (2) (1).
Assume the pressure to follow a simpler model and introduce a length scale, L
Pore pressure satisfies diffusion equation in lumped parameter model with forcing term .
System of 5 equations with 5 variables
Steady state values of variables are:
System of first order ODEs is solved numerically.
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 anundrainedsystem.
Linearizethe 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.
The systemhas stable slip if allRe(s) < 0 and unstable slip if any Re(s) > 0.
Similar to Ruina’s drained stability analysis, a kcrit is found and is given by
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 θssis ~ 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.
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.
The stiffness of a patch in a continuous media is given by:
The strain is proportional to:
The change in shear stress goes as:
Drained case (p = const)
Undrained case (fluid trapped)
In Sleep and Blanpied (1992) as pσ, (σ – p) 0 and hcrit ∞