1. 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 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 used 1)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.

2. 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  μ ss and ϕ  ϕ ss ) 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

3. Need to 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. 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, d c, makes proposed form physically reasonable Constitutive Equations Based on Experimental Observation Approach 1 Approach 2 Dilatancy coefficient

4. 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 coefficient Frictional 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). 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). (1) (2) (3) (4) (5)

5. (4) A lumped parameter model is derived by combining equations (4)  (3)  (2)  (1). Governing Equations for Fluid: Linking Dilatancy, ϕ, to Pore Pressure, 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) Defining ELASTIC PORE COMPRESSIBILITY Can write as 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. Neglecting poroelastic coupling

6. System of 5 equations with 5 variables Full Set of Governing Equations (1) (2) (3) (4) (5) Steady state values of variables are:

7. 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 v 0 (1 μm/s  10 μm/s  1 μm/s) Confining eff. pressure of 150 MPa a = 0.010 b = 0.006 d c = 0.02 mm ε = 1.7e-4

8. The drained case (p = const) has already been solved (Ruina’s spring slider model yields a k crit 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 F spring = F fric. resistance : Assume solutions of the form plug into the linearized equations, generate a characteristic equation for s, and solve for the roots. The system has stable slip if all Re(s) 0. Stability Analysis of an Undrained System

9. Similar to Ruina’s drained stability analysis, a k crit is found and is given by Linear Stability Analysis Results 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∞/d c is the inverse of θ ss 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 k crit drained. For values of k slightly smaller than k crit a velocity perturbation causes decaying oscillations in stress, sliding velocity, porosity, and pore pressure and the converse for values slightly larger than k crit. Persistent oscillations exist for k = k crit. Undrained Drained

10. Numerical Simulations of Governing Equations Results show that decaying oscillations exist for case A (k/k crit = 1.05)  Stable Slowly growing sustained oscillations exist for case B (k/k crit = 0.95)  Limit cycle Slowly growing sustained oscillations exist for case C (k/k crit = 0.75)  Limit cycle Seeing larger magnitude sustained oscillations for case D (k/k crit = 0.40)  Limit cycle Results show that oscillations rapidly increase for case E (k/k crit = 0.30)  Unstable

11. Nucleation Size h crit in a Continuum: Relation to k crit 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, h crit. D h τ τ Δδ 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) As p increases, (σ – p) decreases and h crit increases If h crit is too large  no instability can occur Undrained case has even larger h crit In Sleep and Blanpied (1992) as p  σ, (σ – p)  0 and h crit  ∞

12. Quantitative Analysis of h crit Sleep and Blanpied (1992)  undrained case Question: if we would like h crit small enough, how small can (σ – p) be? Using the values of parameters from Segall and Rice (1995)