Description¶
Background¶
This notebook simulates dynamic earthquake ruptures and elastic wave propagation in one space dimensions (1D). Spontaneously propagating shear ruptures on a frictional interface in an elastic solid is a useful idealization of natural earthquakes. The conditions relating slip-rate and fault strength are often expressed as nonlinear friction laws.
During an earthquake, two sides of the fault, initially held in contact by a high level of frictional resistance, slip suddenly when that resistance catastrophically decreases, generating strong ground shaking which is carried by seismic waves to remote areas, far away from fault zones
This notebook describes, and implements, a high order accurate finite difference method for enforcing nonlinear friction laws, in a consistent and provably stable manner, suitable for efficient explicit time integration and accurate propagation of seismic waves in heterogeneous media with free surface boundary conditions.
The elastic wave equation¶
Consider two elastic solids separated by a fault at the interface $x = 0$. In each elastic solid, wave motion is described by the elastic wave equation, in velocity-stress formulation:
\begin{align}
{\rho} \frac{ \partial v}{\partial t} = \frac{ \partial \sigma}{\partial x}, {\quad} \frac{1}{\mu} \frac{ \partial \sigma}{\partial t} = \frac{\partial v}{\partial x},
\end{align}
with particle velocity ${v}$, stress field ${\sigma }$, shear modulus $\mu$, and density ${\rho}$.
Denoting the fields in the positive and the negative parts of the fault with the superscripts $\pm$, we introduce the jumps in particle velocities by $\lbrack{v}\rbrack = {v}^{+}-{v}^{-}$, and the total traction by $\sigma = \sigma_{0} + \Delta{\sigma}$, where $\sigma_{0}$ is the initial background shear traction and $\Delta{\sigma}$ is the traction changes on the fault. The conditions on the fault connecting the two elastic solids are
\begin{align}
\text{force balance}&: \quad \sigma^{+} = \sigma^{-} = \sigma, \\
\text{friction laws}&: \quad \sigma = \sigma_n\frac{f(V, \theta)}{V}\lbrack{v}\rbrack, \quad V = \left|\lbrack{v}\rbrack\right|.
\end{align}
Here $\sigma_n > 0$ is the compressive normal stress and $ f(V, \theta) \ge 0 $ is the nonlinear fricition coefficient relating to the fault's shear strength, $\tau = |\sigma| > 0$, and will be described in more detail below.
Friction laws¶
It is necessary to consider the role of friction in all of this because it modulates the stresses and slip speed as the separate sides of the fault slip past each other. The first thing to consider is that as the fault slips it experiences shear strength ${\tau}$ which is exerted by the normal stresses on the fault ${\sigma_n}$. The way these two variables are related is through the friction coefficient ${f}$: ${\quad} {\tau} = f \cdot {\sigma_n}$.
For the purposes of this notebook we consider two widely used models of friction, slip-weakening and rate-and-state.
Slip Weakening Friction¶
Slip weakening friction describes a friction behavior in which friction coefficient weakens linearly with slip. When the fault is at rest the friction coefficient is denoted by static friction coefficient ${f_s}$, when the fault is in motion the friction coefficient is denoted by the dynamic friction coefficient ${f_d}$. An earthquake begins when the load $\tau$ at a finite patch on the fault overcomes the peak frictional strength, $\tau \ge \tau_p$, ${\quad \tau_p = f_s \cdot \sigma_n}$. As the earthquake continues the fault surfaces slide relative to each other and the frictional coefficient evolves from linearly from static friction coefficient $f_s$ to the dynamic friction coefficient ${f_d}$. The fault is fully weakened when the slip reaches the critical slip ${D_c}$. This can be seen in the figure below which plots the friction coefficient as a function of slip.
Rate and State Friction¶
Rate and state friction is a constitutive law that was empirically found and is considered more realistic than slip weakening. It takes the form:
\begin{align} &\tau = \sigma_n \cdot f(V, \theta), \quad f(V, \theta) = [ f_0 + a \cdot ln(\frac {V}{V_0}) + b \cdot ln( \frac{V_0 \theta}{D_c})], \\ &\frac {d \theta}{dt} = G(V, \theta). \end{align}In this equation ${V}$ is the current slip velocity and ${V_0}$ is a reference slip velocity while ${f_0}$ is the steady state friction coefficient at the reference velocity: ${f_{V= V_0}}$. There is the state variable ${\theta}$, and the state evolution law $G(V, \theta)$. We consider specifically the aging law: ${ G(V, \theta) = 1 - \frac {V\theta }{D_c}.\quad}$ At steady state the state variable is proportional to the critical distance divided by velocity ${ \theta_{ss} = \frac {D_c}{V}}$
Here ${a}$ is the direct effect and is used to model how the system responds to velocity changes. ${b}$ is the evolution effect and describes the magnitude of the steady state friction. ${D_c}$ is the critical distance and corresponds to the slip length over which the evolution of ${a \to b}$ occurs. The relationship of ${a}$ and ${b}$ to the slip weakening friction variables is as follows: ${\quad b = \frac {df_s}{d(ln(t))} \quad, \quad (a-b) = \frac {df_d}{d(ln(V))} \quad}$ where ${t}$ is the amount of time of contact without an earthquake. It is interesting to notice that there is a relationship between slip velocity ${V}$ and dynamic friction ${f_d}$ that depends on the difference ${(a-b)}$. If ${(a-b)>0}$ it shows that there is a velocity strengthening relationship ${\frac {df_d}{d(ln(V))} > 0}$. Any rupture that enters this region will arrest and is called a stable region. If ${(a-b)<0}$ it shows that there is a velocity weakening relationship ${\frac {df_d}{d(ln(V))} < 0}$, this is where earthquakes can nucleate and grow and is called unstable. On the border between stable and unstable regions is a zone of conditional stability where ruptures can sustain propagation while not growing or terminating. This notebook only exists in the unstable zone.
The equations are discretized using the SBP-SAT finite difference scheme. Time integration is performed using the classical fourth order accurate Runge-Kutta method. For more elaborate discussions, we refer the reader to the references and the notebooks on the SBP-SAT method. A summary of the numerical implementations used in the this notebook is presentated in the flow-chart below.
Putting it together¶
In the parameter window you can modulate the material properties, run time, domain size, and CFL criterion. In addition, you can choose rate and state friction or slip weakening friction, notice the differences that occur with this choice. In the Calculations and plotting cell the resulting on fault tractions and slip velocities are solved for in the time_integrator.elastic_RK4 function, which also calls on the other associated functions. The bottom cell contains 5 output plots of the rupture behavior.
References¶
Byerlee, J. “Friction of Rocks.” Rock Friction and Earthquake Prediction, vol. 116, 1978, pp. 615–626., doi:10.1007/978-3-0348-7182-2_4.
Duru, Kenneth, and Eric M. Dunham. “Dynamic Earthquake Rupture Simulations on Nonplanar Faults Embedded in 3D
Geometrically Complex,
${\quad}$ Heterogeneous Elastic Solids.” Journal of Computational Physics, vol. 305, 2016, pp. 185–207., doi: 10.1016/j.jcp.2015.10.0215
Gustafsson, Bertil. High Order Difference Methods for Time Dependent PDE. Springer, 2008.
Kozdon, J. E., E. M. Dunham, and J. Nordström (2012), Interaction of waves with frictional interfaces using summation-by-parts difference operators:
${\quad}$ Weak enforcement of nonlinear boundary conditions, Journal of Scientific Computing, 50(2), 341-367, doi:10.1007/s10915-011-9485-3
Leeman, John. “Modeling Rate and State Friction with Python.” SciPy 2016. Austin, USA.
Scholz, Christopher H. “Earthquakes and Friction Laws.” Nature, vol. 391, no. 6662, 1998, pp. 37–42., doi:10.1038/34097.
Exercises¶
Explore the following questions by interacting with the variables in the parameter window.
- 1) Explore the differences between slip weakening and rate and state frictions. How does this alter the slip rate and Traction drop profiles?
- 2) How does altering the CFL change the model time? Does lowering it increase or decrease the amount of numerical artifacts?
- 3) Does changing the density alter the slip rate?
- 4) What do you notice when altering the order of accuracy in terms of computation time and accuracy of the model? Is the trade-off worth it for this model?
- 5) What is the relationship between fault slip and traction in both types of friction?
# Import necessary routines
import numpy as np
import matplotlib.pyplot as plt
import time_integrator
import rate
import utils
import timeit
plt.switch_backend("nbagg") # plots within this notebook
# Parameter cell
# All user defined parameters can be entered in this window.
tend = 2.0 # final time
cfl = 0.5 # CFL number
L = 15.0 # length of the domain (km)
nx = 601 # grid points in x
cs = 3.464 # shear wave velocity (km/s) (can be an array)
rho = 2.6702 # density [g/cm^3]
iplot = 5 # snapshot frequency
order = 6 # order of accuracy
fric_law = 'SW' #'SW = slip weakening', 'RS' = rate and state
# Initialization cell
# Do not alter
# spacing and material parameters
ny = nx # grid points in y
dx = L/(nx-1) # grid increment in x
dy = dx # grid increment in y
rho_l = rho # density [g/cm^3]
mu_l = rho_l*cs**2 # shear modulus [GPa]
Zs_l = rho_l*cs # shear impedance
rho_r = rho # density [g/cm^3]
mu_r = rho_r*cs**2 # shear modulus [GPa]
Zs_r = rho_r*cs # shear impedance
#Initialize the domains L R
y_l = np.zeros((nx, 1))
y_r = np.zeros((nx, 1))
# Initial particle velocity perturbation and discretize the domain L R
for j in range(0, nx):
y_l[j, :] = -L + j*dx # discrete domain
y_r[j, :] = 0 + j*dx # discrete domain
# Time stepping parameters
t=0 # initial time
dt = (cfl/cs)*dx # Time step
nt = int(round(tend/dt)) # number of time steps
# Initialize: particle velocity (v); and shear stress (s); and returns L
v_l = np.zeros((nx,1))
s_l = np.zeros((nx,1))
U_l = np.zeros((nx,1))
V_l = np.zeros((nx,1))
U_t_l = np.zeros((nx,1))
V_t_l = np.zeros((nx,1))
U_x_l = np.zeros((nx,1))
V_x_l = np.zeros((nx,1))
# Initialize: particle velocity (v); and shear stress (s); and returns R
v_r = np.zeros((nx,1))
s_r = np.zeros((nx,1))
U_r = np.zeros((nx,1))
V_r = np.zeros((nx,1))
U_t_r = np.zeros((nx,1))
V_t_r = np.zeros((nx,1))
U_x_r = np.zeros((nx,1))
V_x_r = np.zeros((nx,1))
# Initialize slip parameters
slip = np.zeros((1, 1))
psi = np.zeros((1, 1))
# Initial particle velocity perturbation and discretize the domain
for j in range(0, nx):
y_l[j, :] = -L + j*dx # discrete domain
y_r[j, :] = 0 + j*dx
# Boundary condition reflection coefficients L
r0_l = 1 # r=0:absorbing, r=1:free-surface, r=-1: clamped
r1_l = 1 # r=0:absorbing, r=1:free-surface, r=-1: clamped
# penalty parameters L
tau_11_l = 1
tau_12_l = 1
tau_21_l = 1
tau_22_l = 1
# Boundary condition reflection coefficients R
r0_r = 1 # r=0:absorbing, r=1:free-surface, r=-1: clamped
r1_r = 1 # r=0:absorbing, r=1:free-surface, r=-1: clamped
# penalty parameters R
tau_11_r = 1
tau_12_r = 1
tau_21_r = 1
tau_22_r = 1
if fric_law not in ('SW', 'RS'):
# Choose friction law: fric_law
# We use linear (LN: T = alpha*v, alpha >=0)
# Slip-weakening (SW)
# Rate-and-state friction law (RS)
print('friction law not implemented. choose fric_law = SW or fric_law = RS')
exit(-1)
if fric_law in ('SW'):
alpha = 1e1000000 # initial friction coefficient
slip[:,:] = 0.0 # initial slip (in m)
Tau_0 = 81.24+1.0*0.36 # initial load (81.24 in MPa), slight increase will unlock the fault
alp_s = 0.677 # stastic friction
alp_d = 0.525 # dynamic friction
D_c = 0.4 # critical slip
sigma_n = 120.0 # normal stress
# These are not needed for the slip weakening case
psi[:,:] = 0.0 # initial condition for the state variable in friction law
L0 = 1.0 # state evolution distance
f0 = 1.0 # referance friction coeff
a = 1.0 # direct effect
b = 1.0 # evolution parameter
V0 = 1.0 # reference slip rate
if fric_law in ('RS'):
alpha = 1e1000000 # initial friction coefficient
slip[:,:] = 0.0 # initial slip (in m)
Tau_0 = 81.24+0.1*0.36 # initial load (81.24 in MPa), slight increase will unlock the fault
psi[:,:] = 0.4367 # initial condition for the state variable in friction law
L0 = 0.02 # state evolution distance
f0 = 0.6 # referance friction coeff
a = 0.008 # direct effect
b = 0.012 # evolution parameter
V0 = 1.0e-6 # reference slip rate
sigma_n = 120.0 # background normal stress
# These are not needed for the rate and state case
alp_s = 1.0 # stastic friction
alp_d = 1.0 # dynamic friction
D_c = 1.0 # critical slip
friction_parameters = [fric_law, alpha, Tau_0, L0, f0, a, b, V0, sigma_n, alp_s, alp_d, D_c]
# 0 1 2 3 4 5 6 7 8 9 10 11
# Initial variables
Vd = [0] # initial velocity
Sd = [Tau_0] # background stress
T = [t] # time
Slip = [0]
# setting slip, slip rate, traction, and time vectors to empty sets
traction_vector = []
slip_vector = []
time_vector = []
sliprate_vector = []
# Calculations and plottings for on fault slip velocity and stress
# Initialize animated plot for velocity and stress
fig1 = plt.figure(figsize=(10,10))
ax1 = fig1.add_subplot(5,1,1)
line1 = ax1.plot(y_l, v_l, 'r', y_r, v_r, 'g')
plt.title('Off fault velocity')
plt.xlabel('x[km]')
plt.ylabel('velocity [m/s]')
ax2 = fig1.add_subplot(5,1,2)
line2 = ax2.plot(y_l, s_l, 'r', y_r, s_r, 'g')
plt.title('Off fault stress')
plt.xlabel('x[km]')
plt.ylabel('stress [MPa]')
ax3 = fig1.add_subplot(5,1,3)
line3 = ax3.plot(0,0,'g')
plt.title('Slip')
plt.xlabel('Time [s]')
plt.ylabel('Slip [m]')
ax4 = fig1.add_subplot(5,1,4)
line4 = ax4.plot(0,0,'g')
plt.title('Slip rate')
plt.xlabel('Time [s]')
plt.ylabel('Slip rate[m/s]]')
ax5 = fig1.add_subplot(5,1,5)
line5 = ax5.plot(0,0,'g')
plt.title('Traction')
plt.xlabel('Time [s]')
plt.ylabel('stress [MPa]')
plt.tight_layout()
plt.ion()
plt.show()
# Loop through time and evolve the wave-fields using ADER time-stepping scheme of N+1 order of accuracy
start = timeit.default_timer()
n = 0 # counter
for t in utils.drange (0.0, tend+dt,dt):
n = n+1
# compute numerical solution
time_integrator.elastic_RK4(v_l, s_l, v_l, s_l, rho_l, mu_l, nx, dx, order, y_l, t, dt, r0_l, r1_l, tau_11_l,\
tau_21_l, tau_12_l, tau_22_l,v_r, s_r, v_r, s_r, rho_r, mu_r, nx, dx, order,\
y_r, t, dt, r0_r, r1_r, tau_11_r, tau_21_r, tau_12_r, tau_22_r,\
slip, psi, slip, psi,friction_parameters)
# appending slip, slip rate, traction, and time vector through each time step.
traction_vector.append(Tau_0+s_l[-1,0])
slip_vector.append(slip[0,0])
time_vector.append(n*dt)
sliprate_vector.append(np.abs(v_r[0,0]-v_l[-1,0]))
# Updating plots
if n % iplot == 0:
for l in line1:
l.remove()
del l
for l in line2:
l.remove()
del l
for l in line3:
l.remove()
del l
# Display lines
line1 = ax1.plot(y_r, v_r, 'r',y_l, v_l, 'g')
ax1.legend(iter(line1),('Right', 'Left'))
line2 = ax2.plot(y_r, s_r+Tau_0, 'r',y_l, s_l+Tau_0, 'g')
ax2.legend(iter(line2),('Right', 'Left'))
line3 = ax3.plot(time_vector,slip_vector,'g')
ax3.legend(iter(line3),('slip'))
line4 = ax4.plot(time_vector,sliprate_vector,'g')
ax4.legend(iter(line4),('slip rate'))
line5 = ax5.plot(time_vector,traction_vector,'g')
ax5.legend(iter(line5),('traction'))
plt.gcf().canvas.draw()
plt.ioff()
plt.show()
# Simulation end time
stop = timeit.default_timer()
print('total simulation time = ', stop - start) # print the time required for simulation
print('spatial order of accuracy = ', order) # print the polynomial degree used
print('number of grid points = ', nx) # print the degree of freedom
