Chapter 7

COULOMB STRESS CALCULATIONS

Here we calculate static stress changes caused by the displacement of a fault or dike or point source (we refer to these as ‘source faults’), derived from the strain field shown in previous sections. We resolve the shear and normal components of the stress change on grid node points or on specified ‘receiver’ fault planes. Receiver faults are planes with specified strike, dip, and rake, on which the stress changes caused by the source faults are resolved. The shear stress change (an increase or decrease) is dependent on the position, geometry, and slip of the source fault, and on the position and geometry of the receiver fault geometry (including its rake). The normal stress change (clamping or unclamping) alone is independent of the receiver fault rake.

We use the Coulomb failure criterion, Ds_f = D t_s + m’ D s_n, in which failure is hypothesized to be promoted when the Coulomb stress change is positive. Here, Ds_f is the change in failure stress on the receiver fault caused by slip on the source fault(s), D t_s is the change in shear stress (reckoned positive when sheared in the direction of fault slip), D s_n is change in normal stress (positive in unclamping of the fault), and m’ is effective coefficient of friction on the fault.

We’ll explore three kinds of receiver faults: (1) receiver faults listed in the input file with no slip, (2) focal mechanism files, and (3) faults optimally-oriented for failure. Their optimal orientations are a function of the regional stress, the stress change associated with the source fault, and the assumed friction coefficient.

On the following page are Figures 2 and 3 from King et al. [1994], which graphically present the Coulomb stress change resolved on vertical strike-slip faults parallel to the source fault (Fig. 2a); and resolved on optimally oriented planes (Fig. 2b) for a given regional uniaxial compression and friction coefficient. In Figure 3, the influence of the regional stress magnitude is seen on the orientation of the optimal planes, and on the stress change resolved on these planes. These figures were made in a primordial version of Coulomb.

Illustration of the Coulomb stress change (Fig. 2 from King et al [1994]). The panels show a map view of a vertical strike-slip fault embedded in an elastic halfspace, with imposed slip that tapers toward the fault ends. Stress changes are depicted by graded colors; green represents no change in stress. (A) Graphical presentation of equation 8 of King et al (1994), a “specified fault” calculation. (B) Graphical presentation of equation 13 of King et al (1994), for optimally-oriented strike-slip (“opt strike-slip”) faults.

Dependence of the Coulomb stress change on the regional stress magnitude, s^r, for a given earthquake stress drop, Dt (Fig. 3 from King et al. [1994]). If the earthquake relieves all of the regional stress (left panel), resulting optimum slip planes rotate near the fault. If the regional deviatoric stress is much larger than the earthquake stress drop (right panel), the orientations of the optimum slip planes are more limited, and regions of increased Coulomb stress diminish in size and become more isolated from the source fault. In this and subsequent plots, the maximum and minimum stress changes exceed the plotted color bar range (in other words, the scale is saturated).

7.1 Coulomb stress change on specified receiver faults (‘Specified faults’)

This is the simplest calculation, and is widely used by researchers. To resolve the stress, you need to specify the fault strike, dip, and rake of the receiver planes following the Aki & Richards convention, shown graphically in the next page of this manual.

1. Launch Matlab/Coulomb 3.1.

2. Choose Input > Open existing input file. Then in the Open input file window, choose the “input_files” sub-folder within the “Coulomb 3.1” folder, and select “Example-2(LL)-lonlat.inp”

3. Choose Functions > Stress > Coulomb stress change.

4. You will see the pop-up Stress control panel.

5. To calculate stress changes on specified receiver fault planes, click “Specified faults”. Coulomb averages the information on all input fault patches listed in the input file and puts the values in the boxes, but you can change them. Strike, dip, and rake are defined following the conventions of Aki & Richards (1980, 2002):

For practice, choose a strike/dip/rake of 360°/90°/0°, change the friction coefficient to 0.0, set the stress-change color saturation to ±5 bars, and hit ‘Calc. & View’.

The Coulomb stress changes default using our “Anatolia” color scheme. Other schemes (“Rainbow” or “Black & White”) can be changed by pulling down the Input > Preferences menu bar and clicking on “Color Map”. Then re-run the input file.

7.2 Using the strike/dip/rake/friction slider (Specified slip control panel)

To vary these parameters on the fly, hit the > button in the Stress control panel, and the Specified slip control panel pops up (see it above). Slide the balls to explore how the stress pattern changes.

7.3 Saving the graphic and numerical output of stress calculations

To save this graphic, File > Save As… > choose a .pdf format and rename it; its only 40 kb but is a full vector image. Stress change is calculated in the lower left corner of each grid square at the target depth specified in the input file. Calculated values of stress change may saturate (exceed the plotted range), so experiment with the slider to see more subtle features. The stress-scale legend is to the right of the plot.

Every time you click “Cal. & View” in the Stress control panel, a numerical output file called “dcff.cou” will be created or updated and saved in a sub-folder you designate in the Preferences menu. Remember to rename “dcff.cou” if you want to save the numerical file; otherwise they will be overwritten.

7.4 The importance of the receiver fault geometry in Coulomb modeling

In the figure below, from Lin & Stein (JGR, 2004), the source is an idealized M~7.9 right-lateral southern San Andreas rupture. The most familiar stress pattern is the case of source and receiver faults with the same strike, dip and rake (a). But look how strongly the pattern changes when the receiver faults are nearby thrusts (b-c) or left-lateral faults (d). Thus, a stress shadow for one receiver may be a stress trigger zone for another receiver fault. Note also that we tend to assume a high coefficient of friction (~0.8) for continental thrust faults, moderate friction (0.4) for strike-slip or unknown faults, and very low friction (>0.2) for major transforms, such as the San Andreas.

7.5 Adding Coulomb stress to an overlay plot and viewing it in 3D

1. Launch ‘Example-SFBayArea.inp’. Overlay > Coastlines. Open the ‘coastline data’ folder in the Coulomb30 folder, and select ‘california_coastline_di_neg.dat’ Answer the pop-up question, ‘no’ (some datasets treat western longitudes as negative; others do not). Now, Overlay > Active faults. Choose ‘California faults_longlat_dat’.

2. You can now add any kind of stress changes to this plot without losing any of the overlays. Just hit Functions > Stress > Coulomb stress change, and select a stress component and option, and you will see a map view image such as that below left.

3. After completing any Coulomb stress calculation made over the grid, such as Coulomb stress change, Shear stress change, or Normal stress change, just return to the MATLAB window and type in "coulomb_3d_view". This calls a plugin (which can be found in the plugin folder in Coulomb30 folder). It executes a 3D view that you can manipulate and save as a pdf file, such as that above right.

7.6 Coulomb stress cross-sections

1. Choose Input > Open existing input file > Example-2(TH).inp. Next, choose Functions > Stress > Coulomb stress change. Change the color saturation to 3.0 bars, and hit ‘Calc. & View’ in the Stress control panel. After the mapview plot appears, hit the Cross section button on the Stress control panel, which displays the cross section parameters from your input file (left plot below). Click “Calc. & View” and you will see the image below left. Now change the saturation to ±10 bars and chose interpolated shading, and hit “Calc. & View” again (right panel below).

Mosaic, ±3 bars Interpolated, ±10 bars

2. When building a cross-section, the “Start (A)” point should always be on the left side of the “Finish (B)” point (see below right). You can overlay seismicity in cross-sections if you first plot the mapview with earthquakes and then make the section. The default setting plots seismicity within ±20 km of the section line. To change this, print EQPICK_WIDTH=5 for ±5 km, etc, in the command window.

2. You can also change the dip angle of the cross section by entering a new value in the “Dip” box in the above panel. The blue dashed line is the depth at which stress was sampled in the map view. The red line is the intersection of the fault plane with the cross-section line. These can be seen in map view, which has a blue cross-section line, a red fault perimeter, a green line where the fault projects to the ground surface, and a black line where the fault intersects the depth at which stress is being sampled. You can change the depth, and many other parameters, in the Stress control panel.

4. You can calculate the maximum or mean values of Coulomb stress changes between various depth ranges. Click “Depth range” and then “Cal. & View” in the Stress control panel, you will see Depth control panel. In the Depth control panel, enter depths of the top and bottom surfaces, as well as the depth increment, for which you want to perform the comparisons. You can calculate either “Maximum values” of Coulomb stress changes over the given depth range (below left), or the “Mean values” over the range (below, right). The numerical output file shows the maximum value at each grid point. The maps below were made using the left-lateral strike-slip fault input file, Example-2(LL)-lonlat.inp.

Max stress change over a depth range Mean stress change over a depth range

5. You can plot the orientation of the ‘specified fault’ strike in map view. Choose Input > Open existing input file > Example-2(LL).inp. Next, choose Functions > Stress > Coulomb stress change. Check the “Strike line” box in the Stress control panel, and for practice, change friction to 0.0, shading to interpolatedand then “Calc. & View”, resulting in the following image on the left. Notice that the lines strike 41° as specified. Stresses and slip lines are plotted on the lower left corner of each grid box.

7.7 To display the principal stress axes rotated by the earthquake stress change

Coulomb makes the tensor addition of the earthquake stress change, which diminishes with distance from the source fault, and the regional stress, which is assumed uniform, to calculate the total principal stress axes. Once the total principal stress axes are determined, Coulomb uses this and the assumed friction coefficient to determine the optimum planes.

Choose Input > Open existing input file > Example-2(LL).inp. Then, Functions > Change parameters > Grid, and change the grid to 5.0 x 5.0 km so the axes are not too crowded (this is not a requirement; you can choose any grid spacing of interest). Now, Functions > Stress > Coulomb stress change. In the pop-up Stress control panel, choose “Opt. Strike S.”, “Principal Stress”, and “Interpolating”, and click “Calc. & View”. You will see the image on left below. Using Tools > Zoom In, you will see the right image below.

Normal view with principal axes Zoomed view

The axes are not scaled by their stress magnitudes, since too many would be invisible. Notice that the axes rotate in 3D, not just in the horizontal plane, and so s₃ deviates from being vertical close to the source, as a result of the stress changes imparted by the earthquake. This means that s₃ can be seen in the lower left corner of the zoomed image.

7.8 Using the regional or ‘tectonic’ stress in optimally-oriented stress calculations

The regional stress is used only when you choose Functions > Stress > Coulomb stress changes AND choose one of the optimally-oriented (“Opt.”) stress changes in the Stress control panel. Otherwise the regional stresses are ignored (see below). See King et al (BSSA, 1994) for more on this topic.