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Lake Loading Impact on the San Andreas Fault

SIO 234 - GeodynamicsBy: Bill Savran and Chris Trautner12 December 2012

• Lake Cahuilla (a.k.a Lake LeConte or the Blake Sea)– Periodically-filled freshwater lake extending from Indio to Cerro

Prieto (Mexico):– About 180km x 50km

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Background

(Luttrell et. al 2007)

Background

• Lake Cahuilla has undergone significant size and depth changes in the relatively recent past– Salton sea: current version – Caused by changes in the path of the Colorado river– Lake levels studied by Waters (1983) and Gurrola and

Rockewell (1996), among other researchers

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Problem Description

• The recurrence of the lake is similar to the recurrence of rupture on the SA and SJ faults:

• Could the Columb stress perturbations associated with the lake cycle induce seismcity?

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Flexural Model

• Stress perturbations due to lake filling may be due to:– Changes in flexural stresses– Changes in pore pressures

• The Luttrell study used a three dimensional model of an elastic plate overlying a viscoelastic halfspace– Previous work by Larson (1990) used point load solution for an

elastic halfspace from Love (1929)– Halfspace model showed poor fit to rebound data from current

shoreline measurements– 3D model in current work is verified by solution to 2-D flexure

problem

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Derivation of Elastic Plate Solution

• Want the response of the lithosphere to a boxcar-type water load as a check of the 3-D model:

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• Two methods to derive response:– Integration of the line load solution (T&S Eq. 3-130) – Solution of the differential equation for a step function loading

Symmetry

Derivation of Elastic Plate Solution

Line Load Approach

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Differential Equation Sol’n

H(x)V0

Σ

T&S Eq 3-130 (line load): Flexural differential equation:

Take Fourier:

Reorganize result for W(k):

In terms of a sum, then an integral:

a

V1=V0a

V0

Derivation of Elastic Plate Solution

Line Load Approach

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Differential Equation Sol’n

H(x)

Σ

Substitute definition for α:

To get:

Inverse transform:

Go surfing…..

Derivation of Elastic Plate Solution

Line Load Approach

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Differential Equation Sol’n

H(x)

Σ

Can expand the denominator:

Using Cauchy residue theorem for the relevant poles:

Derivation of Elastic Plate Solution

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Differential Equation Sol’n:

Simplifying:

For x>0

Extending solution into x<0, and recognizing solution should be antisymmetric (continuous slope at x=0):

Derivation of Elastic Plate Solution

• Bending moment is related to second derivative of vertical displacement

• Once displacement solution is known, stresses follow from T&S Eq. 3-86:

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Comparison to Thick Plate 3-D Solution

Results from Luttrell

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Presentation Results

Pore Pressure

• Water from Lake Cahuilla interacts earth’s surface creating additional stresses.

• If lake level increases by h, the resulting change in normal stress is:

• This gives a change in Coloumb stress of:

• 90m of lake level change seen in Lake Cahuilla results in Δσn= 500 kPa.

n wgh

c f wgh

Total Effect of Coulomb Stress

• We need to consider both:– Direct pore pressure– Indirect plate bending effects

• Fault location with relation to lake is also important.– Beneath the lake, ΔP causes a step increase in σc, but it will

decay after several viscoelastic relaxation times– Faults far away from the lake will only experience indirect effects

of Δσn.

– Faults running between high and low shorelines will experience time-dependent stress changes due to changes in water level.

Constraining Best-Fitting Model

• Several models were tested with varying plate thickness (H) and relaxation times (τm).– 0 km ≤ H ≤ 100 km

– 20 yr ≤ τm ≤ 200 yr

• Models are constrained with 21 GPS elevation measurements– WGS84 reference ellipsoid– Corrected to height above sea-level with GEOID03

• No single best-fitting model!

Additional Constraints

• Goodness of fit tests reveal a trade off between H and τm

– Thinner plate with more viscous half-space– Thicker plate with less viscous half-space

• How can we constrain these models more?– Examine vertical velocity each model predicts for present day.

• Anything more than 5 mm/yr would be observable at the SCIGN GPS sites.– Leaves two models:

(H = 25 km, τm= 70 yrs) & (H = 35 km, τm= 30 yrs)

Nice stats….

Stress at Lake Cahuilla

• Coloumb stress perturbations due to a single cycle of lake rise and fall.

• Lake starts to fill at 0τm

and falls at 10τm

• At 0τm pore pressure effect from lake loading is dominant signal.

• Plate bending response is fully relaxed at 6τm.

• Lithosphere is fully relaxed by 16τm.

• σc rate much more significant than Δσc.

So much stress…

Stress evolution

Main Differences between model predictions:

1.) Magnitudes tend to be higher for the shorter relaxation time

2.) Plate deformation effect is broader for the thicker plate.

Discussion of Relevance

• Coulomb stress change due to Cahuilla Lake loading is about 10 times smaller than tectonic loading between major earthquakes.

• However, it is of reasonable size to trigger an earthquake.

• We see Δσc=0.1-0.2 MPa triggered 1999 Hector Mine earthquake. (Freed and Lin, 2001).

• TL;DR: Lake loading is not large enough to make unloaded fault fail, but can trigger events late in cycle.

Pull the trigger…

• When do we expect to see events to be triggered?– Events are triggered at times of increasing Coloumb stress

(loading = stress from ~1 recurrence interval).– When could this happen?

• 1.) Shortly after lake level rise, due to pore pressure.

• 2.) In the time after a lake fall, due to fault normal extension.

• Does this actually mean the lake influenced the SAF?

Well… let us see.

• First, we need to know when the earthquake occurred relative to lake loading and unloading.

• Also, it’s expected the Δσc will cause smaller events (M < 5) in addition to the large events (M > 7)

• Unfortunately our paleoseismic data is not that accurate.

• HYPOTHESIS: “…that lake loads modulate the earthquake cycle is a tenable explanation rooted in well-understood physics and could be extended to include other vertical loads…” (Luttrell et. al)

What about the Salton Sea?

• Salton Sea Authority has proposed a dam to isolate the northern and southern sections of the lake. – Why is this?

• Control salinity

• Control water elevation

• This would cause drawdown of lake level. • Note to SSA: Think about earthquakes as well as

other hazards of a dam near a fault.

Conclusions

• This model is a drastic over simplification of real-earth rheology.– No depth dependent viscosity.– No variations in plate thickness, as expected in a complex

tectonic region.

• Over the last 1300 years lake cycles on Lake Cahuilla cause perturbed Coulomb stress 0.2-0.6 MPa within the lake and 0.1-0.2 MPa outside the lake.

• Coulomb stresses are an order of magnitude smaller than tectonic loading.

• Therefore, lake cycling is likely to modulate the earthquake cycle when faults are near failure.

References

1. Larson, K. M. (1990), Precision, accuracy, and tectonics from the Global Positioning System, Ph.D. thesis, 269 pp., Univ. of Calif., San Diego, La Jolla, Calif.

2. Love, A. E. H. (1929), The stress produced in a semi-infinite solid by pressure on part of the boundary, Proc. Roy. Soc. Lond. A, 228, 377– 420.

3. Luttrell, K., D. Sandwell, B. Smith-Konter, B. Bills, and Y. Bock (2007), Modulation of the earthquake cycle at the southern San Andreas fault by lake loading, J. Geophys. Res., 112, B08411, doi:10.1029/2006JB004752.

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