Collective properties of injection‐induced earthquake sequences: 2. Spatiotemporal evolution and magnitude frequency distributions

The problem

Injecting fluid underground can trigger earthquakes on faults that are close to failure (Raleigh et al., 1976). Seismic hazard assessment requires an estimate of how many of these earthquakes are large compared with small ones.

This is usually described by the Gutenberg-Richter model, in which the b value sets the relative number of small and large earthquakes. For b = 1, there are ten times fewer earthquakes for each step of one in magnitude. Tectonic earthquakes have b values of 0.8 to 1.2 (Schorlemmer et al., 2005). Some induced sequences have higher values, up to 2.0 during hydraulic fracturing of shales (Zhou et al., 2013). However, some wastewater disposal sequences show lower b values during injection (Lei et al., 2008; Huang and Beroza, 2015). Our goal was to determine how these differences arise from the interaction between rising fluid pressure and rupture on faults with uneven stress.

What we did

We used the earthquake sequence model developed in the first paper of this two-part series (Dempsey and Suckale, 2016). Each fault is a line, with shear stress that varies along it according to a fractal model. An earthquake starts where shear stress first reaches the static strength of the fault, as stress rises or as fluid pressure lowers the strength. Two rupture tips then spread outward and stop where the energy released is less than the energy needed to continue.

A tectonic reference catalogue, loaded by a uniform increase in shear stress, gave b = 1.15. We compared it with a uniform rise in fluid pressure, with injection directly into a fault, and with injection into an aquifer above impermeable basement rock cut by a permeable fault. We then modelled the 2010–2011 earthquakes near Guy and Greenbrier, Arkansas, linked to wastewater disposal at two wells.

What we found

Triggering the same earthquakes by fluid pressure in place of tectonic stress raised the b value from 1.15 to 1.53. This is because fluid pressure lowers the static strength, which controls when slip starts, faster than the residual strength, which controls how much stress slip releases. With less stress available to drive each rupture, ruptures are shorter and the earthquakes smaller.

As pressure first rose, the b value fell from 1.8 to 1.2. This is because the stress available to drive rupture is small at first and grows as pressure increases.

Faults with lower initial shear stress gave higher b values. Faults poorly oriented for slip in the regional stress field experience lower shear stress. We therefore expect them to have higher b values.

Injection directly into a fault with constant permeability gave b of about 2, with fewer large earthquakes than a Gutenberg-Richter model predicts. When we recomputed the same ruptures with pressure uniform along the fault, the deficit disappeared and b fell to about 1. This is because rupture is easier inside the pressurised zone than outside it. However, injection into an aquifer, the setting at Guy-Greenbrier, produced no deficit of large earthquakes. The model therefore does not explain the deficit observed there (Huang and Beroza, 2015).

The Guy-Greenbrier sequence (Horton, 2012) included 1317 catalogued earthquakes. Well 1 injected 3.9 × 10⁵ m³, a cube of water about 73 m on each side, into the top of the Ozark Aquifer. Well 5 injected 0.7 × 10⁵ m³ deeper in the same aquifer. Because Well 1 also injected into an overlying aquifer, we used two scenarios, with 100% or 10% of its volume reaching the Ozark Aquifer. An aquifer permeability of 10⁻¹³ m², ten times our value for a generic disposal aquifer, matched the roughly 100-day delay before seismicity began. In the 100% scenario, a fault permeability of 5 × 10⁻¹² m², 50 times higher again, matched the southwest migration of seismicity away from Well 1. The 10% scenario matched the migration less well.

However, the model did not reproduce the low b value of about 0.8 observed during injection. We matched it only by raising the average initial shear stress on the fault.

We also modelled each well alone, keeping the overall earthquake rate set to give 743 earthquakes of magnitude 1.6 or above with both wells. In the 100% scenario, Well 1 alone triggered 578 and Well 5 alone none. In the 10% scenario, Well 1 alone triggered 26 and Well 5 alone 358. Modelled pressure from the two wells is the sum of the pressure from each. However, an earthquake is triggered only once pressure exceeds a threshold, here 0.015 MPa, about the water pressure at the bottom of a 1.5 m deep pool. In our model, the seismicity from both wells together is therefore greater than the sum from each well alone. We suggest that the combined effect of neighbouring disposal wells be considered in assessing induced seismicity risk.

Limitations

Our model is one-dimensional. On a real fault, a rupture can pass around a low-stress barrier that would stop it on a line. We therefore draw qualitative conclusions only.

To obtain Gutenberg-Richter scaling we needed a fractal exponent of 0.25, against estimates of 0.7 to 1.0 for real faults. A higher exponent makes stress more similar over long distances and favours large earthquakes. One possible reason for needing the low exponent is that our model holds constant the energy needed to break the fault. This energy appears to increase with rupture size (Abercrombie and Rice, 2005). Including that increase would make large ruptures harder to grow and may allow a more realistic exponent.

Glossary

An induced earthquake is an earthquake triggered by human activity, here by raising fluid pressure in a fault.

The Gutenberg-Richter model states that the logarithm of the number of earthquakes above a magnitude decreases linearly with that magnitude.

The b value is the slope of the Gutenberg-Richter model. A higher b value means a higher proportion of small earthquakes.

Shear stress is the force per unit area acting to slide the two sides of a fault past each other.

Static strength is the shear stress needed to start slip on a fault.

Residual strength is the reduced strength of a fault while it slides during an earthquake.

A fractal model describes variation that has a similar pattern at every length scale.

A catalogue is a list of earthquakes with their properties, here modelled.

Permeability measures how easily fluid flows through rock.

An aquifer is a rock layer that fluid can flow through and be injected into.

Basement is the crystalline rock beneath the aquifer, here impermeable except where it is cut by a permeable fault.

The fractal exponent sets how similar stress is over long distances along a fault. A higher value favours large earthquakes.