Collective properties of injection‐induced earthquake sequences: 1. Model description and directivity bias

The problem

Injecting fluid underground can trigger earthquakes on nearby faults, and several induced earthquakes in the years before 2016 exceeded magnitude 5. Raising the fluid pressure inside a fault lowers its frictional strength. The fault slips once that strength falls to the shear stress, the force per unit area pushing its two sides past each other. This condition is the Mohr-Coulomb failure criterion (Hubbert and Rubey, 1959; Raleigh et al., 1976).

We focused on directivity, which describes whether a rupture spreads in one direction from its starting point (unilateral) or in both directions (bilateral). We measured it with the directivity ratio, D, which takes values from −1 to 1, with 0 for a perfectly bilateral rupture (McGuire et al., 2002). We gave D a sign relative to the injection well: negative for ruptures travelling towards it, positive for ruptures travelling away. Our goal was to determine whether induced earthquakes prefer one direction, and how many observations it would take to detect that preference.

What we did

We represented a fault as a line, with shear stress that varies randomly along it according to a fractal model. An earthquake starts at the point of highest stress for tectonic loading, or at the first point where rising fluid pressure has weakened the fault enough. From this hypocentre, two rupture tips spread in opposite directions. A tip stops when it enters a low-stress section of fault that cannot supply the energy needed for further propagation.

We used three alternative scenarios for fluid pressure, the first two from reservoir simulations. In the first, fluid was injected directly into the fault and spread outward with constant permeability. In the second, a well injected into an aquifer. Below the aquifer was impermeable crystalline basement cut by a permeable fault, similar to wastewater disposal in the central United States. In the third, pressure was high inside a growing zone of fault whose permeability had been enhanced by injection, and fell sharply at its edge. We used this advancing front to approximate stimulation of an Enhanced Geothermal System.

We compared catalogues of induced earthquakes against a tectonic reference catalogue of one earthquake on each of 100,000 simulated faults. Magnitudes were scaled so that rupture of a whole fault gives magnitude 4.5. This value is within the range of interest for induced seismicity.

What we found

Ruptures in the tectonic catalogue clustered near D = −1 and D = 1. This agrees with directivity ratios reported for 42 large shallow earthquakes (McGuire et al., 2002), although the modelled preference is weaker. The distribution is symmetric about zero. This is because tectonic loading is uniform along the fault.

Hypocentres in the tectonic catalogue cluster near the middle of their ruptures. However, a hypocentre 40% of the way along a rupture gives D of about 0.53. For the same rupture, the proportion of unilateral rupture, E, which changes linearly with hypocentre position, is only 0.2. A small offset of the hypocentre from the middle therefore gives a large shift towards unilateral rupture.

Injection directly into a fault made the distribution asymmetric. More earthquakes started near the well, and hypocentres tended to be located at the end of the rupture furthest from the well. Ruptures therefore tended to propagate back towards the well. The average D of the catalogue, which we call the directivity bias, became negative. This is because high fluid pressure lowers the residual strength of the fault, the strength at which it slides during an earthquake. A rupture inside the high-pressure zone releases more stress and is more likely to continue propagating.

The bias was strongest for the advancing front, which raises pressure most between the hypocentre and the well. It was weaker for radial flow, and stronger for larger earthquakes. For a well off the fault, the bias was small and may be too weak to detect. Because the advancing-front bias depended on injection pressure relative to how close the fault was to failure, we suggest that a well-constrained bias gives an approximate measure of that closeness.

We tested detection by comparing random samples from each induced catalogue with the tectonic distribution, using a Kolmogorov-Smirnov test. For the advancing front, a sample of three modelled ruptures was enough to reject the tectonic distribution at 95% confidence. For radial flow, samples of 100 ruptures of all sizes were not enough. Restricting the sample to earthquakes above magnitude 2.4 reduced the number needed to 20. We therefore suggest that efforts to detect the bias focus on larger earthquakes, whose directivity is also easier to measure. Jonas et al. (2015) analysed three of the larger earthquakes triggered during the Basel Enhanced Geothermal System stimulation. These appear to have started at the edge of the stimulated volume and propagated back towards the well. This behaviour is consistent with our model.

Limitations

Our model is one-dimensional. On a real two-dimensional fault, a rupture can pass around a low-stress barrier that would stop it on a line. We therefore expect the model to agree qualitatively with real faults, although quantitative agreement is less likely.

The detection test assumes that tectonic directivity is symmetric. However, faults with different rock on each side may have a preferred rupture direction (Ben-Zion, 2001). Uneven loading, or the ground surface above a dip-slip fault, could also break the symmetry. In those cases the test could wrongly indicate induced seismicity. Applying it to dip-slip earthquakes would first require a better description of directivity in natural dip-slip earthquakes.

Glossary

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

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

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

The hypocentre is the point where an earthquake rupture starts.

Directivity describes whether a rupture spreads from its hypocentre in one direction (unilateral) or in two (bilateral).

The directivity ratio, D, measures directivity on a scale from −1 to 1, with 0 for a perfectly bilateral rupture.

The proportion of unilateral rupture, E, is a second directivity measure that varies linearly with the position of the hypocentre inside the rupture.

Directivity bias is the average directivity ratio of a catalogue, which is zero when no direction is preferred.

The Mohr-Coulomb failure criterion states that a fault slips when the shear stress on it reaches its frictional strength.

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

Permeability measures how easily fluid flows through rock.

An Enhanced Geothermal System is a geothermal reservoir created by injecting fluid at high pressure to make hot rock more permeable.

A Kolmogorov-Smirnov test checks whether a set of observations could have come from a given probability distribution.

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