Computational Earthquake Physics

Geodynamics

"The crucial point that was finally understood by the geological community is that both viscous (i.e., fluid-like) and elastic (i.e., solid-like) behaviour is a characteristic of the Earth depending on the time scale of deformation. The Earth’s mantle, which is elastic on a human time scale, is viscous on geological time scales (>10 000 years) and can be strongly internally deformed due to solid-state creep. – (Gerya 2019)"

The dual viscous-elastic behaviour of the Earth can be demonstrated using the "silly putty", which jumps up like a rubber ball (acts like solid) when we drop it on the floor for a very short timescale, but demonstrates more fluid-like behaviors in a longer time period (few days/weeks).

See demonstration of the visco-elasticity here

Rock rheology

Rheology: the composite physical property characterizing deformation behavior of a material.

Rock rheology: includes several different deformation mechanisms, and is in general visco-elasto-plastic. Elastic properties are important to be taken into account on a relatively short time scale ($<10^4$ years) for fast processes like magma intrusion. On the other hand, at low temperature rocks can be subjected to localized brittle and plastic deformation.

Followingly is a small summary of different types of rheology. The major difference between them is the composition of the bulk deviatoric strain rate $\dot{\epsilon}_{ij}'$. For example, it is decomposed into 2 respective components for visco-plastic rheology as $\dot{\epsilon}_{ij}' = \dot{\epsilon}_{ij\text{(viscous)}}' + \dot{\epsilon}_{ij\text{(plastic)}}'$

• visco-plastic: a strain rate based formation is more suitable. Simplification by assuming that elastic effects are negligible and can be ignored on the long time scales.

• visco-elastic: a stress-based formulation is more suitable. Viscous and elastic rheological relations are combined under certain physical assumptions. Maxwell visco-elastic rheology is the most commenly used type. Definitions of shear and bulk moduli are modified in the equations. More see section 12.4 in (Gerya 2019)

• visco-elasto-plastic: a stress-based formulation is more suitable. It characterises the non-linear instantaneous response at higher stress levels or temperature

• poro-elastic: Biot's model and its validity at low stress level and negligible viscous relaxation

• poro-elasto-plastic:

• thermo-hydro-chemico-mechanical (THCM): thermal and chemical couplings to deformation (mechanics)

NOTE: in general, coupling among processes triggers non-linear interactions that may result in significant and spontaneous localization of flow, heat and deformation.

Plastic rheology

Assumption: Existence of an absolute shear stress limit $\sigma_\text{yield}$ for a body and after reaching this limit plastic yielding occurs.

Formulations based on the simplified Griffith theory:

Case 1: Dry rocks

\[\sigma_\text{yield} = \begin{cases} \sigma_c + \gamma_\text{int}P, & P > \frac{\sigma_c - \sigma_t}{1-\gamma_\text{int}} \text{ (confined fractures)} \\ \sigma_t + P, & P < \frac{\sigma_c - \sigma_t}{1-\gamma_\text{int}} \text{ (tensile fractures)} \end{cases}\]

Or more compactly

\[\sigma_\text{yield} = \max [0, \min(\sigma_c + \gamma_\text{int} P^{[t]}, \sigma_t + \gamma_\text{int} P^{[t]})]\]

Case 2: Fluid-bearing rocks

\[\sigma_\text{yield} = \begin{cases} \sigma_c + \gamma_\text{int}(P^{[t]}-P^{[f]}), & (P^{[t]}-P^{[f]}) > \frac{\sigma_c - \sigma_t}{1-\gamma_\text{int}} \text{ (confined fractures)} \\ \sigma_t + (P^{[t]}-P^{[f]}), & (P^{[t]}-P^{[f]}) < \frac{\sigma_c - \sigma_t}{1-\gamma_\text{int}} \text{ (tensile fractures)} \end{cases}\]

where $\sigma_c$ and $\sigma_t$ are respectively compressive length and tensile length such that $2 \leq \frac{\sigma_c}{\sigma_t} \leq 8$

Or using effective pressure for the formulation

\[\sigma_\text{yield} = \max [0, \min(\sigma_c + \gamma_\text{int} P_\text{eff}, \sigma_t + \gamma_\text{int} P_\text{eff})]\]

NOTE: Yield point - the point on a stress-strain curve that indicates the limit of elastic behavior and the beginning of plastic behavior wiki