Emergence and seismological implications of phase transition and universality in a system with interaction between thermal pressurization and dilatancy

A dynamic earthquake source process is modeled by assuming interaction among frictional heat, fluid pressure, and inelastic porosity. In particular, fluid pressure increase due to frictional heating (thermal pressurization effect) and fluid pressure decrease due to inelastic porosity increase (dilatancy effect) play important roles in this process. Two nullclines become exactly the same in the system of governing equations, which generates non-isolated fixed points in the phase space. These lead to a type of phase transition, which produces a universality described by the power law between the initial value of one variable and the final value of the other variable. The universal critical exponent is found to be $1/2$, which is independent of the details of the porosity evolution law. We can regard the dynamic earthquake slip process as a phase transition by considering the final porosity or slip as the order parameter. Physical prediction of phase emergence is difficult because the porosity evolution law has uncertainties, and the final slip amount is difficult to predict because of the universality. Finally, nonlinear mathematical application of the result is also discussed.

Figure 1

Model setup. The thermoporoelastic medium moves as symbols show. (See also Fig. 1 in Ref. [6].)

Figure 2

Solution orbits. The blue and red curves are absorbed into the line and point attractors, respectively. Small arrows denote the directions of solution movement with increasing time. Points $(0,v_{01})–(0,v_{04})$ are those where the solution orbits cross the $v$ axis.

Figure 3

Solution orbits. The blue and red curves are absorbed into the line and point attractors, respectively. (a) Case wherein the nullcline $C^{\mathrm{crit}}$ crosses the $\varphi$ axis once. (b) Case wherein the nullcline $C^{\mathrm{crit}}$ crosses the $\varphi$ axis more than once. Points $(0,v_{01})–(0,v_{07})$ are those where the solution orbits cross the $v$ axis. The dotted rectangles A and B describe the examples treated in Figs. 4 and 4, respectively.

Figure 4

$\mathrm{Down}\text{−}\mathrm{up}$ and $\mathrm{up}\text{−}\mathrm{down}$ transitions of $C^{\mathrm{crit}}$. Green lines describe line attractors, and a point such that indicated in green color represents a converged one. (a) The $\mathrm{down}\text{−}\mathrm{up}$ transition emerges on the underside of the $\varphi$ axis, and $C^{\mathrm{crit}}$ crosses the $\varphi$ axis. (b) The $\varphi$ axis is tangential to the $\mathrm{down}\text{−}\mathrm{up}$ transition. (c) The $\mathrm{down}\text{−}\mathrm{up}$ transition occurs above the $\varphi$ axis. (d) The $\mathrm{up}\text{−}\mathrm{down}$ transition emerges on the upside of the $\varphi$ axis, and $C^{\mathrm{crit}}$ crosses the $\varphi$ axis. (e) The $\varphi$ axis is tangential to the $\mathrm{up}\text{−}\mathrm{down}$ transition. (f) The $\mathrm{up}\text{−}\mathrm{down}$ transition occurs on the underside of the $\varphi$ axis.

Figure 5

Cases wherein the $\varphi$ axis is tangential to $C^{\mathrm{crit}}$. Green lines describe line attractors, and a point such that indicated in green color represents a converged one. The values $n$ and $g^{\left(n\right)}\left({\varphi }_c^m\right)(\ne 0)$ are (a) odd and negative, (b) odd and positive, (c) even and positive, and (d) even and negative. The points $({\varphi }_c^m,0)$ are (a) the left end point of the line attractor, (b) the right end point of the line attractor, (c) the converged line attractor, and (d) neither the left nor the right end points of the line attractor.

Figure 6

General form of $C^{\mathrm{crit}}$. The $m\mathrm{th}$ smallest positive solution of $g\left(\varphi \right)=0$ is described as ${\varphi }_c^m$. The equation is assumed to have $N$ different solutions, where $N$ is a natural number. In this example, ${\varphi }_c^3$ is a multiple root of $g\left(\varphi \right)=0$.

Figure 7

Physically prohibited region. The light blue curves cross $({\varphi }_c^m,0)$. The solid parts of the orbits are physically meaningful, whereas the dotted parts represent intervals of the mathematical solution which do not have a real physical meaning. The physically meaningful orbits cannot approach the region labeled as PPR, even though the region is on the line attractor.

Figure 8

Solution orbits whose configuration is an enlarged version of Fig. 2 in the vicinity of $({\varphi }_{\mathrm{right}}^1,0)$. The light blue curve represents the critical manifold. The blue curves illustrate the solution orbit absorbed into the first line attractor. The red curve indicates the solution orbit absorbed into the point attractor. Short black arrows show the direction of solution movement with increasing time. Two variables, $\delta v_{+}^1$ and $\delta {\varphi }_{+}^1$, for two orbits are also shown. See details in the text.

Figure 9

Power law. (a) Relations between $\delta v$ and $\delta u$, and their dependence on the values of $(S_u,T_a)$. The values of $(S_u,T_a)$ are $(1.5,2)$ (red), $(2,3)$ (blue), and $(2.5,6)$ (green). The solid lines are numerically obtained relations, whereas the dotted lines describe Eq. (41). The numerical curves nearly overlap the analytical ones. (b) Relations between $\delta v/v_c$ and $\delta u/u_c$. Values of $v_c$ and $u_c$ are calculated based on Eqs. (38) and (40), respectively. (c) Log-log scale version of (b).

Figure 10

The case wherein the solution orbits move vertically on the straight line $\varphi =1$. The meanings of the red and blue curves and small arrows are the same as those in Fig. 2.

Figure 11

Left end point. The solution orbit illustrated by the light blue curve indicates a solution orbit crossing the point $({\varphi }_{\mathrm{left}}^2,0)$. The dashed parts of the orbits correspond to mathematical and unphysical solutions.