Presentation information

[E] Oral

S (Solid Earth Sciences ) » S-CG Complex & General

[S-CG48] Science of slow earthquakes: Toward unified understandings of whole earthquake process

Thu. May 30, 2019 1:45 PM - 3:15 PM Convention Hall A (2F)

convener:Satoshi Ide(Department of Earth an Planetary Science, University of Tokyo), Hitoshi Hirose(Research Center for Urban Safety and Security, Kobe University), Kohtaro Ujiie(Graduate School of Life and Environmental Sciences, University of Tsukuba), Takahiro Hatano(Earthquake Research Institute, University of Tokyo), Chairperson:Takahiro Hatano, Takehito Suzuki

3:00 PM - 3:15 PM

[SCG48-32] Solution Orbit for k and epsilon Model for Turbulent Flow in Porous Media

Keywords:porous media, k-epsilon model, solution orbit

Turbulent flow within porous media such as fault rocks can affect dynamic earthquake rupture processes and has attracted interests of many researchers. Among models treating the turbulent flow, the k-epsilon model has widely been employed, where k is the turbulent kinetic energy and epsilon is the dissipation rate of k. However, analytical treatment associated with the effect of the initial values of k and epsilon on the final state has not been performed.

We assume a one-dimensional homogeneous porous medium and isotropic turbulence. We also assume that the density of the fluid, rho, and the porosity, phi, are constant, and the ensemble averages of phi k and phi epsilon will be written as k and epsilon below, respectively. With these assumptions, it should be emphasized that exactly the same straight line given by the equation epsilon=ck uD k /\sqrt{K} is a nullcline common to both variables on the k-epsilon phase space, where ck is a constant number, uD is averaged Darcy velocity, and K is permeability. We can regard this line as a line attractor or repeller as observed in other systems such as Suzuki (2017). Moreover, the straight line epsilon=0 (k axis) is also a nullcline for epsilon.

Actually, the analytical form of solution orbits is given by epsilon=epsilon0 (k/k0)^C2, where C2 is a constant, 1.9. Using this, we show below that the common nullcline epsilon=ck phi uD k/\sqrt{K} is a line attractor for this solution orbit. First, we define Region I as the region 0< epsilon < ck uD k /\sqrt{K} on the phase space, and Region II as the region epsilon > ck uD k/\sqrt{K}>0. We also define the point (kf, epsilonf) as the point where the solution orbit passing the point (k0, epsilon0) crosses the nullcline. With these definitions, we have relations k0f and epsilon0f if (k0, epsilon0) is in the Region I. This occurs because epsilon is proportional to k on the nullcline, while epsilon is proportional to k^C2 on the solution orbit and C2 >1. We can also conclude k0>kf and epsilon0>epsilonf if (k0, epsilon0) is in the Region II. Second, we should emphasize that k and epsilon increase (decrease) with increasing time in the Region I (II), since \partial k/\partial t and \partial epsilon/\partial t are positive (negative). Therefore, if (k0, epsilon0) is in the Region I (II), the solution moves to the upper right (lower left), and is absorbed into the nullcline with the limit t \to \infty. We can conclude that the nullcline is a line attractor, not a repeller. The steady stable solution is given by (k, epsilon)=(kf, epsilonf).

Note that k and epsilon vanish with the limit t \to \infty for usual isotropic turbulent flow, even though the solution orbit for such flow is the same as one obtained here. Actually, the usual turbulent flow is described by the limit K \to \infty in the present model. The nullcline is k axis in such a case, and the Region I vanishes. Therefore, all the solutions are absorbed into the origin. The finite K enables the turbulence to be survive with t \to \infty for the homogeneous state.