Cross-diffusion waves in hydro-poro-mechanics

Abstract. We propose a non-local, meso-scale approach for coupling multiphysics processes across scale. The physics is based on discrete phenomena, triggered by local Thermo-Hydro-Mechano-Chemical (THMC) instabilities, that cause cross-diffusion (quasi-soliton) acceleration waves. These waves nucleate when the overall stress field is incompatible with accelerations from local feedbacks of generalized THMC thermodynamic forces that trigger generalized thermodynamic fluxes of another kind. Cross-diffusion terms in the 4 × 4 THMC diffusion matrix are shown to lead to multiple diffusional P- and S-wave equations as coupled THMC solutions. Uncertainties in the location of meso-scale material instabilities are captured by a wave-scale correlation of probability amplitudes. Cross-diffusional waves have unusual dispersion patterns and, although they assume a solitary state, do not behave like solitons but show complex interactions when they collide. Their characteristic wavenumber and constant speed define mesoscopic internal material time-space relations entirely defined by the coefficients of the coupled THMC reaction-cross-diffusion equations. For extreme conditions, cross-diffusion waves can lead to an energy cascade connecting large and small-scales and cause solid-state turbulence.

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Quasi-solitary multiscale cross-diffusion waves as a precursor to Earth instabilities

10.5194/egusphere-egu21-3584 ◽

2021 ◽

Author(s):

Klaus Regenauer-Lieb ◽

Manman Hu ◽

Qinpei Sun ◽

Christoph Schrank

Keyword(s):

Wave Energy ◽

Fluid Pressure ◽

Reaction Diffusion ◽

High Energy ◽

P Wave ◽

S Wave ◽

Cross Diffusion ◽

Diffusion Waves ◽

The Cross ◽

Meso Scale

We propose a mesoscopic thermodynamics approach for coupling multiphysics processes across scales in porous or multiphase media. In this multiscale reaction-diffusion formalism interactions of discrete phenomena at the local scale are seen as being subject to a larger scale Thermo-Hydro-Mechano-Chemical (THMC) thermodynamic force. When local interactions are incompatible with the large-scale thermodynamic stress field incompatibilities can arise which trigger accelerations resulting in meso-scale generalized thermodynamic fluxes of another (THMC) kind. The classical acoustic tensor localization criterion in plasticity theory is here understood as a standing wave solution of such acceleration waves. These classical zero-speed acceleration wave solutions are solitary waves, also known as solitons, and are interpreted in the reaction-diffusion formalism as self-diffusion dominated by harvesting all available energy from the cross-diffusional tails.The more general case of non-zero traveling wave speed solutions is related to the cross-diffusion coefficients between different macro- and meso-scale thermodynamic THMC forces and fluxes. These cross-diffusion terms in the 4 x 4 THMC diffusion matrix are shown to lead to multiple diffusional P- and S-wave equations as THMC coupled, time-resolved dynamic solutions of the equation of motion. We show that the off-diagonal cross-diffusivities can give rise to a new class of waves also known as cross-diffusion waves or quasi-solitons. Their unique property is that for critical conditions cross-diffusion waves can funnel wave energy into a soliton wave focus.Mathematically these solutions can be compared to events in ocean waves and optical fibers known as 'rogue waves' or 'high energy pulses of light' in lasers. In the context of hydromechanical coupling, a rogue wave would appear as a sudden fluid pressure spike on the future fault plane. This hydromechanically coupled fluid pressure P-wave instability is here interpreted as a trigger for the S-wave seismic moment release of a double couple dominated earthquake event. The proposed multiscale cascade of wave energy may apply to many other material instabilities.&#160;&#160;

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Cross-diffusion waves resulting from multiscale, multiphysics instabilities: application to earthquakes

Solid Earth ◽

10.5194/se-12-1829-2021 ◽

2021 ◽

Vol 12 (8) ◽

pp. 1829-1849

Author(s):

Klaus Regenauer-Lieb ◽

Manman Hu ◽

Christoph Schrank ◽

Xiao Chen ◽

Santiago Peña Clavijo ◽

...

Keyword(s):

Wave Energy ◽

Ocean Waves ◽

High Energy ◽

Reaction Diffusion Equations ◽

Shear Instability ◽

Small Scale ◽

Cross Diffusion ◽

New Class ◽

Diffusion Waves ◽

Theoretical Approaches

Abstract. Theoretical approaches to earthquake instabilities propose shear-dominated source mechanisms. Here we take a fresh look at the role of possible volumetric instabilities preceding a shear instability. We investigate the phenomena that may prepare earthquake instabilities using the coupling of thermo-hydro-mechano-chemical reaction–diffusion equations in a THMC diffusion matrix. We show that the off-diagonal cross-diffusivities can give rise to a new class of waves known as cross-diffusion or quasi-soliton waves. Their unique property is that for critical conditions cross-diffusion waves can funnel wave energy into a stationary wave focus from large to small scale. We show that the rich solution space of the reaction–cross-diffusion approach to earthquake instabilities can recover classical Turing instabilities (periodic in space instabilities), Hopf bifurcations (spring-slider-like earthquake models), and a new class of quasi-soliton waves. Only the quasi-soliton waves can lead to extreme focussing of the wave energy into short-wavelength instabilities of short duration. The equivalent extreme event in ocean waves and optical fibres leads to the appearance of “rogue waves” and high energy pulses of light in photonics. In the context of hydromechanical coupling, a rogue wave would appear as a sudden fluid pressure spike. This spike is likely to cause unstable slip on a pre-existing (near-critically stressed) fault acting as a trigger for the ultimate (shear) seismic moment release.

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Cross-diffusion waves resulting from multiscale, multi-physics instabilities: theory

Solid Earth ◽

10.5194/se-12-869-2021 ◽

2021 ◽

Vol 12 (4) ◽

pp. 869-883

Author(s):

Klaus Regenauer-Lieb ◽

Manman Hu ◽

Christoph Schrank ◽

Xiao Chen ◽

Santiago Peña Clavijo ◽

...

Keyword(s):

Wave Equations ◽

Companion Paper ◽

Reaction Cross ◽

Multiscale Approach ◽

S Wave ◽

Cross Diffusion ◽

Diffusion Waves ◽

Complex Interactions ◽

Time Space ◽

Acceleration Waves

Abstract. We propose a multiscale approach for coupling multi-physics processes across the scales. The physics is based on discrete phenomena, triggered by local thermo-hydro-mechano-chemical (THMC) instabilities, that cause cross-diffusion (quasi-soliton) acceleration waves. These waves nucleate when the overall stress field is incompatible with accelerations from local feedbacks of generalized THMC thermodynamic forces that trigger generalized thermodynamic fluxes of another kind. Cross-diffusion terms in the 4×4 THMC diffusion matrix are shown to lead to multiple diffusional P and S wave equations as coupled THMC solutions. Uncertainties in the location of meso-scale material instabilities are captured by a wave-scale correlation of probability amplitudes. Cross-diffusional waves have unusual dispersion patterns and, although they assume a solitary state, do not behave like solitons but show complex interactions when they collide. Their characteristic wavenumber and constant speed define mesoscopic internal material time–space relations entirely defined by the coefficients of the coupled THMC reaction–cross-diffusion equations. A companion paper proposes an application of the theory to earthquakes showing that excitation waves triggered by local reactions can, through an extreme effect of a cross-diffusional wave operator, lead to an energy cascade connecting large and small scales and cause solid-state turbulence.

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Cross-Diffusion Waves as a trigger for multiscale, multiphysics Instabilities: Application to earthquakes

10.5194/se-2020-149 ◽

2020 ◽

Author(s):

Klaus Regenauer-Lieb ◽

Manman Hu ◽

Christoph Schrank ◽

Xiao Chen ◽

Santiago Peña Clavijo ◽

...

Keyword(s):

Fluid Pressure ◽

Ocean Waves ◽

High Energy ◽

Reaction Diffusion Equations ◽

Shear Instability ◽

Critical Conditions ◽

Small Scale ◽

Cross Diffusion ◽

Diffusion Waves ◽

Theoretical Approaches

Abstract. Theoretical approaches to earthquake instabilities propose shear-dominated instabilities as a source mechanism. Here we take a fresh look at the role of possible volumetric instabilities preceding a shear instability. We investigate the phenomena that may prepare earthquake instabilities using the coupling of Thermo-Hydro-Mechano-Chemical reaction-diffusion equations in a THMC diffusion matrix. We show that the off-diagonal cross-diffusivities can give rise to a new class of waves known as cross-diffusion waves. Their unique property is that for critical conditions cross-diffusion waves can funnel wave energy into a quasi-stationary wave focus from large to small-scale. The equivalent extreme event in ocean waves and optical fibres leads to the appearance of rogue waves and high energy pulses of light in lasers. In the context of hydromechanical coupling, a rogue wave would appear as a sudden fluid pressure spike on the future fault plane. This is here interpreted as a trigger for the ultimate (shear) seismic moment release.

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