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

The theory presented in this paper grew out of the conference series dedicated to understanding coupled thermo-hydro-mechanical-chemical (THMC) in Geosystems (GEOPROC). The 7th International Conference on Coupled THMC Processes was held in 2019 in Utrecht and focussed on earthquake and faulting mechanics (as does this special issue). Integration of mechanical, hydrodynamical, thermal, and chemical processes covers, however, a much wider field from the pore to plate-tectonic scale for a wide range of natural and engineering problems in geological systems discussed in focus topics at earlier GEOPROC conferences. These problems include nuclear waste disposal, coal seam gas, enhanced oil and gas recovery, geothermal energy, mineral deposits, tailing dam collapse, landslides, and many others. The individual problems may have their own characteristics. However, the common scientific issue of multiscale feedback of THMC processes remains the same.

The GEOPROC theme seeks to foster the urgently needed growth of experimental, numerical, and theoretical studies on multi-physics (THMC) and multiscale framework studies in earth sciences. The current practice is to still use engineering solutions based on empirical material laws to address specific natural and engineering problems in geological systems and energy production in geothermal energy, nuclear waste disposal, reservoir engineering for oil and gas, the formation of mineral deposits, induced seismicity, natural hazards, and

Part of the reasons for the lack of a wider adoption of coupled THMC approaches in the community is a lack of a theoretical basis on which to assess the rich solution space that arises from a coupling of the four (THMC) partial differential reaction–diffusion equations. While parallel numerical tools for modelling fully coupled non-linear systems of THMC equations have become available through pioneering work in nuclear engineering

Before discussing a possible application of the new theory to the processes of earthquakes and faulting in our companion article

While applied mathematical solutions exist, the preference in geosciences is to address the problem of unbounded growth by explicit consideration of additional physics. A specific case was shown where the infinite response can be captured by postulating a carefully chosen chemical reaction

The dynamic field is of special interest to the researcher in the area of earthquake and faulting instabilities. The state of the art in this field is defined by the influential experimental work of

In this paper, we introduce the classical approach of acceleration waves in plasticity theory to the seismology community by starting with the Helmholtz decomposition of the seismic wave equation into P and S waves (see Sect.

There are two opposite starting points for the derivation of the approach. Here, we investigate the meso-scale from the conventional mechanical quasi-steady-state (infinite timescale) solution of the macro-scale. Although the present paper uses the macro-scale perspective, i.e. the classical mechanical viewpoint for the investigation of the physics of acceleration waves

In order to define the separation between the meso- and macro-scale of a coupled THMC problem, we propose that the scale for each of the THMC processes is defined by its own characteristic diffusion timescales and length scales

A particular challenge for deriving dynamic THMC coupled wave solutions is the discrete nature of the cascade of steady-state solutions defined by the standing-wave solutions of thermomechanics, which leads to a discrete material behaviour as discussed in the next section. Standard probability theory is therefore not suitable as this assumes a continuum of wave functions

The nucleation mechanism of these waves relies on the meso-scale open-system behaviour where the overall macro-scale thermodynamic forces can become incompatible with accelerations from local thermodynamic fluxes. These incompatibilities radiate wave energy away from its source in the form of “cross-diffusion” waves. The emergence of cross-diffusion waves can be perhaps best understood from a chemical viewpoint

The concept was introduced first in chemical and biological systems where morphogenic patterns

We propose here a generalized approach to cross-diffusion that is known in bio-physics as taxis

However, except for the seminal early work by Ortoleva and co-workers

By analogy to the mathematically similar biological and chemical systems, we propose here that earthquake instabilities are preceded (and followed in the post-seismic stage) by propagating THMC dissipative waves which could enable new detection methods if they can be resolved by sensors. We will discuss such possible precursor phenomena for earthquakes in the companion article

Dissipative patterns in chemical and biological systems: simulated and real patterns on pufferfish

While chemical oscillations thus appear to be well understood, the phenomenon of an oscillatory response is less well established in other THMC systems. However, these systems show the same transitions from a simple continuum response to a highly localized state. The existence of a discrete, particle-like nature has also been discovered in fluids when they are driven far from equilibrium. If driven far from equilibrium by surface forces, fluids clearly show (see Fig.

Dissipative patterns in fluid systems: water molecules exhibit a discrete quantum-like solitary state when forced by a mechanical shaker at a critical condition (here 41 Hz). Periodic finger-like solitary states travel from right to left at a constant velocity. Each snapshot shows 20 ms intervals. Unlike classical solitons their appearance is particle-like. They can pass through each other with a slight loss of amplitude, or “collide” to create a new state whose direction of propagation is at an angle to that of the original states or disintegrate upon collision (image from

For the case of deforming geomaterials, a theory for localization phenomena, characterized by a sudden transition from continuum deformation behaviour to a highly localized state, is well established

The dynamics of the formation of these mechanical dissipative patterns can therefore only be investigated using analogue materials in the laboratory. Analogue experiments have been performed in granular, brittle matter compressed uniaxially

Experiments with highly porous carbonates have been performed

We propose here that the dissipative wave phenomenon is universal for THMC reaction–diffusion systems that are driven far from equilibrium. The approach allows an interpretation of observations in nature and the laboratory in terms of propagating particle-like states which emerge as stationary Turing patterns for long-timescale standing-wave solutions of a THMC cross-diffusion formulation. In order to recover the dissipative wave equations, we present in the following the standard constitutive assumptions for any generic thermodynamic fluid or solid mechanical system and describe how the physics of THMC feedbacks can be implemented to resolve the phenomenon of propagating dissipative waves in these systems.

The fundamental equation of motion is

For an isotropic elastic medium, for instance, accelerations in Eq. (

Similarly, by allowing the material to deform in a viscous manner, acceleration can be monitored by a local change in velocity

We emphasize here, that although the Helmholtz decomposition can be performed in a similar way to derive volumetric and shear moduli that describe dissipative material behaviour, their response to infinitesimal perturbation is generally to dampen propagating elastic waves. One could, therefore, come to the erroneous conclusion that their contribution to precursory wave phenomena and to macroscopic failure is an overall suppression of instabilities.

For the elasto-viscoplastic case, we have the equivalent fourth-order elastic-viscoplastic stiffness tensor

They are an entirely different class of waves as they are based on dissipation in active kinetic systems in contrast to waves in conservative systems. For simplicity, we only discuss the slow viscoplastic wave phenomenon allowing the investigation of conservative and dissipative waves as different processes. For decoupling elastic and dissipative waves, we need to assume large differences in the propagation speed of the waves. This is done by assuming Maxwellian rheology, implying a separation of elastic and viscoplastic wave timescales in the context of an additive strain-rate decomposition of Eqs. (

The simplest implementation of the non-equilibrium approach of

Acceleration waves can be described in two ways. One can use two coordinate systems, one for the reference state and one for the current state. A more elegant way is to consider convective coordinate systems by formulating the constitutive law in terms of stress rate. For this we consider the space derivative normal to the moving wavefront (see Fig.

Considering the traction (load per unit area) in the direction normal to the wavefront as

Acceleration waves can originate at a body surface when the existing internal stress gradient is dynamically incompatible with accelerations imposed on particles of the surface. A propagating plane-wave front is shown here for reference, but a plane-wave is not a necessary restriction. Across these surfaces particle accelerations and spatial gradients of velocity are momentarily discontinuous while the velocity itself is continuous.

Acceleration waves form the basis of localization criteria in plasticity theory. The criterion for instability is derived from the equivalent theory in elastodynamics where for an elastoplastic body the acoustic tensor

These waves are interpreted as stationary (standing) waves when the determinant of the acoustic tensor, and consequently the wave speed is zero:

However, to date, no generally accepted localization criteria for the transition from a dynamic to quasi-static rate-dependent solution of Eq. (

We assume creeping flow in Eq. (

Hadamard's jump conditions state that if time derivatives (

Equation (

So far we have only discussed the mechanical reaction–diffusion equation, where the shear and bulk viscosities control the diffusion of stress. For the multi-physics implementation, it is convenient to think of diffusion of momentum and use the momentum diffusivity (kinematic viscosity) instead of the dynamic viscosity. We, therefore, denominate

In these formulations, the viscous (M) mechanical pressure diffusion equation finds its counterparts in the equivalent thermal (T) Fourier, (H) Darcy, and (C) Fick diffusion laws where the diffusion coefficients are indicated by the associated THMC subscript. The corresponding reaction rates are the local hidden-variable reaction rate

In the adiabatic limit we obtain similar reaction–diffusion equations across a vast range of THMC diffusion length scales as tabulated in Table 1. The reaction rates most often stem from different micro-processes at lower scale inside the considered continuum element which introduces cross-scale diffusion fluxes as shown in the next section.

In order to generalize the approach, we propose that all reaction–diffusion equations in Table 1 are strongly coupled. We construct a composite multiscale THMC diffusion wave operator

The wave operator

The discussion of these waves and their unusual properties will be the subject of the remainder of the paper. For introduction and completeness, however, the arguments for non-local reaction–diffusion equations with cross-diffusion terms

For the following discussion, we also simplify further and neglect the deviatoric terms in Eq. (

In the following we generalize the discussion of the meso-scale THMC mass exchange processes using mixture theory applied to HM coupling as presented in

We consider two mass fractions A and B for mass exchange denoted by the

Generalized thermodynamic fluxes and forces in a THMC coupled system.

In a saturated porous medium, a straightforward interpretation of

To illustrate the point of choosing a particular time–space scale of observation of THMC waves we first consider the simple homo-entropic flow assumption and choose a classical mechanics point of view. Density is then defined as a function of pressure, temperature, and chemical concentrations by the

The concept of cross-diffusion is well known in chemistry. In a chemical system with just two species A and B, for instance, cross-diffusion is the phenomenon in which a flux of species A is induced by a gradient of species B

Following Eq. (

A detailed discussion of the criterion for nucleation of cross-diffusion waves and their waveforms can be found in Tsyganov and Biktashev

The criterion for nucleation of cross-diffusion waves relies on assessing the dispersion relation of the eigenvalues of the characteristic matrix of a perturbed cross-diffusion–reaction equation

Since cross-diffusion waves in geomaterials are largely unexplored due to the extreme length scales and timescales encountered in a geosystem, an appreciation of their complex characteristics can be obtained from mathematically similar systems such as waves in oceans, lasers, and ice. There is an important difference between solitonic waves and quasi-solitonic cross-diffusion waves. We follow Zakharov et al.'s

The following dynamic properties of quasi-solitons have been identified

The discrete particle-like behaviour can be explained by their unusual dispersion relation. Quasi-solitons travel with a constant group velocity

In photonics, optical turbulence in the form of sporadic bursts of light

The independent choice of a reference system such as discussed in the convolution filter analogy also applies to the energy carried by the waves. If we choose for instance an observer of hydro-mechanical waves, the inverse energy cascade from THMC wave action from small to large scale and the direct energy cascade from large to small scales

This paper has introduced three important innovations for modelling THMC instabilities: (i) a multiscale extension of the theory of thermodynamics of irreversible processes to include dynamic events by using a meso-macro-scale model; (ii) a generalization of the theory of cross-diffusion waves from chemical systems to generalized THMC thermodynamic-force flux pairs; (iii) a transfer of knowledge from classical quantum mechanics to characterize any system at a larger scale in order to deal with the discreteness of multiscale material behaviour.

We have shown that cross-diffusion waves in THMC systems can be decomposed into cross-diffusional S and P acceleration waves and have discussed a THMC multi-physics implementation, where cross-diffusion waves appear as quasi-soliton waves for critical conditions identified from a perturbed Eq. (

The multiscale approach can be encapsulated in a concise fully populated self- and cross-diffusion matrix (Eq.

Cross-diffusion waves have first been discovered for interactions in quantum mechanics such as in photonics where they show anomalous dispersion patterns that, unlike solitons, radiate energy in the form of oscillatory (Cherenkov) tails

KRL together with CS secured research funding. KRL, MH, and CS developed the research plan. KRL prepared the first draft of the paper, and all authors contributed to subsequent revisions and interpretations of the proposed theory.

The authors declare that they have no conflict of interest.

This article is part of the special issue “Thermo-hydro-mechanical–chemical (THMC) processes in natural and induced seismicity”. It is a result of the 7th International Conference on Coupled THMC Processes, Utrecht, Netherlands, 3–5 July 2019.

We thank Emily and Anders Crofoot of Castlepoint Station (NZ) for their hospitality and access to the beautiful rocks on their property. Finally, we would like to acknowledge support from the Central Analytical Research Facility (CARF) of QUT.

This research has been supported by the Australian Research Council (grant no. ARC DP170104550), the Australian Research Council (grant no. DP170104557), and the University of New South Wales (grant no. SPF01).

This paper was edited by Andre R. Niemeijer and reviewed by Angelo De Santis, Jean Paul Ampuero, and one anonymous referee.