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

Solid Earth ◽  
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.

scholarly journals Cross-Diffusion Waves as a trigger for Multiscale, Multi-Physics Instabilities: Theory

2020 ◽  
Author(s):  
Klaus Regenauer-Lieb ◽  
Manman Hu ◽  
Christoph Schrank ◽  
Xiao Chen ◽  
Santiago Peña Clavijo ◽  
...  
Keyword(s):  
Wave Equations ◽  
Reaction Cross ◽  
S Wave ◽  
Cross Diffusion ◽  
Diffusion Waves ◽  
Complex Interactions ◽  
Time Space ◽  
Acceleration Waves ◽  
Non Local ◽  
Meso Scale

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.

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

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

<p>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.</p><p>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.</p><p>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.</p><p> </p><p> </p>

Pure P- and S-Wave Equations in Anisotropic Media

2020 ◽  
Author(s):  
A. Stovas ◽  
T. Alkhalifah ◽  
U. Bin Waheed
Keyword(s):  
Wave Equations ◽  
Anisotropic Media ◽  
S Wave

Wave characteristics in elliptical orthorhombic media

Geophysics ◽  
2021 ◽  
pp. 1-52
Author(s):  
Alexey Stovas ◽  
Yuriy Roganov ◽  
Vyacheslav Roganov
Keyword(s):  
Anisotropic Medium ◽  
Wave Equations ◽  
Reference Model ◽  
Relative Magnitude ◽  
P Wave ◽  
S Wave ◽  
Wave Characteristics ◽  
Singularity Point ◽  
Wave Properties ◽  
Wave Modes

An elliptical anisotropic medium is defined as a simplified representation of anisotropy in which the anelliptic parameters are set to zero in all symmetry planes. Despite of the fact that this model is rather seldom observed for real rocks, it is often used as a reference model. The P-wave equations for an elliptical anisotropic medium is well known. However, the S-wave equations have not been derived. Thus, we define all wave modes in elliptical orthorhombic models focusing mostly on the S-wave properties. We show that all wave modes in elliptical orthorhombic model are generally coupled and analyze the effect of additive coupling term. As the result, there is an S wave fundamental singularity point located in one of the symmetry planes depending on the relative magnitude of S wave stiffness coefficients.

Time-space-domain implicit finite-difference methods for modeling acoustic wave equations

Geophysics ◽  
2018 ◽  
Vol 83 (4) ◽  
pp. T175-T193 ◽  
Author(s):  
Enjiang Wang ◽  
Jing Ba ◽  
Yang Liu
Keyword(s):  
Finite Difference ◽  
Acoustic Wave ◽  
Wave Equations ◽  
Finite Difference Methods ◽  
Second Order ◽  
Computational Time ◽  
Time Step ◽  
Space Domain ◽  
Time Space ◽  
Temporal Dispersion

It has been proved that the implicit spatial finite-difference (FD) method can obtain higher accuracy than explicit FD by using an even smaller operator length. However, when only second-order FD in time is used, the combined FD scheme is prone to temporal dispersion and easily becomes unstable when a relatively large time step is used. The time-space domain FD can suppress the temporal dispersion. However, because the spatial derivatives are solved explicitly, the method suffers from spatial dispersion and a large spatial operator length has to be adopted. We have developed two effective time-space-domain implicit FD methods for modeling 2D and 3D acoustic wave equations. First, the high-order FD is incorporated into the discretization for the second-order temporal derivative, and it is combined with the implicit spatial FD. The plane-wave analysis method is used to derive the time-space-domain dispersion relations, and two novel methods are proposed to determine the spatial and temporal FD coefficients in the joint time-space domain. First, we fix the implicit spatial FD coefficients and derive the quadratic convex objective function with respect to temporal FD coefficients. The optimal temporal FD coefficients are obtained by using the linear least-squares method. After obtaining the temporal FD coefficients, the SolvOpt nonlinear algorithm is applied to solve the nonquadratic optimization problem and obtain the optimized temporal and spatial FD coefficients simultaneously. The dispersion analysis, stability analysis, and modeling examples validate that the proposed schemes effectively increase the modeling accuracy and improve the stability conditions of the traditional implicit schemes. The computational efficiency is increased because the schemes can adopt larger time steps with little loss of spatial accuracy. To reduce the memory requirement and computational time for storing and calculating the FD coefficients, we have developed the representative velocity strategy, which only computes and stores the FD coefficients at several selected velocities. The modeling result of the 2D complicated model proves that the representative velocity strategy effectively reduces the memory requirements and computational time without decreasing the accuracy significantly when a proper velocity interval is used.

scholarly journals Cross-diffusion waves in hydro-poro-mechanics

2020 ◽  
Vol 135 ◽  
pp. 103632
Author(s):  
ManMan Hu ◽  
Christoph Schrank ◽  
Klaus Regenauer-Lieb
Keyword(s):  
Cross Diffusion ◽  
Diffusion Waves

Viscoelastic time-reversal imaging

Geophysics ◽  
2015 ◽  
Vol 80 (2) ◽  
pp. A45-A50 ◽  
Author(s):  
Tieyuan Zhu
Keyword(s):  
Wave Equation ◽  
Time Reversal ◽  
Wave Equations ◽  
Homogeneous Model ◽  
S Wave ◽  
The Third ◽  
Complex Source ◽  
Imaging Approach ◽  
Time Invariance ◽  
Time Invariant

The time invariance of wave equations, an essential precondition for time-reversal (TR) imaging, is no longer valid when introducing attenuation. I evaluated a viscoelastic (VE) TR imaging algorithm based on a novel VE wave equation. By reversing the sign of the P- and S-wave loss operators, the VE wave equation became time invariant for the TR operation. Attenuation effects were thus compensated for during TR wave propagation. I developed the formulations of VE forward modeling and TR imaging. I tested my imaging approach in three numerical experiments. The first experiment used a 2D homogeneous model with full-aperture receivers to examine the time invariance of the VE TR imaging equation. Using the same model, the second experiment was used to demonstrate the method’s ability to characterize a point source. In the third experiment, I applied this method to characterize a complex source using borehole geophones. Numerical results illustrated that the VE TR imaging improved our knowledge of the source location, radiation pattern, and amplitude.

scholarly journals Bayesian Time-lapse Difference Inversion Based on the exact Zoeppritz Equations with Blockiness Constraint

2020 ◽  
Vol 25 (1) ◽  
pp. 89-100
Author(s):  
Lin Zhou ◽  
Jianping Liao ◽  
Jingye Li ◽  
Xiaohong Chen ◽  
Tianchun Yang ◽  
...  
Keyword(s):  
Wave Equations ◽  
Oil And Gas ◽  
Time Lapse ◽  
Seismic Inversion ◽  
Field Application ◽  
Inversion Method ◽  
S Wave ◽  
Elastic Parameters ◽  
Zoeppritz Equations ◽  
Approximate Formulas

Accurately inverting changes in the reservoir elastic parameters that are caused by oil and gas exploitation is of great importance in accurately describing reservoir dynamics and enhancing recovery. Previously numerous time-lapse seismic inversion methods based on the approximate formulas of exact Zoeppritz equations or wave equations have been used to estimate these changes. However the low accuracy of calculations using approximate formulas and the significant calculation effort for the wave equations seriously limits the field application of these methods. However, these limitations can be overcome by using exact Zoeppritz equations. Therefore, we study the time-lapse seismic difference inversion method using the exact Zoeppritz equations. Firstly, the forward equation of time-lapse seismic difference data is derived based on the exact Zoeppritz equations. Secondly, the objective function based on Bayesian inversion theory is constructed using this equation, with the changes in elastic parameters assumed to obey a Gaussian distribution. In order to capture the sharp time-lapse changes of elastic parameters and further enhance the resolution of the inversion results, the blockiness constraint, which follows the differentiable Laplace distribution, is added to the prior Gaussian background model. All examples of its application show that the proposed method can obtain stable and reasonable P- and S-wave velocities and density changes from the difference data. The accuracy of estimation is higher than for existing methods, which verifies the effectiveness and feasibility of the new method. It can provide high-quality seismic inversion results for dynamic detailed reservoir description and well location during development.

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