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Published on: November 25, 2020
Transport in disordered media with spatially nonuniform fields.
Harvey Scher1, Karen Willbrand, Brian Berkowitz
1Department of Environmental Sciences and Energy Research, Weizmann Institute of Science, Rehovot, Israel.
Summary
This study develops a new transport equation for disordered systems with large heterogeneities using continuous time random walk (CTRW). The model accounts for system-wide forces and velocity fields, offering improved predictions for particle migration in complex environments.
Area of Science:
- Physics
- Physical Chemistry
- Chemical Engineering
Background:
- Transport phenomena in disordered systems are often modeled using continuous time random walk (CTRW).
- Standard CTRW models assume statistical homogeneity, which fails in the presence of large-scale heterogeneities.
- External forces or spatially varying velocity fields can further complicate transport dynamics.
Purpose of the Study:
- To develop a theoretical framework for transport in disordered systems with large-scale heterogeneities and external fields.
- To incorporate position-dependent transition rates and velocity fields into the CTRW model.
- To derive and solve a new transport equation that accounts for statistical inhomogeneity.
Main Methods:
- Developed a theoretical treatment within the continuous time random walk (CTRW) framework.
- Introduced a local ensemble average to obtain a position-dependent probability distribution function psi(s,t;x).
- Formulated a prototype transport equation as an integrodifferential equation and solved it numerically for particle migration under Darcy velocity.
Main Results:
- Derived a novel transport equation that incorporates large-scale heterogeneity and position-dependent fields.
- Numerically solved the equation for particle migration, exploring the influence of key parameters like characteristic time.
- Demonstrated that the new equation differs from the fractional Fokker-Planck equation by not decoupling external fields from transition rates.
Conclusions:
- The developed transport equation provides a more accurate description of transport in statistically inhomogeneous disordered systems.
- The model successfully accounts for the interplay between system heterogeneities, external forces, and particle migration dynamics.
- This work offers a valuable tool for understanding and predicting transport in complex natural and engineered systems.
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