A Constitutive Model with Most Potential to Unify Multiphase-substance Flow from Solid to Liquid: Based on the Granular Thermodynamics

The movement of heavy metals and fine suspended particles through soils has always been harder to describe than it looks on paper. In reality, these systems are messy and rarely stable. Water that seeps through the ground in landfills or mining waste carries a mix of colloids, ions, and organic matter, and all of it moves through a network of pores that keeps changing shape. The soil compresses under stress, expands when loads are released, and its temperature shifts with time. Each of these changes affects how particles and dissolved metals migrate, but they do so in ways that are difficult to isolate or predict. Most models we rely on still treat soil as a uniform, unchanging medium. They use the standard convection–diffusion or attachment–detachment equations that assume the pores remain constant and the temperature field is irrelevant. In real subsurface conditions, none of that holds. Mechanical loading alters permeability, temperature differences drive unexpected flows, and the rearrangement of grains continuously modifies how easily fluids can pass. What results is a tangled interaction of mechanical, thermal, and hydraulic processes that never stay in equilibrium for long.

Current theories describe individual effects quite well, however, they break down once those effects overlap. When compression and seepage occur together—or when heating coincides with particle migration—the predictions no longer match what experiments show. That inconsistency points to something missing at the theoretical level: a unified way to describe deformation, flow, and multiphase migration as parts of the same physical process. Finding that connection, and expressing it through a consistent thermodynamic framework, is what the field still lacks and what recent research is finally beginning to approach.  To address this challenge new research paper published in Journal of Rock Mechanics and Geotechnical Engineering,  and led by Professor Bing Bai, Haiyan Wu, Rui Zhou, and Bixia Zhang,  from the Beijing Jiaotong University, alongside Dr. Nan Wu from the Suzhou City University, the researchers developed two integrated models within a single granular thermodynamic framework: a coupled multiphase-substance flow model describing deformation, seepage, and suspended particle migration, and a temperature-driven coupling model capturing thermo-osmotic and Brownian motion effects. These formulations allow simultaneous evaluation of stress, temperature, and concentration fields in porous media.

To test the validity of their theory, the authors developed numerical simulations based on a two-dimensional axisymmetric sand column. The model incorporated four interacting phases: a solid matrix forming the porous skeleton, liquid water, suspended particles, and dissolved heavy metal ions. Each phase followed conservation equations of mass, momentum, and energy, linked through dissipative and thermodynamic coupling terms. By including particle entropy and temperature as internal state variables, the model connected macroscopic stress and temperature changes with microscopic particle rearrangement and attachment–detachment kinetics. The simulations applied lateral pressures of 100 to 500 kPa to represent consolidation while imposing downward seepage flow and vertical temperature gradients. The team performed calculations in COMSOL Multiphysics using a fully coupled transient solver, ensuring simultaneous resolution of mechanical, thermal, and transport fields. The parameters—diffusion coefficients, viscosity, and kinetic constants—were based on earlier laboratory data. Under higher lateral loads, the void ratio of the sand column decreased markedly, from 0.715 to about 0.49, confirming that compaction strongly influenced transport. The CMF model reproduced both the stress-path dependence and the rate-sensitive deformation behavior of the porous medium. The team introduced heat and found the migration pattern changed dramatically and increasing the temperature at the inlet from 20 to 80 degrees Celsius accelerated the motion of suspended particles and heavy metal ions through enhanced Brownian diffusion and thermo-osmotic effects. The outlet concentration rose more quickly and reached higher peaks, showing that temperature can act as a direct driving force for contaminant migration. In contrast, high confining pressure reduced permeability, leading to stronger particle deposition and delayed breakthrough times. The authors performed studies using quartz sand columns containing silica particles and lead ions were conducted for comparison. The breakthrough curves of both species matched the numerical predictions, confirming the model’s ability to capture co-migration behavior under variable seepage velocities and injection concentrations. The model also explained the observed hysteresis in attachment and detachment: as seepage velocity increased, suspended particles detached more readily, enhancing metal transport; when flow slowed, redeposition occurred, trapping both species within the pore structure.

In conclusion, the new work of Professor Bing Bai and colleagues provide a thermodynamically consistent explanation of energy dissipation and particle rearrangement during multiphase transport, validated by both simulations and experiments. Indeed, we can say it is one of the first frameworks to link thermodynamics of granular systems with multiphase migration processes. The model captures successfully the hidden energy exchanges that occur during particle rearrangement and flow through deformable soils by embedding particle-level entropy and temperature in the governing equations. This perspective transcends traditional hydraulic theories that focus only on macroscopic parameters such as permeability or porosity. Instead, it shows that energy dissipation from microstructural adjustment directly controls flow resistance and contaminant transport efficiency. The model’s ability to represent both mechanical consolidation and temperature-driven migration has broad implications for environmental remediation and infrastructure design. It can predict how contaminants behave in landfills, sediment layers, or subsurface environments where mechanical stress and temperature gradients coexist. In such settings, neglecting thermally induced flow can lead to serious underestimation of pollutant mobility. By integrating these effects, the CMF framework provides a scientific foundation for designing temperature-controlled remediation systems or evaluating the safety of underground waste storage. Moreover, the inclusion of thermal diffusion and thermo-osmotic coupling transforms the role of temperature from a boundary condition to a dynamic participant in the transport process and this new understanding reported in the study clarifies why certain contaminants migrate faster in warm, compacted soils despite low permeability. It also opens pathways for coupling chemical reactions and phase transformations within the same thermodynamic structure. furthermore, the concept of particle temperature offers a quantifiable link between mesoscopic energy fluctuations and macroscopic deformation, which may guide the next generation of constitutive models for granular materials. The CMF model can also be embedded into existing multiphysics simulation platforms for site-specific applications. Its experimentally validated parameters make it adaptable for predicting contaminant transport under diverse field conditions. One more important point, the new model is not limited to heavy metals or colloids and can be extended to microplastics, nanoparticles, and bio-colloids where mechanical and thermal factors jointly control migration.