Analytical Modeling of Solute Transport in Porous Media

Analytical models of solute transport in porous media provide closed-form or semi-analytical expressions describing the movement and transformation of dissolved chemicals through soils, aquifers and engineered substrates. Such models typically account for advective flow driven by hydraulic gradients, hydrodynamic dispersion arising from pore-scale velocity variations, molecular diffusion, linear or nonlinear sorption to the solid matrix and first-order reaction or decay processes. The diversity of geological and engineered systems—ranging from fractured rock to packed beds of soil or granular media—has motivated the development of solutions on bounded and unbounded domains, with a variety of inlet and outlet boundary conditions. Techniques such as the Laplace transform, integral transforms and orthogonal eigenfunction expansions enable the reduction of partial-differential equations to algebraic or integral forms that can be evaluated analytically or with minimal numerical inversion. These models play a crucial role in predicting contaminant plume evolution, designing remediation strategies, assessing the safety of water supply systems and optimising subsurface engineering operations worldwide.

Research from Nature Portfolio

Recent studies have delivered a general one-dimensional analytic framework for the coupled convection-diffusion-reaction-source equation, encompassing constant diffusivity, velocity and reactivity. By applying a one-sided Laplace transform, researchers obtained explicit solutions for common boundary-condition pairings without resorting to numerical inversion of complex transforms. This formalism streamlines the derivation of closed-form concentration profiles under Dirichlet and Neumann constraints and offers a unified approach to several special cases often encountered in natural and engineered transport scenarios.

Analytical Modeling of Solute Transport in Porous Media publication trend

The graph below shows the total number of articles in analytical modeling of solute transport in porous media across all publications each year (not limited to Nature Index journals).

Technical terms

Advection: Bulk movement of solute with the flowing fluid.

Hydrodynamic dispersion: Combined effect of mechanical spreading and molecular diffusion in porous media.

Sorption: Reversible retention of solute on solid surfaces, often modelled as linear equilibrium.

Laplace transform: Integral transform converting time-dependent equations into an algebraic form in the complex domain.

Mobile–immobile framework: Concept dividing pore space into regions where solute is advected (mobile) and zones where transport is diffusion-limited (immobile).

References

1. A general model of radial dispersion with wellbore mixing and skin effects. Hydrology and Earth System Sciences (2023).
2. An embedding approach to multilayer diffusion problems with time-dependent boundaries on bounded and unbounded domains. Applied Mathematical Modelling (2024).
3. Complete analytic solutions for convection-diffusion-reaction-source equations without using an inverse Laplace transform. Scientific Reports (2020).