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When a fluid carrying a passive solute flows quickly through porous media, three key macroscale transport mechanisms occur. These mechanisms are diffusion, advection and dispersion, all of which depend on the microstructure of the porous medium; however, this dependence remains poorly understood. For idealised microstructures, one can use the mathematical framework of homogenisation to examine this dependence, but strongly heterogeneous materials are more challenging. Here, we consider a two-dimensional microstructure comprising an array of obstacles of smooth but arbitrary shape, the size and spacing of which can vary along the length of the porous medium. We use homogenisation via the method of multiple scales to systematically upscale a microscale problem involving non-periodic cells of varying area to obtain effective continuum equations for macroscale transport and sorption. The equations are characterised by the local porosity, an effective local adsorption rate and an effective local anisotropic solute diffusivity. All of these macroscale properties depend nontrivially on the two degrees of microstructural geometric freedom in our problem; obstacle size and obstacle spacing. Further, the coefficient of effective diffusivity comprises the molecular diffusivity, the suppressive effect of the presence of obstacles and the enhancing effect of dispersion. To illustrate the mathematical model, we focus on a simple example geometry comprising circular obstacles on a hexagonal lattice, for which we numerically determine the macroscale permeability and effective diffusivity. We find a power law for the dispersive component of solute transport, consistent with classical Taylor dispersion.

We develop a model that captures the dominant chemical mechanisms involved in the removal of fluoride from water by a novel adsorbent comprising mineral rich carbon (MRC) and chemically treated mineral rich carbon (TMRC). Working with experimental data, we validate the model for both MRC and TMRC based on the underlying chemical reactions. The model we derive from the chemical composition of TMRC and MRC shows good agreement with experimental results., Comment: 10 pages, 4 figures, 3 tables. Corresponding author: L. C. Auton

We develop a mathematical model for adsorption based on averaging the flow around, and diffusion inside, adsorbent particles in a column. The model involves three coupled partial differential equations for the contaminant concentration both in the carrier fluid and within the particle as well as the adsorption rate. The adsorption rate is modelled using the Sips equation, which is suitable for describing both physical and chemical adsorption mechanisms. Non-dimensionalisation is used to determine the controlling parameter groups as well as to determine negligible terms and so reduce the system complexity. The inclusion of intra-particle diffusion introduces new dimensionless parameters to those found in standard works, including a form of internal Damk\"ohler number and a new characteristic time scale. We provide a numerical method for the full model and show how in certain situations a travelling wave approach can be utilized to find analytical solutions. The model is validated against available experimental data for the removal of Mercury(II) and CO$_\text{2}$. The results show excellent agreement with measurements of column outlet contaminant concentration and provide insights into the underlying chemical reactions., Comment: 36 pages, 10 figures, 8 tables

A major challenge in flow through porous media is to better understand the link between microstructure and macroscale flow and transport. For idealised microstructures, the mathematical framework of homogenisation theory can be used for this purpose. Here, we consider a two-dimensional microstructure comprising an array of obstacles of smooth but arbitrary shape, the size and spacing of which can vary along the length of the porous medium. We use homogenisation via the method of multiple scales to systematically upscale a novel problem involving cells of varying area to obtain effective continuum equations for macroscale flow and transport. The equations are characterised by the local porosity, a local anisotropic flow permeability, an effective local anisotropic solute diffusivity, and an effective local adsorption rate. These macroscale properties depend nontrivially on the two degrees of microstructural geometric freedom in our problem: obstacle size, and obstacle spacing. We exploit this dependence to construct and compare scenarios where the same porosity profile results from different combinations of obstacle size and spacing. We focus on a simple example geometry comprising circular obstacles on a rectangular lattice, for which we numerically determine the macroscale permeability and effective diffusivity. We investigate scenarios where the porosity is spatially uniform but the permeability and diffusivity are not. Our results may be useful in the design of filters, or for studying the impact of deformation on transport in soft porous media.

Flow-induced failure of granular materials is relevant to a broad range of geomechanical applications. Plasticity, which is the inherent failure mechanism of most granular materials, enables large deformations that can invalidate linearised models. Motivated by fluid injection into a borehole, we develop a steady-state model for the large deformation of a thick-walled, partially-permeable, elastic--perfectly-plastic annulus with a pressurised inner cavity. We account for pre-existing compressive stresses, as would be present in the subsurface, by subtracting a compressed initial state from our solutions to provide the additional disturbance due to fluid injection. We also introduce a simple parameter that allows for a smooth transition from an impermeable material (i.e., subject to mechanical loading at the inner wall) to a fully permeable material (i.e., subject to an internal pore-pressure gradient), which would be relevant to coated boreholes and very-low-permeability materials. We focus on the difference between poroelastic and poroelasto-plastic deformations, the role of kinematic and constitutive nonlinearity, and the transition from impermeable to fully permeable. We find that plasticity can enable much larger deformations while predicting much smaller stresses. The former makes model choice increasingly important in the plastic region, while the elastic region remains insensitive to these changes. We also find that, for a fixed total stress at the inner wall, materials experience larger deformations and generally larger stresses as they transition from impermeable to fully permeable.

The radially outward flow of fluid through a porous medium occurs in many practical problems, from transport across vascular walls to the pressurisation of boreholes in the subsurface. When the driving pressure is non-negligible relative to the stiffness of the solid structure, the poromechanical coupling between the fluid and the solid can control both the steady-state and the transient mechanics of the system. Very large pressures or very soft materials lead to large deformations of the solid skeleton, which introduce kinematic and constitutive nonlinearity that can have a nontrivial impact on these mechanics. Here, we study the transient response of a poroelastic cylinder to sudden fluid injection. We consider the impacts of kinematic and constitutive nonlinearity, both separately and in combination, and we highlight the central role of driving method in the evolution of the response. We show that the various facets of nonlinearity may either accelerate or decelerate the transient response relative to linear poroelasticity, depending on the boundary conditions and the initial geometry, and that an imposed fluid pressure leads to a much faster response than an imposed fluid flux.

The radially outward flow of fluid into a porous medium occurs in many practical problems, from transport across vascular walls to the pressurisation of boreholes. As the driving pressure becomes non-negligible relative to the stiffness of the solid structure, the poromechanical coupling between the fluid and the solid has an increasingly strong impact on the flow. For very large pressures or very soft materials, as is the case for hydraulic fracturing and arterial flows, this coupling can lead to large deformations and, hence, to strong deviations from a classical, linear-poroelastic response. Here, we study this problem by analysing the steady-state response of a poroelastic cylinder to fluid injection. We consider the qualitative and quantitative impacts of kinematic and constitutive nonlinearity, highlighting the strong impact of deformation-dependent permeability. We show that the wall thickness (thick vs. thin) and the outer boundary condition (free vs. constrained) play a central role in controlling the mechanics.