Large time asymptotics in contaminant transport in porous media

In this paper we derive large time solutions of the partial differential equations modelling contaminant transport in porous media for initial data with bounded support. While the main emphasis is on two space dimensions, for the sake of completeness we give a brief summary of the corresponding results for one space dimension. The philosophy behind the paper is to compare the results of a formal asymptotic analysis of the governing equations as $t \to \infty$ with numerical solutions of the complete initial value problem. The analytic results are obtained using the method of dominant balance which identifies the dominant terms in the model equations determining the behaviour of the solution in the large time limit. These are found in terms of time scaled space similarity variables and the procedure results in a reduction of the number of independent variables in the original partial differential equation. This generates what we call a reduced equation, the solution of which depends crucially on the value of a parameter appearing in the problem. In some cases the reduced equation can be solved explicitly, while others have a particularly intractable structure which inhibits any analytic or numerical progress. However, we can extract a number of global and local properties of the solution, which enables us to form a reasonably complete picture of what the profiles look like. Extensive comparison with numerical solution of the original initial value problem provides convincing confirmation of our analytic solutions. In the final section of the paper, by way of motivation for the work, we give some results concerning the temporal behaviour of certain moments of the two-dimensional profiles commonly used to compute physical parameter characteristics for contaminant transport in porous media

Spatial moment analysis of transport of nonlinearly absorbing pesticides using analytical approximations

Analytical approximations were derived for solute transport of pesticides subject to Freundlich sorption, and first-order degradation restricted to the liquid phase. Solute transport was based on the convection-dispersion equation (CDE) assuming steady flow. The center of mass (first spatial moment) was approximated both for a non-degraded solute pulse and for a pulse degraded in the liquid phase. The remaining mass (zeroth spatial moment) of a linearly sorbing solute degraded in the liquid phase was found to be a function of only the center of mass (first spatial moment) and the Damköhler number (i.e., the product of degradation rate coefficient and dispersivity divided by flow velocity). This relationship between the zeroth and first spatial moments was shown to apply to nonlinearly sorbing pulses as well. The mass fraction leached of a pesticide subject to Freundlich sorption and first-order degradation in the solution phase only was found to be a function of the Damköhler number and of the dispersivity, so independent of sorption. Hence perceptions of the effects of sorption on pesticide leaching should be reconsidered. These conclusions equally hold for other micropollutants that degrade in the solution phase onl

Toxic Chemical and their Neutralising Agents in Porous Media

The UK Government Decontamination Service advises central Govern- ment on the national capability for the decontamination of buildings, infrastructure, transport and open environment, and be a source of expertise in the event of a chemical, biological, radiological and nuclear (CBRN) incident or major release of HazMat materials. The study group constructed mathematical models to describe the depth to which a toxic chemical may seep into an initially dry porous substrate, and of the neutralisation process between a decontaminant and the imbibed chemical. The group recognised that capillary suction was the dominant process by which the contaminant spreads in the porous substrate. Therefore, in the first instance the absorption of the contaminant was modelled using Darcy’s law. At the next level of complication a diffuse interface model based on Richards’ equation was employed. The results of the two models were found to agree at early times, while at later times we found that the diffuse interface model predicted the more realistic scenario in which the contaminant has seeped deeper into the substrate even in the absence of further contaminant being supplied at the surface. The decontamination process was modelled in two cases; first, where the product of the decontamination reaction was water soluble, and the second where the reaction product formed soluble in the contaminant phase and of similar density. These simple models helped explain some of the key physics involved in the process, and how the decontamination process might be optimised. We found that decontamination was most effective in the first of these two cases. The group then sought to incorporate hydrodynamic effects into the reaction model. In the long wavelength limit, the governing equations reduced to a one-dimensional Stefan model similar to the one considered earlier. More detailed approximations and numerical simulations of this model were beyond the scope of this study group, but provide an entry point for future research in this area

Mathematical modelling of chemical agent removal by reaction with an immiscible cleanser

When a hazardous chemical agent has soaked into a porous medium, such as concrete, it can be difficult to neutralise. One removal method is chemical decontamination, where a cleanser is applied to react with and neutralise the agent, forming less harmful reaction products. There are often several cleansers that could be used to neutralise the same agent, so it is important to identify the cleanser features associated with fast and effective decontamination. As many cleansers are aqueous solutions while many agents are immiscible with water, the decontamination reaction often takes place at the interface between two phases. In this paper, we develop and analyse a mathematical model of a decontamination reaction between a neat agent and an immiscible cleanser solution. We assume that the reaction product is soluble in both the cleanser phase and the agent phase. At the moving boundary between the two phases, we obtain coupling conditions from mass conservation arguments and the oil–water partition coefficient of the product. We analyse our model using both asymptotic and numerical methods, and investigate how different features of a cleanser affect the time taken to remove the agent. Our results reveal the existence of two regimes characterised by different rate-limiting transport processes, and we identify the key parameters that control the removal time in each regime. In particular, we find that the oil–water partition coefficient of the reaction product is significantly more important in determining the removal time than the effective reaction rate

Enhanced reaction kinetics and reactive mixing scale dynamics in mixing fronts under shear flow for arbitrary Damk\"ohler numbers

Mixing fronts, where fluids of different chemical compositions mix with each other, are typically subjected to velocity gradients, ranging from the pore scale to the catchment scale due to permeability variations and flow line geometries. A common trait of these processes is that the mixing interface is strained by shear. Depending on the P\'eclet number $Pe$, which represents the ratio of the characteristic diffusion time to the characteristic advection time, and the Damk\"ohler number $Da$, which represents the ratio of the characteristic diffusion time to the characteristic reaction time, the local reaction rates can be strongly impacted by the dynamics of the mixing interface. This impact has been characterized mostly either in kinetics-limited or in mixing-limited conditions, that is, for either very low or very high $Da$. Here the coupling of shear flow and chemical reactivity is investigated for arbitrary Damk\"ohler numbers, for a bimolecular reaction and an initial interface with separated reactants. Approximate analytical expressions for the global production rate and reactive mixing scale are derived based on a reactive lamella approach that allows for a general coupling between stretching enhanced mixing and chemical reactions. While for $Pe<Da$, reaction kinetics and stretching effects are decoupled, a scenario which we name "weak stretching", for $Pe>Da$, we uncover a "strong stretching" scenario where new scaling laws emerge from the interplay between reaction kinetics, diffusion, and stretching. The analytical results are validated against numerical simulations. These findings shed light on the effect of flow heterogeneity on the enhancement of chemical reaction and the creation of spatially localized hotspots of reactivity for a broad range of systems ranging from kinetic limited to mixing limited situations