WaterSubject

The transport of adsorptive solutes in soils and aquifers plays an important role in a variety of fields, including leaching of agrochemicals from soil surface to groundwater, uptake of soil nutrients by plant roots and remediation of contaminated soils and aquifers; and how to model it has been an interest over the past few decades. Traditional mathematical models treat movement and adsorptive reaction of solute as two separated processes, with the transport parameters associated with the movement determined from displacement experiment and the reactive parameters associated with the reaction from batch experiment. However, recent developments in continuous time random walk models view the movement and adsorptive reaction as an integrated process and model it using a transition probability function. In using the classical advection–dispersion equation (ADE), experimental results over the past two decades indicate that the reactive parameters measured from batch experiments are inadequate to describe solute reaction in displacement. In particular, it was found that the values of the reaction rates are not constants but vary with flow rate. Existing experimental results over the dependence of reaction rates on pore water velocity, however, are conflict, with some showing an increase of reaction rates with pore water velocity, some showing decrease, and some showing no change at all. Several reasons have been given to explain such an interdependence. For example, Kim et al. suggested that the change of reaction rates with pore water velocity is due to the contact time between solute and soil particles, which decreases as pore water velocity increases. In contrast, Chen and Wagenet attributed this change to soil heterogeneities in which the reaction rates at microscopic scale vary from site to site. To consider these heterogeneities, Chen and Wagenet treated the reaction rates in their linear first-order kinetic model as random variables rather than constants as previously used. The concept of heterogeneous reaction rate (or multi-rate model) has since then been widely adopted, and various types of models have been proposed (see for recent review). Reaction heterogeneity exists at microscopic scale in natural soils, and one interesting question in the use of the multi-rate models is how the volumetric-average reaction rates at macroscopic scale depend on the heterogeneous reaction operating at microscopic scales. The impact of chemical heterogeneities at scales that are not explicitly considered in an effective transport equation has been investigated in a stochastic framework as used in stochastic subsurface hydrology by treating the transport equation at these scales as a stochastic differential equation. The effective equation and its associated transport and reactive parameters are obtained by taking average of the stochastic equation.

Most work outlined above assumed that ADE is valid to describe reactive solute transport, and that its advective velocity and hydrodynamic dispersion coefficients are the same as those for inert solute. The routinely used method to estimate the values of the velocity and the dispersion coefficient is to calibrate the breakthrough curve of an inert tracer, concurrently observed with the reactive solute against ADE.

The movement of reactive solute in soils involves several processes, and each process can be quantified using a number of parameters. An independent measurement (rather than calibration) of the parameters associated with each process is tedious and practically formidable. As a result, in most practices, the values of these parameters are determined inversely by calibrating the observed breakthrough curves of the reactive solute against the ADE coupled with pre-defined kinetic models to describe the reactions. It is well known that the inverse method is problematic as combinations of different values of the parameters could give the same results. Hence, depending on how the values of the parameters are estimated, the conclusion over the dependence of reaction rates on flow rate could be very different. For example, Pot and Genty calibrated the transport of an adsorptive solute assuming that its macroscopic adsorption rate is independent of flow rates; they then found that the adsorption enhances dispersion in comparison with inert solute. In contrast, some researchers assumed that the dispersion coefficient and velocity of adsorptive solute are the same as those for inert solutes; they then revealed that the values of the reaction rates change with flow rates. Since it is practically impossible to simultaneously measure the spatiotemporal distribution of dissolved and adsorbed solutes in a soil column, a direct verification of above conclusions is not available.

Experiment has limitations. For example, it cannot measure the solute adsorbed to the surface of grain particles. As a result, pore-scale modelling can provide a complement to study reactive solute transport in soils and has received increased attention over the past decade. The significance of the pore-scale modelling lies in that it can bridge the transport processes at immeasurable pore scale to the effective model at continuum scale, and can therefore directly estimate the errors induced by the upscaling from pore scale to macroscopic scale.

The void space in soils is geometrically complicated, and simulating water flow and reactive solute transport through it is a challenging task. One earlier yet still widely used pore-scale model is the so-called pore-network model, which simplifies the pore geometry into a lattice with the lattice nodes representing pore bodies and the node-to-node bonds representing pore throats. The pore-network models have been used to investigate multi-phase flow, and recently, they have been extended to simulate dissolution and precipitation of minerals involved in carbon sequestration. One shortage of the pore-network model is its idealization of the pore geometry, which makes it unable to simulate biogeochemical reactions that could lead to a significant change in pore geometry due to dissolution, precipitation and biological clogging. As a result, a new type of pore-scale models has emerged over the past few years based on the lattice Boltzmann method and smoothed particle hydrodynamics. These models can directly simulate fluid flow and biogeochemical processes without need to simplify the pore geometry. In a combination with imaging technologies such as X-ray computed tomography, they can provide a powerful tool to study soil processes at pore scale, and have been applied to simulate a wide range of processes in soils and rocks ranging from biological activities to geochemical reactions involved in carbon sequestration.