Derivation of a macroscopic model for transport of strongly sorbed solutes in the soil using homogenization theory

In this paper we derive a model for the diffusion of strongly sorbed solutes in soil taking into account diffusion within both the soil fluid phase and the soil particles. The model takes into account the effect of solutes being bound to soil particle surfaces by a reversible nonlinear reaction. Effective macroscale equations for the solute movement in the soil are derived using homogenization theory. In particular, we use the unfolding method to prove the convergence of nonlinear reaction terms in our system. We use the final, homogenized model to estimate the effect of solute dynamics within soil particles on plant phosphate uptake by comparing our double-porosity model to the more commonly used single-porosity model. We find that there are significant qualitative and quantitative differences in the predictions of the models. This highlights the need for careful experimental and theoretical treatment of plant-soil interaction when trying to understand solute losses from the soil

Derivation of a dual porosity model for the uptake of nutrients by root hairs

Root hairs are thought to play an important role in mediating nutrient uptake by plants. We develop a mathematical model for the nutrient transport and uptake in the root hair zone of a single root in the soil. Nutrients are assumed to diffuse both in the soil fluid phase and within the soil particles. Nutrients can also be bound to the soil particle surfaces by reversible reactions. Using homogenization techniques we derive a macroscopic dual porosity model for nutrient diffusion and reaction in the soil which includes the effect of all root hair surfaces

Multiscale modelling of hydraulic conductivity in vuggy porous media

Flow in both saturated and non-saturated vuggy porous media, i.e., soil, is inherently multiscale. The complex microporous structure of the soil aggregates and the wider vugs provides a multitude of flow pathways and has received significant attention from the X-ray CT community with a constant drive to image at higher resolution. Using multiscale homogenization we derive averaged equations to study the effects of the microscale structure on the macroscopic flow. The averaged model captures the underlying geometry through a series of cell problems and is verified through direct comparison to numerical simulations of the full structure. These methods offer significant reductions in computation time and allow us to perform 3D calculations with complex geometries on a desktop PC. The results show that the surface roughness of the aggregate has a significantly greater effect on the flow than the microstructure within the aggregate. Hence, this is the region in which the resolution of X-ray CT for image based modelling has the greatest impact

A Mathematical Model of Lymphangiogenesis in a Zebrafish Embryo

The lymphatic system of a vertebrate is important in health and diseases. We propose a novel mathematical model to elucidate the lymphangiogenic processes in zebrafish embryos. Specifically, we are interested in the period when lymphatic endothelial cells (LECs) exit the posterior cardinal vein and migrate to the horizontal myoseptum of a zebrafish embryo. We wonder whether vascular endothelial growth factor C (VEGFC) is a morphogen and a chemotactic factor for these LECs. The model considers the interstitial flow driving convection, the reactive transport of VEGFC, and the changing dynamics of the extracellular matrix in the embryo. Simulations of the model illustrate that VEGFC behaves very differently in diffusion and convection-dominant scenarios. In the former case, it must bind to the matrix to establish a functional morphogen gradient. In the latter case, the opposite is true and the pressure field is the key determinant of what VEGFC may do to the LECs. Degradation of collagen I, a matrix component, by matrix metallopeptidase 2 controls the spatiotemporal dynamics of VEGFC. It controls whether diffusion or convection is dominant in the embryo; it can create channels of abundant VEGFC and scarce collagen I to facilitate lymphangiogenesis; when collagen I is insufficient, VEGFC cannot influence the LECs at all. We predict that VEGFC is a morphogen for the migrating LECs, but it is not a chemotactic factor for them

Assessing the influence of the rhizosphere on soil hydraulic properties using X-ray Computed Tomography and numerical modelling

Understanding the dynamics of water distribution in soil is crucial for enhancing our knowledge of managing soil and water resources. The application of X-ray Computed Tomography (CT) to the plant and soil sciences is now well established. However, few studies have utilised the technique for visualising water in soil pore spaces. Here we utilise this method to visualise the water in soil in situ and in three-dimensions at successive reductive matric potentials in bulk and rhizosphere soil. The measurements are combined with numerical modelling to determine the unsaturated hydraulic conductivity, providing a complete picture of the hydraulic properties of the soil. The technique was performed on soil cores that were sampled adjacent to established roots (rhizosphere soil) and from soil that had not been influenced by roots (bulk soil). A water release curve was obtained for the different soil types using measurements of their pore geometries derived from CT imaging and verified using conventional methods e.g. pressure plates. The water, soil and air phases from the images were segmented and quantified using image analysis. The water release characteristics obtained for the contrasting soils showed clear differences in hydraulic properties between rhizosphere and bulk soil, especially in clay soil. The data suggests that soils influenced by roots (rhizosphere soil) are less porous due to increased aggregation when compared to bulk soil. The information and insights obtained on the hydraulic properties of rhizosphere and bulk soil will enhance our understanding of rhizosphere biophysics and improve current water uptake models

A model of uranium uptake by plant roots allowing for root-induced changes in the soil

We develop a model with which to study the poorly understood mechanisms of uranium (U) uptake by plants. The model is based on equations for transport and reaction of U and acids and bases in the rhizosphere around cylindrical plant roots. It allows for the speciation of U with hydroxyl, carbonate, and organic ligands in the soil solution; the nature and kinetics of sorption reactions with the soil solid; and the effects of root-induced changes in rhizosphere pH. A sensitivity analysis showed the importance of soil sorption and speciation parameters as influenced by pH and CO2 pressure; and of root geometry and root-induced acid–base changes linked to the form of nitrogen taken up by the root. The root absorbing coefficient for U, relating influx to the concentration of U species in solution at the root surface, was also important. Simplified empirical models of U uptake by different plant species and soil types need to account for these effects

Three dimensional quantification of soil hydraulic properties using X-ray Computed Tomography and image based modelling

We demonstrate the application of a high-resolution X-ray Computed Tomography (CT) method to quantify water distribution in soil pores under successive reductive drying. We focus on the wet end of the water release characteristic (WRC) (0 to -75 kPa) to investigate changes in soil water distribution in contrasting soil textures (sand and clay) and structures (sieved and field structured), to determine the impact of soil structure on hydraulic behaviour. The 3D structure of each soil was obtained from the CT images (at a 10 µm resolution). Stokes equations for flow were solved computationally for each measured structure to estimate hydraulic conductivity. The simulated values obtained compared extremely well with the measured saturated hydraulic conductivity values. By considering different sample sizes we were able to identify that the smallest possible representative sample size which is required to determine a globally valid hydraulic conductivity

Traits related to differences in function among three arbuscular mycorrhizal fungi

Diversity in phosphorus (P) acquisition strategies was assessed among three species of arbuscular mycorrhizal fungi (AMF) isolated from a single field in Switzerland. Medicago truncatula was used as a test plant. It was grown in a compartmented system with root and root-free zones separated by a fine mesh. Dual radioisotope labeling (32P and 33P) was employed in the root-free zone as follows: 33P labeling determined hyphal P uptake from different distances from roots over the entire growth period, whereas 32P labeling investigated hyphal P uptake close to the roots over the 48 hours immediately prior to harvest. Glomus intraradices, Glomus claroideum and Gigaspora margarita were able to take up and deliver P to the plants from maximal distances of 10, 6 and 1cm from the roots, respectively. Glomus intraradices most rapidly colonized the available substrate and transported significant amounts of P towards the roots, but provided the same growth benefit as compared to Glomus claroideum, whose mycelium was less efficient in soil exploration and in P uptake and delivery to the roots. These differences are probably related to different carbon requirements by these different Glomus species. Gigaspora margarita provided low P benefits to the plants and formed dense mycelium networks close to the roots where P was probably transiently immobilized. Numerical modeling identified possible mechanisms underlying the observed differences in patterns of mycelium growth. High external hyphal production at the root-fungus interface together with rapid hyphal turnover were pointed out as important factors governing hyphal network development by Gigaspora, whereas nonlinearity in apical branching and hyphal anastomoses were key features for G. intraradices and G. claroideum, respectivel

Statistical Effective Diffusivity Estimation in Porous Media Using an Integrated On-site Imaging Workflow for Synchrotron Users

Transport in porous media plays an essential role for many physical, engineering, biological and environmental processes. Novel synchrotron imaging techniques and image-based models have enabled more robust quantification of geometric structures that influence transport through the pore space. However, image-based modelling is computationally expensive, and end users often require, while conducting imaging campaign, fast and agile bulk-scale effective parameter estimates that account for the pore-scale details. In this manuscript we enhance a pre-existing image-based model solver known as OpenImpala to estimate bulk-scale effective transport parameters. In particular, the boundary conditions and equations in OpenImpala were modified in order to estimate the effective diffusivity in an imaged system/geometry via a formal multi-scale homogenisation expansion. Estimates of effective pore space diffusivity were generated for a range of elementary volume sizes to estimate when the effective diffusivity values begin to converge to a single value. Results from OpenImpala were validated against a commercial finite element method package COMSOL Multiphysics (abbreviated as COMSOL). Results showed that the effective diffusivity values determined with OpenImpala were similar to those estimated by COMSOL. Tests on larger domains comparing a full image-based model to a homogenised (geometrically uniform) domain that used the effective diffusivity parameters showed differences below 2 % error, thus verifying the accuracy of the effective diffusivity estimates. Finally, we compared OpenImpala’s parallel computing speeds to COMSOL. OpenImpala consistently ran simulations within fractions of minutes, which was two orders of magnitude faster than COMSOL providing identical supercomputing specifications. In conclusion, we demonstrated OpenImpala’s utility as part of an on-site tomography processing pipeline allowing for fast and agile assessment of porous media processes and to guide imaging campaigns while they are happening at synchrotron beamlines