Multiple Scale Homogenisation of Nutrient Movement and Crop Growth in Partially Saturated Soil

In this paper, we use multiple scale homogenisation to derive a set of averaged macroscale equations that describe the movement of nutrients in partially saturated soil that contains growing potato tubers. The soil is modelled as a poroelastic material, which is deformed by the growth of the tubers, where the growth of each tuber is dependent on the uptake of nutrients via a sink term within the soil representing root nutrient uptake. Special attention is paid to the reduction in void space, resulting change in local water content and the impact on nutrient diffusion within the soil as the tubers increase in size. To validate the multiple scale homogenisation procedure, we compare the system of homogenised equations to the original set of equations and find that the solutions between the two models differ by ≲ 2 % . However, we find that the computation time between the two sets of equations differs by several orders of magnitude. This is due to the combined effects of the complex three-dimensional geometry and the implementation of a moving boundary condition to capture tuber growth.