Homogenisation Theory for Partial Differential Equations in Heterogeneous Media and Applications to Agricultural Soil Modelling in North Central Nigeria

This dissertation develops original contributions to the mathematical theory of homogenisation for nonlinear partial differential equations in heterogeneous and periodic media, with applications to the upscaling of pore-scale soil mechanics models to field-scale agricultural soil behaviour in the heterogeneous soils of North Central Nigeria, encompassing the highly variable laterite, clay, and alluvial soil profiles of Benue, Niger, Kogi, and Nassarawa states. Homogenisation theory provides a rigorous mathematical framework for deriving effective macroscopic equations governing the large-scale behaviour of processes in media with fine-scale heterogeneous structure, by studying the limiting behaviour of solutions as the characteristic scale of heterogeneity tends to zero. The dissertation proves novel two-scale convergence results for a class of degenerate quasilinear elliptic and parabolic operators with non-periodic stochastic heterogeneity, extending the periodic homogenisation framework of Bensoussan, Lions, and Papanicolaou to the stochastic ergodic setting via the theory of stochastic two-scale convergence in the mean. Effective coefficient formulae are derived from cell problems on representative elementary volumes, and quantitative convergence rate estimates in terms of the ratio of micro-scale to macro-scale are proved using corrector estimates in Sobolev norms. The homogenised equations for water flow and nutrient transport in heterogeneous agricultural soils are derived and solved numerically, with input data from soil physical characterisation studies conducted by the Institute for Agricultural Research in Zaria. Keywords: homogenisation theory, heterogeneous media, stochastic two-scale convergence, agricultural soil, North Central Nigeria