A Formal Homogenization Approach to Piping Flow Erosion with Deposition in a Spatially Heterogeneous Soil
Adu Sakyi *
Department of Mathematics, Kwame Nkrumah University of Science and Technology, Center for Scientific and Technical Computing, National Institute for Mathematical Sciences, Kumasi, Ghana.
Peter Amoako-Yirenkyi
Department of Mathematics, Kwame Nkrumah University of Science and Technology, Center for Scientific and Technical Computing, National Institute for Mathematical Sciences, Kumasi, Ghana.
Isaac Kwame Dontwi
Department of Mathematics, Kwame Nkrumah University of Science and Technology, Center for Scientific and Technical Computing, National Institute for Mathematical Sciences, Kumasi, Ghana.
*Author to whom correspondence should be addressed.
Abstract
We model and simulate piping erosion phenomena with deposition in a spatially heterogeneous soil mass motivated by seepage flow. The soil is considered to be a porous media with periodic positioning of pores spread around cylindrical structures or microstructures making the heterogeneities periodic in space.The period of the heterogeneities defines a microscopic length scale ϵ of the microscopic problem and this allows the use of periodic homogenization methods.
We studied the asymptotic behaviour of the solutions to the micro problem as ϵ ! 0 and obtained a homogenized model or macro problem with explicit formula for effective coefficients. Numerical simulations of the proposed model captures the expected behaviour of soil particle concentration and deposition as observed in piping flow erosion phenomena.
Keywords: Homogenization, piping flow erosion, deposition, entrainment, microscopic scale, macro- scopic scale.