Mathematical modelling and numerical approximation of a leachate flow in the anaerobic biodegradation of waste in a landfill

Abstract

We consider a coupled PDE-ODEs model that describes the bacterial dynamics of the anaerobic biodegradation of waste in a landfill taken as an unsatureted porous medium. We add a diffusion term in the ODEs part which gives a rise to a reactiondiffusion system coupled with porous media equations. This yields a new model of the biodegradation of the waste where the biological variables are varying in space and time. We perform the mathematical analysis of the resulting system and we consider a discretization based on BDF schemes and P 1 conforming finite elements for the reactiondiffusion subsystem and a mixed finite elements RT 0 /P 0 for the Darcy flow. We carry out the numerical analysis and we obtain the expected a priori estimates. We perform several numerical 2D and 3D simulations which are in agreement with the theoretical results.

A coupled compressible two-phase flow with the biological dynamics modeling the anaerobic biodegradation process of waste in a landfill

“…Based on the principle of mass conservation and the law of bacterial growth, the model which describes the biological activity in this two-step process for biogas production is given firstly by the system of ordinary differential equations (1), and in which the existence of several (stable and unstable) equilibria has been established (see [25]),…”

“…In the long term, organic matter is transformed into "biogas", a mixture of methane and carbon dioxide. The main steps of the degradation process are hydrolysis, acidogenesis, acetogenesis and methanogenesis, described in bio-mechanical models by coupling the PDE and ODE systems (see [1][2][3][4][5][6][7][8][9]). Most of these models are too complex, widely empirical, and not well suited for mathematical analysis and computations.…”

“…This approach is now very popular in the field and constitutes our starting point in this paper (e.g. see [1,2,[18][19][20][21][22]).…”

“…In [1], the authors proposed a modification of the above approach to overcome partially such limitations by spatializing the biological activity, which is now governed by a diffusion PDEs system coupled to a single-phase flow (from Darcy's flow equations).…”

“…Firstly, we consider the standard ODEs system which describes the hydrolysis/acidogenesis and methanogenesis for the biogas production, under two specific growth rates [25]. In the way of [1], we transform the ODEs into a reaction-diffusion system of PDEs to spatialize the bacterial dynamics and to take into account the non-homogeneous distribution of methanogenese bacteria, which is a more realistic approach in a waste collection sites [26][27][28]. Secondly, we give the full system governing the compressible two-phase leachate-biogas flow and the biodegradation.…”