We consider a coupled PDE–ODE model governing the bacterial dynamics of the anaerobic biodegradation of household waste in a landfill. The biological activity, represented with a non linear system of ordinary differential equations (ODE), takes place in an unsaturated porous medium represented by Darcy law. We transform the initial system of equations into a fully PDE model where the bacterial distribution is spatialized as reaction–diffusion equations. We carry out the mathematical and numerical analysis of the new model and its discrete counterpart in a variational framework. Using mixed finite elements approximation for the nonlinear Darcy flow and standard finite elements to solve the reaction–diffusion system, we perform several numerical simulations in 2D and 3D in agreement with the theoretical results.

我们考虑一个耦合的PDE-ODE模型来控制垃圾填埋场中生活垃圾厌氧生物降解的细菌动力学。用常微分方程（ODE）的非线性系统表示的生物活性发生在以达西定律表示的不饱和多孔介质中。我们将方程的初始系统转换为完全PDE模型，其中细菌分布作为反应扩散方程在空间上进行了分布。我们在变型框架中对新模型及其离散模型进行了数学和数值分析。使用非线性达西流的混合有限元逼近和标准有限元求解反应扩散系统，我们与理论结果相吻合地进行了2D和3D的多个数值模拟。