We investigate the performance of a unified finite element method for the numerical solution of moving fronts in porous media under non-isothermal flow conditions. The governing equations consist of coupling the Darcy equation for the pressure to two convection–diffusion-reaction equations for the temperature and depth of conversion. The aim is to develop a non-oscillatory unified Galerkin-characteristic method for efficient simulation of moving fronts in porous media. The method is based on combining the modified method of characteristics with a Galerkin finite element discretization of the governing equations. The main feature of the proposed unified finite element method is that the same finite element space is used for all solutions to the problem including the pressure, velocity, temperature and concentration. Analysis of convergence and stability is also presented in this study and error estimates in the $L^2$-norm are established for the numerical solutions. In addition, due to the Lagrangian treatment of convection terms, the standard Courant–Friedrichs–Lewy condition is relaxed and the time truncation errors are reduced in the diffusion-reaction part. We verify the method for the benchmark problem of moving fronts around an array of cylinders. The numerical results obtained demonstrate the ability of the proposed method to capture the main flow features.

我们研究非等温流动条件下多孔介质中移动前沿数值解的统一有限元方法的性能。控制方程包括将压力的达西方程与用于温度和转换深度的两个对流扩散反应方程耦合。目的是开发一种非振荡统一的Galerkin特征方法，用于有效模拟多孔介质中的运动前沿。该方法基于改进的特征方法与控制方程的Galerkin有限元离散化方法。所提出的统一有限元方法的主要特征是，对所有问题的解决方案，包括压力，速度，温度和浓度，都使用相同的有限元空间。${\mathrm{大号}}^2$建立了数值解的-范数。此外，由于对流项的拉格朗日处理，标准的库兰特-弗里德里希斯-路易​​条件得到了放宽，并且扩散反应部分的时间截断误差减小了。我们验证了围绕圆柱体移动前沿的基准问题的方法。获得的数值结果证明了该方法捕获主要流动特征的能力。