Numerical Methods For Non-equilibrium Solute Transport In Porous Media

In recent years, with the development of agriculture and industry, the discharge of large amount of waste gas, waste water and industrial residue, the widely use of pesticides and fertilizer, the increasing growth of radioactive wastes when using the nuclear energy, the increasing garbage caused by population growth and urbanization are causing more and more damages. The transport and pollution of these contaminants in soil and water have become worldwide problems. The treatment of all these problems are based on theory about the water movement and solute transport in soil.Convection-Dispersion Equation is one of basic motion equations about solute transport in soil. This simple model have a solid physics foundation and could describe the transport of solute, heat, reaction diffusion process, etc. However, according to the complexity of the practical conditions, this model cannot be used in the field of large scale solute transport problems. The non-equilibrium solute transport two-region model (TRM) is based on the convection-dispersion model. Two-Region Model have taken the immobile water, influence of absorption and degradation into account, which is more accurately to describe the solute transport process in soil.For one-dimensional simple models, Van Genuchten had given some forms of analytical solutions for different initial-boundary value problems. For more complex problems, it is difficult to present the analytical solutions and numerical simulation is a kind of effective method. The basic numerical simulation method is FDM and FEM, but the traditional FDM are not efficient for some models with convection term. Due to the hyperbolic properties of convection-dominate dispersion equations, the central difference formula often cause numerical dispersion and oscillation even it has two-order precision in space. The characteristics combing the difference or finite element method can be better to reflect the first-order hyperbolic properties of convection-dispersion equations. The characteristic methods, discrete the equation along the characteristic line according to physical and mechanical properties, can effectively overcome nonphysical oscillation and reduce truncation errors and greatly improve the calculation accuracy. Cui Ming and ZhangDe sheng had impled the characteristic finite element method to two-region model and made an assay of numerical calculation. In the second chapter of this paper, we applied the characteristic difference method to non-equilibrium solute transport models. We first propose the discrete scheme of this method, then give rigorous proof of optimal l~2 error estimate. In the end of this section, we compared numerical discrete schemes with analytical solutions under special conditions to test the efficiency of characteristic difference method and ensure the correctness of the numerical model.Although the method of characteristics is efficiently to solve convection-dominate problems, the mathematical scheme is quite complicated in practical application, because the need for special treatment when exceeding the border and temporal and spatial interpolation are also needed especially for computing boundary. Prof. Yuan Yi-rang had presented a class of modified second-order upwind difference schemes to compute compressible two-phase displacement problem. This method could efficiently overcome the numerical oscillation and improve the calculation accuracy in spatial to second order. In chapter three, we compute the model using the modified upwind difference method and derive optimal discrete l~2 error estimates. Compared with the exact solutions, we test the correctness of discrete scheme. For each schemes (characteristics and modified upwind difference), the error in calculation is small and wonderful numerical results are achieved.