考虑尺度效应的瞬时输入溶质运移模型及解析解

Analytical solution for scale-dependent solute transport model with instantaneous source

为探索土壤环境中尺度效应对溶质运移的影响,建立了瞬时输入条件下考虑尺度效应的溶质运移模型。通过Laplace变换和复变函数理论得到了模型的解析解,并利用解析解分析了弥散尺度效应对溶质运移过程的影响。结果表明:随着土壤弥散尺度效应的增强,土壤中溶质浓度分布范围越广,浓度峰运移的距离越大,但浓度峰值越小;随着入口弥散系数(D0)的增加,溶质运移的范围更大,溶质浓度峰值越小,但浓度峰运移的距离几乎没有变化;随着入口孔隙水流速度(v0)的增加,溶质浓度峰运移的距离越大,溶质的运移锋面越远,而溶质浓度峰值及溶质浓度分布范围几乎没有变化。用一维8 m长土柱中的溶质运移试验资料对所推解析解进行验证,模拟结果与试验结果吻合较好,决定系数可达0.95以上。结果表明所推得的解析解可用来模拟预测较大尺度上溶质运移过程,为土壤环境治理等实际工程提供理论依据。

Solute transport in soil and groundwater is important for water resources management, crop production, and groundwater quality control. Accurate prediction of the transport of solutes is crucial to the effective management of these processes. Traditionally, transport is described with the convection-dispersion equation （CDE） with constant coefficients, but practical problems generally involve non-uniform velocity fields and variable diffusion coefficients due to the complex nature of medium heterogeneity on different scales. Therefore, this study investigated the solute transport behavior along an unsteady flow domain through heterogeneous porous media, and presented a one-dimensional scale-dependent solute transport model, considering linear equilibrium sorption with an instantaneous source in a semi-infinite porous media in order to get a better understanding of the solute transport in soil. The proposed model was analytically solved by using the Laplace transformation technique, with the inverse transformation based on the complex formulation theory. The model has an empirical constant （a）, which represents different media having different heterogeneous natures. The effects of a, the inlet dispersion coefficient （D0）, and the inlet velocity （v0） at the origin of the soil column on the solute transport were analyzed. The results indicated that the spreading of the solute concentration distribution curves increased with the increase of a or D0 and the peak concentrations decreased with the increase of a or D0 with little change to the v0. The arrival distances of the peak concentrations of the solute concentration distribution curves increased with the increase of a or v0 with little change to the D0. The applicability of the analytical solution was verified experimentally in the analysis of the measured data of conservative solute transport obtained from an 8-m-long soil column in the literature. In order to avoid the effect of the exit boundary on the solute transport parameters estimation, observed and model-simulated breakthrough curves （BTC） at the 6 m location were compared, and the optimized parameter values were estimated. The fitting result of the analytical solution at the 6 m location was in satisfactory agreement with the measured concentrations, and the values of determination coefficient R2, root mean square error RMSE, and mean relative error P were 0.9709, 0.0101, and 20.82%, respectively. The parameters estimated at the 6 m location could be used to predict the solute transport process. The BTCs at the 2, 4 and 8 m locations were simulated using the estimated parameter values and compared with experimental data. The statistical analysis indicated that the R2, RMSE, and P between the measured and estimated BTCs were 0.8951, 0.0140, and 28.24%at 8 m, 0.9557, 0.0148, and 22.38%at 4 m, and 0.1549, 0.0931, and 47.36%at 2 m, respectively. The results showed that the prediction from the proposed analytical solution was rather similar with the experimental data at the 4 m location. The prediction was similar at the 8 m location due to the effect of the exit boundary and poor at the 2 m location due to the greater impact of the lower flow rate at the early stages of the experiment on the breakthrough curves at a smaller distance. The proposed analytical solution will be valuable for assessing the stability of numerical solutions in more realistic dispersion problems or describing solute transport at relatively large scales in heterogeneous porous media. The study provides valuable information for in practical applications of the reasonable fertilization and soil water environmental pollution forecasting.