王娇(1989—),女,山东滨州人,助理研究员,主要从事土壤物理研究。Email:
边界层方法为土壤溶质迁移研究中描述一维稳态流条件下,仅考虑吸附作用的对流-弥散方程(Convection-dispersion equation,CDE)的求解和参数估算提供了简单可靠的新手段,利用该方法可以有效避免参数反演的不唯一性。对不同边界层解(多项式解、指数解、复合解、对数解和微小通量解)在预测溶质浓度分布和估算溶质迁移参数方面的精度进行了对比研究,归纳了边界层解的适用范围和选取方法。结果表明:边界层解的精度受平均孔隙水流速、弥散系数和延迟因子共同影响,当溶质浓度随剖面深度的变化率呈不断减小趋势时边界层解与精确解最接近;边界层解预测溶质分布误差随时间推移先减小后增加,溶质迁移过程初期三次多项式解精度最高,后期指数解表现最好;边界层方法对延迟因子的估算结果优于弥散系数,且不同边界层解对延迟因子的估算值近似,三次多项式解和对数解估算弥散系数最为准确。
Understanding the behaviours of solute transport in soils is of great importance to agricultural management, resource utilitzation and environmental protection. Introducing boundary-layer theory to solve solute transport problems provides a simple and accurate alternative method to predict solute profile distribution and estimate transport parameters. Appropriate selection of boundary layer solutions requires an overall understanding of the characteristics of boundary layer solution accuracy under different conditions.
This study compared the accuracy of the polynomial solution, exponential solution, combined solution, logarithmic solution and small flux solution based on multiple parameter combinations. Solute front movement with time in soil column experiments was further used to evaluate the performance of boundary layer solutions for parameter estimation.
The accuracy of boundary layer solutions for predicting solute concentration profile increased first and then decreased with time. Comparison of different boundary layer solutions indicated that the cubic polynomial solution was optimal at the beginning while the exponential solution turned to be better afterwards for most cases. Importantly, the boundary layer methods performed better in estimating retardation factor than dispersion coefficient. The retardation factor obtained from boundary layer solutions was almost the same but with an exception of the small flux solution. The dispersion coefficient was greater than the breakthrough curve fitting method and varied between boundary layer solutions. The cubic polynomial and logarithmic solutions had a minimum error in the determination of the dispersion coefficient.
Boundary layer solutions can be used to accurately predict solute profile distribution for the early stage of solute transport processes. Cubic and exponential solutions had better performance than other solutions. Neverthless, cubic and logarithmic solutions can be the first choice for estimating parameters.
了解溶质在土壤中的迁移规律对农业生产、资源利用和环境污染防治等具有重要意义,数学模型是描述溶质迁移过程和预测溶质动态分布的有效手段[
溶质锋深度随时间变化的实测数据来源于郑纪勇[
土柱实验基本设置及基本物理参数
Experimental setup and basic physical parameters of soil columns
土壤 |
土柱高度 |
容重 |
探针数量 |
平均孔隙水流速 |
黄绵土Loess soil | 46 | 1.30 | 4 | 2.14 |
塿土Lou soil | 48 | 1.27 | 4 | 1.25 |
风沙土Sandy soil | 68 | 1.34 | 6 | 1.05 |
红壤Red soil | 35 | 1.43 | 4 | 3.96 |
稳态流条件下,考虑对流、弥散和线性等温吸附作用时,描述一维溶质迁移的CDE方程可写为[
式中,
(1)多项式边界层解。在假设溶质浓度剖面分布可以用二次函数近似表示的基础上,Shao等[
进一步利用三次函数近似表示浓度剖面,得到三次多项式解为:
通过推求更高阶数方程,Wang等[
(2)指数边界层解。魏峰和王全九[
通过分析不同情形下指数边界层解的精度发现,指数解在孔隙流速较大、延迟作用较弱时对参数
(3)复合函数边界层解。为探究指数函数与多项式组合是否具有更高精度,Wang和Shao[
(4)微小通量解。上述方法的求解条件之一是溶质锋处溶质浓度梯度为0,但这与实际情况可能存在一定差异,因此刘春平和邵明安[
研究发现
(5)对数边界层解。同样地,考虑到溶质锋处浓度梯度不为0的情况,Wang等[
利用相对均方根误差(Relative root mean square error,RRMSE)评价各边界层解在预测溶质浓度剖面,其计算方法如下:
式中,
不同边界层解是基于不同的剖面浓度分布假设所获得的,其预测精度之间存在一定的差异。本研究依据文献中的室内及田间观测数据,分别以
边界层解和精确解方法预测1 h溶质浓度剖面对比
Comparison between concentration profiles obtained from boundary layer and exact solution at 1 h
边界层解和精确解方法预测5 h溶质浓度剖面对比
Comparison between concentration profiles obtained from boundary layer and exact solution at 5 h
上述结果表明,边界层解可以用于预测溶质在土壤中的迁移过程和动态分布,但在水流流速较快的情形下精度相对较低,此时溶质主要发生对流运移,由此可知边界层解更适用于弥散作用为主要机制的情形,因此应尽量在溶质流速慢、弥散作用或溶质吸附性较强时选用。这主要因为由多项式函数、指数函数、复合函数以及对数函数获得的溶质浓度剖面分布始终呈现浓度变化率随深度增加而不断减小的趋势,而当平均孔隙水流速大、溶质不被土壤颗粒吸附且弥散作用较弱时,溶质浓度变化在初始阶段表现为随深度增加而不断减小,但剖面上层溶质累积迅速,经过较短时间后溶质浓度变化率就出现随深度先增大后减小的现象,边界层方法中假设的浓度分布曲线难以准确描述此时的溶质分布,因此由边界层方法获取的浓度分布与实际情况存在较大差异。此外,由于填装均质土柱弥散度通常较小,因而在水流流速小的情形下使用边界层解更可能得到准确的结果;田间试验条件下边界层解用于较大尺度土体研究的精度相对更高。
为进一步了解边界层解精度在不同参数条件下的变化规律,
不同参数条件下边界层解预测浓度剖面的相对均方根误差(RRMSE)
The relative root mean square errors of boundary layer solutions for predicting concentration profiles under different conditions
V |
D |
R | 二次多项式解Parabolic solution | 三次多项式解 |
指数解 |
复合解 |
对数解 |
|||||||||
1 h | 5 h | 1 h | 5 h | 1 h | 5 h | 1 h | 5 h | 1 h | 5 h | |||||||
1 | 1 | 1 | 0.0896 | 0.2750 | 0.1709 | 0.3690 | 0.0607 | 0.2373 | 0.1029 | 0.2883 | 0.1305 | 0.2784 | ||||
1 | 10 | 1 | 0.0827 | 0.0485 | 0.0352 | 0.1110 | 0.1284 | 0.0580 | 0.0679 | 0.0555 | 0.0753 | 0.0952 | ||||
5 | 5 | 1 | 0.2750 | 0.4803 | 0.3690 | 0.5895 | 0.2373 | 0.4428 | 0.2883 | 0.4932 | 0.2784 | 0.4556 | ||||
5 | 50 | 1 | 0.0485 | 0.1862 | 0.1110 | 0.2756 | 0.0580 | 0.1484 | 0.0555 | 0.2000 | 0.0952 | 0.2055 | ||||
1 | 1 | 20 | 0.1008 | 0.0534 | 0.0292 | 0.0679 | 0.1487 | 0.0907 | 0.0844 | 0.0457 | 0.0790 | 0.0783 | ||||
1 | 10 | 20 | 0.1322 | 0.1141 | 0.0459 | 0.0332 | 0.1830 | 0.1633 | 0.1143 | 0.0970 | 0.0913 | 0.0834 | ||||
5 | 5 | 20 | 0.0534 | 0.1099 | 0.0679 | 0.1939 | 0.0907 | 0.0762 | 0.0457 | 0.1236 | 0.0783 | 0.1460 | ||||
5 | 50 | 20 | 0.1141 | 0.0759 | 0.0332 | 0.0406 | 0.1633 | 0.1204 | 0.0970 | 0.0619 | 0.0834 | 0.0748 |
不同参数条件下各边界层解的精度存在差异,在
边界层解对溶质迁移过程不同阶段剖面溶质分布的预测精度是不同的,为进一步掌握各边界层解精度的变化规律,本文分析了各边界层解对剖面溶质分布的预测误差随时间的变化,结果见
各边界层解预测浓度剖面的相对均方根误差随时间变化
Variation of RRMSE of boundary layer solutions with time on predicting solute concentration profiles under different conditions
对比边界层解和精确解的浓度表达式可知,在溶质迁移进行较长时间后,由边界层方法计算得到的剖面浓度分布与实际情况将存在很大误差,因此利用该方法预测剖面溶质浓度和估算溶质迁移参数时,选取适宜的边界层解和测定时长对于准确获取结果起着关键作用。在
分别利用各边界层解对黄绵土、塿土、风沙土、红壤等4种不同土壤在土柱实验中的溶质迁移参数进行估算,结果见
不同边界层解方法获得的溶质迁移参数
Solute transport parameters estimated by different boundary layer solutions
黄绵土Loess soil | 塿土Lou soil | 风沙土Sandy soil | 红壤Red soil | ||||||||
①Parabolic solution,②Cubic solution,③Exponential solution,④Combined solution,⑤Logarithmic solution,⑥Small flux solution | |||||||||||
二次多项式解① | 6.59 | 1.15 | 6.60 | 0.89 | 4.24 | 0.98 | 0.02 | 1.55 | |||
三次多项式解② | 6.27 | 1.15 | 6.28 | 0.89 | 3.94 | 0.98 | 0.22 | 1.53 | |||
指数解③ | 6.81 | 1.15 | 6.83 | 0.89 | 4.46 | 0.98 | NA | 1.55 | |||
复合解④ | 6.51 | 1.15 | 6.55 | 0.89 | 4.17 | 0.98 | 0.07 | 1.54 | |||
对数解⑤ | 6.29 | 1.15 | 6.30 | 0.89 | 3.96 | 0.98 | 0.21 | 1.54 | |||
微小通量解⑥ | 0.40 | 1.03 | 0.29 | 0.86 | 0.28 | 0.81 | 0.50 | 0.97 | |||
CXTFIT | 4.44 | 1.16 | 1.47 | 0.75 | 2.09 | 0.93 | 0.62 | 1.01 |
基于CXTFIT拟合获得的参数,分别利用不同边界层解计算下边界层深度并与精确解结果进行比较分析,各边界层解的相对误差如
边界层解计算溶质锋深度的相对误差
Relative error of boundary layer solutions in determining the solute front depth
精确解和溶质浓度剖面对比
Comparison of solute concentration profiles obtained by the boundary layer and exact solutions
边界层方法能够准确描述早期的溶质剖面分布情况,但不适宜用于研究以对流迁移为主的溶质迁移过程,即溶质流速较大、弥散作用和吸附性均较弱的情形。利用边界层解计算短历时内的溶质浓度剖面分布时,多数情况下可优先选择三次多项式解,流速快且弥散作用弱的情形选用指数解或复合解最优,计算剖面上部溶质浓度分布时可选用对数解。边界层解能够更加准确地估算延迟因子,且不同边界层解获得的延迟因子数值近似,估算弥散系数时应选用三次多项式解和对数解。溶质锋深度测量误差是边界层方法误差产生的主要原因之一,因此测定时应尽量选用检测灵敏度较高的仪器或对测定深度进行校正。
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