MATHEMATICAL ANALYSIS OF ONE-DIMENSIONAL SOLUTE TRANSPORT IN A LAYERED SOIL PROFILE
MATHEMATICAL ANALYSIS OF ONE-DIMENSIONAL SOLUTE TRANSPORT IN A LAYERED SOIL PROFILE
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DOI：
10.2136/sssaj1991.03615995005500040008x
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发表时间：
1991-07
影响因子：
2.9
通讯作者：
F. Leij;M. Genuchten;J. H. Dane
中科院分区：
文献类型：
--
作者：
F. Leij;M. Genuchten;J. H. Dane
涉及层状介质的溶质运移研究对于研究限制层如何影响土壤剖面中溶质迁移速率非常重要，更广泛地说，对于检查土壤异质性对溶质运移的影响非常重要。借助用于两层土壤剖面传输的拉普拉斯变换，获得了一维平流-弥散方程 (ADE) 的解析解。假设这些层实际上是半无限的，则在入口边界和两层界面处获得第一类（恒定浓度）和第三类（恒定通量）条件的解。还通过拉普拉斯变换解的数值反演获得了有限第一层的浓度分布，在入口处使用第三类型条件，并且同时在界面处使用第一和第三类型条件。所有情况下均使用体积平均浓度或居民类型浓度。第一类条件不符合我们的质量守恒标准，而第三类条件导致具有不同传输参数的层界面处的浓度不连续。当在界面处同时施加第一类和第三类条件时，发现界面处的浓度是连续的，并且没有发生质量平衡误差。几个示例计算显示了土壤分层对一维土壤剖面中溶质迁移的影响。 I 土壤中化学品的运输是由于农业和其他化学品（化肥、农药、工业废物）从土壤表面通过非饱和带向地下水位移动的潜力。非饱和带中溶质的淋滤可能会因土层的存在而受到很大影响。土壤分层是许多土壤常见的自然现象。此外，人工屏障（例如粘土衬里）通常用于减缓或阻止某些化学品的移动。最后，土壤分层的宏观影响对于土壤异质性和各向异性对溶质迁移影响的更一般研究非常重要。例如，异质土壤剖面通常由一系列均质土层来近似。尽管也存在替代的随机方法，但通常使用 ADE 确定性地描述土壤中的溶质迁移（Jury，1982）。现在，对于有限的几种情况，可以使用层状土壤中传输的精确和近似分析解决方案。然而，在大多数情况下，需要采用数值方法。不幸的是，与解析解相比，数值解通常无法深入了解物理和化学过程对溶质输运的影响，并且通常还容易受到计算错误的影响。因此，解析解对于各种应用仍然有用，包括数值解的验证。 F.J. Leij 和 M.Th. van Genuchten，美国盐度实验室，4500 Glenwood Drive, Riverside, CA 92501；和 J.H.戴恩，部门。奥本大学农学和土壤学系，奥本，AL 36849。阿拉巴马州农业。过期。站。期刊编号3-892244P。收稿日期：1990 年 4 月 23 日。‘通讯作者。发表于土壤科学。苏克。是。 J.55：944-953（1991）。过去关于层状土壤中迁移的研究主要集中在层状对溶质突破曲线的影响上。 Shamir 和 Harleman (1966)、Selim 等人报道了实验结果。 （1977）和莫兰维尔等人。 （1977）。 Kreft (198 la) 以及 Barry 和 Parker (1987) 使用传递函数和时间矩对溶质突破曲线进行了理论分析。然而，在许多情况下，了解分层分布中的实际溶质分布也很重要。 Shamir 和 Harleman (1967) 以及 Al-Niami 和 Rushton (1979) 报道了溶质迁移的分析解决方案。两项研究均使用第一类（恒定浓度）入口条件。 Shamir 和 Harleman (1967) 假设剖面由半无限层组成，而 Al-Niami 和 Rushton (1979) 使用有限层。由于只有入口和界面处的浓度是连续的（狄利克雷条件），因此两种解决方案都不能满足界面处的质量守恒原理，因此，所施加的溶质质量不等于土壤剖面中累积的质量。 Al-Niami 和 Rushton (1979) 还强加了物理上不切实际的假设，即层界面处的浓度梯度为零。本研究的目的是使用数值和分析方法模拟层状土壤中的一维溶质运移。与大多数先前的研究一样，层状轮廓近似于一系列均匀层，每个层都有其独特的物理和化学特性。使用入口和界面条件的三种不同组合获得了两层介质的分析解。由于边界和界面条件在层状介质的 ADE 求解中起着重要作用，因此简要回顾了适用的边界条件。使用解析解和数值解定性地说明了分层对分层土壤剖面中计算的溶质浓度的影响。理论 两层介质的分析解 图 1 示意性地显示了在垂直于两层界面的稳定水流期间由两个均匀层组成的分层土壤。第一层位于 x = 0 和 x = L 之间，第二层位于 x = L 和无穷大之间。两层中线性交换溶质的传输用 ADE 描述： dCk dx £* dx t> 0 [1] 944 其中 R 是延迟因子，C 是以每体积溶液中溶质质量表示的驻留浓度 [M L-]，t 是时间 [T]，D 是分散系数 [L T-']，v 是平均孔隙水速度 [L T"]，x 是距离 沿水流方向 [L]，下标 k 指土层 (k = 1,2)。我们强调，在本研究中，C 代表体积平均或驻留浓度（van Genuchten 和 Parker，1984），而不是 Barry 和 Parker（1987）所采用的通量平均浓度。体积和通量平均浓度根据（例如，Kreft 和 Zuber， 1978) LEU 等人：一维溶质输运 945
Solute transport studies involving layered media are important for investigating how restricting layers affect rates of solute migration in the soil profile and, more generally, for examining the influence of soil heterogeneity on solute transport. Analytical solutions of the one-dimensional advection-dispersion equation (ADE) were obtained with the help of Laplace transforms for transport in a two-layered soil profile. Assuming that the layers are, in effect, semi-infinite, solutions were obtained for first-type (constant concentration) and third-type (constant flux) conditions at both the inlet boundary and the interface of the two layers. Concentration profiles were also obtained for a finite first layer via numerical inversion of the Laplace transform solution, using a third-type condition at the inlet, and, simultaneously, a firstand third-type condition at the interface. Volume-averaged or resident-type concentrations were used in all cases. First-type conditions did not meet our criterion of mass conservation, whereas third-type conditions caused discontinuities in the concentration at the interfaces of layers with differing transport parameters. The concentration at the interface was found to be continuous, and no mass-balance error occurred, when firstand thirdtype conditions were imposed simultaneously at the interface. Several example calculations show the effect of soil layering on solute transport in a one-dimensional soil profile. I IN THE TRANSPORT of chemicals in soils is motivated by the potential of agricultural and other chemicals (fertilizers, pesticides, industrial wastes) to move from the soil surface through the unsaturated zone toward the groundwater table. The leaching of solutes in the unsaturated zone may be affected greatly by the presence of soil layers. Soil stratification is a natural phenomenon that is common to many soils. Also, artificial barriers (e.g., clay liners) are often used to slow down or prevent the movement of certain chemicals. Finally, the macroscopic impact of soil layering is important in more general studies of the effect of soil heterogeneity and anisotropy on solute transport. For example, a heterogeneous soil profile is often approximated by a series of homogeneous soil layers. Solute transport in soils is usually described deterministically with the ADE, although alternative stochastic approaches also exist (Jury, 1982). Exact and approximate analytical solutions for transport in layered soils are now available for a limited number of situations. In most cases, however, numerical methods need to be employed. Unfortunately, compared with analytical solutions, numerical solutions frequently do not provide as much insight into the effects of physical and chemical processes on solute transport, and often are also susceptible to computational errors. Hence, analytical solutions remain useful for a variety of applications, including verification of numerical solutions. F.J. Leij and M.Th. van Genuchten, U.S. Salinity Lab., 4500 Glenwood Drive, Riverside, CA 92501; and J.H. Dane, Dep. of Agronomy and Soils, Auburn University, Auburn, AL 36849. Alabama Agric. Exp. Stn. Journal no. 3-892244P. Received 23 Apr. 1990. 'Corresponding author. Published in Soil Sci. Soc. Am. J. 55:944-953 (1991). Past research on transport in layered soils has focused mainly on the effects of layering on solute breakthrough curves. Experimental results have been reported by Shamir and Harleman (1966), Selim et al. (1977), and Moranville et al. (1977). Theoretical analyses of solute breakthrough curves, using transfer functions and time moments, were carried out by Kreft (198 la) and Barry and Parker (1987). In many cases, however, it is also important to know the actual solute distribution in a layered profile. Analytical solutions for solute transport have been reported by Shamir and Harleman (1967), and Al-Niami and Rushton (1979). Both studies used a first-type (constant concentration) inlet condition. Shamir and Harleman ( 1 967) assumed that the profile consisted of semi-infinite layers, while Al-Niami and Rushton (1979) used finite layers. Since only the concentration was continuous at the inlet and interfaces (Dirichlet conditions), neither solution will satisfy the principle of mass conservation at interfaces and, as a result, the applied solute mass is not equal to the mass accumulated in the soil profile. Al-Niami and Rushton (1979) also imposed the physically unrealistic assumption that the concentration gradient is zero at the interfaces of the layers. The objective of this study was to simulate onedimensional solute transport in a layered soil using both numerical and analytical methods. As with most previous studies, the layered profile was approximated by a series of homogeneous layers, each having its unique physical and chemical characteristics. Analytical solutions were obtained for a two-layer medium using three different combinations of inlet and interface conditions. Since the boundary and interface conditions play an important role in the solution of the ADE for layered media, the applicable boundary conditions are briefly reviewed. The effect of layering on calculated solute concentrations in a layered soil profile is illustrated qualitatively, using both analytical and numerical solutions. THEORY Analytical Solution for a Two-Layer Medium Figure 1 schematically shows a layered soil, consisting of two homogeneous layers, during steady water flow perpendicular to the interface of the two layers. The first layer is located between x = 0 and x = L, and the second layer between x = L and infinity. Transport of a linearly exchanging solute in both layers is described with the ADE: dCk dx ££* dx t> 0 [1] 944 where R is the retardation factor, C is the resident concentration expressed in mass of solute per volume of solution [M L-], t is time [T], D is the dispersion coefficient [L T-'], v is the mean pore-water velocity [L T"], x is distance in the direction of flow [L], and the subscript k refers to the soil layer (k = 1,2). We emphasize that, in this study, C represents volume-averaged or resident concentrations (van Genuchten and Parker, 1984) rather than flux-averaged concentrations as employed by Barry and Parker (1987). Volumeand flux-averaged concentrations are related according to (e.g., Kreft and Zuber, 1978) LEU ET AL: ONE-DIMENSIONAL SOLUTE TRANSPORT 945