A new semi-analytical solution to the advection–dispersion–reaction equation for modelling solute transport in layered porous media is derived using the Laplace transform. Our solution approach involves introducing unknown functions representing the dispersive flux at the interfaces between adjacent layers, allowing the multilayer problem to be solved separately on each layer in the Laplace domain before being numerically inverted back to the time domain. The derived solution is applicable to the most general form of linear advection–dispersion–reaction equation, a finite medium comprising an arbitrary number of layers, continuity of concentration and dispersive flux at the interfaces between adjacent layers and transient boundary conditions of arbitrary type at the inlet and outlet. The derived semi-analytical solution extends and addresses deficiencies of existing analytical solutions in a layered medium, which consider analogous processes such as diffusion or reaction–diffusion only and/or require the solution of complicated nonlinear transcendental equations to evaluate the solution expressions. Code implementing our semi-analytical solution is supplied and applied to a selection of test cases, with the reported results in excellent agreement with a standard numerical solution and other analytical results available in the literature.

中文翻译：

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多层多孔介质中对流-弥散方程的新半解析解

使用拉普拉斯变换推导出对流-弥散-反应方程的一种新的半解析解，用于模拟层状多孔介质中的溶质输运。我们的解决方法包括在相邻层之间的界面引入表示色散通量的未知函数，允许在拉普拉斯域中的每一层上分别解决多层问题，然后再数值反演回时域。推导出的解适用于线性平流-弥散-反应方程的最一般形式、包含任意层数的有限介质、相邻层之间界面处的浓度和色散通量的连续性以及任意类型的瞬态边界条件入口和出口。派生的半解析解扩展并解决了分层介质中现有解析解的缺陷，这些解仅考虑类似的过程，例如扩散或反应扩散，和/或需要复杂非线性超越方程的解来评估解表达式。提供了实现我们的半解析解决方案的代码，并将其应用于一系列测试用例，报告的结果与标准数值解决方案和文献中提供的其他解析结果非常吻合。