Source code of analytical solution

Existing analytical models for tidal wave propagation in coastal leaky aquifers remain limited to one- or two-layer systems with either homogeneity or oversimplified heterogeneityrepresentations. Complex heterogeneity patterns, such as the prevalent occurrence of aquitard preferential conduits in multilayered systems, have not been addressed analytically. This study develops a generalized analytical model that accommodatesan arbitrary number ofvertically stratified aquifer and aquitard layers and horizontally segmented zones with distinct hydraulic properties. The model’s generality is enhanced by incorporating four key features, including the finite lateral extent representation, mixed-form lateral boundary conditions, zone-specific tidal forcing at the top boundary, and zone-specific tidal loading effects on both aquifer and aquitard layers. Analytical solution development begins with the establishment of a coupled system of ordinary differential equations governing zone-specific steady-state and periodic components of tidal wave propagation in aquifer layers.The solution is derived by employing matrix eigenvalue analysis to obtain the zone-specific general solution, followed by the recursive formulation implementation to enforce hydraulic head and flux continuity across aquifer zone interfaces.Under equivalent boundary conditions and system configurations, the newly derived analytical solution is shown to subsume multiple existing analytical solutions as special cases. Theoretical investigations reveal that aquitard preferential conduits can reverse the typical landward amplitude decay and phase shift increase in the less permeable adjacent aquifer. Such a reversed tidal wave propagation pattern is pronounced with strong aquifer permeability contrasts, high conduit-to-aquitard permeability ratios, and sizable conduit widths, and exhibits remarkable period dependence. For conduit identification through multi-frequency tidal analysis, greater weight should be assigned to higher-frequency components.

现有针对海岸越流含水层（coastal leaky aquifer）的潮汐波传播解析模型，仅局限于单层或双层含水层系统，且仅能处理均质化或过度简化的非均质性表征。诸如多层系统中普遍存在的弱透水层（aquitard）优先导水通道（preferential conduits）这类复杂非均质性模式，目前尚未有解析解研究涉及。本研究构建了一套广义解析模型，可适配任意数量的垂向分层含水层与弱透水层，以及具有不同水力特性的水平分区系统。该模型通过纳入四大核心特性进一步提升了通用性，包括有限侧向范围表征、混合形式侧向边界条件、顶边界分区潮汐强迫（tidal forcing），以及针对含水层与弱透水层的分区潮汐加载效应（tidal loading effects）。解析解的推导首先建立了耦合常微分方程组，用于描述含水层中潮汐波传播的分区稳态与周期分量。求解过程首先通过矩阵特征值分析得到分区通解，随后采用递推公式实现含水层分区界面处的测压水头（hydraulic head）与通量连续性（flux continuity）约束。在等效边界条件与系统构型下，本研究新推导的解析解可涵盖多种现有解析模型作为其特例。理论研究表明，弱透水层优先导水通道可逆转弱透水相邻含水层中典型的向陆振幅衰减与相位偏移递增现象。这种反向潮汐波传播模式在含水层渗透性差异显著、通道与弱透水层渗透率比值较高、通道宽度较大的条件下表现尤为明显，且呈现出显著的周期依赖性。若通过多频潮汐分析识别导水通道，应赋予高频分量更高的权重。