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.现有针对滨海越流含水层（coastal leaky aquifers）中潮汐波传播（tidal wave propagation）的解析模型，仅局限于采用均质或过度简化非均质性表征的单层或双层系统。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，尚未通过解析方法开展相关研究。本研究构建了一款广义解析模型（generalized analytical model），可兼容任意数量的垂向分层含水层与弱透水层（vertically stratified aquifer and aquitard layers），以及具备不同水力特性（hydraulic properties）的水平分带（horizontally segmented zones）。该模型通过四项关键特征进一步提升通用性：有限侧向范围（finite lateral extent）表征、混合形式侧向边界条件（mixed-form lateral boundary conditions）、分带顶部边界的专属潮汐强迫（zone-specific tidal forcing），以及作用于含水层与弱透水层的分带专属潮汐加载效应（tidal loading effects）。解析解的推导始于建立常微分方程组（ordinary differential equations）耦合系统，用以描述含水层内各分带的潮汐波传播稳态分量与周期分量。求解过程首先采用矩阵特征值分析（matrix eigenvalue analysis）得到各分带的通解，随后通过递归格式实施约束，以满足含水层分带界面间的水头与通量连续性（hydraulic head and flux continuity）。在等效边界条件与系统配置下，新推导的解析解可涵盖多款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渗透性更低的邻近含水层中典型的向陆侧振幅衰减与相移递增现象。这种反向潮汐波传播模式在含水层渗透系数差异显著、导水通道与弱透水层的渗透系数比值较高、导水通道宽度较大时尤为显著，且表现出显著的周期依赖性。针对通过多频潮汐分析（multi-frequency tidal analysis）识别导水通道的场景，应赋予高频分量更高的权重。