Cylindrical Coordinate Analytical Solution for Axisymmetric Consolidation of Unsaturated Soils: Dual Bessel–Trigonometric Orthogonal Expansion Approach to Radial–Vertical Composite Seepage Systems

This study develops a novel analytical solution for three-dimensional axisymmetric consolidation of unsaturated soils incorporating radial–vertical composite seepage mechanisms and anisotropic permeability characteristics. A groundbreaking dual orthogonal expansion framework is established, utilizing innovative Bessel–trigonometric function coupling to solve the inherently complex spatiotemporal coupled partial differential equations in cylindrical coordinate systems. The mathematical approach synergistically combines modal expansion theory with Laplace transform methodology, achieving simultaneous spatial expansion of gas–liquid two-phase pressure fields through orthogonal function series, thereby transforming the three-dimensional problem into solvable ordinary differential equations. Rigorous validation demonstrates exceptional accuracy with coefficient of determination R2 exceeding 0.999 and relative errors below 2% compared to numerical simulations, confirming theoretical correctness and practical applicability. The analytical solutions reveal four critical findings with quantitative engineering implications: (1) dual-directional drainage achieves 28% higher pressure dissipation efficiency than unidirectional drainage, providing design optimization criteria for vertical drainage systems; (2) normalized matric suction variation exhibits characteristic three-stage evolution featuring rapid decline, plateau stabilization, and slow recovery phases, while water phase follows bidirectional inverted S-curve patterns, enabling accurate consolidation behavior prediction under varying saturation conditions; (3) gas-water permeability ratio ka/kw spanning 0.1 to 1000 produces two orders of magnitude time compression effect from 10−2 s to 10−4 s, offering parametric design methods for construction sequence control; (4) initial pressure gradient parameters λa and λw demonstrate opposite regulatory mechanisms, where increasing λa retards consolidation while λw promotes the process, providing differentiated treatment strategies for various geological conditions. The unified framework accommodates both uniform and gradient initial pore pressure distributions, delivering theoretical support for refined embankment engineering design and construction control.