## 1 Introduction  
The PVD technique is a traditional foundation treatment method for its high efficiency to accelerate the drainage of air and water in soil [1, 2, 3, 4, 5], and thus shorten the consolidation process of a foundation. In fact, the shorter drainage paths along the horizontal direction will be produced when a PVD is installed in a soil layer, resulting in the efficient radial dissipation of excess pore pressures. For the PVD, however, the construction process itself is an external disturbance, densifying the soil around the drain (i.e., smear effect). Besides, drain resistance will occur even though the drain is filled with highly permeable materials, which have been studied comprehensively in saturated soils [6, 7, 8, 9, 10, 11, 12, 13, 14]. Nevertheless, the soils in semi-arid regions around the world are always under a partially saturated state, whose consolidation results calculated with saturated consolidation theory could directly produce unacceptped errors. In drought regions, the soils of foundation are always under the unsaturated state because of the low groundwater levels [15]. Under this circumstance, the designs of the foundations based on the consolidation theory in saturated soils may be uneconomical, because the dissipation time of excess pore pressures is misestimated. Therefore, [PERSON] and his colleagues [16, 17, 18] researched the consolidation theory of unsaturated soils by the two-stress variables constitutive model, based on which the consolidation characteristics were analyzed by later scholars. For example, the variations of the excess pore-air and pore-water pressures were first researched by [PERSON] et al. [19], in which the Laplace transform method is adopted to solve the partial differential equations of the unsaturated soil layer. [PERSON] et al. [20] proposed the semi-analytical solutions considering that the soil layer is multi-layered, and the boundary is time-dependent. Later, [PERSON] et al. [21; 22; 23] analyzed the one-dimensional consolidation patterns of the soil layer under the unsaturated condition by considering that the foundation is single- and multi-layered, and the mixed boundary condition was investigated. With the consideration of various initial and boundary conditions, [PERSON] et al. [24; 25; 26] proposed more generalized investigations of the one-dimensional consolidation. In real foundation engineering, the upper surfaces are not ideally permeable or impermeable, and [PERSON] et al. [27; 28] comprehensively investigated the third type of boundary condition in unsaturated soils with semi-analytical methods. Afterwards, [PERSON] and [PERSON] [29] and [PERSON] et al. [30; 31] extended the solutions of consolidation from one-dimensional to two-dimensional situations. By taking the unsaturated and saturated conditions into account, [PERSON] et al. [32] presented semi-analytical solutions to illustrate the distributions of excess pore pressures in depth. In 2010, a vertical drain foundation modeling under the partially saturated condition with the radial dissipation of air and water phases was first put forward by [PERSON] et al. [33], in which the Laplace transform and the Crump method were applied to obtain the semi-analytical solution. [PERSON] presented the numerical solutions [34] (with the utilization of the differential quadrature method), and the analytical solutions [13] (by the decoupling method) of the unsaturated vertical drain foundation considering the radial dissipation. According to the difference of the initial values of the excess pore pressures along the depth, [PERSON] et al. [35] derived the analytical solution to explore the relatively more practical consolidation characteristics with the linear-distributed initial excess pore pressures. Moreover, [PERSON] and [PERSON] [36]; [PERSON] and [PERSON] [37] researched the consolidation of an unsaturated vertical drain foundation considering the vertical dissipation, free-strain and equal-strain assumptions with analytical methods. More practically, [PERSON] et al.