# One-Dimensional Consolidation of Multi-Layered Unsaturated Soil with Impeded Drainage Boundaries
## 1 Introduction  
In geotechnical engineering, the consolidation of unsaturated soil is one of the most important subjects. Because of having both an air and water phase, unsaturated soil shows much more complex properties [1; 2; 3]. In order to describe the behavior of unsaturated soil undergoing compression and consolidation, [PERSON] et al. [4] proposed a constitutive model of unsaturated soil on the basis of framework of hardening plasticity with two independent sets of stress variables. [PERSON] and [PERSON] [5] developed an elastoplastic framework to account for the volumetric behavior of unsaturated soil. [PERSON] et al. [6] discussed the coefficient of consolidation of unsaturated soil on the basis of a set of experimental results. For the consolidation of unsaturated soil, the past few decades have witnessed significant progress in developing various consolidation theories [7; 8; 9] and the most popular one was propounded by [PERSON] and [PERSON] [10], which used a second-order partial differential system to characterize the dissipating phenomenon of excess pore-air pressure and that of excess pore-water pressure simultaneously. [PERSON] and [PERSON] [11] further employed the assumption that all the soil parameters do not change during the consolidation process to develop a simplified form of the previous consolidation system. Afterwards, this consolidation system was expanded to the two- and three-dimensional situations [12; 13] and it has also extensively inspired a noticeable number of studies on the consolidation behavior of unsaturated soil [14; 15; 16]. However, most studies treated both the top surface and bottom base as fully drained or absolutely undrained boundaries for seeking thetheoretical solutions more easily. In reality, these boundaries are commonly impeded drainage [17; 18; 19; 20]. To capture this feature, [PERSON] [21] defined a novel impeded drainage boundary by taking advantage of the third kind of boundary condition. This boundary can approach the fully drained one or undrained one by changing the drainage parameter. Using this boundary, [PERSON] and [PERSON] [22] proposed a numerical solution for the 1D consolidation system of unsaturated soil on the basis of the differential quadrature method. [PERSON] et al. [23; 24] derived semi-analytical solutions for this topic by considering the impeded drainage boundary conditions in one way or the double way. However, these studies are only confined to a simple homogenous soil stratum. In nature, the soil stratum generally consists of multi-layers due to the complex sedimentation. Under this situation, it seems to be unreasonable to treat this type of soil stratum as a homogenous one. Therefore, it is significant to address the consolidation problem of multi-layered unsaturated soil. To the best knowledge of the authors, however, the studies on the consolidation of multi-layered unsaturated soil are rare. [PERSON] et al. [25] focused on the 1D consolidation system of multi-layered unsaturated soil only catering to the situation that all the soil layers have the same coefficient of air volume change and also have the same coefficient of water volume change. Actually, these coefficients generally vary with increasing depth. Moreover, they only considered ideal single-drainage boundary condition. For the case of the impeded drainage boundary condition, [PERSON] et al. [26] introduced the well known incremental differential quadrature method (IDQM) to formulate a numerical method for assessing the 1D consolidation behavior of multi-layered unsaturated soil. This kind of numerical solution shows a good accuracy and efficiency, but there are still shortcomings in the application to mixed boundary [27; 28]. In particular, because of the cumbersome and complex mathematical derivations, it is very difficult to be used by practicing engineers. Therefore, it is crucial to derive an accurate and simple solution for assessing the 1D consolidation behavior of multi-layered unsaturated soil.