## 1 Introduction  
As a branch of mathematics, the fractional calculus deals with the generalization of integrals and derivatives to all real (and even complex) orders [1, 2]. People have discovered that the fractional calculus can provide more accurate models than the integer calculus to describe the hereditary and memory characteristics of various complex processes and natural phenomena [3]. During the last two decades, the fractional calculus has been widely applied to model the complex processes in the fields of science, engineering and mathematics [1, 4, 5, 6, 7], and so on. In particular, since the fractional calculus was firstly used to describe the viscoelasticity and rheology in 1980s, the fractional-derivative model has achieved great success in simulating the viscoelastic behavior of real materials [1, 8]. However, the application of the fractional-derivative to the one-dimensional consolidation of the viscoelastic soils is scarce. In the geotechnical community, the soil consolidation is one of the most common issues [9, 10, 11, 12]. After the seminal work of [PERSON] [9], a considerable number of one-dimensional consolidation models have been proposed to simulate the consolidation process of the soil [13, 14, 15, 16, 17]. For obtaining a simplified solution, the one-dimensional consolidation models put forward in the early days treated the soil as a linear elastic material. In reality, the soil generally shows obvious rheological properties during the consolidation process. Afterwards, in order to capture the rheological properties of the soil, some viscoelastic models including Maxwell model, Kelvin-Voigt model, Merchant model, four-element model and generalized Kelvin-Voigt model have been introduced to extend the one-dimensional consolidation model of the soil [13, 18, 19, 20, 21]. These consolidation models simulated the rheological properties of the soil using the viscoelastic models with integer derivatives (namely standard viscoelastic model). Recently, the concept of fractional calculus has been introduced to develop the viscoelastic model for describing the rheological phenomena of materials. [PERSON] [22] put forward a series of fractional-derivative models to study the rheological phenomena. [PERSON] [2] derived the creep and relaxation functions of the fractional-derivative models. In the geotechnical community, [PERSON] et al. [23, 24] proposed a fractional-derivative model to describe the viscoelasticity of the soil samples under triaxial conditions. [PERSON] et al. [8] adopted a fractional-derivative model to predict the compression properties of the soft clay. Inspired by the above work, [PERSON] et al. [25] introduced a fractional-derivative model to extend the one-dimensional consolidation theory of viscoelastic soil. However, the available fractional-derivative models are usually defined by the Caputo or Riemann-Liouville fractional derivatives with a kernel of power function [26, 27, 28]. The kernel of power function is investigated to be singular at the end point of the interval [26]. To solve the singularity of the kernel of power function, [PERSON] and [PERSON] recently put forward a new fractional derivative using the kernel of exponential function [27, 28]. This fractional derivative (namely Caputo-Fabrizio fractional derivative) has no singularity and it has been proved to be suitable for some classical physical problems. Using the Caputo-Fabrizio fractional derivative, this paper attempts to establish one-dimensional consolidation model of viscoelastic soils. In order to capture the rheological properties of the soils, the Caputo-Fabrizio operator is adopted to develop a novel four-element fractional-derivative model, and one-dimensional consolidation model is establish to simulate the complex consolidation process of the viscoelastic soils. Using the techniques of eigenfunction expansion and Laplace transform, a series of analytical solutions are derived to calculate the excess pore-water pressure and the average degree of consolidation of the viscoelastic soils.