# A Fractional-Order Creep Model of Water-Immersed Coal
## 1 Introduction  
The coal pillar dam is an important part of coal mine underground reservoirs (CUR) [1; 2; 3], which isolate the various water purification areas and support the overburden rock [4; 5]. In such a complex environment, the coal pillars are usually subjected to a hydro-mechanical-chemical coupled process. Since coal has significant creep behavior, especially in the water-immersed conditions [6; 7; 8], if creep-induced instability occurred in coal pillars, the supporting and isolating effects of the coal pillar will disappear, consequently destroying the CUR; therefore, the long-term stability of the coal pillar dam is a serious concern for CUR. Research on the short-term stability of CUR is numerous and great success has been achieved using innovative techniques [9; 10; 11; 12]. In recent years, the long-term performance of CUR, especially concerning the coal pillars of CUR, aroused great interest in the research community because of the safety issue brought by the impact of circular flooding and water-immersion to coal pillars [13]. Evidence from the laboratory suggests that water immersion has a significant influence on the mechanical performance of coal [14; 15] because of the water-softening effect on the coal rock [16; 17]. Based on the acoustic emission features of the coal-rock combined samples under different immersion durations, refs. [18; 19; 20; 21] found that water has a deteriorating effect on the mechanical properties of the coal-rock combined samples as well. In the work of a stability assessment of the coal pillars in a CUR, a proper creep model of coal rock is a priority for numerical modeling or calculating works. Currently, the majority of research is focused on how the water content affects coal strength, whereas there is still a lack of studies on the coal creep process in water-immersion conditions [22; 23]. In practical engineering, coal pillar dams are often exposed to groundwater immersion for a long period of time, leading to creep under the overburdened rock [24; 25; 26]. In order to accurately describe the creep performance of coal columns under the influence of the softening effect of water immersion, it is necessary to establish a suitable creep model for water-immersed coal, which reflects the deformation characteristics of coal at various stages. According to the summary of [27], the creep constitutive model for rock materials has three types, namely, empirical models, component models, and mechanism-based creep constitutive models. Empirical models are derived from experimental observations, which are usually expressed by a power law or an exponential function [28]. However, the difference in time scale between an experimental study and a practical engineering project leads to an error between the predicted and measured creep deformations [29; 30]. The classic component combination models include the Maxwell model, the Kelvin model, the Bingham model, the Burgers model, and the Nishihara model, with clear physical meanings [31; 32]. Considering the creep damage accumulation, mechanism-based creep models are commonly established according to the viscoelastic and viscoplastic behavior of rock during the creep process [33; 34; 35]. On account of the complex nonlinear behavior of creep deformation in rock materials, the fractional derivative theory was introduced into the creep model, where [PERSON] et al. [27; 36] proposed a creep constitutive model based on the fractional derivative with an Abel damper instead of a Newton damper, which well reflects the three stages of salt rock, especially the accelerated creep stage. [PERSON] et al. [37] developed a fractional derivative constitutive model to describe the creep process of deep coal rock, which depicts well the accelerated creep stage. [PERSON] et al. [38] proposed a fractional-order creep model of coal and rock to characterize the changes in each loading stage.