## 1 Introduction  
Cryogenic fracturing [1; 2; 3; 4] is a waterless reservoir stimulation technique which uses cryogenic fluids such as liquid carbon dioxide and liquid nitrogen for fracturing purpose. The technique takes advantage of the effect of thermal shock which presents as a sharp thermal gradient introduced by the cryogen on the rocks. It could enhance the fractures initiation and propagation in the rock stratum, viz. enhancing fracturing. Recently, as the disadvantages of the conventional hydraulic fracturing technique (heavy water consumption, soil pollution, formation damage, etc.) are widely recognized, cryogenic fracturing is gaining an increasing attention in the field of petroleum engineering. Quite a few researches on cryogenic fracturing [5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20] have been carried out by scholars and engineers, in which significant efforts are devoted to investigating the feasibility of cryogenic fracturing for engineering application and figuring out the damage mechanism of cryogenic fracturing. These works are largely comprised of experimental studies in the laboratory. The porous structure of the reservoir stratum makes the fracturing process complicated and it enhances the difficulty of investigation. Nuclear magnetic resonance (NMR) [7; 8], scanning electron microscope (SEM) [9; 10], computed tomography (CT) [11; 12] are widely used to capture the micro-structural changes in the specimens. Fractures are characterized with the aid of acoustic transmission, pressure-decay measurements [12], which enables further quantitative analysis. A series of cryogenic quenching tests (into liquid nitrogen) of specimens [13; 14] were conducted and the evolution of permeability (pore distribution) [15; 16] and mechanical properties (elastic modulus, tensile/compressive strength) [9; 17; 18] were studied. It was found that liquid nitrogen could induce damage to the specimens more efficiently by comparing the methods such as air cooling and water cooling [10]. The strength and brittleness of the shale could be reduced through the treatment of liquid nitrogen cooling, and it could help in decreasing the initiation pressure of reservoir stimulation [19]. It was also revealed that a repeated cooling-heating treatment could be more favorable in fracturing in comparison to a single cooling treatment [4; 7; 8]. Various factors, such as the cooling rate, initial temperature of specimens, the water saturation of specimens, the period of freeze thaw cycle [4], are found to affect the fracturing effect. Besides the experimental works, attempts were made to introduce numerical methods into the research on cryogenic fracturing in recent studies. [PERSON] [21] simulated the cryogenic fracturing process in unconventional reservoirs and evaluated the effect of thermal stresses on fracture development. [PERSON] and co-workers [20] simulated the cryogenic fracturing process in Niobrara shale sample with the modified TOUGH2-EGS. A numerical scheme was developed by [PERSON] and co-workers [14] which could simulate the temperature history and the rewetting front positions in a laboratory quenching test. [PERSON] and co-workers [9] numerically analyzed the influences of the cooling rate on rock degradation by conducting a three dimensional simulation using ABAQUS. Most of the previous researches are focusing on the mechanical changes in the reservoir stratum caused by cryogenic stimulation (initiation and propagation of fracture, rock deformation, evolution of permeability etc.). Nevertheless, only few attentions are aroused to the heat and mass transfer process during cryogenic fracturing. The heat and mass transfer during fracturing would significantly influence the temperature distribution in the reservoir, further affecting the occurrence of thermal shock. Consequently, the heat and mass transfer process would significantly influence the cryogenic fracturing process which should be carefully considered.