## 1 Introduction  
Being a green energy source, natural gas has recently attracted increased scientific and industrial interest, and shale gas accounts for more than 63% of the total global sources of unconventional gas [1]. The economic exploitation of shale gas therefore has the ability to satisfy the increasing world energy consumption [2]. Although the extremely low permeability of the shale matrix obstructs gas migration from the matrix to the production wells, some reservoir-enhancement techniques, such as hydro-fracturing, can significantly enhance permeability by creating a fracture network between the production wells and the shale matrix, that eventually enhance shale gas productivity [3; 4; 5]. For the hydraulic fracturing process, water-based fracturing fluids are currently used. However, this has raised many issues, including formation damage, long clean-up times, excessive water consumption and wastewater generation [6; 7; 8; 9]. Liquid CO\({}_{2}\) with low viscosity can be used as an alternative to water-based fracturing fluids [10], which avoids most of the issues created by water-based fluids. The use of CO\({}_{2}\) as a fracturing fluid also contributes to CO\({}_{2}\) geo-sequestration, which has already been successfully verified in some shale formations [11]. During the fracturing process, CO\({}_{2}\) is often injected into reservoir formations in its liquid or super-critical state in a short time, and high pressurized CO\({}_{2}\) can fracture the shale matrix or open up existing fractures [12]. However, the low viscosity of CO\({}_{2}\) and its high penetration capacity cause very fast leak-off rates [13], and insufficient leak-off control can cause unacceptably high injection rates and excessive CO\({}_{2}\) consumption, challenging the capacity of the fracturing pump. Consequently, treatment using CO\({}_{2}\) may not be applicable to relatively high permeable reservoirs for hydraulic fracturing. After the completion of the fracturing process, CO\({}_{2}\) flows from the shale matrix to the wellhead through the fracturing wells, where CO\({}_{2}\) is easily subjected to phase transition among gaseous, liquid and super-critical states because the physical properties of CO\({}_{2}\) are highly pressure- and temperature-dependent [14]. The huge pressure gradient induced near the fracture points upon the release of injection pressure may cause fine particle migration, which can move from the reservoir to the well along the CO\({}_{2}\) flow [15]. Furthermore, since injecting CO\({}_{2}\) into depleted oil and gas reservoirs and saline aquifers also contributes to the geological storage of CO\({}_{2}\), the safe storage of injected CO\({}_{2}\) in these formations depends on the ultralow permeable rock layers lying above the formation [16; 17]. As the primary security key for CO\({}_{2}\)-based hydraulic fracturing and the associated geological storage of CO\({}_{2}\), a full understanding of CO\({}_{2}\) flow behavior in low permeable formations under different in situ environments is therefore essential, otherwise, erroneous leak-off rate estimations based on a vague understanding of CO\({}_{2}\) flow behavior in gas reservoirs can lead to inaccurate estimation of total CO\({}_{2}\) usage and prediction of the CO\({}_{2}\) injection rates required for fracturing. On the other hand, the complicated thermodynamic properties of CO\({}_{2}\) increase the complexity of precisely estimating the apparent permeability of shale for CO\({}_{2}\). Various methods have been developed to measure the permeability of reservoir rock with ultralow permeability and studies have been performed under both steady-state and transient conditions [18].