# Study on the Wellbore Flow for Carbon Dioxide Fracturing
## 1 Introduction  
Carbon dioxide (CO\({}_{2}\)) is an environmentally friendly waterless working fluid for fracturing [1]. CO\({}_{2}\) fracturing has been studied since the 1960s under the background of growing oil and gas demand. So far, CO\({}_{2}\) fracturing has been applied in many countries such as America, Canada, Germany, and China [2; 3; 4], showing unexceptionable advantages especially in unconventional oil and gas resources. Firstly, the mechanical properties of reservoir rock are changed after CO\({}_{2}\) immersion [5], which can reduce the breakdown pressure and facilitate secondary cracking. Secondly, the waterless CO\({}_{2}\) inhibits the swelling of clay mineral and eliminates water sensitivity [6], improving the permeability of the reservoir to some extent, which gives this technology enormous prospects. Thirdly, the low additive content in CO\({}_{2}\) fracturing fluid leads to the high flowback rate. Last but not least, CO\({}_{2}\) can increase the mobility of oil and replace the adsorbed CH\({}_{4}\)[7; 8; 6], enhancing oil and gas recovery (EOR). Furthermore, the mitigation of greenhouse gases such as CO\({}_{2}\) has become an increasing global demand in recent years [9], and CO\({}_{2}\) fracturing has the potential of improving the development of carbon capture [10; 11; 12; 13], utilization, and storage (CCUS), which has both environmental and economic benefits. Compared with traditional hydraulic fracturing, the pressure at the wellhead of CO\({}_{2}\) fracturing increases slowly and fluctuates, as shown in Figure 1, which is related to the compressibility of COThe temperature-pressure condition and the properties of CO\({}_{2}\) are coupled in the wellbore due to the compressibility of CO\({}_{2}\). Therefore, the wellbore flow and heat transfer behaviors are essential conditions to optimize the operating parameters during CO\({}_{2}\) fracturing. Many studies have been conducted on fluid flow and heat transfer in the wellbore since the 1950s. [PERSON] [14] proposed a semisteady model which only considered the unsteady heat transfer in the stratum outside the wellbore and first used the overall heat transfer coefficient to solve the equation. A number of later semisteady wellbore flow models were mostly improvements to [PERSON]'s model. [PERSON] [15] developed a nonsteady model for temperature distribution during drilling fluid circulation, which was the basis of later nonsteady wellbore flow models. [PERSON] [16] presented a finite difference model for the early transient temperature performance in the wellbore during injection and production, appropriate for short-term operations such as fracturing. [PERSON] and [PERSON] [17; 18; 19] presented a solution to continuity, momentum, and energy conservation equations, and developed wellbore heat transfer models for various situations. [PERSON] et al. [20] quantified the heat generated by the flow friction of high-yield production wells. As for CO\({}_{2}\) flow in the wellbore, the sensitivity of CO\({}_{2}\) properties to temperature and pressure results in the complexity of flowing temperature and pressure calculation. Plenty of research is devoted to the coupling of temperature and pressure during the CO\({}_{2}\) injection or circulation process in the wellbore. [PERSON] [21] developed a numerical model of CO\({}_{2}\) nonisothermal flow for production or injection wellbores, which matched single- or two-phase flow of CO\({}_{2}\). [PERSON] [22] first proposed drilling with supercritical CO\({}_{2}\) and established a simplified model of CO\({}_{2}\) circulation in the wellbore. [PERSON] [23] quoted the high-precision Span-Wagner state equation for thermodynamic properties of CO\({}_{2}\) and improved the heat transfer model for supercritical CO\({}_{2}\) wellbore circulation.