Production analysis for fractured vertical well in rectangular coal reservoirs

As one kind of unconventional natural gas, coalbed methane is an important energy resource that is subject to active research. Gas exists in coalbed methane reservoirs in two forms: free gas and adsorbed gas. In the course of coalbed methane production, the reservoir experiences pressure decrease, desorption, diffusion, and seepage. Previous models of coalbed methane production were mainly concerned with circular boundaries. However, field tests revealed that some fractured wells possess the characteristics of rectangular boundaries. For fractured rectangular coalbed methane reservoirs, it is necessary to deal with the four boundaries with mirror image theory, which complicates calculations. In addition, the desorption and adsorption process of coalbed methane exerts a strong effect on the seepage process. Furthermore, the complexity of the rectangular coal seam embedded with the finite-conductivity fracture results in a significant computational challenge. For the first time, this paper presented a fast analytical solution for a finite-conductivity fractured vertical well model with either rectangular closed or constant-pressure boundaries in the coal seam. On the basis of the Fick diffusion law and the Darcy seepage law, a mathematical model that considers diffusion in matrix and seepage within natural fractures was established. Then, we integrated the fracture conductivity function method with the hydraulic fracture model to greatly increase computational efficiency. The analytical solutions were validated against a numerical simulation. Parameter sensitivity analysis reveals that interporosity coefficient and storage coefficient, respectively, affect the appearance time and degree of desorption and diffusion. Desorption coefficient mainly describes the capacity of desorption and diffusion. Well storage coefficient, conductivity factor, and skin factor mainly affect the early stage of production. Finally, the proposed solutions were applied to field history match. The model developed is applicable to production analysis and well testing for coalbed methane reservoirs. The new proposed model extended flow mechanism of coalbed methane, and provided a better production and pressure forecast for coalbed methane reservoirs. In addition, the analytical solutions can be used to generate type curves for fractured vertical wells with finite conductivity and in the rectangular boundary, and provide a sound theoretical basis for well tests in the coal seam. The model is also applicable to other types of unconventional gas reservoirs, such as gas shales, in which the same processes are present.