A Semi-Analytical Fractal-Fractional Mathematical Model for Multi-Fractured Horizontal Wells in Coalbed Methane Reservoirs

Abstract

Abstract Coalbed methane (CBM) which is clean energy has received great emphasis recently, and the multi-fracturing technology is widely applied in the exploitation of CBM. Due to the complexity, the randomness, and the anisotropism of the porous medium and the anomalous diffusion process, the fractal theory and fractional calculus are utilized to establish a semi-analytical fractal-fractional mathematical model considering the stress sensitivity of the cleat system for multi-fractured horizontal wells in CBM reservoirs. Through line-sink theory, Pedrosa transformation, perturbation theory, Laplace transformation, element discretization, superposition principle, and Stehfest numerical inversion, the pressure-transient analysis curves are plotted in the double logarithmic coordinates. By comparing with the existing model, the validation of the proposed model is illustrated. Also, nine flowing stages are identified according to different characteristics. Then, sensitivity analysis is conducted and influence laws are summarized. At last, a field application is introduced to furtherly verify the reliability of the proposed model. The relevant results analysis can provide some new significant guidance for interpreting the field data more precisely.