Fractional Order Derivative Models of Porosity on Physical Fractal Spaces

Rock pore–fracture systems exhibit inherent fractal characteristics, which exert a significant influence on fluid transport. In this study, coal rock is selected as the representative medium. Based on fractional calculus in physical fractal space, and by integrating operator algebra with the force–electric analogy method, a fractional order control equation is derived. To validate the proposed model, porosity measurements of coal and limestone were performed using the two-compartment Boyle’s law method and compared with conventional porosity calculation approaches. The results demonstrate that the fractional order model achieves a coefficient of determination (R2) of up to 0.99 for porosity and 0.98 for pressure, representing an improvement of approximately 0.07 over the exponential model. Moreover, the root mean square error (RMSE) of porosity is as low as 0.0008, while the RMSE of pressure is 0.0715, both significantly lower than those obtained using the exponential model. These results indicate that the fractional order model more effectively captures the non-Darcy flow behavior and the temporal evolution of porosity, providing substantially improved fitting accuracy. Further analysis reveals that the porosity–time relationship is jointly governed by fluid compressibility and pore compressibility under effective stress conditions. Comparative results across different lithologies reveal that the pore compressibility coefficient increases with porosity; for the same rock type, a higher coefficient implies a more complex pore structure and a longer equilibration time. Overall, the proposed fractional order framework provides a more accurate description of the fractal pore structures in rocks, establishing a clear link between microscale fractal geometry and macroscale fractional order response.