Parameter estimation for the fractional fractal diffusion model based on its numerical solution

Abstract

In this paper, we mainly consider a problem of parameter estimation for the fractional fractal diffusion model used to describe the anomalous diffusion in porous media. The Bayesian method is proposed to estimate parameters for the fractional fractal diffusion model based on its numerical solution. Firstly, the center Box difference method is employed to solve the fractional fractal diffusion equation subject to the initial–boundary conditions. Then we apply the Bayesian method to estimate three parameters for the model simultaneously, that is the fractional order α, the fractal dimension df and the structure parameter θ. To testify the validity of the results for the model’s parameter estimation, experimental data from the fast desorption process of methane in coal are used. It is shown that the numerical results modeled with parameters given by the Bayesian method coincide well with the desorption experimental data, indicating that the Bayesian method is efficient and valid in identifying multi-parameter for the fractional fractal diffusion model. Besides, the fractional fractal diffusion model is demonstrated to be more capable than the classical Fick’s model to describe the anomalous diffusion behavior of gas in coal. To clarify the parameters’ effect on the relative diffusion amount of methane for the fractional fractal diffusion model, parameter sensitivity analysis is also carried out. This paper provides a specific and efficient parameter estimation method for the fractional fractal diffusion model.