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<p>Accurate estimation of coalbed methane content is constrained by anomalous diffusion characteristics induced by complex pore networks, while classical Fick's models fail to describe this process. Therefore, the space–time fractional-order derivative was introduced in this study based on the classical Fick diffusion model. The fractal dimension was introduced into the model, and a gas fractal-based sub-diffusion model was constructed. To enhance the model’s engineering applicability, the study derived an approximate analytical solution based on the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\sqrt{\text{t}}\)</EquationSource> <EquationSource Format="MATHML"><math> <msqrt> <mtext>t</mtext> </msqrt> </math></EquationSource> </InlineEquation> model by focusing on the mechanism that induced anomalous mean square displacement of methane molecules (<i>Q</i><sub><i>t</i></sub><i>/Q</i><sub><i>∞</i></sub><i> &lt; </i>0.5). By comparing the fitting data of four models against desorption data, it was found that the fitting effect of the gas fractal-based sub-diffusion model was good (<i>R</i><sup>2</sup> &gt; 0.97), and there was no infinite series, which greatly enhanced the practicability of the model. Moreover, gas diffusion in such anomalous space–time exhibited sub-diffusive behavior, with a rapidly decaying diffusion coefficient and heavy-tailed desorption curves. Coal samples with different degrees of metamorphism exhibited exponential and power-law decay. The study also found that the time fractional derivative (<i>ξ</i>) effectively captures the heavy-tail phenomenon. When <i>ξ → </i>0, the tailing effect in gas diffusion becomes more pronounced. When <i>ξ</i> equals 1, the diffusion process can be idealized as classical Fick diffusion.</p>

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Kinetic Characteristics and Migration Behavior of Fractal-Based Sub-Diffusion of Coal Reservoir Gas under Anomalous Spatio-Temporal Conditions

  • 期刊论文

  • 发表日期：

    2026年7月31日

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  • Zhenyang Wang,
  • Shuo Zhang,
  • Congmeng Hao,
  • Lu Bai,
  • Rong Zhang,
  • Guanhua Ni,
  • Zhen Liu,
  • Zhao Li,
  • Jiaxuan Liu

摘要

Accurate estimation of coalbed methane content is constrained by anomalous diffusion characteristics induced by complex pore networks, while classical Fick's models fail to describe this process. Therefore, the space–time fractional-order derivative was introduced in this study based on the classical Fick diffusion model. The fractal dimension was introduced into the model, and a gas fractal-based sub-diffusion model was constructed. To enhance the model’s engineering applicability, the study derived an approximate analytical solution based on the \(\sqrt{\text{t}}\) t model by focusing on the mechanism that induced anomalous mean square displacement of methane molecules (Qt/Q∞ < 0.5). By comparing the fitting data of four models against desorption data, it was found that the fitting effect of the gas fractal-based sub-diffusion model was good (R2 > 0.97), and there was no infinite series, which greatly enhanced the practicability of the model. Moreover, gas diffusion in such anomalous space–time exhibited sub-diffusive behavior, with a rapidly decaying diffusion coefficient and heavy-tailed desorption curves. Coal samples with different degrees of metamorphism exhibited exponential and power-law decay. The study also found that the time fractional derivative (ξ) effectively captures the heavy-tail phenomenon. When ξ → 0, the tailing effect in gas diffusion becomes more pronounced. When ξ equals 1, the diffusion process can be idealized as classical Fick diffusion.