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This study reveals a new relationship between fluid infiltration exponents and fractal dimensions in porous media. The findings explain anomalous diffusion in Sierpinski carpets and Menger sponges, offering insights into fluid flow in complex geometries.

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Area of Science:

  • Physics
  • Complex Systems
  • Fluid Dynamics

Background:

  • Previous work showed fluid infiltration in Sierpinski carpets (SCs) exhibits anomalous diffusion (exponent n < 1/2).
  • The infiltration exponent (n) differed from single-particle diffusion exponents (ν) in SCs.
  • This discrepancy highlighted a need to understand the relationship between macroscopic infiltration and microscopic diffusion in fractal geometries.

Purpose of the Study:

  • To establish a scaling relationship between the fluid infiltration exponent (n) and fractal dimensions of the bulk (D_F) and border (D_B).
  • To validate this relationship through numerical simulations in various fractal geometries.
  • To investigate log-periodic oscillations in single-particle diffusion and diffusion front roughness.

Main Methods:

  • Developed a scaling approach to relate infiltration and diffusion exponents: n = ν(D_F - D_B).
  • Performed numerical simulations of single-particle random walks (RWs) and diffusive infiltration models.
  • Utilized Sierpinski carpets (SCs) and generalized Menger sponges (MSs), including one with a fractal external border.

Main Results:

  • Confirmed the scaling relation n = ν(D_F - D_B) in SCs and MSs.
  • Observed log-periodic oscillations in the mean-square displacement of RWs and diffusion front roughness, linked to discrete scale invariance.
  • Demonstrated that the exponent relation explains the quadratic dependence of n on D_F in SCs.

Conclusions:

  • The study provides a unified scaling theory for anomalous diffusion in fluid infiltration within fractal porous media.
  • Log-periodic oscillations in diffusion processes are a signature of discrete scale invariance in fractals.
  • The findings enhance understanding of fluid transport in complex geological and synthetic materials.