Simulation of the impregnation in the porous media by the Self- organized Gradient Percolation method

Many processes can correspond to reactive impregnation in porous solids. These processes are usually numerically computed by classical methods like finite element method, finite volume method, etc. The disadvantage of these methods remains in the computational time. The convergence and accuracy require a small step-time and a small mesh size, which is expensive in computational time and can induce a spurious oscillation. In order to avoid this problem, we propose a Self-organized Gradient Percolation algorithm. This method permits to reduce the CPU time drastically.

💡 Research Summary

The paper addresses the computational inefficiency inherent in conventional numerical methods (finite element, finite volume) for simulating non‑reactive impregnation in porous media. Traditional approaches solve coupled Richards’ equation and Darcy’s law, which demand very small time steps and fine spatial meshes to achieve convergence and avoid spurious oscillations, leading to prohibitive CPU times. To overcome these limitations, the authors propose a Self‑organized Gradient Percolation (SGP) algorithm that recasts the capillary pressure–saturation relationship as a probability density function (PDF).

In the SGP framework, the initial capillary pressure curve is represented by a PDF (either Gaussian or Laplace, depending on a parameter m). Each lattice site of a square grid receives a random value drawn from this PDF, and the local saturation is obtained by convolving the random field with the local average porosity. Time evolution is modeled solely by increasing the variance of the PDF; the increment of variance is directly linked to the physical driving force (gravity, fluid density, capillary pressure, viscosity, etc.) through equations (13)–(17). Consequently, the model bypasses the need to solve a nonlinear partial differential equation at every time step; instead, it updates a statistical descriptor (variance) and recomputes the saturation field via simple stochastic sampling and convolution.

The algorithm proceeds as follows: (1) define the initial capillary pressure curve as a PDF; (2) assign random values to each lattice node to obtain the initial saturation distribution; (3) at each subsequent time step, compute the variance increment from the physical parameters; (4) generate a new PDF with the updated variance and re‑sample the lattice, producing a new saturation field. This process yields a continuous saturation profile that respects boundary conditions thanks to the convolution operator.

Experimental validation involved two impregnation tests: (i) alumina (99 % purity) with glycerine, and (ii) carbon with LCC oil. Cylindrical specimens (30 mm height, 20 mm diameter) were hung over a liquid reservoir, and mass uptake was recorded over time. For comparison, the authors built two finite‑element models using the widely adopted Van Genuchten and Brooks‑Corey capillary pressure relationships, implemented in the ASTER code. The FEM simulations reproduced the experimental mass gain curves but required substantial computational effort (≈70 s for test 1, ≈120 s for test 2).

The SGP model, calibrated with the same initial saturation height, maximum saturation, variance, and distribution type (parameter m), reproduced the capillary pressure curves and mass‑gain histories with comparable fidelity. CPU times dropped dramatically to ≈7 s (test 1) and ≈12 s (test 2), representing speed‑ups of roughly 10‑fold and 100‑fold, respectively. The authors emphasize that by adjusting the distribution type (through m) the SGP can emulate both Van Genuchten and Brooks‑Corey shapes, offering a unified, physics‑based stochastic framework.

In conclusion, the Self‑organized Gradient Percolation method provides a novel, statistically driven alternative to deterministic PDE solvers for porous‑media impregnation. It achieves orders‑of‑magnitude reductions in computational cost while preserving accuracy in both capillary pressure evolution and overall mass uptake. The paper acknowledges remaining challenges: systematic mesh and time‑step sensitivity studies, quantitative linking of the three SGP parameters (m, variance, height) to material properties, and extension to anisotropic or multiphase flow scenarios. Future work will need to address these issues to fully establish SGP as a robust tool for porous‑media transport modeling.