The diffusion progress of grout is hindered by groundwater, which means the diffusion distance cannot reach the designed values required in engineering for water plugging or reinforcement. In this study, based on the generalized Darcy’s law and the continuity equation of steady column penetration, a column diffusion mechanism for Bingham fluid, considering the displacement effect of grout on groundwater, is proposed. This diffusion mechanism is then validated by the penetration grouting experiments that have been previously performed. The influences of the grouting pressure, groundwater pressure, water–cement ratio and penetration coefficient of porous media on the diffusion radius are analyzed. Based on the Comsol Multiphysics platform, a three-dimensional numerical simulation program for this mechanism is developed using computer programming technology. Numerical simulations of the penetration and diffusion morphology of Bingham cement grout in porous media are then carried out. The results show that the theoretical calculation values of diffusion radius obtained from this mechanism are closer to the experimental values than those obtained from the column penetration grouting theory of Bingham fluid, without considering the displacement effect. The results of this study can provide theoretical support for practical grouting engineering.

Porosity and particle arrangement are important parameters affecting soil tortuosity, so it is of great significance to determine the intrinsic relationship between them when studying soil permeability characteristics. Theoretical derivation and geometric analysis methods are used to derive a two-dimensional geometric tortuosity model. The model is a function of particle arrangement parameters ($m$ and $\theta$) and porosity. An analysis of the model and its parameters shows that: (1) The arrangement of particles is one of the reasons for the different functional relationship between tortuosity and porosity, which proved that the tortuosity is not only related to the porosity but also affected by the particle arrangement. (2) The greater the anisotropy parameter $m$ is, the greater the tortuosity is, indicating $m$ varies when fluid passes through the soil from different sides resulting in different values of permeability. (3) The tortuosity increases with the increase in the blocking parameters $\theta$. (4) With increasing porosity, the influence of the parameters $m$ and $\theta$ on the tortuosity gradually decreases, suggesting that the influence of particle arrangement on tortuosity gradually decreases. The results presented here increase the understanding of the physical mechanisms controlling tortuosity and, hence, the process of fluid seepage through soil.