Why 2D Models Fail to Capture Flow Properties in 3D Heterogeneous Media: A Connectivity Perspective
We analyze how the equivalent hydraulic conductivity, $K_{eq}$, varies with the coarsening scale $λ/I$ in 2D and 3D heterogeneous media. Their local hydraulic conductivity, $k({\bf r})$, follows a lognormal distribution in all cases, while spatial connectivity of $k({\bf r})$ ranges from high (HCS), through intermediate or multi-Gaussian (ICS) and low (LCS), to unstructured (NS). Using a stochastic approach, we characterize the full distribution $P[\log(K_{eq})]$, and its moments: mean $\langle K_{eq}\rangle$, log-variance $σ_{\log(K_{eq})}^{2}$, and log-skewness $γ_{\log(K_{eq})}$, as a function of $λ/I$. Significant contrasts appear between 2D and 3D, which we interpret in terms of two key factors associated with spatial dimensionality. We observe that, compared with ICS, LCS and HCS exhibit two distinct features: existing analytical expressions for coarse-graining of $k({\bf r})$ in multi-Gaussian media do not apply, while $P[\log(K_{eq})]$ deviates from Gaussian at all scales $λ/I$. Critical path analysis is then used to quantify the connectivity of our samples, and show that $K_{eq}$ exhibits a power-law dependence on it. Our results reveal that connectivity can lead to substantial differences between 2D and 3D macroscopic flow properties, even for the same geostatistical parameters, highlighting the severe limitations of using 2D models to represent 3D flows.