Scale-dependent permeability in geologic formations: Renormalization group theory and finite-size scaling analysis

Understanding the scale dependence of permeability ($k$) of geologic formations at field scales is essential for precise modeling of flow and transport in subsurface, particularly for underground energy storage. In this study, we conducted extensive computations and investigated the scale dependence of $k$ in random and heterogeneous formations by employing renormalization group theory (RGT) and finite-size scaling analysis. Based on the random permeability field and following the Gaussian distribution of $\text{ln}\left(k\right)$, we first generated ten formations with different levels of heterogeneity at five dyadic domain sizes, $L$ = 2ⁱ, $i\in \left\{3,\dots ,7\right\}$. The first formation reflects the permeability distribution of an actual reservoir, while the others were generated with varying degrees of heterogeneity. We then applied the RGT to determine the effective permeability ($k_{eff}$) of each formation at various occupation probabilities $p$ = 0.5, 0.6, 0.7, 0.8, 0.9 and 1. We next used finite-size scaling theory to further analyze the scale-dependent $k_{eff}$. The $k_{eff}-L$ plot for each formation was scattered. However, by applying the finite-size scaling analysis the data collapsed onto a single quasi-universal curve. It means that finite-size scaling theory could successfully incorporate the effect of large-scale heterogeneities in the scale dependence of permeability.