Investigating Scale-Dependent Clustering and Flow Behaviour in Multifractal Permeability Fields

Lacunarity analysis has proven effective in distinguishing between multifractal grayscale patterns that share the same correlation dimension but exhibit differing degrees of scale-dependent clustering. In our earlier studies, we demonstrated that such scale-dependent clustering significantly influences fluid flow behavior in fractal-fracture networks, even when the fractal dimensions remain constant. Building on this foundation, the present research explores whether lacunarity, a metric quantifying heterogeneity and spatial clustering, can serve as a predictor of fluid flow behavior in multifractal permeability fields. To investigate this, we considered a set of four multifractal patterns (three deterministic and one random), each characterized by identical correlation dimensions and defined by the parameters b = 7, p = 40/49, and three iterations. These patterns are subjected to lacunarity analysis using the gliding-box algorithm to quantify their scale-dependent clustering, where lacunarity (L) is related to the correlation dimension, D₂, by the equation dlog(L)/dlog(r) = D₂ - 2. We empirically tested this equation using two-dimensional multifractal grayscale patterns with known correlation dimensions. Subsequently, flow simulations are performed on these permeability fields using Trace3D: a streamline-based simulator that generates streamlines in 3D space and then solves the 1D equations analytically or numerically along the streamlines. In these simulations, the occupied regions of the multifractal fields are assumed to be highly porous and permeable, enabling all fluid flow to occur exclusively through these zones. The simulation results reveal a strong positive correlation of 95% between lacunarity and fluid recovery, indicating that increased clustering (as measured by lacunarity) enhances flow efficiency in such heterogeneous systems. These findings suggest that lacunarity is a valuable parameter for predicting fluid flow behavior in multifractal permeability fields and can potentially generate informed decisions on reservoir characterization and field development modeling strategies.