Relationship between percolation–fractal properties and permeability of 2-D fracture networks

The concepts of percolation theory and fractal geometry are combined to define the connectivity characteristics of 2-D fracture networks and a new approach to estimate the equivalent fracture network permeability (EFNP) is introduced. In this exercise, the fractal dimensions of different fracture network features (intersection points, fracture lines, connectivity index, and also fractal dimensions of scanning lines in X- and Y-directions), and the dimensionless percolation density of fracture networks are required. The method is based on the proposed correlations between the EFNP and a percolation term, (ρ′–ρ′c). The first parameter in this term is the dimensionless density, and the second one is the percolation threshold (a constant number). This term is obtained using the relationships with the properties of fracture networks mentioned above. It was found that the highest correlation coefficient between the actual and the predicted EFNP could be obtained using the percolation term, (ρ′–ρ′c), calculated from the fractal dimension of fracture lines by the box-counting technique. The method introduced is validated using different fracture patterns representing a wide range of fracture and length values. In addition, a correlation between the number of fractures in the domain and the minimum size of the fracture length is presented to estimate the shortest or minimum fracture length required for percolation for a given number of fractures in the domain.

► Correlations to estimate fracture network permeability using 2-D fractal and percolation properties.
► Box-counting dimension is more accurate in the estimation compared to the fractal dimension of other properties.
► Validation exercise confirmed the use of new correlations practically.
► A correlation to quickly estimate the possibility of connectivity (percolation).