Hydraulic properties of two-dimensional random fracture networks following a power law length distribution 2. Permeability of networks based on lognormal distribution of apertures

The broad length and aperture distributions are two characteristics of the heterogeneity of fractured media that make difficult, and even theoretically irrelevant, the application of homogenization techniques. We propose a numerical and theoretical study of the consequences of these two properties on the permeability of bidimensional synthetic fracture networks. We use a power law for the model of length distribution and a lognormal model for aperture distribution. We have especially studied the two endmost models for which length and aperture are (1) independent and (2) perfectly positively correlated. For the model without correlation between length and aperture we show that the permeability can be adequately characterized by a power-averaging function whose parameters are detailed in the text. In contrast, for the model with correlation we show that the prevailing parameter is the correlation when the power law length exponent a is lower than 3, whereas the random structure of the network is a second-order parameter. We also determine the permeability scaling and the scale dependence of the flow pattern structure. Three types of scale effects are found, depending exclusively on the geometrical properties of the network, i.e., on the length distribution parameter a. For a larger than 3, permeability decreases for scales below a definite correlation length and becomes constant above. We show in this case that a correlation between length and aperture does not fundamentally change the permeability model. In all other cases the correlation entails much larger-scale effects. For a in the range 1-3 in the case of an absence of correlation and for a in the range 2-3 in the case of correlation, permeability increases and tends to a limit, whereas the flow structure is channeled when permeability increases and tends to homogenize when permeability tends to its limit. We note that this permeability model is consistent with natural observations of permeability scaling. For a in the range 1-2, in the case of correlation, permeability increases with scale with no apparent limit. We characterize the channeled flow pattern, and we show that permeability may increase even when flow is distributed in several independent structures.