Hydraulic properties of two-dimensional random fracture networks followinga 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 theoreticall y irrelevant, the application of homogenization techniques. We propose a nu merical and theoretical study of the consequences of these two properties o n the permeability of bidimensional synthetic fracture networks. We use a p ower law for the model of length distribution and a lognormal model for ape rture distribution. We have especially studied the two endmost models for w hich length and aperture are (1) independent and (2) perfectly positively c orrelated. For the model without correlation between length and aperture we show that the permeability can be adequately characterized by a power-aver aging function whose parameters are detailed in the text. In contrast, for the model with correlation we show that the prevailing parameter is the cor relation 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 determ ine the permeability scaling and the scale dependence of the flow pattern s tructure. 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 c ase that a correlation between length and aperture does not fundamentally c hange the permeability model. In all other cases the correlation entails mu ch larger-scale effects. For a in the range 1-3 in the case of an absence o f correlation and for a in the range 2-3 in the case of correlation, permea bility increases and tends to a limit, whereas the flow structure is channe led when permeability increases and tends to homogenize when permeability t ends 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 th e case of correlation, permeability increases with scale with no apparent l imit. We characterize the channeled flow pattern, and we show that permeabi lity may increase even when flow is distributed in several independent stru ctures.