Gennady Margolin

Structure, flow, and generalized conductivity scaling in fracture networks

We present a three-dimensional (3-D) model of fractures that within the same framework, allows a systematic study of the interplay and relative importance of the two key factors determining the character of flow in the system. The two factors of complexity are () the geometry of fracture plane structure and interconnections and (2) the aperture variability within these planes. Previous models have concentrated on each separately. We introduce anisotropic percolation to model a wide range of fracture structures and networks. The conclusion is that either of these elements, fracture geometry and aperture variability, can give rise to channeled flow and that the interplay between them is especially important for this type of flow. Significant outcomes of our study are (1) a functional relationship that quantifies the dependence of the effective hydraulic conductivity on aperture variability and on the network structure and fracture element density, (2) a relation between aperture variability and the Peclet number, and (3) a basis for a new explanation for the field-length dependence of permeability observed in fractured and heterogeneous porous formations.

Towards a unified framework for anomalous transport in heterogeneous media

Abstract We develop a unified framework to model the anomalous transport of tracers in highly heterogeneous media. While the framework is general, our working media in this study are geological formations. The basis of our approach takes into account the different levels of uncertainty, often associated with spatial scale, in characterizing these formations. The effects on the transport of smaller spatial scale heterogeneities are treated probabilistically with a model based on a continuous time random walk (CTRW), while the larger scale variations are included deterministically. The CTRW formulation derives from the ensemble average of a disordered system, in which the transport in each realization is described by a Master Equation. A generic example of such a system – a 3D discrete fracture network (DFN) – is treated in detail with the CTRW formalism. The key step in our approach is the derivation of a physically based ψ( s ,t) , the joint probability density for a displacement s with an event-time t . We relate the ψ( s ,t) to the velocity spectrum Φ(ξ) (|ξ|=1/v, ξ = v ) of the steady flow-field in a fluid-saturated DFN. Heterogeneous porous media are often characterized by a log-normal permeability distribution; the Φ ( ξ ) we use in this case is an analytic form approximating the velocity spectrum derived from this distribution. The common approximation of ψ( s ,t)∼p( s )t −1−β with a constant β , is evaluated in these cases. For the former case it is necessary to include s −t coupling while the latter case points to the presence of an effective t -dependent β . The full range of these features can be included in the CTRW solution but, as is shown, not in the fractional-time derivative equation (FDE) formulation of CTRW. Finally, the methods used for the unified framework are critically examined.