Relation Between First Arrival Time and Permeability in Self-Affine Fractures with Areas in Contact

We demonstrate that the first arrival time in dispersive processes in self-affine fractures are governed by the same length scale characterizing the fractures as that which controls their permeability. In one-dimensional channel flow this length scale is the aperture of the bottle neck, i.e., the region having the smallest aperture. In two dimensions, the concept of a bottle neck is generalized to that of a minimal path normal to the flow. The length scale is then the average aperture along this path. There is a linear relationship between the first arrival time and this length scale, even when there is strong overlap between the fracture surfaces creating areas with zero permeability. We express the first arrival time directly in terms of the permeability. Due to their role in the flow properties of tight and low permeability reservoirs such as shale gas reservoirs and carbonate reservoirs, and on contaminant transport e.g. in connection with waste storage, the study of transport in fractures is still a very vigorous field [1–4]. Most present theoretical efforts attempts to relate the transport properties of fractures to the statistics of the aperture fields through analytical models based on statistical averages, weak disorder perturbation expansions [5], mean-field approximations or simplified aperture models [6]. We also mention the work of Zhan and Yortsos [7] where a method to deduce the heterogenities of a permeability distribution from the concentration arrival time field was proposed. Due to the surface roughness, i.e., the heterogenities of the aperture field, these relations provide satisfactory results only over a finite range of conditions and do not permit to predict the behavior of a fracture with large heterogenities in aperture field. One of the main difficulties is to correctly take into account the increasing influence of the contact area as the fracture aperture is decreased [8–11]. We analyze in this Letter the dispersion problem at finite Péclet number and identify the proper aperture measure for this problem, taking into account severe heterogeneities such as large contact zones. Our main focus is on the breakthrough time, i.e., the time at which the tracer appears at a given position. The