## 1 Introduction

[2] A non-intrusive geophysical technique to probe the hydraulic properties of rock fractures has long been sought by scientists and engineers. Such a technique would provide a new method to ascertain the effectiveness of subsurface projects such as the extraction of drinkable water, production of oil and petroleum, installation and monitoring of subsurface infrastructure, and the storage of anthropogenic byproducts (CO_{2}, nuclear waste, etc.) in subsurface reservoirs. Extensive research has been performed on the laboratory scale to examine fluid flow through fractures, fracture geometry, and deformation under stress as well as the seismic response of fractures. However, one of the fundamental tasks in geophysics is to relate fracture properties and processes at one length scale to properties and processes at other length scales. For example, in the laboratory, measurements are performed on fractured rock samples that range in size from 10^{−2} to 10^{−1} m with fracture apertures on the order of 10^{−6} to 10^{−4} m using seismic wavelengths on the order of 10^{−3} to 10^{−2} m. Conversely, at field scales, seismic frequencies from 1 Hz to 1 kHz illuminate regions on the order of 10^{3} to 10^{1} m. Thus, the development of seismic methods that can delineate and characterize the hydraulic properties of fractures requires a fundamental understanding of the relationship between the hydraulic and mechanical properties of fractures and how this relationship scales with the size of the sampled region.

[3] The ability to relate and scale the hydromechanical properties of fractures requires that both hydraulic and mechanical processes are controlled at similar length scales associated with fracture geometry (e.g., size and spatial distributions of aperture and contact area, surface roughness, fracture length, etc.). There have been many attempts to quantify the role of these geometric quantities with regard to fluid flow and deformation as a function of stress. For instance, *Witherspoon et al.* [1980] showed that the flow rates associated with fractures under normal load have three distinct behaviors as a function of stress. At low stresses, flow rates obey the “cubic” law. However, as normal stress increases, the flow rate deviates from the cubic-law aperture dependence. Deviations from the cubic law were partially explained by using the dominant surface roughness wavelength to approximate the hydraulic aperture [*Zimmerman et al*., 1990; *Zimmerman and Bodvarsson*, 1996]. Alternatively, a correction factor was constructed from the ratio of the first and second moments of the aperture distribution [*Renshaw*, 1995]. While these approaches focused on the void areas across the fracture plane, the contact area provides another approach. The fracture was modeled as a system of interacting circular obstructions confined to a plane [*Walsh*, 1981]. The analytic solution for the flow around a circular obstruction was used to compute the total flow rate through the fracture. This approach provided a stress-dependent flow rate, but the contact area was assumed to increase linearly with stress [*Walsh and Grosenbaugh*, 1979].

[4] It has been shown experimentally that, at high stresses, the flow exponent deviates from the “cubic” law due to the deformation of the fracture void geometry. Metal castings of natural granite fractures were made at stresses as high as 85 MPa. The castings showed large regions of void space connected by narrow tortuous channels [*Pyrak-Nolte et al.*, 1987; *Jaeger et al.*, 2007]. This experiment found that the large void spaces deformed significantly as the normal load increased, while narrow channels remained open because they were supported by adjacent contact area. From these observations, the authors concluded that once the narrow paths dominate the fluid flow, the flow becomes approximately independent of stress. Following this study, a more unified numerical approach was taken that included both mechanical deformation and fluid flow [*Pyrak-Nolte and Morris*, 2000]. Experimental flow-stiffness data for fractures that ranged in length from 0.05 m to 0.3 m suggested an empirical relationship between the hydraulic and mechanical properties that appeared to be controlled by the geometry of the void spaces and the contact area in the fracture. A strong dependence of flow on stiffness was observed, but the samples had different aperture distributions and scale. An outstanding question is whether there exists a scaling relationship between flow and stiffness when appropriate geometric length scales are taken into consideration.

[5] In this letter, a finite-size scaling approach is presented that quantifies the scaling relationship between fluid flow and fracture specific stiffness for single fractures with weakly correlated random aperture distributions. The scale-dependence is removed by finding the critical transport scaling exponent that yields a scaling function.