Universal Scaling of Single Fracture Hydromechanics

Using non-intrusive geophysical techniques to probe the hydraulic properties of fractures is a long sought goal. While seismic techniques have been developed to probe the mechanical properties of fractures, i.e. fracture specific stiffness, there are no techniques to characterize remotely their hydraulic properties. Linking these two properties would provide a new, non-intrusive method to ascertain risk in many subsurface projects such as the extraction of drinkable water, production of oil & petroleum, subsurface constructions and the storage of anthropogenic byproducts (CO2) in subsurface reservoirs. However, this relationship is very complicated because the subsurface is composed of a hierarchy of structures and processes that span a large range in length and temporal scales. In this study, an empirical finite sized scaling approach was used to understand the hydromechanical relationship of single fractures. Fractures ranging in size from 0.0625 to 1 m on an edge were generated numerically using the Stratified Percolation Method to allow over an order of magnitude change in scale. The fractures were constructed with spatially uncorrelated apertures with an approximately log-normal distribution. The fractures were deformed, numerically, under normal loads up to 20 MPa. The deformation solver was setup to use periodic boundary conditions so that it modeled a subsection of a much larger fracture. The flow rate was computed at each step of normal load. From these calculations, the hydromechanical scaling laws were investigated. From the data from this computational study, full data collapse was achieved for the Flow-Stiffness relationship. This was accomplished by taking a finite-size scaling approach, which means that the flow equations are separated into two parts: a power law dependence on scale and a universal scaling function. Near threshold, the universal scaling function reduces to a constant leaving scale as the dominating factor in flow. To write fluid flow in this fashion, the extraction of two global parameters of the system, the critical threshold value and the critical flow exponent, was required. The critical threshold was extracted by extrapolating each scale's spanning probabilities fixed-point to the infinite limit. The critical threshold was found to be when 43.2% of the fracture was in contact. The critical flow exponent was found as well by extracting the flow rate at threshold for each scale. These critical flow values were plotted as a function of scale and, as expected, this relation displayed a power law form. The dynamic flow exponent was found to be 3.17. From these two macroscopic parameters, the scale dependence near threshold in the flow-stiffness relationship was removed during data collapse. This means that fluid flow was represented at all scales by a single universal scaling function near threshold. Acknowledgments: This work is supported by the Geosciences Research Program, Office of Basic Energy Sciences US Department of Energy (DEFG02-97ER14785 08), by the Geo-mathematical Imaging Group at Purdue University, and the Purdue Research Foundation.