A new geometric model for non-linear flow in rough-walled fractures based on the cubic law

The wall roughness causes a non-linear flow in rock fractures. For analyzing these issues,the cubic law (CL) is most convenient but maybe not adequate. The authors propose a geometric model applicable for non-linear flow in rock fractures based on the simple CL. An order-of-magnitude analysis of the Navier-Stokes equation yields three conditions for the applicability of the CL:sufficient small roughness, sufficient small ratio between length and ap- erture,and low enough flow inertial effects. These conditions may not be met in many real rock fractures because of their extended length with rough walls. To solve this problem,a new model for rough fractures on the basis of the geom- etry discretizing equivalent is developed. In applying this model to a 2-D rock fracture,the fracture is ‘ broken’ into shorter segments appropriately to meet the CL conditions. The method of rough segment equivalent to parallel plate is derived,and the CL is applied in local segments (local cubic law,LCL),thus a rock fracture can be equivalent to a series of unit parallel plate elements. Moreover,the model has been evaluated with the numerical simulation of a rough fracture,the results show that the equivalent fracture consisting of parallel plates has the similar flow behaviors with the original rough fracture. As a preliminary test,the theoretical flux values,calculated with LCL on the proposed model, are compared with experimental ones for four different fractures and found to be in good agreement. This model,togeth- er with the solution of the flow through it,can be used for modeling the processes which take place in real rock frac- tures.