A Simple Statistical Model for Transmissivity Characteristics Curve for Fluid Flow Through Rough-Walled Fractures

The fracture transmissivity characteristics curve (Witherspoon et al. in Water Resour. Res. 16(6):1016-1024, 1980) is found to deviate from cubic law as aperture decreases and still have residual transmissivity when aperture is very small. The existing models can partly explain the deviation from cubic law (e.g., Sisavath et al. in PAGEOPH 160:1009-1022, 2003), or the residual transmissivity due to irreducible flow (e.g., Nolte et al. in PAGEOPH 131(1/2):111-138, 1989). In order to predict the transmissivity curve with both the above characteristics, in this study, a simple statistical model is employed with the following assumptions: (1) fracture boundaries are assumed parallel flat at global scale, but with normally distributed aperture variations at local scale (like frosted glasses); and (2) in this case, the flow field is assumed regular with straight head-contours and flow-lines. Then the equivalent transmissivity can be approximated as a series of parallel-connected local transmissivities. The transmissivity curve can be fitted very well with both the above characteristics. It is suggested that the reason for the deviation from cubic low is possibly due to the variations of local apertures which induce redistribution of hydraulic gradients, and the residual foot is because of residual open apertures or micro-fractures in the fracture surfaces.