Estimation of closure of a fracture under normal stress based on aperture data

The closure of 41-mm hydraulic fractures under normal stress in both loading and elastic closure (unloading) was estimated according to the formula proposed by Brown and Scholz and based on data measured for the initial aperture. By introducing the concept of an effective/ineffective initial aperture and by assuming Gaussian and χ2 probability density functions (PDFs) of the initial aperture, the normal stress versus closure curve was determined from the standard deviation (SD) and the spectral moments of the initial aperture and the ratio of the mean effective initial aperture to the SD of the initial aperture. The results showed that the non-linearity in the normal stress versus closure curve at large normal stresses was reproduced better by the χ2 PDF of the initial aperture than the Gaussian PDF for both loading and elastic closure. Furthermore, based on the ratio of the mean effective initial aperture to the SD of the initial aperture determined for the hydraulic fractures, the effect of size on the normal stress versus closure curve was estimated for fracture areas in a tensile fracture of 1 m. The results showed that closure increases with the size of the fracture area, and that the effect of size on the closure of the fracture is governed by the SD of the initial aperture.