Extended analysis of constant-height hydraulic fractures for the estimation of in-situ crack-opening modulus from bottomhole pressure records

Hydraulic fractures created in oil and gas bearing rock formations can be made to propagate for a limited time at approximately constant height if favorable stress, deformation modulus or fracture toughness barriers to height growth exist and if the fracture design is suitably optimized to exploit these favorable conditions and reduce height growth. In this report, a unified theoretical formulation for the Perkins-Kern-Nordgren (PKN) and Christianovitch-Geertsma-De Klerk-Daneshy (CGDD) constant height fracture models is first presented. For a fracture fluid injection rate that varies as an arbitrary power of time, growth laws for fracturing fluid pressure, fracture width, and flow rate are rigorously derived for PKN and CGDD types of fractures. These similarity solutions account for non-Newtonian power-law fluid flow, transient fluid storage and generalized power-law fluid leak-off to the rock formation. They include and extend the results currently available in the literature for PKN and CGDD fractures. The results for PKN and CGDD fractures are then generalized to obtain an approximate hybrid CGDD-PKN fracture model that can be applied to constant height fractures of arbitrary length/height aspect ratio and arbitrary cross-sectional shape. Characteristic times for fracture extension are identified and estimates are given for the transition times when the fracturemore » evolves from a CGDD-type fracture at small aspect ratio to a PKN-type fracture at large aspect ratio. These results are useful for interpreting fracturing data and for designing fractures for crack-opening modulus measurements.« less