Scaling laws for hydraulic fractures with constricted opening

Abstract

Mode I fractures in rocks and other heterogeneous materials do not grow straight – their traces have interruptions and overlappings caused by the fracture avoiding tough zones (e.g., strong inclusions or the compressive zones of a random stress field). The interruptions and overlappings create bridges working as columns or beams connecting the opposite faces of the fracture. The bridges are distributed over the whole fracture and can constrict the fracture opening. We model the constriction by introducing a Winkler layer of the stiffness proportional to the average bridge stiffness and the number of bridges per unit area of the fracture. Constriction is characterised by the constriction length − the ratio of Young’s modulus of the rock and the Winkler layer stiffness. Two asymptotic limits exist: fractures much smaller than the constriction length (they are the conventional tensile fractures) and fractures much larger than the constriction length. Interpolation formulae are developed to account for intermediate fracture lengths. Breakage of bridges contributes to the microseismisity associated with the fracture growth. Three basic fracture geometries, disc-like fractures, 2D (KGD) fractures and elongated elliptical fractures are considered. For large fracture growth times power scalings exist for fracture size and width. For toughness-dominated fractures, depending on fracture geometry, the constriction either increases the time scaling exponent of the fracture size and width or leaves them unaffected. For volume-balance fractures the constriction does not affect scalings of the fracture size. Constriction strengthens the time scaling for the fracture width of disc-like and 2D fractures but does not change scaling for the elongated elliptical fracture. Constriction does not affect the time scaling for the PKN fractures. The knowledge of scaling of fracture size and width may add to the determination of the type (geometry and constriction) of the hydraulic fracture. Comparison of the theoretical results with field data shows the significance of bridges which can control the behaviour of hydraulic fractures.