Theoretical Constraints on Fracture Aperture Based on Linear Elastic Fracture Mechanics: the importance of stress concentration and lithology

One of the major unknowns in fractured aquifer or fractured reservoir studies is the magnitude of fracture aperture. Since fluid flow through fractures scales with the third power of fracture aperture (cubic law), small variations in aperture have a large influence on the permeability. Fracture apertures, however, are difficult to measure in-situ since the very measurement itself often changes the stress state and therefore the fracture aperture. We used Linear Elastic Fracture Mechanical theory to calculate in-situ apertures for fractures with different dimensions, for a variety of rock types, under different tensional stress states. We calculated the displacement field around a penny-shaped discontinuity in a homogeneous, anisotropically stressed elastic material. We assume that penny-shaped discontinuities are present a priori in the homogeneous rock and that the cracks do not grow, they just open up in the direction of the minimum compressive stress in a way that the tensile stresses around the crack and especially at the crack’s tip do not exceed the tensile strength of the rock. The displacement field around these cracks depends on the position with respect to the centre of the crack, the magnitude of the tensile stress, and the material properties such as Young’s modulus and Poisson's ratio. The maximum displacement in the center of the crack can be regarded as the maximum theoretical aperture for fluid flow. The aperture tapers out towards the crack tip where aperture reduces to zero. The Poisson’s ratio is kept as 0.25 for all rock types and the Young’s modulus varies according to lithology. For shale, marl and chalk, anhydrite, sandstone, dolomite, limestone, gypsum, rock salt and conglomerate curves are constructed that show the relationship between fracture length and fracture aperture. These calculated curves show that the longer the fractures, the larger the maximum possible aperture at its center. In limestone for example, a one meter long fracture loaded by 0.5 MPa tensional stress has a maximum aperture of 20 μm and a 10 meter long fracture has a maximum aperture of 150 μm. Fractures in shale, marl and chalk tend to have larger apertures for the same fracture length than gypsum, rock salt and conglomerate which in turn have larger openings than limestone, dolomite sandstone and anhydrite. For example a fracture with a length of 1 m has a maximum aperture of 10 000 μm in shale but of only 20 μm in anhydrite.