Determination of the transmissivity of a heterogeneous anisotropic fracture in slip flow conditions

Rough fractures often exhibit a broad spectrum of defect length scales ranging from the microscopic (roughness) scale to a macroscopic one (waviness) and further to the megascopic scale corresponding to the entire fracture. The influence of these multiple scales and their reciprocal interactions are expected to play a significant role on the transport properties at the megascale. Focusing on the pressure-driven slightly compressible gas slip flow, a two-scale method is presented allowing the determination of the global transmissivity of a fracture on the basis of an upscaled Reynolds model. This model is applied on a tessellation of the fracture, each tile being affected by a macroscopic transmissivity tensor which encompasses the microscale transport information as a result of the first upscaling process. Then, the megascale flow problem in this structure, made of a set of tiles characterized by a heterogeneous and anisotropic transmissivity tensor field, is solved using a boundary element method. Numerical results obtained with this two-scale method are compared to the transmissivity computed with direct simulations carried out at the microscale on the whole fracture. This is performed on two model rough fractures, namely, a spiral groove and a fractal fracture, while varying their mean apertures to investigate a wide range of the average Knudsen number characteristic of the flow at the megascale. A good agreement is obtained between the two approaches showing the robustness of the two-scale method to determine the global transmissivity of the fracture while significantly reducing the overall computational time