ON SUSPENSION TRANSPORT IN A FRACTURE - FRAMEWORK FOR A GLOBAL-MODEL

We consider a hydraulically induced fracture to define a two-dimensional surface; for simplicity the analysis given here treats it as plane. Associated with the surface we define a set of two-dimensional dynamical variables obtained by integration of the usual three-dimensional field variables across the fracture width to describe suspension transport in the fracture. This has strong analogies with the well-known lubrication approximation. We then derive a set of dynamical equations for these variables, noting what approximations seem reasonable, and discuss the connection between the solutions of the full equations of motion for a sedimenting suspension in a complex fluid and the associated constitutive relations that arise in the two-dimensional description. We conclude that, if a fully developed lubrication flow covering heat transport as well as migration and sedimentation cannot be used as a satisfactory approximation, a three-dimensional problem based on suitably stretched variables has to be solved.