Modeling Forchheimer fractures as interfaces

Modeling flow and transport of contaminants in porous media is made difficult by the presence of heterogenieties in the characteristics of the medium occurring at scales quite different from those describing the average characteristics of the medium. One particular instance of this phenomena is the occurrence of fractures in the medium, regions very small in width but very important for modeling flow because of their much higher (or possibly much lower ) permeability. Fine networks of interconnected fractures occurring with some degree of regularity are often taken into account by double porosity models. Here however we are concerned with larger fractures or faults of known location that need to be included specifically in the model. In earlier works, a model was introduced in which the fracture was treated as a lower dimensional domain, as an interface between two subdomains, and domain decomposition techniques with nonlocal interface conditions were used to solve the equations. The porous medium was divided into subdomains with some of the interfaces representing fractures. At these fracture interfaces, the flux continuity condition was replaced by an equation representing Darcy flow along the interface. When the flow in the fracture is sufficiently rapid however, inertial effects need to be taken into account, and Forchheimer's law describes the flow in the fracture more accurately than does Darcy's law. The object of this presentation is to extend the above model to the case in which the flow along the fracture is governed by Forchheimer's law. As Forchheimer's law is given by a nonlinear equation, this model is more complex. It is derived by averaging across the fracture under the assumption that the flow in the direction normal to the fracture is less important than in directions tangential to the fracture. Domain decomposition type techniques can still be used as the nonlinearities involve only the interface unknowns. A quasi-Newton method can then be used to solve the resulting nonlinear problem on the interfaces. In this presentation the numerical model will be derived and numerical results will be presented.