Mass Transport in Fractures
When thin fractures occur in porous media, fluid flow tends to move faster along the fracture than in the surrounding media. The transport of chemical species therefore also occur also faster in the direction of the fractures.
The fluid flow in a fracture can be modeled using Darcy’s law formulated in a thin sheet of porous medium (a fracture):
Here u is the tangential Darcy velocity, κ is the fracture permeability, μ the fluid’s dynamic viscosity, and ∇tp is the tangential gradient of the fluid pressure.
The equation to solve for mass transport of species ci in a thin fracture, embedded in a porous media, is derived from Equation 3-12. The resulting equation is:
(3-20)
Here dfr is the fracture thickness, cP, i the amount of species adsorbed to (or desorbed from) the porous matrix (moles per unit dry weight of the solid), εp is the fracture porosity, and De is the effective diffusivity. The first two terms on the right hand side represent source terms from reactions, and n0 corresponds to out-of plane flux from the adjacent porous domain.
In order to arrive at the tangential differential equation, the gradient is split into the contributions normal and tangential to the fracture:
The normal gradient is defined in the direction normal to the boundary representing the fracture and the tangential gradient is defined along the boundary. Assuming that the variations in the normal (thin) direction of the fracture are negligible compared to those in the tangential direction, the gradient is simplified as:
Using The Transport of Diluted Species in Fractures Interface, the transport along fracture boundaries alone is solved for. In this case the transport in the surrounding porous media neglected and the out-of plane flux n0 vanishes.
See Fracture for more information about the boundary feature solving Equation 3-20. See The Transport of Diluted Species in Fractures Interface for more information about the physics interface solving the equation on boundaries only.