Fracture
When fractures occur in porous media, fluid flow tends to move faster than in the bulk medium. The transport of heat occurs faster in the fractures that in the surrounding medium, so in this sense, heat transfer in fractures filled with fluids is more similar to a highly conductive layer than to a thin thermally resistive layer.
The mass transport in fractures can be modeled as Darcy’s law in a thin sheet of porous medium:
where u is the tangential Darcy’s velocity (SI unit: m/s), κ is the fracture permeability (SI unit: m2), μ the fluid’s dynamic viscosity (SI unit: Pa⋅s), and ∇tp the tangential gradient of the fluid’s pressure.
Typically, Darcy’s Law with tangential derivatives is solved to compute mass transport, so in addition to the fluid properties, the fracture should define its own permeability (or hydraulic conductivity in case the fluid is water), porosity, and fracture thickness.
For heat transfer in fractures, the fracture also needs to define the density of the porous sheet, heat capacity, and thermal conductivity. The effective thermal conductivity of the fracture must be adjusted to the fracture porosity and thermal conductivity of the fluid. In rocks and geological formations, the fracture might also contain highly conductive material, different than the bulk porous matrix.
The equation to solve for computing heat transfer in fractures is derived from Equation 4-42 to Equation 4-44 and using the procedure detailed in Theory for Heat Transfer in Porous Media to apply the mixture rule on solid and fluid internal energies. The resulting equations are:
(4-64)
(4-65)
Here (ρCp)eff is the effective heat capacity at constant pressure of the fracture-fluid volume, ρ is the fluid’s density, Cp is the fluid’s heat capacity at constant pressure, qfr is the conductive heat flux in the fracture-fluid volume, keff is the effective thermal conductivity of the fluid-fracture mixture, and Q is a possible heat source.
From the point of view of the domain, the following heat source, derived from Equation 4-44, is received from the fracture:
(4-66)
See Fracture (Heat Transfer Interface) and Porous Medium (Heat Transfer in Shells Interface) for more information about the boundary feature solving Equation 4-66. See The Heat Transfer in Fractures Interface for more information about the physics interface solving Equation 4-64.