When is integrated over a direction normal to the fracture, the result is. The one-dimensional Dirac delta function,, has SI units m, and is zero everywhere except on the fracture. For example, if the fracture occupies the plane, then would be a suitable function. Suppose the position of the fracture is given by a level-set function, (where, and are the Cartesian coordinates): that is, the fracture is at points where, but not at points where. In order to specify the coupling between the fracture and matrix system precisely, let us introduce some notation. If tensorial quantities such as or the permeability tensor are anisotropic, they need to be expressed in 3D space. That is, in the MOOSE input file governing the 2D physics, the coefficient of the CoefTimeDerivative Kernel is, and the coefficient of the AnisotropicDiffusion is. The preceeding equation implies all the Kernels for heat and mass flow in the fracture system must be multiplied by the fracture aperture. This equation is integrated (by MOOSE) over the fracture-element 2D area in the finite-element method. Where is the fracture thickness (aperture), and are derivatives transverse to the fracture. Therefore, the integral over the thickness may be done explicitly, and the fracture modelled as a 2D object, with heat equation reading The fracture is actually a 3D object, but it is so thin that temperature (and and ) do not vary appreciably over its thickness (aperture). This equation (multiplied by the test functions) gets integrated over the elemental volume in the finite-element scheme. Is the thermal conductivity, with SI units J.s. Is time and is the vector of spatial derivatives Is the volumetric heat capacity of the medium (which is the product of specific heat capacity and density), with SI units J.m. The heat equation in fracture-matrix systems Uncoupled equations MultiApp primer: the diffusion equation with no fractures, and quantifying the errors introduced by the MultiApp approach This page is part of a set that describes a MOOSE MultiApp approach to simulating such models: The fundamental premise is the fractures can be considered as lower dimensional entities within the higher-dimensional porous media. PorousFlow can be used to simulate flow through fractured porous media in the case when the fracture network is so complicated it cannot be incorporated into the porous-media mesh. cpp files.Fracture flow using a MultiApp approach: Equations Background Doxygen only generates documentation from.
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