13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- 2.2. Averaged permeability of a RVE The homogenised permeability for a given local permeability distribution within the RVE can be evaluated using an upscaling scheme similar to the one developed for heat conduction in [2]. For fluid flow, the mass conservation equation at the scale of the components has to be solved. Expressing the fine scale flux is expressed in terms of the local permeability distribution and with a fluid of dynamic viscosity , mass conservation reads !− # $ 0 (4) A periodic fluctuation # of the pressure field is assumed [1,2] to describe the pressure variation inside the RVE according to # # & '#'.( − &) # (5) where '#' is the macroscopic pressure gradient to be applied in an average sense on the RVE, and where # & is the pressure of an arbitrary point in the RVE. An averaging relation for the pressure gradient is required as '#' 1 # (6) Assuming the periodicity of the pressure field fluctuation and using the Green-Gauss theorem, the average of the fine-scale pressure gradient can be expressed as 1 # '#' 1 * # + , (7) With the periodicity of # , the last integral of this relationship vanishes and the relationship prescribing the consistency between the macroscopic and fine scale pressure gradients is satisfied. In a computational treatment, the periodicity constraint on the pressure fluctuation field requires that relationships of the form # - −# ' '#'. - − ' (8) be satisfied between the master M and slave S nodes of opposite faces of the RVE. An additional averaging relationship is used to impose consistency between scales of the product of the pressure gradient by the flux '#'.. ' 1 # .. (9) Combining the pressure gradient averaging with this relationship, it can be shown that the macroscopic flux is automatically obtained as the RVE average of the fine-scale fluxes [2] . ' 1 . (10) Details of the derivation of this last averaging relationship can be found in [1]. The constraints on the pressures at the boundary can be enforced using control nodes to apply the macroscopic pressure gradients to the RVE. The averaged permeability tensor of the RVE can be identified from the link between the applied macroscopic pressure gradient and the fluxes developing as a reaction
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