Eulerian kinematics of flow through spatially periodic models of porous media

For spatially periodic models of porous media, and incompressible fluids, we consider here in detail thephysical interpretation of averaged velocity fields in terms of a purely continuum-level Eulerian seepage flux relation involving the mass flow across an arbitrarily oriented ‘macroscale’ surface element. We prove that the appropriately averaged microscale velocities do indeed satisfy a macroscale Eulerian flux relation (albeit to some approximation), provided that the surface ‘element’ is large compared with the pore lengthscale (microscale). A quantitative discussion of an ‘intermediate scale’ (lying between the pore scale and a characteristic macroscale) is achieved through the use of locally periodic velocity fields. Within the context of seepage velocity kinematics, our developments provide a conceptual foundation that renders continuum theories of flow of incompressible fluids through (periodic) porous media entirely selfcontained on the macroscale—free from any details of the discrete microscale structure (inaccessible to macroscopic observers) from which this continuum theory sprang. The error bounds and scaling laws derived are of special interest when the macroscale/microscale disparity is not very ‘wide’. With reference to current rational mechanical theories of ‘immiscible mixtures,’ our analysis provides aconstructive existence proof, at least for periodic models of porous media, for phase-specific velocities; heretofore, such velocities were usually accepted, in a Lagrangian sense, on an axiomatic basis. Thus, clarification and quantification of thekinematical aspects of the continuum hypothesis for certain classes of multiphase systems can be effecteda priori by our arguments, independently of any subsequent macroscaledynamical modelling. Our analysis, whereby we systematically proceed from a discrete picture of the underlying geometry to a purely continuum picture, may be likened to that achieved in classical statistical mechanics. As in that case, we hope that the general results obtained transcend the underlying discrete (i.e. spatially periodic) model of the subcontinuum structure.