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Up-scaling Flow in Fractured Media: Equivalence Between the Large Scale Averaging Theory and the Continuous Time Random Walk Method 总被引:1,自引:0,他引:1
In a recent paper, (McNabb, 1978), we set up a method allowing to compute both the transient and steady-state exchange terms between the matrix and fractured regions of a naturally fractured porous medium using the continuous time random walk method (CTRW). In particular, the exchange coefficient parametrizing the large-scale exchange term was computed on physical grounds using a time integration of the so-called time correlation function corresponding to the particle presence in the fractures. On the other hand, the large scale averaging theory (LSAT) as developed by Quintard and Whitaker (Quintard and Whitaker, 1996) gives another definition for this exchange coefficient . It also provides a so-called closure problem allowing to compute from the solution of a well-defined steady state boundary value problem, to be solved over a representative volume of the high resolution fractured map. The goal of the present paper is to show analytically that both definitions coincide, yielding a unique and well defined value of the coefficient. This provides an unification of two approaches whose respective backgrounds are very different at the first glance. 相似文献
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Permeability up-scaling using Haar Wavelets 总被引:1,自引:0,他引:1
In the context of flow in porous media, up-scaling is the coarsening of a geological model and it is at the core of water
resources research and reservoir simulation. An ideal up-scaling procedure preserves heterogeneities at different length-scales
but reduces the computational costs required by dynamic simulations. A number of up-scaling procedures have been proposed.
We present a block renormalization algorithm using Haar wavelets which provide a representation of data based on averages
and fluctuations.
In this work, absolute permeability will be discussed for single-phase incompressible creeping flow in the Darcy regime, leading
to a finite difference diffusion type equation for pressure. By transforming the terms in the flow equation, given by Darcy’s
law, and assuming that the change in scale does not imply a change in governing physical principles, a new equation is obtained,
identical in form to the original. Haar wavelets allow us to relate the pressures to their averages and apply the transformation
to the entire equation, exploiting their orthonormal property, thus providing values for the coarse permeabilities.
Focusing on the mean-field approximation leads to an up-scaling where the solution to the coarse scale problem well approximates
the averaged fine scale pressure profile. 相似文献
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Two-Phase Inertial Flow in Homogeneous Porous Media: A Theoretical Derivation of a Macroscopic Model
The purpose of this article is to derive a macroscopic model for a certain class of inertial two-phase, incompressible, Newtonian
fluid flow through homogenous porous media. Starting from the continuity and Navier–Stokes equations in each phase β and γ, the method of volume averaging is employed subjected to constraints that are explicitly provided to obtain the macroscopic
mass and momentum balance equations. These constraints are on the length- and time-scales, as well as, on some quantities
involving capillary, Weber and Reynolds numbers that define the class of two-phase flow under consideration. The resulting
macroscopic momentum equation relates the phase-averaged pressure gradient to the filtration or Darcy velocity in a coupled nonlinear form explicitly given by
or equivalently
In these equations, and are the inertial and coupling inertial correction tensors that are functions of flow-rates. The dominant and coupling permeability tensors and and the permeability and viscous drag tensors and are intrinsic and are those defined in the conventional manner as in (Whitaker, Chem Eng Sci 49:765–780, 1994) and (Lasseux
et al., Transport Porous Media 24(1):107–137, 1996). All these tensors can be determined from closure problems that are to
be solved using a spatially periodic model of a porous medium. The practical procedure to compute these tensors is provided. 相似文献
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