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By M. Panfilov
The The booklet publication was once used to be deliberate deliberate in in such any such a fashion demeanour that that easy easy ambitions pursuits may will be be reached. reached. On at the the only one hand, hand, the the aim target was once used to be to to teach express a few a few new new effects ends up in within the the sector box of of modeling modeling delivery shipping via via hugely hugely heterogeneous heterogeneous media, media, dependent according to at the the homogenization homogenization conception. thought. a number of a number of new new mathematical mathematical types types of of delivery delivery are are provided offered herein, herein, learning learning their their houses, houses, constructing constructing tools the right way to to compute compute powerful potent parameters parameters of of the the averaged averaged media, media, simulation simulation of of mobilephone phone difficulties, difficulties, utilizing utilizing new new versions types to to simulate simulate a few a few useful functional difficulties. difficulties. excessive excessive heterogeneity heterogeneity being being subjected subjected to to the the homogenization homogenization technique, approach, generates generates non-local non-local phenomena phenomena and after which then provides supplies a available threat to to boost enhance a a brand new, new, non-local non-local (or (or "dynamic"), "dynamic"), conception conception of of delivery shipping in in porous porous media. media.
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Extra info for Macroscale Models of Flow Through Highly Heterogeneous Porous Media
Sample text
C, i=1, ... , s. 6) get the form: :=1 lap a ( I ap ) cpb aT - aYi aij aYj 2 = = c f*, = c2 f c2 bII ap _ ~ (aU a p ) = aT aYi ~J aYj *, yEy l yEyII A new parameter arises: c2 c 2 Wm c =--=p WK A Its order determines the difference in the equations on the blocks and on the matrix. The parameter cp is the measure of the rate of perturbation propagation through a block. In fact, if fP/ / a:/ I =WK / Wm is a ratio of the block and the matrix piezo-conductivity rei , than the coefficient II )) is the ratio of perturbation propagation time for one cp={l2 /re l )/{L 2/re II block to that for the overall matrix.
5-b. 'I'ranslation-Type Translation- Type Media (re-Homogeneous) rv c2 . The heterogeneity degrees These media corresponds to the case of cp cpI"Vc for porosity and permeability are equivalent: WKrvw WKrvWm , Arv1. 3) are equivalent and quasistationary, such that the non-stationarity is displayed only in a time boundary layer of the size rvc rv£22 at vicinity of the perturbation moment. The medium is trivially heterogeneous. The problem refers to coefficiently averaged. ~~ ~~ .. 5. II b Source flow (a) and Translation flow (b) through a cell and to the permeability.
If we reduce the heterogeneity degree, we get an intermediate case, when the homogenized model exists, but appears to be not a best limit of upscale process. More strong convergence is ensured to other limit, which does not coincide with the simple homogenized value. At least, in highly heterogeneous media, upscaling process is not equivalent to the usual homogenization. As consequence, usual methods of upscaling should be modified. Hence, the problem of highly heterogeneous media consists in two items: - 1) what is the limit of upscaling ?