By John H. Cushman

Porous media are ubiquitous all through nature and in lots of glossy applied sciences. as a result of their omnipresent nature, porous media are studied to 1 measure or one other in just about all branches of technological know-how and engineering. this article is an outgrowth of a two-semester graduate path on multiscale porous media provided to scholars in utilized math, physics, chemistry, engineering (civil, chemical, mechanical, agricultural), and environmental and soil technology. The textual content is essentially according to Dr Cushmans' teams efforts to construct a rational method of learning porous media over a hierarchy of spatial and temporal scales. No different textual content covers porous media on scales starting from angstroms to miles. Nor does the other textual content increase and use this type of range of instruments for his or her learn. The textual content is designed to be self-contained, because it offers all proper mathematical and actual constructs.

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13) i 0. fif ~ ~ and ~ For brevity we sometimes write =

It is also postulated that any two such states can be connected by an adiabatic work process. The internal energy E of a system in any state is defined as the adiabatic work done on the system in taking it from some arbitrarily chosen reference state to the state in question. E is clearly a state function of the independent state variables, and it is extensive (it depends on the amount of material present). W. W. 6. 4) THE PHYSICS OF FLUIDS IN HEIRARCHICAL POROUS MEDIA 23 is called the heat supplied to the system during the process.

The functions I a,N+1 and I a,N are graphed in Figs. 4. 3. 000 Figure 3 of [37). 4. 000 = 2, 1; adapted from Figure 4 of [37]. 39 40 CHAPTER I In a N scale transport problem, it may be advantageous to successively filter out lower noise scales (the drilling mud example was a two-scale problem on which these concepts should be applied). To do this, we form the sequential averages (1/J) N _2~ ~ ~ =(27r) 2fa,l*fa,2*···*fa,N*1/J. 22) Note that (1/J)N. = fa,l · · · fa,N,(fi. a,da,2 is plotted in Figs.

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