Bear - Hydraulics of Groundwater.

(A-1)

(A-2)

(A-3)

(A-4)

(A-5)

(A-6)

(A-7)

Let P(x) be a point in a multiphase porous medium domain through which flow takes place. Similar to the definition of a Representative Elementary Volume (Sec. 2-5), we may also define at every point x a Representative Elementary Area

(REA) A 0, such that an average of a property over it will represent a meaningful value of that property at its centroid. The area A 0 j () A 0 j at P is facing the direction Ij.

In order to interpret the meaning of the product at point P(x), let U 0 have the shape of a straight cylinder having a cross-sectional area A 0 j facing the direction 1j and length L j . Let V j be the component in the direction 1j of the local velocity at all points in the phase, that is, V j = 0 outside the phase. Then, from (A-7)

(A-8)

Now, the expression in the square brackets in (A-8) is the specific discharge , that is, the total discharge per unit area of A 0 j

(A-9)

so that is the average of over parallel cross-sectional areas of the REV (here from x - ( L j l2 ) 1j to x + ( L j /2) 1j). For the sake of simplicity, we have assumed that the REV has a cylindrical shape, with U 0 = L j A 0 j .

We have thus shown that the macroscopic velocity defined by (A-7) has the meaning of the specific discharge vector q of the phase at that point. It is of interest to note that q is a function of (x, t ) only, independent of the direction chosen for Ij.

A-3 SOME AVERAGING RULES (BACHMAT, 1972; GRAY AND LEE, 1977)

( a ) Let 0 denote the total surface area between the phase and the other phases within U 0. We define the specific surface of the phase by

(A-10)

and the average of G (x, t ) over 0 for a given ( U 0) by

(A-11)

Also

(A-12)

(b) Average of a product

(A-13)

Also

Hence

(A-14)

(c) Average of a time derivative

(A-15)

where u n is the speed of displacement, along the outward normal, of a point on the interface between the phase and the other phases within U 0. Cases of interest:

1. G 1. Then = land

/ t = U n

2. U n = 0, that is, the configuration of the phase is nondeformable. Then

( G / t ) = / t

(d) Average of a spatial derivative

(A-16)

where n i is the i -component of a unit vector normal to and pointing outwards.

In (A-16) we have to consider only the multiply connected portion of the domain.

Case of special interest: