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# FORMATION FACTOR BOUNDS

To obtain some useful bounds, I again consider the form of (1)
 (4)
For reasons that will become apparent I want to compare the values of G(g1+2g0,g2+2g0) and G(g1,g2)+2g0, where g0 can take any positive value, but g0 is limited in the negative range by the limitations that both g1+2g0 and g2+2g0 must always be nonnegative. A straightforward, but somewhat tedious calculation shows that
 (5)
The right hand side of this equation is always positive whenever g0 >0 and . It vanishes when g0 = 0 or g1 = g2. If g1 < g2, then for negative values of the parameter g0, allowed values of g0 lie in the range . For such values of g0, the right hand side of (5) is strictly negative.

The limiting case obtained by taking is most useful because, in this limit, -- thus eliminating the unknown functional from this part of the expression. Then, (5) shows that
 (6)
which is a general lower bound on G(g1,g2) without any further restrictions on the measurable quantities , and F2.

A second bound can be obtained (again in the limit 2g0 = -g1) by noting that
 (7)
and then recalling that
 (8)
Substituting (7) into (5) produces an upper bound on G(g1,g2). By subsequently substituting (8) and then rearranging the result, the final bound is
 (9)

Comparing (6) and (9), I see consistency requires that
 (10)
must be true. Rearranging this expression gives the condition
 (11)
the validity of which I need to check. In the limit g1 = g2 = 1, a sum rule follows from (4), and from this I have:
 (12)
This shows explicitly that (11) is always satisfied as long as . If this inequality does not hold, then the sense of the bounding inequalities is changed, so the expressions for the upper and lower bounds trade places.

When and g1 varies (as would be expected in a series of thermal conductivity experiments with different fluids in the same porous medium), then (6) and (9) are both straight lines that cross at g1 = g2. The general bounds are therefore
 (13)
where S1 and S2 were defined in (6) and (9).

Next: SECOND DERIVATION Up: Berryman: Bounds on transport Previous: THE ANALTYICAL FORMULATION
Stanford Exploration Project
10/23/2004