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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*(*g*_{1}+2*g*_{0},*g*_{2}+2*g*_{0}) and *G*(*g*_{1},*g*_{2})+2*g*_{0}, where *g*_{0} can take any
positive value, but *g*_{0} is limited in the negative range by the
limitations that both *g*_{1}+2*g*_{0} and *g*_{2}+2*g*_{0} must always be
nonnegative.
A straightforward, but somewhat tedious calculation shows that
| |
(5) |

The right hand side of this equation is always positive whenever *g*_{0}
>0 and . It vanishes when *g*_{0} = 0 or *g*_{1} = *g*_{2}.
If *g*_{1} < *g*_{2}, then for negative values of the parameter *g*_{0}, allowed
values of *g*_{0} lie in the range . For such values
of *g*_{0}, 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*(*g*_{1},*g*_{2}) without any further
restrictions on the measurable quantities , and *F*_{2}.
A second bound can be obtained (again in the limit 2*g*_{0} = -*g*_{1})
by noting that

| |
(7) |

and then recalling that
| |
(8) |

Substituting (7) into (5) produces an upper
bound on *G*(*g*_{1},*g*_{2}). 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 *g*_{1} = *g*_{2} = 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 *g*_{1} 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 *g*_{1} = *g*_{2}. The general bounds
are therefore

| |
(13) |

where *S*_{1} and *S*_{2} were defined in (6) and
(9).

** Next:** SECOND DERIVATION
** Up:** Berryman: Bounds on transport
** Previous:** THE ANALTYICAL FORMULATION
Stanford Exploration Project

10/23/2004