In this third example, LVPM was used to model the effects
of a vuggy porous oil-saturated dolomite on neutron and
density logs - i.e. finite pore size effects.  Bed
thickness is not used.  The focus is on pore size effects
at a fixed porosity.

In Figure 1, LVPM heterogeneous bulk density decreases
linearly from 2.510 g/cc to 2.471 g/cc as pore size varies
from 0.0001 cm to 1.0 cm. The slope is –0.0386 g/cc per cm.
From Figure 2, note that the corresponding heterogeneous
density porosity increases from 0.117 pu to 0.140 pu at a
rate of 0.0229 pu per cm over the same pore size interval.

In Figure 3 the heterogeneous neutron slowing down length
increases from 9.954 cm to 11.464 cm as the pore size varies
from 0.0001 cm to 1.0 cm.  This leads to the dramatic
decrease in heterogeneous neutron porosity from 0.364 pu to
0.177 pu shown in Figure 3, as the LVPM proxy model uses
this slowing down length to compute neutron porosity.

Figure 4 shows that the heterogeneous neutron capture cross
section (SIGMA) decreases linearly as the pore size increases;
the slope is –0.0329 cu/cm.  Recall that SIGMA is interrelated
with the thermal neutron diffusion length (L) and the thermal
neutron diffusion coefficient (D) by the expression:

              D = (L**2)*SIGMA.

Figure 5 reveals the variation in heterogeneous thermal
neutron diffusion length and thermal neutron diffusion
coefficient as the pore size varies.  Although the change in
capture cross section (SIGMA) with pore size is not large
in this example, these variations of D and L with pore size
are an important aspect of the correction of SIGMA for
thermal neutron diffusion.

These variations with pore size also have important
implications for porosity logging based on the thermal
neutron diffusion coefficient [see U. S. Patent 3,818,225 and
also the excellent review article “Nuclear Geophysics in
Prospecting, Exploration and Development of Oil and Gas
Fields” by E. V. Karus and Yu. S. Shimelevich, in All-Union
Research Institute of Geophysics, 8 Warshavskoye Shosse,
M-105, Moscow, USSR; also published in 1983: International
Journal of Applied Radiation and Isotopes, v. 34, no. 1,
p. 95-117, by Elsevier Science Ltd.]