Nuclear Micro-Geophysical Model LPVM
Results Involving FRACTAL DIMENSION
for Porous Media
In this site's first example with fractal dimension, LVPM was
used to describe a 20% porosity pure limestone with a matrix
density of 2.71 g/cc and a matrix capture cross section of 7.1 CU;
its pores were 100% saturated with fresh water. The maximum
pore radius (Rmax) was fixed at 0.20 cm and the minimum pore
radius (Rmin) was allowed to vary while remaining consistent both
with the current input fractal dimension and the current input
porosity.
In the left-hand figure below, apparent density and apparent
neutron porosity, as obtained from standard nuclear logging tools,
are plotted on the left axis while the minimum pore radius is
plotted on the right axis; all are shown versus the limestone fractal
dimension (D) as outputs from the nuclear micro-geophysical
forward model LVPM. Similarly, the right-hand figure below
shows the apparent logging bulk density and apparent fast neutron
slowing down length versus limestone fractal dimension. These
physical quantities drive the apparent density and apparent neutron
porosity values previously discussed.
The apparent thermal neutron diffusion length and apparent
thermal neutron diffusion coefficient are also plotted versus the
limestone fractal dimension for the exact same conditions as the
other figures of this first example. Finally, the apparent
photoelectric factor is shown versus fractal dimension: over the full
range of fractal dimensions from 1.001 to 2.999, barely a 1%
change in Pe results for this textbook limestone!
At lower fractal dimensions, apparent density porosities remain
more than 1 PU above the true porosity while the apparent neutron
porosities are more than 2 PU below the true porosity. At these low
fractal dimensions, pore sizes are relatively large and tightly
clustered just below Rmax (=0.20 cm): this behavior is analogous
to previous LVPM results in vuggy porous media with a fixed
pore size.
At higher fractal dimensions, the minimum pore radius
dramatically reduces and both the density and neutron
apparent porosities begin their approach to the true porosity
at 0.20. At a fractal dimension of 2.99, the apparent neutron
porosity is 0.198 and the apparent density porosity is 0.201.
Neutron porosity logging measurements in both the wireline and
MWD commercial operations are based on four test tanks located
at the University of Houston: (1) Carthage Marble, (2) Indiana
Limestone, (3) Austin Chalk, and (4) 100% freshwater. For these
so-called "gold standard" calibrators there exist no supporting
measurements of sigma matrix and no measurements of pore
sizes, pore size distributions, or fractal dimensions. Both sigma
matrix and fractal dimension corrections to neutron porosity
are supported by the nuclear micro-geophysical forward model
LVPM!
