Nuclear Micro-Geophysical Model LVFPM Results Involving FRACTAL DIMENSION for Porous Media
In this site's first example with fractal dimension, LVFPM was used to describe a 20% porosity pure limestone with a matrixcdensity 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 LVFPM. 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 LVFPM!