[page 1, §1]

[1.1.1.1] Accurate prediction and understanding
of material parameters for
disordered systems such as rocks [1],
soils [2], papers [3], clays [4],
ceramics [5], composites [6], microemulsions [7]
or complex fluids
require geometrical microstructures as a starting point
as emphasized by Landauer [8] and numerous authors
[9, 10, 11, 12].
[1.1.1.2] Digital three dimensional images of unprecedented
size and accuracy have been prepared for the case
of Fontainebleau sandstone,
and are being made available to the scientific
community in this brief report.

[1.1.2.1] Multiscale modelling of disordered media
has recently become a research focus in mathematics
and physics of complex materials and
porous media [13, 14, 15, 16, 17, 18].
[1.1.2.2] Accurate prediction of physical observables
for multiscale heterogeneous media
is a perennial problem [9, 8].
[1.1.2.3] It requires knowledge
of the three dimensional disordered microstructure [11].
[1.1.2.4] Our objective in this brief report is to
provide to the scientific public a sequence of fully three dimensional
digital images with a realistic strongly correlated
microstructure typical for sandstone.
[1.1.2.5] Resolutions from

[1.1.3.1] Despite the impressive progress in fully three dimensional
high resolution X-ray and synchrotron computed tomography
of porous media in recent years [20, 21]
acquisition times for 1500 radiograms needed for a

[1.1.4.1] The continuum multiscale modeling technology for carbonates
developed in [23, 24, 25] was applied to Fontainebleau
sandstone in [26], to create
a synthetic, non-experimental image at very high resolution.
[1.2.0.1] A laboratory sized cubic sample of sidelength

[1.2.1.1] The continuum sample generated and characterized
in [26] is the starting point for the
work reported here.
[1.2.1.2] To eliminate boundary effects a centered cubic sample,
denoted by

[1.2.2.1] The sample region

(1) |

which is the porosity inside the voxel

128 | 256 | 512 | 1024 | 2048 | 4096 | 8192 | 16384 | 32768 | |

1 | 1 | 1 | 1 | 8 | 64 | 512 | 4096 | 32768 | |

1.6 | 3.2 | 6.5 | 11 | 88 | 704 | 5632 | 45056 | 360448 |

[2.1.1.1] At the lowest resolution

[2.1.2.1] All computations were performed on the HLRS’s bwGRID cluster at the Universität Stuttgart consisting of 498 compute nodes each holding two Intel Xeon CPU’s capable of 11.32 GFLOPs. [2.1.2.2] The peak performance (Linpack) is 38 TFLOP [28]. For the calculations performed in this work 256 nodes (=512 CPU’s) were used in parallel. [2.1.2.3] The discretization algorithm was parallelized, but not optimized. [2.1.2.4] Every discretized volume element (voxel) requires one byte of storage. [2.1.2.5] The storage requirements without compression amount to roughly 40 Terabytes.

[2.1.3.1] The results can be used to calculate resolution
dependent geometric and physical properties.
[2.1.3.2] Because the underlying continuum microstructure
is available with floating point precision, the
resolution can be changed over many decades.
[2.1.3.3] Resolution dependent geometrical or
physical properties can be compared with,
or extrapolated to, the continuum result.
[2.1.3.4] This is illustrated with the porosity

[2.1.4.1] The exact values of

[2.1.5.1] The discretized samples approximate
the exact geometry of the continuum
model.
[2.1.5.2] Let

(2) |

is the number of voxels with grey value

(3) |

is an estimate for the porosity
based on a segmentation of its
discretization
at resolution

(4) |

for all

[2.2.1.1] Figure 1 shows the
porosity

[2.2.2.1] Figure 2 shows the porosity
bounds

(5) |

where

(6) |

for

-0.004944104133756 | 0.131677153054625 | 0.006128869055829 | 0.133806445926894 | |

-0.005437994277600 | 0.133652713629999 | 0.006038194581379 | 0.13416914382469 | |

-0.005680456774371 | 0.134137638623542 | 0.005981708264244 | 0.134282116458962 | |

-0.005843877976029 | 0.134301059825200 | 0.005942813391024 | 0.13432101133218 |

[3.1.1.1] Figure 3 shows the mean values

(7) |

plotted against the differences

(8) |

as data points. [3.1.1.2] It also shows a linear fit to the data as a dashed line. [3.1.1.3] The linear fit extrapolates to the value

(9) |

where the uncertainty is from the residues of the fit.

8 | 4 | 2 | 1 | 0.5 | |
---|---|---|---|---|---|

20.039 | 20.530 | 20.774 | 20.904 | 20.946 | |

9.6371 | 9.8774 | 9.9978 | 10.061 | 10.065 | |

6.0407 | 6.1915 | 6.2673 | 6.3068 | 6.3096 | |

9.168 |
9.825 |
10.159 |
10.355 |

[3.1.2.1] We now turn to an estimate for the specific internal surface

(10) |

and

(11) |

is the volume of the spherical cap. [3.2.0.2] Introducing the voxel porosity

(12) |

and solving the last equation for

(13) |

as a function of voxel porosity, where

(14) |

selects the
solution with

(15) |

is the relation between the circular base area and the
volume fraction of a spherical cap cut from a sphere of radius

(16) |

for the specific internal surface inside a voxel
with resolution

[4.1.1.1] For the lower bound
we use the circular base area obtained by
the intersection with the inscribed ball

(17) |

holds for

(18) |

where b=min resp. b=max.
[4.1.1.7] Table 3 lists the results for

[4.1.2.1] Table 3 shows that the bounds for the specific
surface are rather wide.
[4.1.2.2] It is then of interest to investigate the following
simple geometric estimate:
In the limit

[4.1.3.1] Let

(19) |

and the area of the triangle is

(20) |

[4.2.0.2] For the range

(21) |

and

(22) |

[4.2.0.4] The specific surface area of a voxel with grey value

(23) |

computed from
eqs. (19) and (20)
for

[4.2.1.1] The last line in Table 3 gives the specific internal
surface