IEEE Signal Processing - March 2018 - 128

ments typically do not use the same rock volumes. Printing
3-D models of digital rock scans overcomes this mismatch
between physical and digital experiments. It can also provide a
useful tool for the validation of various theoretical models used
in estimating rock properties. In [35], 3-D-printed models were
used to verify theoretical models used for permeability estimation. An example of these models is the permeability-tortuosity
relationship. Assuming that fluid flows, in a cylindrical sample
with cross-sectional area A, within a tube of radius r such that
the flow path has tortuosity, x, the permeability of the sample,
k, can be computed using the following equation:

Fontainebleau Sample

40
35

Bulk Modulus (GPa)

30
25
2-D Trend
20
15

2
k = rr ,
8Ax

10
5
0

0

0.05

0.1

0.15
Porosity

0.2

0.25

3

Sample A
Sample B
Sample C
2-D Shifted with Power Law (m = 0.4)
2-D Shifted with Power Law (m = 0.5)
2-D Shifted with Power Law (m = 0.6)
(a)
106

LBM-Computed 2-D

Permeability (mDarcy)

104

Porosity Model (ρ = 1-2)

102

Radius Variation
Model (δ = 0.3-0.5)

LBM-Computed 3-D

100

10-2

where tortuosity, x, is defined as the ratio between the length
of the flow line between two points to the straight line distance
between the points.
Four different cylindrical samples were perforated with
four helical tubes with diameters of 600 nm. The number of
turns varied from two to five to test different tortuosity values.
Figure 10 shows illustrations of two helical tubes along with
comparisons between permeability measurements as calculated using (14), simulated using LBM, and measured using
experimental laboratory techniques. Shaded regions cover
a 5% margin of deviations between the model used in computations and the 3-D-printed objects. These differences are
caused by the limited resolution of the 3-D printer (minimum
layer thickness of 25 nm) . Laboratory measurements are consistently below simulated and calculated values. This is likely
caused by the incomplete removal of the wax used by 3-D
printers to fill cavities such as rock pores.
Another application of 3-D printing was shown in [37] in
which a fracture network was 3-D printed. The use of 3-D-printed
networks can help validate and improve current flow modeling
techniques in fractured media. Experimental and numerical
methods for evaluating effective permeability in the printed
fractured sample were compared. Experimental values were
shown to be consistent with numerical techniques.

Applications in the coal industry
0

0.02

0.04
0.06
Porosity

0.08

0.1

(b)

Figure 9. Estimating 3-D properties from 2-D thin sections: (a) Bulk modulus estimates are shown for the 3-D rock samples and the corresponding
thin sections (smaller shapes). The power-low (10) with m K = 0.5 correctly matches the 3-D porosity-permeability trend [33]. (b) The average
permeability from thin sections is used to correctly predict 3-D permeability using (12) and (13). (Figure used with permission from [34].)

3-D printing
Several researchers have proposed the use of 3-D printers for
building synthetic rocks from CT scans [35], [36]. This new
technology can potentially be used effectively to match laboratory and digital experiments. Laboratory and digital experi128

(14)

Coal bed methane (CBM) is a natural gas commonly extracted
from unconventional gas reservoirs. Gas permeability is a key
factor in the permeability of such carbon reservoirs. These
reservoirs are dual-porosity reservoirs that are composed of
a porous matrix surrounded by a larger-scale fracture system
(cleats). Permeability of the reservoir is effectively controlled
by the cleat system since the permeability of the coal matrix is
much lower than that of cleat networks [38].
A major difficulty in imaging cleat networks is the very high
resolution required to resolve them in CT scans. The contrast
between cleat networks and the coal matrix is often low due to
partial volume effects. Ramandi et al. [39] used a contrast agent to
enhance the appearance of cleats. A dry sample was first imaged
using an X-ray scanner. Next, a brine fluid of potassium chloride
and sodium iodide was used to saturate all cleats. The saturated
core was then imaged using a micro-CT scanner.

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