[68.1.2.1] This section returns to eq. (3.2)
and presents numerical solutions.
[68.1.2.2] This is intended as a case study exploring the relationship
between the statistics of local geometries
and bulk dielectric behavior.
[68.1.2.3] The main focus will be on dielectric enhancement.
[68.1.2.4] To solve eq. (3.2) for

[68.1.3.1] The hypothesis of local simplicity states that the local geometries are simple and that the effective local dielectric constants are insensitive to geometrical details other than local porosity. [68.1.3.2] The simplest isotropic local geometry is spherical. [68.2.0.1] For conducting local geometries, a water-coated spherical rock grain will serve as the local model. [68.2.0.2] For blocking geometries a rock-coated spherical water pore is employed. [68.2.0.3] In the notation of Sec. IV, this means

(6.1) | |||

(6.2) |

[68.2.0.4] In the low-frequency limit, one obtains for the conducting geometry

(6.3) |

thereby identifying

(6.4) |

[68.2.0.6] For the blocking geometry, the dc limit gives

(6.5) |

for

[68.2.1.1] It was mentioned repeatedly that no experimental data for

[68.2.2.1] For the subsequent calculations,
a simple mixture of two

(6.6) |

where

(6.7) |

[68.2.2.4] For

[68.2.3.1] Eight different local porosity distributions
are compared in the calculations.
[68.2.3.2] All of them are chosen such that they give
the same bulk porosity

Curve No. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|---|

1 | 1 | 2/3 | 1 | 1 | 1 | 2/3 | 2/3 | |

1.0 | 360.0 | 191.1 | 7.2 | 4.5 | 1.8 | 28.8 | 58.6 | |

1.000 | 40.000 | 3.900 | 0.800 | 0.500 | 0.200 | 0.087 | 0.176 | |

1423.0 | 13.9 | 2.24 | ||||||

500.00 | 6.00 | 0.96 | ||||||

0.1 | 0.1 | 0.02 | 0.1 | 0.1 | 0.1 | 0.003 | 0.003 | |

0.26 | 0.294 | 0.294 | ||||||

0.00333 | 0.00024 | 0.00010 | 0.00010 | 0.01500 | 0.03000 | 0.00010 | 0.00005 | |

0.00010 | 0.01000 | 0.05000 | ||||||

0.1 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | |

Variance | 0.00333 | 0.00024 | 0.01290 | 0.01000 | 0.01500 | 0.03000 | 0.02213 | 0.03541 |

Skewness | 0 | 0.2657 | 0.6993 | 1.6004 | 1.8679 | 2.3124 | 1.1588 | 2.1176 |

[69.1.1.1] The choices presented in Fig.

[69.2.1.1] The local percolation probabilities

[69.2.2.1] Three simple models will be compared in the calculations: the uniformly connected model (UCM), the central pore model (CPM), and the grain consolidation model (GCM).

[69.2.3.1] In these models

(6.8) |

[69.2.3.2] In the simplest case, the fully connected model,

[70.1.1.1] Consider a cubic cell of volume 1 filled with rock.
[70.1.1.2] Inside the cubic cell a centered cubic pore
of side length

[70.1.2.1] The result of the process described above
is a cubic cell whose porosity can be expressed
in terms of the side length

(6.9) |

[70.1.2.2] According to the definitions in Section II,
the cell is called percolating
if there exist at least one path
within the pore space connecting a face to a face
different than itself.
[70.1.2.3] To obtain

(6.10) |

[70.1.2.6] Thus, in the central pore model,

(6.11) |

where

(6.12) |

where

[70.1.3.1] The grain consolidation model was proposed as a simple geometrical model for diagenesis.[11][70.1.3.2] Its main observation is the existence and smallness of the percolation threshold in regular and random bead packings when the bead radii are increased. [70.1.3.3] In fact, the model has recently been modified such that the critical porosity at which conduction ceases can be arbitrarily small.[12][70.2.0.1] For regular bead packings, this implies

(6.13) |

[70.2.0.2] For random packings

[70.2.1.1] The most important aspect of

[70.2.2.1] Numerical solutions to eq. (3.2)
were obtained using an iterative technique.
[70.2.2.2] The iteration was stopped whenever

[70.2.3.1] It is obvious from Figs. 2-5 [especially part (b)]
that the low-frequency dielectric response
depends sensitively on the details of

[71.1.1.1] The absolute dispersion for all figures is collected
in Table 2.
[71.1.1.2]

[71.2.1.1] Before discussing the three mechanisms,
it is important to note that the bulk porosity

[page 72, §0]

[page 73, §0]
[73.1.1.1] A second observation is that in all figures
the high-frequency real dielectric constant

[73.1.2.1] The first mechanism, dispersion from the form of

[73.1.3.1] Consider the thin-plate mechanism.
[73.1.3.2] It requires the presence of blocking geometries
of high porosity.
[73.1.3.3] Mathematically, this means

[73.2.1.1] The percolation mechanism is responsible
for strong dielectric disperion
in figures 4 and 5.
[73.2.1.2] There is essentially no dispersion
from thin-plate mechanism in these cases
because in both cases

Curve No. 1 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|---|

0.798 | 0.053 | 3.716 | 1.863 | 3.147 | 9.261 | 28.793 | 20.484 | |

0.298 | 0.035 | 1.067 | 0.619 | 0.889 | 1.545 | 2.400 | 2.029 | |

19.197 | 13.819 | 21.956 | 25.162 | 34.950 | 96.046 | 186.010 | 115.270 | |

1.585 | 1.370 | 1.664 | 1.674 | 1.741 | 1.648 | 1.751 | 1.489 | |

5.996 | 3.835 | 16.616 | 9.639 | 14.204 | 52.196 | 259.700 | 3932.522 | |

1.105 | 0.801 | 1.974 | 1.455 | 1.708 | 2.176 | 2.877 | 2.283 | |

2.119 | 0.053 | 371.525 | 5.498 | 11.430 | 291.481 | 277.396 | 37.691 | |

0.595 | 0.035 | 3.025 | 1.137 | 1.604 | 2.565 | 3.043 | 2.436 |

[73.2.2.1] The complexity and variability of

Curve No. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|---|

CPM | 0.6542 | 0.6940 | 0.5420 | 0.5858 | 0.5341 | 0.4059 | 0.3643 | 0.3334 |

GCM | 0.7500 | 1.0000 | 0.3399 | 0.6048 | 0.4962 | 0.3415 | 0.3376 | 0.2841 |

[73.2.3.1] The present paper deals only
with simple homogeneous and isotropic porous media.
[73.2.3.2] Real rocks are highly inhomogeneous,
but they can also be discussed
within the present framework.
[page 74, §0]
[74.1.0.1] Sedimentary rocks exhibit two main types of porosity.
[74.1.0.2] Primary interparticle porosity
is the porosity between the grains
of the original sediment.
[74.1.0.3] Often, this pore space is changed
during diagenesis of the sediment.
[74.1.0.4] In particular cements between the grains
can exhibit a qualitatively different secondary porosity.[33]
[74.1.0.5] This situation can be treated within
the present formalism
by replacing the dielectric constant