Grechka and Kachanov (2006a,b) have generated two distinct numerical data sets
for crack densities , 0.10, 0.15, and 0.20. (Crack
density is defined as , where *n* is the number density
per unit volume. Radius of a typical crack is *a*.
If there is a distribution of crack sizes, then *na ^{3}* is replaced by
an appropriate average value.) Cracks are all penny-shaped, but not necessarily flat. We assume that the
aspect ratio is small (but finite,

The two models considered have very different Poisson's ratios for the
isotropic background media: (1) and
(2) . We will call these two models,
respectively, the first model and the second model.
The first model corresponds to a rigid medium very stiff under shear,
while the second model corresponds to a medium much more compliant
in shear.
The two models approximately bracket the expected range of
behavior in earth systems, for which a typical value might be
in reservoir rocks.
The first model has background stiffness matrix values
*C _{11}* =

The numerical modeling of Grechka and Kachanov (2006a,b) for the first model resulted in 110 examples: 23 for , 21 for ,35 for , and 31 for . Of these 110 cases, we used 79 (selected randomly for the three smaller crack densities): 8 for , 16 for , 24 for , and all 31 for .

The numerical modeling of Grechka and Kachanov (2005,2006a,b) for the second model resulted in 100 examples: 25 each for all four cases . Of these 100 cases, we used 73 (selected randomly for the three smaller crack densities): 8 for , 16 for , 24 for , and all 25 for .

For both numerical data sets, the output of the modeling for each
example was a stiffness matrix. All these matrices were observed to
be essentially orthotropic in character (*i.e.*, depending
significantly only on the nine stiffness constants
*C _{11}*,

We also show two other estimates: SMP for ``sample,'' and GR for ``grain.''
Since these computations were all done using displacement boundary
conditions, we have made use of results from Huet (1990) on sampling -- the pertinent result being that effective overall stiffness
*C*_{ij}^{*} of a random medium (one having a well-defined statistical ensemble
associated with it) satisfies
, where

(1) |

One class of approximations that has been found very useful for analysis of these cracked systems by Grechka and Kachanov (2006a,b) is the non-interaction approximation (NIA). Since our results will be presented in the form of plots of effective bulk and shear moduli, it is useful to consider the corresponding NIA formulas for bulk and shear moduli. Zimmerman (1991) gives such formulas, and in our present notation these formulas are:

(2) |

(3) |

Another method that we use here involves a crack-influence
decomposition method of Sayers and Kachanov (1991); see Appendix B for details.
When all cracks in the system have the same vertical (*z*-)axis
of symmetry, then the cracked/fractured system is not isotropic,
and we have the compliance correction matrix

(4) |

(5) |

(6) |

(7) |

(8) |

(9) |

(10) |

(11) |

(12) |

The crack influence decomposition parameters and can be evaluated using the formulas presented for any convenient value of the crack density . But this procedure is seen to be most useful if we evaluate the parameters at small , since in that limit all the standard methods should give essentially the same results. This approach has been tested and found to be correct.

Once it is known that it makes little difference at low crack densities which theoretical methods we use to estimate the crack-influence parameters, we might as well consider the simplest one, which is surely the non-interaction approximation. Combining (2) and (3) with (11) and (12), we find easily that, within the NIA,

(13) |

(14) |

(15) |

(16) |

Once values of and are known in this way (using NIA as shown or some other method), we can use the Sayers and Kachanov (1991) method as one convenient way to study and evaluate anisotropic behavior in cracked systems -- thus, providing a simple method of extending the non-interaction approximation results, as quoted by Zimmerman (1991), to nonisotropic systems. We then introduce some crack-crack interactions here in a novel way by making use of the polycrystal of cracked-grains model.

Appendix B summarizes the main analysis using the crack-influence
parameter approach of the Sayers and Kachanov (1991) method.
We find that, in addition to the two parameters
and that are easily found using effective medium
theories such as NIA or DS (differential scheme), there are three more
parameters that are expected to play a role in our results at higher
crack densities: , , and . These are the only
significant quadratic corrections to (4). The analysis shows
[see, for example, Eq. (40)] that to second
order in the crack
density , we need to replace by and by . In addition, the parameter
comes into play, but perturbs only the *S _{33}* component of
the compliance. The Reuss averages of shear modulus and bulk modulus
are now given by

(17) |

(18) |

Also, recall that we are focusing here on penny-shaped (and therefore ellipsoidal) cracks. Phenomenology for other crack shapes may differ somewhat from the discussion presented here (Mavko and Nur, 1978). However, other shapes are beyond our present scope and so will necessarily be treated elsewhere. The numerical experiments considered here all used ellipsoidal cracks.

1/16/2007