The S&M group-theoretic formalism is used to fracture the matrix rock. Fractures are treated as very thin compliant layers, which are added to the matrix rock by summing their group representations. Also, rocks are ``uncracked'' by subtracting their group representations.
I also use Hudson's first-order model in crack density to build penny-shaped cracks that are added to the matrix rock using S&M's group-theoretic formalism Nichols (1989). Both Hudson's and S&M's formulations are valid under conditions of small volume density of cracks, small crack aspect ratio, and in the long-wavelength limit.
In the first part of next section I use the S&M group theoretic formalism to add dry, wet, and air-filled cracks to the isotropic chalk. Adding dry cracks is done in the same way as for water- and air-filled cracks. I compute the fracture compliance matrix of the filling material, add it to the group representation of the matrix rock, and invert for the elastic constants of the resulting equivalent fractured medium. In the case of dry cracks, it happens that the filling material of the fracture is equal to the background material.
Since the group elements of any anisotropic system form an Abelian group, we know that the resulting elastic moduli is a stable physical ``rock.'' Yet we must be aware of the limitations of the S&M formalism: we can only add fractures as a small excess percentage of the matrix rock, since the elastic moduli of the fracture are much smaller than a typical nonzero modulus of the background material, that is,
In the second part of next section, I use Hudson's model of fracturing to vary the aspect ratio and the volume density of cracks. I have modeled three different types of cracks. Model A describes cracks with large aspect ratio and medium volume density, with a fluid as the crack filling material. Model B maintains the filling material but represents the other extreme, a small aspect ratio and large volume density of cracks. Model C is the chalk matrix, with a medium aspect ratio and a small volume density of cracks.
The numerical results for the PP reflection amplitudes are computed in a grid mesh as a function of horizontal slownesses. The grid's minimum and maximum x and y slowness values are defined from the critical angle of the isotropic chalk.
Fracture direction is considered constant for all models along the x axis (see Figure 2); thus the symmetry axis of the equivalent transversely isotropic medium lies along the y axis ().