Schoenberg's angle on fractures and anisotropy: A study in orthotropy

DISCUSSION OF VARIOUS FRACTURE MODELS

Among others, there are two main methods that are typically used in the seismic exploration literature for modeling the effects of fractures on seismic wave propagation: One is the method introduced originally by Schoenberg (1980) and the other was introduced at about the same time by Kachanov (1980). These two methods have both been used extensively in the exporation community, especially since the work of Sayers and Kachanov (1991) and Schoenberg and Sayers (1995). Connections, including many similarities and a few differences, are discussed in an overview by Schoenberg and Sayers (1995). I have some personal preference for the analytical version of Sayers and Kachanov (1991), because it is so explicit and also permits deep connections to be made to effective medium theory, especially through the work of Eshelby (1957). In particular, some recent and related work on fractures in outcrops using an effective medium theory approach by Berryman and Aydin (2009) has made use of the approach put forward by Sayers and Kachanov (1991) for modeling higher fracture density media. This study might have been more difficult to carry through using the original Schoenberg (1980) approach. But, once these higher fracture density results were known, it became straightforward to incorporate them into the layer-averaging approach emphasized by Backus (1962) and Schoenberg and Muir (1989). Thus, the effective medium theory for layering was found to be most useful for providing a means of determining fracture-fracture interactions when the fracture sets are close, but not actually intersecting.

Effects of fluids on the fracture behavior have not been emphasized here, but this issue is obviously a very important one for our applications, and it has been treated in other recent work by Daley et al. (2006) and Berryman (2006). Effects that liquids have on elastic moduli may be incorporated fairly easily using results of Gassmann (1951) and Skempton (1954), as shown by Berryman (2006).

Schoenberg's angle on fractures and anisotropy: A study in orthotropy