Effective elastic properties of heavily faulted structures

Recent results have shown how to construct the smoothed transmission proper ties of a plane fault from the parameters of its microstructure in two part icular cases. In the first, the fault is modelled as a plane distribution o f approximately circular cracks while elsewhere the faces of the fault are held together by the ambient pressure and friction. In the second, the mode l consists of a plane distribution of approximately circular stuck regions within an area where the faces are separated as for a crack. The averaging method for a sequence of such slip planes enables the construction of overa ll properties of a. material weakened by a series of parallel faults. With the first model, where the distribution of cracks is sparse, this approach leads to exactly the same expressions to first order in the number density as for dilute volume distributions of cracks. The higher-order terms do not agree since they refer to crack-crack interactions and in the Schoenberg-D ouma averaging process only the overall interactions between faults are all owed for, not individual interactions between cracks on different faults. A pplication of this procedure to the second model, in which the fracture den sity is high, gives for the first time an exact first-order formula for the overall properties of heavily cracked material, the cracks being aligned a nd confined to the fault planes. These expressions are first order in the ( small) parameter, denoting the proportion of each slip plane that is welded . The unwelded part may be free (any cracks) or filled with an incompressib le inviscid fluid. An alternative approach in either case is to replace eac h fault or slip plane by an equivalent thin layer of material whose propert ies are related, at least in part, to the structure of the fault. The corre sponding process of averaging over the layers is, in this case, the origina l Backus method. Comparison between the properties of the equivalent layers for dilute cracks and for extended cracking leads to an extension of the s lip relations on a single heavily cracked fault to cases where the cracks c ontain secondary material with arbitrary elastic properties. Finally, resul ts for a stack of parallel, heavily cracked faults is identical, to first o rder in the number density of the contact regions on the faults, to those f or a cubical packing of spheres. This further reveals the insensitivity of first-order results to many of the details of the microstructure.