Effective elasticity of fractured rocks: A snapshot of the work in progress

Exploration and development of naturally fractured reservoirs rely on understanding and interpretation of certain signatures associated with seismic waves propagating through cracked rocks. This understanding comes primarily from the effective media theories that predict an overall elastic behavior of a solid containing many inhomogeneities (cracks, in particular) whose sizes are too small to be seen individually by the waves. To model seismic responses of fractured formations, a geophysicist typically has a choice between the effective media schemes of Hudson and Schoenberg. While the two predictions usually deviate slightly for liquid-filled cracks, the differences are significant when the fractures are dry. Explaining the origin of these differences and selecting a more accurate scheme is the first goal of this tutorial. Our second, more challenging task, is to prove that simply adding the compliance contributions of cracks as if they were isolated and noninteracting remains sufficiently accurate even for fractures that grossly violate the basic theoretical assumption, of penny-shaped cracks. Real fractures have notoriously irregular shapes, might be partially closed, and often form interconnected networks. Yet, these details of fracture microgeometry turn out to be unimportant for the effective elasticity given a typical noise level in seismic data. No closed-form theory exists for irregular fracture shapes. However, take into account finite-element simulations on so-called digital rocks demonstrate which features of the crack geometry have to be taken into account because they influence propagation of long (compared to the size of fractures) seismic waves and which features can be ignored.