Carbonate reservoirs pose a scientific and engineering challenge to geophysical prediction and monitoring of fluid flow in the subsurface. Difficulties in interpreting hydrological, reservoir and other exploration data arise because carbonates are composed of a hierarchy of geological structures, constituents and processes that span a wide spectrum of length and time scales. What makes this problem particularly challenging is that length scales associated with physical structure and processes are often not discrete, but overlap, preventing the definition of discrete elements at one scale to become the building blocks of the next scale. This is particularly true for carbonates where complicated depositional environments, subsequent post-deposition diagenesis and geochemical interactions result in pores that vary in scale from submicron to centimeters to fractures, variation in fabric composition with fossils, minerals and cement, as well as variations in structural features (e.g., oriented inter- and intra layered - interlaced bedding and/or discontinuous rock units). In addition, this complexity is altered by natural and anthropogenic processes such as changes in stress, fluid content, reactive fluid flow, etc. Thus an accurate geophysical assessment of the flow behavior of carbonate reservoirs requires a fundamental understanding of the interplay of textural and structural features subjected to physical processes …