Core sample from cavernous fractured carbonate reservoir. In the paper we present an approach to reconstruction of the structure of the cavernous-fractured reservoirs, based on the accurate numerical simulation of seismic waves' interaction with their fine structure resulting in generation of scattered waves. The main challenge with a full scale simulation of cavernous/fractured (carbonate) reservoirs in a realistic environment is that one should take into account both the macro-and microstructures. A straightforward implementation of finite difference techniques provides oversampled reference model. From the computational point of view, this means a huge amount of memory required for the simulation and, therefore, extremely high computer cost. In particular, a in order to simulate a model of dimension 10km  10km  10 km, which is common for seismic explorations, with a cell size of

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... of the key challenges in modern seismic processing is to recover the microstructure of the hydrocarbon reservoirs, especially in carbonate environment. These reservoirs hold clusters of caverns conjunct by network of fractures. Caverns and fractures are scaled within the range of 0.001 -0.1 meters, while their clusters (fracture corridors) can be spread for hundreds meters and more. Capacity of reservoir and its hydrocarbons recovery is tightly connected with distribution of caverns, while its recoverability governs in main by fractures because they control hydraulic flow as conductors (open fractures) or barriers (mineralized fractures) (Figure 1). Recently various techniques have been developed to perform imaging of fractures on the base of separation of reflected and scattered waves and the imaging of the latter. Among them, the Focusing Transformation ( Pozdnyakov and Tcheverda, 2005), the Scattering Index ( Willis et al., 2006), separation on the base of different smothness and continuity of local events (Fomel, Landa and Taner, 2007) and a variety of the imaging techniques developed under the generic name of interferometry (see e.g. book of G. Schuster, 2009). Unfortunately up to now all of them provide qualitative knowledge only indicating the presence of scatterers within a target area, but do not give any knowledge about fine structure of the reservoir and its capacity. In the paper we present an approach to reconstruction of the structure of the cavernous-fractured reservoirs, based on the accurate numerical simulation of seismic waves' interaction with their fine structure resulting in generation of scattered waves. The main challenge with a full scale simulation of cavernous/fractured (carbonate) reservoirs in a realistic environment is that one should take into account both the macro-and microstructures. A straightforward implementation of finite difference techniques provides oversampled reference model. From the computational point of view, this means a huge amount of memory required for the simulation and, therefore, extremely high computer cost. In particular, a in order to simulate a model of dimension 10km  10km  10 km, which is common for seismic explorations, with a cell size of 0.5m claims 12 10 8  cells and needs  350 Tb of RAM. The popular approach to overcome these troubles is to refine a grid in space only and there are many publications dealing with its implementation (see (Kristek, Moczo and Galis, 2010) for a detailed review), but it has at least two drawbacks: 1) To ensure stability of the finite-difference scheme the time step must be very small everywhere in the computational domain; 2) Unreasonably small Courant ratio in the area with a coarse spatial grid leads to a noticeable increase in numerical dispersion. Our solution to this issue is to use a mutually agreed local grid refinement in time and space: spatial and time steps are refined by the same factor (Lisitsa, Reshetova and Tcheverda, 2011 ...