A new mathematical model for the miscible displacement in fractured-porous reservoirs is developed. A model is obtained by the upscaling of the traditional miscible displacement equations from the scale, which is lower than the fracture opening, up to the scale, which is much larger than the block size.
The model is based on simultaneous use of different mathematical methods: areal averaging of fluid fluxes, asymptotical averaging in systems with periodical heterogeneity, analytical solutions of the problem of interaction between the flux via fractures and the single block. The model takes into account diffusive, gravitational and advective mechanisms of the mass exchange between blocks and fractures and also hydraulic interaction between fluxes via systems of blocks and fractures.
The formula for the modified fractional flow functions which depends on the geometric tortuosity of the system of fractures are proposed. The tortuosity coefficient can be found from the 1-D laboratory displacement data.
The model developed has been validated by comparison with the number of laboratory studies of the miscible displacement in fractured-porous media.
Miscible gas injection into fractured-porous oil and gas-condensate reservoirs is the effective enhanced hydrocarbon recovery method. Nevertheless the efficiency of this IOR method is strongly dependent on parameters of fractured-porous system and on properties of reservoirs and of injected fluids.
The major oil reserves are located in the block system, so the recovery efficiency during the miscible gas injection is determined by mechanisms of the 'fracture-block' mass exchange and by the resulting displacement from blocks.
The displacement efficiency in fractured-porous media with the large opening and the high permeability of fractures and the low permeability of blocks is poor due to the fast breakthrough of the injected gas through the fractured system and low recovery from blocks. For the fractured-porous media with the less contrast between the fracture and the block conductivity the recovery is high.
Nevertheless, the recovery is determined not by the heterogeneity of the fractured-porous system only. If the fractured system conductivity is significantly higher then the one for the block system, but the blocks are small, the recovery could be still high.
The diffusive, gravitational and advective (convective) mechanisms of the fracture-block mass exchange during the two-phase partly miscible flow have been distinguished. The intensity of these three mechanisms are affected differently by the variation of the displacement velocity and also by the viscosity and the density of the injected fluid. Therefore, the comprehensive mathematical model which takes into account all the above mentioned recovery mechanisms is required for the optimization of the miscible gas injection into fractured-porous reservoirs.
Mathematical description of the flow in fractured-porous media are based on the image of the mass transfer in the double-porosity system (two fluxes via systems of fractures and of blocks interact with each other), by G. Barenblatt, Y. Zheltov and I. Kochina. In some other models the flux in blocks is neglected and blocks are treated as the source-sink terms with the given law of the mass exchange.
Another approach is based on the consideration of the fractured-porous media as a periodically heterogeneous media, and the upscaled model is obtained by the asymptotic averaging in periodical systems. The method allows to derive more complex formulae for the fracture-block mass exchange and to obtain the explicit formulae for effective permeabilities for systems of fractures and of blocks.
Diffusive mechanism of the fracture-block mass exchange corresponds to the linear term proportional to the concentration difference in blocks and fractures.