Permeability Anisotropy Developed From the Up-scaling of Anisotropic Channels. Impact on Waterflooding Simulation Results.
Authors: Massonnat G. J. , Gommard D. 
Published: December 2007

Abstract
Geological systems, and hence associated permeability fields, are inherently complex, being characterized by nested levels of heterogeneity. These geologically complex volumes are represented in the reservoir models for flow simulation by equivalent homogeneous medium grid blocks.
Traditionally, this necessary simplification leads to underestimate geological and petrophysical heterogeneity, as the reservoir is dicretized in coarse grid-blocks which size and shape owe more to limitations of numerical simulation procedures than to any consideration of geological heterogeneity.
Recent advances in reservoir characterization and stochastic modelling have vastly improved the representation of complex geology and petrophysics in unknown interwell regions. The high level of detail provided by such models often requires several million grid blocks. This degree of detail cannot, however, be utilized by flow simulators, and hence, scaling-up is necessary.
Scaling up absolute permeabilities is a delicate point since averaging heterogenous values, even isotrope at fine scale, develops anisotropy at the larger scale. The memory of the pre-up scaled heterogeneities is, therefore, included into all the members of the permeability tensor (diagonal and crossed members).
However, most of the flow simulators work only with the diagonal members of the permeability tensor and, in this case, a whole part of the information concerning heterogeneity is "lost" in the up-scaling phase. Then, the difficulty in scaling–up of rock-property data are based on the conflicting requirement of significantly reducing the number of grid blocks, whilst maintaining a reasonable degree of detail in the representation of heterogeneity. This is critical for all fields for which the driving heterogeneties in a coarse grid model must be preserved in order to correctly represent the flooding mechanism and the swept area.
The problem is more complex in the case of reservoirs for which the fundamental heterogeneity has a very anisotropic shape, such as channelled reservoirs deposited in alluvial or turbiditic fans.
When modelling reservoirs with such a sedimentary origin, a strategy must be elaborated to minimize heterogeneity lost when up-scaling absolute permeability for flow simulation.
The aggregation rate, that means the number of fine grid blocks to be associated to constitute a coarse mesh, is one of the parameters used to optimize the up-scaling process.
To analyse whether grid blocks adequately represent such complex heterogeneties, geological realistic channel levee complexes are constructed using an object modelling (boolean) approach. Then, using these known heterogenous geological images, the impact of varying the size and the shape of up-scaled blocks is investigated.
Results are interpreted first in terms of representation of the fine scale geological heterogeneity by anisotropic permeability values in coarse cells, then through the date of the water breakthrough in a water flooding simulation.
In order to quantify the data loss, and to evaluate the up-scaling process quality, a distance-related criterion is proposed. The application of this criterion shows that, for each direction, the accurate representation of heterogeneity breaks down at certain scales, with significant detail being lost.
Perpendicularly to the direction of the channels, square cells rapidly average across different facies and heterogeneity is averaged-out.
As this effect is delayed in the direction of channels (the correlation length being higher), horizontal permeability anisotropy is created as the up-scaling is increased.
Similar results are obtained when the aggregation rate becomes very high on the vertical axis.
The shape of the grid blocks is shown to impact the representation of heterogeneity. Indeed, the use of non square (x, y) grid blocks, aligned to geological anisotropy, can be seen as a criterious for maximizing the efficiency of the gridding; that is, minimizing the number of grid-blocks, whilst guarding the reservoir heterogeneity and permeability anisotropy for flow.
A reference flow simulation is performed on the stochastic fine-gridded model, with isotropic values of permeability. Two oil producers are simulated in a black oil model, with an active aquifer. Other simulations are performed with up-scaled permeabilities, but using the fine-gridded model in order to avoid numerical dispersion.
Comparison between the results and the reference simulation (form t = 0 to t1 = time of water cut = 50 % at the second well) shows that the water breakthrough is delayed and the evolution of Sw too smooth when the simulation is performed on the permeability up-scaled model.
The delay increases with the aggregation rate; but for a same aggregation rate, results are better with anisotropic cells than with square grid blocks. Similarly, results are poorer when a more traditional approach is retained, with isotropic values for horizontal permeability (calculated from upscaled values of Kx and Ky).
This shows clearly that the way absolute permeabilities are upscaled plays a significant role in the representation of geological heterogeneities, and hence in the simulation results of a reservoir flooding.