A Fast Algorithm for the Estimation of the Equivalent Permeability in Heterogeneous Formations

Liao, Qinzhuo (KFUPM)



Reservoir simulation is an important component of reservoir development and management. Due to the heterogeneity in the subsurface formation, the accurate representation of the reservoir requires high-resolution geostatistical modeling with extremely large numbers of grid blocks in realistic models, which can be computationally prohibitive. This motivates the development of upscaling methods from fine-scale to coarse-scale by estimating the equivalent permeability. In this work, we have developed a rapid analytical method to increase the absolute permeability of heterogeneous reservoirs.

The equivalent permeability for a uniform flow in a given direction is bounded by (1) the harmonic mean of the arithmetic means of the local permeabilities, calculated over each slice of cells perpendicular to the given direction (upper bound), and (2) the arithmetic mean of the harmonic means of the local permeabilities, calculated on each line of cells parallel to the given direction (lower bound). The idea is to take a value between these two bounds, and the weighting coefficient is the key for accurate results. We presented a fast algorithm to estimate the weighting coefficient in the sense of probability expectation.

We compared the pressures and velocities calculated from three approaches, the fine-scale model, the coarse-scale model by numerical upscaling, and the coarse-scale model by analytical upscaling. We considered various conditions, including uncorrelated and correlated, isotropic and anisotropic, the effects of permeability variance and grid block geometry. We found that the pressures and velocities calculated from the coarse-scale model by analytical upscaling are very close to those from the coarse-scale model by numerical upscaling, i.e., the analytical method is as accurate as the numerical method, while the former can be O(10) times faster than the latter.


Large-scale reservoir simulation requires sophisticated geological modeling and accurate numerical simulations. The number of grids that conventional reservoir simulators can handle is typically in the range of hundreds of thousands to millions, depending on the type of model (such as black oil model or component model) and on different simulator performance. The geological structure and parameters obtained through seismic analysis and well logging analysis have a very large number of grids, and the order of magnitude is up to 100 million. This type of fine model clearly exceeds the ability of the reservoir simulator to handle. At the same time, in order to analyze uncertainty and assess risks, it is often necessary to use geostatistical methods to generate a large number of samples, further increasing the amount of calculations. Therefore, under the premise of keeping the simulation results such as pressure and saturation as accurate as possible, by upscaling the fine grid to the coarse grid and reducing the number of grids, it is an important research content for the numerical simulation of modern reservoirs.