Two-dimensional upscaling of reservoir data using adaptive bandwidth in the kernel function

Document Type : Research Paper


1 Phd Student of Shahrood university of Technology

2 Faculty of Mining, Petroleum & Geophysics Engineering, Shahrood University of Technology, Shahrood, Iran

3 Faculty of Mathematical Science, Shahrood University of Technology, Shahrood, Iran


In this paper, a new method called adaptive bandwidth in the kernel function has been used for two-dimensional upscaling of reservoir data. Bandwidth in the kernel can be considered as a variability parameter in porous media. Given that the variability of the reservoir characteristics depends on the complexity of the system, either in terms of geological structure or the specific feature distribution, variations must be considered differently for upscaling from a fine model to a coarse one. The upscaling algorithm, introduced in this paper, is based on the kernel function bandwidth, written in combination with the A* search algorithm and the first-depth search algorithm. In this algorithm, each cell in its x and y neighborhoods as well as the optimal bandwidth, obtained in two directions will be able to be merged with its adjacent cells. The upscaling process is performed on artificial data with 30×30 grid dimensions and SPE-10 model as real data. Four modes are used to start the point of upscaling and the process is performed according to the desired pattern, and in each case, the upscaling error and the number of final upscaled blocks are obtained. Based on the number of coarsen cells as well as the upscaling error, the first pattern is selected as the optimal pattern for synthetic data and the second pattern is selected as the optimal simulator model for real data. In this model, the number of cells was 236 and 3600, and the upscaling errors for synthetic and real data were 0.4183 and 12.2, respectively. The results of the upscaling in the real data were compared with the normalization method and showed that the upscaling error of the normalization method was 15 times the upscaling error of the kernel bandwidth algorithm.


[1] Christie, M. A., (1996). Upscaling for Reservoir Simulation. Journal of Petroleum Technolog, 48(11), 1-4.
[2] Christie, M. A., & Blunt, M. J. (2001). Tenth SPE Comparative Solution Project: A Comparison of Upscaling Techniques. Journal of Petroleum Technology. 4(4), 1-10.
[3] Chen, T., Clauser, C., Marquart, G., Willbrand, K. and Mottaghy, D. (2015). A new upscaling method for fractured porous media. journal of Advances in Water Resources, 80, 60-68.
[4] Farmer CL., (2000). Upscaling: a review. International Journal Numerical Methods Fluids. 40, 63 – 78.
[5] Dadvar, M., & Sahimi, M., (2007). The effective diffusivities in porous media with and without nonlinear reactions. Chemical engineering science. 62, 1466-1476.
[6] Hochstetler, D. L., & Kitanidis, P. K., (2013). The behavior of effective rate constants for bimolecular reactions in an asymptotic transport regime. Journal of contaminant hydrology. 144, 88-98.
[7] Pereira, J. M. C., Navalho, J. E. P., Amador, A. C. G., & Pereira, J. C. F., (2014). Multi-scale modeling of diffusion and reaction–diffusion phenomena in catalytic porous layers: comparison with the 1D approach. Chemical Engineering Science. 117, 364-375.
[8] Ratnakar, R. R., Bhattacharya, M., & Balakotaiah, V., (2012). Reduced order models for describing dispersion and reaction in monoliths. Chemical Engineering Science. 83, 77-92.
[9] Chen, T., Clauser, C., Marquart, G., Willbrand, K. and Hiller, T., (2018). Upscaling permeability for three-dimensional fractured porous rocks with the multiple boundary method, Hydrogeology Journal, 1-14. 018-1744-z.
[10] Ding, D. Y., (2004). Near-Well Upscaling for Reservoir Simulations. Oil & Gas Science and Technology. 59(2), 157- 165.
[11] Gholinezhad, S., Jamshidi, S. and Hajizadeh, A., (2015). QuadTree decomposition method for areal upscaling of heterogeneous reservoirs: Application to arbitrary shaped reservoirs. Fuel, 139, 659-670.
[12] Moslehi, M., Felipe P.J., de Barros, Ebrahimi, F. and Sahimi, M.,(2016). Upscaling of solute transport in disordered porous media by wavelet transformations, Advances in Water Resources, 96, 180-189.
[13] King, P.R. (1989). The Use of Renormalization for Calculating Effective Permeability, Transport in Porous Media, 4, 37-58.
[14] Silverman, B.W. (1986). Density Estimation for Statistics and Data Analysis. Published in Monographs on Statistics and Applied Probability, London: Chapman and Hall, 176 P.
[15] Scott, D. W., 1992. Multivariate Density Estimation: Theory, Practice, and Visualization. New York: John Wiley & Sons Inc., pp. 384.
[16] Raykar, V.C., Duraiswami, R., Zhao, L.H., (2010). Fast computation of kernel estimators. Journal of Computational and Graphical Statistics, 19 (1), 205–220.
[17] Hardle, W. K. K., Muller, M. and Sperlich., (2004). Nonparametric and semiparametric models, Springer Series in Statistics, 277 P.
[18] Delling, D., Sanders, P., Schultes, D. and Wagner, D., (2009). Algorithmic of Large and Complex Networks: Design, Analysis, and Simulation. Springer-Verlag Berlin Heidelberg. 401 P.
[19] Even, S., (2011), Graph Algorithms (2nd Ed.). Cambridge University Press, 48 P.