Coarsening of computational spatial grids is one of the main ways to reduce the cost of computing resources in geological and hydrodynamic modeling of hydrocarbon reservoirs. The procedure of overriding reservoir properties in an upsized cell of the computational grid is called upscaling (averaging). The quality of this procedure is determined by degree of prognostic capability decreasing of applied models. The traditional way for determining the average value as the arithmetic mean is not always applicable in practice, since it does not take into account the spatial heterogeneity of the averaged values distribution. In this paper, we consider the case of a reservoir with formation reservoir properties (permeability and porosity) values close to power functions of the spatial variable. Proximity of reservoir properties to power function indicates to a fractal inhomogeneity of the porous medium. The power-law upscaling procedure is proposed for this case. An initial-boundary-value problem for a one-dimensional fractal model of unsteady-state filtration is considered. The identification procedure of fractal quantities of this model is proposed and investigated. The proposed methods tested on data from one of the fields in Western Siberia. A comparative analysis with the arithmetic mean method is performed on permeability data. The proposed techniques have a potential for use in reservoir engineering and monitoring.

References

1. Mandelbrot B.B., The fractal geometry of nature, San Francisco: Freeman, 1992, 750 p.

2. Mirzadzhanzade A.Kh., Khasanov M.M., Bakhtizin R.N., Modelirovanie protsessov neftegazodobychi. Nelineynost’, neravnovesnost’, neopredelennost’ (Modelling of oil and gas production processes. Nonlinearity, disequilibrium, uncertainty), Moscow-Izhevsk: Publ. of Institute of Computer Science, 2004, 368 p.

3. Feder J., Fractals, Springer Science & Business Media, 2013, 283 p.

4. Uchaykin V.V., Metod drobnykh proizvodnykh (Fractional derivative method), Ul'yanovsk: Artishok Publ., 2008, 512 p.

5. Barabanov V.L., The rocks capillary imbition – The primary stage; fractal modelling (In Russ.), Aktual'nye problemy nefti i gaza, 2016, no. 1(13), pp. 5/1-16.

6. Yu B., Analysis of flow in fractal porous media, Applied Mechanics Reviews, 2008, V. 61, no. 5, pp. 1–19.

7. O'Shaughnessy B., Procaccia I., Analytical solutions for diffusion on fractal objects, Physical Review Letters, 1985, V. 54, no. 5, pp. 455–458.

8. Bagmanov V.Kh., Baykov V.A., Latypov A.R., Vasil'ev I.B., The technique of interpretation and determination of the parameters of the filtration equation in a porous medium with fractal properties (In Russ.), Vestnik UGATU, 2006, V. 7, no. 2, pp. 146–149.

9. Xu P., Yu B., Developing a new form of permeability and Kozeny–Carman constant for homogeneous porous media by means of fractal geometry, Advances in water resources, 2008, V. 31, no. 1, pp. 74-81.

10. Barenblatt G.I., Entov V.M., Ryzhik V.M., Teoriya nestatsionarnoy fil'tratsii zhidkosti i gaza (The theory of non-stationary filtration of liquid and gas), Moscow: Nedra Publ., 1972, 288 p.

11. Kostin A.B., Recovery of the coefficient of u t in the heat equation from a condition of nonlocal observation in time (In Russ.), Zhurnal vychislitel'noy matematiki i matematicheskoy fiziki, 2015, V. 55, no. 1, pp. 89–104.

12. Bakhvalov N.S., Panasenko G.P., Osrednenie protsessov v periodicheskikh sredakh: Matematicheskie zadachi mekhaniki kompozitsionnykh materialov (Averaging of processes in batch media: Mathematical problems in the mechanics of composite materials), Moscow: Nauka Publ., 1984, 352 p.

13. Baykov V.A., Zhonin A.V., Konovalova S.I. et al., Petrophysical modeling of complex terrigenous reservoirs (In Russ.), Territoriya NEFTEGAZ, 2018, no. 11, pp. 34–38.

Coarsening of computational spatial grids is one of the main ways to reduce the cost of computing resources in geological and hydrodynamic modeling of hydrocarbon reservoirs. The procedure of overriding reservoir properties in an upsized cell of the computational grid is called upscaling (averaging). The quality of this procedure is determined by degree of prognostic capability decreasing of applied models. The traditional way for determining the average value as the arithmetic mean is not always applicable in practice, since it does not take into account the spatial heterogeneity of the averaged values distribution. In this paper, we consider the case of a reservoir with formation reservoir properties (permeability and porosity) values close to power functions of the spatial variable. Proximity of reservoir properties to power function indicates to a fractal inhomogeneity of the porous medium. The power-law upscaling procedure is proposed for this case. An initial-boundary-value problem for a one-dimensional fractal model of unsteady-state filtration is considered. The identification procedure of fractal quantities of this model is proposed and investigated. The proposed methods tested on data from one of the fields in Western Siberia. A comparative analysis with the arithmetic mean method is performed on permeability data. The proposed techniques have a potential for use in reservoir engineering and monitoring.

References

1. Mandelbrot B.B., The fractal geometry of nature, San Francisco: Freeman, 1992, 750 p.

2. Mirzadzhanzade A.Kh., Khasanov M.M., Bakhtizin R.N., Modelirovanie protsessov neftegazodobychi. Nelineynost’, neravnovesnost’, neopredelennost’ (Modelling of oil and gas production processes. Nonlinearity, disequilibrium, uncertainty), Moscow-Izhevsk: Publ. of Institute of Computer Science, 2004, 368 p.

3. Feder J., Fractals, Springer Science & Business Media, 2013, 283 p.

4. Uchaykin V.V., Metod drobnykh proizvodnykh (Fractional derivative method), Ul'yanovsk: Artishok Publ., 2008, 512 p.

5. Barabanov V.L., The rocks capillary imbition – The primary stage; fractal modelling (In Russ.), Aktual'nye problemy nefti i gaza, 2016, no. 1(13), pp. 5/1-16.

6. Yu B., Analysis of flow in fractal porous media, Applied Mechanics Reviews, 2008, V. 61, no. 5, pp. 1–19.

7. O'Shaughnessy B., Procaccia I., Analytical solutions for diffusion on fractal objects, Physical Review Letters, 1985, V. 54, no. 5, pp. 455–458.

8. Bagmanov V.Kh., Baykov V.A., Latypov A.R., Vasil'ev I.B., The technique of interpretation and determination of the parameters of the filtration equation in a porous medium with fractal properties (In Russ.), Vestnik UGATU, 2006, V. 7, no. 2, pp. 146–149.

9. Xu P., Yu B., Developing a new form of permeability and Kozeny–Carman constant for homogeneous porous media by means of fractal geometry, Advances in water resources, 2008, V. 31, no. 1, pp. 74-81.

10. Barenblatt G.I., Entov V.M., Ryzhik V.M., Teoriya nestatsionarnoy fil'tratsii zhidkosti i gaza (The theory of non-stationary filtration of liquid and gas), Moscow: Nedra Publ., 1972, 288 p.

11. Kostin A.B., Recovery of the coefficient of u t in the heat equation from a condition of nonlocal observation in time (In Russ.), Zhurnal vychislitel'noy matematiki i matematicheskoy fiziki, 2015, V. 55, no. 1, pp. 89–104.

12. Bakhvalov N.S., Panasenko G.P., Osrednenie protsessov v periodicheskikh sredakh: Matematicheskie zadachi mekhaniki kompozitsionnykh materialov (Averaging of processes in batch media: Mathematical problems in the mechanics of composite materials), Moscow: Nauka Publ., 1984, 352 p.

13. Baykov V.A., Zhonin A.V., Konovalova S.I. et al., Petrophysical modeling of complex terrigenous reservoirs (In Russ.), Territoriya NEFTEGAZ, 2018, no. 11, pp. 34–38.