Approximating the derivative of the Buckley – Leverett function

UDK: 622.276.038:532.5
DOI: 10.24887/0028-2448-2019-2-44-48
Key words: approximation, the function of the Buckley-Leverett, forecast, production, oil
Authors: Am.At. Khayrullin(Tyumen Industrial University, Tyumen), S.I. Grachev (Tyumen Industrial University, Tyumen), Az.Am. Khayrullin (Tyumen Industrial University, Tyumen)

The displacement of oil by water from a layered reservoir can be thought of as a frontal drive. In this case, all the interlayers are ordered in such a way that their absolute permeability changes sequentially from lowest to highest. At the bottom of this layered reservoir is the most permeable interbed, at the top - the least permeable. According to the probabilistic-statistical model of a layered heterogeneous reservoir, in accordance with the law of permeability distribution, it is possible to determine the total thickness of the layers, the permeability of the most permeable of which is not lower than some permeability k. To describe the process of displacement from a layered heterogeneous reservoir, we can use the two-phase frontal drive model. In the case of a two-phase non- frontal drive, the Buckley-Leverett function and the generalized Darcy law are used, including relative permeabilities that depend on water saturation and viscosity of water and oil, without taking into account capillary effects.

Frontal water drive implies a clear interface, in contrast to non-frontal displacement. When describing a non-frontal drive, the water saturation distribution according to the Buckley – Leverett model contains a derivative of the Buckley – Leverett function, whose form is similar to a pseudo-gamma distribution.

The article gives an example of using the pseudo-gamma distribution in approximating the derivative of the Buckley – Leverett function. It is noted that the extrapolation of production data by different models is very sensitive to small errors contained in the initial information. In this regard, it is of great importance to make an informed choice of the type of equations that approximate the dynamics of cumulative oil production. It is shown that the proposed approximation in the form of a pseudo-gamma distribution allows to describe the processes in which the growth and decrease in characteristic indicators occur.

References

1. Zheltov Yu.P., Razrabotka neftyanykh mestorozhdeniy (The oil fields development), Moscow: Nedra Publ., 1986, 332 p.

2. Khayrullin Am.At., Khayrullin Az.Am., Plotnostʹ raspredeleniya neodnorodnostey kollektorskikh svoystv porod (Distribution density of heterogeneity of reservoir properties of rocks), Proceedings of International Scientific-practical seminar "Rassokhin reading", Ukhta: Publ. of Ukhta State Technical University, 2015, 212 р.

3. 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.

The displacement of oil by water from a layered reservoir can be thought of as a frontal drive. In this case, all the interlayers are ordered in such a way that their absolute permeability changes sequentially from lowest to highest. At the bottom of this layered reservoir is the most permeable interbed, at the top - the least permeable. According to the probabilistic-statistical model of a layered heterogeneous reservoir, in accordance with the law of permeability distribution, it is possible to determine the total thickness of the layers, the permeability of the most permeable of which is not lower than some permeability k. To describe the process of displacement from a layered heterogeneous reservoir, we can use the two-phase frontal drive model. In the case of a two-phase non- frontal drive, the Buckley-Leverett function and the generalized Darcy law are used, including relative permeabilities that depend on water saturation and viscosity of water and oil, without taking into account capillary effects.

Frontal water drive implies a clear interface, in contrast to non-frontal displacement. When describing a non-frontal drive, the water saturation distribution according to the Buckley – Leverett model contains a derivative of the Buckley – Leverett function, whose form is similar to a pseudo-gamma distribution.

The article gives an example of using the pseudo-gamma distribution in approximating the derivative of the Buckley – Leverett function. It is noted that the extrapolation of production data by different models is very sensitive to small errors contained in the initial information. In this regard, it is of great importance to make an informed choice of the type of equations that approximate the dynamics of cumulative oil production. It is shown that the proposed approximation in the form of a pseudo-gamma distribution allows to describe the processes in which the growth and decrease in characteristic indicators occur.

References

1. Zheltov Yu.P., Razrabotka neftyanykh mestorozhdeniy (The oil fields development), Moscow: Nedra Publ., 1986, 332 p.

2. Khayrullin Am.At., Khayrullin Az.Am., Plotnostʹ raspredeleniya neodnorodnostey kollektorskikh svoystv porod (Distribution density of heterogeneity of reservoir properties of rocks), Proceedings of International Scientific-practical seminar "Rassokhin reading", Ukhta: Publ. of Ukhta State Technical University, 2015, 212 р.

3. 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.


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