Hydrodynamic researches of nonlinear filtrations in low-permeable reservoirs

UDK: 622.276:532.5
DOI: 10.24887/0028-2448-2021-1-32-37
Key words: low-permeability formations, non-linear Darcy law, well-test analysis
Authors: A.M. Svalov (Oil and Gas Research Institute of RAS, RF, Moscow)

Currently, the share of low-permeability oil and gas reservoirs among newly discovered fields is steadily growing and even becoming decisive. In this regard, analytical studies of nonlinear filtering processes are of particular importance. As shown in the paper, the nonlinearity of this equation fundamentally changes the form of analytical dependencies describing the form of pressure curves during well-test analysis and this means that the use of currently accepted methods for processing field research data will inevitably lead to erroneous conclusions regarding the characteristics of low-permeability productive layers. Based on the analysis of the properties of generalized self-similar solutions, the dependence of the well production rate with time at a constant value of depression is obtained, as well as the time dependence of the pressure in the wellbore when it is put into operation with a constant production rate. It is shown that under the Darcy law with a power-law dependence of the filtration rate on the pressure gradient, the flow rate of the well with constant depression, and the pressure in the wellbore with constant selection of the reservoir liquids are represented by power functions of time. Compared to the logarithmic functions of time, power-law functions are characterized by faster rates of change over time, which means that in low-permeability reservoirs, quasi-stationary well operation modes are almost impossible. Such features of the operation of production wells in low-permeability reservoirs may be erroneously evaluated as evidence of the existence of limited-sized oil-saturated lenses around these wells. From the physical point of view, this feature is due to the fact that the size of the depression funnel around the borehole with a power-law form of the Darcy law grows very slowly with time and, moreover, there is a moving boundary separating the region of the perturbed filtration flow around the borehole from resting reservoir fluid away from the well.

The analytical dependences presented in the work were compared with the results of the numerical solution of the corresponding problems, and such a comparison confirmed the validity of the obtained dependencies. Thus, the analytical results obtained in the work allow to explain some features of low-permeability reservoirs development, and more correctly interpret the results of well-test analysis in such reservoirs.

References

1. Baykov V.A., Galeev R.R., Kolonskikh A.V., Makatrov A.K. et al.,  Nonlinear filtration in low-permeability reservoirs. Analisys and interpretation of laboratory core examination for Priobskoye oilfield (In Russ.), Nauchno-tekhnicheskiy vestnik OAO “NK “Rosneft'”, 2013, no. 2, pp.  8–12.

2. Xu J., R., Jiang L., Xie M. Yang et al., Transient pressure behavior for dual porosity low permeability reservoir based on modified Darcy’s equation, SPE-153480-MS, 2012, https://doi.org/10.2118/153480-MS

3. Barenblatt G.I., Entov V.M., Ryzhik V.M., Dvizhenie zhidkostey i gazov v prirodnykh plastakh (Movement of liquids and gases in natural reservoirs), Moscow: Nedra Publ., 1982, 211 p.

4. Rebinder P.A., Kusakov M.M, Zinchenko K.E., Surface phenomena in filtration processes (In Russ.), Doklady AN SSSR, 1940, V. 28, no. 5, pp. 42–49. 

5. Ar'e A.G., Fizicheskie osnovy fil'tratsii podzemnykh vod (Physical principles of groundwater filtration), Moscow: Nedra Publ., 1984, 101 p.

6. Carslaw H., Jaeger J., Conduction of heat in solids, Oxford University Press, USA, 1959, 510 p.

7. Courant R., Hilbert D., Methods of mathematical physics, V. 2: Partial differential equations, Wiley, 1966, 811 p.

Currently, the share of low-permeability oil and gas reservoirs among newly discovered fields is steadily growing and even becoming decisive. In this regard, analytical studies of nonlinear filtering processes are of particular importance. As shown in the paper, the nonlinearity of this equation fundamentally changes the form of analytical dependencies describing the form of pressure curves during well-test analysis and this means that the use of currently accepted methods for processing field research data will inevitably lead to erroneous conclusions regarding the characteristics of low-permeability productive layers. Based on the analysis of the properties of generalized self-similar solutions, the dependence of the well production rate with time at a constant value of depression is obtained, as well as the time dependence of the pressure in the wellbore when it is put into operation with a constant production rate. It is shown that under the Darcy law with a power-law dependence of the filtration rate on the pressure gradient, the flow rate of the well with constant depression, and the pressure in the wellbore with constant selection of the reservoir liquids are represented by power functions of time. Compared to the logarithmic functions of time, power-law functions are characterized by faster rates of change over time, which means that in low-permeability reservoirs, quasi-stationary well operation modes are almost impossible. Such features of the operation of production wells in low-permeability reservoirs may be erroneously evaluated as evidence of the existence of limited-sized oil-saturated lenses around these wells. From the physical point of view, this feature is due to the fact that the size of the depression funnel around the borehole with a power-law form of the Darcy law grows very slowly with time and, moreover, there is a moving boundary separating the region of the perturbed filtration flow around the borehole from resting reservoir fluid away from the well.

The analytical dependences presented in the work were compared with the results of the numerical solution of the corresponding problems, and such a comparison confirmed the validity of the obtained dependencies. Thus, the analytical results obtained in the work allow to explain some features of low-permeability reservoirs development, and more correctly interpret the results of well-test analysis in such reservoirs.

References

1. Baykov V.A., Galeev R.R., Kolonskikh A.V., Makatrov A.K. et al.,  Nonlinear filtration in low-permeability reservoirs. Analisys and interpretation of laboratory core examination for Priobskoye oilfield (In Russ.), Nauchno-tekhnicheskiy vestnik OAO “NK “Rosneft'”, 2013, no. 2, pp.  8–12.

2. Xu J., R., Jiang L., Xie M. Yang et al., Transient pressure behavior for dual porosity low permeability reservoir based on modified Darcy’s equation, SPE-153480-MS, 2012, https://doi.org/10.2118/153480-MS

3. Barenblatt G.I., Entov V.M., Ryzhik V.M., Dvizhenie zhidkostey i gazov v prirodnykh plastakh (Movement of liquids and gases in natural reservoirs), Moscow: Nedra Publ., 1982, 211 p.

4. Rebinder P.A., Kusakov M.M, Zinchenko K.E., Surface phenomena in filtration processes (In Russ.), Doklady AN SSSR, 1940, V. 28, no. 5, pp. 42–49. 

5. Ar'e A.G., Fizicheskie osnovy fil'tratsii podzemnykh vod (Physical principles of groundwater filtration), Moscow: Nedra Publ., 1984, 101 p.

6. Carslaw H., Jaeger J., Conduction of heat in solids, Oxford University Press, USA, 1959, 510 p.

7. Courant R., Hilbert D., Methods of mathematical physics, V. 2: Partial differential equations, Wiley, 1966, 811 p.



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