Release:2020. Vol. 6. № 3 (23)
About the authors:Aleksandr I. Filippov, Dr. Sci. (Tech.), Professor, Department of General and Theoretical Physics, Sterlitamak branch of the Bashkir State University; eLibrary AuthorID, email@example.com
Due to the problems of hydrocarbon production, the study of pressure fields in natural reservoirs is the primary task of the filtration theory and special interest to the practitioners. Because of the variety of natural conditions necessary to account for in setting, the number of tasks is steadily increasing. At present, several analytical solutions of problems for homogeneous reservoirs with the simplest boundary conditions have been constructed. Studying the influence of borehole conditions on pressure fields in productive formations presents interest, as in this case, it is expressed as a boundary condition in the form of an integro-differential equation, and the corresponding class of problems is not sufficiently studied in the fundamental sections of the mathematical physics.
In contrast to the existing works, this article constructs an analytical solution of the problem about the pressure field in the formation, represented by the classical equation of piezo-pipeline, for the case when the boundary condition in the form of the integro-differential equation describes the process of fluid extraction from the well with the help of a pump of the given productivity. The authors study the oil-saturated porous formation, opened by the producing well for the whole thickness. The pressure field in the reservoir is described by the classical piezoelectricity equation in the cylindrical coordinate system in the assumption of axial symmetry. The function of pressure perturbation distribution across the reservoir is determined considering the influence of the well. The relationship of pressures in the well and formation is established based on the law of mass conservation and Pascal’s law.
The dimensionless criteria characterizing the filtration process and allowing to simplify the task definition are defined. It is shown, that all parameters, characterizing influence of well and formation, are grouped in two interrelated criteria, which define filtration process in such conditions.
The authors have presented an exact solution of the problem in the space of Laplace — Carson integral transformation with an analytical transition to the original space. In addition, they have created a numerical conversion program, based on Den Iseger’s algorithm, and developed a finite-difference scheme for the task at hand. Having calculated the space-time distributions of pressure fields, the authors have compared the graphical pressure dependencies based on an analytical formula and a numerical program with the results of finite-difference calculations. That increases the reliability of the results obtained, which seems to be of great importance, since the existence and uniqueness theorems are not proved for this class of the problem in question.
The analysis of the calculation results allowed determining the contribution of the well and reservoir parameters to the pressure field. The results show that in the relaxation mode, the parameters of the well and pump have a significant influence on the formation of the pressure field. In the stabilization mode, the contribution of its physical parameters is predominant.
Buzinov S. N., Umrihin I. D. 1964. Research of Formations and Wells in the Elastic Regime of Filtration. Moscow: Nedra. 272 pp. [In Russian]
Ditkin V. A., Prudnikov A. P. Operational Calculus. Moscow: Vysshaya shkola. 406 pp. [In Russian]
Entov V. M., Chekhonin E. M. 2007. “Pressure field around a well in a layered-inhomogeneous formation”. Izvestiya Rossiyskoy akademii nauk. Mekhanika zhidkosti i gaza, no. 1, pp. 83-90. [In Russian]
Zanochuev S. A., Shabarov A. B. 2019. “An experimental method for predicting the composition and properties of a produced fluid under the conditions of a two-phase filtration of a gas-liquid mixture during field development for depletion”. Tyumen State University Herald. Physical and Mathematical Modeling. Oil, Gas, Energy, vol. 5, no. 4 (20), pp. 21-36. DOI: 10.21684/2411-7978-2019-5-4-21-36 [In Russian]
Karslou G., Eger D. 1964. Thermal Conductivity of Solids. Moscow: Nauka. 488 pp. [In Russian]
Kuznecov D. S. 1965. Special Functions. Moscow: Vysshaya shkola. 273 pp. [In Russian]
Muskat M. 2004. Flow of Homogeneous Liquids in a Porous Medium. Moscow; Izhevsk: IKI. 628 pp. [In Russian]
Nikolaevskiy V. N., Basniev K. S., Gorbunov A. T., Zotov G. A. 1970. Mechanics of Saturated Porous Media. Moscow: Nedra.339 pp. [In Russian]
Pyatibrat V. P., Sokolov V. A., Burakov Yu. G., Arefyev V. V., Ckhadaya N. D., Bannikova A. T. 2009. “Mathematical modeling of the pressure recovery process at the bottom of the gallery in the case of an inhomogeneous thickness of a limited linear formation”. Avtomatizatsiya, telemekhanizaciya i svyaz v neftyanoy promyshlennosti, no. 6, pp. 14-18. [In Russian]
Rubinshtejn L. I. 1972. Temperature Fields in Oil Reservoirs. Moscow: Nedra. 275 pp. [In Russian]
Filippov A. I., Korotkova K. N. 2009. “Wave fields of pressure in the reservoir and well”. Fizika volnovykh protsessov i radiotekhnicheskie sistemy, vol. 12, no. 1, pp. 48-53. [In Russian]
Filippov A. I., Akhmetova O. V., Kovalskiy A. A. 2018. “Coefficient averaging method in the problem of laminar gas flow in a well”. Prikladnaya mekhanika i tekhnicheskaya fizika, vol. 59, no. 1 (347), pp. 71-82. [In Russian]
Filippov A. I., Mihajlov P. N., Ahmetova O. V. 2006. “The main objective of thermo logging”. Teplofizika vysokikh temperature, vol. 44, no. 5, pp. 747-755. [In Russian]
Charnyy I. A. 1963. Underground Hydrodynamics. Moscow: Gostoptekhizdat. 397 pp. [In Russian]
Shabarov A. B., Shatalov A. V., Markov P. V., Shatalova N. V. 2018. “Relative permeability calculation methods in multiphase filtration problems”. Tyumen State University Herald. Physical and Mathematical Modeling. Oil, Gas, Energy, vol. 4, no. 1, pp. 79-109. DOI: 10.21684/2411-7978-2018-4-1-79-109 [In Russian]
Den Iseger P. 2006. “Numerical transform inversion using Gaussian quadrature”. Probability in the in Engineering and Informational Sciences, no. 20, pp. 1-44.