Presenting filtration-wave fields in a layered anisotropic medium as a plane wave (part I)

About the authors:

Aleksandr I. Filippov, Dr. Sci. (Tech.), Professor, Chief Researcher, Sterlitamak Branch of Ufa University of Science and Technology, Sterlitamak, Russia; filippovai1949@mail.ru, https://orcid.org/0000-0002-0964-9805

Oksana V. Akhmetova, Dr. Sci. (Phys.-Math.), Professor, Department of General and Theoretical Physics, Sterlitamak branch of the Bashkir State University; eLibrary AuthorID, ahoksana@yandex.ru

Abstract:

The first part of the article describes the attempt to present the filtration wave process in a three-layered anisotropic medium as an equivalent plane wave in the central layer. A modification of “the average precision” asymptotic method is employed for this purpose. The central area is represented by a semi-infinite layer bounded by two parallel half-planes. At the border of this layer pressure perturbations are set which produce pressure waves in a half-space, while at the borders of the environments no perturbations are observed. Each of the three media is homogeneous as its physical properties are independent of the spatial coordinates. Yet these properties of all the three media depend on the direction and therefore the media are anisotropic. At the boundaries of the contact areas the pressure and flow of the fluid equalities are set. The task is to determine the pressure field in each of the media. To determine the pressure wave fields the unknown functions are presented in the form of asymptotic formulas so the original task of conjugation is reduced to easier tasks for the coefficients of the asymptotic expansion. It is shown that the zero coefficient of the asymptotic expansion describes the dependence of the amplitude of the said plane wave on its spatial coordinates as well as the physical properties of the wave and the medium.

References:

1. Gilmiyev, D.R., Shabarov, A.B. The effectiveness of hydraulic fracturing in the row system of alignment of wells // Tyumen State University Herald. № 7. Tyumen, 2013. Pр. 54–63. (in Russian).

2. Kuznetsova, E.I. Fluid filtration in the two-band fractured porous formation // Tyumen State University Herald. № 4. Tyumen, 2012. Pр. 80–86. (in Russian).

3. Bakhtiy, N.S., Kutrunov, V.N. Fluid flow to an imperfect well from the radial formation // Tyumen State University Herald. № 6. Tyumen, 2010. Pр. 134–139. (in Russian).

4. Filippov, A.I., Korotkova, K.N. Wave pressure fields in the reservoir and the wellbore // Physics of Wave Processes and Radio Systems. Vol. 12. № 1. 2009. Pp. 48–53. (in Russian).

5. Polenov, V.S., Chigarev, A.V. Wave propagation in fluid saturated inhomogeneous porous medium // Applied Mathematics and Mechanics. Vol. 74. № 2. 2010. Pp. 276–284. (in Russian).

6. Panin, V.I., Startsev, Yu.A. Controlling the dynamics of the stress-deformed geologicalenvironment in mining operations by seismic tomography // Information-Analytical Bulletin of Mining (Scientific and Technical Journal). № 9. 2011. Pp. 223–230. (in Russian).

7. Bolgarov, A.G., Roslov, Ju.V. Cross-well seismic tomography for geotechnical problems // Seismic Technologies. № 1. 2009. Pp. 105–111. (in Russian).

8. Akhmetova, O.V., Filippov, A.I., Filippov, I.M. Quasi-steady pressure fields in linear filtration of a non-uniform anisotropic formation in the asymptotic approximation // Russian Academy of Sciences Herald. Fluid and Gas Mechanics. № 3. 2012. Pp. 89–100. (in Russian).

9. Filippov, A.I., Akhmetova, O.V., Zamanova, G.F. Asymptotic representations of elastic wave fields in permeable formations // Acoustics Journal. Vol. 5. № 5. 2013. Pp. 548–558. (in Russian).

10. Filippov, A.I., Akhmetova, O.V., Kovalsky, A.A., Zamanova, G.F. Filtration waves in weak anisotropic medium // Bashkir State University Herald. Vol. 18. № 4. 2013. Pp. 1004–1005. (in Russian).