Determination of stress state in coal seam based on inverse problem solution using acoustic sounding data
| 1 | 071400 VKO Semey, Kazakhstan, st. Mangilik El 11, Kazakh Humanitarian Juridical Innovative University |
| 2 | 010008 Nur-Sultan, Kazakhstan, korpus 2, st. Pushkina 11, L.N. Gumilyov Eurasian National University |
CORRESPONDING AUTHOR
Mels Kudaibergenovich Kudaibergenov
071400 VKO Semey, Kazakhstan, st. Mangilik El 11, Kazakh Humanitarian Juridical Innovative University
071400 VKO Semey, Kazakhstan, st. Mangilik El 11, Kazakh Humanitarian Juridical Innovative University
Mining Science 2019;26:173–187
KEYWORDS
TOPICS
Application of fundamental science to miningMining geomechanics and geotechnicsGeodesy, geoinformatics and mine surveying
ABSTRACT
The paper presents an algorithm for calculating the horizontal, vertical and tangential stresses in a horizontal coal seam lying between galleries. These stresses are expressed in terms of the Airy function, which satisfies a homogeneous biharmonic equation. For its numerical solution it is necessary to set boundary conditions. Practical limitations do not allow us to determine the tangential stresses on horizontal boundaries of the coal seam. Calculations of stresses are proposed to be carried out in two steps. The first step is to solve the inverse problem for the biharmonic equation to find unknown tangential stresses on horizontal boundaries of the coal seam. The inverse problem is solved by minimizing the residual functional. Its strong convexity is proved, which implies the existence and uniqueness of the solution of the inverse problem. The second step is to solve the boundary value problem for the biharmonic equation to calculate stresses in the coal seam. The result of a numerical experiment is presented.
REFERENCES (47)
1.
ALEKSANDROV A.V., POTAPOV V.D., 1990, Fundamentals of the theory of elasticity and plasticity, Vyssh. Shk., Moscow.
3.
BEZUKHOV N.I., 1968, Fundamentals of the theory of elasticity, plasticity and creep, Vyssh. Shk., Mos-cow.
4.
BULAT A.F., Zvyagil’skii, E.L., Lukinov V.V., Perepelitsa V.G., Pimonenko L.I. Shevelev G.A., 2008. Coal-bearing massif of Donbass as a heterogeneous medium. Kiev, Naukova dumka.
5.
DJADKOV P.G., NAZAROV L.A., NAZAROVA L.A., 1997, Numerical modeling of the stressed state of the Earth’s crust and conditions of dynamic instability of seismoactive faults during rifting, Geologiya i Geofizika, Vol. 38, No. 12, 2001–2010.
6.
DOROKHOV D.V., SIVOKHIN V.I., KOSTYUK I.S., PODTYKALOV A.S., 1997, Technology under-ground mining of bedded deposits of minerals, DonGU, Donetsk.
8.
JAEGER J.C., COOK N.G.W., ZIMMERMAN R.W., 2007, Fundamentals of Rock Mechanics, 4th ed., Wiley-Blackwell Publishing.
9.
KARCHEVSKY A.L., 2016, Calculation of Stresses in a Coal Seam in Presence of Gas Diffusion, Journal of Applied and Industrial Mathematics, Vol. 10, No. 4, 482–493, https://doi.org/10.1134/.
11.
KARCHEVSKY A.L., 2017, Determination of the Possibility of Rock Burst in a Coal Seam, Journal of Applied and Industrial Mathematics, Vol. 11, No. 4, 527–534, https://doi.org/10.1134/S19904....
12.
KARCHEVSKY A.L., NAZAROVA L.A., ZAKHAROV V.N., NAZAROV L.A., 2017, Stress state estimation in coal bed under random conditions in contact zone with enclosing rocks based on inverse problem solution, Gornyi Zhurnal, Vol. 11, 37–40, https://doi.org/10.17580/gzh.2....
13.
KISELYOV V.A., 1976, The plane problem of elasticity theory, Vyssh. Shk., Moscow.
14.
KHALILOV S.A., 1977, About a System of Coordinate Functions for Solving Boundary Value Problems of Theory of Plates and Shells. In: Strength of Aircraft Structures, Vol. 4, Khar’kov. Aviats. Inst., Khar’kov, 60–65.
15.
KHALILOV S.A., 1978, New Systems of Orthonormal Polynomials, Some Their Properties and Applica-tions. In: Strength of Aircraft Structures, Vol. 5, Khar’kov. Aviats. Inst., Khar’kov, 46–56.
16.
KHALILOV S.A., 1982, The solution in the rectangle static problems of the theory of elasticity given on the boundary stresses. In: Aircraft design issues, Vol. 3, Khar’kov. Aviats. Inst., Khar’kov, 120–127.
17.
KHALILOV S.A., 1984, Calculation of Some Definite Integrals Containing Associated Legendre Func-tions of the Second and Fourth Orders. In: Strength of Aircraft Structures, Vol. 7, Khar’kov. Aviats. Inst., Khar’kov, 158–165.
18.
KHALILOV S.A., MINTYUK V.B., TKACHENKO D.A., 2010a, Construction and Investigation of the Approximate Analytical Solution of a Biharmonic Problem in aRectangle with Homogeneous Main Boundary Conditions, Aviats.-Kosmich. Tekhnika i Tekhnologiya, Vol. 2, 40–49.
19.
KHALILOV S.A., MINTYUK V.B., TKACHENKO D.A., 2010b, An Approximate Analytic Solution of a Biharmonic Problem in the Rectangle with the Main Boundary Conditions Homogeneous on Two Op-posite Sides and Arbitrary, on the Others, Aviats.-Kosmich. Tekhnika i Tekhnologiya, Vol. 5, 40–49.
20.
KHALILOV S.A., MINTYUK V.B., TKACHENKO D.A., 2011, Construction and Investigation of an Analytical-Numerical Solution of the Problem on Bending a Rectangular Plate Rigidly Fixed, Otkrytye Inform. i Kompyut. Integrir. Tekhnologii, Vol. 49, 81–94.
21.
KHALILOV S.A., MINTYUK V.B., TKACHENKO D.A., KOPYCHKO V.V., 2014, Spectrum of a Biharmonic Operator with Main Boundary Conditions in a Rectangle, Aviats.-Kosmich. Tekhnika i Tekh-.
23.
KHALILOV S.A., KRIVTSOV V.S., MINTYUK V.B., TKACHENKO D.A., 2015, The Green’s Function of the Main Boundary Value Problem for the Biharmonic Operator in a Rectangle, Aviats.-Kosmich. Tekhnika i Tekhnologiya, Vol. 6, 12–22.
24.
KUDAIBERGENOV M.K., KARCHEVSKY A.L., ISKAKOV K.T., 2018, Stress-strain state horizontal coal seam of finite length, Bulletin of the Karaganda University, Mathematics series, Vol. 2, 133–142.
25.
LUXBACHER K., WESTMAN E., SWANSON P., KARFAKIS M., 2008, Three-Dimensional Time-Lapse Velocity Tomography of an Underground Longwall Panel, International Journal of Rock Mechanics and Mining Sciences, Vol. 45, No. 4, 478–485, https://doi.org/10.1016/j.ijrm....
26.
MINTYUK V.B., 2007, Orthonormal Basis of One-Dimensional Boundary Value Problems. Aviats.-.
30.
NAZAROV L.A., NAZAROVA L.A., YAROSLAVTSEV A.F., MIROSHNICHENKO N.A., VASIL’EVA E.V., 2011, Evolution of stress fields and induced seismicity in operating mines, Journal of Mining Science, Vol. 47, No. 6, 707–713, https://doi.org/10.1134/S10627....
31.
NAZAROV L.A., NAZAROVA L.A., KARCHEVSKY A.L., MIROSHNICHENKO N.A., 2013a, Pressure Distribution in a Hydrocarbon-Bearing Formation Based on the Daylight Surface Movement Measurements, Journal of Mining Science, Vol. 49, No. 6, 854–861. https://doi.org/10.1134/S10627....
32.
NAZAROV L.A., NAZAROVA L.A., KARCHEVSKII A.L., PANOV A.V., 2013b, Estimation of Stresses and Deformation Properties of Rock Masses which Is Based on the Solution of an Inverse Problem From the Measurement Data of the Free Surface Displacement, Journal of Applied and In-dustrial Mathematics, Vol. 7, No. 2, 234–240, https://doi.org/10.1134/S19904....
33.
NAZAROVA L.A., NAZAROV L.A., KARCHEVSKY A.L., VANDAMME M., 2014, Estimating diffusion-capacity parameters of a coal bed using the gas pressure measured in a hole and the solution of an inverse problem, Journal of Applied and Industrial Mathematics, Vol. 8, No. 2, 267–273, https://.
35.
NAZAROVA L.A., NAZAROV L.A., KARCHEVSKY A.L., VANDAMME M., 2015, Determining Kinetic Parameters of a Block Coal Bed Gas by Solving Inverse Problem Based on Data of Borehole Gas Measurements, Journal of Mining Science, Vol. 51, No. 4, 666–672, https://doi.org/10.1134/.
37.
NAZAROVA L.A., NAZAROV L.A., PROTASOV M.I., 2016, Reconstruction of 3D stress field in coal–rock mass by solving inverse problem using tomography data, Journal of Mining Science, Vol. 52, No. 4, 623–631. https://doi.org/10.1134/S10627....
39.
POLAK E., 1971, Computational methods in optimization: a unified approach, Academic Press, New York.
40.
SHKURATNIK V.L., NIKOLENKO P.V., 2012, Methods for determining the stress-strain state of rock mass, Gornaya Kniga, Moscow.
41.
SAMUL’ V.A., 1982, Fundamentals of Elasticity and Plasticity Theory, Vyssh. Shk., Moscow.
42.
TKACHENKO D.A., 2014, An Orthonormal Basis in the Energetic Space of a Biharmonic Operator in a Rectangle with the Homogeneous Main Boundary Conditions at the Border, Aviats.-Kosmich. Tekhnika i Tekhnologiya, Vol. 3, 41–51.
43.
TURCHANINOV I.A., IOFIS V.A., KASPAR’YAN E.V., 1989, Fundamentals of rock mechanics, Nedra, Leningrad.
44.
VASIL’EV F.P., 1988, Numerical Methods for Solving Extremal Problems, Nauka, Moscow.
45.
ZAKHAROV V.N., MALINNIKOVA O.N., AVERIN A.P., 2016, Modeling mining-induced vibrations in production face area in coal-rock mass, Gornyi Zhurnal, Vol. 12, 28–32. https://doi.org/10.17580/.
47.
ZAKHAROV V.N., 2002, Seismic-Acoustic Prediction and Control of the State and Properties of Rocks in Development of Coal Deposits, Inst. Gorn. Dela, Moscow.
