The use of full-wave numerical simulation for the investigation of fractured zones

A. V. Favorskaya ; I. B. Petrov

Matematičeskoe modelirovanie, Tome 30 (2018) no. 11, pp. 105-126.

Résumé

In this paper we describe the stages and results of the investigation of the features of oilsaturated fracturing zones by applying the analysis of spatial dynamical wave patterns, obtained as a result of supercomputer modeling by the grid-characteristic method. Fullwave modeling is used in geophysics to construct synthetic seismograms and as part of solving inverse problems. In this paper we demonstrate that it is possible to derive conclusions that can later be useful in carrying out geophysical studies by analyzing the calculated spatial dynamic wave patterns. The proposed approach of wave patterns analyzing simplifies the study of the dynamics of different wave types in comparison with the methods of analyzing and interpreting the seismograms, and is more accurate than the ray-tracing method and the geometric approximation. Three types of fractured clusters are considered: "Solid", "Intermittent" and "Chess". As a result of the research, characteristic regularities were obtained, for example, the dependence of the angle of scattering seismic waves on the frequency used and on the geometrical features of location of the fractures into the clusters and the dependence on the source frequency of the trajectory and the velocity of motion of the point of separation of the longitudinal head wave from the S-wave. These regularities can subsequently be adapted to optimize the process of seismic prospecting of hydrocarbons and fractured zones investigation, for example, for the selection of the optimal equipment and the method of seismic survey. Also, we discuss the importance of studying of the spatial dynamic wave patterns when developing and testing of numerical methods, interface and boundary conditions, including the absorbing ones. Also, we propose the approach to construct a nonlinear scale that allows simultaneous analysis of spatial dynamic wave processes whose amplitudes differ by more than 20 times.

Détail

Keywords: wave dynamics, elastic waves, fractured zones, supercomputer modeling, grid-characteristic method, seismic survey.

Bibliographie

[1] K.M. Magomedov, A.S. Kholodov, “The construction of difference schemes for hyperbolic equations based on characteristic relations”, USSR Comp. Math. and Math. Physics, 9:2 (1969), 158–176 | DOI | MR

[2] K.M. Magomedov, A.S. Kholodov, Setochno-harakteristicheskie chislennye metody, Nauka, M., 1988, 287 pp. | MR

[3] A.S. Kholodov, “Construction of difference schemes with positive approximation for hyperbolic equations”, USSR Comp. Math. and Math. Physics, 18:6 (1978), 116–132 | DOI | MR

[4] A.S. Kholodov, “The construction of difference schemes of increased order of accuracy for equations of hyperbolic type”, USSR Comp. Math. and Math. Physics, 20:6 (1980), 234–253 | DOI | MR | Zbl

[5] I.B. Petrov, A.S. Kholodov, “Regularization of discontinuous numerical solutions of equations of hyperbolic type”, USSR Comp. Math. and Math. Physics, 24:4 (1984), 128–138 | DOI | MR | Zbl

[6] I.E. Petrov, A.S. Kholodov, “Numerical study of some dynamic problems of the mechanics of a deformable rigid body by the mesh-characteristic method”, USSR Comp. Math. and Math. Physics, 24:3 (1984), 61–73 | DOI | MR | Zbl

[7] V.I. Kondaurov, I.B. Petrov, A.S. Kholodov, “Numerical modeling of the process of penetration of a rigid body of revolution into an elastoplastic barrier”, Journal of Applied Mechanics and Technical Physics, 25:4 (1984), 625–632 | DOI | MR

[8] A.V. Favorskaya, I.B. Petrov, “Grid-characteristic method”, Innovations Wave Modelling and Decision Making, Chapter 7, SIST Series, 90, Springer Switzerland, 2018, 117–160 | MR

[9] V.A. Biryukov, V.A. Miryakha, I.B. Petrov, N.I. Khokhlov, “Simulation of elastic wave propagation in geological media: Intercomparison of three numerical methods”, Comp. Mathematics and Mathematical Physics, 56:6 (2016), 1086–1095 | DOI | MR | Zbl

[10] I.B. Petrov, A.V. Favorskaya, N.I. Khokhlov, V.A. Miryakha, A.V. Sannikov, V.I. Golubev, “Monitoring the state of the moving train by use of high performance systems and modern computation methods”, Math. Models and Comp. Simulations, 7:1 (2015), 51–61 | DOI | MR

[11] I. Petrov, A. Vasyukov, K. Beklemysheva, A. Ermakov, A. Favorskaya, “Numerical Modeling of Non-destructive Testing of Composites”, Proc. Comp. Science, 96 (2016), 930–938 | DOI

[12] A. Favorskaya, I. Petrov, N. Khokhlov, “Numerical Modeling of Wave Processes during Shelf Seismic Exploration”, Procedia Computer Science, 96 (2016), 920–929 | DOI

[13] A.V. Favorskaya, N.I. Khokhlov, V.I. Golubev, A.V. Ekimenko, Yu.V. Pavlovskiy, I.Yu. Khromova, I.B. Petrov, “Wave processes modelling in geophysics”, Innovations Wave Modelling and Decision Making, Chapter 7, SIST Series, 90, Springer Switzerland, 2018, 187–218 | MR

[14] A.V. Favorskaya, I.B. Petrov, “Wave responses from oil reservoirs in the Arctic shelf zone”, Doklady Earth Sciences, 466:2 (2016), 214–217 | DOI | MR

[15] A.V. Favorskaya, I.B. Petrov, “Theory and practice of wave processes modelling”, Innovations Wave Modelling and Decision Making, Chapter 1, SIST Series, 90, Springer Switzerland, 2018, 1–6 | MR

[16] A.V. Favorskaya, I.B. Petrov, “Numerical modeling of dynamic wave effects in rock masses”, Doklady Mathematics, 95:3 (2017), 287–290 | DOI | MR | Zbl

[17] M. Dumbser, M. Kaser, “An arbitrary high-order discontinuous Galerkin method for elastic waves on unstructured meshes-II. The three-dimensional isotropic case”, Geophysical Journal International, 167:6 (2006), 319–336 | DOI

[18] D. Komatitsch, J.P. Vilotte, R. Vai, J.M. Castillo-Covarrubias, F.J. Sanchez-Sesma, “The spectral element method for elastic wave equations-application to 2-D and 3-D seismic problems”, International Journal for numerical methods in engineering, 45:9 (1999), 1139–1164 | 3.0.CO;2-T class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | Zbl

[19] E. Faccioli, F. Maggio, R. Paolucci, A. Quarteroni, “2D and 3D elastic wave propagation by a pseudo-spectral domain decomposition method”, J. of Seismol., 1:3 (1997), 237–251 | DOI

[20] P. Moczo, J.O. Robertsson, L. Eisner, “The finite-difference time-domain method for modeling of seismic wave propagation”, Advances in geophysics, 48 (2007), 421–516 | DOI

[21] T. Wang, X. Tang, “Finite-difference modeling of elastic wave propagation: A nonsplitting perfectly matched layer approach”, Geophysics, 68:5 (2003), 1749–1755 | DOI

[22] R.W. Graves, “Simulating seismic wave propagation in 3D elastic media using staggered-grid finite differences”, Bulletin of the Seismological Society of America, 86:4 (1996), 1091–1106 | MR

[23] M.S. Zhdanov, Geophysical inverse theory and regularization problems, Elsevier, Amsterdam, 2002

[24] P.V. Krauklis, L.A. Krauklis, “One type of waves in media with loosely bonded interface”, Journal of Mathematical Sciences, 55:3 (1991), 1725–1732 | MR | Zbl | Zbl

[25] V.B. Levjant, V.A. Mirjaha, M.V. Muratov, I.B. Petrov, “Ocenka vlijanija na sejsmicheskij otklik stepeni raskrytosti treshhiny i doli ploshhadi lokal'nyh kontaktov k ee poverhnosti”, Tehnologii sejsmorazvedki, 2015, no. 3, 16–30

[26] J. Zhang, “Elastic wave modeling in fractured media with an explicit approach”, Geophysics, 70:5 (2005), T75-T85 | DOI

[27] R. LeVeque, Finite volume methods for hyperbolic problems, Cambridge University Press, 2002 | MR | Zbl

[28] A.V. Favorskaya, I.B. Petrov, “A study of high-order grid-characteristic methods on unstructured grids”, Numerical Analysis and Applications, 9:2 (2016), 171–178 | DOI | MR | Zbl

[29] V.M. Babich, “Mnogomernyj metod VKB ili luchevoj metod. Ego analogi i obobshhenija”, Itogi nauki i tehniki. Ser. «Sovremennye problemy matematiki. Fundamental'nye napravlenija», 34, 1988, 93–134

[30] A.S. Glassner (ed.), An introduction to ray tracing, Elsevier, 1989

[31] A.G. Kulikovskij, N.V. Pogorelov, A.Ju. Semenov, Matematicheskie voprosy chislennogo reshenija giperbolicheskih sistem uravnenij, Fizmatlit, M., 2001, 607 pp.

[32] M.A. Il'gamov, A.N. Gil'manov, Neotrazhajushhie uslovija na granicah raschjotnoj oblasti, Fizmatlit, M., 2003

[33] Q.H. Liu, B.K. Sinha, “A 3D cylindrical PML/FDTD method for elastic waves in fluid-filled pressurized boreholes in triaxially stressed formations”, Geophysics, 68:5 (2003), 1731–1743 | DOI

[34] F.H. Drossaert, A. Giannopoulos, “A nonsplit complex frequency-shifted PML based on recursive integration for FDTD modeling of elastic waves”, Geophysics, 72:2 (2007), T9–T17 | DOI | MR

[35] N. Hamdan, O. Laghrouche, P.K. Woodward, A. El-Kacimi, “Combined paraxial-consistent boundary conditions finite element model for simulating wave propagation in elastic halfspace media”, Soil Dynamics and Earthquake Engineering, 70 (2015), 80–92 | DOI

[36] G. Festa, S. Nielsen, “PML absorbing boundaries”, Bulletin of the Seismological Society of America, 93:2 (2003), 891–903 | DOI | MR