1 / 51

Computational electrodynamics in geophysical applications

Computational electrodynamics in geophysical applications. Epov M . I . , Shurina E.P., Arhipov D.A., Mikhaylova E.I., Kutisheva A. Yu., Shtabel N.V. The main features of the geological media. Heterogeneous media , fluid-saturated rocks . The complex geometry of objects .

dante-stuart
Download Presentation

Computational electrodynamics in geophysical applications

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Computational electrodynamics in geophysical applications Epov M. I., Shurina E.P., Arhipov D.A., Mikhaylova E.I., Kutisheva A. Yu., Shtabel N.V.

  2. The main features of the geological media • Heterogeneous media, fluid-saturated rocks. • The complex geometry of objects. • The complex configuration of interface boundary. • The electrophysical properties:the contrast between separate fragments of media, anisotropy, polarization, the dispersionof the conductivity, permittivity andpermeability.

  3. The Maxwell’s equations The Faraday's law The Maxwell – Ampere law The Gauss’s laws for electric and magnetic flux densities

  4. The Second order equations Hyperbolic equation Parabolic equation

  5. Frequency domain.Helmholtz equation The charge conservation law The boundary conditions

  6. The interface conditions

  7. The functional spaces

  8. The functional subspaces and de Rham’s complex

  9. such For find that the following is held For find such that the following is held Variational Formulations Hyperbolic equation Parabolic equation

  10. We introduce the following partition of the time where is a solution onj-th time step step onj-th step of time scheme. Time Approximation and function on -th time step Then the function of interest is

  11. where the value of right hand side on j-th time step, parameter of the scheme. Newmark-beta Scheme

  12. The variational formulation For to find such that the following is held The following property allows to fulfill the variational analog of the charge conservation law

  13. Geometric domain decomposition Difficulties: • Local source of the field (the source should be in one subdomain and can’t touch its boundaries) • Balancing the dimensions of subdomains matrices(CPU time should be comparable in different subdomains) • The geometry of the computational domain should be taken into account Decomposition approaches: • Custom decomposition (effective, but time-consuming) • Automatic decomposition

  14. Automatic Decomposition • Decomposition by enclosed “spheres” • Decomposition by layers

  15. EM Logging

  16. Borehole - Inclined bed 1 – 3-coil probe, 2 – borehole with mud, 3 – host formation, 4 – low-conductive bed, Г – generator coil, И1, И2 – receiver coils

  17. ElectroPhysical Properties Operating frequency 14МHz, amperageJ=1 А.

  18. Re Ex (X0Y) Zenith angle 00 450 750

  19. Re Ez(X0Y) Zenith angle 00 450 750 Ez=0

  20. Surface Soundings

  21. Transmitter loop 40 x 40 m² Receiver loop 20 x 20 m² Impulse length 5 µs Simulation time 10 ms Mesh: 335666 edges, 49244 nodes, 281342 tetrahedrons Computation one time step 30 sec, after current is turn off Solver: Multilevel iterative solver with V-cycle Anisotropic layer Isotropic layer Zenith Angle 0, 30, 60, 90

  22. Ey, z=0 Ex, z=0 Ez, z=-50 Ex, z=-50 Ey, z=-50 Transversal isotropic medium θ=0°

  23. Ey z=0 Ex z=0 Ex z=-50 Ey z=-50 Ez z=-50 Transversal isotropic medium rotated for zenith angle θ=60°

  24. The anisotropic object in the isotropic halfspace

  25. Re Ex, Ez for vertical object The isotropic object The anisotropic object The cross-section x=3.4 m The conductivity of the medium is =0.01 Sm

  26. Re Ex, Ez for horizontal object The isotropic object The anisotropic object The cross-section z= -1 m The conductivity of the medium is =0.01 Sm

  27. The multiscale modeling in media with microinclusions

  28. Problem definition The problem is stated in the domain and governed by the following equation:

  29. Variational problem We introduce the Hilbert space Then the variational problem of the homogeneous elliptic problem states:

  30. Discrete variational problem Let's considera partition in the area Ω. Element is atetrahedron. Let's introduce the spaces Then the variational problem of the homogeneous elliptic problem states:

  31. Discrete variational problem Taking into account the partition we introduce the following statements: whereand – quadrature points and weights respectively.

  32. Heterogeneous Finite Element Method The basic principles The global multiscale “form functions” Assemble according degrees offreedom associated with nodes of the coarse mesh The local multiscale “form functions” FEM The local functions The integration points

  33. Scalability 40 mm 15 mm Z Y 0 15 mm X CPU time (sec)

  34. Comparison with the physical experiment

  35. Comparison with the physical experiment

  36. The cylinder with inclusions г) spheres a) vertical В)horizontal b) arbitrary directed The size of the inclusions:

  37. The influence of the geometry and orientation of the inclusions Horizontal plates Arbitrary oriented plates Vertical plates Spheres Horizontal plates Arbitrary oriented plates Vertical plates Spheres

  38. The percolation The size of the inclusions

  39. The calculation of the effective tensor coefficients

  40. The main steps of the algorithm E. Shurina, M. Epov, N. Shtabel and E. Mikhaylova. The Calculation of the EffectiveTensor Coefficient of the Medium for the Objects with Microinclusions // Engineering, Vol. 6 No. 3, 2014, pp. 101-112.

  41. The direct problem Mathematical model The Helmholtz equation inΩ is the wave number Boundary conditions

  42. Calculation of the effective coefficient Scalar Tensor Z is a complex-valued second rank tensor, which can be interpreted as the analog of

  43. The 1-st method of calculating tensor Z where

  44. The 2-nd method of calculating tensor Z where FieldsEand rot Hare calculated in Npoints of the domain (for example, inbarycentres of tetrahedral finite elements). We obtain the set of tensors Z {Zm, m=1,..,N-2}, by runningover the points xi, xj, xk. The effective tensor coefficient of the medium is calculated as an average of {Zm, m=1,..,N-2}.

  45. Variational formulation Find such that the following is held The problem in anisotropic media Helmholtz equation in anisotropic media Variational formulation:

  46. Boundary conditions The domain with one sideboundary conditions The domain with boundary conditions given by the closed path

  47. Domains • The size of the computational domain: 15 mm40 mm  15 mm • The diameter of the inclusions d = 2 mm • The number of the inclusions is different

  48. The electrophysical properties of the computational domain ε0= 8,85 ×10-12 F/mµ0 = 4π ×10-7H/m The results of numerical experiment The mesh (40 inclusions)

  49. The homogeneous medium. The one size boundary conditions. The frequency 10 kHz EzR– ReEz computed for homogeneous medium (=0.001Sm/m) with inclusions EzRtensor – Re Ez computed for the medium with tensor coefficientZ2

  50. The homogeneous medium. The one size boundary conditions. The frequency 7 GHz EzR– Re Ez computed in homogeneous medium (=0.001Sm/m) with inclusions EzRtensor - ReEz computed in the medium with tensor coefficientZ2

More Related