1. Finite element modeling of the electric field for geophysical application Trofimuk Institute of Petroleum Geology and Geophysics SB RAS Shtabel Nadezhda, Antonov Eugeniy

2. Outline 1.The types of the geophysical problems: mathematical models 2.Finite element method: spaces, variational formulations, discretization 3.The impulse sounding problem in time domain 4.The sounding problem in frequency domain

3. The types of the geophysical problems Problems in time domain Problems in frequency domain The surface sounding The marine geoelectric The impulse sounding The borehole logging The frequency sounding Time dependent second order equations for electric field Helmholtz equation for electric field

4. Second order equations Hyperbolic equation Parabolic equation

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

6. Advantages of applying the vector finite element method to the electromagnetic problems Solves equations in nature quantities as 3D vector field Keeps the tangential component of the electric field continuous on the interface boundary The normal component of the electric field has jump on the interface boundary The FEM solution fulfils to the charge conservation law Any type and geometry of the field source are taking into account

7. The functional spaces

8. The functional subspaces and de Rham’s complex

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

10. The variational formulation The following property allows to fulfill the variational analog of the charge conservation law Forto findsuch thatthe following is held

11. Discretization by time and space Basic function of the space H (rot,Ω)

12. Discretization of Helmholtz equation Basic function of the space H (rot,Ω) O.V. Nechaev, E.P. Shurina, M.A. Botchev. Multilevel iterative solvers for the edge finite element solution of the 3D Maxwell equation // Computers and Mathematics with Applications. - 2008 - Vol. 55 - Pp. 2346-2362

13. Mesh generation Complex structure of the investigated media requires to use meshes that have good approximation of the curvilinear boundaries, have local thickening. C. Geuzaine and J.-F. Remacle. Gmsh: a three- dimensional finite element mesh generator with built-in pre- and post-processing facilities. International Journal for Numerical Methods in Engineering 79(11), pp. 1309- 1331, 2009.

14. Subdomain Ω1: air σ = 1е-6 1/(Оm·m) Subdomain Ω2: Conductive ground Field source – the current loop with impulse signal Impulse sounding

15. EMF, V Time, s  =10  =1  =0,1 The features of the EMF graphs for the impulse sounding problem

16. 1.Domain 5000 m х 5000 m – 50 sizes of source 2.Domain 10000 м х 10000 м – 100 sizes of source 3.Domain 25000 м х 25000 м – 250 sizes of source The size of the domain should be not less then 25 km for the source loop with the size 500 m The limitation of the computational domain EMF, V t

17. The limitation for computational domain Domain: 25 km х 25 km х 10 km Source loop: 500 m х 500 m Layers – 5 km, 300 m, 3.5 km The size of the mesh: 332`760 nodes 2`288`590 edges Domain: 25 km х 25 km х 16 km Source loop: 500 м х 500 м Layers – 5 km, 1.5 km, 2.5 km, 1 km, 2 km, 4 km The size of the mesh: 84`355 nodes 574`035 edges

18.  =10  =5  =1000 Hz w The features of the EMF graphs for the frequency impulse sounding problem

19. Hz w The features of the EMF graphs for the frequency impulse sounding problem

20. Hz w The features of the EMF graphs for the frequency impulse sounding problem