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

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

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

Second order equations Hyperbolic equation Parabolic equation

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

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

The functional spaces

The functional subspaces and de Rham’s complex

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

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

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

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 Vol Pp

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 , 2009.

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

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

1.Domain 5000 m х 5000 m – 50 sizes of source 2.Domain м х м – 100 sizes of source 3.Domain м х 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

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

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

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

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