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. The complex configuration of interface boundary. The electrophysical properties: the contrast between separate fragments of media, anisotropy, polarization, the dispersion of the conductivity, permittivity and permeability.

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

The Second order equations Hyperbolic equation Parabolic equation

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

The interface conditions

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

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

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

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

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

Automatic Decomposition Decomposition by enclosed “spheres” Decomposition by layers

EM Logging

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

ElectroPhysical Properties Operating frequency 14 МHz, amperage J=1 А. Domain

Re Ex (X0Y) 0 Zenith angle

Re Ez (X0Y) 0 Zenith angle E z =0

Surface Soundings

Transmitter loop 40 x 40 m² Receiver loop 20 x 20 m² Impulse length 5 µs Simulation time 10 ms Mesh: edges, nodes, 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

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

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

The anisotropic object in the isotropic halfspace 24

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

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

The multiscale modeling in media with microinclusions

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

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

Discrete variational problem Let's consider a partition in the area Ω. Element is a tetrahedron. Let's introduce the spaces Then the variational problem of the homogeneous elliptic problem states:

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

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

0 15 mm 40 mm X Z Y 15 mm Scalability Inclusions Volume of inclusions, % Number of Cores 124 5х10х CPU time (sec)

Method The error Physical experiment Maxwell's approach % Bruggeman's approach % Approach of coherent potential % Numerical Modeling % Comparison with the physical experiment

MethodThe error Physical experiment Numerical Modeling % Comparison with the physical experiment

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

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

The percolation The size of the inclusions

The calculation of the effective tensor coefficients

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

Mathematical model The Helmholtz equation in Ω Boundary conditions is the wave number The direct problem

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

The 1-st method of calculating tensor Z where

The 2-nd method of calculating tensor Z Fields E and rot H are calculated in N points of the domain (for example, in barycentres of tetrahedral finite elements). We obtain the set of tensors Z {Z m, m=1,..,N-2}, by running over the points xi, xj, xk. The effective tensor coefficient of the medium is calculated as an average of {Z m, m=1,..,N-2}.

Variational formulation Helmholtz equation in anisotropic media Variational formulation: The problem in anisotropic media Findsuch that the following is held

Boundary conditions The domain with one side boundary conditions The domain with boundary conditions given by the closed path

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 Domains

The electrophysical properties of the computational domain The matrixThe inclusions ε [F/m]4.5 ε 0 1 ε01 ε0 σ [Sm/m] µ [H/m]1 µ 0 ε 0 = 8,85 × F/m µ 0 = 4π ×10 -7 H/m The mesh (40 inclusions) The results of numerical experiment Number of the inclusions Volume of the inclusions The size of SLAE 40 regular  2% chaotically  2% regular  27%

The homogeneous medium. The one size boundary conditions. The frequency 10 kHz EzR – Re Ez computed for homogeneous medium (  =0.001Sm/m) with inclusions EzR tensor – Re Ez computed for the medium with tensor coefficient Z 2

The homogeneous medium. The one size boundary conditions. The frequency 7 GHz EzR – Re Ez computed in homogeneous medium (  =0.001Sm/m) with inclusions EzR tensor - Re Ez computed in the medium with tensor coefficient Z 2

176 inclusions 10 kHz7 GHz In the medium with inclusions In anisotropic medium In uniform medium, Sm/m In the medium with inclusions In anisotropic medium In uniform medium, Sm/m In uniform medium, 0.1 Sm/m