Earth Resources Laboratory :: Abstracts

Couple Electromagnetic and Acoustic Wavefield Modeling in
Poro-Elastic Media and its Applications in Geophysical Exploration

Matthijs W. Haartsen

Submitted to the Department of Earth, Atmospheric, and Planetary Sciences on June, 1995 in partial fulfillment of the requirements for the degree of Doctor of Philosophy

Abstract

This thesis investigates the macroscopic dynamics of two-phase (fluid and solid) porous media possessing continuously distributed phases. A fluid saturated sedimentary rock is an example of such a material. When a mechanical disturbance propagates through it, a small amount of relative motion is induced between the fluid and solid phases. The driving force for this relative flow is a combination of pressure gradients set up by the peaks and troughs of a compressional and shear waves. Mechanical waves, therefore, act as a current source for electromagnetic disturbances.

First, the governing equations that control the coupled electromagnetics and acoustics of porous media, including the transport equations through which all coupling occurs, are used to study electrical streaming currents through a homogeneous porous medium induced by seismic point sources. Then the coupled equations are solved in heterogeneous, i.e. layered porous media. Electroseismic boundary conditions at singular jumps in the macroscopic material properties are defined, and a macroscopic energy transfer consistent field-vector-formalism is derived and solved numerically in a layered medium. Different seismic point sources applied on either part of the two phase medium are derived in a poro-elastic medium. A method is presented to simulate the propagation of seismic and converted electromagnetic waves generated by a mechanical borehole source embedded in a layered poro-elastic medium. Electroseismic field experiments in shallow subsurface environments and in boreholes are analyzed using the developed electroseismic numerical algorithms.

The coupled transport equations, flux/force relations that relate the current relative flow to potential and pressure gradients, are simplified to solve for the mechanically induced electrical streaming currents generated by seismic point sources in homogeneous porous media. The mechanical wave behavior is decoupled from the electromagnetic wave behavior to simplify the analysis of induced fluxes (relative fluid flow and current flow). The mechanically induced relative flow is determined by Green's function solution. The stationary-phase method is used to calculate streaming current radiation patterns for an explosive and vertical point source acting on the bulk phase and a pressure source acting on the fluid phase. The dynamic streaming current amplitude behavior with respect to porosity, permeability, and fluid chemistry is investigated at three different source center frequencies
From the macroscopic coupled equations of motion, electromagnetic equations, and constitutive equations the acoustic-electromagnetic power balance in its global form in a porous medium is derived. The time rate at which the sources deliver mechanical and electromagnetic work to the coupled acoustic, electromagnetic disturbances is shown to equal the time rate of kinetic energy plus the time rate of deformation energy plus the time rate of coupled field energy plus the time rate of total stored electromagnetic energy in a volume plus the acoustic-electromagnetic powerflow through a surface bounding the volume.

Conditions are derived for uniqueness of solution of the coupled field equations. Particular attention is paid to continuity requirements at an interface where a jump in macroscopic medium parameters occurs. Electroseismic boundary conditions at singular jumps in the macroscopic material properties are determined. This defies a displacement-stress-EM wavefield vector whose components are continuous through a contrast. The governing electroseismic equations are transformed into a field-vector-formalism using a plane wave solution procedure. Wavefield eigenvectors and associated eigenvalues (wavefield slownesses) are derived for up and down going waves in porous media (i.e. fast compressional wave, slow compressional wave, rotational wave and electromagnetic waves). To guarantee a consistent macroscopic theory of energy transfer, conservation of electromagnetic and poro-elastic Poynting power upon crossing an interface is imposed by energy normalization of the up and down going vectors.

The macroscopic governing equations, transformed into a field vector formalism, are numerically solved using the very stable global matrix method. The effects of both pressure and shear seismic waves traversing a mechanical and/or electrical contrast are studied. When seismic waves traverse contrast, dynamic current imbalances are induced which generate radiating electromagnetic disturbances. Amplitude versus offset and sensitivity to macroscopic medium properties of the seismic to elecrtomagnetic signals are studied.

Relative flow and displacement Green's functions in dyadic form are derived. The final dyadic form is similar in form to the isotropic elastic dyadic Green's function. Source jump representations are derived for an explosive and vertical point source using the developed Green's tensors together with the deformation equations of the two phases. The explosive point source, which generates both P and SV waves, are compared with respect to their conversion behavior into electromagnetic disturbances in layered porous medium.

The Biot-Rosenbaum model is extended by including the effect of a heterogeneous porous formation surrounding the borehole and by including the conversions of mechanical into electromagnetic waves at the mechanical and/or electrical contrasts in the poro-elastic formation. The method to solve the extended Biot-Rosenbaum problem is formulated as a boundary element technique. The singular properties of the Green's functions are determined analytically using static Green's functions to regularize the integrals. This is necessary to calculate the element's self interaction.

Two real electroseismic datasets are interpreted using the developed electroseismic theory and numerical algorithms. Synthetics in Transpose Vertical ElectroSeismic Profiling (TV ESP) geometry are generated to identify the electroseismic responses in borehole geometry from a soil-glacial till and glacial-till-bedrock interface. The synthetics in surface seismic geometry are calculated to extract converted electrical field amplitudes versus dipole antenna offset which is compared against the real measured data result.