Simulation Of The Coupled Poroelastic And Electromagnetic Wavefields In Seismoelectric Logging

Analytical and numerical methods are proposed to model the coupled poroelastic (or acoustic) and electric fields in fluid-saturated porous media. The poroelastic field itself is a coupled field of the solid strain and the pore fluid filtration, and it is coupled with the electromagnetic field due to the net charged ions in the pore fluid. This double coupling is called as seismoelectric effects. Seismoelectric exploration and seismoelectric well logging have caught the attention of geophysics and petroleum research engineers due to their potential usage in petroleum exploration. It is important to have a thorough understanding of the characteristics of the seismoelectric wave-fields and their relations to the formation parameters. This cannot be achieved without modeling the seismoelectric wave-fields correctly and efficiently. Numerical methods that can be applied in heterogeneous porous media are especially important and must be explored. But the simulation of seismoelectric wave-fields is a challenging task due to the difficulty in solving the coupling field.In this thesis, the acoustic field and its converted electromagnetic field of the seismoelectric logs excited by a point acoustic source in the fluid-filled borehole are modeled. In addition, the converted acoustic field of the electroseismic logs excited by an electric source in the borehole is simulated.The first two chapters are devoted to the semi-analytical methods for simulation of seismoelectric and electroseismic logging. By supposing the acoustic field is not influenced on its induced electromagnetic field, the analytical expressions for the induced electromagnetic field are derived by introducing Hertz vectors in Chapter 2. The full waveforms of the acoustic field and the electromagnetic field in the borehole are calculated when the borehole acoustic sources are respectively a monopole, a dipole and a quadrupole. The influence of formation parameters, such as the permeability, porosity, tortuosity, pore fluid viscosity and salinity, on the waveforms are investigated,In Chapter 3, two different methods are proposed to derive the analytical expressions for the electromagnetic field and its induced acoustic field of the electroseismic logs excited by a vertical electric dipole in the borehole. In the coupled method, the EM field and the acoustic field are modeled using Pride's model, which couples Maxwell's equations and Biot's equations. In the uncoupled method, the EM field is uninfluenced by the converted acoustic field, resulting in separate acoustic formulation with an electrokinetic source term derived from the primary EM field. The difference of the transient full waveforms between the above two methods is remarkably small for all examples, thus confirming the validity of using the computationally simpler uncoupled method. The waveform characteristics of the electric-induced acoustic field and the influence of formation parameters on the waveforms are investigated.For the modeling of the electroseismic logs excited by a horizontal current loop, the analytical expressions for the electric field and its induced acoustic field are derived by solving Pride's equations. In this logging problem, the transverse electric wave is coupled with the induced shear horizontal wave. Numerical simulations of the logs are calculated in different fluid-saturated porous formations, and the shear wave velocities are directly extracted from the simulated full waveforms. This logging method has potential application value in shear velocity logging.In the two subsequent chapters, I adapt the finite deference (FD) method to cater for acoustic, seismoelectric, and electroseismic wave propagation problems in heterogeneous porous media, with special efforts to treat the inner and outer boundary conditions.A velocity-stress finite-difference time-domain (FDTD) algorithm is proposed in Chapter 4 to simulate the acoustic fields in a fluid filled borehole embedded in fluid-saturated porous media. This algorithm considers both the low-frequency viscous force and the high-frequency inertial force in poroelastic media, extending its application to a wider frequency range compared to existing algorithms which are only valid in the low-frequency limit. The perfectly matched layer (PML) is applied as an absorbing boundary condition to truncate the computational region. A PML technique without splitting the fields is extended to the poroelastic wave problem. In the FDTD algorithm, two different method are respectively used to discrete the field quantities on the borehole wall between the fluid and porous media. One is to derive the discrete equations by using the boundary conditions. The other is to formulate in the same forms as those in the homogeneous media, but the average of the medium parameters on both sides of the interface are used instead, this method is called as parameter averaging technique. The FDTD algorithm is validated by comparisons against the analytical method in a variety of formations with different velocities and permeabilities. The acoustic logs in a horizontally stratified porous formation are simulated with the proposed FDTD algorithm.The validity of the parameter averaging technique used for the interface between two different porous media is proved by deriving the discrete equations with average parameters from the continuity conditions on the interface. By using a set of generic equations to express elastic waves in solid, fluid and porous media, this parameter averaging technique is applied to deal with not only the interfaces between two media of the same type but also those between solid, fluid, and porous media. And the 2-D FDTD algorithm in axisymmetric cylindrical coordinates is presented to simulate elastic wave propagation in combined structures with solid, fluid and porous subregions. This algorithm has important significance to normalize and popularize the FDTD method for the modeling of elastic waves in porous media.In the last chapter, FD algorithms are proposed to simulate the induced acoustic field of the electroseismic logging and the induced electric field of the seismoelectric logging in horizontally stratified porous formations. For the modeling of electroseismic logging, the EM field excited by the electric dipole is calculated separately by FD first, and is considered as a distributed exciting source term in a set of extended Biot's equations for the induced acoustic wavefield in the formation. This set of equations is solved by a modified FDTD algorithm. For the modeling of seismoelectric logging, the acoustic field excited by the acoustic source is calculated by the FDTD algorithm first, and is considered as a wave-like exciting source term in the Poisson equation for the induced electric potential. This Poisson equation is solved by FD. The FD algorithms are validated by comparisons against the analytical method in homogeneous porous formation. The electroseismic and seismoelectric logs in heterogeneous porous formations with a horizontal interface are simulated with the FD algorithms.