The Perfectly Matched Layer (PML) approach is widely used to implement the absorbing boundary conditions for coupled multi-physics wave propagation problems. However, it has been recognized that the solution in the PML absorbing layer can become unstable in the presence of back-propagating modes. This paper analyzes the spectral location of those modes excited by monopole acoustic sources in logging-while-drilling cylindrical tools. To recover the stability of the solution for such a class of problems involving the modeling of elastic cylindrical waveguides, we propose the implementation of artificial attenuation in the waveguide to dampen undesirable modes, thereby making their amplitude negligible in the PML absorbing layer due to the limited numerical accuracy and round-off errors.
A model was recently 26 introduced to describe the complex electrical conductivity and high frequency dielectric constant of isotropic clayey porous materials. That approach is generalized here to the case of anisotropic and tight hydrocarbon-bearing shales and mudrocks by introducing tensorial versions of both formation factor and tortuosity. It is shown that in-phase and quadrature conductivity tensors have common eigenvectors, but that the eigenvectors of the dielectric tensor may be different due to influence of the solid phase at high requencies. In-phase and quadrature contributions to complex electrical conductivity depend on saturation, salinity, porosity, temperature, and cation exchange capacity (alternatively, specific surface area) of the porous material. Kerogen is likely to have a negligible contribution to the cation exchange capacity of the material because all exchangeable sites in the functional groups of organic matter may have been polymerized during diagenesis. An anisotropic experiment is performed to validate some of the properties described by the proposed model, especially to verify that the electrical anisotropy factor is the same for both in-phase and quadrature conductivities. We use two samples from the Bakken formation. Experimental data confirm the validity of the model. It is also found that the range of values for cation exchange capacity determined when implementing the new model with experimental data agree with the known range of cation exchange capacity for the Bakken shale. Measurements indicate that the bulk-space tortuosity in the direction normal to bedding plane can be higher than 100.