Equivalent Isotropic Media Research Articles

Numerical study of water inflow into tunnels in stratified rock masses with a dual permeability model

Stratification planes form an anisotropic structural setting in the stratified rock mass, and strongly influence the water inflow into tunnels. However in the numerical simulation studies, the equivalent porous model pays little regard for the anisotropic structural setting and the discrete fracture network model costs too much computing power. In this study, a dual permeability model is utilized to calculate the water inflow into tunnels and the anisotropic flow behavior with finite element numerical simulations. The dual permeability model takes the stratification planes into artery fractures, and consider the rock matrix with other fractures as an equivalent isotropic medium. Parameters including the piezometric head above the tunnel center, the artery fracture angle, and the artery fracture spacing are studied and discussed. It is shown that both the piezometric head above the tunnel center and the artery fracture spacing affect the water inflow a lot. In this research, the influence of the artery fracture spacing on water inflow is larger than that of the piezometric head above the tunnel center when the artery fracture spacing 4.62 m. What is more, the artery fracture angle affects the water inflow slightly but it controls the flow direction. The water inflow case of the Huangjiagou Tunnel is used to verify the proposed model. The results show that the dual permeability model could give good predictions to the Huangjiagou Tunnel and describe the anisotropic flow behavior in stratified rock mass tunnels.

Characterization of horizontal transmissivity anisotropy using cross-hole slug tests

Traditional methods for characterizing horizontally anisotropic aquifers are all based on pumping tests. In this paper, we present a new method for the identification of horizontal anisotropy using cross-hole slug tests, which is much more easily accessible comparing to pumping tests. Through scaler transform, an anisotropic medium was converted into an equivalent isotropic medium. When applying the analytical solutions derived for isotropic media to interpret cross-hole slug tests performed in anisotropic media, the estimated transmissivity is the geometrical mean of the anisotropic transmissivity tensor, regardless of sampling direction. However, the apparent storativity calculated from different observation wells, is equal to the true storativity scaled by a factor defined by the formation anisotropy. Thus, anisotropy can be resolved through apparent storativity that bears directional information. The proposed method was first validated by a numerical experiment and then applied to field data. It was found that the tensor results obtained by this new method with cross-hole slug tests are consistent with that obtained by applying the classical Papadopulos’ method with cross-hole pumping tests conducted in the same wells. When applied to heterogeneous media, tensor results produced by the new method may be subject to considerable errors. This is because the results are strongly sensitive to the connectivity between measurement boreholes and the formation medium. To reduce the error level, responses from a large number of observation wells located in various directions around the source well are needed. In addition to the adopted KGS solution for confined aquifers, the proposed methodology can be applied to incorporate other analytical methods for a variety of aquifer types, as long as the characteristic relationship between the measure scale and the aquifer storativity can be established.

Homogenization of Periodic Structured Materials With Chiral Properties

This paper presents a new and efficient method to compute the quasi-static homogenized constitutive parameters of biperiodic chiral artificial material. In this method, the studied domain is reduced to an elementary cell with pseudo-periodic conditions on the lateral sides and the local electromagnetic properties are computed by using the finite-element method. This chiral media is decomposed into two equivalent isotropic media and each one is treated separately. Indeed, the homogenized constitutive parameters of each isotropic media are expressed as a function of the macroscopic electromagnetic properties which are obtained by averaging the local electromagnetic properties. The homogenized constitutive parameters of the initial chiral material are expressed using the two equivalent isotropic materials. The validation of the numerical results is presented in the case of the lattices with 3-D inclusions. In addition, the obtained results are compared to existing results in literature. Finally, the incidence wave angle impact on the homogenized parameters is studied.

A simplified multiscale damage model for the transversely isotropic shale rocks under tensile loading

A simplified multiscale damage model is proposed for the transversely isotropic shale rocks under tensile loading. In this framework, the multiscale representations for the shale rocks are presented by introducing the microcrack-weakened equivalent solid with hierarchical microstructures, whose transversely isotropic properties are obtained by performing multilevel homogenization procedures. To simplify the calculation process for the damage-induced properties, the equivalent isotropic medium is attained by applying the Voigt–Reuss–Hill averaging process to the transversely isotropic solid. Subsequently, the microcrack-induced inelastic compliances are approximately derived in terms of microcrack opening displacements in the equivalent isotropic medium of the shale rock under tensile loading. The sizes and orientations of microcracks are taken as random variables. Both stationary and evolutionary damage models are considered. Microcrack kinetic equations are characterized through the use of a fracture mechanics-based stability criterion and microcrack geometry within a representative volume element. Numerical examples including experimental validations and comparisons with existing micromechanical models are presented to verify the proposed multiscale damage model. Finally, the influences of the silt inclusions and porosity on the material intrinsic and damage-induced properties are discussed.

Propagation and annihilation of vortex pairs in a smoothly inhomogeneous Bose-Einstein condensate

An asymptotic theory has been developed to describe the propagation and annihilation of vortex pairs in a smoothly inhomogeneous Bose-Einstein condensate with the repulsive interaction between atoms. It has been shown that the trajectories of vortex pairs coincide with geometric-optical rays in an equivalent isotropic medium with the refractive index depending both on the density of the unperturbed inhomogeneous condensate and on the energy of structures under study. The transformation of vortex pairs to vortex-free quasisolitons and back has been described.

Scattering by Conducting Bodies Coated With Bi-Isotropic Materials

Electromagnetic scattering by arbitrarily shaped conducting bodies coated with general bi-isotropic materials is formulated in terms of the surface integral equation method. In order to facilitate the implementation of the surface equivalence principle, a field decomposition scheme is utilized to split a bi-isotropic media into two equivalent isotropic media. By enforcing the boundary condition on the interfaces of the body, a set of coupled integral equations is finally obtained for the unknown surface currents and then numerically solved using the moment methods combined with the vector triangular basis function. The fast multipole technique has been embedded into the algorithm to accelerate the solution process. The validity of theoretical formulations is verified by numerical results and their comparisons. The calculated results for bi-isotropically coated conducting spheres and oblate spheroids are compared with the exact solution and the existing data, and excellent agreements are observed.

BI-FDTD: A Novel Finite-Difference Time-Domain Formulation for Modeling Wave Propagation in Bi-Isotropic Media

This paper presents a newly developed finite-difference time-domain (FDTD) technique, referred to as BI-FDTD, for modeling electromagnetic wave interactions with bi-isotropic (BI) media. The theoretical foundation for the BI-FDTD method will be developed based on a wavefield decomposition. The main advantage of this approach is that the two sets of wavefields are uncoupled and can be viewed as propagating in an equivalent isotropic medium, which makes it possible to readily apply conventional FDTD analysis techniques. The BI-FDTD scheme will also be extended to include the dispersive nature of chiral media, an important subclass of bi-isotropic media. This extension represents the first of its kind in the FDTD community. Validations of this new model are demonstrated for a chiral half-space and a chiral slab.

Borehole flexural modes in anisotropic formations

A perturbation method of solution is an efficient way of analyzing elastic wave propagation along a borehole in anisotropic formations. The perturbation model allows us to calculate changes in the modal dispersion curves caused by the differences in elastic constants between the anisotropic formation of interest and a reference, or unperturbed, isotropic formation. The equivalent isotropic constants in the reference formation are obtained from the appropriate compressional‐and shear‐wave velocities for the selected propagation and polarization directions of the flexural mode. This choice of the unperturbed solution means that the required perturbation is minimal, resulting in enhanced accuracy of the perturbed solution. Computational results are presented for the dispersion curves of borehole flexural waves in a transversely isotropic (TI) formation as a function of borehole deviation from the TI symmetry axis. In addition, radial distributions of displacement and stress fields associated with the flexural wave are obtained as a function of frequency. These provide qualitative information on the radial depth of investigation with flexural wave logging. The flexural wave excitation function is a measure of the energy that a source converts to flexural motion. We deduce an expression for the flexural wave excitation and show that its bandlimited characteristic is influenced by both the borehole diameter and formation parameters. From the dispersion curves and excitation functions, we can compute the flexural waveforms caused by a dipole source with arbitrary orientation in the borehole. In the numerical computations, we have used the unperturbed mode shapes for an equivalent isotropic medium together with the perturbed dispersion relations caused by the formation anisotropy.

Flow under a Weir with a Sloping Step on Anisotropic Soil of Finite Depth

An analytical solution is presented, making use of the Schwartz-Christoffel transformation, to determine the seepage characteristics for the problem of flow under a weir having an inclined stepping, with a sheet pile at the downstream side, embedded in an anisotropic porous medium of finite thickness. The effect of anisotropy is studied by analyzing the problem in an equivalent isotropic medium. Numerical results are presented for quantity of seepage and exit gradient distribution in the equivalent isotropic medium.