Reflection and transmission approximations for monoclinic media with a horizontal symmetry plane

Geophysics ◽  
2020 ◽  
Vol 85 (1) ◽  
pp. C13-C36 ◽  
Author(s):  
Song Jin ◽  
Alexey Stovas
Keyword(s):  
Order Approximation ◽  
Symmetry Plane ◽  
Second Order ◽  
Model Parameters ◽  
Reflection And Transmission ◽  
Model Parameter ◽  
First Order ◽  
Background Medium ◽  
Horizontal Symmetry ◽  
Parametric Analyses

We have considered a horizontal plane interface bounded by two monoclinic half-spaces to approximate the reflection and transmission (R/T) responses normalized by the vertical energy flux. We assume the monoclinic media have a horizontal symmetry plane, and the exact R/T coefficients can be analytically obtained for P-, S1-, and S2-waves. The exact R/T coefficients depend on the absolute model parameters of both half-spaces, whereas the R/T responses indicate the heterogeneity at the interface and, thus, can be characterized with model parameter contrasts across the interface. Compared with the exact R/T solutions, appropriate approximations with desirable accuracy can be determined by fewer model parameter contrasts and facilitate the parametric analyses and inversions. We first consider the weak-contrast assumption and use the perturbation method to obtain first-order approximations based on the homogeneous monoclinic background medium. To accommodate the strong-contrast interface, the published second-order approximations are then revised for the monoclinic media. For weakly anisotropic media, the first- and second-order approximations are proposed based on the isotropic background medium. Two pseudowaves are introduced as intermediate waves to legitimize our approximations for S1, S2, and converted waves in the applications. The derived approximations are verified on monoclinic models numerically. The second-order approximation method based on the monoclinic background medium is proven to have the overall best accuracy.

scholarly journals Exact and approximate reflection/transmission responses from a layer containing vertical fractures

2020 ◽  
Vol 222 (1) ◽  
pp. 260-288
Author(s):  
Song Jin ◽  
Alexey Stovas
Keyword(s):  
Plane Wave ◽  
Layer Thickness ◽  
Finite Thickness ◽  
Subsurface Layer ◽  
Symmetry Plane ◽  
Middle Layer ◽  
Second Order ◽  
First Order ◽  
Background Medium ◽  
Vertical Fractures

SUMMARY Analyses of vertical fractures are of great interest in characterizing the fluid flow and minimum in situ stress direction in reservoirs. Long-wavelength equivalent orthorhombic (ORT) media typically characterize the anisotropy induced by a set of vertical parallel fractures or two sets of vertical and mutually orthogonal fractures embedded into a transversely isotropic medium with a vertical symmetry axis (VTI). Reflection and transmission (R/T) responses quantify wave amplitude variations in 1-D media and help to reveal the model property enclosing the heterogeneity. Conventionally, the R/T responses are analysed for an interface bounded by two half-spaces. However, for a plane wave travelling through a subsurface layer, the wave scattering effects at the top and bottom of the layer interact with each other. For a continuous infinite ORT space cut in two halves along the horizontal symmetry plane, we focus on the plane wave R/T responses from an ORT layer that is placed between the two halves, where the azimuths of the vertical symmetry plane in the layer and in the upper and lower half-spaces are identical. The R/T coefficient modelling method can be found in many publications for the ORT layer with an arbitrary finite thickness. We decompose the exact R/T coefficients into series expansions that correspond to different orders of intrabed multiples in the ORT layer. Under the weak-contrast assumption for the ORT half-spaces and the ORT layer, we use the anisotropic background medium to obtain the first-order R/T coefficient approximations and second-order reflectivity approximations. There is no constraint for the middle layer thickness in the obtained first-order reflectivity approximations. In the proposed first-order transmissivity approximations and second-order reflectivity approximations, the layer thickness is assumed to be thin to obtain appropriate approximations for a few wave modes. The isotropic background medium is also considered for weakly anisotropic models to obtain simpler approximations that facilitate parametric analyses. For the ORT layer with its thickness much smaller than the propagating wave's wavelength, the influences of the layer thickness on R/T coefficients can be inspected conveniently from the derived approximations. Particularly, the R/T coefficients are analysed for the model which would be a homogeneous VTI medium, if the vertical parallel fractures were absent from the middle layer. Numerical tests demonstrate that the proposed R/T coefficient approximations perform well for the thin ORT layer. The approximation accuracy decreases when the thin layer thickness increases.

Reflection and transmission approximations for weak contrast orthorhombic media

Geophysics ◽  
2020 ◽  
Vol 85 (2) ◽  
pp. C37-C59 ◽  
Author(s):  
Song Jin ◽  
Alexey Stovas
Keyword(s):  
Perturbation Theory ◽  
Symmetry Plane ◽  
Azimuthal Anisotropy ◽  
Coordinate Frame ◽  
Model Parameters ◽  
Weak Anisotropy ◽  
Reflection And Transmission ◽  
Vertical Coordinate ◽  
Coordinate Frames ◽  
Model Parameter

Subsurface media are in general anisotropic, and this fact should be taken into account for analyzing reflection and transmission (R/T) coefficients. Orthorhombic (ORT) media are commonly regarded as a practical symmetry system to account for polar anisotropy and azimuthal anisotropy. We have focused on the model made up of two welded ORT half-spaces to analyze the R/T coefficients normalized by vertical energy flux. The two half-spaces have azimuthally nonaligned vertical symmetry planes and are parameterized in two local 3D Cartesian coordinate frames, respectively. The vertical coordinate planes in each local frame coincide with the vertical symmetry planes for the corresponding ORT half-spaces. Under the weak contrast assumption for the two half-spaces, this model is taken as the perturbed model in R/T approximations with the perturbation theory. The unperturbed model is also composed of two unperturbed ORT half-spaces with their different vertical symmetry plane orientations inheriting the counterparts for the perturbed half-spaces above and below the interface, respectively. The unperturbed ORT half-spaces above and below the interface have the same model parameters defined in the two local coordinate frames, respectively. With the perturbations respectively evaluated in the local coordinate frames above and below the interface, the azimuth angle that indicates the local frames’ azimuthal difference is decoupled from the model parameter contrasts. Compared with the traditional approximation method with the perturbation theory in a global coordinate frame, the proposed R/T approximations depend on fewer model parameter discontinuities. We also consider the isotropic background medium under the weak anisotropy assumption. Influences of S-wave singularity points are mitigated by introducing pseudowaves for approximations. Numerical tests are implemented to demonstrate the accuracy.

A Perturbation Analysis of the Wavemaking of a Ship, with an Interpretation of Guilloton’s Method

1975 ◽  
Vol 19 (03) ◽  
pp. 140-148
Author(s):  
F. Noblesse
Keyword(s):  
Approximate Solution ◽  
Free Surface ◽  
Perturbation Analysis ◽  
Order Approximation ◽  
Second Order ◽  
Regular Perturbation ◽  
First Order ◽  
Perturbation Problem ◽  
Sinkage And Trim ◽  
Basic Ideas

A thin-ship perturbation analysis, suggested by Guilloton's basic ideas, is presented. The analysis may be regarded as an application of Lighthill's method of strained coordinates to a regular perturbation problem. An inconsistent second-order approximation in which the Laplace equation is satisfied to first order, and the boundary conditions both at the free surface and on the ship hull are satisfied to second order, is derived. When sinkage and trim, incorporated into the present analysis, are ignored, this approximate solution is shown to be essentially equivalent to the method of Guilloton.

scholarly journals Generalized second-order slip for unsteady convective flow of a nanofluid: a utilization of Buongiorno’s two-component nonhomogeneous equilibrium model

2020 ◽  
Vol 9 (1) ◽  
pp. 156-168
Author(s):  
Seyed Mahdi Mousavi ◽  
Saeed Dinarvand ◽  
Mohammad Eftekhari Yazdi
Keyword(s):  
Boundary Layer ◽  
Equilibrium Model ◽  
Second Order ◽  
Model Parameters ◽  
Slip Parameter ◽  
Two Phase ◽  
Convective Boundary ◽  
First Order ◽  
Special Cases ◽  
Two Component

AbstractThe unsteady convective boundary layer flow of a nanofluid along a permeable shrinking/stretching plate under suction and second-order slip effects has been developed. Buongiorno’s two-component nonhomogeneous equilibrium model is implemented to take the effects of Brownian motion and thermophoresis into consideration. It can be emphasized that, our two-phase nanofluid model along with slip concentration at the wall shows better physical aspects relative to taking the constant volume concentration at the wall. The similarity transformation method (STM), allows us to reducing nonlinear governing PDEs to nonlinear dimensionless ODEs, before being solved numerically by employing the Keller-box method (KBM). The graphical results portray the effects of model parameters on boundary layer behavior. Moreover, results validation has been demonstrated as the skin friction and the reduced Nusselt number. We understand shrinking plate case is a key factor affecting non-uniqueness of the solutions and the range of the shrinking parameter for which the solution exists, increases with the first order slip parameter, the absolute value of the second order slip parameter as well as the transpiration rate parameter. Besides, the second-order slip at the interface decreases the rate of heat transfer in a nanofluid. Finally, the analysis for no-slip and first-order slip boundary conditions can also be retrieved as special cases of the present model.

A Theory of Substructure Modal Synthesis

1982 ◽  
Vol 49 (4) ◽  
pp. 903-909 ◽  
Author(s):  
K. Kubomura
Keyword(s):  
Degrees Of Freedom ◽  
Order Approximation ◽  
Second Order ◽  
Natural Mode ◽  
Frequency Range ◽  
Hybrid Mode ◽  
Second Order Approximation ◽  
First Order ◽  
Modal Synthesis ◽  
Modal Coordinates

A theory is presented for representing the displacements of a substructure finite-element mathematical model with a reduced number of degrees of freedom. A first or second-order approximation is used for the substructure’s modal coordinates associated with significantly larger or smaller eigenvalues than the system eigenvalues of excitation. The derived representations of the substructure displacements are capable of employing any type of substructure natural mode; free-free, cantilever or hybrid mode, and of retaining the dynamic behavior of any frequency range. It is shown that the present representations compute the system eigenvalues of interest with satisfactory accuracy, and it appears that the second-order approximation methods compute the system eigenvalues with greater accuracy than the first-order methods.

SECOND ORDER HIGH DENSITY EXPANSION FOR FREE ENERGY AND ORDER PARAMETER

1991 ◽  
Vol 05 (18) ◽  
pp. 2935-2949
Author(s):  
M. BARTKOWIAK ◽  
K.A. CHAO
Keyword(s):  
Free Energy ◽  
Order Parameter ◽  
Order Approximation ◽  
Local Approximation ◽  
Second Order ◽  
High Density ◽  
First Order ◽  
First Order Approximation ◽  
Consistent Equation ◽  
Density Expansion

The self-consistently renormalized high-density expansion (SHDE) is first used to determine temperature dependence of order parameter. Free energy and magnetization of the Ising model has been calculated to the second order. It is shown that the unphysical discontinuity of the order parameter as a function of temperature, which appears in the first-order approximation, still remains in the second-order calculation. Based on the 1/d expansion, we then construct a method to select (1/z)i contributions from the high density expansion terms. This method is applied to the first and second-order self-consistent equation for magnetization. Selection of the first order in 1/z contributions within the first order of the SHDE leads to considerable improvement of the behavior of magnetization as a function of temperature, and application of the local approximation to the second order of the SHDE term gives an acceptable single-value behavior of the order parameter.

Interaction of a Second-Order Solitary Wave With an Array of Four Vertical Cylinders

2006 ◽  
Author(s):  
A. Basmat ◽  
M. Markiewicz ◽  
S. Petersen
Keyword(s):  
Solitary Wave ◽  
Differential Equations ◽  
Order Approximation ◽  
Boussinesq Equations ◽  
Second Order ◽  
Nonlinearity Parameter ◽  
Wave Forces ◽  
Weakly Nonlinear ◽  
First Order ◽  
Vertical Cylinders

In this paper the interaction of a plane second order solitary wave with an array of four vertical cylinders is investigated. The fluid is assumed to be incompressible and inviscid. The diffraction analysis assumes irrotationality, which allows for the use of Boussinesq equations. A simultaneous expansion in a small nonlinearity parameter (wave amplitude/depth) and small dispersion parameter (depth/horizontal scale) is performed. Boussinesq models, which describe weakly nonlinear and weakly dispersive long waves, are characterized by the assumption that the nonlinearity and dispersion are both small and of the same order. An incident plane second order solitary wave is the Laitone solution of Boussinesq equations. The representation of variables as the series of small nonlinearity parameters leads to the sequence of linear boundary value problems of increasing order. The first order approximation can be determined as a solution of homogeneous differential equations and the second order approximation follows as a solution of non-homogeneous differential equations, where the right hand sides may be computed from the first order solution. For the case of a single cylinder an analytical solution exists. However, when dealing with more complex cylinder configurations, one has to employ numerical techniques. In this contribution a finite element approach combined with an appropriate time stepping scheme is used to model the wave propagation around an array of four surface piercing vertical cylinders. The velocity potential, the free surface elevation and the subsequent evolution of the scattered field are computed. Furthermore, the total second order wave forces on each individual cylinder are determined. The effect of the incident wave angle is discussed.

scholarly journals Development of FOPDT and SOPDT model from arbitrary process identification data using the properties of orthonormal basis function

2018 ◽  
Vol 7 (2.21) ◽  
pp. 77 ◽  
Author(s):  
Lalu Seban ◽  
Namita Boruah ◽  
Binoy K. Roy
Keyword(s):  
Basis Function ◽  
Delay Time ◽  
Orthonormal Basis ◽  
Control Point ◽  
Second Order ◽  
Step Test ◽  
Step Response ◽  
Order System ◽  
Model Parameters ◽  
First Order

Most of industrial process can be approximately represented as first-order plus delay time (FOPDT) model or second-order plus delay time (FOPDT) model. From a control point of view, it is important to estimate the FOPDT or SOPDT model parameters from arbitrary process input as groomed test like step test is not always feasible. Orthonormal basis function (OBF) are class of model structure having many advantages, and its parameters can be estimated from arbitrary input data. The OBF model filters are functions of poles and hence accuracy of the model depends on the accuracy of the poles. In this paper, a simple and standard particle swarm optimisation technique is first employed to estimate the dominant discrete poles from arbitrary input and corresponding process output. Time constant of first order system or period of oscillation and damping ratio of second order system is calculated from the dominant poles. From the step response of the developed OBF model, time delay and steady state gain are estimated. The parameter accuracy is improved by employing an iterative scheme. Numerical examples are provided to show the accuracy of the proposed method. 

Numerical Study of the Influence of Waving Bottom on the Fluid Surface Wave

2013 ◽  
Vol 275-277 ◽  
pp. 442-445
Author(s):  
Zheng Ren Wu ◽  
Chong Yuan Mo ◽  
Yong Xin Zhu
Keyword(s):  
Surface Wave ◽  
Multiple Scales ◽  
Numerical Study ◽  
Order Approximation ◽  
Approximate Equation ◽  
Second Order ◽  
First Order ◽  
Fluid Surface ◽  
Basic Equations ◽  
Multiple Scales Perturbation Method

The effect of waving bottom on the fluid surface wave was investigated. Starting from the basic equations of potential flow theory and boundary conditions, we used the multiple scales perturbation method to deduce fluid surface waves satisfy the first-order approximate equation and second-order approximate equation. Under the second-order approximation, the fluid surface waveform was simulated with the Matlab in the presence of different waving bottom form. The results show that there are three solitary waves on the surface of the fluid. With the development of time, the amplitude of each solitary wave has not changed. It seems that they are not affected each other and propagate independently. So it suggests that the waving bottom is effective for maintaining surface wave energy balance income and expenditure in spreading process.

Sign in / Sign up

Export Citation Format

Share Document