scholarly journals A numerical simulation of the acoustic and elastic wavefields radiated by a source on a fluid‐filled borehole embedded in a layered medium

Geophysics ◽  
1993 ◽  
Vol 58 (4) ◽  
pp. 475-475 ◽  
Author(s):  
Michel Bouchon
Keyword(s):  
Layered Medium ◽  
Borehole Wall ◽  
Stoneley Wave ◽  
Wave Reflections ◽  
Volume Source ◽  
Linear System Of Equations ◽  
Method Of Calculation ◽  
Secondary Sources ◽  
Element Technique ◽  
Discrete Wavenumber Method

We present a method of calculation to simulate the propagation of acoustic and elastic waves generated by a borehole source embedded in a layered medium. The method is formulated as a boundary element technique where the Green’s functions are calculated by the discrete wavenumber method. The restrictive assumptions are that the borehole is cylindrical and that its axis runs normal to the layer interfaces. The physics of the method rely on Huygens’s principle that states that a diffracting boundary—the borehole wall in the present case—can be represented as a distribution of secondary sources. The borehole is discretized into small cylindrical elements and each element is represented by three sources: a volume source representing the wavefield diffracted in the fluid and two surface forces that give rise to the elastic wavefield radiated outside the borehole. The strength of each source is obtained by solving the linear system of equations that describes the boundary conditions at the borehole wall. The method is used to generate synthetic acoustic logs and to investigate the wavefield radiated into the formation. The simulations considered display the Stoneley wave reflections at the bed boundaries and show the importance of the diffraction that takes place where the borehole wall intersects the layer interfaces.

Full‐wave acoustic logging in an irregular borehole

Geophysics ◽  
1989 ◽  
Vol 54 (6) ◽  
pp. 758-765 ◽  
Author(s):  
Michel Bouchon ◽  
Denis P. Schmitt
Keyword(s):  
Wave Field ◽  
Boundary Integral ◽  
Borehole Diameter ◽  
Borehole Wall ◽  
Source Distribution ◽  
Small Scale ◽  
Stoneley Wave ◽  
Linear System Of Equations ◽  
Integral Equation Formulation ◽  
Elastic Rock

A new boundary integral equation formulation for wave propagation in a borehole of irregular cross‐section represents the wave field diffracted at the borehole‐rock interface by the radiation from a distribution of surface sources applied along the borehole wall. The wave fields in the borehole fluid and in the elastic rock are then expressed using the discrete wavenumber method. Application of boundary conditions at discretized locations along the borehole wall leads to a linear system of equations, whose inversion yields the required source distribution. We have used the method to investigate the effect of changes in borehole diameter on the pressure wave field inside the borehole. When the change is smooth, records obtained ahead of the discontinuity location are not affected by its presence. In the case of a steep variation, however, a significant amount of the Stoneley‐wave energy is reflected. When the borehole diameters are different at the source and receiver levels, the microseismograms obtained are somewhat of an average of those that would have been recorded in constant‐radius source and receiver boreholes, respectively. Small‐scale fluctuations in borehole diameter decrease the velocity of the Stoneley wave and of the pseudo‐Rayleigh wave.

Wave Propagation in a Two-Layered Medium

1964 ◽  
Vol 31 (2) ◽  
pp. 213-222 ◽  
Author(s):  
J. P. Jones
Keyword(s):  
Wave Propagation ◽  
Elastic Wave ◽  
Rayleigh Waves ◽  
Layered Medium ◽  
Elastic Wave Propagation ◽  
Flexural Wave ◽  
Stoneley Wave ◽  
Sh Waves ◽  
The Waves

Elastic wave propagation in a medium consisting of two finite layers is considered. Two types of solutions are treated. The first is a Rayleigh train of waves. It is seen that for this case, when the wavelength becomes short, the waves approach two Rayleigh waves plus a possible Stoneley wave. When the wavelength becomes large, there are two waves; i.e., a flexural wave and an axial wave. Calculations are presented for this case. The propagation of SH waves is treated, but no calculations are presented.

scholarly journals NUMERICAL CALCULATION OF SEICHE MOTIONS IN HARBOURS OF ARBITRARY SHAPE

1982 ◽  
Vol 1 (18) ◽  
pp. 11
Author(s):  
P. Gaillard
Keyword(s):  
Finite Element ◽  
Numerical Calculation ◽  
Arbitrary Shape ◽  
Wave Pattern ◽  
Offshore Structures ◽  
Experimental Investigations ◽  
Far Field ◽  
Method Of Calculation ◽  
Long Wave ◽  
Element Technique

A new method of calculation of wave diffraction around islands, offshore structures, and of long wave oscillations within offshore or shore-connected harbours is presented. The method is a combination of the finite element technique with an analytical representation of the wave pattern in the far field. Examples of application are given, and results are compared with other theoretical and experimental investigations.

scholarly journals Numerical simulation of flood waves and calculation of exerted forces on the cylindrical piers in contraction channels with different cross section profiles

2011 ◽  
Vol 14 (2) ◽  
pp. 366-385 ◽  
Author(s):  
Parviz Ghadimi ◽  
Arsham Reisinezhad
Keyword(s):  
Free Surface ◽  
Cross Section ◽  
Time Discretization ◽  
Surface Elevation ◽  
Wave Forces ◽  
Free Surface Elevation ◽  
Numerical Work ◽  
Linear System Of Equations ◽  
Element Technique ◽  
Narrow Section

A numerical model based on two-dimensional shallow water equations is presented. The depth-averaged velocity components with free-surface elevation have been used as independent variables in the model. The finite element technique is applied to discretize the spatial derivatives. Triangular elements with quadratic and linear interpolating functions are employed for two horizontal velocity components and the free-surface elevation, respectively. The standard Galerkin method is applied for discretization of the governing equations. Time discretization is performed using an implicit scheme. The resulting linear system of equations is solved by the GMRES method. The model is validated using three test cases and the results are compared with an analytical solution, the result of numerical work and experimental data, respectively. Favorable agreement was achieved in all three cases. Subsequently, the developed model is applied to simulate free-surface elevation through a channel contraction. The effects of width of the narrow section as well as the profile of the cross section of the channel on the wave forces exerted on a circular cylinder were studied. This was done in a channel with a quartic narrow section. Plots of time histories of the drag coefficient on the cylinder were produced, demonstrating the effects of the mentioned parameters.

A 3D MODEL TO SIMULATE VIBRATIONS IN A LAYERED MEDIUM WITH STOCHASTIC MATERIAL PARAMETERS

2011 ◽  
Vol 19 (02) ◽  
pp. 139-154 ◽  
Author(s):  
W. KREUZER ◽  
H. WAUBKE ◽  
G. RIECKH ◽  
P. BALAZS
Keyword(s):  
Stochastic System ◽  
3D Model ◽  
Layered Medium ◽  
Polynomial Chaos ◽  
Fourier Domain ◽  
Deterministic System ◽  
Linear System Of Equations ◽  
Material Parameters ◽  
The Fourier Transform ◽  
Present Simulation

A major problem for the simulation of the propagation of vibrations in ground layers is the fact that it is almost impossible to determine the material parameters needed for a numerical model exactly. In this work, we present a 3D model for layered soil, where in each layer the shear modulus is modeled as a stochastic process. Using the Karhunen Loeve expansion, the polynomial chaos expansion, and the Fourier transform, the stochastic system can be transformed into a linear system of equations in the wavenumber frequency domain. Unfortunately, the size of this system becomes very large and — contrary to a deterministic system — the stochastic system can no longer be decoupled for every wavenumber in the spatial Fourier domain. To solve this system efficiently, we propose an iteration procedure where the system is split into a deterministic and a stochastic part. As an external load on top of the ground, we use a vibrating box load moving along the x-axis. We discuss implementational details and present simulation results.

scholarly journals Numeric Simulation of Acoustic-Logging of Cave Formations

Energies ◽  
2020 ◽  
Vol 13 (15) ◽  
pp. 3908
Author(s):  
Fanghui Xu ◽  
Zhuwen Wang
Keyword(s):  
Excitation Frequency ◽  
Scattered Wave ◽  
P Wave ◽  
Borehole Wall ◽  
Stoneley Wave ◽  
S Wave ◽  
P Waves ◽  
Acoustic Logging ◽  
Source Distance ◽  
First Arrival

The finite difference (FD) method of monopole source is used to simulate the response of full-wave acoustic-logging in cave formations. The effect of the cave in the formation of borehole full-waves was studied. The results show that the radius of cave is not only linearly related to the first arrival of the compressional wave (P-wave), but also to the energy of the shear wave (S-wave). The converted S (S–S wave) and P-waves (S–P wave) are formed when the S-wave encounters the cave. If the source distance is small, the S–S and S–P waves are not separated, and the attenuation of the S-wave is not large, due to superposition of the converted waves. The S–P wave has been separated from the S-wave when the source distance is large, so the attenuation of the S-wave increases. The amplitude of the P and S–waves changes most when the distance of the cave to the borehole wall reaches a certain value; this value is related to the excitation frequency. The amplitude of the Stoneley wave (ST wave) varies directly with the radius of cave. If the radius of the cave is large, the energy of ST wave is weak. The scattered wave is determined by the radius and position of the cave. The investigation depth of a monopole source is limited. When the distance of the cave to the borehole wall exceeds the maximum investigation depth, the borehole acoustic wave is little affected by the cave. In actual logging, the development of the cave can be evaluated by using the first arrival of the P-wave and the energy of the S and ST waves.

The interaction of tube waves with borehole fractures, Part II: Analytical models

Geophysics ◽  
1998 ◽  
Vol 63 (3) ◽  
pp. 809-815 ◽  
Author(s):  
Sergio Kostek ◽  
David Linton Johnson ◽  
Kenneth W. Winkler ◽  
Brian E. Hornby
Keyword(s):  
Finite Difference ◽  
Analytical Models ◽  
Stoneley Wave ◽  
Single Fracture ◽  
Elastic Model ◽  
Formation Model ◽  
Wave Reflections ◽  
Fracture Width ◽  
Fracture Apertures

We develop a series of analytical models that can be used to interpret Stoneley‐wave reflections from fractures intersecting a borehole, thus facilitating the determination of effective fracture apertures from logs. The first model considers the combined effects of borehole enlargements (e.g., washouts) and fractures on the reflection coefficient of Stoneley waves. The result is expressed in terms of the washout volume, which can be obtained from a caliper log, as well as the fracture width. The predictions of this model are in excellent agreement with finite‐difference calculations. Next we develop an analytical elastic model that generalizes the rigid formation model. It also agrees with finite‐difference calculations. Finally, we establish the equivalence between a closely spaced multiple fracture and a permeable medium. In all the examples, the generalization of the result for the single fracture in a rigid formation can be accomplished with standard logging measurements.

Experimental studies of electrokinetic conversions in fluid‐saturated borehole models

Geophysics ◽  
1999 ◽  
Vol 64 (5) ◽  
pp. 1349-1356 ◽  
Author(s):  
Zhenya Zhu ◽  
Matthijs W. Haartsen ◽  
M. Nafi Toksöz
Keyword(s):  
Seismic Wave ◽  
Saturated Porous Medium ◽  
Electric Pulse ◽  
Experimental Studies ◽  
Ground Surface ◽  
Well Logging ◽  
Borehole Wall ◽  
Stoneley Wave ◽  
Apparent Velocity ◽  
Homogeneous Formation

Experimental and theoretical studies show that there are electromagnetic (EM) fields generated by seismic waves with two kinds of conversion mechanisms in a fluid‐saturated, porous medium. Within a homogeneous formation, the seismic wave generates a seismoelectric field that exists only in the area disturbed by the seismic wave and whose apparent velocity is that of the seismic wave. At an interface between differing formation properties, the generated seismoelectric wave is a propagating EM wave that can be detected everywhere. An electrode, used as a receiver on the ground surface, can detect the propagating EM wave generated at an interface, but cannot detect the seismoelectric field generated in a homogeneous formation. When the electrode is in a borehole and close to a porous formation, it can detect both the EM waves and the seismoelectric field. In this paper, electrokinetic measurements are performed with borehole models made of natural rocks or artificial materials. Experimental results show that the Stoneley wave and other acoustic modes, excited by a monopole source in the borehole models, generate seismoelectric fields in fluid‐saturated formations. The electric components of the seismoelectric fields can be detected by an electrode in the borehole or on the borehole wall. The amplitude and frequency of the seismoelectric fields are related not only to the seismic wave, but also to formation properties such as permeability, conductivity, etc. Comparison between the waveforms of the seismoelectric signals and acoustic logging waves suggests that seismoelectric well logging may explore the different properties of the formation. Electroseismic measurements are also performed with these borehole models. The electric pulse through the electrode in the borehole or on the borehole wall induces Stoneley waves in fluid‐saturated models that can be received by a monopole transducer in the same borehole. These measurement methods (seismoelectric logging or electroseismic logging) might directly apply to well logging to investigate formation properties related to the pore fluid flow.

scholarly journals Complex image method for calculating electric and magnetic fields produced by an auroral electrojet of finite length

1998 ◽  
Vol 16 (11) ◽  
pp. 1434-1444 ◽  
Author(s):  
R. Pirjola ◽  
A. Viljanen
Keyword(s):  
Electromagnetic Field ◽  
Current System ◽  
Image Method ◽  
Horizontal Line ◽  
Geomagnetically Induced Currents ◽  
Ionospheric Currents ◽  
Method Of Calculation ◽  
Secondary Sources ◽  
The Earth ◽  
Complex Image

Abstract. The electromagnetic field due to ionospheric currents has to be known when evaluating space weather effects at the earth's surface. Forecasting methods of these effects, which include geomagnetically induced currents in technological systems, are being developed. Such applications are time-critical, so the calculation techniques of the electromagnetic field have to be fast but still accurate. The contribution of secondary sources induced within the earth leads to complicated integral formulas for the field at the earth's surface with a time-consuming computation. An approximate method of calculation based on replacing the earth contribution by an image source having mathematically a complex location results in closed-form expressions and in a much faster computation. In this paper we extend the complex image method (CIM) to the case of a more realistic electrojet system consisting of a horizontal line current filament with vertical currents at its ends above a layered earth. To be able to utilize previous CIM results, we prove that the current system can be replaced by a purely horizontal current distribution which is equivalent regarding the total (=primary + induced) magnetic field and the total horizontal electric field at the earth's surface. The latter result is new. Numerical calculations demonstrate that CIM is very accurate and several magnitudes faster than the exact conventional approach.Key words. Electromagnetic theory · Geomagnetic induction · Auroral ionosphere

scholarly journals Borehole seismic‐source radiation in layered isotropic and anisotropic media: Boundary element modeling

Geophysics ◽  
1995 ◽  
Vol 60 (3) ◽  
pp. 735-747 ◽  
Author(s):  
Wenjie Dong ◽  
Michel Bouchon ◽  
M. Nafi Toksöz
Keyword(s):  
Boundary Element ◽  
Anisotropic Media ◽  
Seismic Source ◽  
Borehole Wall ◽  
Cement Interface ◽  
Secondary Sources ◽  
Source Radiation ◽  
Guided Modes ◽  
Borehole Seismic ◽  
Ti Media

In modeling waves radiated from a borehole seismic source in layered isotropic or anisotropic media, the commonly used numerical methods (e.g., finite difference and finite element) encounter difficulties because of the large scale difference between the borehole diameter and the formation extent. To get around this problem, we apply the indirect boundary element method to establish a general algorithm for modeling source radiation from open and cased boreholes in layered transversely isotropic (TI) media. The essence of the algorithm is to use discrete secondary sources (unknowns) on both sides of the borehole wall (formation/cement interface) to represent the influence of the interface on wave scattering, so that wave propagation inside and outside the borehole can be carried out by Green’s functions. The discrete distribution of the secondary sources is determined by matching boundary conditions on the borehole wall. Comparison with the discrete wavenumber method validates the implementation. Applications to fluid‐filled open and cased boreholes in three‐layer media demonstrate the creation of guided modes in low velocity layers. Presence of anisotropy complicates the guided modes as a result of dispersion and P‐ and SV‐waves coupling in homogeneous TI media. Presence of casing and cement enhances the visibility of the guided modes.

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