Finite‐difference and frequency‐wavenumber modeling of seismic monopole sources and receivers in fluid‐filled boreholes

Geophysics ◽  
1994 ◽  
Vol 59 (7) ◽  
pp. 1053-1064 ◽  
Author(s):  
A. L. Kurkjian ◽  
R. T. Coates ◽  
J. E. White ◽  
H. Schmidt
Keyword(s):  
Finite Difference ◽  
Guided Waves ◽  
Computational Cost ◽  
Seismic Source ◽  
Low Velocity ◽  
Imaging Geometry ◽  
Receiver Array ◽  
Single Well ◽  
Tube Wave ◽  
Effective Source

In borehole seismic experiments the presence of the borehole has a significant effect on observations. Unfortunately, including boreholes explicitly in modeling schemes excludes the use of some methods (e.g., frequency‐wavenumber) and adds prohibitively to the cost of others (e.g., finite difference). To overcome this problem, we use the concept of an effective source/receiver array to replace the explicit representation of the borehole by a distributed seismic source/receiver. This method mimics the presence of the borehole at seismic frequencies under a wide variety of conditions without adding a significant computational cost. It includes the effects of dispersive and attenuative tube wave propagation, the generation of secondary sources at interfaces and caliper changes, and the generation of conical waves in low‐velocity layers. Comparison with a finite‐difference scheme with an explicit borehole representation validates the approach. The modeling method applied to a continuity logging geometry demonstrates that the presence of guided waves does not uniquely imply bed connectivity. Results for a single‐well imaging geometry emphasize the dominance of the tube wave in the hydrophone synthetics and demonstrates the necessity of using clamped geophones for single‐well experiments. The concept of an effective source/receiver array is an efficient way of including borehole phenomena in seismic modeling methods at minimal extra computational cost.

Hybrid finite-difference integral equation solver for 3D frequency domain anisotropic electromagnetic problems

Geophysics ◽  
2011 ◽  
Vol 76 (2) ◽  
pp. F123-F137 ◽  
Author(s):  
M. Zaslavsky ◽  
V. Druskin ◽  
S. Davydycheva ◽  
L. Knizhnerman ◽  
A. Abubakar ◽  
...  
Keyword(s):  
Integral Equation ◽  
Finite Difference ◽  
Condition Number ◽  
Stiffness Matrix ◽  
Computational Cost ◽  
Integral Equation Method ◽  
Condition Numbers ◽  
Computational Domain ◽  
Electromagnetic Problems ◽  
Single Well

The modeling of the controlled-source electromagnetic (CSEM) and single-well and crosswell electromagnetic (EM) configurations requires fine gridding to take into account the 3D nature of the geometries encountered in these applications that include geological structures with complicated shapes and exhibiting large variations in conductivities such as the seafloor bathymetry, the land topography, and targets with complex geometries and large contrasts in conductivities. Such problems significantly increase the computational cost of the conventional finite-difference (FD) approaches mainly due to the large condition numbers of the corresponding linear systems. To handle these problems, we employ a volume integral equation (IE) approach to arrive at an effective preconditioning operator for our FD solver. We refer to this new hybrid algorithm as the finite-difference integral equation method (FDIE). This FDIE preconditioning operator is divergence free and is based on a magnetic field formulation. Similar to the Lippman-Schwinger IE method, this scheme allows us to use a background elimination approach to reduce the computational domain, resulting in a smaller size stiffness matrix. Furthermore, it yields a linear system whose condition number is close to that of the conventional Lippman-Schwinger IE approach, significantly reducing the condition number of the stiffness matrix of the FD solver. Moreover, the FD framework allows us to substitute convolution operations by the inversion of banded matrices, which significantly reduces the computational cost per iteration of the hybrid method compared to the standard IE approaches. Also, well-established FD homogenization and optimal gridding algorithms make the FDIE more appropriate for the discretization of strongly inhomogeneous media. Some numerical studies are presented to illustrate the accuracy and effectiveness of the presented solver for CSEM, single-well, and crosswell EM applications.

Elastic finite-difference modeling of cross-hole seismic data

1988 ◽  
Vol 78 (5) ◽  
pp. 1796-1806 ◽  
Author(s):  
Liang-Zie Hu ◽  
George A. McMechan ◽  
Jerry M. Harris
Keyword(s):  
Finite Difference ◽  
Seismic Data ◽  
Guided Waves ◽  
Fixed Time ◽  
Seismic Profile ◽  
Two Dimensional ◽  
Profile Data ◽  
Low Velocity ◽  
Vertical Seismic Profile ◽  
Surface Survey

Abstract Cross-hole seismic data exhibit unique characteristics not seen in surface survey data or even in vertical seismic profile data. These are, to a large extent, due to the near-horizontal propagation involved. Transmitted, reflected, evanescent, guided, and converted waves are all prominent; these require an elastic algorithm for realistic simulation. Elastic finite-differences are used to synthesize responses (both fixed-time snapshots and seismogram profiles) for a series of two-dimensional models of increasing complexity. Special emphasis is given to guided waves in continuous and segmented low-velocity zones.

An eigenfunction representation of deep waveguides with application to unconventional reservoirs

Geophysics ◽  
2021 ◽  
Vol 86 (6) ◽  
pp. T509-T521
Author(s):  
Owen Huff ◽  
Bin Luo ◽  
Ariel Lellouch ◽  
Ge Jin
Keyword(s):  
Finite Difference ◽  
Energy Distribution ◽  
Wave Energy ◽  
High Frequency ◽  
Guided Waves ◽  
Guided Wave ◽  
Low Velocity ◽  
Eagle Ford ◽  
Low Velocity Zone ◽  
Velocity Zone

Guided waves that propagate in deep low-velocity zones can be described using the displacement-stress eigenfunction theory. For a layered subsurface, these eigenfunctions provide a framework to calculate guided-wave properties at a fraction of the time required for fully numerical approaches for wave-equation modeling, such as the finite-difference approach. Using a 1D velocity model representing the low-velocity Eagle Ford Shale, an unconventional hydrocarbon reservoir, we verify the accuracy of the displacement eigenfunctions by comparing with finite-difference modeling. We use the amplitude portion of the Green’s function for source-receiver eigenfunction pairs as a proxy for expected guided-wave amplitude. These response functions are used to investigate the impact of the velocity contrast, reservoir thickness, and receiver depth on guided-wave amplitudes for discrete frequencies. We find that receivers located within the low-velocity zone record larger guided-wave amplitudes. This property may be used to infer the location of the recording array in relation to the low-velocity reservoir. We also study guided-wave energy distribution between the different layers of the Eagle Ford model and find that most of the high-frequency energy is confined to the low-velocity reservoir. We corroborate this measurement with field microseismic data recorded by distributed acoustic sensing fiber installed outside of the Eagle Ford. The data contain high-frequency body-wave energy, but the guided waves are confined to low frequencies because the recording array is outside the waveguide. We also study the energy distribution between the fundamental and first guided-wave modes as a function of the frequency and source depth and find a nodal point in the first mode for source depths originating in the middle of the low-velocity zone, which we validate with the same field data. The varying modal energy distribution can provide useful constraints for microseismic event depth estimation.

scholarly journals Deep learning for fast simulation of seismic waves in complex media

Solid Earth ◽  
2020 ◽  
Vol 11 (4) ◽  
pp. 1527-1549 ◽  
Author(s):  
Ben Moseley ◽  
Tarje Nissen-Meyer ◽  
Andrew Markham
Keyword(s):  
Deep Learning ◽  
Finite Difference ◽  
Seismic Response ◽  
Network Architecture ◽  
Seismic Waves ◽  
Computational Cost ◽  
Seismic Source ◽  
Spectral Element ◽  
Seismic Inversion ◽  
Order Of Magnitude

Abstract. The simulation of seismic waves is a core task in many geophysical applications. Numerical methods such as finite difference (FD) modelling and spectral element methods (SEMs) are the most popular techniques for simulating seismic waves, but disadvantages such as their computational cost prohibit their use for many tasks. In this work, we investigate the potential of deep learning for aiding seismic simulation in the solid Earth sciences. We present two deep neural networks which are able to simulate the seismic response at multiple locations in horizontally layered and faulted 2-D acoustic media an order of magnitude faster than traditional finite difference modelling. The first network is able to simulate the seismic response in horizontally layered media and uses a WaveNet network architecture design. The second network is significantly more general than the first and is able to simulate the seismic response in faulted media with arbitrary layers, fault properties and an arbitrary location of the seismic source on the surface of the media, using a conditional autoencoder design. We test the sensitivity of the accuracy of both networks to different network hyperparameters and show that the WaveNet network can be retrained to carry out fast seismic inversion in the same media. We find that are there are challenges when extending our methods to more complex, elastic and 3-D Earth models; for example, the accuracy of both networks is reduced when they are tested on models outside of their training distribution. We discuss further research directions which could address these challenges and potentially yield useful tools for practical simulation tasks.

An explicit method to calculate implicit spatial finite differences

Geophysics ◽  
2021 ◽  
pp. 1-71
Author(s):  
Hongwei Liu ◽  
Yi Luo
Keyword(s):  
Finite Difference ◽  
High Performance ◽  
Computational Cost ◽  
New Method ◽  
Seismic Exploration ◽  
Explicit Method ◽  
Lu Decomposition ◽  
Implicit Methods ◽  
Band Matrices ◽  
Explicit Finite Difference

The finite-difference solution of the second-order acoustic wave equation is a fundamental algorithm in seismic exploration for seismic forward modeling, imaging, and inversion. Unlike the standard explicit finite difference (EFD) methods that usually suffer from the so-called "saturation effect", the implicit FD methods can obtain much higher accuracy with relatively short operator length. Unfortunately, these implicit methods are not widely used because band matrices need to be solved implicitly, which is not suitable for most high-performance computer architectures. We introduce an explicit method to overcome this limitation by applying explicit causal and anti-causal integrations. We can prove that the explicit solution is equivalent to the traditional implicit LU decomposition method in analytical and numerical ways. In addition, we also compare the accuracy of the new methods with the traditional EFD methods up to 32nd order, and numerical results indicate that the new method is more accurate. In terms of the computational cost, the newly proposed method is standard 8th order EFD plus two causal and anti-causal integrations, which can be applied recursively, and no extra memory is needed. In summary, compared to the standard EFD methods, the new method has a spectral-like accuracy; compared to the traditional LU-decomposition implicit methods, the new method is explicit. It is more suitable for high-performance computing without losing any accuracy.

scholarly journals CMMSE: A technique for generating adapted discretizations to solve partial differential equations with the generalized finite difference method

2021 ◽  
Author(s):  
Augusto César Ferreira ◽  
Miguel Ureña ◽  
HIGINIO RAMOS
Keyword(s):  
Partial Differential Equations ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Differential Equations ◽  
Difference Method ◽  
Meshless Method ◽  
Computational Cost ◽  
Partial Differential

The generalized finite difference method is a meshless method for solving partial differential equations that allows arbitrary discretizations of points. Typically, the discretizations have the same density of points in the domain. We propose a technique to get adapted discretizations for the solution of partial differential equations. This strategy allows using a smaller number of points and a lower computational cost to achieve the same accuracy that would be obtained with a regular discretization.

Finite‐difference modeling of faults and fractures

Geophysics ◽  
1995 ◽  
Vol 60 (5) ◽  
pp. 1514-1526 ◽  
Author(s):  
Richard T. Coates ◽  
Michael Schoenberg
Keyword(s):  
Finite Difference ◽  
Seismic Wave ◽  
Computation Time ◽  
Anisotropic Materials ◽  
Model Material ◽  
Material Behavior ◽  
Interface Condition ◽  
Medium Theory ◽  
Difference Grid ◽  
Receiver Array

For the purposes of seismic propagation, a slip fault may be regarded as a surface across which the displacement caused by a seismic wave is discontinuous while the stress traction remains continuous. The simplest assumption is that this slip and the stress traction are linearly related. Such a linear slip interface condition is easily modeled when the fault is parallel to the finite‐difference grid, but is more difficult to do for arbitrary nonplanar fault surfaces. To handle such situations we introduce equivalent medium theory to model material behavior in the cells of the finite‐ difference grid intersected by the fault. Virtually identical results were obtained from modeling the fault by (1) an explicit slip interface condition (fault parallel to the grid) and (2) using the equivalent medium theory when the finite‐difference grid was rotated relative to the fault and receiver array. No additional computation time is needed except for the preprocessing required to find the relevant cells and their associated moduli. The formulation is sufficiently general to include faults in and between arbitrary anisotropic materials with slip properties that vary as a function of position.

Topology of ray surfaces in low-velocity zones

1979 ◽  
Vol 69 (2) ◽  
pp. 369-378
Author(s):  
George A. McMechan
Keyword(s):  
Guided Waves ◽  
Turning Points ◽  
Three Dimensional ◽  
Focal Depth ◽  
Low Velocity ◽  
Low Velocity Zone ◽  
Ray Trajectories ◽  
Definition Of ◽  
Single Set ◽  
Velocity Zone

abstract Plotting of three-dimensional ray surfaces in p-Δ-z space provides a means of determining p-Δ curves for any focal depth. A region of increasing velocity with depth is represented in p-Δ-z space by a trough, and a region of decreasing velocity, by a crest. Two sets of ray trajectories, the arrivals refracted outside a low-velocity zone, and the guided waves inside the zone, can be merged into a single set along the ray that splits into two at the top of the low-velocity zone. This ray is common to both sets. This construction provides continuity of the locus of ray turning points through the low-velocity zone and thus allows definition of p-Δ curves inside as well as outside the low-velocity zone.

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