FINITE‐DIFFERENCE RESISTIVITY MODELING FOR ARBITRARILY SHAPED TWO‐DIMENSIONAL STRUCTURES

Geophysics ◽  
1976 ◽  
Vol 41 (1) ◽  
pp. 62-78 ◽  
Author(s):  
Irshad R. Mufti
Keyword(s):  
Finite Difference ◽  
Potential Field ◽  
Line Source ◽  
Simulation Models ◽  
Iterative Solution ◽  
Single Step ◽  
Two Dimensions ◽  
Discrete Version ◽  
Grid Spacing ◽  
Source Dipole

Resistivity surveying is commonly done by using a point‐source dipole. Consequently, a finite‐difference evaluation of apparent resistivity curves implies the use of three‐dimensional simulation models which necessitate prohibitive computer costs. However, if we assume variation of resistivity only in two dimensions and use a line‐source dipole for setting up the finite‐difference model of a given structure, the potential field can be evaluated easily. A discrete version of the resistivity problem in two dimensions, which takes into account nonuniform grid spacing, is presented as a system of self‐adjoint difference equations. Since the iterative solution of such a system does not require grid spacing to be less than a certain critical value, it was successfully used for the development of fast‐convergence finite‐difference models. By examining in detail the characteristics of the matrix associated with the evaluation of the potential field, it is demonstrated that the proposed modeling procedure will remain stable for all conceivable geometries and resistivity distributions. It was used for the investigation of certain models for which the corresponding results could also be computed analytically. A direct superposition of results obtained in the two cases shows that they are virtually identical. By making use of the reciprocity theorem, a computational short‐cut, which provides the evaluation of vertical sounding curves for a line‐source dipole in a single step, is put forward. Special problems related to the optimization of acceleration parameters as well as the estimation of the potential function along the subsurface boundaries of the model are discussed. It is concluded that by surrounding the model by a termination strip of very large effective width, either Neumann‐ or Dirichlet‐type boundary conditions can be used for simulating a semiinfinite medium without introducing signficant errors in the results.

A PRACTICAL APPROACH TO FINITE‐DIFFERENCE RESISTIVITY MODELING

Geophysics ◽  
1978 ◽  
Vol 43 (5) ◽  
pp. 930-942 ◽  
Author(s):  
Irshad R. Mufti
Keyword(s):  
Finite Difference ◽  
Point Source ◽  
Potential Field ◽  
Line Source ◽  
Sparse Matrix ◽  
Analytical Data ◽  
Parallel Layer ◽  
The Matrix ◽  
The Given ◽  
Resistivity Modeling

Highly efficient finite‐difference resistivity modeling algorithms which yield accurate results are put forward. The given medium is discretized and divided into rectangular blocks by using a very coarse system of vertical and horizontal grid lines, whose distance from the source(s) increases logarithmically. Expressions are derived to compute the longitudinal conductance and transverse resistance associated with each of these blocks for a parallel‐layer medium followed by a generalized treatment to accommodate arbitrarily shaped structures. Since the values of Dar Zarrouk parameters are derived from the exact resistivity distribution of the given medium, fine features such as a thin but anomalously resistive bed which ordinarily would be missed entirely in coarse discretization can be taken into account. Further reduction in the size of the model is achieved by making use of a symmetry wherever possible. In most cases the computation of the potential field which involves the inversion of a small sparse matrix requires about 0.5 sec of computer time. Moreover, changes in geology affect neither the size nor the zero structure of the matrix. Therefore, when more than one model is to be computed, the factorization of the matrix can be done symbolically only once for all models, followed by numeric factorization for each individual model. The coarse grid algorithm was applied to a number of horizontally layered models involving a point source. The results obtained for each model were in excellent agreement with the corresponding analytical data. Finite‐difference investigation of the potential field for two‐dimensional structures and a line source dipole indicates that as long as one is interested only in the evaluation of the Schlumberger‐type apparent resistivity curves, the line‐source results may be a much better approximation to the corresponding point‐source data than is commonly believed.

A High-Accuracy Finite-Difference Technique for Treating the Convection-Diffusion Equation

1975 ◽  
Vol 15 (06) ◽  
pp. 517-531 ◽  
Author(s):  
D.D. Laumbach
Keyword(s):  
Finite Difference ◽  
Linear Form ◽  
Large Time ◽  
High Accuracy ◽  
Oscillatory Behavior ◽  
Two Dimensions ◽  
Grid Size ◽  
Grid Spacing ◽  
Time Step ◽  
Convection Diffusion

Abstract The convection-diffusion (C-D) equation arises from the conservation equations used in thermal recovery and miscible flooding. When convection predominates, this equation is very difficult to predominates, this equation is very difficult to represent numerically. The difficulty arises due to the hyperbolic character assumed as the Peclet number becomes large; consequently, the method presented here aims at providing the highest order presented here aims at providing the highest order of accuracy within this limit. The rationale underlying the treatment is to cancel a portion of the error in the convection term with that in the accumulation term. Thus, the technique presented is referred to as the truncation cancellation procedure (TCP). procedure (TCP). The application of this technique results in a new finite-difference representation of the C-D equation that is correct to the fourth order when the Peclet number approaches infinity. In this limit, when the dimensionless time step equals the dimensionless spatial increment, the discretization is exact. For very small time steps the method reduces to one previously considered to be one of the best semi-implicit discretizations of the C-D equation. For larger time steps, it yields significantly better results. The technique is applied to a linear form of the C-D equation in an equal-size grid block system; however, it is expected that the method should give favorable results in nonlinear and variable grid systems.The method is a semi-implicit one based on a three-point spatial and two-level time approximation. Thus, in one dimension, a set of difference equations is obtained that can be treated by solving a simple tridiagonal matrix. The extension of the TCP to two dimensions using a five-spatial-point star is demonstrated through an alternating-direction implicit solution method. The numerical results presented show that the TCP discretization yields presented show that the TCP discretization yields an excellent approximation in both Cartesian and radial coordinates. Comparisons with exact analytic solutions, conventional numerical techniques, and other high-accuracy numerical methods attest to the method's superiority over other formulations based on two time levels and three spatial locations. Introduction The purpose of this paper is to present a high-accuracy, finite-difference formulation for the convection-diffusion (C-D) equation. The discretization method presented is applied to a linear form of the C-D equation, with the expectation that the demonstrated improvements over existing methods will apply also to nonlinear forms, particularly those that are weakly nonlinear. The method is developed for equal grid spacing; however, it should give favorable results for variable grid spacing provided the grid size does not change too abruptly. provided the grid size does not change too abruptly. The difficulties associated with obtaining sufficient accuracy from conventional numerical representations of this equation were outlined quite clearly by Peaceman and Rachford. Improved methods for treating the C-D equation have been presented by Stone and Brian, Garder et al., presented by Stone and Brian, Garder et al., Price et al., Lantz, and Chaudhari. Price et al., Lantz, and Chaudhari. A significant improvement in the numerical treatment of the C-D equation was achieved by Stone and Brian. Unfortunately, the extension of this method to two dimensions has not been clear. Also, their method still possesses some oscillatory behavior in the vicinity of large gradients in the dependent variable when convection is strongly predominant This is particularly true for large time predominant This is particularly true for large time steps. The method proposed by Garder et al. uses the method of characteristics. Here the diffusion calculation is, in effect, an explicit one that consequently imposes a stability time-step limitation. Also, calculations are somewhat complicated by the moving points that must be tracked. Price et al. motivated a number of high-accuracy discretizations through the use of Galerkin's method. The discretization that they obtained through the use of chapeau basis functions is shown below to coincide with the recommended discretization of Stone and Brian. Consequently, this difference form suffers from the same oscillatory behavior in the vicinity of sharp fronts for large time steps. SPEJ P. 517

Controlling Numerical Dispersion by Variably Timed Flux Updating in Two Dimensions

1982 ◽  
Vol 22 (03) ◽  
pp. 409-419 ◽  
Author(s):  
R.G. Larson
Keyword(s):  
Finite Difference ◽  
Pressure Gradient ◽  
Single Point ◽  
Companion Paper ◽  
Two Dimensions ◽  
Independent Component ◽  
Numerical Dispersion ◽  
Coarse Mesh ◽  
Fixed Concentration ◽  
Order Of Magnitude

Abstract The variably-timed flux updating (VTU) finite difference technique is extended to two dimensions. VTU simulations of miscible floods on a repeated five-spot pattern are compared with exact solutions and with solutions obtained by front tracking. It is found that for neutral and favorable mobility ratios. VTU gives accurate results even on a coarse mesh and reduces numerical dispersion by a factor of 10 or more over the level generated by conventional single-point (SP) upstream weighting. For highly unfavorable mobility ratios, VTU reduces numerical dispersion. but on a coarse mesh the simulation is nevertheless inaccurate because of the inherent inadequacy of the finite-difference estimation of the flow field. Introduction A companion paper (see Pages 399-408) introduced the one-dimensional version of VTU for controlling numerical dispersion in finite-difference simulation of displacements in porous media. For linear and nonlinear, one- and two-independent-component problems, VTU resulted in more than an order-of-magnitude reduction in numerical dispersion over conventional explicit. SP upstream-weighted simulations with the same number of gridblocks. In this paper, the technique is extended to two dimensional (2D) problems, which require solution of a set of coupled partial differential equations that express conservation of material components-i.e., (1) and (2) Fi, the fractional flux of component i, is a function of the set of s - 1 independent-component fractional concentrations {Ci}, which prevail at the given position and time., the dispersion flux, is given by an expression that is linear in the specie concentration gradients. The velocity, is proportional to the pressure gradient,. (3) where lambda, in general, can be a function of composition and of the magnitude of the pressure gradient. The premises on which Eqs. 1 through 3 rest are stated in the companion paper. VTU in Two Dimensions The basic idea of variably-timed flux updating is to use finite-difference discretization of time and space, but to update the flux of a component not every timestep, but with a frequency determined by the corresponding concentration velocity -i.e., the velocity of propagation of fixed concentration of that component. The concentration velocity is a function of time and position. In the formulation described here, the convected flux is upstream-weighted, and all variables except pressure are evaluated explicitly. As described in the companion paper (SPE 8027), the crux of the method is the estimation of the number of timesteps required for a fixed concentration to traverse from an inflow to an outflow face of a gridblock. This task is simpler in one dimension, where there is only one inflow and one outflow face per gridblock, than it is in two dimensions, where each gridblock has in general multiple inflow and outflow faces. SPEJ P. 409^

PERFORMANCE LIMITS OF SIMULATION MODELS FOR NOISE CHARACTERIZATION OF MM WAVE DEVICES

2007 ◽  
Vol 07 (03) ◽  
pp. L299-L312
Author(s):  
ALI ABOU-ELNOUR
Keyword(s):  
Boltzmann Transport Equation ◽  
Simulation Models ◽  
Two Dimensions ◽  
Boltzmann Transport ◽  
Performance Limits ◽  
Drift Diffusion ◽  
Model Equations ◽  
Accuracy Of Results ◽  
Noise Characterization

Based on Boltzmann transport equation, the drift-diffusion, hydrodynamic, and Monte-Carlo physical simulators are accurately developed. For each simulator, the model equations are self-consistently solved with Poisson equation, and with Schrödinger equation when quantization effects take place, in one and two-dimensions to characterize the operation and optimize the structure of mm-wave devices. The effects of the device dimensions, biasing conditions, and operating frequencies on the accuracy of results obtained from the simulators are thoroughly investigated. Based on physical understanding of the models, the simulation results are analyzed to fully determine the limits at which a certain device simulator can be accurately and efficiently used to characterize the noise behavior of mm-wave devices.

Efficient Two-dimensional Acoustic Wave Finite-Difference Numerical Simulation in Strongly Heterogeneous Media Using the Adaptive Mesh Refinement (AMR) Technique

Geophysics ◽  
2021 ◽  
pp. 1-76
Author(s):  
Chunli Zhang ◽  
Wei Zhang
Keyword(s):  
Finite Difference ◽  
Acoustic Wave ◽  
Adaptive Mesh Refinement ◽  
Heterogeneous Media ◽  
Mesh Refinement ◽  
Adaptive Mesh ◽  
Grid Spacing ◽  
Time Step ◽  
Velocity Models ◽  
The Stability

The finite-difference method (FDM) is one of the most popular numerical methods to simulate seismic wave propagation in complex velocity models. If a uniform grid is applied in the FDM for heterogeneous models, the grid spacing is determined by the global minimum velocity to suppress dispersion and dissipation errors in the numerical scheme, resulting in spatial oversampling in higher-velocity zones. Then, the small grid spacing dictates a small time step due to the stability condition of explicit numerical schemes. The spatial oversampling and reduced time step will cause unnecessarily inefficient use of memory and computational resources in simulations for strongly heterogeneous media. To overcome this problem, we propose to use the adaptive mesh refinement (AMR) technique in the FDM to flexibly adjust the grid spacing following velocity variations. AMR is rarely utilized in acoustic wave simulations with the FDM due to the increased complexity of implementation, including its data management, grid generation and computational load balancing on high-performance computing platforms. We implement AMR for 2D acoustic wave simulation in strongly heterogeneous media based on the patch approach with the FDM. The AMR grid can be automatically generated for given velocity models. To simplify the implementation, we employ a well-developed AMR framework, AMReX, to carry out the complex grid management. Numerical tests demonstrate the stability, accuracy level and efficiency of the AMR scheme. The computation time is approximately proportional to the number of grid points, and the overhead due to the wavefield exchange and data structure is small.

GPU-based pseudo-transient finite difference solution for power-law viscous flow in cartesian, polar and spherical coordinates

2021 ◽  
Author(s):  
Emilie Macherel ◽  
Yuri Podladchikov ◽  
Ludovic Räss ◽  
Stefan M. Schmalholz
Keyword(s):  
High Resolution ◽  
Finite Difference ◽  
Viscous Flow ◽  
Power Law ◽  
High Performance ◽  
Viscoelastic Behavior ◽  
Iterative Solution ◽  
Circular Inclusion ◽  
Lithospheric Flexure ◽  
Gpu Architectures

<p>Power-law viscous flow describes well the first-order features of long-term lithosphere deformation. Due to the ellipticity of the Earth, the lithosphere is mechanically analogous to a shell, characterized by a double curvature. The mechanical characteristics of a shell are fundamentally different to the characteristics of plates, having no curvature in their undeformed state. The systematic quantification of the magnitude and the spatiotemporal distribution of strain, strain-rate and stress inside a deforming lithospheric shell is thus of major importance: stress is for example a key physical quantity that controls geodynamic processes such as metamorphic reactions, decompression melting, lithospheric flexure, subduction initiation or earthquakes.</p><p>Stress calculations in a geometrically and mechanically heterogeneous 3-D lithospheric shell require high-resolution and high-performance computing. The pseudo-transient finite difference (PTFD) method recently enabled efficient simulations of high-resolution 3-D deformation processes, implementing an iterative implicit solution strategy of the governing equations for power-law viscous flow. Main challenges for the PTFD method is to guarantee convergence, minimize the required iteration count and speed-up the iterations.</p><p>Here, we present PTFD simulations for simple mechanically heterogeneous (weak circular inclusion) incompressible 2-D power-law viscous flow in cartesian and cylindrical coordinates. The flow laws employ a pseudo-viscoelastic behavior to optimize the iterative solution by exploiting the fundamental characteristics of viscoelastic wave propagation.</p><p>The developed PTFD algorithm executes in parallel on CPUs and GPUs. The development was done in Matlab (mathworks.com), then translated into the Julia language (julialang.org), and finally made compatible for parallel GPU architectures using the ParallelStencil.jl package (https://github.com/omlins/ParallelStencil.jl). We may unveil preliminary results for 3-D spherical configurations including gravity-controlled lithospheric stress distributions around continental plateaus.</p>

Optimized Icosahedral Grids: Performance of Finite-Difference Operators and Multigrid Solver

2013 ◽  
Vol 141 (12) ◽  
pp. 4450-4469 ◽  
Author(s):  
Ross P. Heikes ◽  
David A. Randall ◽  
Celal S. Konor
Keyword(s):  
Finite Difference ◽  
Convergence Analysis ◽  
Optimization Methods ◽  
Difference Operators ◽  
Grid Spacing ◽  
Multigrid Solver ◽  
Poisson Equations ◽  
The Earth ◽  
Grid Optimization ◽  
Accuracy Performance

Abstract This paper discusses the generation of icosahedral hexagonal–pentagonal grids, optimization of the grids, how optimization affects the accuracy of finite-difference Laplacian, Jacobian, and divergence operators, and a parallel multigrid solver that can be used to solve Poisson equations on the grids. Three different grid optimization methods are compared through an error convergence analysis. The optimization process increases the accuracy of the operators. Optimized grids up to 1-km grid spacing over the earth have been created. The accuracy, performance, and scalability of the multigrid solver are demonstrated.

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