Controlling Numerical Dispersion by Variably Timed Flux Updating in One Dimension

1982 ◽  
Vol 22 (03) ◽  
pp. 399-408 ◽  
Author(s):  
R.G. Larson
Keyword(s):  
Finite Difference ◽  
Truncation Error ◽  
Hyperbolic Equations ◽  
Numerical Technique ◽  
Material Balance ◽  
Three Dimensions ◽  
Numerical Dispersion ◽  
Point Tracking ◽  
Multidimensional Problems ◽  
Grid Orientation

Abstract The one-dimensional (1D) material balance equations for multiphase multicomponent transport in porous media can be cast into forms, analogous to characteristic equations, that express explicitly the velocities at which fixed values of concentration are propagated. Use of these concentration-velocity equations to control the frequency with which component fluxes from finite-difference gridblocks ate updated leads to greatly reduced numerical dispersion, as demonstrated in miscible flooding, waterflooding, surfactant flooding, and other example problems. Introduction Accurate numerical simulation of enhanced oil-recovery processes, such as CO2, surfactant, thermal, or caustic flooding can involve calculations of phase behavior, interfacial tension, relative permeabilities, viscosities, heat and mass transfer, and even chemical reactions, thereby requiring considerable computational effort for each meshpoint or gridblock at each timestep. It is therefore impractical to resolve the steep concentration or thermal gradients often present in these processes by resorting to ultrafine meshes. Because the mathematical description of such processes is often unavoidably complex, it is important that the numerical technique be simple and ruggedly insensitive to the details of the process description, if one is to avoid becoming ensnarled in cumbersome and tedious programming and debugging.Although the finite-difference method's simplicity is its great advantage, its accuracy is seriously deficient, at least when one is using the simplest and most obvious discretizations. Central-difference discretization leads to artificial oscillations and overshoot, and upstream differencing leads to artificial smearing of sharp fronts-i.e., numerical dispersion or truncation error. Upstream difference solutions in two or three dimensions often show a significant dependence on grid orientation. Suggested improvements in the finite-difference technique, such as "transfer of overshoot," "truncation error analysis," or "two-point upstream weighting," still have significant numerical dispersion, grid orientation or oscillation errors.The method of characteristics, or point tracking, incurs no numerical dispersion or overshoot errors, but for general multicomponent, multidimensional problems, computer programs based on these techniques can become labyrinthine in their complexity.The finite-element, or variational, methods hold the potential of significantly reducing overshoot and/or numerical dispersion below that produced by finite difference, but implementation is considerably more complicated and time-consuming.The method of random choice, a technique developed for solving sets of multidimensional hyperbolic equations that appear in gas dynamics, recently has been employed in reservoir simulation. This method is somewhat akin to point tracking, propagating discontinuous fronts without smearing or overshoot errors.A new numerical technique is presented here that has the form and simplicity of finite difference, but utilizes variably timed flux updating (VTU) to gain a considerable improvement in accuracy. The technique is potentially applicable to general multicomponent, multidimensional problems. In this and a companion paper (see Pages 409-419), however, the technique is restricted to problems governed by the following equations. SPEJ P. 399^

Multidimensional Numerical Dispersion

1983 ◽  
Vol 23 (01) ◽  
pp. 143-151 ◽  
Author(s):  
John R. Fanchi
Keyword(s):  
Finite Difference ◽  
Dispersion Equation ◽  
Truncation Error ◽  
Nonlinear Partial Differential Equations ◽  
Three Dimensions ◽  
Numerical Dispersion ◽  
Constant Coefficients ◽  
Finite Difference Solution ◽  
Taylor’S Series ◽  
Difference Solution

Abstract Numerical dispersion can cause a smearing of otherwise sharp saturation fronts. The usual methods of estimating the magnitude of the smearing effect in one dimension (1D) are shown to apply in two and three dimensions (2 and 3D) as well. Besides the smearing effect, numerical dispersion affects the finite-difference solution of a multidimensional flow problem by rotating the principal flow axes. A method for determining the importance of the rotation effect is discussed. Numerical illustrations are included. Introduction Most reservoir simulation models available today obtain solutions to fluid flow equations-usually nonlinear partial differential equations-by replacing derivatives with finite-difference approximations. The use of these approximations, derived by manipulating Taylor's series, introduces an error known as truncation error. For many problems, the error is small and the approximate solutions of the subsequent finite-difference equations are sufficiently accurate for engineering purposes. However, truncation error can cause significant solution inaccuracies for certain types of problems. Examples of these problems include miscible floods and immiscible floods in which viscous forces are much larger than capillary forces. The most common example of the latter is the Buckley-Leverett problem with capillary pressure set to zero. A relatively simple equation that exhibits the effect of truncation error is the 1D convection-dispersion equation, (1) where the constant coefficients phi, K, and v are porosity, the dispersion coefficient, and velocity, respectively. The solution, S, of Eq. 1 may be saturation or concentration. The finite-difference solution of Eq. 1 introduces truncation error that can smear an otherwise sharp saturation front as if additional physical dispersion were present. This smearing, caused by truncating Taylor's series, often is called numerical dispersion or numerical diffusion. Truncation error studies often begin with a 1D convection-dispersion equation, such as Eq. 1, after the space and time coordinates (x and t) are redefined to remove two of the three constant coefficients (phi, K, and v). It appears that the effects of numerical dispersion in more than 1D have not been studied analytically, though attempts to minimize numerical dispersion in 2D -- particularly the grid orientation effect -- are numerous. All the attempts in more than 1D are empirical in the sense that the degree of success of the proposed numerical dispersion reduction method is determined relative to a case that does not include the proposed method. It is the purpose of this paper to present an analysis of the effects of numerical dispersion on the finite-difference numerical solution of the multidimensional convection-dispersion equation. The analytic results will provide a better understanding of the role that numerical dispersion plays in 2- or 3D; they can be used for estimating numerical dispersion effects, and they can provide a standard analytic basis for evaluating the degree of success of proposed numerical dispersion reduction methods. Analytical expressions for multidimensional numerical dispersion (MND) coefficients, derived by performing a truncation error analysis on the 3D convection-dispersion equation, are presented. This analysis is analogous to that used by Lantz in 1D. The significance of the results then is examined. SPEJ P. 143^

Reduction of Grid-Orientation Effects in Reservoir Simulation With Generalized Upstream Weighting

1985 ◽  
Vol 25 (06) ◽  
pp. 902-908 ◽  
Author(s):  
James C. Frauenthal ◽  
Roland B. di Franco ◽  
Brian F. Towler
Keyword(s):  
Finite Difference ◽  
Reservoir Simulation ◽  
Computation Time ◽  
Three Dimensions ◽  
Orientation Effect ◽  
Grid Line ◽  
Orientation Effects ◽  
Grid Orientation ◽  
Upstream Weighting ◽  
Grid Orientation Effect

Abstract A generalization of upstream weighting is proposed as a method for reducing grid-orientation effects in reservoir simulation. For the two sample problems studied,. a piston-flow waterflood and a realistic gas injection, the piston-flow waterflood and a realistic gas injection, the grid-orientation effect was almost completely eliminated. The new generalized upstream weighting (GUW) method is particularly attractive because it is fast and accurate, and particularly attractive because it is fast and accurate, and can be added easily to an existing simulator that uses upstream weighting. Introduction The grid-orientation effect is a well-known phenomenon in finite-difference reservoir simulation. Numerical results are highly dependent on the orientation of the finite-difference grid imposed on the model. In practice it occurs whenever one has a strongly adverse mobility ratio. This happens when one tries to push a viscous oil with a highly mobile fluid, such as steam or hydrocarbon gas. This paper presents a technique for reducing grid-orientation effects that is fast, flexible, and easily added to an existing simulator. A good survey of the research in this area was recently published. With this in mind, we will give an published. With this in mind, we will give an idiosyncratic interpretation of some of the techniques suggested by others. The main numerical difficulty in petroleum reservoir simulation is largely a consequence of the need to estimate individual phase mobilities halfway between finite-difference gridpoints. Because averaging the values from adjacent gridpoints is numerically unstable, the midgridpoint typically is assigned the value at the next upstream point. The idea of looking upstream for information point. The idea of looking upstream for information is found throughout much of computational fluid dynamics. Many improvements on one-point upstream weighting have been proposed in the reservoir simulation literature. The principal attractions of these techniques are that they can be interchanged easily within existing computer codes and do not add significantly to computation time. We found that the upstream weighting procedures have a common feature. If the midgridpoint in procedures have a common feature. If the midgridpoint in question lies, for example, on a grid line in the x direction, these techniques consider only other points on this same grid line in the extrapolation/interpolation process. A second body of literature developed around the idea of using a nine-point (instead of the standard five-point) finite-difference scheme to represent two-dimensional (2D) second derivatives. Because the nine-point scheme is a weighted superposition of two 5-point grids with a common center point and a 45 * relative rotation, the procedure averages away the grid-orientation effect to some extent without explaining it. Nevertheless, the nine-point grid schemes include one attractive feature absent from the upstream schemes: the weighting parameter can be tuned to improve the quality of the results. parameter can be tuned to improve the quality of the results. Perhaps the biggest fault of these procedures is that they Perhaps the biggest fault of these procedures is that they do not extend easily to three dimensions. The widening of the matrix bandwidth also increases the computation time. Our proposed technique is a modification of a procedure used successfully in the convective-heat transfer literature. It amounts to a generalization of one-point upstream weighting, accomplished by the introduction of mobility values from nearby points that lie in the true upstream direction rather than along a single grid line. This is explained in more detail in the next section. Note that the technique requires very little computer time. In fact, because most reservoir simulators use an automatic timestep adjustment, the improved stability of the technique, relative to standard upstream procedures, allows larger timesteps to be taken. Also, two adjustable parameters that permit the grid-orientation effect to be almost parameters that permit the grid-orientation effect to be almost completely eliminated are introduced. Finally, because the procedure works well with the standard five-point finite-difference representation of 2D second derivatives, it generates easily to three dimensions and is completely compatible with most reservoir simulators. Governing Equations The conservation equations for multiphase fluid flow in porous media are well known. However, the porous media are well known. However, the equations for three-phase flow are listed below for completeness. The continuity equations are as follows. SPEJ P. 902

A comparison of the dispersion relations for anisotropic elastodynamic finite-difference grids

Geophysics ◽  
2011 ◽  
Vol 76 (3) ◽  
pp. WA43-WA50 ◽  
Author(s):  
Henrik Bernth ◽  
Chris Chapman
Keyword(s):  
Finite Difference ◽  
Wave Equations ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Numerical Dispersion ◽  
Staggered Grid ◽  
Isotropic Case ◽  
Anisotropic Elastic ◽  
Memory Accesses ◽  
Average Dispersion

Several staggered grid schemes have been suggested for performing finite-difference calculations for the elastic wave equations. In this paper, the dispersion relationships and related computational requirements for the Lebedev and rotated staggered grids for anisotropic, elastic, finite-difference calculations in smooth models are analyzed and compared. These grids are related to a popular staggered grid for the isotropic problem, the Virieux grid. The Lebedev grid decomposes into Virieux grids, two in two dimensions and four in three dimensions, which decouple in isotropic media. Therefore the Lebedev scheme will have twice or four times the computational requirements, memory, and CPU as the Virieux grid but can be used with general anisotropy. In two dimensions, the rotated staggered grid is exactly equivalent to the Lebedev grid, but in three dimensions it is fundamentally different. The numerical dispersion in finite-difference grids depends on the direction of propagation and the grid type and parameters. A joint numerical dispersion relation for the two grids types in the isotropic case is derived. In order to compare the computational requirements for the two grid types, the dispersion, averaged over propagation direction and medium velocity are calculated. Setting the parameters so the average dispersion is equal for the two grids, the computational requirements of the two grid types are compared. In three dimensions, the rotated staggered grid requires at least 20% more memory for the field data and at least twice as many number of floating point operations and memory accesses, so the Lebedev grid is more efficient and is to be preferred.

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^

Synthetic reflection seismograms in three dimensions by a locked‐mode approximation

Geophysics ◽  
1989 ◽  
Vol 54 (3) ◽  
pp. 350-358 ◽  
Author(s):  
G. Nolet ◽  
R. Sleeman ◽  
V. Nijhof ◽  
B. L. N. Kennett
Keyword(s):  
Finite Difference ◽  
Born Approximation ◽  
Large Scale ◽  
Layered Medium ◽  
Three Dimensional ◽  
Simple Algorithm ◽  
Realistic Model ◽  
Three Dimensions ◽  
Acoustic Response ◽  
Many Sources

We present a simple algorithm for computing the acoustic response of a layered structure containing three‐dimensional (3-D) irregularities, using a locked‐mode approach and the Born approximation. The effects of anelasticity are incorporated by use of Rayleigh’s principle. The method is particularly attractive at somewhat larger offsets, but computations for near‐source offsets are stable as well, due to the introduction of anelastic damping. Calculations can be done on small minicomputers. The algorithm developed in this paper can be used to calculate the response of complicated models in three dimensions. It is more efficient than any other method whenever many sources are involved. The results are useful for modeling, as well as for generating test signals for data processing with realistic, model‐induced “noise.” Also, this approach provides an alternative to 2-D finite‐difference calculations that is efficient enough for application to large‐scale inverse problems. The method is illustrated by application to a simple 3-D structure in a layered medium.

Solution of Navier’s Equation in Cylindrical Curvilinear Coordinates

1978 ◽  
Vol 45 (4) ◽  
pp. 812-816 ◽  
Author(s):  
B. S. Berger ◽  
B. Alabi
Keyword(s):  
Fourier Transform ◽  
Finite Difference ◽  
Boundary Conditions ◽  
Half Space ◽  
Coordinate Transformation ◽  
Numerical Results ◽  
Three Dimensions ◽  
Curvilinear Coordinates ◽  
Rectangular Region

A solution has been derived for the Navier equations in orthogonal cylindrical curvilinear coordinates in which the axial variable, X3, is suppressed through a Fourier transform. The necessary coordinate transformation may be found either analytically or numerically for given geometries. The finite-difference forms of the mapped Navier equations and boundary conditions are solved in a rectangular region in the curvilinear coordinaties. Numerical results are given for the half space with various surface shapes and boundary conditions in two and three dimensions.

Finite Difference Time Domain Simulations of Hybrid Neurotransducers Based Optical Microlasers

2020 ◽  
Author(s):  
Maurizio Manzo ◽  
Omar Cavazos
Keyword(s):  
Finite Difference ◽  
Chronic Traumatic Encephalopathy ◽  
Numerical Technique ◽  
Well Being ◽  
Infrared Light ◽  
Neuron Activity ◽  
Living Matter ◽  
Brain Functions ◽  
Laser Modes ◽  
The Brain

Abstract Different pathologies such as Alzheimer’s, Parkinson’s, Wilson’s diseases, and chronic traumatic encephalopathy due to blasts and impacts affect the brain functions altering the neuronal electrical activity. An important aspect of the brain study is the use of non-invasive, non-surgical methodologies that are suitable to the well-being of the patients. Only a portion of the electromagnetic field can be detected by applying sensors outside the scalp; in addition, surgery is often involved if sensors are applied in the subcutaneous region of the skull. Optical techniques applied to biomedical research and diagnostics have been spread during the last decades. For example, near infrared light (NIR) of spectral range goes from 800 nm to 1300 nm, it is harmless radiation for the living tissue, and can penetrate the living matter in depth as, it turns out that most of the living matter is transparent to the NIR light. Optical microlasers have been recently proposed as neurotransducers for minimally invasive neuron activity detection for the next generation of brain-computer interface (BCI) systems. They are lightweight, require low power consumption and exhibit low latency. This novel sensor that can be made of biocompatible material is coupled with a voltage sensitive dye; the fluorescence of the dye, which is excited by an external light source, is used to generate optical (laser) modes. Any variation in the neurons’ membrane electric potential via evanescent field’s perturbation turn affect the shifting of these laser modes. In order to reduce the energy required to power these devices and to improve their optical emission, metal nanoparticles can be coupled in order to use their plasmonic effect. In this paper, finite-difference timedomain (FDTD) numerical technique is used to analyze the performances on a dye-doped microlaser. Purcell effect and resonant wavelengths are observed.

Fourth‐order finite‐difference P-SV seismograms

Geophysics ◽  
1988 ◽  
Vol 53 (11) ◽  
pp. 1425-1436 ◽  
Author(s):  
Alan R. Levander
Keyword(s):  
Free Surface ◽  
Finite Difference ◽  
Fourth Order ◽  
Equations Of Motion ◽  
Dispersion Analysis ◽  
Numerical Dispersion ◽  
Staggered Grid ◽  
Order System ◽  
Surface Boundary ◽  
Lamb’S Problem

I describe the properties of a fourth‐order accurate space, second‐order accurate time, two‐dimensional P-SV finite‐difference scheme based on the Madariaga‐Virieux staggered‐grid formulation. The numerical scheme is developed from the first‐order system of hyperbolic elastic equations of motion and constitutive laws expressed in particle velocities and stresses. The Madariaga‐Virieux staggered‐grid scheme has the desirable quality that it can correctly model any variation in material properties, including both large and small Poisson’s ratio materials, with minimal numerical dispersion and numerical anisotropy. Dispersion analysis indicates that the shortest wavelengths in the model need to be sampled at 5 gridpoints/wavelength. The scheme can be used to accurately simulate wave propagation in mixed acoustic‐elastic media, making it ideal for modeling marine problems. Explicitly calculating both velocities and stresses makes it relatively simple to initiate a source at the free‐surface or within a layer and to satisfy free‐surface boundary conditions. Benchmark comparisons of finite‐difference and analytical solutions to Lamb’s problem are almost identical, as are comparisons of finite‐difference and reflectivity solutions for elastic‐elastic and acoustic‐elastic layered models.

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