Investigation of numerical dispersion with time step of the FDTD methods: avoiding erroneous conclusions

Abstract Numerical simulation of surfactant flooding using conventional reservoir simulation models can lead to unreliable forecasts and bad decisions due to the appearance of numerical effects. The simulations give approximate solutions to systems of nonlinear partial differential equations describing the physical behavior of surfactant flooding by combining multiphase flow in porous media with surfactant transport. The approximations are made by discretization of time and space which can lead to spurious pulses or deviations in the model outcome. In this work, the black oil model was simulated using the decoupled implicit method for various conditions of reservoir scale models to investigate behaviour in comparison with the analytical solution obtained from fractional flow theory. We investigated changes to cell size and time step as well as the properties of the surfactant and how it affects miscibility and flow. The main aim of this study was to understand pulse like behavior that has been observed in the water bank to identify cause and associated conditions. We report for the first time that the pulses occur in association with the simulated surfactant water flood front and are induced by a sharp change in relative permeability as the interfacial tension changes. Pulses are diminished when the adsorption rate was within the value of 0.0002kg/kg to 0.0005kg/kg. The pulses are absent for high resolution model of 5000 cells in x direction with a typical cell size as used in well-scale models. The growth or damping of these pulses may vary from case to case but in this instance was a result of the combined impact of relative mobility, numerical dispersion, interfacial tension and miscibility. Oil recovery under the numerical problems reduced the performance of the flood, due to large amounts of pulses produced. Thus, it is important to improve existing models and use appropriate guidelines to stop oscillations and remove errors.

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A New Unconditionally Stable Time Integration Method for Analysis of Nonlinear Structural Dynamics

Journal of Applied Mechanics ◽

10.1115/1.4007682 ◽

2013 ◽

Vol 80 (2) ◽

Cited By ~ 6

Author(s):

Ali Akbar Gholampour ◽

Mehdi Ghassemieh ◽

Mahdi Karimi-Rad

Keyword(s):

Time Integration ◽

Second Order ◽

Numerical Dispersion ◽

Computational Time ◽

Numerical Dissipation ◽

Integration Scheme ◽

Time Step ◽

Similar Time ◽

Unconditionally Stable ◽

Average Acceleration

A new time integration scheme is presented for solving the differential equation of motion with nonlinear stiffness. In this new implicit method, it is assumed that the acceleration varies quadratically within each time step. By increasing the order of acceleration, more terms of the Taylor series are used, which are expected to have responses with better accuracy than the classical methods. By considering this assumption and employing two parameters δ and α, a new family of unconditionally stable schemes is obtained. The order of accuracy, numerical dissipation, and numerical dispersion are used to measure the accuracy of the proposed method. Second order accuracy is achieved for all values of δ and α. The proposed method presents less dissipation at the lower modes in comparison with Newmark's average acceleration, Wilson-θ, and generalized-α methods. Moreover, this second order accurate method can control numerical damping in the higher modes. The numerical dispersion of the proposed method is compared with three unconditionally stable methods, namely, Newmark's average acceleration, Wilson-θ, and generalized-α methods. Furthermore, the overshooting effect of the proposed method is compared with these methods. By evaluating the computational time for analysis with similar time step duration, the proposed method is shown to be faster in comparison with the other methods.

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TWO EFFICIENT UNCONDITIONALLY-STABLE FOUR-STAGES SPLIT-STEP FDTD METHODS WITH LOW NUMERICAL DISPERSION

Progress In Electromagnetics Research B ◽

10.2528/pierb12103011 ◽

2013 ◽

Vol 48 ◽

pp. 1-22 ◽

Cited By ~ 2

Author(s):

Yong-Dan Kong ◽

Qing-Xin Chu ◽

Rong-Lin Li

Keyword(s):

Numerical Dispersion ◽

Unconditionally Stable ◽

Fdtd Methods

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Conformal FDTD-methods to avoid time step reduction with and without cell enlargement

Journal of Computational Physics ◽

10.1016/j.jcp.2007.02.002 ◽

2007 ◽

Vol 225 (2) ◽

pp. 1493-1507 ◽

Cited By ~ 31

Author(s):

Igor Zagorodnov ◽

Rolf Schuhmann ◽

Thomas Weiland

Keyword(s):

Cell Enlargement ◽

Time Step ◽

Fdtd Methods

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Computationally efficient methods for modelling laser wakefield acceleration in the blowout regime

Journal of Plasma Physics ◽

10.1017/s0022377812000517 ◽

2012 ◽

Vol 78 (4) ◽

pp. 469-482 ◽

Cited By ~ 20

Author(s):

B. M. COWAN ◽

S. Y. KALMYKOV ◽

A. BECK ◽

X. DAVOINE ◽

K. BUNKERS ◽

...

Keyword(s):

Initial Conditions ◽

Cylindrical Symmetry ◽

Small Time ◽

Numerical Dispersion ◽

Computationally Efficient ◽

Plasma Bubble ◽

Time Step ◽

Simulation Code ◽

Particle In Cell ◽

Comput Phys

AbstractElectron self-injection and acceleration until dephasing in the blowout regime is studied for a set of initial conditions typical of recent experiments with 100-terawatt-class lasers. Two different approaches to computationally efficient, fully explicit, 3D particle-in-cell modelling are examined. First, the Cartesian code vorpal (Nieter, C. and Cary, J. R. 2004 VORPAL: a versatile plasma simulation code. J. Comput. Phys.196, 538) using a perfect-dispersion electromagnetic solver precisely describes the laser pulse and bubble dynamics, taking advantage of coarser resolution in the propagation direction, with a proportionally larger time step. Using third-order splines for macroparticles helps suppress the sampling noise while keeping the usage of computational resources modest. The second way to reduce the simulation load is using reduced-geometry codes. In our case, the quasi-cylindrical code calder-circ (Lifschitz, A. F. et al. 2009 Particle-in-cell modelling of laser-plasma interaction using Fourier decomposition. J. Comput. Phys.228(5), 1803–1814) uses decomposition of fields and currents into a set of poloidal modes, while the macroparticles move in the Cartesian 3D space. Cylindrical symmetry of the interaction allows using just two modes, reducing the computational load to roughly that of a planar Cartesian simulation while preserving the 3D nature of the interaction. This significant economy of resources allows using fine resolution in the direction of propagation and a small time step, making numerical dispersion vanishingly small, together with a large number of particles per cell, enabling good particle statistics. Quantitative agreement of two simulations indicates that these are free of numerical artefacts. Both approaches thus retrieve the physically correct evolution of the plasma bubble, recovering the intrinsic connection of electron self-injection to the nonlinear optical evolution of the driver.

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Methods for Increased Accuracy in Numerical Reservoir Simulators

Society of Petroleum Engineers Journal ◽

10.2118/3516-pa ◽

1972 ◽

Vol 12 (06) ◽

pp. 515-530 ◽

Cited By ~ 83

Author(s):

M.R. Todd ◽

P.M. O'Dell ◽

G.J. Hirasaki

Keyword(s):

Relative Permeability ◽

Single Point ◽

Numerical Dispersion ◽

Time Step ◽

Time Step Size ◽

Point Approximation ◽

Grid Block ◽

Grid Orientation ◽

Grid Points ◽

Upstream Weighting

Abstract This paper proposed the use of two-point upstream weighting of fluid mobility as an alternative to the generally employed single-point approximation Use of the two-point formula results in the reduction of both numerical dispersion of flood fronts and the sensitivity of predicted areal displacement performance to grid orientation. Stability analysis performance to grid orientation. Stability analysis provides the time-step limitation for control of provides the time-step limitation for control of solution oscillations. This together with limitations for control of overshoot and truncation error provides a practical basis for the automatic selection of time steps. Introduction As an indication of the growing concern for controlling the total cost of large-scale reservoir simulations, the emphasis of a number of recent publications has been directed toward increasing publications has been directed toward increasing computing efficiency. In this paper, two methods to increase the computing efficiency of reservoir simulators are described. The use of two-point upstream weighting of fluid mobility is described and compared with the commonly used single-point upstream approximation. The two-point approximation generally requires fewer grid blocks to obtain a given accuracy than does the single-point approximation. In addition, the calculated performance of areal models is less sensitive to grid performance of areal models is less sensitive to grid orientation when using the two-point approximation. Computing efficiency is also increased with the use of an automatic time-step selector. Time-step limitations are described in this paper for controlling stability, overshoot (negative saturations), and truncation error. In general, these limitations change each time step as conditions change. If any of the limitations is exceeded, the results of the simulation may be meaningless. An automatic time-step selector detects and avoids running difficulties by using the proper time-step size. Using these methods, simulation proper time-step size. Using these methods, simulation results are obtained with less expenditure of engineering and computer time. TWO-POINT APPROXIMATIONS FOR FLUID MOBILITY The majority of general-purpose reservoir simulators reportedly in use today are based on the solution of finite-difference analogs to the conservation equations describing multiphase flow in porous media. Thus, the continuous domain of a reservoir is divided into a number of discrete blocks, and solutions for pressure and saturations are obtained at the grid block centers (or grid points). Central-difference approximations are normally used for the spatial derivatives in the discrete formulation of the conservation equations. As described below, this scheme necessitates the evaluation of flow coefficients (kk,/muB) at the planes separating adjacent grid blocks. As fluid and planes separating adjacent grid blocks. As fluid and reservoir properties are only defined at grid points, some method must be devised for approximating interblock flow coefficients based on values at the grid points. Of the terms that make up the flow coefficients, only the saturation-dependent relative permeability changes rapidly enough from grid block to grid block to cause significant difficulty. Although several weighting schemes have been employed in the past for evaluating the relative permeability at a block face, only single-point upstream weighting appears to be in general use. Unfortunately, use of this weighting scheme is well known to cause excessive numerical dispersion of flood fronts. In addition, areal displacement performance is found to be quite sensitive to the grid orientation for grid meshes of practical extent for large-scale reservoir simulations. This has been demonstrated qualitatively by Garrett and will be described both qualitatively and quantitatively later in this paper. As an alternative to single-point weighting of relative permeability, a two-point weighting of relative permeability, a two-point scheme is now described which results in both reduced numerical dispersion of flood fronts and decreased sensitivity of predicted areal performance to grid orientation. SPEJ P. 515

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