Methods for Increased Accuracy in Numerical Reservoir Simulators

1972 ◽  
Vol 12 (06) ◽  
pp. 515-530 ◽  
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

Analysis of the Sources and the Modal Content of the Acoustic Field in a Radial Compressor Outflow

2016 ◽  
Author(s):  
Marius C. Banica ◽  
Peter Limacher ◽  
Heinz-Jürgen Feld
Keyword(s):  
Experimental Data ◽  
Pressure Ratio ◽  
Single Point ◽  
Dipole Source ◽  
Grid Refinement ◽  
Step Size ◽  
Time Step ◽  
Vaned Diffuser ◽  
Radial Compressor ◽  
Time Step Size

In large modern turbochargers, compressors often constitute the main source of noise, with a frequency spectrum typically dominated by tonal noise at the blade passing frequency (BPF) and its harmonics. In transonic operation, inflow BPF noise is mainly generated by rotor locked shock fronts. These and the resulting acoustic fields can be predicted numerically with reasonable accuracy. Outflow noise, while also dominated by BPF tones, is linked to more complex source mechanisms. Its modal structure and the relationships between sources and modal sound pressure levels (SPL) are less well understood. Perhaps this is linked to the intrinsically non-axisymmetric geometries, which results in the need for full stage simulations if high accuracy is of paramount importance. In order to shed some light on outflow noise generation, a transient simulation of a 360° model of a radial compressor stage, including a vaned diffuser and a volute, was carried out using state-of-the-art CFD. Additionally, experimental data was gathered at a multitude of data points downstream of the volute exit for post processing and modal analysis. The sources and the propagation were calculated directly. Optimized values for tempo-spatial acoustic wave resolution and buffer layer design were chosen, based on extensive studies on simplified models. Two grid refinement levels were used to check grid convergence and time step size independence of the results was ensured. Numerical and experimental data match within 1% for total pressure ratio, volume flow and exit total temperature for the studied operating point. Both show the same modal content at the 1st BPF and indicate the presence of the same single dominating mode. The numerical results underpredict overall sound power levels (PWL) at the 1st BPF by 6.6dB. This difference is expected to decrease with further grid refinement and improved accounting for numerical damping. At the 2nd BPF, the experimental data show a significant broadening of the modal content with homogeneous modal PWL distributions. The multitude of modes leads to the generation of complex interference patterns, which shows that single-point acoustic measurements are often inadequate for component noise qualification and should be substituted by modal techniques. The dominating dipole sound sources are found in narrow areas around the vane leading edges and the rotor blade trailing edges. Because of the non-axisymmetric geometry, vane dipole source strengths become a function of circumferential position. The unsteady shedding of vortices from the vane suction surfaces is identified as a further possible source mechanism. However, the contributions of structural vibrations and mode scattering due to small manufacturing imperfections remain unclear.

Investigation on multi-body dynamics based approach to the toolpath generation for ultraprecision machining of freeform surfaces

2019 ◽  
Vol 234 (3) ◽  
pp. 571-583 ◽  
Author(s):  
Ali Khaghani ◽  
Kai Cheng
Keyword(s):  
Rotational Symmetry ◽  
Single Point ◽  
Equations Of Motion ◽  
Critical Role ◽  
Diamond Turning ◽  
Ultraprecision Machining ◽  
Time Step ◽  
Freeform Surfaces ◽  
Time Step Size ◽  
Toolpath Generation

This article presents an innovative approach to toolpath generation for ultraprecision machining of freeform optic surfaces based on the principle of Automatic Dynamics Analysis of Mechanical Systems. As components with freeform surfaces often have non-rotational symmetry, there are potential challenges facing their ultraprecision machining through single-point diamond turning, such as the projected points in complex large sag surfaces, which likely find it difficult to communicate with the control system and, thus, do not perform successfully. In ultraprecision machining, to achieve the highest performance in freeform surface resolution, the factors of dynamics, material and mechanical stiffness, frictions, tooling and accuracy of the servo component should be considered. The investigation is focused on an integrated approach and the associated scientific understanding of precision engineering design, ultraprecision machining and metrology of freeform surfaces as well as their application perspective. In this approach, the toolpath for very complex freeform surfaces can be generated using the Newton–Raphson method to solve the kinematics and dynamics equations of motion. The effect of friction and contact force are also investigated for accurate toolpath curve generation. Moreover, the Gear stiff (GSTIFF)/ Wielenga stiff (WSTIFF) integrator for solving the non-linear equations of motion is employed, and the result shows the time step size, playing a critical role in generating toolpath curves with a higher accuracy and resolution.

scholarly journals An Efficient Numerical Formulation for Wave Propagation in Magnetized Plasma Using PITD Method

Electronics ◽  
2020 ◽  
Vol 9 (10) ◽  
pp. 1575 ◽  
Author(s):  
Zhen Kang ◽  
Ming Huang ◽  
Weilin Li ◽  
Yufeng Wang ◽  
Fang Yang
Keyword(s):  
Wave Propagation ◽  
Time Domain ◽  
Fdtd Method ◽  
Magnetized Plasma ◽  
Numerical Dispersion ◽  
Step Size ◽  
Time Step ◽  
Precise Integration ◽  
Dispersion Error ◽  
Time Step Size

A modified precise-integration time-domain (PITD) formulation is presented to model the wave propagation in magnetized plasma based on the auxiliary differential equation (ADE). The most prominent advantage of this algorithm is using a time-step size which is larger than the maximum value of the Courant–Friedrich–Levy (CFL) condition to achieve the simulation with a satisfying accuracy. In this formulation, Maxwell’s equations in magnetized plasma are obtained by using the auxiliary variables and equations. Then, the spatial derivative is approximated by the second-order finite-difference method only, and the precise integration (PI) scheme is used to solve the resulting ordinary differential equations (ODEs). The numerical stability and dispersion error of this modified method are discussed in detail in magnetized plasma. The stability analysis validates that the simulated time-step size of this method can be chosen much larger than that of the CFL condition in the finite-difference time-domain (FDTD) simulations. According to the numerical dispersion analysis, the range of the relative error in this method is 10−6 to 5×10−4 when the electromagnetic wave frequency is from 1 GHz to 100 GHz. More particularly, it should be emphasized that the numerical dispersion error is almost invariant under different time-step sizes which is similar to the conventional PITD method in the free space. This means that with the increase of the time-step size, the presented method still has a lower computational error in the simulations. Numerical experiments verify that the presented method is reliable and efficient for the magnetized plasma problems. Compared with the formulations based on the FDTD method, e.g., the ADE-FDTD method and the JE convolution FDTD (JEC-FDTD) method, the modified algorithm in this paper can employ a larger time step and has simpler iterative formulas so as to reduce the execution time. Moreover, it is found that the presented method is more accurate than the methods based on the FDTD scheme, especially in the high frequency range, according to the results of the magnetized plasma slab. In conclusion, the presented method is efficient and accurate for simulating the wave propagation in magnetized plasma.

Direct Methods in Reservoir Simulation

1974 ◽  
Vol 14 (03) ◽  
pp. 295-308 ◽  
Author(s):  
H.S. Price ◽  
K.H. Coats
Keyword(s):  
Iterative Methods ◽  
Gaussian Elimination ◽  
Direct Solution ◽  
Step Size ◽  
Time Step ◽  
Property Variation ◽  
Time Step Size ◽  
Grid Points ◽  
And Storage ◽  
Efficiency And Reliability

Abstract During the past decade, efforts in reservoir modeling have focused on the three areas of capability efficiency, and reliability. Capability means the ability to handle larger and more complex problems where complexity includes physical problems where complexity includes physical phenomena, such as gas percolation and variable phenomena, such as gas percolation and variable PVT properties, and severe heterogeneity due to PVT properties, and severe heterogeneity due to property variation or geometry, or both. Efficiency property variation or geometry, or both. Efficiency is increased by improving model formulations anti solution techniques to increase tolerable time-step size and reduce computer time per time step. Reliability refers to ease of use and minimum burden in selecting or experimenting with time-step size, solution technique options, iteration parameters, and closure tolerances. parameters, and closure tolerances. The single facet of a reservoir simulator that has the greatest combined influence in all three categories is the technique used to solve the large systems of equations arising from the numerical approximation of the nonlinear fluid flow equations. Available techniques include both direct solution and iterative methods such as ADIP, SOR, and SIP. Iterative methods are currently used almost to the exclusion of direct solution because of the significantly higher computer storage and time requirements of the latter. This paper describes some new ordering schemes for Gaussian elimination that reduce computing time and storage requirements by factors as large as 6 and 3, respectively, relative to more standard orderings. Computational work estimates are given for these methods, for the standard Gaussian ordering, and for several iterative methods. These work estimates are checked by comparisons of actual run times using different solution techniques. Numerical examples are given to illustrate the increased efficiency and reliability that can be achieved in many cases through use of the new direct solution methods. Introduction It is well known that the way we number or order the unknowns of a sparse system of linear algebraic equations can drastically affect the amount of computation and storage for a direct solution. However, until recently the best ordering scheme that appeared in the literature numbered the points of a three-dimensional grid first along the shortest direction - i.e., the dimension with the fewest number of grid points - then in the next shortest direction, and finally in the longest direction. This ordering, which we shall call the standard ordering for Gaussian elimination is still widely used even though it is substantially slower than many other orderings. Ogbuobiri et al. present a survey of the literature related to ordering schemes that exploit matrix sparsity. These schemes are grouped into the two classes of matrix-banding schemes and optimal or pseudo-optimal schemes. The latter schemes pseudo-optimal schemes. The latter schemes purport to yield generally greater efficiency. purport to yield generally greater efficiency. In a recent paper, Georges has shown that for five-point difference approximations on square n x n two-dimensional grids, the total work for certain orderings of the grid points is less than C1n3 and the storage is less than C2n2 log n, compared with n4 and n3, respectively, for the standard ordering. Moreover, George has shown that no ordering scheme can require less work than the order of n3. For the special case of n - 21 he shows that work W is less than 10n3 and the storage S is less than 8ln2 for symmetric matrices. For nonsymmetric matrices these results become W less than 20n3 and S less than 16ln2, respectively. In this paper we describe some specific orderings in the matrix-banding class. Analyses of work and storage requirements are given for these orderings as applied to the diffusivity-type pressure equation that arises in reservoir simulation problems. These work and storage requirements are compared with those of the standard Gaussian ordering and of some iterative methods. These comparisons are performed for problems ranging from simple performed for problems ranging from simple homogeneous squares to practical reservoir problems of typical heterogeneity and irregular problems of typical heterogeneity and irregular geometry. The work requirements of the orderings presented here are also compared experimentally with those of one of the leading pseudo-optimal schemes. SPEJ P. 295

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^

scholarly journals Analysis of a Decoupled Time-Stepping Scheme for Evolutionary Micropolar Fluid Flows

2016 ◽  
Vol 2016 ◽  
pp. 1-13 ◽  
Author(s):  
S. S. Ravindran
Keyword(s):  
Micropolar Fluid ◽  
Stokes Equations ◽  
Fluid Model ◽  
Navier Stokes ◽  
Optimal Order ◽  
Step Size ◽  
Time Step ◽  
Order Error ◽  
Time Step Size ◽  
Time Stepping

Micropolar fluid model consists of Navier-Stokes equations and microrotational velocity equations describing the dynamics of flows in which microstructure of fluid is important. In this paper, we propose and analyze a decoupled time-stepping algorithm for the evolutionary micropolar flow. The proposed method requires solving only one uncoupled Navier-Stokes and one microrotation subphysics problem per time step. We derive optimal order error estimates in suitable norms without assuming any stability condition or time step size restriction.

FINITE ELEMENT METHODS FOR NONLINEAR ACOUSTICS IN FLUIDS

2007 ◽  
Vol 15 (03) ◽  
pp. 353-375 ◽  
Author(s):  
TIMOTHY WALSH ◽  
MONICA TORRES
Keyword(s):  
Finite Element ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Nonlinear Acoustics ◽  
Time Step ◽  
Partial Differential ◽  
Time Step Size ◽  
Finite Element Discretizations ◽  
Acoustic Fluid ◽  
Element Discretizations

In this paper, weak formulations and finite element discretizations of the governing partial differential equations of three-dimensional nonlinear acoustics in absorbing fluids are presented. The fluid equations are considered in an Eulerian framework, rather than a displacement framework, since in the latter case the corresponding finite element formulations suffer from spurious modes and numerical instabilities. When taken with the governing partial differential equations of a solid body and the continuity conditions, a coupled formulation is derived. The change in solid/fluid interface conditions when going from a linear acoustic fluid to a nonlinear acoustic fluid is demonstrated. Finite element discretizations of the coupled problem are then derived, and verification examples are presented that demonstrate the correctness of the implementations. We demonstrate that the time step size necessary to resolve the wave decreases as steepening occurs. Finally, simulation results are presented on a resonating acoustic cavity, and a coupled elastic/acoustic system consisting of a fluid-filled spherical tank.

On the Influence of Inflow Model Selection for Time-Domain Tiltrotor Aeroelastic Analysis

2021 ◽  
Author(s):  
Ethan Corle ◽  
Matthew Floros ◽  
Sven Schmitz
Keyword(s):  
Experimental Data ◽  
Particle Method ◽  
Comprehensive Analysis ◽  
Solution Stability ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Vortex Particle Method ◽  
Whirl Flutter ◽  
Vortex Particle

The methods of using the viscous vortex particle method, dynamic inflow, and uniform inflow to conduct whirl-flutter stability analysis are evaluated on a four-bladed, soft-inplane tiltrotor model using the Rotorcraft Comprehensive Analysis System. For the first time, coupled transient simulations between comprehensive analysis and a vortex particle method inflow model are used to predict whirl-flutter stability. Resolution studies are performed for both spatial and temporal resolution in the transient solution. Stability in transient analysis is noted to be influenced by both. As the particle resolution is refined, a reduction in simulation time-step size must also be performed. An azimuthal time step size of 0.3 deg is used to consider a range of particle resolutions to understand the influence on whirl-flutter stability predictions. Comparisons are made between uniform inflow, dynamic inflow, and the vortex particle method with respect to prediction capabilities when compared to wing beam-bending frequency and damping experimental data. Challenges in assessing the most accurate inflow model are noted due to uncertainty in experimental data; however, a consistent trend of increasing damping with additional levels of fidelity in the inflow model is observed. Excellent correlation is observed between the dynamic inflow predictions and the vortex particle method predictions in which the wing is not part of the inflow model, indicating that the dynamic inflow model is adequate for capturing damping due to the induced velocity on the rotor disk. Additional damping is noted in the full vortex particle method model, with the wing included, which is attributed to either an interactional aerodynamic effect between the rotor and the wing or a more accurate representation of the unsteady loading on the wing due to induced velocities.

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