Some Practical Considerations in the Numerical Solution of Two-Dimensional Reservoir Problems

1968 ◽  
Vol 8 (02) ◽  
pp. 185-194 ◽  
Author(s):  
J.E. Briggs ◽  
T.N. Dixon
Keyword(s):  
Finite Difference ◽  
High Speed ◽  
Three Dimensional ◽  
Simultaneous Equations ◽  
Two Dimensional ◽  
Time Step ◽  
Finite Difference Approximations ◽  
Large Sets ◽  
Adi Methods ◽  
Three Dimensional Models

Abstract A study was made of numerical techniques for solving the large sets of simultaneous equations that arise in the mathematical modeling of oil reservoir behavior. It was found that noniterative techniques, such as the Alternating Direction Implicit (ADI) method, as well as some other finite difference approximations, produce oscillatory or unsmooth results for large time steps. Estimates of time step sizes sufficient to avoid such behavior are given. A comparison was made of the Point Successive Over-Relaxation (PSOR), Two-Line Cyclic Chebyshev Semi-Iterative SOR (2LCC), and iterative ADI methods, with respect to speed of solution of a test problem. It was found that, when applicable, iterative ADI is fastest for problems involving many points, while 2LCC is preferable for smaller problems. Introduction With the advent of high speed, large memory, digital computers, there has been an increasing emphasis on the development of improved methods for simulating and predicting reservoir performance. Two-dimensional, three-phase reservoir models with various combinations of PVT effects, as well as gravity and capillary forces, are common throughout the industry. Such models are also available through consulting firms, to anyone desiring to use them. Three-dimensional models will probably be practical in only a few years. We conducted a study of some of the numerical methods used for solving the large sets of simultaneous equations that arise in such models. A typical set of equations for a reservoir model is shown below: ..............(1) .............(2) .............(3) ..............(4) where a = 5.615 cu ft/bbl. In addition to Eqs. 1 through 4, one also would have to specify the conditions at the boundaries of the reservoir or aquifer being studied. Equations such as these are normally approximated by finite difference techniques and solved numerically because of their complexity. In deciding how to solve such equations, a number of decisions must be made. It is not our intention to cover all facets of the problem, but rather to concentrate on one of the important aspects, such as solving Eq. 1. SPEJ P. 185ˆ

Sight distance red zones on combined horizontal and sag vertical curves

1998 ◽  
Vol 25 (4) ◽  
pp. 621-630 ◽  
Author(s):  
Yasser Hassan ◽  
Said M Easa
Keyword(s):  
Quantitative Analysis ◽  
Three Dimensional ◽  
The Other ◽  
Two Dimensional ◽  
Horizontal Curve ◽  
Sight Distance ◽  
Stopping Sight Distance ◽  
Three Dimensional Models ◽  
Highway Alignment ◽  
Current Standards

Coordination of highway horizontal and vertical alignments is based on subjective guidelines in current standards. This paper presents a quantitative analysis of coordinating horizontal and sag vertical curves that are designed using two-dimensional standards. The locations where a horizontal curve should not be positioned relative to a sag vertical curve (called red zones) are identified. In the red zone, the available sight distance (computed using three-dimensional models) is less than the required sight distance. Two types of red zones, based on stopping sight distance (SSD) and preview sight distance (PVSD), are examined. The SSD red zone corresponds to the locations where an overlap between a horizontal curve and a sag vertical curve should be avoided because the three-dimensional sight distance will be less than the required SSD. The PVSD red zone corresponds to the locations where a horizontal curve should not start because drivers will not be able to perceive it and safely react to it. The SSD red zones exist for practical highway alignment parameters, and therefore designers should check the alignments for potential SSD red zones. The range of SSD red zones was found to depend on the different alignment parameters, especially the superelevation rate. On the other hand, the results showed that the PVSD red zones exist only for large values of the required PVSD, and therefore this type of red zones is not critical. This paper should be of particular interest to the highway designers and professionals concerned with highway safety.Key words: sight distance, red zone, combined alignment.

scholarly journals TSUNAMI GENERATION AND PROPAGATION

1972 ◽  
Vol 1 (13) ◽  
pp. 146
Author(s):  
Joseph L. Hammack ◽  
Frederic Raichlen
Keyword(s):  
Linear Theory ◽  
Source Region ◽  
Three Dimensional ◽  
Frequency Dispersion ◽  
Nonlinear Effects ◽  
Its Region ◽  
Experimental Results ◽  
Specific Class ◽  
Two Dimensional ◽  
Three Dimensional Models

A linear theory is presented for waves generated by an arbitrary bed deformation {in space and time) for a two-dimensional and a three -dimensional fluid domain of uniform depth. The resulting wave profile near the source is computed for both the two and three-dimensional models for a specific class of bed deformations; experimental results are presented for the two-dimensional model. The growth of nonlinear effects during wave propagation in an ocean of uniform depth and the corresponding limitations of the linear theory are investigated. A strategy is presented for determining wave behavior at large distances from the source where linear and nonlinear effects are of equal magnitude. The strategy is based on a matching technique which employs the linear theory in its region of applicability and an equation similar to that of Korteweg and deVries (KdV) in the region where nonlinearities are equal in magnitude to frequency dispersion. Comparison of the theoretical computations with the experimental results indicates that an equation of the KdV type is the proper model of wave behavior at large distances from the source region.

scholarly journals Quasi-two-dimensional approximation for numerical simulation of flows in engine ducts

2019 ◽  
Author(s):  
V. Vlasenko ◽  
A. Shiryaeva
Keyword(s):  
High Speed ◽  
Fuel Injection ◽  
Three Dimensional ◽  
Preliminary Design ◽  
Two Dimensional ◽  
Combustion Chambers ◽  
New Approach ◽  
One Dimensional ◽  
Dimensional Approximation ◽  
3D Flows

New quasi-two-dimensional (2.5D) approach to description of three-dimensional (3D) flows in ducts is proposed. It generalizes quasi-one-dimensional (quasi-1D, 1.5D) theories. Calculations are performed in the (x; y) plane, but variable width of duct in the z direction is taken into account. Derivation of 2.5D approximation equations is given. Tests for verification of 2.5D calculations are proposed. Parametrical 2.5D calculations of flow with hydrogen combustion in an elliptical combustor of a high-speed aircraft, investigated within HEXAFLY-INT international project, are described. Optimal scheme of fuel injection is found and explained. For one regime, 2.5D and 3D calculations are compared. The new approach is recommended for use during preliminary design of combustion chambers.

Numerical Methods of Higher-Order Accuracy for Diffusion- Convection Equations

1968 ◽  
Vol 8 (03) ◽  
pp. 293-303 ◽  
Author(s):  
H.S. Price ◽  
J.C. Cavendish ◽  
R.S. Varga
Keyword(s):  
Finite Difference ◽  
Boundary Conditions ◽  
Variational Methods ◽  
Miscible Displacement ◽  
Two Dimensional ◽  
Order Accuracy ◽  
One Dimensional ◽  
Finite Difference Approximations ◽  
Difference Approximations ◽  
Diffusion Convection

Abstract A numerical formulation of high order accuracy, based on variational methods, is proposed for the solution of multidimensional diffusion-convection-type equations. Accurate solutions are obtained without the difficulties that standard finite difference approximations present. In addition, tests show that accurate solutions of a one-dimensional problem can be obtained in the neighborhood of a sharp front without the need for a large number of calculations for the entire region of interest. Results using these variational methods are compared with several standard finite difference approximations and with a technique based on the method of characteristics. The variational methods are shown to yield higher accuracies in less computer time. Finally, it is indicated how one can use these attractive features of the variational methods for solving miscible displacement problems in two dimensions. Introduction The problem of finding suitable numerical approximations for equations describing the transport of heat or mass by diffusion and convection simultaneously has been of interest for some time. Equations of this type, which will be called diffusion-convection equations, arise in describing many diverse physical processes. Of particular interest here is the equation describing the process by which one miscible liquid displaces another liquid in a one-dimensional porous medium. The behavior of such a system is described by the following parabolic partial differential equation: (1) where the diffusivity is taken to be unity and c(x, t) represents a normalized concentration, i.e., c(x, t) satisfied 0 less than c(x, t) less than 1. Typical boundary conditions are given by ....................(2) Our interest in this apparently simple problem arises because accurate numerical approximations to this equation with the boundary conditions of Eq. 2 are as theoretically difficult to obtain as are accurate solutions for the general equations describing the behavior of two-dimensional miscible displacement. This is because the numerical solution for this simplified problem exhibits the two most important numerical difficulties associated with the more general problem: oscillations and undue numerical dispersion. Therefore, any solution technique that successfully solves Eq. 1, with boundary conditions of Eq. 2, would be excellent for calculating two-dimensional miscible displacement. Many authors have presented numerical methods for solving the simple diffusion-convection problem described by Eqs. 1 and 2. Peaceman and Rachford applied standard finite difference methods developed for transient heat flow problems. They observed approximate concentrations that oscillated about unity and attempted to eliminate these oscillations by "transfer of overshoot". SPEJ P. 293ˆ

Experimental Study on Gravity Driven Discharging of Quasi-Two-Dimensional Pebble Bed Based on Mathematical Morphology

2020 ◽  
Author(s):  
Yujia Liu ◽  
Sifan Peng ◽  
Nan Gui ◽  
Xingtuan Yang ◽  
Jiyuan Tu ◽  
...  
Keyword(s):  
Experimental Study ◽  
Mathematical Morphology ◽  
High Speed ◽  
Three Dimensional ◽  
Reactor Core ◽  
Engineering Application ◽  
Discharge Process ◽  
Two Dimensional ◽  
Pebble Bed ◽  
Gravity Driven

Abstract The pebbles flow is a fundamental issue for both academic investigation and engineering application in reactor core design and safety analysis. In general, experimental methods including spiral X-ray tomography and refractive index matched scanning technique (RIMS) are applied to obtain the identification of particles’ positions within a three-dimensional pebble bed. However, none of the above methods can perform global bed particles’ position identification in a dynamically discharging pebble bed, and the corresponding experimental equipment is difficult to access due to the complication and high expense. In this research, the experimental study is conducted to observe the gravity driven discharging process in the quasi two-dimensional silos by making use of the high-speed camera and the uniform backlight. A mathematical morphology-based method is applied to the pre-processing of the captured results. After being increased the gray value gradient by the threshold segmentation, the edges of the particles are identified and smoothed by the Sobel algorithm and the morphological opening operation. The particle centroid coordinates are identified according to the Hough circle transformation of the edges. For the whole pebble bed, the self-programmed process has a particle recognition accuracy of more than 99% and a particle centroid position deviation of less than 3%, which can accurately obtain the physical positions of all particles in the entire dynamically discharge process. By analyzing the position evolution of individual particles in consecutive images, velocity field and motion events of particles are observed. The discharging profiles of 5 conditions with different exit are analyzed in this experiment. The results make a contribution to improving the understanding of the mechanism of pebbles flow in nuclear engineering.

Three-Dimensional Mesh Model Watermarking Algorithm

2012 ◽  
Vol 198-199 ◽  
pp. 1481-1486
Author(s):  
Xin Li ◽  
Yi Ping Tian
Keyword(s):  
Wavelet Transform ◽  
Copyright Protection ◽  
Control Point ◽  
Three Dimensional ◽  
Digital Watermark ◽  
Two Dimensional ◽  
Mesh Model ◽  
Watermark Embedding ◽  
Three Dimensional Models ◽  
Watermark Algorithm

Watermark information is embedded in three-dimensional mesh model through three-dimensional watermarking algorithm for effective copyright protection. The widely use of three-dimensional grid model attracts more attention on the copyright protection. The digital watermark algorithm with the NURBS model based on the wavelet transform aims to get the virtual grayscale images using the control point coordinate. Then we can embed the watermark into the virtual gray image watermark. It can change the three-dimensional models into two-dimensional images. And this algorithm can enhance the operability and simplicity of the watermark embedding. Experiments show that the proposed algorithm is easy to implement, simple in principle, and the extracted watermark is clearly visible, moreover, the model does not need to be directly modified, so it has good robustness. Watermarked model does not change in the visual, it has good invisibility.

A Study of Spike and Modal Stall Phenomena in a Low-Speed Axial Compressor

1997 ◽  
Author(s):  
T. R. Camp ◽  
I. J. Day
Keyword(s):  
High Speed ◽  
Three Dimensional ◽  
Axial Compressor ◽  
Operating Conditions ◽  
Two Dimensional ◽  
Stability Criteria ◽  
Stall Inception ◽  
Low Speed ◽  
The Stability ◽  
Dimensional Instability

This paper presents a study of stall inception mechanisms a in low-speed axial compressor. Previous work has identified two common flow breakdown sequences, the first associated with a short lengthscale disturbance known as a ‘spike’, and the second with a longer lengthscale disturbance known as a ‘modal oscillation’. In this paper the physical differences between these two mechanisms are illustrated with detailed measurements. Experimental results are also presented which relate the occurrence of the two stalling mechanisms to the operating conditions of the compressor. It is shown that the stability criteria for the two disturbances are different: long lengthscale disturbances are related to a two-dimensional instability of the whole compression system, while short lengthscale disturbances indicate a three-dimensional breakdown of the flow-field associated with high rotor incidence angles. Based on the experimental measurements, a simple model is proposed which explains the type of stall inception pattern observed in a particular compressor. Measurements from a single stage low-speed compressor and from a multistage high-speed compressor are presented in support of the model.

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