scholarly journals 2D TESTING OF A FINITE DIFFERENCE SCHEME FOR MODELLING OF A VISCOUS DIFFUSION PROCESS IN COMPRESSIBLE GAS

2020 ◽  
Vol 22 (5) ◽  
pp. 128-131
Author(s):  
V.V. Nikonov ◽  
Keyword(s):  
Numerical Simulation ◽  
Finite Difference ◽  
Initial Conditions ◽  
Stokes Problem ◽  
Velocity Fields ◽  
Two Dimensional ◽  
Dimensional Problem ◽  
Time Step ◽  
Acceptable Accuracy ◽  
Compressible Gas

Viscous subproblem of direct numerical simulation of compressible gas is solved. This subproblem is tested on the two-dimensional problem of impulse start of a flat plate (Stokes’ problem). Three calculations were made with the different initial conditions and velocity fields were obtained. The numerical results are compared with the solution of Stokes’ problem. Analyzing the results, we can conclude that in order to achieve acceptable accuracy, it suffices to choose a time step according to the rule that the author formulated in his earlier works.

scholarly journals A new fourth-order explicit group method in the solution of two-dimensional fractional Rayleigh–Stokes problem for a heated generalized second-grade fluid

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Muhammad Asim Khan ◽  
Norhashidah Hj. Mohd Ali ◽  
Nur Nadiah Abd Hamid
Keyword(s):  
Finite Difference ◽  
Stokes Problem ◽  
Stability And Convergence ◽  
Second Grade ◽  
Group Method ◽  
Two Dimensional ◽  
Second Grade Fluid ◽  
The Matrix ◽  
Grade Fluid ◽  
Generalized Second Grade Fluid

Abstract In this article, a new explicit group iterative scheme is developed for the solution of two-dimensional fractional Rayleigh–Stokes problem for a heated generalized second-grade fluid. The proposed scheme is based on the high-order compact Crank–Nicolson finite difference method. The resulting scheme consists of three-level finite difference approximations. The stability and convergence of the proposed method are studied using the matrix energy method. Finally, some numerical examples are provided to show the accuracy of the proposed method.

A Numerical Coning Model

1970 ◽  
Vol 10 (04) ◽  
pp. 418-424 ◽  
Author(s):  
J.P. Letkeman ◽  
R.L. Ridings
Keyword(s):  
Numerical Simulation ◽  
Finite Difference ◽  
Two Dimensions ◽  
Solution Technique ◽  
Production Rates ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Size Limitation ◽  
Single Well

Abstract The numerical simulation of coning behavior bas been one of the most difficult applications of numerical analysis techniques. Coning simulations have generally exhibited severe saturation instabilities in the vicinity of the well unless time-step sizes were severely restricted. The instabilities were a result of using mobilities based on saturations existing at the beginning of the time step. The time-step size limitation, usually the order of a few minutes, resulted in an excessive amount of computer time required to simulate coning behavior. This paper presents a numerical coning model that exhibits stable saturation and production behavior during cone formation and after breakthrough. Time-step sizes a factor of 100 to 1,000 times as large as those previously possible may be used in the simulation. To ensure stability, both production rates and mobilities are extrapolated production rates and mobilities are extrapolated implicitly to the new time level. The finite-difference equations used in the model are presented together with the technique for incorporating the updated mobilities and rates. Example calculations which indicate the magnitude of the time-truncation errors are included. Various factors which affect coning behavior are discussed. Introduction The usual formulation of numerical simulation models for multiphase flow involves the evaluation of flow coefficient terms at the beginning of a time step and assumes that these terms do not change over the time step. These assumptions are valid only if the values of pressure and saturation in the system do not change significantly over the time step. The design of a finite-difference model to evaluate coning behavior of gas or water in a single well usually results in a model which uses radial coordinates. A two-dimensional single-well model is illustrated in Fig. 1. This type of model will often produce finite-difference blocks with pore volumes less than 1 bbl near the wellbore while producing large blocks with pore volumes greater producing large blocks with pore volumes greater than 1 million bbl near the external radius. If one chooses to use a reasonable time-step size of, say, 1 to 10 days, then normal well rates would result in a flow of several hundred pore volumes per time step through blocks near the wellbore. Therefore the assumption that saturations remain constant, for the purpose of coefficient evaluation, is not valid. Welge and Weber presented a paper on water coning which recognized the limitation of using explicit coefficients and applied an arbitrary limitation on the maximum saturation change over a time step. While this method is workable for a certain class of problems, it is not rigorous and is not generally applicable. In 1968, Coats proposed a method to solve the gas percolation problem which is similar in that it also results from explicit mobilities. This proposal involved adjusting the relative permeability to gas at the beginning of the time step so that an individual block would not be over-depleted of gas during a time step. This method is not conveniently extended to two dimensions nor to coning problems where a block is voided many times during a time step. Blair and Weinaug explored the problems resulting from explicitly determined coefficients and formulated a coning model with implicit mobilities and a solution technique utilizing Newtonian iteration. While this method is rigorous, achieving convergence on certain problems is difficult and, in many cases, time-step size is still severely restricted. In addition to the problems resulting from explicit flow-equation coefficients in coning models, the specification of rates requires attention to ensure that the saturations remain stable in the vicinity of the producing block. SPEJ P. 418

scholarly journals A modification of the CABARET scheme for numerical simulation of multicomponent gaseous flows in two-dimensional domains

2015 ◽  
pp. 436-445
Author(s):  
А.В. Данилин ◽  
А.В. Соловьев ◽  
А.М. Зайцев
Keyword(s):  
Numerical Simulation ◽  
Initial Conditions ◽  
Experimental Studies ◽  
Two Dimensional ◽  
Governing Equations ◽  
Cabaret Scheme ◽  
Gaseous Flows ◽  
Multicomponent Gas ◽  
Good Agreement ◽  
Made In

Предложен явный численный алгоритм для расчета течений смесей идеальных газов в двумерных областях. Приведены физическая модель и уравнения движения смеси в консервативной и характеристической формах. Дискретизация уравнений движения произведена по методике Кабаре. Алгоритм испытан на задачах о прохождении ударной волны в воздухе через неоднородности из легкого и тяжелого газов, начальные условия для которых адаптированы из рассмотренных другими авторами натурных и численных экспериментов. Показано хорошее совпадение расчетов по предложенному алгоритму с результатами этих экспериментов. An explicit numerical algorithm for calculation of two-dimensional motion of multicomponent gas mixtures is proposed. A physical model as well as conservative and characteristic forms of governing equations are given. The discretization of the governing equations is made in accordance with the CABARET (Compact Accurately Boundary Adjusting-REsolution Technique) approach. The proposed algorithm is tested on problems of air shock waves passing through dense and dilute volume inhomogeneities with initial conditions adopted from numerical and experimental studies of other authors. A good agreement between the results of these studies and those obtained by the CABARET approach is shown.

scholarly journals Explicit Stencil Computation Schemes Generated by Poisson's Formula for the 2D Wave Equation

2019 ◽  
Author(s):  
Naum Khutoryansky
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Integral Representation ◽  
Initial Conditions ◽  
Representation Formula ◽  
Time Step ◽  
Stencil Computation ◽  
Integral Representation Formula ◽  
Time Marching ◽  
First Time

An approach to building explicit time-marching stencil computation schemes for the transient 2D acoustic wave equation without using finite-difference approximations is proposed and implemented. It is based on using the integral representation formula (Poisson's formula) that provides the exact solution of the initial-value problem for the transient 2D scalar wave equation at any time point through the initial conditions. For the purpose of constructing a two-step time-marching algorithm, a modified integral representation formula involving three time levels is also employed. It is shown that integrals in the two representation formulas are exactly calculated if the initial conditions and the sought solution at each time level as functions of spatial coordinates are approximated by stencil interpolation polynomials in the neighborhood of any point in a 2D Cartesian grid. As a result, if a uniform time grid is chosen, the proposed time-marching algorithm consists of two numerical procedures: 1) the solution calculation at the first time-step through the initial conditions; 2) the solution calculation at the second and next time-steps using a generated two-step numerical scheme. Three particular explicit stencil schemes (with five, nine and 13 space points) are built using the proposed approach. Their stability regions are presented. The obtained stencil expressions are compared with the corresponding finite-difference schemes available in the literature. Their novelty features are discussed. Simulation results with new and conventional schemes are presented for two benchmark problems that have exact solutions. It is demonstrated that using the new first time-step calculation procedure instead of the conventional one can provide a significant improvement of accuracy even for later time steps.

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

Sign in / Sign up

Export Citation Format

Share Document