scholarly journals Two-dimensional shallow flow equations for partially dry areas

2000 ◽  
Vol 36 (11) ◽  
pp. 3251-3264 ◽  
Author(s):  
Andrea Defina
Keyword(s):  
Two Dimensional ◽  
Shallow Flow ◽  
Flow Equations ◽  
Dry Areas

Treatment of Individual Wells and Grids in Reservoir Modeling

1968 ◽  
Vol 8 (04) ◽  
pp. 341-346 ◽  
Author(s):  
H.K. Van Poollen ◽  
E.A. Breitenbach ◽  
D.H. Thurnau
Keyword(s):  
Three Dimensional ◽  
Computer Time ◽  
Small Time ◽  
Economic Consequences ◽  
Productivity Index ◽  
Reservoir Modeling ◽  
Two Dimensional ◽  
Flow Equations ◽  
General Utility ◽  
Partially Penetrating Well

Abstract Reservoir modeling, mathematical modeling, or simulation of a petroleum or natural gas reservoir enables the engineer to examine and evaluate the physical a-nd economic consequences of various physical a-nd economic consequences of various alternative production policies. Approximations are inherent in all workable, economical simulators. This paper describes three workable, useful approximations. (1)a method to compare observed field pressures with those calculated by a numerical simulator, (2) a method to reduce three-dimensional problems to two space dimensions with pseudo-third-dimensional features, and (3) a method to calculate the productivity index (PI) and the water-oil ratio (WOR) in a partially penetrating well partially penetrating well These methods, although admittedly approximations, are workable and have been found to be very useful. Their general utility will, however, depend upon the extent to which any underlying assumptions used in their formulation apply to a particular problem. particular problem Introduction The objectives, applications and mathematical background of reservoir modeling have been described in other works. Ideally networks should be as shown in Fig. 1. Here, the grids are smaller near the wellbore than farther away. However, the number of grid points becomes large, even in a two-dimensional grid. Also, the small block sizes force one to use very small time steps, which can increase the computer time to the point of rendering the study economically unfeasible. Fig. 1 shows an example where the wells are located on a regular pattern. If that pattern becomes irregular enough, all cells pattern becomes irregular enough, all cells eventually will have to be small. In order to proceed with a study, modelers are forced to use linger grid sizes, as shown in Fig. 2. We realize that, by using large grid sizes, the fundamental flow equations are not truly represented. The network approaches a set of interconnected material balances with flow terms as a function of pressures and saturations. This paper describes the present method of handling wellbores in models with grid sizes many times the wellbore diameters. A method to compare pressures observed in the field with those calculated in the model is presented. A method also is given to reduce three- dimensional problems to two-dimensional grids. SPEJ P. 341

A rapidly convergent procedure for solving the equations of subsonic potential flow. I. Numerical solutions

1960 ◽  
Vol 255 (1280) ◽  
pp. 101-113 ◽  
Keyword(s):  
Potential Flow ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Local Density ◽  
Relaxation Method ◽  
Two Dimensional ◽  
Approximation Solution ◽  
Flow Equations ◽  
Bernoulli’S Equation ◽  
First Approximation

The subsonic potential flow equations for a perfect gas are transformed by means of dependent variables s = ( ρ / ρ 0 ) n q/ a 0 and σ = 1/2 In ( ρ 0 / ρ ), where q is the local velocity, ρ and a the local density and speed of sound, and the suffix 0 indicates stagnation conditions, n is a parameter which is to be chosen to optimize the approximations. Bernoulli’s equation then becomes a relation between s 2 and σ which is independent of initial conditions. A family of first-approximation solutions in terms of the incompressible solution is obtained on linearizing. It is shown that for two-dimensional flow, the choice n = 0∙5 gives results as accurate as those obtained with the Karman—Tsien solution. The exact equations are then transformed into the plane of the incompressible velocity potential and stream function and the first-approximation results substituted in the non ­linear terms. The resulting second-approximation equations can then be solved by a relaxation method and the error in this approximation estimated by carrying out the third-approximation solution. Results are given for a circular cylinder at a free-stream Mach number, M ∞ = 0∙4, and a sphere at M ∞ = 0∙5. The error in the velocity distribution is shown to be less than ±1 % in the two-dimensional case. A rough and ready compressibility rule is formulated for axisymmetric bodies, dependent on their thickness ratios.

scholarly journals A NOTE ON THE QUANTUM-MECHANICAL RICCI FLOW

2009 ◽  
Vol 24 (27) ◽  
pp. 4999-5006
Author(s):  
JOSÉ M. ISIDRO ◽  
J. L. G. SANTANDER ◽  
P. FERNÁNDEZ DE CÓRDOBA
Keyword(s):  
Quantum Mechanics ◽  
Configuration Space ◽  
Ricci Flow ◽  
Riemannian Metric ◽  
Quantum Mechanical ◽  
Two Dimensional ◽  
Conformally Flat ◽  
Flow Equations ◽  
Emergent Quantum Mechanics

We obtain Schrödinger quantum mechanics from Perelman's functional and from the Ricci-flow equations of a conformally flat Riemannian metric on a closed two-dimensional configuration space. We explore links with the recently discussed emergent quantum mechanics.

The shallow flow equations solved on adaptive quadtree grids

2001 ◽  
Vol 37 (6) ◽  
pp. 691-719 ◽  
Author(s):  
A. G. L. Borthwick ◽  
S. Cruz León ◽  
J. Józsa
Keyword(s):  
Shallow Flow ◽  
Flow Equations

Design Methods of Volute Casings for Turbocharger Turbine Applications

1996 ◽  
Vol 210 (2) ◽  
pp. 149-156 ◽  
Author(s):  
H Chen
Keyword(s):  
Experimental Data ◽  
Potential Flow ◽  
Design Method ◽  
Three Dimensional ◽  
Design Methods ◽  
Two Dimensional ◽  
Aerodynamic Design ◽  
3D Design ◽  
Flow Equations ◽  
Good Agreement

This paper discusses aerodynamic design methods of volute casings used in turbocharger turbines. A quasi-three-dimensional (Q-3D) design method is proposed in which a group of extended two-dimensional potential flow equations and the streamline equation are numerically solved to obtain the geometry of spiral volutes. A tongue loss model, based on the turbulence wake theory, is also presented, and good agreement with experimental data is shown.

von Neumann Stability Analysis of a Segregated Pressure-Based Solution Scheme for One-Dimensional and Two-Dimensional Flow Equations

2016 ◽  
Vol 138 (10) ◽  
Author(s):  
Santosh Konangi ◽  
Nikhil K. Palakurthi ◽  
Urmila Ghia
Keyword(s):  
Stability Analysis ◽  
Euler Equations ◽  
Two Dimensional ◽  
Von Neumann ◽  
One Dimensional ◽  
Flow Equations ◽  
Solution Scheme ◽  
Von Neumann Stability ◽  
The Stability ◽  
Stability Limits

The goal of this paper is to derive the von Neumann stability conditions for the pressure-based solution scheme, semi-implicit method for pressure-linked equations (SIMPLE). The SIMPLE scheme lies at the heart of a class of computational fluid dynamics (CFD) algorithms built into several commercial and open-source CFD software packages. To the best of the authors' knowledge, no readily usable stability guidelines appear to be available for this popularly employed scheme. The Euler equations are examined, as the inclusion of viscosity in the Navier–Stokes (NS) equation serves to only soften the stability limits. First, the one-dimensional (1D) Euler equations are studied, and their stability properties are delineated. Next, a rigorous stability analysis is carried out for the two-dimensional (2D) Euler equations; the analysis of the 2D equations is considerably more challenging as compared to analysis of the 1D form of equations. The Euler equations are discretized using finite differences on a staggered grid, which is used to achieve equivalence to finite-volume discretization. Error amplification matrices are determined from the stability analysis, stable and unstable regimes are identified, and practical stability limits are predicted in terms of the maximum allowable Courant–Friedrichs–Lewy (CFL) number as a function of Mach number. The predictions are verified using the Riemann problem, and very good agreement is obtained between the analytically predicted and the “experimentally” observed CFL values. The successfully tested stability limits are presented in graphical form, as compared to complicated mathematical expressions often reported in published literature. Since our analysis accounts for the solution scheme along with the full system of flow equations, the conditions reported in this paper offer practical value over the conditions that arise from analysis of simplified 1D model equations.

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