Theory of polarographic kinetic currents for second-order regeneration reactions at spherical electrodes. I. Numerical solution of finite-difference equations

1969 ◽  
Vol 41 (6) ◽  
pp. 737-746 ◽  
Author(s):  
D. T. Pence ◽  
Joseph R. Delmastro ◽  
G. L. Booman
Keyword(s):  
Finite Difference ◽  
Numerical Solution ◽  
Difference Equations ◽  
Second Order ◽  
Finite Difference Equations

Solution of the Immiscible Fluid Flow Simulation Equations

1969 ◽  
Vol 9 (02) ◽  
pp. 155-169 ◽  
Author(s):  
E.A. Breitenbach ◽  
D.H. Thurnau ◽  
H.K. Van Poolen
Keyword(s):  
Finite Difference ◽  
Numerical Solution ◽  
Difference Equations ◽  
Computing Time ◽  
Flow Simulation ◽  
Computational Technique ◽  
Immiscible Fluid ◽  
Finite Difference Equations ◽  
Gauss Elimination ◽  
Point Form

American Institute of Mining, Metallurgical and Petroleum Engineers, Inc. Abstract This paper presents the methods used to solve the finite difference equations which we developed in a companion paper (1). Various possible methods of solution are discussed. Experience has narrowed the numb of suitable numerical methods that are practical to three: Gauss elimination, successive practical to three: Gauss elimination, successive overrelaxation, and the iterative alternating direction implicit process. The final sections of the paper are devoted to a presentation of computational technique which are vital to actual use of each of the above-mentioned methods. FINITE DIFFERENCE EQUATIONS, THE MATRIX AND DEFINITIONS The final finite difference equation for pressure developed in Reference (1) is: pressure developed in Reference (1) is: ..........................................(1) All the terms are defined in the paper. Here, however, we have dropped the subscript denoting the pressure, p, as an oil pressure. Further breakdown requires definition of the numerical solution to be used. This paper describes the breakdown and solution processes most often used in the MUFFS program. Sufficient detail is given so that computer programming can be done. Contrary to popular opinion, economic simulation has been found to require the development of several solution methods, rather than relying on a single one. This requires that the computer subprogram for generating coefficients (A's and O's) be written as a distinct, separate entity to supply the coefficients in Equation (1). Furthermore, it is necessary to be able to obtain these coefficients automatically in column-by-column, row-by-raw, or point-by-point form, in any order required by a point-by-point form, in any order required by a numerical solution. Columns, rows, and points refer to the columns, rows, and points of the finite difference grid. A program that can generate coefficients in several forms is a simple but important concept, for it allows the easy insertion and modification of experimental methods. The computing inefficiencies that may be incurred within a general coefficient generator are small in comparison to the computing time saved by using the fastest of several solution techniques.

Pressure Drawdown and Buildup in the Presence of Radial Discontinuities

1967 ◽  
Vol 7 (03) ◽  
pp. 301-309 ◽  
Author(s):  
H.C. Bixel ◽  
H.K. Van Poollen
Keyword(s):  
Finite Difference ◽  
Numerical Solution ◽  
Difference Equations ◽  
Analytical Solutions ◽  
Sudden Change ◽  
Mathematical Treatment ◽  
Dimensionless Time ◽  
Transient Pressure ◽  
Reservoir Properties ◽  
Finite Difference Equations

Abstract A treatment is given of the transient pressure behavior of a well located at the center of a circular region surrounded by a radial discontinuity. On either side of the discontinuity, the values of permeability, viscosity, compressibility and porosity are uniform but may be different from those on the other side of the discontinuity. The results are obtained by solving a pair of finite difference equations. The numerical solution to these equations is obtained using a digital computer. The results are compared with previously published, approximated analytical solutions to the same problem. Solutions, presented graphically, show pressure decline for constant rate fluid production. The range of variables studied include dimensionless time from 0.001 to 100 and storage capacity ratio from 0.001 to 1,000. The well behaves as if it were in an infinite reservoir for dimensionless times less than 0.25. Reservoir properties near the well can be estimated in the usual manner. An overlay technique is used to match an experimental curve with one of the theoretical curves. It is possible to estimate the distance to a discontinuity by substituting the actual time t and the corresponding dimensionless time tD at which a match occurs into the equation a=(0.000264 tkI/tD fI µIcI)1/2, where kI/fIµIcI is the diffusivity near the well, and may be estimated from data taken at early time. Several buildup curves are computed. These curves show that for early shut-in times, correct values for transmissibility are obtained from conventional analysis. However, erroneous values of static reservoir pressure are obtained unless data at large shut-in times are taken. INTRODUCTION A mathematical treatment of the transient pressure response of a well located at the center of a region bounded by a circular discontinuity is given. Within a region (Fig. 20) the properties of both the rock and the fluid are considered to be constant on either side of the discontinuity. While these properties are considered constant, they may be different on opposite sides of the discontinuity. A discontinuity of this type could be a fluid-fluid contact or a sudden change in rock characteristics such as thickness, porosity or permeability. Analytical solutions to this problem are available.1-3 However, they are so involved that they are of little practical use. An approximate solution1 is available but the range of times over which it is valid has not been specified. A numerical solution to a pair of finite difference equations is used to obtain the solution given in this paper. PREVIOUS WORK One of the first solutions to the problem was published by William Hurst.1 Hurst considered unsteady-state flow of fluids through two sands in series with different mobilities in each sand. He used Laplace transforms to obtain a solution for a single well located at the center of the circle enclosing the first of the two sands. In this case, the solution for the pressure change in Region I (Fig. 20) is2Equation 1 for r a.

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