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

Development and Application of Variational Methods for Simulation of Miscible Displacement in Porous Media

1977 ◽  
Vol 17 (03) ◽  
pp. 228-246 ◽  
Author(s):  
A. Settari ◽  
H.S. Price ◽  
T. Dupont
Keyword(s):  
Finite Difference ◽  
Miscible Displacement ◽  
High Order ◽  
Numerical Dispersion ◽  
Convection Diffusion ◽  
Finite Difference Approximations ◽  
Variational Approximations ◽  
Difference Approximations ◽  
Low Dispersion ◽  
Finite Difference Techniques

Abstract Many reservoir engineering problems involve solving fluid flow equations whose solutions are characterized by sharp fronts and low dispersion levels. This is particularly important in tracking small slugs that are characteristic of chemical floods, polymer floods, first- and multiple-contact hydrocarbon miscible polymer floods, first- and multiple-contact hydrocarbon miscible displacements, and most thermal processes. The use of finite-difference approximations to solve these problems when low dispersion levels and small slugs need to be modeled accurately may be prohibitively expensive. This paper shows that the use of high-order variational approximations is a very effective means for economically solving these problems. This paper presents some numerical results that demonstrate that high-order variational methods can be used to solve two-dimensional reservoir engineering problems where finite-difference approximations would require 104 problems where finite-difference approximations would require 104 to 105 blocks. The variational solutions are shown to be essentially insensitive to grid orientation for unfavorable mobility ratios up to M = 100. Introduction The equations describing miscible displacement in a porous medium (convection-diffusion equations) are among the more difficult to solve by numerical means. The character of the concentration equation ranges from parabolic to almost hyperbolic depending on the ratio of convection to diffusion (Peclet number). Consequently, the finite-difference techniques developed for solving the convection diffusion problem can be divided into two categories: those solving the problem as parabolic and those treating the problem as hyperbolic. The parabolic techniques are unsatisfactory when the diffusion becomes small compared with the convection. The methods using central difference approximations for the convection terms oscillate. Price et al. have shown that these oscillations can be eliminated only by using small spatial increments. Methods using upstream difference approximations do not oscillate, but they introduce large truncation errors that have the character of a large diffusion term. Lantz has shown that for many practical problems, reducing the magnitude of numerical dispersion problems, reducing the magnitude of numerical dispersion sufficiently so that it does not mask the physical dispersion will force an impractically fine grid. Several improvements have been suggested, such as transfer of overshoots truncation-error cancellation, and two-point upstream approximations; but none of these is quite satisfactory in the general case. The hyperbolic methods (method of characteristics, point tracking, etc.) also pose many practical problems. These include the complex treatment required for sources and sinks, the need to redistribute points continually when modeling converging and diverging flow, the problem of maintaining a material balance, problems created by complex geometries, and the practical limitation problems created by complex geometries, and the practical limitation of the time-step size. Moreover, these schemes cannot be shown to converge, thereby making the choice of grid size and point distribution fairly arbitrary. Finally, many nonlinearities, such as reactions and adsorption, need to be treated point-by-point, requiring large amounts of computer time and storage. Because the major difficulty in solving the miscible displacement problem is the determination of an accurate approximation to a very sharp concentration front, one of the most promising alternatives to the schemes mentioned above is the use of promising alternatives to the schemes mentioned above is the use of high-order variational approximations, such as those proposed by Ciarlet et al. These methods (which include Galerkin and finite-element methods) are potentially far more accurate for a given amount of computation than the standard finite-difference techniques and, therefore, more able to solve problems that would otherwise be impractical. SPEJ P. 228

Solution of the Equations for Multidimensional, Two-Phase, Immiscible Flow by Variational Methods

1977 ◽  
Vol 17 (01) ◽  
pp. 27-41 ◽  
Author(s):  
A. Spivak ◽  
H.S. Price ◽  
A. Settari
Keyword(s):  
Finite Difference ◽  
Variational Methods ◽  
Piecewise Linear ◽  
Basis Functions ◽  
Two Dimensional ◽  
Immiscible Flow ◽  
Two Phase ◽  
One Dimensional ◽  
Linear Basis ◽  
Dependent Variables

Abstract This paper describes the solution of the equations for two-dimensional, two-phase, immiscible flow by variational methods. The formulation of the equations and the Galerkin procedure for solving the equations are given. procedure for solving the equations are given. The results of numerical experiments for one-dimensional, two-dimensional areal, and two-dimensional cross-sectional examples are presented. In each case, the results are compared with finite-difference solutions for the same problem. The ability to track sharp fronts is demonstrated by the variational approach. The time approximation used is shown to be stable for difficult problems such as converging flow and gas percolation. Also, the variational solution is shown to be percolation. Also, the variational solution is shown to be insensitive to grid orientation. Introduction In practical applications in the petroleum industry, the nonlinear, partial differential equations for fluid flow through a porous medium are currently solved almost exclusively by finite-difference methods. Variational or Galerkin (the terms are used interchangeably here) methods for solving these equations offer the potential advantage of higher-order accuracy at lower computational cost.This paper describes research on the solution of the equations for two-phase immiscible fluid flow using variational methods. The literature on the application of these methods to immiscible fluid flow is sparse. Douglas et al. describe solution of the one-dimensional immiscible displacement problem using cubic-spline basis functions and solving simultaneously for pressure and saturation as the dependent variables. They concluded pressure and saturation as the dependent variables. They concluded that the method was practical and that better answers are obtained with the same computational effort than by finite-difference methods. They also concluded that their choice of basis functions was probably not optimal. Verner et al. discuss the solution to the one-dimensional problem using "parabolic basis elements" (C degrees quadratic-basis problem using "parabolic basis elements" (C degrees quadratic-basis elements). Using the same data as was used by Douglas et al., they concluded that the parabolic, finite-element, spatial approximation gives results similar to the cubic splines for the same number of degrees of freedom. McMichael and Thomas solved the equations for three-phase, multidimensional immiscible flow. They solved simultaneously for the three-phase potentials as dependent variables. Although they stated that a general three-dimensional program with variable-basis function capability was developed, program with variable-basis function capability was developed, the examples they presented were two-dimensional areal. Also, piecewise linear basis (Chapeau) functions were used in their piecewise linear basis (Chapeau) functions were used in their example problems. The numerical experiments presented by McMichael and Thomas were limited to two relatively simple problems. They concluded that the Galerkin method requires significantly more work per time step than a finite-difference model, but that larger time steps could be taken. Vermuelen discussed the solution of the two-phase immiscible flow equations by simultaneously solving for the wetting- and nonwetting-phase pressures using a semi-implicit, first-order time approximation. Vermuelen's example problems used piecewise linear-basis functions. Based on one of these examples, piecewise linear-basis functions. Based on one of these examples, he concluded that the Galerkin technique appears to be less accurate than the finite-difference method for problems of water tongue displacement. In addition to the above work on two-phase immiscible flow through porous media, several authors have discussed the application of variational methods to miscible displacement problems and single-phase flow problems. SPEJ P. 27

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