An Accurate Numerical Technique for Solution of Convection-Diffusion Equations Without Numerical Dispersion

Abstract

Summary A technique is presented that optimizes the accuracy of numerical solutions to convection-diffusion equations while eliminating the detrimental effects of numerical dispersion. Such equations arise in the simulation of EOR processes where front smearing as a result of numerical dispersion can distort the sharp displacement fronts that are such an important part of the processes. Such front smearing can also lead to false predictions processes. Such front smearing can also lead to false predictions of breakthrough times. More accurate, and hence more reliable, simulation of oil recovery processes should facilitate the assessment of the economics of EOR processes. Our technique uses a parameterized, third order-correct algorithm and Fourier analysis. parameterized, third order-correct algorithm and Fourier analysis. The Fourier analysis allows for analytical solutions of both the differential equation and the algorithm. This enables the formulation of a concise method for requiring that the numerical solution conforms as closely as possible to the correct solution. The method is illustrated by performing the optimization on several examples with different degrees of convection and dispersion. The solutions have no numerical dispersion and little truncation error. Our method can serve as a starting point for generalization to the solution of more complicated equations that occur in multicomponent, multiphase reservoir simulators. Suggestions for such generalizations are presented. Introduction Virtually all reservoir simulators obtain solutions to fluid flow equations, usually nonlinear partial-differential equations, by making discrete approximations to derivatives. Whether finite-difference or finite-element methods are used, these approximations always introduce truncation errors that often can distort the accuracy and stability of the solution. The truncation error is often referred to as numerical dispersion because, to lowest order, it can be represented as a second spatial derivative term added to any true dispersion term in the problem. Distortion of the numerical solution is most significant in the simulation of EOR processes where sharp displacement, concentration, and/or temperature fronts are an important part of the efficiency of the processes, and artificial smearing as a result of numerical dispersion can render the simulation meaningless. For processes involving CO2, surfactant, polymer, or steam, it is also necessary to perform calculations of phase behavior, relative permeabilities, viscosities, interfacial tension, heat and mass transfer, and even chemical reactions at each mesh point and timestep. Therefore, resolving the sharp fronts merely by resorting to very fine meshes is impractical. Hence it is essential to be able to derive solution algorithms that have minimal truncation error for a given mesh size. A simple equation that exhibits many of the features of reservoir simulator equations is the one-dimensional (1D) linear convection-diffusion equation (1) where v is the convection rate (velocity) and D is the diffusion coefficient. This equation can be viewed as a good approximation to oil recovery by miscible displacement, immiscible displacement, or thermal methods. Therefore, most discussions of the effects of numerical dispersion have used Eq. 1 as a prototype, although some have preferred to work with specific simulator equations. Lantz was the first to quantify numerical dispersion in ID equations like Eq. His formulation was used by Peaceman to characterize numerical solutions to such equations and proved useful in interpreting the results of a simple chemical-flood model, where the predicted oil recovery was strongly dependent on gridblock size. Fanchi generalized Lantz's formulation to three dimensions. Larson, Jensen and Finlayson, and Laprea-Bigott and Morse used different forms of front-tracking schemes to resolve sharp fronts. Jensen and Finlayson actually used a moving coordinate system and finite-element methods in tracking fronts; the resultant moving boundary conditions were the price paid for the effective front tracking. price paid for the effective front tracking. SPERE p. 373

Modelling Physical Dispersion in Miscible Displacement-Part 1: Theory and the Proposed Numerical Scheme

“…Applying a Taylor series expansion of mole fractions at A, E, I about node 1 and retaining the first order terms gives: The following variables, for u = x, y, and z are now defined: The expression for mole fraction gradients from Equation (13) are inserted into the expression for dispersive velocities in Equation (12). The resulting expressions are incorporated into the constraint equations, Equations (14) - (16). The coefficients of mole fractions in the dispersive velocity equations are then re-arranged to form the following matrix equation: The numerical dispersive flux (in length/unit time) across interface A at the mid-point of the interface in the sub-region 1 is given by: ........................................................................... (23) For an interaction region located inside the reservoir, away from the boundaries, the mole fraction gradients from Equation (13) …”

“…Equations (14) - (16) will, in that case, become dispersive flux constraints equating dispersive flux on either side of the interaction interface; i.e. the equivalent equation for Equations (14) -(16) would be: (29) and so on.…”

“…A number of techniques (11)(12)(13)(14)(15)(16)(17)(18)(19)(20) to reduce the numerical dispersion, ranging from the addition of a negative dispersion term based on flow velocity and grid size, variably timed flux updating methods, high-order Godunov schemes, total variation diminishing (TVD) schemes to the method of characteristics, have appeared in the literature. These methods have had limited applicability to full field simulation of reservoirs with complex geometries.…”

A Novel Approach for Incorporating Physical Dispersion in Miscible Displacement

“…Techniques [16][17][18][19][20][21][22][23][24][25][26][27] for reducing numerical dispersion include the addition of a negative dispersion term based on flow velocity and grid size, variably timed flux updating methods, high-order Godunov schemes, total variation diminishing (TVD) schemes, solutions using the method of characteristics. These methods have had limited applicability to full field simulation of reservoirs with complex geometries.…”