Abstract
Numerical diffusion (truncation error) can limit the usefulness of numerical finite-difference approximations to solve partial differential equations. Many reservoir simulation users are aware of these limitations but are not as familiar with actually quantifying the magnitude of the truncation error. This paper illustrates that, over a wide range of block size and time step, the truncation error expressions for convective-diffusion partial differential equations are quantitative. Since miscible, thermal, and immiscible processes can be of the convective-diffusion equation form, the truncation error expressions presented can provide guidelines for choosing block size-time step combinations that minimize the effect of numerical diffusion.
Introduction
Truncation error limits the use of numerical finite-difference approximations to solve partial differential equations. In the solution of convection-diffusion equations, such as occur in miscible displacement and thermal transport, truncation error results in an artificial dispersion term often denoted as numerical diffusion. The differential equations describing two-phase fluid flow can also be rearranged into a convection-diffusion form. And, in fact, miscible and immiscible differential equations have been shown to be completely analogous. In this form, it is easy to infer that numerical diffusion will result in an additional term resembling flow due to capillarity. Many users of numerical programs, and probably all numerical analysts, recognize that the magnitude of the numerical diffusivity for convection-diffusion equations can depend on both block size and time step. Most expressions developed in the literature have been used primarily to determine the order of the error rather than to quantify it. The primary purpose of this paper is to give the user more than just a qualitative feel for the importance of truncation error. In this paper, insofar as possible, analytical expressions for truncation error are compared by experiment to computed values for the numerical diffusivity. Consequently, the reservoir simulator user can observe that these expressions are quantitative and can use them as guidelines for choosing block sizes and time steps that keep the numerical diffusivity small.
DEVELOPMENT OF EXPRESSIONS FOR TRUNCATION ERROR
APPLICATION TO CONVECTION-DIFFUSION EQUATION
To illustrate the method of quantifying numerical diffusivity, consider a convective-diffusion equation of the form:
..............(1)
Symbols are defined in the Nomenclature. The first term on the right-hand side represents the diffusion, and the second term represents convection. Such an equation describes the flow of either a two-component miscible mixture or heat in one dimension with constant diffusivity.
EXPLICIT DIFFERENCE FORMS
An explicit expression for the truncation error (the space derivatives are approximated at a known time level) can be developed by examining the Taylor's series expansion representing first- and second-order derivatives. For the time derivative:
.....(2)
SPEJ
P. 315
Robust Low-Rank Discovery of Data-Driven Partial Differential Equations