Prediction of Molecular Diffusion At Reservoir Conditions. Part II - Estimating the Effects Of Molecular Diffusion And Convective Mixing In Multicomponent Systems
Authors: Sigmund Phillip M. 
Published: March 2010

Abstract
It is shown how multicomponent diffusion theory and the empirical corelation for binary diffusion described previously in Part I of this study may be used for predicting diffusion coefficients in multicomponent systems at reservoir conditions. It is also demonstrated how the effects of convection and diffusion may be combined to calculate concentration profiles for fluid mixtures flowing in a porous media.

Theory of Diffusion in Multicomponent Systems

The Continuity and Flux Equations
The fundamental equation used to express the composition change of each component in a diffusing mixture with respect to time and space is the continuity equation. In an n + 1 component mixture, this equation may be written for each of the independent components as (1, 2, 3).
Equation (1) Available In Full Paper.
At constant temperature and pressure, in a system in which all components have a constant partial molar volume, the flux equations which express diffusion rates with respect to a coordinate moving at the reference velocity u may written as(4, 5)
Equation (2) Available In Full Paper.
The reference velocity u may be taken as mass, molar or volume average(2, 5). For an n + 1 component system, equation (2) defines a (n × n) matrix of diffusion coefficients [D] of the type developed by Onsager(2, 10). The n2 elements Dlk of the matrix [D] are termed the multi component diffusion coefficients. The off-diagonal diffusion coefficients Dlk (i ≠k) have been termed the "cross diffusion coefficients" and the diagonal values (Dll) the "main diffusion coefficients"(2). The magnitudes of the cross coefficients, Dlk, are a measure of the coupling or interaction that takes place between the n + 1 diffusing species. It has been suggested(1, 17) that the effects of coupling can e conveniently taken into account, for some calculations, by defining an effective diffusion coefficient Dlm for each component such that
Equation (3) Available In Full Paper.
Methods of estimating the multicomponent diffusion coefficents Dlk and the effective diffusion coefficients Dlm are considered in subsequent discussions.

Concentration Distributions Resulting From Diffusion In Multicomponent Systems
If the values of Dlk can be estimated either by measurement or by theory, then equations (1) and (2) can be solved(2, 3). Analytic solutions can be obtained by assuming concentration independence of [D]. An examination of the errors which result from not accounting for concentration dependence has been made previously(1, 7). A thorough discussion of the linearized treatment of equations (1) and (2) used to calculate multicomponent concentration profiles appears elsewhere(2, 3). The solution for a ternary system in one dimension can be written as(3, 3, 8)
Equation (4a) Available In Full Paper.
Equation (4b) Available In Full Paper.
where
Equation (4c) Available In Full Paper.
and where Dk are the eigenvalues of the matrix [D]. For the ternary case they are given by(2, 8)
Equation (5a) Available In Full Paper.