Modeling of geochemical processes
                                      Reactive-Transport Models

                                             J. Faimon

                                 Modeling of geochemical processes
                                           Dynamic models

                                 Modeling of geochemical processes
                                           Dynamic models



Mass flux

The flux j[i] into the volume element V. The flux j[i] relates to an increment of content m[i] in
the volume element V:



                                 Modeling of geochemical processes
                                           Dynamic models

The increment of concentration dc[i] in the volume element dV in time dt equals to gradient of flux
j[i]^D in direction x:

                                 Modeling of geochemical processes
                                           Dynamic models

                                 Modeling of geochemical processes
                                           Dynamic models

The complete diffusion equation is:

                                 Modeling of geochemical processes
                                           Dynamic models

                                 Modeling of geochemical processes
                                           Dynamic models

                                 Modeling of geochemical processes
                                           Dynamic models

                                 Modeling of geochemical processes
                                           Dynamic models

The change in concentration due to

(1) transport,

(2) dispersion (diffusion) and

(3) reaction

                                 Modeling of geochemical processes
                                           Dynamic models

Diffusion and reaction 

                                 Modeling of geochemical processes
                                           Dynamic models

               Diffusion of aqueous Fe through alkaline solution in pore environment

                                 Modeling of geochemical processes
                                           Dynamic models

                                 Modeling of geochemical processes
                                           Dynamic models

                                 Modeling of geochemical processes
                                           Dynamic models

The analytical solution

The Gaussian function



where s is referred to as the spread or standard deviation and A is a constant

The function can be normalized

   yielding the normalized Gaussian:



The Error function



The relation between the normalized Gaussian

distribution and the error function equals:



A series approximation for small value of x

is given by:

                                 Modeling of geochemical processes
                                           Dynamic models

An approximate expression for large values of x can be obtained from:





The Complementary Error function

The complementary error function equals one minus the error function yielding:





which, combined with the series expansion of the error function listed above, provides approximate
expressions for small and large values of x:

                                 Modeling of geochemical processes
                                           Dynamic models

Numeric solution of diffusion equation

The Method of Finite Differential

Creation of a uniform net by discretization of the variables x and t

*       Let Dx and Dt are discrete steps on variables x and t, respectively. The variables x and t
are defined as   x = i Dx  and   t = j Dt, respectively.

*       The index i or j relates to a step number. i and j are 0,1,2...n!

*       The concentration  relates to a point in the two-dimensional net with coordinates x = iDx 
a  t = jDt.

                                 Modeling of geochemical processes
                                           Dynamic models

Differentiation with respect x (in the point j) can be replaced by finite difference:



                                                             and

                                 Modeling of geochemical processes
                                           Dynamic models

The complete diffusion equation is: