The spreading of mass, as a fluid travel through a porous medium is caused by several contributing effects. Local variations in fluid velocity lead to mechanical mixing referred to as dispersion occurs because the fluid in the pore space flows around solid particles, so the velocity field varies within pore channels. The spreading in the direction parallel to the flow, or longitudinal dispersivity, typically exceeds the transverse dispersivity from up to an order of magnitude. Being driven by the concentration gradient alone, molecular diffusion rate is low relative to the mechanical dispersion, except at very low fluid velocities.

The dispersion is controlled through the dispersion tensor DD. The tensor components can either be given by user-defined values or expressions or derived from the directional dispersivities.

Using the longitudinal and transverse dispersivities in 2D, the dispersivity tensor components are (Ref. 9):

In these equations, DDii (SI unit: m2/s) are the principal components of the dispersivity tensor, and DDji and DDji are the cross terms. The parameters αL and αT (SI unit: m) specify the longitudinal and transverse dispersivities; and ui (SI unit: m/s) stands for the velocity field components.

In order to facilitate modeling of stratified porous media in 3D, the tensor formulation by Burnett and Frind (Ref. 10) can be used. Consider a transverse isotropic media, where the strata are piled up in the z direction, the dispersivity tensor components are:

In Equation 3-17 the fluid velocities u, v, and w correspond to the components of the velocity field u in the x, y, and z directions, respectively, and α1 (SI unit: m) is the longitudinal dispersivity. If z is the vertical axis, α2 and α3 are the dispersivities in the transverse horizontal and transverse vertical directions, respectively (SI unit: m). Setting α2 = α3 gives the expressions for isotropic media shown in Bear (Ref. 9 and Ref. 11).