Parameter Estimation in a Model for Tracer Transport in One-Dimensional Fractals - Selected Topics of Computational and Experimental Fluid Mechanics - page 232

Fig. 1 Physical description
of the fractal system with
uniform fluid flow

where ˆ is the porosity (added here to the Ostaja-Starzewski model) and α is the

fractal dimension of length along the x -direction with 0 <α ≤ 1. It is worth to

mention that the fluid can flow in the x -direction even if 1 since we have a

3D system, where the basic flowing units (fracture planes) can connect outside the
x -line. If we consider stationary fluid injection and a fluid velocity pointing in the
x -direction, whose magnitude depends on x only, i.e. u

α<

= u ( x ) dž x , the corresponding

fluid conservation equation is

1
c 1 ∇· (ˆˁ

u ) = S F . (3)

The tracer is introduced in the injection plate and extracted in the extraction plate.
An expression for the velocity can be derived by integrating it over a volume that
contains the fluid source and a part of fractal continuum. It results that

M F
A cs ˆˁ(

u x ( x ) =

) ,

(4)

x

where M F is the amount of fluid mass injected into the fractal porous medium per
unit of time, A cs is a cross section perpendicular to the x -direction in which the fluid
is injected,

ˆ is the porosity and ˁ( x ) is the fluid density. Note that for constant fluid

density the velocity is constant.

2.2 The Tracer Advection-Dispersion Equation

The tracer pulse dynamics is obtained from the conservation equation

∂ C

∂ t +∇·

J = S T . (5)

Here the tracer flux contains the advective part due to the process carrying fluid and
the dispersive part. This is J

= − ∇

C . According to percolation theory (Gefen
et al. 1983 ; Orbach 1986 ; Sahimi 1993 ), we introduce the dispersion coefficient as a

U C D