Characterization of mixing and spreading in a bounded stratified media

CHARACTERIZATION OF MIXING AND SPREADING IN A BOUNDED STRATIFIED MEDIUM.
Vanessa Zavala, Marco Dentz, Xavier Sánchez-Vila 13 marzo 2008

Contents
Objective. Basics.
Model. StochasticFramework. Transport coefficients.

Transport Coefficients in a bounded stratified medium.
Motivation. Results

Conclusions.

Objective
Point injection at y=0 50

l

25

2a
y

0

-25Investigate solute mixing and spreading in a vertically bounded random medium in terms of effective dispersion coefficients.
0 50 x
Uniform line injection 50

-50 -50

100

150

25

Weinvestigate two initial conditions: 1) Point like injection. 2) Uniform line injection.

0

y

-25

-50 -40

-20

0

20

40 x

60

80

100

120

Basics
•Advection dispersionequation. Homogeneous case:

Center of mass position
flow •Center of mass position:

•Boundary conditions: •Solution:

Instantaneous injection at t=0.

width

•Width:

Basics
Model.Point injection at y=0 50 25

Center of mass position

•We have a constant gradient: Je1. •Incompressibility condition •Exact Solution:

0

y

-25

e2

-50 -50

flow
0

width
50 x100 150

e1

?

Basics.
Stochastic Framework

We do not know the heterogeneity deterministically !!

Basics
Solution of the transport differential equation:
Solve it in terms of knownparameters:
Axial Moment Equations:

•Diffusion equation. Onedimensional projection of the advection dispersion equation.

Transport Coefficients
Point like injection
•Dentz and Carrera(2007): Defined effective dispersion coefficients for a single realization in terms of local moments.

Single realization

Ensemble

Uniform line injection

Transport Coefficients
Singlerealization

Ensemble

Transport Coefficients

Bounded stratified media
Why are we interested in stratified media models?
Effective and Ensemble Dispersion Coefficients
1000

100 DEFF DENS 10...