1. Advective Transport
Advection (convection) is the transport of dissolved or suspended material by motion of the host fluid.
Requires knowledge of the fluid velocity field (the velocity of a fluid particle)
In porous media, the pore velocity, or average linear velocity is the required.

2. Mean Section Velocity D t=L*A/Q L A Q L t=0 Pipe Segment: Volume = L*A
Flow Rate = Q
t=0.2 D t
Displacement of one segment
volume takes a certain time, Dt
t=0.4 D t
D t=L*A/Q
t=0.6 D t
Distance traveled by marker is segment length, L;
Marker velocity is distance/time
t=0.8 D t
t=1.0 D t

3. Average Linear VelocityQ L
t=0
Sand Filled Segment:
Segment Volume = L*A
Pore Volume = w*L*A
Flow Rate = Q
t=0.2Dt
Displacement of one pore
volume takes a certain time, Dt
t=0.4Dt
D t=wL*A/Q
t=0.6Dt
Distance traveled by marker is segment length, L;
Marker velocity is distance/time
t=0.8Dt
t=1.0Dt wL

4. Mass Flux wL Suppose one “blue” pore volume enters the sand pack. t=0
The mass of “blue” per unit volume is the concentration of blue.
Let one pore volume enter the sand pack.
Total mass of blue in the pore volume is the concentration*fluid volume
Rate of blue entering the sand pack is mass/time
t=0
t=0.5Dt
t=1.0Dt

5. Mass Balance = + Now consider a small portion of the sand pack. 1 2 DL
Mass flow into segment.
Mass flow out of segment.
Rate of accumulation in segment. = +

6. Balance Equations
For a non-deforming medium this mass balance is expressed as:
Substituting the definition of average linear velocity:
Taking the limit as DL vanishes produces the fundamental equation governing convective transport.

7. 3D Generalization
Observe the obvious dependence on the velocity field (u,v,w).
In order to compute any mass fluxes we must first determine the velocity values in the domain of interest.

8. Analytical Model
Water at a constant pore velocity, v, is flowing through the zone carrying the dissolved component at a specific concentration, Co.
There is no degradation of the component, no dispersion of the component, nor is there any interaction with the solid phase (adsorbtion).
The zone simply translates in space at a rate determined by the pore water velocity.
The contaminant is completely dissolved, and does not alter the density of the flowing water in any significant fashion.
The contaminant is assumed to be uniformily mixed in contaminated zone.

9. Governing Equations, Initial, and Boundary Conditions
The governing equation of mass transport for this case is:
The initial conditions throughout the aquifer are:
The boundary conditions at the source are:
The solution for this case is:

11. Cell Balance Model
Suppose at t=0 the concentration in the inlet cell is Co. We want to determine the concentration in the porous medium at future times.
We will assume the pore velocity is identical throughout the column.
A simple modeling approach is to treat each cell as completely mixed. This means that the concentration at the cell exit is identical to the concentration in the cell.
Inlet
Cell 1 DL
Cell 2
Cell 3
Cell 4
Outlet
Cell 5

12. c=0
Cell 1
c=c1
Cell 2 DL
c=c2
Cell 3
c=c3
Cell 4
c=c4
Cell 5
c=c5

14. Timed Release Case
Water at a constant pore velocity, v, is flowing through the aquifer.
At the origin (x=0) a contaminant is added to the flowing water at fixed concnetration Co for a period of time t.
At the end of the time period the contaminant addition is stopped.
By the end of the time period a “zone” of contaminated aquifer is created.
No degradation, no dispersion, nor interaction with the solid phase (adsorbtion).
Contaminant is completely dissolved, and does not alter the density of the flowing water in any significant fashion.
Mechanism of release does not disturb the local flow field in any fashion.
Contaminant is assumed to be uniformily mixed in the y and z directions.

15. Solution Solution identical to first case.
Substitute vt = L into the previous solution.