1. Advection-Dispersion Equation (ADE)
Assumptions
Equivalent porous medium (epm)
(i.e., a medium with connected pore space
or a densely fractured medium with a single
network of connected fractures)
Miscible flow
(i.e., solutes dissolve in water; DNAPL’s and
LNAPL’s require a different governing equation.
See p. 472, note 15.5, in Zheng and Bennett.)
3. No density effects
(density dependent flow requires a different
governing equation, Z&B, Ch. 15)

2. Dual Domain Models Fractured Rock Heterogeneous porous media
Note the presence of
“mobile” domains (fractures/high K units) and
“immobile” domains (matrix/low K units)
Each domain has a different porosity such that:  = m + im
Z&B Fig. 3.25

3. Note: model allows for a different porosity for each domain
Governing Equations – no sorption
Immobile domain
mass transfer rate between the 2 domains
Note: model allows for a different porosity for each domain
 = m + im

4. (MT3DMS manual,
p. 2-14)

5. Sensitivity to the mass transfer rate Sensitivity to the
porosity ratio
Z&B, Fig. 3.26

6. Sensitivity to Dispersivity
Dual domain model
Advection-dispersion
model

7. Governing Equations – with linear sorption

8. Dual Domain/Dual Porosity Models
Summary
“New” Parameters
Porosities in each domain: m ; im ( = m + im)
Mass transfer rate: 
Fraction of sorption sites: f = m /  (hard-wired into MT3DMS)
Porosities
Mass transfer rate
Treated as calibration parameters

9. Shapiro (2001)
WRR
Tracer results in fractured
rock at Mirror Lake, NH

10. MADE-2 Tracer Test
Injection Site

11. Advection-dispersion model
(One porosity value for entire model)
kriged hydraulic conductivity field
stochastic hydraulic conductivity field
Observed

12. Dual domain model with a
kriged hydraulic conductivity field
Observed

13. Dual domain model with a
stochastic hydraulic conductivity field
Observed

14. Results with a stochastic K field
Feehley & Zheng,
2000, WRR
Results with a stochastic K field

15. Feehley & Zheng (2000)
WRR

16. Ways to handle unmodeled heterogeneity
Large dispersivity values
Stochastic hydraulic conductivity field and “small”
macro dispersivity values
Stochastic hydraulic conductivity field with even
smaller macro dispersivity values & dual domain porosity
and mass exchange between domains
Alternatively, you can model all the relevant heterogeneity
Statistical model of geologic facies with dispersivity
values representative of micro scale dispersion

17. Stochastic GWV

19. Stochastic GWV