1. Advection-Dispersion Equation (ADE) Assumptions 1.Equivalent porous medium (epm) (i.e., a medium with connected pore space or a densely fractured medium with a single network of connected fractures) 2.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 Z&B Fig. 3.25 Note the presence of “mobile” domains (fractures/high K units) and “immobile” domains (matrix/low K units) Fractured RockHeterogeneous porous media Each domain has a different porosity such that:  =  m +  im

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

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

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

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

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. Injection Site MADE-2 Tracer Test

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