Buoyancy Flow with Darcy’s Law - the Elder (1967) problem for saltwater concentrations

1. Buoyancy Flow with Darcy’s Law - the Elder (1967) problemfor saltwater concentrations

2. Density driven flow • Fluids pick up contaminants (natural or otherwise) in travel through the subsurface. Fluid density can vary with the contaminant concentrations producing buoyancy flow. • Conventional flow/transport packages deal with fluids of constant density so adding density driven flow typically means rebuilding your model with another specialized software. • In COMSOL Multiphysics, it is straightforward to add density variations to most flow/transport models. • Methods shown here for solute concentrations apply to density variations brought about by other factors, including temperature, for example.

3. Density driven flow – the Elder problem • Originally this density driven flow example was set up for heat transfer by Elder (1967). • Recast for salt concentrations by Voss and Souza (1987). • Used as a benchmark for testing many salt-water transport codes; e.g., SEAWAT/MODFLOW, SUTRA ... • The Elder problem is notoriously sensitive to nuances in the mesh and solution method.

4. Geometry and boundary conditions 150 m 150 m c=csalt p=p0 at points p=rgD at t=0 150 m c=0 at t=0 no flux all others c=0 600 m

5. Geometry and boundary conditions 150 m 150 m c=csalt p=p0 at points p=rgD att=0 150 m c=0att=0 no flux all others c=0 600 m symmetric

6. 2-way coupling between flow & transport • Density dependent fluid flow - Darcy’s Law • Salt concentration – Saturated solute transport • r varies with c

7. Density driven flow (typically) • Darcy’s law with density terms Density appears as a scaling coefficient Accounts for change in storage from concentration p = pressure c = concentration • = density (varies with concentration) cf, cs = compressibility of solid and fluid q = porosity k = permeability h = dynamic viscosity g = gravity D = elevation

8. Density driven flow (the Elder problem) fluid velocityu • Density driven fluid flow with Darcy’s law 0 0 • Implementation: • Physics>Subdomain settings: • Storage coefficient is user defined as the very small number eps • Density is a scaling coefficient on Scaling Terms tab • Physics>Equation systems>Subdomain Settings: • New term in da matrix accounts for storage change related to time rate change in concentration • Options>Expressions>Scalar Expressions: • Density is a function of concentration • Directional velocities defined because divergence operator now includes extra density term

9. Non-reactive transport (typically) c = concentration q = porosity D = hydrodynamic dispersion tensor (see below) u = vector of directional velocities (from flow equation) • Dispersion consists of mechanical spreading plus molecular diffusion aL, aT = longitudinal and transverse dispersivities Dm = molecular diffusion; t = tortuosity factor (t < 1)

10. Salt transport (the Elder Problem) 0 0 0 Dispersion here is molecular diffusion only • Implementation: • Physics>Subdomain settings: • Flow and Media Tab: directional velocities are the scalar expressions u and v • Liquid Tab: aL aT set to zero • Physics>Equation systems>Subdomain Settings: • Variables tab: Set thDxx and thDyy to the diffusion component only Set thDxy and thDyx to zero defining thD as a lumped isotropic molecular diffusion

11. Density driven flow – Concentration Snapshots year 1 year 10 year 2 year 15 year 20 year 3

12. Density driven flow – Animation of Concentrations • As the water becomes increasingly saline it sinks. When the dense salty water sinks it displaces relatively fresh water, which rises to the surface.

13. Elder, SUTRA, SEWAT Results • The COMSOL Multiphysics results give an excellent match with the Elder results. • Differences between the COMSOL Multiphysics and SUTRA concentrations occur because COMSOL Multiphysics solves for the dependent variable and its gradients simultaneously. • figure from SEWAT/MODFLOW manual (Guo and Langevin, 2002)

14. Density driven flow – Animation of Streamlines • Concentrations (surface) and velocities (streamlines) show the development of several convection cells over the course of the 20-year simulation period.

15. References • Elder, J.W. (1967). Transient convection in a porous medium: Journal of Fluid Mechanics, v. 27, no. 3, p. 609-623. • Guo, W. and Langevin, C.D. (2002). User’s Guide to SEAWAT: A Computer Program for Simulation of Three-Dimensional Variable-Density Ground-Water Flow: U.S. Geological Survey Techniques of Water-Resources Investigations 6-A7. • Voss, C. I. and Souza,W. R. (1987). Variable density flow and solute transport simulation of regional aquifers containing a narrow freshwater-saltwater transition zone: Water Resources Research, v. 23, no. 10, p. 1851-1866. • Voss, C.I. (1984). A finite-element simulation model for saturated-unsaturated, fluid-density-dependent ground-water flow with energy transport or chemically-reactive single-species solute transport: U.S. Geological Survey Water-Resources Investigation Report 84-4369, 409 p.