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

for saltwater concentrations

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.

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.

Geometry and boundary conditions

p=p0 at pointsp=rgD at t=0

c=csalt

c=0

c=0 at t=0

600 m

150

m

150 m

no flux all others

150 m

Geometry and boundary conditions

p=p0 at pointsp=gD at t=0

c=csalt

c=0

c= at t=0

600 m

150

m

150 m

no flux all others

150 m

symmetric

2-way coupling between flow & transport

0

CcDt

cu

0)(/])1([

gDpt

c

ct

p

)( 0cc

• Density dependent fluid flow - Darcy’s Law

• Salt concentration – Saturated solute transport

• varies with c

Density driven flow (typically)

0)(/])1([

gDpt

c

ct

pfs

• Darcy’s law with density terms

p = pressure

c = concentration = density (varies with concentration)

f, s = compressibility of solid and fluid

= porosity

= permeability

= dynamic viscosity

g = gravity

D = elevation

Density appears as a scaling coefficientAccounts for change in storage from concentration

0)(/])1([

gDpt

c

ct

pfs

Density driven flow (the Elder problem)

• Density driven fluid flow with Darcy’s law

• 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

fluid velocity u

0 0

Non-reactive transport (typically)

0

ccDt

cu

c = concentration

= porosity

D = hydrodynamic dispersion tensor (see below)

u = vector of directional velocities (from flow equation)

mj

Ti

Lii Duu

D |u||u|

22

• Dispersion consists of mechanical spreading plus molecular diffusion

|u|)( ji

TLij

uuD

L, T = longitudinal and transverse dispersivities

Dm = molecular diffusion; = tortuosity factor ( < 1)

• 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 onlySet thDxy and thDyx to zero defining thD as a lumped isotropic molecular diffusion

Salt transport (the Elder Problem)

0

ccDt

cu m

jT

iLii D

uuD

|u||u|

22

|u|)( ji

TLij

uuD

0 0 0

Dispersion here is molecular diffusion only

year 2

year 1

year 3

year 10

year 15

year 20

Density driven flow – Concentration Snapshots

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.

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)

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.

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.