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.
