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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. - PowerPoint PPT Presentation
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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.
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