Variably Saturated Flow and Transport: Sorbing Solute

• With variably saturated flow, fluids fill only part of the pore space.

• Flow properties depend on degree of saturation, making Richards’ equation nonlinear

• Often researchers use analytic expressions (e.g., van Genuchten or Brooks & Corey) to describe how material properties vary with the solution.

• This example also shows how to incorporate experimental data directly into the COMSOL Multiphysics model.

• Example based upon Hydrus2d Manual (Simunek and Van Genuchten, 1992)

Variably saturated flow

Fluid in column moves into “disc” where it is distributed over given radius.

Fluid moves from disc into dry soil.

With good control on fluid (and contaminant) coming fromdisc, researchers analyze Subsurface properties and behaviors.

Image fromDepartment of Agriculture and Soil ScienceUniversity of Sidney, Australiahttp://www.usyd.edu.au/su/agric/ACSS/sphysic/infiltration.html

Disc permeameter

Axisymmetry

1.3 m

Ground surface

Inlet “just ponding” at known water height

2-layersoil

column

initially unsaturatedto depth of

about 1.2 m

Extremely low permeability

Upper soil layer

Lower soil layer

Problem set up

0

DHKt

HC

t

HSSe pHp

pHp

pHp

Hp = pressure headSe = effective saturationS = specific storageC = specific moisture capacityK = hydraulic conductivity D = elevation = fluid volume fraction (constitutive relation)

Hp=0

, Se, C, K

- Hp +

Variably saturated flow equation

Hp subscripts to denote dependency on Hp NONLINEAR

We can set up the permeability and retention formulae three ways:

(1) Using analytic formulae predefined from van Genuchten or Brooks & Corey

(2) Defining your own expressions

(3) By interpolating between experimental data

Variably saturated flow equation

00

0)1()()1/( /1/1

p

pmmm

rs

Hif

HifSeSemmC

01

011 2/1

p

p

mmL

rHif

HifSeSekkr Relative permeability

C Specific capacity

Se Effective saturation

Volume liquid fraction

01

01

1

p

pmn

p

Hif

HifHSe

0

0)(

ps

prsr

Hif

HifSe

(1) … van Genuchten (shown in “Sorbing Solute” model from ES Library)

kr Relative permeability

C Specific capacity

Se Effective saturation

Volume liquid fraction

01

0

p

prs

r

Hif

HifSe

01

0

p

p

r Hif

HifvaluesedInterpolatk

0

0

ps

p

Hif

HifvaluesedInterpolat

(3) … Interpolation from experimental data (shown in

“Interpolation” model from ES Library)

00

0

p

pp

Hif

HifHC

Hp = 0

Hp = Hp(x,z,0)

0)( zHpKn

0)( zHpKn

100/)( sKzHpKn

Axisymmetry

No flow

No flow

Leaky

Specified pressure head

Flow: Boundary and Initial Conditions

1 day 5 days

10 days

Day 5:Soil wetting up, stilldry at surface far from disc

Day 10:Almost all porespace filled with water.

Day 1:Mostly unsaturated (Hp<0)Notice wetting front

Flow Snapshots

Flow Movie

c = concentration = liquid volume fractionb = bulk densitykp = linear sorption coefficientC = specific moisture capacityHp = pressure headD = dispersion tensor q = specific discharge = decay rate

ckcccDt

HC

t

ck ppbLij

ppb

u

Variably saturated solute transport

liquid concentrationc

solid concentrationcp = b kp c

c (r,z,0)=0

Transport: Boundary and Initial Conditions

Axisymmetry

Free advection

No flux

Specified concentrationc=1

0)( ccDn ij u

0)( cDn ij

Free advection

0)( cDn ij

Initially pristine

Transport Concentration Snapshots

Surface is liquid concentration; contours are pressure head

1 day 5 days

Transport Retardation Factor

Sorption slows contaminantmovement relative to water

usolute=uwater/RF

Retardation greatest where pore space is emptiest

Transport Concentration Movie