Document

Download Report

Transcript Document

Part 5: 151-199 SHORT COURSE ON

MODELING FLOW AND SOLUTE TRANSPORT IN THE SUBSURFACE

by JACOB BEAR

Professor Emeritus, Technion—Israel Institute of Technology, Haifa, Israel

Lectures presented at the Instituto de Geologia, UNAM, Mexico City, Mexico, December 6--8, 2003

01-01-01 JB/SWICA 1

STATICS and DISTRIBUTION OF PHASES IN THE UNSATURATED ZONE

FLUID CONTENT at a point

and time t :   

volume of



-fluid in REV volume of REV

Vadose zone 0      ,



    .

With:    

 .

SATURATION at a point

 

and time  t :

volume of void space in REV

01-01-01 0 

  1, 

  1.

JB/SWICA 151

We shall discuss: The distribution of moisture (water) and other phases (gas and possibly another liquid phase, which is not water) within the unsaturated zone.

The movement of fluid phases in the unsaturated zone.

Mass balance equations and, eventually, complete models, of flow of fluid phases within the unsaturated zone.

Both two and three fluid phases within the void space.

01-01-01 JB/SWICA 151

TWO OR THREE FLUID PHASES OCCUPY THE VOID SPACE IN THIS ZONE.

How are the phases distributed within the void space in this zone?

How are phase pressures distributed within this zone?

CONCEPTS: MISCIBLE and IMMISCIBLE FLUID PHASES .

(In reality: very low miscibility) The actual distributions and interphase interactions take place at the microscopic level .

We need the macroscopic description, obtained by averaging.

A PHASE is defined as a homogeneous spatial domain that is separated from other such domains by a well defined, sharp, physical boundary (= INTERFACE ).

01-01-01 JB/SWICA 152

HYDROSTATICS IN THE UNSATURATED ZONE

Vadose Two immiscible liquids , or a liquid and a gas, that together occupy the entire void space, are separated by a sharp interface. The properties of the fluids

and

close to

this surface differ significantly from those

within

the respective fluid bodies.

Interface transition Transition zone and (a sharp) interface between phases.

A surface between two liquids, between a liquid and a gas, or between a liquid and its vapor.

01-01-01 JB/SWICA 153

Molecules of a fluid are attracted to each other by an attractive force.

Resultant attractive force on a molecule in the interior is zero.

A molecule belonging to the surface is subjected to a resultant attractive force towards the INTERIOR of the liquid . As a consequence of the pull towards the liquid, work must be performed in order to increase the surface of the interface by bringing liquid molecules from the interior to the interface.

Left alone, the interface always tends to contract to the smallest area possible, seeking the shape that corresponds to MINIMUM ENERGY,

under the prevailing conditions

e.g., the proximity of a solid surface. 01-01-01 JB/SWICA 154

Actually, at the molecular level, no SHARP interface exists . Instead, a transition takes place across a thin zone. The transition zone is CONCEPTUALLY replaced, as an approximation, by a sharp interface that is assumed to separate the two fluids.

SURFACE ENERGY

, associated with this surface, per unit area, equals the difference between the energy of all molecules (of both phases) in the transition zone and the energy that they would have possessed had they been in the interior of their respective domains.

Interface area can be increased by bringing molecules from the interior. WORK must be done against the net cohesive force among the molecules in the two fluids.

Energy is gained when the area of the interface is reduced .

01-01-01 JB/SWICA 155

SURFACE (or INTERFACIAL) FREE ENERGY = work required to increase the area of an interface by one unit.

Molecules at the behave

AS IF

they belong to a thin, skin-like membrane under tension , that adjusts its geometry to give the smallest possible surface area. A real membrane behaves differently .

Interfacial tension within the "MEMBRANE" is measured as energy per unit area ( erg/cm 2 ). The same surface tension is also measured As FORCE PER UNIT LEGTH ( dyne/cm ) of a line within the surface.

The

interfacial tension

g

b = amount of work that must be Performed in order to separate a unit area of substance



substance 01-01-01

b,

or to increase their interface by a unit area. from JB/SWICA 156

For air (

) and water (

) at 200 C,

g

(equivalently, as force unit length,

g

= 72 erg/cm 2 = 72 dyne/cm ).

Surface tension,

g

 = interfacial tension between an  -substance and its own vapor.

The magnitude of the surface tension,

g

b , depends on the temperature, composition, and pressure of the fluids CONTACT ANGLE ,



= angle between the solid surface and the fluid-fluid interface, measured through the denser fluid. It expresses the affinity of the 01-01-01 fluids for the solid.

JB/SWICA 157

HOWEVER, there must exist equilibrium ALSO normal to the solid.

PREFERENTIAL WETTING

FOR ANY TWO FLUIDS AND A SOLID:  

Wetting fluid. Nonwetting fluid.

90 0  

90

0 Additives, called surfactants tend to accumulate in the liquid close to and at the interface. They reduce the interfacial tension, sometimes significantly, and may alter contact angle 01-01-01 JB/SWICA 158

For THREE FLUID PHASES , When a LENS of the intermediate wettability B-liquid will be formed.

FILMS In two-phase flow, molecular forces prevent complete drainage of the wetting fluid from the void space. A thin film of adsorbed wetting fluid will always remain on the solid in the void space.

The film may play a role in contaminant transport.

Solid may be L-fluid-wet, or a G-fluid-wet. In the unsaturated (air--water) zone in the soil, water is, usually, the wetting phase. The air is the non-wetting fluid.

01-01-01 JB/SWICA 159

As the soil is being drained, water is drained from the void space, and air fills up the evacuated space. As water leaves the void space, the

LARGE PORE DRAIN FIRST!

As water is being drained, water--air interfaces (menisci) are being formed. The radius of curvature at every point on a meniscus depends on the (local, microscopic) capillary pressure , i.e., on the pressure jump across the meniscus, with this radius becoming smaller as the capillary pressure increases.

To be discussed later 01-01-01 JB/SWICA 160

Pendular water Insular air As saturation decreases water leaves the larger pores, remaining in the smaller ones,

on the average!

Eventually,

at a certain low saturation

, the remaining water occupies he "smallest" pores-----

rings around grain contact points---

called pendular rings.

Sometimes, a number of adjacent pendular rings coalesce..

01-01-01 JB/SWICA 161

CAPILLARY PRESSURE and RETENTION CURVES

The jump in the pressure between the nonwetting fluid (air) and the wetting one (water) is called CAPILLARY PRESSURE.

A discontinuity in pressure exists across the interface separating the two fluids . This is the consequence of the interfacial tension, which exists between any two adjacent phases The difference between the pressure in the air (or in general, the non-wetting phase),

p air

, and in the water (the wetting phase), is called CAPILLARY PPRESSURE,

p' c

p c

' 



p wetting



p air



p p water

water

01-01-01 Microscopic level!

JB/SWICA 162

Microscopic pressures as the interface is approached from both sides at a point.

Radius of curvature The magnitude of pressure jump depends on the curvature and on the surface tension.

The interface is visualized as a (two-dimensional) material body (surface) which has rheological properties of its own -- It behaves as a `stretched membrane' under tension.

01-01-01 JB/SWICA 163

Force balance across a ` stretched membrane'

The Laplace equation

Microscopic capillary pressure.

p

' 

p



p



g

r’,r’’ : Two principal radii of curvature.

r* : Mean radius of curvature 01-01-01

1 r

'

 1

' 

r

1 ''

 

2 r

g

.

2 *

JB/SWICA 164

Reminder: Immiscible fluids in a capillary tube.

Wetting fluid Nonwetting fluid 01-01-01

h c

'  2 g 

aw w

cos 

o r

h c

'  2 g



w gr

*  

p c

w g

* 

co s  JB/SWICA 165

FLUID RETENTION RELATIONS

MACROSCOPIC CAPILLARY PRESSURE Microscopic capillary pressure --- at a point on a meniscus.

Macroscopic relations --- by averaging…..,but…..over surface Within REV!

Static and dynamic equilibrium.

MACROSCOPIC CAPILLARY PRESSURE

p c



p n n



p w w

In the unsaturated zone, the wetting fluid is water, while the Non-wetting one is air. It is often ASSUMED that the air is at a constant, atmospheric pressure, often taken as zero, i.e., 01-01-01

p n n



p a a

 0.

p c

 0,

p w w

 0.

JB/SWICA 166

p n n



p a a

 0.

p c

 0,

p w w

 0.

i.e., the pressure in the water is LESS THAN ATMOSPHERIC!

Often we use capillary pressure head

h c

 

p w w g

as variable.

In soil science, we often use SUCTION or MATRIC SUCTION  

w p w



w g

(  0) Definitions should be employed only when the average water 01-01-01 density is constant.

p a a

 0,  

p a a





p w w w g

 

w p c w g

.

JB/SWICA 167

CAPILLARY PESSURE CURVES

In analogy to the microscopic Laplace formula, we may write a macroscopic Laplace formula in the form: Relationship AT THE MACROSCOPIC LEVEL , between capillary pressure and saturations of two fluid phases.

LARGE PORES DRAIN FIRST!

What is a PORE ? Suggested definition of a “pore”.

01-01-01 JB/SWICA 168

DRAINAGE AND IMBIBITION PROCESSES

AIR--WATER DISTRIBUTIONS: Start with full water saturation.

Intermediate water saturations--- FUNICULAR WATER At low water saturation - PENDULAR RINGS . Water is discontinuous .

Trapped air upon rewetting.

INSULAR air saturation. Air is discontinuous.

Curvature of meniscus depends on size of pore. Hence, the relationship between capillary pressure and saturation.

01-01-01 JB/SWICA 169

The relationship between the quantity of water in the void space (within an REV), and prevailing capillary pressure is expressed as CAPILLARY PRESSURE CURVE , ,

p c = p c (S w )

, Or SUCTION CURVE :

  

(S w )

In soil science: The RETENTION CURVE shows how much water is retained in the soil by capillary forces (against gravity). Actually: This is the capillary pressure curve.

The shape of the RETENTION CURVE depends on : 01-01-01

p



p



Simply:

p

c w c

,

 g

w aw

, ,

)

) )).

only .

T c

g = temperature, = concentration of species in the fluids.

JB/SWICA 170

TYPICAL CAPILLARY PRESSURE CURVES DURING DRAINAGE Point A indicates the threshold capillary pressure head . (= Bubbling pressure, or air entry pressure ), corresponding to the largest pore size .

Starting from a fully water saturated sample, , no air will penetrate the sample until a critical (threshold) capillary head 01-01-01 is reached. JB/SWICA 171

The shape of the retention curve depends on the distributions of pore sizes and shapes.

As water drainage progresses, some water remains even at very high values of

p c

films. as isolated pendular rings and relatively immobile thin Notation:

S wr

= IRREDUCIBLE or RESIDUAL WATER SATURATION.



= RESIDUAL WATER CONTENT , with





S wr

(or moisture)

S ar

= RESIDUAL AIR SATURATION , when air saturation takes the form of isolated air bubbles only.

IRREDUCIBLE ?

01-01-01 JB/SWICA 172

“IRREDUCIBLE” ?

Further reduction is possible by water evaporation & air dissolving in water 01-01-01 JB/SWICA 173

Usually, use the term "

retention curve

" The retention curve is an expression of the pore-size distribution of a soil. Recall Laplace formula: where

r* p c n p w

* g

is a measure of the pores occupied by the wetting fluid.

For the same fluid,

p c

(

S w

)

p c

(

S w

)



2 *

1 * (Scaling) For the same porous medium, the scaling is by the fluid's interfacial tension : 01-01-01

S w

(

p c

b 12  g g 2 ) , 1 co s   cos 

S w

(

p c

2 , b 12 , 2 ) 

Scaling fa ctor.

JB/SWICA 174

tensiometer

used for determining the moisture of the soil

in the field

. Actually, the instrument determines the suction, and then a retention curve (see below), prepared for the same soil, is used to determine the moisture content.

Other methods : from cores, Neutron attenuation device, Electrical resistance, or dielectric cells,… 01-01-01 JB/SWICA 175

TENSIOMETER The instrument used for measuring the capillary pressure (or the suction) in the unsaturated zone.

porous cup

, made of unglazed earthware, or ceramics). The role: Provide a contact between the water in the tensiometer , but…. To prevent air from being sucked through the porous cup into the manometer, thus severing Recall

: p c = 2

g

aw /r*

JB/SWICA 176

DETERMINING THE RETENTION CURVE

Gradual drainage by producing suction.

Make sure that equilibrium is reached at each step. Sometimes a very long time!

Range limited by entry value of porous plate.

The pressures are atmospheric or controlled.

01-01-01 JB/SWICA 177

ENTRAPPED AIR Upon re-wetting, or imbibition, the capillary pressure curve,

p c = p c (S w )

, differs from that obtained during drainage. Upon re-wetting, or imbibition, the capillary pressure curve,

p c = p c (S w )

, differs from that obtained during drainage. We start from a saturated soil sample (

S w = 1.0

). During DRAINAGE , we move to the left along the drainage curve. If we stop the drainage process, and switch to imbibition , i.e., to wetting the sample (i.e., moving to the right), we note two interesting phenomena: 01-01-01 JB/SWICA 178

…two interesting phenomena: WE DO NOT MOVE BACK ALONG THE SAME DRAINAGE CURVE.

Instead, we move, now from left to right, along a different curve.

WE DO NOT RETURN TO THE INITIAL POINT OF FULL SATURATION Although we return to

p c = 0

, we do not return to

S w = 1.0.

We have

entrapped

air

that remains in the void space and cannot be removed by the process of imbibition 01-01-01 JB/SWICA 179

HYSTERESIS and ENTRAPPED AIR in a CAPILLARY PRESSURE CURVE 01-01-01 DRAINAGE AND IMBIBITION SCANNING CURVES: JB/SWICA 180

HYSTERESIS: Ink bottle effect . results from the shape of the pore space, with interchanging narrow (throats) and wide passages. We note the same meniscus curvature at different elevations., Raindrop effect is due to the fact that the contact angle at the advancing trace of a water--air interface on a solid surface is larger than at the receding one. 01-01-01  2 JB/SWICA 181

ANALYTICAL EXPRESSIONS FOR RETENTION CURVES

. METHODS BASED ON GRAIN-SIZE DISTRIBUTION.

Capillary pressure curve may be inferred from the pore size distribution function via

Laplace's capillary formula

E ffective (water) saturation

S e



 

S S wr wr

,   

Sometimes

S e

Brooks and Corey

S e

  (1964)

p p b c

   f or

p c

  1 

S w S



wr p b

,   1 = pore size distribution index for

p c



p b



wr S ar

  

p b

= bubbling pressure (minimum value of 01-01-01

p c

on a drainage capillary 182

VAN GENUCHTEN (1978):

S e

 1  1 (

 )



 for   0, 1 for   0.

A, B, C ( = 1 - 1/B) = curve fitting coefficients VAN GENUCHTEN (1980): The same as the 1978 formulation, but

S



1



S



S



S

ar S ar S wr

= residual air saturation, due to entrapped air.

= irreducible (= residual) water saturation.

01-01-01 JB/SWICA 183

SATURATION DISTRIBUTIONALONG THE VERTICAL

Homogeneous soil.

Deep water table.

No flow -- constant piezometric head,’ AIR_WATER system,



= a,w

By equating piezometric heads at points

m =1,2: h



z m



p g



and



"No flow" means : Hence

2 

1 

h w

,1 

h w

g p c

,1 ( 

 

p c

,2 

) .

z m



and

p g



h a

,1 

h a

,2 01-01-01 JB/SWICA 184

2   1

g p c

,1 ( 

 

p c

,2 

) .

Choose datum such that

S w =1.0

z 1 =0

Then:

S w,1 =1.0 , S a,1 =0 , p c,1 =0 ,

and with



. Hence:

z

2 

g

(

p

,2  



)



g p

,2 



h

.

CONCLUSION: The curves

h c p w = 0

, with

p a = h c = 0 (S

w )

and

z = z(S

are the same .

w )

above the water table, 01-01-01 JB/SWICA 185

SATURATION JUMP AT A POROUS MEDIUM DISCONTINUITY Assume a sharp boundary, in the continuum domain, between porous media,

and

II.

Two conditions at such boundary Continuity of fluxes continuity of pressure.

: Capillary pressure curves, and saturation discontinuities Continuity of pressure

 ,

 

p p

 ,

, 

  01-01-01 JB/SWICA 186

SPECIFIC YIELD , 

, and FIELD CAPACITY A concept often used in modeling drainage of agricultural lands and in discussing water table drawdown in phreatic aquifers SPECIFIC YIELD : Volume of water drained from a soil column of a unit horizontal cross-sectional area that extends from the soil surface down to the underlying phreatic surface, per unit drawdown of the latter.

A time-dependent quantity SPECIFIC RETENTION, 



y +



r =

 Same definition as “ storativity of a phreatic aquifer” .

01-01-01 JB/SWICA 187

Moisture distributions.

Deep water table For deep water table: 

( ''

t

') 1

Shallow water table    water table's depths,



0 '  '

t’

and

d'‘

'



' 0  



'' 01-01-01

t’’

. JB/SWICA

''

   188



01-01-01

t

1

   

1



h

  

h



0  ' 

  

'



' 0  



.

''

   JB/SWICA 189

SPECIFIC YIELD depends on homogeneity of profile, time, And depth to water table.

Usually, specific yield is used to indicate the volume of water drained during a sufficiently long time following a rise or drop in phreatic surface . Then:

  

y wr







S wr

In reality, equilibrium conditions will rarely, if ever, be reached.

FIELD CAPACITY: In soil science, approximately, the water content remaining in a unit volume of soil after gravity drainage has (practically) ceased.

Depends on sample elevation. Hence, useful when soil is located both sufficiently high above a water table, and at a sufficient depth below ground surface.

01-01-01 JB/SWICA 190

THREE FLUID PHASES

Background: 01-01-01 NAPL LNAPL DNAPL JB/SWICA 191

THREE FLUID PHASES

AIR, WATER and NONAQUEOUS PHASE LIQUID (NAPL) Subsurface contamination (leaks, spills) by: hydrocarbons, organic solvents. volatile components. components that dissolve in water.} Small, medium and large spills FREE PRODUCT Sharp (microscopic) interfaces between the NAPL and both the gas and the water within the void space.

LNAPL ….. Lighter than water, e.g.,gasoline.

DNAPL (HNAPL )….. Denser than water, e.g. chlorinated solvents.

NAPL components dissolve in water in small (ppb) (yet dangerous) quantities. Solubility of TCE (= Trichloroethene), aprox. 1.1 x 10 01-01-01 6 ppb at 20 o C.

JB/SWICA 192

THREE PHASE CAPILLARY PRESSURE RELATIONS

For any set of three phases and a solid: Wetting phase (

Intermediate wetting phase (

, or

Nonwetting phase (

(A mineral surface (with a high organic content) may be oil-wet.} SATURATIONS:

S

+ S

= 1.0

Oil---A nonaqueous phase---is the intermediate wetting phase. Air---a gaseous phase---is the nonwetting phase. It tries to be as far as possible from the solid. 01-01-01 JB/SWICA 193

CONCEPT OF CAPILLARY PRESSURE EXTENDED TO THREE FLUID PHASES.

LAPLACE's FORMULA for water(

), oil(

), air(

p cow , p cao

= oil--water & air—oil capillary pressures.

r* ow , r* ao

= radii of air--water & air—oil interfaces.

g

ow ,

g

= interfacial tensions.

Capillary pressure curves depend on pore-size distribution .

01-01-01 JB/SWICA 194

CUMULATIVE PORE-SIZE DISTRIBUTION FUNCTION

R* = R*(r*):

= cumulative fraction of the void space with pores smaller than

r*.

Disregarding hysteresis and fluid entrapment: *

r

 *

r

(

S

)



r

,

r

S

  *

r

(

S

)



r S



S

,



total liquid saturation.

r* ow

is a function of

S w

than

r* ow

only, since all pore with radii smaller are assumed to be occupied by water only .

01-01-01 JB/SWICA 195

All pore with radii smaller than

r* ow

are assumed to be occupied by water only .

All pore with radii smaller than

r* ao

are assumed to be occupied by both water and oil.

With respect to air, the two liquids behave as a single wetting fluid.

p cow

(

S w

)



2 ,

w p caw

(

S l

)



2 ,

CONCLUSION:

For a given pair of fluids

, S w S l

is a function of

p cow

is a function of

p caw

In a three-fluid phase system, if

S w

(

p cow

) only, and only. is independent of air pressure, it should be IDENTICAL to the function

S w

(

p cow

) oil-water, system.

01-01-01 for a two-phase , JB/SWICA 196

Repeat: In a three-fluid phase system, if

S w

(

p cow

) is independent of air pressure, it should be IDENTICAL to the function

S w

(

p cow

) for a two-phase , oil-water, system.

S III w



p cow

 

S w II



p cow

 ,

S l III



p caw

 

S l II



p caw

 .

II, III

denote 2- and 3-phase systems 01-01-01 JB/SWICA 197

VERTICAL EQUILIBRIUM SATURATION DISTRIBUTION OF THREE FLUID PHASES

Vertical-equilibrium (VE-) hypothesis

01-01-01 JB/SWICA 198

For a sufficiently large (one time) spill of LNAPL, such that the percolating LNAPL will reach and accumulate on an underlying water table in the form of a floating lens that spreads out laterally.

The lens will also move in the direction of the sloping water table. The assumption that the flow of both the LNAPL and water are essentially horizontal is equivalent to stating that

the vertical pressure distribution within each phase is hydrostatic

. 01-01-01 End of part 5 JB/SWICA 199

MODELING FLOW AND SOLUTE TRANSPORT IN THE SUBSURFACE

by JACOB BEAR WORKSHOP I 01-01-01 Part 5:slides 150-199 JB/SWICA 150

01-01-01 JB/SWICA