CAPILLARY PRESSURE - UPC

Daniel Fernàndez Garcia 

UPC – Constitutive eqs. 

CAPILLARY PRESSURE 

Pressure difference between the non-wetting fluid and the wetting fluid. 

P 

c 

= 

P 

nw 

(non - 

wetting) 

− 

P 

w 

(wetting) 

Gas-Water 

P 

cgw 

= 

P 

g 

− 

P 

w 

Gas-NAPL 

P 

cgn 

= 

P 

g 

− 

P 

n 

Order of Wettability: WATER > NAPL > GAS 

General YOUNG-LAPLACE 

P 

c 

⎛ 

= σ 

⎜ 

⎝ 

1 

r 

1 

+ 

1 

r 

2 

⎞ 

⎟ 

⎠ 

Capillary tub YOUNG-LAPLACE 

P c 

= 

2σcos( 

θ) 

r 

1



CAPILLARY PRESSURE 

A decrease of the meniscus radii corresponds to an increase of the capillary pressure. 

If a water-saturated soil is drained , the wetting phase retreats to smaller pores 

An increase of the meniscus radii corresponds to a decrease of the capillary pressure 

If imbibition occurs, the wetting phase penetrates into larger pores 

P c2 



R 2 

> R 1 

2



CAPILLARY PRESSURE - SATURATION 

If we replace the meniscus radii by the pore radius, it becomes clear that 

an increase of the saturation of the non-wetting phase must also lead to 

an increase of the capillary pressure 

P 

c 

⎛ 

= σ 

⎜ 

⎝ 

1 

r 

1 

+ 

1 

r 

2 

⎞ 

⎟ 

⎠ 

P c 

= 

2σcos( 

θ) 

r 

The relationship between the capillary pressure and saturation is known 

as the retention curve 

P 

cgw 

= 

P 

cgw 

(S 

w 

,S 

g 

) 

3



TYPICAL RETENTION CURVES 

The shape of the retention curve depends on the material 

4



HISTERESIS 

The relationship for p c (S w ) depends on the pressure (or saturation) history. 

Different relationship for drainage and imbibition 

5


a) Geometry of pores 

HISTERESIS 


b) Contact angle depends on the process (drainage or imbibition) 

c) Trapped air reduces the amount of water available 

d) Others: swelling of soil. 

6



RETENTION CURVES 

HISTERESIS 

Medium sand 

Fine sand 

7



PUNTS DE LA CORBA DE RETENCIÓ 

Punt de marciment (θ w 

): θ en que les plantes no poden extreure l’aigua del subsòl. Les forces de 

succió de les plantes és més petita que les forces de retenció del terreny 

Capacitat de camp (θ f 

): θ després de drenar l’aigua gravífica. Representa l’aigua que, després de 

tres dies de ploure o regar, es queda en el terreny 

Capacitat de retenció (∆θ p 

⇒ θ f 

- θ w 

): θ disponible per les plantes 

Porositat drenable (S y 

= φ - θ f 

) 

Pressió d’entrada (Pd) 

Contingut volumètric residual (θ r 

) 

P c 

θ r 

∆θ p S y 

P d 

θ → 

θ w φ 

θ f 

8


RETENTION CURVE MODELS 


Brooks and Corey (1964) 

S −S ⎛p 

⎞ 

S (p ) = = if p ≥p 

− ⎝ ⎠ 

w wr d 

e c ⎜ ⎟ 

c d 

Sm Swr pc 

S −S 

S (p ) = = 1 if p < p 

w wr 

e c c d 

Sm 

−Swr 

Van Genutchen (1980) 

S −S 

S (p ) = = 1 + ( α⋅ p ) if p > 0 

λ 

n 

− 

( ) m 

w wr 

e c c c 

Sm 

−Swr 

S 

λ : pore -size index → values between 0.3 and 3 

p 

wr 

d 

: residual water saturation 

: air entry pressure 

α: constant 

n: constant 

m:constant 

Physical meaning: 

λ : slope → granulometry 

p : related to the largest pores 

d 

α: 

interfacial properties of the fluids 

9



EFFECTIVE SATURATION 

S(p) 

e 

c 

= 

S 

S 

w 

m 

− S 

− S 

wr 

wr 

S m 

= 1 when DNAPL is invading into a 

water saturated porous medium 

S m 

< 1 when water displaces DNAPL 

because of the entrapment of DNAPL 

s m 

s m : Maximum wetting phase saturation 

10


RETENTION CURVE MODELS 


Correlation of Lenhard et al. (1989): 

1 

m= 1− n ∈[ 

λ,1] 

n 

m 

λ= − 

1− 

m 

x 

( 1 S 

1/m 

) we 

S = 0.72 −0.35e 

Sx 

S 1 

d 

( 

1/m 

) 

x 

α= − 

p 

−n 

4 

1−m 

11



LEVERETT J-FUNCTION (1941) 

The p c -S-relations in NAPL-water or gas-NAPL systems differ from 

gas-water due to the interdependence of the interfacial tension and 

the contact angle 

Dimensionless form of the p c -S-relations 

Leverett (1941) 

Richardson (1961) 

J(S) 

J(S) 

Pc 

= κ / φ 

σ 

= Pc 

κ φ 

f ( θ) 

σ 

/ 

Influence of contact angle not negligible for drainage or wetting 

processes with θ =35-40 or 15-25, respectively. 

Contact angle increases with decreasing surface tension so that it is 

possible to estimate a range for σ (not negligible for < 50 mN/m) 

12



Pc-S relation for water-NAPL-gas system 

Order of Wettability: WATER > NAPL > GAS 

Gas-Water 

Gas-NAPL 

Pn − Pw 

= 

P 

cnw 

(S 

Pg − Pn 

= Pcgn 

(Sg 

) = Pcgn 

(Sn 

+ S 

w 

) 

w 

) 

GAS 

NAPL 

Parker et al Model (1987) 

WATER 

S + S −S 

n w wr 

1−S 

wr 

( 

n 

) 

− 

1 ( m 

gn 

p 

cgn 

) 

= + α⋅β ⋅ 

β 

gn 

= 

σ 

σ 

gw 

gn 

S 

w 

−S 

1− 

S 

wr 

wr 

( 

n 

) 

− 

1 ( m 

nw 

p 

cnw 

) 

= + α⋅β ⋅ 

β 

nw 

= 

σ 

σ 

gw 

nw 

13



EFFECTIVE SATURATION 

S(p) 

e 

c 

14



TYPICAL PARAMETERS FOR VAN GENUTCHEN 

15



ENERGY STATUS OF THE PORE WATER 

The soil-water system has energy associated to it. 

E = Energy [N-m], [Kg m 2 s -2 ] 

h = hydraulic head = Energy per unit weight 

h 

s 

= −∫ 

0 

F(s) 

m(s) 

ds 

E = E + E + E + ... = PV + mgz + 1/2mv 

T p g c 

h = E /mg = P / ρ g + z + ... 

T 

T 

2 

We can define energy in other basis. For example, in many soil 

physics texts: 

"Soil Water Potential" ( ψ) 

ψ = Energy per unit volume (units of pressure) 

s 

ψ=− ρ ∫ 

0 

(s) 

F(s) 

ds 

m(s) 

16



COMPONENTS OF SOIL WATER POTENTIAL 

We continue to neglect kinetic energy 

Gravimetric head: Energy required to move pure free water from a 

reference elevation to the elevation z 

Gas, Water or NAPL Pressure head: Energy associated with 

pressure exerted on a point by surrounding water. 

h 

g 

s 

pg 

= ∫ ds ρ g 

0 

g 

h 

w 

s 

p 

= ∫ ρ 

0 

Capillary pressure head: Energy associated with the curvature of 

the air-water (or NAPL-air-water) interface, and the wettability of the 

soil, for the phases of interest. 

w 

ds 

g 

w 

hz 

h 

= 

n 

z 

s 

pn 

= ∫ ds ρ g 

0 

n 

s 

s 

pw − pg pc 

hc 

= ∫ ds=− ds< 

0 

ρ g 

∫ρ 

g 

0 w 

0 

w 

17




Not considered potentials: 

Osmotic head (Solute Potential): Change in energy associated with 

solutes (dissolved chemicals) in the soil water (or gas) compared to 

clean water, pure water. Measures the capability of a solution to suck 

water in. 

Overburden head: Energy associated with changing the mechanical 

pressure exerted by the unsupported solid material on the soil water 

from zero to some value P env . For rigid soil P env =0. For swelling or 

compacting sois P env >0. 

Other grouped potentials terms: 

h o 

h b 

Matric Potental and head: Soil-water systems only. Capillary 

pressure or suction but may include overburden or intergranular forces 

18




FOR VARIOUS APPLICATIONS 

Swelling Soil: Containing Water and Gas 

h = h + h + h + h + 

T 

z 

o 

cgw 

g 

h 

b 

Rigid Soil without osmotic process: Containing Water and Gas 

h = h + h + 

T 

z 

cgw 

h 

g 

19



MEASUREMENTS OF Pc-S 

LABORATORY 

LONG COLUMNS 

A long porous media is allowed to reach equilibrium with the source of wetting fluid at its base 

HANGING COLUMN 

Small porous plate connected to a water column that ends in a burette. The position of the 

burette can be changed to increase suction. 

CENTRIFUGE 

Saturated short column is placed with its long axis horizontal in a centrifuge. 

PRESSURE PLATE 

Similar to hanging column but sample is in chamber with controlled external pressures. 

FIELD 

TENSIOMETER 

20



Pc-S LABORATORY MEASUREMENTS 

“ HANGING COLUMN FOR WATER-AIR SYSTEMS” 

Cover to prevent 

evaporation 

-300 cm < -h c 

< 0 

Drainage of a saturated sample. 

After drainage step the system us 

taken to equilibrium. 

Equilibrium time depends on 

microporous cup material. 

21



Pc-S LABORATORY MEASUREMENTS 

“ HANGING COLUMN FOR DNAPLs” 

DNAPL pressure increased 

incrementally. 

Porous plate allows water to flow 

but not DNAPL. 

Water pressure is indicated by 

the water level in the outflow 

burette, and the DNAPL 

pressure is calculated from the 

fluid levels in the DNAPL 

reservoir. 

22



EXAMPLE PCE-WATER SYSTEM 

CAPILLARY 

PRESSURE 

[cm-water] 

23



Pc MEASUREMENTS IN THE FIELD 

Pressure 

transducer or 

manometer 

Water 

TENSIOMETER 

L 

Contact between water-soil and 

tensiometer through microporous cup 

Microporous cup allows contact but not 

air penetration into the tensimeter 

Equilibrium time depends on flux 

resistance through microporous cup 

Capillary head = manometer – pressure 

head tensiometer 

h 

⎛p 

manometer 

c 

= ⎜ − 

ρwg 

⎝ 

⎞ 

L⎟ 

⎠ 

Maximum suction 800-900 milibars o cmaigua 

Microporous cup (5-10 cm long and 1-2 cm 

in diametre) 

24



APLICATION - AGRICULTURE 

TENSIOMETER 

Can be automated 

Tensiometer indicates when to apply 

water but not how much. 

Water quantity calculated through 

retention curve 

25



BUNDLE OF TUBES MODEL FOR 

RETENTION CURVE 

h i 

26



MODELS OF RELATIVE PERMEABILITY 

Relationship between relative permeabiility and capillary pressure 

κ = S 

⎡ 

⎢ 

⎢ 

⎢ 

⎢ 

⎣ 

Se 

∫ 

A 0 

rw e 1.0 

∫ 

0 

[ p(S) ] 

c 

[ p(S) ] 

c 

−B 

−B 

⎤ 

dS⎥ 

⎥ 

⎥ 

dS⎥ 

⎦ 

C 

( 1 S ) 

κ = − 

Burdine amb Brooks and Corey 

A 

⎡ 

⎢ 

⎢ 

⎢ 

⎢ 

⎢⎣ 

1.0 

∫ 

rnw e 1.0 

S 

e 

∫ 

0 

[ p (S)] 

c 

[ p (S)] 

c 

−B 

−B 

⎤ 

dS⎥ 

⎥ 

⎥ 

dS⎥ 

⎥⎦ 

C 

MODEL A 

BURDINE 2 

MUALEM 0.5 

S 

e 

S−S 

= 

1 − S 

r 

r 

B 

2 

1 

C 

1 

2 

κ = 

rw 

( S ) 2 3 

λ 

e 

+ λ 

⎛ 

κ = − ⎜ − 

⎝ 

2+ 3λ 

2 

λ 

rnw ( 1 Se ) 1 Se 

⎞ 

⎟ 

⎠ 

S: saturation 

λ: pore size index 

S : residual saturation 

r 

Mualem amb Van Genutchen 

( ) ( 

1/m 

S 1 1 S ) 

ε 

⎡ 

rw e e 

κ = 

⎢ 

− − 

⎣ 

m 

⎤ 

⎥⎦ 

2 

( ) 

1/m 

rnw 

1 S ε ⎡ 

e 

1 Se 

κ = − ⎣ − 

⎤ 

⎦ 

2m 

ε: connectivity of the pores ≈ 

1 

2 

27



28




NAPL-WATER SYSTEM 

29




NAPL-GAS SYSTEM 

30




WATER-NAPL-GAS SYSTEM 

We need to define 

the region where 

NAPL is mobile 

31




3-PHASE SYSTEMS 

Parker et al. (1987) 

WATER 

NAPL 

⎡ 

⎢ 

⎢ 

⎢ 

⎢ 

⎢⎣ 

S 

w 

A 

0 

rw 

Sw 

1 

κ = 

∫ 

∫ 

0 

( S Sw 

) 

κ = − 

[ p(S) ] 

c 

[ p(S) ] 

c 

A 

⎡ 

⎢ 

⎢ 

⎢ 

⎢ 

⎢ 

⎣ 

rn L 1 

S 

L 

∫ 

S 

w 

∫ 

0 

−B 

−B 

⎤ 

dS⎥ 

⎥ 

⎥ 

dS⎥ 

⎥⎦ 

C 

[ p(S) ] 

[ p(S) ] 

c 

c 

−B 

−B 

⎤ 

dS⎥ 

⎥ 

⎥ 

dS⎥ 

⎥ 

⎦ 

C 

S 

w 

S 

L 

S 

= 

S 

w 

wm 

−S 

−S 

wr 

wr 

Total liquid (NAPL+water) effective saturation 

Sw + Sn −Swr −Snr 

= 

S + S −S −S 

wm nm wr nr 

AIR 

⎡ 

⎢ 

⎢ 

⎢ 

⎢ 

⎢⎣ 

1 

A S L 

ra 

Sa 

1 

κ = 

∫ 

∫ 

0 

[ p(S) ] 

c 

[ p(S) ] 

c 

−B 

−B 

⎤ 

dS⎥ 

⎥ 

⎥ 

dS⎥ 

⎥⎦ 

C 

S 

a 

= 

S 

S 

a 

am 

−S 

−S 

ar 

ar 

32





Parker et al. (1987) with Mualem / Van Genucthen Model 

WATER 

1/2 

w 

κ 

rw 

= S ⎡ 

⎣1 −(1−S ) 

1/m m 

w 

⎤ 

⎦ 

2 

S 

w 

= 

S 

S 

w 

wm 

−S 

−S 

wr 

wr 

NAPL 

( ) 1/2 1/m m 1/m m 

2 

S S w ⎡(1 S ) (1 S ) ⎤ 

κ = − ⎣ − − − 

rn L w L 

( ) 1/2 2m 

1 S ⎡1 S ⎤ 

κ = − − 

1/m 

AIR ra 

L 

⎣ L ⎦ 

Sa −Sar 

a 

OBSERVATION: 

K rw depends only on S w 

K ra depends only on S a (or S L ) 

⎦ 

S 

S 

L 

Sw + Sn −Swr −Snr 

= 

S + S −S −S 

= 

S 

wm nm wr nr 

am 

−S 

ar 

K rn depends on both S a and S w 

33





Stone’s 1 model (1970) – Estimates of K rn based on two-phase systems 

First K rw (S w ) is obtained as for water-NAPL. Then K rg (S g ) is 

determined for gas-NAPL phase system. 

NAPL 

κ 

⎛ 

= ⎜ 

S ⎞ 

⎟κ κ 

⎝ 

⎠ 

n 

rn 

⎜ 

rnw rng 

(1−S w)(1−S g) 

⎟ 

κ 

rnw 

=κrn (S 

w 

) in a water-NAPL system 

κ 

= κ 

rng rg g 

(S ) in a gas-NAPL system 

34





Stone’s 2 model (1973) – Estimates of K rn based on two-phase systems 

First K rw (S w ) is obtained as for water-NAPL. Then K rg (S g ) is 

determined for gas-NAPL phase system. 

⎡⎛κ 

⎞⎛ κ ⎞ 

⎢⎜ ⎟⎜ ⎟ 

⎣⎝ ⎠⎝ ⎠ 

NAPL * rnw 

rng 

κ =κ +κ +κ −( κ +κ ) 

rn rnw * rw * rg rw rg 

κrnw 

κrnw 

⎤ 

⎥ 

⎦ 

κ =κ (S = S ) in a water-NAPL system 

* 

rnw rn w wr 

κ 

rnw 

=κrn (S 

w 

) in a water-NAPL system 

κ 

= κ 

rng rg g 

(S ) in a gas-NAPL system 

35





Interpolation model 

κ 

rw 

=κrw (S 

w 

) 

κ 

= κ 

(S ) 

ra ra a 

κ =κ (S ,S ) = 

rn rn g w 

S κ (S ) + S κ (S ) 

g rng g w rnw w 

S 

g 

+ S 

w 

36

CAPILLARY PRESSURE - UPC

Create successful ePaper yourself

Delete template?

Save as template?