pore and sample scale unsaturated hydraulic conductivity for homogeneous porous media dani or and...

Pore and Sample Scale Unsaturated Hydraulic

Conductivity for Homogeneous Porous Media

Dani Or and Markus Tuller

Dept. of Plants, Soils and Biometeorology, Utah State University

Tuller, M. and D. Or, 2001, Hydraulic conductivity of variably saturated porous media - Film and corner flow in angular pore space, Water Resour. Res. (next issue)

Review of pore scale hydrostatics

i. Unitary approach for capillarity and adsorption:

ii. Liquid films and the disjoining pressure concept.

iii. New model for basic pore geometry.

iv. Liquid configurations for different potentials are

obtained by simplified AYL equation where

interface curvature r() in pore corners is

shifted by film thickness h().

v. Expressions for unit pore retention and liquid-

vapor interfacial areas were statistically upscaled

to represent Gamma distributed pore populations.

vi. Limitations: 2-D representation of a 3-D system;

advantages: improved representation of physical

processes (A & C), angular geometry, and

quantifying L-V and S-W interfacial areas.

3 svl

6

A)(h

)(r

)(C)h(A

Degree of Saturation

0.0 0.2 0.4 0.6 0.8 1.0 1.2-

Ch

emic

al P

ote

nti

al [

J/kg

]10-2

100

102

104

106

AdsoroptionCapillarityNew ModelVG-ModelMeasurements

Millville Silt Loam

films corners/full pores

Clarification of a few open issues discussed in session of pore scale

hydrostatics

Additional “snap-off” mechanism is associated with passage of invading non-wetting phase “finger” in the 3rd dimension (i.e. Main Terminal Meniscus) which would appear as a “hole” in the 2-D plane.

Note the rapid rupture of the liquid vapor interface and the formation of a new configuration after snap-off.

Equivalency between the van Genuchten (1980) model parameters and the new model parameters may be established as follows: Lmax vG

-1 (largest pore determines air entry value).

s (porosity and saturated water content).

(or SA) r (surface area x film determine residual water content).

n (both parameters determine the shape of statistical PSD).

The “trapped” microbe problem

1

tan1

1

1tan

11

)(rr

2

2*

r() r*

tan

)(rx

tan

rx

**

2tan

11)(rL

1.E-07

1.E-06

1.E-05

1.E-04

0 20 40 60 80 100

Matric Potential (J/kg or kPa)

Ra

diu

s o

f In

scrib

ed

Mic

rob

e (

m)

30

90

r_capil.

More angular

The ratio r*/r() is 0.17, 0.33, 0.59 for corner angles (2) of 30o, 600 and 900, respectively.

More angular porous media delay l-v interfacial constraints for microbial aquatic habitats.

Outline – section 3 Hydrodynamics in homogeneous porous media

• Estimation of capillary size distribution for hydrodynamic

considerations

• Statistical application of Poiseuille’s law and the standard BCC

approach (cross-section only!)

• Coupling flow in tubes with Darcy’s macroscopic flow equation

[Childs and Collis-George, 1950; Fatt and Dykstra, 1951].

• Flow regimes in angular pores and slits (cross-section only!)

• The assumption of interfacial stability for slow laminar flow.

• Assembly of K() for a unit cell.

• Upscaling to a population of unit cells (parallel flow pathways!)

• Input parameters and upscaling procedure.

• Examples – the role of film flow

BCC-based prediction of unsaturated hydraulic conductivity K()

1. Extraction of radii distribution of

capillary radii of the BCC (from

retention data).

2. Application of hydrodynamic

considerations, i.e., the volumetric

discharge in a cylindrical tube is

proportional to the 4th power of

tube radius (Poiseuille’s law).M

atri

c S

uct

ion

-

[m

]

Water Content - - [m3 m-3]

i

i

i

i g2

r

2

i

Vi

rn

(1)

L

P

8

rQ

0

4

(2)

P1P2

Lr

BCC-based prediction of unsaturated hydraulic conductivity K() (cont.)

3. Statistical application of Poiseuille’s law

for a bundle of capillaries coupled with

Darcy’s macroscopic flow equation

[Childs and Collis-George, 1950; Fatt

and Dykstra, 1951; Burdine, 1953;

Mualem, 1976; van Genuchten, 1980].

4. K() function is constructed by

summation of the discharge (for a unit-

gradient) over all “tubes” that are liquid-

filled at a given potential () divided by

total sample cross-sectional area (voids

and solids).

dz

dhgk

A

QJ

0total

w

time

lengthgkK

0

z

H

L

L

h

1

g8r

rL8

Hg

A

QJ

c

M

1j2j

24

M

1j2jc

w

Ksnj

BCC-based prediction of unsaturated hydraulic conductivity K() (cont.)

(Mualem, 1976)

5. Geometrical and hydrodynamic

aspects of real porous media were

introduced into the BCC by

consideration of a more complex

capillary structures, for example

cut-and-randomly rejoin concepts;

or the effective flow through a pair

of unequal capillaries such as

treated by Mualem, [1976].

6. The concept of tortousity (Lc/L)

improves BCC model predictions. Lc L

Hydrodynamic Considerations for Angular Pores

Equilibrium liquid-vapor interfacial configurations at various potentials serve as fixed boundaries for the definition of flow regimes (laminar) in angular pore space (film and corner flows).

The simple cell geometry and well-defined boundary conditions permit solution of the Navier-Stokes equations for average liquid velocity for each flow regime (i.e., geometrical feature).

Analogy with Darcy’s law is invoked to identify the coefficient of proportionality between flux and hydraulic gradient as the hydraulic conductivity for each flow regime under consideration.

dz

dp

g

K

A

Qv

Primary Flow Regimes in a Unit Cell

(1) Flow in ducts and between parallel plates for completely liquid-filled pores and slits.

(2) Flow in thin liquid films lining flat surfaces following pore

and slit snap-off.

(3) Flow in corners (bounded by l-v interface) of the central pore.

Corner

Film

Full ductParallelPlates

Hydraulic conductivity of full ducts

•Square Duct

with Bs (L1=L2) given as:

2

0

S2SS L

4

BgLKdKD

1n55S )1n2(

2)1n2(

tanh64

3

1B

2

0

2TT L

80

gLKdKD

•Triangular Duct

2

0

n2CC L

8

AgLKdKD

•Rhombic Duct ( tube)

Hydraulic conductivity in high-order polygons/rhombic pores

2

0

n2CC L

8

AgLKdKD

Rhombic Duct (tube)

n

cot4

nAn

The area constant An is given as:

As n

4n

n/1

4n

)n/sin()n/cos(

4n

A2

n

Pore radius r and edge size L for large n are related by: n

r2L

0

2

2

222

0C 8

gr

n

r4

4

n

8

gKD

Substitution into the rhombic duct equation “recovers” Poiseuille’s law for mean velocity (unit gradient) in a cylindrical tube:

Expressions for flow in partially-filled corners as a function of chemical potential , and corner angle , were based on Ransohoff and Radke (1988) solution to the Navier-Stokes equation:

r()

100

102

104

0 40 80 120 160

Corner Angle [degree]

Flo

w R

esis

tan

ce

Tabulated values of the dimensionless flow resistance as a function of corner angle were parameterized.

dz

dP)(rv

2

Flow in corners bounded by a liquid-vapor interface

c1

dbexp)(

The key lies in the explicit dependence on radius of interfacial curvature r() .

Expressions for thin film flow considering modified viscosity near the solid surface (for thin films h<10 nm) were developed:

with:

For thicker films (h>10 nm) the standard relationships for mean flow velocity vs. film thickness and constant viscosity are used:

for exponential viscosity profile.

dz

dP

)(h12

)(Av

0

)(h

aEi)(ha6a

)(h

aexp)(h4)(ha5)(ha)(A 23322

dz

dP

3

)(hv

0

2

Laminar flow in thin liquid films

Exponential viscosity profile near the solid clay surface was measured by Low (1979).

Flow is thin films (h<10 nm) is strongly modified.

Implications for hydraulic conductivity and flow rates through clay layers.

Modified water viscosity near clay surfaces

1

2

2

2

1

2

1

h

h

k

k

k1,k2 permeabilities [m2]h1,h2 slit spacing [m]1,2 viscosities [Pa s]


Dimensionless flow resistance Viscosity of bulk liquidA() Function for modified viscosityP Hydraulic pressure

dz

dP

3

)(hv

2

dz

dP

)(h12

)(Av

Film Flow h()> 10nm

Film Flow h() 10nm

dz

dP)(rv

2

Corner Flow

Interfacial stability - A critical assumption A critical assumption regarding

stability of equilibrium liquid-vapor interfacial configurations under slow laminar flow...

Indirect “evidence” Time sequence photographs of

water drop formation and detachment from a vertical v-shaped groove. Note l-v interface above the drop remains constant during flow! (Or and Ghezzehei, 1999).

The capillary number (Ca) is a measure of the relative importance of viscous to capillary forces – typical values are in the range of Ca=10-5 for soils (Friedman, 2000)

- 0.26 J/kg - 0.16 J/kg - 0.17 J/kg- 0.15 J/kg - 0.25 J/kg

0

1

[mm

]

- 0.14 J/kg - 0.18 J/kg - 0.25 J/kg - 0.50 J/kg - 1.20 J/kg

4 mm

4 s 0 s 10 s 12 s

cos

vCa


(1) Flow in ducts and between parallel plates for completely liquid-filled pores and slits.

(2) Flow in thin liquid films lining flat surfaces following pore

and slit snap-off.

(3) Flow in corners (bounded by l-v interface) of the central pore.

Corner

Film

Full ductParallelPlates


Parallel Plates (slits)2

0

22 L

12

gLKsKS

2

0

2TT L

80

gLKdKD

2

0

S2SS L

4

BgLKdKD

2

0

n2CC L

8

AgLKdKD

3

)(hg)(KF

2

0

)(h

)(Bg)(KF

120

2

0

)(rg)(KC

Thick Film Flow (h()> 10nm)

Triangular Duct

Square Duct

Circular Duct (tube)

Corner (bounded by l-v)

Thin Film Flow (h()< 10nm)

dz

dp

g

K

A

Qv

Hydraulic Conductivity for a Unit Cell

Saturated and unsaturated hydraulic conductivity for the unit cell was derived by weighting the conductivities of each flow regime over the liquid-occupied cross-sectional areas and dividing by total cross-sectional area (AT) including the solid shell.

Saturated Hydraulic Conductivity

T

2n

2

sat A

KDLAKSL2K

KS Slit hydraulic conductivity

KD Duct hydraulic conductivity (e.g., triangular is given by:

AT Cross sectional area:

2

nT

LA2A

2

0T L

80g

KD

2

0

2

L12

gKS

Full ductFull slits

Unsaturated Hydraulic Conductivity for a Unit Cell

Unsaturated hydraulic conductivity for the unit cell was derived by weighting the conductivities of each flow regime over the liquid-occupied cross-sectional areas and dividing by total cross-sectional area.

After Pore Snap-Off

T

n22

A

)(KCF)(r)(KF)2tan(

)(r2Ln)(hKSL2

)(K

T

n2

A

)(KCF)(r)(KF)2tan(

)(r2LnL4)(h

)(K

After Slit Snap-Off

KS Slit hydraulic conductivityKD Duct hydraulic conductivityKF(m) Film hydraulic conductivity

KC(m) Corner hydraulic conductivity

Corner

Film

- Chemical Potential [J/kg]

10-1 103 107

Rel

ativ

e H

ydra

ulic

Con

duct

ivity

10-15

10-10

10-5

100

FilmCornerTotal

Pore snap-off

Slit snap-off

Unsaturated Hydraulic Conductivity for a Unit Cell

Hydraulic Functions for a Single Unit CellFitted to Measured Data [Hygiene Sandstone]


10-1 100 101 102 103 104

De

gre

e o

f S

atu

rati

on

0.00

0.25

0.50

0.75

1.00

Liquid SaturationRelative Hydraulic

Conductivity


10-1 100 101 102 103 104

Re

lati

ve

Hyd

rau

lic C

on

du

cti

vit

y

10-5

10-4

10-3

10-2

10-1

100

L=0.033 mm, =0.0012, =0.0001

Ks=3.7 m/day(measuredKs=1.1 m/day)

0.00E+00

2.00E+04

4.00E+04

6.00E+04

8.00E+04

1.00E+05

1.20E+05

L1

L2

L3

L4

L5

L6

A statistical approach using Gamma distributed cell sizes (L) is employed for upscaling unit cell expressions for liquid retention and hydraulic conductivity to represent a sample of porous medium.

Upscaled equations for liquid retention were fitted to measured SWC data subject to porosity and SA area constraints.

The resulting best fit parameters are used to predict sample scale saturated and unsaturated hydraulic conductivities.

WetDryL1

L2

L3

L4

L5

21 3

L6

2:withLExp!L)L(f

1

Gamma Distribution for L

f(L)

Slits

Upscaling from pore- to sample-scale

Model input parameters for upscaling

Equivalency between the van Genuchten (1980) model parameters and the new model parameters: Lmax vG

-1 (largest pore determines air entry value).

s (porosity and saturated water content).

r (surface area x film determine residual water content).

n (both parameters determine the shape of statistical PSD).

1) Choice of unit cell shape2) Use of liquid retention data, constrained by:3) Soil porosity, and4) Specific surface area

Possibility of using other types of distributions

Upscaling Results for a Clay Loam Soil

[Source: Pachepsky et al., 1984]


10-1 100 101 102 103 104 105 106

Rel

ativ

e H

ydra

ulic

Co

nd

uct

ivit

y

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

New ModelFilm Flow Corner FlowVG - MualemMeasurements


10-1 100 101 102 103 104 105 106

De

gre

e o

f S

atu

rati

on

0.00

0.25

0.50

0.75

1.00 New ModelCapillary CurveAdsorption CurveVG ModelMeasurements

Relative Hydraulic Conductivity

Liquid Saturation

Upscaling Results for a Sandy Loam Soil[Source: Pachepsky et al., 1984]


10-1 100 101 102 103 104 105 106

Re

lati

ve

Hyd

rau

lic C

on

du

cti

vit

y

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

New ModelFilm Flow Corner FlowVG - MualemMeasurements


10-1 100 101 102 103 104 105 106

De

gre

e o

f S

atu

rati

on

0.00

0.25

0.50

0.75

1.00 New ModelCapillary CurveAdsorption CurveVG ModelMeasurements

Relative Hydraulic Conductivity

Liquid Saturation


10-1 100 101 102 103 104

Re

lati

ve

Hyd

rau

lic C

on

du

cti

vit

y

10-4

10-3

10-2

10-1

100


10-1 100 101 102 103 104

De

gre

e o

f S

atu

rati

on

0.00

0.25

0.50

0.75

1.00New Model = 2New Model = 6VG ModelMeasurements

Fitted Saturation Predicted K(h)

Upscaling Touchet Silt Loam with Variable

[Source: van Genuchten, 1980]

LExp!

L)L(f1

Sample Scale Parameter Estimation Scheme

Input Information

Parameter EstimationResulting Hydraulic

Functions

Hydrodynamic Considerations

An alternative framework for hydraulic conductivity modeling in partially saturated porous media, considering film and corner flow phenomena was developed.

Equilibrium liquid-vapor interfacial configurations for various chemical potentials were used as boundary conditions to solve the Navier-Stokes equations for average velocities in films, corners, ducts, and parallel plates.

Analogy to Darcy’s law was invoked to derive proportionality coefficients between flux and hydraulic gradient representing average hydraulic conductivity for the various flow regimes.

Pore scale expressions were statistically upscaled to represent conductivity of a sample of partially-saturated porous medium.

Summary

pore and sample scale unsaturated hydraulic conductivity for homogeneous porous media dani or and...

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