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)
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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;
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.
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)
(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
Primary Flow Regimes in a Unit Cell
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]
- Chemical Potential [J/kg]
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
- Chemical Potential [J/kg]
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]
- Chemical Potential [J/kg]
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
- Chemical Potential [J/kg]
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]
- Chemical Potential [J/kg]
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
- Chemical Potential [J/kg]
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
- Chemical Potential [J/kg]
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
- Chemical Potential [J/kg]
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.