Title: 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
1Pore 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)
2Review of pore scale hydrostatics
- Unitary approach for capillarity and adsorption
- Liquid films and the disjoining pressure concept.
- New model for basic pore geometry.
- Liquid configurations for different potentials
are obtained by simplified AYL equation where
interface curvature r(?) in pore corners
isshifted by film thickness h(?). - Expressions for unit pore retention and
liquid-vapor interfacial areas were statistically
upscaled to represent Gamma distributed pore
populations. - 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.
films
corners/ full pores
3Clarification 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).
4The trapped microbe problem
- 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.
5Outline section 3 Hydrodynamics in homogeneous
porous media
- Estimation of capillary size distribution for
hydrodynamic considerations - Statistical application of Poiseuilles law and
the standard BCC approach (cross-section only!) - Coupling flow in tubes with Darcys 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
6BCC-based prediction of unsaturated hydraulic
conductivity K(?)
- Extraction of radii distribution of capillary
radii of the BCC (from retention data). - Application of hydrodynamic considerations, i.e.,
the volumetric discharge in a cylindrical tube is
proportional to the 4th power of tube radius
(Poiseuilles law).
(1)
(2)
L
r
P2
P1
7BCC-based prediction of unsaturated hydraulic
conductivity K(?) (cont.)
- Statistical application of Poiseuilles law for a
bundle of capillaries coupled with Darcys
macroscopic flow equation Childs and
Collis-George, 1950 Fatt and Dykstra, 1951
Burdine, 1953 Mualem, 1976 van Genuchten,
1980. - 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).
Ks
nj
8BCC-based prediction of unsaturated hydraulic
conductivity K(?) (cont.)
- 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. - The concept of tortousity (Lc/L) improves BCC
model predictions.
(Mualem, 1976)
Lc
L
9Hydrodynamic 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 Darcys law is invoked to identify
the coefficient of proportionality between flux
and hydraulic gradient as the hydraulic
conductivity for each flow regime under
consideration.
10Primary 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.
11Hydraulic conductivity of full ducts
with Bs (L1L2) given as
12Hydraulic conductivity in high-order
polygons/rhombic pores
Rhombic Duct (tube)
The area constant An is given as
As n??
Pore radius r and edge size L for large n are
related by
Substitution into the rhombic duct equation
recovers Poiseuilles law for mean velocity
(unit gradient) in a cylindrical tube
13Flow in corners bounded by a liquid-vapor
interface
- 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
- The key lies in the explicit dependence on radius
of interfacial curvature r(?) .
- Tabulated values of the dimensionless flow
resistance ? as a function of corner angle ?
were parameterized.
14Laminar flow in thin liquid films
- Expressions for thin film flow considering
modified viscosity near the solid surface (for
thin films hlt10 nm) were developed - with
for exponential viscosity profile.
- For thicker films (hgt10 nm) the standard
relationships for mean flow velocity vs. film
thickness and constant viscosity are used
15Modified water viscosity near clay surfaces
- Exponential viscosity profile near the solid clay
surface was measured by Low (1979). - Flow is thin films (hlt10 nm) is strongly
modified. - Implications for hydraulic conductivity and flow
rates through clay layers. -
k1,k2 permeabilities m2 h1,h2 slit spacing
m ?1,?2 viscosities Pa s
16Primary Flow Regimes in a Unit Cell
Film Flow h(m )gt 10nm
Film Flow h(m )? 10nm
e Dimensionless flow resistance h Viscosity of
bulk liquid A(m) Function for modified
viscosity P Hydraulic pressure
17Interfacial 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
Ca10-5 for soils (Friedman, 2000)
18Primary 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.
19Primary Flow Regimes in a Unit Cell
Parallel Plates (slits)
Triangular Duct
Square Duct
Circular Duct (tube)
Corner (bounded by l-v)
Thick Film Flow (h(m )gt 10nm)
Thin Film Flow (h(m )lt 10nm)
20Hydraulic 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
KS Slit hydraulic conductivity KD Duct hydraulic
conductivity (e.g., triangular is given
by AT Cross sectional area
21Unsaturated 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.
Corner
After Pore Snap-Off
After Slit Snap-Off
Film
KS Slit hydraulic conductivity KD Duct hydraulic
conductivity KF(m) Film hydraulic conductivity
KC(m) Corner hydraulic conductivity
22Unsaturated Hydraulic Conductivity for a Unit Cell
23Hydraulic Functions for a Single Unit CellFitted
to Measured Data Hygiene Sandstone
Relative Hydraulic Conductivity
Liquid Saturation
L0.033 mm, ?0.0012, ?0.0001
Ks3.7 m/day (measured Ks1.1 m/day)
24Upscaling from pore- to sample-scale
- 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.
Gamma Distribution for L
Slits
f(L)
Wet
Dry
L1
L2
L3
L4
L5
L6
m2
m1
m3
25Model input parameters for upscaling
- Choice of unit cell shape
- Use of liquid retention data, constrained by
- Soil porosity, and
- Specific surface areaPossibility of using other
types of distributions
- 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).
26Upscaling Results for a Clay Loam SoilSource
Pachepsky et al., 1984
Relative Hydraulic Conductivity
Liquid Saturation
27Upscaling Results for a Sandy Loam SoilSource
Pachepsky et al., 1984
Relative Hydraulic Conductivity
Liquid Saturation
28Upscaling Touchet Silt Loam with Variable
?Source van Genuchten, 1980
Fitted Saturation
Predicted K(h)
29Sample Scale Parameter Estimation Scheme
30Summary
- 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 Darcys 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.