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 - PowerPoint PPT Presentation

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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

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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


1
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)
2
Review of pore scale hydrostatics
  1. Unitary approach for capillarity and adsorption
  2. Liquid films and the disjoining pressure concept.
  3. New model for basic pore geometry.
  4. Liquid configurations for different potentials
    are obtained by simplified AYL equation where
    interface curvature r(?) in pore corners
    isshifted by film thickness h(?).
  5. Expressions for unit pore retention and
    liquid-vapor interfacial areas were statistically
    upscaled to represent Gamma distributed pore
    populations.
  6. 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
3
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).

4
The 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.

5
Outline 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

6
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
    (Poiseuilles law).

(1)
(2)
L
r
P2
P1
7
BCC-based prediction of unsaturated hydraulic
conductivity K(?) (cont.)
  1. 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.
  2. 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
8
BCC-based prediction of unsaturated hydraulic
conductivity K(?) (cont.)
  1. 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.
  2. The concept of tortousity (Lc/L) improves BCC
    model predictions.

(Mualem, 1976)
Lc
L
9
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 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.

10
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.
11
Hydraulic conductivity of full ducts
  • Triangular Duct
  • Square Duct

with Bs (L1L2) given as
  • Rhombic Duct (? tube)

12
Hydraulic 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
13
Flow 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.

14
Laminar 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

15
Modified 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
16
Primary 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
17
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
    Ca10-5 for soils (Friedman, 2000)

18
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.
19
Primary 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)
20
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
KS Slit hydraulic conductivity KD Duct hydraulic
conductivity (e.g., triangular is given
by AT Cross sectional area
21
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.

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
22
Unsaturated Hydraulic Conductivity for a Unit Cell
23
Hydraulic 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)
24
Upscaling 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
25
Model input parameters for upscaling
  1. Choice of unit cell shape
  2. Use of liquid retention data, constrained by
  3. Soil porosity, and
  4. 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).

26
Upscaling Results for a Clay Loam SoilSource
Pachepsky et al., 1984
Relative Hydraulic Conductivity
Liquid Saturation
27
Upscaling Results for a Sandy Loam SoilSource
Pachepsky et al., 1984
Relative Hydraulic Conductivity
Liquid Saturation
28
Upscaling Touchet Silt Loam with Variable
?Source van Genuchten, 1980
Fitted Saturation
Predicted K(h)
29
Sample Scale Parameter Estimation Scheme
30
Summary
  • 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.
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