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Welcome toWelcome toBAE 558BAE 558

Fluid Mechanics ofFluid Mechanics ofPorous MediaPorous Media

Williams, 2008 http://www.its.uidaho.edu/BAE558Modified after Selker, 2000 http://bioe.orst.edu/vzp/

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Outline - IntroductionOutline - Introduction

Introduction to CourseIntroduction to CourseRequired and Related TextsRequired and Related TextsDefinitions: Immiscible Fluids, Phase Definitions: Immiscible Fluids, Phase

Boundaries, Vadose Zone Boundaries, Vadose Zone Related Areas of StudyRelated Areas of StudyHistory of Investigation of Vadose History of Investigation of Vadose

ProcessesProcessesRelationship to Saturated MediaRelationship to Saturated Media

Introduction to CourseIntroduction to CourseRequired and Related TextsRequired and Related TextsDefinitions: Immiscible Fluids, Phase Definitions: Immiscible Fluids, Phase

Boundaries, Vadose Zone Boundaries, Vadose Zone Related Areas of StudyRelated Areas of StudyHistory of Investigation of Vadose History of Investigation of Vadose

ProcessesProcessesRelationship to Saturated MediaRelationship to Saturated Media

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Course OutlineCourse Outline1. An Introduction to the Vadose Zone (3-4 lect.)

• History of investigation• Modern concerns• Relationship to saturated media• Primer on soils

2. Physical & Hydraulic Properties of Unsaturated Media (8 lect.)

• Basic definitions • Hydrostatics (Surface tension;Characteristic curves; Hysteresis)• Hydrodynamics in porous media (Darcy's law; Richards equation)

3. Flow of Water in the Vadose Zone (10 lect.)• The classic solutions (Green & Ampt; Evaporation from Water Table).• Solution for capillary barriers• Miller and Miller scaling• Characterization of soil hydraulic properties

1. An Introduction to the Vadose Zone (3-4 lect.)• History of investigation• Modern concerns• Relationship to saturated media• Primer on soils

2. Physical & Hydraulic Properties of Unsaturated Media (8 lect.)

• Basic definitions • Hydrostatics (Surface tension;Characteristic curves; Hysteresis)• Hydrodynamics in porous media (Darcy's law; Richards equation)

3. Flow of Water in the Vadose Zone (10 lect.)• The classic solutions (Green & Ampt; Evaporation from Water Table).• Solution for capillary barriers• Miller and Miller scaling• Characterization of soil hydraulic properties

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Course Outline ContinuedCourse Outline Continued4. Vadose Biogeochemical Processes (6 lect.)

• Kinetics, Thermodynamics, Equilibria• Biological Processes• Acid Consumptive Processes (Fluid-Rock interactions, ARD,

etc.)

5. Solute Transport in the Vadose Zone (6 lect.)• Processes - Advection, adsorption, diffusion, degradation. • Advective Diffusive Equation (Linearity, superposition,

solutions).

6. Heterogeneity in the Vadose Zone (2 lect.)

4. Vadose Biogeochemical Processes (6 lect.)• Kinetics, Thermodynamics, Equilibria• Biological Processes• Acid Consumptive Processes (Fluid-Rock interactions, ARD,

etc.)

5. Solute Transport in the Vadose Zone (6 lect.)• Processes - Advection, adsorption, diffusion, degradation. • Advective Diffusive Equation (Linearity, superposition,

solutions).

6. Heterogeneity in the Vadose Zone (2 lect.)

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IntroductionsIntroductions

Name Title/Student StatusWork/Research Focus at this time Barbara will introduce the Engineering Outreach

People later in the semester

Note: We will have the emails of all class participants (who agree) listed on the web so that students can communicate among themselves

Name Title/Student StatusWork/Research Focus at this time Barbara will introduce the Engineering Outreach

People later in the semester

Note: We will have the emails of all class participants (who agree) listed on the web so that students can communicate among themselves

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Context re: DisciplinesContext re: Disciplines

Required and Related Texts

Definition/importance of Vadose Zone

Related areas of study

Required and Related Texts

Definition/importance of Vadose Zone

Related areas of study

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Go to other resources…..Go to other resources…..Go to other resources…..Go to other resources…..

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What is a porous medium?What is a porous medium?

Definition of porous mediumDefinition of porous medium

Definition of porosityDefinition of porosity

Fun question….Fun question….

Definition of porous mediumDefinition of porous medium

Definition of porosityDefinition of porosity

Fun question….Fun question….

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Definition of Porous MediumDefinition of Porous Medium

A solid (often called matrix) A solid (often called matrix) permeated by interconnected permeated by interconnected network of pores (voids) filled with a network of pores (voids) filled with a fluid (liquid or gas). fluid (liquid or gas).

Usually both the solid matrix and the Usually both the solid matrix and the pore space are assume to be pore space are assume to be continuous….continuous….

A solid (often called matrix) A solid (often called matrix) permeated by interconnected permeated by interconnected network of pores (voids) filled with a network of pores (voids) filled with a fluid (liquid or gas). fluid (liquid or gas).

Usually both the solid matrix and the Usually both the solid matrix and the pore space are assume to be pore space are assume to be continuous….continuous….

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Definition of PorosityDefinition of Porosity

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QuestionQuestion

Which of these has the largest porosity?

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HISTORY OF INVESTIGATIONHISTORY OF INVESTIGATION

It’s worthwhile to understand the It’s worthwhile to understand the historical context of the study of historical context of the study of unsaturated flow:unsaturated flow:

Variably saturated / vadose zone fluid Variably saturated / vadose zone fluid mechanics is quite a young field still in mechanics is quite a young field still in conceptual developmentconceptual development

Provides a preview of the topics covered Provides a preview of the topics covered in the coursein the course

It’s worthwhile to understand the It’s worthwhile to understand the historical context of the study of historical context of the study of unsaturated flow:unsaturated flow:

Variably saturated / vadose zone fluid Variably saturated / vadose zone fluid mechanics is quite a young field still in mechanics is quite a young field still in conceptual developmentconceptual development

Provides a preview of the topics covered Provides a preview of the topics covered in the coursein the course

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Review: First quantitative Review: First quantitative understanding of saturated flow understanding of saturated flow Darcy 1856 study of the aquifers under Darcy 1856 study of the aquifers under

Dijon; Introduced the concept of potential Dijon; Introduced the concept of potential flowflow

Water moves in direct proportion to:Water moves in direct proportion to:

the gradient of potential energythe gradient of potential energy

the permeability of the mediathe permeability of the media

Darcy 1856 study of the aquifers under Darcy 1856 study of the aquifers under Dijon; Introduced the concept of potential Dijon; Introduced the concept of potential flowflow

Water moves in direct proportion to:Water moves in direct proportion to:

the gradient of potential energythe gradient of potential energy

the permeability of the mediathe permeability of the media

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First quantitative application to First quantitative application to unsaturated flowunsaturated flow

1870’s Bousinnesq extended Darcy’s 1870’s Bousinnesq extended Darcy’s law with two approximations (Dupuit-law with two approximations (Dupuit-Forcheimer) to deal with drainage and Forcheimer) to deal with drainage and filling of media.filling of media.

““Free water surface” problems. Free water surface” problems. Useful solutions for dikes land drainage, Useful solutions for dikes land drainage,

etc. (all as a footnote in his book)etc. (all as a footnote in his book)Bousinnesq equation is strongly Bousinnesq equation is strongly

nonlinear: much tougher to solve! nonlinear: much tougher to solve!

1870’s Bousinnesq extended Darcy’s 1870’s Bousinnesq extended Darcy’s law with two approximations (Dupuit-law with two approximations (Dupuit-Forcheimer) to deal with drainage and Forcheimer) to deal with drainage and filling of media.filling of media.

““Free water surface” problems. Free water surface” problems. Useful solutions for dikes land drainage, Useful solutions for dikes land drainage,

etc. (all as a footnote in his book)etc. (all as a footnote in his book)Bousinnesq equation is strongly Bousinnesq equation is strongly

nonlinear: much tougher to solve! nonlinear: much tougher to solve!

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Rigorous foundation for Rigorous foundation for Darcy’s LawDarcy’s Law

First encyclopedic source of practical First encyclopedic source of practical solutions based on pore-scale analysissolutions based on pore-scale analysis

1899 Schlichter “Theory of Flow Through 1899 Schlichter “Theory of Flow Through Porous Media”Porous Media”

Exact solutions for multiple pumped wellsExact solutions for multiple pumped wells

Basis of aquifer testing.Basis of aquifer testing.

First encyclopedic source of practical First encyclopedic source of practical solutions based on pore-scale analysissolutions based on pore-scale analysis

1899 Schlichter “Theory of Flow Through 1899 Schlichter “Theory of Flow Through Porous Media”Porous Media”

Exact solutions for multiple pumped wellsExact solutions for multiple pumped wells

Basis of aquifer testing.Basis of aquifer testing.

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Extension of Darcy’s Law to Extension of Darcy’s Law to UnsaturatedUnsaturated Conditions Conditions

1907 Buckingham (of Buckingham-pi 1907 Buckingham (of Buckingham-pi fame) Darcy for steady flow with:fame) Darcy for steady flow with:

Conductivity a function of moisture contentConductivity a function of moisture content

Potential includes capillary pressures Potential includes capillary pressures

1907 Buckingham (of Buckingham-pi 1907 Buckingham (of Buckingham-pi fame) Darcy for steady flow with:fame) Darcy for steady flow with:

Conductivity a function of moisture contentConductivity a function of moisture content

Potential includes capillary pressures Potential includes capillary pressures

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Extension of Darcy’s Law Extension of Darcy’s Law (cont.)(cont.)

Rule: Folks who write equations Rule: Folks who write equations are remembered for eternity, are remembered for eternity, while the poor work-a-days who while the poor work-a-days who solve them are quickly forgotten. solve them are quickly forgotten.

Exception: Green and Ampt, Exception: Green and Ampt, 1911. Key problem of infiltration.1911. Key problem of infiltration.

Modeled as a capillary tubes which Modeled as a capillary tubes which filled in parallel, from dry to saturation.filled in parallel, from dry to saturation.

Still most widely used infiltration Still most widely used infiltration model.model.

Rule: Folks who write equations Rule: Folks who write equations are remembered for eternity, are remembered for eternity, while the poor work-a-days who while the poor work-a-days who solve them are quickly forgotten. solve them are quickly forgotten.

Exception: Green and Ampt, Exception: Green and Ampt, 1911. Key problem of infiltration.1911. Key problem of infiltration.

Modeled as a capillary tubes which Modeled as a capillary tubes which filled in parallel, from dry to saturation.filled in parallel, from dry to saturation.

Still most widely used infiltration Still most widely used infiltration model.model.

d

L

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Time passes... Time passes... We need a few tools!!We need a few tools!!

Early 1920’s, W. Gardner’s lab develop the Early 1920’s, W. Gardner’s lab develop the tensiometer: direct measurement of the capillary tensiometer: direct measurement of the capillary pressurepressure

L.A. Richards extended idea to tension plate: L.A. Richards extended idea to tension plate: measure moisture content as a function of measure moisture content as a function of capillary pressurecapillary pressure

And then...And then...

1931, 1931, Richards derived equation for unsaturated Richards derived equation for unsaturated flowflow. (note: Richards just died in late 90’s).. (note: Richards just died in late 90’s).

Early 1920’s, W. Gardner’s lab develop the Early 1920’s, W. Gardner’s lab develop the tensiometer: direct measurement of the capillary tensiometer: direct measurement of the capillary pressurepressure

L.A. Richards extended idea to tension plate: L.A. Richards extended idea to tension plate: measure moisture content as a function of measure moisture content as a function of capillary pressurecapillary pressure

And then...And then...

1931, 1931, Richards derived equation for unsaturated Richards derived equation for unsaturated flowflow. (note: Richards just died in late 90’s).. (note: Richards just died in late 90’s).

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Moisture contents depends on Moisture contents depends on history of wettinghistory of wetting

Haines (1930) wetting Haines (1930) wetting proceeds as “jumps”proceeds as “jumps”

Still largely ignored, but Still largely ignored, but essential to unsaturated essential to unsaturated flow processes.flow processes.

Haines (1930) wetting Haines (1930) wetting proceeds as “jumps”proceeds as “jumps”

Still largely ignored, but Still largely ignored, but essential to unsaturated essential to unsaturated flow processes.flow processes.

Soil

Porous Plate

r1

2r2

stage 1 stages 2 and 3

stages 4 and 5 stages 6 and 7

-2r 2

-2r 1

Moisture Content sat0

0 r

(2)

(5)

(6)

(7)

(4)

(1)

(3)

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Time passes ... time passesTime passes ... time passesTurns out that Richards equation is a bear to solve! Turns out that Richards equation is a bear to solve! Depends on three non-linear variables: q, y, K Depends on three non-linear variables: q, y, K

First big break for R’s Eq.First big break for R’s Eq.

1952, Klute rewrote Richards equation in terms of 1952, Klute rewrote Richards equation in terms of moisture content alonemoisture content alonediffusion equation (AKA: Fokker-Plank eq.)diffusion equation (AKA: Fokker-Plank eq.)

Klute gave solution to 1-D capillary infiltrationKlute gave solution to 1-D capillary infiltration

Turns out that Richards equation is a bear to solve! Turns out that Richards equation is a bear to solve! Depends on three non-linear variables: q, y, K Depends on three non-linear variables: q, y, K

First big break for R’s Eq.First big break for R’s Eq.

1952, Klute rewrote Richards equation in terms of 1952, Klute rewrote Richards equation in terms of moisture content alonemoisture content alonediffusion equation (AKA: Fokker-Plank eq.)diffusion equation (AKA: Fokker-Plank eq.)

Klute gave solution to 1-D capillary infiltrationKlute gave solution to 1-D capillary infiltration

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Analytical vs. NumericalAnalytical vs. Numerical

Since 1952, more analytical solutions have Since 1952, more analytical solutions have been presented, BUT non-linearity limited to been presented, BUT non-linearity limited to special conditions.special conditions.

What is the use of Analytical results?What is the use of Analytical results?They let you see the implications of the physical They let you see the implications of the physical

parameters parameters

computers allow solution of individual problems: computers allow solution of individual problems: tough to generalizetough to generalize

Since 1952, more analytical solutions have Since 1952, more analytical solutions have been presented, BUT non-linearity limited to been presented, BUT non-linearity limited to special conditions.special conditions.

What is the use of Analytical results?What is the use of Analytical results?They let you see the implications of the physical They let you see the implications of the physical

parameters parameters

computers allow solution of individual problems: computers allow solution of individual problems: tough to generalizetough to generalize

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Then things took off! Then things took off!

Lots of great stuff in the 50’s and early Lots of great stuff in the 50’s and early 60’s60’s1956: Miller and Miller: relationship of 1956: Miller and Miller: relationship of

grain size to fluid propertiesgrain size to fluid properties

Lots of great stuff in the 50’s and early Lots of great stuff in the 50’s and early 60’s60’s1956: Miller and Miller: relationship of 1956: Miller and Miller: relationship of

grain size to fluid propertiesgrain size to fluid properties

Degree of Saturation

Mat

ric

Pot

enti

al (

cm H

2O)

0

10

20

30

40

50

60

70

80

0 0.2 0.4 0.6 0.8 1

12/20

20/30

30/40

40/50

Degree of Saturation

Sca

led

Pot

enti

al

0

10

20

30

40

50

60

0 0.2 0.4 0.6 0.8 1

12/20

20/30

30/40

40/50

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More 50’s and 60’sMore 50’s and 60’s

1957: Philip start to deal with infiltration1957: Philip start to deal with infiltration

1962: Poulovassilis: independent 1962: Poulovassilis: independent domain model of hysteresis (finally domain model of hysteresis (finally Haines stuff can be included)Haines stuff can be included)

1957: Philip start to deal with infiltration1957: Philip start to deal with infiltration

1962: Poulovassilis: independent 1962: Poulovassilis: independent domain model of hysteresis (finally domain model of hysteresis (finally Haines stuff can be included)Haines stuff can be included)

h

(Fill

ing

Pressu

re)

00

h (Emptying Pressure)

e

f

45o

f

(Fra

ctio

n of

Moi

stur

e C

onte

nt)

he h +he e

hf

h +hf f

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1970’s – to now: limitations of 1970’s – to now: limitations of the assumptions the assumptions

Biggar & Nielson (1970)Biggar & Nielson (1970) field scale heterogeneity field scale heterogeneity

Hill & Parlange (1972)Hill & Parlange (1972)fingered flow fingered flow

Others: Others: macroporesmacroporesKung (1988): Funnel FlowKung (1988): Funnel FlowStochastics – small-scale to large-scaleStochastics – small-scale to large-scale

Biggar & Nielson (1970)Biggar & Nielson (1970) field scale heterogeneity field scale heterogeneity

Hill & Parlange (1972)Hill & Parlange (1972)fingered flow fingered flow

Others: Others: macroporesmacroporesKung (1988): Funnel FlowKung (1988): Funnel FlowStochastics – small-scale to large-scaleStochastics – small-scale to large-scale

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Relationship to saturated mediaRelationship to saturated media

While the similarity has been very useful, it While the similarity has been very useful, it is a source of many errors is a source of many errors

Main distinctions in three areas. Main distinctions in three areas. Capillarity (lateral, upward flow)Capillarity (lateral, upward flow)Heterogeneity into the temporal domainHeterogeneity into the temporal domainBiochemical activityBiochemical activity

Diffusion is two orders of magnitude fasterDiffusion is two orders of magnitude fasterAmple oxygenAmple oxygen

Take-home message: be very careful!Take-home message: be very careful!

While the similarity has been very useful, it While the similarity has been very useful, it is a source of many errors is a source of many errors

Main distinctions in three areas. Main distinctions in three areas. Capillarity (lateral, upward flow)Capillarity (lateral, upward flow)Heterogeneity into the temporal domainHeterogeneity into the temporal domainBiochemical activityBiochemical activity

Diffusion is two orders of magnitude fasterDiffusion is two orders of magnitude fasterAmple oxygenAmple oxygen

Take-home message: be very careful!Take-home message: be very careful!

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Issue Vadose Zone Saturated ZoneConductivity at apoint

A nonlinear function ofmoisture content.

Constant.

Density Effects Negligible influence fortemperature and solutebased changes.

Temperature and solutebased density differences candominate both static anddynamic disposition.

Spatial variability Lognormal distributionand a function of moisturecontent and hence time.

Lognormal distribution fixedin time.

Biological andchemical activity

Often high in carbon andoxygen, leading to rapidmicrobial metabolism.

Typically anoxic with sparsecarbon: comparatively slowmicrobial activity.

Transportmechanisms

Advection 0-10 cm/dayDispersivity 0.5-20 cmDiffusion 0.1-.3 cm2/s

Advection 0-100 cm/dayDispersivity 0.5-20 cmDiffusion 0.00002 cm2/s

Similarities:• Both have governing equations for flow that are

linear in the local potential gradient• They share similar constitutive media (with

particles ranging from clay to gravel)

Differences

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Contemporary Concerns with Contemporary Concerns with the Vadose Zonethe Vadose Zone

Water conservation (how to use Water conservation (how to use minimum water to irrigate crops)minimum water to irrigate crops)

Nutrient storage and transportNutrient storage and transportContaminant degradation and Contaminant degradation and

movementmovementWater budget for climatic modelingWater budget for climatic modelingBulk petroleum and organic contaminant Bulk petroleum and organic contaminant

transport (vapor and liquid): Industrial transport (vapor and liquid): Industrial contaminationcontamination

Water conservation (how to use Water conservation (how to use minimum water to irrigate crops)minimum water to irrigate crops)

Nutrient storage and transportNutrient storage and transportContaminant degradation and Contaminant degradation and

movementmovementWater budget for climatic modelingWater budget for climatic modelingBulk petroleum and organic contaminant Bulk petroleum and organic contaminant

transport (vapor and liquid): Industrial transport (vapor and liquid): Industrial contaminationcontamination