Welcome to BAE 558 Fluid Mechanics of Porous Media

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Welcome toBAE 558Fluid Mechanics ofPorous Media
Williams, 2008
http//www.its.uidaho.edu/BAE558 Modified after
Selker, 2000 http//bioe.orst.edu
/vzp/
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Outline - Introduction

• Introduction to Course
• Required and Related Texts
• Definitions Immiscible Fluids, Phase Boundaries,
  Vadose Zone
• Related Areas of Study
• History of Investigation of Vadose Processes
• Relationship to Saturated Media

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Course Outline

• 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 tensionCharacteristic
  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 Continued

• 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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Introductions

• Name
• Title/Student Status
• Work/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 Disciplines

• Required and Related Texts
• Definition/importance of Vadose Zone
• Related areas of study

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• Go to other resources..

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What is a porous medium?

• Definition of porous medium
• Definition of porosity
• Fun question.

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Definition of Porous Medium

• A solid (often called matrix) permeated by
  interconnected network of pores (voids) filled
  with a fluid (liquid or gas).
• Usually both the solid matrix and the pore space
  are assume to be continuous.

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Definition of Porosity
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Question
Which of these has the largest porosity?
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HISTORY OF INVESTIGATION

• Its worthwhile to understand the historical
  context of the study of unsaturated flow
• Variably saturated / vadose zone fluid mechanics
  is quite a young field still in conceptual
  development
• Provides a preview of the topics covered in the
  course

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Review First quantitative understanding of
saturated flow

• Darcy 1856 study of the aquifers under Dijon
  Introduced the concept of potential flow
• Water moves in direct proportion to
• the gradient of potential energy
• the permeability of the media

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First quantitative application to unsaturated flow

• 1870s Bousinnesq extended Darcys law with two
  approximations (Dupuit-Forcheimer) to deal with
  drainage and filling of media.
• Free water surface problems.
• Useful solutions for dikes land drainage, etc.
  (all as a footnote in his book)
• Bousinnesq equation is strongly nonlinear much
  tougher to solve!

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Rigorous foundation for Darcys Law

• First encyclopedic source of practical solutions
  based on pore-scale analysis
• 1899 Schlichter Theory of Flow Through Porous
  Media
• Exact solutions for multiple pumped wells
• Basis of aquifer testing.

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Extension of Darcys Law to Unsaturated Conditions

• 1907 Buckingham (of Buckingham-pi fame) Darcy for
  steady flow with
• Conductivity a function of moisture content
• Potential includes capillary pressures

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Extension of Darcys Law (cont.)

• Rule Folks who write equations are remembered
  for eternity, while the poor work-a-days who
  solve them are quickly forgotten.
• Exception Green and Ampt, 1911. Key problem of
  infiltration.
• Modeled as a capillary tubes which filled in
  parallel, from dry to saturation.
• Still most widely used infiltration model.

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

• Early 1920s, W. Gardners lab develop the
  tensiometer direct measurement of the capillary
  pressure
• L.A. Richards extended idea to tension plate
  measure moisture content as a function of
  capillary pressure
• And then...
• 1931, Richards derived equation for unsaturated
  flow. (note Richards just died in late 90s).

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Moisture contents depends on history of wetting

• Haines (1930) wetting proceeds as jumps
• Still largely ignored, but essential to
  unsaturated flow processes.

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Time passes ... time passes

• Turns out that Richards equation is a bear to
  solve! Depends on three non-linear variables q,
  y, K
• First big break for Rs Eq.
• 1952, Klute rewrote Richards equation in terms of
  moisture content alone
• diffusion equation (AKA Fokker-Plank eq.)
• Klute gave solution to 1-D capillary infiltration

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Analytical vs. Numerical

• Since 1952, more analytical solutions have been
  presented, BUT non-linearity limited to special
  conditions.
• What is the use of Analytical results?
• They let you see the implications of the physical
  parameters
• computers allow solution of individual problems
  tough to generalize

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Then things took off!

• Lots of great stuff in the 50s and early 60s
• 1956 Miller and Miller relationship of grain
  size to fluid properties

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More 50s and 60s

• 1957 Philip start to deal with infiltration
• 1962 Poulovassilis independent domain model of
  hysteresis (finally Haines stuff can be included)

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1970s to now limitations of the assumptions

• Biggar Nielson (1970)
• field scale heterogeneity
• Hill Parlange (1972)
• fingered flow
• Others
• macropores
• Kung (1988) Funnel Flow
• Stochastics small-scale to large-scale

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Relationship to saturated media

• While the similarity has been very useful, it is
  a source of many errors
• Main distinctions in three areas.
• Capillarity (lateral, upward flow)
• Heterogeneity into the temporal domain
• Biochemical activity
• Diffusion is two orders of magnitude faster
• Ample oxygen
• Take-home message be very careful!

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Differences
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Contemporary Concerns with the Vadose Zone

• Water conservation (how to use minimum water to
  irrigate crops)
• Nutrient storage and transport
• Contaminant degradation and movement
• Water budget for climatic modeling
• Bulk petroleum and organic contaminant transport
  (vapor and liquid) Industrial contamination