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## 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/**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**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 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**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.)**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**Context re: Disciplines**• Required and Related Texts • Definition/importance of Vadose Zone • Related areas of study**What is a porous medium?**• Definition of porous medium • Definition of porosity • Fun question….**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….**Question**Which of these has the largest porosity?**HISTORY OF INVESTIGATION**• It’s 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**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**First quantitative application to unsaturated flow**• 1870’s Bousinnesq extended Darcy’s 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!**Rigorous foundation for Darcy’s 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.**Extension of Darcy’s 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**Extension of Darcy’s 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.**Time passes... We need a few tools!!**• Early 1920’s, W. Gardner’s 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 90’s).**Moisture contents depends on history of wetting**• Haines (1930) wetting proceeds as “jumps” • Still largely ignored, but essential to unsaturated flow processes.**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 R’s 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**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**Then things took off!**• Lots of great stuff in the 50’s and early 60’s • 1956: Miller and Miller: relationship of grain size to fluid properties**More 50’s and 60’s**• 1957: Philip start to deal with infiltration • 1962: Poulovassilis: independent domain model of hysteresis (finally Haines stuff can be included)**1970’s – 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**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!**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