1 Welcome to BAE 558 Fluid Mechanics of Porous Media Williams, Modified after Selker,

2 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  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

3 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 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

4 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.) 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.)

5 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 s of all class participants (who agree) listed on the web so that students can communicate among themselves 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 s of all class participants (who agree) listed on the web so that students can communicate among themselves

6 Context 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

7 Go to other resources…..

8 What is a porous medium? Definition of porous medium Definition of porosity Fun question…. Definition of porous medium Definition of porosity Fun question….

9 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…. 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….

10 Definition of Porosity

11 Question Which of these has the largest porosity?

12 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 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

13 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  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

14 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!  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!

15 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. 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.

16 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  1907 Buckingham (of Buckingham-pi fame) Darcy for steady flow with:  Conductivity a function of moisture content  Potential includes capillary pressures

17 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, Key problem of infiltration.  Exception: Green and Ampt, Key problem of infiltration.  Modeled as a capillary tubes which filled in parallel, from dry to saturation.  Still most widely used infiltration model.  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, Key problem of infiltration.  Exception: Green and Ampt, Key problem of infiltration.  Modeled as a capillary tubes which filled in parallel, from dry to saturation.  Still most widely used infiltration model.

18 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).  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).

19 Moisture contents depends on history of wetting  Haines (1930) wetting proceeds as “jumps”  Haines (1930) wetting proceeds as “jumps”  Still largely ignored, but essential to unsaturated flow processes.  Haines (1930) wetting proceeds as “jumps”  Haines (1930) wetting proceeds as “jumps”  Still largely ignored, but essential to unsaturated flow processes.

20 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 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

21 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  They let you see the implications of the physical parameters  computers allow solution of individual problems: tough to generalize 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  They let you see the implications of the physical parameters  computers allow solution of individual problems: tough to generalize

22 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 Lots of great stuff in the 50’s and early 60’s  1956: Miller and Miller: relationship of grain size to fluid properties

23 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)  1957: Philip start to deal with infiltration  1962: Poulovassilis: independent domain model of hysteresis (finally Haines stuff can be included)

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

25 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! 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!

26 Differences

27 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  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