Red: Ch 2. nd 2.2 PART 4 Flow in porous medi Drcy s lw Imgine point (A) in column of wter (figure below); the point hs following chrcteristics: () elevtion z (2) pressure p (3) velocity v (4) density ρ Point A hs some energy which cn be described s sum of potentil, kinetic nd elstic energies: z A ρ, v, p ψ h dtum, z, pp (p tmospheric) Potentil energy mgz Kinetic energy 2 --mv2 Elstic energy p m dp m ---- --- ( p p ρ ρ ) p [ner equlity in (3) is true under ssumption of incompressible fluid]. Totl energy mgz + --mv 2 + m 2 --- ( ρ p p ) Hydrogeology, 43/53 - University of Arizon - Fll 22 Dr. Mrek Zred
Flow in porous medi 6 Energy per unit mss is clled fluid potentil: Φ gz 2 --v 2 p + + --- p ρ Becuse velocity (v) is low in porous medi, the kinetic energy term is smll, tht is, mv 2, nd we cn write Φ p p gz + --- ρ Pressure t point A is p ρgψ + p where ψ iswter column bove A nd p is the tmospheric pressure. We now hve ρgψ + p Φ gz p + --- gz + g( h z) gz+ gh gz ρ Φ gh Fluid potentil Fluid t point A hs potentil Φ gh. Fluid will flow from point of higher potentil to point of lower potentil. Dividing by g (which cn be ssumed constnt), we get hydrulic hed: h ψ + z Hydrulic hed Hydrulic hed hs two components: ψ pressure hed (due to pressure of wter bove point A) z elevtion hed (due to elevtion of point A bove the dtum) Hydrogeology, 43/53 - University of Arizon - Fll 22 Dr. Mrek Zred
Flow in porous medi 7 Drcy s experiment Drcy observed tht flow is: h 2 z, p ; y ;; yy ;; ; ; A cross-sectionl re h () proportionl to hed difference dh (2) inversely proportionl to column length dl (3) proportionl to cross-sectionl re A dl dtum: z z 2 z 2, p 2 Add proportionlity constnt (K), which depends on properties of fluid nd properties of soil (porous medium), to get Drcy s lw: KA- dh KA h dl Drcy s lw h hydrulic hed [L] A column cross-sectionl re [L 2 ] L column length [L] K hydrulic conductivity [LT - ] flow rte [L 3 T - ] hydrulic grdient -dh/dl [-] Drcy s lw per unit re: q --- K- dh K h A dl q specific dischrge [LT - ] Is trvel time t distnce/q? Why not? Flow velocities re fster thn the specific dischrge becuse flow occurs in pores only. Therefore, we hve seepge (liner) velocity: q --- K- dh K h A dl This is verge mcroscopic velocity of wter. Hydrogeology, 43/53 - University of Arizon - Fll 22Dr. Mrek Zred
Flow in porous medi 8 Components of hydrulic hed h ψ + z totl hed pressure hed + elevtion hed Go bck to our Drcy s experiment. Plot ψ, z, h s function of l: h, ψ, z h z ψ 2, h 2 ψ z 2 h z ψ l l l 2 l l 2 h h h 2 z z z 2 ψ h - z h 2 - z 2 Procedure: determine h, determine z, clculte ψ. We ssumed the following: () uniform medi (2) incompressible fluid (3) wter only (single phse flow) (4) isotherml fluid (5) constnt cross-sectionl re (6) stedy flow (7) lminr flow Vlidity of Drcy s lw Upper limit of Drcy s lw Look t lminr flow ssumption (determined using the Reynolds number R e ): R e inertil forces forces due to ccelertion --- viscous forces forves due to friction inerti mss ccelertion ccelertion velocity / time inerti mss velocity / time [M L T -2 ] mss density length 3 Hydrogeology, 43/53 - University of Arizon - Fll 22 Dr. Mrek Zred
Flow in porous medi 9 viscous forces viscosity velocity length [M L T -2 ] density length 2 velocity R e - time dynmic viscosity velocity length density length velocity ---- dynmic viscosity R e Lvρ ---- µ dimensionless number L some chrcteristic length of pore size, such s d 5 (medin grin size), d (-th percentile), or k, nd v wter velocity. Rules: Re lminr flow; Drcy s lw pplies q < Re trnsition zone; Drcy s lw is questionble Re > turbulent flow; Drcy s lw does not pply tn K Lower limit of Drcy s lw R e R e dh/dl In fine soils, such s clys, there exists minimum grdient below which flow does not occur. q Explntions: () smll pores; wter molecules strongly influenced by electricl chrge on solid prticles; leds to incresed effective viscosity. (2) streming potentil; wter crries ctions which re ttrcted to solid surfces nd slow the movement of wter. (3) non-newtonin fluid; only in cpillry spces. (4) electrosttic counterflow. minimum grdient dh/dl Hydrogeology, 43/53 - University of Arizon - Fll 22Dr. Mrek Zred
Flow in porous medi 2 Nvier-Stokes eqution for flow in tube p p 2 A(r, u, x) x r u dx Tube long x Rdius Rdil (cylindricl) coordintes: rdil distnce r, ngle u, (xil distnce x) Assume flow prllel to x nd stedy stte; no r nd u components. Nvier-Stokes equtions (Ber, 972 ; de Mrsily, 986 2 ) reduce to: Multiply both sides by r to get: p µ -- V x + r r -- r r p r µ V x + r -- r r -D, stedy stte flow BC: velocity is zero t the wll of the tube (due to viscosity nd dhesive wter) V x @ r BC2: velocity is mximum in the center of the tube (due to symmetry of V x in the tube) V x / r @ r. Ber, J., 972, Dynmics of fluids in porous medi, Elsevier (reprinted by Dover). 2. de Mrsily, G., 986, untittive hydrogeology, Acdemic Press. Hydrogeology, 43/53 - University of Arizon - Fll 22 Dr. Mrek Zred
Optionl reding: J. Ber, 972, Dynmics of fluids in Flow in porous medi porous medi, Ch. 5.; G. de Mrsily, 986, Ch. 3. 2 First integrtion gives: Use BC2: + C then C r 2 p 2 µr + -- Vx C r Plug C in nd divide by r, nd the eqution becomes: Second integrtion gives: Use BC: substitute for C 2 : nd the solution becomes: r p 2 µ + -- Vx r r 2 p + µv 4 x C 2 2 p C 4 2 r 2 p + µv 4 x 2 p 4 V x - ( r 2 2 ) p rdil distribution of velocity 4µ Note: some informtion hs been removed so tht you would hve to do your own derivtions. Hydrogeology, 43/53 - University of Arizon - Fll 22Dr. Mrek Zred
Flow in porous medi 22 The dischrge through the tube is clculted by integrting the velocity over the tube re: V x da V x d( πr 2 ) V x 2πrdr 2πr- ( r 2 2 ) p π dr 4µ 2µ - p ( r 3 2 r) dr π - 2µ p --r 4 4 2 --2 r 2 - π 2µ p -- 4 4 2 --4 - π p -- 4 2µ 4 π 8µ - 4 p Totl dischrge through tube of rdius Hydrogeology, 43/53 - University of Arizon - Fll 22 Dr. Mrek Zred
Flow in porous medi 23 We cn clculte dischrge per unit re, or specific dischrge, q. It is often used insted of totl dischrge becuse it is independent of the size of the nlyzed system nd thus, it is useful in comprison of different flow systems (for exmple different quifers or different lbortory columns). q --- A --- π 2 q -- p specific dischrge 8µ 2 Significnce of this result: () Flow depends on 2, /µ nd p/ (2) Velocity is zero t tube wll (3) Velocity is mximum t tube center Exercise: Compre the bove expression for q with tht from Drcy s lw (p. 7). Look t ech term (do term-by-term comprison). Wht is the hydrulic conductivity (K) in the solution of the Nvier- Stokes eqution? Wht re components of hydrulic conductivity? Hint: write pressure (p) in terms of hydrulic hed (h) to obtin comprble expressions for q from Drcy s lw nd from the Nvier-Stokes eqution. Hydrogeology, 43/53 - University of Arizon - Fll 22Dr. Mrek Zred
Flow in porous medi 24 Fctors controlling flow in porous medi Bck to solution of Nvier-Stokes eqution in its form integrted over cross-sectionl re: π 4 - p 8µ Now, mke medium from m prllel tubes in cube with side length b. Side re:bb 2 b b Are of tubes:a mπ 2 Define porosity:n void spce / totl spce n A/B mπ 2 /B Then:m nb/π 2 Totl flux: T m nb/π 2 T nb --- π 2 π 4 p 8µ T B Observe tht flow is proportionl to: () totl re of the medium (2) porosity (3) pore size (4) externl grdient (5) inverse of viscosity (6) /8, which is constnt vlid for circulr tube only. Other pore geometries will hve different constnts. n 2 -- 8µ p Hydrogeology, 43/53 - University of Arizon - Fll 22 Dr. Mrek Zred