Updated: 25 October 2015 Print version CEE 370 Environmental Engineering Principles Lecture #22 Water Resources & Hydrology II: Wells, Withdrawals and Contaminant Transport Reading: Mihelcic & Zimmerman, Chapter 7 David Reckhow CEE 370 L#22 1
Darcy s Law Groundwater flow, or flow through porous media Used to determine the rate at which water or other fluids flow in the sub-surface region Also applicable to flow through engineered system having pores Air Filters Sand beds Packed towers David Reckhow CEE 370 L#23 2
Groundwater flow Balance of forces, but frame of reference is reversed Water flowing though a field of particles David Reckhow CEE 370 L#23 3
Terminology Head Height to which water rises within a well At water table for an unconfined aquifer Above water table for confined aquifers Hydraulic Gradient The difference in head between two points in a aquifer separated in horizontal space Hydraulic Gradient dh dx David Reckhow CEE 370 L#23 4
Terminology Porosity The fraction of total volume of soil or rock that is empty pore space Typical values 5-30% for sandstone rock 25-50% for sand deposits 5-50% for Karst limestone formations 40-70% for clay deposits η volumes of pores total volume David Reckhow CEE 370 L#23 5
Darcy s Law Obtained theoretically by setting drag forces equal to resistive forces Determined experimentally by Henri Darcy (1803-1858) L, or Q KA dh dx h K L A Flow per unit crosssectional area is directly proportional to the hydraulic gradient David Reckhow CEE 370 L#23 6
Hydraulic Conductivity, K Proportionality constant between hydraulic gradient and flow/area ratio A property of the medium through which flow is occurring (and of the fluid) Very High for gravel: 0.2 to 0.5 cm/s High for sand: 3x10-3 to 5x10-2 cm/s Low for clays: ~2x10-7 cm/s Almost zero for synthetic barriers: <10-11 for high density polyethylene membranes Measured by pumping tests David Reckhow CEE 370 L#23 7
Hydraulic Conductivity - Table David Reckhow CEE 370 L#23 8
Darcy Velocity just a re-arrangement of Darcy s Law Q dh Q v d K or v A dx A Not the true (or linear or seepage) velocity of groundwater flow because flow can only occur in pores combining v true L τ L ηv Q QL ηv 1 η Q A h K L 1 ' 1 v true v d or v water v η η D&M 2 nd ed David Reckhow CEE 370 L#23 9
Velocities Illustrated Pipe with soil core Empty Soil Empty Q Q Darcy Velocity v True Velocity v η Darcy Velocity v Water Velocity Distance David Reckhow CEE 370 L#22 10
Alternative illustration David Reckhow CEE 370 L#23 11
Example 7-9 An aquifer material of coarse sand has piezometric surfaces of 10 cm and 8 cm above a datum and these are spaced 10 cm apart. If the cross sectional area is 10 cm 2, what is the linear velocity of the water? h 10cm 8cm Hydraulic gradient: 0.2cm L 10cm cm From Table 7-4, K for coarse sand is 5.2 x 10-4, so the Darcy velocity is: v K h L ( ) 4 m cm 4 5.2x10 0.2 1.04x10 m s cm s Assuming that the porosity is 30% or 0.3 (Table 7-4): 4 m v 1.04x10 s 3.47x10 m η 0.3 s Note a different K was used in text ' 4 v water David Reckhow CEE 370 L#22 12
Definitions Specific Yield the fraction of water in an aquifer that will drain by gravity Less than porosity due to capillary forces See Table 7-5 in D&M for typical values Transmissibility (T) flow expected from a 1 m wide cross section of aquifer (full depth) when the hydraulic gradient is 1 m/m. TK/D Where D is the aquifer depth and K is hydraulic conductivity David Reckhow CEE 370 L#22 13
Drawdown I Unconfined aquifer D&M: Figure 7-31a Showing cone of depression David Reckhow CEE 370 L#22 14
Drawdown II Confined aquifer D&M: Figure 7-31b David Reckhow CEE 370 L#22 15
Cones of Depression Conductivity Low K Deep, shallow cone overlapping David Reckhow CEE 370 L#22 16
Flow Model Well in confined aquifer 2 KD( h2 h1 ) Q π ln( r / r ) 2 1 Where: h x is the height of the piezometric surface at distance r x from the well In an unconfined aquifer D is replaced by average height of water table (h 2 +h 1 )/2, so: Q K π ln ( 2 2 h h ) 2 ( r / r ) 2 1 1 See examples: 7-10 and 7-11 in D&M David Reckhow CEE 370 L#22 17
Contaminant Flow Separate Phase flow low solubility compounds Low density: LNAPL light non-aqueous phase liquid High density: HNAPL Dissolved contaminant Flows with water, but subject to retardation See D&M section 9-7, pg.389-393 Caused by adsorption to aquifer materials David Reckhow CEE 370 L#22 18
Adsorption in Groundwater Based on relative affinity of contaminant for aquifer to water Defined by partition coefficient, K d : K d C C w s ( moles ( moles adsorbed dissolved / kg soil) / L water) And more fundamentally the Kd can be related to the soil organic fraction (f oc ) and an organic partition coefficient (K OC ): K K f See also pg 392 in d oc oc Equ 2-89, pg 76 in D&M 2 nd ed. D&M 2 nd ed. David Reckhow CEE 370 L#22 19
Relative Velocities The retardation coefficient, R, is defined as the ratio of water velocity to contaminant ' velocity ν R water ν ' cont Equ 9-42, pg 391 in D&M 2 nd ed. And since only the dissolved fraction of the contaminant actually moves ν ' cont ' ν water moles molesdissolved + moles dissolved adsorbed David Reckhow CEE 370 L#22 20
Relating R to K d So ν ' cont ν ' water moles molesdissolved + moles dissolved adsorbed And therefore ν moles + moles ' water dissolved adsorbed R 1+ ' ν cont molesdissolved And we can parse the last term: moles moles adsorbed dissolved moles moles adsorbed dissolved C C w s ( moles ( moles adsorbed dissolved / kg soil) / L water) Y ( L aquifer / L water) X ( L aquifer / kg soil) David Reckhow CEE 370 L#22 21
cont Note that the fundamental partition coefficient is: C ( moles / kg K d C w s ( moles adsorbed dissolved / L soil) water) So: moles moles adsorbed dissolved K d Y ( L aquifer / L X ( L aquifer / kg water) soil) And then R 1+ K d Y X David Reckhow CEE 370 L#22 22
cont where: Where: ρ s is density of soil particles without pores usually ~2-3 g/cm 3 ρ b is the bulk soil density with pores So, then Y 1 L aquifer X ( kg soil) ( L aquifer ) L water η R 1+ K d 1 K η ρs ρb + d ( 1 η) η 1 η 1 ρ ρ Compare to Equ 9-43, pg 391 in D&M 2 nd ed. s b David Reckhow CEE 370 L#22 23
cont Retardation in Groundwater & solute movement R 1+ ρ η b Kd ρsoil bulk mass density η void fraction David Reckhow CEE 370 L#29 24
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To next lecture David Reckhow CEE 370 L#22 26