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Darcys law Groundwater Hydraulics Daene C. McKinney

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Outline Properties – Aquifer Storage Darcys Law Hydraulic Conductivity Heterogeneity and Anisotropy Refraction of Streamlines Generalized Darcys Law

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Aquifer Storage Storativity (S) - ability of an aquifer to store water Change in volume of stored water due to change in piezometric head. Volume of water released (taken up) from aquifer per unit decline (rise) in piezometric head. Unit area Unit decline in head Released water

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Aquifer Storage Fluid Compressibility ( ) Aquifer Compressibility ( ) Confined Aquifer – Water produced by 2 mechanisms 1.Aquifer compaction due to increasing effective stress 2.Water expansion due to decreasing pressure Unconfined aquifer – Water produced by draining pores

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Unconfined Aquifer Storage Storativity of an unconfined aquifer (S y, specific yield) depends on pore space drainage. Some water will remain in the pores - specific retention, S r S y = – S r Unit area Unit decline in head Released water

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Porosity, Specific Yield, & Specific Retention

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Confined Aquifer Storage Storativity of a confined aquifer (S s ) depends on both the compressibility of the water ( ) and the compressibility of the porous medium itself ( ). Unit area Unit decline in head Released water

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Example Storage in a sandstone aqufier = 0.1, = 4x10 -7 ft 2 /lb, = 2.8x10 -8 ft 2 /lb, = 62.4 lb/ft 3 2.5x10 -5 ft -1 and 1.4x10 -7 ft -1 Solid Fluid 2 orders of magnitude more storage in solid b = 100 ft, A = 10 mi 2 = 279,000,000 ft 2 S = S s *b = 2.51x10 -3 If head in the aquifer is lowered 3 ft, what volume is released? V = SA h = 2.1x10 -6 ft 3

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Darcy http://biosystems.okstate.edu/Darcy/English/index.htm

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Darcys Experiments Discharge is Proportional to – Area – Head difference Inversely proportional to – Length Coefficient of proportionality is K = hydraulic conductivity

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Darcys Data

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Hydraulic Conductivity Has dimensions of velocity [L/T] A combined property of the medium and the fluid Ease with which fluid moves through the medium k= cd 2 intrinsic permeability ρ= density µ= dynamic viscosity = specific weight Porous medium property Fluid properties

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Hydraulic Conductivity

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Groundwater Velocity q - Specific discharge Discharge from a unit cross- section area of aquifer formation normal to the direction of flow. v - Average velocity Average velocity of fluid flowing per unit cross- sectional area where flow is ONLY in pores.

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dh = (h 2 - h 1 ) = (10 m – 12 m) = -2 m J = dh/dx = (-2 m)/100 m = -0.02 m/m q = -KJ = -(1x10 -5 m/s) x (-0.02 m/m) = 2x10 -7 m/s Q = qA = (2x10 -7 m/s) x 50 m 2 = 1x10 -5 m 3 /s v = q/ = 2x10 -7 m/s / 0.3 = 6.6x10 -7 m/s / / h 1 = 12mh 2 = 12m L = 100m 10m 5 m Flow Porous medium Example K = 1x10 -5 m/s = 0.3 Find q, Q, and v

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Hydraulic Gradient Gradient vector points in the direction of greatest rate of increase of h Specific discharge vector points in the opposite direction of h

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Well Pumping in an Aquifer Aquifer (plan view) y h 1 < h 2 < h 3 x h1h1 h2h2 h3h3 Well, Q q h Circular hydraulic head contours K, conductivity, Is constant Hydraulic gradient Specific discharge

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Validity of Darcys Law We ignored kinetic energy (low velocity) We assumed laminar flow We can calculate a Reynolds Number for the flow q = Specific discharge d 10 = effective grain size diameter Darcys Law is valid for N R < 1 (maybe up to 10)

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Specific Discharge vs Head Gradient q Re = 10 Re = 1 Experiment shows this tan -1 ( )= (1/K) Darcy Law predicts this

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Estimating Conductivity Kozeny – Carman Equation Kozeny used bundle of capillary tubes model to derive an expression for permeability in terms of a constant (c) and the grain size (d) So how do we get the parameters we need for this equation? Kozeny – Carman eq.

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Measuring Conductivity Permeameter Lab Measurements Darcys Law is useless unless we can measure the parameters Set up a flow pattern such that – We can derive a solution – We can produce the flow pattern experimentally Hydraulic Conductivity is measured in the lab with a permeameter – Steady or unsteady 1-D flow – Small cylindrical sample of medium

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Measuring Conductivity Constant Head Permeameter Flow is steady Sample: Right circular cylinder – Length, L – Area, A Constant head difference (h) is applied across the sample producing a flow rate Q Darcys Law Continuous Flow Outflow Q Overflow A Sample head difference flow

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Measuring Conductivity Falling Head Permeameter Flow rate in the tube must equal that in the column Outflow Q Sample flow Initial head Final head

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Heterogeneity and Anisotropy Homogeneous – Properties same at every point Heterogeneous – Properties different at every point Isotropic – Properties same in every direction Anisotropic – Properties different in different directions Often results from stratification during sedimentation www.usgs.gov

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Layered Porous Media (Flow Parallel to Layers) W b h h2h2 h1h1 Piezometric surface Q datum

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Layered Porous Media (Flow Perpendicular to Layers) Q b Q L L3L3 L2L2 L1L1 h 1 Piezometric surface h 2 h 3 h

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Anisotrpoic Porous Media General relationship between specific discharge and hydraulic gradient

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Principal Directions Often we can align the coordinate axes in the principal directions of layering Horizontal conductivity often order of magnitude larger than vertical conductivity

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Groundwater Flow Direction Water level measurements from three wells can be used to determine groundwater flow direction

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Groundwater Flow Direction Set of linear equations can be solved for a, b and c given (x i, h i, i=1, 2, 3) 3 points can be used to define a plane Equation of a plane in 2D

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Groundwater Flow Direction Negative of head gradient in x direction Negative of head gradient in y direction Magnitude of head gradient Direction of flow

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x = -5.3 deg Well 2 (200, 340) 55.11 m Well 1 (0,0) 57.79 m Well 3 (190, -150) 52.80 m Example

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Contour Map of Groundwater Levels Contours of groundwater level (equipotential lines) and Flowlines (perpendicular to equipotiential lines) indicate areas of recharge and discharge

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Refraction of Streamlines Vertical component of velocity must be the same on both sides of interface Head continuity along interface So Upper Formation y x Lower Formation

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Summary Properties – Aquifer Storage Darcys Law – Darcys Experiment – Specific Discharge – Average Velocity – Validity of Darcys Law Hydraulic Conductivity – Permeability – Kozeny-Carman Equation – Constant Head Permeameter – Falling Head Permeameter Heterogeneity and Anisotropy – Layered Porous Media Refraction of Streamlines Generalized Darcys Law

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