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**Groundwater Hydraulics Daene C. McKinney**

Darcy’s law Groundwater Hydraulics Daene C. McKinney

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**Outline Properties – Aquifer Storage Darcy’s Law**

Hydraulic Conductivity Heterogeneity and Anisotropy Refraction of Streamlines Generalized Darcy’s 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 (b) Aquifer Compressibility (a)**

Confined Aquifer Water produced by 2 mechanisms Aquifer compaction due to increasing effective stress 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 (Sy, specific yield) depends on pore space drainage. Some water will remain in the pores - specific retention, Sr Sy = f – Sr 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 (Ss) depends on both the compressibility of the water (b) and the compressibility of the porous medium itself (a). Unit area Unit decline in head Released water

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**Example Storage in a sandstone aqufier**

f = 0.1, a = 4x10-7 ft2/lb, b = 2.8x10-8 ft2/lb, g = 62.4 lb/ft3 ga = 2.5x10-5 ft-1 and gbf = 1.4x10-7 ft-1 Solid Fluid 2 orders of magnitude more storage in solid b = 100 ft, A = 10 mi2 = 279,000,000 ft2 S = Ss*b = 2.51x10-3 If head in the aquifer is lowered 3 ft, what volume is released? DV = SADh = 2.1x10-6 ft3

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Darcy

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**Darcy’s Experiments Discharge is Coefficient of proportionality is**

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

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Darcy’s 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 = cd2 intrinsic permeability ρ = density µ = dynamic viscosity g = specific weight Porous medium property Fluid properties

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**Hydraulic Conductivity**

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**Groundwater Velocity q - Specific discharge v - Average velocity**

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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**Example K = 1x10-5 m/s f = 0.3 /” Find q, Q, and v Flow**

h1 = 12m h2 = 12m K = 1x10-5 m/s f = 0.3 Find q, Q, and v /” Flow 10m Porous medium 5 m L = 100m dh = (h2 - h1) = (10 m – 12 m) = -2 m J = dh/dx = (-2 m)/100 m = 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 m2 = 1x10-5 m3/s v = q/f = 2x10-7 m/s / 0.3 = 6.6x10-7 m/s

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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**

Hydraulic gradient y h1 < h2 < h3 x h1 h2 h3 Well, Q q Dh Circular hydraulic head contours K, conductivity, Is constant Specific discharge Aquifer (plan view)

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**Validity of Darcy’s Law**

We ignored kinetic energy (low velocity) We assumed laminar flow We can calculate a Reynolds Number for the flow q = Specific discharge d10 = effective grain size diameter Darcy’s Law is valid for NR < 1 (maybe up to 10)

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**Specific Discharge vs Head Gradient**

q Re = 10 Re = 1 Experiment shows this a tan-1(a)= (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**

Darcy’s 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 Darcy’s Law Continuous Flow Outflow Q Overflow head difference flow A Sample

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**Measuring Conductivity Falling Head Permeameter**

Flow rate in the tube must equal that in the column Outflow Q Initial head Final head flow Sample

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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

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**Layered Porous Media (Flow Parallel to Layers)**

Piezometric surface Dh h1 h2 datum Q b W

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**Layered Porous Media (Flow Perpendicular to Layers)**

Piezometric surface Dh1 Dh2 Dh Dh3 Q b Q L1 L2 L3 L

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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**

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

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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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**Example x q = -5.3 deg Well 2 (200, 340) 55.11 m Well 1 (0,0) 57.79 m**

(190, -150) 52.80 m x q = -5.3 deg

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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**

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

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**Summary Properties – Aquifer Storage Darcy’s Law**

Darcy’s Experiment Specific Discharge Average Velocity Validity of Darcy’s Law Hydraulic Conductivity Permeability Kozeny-Carman Equation Constant Head Permeameter Falling Head Permeameter Heterogeneity and Anisotropy Layered Porous Media Refraction of Streamlines Generalized Darcy’s Law

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