# Groundwater Hydraulics Daene C. McKinney

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Groundwater Hydraulics Daene C. McKinney
Darcy’s law Groundwater Hydraulics Daene C. McKinney

Outline Properties – Aquifer Storage Darcy’s Law
Hydraulic Conductivity Heterogeneity and Anisotropy Refraction of Streamlines Generalized Darcy’s Law

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

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

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

Porosity, Specific Yield, & Specific Retention

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

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

Darcy

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

Darcy’s Data

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

Hydraulic Conductivity

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.

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

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

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)

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)

q Re = 10 Re = 1 Experiment shows this a tan-1(a)= (1/K) Darcy Law predicts this

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.

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

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

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

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

Layered Porous Media (Flow Parallel to Layers)
Piezometric surface Dh h1 h2 datum Q b W

Layered Porous Media (Flow Perpendicular to Layers)
Piezometric surface Dh1 Dh2 Dh Dh3 Q b Q L1 L2 L3 L

Anisotrpoic Porous Media
General relationship between specific discharge and hydraulic gradient

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

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

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)

Groundwater Flow Direction

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

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

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

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