darcys law groundwater hydraulics daene c. mckinney
TRANSCRIPT
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Darcy’s lawGroundwater 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
1. Aquifer compaction due to increasing effective stress
2. Water expansion due to decreasing pressure
• Unconfined aquifer– Water produced by draining
pores
gV
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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 Retentionyr SS
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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
http://biosystems.okstate.edu/Darcy/English/index.htm
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Darcy’s Experiments
• Discharge is Proportional to – Area– Head differenceInversely proportional to – Length
• Coefficient of proportionality is K = hydraulic conductivity
L
hhAQ 21
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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 = densityµ = dynamic viscosityg = specific weight
Porous medium property
Fluid properties
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Hydraulic Conductivity
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Groundwater Velocity
• q - Specific dischargeDischarge from a unit cross-section area of aquifer formation normal to the direction of flow.
• v - Average velocityAverage velocity of fluid flowing per unit cross-sectional area where flow is ONLY in pores. A
A
Qqv
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dh = (h2 - h1) = (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 m2 = 1x10-5 m3/s
v = q/f = 2x10-7 m/s / 0.3 = 6.6x10-7 m/s
/”
h1 = 12m h2 = 12m
L = 100m
10m
5 m
FlowPorous medium
Example
K = 1x10-5 m/sf = 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
h1 < h2 < h3
x
h1
h2 h3
Well, Q
q
Dh
Circular hydraulic head contours
K, conductivity, Is constant
Hydraulic gradient
Specific discharge
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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 discharged10 = 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 ConductivityKozeny – 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?
22
32
)1(180dcdk
Kozeny – Carman eq.
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Measuring ConductivityPermeameter 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 ConductivityConstant 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
OutflowQ
Overflow
A
Sample
head difference
flow
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Measuring ConductivityFalling Head Permeameter
• Flow rate in the tube must equal that in the column
OutflowQ
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
verticalhorizontal KK
www.usgs.gov
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Layered Porous Media(Flow Parallel to Layers)
3K
2K
1K
W
b
1b
2b
3b
1Q
2Q
3Q
Dh
h2
h1
Piezometric surface
Q
datum
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Layered Porous Media(Flow Perpendicular to Layers)
Q
3K2K1K
bQ
L
L3L2L1
Dh1
Piezometric surface
Dh2
Dh3
Dh
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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 (xi, hi, 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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xq = -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
2K
1KUpper Formation
12 KK
y
x
1
2
2q
1q
Lower Formation
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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