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Darcy’s lawGroundwater 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
1. Aquifer compaction due to increasing effective stress
2. Water expansion due to decreasing pressure
• Unconfined aquifer– Water produced by draining
pores
gV
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 Retentionyr SS
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
http://biosystems.okstate.edu/Darcy/English/index.htm
Darcy’s Experiments
• Discharge is Proportional to – Area– Head differenceInversely proportional to – Length
• Coefficient of proportionality is K = hydraulic conductivity
L
hhAQ 21
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 = densityµ = dynamic viscosityg = specific weight
Porous medium property
Fluid properties
Hydraulic Conductivity
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
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
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
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
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)
Specific Discharge vs Head Gradient
q
Re = 10
Re = 1
Experiment shows this
a
tan-1(a)= (1/K)
Darcy Law predicts this
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.
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
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
Measuring ConductivityFalling Head Permeameter
• Flow rate in the tube must equal that in the column
OutflowQ
Sample
flow
Initial head
Final head
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
Example
• a = ???, b = 4.673x10-10 m2/N, g = 9798 N/m3, • S = 6.8x10-4, b = 50 m, f = 0.25, • Saquifer = gabb = ???
• Swater = gbfb
• % storage attributable to water expansion • • %storage attributable to aquifer expansion •
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
Layered Porous Media(Flow Perpendicular to Layers)
Q
3K2K1K
bQ
L
L3L2L1
Dh1
Piezometric surface
Dh2
Dh3
Dh
Example
• Find average K
Flow Q
Example
• Find average K
Flow Q
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
GroundwaterContours
GroundwaterFlow, Q
x
y
z
Head Gradient, J
hk
hjhi
hi > hj > hk
h1(x1,y1)h3(x3,y3)
h2(x2,y2)
Groundwater Flow Direction
Magnitude of head gradient =
Angle of head gradient =
Head gradient =
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 2DGroundwaterFlow, Q
x
y
z
Head Gradient, J
h1(x1,y1)h3(x3,y3)
h2(x2,y2)
Groundwater Flow Direction
Negative of head gradient in x direction
Negative of head gradient in y direction
Magnitude of head gradient
Direction of flow
x
Well 2(200 m, 340 m)55.11 m
Well 1(0 m,0 m)57.79 m
Well 3(190 m, -150 m)52.80 m
ExampleFind:
Magnitude of head gradient
Direction of flow
y
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• 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
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
Darcy’s LawExamples
Example
• a = ???, b = 4.673x10-10 m2/N, g = 9798 N/m3, • S = 6.8x10-4, b = 50 m, f = 0.25, • Saquifer = gabb = ???
• Swater = gbfb = (9798 N/m3)(4.673x10-10 m2/N)(0.25)(50 m)
• = 5.72x10-5
• percent of storage coefficient attributable to water expansion • = Swater /S = 5.72x10-5 /6.8x10-4 *100 = 8.4%• percent of storage coefficient attributable to aquifer
expansion • = Saquifer /S = 1 – (Swater /S ) = 91.6%
Example
Flow Q
ExampleFlow Q
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
