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

2. Outline Darcy’s Law Hydraulic Conductivity
Heterogeneity and Anisotropy
Refraction of Streamlines
Generalized Darcy’s Law

3. Darcy

4. 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

5. Darcy’s Data

6. 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

7. Hydraulic Conductivity

8. 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.

9. 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

10. 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

11. 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)

12. 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)

13. Specific Discharge vs Head Gradientq
Re = 10
Re = 1
Experiment
shows this a
tan-1(a)= (1/K)
Darcy Law
predicts this

14. 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.

15. 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

16. 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

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

18. 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

19. 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

20. Layered Porous Media (Flow Parallel to Layers)
Piezometric surface Dh h1 h2
datum Q
𝑖=1 3 𝑏 𝑖 𝐾 𝑖 =𝑏 𝐾 b W

21. Layered Porous Media (Flow Perpendicular to Layers)
Piezometric surface
Dh1
Dh2 Dh
Dh3 Q b Q
𝑖=1 3 𝐿 𝑖 𝐾 𝑖 = 𝐿 𝐾 L1 L2 L3 L

22. Example
Find average K
Flow Q

23. Example
Flow Q
Find average K

24. Anisotrpoic Porous Media
General relationship between specific discharge and hydraulic gradient

25. 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

26. Flow between 2 adjacent flow lines
𝑞=𝐾 𝑑ℎ 𝑑𝑠 𝑑𝑚
For the squares of the flow net
𝑑𝑠=𝑑𝑚 so
𝑞=𝐾𝑑ℎ
For entire flow net, total head loss h is divided into n squares
𝑑ℎ= ℎ 𝑛
If flow is divided into m channels by flow lines
𝑄=𝑚𝑞=𝐾 𝑚ℎ 𝑛

27. Flow lines are perpendicular to water table contours
Flow lines are parallel to impermeable boundaries
KU/KL = 1/50
KU/KL = 50

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

29. Groundwater Flow Direction
Water level measurements from three wells can be used to determine groundwater flow direction
Groundwater
Contours
hi > hj > hk
Head Gradient, J hi hj hk
h1(x1,y1)
h3(x3,y3) z y
Groundwater
Flow, Q
h2(x2,y2) x

30. Groundwater Flow Direction
Head gradient =
Magnitude of head gradient =
Angle of head gradient =

31. Groundwater Flow Direction
Head Gradient, J
h1(x1,y1)
h3(x3,y3) z
Equation of a plane in 2D y
Groundwater
Flow, Q
3 points can be used to define a plane
h2(x2,y2) x
Set of linear equations can be solved for a, b and c given (xi, hi, i=1, 2, 3)

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

33. Example Find: y Magnitude of head gradient Direction of flow x Well 2
(200 m, 340 m)
55.11 m
Magnitude of head gradient
Direction of flow
Well 1
(0 m,0 m)
57.79 m x
Well 3
(190 m, -150 m)
52.80 m

34. 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

35. 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

36. 𝐾 1 𝐾 2 = 𝑡𝑎𝑛𝜃 1 𝑡𝑎𝑛𝜃 2 0.052 𝑚/𝑑 4.5 𝑚/𝑑 = 𝑡𝑎𝑛 5 𝑜 𝑡𝑎𝑛𝜃 2
Consider a leaky confined aquifer with 4.5 m/d horizontal hydraulic conductivity is overlain by an aquitard with m/d vertical hydraulic conductivity. If the flow in the aquitard is in the downward direction and makes an angle of 5o with the vertical, determine q2.
𝐾 1 𝐾 2 = 𝑡𝑎𝑛𝜃 1 𝑡𝑎𝑛𝜃 2
0.052 𝑚/𝑑 4.5 𝑚/𝑑 = 𝑡𝑎𝑛 5 𝑜 𝑡𝑎𝑛𝜃 2
𝜃 2 = 82.5 𝑜

37. 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

38. Example
Flow Q

39. Example
Flow Q