1. Groundwater Pollution Remediation (NOTE 2)
Joonhong Park
Yonsei CEE Department
CEE3330 Y2013 WEEK3

2. K= hydraulic conductivity
Darcy’s Experiment (1856)
Flow of water in homogeneous sand filter under steady conditions
A: cross area
Sand Porous Medium h2 L h1
Datum
Q = - K * A * (h2-h1)/L
K= hydraulic conductivity
CEE May 8, 2007 Joonhong Park Copy Right

3. Darcy’s Law Q = - K * A * (Φ2 - Φ1)/L
Φ piezometric head
In a 1-D differential form, Darcy’s law may be:
Darcy’s velocity: q = Q/A = dV/[A*dt] = - K * [dΦ/dL]
Hydraulic Conductivity, K (L/T)
K Ξ k * ρ * g / μ
Here, k = intrinsic permeability (L2)
ρ: fluid density (M L-3); g: gravity (LT-2) μ: fluid dynamic viscosity (M L-1 T-1)
CEE May 8, 2007 Joonhong Park Copy Right

4. Modeling of Water Flow in Porous Media
Micro-scale modeling: the Navier-Stokes equation (flow through the void spaces in aquifers; fluid elements are described by differential equations )
Macro-scale modeling: the Darcy’s equation
(Darcy’s velocity: a volume flux defined as the volume of discharge per unit of bulk area)
(What is seepage velocity? Velocity of a fluid element [v] vs Average v [q/n])
Discussion
(Differences? Advantages/Disadvantages?)

5. Forces on Fluids in Porous Media (I)
Driving forces: pressure (p) and a body force due to gravity
Resistance forces (F) are involved in fluid motion in porous media
(p+dl*dp/dl)*n*dA z dz dl F
ρ:density of fluid
g:gravity constant
n:porosity
p:pressure
p*n*dA dA
ρ*g*n*dA*dl l
p*n*dA -
(p+dl*dp/dl)*n*dA =
ρ*g*n*dA*dl * (dz/dl) + F (at Equilibrium)
Macro-scale
F/(n*dA*dl) = - (dp/dl + ρ*g*dz/dl)

7. Forces on Fluids in Porous Media (II)
Meanwhile, from Exact Solution of N-S Equation
1) 8*μ*ave. v/R^2 = - (dp/dl + ρ*g*dz/dl)
for a cylindrical tube of small radius R
2) 3*μ*ave v/d^2 = - (dp/dl + ρ*g*dz/dl)
for a thin film of thickness d
3) 12*μ*ave v/b^2 = - (dp/dl + ρ*g*dz/dl)
for between two plates spaced a distance b apart
Micro-scale
Resistance forces per unit volume (F/[dA*dl])

8. Forces on Fluids in Porous Media (III)
F/(n*dA*dl) = (C*μ/[characteristic length^2])*q
Here: q= ave v/n
The effects of the tortuous path traversed by fluid elements
in a porous medium are Included in the parameters of characteristic length
and a dimensionless number (C). WHY?
q = - (characteristic length^2/ [C*μ]) * (dp/dl + ρ*g*dz/dl)
= - (k/μ)*(dp/dl + ρ*g*dz/dl) = - (k ρ g/μ)*(dФ/dl)
Fundamental Background for the1-D Darcy’s Law

9. Effect of turbulence q = - (k/μ)*(dp/dl + ρ*g*dz/dl)
QUESTION: When can the linearity maintain or when cannot?
(1) F/(n*dA*dl) = (μ/k)*q + ρ*q^2/([k/C]^0.5) = - (dФ/dl)
(The Forchheimer’s equation) (q^2 is the inertial forces)
(2) -([k/C]^0.5/[ρ*q^2])*(dФ/dl) = μ/(ρ*q*([k*C]^0.5) + 1
(3) f = 1/Re + 1 (f=the friction factor)
when Re < 0.02 [<0.1], Darcy’s law is extremely exact [probably acceptable]

10. Effects of change in fluid density
q = - (k/μ)*(dp/dl + ρ*g*dz/dl) (Eq.3.10)
A rather general form of Darcy’s Law which applies for fluids with either constant or variable density contained in porous media whose intrinsic permeability may depend upon both direction and location.
Density of water is fairly constant. Therefore, the Eq.3.10 can be rewritten into the following equation.
q = - (k*ρ*g/μ)*d(p/ρ*g + z)/dl = - (k ρ g/μ)*(dh/dl) (Eq.3.15).
Here (p/ρ*g + z) is a scalar force potential or piezometric head (h).

11. 3-D Differential Form of Darcy’s Equation
q = - (k ρ g/μ) * ∇h (Eq.3.17)
∇ = ∂/∂x * i + ∂/∂y * j + ∂/∂z * k (the gradient operator)
i, j, and k are the unit vectors in the x, y, and z coordinate directions, respectively.
Piezometric head is a scalar. Its negative gradient is a vector representing the force per unit weight acting on the fluid. (force potential)
q = - (k ρ g/μ) * ∇h = -K * ∇h (Eq.3.20)
Barotropic fluids (ρ = function of p). However, constant density of water in most of groundwater is a good assumption. Of course, there are often exceptions.
Suppose K is constant (homogeneous). Then it is permissible to define Ф = K*h
q = -∇ Ф (Eq.3.21)

12. Laboratory Determination of K
The Fair-Hatch formula Eq.3-25 at p.81.
k = 1/{A*[(1-n)^2/n^3]*[(B/100)* ∑(F/dm)]^2}
n:porosity
A: a dimensionless packing factor (~5)
B: a particle shape factor (ex. 6 for spherical particles and 7.7 for highly angular ones)
F: the percent by weight of the sample between two arbitrary particle sizes
dm:the geometric mean of the particle sizes corresponding to F.
Harleman et al.’ formula: k = (6.54 x ) * d^2
d:characteristic grain size
The formula is nearly valid for materials of very uniform particle size and shape.

13. Carman-Kozeny Equation
k = Co * [n3/(1-n)2] * (1/SS2)
n: porosity
SS: specific surface area
or empirically,
k = [n3/(1-n)2] * (dM2/180)
dM: grain size for 50 percentile

14. Reading assignments
Please read Darcy’s Law and the Equations of Groundwater Motion, p including
Example 3-1
Example 3-2
Example 3-3
Example 3-4
Example 3-5
Example 3-6
Example 3-7

15. Non-Homogeneity Homogeneous: K is a scalar
Heterotrophic: K is a function of positions at x, y, and z.
See p
K-a/K-b = tan (α-a) / tan (α-b)
q-a
α-a dl ∇
K-a
q-b ∇
α-b
K-b

16. Anisotropy
-∇h
q-x q
q-y

17. Reading assignments
Please read Darcy’s Law and the Equations of Groundwater Motion, p including
Flow parallel to the layers in a stratified aquifer
Flow through beds in series
Figure 3-12 and Eq.3.39 to 3.44
See p in the reading material

18. 3D Generalization of Darcy’s Law
Heterotrophic Isotropic: q = - K (x,y,z) ∇ h
For homogenous case, can rewrite as
q = ~ ~
- ∇ [K * h] = -
∇ Φ ~ ~ ~
Anisotropic: q = - K ∇h ~ = ~
Kxx Kxy Kxz
Kyx Kyy Kyz
Kzx Kzy Kzz
K = =

20. General form of Darcy’s law
Valid for multi-dimensions, all Newtonian fluids – incompressible or compressible.

21. Flow in Aquifer Qz+dz A Differential Mass Balance Qy △Z Qx Qx+dx △X △Y
(x,y,z)
Qy+dy Qz

22. Reading assignments Please read p.58-63 in the reading material
Governing Equation for Confined Aquifers
Governing Equation for Unconfined Aquifers
Governing Equation for Aquitards
The Duipuit-Forchheimer Approximation
The Boussinesq Equation
Also read p. 72

23. GW Flow Eq: Confined aquifer with leakage
qz-t
Assumptions:
Horizontal flow
Constant width into paper, W (a fixed y-value)
Aquifer thinkness at a point: B(X)
B(X) Z ΔX X
qz-b

24. GW Flow Eq: Confined aquifer with leakage
K’: hydraulic conductivity for aquitard
b’: thickness of aquitard
Aquitard
K: hydraulic conductivity for aquifer
b: the thickness of aquifer A
Impermeable rock x
Assumptions: Homogeneous formation
Steady-state
Constant thickness
Φ = Φ A at left boundary, Φ = Φo in overlying formation
Semi-infinite system