Groundwater Pollution
Remediation (NOTE 2)
Joonhong Park
Yonsei CEE Department
2015. 10. 05.
CEE3330 Y2013 WEEK3
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
CEE3330-01
May 8, 2007
Joonhong Park Copy
K= hydraulic
conductivity
Right
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)
KΞk*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)
CEE3330-01 May 8, 2007 Joonhong Park Copy
Right
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?)
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
F
p*n*dA
ρ*g*n*dA*dl
ρ:density of fluid
g:gravity constant
n:porosity
p:pressure
l
Macro-scale
p*n*dA - (p+dl*dp/dl)*n*dA = ρ*g*n*dA*dl * (dz/dl) + F (at Equilibrium)
F/(n*dA*dl) = - (dp/dl + ρ*g*dz/dl)
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
Resistance forces per unit volume (F/[dA*dl])
Micro-scale
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
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]
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).
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)
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 0.0001) * d^2
 d:characteristic grain size
 The formula is nearly valid for materials of very uniform particle size and shape.
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
Reading assignments
Please read Darcy’s Law and the Equations of Groundwater Motion,
p.65-82 including
Example
Example
Example
Example
Example
Example
Example
3-1
3-2
3-3
3-4
3-5
3-6
3-7
Non-Homogeneity
Homogeneous: K is a scalar
Heterotrophic: K is a function of positions at x, y, and z.
K-a/K-b = tan (α-a) / tan (α-b)
q-a
α-a
dl
q-b
K-b
See p. 84-87
α-b
K-a
Anisotropy
-∇h
q-x
q
q-y
Reading assignments
Please read Darcy’s Law and the Equations of Groundwater Motion,
p.82-90 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. 70-71 in the reading material
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
K=
=
Kyx Kyy Kyz
Kzx Kzy Kzz
General form of Darcy’s law
Valid for multi-dimensions, all Newtonian
fluids – incompressible or compressible.
.
q=-k/µ [ P -ρg]
=
~
~
~
Flow in Aquifer
Qz+dz
A Differential Mass Balance
△Z
Qx
Qy
△X
(x,y,z)
Qy+dy
Qz
Qx+dx
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
GW Flow Eq: Confined aquifer with leakage
qz-t
Assumptions:
B(X)
Horizontal flow
Constant width into paper, W
(a fixed y-value)
Z
ΔX
X
qz-
Aquifer thinkness at a point: B(X)
GW Flow Eq: Confined aquifer with leakage
ФA
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
formation
A
at left boundary, Φ = Φo in overlying