groundwater pollution remediation (note 2) joonhong park yonsei cee department 2015. 10. 05. cee3330...
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Groundwater Pollution Remediation (NOTE 2)
Joonhong Park
Yonsei CEE Department
2015. 10. 05.
CEE3330 Y2013 WEEK3
CEE3330-01 May 8, 2007 Joonhong Park Copy Right
Darcy’s Experiment (1856)
Flow of water in homogeneous sand filter under steady conditions
Datum
h1
h2Sand
Porous Medium L
A: cross area
Q = - K * A * (h2-h1)/L K= hydraulic conductivity
CEE3330-01 May 8, 2007 Joonhong Park Copy 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)
K Ξ k * ρ * g / μHere, k = intrinsic permeability (L2)
ρ: fluid density (M L-3); g: gravity (LT-2) μ: fluid dynamic viscosity (M L-1 T-1)
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
ρ:density of fluidg:gravity constantn:porosityp:pressure
ρ*g*n*dA*dl
dA
dl
z
l
dz
p*n*dA
(p+dl*dp/dl)*n*dA
F
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) Macro-scale
Forces on Fluids in Porous Media (II)
1) 8*μ*ave. v/R^2 = - (dp/dl + ρ*g*dz/dl) for a cylindrical tube of small radius R
Meanwhile, from Exact Solution of N-S Equation
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-scaleResistance forces per unit volume (F/[dA*dl])
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:porosityA: 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 sizesdm:the geometric mean of the particle sizes corresponding to F.
Harleman et al.’ formula: k = (6.54 x 0.0001) * d^2d:characteristic grain sizeThe 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 3-1Example 3-2Example 3-3Example 3-4Example 3-5Example 3-6Example 3-7
Non-Homogeneity
Homogeneous: K is a scalar
Heterotrophic: K is a function of positions at x, y, and z.
See p. 84-87
∇
∇
dl
q-a
q-b
K-a
K-b
α-a
α-b
K-a/K-b = tan (α-a) / tan (α-b)
Anisotropy
-∇h
q-x
q-y
q
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] = - ~
Anisotropic: q = - K ∇h
~ ~=
K ==
Kxx Kxy Kxz
Kyx Kyy Kyz
Kzx Kzy Kzz
~ ∇ Φ
~
General form of Darcy’s lawValid for multi-dimensions, all Newtonian fluids – incompressible or compressible.
q = - k / µ . [ P - ρ g ]= ~~~
Flow in Aquifer
A Differential Mass Balance
△ X
△ Y
△Z
(x,y,z)
QxQx+dx
Qy
Qy+dy
Qz
Qz+dz
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
X
ΔX
qz-t
qz-b
Z
Assumptions:
Horizontal flow
Constant width into paper, W (a fixed y-value)
Aquifer thinkness at a point: B(X)
B(X)
GW Flow Eq: Confined aquifer with leakage
Aquitard
Impermeable rockx
Assumptions: Homogeneous formation
Steady-state
Constant thickness
Φ = Φ A at left boundary, Φ = Φo in overlying formation
Semi-infinite system
A
ФAb’: thickness of aquitard
b: the thickness of aquifer
K’: hydraulic conductivity for aquitard
K: hydraulic conductivity for aquifer