Download Solute Transport (5/1/05)

Our purpose of well studies
• Compute the decline in the water level, or
drawdown, around a pumping well whose
hydraulic properties are known.
• Determine the hydraulic properties of an
aquifer by performing an aquifer test in
which a well is pumped at a constant rate
and either the stabilized drawdown or the
change in drawdown over time is measured.
Drawdown
• T = Q/ 4(h0-h)G(u)
• G(u) =
W(u)
- completely confined.
W(u,r/B) – leaky, confined, no storage.
H(u,) – leaky, confined, with storage.
W(uA,uB,) - unconfined.
Aquifer test
• Steady-state conditions.
Cone of depression stabilizes.
• Nonequilibrium flow conditions.
Cone of depression changes.
Needs a pumping well and at least one
observational well.
Aquifer tests
• T = Q/ 4(h0-h)G(u)
• G(u) =
W(u)
- completely confined.
W(u,r/B) – leaky, confined, no storage.
H(u,) – leaky, confined, with storage.
W(uA,uB,) - unconfined.
Slug test
• Overdamped
– water level recovers to the initial static level in a
smooth manner that is approximately exponential.
• Underdamped
– water level oscillates about the static water level
with the magnitude of oscillation decreasing with
time until the oscillations cease.
Cooper-Bredehoeft-Papadopulos
Method (confined aquifer)
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•
•
•
•
H/H0 = F(,)
H – head at time t.
H0 – head at time t = 0.
 = T t/rc2
 = rs2S/rc2
Underdamped Response Slug
Test
• Van der Kamp Method – confined aquifer
and well fully penetrating.
• H(t) = H0 e-t cos t
H(t) - hydraulic head (L) at time t (T)
H0 - the instantaneous change in head (L)
 - damping constant (T-1)
 - an angular frequency (T-1)
 = 2/(t2-t1)
 = ln[H(t1)/H(t2)]/ (t2 – t1)
Underdamped Response Slug
Test (cont.)
• T = c + a ln T
c = -a ln[0.79 rs2S(g/L)1/2]
a = [rc2(g/L)1/2] / (8d)
d = /(g/L)1/2
L = g / (2 + 2)
Confined
x = -y/tan(2Kbiy/Q)
Q - pumping rate
K - conductivity
b – initial thickness
i – initial h gradient
x0 = -Q/tan(2Kbi)
ymax =  Q/(2Kbi)
Capture Zone Analysis
(unconfined aquifer)
• x = -y / tan[K[h12-h22)y/QL]
• x0 = -QL/[K(h12-h22)]
• ymax =  QL/[K (h12-h22)]
Static fresh and slat water
Ghyben-Herzberg principle
Total Dissolved Solids (TDS)
• Total dissolved solids (TDS) is the total
amount of solids, in milligrams per liter,
that remain when a water sample is
evaporated to dryness.
Solid Constituents
• Major constituents: Calcium, magnesium,
sodium, and potassium (cations); Chloride,
sulfate, carbonate, and bicarbonate (anions).
• Minor constituents: iron, manganese,
fluoride, nitrate, strontium, and Boron.
• Trace elements: arsenic, lead, cadmium, and
Chromium.
Dissolved Gases
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Oxygen.
Carbon dioxide.
Nitrogen.
Hydrogen sulfide
Methane.
Mass transport of solutes
• Diffusion – both ionic and molecular
species dissolved in water move from area
of higher concentration (chemical activity)
to areas of lower concentration.
• Advection – moving water carries it
dissolved solutes.
Diffusion – Fick’s laws
• Fick’s first law
F = -D dC/dx
F = mass flux of solute per unit area per unit time.
D = diffusion coefficient (area/time)
C = solute concentration (mass/volume)
dC/dx = concentration gradient
(mass/volume/distance).
• D ranges from 1 x 10-9 to 2 x 10-9 m2/s, for the
major cations and anions.
Diffusion – Fick’s laws (cont.)
• Fick’s second law
C/t = D 2C/x2
D = diffusion coefficient (area/time)
C = solute concentration (mass/volume)
t = time
Effective diffusion coefficient
• D* = wD.
D* = effective diffusion coefficient.
w = empirical coefficient.
Advection
• Advecting contaminants travel at the same
rate as the average linear velocity of ground
water
vx = -(K/ne) dh/dl
vx = average linear velocity
K = hydraulic conductivity
ne = effective porosity
dh/dl = hydraulic gradient
Mechanical Dispersion
• Dispersion is a process that a contaminated
fluid dilutes as it mixs with
noncontaminated water when passing
through a porous medium.
Mechanical Dispersion
• Longitudinal dispersion: the mixing occurs
along the pathway of fluid flow
Mechanical Dispersion
• Longitudinal dispersion: if the mixing occurs
along the pathway of fluid flow
- it moves faster through the center of the pore;
- some of the fluid will travel in longer pathways;
- fluid travels faster through larger pore.
• Transverse or lateral dispersion: if the mixing
occurs normal to the pathway of fluid flow.
- flow paths can split and branch out to the side.
Mechanical Dispersion
• Mechanical dispersion = aLvx
aL = dynamic dispersivity
vx = average linear velocity
Hydrodynamic Dispersion
• Hydrodynamic dispersion:
DL = D* + aLvx
DL = longitudinal coefficient of
hydrodynamic dispersion
D* = effective molecular diffusion
coefficient
aL = dynamic dispersivity
vx = average linear ground-water velocity
Advection-dispersion Equation
• DL2C/x2 – vxC/x = C/t
DL2C/x2 – dispersion (diffusion +
dispersivity).
vxC/x – Advection
Solute Transport by AdvectionDispersion
• C = C0/2{erfc[(L-vxt)/2(DLt)1/2] +
exp(vxL/DL)erfc[(L-vxt)/2(DLt)1/2] }
C = solute concentration (M/L3, mg/L)
C0 = initial concentration (M/L3, mg/L)
L = flow path length (L; ft/m)
vx = average ground velocity (L/T)
t = time since release of the solute (T)
DL = longitudinal dispersion coefficient (L2/T)
Apparent longitudinal dynamic
dispersivity
• aL = 0.83(log L)2.414
• aL = apparent longitudinal dynamic
dispersivity (L; ft/m)
• L = length of the flow path (L; ft or m).
Ground water flow
Continuous source
Ground water flow
Continuous source
Retardation
• Adsorption is a process for a negative (positive)
charge to adsorbing a charged cation (ion).
Retardation – adsorption isotherm
• A graphic plot of C as a function of C*
• C = mass of solute adsorbed per bulk unit dry
mass of soil
C* = equilibrium solute concentration
Retardation - Freundlich equation
• log C* = j log C + log Kf or C* = KfCj
C = mass of solute adsorbed per bulk unit dry
mass of soil
C* = equilibrium solute concentration
Kf, j = coefficients
• If C vs C* is a straight line: Kd = dC*/dC
(distribution coefficient)
C* mass adsorbed per unit weight of soil
C equilibrium concentration of solute remaining
in solution
Adsorption isotherm
Langmuir Adsorption Isotherm
• If C/C* vs. C is a straight line:
C/C* = 1/(12) + C/2
C = equilibrium concentration of the ion in contact
with the soil (mg/L)
C* = amount of the ion adsorbed perl unit weight of
soil (mg/g)
1 = an adsorption constant related to the binding
energy
2 = an adsorption maximum for the soil.
Retardation Factor
• Retardation factor = 1 + (b/)(Kd)
b = dry bulk mass density of the soil (M/L3;
gm/cm3)
 = volumetric moisture content of the soil
(dimensionless).
Kd = distribution coefficient for solute with
the soil (L3/M; mL/g)
Solute Movement with
Retardation
• vc = vx/[1+ (b/)(Kd)]
vc = velocity of the solute front. In onedimensional column the solute
concentration is one-half of the original
value (L/T; ft/day or m/day).
vx = average linear velocity (L/T; ft/day or
m/day).
Mass transport of solutes
• Diffusion – both ionic and molecular
species dissolved in water move from area
of higher concentration (chemical activity)
to areas of lower concentration.
• Advection – moving water carries it
dissolved solutes.
Retardation Factor
• Retardation factor = 1 + (b/)(Kd)
b = dry bulk mass density of the soil (M/L3;
gm/cm3)
 = volumetric moisture content of the soil
(dimensionless).
Kd = distribution coefficient for solute with
the soil (L3/M; mL/g)