Solutions to the AdvectionDispersion Equation

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Solutions to the Advection-Dispersion Equation
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Road Map to Solutions

• We will discuss the following solutions
• Instantaneous injection in infinite and
  semi-infinite 1-dimensional columns
• Continuous injection into semi-infinite 1-D
  column
• Instantaneous point source solution in
  two-dimensions (line source in 3-D)
• Instantaneous point source in 3-dimensions
• Keep an eye on
• the initial assumptions
• symmetry in space, asymmetry in time

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Recall the Governing Equation

• What have we assumed thus far?
• Dispersion can be expressed as a Fickian process
• Diffusion and diffusion can be folded into a
  single hydrodynamic dispersion
• First order decay
• What do we need next?
• More Assumptions!

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Adding Sorption

• Thus far we have addressed only the solute
  behavior in the liquid state.
• We now add sorption using a linear isotherm
• Recall the linear isotherm relationshipwhere
  cl and cs are in mass per volume of water and
  mass per mass of solid respectively
• The total concentration is then

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Retardation factor

• We have
• Which may be written as
• where we have defined the retardation factor R to
  be

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Putting this all together

• The ADE with 1st order decay linear isotherm
• What do we need now? More assumptions!
• ? constant in space (pull from derivatives
  cancel)
• D constant in space (slide it out of derivative)
• R constant in time (slide it out of derivative)
• Use the chain rulethe divergence of
• a scalar gradientdivergence of a constant is
  zero

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Applying the previous assumptions

• The divergence operators turn into gradient
  operators since they are applied to scalar
  quantities.
• What does this give us? The new ADE

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Looking at 1-D case for a moment

• To see how this retardation factor works, take t
  t/R, and ? 0. With a little algebra,
• The punch line
• the spatial distribution of solutes is the same
  in the case of non-adsorbed vs. adsorbed
  compounds!
• For a given boundary condition and time t, the
  solution is unique and independent of R

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1-D infinite column Instantaneous Point Injection

• Column goes to ??and -?
• Area A
• steady velocity u
• mass M injected at x o and t0(boundary
  condition)
• initially uncontaminated column. i.e. c(x,0) 0
  (initial condition)
• linear sorption (retardation R)
• first order decay (?)

velocity u
x 0
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• Features of solution
• Gaussian, symmetric in space, ?2 Dt/R
• Exponential decay of pulse
• Except for decay, R only shows up as t/R

Upstream solutes
Peak at 1.23 hr
Center of Mass 2 hr
Spatial Distribution
Temporal Distribution
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1-D semi-infinite Instantaneous Point Injection

• How do we handle a surface application?
• Use the linearity of the simplified ADE
• Can add any two solutions, and still a solution
• By uniqueness, any solution which satisfies
  initial and boundary conditions is THE solution
• Boundary and initial conditions

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Semi-infinite solution

• Upward pulse and downward.
• Only include region of x gt 0 in domain
• Solution symmetric about x 0, therefore slope
  of dc(x0,t)/dx 0 for all t, as required
• Compared to infinite column, c starts twice as
  high, but in time goes to same solution

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Continuous injection, 1-D

• Since the simplified ADE is linear, we use
  superposition. Basically get a continuous
  injection solution by adding infinitely many
  infinitely small Gaussian plumes.
• Use the complementary error function erfc

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Plot of erfc(x)
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Solution for continuous injection, 1-D
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Plot of solution

? 0.1, R 1, ?? 0.02, u 1.0, and m 1
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1-D, Cont., simplified

• With no sorption or degradation this reduces to

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2-D and 3-D instantaneous solutions
Note - Same Gaussian form as 1-D - Note
separation of longitudinal and transverse
dispersion
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Review of Assumptions

• Assumption Effects if Violated
• ? constant in space -R higher where ??lower
• -Velocity varies inversely with ?
• D constant in space -Increased overall
  dispersion due
• to heterogeneity
• D independent of scale - Plume will grow more
  slowly at
• first, then faster.
• Reversible Sorption - Increase plume spreading
  and
• overall region of contamination
• Equilibrium Sorption - Increased tailing and
  spreading
• Linear Sorption - Higher peak C and faster
  travel
• Anisotropic media - Stretching smearing along
  beds
• Heterogeneous Media - Greater scale effects of D
  and ALL
• EFFECTS DISCUSSED ABOVE