Approximate Analytical/Numerical Solutions to the Groundwater Transport Problem

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Approximate Analytical/Numerical Solutions to the
Groundwater Transport Problem

• CWR 6536
• Stochastic Subsurface Hydrology

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3-D Saturated Groundwater Transport

• vi(x,y,z) spatial random velocity field
• c(x,y,z, t) spatiotemporal random concentration
  field
• No analytic solution exists to this problem
• 3-D Monte Carlo very CPU intensive
• Look for approximate analytical/numerical
  solutions to the 1st and 2nd ensemble moments of
  the conc field

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System of Approximate Moment Eqns

• Use as
  best estimate of c(x,t)
• Use sc2(x,t)Pcc(x,tx,t) as measure of
  uncertainty
• Use Pcv(x,tx) and Pcc(x,tx,t) to optimally
  estimate c or v based on field observations

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Solution Techniques

• Fourier Transform Techniques (Gelhar et al)
• Finite Difference/Finite Element Techniques
  (Graham and McLaughlin)
• Greens Function or Impulse Response Techniques
  (Neuman et al, Cushman et al, Li and Graham)

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Fourier Transform Techniques(Gelhar et al)

• Require an infinite domain
• Require coefficients in pdes for Pcvi and Pcc to
  be constant
• Require input covariance function to be
  stationary
• Convert pdes for covariance functions Pcvi and
  Pcc into algebraic expressions for Scvi and Scc.

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Spectral Solution for Steady-State
Macrodispersive Flux (Pcvi(x,x) )

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Spectral Solution for Steady-State
Macrodispersive Flux (Pcvi(x,x) )

• Therefore mean equation looks like
• Aij determined from spectral relationship
  between Svic and Svivj
• Svivj is the inverse Fourier transform of Pvivj
  determined from Pff, flow equation and Darcys
  law.

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Results

• Assuming 3-D isotropic negative exponential for
  Pff, and al, al ltltl (Gelhar and Axness, 1981)
• Longitudinal macrodispersivity increases with
  variance and correlation scale of log
  conductivity
• Transverse macrodispersivity increase with
  variance of log conductivity, independent of
  correlation scale and depends on local
  dispersivity
• Fickian relationship emerges as a result of
  constant conc. gradient assumption

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Spectral Solution for Steady-StateConcentration
Variance

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Results

• Assuming hole-type isotropic negative exponential
  for Pff, and al, al ltltl (Vomvoris and Gelhar,
  1990)
• Simpler negative exponential spectrum gives
  infinite concentration variance (caused by small
  wave number energy, i.e. high wave length
  variations at a scale large than plume scale)
• Concentration variance increases with increasing
  log hydraulic conductivity mean and variance,
  increasing mean concentration gradient, and
  decreasing local dispersivity

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Numerical Solution

• Solve coupled pdes using finite element or finite
  difference technique
• Does not require an infinite domain
• Does not require coefficients in pdes for Pcvi
  and Pcc to be constant
• Does not require input covariance functions to be
  stationary
• Does not require any special form of the input
  covariance function
• Requires lots of computer time and memory