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Approximate Analytical Solutions to the Groundwater Flow Problem

Approximate Analytical Solutions to the Groundwater Flow Problem. CWR 6536 Stochastic Subsurface Hydrology. 3-D Steady Saturated Groundwater Flow. K(x,y,z) random hydraulic conductivity field f (x,y,z) random hydraulic head field

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Approximate Analytical Solutions to the Groundwater Flow Problem

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  1. Approximate Analytical Solutions to the Groundwater Flow Problem CWR 6536 Stochastic Subsurface Hydrology

  2. 3-D Steady Saturated Groundwater Flow • K(x,y,z) random hydraulic conductivity field • f (x,y,z) random hydraulic head field • want approximate analytical solutions to the 1st and 2nd ensemble moments of the head field

  3. System of Approximate Moment Eqns to order e2 • Use f0(x), as best estimate of f(x) • Use sf2=Pff(x,x) as measure of uncertainty • Use Pff(x,x’) and Pff(x,x’) for cokriging to optimally estimate f or f based on field observations

  4. Possible Solution Techniques • Fourier Transform Methods (Gelhar et al.) • Greens Function Methods (Dagan et al.) • Numerical Techniques (McLaughlin and Wood, James and Graham)

  5. Fourier Transform Methods Require • A solution that applies over an infinite domain • Coefficients in equations that are constant, or can be approximated as constants or simple functions • Stationarity of the input and output covariance functions (guaranteed for constant coefficients) • All Gelhar solutions use a special form of Fourier transform called the Fourier-Stieltjes transform.

  6. Recall Properties of the Fourier Transform • In N-Dimensions (where N=1,3): • Important properties:

  7. Recall properties of the spectral density function • Spectral density function describes the distribution of the variation in the process over all frequencies: • Eg

  8. Look at equation for Pff(x,x’) • Are coefficients constant? • Can input statistics be assumed stationary? • If so output statistics will be stationary. • Assume ; substitute

  9. Solve equation for Pff(x,x’) • Expand equation • Take Fourier Transform

  10. Solve equation for Pff(x,x’) • Rearrange • Align axes with mean flow direction and let

  11. Look at equation for Pff(x,x’) • Are coefficients constant? • Can input statistics be assumed stationary? • If so output statistics will be stationary. • Assume ; substitute

  12. Solve equation for Pff(x,x’) • Expand equation • Take Fourier Transform

  13. Solve equation for Pff(x,x’) • Rearrange • Align axes with mean flow direction and let

  14. Procedure • Given Pff(x) Fourier transform to get Sff(k) • Use algebraic relationships to get Sff1(k) and Sf1f1(k) • Inverse Fourier transform to get Pff1(x) and Pf1f1(x) • Then multiply each by e2=slnK2 to get Pff(x) and Pff(x)

  15. Results • Head Variance: • Head Covariance

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