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Approximate Analytical Solutions to the Groundwater Flow Problem

This paper explores approximate analytical solutions to the groundwater flow problem, focusing on stochastic subsurface hydrology. It utilizes 3-D steady saturated flow with random hydraulic conductivity and head fields. Key techniques discussed include Fourier Transform and Green's Function methods for estimating the ensemble moments of head fields, crucial for risk assessments in hydrology. The work emphasizes stationarity assumptions, covariance functions, and cokriging for optimal estimation based on field observations, providing essential insights for groundwater management.

yvonne-lott
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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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