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lecture18 - Approximate Analytical Solutions to the...

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Approximate Analytical Solutions to the Groundwater Flow Problem CWR 6536 Stochastic Subsurface Hydrology
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3-D Steady Saturated Groundwater Flow K(x,y,z) random hydraulic conductivity field φ (x,y,z) random hydraulic head field want approximate analytical solutions to the 1st and 2nd ensemble moments of the head field + + = z K z y K y x K x φ φ φ 0
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System of Approximate Moment Eqns to order ε 2 • Use φ 0 (x), as best estimate of φ (x) • Use σ φ 2 =P φφ (x,x) as measure of uncertainty Use P φφ (x,x’) and P f φ (x,x’) for cokriging to optimally estimate f or φ based on field observations ) ( ) ' , ( ) ' , ( ) ( ) ' , ( 0 ) ' ( ) ' , ( ) ' , ( ) ' ( ) ' , ( 0 ) ( ) ( ) ( 0 0 2 0 ' ' ' ' 2 ' 0 0 2 1 1 1 1 1 1 1 x x x P x x P x F x x P x x x P x x P x F x x P x x F x f x ff x f x x f x φ φ φ φ φ φ φ φ φ φ φ + + = + + = + =
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Possible Solution Techniques Fourier Transform Methods (Gelhar et al.) Greens Function Methods (Dagan et al.) Numerical Techniques (McLaughlin and Wood, James and Graham)
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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
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