csh_lecture11_moments

csh_lecture11_moments - CWR 6537 Subsurface Contaminant...

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CWR 6537 Subsurface Contaminant Hydrology Lecture 11 1 METHOD OF MOMENTS A. Lecture Goals To present the concepts of spatial and temporal moment analysis as they are applied to contaminant transport. To present the concepts of Fourier and Laplace transforms and how they can facilitate moment analysis. B. Introduction For a given distribution of data, a set of statistical parameters called moments can define the appearance of the plotted data. For example, the appropriate forms of the first, second, and third moments will describe the mean, spread, and skew, respectively, of the distribution. For our purposes, the distribution of interest is the dissolved solute concentration distributed either in space or time. For this lecture, resident (volume-averaged) concentrations are assumed. C. Spatial Moment Analysis Recall the 1-dimensional advection-dispersion equation (with R = 1): c t v c x D c x =− + 2 2 ( 1 ) where c = solute concentration (ML -3 ), v = q/ θ = porewater velocity (LT -1 ), and D = hydrodynamic dispersion in x -direction (L 2 T -1 ). Consider the problem of concentrations observed at various points in space at a specific point in time. What are the values of the parameters v and D which correspond to the observed concentration distribution? One way to solve this problem is through the use of spatial moment analysis. x c at time = t
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CWR 6537 Subsurface Contaminant Hydrology Lecture 11 2 One-Dimensional Spatial Moments A b s o l u t e M o m e n t s U n i t s zeroth moment == −∞ mc d x 0 [ M L -2 ] (2) first moment −∞ x d x 1 [ M L -1 ] (3) second moment −∞ x d x 2 2 [ M ] (4) nth moment −∞ x d x n n [ M L n-2 ] (5) N o r m a l i z e d M o m e n t s U n i t s µ n n n m m cx dx cdx ' −∞ −∞ 0 [ L n ] (6) C e n t r a l M o m e n t s U n i t s n n cx d x = −∞ −∞ (' ) 1 [ L n ] (7) The central moments can be expressed as a function of the normalized moments. This facilitates the numerical calculation of central moments from the data. Expanding (7), with n =2:
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CWR 6537 Subsurface Contaminant Hydrology Lecture 11 3 µ µµ 2 1 22 11 2 2 = = −+ −∞ −∞ −∞ −∞ cx d x cdx x d x (' ) ( '' ) 2 2 2 2 =− + −∞ −∞ −∞ −∞ −∞ −∞ cx dx cxdx 1 2 1 2 21 2 2 + ' ' ' ' ' (8) Similarly, for n = 3 : 33 1 2 1 3 32 + ' ' ' ' ( 9 ) Meaning of the different moments: zeroth absolute moment, m 0 = applied mass of solute (area under c vs. x curve) first normalized moment, 1 ' = the x-location for the center of plume mass (mean) second central moment, 2 = a measure of the spread of the plume about the location of the center of mass (variance, 2 = σ 2 ) third central moment, 3 = the skew of the concentration distribution 3 > 0 : skewed to the right 3 < 0 : skewed to the left The next step in our analysis involves the determination of the analytical equations that relate the numerical moments (which can be estimated from the spatial distribution curve) to the system parameters ( v, D ). If we knew the exact analytical solution for c(x,t), we could determine the
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csh_lecture11_moments - CWR 6537 Subsurface Contaminant...

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