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# csh_lecture7_advection - CWR 6537 Subsurface Contaminant...

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CWR 6537 Subsurface Contaminant Hydrology Lecture 7 1 SOLUTE TRANSPORT: ADVECTION OR MASS FLOW A. Lecture Goals: To understand mass flow transport and, ultimately, mechanical dispersion, a known manifestation of spatial variations in advective transport. B. Advective Transport: Let us consider solute transport by advective flow alone; ignore diffusive transport for the moment. The total amount of solute transported per unit time across a unit cross section, designated by J c , is given by the product of volumetric flux (L 3 T -1 ) and solution concentration (ML -3 ). Thus, UnitArea 1 tion VolumeSolu MassSolute UnitTime tion VolumeSolu = J a (1) C q = C A Q = J L L a (2) where, ¯q is a vector equal to Darcy's flux (LT -1 ). Note that J a has units of mass per time per unit area, (ML -2 T -1 ). Considering transport in one spatial direction, the continuity equation for simultaneous transport of water and solute is x ) C q ( - = x J - = t ) C ( m L x m a L θ (3) where θ C L represents the total solute mass per unit soil volume. For steady one-dimensional water flow alone, θ and q x are constants. Thus, Eq. (3) can be restated as x C q - = t C m L x L θ (4) Dividing both sides by θ , and defining v x = (q x / θ ),

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CWR 6537 Subsurface Contaminant Hydrology Lecture 7 2 x C v - = t C m L x L (5) Note that v x is the linear flow velocity within the pores, while q x is water flux per unit area (L 3 ·L -2 ·T -1 ); v x is referred to as the pore-water velocity and has the units of (L·T -1 ). In order to solve Eq.[5], we use a MOVING COORDINATE system (rather than a FIXED COORDINATE system). Let x 1 and t 1 be the variables in the new coordinate system, where t v - x = x x m 1 (6) t = t 1 (7) Notice the following relationships: v - = t x x 1 (8) 1 = t t 1 (9) 1 = x x m 1 (10) t t t C + t x x C = t C 1 1 L 1 1 L L (11) Now, we need to transform C L (x m ,t) to C L (x 1 ,t 1 ). The total change of C L with respect to time can be written using the chain rule: Substituting (8) and (9) into (11) gives t C + x C v - = t C 1 L 1 L x L (12) Then from using the chain rule with Eq. (10),
CWR 6537 Subsurface Contaminant Hydrology Lecture 7 3 x C = x x x C = x C 1 L m 1 1 L m L (13) recalling Eq.(5), x C v - = t C m L x L (14) and combining it with equation (13) gives x C v - = t C 1 L x L (15) A comparison of equations (12) and (15) reveals the governing differential equation for the moving coordinate system: 0 = t d ) t , x ( C d 1 1 1 L (16) or, C L (x 1 ,t 1 ) = constant in the moving coordinate system. Now, transforming back to the FIXED coordinate system, t v < x ; C = t) , x ( C x m o m L (17) t v > x ; 0 = t) , x ( C x m m L (18) Equations (17) and (18) reflect a C 0 Heaviside function (also called a unit step function). x m = v x t x m C o

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CWR 6537 Subsurface Contaminant Hydrology Lecture 7 4 We note that the solute concentration (C L ) in the medium is sharp with an abrupt change from C L = C o to C L = 0 at x m = v x t. Thus, the displacing fluid piston displaces the displaced fluid.
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csh_lecture7_advection - CWR 6537 Subsurface Contaminant...

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