csh_lecture4_unsaturated

csh_lecture4_unsaturated - CWR 6537 Subsurface Contaminant...

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CWR 6537 Subsurface Contaminant Hydrology Lecture 4 1 UNSATURATED SUBSURFACE FLOW A. Lecture Goals Develop equations describing capillary rise, soil-water pressure saturation relationships, capillary flow, and soil hydraulic properties. Our focus in this section will be on fluid flow in the vadose zone . The vadose zone is defined as the space between the ground surface and the water table (note that this includes the capillary fringe ). Why is it called “vadose”? B. General 3-Dimensional Unsaturated Subsurface Flow Recall from Lecture 3, the general continuity equation for incompressible flow. (1) where C ( R ) = the specific moisture capacity [L -1 ]; S w = the saturated fraction; and S s = specific storage [L -1 ]; R = pressure [L]; t = time [T]; and q P = the Darcy-Buckingham discharge vector [L A T -1 ]. For an unsaturated system C ( R ) $ 0 and S w S s . 0; thus, (2) Recalling that the specific discharge vector, q , has three components: (3) where i , j , and k are unit vectors. For a system in which the primary axes of permeability correspond with the modeling axes, the magnitudes q x , q y , and q z of q components are defined as follows: (4a)
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CWR 6537 Subsurface Contaminant Hydrology Lecture 4 2 (4b) (4c) Combining Eqs. (2) - (4) we obtain Richards' Equation : (5) To solve Eq. (5) we need to define hydraulic conductivities, K ( R ) xx , K ( R ) yy , and K ( R ) zz as a function pressure R . To achieve this, we consider hydraulic conductivity varying with respect to R , because the soil-water content, 2 , varies with respect to R . To understand the relationship between R and 2 we consider first the concept of capillary rise in a single capillary tube and the expand this idea to soils conceptualized as a bundle of capillary tubes having a distribution of tube sizes (diameters). C. Capillary Rise: Consider a capillary tube of radius r , in which water has risen to an elevation h , above a datum z = 0 . R is the radius of curvature and is obtained from r /cos $ . The upward force, F c , that supports the column of water is defined: (6) where F = surface tension [M A T -2 ], and $ = the contact angle. The upward force holding the column of water is balanced by a gravitational force, F
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CWR 6537 Subsurface Contaminant Hydrology Lecture 4 3 (7) Equating these two forces we obtain a relationship for elevation of the meniscus, h. (8) D. CAPILLARY FLOW: Now we consider laminar flow in a horizontal capillary tube. This will lead us to see that an approximation of horizontal 1-dimensional infiltration is mathematically analogous to horizontal capillary flow. Consider first that a porous medium is an interconnected structure of tiny conduits of variable shapes and sizes. The Hagen-Poiseuille equation for steady laminar flow in a circular conduit is: (9) in which Q = the volumetric discharge through the conduit [L 3 A T -1 ]; A = cross-sectional area of the conduit perpendicular to the direction of flow [L 2 ]; : = is the dynamic viscosity of water [M
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This note was uploaded on 09/24/2011 for the course CWR 6537 taught by Professor Jawitz during the Spring '08 term at University of Florida.

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csh_lecture4_unsaturated - CWR 6537 Subsurface Contaminant...

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