Analytical Chem Soil HW Solutions 32

Analytical Chem Soil HW Solutions 32 - Q3.4 Solution You...

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32 Q3.4 Solution You will need to derive the formulae (main text Equations 3.21 and 3.22) for the equivalent bulk horizontal and vertical permeabilities of an alternating layer system, as in the main text Section 3.6. For horizontal flow (ie flow parallel to the laminations), the hydraulic gradient between two vertical sections A and B is the same for both layers (see main text Figure 3.16). The total flowrate q T is the sum of the flowrates through the individual layers. We seek an expression of the form q T = A T .k H .i, where k H is the overall (bulk) permeability in the horizontal direction and A T is the total area available for flow. For a unit depth perpendicular to the plane of the paper, A T =d 1 + d 2 Applying Darcy's law to each layer in turn, q 1 = d 1 .k 1 .i and q 2 = d 2 .k 2 .i, hence q T = q 1 + q 2 = (d 1 .k 1 + d 2 .k 2 ).i, and by comparison with the initial expression q T = A T .k H .i, the horizontal permeability is given by k H = (d 1 .k 1 + d 2 .k 2 )/(d 1 + d 2 ) (main text Equation 3.21) In vertical flow (ie flow perpendicular to the laminations), the same flow passes through each
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