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EAS446lec9 - EAS 44600 Groundwater Hydrology Lecture 9 Flow...

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EAS 44600 Groundwater Hydrology Lecture 9: Flow Nets Dr. Pengfei Zhang Specific Discharge and Average Linear Velocity Recall Darcy’s law: = = dl dh KA L h h KA Q b a (9-1) It states that the flow (discharge) will be proportional to the hydraulic gradient ( dh/dl ), the hydraulic conductivity ( K ), and the cross-sectional area of flow ( A ). The specific discharge ( Darcian velocity ), q , of groundwater is equal to Q/A : = dl dh K q (9-2) The specific discharge is an apparent velocity only, since flow of water does not occur through the entire cross section (Figure 9-1A). Remember that flow occurs only in the void spaces between grains (Figure 9-1A). Hence, the specific discharge given above underestimates the actual velocity of water moving through the void spaces. B A Flow No flow Figure 9-1. Cross section of an R.E.V. (A) and average linear velocity (B). To find the actual average linear velocity , v , at which the water is moving, one divides the specific discharge by the porosity ( n ) to account for the actual open space available for flow: = = dl dh n K n q v (9-3) in terms of units: 2 3 1 REV w L T L A Q q = = (9-4) 9-1
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T L L L L T L n A Q v REV w REV REV w = = = 3 3 2 3 1 1 (9-5) Notice that in the above units for porosity it is assumed that all voids are filled with water ( versus ), in other words, we are talking about groundwater, not the unsaturated zone. 3 3 / REV w L L 3 3 / REV voids L L The units are useful because they help us to distinguish between specific discharge ( L 3 w /L 2 REV T ) and the average linear velocity ( L REV /T ). The specific discharge concerns the volume of water that passes by a particular cross section of the R.E.V. in a given amount of time, whereas the average linear velocity concerns the length of R.E.V. that the average molecule of water travels in a given amount of time. The specific discharge is less straightforward to understand physically, and the units show that. The average linear velocity is an average because the water molecules follow various tortuous paths through the porous medium (Figure 9-1B). Some water molecules travel through paths of greater tortuosity, resulting in lower linear velocities, whereas other water molecules follow more straightforward paths, resulting in higher linear velocities.
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