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Analytical Chem Soil HW Solutions 37

Analytical Chem Soil HW Solutions 37 - initial groundwater...

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37 Q3.6 Solution The derivation of the equation needed to solve this problem is given in full in main text Section 3.5.1. Consider the flowrate q through an annular ring around the well, concentric with the well and having a general radius r. The flow area at a general radius r is 2 π rD, where D is the thickness of the aquifer. The hydraulic gradient i = -dh/dr where h is the head measured above some convenient datum. Applying Darcy's Law, q = Aki = 2 rD.k.dh/dr (main text Equation 3.12) The negative sign has been omitted from the hydraulic gradient in Equation 3.12, because we are interested in the flow towards the well, which is in the r negative direction. Rearranging Equation 3.12 and integrating between limits of (h = h w , r = r w ) at the perimeter of the well and a general point (h, r) at radius r, = h h r r w w dh q kD r dr 2 hence ln(r/r w ) = (2 Dk/q).(h-h w ) or k = [q.ln(r/r w )] ÷ [2 D.(h-h w )] We need to think about the relationships between heads and drawdowns. Taking the datum for the measurement of head h at the bottom of the aquifer, 24 m below ground level, the
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Unformatted text preview: initial groundwater level (at a depth of 2 m below the ground surface) corresponds to a head of 24 m – 2 m = 22 m. The drawdown just outside the well of 2 m corresponds to a head measured from the base of the aquifer of 22 m – 2 m = 20m, and the drawdown of 0.2 m at a distance of 100 m from the well corresponds to a head of 22 m – 0.2 m = 21.8 m. Substituting in the values h = 21.8 m at r = 100 m; h w = 20 m at r w = 0.1 m; D = 20 m and q = 1.637 × 10-3 m 3 /sec gives k = [1.637 10-3 m 3 /sec ln (100/0.1)] [2 20 m (21.8 m – 20 m)] ⇒ k = 5 10-5 m/s If the drawdown inside the well were increased to 4 m, the aquifer would become unconfined (or strong vertical flow would occur) near the well and the analysis will not strictly be valid. In reality, the error will probably be small, but will increase with increasing drawdown....
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