Results from the permeability test of rock samples, E, F, and G are shown. The correct order of hydraulic conductivity (k) of the rock samples is Note: Q is water flow rate, A is area of cross-section of the sample, h is water head, L is length of the sample.
Show Hint
The slope of the Q versus Ah/L line for each sample equals its hydraulic conductivity by Darcy's law.
Step 1: Recall Darcy's law.
For flow through a rock sample, Darcy's law states
\[ Q = k \, i \, A \]
where \(Q\) is the flow rate, \(k\) is the hydraulic conductivity, \(i\) is the hydraulic gradient and \(A\) is the cross section area.
The hydraulic gradient is \(i = h/L\), where \(h\) is the head loss and \(L\) is the length of the sample, so the law can be written as
\[ Q = k \left( \frac{Ah}{L} \right) \]
Step 2: Read what the graph plots.
The graph plots \(Q\) on the vertical axis against \(\frac{Ah}{L}\) on the horizontal axis for each sample.
Comparing this to \(Q = k \left( \frac{Ah}{L} \right)\), the plotted line for each sample is a straight line through the origin whose slope is exactly \(k\) for that sample.
Step 3: Compare the slopes.
A steeper line means a larger \(k\), because the sample lets more flow \(Q\) pass for the same value of \(\frac{Ah}{L}\).
Reading the three lines drawn from the origin, line E rises the steepest, line F is next, and line G is the flattest of the three.
So the hydraulic conductivities rank in the same order as the slopes: \(k_E > k_F > k_G\).
Final Answer:
Sample E is the most permeable and sample G is the least permeable.
\[ \boxed{k_E > k_F > k_G} \]
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