Problem 56205. Measure the hydraulic conductivity with a falling-head permeameter

A falling-head permeameter is another device for measuring the hydraulic conductivity K of a soil sample. In this problem the sample is placed in a cylinder of length L and cross-sectional area A_c. Unlike the constant-head permeameter, in which the water level in a standpipe (or tube) is kept constant, the falling-head permeameter has a water level in the tube (of cross-sectional area A_t) that falls. In other words, the head difference δ, or the difference between the water levels in the tube and the outlet, decreases in time.
The hydraulic conductivity can be determined from a statement of conservation of mass (i.e., water volume) by equating the rate of change of volume in the tube to the flow out of the soil sample. The outflow can be computed with Darcy’s law, which states in general
Q = -K(dh/dx)A
Darcy’s law applies when a Reynolds number based on the specific discharge v = Q/A and representative diameter d of the soil grains is less than (approximately) 1—that is,
Re = vd/nu < 1
where ν is the kinematic viscosity of the fluid.
Derive and solve an ordinary differential equation for the head difference δ. Then write a function that takes as input measurements of head difference as a function of time, as well as the soil’s porosity, diameter of the tube, and length and diameter of the cylinder holding the soil sample. The function should compute the hydraulic conductivity by fitting the solution to the ordinary differential equation to the data and using Darcy’s law regardless of its validity. Also return a flag indicating whether Darcy’s law is valid throughout the experiment; to assess the validity, relate the hydraulic conductivity to the representative grain diameter with the Kozeny-Carman equation, as described in the previous problem. Use g = 981 cm/s^2 and nu = 10^{-2} cm^2/s.
falling-head permeameter

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Last Solution submitted on Sep 23, 2023

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