01 Exercises flow

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Sustainable Use of Underground Energy Resources 2023-2024 Darcy’s law and use of vertically averaged PDE’s to represent flow in porous media 1) Consider a 100 cm long vertical column undergoing steady -state saturated flow. The hydraulic heads at the top and bottom of the column are 10 cm and 0 cm, respectively. The hydraulic conductivity is 10-5 cm/s. • What is the gradient, J, through the column? • What is the darcy flux, q, through the column? • What are the gradient, J, and the flux, q, if the hydraulic conductivity is 5 × 10-5 cm/s? • Plot the equipotential lines. • Plot the hydraulic head profile. 2) Consider a vertical column filled with three soils. Assume that the hydraulic head at the bottom of the column, h4, is known, and the flow is in steady - state. Water is poured into the top of the column at a constant flow rate, Q. The area of the column, A, is known. Determine h1 and the hydraulic head gradient across each soil. 3) Consider the same column as in exercise 1. Now, the upper half of the column, has hydraulic conductivity K 1 = 10-5 cm/s and the lower half has a hydraulic conductivity of K2 = 5 × 10-5 cm/s. • What is the equivalent hydraulic conductivity of the column? • What is the steady state flux through the column? • What is the gradient J in the upper half of the column? • What is the gradient J in the lower half of the column? • What is the equivalent hydraulic conductivity of the column if we invert the two layers of soils? 4) An experiment is performed on a 30 cm long cylindrical sand pack with a cross-sectional area of 3 cm2. The pressure drop across the pack is 6000 Pa, and the flow rate is 0.1 cm3/s. You may assume that the viscosity of water is 10-3 Pa s. The pack is held at an angle of 30 degrees from horizontal, so that the water is…