Energy Engineering - Sustainable use of underground energy resources

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 K 2 = 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 cm 2. The pressure drop across the pack is 6000 Pa, and the flow rate is 0.1 cm 3/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 flowing uphill. The water density is 1000 kg/m 3, and the acceleration due to gravity g = 9.81 m/s 2. • What is the gradient, J, through the column? • What is the hydraulic conductivity of the column? • What is the permeability of the column? 5) An experiment is performed on a 30 cm long rectangular sand pack with a cross -sectional area of 3 cm 2. The pressure drop across the pack is 6000 Pa. The water is flowing from left to right. The column is horizontal, and it is filled with two layers of soil as depicted in the figure below (consider unit thickness of the column). The hydraulic conductivity of the soils are respectively K 1 = 10 -2 cm/s and K 2 = 5×10 -2 cm/s. Consider the density of the water 1000 kg/m 3. • What is the equivalent hydraulic conductivity of the column? • What is the darcy flux through the column? • What are the darcy fluxes through the upper and the lower layer of the column? • What are the flow rate through the column and the flow rate through the upper and lower layer of the column? 6) We consider a confined aquifer (assume 1D flow). The hydraulic conductivities of the porous materials are: K1 = 8,5 10 -2 cm/s; K2 = 6,5 10 -3 cm/s ; K 3 = 3,0 10 -2 cm/s. The piezometric heads are prescribed at the two boundaries: 18m upstream, 12m downstream. • Compute Darcy’s velocity in the aquifer • What are the head values at piezometer A and B ? 7) A channel is supplying a drainage system through an aquifer. Groundwater flow is partly confined and partly unconfined. Hydraulic conductivity is K = 4.5 x 10 -4 m/s. Compute the volumetric flow rate in the aquifer and the location where the groundwater changes from confined to unconfined state. 8) The hydraulic conductivity of the aquifer is K=10 -4 m/s. The infiltration rate is 684 mm/year. The net recharge is 35% of the infiltration. Find: location of the highest piezometric head and the value of such head. What is the flow rate arriving at each river? Compute the recharge rate that sets the volumetric flowrate in the upstream river equal to 0. 9) Water infiltrates from a channel into a free aquifer which supplies a river. The hydraulic conductivities are K1 = 5 x 10 -3 m/s over a length of L1 = 170 m and K2 = 1.7 x 10 -4 m/s over a length of L2 = 45 m. Consider steady -state conditions. • What is the piezometric head at A? • What is the flow rate which supplies the river?