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Water flow in saturated soil D A Cameron Civil Engineering Practice 1 DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 1 SEEPAGE – water pressures Water flows from points of high to low TOTAL head WATER HEADS [“height of water”] x [w] = water pressure, u Total head = [elevation head + pressure head] i.e h = hT = [he + hp] Kinetic head is ignored in soils DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 2 Head of Water Pressure head = height water rises to in a standpipe above the point No loss of Water table level head, h, in soil hp mass, h so no flow - Steady State he Arbitrary datum Element of soil within soil mass DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 3 Confined Aquifer A water bearing layer, overlain and underlain by far less permeable soils. Water level in aquifer standpipe Clay, silt - no free water x Clay, silt DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 4 Steady flow in soils – Laminar flow Assumptions to theory: • Uniform soil, homogeneous & isotropic • Continuous soil media • Small seepage flow (non turbulent flow) Darcy’s Law of 1850 – a Frenchman DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 5 Darcy’s Law q = kiA where q = rate of flow (m3/s) i = hydraulic gradient A = area normal to flow direction (m2) k = coefficient of permeability (m/s) DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 6 Hydraulic Gradient, i h Area of flow, A Flow rate, q Length of flow, l DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 7 Hydraulic Conductivity • Coefficient of permeability or just “permeability” • SATURATED soil permeability Hazen’s formula, for clean, almost uniform sands: 2 d10 k 100 m/sec if particle size in mm DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 8 TYPICAL PERMEABILITIES Clean gravels > 10-1 m/s Clean sands, sand-gravel 10-4 to 10-2 m/s Fine sands, silts 10-7 to 10-4 m/s Intact clays, clay-silts 10-10 to 10-7 m/s DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 9 Measuring Permeability [A] Laboratory [A] Laboratory • Constant head test How good is the • Falling head test sample? • Other [B] Field [B] Field • Pumping tests Need to know soil • Borehole infiltration profile (incl. WT) & boundary conditions tests DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 10 Lab Test 1: Constant head test • Cylinder of saturated coarse grained soil • Water fed under constant head ─ elevated water tank with overflow • Rate of outflow measured Repeat the above after raising the water tank DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 11 1. Constant head permeameter Water tank - moveable ht q A hpB hpC B l he C D soil DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 12 Constant head test Suitable for clean sands and fine gravels EXAMPLE: • If the sample area is 4500 mm2, • the vertical distance between the 2 standpipe points is 100 mm, • h is 75 mm • Outflow is 1 litre every minute What is the coefficient of permeability? DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 13 Solution • 1000 cm3/min OR q = 16.7 cm3/sec = 16.7x10-6 m3/sec • i = 75/100 = 0.75 • k = q/(iA) = (16.7x10-6)/(0.75x4500x10-6) m/sec k = 5 x 10-3 m/sec Typical permeability of a clean sand or gravel DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 14 Test 2: Falling head permeameter For fine sands, silts, & maybe clays • Rate of water penetration into cylindrical sample from loss of head in feeder tube • Must ensure: − no evaporation − sufficient water passes through A slow procedure DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 15 2. Falling Head Permeameter Level at time, t1 Level at time, t2 Tube of cross-sectional area 'a' h1 h2 To permeameter cell Level of cell outflow DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 16 Falling head test • Soil sample length, L, & area, A • Flow in the tube = flow in the soil – tube has area “a” a L h1 k ln A (t 2 t1 ) h 2 DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 17 3. Field testing – drawdown test Pumping well Water q r2 table r1 h2 h1 Impermeable boundary Drawdown – phreatic or flow line DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 18 Drawdown test Needs 1. a well-defined water table and 2. a confining boundary Must be able to 1. pull down water table and 2. create flow (phreatic line = uppermost flow line) DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 19 Solution Axi-symmetric problem By integration of Darcy’s Law, q r2 k ln π(h 2 h 2 ) r 2 1 1 DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 20 TUTORIAL PROBLEMS A canal and a river run parallel, an average of 60 m apart. The elevation of water in the canal is 200 m and the river 193 m. A stratum of sand intersects both the river and canal below the water levels The sand is 1.5 m thick and is sandwiched between strata of impervious clay Compute the seepage loss from the canal in m3/s per km length of the canal, given the permeability of the sand is 0.65 mm/s DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 21 THE PROBLEM Sand seam RL 200 m RL 193 m canal river 60 m DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 22 SOLUTION q = kiA k = 0.65 mm/s = 0.65 x 10-3 m/s h = 7 m q = 0.65 x 10-3 x 0.117 x 1.5 m2/m length q = 0.114 x 10-3 m3/sec /m length q = 0.114 m3/sec/km length DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 23 Hydraulic gradient, i = 0.117 RL 200 m RL 193 m h = 7 m l = 60 m DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 24 Flow Lines – shortest paths for water to exit Equipotential lines hp1 h Flow tube hp2 h1e1 l he2 Elevation head reference line DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 25 The Flow Net - FLOW LINES Run parallel to impervious boundaries (impermeable walls or “cut-offs”) and the phreatic surface The “Phreatic surface” is the top flow line 2 consecutive flow lines constitute a “flow tube” DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 26 h 5 Flow Lines Impervious boundary DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 27 The Flow Net - EQUIPOTENTIALS • Are lines of equal total head • The total head loss between consecutive equipotentials is constant • Equipotentials can be derived from boundary conditions and flow lines DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 28 Flownet Basics Water flow follows paths of maximum hydraulic gradient, h i imax i m ax b m in flow lines and equipotentials must cross at 90°, since: DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 29 Since q is the same, ratio of sides will be constant for all the “squares” along the flow tube h M Equi- potential lines Impervious boundary DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 30 h i Flow q q k a per m b q constant,if a : b is constant hi head lost between equipotent ials b a Common convention draw “squares” with a = b “square, M”, a x b DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 31 Discharge in flow direction, Equipotentials = q per “flow tube” h3 90º l h2 Flow lines h1 DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 32 Flownet Construction DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 33 Flow Net Calculations Nd equal potential drops along length of flow? Then the head loss from one line to another is: h1-2 = (h) = h / Nd From Darcy’s Law, flow rate in a flow tube, Δh i ΔqkiA k a1 b OR Δh a N b Δqk d DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 34 Flow Net Calculations BUT a=b AND total flow for Nf “flow channels”, per unit width is: Nf qk Δh Nd But only for “squares”! DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 35 Example: if k = 10-7 m/sec, what would be the flow per day over a 100 m length of wall? Dam cutoff 50 m of water 5 m of water Low permeability rock DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 36 Calculations Nf = 5 Answer: Nd = 14 h = 45 m = 10-7(5/14)45 x 100 m length k = 10-7 m/sec = 0.000161 m3/sec = 13.9 m3/day DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 37 Example: what is the hydraulic gradient in the “square” C? Dam cutoff 50 m of water 5 m of water Low permeability rock DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 38 Calculations [h / Nd] = 45/14 Answer: 1.1 and = 3.2 m therefore head per drop dangerous! Average length of flow is about 3 m DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 39 Critical hydraulic gradient, ic The value of i for which the effective stress in the saturated system becomes ZERO! Consequences: no stress to hold granular soils together soil may flow “boiling” or “piping” = EROSION DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 40 Seepage Condition – upward flow of water = satz = total stress u due to seepage, B = i(z)(w) z (represents proportion of h A occurring over length AB) = - u = (satz) – (wz + i(z)w) = z – i(z)w = 0, when z = i(z)w OR i = (/ w) DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 41 Likelihood of Erosion GRANULAR SOILS chiefly! When the effective stress becomes zero, no stress is carried by the soil grains Note: when flow is downwards, the effective stress is increased! So the erosion problem and ensuing instability is most likely for upward flow, i.e. water exit points through the foundations of dams and cut-off walls DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 42 DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 43 Minimising the risk of erosion 1. Add more weight at exit points permeable concrete mats? DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 44 Lengthen flow path? 1. Deeper cut-offs 2. Horizontal barriers 3. Impermeable blanket on exit surface DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 45 Simple cut-offs (FESEEP) Nf = 5 Nf Nd =10= 5 NN=115 d f= Nd =13 DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 46 “Impermeable” Clay Blanket DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 47 Summary: Key Points • Heads in soil • Darcy’s Law • Coefficient of permeability • Measurement of permeability • Flownets • Flownet rules • Seepage from flownets • Piping, boiling or erosion • Critical hydraulic gradient DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 48 Exercises a) Draw a flow net for seepage under a vertical sheet pile wall penetrating 10 m into a uniform stratum of sand 20 m thick. b) If the water level on one side of the wall is 11 m above the sand and on the other side 1.5 m above the sand, compute the quantity of seepage per unit width of wall. [k = 3 10-5 m/s] c) What is the factor of safety against developing the “quick” condition on the outflow side of the wall? [sat= 21 kN/m3] DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 49 Finite Difference spreadsheet solution and other numerical approaches Authors: Mahes Rajakaruna (ex UniSA) & University of Sydney (FESEEP) DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 50 Finite Difference approach to flow nets - flow line set up ROWCO A B C D E F G H I J K L M N O P Q R S T U V W L 1 100 Soil level 104 2 100 104 3 100 104 4 100 Cell H5 104 5 100 104 6 100 104 7 100 104 8 100 104 9 100 Interior cell value = 104 104 104 104 104 104 104 10 100 (H4+I5+H6+G5)/4 11 100 12 100 Impermeable boundary 13 100 14 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 51 Flow lines from finite difference program (spreadsheet) 98-99 99-100 100-101 101-102 102-103 DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 52 Equipotentials from finite difference program (spreadsheet) 113-114 112-113 111-112 110-111 109-110 108-109 107-108 106-107 105-106 104-105 103-104 102-103 101-102 100-101 DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 53 FESEEP: University of Sydney cutoff Mesh of foundation soil DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 54 FESEEP Output (University of Sydney) flownet pore pressures increasing DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 55 Dam cutoff 50 m of water 5 m of water Low permeability rock DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 56