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Unformatted text preview: FLOW OF WATER THROUGH POROUS MEDIA FLOW CHARACTERISTICS Flow Classification Steady flow the flow conditions are constant over time Unsteady flow the flow conditions change over time Flow Types One dimensional flow, the flow parameters (pressure, velocity, temperature, etc.) vary in 1-D. Two dimensional flow, the parameters vary in 2-D. Three dimensional flow, the parameters vary in 3-D. FLOW CHARACTERISTICS Flow State Laminar the fluid flows in parallel layers without mixing. Turbulent the fluid mixes during the flow. Transitional a state of fluid flow between laminar and turbulent. The state of flow is a function of the fluid velocity For most soils, the velocity is so small that the flow can be considered laminar. FLOW CHARACTERISTICS Hydraulic gradient, i Transitional Laminar Turbulent Flow velocity, v HYDRAULIC GRADIENT The hydraulic gradient (i) is defined as the total head loss (h) per unit distance of travel (l). h i= l For laminar flow, the flow velocity (v) is linearly proportional to the hydraulic gradient (i), for soil, the constant of proportionality was found by Darcy in 1856 as the soil permeability (k) as stated below by the Darcy's law/equation h v = ki = k l Darcy's law is valid for v < 1 cm/sec QUANTITY OF FLOW The quantity of flow (q) through a cross-sectional area (A) can be calculated as q = Av = Aki = Ak(h/l) In soil, the total cross-sectional area is typically used, if the area of the void (Av) is used, the velocity is called the seepage velocity (vs) The relationship between v and vs can be found using phase diagram and the soil porosity n n = vv/v = Av/A Av = nA Hence, q = Av = Avv s = Avsn; thus, v = v n SOIL PERMEABILITY Range of Permeability (cm/sec) Clean sand k > 1.0 Sand with < 5% passing no 200 sieve 1.0 > k > 10 LAB PERMEABILITY TESTS Lab permeability tests are typically conducted using constant head or falling head test Constant head dh Falling head h Soil h Soil l l LAB PERMEABILITY TESTS Constant head test h Ql Q = Avt = Akit = Ak t; k = l Aht Falling head test dh qin = -a p v p = -a p ; a p = area of pipe dt v p = velocity of water in pipe h dh h qout = kiA = k A; qin = qout -a p =k A l dt l -a h1 h2 dh A al h1 = k dt k = ln h l At h2 t1 t2 FIELD PERMEABILITY TESTS Field pumping test Field dissipation test FIELD PUMPING TEST One pumping central well and two observation wells to measure the drawdown. According to Dupuit's assumption, the hydraulic gradient is: i = dz/dr Initial GWT Drawdown curve Impervious layer FIELD PUMPING TEST i = dz/dr The flow area A = 2rz The quantity of flow: r2 qv ln r 1 k= 2 2 h2 - h1 q = 2 rzk(dz/dr) ( ) MULTI-LAYER SYSTEM (Horizontal Flow) Flow parallel to soil layers (horizontal direction) qv = Aki = (1)( H o )(k x ) eq i qv = qvi = z1k x1i + z2 k x 2i + z3k x3i 1 ( z1k x1 + z2k x 2 + z3k x3 ) (k x )eq = Ho Z1 HO Z2 Z3 k1 k2 k3 MULTI-LAYER SYSTEM (Vertical Flow) For a flow normal h3 H h1 h2 to the soil layers (k z ) eq = k z1 + kz2 + k z3 (vertical direction), Ho z1 z2 z3 the total head loss Ho (H) is the sum of ( k ) = z eq z3 z1 z2 the head losses in + + all layers k z1 k z 2 k z 3 Z1 HO Z2 Z3 k1 k2 k3 H = h1 + h2 + h3 EXAMPLE 1 .4m Soil 1m .5m .5m .5m .5m .75m C Exit Datum .5m d 2 (10) 2 2 = Soil cross-sectional area A= = = 78.54 cm 4 4 d ( 0.5) a= = = 0.196 cm 2 = Standpipe cross-sectional area 4 4 aL h1 (0.196)(15) 680 -5 k= ln = ln = 3.9(10) cm / sec At h2 (78.54)(240) 530 2 2 EXAMPLE 2 10 m 2m 10 m k = 2.3 x 10-2 cm/sec k = 5.7 x 10-4 cm/sec k = 9.2 x 10-7 cm/sec The equivalent permeability can be calculated as follows: k= 10 2.3(10) -2 + 22 2 5.7(10) -4 + 10 9.2(10) -7 k = 2(10) - 6 cm / sec FLOW NET FLOW NET h b T b S SOIL k Bedrock ...
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