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 1D. Two dimensional flow, the parameters vary in 2D. Three dimensional flow, the parameters vary in 3D. 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 crosssectional area (A) can be calculated as q = Av = Aki = Ak(h/l) In soil, the total crosssectional 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) ( ) MULTILAYER 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 MULTILAYER 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 crosssectional area A= = = 78.54 cm 4 4 d ( 0.5) a= = = 0.196 cm 2 = Standpipe crosssectional 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 102 cm/sec k = 5.7 x 104 cm/sec k = 9.2 x 107 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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 Spring '08
 Baladi
 Fluid Dynamics, Darcy, hydraulic gradient, lab permeability tests

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