Period #8: Fluid Flow in Soils (II)

Transcription

1 Period #8: Fluid Flow i Soils (II) 53:030 Class Notes; C.C. Swa, Uiversity of Iowa A. Measurig Permeabilities i Soils 1. The Costat Head Test (For Coarse Graied Soils): Upstream ad dowstream head elevatios are maitaied at costat levels. The head differece across the soil is a costat value h. The hydraulic gradiet i across the sample is also costat. i = hydraulic gradiet i the soil ( h/l) Q = vat = volumetric flow through the soil over a elapsed time T. q = Q/T = va = rate of volume flow v = q/a = the so called discharge velocity A = the cross sectioal area of the soil sample Recall from Darcy s Law that: v = ki From a costat head test, soil permeability k ca be computed as: k = Q/(AiT) where Soil L h 1

2 2. The Fallig Head Test (relatively impermeable soils). The total head differece h(t) across the sample chages with time. > The flow rate through the soil is ot costat. q i = Flow rate ito the soil = a dh(t)/dt q out = Flow rate out of the soil = ka * i(t) = ka * h(t)/l a Coservatio of fluid mass gives: q i = q out a dh(t)/dt = ka * h(t)/l This represets a first order ODE to be solved for h(t): Soil L h(t) Solutio: l(h(t)) l(h o ) = kat/ (al) The permeability k of the soil ca thus be determied from this test as follows: k = l(h(t)/h o ) al/[at] 2

3 3. The well drawdow test: used to measure i situ permeabilities of soils Procedure: a) drill a test well ad two observatio wells; b) cotiuously pump water out of the test well util water levels i all three wells achieve equilibrium levels; c) oce steady state flow is achieved, the radial flow rate q(r) is costat; q = ki(r)a(r) = costat as fuctio of r where: k = soil permeability i = local gradiet = dh/dr A = cross sectioal area of flow = 2πrh(r) Assumptios: dh/dr <<1 (i.e. the gradiet is small) flow is therefore approximately horizotal homogeeous soil cylidrical symmetry about axis of pumpig well q = 2πkrh*dh/dr First order ODE describig the radial flow rate ito the pumpig well. Origial groudwater level observatio well #1 h1 pumpig well r 1 r 2 observatio well #2 h 2 Solutio: (πk/q) [h 2 2 h 1 2 ] = l(r2 /r 1 ) Sice q, r 1, r 2, h 1, h 2 ca all be measured, we ca solve for the soil s permeability k as: k = (q/π) l(r 2 /r 1 ) [h 2 2 h ] r 1 r 2 3

4 B. Hydraulic Coductivity Values for Soils: A joit property of both the soil ad the fluid; Hydraulic coductivity k has uits of (L/T) such as m/s, ft/mi, cm/day, etc. Tabulated Hydraulic Coductivities (pore fluid is water) SOIL TYPE k(mm/sec) Relative Permeability coarse gravel, joited rock sad, fie sad silty sad, dirty sad silt, fie sadstoe clay, mudstoe w/o joits > < 10 6 high medium low very low impermeable Observatio: As the grai size of the soil decreases, the coductivity decreases sigificatly. This is due to the higher SSA of fie graied soils. Relatioship betwee coductivity ad fluid properties: k = K * [γ f / η f ], where: K is the soil s absolute permeability (L 2 ); γ f is the uit weight of the fluid (FL 3 ), η f is the viscosity of the fluid (FTL 2 ) 4

5 5 C. Effective Coductivities for Flows i Stratified Soils 1. Case #1: Steady Flow Parallel to Soil Layers Assume that we wated to compute the rate of horizotal fluid flow i this soil deposit. How would we do it? Impermeable Soil Impermeable Soil A C B 25m 75m 10m 6m h c coarse sad, k=10 4 m/s med sad, k=0.5*10 4 m/s coarse sad, k=2.0*10 4 m/s 6m 4m 3m 53:030 Class Notes; C.C. Swa, Uiversity of Iowa

6 Sice the soil layers are bouded above ad below by impermeable layers, the flow ca oly be parallel to the soil layers. Sice the head falls as we proceed from left to right, the water will also flow i this directio. Give: The thickess of each layer (H j ) The coductivity of each layer (k j ) The hydraulic gradiet i the soil deposit: parallel to the layers, i = h/l orthogoal to the layers, i = 0 ( sice there is o flow i that directio) Solutio: Total flow = sum of flows i the layers : q = Σ q j = Σ v j H j = Σ k j i j H j = i Σ k j H j If we write a expressio for the total flow as: q = k equiv i H = i Σ k j H j, the it is clear that k equiv = (1/H) Σ k j H j : The effective coductivity of the layered soil 6

7 Example Problem: Compute the flow rate i the three layered medium o page 4. Solutio: for the problem posed: i = 4m/100m = 0.04; k equiv = (1/H) k j H j Σ = (1/13m)[6m*10 4 m/s + 4m*0.5*10 4 m/s + 3m*2.0*10 4 m/s] = 1.077*10 4 m/s q = k equiv i H = (1.077*10 4 m/s)(0.04)(13m) = 5.60*10 5 m 2 /s 7

8 2. Case #2: Steady Flow Orthogoal to Layers h H 1 k 1 v 1 H H 2 H 3 k 2 k 3 v 2 v 3 H 4 k 4 v 4 Observatios: The total head loss across all of the layers is kow to be h; The average hydraulic gradiet i = h/h The thickess of each layer is H j The coductivity of each layer is k j Also: From cotiuity cosideratios the discharge velocity i each layer must be the same. That is, v 1 = v 2 = v 3 = v 4 = v Darcy s Law holds i each layer: v j =k j ( h j /H j ) where h j is the head loss across the j th layer. 8

9 Across the stratified soil deposit, we ca write a form of Darcy s Law usig the equivalet or effective coductivity of the layered soils: v = k equiv i = k equiv ( h/h) where: k equiv is the effective coductivity of the soil deposit; h is the total head loss across the deposit; H is the total thickess of the deposit. Our objective is to fid a expressio for the equivalet or effective permeability. Note that the total head loss across the deposit h is the sum of the head losses across all of the idividual layers. Σ Σ That is, h = h j = v j H j /k j = v (H j /k j ) Now, recallig that v = k equiv ( h/h) = k equiv (v/h) (H j /k j ) Equatig the first ad last terms i the precedig expressio ad simplifyig it thus follows that k equiv = H[ Σ (H j /k j )] 1 Σ Σ 9

10 Numerical Example: Cosider the stratified soil placed i the U tube show below. Give: All soil properties ad dimesios of the idividual layers Fid: q, the flow rate through the soil (per uit width), ad p w at poit C. L H i L 1 L 2 L 3 L 4 C(x *,z * ) k 1 k 2 k 3 k 4 T H o 10

11 This page is left blak for solutio of the precedig problem. 11

12 3. Coclusios about stratified soils I geeral, for stratified soils, the (k equiv ) parallel is ot equal to (k equiv ) orthogoal. I cases where a soil deposit s permeabilities or coductivities are ot the same i all directios, we saythat the properties are aisotropic. If the properties are the same i all directios, the it is said to be isotropic. Layered soil deposits typically have aisotropic effective or equivalet coductivities. Very Importat Poits to Remember: I fluid is ot flowig i soil, the pore pressure ca be computed directly usig hydrostatics. But whe fluid is flowig i soils, we must first compute the value of head h ad the compute pressure usig the defiitio of head. 12