CIV3248 Flow Net Workshop 1

CIV3248Groundwater,seepage and environmental engineering

•Workshop (4)

•Flownet sketching

•Keith H McKenry

CIV3248 Flow Net Workshop 2

Intention:

•Mathematical insight.

•Practise sketching skills.

•Flow net compared with SEEPW output.

•Interpretation of flow nets

CIV3248 Flow Net Workshop 3

Mathematics.

• Steady state: conservation of mass, continuity equation, stream function (), streamlines.

• Energy balance: Potential function (), total head lines, irrotational flow (flow around a curve by distortion rather than rotation).

• Laplaces Equation for & .

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Streamlines

• Seepage velocity vector is tangent to streamline.

• Streamlines cannot cross.

• Commence on inflow boundary, finish at outflow boundary.

• Streamlines have constant stream function value.

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Streamlines

Seepage

velocity

stream function value

dQ =

Stream tube

dQ

Unconfined flow: is a boundary streamline

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Streamlines and stream function (x,y),

• Two streamlines define a streamtube.

• Streamtube width is inversely proportional to local seepage velocity.

• Difference in in stream function value equals the discharge between two streamlines defining the stream tube.

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Total head lines

• Total head = - / k

• Lines of constant value of potential function.

• Zero hydraulic gradient zero flow

• orthogonal to streamlines (cross at right angles).

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Equipotential / Total head lines

equipotential value = - H*k m3/s/m

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Flownet definition…1

• Selection of streamlines and total head lines.

• Constant difference in stream function values between all streamlines.

• Constant difference in potential function values between all total head lines.

• Same difference value for both stream function and potential function.

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Flownet definition…2

Streamlines

Equipotential lines

(constant Total Head)

dq = 3

dq = 3

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Flownet features

• Orthogonal net.

• Inscribed circles.

• Diagonals to curvilinear squares form another orthogonal net.

• Ratio of number of streamtubes to number of potential drops is constant for a given flow boundary.

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Flownet Examples

• Based on explicit functions for total head and stream function.

• Prior to the advent of cheap computers much mathematical effort was expended on techniques to allow explicit equations to fit a wide range of boundary geometries.

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Basis

• For a given boundary geometry, only ONE true flownet exists.

• Find the true flow net by “trial and correct”.

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Strategy• Identify the two streamlines and two total

total head lines that define the problem.

• For 4 or 5 STREAM TUBES draw the 3 or 4 internal streamlines.

• Draw TOTAL HEAD lines to attempt to form curvilinear squares (a true flownet).

• Adjust the trial lines to get better curvilinear squares.

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Confined Flow example

“Impermeable”

“Pervious”

Boundary streamlines

Boundary total head lines Total head difference

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Cross-section to scale: 10 mm = 1 m. W1

A

C D EF

B

Datum RL 60m

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Cross-section to scale: 10 mm = 1 m. W2

A

C D EF

B

Datum RL 60m

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Cross-section to scale: 10 mm = 1 m. W3

A

C D EF

B

Datum RL 60m

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Cross-section to scale: 10 mm = 1 m. W4

A

C D EF

B

Datum RL 60m

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Flownet Interpretataion

• Discharge (m3/sec/m)

• Pore pressure

• Hydraulic gradient.

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Discharge per unit width (Q m3/sec/m)

Q = k.HL.(N /N)

• k - coefficient of permeability.

• HL - drop in Total Head across flow.

• N - number of stream tubes.

• N - number of potential drops.

• Note N might not be an integer.

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HL = 4 m

N = 3

N = 8

Q = k.HL.(N /N) = 7 x 10-5 . 4 .(3/8)

= 10.5 x 10-5 m3/s/m

K = 7 x 10-5 m/s

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Pore Pressure “u” at a point

• Identify total head value of all constant total head lines relative to datum.

• Find total head at point by interpolation “H”.

• Identify elevation head of point “z”.

• u = (H - z) * w (w = unit weight of

water)

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RL95m

RL91m

RL72m

H=95m

H=94.5m

H=94mH=93.5m

H=93m H=92.5m

H=92m

H=91.5m

H=91m

Total head datum = RL 0m

RL80m

RL94m

RL83m

RL92.5m

A

B

u = (H - z) * w

uA =(94 – 80)*9.8 = 137 kPa

uBD =(94 – 75)*9.8 = 186 kPa uB =(92.5 – 83)*9.8 = 93 kPa

DRL75m

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RL93m

RL91m

RL72m

H=93m

H=92.75m

H=92.5m

H=92.25mH=92m H=91.75m

H=91.5m

H=91.25m

H=91m

Total head datum = RL 0m

RL80m

RL92.5m

RL83m

RL91.75m

A

B

u = (H - z) * w

uA =(92.5 – 80)*9.8 = 123 kPauB =(91.75 – 83)*9.8 = 86 kPa

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Hydraulic gradient at a point - “i”

• Interpolate streamline through the point.• Determine difference in total head

between constant total head lines either side of point “dH”.

• Measure real distance between these constant total head lines along the interpolated streamline “dL”

• i = dH /dL

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RL95m

RL91m

RL72m

H=95m

H=94.5m

H=94m

H=93.5mH=93m H=92.5m

H=92m

H=91.5m

H=91m

Total head datum = RL 0m

C

dL = 4.5m

dH = 0.5m(-)iC = dH /dL = 0.5 / 4.5 = 0.11

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Upward seepage flow:Critical hydraulic gradient at exit - Icrit

• Liquifaction of soil may occur if the exit hydraulic gradient exceeds a critical value such that effective stress equals zero.

• Icrit = (Gs - 1)/(e +1)

Gs - Specific gravity of soil particles

e - Void Ratio

• Icrit ~ 1.0

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A

5.81

20e-0

03

Feet

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 1500

10

20

30

40

50

60

70

Seepw solution

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Conclusion: flow net sketching

• Limited to isotropic soils, homogeneous flow

• Quick, simple equipment,

• Analysis of multiple geometry for design.

• Check on computer packages.