14.330 flow nets - faculty server...

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14.333 GEOTECHNICAL LABORATORYFlow Nets

LAPLACE'S EQUATION OF CONTINUITYSteady-State

Flow around an impervious

Sheet Pile Wall

Consider water flow at Point A:

vx = Discharge Velocity in x Direction

vz = Discharge Velocity in z Direction

Y Direction Out Of PlaneFigure 5.11. Das FGE (2005).

x

z

y



dxdydzzvv

dzdydxxv

v

zz

xx

Consider water flow at Point A(Soil Block at Pt A shown left)

Rate of water flow into soil block in x direction:

vxdzdyRate of water flow into soil block in z direction:

vzdxdy

Figure 5.11. Das FGE (2005).

Rate of water flow out of soil block in x,z directions:

LAPLACE'S EQUATION OF CONTINUITY



0

0

zv

xv

ordxdyvdzdyv

dxdydzzvvdzdydx

xv

v

zx

zx

zz

xx



Total Inflow = Total Outflow




02

2

2

2

zhk

xhk

zhkikv

xhkikv

zx

zzzz

xxxx



Using Darcy’s Law (v=ki)




FLOW NETS: DEFINITION OF TERMSFlow Net: Graphical Construction used to calculate groundwater flow through soil. Comprised of Flow Lines and Equipotential Lines.Flow Line: A line along which a water particle moves through a permeable soil medium. (a.k.a. streamline).Flow Channel: Strip between any two adjacent Flow Lines.Equipotential Lines: A line along which the potential head at all points is equal.

NOTE: Flow Lines and Equipotential Lines must meet at right angles!



Figure 5.12a. Das FGE (2005).

FLOW NETSFLOW AROUND

SHEET PILE WALL



Figure 5.12b. Das FGE (2005).

FLOW NETSFLOW AROUND

SHEET PILE WALL



1.The upstream and downstream surfaces of the permeable layer (i.e. lines ab and de in Figure 12b Das FGE (2005)) are equipotential lines.

2.Because ab and de are equipotential lines, all the flow lines intersect them at right angles.

3.The boundary of the impervious layer (i.e. line fg in Figure 12b Das FGE (2005)) is a flow line, as is the surface of the impervious sheet pile (i.e. line acd in Figure 12b Das FGE (2005)).

4.The equipontential lines intersect acd and fg(Figure 12b Das FGE (2005)) at right angles.

FLOW NETS: BOUNDARY CONDITIONS




FLOW NETSFLOW UNDER ANIMPERMEABLE

DAM



nqqqq ...321


q k h1 h2

l1

l1 k h2 h3

l2

l2 k h3 h4

l3

l3 ...

h1 h2 h2 h3 h3 h4 ... HNd

Rate of Seepage ThroughFlow Channel (per unit length):

Using Darcy’s Law (q=vA=kiA)

Potential DropWhere:

H = Head DifferenceNd = Number of Potential Drops

FLOW NETS: DEFINITION OF TERMS



FLOW NETS: RULES FOR CREATINGFLOW NETS (FROM UTEXAS)

1. Head drops between adjacent equipotential lines must be constant (or, in those rare cases where this is not desirable, clearly stated, just as in topographic contour maps)!

2. Equipotential lines must match known boundary conditions.

3. Flow lines can never cross.

Flow Line

Equi.




4. Refraction of flow lines must account for differences in hydraulic conductivity.

5. For isotropic media (what you have).a) Flow lines must intersect equipotential

lines at right angles.b) The flow line-equipotential polygons

should approach curvilinear squares, as shown in the Figure to the right.

Flow Line

Equi.




6. The quantity of flow between any two adjacent flow lines must be equal.

7. The quantity of flow between any two stream lines is always constant.

Flow Line

Equi.



FLOW NETS: DRAWING PROCEDURE(AFTER HARR (1962, P. 23)

1. Draw the boundaries of the flow region to scale so that all equipotential lines and flow lines that are drawn can be terminated on these boundaries.

2. Sketch lightly three or four flow lines, keeping in mind that they are only a few of the infinite number of curves that must provide a smooth transition between the boundary flow lines. As an aid in spacing of these lines, it should be noted that the distance between adjacent flow lines increases in the direction of the larger radius of curvature.

3. Sketch the equipotential lines, bearing in mind that they must intersect all flow lines, including the boundary streamlines, at right angles and that the enclosed figures must be (curvilinear) squares.



FLOW NETS: DRAWING PROCEDURE(FROM HARR (1962, P. 23)

4. Adjust the locations of the flow lines and the equipotential lines to satisfy the requirements of step 3. This is a trail-and-error process with the amount of correction being dependent upon the position of the initial flow lines. The speed with which a successful flow net can be drawn is highly contingent on the experience and judgment of the individual. A beginner will find the suggestions in Casagrande (1940) to be of assistance.

5. As a final check on the accuracy of the flow net, draw the diagonals of the squares. These should also form smooth curves that intersect each other at right angles.



FLOW NETS: EXAMPLES

Unconfined groundwater flow nets on a slope

Wrong Wrong

Correct!



FLOW NETS: EXAMPLES

Cross-sectional flow net of a homogeneous and isotropic aquifer (Hubbert, 1940).



FLOW NETS: EXAMPLES

Contour map of the piezometric surface near Savannah, Georgia, 1957, showing closed contours resulting from heavy local groundwater pumping (from Bedient, after USGS Water-Supply Paper 1611).



FLOW NETS: DAM EXAMPLES



Figure 5.12b. Das FGE (2005).

d

f

NHN

kq

If Number of Flow Channels = Nf, then the total flow for all channels per unit length is:

Therefore, flow through one channel is:

dNHkq

FLOW NETSFLOW AROUND SHEET PILE WALL EXAMPLE



GIVEN:

Flow Net in Figure 5.17.Nf = 3Nd = 6kx=kz=5x10-3 cm/sec

DETERMINE:

a. How high water will rise in piezometers at points a, b, c, and d.

b. Rate of seepage through flow channel II.

c. Total rate of seepage.





mmm 56.06

)67.15(

SOLUTION:

Potential Drop = dN

H

At Pt a:Water in standpipe =(5m – 1x0.56m) = 4.44m

At Pt b:Water in standpipe =(5m – 2x0.56m) = 3.88m

At Pts c and d:Water in standpipe =(5m – 5x0.56m) = 2.20m






SOLUTION:

dNHkq

k = 5x10-3 cm/seck = 5x10-5 m/sec

q = (5x10-5 m/sec)(0.56m)q = 2.8x10-5 m3/sec/m

fd

f qNN

HNkq

q = (2.8x10-5 m3/sec/m) * 3q = 8.4x10-5 m3/sec/m


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