Civil Engineering-Construction of equipotential maps

7/30/2019 Civil Engineering-Construction of equipotential maps.

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GEO 476K & 191 LAB 4

FLOW NETS

OBJECTIVE:

The objective of this laboratory is to introduce you to flow nets and the construction of equipotential

maps. From such maps, flow directions, rates of flow, and the hydrostratigraphy can often be inferred.

BACKGROUND:

A flow net is a 2-D diagram of equipotentials (lines of equal head) and flow lines. They are built from

field observations and/or theoretical constraints. Equipotentials are the loci of points of equal potential

(or head), and flow lines (or stream lines) correspond to directions of groundwater flow. A

potentiometric map typically represents a family of equipotentials, which may or may not have flow

lines depicted. A water-table map is an example of a potentiometric map, as it represents the lines of

equal head that intersect the water table. The equipotentials must be spaced so that equal head

drops occur between adjacent equipotentials and equal volumetric flow occurs between adjacent flow

lines, as shown in Figure 1.

Figure 1. Diagram of a basic flow net.

Equipotentials are usually denoted by h1, h2, ... (or1, 2, ...) and flow lines by 1, 2, .... Thus we

can determine the discharge between two flow lines


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21 QQ

AL

hKCONSTANTQ

=

==

where

Q1 = flow rate between 1, and 2

A = area of the flow tube

K = hydraulic conductivity

L1 = distance between equipotentials 1, and 2

Figure 2: Diagram depicting how to determine Q between 2 flow lines.

The rates of flow can be inferred from flow nets if the hydraulic conductivity is known. Often the total

volumetric rate of flow (Q) can be calculated by

KHN

NHKNQ

e

f

f ==

where

Nf= number of flow channels

Ne = number of equipotential drops

H = total head drop (H = Neh)


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RULES FOR CREATING FLOW NETS:

1. Head drops between adjacent equipotentials must be constant (or, in those rare cases

where this is not desirable, clearly stated, just as in topographic contour maps)!

2. Equipotentials must match known boundary conditions.

3. Flow lines can never cross.

4. Refraction of flowlines must account for differences in hydraulic conductivity.

5. For isotropic media

a. Flow lines must intersect equipotentials at right angles.

b. The flow line-equipotential polygons should approach curvilinear squares, as shown

below (Figure 3).

Figure 3. Flow net indicating how polygons approach curvilinear squares.

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.


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PROCEDURE:

From Harr (1962, p.23)

1. Draw the boundaries of the flow region to scale so that all equipotential lines and lines that

are drawn can be terminated on these boundaries.

2. Sketch lightly three or four streamlines, keeping in mind that they are only a few of the

infinite number of curves that must provide a smooth transition between the boundary

streamlines. As an aid in spacing of these lines, it should be noted that the distance

between adjacent streamlines increases in the direction of the larger radius of curvature.

3. Sketch the equipotential lines, bearing in mind that they must intersect all streamlines,

including the boundary streamlines, at right angles and that the enclosed figures must be

(curvilinear) squares.

4. Adjust the locations of the streamlines and the quipotential 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 streamlines. The speed with which a successful flow net can

be drawn is highly contingent on the experience and judgement 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. Theseshould also form smooth curves that intersect each other at right angles.


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EXAMPLES:

Figure 4. Unconfined groundwater flow nets on a slope.(a) and (b) are incorrect interpretations, and (c) is correct.

Figure 5. Flow net showing the topographic control of groundwater flow (Hubbert, 1940).

Wrong!

Wrong!

Correct!


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Figure 6. Influence of topography on flow nets (Freeze & Witherspoon, 1967).

Figure 7. Examples of flow nets (a) flow in the x-z plane of a pervious stratum underlying a dam (b)flow toward a discharging well in the x-y plane as influenced by a line source (constanthead boundary) (Domenico & Schwartz).


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Figure 8. Flow conditions in the vicinity of a lake demonstrating the effect of a high permeability layerat depth (Winter, 1976).

Figure 9. Some common errors include (a) equipotentials entering or exiting a no-flow boundary, and(b) disappearing flow lines.

Wrong!

Wrong!


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HETEROGENOUS SYSTEMS:

For flow in inhomogeneous systems refraction of flow lines is required. Recall that

the refraction can be approximated by

As can be readily inferred from the relative permeabilities of confining units vis--vis aquifers, the flow

in confining units will almost always be perpendicular to their stratification. This is depicted in figures10 thru 12.

Figure 10. Regional flow showing the effect of permeability contrasts (a) and (b) in adjacent layers (c)and (d) effect of a high permeability lens (Freeze & Witherspoon, 1967).

)tan(

)tan(

2

1

2

1

=K

K


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Figure 11. Hydrogeologic regime in a thrusted sedimentary environment. The potentiometric lineindicates hydraulic-head values on base of unit A (after Hodge and Freeze, 1977).

Figure 12. Regional groundwater flow in confined aquifers: (a) Aquifer confined by a sloping confinedlayer. (b) Aquifer confined by a flat-lying confining layer (Freeze & Witherspoon, 1967).


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ANISOTROPIC SYSTEMS:

Flow nets in anisotropic systems are a little beyond the scope of this lab. Your scientific judgment will

allow you to make qualitative interpretations. When the hydraulic conductivity is anisotropic, we can

transform the x-z coordinate system into an X-Z system, which we can treat as isotropic. Then we

draw the flow net and reverse the transformation to obtain the final flow net.

Figure 13: a) Flow problem in an homogeneous, anisotropic region with Kx0.5/Ky0.5 = 4.0.b) Flow net into a transformed, isotropic system.c) Flow net in the actual naysotropic system (T means transformation and I means

inversion). See Freeze and Cherry (1979, p. 174-178) or Harr (1962, p. 26-35) for more

details.


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Fractured media will almost always have a considerable degree of anisotropy. Finally, even though

considerable assumptions might often be needed for flow net construction, the insights that can be

gained are great. This is true even where exact flow directions are unobtainable.

REFERENCES:

Casagrande, A., 1940, Seepage through dams in Contributions to Soil Mechanics: 1925-1940, Boston

Soc. Civil Engineers.

Cedergren, H., 1968, Seepage, drainage, and flow nets: Wiley, New York.

Freeze, R. A., and Cherry, J. A., 1979, Groundwater: Prentice-Hall, Englewood Cliffs, N.J., p. 174-178.

Freeze, R. A., and Witherspoon, P.A., 1967, Theoretical analysis of regional groundwater flow, 2.

Effect of water-table configuration and subsurface permeability variation: Water Resources

Res., v. 3, p. 623-634.

Harr, M. E., 1962, Groundwater and seepage: McGraw-Hill, New York, 315 p.

Hodge, R. A. L., and Freeze, R. A., 1977, Groundwater flow systems and slope stability: Canadian

Geotechnical Jour., v. 14, p. 466-476.

Patton, F. D., and Hendron, A. J., Jr., 1974, General report on mass movement: Proc. Second Intern.

Congr., Intern. Assoc. Engr. Geol., Sao Paulo, Brazil, v. 2, p. VGR.1 - V-GR. 57.


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EXERCISES:

Problems are worth 4 points each.

1. Given the flow nets below, what are the relative permeabilities between zones (1)and (2)? Use a scale or a ruler.

2. Construct a potentiometric map of the water table in the unconfined sandstone aquifer underlyingthe Belagoa city well field area. Sketch in a few flow lines (dont try for curvilinear squares). The headsare measured in meters at several observation and supply wells over the area. Sketch the flow net incross-section (x-y plane) from A to A.

A A

Sandstone aquifer____________________________________________________________Low permeability material


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3. For the concrete gravity dam below, construct cross-section flow net. If the hydraulic conductivity is10-4 cm/sec, what is the total flux under the dam, and what is the darcian velocity at point A? Assumethat the length of the base of the dam is 15 m.


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4. Complete the following table for each of the 9 combinations of shale (material 1) and sandstone(material 2). Compare and comment on the results.

1 2 K1 (m/s) K2 (m/s)

_____ 10 10-7 10-2

_____ 35 10-7

10-2

_____ 70 10-7 10-2

_____ 70 10-7 10-3

_____ 70 10-7 10-4

5 85 _____ 10-3

10 10 _____ 10-3

10 85 10-11

_____

1 88 10-7 _____

5. What are the scaling coordinates (X-Z) for the following two anisotropic systems?a. Kx = 5 Kz

b. Kx = 10Kz

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Civil Engineering-Construction of equipotential maps