SOIL PERMEABILITY AND SEEPAGE
4.15 FLO W NE T CONSTRUCTIO N
Properties o f a Flo w Ne t
The properties o f a flow net can be expressed a s given below:
1. Flo w an d equipotential lines are smooth curves .
2. Flo w line s and equipotential lines meet at right angles to each other .
Soil Permeability an d Seepage 11 7 3. N o two flow line s cross each other .
4. N o two flow or equipotential lines start from th e same point.
Boundary Condition s
Flow o f wate r throug h eart h masse s i s i n genera l thre e dimensional . Sinc e th e analysi s o f three-dimensional flow i s too complicated, the flow problems are solved on the assumption that the flow i s two-dimensional. All flow line s in such a case are parallel to the plane of the figure, and the condition is therefore known as two-dimensional flow. All flow studie s dealt with herein are for the steady stat e case. The expressio n fo r boundar y conditions consists o f statement s of hea d o r flo w conditions at all boundary points. The boundary conditions are generally four in number though there are only three in some cases. The boundary conditions for the case shown in Fig. 4. 16 are as follows:
1 . Lin e ab is a boundary equipotential line along which the head is h(
2. Th e line along the sheet pile wall is a flow boundary
3. Th e line xy is a boundary equipotentia l line along which the head is equal to hd 4. Th e line m n is a flow boundary (at depth H below bed level).
If we consider any flow line , say, p1 p2 p 3 i n Fig. 4.16, the potential head at p{ i s h( and at p3 i s hd. The total head lost as the water flows along the line is h which is the difference between the upstream an d downstrea m head s o f water . The hea d los t a s th e wate r flow s fro m p l t o equi- potential line k is Ah which is the difference between the heads shown by the piezometers. This loss of head Ah is a fraction o f the total head lost.
Flow Ne t Constructio n
Flow nets are constructed in such a way as to keep the ratio of the sides of each block bounded by two flow lines and two equipotential lines a constant. If all the sides of one such block are equal, then the flow ne t must consist of squares. The square block referred to here does not constitute a square according to the strict meaning of the word, it only means that the average width of the square blocks are equal. For example, in Fig. 4.16, the width al of block 1 is equal to its length b}.
The area bounded by any two neighboring flow line s is called a/low channel. If the flow net is constructed in such a way that the ratio alb remains the same for all blocks, then it can be shown that there is the same quantity of seepage in each flow channel. In order to show this consider two blocks 1 and 2 i n on e flo w channe l an d anothe r block 3 i n anothe r flo w channe l a s show n in Fig. 4.16. Block 3 is chosen in such a way that it lies within the same equipotential lines that bound the block 2. Darcy's law for the discharge through any block such as 1 per unit length of the section may be written as
Aq = kia = —Ah ab b a = kAh —
where Ah represents the head loss in crossing the block. The expressions in this form for each of the three blocks under consideration ar e
Aq{ = kAh—, Aq 2 = kAh2 —
b\ b 2
In th e abov e equatio n th e valu e o f hydrauli c conductivity k remain s th e sam e fo r al l th e blocks. If the blocks are all squares then
b2
Piezometer tube s
(a) Flow ne t
Piezometer
Flow line
Equipotential line
(b) Flow through a prismatic element
Figure 4.16 Flo w through a homogeneous stratu m of soi l
Since blocks 1 and 2 are in the same flow channel, we have &ql = Ag2. Since blocks 2 and 3 are within the same equipotential lines we have A/z2 = A/?3. If these equations are inserted we obtain the following relationship:
A#j = Ag2 and A/Z J = A/z2
This proves that the same quantity flows through each block and there is the same head drop in crossin g each bloc k i f all the blocks ar e square s or possess th e sam e rati o alb. Flo w net s are constructed by keeping the ratio alb the same in all figures. Squar e flow net s are generally used in practice as this is easier to construct.
Soil Permeability an d Seepage 11 9
There ar e man y method s tha t ar e i n us e fo r th e constructio n o f flo w nets . Som e o f th e important methods are
1. Analytica l method, 2. Electrica l analo g method, 3. Scale d model method, 4. Graphica l method .
The analytica l method , base d o n the Laplac e equatio n althoug h rigorousl y precise , i s not universally applicabl e i n al l case s becaus e o f th e complexit y o f th e proble m involved . Th e mathematics involved even in some elementary cases is beyond the comprehension of many design engineers. Although this approac h i s sometime s usefu l i n th e checkin g o f othe r methods , i t i s largely of academic interest .
The electrical analog y method has been extensivel y made us e of in many important design problems. However , i n most o f the case s i n the fiel d o f soi l mechanic s wher e th e estimatio n of seepage flow s an d pressures ar e generally required , a more simple metho d suc h as the graphica l method is preferred.
Scaled model s ar e ver y usefu l t o solv e seepag e flo w problems . Soi l model s ca n b e constructed to depict flow of water below concrete dam s or through earth dams. These models ar e very useful t o demonstrate the fundamentals of fluid flow, but their use in other respects i s limited because of the large amount of time and effort require d to construct such models .
The graphical method developed b y Forchheimer (1930 ) has been found to be very useful in solving complicated flo w problems. A. Casagrande (1937) improved thi s method by incorporating many suggestions. Th e main drawback of this method i s that a good dea l of practice an d aptitude are essential to produce a satisfactory flow net. In spite of these drawbacks, the graphical method is quite popular among engineers .
Graphical Metho d
The usual procedure for obtaining flow nets is a graphical, trial sketching method, sometimes calle d the Forchheimer Solution. This method of obtaining flow nets is the quickest and the most practical of al l the available methods. A . Casagrande (1937 ) ha s offered man y suggestions t o the beginne r who is interested in flow net construction. Some of his suggestions are summarized below:
1. Stud y carefully the flow ne t pattern of well-constructed flo w nets . 2. Tr y to reproduce th e same flow nets without seeing them.
3. A s a first trial, use not more than four to five flow channels. Too many flow channels would confuse the issue.
4. Follo w th e principle of 'whole to part', i.e., one has to watch the appearance o f the entire flow ne t and when once th e whole net is found approximatel y correct , finishing touches can be given to the details.
5. Al l flo w an d equipotentia l line s shoul d b e smoot h an d ther e shoul d no t b e an y shar p transitions between straight and curved lines.
The abov e suggestions , thoug h quite useful fo r drawing flow nets , ar e no t sufficien t fo r a beginner. In order t o overcome thi s problem, Taylor (1948 ) propose d a procedure know n as the procedure by explicit trials. Some of the salient features of this procedure ar e given below:
1. A s a first step in the explicit trial method, one trial flow line or one trial equipotential line is sketched adjacen t to a boundary flow line or boundary equipotential.
2. Afte r choosin g the first trial line (say it is a flow line) , the flow path between th e line and the boundary flow line is divided into a number of squares by drawing equipotential lines.
These equipotential lines are extended to meet the bottom flow line at right angles keeping in view that the lines drawn should be smooth without any abrupt transitions.
3. Th e remaining flow line s are next drawn, adhering rigorously to square figures.
4. I f th e firs t tria l i s chose n property , th e ne t draw n satisfies al l th e necessar y conditions . Otherwise, th e last drawn flow lin e will cross th e bottom boundar y flow line , indicating that the trial line chosen is incorrect and needs modification.
5. I n such a case, a second tria l line should be chosen and the procedure repeated .
A typica l exampl e o f a flow ne t under a sheet pil e wal l is given i n Fig. 4.16. I t should b e understood tha t the number of flow channel s will be an integer only by chance. That means , th e bottom flo w line sketched migh t not produce ful l square s with the bottom boundary flow line . In such a case the bottom flow channel will be a fraction of a full flow channel. It should also be noted that the figure formed by the first sketched flow line with the last equipotential line in the region is of irregular form. This figure is called a singular square. The basic requirement for such squares, as for al l th e othe r squares , i s that continuous sub-division of th e figure s giv e a n approac h t o true squares. Suc h singula r squares ar e forme d a t th e tip s of shee t pil e wall s also. Square s mus t b e thought of as valid only where the Laplace equation applies. The Laplace equation applies to soils which are homogeneous and isotropic. When the soil is anisotropic, the flow net should be sketched as before o n the transformed section. The transforme d sectio n ca n b e obtaine d fro m th e natural section explaine d earlier.