Flownet diagrams —the use of finite differencesand a spreadsheet to determine potential headsby BP WILLIAMS, AG SMYRELL and PJ LEWIS

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IntroductionA flownet diagram is frequently used to model the seepage of waterthrough the ground and to estimate the flow of water through theground and the pore water pressure regime.

Flownets may be drawn by hand but require practice to produceacceptable results. Specialist software programs for use withdesktop computers which will calculate and draw flownets do exist,but not all engineers have ready access to them. Spreadsheetprograms are more generally available, and this paper sets out toshow how they may be used to construct flownets.

As with the flow of heat, the seepage of water is governed by theLaplace equation:

Figure 1.Traditional five point scheme.

Bh 8'h—+—=0ax'z'o

= 4(h, +h +h +h4) (2)

where h represents the potential head of water. The 6nitedifference representation of equation (1) is derived in a number iftextbooks; that presented by Das (1985) is particularly to berecommended. Smyrell (1991)showed how the finite differencesolution for the related Poisson equation may be obtained by the useof a modern spreadsheet on a desktop computer and gave examplesin its application to torsional stiffness.

Application to flownetsIn applying Smyrell's method to flownet problems the traditional 6vepoint scheme may be used, which yields the following equation:

Yy~/

r4

32

This is often displayed in 'molecular'orm as shown in Figure 1.The schemes used to model the impervious boundary conditions areshown in Figure 2. Where there is an unsymetrical flow around thetoe of sheet piling a slightly modi6ed scheme is necessary, as shownin Figure 3.Any spreadsheet capable of iterative calculation maybe used. In the following examples Symphony 1.2 and Excel 4.0have been used. This demonstrates that almost any spreadsheetmay be used, since the version of Symphony used is relatively old.

Formulae for the various boundary conditions are entered into thetop area of the spreadsheet in suitably named and situated cells as inFigure 4. They can then be stored in a master Se for repeated use.

The area of ground under consideration is divided up into a squaregrid and cells in the spreadsheet are allocated for each node point ofthe grid. The cells are Sled with either formulae to calculate thepotential head or values of the known potential head at thecorresponding point, normally at the recharge and dischargesurfaces or along a line of symmetry.

Where the flow is around the toe of sheet piling, the vertical line ofthe grid at the line of piling must be represented by two columns onthe spreadsheet as in Figure 5.It must be remembered that thesetwo columns represent only one vertical line on a true scale drawing ofthe flownet.

The calculation mode in the spreadsheet should be set to manual

GROUND ENGINEERING JUNE 1993

Impermeableboundary

Figure 2. Schemes for boundary conditions.

Sheet pilingLeft sideof pile

Normal four pointscheme at impermecoleboundary on sideof pile

C i)

Sheet

pileI

0

oe

Right sideof pile

Six point scheme at toe of pile Toe of pile

Figure 3.Six point scheme at the toe of sheet piling.

with iteration. The number of iterations necessary will depend onthe size of the grid, but should be continued until there is nosignificant change in the calculated values. Some of the moremodern spreadsheets allow this process to be fully automated byspecifying either the maximum number of iterations and/or theiteration closing tolerance. It is the experience of the authors that aniteration tolerance of 0.001 gives good results.

C

Five point scheme forinternal points undertoe of pile

C /'N

0

QExampleConsider the situation shown in Figure 6with a cofferdam 12m wideand Sm deep. With water levels at ground level outside and atexcavation level inside the differential head is 8m. Assume that thishead will be lost within the sand layer as the gravel is very much morepermeable. Set up a model of the sand layer with a grid spacing of1.5m as in Figure 7, allowing two columns for the line of sheet piling.The recharge and discharge surfaces will be at the gravel/sandinterface and excavation level respectively.

The model will be symetrical about the centre line which can thenbe regarded as an impermeable boundary, as will the sand/clayinterface.

The grid outside outside the cofferdam must be extended to aboundary where any change in its position will have little effect on

Four point scheme forboundary point under

toe of pile

/'xr

Impermeable

Figure 5.Scheme for various grid positions on the left sideof a sheet pile (mirror image on the right side).

I~~'~ ~~~~~~ e) T~ t/~ X y2

4 Description567891$1112151415161718192$

Upper lef t cornerUpper boundaryUpper r'.ight cornerRight boundaryBottom !right cornerBottom .boundaryBottom lef t cornerLeft boundaryInternal pointsUpper right pileUpper left pileInternal left pileInternal right pileBottom lef t pileBottom right pile

~ Cofferdam

//$ ,~7N%W/s,IV

I WaterGround level level

Gravel

[ the calculated potentials near the cofferdam, in this case say 30mand the boundary is assumed as impermeable.

The required formulae from the named cells are copied into thespreadsheet cells as in the model.

The values of 100 and zero are entered for the recharge anddischarge surfaces as the calculated potentials will reflect thepercentage values of the potential head at each point in the grid andwill facilitate the calculation of pore pressures.

A column width of three spaces has been used in the spreadsheetas this gives a reasonable number of columns visible on the screen atany time, also if a suitable character and line spacing is arranged theprintout of the results will give a near square grid, see Figure 8.With the format set to zero (ie values rounded to integers) theresults are presented to a satisfactory degree of accuracy forflownets. The only small disadvantage of Symphony is that the valueof 100 is shown as ***since the spreadsheet requires one extraspace for printing a value.

Symphony allows a maximum of 50 iterations for each time theCALC key is pressed. The computer used had a 386 microprocessor

Excavated level tr Water levelPAW

Sand

Impermeable clay

Figure 6. Example —a 12m wide cofferdam.

8 8LB +LB +LB tLB +LB +LB +LB +LB +BLCBB

188188188188188188188188188188188188188188188188188188188188188+ + + + + + + + + + + t + + + + + + + RB

LB + + + + + + + + + + + + + + + + + + tLB+++++++++++++++++-t +8 8 8 LB + t + + + + + + + + + + + + + + t + ++ + RB LB + + + + + + + + + + + + + + + + + + + RB+ + RB LB + + + + + + + + + + + + + + + + + + + RB+ + RB LB + t + t + + + + + + + + + + + + + + + RB+ + RB LB + + + + + + + + + + + t + + + + + + + RB+ + ULPORP + + + + + + + + + + t t + + + + + + ++ + ILPZRP + + + + + + + + + + + + + t + + + + + RB+ + ILPZRP + + + + + + + + + + + + + + + + + + + RB+ + ILPZRP + + + t + + + + + + + + + + + + + + + RB

BB BB BLPBRPBB BB BB BB BB BB BB BB BB BB BB BB BB BB BB BB BB BB BB BRC

Note: a) See Figure 4 for details of the formula for each cell nameb) For all internal points + use INTc) Values of 188 placed at, recharge boundaryd) Values of zero placed at discharge boundary

Figure 7. 1.5m square grid to model example in figure 6.

34

at 16Mhz but no math co-processor. The time taken to reach thestage at which the results stabilised was 4 minutes 20 seconds, andrequired 1000 iterations. This was determined by observation of thechanges in the displayed values. Unfortunately, having the figuresdisplayed during iteration makes the process slower since theprogram must continuously update the screen. For this reason amore modern spreadsheet which automates the iterative process isto be preferred. In this case, the figures may be scrolled off thescreen and the computer allowed to seek convergence withoutupdating the screen. The time taken is then of the order of a fewseconds rather than a few minutes.

The results are shown in Figure 8 where the dimensions andequipotential lines have been added.

A reduction of the distance outside the cofferdam to the right handboundary from 30m to 25.5m reduced some of the values adjacent tothe toe of the pile by one. Treating the right hand boundary as arecharge surface, ie potential equal to 100, made no difference tothe values adjacent to the pile.

GROUND ENGINEERING JUNE 1993

A true scale drawing of the equipotential line can be made fromthese results and the flow lines added by hand to give the completefiownet (Figure 9). Remember that the two columns for the sheetpiling must be reduced to one for the scale drawing.

Potentials on each face of the sheet piling and across thecofferdam at the level of the toe of the piles can be read directly fromthe spreadsheet which allow a rapid determination of:a) The pore water pressures affecting the piles which is essential

for design particularly where the width is narrow.b) The factors of safety against piping and base heave.Figure 10 shows the same problem analysed using the finiteelement package Lusas. As can be seen, the comparison of potentialhead contours is sufficiently accurate for practical purposes.

Orthotropy, grid grading and soil layersFor many situations, the above finite difference equations will proveto be sufficient. However, they may be modified in a number of waysto reduce calculation time and allow the solution of more complex

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144

Sand

Impermeable clay

Figure 9. Completed flow net.GROUND ENGINEERING JUNE 1993

35

MYSTRO: 10.2-3 DATE: 6-02-93

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TITLE:

Figure 10.Finite element results for cofferdam example.

problems. In the case of the example shown in Figures 6 to 10,which uses a square grid of points, in those areas where the rate ofchange of head is low, it would improve the time to solution by usinga rectangular grid, thus reducing the number of nodes. Similarly,orthotropic soil properties, or the existence of horizontal and/orvertical strata may be dealt with by modifying the molecule used.The simple five-point molecule shown in Figure 1 can easily bemodified to allow all of these possibilities to be modelled. Thenecessary generalised molecule is shown in Figure 11This leads tothe following equation, a full derivation of which is given inappendix A:

a,h, =a,h, +a,h, +a,h, +a,h,

—k„yxor

The permeability terms have been stated with reference to therelevant quadrant of the molecule, since this allows the formula to beapplied more easily to impervious boundaries and soil layers, wherethe properties change. At impervious boundaries, the relevantquadrant terms are removed, while at soil layers, the correctquadrant permeabilities are inserted. In all cases, the single generalformula can be used, no special boundary formulae need to beremembered. Notice that each of the terms is of the form:

where: where x and y are the horizontal and vertical dimensions ofquadrant Q.

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II'l 'I 'ilu Ua, = —k,a+ kccm

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a, = —k~+ —k~m nU U

JIl I iM4

and: Permeability= k„

36 a, = a, + a, + a, + a,GROUND ENGINEERING . JUNE 1993

Figure 11.Generalised five point scheme.

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Figure 12. Concrete dam with imperfect clay cut-off wall.

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Figure 13.Finite difference mesh used and contours of percentage head.

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Figure 14.Spreadsheet solution to cut-off wall example.GROUND ENGINEERING JUNE 1993

37

IMYSTRO: 10 ' 3I

DATE: 29-01-93

Figure 15.Finite element output for cut-off wall example.

As an example ofusing equation (3), consider the situation shown inFigure 12, which is used to test the efficiency of an imperfect, claycut-off wall underneath what is taken to be an impermeable concretedam. Figure 13shows the Finite Difference grid used. Figure 14shows the spreadsheet results after approximately 1000 iterations.Note that the grid dimensions are entered in the leftmost column andin the row underneath the grid. The cell formulae use a combination ofabsolute and relative referencing to obtain the correct values of m, n,u, and v. In other words, to obtain u or v, the formula refers to theleftmost column absolutely but to the row in a relative fashion; forvalues ofm and n, the opposite is true.

Care must be taken to eliminate the correct quadrant terms if thesolution is to be accurate. The process is made much simpler if thespreadsheet used allows cells to be given variable names, as shownat the top of Figure 14. These names are more easily spotted in theformulae, making it more difficult to remove the wrong termsaccidentally. In addition, it is best to leave the boundary cells unfilleduntil the last moment before iteration, since it is all too easy togenerate 'division by zero'rrors. These are due to the formulareferring to the cells containing zero values for m, n, u, and voutside the boundary of the problem.

Figure 15 shows a finite element solution of the same problem.The finite element mesh was chosen to be exactly that of the finitedifference grid so that nodal values could be compared more easily.In addition, it allows a speedier comparison of the contour positions.These can be seen to compare very favourably. Normally, fewerfinite elements would be used. Quadratic, eight-noded finiteelements were used and analysed using Lusas.

The efficiency of the cut-off wall can be assessed by calculatingthe water flow rate from the model. It is observed that at the toe ofthe dam, the flow is almost vertical, thus the flow rate is given by:

ReferencesBrala M Das (1965L 'Advanced sod mechanics'. pp 122-190 pub Mcnravv-Htg.AO SmyreU (1991).'The denvatton of torsional stiffness using a modern spreadsheet and unprovedfintte difference scheme'. The Structural Engmeer, Vol 69 bfo24.Symphony is produced by Lotus Development Corporation.Microsoft Excel 4.0 ts produced by Microsoft Corporation.Mystro and Lusas are produced by FEA, Forge House, 66 High St, Kingston-upon-Thames, London.

Appendix A —Derivation of thegenerabsed fltnite difference schemeFor unidirectional flow in a soil with constant head gradient, thevolume of water per second is given by:

A = Cross-sectional area of the flow channelk = Permeability coefficienti = Potential head gradient

The generahsed finite difference scheme is derived byconsidering the shaded flow channels in Figure 11.Smce thesechannels overlap at point 0, they only extend to half the width ofthe finite difference cell; the white areas represent the commonareas and these are subtracted from the overall cell to aflow forthe overlap. Using equation (Al), and assuming unit ~ss,the inward flow to point 0 can be obtained for aII four channels as:

Q = Ak„i.„„~, (4l qto ktvt + katt

38

where i„„ag,is the average exit head gradient taken from the top tworows of cells on the spreadsheet, which represent the toe of the dam.This gives Q = 6.695 x 10 6ms/s/mrun compared with the Lusasanalysis value of Q = 6.453 x 10 m /s/mrun. If the analysis is donewithout the presence of the cut-of'f wall, the flow rate is given byQ = 9.44 x 10 bm'/s/mrun. Performing this analysis is a simplematter of copying formulae from the left of the cut-off wall.

ConclusionsThis paper has shown that seepage problems can be solved quicklyand simply on a modern spreadsheet, using the method of finitedifferences. It has been shown that the traditional five-point schemecan be applied to simple situations, but that a slightly modifiedmolecule can be used to model more complex problems. It issuggested that the engineer can gain experience and confidenceusing the traditional scheme before moving up to the more powerfulgeneralised scheme. The technique has been shown to besufficiently accurate for practical purposes.

GROUND ENGINEERING JUNE 1993

q~ = —N+-~~

The net inflow must be zero, hence:

q«+qso+qan+qo"0

Combining equations (A2) and {A3)leads directly to thegeneralised formula quoted in equation (3).

(A3}