BRAND, E. W. & PFCEMCHITT, J. (1980). GCotechnique 30, No. 4, 369-384

Shape factors of cylindrical piezometers

E. W. BRAND* and J. PREMCHITTt

This Paper describes a liquid electric analogue model and a finite difference model used to establish reliable values for the shape factors of cylindrical piezometers with length/diameter ratios of up to 15. For piezometers with length/diameter ratios greater than 4, it was found that the shape factor is given with negligible error by the relationship: F = 7d + 1.651. An examination of the effect on the shape factor of piezometer proximity to the water- table showed this to be negligible for practical purposes, except where the piezometer is within a few piezometer lengths of the water-table. Shape factors were also established for cylindrical piezometers embedded in cylin- drical soil specimens subjected to all five possible boun- dary conditions. The measured shape factors were higher or lower than the infinite values depending upon the boundary conditions and the proximity of the boundaries to the piezometer. These data give some guidance to the performance of pore pressure probes used in laboratory specimens, and they are useful in the analysis of steady state flow conditions and for the assessment of response studies of piezometer systems.

Cet Article d&it un modtle analogique tlectrique fluide et un modele a difference finie servant a itablir des valeurs fiables pour les coefficients de forme de pitzometres cyhndriques dont les rapports longueur/diamttre peuvent atteindre 15. Pour les piezometres dont le rapport longueur/daimetre est superieur a 4, on a trouve que le coefficient de forme etait donne, avec une erreur negligeable, par la relation F = 7d + 1.65. Linfluence sur le coefficient de forme de la proximite de la nappc phreatique est pratiquement negligeable sauf lorsque le piezomttre itait a quelques longueurs de piezomttre de la nappe phreatique. Les coefficients de forme ont igalement tte Ctablis pour des piezometres cylindriques enfonces dans des ichantillons de sols cylindriques soumis aux cinq conditions aux limites possibles. Les coefficients de forme mesurts Ctaient superieurs ou inferieurs aux valeurs infinies dependant des conditions aux hmites, et de la proximite des limites par rapport au piezometre. Ces donnees permettent de se faire une certaine idte de la performance de sondes de pression interstitielle utilisees dans des tchantillons de laboratoire et sont utiles pour lanalyse des conditions dicoulement stationnaire ainsi que pour Ievaluation des rtponses des pitzometres.

INTRODUCTION

Piezometers are widely used for measuring in situ pore pressure and for the insitu determination of certain soil properties. A piezometer system with a quick response is often important for the accurate measurement of pore pressure, and a complete understanding of the factors which govern the response characteristics is crucial to accurate determinations of soil properties.

Hvorslev (195 1) derived the theoretical time lag for a piezometer system in an incompressible soil by using the basic differential equation that governs the saturated flow through a falling head permeameter. The pore pressure u, at any time C, in a piezometer in a soil of permeability k was shown to be related to the initial pore pressure a,, in the piezometer and the equalization pore pressure u, by

U -U m = exp ( - Fkt/Vy,) 4c -uo

where yw is the unit weight of the water, and F and Vare respectively the shape factor (or intake factor) of the piezometer and the volume factor of the piezometer system.

The shape factor and the volume factor, together with the soil permeability, govern the response characteristics of a piezometer system in an incompressible soil and can be thought of as the piezometer system characteristics. These characteristics can be defined as follows.

Discussion on this Paper closes 1 March, 1981. For further details see inside back cover. * Public Works Department, Hong Kong. t Asian Institute of Technology, Bangkok.

370 E. W. BRAND AND J. PREMCHITT

Shapefactor F is a physical dimension of the piezometer which determines the rate of flow q into the piezometer under a fixed head drop H as

q=FkH (2)

ViZumefactor Vis the volume of water required to flow into or out of the piezometer system to equalize a unit pressure difference between the piezometer system and the surrounding soil.

In compressible soils, the consolidation and swelling of the soil surrounding a piezometer plays a major part in piezometer response (Gibson, 1963), and equation (1) does not govern the equalization process. Nevertheless, shape factor and volume factor play equally important parts for piezometers in compressible and incompressible soils.

The volume factor represents the hydraulic flexibility of the piezometer system, which comprises the piezometer itself, the measuring system and all connecting tubes and valves. Its absolute value can be measured fairly easily for a given system, or it may be computed from the volume flexibilities of the separate components.

The shape factor of a piezometer is a function ofits physical dimensions, and this controls the flow pattern in the soil surrounding the piezometer. It is independent of the soil permeability. The shape factor is generally a characteristic of an axisymmetrical flow net, since the porous element of a piezometer is nearly always axisymmetrical in shape. Because the flow net is affected by the shape and size of the body of soil in which the piezometer is placed, the value of F is also affected by the physical dimensions of the flow regime and by the conditions at its boundaries. For a spherical pizometer in a spherical or infinite body of soil, it is possible to integrate directly the governing equation of flow to obtain a closed-form solution for shape factor. For piezometers of practical shape however the partial differential equation which governs the flow cannot usually be solved by analytical means, and no closed-form solutions are available for the commonly used piezometers.

Apart from their use in predictions of response times of piezometer systems, precise values of shape factor are vital to the accurate interpretation of in situ methods to determine coefficients of permeability, consolidation and earth pressure (see Hvorslev, 1951; Gibson, 1963; Bishop & Al-Dhahir, 1969; Wilkinson, Barden & Rocke, 1969; Bjerrum & Andersen, 1972; Penman, 1975). Very little attention has been given to the accurate determination of shape factor for cylindrical piezometers and less still to those for other shapes. Various approaches have been used by several investigators to obtain numerical solutions for some cylindrical piezometers, but the values they obtained are not in general agreement, and objections can be raised to some of the experimental techniques employed. For these reasons, and because of the importance of shape factors to in situ measurements, the Authors set out to establish reliable values for cylindrical piezometers, which they consider have not been available hitherto. Because piezometers have been the subject of response studies (Penman, 1961; Brooker & Lindberg, 1965) and are used for laboratory measurements, an examination was also undertaken of the effects of boundary proximity and boundary conditions on measured shape factors. An electric analogue model, in the form of an electrolytic tank, and a finite difference model were both employed to meet the objectives of the study.

PREVIOUS SHAPE FACTOR DETERMINATIONS

To assign a value of shape factor to a piezometer, it is necessary to determine the theoretical flow rate into the piezometer. The Laplace equation (V2 u = 0) which governs the steady state flow in a porous medium can be solved in closed-form only for a spherical piezometer, for which it can readily be shown that F = 2nd, where d is the diameter of the sphere. Piezometers used in

ed.

SHAPE FACTORS OF CYLINDRICAL PIEZOMETERS 371

practice are virtually all axisymmetrical, and the steady state flow to these is governed by the equation

la 1au (i%=O __ _- () rar rar +a22

where the pore pressure u is expressed in terms of the cylindrical co-ordinates (I, z). No closed- form solution is available for a cylindrical cavity in an infinite medium. Dachler (1936) derived a solution for the flow from a line source for which the equipotential surface was a hemispheroid, and Hvorslev (1951) applied this solution to a cylindrical piezometer by representing the cylinder by its inscribed prolate spheroid, to obtain the shape factor as

2nl F = log(I/d+J[1+(l/d)2]}

(4)

where 1 and dare the length and diameter of the piezometer. This equation is only approximate, the error in the shape factor calculated by equation (4) increasing as l/d decreases.

Kallstenius & Wallgren (1956) presented an alternative derivation of equation (4) by considering the steady state flow to a spherical piezometer. They suggested that a piezometer of any shape could be represented by a spherical piezometer with the same surface area. This leads to the simple expression for shape factor: F = 2,/(nS), where S is the surface area. For cylindrical piezometers

F = 2nJ(ld) (5)

This expression predicts F values much below those of Hvorslevs values for l/d< 1 and for l/d> 15. In practice cylindrical piezometers generally have l/d ratios between 4 and 10, and for this range, equation (5) gives F values which are within f 5% of those given by equation (4).

Wilkinson (1968) noted that Hvorslevs expression slightly underestimated the value of F, and he suggested that a more accurate value could be obtained by representing the cylinder by an inscribed prolate spheroid with its major axis adjusted so that its volume was equal to that ofthe cylinder. This empirical adjustment results in

3lrl F = log {la/d+JCl +UW421} (6)

which is equivalent to equation (4) with l/d replaced by 1.51/d. Smiles & Youngs (1965) employed an electric analogue model to measure directly shape

factors for cylindrical cavities with l/d ratios in the range &4. A square tank, 1000 x 1000 x 250 mm deep filled with water, simulated the soil medium, while brass rods represented the piezometers. The shape factor was determined from the resistance between the brass rod and the brass sheet at the boundary of the tank as measured by an AC bridge with a cathode ray oscilloscope as the null indicator. The values of shape factor measured in this way were generally about 15% higher than those predicted by equation (4).

Numerical analysis for shape factors of cylindrical piezometers has been carried out using finite differences by only Al-Dahir & Morgenstern (1969) and Raymond & Azzouz (1969). Al- Dhahir & Morgenstern employed the Gauss-Siedel successive over-relaxation technique to examine the flow towards cylindrical piezometers in an infinite porous medium (with a diameter 50 times the piezometer diameter); their results were within about 5% of those obtained experimentally by Smiles dz Youngs (1965) for l/d ratios of &4. Using a similar numerical technique, however, Raymond & Azzouz (1969) arrived at shape factor values which averaged only about 70% of those reported by Al-Dhahir & Morgenstern (1969); unfortunately, they gave no details of their finite difference method.

312

X Smiles 8 Youngs (1965)

0 Al-Dhohir 8 Morgenstern (1969)

E. W. BRAND AND J. PREMCHITT

0 2 4 6 a IO 12 14 16

Length/diameter (P/d)

Fig. 1. Previously published shape factor measurements for cylindrical piezometers, and the relationships proposed between shape factor and length/diameter ratio

The results of the direct determinations of shape factor by Smiles & Youngs (1965) Al-Dhahir & Morgenstern (1969) and Raymond & Azzouz (1969) are plotted in Fig. 1 in terms of the ratio F/d for cylindrical piezometers with l/d ratios from (r20. These may be compared with the closed-form solutions of Hvorslev (1951), Kallstenius & Wallgren (1956) and Wilkinson (1968). Figure 1 illustrates clearly the present uncertainty which exists over the numerical values of shape factor for cylindrical piezometers; it is also apparent that reliable measurements have been limited to an l/d ratio of less than 4.

PROGRAMME OF SHAPE FACTOR STUDIES

The three express objectives of the Authors programme of shape factor studies were

(a) to establish for field use accurate values of shape factor for cylindrical piezometers with l/d ratios from 2 to 15

(b) to evaluate the effect on its shape factor of the proximity of a piezometer to the water- table

(c) to examine the influence of boundary conditions on the shape factors of piezometers used in laboratory test specimens

To achieve these objectives, a circular electrolytic tank model (liquid analogue) was used, all the results obtained from part (c) being checked against those obtained by means of a finite difference model.

Shape factors for cylindrical piezometers in an infinite medium were established by means of the liquid analogue alone after calibration with spherical model piezometers to ensure its validity and to enable the cylindrical piezometers to be modelled at a scale that ensured the accurate representation of an infinite flow regime. A secondary calibration of the model was achieved by carrying out the shape factor measurements with two separate boundary conditions to ensure that no boundary effects existed. A boundary of a flow regime can theoretically be a recharge (or drainage) boundary, where the pore pressure u = constant (i.e. an equipotential surface); or an impermeable boundary, where au/az = 0 or where au/& = 0 (i.e. a flow surface).

SHAPE FACTORS OF CYLINDRICAL PIEZOMETERS 373

Bl (a) Boundary conditions examined B2

1 I 7 Brass model

I . A{ .-piezometers

I

I. \rll 81

.-Top water -

1 T i &/boss mesh-- y

--_ -_-- I

(b) Simulation of boundary conditions in liquid analogue

Fig. 2. Boundary conditions Bl and B2 used with the liquid electric analogue model for measurements of shape factors for cyliklrical piezometers in an infinite medium

In the field, the flow regime in the region of a piezometer is generally of finite extent in a vertical direction, the soil stratum in question being bounded by a relatively impermeable surface below and a water-table (equipotential surface) above. These boundary conditions, designated as Bl in Fig. 2, were first applied to the model (in which the water-table is inverted), and the shape factor measurements were repeated for boundary conditions B2.

Frevert & Kirkham (1948) reported that the proximity of an open-ended tube to the water- table had a significant effect on the measured shape factor. This might be important for the determination of in situ soil properties, and it was investigated for cylindrical piezometers in the liquid analogue.

Where piezometers are used for laboratory measurements, they are most commonly placed in cylindrical soil specimens with length twice the diameter, the side and end boundaries of which are either impermeable surfaces (rubber membranes and solid end platens) or drainage surfaces (filter paper drains or porous end platens). The five possible sets of boundary conditions are illustrated in Fig. 3, where they are designated as Bl-B5. These five boundary conditions represent commonly used drainage mechanisms for biaxial specimens, and all were examined in the shape factor studies. Both the electric analogue and the finite difference model were used to determine the shape factors, the variables examined being the boundary conditions, the ratio of specimen diameter D to piezometer diameter d, the l/d ratio for the piezometer, and the vertical position of the piezometer relative to the end platens.

THE ELECTRIC ANALOGUE MODEL

The electric analogue model was constructed specially for the shape factor measurements. A cylindrical electrolytic tank, 900mm diameter and 500mm deep, was made of 3 mm thick perspex with a 6mm perspex bottom plate. The electrolyte used was water. The model piezometers were made of brass and were located on the axis of the tank. The recharge boundaries at the sides and bottom of the tank were represented by a BS No. 30 brass wire mesh.

314 E. W. BRAND AND J. PREMCHITT

Bl B2 83 B4 B5

- Recharge or drainage boundary

- Impermeable boundary

Fig. 3. Boundary conditions BlLB5 used for measurements of shape factors for cylindrical piezometers in a cylindrical soil specimen with length/diameter ratio of 2

For the tests conducted to examine boundary effects in laboratory specimens, 1 mm brass plate was formed into cylinders which were placed symmetrically in the tank. An AC bridge was used for measurements between the two electrodes.

The cylindrical piezometers used for shape factor measurements in an infinite porous medium were represented by a 3 mm diameter brass rod wrapped with PVC insulating tape except along an exposed length which was varied to achieve the desired range of l/d ratios. The ratio of piezometer diameter d to tank diameter D therefore was 300, and this was found to satisfy the condition for an infinite porous medium (see below). The insulated rod was held in a screw clamp attached to a vertical lory which was, in turn, mounted on a horizontal lory positioned across the tank diameter. For the simulation of piezometers in laboratory specimens, 12.7 mm diameter brass models were attached to an insulated brass rod. By means of the lory system, any piezometer could be positioned in the electrolytic tank to an accuracy of 01 mm.

The measurement ofshape factor was carried out in much the same way as that used by Smiles & Youngs (1965), but some small refinements were adopted. The resistance of the electrolyte between the electrodes was determined by means of an AC bridge circuit energized by a constant 9 volts 50 Hz power supply. Resistances were measured to an accuracy of 1% by using a digital voltmeter as a null indicator. Measurements were made quickly to avoid polarization. The shape factor F was calculated directly from the measured resistance R as

F = l/oR

where cr is the specific conductivity of the electrolyte.

(7)

The conductivity ofthe tap water used as the electrolyte was measured in a cylindrical perspex conductivity cell 140 mm diameter and 400 mm long. Two brass discs in the cell were used as electrodes in the same AC bridge employed with the electrolytic tank, and the resistance between the discs was measured for various lengths ofelectrolyte column by varying the spacing between the discs. The conductivity of the tap water was found to vary significantly with temperature (Fig. 4), a fact which has not been mentioned previously by those working with liquid analogue models. The variation of about 1.4 x 10e6 mho/mm (or about 2%) per degree Celsius at 25 C would cause appreciable errors in shape factor measurements in conditions where a controlled temperature environment was not available and where the conductivity

SHAPE FACTORS OF CYLINDRICAL PIEZOMETERS 375

80 I I I I I I I

75 -

70 - 0 Sample 1 0 2 v n 3 x . 4

65 I I I I 20 21 22 23 24 25 26 27 28

Temperature, C

Fig. 4. Measured variation with temperature of tbe specific conductivity of tap water

Table 1. Calibration of liquid electric analogue by brass spheres

Sphere diameter d: Ratio diameter tank

mm diameter sphere

50.8 18 25.4 35 19.05 41 12-7 71

Measured F/d

6.78 6.48 6.43 6.35

Theoretical F/d

6.28

variations were not appreciated. The Authors measurements were carried out in an environment where the temperature varied only a few degrees, and a thermometer was used constantly to measure the temperature of the water in the electrolytic tank.

In order to calibrate the electrolytic tank, measurements were first made of shape factors for spherical piezometers, for which F = 2nd or F/d = 6.28. Brass spheres with diameters 12.5, 19, 25 and 50 mm were used for this purpose, each being suspended mid-depth in the water by a nylon sling attached to the horizontal lory over the tank. The results are shown in Table 1, where it can be seen that the value of F/d approached 6.28 as the ratio of the tank diameter to the piezometer diameter increased. For the largest diameter ratio of 71, the F/d value was only about 1% above the theoretical value for an infinite medium.

THE FINITE DIFFERENCE MODEL

The finite difference form of equation (3) can be written for node (i,j) as

Ui+,,j+Ui_~,j+(1+h/2rj)Ui,j+~+(l-h/2rj)Ui,j-,-4Ui,j=O (8)

where h is the node spacing (mesh size), and r is the radial distance of node (i,j) from the axis; i and j increase in the z and r directions respectively. The most powerful means for solving the system of finite difference equations is the Gauss-Siedel iterative method with over-relaxation (Forsythe & Wasow, 1960), and this was adopted by the Authors, as it had been by Al-Dhahir & Morgenstern (1969) for their work on shape factors. The piezometer was treated as a sink (u = 0)

E. W. BRAND AND J. PREMCHI-M

A

1.84 1.86 1.88 1.90 1.92 1.94 1.96 I .98 2.00

Over- relaxation parameter (Cd)

Fig. 5. Effect of the value of the over-relaxation parameter on the rate of convergence of the finite difference solutions for boundary conditions B2 and B3

and the recharge boundaries were assigned the value u = 10. The iteration was terminated when the maximum difference between two successive values of the pore pressure at any node was less than 0N105.

Although convergence is ensured by the nature of the Laplace equation, the rate of convergence was found to depend on the boundary conditions and the value of the over-relaxation parameter, w. At first, the Authors used the value w = 1.9, shown to be the theoretical optimum by Forsythe & Wasow (1960), and this resulted in rapid convergence for boundary conditions Bl, B4 and B5 (Fig. 3). Convergence for B3 however was appreciably slower and B2 required almost 400 iterations. Al-Dhahir & Morgenstern had experienced the same difficulty. An examination was made of the effect of the over-relaxation parameter on the rate of convergence for boundary conditions B2 and B3, with the result shown in Fig. 5. The optimum value of w for B3 is confirmed as being in the region of 1.90; for B2 the optimum value of w = 1.96 resulted in convergence being achieved in 200 iterations.

As in the liquid analogue tests, the cylindrical piezometers were simulated as being permeable only over their curved surface. The piezometer ends were assumed to be impermeable, and the boundary condition au/& = 0 was assigned. A singularity existed at each of the four corners of the piezometer where it was necessary for the node to satisfy the condition u = 0 for the curved permeable surface and au/dz = 0 for the impermeable end cap. Al-Dhahir & Morgenstern (1969) showed that this difficulty has a critical effect on shape factors calculated by finite differences, and they dealt with the singularity by calculating F values for both boundary conditions for a range of mesh sizes. The two values of F converged to a unique value at zero mesh size, thus eliminating the effects of both singularity and mesh size. The Authors adopted this same technique, a typical example of which is illustrated in Fig. 6.

When convergence had been achieved for the finite difference solution to the pore pressure distribution in the flow regime, the flow rate to the piezometer was calculated by the contour integration method. For steady state flow, the flow rate through any closed surface around the piezometer should be the same and should be equal to the flow rate into the piezometer. Three

SHAPE FACTORS OF CYLINDRICAL PIEZOMETERS 377

Mesh size / piezometer diameter

Fig. 6. Effect of mesh size and singularity condition on the value of shape factor determined from the finite difference model (I/d = 2, D/d = 8)

Table 2. Measured shape factors for cylindrical piezometers in infinite soil body

1

Ratio d

Measured F/d for boundary conditions Bl

Measured F/d for boundary conditions B2

2 9.10 8.80 4 13.51 13.36 6 17.21 16.95 8 20.30 20.10

12 26.77 26.41 15 30.74 30.69

Ratio gi

1.03 1.01 1.01 1.01 1.01 1.00

cylindrical surfaces were selected in each case, the shape factor being computed from the flow rates through each of these. Shape factors so determined were accepted as accurate if the three values were all within 1% of each other.

The results obtained from the finite difference model were all checked independently by measurements carried out in the liquid analogue model. In no case did the discrepancy between the two values of shape factor exceed 2%. The Authors are confident, therefore, that their results are completely reliable.

SHAPE FACTORS OF CYLINDRICAL PIEZOMETERS IN INFINITE SOIL

The calibration of the electrolytic tank by spherical piezometers verified that infinite values of shape factor would be determined for pizometers located at mid-depth in the centre of the tank. The values of F/d determined in this way, with the ratio of the tank diameter to piezometer diameter of 300, are listed in Table 2 for the two boundary conditions Bl and B2 (Fig. 2). It can be seen that the values are very nearly the same for the two boundary conditions, which verifies that the distance to the boundaries was sufficiently large not to affect the measured shape factors.

E. W. BRAND AND J. PREMCHITT

I I I I I I 7

_*

x 0 Al - Dhohir 8 Morgenstern(l969 i A Roymond 8 Azzoui (1969) I I I I I I I

z 4 ti IO I2 14 16

Length / diameter (l/d)

Fig. 7. Measured shape factors for cylindrical piezometers in an infinite medium as a function of length/diameter ratio

The measured infinite shape factors (for boundary conditions Bl) are plotted in Fig. 7 in terms ofthe ratio F/d versus the ratio l/d, and these are compared with the results obtained by previous investigators. The Authors results are seen to be in good agreement with those of Smiles & Youngs (1965) and Al-Dhahir & Morgenstern (1969) all of which give shape factors appreciably higher than those determined by Raymond & Azzouz (1969). The empirical closed-form expressions for F/d proposed by Hvorslev (1951) Kallstenius & Wallgren (1956) and Wilkinson (1968) do not fit the experimental results well. A good fit is obtained however if l/d in Hvorslevs equation (4) is replaced by 1.21/d to give

2.4nl F = log {1.21/d+& +wv021~ (9)

For piezometers with l/d 2 4, the measured shape factors can be approximated with negligible error by the relationship

F = 7d+ 1,651 (10)

VARIATION OF SHAPE FACTOR WITH DEPTH BELOW WATER-TABLE

The effect of the proximity of the water-table on the shape factor of a piezometer is a boundary effect, but it is worth consideration separate from other boundary effects because of its practical implications.

For boundary conditions Bl and B2, the electrolytic tank was used to measure the variations in F values with depth below the water-table for a whole range of vertical boundary proximities. When a piezometer was placed in a narrow flow regime, the effects of depth were very marked, but this decreased as the regime widened. The only significant results were those obtained with boundary conditions Bl applied to the semi-infinite condition (i.e. where the ratio of tank diameter to piezometer diamehr was 300 and the piezometer approached the water-table). These results are shown in Fig. 8. In Fig. 8(a), the measured values of F/d are plotted against the ratio of the piezometer depth to piezometer diameter z/d; it is apparent that the effect of the

SHAPE FACTORS OF CYLINDRICAL PIEZOMETERS 379

12.5 A I I I

X

n $d

10.0 - 0 4 A 6

0 0 0 X 12

x5- ox * 0 L A

,e I ,8

IL I 5.0

8 - 0

l

2.5- I A

0

0 I I I I

0 I 2 3 4 5

Depth below water- table/piezometer length (z/l ) 03

30

20

IO F

0

r

a/d = 4 0

P/d=2

I I I I I I Illll[

4 6 8 IO 20 40 60 80 100

Depth below water - table /

pietometer diameter (z/d 1

(4

Fig. 8. Variation with depth below water-table of measured shape factors for cylindrical piezometers (a) shape factors expressed in terms of the ratio of depth below water-table z to piezometer diameter d, (b) increase in shape factor F from infinite value F, as water-table is approached (depth z is measured from piezometer centre)

380

B2

tz

E. W. BRAND AND .I. PREMCHITT

I B4

! ( F/d = 11.19)

Fig. 9. Equipotential lines for steady state flow to a piezometer (l/d = 2) in a soil specimen (L/D = 2, D/d = 8) to which boundary conditions B2, B3 and 84 are applied (the horizontal scale is 1.2 times the natural scale)

water-table is insignificant except for long piezometers very close to it. In Fig. 8(b), the results are shown in terms of the deviation of the shape factor from the infinite value for piezometers within five lengths of the water-table; the deviation is within 10% until the piezometer is within one length of the water-table. For most practical purposes the water-table effect is negligible.

SHAPE FACTORS FOR PIEZOMETERS IN LABORATORY SPECIMENS

When steady state flow occurs into or out of a piezometer located in a soil specimen, the distribution of pore pressures (potentials) in the specimen is governed by the geometries of the piezometer and the specimen and by the drainage conditions at the specimen boundaries. The pore pressure distribution in turn controls the rate of flow and hence the shape factor of the piezometer under those particular conditions. It is clear that there is an infinite number of shape factors for cylindrical piezometers in cylindrical soil specimens because of the infinite number of flow regime geometries which are possible. For this reason, it is worthwhile for only typical results ofthe Authors measurements to be presented to illustrate the main factors which control the rates of flow and hence the shape factors of the piezometers. The shape factor values computed for boundary conditions Bl, B4 and B5 (Fig. 3) were almost the same; typical results will be given to show the effects of boundary conditions B2, B3 and B4, and the proximity of the boundaries to the piezometer in terms of the ratios of: specimen diameter to piezometer diameter D/d; specimen length to piezometer length L/l; and depth of piezometer below end cap to piezometer diameter z/d.

SHAPE FACTORS OF CYLINDRICAL PIEZOMETERS 381

0 4 8 I2 I6 20 24 28

Specimen diameter/ piezometer diameter (D/d)

Fig. 10. Effect of boundary proximity and bwndary conditions on shape factors for cylindrical piezometers in soil specimens for which l/d = L/D = 2

Piezometer / length diameter U/d)

2 4 6 8 IO I2 I4

30 21: /!!jy

OO I I I I I I 4 8 12 I6 20 24 , 1

Specimen diameter/ piezometer diameter ( D/d )

Fig. 11. Variation in shape factor with piezometer diameter for piezometers of constant length embedded in a soil specimen (L/I = 4)

The effect of the boundary conditions on the rate of flow into a piezometer at the centre of a specimen is illustrated directly in Fig. 9; here the pore pressure distributions (in terms of equipotential lines) are compared for boundary conditions B2, B3 and B4 for the situation where L/l = D/d = 8 and L/D = l/d = 2. Plotted in Fig. 10 are the shape factors measured for a range of values of L/l for piezometers with the same proportions as the specimen, i.e. l/d = L/D = 2. As would be expected, boundary conditions B4 result in higher measured shape factors than do conditions B2 and B3, but the B4 values are always higher than the infinite shape

382 E. W. BRAND AND I. PREMCHIIT

Piezometer length /diameter (P/d 1 12 6 4 3 2

OO I I I I I I

4 8 12 I6 20 24 28

Specimen length / piezometer length (L/e)

Fig. 12. Variation in shape factor with piezometer length for piezometers of constant diameter embedded in a soil specimen (D/d = 24)

factors, whereas the B2 and B3 values are always lower. As the distance from the boundaries to the piezometer increases, the infinite F value is approached in each case; but even when D/d = 24 the B2 value reaches less than 65% of the infinite value.

Measured shape factors are shown in Fig. 11 for piezometers of constant length (L/Z = 4) but variable diameter, and in Fig. 12 for piezometers of constant diameter (D/d = 24) but variable length. The dominant effect of the specimen ends is apparent where the piezometer length approaches that of the specimen; this is the same as the water-table effect already noted above. The effect of the depth of the piezometer in the soil specimen is illustrated in Fig. 13 for the two situations where D/d = 24 and 11; only the two extreme boundary conditions B2 and B4 are considered. The shape factors for boundary conditions B2 are greatly affected by the piezometers position in the specimen, whereas little effect was measured for condition B4.

The results depicted in Figs 9-13 provide guidance to the performance of pore pressure probes in soil specimens and to the efficiency of the various methods of draining triaxial specimens during consolidation. They can also be used to determine accurately the permeability of cylindrical soil specimens directly from constant head tests in which steady state conditions have been achieved.

The results in Figs 9-13 are also of relevance in the interpretation of the data obtained by Penman (1961) and Brooker & Lindberg (1965), both of whom examined the performance characteristics of field piezometers by installing them in large cylindrical soil specimens in the laboratory. Both investigations were concerned entirely with the response rates of the piezometers. Penman concluded that the time to 100% response was inversely proportional to the shape factor, as predicted by Hvorslevs theory (equation (1)) but that the theoretically predicted response curves were quite different from those measured. Brooker & Lindberg also found that Hvorslevs theory only gave good predictions of response times for responses greater than 90%. Since both investigations employed a clay soil, it is not surprising that Hvorslevs

SHAPE FACTORS OF CYLINDRICAL PIEZOMETERS

i5

6 (a)D/d=24

0 -B4, -- 82, I I

1, I I 1 P/d = 12

- 84, -- 82, I I _

0 5 IO 15 20 0 5 IO 15 20 25

383

Distance of piezometer from drainage platten/piezometer diameter(z/d)

Fig. 13. Variation in shape factor with depth of embedment for piezometers of constant diameter embedded in soil specimens subjected to boundary conditions B2 and B4

theory for incompressible soils provided poor agreement with experimental response times during the early stages of equalization. The theory of piezometer response in compressible soils (Gibson, 1963) was not available when Penman conducted his experiments, but Brooker & Lindberg did compare their results with those predicted by Gibsons theory; they reached the surprising conclusion that Hvorslevs theory gave better predictions of the time-response relationships.

It is probable that the conclusions drawn from their experimental results by Penman (1961) and Brooker & Lindberg (1965) are unsound for a number of reasons, one of these being their failure to appreciate that the shape factor of a piezometer in a restricted flow regime is not a unique quantity but is a function of the geometry of the flow regime and the boundary conditions. At the time that the two investigations were carried out, not even reasonably accurate infinite shape factors for piezometers were available, and the investigators relied upon Hvorslevs empirical equation (equation (4)) for the interpretation of their results.

ACKNOWLEDGEMENTS

The work described in this Paper was carried out in the Geotechnical & Transportation Engineering Division of the Asian Institute of Technology, as part of a continuing programme of research into the engineering behaviour of soft clays.

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