Laboratory Investigation of Radial Flow Dynamics

Laboratory Investigation of

Radial Flow Dynamics

Thomas Gallop

Laboratory Investigation of Radial Flow Dynamics

Acknowledgements

Work on this project has taken up the best part of a year of my life and now that is

completed I realise that what I have achieved would not have been possible without great

support from the people around me.

Firstly I would like to thank Dr. Prabhakar Clement for being a great supervisor and

providing the inspiration and enthusiasm required for this project. At same time I would

like to acknowledge the work of Matthew Simpson, whose amazing patience and support

throughout the year made the work so much easier.

Also worthly of thanks is the group of final students from the Centre for Water Research.

Studying with you guys over the past year has been an absolute pleasure and I wish you

all the best of luck for the future. At the same time I wish to acknowledge all my friends

outside of department for their constant support. I am in debt to all of you.

Last, but not least I wish to acknowledge my family, Dad, Mum and Leo for providing so

much support throughout the year. I realise that I have not been the easiest person to live

with and I thank you all for your patience and inspiration.


Abstract

An understanding of unconfined radial flow is crucial for predicting the flow dynamics of

groundwater systems. Two theories used to simulate unconfined radial flow are the

Dupuit-Forchheimer flow model and the variably-saturated flow model. Simplifying

assumptions allow the Dupuit-Forchheimer flow model to reduce the governing flow

equation to an ordinary differential equation that can be solved analytically; while

solution of the variably-saturated flow model requires numerical approximation of a non-

linear partial differential equation.

Using a physical model of a radial flow system, head differentials were set up and

measurements were taken of travel time, flow, streamlines and pressure heads. These data

sets were compared to the predictions of the two flow models described above.

Measurement of particle travel times using dye showed that travel times for dye reaching

a downstream well increased as the height of release at the upstream boundary was

increased. This phenomenon is caused by the increase in the length of travel paths as the

height of release is increased together with the fall in average flow velocities with

increasing height. The Dupuit-Forchheimer flow model assumes that there should be no

variation in travel times with height. For the systems considered here this assumption

does not hold, with the Dupuit-Forchheimer model underestimating the actual travel

times with an average error of 140%. The variably-saturated flow model on the other

hand was found to give a good approximation to the actual travel times with an average

error of 8%. Measurement of travel times also enabled the observation of downstream

velocity profiles for the radial flow system. The profiles showed a general increase in

flow velocity moving down the downstream end, reaching a maximum value at the height

of the downstream well and then decreasing.


Contents

1.0 LITERATURE REVIEW .................................................................................................................... 9

2.0 INTRODUCTION............................................................................................................................... 20

3.0 METHODS .......................................................................................................................................... 21

3.1 TANK PREPARATION......................................................................................................................... 21

3.1.1 Tank Screens ............................................................................................................................... 24

3.1.2 Porous Medium ........................................................................................................................... 25

3.2 SATURATED HYDRAULIC CONDUCTIVITY ....................................................................................... 25

3.3 PHYSICAL MEASUREMENTS ............................................................................................................. 27

3.3.1 Steady-State Conditions.............................................................................................................. 27

3.3.2 Flow............................................................................................................................................. 30

3.3.3 Travel times ................................................................................................................................. 30

3.3.4 Downstream velocity profile....................................................................................................... 32

3.3.5 Streamlines .................................................................................................................................. 33

3.3.6 Potentiometric levels................................................................................................................... 34

3.4 NUMERICAL MODELS........................................................................................................................ 34

3.4.1 Variably-saturated flow equation............................................................................................... 34

3.4.2 Dupuit-Forchheimer flow equation: Evaluation of travel times............................................... 35

4.0 ANALYSIS AND SYNTHESIS......................................................................................................... 36

4.1 SATURATED SAND HYDRAULIC CONDUCTIVITY............................................................................. 36

4.2 FLOW................................................................................................................................................. 36

4.2.1 Variably-saturated flow .............................................................................................................. 37

4.2.2 Analysis of physical results......................................................................................................... 38

4.3 TRAVEL TIMES .................................................................................................................................. 42

4.4 DOWNSTREAM VELOCITY PROFILE................................................................................................... 56

4.5 STREAMLINES ................................................................................................................................... 60

4.6 POTENTIOMETRIC LEVELS................................................................................................................ 68

5.0 CONCLUSIONS ................................................................................................................................. 74

6.0 RECOMMENDATIONS ................................................................................................................... 76

7.0 REFERENCES.................................................................................................................................... 77


APPENDIX A: NUMERICAL MODEL OF TWO-DIMENSIONAL, VARIABLY-SATURATED

FLOW.................................................................................................................................................................. 80

A.1 BOUNDARY CONDITIONS.................................................................................................................. 81

A.2 SOIL PROPERTIES.............................................................................................................................. 84

APPENDIX B: RADIAL FLOW MODEL PLANS ...................................................................................... 86

APPENDIX C: MEASURING SATURATED HYDRAULIC CONDUCTIVITY ................................... 92

APPENDIX D: PHYSICAL RESULTS.......................................................................................................... 95


Figures

Figure 1: Confined and unconfined aquifer flow (Fetter, 1994)........................................9

Figure 2: Phreatic surfaces predicted by the Dupuit-Forchheimer and variably-saturated

flow models (Modified from Clement et al. (1996)) ...............................................13

Figure 3: Velocity distributions along the outflow face of a two-dimensional radial flow

system (Muskat, 1937)...........................................................................................15

Figure 4: Top view of system under consideration. ........................................................22

Figure 5: Piezometer positions (all values in cm). ..........................................................23

Figure 6: Analysis grid for tank (r = distance from upstream end, z = elevation) ............24

Figure 7: Constant Head Permeameter...........................................................................26

Figure 8: Falling Head Permeameter..............................................................................27

Figure 9: Time for achievement of steady state. .............................................................29

Figure 10: Pore drainage areas for different head gradients ............................................30

Figure 11: Dye traveling through radial tank..................................................................32

Figure 12: Flow observed at different head differentials with hydraulic conductivity of 48

m/day (upstream head = 90cm) ..............................................................................39

Figure 13: Flow rate predictions using a saturated hydraulic conductivity of 67m/day. ..41

Figure 14: Dye released at heights of 80cm, 50cm and 20cm for a 90:10 head gradient .43

Figure 15: Total travel times for 90:20 head differential ................................................44




Figure 19: Comparison of total travel times for changing head differentials ...................46

Figure 20: Comparison of total travel times for dye released at 90cm and dye released at

15cm......................................................................................................................47

Figure 21: Predicted travel times for four head differentials together with their quadratic

curves of best fit ....................................................................................................51

Figure 22: Dye travel times for 90:20 head differential through tank at different heights of

release ...................................................................................................................53



release ...................................................................................................................54


release ...................................................................................................................55


release ...................................................................................................................56

Figure 26: Observed downstream velocities for the 90:10 head differential....................57

Figure 27: Observed downstream velocities for the 90:75 head differential....................58

Figure 28: Correct pump and treat design ......................................................................60

Figure 29: Streamlines observed for the 90:20 head differential.....................................61



Figure 32: Streamlines observed for 90:70 head differential...........................................64

Figure 33: Streamlines observed for a 90:10 head differential........................................65

Figure 34: Seepage face heights for upstream well height of 90cm ................................67

Figure 35: Piezometric levels for the 90:20 head differential..........................................69





Tables

Table 1: Observed and measured flow rates for various head differentials .....................37

Table 2: Flow rate predicted by variably-saturated flow model at different grid sizes for a

90:75 head differential ...........................................................................................38

Table 3: Factor of difference between observed flow and predicted flow from the

variably-saturated flow model................................................................................40

Table 4: Percentage errors in flow measurements for Dupuit-Forchheimer and variably-

saturated flow models ............................................................................................41

Table 5: Comparison of errors in travel times predicted by the variably-saturated flow

model and those physically observed. ....................................................................49

Table 6: Comparison of travel times predicted by the Dupuit-Forchheimer flow model

and those physically observed. ...............................................................................50

Table 7: Comparison of observed travel times with those predicted by an empirical

relation. .................................................................................................................52


1.0 Literature Review

The flow of groundwater towards a pumping well can be divided into two main areas:

confined aquifer and unconfined (gravity) aquifer flow.

Figure 1: Confined and unconfined aquifer flow (Fetter, 1994)

Confined aquifer flow applies to the flow of water between two confining layers (Figure

1). The distance between the confining layers governs saturated thickness of the confined

flow, with the porous medium between the layers being completely saturated.

Unconfined aquifer flow on the other hand applies to flow with only a single confining

layer at the base of the aquifer. As a result this makes the study of unconfined flow more

complex as the saturated thickness of the porous medium can vary spatially. Unconfined

aquifer flow also enables the occurrence of unsaturated flow above the saturated area of

the aquifer. Due to these factors unconfined flow exhibits a non-linear nature, which


leads to complex governing equations that are difficult to solve. Accurate solutions

however are often necessary as unconfined flow is the most commonly occurring flow in

nature. Therefore a greater understanding of unconfined flow is required for

understanding problems involving groundwater contamination.

At the core of any argument involving the flow of water in porous media systems is

Darcy’s law, which was developed in 1856. Darcy was able to show experimentally that

the discharge of water through a pipe filled with porous medium was proportional to the

difference in the height of water at the two ends of the pipe and inversely proportional to

the length of the pipe. Darcy’s law takes the general form:

√↵

−=dl

dhKAQ

(1)

where K is the hydraulic conductivity [L/T], A is the cross sectional area of the porous

medium [L2] and dl

dh is the hydraulic gradient across the porous medium.

Dupuit (1863) was able to use the work of Darcy as a basis for deriving the earliest

equation to describe steady-state unconfined aquifer flow. Dupuit worked on the basic

premise that the velocity in any vertical section is uniform and horizontal, and

proportional to the slope of the free surface (DeWiest, 1965). He also implicitly made the

assumption of no flow through the vadose zone, as demonstrated by Clement et al.

(1996). The result was the following equation used to describe flow for a radial gravity

flow system (Bear, 1979, pp. 308-311):

( )( )WR

WRss rr

hhK

dr

dhrhKQ

/ln2

22 −=−= ππ

(2)


where Q [L3T-1] is the total discharge through the system with boundary conditions, hw

[L] the water table elevation at the radius of the well, rw [L], and hR [L] the height of the

water table at the radius of influence, rR [L]. Integration of equation (2) yields:

( ) ( ) ( )( )

2/1

222

/ln

/ln(

??−+=

WR

wWRW rr

rrhhhrh

(3)

where h is the height of the water above the impermeable bed of the system at radius r.

In 1886, Ph. Forchheimer used the assumptions made by Dupuit to derive the governing

equation for unconfined flow. This enabled an extension of the solution produced by

Dupuit to solve depth averaged, two-dimensional cartesian flow problems. The governing

equation for steady state flow is:

02

22

2

2222 =

ƒƒ+

ƒƒ=

y

h

x

hh

(4)

These equations form the basis for the Dupuit-Forchheimer theory of unconfined flow

systems.

In 1931, L.A. Richards developed an equation that could be used to model one-

dimensional flow in unsaturated soil columns. Extension of the Richards’ equation to

radial coordinates in two dimensions led to the steady-state form of the variably-saturated

flow equation as shown in Clement et al. (1996):

0)(

)()(1 =++

z

K

zK

zrrK

rr ƒθƒ

ƒƒψθ

ƒƒ

ƒƒψθ

ƒƒ (5)

where r [L] is the radial coordinate, ψ [L] the pressure head, K(θ) [LT-1] the hydraulic

conductivity and θ is the moisture content; x[L] and z[L] are the Cartesian coordinates in


the horizontal and vertical directions respectively. This equation assumes that the

dynamics of the air phase do not affect those in the water phase, the density of the water

is only a function of the pressure and the spatial gradient of the water density is

negligible.

The variably-saturated flow equation (5) allows for both vertical variation in pressure (as

zƒƒ

is non-zero), and changes in unsaturated hydraulic conductivity (as K = K(θ)). As a

result, this approximation avoids the assumptions used in the Dupuit-Forchheimer model.

The solution of the variably-saturated flow equation leads to the development of seepage

face at the downstream end of the model. The seepage face refers to the area at the

downstream face between the intersection of the phreatic surface at the downstream end

of the domain and the water level in the downstream well, where the water pressure is

atmospheric (Clement et al., 1994). Figure 2 compares the phreatic surfaces predicted by

the two models, showing the formation of a seepage face in the variably-saturated flow

model.


Figure 2: Phreatic surfaces predicted by the Dupuit-Forchheimer and variably-saturated

flow models (Modified from Clement et al. (1996))

The formation of a seepage face in the variably-saturated flow model is a two-

dimensional flow phenomena, rather than an unsaturated flow phenomena (Clement et

al., 1996). Neglecting vertical flow and then solving equation (4) produces the Dupuit-

Forchheimer solution, where no seepage face is observed. However, allowing for vertical

flow and solving for equation (4) produces a solution where a seepage face is formed

with a length slightly smaller than that predicted by the variably saturated flow model

(Clement et al., 1996). This demonstrates that unsaturated flow is not necessary for the

formation of a seepage face.

For the Dupuit-Forchheimer model the phreatic surface represents an upper flow

boundary, separating the saturated from the unsaturated zone. For the variably-saturated

flow equation it is not a boundary to the flow but rather a locus of points where the water

pressure is atmospheric (Clement et al., 1996). Flow is possible both up through the

phreatic surface into the unsaturated zone, and back through the phreatic surface from the

unsaturated zone.

R coordinate

Ele

vati

on

Dupuit-Forchheimer Model

Variably Saturated ModelS

eep

age

Fac

e

Downstream water level

Upstream water level


The Dupuit-Forchheimer model has a distinct advantage in that the equation used to

model the flow is one dimensional and can be solved analytically. On the other hand, the

variably-saturated flow equation is a non-linear partial differential equation, where

analytical solutions cannot be produced. The Dupuit-Forchheimer model also has the

advantage of easily identifiable boundary conditions, with only upstream and

downstream water levels required. Boundary conditions for the variably-saturated flow

equation are not clearly identifiable as the height of the seepage face is unknown. The

variably-saturated flow equation however is more accurate, especially in conditions

where vertical flows and flow in the unsaturated zone become important. These

conditions are generally produced by large hydraulic gradients in a porous medium that

exhibits strong capillary effects.

Wyckoff et al. (1932) conducted an experiment using the sector of a radial tank to gain a

greater understanding of radial flow. The authors used the tank to show that outflow is

proportional to the square of the differences in fluid heads (as measured from the tank

bottom). The Dupuit formula gives a similar result with fluid heads replaced by phreatic

surface heights. Some findings were also made that contradicted the assumptions of the

Dupuit formula. The authors were one of the first to observe the formation of a seepage

face, with the size of the seepage face increasing as the upstream height was increased. A

reduction in fluid head, when compared to fluid level, at the downstream end of the

model highlighted seepage face formation. Streamlines from the model also showed that

significant vertical flow was present which further contradicted the Dupuit-Forchheimer

assumptions. At the upstream end, flow moved upwards into the capillary zone while at

the downstream end, downward flow out of the capillary zone was observed. Under

conditions where the capillary zone was large it was found that the Dupuit-Forchheimer

model for predicting flow was in fact incorrect and an extra term for capillary flow was

required to describe the flow. The introduction of this extra term into the Dupuit flow

equation (2) is highlight in equation (6):


( )( )( )WR

CWRWRs rr

hhhhhKQ

/ln

++−=π

(6)

where hc was the capillary rise of the fluid. The observation of seepage faces, vertical and

capillary flow highlighted the deficiencies of some of the Dupuit-Forchheimer

approximations and that further work in the area was required.

Later Muskat (1937), a colleague of Wyckoff, investigated various situations of water

flow through porous media. One of the situations investigated was radial gravity flow.

Muskat was able to predict the position of the free surface, as well as the velocities along

the inflow and outflow faces using hodographs. He showed that along the inflow face

there would be a steady reduction in the velocity as height increased, reaching zero at the

top. On the outflow face there was expected to be zero velocity at the free surface height,

increasing to a maximum value in going down to the downstream water level, and then

decreasing uniformly until the base of the model was reached (Figure 3).

Figure 3: Velocity distributions along the outflow face of a two-dimensional radial flow

system (Muskat, 1937)


Muskat explained how despite the obvious limitations of the Dupuit approximations, the

Dupuit-Forchheimer formula still led to a reasonable calculation of the total flow rate for

the system, thus attempting to explain the observations of Wycoff et al. (1932). He

achieved this by using the same boundary conditions as Dupuit but without the free

surface boundary. Muskat also described an experiment undertaken by Wycoff et al.

(1935) where this gravity flow system was modelled using an electric circuit. The model

supported what had earlier been observed in the laboratory, with the formation of large

seepage faces and significant vertical velocities.

In 1955, Hall used modelling procedures similar to those used by Wycoff et al. (1932) to

conduct a series of physical experiments. Hall particularly concentrated on observation

close to the downstream gravity well of the system where the slope of the free surface

becomes steep and the Dupuit-Forchheimer assumptions are expected to be invalid. Hall

observed the formation of streamlines in the tank and took pressure readings from

piezometers connected to the tank. From this data he was able to produce flow patterns

for each head gradient showing streamlines and equipotential lines. Hall’s results were

used to validate numerical results produced by a relaxation method. This method used

equation (4) to define a pair of empirical equations that may be used to locate the phreatic

surface on a typical radial cross section. The method however did not account for

capillary effects. Hall was the first to physically observe a reduction in travel time with

falling height of release. He also noted that travel times for dyes released in the capillary

layer were considerably slower than those just slightly below this zone. Some rough

measurements of travel time were taken however these results were only extensive

enough for qualitative analysis. These velocities however did enable Hall to make

estimations of the contribution of the capillary layer relative to the total flow.

From the physical observations of Wycoff et al. (1932) and Hall (1955) it was clear that

the Dupuit-Forchheimer formula could be inaccurate at the downstream end of the flow

problem due to the formation of a seepage face. Borelli (1955) investigated situations

where the Dupuit-Forchheimer model could still be used accurately. By comparing


results gained from the relaxation method with those of Dupuit-Forchheimer, he was able

to estimate a ratio for radius r to the height of the phreatic surface h (see Figure 1 for

unconfined aquifer), above which the Dupuit-Forchheimer model would predict the

position of the phreatic surface to within 1%. Using this estimate he developed a formula

to calculate the position of the phreatic surface at positions closer to the downstream

well, where the Dupuit-Forchheimer assumptions are not valid. This approach however

has limited use as it is related to a particular flow system.

Emphasis on modelling the variably-saturated flow equation (5) using numerical schemes

was instigated by Rubin (1968). Rubin used a finite difference method to solve the two-

dimensional variably-saturated flow equation. Other work included that of Neuman

(1973), who used a finite element approach to solve the two-dimensional variably-

saturated flow equation. Narasimhan and Witherspoon (1976) used an integrated finite

difference method, while Cooley (1983) used a sub-domain finite-element approach to

solve a similar flow problem.

Vachaud and Vauclin (1975) performed a series of laboratory experiments on a cartesian

flow tank. By combining measurements from pressure transducers on the tank with

Darcy’s Law they were able to measure water fluxes in the system. From these results

they observed a downstream velocity profile similar to that predicted by Muskat (1937).

Shamsai and Naraisimhan (1991) used a numerical model for the variably-saturated flow

equation, previously developed by Naraisimhan and Witherspoon (1978), to model some

of the experimental observations documented by Hall (1955). The model was found to be

accurate in calculating the phreatic surface and the distribution of potentials within the

phreatic zone observed by Hall (1955). Work was also undertaken to compare the

discharge levels observed by the model with those estimated by the Dupuit-Forchheimer

model under different situations. The results showed that discharge estimates from the

Dupuit-Forchheimer model were in error by up to 12-20% for both radial and cartesian

flow situations. These are much higher levels than the 1-2% predicted by several other


investigators including Muskat (1937), thus indicating that one cannot always neglect the

role of the unsaturated zone when dealing with unconfined flow problems.

Clement et al. (1994) developed a numerical algorithm capable of solving a variety of

two-dimensional, variably-saturated flow problems. Details of this numerical model are

included in Appendix A. The model was shown to accurately predict several known

experimental data sets, including the water table data observed by Hall (1955).

Clement et al. (1996) compared the effectiveness of different models for predicting

steady state, unconfined flow for different soil properties, problem dimensions and flow

geometries. Using a previously developed numerical model (Clement et al., 1994) a

comparison of results produced by the Dupuit-Forchheimer equation and the variably-

saturated flow equation was completed. Observations showed that the position of the

phreatic surface was relatively insensitive to the soil parameters used in the variably-

saturated flow equation. This explains why the experiment conducted by Hall (1955) was

reproducible by numerical means e.g. Shamsai and Naraisimhan (1991), despite the fact

that Hall did not report any soil parameters. Comparison of radial to cartesian problems

revealed that the effect of flow through the vadose zone is relatively less important in

radial than in cartesian systems. The radial flow case was also found to produce a more

pronounced seepage face. This can be explained by the convergent nature of radial flow,

where a larger seepage area is required to accommodate the induced flow. The radial

flow problem was much less sensitive to changes in the scale of the model compared to

cartesian flow problems, highlighting the persistent nature of seepage faces in radial flow

problems.

The literature indicates that although extensive study has been undertaken in the area of

radial unconfined flow, further investigation is still required, particularly in relation to the

observation of flow velocities and internal flow patterns near a seepage face boundary. It

has been observed by Hall (1955) that travel times increase with increasing height of

release, however this observation was made from only qualitative results for a simple

case. The opportunity is therefore present for quantitative physical study of the travel


times in a radial flow situation. The Dupuit-Forchheimer model for radial flow systems

predicts that the travel times observed should be independent of height of release.

Detailed observation of travel times over a number of head differentials will enable the

extent to which this assumption fails to be physically documented. Measurement of travel

times will also allow measurement of the downstream velocity profile. This typical

profile shown in Figure 3 has been observed in a physical cartesian system however it has

never been observed physically in a radial system. Comparison of the results with those

predicted by a numerical model will help validate the effectiveness of the model to

predict flow velocities for an unconfined radial flow system. A greater understanding of

unconfined radial flow will lead to a greater understanding radial flow dynamics, thus

allowing for the implementation of more efficient pumping practices.


2.0 Introduction

This study focused on the physical observation of flow through a two-dimensional radial

system. This involved using a radial tank and experimental design developed by Matthew

Simpson for the Centre for Water Research as part of his PhD thesis. The design details

of the tank are included in Appendix B.

This tank allowed a constant head differential to be set up between the two ends of the

system so that steady state flow conditions could be achieved. At this stage observation

of travel times, streamlines, and pressure levels at various positions in the tank took

place, together with a measure of total flow through the tank. The process was repeated at

a number of head differentials. A comparison of these results with those from the

numerical model for variably-saturated flow (Appendix A) will enable further

understanding of unconfined flow near a seepage face boundary.

The laboratory efforts were focused on observing travel times for flow in the tank.

Observation of changing travel times with height of release had been insufficient in

previous studies, and quantitative physical observation was required to validate much of

the seepage face theory, including that of Clement et al. (1996). Some study also focused

on measuring a downstream velocity profile, and comparing it to the theoretical profile.


3.0 Methods

3.1 Tank Preparation

The model used to observe flows was a 15o cut of a radial system with a height of 100 cm

and distance between well centres of 129.56 cm (Simpson, 2000). The 15o cut was

assumed to represent a portion of a complete radial flow system. The model was divided

into three chambers: the inlet flow chamber (or upstream well), the porous medium and

the outlet flow chamber (or downstream well). The internal diameter of the upstream well

was 20 cm and the internal diameter of the downstream well was 10 cm. This enabled the

investigation of a convergent radial flow situation. The model used was constructed of

pexiglass with a metal frame attached to avoid deflection of the sidewalls. Two Pexiglass

screens were used at the upstream and downstream ends to hold in the sand medium.

Although we assume that the system we are considering is radial, this is not completely

true due to the presence of flat screens (Figure 4). However, this is an adequate

assumption as the small value chosen for the angle of the cut (15o) means that the

difference from the ideal system is also small.


Figure 4: Top view of system under consideration.

Flow of water into the tank was controlled via a hose connected to the mains water

supply. Water outflow was controlled by poly vinyl chloride (PVC) pipes present in the

upstream and downstream wells. Water was able to flow out of the pipes and into the

drain via tubing connected to the bottom of the pipe. Connections at the bottom of the

tank allowed these pipes to be moved up and down in the wells, thus controlling the

water levels in the upstream and downstream ends of the tank. A valve was also present

at the bottom of the tank allowing the water flow through these pipes to be stopped at any

time.

10cm

Model system

True radial flow system

109.5cm


One side of the tank contained holes drilled such that plastic piping could be attached.

These holes were covered with small pieces of geofabric on the inside of the tank to

prevent sand from entering the piping. Each piece of piping acted as a piezometer

enabling the measurement of pressure heads in the tank. Figure B3 (Appendix B) shows

the piezometers on the side of the tank and Figure 5 shows their positions relative to the

upstream and downstream wells.

Figure 5: Piezometer positions (all values in cm).

Ups

trea

m w

ell s

cree

n

Dow

nstr

eam

wel

l scr

een

15.5 20 20 19.5 20 15

5

20

20

20

15

15

125

125

Side View

Top View


On the other side of the tank the glass area was divided into a grid with squares 5 cm × 5

cm (Figure 6). The grid enabled easier monitoring of the streamlines and flow velocities

in the tank.

z

r

4070 105

Figure 6: Analysis grid for tank (r = distance from upstream end, z = elevation)

3.1.1 Tank Screens

The sand was held within the tank by two pexiglass screens, with holes drilled at regular

intervals to allow the flow of water through the screens. The design of the screens is seen

in Figure B4 (Appendix B). The holes were positioned such that the screen had maximum

strength, and that water could enter and exit the tank at all heights along the screen. The


screens were wrapped initially with a rubber mesh and then with geofabric to prevent the

movement of porous media into the upstream and downstream ends of the tank.

3.1.2 Porous Medium

The porous medium used to pack the radial tank was a 1±0.5 mm medium sand (Cook

Industrial Minerals Pty. Ltd.). The sand was added to the tank in layers of ~2 cm with the

water level always maintained above that in the sand. After each addition, the sand in the

tank was mixed thoroughly using a piece of PVC piping, allowing for removal of air

bubbles trapped in the sand pores. This was continued until the height of the sand in the

tank was 95 cm. So as to prevent air bubbles from entering the porous media and

disrupting the results between experimentation, the water level in the tank was

maintained above the height of the sand. After packing the tank with porous medium it

was also noted that significant deflection of the upstream well screen into the upstream

well occurred. As a result, the system in the model moved closer to the true radial flow

system described in Figure 4.

3.2 Saturated Hydraulic Conductivity

The saturated hydraulic conductivity of the porous medium was measured using a

constant head test. Two sets of sand packing conditions were used, one where the sand

was packed tightly and one where no attempt was made to pack the sand in the

permeameter. A constant head differential was then set up between the two ends of the

permeameter and a flow rate was measured (Figure 7).


Outlet

Inlet

Connectingpipes

Water supply

Figure 7: Constant Head Permeameter

The process was repeated for different head gradients and the results were used to

estimate a value of saturated hydraulic conductivity for each different packing condition.

The constant head test is generally the first test used to determine sand saturated

hydraulic conductivity (Fetter, 1994). To verify the results from the constant head test, a

falling head test was also conducted on the sand. The falling head test is generally more

accurate for more impermeable materials like clays, however it does enable the

production of a set of results that can be compared with those from the constant head

permeameter.

Validation via the falling head permeameter was undertaken on a tightly packed sample

of the porous medium. The water level was then observed to fall between two heights,

with the time recorded (Figure 8).


Figure 8: Falling Head Permeameter

This process was repeated four times with an average taken to produce a value for

saturated hydraulic conductivity for the tightly packed sand sample.

3.3 Physical Measurements

3.3.1 Steady-State Conditions

All experiments were conducted under conditions of steady-state flow. For steady-state

conditions are there should be no fluctuation in the physical model with time. For the


case of the radial tank this involves leaving the tank to run at a certain head differential

for a period of time before any experimentation is undertaken.

Maintaining a constant head differential across the tank involved having a constant input

of water into the upstream end via a hose connected to the mains water supply. The pipes

in the upstream and downstream ends were then set to the appropriate levels and their

valves were opened. The constant input of water via the hose allowed the water levels to

be maintained, while the pipes in the upstream and downstream wells prevented the water

levels from rising above the required values. This system was then left to run until steady

state conditions were achieved.

To estimate the time taken for steady-state conditions to be reached a head gradient of 90

cm at the upstream end and 10 cm at the downstream end (90:10) was set up. Water

levels in the upstream and downstream ends were originally maintained constant across

the tank and then they were dropped to 90cm and 10cm respectively, with readings being

taken from the piezometer at the base of the tank closest to the downstream end, at half

hourly intervals. It is this piezometer in which the greatest fall in head occurs, so

consequently it should be the slowest to reach a steady level. A constant level in this

piezometer gave a good indication that steady state conditions had been achieved.

Assuming that steady height was achieved after six hours, the results show that there is

only a change in height of 0.4 cm between the reading after 1 hour and the steady height

(Figure 9).


0 1 2 3 4 5 6 7−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Time (hr)

Diff

eren

ce fr

om s

tead

y he

ight

(cm

)

Figure 9: Time for achievement of steady state.

A deviation of 0.4 cm will mean negligible change to the flow pattern in the tank, so

consequently a time period of 1 hour was deemed as sufficient for the achievement of

steady-state conditions. The choice of the 90:10 gradient to conduct this test meant that

the value of 1 hour to achieve steady-state could be applied to all head gradients used, as

the 90:10 gradient was the strongest head gradient used in the experimentation. Figure 10

shows the volume of pore drainage for the 90:10 head gradient, with this volume

reducing as the size of the head gradient is reduced. As the 90:10 head gradient requires

the largest pore drainage volume, it should require the longest time to achieve steady-

state conditions.


R coordinate

Ele

vati

on

Downstream water level

Upstream water level

Phreatic surface

Volume of pore drainage required

Figure 10: Pore drainage areas for different head gradients

3.3.2 Flow

Total flow through the tank was measured by collecting the flow at the downstream end

of the flow tank. The time taken for this flow to fill a 1 L container was measured in

order to calculate a flow rate for the system. This process was then replicated twice with

an average value used as a measure for flow rate. Total flow was measured for six head

differentials: 90:10, 90:20, 90:40, 90:60, 90:70 and 90:75.

3.3.3 Travel times

Travel times in the tank were measured by tracking the paths of coloured food dye

injected at various intervals. The dye was injected using a thin piece of metal pipe

(Diameter ~ 1 mm) that was attached to a 10 mL syringe. Once steady state conditions


were achieved the syringe was pushed slowly through the sand layer along the sidewall of

the tank, 5cm from the upstream end. Once the appropriate height of release was reached

a very small portion of dye was injected (~0.1 mL) and the stopwatch was started. The

dye was then tracked through the grid on the side of the tank. As the dye cloud moved

through the tank it was observed to undergo spreading. The small initial dye cloud

injection into the tank increased in size considerably as it moved through the sand.

To highlight differences in travel times with height a 90:10 head differential was set up

and dye was injected at heights of 80cm, 50cm and 20cm at the upstream end of the tank.

Photos of the tank were then taken at five-minute intervals, until the dye exited the tank.

The choice of a strong hydraulic gradient (90:10) produced large differences in travel

time with height of release, which could be easily identified through the photographs.

Measurements were then made of the time necessary for the front, centre and the tail of

the plume to cross positions 40cm, 70cm and 105cm from the upstream end of the tank

(Figure 6). The centre of the plume provided the best measure of travel time as this

allows spreading effects to be ignored (Figure 11).


Front

Centre

Tail

Figure 11: Dye traveling through radial tank

Travel times were measured from five different release heights: 90, 75, 60, 45 and 15cm.

This was repeated for four head differentials 90:20, 90:40, 90:60 and 90:70. These head

differentials were chosen as they provided a series ranging from large hydraulic gradients

(90:20) to smaller hydraulic gradients (90:70).

3.3.4 Downstream velocity profile

For the 90:10 and 90:75 head differentials downstream travel times were taken. This

involved injecting dye 95 cm from the upstream end and measuring the time to move

5cm downstream. These travel times (t) were measured at 5 cm height intervals from a

height of 90 cm to a height of 10 cm and were then converted to horizontal velocities via:


tcmVelocity

5sec)/( =

(7)

The 90:10 head differential was chosen as it results in the formation of a large seepage

face, thus allowing for the observation of a broad velocity profile above the downstream

well height. The 90:75 head differential was also studied in conjunction; although it only

results in the formation of a small seepage face, it does enable observation of flow

velocities over a broad range below the downstream well height. The 90:10 head

differential does not allow this observation. The flow velocities measured here are not

true downstream flow velocities, as the design of the model restricts observation of dye

any closer to the downstream end of the tank. The observations however are close to the

general profile observed at the downstream end.

3.3.5 Streamlines

Streamlines from the tank were measured simultaneously with the travel times. As the

injected dye plume was observed to move through the tank the height of the plume centre

was marked at 5cm intervals along the tank, corresponding to the vertical lines of the grid

on the tank. This was repeated for the same head differentials as described in Section

3.3.3, at release heights of 90, 75, 60, 45, 30 and 15 cm.

A set of streamlines was also produced for the 90:10 head differential. For these

streamlines dye was regularly injected at the same height. This allowed for the formation

of a complete streamline moving from the upstream to the downstream end of the tank.

Photographs were taken of streamlines observed at 5cm intervals from a height of 10cm

to a height of 90cm. The 90:10 head differential was chosen to observe these streamlines

as it involves an extreme hydraulic gradient being applied to the tank. This extreme

hydraulic gradient results in large vertical flows and hence the greatest shift from the

Dupuit-Forchheimer model.


3.3.6 Potentiometric levels

To measure potentiometric levels fluid height in each of the piezometers was noted for

the four head differentials described in Section 3.3.3. For piezometers located in the

unsaturated zone no fluid levels were observed.

3.4 Numerical models

3.4.1 Variably-saturated flow equation

The numerical model developed by Clement (1993), and described in Appendix B, was

used to solve the variably-saturated flow equation. Parameters required by the model

include the saturated hydraulic conductivity of the porous medium Ks, the Van

Genuchten parameters αv and nv and the saturated (θs) and residual (θr) water contents.

The Van Genuchten parameters are used to describe the soil properties, with αv providing

a measure of the first moment of the pore size density function [L-1] and nv an inverse

measure of the second moment of the pore size density function (Wise, 1991; Wise et al.,

1994).

The 120 cm × 95 cm domain was discretised using a 39 node × 39 node grid. The

following parameters were used in the model: Ks = 48 m/day, θs = 0.3, θr = 0, α v = 2.0

and nv = 2.0. The value for Ks was calculated using a constant head permeameter. This

value was then validated using a fixed head permeameter. The other soil parameters θs,

θr, αv and nv were not calculated but instead typical values were taken. The effects of soil

parameters θs, θ r, α v and nv are not of particular concern as both Shamsai and

Naraisimhan (1991) and Clement et al. (1996) have shown that their effects are negligible

for radial flow systems.


The variably-saturated flow model enables prediction of the pressure heads, flow

velocities and the position of the phreatic surface for the radial flow system. The model

also calculates the flow into the system at the upstream end and the flow out of the

system at the downstream end. These two values are expected to be close for mass

balance, however some error occurs. As a result, the flow rate of the system is measured

by taking an average of the inlet and outlet values. The output from the variably-saturated

model was used in conjunction with a particle tracking model, as shown in Simpson and

Clement (2001), to predict travel times and streamlines.

3.4.2 Dupuit-Forchheimer flow equation: Evaluation of travel times

Based on Dupuit-Forchheimer theory, application of equations (2) and (3) directly

computes total flow and head distributions. To calculate the total travel time using the

Dupuit-Forchheimer model, equation (3) was differentiated with respect to radius and

then Darcy’s Law was applied:

( )( ) ( ) ( )

( )

d

t

os

s

ssr

WR

WWRW

WRW

WR

tdtrf

drKn

dtrf

drKn

rfn

K

dr

dh

n

K

dt

drv

rfrr

rrhhh

rrrr

hh

dr

dh

d

==

=

===

=??−+

−=

−

10

110

2/1

22222

)(

)(

)(

)(/ln

/ln(

/ln2

(8)

The integral on the left-hand side of the equation was evaluated numerically using the

trapezoidal method.


4.0 Analysis and Synthesis

4.1 Saturated Sand Hydraulic Conductivity

The method for calculating the value for saturated hydraulic conductivity is included in

Appendix C. The test measured a saturated hydraulic conductivity of 48 m/day for the

tightly packed sample and 86 m/day for the unpacked sample of the porous medium.

The falling head test was then used as a method to validate these results. The falling head

test predicted a saturated hydraulic conductivity of 49 m/day for the tightly packed

sample thus supporting the value observed in the constant head test. The procedure used

for the falling head test is also included in Appendix C.

As the packing conditions in the tank itself will lie between the tightly packed and

unpacked conditions we can infer that the saturated hydraulic conductivity of the porous

medium lies between 48 and 86 m/day. A more accurate value is difficult to quantify due

to the difficulties in reproducing the packing conditions in the tank with those in the

constant head permeameter.

4.2 Flow

The total flow rate through the system measured for six head differentials is included

below, together with the values predicted by the Dupuit-Forchheimer and variably-

saturated flow models for the hydraulic gradient of 48 m/day (Table 1).


Table 1: Observed and measured flow rates for various head differentials

4.2.1 Variably-saturated flow

Table 1 includes the flow predicted by the variably-saturated flow model at both the

inflow and outflow faces, together with the percentage difference between the flow at the

inflow and outflow faces. The results reveal a percentage difference of less than 1% for

largest four head gradients, while for the smallest head gradients 90:70 and 90:75 large

errors in flow between the inflow and outflow faces are observed. The relatively coarse

grid (39 nodes × 39 nodes) used to study the problem results in these errors. For smaller

head gradients seepage faces still develop however they are not as large as those for

larger head gradients. For the 90:70 and 90:75 head differentials the variably-saturated

flow model predicted that no seepage faces would be formed which is in fact incorrect, as

some small seepage faces would still form. The seepage face is important when

calculating the flow out of the system and consequently its incorrect estimation explains

the deviation that exists between the flows observed for the inflow and outflow faces.

Figure 3 shows the high outflow velocities predicted along the seepage face. The use of a

finer grid size enables more accurate prediction of the seepage face height, and as a result

more accurate prediction of the flow rate. To highlight the importance of the grid size to

the flow rate, calculation was repeated for two finer grids, 77 nodes × 77 nodes and 153

nodes × 153 nodes at the head differential of 90:75 (Table 2).

Flow (m^3/day) 10 20 40 60 70 75Observed 3.05 2.74 2.56 1.76 1.14 0.76

Dupuit-Forchheimer 2.00 1.93 1.63 1.13 0.80 0.62Variably saturated in 2.12 2.03 1.71 1.17 0.73 0.58

Variably saturated out 2.10 2.02 1.70 1.18 0.86 0.66Variably saturated ave 2.11 2.03 1.71 1.18 0.79 0.62Variably saturated %

difference 0.83 0.58 0.26 0.92 18.51 15.33

Downstream well height (cm), upstream well height = 90cm


Table 2: Flow rate predicted by variably-saturated flow model at different grid sizes for a

90:75 head differential

The results show that refinement of the grid leads to a reduction in the percentage flow

differences between the outflow and inflow faces. The table also shows that the grid

refinement leads to a convergence onto a flow value, with the average value of flow from

the inflow and outflow faces remaining relatively consistent for all three grids. As a

consequence, the use of the average flow from the 39 node × 39 node grid was deemed as

an appropriate measure of flow for the variably-saturated flow model.

4.2.2 Analysis of physical results

The flow observed in the model, together with flow predicted by the Dupuit-Forchheimer

and variably saturated flow models for a hydraulic conductivity of 48 m/day is included

in Figure 12.

Flow (m^3/day) 39*39 77*77 153*153Variably saturated in 0.58 0.59 0.60

Variably saturated out 0.66 0.65 0.64Variably saturated ave 0.62 0.62 0.62Variably saturated %

difference 15.33 8.87 6.56

Number of nodes


0 10 20 30 40 50 60 70 800

0.5

1

1.5

2

2.5

3

3.5

Downstream water level (cm)

Flo

w (

m3 /d

ay)

Observed Dupuit−ForchheimerVariably Saturated

Figure 12: Flow observed at different head differentials with hydraulic conductivity of 48

m/day (upstream head = 90cm)

The results from the three methods of calculating the flow all show a very similar pattern,

with a reduction in the amount of flow with an increase in downstream water level. As

expected a factor exists which separates the values predicted by the variably-saturated

and Dupuit-Forchheimer flow models from those observed in the tank. This factor can be

directly related to the hydraulic conductivity used in the numerical model as the value

used was only an estimate. As flow is directly proportional to hydraulic conductivity the

factor of difference between the observed flow values and the predicted values can be

used to gain an improved estimate of hydraulic conductivity of the porous medium used

in the tank. The factor of difference was calculated between observed flow and the flow

predicted by the variably saturated flow model, as the variably saturated flow model


provides the most accurate prediction of total flow. The calculated factor of difference for

each of the head differentials is included in Table 3.

Table 3: Factor of difference between observed flow and predicted flow from the

variably-saturated flow model

As flow is directly proportional to the hydraulic conductivity multiplication of the

average factor by the previous value for hydraulic conductivity (48 m/day) enables the

calculation of an improved estimate for hydraulic conductivity (67 m/day). This value

lies within the range of 48 – 86 m/day predicted by the constant head test thus supporting

its application to the hydraulic conductivity of the tanks porous medium. Substitution of

67m/day for the hydraulic conductivity reveals that the variably-saturated and Dupuit-

Forchheimer models fit the observed flow data well (Figure 13).

Downstream head (cm) 10 20 40 60 70 75

Observed 3.05 2.74 2.56 1.76 1.14 0.76Variably saturated 2.11 2.03 1.71 1.18 0.79 0.62 Average

Factor 1.44 1.35 1.50 1.50 1.44 1.23 1.41


0 10 20 30 40 50 60 70 800

0.5

1

1.5

2

2.5

3

3.5


Flo

w (

m3 /d

ay)

Observed Dupuit−ForchheimerVariably Saturated

Figure 13: Flow rate predictions using a saturated hydraulic conductivity of 67m/day.

Using the new value for hydraulic conductivity the percentage deviation of the Dupuit-

Forchheimer and variably-saturated flow models from the observed flow rates was

calculated and is included in Table 4.

Table 4: Percentage errors in flow measurements for Dupuit-Forchheimer and variably-

saturated flow models

% Error 10 20 40 60 70 75 AverageDupuit-Forchheimer 8.37 1.09 11.94 11.33 1.32 12.67 7.79Variably saturated 2.87 3.77 6.89 6.81 2.37 12.59 5.88

Downstream well height (cm), upstream well height = 90cm


The variably-saturated flow model generally provides a better prediction of the actual

flow rate in the system (6% error) than the Dupuit-Forchheimer model (8% error). This is

especially highlighted for the 90:10, 90:40 and 90:60 head differentials where unsaturated

flows are larger due to the larger hydraulic gradients involved. At head differentials

where the Dupuit-Forchheimer model provides a better prediction of the total flow

(90:20, 90:70 and 90:75) the error differences between the variably-saturated and Dupuit-

Forchheimer model predictions are small and can be accounted for as experimental error.

Large errors in prediction of flow by both models for the 90:75 head differential indicate

that the observed value may have been incorrectly measured. Overall the errors between

the two data sets are small and the use of 67m/day for the hydraulic conductivity is

deemed appropriate.

4.3 Travel times

A clear indication of the variation of travel time with height is shown in Figure 14 where

for a 90:10 head differential photos were taken of dye released at heights of 80cm, 50cm

and 20cm at five minute intervals.


Figure 14: Dye released at heights of 80cm, 50cm and 20cm for a 90:10 head gradient

The travel times measured for the 90:20, 90:40, 90:60 and 90:70 head differentials are

included in Appendix D. Due to the physical constraints of the tank, observation of the

dye at the upstream and downstream ends of the tank was not possible. Total travel times

were taken as the time for the dye to move from x = 5cm to x = 105cm. It was observed

for each of the head differentials that the observed travel times increased with height of

t = 5

t = 15

t = 25

t = 10

t = 20

t = 30


release (Figures 15-18). Figures 15-18 also include the travel times predicted by the

Dupuit-Forchheimer model, as well as times predicted by variably-saturated model for a

hydraulic conductivity of 67m/day.

0 20 40 60 80 1000

5

10

15

20

25

30

35

Height of release (cm)

Tra

vel t

ime

(min

)Observed Variably SaturatedDupuit−Forchheimer

Figure 15: Total travel times for 90:20 head differential

0 20 40 60 80 1000

5

10

15

20

25

30

35

40

45

50


Tra

vel t

ime

(min

)

Observed Variably SaturatedDupuit−Forchheimer



0 20 40 60 80 1000

10

20

30

40

50

60


Tra

vel t

ime

(min

)



0 20 40 60 80 1000

10

20

30

40

50

60

70

80

90

100


Tra

vel t

ime

(min

)



A comparison of the total travel times observed for the four head differentials is included

in Figure 19.


0 20 40 60 80 1000

10

20

30

40

50

60

70

80

90

100


Tra

vel t

ime

(min

)

90:2090:4090:6090:70

Figure 19: Comparison of total travel times for changing head differentials

The reason for the variation of travel time with height is the combination of two factors.

The first being that as height increases so does the length of the travel path to the

downstream end, with a longer path meaning that it takes the dye longer to move to the

downstream end. Longer travel paths with height are due to paths at higher release

heights showing larger curvature, while paths at lower release heights are flatter and

therefore shorter. Secondly, the reduction of pressure with height leads to stronger

hydraulic gradients and thus stronger flow velocities towards the bottom of the tank.

Factorial differences in travel time of dye released at a height of 90cm and dye released at

a height of 15cm range from 1.91 times for a 90:20 head differential to 1.55 for a 90:70

head differential (Figure 20).


0 10 20 30 40 50 60 70 800

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Downstream well height (cm)

Tim

e(90

)/T

ime(

15)

Figure 20: Comparison of total travel times for dye released at 90cm and dye released at

15cm.

These are significant deviations in travel time especially considering that the Dupuit-

Forchheimer model predicts that travel times should not change with height. Figures 15-

18 also show that the Dupuit-Forchheimer model underestimates the total travel time in

all cases studied. Using the Dupuit-Forchheimer model could therefore be very costly in

cases involving contamination transport, as underestimation of the travel time for a radial

pumping situation could result in underestimation of pumping times required to remove

contaminants from the system.

These results also show that as the height of release increases the change in travel time

for a further increase in height of release becomes greater. This is shown by the points

curving upwards as height of release is increased for Figure 19. This result is explained


by the changes in the length of travel paths and changes in the flow velocity for different

heights of release. At lower release heights the differences between the paths at different

release heights are relatively small, explaining why there are only small deviations in

travel time between release heights. At higher points of release the lengths of flow paths

are much more dependent on height of release due to the larger curvature of flow for

these paths, and hence so is travel time. Flow velocities are also seen to generally reduce

as height of release is increased. For higher points of release larger reductions in velocity

are observed for a further increase in release height. This affect adds to the curvature of

the travel time graphs in Figure 19. Flow through the unsaturated zone is also important

in order to describe the variation of travel time with height. This is particularly valid for a

release height of 90cm where the dye crosses the phreatic surface into the unsaturated

zone. Flow through this zone is considerably slower than flow through the saturated zone

so consequently this affect leads to further increases in travel time. This explains further

the curving upwards of the points in Figure 19 for all four head differentials.

Figure 19 also shows that the travel times are all generally seen to reduce as the hydraulic

gradient is increased for all heights of release. This is as would be expected as larger

hydraulic gradients produce larger flow velocities. From Figure 19 it can also be noted

that there is a much smaller difference in travel times between the 90:20 and 90:40 head

differentials than there is between the 90:40 and 90:60 head gradients. Similar can be said

between the 90:40 and 90:60 and the 90:60 and 90:70 head differentials. This shows that

travel times are much more sensitive to changes in head gradient at smaller head

gradients (e.g. 90:70) than for larger head gradients (e.g. 90:20).

Figures 15-18 indicate that the variably-saturated flow equation provides a better fit of

the observed travel times for situations of larger hydraulic gradient (e.g. 90:20, 90:40)

than for smaller hydraulic gradients (e.g. 90:70). It is also noticeable that better

approximations of total travel time are provided for dye released at lower heights. As

height of release is increased the deviation between the variably-saturated model results

and the observed results also increases. Where poor predictions of travel times occur:

small head gradients and high heights of release, the slowest travel times observed. The


slower the travel time the larger the amount of spreading of the dye plume during its path.

Greater spreading makes it more difficult to identify the centre of the plume and

consequently errors in measuring the travel times are larger. Another reason for this

deviation lies in the fact that as the travel time increases so does the number of

heterogeneities encountered dye plume as it moves through the tank. In the variably-

saturated flow model we assume a homogeneous medium, however in reality some small

heterogeneities still exist. For the shorter travel times the affect of heterogeneities is not

as pronounced, however as travel times increase so do the errors from the variably-

saturated flow model. The largest errors in the prediction of dye travel times occur for

dye released at a height of 90cm. This height corresponds to the top of the water table at

the upstream end, where the length of the travel path is at a maximum and where flow

moves through unsaturated zone. The longer travel paths and the slowing of flow in the

unsaturated zone results in higher travel times for dye released at this height, making

prediction of travel times difficult. This explains the failure of the variably-saturated flow

model to accurately predict travel times at this height of release. Omitting the results

from the 90:70 head differential where some experimental errors occur, the results show

that the variably-saturated model provides a reasonable prediction of total travel time for

the radial flow system with an average error of 8% (Table 5).

Table 5: Comparison of errors in travel times predicted by the variably-saturated flow

model and those physically observed.

Height (cm) 20 40 60 Average

90 9.78 13.68 16.33 13.2675 3.14 4.95 15.04 7.7160 7.55 1.56 8.42 5.8445 9.86 2.25 2.34 4.8215 15.70 4.83 4.21 8.25

Average 9.21 5.45 9.27 7.98

% Error


For the same data the Dupuit-Forchheimer solution produces an average error of 140%

when estimating travel times (Table 6).

Table 6: Comparison of travel times predicted by the Dupuit-Forchheimer flow model

and those physically observed.

These results highlight the inability of the Dupuit-Forchheimer model to predict travel

times in particular unconfined systems, with the variably-saturated flow equation

providing much more accurate predictions of observed values.

Although the variably-saturated flow equation produces results that enable an accurate fit

of the results observed, this method of analysis is very complex and quick calculations of

travel times are unobtainable. As a result an empirical formula to calculate travel times

was produced, allowing these calculations to be performed in a much simpler manner.

The empirical formula was derived using the values obtained for travel times from the

variably-saturated flow model. Figure 21 shows the predicted travel times for the four

head differentials studied at the five heights of release, together with curves of best fit.

Height (cm) 20 40 60 Average90 230.04 74.70 274.04 192.9375 198.48 68.27 234.85 167.2060 149.61 63.50 196.64 136.5845 118.77 58.95 167.92 115.2115 82.34 54.56 142.13 93.01

Average 155.85 64.00 203.12 140.99

% Error


Figure 21: Predicted travel times for four head differentials together with their quadratic

curves of best fit

A second-degree polynomial (quadratic) fit was used. This revealed an almost perfect fit

for all four head differentials (R2~1 in all cases). Considering the general equation for a

quadratic of y=Ax2+Bx+C we observed that the value for A is the same for the 90:20,

90:40 and 90:60 head differentials. The value for the 90:70 head differential was

observed to be slightly different so as a consequence this data was neglected when

finding the empirical formula. The values for B for the 90:20, 90:40 and 90:60 head

differentials showed a general reduction in the value of B with downstream well height.

The values of B were plotted against the downstream well heights and a linear curve was

fitted, giving the relation: B = -0.0007H - 0.0589 where H is the downstream well height

and R2 = 0.9633. The C values were observed to increase as H was increased, with a plot

of C against H producing the linear relationship: C = 0.488H + 9.3673 with R2 = 0.9375.

Substitution of the expressions for B and C into the general expression produced the

y = 0.0015x2 - 0.0633x + 56.609

R2 = 0.9886

y = 0.0023x2 - 0.1008x + 40.101

R2 = 0.9963

y = 0.0024x2 - 0.0912x + 25.976

R2 = 0.9983

y = 0.0024x2 - 0.0718x + 20.582

R2 = 0.9988

0

10

20

30

40

50

60

70

0 10 20 30 40 50 60 70 80 90 100

Distance from downstream end (cm)

Tra

vel t

ime

(min

)

Head Differential 90:20 Head Differential 90:40 Head Differential 90:60Head Differential 90:70 Poly. (Head Differential 90:70) Poly. (Head Differential 90:60)Poly. (Head Differential 90:40) Poly. (Head Differential 90:20)


following expression relating travel time (t) to the height of release (h) and the

downstream well height (H):

( ) 3673.9H488.0h0589.0H0007.0h0024.0t 2 ++−−+= (9)

It should be stressed that this relationship is simply gained from relating the data to a

simple equation and the coefficients involved have no physical meaning. Comparison of

the travel times predicted by this relationship with those observed for the radial flow tank

produce an average error of 8% for the 90:20, 90:40 and 90:60 head differentials (Table

7).

Table 7: Comparison of observed travel times with those predicted by an empirical

relation.

This is a good prediction especially compared to the Dupuit-Forchheimer model, which

gave an error of 140% for the same data. However, its usefulness outside the data

observed is questionable. For a more accurate relationship more data is required so that

the relationship can be validated.

H hPredicted t

(min)Observed

t (min) % Error20 90 32.01 30.50 4.9620 75 27.17 27.58 1.5120 60 23.40 23.07 1.4520 45 20.71 20.22 2.4620 15 18.58 16.85 10.2840 90 40.51 42.88 5.5340 75 35.88 34.20 4.9040 60 32.32 29.73 8.7040 45 29.84 26.43 12.9040 15 28.13 23.88 17.7960 90 49.01 57.75 15.1360 75 44.59 51.70 13.7660 60 41.24 45.80 9.9560 45 38.97 41.37 5.7860 15 37.68 37.38 0.80

Average 7.73


For each of the head differentials the time for the centre of the dye plume to reach certain

points in the tank is shown below (Figures 22-25).

0 20 40 60 80 100 1200

5

10

15

20

25

30

35

Distance from upstream end (cm)

Tim

e (m

in)

H=90cmH=75cmH=60cmH=45cmH=15cm


release


0 20 40 60 80 100 1200

5

10

15

20

25

30

35

40

45


Tim

e (m

in)



release


0 20 40 60 80 100 1200

10

20

30

40

50

60


Tim

e (m

in)



release


0 20 40 60 80 100 1200

10

20

30

40

50

60

70

80

90

100


Tim

e (m

in)



release

Figures 22-25 are observed to flatten out as you move towards the downstream end of the

tank indicating that the dye increases in velocity as it moves through the tank. This

increase can be explained by rising pressure gradients when moving towards the

downstream end of the tank.

4.4 Downstream velocity profile

The downstream flow velocity for the 90:10 head differential is shown in Figure 26.


0 0.2 0.4 0.6 0.8 1 1.20

10

20

30

40

50

60

70

80

90

100

Horizontal Velocity (cm/s)

Hei

ght o

f rel

ease

(cm

)Observed Predicted

Figure 26: Observed downstream velocities for the 90:10 head differential

The downstream flow velocity for the 90:75 head differential is shown in Figure 27.


0 0.01 0.02 0.03 0.04 0.05 0.060

10

20

30

40

50

60

70

80

90

100

Horizontal Velocity (cm/s)

Hei

ght o

f rel

ease

(cm

)Observed Predicted

Figure 27: Observed downstream velocities for the 90:75 head differential

For the 90:10 head differential the observed downstream velocities are seen to generally

increase in magnitude in moving towards the downstream well height (Figure 26).

Although the observed values were greater than the predicted values for all heights this

was due to the observed values being measured close to, but not at, the downstream well

height. The observed outflow velocities are seen to peak just above the downstream well

height, rather than at the downstream well height, which was the numerical models

prediction. This difference can once again be explained by the position where flow

velocities were measured. The profile however does fit the general profile expected by

the numerical model, with outflow velocity increasing with decreasing height until close

to the downstream well height. Observations of flow velocities below the downstream

well height were not possible at this head differential due to the tank design.


The 90:75 head differential also shows a profile supporting the numerical data with the

outflow velocity increasing with depth, reaching a peak close to the downstream well

height and then reducing to a steady value below the downstream well height (Figure 27).

Although the velocities observed did not accurately imitate the numerically predicted

profile they do follow the general profile predicted by the numerical model. As for the

90:10 head differential the observed values were larger than the predicted values for all

observation heights. The variably-saturated flow model predicted that the outflow

velocity would reach a steady value much closer to the downstream water level (~50cm)

than the observed values, which stabilise at a lower point (~30cm). As for the 90:10 head

differential measurement of dye velocities close to, but not exactly at the downstream end

of the tank can explain these differences.

These observations highlight the first case where the theoretical downstream flow

velocity profile observed in Figure 3, has been observed physically in a radial system.

This is despite there being difficulties in accurately measuring the downstream flow

velocities, due to both the tank design and errors resulting from the use of dyes to

measure such small flow velocities. Vachaud and Vauclin (1975) had previously

observed this profile in a physical cartesian system, where a similar profile to the radial

system was expected. However, these observations were made from application of

Darcy’s Law to pressure heads observed in piezometers attached to the tank, rather than

by actually observing flow velocities using dye.

These results have particular relevance for contaminant transport as they show that

contamination closer to the downstream height of the well is likely to enter the well faster

than at other heights. As a consequence pump and treat schemes should be designed such

that the level in the pumping well be located as close to the centre of the contaminant

plume as possible, where inflow velocities into the well are maximum. Figure 28 shows a

correct pump and treat design, with the extraction well height at the same height as the

centre of the contaminant plume. At this height the maximum flow velocity into the well

occurs, allowing for the most efficient extraction of the contaminant.


Figure 28: Correct pump and treat design

4.5 Streamlines

Streamlines observed for the 90:20, 90:40, 90:60 and 90:70 head differentials are

included below, together with the values predicted by the variably-saturated flow model

(Figures 29-32). The data for the streamlines is included in Appendix D.

Flow velocity profile

Extraction well water

level

Contaminant


Figure 29: Streamlines observed for the 90:20 head differential


Figure 32: Streamlines observed for 90:70 head differential.

Photographs of streamlines observed for the 90:10 head differential are included below

(Figure 33).


Figure 33: Streamlines observed for a 90:10 head differential


Figures 29-32 show the predicted streamlines corresponding to the observed values well,

particularly for streamlines observed from the lower heights of release. As height of

release is increased and the influence of the capillary layer becomes more important the

streamlines from the tank are observed to discharge at positions higher than those

predicted by the variably-saturated flow equation. This indicates that the affect of the

capillary zone is more pronounced for the physical than it is for the numerical model.

This affect is compounded by the fact that dye released higher in the tank generally

undergoes more dispersion, making it more difficult to accurately track the dye’s flow

path. Some heterogeneities are also present in the tank, with this fact being highlighted by

the upward flow movement observed for the 90:70 head differential.

The predicted lines show the smallest error when compared to the observed flow lines

closest to the upstream end of the flow model. In this area the dye had undergone little

diffusion and consequently accurate tracking of the dye’s path was much easier.

Streamlines observed for the radial flow system follow the laboratory observations of

Wycoff (1932) and Hall (1955), with significant vertical flow occurring and the

formation of large seepage faces at the downstream end of the radial flow model. The

height of the seepage face was approximated by the position at which dye released at a

height of 90cm was observed 5cm from the downstream end. Ideally this observation

would have been taken at the downstream end, however physical constraints of the tank

prevented this. Observation at the downstream end gives a good measure of the seepage

face height as the upstream water level lies at a height of 90cm and water released at the

upstream water level is expected to discharge at the highest point at the downstream end

of the radial tank, which corresponds to the seepage height. The predicted seepage face

height from the variably-saturated model is taken as the highest point at the downstream

well where pressure head is observed to be zero. The results show that in general the

model underestimates the seepage face height, with the pattern observed between the two

sets of data being very similar (Figure 34). This pattern shows the seepage face height

increasing as the downstream water level increases, with the increase becoming smaller

as the downstream water height becomes larger (Figure 34).


0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

80

90


See

page

face

hei

ght (

cm)

Observed Predicted

Figure 34: Seepage face heights for upstream well height of 90cm

The deviation between the observed and predicted data is due to the observed data being

taken at a distance of 5cm from the downstream end, rather than at the downstream end.

This was the closest that seepage height could be observed due to physical restraints

placed by the design of the tank. Consequently observed seepage height was higher than

the predicted seepage face height. We expect the seepage height to fall in this last 5cm

leading to a set of seepage heights that lie closer to the values observed by the variably-

saturated model. One area of interest from the model occurs at a downstream water level

of 70cm where the variably-saturated model predicts that there will be no seepage face.

This is the smallest head differential studied, and coupling this to the fact that the sand

chosen for the experiment was coarse, and hence exhibited little capillary tension, this

result is of little surprise. Some seepage face would still exist in an experimental


situation, with the ability of the numerical model to predict this result being limited by

the size of the grid chosen.

4.6 Potentiometric Levels

The potentiometric levels in each of the standpipes were measured from the tank and

compared to the values predicted by the variably-saturated model (Figures 35-38). In

Figures 35-38, the stars represent the observed values and the lines the values predicted

by the variably-saturated flow model, while the values for H represent the height in the

tank where these pressures were observed. The data for piezometric levels is included in

Appendix D.


H=70cm

H=15cm

H=30cm

H=50cm

H=0cm

Figure 35: Piezometric levels for the 90:20 head differential


H=50cm

H=15cm

H=30cm

H=0cm

H=70cm



H=0cm

H=15cm

H=30cm

H=50cm

H=70cm



H=70cm

H=50cm

H=30cm

H=15cm

H=0cm



The variably-saturated flow model accurately predicts the piezometric levels for all of the

head differentials observed, with the accuracy of the prediction increasing with reducing

hydraulic gradient. The largest deviations between the observed and predicted values

occurred with the values taken from the piezometers approximately 25cm from the

downstream end at a height of 0cm. In this area the drop in the piezometric head is

largest, with the numerical model predicting lower piezometric heads than those observed

in the tank.


5.0 Conclusions

A detailed understanding of radially driven gravity flow is required to understand well-

flow hydraulics and transfer near pumping wells. Although extensive literature is

available in this area, detailed data on flow velocities and internal flow profiles has been

limited. Using this as a motivation, a number of physical experiments were performed

using a 15o cut of a radial flow system to simulate radial unconfined flow. Using dye

tracers, travel times were measured for four different head differentials, with the results

showing a clear increase in travel time with increasing height of release. This is the first

time where this relationship has been quantitatively observed in a physical system. The

physical results were compared to theoretical values computed from the Dupuit-

Forchheimer and variably-saturated flow models. The Dupuit-Forchheimer flow model

predicts that travel times should not change with height, an assumption that does not

hold. The Dupuit-Forchheimer model was also found to underestimate the travel times

for all heights of release with an average error of 140%. The variably-saturated flow

model however was able to better predict the travel times observed in the physical model,

with an average error of 8%. The increase in travel times with height of release is due to

longer travel paths for increased height of release and reduced flow velocities for

increased height of release.

Observation of travel times from the radial tank also enabled the measurement of

downstream velocity profiles. The velocity profiles for two head differentials revealed an

increase in velocity in moving from the top of the seepage face, with a maximum value

being reached near the downstream water level. Below this level the velocity was seen to

reduce, reaching a steady value below the downstream water level. This is the first case

where the theoretical velocity profile for a radial flow system has been reproduced in a

physical radial model.

Measurements of total flow, streamlines and pressure heads were also taken from the

radial flow model. As expected, comparison of observed total flow with predicted values

from the variably-saturated flow model produced smaller errors (6%) than with the


Dupuit-Forchheimer flow model (8%). This discrepancy is due to unsaturated flow,

which is ignored by the Dupuit-Forchheimer flow model. These errors also show that

despite the Dupuit-Forchheimer providing poor predictions of the travel times, its

predictions for flow are still relatively accurate. The streamlines and pressure heads

observed physically were predicted with good accuracy by the variably-saturated flow

model. The streamlines observed showed significant vertical flow, with the production of

large seepage faces supporting previous work in the area.

These results have particular relevance to the field of contaminant transport where

improved knowledge can help alleviate environmental problems. Application of the

Dupuit-Forchheimer flow model to measure travel times in radial flow systems has been

observed to lead to an underestimation of travel times. In a pump and treat problem this

underestimation could mean underestimation of required pumping periods, with

contaminants remaining in the aquifer after pumping. Application of the variably-

saturated flow model to a pump and treat problem could be much more beneficial.

Applying this model to the highest point of the contaminant plume, where travel times

are generally slowest, should produce an accurate lower bound for the pumping time

required to remove the contaminant plume from the aquifer. It has also been observed

that for most efficient removal of contaminants the well height should be at the same

level as the centre of the contaminant plume. Maximum inflow into the well is observed

at the well height so consequently contaminants at this height will exit the well at the

fastest rate. Application of the concepts discussed here will enable the implementation of

more efficient pumping practices for the removal of contaminants from groundwater

systems.


6.0 Recommendations

During this study an extensive data set was collected from a physical model of radial

flow that included travel times, flow, streamlines and pressure heads. Using this

apparatus there are many areas in which future research could be undertaken.

Further work could enable a reliable empirical relationship for calculating travel times

from height of release and head gradient to be identified. In this study an attempt was

made to derive a similar relationship, however the size of the data set was not large

enough for an effective relationship to be identified. Collection of travel times for

additional hydraulic gradients using the same porous medium would allow for a reliable

relationship to be obtained.

All data collected in this study was for a single porous media. Collection of data over a

number of different porous media would help to gain a greater understanding of the affect

of porous media and unsaturated flow on two-dimensional radial flow systems.

Future studies could also concentrate on measurements made under transient, as well as

steady-state conditions. Performing transient tests, such as a slug test, would enable a

greater understanding of the flow dynamics observed for radial flow systems.


7.0 References

Bear, J., 1979. Hydraulics of Groundwater. McGraw-Hill, New York.

Borelli, M. 1955. Free-surface flow toward partially penetrating wells. Transactions of

the AGU 36: 664-672.

Clement, T.P., 1993. Numerical modeling of variably-saturated seepage face boundaries.

Auburn University, Alabama

Clement, T.P., Wise, W.R. and Molz, F.J., 1994. A physically based, two-dimensional,

finite-difference algorithm for modeling variably-saturated flow. J. Hydrol., 161: 71-90.

Clement, T.P., Wise, W.R., Molz, F.J. and Wen, M., 1996. A comparison of modeling

approaches for steady-state unconfined flow. J. Hydrol., 181: 189-209.

Cooley, R.L., 1983. Some new procedures for numerical solution of variably-saturated

flow problems. Water Resour. Res. 19: 1271-1285.

Dewiest, R.J.M., 1965. History of the Dupuit-Forchheimer Assumptions on Groundwater

Hydraulics. Transactions of the ASAE, Paper 64-717: 508-509

Dupuit, J., 1863. Etudes theoriques et pratiques sur le mouvement des eaux dans les

canaux decouverts et a travers les terrains permeables. (2nd Edition). Dunod, Paris.

Fetter, C.W. 1994. Applied Hydrogeology (Third Edition). Prentice Hall, Upper Saddle

River, New Jersey.

Huyakorn, P.S., Thomas, S.D. and Thompson, B.M., 1984. Techniques for making finite

elements competitive in modeling flow in variably-saturated media. Water Resour. Res.,

20: 1099-1115


Huyakorn, P.S., Springer, E.P., Guvanasen, V. and Wadsworth, T.D., 1986. A three

dimensional finite-element model for simulating water flow in variably-saturated porous

media. Water Resour. Res., 22: 1790-1808.

Mualem, Y., 1976. A new model for predicting hydraulic conductivity of unsaturated

porous media. Water Resour. Res., 12: 513-522

Muskat, M., 1937. The Flow of Homogeneous Fluids through Porous Media. McGraw-

Hill, New York.

Narasimhan, T.N. and Witherspoon, P.A., 1976. An integrated finite difference method

for analyzing fluid flow in porous media, Water Resour. Res., 12(1): 57-64.

Narasimhan, T.N. and Witherspoon, P.A., 1978. An integrated finite difference method

for analyzing fluid flow in deformable porous media, 3, Applications, Water Resour.

Res., 14(6): 1017-1034.

Peterson, D.F., 1957. Hydraulics of wells. Trans. ASCE, 122: 502-517

Pinder, G.F. and Gray, W.G., 1977. Finite Element Simulation in Surface and Subsurface

Hydrology. Academic Press, New York.

Richards, L.A., 1931. Capillary conduction of liquids through porous mediums. Physics,

1: 318:333.

Rubin, J., 1968. Theoretical analysis of two-dimensional, transient flow of water in

unsaturated and partially saturated soils. Soil Sci. Soc. Am. Proc., 32: 607-615.


Shamsai, A. and Narashimhan, T.N., 1991. A numerical investigation of free surface-

seepage face relationship under steady state flow conditions. Water Resour. Res., 27(3):

409-421.

Simpson, M. 2000. Experimental and Numerical Investigation of Unconfined Flow about

a Density Interface. Unpublished thesis proposal. Centre for Water Research, Western

Australia.

Simpson, M.J. and Clement, T.P., 2001. Implications of Dupuit-Forchheimer

approximations on solute transport. Proceedings of the 2nd Australia and New Zealand

GeoEnvironment Conference, Newcastle.

Van Genuchten, M.Th., 1980. A closed-form equation for predicting the hydraulic

conductivity of unsaturated soils. Soil Sci. Soc. Am. J., 44: 892-898.

Vachaud, G. and Vauclin, M., 1975. Comments on ‘A Numerical Model Based on

Coupled One-Dimensional Richards and Boussinesq Equations’ by Mary F. Pikul,

Robert L. Street, and Irwin Remson. Water Resour. Res., 11(3): 506-509.

Wise, W.R., 1991. Discussion of “On the relation between saturated and capillary

retention characteristics”, by S. Mishra and J.C. Parker, September-October 1990 issue,

v. 28, no. 5, pp. 775-777. Ground Water, 29(2): 272-273.

Wise, W.R., Clement, T.P. and Molz, F.J., 1994. Variably-saturated modeling of

transient drainage: sensitivity to soil properties. J. Hydrol., 161: 91-108.

Wycoff, R.D., Botset, H.G. and Muskat, M., 1932. Flow of liquids through porous media

under the action of gravity. Physics, 2: 90-113.

Wycoff, R.D. and Reed, D.W. 1935. Electrical Conduction Models for the Solution of

Water Seepage Problems. Physics, 6: 395-401


APPENDIX A: Numerical Model of two-dimensional, variably-

saturated flow

A two-dimensional finite difference approximation to the variably-saturated flow

equation is used to numerically solve the problem of radial gravity flow. The method of

approximation developed by Clement et al. (1994) is included below:

( )

( )

√√

↵

+−√

√

↵

+

∆+

√√

↵

∆

−√√

↵

+−

√√

↵

∆

−√√

↵

+

∆+

√√↵

∆+∆

−√√

↵

+−

√√↵

∆+∆

−√√

↵

+

∆

≅++

−+

−−

++

−

−

+

+

++

++

−

++

+

++

22

1

2

21

2

2

)()(

2

2

)()(

11

)()()(

1

11

11

11

1

1

1

1

11

11

1

11

1

11

mmmm

mmmm

mmmm

jiij

mm

mm

ijji

mm

mm

ijij

ijijijij

ijijijij

ijijijij

jiij

jiij

ijji

jiij

KKKK

z

z

KK

z

KK

z

rr

rKrK

rr

rKrK

rr

z

K

zK

zrrK

rr

ψψ

ψψ

ψψ

ψψ

ƒθƒ

ƒƒψθ

ƒƒ

ƒƒψθ

ƒƒ (A1)

where Kij is the value of hydraulic conductivity at node ij, ψij is the pressure head at the

node ij and m is the Picard iteration level. The non-linear functions ψ and K(θ) are

linearised using the Picard iteration.

It is possible to rearrange equation (A1) into the form

gedcba mmmmm

ijjiijjiij=++++ +++++

++−−

11111

1111ψψψψψ (A2)

where coefficients a, b, c, d, e and g are defined as


√√↵

∆

+−=

√√↵

∆

+=

√√↵

∆+∆

+√√↵

∆

=

√√↵

∆+∆

+√√↵

∆

=√√↵

∆+

=

+

++

−

+

+

−

−−

z

KKg

z

KKe

rr

KrKr

rrd

rr

KrKr

rrb

z

KKa

mm

mm

ijji

mij

mji

ijij

jiij

mji

mij

ijij

mij

mij

ijij

ijijijji

jiij

2

2,

1

1,

2

1

11

1

21

1

1

1

2

1

and c = -[a + b + d + e]

(A3)

Equation (A2) can be modified to reflect the models boundary conditions leaving a set of

linear equations that can be written in matrix notation as:

bA ˆˆ =ψ (A4)

where

ψ is the vector of unknown pressure heads, ijψ and b is the forcing vector.

Solving equation (A4) yields the unknown pressure heads that can be used to describe the

position of the phreatic surface and flow velocities for the problem.

A.1 Boundary Conditions

Appropriate boundary conditions necessary to solve equation (A4).

Dirichlet nodes

Points where pressure heads are known. These points are represented by ijij ψψ Γ= where

ijψΓ are known pressure heads for nodes ij. For these nodes all coefficients of the matrix

in equation (A2) are zero except c, which is unity. Because of this the known pressure


heads, ijψΓ are equal to the ij terms in b . For the radial gravity flow problem these

include nodes along the upstream end below the upstream water level and nodes along

the downstream end below the seepage face height.

Neumann boundaries

Nodes where the normal fluxes, qn are known. At the right boundary Darcy flux in the x

direction at the node ij can be described by:

∆−+

−∪ ++

r

KKq ijjiijji

r

ψψ 11

2

(A5)

If the right boundary is specified as no-flow then using (A5), with qr = 0 gives:

ijji ψψ =+1(A6)

meaning that equation (A2) reduces to:

ge)dc(ba 1ijijj1i1ij =ψ+ψ++ψ+ψ +−−(A7)

Using the same method this can be applied to the left no flow boundary enabling equation

(A2) to be reduced to:

ged)cb(a 1ijj1ij1i1ij =ψ+ψ+ψ++ψ ++−− (A8)

In the vertical direction Darcy’s law can be expressed by:

−∆−+

−∪ ++ 12

11

z

KKq ijijijij

z

ψψ (A9)


Equation (A9) can be used to modify equation (A2) for vertical flux boundary nodes.

The radial gravity flow problem has specified no flux boundaries at the top and the

bottom of the flow domain, and above the upstream water level and seepage face height.

Seepage-face boundaries

Along the seepage face boundaries the nodes are treated as Dirichlet nodes with ψ = 0,

with all nodes above the seepage face specified as areas of no flow (Figure A1).

Figure A1: Pressure levels observed along downstream end of the radial flow problem

The position of the seepage face is initially unknown, with an initial guess of its position

being taken and an iterative process used to determine its actual position. The initial

guess overestimates the position of the seepage, thus ensuring that the node directly

above the seepage face shows a negative ψ value. The process then automatically moves

down one node and recalculates the pressure at the node above the seepage face. This

process is repeated until a positive value for ψ is observed in the node above the seepage

ψ positive

ψ = 0

ψ negative

Seepage face

Downstreamwater level


face. At this time the process is terminated, with the top of the seepage face taken as the

last node where a negative ψ was observed in the node directly above it.

A.2 Soil Properties

The soil properties are described using Van Genuchten’s (1980) closed form of the soil

water retention curve and Mualem’s (1976) unsaturated hydraulic conductivity function.

Pressure head (under tension) and water content are related according to (Van Genuchten,

1980):

Θ =1

1+ (α v ψ nv

? ?

mv

(A10)

where αv, nv and mv = 1-(1/nv) are the Van Genuchten parameters which depend on the

soil properties. αv increases with the first moment and nv increases as the width of the

pore size density function reduces (Wise, 1991; Wise et al., 1994). The parameter Θ is

the effective saturation, described by:

rs

r

θθθθ

−−

=Θ (A11)

where θs describes the saturated water content and θr the residual water content of the soil

under consideration. Using a model produced by Mualem (1976) this water content is

related to hydraulic conductivity (Van Genuchten, 1980):

( ){ }{ } )2/1(2

/111)( ΘΘ−−=Θ vvmm

sKK (A12)


where Ks describes the saturated hydraulic conductivity. One point from this formula that

should be noted is that the relative permeability, K(Θ)/Ks is independent of αv (Clement

et al., 1994).


APPENDIX B: Radial Flow Model Plans

The radial flow tank used was designed by Matthew Simpson for the Centre for Water

Research as part of his PhD (Simpson, 2000). Figures B1, B2 and B3 show the physical

model of the radial flow tank:


Figure B1: Top view of the radial flow tank (Simpson, 2000)


Figure B2: Side view of the radial flow tank.


Figure B3: Piezometers attached to the radial flow tank.


100

33.2

2

10

154.2

0.5

Upstream Screen

Enlargement


Figure B4: Screens used for the radial flow tank.

100

52

1.6

0.6

Enlargement

Downstream Screen

0.5

Enlargement


APPENDIX C: Measuring saturated hydraulic conductivity

A constant head test was used to measure the hydraulic conductivity of the porous

medium used in the radial flow tank. The test used a cylindrical permeameter that was

filled with the porous sample (Figure 7). Water was able to enter the cylinder through a

connection at its base and exit through an outlet pipe at the top of the cylinder. On the

side of the permeameter nine tubes were connected enabling the measurement of

piezometric heads at nine points in the cylinder (Figure C1).

Figure C1: Piezometers measuring heads in constant head permeameter

Head differentials were set up using an elevated fixed head water supply that was

connected to the permeameter via the inlet pipe. Using the constant head permeameter the

time to fill 300mL of a 1L flask was calculated for five different hydraulic gradients.

Table C1 shows the results for the tightly packed sand sample.


Table C1: Constant head test results for tightly packed sample

Using the results and equation C1, given by Darcy’s Law, the saturated hydraulic

conductivity was identified as 48 m/day for the tightly packed sample (Fetter, 1994).

Ath

VLK =

(C1)

Table C2 shows the results from the constant head test on an unpacked sand sample.

Table C2: Constant head test results for unpacked sample

The results predict a hydraulic conductivity of 86 m/day for the unpacked sand sample.

Using a different permeameter, the falling head test involved measuring the time for the

water level to drop between two head intervals to measure hydraulic conductivity. The

process was repeated four times, with the results for a tightly packed sand sample

included in Table C3.

500 0.0084541

1 150 300 401 7.48E-07 0.30 25.492 200 300 174 1.72E-06 0.40 44.053 250 300 111 2.70E-06 0.50 55.244 300 300 91 3.30E-06 0.60 56.155 350 300 77 3.90E-06 0.70 56.88

47.56

TestTotal Head

h (mm)

Water Volume V

(mL)

Area A m2

Flow V/t

m3/sec

Time t (s) Total

gradient h/L k (m/day)

Sample Height L (mm)

500 0.0084541

1 150 300 103 2.91E-06 0.30 99.222 200 300 85 3.53E-06 0.40 90.183 250 300 74 4.05E-06 0.50 82.864 300 300 63 4.76E-06 0.60 81.115 350 300 58 5.17E-06 0.70 75.52

85.78

Time t (s) Total

gradient h/L k (m/day)

Flow V/t

m3/sec

Area A m2

TestTotal Head

h (mm)

Water Volume V

(mL)

Sample Height L (mm)


Table C3: Falling head test results for tightly packed sand sample

The results were used together with equation C2, given by Darcy’s Law and the principle

of continuity, to identify a hydraulic conductivity of 49 m/day for the tightly packed sand

sample (Fetter, 1994).

√↵

=h

h

td

LdK

c

t 02

2

ln(C2)

h1 (cm) 80.5 dt 2.5h2 (cm) 50.5 dc 10.2h3 (cm) 20.5 L (cm) 11.8

Trail t1 (s) tfinal (s) K1

(m/day)K2

(m/day)1 5.90 17.26 48.40 48.612 5.88 17.50 48.57 47.523 6.07 17.32 47.05 49.084 5.93 16.74 48.16 51.08

Average 48.04 49.07 48.56


APPENDIX D: Physical results

The travel times for the four head differentials studied are included below (Table D1). In

the table front, centre and tail indicate the front, centre and tail of the plume and x is the

distance from the upstream end of the tank.


Table D1: Observed travel times

The six streamlines produced (A, B, C, D, E and F) for each of the four head differentials

studied is included below (Table D2).

Head Differential 90:20x (cm) 40 40 40 70 70 70 105 105 105Height (cm)

t front (min:sec)

t centre (min:sec)

t tail (min:sec)

t front (min:sec)

t centre (min:sec)

t tail (min:sec)

t front (min:sec)

t centre (min:sec)

t tail (min:sec)

90 14:08 22:00 26:22 30:30 33:5975 14:25 22:28 23:55 27:35 30:1160 12:35 18:40 19:37 23:04 25:4745 10:36 16:37 16:37 20:13 22:4315 09:18 13:57 13:32 16:51 20:14


t front (min:sec)

t centre (min:sec)

t tail (min:sec)

t front (min:sec)

t centre (min:sec)

t tail (min:sec)

t front (min:sec)

t centre (min:sec)

t tail (min:sec)

90 18:05 24:00 28:30 33:55 41:38 37:02 42:53 49:1175 15:20 18:23 24:00 24:51 28:06 36:30 29:44 34:12 39:0860 12:18 16:29 20:25 20:33 24:33 29:28 25:06 29:44 33:3545 10:55 14:15 17:30 18:50 21:52 27:28 22:50 26:26 30:0015 10:21 13:39 20:00 17:20 20:30 25:41 20:30 23:53 28:12


t front (min:sec)

t centre (min:sec)

t tail (min:sec)

t front (min:sec)

t centre (min:sec)

t tail (min:sec)

t front (min:sec)

t centre (min:sec)

t tail (min:sec)

90 26:22 34:04 48:00 40:57 48:18 1:01:45 50:29 57:45 1:07:3175 23:16 28:19 36:12 37:10 43:17 54:10 44:30 51:42 57:4560 20:52 27:12 34:29 32:35 38:55 50:57 38:33 45:48 54:5845 17:02 22:41 31:48 29:20 35:13 44:57 36:06 41:22 49:4215 16:26 21:30 29:14 25:54 32:25 39:46 31:33 37:23 46:48


t front (min:sec)

t centre (min:sec)

t tail (min:sec)

t front (min:sec)

t centre (min:sec)

t tail (min:sec)

t front (min:sec)

t centre (min:sec)

t tail (min:sec)

90 0:39:03 0:51:21 1:17:54 1:01:10 1:13:35 1:39:15 1:15:54 1:35:40 1:45:0575 0:39:33 0:50:30 1:11:30 1:00:45 1:11:06 1:30:41 1:11:52 1:27:06 1:38:2560 0:38:04 0:47:35 1:09:30 0:57:26 1:06:30 1:21:33 1:06:40 1:18:53 1:29:3245 0:31:25 0:37:50 0:50:09 0:48:52 0:58:00 1:12:41 1:01:11 1:10:08 1:18:3815 0:27:40 0:33:11 0:41:04 0:45:11 0:51:35 0:57:31 1:01:48 1:05:42


Table D2: Observed streamlines

Head Differential 90:20Distance from downstream

end (cm)Height A

(cm)Height B

(cm)Height C

(cm)Height D

(cm)Height E

(cm)Height F

(cm)115 90 75 60 45 30 15110 89.1 74.4 59.9 45 30.6 14.6105 88.3 74.4 59.9 45.5 30.1 13.9100 87.9 73.8 59.9 45 29.2 13.395 87.7 73.5 59.9 44.7 28.8 12.890 87.4 73.4 59.5 43.5 28.1 12.385 87.1 72.9 58.6 43 27.7 1280 86.9 72.4 57.6 41.4 27 11.875 86.9 71.9 55.8 40.2 26.3 11.470 86.5 71.3 54.6 39.1 25.6 11.265 86.2 69.3 53.3 38 24.6 10.760 85.1 67.5 52.1 37 24 10.655 85 66.4 50.7 35.7 23.3 10.550 84.1 64.1 49.2 35.6 22.4 1045 83.3 62.2 47.9 34.3 21.9 1040 81.1 60.4 46.2 33.4 21.1 9.735 78 58.9 44.6 32.3 20.3 9.530 76 57.3 43.3 31.2 19.6 8.825 72.8 55.7 41.9 30 18.9 8.520 70.5 54.2 40.7 28.8 17.8 8.315 66.9 52 39.9 28.2 16.9 7.9



end (cm)Height A

(cm)Height B

(cm)Height C

(cm)Height D

(cm)Height E

(cm)Height F

(cm)115 90 75 60 45 30 15110 90 74.5 59.7 45 30.4 14.1105 89.1 74.7 59.8 45.1 30.4 13.5100 88.6 74.3 60 45.3 30.2 13.195 88.6 74.3 60.2 45.3 29.8 12.790 88.3 73.9 60 44.7 29.3 12.385 88.1 73.6 59.7 44.5 28.8 12.180 87.6 73.1 58.8 43.4 28.2 1275 87.6 73 57.7 42.1 27.8 11.970 87.6 72.5 56.7 41.4 27.5 11.965 86.9 72.3 55.5 40.9 26.9 11.760 86.4 70.6 54.6 40.2 26.6 11.755 85.9 69.4 54 39.3 26.3 11.950 85 68.6 52.9 38.7 25.8 11.645 83.9 66.9 52 38.1 25.5 11.640 83 65 51.4 37.5 25.1 11.735 81.8 63.9 50.5 37 24.6 11.130 80 62.8 49.7 36.3 24.3 10.725 77.9 61.1 49.3 35.9 23.8 10.320 75.9 59.8 48.5 35.9 22.9 1015 71.9 58 47.5 35.9 22.3 9.9



end (cm)Height A

(cm)Height B

(cm)Height C

(cm)Height D

(cm)Height E

(cm)Height F

(cm)115 90 75 60 45 30 15110 89.3 74.7 61 45.7 30.5 13.6105 88.7 74.7 61.3 46.1 30.2 13100 88.4 74.6 61.8 46.4 30 12.395 88.5 74.2 62.2 46.3 29.9 1290 88.2 74.5 62.4 45.8 29.8 11.685 87.9 74.5 62.1 45.5 29.5 11.580 87.9 74.4 61.9 45.3 29.1 11.975 87.9 74.7 61 44.4 28.7 11.670 87.9 74.5 60.3 43.8 28.5 11.865 87.7 74.1 59.6 43.6 28.5 11.960 87.1 72.6 59.5 43 28.5 11.755 86.9 72 59.3 42.7 28.6 12.150 86.5 71.2 58.5 42.5 28.1 1245 85.7 70.8 58 42.4 28.3 12.340 85 69.8 57.8 42.2 28.3 12.135 83.9 69 57.4 42.3 27.6 11.930 82.8 68.5 57.2 42.5 27.6 11.725 82 68.1 57.5 43 27.6 10.820 80.2 67.7 57.5 43 27.4 10.815 76.7 67 57.3 43.8 27.5 10.5


The levels recorded in each of the standpipes is recorded below for the four head

differentials studied (Table D3).


end (cm)Height A

(cm)Height B

(cm)Height C

(cm)Height D

(cm)Height E

(cm)Height F

(cm)115 90 75 60 45 30 15110 87 72.2 60.8 47.4 28.8 14.2105 86.6 71.2 61.8 47.6 28.7 13100 86.2 70.9 61.9 48.5 28.5 11.895 86.6 71.4 62.5 48.7 28.7 11.290 86.6 71.5 62.8 49.3 28.8 10.985 86.6 71.5 63.1 49.8 29.1 10.980 86.6 71.5 62.8 49.8 29.1 11.275 86.7 71.8 62.9 49.3 29.4 12.170 86.7 71.9 62.2 49.1 29.6 12.565 86.5 70.6 62.1 49 29.7 12.360 86.5 70 61.7 49 29.6 1355 86.3 69.6 61.6 49.1 29.7 12.950 86.2 69.4 61.9 48.9 30 13.545 85.7 69.4 61.6 49.2 30.5 13.840 85.5 68.5 61.5 49.3 30.3 13.935 84.7 67.5 61.5 49.9 30.5 13.730 84.2 67.9 61.9 50.7 30.3 13.525 83.2 68.1 62.5 51.3 30.2 12.920 81.4 68.2 63 52.2 30 11.915 78.2 68.4 63.8 53 30.4 11.6


Table D3: Observed piezometric levels (nwl – no water level)

Head Differential 90:20Distance to downstream end (cm) 104.5 84.5 64.5 45.1 25.1 10

Height of 90 nwl nwl nwl nwl nwl nwlstandpipe 70 89.35 87.15 84.15 80.35 74.75 nwl(cm) 49.5 89.45 86.75 83.05 78.75 70.75 49.5

30.6 89.05 86.15 82.45 76.95 68.1 30.615.6 89.85 85.35 81.75 76.25 65.2 15.6

0 88.95 85.65 81.55 76.45 65.1 10


Height of 90 nwl nwl nwl nwl nwl nwlstandpipe 70 89.35 87.75 84.95 82.05 76.2 nwl(cm) 49.5 89.45 87.75 84.75 81.25 75 49.5

30.6 89.55 87.05 84.85 80.75 74.1 30.615.6 90.35 86.95 84.25 80.05 72.7 15.6

0 89.65 87.05 84.05 79.95 72.4 10


Height of 90 nwl nwl nwl nwl nwl nwlstandpipe 70 89.65 88.45 86.55 84.25 80.4 nwl(cm) 49.5 89.65 88.65 86.25 83.85 79.5 60

30.6 89.65 88.25 86.45 84.15 80 6015.6 89.65 88.05 85.95 84.05 79.2 60

0 89.65 87.95 85.95 83.55 79 60


Height of 90 nwl nwl nwl nwl nwl nwlstandpipe 70 89.95 89.25 87.55 87.05 83.8 70(cm) 49.5 89.95 89.25 87.55 87.05 83.8 70

30.6 89.95 89.25 87.55 87.25 83.7 7015.6 89.95 89.25 87.35 86.25 83.5 70

0 89.95 89.05 87.35 86.85 83.2 70

Laboratory Investigation of Radial Flow Dynamics

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