Braunschweig Technical University Institute of GeoEcology · 2006. 10. 16. · Braunschweig Technical University Institute of GeoEcology Measuring soil hydraulic properties of FIMOTUM

Braunschweig Technical UniversityInstitute of GeoEcology

Measuring soil hydraulic properties of FIMOTUM porous media:Results for medium sand

- Report-

Andre Peters and Wolfgang Durner16.10.2006

Contents

1 Materials and Methods 6

1.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2 Analysis of Data and Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . 9

1.2.1 Inversion of Richard’s Equation . . . . . . . . . . . . . . . . . . . . . . . .9

1.2.2 Fit of Equilibrium Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Results 11

2.1 Saturated Water Content, Porosity, Bulk Density and Saturated HydraulicConductivity 11

2.2 Inversion of Richard’s Equation . . . . . . . . . . . . . . . . . . . . . . . . .. . . 11

2.2.1 First Outflow Branch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.2 First Imbibition Branch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.3 Second Outflow Branch . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.4 Hysteresis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.3 Fit of Equilibrium Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2

List of Tables

1.1 Matric head at lower boundary of the ceramic plate,hLB, for the multistep out-

flow/inflow experiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1 Estimated parameters for first drainage branch - column 1. . . . . . . . . .. . . . . 16

2.2 Estimated parameters for first drainage branch - column 2. . . . . . . . . .. . . . . 17

2.3 Estimated parameters for first imbibition branch - column 1. . . . . . . . . . . . .. 17

2.4 Estimated parameters for first imbibition branch - column 2. . . . . . . . . . . . .. 21

2.5 Estimated parameters for second drainage branch - column 1. . . . . . . .. . . . . . 25

2.6 Estimated parameters for second drainage branch - column 2. . . . . . . .. . . . . . 25

2.7 Data for equilibrium fits. First outflow branch. . . . . . . . . . . . . . . . . .. . . . 27

2.8 Data for equilibrium fits. Second outflow branch. . . . . . . . . . . . . . . .. . . . 27

2.9 Estimated parameters of equilibrium data - first drainage branch. . . . . .. . . . . . 28

2.10 Estimated parameters of equilibrium data - second drainage branch. . .. . . . . . . 28

3

List of Figures

1.1 Experimetal setup of the multistep outflow/inflow experiments . . . . . . . . . . . .7

1.2 Matric head at lower boundary of the ceramic plate,hLB, for the multistep out-

flow/inflow experiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1 First drainage branch, column 1: Measured data and fitted with 4 different hydraulic

properties models. Top: Matric head; Middle: Cumulative outflow; Bottom: Cumu-

lative outflow forhLB ≥ −25 to show the better performance of the more flexible

models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 First drainage branch, column 2: Measured data and fitted with 4 different hydraulic



models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3 First drainage branch, column 1: Hydraulic properties. . . . . . . . . .. . . . . . . 15

2.4 First drainage branch, column 2: Hydraulic properties. . . . . . . . . .. . . . . . . 15

2.5 First outflow branch, both columns: Hydraulic properties. solid lines: column 2;

dashed lines: column 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.6 First imbibition branch, column 1: Measured data and fitted with 4 differenthydraulic

properties models. Top: Matric head; Bottom: Cumulative outflow. . . . . . . . .. . 18

2.7 First imbibition branch, column 2: Measured data and fitted with 4 differenthydraulic



models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.8 First imbibition branch, column 1: Hydraulic properties. . . . . . . . . . . . .. . . 20

2.9 First imbibition branch, column 2: Hydraulic properties. . . . . . . . . . . . .. . . 20

2.10 Second outflow branch, column 1: Measured data and fitted with 4 different hydraulic



models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4

LIST OF FIGURES 5

2.11 Second outflow branch, column 2: Measured data and fitted with 4 different hydraulic



models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.12 Second outflow branch, column 1: Hydraulic properties. . . . . . . . .. . . . . . . 24

2.13 Second outflow branch, column 2: Hydraulic properties. . . . . . . . .. . . . . . . 24

2.14 Second outflow branch, both columns: Hydraulic properties. solid lines: column 1;

dashed lines: column 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.15 Hysteresis of Hydraulic Properties, Column 1. Solid lines: first outflow; dots: first

imbibition; dashed lines: second outflow. . . . . . . . . . . . . . . . . . . . . . . . 26

2.16 Hysteresis of Hydraulic Properties, Column 2. Solid lines: first outflow; dots: first

imbibition; dashed lines: second outflow. . . . . . . . . . . . . . . . . . . . . . . . 26

2.17 Retention functions resulting from equilibrium fit. Left: column 1, right: column 2.

Solid lines: first outflow brach; dashed lines: second outflow branch. .. . . . . . . 27

Chapter 1

Materials and Methods

1.1 Experimental Setup

The air dry sand was packed in a 7.3 cm high column with a diameter of 9.4 cm. In order to obtain a

horizontally homogeneous packed soil column, the sand rinsed through a device with 5 consecutive

sieves before it reached the column. Whereas the packing procedure the column was slowly lowered

underneath the packing device so that the distance between device outlet and soil surface was alway

≈ 5 cm. This allowed also for a vertically homogeneous packed soil column. The column was

placed on a porous ceramic plate with a height of 0.7 cm and a saturated hydraulic conductivity

of 6.0 cm h−1. The ceramic plate in turn was in contact with a burette (Fig. 1.1). Details of the

experimental setup are given by Zurmühl (1998).

One pressure transducer tensiometer was placed in the center of the column, i. e. at a height of 3.65

cm above the plate. To prevent the building of macropores during the experiment due to “escaping

air bubbles” the soil surface was covered with a metal grid and a weight ofapproximately 1.5 kg.

The width of the grid was> 3 mm, so that no “extra pores” due to the grid were created. During the

experiment no setting or building of macropores was observed.

Before the Multistep-Outflow/Inflow (MSO/I) experiment was started the whole system was slowly

saturated and then flushed with approximately 5 liters (approximately 25 pore volumes) of degassed

water from the bottom. Simultaneously the saturated hydraulic conductivity,Ks, was measured. To

avoid evaporation at the upper boundary the column was covered with a lid.Contact to the atmosphere

was allowed by a thin needle. For the MSO/I experiment the pressure at the lower boundary was

controlled as shown in Fig. 1.2 and documented in Tab. 1.1. At each step the pressure change was

linearly within 30 seconds. In the outflow branches the hydraulic equilibriumwas achieved until the

pressure at the lower boundary of the ceramic plate,hLB, was−40 cm. This was done to be able to

predict point values of the pressure heads and water contents without measuring the water contents at

certain depths.

The cumulative outflow/inflow was determined by measuring the water volume in theburette. Cumu-

6

1.1. EXPERIMENTAL SETUP 7

tensiometer(3.65 cm)

sand

pressurecontrol

valve

vacuumreservoir

9.4 cm7

.3

pressuretransducer

positivepressurereservoir

pressurereservoir

pressuretransducer

burette

porousplate

air inlet

top cover

referencepressure

line

metal weightmetal grid

Figure 1.1: Experimetal setup of the multistep outflow/inflow experiments

0 50 100 150 200 250 300

−50

−40

−30

−20

−10

0

10

time [h]

h LB [c

m]

Figure 1.2: Matric head at lower boundary of the ceramic plate,hLB , for the multistep outflow/inflow experi-ments.

8 CHAPTER 1. MATERIALS AND METHODS

Table 1.1: Matric head at lower boundary of the ceramic plate,hLB , for the multistep outflow/inflow experi-ments.

∆t[h] t[h] hUR[cm]

0.25 0.25 7.50

0.25 0.50 3.75

0.50 1.00 0.00

0.75 1.75 -5.00

1.00 2.75 -10.00

1.50 4.25 -15.00

2.75 7.00 -20.00

4.00 11.00 -25.00

6.00 17.00 -30.00

10.00 27.00 -35.00

18.00 45.00 -40.00

24.00 69.00 -45.00

24.00 93.00 -50.00

24.00 117.00 -45.00

24.00 141.00 -40.00

24.00 165.00 -35.00

18.00 183.00 -30.00

10.00 193.00 -25.00

6.00 199.00 -20.00

4.00 203.00 -15.00

2.75 205.75 -10.00

1.50 207.25 -5.00

1.00 208.25 0.00

0.75 209.00 3.75

0.50 209.50 7.50

0.50 210.00 7.50

1.2. ANALYSIS OF DATA AND PARAMETER ESTIMATION 9

lative outflow and the suction in the center of the column were measured with a frequency of 10 Hertz

and logged with a frequency of 1 Hertz. The measurements were conducted in two replications.

1.2 Analysis of Data and Parameter Estimation

The measured data were split into three branches:

1. first outflow

2. first inflow

3. second outflow

1.2.1 Inversion of Richard’s Equation

The data were thinned out, so that approximately 600 data points were left for evaluation. To obtain

a maximum of information in the remaining data, the data density was highest close to the time when

the boundary changes occurred.

The governing equation for the flow simulation was the one dimensional Richards equation:

dθ

dh

∂h

∂t−

∂

∂z

[

K(h)

(

∂h

∂z− 1

)]

= 0 , (1.1)

whereh [cm] is the matric head,θ [−] is the volumetric water content,z [cm] is the vertical length

unit, t [h] is time andK(h) [cm h−1] is the hydraulic conductivity function.

The constitutive relationships, i.e. the soil water retention function,θ(h), and the hydraulic conductiv-

ity function,K(h), were described by parametric models of different flexibility.θ(h) was described

by:

1. the constrained van Genuchten model (Genuchten, 1980) (vG cons):

Se(h) = (1 + (α|h|)n)−m (1.2)

with m = 1 − 1n whereSe is the water saturation, defined by

Se =θ − θrθs − θr

(1.3)

andα[cm−1], n [−] andm [−] are curve shape parameters.

2. the unconstrained van Genuchten model withm as an independent fitting parameter( vG un-

cons)

10 CHAPTER 1. MATERIALS AND METHODS

3. the bimodal van Genuchten model (Durner, 1994) (vG bimod)

Se(h) =2

∑

i=1

wi

[

1

1 + (αi|h|, )ni

]1− 1ni

(1.4)

wherewi [−] are the weighting factors for the subfunctions, subject to0 < wi < 1 and∑

wi =

1.

4. and free form hermite spline functions (Bitterlichet al. , 2004) (ff).

The hydraulic conductivity function was for the first 3 cases describedby the Mualem model

(Mualem, 1976):

Kr(Se(h)) = Sτe

Se∫

0

h−1dS∗

e (h)

1∫

0

h−1dS∗

e (h)

2

, (1.5)

that has analytical solutions for cases 1 and 3 (Priesack & Durner, 2006):

K(h) =

[

k∑

i=1

wi [1 + (αi|h|)ni ]−mi

]τ

k∑

i=1wiαi

[

1 − (αi|h|)ni [1 + (αi|h|)

ni ]−mi]

k∑

i=1wiαi

2

, (1.6)

wherek is the modality of the function, i.e.k = 1 for case one and 2 for case 3. For the second case

(independentm), the mualem model was solved numerically.

In the free form caseθ(h) andK(h) are treated as completely independent spline functions. The

complexity of the models increases from case one to case 4.

Equation 1.1 was fitted with the four different models for the constitutive relationships to the three

data branches of both columns, so that 24 parameter sets forθ(h) andK(h) were obtained. For each

model and soil column we got a primary drainage curve (first outflow), a “primary” imbibition curve

(first inflow) and a secondary drainage curve (second outflow).

The fitting procedure was carried out with the shuffeld complex evolution algorithm (Duanet al. ,

1992) which is a global optimizer. The software tool was developed in our department and is called

RIEHP (Robust Inverse Estimation of Hydraulic Properties).

1.2.2 Fit of Equilibrium Data

The hydraulic equilibrium data (values at the end of each pressure step until hLB = −40 cm) for the

4 outflow branches we also fitted with the three parametric models as describedby (Peters & Durner,

2006) with the software tool SHYPFIT2.0 (Soil Hydraulic Properties Fitting).

Chapter 2

Results

2.1 Saturated Water Content, Porosity, Bulk Density and Saturated

Hydraulic Conductivity

The saturated hydraulic conductivity,Ks, was 144 and 68.3cm h−1, and the saturated water content,

θs, was 0.396 and 0.385, for column 1 and 2, respectively. For the imbibition branch and the second

outflow branch theθs was assumed to be equal to the water content at the end of the measurement

(hLB = +7.5 cm). Hereθs was 0.351 and 0.343. The bulk density,ρb was very similar with 1.567

and 1.563g cm−3 for the two columns. The porosity,φ, was calculated according toφ = ρb/ρswhereρs is the particle density. The value ofρs was assumed to be 2.65g cm−3. The results are

summarized in Tables 2.1 to 2.6.

2.2 Inversion of Richard’s Equation

2.2.1 First Outflow Branch

The measured data and model fits are shown in Figures 2.1 and 2.2. For the last two steps

(hLB ≤ −45 cm) hydraulic equilibrium was not achieved. For all other steps the point water con-

tents and pressures may be predicted at any height. Although we assume a homogeneous soil without

any macropores we observed that the soil column drained already slightly at very high values for

hLB. That behavior can only be described by very flexible hydraulic models such as the bimodal van

Genuchten model or free form approaches.

The two outflow plateaus athLB = −25 cm in column 2 (Fig. 2.1 bottom, time 10 to 15 hours)

may be explained by a break of an instability after some fluctuations at the lowerboundary occurred.

These fluctuations have no effect on the matric head in the middle of the column.

The resulting hydraulic properties functions are plotted in Figures 2.3 to 2.5.The shapes of the

different models look for both columns quite similar (Fig. 2.5), so that we regard the experiment as

well reproducible. The corresponding parameter values and goodness of fits are listed in Tables 2.1

11

12 CHAPTER 2. RESULTS

and 2.2. Since the parameter correlation for the bimodal van Genuchten model is very high, the single

parameter values should not be interpreted. The more valuable information isthe mere shape of the

curves curve.

2.2. INVERSION OF RICHARD’S EQUATION 13

0 20 40 60 80 100

−50

−40

−30

−20

−10

0

time [h]

mat

ric h

ead

[cm

]

measuredvG consvG unconsvG bimodfree form

0 20 40 60 80 100

−2

−1.5

−1

−0.5

0

time [h]

cum

ulat

ive

outfl

ow [c

m]


0 3 6 9 12 15

−1

−0.8

−0.6

−0.4

−0.2

0

time [h]

cum

ulat

ive

outfl

ow [c

m]


Figure 2.1: First drainage branch, column 1: Measured data and fitted with 4 different hydraulic propertiesmodels. Top: Matric head; Middle: Cumulative outflow; Bottom: Cumulative outflow forhLB ≥ −25 to show

the better performance of the more flexible models.


0 20 40 60 80 100

−50

−40

−30

−20

−10

0

time [h]

mat

ric h

ead

[cm

]measuredvG consvG unconsvG bimodfree form

0 20 40 60 80 100−2.5

−2

−1.5

−1

−0.5

0

time [h]

cum

ulat

ive

outfl

ow [c

m]


0 5 10 15−1

−0.8

−0.6

−0.4

−0.2

0

time [h]

cum

ulat

ive

outfl

ow [c

m]


Figure 2.2: First drainage branch, column 2: Measured data and fitted with 4 different hydraulic propertiesmodels. Top: Matric head; Middle: Cumulative outflow; Bottom: Cumulative outflow forhLB ≥ −25 to show



0 10 20 30 40 50 60

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

h [cm]

vol.

wat

er c

onte

nt [−

]

vG consvG unconsvG bimodfree form

0 10 20 30 40 50 60−5

−4

−3

−2

−1

0

1

2

3

h [cm]

log 1

0 (K

in c

m h

−1 )


Figure 2.3: First drainage branch, column 1: Hydraulic properties.

0 10 20 30 40 50 60

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

h [cm]

vol.

wat

er c

onte

nt [−

]


0 10 20 30 40 50 60−5

−4

−3

−2

−1

0

1

2

3

h [cm]

log 1

0 (K

in c

m h

−1 )


Figure 2.4: First drainage branch, column 2: Hydraulic properties.

0 10 20 30 40 50 60

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

h [cm]

vol.

wat

er c

onte

nt [−

]


0 10 20 30 40 50 60−5

−4

−3

−2

−1

0

1

2

3

h [cm]

log 1

0 (K

in c

m h

−1 )


Figure 2.5: First outflow branch, both columns: Hydraulic properties. solid lines: column 2; dashed lines:column 2.


Table 2.1: Estimated parameters for first drainage branch - column 1.

Parameter unit measured

ρb g cm−3 1.567

φ 1 - 0.409

Ks cm h−1 144

θs - 0.396

estimated

vG cons vG uncons vG bimod ff r=10

α1 cm−1 0.0340 0.0357 0.0330 −

n1 − 11.42 15.13 12.77 −

m1 − = 1 − 1/n1 0.479 = 1 − 1/n1 −

θr − 0.076 0.070 0.072 −

α2 cm−1

− − 0.076 −

n2 − − − 1.87 −

m2 − − − = 1 − 1/n2 −

w2 − − − 0.061 −

τ − 0.625 1.039 0.902 −

RMSEh2 cm 5.987E-01 5.763E-01 5.849E-01 5.780E-01

RMSEQ3 cm 4.178E-02 3.948E-02 3.419E-02 2.040E-02

1 calculated with assumption thatρs = 2.65 g cm−3: φ = 1 − ρb/ρs2 root mean square error of the measurement and model prediction ofmatric head,h3 root mean square error of the measurement and model prediction ofcumulative outflow,Q

2.2.2 First Imbibition Branch

In Figures 2.6 and 2.7 the inflow data and corresponding fits are shown. In column 1 hydraulic

equilibrium is only reached afterhLB ≥ −10 cm. This late response on the change of the pressure at

the lower boundary could possibly be explained by the existence of some small air bubbles underneath

the ceramic plate which cause contact problems of the water phase. These air bubbles were not visible.

Due to this late response almost the complete inflow happened at the last coupleof pressure steps,

that can only be described by a very steep retention function close to saturation. The fit with the both

unimodal models failed completely. The data for column 2 look more reasonable.But also here the

system behavior can only adequately be described by the more flexible models, i.e. the bimodal or

the free form approaches.

The hydraulic properties functions and the corresponding parameter values are shown in Figures 2.8

and 2.9 and Tables 2.3 and 2.4. Due to the assumption that the imbibition measurement for column 1

was influenced by some measurement artifacts we suggest that the hydraulic properties functions for

the imbibition branch are interpreted only from column 2.

2.2.3 Second Outflow Branch

The measured data and resulting fits are shown in Figures 2.10 and 2.11. Again, the behavior close to

saturation can only adequately be described by the more flexible functions.


Table 2.2: Estimated parameters for first drainage branch - column 2.


ρb g cm−3 1.563

φ 1 - 0.410

Ks cm h−1 68.3

θs - 0.385

estimated


α1 cm−1 0.0327 0.0329 0.0324 −

n1 − 10.91 11.05 12.04 −

m1 − = 1 − 1/n1 0.870 = 1 − 1/n1 −

θr − 0.095 0.0946 0.0976 −

α2 cm−1

− − 0.107 −

n2 − − − 4.929 −

m2 − − − = 1 − 1/n2 −

w2 − − − 0.032 −

τ − 0.370 0.408 0.216 −

RMSEh2 cm 2.738E-01 2.730E-01 2.895E-01 3.273E-01

RMSEQ3 cm 3.528E-02 3.527E-02 2.207E-02 7.565E-03


Table 2.3: Estimated parameters for first imbibition branch - column 1.


ρb g cm−3 1.567

φ 1 - 0.409

Ks cm h−1 144

θs - 0.351

estimated


α1 cm−1 0.0643 0.0628 0.062 −

n1 − 14.45 13.22 15.02 −

m1 − = 1 − 1/n1 1.031 = 1 − 1/n1 −

θr − 0.120 0.119 0.02 −

α2 cm−1

− − 0.450 −

n2 − − − 1.104 −

m2 − − − = 1 − 1/n2 −

w2 − − − 0.366 −

τ − 1.606 1.031 2.02 −

RMSEh3 cm 5.346E+00 5.512E+00 2.081E+00 7.686E-01

RMSEQ4 cm 9.324E-02 9.093E-02 7.856E-02 5.023E-02

1 calculated with assumption thatρs = 2.65 g cm−3: φ = 1 − ρb/ρs2 estimated parameter reached boundary of parameter space3 root mean square error of the measurement and model prediction ofmatric head,h4 root mean square error of the measurement and model prediction ofcumulative outflow,Q


0 20 40 60 80 100

−50

−40

−30

−20

−10

0

time [h]

mat

ric h

ead

[cm

]


0 20 40 60 80 100

0

0.5

1

1.5

2

time [h]

cum

ulat

ive

outfl

ow [c

m]


Figure 2.6: First imbibition branch, column 1: Measured data and fitted with 4 different hydraulic propertiesmodels. Top: Matric head; Bottom: Cumulative outflow.


0 20 40 60 80 100

−50

−40

−30

−20

−10

0

time [h]

mat

ric h

ead

[cm

]

measuredvG consvG unconsvG bimod

0 20 40 60 80 1000

0.5

1

1.5

2

2.5

time [h]

cum

ulat

ive

outfl

ow [c

m] measured

vG consvG unconsvG bimod

75 80 85 90 950

0.5

1

1.5

2

2.5

time [h]

cum

ulat

ive

outfl

ow [c

m] measured


Figure 2.7: First imbibition branch, column 2: Measured data and fitted with 4 different hydraulic propertiesmodels. Top: Matric head; Middle: Cumulative outflow; Bottom: Cumulative outflow forhLB ≥ −25 to show



0 10 20 30 40 50 60

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

h [cm]

vol.

wat

er c

onte

nt [−

] vG consvG unconsvG bimodfree form

0 10 20 30 40 50 60−5

−4

−3

−2

−1

0

1

2

3

h [cm]

log 1

0 (K

in c

m h

−1 )


Figure 2.8: First imbibition branch, column 1: Hydraulic properties.

0 10 20 30 40 50 60

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

h [cm]

vol.

wat

er c

onte

nt [−

] vG consvG unconsvG bimod

0 10 20 30 40 50 60−5

−4

−3

−2

−1

0

1

2

3

h [cm]

log 1

0 (K

in c

m h

−1 )


Figure 2.9: First imbibition branch, column 2: Hydraulic properties.


Table 2.4: Estimated parameters for first imbibition branch - column 2.


ρb g cm−3 1.563

φ 1 - 0.410

Ks cm h−1 68.3

θs - 0.343

estimated


α1 cm−1 0.0556 0.0686 0.049 −

n1 − 4.93 8.338 10.28 −

m1 − = 1 − 1/n1 0.278 = 1 − 1/n1 −

θr − 0.130 0.122 0.094 −

α2 cm−1

− − 0.088 −

n2 − − − 1.972 −

m2 − − − = 1 − 1/n2 −

w2 − − − 0.4567 −

τ − 1.084 2.02 2.02 −

RMSEh3 cm 1.906E+00 1.534E+00 6.523E-01 xxx

RMSEQ4 cm 4.751E-02 5.311E-02 2.180E-02 xxx

1 calculated with assumption thatρs = 2.65 g cm−3: φ = 1 − ρb/ρs2 estimated parameter reached boundary of parameter space3 root mean square error of the measurement and model prediction ofmatric head,h4 root mean square error of the measurement and model prediction ofcumulative outflow,Q

The resulting hydraulic properties functions are plotted in Figures 2.12 to 2.14. The shapes of the

different models look for both columns quite similar (Fig. 2.14), so that we regard the experiment as

well reproducible. The corresponding parameter values and goodness of fits are listed in Tables 2.1

and 2.2.


0 20 40 60 80 100

−50

−40

−30

−20

−10

0

time [h]

mat

ric h

ead

[cm

]


0 20 40 60 80 100

−2

−1.5

−1

−0.5

0

time [h]

cum

ulat

ive

outfl

ow [c

m]


0 5 10 15−1

−0.8

−0.6

−0.4

−0.2

0

time [h]

cum

ulat

ive

outfl

ow [c

m]


Figure 2.10: Second outflow branch, column 1: Measured data and fitted with4 different hydraulic propertiesmodels. Top: Matric head; Middle: Cumulative outflow; Bottom: Cumulative outflow forhLB ≥ −25 to show



0 20 40 60 80 100 120

−50

−40

−30

−20

−10

0

time [h]

mat

ric h

ead

[cm

]measuredvG consvG unconsvG bimodfree form

0 20 40 60 80 100 120−2.5

−2

−1.5

−1

−0.5

0

time [h]

cum

ulat

ive

outfl

ow [c

m]


0 5 10 15−1

−0.8

−0.6

−0.4

−0.2

0

time [h]

cum

ulat

ive

outfl

ow [c

m]


Figure 2.11: Second outflow branch, column 2: Measured data and fitted with4 different hydraulic propertiesmodels. Top: Matric head; Middle: Cumulative outflow; Bottom: Cumulative outflow forhLB ≥ −25 to show



0 10 20 30 40 50 60

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

h [cm]

vol.

wat

er c

onte

nt [−


0 10 20 30 40 50 60−5

−4

−3

−2

−1

0

1

2

3

h [cm]

log 1

0 (K

in c

m h

−1 )


Figure 2.12: Second outflow branch, column 1: Hydraulic properties.

0 10 20 30 40 50 60

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

h [cm]

vol.

wat

er c

onte

nt [−


0 10 20 30 40 50 60−5

−4

−3

−2

−1

0

1

2

3

h [cm]

log 1

0 (K

in c

m h

−1 )


Figure 2.13: Second outflow branch, column 2: Hydraulic properties.

0 10 20 30 40 50 60

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

h [cm]

vol.

wat

er c

onte

nt [−

]


0 10 20 30 40 50 60−5

−4

−3

−2

−1

0

1

2

3

h [cm]

log 1

0 (K

in c

m h

−1 )


Figure 2.14: Second outflow branch, both columns: Hydraulic properties.solid lines: column 1; dashed lines:column 2.


Table 2.5: Estimated parameters for second drainage branch - column 1.


ρb g cm−3 1.567

φ 1 - 0.409

Ks cm h−1 144

θs - 0.351

estimated


α1 cm−1 0.0337 0.0311 0.0332 −

n1 − 8.016 7.148 9.467 −

m1 − = 1 − 1/n1 1.372 = 1 − 1/n1 −

θr − 0.062 0.068 0.0625 −

α2 cm−1

− − 0.0672 −

n2 − − − 2.276 −

m2 − − − = 1 − 1/n2 −

w2 − − − 0.097 −

τ − 1.380 1.018 1.315 −

RMSEh2 cm 5.758E-01 5.922E-01 5.689E-01 5.727E-01

RMSEQ3 cm 3.042E-02 2.857E-02 1.630E-02 1.363E-02


Table 2.6: Estimated parameters for second drainage branch - column 2.


ρb g cm−3 1.563

φ 1 - 0.410

Ks cm h−1 68.3

θs - 0.343

estimated


α1 cm−1 0.0336 0.0333 0.0332 −

n1 − 8.77 8.661 9.900 −

m1 − = 1 − 1/n1 0.931 = 1 − 1/n1 −

θr − 0.061 0.062 0.064 −

α2 cm−1

− − 0.150 −

n2 − − − 2.350 −

m2 − − − = 1 − 1/n2 −

w2 − − − 0.0464 −

τ − 0.783 0.744 0.561 −

RMSEh2 cm 3.447E-01 3.462E-01 3.456E-01 3.786E-01

RMSEQ3 cm 3.344E-02 3.340E-02 2.043E-02 1.110E-02



0 10 20 30 40 50 60

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

h [cm]

vol.

wat

er c

onte

nt [−

]


0 10 20 30 40 50 60−5

−4

−3

−2

−1

0

1

2

3

h [cm]

log 1

0 (K

in c

m h

−1 )


Figure 2.15: Hysteresis of Hydraulic Properties, Column 1. Solid lines:first outflow; dots: first imbibition;dashed lines: second outflow.

0 10 20 30 40 50 60

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

h [cm]

vol.

wat

er c

onte

nt [−

]


0 10 20 30 40 50 60−5

−4

−3

−2

−1

0

1

2

3

h [cm]

log 1

0 (K

in c

m h

−1 )


Figure 2.16: Hysteresis of Hydraulic Properties, Column 2. Solid lines:first outflow; dots: first imbibition;dashed lines: second outflow.

2.2.4 Hysteresis

In Figures 2.15 and 2.16 the hydraulic properties functions are combined for both columns. As

expected the imbibition curves show a air entry closer to saturation. The retention functions of the

imbibition branch for column 2 are also less steep.

2.3 Fit of Equilibrium Data

The model fits of the equilibrium data gave very similar results as the inversion of Richards Equation.

This is remarkable since only the data up tohLB ≥ −40 cm could be used. With that restriction

10 data points were available for evaluation (Tables 2.7 and 2.8). The resulting parameter values are

given in Tables 2.9 and 2.10. The retention functions are plotted in Figure 2.17.

2.3. FIT OF EQUILIBRIUM DATA 27

Table 2.7: Data for equilibrium fits. First outflow branch.column 1 column 2

hm θ̄ hLB θ̄

cm [−] cm [−]

-0.810 0.396 -0.680 0.385

-4.710 0.395 -4.430 0.384

-9.460 0.393 -9.590 0.381

-14.550 0.388 -14.620 0.376

-19.450 0.381 -19.340 0.370

-24.670 0.363 -24.370 0.356

-29.430 0.254 -29.440 0.268

-34.420 0.147 -34.420 0.152

-39.400 0.105 -39.480 0.102

-43.720 0.089 -44.410 0.084

1 rmean of matric head in soil column at each pressure step.hm = hLB − L/2, whereL [cm] is the height of the soil column with

cermaic plate.

Table 2.8: Data for equilibrium fits. Second outflow branch.column 1 column 2

hm θ̄ hLB θ̄

cm [−] cm [−]

-0.620 0.351 -0.730 0.343

-4.570 0.351 -4.430 0.341

-9.450 0.347 -9.550 0.337

-14.430 0.341 -14.490 0.333

-19.400 0.332 -19.390 0.326

-24.340 0.298 -24.460 0.298

-29.400 0.224 -29.500 0.216

-34.340 0.142 -34.440 0.136

-39.410 0.102 -39.390 0.097

-44.420 0.084 -44.500 0.078

1 rmean of matric head in soil column at each pressure step.hm = hLB − L/2, whereL [cm] is the height of the soil column with

cermaic plate.

0 10 20 30 40 50 600.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

h [cm]

volu

met

ric w

ater

con

tent

[−]


0 10 20 30 40 50 600.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

h [cm]

volu

met

ric w

ater

con

tent

[−]


Figure 2.17: Retention functions resulting from equilibrium fit. Left: column 1, right: column 2. Solid lines:first outflow brach; dashed lines: second outflow branch.


Table 2.9: Estimated parameters of equilibrium data - first drainage branch.

Parameter unit column 1 column 2

vG cons vG uncons vG bimod vG cons vG uncons vG bimod

α1 cm−1 0.0337 0.0346 0.0335 0.0328 0.0320 0.0325

n1 − 11.85 12.78 13.11 10.88 10.31 12.15

m1 − = 1 − 1/n1 0.727 = 1 − 1/n1 = 1 − 1/n1 1.089 = 1 − 1/n1θr − 0.086 0.083 0.089 0.0741 0.0757 0.0771

α2 cm−1

− − 0.0940 − − 0.1038

n2 − − − 11.43 − − 7.83

m2 − − − = 1 − 1/n2 − − = 1 − 1/n2

w2 − − − 0.0294 − − 0.0337

RMSE 1 cm 4.877E-03 4.853E-03 3.048E-03 5.188E-03 5.166E-03 2.413E-03

1 root mean square error of the measurement and model prediction ofmean water content,̄θ

Table 2.10:Estimated parameters of equilibrium data - second drainagebranch.

Parameter unit column 1 column 2

vG cons vG uncons vG bimod vG cons vG uncons vG bimod

α1 cm−1 0.0342 0.0217 0.0344 0.0340 0.0281 0.1258

n1 − 7.57 5.87 2.5409 8.45 6.95 4.07

m1 − = 1 − 1/n1 9.456 = 1 − 1/n1 = 1 − 1/n1 2.6609 = 1 − 1/n1θr − 0.063 0.084 0.0356 0.0643 0.0766 0.0718

α2 cm−1

− − 0.0336 − − 0.0336

n2 − − − 10.52 − − 9.87

m2 − − − = 1 − 1/n2 − − = 1 − 1/n2

w2 − − − 0.6842 − − 0.9567

RMSE 1 cm 3.992E-03 2.969E-03 8.035E-04 4.493E-03 4.155E-03 1.100E-03

1 root mean square error of the measurement and model prediction ofmean water content,̄θ

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media.Water Resour. Res., 12(3), 513–521.

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29

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Braunschweig Technical University Institute of GeoEcology · 2006. 10. 16. · Braunschweig Technical University Institute of GeoEcology Measuring soil hydraulic properties of FIMOTUM